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Manuals and Guides 56

Intergovernmental Oceanographic Commission

The international thermodynamic
equation of seawater – 2010:
Calculation and use of thermodynamic properties

The Intergovernmental Oceanographic Commission (IOC) of UNESCO celebrates
its 50th anniversary in 2010. Since taking the lead in coordinating the International
Indian Ocean Expedition in 1960, the IOC has worked to promote marine research,
protection of the ocean, and international cooperation. Today the Commission is also
developing marine services and capacity building, and is instrumental in monitoring
the ocean through the Global Ocean Observing System (GOOS) and developing
marine-hazards warning systems in vulnerable regions. Recognized as the UN
focal point and mechanism for global cooperation in the study of the ocean, a key
climate driver, IOC is a key player in the study of climate change. Through promoting
international cooperation, the IOC assists Member States in their decisions towards
improved management, sustainable development, and protection of the marine
environment.

ManualsȱandȱGuidesȱ56ȱ
IntergovernmentalȱOceanographicȱCommissionȱ

Theȱinternationalȱthermodynamicȱȱ
equationȱofȱseawaterȱ–ȱ2010:ȱȱ
Calculation and use of thermodynamic properties

ȱ

The authors are responsible for the choice and the presentation of the facts contained in this publication
and for the opinions expressed therein, which are not necessarily those of UNESCO, SCOR or IAPSO and
do not commit those Organizations.

The photograph on the front cover of a CTD and lowered ADCP hovering just below the sea surface was
taken south of Timor from the Southern Surveyor in August 2003 by Ann Gronell Thresher.

For bibliographic purposes, this document should be cited as follows:
IOC, SCOR and IAPSO, 2010: The international thermodynamic equation of seawater – 2010: Calculation and
use of thermodynamic properties. Intergovernmental Oceanographic Commission, Manuals and Guides No.
56, UNESCO (English), 196 pp.

Printed by UNESCO
(IOC/2010/MG/56 Rev.)

© UNESCO/IOC et al. 2010

iii

  
  

Table  of  contents  
  
  
  
Acknowledgements   ……………………………………………………………………...   vii  
Foreword   ……………………………………………………………………………….……   viii  
Abstract   ………………………………………………………………………………...………   1  

  

  
  

1.   Introduction   ………………………………………………………………….…..…   2  
  

1.1  Oceanographic  practice  1978  -­‐‑  2009  …………………………………………………….  2  
1.2  Motivation  for  an  updated  thermodynamic  description  of  seawater  ………….…  2  
1.3   SCOR/IAPSO   WG127   and   the   approach   taken   ………………....……………….…   3  
1.4  A  guide  to  this  TEOS-­‐‑10  manual  ……………………………………………………….  6  
1.5  A  remark  on  units  …………………………………………………………………..……  7  
1.6  Recommendations  …………………………………………………………………..……  7  

  
  

2.  Basic  Thermodynamic  Properties   ……………….………………..….   9  
  
2.1  ITS-­‐‑90  temperature  ………………………………...…..…………………………..…….  9  
2.2   Sea   pressure   ………………………………….…………………..……….…………….   9  
2.3  Practical   Salinity   …………………………..…………………………………………..…   9  
2.4   Reference   Composition   and   the   Reference-­‐‑Composition   Salinity   Scale   …..…….   10  
2.5   Absolute   Salinity   ……………………………………………………………………….   11  
2.6   Gibbs   function   of   seawater   .…….…………………………………………………   15  
2.7   Specific   volume   ………….…….……………………….…….………………………   18  
2.8   Density   ………...…………………...……………………………………………….…   18  
2.9   Chemical   potentials   ……..………………………………..…………………………   19  
2.10   Entropy   …………………………………………………….............................………   20  
2.11   Internal   energy   …………………………………………………..............….………   20  
2.12   Enthalpy   ………..………………………………………………………….…...……   20  
2.13   Helmholtz   energy   ….…………………………………………………….....………   21  
2.14   Osmotic   coefficient   ….………………………………………………….….........…   21  
2.15   Isothermal   compressibility   ..…….………………………………………………...   21  
2.16   Isentropic   and   adiabatic   compressibility   …..…………….………………………   22  
2.17   Sound   speed   ……………………….……………………………………………..…   22  
2.18   Thermal   expansion   coefficients   ……...…………………………………………....   22  
2.19   Saline   contraction   coefficients   ……………………………………………….……   23  
2.20   Isobaric   heat   capacity   ………..……………………………………………………   24  
2.21   Isochoric   heat   capacity   ……….……………………………………………………   24  
2.22   Adiabatic   lapse   rate   ………..………………………………………………………   25  

  

IOC Manuals and Guides No. 56

iv

3.  Derived  Quantities   …………………………………..……….……………….   26  
  

  

3.1  Potential  temperature   ………………………………………………………………….   26  
3.2   Potential   enthalpy   ………………………………………....…………………………   27  
3.3   Conservative   Temperature   ……………….………………….……………………….   27  
3.4   Potential   density   ……………………………………………….………………………   28  
3.5   Density   anomaly   …………………….………………………….……………………   28  
3.6   Potential   density   anomaly   ……….…………………………….……………………   29  
3.7   Specific   volume   anomaly   ………………………………………….………………….   29  
3.8  Thermobaric  coefficient  ………….…………………………………………………….  30  
3.9   Cabbeling   coefficient   ………….…………………………………………………....….   31  
3.10   Buoyancy   frequency   ……….…………………………………………….……..….…   32  
3.11   Neutral   tangent   plane   …….……………………………………………………..….   32  
3.12   Geostrophic,   hydrostatic   and   “thermal   wind”   equations   …….……………….   34  
3.13   Neutral   helicity   …………….…………………………………………………..….….   35  
3.14  Neutral  Density  ….…………………………………………………………..….……..  39  
3.15   Stability   ratio   …..………………………………………………………………...….   39  
3.16   Turner   angle   ….……………………………………………………………………….   39  
3.17   Property   gradients   along   potential   density   surfaces   ……………………………   40  
3.18  Slopes  of  potential  density  surfaces  and  neutral  tangent  planes  compared  ..…  40  
3.19  Slopes  of  in  situ  density  surfaces  and  specific  volume  anomaly  surfaces  …..…  41  
3.20  Planetary  potential  vorticity  …………………………………….......……………….  42  
3.21   Vertical   velocity   through   the   sea   surface   …….……………………………………   45  
3.22   Freshwater   content   and   freshwater   flux   ………………………………………….   46  
3.23   Heat   transport   …………………….…………………………………………………..   46  
3.24   Geopotential   ………….………………………………………………………………..   47  
3.25   Total   energy   …………….……………………………………………………………..   47  
3.26   Bernoulli   function   ……….……………………………………………………………   47  
3.27   Dynamic   height   anomaly   ……………………………………………………………   48  
3.28   Montgomery   geostrophic   streamfunction   ………….……………………………   49  
3.29   Cunningham   geostrophic   streamfunction   ……….………………………………   50  
3.30   Geostrophic   streamfunction   in   an   approximately   neutral   surface   ….…………   51  
3.31   Pressure-­‐‑integrated   steric   height   ……..……………………………………………   51  
3.32  Pressure  to  height  conversion  …….…………………………………………………  52  
3.33  Freezing  temperature  ……….…………………………………………………….…..  53  
3.34   Latent   heat   of   melting   ….…………………………………………..………….……   55  
3.35   Sublimation   pressure   …………………………………………………………….…   56  
3.36   Sublimation   enthalpy   …………………………………………………………….…   57  
3.37   Vapour   pressure   ……………………………………………………………….……   59  
3.38   Boiling   temperature   ……….……………………………………….…………….…   60  
3.39   Latent   heat   of   evaporation   …………………………………………………………   60  
3.40   Relative   humidity   and   fugacity   ……………………………………………………   62  
3.41   Osmotic   pressure   ……………………………………………………………………   65  
3.42   Temperature   of   maximum   density   …….….………………………………………   65  
  

4.   Conclusions   ………………………………………….……..………………………   67  
  

IOC Manuals and Guides No. 56

v

  

Appendix  A:  Background  and  theory  underlying  the  use  of  the    
                                                  Gibbs  function  of  seawater  ………………..……………......   69  
  
A.1   ITS-­‐‑90   temperature   …………………………………………………………………...   69  
A.2   Sea   pressure,   gauge   pressure   and   Absolute   Pressure   …………….…………....   73  
A.3  Reference  Composition  and  the  Reference-­‐‑Composition  Salinity  Scale  …….…...  74  
A.4   Absolute   Salinity   ……………………………………………….……………………...   76  
A.5  Spatial  variations  in  seawater  composition  ……………………………….………...  82  
A.6  Gibbs  function  of  seawater  ………………………………….…………….……..…...  86  
A.7  The  fundamental  thermodynamic  relation  ………………………………….……...  87  
A.8  The  “conservative”  and  “isobaric  conservative”  properties  ……………….…..….  87  
A.9   The   “potential”   property   ……………….……….………………………………......   90  
A.10  Proof  that   θ = θ ( SA ,η )   and   Θ= Θ ( SA ,θ )   .…………………….…………………...  92  
A.11   Various   isobaric   derivatives   of   specific   enthalpy   ……………...…..……………   92  
A.12  Differential  relationships  between   η, θ , Θ   and   S A   …………...……...….……….  95  
A.13   The   First   Law   of   Thermodynamics   ………………….……………………...…......   95  
A.14  Advective  and  diffusive  “heat”  fluxes  ………………….………………..……......  99  
A.15  Derivation  of  the  expressions  for   α θ , β θ , α Θ   and   β Θ   ………………….……..  101  
A.16  Non-­‐‑conservative  production  of  entropy  ………………………...…………….....  102  
A.17   Non-­‐‑conservative   production   of   potential   temperature   ………………….…...   106  
A.18   Non-­‐‑conservative   production   of   Conservative   Temperature   ………………..   108  
A.19  Non-­‐‑conservative  production  of  specific  volume  …………………………..…...  111  
A.20  The  representation  of  salinity  in  numerical  ocean  models  ………………...........  112  
A.21  The  material  derivatives  of   S* ,    SA ,    S R   and   Θ   in  a  turbulent  ocean  ………....  117  
A.22  The  material  derivatives  of  density  and  of  locally-­‐‑referenced    
                        potential  density;  the  dianeutral  velocity   e   …………….……............................  121  
A.23  The  water-­‐‑mass  transformation  equation  …………….……..................................  123  
A.24  Conservation  equations  written  in  potential  density  coordinates  ………..…….  125  
A.25  The  vertical  velocity  through  a  general  surface  …………….…………………....  126  
A.26   The   material   derivative   of   potential   density   ………………………..….…….....   127  
A.27  The  diapycnal  velocity  of  layered  ocean  models  (without  rotation    
                        of  the  mixing  tensor)  ……………………………………………….….….…….....  128  
A.28  The  material  derivative  of  orthobaric  density  ………………………..….…….....  128  
A.29  The  material  derivative  of  Neutral  Density  …………………………….…….....  129  
A.30  Computationally  efficient  75-­‐‑term  expression  for  the  specific  volume    
                        of   seawater   in   terms   of   Θ …………………….....………..……………………..   130  

  

  
  
Appendix  B:  Derivation  of  the  First  Law  of  Thermodynamics   ….…….……  132  
  
  
Appendix  C:  Publications  describing  the  TEOS-­‐‑10  thermodynamic      
                                                descriptions  of  seawater,  ice  and  moist  air   ……...………..…..….  140  
  
  
Appendix  D:  Fundamental  constants   …….…………………..……………………...  143  
  

IOC Manuals and Guides No. 56

vi

Appendix  E:  Algorithm  for  calculating  Practical  Salinity  ……….……………..  147  

  
E.1  Calculation  of  Practical  Salinity  in  terms  of  K15  ….……...........................................  147  
E.2  Calculation  of  Practical  Salinity  at  oceanographic  temperature  and  pressure  .....  147  
E.3  Calculation  of  conductivity  ratio  R  for  a  given  Practical  Salinity  ..........................  148  
E.4  Evaluating  Practical  Salinity  using  ITS-­‐‑90  temperatures  .........................................  149  
E.5  Towards  SI-­‐‑traceability  of  the  measurement  procedure  for  Practical  Salinity    
                and  Absolute  Salinity  ..................................................................................................  149  

  
  
Appendix  F:  Coefficients  of  the  IAPWS-­‐‑95  Helmholtz  function  of    
                                                fluid  water  (with  extension  down  to  50K)  …………...………...…  152  
  
  
Appendix  G:  Coefficients  of  the  pure  liquid  water  Gibbs  function    
                                                  of  IAPWS-­‐‑09  ………………………...……………………….…...……….  155  
  
  
Appendix  H:  Coefficients  of  the  saline  Gibbs  function  for  seawater    
                                                  of  IAPWS-­‐‑08  ………………………………………………..…….……….  156  
  
  
Appendix  I:  Coefficients  of  the  Gibbs  function  of  ice  Ih  of  IAPWS-­‐‑06  …...  157  
  
  
Appendix  J:  Coefficients  of  the  Helmholtz  function  of  moist  air    
                                              of   IAPWS-­‐‑10   ……………………………..…….…………………………   159  
  
  
Appendix  K:  Coefficients  of  the  75-­‐‑term  expression  for  the  specific        
                                                  volume   of   seawater   in   terms   of   Θ ……………….….…………....   163  
  
  
Appendix  L:  Recommended  nomenclature,  symbols  and      
                                                units   in   oceanography   …………………………………………………   165  
  
  
Appendix  M:  Seawater-­‐‑Ice-­‐‑Air  (SIA)  library  of  computer  software  ………..  171  
  
  
Appendix  N:  Gibbs-­‐‑SeaWater  (GSW)  Oceanographic  Toolbox   .....................  182  
  
  
Appendix  O:  Checking  the  Gibbs  function  of  seawater  against  the  
                                                  original  thermodynamic  data   ……………………….………….……  188  
  
  

Appendix  P:  Thermodynamic  properties  based  on   g SA ,t, p , h SA ,η , p ,     

(

)

(

)

(

) (

)

                                                     h SA ,θ , p    and   ĥ SA ,Θ, p    …………………….…………………...   191  
  

References  ……………………………..……………………………….………….…….…  195  
  

Index   ……………………………………..……………………………….…………….….…  205  
  
Changes  made  to  the  TEOS-­‐‑10  Manual  since  13th  April  2010   .…………......…  207  
IOC Manuals and Guides No. 56

vii

  
  

Acknowledgements    
  
This   TEOS-­‐‑10   Manual   reviews   and   summarizes   the   work   of   the   SCOR/IAPSO   Working  
Group  127  on  the  Thermodynamics  and  Equation  of  State  of  Seawater.    Dr  John  Gould  and  
Professor   Paola   Malanotte-­‐‑Rizzoli   played   pivotal   roles   in   the   establishment   of   the  
Working   Group   and   we   have   enjoyed   rock-­‐‑solid   scientific   support   from   the   officers   of  
SCOR,   IAPSO   and   IOC.      TJMcD   wishes   to   acknowledge   fruitful   discussions   with   Drs  
Jürgen  Willebrand  and  Michael  McIntyre  regarding  the  contents  of  appendix  B.    We  have  
benefited  from  extensive  comments  on  drafts  of  this  manual  by  Dr  Stephen  Griffies  and  Dr  
Allyn   Clarke.      Dr   Harry   Bryden   is   thanked   for   valuable   and   timely   advice   on   the  
treatment  of  salinity  in  ocean  models.     Louise  Bell  of  CSIRO  provided  much  appreciated  
advice   on   the   layout   of   this   document.      Dr   Paul   Barker   is   thanked   for   carefully   proof-­‐‑
reading   this   TEOS-­‐‑10   Manual,   for   writing   most   of   the   GSW   Oceanographic   Toolbox  
functions   and   for   designing   the   TEOS-­‐‑10   web   site,   www.TEOS-10.org.    This   document   is  
based   on   work   partially   supported   by   the   U.S.   National   Science   Foundation   to   SCOR  
under  Grant  No.  OCE-­‐‑0608600.    FJM  wishes  to  acknowledge  the  Oceanographic  Section  of  
the   National   Science   Foundation   and   the   National   Oceanic   and   Atmospheric  
Administration  for  supporting  his  work.          
  
  
This  document  has  been  written  by  the  members  of  SCOR/IAPSO  Working  Group  127,    

Trevor  J.  McDougall,  (chair),  University  of  New  South  Wales,  Sydney,  Australia    
Rainer  Feistel,  Leibniz-­‐‑Institut  fuer  Ostseeforschung,  Warnemuende,  Germany    
Daniel  G.  Wright+,  formerly  of  Bedford  Institute  of  Oceanography,  Dartmouth,  Canada    
Rich  Pawlowicz,  University  of  British  Columbia,  Vancouver,  Canada    
Frank  J.  Millero,  University  of  Miami,  Florida,  USA    
David  R.  Jackett++,  formerly  of  CSIRO,  Hobart,  Australia    
Brian  A.  King,  National  Oceanography  Centre,  Southampton,  UK    
Giles  M.  Marion,  Desert  Research  Institute,  Reno,  USA    
Steffen  Seitz,  Physikalisch-­‐‑Technische  Bundesanstalt  (PTB),  Braunschweig,  Germany    
Petra  Spitzer,  Physikalisch-­‐‑Technische  Bundesanstalt  (PTB),  Braunschweig,  Germany    
C-­‐‑T.  Arthur  Chen,  National  Sun  Yat-­‐‑Sen  University,  Taiwan,  R.O.C.    
  
  

March  2010    
  

    +  deceased,  8th  July  2010      

++  deceased,  31st  March  2012      

  
  

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viii

Foreword  
  
This  document  describes  the  International  Thermodynamic  Equation  Of  Seawater  –  2010  
(TEOS-­‐‑10  for  short).    TEOS-­‐‑10  defines  the  thermodynamic  properties  of  seawater  and  of  
ice  Ih  and  has  been  adopted  by  the  Intergovernmental  Oceanographic  Commission  at  its  
25th  Assembly  in  June  2009,  replacing  EOS-­‐‑80  as  the  official  description  of  seawater  and  
ice  properties  in  marine  science.      
Fundamental   to   TEOS-­‐‑10   are   the   concepts   of   Absolute   Salinity   and   Reference  
Salinity.      These   variables   are   described   in   detail   here,   emphasising   their   relationship   to  
Practical  Salinity.      
The   science   underpinning   TEOS-­‐‑10   has   been   described   in   a   series   of   papers  
published   in   the   refereed   literature   (see   appendix   C).      The   present   document   may   be  
called   the   “TEOS-­‐‑10   Manual”   and   acts   as   a   guide   to   those   published   papers   and  
concentrates   on   how   the   thermodynamic   properties   obtained   from   TEOS-­‐‑10   are   to   be  
used  in  oceanography.      
In  addition  to  the  thermodynamic  properties  of  seawater,  TEOS-­‐‑10  also  describes  the  
thermodynamic  properties  of  ice  and  of  humid  air,  and  these  properties  are  summarised  
in   this   document.      The   TEOS-­‐‑10   computer   software,   this   TEOS-­‐‑10   Manual   and   other  
documents   may   be   obtained   from   www.TEOS-10.org. In   particular,   there   are   two  
introductory   articles   about   TEOS-­‐‑10   on   this   web   site,   namely   “What   every  
oceanographer   needs   to   know   about   TEOS-­‐‑10   (The   TEOS-­‐‑10   Primer)”   (Pawlowicz,  
2010b)   and   “Getting   started   with   TEOS-­‐‑10   and   the   Gibbs   Seawater   (GSW)  
Oceanographic   Toolbox”   (McDougall   and   Barker,   2011).      An   historical   account   of   how  
TEOS-­‐‑10  was  developed  has  appeared  in  Pawlowicz  et  al.  (2012).        
When   referring   to   the   use   of   TEOS-­‐‑10,   it   is   the   present   document   that   should   be  
referenced   as   IOC   et   al.   (2010a)   [IOC,   SCOR   and   IAPSO,   2010:   The   international  
thermodynamic   equation   of   seawater   –   2010:   Calculation   and   use   of   thermodynamic   properties.    
Intergovernmental   Oceanographic   Commission,   Manuals   and   Guides   No.   56,   UNESCO  
(English),  196  pp.].      
  
  
  
This  version  of  the  TEOS-­‐‑10  Manual  includes  corrections  up  to  20th  December  2016.      
  
  

  

IOC Manuals and Guides No. 56

TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

1

  

Abstract    
  

  
This   document   outlines   how   the   thermodynamic   properties   of   seawater   are   evaluated  
using   the   International   Thermodynamic   Equation   Of   Seawater   –   2010   (TEOS-­‐‑10).      This  
thermodynamic   description   of   seawater   is   based   on   a   Gibbs   function   formulation   from  
which   thermodynamic   properties   such   as   entropy,   specific   volume,   enthalpy   and  
potential   enthalpy   are   calculated   directly.      When   determined   from   the   Gibbs   function,  
these  quantities  are  fully  consistent  with  each  other.    Entropy  and  enthalpy  are  required  
for  an  accurate  description  of  the  advection  and  diffusion  of  heat  in  the  ocean  interior  and  
for  quantifying  the  ocean’s  role  in  exchanging  heat  with  the  atmosphere  and  with  ice.    The  
Gibbs  function  is  a  function  of  Absolute  Salinity,  temperature  and  pressure.    In  contrast  to  
Practical  Salinity,  Absolute  Salinity  is  expressed  in  SI  units  and  it  includes  the  influence  of  
the  small  spatial  variations  of  seawater  composition  in  the  global  ocean.    Absolute  Salinity  
is   the   appropriate   salinity   variable   for   the   accurate   calculation   of   horizontal   density  
gradients  in  the  ocean.    Absolute  Salinity  is  also  the  appropriate  salinity  variable  for  the  
calculation  of  freshwater  fluxes  and  for  calculations  involving  the  exchange  of  freshwater  
with  the  atmosphere  and  with  ice.    Potential  functions  are  included  for  ice  and  for  moist  
air,  leading  to  accurate  expressions  for  numerous  thermodynamic  properties  of  ice  and  air  
including   freezing   temperature   and   latent   heats   of   melting   and   of   evaporation.      This  
TEOS-­‐‑10  Manual  describes  how  the  thermodynamic  properties  of  seawater,  ice  and  moist  
air   are   used   in   order   to   accurately   represent   the   transport   of   heat   in   the   ocean   and   the  
exchange  of  heat  with  the  atmosphere  and  with  ice.      

  
  

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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

  
  

1.  Introduction    
  
  
  

1.1  Oceanographic  practice  1978  -­‐‑  2009    

  
The   Practical   Salinity   Scale,   PSS-­‐‑78,   and   the   International   Equation   of   State   of   Seawater  
(UNESCO   (1981))   which   expresses   the   density   of   seawater   as   a   function   of   Practical  
Salinity,  temperature  and  pressure,  have  served  the  oceanographic  community  very  well  
for   thirty   years.      The   Joint   Panel   on   Oceanographic   Tables   and   Standards   (JPOTS)  
(UNESCO   (1983))   also   promulgated   the   Millero,   Perron   and   Desnoyers   (1973)   algorithm  
for  the  specific  heat  capacity  of  seawater  at  constant  pressure,  the  Chen  and  Millero  (1977)  
expression  for  the  sound  speed  of  seawater  and  the  Millero  and  Leung  (1976)  formula  for  
the  freezing  point  temperature  of  seawater.    Three  other  algorithms  supported  under  the  
auspices  of  JPOTS  concerned  the  conversion  between  hydrostatic  pressure  and  depth,  the  
calculation   of   the   adiabatic   lapse   rate,   and   the   calculation   of   potential   temperature.      The  
expressions   for   the   adiabatic   lapse   rate   and   for   potential   temperature   could   in   principle  
have  been  derived  from  the  other  algorithms  of  the  EOS-­‐‑80  set,  but  in  fact  they  were  based  
on  the  formulas  of  Bryden  (1973).    We  shall  refer  to  all  these  algorithms  jointly  as  ‘EOS-­‐‑80’  
for  convenience  because  they  represent  oceanographic  best  practice  from  the  early  1980s  to  
2009.      

  

1.2  Motivation  for  an  updated  thermodynamic  description  of  seawater    

  
In  recent  years  the  following  aspects  of  the  thermodynamics  of  seawater,  ice  and  moist  air  
have   become   apparent   and   suggest   that   it   is   timely   to   redefine   the   thermodynamic  
properties  of  these  substances.      
• Several   of   the   polynomial   expressions   of   the   International   Equation   of   State   of  
Seawater  (EOS-­‐‑80)  are  not  totally  consistent  with  each  other  as  they  do  not  exactly  
obey   the   thermodynamic   Maxwell   cross-­‐‑differentiation   relations.      The   new  
approach  eliminates  this  problem.      
• Since   the   late   1970s   a   more   accurate   and   more   broadly   applicable   thermodynamic  
description   of   pure   water   has   been   developed   by   the   International   Association   for  
the  Properties  of  Water  and  Steam,  and  has  appeared  as  an  IAPWS  Release  (IAPWS-­‐‑
95).      Also   since   the   late   1970s   some   measurements   of   higher   accuracy   have   been  
made  of  several  properties  of  seawater  such  as  (i)  heat  capacity,  (ii)  sound  speed  and  
(iii)   the   temperature   of   maximum   density.      These   can   be   incorporated   into   a   new  
thermodynamic  description  of  seawater.    
• The   impact   on   seawater   density   of   the   variation   of   the   composition   of   seawater   in  
the   different   ocean   basins   has   become   better   understood.      In   order   to   further  
progress   this   aspect   of   seawater,   a   standard   model   of   seawater   composition   is  
needed   to   serve   as   a   generally   recognised   reference   for   theoretical   and   chemical  
investigations.      
• The   increasing   emphasis   on   the   ocean   as   being   an   integral   part   of   the   global   heat  
engine   points   to   the   need   for   accurate   expressions   for   the   entropy,   enthalpy   and  
internal  energy  of  seawater  so  that  heat  fluxes  can  be  more  accurately  determined  in  

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•

•

  

3

the   ocean   and   across   the   interfaces   between   the   ocean   and   the   atmosphere   and   ice  
(entropy,  enthalpy  and  internal  energy  were  not  available  from  EOS-­‐‑80).      
The  need  for  a  thermodynamically  consistent  description  of  the  interactions  between  
seawater,   ice   and   moist   air;   in   particular,   the   need   for   accurate   expressions   for   the  
latent   heats   of   evaporation   and   melting,   both   at   the   sea   surface   and   in   the  
atmosphere.      
The  temperature  scale  has  been  revised  from  IPTS-­‐‑68  to  ITS-­‐‑90  and  revised  IUPAC  
(International  Union  of  Pure  and  Applied  Chemistry)  values  have  been  adopted  for  
the  atomic  weights  of  the  elements  (Wieser  (2006)).      

1.3  SCOR/IAPSO  WG127  and  the  approach  taken  
In   2005   SCOR   (Scientific   Committee   on   Oceanic   Research)   and   IAPSO   (International  
Association  for  the  Physical  Sciences  of  the  Oceans)  established  Working  Group  127  on  the  
“Thermodynamics  and  Equation  of  State  of  Seawater”  (henceforth  referred  to  as  WG127).    
This   group   has   now   developed   a   collection   of   algorithms   that   incorporate   our   best  
knowledge  of  seawater  thermodynamics.    The  present  document  summarizes  the  work  of  
SCOR/IAPSO  Working  Group  127.      
To  compute  all  thermodynamic  properties  of  seawater  it  is  sufficient  to  know  one  of  its  
so-­‐‑called  thermodynamic  potentials  (Fofonoff  1962,  Feistel  1993,  Alberty  2001).    It  was  J.W.  
Gibbs  (1873)  who  discovered  that  “an  equation  giving  internal  energy  in  terms  of  entropy  and  
specific  volume,  or  more  generally  any  finite  equation  between  internal  energy,  entropy  and  specific  
volume,  for  a  definite  quantity  of  any  fluid,  may  be  considered  as  the  fundamental  thermodynamic  
equation  of  that  fluid,  as  from  it…  may  be  derived  all  the  thermodynamic  properties  of  the  fluid  (so  
far  as  reversible  processes  are  concerned).”      
The   approach   taken   by   WG127   has   been   to   develop   a   Gibbs   function   from   which   all  
the   thermodynamic   properties   of   seawater   can   be   derived   by   purely   mathematical  
manipulations   (such   as   differentiation).      This   approach   ensures   that   the   various  
thermodynamic   properties   are   self-­‐‑consistent   (in   that   they   obey   the   Maxwell   cross-­‐‑
differentiation  relations)  and  complete  (in  that  each  of  them  can  be  derived  from  the  given  
potential).      
The   Gibbs   function   (or   Gibbs   potential)   is   a   function   of   Absolute   Salinity   S A    (rather  
than  of  Practical  Salinity   S P ),  temperature  and  pressure.    Absolute  Salinity  is  traditionally  
defined   as   the   mass   fraction   of   dissolved   material   in   seawater.      The   use   of   Absolute  
Salinity  as  the  salinity  argument  for  the  Gibbs  function  and  for  all  other  thermodynamic  
functions  (such  as  density)  is  a  major  departure  from  present  practice  (EOS-­‐‑80).    Absolute  
Salinity   is   preferred   over   Practical   Salinity   because   the   thermodynamic   properties   of  
seawater   are   directly   influenced   by   the   mass   of   dissolved   constituents   whereas   Practical  
Salinity  depends  only  on  conductivity.    Consider  for  example  exchanging  a  small  amount  
of  pure  water  with  the  same  mass  of  silicate  in  an  otherwise  isolated  seawater  sample  at  
constant   temperature   and   pressure.      Since   silicate   is   predominantly   non-­‐‑ionic,   the  
conductivity   (and   therefore   Practical   Salinity   S P )   is   almost   unchanged   but   the   Absolute  
Salinity  is  increased,  as  is  the  density.    Similarly,  if  a  small  mass  of  say  NaCl  is  added  and  
the   same   mass   of   silicate   is   taken   out   of   a   seawater   sample,   the   mass   fraction   absolute  
salinity   will   not   have   changed   (and   so   the   density   should   be   almost   unchanged)   but   the  
Practical  Salinity  will  have  increased.      
The   variations   in   the   relative   concentrations   of   seawater   constituents   caused   by  
biogeochemical   processes   actually   cause   complications   in   even   defining   what   exactly   is  
meant  by  “absolute  salinity”.    These  issues  have  not  been  well  studied  to  date,  but  what  is  
known   is   summarized   in   section   2.5   and   appendices   A.4,   A.5   and   A.20.      Here   it   is  
sufficient   to   point   out   that   the   Absolute   Salinity   S A    that   is   the   salinity   argument   of   the  
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

TEOS-­‐‑10  Gibbs  function  is  the  version  of  absolute  salinity  that  provides  the  best  estimate  
of  the  density  of  seawater;  another  name  for   S A   is  “Density  Salinity”.      
The  Gibbs  function  of  seawater,  published  as  Feistel  (2008),  has  been  endorsed  by  the  
International  Association  for  the  Properties  of  Water  and  Steam  as  the  Release  IAPWS-­‐‑08.    
This  thermodynamic  description  of  seawater  properties,  together  with  the  Gibbs  function  
of   ice   Ih,   IAPWS-­‐‑06,   has   been   adopted   by   the   Intergovernmental   Oceanographic  
Commission  at  its  25th  Assembly  in  June  2009  to  replace  EOS-­‐‑80  as  the  official  description  
of  seawater  and  ice  properties  in  marine  science.    The  thermodynamic  properties  of  moist  
air   have   also   recently   been   described   using   a   Helmholtz   function   (Feistel   et   al.   (2010a),  
IAPWS   (2010))   so   allowing   the   equilibrium   properties   at   the   air-­‐‑sea   interface   to   be   more  
accurately  evaluated.    The  new  approach  to  the  thermodynamic  properties  of  seawater,  of  
ice   Ih   and   of   humid   air   is   referred   to   collectively   as   the   “International   Thermodynamic  
Equation  Of  Seawater  –  2010”,  or  “TEOS-­‐‑10”  for  short.    Appendix  C  lists  the  publications  
which  lie  behind  TEOS-­‐‑10.      
A   notable   difference   of   TEOS-­‐‑10   compared   with   EOS-­‐‑80   is   the   adoption   of   Absolute  
Salinity   to   be   used   in   journals   to   describe   the   salinity   of   seawater   and   to   be   used   as   the  
salinity   argument   in   algorithms   that   give   the   various   thermodynamic   properties   of  
seawater.      This   recommendation   deviates   from   the   current   practice   of   working   with  
Practical   Salinity   and   typically   treating   it   as   the   best   estimate   of   Absolute   Salinity.      This  
practice   is   inaccurate   and   should   be   corrected.      Note   however   that   we   strongly  
recommend   that   the   salinity   that   is   reported   to   national   data   bases   remain   Practical  
Salinity   as   determined   on   the   Practical   Salinity   Scale   of   1978   (suitably   updated   to   ITS-­‐‑90  
temperatures  as  described  in  appendix  E  below).    
There  are  three  very  good  reasons  for  continuing  to  store  Practical  Salinity  rather  than  
Absolute  Salinity  in  such  data  repositories.    First,  Practical  Salinity  is  an  (almost)  directly  
measured  quantity  whereas  Absolute  Salinity  is  generally  a  derived  quantity.    That  is,  we  
calculate  Practical  Salinity   directly  from  measurements  of  conductivity,  temperature  and  
pressure,   whereas   to   date   we   derive   Absolute   Salinity   from   a   combination   of   these  
measurements  plus  other  measurements  and  correlations  that  are  not  yet  well  established.    
Practical   Salinity   is   preferred   over   the   actually   measured   in   situ   conductivity   value  
because  of  its  conservative  nature  with  respect  to  changes  of  temperature  or  pressure,  or  
dilution  with  pure  water.    Second,  it  is  imperative  that  confusion  is  not  created  in  national  
data  bases  where  a  change  in  the  reporting  of  salinity  may  be  mishandled  at  some  stage  
and   later   be   misinterpreted   as   a   real   increase   in   the   ocean’s   salinity.      This   second   point  
argues  strongly  for  no  change  in  present  practice  in  the  reporting  of  Practical  Salinity   S P   
in  national  data  bases  of  oceanographic  data.    Thirdly,  the  algorithms  for  determining  the  
"ʺbest"ʺ   estimate   of   Absolute   Salinity   of   seawater   with   non-­‐‑standard   composition   are  
immature   and   will   undoubtedly   change   in   the   future,   so   we   cannot   recommend   storing  
Absolute  Salinity  in  national  data  bases.    Storage  of  a  more  robust  intermediate  value,  the  
Reference  Salinity,   S R   (defined  as  discussed  in  appendix  A.3  to  give  the  best  estimate  of  
Absolute  Salinity  of  Standard  Seawater)  would  also  introduce  the  possibility  of  confusion  
in  the  stored  salinity  values  without  providing  any  real  advantage  over  storing  Practical  
Salinity   so   we   also   avoid   this   possibility.      Values   of   Reference   Salinity   obtained   from  
suitable   observational   techniques   (for   example   by   direct   measurement   of   the   density   of  
Standard   Seawater)   should   be   converted   to   corresponding   numbers   of   Practical   Salinity  
for  storage,  as  described  in  sections  2.3  -­‐‑  2.5.      
Note   that   the   practice   of   storing   one   type   of   salinity   in   national   data   bases   (Practical  
Salinity)  but  using  a  different  type  of  salinity  in  publications  (Absolute  Salinity)  is  exactly  
analogous  to  our  present  practice  with  temperature;  in  situ  temperature   t   is  stored  in  data  
bases   (since   it   is   the   measured   quantity)   but   the   temperature   variable   that   is   used   in  

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publications  is  a  calculated  quantity,  being  potential  temperature   θ   under  EOS-­‐‑80  and  is  
now  Conservative  Temperature   Θ   under  TEOS-­‐‑10.      
In  order  to  improve  the  determination  of  Absolute  Salinity  we  need  to  begin  collecting  
and   storing   values   of   the   salinity   anomaly   δ SA = SA − SR    based   on   measured   values   of  
density  (such  as  can  be  measured  with  a  vibrating  tube  densimeter,  Kremling  (1971)).    The  
4-­‐‑letter   GF3   code   (IOC   (1987))   DENS   is   currently   defined   for   in   situ   measurements   or  
computed  values  from  EOS-­‐‑80.    It  is  recommended  that  the  density  measurements  made  
with   a   vibrating   beam   densimeter   be   reported   with   the   GF3   code   DENS   along   with   the  
laboratory  temperature  (TLAB  in   ° C )  and  laboratory  pressure  (PLAB,  the  sea  pressure  in  
the   laboratory,   usually   0   dbar).      From   this   information   and   the   Practical   Salinity   of   the  
seawater   sample,   the   Absolute   Salinity   Anomaly   δ SA = SA − SR    can   be   calculated   using  
an  inversion  of  the  TEOS-­‐‑10  equation  for  density  to  determine   SA .     For  completeness,  it  is  
advisable  to  also  report   δ SA   under  the  new  GF3  code  DELS.        
The  thermodynamic  description  of  seawater  and  of  ice  Ih  as  defined  in  IAPWS-­‐‑08  and  
IAPWS-­‐‑06   has   been   adopted   as   the   official   description   of   seawater   and   of   ice   Ih   by   the  
Intergovernmental   Oceanographic   Commission   in   June   2009.      These   new   international  
standards   were   adopted   while   recognizing   that   the   techniques   for   estimating   Absolute  
Salinity   will   likely   improve   over   the   coming   decades,   and   the   algorithm   for   evaluating  
Absolute   Salinity   in   terms   of   Practical   Salinity,   latitude,   longitude   and   pressure   will   be  
updated  from  time  to  time,  after  relevant  appropriately  peer-­‐‑reviewed  publications  have  
appeared,  and  that  such  an  updated  algorithm  will  appear  on  the  www.TEOS-10.org web  
site.    Users  of  this  software  should  always  state  in  their  published  work  which  version  of  
the  software  was  used  to  calculate  Absolute  Salinity.      
        The  more  prominent  advantages  of  TEOS-­‐‑10  compared  with  EOS-­‐‑80  are      
•

The   Gibbs   function   approach   allows   the   calculation   of   internal   energy,   entropy,  
enthalpy,  potential  enthalpy  and  the  chemical  potentials  of  seawater  as  well  as  the  
freezing   temperature,   and   the   latent   heats   of   melting   and   of   evaporation.      These  
quantities   were   not   available   from   the   International   Equation   of   State   1980   but   are  
essential   for   the   accurate   accounting   of   “heat”   in   the   ocean   and   for   the   consistent  
and   accurate   treatment   of   air-­‐‑sea   and   ice-­‐‑sea   heat   fluxes.      For   example,   the   new  
TEOS-­‐‑10   temperature   variable,   Conservative   Temperature,   Θ ,   is   defined   to   be  
proportional   to   potential   enthalpy   and   is   a   very   accurate   measure   of   the   “heat”  
content  per  unit  mass  of  seawater;   Θ   is  two  orders  of  magnitude  more  conservative  
than  potential  temperature   θ .      

•

For  the  first  time  the  influence  of  the  spatially  varying  composition  of  seawater  can  
systematically   be   taken   into   account   through   the   use   of   Absolute   Salinity.      In   the  
open  ocean,  this  has  a  non-­‐‑trivial  effect  on  the  horizontal  density  gradient  computed  
from  the  equation  of  state,  and  thereby  on  the  ocean  velocities  and  heat  transports  
calculated  via  the  “thermal  wind”  relation.    

•

The   thermodynamic   quantities   available   from   the   new   approach   are   totally  
consistent  with  each  other.      

•

The  new  salinity  variable,  Absolute  Salinity,  is  measured  in  SI  units.    Moreover  the  
treatment   of   freshwater   fluxes   in   ocean   models   will   be   consistent   with   the   use   of  
Absolute  Salinity,  but  is  only  approximately  so  for  Practical  Salinity.      

•

The  Reference  Composition  of  standard  seawater  supports  marine  physicochemical  
studies  such  as  the  solubility  of  sea  salt  constituents,  the  alkalinity,  the  pH  and  the  
ocean  acidification  by  rising  concentrations  of  atmospheric  CO2.      

  

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1.4  A  guide  to  this  TEOS-­‐‑10  manual      

  
The   remainder   of   this   manual   begins   by   listing   (in   section   2)   the   definitions   of   various  
thermodynamic   quantities   that   follow   directly   from   the   Gibbs   function   of   seawater   by  
simple   mathematical   processes   such   as   differentiation.      These   definitions   are   then  
followed   in   section   3   by   the   discussion   of   several   derived   quantities.      The   computer  
software  to  evaluate  these  quantities  is  available  from  two  separate  libraries,  the  Seawater-­‐‑
Ice-­‐‑Air  (SIA)  library  and  the  Gibbs-­‐‑SeaWater  (GSW)  Oceanographic  Toolbox,  as  described  
in  appendices  M  and  N.    The  functions  in  the  SIA  library  are  generally  available  in  basic-­‐‑SI  
units  ( kg kg −1 ,  kelvin  and  Pa),  both  for  their  input  parameters  and  for  the  outputs  of  the  
algorithms.      Some   additional   routines   are   included   in   the   SIA   library   in   terms   of   other  
commonly   used   units   for   the   convenience   of   users.      The   SIA   library   takes   significantly  
more   computer   time   to   evaluate   most   quantities   (approximately   a   factor   of   65   more  
computer   time   for   many   quantities,   comparing   optimized   code   in   both   cases)   and  
provides  significantly  more  properties  than  does  the  GSW  Toolbox.    The  SIA  library  uses  
the   world-­‐‑wide   standard   for   the   thermodynamic   description   of   pure   water   substance  
(IAPWS-­‐‑95).    Since  this  is  defined  over  extended  ranges  of  temperature  and  pressure,  the  
algorithms   are   long   and   their   evaluation   time-­‐‑consuming.      The   GSW   Toolbox   uses   the  
Gibbs  function  of  Feistel  (2003)  (IAPWS-­‐‑09)  to  evaluate  the  properties  of  pure  water,  and  
since  this  is  valid  only  over  the  restricted  ranges  of  temperature  and  pressure  appropriate  
for   the   ocean,   the   algorithms   are   shorter   and   their   execution   is   faster.      The   GSW  
Oceanographic   Toolbox   is   not   as   comprehensive   as   the   SIA   library;   for   example,   the  
properties  of  moist  air  are  only  available  in  the  SIA  library.    In  addition,  a  computationally  
efficient   expression   for   density   specific   volume   in   terms   of   Conservative   Temperature  
(rather  than  in  terms  of  in  situ  temperature)  involving  just  75  coefficients  is  also  available  
and  is  described  in  appendix  A.30  and  appendix  K.      
The   input   and   output   parameters   of   the   GSW   Oceanographic   Toolbox   are   in   units  
which   oceanographers   will   find   more   familiar   than   basic   SI   units.      We   expect   that  
oceanographers   will   mostly   use   this   GSW   Toolbox   because   of   its   greater   simplicity   and  
computational   efficiency,   and   because   of   the   more   familiar   units   compared   with   the   SIA  
library.    The  name  GSW  (Gibbs-­‐‑SeaWater)  has  been  chosen  to  be  similar  to,  but  different  
from  the  existing  “sw”  (Sea  Water)  library  which  is  already  in  wide  circulation.    Both  the  
SIA  and  GSW  libraries,   together  with  this  TEOS-­‐‑10  Manual  are  available  from  the  website  
www.TEOS-10.org. Initially   the   SIA   library   is   being   made   available   in   Visual   Basic   and  
FORTRAN  while  the  GSW  library  is  in  MATLAB  with  some  functions  in  FORTRAN  and  C.        
After   these   descriptions   in   sections   2   and   3   of   how   to   determine   the   thermodynamic  
quantities   and   various   derived   quantities,   we   end   with   some   conclusions   (section   4).    
Additional   information   on   Practical   Salinity,   the   Gibbs   function,   Reference   Salinity,  
composition   anomalies,   Absolute   Salinity,   and   some   fundamental   thermodynamic  
properties  such  as  the  First  Law  of  Thermodynamics,  the  non-­‐‑conservative  nature  of  many  
oceanographic   variables,   a   list   of   recommended   symbols,   and   succinct   lists   of  
thermodynamic   formulae   are   given   in   the   appendices.      Much   of   this   work   has   appeared  
elsewhere   in   the   published   literature   but   is   collected   here   in   a   condensed   form   for   the  
users'ʹ  convenience.      
Two  introductory  articles  about  TEOS-­‐‑10,  namely  “What  every  oceanographer  needs  
to   know   about   TEOS-­‐‑10   (The   TEOS-­‐‑10   Primer)”   (Pawlowicz,   2010b),   and   “Getting  
started   with   TEOS-­‐‑10   and   the   Gibbs   Seawater   (GSW)   Oceanographic   Toolbox”  
(McDougall   and   Barker,   2011)   are   available   from   www.TEOS-10.org. An   introductory  
article,  Pawlowicz  et  al.  (2012),  describes  the  multi-­‐‑year  scientific  puzzles  with  which  we  
wrestled  during  the  development  of  TEOS-­‐‑10.      
  
  

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1.5  A  remark  on  units      

  
The  SIA  software  library  of  TEOS-­‐‑10  is  written  in  terms  of  Absolute  Salinity   S A   in  units  of  
kg   kg-­‐‑1,   Absolute   Temperature   T   in   K,   and   Absolute   Pressure   P    in   Pa,   however  
oceanographic   practice   to   date   has   used   non-­‐‑basic-­‐‑SI   units   for   many   variables,   in  
particular,   temperature   is   usually   measured   on   the   Celsius   ( °C )   scale,   pressure   is   sea  
pressure   quoted   in   decibars   relative   to   the   pressure   of   a   standard   atmosphere   (10.1325  
dbar),   while   salinity   has   had   its   own   oceanography-­‐‑specific   scale,   the   Practical   Salinity  
Scale   of   1978.      In   the   GSW   Oceanographic   Toolbox   of   TEOS-­‐‑10   we   adopt   °C    for   the  
temperature  unit,  pressure  is  sea  pressure  in  dbar  and  Absolute  Salinity   S A   is  expressed  
in   units   of   g   kg−1   so   that   it   takes   numerical   values   close   to   those   of   Practical   Salinity.    
Adopting   these   non-­‐‑basic-­‐‑SI   units   does   not   come   without   a   penalty   as   there   are   many  
thermodynamic   formulae   that   are   more   conveniently   manipulated   when   expressed   in   SI  
units.    As  an  example,  the  freshwater  fraction  of  seawater  is  written  correctly  as   (1 − SA ) ,  
but   it   is   clear   that   in   this   instance   Absolute   Salinity   must   be   expressed   in   kg kg −1   not   in  
g kg −1.     Thus  there  are  cases  within  the  GSW  Toolbox  in  which  SI  units  are  required  and  
this  may  occasionally  cause  some  confusion.    A  common  example  of  this  issue  arises  when  
a  variable  is  differentiated  or  integrated  with  respect  to  pressure.    Nevertheless,  for  many  
applications  it  is  deemed  important  to  remain  close  to  present  oceanographic  practice  even  
though  it  means  that  one  has  to  be  vigilant  to  detect  those  expressions  that  need  a  variable  
to  be  expressed  in  the  less-­‐‑familiar  SI  units.      
  
  

1.6  Recommendations      

  
In   accordance   with   resolution   XXV-­‐‑7   of   the   Intergovernmental   Oceanographic  
Commission  at  its  25th  Assembly  in  June  2009,  and  the  several  Releases  and  Guidelines  of  
the   International   Association   for   the   Properties   of   Water   and   Steam,   the   TEOS-­‐‑10  
thermodynamic  description  of  seawater,  of  ice  and  of  moist  air  has  been  adopted  for  use  
by   oceanographers   in   place   of   the   International   Equation   Of   State   –   1980   (EOS-­‐‑80).      The  
software  to  implement  this  change  is  available  at  the  web  site  www.TEOS-10.org.   
Under   TEOS-­‐‑10   it   is   recognized   that   the   composition   of   seawater   varies   around   the  
world   ocean   and   that   the   thermodynamic   properties   of   seawater   are   more   accurately  
represented  as  functions  of  Absolute  Salinity   S A   than  of  Practical  Salinity   S P .    It  is  useful  
to  think  of  the  transition  from  Practical  Salinity  to  Absolute  Salinity  in  two  steps.     In  the  
first  step  a  seawater  sample  is  effectively  treated  as  though  it  is  Standard  Seawater  and  its  
Reference   Salinity   S R    is   calculated;   Reference   Salinity   may   be   taken   to   be   simply  
proportional  to  Practical  Salinity.    Reference  Salinity  has  SI  units  (for  example,   g kg −1 )  and  
is  the  natural  starting  point  to  consider  the  influence  of  any  variation  in  composition.    In  
the   second   step   the   Absolute   Salinity   Anomaly   is   evaluated   using   one   of   several  
techniques,   the   easiest   of   which   is   via   a   computer   algorithm   that   essentially   interpolates  
between  a  spatial  atlas  of  these  values.    Then  Absolute  Salinity  is  estimated  as  the  sum  of  
Reference   Salinity   and   Absolute   Salinity   Anomaly.      Of   the   four   possible   versions   of  
absolute   salinity,   the   one   that  is  used  as  the  argument  for  the  TEOS-­‐‑10   Gibbs   function   is  
designed  to  provide  accurate  estimates  of  the  density  of  seawater.      
It   is   recognized   that   our   knowledge   of   how   to   estimate   seawater   composition  
anomalies   and   their   effect   on   thermodynamic   properties   is   limited.      Nevertheless,   we  
should   not   continue   to   ignore   the   influence   of   these   composition   variations   on   seawater  
properties   and   on   ocean   dynamics.      As   more   knowledge   is   gained   in   this   area   over   the  
coming  decade  or  so,  and  after  such  knowledge  has  been  duly  published  in  the  scientific  
literature,  any  updated  algorithm  to  evaluate  the  Absolute  Salinity  Anomaly  will  be  made  
available  (with  its  version  number)  from  www.TEOS-10.org.   

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The   salinity   that   is   stored   in   national   data   bases   should   continue   to   be   Practical  
Salinity,   as   this   will   maintain   continuity   of   this   important   time   series.      Oceanographic  
databases   label   stored,   processed   or   exported   parameters   with   the   GF3   code   PSAL   for  
Practical   Salinity   and   SSAL   for   salinity   measured   before   1978   (IOC,   1987).      In   order   to  
avoid   possible   confusion   in   data   bases   between   different   types   of   salinity   it   is   very  
strongly   recommended   that   under   no   circumstances   should   either   Reference   Salinity   or  
Absolute  Salinity  be  stored  in  national  data  bases.      
In   order   to   accurately   calculate   the   thermodynamic   properties   of   seawater,   Absolute  
Salinity  must  be  calculated  by  first  calculating  Reference  Salinity  and  then  adding  on  the  
Absolute  Salinity  Anomaly.    Because  Absolute  Salinity  is  the  appropriate  salinity  variable  
for  use  with  the  equation  of  state,  Absolute  Salinity  is  the  salinity  variable  that  should  be  
published   in   oceanographic   journals.      The   version   number   of   the   software,   or   the   exact  
formula,  that  was  used  to  convert  Reference  Salinity  into  Absolute  Salinity  should  always  
be  stated  in  publications.    Nevertheless,  there  may  be  some  applications  where  the  likely  
future   changes   in   the   algorithm   that   relates   Reference   Salinity   to   Absolute   Salinity  
presents  a  concern,  and  for  these  applications  it  may  be  preferable  to  publish  graphs  and  
tables   in   Reference   Salinity.      For   these   studies   or   where   it   is   clear   that   the   effect   of  
compositional   variations   are   insignificant   or   not   of   interest,   the   Gibbs   function   may   be  
called   with   S R    rather   than   S A .      When   this   is   done,   it   should   be   clearly   stated   that   the  
salinity  variable  that  is  being  graphed  is  Reference  Salinity,  not  Absolute  Salinity.      
The   TEOS-­‐‑10   approach   of   using   thermodynamic   potentials   to   describe   the   properties  
of   seawater,   ice   and   moist   air   means   that   it   is   possible   to   derive   many   more  
thermodynamic   properties   than   were   available   from   EOS-­‐‑80.      The   seawater   properties  
entropy,  internal  energy,  enthalpy  and  particularly  potential  enthalpy  were  not  available  
from  EOS-­‐‑80  but  are  central  to  accurately  calculating  the  transport  of  “heat”  in  the  ocean  
and  hence  the  air-­‐‑sea  heat  flux  in  the  coupled  climate  system.      
Under  EOS-­‐‑80  the  observed  variables   ( SP , t , p )   were  first  used  to  calculate  potential  
temperature   θ    and   then   water   masses   were   analyzed   on   the   SP − θ    diagram.      Curved  
contours   of   potential   density   ρ θ    could   also   be   drawn   on   this   same   SP − θ    diagram.    
Under   TEOS-­‐‑10,   since   density   and   potential   density   are   now   not   functions   of   Practical  
Salinity   SP    but   rather   are   functions   of   Absolute   Salinity   S A ,   it   is   no   longer   possible   to  
draw   isolines   of   potential   density   on   a   SP − θ    diagram.      Rather,   because   of   the   spatial  
variations  of  seawater  composition,  a  given  value  of  potential  density  defines  an  area  on  
the   SP − θ   diagram,  not  a  curved  line.      
Under   TEOS-­‐‑10,   the   observed   variables   ( SP , t , p ) ,   together   with   longitude   and  
latitude,  are  first  used  to  form  Absolute  Salinity   S A ,  and  then  Conservative  Temperature  
Θ   is  evaluated.    Oceanographic  water  masses  are  then  analyzed  on  the   SA − Θ   diagram,  
and   potential   density   ρ Θ    contours   can   also   be   drawn   on   this   SA − Θ    diagram.      The  
computationally-­‐‑efficient   75-­‐‑term   expression   for   the   specific   volume   of   seawater   (of  
appendix   K)   is   a   convenient   and   accurate   equation   of   state   for   observational   and  
theoretical   studies   and   for   ocean   modelling.      Preformed   Salinity   S*    is   used   internally   in  
numerical  ocean  models  where  it  is  important  that  the  salinity  variable  be  conservative.      
When   describing   the   use   of   TEOS-­‐‑10,   it   is   the   present   document   (the   TEOS-­‐‑10  
Manual)  that  should  be  referenced  as  IOC  et  al.  (2010)  [IOC,  SCOR  and  IAPSO,  2010:  The  
international  thermodynamic  equation  of  seawater  –  2010:  Calculation  and  use  of  thermodynamic  
properties.    Intergovernmental  Oceanographic  Commission,  Manuals  and  Guides  No.  56,  
UNESCO  (English),  196  pp].   Two  introductory  articles  about  TEOS-­‐‑10,  namely  “Getting  
started   with   TEOS-­‐‑10   and   the   Gibbs   Seawater   (GSW)   Oceanographic   Toolbox”  
(McDougall   and   Barker,   2011),   and   “What   every   oceanographer   needs   to   know   about  
TEOS-­‐‑10:-­‐‑   The   TEOS-­‐‑10   Primer”   (Pawlowicz,   2010b),   are   available   from   www.TEOS10.org.   

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2.  Basic  Thermodynamic  Properties    
  
  
  

2.1  ITS-­‐‑90  temperature    

  
In   1990   the   International   Practical   Temperature   Scale   1968   (IPTS-­‐‑68)   was   replaced   by   the  
International   Temperature   Scale   1990   (ITS-­‐‑90).      There   are   two   main   methods   to   convert  
between  these  two  temperature  scales;  Rusby’s  (1991)  8th  order  fit  valid  over  a  wide  range  
of   temperatures,   and   Saunders’   (1990)   1.00024   scaling   widely   used   in   the   oceanographic  
community.      The   two   methods   are   formally   indistinguishable   in   the   oceanographic  
temperature   range   because   they   differ   by   less   than   either   the   uncertainty   in  
thermodynamic   temperature   (of   order   1   mK),   or   the   practical   application   of   the   IPTS-­‐‑68  
and   ITS-­‐‑90   scales.      The   differences   between   the   Saunders   (1990)   and   Rusby   (1991)  
formulae  are  less  than  1  mK  throughout  the  temperature  range  -­‐‑2  °C  to  40  °C  and  less  than  
0.03mK  in  the  temperature  range  between  -­‐‑2  °C  and  10  °C.    Hence  we  recommend  that  the  
oceanographic  community  continues  to  use  the  Saunders  formula    

(t68 /°C)

= 1.00024 (t90 /°C) .   

(2.1.1)  

One  application  of  this  formula  is  in  the  updated  computer  algorithm  for  the  calculation  of  
Practical   Salinity   (PSS-­‐‑78)   in   terms   of   conductivity   ratio.      The   algorithms   for   PSS-­‐‑78  
require   t68   as  the  temperature  argument.    In  order  to  use  these  algorithms  with   t90   data,  
t68   may  be  calculated  using  (2.1.1).      
An  extended  discussion  of  the  different  temperature  scales,  their  inherent  uncertainty  
and  the  reasoning  for  our  recommendation  of  (2.1.1)  can  be  found  in  appendix  A.1.      
  
  

2.2  Sea  pressure    

  
Sea  pressure   p   is  defined  to  be  the  Absolute  Pressure   P   less  the  Absolute  Pressure  of  one  
standard  atmosphere,   P0 ≡ 101 325Pa;   that  is    

p ≡ P − P0 .   

(2.2.1)  

It  is  common  oceanographic  practice  to  express  sea  pressure  in  decibars  (dbar).    Another  
common  pressure  variable  that  arises  naturally  in  the  calibration  of  sea-­‐‑board  instruments  
is   gauge   pressure   p gauge    which   is   Absolute   Pressure   less   the   Absolute   Pressure   of   the  
atmosphere   at   the   time   of   the   instrument’s   calibration   (perhaps   in   the   laboratory,   or  
perhaps  at  sea).    Because  atmospheric  pressure  changes  in  space  and  time,  sea  pressure   p   
is   preferred   as   a   thermodynamic   variable   as   it   is   unambiguously   related   to   Absolute  
Pressure.    The  seawater  Gibbs  function  in  the  GSW  Toolbox  is  expressed  as  a  function  of  
sea  pressure   p   (functionally  equivalent  to  the  use  of  Absolute  Pressure   P   in  the  IAPWS  
Releases  and  in  the  SIA  library);  that  is,   g   is  a  function  of   p ,  it  is  not  a  function  of   p gauge .      
  
  

2.3  Practical  Salinity  
  

Practical   Salinity   S P    is   defined   on   the   Practical   Salinity   Scale   of   1978   (UNESCO   (1981,  
1983))   in   terms   of   the   conductivity   ratio   K15    which   is   the   electrical   conductivity   of   the  
sample  at  temperature   t68   =  15  °C  and  pressure  equal  to  one  standard  atmosphere  ( p   =  0  
dbar   and   Absolute   Pressure   P    equal   to   101   325   Pa),   divided   by   the   conductivity   of   a  

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standard   potassium   chloride   (KCl)   solution   at   the   same   temperature   and   pressure.      The  
mass  fraction  of  KCl  (i.e.,  the  mass  of  KCl  per  mass  of  solution)  in  the  standard  solution  is  
32.4356 × 10−3 .      When   K15    =   1,   the   Practical   Salinity   S P    is   by   definition   35.      Note   that  
Practical   Salinity   is   a   unit-­‐‑less   quantity.      Though   sometimes   convenient,   it   is   technically  
incorrect   to   quote   Practical   Salinity   in   “psu”;   rather   it   should   be   quoted   as   a   certain  
Practical   Salinity   “on   the   Practical   Salinity   Scale   PSS-­‐‑78”.      The   formula   for   evaluating  
Practical  Salinity  can  be  found  in  appendix  E  along  with  the  simple  change  that  must  be  
made   to   the   UNESCO   (1983)   formulae   so   that   the   algorithm   for   Practical   Salinity   can   be  
called   with   ITS-­‐‑90   temperature   as   an   input   parameter   rather   than   the   older   t68   
temperature  in  which  the  PSS-­‐‑78  algorithms  were  defined.    The  reader  is  also  directed  to  
the   CDIAC   chapter   on   “Method   for   salinity   (conductivity   ratio)   measurement”   which  
describes  best  practice  in  measuring  the  conductivity  ratio  of  seawater  samples  (Kawano  
(2009)).      
Practical  Salinity  is  defined  only  in  the  range   2 < SP < 42.     Practical  Salinities  below  2  
can   be   evaluated   from   conductivity   using   the   PSS-­‐‑78   extension   of   Hill   et   al.   (1986).      We  
have  modified  this  Hill  et  al.  (1986)  extension  to  make  the  result  a  continuous  function  of  is  
arguments;   this   function   is   available   as   gsw_SP_from_C   in   the   GSW   Oceanographic  
Toolbox.    Samples  exceeding  a  Practical  Salinity  of  42  must  be  diluted  to  the  valid  salinity  
range  and  the  measured  value  should  be  adjusted  based  on  the  added  water  mass  and  the  
conservation  of  sea  salt  during  the  dilution  process  (as  discussed  in  appendix  E).      
Data  stored  in  national  and  international  data  bases  should,  as  a  matter  of  principle,  be  
measured   values   rather   than   derived   quantities.      Consistent   with   this,   we   recommend  
continuing   to   store   the   measured   (in   situ)   temperature   rather   than   the   derived   quantity,  
Conservative   Temperature.      Similarly   we   strongly   recommend   that   Practical   Salinity   S P   
continue   to   be   the   salinity   variable   that   is   stored   in   such   data   bases   since   S P    is   closely  
related   to   the   measured   values   of   conductivity.      This   recommendation   has   the   very  
important   advantage   that   there   is   no   change   to   the   present   practice   and   so   there   is   less  
chance  of  transitional  errors  occurring  in  national  and  international  data  bases  because  of  
the  adoption  of  Absolute  Salinity  in  oceanography.      
  
  

2.4  Reference  Composition  and  the  Reference-­‐‑Composition  Salinity  Scale    

  
The  reference  composition  of  seawater  is  defined  by  Millero  et  al.  (2008a)  as  the  exact  mole  
fractions   given   in   Table   D.3   of   appendix   D   below.      This   composition   was   introduced   by  
Millero  et  al.  (2008a)  as  their  best  estimate  of  the  composition  of  Standard  Seawater,  being  
seawater   from   the   surface   waters   of   a   certain   region   of   the   North   Atlantic.      The   exact  
location  for  the  collection  of  bulk  material  for  the  preparation  of  Standard  Seawater  is  not  
specified.      Ships   gathering   this   bulk   material   are   given   guidance   notes   by   the   Standard  
Seawater  Service,  requesting  that  water  be  gathered  between  longitudes  50°W  and  40°W,  
in   deep   water,   during   daylight   hours.      Reference-­‐‑Composition   Salinity   S R    (or   Reference  
Salinity  for  short)  was  designed  by  Millero  et  al.  (2008a)  to  be  the  best  estimate  of  the  mass-­‐‑
fraction   Absolute   Salinity   SA    of   Standard   Seawater.      Independent   of   accuracy  
considerations,   it   provides   a   precise   measure   of   dissolved   material   in   Standard   Seawater  
and  is  the  correct  salinity  argument  to  be  used  in  the  TEOS-­‐‑10  Gibbs  function  for  Standard  
Seawater.      
For   the   range   of   salinities   where   Practical   Salinities   are   defined   (that   is,   in   the   range  
2 < SP < 42 )  Millero  et  al.  (2008a)  show  that    

SR ≈ uPS SP               where             uPS ≡ (35.165 04 35) g kg −1 .  

(2.4.1)  

In  the  range   2 < SP < 42 ,  this  equation  expresses  the  Reference  Salinity  of  a  seawater  sample  
on   the   Reference-­‐‑Composition   Salinity   Scale   (Millero   et   al.   (2008a)).      For   practical  

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purposes,  this  relationship  can  be  taken  to  be  an  equality  since  the  approximate  nature  of  
this   relation   only   reflects   the   extent   to   which   Practical   Salinity,   as   determined   from  
measurements   of   conductivity   ratio,   temperature   and   pressure,   varies   when   a   seawater  
sample   is   heated,   cooled   or   subjected   to   a   change   in   pressure   but   without   exchange   of  
mass   with   its   surroundings.      The   Practical   Salinity   Scale   of   1978   was   designed   to   satisfy  
this   property   as   accurately   as   possible   within   the   constraints   of   the   polynomial  
approximations  used  to  determine  Chlorinity  (and  hence  Practical  Salinity)  in  terms  of  the  
measured  conductivity  ratio.      
From   Eqn.   (2.4.1),   a   seawater   sample   of   Reference   Composition   whose   Practical  
Salinity   S P    is   35   has   a   Reference   Salinity   S R    of   35.165 04 g kg−1 .      Millero   et   al.   (2008a)  
estimate   that   the   absolute   uncertainty   in   this   value   is   ± 0.007 g kg −1 .      The   difference  
between   the   numerical   values   of   Reference   and   Practical   Salinities   can   be   traced   back   to  
the   original   practice   of   determining   salinity   by   evaporation   of   water   from   seawater   and  
weighing   the   remaining   solid   material.      This   process   also   evaporated   some   volatile  
components  and  most  of  the   0.165 04 g kg−1   salinity  difference  is  due  to  this  effect.        
Measurements  of  the  composition  of  Standard  Seawater  at  a  Practical  Salinity   S P   of  35  
using   mass   spectrometry   and/or   ion   chromatography   are   underway   and   may   provide  
updated  estimates  of  both  the  value  of  the  mass  fraction  of  dissolved  material  in  Standard  
Seawater   and   its   uncertainty.      Any   update   of   this   value   will   not   change   the   Reference-­‐‑
Composition  Salinity  Scale  and  so  will  not  affect  the  calculation  of  Reference  Salinity  nor  
of   Absolute   Salinity   as   calculated   from   Reference   Salinity   plus   the   Absolute   Salinity  
Anomaly.      
Oceanographic  databases  label  stored,  processed  or  exported  parameters  with  the  GF3  
code  PSAL  for  Practical  Salinity  and  SSAL  for  salinity  measured  before  1978  (IOC,  1987).    
In   order   to   avoid   possible   confusion   in   data   bases   between   different   types   of   salinity,  
under  no  circumstances  should  either  Reference  Salinity  or  Absolute  Salinity  be  stored  in  
national  data  bases.      
Detailed   information   on   Reference   Composition   and   Reference   Salinity   can   be   found  
in  Millero  et  al.  (2008a).    For  the  user'ʹs  convenience  a  brief  summary  of  information  from  
Millero   et   al.   (2008a),   including   the   precise   definition   of   Reference   Salinity   is   given   in  
appendix  A.3  and  in  Table  D3  of  appendix  D.      
  
  

2.5  Absolute  Salinity    

  
Absolute   Salinity   is   traditionally   defined   as   the   mass   fraction   of   dissolved   material   in  
seawater.      For   seawater   of   Reference   Composition,   Reference   Salinity   gives   our   current  
best   estimate   of   Absolute   Salinity.      To   deal   with   composition   anomalies   in   seawater,   we  
need   an   extension   of   the   Reference-­‐‑Composition   Salinity   S R    that   provides   a   useful  
measure   of   salinity   over   the   full   range   of   oceanographic   conditions   and   agrees   precisely  
with   Reference   Salinity   when   the   dissolved   material   has   Reference   Composition.      When  
composition   anomalies   are   present,   no   single   measure   of   dissolved   material   can   fully  
represent  the  influences  on  seawater  properties  on  all  thermodynamic  properties,  so  it  is  
clear   that   either   additional   information   will   be   required   or   compromises   will   have   to   be  
made.    In  addition,  we  would  like  to  introduce  a  measure  of  salinity  that  is  traceable  to  the  
SI   (Seitz   et   al.,   2011)   and   maintains   the   high   accuracy   of   PSS-­‐‑78   necessary   for  
oceanographic  applications.    The  introduction  of  "ʺDensity  Salinity"ʺ   SAdens   addresses  both  of  
these  issues;  it  is  this  type  of  absolute  salinity  that  in  TEOS-­‐‑10  parlance  is  labeled   S A   and  
called  Absolute  Salinity.    In  this  section  we  explain  how   S A   is  defined  and  evaluated,  but  
first  we  outline  other  choices  that  are  available  for  the  definition  of  absolute  salinity  in  the  
presence  of  composition  variations  in  seawater.      

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The  most  obvious  definition  of  absolute  salinity  is  “the  mass  fraction  of  dissolved  non-­‐‑
H2O   material   in   a   seawater   sample   at   its   temperature   and   pressure”.      This   seemingly  
simple   definition   is   actually   far   more   subtle   than   it   first   appears.      Notably,   there   are  
questions  about  what  constitutes  water  and  what  constitutes  dissolved  material.    Perhaps  
the  most  obvious  example  of  this  issue  occurs  when  CO2  is  dissolved  in  water  to  produce  a  
mixture   of   CO2,   H2CO3,   HCO3-­‐‑,   CO32-­‐‑,   H+,   OH-­‐‑   and   H2O,   with   the   relative   proportions  
depending  on  dissociation  constants  that  depend  on  temperature,  pressure  and  pH.    Thus,  
the   dissolution   of   a   given   mass   of   CO2   in   pure   water   essentially   transforms   some   of   the  
water  into  dissolved  material.    A  change  in  the  temperature  and  even  an  adiabatic  change  
in   pressure   results   in   a   change   in   absolute   salinity   defined   in   this   way   due   to   the  
dependence   of   chemical   equilibria   on   temperature   and   pressure.      Pawlowicz   et   al.   (2010)  
and  Wright  et  al.  (2011)  address  this  second  issue  by  defining  “Solution  Absolute  Salinity”  
(usually  shortened  to  “Solution  Salinity”),   S Asoln ,  as  the  mass  fraction  of  dissolved  non-­‐‑H2O  
material  after  a  seawater  sample  is  brought  to  the  constant  temperature   t = 25°C   and  the  
fixed  sea  pressure  0  dbar  (fixed  Absolute  Pressure  of  101  325  Pa).      
Another  measure  of  absolute  salinity  is  the  “Added-­‐‑Mass  Salinity”   SAadd   which  is   S R   
plus  the  mass  fraction  of  material  that  must  be  added  to  Standard  Seawater  to  arrive  at  the  
concentrations  of  all  the  species  in  the  given  seawater  sample,  after  chemical  equilibrium  
has   been   reached,   and   after   the   sample   is   brought   to   the   constant   temperature   t = 25°C   
and   the   fixed   sea   pressure   of   0   dbar.      The   estimation   of   absolute   salinity   SAadd    is   not  
straightforward   for   seawater   with   anomalous   composition   because   while   the   final  
equilibrium  state  is  known,  one  must  iteratively  determine  the  mass  of  anomalous  solute  
prior   to   any   chemical   reactions   with   Reference-­‐‑Composition   seawater.      Pawlowicz   et   al.  
(2010)   provide   an   algorithm   to   achieve   this,   at   least   approximately.      This   definition   of  
absolute   salinity,   SAadd ,   is   useful   for   laboratory   studies   of   artificial   seawater   and   it   differs  
from   S Asoln   because  of  the  chemical  reactions  that  take  place  between  the  several  species  of  
the   added   material   and   the   components   of   seawater   that   exist   in   Standard   Seawater.    
Added-­‐‑Mass   Salinity   may   be   the   most   appropriate   form   of   salinity   for   accurately  
accounting   for   the   mass   of   salt   discharged   by   rivers   and   hydrothermal   vents   into   the  
ocean.      
“Preformed   Absolute   Salinity”   (usually   shortened   to   “Preformed   Salinity”),   S* ,   is   a  
different  type  of  absolute  salinity  which  is  specifically  designed  to  be  as  close  as  possible  
to   being   a   conservative   variable.      That   is,   S*    is   designed   to   be   insensitive   to  
biogeochemical   processes   that   affect   the   other   types   of   salinity   to   varying   degrees.    
Preformed   Salinity   S*    is   formed   by   first   estimating   the   contribution   of   biogeochemical  
processes   to   one   of   the   salinity   measures   S A ,   S Asoln ,   or   SAadd ,   and   then   subtracting   this  
contribution  from  the  appropriate  salinity  variable.    In  this  way  Preformed  Salinity   S*   is  
designed   to   be   a   conservative   salinity   variable   which   is   independent   of   the   effects   of   the  
non-­‐‑conservative   biogeochemical   processes.      S*    will   find   a   prominent   role   in   ocean  
modeling.      The   three   types   of   absolute   salinity   S Asoln ,   SAadd    and   S*    are   discussed   in   more  
detail   in   appendices   A.4   and   A.20,   where   approximate   relationships   between   these  
variables  and   SA ≡ SAdens   are  presented,  based  on  the  work  of  Pawlowicz  et  al.  (2010)  and  
Wright   et   al.   (2011).      Note   that   for   a   sample   of   Standard   Seawater,   all   of   the   five   salinity  
variables   S R ,   S A ,   S Asoln ,   SAadd   and   S*   and  are  equal.    
There   is   no   simple   means   to   measure   either   S Asoln    or   SAadd    for   the   general   case   of   the  
arbitrary  addition  of  many  components  to  Standard  Seawater.    Hence  a  more  precise  and  
easily   determined   measure   of   the   amount   of   dissolved   material   in   seawater   is   required  
and   TEOS-­‐‑10   adopts   “Density   Salinity”   for   this   purpose.      “Density   Salinity”   SAdens    is  
defined  as  the  value  of  the  salinity  argument  of  the  TEOS-­‐‑10  expression  for  density  which  
gives   the   sample’s   actual   measured   density   at   the   temperature   t = 25°C    and   at   the   sea  
pressure   p   =  0  dbar.    When  there  is  no  risk  of  confusion,  “Density  Salinity”  is  also  called  

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Absolute   Salinity   with   the   label   S A ,   that   is   SA ≡ SAdens .      Usually   we   do   not   have   accurate  
measurements   of   density   but   rather   we   have   measurements   of   Practical   Salinity,  
temperature   and   pressure,   and   in   this   case,   Absolute   Salinity   may   be   calculated   using  
Practical   Salinity   and   the   computer   algorithm   of   McDougall,   et   al.   (2012)   which   provides  
an  estimate  of   δ SA = SA − SR .    This  computer  program  was  formed  as  follows.      
In  a  series  of  papers  (Millero  et  al.  (1976a,  1978,  2000,  2008b),  McDougall  et  al.  (2012)),  
accurate   measurements   of   the   density   of   seawater   samples,   along   with   the   Practical  
Salinity  of  those  samples,  gave  estimates  of   δ SA = SA − SR   from  most  of  the  major  basins  of  
the   world   ocean.      This   was   done   by   first   calculating   the   “Reference   Density”   from   the  
TEOS-­‐‑10  equation  of  state  using  the  sample’s  Reference  Salinity  as  the  salinity  argument  
(this   calculation   essentially   assumes   that   the   seawater   sample   has   the   composition   of  
Standard   Seawater).      The   difference   between   the   measured   density   and   the   “Reference  
Density”  was  then  used  to  estimate  the  Absolute  Salinity  Anomaly   δ SA = SA − SR   (Millero  
et  al.  (2008a)).    The  McDougall  et  al.  (2012)  algorithm  is  based  on  the  observed  correlation  
between  this   SA − SR   data  and  the  silicate  concentration  of  the  seawater  samples  (Millero  
et   al.   ,   2008a),   with   the   silicate   concentration   being   estimated   by   interpolation   of   a   global  
atlas  (Gouretski  and  Koltermann  (2004)).      
The  algorithm  for  Absolute  Salinity  takes  the  form    

SA = SR + δ SA = SA ( SP , φ , λ , p ) ,   

(2.5.1)  

Where   φ    is   latitude   (degrees   North),   λ    is   longitude   (degrees   east,   ranging   from   0°E   to  
360°E)  while   p   is  sea  pressure.      
Heuristically   the   dependence   of   δ SA = SA − SR    on   silicate   can   be   thought   of   as  
reflecting   the   fact   that   silicate   affects   the   density   of   a   seawater   sample   without  
significantly   affecting   its   conductivity   or   its   Practical   Salinity.      In   practice   this   explains  
about   60%   of   the   effect   and   the   remainder   is   due   to   the   correlation   of   other   composition  
anomalies  (such  as  nitrate)  with  silicate.    In  the  McDougall  et  al.  (2012)  algorithm  the  Baltic  
Sea  is  treated  separately,  following  the  work  of  Millero  and  Kremling  (1976)  and  Feistel  et  
al.   (2010c,   2010d),   because   some   rivers   flowing   into   the   Baltic   are   unusually   high   in  
calcium  carbonate.      
  

  

  
Figure   1.      A   sketch   indicating   how   thermodynamic   quantities  
such  as  density  are  calculated  as  functions  of  Absolute  Salinity.    
Absolute   Salinity   is   found   by   adding   an   estimate   of   the  
Absolute  Salinity  Anomaly   δ SA   to  the  Reference  Salinity.    

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Since   the   density   of   seawater   is   rarely   measured,   we   recommend   the   approach  
illustrated   in   Figure   1   as   a   practical   method   to   include   the   effects   of   composition  
anomalies   on   estimates   of   Absolute   Salinity   and   density.      When   composition   anomalies  
are  not  known,  the  algorithm  of  McDougall  et  al.  (2012)  may  be  used  to  estimate  Absolute  
Salinity   in   terms   of   Practical   Salinity   and   the   spatial   location   of   the   measurement   in   the  
world  oceans.      
The   difference   between   Absolute   Salinity   and   Reference   Salinity,   as   estimated   by   the  
McDougall  et  al.  (2012)  algorithm,  is  illustrated  in  Figure  2  (a)  at  a  pressure  of  2000  dbar,  
and  in  a  vertical  section  through  the  Pacific  Ocean  in  Figure  2  (b).      
Of   the   approximately   800   samples   of   seawater   from   the   world   ocean   that   have   been  
examined   to   date   for   δ SA = SA − SR    the   standard   error   (square   root   of   the   mean   squared  
value)  of   δ SA = SA − SR   is  0.0107  g  kg-­‐‑1.    That  is,  the  “typical”  value  of   δ SA = SA − SR   of  the  
811  samples  taken  to  date  is  0.0107  g  kg-­‐‑1.    The  standard  error  of  the  difference  between  the  
measured  values  of   δ SA = SA − SR   and  the  values  evaluated  from  the  computer  algorithm  
of   McDougall   et   al.   (2012)   is   0.0048   g   kg-­‐‑1.      The   maximum   values   of   δ SA = SA − SR    of  
approximately  0.025  g  kg-­‐‑1  occur  in  the  North  Pacific.      

  

Figure  2  (a).    Absolute  Salinity  Anomaly   δ SA   at   p   =  2000  dbar.  
  

Figure  2  (b).    A  vertical  section  of  Absolute  Salinity  
Anomaly   δ SA   along  180oE  in  the  Pacific  Ocean.      

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The  thermodynamic  description  of  seawater  and  of  ice  Ih  as  defined  in  IAPWS-­‐‑08  and  
IAPWS-­‐‑06   has   been   adopted   as   the   official   description   of   seawater   and   of   ice   Ih   by   the  
Intergovernmental   Oceanographic   Commission   in   June   2009.      These   thermodynamic  
descriptions   of   seawater   and   ice   were   endorsed   recognizing   that   the   techniques   for  
estimating  Absolute  Salinity  will  likely  improve  over  the  coming  decades.    The  algorithm  
for   evaluating   Absolute   Salinity   in   terms   of   Practical   Salinity,   latitude,   longitude   and  
pressure,   will   likely   be   updated   from   time   to   time,   after   relevant   appropriately   peer-­‐‑
reviewed  publications  have  appeared,  and  such  an  updated  algorithm  will  appear  on  the  
www.TEOS-10.org web   site.      Users   of   this   software   should   state   in   their   published   work  
which  version  of  the  software  was  used  to  calculate  Absolute  Salinity.      
The  present  computer  software   which  evaluates  Absolute  Salinity   S A   given  the  input  
variables   Practical   Salinity S P ,   longitude   λ ,   latitude   φ    and   pressure   is   available   at  
www.TEOS-10.org. Absolute   Salinity   is   also   available   as   the   inverse   function   of   density  
SA (T , P, ρ )    in   the   SIA   library   of   computer   algorithms   as   the   algorithm   sea_sa_si   (see  
appendix  M)  and  in  the  GSW  Toolbox  as  the  algorithm  gsw_SA_from_rho_t_exact.      
  
  

2.6  Gibbs  function  of  seawater    

  
The   Gibbs   function   of   seawater   g ( SA , t , p )    is   related   to   the   specific   enthalpy   h    and  
entropy   η ,    by   g = h − (T0 + t )η    where   T0 = 273.15K    is   the   Celsius   zero   point.      TEOS-­‐‑10  
defines  the  Gibbs  function  of  seawater  as  the  sum  of  a  pure  water  part  and  the  saline  part  
(IAPWS-­‐‑08)    

g ( SA , t , p ) = g W ( t , p ) + g S ( S A , t , p ) .  

(2.6.1)  

The   saline   part   of   the   Gibbs   function,   g S ,    is   valid   over   the   ranges   0   < S A <   42   g   kg–1,    
–6.0   °C   <   t    <   40   °C,   and   0 < p < 104 dbar ,   although   its   thermal   and   colligative   properties  
are  valid  up  to   t   =  80  °C  and   S A   =  120  g  kg–1  at   p   =  0.      
                        The  pure-­‐‑water  part  of  the  Gibbs  function,   g W ,   can  be  obtained  from  the  IAPWS-­‐‑95  
Helmholtz  function  of  pure-­‐‑water  substance  which  is  valid  from  the  freezing  temperature  
or  from  the  sublimation  temperature  to  1273  K.    Alternatively,  the  pure-­‐‑water  part  of  the  
Gibbs   function   can   be   obtained   from   the   IAPWS-­‐‑09   Gibbs   function   which   is   valid   in   the  
oceanographic   ranges   of   temperature   and   pressure,   from   less   than   the   freezing  
temperature   of   seawater   (at   any   pressure),   up   to   40 °C    (specifically   from  
− (2.65 + ( p + P0 ) × 0.0743 MPa −1 ) °C    to   40   °C),   and   in   the   pressure   range   0 < p < 104 dbar .    
For  practical  purposes  in  oceanography  it  is  expected  that  IAPWS-­‐‑09  will  be  used  because  
it   executes   approximately   two   orders   of   magnitude   faster   than   the   IAPWS-­‐‑95   code   for  
pure   water.      However   if   one   is   concerned   with   temperatures   between   40 °C    and   80 °C   
then   one   must   use   the   IAPWS-­‐‑95   version   of   g W    (expressed   in   terms   of   absolute  
temperature  (K)  and  Absolute  Pressure  (Pa))  rather  than  the  IAPWS-­‐‑09  version.      
The  thermodynamic  properties  derived  from  the  IAPWS-­‐‑95  (the  Release  providing  the  
Helmholtz  function  formulation  for  pure  water)  and  IAPWS-­‐‑08  (the  Release  endorsing  the  
Feistel   (2008)   Gibbs   function)   combination   are   available   from   the   SIA   software   library,  
while   that   derived   from   the   IAPWS-­‐‑09   (the   Release   endorsing   the   pure   water   part   of  
Feistel   (2003))   and   IAPWS-­‐‑08   combination   are   available   from   the   GSW   Oceanographic  
Toolbox.      The   GSW   Toolbox   is   restricted   to   the   oceanographic   standard   range   in  
temperature   and   pressure,   however   the   validity   of   results   extends   at   p    =   0   to   Absolute  
Salinity   up   to   mineral   saturation   concentrations   (Marion   et   al.   2009).      Specific   volume  
(which   is   the   pressure   derivative   of   the   Gibbs   function)   is   presently   an   extrapolated  
quantity  outside  the  Neptunian  range  (i.  e.  the  oceanographic  range)  of  temperature  and  
Absolute   Salinity   at   p    =   0,   and   exhibits   errors   there   of   up   to   3%.      We   emphasize   that  
models   of   seawater   properties   that   use   a   single   salinity   variable,   SA ,    as   input   require  

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approximately  fixed  chemical  composition  ratios  (e.g.,  Na/Cl,  Ca/Mg,  Cl/HCO3,  etc.).    As  
seawater  evaporates  or  freezes,  eventually  minerals  such  as  CaCO3  will  precipitate.    Small  
anomalies  are  reasonably  handled  by  using   S A   as  the  input  variable  (see  section  2.5)  but  
precipitation   may   cause   large   deviations   from   the   nearly   fixed   ratios   associated   with  
standard  seawater.    Under  extreme  conditions  of  precipitation,  models  of  seawater  based  
on  the  Millero  et  al.  (2008a)  Reference  Composition  will  no  longer  be  applicable.    Figure  3  
illustrates   SA − t   boundaries  of  validity  (determined  by  the  onset  of  precipitation)  for  2008  
(pCO2  =  385   µ atm )  and  2100  (pCO2  =  550   µ atm )  (from  Marion  et  al.  (2009)).      
  

Figure  3.    The  boundaries  of  validity  of  the  Millero  et  al.  (2008a)  
composition  at   p   =  0  in  Year  2008  (solid  lines)  and  potentially  
in  Year  2100  (dashed  lines).    At  high  salinity,  calcium  carbonate  
saturates   first   and   comes   out   of   solution;   thereafter   the  
Reference   Composition   of   Standard   Seawater   of   Millero   et   al.  
(2008a)  does  not  apply.      

  

  
The  Gibbs  function  (2.6.1)  contains  four  arbitrary  constants  that  cannot  be  determined  
by   any   set   of   thermodynamic   measurements.      These   arbitrary   constants   mean   that   the  
Gibbs   function   (2.6.1)   is   unknown   and   unknowable   up   to   the   arbitrary   function   of  
temperature  and  Absolute  Salinity  (where   T0   is  the  Celsius  zero  point,  273.15   K )    

⎡⎣a1 + a2 (T0 + t )⎤⎦ + ⎡⎣a3 + a4 (T0 + t )⎤⎦ SA   

(2.6.2)  

(see  for  example  Fofonoff  (1962)  and  Feistel  and  Hagen  (1995)).    The  first  two  coefficients  
a1    and   a2    are   arbitrary   constants   of   the   pure   water   Gibbs   function   g W ( t , p )    while   the  
second  two  coefficients   a3   and   a4   are  arbitrary  coefficients  of  the  saline  part  of  the  Gibbs  
function   g S ( SA , t , p ) .       Following   generally   accepted   convention,   the   first   two   coefficients  
are  chosen  to  make  the  entropy  and  internal  energy  of  liquid  water  zero  at  the  triple  point    
and    

η W ( tt , pt ) = 0   

(2.6.3)  

u W ( tt , pt ) = 0   

(2.6.4)  

as   described   in   IAPWS-­‐‑95   and   in   more   detail   in   Feistel   et   al.   (2008a)   for   the   IAPWS-­‐‑95  
Helmholtz   function   description   of   pure   water   substance.      When   the   pure-­‐‑water   Gibbs  
function   g W ( t , p )   of  (2.6.1)  is  taken  from  the  fitted  Gibbs  function  of  Feistel  (2003),  the  two  
arbitrary  constants   a1   and   a2   are  (in  the  appropriate  non-­‐‑dimensional  form)   g 00   and   g10   
of   the   table   in   appendix   G   below.      These   values   of   g 00    and   g10    are   not   identical   to   the  
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17

values   in   Feistel   (2003)   because   the   present   values   have   been   taken   from   IAPWS-­‐‑09   and  
have  been  chosen  to  most  accurately  achieve  the  triple-­‐‑point  conditions  (2.6.3)  and  (2.6.4)  
as  discussed  in  Feistel  et  al.  (2008a).      
The  remaining  two  arbitrary  constants   a3   and   a4   of  (2.6.2)  are  determined  by  ensuring  
that  the  specific  enthalpy   h   and  specific  entropy   η   of  a  sample  of  standard  seawater  with  
standard-­‐‑ocean   properties   ( SSO , tSO , pSO ) = (35.165 04 g kg −1, 0 °C, 0 dbar)    are   both   zero,  
that  is  that    
(2.6.5)  
h ( SSO , tSO , pSO ) = 0   
and    

η ( SSO , tSO , pSO ) = 0.  

(2.6.6)  

In  more  detail,  these  conditions  are  actually  officially  written  as  (Feistel  (2008),  IAPWS-­‐‑08)    
and    

hS ( SSO , tSO , pSO ) = u W (tt , pt ) − h W ( tSO , pSO )   

(2.6.7)    

η S ( SSO , tSO , pSO ) = η W (tt , pt ) − η W ( tSO , pSO ) .  

(2.6.8)    

Written  in  this  way,  (2.6.7)  and  (2.6.8)  use  properties  of  the  pure  water  description  (the  
right-­‐‑hand  sides)  to  constrain  the  arbitrary  constants  in  the  saline  Gibbs  function.    While  
the  first  terms  on  the  right-­‐‑hand  sides  of  these  equations  are  zero  (see  (2.6.3)  and  (2.6.4)),  
these   constraints   on   the   saline   Gibbs   function   are   written   this   way   so   that   they   are  
independent   of   any   subsequent   change   in   the   arbitrary   constants   involved   in   the  
thermodynamic   description   of   pure   water.      While   the   two   slightly   different  
thermodynamic   descriptions   of   pure   water,   namely   IAPWS-­‐‑95   and   IAPWS-­‐‑09,   both  
achieve   zero   values   of   the   internal   energy   and   entropy   at   the   triple   point   of   pure   water,  
the   values   assigned   to   the   enthalpy   and   entropy   of   pure   water   at   the   temperature   and  
pressure  of  the  standard  ocean,   h W ( tSO , pSO )   and   η W ( tSO , pSO )   on  the  right-­‐‑hand  sides  of  
(2.6.7)   and   (2.6.8),   are   slightly   different   in   the   two   cases.      For   example   h W ( tSO , pSO )    is  
3.3x10−3    J kg −1   from  IAPWS-­‐‑09  (as  described  in  the  table  of  appendix  G)  compared  with  
the   round-­‐‑off   error   of   2 x10−8    J kg −1    when   using   IAPWS-­‐‑95   with   double-­‐‑precision  
arithmetic.    This  issues  is  discussed  in  more  detail  in  section  3.3.      
The  polynomial  form  and  the  coefficients  for  the  pure  water  Gibbs  function   g W ( t , p )   
from   Feistel   (2003)   and   IAPWS-­‐‑09   are   given   in   appendix   G,   while   the   combined  
polynomial   and   logarithmic   form   and   the   coefficients   for   the   saline   part   of   the   Gibbs  
function   g S ( S A , t , p )   (from  Feistel  (2008)  and  IAPWS-­‐‑08)  are  reproduced  in  appendix  H.      
SCOR/IAPSO  Working  Group  127  has  independently  checked  that  the  Gibbs  functions  
of   Feistel   (2003)   and   of   Feistel   (2008)   do   in   fact   fit   the   underlying   data   of   various  
thermodynamic  quantities  to  the  accuracy  quoted  in  those  two  fundamental  papers.    This  
checking  was  performed  by  Giles  M.  Marion,  and  is  summarized  in  appendix  O.    Further  
checking   of   these   Gibbs   functions   has   occurred   in   the   process   leading   up   to   IAPWS  
approving  these  Gibbs  function  formulations  as  the  Releases  IAPWS-­‐‑08  and  IAPWS-­‐‑09.      
Discussions  of  how  well  the  Gibbs  functions  of  Feistel  (2003)  and  Feistel  (2008)  fit  the  
underlying  (laboratory)  data  of  density,  sound  speed,  specific  heat  capacity,  temperature  
of  maximum  density  etc  may  be  found  in  those  papers,  along  with  comparisons  with  the  
corresponding   algorithms   of   EOS-­‐‑80.      The   IAPWS-­‐‑09   release   discusses   the   accuracy   to  
which   the   Feistel   (2003)   Gibbs   function   fits   the   underlying   thermodynamic   potential   of  
IAPWS-­‐‑95;   in   summary,   for   the   variables   density,   thermal   expansion   coefficient   and  
specific   heat   capacity,   the   rms   misfit   between   IAPWS-­‐‑09   and   IAPWS-­‐‑95,   in   the   region   of  
validity  of  IAPWS-­‐‑09,  are  a  factor  of  between  20  and  100  less  than  the  corresponding  error  
in  the  laboratory  data  to  which  both  thermodynamic  potentials  were  fitted.    Hence,  in  the  
oceanographic  range  of  parameters,  IAPWS-­‐‑09  and  IAPWS-­‐‑95  may  be  regarded  as  equally  
accurate  thermodynamic  descriptions  of  pure  liquid  water.      
The  Gibbs  function   g   has  units  of   J kg −1   in  both  the  SIA  and  GSW  software  libraries.      
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

2.7  Specific  volume    

  
The   specific   volume   of   seawater   v    is   given   by   the   pressure   derivative   of   the   Gibbs  
function  at  constant  Absolute  Salinity   S A   and  in  situ  temperature   t ,   that  is    

v = v ( SA , t, p ) = g P = ∂g ∂P S

A ,T

.   

(2.7.1)  

Notice  that  specific  volume  is  a  function  of  Absolute  Salinity   S A   rather  than  of  Reference  
Salinity   S R   or  Practical  Salinity   S P .     The  importance  of  this  point  is  discussed  in  section  
2.8.    When  derivatives  are  taken  with  respect  to  in  situ  temperature,  or  at  constant  in  situ  
temperature,  the  symbol   t   is  avoided  as  it  can  be  confused  with  the  same  symbol  for  time.    
Rather,  we  use   T   in  place  of   t   in  the  expressions  for  these  derivatives.      
For   many   theoretical   and   modeling   purposes   in   oceanography   it   is   convenient   to  
regard   the   independent   temperature   variable   to   be   Conservative   Temperature   Θ    rather  
than  in  situ  temperature   t .     We  note  here  that  the  specific  volume  is  equal  to  the  pressure  
derivative  of  specific  enthalpy  at  fixed  Absolute  Salinity  when  any  one  of   η, θ   or   Θ   is  also  
held  constant,  as  follows  (from  appendix  A.11)    

∂h ∂P S

A ,η

= ∂h ∂P S

A,Θ

= ∂h ∂P S

A ,θ

= v .   

(2.7.2)  

The  use  of   P   in  these  equations  emphasizes  that  it  must  be  in   Pa   not   dbar.     Specific  
volume   v   has  units  of   m3 kg −1   in  both  the  SIA  and  GSW  software  libraries.      
  
  

2.8  Density    

  
The   density   of   seawater   ρ    is   the   reciprocal   of   the   specific   volume.      It   is   given   by   the  
reciprocal  of  the  pressure  derivative  of  the  Gibbs  function  at  constant  Absolute  Salinity   S A   
and  in  situ  temperature   t ,   that  is    

ρ = ρ ( SA , t , p ) = ( g P )

−1

(

= ∂g ∂P S

A ,T

)

−1

.   

(2.8.1)  

Notice  that  density  is  a  function  of  Absolute  Salinity   S A   rather  than  of  Reference  Salinity  
S R    or   Practical   Salinity   S P .       This   is   an   extremely   important   point   because   Absolute  
Salinity   S A    in   units   of   g kg −1    is   numerically   greater   than   Practical   Salinity   by   between  
0.165   g kg −1    and   0.195   g kg −1    in   the   open   ocean   so   that   if   Practical   Salinity   were  
inadvertently  used  as  the  salinity  argument  for  the  density  algorithm,  a  significant  density  
error  of  between   0.12 kg m−3   and   0.15 kg m−3   would  result.      
For   many   theoretical   and   modeling   purposes   in   oceanography   it   is   convenient   to  
regard   density   to   be   a   function   of   Conservative   Temperature   Θ    rather   than   of   in   situ  
temperature   t .     That  is,  it  is  convenient  to  form  the  following  functional  form  of  density,    

ρ = ρˆ ( SA , Θ, p ) ,   

(2.8.2)  

where   Θ    is   Conservative   Temperature.      We   will   adopt   the   convention   (see   Table   L.2   in  
appendix   L)   that   when   enthalpy   h,    specific   volume   v    or   density   ρ    are   taken   to   be  
functions  of  potential  temperature  they  attract  an  over-­‐‑tilde  as  in   v   or   ρ ,   and  when  they  
are  taken  to  be  functions  of  Conservative  Temperature  they  attract  a  caret  as  in   v̂   and   ρˆ .     
With   this   convention,   expressions   involving   partial   derivatives   such   as   (2.7.2)   can   be  
written  more  compactly  as  (from  appendix  A.11)    

(2.8.3)  
hP = hP = ĥP = v = ρ −1   
since   the   other   variables   are   taken   to   be   constant   during   the   partial   differentiation.    
Appendix   P   lists   expressions   for   many   thermodynamic   variables   in   terms   of   the  
thermodynamic  potentials    

h = h SA ,η , p ,   h = h SA ,θ , p   and   h = hˆ ( SA , Θ, p ) .   
(2.8.4)  

(

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19

Density   ρ   has  units  of   kg m −3   in  both  the  SIA  and  GSW  software  libraries.      
The   computationally   efficient   expression   for   v̂ SA ,Θ, p    involving   75   coefficients  
(Roquet  et  al.  (2015))  is  described  in  appendix  A.30  and  appendix  K  and  is  available  in  the  
GSW   computer   software   library   as   the   function   gsw_specvol(SA,CT,p).      Note   that  
potential   density   with   respect   to   reference   pressure   p_ref    is   calculated   using   the   same  
function,  as  gsw_rho(SA,CT,p_ref).    Note  that   v̂ SA ,Θ, p   can  be  integrated  with  respect  to  
pressure  to  provide  a  closed  expression  for   hˆ ( S A , Θ, p )   (see  Eqns.  (2.8.3)  and  (3.2.1))  which  
is  available  as  the  function  gsw_enthalpy(SA,CT,p).      
  
  

(

(

)

)

2.9  Chemical  potentials    

  
As   for   any   two-­‐‑component   thermodynamic   system,   the   Gibbs   energy,   G,    of   a   seawater  
sample  containing  mass  of  water   mW   and  mass  of  salt   mS   at  temperature   t   and  pressure  
p   can  be  written  in  the  form  (Landau  and  Lifshitz  (1959),  Alberty  (2001),  Feistel  (2008))      

G ( mW , mS , t , p ) = mW µ W + mS µ S   

(2.9.1)  

where   the   chemical   potentials   of   water   in   seawater   µ W    and   of   salt   in   seawater   µ S    are  
defined  by  the  partial  derivatives    

µW =

∂G
∂mW

,      and     µ S =
mS , T , p

∂G
∂mS

.  

(2.9.2)  

mW ,T , p

Identifying   absolute   salinity   with   the   mass   fraction   of   salt   dissolved   in   seawater,  
SA = mS / ( mW + mS )   (Millero  et  al.  (2008a)),  the  specific  Gibbs  energy   g   is  given  by    

(

)

G
(2.9.3)  
= (1 − SA ) µ W + SA µ S = µ W + SA µ S − µ W .  
mW + mS
Note  that  this  expression  for   g   as  the  sum  of  a  water  part  and  a  saline  part  is  not  the  same  
as  the  pure  water  and  the  saline  split  in  Eqn.  (2.6.1)  ( µ W   is  the  chemical  potential  of  water  
in   seawater;   it   does   not   correspond   to   a   pure   water   sample   as   g W    does   in   Eqn.   (2.6.1)).    
This  Gibbs  energy   g   is  used  as  the  thermodynamic  potential  function  (Gibbs  function)  for  
seawater.    The  above  three  equations  can  be  used  to  write  expressions  for   µ W   and   µ S   in  
terms  of  the  Gibbs  function   g   of  seawater  as    
g ( SA , t , p ) =

∂ ⎡( mW + mS ) g ⎦⎤
µW = ⎣
∂mW

= g + ( mW + mS )
mS ,T , p

∂g
∂SA

T,p

∂SA
∂mW

T,p

∂SA
∂mS

= g − SA
mS

∂g
∂SA

  

(2.9.4)  

T,p

and  for  the  chemical  potential  of  salt  in  seawater,    

∂ ⎡( mW + mS ) g ⎦⎤
µS = ⎣
∂mS

= g + ( mW + mS )
mW ,T , p

∂g
∂SA

= g + (1 − S A )
mW

∂g
∂SA

  

(2.9.5)  

T,p

The  relative  chemical  potential   µ   (commonly  called  the  “chemical  potential  of  seawater”)  
follows  from  (2.9.4)  and  (2.9.5)  as    

µ = µS − µ W =

∂g
∂SA

,   

(2.9.6)  

T,p

and   describes   the   change   in   the   Gibbs   energy   of   a   parcel   of   seawater   of   fixed   mass   if   a  
small   amount   of   water   is   replaced   by   salt   at   constant   temperature   and   pressure.      Also,  
from   the   fundamental   thermodynamic   relation   (Eqn.   (A.7.1)   in   appendix   A.7)   it   follows  
that   the   chemical   potential   of   seawater   µ    describes   the   change   of   enthalpy   dh   if   at  
constant   pressure   and   entropy,   a   small   mass   fraction   of   water   is   replaced   by   salt,   dSA .     
Equations   (2.9.4)   –   (2.9.6)   serve   to   define   the   three   chemical   potentials   in   terms   of   the  
Gibbs   function   g    of   seawater.      Note   that   the   weights   of   the   sums   that   appear   in   Eqns.  

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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

(2.9.1)  –  (2.9.5)  are  strictly  the  mass  fractions  of  salt  and  of  pure  water  in  seawater,  so  that  
for   a   seawater   sample   of   anomalous   composition   these   mass   fractions   would   be   more  
accurately  given  in  terms  of   S Asoln   than  by   SA ≡ SAdens .    In  this  regard,  the  Gibbs  energy  in  
Eqn.   (2.9.1)   should   strictly   be   the   weighted   sum   of   the   chemical   potentials   of   all   the  
constituents   in   seawater.      However,   practically   speaking,   the   vapour   pressure,   the   latent  
heat  and  the  freezing  temperature  are  all  rather  weakly  dependent  on  salinity,  and  hence  
the  use  of   S A   in  this  section  is  recommended.      
Note   that   both   µ    and   µ S    have   singularities   at   SA = 0 g kg −1    while   µ W    is   well-­‐‑
behaved  there.      
The  SIA  computer  software  library  (appendix  M)  predominantly  uses  basic  SI  units,  so  
that   S A    has   units   of   kg kg −1   and   g , µ , µ S    and   µ W    all   have   units   of   J kg−1.       In   the   GSW  
Oceanographic  Toolbox  (appendix  N)   S A   has  units  of   g kg −1   while   µ , µ S   and   µ W   all  have  
units  of   J g −1.     This  adoption  of  oceanographic  (i.e.  non-­‐‑basic-­‐‑SI)  units  for   S A   means  that  
special  care  is  needed  in  evaluating  equations  such  as  (2.9.3)  and  (2.9.5)  where  in  the  term  
(1 − SA )   it  is  clear  that   SA   must  have  units  of   kg kg −1.    The  adoption  of  non-­‐‑basic-­‐‑SI  units  
is  common  in  oceanography,  but  often  causes  some  difficulties  such  as  this.    To  be  specific,  
the  use  of  oceanographic  units  for  Absolute  Salinity  (such  as  in  the  GSW  Oceanographic  
Toolbox)  means  that  the  above  equations  (2.9.4)  –  (2.9.5)  are  evaluated  as    

µW =

S ∂g
g
− A
R ∂SA
R

,                    and                   µ S =
T ,p

⎛
S ⎞ ∂g
g
+ ⎜1 − A ⎟
R ⎠ ∂SA
R ⎝

  

(2.9.7)  

T ,p

where  the  constant   R   is  defined  as   R = 1000 g kg −1 ,  while  Eqn.  (2.9.6)  is  unchanged.    
  
  
2.10  Entropy    
  
The  specific  entropy  of  seawater   η   is  given  by    

η = η ( SA , t, p ) = − gT = − ∂g ∂T S

A, p

.   

(2.10.1)  

When  taking  derivatives  with  respect  to  in  situ  temperature,  the  symbol   T   will  be  used  for  
temperature  in  order  that  these  derivatives  not  be  confused  with  time  derivatives.    
Entropy   η   has  units  of   J kg −1 K −1   in  both  the  SIA  and  GSW  software  libraries.      
  
  

2.11  Internal  energy    

  
The  specific  internal  energy  of  seawater   u   is  given  by  (where   T0   is  the  Celsius  zero  point,  
273.15   K   and   P0 = 101 325Pa   is  the  standard  atmosphere  pressure)    
∂g
∂g
.    (2.11.1)  
u = u ( SA , t, p ) = g + (T0 + t )η − ( p + P0 ) v = g − (T0 + t )
− ( p + P0 )
∂T SA , p
∂P SA , T
This   expression   is   an   example   where   the   use   of   non-­‐‑basic   SI   units   presents   a   problem,  
because   in   the   product   − ( p + P0 ) v ,   ( p + P0 ) = P    must   be   in   Pa   if   specific   volume   has   its  
regular  units  of   m3 kg −1 :-­‐‑   hence  here  sea  pressure   p   must  be  expressed  in   Pa .    Also,  the  
pressure  derivative  in  Eqn.  (2.11.1)  must  be  done  with  respect  to  pressure  in   Pa .      
Specific   internal   energy   u    has   units   of   J kg −1    in   both   the   SIA   and   GSW   software  
libraries.      
  
  

2.12  Enthalpy    

  
The  specific  enthalpy  of  seawater   h   is  given  by    

h = h ( SA , t, p ) = g + (T0 + t )η = g − (T0 + t )

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∂g
∂T

.   
SA , p

(2.12.1)  

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21

Specific  enthalpy   h   has  units  of   J kg −1   in  both  the  SIA  and  GSW  software  libraries.    Also,  
note   that   potential   enthalpy   is   defined   in   section   3.2   below,   and   dynamic   enthalpy   is  
defined  as  enthalpy  minus  potential  enthalpy  (Young,  2010).      
  
  

2.13  Helmholtz  energy    

  
The  specific  Helmholtz  energy  of  seawater   f   is  given  by    

f = f ( SA , t, p ) = g − ( p + P0 ) v = g − ( p + P0 )

∂g
.   
∂P SA , T

(2.13.1)  

This  expression  is  another  example  where  the  use  of  non-­‐‑basic  SI  units  presents  a  problem,  
because  in  the  product   − ( p + P0 ) v ,   p   must  be  in  Pa  if  specific  volume  has  its  regular  units  of  
m3 kg −1.       The   specific   Helmholtz   energy   f    has   units   of   J kg −1    in   both   the   SIA   and   GSW  
computer  software  libraries.      
  
  

2.14  Osmotic  coefficient    

  
The  osmotic  coefficient  of  seawater   φ   is  given  by    

⎛

φ = φ ( SA , t , p ) = − ⎜ g ( SA , t , p ) − g ( 0, t , p ) − S A
⎜
⎝

∂g
∂SA

⎞
−1
⎟ ( mSW R (T0 + t ) ) .   
⎟
T, p ⎠

(2.14.1)  

The  osmotic  coefficient  of  seawater  describes  the  change  of  the  chemical  potential  of  water  
per   mole   of   added   salt,   expressed   as   multiples   of   the   thermal   energy,   R (T0 + t )    (Millero  
and  Leung  (1976),  Feistel  and  Marion  (2007),  Feistel  (2008)),    

µ W ( 0, t, p ) = µ W ( SA , t , p ) + mSW R (T0 + t ) φ .  

(2.14.2)  

Here,   R   =  8.314  472   J mol −1 K −1   is  the  universal  molar  gas  constant.    The  molality   mSW   is  
the  number  of  dissolved  moles  of  solutes  (ions)  of  the  Reference  Composition  as  defined  
by   Millero   et   al.   (2008a),   per   kilogram   of   pure   water.      Note   that   the   molality   of   seawater  
may  take  different  values  if  neutral  molecules  of  salt  rather  than  ions  are  counted  (see  the  
discussion   on   page   519   of   Feistel   and   Marion   (2007)).      The   freezing-­‐‑point   lowering  
equations   (3.33.1,   3.33.2)   or   the   vapour-­‐‑pressure   lowering   can   be   computed   from   the  
osmotic  coefficient  of  seawater  (see  Millero  and  Leung  (1976),  Bromley  et  al.  (1974)).      
  
  

2.15  Isothermal  compressibility    

  
The  thermodynamic  quantities  defined  so  far  are  all  based  on  the  Gibbs  function  itself  and  
its  first  derivatives.    The  remaining  quantities  discussed  in  this  section  all  involve  higher  
order  derivatives.      
The  isothermal  and  isohaline  compressibility  of  seawater   κ t   is  defined  by    

κ t = κ t ( SA , t, p ) = ρ −1

∂ρ
∂v
g
= − v −1
= − PP   
∂P SA , T
∂P SA , T
gP

(2.15.1)  

where  the  second  derivative  of   g   is  taken  with  respect  to  pressure  (in   Pa )  at  constant   S A   
and   t.     The  use  of   P   in  the  pressure  derivatives  in  Eqn.  (2.15.1)  serves  to  emphasize  that  
these   derivatives   must   be   taken   with   respect   to   pressure   in   Pa    not   in   dbar .      The  
isothermal  compressibility  of  seawater   κ t   produced  by  both  the  SIA  and  GSW  computer  
software  libraries  (appendices  M  and  N)  has  units  of   Pa −1.       
  
  

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2.16  Isentropic  and  isohaline  compressibility    

  
When  the  entropy  and  Absolute  Salinity  are  held  constant  while  the  pressure  is  changed,  
the  isentropic  and  isohaline  compressibility   κ   is  obtained:    

κ = κ ( SA , t , p ) = ρ −1

(g

=

∂ρ
∂P

2
TP

= − v −1
SA ,η

− gTT g PP
g P gTT

∂v
∂P

).

= ρ −1
SA ,η

∂ρ
∂P

= ρ −1
SA , θ

∂ρ
∂P

SA , Θ

  

(2.16.1)  

The  isentropic  and  isohaline  compressibility   κ   is  sometimes  called  simply  the  isentropic  
compressibility   (or   sometimes   the   “adiabatic   compressibility”),   on   the   unstated  
understanding   that   there   is   also   no   transfer   of   salt   during   the   isentropic   or   adiabatic  
change  in  pressure.    The  isentropic  and  isohaline  compressibility  of  seawater   κ   produced  
by  both  the  SIA  and  GSW  software  libraries  (appendices  M  and  N)  has  units  of   Pa −1.     
  
  

2.17  Sound  speed    

  
The  speed  of  sound  in  seawater   c   is  given  by    

(

)

0.5

c = c ( SA , t , p ) = ∂P ∂ρ S

= ( ρκ )

(

−0.5

)

0.5

2
.  
(2.17.1)  
= g P gTT ⎡⎣ gTP
− gTT g PP ⎤⎦
A ,η
Note  that  in  these  expressions  in  Eqn.  (2.17.1),  since  sound  speed  is  in   m s−1   and  density  
has  units  of   kg m −3   it  follows  that  the  pressure  of  the  partial  derivatives  must  be  in  Pa  and  
the  isentropic  compressibility   κ   must  have  units  of   Pa −1 .    The  sound  speed   c   produced  
by  both  the  SIA  and  the  GSW  software  libraries  (appendices  M  and  N)  has  units  of   m s−1 .      
  
  

2.18  Thermal  expansion  coefficients    

  
The  thermal  expansion  coefficient   α t   with  respect  to  in  situ  temperature   t ,   is    

α t = α t ( SA , t , p ) = −

1 ∂ρ
ρ ∂T

=
SA , p

1 ∂v
v ∂T

=
SA , p

gTP
.   
gP

(2.18.1)  

The  thermal  expansion  coefficient   α θ   with  respect  to  potential  temperature   θ ,   is  (see  
appendix  A.15)    

α θ = α θ ( SA , t, p, pr ) = −

1 ∂ρ
ρ ∂θ

=
SA , p

1 ∂v
v ∂θ

=
SA , p

gTP gTT ( SA ,θ , pr )
,   
gP
gTT

(2.18.2)  

where   pr   is  the  reference  pressure  of  the  potential  temperature.    The   gTT   derivative  in  the  
numerator   is   evaluated   at   ( SA ,θ , pr )    whereas   the   other   derivatives   are   all   evaluated   at  
( SA , t, p ) .       
The  thermal  expansion  coefficient   α Θ   with  respect  to  Conservative  Temperature   Θ ,   is  
(see  appendix  A.15)    

α Θ = α Θ ( SA , t , p ) = −

c0p
1 ∂ρ
1 ∂v
g
=
= − TP
.   
ρ ∂Θ SA , p v ∂Θ SA , p
g P (T0 + θ ) gTT

(2.18.3)  

Note   that   Conservative   Temperature   Θ    is   defined   only   with   respect   to   a   reference  
pressure   of   0   dbar   so   that   the   θ    in   Eqn.   (2.18.3)   is   the   potential   temperature   with  
pr = 0 dbar.       All   the   derivatives   on   the   right-­‐‑hand   side   of   Eqn.   (2.18.3)   are   evaluated   at  
( SA , t, p ) .     The  constant   c0p   is  defined  in  Eqn.  (3.3.3)  below.      

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23

2.19  Saline  contraction  coefficients    

  
The   saline   contraction   coefficient   β t    (sometimes   also   called   the   haline   contraction  
coefficient)  at  constant  in  situ  temperature   t ,   is    

β t = β t ( SA , t , p ) =

1 ∂ρ
ρ ∂SA

= −
T, p

1 ∂v
v ∂SA

= −
T, p

gS

AP

gP

.   

(2.19.1)  

The   saline   contraction   coefficient   β θ    at   constant   potential   temperature   θ ,    is   (see  
appendix  A.15)    

β θ = β θ ( S A , t , p , pr ) =
=

1 ∂ρ
ρ ∂SA

=−
θ, p

1 ∂v
v ∂SA

θ, p

gTP ⎡⎣ g SAT − g SAT ( SA , θ , pr )⎤⎦ − gTT g SA P
g P gTT

  

(2.19.2)  

,

where   pr   is  the  reference  pressure  of   θ .     One  of  the   gS T   derivatives  in  the  numerator  is  
A
evaluated  at   ( SA ,θ , pr )   whereas  all  the  other  derivatives  are  evaluated  at   ( SA , t, p ) .       
The  saline  contraction  coefficient   β Θ   at  constant  Conservative  Temperature   Θ ,   is  (see  
appendix  A.15)    

β Θ = β Θ ( SA , t , p ) =

1 ∂ρ
ρ ∂SA

=−
Θ, p

1 ∂v
v ∂SA

Θ, p

gTP ⎡ g SAT − (T0 + θ ) g SA ( SA ,θ ,0 )⎤ − gTT g SA P
⎣
⎦
=
.
g P gTT
−1

  

(2.19.3)  

Note   that   Conservative   Temperature   Θ    is   defined   only   with   respect   to   a   reference  
pressure   of   0   dbar   as   indicated   in   this   equation.      The   gS    derivative   in   the   numerator   is  
A
evaluated  at   ( SA ,θ , 0)   whereas  all  the  other  derivatives  are  evaluated  at   ( SA , t, p ) .       
In  the  SIA  computer  software  (appendix  M)  all  three  saline  contraction  coefficients  are  
produced  in  units  of   kg kg −1   while  in  the  GSW  Oceanographic  Toolbox  (appendix  N)  all  
three   saline   contraction   coefficients   are   produced   in   units   of   kg g−1    consistent   with   the  
preferred  oceanographic  unit  for   S A   being   g kg −1.       
  
  

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2.20  Isobaric  heat  capacity    

  
The   specific   isobaric   heat   capacity   c p    is   the   rate   of   change   of   specific   enthalpy   with  
temperature  at  constant  Absolute  Salinity   S A   and  pressure   p,   so  that    

c p = c p ( SA , t , p ) =

∂h
∂T

= − (T0 + t ) gTT .   

(2.20.1)  

SA , p

The  isobaric  heat  capacity   c p   varies  over  the   SA − Θ   plane  at   p   =  0  by  approximately  5%,  
as  illustrated  in  Figure  4.      

  

Figure   4.      Contours   of   isobaric   specific   heat   capacity   c p    of   seawater    
                                            (in J kg −1 K −1 ),  Eqn.  (2.20.1),  at   p   =  0.      

  

The   isobaric   heat   capacity   c p    has   units   of   J kg −1 K −1    in   both   the   SIA   and   GSW  
computer  software  libraries.      
  
  

2.21  Isochoric  heat  capacity    

  
The  specific  isochoric  heat  capacity   cv   is  the  rate  of  change  of  specific  internal  energy   u   
with  temperature  at  constant  Absolute  Salinity   S A   and  specific  volume,   v,   so  that    

cv = cv ( SA , t, p ) =

∂u
∂T

(

2
= − (T0 + t ) gTT g PP − gTP
SA , v

)

g PP .   

(2.21.1)  

Note  that  the  isochoric  and  isobaric  heat  capacities  are  related  by    

cv = c p −

(T0 + t ) (α t )

(ρ κ )
t

2

,         and  by       cv = c p

κ
.   
κt

(2.21.2)  

The   isochoric   heat   capacity   cv    has   units   of   J kg −1 K −1    in   both   the   SIA   and   GSW  
computer  software  libraries.      
  
  

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25

2.22  The  adiabatic  lapse  rate    

  
The  adiabatic  lapse  rate   Γ   is  the  change  of  in  situ  temperature  with  pressure  at  constant  
entropy  and  Absolute  Salinity,  so  that  (McDougall  and  Feistel  (2003))    

∂t
Γ =
∂P S

=

A

∂t
=
∂P S
,η

(T0 + θ )α θ
ρ c p ( SA ,θ ,0 )

A

η
g
∂v
= − P = − TP =
ηT
gTT
∂η
,Θ

=

(T0 + θ )
c0p

∂v
∂Θ S

=
A, p

SA

∂2 h
=
∂η ∂P
,p

(T0 + θ )
c0p

∂2 h
∂Θ ∂P

=
SA

=
SA

(T0 + t )α t
ρ cp

(T0 + θ )α Θ .

    

(2.22.1)  

ρ c0p

The  adiabatic  (and  isohaline)  lapse  rate  is  commonly  (and  incorrectly)  explained  as  being  
proportional   to   the   work   done   on   a   fluid   parcel   as   its   volume   changes   in   response   to   an  
increase  in  pressure.    According  to  this  explanation  the  adiabatic  lapse  rate  would  increase  
with   both   pressure   and   the   fluid’s   compressibility,   but   this   is   not   the   case.      Rather,   the  
adiabatic   lapse   rate   is   proportional   to   the   thermal   expansion   coefficient   and   is  
independent  of  the  fluid’s  compressibility.    Indeed,  the  adiabatic  lapse  rate  changes  sign  at  
the  temperature  of  maximum  density  whereas  the  compressibility   κ   and  the  work  done  
by  compression  is  always  positive.    McDougall  and  Feistel  (2003)  show  that  the  adiabatic  
lapse   rate   is   independent   of   the   increase   in   the   internal   energy   that   a   parcel   experiences  
when   it   is   compressed.      Rather,   the   adiabatic   lapse   rate   represents   that   change   in  
temperature  that  is  required  to  keep  the  entropy  (and  also   θ   and   Θ )  of  a  seawater  parcel  
constant  when  its  pressure  is  changed  in  an  adiabatic  and  isohaline  manner.    The  reference  
pressure  of  the  potential  temperature   θ   that  appears  in  the  last  four  expressions  in  Eqn.  
(2.22.1)  is   pr = 0 dbar.       
The  adiabatic  lapse  rate   Γ   in  the  GSW  computer  software  library  is  evaluated  via  the  
functions   gsw_adiabatic_lapse_rate_from_t   and   gsw_adiabatic_lapse_rate_from_CT  
(depending   on   whether   the   input   temperature   is   in   situ   temperature   or   Conservative  
Temperature).    In  both  cases  the  expression  used  is   − gTP gTT = −η P ηT   (see  the  top  line  
of   Eqn.   (2.22.1))   calculated   directly   from   the   Gibbs   function   of   seawater   g SA ,t, p   
(IAPWS-­‐‑08   and   IAPWS-­‐‑09).      This   is   consistent   with   the   exact   use   of   η = η SA ,t, p   
throughout   the   GSW   Toolbox   to   convert   between   in   situ   temperature   and   potential  
temperature.      An   alternative   option   for   calculating   Γ    would   be   to   use   the   75-­‐‑term  
expression  for  specific  volume  in  the  expressions  in  the  second  line  of  Eqn.  (2.22.1).    This  
option   is   not   adopted   as   it   would   mean   that   the   small   errors   in   the   thermal   expansion  
coefficient   α Θ   would  cause  an  rms  error  in  the  adiabatic  lapse  rate   Γ   of   4.7x10−12 K Pa −1 .    
This   error,   while   small,   would   then   conflict   with   the   exact   relationships   that   have   been  
chosen   to   relate   in   situ   temperature,   potential   temperature,   Conservative   Temperature,  
entropy  and  the  adiabatic  lapse  rate.      
The   adiabatic   lapse   rate   Γ    output   of   both   the   SIA   and   the   GSW   computer   software  
libraries  is  in  units  of   K Pa −1 .      
  
  

(
(

)
)

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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

3.  Derived  Quantities    
  
  
  

3.1  Potential  temperature    

  
The  very  useful  concept  of  potential  temperature  was  applied  to  the  atmosphere  originally  
by  Helmholtz  (1888),  first  under  the  name  of  ‘heat  content’,  and  later  renamed  ‘potential  
temperature’   (Bezold   (1888)).      These   concepts   were   transferred   to   oceanography   by  
Helland-­‐‑Hansen   (1912).      Potential   temperature   is   the   temperature   that   a   fluid   parcel  
would  have  if  its  pressure  were  changed  to  a  fixed  reference  pressure   pr   in  an  isentropic  
and   isohaline   manner.      The   phrase   “isentropic   and   isohaline”   is   used   repeatedly   in   this  
document.      To   these   two   qualifiers   we   should   really   also   add   “without   dissipation   of  
kinetic   energy”.      A   process   that   obeys   all   three   restrictions   is   a   thermodynamically  
reversible  process.    Note  that  one  often  (falsely)  reads  that  the  requirement  of  a  reversible  
process   is   that   the   process   occurs   at   constant   entropy.      However   this   statement   is  
misleading   because   it   is   possible   for   a   fluid   parcel   to   exchange   some   heat   and   some   salt  
with   its   surroundings   in   just   the   right   ratio   so   as   to   keep   its   entropy   constant,   but   the  
processes  is  not  reversible  (see  Eqn.  (A.7.1)).      
Potential  temperature  referred  to  reference  pressure   pr   is  often  written  as  the  pressure  
integral  of  the  adiabatic  lapse  rate  (Fofonoff  (1962),  (1985))    
Pr

θ = θ ( SA , t, p, pr ) = t + ∫ Γ ( SA ,θ [SA , t, p, p′], p′) dP′.   

(3.1.1)  

P

Note  that  this  pressure  integral  needs  to  be  done  with  respect  to  pressure  expressed  in   Pa   
not   dbar .      
The   algorithm   that   is   used   with   the   TEOS-­‐‑10   Gibbs   function   approach   to   seawater  
equates  the   specific   entropies   of   two   seawater   parcels,   one   before   and   the   other   after   the  
isentropic   and   isohaline   pressure   change.      In   this   way,   θ    is   evaluated   using   a   Newton-­‐‑
Raphson  iterative  solution  technique  to  solve  the  following  equation  for   θ     
(3.1.2)  
η ( SA ,θ , pr ) = η ( SA , t, p ) ,   
or,  in  terms  of  the  Gibbs  function,   g ,     

− gT ( SA ,θ , pr ) = − gT ( SA , t, p ) .   

(3.1.3)  

This  relation  is  formally  equivalent  to  Eqn.  (3.1.1).    In  the  GSW  Oceanographic  Toolbox   θ   
is  found  to  machine  precision  (  10−14 °C )  in  two  iterations  of  a  modified  Newton-­‐‑Raphson  
method  (McDougall  and  Wotherspoon  (2014)),  using  a  suitable  initial  value.      
Note   that   the   difference   between   the   potential   and   in   situ   temperatures   is   not   due   to  
the  work  done  in  compressing  a  fluid  parcel  on  going  from  one  pressure  to  another:-­‐‑   the  
sign  of  this  work  is  often  in  the  wrong  sense  and  the  magnitude  is  often  wrong  by  a  few  
orders  of  magnitude  (McDougall  and  Feistel  (2003)).    Rather,  the  difference  between  these  
temperatures   is   what   is   required   to   keep   the   entropy   constant   during   the   adiabatic   and  
isohaline  pressure  change.    The  potential  temperature   θ   output  of  the  SIA  software  is  in  
units  of  K  while  the  output  from  the  GSW  Toolbox  is  in   °C .      
  
  

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27

3.2  Potential  enthalpy    

  
Potential   enthalpy   h 0    is   the   enthalpy   that   a   fluid   parcel   would   have   if   its   pressure   were  
changed  to  a  fixed  reference  pressure   pr   in  an  isentropic  and  isohaline  manner.    Because  
heat   fluxes   into   and   out   of   the   ocean   occur   mostly   near   the   sea   surface,   the   reference  
pressure   for   potential   enthalpy   is   always   taken   to   be   pr    =   0   dbar   (that   is,   at   zero   sea  
pressure).    Potential  enthalpy  can  be  expressed  as  the  pressure  integral  of  specific  volume  
as  (from  McDougall  (2003)  and  see  the  discussion  below  Eqn.  (2.8.2))    

(

)

(

)

(

)

(

)

(

)

(

)

(

)

P

(

)

h0 SA ,t, p = h SA ,θ ,0 = h 0 SA ,θ = h SA ,t, p − ∫ v SA ,θ ⎡⎣ SA ,t, p, p′ ⎤⎦ , p′ dP′
P0

(

)

P

= h SA ,t, p − ∫ v SA ,η , p′ dP′
P0
P

(

)

(

)

  

(3.2.1)  

= h SA ,t, p − ∫ v SA ,θ , p′ dP′
P0
P

= h SA ,t, p − ∫ v̂ SA ,Θ, p′ dP′ ,
P0

and  we  emphasize  that  the  pressure  integrals  here  must  be  done  with  respect  to  pressure  
expressed  in   Pa   rather  than   dbar.     In  terms  of  the  Gibbs  function,  potential  enthalpy   h 0   is  
evaluated  as    

h 0 ( SA , t , p ) = h ( SA , θ , 0 ) = g ( SA ,θ , 0 ) − (T0 + θ ) gT ( SA ,θ , 0 ) .   

(3.2.2)  

Also,  note  that  dynamic  enthalpy  is  defined  as  enthalpy  minus  potential  enthalpy  (Young,  
2010)  and  is  available  as  the  function  gsw_dynamic_enthalpy  in  the  GSW  Toolbox.      
  
  

3.3  Conservative  Temperature    

  
Conservative  Temperature   Θ   is  defined  to  be  proportional  to  potential  enthalpy,    

(

)

(

)

(

)

(

 S ,θ = h0 S ,t, p c0 = h 0 S ,θ
Θ SA ,t, p = Θ
A
A
p
A

)

c0p   

(3.3.1)  

where  the  value  that  is  chosen  for   c 0p   is  motivated  in  terms  of  potential  enthalpy  evaluated  
at  an  Absolute  Salinity  of   SSO = 35 uPS = 35.165 04 g kg−1   and  at   θ = 25 °C   by    

⎡⎣ h ( SSO , 25 °C, 0 ) − h ( SSO , 0 °C, 0 )⎤⎦
≈ 3991.867 957 119 63 J kg −1 K −1 ,   
(25 K)

(3.3.2)  

noting  that   h ( SSO , 0 °C, 0dbar )   is  zero  according  to  the  way  the  Gibbs  function  is  defined  
in  (2.6.5).    We  adopt  the  exact  definition  for   c 0p   to  be  the  15-­‐‑digit  value  in  (3.3.2),  so  that    

c0p ≡ 3991.867 957 119 63 J kg −1 K −1 .  

(3.3.3)  

When  IAPWS-­‐‑95  is  used  for  the  pure  water  part  of  the  Gibbs  function,   Θ ( SSO ,0 °C,0)   and  
Θ ( SSO ,25 °C,0)    differ   from   0   °C   and   25   °C   respectively   by   the   round-­‐‑off   amount   of  
5 × 10−12 °C.     When  IAPWS-­‐‑09  (which  is  based  on  the  paper  of  Feistel  (2003),  see  appendix  
G)  is  used  for  the  pure  water  part  of  the  Gibbs  function,   Θ ( SSO ,0 °C,0)   differs  from  0  °C  
by   −8.25 × 10−8 °C    and   Θ ( SSO ,25 °C,0)    differs   from   25   °C   by   9.3 × 10−6 °C.       Over   the  
temperature  range  from   0 ° C   to   40 ° C   the  difference  between  Conservative  Temperature  
using   IAPWS-­‐‑95   and   IAPWS-­‐‑09   as   the   pure   water   part   is   no   more   than   ± 1.5 × 10−5 °C ,   a  
temperature  difference  that  will  be  ignored.      
The  value  of   c 0p   in  (3.3.3)  is  very  close  to  the  average  value  of  the  specific  heat  capacity  
c p    at   the   sea   surface   of   today’s   global   ocean.      This   value   of   c 0p    also   causes   the   average  
value  of   θ − Θ   at  the  sea  surface  to  be  very  close  to  zero.    Since   c 0p   is  simply  a  constant  of  
proportionality   between   potential   enthalpy   and   Conservative   Temperature,   it   is   totally  

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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

arbitrary,   and   we   see   no   reason   why   its   value   would   need   to   change   from   (3.3.3)   even  
when  in  future  decades  an  improved  Gibbs  function  of  seawater  is  agreed  upon.      
McDougall   (2003),   Graham   and   McDougall   (2013)   and   appendix   A.18   outline   why  
Conservative   Temperature   gets   its   name;   it   is   approximately   two   orders   of   magnitude  
more  conservative  compared  with  either  potential  temperature  or  entropy.      
The   SIA   and   GSW   software   libraries   both   include   an   algorithm   for   determining  
Conservative   Temperature   Θ    from   values   of   Absolute   Salinity   S A    and   potential  
temperature   θ    referenced   to   p = 0 dbar .      These   libraries   also   have   an   algorithm   for  
evaluating   potential   temperature   (referenced   to   0 dbar )   from   S A    and   Θ .      This   inverse  
algorithm,   θˆ ( SA , Θ ) ,   has   an   initial   seed   based   on   a   rational   function   approximation   and  
finds  potential  temperature  to  machine  precision  (  10−14 °C )  in  one  and  a  half  iterations  of  
a  modified  Newton-­‐‑Raphson  technique.      
  
  

3.4  Potential  density    

  
Potential   density   ρ θ    is   the   density   that   a   fluid   parcel   would   have   if   its   pressure   were  
changed  to  a  fixed  reference  pressure   pr   in  an  isentropic  and  isohaline  manner.    Potential  
density   referred   to   reference   pressure   pr    can   be   written   as   the   pressure   integral   of   the  
isentropic  compressibility   κ   as    
Pr

ρ θ ( SA , t, p, pr ) = ρ ( SA , t, p ) + ∫ ρ ( SA ,θ [SA , t, p, p′], p′) κ ( SA ,θ [SA , t, p, p′], p′) dP′.    (3.4.1)    
P

The  simpler  expression  for  potential  density  in  terms  of  the  Gibbs  function  is    

ρ θ ( SA , t, p, pr ) = ρ ( SA ,θ [SA , t, p, pr ], pr ) = g P−1 ( SA ,θ [SA , t, p, pr ], pr ).   

(3.4.2)  

Using  the  functional  forms  of  Eqn.  (2.8.2)  and  (2.8.3)  for  in  situ  density,  that  is,  either  
ρ = ρ SA ,θ , p   or   ρ = ρˆ ( SA , Θ, p ) ,   potential  density  with  respect  to  reference  pressure   pr   
(e.  g.  1000  dbar)  can  be  easily  evaluated  as    

ρθ SA ,t, p, pr = ρ Θ SA ,t, p, pr = ρ SA ,η , pr = ρ SA ,θ , pr = ρ̂ SA ,Θ, pr ,   
(3.4.3)  

(

(

)

)

(

)

(

)

(

)

(

)

where   we   note   that   the   potential   temperature   θ    in   the   penultimate   expression   is   the  
potential   temperature   with   respect   to   0 dbar.       Once   the   reference   pressure   is   fixed,  
potential  density  is  a  function  only  of  Absolute  Salinity  and  Conservative  Temperature  (or  
equivalently,   of   Absolute   Salinity   and   potential   temperature).      Note   that   it   is   equally  
correct  to  label  potential  density  as   ρ θ   or   ρ Θ   (or  indeed  as   ρ η )  because   η ,   θ   and   Θ   are  
constant  during  the  isentropic  and  isohaline  pressure  change  from   p   to   pr ;  that  is,  these  
variables  posses  the  “potential”  property  of  appendix  A.9.      
Following   the   discussion   after   Eqn.   (2.8.2)   above,   potential   density   may   also   be  
expressed   in   terms   of   the   pressure   derivative   of   the   expressions   h = h SA ,θ , p    and  
h = hˆ ( SA , Θ, p )   for  enthalpy  as  (see  also  appendix  P)    

(

ρθ ( SA ,t, p, pr ) = ρ Θ ( SA ,t, p, pr ) = ⎡⎣ hP ( SA ,θ , p = pr ) ⎤⎦

  
  

−1

(

)

−1

= ⎡⎣ ĥP SA ,Θ, p = pr ⎤⎦ .   

)

(3.4.4)  

3.5  Density  anomaly    

  
Density   anomaly   σ t    is   an   old-­‐‑fashioned   density   measure  that  is  now   seldom  used.      It   is  
the   density   evaluated   at   the   in   situ   temperature   but   at   zero   sea   pressure,   minus   1000  
kg m−3 ,   that  is,    

σ t ( SA , t, p ) = ρ ( SA , t,0 ) − 1000 kg m −3 = g P−1 ( SA , t,0 ) − 1000 kg m −3.   
θ

(3.5.1)  

σ    was   used   as   an   approximation   to   σ    which   avoided   the   computational   demand   of  
evaluating   θ .    Density  anomaly   σ t   is  not  provided  in  the  TEOS-­‐‑10  software  libraries.      
t

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29

3.6  Potential  density  anomaly    

  
Potential  density  anomaly,   σ θ   or   σ Θ ,   is  simply  potential  density  minus  1000  kg  m–3,    

σ θ ( SA , t, p, pr ) = σ Θ ( SA , t, p, pr ) = ρ θ ( SA , t , p, pr ) − 1000 kg m −3
= ρ Θ ( SA , t , p, pr ) − 1000 kg m −3

  

(3.6.1)  

= g P−1 ( SA ,θ [SA , t , p, pr ], pr ) − 1000 kg m −3.
Note  that  it  is  equally  correct  to  label  potential  density  anomaly  as   σ θ   or   σ Θ   because  both  
θ   and   Θ   are  constant  during  the  isentropic  and  isohaline  pressure  change  from   p   to   pr .       
  
  

3.7  Specific  volume  anomaly    

  
The  specific  volume  anomaly   δ   is  defined  as  the  difference  between  the  specific  volume  
and  a  given  function  of  pressure.    Traditionally   δ   has  been  defined  as    

δ ( SA , t, p ) = v ( SA , t, p ) − v ( SSO ,0°C, p )   

(3.7.1)  

(where   the   traditional   value   of   Practical   Salinity   of   35   has   been   updated   to   an   Absolute  
Salinity  of   SSO = 35 uPS = 35.16504 g kg −1  in  the  present  formulation).    Note  that  the  second  
term,   v ( SSO ,0°C, p ) ,   is  a  function  only  of  pressure.    In  order  to  have  a  surface  of  constant  
specific  volume  anomaly  more  accurately  approximate  neutral  tangent  planes  (see  section  

S
  and  
  with  more  general  values  
3.11),  it  is  advisable  to  replace  the  arguments  
0
°
C
S
SO
A   

and   t    that   are   carefully   chosen   (as   say   the   median   values   of   Absolute   Salinity   and  
temperature   along   the   surface)   so   that   the   more   general   definition   of   specific   volume  
anomaly  is    

 
 
δ SA ,t, p = v SA ,t, p − v SA , t , p = g P SA ,t, p − g P SA , t , p .   
(3.7.2)  

(

)

(

(

)

)

(

(

)

)

The  last  terms  in  Eqns.  (3.7.1)  and  (3.7.2)  are  simply  functions  of  pressure  and  one  has  
the   freedom   to   choose   any   other   function   of   pressure   in   its   place   and   still   retain   the  
dynamical  properties  of  specific  volume  anomaly.    In  particular,  one  can  construct  specific  
volume   and   enthalpy   to   be   functions   of   Conservative   Temperature   (rather   than   in   situ  
temperature)   as   vˆ ( SA , Θ, p )    and   hˆ ( SA , Θ, p )    and   write   a   slightly   different   definition   of  
specific  volume  anomaly  as    

(

)

(

)


 
 
δ ( SA ,Θ, p ) = v̂ ( SA ,Θ, p ) − v̂ SA , Θ,
p = ĥP ( SA ,Θ, p ) − ĥP SA , Θ,
p .   

(3.7.3)  

This  is  the  form  of  specific  volume  anomaly  adopted  in  the  GSW  Oceanographic  Toolbox  

   are   S = 35.165 04 g kg −1   and  
where  the  default  values  of  the  reference  values   SA   and   Θ
SO
0°C   respectively.    The  same  can  also  be  done  with  potential  temperature  so  that  in  terms  
of   the   specific   volume   v SA ,θ , p    and   enthalpy   h SA ,θ , p    we   can   write   another   form   of  
the  specific  volume  anomaly  as    

(

(

)

)

(

(

)

)

(

)

(

)

 
 
v SA ,θ , p − v SA ,θ , p = hP SA ,θ , p − hP SA ,θ , p .   

(3.7.4)  

These  expressions  exploit  the  fact  that  (see  appendix  A.11)    

∂h ∂P S

A ,η

= ∂h ∂P S

A,Θ

= ∂h ∂P S

A ,θ

= v .   

(3.7.5)  

  
  

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3.8  Thermobaric  coefficient    

  
The  thermobaric  coefficient  quantifies  the  rate  of  variation  with  pressure  of  the  ratio  of  the  
thermal   expansion   coefficient   and   the   saline   contraction   coefficient.      With   respect   to  
potential  temperature   θ   the  thermobaric  coefficient  is  (McDougall  (1987b))    

Tbθ = Tbθ ( SA , t , p ) = β θ

(

∂ αθ βθ

)

∂P

∂α θ
∂P

=
S A ,θ

−
S A ,θ

α θ ∂β θ
β θ ∂P

.   

(3.8.1)  

S A ,θ

This  expression  for  the  thermobaric  coefficient  is  most  readily  evaluated  by  differentiating  
an  expression  for  density  expressed  as  a  function  of  potential  temperature  rather  than  in  
situ  temperature,  that  is,  with  density  expressed  in  the  functional  form   ρ = ρ SA ,θ , p .       
With  respect  to  Conservative  Temperature   Θ   the  thermobaric  coefficient  is    

(

TbΘ

=

TbΘ

( SA , t , p ) = β

Θ

(

∂ αΘ βΘ

)

∂P

=
SA , Θ

∂α Θ
∂P

−
SA , Θ

α Θ ∂β Θ
β Θ ∂P

)

.   

(3.8.2)  

SA , Θ

This  expression  for  the  thermobaric  coefficient  is  most  readily  evaluated  by  differentiating  
an  expression  for  density  expressed  as  a  function  of  Conservative  Temperature  rather  than  
in  situ  temperature,  that  is,  with  density  expressed  in  the  functional  form   ρ = ρˆ ( SA , Θ, p ) .       
The   thermobaric   coefficient   enters   various   quantities   to   do   with   the   path-­‐‑dependent  
nature   of   neutral   trajectories   and   the   ill-­‐‑defined   nature   of   neutral   surfaces   (see   (3.13.1)   –  
(3.13.7)).    The  thermobaric  dianeutral  advection  associated  with  the  lateral  mixing  of  heat  
and   salt   along   neutral   tangent   planes   is   given   by   eTb = − gN −2 K Tbθ ∇nθ ⋅ ∇n P    or  
eTb = − gN −2 K TbΘ∇n Θ⋅ ∇n P    where   ∇nθ    and   ∇n Θ    are   the   two-­‐‑dimensional   gradients   of  
either   potential   temperature   or   Conservative   Temperature   along   the   neutral   tangent  
plane,   ∇n P    is   the   corresponding   epineutral   gradient   of   Absolute   Pressure   and   K    is   the  
epineutral   diffusion   coefficient.      Note   that   the   thermobaric   dianeutral   advection   is  
proportional   to   the   mesoscale   eddy   flux   of   “heat”   along   the   neutral   tangent   plane,  
− c0p K ∇n Θ ,   and  is  independent  of  the  amount  of  small-­‐‑scale  (dianeutral)  turbulent  mixing  
and   hence   is   also   independent   of   the   dissipation   of   kinetic   energy   ε    (Klocker   and  
McDougall   (2010a)).      It   is   shown   in   appendix   A.14   below   that   while   the   epineutral  
diffusive  fluxes   − K ∇nθ   and   − K ∇n Θ   are  different,  the  product  of  these  fluxes  with  their  
respective   thermobaric   coefficients   is   the   same,   that   is,   Tbθ ∇nθ = TbΘ∇n Θ .       Hence   the  
thermobaric   dianeutral   advection   e Tb    is   the   same   whether   it   is   calculated   as  
− gN −2 K Tbθ ∇nθ ⋅ ∇n P   or  as   − gN −2 K TbΘ∇n Θ⋅ ∇n P.     Expressions  for   Tbθ   and   TbΘ   in  terms  of  
enthalpy  in  the  functional  forms   h SA ,θ , p   and   hˆ ( SA , Θ, p )   can  be  found  in  appendix  P.      
Interestingly,   for   given   magnitudes   of   the   epineutral   gradients   of   pressure   and  
Conservative   Temperature,   the   dianeutral   advection,   eTb = − gN −2 K TbΘ∇n Θ⋅ ∇n P ,   of  
thermobaricity   is   maximized   when   these   gradients   are   parallel,   while   neutral   helicity   is  
maximized  when  these  gradients  are  perpendicular,  since  neutral  helicity  is  proportional  
to   TbΘ ( ∇n P × ∇n Θ ) ⋅ k   (see  Eqn.  (3.13.2)).      
This   thermobaric   vertical   advection   process,   e Tb ,   is   absent   from   standard   layered  
ocean  models  in  which  the  vertical  coordinate  is  a  function  only  of   S A   and   Θ   (such  as   σ 2 ,   
potential   density   referenced   to   2000   dbar).      As   described   in   appendix   A.27   below,   the  
isopycnal  diffusion  of  heat  and  salt  in  these  layered  models,  caused  by  both  parameterized  
diffusion   along   the   coordinate   and   by   eddy-­‐‑resolved   motions,   does   give   rise   to   the  
cabbeling   advection   through   the   coordinate   surfaces   but   does   not   allow   the   thermobaric  
velocity   e Tb   through  these  surfaces  (Klocker  and  McDougall  (2010a)).    
In   both   the   SIA   and   GSW   computer   software   libraries   the   thermobaric   parameter   is  
output  in  units  of   K −1 Pa −1 .      

(

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)

TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

31

3.9  Cabbeling  coefficient    

  
The   cabbeling   coefficient   quantifies   the   rate   at   which   dianeutral   advection   occurs   as   a  
result  of  mixing  of  heat  and  salt  along  the  neutral  tangent  plane.    With  respect  to  potential  
temperature   θ   the  cabbeling  coefficient  is  (McDougall  (1987b))    

∂α θ
Cb = Cb ( SA , t , p ) =
∂θ
θ

θ

SA , p

α θ ∂α θ
+2 θ
β ∂SA

θ, p

⎛ αθ
− ⎜⎜ θ
⎝β

2

⎞ ∂β θ
⎟⎟
⎠ ∂SA

.  

(3.9.1)  

θ, p

This  expression  for  the  cabbeling  coefficient  is  most  readily  evaluated  by  differentiating  an  
expression  for  density  expressed  as  a  function  of  potential  temperature  rather  than  in  situ  
temperature,  that  is,  with  density  expressed  in  the  functional  form   ρ = ρ SA ,θ , p .       
With  respect  to  Conservative  Temperature   Θ   the  cabbeling  coefficient  is    

(

CbΘ

=

CbΘ

∂α Θ
( SA , t , p ) =
∂Θ

SA

α Θ ∂α Θ
+2 Θ
β ∂SA
,p

)

2

Θ, p

⎛ α Θ ⎞ ∂β Θ
− ⎜⎜ Θ ⎟⎟
⎝ β ⎠ ∂SA

.   

(3.9.2)  

Θ, p

This  expression  for  the  cabbeling  coefficient  is  most  readily  evaluated  by  differentiating  an  
expression  for  density  expressed  as  a  function  of  Conservative  Temperature  rather  than  in  
situ  temperature,  that  is,  with  density  expressed  in  the  functional  form   ρ = ρˆ ( SA , Θ, p ) .       
The  cabbeling  dianeutral  advection  associated  with  the  lateral  mixing  of  heat  and  salt  
along  neutral  tangent  planes  is  given  by   eCab = − gN −2 K CbΘ∇n Θ⋅ ∇n Θ   (or  less  accurately  by  
eCab ≈ − gN −2 K Cbθ ∇nθ ⋅ ∇nθ )   where   ∇nθ    and   ∇n Θ    are   the   two-­‐‑dimensional   gradients   of  
either  potential  temperature  or  Conservative  Temperature  along  the  neutral  tangent  plane  
and   K    is   the   epineutral   diffusion   coefficient.      The   cabbeling   dianeutral   advection   is  
proportional   to   the   mesoscale   eddy   flux   of   “heat”   along   the   neutral   tangent   plane,  
− K ∇n Θ ,    and   is   independent   of   the   amount   of   small-­‐‑scale   (dianeutral)   turbulent   mixing  
and  hence  is  also  independent  of  the  dissipation  of  kinetic  energy  (Klocker  and  McDougall  
(2010a)).      It   is   shown   in   appendix   A.14   that   Cbθ ∇nθ ⋅ ∇nθ ≠ CbΘ∇n Θ⋅ ∇n Θ    so   that   the  
estimate  of  the  cabbeling  dianeutral  advection  is  different  when  calculated  using  potential  
temperature   than   when   using   Conservative   Temperature.      The   estimate   using   potential  
temperature   is   slightly   less   accurate   because   of   the   non-­‐‑conservative   nature   of   potential  
temperature.      
When   the   cabbeling   and   thermobaricity   processes   are   analyzed   by   considering   the  
mixing  of  two  fluid  parcels  one  finds  that  the  density  change  is  proportional  to  the  square  
of   the   property   ( Θ    and/or   p )   contrasts   between   the   two   fluid   parcels   (for   the   cabbeling  
case,  see  Eqn.  (A.19.2)  in  appendix  A.19).    This  leads  to  the  thought  that  if  an  ocean  front  is  
split   up   into   a   series   of   many   less   intense   fronts   then   the   effects   of   cabbeling   and  
thermobaricity  might  be  reduced  in  proportion  to  the  number  of  such  fronts.    This  is  not  
the   case.      Rather,   the   total   dianeutral   transport   across   a   frontal   region   depends   on   the  
product   of   the   lateral   flux   of   heat   passing   through   the   front   and   the   contrast   in  
temperature   and/or   pressure   across   the   front,   but   is   independent   of   the   sharpness   of   the  
front  (Klocker  and  McDougall  (2010a)).    This  can  be  understood  by  noting  from  above  that  
the  dianeutral  velocity  due  to  cabbeling,   eCab = − gN −2 K CbΘ∇n Θ⋅ ∇n Θ,   is  proportional  to  the  
scalar   product   of   the   epineutral   flux   of   heat   − c0p K ∇n Θ    and   the   epineutral   temperature  
gradient   ∇n Θ .      Spatially   integrating   this   product   over   the   area   of   the   frontal   region,   one  
finds   that   the   total   dianeutral   transport   is   proportional   to   the   lateral   heat   flux   times   the  
difference   in   temperature   across   the   frontal   region   (in   the   case   of   cabbeling)   or   the  
difference  in  pressure  across  the  frontal  region  (in  the  case  of  thermobaricity).      
In  both  the  SIA  and  GSW  software  libraries  the  cabbeling  parameter  is  output  in  units  
of   K −2 .      Expressions   for   Cbθ    and   C bΘ    in   terms   of   enthalpy   in   the   functional   forms  
h SA ,θ , p   and   hˆ ( SA , Θ, p )   can  be  found  in  appendix  P.      

(

)

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32

TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

3.10  Buoyancy  frequency    

  
The  square  of  the  buoyancy  frequency  (sometimes  called  the  Brunt-­‐‑Väisälä  frequency)   N 2   
is   given   in   terms   of   the   vertical   gradients   of   density   and   pressure,   or   in   terms   of   the  
vertical   gradients   of   Conservative   Temperature   and   Absolute   Salinity   (or   in   terms   of   the  
vertical   gradients   of   potential   temperature   and   Absolute   Salinity)   by   (the   g    on   the   left-­‐‑
hand   side   is   the   gravitational   acceleration,   and   x,   y   and   z   are   the   spatial   Cartesian  
coordinates)    

(

g −1N 2 = − ρ −1ρ z + κ Pz = − ρ −1 ρ z − Pz / c 2
= α θ θ z − β θ ∂SA ∂z x , y

)

  

(3.10.1)  

= α ΘΘ z − β Θ ∂SA ∂z x , y .
For   two   seawater   parcels   separated   by   a   small   distance   Δz    in   the   vertical,   an   equally  
accurate  method  of  calculating  the  buoyancy  frequency  is  to  bring  both  seawater  parcels  
adiabatically  and  without  exchange  of  matter  to  the  average  pressure  and  to  calculate  the  
difference   in   density   of   the   two   parcels   after   this   change   in   pressure.      In   this   way   the  
potential   density   of   the   two   seawater   parcels   are   being   compared   at   the   same   pressure.    
This  common  procedure  calculates  the  buoyancy  frequency   N   according  to    

(

N 2 = g α Θ Θ z − β Θ SA z

)

≈−

g Δρ Θ
g 2 Δρ Θ
,       or       N 2 = g 2 ρ β Θ SAP − α ΘΘP ≈
,     (3.10.2)  
ρ Δz
ΔP

(

)

where   Δρ Θ    is   the   difference   between   the   potential   densities   of   the   two   seawater   parcels  
with  the  reference  pressure  being  the  average  of  the  two  original  pressures  of  the  seawater  
parcels.    Eqn.  (3.10.2b)  has  made  use  of  the  hydrostatic  relation   Pz = − g ρ .      
  
  

3.11  Neutral  tangent  plane    

  
The  neutral  plane  is  that  plane  in  space  in  which  the  local  parcel  of  seawater  can  be  moved  
an   infinitesimal   distance   without   being   subject   to   a   vertical   buoyant   restoring   force;   it   is  
the  plane  of  neutral-­‐‑  or  zero-­‐‑  buoyancy.    The  normal  vector  to  the  neutral  tangent  plane   n   
is  given  by    

(

g −1 N 2 n = − ρ −1∇ρ + κ∇P = − ρ −1 ∇ρ − ∇P / c 2
θ

θ

= α ∇θ − β ∇SA
Θ

)

  

(3.11.1)  

Θ

= α ∇Θ − β ∇SA .
As  defined,   n   is  not  quite  a  unit  normal  vector,  rather  its  vertical  component  is  exactly   k ,   
that  is,  its  vertical  component  is  unity.    It  is  clear  that   α θ ∇θ − β θ ∇SA   is  exactly  equal  to  
α Θ∇Θ − β Θ ∇SA .       Interestingly,   both   α θ ∇θ    and   β θ ∇SA    are   independent   of   the   four  
arbitrary  constants  of  the  Gibbs  function  (see  Eqn.  (2.6.2))  while  both   α Θ∇Θ   and   β Θ ∇SA   
contain   an   identical   additional   arbitrary   term   proportional   to   a3 ∇SA ;   terms   that   exactly  
cancel  in  their  difference,   α Θ∇Θ − β Θ ∇SA ,   in  Eqn.  (3.11.1).      
Expressing  the  two-­‐‑dimensional  gradient  of  properties  in  the  neutral  tangent  plane  by  
∇n ,   the  property  gradients  in  a  neutral  tangent  plane  obey    

(

)

− ρ −1∇ n ρ + κ∇ n P = − ρ −1 ∇ n ρ − ∇ n P / c 2 = α θ ∇ nθ − β θ ∇ n SA
= α Θ∇ n Θ − β Θ ∇ n SA   

(3.11.2)  

= 0.
Here   ∇n   is  an  example  of  a  projected  gradient    

∇rτ ≡

IOC Manuals and Guides No. 56

∂τ
∂x r

i +

∂τ
∂y r

j + 0 k ,   

(3.11.3)  

TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

33

that   is   widely   used   in   oceanic   and   atmospheric   theory   and   modelling.      Horizontal  
distances  are  measured  between  the  vertical  planes  of  constant  latitude  x  and  longitude  y  
while   the   values   of   the   property   τ    are   evaluated   on   the   r    surface   (e.   g.   an   isopycnal  
surface,  or  in  the  case  of   ∇n ,  a  neutral  tangent  plane).    This  coordinate  system  is  described  
by  Sutcliffe  (1947),  Bleck  (1978),  McDougall  (1987b),  McDougall  (1995)  and  Griffies  (2004).    
Note  that   ∇rτ   has  no  vertical  component;  it  is  not  directed  along  the   r   surface,  but  rather  
it  points  in  exactly  the  horizontal  direction.      
Finite   difference   versions   of   Eqn.   (3.11.2)   such   as   α ΘΔΘ − β Θ ΔSA ≈ 0    are   also   very  
accurate.    Here   α Θ   and   β Θ   are  the  values  of  these  coefficients  evaluated  at  the  average  
values  of   Θ, SA   and   p   of  two  parcels   SA1 , Θ1 , p1   and   SA2 , Θ2 , p2   on  a  “neutral  surface”  
and   ΔΘ    and   ΔSA    are   the   property   differences   between   the   two   parcels.      The   error  
involved  with  this  finite  amplitude  version  of  Eqn.  (3.11.2),  namely    

(

)

− TbΘ ∫ ( P − P ) d Θ ,   
2

(

)

(3.11.4)  

1

is  described  in  section  2  and  appendix  A(c)  of  Jackett  and  McDougall  (1997).    An  equally  
accurate  finite  amplitude  version  of  Eqn.  (3.11.2)  is  to  equate  the  potential  densities  of  the  
two  fluid  parcels,  each  referenced  to  the  average  pressure   p = 0.5 ( p1 + p2 ) .      
The  reason  why  oceanographers  take  the  strong  lateral  mixing  of  mesoscale  eddies  to  
be   directed   along   the   neutral   tangent   plane   is   because   of   the   smallness   of   the   observed  
dissipation   of   kinetic   energy   ε    in   the   ocean   interior.      If   the   lateral   diffusivity  
K ≈ 102 − 103 m 2 s −1    of   mesoscale   dispersion   and   subsequent   molecular   diffusion   were   to  
occur   along   a   surface   that   differed   in   slope   from   the   neutral   tangent   plane   by   an   angle  
whose   tangent   was   s,   then   the   individual   fluid   parcels   would   be   transported   above   and  
below  the  neutral  tangent  plane  and  would  need  to  subsequently  sink  or  rise  in  order  to  
attain  a  vertical  position  of  neutral  buoyancy.    

  
Figure  5.    Sketch  of  the  consequences  of  the  adiabatic  movement  followed  by  
release  of  fluid  parcels  along  a  plane  that  is  different  to  a  neutral  tangent  plane.      

  
This   vertical   motion   would   either   (i)   involve   no   small-­‐‑scale   turbulent   mixing,   in   which  
case  the  combined  process  is  equivalent  to  epineutral  mixing,  or  (ii),  the  sinking  and  rising  
parcels   would   mix   with   and   entrain   the   surrounding   ocean   in   a   plume-­‐‑like   fashion   (see  
Figure  5),  so  suffering  irreversible  diffusion.    In  this  second  case,  the  dissipation  of  kinetic  
energy   associated   with   the   diapycnal   mixing   would   be   observed.      But   in   fact   the  
dissipation   of   kinetic   energy   in   the   main   thermocline   is   consistent   with   a   diapycnal  
diffusivity  of  only   10−5 m 2 s −1 .    This  small  value  of  the  diapycnal  (vertical)  diffusivity  has  
been  confirmed  by  purposely  released  tracer  experiments.      
When  lateral  diffusion  with  diffusivity   K   is  taken  to  occur  along  a  surface  other  than  
a   neutral   tangent   plane,   some   dianeutral   diffusion   occurs,   and   the   amount   of   this  
dianeutral  diffusion  is  the  same  as  achieved  by  a  vertical  diffusivity  of   s 2 K   where   s 2   is  
the   square   of   the   vector   slope   ∇r z − ∇n z    between   the   mixing   direction   and   the   neutral  
tangent  plane.    This  result  is  proven  as  follows.      

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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

The  lateral  flux  of  Neutral  Density  along  the  direction  of  mixing,  the   r   surface  is    

− K∇ rγ = − K γ z ( ∇r z − ∇n z )   ,  

(3.11.5)  

and  the  component  of  this  lateral  flux  across  the  neutral  tangent  plane  is    

− K∇ rγ ⋅ ( ∇r z − ∇n z ) = − K γ z ( ∇r z − ∇n z )   .  
2

(3.11.6)  

Dividing  by  minus  the  vertical  gradient  of  Neutral  Density,   −γ z ,  shows  that  this  flux  is  the  
same   as   that   caused   by   the   positive   fictitious   vertical   diffusivity   of   density  
2
∇r z − ∇n z K = s 2 K .      
Hence  if  all  of  this  observed  diapycnal  diffusivity  (based  on  the  observed  dissipation  
of   turbulent   kinetic   energy   ε )   were   due   to   mesoscale   eddies   mixing   along   a   direction  
different   to   neutral   tangent   planes,   the   (tangent   of   the)   angle   between   this   mesoscale  
mixing  direction  and  the  neutral  tangent  plane,  s,  would  satisfy   10−5 m 2 s −1 = s 2 K .    Using  
K ≈ 103 m 2 s −1   gives  the  maximum  value  of  s  to  be   10−4 .     Since  we  believe  that  bona  fide  
interior   diapycnal   mixing   processes   (such   as   breaking   internal   gravity   waves)   are  
responsible   for   the   bulk   of   the   observed   diapycnal   diffusivity,   we   conclude   that   the  
angular   difference   s   between   the   direction   of   mesoscale   eddy   mixing   and   the   neutral  
tangent  plane  must  be  substantially  less  than   10−4 ;  say   2x10−5   for  argument’s  sake.    
  
  

(

)

3.12  Geostrophic,  hydrostatic  and  “thermal  wind”  equations    

  
The  geostrophic  approximation  to  the  horizontal  momentum  equations  (Eqn.  (B9)  below)  
equates  the  Coriolis  term  to  the  horizontal  pressure  gradient   ∇ z P   so  that  the  geostrophic  
equation  is    
(3.12.1)  
f k × ρ u = −∇z P               or               fv = ρ1 k × ∇ z P = g k × ∇p z ,  

where   u    is   the   three   dimensional   velocity   and   v = − k × ( k × u )    is   the   horizontal   velocity  
where   k    is   the   vertical   unit   vector   (pointing   upwards)   and   f    is   the   Coriolis   parameter.    
The  last  part  of  the  above  equation  has  used   ∇ z P = − Pz ∇ p z   from  Eqn.  (3.12.4b)  below  and  
the   hydrostatic   approximation,   which   is   the   following   approximation   to   the   vertical  
momentum  equation  (B9),    
(3.12.2)  
Pz = − g ρ .   
The  use  of   P   in  these  equations  rather  than   p   serves  to  remind  us  that  in  order  to  retain  
the   usual   units   for   height,   density   and   the   gravitational   acceleration,   pressure   in   these  
dynamical  equations  must  be  expressed  in   Pa   not   dbar.     
The   so   called   “thermal   wind”   equation   is   an   equation   for   the   vertical   gradient   of   the  
horizontal   velocity   under   the   geostrophic   approximation.      Vertically   differentiating   Eqn.  
(3.12.1)  and  using  the  hydrostatic  equation  Eqn.  (3.12.2),  the  thermal  wind  can  be  written    

f vz =

( ) k ×∇ P +
1

ρ z

z

1

ρ

k ×∇ z ( Pz ) = −

g

ρ

k ×∇ p ρ =

N2
gρ

k ×∇ n P,   

(3.12.3)  

where   ∇ p    is   the   projected   lateral   gradient   operator   in   the   isobaric   surface   (see   Eqn.  
(3.11.3)).      The   last   part   of   this   equation   relates   the   “thermal   wind”,   f v z ,   to   the   pressure  
gradient   in   the   neutral   tangent   plane   (McDougall,   1995).      Note   that   the   Boussinesq  
approximation   has   not   been   made   to   derive   any   part   of   Eqn.   (3.12.3).      Under   the  
Boussinesq   approximation,   ∇ p ρ    is   approximated   by   ∇ z ρ ,   and   the   last   term   in   Eqn.  
(3.12.3)   is   approximated   as   − N 2 k ×∇ n z .      The   derivation   of   Eqn.   (3.12.3)   proceeds   as  
follows.    To  go  from  the  second  part  of  Eqn.  (3.12.3)  to  the  third  part  use  is  made  of    

∇ p ρ = ∇ z ρ + ρ z ∇ p z         and         ∇ p P = 0 = ∇ z P + Pz ∇ p z .  

(3.12.4a,b)  

To  go  from  the  third  part  of  Eqn.  (3.12.3)  to  the  final  part,  use  is  made  of  Eqn.  (3.12.4a)  and  
∇ n ρ = ∇ z ρ + ρ z ∇ n z ,  which,  when  combined  gives  

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∇ p ρ = ∇ n ρ − ρ z ∇ n z − ∇ p z .  

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Now  Eqn.  (3.12.4b)  is  used  together  with   ∇ n P = ∇ z P + Pz ∇ n z   to  find    

(

)

∇ n P = Pz ∇ n z − ∇ p z ,  

(3.12.6)  

and  this  is  substituted  into  Eqn.  (3.12.5)  to  find    

∇ p ρ = ∇ n ρ − ρ z ∇ n P Pz .  

(3.12.7)  

Now  along  a  neutral  tangent  plane  we  know  that   ∇ n ρ = ρκ ∇ n P   ( κ   is  the  isentropic  and  
isohaline  compressibility  of  seawater)  and  substituting  this  into  Eqn.  (3.12.7)  leads  to  the  
2
final   expression   of   Eqn.   (3.12.3),   namely   Ng ρ k × ∇ n P    (recognizing   that   the   buoyancy  
frequency  is  defined  by   N 2 = g κ Pz − ρ1 ρ z ).      
The   rotation   of   the   horizontal   velocity   vector   with   height   can   be   determined   as  
follows.      Let   the   angle   of   the   horizontal   velocity   v    with   respect   to   due   east   (measured  
counter-­‐‑clockwise)   be   ϕ    so   that   v = v cosϕ , sin ϕ .      Vertically   differentiating   this  
2
equation  and  taking  the  cross  product  with   v   leads  to   v × v z = k ϕ z v   which  shows  that  
the  rate  of  spiraling  of  the  horizontal  velocity  vector  in  the  vertical   ϕ z   is   proportional   to  
the   amount   by   which   this   velocity   is   not   parallel   to   the   direction   of   the   “thermal   wind”  
shear   v z .    The  last  relation  can  be  rewritten  as    

(

)

(

ϕz v

2

)

= k ⋅ v × v z = uvz − vuz = − v ⋅k × v z = − v ⋅∇ × v ,  

(3.12.8)  

which  demonstrates  that  the  rotation  of  the  horizontal  velocity  with  height  is  proportional  
to  the  helicity  of  the  horizontal  velocity,   v ⋅∇ × v .      
Now,  substituting  Eqn.  (3.12.3)  for  the  “thermal  wind”   v z ,  into  Eqn.  (3.12.8)  we  find    

ϕz v

2

= − v ⋅k × v z =

N2
fg ρ

( )

Under   the   usual   Boussinesq   approximation   − g ρ
neutral  tangent  plane,   ∇n z ,  so  that  we  have    

ϕz v

2

≈ −

N2
f

v ⋅∇n P .  
−1

(3.12.9)  

∇n P    is   set   equal   to   the   slope   of   the  

v ⋅∇n z ,  

(3.12.10)  

and   since   the   vertical   velocity   through   geopotentials,   w ,   is   given   by   the   simple  
geometrical  relationship   w = zt + v ⋅∇n z + e   (where   e   is  the  dianeutral  velocity,  that  is,  
n
the  vertical  velocity  through  the  neutral  tangent  plane),  we  have    

ϕz v

2

≈ −

N2
f

( w − e − z ),   

(3.12.11)  

t n

showing   that   the   rotation   of   the   horizontal   velocity   vector   with   height   is   not   simply  
proportional  to  the  vertical  velocity  of  the  flow  but  rather  only  to  the  sliding  motion  along  
the  neutral  tangent  plane,   v ⋅∇n z .      
  
  

3.13  Neutral  helicity    

  
The  neutral  tangent  plane  was  defined  in  section  3.11  as  the  plane  in  which  parcels  can  be  
moved   in   an   adiabatic   and   isohaline   manner   without   experiencing   a   vertical   buoyant  
force.    The  normal   n   to  the  neutral  tangent  plane  is  given  by  Eqn.  (3.11.1)  and  it  is  natural  
to  think  that  all  these  little  tangent  planes  would  link  up  and  form  a  well-­‐‑defined  surface,  
but   this   is   not   actually   the   case   in   the   ocean.      In   order   to   understand   why   the   ocean  
chooses  to  be  so  ornery  we  need  to  understand  what  property  the  normal   n   to  a  surface  
must  fulfill  in  order  that  the  surface  exists.      
  
In   general,   for   a   surface   to   exist   in   x, y, z    space   there   must   be   a   function   φ x, y, z   
that  is  constant  on  the  surface  and  whose  gradient   ∇φ   is  in  the  direction  of  the  normal  to  
the   surface,   n .      That   is,   there   must   be   an   integrating   factor   b x, y, z    such   that   ∇φ = bn .    
Assuming   now   that   the   surface   does   exist,   consider   a   line   integral   of   bn    along   a   closed  
curved   path   in   the   surface.      Since   the   line   element   of   the   integration   path   is   everywhere  
normal  to   n ,  the  closed  line  integral  is  zero,  and  by  Stokes’s  theorem,  the  area  integral  of  

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∇ × bn    must   be   zero   over   the   area   enclosed   by   the   closed   curved   path.      Since   the   area  
element  of  integration   dA   is  in  the  direction   n ,  it  is  clear  that   ∇ × bn ⋅ dA   is  proportional  
to   ∇ × bn ⋅ n .      The   only   way   that   this   area   integral   can   be   guaranteed   to   be   zero   for   all  
such   closed   paths   is   if   the   integrand   is   zero   everywhere   on   the   surface,   that   is,   if  
∇ × bn ⋅ n = ∇b × n ⋅ n + b ∇ × n ⋅ n = 0 ,   that   is,   if   n ⋅∇ × n = 0    at   all   locations   on   the  
surface.      
  
For   the   case   in   hand,   the   normal   to   the   neutral   tangent   plane   is   in   the   direction  
α Θ∇Θ − β Θ ∇SA    and   we   define   the   neutral   helicity   H n    as   the   scalar   product   of  
α Θ∇Θ − β Θ ∇SA   with  its  curl,    

( )

( )

( )

(

)

(

)

(

)

(

)

H n ≡ α Θ∇Θ − β Θ ∇SA ⋅ ∇ × α Θ∇Θ − β Θ ∇SA   .  

(3.13.1)  

Neutral   tangent   planes   (which   do   exist)   do   not   link   up   in   space   to   form   a   well-­‐‑defined  
neutral  surface  unless  the  neutral  helicity   H n   is  everywhere  zero  on  the  surface.      
  
Recognizing   that   both   the   thermal   expansion   coefficient   and   the   saline   contraction  
coefficient   are   functions   of   SA ,Θ, p ,   neutral   helicity   H n    may   be   expressed   as   the  
following  four  expressions,  all  of  which  are  proportional  to  the  thermobaric  coefficient   TbΘ   
of  the  equation  of  state,    

(

Hn =
=

)

β Θ TbΘ ∇P ⋅ ∇SA × ∇Θ

(

)

Pz β Θ TbΘ ∇ p SA × ∇ p Θ ⋅ k

= g −1 N 2TbΘ ( ∇n P × ∇n Θ ) ⋅ k

  

(3.13.2)  

≈ g −1 N 2TbΘ ( ∇a P × ∇a Θ) ⋅ k
where   Pz   is  simply  the  vertical  gradient  of  pressure  ( Pa m−1 )  and   ∇n Θ   and   ∇ p Θ   are  the  
two-­‐‑dimensional   gradients   of   Θ    in   the   neural   tangent   plane   and   in   the   horizontal   plane  
(actually  the  isobaric  surface)  respectively.    The  gradients   ∇a P   and   ∇a Θ   are  taken  in  an  
approximately  neutral  surface.    
Since   α θ ∇θ − β θ ∇SA    and   α Θ∇Θ − β Θ ∇SA    are   exactly   equal,   neutral   helicity   can   be  
defined   in   Eqn.   (3.13.1)   as   the   scalar   product   of   this   vector   with   its   curl   based   on   either  
formulation,   so   that   (from   the   third   line   of   Eqn.   (3.13.2),   and   bearing   in   mind   that   ∇n Θ   
and   ∇nθ   are  parallel  vectors)  we  see  that   Tbθ ∇nθ = TbΘ∇n Θ,   a  result  that  we  use  in  section  
3.8  and  in  appendix  A.14.    Neutral  helicity  has  units  of   m −3 .       
Interestingly,   for   given   magnitudes   of   the   epineutral   gradients   of   pressure   and  
Conservative   Temperature,   neutral   helicity   is   maximized   when   these   gradients   are  
perpendicular   since   neutral   helicity   is   proportional   to   TbΘ ( ∇n P × ∇n Θ ) ⋅ k    (see   Eqn.  
(3.13.2)),   while   the   dianeutral   advection   of   thermobaricity,   eTb = − gN −2 K TbΘ∇n Θ⋅ ∇n P ,   is  
maximized  when   ∇n Θ   and   ∇n P   are  parallel  (see  section  3.8).      
Because  of  the  non-­‐‑zero  neutral  helicity  in  the  ocean,  lateral  motion  following  neutral  
tangent   planes   has   the   character   of   helical   motion.      That   is,   if   we   ignore   the   effects   of  
diapycnal  mixing  processes  (as  well  as  ignoring  cabbeling  and  thermobaricity),  the  mean  
flow  around  ocean  gyres  still  passes  through  any  well-­‐‑defined  “density”  surface  because  
of  the  helical  nature  of  neutral  trajectories,  caused  in  turn  by  the  non-­‐‑zero  neutral  helicity.    
This   dia-­‐‑surface   flow   is   expressed   in   Eqns.   (A.25.4)   and   (A.25.6)   in   terms   of   the  
appropriate  mean  horizontal  velocity  and  the  difference  between  the  slope  of  the  neutral  
tangent  plane  and  the  slope  of  a  well-­‐‑defined  “density”  surface.      
Neutral  helicity  in  the  world  ocean  is  observed  to  be  small  in  some  sense.    One  way  of  
visualizing   this   smallness   of   H n    is   to   examine   all   the   hydrographic   data   in   SA ,Θ, p   
space.      When   this   is   done   for   an   entire   ocean   basin   (for   example,   the   whole   of   the  
combined   North   and   South   Atlantic   oceans),   and   the   data   is   spun   in   this   three-­‐‑
dimensional   SA ,Θ, p    space,   it   is   clear   that   the   ocean   hydrography   lies   close   to   a   single  
surface  in  this   SA ,Θ, p   space.    We  will  now  show  that  if  all  the   SA ,Θ, p   data  from  the  
ocean  lie  exactly  on  a  single  surface   f SA ,Θ, p = 0   in   SA ,Θ, p   space,  then  this  requires  

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∇SA × ∇Θ ⋅∇P = 0   everywhere  in  physical   x, y, z   space.    That  is,  we  will  prove  that  the  
“skinny”  nature  of  the  ocean  hydrography  in   SA ,Θ, p   space  is  a  direct  indication  of  the  
smallness  of  neutral  helicity   H n .      
Taking   the   spatial   gradient   of   f SA ,Θ, p = 0    in   physical   x, y, z    space   we   have  
∇f = 0   since   f   is  zero  at  every  point  in  physical   x, y, z   space.    Expanding   ∇f   in  terms  
of  the  spatial  gradients   ∇SA ,   ∇Θ ,  and   ∇P ,  and  taking  the  scalar  product  with   ∇SA × ∇Θ   
gives    

(

fP

SA ,Θ

(

)

(

)

)

(

)

∇SA × ∇Θ ⋅∇P = 0 .  

(3.13.3)  

In  the  general  case  of   f P ≠ 0 ,  the  result   ∇SA × ∇Θ ⋅∇P = 0   is  proven.    In  the  special  case  
f P = 0 ,   f    is   independent   of   P    so   that   there   is   a   simpler   equation   for   the   surface   f ,  
being   f SA ,Θ = 0 ,  which  is  the  equation  for  a  single  line  on  the   SA ,Θ   diagram;  a  single  
“water-­‐‑mass”   for   the   whole   world   ocean.      In   this   case,   changes   in   SA    are   locally  
proportional   to   those   of   Θ    so   that   ∇SA × ∇Θ = 0    which   guarantees   ∇SA × ∇Θ ⋅∇P = 0 .    
Hence  we  have  proven  that  the  “skinniness”  of  the  ocean  hydrography  in   SA ,Θ, p   space  
is  a  direct  indication  of  the  smallness  of  neutral  helicity   H n .      
The   “skinny”   nature   of   the   North   and   South   Atlantic   hydrography   is   illustrated   in  
Figure   6,   which   shows   all   the   hydrographic   data   on   the   SA − Θ    diagram   at   a   pressure   of  
500   dbar .      This   cut   at   constant   pressure   through   the   hydrographic   data   in   three-­‐‑
dimensional   SA ,Θ, p    space,   and   similar   cuts   at   different   fixed   pressures,   show   that   the  
data  from  the  whole  physical   x, y, z   volume  of  the  North  and  South  Atlantic  lie  close  to  a  
single   surface   in   the   three-­‐‑dimensional   SA ,Θ, p    space.      Figure   6   also   illustrates   the  
method  of  formation  of  one  of  Reid  and  Lynn’s  (1971)  “isopycnals”  and  how  the  potential  
density   anomaly   with   respect   to   the   sea   surface,   σ Θ ,   of   27.3 kg m −3    is   matched   to   σ 1    of  
31.938 kg m −3    in   the   Southern   Ocean   but   to   a   different   σ Θ    of   27.44 kg m −3    in   the   North  
Atlantic.      

(

)

(

)

(

(

  

)

(

)

(

)

)

  
Figure   6.      Hydrographic   data   from   the   ocean   atlas   of   Gouretski   and  
Koltermann   (2004)   for   the   North   and   South   Atlantic   at   a   pressure   of   500   dbar.    
The   colour   of   the   data   points   indicates   the   latitude,   from   blue   in   the   south  
through  green  at  the  equator  to  red  in  the  north.      

Neutral   helicity   is   proportional   to   the   component   of   the   vertical   shear   of   the  
geostrophic  velocity  ( v z ,   the  “thermal  wind”)  in  the  direction  of  the  temperature  gradient  
along  the  neutral  tangent  plane   ∇n Θ ,   since,  from  Eqn.  (3.12.3)  and  the  third  line  of  (3.13.2)  
we  find  that    

H n = ρTbΘ fv z ⋅ ∇n Θ.   

(3.13.4)  

In   the   evolution   equation   of   potential   vorticity   defined   with   respect   to   potential  
density   ρ θ    there   is   the   baroclinic   production   term   ρ −2∇ρ θ ⋅∇ρ × ∇P    (Straub   (1999))   and  

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the   first   term   in   a   Taylor   series   expansion   for   this   baroclinic   production   term   is  
proportional  to  neutral  helicity  and  is  given  by  (McDougall  and  Jackett  (2007))    

ρ −2∇ρ θ ⋅∇ρ × ∇P ≈ ( Pr − P ) H n   

(3.13.5)  

where   Pr    is   the   reference   pressure   of   the   potential   density.      Similarly,   the   curl   in   a  
potential   density   surface   of   the   horizontal   pressure   gradient   term   in   the   horizontal  
momentum  equation,   ∇σ × ρ1 ∇z p ,   is  given  by  (McDougall  and  Klocker  (2010))      

(

)

−1

⎛ ∂ρ Θ ⎞
(3.13.6)  
∇σ × ρ ∇ z P ⋅ k = H ( Pr − P ) ⎜ −
⎟ .   
⎝ ∂z ⎠
The  fact  that  this  curl  is  nonzero  proves  that  a  geostrophic  streamfunction  does  not  exist  in  
a  potential  density  surface.      
The   absolute   velocity   vector   in   the   ocean   can   be   written   as   a   closed   expression  
involving   neutral   helicity,   and   this   expression   is   derived   as   follows.      First   the   Eulerian-­‐‑
mean   horizontal   velocity   is   related   directly   to   mixing   processes   by   invoking   the   water-­‐‑
mass  transformation  equation  (A.23.1),  so  that    

(

1

(

)

n

)

(

v ⋅∇ nΘ̂ = γ z ∇ n ⋅ γ z−1 K∇ nΘ̂ + KgN −2Θ̂ z CbΘ∇ nΘ̂ ⋅∇ nΘ̂ + TbΘ∇ nΘ̂ ⋅∇ n P
+ Dβ Θ gN −2Θ̂3z

2

d ŜA

− Ψz ⋅∇ nΘ̂ − Θ̂t ,

)

  

(3.13.7)  

n
d Θ̂ 2
where  the  thickness-­‐‑weighted  mean  velocity  of  density-­‐‑coordinate  averaging,   v̂ ,  has  been  
written  as   v̂ = v + Ψz ,  that  is,  as  the  sum  of  the  Eulerian-­‐‑mean  horizontal  velocity   v   and  
the  quasi-­‐‑Stokes  eddy-­‐‑induced  horizontal  velocity   Ψz   (McDougall  and  McIntosh  (2001)).    
The   quasi-­‐‑Stokes   vector   streamfunction   Ψ    is   usually   expressed   in   terms   of   an   imposed  
lateral  diffusivity  and  the  slope  of  the  locally-­‐‑referenced  potential  density  surface  (Gent  et  
al.,  (1995)).    More  generally,  at  least  in  a  steady  state  when   Θ̂t   is  zero,  the  right-­‐‑hand  side  
n
of   Eqn.   (3.13.7)   is   due   only   to   mixing   processes   and   once   the   form   of   the   lateral   and  
vertical   diffusivities   are   known,   these   terms   are   known   in   terms   of   the   ocean’s  
hydrography.    Eqn.  (3.13.7)  is  written  more  compactly  as    

v ⋅ τ = v ⊥                         where                       τ ≡ ∇ nΘ̂ ∇ nΘ̂ ,  

(3.13.8)  

and   v ⊥   is  interpreted  as  being  due  to  mixing  processes.      
Following   Needler   (1985)   and   McDougall   (1995)   the   mean   horizontal   velocity   v    is  
split  into  components  along  and  across  the  contours  of   Θ̂   on  the  neutral  tangent  plane,    

v = v τ × k + v ⊥ τ   ,  

(3.13.9)  



where   v = v ⋅ τ × k .      Note   that   if   τ    points   northwards   then   τ × k    points   eastward.      The  
expression   v ⋅ τ = v ⊥   of  Eqn.  (3.13.8)  is  now  vertically  differentiated  to  obtain    

v ⋅ τ z = − v z ⋅ τ + vz⊥

= −

N2
fg ρ

k × ∇n P ⋅ τ + vz⊥ ,  

(3.13.10)  
2

where  we  have  used  the  “thermal  wind”  equation  (3.12.3),   v z = Nfg ρ k × ∇n P .    We  will  now  
show  that  the  left-­‐‑hand  side  of  this  equation  is   − φ z v   where   φ z is  the  rate  of  rotation  of  the  
direction  of  the  unit  vector   τ   with  respect  to  height  (in  radians  per  metre).    By  expressing  
the  two-­‐‑dimensional  unit  vector   τ   in  terms  of  the  angle   φ   (measured  counter-­‐‑clockwise)  
of   τ    with   respect   to   due   east   so   that   τ = cos φ , sin φ ,   we   see   that   τ × k = sin φ , − cos φ ,  
τ z = − φ z τ × k   and   k ⋅ τ × τ z = φ z .    Interestingly,   φ z   is  also  equal  to  minus  the  helicity  of   τ   
(and   to   minus   the   helicity   of   τ × k ),   that   is,   φ z = − τ ⋅∇ × τ = − τ × k ⋅∇ × τ × k ,   where  
the  helicity  of  a  vector  is  defined  to  be  the  scalar  product  of  the  vector  with  its  curl.    From  
the   velocity   decomposition   (3.13.9)   and   the   equation   τ z = − φ z τ × k    we   see   that   the   left-­‐‑
hand  side  of  Eqn.  (3.13.10),   v ⋅ τ z ,  is   − φ z v ,  hence   v   can  be  expressed  as    

(

)

(

(

v =

)

(

v⊥
vz⊥
N 2 k ⋅∇n P × τ
Hn
− z                       or                     v =
−
,   
fg ρ
φz
φz
φz
φ z ρ f TbΘ ∇ nΘ̂

IOC Manuals and Guides No. 56

)

)

(3.13.11)  

TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

39

where  we  have  used  the  definition  of  neutral  helicity   H n ,  Eqn.  (3.13.2).    The  expression  for  
both  horizontal  components  of  the  Eulerian-­‐‑mean  horizontal  velocity  vector   v   is    

⎧⎪ N 2 k ⋅∇ P × τ
v ⊥ ⎫⎪
n
v = ⎨
− z ⎬ τ × k + v ⊥ τ ,   
(3.13.12)  
φz
φ z ⎪⎭
⎪⎩ fg ρ
and  the  horizontal  velocity  due  to  solely  the  two  mixing  terms  can  be  expressed  as    

vz⊥
(v ⊥ )2 ⎛ τ × k ⎞
1 ⊥
,  whose  magnitude  is  
−
τ × k + v⊥ τ =
v τ
φz
φ z ⎜⎝ v ⊥ ⎟⎠ z
φz

( )

z

=

(v τ )
⊥

φ

.    (3.13.13)  

  
Equation   (3.13.12)   for   the   Eulerian-­‐‑mean   horizontal   velocity   v    shows   that   in   the  
absence   of   mixing   processes   (so   that   v ⊥ = vz⊥ = 0 )   and   so   long   as   (i)   the   epineutral   Θ̂   
contours  do  spiral  in  the  vertical  and  (ii)   ∇ nΘ̂   is  not  zero,  then  neutral  helicity   H n   (which  
is  proportional  to   k ⋅∇n P × τ )  is  required  to  be  non-­‐‑zero  in  the  ocean  whenever  the  ocean  is  
not   motionless.      Neutral   helicity   arises   in   this   context   because   it   is   proportional   to   the  
component   of   the   thermal   wind   vector   v z    in   the   direction   across   the   Θ̂    contour   on   the  
neutral   tangent   plane   (see   Eqn.   (3.13.4)).      This   derivation   of   the   expression   for   the   mean  
absolute  horizontal  velocity  vector   v   is  based  on  McDougall  (1995)  and  Zika  et  al.  (2010a).      
  
  

3.14  Neutral  Density    

  
Neutral   Density   is   the   name   given   to   a   density   variable   that   results   from   the   computer  
software  described  in  Jackett  and  McDougall  (1997).    Neutral  Density  is  given  the  symbol  
γ n    but   it   is   not   a   thermodynamic   variable   as   it   is   a   function   not   only   of   salinity,  
temperature   and   pressure,   but   also   of   latitude   and   longitude.      Because   of   the   non-­‐‑zero  
neutral   helicity   H n    in   the   ocean   it   is   not   possible   to   form   surfaces   that   are   everywhere  
osculate   with   neutral   tangent   planes   (McDougall   and   Jackett   (1988)).      Neutral   Density  
surfaces  minimize  in  some  sense  the  global  differences  between  the  slopes  of  the  neutral  
tangent  plane  and  the  Neutral  Density  surface.    This  slope  difference  is  given  by    

(

)

s = ∇ n z − ∇ a z = − gN −2 α Θ∇ aΘ − β Θ∇ a SA ,  

(3.14.1)  

where   ∇n z   is  the  slope  of  the  neutral  tangent  plane,   ∇a z   is  the  slope  of  the  approximately  
neutral   surface   and   ∇a    is   the   two-­‐‑dimensional   gradient   operator   in   the   approximately  
neutral  surface  (of  which  a  Neutral  Density  surface  is  one  example).    The  vertical  velocity  
through   an   approximately   neutral   surface   due   to   lateral   motion   along   a   neutral   tangent  
plane  is  the  scalar  product   v ⋅ s   where   v   is  the  horizontal  velocity  (see  Eqn.  (A.25.4)).    
  
  

3.15  Stability  ratio    

The   stability   ratio   Rρ    is   the   ratio   of   the   vertical   contribution   from   Conservative  
Temperature  to  that  from  Absolute  Salinity  to  the  static  stability   N 2   of  the  water  column.    
From  (3.10.1)  above  we  find    

Rρ =

α ΘΘ z

β Θ ( SA )z

.   

(3.15.1)  

Rρ   is  available  in  the  GSW  Oceanographic  Toolbox  as  the  function  gsw_Turner_Rsubrho.      

  
  

3.16  Turner  angle    
The   Turner   angle   Tu ,   named   after   J.   Stewart   Turner,   is   defined   as   the   four-­‐‑quadrant  
arctangent  (Ruddick  (1983)  and  McDougall  et  al.  (1988),  particularly  their  Figure  1)    

(

( )z

( )z ) ,  

Tu = tan −1 α ΘΘ z + β Θ SA , α ΘΘ z − β Θ SA

(3.16.1)  

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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

where   the   first   of   the   two   arguments   of   the   arctangent   function   is   the   “y”-­‐‑argument   and  
the   second   one   the   “x”-­‐‑argument,   this   being   the   common   order   of   these   arguments   in  
Fortran   and   MATLAB.      The   Turner   angle   Tu    is   quoted   in   degrees   of   rotation.      Turner  
angles   between   45°   and   90°   represent   the   “salt-­‐‑finger”   regime   of   double-­‐‑diffusive  
convection,   with   the   strongest   activity   near   90°.      Turner   angles   between   − 45°    and   − 90°   
represent  the  “diffusive”  regime  of  double-­‐‑diffusive  convection,  with  the  strongest  activity  
near  −90°.    Turner  angles  between   − 45°   and  45°  represent  regions  where  the  stratification  
is   stably   stratified   in   both   Θ    and   SA .       Turner   angles   greater   than   90°   or   less   than   − 90°   
characterize   a   statically   unstable   water   column   in   which   N 2 < 0.       As   a   check   on   the  
calculation  of  the  Turner  angle,  note  that   Rρ = − tan (Tu + 45° ) .     The  Turner  angle  and  the  
stability   ratio   are   available   in   the   GSW   Oceanographic   Toolbox   from   the   function  
gsw_Turner_Rsubrho.      
  
  

3.17  Property  gradients  along  potential  density  surfaces    
  

The   two-­‐‑dimensional   gradient   of   a   scalar   ϕ    along   a   potential   density   surface   ∇σ ϕ    is  
related  to  the  corresponding  gradient  in  the  neutral  tangent  plane   ∇nϕ   by    

∇σ ϕ = ∇nϕ +

ϕ z Rρ [r − 1]

(3.17.1)  
∇n Θ   
Θz ⎡⎣ Rρ − r ⎤⎦
(from  McDougall  (1987a)),  where   r   is  the  ratio  of  the  slope  on  the   SA − Θ   diagram  of  an  
isoline  of  potential  density  with  reference  pressure   pr   to  the  slope  of  a  potential  density  
surface  with  reference  pressure   p ,  and  is  defined  by    
α Θ ( SA , Θ, p ) β Θ ( SA , Θ, p )
.  
(3.17.2)  
r = Θ
α ( SA , Θ, pr ) β Θ ( SA , Θ, pr )
Substituting   ϕ = Θ    into   (3.17.1)   gives   the   following   relation   between   the   (parallel)  
isopycnal  and  epineutral  gradients  of   Θ     

r ⎡⎣ Rρ − 1⎤⎦

∇n Θ = G Θ∇n Θ   
⎡⎣ Rρ − r ⎤⎦
where  the  “isopycnal  temperature  gradient  ratio”    
∇σ Θ =

(3.17.3)  

⎡⎣ Rρ − 1⎤⎦
  
⎡⎣ Rρ r − 1⎤⎦

(3.17.4)  

⎡ Rρ − 1⎤⎦
GΘ
∇σ SA = ⎣
∇n SA =
∇n SA .   
r
⎡⎣ Rρ − r ⎤⎦

(3.17.5)  

GΘ ≡

has   been   defined   as   a   shorthand   expression   for   future   use.      This   ratio   G Θ    is   available   in  
the  GSW  Toolbox  from  the  algorithm  gsw_isopycnal_vs_ntp_CT_ratio,  while  the  ratio   r   
of  Eqn.  (3.17.2)  is  available  there  as  gsw_isopycnal_slope_ratio.    Substituting   ϕ = SA   into  
Eqn.   (3.17.1)   gives   the   following   relation   between   the   (parallel)   isopycnal   and   epineutral  
gradients  of   S A     

  
  

3.18  Slopes  of  potential  density  surfaces  and  neutral  tangent  planes  compared    

  
The   two-­‐‑dimensional   slope   of   a   surface   is   defined   as   the   two-­‐‑dimensional   gradient   of  
height   z   of   that   surface.      The   two-­‐‑dimensional   slope   of   a   surface   is   an   exactly   horizontal  
gradient   vector;   it   has   no   vertical   component.      The   slope   difference   between   the   neutral  
tangent   plane   and   a   potential   density   surface   with   reference   pressure   pr    is   given   by  
(McDougall  (1988))    

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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

∇n z − ∇σ z =

Rρ [1 − r ] ∇n Θ
=
⎣⎡ Rρ − r ⎦⎤ Θz

(1 − G ) ∇ΘΘ
Θ

n

=

z

41

Rρ [1 − r ] ∇σ Θ
∇n Θ − ∇σ Θ
=
.   (3.18.1)  
Θz
r ⎣⎡ Rρ − 1⎦⎤ Θz

While   potential   density   surfaces   have   been   the   most   commonly   used   surfaces   with  
which   to   separate   “isopycnal”   mixing   processes   from   vertical   mixing   processes,   many  
other  types  of  density  surface  have  been  used.    The  list  includes  specific  volume  anomaly  
surfaces,   patched   potential   density   surfaces   (Reid   and   Lynn   (1971)),   Neutral   Density  
surfaces   (Jackett   and   McDougall   (1997)),   orthobaric   density   surfaces   (de   Szoeke   et   al.  
(2000))  and  some  polynomial  fits  of  Neutral  Density  as  function  of  only  salinity  and  either  
θ    or   Θ    (Eden   and   Willebrand   (1999),   McDougall   and   Jackett   (2005b)).      The   most   recent  
method  for  forming  approximately  neutral  surfaces  is  that  of  Klocker  et  al.  (2009a,b).    This  
method   is   relatively   computer   intensive   but   has   the   benefit   that   the   remnant   mis-­‐‑match  
between   the   final   surface   and   the   neutral   tangent   plane   at   each   point   is   due   only   to   the  
neutral  helicity  of  the  data  through  which  the  surface  passes.    The  relative  skill  of  all  these  
surfaces   at   approximating   the   neutral   tangent   plane   slope   at   each   point   has   been  
summarized  in  the  equations  and  histogram  plots  in  the  papers  of  McDougall  (1989,  1995),  
McDougall  and  Jackett  (2005a,  2005b),  and  Klocker  et  al.  (2009a,b).      
When  lateral  mixing  with  isopycnal  diffusivity   K   is  imposed  along  potential  density  
surfaces   rather   than   along   neutral   tangent   planes,   a   fictitious   diapycnal   diffusivity   arises  
which   is   often   labeled   the   “Veronis   effect”   after   Veronis   (1975)   (who   considered   the   ill  
effects  of  exactly  horizontal  versus  isopycnal  mixing).    This  fictitious  diapycnal  diffusivity  
of  density  is  equal  to   K   times  the  square  of  the  slope  error,  Eqn.  (3.18.1).      
  
  

3.19  Slopes  of  in  situ  density  surfaces  and  specific  volume  anomaly  surfaces    
  
The  vector  slope  of  an  in  situ  density  surface,   ∇ρ z ,   is  defined  to  be  the  horizontal  vector    

∇ρ z =

∂z
∂x ρ

i +

∂z
∂y ρ

j + 0 k ,   

(3.19.1)  

representing   the   “dip”   of   the   surface   in   both   horizontal   directions   (note   that   height   z    is  
defined  positive  upwards).    The  difference  between  this  vector  slope  and  the  (very  small)  
slope   of   an   isobaric   surface   ∇ p z    can   be   related   to   the   slope   of   the   neutral   tangent   plane  
with   respect   to   the   isobaric   surface,   ∇ n z − ∇ p z ,   by   ( g    is   the   gravitational   acceleration)  
(McDougall  (1989))    
−1

⎡
g 2 c2 ⎤
(3.19.2)  
∇ρ z − ∇ p z = ∇n z − ∇ p z ⎢1 +
⎥ ,   
N2 ⎦
⎣
where   c    is   the   speed   of   sound   and   N    is   the   buoyancy   frequency.      In   the   upper   water  
column   where   the   square   of   the   buoyancy   frequency   is   significantly   larger   than  
g 2 c 2 ≈ 4.3x10−5 s −2 ,   the   in   situ   density   surface   has   a   similar   slope   to   the   neutral   tangent  
plane   ∇n z .       In   the   deep   ocean   N 2    is   only   about   1%   of   g 2 c 2    and   so   the   surfaces   of  
constant  in  situ  density  have  a  slope  of  only  1%  of  the  slope  of  the  neutral  tangent  plane.    
At  a  pressure  of  about  1000  dbar  where   N 2 ≈ 10−5 s−2 ,  the  slope  of  an  in  situ  density  surface  
is   only   about   one   fifth   that   of   the   neutral   tangent   plane.      Neutrally   buoyant   floats   in   the  
ocean   are   usually   metal   cylinders   that   are   much   less   compressible   than   seawater.      These  
floats   have   a   constant   mass   and   an   almost   constant   volume.      Hence   these   floats   have   an  
almost   constant   in   situ   density   and   their   motion   approximately   occurs   on   surfaces   of  
constant  in  situ  density  which  at  mid  depth  in  the  ocean  are  much  closer  to  being  isobaric  
surfaces  than  being  locally-­‐‑referenced  potential  density  surfaces.    This  is  why  these  floats  
are   sometimes   described   as   “isobaric   floats”,   and   is   the   reason   why   a   “compressee”   is  
sometimes  added  to  a  float  so  that  its  compressibility  approximates  that  of  seawater.      
The  slope  of  a  specific  volume  anomaly  surface,   ∇δ z ,   can  be  expressed  as  

(

)

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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
−1

⎡
g 2 c2
g 2 c 2 ⎤
(3.19.3)  
∇δ z − ∇ p z = ∇ n z − ∇ p z ⎢1 +
−
⎥ ,   
N2
N 2 ⎥⎦
⎢⎣
 
where   c   is  the  sound  speed  of  the  reference  parcel   SA , Θ
  at  pressure   p.     This  expression  
confirms  that  where  the  local  seawater  properties  are  close  to  those  of  the  reference  parcel,  
the   specific   volume   anomaly   surface   can   closely   approximate   the   neutral   tangent   plane.    
The  square  bracket  in  Eqn.  (3.19.3)  is  equal  to   ρ gN −2 ∂δ ∂z   (from  section  7  of  McDougall  
(1989)  where   δ   is  specific  volume  anomaly).      
  
  

(

)

( )

3.20  Planetary  potential  vorticity    

  
Planetary   potential   vorticity   is   the   Coriolis   parameter   f    times   the   vertical   gradient   of   a  
suitable  variable.    Potential  density  is  often  used  for  that  variable  but  its  use  (i)  involves  an  
inaccurate   separation   between   lateral   and   diapycnal   advection   because   potential   density  
surfaces   are   not   a   good   approximation   to   neutral   tangent   planes   and   (ii)   incurs   the   non-­‐‑
conservative   baroclinic   production   term   of   Eqn.   (3.13.5).      Using   approximately   neutral  
surfaces,   “ans”,   (such   as   Neutral   Density   surfaces)   provides   an   optimal   separation  
between  the  effects  of  lateral  and  diapycnal  mixing  in  the  potential  vorticity  equation.    In  
this   case   the   potential   vorticity   variable   is   proportional   to   the   reciprocal   of   the   thickness  
between  a  pair  of  closely  spaced  approximately  neutral  surfaces.      
  
The  evolution  equation  for  planetary  potential  vorticity  is  derived  by  first  taking  the  
epineutral   “divergence”   ∇n ⋅    of   the   geostrophic   relationship   from   Eqn.   (3.12.1),   namely  
fv = g k × ∇p z .      The   projected   “divergences”   of   a   two-­‐‑dimensional   vector   a    in   the  
neutral   tangent   plane   and   in   an   isobaric   surface,   are   ∇n ⋅a = ∇z ⋅a + a z ⋅∇n z    and  
∇ p ⋅a = ∇z ⋅a + a z ⋅∇ p z   from  which  we  find  (using  Eqn.  (3.12.6),   ∇ n z − ∇ p z = ∇ n P Pz )    

∇n ⋅a = ∇p ⋅a + a z ⋅∇n P Pz .  

(3.20.1)  

Applying  this  relationship  to  the  two-­‐‑dimensional  vector   fv = g k × ∇p z   we  have    

(

∇n ⋅ ( fv ) = g ∇p ⋅ k × ∇p z

) + fv

z

⋅∇n P Pz = 0 .  

(3.20.2)  

The   first   part   of   this   expression   can   be   seen   to   be   zero   by   simply   calculating   its  
components,  and  the  second  part  is  zero  because  the  thermal  wind  vector   v z   points  in  the  
direction   k × ∇n P   (see  Eqn.  (3.12.3)).    It  can  be  shown  that   ∇r ⋅ fv   is  zero  in  any  surface  
r = r ρ , P   which  is  defined  as  a  function  of  in  situ  density  and  pressure.      
  
Eqn.   (3.20.2),   namely   ∇n ⋅ fv = 0 ,   can   be   interpreted   as   the   divergence   form   of   the  
evolution  equation  of  planetary  potential  vorticity  since    

(

( )

)

( )

⎛ qv⎞
∇n ⋅ ( fv ) = ∇n ⋅ ⎜ ⎟ = 0   ,  
⎝γz ⎠

(3.20.3)  

where   q = f γ z   is  the  planetary  potential  vorticity,  being  the  Coriolis  parameter  times  the  
vertical   gradient   of   Neutral   Density.      This   instantaneous   equation   can   be   averaged   in   a  
thickness-­‐‑weighted  sense  in  density  coordinates  yielding    

⎛ q̂ v̂ ⎞
⎛ v ′′q′′ ⎞
−1
∇n ⋅ ⎜
= − ∇n ⋅ ⎜
⎟
⎟ = ∇n ⋅ γ z K∇ nq̂   ,  

γ
γ
⎝ z⎠
⎝ z ⎠

(

)

(3.20.4)  

where   the   double-­‐‑primed   quantities   are   deviations   of   the   instantaneous   values   from   the  
thickness-­‐‑weighted  mean  quantities.    Here  the  epineutral  eddy  flux  of  planetary  potential  
vorticity   per   unit   area   has   been   taken   to   be   down   the   epineutral   gradient   of   q̂    with   the  
epineutral  diffusivity   K .    The  thickness-­‐‑weighted  mean  planetary  potential  vorticity  is    

⎛ q⎞
q̂ ≡ γ z ⎜ ⎟ = f γ z   ,  
⎝γ z⎠ γ

IOC Manuals and Guides No. 56

(3.20.5)  

TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

43

and   the   averaging   in   the   above   equations   is   consistent   with   the   difference   between   the  
thickness-­‐‑weighted   mean   velocity   and   the   velocity   averaged   on   the   Neutral   Density  
surface,   v̂ − v    (the   bolus   velocity),   being   v̂ − v = K ∇n ln q̂ ,   since   Eqn.   (3.20.4)   can   be  
written  as   ∇n ⋅ f v̂ = ∇n ⋅ γ z−1 K∇ nq̂   while  the  epineutral  temporal  average  of  Eqn.  (3.20.3)  
is   ∇n ⋅ f v = 0 .      
  
The   divergence   form   of   the   mean   planetary   potential   vorticity   evolution   equation,  
Eqn.  (3.20.4),  is  quite  different  to  that  of  a  normal  conservative  variable  such  as  Absolute  
Salinity   or   Conservative   Temperature   in   that   (i)   neither   the   vertical   diffusivity   nor   the  
dianeutral   velocity   makes   an   appearance,   and   (ii)   there   is   no   temporal   tendency   term   in  
the  equation.      
  
The  mean  planetary  potential  vorticity  equation  (3.20.4)  may  be  put  into  the  advective  
form  by  subtracting   q̂   times  the  mean  continuity  equation,    

( )

(

( )

()

)

⎛ 1 ⎞
⎛ v̂ ⎞
ez
= 0   ,  
⎜
⎟ + ∇n ⋅ ⎜ ⎟ +
γ z
⎝ γ z ⎠
⎝ γ z n ⎠ t
from  Eqn.  (3.20.4),  yielding  ( γ z−1   times)    

q̂t

n

(

(3.20.6)  

)

+ v̂ ⋅ ∇nq̂ = γ z ∇n ⋅ γ z−1 K∇ nq̂ + q̂ez   ,  

or    

(

(3.20.7)  

)

dq̂
(3.20.8)  
= γ z ∇n ⋅ γ z−1 K∇ nq̂ + ( q̂e )z   .  
dt
In   this   form,   it   is   clear   that   planetary   potential   vorticity   behaves   like   a   conservative  
variable   as   far   as   epineutral   mixing   is   concerned,   but   it   is   quite   unlike   a   normal  
conservative  variable  as  far  as  vertical  mixing  is  concerned.      
  
If   q̂    were   a   normal   conservative   variable   the   last   term   in   Eqn.   (3.20.8)   would   be  
( Dq̂z )z where   D   is  the  vertical  diffusivity.    The  term  that  actually  appears  in  Eqn.  (3.20.8),  
( q̂e )z ,   is   different   to   ( Dq̂z )z    by   ( q̂e − Dq̂z )z = f ( eγ z − Dγ zz )z .      Equation   (A.22.4)   for   the  
mean  dianeutral  velocity   e   can  be  expressed  as   e ≈ Dz + D γ zz γ z   if  the  following  three  
aspects   of   the   non-­‐‑linear   equation   of   state   are   ignored;   (1)   cabbeling   and   thermobaricity,  
(2)   the   vertical   variation   of   the   thermal   expansion   coefficient   and   the   saline   contraction  
coefficient,  and  (3)  the  vertical  variation  of  the  integrating  factor   b ( x, y, z )   of  Eqns.  (3.20.10)  
-­‐‑   (3.20.15)  below.    Even  when  ignoring  these  three  different  implications  of  the  nonlinear  
equation   of   state,   the   evolution   equations   (3.20.7)   and   (3.20.8)   of   q̂    are   unlike   normal  
conservation  equations  because  of  the  extra  term    
q̂t

n

+ v̂ ⋅ ∇nq̂ + eq̂z =

( q̂e − Dq̂z )z

= f ( eγ z − Dγ zz )z ≈ f ( Dzγ z )z =

( Dz q̂ )z     

(3.20.9)  

on  their  right-­‐‑hand  sides.    This  presence  of  this  additional  term  can  result  in  “unmixing”  
of   q̂   in  the  vertical.    Consider  a  situation  where  both   q̂   and   Θ̂   are  locally  linear  functions  
of   ŜA    down   a   vertical   water   column,   so   that   the   ŜA − q̂    and   ŜA − Θ̂    diagrams   are   both  
locally  straight  lines,  exhibiting  no  curvature.    Imposing  a  large  amount  of  vertical  mixing  
at  one  height  (e.  g.  a  delta  function  of   D )  will  not  change  the   ŜA − Θ̂   diagram  because  of  
the   zero   ŜA − Θ̂    curvature   (see   the   water-­‐‑mass   transformation   equation   (A.23.1)).    
However,  the  additional  term   Dz q̂ z   of  Eqn.  (3.20.9)  means  that  there  will  be  a  change  in  
q̂   of   Dz q̂ z = q̂Dzz + q̂z Dz ≈ q̂Dzz   along  the  neural  tangent  plane  (that  is,  in  Eqn.  (3.20.7)).    
This  is   q̂   times  a  negative  anomaly  at  the  central  height  of  the  extra  vertical  diffusion,  and  
is   q̂   times  a  positive  anomaly  on  the  flanking  heights  above  and  below  the  central  height.    
In   this   way,   a   delta   function   of   extra   vertical   diffusion   induces   structure   in   the   initially  
straight   ŜA − q̂   line  which  is  a  telltale  sign  of  “unmixing”.      
  
This   planetary   potential   vorticity   variable,   q̂ = f γ z ,   is   often   mapped   on   Neutral  
Density  surfaces  to  give  insight  into  the  mean  circulation  of  the  ocean  on  density  surfaces.    
The  reasoning  is  that  if  the  influence  of  dianeutral  advection  (the  last  term  in  Eqn.  (3.20.7))  
is  small,  and  the  epineutral  mixing  of   q̂   is  also  small,  then  in  a  steady  ocean   v̂ ⋅ ∇nq̂ = 0   

(

)

(

)

IOC Manuals and Guides No. 56

44

TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

and   the   thickness-­‐‑weighted   mean   flow   on   density   surfaces   v̂    will   be   along   contours   of  
thickness-­‐‑weighted  planetary  potential  vorticity   q̂ = f γ z .      
  
Because  the  square  of  the  buoyancy  frequency,   N 2 ,  accurately  represents  the  vertical  
static  stability  of  a  water  column,  there  is  a  strong  urge  to  regard   fN 2   as  the  appropriate  
planetary  potential  vorticity  variable,  and  to  map  its  contours  on  Neutral  Density  surfaces.    
This   urge   must   be   resisted,   as   spatial   maps   of   fN 2    are   significantly   different   to   those   of  
q̂ = f γ z .    To  see  why  this  is  the  case  the  relationship  between  the  epineutral  gradients  of  
q̂   and   fN 2   will  be  derived.      
  
For  the  present  purposes  Neutral  Helicity  will  be  assumed  sufficiently  small  that  the  
existence  of  neutral  surfaces  is  a  good  approximation,  and  we  seek  the  integrating  factor  
b = b x, y, z    which   allows   the   construction   of   Neutral   Density   surfaces   ( γ    surfaces)  
according  to    
⎞
⎛ ∇ρ
∇γ
= b β Θ∇SA − α Θ∇Θ = b ⎜
− κ ∇P⎟ .  
(3.20.10)  
γ
⎠
⎝ ρ

(

)

(

)

Taking  the  curl  of  this  equation  gives    

∇ρ ⎞
∇b ⎛
= − ∇κ × ∇P .  
× ⎜ κ ∇P −
ρ ⎟⎠
b ⎝

(3.20.11)  

The  bracket  on  the  left-­‐‑hand  side  is  normal  to  the  neutral  tangent  plane  and  points  in  the  
direction   n = − ∇ n z + k   and  is   g −1 N 2 −∇ n z + k .    Taking  the  component  of  Eqn.  (3.20.11)  
in  the  direction  of  the  normal  to  the  neutral  tangent  plane,   n ,  we  find    

(

(∇ nκ

0 = ∇κ × ∇P ⋅ n =

)

+ κ z n ) × ( ∇ n P + Pz n ) ⋅ n

(

)

= ∇ nκ × ∇ n P ⋅ n = ∇ nκ × ∇ n P ⋅k = κ S ∇ n SA + κ Θ∇ nΘ × ∇ n P ⋅k   
A

(3.20.12)  

= TbΘ∇ n P × ∇ nΘ ⋅k = g N −2 H n ,
which   simply   says   that   the   neutral   helicity   H n    must   be   zero   in   order   for   the   dianeutral  
component   of   Eqn.   (3.20.11)   to   hold,   that   is,   ∇ n P × ∇ nΘ ⋅k    must   be   zero.      Here   the  
  have  been  used.      
equalities   κ S = β PΘ   and   κ Θ = − α Θ
P

  

A

Writing   ∇b   as   ∇ nb + bz n ,  Eqn.  (3.20.11)  becomes    

(

)

g −1 N 2 ∇n ln b × ( −∇ n z + k ) = − Pz ∇p κ × −∇ p z + k ,  

(

)

(

)

(3.20.13)  

where   ∇P = Pz −∇ p z + k    has   been   used   on   the   right-­‐‑hand   side,   −∇ p z + k    being   the  
normal   to   the   isobaric   surface.      Concentrating   on   the   horizontal   components   of   this  
equation,   g −1 N 2 ∇n ln b = − Pz ∇p κ ,  and  using  the  hydrostatic  equation   Pz = − g ρ   gives    

(

∇n ln b = ρ g 2 N −2 ∇p κ = − ρ g 2 N −2 α ΘP ∇p Θ − β PΘ∇p SA

(

)

)

= − ρ g 2 TbΘ N −2 ∇n Θ − gN −2 α ΘP Θ z − β PΘ SA z ∇n P.

  

(3.20.14)  

The   second   line   of   this   equation   has   used   the   geometric   relationship  
∇p Θ = ∇ nΘ − Θ z ∇ n P Pz   and  the  corresponding  equation  for  Absolute  Salinity.      
defined  
by  
Eqn.  
(3.20.10),  
that  
is  
The  
integrating  
factor  
b   
b ≡ −gN −2γ −1∇γ ⋅ n (n ⋅ n) = −gN −2γ −1∇γ ⋅ n (1 + ∇n z ⋅∇n z) ,   allows   spatial   integrals   of  
b ( β Θ∇SA − α Θ∇Θ) ≈ ∇ ln γ   to  be  approximately  independent  of  path  for  “vertical  paths”,  
that  is,  for  paths  in  planes  whose  normal  has  zero  vertical  component.      
By  analogy  with   fN 2 ,  Neutral  Surface  Potential  Vorticity  ( NSPV )  is  defined  as   −gγ −1   
times   q̂ = f γ z ,   so   that   NSPV = b fN 2    (from   the   vertical   component   of   Eqn.   (3.20.10)),   so  
that  the  ratio  of   NSPV   to   fN 2   is  found  by  spatially  integrating  Eqn.  (3.20.14)  to  be    

NSPV
fN 2

=b =−

( )z = exp ⎧⎪−

g ln γ
N2

IOC Manuals and Guides No. 56

⎫⎪
⎧⎪
⎫⎪
ρ TbΘ
α ΘP Θ z − β PΘ SA z
∇
P
⋅
dl
⎨ ∫ans −2 2 ∇n Θ ⋅ dl ⎬ exp ⎨− ∫ans
⎬ .  (3.20.15)  
n
g N
g −1 N 2
⎩⎪
⎭⎪
⎩⎪
⎭⎪

TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

45

The   integral   here   is   taken   along   an   approximately   neutral   surface   (such   as   a   Neutral  
Density  surface)  from  a  location  where   NSPV   is  equal  to   fN 2 .       
The  deficiencies  of   fN 2   as  a  form  of  planetary  potential  vorticity  have  not  been  widely  
appreciated.    Even  in  a  lake,  the  use  of   fN 2   as  planetary  potential  vorticity  is  inaccurate  
since  the  right-­‐‑hand  side  of  (3.20.14)  is  then    

− ρ g 2 N −2 α ΘP ∇p Θ = ρ g 2 N −2 α ΘP Θ z ∇Θ P Pz = −

α ΘP

∇Θ P ,  
(3.20.16)  
αΘ
where   the   geometrical   relationship   ∇p Θ = − Θ z ∇Θ P Pz    has   been   used   along   with   the  
hydrostatic  equation.    The  mere  fact  that  the  Conservative  Temperature  surfaces  in  a  lake  
have  a  slope  (i.  e.   ∇Θ P ≠ 0 )  means  that  the  spatial  variation  of  contours  of   fN 2   will  not  be  
the  same  as  that  of  the  contours  of   NSPV   on  a   Θ   surface  in  a  lake.      
In   the   situation   where   there   is   no   gradient   of   Conservative   Temperature   along   a  
Neutral   Density   surface   ( ∇γ Θ = 0 )   the   contours   of   NSPV    along   the   Neutral   Density  
surface   coincide   with   those   of   isopycnal-­‐‑potential-­‐‑vorticity   ( IPV ),   the   potential   vorticity  
defined   with   respect   to   the   vertical   gradient   of   potential   density   by   IPV = − fg ρ −1ρ zΘ .    
IPV   is  related  to   fN 2   by  (McDougall  (1988))    

β Θ ( pr )
IPV
− g ρ −1ρ zΘ
≡
=
fN 2
N2
β Θ ( p)

⎡⎣ Rρ r − 1⎤⎦
β Θ ( pr ) 1
1
=
≈ Θ ,   
Θ
Θ
⎡⎣ Rρ − 1⎤⎦
G
β ( p) G

(3.20.17)  

so  that  the  ratio  of   NSPV   to   IPV ,  evaluated  on  an  approximately  neutral  surface,  is    

( )
( )

Θ
⎧⎪
⎫⎪
⎧⎪
⎫⎪
ρTΘ
α ΘΘ − β Θ S
NSPV β p ⎡⎣ Rρ −1⎤⎦
= Θ
exp ⎨− ∫ans −2 b 2 ∇n Θ ⋅ dl ⎬ exp ⎨− ∫ans P z −1 2P A z ∇n P ⋅ dl ⎬ .   (3.20.18)  
IPV
β pr ⎡ Rρ ⎤
g N
g N
⎪⎩
⎪⎭
⎪⎩
⎪⎭
⎢ −1⎥
⎢⎣ r
⎥⎦
You   and   McDougall   (1991)   show   that   because   of   the   highly   differentiated   nature   of  
potential   vorticity,   isolines   of   IPV    and   NSPV    do   not   coincide   even   at   the   reference  
pressure   pr    of   the   potential   density   variable   (see   equations   (14)   –   (16)   and   Figure   14   of  
that   paper).      NSPV ,   fN 2    and   IPV    have   the   units   s −3 .       The   ratio   IPV fN 2 ,   evaluated  
according   to   the   middle   expression   in   Eqn.   (3.20.17),   is   available   in   the   GSW  
Oceanographic  Toolbox  as  the  function  gsw_IPV_vs_fNsquared_ratio.    
  
  

3.21  Vertical  velocity  through  the  sea  surface    

  
There  has  been  confusion  regarding  the  expression  that  relates  the  net  evaporation  at  the  
sea   surface   to   the   vertical   velocity   in   the   ocean   through   the   sea   surface.      Since   these  
expressions  have  often  involved  the  salinity  (through  the  factor   (1 − SA ) )  and  so  appear  to  
be  thermodynamic  expressions,  here  we  present  the  correct  equation  which  we  will  see  is  
merely  kinematics,  not  thermodynamics.    Let   ρ W ( E − P )   be  the  vertical  mass  flux  through  
the  air-­‐‑sea  interface  on  the  atmospheric  side  of  the  interface  (where   ( E − P )   is  the  notional  
vertical  velocity  of  freshwater  through  the  air-­‐‑sea  interface  with  density   ρ W ;  this  density  
being  that  of  pure  water  at  the  sea  surface  temperature  and  at  atmospheric  pressure).    The  
same   mass   flux   ρ W ( E − P )    must   flow   through   the   air-­‐‑sea   interface   on   the   ocean   side   of  
the   interface   where   the   density   is   ρ = ρ ( SA , t,0) .       The   vertical   velocity   through   an  
arbitrary   surface   whose   height   is   z = η ( x, y, t )    can   be   expressed   as   w − v ⋅∇η − ∂η ∂t   
(where   w    is   the   vertical   velocity   through   the   geopotential   surface,   see   section   3.24,   and  
note   that   t    is   time   in   this   context)   and   the   mass   flux   associated   with   this   dia-­‐‑surface  
vertical  velocity  component  is  this  vertical  velocity  times  the  density  of  the  seawater,   ρ .     
By   equating   the   two   mass   fluxes   on   either   side   of   the   air-­‐‑sea   interface   we   arrive   at   the  
vertical  ocean  velocity  through  the  air-­‐‑sea  interface  as  (Griffies  (2004),  Warren  (2009))    

(

)

w − v ⋅∇η − ∂η ∂t = ρ −1ρ W E − P .   

(3.21.1)  

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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

3.22  Freshwater  content  and  freshwater  flux    

  
Oceanographers   traditionally   call   the   pure   water   fraction   of   seawater   the   “freshwater  
fraction”  or  the  “freshwater  content”.    This  can  cause  confusion  because  in  some  science  
circles  “freshwater”  is  used  to  describe  water  of  low  but  non-­‐‑zero  salinity.    Nevertheless,  
here   we   retain   the   oceanographic   use   of   “freshwater”   as   being   synonymous   with   pure  
water   (i.   e.   SA = 0 ,   this   pure   water   being   in   liquid,   gaseous   or   solid   ice   forms).      The  
freshwater   content   of   seawater   is   (1 − SA ) = 1 − 0.001SA / (g kg −1 ) .       The   first   expression  
here  clearly  requires  that  Absolute  Salinity  is  expressed  in  kg  of  sea  salt  per  kg  of  solution.    
Note   that   the   freshwater   content   is   not   based   on   Practical   Salinity,   that   is,   it   is   not  
(1− 0.001 SP ) .       
The  advective  flux  of  mass  per  unit  area  is   ρ u   where   u   is  the  fluid  velocity  through  
the  chosen  area  element  while  the  advective  flux  of  sea  salt  is   ρ SAu .     The  advective  flux  of  
freshwater  per  unit  area  is  the  difference  of  these  two  mass  fluxes,  namely ρ (1 − SA ) u .     As  
outlined  in  section  2.5  and  appendices  A.4  and  A.20,  for  water  of  anomalous  composition  
there   are   four   types   of   absolute   salinity   that   might   be   relevant   to   this   discussion   of  
freshwater  fluxes;  Density  Salinity   SAdens ≡ SA ,  Solution  Salinity   S Asoln ,  Added-­‐‑Mass  Salinity  
SAadd ,  and  Preformed  Salinity   S* .    Since  Preformed  Salinity  is  designed  to  be  a  conservative  
variable  with  a  zero  flux  air-­‐‑sea  boundary  condition,  probably  the  best  form  of  freshwater  
content,  at  least  in  the  context  of  an  ocean  model,  is   (1 − S* ) = 1 − 0.001S* / (g kg −1 ) .     
  
  

(

)

(

)

3.23  Heat  transport    

  
A  flux  of  heat  across  the  sea  surface  at  a  sea  pressure  of  0  dbar  is  identical  to  the  flux  of  
potential   enthalpy   which   in   turn   is   exactly   equal   to   c 0p    times   the   flux   of   Conservative  
Temperature   Θ ,  where   c 0p   is  given  by  (3.3.3).    By  contrast,  the  same  heat  flux  across  the  
sea  surface  changes  potential  temperature   θ   in  inverse  proportion  to   c p ( SA ,θ , 0 )   and  this  
heat  capacity  varies  by  5%  at  the  sea  surface,  depending  mainly  on  salinity.      
The   First   Law   of   Thermodynamics,   namely   Eqn.   (A.13.1)   of   appendix   A.13,   can   be  
approximated  as    
dΘ
S
ρ c0p
≈ − ∇ ⋅F R − ∇ ⋅FQ + ρε + hS ρ S A,   
(3.23.1)  
A
dt
with   an   error   in   Θ    that   is   approximately   one   percent   of   the   error   incurred   by   treating  
either   c0p θ   or   c p ( SA ,θ , 0 ) θ   as  the  “heat  content”  of  seawater  (see  McDougall  (2003)  and  
appendices  A.13  and  A.18).    Equation  (3.23.1)  is  exact  at  0  dbar  while  at  great  depth  in  the  
ocean   the   error   with   the   approximation   (3.23.1)   is   no   larger   than   the   neglect   of   the  
dissipation  of  kinetic  energy  term   ρε   in  this  equation  (see  appendix  A.21).      
Because   the   left-­‐‑hand   side   of   the   First   Law   of   Thermodynamics,   Eqn.   (3.23.1),   can   be  
written  as  density  times  the  material  derivative  of   c0p Θ   it  follows  that   Θ   can  be  treated  as  
a  conservative  variable  in  the  ocean  and  that   c0p Θ   is  transported  by  advection  and  mixed  
by   turbulent   epineutral   and   dianeutral   diffusion   as   though   it   is   the   “heat   content”   of  
seawater.      For   example,   the   advective   meridional   flux   of   “heat”   is   the   area   integral   of  
ρ vh0 = ρ vc0p Θ   (here   v   is  the  northward  velocity).    The  error  in  comparing  this  advective  
meridional   “heat   flux”   with   the   air-­‐‑sea   heat   flux   is   approximately   1%   of   the   error   in   so  
interpreting   the   area   integral   of   either   ρ vc0pθ    or   ρ v c p ( S A , θ ,0 ) θ .      Similarly,   turbulent  
diffusive  fluxes  of  “heat”  are  accurately  given  by  a  turbulent  diffusivity  times  the  spatial  
gradient   of   c0p Θ    but   are   less   accurately   approximated   by   the   same   turbulent   diffusivity  
times  the  spatial  gradient  of   c0pθ   (see  appendix  A.14  for  a  discussion  of  this  point).      
Warren   (1999,   2006)   has   argued   that   because   enthalpy   is   unknown   up   to   a   linear  
function  of  salinity,  it  is  only  possible  to  talk  of  a  flux  of  “heat”  through  an  ocean  section  if  
the  fluxes  of  mass  and  salt  through  the  ocean  section  are  both  zero.    This  opinion  seems  to  
be  widely  held,  but  it  is  incorrect.    Because  enthalpy  is  unknown  and  unknowable  up  to  a  

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linear  function  of   S A   (i.  e.  up  to  the  arbitrary  function   a1 + a3SA   in  terms  of  the  constants  
defined   in   Eqn.   (2.6.2)),   the   left-­‐‑hand   side   of   Eqn.   (3.23.1)   is   unknowable   to   the   extent  
S
a3ρ dSA dt .     It  is  shown  in  appendix  B  that  the  terms   −∇ ⋅FQ + hSA ρ S A   on  the  right-­‐‑hand  
side  of  Eqn.  (3.23.1)  are  also  unknowable  to  the  same  extent  so  that  the  effect  of   a3   cancels  
from  Eqn.  (3.23.1).    Hence  the  fact  that   c0p Θ   is  unknowable  up  to  a  linear  function  of   S A   
does  not  affect  the  usefulness  of   h 0   or   c0p Θ   as  measures  of  “heat  content”.    Similarly,  the  
difference  between  the  meridional  fluxes  of   c0p Θ   across  two  latitudes  is  equal  to  the  area-­‐‑
integrated   air-­‐‑sea   and   geothermal   heat   fluxes   between   these   latitudes   (after   allowing   for  
any  unsteady  accumulation  of   c0p Θ   in  the  volume),  irrespective  of  whether  there  are  non-­‐‑
zero  fluxes  of  mass  or  salt  across  the  sections.    This  powerful  result  follows  directly  from  
the   fact   that   c0p Θ    is   taken   to   be   a   conservative   variable,   obeying   the   simple   conservation  
statement  Eqn.  (3.23.1).    This  issue  is  discussed  at  greater  length  in  section  6  of  McDougall  
(2003).      
  
  

3.24  Geopotential    
  

The  geopotential   Φ   is  the  gravitational  potential  energy  per  unit  mass  with  respect  to  the  
height   z   =  0.    Allowing  the  gravitational  acceleration  to  be  a  function  of   z ,   Φ   is  given  by    
z

( )

Φ = ∫ g z ′ dz ′ .   
0

(3.24.1)  

If  the  gravitational  acceleration  is  taken  to  be  constant   Φ   is  simply   gz .     Note  that  height  
and   Φ   are  negative  quantities  in  the  ocean  since  the  sea  surface  (or  the  geoid)  is  taken  as  
the   reference   height   and   z    is   measured   upward   from   this   surface.      In   SI   units   Φ    is  
measured  in   J kg−1 = m2 s−2 .     If  the  ocean  is  assumed  to  be  in  hydrostatic  balance  so  that  
Pz = − g ρ    (or   − g dz′ = v dP′ )   then   the   geopotential   Eqn.   (3.24.1)   may   be   expressed   as   the  
vertical  pressure  integral  of  the  specific  volume  in  the  water  column,    
P

( )

Φ = Φ0 − ∫ v p′ dP′ ,   
P0

(3.24.2)  

where   Φ0    is   the   value   of   the   geopotential   at   zero   sea   pressure,   that   is,   the   gravitational  
acceleration   times   the   height   of   the   free   surface   above   the   geoid.      Note   that   the  
gravitational  acceleration  has  not  been  assumed  to  be  constant  in  Eqn.  (3.24.2).      
  
  

3.25  Total  energy    
  

The  total  energy   E   is  the  sum  of  specific  internal  energy   u,   kinetic  energy  per  unit  mass  
0.5 u ⋅ u   (  where   u     is  the  three-­‐‑dimensional  velocity  vector)  and  the  geopotential   Φ ,     

E = u + Φ + 12 u ⋅ u .   

(3.25.1)  

Total   energy   E    is   not   a   function   of   only   ( SA , t, p )    and   so   is   not   a   thermodynamic  
quantity.      
  
  

3.26  Bernoulli  function    

  
The   Bernoulli   function   is   the   sum   of   specific   enthalpy   h,    kinetic   energy   per   unit   mass  
0.5 u ⋅ u ,  and  the  geopotential   Φ ,     

B = h +Φ + 12 u ⋅ u.   

(3.26.1)  

Using   the   expression   (3.2.1)   that   relates   enthalpy   and   potential   enthalpy,   together   with  
Eqn.  (3.24.2)  for   Φ ,   the  Bernoulli  function  (3.26.1)  may  be  written  as    

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P

B = h0 + Φ 0 + 12 u ⋅ u − ∫ vˆ ( p′ ) − vˆ ( S A , Θ, p ′ ) dP′ .   

(3.26.2)  

P0

The   pressure   integral   term   here   is   a   version   of   the   dynamic   height   anomaly   (3.27.1),   this  
time   for   a   specific   volume   anomaly   defined   with   respect   to   the   Absolute   Salinity   and  
Conservative   Temperature   (or   equivalently,   with   respect   to   the   Absolute   Salinity   and  
potential   temperature)   of   the   seawater   parcel   in   question   at   pressure   P .      This   pressure  
integral  is  equal  to  the  Cunningham  geostrophic  streamfunction,  Eqn.  (3.29.2).      
The   Bernoulli   function   B    is   not   a   function   of   only   ( SA , t, p )    and   so   is   not   a  
thermodynamic  quantity.    
The  Bernoulli  function  is  dominated  by  the  contribution  of  enthalpy   h   to  (3.26.1)  and  
by  the  contribution  of  potential  enthalpy   h 0   to  (3.26.2).    The  variation  of  kinetic  energy  or  
the  geopotential  following  a  fluid  parcel  is  typically  several  thousand  times  less  than  the  
variation  of  enthalpy  or  potential  enthalpy  following  the  fluid  motion.      
The   definition   of   specific   volume   anomaly   given   in   Eqn.   (3.7.3)   has   been   used   by  
Saunders   (1995)   to   write   (3.26.2)   as   (with   the   dynamic   height   anomaly   Ψ    defined   in  
(3.27.1))    

(

)

B = h0 + Φ0 + 12 u ⋅ u + Ψ + ∫ v̂ ( SA ,Θ, p′ ) − v̂ SSO ,0°C, p′ dP′
P

P0

(
) (
) (
u ⋅ u + Ψ − ĥ ( SSO ,0°C, p ) + ĥ ( SA ,Θ, p ) − c0pΘ ,

)

(

)

= h0 + Φ0 + 12 u ⋅ u + Ψ − ĥ SSO ,0°C, p + ĥ SSO ,0°C,0 + ĥ SA ,Θ, p − ĥ SA ,Θ,0         (3.26.3)    
= h +Φ +
0

0

1
2

where  the  last  line  has  used   hˆ ( SSO ,0°C,0 ) = 0   and   hˆ ( SA , Θ,0) = c0p Θ .    The  sum  of  the  last  
two  terms  in  this  equation,  namely   ĥ SA ,Θ, p − c0pΘ ,  is  dynamic  enthalpy.      
  
  

(

)

3.27  Dynamic  height  anomaly    

  
The  dynamic  height  anomaly   Ψ   with  respect  to  the  sea  surface  is  given  by    
P

(

)

Ψ = − ∫ δˆ SA ⎡⎣ p′ ⎤⎦ ,Θ ⎡⎣ p′ ⎤⎦ , p′ dP′,     where     δˆ ( SA , Θ, p ) = vˆ ( SA , Θ, p ) − vˆ ( S SO ,0°C, p ) .    (3.27.1)  
P0

This  is  the  geostrophic  streamfunction  for  the  flow  at  pressure   P   with  respect  to  the  flow  
at   the   sea   surface   and   δˆ    is   the   specific   volume   anomaly.      Thus   the   two-­‐‑dimensional  
gradient   of   Ψ    in   the   P    pressure   surface   is   simply   related   to   the   difference   between   the  
horizontal  geostrophic  velocity   v   at   P   and  at  the  sea  surface v 0   according  to    

k × ∇P Ψ = fv − fv0 .  

(3.27.2)  

Dynamic   height   anomaly   is   also   commonly   called   the   “geopotential   anomaly”.      The  
specific   volume   anomaly,   δˆ    in   the   vertical   integral   in   Eqn.   (3.27.1)   can   be   replaced   with  
specific  volume   v̂   without  affecting  the  isobaric  gradient  of  the  resulting  streamfunction.    
That   is,   this   substitution   does   not   affect   Eqn.   (3.27.2)   because   the   additional   term   is   a  
function  only  of  pressure.    Traditionally  it  was  important  to  use   δˆ   in  preference  to   v̂   as  it  
was   more   accurate   with   computer   code   which   worked   with   single-­‐‑precision   variables.    
Since   computers   now   regularly   employ   double-­‐‑precision,   this   issue   has   been   overcome  
and  consequently  either   δˆ   or   v̂   can  be  used  in  the  integrand  of  Eqn.  (3.27.1),  so  making  it  
either   the   “dynamic   height   anomaly”   or   the   “dynamic   height”.      As   in   the   case   of   Eqn.  
(3.24.2),   so   also   the   dynamic   height   anomaly   Eqn.   (3.27.1)   has   not   assumed   that   the  
gravitational   acceleration   is   constant   and   so   Eqn.   (3.27.2)   applies   even   when   the  
gravitational  acceleration  is  taken  to  vary  in  both  the  vertical  and  in  the  horizontal.      
The  dynamic  height  anomaly   Ψ   should  be  quoted  in  units  of   m2 s−2 .    These  are  the  
units   in   which   the   GSW   Toolbox   (appendix   N)   outputs   dynamic   height   anomaly   in   the  
function   gsw_geo_strf_dyn_height(SA,CT,p,p_ref).      When   the   last   argument   of   this  

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49

function,  p_ref,  is  other  than  zero,  the  function  returns  the  dynamic  height  anomaly  with  
respect  to  a  (deep)  reference  pressure  p_ref,  rather  than  with  respect  to   P0   (i.e.  zero   dbar   
sea   pressure)   as   in   Eqn.   (3.27.1).      In   this   case   the   lateral   isobaric   gradient   of   the  
streamfunction   represents   the   geostrophic   velocity   difference   relative   to   the   (deep)   pref   
pressure  surface,  that  is,    
(3.27.3)  
k × ∇ P Ψ = fv − fv ref .   
Note  that  the  integration  in  Eqn.  (3.27.1)  of  specific  volume  anomaly  with  pressure  must  
be   done   with   pressure   in   Pa    (not   dbar )   in   order   to   have   the   resultant   isobaric   gradient,  
∇ P Ψ ,  in  the  usual  units,  being  the  product  of  the  Coriolis  parameter  (units  of   s −1 )  and  the  
velocity  (units  of   m s −1 ).    The  GSW  function  gsw_steric_height(SA,CT,p,p_ref)  returns   Ψ   
divided   by   the   constant   gravitational   acceleration   g0 = 9.7963 ms−2 .      Hence   steric   height  
remains   proportional   to   an   exact   geostrophic   streamfunction   but   the   spatial   variation   of  
the   gravitational   acceleration   ensures   that   it   cannot   be   exactly   equal   to   the   height   of   an  
isobaric  surface  above  a  geopotential  surface.        
  
  

3.28  Montgomery  geostrophic  streamfunction    

  
The  Montgomery  “acceleration  potential”  (Montgomery,  1937)   Ψ M   defined  by    

(

)

P

(

)

Ψ M = P − P0 δˆ − ∫ δˆ SA ⎡⎣ p′ ⎤⎦ ,Θ ⎡⎣ p′ ⎤⎦ , p′ dP′ =
P0

( P − P0 )δˆ + Ψ   

(3.28.1)  

is   the   geostrophic   streamfunction   for   the   flow   in   the   specific   volume   anomaly   surface  
δˆ ( SA , Θ, p ) = δˆ1    relative   to   the   flow   at   P = P0    (that   is,   at   p = 0 dbar ).      Thus   the   two-­‐‑
dimensional  gradient  of   Ψ M   in  the   δˆ   specific  volume  anomaly  surface  is  simply  related  
1

to  the  difference  between  the  horizontal  geostrophic  velocity   v   in  the   δˆ = δˆ1   surface  and  
at  the  sea  surface v 0   according  to    

k × ∇δˆ ΨM = fv − fv0               or             ∇δˆ ΨM = − k × ( fv − fv0 ).   
1

(3.28.2)  

1

The  definition,  Eqn.  (3.28.1),  of  the  Montgomery  geostrophic  streamfunction  applies  to  all  

    in   the   definition,   Eqn.   (3.7.3),   of   the   specific  
choices   of   the   reference   values   SA    and   Θ
volume   anomaly.      By   carefully   choosing   these   reference   values   the   specific   volume  
anomaly   surface   can   be   made   to   closely   approximate   the   neutral   tangent   plane  
(McDougall  and  Jackett  (2007)).      
It   is   not   uncommon   to   read   of   authors   using   the   Montgomery   geostrophic  
streamfunction,   Eqn.   (3.28.1),   as   a   geostrophic   streamfunction   in   surfaces   other   than  
specific   volume   anomaly   surfaces.      This   incurs   errors   that   should   be   recognized.      For  
example,   the   gradient   of   the   Montgomery   geostrophic   streamfunction,   Eqn.   (3.28.1),   in   a  
neutral  tangent  plane  becomes  (instead  of  Eqn.  (3.28.2)  in  the   δˆ = δˆ1   surface)    

∇ n Ψ M = − k × ( fv − fv 0 ) + ( P − P0 ) ∇ nδˆ ,  

(3.28.3)  

where   the   last   term   represents   an   error   arising   from   using   the   Montgomery  
streamfunction  in  a  surface  other  than  the  surface  for  which  it  was  derived.      
Zhang   and   Hogg   (1992)   subtracted   an   arbitrary   pressure   offset,   P − P0 ,   from  
( P − P0 )    in   the   first   term   in   Eqn.   (3.28.1),   so   defining   the   modified   Montgomery  
streamfunction    

(

(

)

Ψ Z-H = P − P δˆ − ∫ δˆ ( S A [ p ′] , Θ [ p ′] , p ′ ) dP′ .   
P

)

(3.28.4)  

P0

The  gradient  of   Ψ Z-H   in  a  neutral  tangent  plane  becomes    

(

)

∇n Ψ Z-H = − k × ( fv − fv 0 ) + P − P ∇ nδˆ ,  

(3.28.5)  

where  the  last  term  can  be  made  significantly  smaller  than  the  corresponding  term  in  Eqn.  
(3.28.3)   by   choosing   the   constant   pressure   P    to   be   close   to   the   average   pressure   on   the  

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surface.    This  term  can  be  further  minimized  by  suitably  choosing  the  constant  reference  



  in  the  definition,  Eqn.  (3.7.3),  of  specific  volume  anomaly   δ   so  that  this  
values   SA   and   Θ
surface   more   closely   approximates   the   neutral   tangent   plane   (McDougall   (1989)).      This  
improvement  is  available  because  it  can  be  shown  that    

(

)

( )


 
 ∇ P.   
ρ ∇ nδ = − ⎡⎢κˆ ( SA ,Θ, p ) − κˆ SA , Θ,
p ⎤ ∇ n P ≈ TbΘ Θ − Θ
n
⎣
⎦⎥

(3.28.6)  

The  last  term  in  Eqn.  (3.28.5)  is  then  approximately    

( P − P ) ∇ nδ

( )

 ∇ ( P − P )2   
(3.28.7)  
ρ −1TbΘ Θ − Θ
n


  can  reduce  the  last  term  in  Eqn.  (3.28.5)  that  
and  hence  suitable  choices  of   P ,   SA   and   Θ
represents   the   error   in   interpreting   the   Montgomery   geostrophic   streamfunction,   Eqn.  
(3.28.4),  as  the  geostrophic  streamfunction  in  a  surface  that  is  more  neutral  than  a  specific  
volume  anomaly  surface.    
The   Montgomery   geostrophic   streamfunction   should   be   quoted   in   units   of   m2 s−2 .    
These   are   the   units   in   which   the   GSW   Toolbox   outputs   the   Montgomery   geostrophic  
streamfunction  in  the  function  gsw_geo_strf_Montgomery(SA,CT,p,p_ref).    When  the  last  
argument  of  this  function,  p_ref,  is  other  than  zero,  the  function  returns  the  Montgomery  
geostrophic   streamfunction   with   respect   to   a   (deep)   reference   sea   pressure   p_ref,   rather  
than  with  respect  to   p = 0 dbar   (i.e.   P = P0 )  as  in  Eqn.  (3.28.1).      
  
  
≈

1
2

3.29  Cunningham  geostrophic  streamfunction    

  
Cunningham   (2000)   and   Alderson   and   Killworth   (2005),   following   Saunders   (1995)   and  
Killworth   (1986),   suggested   that   a   suitable   streamfunction   on   a   density   surface   in   a  
compressible   ocean   would   be   the   difference   between   the   Bernoulli   function   B    and  
potential  enthalpy   h 0 .     Since  the  kinetic  energy  per  unit  mass,   12 u ⋅ u ,  is  a  tiny  component  
of  the  Bernoulli  function,  it  was  ignored  and  Cunningham  (2000)  essentially  proposed  the  
streamfunction   Π + Φ 0   (see  his  equation  (12)),  where    

Π ≡ B − h0 − 12 u ⋅ u − Φ 0
  

= h − h0 + Φ − Φ 0

(3.29.1)  

P

= hˆ( SA , Θ, p) − hˆ( SA , Θ,0) − ∫ vˆ ( SA ( p′), Θ( p′), p′ ) dP′.
P0

The   last   line   of   this   equation   has   used   the   hydrostatic   equation   Pz = − g ρ    to   express  
Φ ≈ gz   in  terms  of  the  vertical  pressure  integral  of  specific  volume  and  the  height  of  the  
sea  surface  where  the  geopotential  is   Φ 0 .     The  difference  between  enthalpy  and  potential  
enthalpy   h − h0   in  this  equation  has  been  named  “dynamic  enthalpy”  by  Young  (2010).      
The  definition  of  potential  enthalpy,  Eqn.  (3.2.1),  is  used  to  rewrite  the  last  line  of  Eqn.  
(3.29.1),  showing  that  Cunningham’s   Π   is  also  equal  to    
P

(

) (

)

Π = − ∫ v̂ SA ( p′ ), Θ( p′ ), p′ − v̂ SA , Θ, p′ dP′
P0

(

)

(

)

= Ψ − ĥ SSO ,0°C, p + ĥ SA ,Θ, p −

c0pΘ

  

(3.29.2)  

.

The   first   line   of   this   equations   appears   very   similar   to   the   expression,   Eqn.   (3.27.1),   for  
dynamic   height   anomaly,   the   only   difference   being   that   in   Eqn.   (3.27.1)   the   pressure-­‐‑
independent   values   of   Absolute   Salinity   and   Conservative   Temperature   were   SSO    and  
0°C   whereas  here  they  are  the  local  values  on  the  surface,   SA   and   Θ .    While  these  local  
values   of   Absolute   Salinity   and   Conservative   Temperature   are   constant   during   the  
pressure   integral   in   Eqn.   (3.29.2),   they   do   vary   with   latitude   and   longitude   along   any  
“density”  surface.      

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51

The  gradient  of   Π   along  the  neutral  tangent  plane  is    

∇n Π ≈

{

1

ρ

}

∇z P −∇Φ0 − 12 ρ −1TbΘ ( P − P0 ) ∇nΘ ,   
2

(3.29.3)  

(from   McDougall   and   Klocker   (2010))   so   that   the   error   in   ∇n Π    in   using   Π    as   the  
2
geostrophic   streamfunction   is   approximately   − 12 ρ −1TbΘ ( P − P0 ) ∇n Θ .      When   using   the  
Cunningham   streamfunction   Π    in   a   potential   density   surface,   the   error   in   ∇σ Π    is  
approximately   − 12 ρ −1TbΘ ( P − P0 )( 2 Pr − P − P0 ) ∇σ Θ .      The   Cunningham   geostrophic  
streamfunction   should   be   quoted   in   units   of   m2 s−2    and   is   available   in   the   GSW  
Oceanographic   Toolbox   as   the   function   gsw_geo_strf_Cunningham(SA,CT,p,p_ref).    
When  the  last  argument  of  this  function,  p_ref,  is  other  than  zero,  the  function  returns  the  
Cunningham   geostrophic   streamfunction   with   respect   to   a   (deep)   reference   sea   pressure  
p_ref,  rather  than  with  respect  to   p = 0 dbar   (i.e.   P = P0 )  as  in  Eqn.  (3.29.1).        
  
  

3.30  Geostrophic  streamfunction  in  an  approximately  neutral  surface    

  
In  order  to  evaluate  a  relatively  accurate  expression  for  the  geostrophic  streamfunction  in  
  
an  approximately  neutral  surface  a  suitable  reference  seawater  parcel   SA , Θ,
p   is  selected  
from   the   approximately   neutral   surface   that   one   is   considering,   and   the   specific   volume  

anomaly   δ   is  defined  as  in  (3.7.3)  above.    The  approximate  geostrophic  streamfunction   ϕ n   
is  given  by  (from  McDougall  and  Klocker  (2010))    

(

ϕn =

1
2

( P − P )δ ( S ,Θ, p) −
A

1
12

(

)(

 P − P
ρ −1TbΘ Θ − Θ

)

2

P

−

)



∫ δ dP′ .   

(3.30.1)  

P0

This   expression   is   more   accurate   than   the   Montgomery   and   Cunningham   geostrophic  
streamfunctions   when   used   in   potential   density   surfaces,   in   the   ω -­‐‑surfaces   of   Klocker   et  
al.  (2009a,b)  and  in  the  Neutral  Density  surfaces  of  Jackett  and  McDougall  (1997).    That  is,  
in  these  surfaces   ∇nϕ n ≈ ρ1 ∇ z P −∇Φ 0 = − k × ( fv − fv 0 )   to  a  very  good  approximation.    In  
Eqn.   (3.30.1)   ρ −1TbΘ    is   taken   to   be   the   constant   value   2.7 x10−15 K −1 (Pa) −2 m2s−2 .      This  
approximate   isopycnal   geostrophic   streamfunction   of   McDougall   and   Klocker   (2010)   is  
available   as   the   function   gsw_geo_strf_isopycnal   in   the   GSW   Toolbox.      When   the   last  
argument   of   this   function,   p_ref,   is   other   than   zero,   the   function   returns   the   isopycnal  
geostrophic   streamfunction   with   respect   to   a   (deep)   reference   sea   pressure   p_ref,   rather  
than  with  respect  to  the  sea  surface  at   p = 0 dbar   (i.e.   P = P0 )  as  in  Eqn.  (3.30.1).      
  
  

3.31  Pressure-­‐‑integrated  steric  height    

  
The   depth-­‐‑integrated   mass   flux   of   the   geostrophic   Eulerian   flow   between   two   fixed  
pressure   levels   can   also   be   represented   by   a   streamfunction.      Using   the   hydrostatic  
relation   Pz = − g ρ ,    and   assuming   the   gravitational   acceleration   to   be   independent   of  
height,   the   depth-­‐‑integrated   mass   flux   ∫ ρ v dz    is   given   by   − g −1 ∫ v dP    and   this   motivates  
taking   the   pressure   integral   of   the   Dynamic   Height   Anomaly   Ψ    (from   Eqn.   (3.27.1))   to  
form   the   Pressure-­‐‑Integrated-­‐‑Steric-­‐‑Height   PISH    (also   called   Depth-­‐‑Integrated   Steric  
Height   DISH   by  Godfrey  (1989)),    
P

( )

P P′′

(

)

PISH = Ψ ′ = − g −1 ∫ Ψ p ′′ dP′′ = g −1 ∫ ∫ δˆ SA ⎡⎣ p ′ ⎤⎦ , Θ ⎡⎣ p ′ ⎤⎦ , p ′ dP′ dP′′
P0

P

(

) (

P0 P0

)

  

(3.31.1)  

= g −1 ∫ P − P′ δˆ SA ⎡⎣ p ′ ⎤⎦ , Θ ⎡⎣ p ′ ⎤⎦ , p ′ dP′.
P0

The   two-­‐‑dimensional   gradient   of   Ψ′    is   related   to   the   depth-­‐‑integrated   mass   flux   of   the  
velocity  difference  with  respect  to  the  velocity  at  zero  sea  pressure,   v 0 ,   according  to    

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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
z ( P0 )

P

z( P )

P0

k × ∇ p Ψ′ = f ∫ ρ ⎡⎣ v ( z′) − v0 ⎤⎦ dz′ = g −1 f ∫ ⎡⎣ v ( p′) − v0 ⎤⎦ dP′.   

(3.31.2)  



S
The  definition,  Eqn.  (3.31.1),  of  
  applies  to  all  choices  of  the  reference  values  
PISH
A , S A   
 

   in  the  definitions,  Eqns.  (3.7.2  –  3.7.4),  of  the  specific  volume  anomaly.      
and   t , θ   or   Θ
Since   the   velocity   at   depth   in   the   ocean   is   generally   much   smaller   than   at   the   sea  
surface,  it  is  customary  to  take  the  reference  pressure  to  be  some  constant  (deep)  pressure  
P1   so  that  Eqn.  (3.27.1)  becomes      

(

P1

)

Ψ = ∫ δˆ SA ⎡⎣ p ′ ⎤⎦ ,Θ ⎡⎣ p ′ ⎤⎦ , p ′ dP′ ,  
P

(3.31.3)  

and   PISH ,  reflecting  the  depth-­‐‑integrated  horizontal  mass  transport  from  the  sea  surface  
to  pressure   P1 ,  relative  to  the  flow  at   P1 ,  is    
P1

(

P1 P1

( )

)

PISH = Ψ ′ = g −1 ∫ Ψ p ′′ dP′′ = g −1 ∫ ∫ δˆ SA ⎡⎣ p ′ ⎤⎦ ,Θ ⎡⎣ p ′ ⎤⎦ , p ′ dP′ dP′′
P0

P1

P0 P′′

) (

(

)

= g −1 ∫ P′ − P0 δˆ SA ⎡⎣ p ′ ⎤⎦ ,Θ ⎡⎣ p ′ ⎤⎦ , p ′ dP′
P0

=

1
2

g

−1

( P1 − P0 )2
∫

0

(

  

(3.31.4)  

)

2
δˆ SA ⎡⎣ p ′ ⎤⎦ ,Θ ⎡⎣ p ′ ⎤⎦ , p ′ d ⎛ ( P′ − P0 ) ⎞ .
⎝
⎠

The   two-­‐‑dimensional   gradient   of   Ψ′    is   now   related   to   the   depth-­‐‑integrated   mass   flux   of  
the  velocity  difference  with  respect  to  the  velocity  at P1 ,   v1,   according  to    
z ( P0 )

P1

z ( P1 )

P0

k × ∇ p Ψ′ = f ∫ ρ ⎡⎣ v ( z′) − v1 ⎤⎦ dz′ = g −1 f ∫ ⎡⎣ v ( p′) − v1 ⎤⎦ dP′.   

(3.31.5)  

The  specific  volume  anomaly   δˆ   in  Eqns.  (3.31.1),  (3.31.3)  and  (3.31.4)  can  be  replaced  with  
specific  volume   v   without  affecting  the  isobaric  gradient  of  the  resulting  streamfunction.    
That   is,   this   substitution   in   Ψ′    does   not   affect   Eqn.   (3.31.2)   or   Eqn.   (3.31.5),   as   the  
additional   term   is   a   function   only   of   pressure.      With   specific   volume   in   place   of   specific  
volume   anomaly,   Eqn.   (3.31.4)   becomes   the   depth-­‐‑integrated   gravitational   potential  
energy   of   the   water   column   (plus   a   very   small   term   that   is   present   because   the  
atmospheric  pressure  is  not  zero,  McDougall  et  al.  (2003)).      
PISH   should  be  quoted  in  units  of   kg s−2   so  that  its  two-­‐‑dimensional  gradient  has  the  
same  units  as  the  depth-­‐‑integrated  flux  of   ρ ⎡⎣ v ( z′) − v1 ⎤⎦   times  the  Coriolis  frequency.      
  
  

3.32  Pressure  to  height  conversion    

  
The  vertical  integral  of  the  hydrostatic  equation  ( g = − v Pz )  can  be  written  as    
0
0
∫ g ( z ′ ) dz ′ = Φ − ∫ v ( p′ ) dP′ = − ∫ v̂ ( SSO ,0°C, p′ ) dP′ + Ψ + Φ
z

P

P

0

P0

P0

(

)

  

(3.32.1)  

= − ĥ SSO ,0°C, p + Ψ + Φ ,
0

where   the   dynamic   height   anomaly   Ψ    is   expressed   in   terms   of   the   specific   volume  
anomaly   δˆ = vˆ ( SA , Θ, p ) − vˆ ( SSO ,0°C, p )   by    
P

Ψ = − ∫ δˆ ( p′ ) dP′ ,  

(3.32.2)  

P0

where   P0 = 101 325Pa    is   the   standard   atmosphere   pressure.      Writing   the   gravitational  
acceleration   of   Eqn.   (D.3)   as   g = g (φ , z ) = g (φ ,0) (1 − γ z ) ,   the   left-­‐‑hand   side   of   Eqn.  
(3.32.1)   becomes   g (φ ,0 ) z − 12 γ z 2 ,   and   using   the   75-­‐‑term   expression   for   the   specific  
enthalpy  of  Standard  Seawater,  Eqn.  (3.32.1)  becomes    

(

(

)

)

( )(

)

ĥ75 SSO , 0°C, p − Ψ − Φ0 + g φ ,0 z − 12 γ z 2 = 0   .  

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(3.32.3)  

TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

53

This  is  the  equation  that  is  solved  for  height   z   in  the  GSW  function  gsw_z_from_p.    It  is  
traditional  to  ignore   Ψ + Φ0   when  converting  between  pressure  and  height,  and  this  can  
be   done   by   simply   calling   this   function   with   only   two   arguments,   as   in  
gsw_z_from_p(p,lat).      Ignoring   Ψ + Φ0    makes   a   difference   to   z    of   up   to   4m   at   5000  
dbar.      Note   that   height   z    is   negative   in   the   ocean.      When   the   code   is   called   with   three  
arguments,  the  third  argument  is  taken  to  be   Ψ + Φ0   and  this  is  used  in  the  solution  of  
Eqn.   (3.32.3).      Dynamic   height   anomaly   Ψ    can   be   evaluated   using   the   GSW   function  
gsw_geo_strf_dyn_height.      The   GSW   function   gsw_p_from_z   is   the   exact   inverse  
function   of   gsw_z_from_p;   these   functions   yield   outputs   that   are   consistent   with   each  
other  to  machine  precision.      
When   vertically   integrating   the   hydrostatic   equation   Pz = − g ρ    in   the   context   of   an  
ocean   model   where   Absolute   Salinity   S A    and   Conservative   Temperature   Θ    are  
piecewise  constant  in  the  vertical,  the  geopotential  (Eqn.  (3.24.2))    
z

( )

P

( )

Φ = ∫ g z ′ dz ′ = Φ0 − ∫ v p′ dP′ ,   
0

P0

(3.32.4)  

can  be  evaluated  as  a  series  of  exact  differences.    If  there  are  a  series  of  layers  of  index   i   
separated   by   pressures   p i    and   p i +1    (with   pi +1 > pi )   then   the   integral   can   be   expressed  
(making  use  of  (3.7.5),  namely   hP
= hˆP = v )  as  a  sum  over   n   layers  of  the  differences  
SA , Θ
in  specific  enthalpy  so  that    
P

( )

n

(

) (

)

Φ = Φ0 − ∫ v p′ dP′ = Φ0 − ∑ ⎡ ĥ SAi ,Θ i , p i+1 − ĥ SAi ,Θ i , p i ⎤ .   
⎦
i =1 ⎣
P
0

(3.32.5)  

The   difference   in   enthalpy   at   two   different   pressures   for   given   values   of   S A    and   Θ    is  
available   in   the   GSW   Oceanographic   Toolbox   via   the   function   gsw_enthalpy_diff.      The  
summation   of   a   series   of   such   differences   in   enthalpy   occurs   in   the   GSW   functions   to  
evaluate   two   geostrophic   streamfunctions   from   piecewise-­‐‑constant   vertical   property  
profiles,  gsw_geo_strf_dyn_height_pc  and  gsw_geo_strf_isopycnal_pc.      
  
  

3.33  Freezing  temperature    

  
Freezing  occurs  at  the  temperature   tf   at  which  the  chemical  potential  of  water  in  seawater  
µ W    equals   the   chemical   potential   of   ice   µ Ih .      Thus,   tf    is   found   by   solving   the   implicit  
equation    

µ W ( SA , tf , p ) = µ Ih ( tf , p )   

(3.33.1)  

or  equivalently,  in  terms  of  the  two  Gibbs  functions,    

g ( SA , tf , p ) − SA gSA ( SA , tf , p ) = g Ih ( tf , p ) .   

(3.33.2)  

The   Gibbs   function   for   ice   Ih,   g ( t , p ) ,    is   defined   by   IAPWS-­‐‑06   (IAPWS   (2009a))   and  
Feistel  and  Wagner  (2006)  and  is  summarized  in  appendix  I  below.    In  the  special  case  of  
zero  salinity,  the  chemical  potential  of  water  in  seawater  reduces  to  the  Gibbs  function  of  
pure  water,   µ W ( 0, t , p ) = g W ( t , p ) .     A  simple  correlation  function  for  the  melting  pressure  
as  a  function  of  temperature  is  available  from  IAPWS  (2008b)  and  has  been  implemented  
in  the  SIA  library.      
At  the  ocean  surface,   p   =  0  dbar,  from  Eqn.  (3.33.1)  the  TEOS-­‐‑10  freezing  point  of  pure  
water  is   tf 0g kg −1 , 0dbar   =  0.002  519  °C  with  an  uncertainty  of  only  2   µK ,  noting  that  the  
triple   point   temperature   of   water   is   exactly   273.16   K   by   definition   of   the   ITS-­‐‑90  
temperature   scale.   The   freezing   temperature   of   the   standard   ocean   is   tf ( SSO , 0dbar )    =    
-­‐‑1.919   °C   with   an   uncertainty   of   2   mK.      Note   that   Eqn.   (3.33.1)   is   valid   for   air-­‐‑free  
water/seawater.      Dissolution   of   air   in   water   lowers   the   freezing   point   slightly;   saturation  
Ih

(

)

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with  air  lowers  the  freezing  temperatures  by  about  2.4   mK   for  pure  water  and  about  1.9  
mK     for  seawater  with  an  Absolute  Salinity  of   SA = SSO = 35.16504 g kg −1 .      
To   estimate   the   effects   of   small   changes   in   the   pressure   or   salinity   on   the   freezing  
temperature,  it  is  convenient  to  consider  a  power  series  expansion  of  (3.33.1).    The  result  in  
the  limit  of  an  infinitesimal  pressure  change  at  fixed  salinity  gives    the  pressure  coefficient  
of  freezing  point  lowering,  as  (Clausius-­‐‑Clapeyron  equation,  Feistel  et  al.  (2010a)),  

g P − SA g PS − g PIh
∂tf
A
= χ P SA , p = −
.   
∂P S
gT − SA g S T − gTIh

(

)

(3.33.3)  

A

A

Its   values,   evaluated   from   TEOS-­‐‑10,   vary   only   weakly   with   salinity   between  
χ p 0g kg −1, 0dbar    =   –0.7429   mK/dbar    for   pure   water   and   χ p ( SSO , 0dbar )    =   –0.7483  
mK/dbar    for   the   standard   ocean.      TEOS-­‐‑10   is   consistent   with   the   most   accurate  
measurement   of   χ p    and   its   experimental   uncertainty   of   0.0015   mK/dbar    (Feistel   and  
Wagner  (2005),  (2006)).  Since  the  value  of   χ p   always  exceeds  that  of  the  adiabatic  lapse  
rate   Γ ,   cold   seawater   may   freeze   and   decompose   into   ice   and   brine   during   adiabatic  
uplift  but  this  can  never  happen  to  a  sinking  parcel.      
In   the   limit   of   infinitesimal   changes   in   Absolute   Salinity   at   fixed   pressure,   we   obtain  
the  saline  coefficient  of  freezing  point  lowering,  as  (Raoult’s  law),  

(

)

∂tf
∂SA

= χ S ( SA , p ) =
p

(

S A g SA SA
gT − SA g SAT − gTIh

.   

(3.33.4)  

)

Typical   numerical   values   are   χ S 0g kg −1 , 0dbar    =   –59.2   mK/(g kg −1 )    for   pure   water   and  
χ S ( SSO , 0dbar )   =  –56.9   mK/(g kg −1 )   for  seawater.      
As   a   raw   practical   estimate,   Eqn.   (3.33.4)   can   be   expanded   into   powers   of   salinity,  
using  only  the  leading  term  of  the  TEOS-­‐‑10  saline  Gibbs  function,   g S ≈ RSTSA ln SA ,  which  
stems   from   Planck’s   ideal-­‐‑solution   theory   (Planck   (1888)).      Here,   RS = R M S    =   264.7599    
J  kg–1  K–1  is  the  specific  gas  constant  of  sea  salt,   R   is  the  universal  molar  gas  constant,  and  
M S    =   31.403   82   g   mol–1   is   the   molar   mass   of   sea   salt   with   Reference   Composition.      The  
denominator   of   Eqn.   (3.33.4)   is   proportional   to   the   melting   heat   LSIp ,   Eqn.   (3.34.7).      The  
convenient  result  obtained  with  these  simplifications  is    

∂tf
∂SA

≈ −
p

RS
2
T + tf ) ≈ − 59 mK/(g kg −1 ) .  
SI ( 0
Lp

(3.33.5)  

where  we  have  used   tf = −2 oC   and   LSIp = 330   J kg −1   as  appropriate  approximations  for  the  
standard   ocean.      This   simple   result   is   only   weakly   dependent   on   these   choices   and   is   in  
reasonable  agreement  with  the  exact  values  from  Eqn.  (3.33.4)  and  with  Millero  and  Leung  
(1976).    The  freezing  temperature  of  seawater  is  always  lower  than  that  of  pure  water.      
When   sea-­‐‑ice   is   formed,   it   often   contains   remnants   of   seawater   included   in   brine  
pockets.      At   equilibrium,   the   salinity   in   these   pockets   depends   only   on   temperature   and  
pressure,   rather   than,   for   example,   on   the   pocket   volume,   and   can   be   computed   in   the  
functional  form   SA ( t, p )   as  an  implicit  solution  of  Eqn.  (3.33.1).    Measured  values  for  the  
brine  salinity  of  Antarctic  sea  ice  agree  very  well  with  those  computed  of  Eqn.  (3.33.1)  up  
to   the   saturation   concentration   of   about   110   g kg −1    at   surface   pressure   (Feistel   et   al.  
(2010b)).    At  high  pressures,  the  validity  of  the  Gibbs  function  of  seawater,  and  therefore  
of  the  computed  freezing  point  or  brine  salinity,  too,  is  limited  to  only  50   g kg −1 .      
We   note   that   in   the   first   approximation,   as   inferred   from   Planck’s   theory   of   ideal  
solutions,  the  above  properties  depend  on  the  number  of  dissolved  particles  regardless  of  
the   particle   sizes,   masses   or   charges.      In   other   words,   they   depend   mainly   on   the   molar  
rather  than  on  the  mass  density  of  the  solute,  in  contrast  to  properties  such  as  the  specific  
volume  of  seawater  and  properties  derived  from  it.    The  properties  considered  in  this  and  
the  following  sections  (sections  3.33  –  3.42)  which  share  this  attribute  are  referred  to  as  the  
colligative  properties  of  seawater.      

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55

3.34  Latent  heat  of  melting    

  
The  melting  process  of  ice  in  pure  water  can  be  conducted  by  supplying  heat  at  constant  
pressure.      If   this   is   done   slowly   enough   that   equilibrium   is   maintained,   then   the  
temperature   will   also   remain   constant.      The   heat   required   per   mass   of   molten   ice   is   the  
latent  heat,  or  enthalpy,  of  melting,   LWI
p .    It  is  found  as  the  difference  between  the  specific  
enthalpy  of  water,   h W ,   and  the  specific  enthalpy  of  ice,   hIh ,   (Kirchhoff’s  law,  Curry  and  
Webster  (1999)):  
W
Ih
LWI
p ( p ) = h (tf , p ) − h (tf , p ) .   

(3.34.1)  

Here,   tf ( p )   is  the  freezing  temperature  of  water,  section  3.33.    The  enthalpies   h W   and   h Ih   
are   available   from   IAPWS-­‐‑95   (IAPWS   (2009b))   and   IAPWS-­‐‑06   (IAPWS   (2009a)),  
respectively.      
In   the   case   of   seawater,   the   melt   water   will   additionally   mix   with   the   ambient   brine,  
thus   changing   the   salinity   and   the   freezing   temperature   of   the   seawater.      Consequently,  
the   enthalpy   related   to   this   phase   transition   will   depend   on   the   particular   conditions  
under  which  the  melting  occurs.      
Here,   we   define   the   latent   heat   of   melting   as   the   enthalpy   increase   per   infinitesimal  
mass   of   molten   ice   of   a   composite   system   consisting   of   ice   and   seawater,   when   the  
temperature   is   increased   at   constant   pressure   and   at   constant   total   masses   of   water   and  
salt,   in   excess   of   the   heat   needed   to   warm   up   the   seawater   and   ice   phases   individually  
(Feistel  and  Hagen  (1998),  Feistel  et  al.  (2010b)).    Mass  conservation  of  both  water  and  salt  
during   this   thermodynamic   process   is   essential   to   ensure   the   independence   of   the   latent  
heat   formula   from   the   unknown   absolute   enthalpies   of   salt   and   water   that   otherwise  
would  accompany  any  mass  exchange.      
The  enthalpy  of  sea  ice,   hSI ,   is  additive  with  respect  to  its  constituents  ice,   hIh ,   with  
the  mass  fraction   wIh ,   and  seawater,   h,   with  the  liquid  mass  fraction   1 − wIh :     

(

(

)

hSI = 1 − wIh h ( SA , t, p ) + wIh h Ih (t , p ) .  

)

(3.34.2)  

Upon  warming,  the  mass  of  melt  water  changes  the  ice  fraction   w Ih   and  the  brine  salinity  
SA .     The  related  temperature  derivative  of  Eqn.  (3.34.2)  is    

∂hSI
∂T

(

= 1 − wIh
p

)

∂h
∂T

(

+ 1 − wIh
SA , p

)

∂h
∂SA

T,p

∂SA
∂T

+ w Ih
p

∂h Ih
∂T

(

+ h Ih − h
p

)

∂w Ih
.  
∂T p

(3.34.3)  

The   rate   of   brine   salinity   change   with   temperature   is   given   by   the   reciprocal   of   Eqn.  
(3.33.4)  and  is  related  to  the  isobaric  melting  rate,   −∂wIh / ∂T ,  by  the  conservation  of  the  
p
total  salt,   1 − wIh SA   =  const,  in  the  form  

(

)

∂SA
∂T

=
p

SA ∂wIh
.  
1 − wIh ∂T p

(3.34.4)  

Using  this  relation,  Eqn.  (3.34.3)  takes  the  simplified  form  

∂hSI
∂T

(

)

SI
= 1 − wIh c p + w Ih c Ih
p − Lp
p

∂w Ih
.  
∂T p

(3.34.5)  

The  coefficient  in  front  of  the  melting  rate,    

LSIp ( SA , p ) = h − S A

∂h
∂SA

− h Ih ,  

(3.34.6)  

T,p

provides   the   desired   expression   for   isobaric   melting   enthalpy,   namely   the   difference  
between   the   partial   specific   enthalpies   of   water   in   seawater   and   of   ice.      As   is   physically  
required  for  any  measurable  thermodynamic  quantity,  the  arbitrary  absolute  enthalpies  of  
ice,   water   and   salt   cancel   in   the   formula   (3.34.6),   provided   that   the   reference   state  

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conditions   for   the   ice   and   seawater   formulations   are   chosen   consistently   (Feistel   et   al.  
(2008a)).    Note  that  because  of   h = g + (T0 + t )η   and  Eqn.  (3.33.2),  the  latent  heat  can  also  
be  written  in  terms  of  entropies   η   rather  than  enthalpies   h,   in  the  form    

⎛
⎞
∂η
(3.34.7)  
− η Ih ⎟ .  
LSIp ( SA , p ) = (T0 + tf ) × ⎜η − SA
⎜
⎟
∂SA T , p
⎝
⎠
Again  the  result  is  independent  of  unknown  (and  unknowable)  constants.      
The   latent   heat   of   melting   depends   only   weakly   on   salinity   and   on   pressure.   At   the  
–1
surface  pressure,  the  computed  value  is   LSIp (0,0) = LWI
p (0)   =  333  426.5  J  kg   for  pure  water,  
SI
–1
and   Lp ( SSO ,0) =  329  928.5  J  kg   for  the  standard  ocean,  with  a  difference  of  about  1%  due  
to   the   dissolved   salt.      At   a   pressure   of   1000   dbar,   these   values   reduce   by   0.6%   to  
SI
–1
–1
LSIp (0,1000dbar ) = LWI
p ( 1000dbar ) =   331  528   J  kg    and   Lp ( SSO ,1000dbar ) =   328   034   J  kg .    
WI
TEOS-­‐‑10  is  consistent  with  the  most  accurate  measurements  of   Lp   and  their  experimental  
uncertainties  of  200  J  kg–1,  or  0.06%  (Feistel  and  Wagner  (2005),  (2006)).    
  
  

3.35  Sublimation  pressure    

  
The   sublimation   pressure   of   ice   Psubl    is   defined   as   the   Absolute   Pressure   P    of   water  
vapour   in   equilibrium   with   ice   at   a   given   temperature   t,   at   or   below   the   freezing  
temperature.    It  is  found  by  equating  the  chemical  potential  of  water  vapour   µ V   with  the  
chemical  potential  of  ice   µ Ih ,   so  it  is  found  by  solving  the  implicit  equation    

(

)

(

)

µ V t, Psubl = µ Ih t, Psubl ,   

(3.35.1)  

or  equivalently,  in  terms  of  the  two  Gibbs  functions,    

(

)

(

)

g V t , Psubl = g Ih t , Psubl .   

(3.35.2)  

The   Gibbs   function   for   ice   Ih,   g Ih ( t , P )    is   defined   by   IAPWS-­‐‑06   and   Feistel   and   Wagner  
(2006)  and  is  summarized  in  appendix  I  below.    Note  that  here  the  Absolute  Pressure   P   
rather  than  the  sea  pressure   p   is  used  because  the  sublimation  pressure  of  ice  at  ambient  
conditions  is  much  lower  than  the  atmospheric  pressure.      
The   Gibbs   function   of   vapour,   g V ( t , P ) ,   is   available   from   the   Helmholtz   function   of  
fluid   water,   as   defined   by   IAPWS-­‐‑95;   for   details   see   for   example   Feistel   et   al.   (2008a),  
(2010a),  (2010b).    The  highest  possible  sublimation  pressure  is  found  at  the  triple  point  of  
water.      The   TEOS-­‐‑10   value   of   the   maximum   sublimation   pressure   (i.e.,   the   triple   point  
pressure)  computed  from  Eqn.  (3.35.1)  is   Psubl = Pt    =  611.655  Pa  and  has  an  uncertainty  of  
0.01  Pa  (IAPWS-­‐‑06,  Feistel  et  al.  (2008a)).      
Reliable   theoretical   values   for   the   sublimation   pressure   are   available   down   to   20   K  
(Feistel   and   Wagner   (2007));   a   simple   correlation   function   for   the   sublimation   pressure  
down   to   50   K   is   provided   by   IAPWS   (2008b)   and   is   included   as   a   function   in   the   SIA  
library.      The   IAPWS-­‐‑95   function   µ V required   for   Eqn.   (3.35.1)   is   only   valid   above   130   K.    
An  extension  to  50  K  was  developed  for  TEOS-­‐‑10  (Feistel  et  al.  (2010a))  and  is  available  as  
the  default  option  in  the  SIA  library.    In  nature,  vapour  cannot  reasonably  be  expected  to  
exist  below  50  K  since  it  has  extremely  low  density,  even  in  the  interstellar  vacuum.    For  
this   reason,   the   ice   of   comets   does   not   evaporate   far   from   the   sun.      The   lowest  
temperatures  estimated  for  the  terrestrial  polar  atmosphere  do  not  go  below  130  K.      
In  the  presence  of  air,  ice  is  under  higher  total  pressure  than  just  its  own  sublimation  
pressure.    The  partial  pressure  of  vapour  in  humid  air,   P vap = xV P ,  is  computed  from  the  
total   Absolute   Pressure   P    and   the   mole   fraction   of   vapour,   xV .       Similar   to   the   Absolute  
Salinity   S A    of   seawater,   the   variable   A    describes   the   mass   fraction   of   dry   air   present   in  
humid  air.    Given   A,   the  mole  fraction  of  vapour  is  computed  from  

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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

xV =

1− A
,  
1 − A (1 − M W / M A )

57

(3.35.3)  

where   M A   is  the  molar  mass  of  dry  air  and   M W   is  the  molar  mass  of  water.      
The  sublimation  pressure,   P subl ( t , P ) = xVsat P ,  of  ice  in  equilibrium  with  humid  air  is  the  
partial   pressure   of   vapour   in   saturated   air.      To   compute   xVsat    from   Eqn.   (3.35.3),   the  
required   air   fraction   at   saturation,   A = Asat ( t , P ) ,    is   found   by   equating   the   chemical  
AV
potential  of  water  vapour  in  humid  air   µW
  with  the  chemical  potential  of  ice   µ Ih ,   so  that  
it  is  found  by  solving  the  implicit  equation      

(

)

AV
µW
Asat , t, P = µ Ih (t, P ) ,   

or  equivalently,  in  terms  of  the  two  Gibbs  functions,    

(

)

(

(3.35.4)  

)

g AV Asat , t, P − Asat g AAV Asat , t , P = g Ih (t , P ) .  

(3.35.5)  

The  Gibbs  function  of  humid  air,   g AV ( A, t , P ) ,  is  defined  by  Feistel  et  al.  (2010a).      
At   t    =   0   °C   and   atmospheric   pressure,   the   sublimation   pressure   of   ice   has   the   value  
Psubl (0  °C,  101  325  Pa)  =  613.745  Pa,  computed  by  solving  Eqn.  (3.35.4)  for   Asat ,   then  using  
(3.35.3)   to   determine   the   corresponding   mole   fraction   and   multiplying   the   atmospheric  
pressure   by   this   quantity.      Similarly,   at   the   freezing   point   of   the   standard   ocean   the  
sublimation  pressure  is   Psubl   (-­‐‑1.919  °C,  101  325  Pa)  =  523.436  Pa.      
The   difference   between   observed   or   modelled   partial   vapour   pressures   and   the  
sublimation   pressure   computed   from   TEOS-­‐‑10   is   an   appropriate   quantity   for   use   in  
parameterizations  of  the  mass  flux  between  ice  and  the  atmosphere.      
  
  

3.36  Sublimation  enthalpy    

  
The  sublimation  process  that  occurs  when  ice  is  in  contact  with  pure  water  vapour  can  be  
conducted   by   supplying   heat   at   constant   t   and   P,   with   t   at   or   below   the   freezing  
temperature.      The   heat   required   per   mass   evaporated   from   the   ice   is   the   latent   heat,   or  
enthalpy,  of  sublimation,   LVI
p .    It  is  found  as  the  difference  between  the  specific  enthalpy  
of  water  vapour,   h V ,   and  the  specific  enthalpy  of  ice,   h Ih :       

(

)

(

)

V
LVI
t , Psubl − h Ih t , Psubl .   
p (t ) = h

(3.36.1)  

Here,   P subl ( t )    is   the   sublimation   pressure   of   ice   at   the   temperature   t ,    section   3.35.      The  
enthalpies   h V   and   h Ih   are  available  from  IAPWS-­‐‑95  and  IAPWS-­‐‑06,  respectively.    Reliable  
values  for  the  sublimation  enthalpy  are  theoretically  available  down  to  20  K  from  a  simple  
correlation  function  (Feistel  and  Wagner  (2007)).    At  the  triple  point  of  water,  the  TEOS-­‐‑10  
–1
–1
sublimation  enthalpy  is   LVI
p ( 0.01°C)   =  2  834  359  J  kg   with  an  uncertainty  of  1000  J  kg ,  or  
0.03%.      
In  the  case  when  air  is  present,  the  vapour  resulting  from  the  sublimation  will  add  to  
the   gas   phase,   thus   increasing   the   mole   fraction   of   vapour   xVsat .      If   for   example   the   total  
pressure   P    is   held   constant,   the   partial   pressure   xVsat P    will   rise,   and   the   ice   must   get  
warmer   to   maintain   equilibrium   at   the   modified   sublimation   pressure   Psubl = xVsat P.     
Consequently,  the  enthalpy  related  to  this  phase  transition  will  depend  on  the  particular  
conditions   under   which   the   sublimation   process   occurs.      These   effects   are   small   under  
ambient  conditions  but  may  be  relevant  at  higher  air  densities.      
Here,   we   define   the   latent   heat   of   sublimation   as   the   enthalpy   increase   per  
infinitesimal  mass  of  sublimated  ice  of  a  composite  system  consisting  of  ice  and  humid  air,  
when   the   temperature   is   increased   at   constant   pressure   and   at   constant   total   masses   of  
water  and  dry  air,  in  excess  of  the  enthalpy  increase  needed  to  warm  up  the  ice  and  humid  
air  phases  individually  (Feistel  et  al.  (2010a)).    Mass  conservation  of  both  total  water  and  
dry  air  during  this  thermodynamic  process  is  essential  to  ensure  the  independence  of  the  

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latent  heat  formula  from  the  unknown  absolute  enthalpies  of  air  and  water  that  otherwise  
would  accompany  any  mass  exchange.      
The  enthalpy  of  ice  air,   h AI ,   is  additive  with  respect  to  its  constituents  ice,   hIh ,   with  
the  mass  fraction   wIh ,   and  humid  air,   h AV ,   with  the  gas  fraction   1 − wIh :     

h

AI

(

= 1− w

Ih

)h

AV

( A, t, p )

+ w h

Ih Ih

(

(t, p ) .  

)

(3.36.2)  

Upon  warming,  the  mass  of  vapour  produced  by  sublimation  reduces  the  ice  fraction   w Ih   
and   increases   the   humidity,   that   is,   decreases   the   relative   dry-­‐‑air   fraction   A    of   the   gas  
phase.  The  related  temperature  derivative  of  Eqn.  (3.36.2)  is  

∂h AI
∂T

(

= 1 − wIh
p

)

∂h AV
∂T

(

+ 1 − wIh

)

A, p

∂h AV
∂A

T,p

∂A
∂T

+ wIh
p

∂h Ih
∂T

(

+ h Ih − h AV
p

)

∂w Ih
.  (3.36.3)  
∂T p

The   air-­‐‑fraction   change   is   related   to   the   isobaric   sublimation   rate,   −∂wIh / ∂T ,   by   the  
p
conservation  of  the  dry  air,   1 − w Ih A   =  const,  in  the  form  

(

)

∂A
∂T

=
p

A ∂w Ih
.  
1 − w Ih ∂T p

(3.36.4)  

Using  this  relation,  Eqn.  (3.36.3)  takes  the  simple  form  

∂h AI
∂T

(

)

Ih Ih
AI
= 1 − wIh c AV
p + w c p − Lp
p

∂wIh
.   
∂T p

(3.36.5)  

The  coefficient  in  front  of  the  sublimation  rate,    
AV
LAI
−A
p ( A, p ) = h

∂h AV
∂A

− h Ih ,  

(3.36.6)  

T,p

provides   the   desired   expression   for   isobaric   sublimation   enthalpy,   namely   the   difference  
between  the  partial  specific  enthalpies  of  vapour  in  humid  air  and  of  ice.    In  the  ideal-­‐‑gas  
approximations   for   air   and   for   vapour,   the   partial   specific   enthalpy   of   vapour   in   humid  
air,   h AV − AhAAV   ,  equals  the  specific  enthalpy  of  vapour,   h V ( t ) ,  as  a  function  of  only  the  
temperature,  independent  of  the  pressure  and  of  the  presence  of  air  (Feistel  et  al.  (2010a)).    
In   this   case,   Eqn.   (3.36.6)   coincides   formally   with   Eqn.   (3.36.1),   except   that   the   two   are  
evaluated  at  the  different  pressures   P   and   Psubl ,   respectively.    As  is  physically  required  
for   any   measurable   thermodynamic   quantity,   the   arbitrary   absolute   enthalpies   of   ice,  
vapour  and  air  cancel  in  the  formula  (3.36.6),  provided  that  the  reference  state  conditions  
for   the   ice   and   humid   air   formulations   are   chosen   consistently   (Feistel   et   al.   (2008a),  
(2010a)).      The   latent   heat   of   sublimation   depends   only   weakly   on   the   air   fraction   and   on  
the  pressure.      
For  saturated  air  over  sea  ice,  the  air  fraction   A = Asat   can  be  computed  from  the  brine  
salinity,   or   from   the   sea   surface   salinity   in   the   case   of   floating   ice,   section   3.38.      At   the  
absolute   surface   pressure   PSO    =   101325   Pa   and   the   freezing   point   tf    =   -­‐‑1.919   °C   of   the  
standard   ocean,   the   TEOS-­‐‑10   value   for   saturated   air   with   ASO = Asat ( tf , PSO )    =   0.996   78   is  
subl
–1
LAI
( tf , PSO )   =  523.436  
p ( ASO , PSO )   =  2  833  006  J  kg .    The  related  sublimation  pressure  is   P
Pa,  see  section  3.35.      
Observational  data  show  that  the  ambient  air  over  the  ocean  surface  is  sub-­‐‑saturated  
in  the  climatological  mean.    Rather  than  being  saturated,  values  for  A  that  correspond  to  a  
relative  humidity  of  75%  –  82%  (see  section  3.40)  may  be  a  more  realistic  estimate  for  the  
marine   atmosphere   (Dai   (2006));   these   values   represent   non-­‐‑equilibrium   conditions   that  
result  in  net  evaporation  as  part  of  the  global  hydrological  cycle.      

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3.37  Vapour  pressure    

  
The   vapour   pressure   of   seawater   P vap ( S A , t )    is   defined   as   the   Absolute   Pressure   P    of  
water  vapour  in  equilibrium  with  seawater  at  a  given  temperature   t   and  salinity   SA .     It  is  
found   by   equating   the   chemical   potential   of   vapour   µ V    with   the   chemical   potential   of  
water  in  seawater µ W   so  that  it  is  found  by  solving  the  implicit  equation    

(

)

(

)

µ V t, P vap = µ W SA , t, P vap ,  
or  equivalently,  in  terms  of  the  two  Gibbs  functions,    

(

) (

)

(3.37.1)  

(

)

g V t, P vap = g SA , t , P vap − SA g SA SA , t , P vap .  

(3.37.2)  

Note  that  here  we  use  the  Absolute  Pressure   P   rather  than  the  sea  pressure   p ;  since  the  
vapour   pressure   of   water   at   ambient   conditions   is   much   lower   than   the   atmospheric  
pressure,   the   corresponding   sea   pressure   (Pvap   –   101325   Pa)   would   be   negative   and   near    
-­‐‑105   Pa.      The   Gibbs   functions   of   vapour   and   seawater,   g V ( t , P )    and   g ( SA , t, P),    are  
available   from   the   Helmholtz   function   of   fluid   water,   as   defined   by   IAPWS-­‐‑95,   and   the  
Gibbs  function  of  seawater,  IAPWS-­‐‑08.      
In   the   case   of   pure   water,   SA = 0,    the   solution   of   Eqn.   (3.37.1)   is   the   so-­‐‑called  
saturation   curve   in   the   t − P    diagram   of   water,   which   connects   the   triple   point   with   the  
critical   point.      The   lowest   possible   vapour   pressure   of   pure   liquid   water   is   found   at   the  
triple   point   of   water.      The   TEOS-­‐‑10   value   of   this   minimum   vapour   pressure,   computed  
from   Eqn.   (3.37.1),   is   P vap (0,   0.01   °C)   =   Pt    =   611.655   Pa   with   an   uncertainty   of   0.01   Pa  
(IAPWS-­‐‑95,   Feistel   et   al.   (2008a)).      For   comparison,   the   vapour   pressure   of   the   standard  
ocean   is     P vap ( SSO ,    0  °C)  =  599.907  Pa.      At  laboratory  temperature  the  related  values  are    
P vap (0,  25  °C)  =  3169.93  Pa  and   P vap ( SSO ,   25  °C)  =  3110.57  Pa.      
The  relatively  small  vapour  pressure  lowering  caused  by  the  presence  of  dissolved  salt  
can   be   computed   from   the   isothermal   salinity   derivative   of   Eqn.   (3.37.1)   in   the   form  
(Raoult’s  law)    
S A g SA SA
∂P vap
.  
(3.37.3)  
=
∂SA T
g P − SA g SA P − g PV
As  a  raw  practical  estimate,  this  equation  can  be  expanded  into  powers  of  salinity,  using  
only  the  leading  term  of  the  TEOS-­‐‑10  saline  Gibbs  function,   g S ≈ RSTSA ln SA ,  which  stems  
from  Planck’s  ideal-­‐‑solution  theory.    Here,   RS = R M S   =  264.7599  J  kg–1  K–1  is  the  specific  
gas  constant  of  sea  salt,   R   is  the  universal  molar  gas  constant,  and   M S   =  31.403  82  g  mol–1  
is   the   molar   mass   of   sea   salt   with   Reference   Composition.      The   specific   volume   of  
seawater,   g p ,    is   neglected   in   comparison   to   that   of   vapour.      The   latter   is   approximately  
considered   as   an   ideal   gas,   g Vp ≈ RT / M W P vap ,   where   M W    =   18.015   268   g   mol–1   is   the  
molar  mass  of  water.    The  convenient  result  obtained  with  these  simplifications  is    

(

∂P vap
∂SA

≈ −
T

)

M W vap
P
≈ − 0.57 × P vap .  
MS

(3.37.4)  

The  vapour  pressure  of  seawater  is  always  lower  than  that  of  pure  water.      
In   the   presence   of   air,   seawater   is   under   a   higher   pressure   P    than   under   its   vapour  
pressure   P vap .     In  this  case,  the  vapour  pressure  of  seawater   P vap ( SA , t , P )   is  defined  as  the  
partial   pressure   of   water   vapour   in   humid   air   that   is   in   equilibrium   with   seawater   at   a  
given   pressure   P,    temperature   t    and   salinity   SA .       It   is   found   by   equating   the   chemical  
V
potential   of   vapour   in   humid   air   µAV
   with   the   chemical   potential   of   water   in   seawater  
W
so  that  it  is  found  by  solving  the  implicit  equation    
µ

(

)

V
µAV
Acond , t, P = µ W ( SA , t , P )   

(3.37.5)  

for   Acond ( S A , t , P ) ,  or  equivalently,  in  terms  of  the  two  Gibbs  functions,    

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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

(

)

(

)

g AV Acond , t, P − Acond g AAV Acond , t , P = g ( SA , t , P ) − SA g SA ( SA , t , P ) .  

(3.37.6)  

Since  the  vapour  pressure  is  lowered  in  the  presence  of  sea  salt  (Eqn.  (3.37.4)),  at  vapour  
pressures  above  the  condensation  point  vapour  condenses  out  of  the  air  at  the  sea  surface,  
even  before  the  saturation  point  (that  is,  relative  humidity  of  100%)  is  reached,  to  maintain  
local   equilibrium   with   the   seawater.      The   larger   scale   equilibration   process   may   involve  
downward  diffusion  of  water  vapour  to  the  sea  surface  rather  than  precipitation  of  dew  or  
fog.      From   the   calculated   sub-­‐‑saturated   air   fraction   of   the   condensation   point,   Acond ,   the  
mole   fraction   of   vapour   xVcond    (3.35.3),   and   in   turn   the   vapour   pressure  
P vap ( SA , t , P ) = xVcond P   are  available  from  straightforward  calculations.    The  Gibbs  function  
of  humid  air   g AV   is  available  from  Feistel  et  al.  (2010a)  and  also  as  the  IAPWS  Guideline,  
IAPWS-­‐‑10  (IAPWS  (2010)).      
The  TEOS-­‐‑10  value  computed  from  Eqn.  (3.37.5)  is   P vap (0,  0  °C,  PSO)  =  613.760  Pa  for  
pure   water   at   surface   air   pressure;   the   vapour   pressure   of   the   standard   ocean   is    
P vap ( SSO ,    0   °C,   PSO )   =   602.403   Pa.      At   laboratory   temperature   the   related   values   are    
P vap (0,  25  °C,   PSO )  =  3183.73  Pa  and   P vap ( SSO ,   25  °C,   PSO )  =  3124.03  Pa.      
  
  

3.38  Boiling  temperature    

  
The  boiling  temperature  of  water  or  seawater  is  defined  as  the  temperature   t boil ( S A , P )   at  
which   the   vapour   pressure   (of   section   3.37)   equals   a   given   pressure   P.       It   is   found   by  
equating   the   chemical   potential   of   vapour   µ V    with   the   chemical   potential   of   water   in  
seawater µ W   so  that  it  is  found  by  solving  the  implicit  equation    

(

)

(

)

µ V t boil , P = µ W SA , t boil , P ,   

(3.38.1)  

for   t boil ( SA , P) ,  or  equivalently  in  terms  of  the  two  Gibbs  functions,    

(

g V t boil , P

)

(

)

(

)

= g SA , t boil , P − SA g SA SA , t boil , P .   

(3.38.2)  

The  TEOS-­‐‑10  boiling  temperature  of  pure  water  at  atmospheric  pressure  is   t boil ( 0, PSO )   =  
99.974   °C.      This   temperature   is   outside   the   validity   range   of   up   to   80   °C   of   the   TEOS-­‐‑10  
Gibbs  function  for  seawater.      
  
  

3.39  Latent  heat  of  evaporation    

  
The   evaporation   process   of   pure   liquid   water   in   contact   with   pure   water   vapour   can   be  
conducted  by  supplying  heat  at  constant   t   and   P.     The  heat  required  per  mass  evaporated  
from   the   liquid   is   the   latent   heat,   or   enthalpy,   of   evaporation,   LVW
p .      It   is   found   as   the  
difference  between  the  specific  enthalpy  of  water  vapour,   h V ,   and  the  specific  enthalpy  of  
liquid  water,   h W :     

(

)

(

)

V
LVW
t , P vap − h W t , P vap .  
p (t ) = h

(3.39.1)  

Here,   P (t )    is   the   vapour   pressure   of   water   at   the   temperature   t   (section   3.37).      The  
enthalpies   h V    and   h W    are   available   from   IAPWS-­‐‑95.      At   the   triple   point   of   water,   the  
–1
TEOS-­‐‑10  evaporation  enthalpy  is   LVW
p ( 0.01°C )   =  2  500  915  J  kg .      
In  the  case  of  seawater  in  contact  with  air,  the  vapour  resulting  from  the  evaporation  
will   add   to   the   gas   phase,   thus   increasing   the   mole   fraction   of   vapour,   while   the   liquid  
water   loss   will   increase   the   brine   salinity,   and   cause   a   change   to   the   seawater   enthalpy.    
Consequently,  the  enthalpy  related  to  this  phase  transition  will  depend  on  the  particular  
conditions  under  which  the  evaporation  process  occurs.      
Here,   we   define   the   latent   heat   of   evaporation   as   the   enthalpy   increase   per  
infinitesimal  mass  of  evaporated  water  of  a  composite  system  consisting  of  seawater  and  
humid   air,   when   the   temperature   is   increased   at   constant   pressure   and   at   constant   total  
vap

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61

masses  of  water,  salt  and  dry  air,  in  excess  of  the  enthalpy  increase  needed  to  warm  up  the  
seawater   and   humid   air   phases   individually   (Feistel   et   al.   (2010a)).      Mass   conservation  
during   this   thermodynamic   process   is   essential   to   ensure   the   independence   of   the   latent  
heat  formula  from  the  unknown  absolute  enthalpies  of  air,  salt  and  water  that  otherwise  
would  accompany  any  mass  exchange.      
The  enthalpy  of  sea  air,   hSA ,   is  additive  with  respect  to  its  constituents,  seawater,   h,   
with  the  mass  fraction   wSW ,   and  humid  air,   h AV ,   with  the  gas  fraction   1 − wSW :     

(

(

)

hSA = 1 − wSW h AV ( A, t, p ) + wSW h ( SA , t , p ) .  

)

(3.39.2)  

Upon   warming,   the   mass   of   water   transferred   from   the   liquid   to   the   gas   phase   by  
evaporation  reduces  the  seawater  mass  fraction   wSW ,   increases  the  brine  salinity   S A   and  
increases  the  humidity,  with  a  corresponding  decrease  in  the  dry-­‐‑air  fraction   A   of  the  gas  
phase.    The  related  temperature  derivative  of  Eqn.  (3.39.2)  is    

∂hSA
∂T

(

= 1 − wSW
p

+w

SW

) ∂∂hT

(

AV

∂h
∂T

+ 1 − wSW
A, p

+w

SW

SA , p

∂h
∂SA

) ∂h∂A

T,p

AV

∂SA
∂T

T,p

∂A
∂T

p

(

+ h−h
p

AV

)

∂wSW
.
∂T p

  

(3.39.3)  

The   isobaric   evaporation   rate   −∂wSW / ∂T    is   related   to   the   air-­‐‑fraction   change   by   the  
p
conservation  of  the  dry  air,   1 − wSW A   =  const,  in  the  form    

(

)

∂A
∂T

=
p

A ∂wSW
,  
1 − wSW ∂T p

(3.39.4)  

and  to  the  change  of  salinity  by  the  conservation  of  the  salt,   wSW SA   =  const,  in  the  form    

∂SA
∂T

= −
p

SA ∂wSW
.   
wSW ∂T p

(3.39.5)  

Using  these  relations,  Eqn.  (3.39.3)  takes  the  simplified  form    

∂hSA
∂T

(

)

= 1 − wSW c AV
+ wSW c p − LSA
p
p
p

∂wSW
.  
∂T p

(3.39.6)  

The  coefficient  in  front  of  the  evaporation  rate,    
AV
LSA
−A
p ( A, S A , t , p ) = h

∂h AV
∂A

− h + SA
T,p

∂h
∂SA

,   

(3.39.7)  

T,p

provides   the   desired   expression   for   isobaric   evaporation   enthalpy,   namely   the   difference  
between  the  partial  specific  enthalpies  of  vapour  in  humid  air  (the  first  two  terms)  and  of  
water   in   seawater   (the   last   two   terms).      In   the   ideal-­‐‑gas   approximations   for   air   and   for  
vapour,   the   partial   specific   enthalpy   of   vapour   in   humid   air,   h AV − AhAAV ,   equals   the  
specific  enthalpy  of  vapour,   h V ( t ) ,  as  a  function  of  only  the  temperature,  independent  of  
the  pressure  and  of  the  presence  of  air  (Feistel  et  al.  (2010a)).    As  is  physically  required  for  
any   measurable   thermodynamic   quantity,   the   arbitrary   absolute   enthalpies   of   water,   salt  
and  air  cancel  in  the  formula  (3.39.7),  provided  that  the  reference  state  conditions  for  both  
the  seawater  and  the  humid-­‐‑air  formulation  are  chosen  consistently  (Feistel  et  al.  (2008a),  
(2010a)).   The   latent   heat   of   evaporation   depends   only   weakly   on   salinity   and   on   air  
fraction,  and  is  an  almost  linear  function  of  the  temperature  and  of  the  pressure.      
Selected  representative  values  for  the  air  fraction  at  condensation,   Acond ,   and  the  latent  
heat  of  evaporation,   LSA
p ,  are  given  in  Table  3.39.1.      
In  the  derivation  of  Eqn.  (3.39.7),  the  value  of   A   is  indirectly  assumed  to  be  computed  
from   the   equilibrium   condition   (3.37.6)   between   humid   air   and   seawater,   A    =   Acond .       At  
this  humidity  the  air  is  still  sub-­‐‑saturated,   Acond > Asat ,   but  its  vapour  starts  condensing  at  
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the   sea   surface.      The   values   of   Acond    and   Asat    coincide   only   below   the   freezing   point   of  
seawater,  or  at  vanishing  salinity,  see  also  the  following  section  3.40.      
The   evaporation   rate,   − ∂wSW / ∂T ,   can   be   computed   from   Eqn.   (3.37.6),   the  
p
equilibrium   condition   between   humid   air   and   seawater,   at   changing   temperature   and  
constant   pressure   (Feistel   et   al.   (2010a)).      In   contrast,   the   derivation   of   LSA
p    using   Eqns.  
(3.39.2)   -­‐‑   (3.39.7)   is   a   mere   consideration   of   mass   and   enthalpy   balances;   no   equilibrium  
condition  is  actually  involved.    Hence,  it  is  physically  evident  that  Eqn.  (3.39.7)  can  also  be  
applied  to  situations  in  which   A   takes  any  given  value  different  from   Acond ,   that  is,  it  can  
be  applied  regardless  of  whether  or  not  the  humid  air  is  actually  at  equilibrium  with  the  
sea  surface.      
  
Table  3.39.1:    Selected  values  for  the  equilibrium  air  fraction,   Acond ,   computed  
from   Eqn.   (3.37.6),   and   the   latent   heat   of   evaporation,   LSA
p ,  
computed  from  Eqn.  (3.39.7),  for  different  sea-­‐‑surface  conditions.    
Note  that  the  TEOS-­‐‑10  formulation  for  humid-­‐‑air  is  valid  up  to  5  
MPa,  i.e.,  almost  500  dbar  sea  pressure.      
  
p   
t   
LSA
SA   
Acond   
p   
Condition  
–1
g  kg   
°C   dbar  
%  
J  kg–1  
Pure  water  
0  
0  
0   99.622  31   2  499  032  
Brackish  water  
10  
0  
0   99.624  27   2  499  009  
Standard  ocean   35.165  04   0  
0   99.629  31   2  498  510  
Tropical  ocean   35.165  04   25  
0   98.059  33   2  438  971  
High  pressure   35.165  04   0   400   99.989  43   2  443  759  
  
  
  

3.40  Relative  humidity  and  fugacity    

  
Parameterised   formulas   for   the   flux   of   water   and   heat   through   the   ocean   surface   are  
usually  expressed  in  terms  of  a  given  relative  humidity  of  the  air  in  contact  with  seawater.    
In  this  section  we  provide  the  formulas  for  the  relative  humidity  and  the  fugacity  from  the  
TEOS-­‐‑10  potential  functions  for  seawater  and  humid  air,  and  we  explain  why  the  relative  
fugacity   with   respect   to   condensation   rather   than   with   respect   to   saturation   should   be  
used  for  oceanographic  flux  estimates  (Feistel  et  al.  (2010a)).    Near  the  saturation  point,  the  
two   flux   formulas   may   even   exhibit   different   signs   (different   flux   directions)   since  
condensation  occurs  at  the  sea  surface  at  sub-­‐‑saturated  values  of  relative  humidity.    
Relative  humidity  is  not  uniquely  defined  in  the  literature,  but  the  common  definitions  
give   the   same   results   in   the   ideal-­‐‑gas   limit   of   humid   air.      Also   in   this   approximation,  
relative   humidity   is   only   a   property   of   fluid   water   at   given   temperature   and   pressure   of  
the  vapour  phase,  independent  of  the  presence  of  air.      
The   CCT1   definition   of   relative   humidity   is   in   terms   of   mole   fraction:   “At   given  
pressure   and   temperature,   [the   relative   humidity   is   defined   as]   the   ratio,   expressed   as   a  
percent,  of  the  mole  fraction  of  water  vapour  to  the  vapour  mole  fraction  which  the  moist  
gas  would  have  if  it  were  saturated  with  respect  to  either  water  or  ice  at  the  same  pressure  
and  temperature.”    Consistent  with  CCT,  IUPAC2  defines  relative  humidity  “as  the  ratio,  
often  expressed  as  a  percentage,  of  the  partial  pressure  of  water  in  the  atmosphere  at  some  
observed   temperature,   to   the   saturation   vapour   pressure   of   pure   water   at   this  
temperature”   (Calvert   (1990),   IUPAC   (1997)).      This   definition   of   the   relative   humidity  
takes  the  form    
1  CCT:  Consultative  Committee  for  Thermometry,  www.bipm.org/en/committees/cc/cct/
2  IUPAC:  International  Union  of  Pure  and  Applied  Chemistry,  www.iupac.org

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RH CCT =

xV
  
xVsat

63

(3.40.1)  

with   regard   to   the   mole   fraction   of   vapour   xV ( A) ,    Eqn.   (3.35.3),   and   the   saturated   air  
fraction   A = Asat ( t , P ) = Acond ( 0, t , P )   either  from  Eqn.  (3.37.6)  with  respect  to  liquid  water,  
at   t   above  the  freezing  point  of  pure  water,  or  from  Eqn.  (3.35.5)  with  respect  to  ice,  at   t   
below  the  freezing  point  of  pure  water.    Here,   Acond ( S A , t , P )   is  the  air  fraction  of  humid  
air  at  equilibrium  with  seawater,  Eqn.  (3.37.5),  which  is  subsaturated  for   SA > 0.       
The   WMO3   definition   of   the   relative   humidity   is   (Pruppacher   and   Klett   (1997),  
Jacobson  (2005)),    

1 / A −1
  
(3.40.2)  
r
1 / Asat − 1
where   r = (1 − A) / A    is   the   humidity   ratio.      If   r    is   small,   we   can   estimate   xV ≈ rM A / M W   
(from   Eqn.   (3.35.3))   and   therefore   RH WMO ≈ RH CCT ,   that   is,   we   find   approximate  
consistency  between  Eqns.  (3.40.1)  and  (3.40.2).      
Sometimes,   especially   when   considering   phase   or   chemical   equilibria,   it   is   more  
convenient   to   use   the   fugacity   (or   activity)   rather   than   partial   pressure   ratio   (IUPAC  
(1997)).    The  fugacity  of  vapour  in  humid  air  is  defined  as    
RH WMO =

r

sat

=

V
V, id ⎫
⎪⎧ µ − µ
⎪
(3.40.3)  
f V ( A, T , P ) = xV P exp ⎨
⎬ .   
⎪⎩ RWT ⎪⎭
Here,   RW = R M W    is   the   specific   gas   constant   of   water,   µ V ( A, T , P ) = g AV − Ag AAV    is   the  
chemical  potential  of  vapour  in  humid  air,  and   µ V, id ( A, T , P )   is  its  ideal-­‐‑gas  limit  which  is  
equal  to  the  true  chemical  potential  in  the  limit  of  very  low  pressure,    
T
x P
T
V, id
V
(3.40.4)  
µ
( A, T , P ) = g0 + ∫ ⎛⎜1 − ⎞⎟ c V,id
(T ' ) dT ' + RWT ln VV .  
p
T'⎠
P
V⎝
0
T
0

The   values   of   g0V ,  
AV

P0V and  

T0V    of  

µ V,id    must   be   chosen   consistently   with   the   adjustable  

id
constants  of   g   (Feistel  et  al.  (2010a)).    The  ideal-­‐‑gas  heat  capacity  of  vapour   c V,
(T )   is  
p
available  from  IAPWS-­‐‑95.    In  the  ideal-­‐‑gas  limit  of  infinite  dilution,   f V   converges  to  the  
partial  pressure  of  vapour  (Glasstone  (1947)),    

lim f V ( A, T , P ) = xV P = P vap .   

(3.40.5)  

P→0

The  saturation  fugacity  is  defined  by  the  equilibrium  between  liquid  water  (or  ice)  and  
vapour  in  air,   µ V ( A, T , P ) = µ W ( 0, T , P ) ,  that  is,    

f Vsat

=

(

)

⎧ µ W ( 0, T , P ) − µ V,id Asat , T , P ⎫
⎪
⎬ ,   
RWT
⎪⎩
⎪⎭

⎪
xVsat P exp ⎨

(3.40.6)  

where   µ W = g ( 0, T , P )    is  the  chemical  potential  of  liquid  water  (or  the  chemical  potential  
of  ice,   µ Ih ).    The  relative  fugacity   ϕ   of  humid  air  is  then  defined,  dividing  Eqn.  (3.40.3)  by  
Eqn.  (3.40.6)  and  making  use  of  Eqn.  (3.40.4),  as  

⎧⎪ µ V ( A, T , P ) − µ W ( 0, T , P ) ⎫⎪
fV
ϕ = sat = exp ⎨
⎬ .   
RWT
fV
⎪⎩
⎪⎭

(3.40.7)  

In   the   ideal-­‐‑gas   limit, µ V = µ V, id ,    and   using   (3.40.3)   we   see   that   the   relative   fugacity   ϕ   
coincides  with  the  relative  humidity,  Eqn.  (3.40.1).      
Taking  Eqn.  (3.40.7)  at  the  condensation  point,   A = Acond ,   Eqn.  (3.37.5),  it  follows  that  
the  relative  fugacity  of  humid  air  at  equilibrium  with  seawater  (“sea  air”  for  short)  is    

3  WMO:  World  Meteorological  Organisation,  www.wmo.int

  

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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

ϕ SA =

⎧⎪ µ W ( SA , T , P ) − µ W ( 0, T , P ) ⎫⎪
f VSA
=
exp
⎨
⎬ .   
RWT
f Vsat
⎩⎪
⎭⎪

(3.40.8)  

The  chemical  potential  difference  in  the  exponent  is  proportional  to  the  osmotic  coefficient  
of   seawater,   φ ,    which   is   computed   from   the   saline   part   of   the   Gibbs   function   as   (Feistel  
and  Marion  (2007),  Feistel  (2008)),    

φ ( SA , T , P ) = −

⎡
∂g
⎢ g ( SA , T , P ) − g ( 0, T , P ) − SA
∂SA
mSW RT ⎢
⎣
1

⎤
⎥ ,   
⎥
T,P ⎦

where   mSW   is  the  molality  of  seawater  (Millero  et  al.  (2008a)),    
SA
mSW =
.   
(1 − SA ) M S

(3.40.9)  

(3.40.10)  

From   the   chemical   potential   of   water   in   seawater,   µW = g − SA g S ,   and   Eqns.   (3.40.8)   -­‐‑
A
(3.40.10)  we  infer  for  the  relative  fugacity  of  sea  air  the  simple  formula    

ϕ SA = exp ( − mSW M W φ ) ,   

(3.40.11)  

which   is   identical   to   the   activity   aW    of   water   in   seawater.      Similar   to   the   ideal   gas  
approximation,   the   relative   fugacity   of   sea   air   is   independent   of   the   presence   or   the  
properties  of  air.    In  Eqn.  (3.40.11),  the  relative  fugacity   ϕ SA ≤ 1  expresses  the  fact  that  the  
vapour   pressure   of   seawater   is   lower   than   that   of   pure   water,   i.e.,   that   humid   air   in  
equilibrium  with  seawater  above  its  freezing  temperature  is  always  sub-­‐‑saturated.      
As   a   raw   practical   estimate,   using   a   series   expansion   of   Eqns.   (3.40.10)   and   (3.40.11)  
with   respect   to   salinity,   we   can   obtain   from   the   molality   mSW = SA / M S + O( SA2 )    and   the  
osmotic  coefficient   φ = 1 + O ( SA )   the  linear  relation    

MW
(3.40.12)  
SA ,   
MS
i.e.,  Raoult’s  law  for  the  vapour-­‐‑pressure  lowering  of  seawater,  Eqn.  (3.37.4).      
Below   the   freezing   temperature   of   pure   water   at   a   given   pressure,   the   saturation   of  
vapour  is  defined  by  the  chemical  potential  of  ice  rather  than  liquid  water,  i.e.  by    

ϕ SA ≈ 1 −

f Vsat

=

(

)

⎧ µ Ih (T , P ) − µ V,id Asat , T , P ⎫
⎪
⎬,
RWT
⎪⎩
⎪⎭

⎪
xVsat P exp ⎨

(3.40.13)

rather  than  Eqn.  (3.40.6).    Then,  the  relative  fugacity  of  sea  air  is    

ϕ

SA

⎧⎪ µ W ( SA , T , P ) − µ Ih (T , P ) ⎫⎪
f VSA
= sat = exp ⎨
⎬ .   
RWT
fV
⎪⎩
⎪⎭

(3.40.14)  

When  the  temperature  is  lowered  further  to  the  freezing  point  of  seawater,  the  exponent  of  
(3.40.14)  vanishes  and  sea  air  is  saturated,   ϕ SA = 1,   for  sea-­‐‑ice  air  at  any  lower  temperature.      
Thermodynamic  fluxes  in  non-­‐‑equilibrium  states  are  driven  by  Onsager  “forces”  such  
as   the   gradient   of   − µ / T (de   Groot   and   Mazur   (1984)).      At   the   sea   surface,   assuming   the  
same   temperature   and   pressure   on   both   sides   of   the   sea-­‐‑air   interface,   the   dimensionless  
Onsager  force   X SA ( A, SA , T , P )    driving  the  transfer  of  water  is  the  difference  between  the  
chemical  potentials  of  water  in  humid  air  and  in  seawater,    
V
µAV
A, T , P )
µ W ( SA , T , P )
⎛ µ ⎞
(
(3.40.15)  
X SA = Δ ⎜
.   
−
⎟ =
RWT
RWT
⎝ RWT ⎠
This   difference   vanishes   at   the   condensation   point,   A = Acond ( SA , T , P ) ,    Eqn.   (3.37.5),  
rather  than  at  saturation.     X SA   can  also  be  expressed  in  terms  of  fugacities,  Eqns.  (3.40.7),  
(3.40.8)  and  (3.40.11),  in  the  form      

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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

X SA = ln

ϕ ( A)
ϕ

SA

( SA )

= mSW M Wφ + ln ϕ ( A) .   

65

(3.40.16)  

Rather  than  the  relative  humidity,  Eqns.  (3.40.1),  (3.40.2),  the  sea-­‐‑air  Onsager  force   X SA ,   in  
conjunction   with   the   formula   (3.39.7),   is   relevant   for   the   parameterization   of   non-­‐‑
equilibrium   latent   heat   fluxes   across   the   sea   surface.      In   the   special   case   of   limnological  
applications,   or   below   the   freezing   point   of   seawater,   it   reduces   to   X SA = ln ϕ ( A) ,   which  
corresponds   to   the   relative   humidity,   ln ( RH CCT ) ,   in   the   ideal-­‐‑gas   approximation.      All  
properties   required   for   the   calculation   of   the   formula   (3.40.16)   are   available   from   the  
TEOS-­‐‑10  thermodynamic  potentials  for  seawater,  ice,  and  humid  air.      
  
  

3.41  Osmotic  pressure    

  
If   pure   water   is   separated   from   seawater   by   a   semi-­‐‑permeable   membrane   which   allows  
water  molecules  to  pass  but  not  salt  particles,  water  will  penetrate  into  the  seawater,  thus  
diluting   it   and   possibly   increasing   its   pressure,   until   the   chemical   potential   of   water   in  
both   boxes   becomes   the   same   (or   the   pure   water   reservoir   is   exhausted).      In   the   usual  
model   configuration,   the   two   samples   are   thermally   coupled   but   may   possess   different  
pressures;   the   resulting   pressure   difference   required   to   maintain   equilibrium   is   the  
osmotic   pressure   of   seawater.      An   example   of   a   practical   application   is   desalination   by  
reverse   osmosis;   if   the   pressure   on   seawater   in   a   vessel   exceeds   its   osmotic   pressure,  
freshwater   can   be   “squeezed”   out   of   solution   through   suitable   membrane   walls  
(Sherwood   et   al.   (1967)).      The   osmotic   pressure   of   seawater   is   very   important   for   marine  
organisms;  it  is  considered  responsible  for  the  small  number  of  species  that  can  survive  in  
brackish  environments.      
The   defining   condition   for   the   osmotic   equilibrium   is   equality   of   the   chemical  
potentials  of  pure  water  at  pressure   p W   and  of  water  in  seawater  at  the  pressure   p,     

(

)

g 0, t , p W = g ( SA , t , p ) − S A

∂g
∂SA

.   

(3.41.1)  

T, p

The   solution   of   this   implicit   relation   for   p    (given   values   of   S A , t    and   p W )   leads   to   the  
osmotic  pressure   p osm     

posm = p − p W .  

(3.41.2)  

An   example   of   the   TEOS-­‐‑10   value   for   the   osmotic   pressure   of   standard   seawater   is  
posm SA = SSO , t = 0 °C, p W = 0dbar =    235.4684   dbar .    Osmotic  pressure  may  be  calculated  
using   the   gsw_osmotic_pressure_t_exact(SA,t,pw)   function   of   the   GSW   Oceanographic  
Toolbox.      
  
  

(

)

3.42  Temperature  of  maximum  density    

  
At  about  4  °C  and  atmospheric  pressure,  pure  water  has  a  density  maximum  below  which  
the  thermal  expansion  coefficient  and  the  adiabatic  lapse  rate  change  their  signs  (Röntgen  
(1892),  McDougall  and  Feistel  (2003)).    At  salinities  higher  than  23.8  g  kg–1  the  temperature  
of  maximum  density   tMD   is  below  the  freezing  point   tf   (Table  3.42.1).    The  seasonal  and  
spatial  interplay  between  density  maximum  and  freezing  point  is  highly  important  for  the  
stratification   stability   and   the   seasonal   deep   convection   for   brackish   estuaries   with  
permanent   vertical   and   lateral   salinity   gradients   such   as   the   Baltic   Sea   (Feistel   et   al.  
(2008b),  Leppäranta  and  Myrberg  (2009),  Reissmann  et  al.  (2009)).      
The   temperature   of   maximum   density   tMD    is   computed   from   the   condition   of  
vanishing  thermal  expansion  coefficient,  that  is,  from  the  solution  of  the  implicit  equation  
for   tMD ( SA , p) ,    

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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

gTP ( SA , tMD , p ) = 0.   

(3.42.1)  

The  temperature  of  maximum  density  is  available  in  the  GSW  Oceanographic  Toolbox  as  
the   function   gsw_t_maxdensity_exact.      Selected   TEOS-­‐‑10   values   computed   from   Eqn.  
(3.42.1)  are  given  in  Table  3.42.1.    
  
Table  3.42.1:  Freezing  temperature   tf   and  temperature  of  maximum  density   tMD     
for  air-­‐‑free  brackish  seawater  with  absolute  salinities   S A   between  0  
and   25 g kg −1,   computed   at   the   surface   pressure   from   TEOS-­‐‑10.    
Values  of   tMD   in  parentheses  are  less  than  the  freezing  temperature.    
  
tf
tf
tf
SA
tMD   
SA
tMD    SA
tMD   
g kg–1
°C
g kg–1
°C
g kg–1
°C
°C
°C
°C
0
+0.003
3.978
8.5
–0.456 2.128 17
–0.912
0.250
0.5
–0.026
3.868
9
–0.483 2.019 17.5
–0.939
0.139
1
–0.054
3.758
9.5
–0.509 1.909 18
–0.966
0.027
1.5
–0.081
3.649
10
–0.536 1.800 18.5
–0.994 –0.085
2
–0.108
3.541
10.5
–0.563 1.690 19
–1.021 –0.196
2.5
–0.135
3.432
11
–0.590 1.580 19.5
–1.048 –0.308
3
–0.162
3.324
11.5
–0.616 1.470 20
–1.075 –0.420
3.5
–0.189
3.215
12
–0.643 1.360 20.5
–1.102 –0.532
4
–0.216
3.107
12.5
–0.670 1.249 21
–1.130 –0.644
4.5
–0.243
2.999
13
–0.697 1.139 21.5
–1.157 –0.756
5
–0.269
2.890
13.5
–0.724 1.028 22
–1.184 –0.868
5.5
–0.296
2.782
14
–0.750 0.917 22.5
–1.212 –0.980
6
–0.323
2.673
14.5
–0.777 0.807 23
–1.239 –1.092
6.5
–0.349
2.564
15
–0.804 0.696 23.5
–1.267 –1.204
7
–0.376
2.456
15.5
–0.831 0.584 24
–1.294 (–1.316)
7.5
–0.403
2.347
16
–0.858 0.473 24.5
–1.322 (–1.428)
8
–0.429
2.238
16.5
–0.885 0.362 25
–1.349 (–1.540)
  

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67

4.  Conclusions    

  
  
The   International   Thermodynamic   Equation   Of   Seawater   –   2010   (TEOS-­‐‑10)   allows   all   the  
thermodynamic  properties  of  pure  water,  ice  Ih,  seawater  and  moist  air  to  be  evaluated  in  
an  internally  self-­‐‑consistent  manner.    For  the  first  time  the  effects  of  the  small  variations  in  
seawater  composition  around  the  world  ocean  are  included,  especially  their  effects  on  the  
density   of   seawater   (which   can   be   equivalent   to   ten   times   the   precision   of   our   Practical  
Salinity  measurements  at  sea).      
Perhaps  the  most  apparent  changes  compared  with  the  International  Equation  of  State  
of   seawater   (EOS-­‐‑80)   are   (i)   the   adoption   of   Absolute   Salinity   S A    instead   of   Practical  
Salinity   S P    (PSS-­‐‑78)   as   the   salinity   argument   for   the   thermodynamic   properties   of  
seawater,   and   (ii)   the   use   of   Conservative   Temperature   Θ    in   place   of   potential  
temperature   θ .      Importantly,   Practical   Salinity   is   retained   as   the   salinity   variable   that   is  
stored  in  data  bases  because  Practical  Salinity  is  virtually  the  measured  variable  (whereas  
Absolute   Salinity   is   a   calculated   variable)   and   also   so   that   national   data   bases   do   not  
become  corrupted  with  incorrectly  labeled  and  stored  salinity  data.      
The   adoption   of   Absolute   Salinity   as   the   argument   for   all   the   algorithms   used   to  
evaluate   the   thermodynamic   properties   of   seawater   makes   sense   simply   because   the  
thermodynamic  properties  of  seawater  depend  on   S A   rather  than  on   S P ;  seawater  parcels  
that   have   the   same   values   of   temperature,   pressure   and   of   S P    do   not   have   the   same  
density  unless  the  parcels  also  share  the  same  value  of   S A .    Absolute  Salinity  is  measured  
in   SI   units   and   the   calculation   of   the   freshwater   concentration   and   of   freshwater   fluxes  
follows  naturally  from  Absolute  Salinity,  but  not  from  Practical  Salinity.      
Absolute  Salinity  is  calculated  from  the  computer  algorithm  of  McDougall  et  al.  (2012)  
or   by   other   means,   as   the   sum   of   Reference   Salinity   and   the   Absolute   Salinity   Anomaly.    
There  are  subtle  issues  in  defining  what  is  exactly  meant  by  “absolute  salinity”  and  at  least  
four   different   definitions   are   possible   when   compositional   anomalies   are   present.      We  
have   chosen   the   definition   that   yields   the   most   accurate   estimates   of   seawater   density  
since  the  ocean  circulation  is  sensitive  to  rather  small  gradients  of  density.    The  algorithm  
that  estimates  Absolute  Salinity  Anomaly  represents  the  state  of  the  art  as  at  2010,  but  this  
area  of  oceanography  is  relatively  immature.    It  is  likely  that  the  accuracy  of  this  algorithm  
will  improve  as  more  seawater  samples  from  around  the  world  ocean  have  their  density  
accurately  measured.    After  such  future  work  is  published  and  the  results  distilled  into  a  
revised   algorithm,   such   an   algorithm   will   be   served   from   www.TEOS-10.org.
Oceanographers  should  publish  the  version  number  of  this  software  that  is  used  to  obtain  
thermodynamic  properties  in  their  manuscripts.      
For   these   reasons   the   TEOS-­‐‑10   salinity   variable   to   appear   in   publications   is   Absolute  
Salinity   S A .    The  version  number  of  the  software  that  is  used  to  convert  Reference  Salinity  
S R   into  Absolute  Salinity   SA   should  always  be  stated  in  publications.    Nevertheless,  there  
may   be   some   applications   where   the   likely   future   changes   in   the   algorithm   that   relates  
Reference   Salinity   to   Absolute   Salinity   presents   a   concern,   and   for   these   applications   it  
may  be  preferable  to  publish  graphs  and  tables  in  Reference  Salinity.    For  these  studies  or  
where   it   is   clear   that   the   effect   of   compositional   variations   are   insignificant   or   not   of  
interest,  the  Gibbs  function  may  be  called  with   S R   rather  than   S A ,  thus  avoiding  the  need  
to  calculate  the  Absolute  Salinity  Anomaly.    When  this  is  done,  it  should  be  clearly  stated  
that  Reference  Salinity  is  being  used,  not  Absolute  Salinity.      

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The   recommended   treatment   of   salinity   in   ocean   models   is   to   carry   evolution  
equations  for  both  Preformed  Salinity   S*   and  another  variable,   F δ ,  which  is  related  to  the  
Absolute  Salinity  Anomaly,  so  that  Absolute  Salinity  can  be  calculated  at  each  time  step  of  
the  model  and  used  to  accurately  evaluate  density  (as  discussed  in  appendix  A.20).      
Potential   temperature   has   been   used   in   oceanography   as   though   it   is   a   conservative  
variable,  and  yet  the  specific  heat  of  seawater  varies  by  5%  at  the  sea  surface,  and  potential  
temperature   is   not   conserved   when   seawater   parcels   mix.      The   First   Law   of  
Thermodynamics  can  be  very  accurately  regarded  as  the  statement  that  potential  enthalpy  
h0    and   Conservative   Temperature   Θ    are   conservative   variables   in   the   ocean.      This,  
together   with   the   knowledge   that   the   air-­‐‑sea   heat   flux   is   exactly   the   air-­‐‑sea   flux   of  
potential   enthalpy   (i.   e.   the   air-­‐‑sea   flux   of   c0p Θ )   means   that   potential   enthalpy   can   be  
treated  as  the  “heat  content”  of  seawater,  and  fluxes  of  potential  enthalpy  in  the  ocean  can  
be   treated   as   “heat   fluxes”.      Just   as   it   is   perfectly   valid   to   talk   of   the   flux   of   salinity  
anomaly   ( SA − constant)    across   an   ocean   section   even   when   the   mass   flux   across   the  
section  is  non-­‐‑zero,  so  it  is  perfectly  valid  to  treat  the  flux  of   c0p Θ   across  an  ocean  section  
as  the  “heat  flux”  even  when  the  fluxes  of  mass  and  of  salt  across  the  section  are  non-­‐‑zero.      
The  temperature  variable  in  ocean  models  has  been  taken  to  be  potential  temperature  
θ ,   but   to   date   the   non-­‐‑conservative   source   terms   that   are   present   in   the   evolution  
equation   of   potential   temperature   have   not   been   included.      To   be   TEOS-­‐‑10   compatible,  
ocean   models   need   to   treat   their   temperature   variable   as   Conservative   Temperature   Θ .     
Ocean   models   should   be   initialized   with   Θ    rather   than   θ ,   the   output   temperature   must  
be   compared   to   observed   Θ    data   rather   than   to   θ    data,   and   during   the   model   run,   any  
air-­‐‑sea  fluxes  that  depend  on  the  sea-­‐‑surface  temperature  (SST)  must  be  calculated  at  each  
model  time  step  using   θ = θˆ ( SA , Θ ) .       
Under  EOS-­‐‑80  the  observed  variables   ( SP , t , p )   were  first  used  to  calculate  potential  
temperature   θ    and   then   water   masses   were   analyzed   on   the   SP − θ    diagram.      Curved  
contours   of   potential   density   ρ θ    could   also   be   drawn   on   this   same   SP − θ    diagram.    
Under   TEOS-­‐‑10,   since   density   and   potential   density   are   now   not   functions   of   Practical  
Salinity   SP    but   rather   are   functions   of   Absolute   Salinity   S A ,   it   is   no   longer   possible   to  
draw   isolines   of   potential   density   on   a   SP − θ    diagram.      Rather,   because   of   the   spatial  
variations  of  seawater  composition,  a  given  value  of  potential  density  defines  an  area  on  
the   SP − θ   diagram,  not  a  curved  line.      
Under   TEOS-­‐‑10,   the   observed   variables   ( SP , t , p ) ,   together   with   longitude   and  
latitude,  are  first  used  to  form  Absolute  Salinity   S A ,  and  then  Conservative  Temperature  
Θ   is  evaluated.    Oceanographic  water  masses  are  then  analyzed  on  the   SA − Θ   diagram,  
and   potential   density   ρ Θ    contours   can   also   be   drawn   on   this   SA − Θ    diagram.      The  
computationally-­‐‑efficient   75-­‐‑term   expression   for   the   specific   volume   of   seawater   (of  
appendix   K)   is   a   convenient   and   accurate   equation   of   state   for   observational   and  
theoretical   studies   and   for   ocean   modelling.      Preformed   Salinity   S*    is   used   internally   in  
numerical  ocean  models  where  it  is  important  that  the  salinity  variable  be  conservative.      
Appendix   L   lists   the   recommended   nomenclature,   symbols   and   units   of  
thermodynamic  quantities  for  use  by  oceanographers.            
When   describing   the   use   of   TEOS-­‐‑10,   it   is   the   present   document   (the   TEOS-­‐‑10  
Manual)  that  should  be  referenced  as  IOC  et  al.  (2010)  [IOC,  SCOR  and  IAPSO,  2010:  The  
international  thermodynamic  equation  of  seawater  –  2010:  Calculation  and  use  of  thermodynamic  
properties.    Intergovernmental  Oceanographic  Commission,  Manuals  and  Guides  No.  56,  
UNESCO  (English),  196  pp].   Two  introductory  articles  about  TEOS-­‐‑10,  namely  “Getting  
started   with   TEOS-­‐‑10   and   the   Gibbs   Seawater   (GSW)   Oceanographic   Toolbox”  
(McDougall   and   Barker,   2011),   and   “What   every   oceanographer   needs   to   know   about  
TEOS-­‐‑10:-­‐‑   The   TEOS-­‐‑10   Primer”   (Pawlowicz,   2010b),   are   available   from  
  www.TEOS-10.org.   

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69

APPENDIX  A:    

  
  
  

Background  and  theory  underlying    
the  use  of  the  Gibbs  function  of  seawater    

A.1 ITS-90 temperature
In  order  to  understand  the  limitations  of  conversion  between  different  temperature  scales,  it  is  
helpful  to  review  the  definitions  of  temperature  and  of  the  international  scales  on  which  it  is  
reported.    
  
  
A.1.1  Definition    
When   considering   temperature,   the   fundamental   physical   quantity   is   thermodynamic  
temperature,   symbol   T.      The   unit   for   temperature   is   the   kelvin.      The   name   of   the   unit   has   a  
lowercase   k.      The   symbol   for   the   unit   is   uppercase   K.      One   kelvin   is   1/273.16   of   the  
thermodynamic  temperature  of  the  triple  point  of  water.    (A  recent  evolution  of  the  definition  
has   been   to   specify   the   isotopic   composition   of   the   water   to   be   used   as   that   of   Vienna  
Standard   Mean   Ocean   Water,   VSMOW.)      The   Celsius   temperature,   symbol   t ,    is   defined   by  
t °C = T K − 273.15,   and  1  °C  is  the  same  size  as  1  K.      
  
  
A.1.2  ITS-­‐‑90  temperature  scale    
The   definition   of   temperature   scales   is   the   responsibility   of   the   Consultative   Committee   for  
Thermometry  (CCT)  which  reports  to  the  International  Committee  for  Weights  and  Measures  
(often  referred  to  as  CIPM  for  its  name  in  the  French  language).    Over  the  last  40  years,  two  
temperature  scales  have  been  used;  the  International  Practical  Temperature  Scale  1968  (IPTS-­‐‑
68),   followed   by   the   International   Temperature   Scale   1990   (ITS-­‐‑90).      These   are   defined   by  
Barber   (1969)   and   Preston-­‐‑Thomas   (1990).      For   information   about   the   International  
Temperature  Scales  of  1948  and  1927  the  reader  is  referred  to  Preston-­‐‑Thomas  (1990).      
In   the   oceanographic   range,   temperatures   are   determined   using   a   platinum   resistance  
thermometer.      The   temperature   scales   are   defined   as   functions   of   the   ratio   W ,    namely   the  
ratio  of  the  thermometer  resistance  at  the  temperature  to  be  measured   R (t ) to  the  resistance  at  
a  reference  temperature   R0 .     In  IPTS-­‐‑68,   R0   is   R ( 0°C ) ,   while  in  ITS-­‐‑90   R0   is   R ( 0.01°C ) .     The  
details   of   these   temperature   scales   and   the   differences   between   the   two   scales   are   therefore  
defined  by  the  functions  of   W   used  to  calculate   T .     For  ITS-­‐‑90,  and  in  the  range  0  °C  <   t90 <  
968.71   °C,   t90    is   described   by   a   polynomial   with   10   coefficients   given   by   Table   4   of   Preston-­‐‑
Thomas  (1990).      
We  note  in  passing  that  the  conversions  from   W   to   T   and  from   T   to   W are  both  defined  
by  polynomials  and  these  are  not  perfect  inverses  of  one  another.    Preston-­‐‑Thomas  points  out  
that  the  inverses  are  equivalent  to  within  0.13mK.    In  fact  the  inverses  have  a  difference  of  0.13  
mK  at  861°C,  and  a  maximum  error  in  the  range  0  °C  <   t90   <  40  °C  of  0.06  mK  at  31  °C.    That  
the   CCT   allowed   this   discrepancy   between   the   two   polynomials   immediately   provides   an  
indication   of   the   absolute   uncertainty   in   the   determination,   and   indeed   in   the   definition,   of  
temperature.      

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A  second  uncertainty  in  the  absolute  realization  of  ITS-­‐‑90  arises  from  what  is  referred  to  
as  sub-­‐‑range  inconsistency.    The  polynomial  referred  to  above  describes  the  behaviour  of  an  
‘ideal’   thermometer.      Any   practical   thermometer   has   small   deviations   from   this   ideal  
behaviour.    ITS-­‐‑90  allows  the  deviations  to  be  determined  by  measuring  the  resistance  of  the  
thermometer  at  up  to  five  fixed  points:  the  triple  point  of  water  and  the  freezing  points  of  tin,  
zinc,   aluminium   and   silver,   covering   the   range   0.01   °C  <   t90   <  961.78   °C.      If   not   all   of   these  
points   are   measured,   then   it   is   permissible   to   estimate   the   deviation   from   as   many   of   those  
points  as  are  measured.    The  melting  point  of  Gallium  ( t90   =  29.7646  °C)  and  the  triple  point  of  
Mercury  ( t90   =  -­‐‑  38.8344  °C)  may  also  be  used  if  the  thermometer  is  to  operate  over  a  smaller  
temperature  range.    Hence  the  manner  in  which  the  thermometer  may  be  used  to  interpolate  
between  the  points  is  not  unique.    Rather  it  depends  on  which  fixed  points  are  measured,  and  
there  are  several  possible  outcomes,  all  equally  valid  within  the  definition.    Sections  3.3.2  and  
3.3.3   of   Preston-­‐‑Thomas   (1990)   give   precise   details   of   the   formulation   of   the   deviation  
function.    The  difference  between  the  deviation  functions  derived  from  different  sets  of  fixed  
points  will  depend  on  the  thermometer,  so  it  not  possible  to  state  an  upper  bound  on  this  non-­‐‑
uniqueness.      Common   practice   in   oceanographic   standards   laboratories   is   to   estimate   the  
deviation   function   from   measurements   at   the   triple   point   of   water   and   the   melting   point   of  
Gallium  ( t90   =  29.7646  °C).    This  allows  a  linear  deviation  function  to  be  determined,  but  no  
higher  order  terms.      
In   summary,   there   is   non-­‐‑uniqueness   in   the   definition   of   ITS-­‐‑90,   in   addition   to   any  
imperfections   of   measurement   by   any   practical   thermometer   (Rudtsch   and   Fischer   (2008),  
Feistel   et   al.   (2008a)).      It   is   therefore   not   possible   to   seek   a   unique   and   perfect   conversion  
between  IPTS-­‐‑68  and  ITS-­‐‑90.      
Goldberg   and   Weir   (1992)   and   Mares   and   Kalova   (2008)   have   discussed   the   procedures  
needed   to   convert   measured   thermophysical   quantities   (such   as   specific   heat)   from   one  
temperature   definition   to   another.      When   mechanical   or   electrical   energy   is   used   in   a  
laboratory   to   heat   a   certain   sample,   this   energy   can   be   measured   in   electrical   or   mechanical  
units  by  appropriate  instruments  such  as  an  ampere  meter,  independent  of  any  definition  of  a  
temperature   scale.      It   is   obvious   from   the   fundamental   thermodynamic   relation   (at   constant  
Absolute   Salinity),   du = Tdη + Pdv,    that   the   same   energy   difference   Tdη    results   in   different  
values   for   the   entropy   η ,    depending   on   the   number   read   for   T    from   a   thermometer  
calibrated  on  the  1990  compared  with  one  calibrated  on  the  1968  scale.    A  similar  dependence  
is  found  for  numbers  derived  from  entropy,  for  example,  for  the  heat  capacity,    
  
c p = T ηT
.   
SA , p

Douglas   (1969)   listed   a   systematic   consideration   of   the   quantitative   relations   between   the  
measured   values   of   various   thermal   properties   and   the   particular   temperature   scale   used   in  
the  laboratory  at  the  time  the  measurement  was  conducted.    Conversion  formulas  to  ITS-­‐‑90  of  
readings  on  obsolete  scales  are  provided  by  Goldberg  and  Weir  (1992)  and  Weir  and  Goldberg  
(1996).      
Any   thermal   experimental   data   that   entered   the   construction   of   the   thermodynamic  
potentials   that   form   TEOS-­‐‑10   were   carefully   converted   by   these   rules,   in   addition   to   the  
conversion   between   the   various   older   definitions   of   for   example   calories   and   joules.      This  
must  be  borne  in  mind  when  properties  computed  from  TEOS-­‐‑10  are  combined  with  historical  
measurements  from  the  literature.  
  
  

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A.1.3  Theoretical  conversion  between  IPTS-­‐‑68  and  ITS-­‐‑90  
Having  understood  that  the  conversion  between  IPTS-­‐‑68  and  ITS-­‐‑90  is  not  uniquely  defined,  
we   review   the   sources   of   uncertainty,   or   even   flexibility,   in   the   conversion   between   t90    and  
t68 .       
Consider  first  why   t90   and   t68   temperatures  differ:    
1)   The   fixed   points   have   new   temperature   definitions   in   ITS-­‐‑90,   due   to   improvements   in  
determining   the   absolute   thermodynamic   temperatures   of   the   melting/freezing   physical  
states  relative  to  the  triple  point  of  water.    
2)  For  some  given  resistance  ratio   W   the  two  scales  have  different  algorithms  for  interpolating  
between  the  fixed  points.      
  
Now  consider  why  there  is  non-­‐‑uniqueness  in  the  conversion:      
3)   In   some   range   of   ITS-­‐‑90,   the   conversion   of   W to   t90    can   be   undertaken   with   a   choice   of  
coefficients   that   is   made   by   the   user   (Preston-­‐‑Thomas   (1990)   Sections   3.3.2.1   to   3.3.3),  
referred  to  as  sub-­‐‑range  inconsistency.      
4)  The  impact  of  the  ITS-­‐‑90  deviation  function  on  the  conversion  is  non-­‐‑linear.    Therefore  the  
size   of   the   coefficients   in   the   deviation   function   will   affect   the   difference,   t90 − t68 .       The  
formal   conversion   is   different   for   each   actual   thermometer   that   has   been   used   to   acquire  
data.    
The  group  responsible  for  developing  ITS-­‐‑90  was  well  aware  of  the  non-­‐‑uniqueness  of  
the  conversion.    Table  6  of  Preston-­‐‑Thomas  (1990)  gives  differences   (t90 − t68 )   with  a  resolution  
of  1  mK,  because    
(a)      the   true   thermodynamic   temperature   T    was   known   to   have   uncertainties   of   order  
1  mK  or  larger  in  some  ranges,      
(b)   the   sub-­‐‑range   inconsistency   of   ITS-­‐‑90   using   the   same   calibration   data   gave   an  
uncertainty  of  several  tenths  of  1  mK.      
Therefore   to   attempt   to   define   a   generic   conversion   of   (t90 − t68 )    with   a   resolution   of   say  
0.1  mK  would  probably  be  meaningless  and  possibly  misleading  as  there  isn’t  a  unique  
generic  conversion  function.      
  
  
A.1.4  Practical  conversion  between  IPTS-­‐‑68  and  ITS-­‐‑90    
Rusby   (1991)   published   an   8th   order   polynomial   that   was   a   fit   to   Table   6   of   Preston-­‐‑Thomas  
(1990).    This  fit  is  valid  in  the  range  73.15  K  to  903.89  K  (-­‐‑200  °C  to  630.74  °C).    He  reports  that  
the  polynomial  fits  the  table  to  within  1  mK,  commensurate  with  the  non-­‐‑uniqueness  of  IPTS-­‐‑
68.      
Rusby’s   8th   order   polynomial   is   in   effect   the   ‘official   recommended’   conversion   between  
IPTS-­‐‑68  and  ITS-­‐‑90.    This  polynomial  has  been  used  to  convert  historical  IPTS-­‐‑68  data  to  ITS-­‐‑
90   for   the   preparation   of   the   new   thermodynamic   properties   of   seawater   that   are   the   main  
subject  of  this  manual.      
As   a   convenient   conversion   valid   in   a   narrower   temperature   range,   Rusby   (1991)   also  
proposed    
(A.1.1)  
(T90 − T68 ) /K = -0.00025 (T68 / K - 273.15)   
in   the   range   260   K   to   400   K   (-­‐‑13   °C   to   127   °C).      Rusby   (1991)   also   explicitly   reminds   readers  
(see  his  page  1158)  that  compound  quantities  that  involve  temperature  intervals  such  as  heat  
capacity   and   thermal   conductivity   are   affected   by   their   dependence   on   the   derivative  
d (T90 − T68 ) /dT68 .     About  the  same  time  that  Rusby  published  his  conversion  from   t68   to   t90 ,   
Saunders   (1990)   made   a   recommendation   to   oceanographers   that   in   the   common  
oceanographic  temperature  range  -­‐‑2  °C  <   t68   <  40  °C,  conversion  could  be  achieved  using  

(t90 /°C) = (t68 /°C) 1.00024.   

(A.1.2)  

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The   difference   between   Saunders   (1990)   and   Rusby   (1991)   arises   from   the   best   slope   being  
1.00024   near   0   °C   and   1.00026   near   100   °C   (recall   that   t68    for   the   boiling   point   of   water   was  
100  °C  while  its   t90   is  99.974  °C).    Thus  Rusby  (1991)  chose  1.00025  over  the  wider  range  of  0  
°C  to  100  °C.      
In  considering  what  is  a  ‘reasonable’  conversion  between  the  two  temperature  scales,  we  
must   recall   that   the   uncertainty   in   conversion   between   measured   resistance   and   either  
temperature   scale   is   of   order   a   few   tenths   of   mK,   and   the   uncertainty   in   the   absolute  
thermodynamic  temperature   T   is  probably  at  least  as  large,  and  may  be  larger  than  1  mK  in  
some   parts   of   the   oceanographic   range.      For   all   practical   purposes   data   converted   using  
Saunders’  1.00024  cannot  be  improved  upon;  conversions  using  Rusby’s  (1991)  8th  order  fit  are  
fully   consistent   with   Saunders’   1.00024   in   the   oceanographic   temperature   range   within   the  
limitations  of  the  temperature  scales.      
  
  
A.1.5  Recommendation  regarding  temperature  conversion    
The   ITS-­‐‑90   scale   was   introduced   to   correct   differences   between   true   thermodynamic  
temperature   T ,  and  temperatures  reported  in  IPTS-­‐‑68.      
There   are   remaining   imperfections   and   residuals   in   T −T90    (Rusby,   pers.   comm.),   which  
may  be  as  high  as  a  couple  of  mK  in  the  region  of  interest.    This  is  being  investigated  by  the  
Consultative   Committee   for   Thermometry   (CCT).      At   a   meeting   in   2000   (Rusby   and   White  
(2003))   the   CCT   considered   introducing   a   new   temperature   scale   to   incorporate   the   known  
imperfections,   referred   to   at   that   time   as   ITS-­‐‑XX.      Further   consideration   by   CCT   WG1   has  
moved   thinking   away   from   the   desirability   of   a   new   scale.      The   field   of   thermometry   is  
undergoing   rapid   advances   at   present.      Instead   of   a   new   temperature   scale,   the   known  
limitations  of  the  ITS-­‐‑90  can  be  addressed  in  large  part  through  the  ITS-­‐‑90  Technical  Annex,  
and   documentation   from   time   to   time   of   any   known   differences   between   thermodynamic  
temperature  and  ITS-­‐‑90  (Ripple  et  al.  (2008)).      
The   two   main   conversions   currently   in   use   are   Rusby’s   8th   order   fit   valid   over   a   wide  
range   of   temperatures,   and   Saunders’   1.00024   scaling   widely   used   in   the   oceanographic  
community.      They   are   formally   indistinguishable   because   they   differ   by   less   than   both   the  
uncertainty  in  thermodynamic  temperature,  and  the  uncertainty  in  the  practical  application  of  
the   IPTS-­‐‑68   and   ITS-­‐‑90   scales.      Nevertheless   we   note   that   Rusby   (1991)   suggests   a   linear   fit  
with  slope  1.00025  in  the  range  -­‐‑13  °C  to  127  °C,  and  that  Saunders’  slope  1.00024  is  a  better  fit  
in  the  range  -­‐‑2  °C  to  40  °C  while  Rusby’s  8th  order  fit  is  more  robust  for  temperatures  outside  
the   oceanographic   range.      The   difference   between   Saunders   (1990)   and   Rusby   (1991)   is   less  
than  1  mK  everywhere  in  the  range  -­‐‑2  °C  to  40  °C  and  less  than  0.03mK  in  the  range  -­‐‑2  °C  to  
10  °C.      
In   conclusion,   the   algorithms   for   PSS-­‐‑78   require   t68    as   the   temperature   argument.      In  
order  to  use  these  algorithms  with   t90   data,   t68   may  be  calculated  using  Eqn.  (A.1.3)  thus    

(t68 /°C)
  
  

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= 1.00024 (t90 /°C) .   

(A.1.3)  

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73

A.2 Sea pressure, gauge pressure and Absolute Pressure
Sea   pressure   p    is   defined   to   be   the   Absolute   Pressure   P    less   the   Absolute   Pressure   of   one  
standard  atmosphere,   P0 ≡ 101 325Pa;   that  is    

p ≡ P − P0 .   

(A.2.1)  

Also,  it  is  common  oceanographic  practice  to  express  sea  pressure  in  decibars  (dbar).    Another  
common  pressure  variable  that  arises  naturally  in  the  calibration  of  sea-­‐‑board  instruments  is  
gauge   pressure   p gauge    which   is   Absolute   Pressure   less   the   Absolute   Pressure   of   the  
atmosphere  at  the  time  of  the  instrument’s  calibration  (perhaps  in  the  laboratory,  or  perhaps  
at  sea).    Because  atmospheric  pressure  changes  in  space  and  time,  sea  pressure   p   is  preferred  
over   p gauge   as  a  thermodynamic  variable  as  it  is  unambiguously  related  to  Absolute  Pressure.    
The   seawater   Gibbs   function   is   naturally   a   function   of   sea   pressure   p    (or   functionally  
equivalently,  of  Absolute  Pressure   P );  it  is  not  a  function  of  gauge  pressure.      
  
Table  A.2.1    Pressure  unit  conversion  table    
  
    

1  Pa  
1  dbar  

  

  

  

  

  

Pascal    

decibar    

bar    

(Pa)  

(dbar)  

(bar)  

≡  1  N/m2  

10−4  

10−5  

104  

≡  105  dyn/cm2  

0.1  
≡  106  dyn/cm2  

  

  

  

  

Technical  
atmosphere   atmosphere  
(at)  

(atm)  

  
torr    
(Torr)  

pound-­‐‑
force  per  
square  inch  
  (psi)  

10.197×10−6   9.8692×10−6   7.5006×10−3   145.04×10−6  
0.101  97  

98.692×10−3  

75.006  

1.450  377  44  

1.0197  

0.986  92  

750.06  

14.503  7744  

1  bar  

100  000  

10  

1  at  

98  066.5  

9.806  65  

0.980  665  

≡  1  kgf/cm2  

0.967  841  

735.56  

14.223  

1  atm  

101  325  

10.1325  

1.013  25  

1.0332  

≡  1  atm  

760  

14.696  

1  torr  

133.322  

1.3332×10−2  

≡  1  Torr  

19.337×10−3  

1.3332×10−3   1.3595×10−3   1.3158×10−3  

1  psi   6  894.757  
0.689  48  
68.948×10−3   70.307×10−3   68.046×10−3  
51.715  
≡  1  lbf/in2  
  
Example:    1  Pa  =  1  N/m2    =  10−4  dbar      =  10−5  bar    =  10.197×10−6  at    =  9.8692×10−6  atm,  etc.  
  
The   difference   between   sea   pressure   and   gauge   pressure   is   quite   small   and   probably  
insignificant  for  many  oceanographic  applications.    Nevertheless  it  would  be  best  practice  to  
ensure  that  the  CTD  pressure  that  is  used  in  the  seawater  Gibbs  function  is  calibrated  on  deck  
to  read  the  atmospheric  pressure  as  read  from  the  ship’s  bridge  barometer,  less  the  Absolute  
Pressure   of   one   standard   atmosphere,   P0 ≡ 101 325Pa.       (When   the   CTD   is   lowered   from   the  
sea  surface,  the  monitoring  software  may  well  display  gauge  pressure,  indicating  the  distance  
from  the  surface.)      
Since   there   are   a   variety   of   different   units   used   to   express   atmospheric   pressure,   we  
present  a  table  (Table  A.2.1)  to  assist  in  converting  between  these  different  units  of  pressure  
(see   ISO   (1993)).     Note   that   one   decibar   (1   dbar)   is   exactly   0.1   bar,   and   that   1   mmHg   is   very  
similar  to  1  torr  with  the  actual  relationship  being  1  mmHg  =  1.000  000  142  466  321...  torr.    The  
torr  is  defined  as  exactly  1/760  of  the  Absolute  Pressure  of  one  standard  atmosphere,  so  that  
one  torr  is  exactly  equal  to  (101  325/760)  Pa.      
  
  

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A.3 Reference Composition and the Reference-Composition Salinity Scale
As  mentioned  in  the  main  text,  the  Reference  Composition  of  seawater  is  defined  by  Millero  et  
al.   (2008a)   as   the   exact   mole   fractions   given   in   Table   D.3   of   appendix   D   below.      This  
composition   model   was   determined   from   the   most   accurate   measurements   available   of   the  
properties   of   Standard   Seawater,   which   is   filtered   seawater   from   the   surface   waters   of   the  
North   Atlantic   as   made   available   by   the   IAPSO   Standard   Seawater   Service.      The   Reference  
Composition  is  perfectly  consistent  with  charge  balance  of  ocean  waters  and  the  most  recent  
atomic   weight   estimates   (Wieser   (2006)).      For   seawater   with   this   reference   composition   the  
Reference-­‐‑Composition   Salinity   S R    as   defined   below   provides   our   best   estimate   of   the  
Absolute  Salinity.      
The   Reference   Composition   includes   all   important   components   of   seawater   having   mass  
fractions  greater  than  about  0.001   g kg −1   (i.  e.  1.0   mg kg −1 )  that  can  significantly  affect  either  
the   conductivity   or   the   density   of   seawater   having   a   Practical   Salinity   of   35.      The   most  
significant   ions   not   included   are   Li +    (~0.18   mg kg −1 )   and   Rb +    (~0.12   mg kg −1 ).      Dissolved  
gases   N 2    (~16   mg kg −1 )   and   O2    (   up   to   8   mg kg −1    in   the   ocean)   are   not   included   as   neither  
have  a  significant  effect  on  density  or  on  conductivity.    In  addition,   N 2   remains  within  a  few  
percent   of   saturation   at   the   measured   temperature   in   almost   all   laboratory   and   in   situ  
conditions.    However,  the  dissolved  gas   CO2   (~0.7   mg kg −1 ),  and  the  ion   OH −   (~0.08   mg kg −1 )  
are  included  in  the  Reference  Composition  because  of  their  important  role  in  the  equilibrium  
dynamics  of  the  carbonate  system.    Changes  in   pH   which  involve  conversion  of   CO2   to  and  
from   ionic   forms   affect   conductivity   and   density.      Concentrations   of   the   major   nutrients  
Si(OH)4 ,   NO3−    and   PO34−    are   assumed   to   be   negligible   in   Standard   Seawater;   their  
concentrations   in   the   ocean   range   from   0-­‐‑16   mg kg −1 ,   0-­‐‑2   mg kg −1 ,   and   0-­‐‑0.2   mg kg −1   
respectively.    The  Reference  Composition  does  not  include  organic  matter.    The  composition  
of  Dissolved  Organic  Matter  (DOM)  is  complex  and  poorly  known.    DOM  is  typically  present  
at  concentrations  of  0.5-­‐‑2   mg kg −1   in  the  ocean.      
Reference-­‐‑Composition   Salinity   is   defined   to   be   conservative   during   mixing   or  
evaporation  that  occurs  without  removal  of  sea  salt  from  solution.    Because  of  this  property,  
the   Reference-­‐‑Composition   Salinity   of   any   seawater   sample   can   be   defined   in   terms   of  
products  determined  from  the  mixture  or  separation  of  two  precisely  defined  end  members.    
Pure  water  and  KCl-­‐‑normalized  seawater  are  defined  for  this  purpose.    Pure  water  is  defined  as  
Vienna   Standard   Mean   Ocean   Water,   VSMOW,   which   is   described   in   the   2001   Guideline   of  
the   International   Association   for   the   Properties   of   Water   and   Steam   (IAPWS   (2005),   BIPM  
(2005));  it  is  taken  as  the  zero  reference  value.    KCl-­‐‑normalized  seawater  (or  normalized  seawater  
for  short)  is  defined  to  correspond  to  a  seawater  sample  with  a  Practical  Salinity  of  35.    Thus,  
any   seawater   sample   that   has   the   same   electrical   conductivity   as   a   solution   of   potassium  
chloride  (KCl)  in  pure  water  with  the  KCl  mass  fraction  of  32.4356  g  kg-­‐‑1  when  both  are  at  the  
ITS-­‐‑90   temperature   t    =   14.996   °C   and   one   standard   atmosphere   pressure,   P    =   101325   Pa   is  
referred   to   as   normalized   seawater.      Here,   KCl   refers   to   the   normal   isotopic   abundances   of  
potassium   and   chlorine   as   described   by   the   International   Union   of   Pure   and   Applied  
Chemistry   (Wieser   (2006)).      As   discussed   below,   any   normalized   seawater   sample   has   a  
Reference-­‐‑Composition  Salinity  of  35.165  04   g kg −1.       
Since   Reference-­‐‑Composition   Salinity   is   defined   to   be   conservative   during   mixing,   if   a  
seawater  sample  of  mass   m1   and  Reference-­‐‑Composition  Salinity   SR1   is  mixed  with  another  
seawater   sample   of   mass   m2    and   Reference-­‐‑Composition   Salinity   S R 2 ,    the   final   Reference-­‐‑
Composition  Salinity   SR12   of  this  sample  is    

m1SR1 + m2 SR2
.  
(A.3.1)  
m1 + m2
Negative   values   of   m1    and   m2 ,    corresponding   to   the   removal   of   seawater   with   the  
appropriate   salinity   are   permitted,   so   long   as   m1 (1 − SR1 ) + m2 (1 − SR2 ) > 0 .      In   particular,   if  
SR2 = 0    (pure   water)   and   m2    is   the   mass   of   pure   water   needed   to   normalize   the   seawater  
SR12 =

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75

sample   (that   is,   m2    is   the   mass   needed   to   achieve   SR12    =   35.165   04   g   kg−1),   then   the   original  
Reference-­‐‑Composition  Salinity  of  sample  1  is  given  by    

SR1 = [1 + (m2 / m1 )] × 35.16504 g kg-1 .  

(A.3.2)  

The   definitions   and   procedures   above   allow   one   to   determine   the   Reference   Salinity   of  
any   seawater   sample   at   the   ITS-­‐‑90   temperature   t    =   14.996   °C   and   one   standard   atmosphere  
pressure.      To   complete   the   definition,   we   note   that   the   Reference-­‐‑Composition   Salinity   of   a  
seawater   sample   at   given   temperature   and   pressure   is   equal   to   the   Reference-­‐‑Composition  
Salinity   of   the   same   sample   at   any   other   temperature   and   pressure   provided   the   transition  
process   is   conducted   without   exchange   of   matter,   in   particular,   without   evaporation,  
precipitation  or  degassing  of  substance  from  the  solution.    Note  that  this  property  is  shared  by  
Practical  Salinity  to  the  accuracy  of  the  algorithms  used  to  define  this  quantity  in  terms  of  the  
conductivity  ratio   R15.       
We   noted   above   that   a   Practical   Salinity   of   35   is   associated   with   a   Reference   Salinity   of  
35.165   04   g kg −1.       This   value   was   determined   by   Millero   et   al.   (2008a)   using   the   reference  
composition  model,  the  most  recent  atomic  weights  (Wieser  (2006))  and  the  relation   S   =  1.806  
55   Cl / (g kg −1 )    which   was   used   in   the   original   definition   of   Practical   Salinity   to   convert  
between   measured   Chlorinity   values   and   Practical   Salinity.      Since   the   relation   between  
Practical  Salinity  and  conductivity  ratio  was  defined  using  the  same  conservation  relation  as  
satisfied  by  Reference  Salinity,  the  Reference  Salinity  can  be  determined  to  the  same  accuracy  
as  Practical  Salinity  wherever  the  latter  is  defined  (that  is,  in  the  range   2< S P < 42 ),  as    

SR ≈ uPS SP               where             uPS ≡ (35.165 04 35) g kg −1 .  

(A.3.3)  

For  practical  purposes,  this  relationship  can  be  taken  to  be  an  equality  since  the  approximate  
nature   of   this   relation   only   reflects   the   accuracy   of   the   algorithms   used   in   the   definition   of  
Practical  Salinity.    This  follows  from  the  fact  that  the  Practical  Salinity,  like  Reference  Salinity,  
is   intended   to   be   precisely   conservative   during   mixing   and   also   during   changes   in  
temperature  and  pressure  that  occur  without  exchange  of  mass  with  the  surroundings.      
The   Reference-­‐‑Composition   Salinity   Scale   is   defined   such   that   a   seawater   sample   whose  
Practical   Salinity   S P    is   35   has   a   Reference-­‐‑Composition   Salinity   S R    of   precisely  
35.165 04 g kg−1 .      Millero   et   al.   (2008a)   estimate   that   the   absolute   uncertainty   associated   with  
using  this  value  as  an  estimate  of  the  Absolute  Salinity  of  Reference-­‐‑Composition  Seawater  is  
± 0.007 g kg−1 .      Thus   the   numerical   difference   between   the   Reference   Salinity   expressed   in  
g kg −1    and   Practical   Salinity   is   about   24   times   larger   than   this   estimate   of   uncertainty.      The  
difference,   0.165 04 ,  is  also  large  compared  to  our  ability  to  measure  Practical  Salinity  at  sea  
(which   can   be   as   precise   as   ± 0.002 ).      Understanding   how   this   discrepancy   was   introduced  
requires   consideration   of   some   historical   details   that   influenced   the   definition   of   Practical  
Salinity.    The  details  are  presented  in  Millero  et  al.  (2008a)  and  in  Millero  (2010)  and  are  briefly  
reviewed  below.      
There  are  two  primary  sources  of  error  that  contribute  to  this  discrepancy.    First,  and  most  
significant,  in  the  original  evaporation  technique  used  by  Sørensen  in  1900  (Forch  et  al.  1902)  
to   estimate   salinity,   some   volatile   components   of   the   dissolved   material   were   lost   so   the  
amount   of   dissolved   material   was   underestimated.      Second,   the   approximate   relation  
determined  by  Knudsen  (1901)  to  determine   S ( ‰ )   from  measurements  of   Cl ( ‰ )   was  based  
on  analysis  of  only  nine  samples  (one  from  the  Red  Sea,  one  from  the  North  Atlantic,  one  from  
the  North  Sea  and  six  from  the  Baltic  Sea).    Both  the  errors  in  estimating  absolute  Salinity  by  
evaporation  and  the  bias  towards  Baltic  Sea  conditions,  where  strong  composition  anomalies  
relative  to  North  Atlantic  conditions  are  found,  are  reflected  in  Knudsen'ʹs  formula,    

SK ( ‰ ) = 0.03 + 1.805 Cl ( ‰ ) .   

(A.3.4)  

When   the   Practical   Salinity   Scale   was   decided   upon   in   the   late   1970s   it   was   known   that  
this  relation  included  significant  errors,  but  it  was  decided  to  maintain  numerical  consistency  

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with  this  accepted  definition  of  salinity  for  typical  mid-­‐‑ocean  conditions  (Millero  (2010)).    To  
achieve   this   consistency   while   having   salinity   directly   proportional   to   Chlorinity,   the   Joint  
Panel   for   Oceanographic   Tables   and   Standards   (JPOTS)   decided   to   determine   the  
proportionality  constant  from  Knudsen'ʹs  formula  at   S K   =  35  ‰  ( Cl =  19.3740   ‰ ),  (Wooster  et  
al.,  1969).    This  resulted  in  the  conversion  formula    

S ( ‰ ) = 1.80655 Cl ( ‰ )   

(A.3.5)  

being   used   in   the   definition   of   the   practical   salinity   scale   as   if   it   were   an   identity,   thus  
introducing   errors   that   have   either   been   overlooked   or   accepted   for   the   past   30   years.      We  
now  break  with  this  tradition  in  order  to  define  a  salinity  scale  based  on  a  composition  model  
for   Standard   Seawater   that   was   designed   to   give   a   much   improved   estimate   of   the   mass-­‐‑
fraction   salinity   for   Standard   Seawater   and   for   Reference-­‐‑Composition   Seawater.      The  
introduction  of  this  salinity  scale  provides  a  more  physically  meaningful  measure  of  salinity  
and   simplifies   the   task   of   systematically   incorporating   the   influence   of   spatial   variations   of  
seawater  composition  into  the  procedure  for  estimating  Absolute  Salinity.      
Finally,   we   note   that   to   define   the   Reference-­‐‑Composition   Salinity   Scale   we   have  
introduced  the  quantity   uPS   in  Eqn.  (A.3.3),  defined  by   uPS ≡ (35.165 04 35) g kg −1 .    This  value  
was   determined   by   the   requirement   that   the   Reference-­‐‑Composition   Salinity   gives   the   best  
estimate  of  the  mass-­‐‑fraction  Absolute  Salinity  (that  is,  the  mass-­‐‑fraction  of  non-­‐‑H2O  material)  
of   Reference-­‐‑Composition   Seawater.      However,   the   uncertainty   in   using   S R    to   estimate   the  
Absolute   Salinity   of   Reference-­‐‑Composition   Seawater   is   at   least   0.007   g kg −1    at   S    =   35  
(Millero   et   al.   (2008b)).      Thus,   although   uPS    is   precisely   specified   in   the   definition   of   the  
Reference-­‐‑Composition   Salinity   Scale,   it   must   be   noted   that   using   the   resulting   definition   of  
the   Reference   Salinity   to   estimate   the   Absolute   Salinity   of   Reference-­‐‑Composition   Seawater  
does   have   a   non-­‐‑zero   uncertainty   associated   with   it.      This   and   related   issues   are   discussed  
further  in  the  next  subsection.      
  
  

A.4 Absolute Salinity
Millero  et  al.  (2008a)  list  the  following  six  advantages  of  adopting  Reference  Salinity   S R   and  
Absolute  Salinity   S A   in  preference  to  Practical  Salinity S P .       
  
1. The  definition  of  Practical  Salinity   S P   on  the  PSS-­‐‑78  scale  is  separate  from  the  system  
of   SI   units   (BIPM   (2006)).      Reference   Salinity   can   be   expressed   in   the   unit    
(g kg −1 )   as  a  measure  of  Absolute  Salinity.    Adopting  Absolute  Salinity  and  Reference  
Salinity  will  terminate  the  ongoing  controversies  in  the  oceanographic  literature  about  
the   use   of   “PSU”   or   “PSS”   and   make   research   papers   more   readable   to   the   outside  
scientific  community  and  consistent  with  SI.      
2. The   freshwater   mass   fraction   of   seawater   is   not   (1   –   0.001   S P ).      Rather,   it   is    
(1  –  0.001   S A /( g kg −1 )),  where   S A   is  the  Absolute  Salinity,  defined  as  the  mass  fraction  
of   dissolved   material   in   seawater.      The   values   of   S A /( g kg −1 )   and   S P    are   known   to  
differ  by  about  0.5%.    There  seems  to  be  no  good  reason  for  continuing  to  ignore  this  
known  difference,  for  example  in  ocean  models.      
3. PSS-­‐‑78  is  limited  to  the  range  2  <   S P   <  42.    For  a  smooth  crossover  on  one  side  to  pure  
water,   and   on   the   other   side   to   concentrated   brines   up   to   saturation,   as   for   example  
encountered  in  sea  ice  at  very  low  temperatures,  salinities  beyond  these  limits  need  to  
be  defined.    While  this  poses  a  challenge  for   S P ,   it  is  trivial  for   SR .       
4. The   theoretical   Debye-­‐‑Hückel   limiting   laws   of   seawater   behavior   at   low   salinities,  
used  for  example  in  the  determination  of  the  Gibbs  function  of  seawater,  can  only  be  
computed  from  a  chemical  composition  model,  which  is  available  for   S R   but  not  for  
S P .       

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5. For  artificial  seawater  of  Reference  Composition,   S R   has  a  fixed  relation  to  Chlorinity,  
independent  of  conductivity,  salinity,  temperature,  or  pressure.    
6. Stoichiometric   anomalies   can   be   specified   accurately   relative   to   Reference-­‐‑
Composition  Seawater  with  its  known  composition,  but  only  uncertainly  with  respect  
to  IAPSO  Standard  Seawater  with  its  unknown  composition.    These  variations  in  the  
composition  of  seawater  cause  significant  (a  few  percent)  variations  in  the  horizontal  
density  gradient.      
  

Regarding  point  number  2,  Practical  Salinity   S P   is  a  dimensionless  number  of  the  order  of  
35   in   the   open   ocean;   no   units   or   their   multiples   are   permitted.      There   is   however   more  
freedom  in  choosing  the  representation  of  Absolute  Salinity   S A   since  it  is  defined  as  the  mass  
fraction  of  dissolved  material  in  seawater.    For  example,  all  the  following  quantities  are  equal  
(see  ISO  (1993)  and  BIPM  (2006)),      
34  g/kg  =  34  mg/g  =  0.034  kg/kg  =  0.034  =  3.4  %  =  34  000  ppm  =  34  000  mg/kg.  

  

In   particular,   it   is   strictly   correct   to   write   the   freshwater   fraction   of   seawater   as   either    
(1   –   0.001   S A /( g kg −1 ))   or   as   (1   –   S A )   but   it   would   be   incorrect   to   write   it   as   (1   –   0.001   S A ).    
Clearly   it   is   essential   to   consider   the   units   used   for   Absolute   Salinity   in   any   particular  
application.      If   this   is   done,   there   should   be   no   danger   of   confusion,   but   to   maintain   the  
numerical   value   of   Absolute   Salinity   close   to   that   of   Practical   Salinity   S P    we   adopt   the   first  
option  above,  namely   g kg −1   as  the  preferred  unit  for   SA ,   (as  in   S A   =  35.165  04  g  kg−1).    The  
Reference  Salinity,   SR ,   is  defined  to  have  the  same  units  and  follows  the  same  conventions  as  
SA .     Salinity  “S‰”  measured  prior  to  PSS-­‐‑78  available  from  the  literature  or  from  databases  is  
usually  reported  in  ‰  or  ppt  (part  per  thousand)  and  is  converted  to  the  Reference  Salinity,  
S R = uPS S‰,   by  the  numerical  factor   uPS   from  (A.3.3).      
Regarding   point   number   5,   Chlorinity   Cl    is   the   concentration   variable   that   was   used   in  
the   laboratory   experiments   for   the   fundamental   determinations   of   the   equation   of   state   and  
other   properties,   but   has   seldom   been   measured   in   the   field   since   the   definition   of   PSS-­‐‑78  
(Millero,   2010).      Since   the   relation   SP = 1.806 55 Cl    for   Standard   Seawater   was   used   in   the  
definition   of   Practical   Salinity   this   may   be   taken   as   an   exact   relation   for   Standard   Seawater  
and   it   is   also   our   best   estimate   for   Reference-­‐‑Composition   Seawater.      Thus,   Chlorinity  
expressed   in   ‰   can   be   converted   to   Reference-­‐‑Composition   Salinity   by   the  
relation, SR = uCl Cl ,    with   the   numerical   factor   uCl = 1.806 55 uPS.       These   constants   are  
recommended  for  the  conversion  of  historical  (pre  1900)  data.    The  primary  source  of  error  in  
using  this  relation  will  be  the  possible  presence  of  composition  anomalies  in  the  historical  data  
relative  to  Standard  Seawater.      
Regarding   point   number   6,   the   composition   of   dissolved   material   in   seawater   is   not  
constant  but  varies  a  little  from  one  ocean  basin  to  another,  and  the  variation  is  even  stronger  
in   estuaries,   semi-­‐‑enclosed   or   even   enclosed   seas.      Brewer   and   Bradshaw   (1975)   and   Millero  
(2000)  point  out  that  these  spatial  variations  in  the  relative  composition  of  seawater  impact  the  
relationship   between   Practical   Salinity   (which   is   essentially   a   measure   of   the   conductivity   of  
seawater  at  a  fixed  temperature  and  pressure)  and  density.    All  the  thermophysical  properties  
of   seawater   as   well   as   other   multicomponent   electrolyte   solutions   are   directly   related   to   the  
concentrations   of   the   major   components,   not   the   salinity   determined   by   conductivity;   note  
that  some  of  the  variable  nonelectrolytes  (e.g.,   Si(OH)4 ,   CO2   and  dissolved  organic  material)  
do   not   have   an   appreciable   conductivity   signal.      It   is   for   this   reason   that   the   TEOS-­‐‑10  
thermodynamic  description  of  seawater  has  the  Gibbs  function   g   of  seawater  expressed  as  a  
function  of  Absolute  Salinity  as   g ( SA , t , p )   rather  than  as  a  function  of  Practical  Salinity   S P   or  
of  Reference  Salinity,   SR .     The  issue  of  the  spatial  variation  in  the  composition  of  seawater  is  
discussed  more  fully  in  appendix  A.5.      

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Regarding  point  number  2,  we  note  that  it  is  debatable  which  of  (1  –  0.001   SAdens /( g kg −1 )),  
(1   –   0.001   S Asoln /( g kg −1 )),   (1   –   0.001   SAadd /( g kg −1 ))   or   (1   –   0.001   S* /( g kg −1 ))   is   the   most  
appropriate   measure   of   the   freshwater   mass   fraction.      (These   different   versions   of   absolute  
salinity   are   defined   in   section   2.5   and   also   later   in   this   appendix.)      This   is   a   minor   point  
compared  with  the  present  use  of  (1  –  0.001   SP )  in  this  context,  and  the  choice  of  which  of  the  
above  expressions  may  depend  on  the  use  for  the  freshwater  mass  fraction.    For  example,  in  
the   context   of   ocean   modelling,   if   S*    is   the   salinity   variable   that   is   treated   as   a   conservative  
variable   in   an   ocean   model,   then   (1   –   0.001   S* /( g kg −1 ))   is   probably   the   most   appropriate  
version  of  freshwater  mass  fraction.      
It   should   be   noted   that   the   quantity   S A    appearing   as   an   argument   of   the   function  
g ( SA , t, p )    is   the   Absolute   Salinity   (the   “Density   Salinity”   SA ≡ SAdens )   measured   on   the  
Reference-­‐‑Composition   Salinity   Scale.      This   is   important   since   the   Gibbs   function   has   been  
fitted   to   laboratory   and   field   measurements   with   the   Absolute   Salinity   values   expressed   on  
this  scale.    Thus,  for  example,  it  is  possible  that  sometime  in  the  future  it  will  be  determined  
that  an  improved  estimate  of  the  mass  fraction  of  dissolved  material  in  Standard  Seawater  can  
be   obtained   by   multiplying   S R    by   a   factor   slightly   different   from   1   (uncertainties   permit  
values   in   the   range   1   ± 0.002).      We   emphasize   that   since   the   Gibbs   function   is   expressed   in  
terms  of  the  Absolute  Salinity  expressed  on  the  Reference-­‐‑Composition  Salinity  Scale,  use  of  
any   other   scale   (even   one   that   gives   more   accurate   estimates   of   the   true   mass   fraction   of  
dissolved   substances   in   Standard   Seawater)   will   reduce   the   accuracy   of   the   thermodynamic  
properties   determined   from   the   Gibbs   function.      In   part   for   this   reason,  we   recommend   that  
the  Reference-­‐‑Composition  Salinity  continue  to  be  measured  on  the  scale  defined  by  Millero  et  
al.  (2008a)  even  if  new  results  indicate  that  improved  estimates  of  the  true  mass  fraction  can  be  
obtained  using  a  modified  scale.    That  is,  we  recommend  that  the  value  of   uPS   used  in  (A.3.3)  
not  be  updated.    If  a  more  accurate  mass  fraction  estimate  is  required  for  some  purpose  in  the  
future,   such   a   revised   estimate   should   definitely   not   be   used   as   an   argument   of   the    
TEOS-­‐‑10  Gibbs  function.      
Finally,  we  note  a  second  reason  for  recommending  that  the  value  assigned  to   uPS   not  be  
modified   without   very   careful   consideration.      Under   TEOS-­‐‑10,   Absolute   Salinity   replaces  
Practical  Salinity  as  the  salinity  variable  in  publications,  and  it  is  critically  important  that  this  
new  measure  of  salinity  remain  stable  into  the  future.    In  particular,  we  note  that  any  change  
in  the  value  of   uPS   used  in  the  determination  of  Reference  Salinity  would  result  in  a  change  in  
reported  salinity  values  that  would  be  unrelated  to  any  real  physical  change.    For  example,  a  
change  in   uPS   from  35.16504/35  to  (35.16504/35)  x  1.001  for  example,  would  result  in  changes  
of  the  reported  salinity  values  of  order  0.035   g kg −1   which  is  more  than  ten  times  the  precision  
of  modern  salinometers.    Thus  changes  associated  with  a  series  of  improved  estimates  of   uPS   
(as  a  measure  of  the  mass  fraction  of  dissolved  salts  in  Standard  Seawater)  could  cause  very  
serious   confusion   for   researchers   who   monitor   salinity   as   an   indicator   of   climate   change.    
Based   on   this   concern,   and   the   fact   that   the   Gibbs   function   is   expressed   as   a   function   of  
Absolute  Salinity  measured  on  the  Reference-­‐‑Composition  Salinity  Scale  as  defined  by  Millero  
et  al.  (2008a),  no  changes  in  the  value  of   uPS   should  be  introduced.      
For   seawater   of   Reference   Composition,   Reference   Salinity   S R    is   the   best   available  
estimate   of   the   mass-­‐‑fraction   of   non-­‐‑H2O   material   in   seawater.      As   discussed   in   sections   2.4  
and   2.5,   under   TEOS-­‐‑10   S R    was   determined   to   provide   the   best   available   estimate   of   the  
mass-­‐‑fraction   of   non-­‐‑H2O   material   in   Standard   Seawater   by   Millero   et   al.   (2008a).    
Subsequently,   Pawlowicz   (2010a)   has   argued   that   the   DIC   content   of   the   Reference  
Composition   is   probably   about   117   µ mol kg −1    low   for   SSW   and   also   for   the   North   Atlantic  
surface  water  from  which  it  was  prepared.    This  difference  in  DIC  causes  a  negligible  effect  on  
both   conductivity   and   density,   and   hence   on   Reference   Salinity   and   Density   Salinity.      The  
influence   on   Solution   Salinity   is   nearly   a   factor   of   10   larger   (Pawlowicz   et   al.,   2011)   but   at  
0.0055   g kg −1    it   is   still   just   below   the   uncertainty   of   0.007   g kg −1    assigned   to   the   estimated  

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Absolute   Salinity   by   Millero   et   al.   (2008a).      In   fact,   the   largest   uncertainties   in   Reference  
Salinity  as  a  measure  of  the  Absolute  Salinity  of  SSW  are  associated  with  uncertainties  in  the  
mass   fractions   of   other   constituents   such   as   sulphate,   which   may   be   as   large   as   0.05   g kg −1   
(Seitz   et   al.,   2010).      Nevertheless,   it   seems   that   the   sulphate   value   of   Reference-­‐‑Composition  
Seawater  lies  within  the  95%  uncertainty  range  of  the  best  laboratory-­‐‑determined  estimates  of  
SSW’s  sulphate  concentration.      
   When   the   composition   of   seawater   differs   from   that   of   Standard   Seawater,   there   are  
several   possible   definitions   of   the   absolute   salinity   of   a   seawater   sample,   as   discussed   in  
section  2.5.    Conceptually  the  simplest  definition  is  “the  mass  fraction  of  dissolved  non-­‐‑ H2O   
material   in   a   seawater   sample   at   its   temperature   and   pressure”.      One   drawback   of   this  
definition   is   that   because   the   equilibrium   conditions   between   H2O    and   several   carbon  
compounds   depends   on   temperature   and   pressure,   this   mass-­‐‑fraction   would   change   as   the  
temperature  and  pressure  of  the  sample  is  changed,  even  without  the  addition  or  loss  of  any  
material  from  the  sample.    This  drawback  can  be  overcome  by  first  bringing  the  sample  to  the  
constant  temperature   t = 25°C   and  the  fixed  sea  pressure  0  dbar,  and  when  this  is  done,  the  
resulting   mass-­‐‑fraction   of   non-­‐‑ H2O    material   is   called   “Solution   Absolute   Salinity”   (usually  
shortened  to  “Solution  Salinity”),   S Asoln .    Another  measure  of  absolute  salinity  is  the  “Added-­‐‑
Mass   Salinity”   SAadd    which   is   S R    plus   the   mass   fraction   of   material   that   must   be   added   to  
Standard   Seawater   to   arrive   at   the   concentrations   of   all   the   species   in   the   given   seawater  
sample,  after  chemical  equilibrium  has  been  reached,  and  after  the  sample  has  been  brought  
to   t = 25°C   and   p =   0  dbar.      
   Another   form   of   absolute   salinity,   “Preformed   Absolute   Salinity”   (usually   shortened   to  
“Preformed   Salinity”),   S* ,      has   been   defined   by   Pawlowicz   et   al.   (2011)   and   Wright   et   al.  
(2011).      Preformed   Salinity   S*    is   designed   to   be   as   close   as   possible   to   being   a   conservative  
variable.    That  is,   S*   is  designed  to  be  insensitive  to  the  biogeochemical  processes  that  affect  
the  other  types  of  salinity  to  varying  degrees.     S*   is  formed  by  first  estimating  the  contribution  
of   biogeochemical   processes   to   one   of   the   salinity   measures   S A ,   S Asoln ,   or   SAadd ,   and   then  
subtracting  this  contribution  from  the  appropriate  salinity  variable.    Because  it  is  designed  to  
be  a  conservative  oceanographic  variable,   S*   will  find  a  prominent  role  in  ocean  modeling.      
Since   S*   is  designed  to  be  a  conservative  salinity  variable,  it  would  appear  to  also  be  the  
best  choice  for  the  salinity  variable  in  inverse  models.    An  argument  can  also  be  made  that   S*   
should  be  the  salinity  variable  that  is  used  as  an  axis  of  the  traditional  “ S − θ   diagram”,  which  
would  then  become  the   S* −Θ   diagram.    However,  this  argument  is  resisted  because  potential  
density   contours   cannot   be   drawn   on   the   S* −Θ    diagram   because   density   is   a   function   of  
Absolute  Salinity,  not  of  Preformed  Salinity.      
   There  are  no  simple  methods  available  to  measure  either   S Asoln   or   SAadd   for  the  general  case  
of   the   arbitrary   addition   of   many   components   to   Standard   Seawater.      Hence   a   more   precise  
and   easily   determined   measure   of   the   amount   of   dissolved   material   in   seawater   is   required,  
and   TEOS-­‐‑10   adopts   “Density   Salinity”   SAdens    for   this   purpose.      “Density   Salinity”   SAdens    is  
defined   as   the   value   of   the   salinity   argument   of   the   TEOS-­‐‑10   expression   for   density   which  
gives   the   sample’s   actual   measured   density   at   the   temperature   t = 25°C    and   at   the   sea  
pressure   p    =   0   dbar.      When   there   is   no   risk   of   confusion,   “Density   Salinity”   is   also   called  
Absolute   Salinity   with   the   label   S A ,   that   is   SA ≡ SAdens .      There   are   two   clear   advantages   of  
SA ≡ SAdens   over  both   S Asoln   and   SAadd .    First,  it  is  possible  to  measure  the  density  of  a  seawater  
sample  very  accurately  and  in  an  SI-­‐‑traceable  manner,  and  second,  the  use  of   SA ≡ SAdens   yields  
the   best   available   estimates   of   the   density   of   seawater.      This   is   important   because   amongst  
various   thermodynamic   properties   in   the   field   of   physical   oceanography,   it   is   density   that  
needs  to  be  known  to  the  highest  relative  accuracy.      
Pawlowicz  et  al.  (2011)  and  Wright  et  al.  (2011)  found  that  while  the  nature  of  the  ocean’s  
composition   variations   changes   from   one   ocean   basin   to   another,   the   five   different   salinity  
measures   S R ,   SAdens ,   S Asoln ,   SAadd    and   S*    are   approximately   related   by   the   following   simple  

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linear  relationships,  (obtained  by  combining  equations  (55)  –  (57)  and  (62)  of  Pawlowicz  et  al.  
(2011))    

S∗ − SR ≈ − 0.35 δ SA ,  

Eqn.  

(A.4.2)  

is  

(A.4.1)  

SAdens − SR ≡ 1.0 δ SA ,  

(A.4.2)  

SAsoln − SR ≈ 1.75 δ SA ,  

(A.4.3)  

   SAadd − SR ≈ 0.78 δ SA .  

(A.4.4)  

simply  

the  

definition  

of  

the  

Absolute  

Salinity  

Anomaly,  

δ SA ≡ δ SRdens ≡ SAdens − SR .      Note   that   here   and   in   many   TEOS-­‐‑10   publications,   the   simpler  
notation   δ SA   is  used  for   δ SRdens ≡ SAdens − SR ,  a  salinity  difference  that  can  now  be  estimated  

from  a  global  atlas  (McDougall  et  al.  (2012)).      
In   the   context   of   ocean   modelling,   it   is   more   convenient   to   cast   these   salinity   differences  
with  respect  to  the  Preformed  Salinity   S∗   as  follows  (using  the  above  equations)    

SR − S∗ ≈ 0.35 δ SA ,  

(A.4.5)  

SAdens − S∗ ≈ 1.35 δ SA ,  

(A.4.6)  

     SAsoln − S* ≈ 2.1 δ SA ,  

(A.4.7)  

   SAadd − S∗ ≈ 1.13 δ SA .  

(A.4.8)  

For   SSW,   all   five   salinity   variables   S R ,   SA ≡ SAdens ,   S Asoln ,   SAadd    and   S*    are   equal.      The  
relationships  (A.4.1),  (A.4.2),  (A.4.5)  and  (A.4.6)  are  illustrated  on  the  number  line  of  salinity  
in  Figure  A.4.1.    It  should  be  noted  that  the  simple  relationships  of  Eqns.  (A.4.1)  –  (A.4.8)  are  
derived   from   simple   linear   fits   to   model   calculations   that   show   more   complex   variations.    
However,  the  variation  about  these  relationships  is  not  larger  than  the  typical  uncertainty  of  
ocean  measurements.    Eqn.  (A.4.6)  provides  a  way  by  which  the  effects  of  anomalous  seawater  
composition  may  be  addressed  in  ocean  models  (see  appendix  A.20).      
  
  

  

  

Figure  A.4.1.    Number  line  of  salinity,  illustrating  the  differences  between    
                                                  Preformed  Salinity   S* ,  Reference  Salinity   S R ,  and  Absolute    
                                                  Salinity   S A   for  seawater  whose  composition  differs  from  that    
                                                  of  Standard  Seawater.          
  
If  measurements  are  available  of  the  Total  Alkalinity,  Dissolved  Inorganic  Carbon,  and  the  
nitrate  and  silicate  concentrations,  but  not  of  density  anomalies,  then  alternative  formulae  are  
available   for   the   salinity   differences   that   appear   on   the   left-­‐‑hand   sides   of   Eqns.   (A.4.1)   –  
(A.4.8).      Pawlowicz   et   al.   (2011)   have   used   a   chemical   model   of   conductivity   and   density   to  
estimate   how   the   many   salinity   differences   introduced   above   depend   on   the   measured  
properties  of  seawater.    The  following  equations  correspond  to  Eqns.  (A.4.1)  –  (A.4.4)  above,  
and  come  from  equations  (51)  –  (54)  and  (59)  of  Pawlowicz  et  al.  (2011).    These  equations  are  
written  in  terms  of  the  values  of  the  nitrate  and  silicate  concentrations  in  the  seawater  sample  
(measured   in   mol kg −1 ),   the   difference   between   the   Total   Alkalinity   ( TA )   and   Dissolved  
Inorganic  Carbon  ( DIC )  of  the  sample  and  the  corresponding  values  of  our  best  estimates  of  
TA    and   DIC    in   Standard   Seawater,   ΔTA    and   ΔDIC ,   both   measured   in   mol kg −1 .      For  
Standard   Seawater   our   best   estimates   of   TA   and   DIC   are   0.0023 ( SP 35)    mol kg −1    and  

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81

0.00208 ( SP 35)    mol kg −1   respectively  (see  Pawlowicz  (2010a),  Pawlowicz  et  al.  (2011)  and  the  

discussion  of  this  aspect  of  SSW  versus  RCSW  in  Wright  et  al.  (2011))).    

(

)

(mol kg −1 ) ,          (A.4.9)  

)

(

)

(mol kg −1 ) ,      (A.4.10)  

)

(

)

(mol kg−1 ) ,      (A.4.11)  

)

(

)

(mol kg−1 ) .        (A.4.12)  

( S* − SR ) / (g kg−1 )

(S

dens
A

(S

− SR / (g kg −1 ) = 55.6 ΔTA + 4.7 ΔDIC+38.9 NO3− + 50.7 Si(OH)4

soln
A

(S

= −18.1 ΔTA − 7.1 ΔDIC − 43.0 NO3− + 0.1 Si(OH)4

add
A

− SR / (g kg −1 ) = 7.2 ΔTA + 47.0 ΔDIC+36.5 NO3− + 96.0 Si(OH)4
− SR / (g kg −1 ) = 25.9 ΔTA + 4.9 ΔDIC+16.1NO3− + 60.2 Si(OH) 4

The  standard  error  of  the  model  fits  in  Eqns.  (A.4.9)  –  (A.4.11)  are  given  by  Pawlowicz  et  al.  
(2011)   at   less   than   10−4 kg m −3    (in   terms   of   density)   which   is   equivalent   to   a   factor   of   20  
smaller  than  the  accuracy  to  which  Practical  Salinity  can  be  measured  at  sea.    It  is  clear  that  if  
measurements   of   TA,   DIC,   nitrate   and   silicate   are   available   (and   recognizing   that   these  
measurements   will   come   with   their   own   error   bars),   these   expressions   will   likely   give   more  
accurate   estimates   of   the   salinity   differences   than   the   approximate   linear   expressions  
presented  in  Eqns.  (A.4.1)  –  (A.4.8).    The  coefficients  in  Eqn.  (A.4.10)  are  reasonably  similar  to  
the   corresponding   expression   of   Brewer   and   Bradshaw   (1975)   (as   corrected   by   Millero   et   al.  
(1976a)):-­‐‑   when  expressed  as  the  salinity  anomaly   SAdens − SR   rather  than  as  the  corresponding  
density  anomaly   ρ − ρ R ,  their  expression  corresponding  to  Eqn.  (A.4.10)  had  the  coefficients  
71.4,  -­‐‑12.8,  31.9  and  59.9  compared  with  the  coefficients  55.6,  4.7,  38.9  and  50.7  respectively  in  
Eqn.  (A.4.10).      
The  salinity  differences  expressed  with  respect  to  Preformed  Salinity   S*   which  correspond  
to   Eqns.   (A.4.5)   –   (A.4.8)   can   be   found   by   linear   combinations   of   Eqns.   (A.4.9)   –   (A.4.12)   as  
follows      

( SR − S* ) / (g kg−1 )

(S
(S

(

= 18.1 ΔTA + 7.1 ΔDIC + 43.0 NO3− − 0.1 Si(OH)4

)

(mol kg −1 ) ,              (A.4.13)  

)

(

)

(mol kg −1 ) ,    (A.4.14)  

)

(

)

(mol kg−1 ) ,      (A.4.15)  

)

(

)

(mol kg−1 ) .      (A.4.16)  

dens
A

− S* / (g kg −1 ) = 73.7 ΔTA + 11.8 ΔDIC+81.9 NO3− + 50.6 Si(OH)4

soln
A

− S* / (g kg −1 ) = 25.3 ΔTA + 54.1 ΔDIC+79.5 NO3− + 95.9 Si(OH)4

(S

add
A

− S* / (g kg −1 ) = 44.0 ΔTA + 12.0 ΔDIC+59.1NO3− + 60.1 Si(OH)4

  
  
  

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A.5 Spatial variations in seawater composition
When  the  oceanographic  data  needed  to  evaluate  Eqn.  (A.4.10)  for   SAdens − SR ≡ SA − SR   is  
not  available,  the  look-­‐‑up  table  method  of  McDougall  et  al.  (2012)  is  recommended  to  evaluate  
δ SA ≡ δ SRdens ≡ SA − SR .    The  following  describes  how  this  method  was  developed.      
In   a   series   of   papers   Millero   et   al.   (1976a,   1978,   2000,   2008b)   and   McDougall   et   al.   (2012)  
have   reported   on   density   measurements   made   in   the   laboratory   on   samples   collected   from  
around  the  world’s  oceans.    Each  sample  had  its  Practical  Salinity  measured  in  the  laboratory  
as   well   as   its   density   (measured   with   a   vibrating   tube   densimeter   at   25   °C   and   atmospheric  
pressure).      The   Practical   Salinity   yields   a   Reference   Salinity   S R    according   to   Eqn.   (A.3.3),  
while   the   density   measurement   ρ meas    implies   an   Absolute   Salinity   SA ≡ SAdens    by   using   the  
equation   of   state   and   the   equality   ρ meas = ρ SAdens , 25 °C, 0dbar .      The   difference   SAdens − SR   
between  these  two  salinity  measures  is  taken  to  be  due  to  the  composition  of  the  sample  being  
different   to   that   of   Standard   Seawater.      In   these   papers   Millero   established   that   the   salinity  
difference   SA − SR    could   be   estimated   approximately   from   knowledge   of   just   the   silicate  
concentration   of   the   fluid   sample.      The   reason   for   the   explaining   power   of   silicate   alone   is  
thought  to  be  that  (a)  it  is  itself  substantially  correlated  with  other  relevant  variables  (e.g.  total  
alkalinity,  nitrate  concentration,  DIC  [often  called  total  carbon  dioxide]),  (b)  it  accounts  for  a  
substantial  fraction  (about  0.6)  of  the  typical  variations  in  concentrations   (g kg −1 )   of  the  above  
species  and  (c)  being  essentially  non-­‐‑ionic;  its  presence  has  little  effect  on  conductivity  while  
having  a  direct  effect  on  density.        
When  the  existing   δ SA   data,  based  on  laboratory  measurements  of  density,  was  regressed  
against   the   silicate   concentration   of   the   seawater   samples,   McDougall   et   al.   (2012)   found   the  
simple  relation    

(

)

(

)

δ SA / (g kg −1 ) = ( SA − SR ) / (g kg −1 ) = 98.24 Si(OH)4 / (mol kg −1 ) .  

Global  (A.5.1)  

This  regression  was  done  over  all  available  density  measurements  from  the  world  ocean,  and  
the  standard  error  of  the  fit  was  0.0054 g kg −1.          
  
The  dependence  of   δ SA   on  silicate  concentration  is  observed  to  be  different  in  each  ocean  
basin,   and   this   aspect   was   exploited   by   McDougall   et   al.   (2012)   to   obtain   a   more   accurate  
dependence   of   δ SA    on   location   in   space.      For   data   in   the   Southern   Ocean   south   of   30oS   the  
best  simple  fit  was  found  to  be    

(

)

δ SA / (g kg −1 ) = 74.884 Si(OH)4 / (mol kg −1 ) ,  

Southern  Ocean  (A.5.2)  

and  the  associated  standard  error  is  0.0026 g kg −1 .      
  
The   data   north   of   30oS   in   each   of   the   Pacific,   Indian   and   Atlantic   Oceans   was   treated  
separately.    In  each  of  these  three  regions  the  fit  was  constrained  to  match  (A.5.2)  at  30oS  and  
the   slope   of   the   fit   was   allowed   to   vary   linearly   with   latitude.      The   resulting   fits   were   (for  
latitudes  north  of  30oS,  that  is  for   λ ≥ − 30° )    

(
)
) = 74.884 (1+ 0.3861[λ / 30°+1]) (Si(OH) / (mol kg ) ) ,  
) = 74.884 (1 +1.0028 [λ / 30°+1]) (Si(OH) / (mol kg ) ) .  

δ SA / (g kg −1 ) = 74.884 (1+ 0.3622 [λ / 30°+1]) Si(OH)4 / (mol kg −1 ) ,  

Pacific  (A.5.3)  

δ SA / (g kg −1

−1

4

Indian  (A.5.4)  

δ SA / (g kg −1

−1

4

Atlantic  (A.5.5)  

These   relationships   between   the   Absolute   Salinity   Anomaly   δ SA = SA − SR    and   silicate  
concentration  have  been  used  by  McDougall  et  al.  (2012)  in  a  computer  algorithm  that  uses  an  
existing   global   data   base   of   silicate   (Gouretski   and   Koltermann   (2004))   and   provides   an  
estimate  of  Absolute  Salinity  when  given  a  seawater  sample’s  Practical  Salinity  as  well  as  its  
spatial  location  in  the  world  ocean.      
Since   version   3.0,   this   computer   algorithm   works   as   follows.      The   values   of   both   the  
Reference   Salinity   and   the   Absolute   Salinity   Anomaly,   calculated   from   the   global   Gouretski  
and  Koltermann  (2004)  hydrographic  atlas  using  Eqns.  (A.5.2)  –  (A.5.5),  were  used  to  form  the  

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83

ratio   Rδ ≡ δ SAatlas S Ratlas    of   these   atlas   values   of   Absolute   Salinity   Anomaly   and   Reference  
Salinity.    These  values  of  the  Absolute  Salinity  Anomaly  Ratio,   Rδ ,  were  stored  as  a  function  
of  latitude,  longitude  and  pressure  on  a  regular   4° × 4°   grid  in  latitude  and  longitude.    These  
values   of   Rδ    are   interpolated   onto   the   latitude,   longitude   and   pressure   of   an   oceanographic  
observation  (the  details  of  the  interpolation  method  can  be  found  in  McDougall  et  al.  (2012))  
and  the  Absolute  Salinity  Anomaly   δ SA   of  an  oceanographic  observation  is  calculated  from    

δ SA = Rδ SR             where           Rδ ≡ δ SAatlas S Ratlas ,  

(A.5.6)  

where   S R    is   the   Reference   Salinity   of   the   oceanographic   observation.      For   the   bulk   of   the  
ocean  this  expression  for   δ SA   is  almost  the  same  as  simply  setting   δ SA   equal  to   δ SAatlas ,  but  
the   use   of   Eqn.   (A.5.6)   is   preferable   in   situations   where   the   sample’s   Reference   Salinity   is  
small,   such   as   in   rivers,   in   estuaries   and   after   a   rain   shower   at   the   sea   surface   in   the   open  
ocean.      In   these   situations   the   influence   of   the   ocean’s   biogeochemical   processes   on   δ SA   
should  approach  zero  as   S R   approaches  zero,  and  this  is  achieved  by  Eqn.  (A.5.6).      
Where  the  nutrient  and  carbon  chemistry  data  are  available  to  evaluate  Eqn.  (A.4.10),  the  
results   obtained   are   similar   although   not   identical   to   those   obtained   from   Eqn.   (A.5.6)   using  
the  McDougall  et  al.  (2012)  algorithm.      
The  relationships  between  the  three  salinity  variables   SA , S*   and   S R   are  found  as  follows.    
First  we  note  the  relationships  between  these  salinities  (from  Eqns.  (A.4.2),  (A.4.1)  and  (A.4.6))    

SA = SR + δ SA ,  

(A.5.7)  

S* = SR − r1δ SA ,      

(A.5.8)  

SA = S* + (1+ r1 )δ SA .  

(A.5.9)  

Substituting   Eqn.   (A.5.6)   into   these   equations   gives   the   following   simple   linear   relationships  
between  the  three  different  salinities,    

(
)
(1 − r Rδ ) ,    
(1 + R ) = S 1+ F       where       F
( )
(1 − r R )

SA = SR 1 + Rδ ,  

(A.5.10)  

S* = SR

(A.5.11)  

1

δ

SA = S*

δ

δ

δ

*

1

=

[1+ r1 ] Rδ

(1 − r Rδ )

.  

(A.5.12)  

1

These   three   equations   are   used   in   the   six   functions   in   the   GSW   Oceanographic   Toolbox   that  
relate  one  salinity  variable  to  another,  where   r1   is  taken  to  be  0.35  while   Rδ   is  obtained  from  
the  look-­‐‑up  table  of  McDougall  et  al.  (2012).      
This   approach   has   so   far   assumed   that   the   Absolute   Salinity   Anomaly   in   fresh   water   is  
zero.    This  is  usually  a  good  assumption  for  rainwater,  but  is  often  not  true  of  water  in  rivers.    
For   example,   the   river   water   flowing   into   the   Baltic   has   an   Absolute   Salinity   Anomaly   of  
approximately   0.087 g kg −1 .    When  one  has  knowledge  of  the  Absolute  Salinity  Anomaly  due  
to  river  water  inflow,  this  can  be  incorporated  as  follows    

δ SA = Rδ SR + δ SAriver ,  
leading  to  (using  Eqn.  (A.5.7))    

(

SA = SR 1 + Rδ

) + δS

river
A

(A.5.13)  
.  

(A.5.14)  

In  turn,  an  estimate  for   δ SAriver   might  be  constructed  in  the  vicinity  of  a  particular  river  from  
prior   knowledge   of   the   Absolute   Salinity   Anomaly   at   the   river   mouth   δ SAriver_mouth    (this   is  

actually  the  Absolute  Salinity  Anomaly  appropriate  for  river  water  extrapolated  to   S R = 0 )  by  
a  formula  such  as  (drawing  inspiration  from  the  formula  for  the  Baltic,  see  below)    

(

)

δ SAriver = δ SAriver_mouth 1 − S R S Ratlas .  

(A.5.15)  

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The  computer  algorithm  of  McDougall  et  al.  (2012)  accounts  for  the  latest  understanding  
of   Absolute   Salinity   in   the   Baltic   Sea,   but   it   is   silent   on   the   influence   of   compositional  
variations  in  other  marginal  seas.    The  Absolute  Salinity  Anomaly  in  the  Baltic  Sea  has  been  
quite  variable  over  the  past  few  decades  of  observation  (Feistel  et  al.  (2010c)).    The  computer  
algorithm   of   McDougall   et   al.   (2012)   uses   the   relationship   found   by   Feistel   et   al.   (2010c)   that  
applies  in  the  years  2006-­‐‑2009,  namely    

SA − SR = δ SA = 0.087 g kg −1 × (1 − SR SSO ) ,  

Baltic  (A.5.16)  

where   SSO   =  35.165  04  g  kg–1  is  the  standard-­‐‑ocean  Reference  Salinity  that  corresponds  to  the  
Practical   Salinity   of   35.      The   Absolute   Salinity   Anomaly   in   the   Baltic   Sea   is   not   due   to  
biogeochemical   activity,   but   rather   is   due   to   the   rivers   bringing   material   of   anomalous  
composition   into   the   Baltic.      Hence   Absolute   Salinity   in   the   Baltic   is   a   conservative   variable  
and  Preformed  Salinity  is  defined  to  be  equal  to  Absolute  Salinity  in  the  Baltic.    That  is,  in  the  
Baltic S* = SA ,  implying  that   r1 = − 1   and   F δ = 0   (see  Eqns.  (A.5.7)  –  (A.5.9)  and  (A.5.12)).      
In   order   to   gauge   the   importance   of   the   spatial   variation   of   seawater   composition,   the  
northward   gradient   of   density   at   constant   pressure   is   shown   in   Fig.   A.5.1   for   the   data   in   a  
world   ocean   hydrographic   atlas   deeper   than   1000m.      The   vertical   axis   in   this   figure   is   the  
magnitude   of   the   difference   between   the   northward   density   gradient   at   constant   pressure  
when  the  TEOS-­‐‑10  algorithm  for  density  is  called  with   SA ≡ SAdens   (as  it  should  be)  compared  
with   calling   the   same   TEOS-­‐‑10   density   algorithm   with   SR    as   the   salinity   argument.      Figure  
A.5.1  shows  that  the  “thermal  wind”  is  misestimated  by  more  than  2%  for  58%  of  the  data  in  
the  world  ocean  below  a  depth  of  1000m  if  the  effects  of  the  variable  seawater  composition  are  
ignored.    When  this  same  comparison  is  done  for  only  the  North  Pacific,  it  is  found  that  60%  
of  the  data  deeper  than  1000m  has  “thermal  wind”  misestimated  by  more  than  10%  if   S R   is  
used  in  place  of   S A .      

  
Figure  A.5.1.    The  northward  density  gradient  at  constant  pressure  (the  horizontal  axis)    
                                                  for  data  in  the  global  ocean  atlas  of  Gouretski  and  Koltermann  (2004)  for    
                                                 p > 1000 dbar.    The  vertical  axis  is  the  magnitude  of  the  difference    
                                                  between  evaluating  the  density  gradient  using   S A   versus   S R   as  the    
                                                  salinity  argument  in  the  TEOS-­‐‑10  expression  for  density.      
  
The  importance  of  the  spatial  variations  in  seawater  composition  illustrated  in  Fig.  A.5.1  
can   be   compared   with   the   corresponding   improvement   achieved   by   the   TEOS-­‐‑10   Gibbs  

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function  for  Standard  Seawater  compared  with  using  EOS-­‐‑80.    This  is  done  by  ignoring  spatial  
variations  in  seawater  composition  in  both  the  evaluation  of  TEOS-­‐‑10  and  in  EOS80  by  calling  
TEOS-­‐‑10   with   S R    and   EOS-­‐‑80   with   SP .      Figure   A.5.2   shows   the   magnitude   of   the  
improvement   in   the   “thermal   wind”   in   the   part   of   the   ocean   that   is   deeper   than   1000m   
through  the  adoption  of  TEOS-­‐‑10  but  ignoring  the  influence  of  compositional  variations.    By  
comparing   Figs.   A.5.1   and   A.5.2   it   is   seen   that   the   main   benefit   that   TEOS-­‐‑10   delivers   to   the  
evaluation  of  the  “thermal  wind”  is  through  the  incorporation  of  spatial  variations  in  seawater  
composition;  the  greater  accuracy  of  TEOS-­‐‑10  over  EOS-­‐‑80  for  Standard  Seawater  is  only  17%  
as   large   as   the   improvement   gained   by   the   incorporation   of   compositional   variations   into  
TEOS-­‐‑10  (i.  e.  the  rms  value  of  the  vertical  axis  in  Fig.  A.5.2  is  17%  of  that  of  the  vertical  axis  of  
Fig.  A.5.1).    If  the  North  Atlantic  were  excluded  from  this  comparison,  the  relative  importance  
of  compositional  variations  would  be  even  larger.      
  

  

  
  
Figure  A.5.2.    The  northward  density  gradient  at  constant  pressure  (the  horizontal  axis)  
                                                  for  data  in  the  global  ocean  atlas  of  Gouretski  and  Koltermann  (2004)  for    
                                                 p > 1000 dbar .    The  vertical  axis  is  the  magnitude  of  the  difference    
                                                  between  evaluating  the  density  gradient  using   S R   as  the  salinity    
                                                  argument  in  the  TEOS-­‐‑10  expression  for  density  compared  with  using   SP     
                                                  in  the  EOS-­‐‑80  algorithm  for  density.      
The   thermodynamic   description   of   seawater   and   of   ice   Ih   as   defined   in   IAPWS-­‐‑08   and  
IAPWS-­‐‑06   has   been   adopted   as   the   official   description   of   seawater   and   of   ice   Ih   by   the  
Intergovernmental   Oceanographic   Commission   in   June   2009.      The   adoption   of   TEOS-­‐‑10   has  
recognized   that   this   technique   of   estimating   Absolute   Salinity   from   readily   measured  
quantities   is   perhaps   the   least   mature   aspect   of   the   TEOS-­‐‑10   thermodynamic   description   of  
seawater.      The   present   computer   software,   in   both   FORTRAN   and   MATLAB,   which   evaluates  
Absolute   Salinity   S A    given   the   input   variables   Practical   Salinity S P ,   longitude   λ ,   latitude   φ   
and  sea  pressure   p   is  available  at  www.TEOS-10.org. It  is  expected,  as  new  data  (particularly  
density  data)  become  available,  that  the  determination  of  Absolute  Salinity  will  improve  over  
the  coming  decades,  and  the  algorithm  for  evaluating  Absolute  Salinity  in  terms  of  Practical  
Salinity,   latitude,   longitude   and   pressure,   will   be   updated   from   time   to   time,   after   relevant  
appropriately  peer-­‐‑reviewed  publications  have  appeared,  and  such  an  updated  algorithm  will  
appear   on   the   www.TEOS-10.org web   site.      Users   of   this   software   should   state   in   their  
published  work  which  version  of  the  software  was  used  to  calculate  Absolute  Salinity.    

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A.6 Gibbs function of seawater
The  Gibbs  function  of  seawater   g ( SA , t , p )   is  defined  as  the  sum  of  the  Gibbs  function  for  pure  
water   g W ( t , p )   and  the  saline  part  of  the  Gibbs  function   g S ( S A , t , p )   so  that      

g ( SA , t , p ) = g W ( t , p ) + g S ( S A , t , p ) .   

(A.6.1)  

In  this  way  at  zero  Absolute  Salinity,  the  thermodynamic  properties  of  seawater  are  equal  to  
those  of  pure  water.    This  consistency  is  also  maintained  with  respect  to  the  Gibbs  function  for  
ice  so  that  the  properties  along  the  equilibrium  curve  can  be  accurately  determined  (such  as  
the   freezing   temperature   as   a   function   of   Absolute   Salinity   and   pressure).      The   careful  
alignment  of  the  thermodynamic  potentials  of  pure  water,  ice  Ih  and  seawater  is  described  in  
Feistel  et  al.  (2008a).      
The   internationally   accepted   thermodynamic   description   of   the   properties   of   pure   water  
(IAPWS-­‐‑95)  is  the  official  pure-­‐‑water  basis  upon  which  the  Gibbs  function  of  seawater  is  built  
according   to   (A.6.1).      This   g W ( t , p )    Gibbs   function   of   liquid   water   is   valid   over   extended  
ranges  of  temperature  and  pressure  from  the  freezing  point  to  the  critical  point  (–22  °C  <  t  <  
374  °C  and    600  Pa  <  p  +  P0  <  1000  MPa)  however  it  is  a  computationally  expensive  algorithm.  
Part   of   the   reason   for   this   computational   intensity   is   that   the   IAPWS-­‐‑95   formulation   is   in  
terms   of   a   Helmholtz   function   which   has   the   pressure   as   a   function   of   temperature   and  
density,  so  that  an  iterative  procedure  is  needed  to  form  the  Gibbs  function   g W ( t , p )   (see  for  
example,  Feistel  et  al.  (2008a)).      
For  practical  oceanographic  use  in  the  oceanographic  ranges  of  temperature  and  pressure,  
from  less  than  the  freezing  temperature  of  seawater  (at  any  pressure),  up  to   40 °C   (specifically  
from   − ⎡2.65 + ( p + P0 ) × 0.0743 MPa −1 ⎤ °C   to  40  °C),  and  in  the  pressure  range   0 < p < 104 dbar   
⎣
⎦
we   also   recommend   the   use   of   the   pure   water   part   of   the   Gibbs   function   of   Feistel   (2003)  
which  has  been  approved  by  IAPWS  as  the  Supplementary  Release,  IAPWS-­‐‑09.    The  IAPWS-­‐‑
09  release  discusses  the  accuracy  to  which  the  Feistel  (2003)  Gibbs  function  fits  the  underlying  
thermodynamic   potential   of   IAPWS-­‐‑95;   in   summary,   for   the   variables   density,   thermal  
expansion   coefficient   and   specific   heat   capacity,   the   rms   misfit   between   IAPWS-­‐‑09   and  
IAPWS-­‐‑95,  in  the  region  of  validity  of  IAPWS-­‐‑09,  are  a  factor  of  between  20  and  100  less  than  
the  corresponding  error  in  the  laboratory  data  to  which  IAPWS-­‐‑95  was  fitted.    Hence,  in  the  
oceanographic   range   of   parameters,   IAPWS-­‐‑09   and   IAPWS-­‐‑95   may   be   regarded   as   equally  
accurate  thermodynamic  descriptions  of  pure  liquid  water.      
Many  of  the  thermodynamic  properties  of  seawater  that  are  described  in  this  Manual  are  
available   as   both   FORTRAN   and   MATLAB   implementations.      These   implementations   are  
available  for   g W ( t , p )   being  IAPWS-­‐‑95  and  IAPWS-­‐‑09,  both  being  equally  accurate  relative  to  
the  laboratory-­‐‑determined  known  properties,  but  with  the  computer  code  based  on  IAPWS-­‐‑09  
being  approximately  a  factor  of  65  faster  than  that  based  on  IAPWS-­‐‑95.      
Most   of   the   experimental   seawater   data   that   were   already   used   for   the   construction   of  
EOS-­‐‑80   were   exploited   again   for   the   IAPWS-­‐‑08   formulation   after   their   careful   adjustment   to  
the   new   temperature   and   salinity   scales   and   the   improved   pure-­‐‑water   reference   IAPWS-­‐‑95.    
Additionally,  IAPWS-­‐‑08  was  significantly  improved  (compared  with  EOS-­‐‑80)  by  making  use  
of   theoretical   relations   such   as   the   ideal-­‐‑solution   law   and   the   Debye-­‐‑Hückel   limiting   law,   as  
well   as   by   incorporating   additional   accurate   measurements   such   as   the   temperatures   of  
maximum   density,   vapour   pressures   and   mixing   heats,   and   implicitly   by   the   enormous  
background   data   set   which   underlies   IAPWS-­‐‑95   (Wagner   and   Pruß   (2002),   Feistel   (2003,  
2008)).    For  example,  Millero  and  Li  (1994)  concluded  that  the  pure-­‐‑water  part  of  the  EOS-­‐‑80  
sound-­‐‑speed  formula  of  Chen  and  Millero  (1977)  was  responsible  for  a  deviation  of  0.5   m s−1   
from  Del  Grosso’s  (1974)  formula  for  seawater  at  high  pressures  and  temperature  below  5   oC.    
Chen  and  Millero  (1977)  only  measured  the  differences  in  the  sound  speeds  of  seawater  and  
pure   water.   The   new   Gibbs   function   in   which   we   use   IAPWS-­‐‑95   for   the   pure-­‐‑water   part   as  
well  as  sound  speeds  from  Del  Grosso  (1974),  is  perfectly  consistent  with  Chen  and  Millero’s  

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(1976)   densities   and   Bradshaw   and   Schleicher’s   (1970)   thermal   expansion   data   at   high  
pressures.      The   accuracy   of   high-­‐‑pressure   seawater   densities   has   increased   with   the   use   of  
IAPWS-­‐‑95,   directly   as   the   pure-­‐‑water   part,   and   indirectly   by   correcting   earlier   seawater  
measurements,   making   them   "ʺnew"ʺ   seawater   data.      In   this   manner   the   known   sound-­‐‑speed  
inconsistency  of  EOS-­‐‑80  has  been  resolved  in  a  natural  manner.      
  
  

A.7 The fundamental thermodynamic relation
The   fundamental   thermodynamic   relation   for   a   system   composed   of   a   solvent   (water)   and   a  
solute  (sea  salt)  relates  the  total  differentials  of  thermodynamic  quantities  for  the  case  where  
the   transitions   between   equilibrium   states   are   reversible.      This   restriction   is   satisfied   for  
infinitesimally   small   changes   of   an   infinitesimally   small   seawater   parcel.      The   fundamental  
thermodynamic  relation  is    

dh − v dP = (T0 + t ) dη + µ dSA .  

(A.7.1)  

A  derivation  of  the  fundamental  thermodynamic  relation  can  be  found  in  Warren  (2006)  (his  
equation   (8)).      The   left-­‐‑hand   side   of   Eqn.   (A.7.1)   is   often   written   as   du + ( p + P0 ) dv    where  
( p + P0 ) = P   is  the  Absolute  Pressure.    Here   h   is  the  specific  enthalpy  (i.e.  enthalpy  per  unit  
mass  of  seawater),   u   is  the  specific  internal  energy,   v = ρ −1   is  the  specific  volume,   (T0 + t ) = T   
is  the  absolute  temperature,   η   is  the  specific  entropy  and   µ   is  the  relative  chemical  potential.    
In  fluid  dynamics  we  usually  deal  with  material  derivatives,   d dt ,  that  is,  derivatives  defined  
following  the  fluid  motion,   d dt = ∂ ∂t + u ⋅ ∇   where   u   is  the  fluid  velocity.    In  terms  of  this  
type   of   derivative,   and   assuming   local   thermodynamic   equilibrium   (i.   e.   that   local  
thermodynamic   equilibrium   is   maintained   during   the   temporal   change),   the   fundamental  
thermodynamic  relation  is    

dh 1 dP
dη
dS
−
= (T0 + t )
+ µ A .   
dt ρ dt
dt
dt

(A.7.2)  

Note   that   the   constancy   of   entropy   in   a   given   situation   does   not   imply   the   absence   of  
irreversible  processes  because,  for  example,  there  can  be  irreversible  changes  of  both  salinity  
and   enthalpy   at   constant   pressure   in   just   the   right   ratio   so   as   to   have   equal   effects   in   Eqns.  
(A.7.1)  or  (A.7.2)  so  that  the  change  of  entropy  in  these  equations  is  zero.      
  
  

A.8 The “conservative” and “isobaric conservative” properties
A  thermodynamic  variable   C   is  said  to  be  “conservative”  if  its  evolution  equation  (that  is,  its  
prognostic  equation)  has  the  form    

dC
(A.8.1)  
= − ∇ ⋅ FC .   
dt
For  such  a  “conservative”  property,  in  the  absence  of  fluxes   FC   at  the  boundary  of  a  control  
volume,  the  total  amount  of  C-­‐‑stuff  is  constant  inside  the  control  volume.    The  middle  part  of  
Eqn.   (A.8.1)   has   used   the   continuity   equation   (which   is   the   equation   for   the   conservation   of  
mass)    
(A.8.2)  
∂ρ ∂t x , y , z + ∇ ⋅ ( ρ u ) = 0 .   

( ρ C )t + ∇ ⋅ ( ρ uC )

= ρ

In  the  special  case  when  the  material  derivative  of  a  property  is  zero  (that  is,  the  middle  part  
of  Eqn.  (A.8.1)  is  zero)  the  property  is  said  to  be  “materially  conserved”.      
The   only   quantity   that   can   be   regarded   as   100%   conservative   in   the   ocean   is   mass  
[equivalent  to  taking   C = 1   and   FC = 0   in  Eqn.  (A.8.1)].    In  fact,  looking  ahead  to  appendices  
A.20  and  A.21,  if  we  strictly  interpret   ρ u   as  the  mass  flux  per  unit  area  of  pure  seawater  (i.  e.  
of   only   pure   water   plus   dissolved   material)   and   specifically,   that   ρ u    excludes   the   flux   of  

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particulate  matter,  then  the  right-­‐‑hand  side  of  the  continuity  equation  (A.8.2)  should  be   ρ S A ,  
the   non-­‐‑conservative   source   of   mass   due   to   biogeochemical   processes.      It   can   be   shown   that  
S
the   influence   of   this  source   term   ρ S A   in   the   continuity   equation  on   the   evolution   equation  
for   Absolute   Salinity   is   less   important   by   the   factor   SˆA 1 − SˆA    than   the   same   source   term  
that  appears  in  this  evolution  equation  for  Absolute  Salinity,  Eqn.  (A.21.8).    Hence  the  current  
practice   of   assuming   that   the   non-­‐‑particulate   part   of   the   ocean   obeys   the   conservative   form  
(A.8.2)   of   the   continuity   equation   is   confirmed   even   in   the   presence   of   biogeochemical  
processes.      
Two   other   variables,   total   energy   E = u + 0.5 u ⋅ u + Φ    (see   Eqn.   (B.15))   and   Conservative  
Temperature   Θ   (or  equivalently,  potential  enthalpy   h 0 )  are  not  completely  conservative,  but  
the   error   in   assuming   them   to   be   conservative   is   negligible   (see   appendix   A.21).      Other  
variables   such   as   Reference   Salinity   SR ,    Absolute   Salinity   SA ,    potential   temperature   θ ,   
enthalpy   h,   internal  energy   u,   entropy   η ,   density   ρ ,   potential  density   ρ θ ,   specific  volume  
anomaly   δ    and   the   Bernoulli   function   B = h + 0.5 u ⋅ u +Φ    (see   Eqn.   (B.17))   are   not  
conservative  variables.      
While  both  Absolute  Salinity  and  Reference  Salinity  are  conservative  under  the  turbulent  
mixing   process,   both   are   affected   in   a   non-­‐‑conservative   way   by   biogeochemical   process.    
Because  the  dominant  variations  of  the  composition  of  seawater  are  due  to  species  which  do  
not  have  a  strong  signature  in  conductivity,  in  some  situations  it  may  be  sufficiently  accurate  
to  take  Reference  Salinity   S R   to  be  a  conservative  variable.    However,  we  note  that  the  error  
involved   with   assuming   that   S R    is   a   conservative   variable   is   a   factor   of   approximately   40  
larger  (in  terms  of  its  effects  on  density)  than  the  error  in  assuming  that   Θ   is  a  conservative  
variable.      Preformed   Salinity   S*    is   constructed   so   that   it   contains   no   signature   of   the  
biogeochemical   processes   that   cause   the   spatial   variation   of   seawater   composition.      In   this  
way   S*    is   specifically   designed   to   be   a   conservative   oceanic   salinity   variable.      Having   said  
that,   the   accuracy   with   which   we   can   construct   Preformed   Salinity   S*    from   ocean  
observations   is   presently   limited   by   our   knowledge   of   the   biogeochemical   processes   (see  
appendices  A.4  -­‐‑  A.5  and  Pawlowicz  et  al.  (2011)).      
Summarizing   this   discussion   thus   far,   the   quantities   that   can   be   considered   conservative  
in   the   ocean   are   (in   descending   order   of   accuracy)   (i)   mass,   (ii)   total   energy  
E = u + 0.5 u ⋅ u + Φ ,  (iii)  Conservative  Temperature   Θ ,  and  (iv)  Preformed  Salinity   S* .      
A  different  form  of  “conservation”  attribute,  namely  “isobaric  conservation”  occurs  when  
the   total   amount   of   the   quantity   is   conserved   when   two   fluid   parcels   are   mixed   at   constant  
pressure  without  external  input  of  heat  or  matter.    This  “isobaric  conservative”  property  is  a  
very   valuable   attribute   for   an   oceanographic   variable.      Any   “conservative”   variable   is   also  
“isobaric   conservative”,   thus   the   four   conservative   variables   listed   above,   namely   mass,  
Conservative  Temperature   Θ ,  Preformed  Salinity   S* ,  and  total  energy   E   are  almost  (but  not  
exactly)  “isobaric  conservative”.    In  addition,  the  Bernoulli  function   B   and  specific  enthalpy  
h    are   also   almost   exactly   “isobaric   conservative”   (see   Eqn.   (B.17)   and   the   discussion  
thereafter).      
Some   variables   that   are   not   “isobaric   conservative”   include   potential   temperature   θ ,   
internal  energy   u,   entropy   η ,   density   ρ ,   potential  density   ρ θ ,   and  specific  volume  anomaly  
δ .       Enthalpy   h    and   Conservative   Temperature   Θ    are   not   exactly   “isobaric   conservative”  
because  enthalpy  increases  when  the  kinetic  energy  of  fluid  motion  is  dissipated  by  molecular  
viscosity   inside   the   control   volume   and   when   there   is   a   salinity   source   term   due   to   the  
remineralization   of   particulate   matter.      However,   these   are   tiny   effects   in   the   First   Law   of  
Thermodynamics  (see  appendix  A.21)  and  traditionally  we  regard  enthalpy   h   as  an  “isobaric  
conservative”  variable.    Note  that  while   h   is  “isobaric  conservative”,  it  is  not  a  “conservative”  
variable.      
Appendices  A.18  and  A.21  show  that  for  all  practical  purposes  we  can  treat   Θ   and   h 0   as  
being   “conservative”   variables   (and   hence   also   “isobaric   conservative”   variables);   doing   so  
S

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ignores  the  dissipation  of  kinetic  energy   ε   and  other  terms  of  similar  or  smaller  magnitude.    
Hence  for  all  practical  purposes  in  oceanography  we  have  mass  and  the  following  three  other  
variables  that  are  “conservative”  and  “isobaric  conservative”;    

  

(1)  Conservative  Temperature   Θ,   (and  potential  enthalpy   h 0 ),    
(2)  Preformed  Salinity   S* ,  and    
(3)  total  energy   E .    

Here  we  comment  briefly  on  the  likely  errors  involved  with  treating  variables  other  than  
S*   and   Θ   to  be  conservative  variables  in  ocean  models.    If  one  took  Absolute  Salinity   SA   as  
an   ocean   model’s   salinity   variable   and   treated   it   as   being   conservative,   the   salinity   error  
would  (after  a  long  spin-­‐‑up  time)  be  approximately  as  large  as  the  Absolute  Salinity  Anomaly  
(as  shown  in  Figure  2),  which  is  larger  than   0.025 g kg −1   in  the  North  Pacific,  implying  density  
errors  of   0.020 kg m−3 .    As  a  measure  of  the  importance  of  this  type  of  density  error,  we  note  
that   if   the   equation   of   state   in   an   ocean   model   were   called   with   S R    instead   of   with   S A ,   the  
northward  density  gradient  at  fixed  pressure  (i.  e.  the  thermal  wind)  would  be  misestimated  
by  more  than  2%   for   more   than   58%  of  the  data  below  a  pressure  of  1000  dbar  in  the  world  
ocean.      It   is   clearly   desirable   to   not   have   this   type   of   systematic   error   in   the   dynamical  
equations   of   the   ocean   component   of   coupled   climate   models.      Appendix   A.20   discusses  
practical  ways  of  including  the  effects  of  the  non-­‐‑conservative  biogeochemical  source  term  in  
ocean  models.    The  recommended  option  is  that  ocean  models  carry  Preformed  Salinity   S*   as  
the   model’s   conservative   salinity   model   variable,   and   that   they   also   carry   an   evolution  
equation   for      as   defined   in   Eqn.   (A.5.12),   as   described   in   section   A.20.1   and   Eqns.   (A.20.4)   –  
(A.20.6).      
The   errors   incurred   in   ocean   models   by   treating   potential   temperature   θ    as   being  
conservative  have  not  yet  been  thoroughly  investigated,  but  McDougall  (2003),  Tailleux  (2010)  
and  Graham  and  McDougall  (2013)  have  made  a  start  on  this  topic.    McDougall  (2003)  found  
that  typical  errors  in   θ   are   ± 0.1°C   while  in  isolated  regions  such  as  where  the  fresh  Amazon  
water  discharges  into  the  ocean,  the  error  can  be  as  large  as   1.4 °C .    The  corresponding  error  
in  the  meridional  heat  flux  appears  to  be  about  0.005  PW  (or  a  relative  error  of  0.4%).    The  use  
of  Conservative  Temperature   Θ   in  ocean  models  reduces  these  errors  by  approximately  two  
orders  of  magnitude.      
If   the   ocean   were   in   thermodynamic   equilibrium,   its   temperature   would   be   the   same  
everywhere  as  would  the  chemical  potentials  of  water  and  of  each  dissolved  species,  while  its  
entropy   and   the   concentrations   of   each   species   would   be   functions   of   pressure.      Turbulent  
mixing  acts  in  the  complementary  direction,  tending  to  homogenize  the  concentration  of  each  
species  and  to  make  entropy  constant,  but  in  the  process  causing  gradients  in  temperature  and  
the  chemical  potentials  as  functions  of  pressure.    That  is,  turbulent  mixing  acts  to  maintain  a  
non-­‐‑equilibrium  state.    This  difference  between  the  roles  of  molecular  versus  turbulent  mixing  
results   from   the   symmetry   breaking   role   of   the   gravity   field;   for   example,   in   a   laboratory  
without  gravity,  turbulent  and  molecular  mixing  would  have  indistinguishable  effects.      
Note  that  the  molecular  flux  of  salt   FS   is  given  by  equation  (58.11)  of  Landau  and  Lifshitz  
(1959)  and  by  Eqn.  (B.23)  below.     FS   consists  not  only  of  the  product  of  the  usual  molecular  
diffusivity   and   −ρ∇SA ,   but   also   contains   two   other   terms   that   are   proportional   to   the  
gradients   of   temperature   and   pressure   respectively.      It   is   these   terms   that   cause   the  
equilibrium  vertical  gradients  of  the  dissolved  solutes  in  a  non-­‐‑turbulent  ocean  to  be  different  
and  non-­‐‑zero;  the  last  term  being  called  the  baro-­‐‑diffusion  effect.    The  presence  of  turbulent  
mixing   in   the   real   ocean   renders   this   process   moot   as   turbulence   tends   to   homogenize   the  
ocean  and  maintains  a  relatively  constant  sea-­‐‑salt  composition.      
Note  that  the  description  “conservation  equation”  of  a  particular  quantity  is  often  used  for  
the  equation  that  describes  how  this  quantity  changes  in  response  to  the  divergence  of  various  
fluxes  of  the  quantity  and  to  non-­‐‑conservative  “source”  terms.    For  example,  it  is  usual  to  refer  
to   the   “conservation   equation”   for   entropy   or   for   “potential   temperature”.      Since   these  

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variables   are   not   conservative   variables   it   seems   unnatural   to   refer   to   their   evolution  
equations   as   “conservation   equations”.      Hence   here   we   will   use   the   term   “conservation  
equation”  only  for  a  variable  that  is  (for  all  practical  purposes)  conserved.    For  other  variables  
we   will   refer   to   their   “evolution   equation”   or   their   “prognostic   equation”   or   their   “local  
balance  equation”.      
  
  

A.9 The “potential” property
Any  thermodynamic  property  of  seawater  that  remains  constant  when  a  parcel  of  seawater  is  
moved   from   one   pressure   to   another   adiabatically,   without   exchange   of   mass   and   without  
interior   conversion   between   its   turbulent   kinetic   and   internal   energies,   is   said   to   possess   the  
“potential”   property,   or   in   other   words,   to   be   a   “potential”   variable.      Prime   examples   of  
“potential”  variables  are  entropy   η   and  all  types  of  salinity.    The  constancy  of  entropy   η   can  
be  seen  from  the  First  Law  of  Thermodynamics  in  Eqn.  (B.19)  below;  with  the  right-­‐‑hand  side  
of  Eqn.  (B.19)  being  zero,  and  with  no  change  in  Absolute  Salinity,  it  follows  that  entropy  is  
also  constant.    Any  thermodynamic  property  that  is  a  function  of  only  Absolute  Salinity  and  
entropy   also   remains   unchanged   by   this   procedure   and   is   said   to   possess   the   “potential”  
property.      Thermodynamic   properties   that   posses   the   “potential”   attribute   include   potential  
temperature   θ ,   potential  enthalpy   h0 ,   Conservative  Temperature   Θ   and  potential  density   ρ θ   
(no  matter  what  fixed  reference  pressure  is  chosen).    Some  thermodynamic  properties  that  do  
not   posses   the   potential   property   are   temperature   t ,    enthalpy   h,    internal   energy   u,    specific  
volume   v,   density   ρ ,   specific  volume  anomaly   δ ,   total  energy   E   and  the  Bernoulli  function  
B .     From  Eqn.  (B.17)  we  notice  that  in  the  absence  of  molecular  fluxes  and  the  source  term  of  
Absolute   Salinity,   the   Bernoulli   function   B    is   constant   following   the   fluid   flow   only   if   the  
pressure   field   is   steady;   in   general   this   is   not   the   case.      The   non-­‐‑potential   nature   of   E    is  
explained  in  the  discussion  following  Eqn.  (B.17).      
Some   authors   have   used   the   term   “quasi-­‐‑material”   to   describe   a   variable   that   has   the  
“potential”  property.    The  name  “quasi-­‐‑material”  derives  from  the  idea  that  the  variable  only  
changes   as   a   result   of   irreversible   mixing   processes   and   does   not   change   in   response   to  
adiabatic  and  isohaline  changes  in  pressure.      
The   word   “adiabatic”   is   traditionally   taken   to   mean   a   process   during   which   there   is   no  
exchange  of  heat  between  the  environment  and  the  fluid  parcel  one  is  considering.    With  this  
definition  of  “adiabatic”  it  is  still  possible  for  the  entropy   η ,  the  potential  temperature   θ   and  
the  Conservative  Temperature   Θ   of  a  fluid  parcel  to  change  during  an  isohaline  and  adiabatic  
process.    This  is  because  the  dissipation  of  kinetic  energy   ε   causes  increases  in   η ,   θ   and   Θ   
(see   the   First   Law   of   Thermodynamics,   Eqns.   (A.13.3)   -­‐‑   (A.13.5)).      While   the   dissipation   of  
kinetic   energy   is   a   small   term   whose   influence   is   routinely   neglected   in   the   First   Law   of  
Thermodynamics   in   oceanography,   it   seems   advisable   to   modify   the   meaning   of   the   word  
“adiabatic”   in   oceanography   so   that   our   use   of   the   word   more   accurately   reflects   the  
properties  we  normally  associate  with  an  adiabatic  process.    Accordingly  we  propose  that  the  
word  “adiabatic”  in  oceanography  be  taken  to  describe  a  process  occurring  without  exchange  
of   heat   and   also   without   the   internal   dissipation   of   kinetic   energy.      With   this   definition   of  
“adiabatic”,   a   process   that   is   both   isohaline   and   adiabatic   does   imply   that   the   entropy   η ,  
potential  temperature   θ   and  Conservative  Temperature   Θ   are  all  constant.      
Using  this  more  restrictive  definition  of  the  word  “adiabatic”  we  can  restate  the  definition  
of   a   “potential”   property   as   follows;   any   thermodynamic   property   of   seawater   that   remains  
constant  when  a  parcel  of  seawater  is  moved  from  one  pressure  to  another  “adiabatically”  and  
without  exchange  of  mass,  is  said  to  possess  the  “potential”  property,  or  in  other  words,  to  be  
a  “potential”  variable.      
In  appendix  A.8  above  we  concluded  that  only  mass,  and  the  three  variables   E ,   S*   and  
Θ    (approximately)   are   “conservative”   (and   hence   also   “isobaric   conservative”).      Since   E   

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does   not   posses   the   “potential”   property,   we   now   conclude   that   only   mass   and   the   two  
variables   S*    and   Θ    posses   all   three   highly   desired   properties,   namely   that   they   are  
“conservative”,   “isobaric   conservative”   and   are   “potential”   variables.      In   the   case   of  
Conservative   Temperature   Θ,    its   “conservative”   (and   therefore   its   “isobaric   conservative”)  
nature  is  approximate:-­‐‑   while Θ   is  not  a  100%  conservative  variable,  it  is  approximately  two  
orders   of   magnitude   closer   to   being   a   totally   conservative   variable   than   is   potential  
temperature.      Similarly,   Preformed   Salinity   S*    is   in   principle   100%   conservative,   but   our  
ability   to   evaluate   S*    from   hydrographic   observations   is   limited   (for   example,   by   the  
approximate  relations  (A.4.1)  or  (A.4.9)).      
  
Table  A.9.1    The  “potential”,  “conservative”,  “isobaric  conservative”  and    
the  functional  nature,  of  various  oceanographic  variables    
Variable

“potential”?

S*

SA
SR , SP
t

x

θ

η
h

Θ, h 0
u
B
E

ρ,v

ρθ
δ

ρv
γn
  

x

“conservative”? “isobaric conservative”?

x
x
x
x
x
x

1
1

x
x
x
x
x

x
x
x

3  Taking  

1

x
x

x
4
4

x
x
x
x
x

x

3

4

x
x
x
x
x

1  The  remineralization  of  organic  matter  changes  
2  Taking  

x

1

2
3

x
x
x
x

function of ( SA , t, p ) ?

x
x

5

x

S R less  than  it  changes   SA .       

ε   and  the  effects  of  remineralization  to  be  negligible.    
ε   and  other  terms  of  similar  size  to  be  negligible  (see  the  discussion  

      following  Eqn.  (A.21.13)).    
4  Taking  the  effects  of  remineralization  to  be  negligible.      
5  Once  the  reference  sound  speed  function  
c0 ( p, ρ )   has  been  decided  upon.      
  
In   Table   A.9.1   various   oceanographic   variables   are   categorized   according   to   whether   they  
posses  the  “potential”  property,  whether  they  are  “conservative”  variables,  whether  they  are  
“isobaric  conservative”,  and  whether  they  are  functions  of  only   ( SA , t, p ) .    Note  that   Θ   is  the  
only   variable   that   achieve   four   “ticks”   in   this   table,   while   Preformed   Salinity   S*    has   ticks   in  
the   first   three   columns,   but   not   in   the   last   column   since   it   is   a   function   not   only   of   ( SA , t, p )   
(since   it   also   depends   on   the   composition   of   seawater).      Hence   Θ    is   the   most   “ideal”  
thermodynamic   variable.      If   it   were   not   for   the   non-­‐‑conservation   of   Absolute   Salinity,   it   too  

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would  be  an  “ideal”  thermodynamic  variable,  but  in  this  sense,  Preformed  Salinity  is  superior  
to   Absolute   Salinity.      Conservative   Temperature   Θ    and   Preformed   Salinity   S*    are   the   only  
two   variables   in   this   table   to   be   both   “potential”   and   “conservative”.      The   last   four   rows   of  
Table   A.9.1   are   for   potential   density,   ρ θ    (see   section   3.4),   specific   volume   anomaly,   δ    (see  
section   3.7),   orthobaric   density,   ρ v    (see  appendix  A.28)  and  Neutral  Density   γ n   (see  section  
3.14  and  appendix  A.29).      
  
  

A.10 Proof that θ = θ ( SA ,η ) and Θ=Θ ( SA ,θ )

Consider   changes   occurring   at   the   sea   surface,   (specifically   at   p = 0   dbar)   where   the  
temperature  is  the  same  as  the  potential  temperature  referenced  to  0  dbar  and  the  increment  
of   pressure   dp    is   zero.      Regarding   specific   enthalpy   h    and   chemical   potential   µ    to   be  
functions  of  entropy   η   (in  place  of  temperature   t ),  that  is,  considering  the  functional  form  of  


h    and   µ    to   be   h = h SA ,η , p    and   µ = µ SA ,η , p ,    it   follows   from   the   fundamental  
thermodynamic  relation  (Eqn.  (A.7.1))  that    



hη SA ,η ,0 dη + hS SA ,η ,0 dSA = T0 + θ dη + µ SA ,η ,0 dSA ,   
(A.10.1)  

(

(

)

)

A

(

(

)

)

(

)

(

)

which   shows   that   specific   entropy   η    is   simply   a   function   of   Absolute   Salinity   S A    and  
potential   temperature   θ ,    that   is   η = η SA ,θ ,   with   no   separate   dependence   on   pressure.      It  
follows  that   θ = θ ( SA ,η ) .       
Similarly,   from   the   definition   of   potential   enthalpy   and   Conservative   Temperature   in  
Eqns.   (3.2.1)   and   (3.3.1),   at   p = 0    dbar   it   can   be   seen   that   the   fundamental   thermodynamic  
relation  (A.7.1)  implies    

(

(

)

)

(

)

c0p dΘ = T0 + θ dη + µ SA ,θ ,0 dSA .   

(A.10.2)  

This  shows  that  Conservative  Temperature  is  also  simply  a  function  of  Absolute  Salinity  and  
potential  temperature, Θ = Θ SA ,θ ,  with  no  separate  dependence  on  pressure.    It  then  follows  
that   Θ    may   also   be   expressed   as   a   function   of   only   S A    and   η .       It   follows   that   Θ    has   the  
“potential”  property.      
  
  

(

)

A.11 Various isobaric derivatives of specific enthalpy
Because  of  the  central  role  of  enthalpy  in  the  transport  and  the  conservation  of  “heat”  in  the  
ocean,  the  derivatives  of  specific  enthalpy  at  constant  pressure  are  here  derived  with  respect  
to  Absolute  Salinity  and  with  respect  to  the  three  “temperature-­‐‑like”  variables   η, θ   and   Θ   as  
well  as  in  situ  temperature   t.       
  
We  begin  by  noting  that  the  three  standard  derivatives  of   h = h ( SA , t, p )   when  in  situ  
temperature   t   is  taken  as  the  “temperature-­‐‑like”  variable  are    

∂h ∂SA T , p = µ ( SA , t , p ) − (T0 + t ) µT ( SA , t , p ) ,   
∂h ∂T

SA , p

∂h ∂P S

A ,T

(A.11.1)  

= c p ( SA , t, p ) = (T0 + t ) ηT ( SA , t, p ) ,   

(A.11.2)  

= v ( SA , t , p ) − (T0 + t ) v T ( SA , t , p ) .   

(A.11.3)  

Now  considering  specific  enthalpy  to  be  a  function  of  entropy  (rather  than  of  temperature  

t ),  that  is,  taking   h = h SA ,η , p ,   the  fundamental  thermodynamic  relation  (A.7.1)  becomes    



hη dη + hS dSA = T0 + t dη + µ dSA     while     ∂ h ∂P
(A.11.4)  
= v ,   

(

)

(

A

so  that    


∂ h ∂η

SA , p

)


= T0 + t             and           ∂ h ∂SA

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(

)

SA ,η

η, p

= µ .   

(A.11.5)  

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93

Now  taking  specific  enthalpy  to  be  a  function  of  potential  temperature  (rather  than  of  in  
situ   temperature   t ),   that   is,   taking   h = h SA ,θ , p ,    the   fundamental   thermodynamic   relation  
(A.7.1)  becomes    

(

(

)

)

hθ dθ + hS dSA = T0 + t dη + µ dSA             while             ∂ h ∂P
A

SA ,θ

= v.   

(A.11.6)  

To  evaluate  the   hθ   partial  derivative,  it  is  first  written  in  terms  of  the  derivative  with  respect  
to  entropy  as    

(A.11.7)  
= hη
ηθ
= T0 + t ηθ ,   
hθ
SA , p

(

SA , p

SA , p

)

SA

where  (A.11.5)  has  been  used.    This  equation  can  be  evaluated  at   p = 0   when  it  becomes  (the  
potential  temperature  used  here  is  referenced  to   pr = 0 )    

hθ

SA , p=0

(

) (

)

= c p SA ,θ ,0 = T0 + θ ηθ

SA

.  

(A.11.8)  

These  two  equations  are  used  to  arrive  at  the  desired  expression  for   hθ   namely    

hθ

SA , p

(

= c p SA ,θ ,0

T +t
) ((T0 +θ))

(

) (

)

= − T0 + t gTT SA ,θ ,0 .   

(A.11.9)  

0

hS   partial  derivative,  we  first  write  specific  enthalpy  in  the  functional  form  
To  evaluate  the  

A
h = h SA ,η SA ,θ , p   and  then  differentiate  it,  finding    


= hS
+ hη
ηS .   
hS
(A.11.10)  

(

(

) )

A θ, p

A η, p

A θ

SA , p

The   partial   derivative   of   specific   entropy   η = − gT    (Eqn.   (2.10.1))   with   respect   to   Absolute  
Salinity,   ηS = − gS T ,   is  also  equal  to   − µT   since  chemical  potential  is  defined  by  Eqn.  (2.9.6)  
A
A
as   µ = gS .      Since   the   partial   derivative   of   entropy   with   respect   to   S A    in   (A.11.10)   is  
A
performed   at   fixed   potential   temperature   (rather   than   at   fixed   in   situ   temperature),   this   is  
equal   to   − µT    evaluated   at   p = 0.       Substituting   both   parts   of   (A.11.5)   into   (A.11.10)   we   have  
the  desired  expression  for   hS   namely    
A

hS

A θ, p

(

) ( ) (
)
  
( SA ,t, p ) − (T0 + t ) gTS ( SA ,θ ,0).

= µ SA ,t, p − T0 + t µT SA ,θ ,0
= gS

A

(A.11.11)  

A

Notice  that  this  expression  contains  some  things  that  are  evaluated  at  the  general  pressure   p   
and  one  that  is  evaluated  at  the  reference  pressure   pr = 0.       
Now   considering   specific   enthalpy   to   be   a   function   of   Conservative   Temperature   (rather  
than  of  in  situ  temperature   t ),  that  is,  taking   h = hˆ ( S A , Θ, p ) ,   the  fundamental  thermodynamic  
relation  (A.7.1)  becomes    

hˆΘ dΘ + hˆSA dSA = (T0 + t ) dη + µ dSA             while             ∂hˆ ∂P

SA , Θ

= v .   

(A.11.12)  

The  partial  derivative   ĥΘ   follows  directly  from  this  equation  as    

hˆΘ

= (T0 + t )ηΘ S

SA , p

A,p

= (T0 + t )ηΘ S .   
A

(A.11.13)  

At   p = 0   this  equation  reduces  to    

hˆΘ

SA , p = 0

= c0p = (T0 + θ )ηΘ S ,   

(A.11.14)  

A

and  combining  these  two  equations  gives  the  desired  expression  for   ĥΘ   namely    

hˆΘ

SA , p

=

(T0 + t ) c0 .
  
(T0 + θ ) p

(A.11.15)  

To   evaluate   the   hˆS    partial   derivative   we   first   write   h    in   the   functional   form  
A
h = h SA ,η SA ,Θ , p   and  then  differentiate  it,  finding  (using  both  parts  of  Eqn.  (A.11.5))    

(

(

) )

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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

hˆSA

Θ, p

= µ ( SA , t, p ) + (T0 + t )ηSA .   

(A.11.16)  

Θ

The  differential  expression  Eqn.  (A.11.12)  can  be  evaluated  at   p = 0   where  the  left-­‐‑hand  side  
is  simply   c0p dΘ   so  that  from  Eqn.  (A.11.12)  we  find  that      

ηSA

Θ

= −

µ ( SA ,θ ,0 )
,   
(T0 + θ )

(A.11.17)  

so  that  the  desired  expression  for   hˆS   is    
A

hˆSA

Θ, p

(T0 + t ) µ S ,θ ,0
(
)
(T0 + θ ) A
  
T0 + t )
(
g SA ( S A , t , p ) −
g ( S ,θ ,0 ).
(T0 + θ ) SA A

= µ ( SA , t , p ) −
=

(A.11.18)  

The   above   boxed   expressions   for   four   different   isobaric   derivatives   of   specific   enthalpy   are  
important   as   they   are   integral   to   forming   the   First   Law   of   Thermodynamics   in   terms   of  
potential  temperature  and  in  terms  of  Conservative  Temperature.    The  partial  derivatives   ĥΘ   
and   hˆS    of   Eqns.   (A.11.15)   and   (A.11.18)   can   be   calculated   using   the   GSW   Oceanographic  
A
Toolbox  function  gsw_enthalpy_first_derivatives_CT_exact.      
The  second  order  partial  derivatives   ĥΘΘ ,   hˆS Θ   and   hˆS S   can  be  written  in  terms  of  the  
A
A A
seawater  Gibbs  function  as  (these  second  order  partial  derivatives  can  be  evaluated  using  the  
GSW  Oceanographic  Toolbox  function  gsw_enthalpy_second_derivatives_CT_exact.)    

hΘΘ

SA , p

= hˆΘΘ =

(c )

2

0
p

(T0 + θ )

2

⎛ (T0 + t )
1
1
−
⎜⎜
,
,0
T
g
S
g
S
θ
θ
+
) TT ( A )
TT ( A , t , p )
⎝( 0

g S AT ( S A , t , p )
⎛ (T0 + t ) g SAT ( SA ,θ ,0 )
−
⎜⎜
gTT ( SA , t , p )
(T0 + θ ) ⎝ (T0 + θ ) gTT ( SA ,θ ,0 )
c0p

hˆSA Θ =

⎞
⎟⎟
⎠

⎞
⎟⎟ ,
⎠

c 0p g SA ( SA ,θ ,0 ) ⎛ (T0 + t )
⎞
1
1
−
−
⎜⎜
⎟,
2
gTT ( SA , t , p ) ⎟⎠
(T0 + θ )
⎝ (T0 + θ ) gTT ( SA ,θ ,0 )
hˆSA SA = g SA SA ( SA , t , p ) −

−

(A.11.20)

(T0 + t ) g
( S ,θ ,0)
(T0 + θ ) SA SA A

(T0 + t ) ⎡⎣ g SAT ( SA ,θ ,0)⎤⎦ − ⎡⎣ g SAT ( SA , t, p )⎤⎦
gTT ( S A , t , p )
(T0 + θ ) gTT ( SA ,θ ,0 )
g S AT ( S A , t , p ) ⎞
2 g SA ( SA , θ ,0 ) ⎛ (T0 + t ) g SAT ( SA , θ ,0 )
−
⎜⎜
⎟
gTT ( SA , t , p ) ⎟⎠
(T0 + θ ) ⎝ (T0 + θ ) gTT ( SA ,θ ,0)
2

+

(A.11.19)

2

(A.11.21)

⎡ g S ( SA , θ ,0 )⎤⎦ ⎛ (T0 + t )
⎞
1
1
−
+ ⎣ A
⎜
⎟.
2
gTT ( SA , t , p ) ⎠
(T0 + θ )
⎝ (T0 + θ ) gTT ( S A , θ ,0 )
2

The  first  order  partial  derivatives   ĥΘ   and   hˆS   evaluated  from  the  75-­‐‑term  expression  for  
A
specific  volume,   v̂ SA ,Θ, p ,  are  available  in  the  GSW  Oceanographic  Toolbox  from  the  
function  gsw_enthalpy_first_derivatives,  while  the  second  order  partial  derivatives   ĥΘΘ ,  
hˆSA Θ   and   hˆSA SA   are  evaluated  from  the  same  75-­‐‑term  expression  for  specific  volume  by  the  
function  gsw_enthalpy_second_derivatives.      

(

)

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95

A.12 Differential relationships between η , θ , Θ and SA
Evaluating  the  fundamental  thermodynamic  relation  in  the  forms  (A.11.6)  and  (A.11.12)  and  
using  the  four  boxed  equations  in  appendix  A.11,  we  find  the  relations    

(T0 + t ) dη + µ ( p ) dSA

=
=

(T0 + t ) c 0 dθ + ⎡ µ p − T + t µ 0 ⎤ dS
( )
⎣ ( ) ( 0 ) T ( )⎦ A
(T0 + θ ) p
  
(T0 + t ) c0 dΘ + ⎡ µ p − (T0 + t ) µ 0 ⎤ dS .
( )⎥ A
⎢ ( )
(T0 + θ ) p
(T0 + θ )
⎢⎣
⎥⎦

(A.12.1)  

The  quantity   µ ( p ) dSA   is  now  subtracted  from  each  of  these  three  expressions  and  the  whole  
equation  is  then  multiplied  by   (T0 + θ ) (T0 + t )   obtaining    

(T0 +θ ) dη

= c p (0) dθ − (T0 + θ ) µT (0) dSA = c0p dΘ − µ (0) dSA .   

(A.12.2)  

From  this  follows  all  the  following  partial  derivatives  between   η, θ , Θ   and   SA ,     
       Θθ
   Θη
     θη

SA

= c p ( SA ,θ ,0) c0p ,                              ΘSA = ⎡⎣ µ ( SA ,θ ,0) − (T0 + θ ) µT ( SA ,θ ,0)⎤⎦ c0p ,   
θ

(A.12.3)  

SA

= (T0 + θ ) c0p ,                                            ΘSA = µ ( SA ,θ ,0) c0p ,   

(A.12.4)  

SA

= (T0 + θ ) c p ( SA ,θ ,0) ,              θ SA = (T0 + θ ) µT ( SA ,θ ,0) c p ( SA ,θ ,0) ,   

(A.12.5)  

η

η

θΘ S = c0p c p ( SA ,θ ,0) ,        θSA = − ⎡⎣ µ ( SA ,θ ,0) − (T0 + θ ) µT ( SA ,θ ,0)⎤⎦ c p ( SA ,θ ,0) ,   (A.12.6)  
A

Θ

         ηθ

SA

= c p ( SA ,θ ,0) (T0 + θ ) ,                ηSA = − µT ( SA ,θ ,0) ,   

(A.12.7)  

   ηΘ

SA

= c0p (T0 + θ ) ,                                            ηSA = − µ ( SA ,θ ,0) (T0 + θ ) .   

(A.12.8)  

θ

Θ

The   three   second   order   derivatives   of   ηˆ ( SA , Θ)    are   listed   in   Eqns.   (P.14)   and   (P.15)   of  
appendix   P.      The   corresponding   derivatives   of   θˆ ( SA , Θ ) ,   namely   θˆΘ ,   θˆS ,   θˆΘΘ ,   θˆS Θ    and  
A
A
θˆS S   can  also  be  derived  using  Eqn.  (P.13),  obtaining    
A A




 Θ

Θ
Θ
Θ
Θ
1
S
θ SA
SA θθ
+
θˆΘ =  ,    θˆS = −  A ,     θˆΘΘ = − θθ 3 ,       θˆS Θ = −
,  
A
A
Θθ
Θθ

 2
 3
Θ
Θ
Θ
θ
θ
θ

( )

( )

( )

(A.12.9a,b,c,d)  



 ⎞2 
⎛Θ
Θ
Θ
Θ
S
θ
S
S
A
A
(A.12.10)  
    and         θˆS S
− ⎜  A ⎟ θθ ,  


A A
⎜
⎟
Θθ Θθ
θ
⎝ Θθ ⎠ Θθ
 ,   Θ
 ,   Θ


 ,   Θ
in  terms  of  the  partial  derivatives   Θ
SA
θ SA   and   Θ SA SA   which  can  be  obtained  by  
θ
θθ
 ( S ,θ )   from  the  TEOS-­‐‑10  Gibbs  function.      
differentiating  the  polynomial   Θ
A
  
  

Θ
S S
= − A A + 2
Θ

A.13 The First Law of Thermodynamics
The  law  of  the  conservation  of  energy  for  thermodynamic  equilibrium  states  was  discovered  
in  the  19th  century  by  Gibbs  (1873)  and  other  early  pioneers.    It  was  formulated  as  a  balance  
between  internal  energy,  heat  and  work,  similar  to  the  fundamental  thermodynamic  equation  
(A.7.1),  and  referred  to  as  the  First  Law  of  Thermodynamics  (Thomson  (1851),  Clausius  (1876),  
Alberty   (2001)).      Under   the   weaker   condition   of   a   local   thermodynamic   equilibrium  
(Glansdorff   and   Prigogine   (1971)),   the   original   thermodynamic   concepts   can   be   suitably  
generalized  to  describe  irreversible  processes  of  fluid  dynamics  which  are  subject  to  molecular  
fluxes  and  macroscopic  motion  (Landau  and  Lifshitz  (1959),  de  Groot  and  Mazur  (1984)).    

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In   some   circles   the   “First   Law   of   Thermodynamics”   is   used   to   describe   the   evolution  
equation   for   total   energy,   being   the   sum   of   internal   energy,   potential   energy   and   kinetic  
energy.      Here   we   follow   the   more   common   practice   of   regarding   the   First   Law   of  
Thermodynamics  as  the  difference  between  the  conservation  equation  of  total  energy  and  the  
evolution   equation   for   the   sum   of   kinetic   energy   and   potential   energy,   leaving   what   might  
loosely  be  termed  the  evolution  equation  of  “heat”,  Eqn.  (A.13.1)  (Landau  and  Lifshitz  (1959),  
de  Groot  and  Mazur  (1984),  McDougall  (2003),  Griffies  (2004)).        
The   First   Law   of   Thermodynamics   can   therefore   be   written   as   (see   Eqn.   (B.19)   and   the  
other  Eqns.  (A.13.3),  (A.13.4)  and  (A.13.5)  of  this  appendix;  all  of  these  equations  are  equally  
valid  incarnations  of  the  First  Law  of  Thermodynamics)    

⎛ dh 1 dP ⎞
S
= − ∇ ⋅F R − ∇ ⋅FQ + ρε + hS ρ S A,   
ρ⎜
−
⎟
A
⎝ dt ρ dt ⎠

(A.13.1)  

where   FR    is   the   sum   of   the   boundary   and   radiative   heat   fluxes   and   FQ    is   the   sum   of   all  
molecular   diffusive   fluxes   of   heat,   being   the   normal   molecular   heat   flux   directed   down   the  
temperature  gradient  plus  a  term  proportional  to  the  molecular  flux  of  salt  (the  Dufour  effect,  
see  Eqn.  (B.24)  below).    Lastly,   ε   is  the  rate  of  dissipation  of  kinetic  energy  per  unit  mass  and  
S
hS ρ S A   is  the  rate  of  increase  of  enthalpy  due  to  the  interior  source  term  of  Absolute  Salinity  
A
caused   by   remineralization.      The   derivation   of   Eqn.   (A.13.1)   is   summarized   in   appendix   B  
below,   where   we   also   discuss   the   related   evolution   equations   for   total   energy   and   for   the  
Bernoulli  function.      
Following  Fofonoff  (1962)  we  note  that  an  important  consequence  of  Eqn.  (A.13.1)  is  that  
when   two   finite   sized   parcels   of   seawater   are   mixed   at   constant   pressure   and   under   ideal  
conditions,  the  total  amount  of  enthalpy  is  conserved.    To  see  this  one  combines  Eqn.  (A.13.1)  
with  the  continuity  equation   ∂ρ ∂t + ∇ ⋅ ( ρ u ) = 0   to  find  the  following  divergence  form  of  the  
First  Law  of  Thermodynamics,  
dP
S
(A.13.2)  
∂ ρ h ∂t + ∇ ⋅ ρ uh −
= − ∇ ⋅F R − ∇ ⋅FQ + ρε + hS ρ S A .   
A
dt
One   then   integrates   over   the   volume   that   encompasses   both   fluid   parcels   while   assuming  
there   to   be   no   radiative,   boundary   or   molecular   fluxes   across   the   boundary   of   the   control  
volume.    This  control  volume  may  change  with  time  as  the  fluid  moves  (at  constant  pressure),  
mixes  and  contracts.    The  dissipation  of  kinetic  energy  by  viscous  friction  and  the  source  term  
due   to   the   production   of   Absolute   Salinity   are   also   commonly   ignored   during   such   mixing  
processes   but   in   fact   the   dissipation   term   does   cause   a   small   increase   in   the   enthalpy   of   the  
mixture  with  respect  to  that  of  the  two  original  parcels  (see  Appendix  A.21).    Apart  from  these  
non-­‐‑conservative   source   terms,   under   these   assumptions   Eqn.   (A.13.2)   reduces   to   the  
statement   that   the   volume   integrated   amount   of   ρ h    is   the   same   for   the   two   initial   fluid  
parcels  as  for  the  final  mixed  parcel,  that  is,  the  total  amount  of  enthalpy  is  unchanged.      
This   result   of   non-­‐‑equilibrium   thermodynamics   is   of   the   utmost   importance   in  
oceanography.      The   fact   that   enthalpy   is   conserved   when   fluid   parcels   mix   at   constant  
pressure   is   the   central   result   upon   which   all   of   our   understanding   of   “heat   fluxes”   and   of  
“heat  content”  in  the  ocean  rests.    The  importance  of  this  result  cannot  be  overemphasized;  it  
must   form   part   of   all   our   introductory   courses   on   oceanography   and   climate   dynamics.      As  
important   as   this   result   is,   it   does   not   follow   that   enthalpy   is   the   best   variable   to   represent  
“heat  content”  in  the  ocean.    Enthalpy  is  a  very  poor  representation  of  “heat  content”  in  the  
ocean   because   it   does   not   posses   the   “potential”   property.      It   will   be   seen   that   potential  
enthalpy   h 0   (referenced  to  zero  sea  pressure)  is  the  best  thermodynamic  variable  to  represent  
“heat  content”  in  the  ocean.      
The  First  Law  of  Thermodynamics,  Eqn.  (A.13.1),  can  be  written  (using  Eqn.  (A.7.2))  as  an  
evolution  equation  for  entropy  as  follows    

( )

(

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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

⎛
dS ⎞
dη
S
ρ ⎜ (T0 + t )
+ µ A ⎟ = − ∇ ⋅F R − ∇ ⋅FQ + ρε + hS ρ S A .   
A
dt
dt ⎠
⎝

97

(A.13.3)  

The   First   Law   of   Thermodynamics   (A.13.1)   can   also   be   written   in   terms   of   potential  
temperature   θ   (with  respect  to  reference  pressure   pr )  by  using  Eqns.  (A.11.9)  and  (A.11.11)  in  
Eqn.  (A.13.1)  as  (from  Bacon  and  Fofonoff  (1996)  and  McDougall  (2003))  

⎛ (T + t )
dS ⎞
dθ
ρ⎜ 0
c p ( pr )
+ ⎡⎣ µ ( p ) − (T0 + t ) µT ( pr ) ⎤⎦ A ⎟ =
dt
dt ⎠
⎝ (T0 + θ )

  

(A.13.4)  

− ∇ ⋅F R − ∇ ⋅FQ + ρε + hS ρ S A ,
S

A

where   T0    is   the   Celsius   zero   point   ( T0    is   exactly   273.15   K),   while   in   terms   of   Conservative  
Temperature   Θ ,   the   First   Law   of   Thermodynamics   is   (from   McDougall   (2003),   using   Eqns.  
(A.11.15)  and  (A.11.18)  above)    

⎛ (T + t ) d Θ ⎡
(T + t ) µ (0)⎤⎥ d SA ⎞⎟ =
ρ⎜ 0
c0p
+ ⎢µ ( p) − 0
⎜⎝ (T0 + θ ) d t ⎢
(T0 + θ ) ⎥⎦ d t ⎟⎠
⎣

  

(A.13.5)  

− ∇ ⋅F R − ∇ ⋅FQ + ρε + hS ρ S A ,
S

A

where   c 0p   is  the  fixed  constant  defined  by  the  exact  15-­‐‑digit  number  in  Eqn.  (3.3.3).      
In  appendices  A.16,  A.17  and  A.18  the  non-­‐‑conservative  production  of  entropy,  potential  
temperature   and   Conservative   Temperature   are   quantified,   both   as   Taylor   series   expansions  
which   identify   the   relevant   non-­‐‑linear   thermodynamic   terms   that   cause   the   production   of  
these   variables,   and   also   on   the   SA − Θ    diagram   where   variables   are   contoured   which  
graphically   illustrate   the   non-­‐‑conservation   of   these   variables.      In   other   words,   appendices  
A.16,   A.17   and   A.18   quantify   the   non-­‐‑ideal   nature   of   the   left-­‐‑hand   sides   of   Eqns.   (A.13.3)   -­‐‑  
(A.13.5).      That   is,   these   appendices   quantify   the   deviations   of   the   left-­‐‑hand   sides   of   these  
equations  from  being  proportional  to   ρ dη dt ,   ρ dθ dt   and   ρ dΘ dt .      
A  quick  ranking  of  these  three  variables,   η ,    θ   and   Θ ,   from  the  viewpoint  of  the  amount  
of   their   non-­‐‑conservation,   can   be   gleaned   by   examining   the   range   of   the   terms   (at   fixed  
pressure)   that   multiply   the   material   derivatives   on   the   left-­‐‑hand   sides   of   the   above   Eqns.  
(A.13.3),   (A.13.4)   and   (A.13.5).      The   ocean   circulation   may   be   viewed   as   a   series   of   adiabatic  
and   isohaline   movements   of   seawater   parcels   interrupted   by   a   series   of   isolated   turbulent  
mixing   events.      During   any   of   the   adiabatic   and   isohaline   transport   stages   every   “potential”  
property   is   constant,   so   each   of   the   above   variables,   entropy,   potential   temperature   and  
Conservative   Temperature   are   100%   ideal   during   these   adiabatic   and   isohaline   advection  
stages.      The   turbulent   mixing   events   occur   at   fixed   pressure   so   the   non-­‐‑conservative  
production   of   say   entropy   depends   on   the   extent   to   which   the   coefficients   (T0 + t )    and   µ    in  
Eqn.   (A.13.3)   vary   at   fixed   pressure.      Similarly   the   non-­‐‑conservative   production   of   potential  
temperature   depends   on   the   extent   to   which   the   coefficients   c p ( pr )(T0 + t ) (T0 + θ )    and  
⎡⎣ µ ( p ) − (T0 + t ) µT ( pr )⎤⎦    in   Eqn.   (A.13.4)   vary   at   fixed   pressure,   while   the   non-­‐‑conservative  
production   of   Conservative   Temperature   depends   on   the   extent   to   which   the   coefficients  
(T0 + t ) (T0 + θ )   and   ⎡⎣ µ ( p ) − µ (0)(T0 + t ) (T0 + θ )⎤⎦   in  Eqn.  (A.13.5)  vary  at  fixed  pressure.      
According  to  this  way  of  looking  at  these  equations  we  note  that  the  material  derivative  of  
entropy  appears  in  Eqn.  (A.13.3)  multiplied  by  the  absolute  temperature   (T0 + t )   which  varies  
by  about  15%  at  the  sea  surface  ( ( 273.15 + 40) 273.15 ≈ 1.146 ),  the  term  that  multiplies   dθ dt   
in  (A.13.4)  is  dominated  by  the  variations  in   c p ( S A , t , pr )   which  is  mainly  a  function  of   S A   and  
which   varies   by   5%   at   the   sea   surface   (see   Figure   4),   while   the   material   derivative   of  
Conservative   Temperature   dΘ dt    in   Eqn.   (A.13.5)   is   multiplied   by   the   product   of   a   constant  
“heat   capacity”   c 0p    and   the   factor   (T0 + t ) (T0 + θ )    which   varies   very   little   in   the   ocean,  
especially  when  one  realizes  that  it  is  only  the  variation  of  this  ratio  at  each  pressure  level  that  
is  of  concern.    This  factor  is  unity  at  the  sea  surface.      

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Graham   and   McDougall   (2013)   have   derived   the   evolution   equations   for   potential  
temperature,   Conservative   Temperature   and   specific   entropy   in   a   turbulent   ocean,  
demonstrating  that  the  non-­‐‑conservative  source  terms  of  potential  temperature  are  two  orders  
of   magnitude   larger   than   those   for   Conservative   Temperature.      They   also   showed   that   the  
dissipation  of  kinetic  energy  is  an  order  of  magnitude  larger  than  the  non-­‐‑conservative  source  
terms  in  the  evolution  equation  for  Conservative  Temperature.    

Figure  A.13.1.    The  difference   θ − Θ   (in   °C )  between  potential  temperature   θ   
and   Conservative   Temperature   Θ    at   the   sea   surface   of   the  
annually-­‐‑averaged  atlas  of  Gouretski  and  Koltermann  (2004).    
  
Fortunately,   Conservative   Temperature   is   not   only   much   more   accurately   conserved   in  
the   ocean   than   potential   temperature   but   it   is   also   relatively   easy   to   use   in   oceanography.    
Because   Conservative   Temperature   also   possesses   the   “potential”   property,   it   is   a   very  
accurate   representation   of   the   “heat   content”   of   seawater.      The   difference   θ − Θ    between  
potential   temperature   θ    and   Conservative   Temperature   Θ    at   the   sea   surface   is   shown   in  
Figure   A.13.1   (after   McDougall,   2003).      If   an   ocean   model   is   written   with   potential  
temperature   as   the   prognostic   temperature   variable   rather   than   Conservative   Temperature,  
and  is  run  with  the  same  constant  value  of  the  isobaric  specific  heat  capacity  ( c 0p   as  given  by  
Eqn.   (3.3.3)),   the   neglect   of   the   non-­‐‑conservative   source   terms   that   should   appear   in   the  
prognostic  equation  for   θ   means  that  such  an  ocean  model  incurs  errors  in  the  model  output.    
These  errors  will  depend  on  the  nature  of  the  surface  boundary  condition;  for  flux  boundary  
conditions  the  errors  are  as  shown  in  Figure  A.13.1.        
This   appendix   has   largely   demonstrated   the   benefits   of   potential   enthalpy   and  
Conservative  Temperature  from  the  viewpoint  of  conservation  equations,  but  the  benefits  can  
also  be  proven  by  the  following  parcel-­‐‑based  argument.    First,  the  air-­‐‑sea  heat  flux  needs  to  be  
recognized  as  a  flux  of  potential  enthalpy  which  is  exactly   c 0p   times  the  flux  of  Conservative  
Temperature.    Second,  the  work  of  appendix  A.18  shows  that  while  it  is  the  in  situ  enthalpy  
that   is   conserved   when   parcels   mix,   a   negligible   error   is   made   when   potential   enthalpy   is  
assumed  to  be  conserved  during  mixing  at  any  depth.    Third,  note  that  the  ocean  circulation  
can   be   regarded   as   a   series   of   adiabatic   and   isohaline   movements   during   which   Θ    is  
absolutely   unchanged   (because   of   its   “potential”   nature)   followed   by   a   series   of   turbulent  
mixing   events   during   which   Θ    is   almost   totally   conserved.      Hence   it   is   clear   that   Θ    is   the  
quantity   that   is   advected   and   diffused   in   an   almost   conservative   fashion   and   whose   surface  
flux  is  exactly  proportional  to  the  air-­‐‑sea  heat  flux.      

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99

A.14 Advective and diffusive “heat” fluxes
In  section  3.23  and  appendices  A.8  and  A.13  the  First  Law  of  Thermodynamics  is  shown  to  be  
practically  equivalent  to  the  conservation  equation  (A.21.15)  for  Conservative  Temperature   Θ.     
We  have  emphasized  that  this  means  that  the  advection  of  “heat”  is  very  accurately  given  as  
the  advection  of   c0p Θ.     In  this  way   c0p Θ   can  be  regarded  as  the  “heat  content”  per  unit  mass  of  
seawater  and  the  error  involved  with  making  this  association  is  approximately  1%  of  the  error  
in   assuming   that   either   c0pθ    or   c p ( SA ,θ , 0dbar )θ    is   the   “heat   content”   per   unit   mass   of  
seawater  (see  also  appendix  A.21  for  a  discussion  of  this  point).      
The  turbulent  flux  of  a  “potential”  property  can  be  thought  of  as  the  exchange  of  parcels  
of  equal  mass  but  contrasting  values  of  the  “potential”  property,  and  the  turbulent  flux  can  be  
parameterized  as  being  down  the  gradient  of  the  “potential”  property.    The  conservative  form  
of   Eqn.   (A.21.15)   implies   that   the   turbulent   flux   of   heat   should   be   directed   down   the   mean  
gradient   of   Conservative   Temperature   rather   than   down   the   mean   gradient   of   potential  
temperature.      In   this   appendix   we   quantify   the   ratio   of   the   mean   gradients   of   potential  
temperature  and  Conservative  Temperature.      
Consider  first  the  respective  temperature  gradients  along  the  neutral  tangent  plane.    From  
Eqn.  (3.11.2)  we  find  that    

(α

θ

)

(

)

β θ ∇nθ = ∇n SA = α Θ β Θ ∇n Θ ,   

(A.14.1)  

so   that   the   epineutral   gradients   of   θ    and   Θ    are   related   by   the   ratios   of   their   respective  
thermal  expansion  and  saline  contraction  coefficients,  namely    

(α
(α θ

Θ

∇nθ =

βΘ
β

θ

) ∇ Θ.   
)

(A.14.2)  

n

This   proportionality   factor   between   the   parallel   two-­‐‑dimensional   vectors   ∇nθ    and   ∇n Θ    is  
readily  calculated  and  illustrated  graphically.    Before  doing  so  we  note  two  other  equivalent  
expressions  for  this  proportionality  factor.      
The  epineutral  gradients  of   θ ,   Θ   and   S A   are  related  by  (using   θ = θˆ ( S A , Θ ) )    

∇nθ = θˆΘ ∇n Θ + θˆSA ∇n SA ,   
and  using  the  neutral  relationship   ∇n SA =

(

(α

Θ

)

(A.14.3)  

β Θ ∇n Θ   we  find    

)

∇nθ = θˆΘ + ⎡⎣α Θ β Θ ⎤⎦ θˆSA ∇n Θ.   

(A.14.4)  

Also,  in  section  3.13  we  found  that   Tbθ ∇nθ = TbΘ∇n Θ,   so  that  we  find  the  expressions    

∇nθ
∇n Θ

(α
(α θ

Θ

=

)
βθ )
βΘ

=

TbΘ
= θˆΘ + ⎡⎣α Θ β Θ ⎤⎦ θˆSA ,   
θ
Tb

(

(A.14.5)  

)

and   it   can   be   shown   that   α Θ α θ = θˆΘ    and   β θ β Θ = 1 + ⎡α Θ β Θ ⎤ θˆS θˆΘ ,   that   is,  
⎣
⎦ A
β θ = β Θ + α Θ θˆSA θˆΘ .       The   ratios   α Θ α θ    and   β θ β Θ    have   been   plotted   by   Graham   and  
McDougall   (2013);   interestingly   α Θ α θ    is   approximately   a   linear   function   of   S A    while  
β θ β Θ   is  approximately  a  function  of  only   Θ .    The  partial  derivatives   θˆΘ   and   θˆSA   in  the  last  
part  of  Eqn.  (A.14.5)  are  both  independent  of  pressure  while   α Θ β Θ   is  a  function  of  pressure.    
The  ratio,  Eqn.  (A.14.5),  of  the  epineutral  gradients  of   θ   and   Θ   is  shown  in  Figure  A.14.1  at  
p = 0 ,  indicating  that  the  epineutral  gradient  of  potential  temperature  is  sometimes  more  that  
1%   different   to   that   of   Conservative   Temperature.      This   ratio   ∇nθ ∇n Θ    is   only   a   weak  
function   of   pressure.      This   ratio,   ∇nθ ∇n Θ    (i.e.   Eqn.   (A.14.5)),   is   available   in   the   GSW  
Oceanographic  Toolbox  as  function  gsw_ntp_pt_vs_CT_ratio.      
Similarly  to  Eqn.  (A.14.3),  the  vertical  gradients  are  related  by    

θ z = θˆΘ Θz + θˆSA SAz ,   

(A.14.6)  

and  using  the  definition,  Eqn.  (3.15.1),  of  the  stability  ratio  we  find  that    

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θ

z
(A.14.7)  
= θˆΘ + Rρ−1 ⎡⎣α Θ β Θ ⎤⎦ θˆSA .   
Θz
For   values   of   the   stability   ratio   Rρ    close   to   unity,   the   ratio   θ z Θz    is   close   to   the   values   of  
∇nθ ∇n Θ   shown  in  Figure  A.14.1.      

  
Figure  A.14.1.    Contours  of   ∇nθ ∇n Θ − 1 × 100%   at   p = 0 ,  showing  the  percentage    
                                                        difference  between  the  epineutral  gradients  of   θ   and   Θ .    The  red  dots  
                                                        are  from  the  ocean  atlas  of  Gouretski  and  Koltermann  (2004)  at   p = 0 .      

(

  

)

As   noted   in   section   3.8   the   dianeutral   advection   of   thermobaricity   is   the   same   when  
quantified  in  terms  of   θ   as  when  done  in  terms  of   Θ .    The  same  is  not  true  of  the  dianeutral  
velocity   caused   by   cabbeling.      The   ratio   of   the   cabbeling   dianeutral   velocity   calculated   using  
potential   temperature   to   that   using   Conservative   Temperature   is   given   by  
Cbθ ∇nθ ⋅ ∇nθ CbΘ∇n Θ ⋅ ∇n Θ   (see  section  3.9)  which  can  be  expressed  as    

(

)(

Cbθ ∇nθ

2

CbΘ ∇n Θ

2

Cθ
= bΘ
Cb

)
(α
(α

Θ

θ

βΘ
β

θ

)
)

2

2

2

=

Cbθ ⎛ TbΘ ⎞
Cθ
= bΘ θˆΘ + ⎡⎣α Θ β Θ ⎤⎦ θˆSA
Θ ⎜ θ ⎟
Cb ⎝ Tb ⎠
Cb

(

) ,   
2

(A.14.8)  

and   this   is   contoured   in   Fig.   A.14.2.      While   the   ratio   of   Eqn.   (A.14.8)   is   not   exactly   unity,   it  
varies  relatively  little  in  the  oceanographic  range,  indicating  that  the  dianeutral  advection  due  
to  cabbeling  estimated  using   θ   or   Θ   are  within  half  a  percent  of  each  other  at   p = 0 .        

(

Figure   A.14.2.      Contours   of   the   percentage   difference   of   Cbθ ∇nθ
                                                      from  unity  at   p = 0   dbar.      

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2

  

) (C

Θ
b

∇n Θ

2

)     

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101

A.15 Derivation of the expressions for α θ , β θ , α Θ and β Θ
This  appendix  derives  the  expressions  in  Eqns.  (2.18.2)  –  (2.18.3)  and  (2.19.2)  –  (2.19.3)  for  the  
thermal  expansion  coefficients   α θ   and   α Θ   and  the  haline  contraction  coefficients   β θ   and   β Θ .       
In  order  to  derive  Eqn.  (2.18.2)  for   α θ   we  first  need  an  expression  for   ∂θ ∂T
.     This  is  
SA , p
found   by   differentiating   with   respect   to   in   situ   temperature   the   entropy   equality  
η ( SA , t, p ) = η ( SA ,θ [SA , t , p, pr ], pr )   which  defines  potential  temperature,  obtaining    

∂θ
∂T

=
SA , p

g ( S , t, p )
ηT ( SA , t , p )
(T +θ ) c p ( SA , t, p ) .   
= TT A
= 0
gTT ( SA ,θ , pr )
ηT ( SA ,θ , pr )
(T0 + t ) c p ( SA ,θ , pr )

(A.15.1)  

This  is  then  used  to  obtain  the  desired  expression  Eqn.  (2.18.2)  for   α θ   as  follows    

1 ∂v
α =
v ∂θ
θ

1 ∂v
=
v ∂T
SA , p

⎛ ∂θ
⎜
⎜
SA , p ⎝ ∂T

⎞
⎟
⎟
SA , p ⎠

−1

=

gTP ( SA , t, p ) gTT ( SA ,θ , pr )
g P ( SA , t , p ) gTT ( SA , t , p )

.  

(A.15.2)  

In  order  to  derive  Eqn.  (2.18.3)  for   α Θ   we  first  need  an  expression  for   ∂Θ ∂T
.     This  is  
SA , p
found   by   differentiating   with   respect   to   in   situ   temperature   the   entropy   equality  
η ( SA , t , p ) = ηˆ ( SA , Θ [ SA , t , p ])   obtaining    

g ( S , t, p )
(T + θ ) c p ( SA , t , p ) ,   
∂Θ
∂Θ
= ηT ( SA , t , p )
= − (T0 + θ ) TT A0
= 0
∂T SA , p
∂η S
cp
c0p
(T0 + t )

(A.15.3)  

A

where  the  second  part  of  this  equation  has  used  Eqn.  (A.12.4)  for   Θη
obtain  the  desired  expression  Eqn.  (2.18.3)  for   α Θ   as  follows    

1 ∂v
1 ∂v
=
α =
v ∂Θ SA , p
v ∂T
Θ

SA

.     This  is  then  used  to  

−1

⎛ ∂Θ
⎞
c0p
gTP ( SA , t, p )
.   
⎜
⎟ = −
⎜
⎟
g P ( SA , t , p ) (T0 + θ ) gTT ( SA , t , p )
SA , p ⎝ ∂T SA , p ⎠

(A.15.4)  

In  order  to  derive  Eqn.  (2.19.2)  for   β θ   we  first  need  an  expression  for   ∂θ ∂SA
.     This  is  
T, p
found   by   differentiating   with   respect   to   Absolute   Salinity   at   fixed   in   situ   temperature   and  
pressure   the   entropy   equality   η ( SA , t , p ) = η ( SA ,θ [SA , t , p, pr ], pr )    which   defines   potential  
temperature,  obtaining    

∂θ
∂SA

= θη
T, p

=
=

SA

⎡ηS ( SA , t, p ) − ηS ( SA ,θ , pr )⎤
A
⎣ A
⎦

(T0 + θ ) ⎡ µ S ,θ , p − µ S , t, p ⎤
  
)⎦
T ( A
r)
T ( A
c p ( S A , θ , pr ) ⎣
⎡ g S T ( SA , t , p ) − g S T ( SA , θ , pr )⎤ gTT ( SA , θ , pr ) ,
A
⎣ A
⎦

(A.15.5)  

where  Eqns.  (A.12.5)  and  (A.12.7)  have  been  used  with  a  general  reference  pressure   pr   rather  
than   with   pr = 0.       By   differentiating   ρ = ρ SA ,θ ⎡⎣ SA ,t, p, pr ⎤⎦ , p    with   respect   to   Absolute  
Salinity  it  can  be  shown  that  (Gill  (1982),  McDougall  (1987a))    

(

βθ =

1 ∂ρ
ρ ∂SA

=
θ, p

1 ∂ρ
ρ ∂SA

)

+ αθ
T, p

∂θ
∂SA

,   

(A.15.6)  

T, p

and  using  Eqn.  (A.15.5)  we  arrive  at  the  desired  expression  Eqn.  (2.19.2)  for   β θ     

βθ = −

( SA , t , p )
g P ( SA , t , p )

gS

AP

+

gTP ( SA , t , p ) ⎡⎣ g SAT ( SA , t , p ) − g SAT ( S A ,θ , pr )⎤⎦
g P ( S A , t , p ) gTT ( S A , t , p )

.   

(A.15.7)  

Note   that   the   terms   in   the   Gibbs   function   in   the   natural   logarithm   of   the   square   root   of  
Absolute   Salinity   cancel   from   the   two   parts   of   the   square   brackets   in   Eqns.   (A.15.5)   and  
(A.15.7).      

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In  order  to  derive  Eqn.  (2.19.3)  for   β Θ   we  first  need  an  expression  for   ∂Θ ∂SA
.     This  is  
T, p
found   by   differentiating   with   respect   to   Absolute   Salinity   at   fixed   in   situ   temperature   and  
pressure   the   entropy   equality   η ( SA , t , p ) = ηˆ ( SA , Θ [ SA , t , p ])    obtaining      (using   Eqns.   (A.12.4)  
and  (A.12.8))    

∂Θ
∂SA

= Θη
T, p

SA

⎡η ( S , t , p ) − ηˆ
⎤
SA
Θ⎦
⎣ SA A

= ⎡⎣ µ ( SA ,θ ,0 ) − (T0 + θ ) µT ( SA , t , p )⎤⎦ c 0p

  

(A.15.8)  

= ⎡⎣ g SA ( SA , θ ,0 ) − (T0 + θ ) g SAT ( SA , t , p )⎤⎦ c 0p .
Differentiating   ρ = ρˆ ( SA , Θ [ SA , t , p ], p )   with  respect  to  Absolute  Salinity  leads  to  

βΘ =

1 ∂ρ
ρ ∂SA

=
Θ, p

1 ∂ρ
ρ ∂SA

+ αΘ
T, p

∂Θ
∂SA

,   

(A.15.9)  

T, p

and  using  Eqn.  (A.15.8)  we  arrive  at  the  desired  expression  (2.19.3)  for   β Θ   namely    

βΘ = −

( SA , t , p )
g P ( SA , t , p )

gS

AP

+

gTP ( SA , t , p ) ⎡⎣ g SAT ( SA , t , p ) − g SA ( SA , θ ,0 ) (T0 + θ )⎤⎦
g P ( SA , t , p ) gTT ( SA , t , p )

.    (A.15.10)    

Note   that   the   terms   in   the   Gibbs   function   in   the   natural   logarithm   of   the   square   root   of  
Absolute   Salinity   cancel   from   the   two   parts   of   the   square   brackets   in   Eqns.   (A.15.8)   and  
(A.15.10).      
  
  

A.16 Non-conservative production of entropy
In   this   and   the   following   three   appendices   (A.16   –   A.19)   the   non-­‐‑conservative   nature   of  
several  thermodynamic  variables  (entropy,  potential  temperature,  Conservative  Temperature  
and  specific  volume)  will  be  quantified  by  considering  the  mixing  of  pairs  of  seawater  parcels  
at  fixed  pressure.    The  mixing  is  taken  to  be  complete  so  that  the  end  state  is  a  seawater  parcel  
that  is  homogeneous  in  Absolute  Salinity  and  entropy.    That  is,  we  will  be  considering  mixing  
to   completion   by   a   turbulent   mixing   process.      In   appendix   A.20   the   non-­‐‑conservative  
production   of   Absolute   Salinity   by   the   remineralization   of   particulate   organic   matter   is  
considered.      This   process   is   not   being   considered   in   appendices   A.16   –   A.19.      The   non-­‐‑
conservative  production  which  is  quantified  in  appendices  A.16  –  A.19  occurs  in  the  absence  
of  any  variation  in  seawater  composition.      
Following   Fofonoff   (1962),   consider   mixing   two   fluid   parcels   (parcels   1   and   2)   that   have  
initially  different  temperatures  and  salinities.    The  mixing  process  occurs  at  pressure   p.     The  
mixing  is  assumed  to  happen  to  completion  so  that  in  the  final  state  Absolute  Salinity,  entropy  
and   all   the   other   properties   are   uniform.      Assuming   that   the   mixing   happens   with   a  
vanishingly   small   amount   of   dissipation   of   kinetic   energy,   the   ε    term   can   be   dropped   from  
the  First  Law  of  Thermodynamics,  (A.13.1),  this  equation  becoming    

( ρ h )t

+ ∇⋅ ( ρ u h ) = − ∇⋅ FR − ∇⋅ FQ .   

at  constant  pressure  (A.16.1)  

Note  that  this  equation  has  the  form  (A.8.1)  and  so   h   is  conserved  during  mixing  at  constant  
pressure,   that   is,   h    is   “isobaric   conservative”.      In   the   case   we   are   considering   of   mixing   the  
two  seawater  parcels,  the  system  is  closed  and  there  are  no  radiative,  boundary  or  molecular  
heat   fluxes   coming   through   the   outside   boundary   so   the   integral   over   space   and   time   of   the  
right-­‐‑hand  side  of  Eqn.  (A.16.1)  is  zero.    The  surface  integral  of   ( ρ u h )   through  the  boundary  
is  also  zero.    Hence  it  is  apparent  that  the  volume  integral  of   ρ h   is  the  same  at  the  final  state  
as  it  is  at  the  initial  state,  that  is,  enthalpy  is  conserved.    Hence  during  the  mixing  process  the  
mass,  salt  content  and  enthalpy  are  conserved,  that  is    

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103

m1 + m2 = m ,   

(A.16.2)  

m1 SA1 + m2 SA2 = m SA ,   

(A.16.3)  

m1 h1 + m2 h2 = mh ,  

(A.16.4)  

while  the  non-­‐‑conservative  nature  of  entropy  means  that  it  obeys  the  equation,    

m1 η1 + m2 η2 + m δη = mη .   

(A.16.5)  

Here   SA , h   and   η   are  the  values  of  Absolute  Salinity,  enthalpy  and  entropy  of  the  final  mixed  
fluid   and   δη    is   the   production   of   entropy,   that   is,   the   amount   by   which   entropy   is   not  
conserved   during   the   mixing   process.      Entropy   η    is   now   regarded   as   the   functional   form  

η = η SA ,h, p   and  is  expanded  in  a  Taylor  series  of   SA   and   h   about  the  values  of   SA   and   h   
of  the  mixed  fluid,  retaining  terms  to  second  order  in   [SA2 − SA1 ] = ΔSA   and  in   [h2 − h1 ] = Δh .     
Then   η1   and   η2   are  evaluated  and  (A.16.4)  and  (A.16.5)  used  to  find        

(

)

δη = −

1
2

m1 m2
m

2

{

2



ηhh ( Δh ) + 2 ηhS Δh ΔSA + ηS
A

A SA

( ΔSA )2 } .   

(A.16.6)  

Graham  and  McDougall  (2013)  have  derived  the  following  evolution  equation  for  entropy  
in   a   turbulent   ocean,   involving   the   epineutral   diffusivity   K    and   the   vertical   turbulent  
diffusivity   D    (see   appendix   A.21   for   the   meaning   of   the   symbols   in   this   thickness-­‐‑weighted  
averaged  equation)    

(

) ( ) + (T ε+ t )

dη̂
∂η̂
∂η̂
=
+ v̂ ⋅∇ nη̂ + e
= γ z ∇ n ⋅ γ z−1 K∇ nη̂ + Dη̂ z
dt
∂t n
∂z
⎛ ĥ
+ K ⎜ η̂Θ ΘΘ
ĥΘ
⎝

z

0

⎛ ĥ
⎞
⎞
ΘSA
− η̂ΘΘ ⎟ ∇ nΘ̂ ⋅∇ nΘ̂ + 2K ⎜ η̂Θ
− η̂ΘS ⎟ ∇ nΘ̂ ⋅∇ n ŜA
A⎟
⎜⎝
ĥΘ
⎠
⎠
⎛ ĥ
⎞
S S
+ K ⎜ η̂Θ A A − η̂S S ⎟ ∇ n ŜA ⋅∇ n ŜA   
A A⎟
⎜⎝
ĥΘ
⎠

(A.16.7)  

⎛ ĥ
⎞
⎛ ĥ
⎞
ΘSA
+ D ⎜ η̂Θ ΘΘ − η̂ΘΘ ⎟ Θ̂ 2z + 2D ⎜ η̂Θ
− η̂ΘS ⎟ Θ̂ z ŜA
A⎟
z
⎜⎝
ĥΘ
ĥΘ
⎝
⎠
⎠
⎛ ĥ
⎞
S S
+ D ⎜ η̂Θ A A − η̂S S ⎟ ŜA
A A⎟
z
⎜⎝
ĥΘ
⎠

( )

2

.

Towards  the  end  of  this  section  the  implications  of  the  production  (A.16.6)  of  entropy  will  
be   quantified,   but   for   now   we   ask   what   constraints   the   Second   Law   of   Thermodynamics  
might   place   on   the   form   of   the   Gibbs   function   g ( SA , t , p )    of   seawater.      The   Second   Law   of  
Thermodynamics   tells   us   that   the   entropy   excess   δη    must   not   be   negative   for   all   possible  
combinations  of  the  differences  in  enthalpy  and  salinity  between  the  two  fluid  parcels.    From  
(A.16.6)  this  requirement  implies  the  following  three  inequalities,  


(A.16.8)  
   ηhh < 0 ,              ηS S < 0 ,   

(η )
hSA

A A

2

 
< ηhh ηS

A SA

,   

(A.16.9)  

where  the  last  requirement  reflects  the  need  for  the  discriminant  of  the  quadratic  in  (A.16.6)  to  
be  negative.    Since  entropy  is  already  a  first  derivative  of  the  Gibbs  function,  these  constraints  
would   seem   to   be   three   different   constraints   on   various   third   derivatives   of   the   Gibbs  
function.    In  fact,  we  will  see  that  they  amount  to  only  two  rather  well-­‐‑known  constraints  on  
second  order  derivatives  of  the  Gibbs  function.      
From   the   fundamental   thermodynamic   relation   (A.7.1)   we   find   that   (where   T    is   the  
absolute  temperature,   T = T0 + t )    

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∂η

     ηh =
∂h

1
  
T

=
SA , p

∂η

ηS =
A
∂SA

(A.16.10)  

µ
,   
T

= −
h, p

(A.16.11)  


and  from  these  relations  the  following  expressions  for  the  second  order  derivatives  of   η   can  
be  found,    
∂2 η

ηhh =
∂h2

ηS


ηS

Ah

A SA

=

=

∂T −1
∂h

=
SA , p

∂2 η
∂h∂SA

∂2 η
∂SA2
h, p

(

p

(

SA , p

∂ −µ T

=

=

=

∂h

∂ −µ T
∂SA

−T −2
,   
cp

)

= −
SA , p

)

−
T,p

(

1 ⎛ µ⎞
,   
c p ⎜⎝ T ⎟⎠ T

∂ −µ T
∂h

(A.16.12)  

)
SA , p

∂h
∂SA

(A.16.13)  

T,p

  
(A.16.14)  
⎡
⎤
⎞
⎛
T
µ
= − A −
⎢
⎥ .
T
c p ⎢⎣⎜⎝ T ⎟⎠ T ⎥⎦



The  last  equation  comes  from  regarding   ηS   as   ηS = ηS SA ,h ⎡⎣ SA ,t, p ⎤⎦ , p .       
A
A
A

The  constraint  (A.16.8)  that   ηhh < 0   simply  requires  (from  (A.16.12))  that  the  isobaric  heat  

capacity   c p   is  positive,  or  that   gTT < 0 .     The  constraint  (A.16.8)  that   ηS S < 0 ,   requires  (from  
A A
(A.16.14))  that    

µS

2

2

(

)

2

⎡⎛ µ ⎞ ⎤
(A.16.15)  
g SA SA
⎢⎜ ⎟ ⎥ ,   
⎣ ⎝ T ⎠T ⎦
that   is,   the   second   derivative   of   the   Gibbs   function   with   respect   to   Absolute   Salinity   gS S   
A A

 
must  exceed  some  negative  number.    The  constraint  (A.16.9)  that   (ηhS )2 < ηhh ηS S   requires  
A
A A
that  (substituting  from  (A.16.12),  (A.16.13)  and  (A.16.14))    
g SA SA
(A.16.16)  
> 0 ,   
T 3c p
T3
> −
cp

and  since  the  isobaric  heat  capacity  must  be  positive,  this  requirement  is  that   gS S > 0 ,   and  
A A
so  is  more  demanding  than  (A.16.15).      
We   conclude   that   while   there   are   the   three   requirements   (A.16.8)   to   (A.16.9)   on   the  

functional  form  of  entropy   η = η SA ,h, p   in  order  to  satisfy  the  constraint  of  the  Second  Law  
of  Thermodynamics  that  entropy  be  produced  when  water  parcels  mix,  these  three  constraints  
are  satisfied  by  the  following  two  constraints  on  the  form  of  the  Gibbs  function   g ( SA , t , p ) ,    
(A.16.17)  
gTT < 0   

(

and    

)

gSASA > 0.   

(A.16.18)  

The   Second   Law   of   Thermodynamics   does   not   impose   any   additional   requirement   on   the  
cross  derivatives   gS T   nor  on  any  third  order  derivatives  of  the  Gibbs  function.      
A
The  constraint  (A.16.18)  can  be  understood  by  considering  the  molecular  diffusion  of  salt,  
which   is   known   to   be   directed   down   the   gradient   of   chemical   potential   µ ( SA , t, p )    (see   Eqn.  
(B.21)).    That  is,  the  molecular  flux  of  salt  is  proportional  to   −∇µ .     Expanding   −∇µ   in  terms  of  
gradients  of  Absolute  Salinity,  of  temperature,  and  of  pressure,  one  finds  that  the  first  term  is  
−µSA ∇SA   and  in  order  to  avoid  an  unstable  explosion  of  salt  one  must  have   µSA = gSASA > 0.     
Hence   the   constraint   (A.16.18)   amounts   to   the   requirement   that   the   molecular   diffusivity   of  
salt  is  positive.      
The  two  constraints  (A.16.17)  and  (A.16.18)  on  the  Gibbs  function  are  well  known  in  the  
thermodynamics   literature.      Landau   and   Lifshitz   (1959)   derive   them   on   the   basis   of   the  

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105

contribution  of  molecular  fluxes  of  heat  and  salt  to  the  production  of  entropy  (their  equations  
58.9  and  58.13).    Alternatively,  Planck  (1935)  as  well  as  Landau  and  Lifshitz  (1980)  in  their  §96  
(this   is   §98   in   editions   before   the   1976   extension   made   by   Lifshitz   and   Pitayevski)   inferred  
such   inequalities   from   thermodynamic   stability   considerations.      It   is   pleasing   to   obtain   the  
same   constraints   on   the   seawater   Gibbs   function   from   the   above   Non-­‐‑Equilibrium  
Thermodynamics   approach   of   mixing   fluid   parcels   since   this   approach   involves   turbulent  
mixing  which  is  the  type  of  mixing  that  dominates  in  the  ocean;  molecular  diffusion  has  the  
complementary  role  of  dissipating  tracer  variance.      
In   addition   to   the   Second   Law   requirements   (A.16.17)   and   (A.16.18)   there   are   other  
constraints   which   the   seawater   Gibbs   function   must   obey.      One   is   that   the   adiabatic   (and  
isohaline)  compressibility  must  be  positive  for  otherwise  the  fluid  would  expand  in  response  
to  an  increase  in  pressure  which  is  an  unstable  situation.    Taking   gP > 0   (since  specific  volume  
needs   to   be   positive)   the   requirement   that   the   adiabatic   (and   isohaline)   compressibility   be  
positive  imposes  the  following  two  constraints  (from  (2.16.1))    

g PP < 0   

(A.16.19)  

and    

( gTP )2

< gPP gTT ,   

(A.16.20)  

recognizing  that   gTT   is  negative  ( gTP   may,  and  does,  take  either  sign).    Equation  (A.16.20)  is  
more   demanding   of   gPP    than   is   (A.16.19),   requiring   gPP    to   be   less   than   a   negative   number  
rather   than   simply   being   less   than   zero.      This   last   inequality   can   also   be   regarded   as   a  
constraint  on  the  thermal  expansion  coefficient   α t ,  implying  that  its  square  must  be  less  than  
g P−2 g PP gTT    or   otherwise   the   relevant   compressibility   ( κ )   would   be   negative   and   the   sound  
speed  complex.      
The  constraints  on  the  seawater  Gibbs  function   g ( SA , t, p )   that  have  been  discussed  above  
are  summarized  as    

g p > 0,      gSASA > 0 ,     gPP < 0 ,    gTT < 0 ,   and   ( gTP ) < g PP gTT .  
2

(A.16.21)  

We   return   now   to   quantify   the   non-­‐‑conservative   production   of   entropy   in   the   ocean.    
When   the   mixing   process   occurs   at   p = 0,    the   expression   (A.16.6)   for   the   production   of  
entropy   can   be   expressed   in   terms   of   Conservative   Temperature   Θ    (since   Θ    is   simply  
proportional   to   h    at   p = 0 )   as   follows   (now   entropy   is   taken   to   be   the   functional   form  
η = ηˆ ( SA , Θ) )    

{

}

m1 m2
2
2
(A.16.22)  
ηˆΘΘ ( ΔΘ) + 2ηˆΘSA ΔΘΔSA + ηˆSASA ( ΔSA ) .   
m2
The   maximum   production   occurs   when   parcels   of   equal   mass   are   mixed   so   that  
1
m m m −2 = 81   and  we  adopt  this  value  in  what  follows.    To  illustrate  the  magnitude  of  this  
2 1 2
non-­‐‑conservation   of   entropy   we   first   scale   entropy   by   a   dimensional   constant   so   that   the  
resulting   variable   (“entropic   temperature”)   has   the   value   25 °C    at   ( SA , Θ) = ( SSO ,25 °C )    and  
then   Θ   is  subtracted.    The  result  is  contoured  in   SA − Θ   space  in  Figure  A.16.1.      
The   fact   that   the   variable   in   Figure   A.16.1   is   not   zero   over   the   whole   SA − Θ    plane   is  
because  entropy  is  not  a  conservative  variable.    The  non-­‐‑conservative  production  of  entropy  
can   be   read   off   this   figure   by   selecting   two   seawater   samples   and   mixing   along   the   straight  
line   between   these   parcels   and   then   reading   off   the   production   (in   °C )   of   entropy   from   the  
figure.    Taking  the  most  extreme  situation  with  one  parcel  at   ( SA , Θ) = 0 g kg −1,0 °C   and  the  
other  at  the  warmest  and  saltiest  corner  of  the  figure,  the  production  of   η   on  mixing  parcels  
of  equal  mass  is  approximately  0.9 °C .      
Since  entropy  can  be  expressed  independently  of  pressure  as  a  function  of  only  Absolute  
Salinity   and   Conservative   Temperature   η = ηˆ ( SA , Θ) ,   and   since   at   any   pressure   in   the   ocean  
both   S A   and   Θ   may  quite  accurately  be  considered  conservative  variables  (see  appendix  A.18  
below),   it   is   clear   that   the   non-­‐‑conservative   production   given   by   (A.16.22)   and   illustrated   in  
Figure  A.16.1  is  very  nearly  equivalent  to  the  slightly  more  accurate  expression  (A.16.6)  which  

δη = −

1
2

(

)

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applies  at  any  pressure.    The  only  discrepancy  between  the  production  of  entropy  calculated  
from  (A.16.22)  and  that  from  (A.16.6)  is  due  to  the  very  small  non-­‐‑conservative  production  of  
Θ    at   pressures   other   than   zero   (as   well   as   the   fact   that   both   expressions   contain   only   the  
second  order  terms  in  an  infinite  Taylor  series).      

Figure  A.16.1.    Contours  (in   °C )  of  a  variable  which  illustrates  the  non-­‐‑conservative  
                  production  of  entropy   η   in  the  ocean.      
  
  

A.17 Non-conservative production of potential temperature
When   fluid   parcels   undergo   irreversible   and   complete   mixing   at   constant   pressure,   the  
thermodynamic   quantities   that   are   conserved   during   the   mixing   process   are   mass,   Absolute  
Salinity  and  enthalpy.    As  in  section  A.16  we  again  consider  two  parcels  being  mixed  without  
external  input  of  heat  or  mass  and  the  three  equations  that  represent  the  conservation  of  these  
quantities  are  again  Eqns.  (A.16.2)  –  (A.16.4).    The  production  of  potential  temperature  during  
the  mixing  process  is  given  by    

m1 θ1 + m2 θ2 + m δθ = mθ .   

(

(A.17.1)  

)

Enthalpy   in   the   functional   form   h = h SA ,θ , p    is   expanded   in   a   Taylor   series   of   S A    and   θ   
about   the   values   S A    and   θ    of   the   mixed   fluid,   retaining   terms   to   second   order   in  
[SA2 − SA1 ] = ΔSA   and  in   [θ2 −θ1 ] = Δθ .     Then   h1   and   h2   are  evaluated  and  Eqns.  (A.16.4)  and  
(A.17.1)  used  to  find    
⎫
hθ S
hS S
m m ⎧⎪ h
2
2⎪
δθ = 12 1 2 2 ⎨ θθ Δθ + 2  A Δθ ΔSA + A A ΔSA ⎬ .   
(A.17.2)  
hθ
hθ
m ⎪ hθ
⎪
⎩
⎭
The   maximum   production   occurs   when   parcels   of   equal   mass   are   mixed   so   that  
1
m m m −2 = 81 .       The   “heat   capacity”   hθ    is   not   a   strong   function   of   θ    but   is   a   stronger  
2 1 2
function   of   SA    so   the   first   term   in   the   curly   brackets   in   Eqn.   (A.17.2)   is   generally   small  
compared   with   the   second   term.      Also,   the   third   term   in   Eqn.   (A.17.2)   which   causes   the   so-­‐‑
called  “dilution  heating”,  is  usually  small  compared  with  the  second  term.    A  typical  value  of  
hθ S   is  approximately  –5.4   J kg−1 K −1 (g kg −1 ) −1   (see  the  dependence  of  isobaric  heat  capacity  
A
on   S A    in   Figure   4   of   section   2.20)   so   that   an   approximate   expression   for   the   production   of  
potential  temperature   δθ   is    

( )

δθ
≈
Δθ

1 
h ΔSA
4 θ SA

(

(

)

)

hθ ≈ − 3.4x10−4 ΔSA / [g kg −1] .   

(A.17.3)  

The  same  form  of  the  non-­‐‑conservative  production  terms  in  Eqn.  (A.17.2)  also  appears  in  the  
following   turbulent   evolution   equation   for   potential   temperature,   in   both   the   epineutral   and  

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107

vertical   diffusion   terms   (Graham   and   McDougall,   2013).      (See   appendix   A.21   for   an  
explanation  of  the  symbols  that  appear  in  this  thickness-­‐‑weighted  averaged  equation.)    
∂θˆ
∂θˆ
dθˆ
=
+ v̂ ⋅∇ nθˆ + e
= γ z ∇ n ⋅ γ z−1 K∇ nθˆ + Dθˆz + ε hθ
z
dt
∂t
∂z

(

) ( )

n

⎛ h
⎞
hθ S
hS S
+ K ⎜ θθ ∇ nθˆ ⋅∇ nθˆ + 2  A ∇ nθˆ ⋅∇ n ŜA + A A ∇ n ŜA ⋅∇ n ŜA ⎟         (A.17.4)  
⎜⎝ hθ
⎟⎠
hθ
hθ
⎛ h
2⎞
hθ S
hS S
+ D ⎜ θθ θˆz2 + 2  A θˆz ŜA + A A ŜA ⎟ .
z
z
⎜⎝ hθ
⎟⎠
hθ
hθ
Since  potential  temperature   θ = θˆ ( S , Θ )   can  be  expressed  independently  of  pressure  as  a  

( )

A

function  of  only  Absolute  Salinity  and  Conservative  Temperature,  and  since  during  turbulent  
mixing  both   S A   and   Θ   may  be  considered  approximately  conservative  variables  (see  section  
A.18   below),   it   is   clear   that   the   non-­‐‑conservative   production   given   by   (A.17.2)   can   be  
approximated  by  the  corresponding  production  of  potential  temperature  that  would  occur  if  
the  mixing  had  occurred  at   p = 0 ,  namely    

δθ ≈

1
2


m1 m2 ⎧⎪ Θ
θθ
Δθ
⎨
2

Θ
m ⎪ θ
⎩

( )

2



Θ
Θ
θS
S S
+ 2  A Δθ ΔSA + A A ΔSA
Θθ
Θθ

(

⎫

)2 ⎪⎬ ,  
⎪⎭

(A.17.5)  

where   the   exact   proportionality   between   potential   enthalpy   and   Conservative   Temperature  
h 0 ≡ c0p Θ    has   been   exploited.      The   maximum   production   occurs   when   parcels   of   equal   mass  
are  mixed  so  that   12 m1 m2 m −2 = 81   and  we  adopt  this  value  in  what  follows.      
Equations  (A.17.2)  or  (A.17.5)  may  be  used  to  evaluate  the  non-­‐‑conservative  production  of  
potential   temperature   due   to   mixing   a   pair   of   fluid   parcels   across   a   front   at   which   there   are  
known   differences   in   salinity   and   temperature.      The   temperature   difference   θ − Θ    is  
contoured  in  Figure  A.17.1  and  can  be  used  to  illustrate  Eqn.  (A.17.5).     δθ   can  be  read  off  this  
figure   by   selecting   two   seawater   samples   and   mixing   along   the   straight   line   between   these  
parcels   (along   which   both   Absolute   Salinity   and   Conservative   Temperature   are   conserved)  
and  then  calculating  the  production  (in   °C )  of   θ   from  the  contoured  values  of   θ − Θ .    Taking  
the   most   extreme   situation   with   one   parcel   at   ( SA , Θ) = 0 g kg −1 ,0 °C    and   the   other   at   the  
warmest  and  saltiest  corner  of  Figure  A.17.1,  the  non-­‐‑conservative  production  of   θ   on  mixing  
parcels   of   equal   mass   is   approximately   -­‐‑0.55 °C .      This   is   to   be   compared   with   the  
corresponding   maximum   production   of   entropy,   as   discussed   above   in   connection   with  
Figure  A.16.1,  of  approximately  0.9   °C .      
If  Figure  A.17.1  were  to  be  used  to  quantify  the  errors  in  oceanographic  practice  incurred  
by  assuming  that   θ   is  a  conservative  variable,  one  might  select  property  contrasts  that  were  
typical  of  a  prominent  oceanic  front  and  decide  that  because   δθ   is  small  at  this  one  front,  that  
the  issue  can  be  ignored  (see  for  example,  Warren  (2006)).    But  the  observed  properties  in  the  
ocean  result  from  a  large  and  indeterminate  number  of  such  prior  mixing  events  and  the  non-­‐‑
conservative  production  of   θ   accumulates  during  each  of  these  mixing  events,  often  in  a  sign-­‐‑
definite   fashion.      How   can   we   possibly   estimate   the   error   that   is   made   by   treating   potential  
temperature  as  a  conservative  variable  during  all  of  these  unknowably  many  past  individual  
mixing   events?      This   seemingly   difficult   issue   is   partially   resolved   by   considering   what   is  
actually   done   in   ocean   models   today.      These   models   carry   a   temperature   conservation  
equation   that   does   not   have   non-­‐‑conservative   source   terms,   so   that   the   model’s   temperature  
variable   is   best   interpreted   as   being   Θ .      This   being   the   case,   the   temperature   difference  
contoured   in   Figure   A.17.1   illustrates   the   error   that   is   made   by   interpreting   the   model  
temperature   as   being   θ .      That   is,   the   values   contoured   in   Figures   A.16.1   and   A.17.1   are  
representative   of   the   error,   expressed   in   °C ,   associated   with   assuming   that   η    and   θ   
respectively   are   conservative   variables.      These   contoured   values   of   temperature   difference  
encapsulate   the   accumulated   non-­‐‑conservative   production   that   has   occurred   during   all   the  

(

)

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many  mixing  processes  that  have  lead  to  the  ocean’s  present  state.    The  maximum  such  error  
for   η    is   approximately   -­‐‑1.0   °C    (from   Figure   A.16.1)   while   for   θ    the   maximum   error   is  
approximately  -­‐‑1.8   °C   (from  Figure  A.17.1).    One  percent  of  the  data  at  the  sea  surface  of  the  
world   ocean   have   values   of   θ − Θ    that   lie   outside   a   range   that   is   0.25   °C    wide   (McDougall  
(2003)),  implying  that  this  is  the  magnitude  of  the  error  incurred  by  ocean  models  when  they  
treat   θ   as  a  conservative  quantity.    To  put  a  temperature  difference  of  0.25   °C   in  context,  this  
is  the  typical  difference  between  in  situ  and  potential  temperatures  for  a  pressure  difference  of  
2500   dbar,   and   it   is   approximately   100   times   as   large   as   the   typical   differences   between   t90   
and   t68    in   the   ocean.      From   the   curvature   of   the   isolines   on   Figure   A.17.1   it   is   clear   that   the  
non-­‐‑conservative  production  of   θ   takes  both  positive  and  negative  signs.      

  
  

Figure  A.17.1.    Contours  (in   °C )  of  the  difference  between  potential  temperature    
and   Conservative   Temperature   θ − Θ .      This   plot   illustrates   the   non-­‐‑
conservative  production  of  potential  temperature   θ   in  the  ocean.      

A.18 Non-conservative production of Conservative Temperature
When   fluid   parcels   undergo   irreversible   and   complete   mixing   at   constant   pressure,   the  
thermodynamic  quantities  that  are  conserved  are  mass,  Absolute  Salinity  and  enthalpy.    As  in  
sections  A.16  and  A.17  we  consider  two  parcels  being  mixed  without  external  input  of  heat  or  
mass,   and   the   three   equations   that   represent   the   conservation   of   these   quantities   are   Eqns.  
(A.16.2)   –   (A.16.4).      Neither   potential   enthalpy   h0    nor   Conservative   Temperature   Θ    are  
exactly  conserved  during  the  mixing  process  and  the  production  of   Θ   is  given  by    
(A.18.1)  
m1 Θ1 + m2 Θ2 + m δΘ = m Θ.   
ˆ
Enthalpy   in   the   functional   form   h = h ( SA , Θ, p )    is   expanded   in   a   Taylor   series   of   S A    and   Θ   
about   the   values   S A    and   Θ    of   the   mixed   fluid,   retaining   terms   to   second   order   in  
[SA2 − SA1 ] = ΔSA   and  in   [Θ2 −Θ1 ] = ΔΘ.     Then   h1   and   h2   are  evaluated  and  Eqns.  (A.16.4)  and  

(A.18.1)  are  used  to  find    

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δΘ =

1
2

⎧⎪ hˆ
⎫
hˆΘSA
hˆSA SA
2
2⎪
ΘΘ
⎨ ˆ ( ΔΘ) + 2 ˆ ΔΘΔSA + ˆ ( ΔSA ) ⎬ .   
hΘ
hΘ
⎩⎪ hΘ
⎭⎪

m1 m2
m2

109

(A.18.2)  

Graham   and   McDougall   (2013)   have   shown   that   the   same   form   of   the   non-­‐‑conservative  
production   terms   in   Eqn.   (A.18.2)   also   appears   in   the   following   turbulent   evolution   equation  
for   Conservative   Temperature,   in   both   the   epineutral   and   vertical   diffusion   terms   (see  
appendix   A.21   for   an   explanation   of   the   symbols   that   appear   in   this   thickness-­‐‑weighted  
averaged  equation),    

(

) (

)

∂Θ̂
dΘ̂
∂Θ̂
=
+ v̂ ⋅∇ nΘ̂ + e
= γ z ∇ n ⋅ γ z−1 K∇ nΘ̂ + DΘ̂ z + ε ĥΘ
z
∂t
dt
∂z
n

⎛ ĥ
⎞
ĥS S
ĥΘS
+ K ⎜ ΘΘ ∇ nΘ̂ ⋅∇ nΘ̂ + 2 A ∇ nΘ̂ ⋅∇ n ŜA + A A ∇ n ŜA ⋅∇ n ŜA ⎟         (A.18.3)  
⎜⎝ ĥΘ
⎟⎠
ĥΘ
ĥΘ
⎛ ĥ
ĥΘS
ĥS S
+ D ⎜ ΘΘ Θ̂ 2z + 2 A Θ̂ z ŜA + A A ŜA
z
z
⎜⎝ ĥΘ
ĥΘ
ĥΘ

2⎞

( ) ⎟⎟⎠ .

The  mesoscale  epineutral  turbulent  fluxes  in  Eqn.  (A.18.3)  have  been  expressed  in  terms  of  the  
epineutral   diffusivity   K .      In   terms   of   the   turbulent   mesoscale   fluxes   themselves   rather   than  
ˆ ⋅∇ Θ
ˆ    and   K ∇ Sˆ ⋅ ∇ Sˆ    are   minus   the   scalar  
their   parameterized   versions,   the   terms   K∇n Θ
n
n A
n A
product  of  the  epineutral  flux  of   Θ   and   S A   with  their  respective  epineutral  gradients,  while  
ˆ ⋅ ∇ Sˆ   is  the  sum  of  minus  the  scalar  product  of  the  epineutral  flux  of   Θ   and   ∇ Sˆ ,  
2 K ∇n Θ
n A
n A
ˆ .      
and  minus  the  scalar  product  of  the  epineutral  flux  of   S A   and   ∇ n Θ
In   order   to   evaluate   the   partial   derivatives   in   Eqns.   (A.18.2)   and   (A.18.3),   we   first   write  
enthalpy  in  terms  of  potential  enthalpy  (i.  e.   c0p Θ )  using  Eqn.  (3.2.1),  as    

h = hˆ ( SA , Θ, p ) = c0p Θ +

∫P vˆ ( SA , Θ, p′) dP′.   
P

(A.18.4)  

0

This  is  differentiated  with  respect  to   Θ   giving    

hΘ S

A, p

= hˆΘ = c0p +

P

∫P α

Θ

ρ dP′.   

(A.18.5)  

0

The  right-­‐‑hand  side  of  Eqn.  (A.18.5)  scales  as   c0p + ρ −1 ( P − P0 )α Θ ,   which  is  more  than   c 0p   by  
only   about   0.0015 c0p    for   ( P − P0 )    of   4 × 107    Pa   (4,000   dbar).      Hence,   to   a   very   good  
approximation,   ĥΘ    in   Eqns.   (A.18.2)   and   (A.18.3)   may   be   taken   to   be   simply   c 0p .      It   is  
interesting   to   examine   why   this   approximation   is   so   accurate   when   the   difference   between  
enthalpy,   h,   and  potential  enthalpy,   h0 ,   as  given  by  Eqns.  (3.2.1)  and  (A.18.4),  scales  as   ρ −1P   
which   is   as   large   as   typical   values   of   potential   enthalpy.      The   reason   is   that   the   integral   in  
Eqns.  (3.2.1)  or  (A.18.4)  is  dominated  by  the  integral  of  the  mean  value  of   ρ −1 ,   so  causing  a  
significant   offset   between   h    and   h 0    as   a   function   of   pressure   but   not   affecting   the   partial  
derivative   ĥΘ   which  is  taken  at  fixed  pressure.    Even  the  dependence  of  density  on  pressure  
alone  does  not  affect   hˆΘ .       
The  second  order  derivatives  of   ĥ   are  needed  in  Eqns.  (A.18.2)  and  (A.18.3),  and  these  can  
be  estimated  by  differentiating  Eqn.  (A.18.4)  or  (A.18.5),  so  that,  for  example,    

hˆΘΘ =

P

∫P vˆΘΘ
0

dP′ =

∫P (α
P

0

Θ

ρ

)

Θ

dP′ ,   

(A.18.6)  

so  that  we  may  write  Eqn.  (A.18.2)  approximately  as  (assuming   m1 = m2 )    

δΘ ≈

( P − P0 )
8c0p

{ vˆ

ΘΘ

( ΔΘ)2 + 2 vˆSAΘ ΔΘ ΔSA

+ vˆSA SA ( ΔSA )

2

} .   

(A.18.7)  

This  equation  is  approximate  because  the  variation  of   v̂ΘΘ , v̂ΘS   and   v̂S S   with  pressure  has  
A
A A
been  ignored.    The  dominant  term  in  Eqn.  (A.18.7)  is  usually  the  term  in   v̂ΘΘ   and  from  Eqn.  
(A.19.2)   below   we   see   that   δΘ    is   approximately   proportional   to   the   non-­‐‑conservative  
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destruction   of   specific   volume   at   fixed   pressure   caused   by   the   “cabbeling”   non-­‐‑linearities   in  
the  equation  of  state  (McDougall,  1987b),  so  that    
( P − P0 ) vˆ ΔΘ 2 ≈ − ( P − P0 ) δ v .   
(A.18.8)  
δΘ ≈
)
ΘΘ (
c0p
8c0p
The   production   of   Θ    causes   an   increase   in   Conservative   Temperature   and   a   consequent  
decrease   in   density   of   − ρα ΘδΘ .      The   ratio   of   this   change   in   density   (using   Eqn.   (A.18.7))   to  
that   caused   by   cabbeling   (from   Eqn.   (A.19.2)   and   using   δρ ≈ − ρ 2 δ v )   is   − ( P − P0 ) α Θ ρ c0p   
which  is  about  0.0015  for  a  value  of   ( P − P0 )   of  40  MPa.    Hence  it  is  clear  that  cabbeling  has  a  
much   larger   effect   on   density   than   does   the   non-­‐‑conservation   of   Θ .       Nevertheless,   it   is  
interesting   to   note   from   Eqn.   (A.18.7)   that   the   non-­‐‑conservative   production   of   Θ    is  
approximately  proportional  to  the  product  of  sea  pressure  and  the  strength  of  cabbeling.      
The  first  term  in  the  bracket  in  Eqn.  (A.18.7)  is  usually  about  a  factor  of  ten  larger  than  the  
other  two  terms  (McDougall  (1987b)),  so  the  production  of  Conservative  Temperature   δΘ   as  a  
ratio   of   the   contrast   in   Conservative   Temperature   ΔΘ = Θ2 −Θ1   may   be   approximated   as  
(since   v̂ΘΘ ≈ − ρ −2 ρˆ ΘΘ ≈ ρ −1αΘΘ )  

δΘ
ΔΘ

≈

( P − P0 )αΘΘ ΔΘ
8ρ c0p

≈ 3.3 × 10−9 ( p dbar )( ΔΘ K ) .   

(A.18.9)  

where   α ΘΘ   has  been  taken  to  be   1.1 × 10−5 K −2   (McDougall,  1987b).      
At   the   sea   surface   Conservative   Temperature   Θ    is   totally   conserved   ( δΘ = 0 ).      The  
expression  for  the  non-­‐‑conservative  production  of  Conservative  Temperature,   δΘ ,   increases  
almost   linearly   with   pressure   (see   Eqn.   (A.18.7))   but   at   larger   pressure   the   range   of  
temperature  and  salinity  in  the  ocean  decreases,  and  from  the  above  equations  it  is  clear  that  
the  magnitude  of   δΘ   is  proportional  to  the  square  of  the  temperature  and  salinity  contrasts.    
McDougall   (2003)   concluded   that   the   production   δΘ    between   extreme   seawater   parcels   at  
each   pressure   is   largest   at   600   dbar.      The   magnitude   of   the   non-­‐‑conservative   production   of  
Conservative  Temperature,   δΘ ,   is  illustrated  in  Figure  A.18.1  for  data  at  this  pressure.    The  
quantity   contoured   on   this   figure   is   the   difference   between   Θ    and   the   following   totally  
conservative  quantity  at   p = 600   dbar.    This  conservative  quantity  was  constructed  by  taking  
the   conservative   property   enthalpy   h    at   this   pressure   and   adding   the   linear   function   of   S A   
which   makes   the   result   equal   to   zero   at  
( SA = 0, Θ= 0 °C)    and   at  
SA = 35.165 04 g kg −1, Θ= 0 °C .       This   quantity   is   then   scaled   so   that   it   becomes   25 °C    at  
SA = 35.165 04 g kg −1, Θ = 25 °C .       In   this   manner   the   quantity   that   is   contoured   in   Figure  
A.18.1  has  units  of   °C   and  represents  the  amount  by  which  Conservative  Temperature   Θ   is  
not   a   totally   conservative   variable   at   a   pressure   of   600   dbar.      The   maximum   amount   of  
production  by  mixing  seawater  parcels  at  the  boundaries  of  Figure  A.18.1  is  about   4 × 10−3 °C   
although   the   range   of   values   encountered   in   the   real   ocean   at   this   pressure   is   actually   quite  
small,  as  indicated  in  Figure  A.18.1.      
From   the   curvature   of   the   isolines   on   Figure   A.18.1   it   is   clear   that   the   non-­‐‑conservative  
production  of  Conservative  Temperature  at   p = 600   dbar  is  positive,  so  that  an  ocean  model  
that   ignores   this   production   of   Conservative   Temperature   will   slightly   underestimate   Θ .    
From  Eqn.  (A.18.2)  one  sees  the  non-­‐‑conservative  production  of  Conservative  Temperature  is  
always   positive   if   hˆΘΘ > 0 ,   hˆS S > 0    and   (hˆΘS )2 < hˆΘΘ hˆS S ,   and   Graham   and   McDougall  
A A
A
A A
(2013)  have  shown  that  these  requirements  are  met  everywhere  in  the  full  TEOS-­‐‑10  ranges  of  
salinity,  temperature  and  pressure  for  the  full  TEOS-­‐‑10  Gibbs  function  and  this  is  also  the  case  
when  using  the  75-­‐‑term  polynomial  for  specific  volume  in  its  range  of  applicability.      
From  Eqns.  (A.18.9)  and  (A.17.3)  we  can  write  the  ratio  of  the  production  of  Conservative  
Temperature  to  the  production  of  potential  temperature  when  two  seawater  parcels  mix  as  the  
approximate  expression    

(
(

)

)

δΘ
≈ −10−5 ( p dbar )( ΔΘ K ) ΔSA / [g kg −1 ]
δθ

(

IOC Manuals and Guides No. 56

)

−1

.   

(A.18.10)  

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111

Taking  a  typical  ratio  of  temperature  differences  to  salinity  differences  in  the  deep  ocean  to  be  
5K / [g kg −1 ] ,  Eqn.  (A.18.10)  becomes   δΘ δθ ≈ − 5 x10−5 ( p dbar ).     At  a  pressure  of  4000  dbar  
this   ratio   is   δΘ δθ ≈ − 0.2    implying   that   Conservative   Temperature   is   a   factor   of   five   more  
conservative  than  potential  temperature  at  these  great  depths.    Note  also  that  the  temperature  
and   salinity   contrasts   in   the   deep   ocean   are   small,   so   the   non-­‐‑conservation   of   both   types   of  
temperature   amount   to   very   small   temperature   increments   of   both   δθ    and   δΘ.       The   largest  
non-­‐‑conservative  increment  of  Conservative  Temperature   δΘ   seems  to  occur  at  a  pressure  of  
about   600   dbar   (McDougall   (2003))   and   this   value   of   δΘ    is   approximately   two   orders   of  
magnitude  less  than  the  maximum  value  of   δθ   which  occurs  at  the  sea  surface.    The  material  
in  appendices  A.16  -­‐‑  A.18  has  closely  followed  McDougall  (2003).      

  
  

Figure  A.18.1.    Contours  (in   °C )  of  a  variable  that  is  used  to  illustrate  the  non-­‐‑  
conservative   production   of   Conservative   Temperature   Θ    at   p = 600    dbar.    
The   cloud   of   points   show   where   most   of   the   oceanic   data   reside   at   p = 600   
dbar.    The  three  points  that  are  forced  to  be  zero  are  shown  with  black  dots.      

A.19 Non-conservative production of specific volume
Following   Graham   and   McDougall   (2013)   specific   volume   is   expressed   as   a   function   of  

Absolute  Salinity   S A ,  specific  enthalpy   h   and  pressure  as   v = v SA ,h, p   and  the  same  mixing  
process  between  two  fluid  parcels  is  considered  as  in  the  previous  appendices.    Mass,  salt  and  
enthalpy  are   conserved   during   the   turbulent   mixing   process   (Eqns.   (A.16.2)  -­‐‑   (A.16.4))   while  
the  non-­‐‑conservative  nature  of  specific  volume  means  that  it  obeys  the  equation,    

(

)

m1 v1 + m2 v2 + m δ v = mv .   

(A.19.1)  

Specific  volume  is  expanded  in  a  Taylor  series  of   S A   and   h   about  the  values  of   S A   and   h   of  
the   mixed   fluid   at   pressure   p ,   retaining   terms   to   second   order   in   [SA2 − SA1 ] = ΔSA    and   in  
[h2 − h1 ] = Δh .    Then   v1   and   v2   are  evaluated  and  (A.19.1)  is  used  to  find    

δv = −

1
2

≈ −

1
2

m1 m2
2

m
m1 m2

{
{

( )

2



vhh Δh + 2 vhS Δh ΔSA + vS

( )

A

2

A SA

( ΔSA )2 }
(

)}

  

(A.19.2)  

2

v̂ΘΘ ΔΘ + 2 v̂ΘS ΔΘ ΔSA + v̂S S ΔSA .
A
A A
m2
The   non-­‐‑conservative   destruction   of   specific   volume   of   Eqn.   (A.19.2)   is   illustrated   in  
Figure   A.19.1   for   mixing   at   p = 0    dbar.      The   quantity   contoured   on   this   figure   is   formed   as  
follows.      First   the   linear   function   of   S A    is   found   that   is   equal   to   specific   volume   at  

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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

( SA = 0, Θ= 0 °C)   

(

)

and   at   SA = 35.165 04 g kg −1, Θ= 0 °C .       This   linear   function   of   S A    is  
subtracted  from   v   and  the  result  is  scaled  to  equal   25 °C   at   SA = 35.165 04 g kg −1 , Θ= 25 °C .     
The   variable   that   is   contoured   in   Figure   A.19.1   is   the   difference   between   this   scaled   linear  
combination   of   v    and   S A ,   and   Conservative   Temperature.      This   figure   allows   the   non-­‐‑
conservative  nature  of  specific  volume  to  be  understood  in  temperature  units.    The  mixing  of  
extreme   fluid   parcels   on   Figure   A.19.1   causes   the   same   decrease   in   specific   volume   as   a  
cooling   of   approximately   10 °C ,   which   is   approximately   4000   times   larger   than   the  
corresponding  non-­‐‑conservative  production  of   Θ   at  600dbar  (from  Figure  A.18.1).      
From   Eqn.   (A.19.2)   it   follows   that   specific   volume   is   always   destroyed   by   turbulent  





mixing   processes   if   vhh > 0 ,   vS S > 0    and   ( vhS )2 < vS S vhh ,   and   Graham   and   McDougall  
A
A A
A A
(2013)  have  shown  that  these  conditions  are  satisfied  over  the  full  TEOS-­‐‑10  ranges  of  salinity,  
temperature  and  pressure  by  the  full  TEOS-­‐‑10  Gibbs  function  and  this  is  also  true  of  the  75-­‐‑
term   expression   for   specific   volume   of   Appendix   K.      Note   that   in   contrast   to   the   case   of  
specific  volume,  the  non-­‐‑conservation  of  density  is  not  sign-­‐‑definite.    That  is,  while  turbulent  
mixing  always  destroys  specific  volume,  it  does  not  always  produce  density   ρ = v −1 .      
The  fact  that  turbulent  mixing  at  constant  pressure  always  destroys  specific  volume  also  
implies  that  internal  energy  is  always  produced  by  this  turbulent  mixing  at  constant  pressure  
(see  the  First  Law  of  Thermodynamics,  Eqn.  (B.19)).      

(

)

Figure  A.19.1.    Contours  (in   °C )  of  a  variable  that  is  used  to  illustrate  the  non-­‐‑  
conservative   production   of   specific   volume   at   p   =   0   dbar.      The  
three  points  that  are  forced  to  be  zero  are  shown  with  black  dots.      
  
  

A.20 The representation of salinity in numerical ocean models
Ocean  models  need  to  evaluate  salinity  at  every  time  step  as  a  necessary  prelude  to  using  the  
equation   of   state   to   determine   density   and   its   partial   derivatives   for   use   in   the   hydrostatic  
relationship  and  in  neutral  mixing  algorithms.    The  current  practice  in  numerical  models  is  to  
treat  salinity  as  a  perfectly  conserved  quantity  in  the  interior  of  the  ocean;  salinity  changes  at  
the   surface   and   at   coastal   boundaries   due   to   evaporation,   precipitation,   brine   rejection,   ice  
melt   and   river   runoff,   and   satisfies   an   advection-­‐‑diffusion   equation   away   from   these  
boundaries.      The   inclusion   of   composition   anomalies   necessitates   several   changes   to   this  
approach.    These  changes  can  be  divided  into  two  broad  categories.    First,  in  addition  to  fresh  
water  inputs  and  brine  rejection,  all  sources  of  dissolved  material  entering  through  the  surface  
and  coastal  boundaries  of  the  model  should  be  considered  as  possible  sources  of  composition  

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113

anomalies.      Second,   within   the   interior   of   the   model,   changes   due   to   the   growth,   decay   and  
remineralization   of   biological   material   must   be   considered.      Here,   we   focus   on   this   second  
issue.      While   the   ultimate   resolution   of   these   issues   will   involve   biogeochemical   models,   in  
this   appendix   we   discuss   some   practical   ways   forward   based   on   the   approximate   relations  
(A.5.7)  -­‐‑   (A.5.12)  between  the  salinity  variables   SR , S*   and   SA = SAdens .    At  the  time  of  writing,  
the   suggested   approaches   here   have   not   been   tested,   so   it   must   be   acknowledged   that   the  
treatment  of  seawater  composition  anomalies  in  ocean  models  is  currently  a  work  in  progress.      
We  begin  by  repeating  Eqns.  (A.5.11)  and  (A.5.12),  namely    
(A.20.1)  

SA =

(A.20.2)  

where    

Rδ ≡

(
)
S (1+ F δ ) ,  

S* = SR 1 − r1Rδ ,  

δ SAatlas
S Ratlas

*

[1+ r1 ] Rδ

            and             F δ =

(1 − r Rδ )

.  

(A.20.3)  

1

δ

Recall   that   the   Absolute   Salinity   Anomaly   Ratio,   R ≡ δ SAatlas S Ratlas ,   is   the   ratio   of   the   atlas  
values   of   Absolute   Salinity   Anomaly   and   Reference   Salinity.      The   stored   values   of   Rδ    are  
interpolated   onto   the   latitude,   longitude   and   pressure   of   an   oceanographic   observation.    
Rδ   is  bounded  between  zero  0.001  in  the  global  ocean.    With   r1   taken  to  be   0.35   we  note  the  
following  approximate  expression   F δ = SA S* − 1 ≈ 1.35 Rδ .      
  
A.20.1    Using  Preformed  Salinity   S*   as  the  conservative  salinity  variable    
Because   Preformed   Absolute   Salinity   S*    (henceforth   referred   to   by   the   shortened   name,  
Preformed   Salinity)   is   designed   to   be   a   conservative   salinity   variable,   blind   to   the   effects   of  
biogeochemical   processes,   its   evolution   equation   will   be   in   the   conservative   form   (A.8.1).    
When  this  type  of  conservation  equation  is  averaged  in  the  appropriate  manner  (see  appendix  
A.21)  the  conservation  equation  for  Preformed  Salinity  becomes  (from  Eqn.  (A.21.7)),    

⎛ ∂ Ŝ ⎞
dŜ*
= γ z ∇ n ⋅ γ z−1 K∇ n Ŝ* + ⎜ D * ⎟ .   
dt
⎝ ∂z ⎠ z

(

)

(A.20.4)  

As   explained   in   appendix   A.21,   the   over-­‐‑tilde   of   Ŝ*    indicates   that   this   variable   is   the  
thickness-­‐‑weighted   average   Preformed   Salinity,   having   been   averaged   between   a   pair   of  
closely   spaced   neutral   tangent   planes.      The   material   derivative   on   the   left-­‐‑hand   side   of   Eqn.  
(A.20.4)   is   with   respect   to   the   sum   of   the   Eulerian   and   quasi-­‐‑Stokes   velocities   of   height  
coordinates  (equivalent  to  the  description  in  appendix  A.21  in  terms  of  the  thickness-­‐‑weighted  
average   horizontal   velocity   and   the   mean   dianeutral   velocity).      The   right-­‐‑hand   side   of   this  
equation   is   the   standard   notation   indicating   that   Ŝ*    is   diffused   along   neutral   tangent   planes  
with  the  diffusivity  K  and  in  the  vertical  direction  with  the  diapycnal  diffusivity  D  (and   γ z−1   is  
proportional  to  the  average  thickness  between  two  closely  spaced  neutral  tangent  planes).      
In   order   to   evaluate   density   during   the   running   of   an   ocean   model,   Absolute   Salinity  
SA = SAdens    must   be   evaluated.      This   can   be   done   from   Eqn.   (A.20.2)   as   the   product   of   the  
model’s   salinity   variable   Ŝ*    and   (1 + F δ ) .      This   could   be   done   by   simply   multiplying   the  
model’s  salinity  by  the  fixed  spatial  map  of   (1 + F δ )   as  observed  today  (using   r1 = 0.35   and  
the  value  of   Rδ   obtained  from  the  computer  algorithm  of  McDougall  et  al.  (2012)).    However  
experience   has   shown   that   even   a   smooth   field   of   density   errors   can   result   in   significant  
anomalies  in  diagnostic  model  calculations,  primarily  due  to  the  misalignment  of  the  density  
errors   and   the   model   bottom   topography.      Indeed,   even   if   the   correct   mean   density   could  
somehow   be   determined,   approximations   associated   with   the   specification   of   the   model  
bottom   topography   would   result   in   significant   errors   in   bottom   pressure   torques   that   can  
degrade   the   model   solution.      One   way   to   minimize   such   errors   is   to   allow   some   dynamical  
adjustment   of   the   specified   density   field   so   that,   for   example,   density   contours   tend   to   align  

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with  bottom  depth  contours  where  the  flow  is  constrained  to  follow  bottom  topography.    This  
simple  idea  is  the  key  to  the  success  of  the  robust  diagnostic  approach  (Sarmiento  and  Bryan  
(1982)).    To  allow  dynamical  adjustment  of  the  salinity  difference   S A − S*   while  not  permitting  
SA − S*    to   drift   too   far   from   the   observed   values,   we   recommend   carrying   an   evolution  
equation  for   F δ   so  that  it  becomes  an  extra  model  variable  which  evolves  according  to    

⎛ ∂F δ ⎞
dF δ
= γ z ∇ n ⋅ γ z−1 K∇ n F δ + ⎜ D
+ τ −1 F δ obs − F δ .  
dt
∂z ⎟⎠
⎝
z

(

)

(

)

(A.20.5)  

Here   the   model   variable   F δ    would   be   initialized   based   on   observations,   F δ obs    (using   Eqn.  
(A.20.3)   with   r1 = 0.35    and   the   interpolated   values   of   Rδ    from   McDougall   et   al.   (2012)),   and  
advected  and  diffused  like  any  other  tracer,  but  in  addition,  there  is  a  non-­‐‑conservative  source  
term   τ −1 F δ obs − F δ    which   serves   to   restore   the   model   variable   F δ    towards   the   observed  
value  with  a  restoring  time   τ   that  can  be  chosen  to  suit  particular  modeling  needs.    It  should  
be  at  least  30  days  to  permit  significant  adjustment,  but  it  might  prove  appropriate  to  allow  a  
much  longer  adjustment  period  (up  to  several  years)  if  drift  from  observations  is  sufficiently  
slow.    The  lower  bound  is  based  on  a  very  rough  estimate  of  the  time  required  for  the  density  
field   to   be   aligned   with   topography   by   advective   processes.      The   upper   bound   is   set   by   the  
requirement   to   have   the   restoring   time   relatively   short   compared   to   vertical   and   basin-­‐‑scale  
horizontal  redistribution  times.      
Ideally  one  would  like  the  non-­‐‑conservative  source  term  to  reflect  the  actual  physical  and  
chemical   processes   responsible   for   remineralization   in   the   ocean   interior,   but   until   our  
knowledge  of  these  processes  improves  such  that  this  is  possible,  the  approach  based  on  Eqn.  
(A.20.5)   provides   a   way   forward.      An   indication   of   how   an   approach   based   on   modeled  
biogeochemical  processes  might  be  implemented  in  the  future  can  be  gleaned  from  looking  at  
Eqn.  (A.4.14)  for   S A − S∗ .    If  a  biogeochemical  model  produced  estimates  of  the  quantities  on  
the  right-­‐‑hand  side  of  this  equation,  it  could  be  immediately  integrated  into  an  ocean  model  to  
diagnose  the  effects  of  the  biogeochemical  processes  on  the  model'ʹs  density  and  its  circulation.      
In  summary,  the  approach  suggested  here  carries  the  evolution  Eqns.  (A.20.4)  and  (A.20.5)  
for   Ŝ*   and   F δ ,  while   Ŝ A   is  calculated  by  the  model  at  each  time  step  according  to    

(

)

(

)

SˆA = Sˆ* 1+ F δ .  

(A.20.6)  

The   model   is   initialized   with   values   of   Preformed   Salinity   using   Eqn.   (A.20.1)   based   on  
observations   of   Reference   Salinity   and   on   the   interpolated   global   database   of   Rδ    from  
McDougall   et   al.   (2012)   using   r1 = 0.35 .      This   approach   applies   to   the   open   ocean,   but   the  
Baltic   Sea   is   to   be   treated   differently.      As   described   in   appendix   A.5,   the   observed   Absolute  
Salinity   Anomaly   δ SA    in   the   Baltic   Sea   is   not   primarily   due   to   non-­‐‑conservative  
biogeochemical   source   terms   but   rather   is   due   to   rivers   delivering   water   to   the   Baltic   with  
much  larger  Absolute  Salinity  than  would  be  expected  from  the  Practical  Salinity  of  the  river  
discharge.      In   the   Baltic   Sea,   SA = S∗ ,   r1 = − 1    and   F δ = 0    (as   discussed   in   appendix   A.5)   so  
that   in   the   Baltic   region   of   an   ocean   model   the   equation   of   state   should   be   called   with   the  
model’s  salinity  variable,  Preformed  Salinity   S∗ .    The  discharges  (mass  fluxes)  of  river  water  
and  of  Absolute  Salinity  should  both  appear  as  source  terms  at  the  edges  of  the  Baltic  Sea.      
  
A.20.2    Including  a  source  term  in  the  evolution  equation  for  Absolute  Salinity    
An  alternative  procedure  would  be  to  carry  an  evolution  equation  for  Absolute  Salinity  rather  
than  for  Preformed  Salinity  in  an  ocean  model.    Using  Eqns.  (A.20.4)  -­‐‑   (A.20.6),  the  following  
evolution  equation  for  Absolute  Salinity  can  be  constructed,    

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⎛ ∂ Ŝ ⎞
∂ Ŝ
dŜA
Ŝ
= γ z ∇ n ⋅ γ z−1 K∇ n ŜA + ⎜ D A ⎟ − 2K∇ n Ŝ* ⋅∇ n F δ − 2DFzδ * + * F δ obs − F δ
dt
∂z
τ
⎝ ∂z ⎠ z

(

(

= γ z∇ n ⋅ γ

)

−1
K∇ n ŜA
z

)

(

⎛ ∂ Ŝ ⎞
S
+ ⎜ D A ⎟ + Sˆ A .
⎝ ∂z ⎠ z

)

115

      (A.20.7)  

Here   the   non-­‐‑conservative   source   term   in   the   evolution   equation   for   Absolute   Salinity   has  
S
been  given  the  label   Sˆ A   for  later  use.    If  the  ocean  model  resolves  mesoscale  eddies  then  the  
δ
term   − 2 K ∇n Sˆ* ⋅ ∇ n F    in   Eqn.   (A.20.7)   becomes   the   scalar   product   of   ∇ n F δ    and   the  
epineutral   flux   of   S*    plus   the   scalar   product   of   ∇n Sˆ*    and   the   epineutral   flux   of   F δ .      In   this  
approach   the   evolution   equation   (A.20.5)   for   F δ    is   also   carried   and   the   model’s   salinity  
variable,   Ŝ A ,   is   used   directly   as   the   argument   of   the   equation   of   state   and   other  
thermodynamic   functions   in   the   model.      The   model   would   be   initialized   with   values   of  
Absolute   Salinity   using   Eqn.   (A.5.10)   (namely   SA = SR 1 + Rδ )   based   on   observations   of  
Reference   Salinity   and   on   the   global   data   base   of   Rδ    from   McDougall   et   al.   (2012).      The  
production   terms   involving   Ŝ*    in   Eqn.   (A.20.7)   would   need   to   be   evaluated   in   terms   of   the  
model’s  salinity  variable   Ŝ A   using  Eqn.  (A.20.6).      
This   approach   should   give   identical   results   to   that   described   in   section   A.20.1   using  
Preformed   Salinity.      One   disadvantage   of   having   Density   Salinity   as   the   model’s   salinity  
variable   is   that   its   evolution   equation   (A.20.7)   is   not   in   the   conservative   form   so   that,   for  
example,  it  is  not  possible  to  perform  easy  global  budgets  of  salinity  to  test  for  the  numerical  
integrity   of   the   model   code.      Another   disadvantage   is   that   the   air-­‐‑sea   flux   of   carbon   dioxide  
and   other   gases   may   need   to   be   taken   into   account   as   the   surface   boundary   condition   of  
Absolute   Salinity.      Such   air-­‐‑sea   fluxes   do   not   affect   Preformed   Salinity.      But   perhaps   the  
largest   disadvantage   of   this   approach   is   the   difficulty   in   evaluating   the   non-­‐‑conservative  
terms   − 2K ∇n Sˆ* ⋅ ∇n F δ − 2DFzδ ∂Sˆ* ∂z   in  Eqn.  (A.20.7),  especially  when  meso-­‐‑scale  eddies  are  
present,  as  discussed  above.      
  
A.20.3    Discussion  of  the  consequences  if  remineralization  is  ignored    
If   an   ocean   model   does   not   carry   the   evolution   equation   for   F δ    (Eqn.   (A.20.5))   and   the  
model’s  salinity  evolution  equation  does  not  contain  the  appropriate  non-­‐‑conservative  source  
term,  is  there  then  any  preference  for  initializing  and  interpreting  the  model’s  salinity  variable  
as   either   Preformed   Salinity,   Absolute   Salinity   or   Reference   Salinity?      That   is,   the   simplest  
method  of  dealing  with  these  salinity  issues  is  to  continue  the  general  approach  that  has  been  
taken  for  the  past  several  decades  of  simply  taking  one  type  of  salinity  in  the  model,  and  that  
salinity  is  taken  to  be  conservative.    Under  this  approximation  the  salinity  that  is  used  in  the  
equation   of   state   to   calculate   density   in   the   model   is   the   same   as   the   salinity   that   obeys   a  
normal  conservation  equation  of  the  form  Eqn.  (A.20.4).    In  this  approach  there  is  still  a  choice  
of  how  to  initialize  and  to  interpret  the  salinity  in  a  model,  and  here  we  discuss  the  relative  
virtues  of  these  options.      
If   the   model   is   initialized   with   a   data   set   of   estimated   Preformed   Salinity   S* ,   then   S*   
should   evolve   correctly,   since   S*    is   a   conservative   variable   and   its   evolution   equation   Eqn.  
(A.20.4)  contains  no  non-­‐‑conservative  source  terms.    In  this  approach  the  equation  of  state  will  
be   called   with   Ŝ*    rather   than   ŜA ,   and   these   salinities   differ   by   approximately   (1 + r1 ) δ SA .    
The   likely   errors   with   this   approach   can   be   estimated   using   the   simple   example   of   Figure  
A.5.1.    The  vertical  axis  in  this  figure  is  the  difference  between  the  northward  density  gradient  
at   constant   pressure   when   the   equation   of   state   is   called   with   ŜA    and   with   Ŝ R .      The   figure  
shows  that  when  using   Ŝ R ,  for  all  the  data  in  the  world  ocean  below  a  depth  of  1000   m ,  58%  
of   this   data   is   in   error   by   more   than   2%.      If   this   graph   were   re-­‐‑done   with   Ŝ*    as   the   salinity  
argument  rather  than   Ŝ R ,  the  errors  would  be  larger  by  the  ratio   (1 + r1 ) ≈ 1.35 .    That  is,  for  
58%  of  the  data  in  the  world  ocean  deeper  than  1000   m ,  the  “thermal  wind”  relation  would  be  

(

)

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misestimated  by   ≈ 2.7%   if   Ŝ*   is  used  in  place  of   ŜA   as  the  salinity  argument  of  the  equation  
of  state.    Also,  these  percentage  errors  in  “thermal  wind”  are  much  larger  in  the  North  Pacific.      
Another   choice   of   the   salinity   data   to   initialize   the   model   is   ŜA .      An   advantage   of   this  
choice   is   that   initially   the   equation   of   state   is   called   with   the   correct   salinity   variable.    
However   at   later   times,   the   neglect   of   the   non-­‐‑conservative   source   terms   in   Eqn.   (A.20.7)  
means  that  the  model’s  salinity  variable  will  depart  from  reality  and  errors  will  creep  in  due  
to   the   lack   of   these   legitimate   non-­‐‑conservative   source   terms.      How   long   might   it   be  
acceptable   to   integrate   such   a   model   before   the   errors   approached   those   described   in   the  
previous   paragraph?      One   could   imagine   that   in   the   upper   ocean   the   influence   of   these  
different  salinity  variables  is  dwarfed  by  other  physics  such  as  air  sea  interaction  and  active  
gyral  motions.    If  one  considered  a  depth  of  1000m  as  being  a  depth  where  the  influence  of  the  
different   salinities   would   be   both   apparent   and   would   make   a   significant   impact   on   the  
thermal   wind   equation,   then   one   might   guess   that   it   would   take   several   decades   for   the  
neglect  of  the  non-­‐‑conservative  source  terms  in  the  evolution  equation  for  Absolute  Salinity  to  
begin  to  be  important.    This  is  not  to  suggest  that  the  relaxation  time  scale   τ   should  be  chosen  
to  be  as  long  as  this,  rather  this  is  an  estimate  of  how  long  it  would  take  for  the  neglect  of  the  
S
non-­‐‑conservative  source  term   Sˆ A   in  Eqn.  (A.20.7)  to  become  significant.      
A  third  choice  is  to  initialize  the  model  with  Reference  Salinity,   Ŝ R .    This  choice  incurs  the  
errors  displayed  in  Figure  A.5.1  right  from  the  start  of  any  numerical  simulation.    Thereafter,  
on   some   unknown   timescale,   further   errors   will   arise   because   the   conservation   equation   for  
Reference   Salinity   is   missing   the   legitimate   non-­‐‑conservative   source   terms   representing   the  
effects  of  biogeochemistry  on  conductivity  and   Ŝ R .    Hence  this  choice  is  the  least  desired  of  
the  three  considered  in  this  subsection.    Note  that  this  choice  is  basically  the  approach  that  has  
been   used   to   date   in   ocean   modeling   since   we   have   routinely   initialized   models   with  
observations  of  Practical  Salinity  and  have  treated  it  as  a  conservative  variable  and  have  used  
it  as  the  salinity  argument  for  the  equation  of  state.      
To  summarize,  the  approaches  of  both  subsections  A.20.1  and  A.20.2  of  this  appendix  can  
each  account  for  the  non-­‐‑conservative  effects  of  remineralization  if   r1   is  a  constant  and  so  long  
as   the   appropriate   boundary   conditions   are   imposed.      The   advantage   of   using   Ŝ*    is   that   it  
obeys  a  standard  conservative  evolution  equation  (A.20.4)  with  no  source  term  on  the  right-­‐‑
hand  side.    If  an  ocean  model  were  to  be  run  without  carrying  the  evolution  equation  for   F δ   
and  hence  without  the  ability  to  incorporate  the  appropriate  non-­‐‑conservative  source  terms  in  
either   Eqns.   (A.20.6)   or   (A.20.7),   then   the   model   must   resort   to   carrying   only   one   salinity  
variable,   and   this   salinity   variable   must   be   treated   as   a   conservative   variable   in   the   ocean  
model.      In   this   circumstance,   we   advise   that   the   ocean’s   salinity   variable   be   interpreted   as  
Absolute  Salinity,  and  initialized  as  such.    In  this  way,  the  errors  in  the  thermal  wind  equation  
will  develop  only  slowly  over  a  time  scale  of  several  decades  or  more  in  the  deep  ocean.      
The   use   of   an   existing   climatology   for   F δ    and   the   introduction   of   a   rather   arbitrary  
relaxation   time   τ    are   less   than   desirable   features   of   this   way   of   treating   salinity   in   ocean  
models.    An  alternative  strategy  is  available  in  an  ocean  model  that  includes  biogeochemical  
processes   and   carries   evolution   equations   for   Total   Alkalinity   (TA),   Dissolved   Inorganic  
Carbon  (DIC)  as  well  as  nitrate  and  silicate  concentrations.    Having  these  quantities  available  
during   the   running   of   an   ocean   model   allows   the   use   of   the   following   equation   (this   is   Eqn.  
(A.4.14),  from  Pawlowicz  et  al.,  2011)  to  evaluate  Absolute  Salinity      

( SA − S* ) / (g kg−1 ) = (73.7 ΔTA +11.8ΔDIC+81.9 NO3− +50.6 Si(OH)4 )

(molkg −1 ) .  

(A.20.8)  

Under   this   approach,   Preformed   Salinity   would   be   carried   as   the   model’s   conservative  
prognostic  salinity  variable  as  in  Eqn.  (A.20.4),  and  the  above  equation  for   SA − S*   in  terms  of  
the   biogeochemical   variables   would   be   used   to   evaluate   Absolute   Salinity   for   use   in   the  
model’s  expression  for  specific  volume.        

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117

A.21 The material derivatives of S* , S A , SR and Θ in a turbulent ocean
Preformed   Salinity   S*    is   designed   to   be   a   conservative   variable   which   obeys   the   following  
instantaneous  conservation  equation  (based  on  Eqn.  (A.8.1))    

dS*
(A.21.1)  
= − ∇ ⋅ FS .   
dt
There   are   several   different   contributions   to   the   molecular   flux   of   salt   FS ,   expressions   for  
which  can  be  seen  at  equation  (58.11)  of  Landau  and  Lifshitz  (1959)  and  in  Eqn.  (B.23)  below.    
For  completeness,  we  repeat  the  continuity  equation  (A.8.2)  here  as    

( ρ S* )t + ∇ ⋅ ( ρ uS* )

= ρ

ρt + ∇ ⋅ ( ρ u ) = 0.   

(A.21.2)  

Temporally  averaging  this  equation  in  Cartesian  coordinates  (i.  e.  at  fixed   x, y , z )  gives    

( )

ρt + ∇⋅ ρ u = 0,   

(A.21.3)  

which   we   choose   to   write   in   the   following   form,   after   division   by   a   constant   density   ρ 0   
(usually  taken  to  be   1035 kg m −3 ,  see  Griffies  (2004))    

( ρ ρ0 )t + ∇ ⋅ u = 0     where       u ≡ ρu

ρ0 .   

(A.21.4)  

This  velocity   u   is  actually  proportional  to  the  average  mass  flux  of  seawater  per  unit  area.      
The   conservation   equation   for   Preformed   Salinity   (A.21.1)   is   now   averaged   in   the  
corresponding  manner  obtaining  (McDougall  et  al.  2002)    

⎛ ρ ⎞
⎛ ρ ρ⎞
⎜⎝ ρ0 S* ⎟⎠ + ∇ ⋅ ⎜⎝ S* u ⎟⎠ =
t

ρ
ρ0

∂S*
∂t

ρ

ρ

+ u ⋅∇S* = −

1
ρ0

∇ ⋅FS −

1
ρ0

(

)

∇ ⋅ ρ S*′′u′′ .   

(A.21.5)  

ρ

Here   the   Preformed   Salinity   has   been   density-­‐‑weighted   averaged,   that   is,   S* ≡ ρ S* ρ ,   and  
the   double   primed   quantities   are   deviations   of   the   instantaneous   quantity   from   its   density-­‐‑
weighted  average  value.    Since  the  turbulent  fluxes  are  many  orders  of  magnitude  larger  than  
molecular  fluxes  in  the  ocean,  the  molecular  flux  of  salt  is  henceforth  ignored.    
The   averaging   process   involved   in   Eqn.   (A.21.5)   has   not   invoked   the   traditional  
Boussinesq   approximation.      The   above   averaging   process   is   best   viewed   as   an   average   over  
many  small-­‐‑scale  mixing  processes  over  several  hours,  but  not  over  mesoscale  time  and  space  
scales.      This   later   averaging   over   the   energetic   mesoscale   eddies   is   not   always   necessary,  
depending  on  the  scale  of  the  piece  of  ocean  or  ocean  model  that  is  under  investigation.    The  
two-­‐‑stage   averaging   process,   without   invoking   the   Boussinesq   approximation,   over   first  
small-­‐‑scale   mixing   processes   (several   meters)   followed   by   averaging   over   the   mesoscale   (of  
order   100   km)   has   been   performed   by   Greatbatch   and   McDougall   (2003),   yielding   the  
prognostic  equation  for  Preformed  Salinity    

h−1

(

ρ!
ρ0

hŜ*

)

t n

+ h−1∇ n ⋅

(

ρ!
ρ0

) (

hv̂Ŝ* +

ρ!
ρ0

e! Ŝ*

)

=
z

ρ!
ρ0

∂ Ŝ*
∂t

= γ! z ∇ n ⋅

+

ρ!
ρ0

v̂ ⋅∇ n Ŝ* +

n

(

γ! z−1 K∇ n Ŝ*

)

ρ!
ρ0

e!

∂ Ŝ*
∂z

⎛ ∂ Ŝ ⎞
+ ⎜D *⎟ .
⎝ ∂z ⎠ z

  

(A.21.6)  

Here   the   over-­‐‑caret   means   that   the   variable   (e.g.   Ŝ* )   has   been   averaged   in   a   thickness-­‐‑and-­‐‑
density-­‐‑weighted   manner   between   a   pair   of   “neutral   surfaces”   a   small   distance   apart   in   the  
vertical,   v̂    is   the   thickness-­‐‑and-­‐‑density-­‐‑weighted   horizontal   velocity,   e    is   the   dianeutral  
velocity  (the  vertical  velocity  that  penetrates  through  the  neutral  tangent  plane)  and   e   is  the  
temporal   average   of   e   on   the   “neutral   surface”   (that   is,   e    is   not   thickness-­‐‑weighted).      The  
turbulent   fluxes   are   parameterized   by   the   epineutral   diffusivity   K   and   the   dianeutral   (or  
vertical)   diffusivity   D .      γ! z    is   the   vertical   gradient   of   a   suitable   compressibility-­‐‑corrected  
density   such   as   Neutral   Density   or   locally-­‐‑referenced   potential   density,   and   the   averaging  
involved   in   forming   γ! z    is   done   to   preserve   the   average   thickness   between   closely-­‐‑spaced  
neutral   tangent   planes;   that   is,   the   averaging   is   performed   on   γ z−1 .      The   issues   of   averaging  
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involved   in   Eqns.   (A.21.5)   and   (A.21.6)   are   subtle,   and   are   not   central   to   our   purpose   in   this  
thermodynamic   manual.      Hence   we   proceed   with   the   more   standard   Boussinesq   approach,  
but   retain   the   over-­‐‑carets   to   remind   the   reader   of   the   thickness-­‐‑weighted   nature   of   the  
variables.      
Having  derived  this  evolution  equation  (A.21.6)  for  Preformed  Salinity  without  invoking  
the   Boussinesq   approximation,   we   now   follow   common   practice   and   invoke   this  
approximation,  finding  the  simpler  expression    

∂ Ŝ*
∂t

+ v̂ ⋅∇ n Ŝ* + e!
n

⎛ ∂ Ŝ ⎞
∂ Ŝ*
= γ! z ∇ n ⋅ γ! z−1 K∇ n Ŝ* + ⎜ D * ⎟ .   
∂z
⎝ ∂z ⎠ z

(

)

(A.21.7)  

The   left-­‐‑hand   side   is   the   material   derivative   of   the   thickness-­‐‑weighted   Preformed   Salinity  
with   respect   to   the   thickness-­‐‑weighted   horizontal   velocity   v̂    and   the   temporally   averaged  
dianeutral   velocity   e    of   density   coordinates.      The   right-­‐‑hand   side   is   the   divergence   of   the  
turbulent  fluxes  of  Preformed  Salinity;  the  fact  that  the  lateral  diffusion  term  is  the  divergence  
of  a  flux  can  be  seen  when  it  is  transformed  to  Cartesian  coordinates.    The  same  conservation  
statement   Eqn.   (A.21.7)   can   be   derived   without   making   the   Boussinesq   approximation   by   a  
simple  reinterpretation  of  the  vertical  coordinate  as  being  pressure,  and  this  interpretation  is  
now  becoming  common  in  ocean  modelling  (see  Bleck  (1978),  Huang  et  al.  (2001),  de  Szoeke  
and  Samelson  (2002),  Losch  et  al.  (2004)  and  Griffies  (2004)).      
We   now   proceed   to   develop   the   corresponding   evolution   equation   for   Absolute   Salinity  
SA .     Note  that   SA   is  the  convenient  generic  symbol  for  Density  Salinity   SAdens ;  unless  there  is  
room   for   confusion   with   the   other   measures   of   absolute   salinity,   S Asoln    and   SAadd ,   it   proves  
convenient  to  use  the  simpler  symbol   S A   rather  than   SAdens   and  to  use  the  description  Absolute  
Salinity  rather  than  Density  Salinity.      
Absolute  Salinity  obeys  the  instantaneous  evolution  equation  (based  on  Eqn.  (A.8.1))    

( ρ SA )t + ∇ ⋅ ( ρuSA )

= ρ

dSA
S
= − ∇ ⋅FS + ρ S A .   
dt

(A.21.8)  

The  source  term   S A   is  described  in  appendix  A.20  (see  eqn.  (A.20.7)).    This  non-­‐‑conservative  
source   term   is   due   to   biogeochemical   processes,   for   example,   the   remineralization   of  
biological   material;   the   turning   of   particulate   matter   into   dissolved   seasalt.      When   this  
equation  is  density-­‐‑weighted  averaged,  we  find    
S

ρ⎞
⎛ ρ ⎞
⎛ ρ
⎜⎝ ρ0 SA ⎟⎠ + ∇ ⋅ ⎜⎝ SA u ⎟⎠ =
t

= −

ρ
ρ0
1
ρ0

∂SA
∂t

ρ

∇ ⋅FS −

+ u ⋅∇SA
1
ρ0

ρ

(

)

∇ ⋅ ρ SA′′ u′′ +

ρ
ρ0

S

SA

  

ρ

(A.21.9)  

,

which  corresponds  to  Eqn.  (A.21.5)  above.    When  averaged  over  the  mesoscale  the  prognostic  
equation  for  Absolute  Salinity  becomes    

h−1

(

ρ!
ρ0

hŜA

)

t n

+ h−1∇ n ⋅

(

ρ!
ρ0

)(

hv̂ŜA +

ρ!
ρ0

e! ŜA

)=
z

ρ!
ρ0

∂ ŜA
∂t

= γ! z ∇ n ⋅

+

ρ!
ρ0

v̂ ⋅∇ n ŜA +

n

(

γ! z−1 K∇ n ŜA

)

ρ!
ρ0

e!

∂ ŜA
∂z

⎛ ∂ Ŝ ⎞
+⎜D A⎟ +
⎝ ∂z ⎠ z

      (A.21.10)  
ρ!
ρ0

ˆ SA

S

,

and  when  the  Boussinesq  approximation  is  made  we  find  the  simpler  expression    

∂ ŜA
∂t

+ v̂ ⋅∇ n ŜA + e!
n

⎛ ∂ Ŝ ⎞
∂ ŜA
S
= γ! z ∇ n ⋅ γ! z−1 K∇ n ŜA + ⎜ D A ⎟ + Sˆ A .   
∂z
⎝ ∂z ⎠ z

(

)

(A.21.11)  

The  left-­‐‑hand  side  is  the  material  derivative  of  the  thickness-­‐‑weighted  Absolute  Salinity  with  
respect   to   the   thickness-­‐‑weighted   horizontal   velocity   v̂    and   the   temporally   averaged  

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119

dianeutral   velocity   e    of   density   coordinates.      Apart   from   the   non-­‐‑conservative   source   term  
S
Sˆ A ,  the  right-­‐‑hand  side  is  the  divergence  of  the  turbulent  fluxes  of  Absolute  Salinity.      
The  corresponding  turbulent  evolution  equation  for  Reference  Salinity  can  be  shown  to  be    
⎛ ∂ Ŝ ⎞
∂ Ŝ
r1
∂ Ŝ R
S
(A.21.12)  
Sˆ A .   
+ v̂ ⋅∇ n Ŝ R + e! R = γ! z ∇ n ⋅ γ! z−1 K∇ n Ŝ R + ⎜ D R ⎟ +
∂t
∂z
1 + r1
⎝ ∂z ⎠

(

)

n

z

(

)

As  discussed  in  appendices  A.4  and  A.20,  given  our  rather  elementary  knowledge  of  the  way  
variations  in  seawater  composition  affect  conductivity,  we  recommend  that   r1   be  taken  to  be  
the   constant   r1 = 0.35.       Hence   we   see   that   Reference   Salinity   is   affected   by   biogeochemical  
processes  at  the  fraction   0.35/ 1.35   ( ≈ 0.26 )  of  the  corresponding  influence  of  biogeochemistry  
on  Absolute  Salinity   S A .      
We   turn   now   to   consider   the   material   derivative   of   Conservative   Temperature   in   a  
turbulent  ocean.    From  Eqns.  (A.13.5)  and  (A.21.8)  the  instantaneous  material  derivative  of   Θ   
is,  without  approximation,    

ρ c0p

(T0 + θ ) − ∇ ⋅F R − ∇ ⋅FQ + ρε + h ρ S S
)
S
(T0 + t ) (
  
⎡ (T0 + θ )
⎤
S
S
− ⎢
µ ( p ) − µ ( 0 ) ⎥ ( −∇ ⋅F + ρ S ) .
⎢⎣ (T0 + t )
⎥⎦

dΘ
=
dt

A

A

(A.21.13)  

A

The  fact  that  the  right-­‐‑hand  side  of  Eqn.  (A.21.13)  is  not  the  divergence  of  a  flux  means  that   Θ   
is  not  a  100%  conservative  variable.    However,  the  finite-­‐‑amplitude  analysis  of  mixing  pairs  of  
seawater   parcels   in   appendix   A.18   has   shown   that   the   non-­‐‑constant   coefficients   of   the  
divergences   of   the   molecular   fluxes   of   heat   − ∇ ⋅ FQ    and   salt   −∇ ⋅ FS    appearing   on   the   right-­‐‑
hand   side   of   Eqn.   (A.21.13)   are   of   no   practical   consequence   as   they   cause   an   error   in  
Conservative   Temperature   of   no   more   than   1.2   mK    (see   Figure   A.18.1).      These   non-­‐‑ideal  
terms  on  the  right-­‐‑hand  side  of  Eqn.  (A.21.13)  in  a  turbulent  ocean  have  been  shown  to  be  an  
order   of   magnitude   less   than   the   dissipation   term   ρε    which   is   also   justifiably   neglected   in  
S
oceanography  (Graham  and  McDougall,  2013).    The  source  term   ρ S A   was  not  considered  in  
the   mixing   of   seawater   parcels   in   appendix   A.18,   and   we   now   show   that   these   terms   also  
make  negligible  contributions  to  Eqn.  (A.21.13).      
The  partial  derivative  of  enthalpy  with  respect  to  Absolute  Salinity,   hS ,   that  appears  in  
A
Eqn.   (A.21.13)   is   about   − 65 J g −1    (i.e.   − 65 J kg−1 (g kg −1 ) −1 )   at   a   temperature   of   10 °C .      This  
value   can   be   deduced   from   Figure   A.17.1   and   also   from   Figure   30(c)   and   Table   12   of   Feistel  
(2003),  albeit  for  the  Gibbs  function  of  seawater  that  immediately  predated  the  TEOS-­‐‑10  saline  
Gibbs   function   of   Feistel   (2008)   and   IAPWS   (2008).      The   spatial   integral   of   the   source   term  
S
ρ S A   from  the  North  Atlantic  to  the  North  Pacific  is  sufficient  to  cause  a  change  in  Absolute  -­‐‑
Salinity  of  0.025   g kg −1 ,  so  the  maximum  contribution  to  an  error  in   Θ   from  the  source  term  
S
hS ρ S A T0 + θ T0 + t    in   Eqn.   (A.21.13),   when   integrated   over   the   whole   ocean,   is  
A
S
approximately   (c0p ) −1 65 J g −1 0.025 g kg −1 ≈ 0.4 mK .      The   other   term   in   ρ S A    in   Eqn.  
(A.21.13)  is  multiplied  by  the  square  bracket  which  from  equation  (27)  of  McDougall  (2003)  is  
equal   to   (T0 + θ ) (T0 + t )    times   approximately   − pβ Θ ρ −1 ,   so   that   this   square   bracket   is  
approximately  30   J g −1   (i.e.   30 J kg−1 (g kg −1 ) −1 )  at  a  pressure   p   of  4000  dbar  (40  MPa)  so  the  
S
contribution   of   this   term   is   less   than   half   that   of   the   term   in   ρ S A    in   the   first   line   of   Eqn.  
S
(A.21.13).      This   confirms   that   the   presence   of   the   two   terms   in   ρ S A    in   the   First   Law   of  
Thermodynamics   has   less   impact   than   even   the   non-­‐‑ideal   nature   of   the   molecular   flux  
divergence  terms  in  Eqn.  (A.21.13)  and  the  dissipation  of  kinetic  energy  in  this  equation.      
Hence  with  negligible  error,  the  right-­‐‑hand  side  of  Eqn.  (A.21.13)  may  be  regarded  as  the  
sum  of  the  ideal  molecular  flux  of  heat  term   − ∇ ⋅ FQ   and  the  term  due  to  the  boundary  and  
radiative  heat  fluxes,   − (T0 + θ ) ∇ ⋅ F R (T0 + t ) .       At   the   sea   surface   the   potential   temperature  
θ   and  in  situ  temperature  t  are  equal  so  that  this  term  is  simply   − ∇ ⋅ FR   so  that  there  are  no  
approximations   with   treating   the   air-­‐‑sea   sensible,   latent   and   radiative   heat   fluxes   as   being  

(

)(

)

(

)(

)

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fluxes  of   c0p Θ.     There  is  an  issue  at  the  sea  floor  where  the  boundary  heat  flux  (the  geothermal  
heat  flux)  affects  Conservative  Temperature  through  the  “heat  capacity”   (T0 + t ) c0p (T0 + θ )   
rather   than   simply   c 0p .       That   is,   the   input   of   a   certain   amount   of   geothermal   heat   flux   will  
cause   a   local   change   in   Θ    as   though   the   seawater   had   the   “specific   heat   capacity”  
(T0 + t ) c0p (T0 + θ )   rather  than   c 0p .     These  two  specific  heat  capacities  differ  from  each  other  
by   no   more   than   0.15%   at   a   pressure   of   4000   dbar.      If   this   small   percentage   change   in   the  
effective   “specific   heat   capacity”   was   ever   considered   important,   it   could   be   corrected   by  
artificially   multiplying   the   geothermal   heat   flux   at   the   sea   floor   by   T0 + θ ⎡⎣T0 + t ⎤⎦ c0p ,   so  
becoming  the  geothermal  flux  of  Conservative  Temperature.      
Graham   and   McDougall   (2013)   have   derived   the   evolution   equation   for   Conservative  
Temperature   for   a   turbulent   ocean   while   retaining   the   non-­‐‑conservative   source   terms   (see  
Eqn.  (A.18.3)  above),  and  have  used  these  terms  to  quantify  the  extent  of  the  non-­‐‑conservation  
of   Θ   in  a  realistic  ocean  model.    This  work  has  confirmed  that  Conservative  Temperature  is  
two   orders   of   magnitude   more   conservative   in   the   ocean   than   is   potential   temperature,   and  
has  also  shown  that  the  neglect  of  the  dissipation  of  kinetic  energy   ε   is  more  than  an  order  of  
magnitude   more   important   than   the   neglect   of   the   non-­‐‑ideal   nature   of   the   Conservative  
Temperature  variable  (that  is,  the  neglect  of  the  last  two  lines  of  Eqn.  (A.18.3)).    We  conclude  
that   for   the   purpose   of   accounting   for   the   transport   of   “heat”   in   the   ocean   it   is   sufficiently  
accurate   to   assume   that   Conservative   Temperature   is   in   fact   conservative   and   that   its  
instantaneous  conservation  equation  is    

(

)

(

)

dΘ
(A.21.14)  
= − ∇ ⋅ FR − ∇ ⋅ FQ .   
dt
Now  we  perform  the  same  two-­‐‑stage  averaging  procedure  as  outlined  above  in  the  case  of  
Preformed  Salinity.    The  Boussinesq  form  of  the  mesoscale-­‐‑averaged  equation  is  (analogous  to  
Eqn.  (A.21.7))    
c0p ( ρ Θ)t + c0p ∇ ⋅ ( ρ Θu ) = ρ c0p

(

) (

)

∂Θ̂
∂Θ̂
+ v̂ ⋅∇ nΘ̂ + e!
= γ! z ∇ n ⋅ γ! z−1 K∇ nΘ̂ + DΘ̂ z − F bound .   
z
∂t
∂z

(A.21.15)    

n

As  in  the  case  of  the   S*   equation  (A.21.7),  the  molecular  flux  of  heat  has  been  ignored  in  
comparison   with   the   turbulent   fluxes   of   Conservative   Temperature.      The   air-­‐‑sea   fluxes   of  
sensible  and  latent  heat,  the  radiative  and  the  geothermal  heat  fluxes  remain  in  Eqn.  (A.21.15)  
in   the   vertical   heat   flux   F bound    which   is   the   sum   of   these   boundary   heat   fluxes   divided   by  
ρ0c0p .     Any  conservative  variable,   C ,   obeys  a  conservation  equation  identical  in  form  to  Eqns.  
(A.21.7)   and   (A.21.15),   with   Ĉ    simply   replacing   Ŝ*    or   Θ̂    in   these   equations,   and   of   course  
with  the  boundary  flux  being  the  boundary  flux  of  property   C .      
The   errors   incurred   in   ocean   models   by   treating   potential   temperature   θ    as   being  
conservative  have  not  yet  been  thoroughly  investigated,  but  McDougall  (2003),  Tailleux  (2010)  
and  Graham  and  McDougall  (2013)  have  made  a  start  on  this  topic.    McDougall  (2003)  found  
that  typical  errors  in   θ   are   ± 0.1°C   while  in  isolated  regions  such  as  where  the  fresh  Amazon  
water  discharges  into  the  ocean,  the  error  can  be  as  large  as   1.4 °C .    The  corresponding  error  
in  the  meridional  heat  flux  appears  to  be  about  0.005  PW  (or  a  relative  error  of  0.4%).    The  use  
of  Conservative  Temperature   Θ   in  ocean  models  reduces  the  non-­‐‑conservative  source  terms  
associated   with   the   use   of   potential   temperature   by   two   orders   of   magnitude   (Graham   and  
McDougall,   2013).      Note   that   the   consequences   for   dynamical   oceanography   of   ignoring   the  
non-­‐‑conservative  source  terms  in  the  potential  temperature  evolution  equation  are  larger  than  
ignoring  the  variations  in  seawater  composition;  a   θ   range  of   0.2 °C   corresponds  to  a  density  
range  of   0.04 kg m−3   which  is  twice  as  large  as  the  density  error  due  to  ignoring  the  maximum  
value  of   SA − SR   of   0.025 g kg −1 .      
The   evolution   equations   of   Preformed   Salinity   (A.21.7)   and   Conservative   Temperature  
(A.21.15)   are   the   underpinning   conservation   equations   for   these   variables   in   ocean   models.    
An   important   issue   for   ocean   models   is   how   to   relate   v̂    to   the   Eulerian-­‐‑mean   horizontal  
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velocity   v .      This   area   of   research   involves   temporal-­‐‑residual-­‐‑mean   theory   and   the   quasi-­‐‑
Stokes   streamfunction   (Gent   and   McWilliams   (1990),   Gent   et   al.   (1995),   McDougall   and  
McIntosh  (2001)  and  Griffies  (2004)).    We  will  not  discuss  this  topic  here.    Suffice  it  to  say  that  
the   mean   advection   can   be   expressed   in   Cartesian   coordinates,   with   for   example,   Eqn.  
(A.21.15)  becoming    

(

) (

)

dΘ̂
= Θ̂ t + v̂ ⋅∇ z Θ̂ + w*Θ̂ z = γ! z ∇ n ⋅ γ! z−1 K∇ nΘ̂ + DΘ̂ z − F bound ,   
z
z
dt

(A.21.16)  

where  the  vertical  velocity   w*   is  related  to   e   by    

w* = zt + v̂ ⋅∇ n z + e .   

(A.21.17)  

n

  
  

A.22 The material derivatives of density and of locally-referenced
potential density; the dianeutral velocity e
Regarding   density   to   be   a   function   of   Conservative   Temperature   (i.   e.   ρ = ρˆ ( SA , Θ, p ) )   and  
taking   the   material   derivative   of   the   natural   logarithm   of   density   following   the   mesoscale-­‐‑
thickness-­‐‑weighted-­‐‑averaged  mean  flow  (as  in  Eqns.  (A.21.15)  or  (A.21.16)),  we  have    

ˆ
dρˆ
dSˆ
dΘ
dP
(A.22.1)  
= β Θ A − αΘ
+ κ
,   
dt
dt
dt
dt
where   ρ̂    is   the   thickness-­‐‑weighted   average   value   of   density.      One   can   continue   to   consider  
the   material   derivative   of   in   situ   density,   and   in   so   doing,   one   carries   along   the   last   term   in  
Eqn.   (A.22.1),   κ dP dt ,   but   it   is   more   relevant   and   more   interesting   to   consider   the   material  
derivative   of   the   logarithm   of   the   locally-­‐‑referenced   potential   density,   ρˆ l ,    since   this   variable   is  
locally  constant  in  the  neutral  tangent  plane.    The  material  derivative  of   ρˆ l   is  given  by    
ˆ
dρˆ l
dρˆ
dP
dSˆ
dΘ
(A.22.2)  
ρˆ −1
= ρˆ −1
−κ
= β Θ A − αΘ
.   
dt
dt
dt
dt
dt
Substituting  from  Eqns.  (A.21.11)  and  (A.21.15)  above,  and  noting  that  both  the  temporal  and  
the   lateral   gradients   of   ρˆ l    vanish   along   the   neutral   tangent   plane   (that   is,  
ˆ − β Θ∇ Sˆ = 0    and   α Θ Θ
ˆ − β Θ Sˆ
α Θ∇n Θ
= 0 ),   the   material   derivative   of   ρˆ l    amounts   to  
t
At
n A
n
n
the  following  equation  for  the  dianeutral  velocity   e   (note  that  the  boundary  heat  flux   F bound   
also  needs  to  be  included  for  fluid  volumes  that  abut  the  sea  surface)    

ρˆ −1

(

e α ΘΘ̂ z − β Θ ŜA

) = α γ ∇ ⋅ (γ
Θ

z

z

n

(

−1
z K∇ nΘ̂

)

) − β γ ∇ ⋅ (γ
Θ

(

+ α Θ DΘ̂ z − β Θ DŜA
z

)

z z

z

n

−1
z K∇ n Ŝ A

S
− β Θ Sˆ A .

)   

(A.22.3)  

The   left-­‐‑hand   side   is   equal   to   e g −1 N 2    and   the   first   two   terms   on   the   right   hand   side   would  
sum   to   zero   if   the   equation   of   state   were   linear.      This   equation   can   be   rewritten   as   the  
following  equation  for  the  temporally  averaged  vertical  velocity  through  the  neutral  tangent  
plane   at   a   given   longitude   and   latitude   (from   McDougall   (1987b),   and   see   Eqns.   (3.8.2)   and  
(3.9.2)  for  the  definitions  of   C bΘ   and   TbΘ )    

(

)

(

)

(

e g −1 N 2 = − K CbΘ∇ nΘ̂ ⋅∇ nΘ̂ + TbΘ∇ nΘ̂ ⋅∇ n P + α Θ DΘ̂ z − β Θ DŜA
z

)

z z

S
− β Θ Sˆ A .   (A.22.4)  

The   cabbeling   nonlinearity   (the   C bΘ    term)   always   causes   “densification”,   that   is,   it   always  
causes   a   negative   dianeutral   velocity,   e ,   while   the   thermobaric   nonlinearity   (the   TbΘ    term)  
can  cause  either  diapycnal  upwelling  or  downwelling.    The  vertical  turbulent  diffusion  terms  
can  be  re-­‐‑expressed  in  terms  of   DN 2   so  that  Eqn.  (A.22.4)  becomes    

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(

)

S
e N 2 = − gK CbΘ∇ nΘ̂ ⋅∇ nΘ̂ + TbΘ∇ nΘ̂ ⋅∇ n P − g β Θ Sˆ A

+

(

)

Rρ
1
ADN 2 − DN 2
z
A
Rρ −1

(

)

⎡ α Θ β Θ 1 ⎤   
⎢ zΘ − zΘ
⎥,
β Rρ ⎥⎦
⎢⎣ α

(A.22.5)  

where  the  area   A = A( z )   of  the  density  surfaces  is  included  (Klocker  and  McDougall  (2010a)).    
This  is  the  complete  equation  relating  upwelling   e   to  diffusion   D   in  the  sense  of  the  “abyssal  
recipes”   of   Munk   (1966)   and   Munk   and   Wunsch   (1998).      In   this   context,   the   Osborn   (1980)  
relation   DN 2 = Γε ≈ 0.2 ε   can  be  used  in  the  second  line  of  Eqn.  (A.22.5)  to  relate  upwelling  to  
the  dissipation  of  turbulent  kinetic  energy,   ε   (Klocker  and  McDougall  (2010a)).      
The   thermobaric   and   cabbeling   dianeutral   advection   processes   are   illustrated   in   Figure  
A.22.1.      Water   parcels   A   and   B   are   brought   together   in   an   adiabatic   and   isohaline   manner  
until   they   meet   at   location   D.      During   this   adiabatic   advection   process   their   values   of  
Absolute   Salinity   and   Conservative   Temperature   are   constant,   and   since   they   meet   at   the  
pressure  at  D,  they  must  have  the  same  value  of  potential  density  with  respect  to  the  pressure  
at  D  (see  this  potential  isopycnal  on  panel  (b)  of  the  figure).    Also,  during  this  adiabatic  and  
isohaline   motion,   both   parcels   A   and   B   fall   off   the   neutral   trajectory   that   links   the   original  
positions   of   the   parcels.      This   vertical   motion   occurs   because   these   parcels   have   a   different  
compressibility   to   the   water   on   the   neutral   trajectory   (because   they   have   different  
temperatures   and   salinities   to   the   corresponding   parcels   on   the   neutral   trajectory).      Once  
parcels   A   and   B   mix   intimately,   the   density   of   the   mixed   parcel   is   greater   than   that   of   the  
original   parcels   and   so   the   combined   parcel   sinks   vertically   from   location   D   to   location   E.    
This  sinking  is  due  to  cabbeling,  that  is,  it  is  due  to  the  potential  density  surfaces  being  curved  
on  the   SA − Θ   diagram.        

Figure  A.22.1.    Sketch  of  the  dianeutral  advection  processes,  thermobaricity  and  cabbeling.    
  
To   summarize   this   appendix   A.22;   we   have   found   that   the   material   derivative   of   in   situ  
density   Eqn.   (A.22.1),   when   adjusted   for   the   dynamically   passive   compressibility   term,  
becomes   the   material   derivative   of   locally-­‐‑referenced   potential   density   Eqn.   (A.22.2)   which  

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123

can   be   interpreted   as   an   expression   Eqn.   (A.22.4)   for   e ,    the   temporally-­‐‑averaged   vertical  
velocity  through  the  local  neutral  tangent  plane.    This  dianeutral  velocity   e   is  not  a  separate  
mixing   process,   but   rather   is   a   direct   result   of   mixing   processes   such   as   (i)   small-­‐‑scale  
turbulent   mixing   as   parameterized   by   the   diffusivity   D,    and   (ii)   lateral   turbulent   mixing   of  
heat   and   salt   along   the   neutral   tangent   plane   (as   parameterized   by   the   lateral   turbulent  
diffusivity   K )  acting  in  conjunction  with  the  cabbeling  and  thermobaric  nonlinearities  of  the  
equation   of   state.      Note   that   a   common   diapycnal   mixing   mechanism,   double-­‐‑diffusive  
convection   (which   actually   comes   in   two   separate   flavors,   a   salt-­‐‑fingering   type   and   a  
“diffusive”   type   of   double-­‐‑diffusive   convection)   is   omitted   from   the   conservation   equations  
(A.21.11)   and   (A.21.15)   and   also   from   the   mean   dianeutral   velocity   equation   (A.22.4).      It   is  
however   straightforward   to   include   these   processes   in   these   conservation   equations   (see   for  
example  McDougall  (1984,  1987b)).    
  
  

A.23 The water-mass transformation equation
It  is  instructive  to  substitute  Eqn.  (A.22.4)  for   e   into  the  expression  (A.21.15)  for  the  material  
derivative   of   Θ̂ ,   thus   eliminating   e    and   obtaining   the   following   equation   for   the   temporal  
and  spatial  evolution  of   Θ̂   along  the  neutral  tangent  plane  (McDougall  (1984))      

(

)

(

∂Θ̂
+ v̂ ⋅∇ nΘ̂ = γ! z ∇ n ⋅ γ! z−1 K∇ nΘ̂ + KgN −2Θ̂ z CbΘ∇ nΘ̂ ⋅∇ nΘ̂ + TbΘ∇ nΘ̂ ⋅∇ n P
∂t

)

n

Θ

+ Dβ gN

−2

Θ̂ 3z

d 2 ŜA
d Θ̂ 2

   (A.23.1)  

β Θ Rρ
S
Sˆ A ,
+ Θ
α Rρ −1

(

)

ˆ β Θ Sˆ .     The  term  involving   D   
where   Rρ   is  the  stability  ratio  of  the  water  column,   Rρ = α ΘΘ
z
Az
ˆ   diagram  of  a  vertical  cast;  this  
has  been  written  as  proportional  to  the  curvature  of  the   SˆA − Θ
ˆ Sˆ − Sˆ Θ
ˆ
term   can   also   be   written   as   Dβ Θ gN −2 Θ
z A zz
A z zz .       The   form   of   Eqn.   (A.23.1)   illustrates  
that  when  analyzed  in  density  coordinates,  Conservative  Temperature  (and  Absolute  Salinity)  
(i)   are   affected   not   only   by   the   expected   lateral   diffusion   process   along   density   surfaces   but  
also   by   the   nonlinear   dianeutral   advection   processes,   cabbeling   and   thermobaricity,   (ii)   are  
ˆ   diagram  is  
affected  by  diapycnal  turbulent  mixing  only  to  the  extent  that  the  vertical   SˆA − Θ
not  locally  straight,  and  (iii)  are  not  influenced  by  the  vertical  variation  of   D   since   Dz   does  
not  appear  in  this  equation.      
Equations   (A.21.11)   and   (A.21.15)  are  the  fundamental  conservation  equations  of  salinity  
and   Conservative   Temperature   in   a   turbulent   ocean,   and   the   pair   of   equations   (A.22.4)   and  
(A.23.1)   are   simply   derived   as   linear   combinations   of   Eqns.   (A.21.11)   and   (A.21.15).      The  
“density”   conservation   equation   (A.22.4)   and   the   “water-­‐‑mass   transformation”   equation  
(A.23.1)  are  in  some  sense  the  “normal  modes”  of  Eqns.  (A.21.11)  and  (A.21.15).    That  is,  Eqn.  
(A.22.4)   expresses   how   mixing   processes   contribute   to   the   mean   vertical   velocity   e    through  
the   neutral   tangent   plane,   while   (A.23.1)   expresses   how   the   tracer   called   “Conservative  
Temperature   measured   along   the   neutral   direction”   is   affected   by   mixing   processes;   this  
equation  does  not  contain   e .      
For   completeness,   the   water-­‐‑mass   conservation   equation   for   Absolute   Salinity   that  
corresponds  to  Eqn.  (A.23.1)  is    

(

(

)

)

(

∂ ŜA
+ v̂ ⋅∇ n ŜA = γ! z ∇ n ⋅ γ! z−1 K∇ n ŜA + K gN −2 ŜA CbΘ∇ nΘ̂ ⋅∇ nΘ̂ + TbΘ∇ nΘ̂ ⋅∇ n P
z
∂t
n

Θ

+ Dα gN

−2

Θ̂ 3z

d 2 ŜA
d Θ̂ 2

+

Rρ

( R −1)

)
   (A.23.2)  

S
Sˆ A ,

ρ

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and  it  easy  to  show  that   α Θ   times  the  right-­‐‑hand  side  of  Eqn.  (A.23.1)  is  equal  to   β Θ   times  the  
right-­‐‑hand  side  of  Eqn.  (A.23.2).      
The   water-­‐‑mass   transformation   rates   of   Absolute   Salinity   and   of   Conservative  
Temperature  are  illustrated  in  Figure  A.23.1  for  an  ocean  in  steady-­‐‑state.    In  this  situation,  the  
water-­‐‑mass  transformation  rates  in  terms  of   SA   and   Θ   (from  Eqns.  (A.23.1)  and  (A.23.2))  are  
v̂ ⋅∇ n ŜA    and   v̂ ⋅∇ nΘ̂    respectively,   and   these   are   illustrated   as   a   vector   in   the   figure,   directed  
along   the   neutral   direction.      By   contrast,   the   material   derivative   of   SA    and   Θ    (from   Eqns.  
(A.21.11)   and   (A.21.15),   also   shown   in   the   figure)   include   contributions   from   the   mean  
dianeutral   velocity   e .      The   contribution   to   the   material   derivatives   from   purely   horizontal  
advection  along  the  local  isobaric  surface  is  also  sketched  in  the  figure.    The  advantage  of  the  
water-­‐‑mass   transformation   approach   using   the   neutral   framework,   namely   v̂ ⋅∇ n ŜA    and  
v̂ ⋅∇ nΘ̂    is   that   it   can   be   observed   in   the   ocean   due   to   spatial   (or   corresponding   temporal)  
changes  along  neutral  density  surfaces.    In  contrast,  one  seldom  has  a  reliable  estimate  of  the  
dianeutral  advection   e   at  any  particular  location  in  the  ocean  and  so  the  material  derivatives  
dSA dt    and   dΘ dt    are   not   observable   quantities.      Moreover,   in   contrast   to   the   isobaric  
gradients,   the   epineutral   gradients   of   “water-­‐‑mass   conversion”,   v̂ ⋅∇ n ŜA    and   v̂ ⋅∇ nΘ̂    are   not  
affected  by  the  passive  vertical  motion  of  a  water  column  caused  by  adiabatic  vertical  heaving  
motion.      

  
Figure  A.23.1.    Sketch  of  the  water-­‐‑mass  transformation,  compared  with  the  material    
                                                    derivative  of  Absolute  Salinity  and  Conservative  Temperature.    Two  vertical  
                                                    casts  at  different  horizontal  locations  are  sketched  in  the  figure.        
  

To  construct  the  water-­‐‑mass  transformation  equation  of  a  conservative  tracer   C ,   the  mean  
dianeutral   velocity   e    is   eliminated   from   the   Ĉ    conservation   equation   (A.24.1)   using   Eqn.  
(A.22.4)  giving  (from  McDougall  (1984))    

(

)

(

∂Ĉ
+ v̂ ⋅∇ nĈ = γ! z ∇ n ⋅ γ! z−1 K∇ nĈ + K gN −2 Ĉz CbΘ∇ nΘ̂ ⋅∇ nΘ̂ + TbΘ∇ nΘ̂ ⋅∇ n P
∂t
n

( )

+ D ŜA

2

z

d 2Ĉ
dŜA2

+

Ĉz
ŜA

z

Θ

Dα gN

−2

Θ̂ 3z

d 2 ŜA
d Θ̂ 2

−2

)

  

(A.23.3)  

Θ

S
+ Ĉz gN β Sˆ A .

This   equation   shows   that   vertical   turbulent   mixing   processes   affect   the   tracer   on   neutral  
tangent   planes   according   to   the   curvatures   of   vertical   casts   as   displayed   on   both   the   SˆA − Cˆ   
ˆ   curves.    The  terms  involving   D   can  also  be  written  as      
and  the   SˆA − Θ
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( )

D SˆA z

2

2ˆ
d 2Cˆ
Cˆ z
Θ
−2 ˆ 3 d S A
+
α
Θ
=
D
gN
z
ˆ2
dSˆA2
SˆA z
dΘ

(

D SˆA z Cˆ zz − SˆA zz Cˆ z

)

Θ

SˆA z + DCˆ zα gN

−2

(Θˆ Sˆ

ˆ
− SˆA z Θ
zz

z A zz

)

  

125

(A.23.4)  

SˆA z .

  
  

A.24 Conservation equations written in potential density coordinates
The   material   derivative   of   a   conservative   quantity   C    can   be   expressed   with   respect   to   the  
Cartesian  reference  frame,  the  neutral  tangent  plane,  or  a  potential  density  reference  frame  so  
that  the  conservation  equation  of  a  conservative  variable  can  be  written  as  (see  Eqn.  (A.21.16),    

∂Ĉ
∂Ĉ
∂Ĉ
+ v̂ ⋅∇ z Ĉ + w*
=
∂t
∂z
∂t
z

+ v̂ ⋅∇ nĈ + e! Ĉz =
n

∂Ĉ
∂t

+ v̂ ⋅∇σ Ĉ + e! d Ĉz
σ

(

) ( )

  

(A.24.1)  

= γ! z ∇ n ⋅ γ! z−1 K∇ nĈ + DĈz ,
z

where   e d    is   the   mean   vertical   component   of   the   total   transport   velocity   that   moves   through  
the  potential  density  surface.    Any  flux  of   C   across  the  ocean  boundaries   F bound   (e.g.,  the  sea  
surface)   would   need   to   be   added   as   the   extra   term   − Fzbound    on   the   last   line   of   Eqn.   (A.24.1).    
Notice   that   the   lateral   diffusion   occurs   along   the   neutral   tangent   plane.      In   this   section   we  
consider  what  terms  are  neglected  if  this  lateral  mixing  term  is  instead  regarded  as  diffusion  
occurring  along  potential  density  surfaces.      
The  temporal  and  lateral  gradients  of  Absolute  Salinity  and  Conservative  Temperature  in  
a  potential  density  surface  are  related  by  (McDougall  (1991))    

ˆ
α Θ ( pr ) Θ
t

σ

− β Θ ( pr ) SˆAt

σ

ˆ − β Θ ( p ) ∇ Sˆ = 0 ,  
= 0       and       α Θ ( pr ) ∇σ Θ
r
σ A

(

(A.24.2)  

)

(

)

ˆ , p    and   β Θ Sˆ , Θ
ˆ , p   
where   α Θ ( pr )    and   β Θ ( pr )    are   shorthand   notations   for   α Θ SˆA , Θ
r
A
r
respectively,  and   pr   is  the  reference  pressure  of  the  potential  density.    Using  Eqns.  (3.17.1)  to  
(3.17.5)   which   relate   the   gradients   of   properties   in   a   potential   density   surface   to   those   in   a  
neutral   tangent   plane,   the   following   form   of   the   conservation   equation   (A.21.15)   for  
Conservative  Temperature  can  be  derived  (see  equation  (26)  of  McDougall  (1991))    
∂Θ̂
∂t

+ v̂ ⋅∇σ Θ̂ + e! d
σ

(

) (

∂Θ̂
= σ! z ∇σ ⋅ σ! z−1 K∇σ Θ̂ + DΘ̂ z
∂z

(

)

z

)

⎛
∇ Θ̂ ⋅∇ nΘ̂ ⎞
− h−1∇ n ⋅ ⎡G Θ −1⎤ hK∇ nΘ̂ − ⎜ G Θ ⎡G Θ −1⎤ K n
⎟ ,
⎣
⎦
⎣
⎦
Θ̂ z
⎝
⎠z

  

(A.24.3)  

where   the   “isopycnal   temperature   gradient   ratio”   G Θ    is   defined   as   (from   Eqn.   (3.17.4))  
G Θ = ∇σ Θ ∇ n Θ = r ⎡⎣ Rρ −1⎤⎦ ⎡⎣ Rρ − r ⎤⎦   and   r   is  defined  in  Eqn.  (3.17.2)  as  the  ratio  of   α Θ / β Θ   
at  the  in  situ  pressure   p   to  that  evaluated  at  the  reference  pressure   pr .      σ z−1   is  the  averaged  
value   of   the   reciprocal   of   the   vertical   gradient   of   potential   density,   while   σ z    is   simply   the  
reciprocal  of   σ z−1 .    The  corresponding  equation  for  Absolute  Salinity  is    

∂ ŜA
∂t

+ v̂ ⋅∇σ ŜA + e! d
σ

(

) (

∂ ŜA
= σ! z ∇σ ⋅ σ! z−1 K∇σ ŜA + DŜA
z
∂z

)

z

S
+ Sˆ A

⎛ ⎡ GΘ
⎞
⎛ GΘ
⎤
∇ Θ̂ ⋅∇ n ŜA ⎞
⎡G Θ −1⎤ K n
− h−1∇ n ⋅ ⎜ ⎢
−1⎥ hK∇ n ŜA ⎟ − ⎜
⎟ .
⎣
⎦
Θ̂ z
⎝ r
⎠z
⎥⎦
⎝ ⎢⎣ r
⎠

  

(A.24.4)  

The   terms   in   the   second   lines   of   Eqns.   (A.24.3)   and   (A.24.4)   arise   because   in   the   first   line   of  
these  equations,  the  lateral  diffusion  is  written  as  being  along  potential  density  surfaces  rather  
than   along   neutral   tangent   planes.      As   explained   in   McDougall   (1991),   these   terms   are   non  
zero  even  at  the  reference  pressure  of  the  potential  density  variable.      

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Multiplying   Eqn.   (A.24.4)   by   β Θ ( pr )    and   subtracting   α Θ ( pr )    times   Eqn.   (A.24.3)   the  
corresponding  expression  for  the  diapycnal  velocity   e d   is  (following  McDougall  (1991))    

e! d

(

)

(

1 ∂ ρ̂ Θ
= β Θ pr σ! z ∇σ ⋅ σ! z−1 K∇σ ŜA − α Θ pr σ! z ∇σ ⋅ σ! z−1 K∇σ Θ̂
ρ̂ Θ ∂z

( )

( )(

+ β Θ pr DŜA

) −α

z z

Θ

( )

( pr )( DΘ̂ z )z + β Θ ( pr ) Sˆ S

)

A

)
(
) ( ) Gr K∇nr ⋅∇nΘ̂
⎛ α Θ ( p )⎞
∇ Θ̂ ⋅∇ nΘ̂
− β Θ ( pr ) ⎜ Θ r ⎟ G Θ ⎡G Θ −1⎤ K n
⎣
⎦
Θ̂ z
⎝ β ( pr ) ⎠ z
⎤
β Θ ( p ) ⎡ GΘ
− Θ r ⎢
− 1⎥ K ( CbΘ∇ nΘ̂ ⋅∇ nΘ̂ + TbΘ∇ nΘ̂ ⋅∇ n P ) .
β ( p ) ⎢⎣ r
⎥⎦
( )(

+ α Θ pr r −1 γ z ∇ n ⋅ γ z−1 K∇ nΘ̂ + α Θ pr

Θ

  

(A.24.5)  

All   the   terms   in   the   last   three   lines   of   this   equation   occur   because   the   first   line   has   lateral  
mixing  along  potential  density  surfaces  rather  than  along  neutral  tangent  planes.    Even  at  the  
reference  pressure  where   G Θ = r =1   these  last  three  lines  do  not  reduce  to  zero  but  rather  to  
ˆ ⋅ ∇ P   showing  that  the  thermobaric  effect  remains.      
TbΘ K ∇n Θ
n
In  summary,  this  section  has  written  down  the  expressions  for  the  material  derivatives  of  
Conservative   Temperature,   Absolute   Salinity   and   potential   density   in   a   form   where   one   can  
identify   the   many   rather   nasty   terms   that   are   neglected   if   one   assumes   that   the   ocean   mixes  
laterally   along   potential   density   surfaces   instead   of   the   physically   correct   neutral   tangent  
planes.    It  is  noted  in  passing  that  the  first  line  of  the  right-­‐‑hand  side  of  Eqn.  (A.24.5)  can  also  
ˆ ⋅∇ Θ
ˆ
be  written  as   CbΘ ( pr ) K ∇σ Θ
σ   (c.f.  the  last  line  of  Eqn.  (A.27.2)  below).      
  
  

A.25 The vertical velocity through a general surface
Consider  a  general  surface  which  we  identify  with  the  label  “a”  (for  example,  this  could  stand  
for   “approximately   neutral   surface”).      The   material   derivative   on   the   left-­‐‑hand   sides   of   the  
conservation   equations   (A.21.11)   and   (A.21.15)   for   Absolute   Salinity   and   Conservative  
Temperature  are  now  written  with  respect  to  this  general  “a”  coordinate  as    

∂ ŜA
∂t

+ v̂ ⋅∇ a ŜA + e! a
a

⎛ ∂ Ŝ ⎞
∂ ŜA
S
= γ! z ∇ n ⋅ γ! z−1 K∇ n ŜA + ⎜ D A ⎟ + Sˆ A ,   
∂z
∂z
⎝
⎠z

(

)

(A.25.1)  

and    

(

) ( )
(Sˆ , Θˆ , p)    and  

∂Θ̂
∂Θ̂
+ v̂ ⋅∇ aΘ̂ + e! a
= γ! z ∇ n ⋅ γ! z−1 K∇ nΘ̂ + DΘ̂ z .   
z
∂t
∂z
a

(A.25.2)  

(

)

ˆ , p    and  
Cross-­‐‑multiplying   these   equations   by   β Θ = β Θ A
α Θ = α Θ SˆA , Θ
subtracting   gives   the   following   equation   for   the   vertical   velocity   through   the   approximately  
neutral  surface,    

(

e a = − g N −2 K CbΘ∇ nΘ̂ ⋅∇ nΘ̂ + TbΘ∇ nΘ̂ ⋅∇ n P

(

)

(

)

)

S
+ g N −2 ⎛ α Θ DΘ̂ z − β Θ DŜA ⎞ − g N −2β Θ Sˆ A
⎝
z z⎠
z

  

(A.25.3)  

⎡ ∂ Ŝ
∂Θ̂ ⎤
⎥.
+ g N −2 v̂ ⋅ ⎡ β Θ∇ a ŜA − α Θ∇ aΘ̂ ⎤ + g N −2 ⎢ β Θ A − α Θ
⎣
⎦
∂t
∂t ⎥
⎢⎣
a ⎦
a
The  terms  in  the  third  line  of  this  equation  represent  the  deviation  of  the  “a”  coordinate  from  
neutrality   and   these   terms   can   be   shown   to   be   (from   Klocker   and   McDougall   (2010b)   and  
from  Eqn.  (3.14.1)  above,  assuming  the  surfaces  are  not  vertical)    

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l
ˆ ⎤ = − vˆ ⋅ ∇a ρˆ = vˆ ⋅ (∇ z − ∇ z ) = vˆ ⋅ s   
g N −2 vˆ ⋅ ⎡ β Θ∇a SˆA − α Θ∇a Θ
n
a
⎣
⎦
ρˆ zl

127

(A.25.4)  

and    

ρˆtl
⎡
ˆ ⎤
∂Sˆ
∂Θ
(A.25.5)  
g N −2 ⎢ β Θ A − α Θ
⎥ = − la = zt n − zt a   
ˆz
∂t a
∂t a ⎥
ρ
⎢⎣
⎦
l
where   ρˆ   is  the  (thickness-­‐‑weighted)  locally-­‐‑referenced  potential  density.      
Combining  these  results  with  Eqn.  (A.22.4)  we  have  the  simple  kinematic  result  that    
e a = e + v̂ ⋅s + zt − zt ,   
n

(A.25.6)  

a

showing   that   the   vertical   velocity   through   a   general   “a”   surface,   e a ,    is   that   through   the  
neutral  tangent  plane   e   plus  that  due  to  the  “a”  surface  having  a  different  slope  in  space  to  
the  neutral  tangent  plane,   vˆ ⋅ s,   plus  that  due  to  the  “a”  surface  moving  vertically  in  time  (at  
fixed  latitude  and  longitude)  at  a  different  rate  than  the  neutral  tangent  plane,   zt n − zt a .     
  
  

A.26 The material derivative of potential density
The   material   derivative   of   the   natural   logarithm   of   potential   density   is   β Θ ( pr )    times   the  
material   derivative   Eqn.   (A.21.11)   of   Absolute   Salinity   minus   α Θ ( pr )    times   the   material  
derivative  Eqn.  (A.21.15)  of  Conservative  Temperature.    Using  the  relationships  Eqn.  (A.24.2)  
that   relate   the   gradients   of   Absolute   Salinity   and   Conservative   Temperature   in   potential  
density   surfaces,   and   taking   the   material   derivative   of   potential   density   with   respect   to  
potential   density   surfaces,   one   finds   that   the   temporal   and   isopycnal   gradient   terms   cancel  
leaving  only  the  term  in  the  mean  diapycnal  velocity   e d   as  follows    

e! d

(

)

( )

(

)

( r)

)

1 ∂ ρ̂ Θ
= β Θ pr γ! z ∇ n ⋅ γ! z−1 K∇ n ŜA − α Θ pr γ! z ∇ n ⋅ γ! z−1 K∇ nΘ̂
ρ̂ Θ ∂z
  
SA
Θ
Θ
Θ
ˆ
+ β p DŜ
− α p DΘ̂ + β p S ,

( )

( r )(

Az

)

( r )(

z

z z

(A.26.1)  

where  the  exact  expression  for  the  vertical  gradient  of  potential  density  has  been  used,      
1 ∂ρˆ Θ
ˆ .   
(A.26.2)  
= β Θ ( pr ) SˆA z − α Θ ( pr ) Θ
z
ρˆ Θ ∂z
Equation  (A.26.1)  can  be  written  more  informatively  as  (following  McDougall,  1991)    

e! d

⎛ D ∂ ρ̂ Θ ⎞
1 ∂ ρ̂ Θ
Θ
ˆ SA
=
⎜ Θ ∂z ⎟ + β pr S
ρ̂ Θ ∂z
ρ̂
⎝
⎠z

( )

{

( )

( )

( )

+ D α ΘΘ pr Θ̂ 2z + 2α SΘ pr Θ̂ z ŜA − β SΘ pr ŜA2
+α

Θ

( )

A

(

z

)

pr ⎡⎣ r −1⎤⎦ γ! z ∇ n ⋅ γ! z−1 K∇ nΘ̂ +

A

β Θ ( pr )
β

Θ

( p)

(

z

}

(A.26.3)  

)

K CbΘ∇ nΘ̂ ⋅∇ nΘ̂ + TbΘ∇ nΘ̂ ⋅∇ n P ,

where   r    is   defined   in   Eqn.   (3.17.2)   as   the   ratio   of   α Θ / β Θ    at   the   in   situ   pressure   p    to   that  
evaluated  at  the  reference  pressure   pr .     If  the  equation  of  state  were  linear,  only  the  first  two  
terms  would  be  present  on  the  right  of  Eqn.  (A.26.3).      
  
  

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A.27 The diapycnal velocity of layered ocean models (without rotation of the mixing
tensor)
Layered   models   of   the   ocean   circulation   have   a   potential   density   variable   (usually   with   a  
reference   pressure   pr    of   2000   dbar)   as   their   vertical   coordinate.      To   date   these   models   have  
not   rotated   the   direction   of   lateral   mixing   to   align   with   the   neutral   tangent   plane   but   have  
mixed   laterally   along   the   potential   density   coordinate   direction.      The   diapycnal   velocity  
e d_model   in  this  class  of  model  obeys  the  equation  (c.f.  Eqn.  (A.26.1)  above)    

e! d_model

(

) ( ) (
( )( )
( )

)

1 ∂ ρ̂ Θ
= β Θ pr σ! z ∇σ ⋅ σ! z−1 K σ ∇σ ŜA − α Θ pr σ! z ∇σ ⋅ σ! z−1 K σ ∇σ Θ̂
ρ̂ Θ ∂z
  
SA
Θ
Θ
Θ
ˆ
+ β p DŜ
− α p DΘ̂ + β p S ,

( )

( r )(

Az

)

r

z

z z

(A.27.1)  

r

where   ∇σ    is   the   gradient   operator   along   the   potential   density   coordinate,   K σ    is   the   lateral  
diffusivity  along  the  layers,   σ z−1   is  the  averaged  value  of  the  reciprocal  of  the  vertical  gradient  
of  potential  density,  while   σ z   is  simply  the  reciprocal  of   σ z−1 .    This  equation  can  be  written  as    

e d_model

⎛ D ∂ ρ̂ Θ ⎞
1 ∂ ρ̂ Θ
Θ
ˆ SA
=
⎜ Θ ∂z ⎟ + β pr S
ρ̂ Θ ∂z
⎝ ρ̂
⎠z

( )

{

( )
( )
K σ CbΘ ( pr ) ∇σ Θ̂ ⋅∇σ Θ̂ .

( )

}

+ D α ΘΘ pr Θ̂ 2z + 2α SΘ pr Θ̂ z ŜA − β SΘ pr ŜA2   
+

A

z

A

z

(A.27.2)  

The  terms  in  the  vertical  turbulent  diffusivity   D   are  identical  to  those  in  the  correct  equation  
(A.26.3)   while   the   diapycnal   velocity   due   to   cabbeling   is   quite   similar   to   that   in   the   correct  
expression   Eqn.   (A.26.3);   the   difference   mostly   being   that   the   cabbeling   coefficient   is   here  
evaluated   at   the   reference   pressure   instead   of   at   the   in   situ   pressure,   and   that   the   lateral  
temperature   gradient   is   here   evaluated   along   the   potential   density   surface   rather   than   along  
the   neutral   tangent   plane   (these   gradients   are   proportional   to   each   other   via   the   relation  
(3.17.3)).    Another  difference  is  that  the  term   α Θ pr ⎡⎣ r −1⎤⎦ γ z ∇ n ⋅ γ z−1 K∇ nΘ̂   in  Eqn.  (A.26.3)  is  
missing  from  Eqn.  (A.27.2).    This  type  of  difference  is  to  be  expected  since  the  direction  of  the  
lateral  mixing  is  different.      
Notice   the   absence   of   the   thermobaric   diapycnal   advection   in   Eqn.   (A.27.2);   that   is,   the  
ˆ ⋅ ∇ P   in  Eqn.  (A.26.3)  does  not  appear  in  Eqn.  (A.27.2);  this  was  
term  proportional  to   K TbΘ∇n Θ
n
first  pointed  out  by  Iudicone  et  al.  (2008).    The  thermobaric  diapycnal  advection  is  significant  
in  the  Southern  Ocean  (Klocker  and  McDougall  (2010a))  and  its  omission  from  layered  ocean  
models  amounts  to  a  non-­‐‑trivial  inherent  limitation  of  this  type  of  ocean  model.    Also  missing  
from   layered   ocean   models   is   the   mean   vertical   advection   vˆ ⋅ s    due   to   the   helical   nature   of  
neutral   trajectories   in   the   ocean   (see   section   3.13,   Eqn.   (A.25.4)   and   Klocker   and   McDougall  
(2010b)),  whereas  this  physical  process  occurs  naturally  in  z-­‐‑coordinate  ocean  models.        
  
  

( )

(

)

A.28 The material derivative of orthobaric density
Orthobaric   density   ρ v ( p, ρ )    has   been   defined   by   de   Szoeke   et   al.   (2000)   as   a   pressure  
corrected   form   of   in   situ   density.      The   construction   of   orthobaric   density   requires   the  
isentropic   compressibility   to   be   approximated   as   a   function   of   pressure   and   in   situ   density.    
While   orthobaric   density   has   the   advantage   of   being   a   thermodynamic   variable,   orthobaric  
density  surfaces  are  often  not  particularly  good  approximations  to  neutral  tangent  planes  (see  
McDougall  and  Jackett  (2005a)  and  Klocker  et  al.  (2009a,b)).    The  material  derivative  of   ρ v   can  
be  expressed  with  respect  to  orthobaric  density  surfaces  as    

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∂ ρv
∂t

+ v̂ ⋅∇ ρ ρv + e

ρv

v

ρv

∂ ρv
ρ ∂ρ
= e v v ,   
∂z
∂z

129

(A.28.1)  

where   the   temporally   averaged   vertical   velocity   through   the   ρ v    surface   is   given   by   (from  
McDougall  and  Jackett  (2005a))    

e

ρv

(

) ( )

 − β Θ S + ψ −1 ⎛ p
= gN −2 α ΘΘ
A
⎝ t

ρv

+ v̂ ⋅∇ ρ p ⎞ pz ,   
⎠
v

(A.28.2)  

where  (from  de  Szoeke  et  al.  (2000))    

(ψ −1)

ˆ Θ ( p, ρ )⎤ ,   
≈ 2 g 2 N −2c0−3Δc ≈ − ρ g 2TbΘ N −2 ⎡Θ−
0
⎣
⎦

(A.28.3)  

and   Δc   is  the  difference  between  the  reference  sound  speed  function   c0 ( p, ρ )   and  the  sound  
speed  of  seawater  which  can  be  expressed  in  the  functional  form   c ( p, ρ , Θ) .     This  difference  in  
the  sound  speed  is  equivalent  to  the  difference  between  the  actual  Conservative  Temperature  
of   a   water   parcel   and   the   reference   value   Θ0 ( p, ρ ) .       Here   SA    is   shorthand   for   the   material  
derivative  of   Ŝ A   and  is  expressed  in  terms  of  mixing  processes  by  the  right-­‐‑hand  side  of  Eqn.  
   is  similarly  shorthand  for  the  material  derivative  of   Θ̂   and  is  given  by  the  right-­‐‑
(A.21.11);   Θ
hand  side  of  Eqn.  (A.21.15).      
The   first   term   on   the   right   of   Eqn.   (A.28.2)   represents   the   effects   of   irreversible   mixing  
ρ
processes   on   the   flow   through   orthobaric   density   surfaces,   and   this   contribution   to   e v    is  
exactly   the   same   as   the   flow   through   neutral   tangent   planes,   e    (Eqn.   (A.22.4)).      The   second  
term   in   Eqn.   (A.28.2)   arises   from   the   non-­‐‑quasi-­‐‑material   (non-­‐‑potential)   nature   of   orthobaric  
density.    This  vertical  advection  arises  from  the  seemingly  innocuous  sliding  motion  along  the  
sloping  orthobaric  density  surface  and  from  the  vertical  heaving  of  these  surfaces.      
  
  

A.29 The material derivative of Neutral Density
Neutral   Density   γ n    is   not   a   thermodynamic   function   since   it   depends   on   latitude   and  
longitude.    The  Neutral  Density  algorithm  finds  the  data  point  in  a  pre-­‐‑labeled  reference  data  
set   that   has   the   same   potential   density   as   the   data   point   that   is   being   labeled;   the   reference  
pressure   of   this   potential   density   is   the   average   of   the   pressures   of   the   two   parcels.      The  
material  derivative  of   γ n   can  be  expressed  as    

∂γ n
∂t

+ v̂ ⋅∇γ γ n + eγ γ zn = eγ γ zn ,   

(A.29.1)  

γ

where   the   temporally   averaged   vertical   velocity   through   the   γ n    surface   is   given   by   (from  
McDougall  and  Jackett  (2005b))    

e

γ

≈

(

(α

α

Θ

Θ

 − β Θ ( p) S
( p) Θ
A

( p)Θ ref
z

−β

Θ

)

( p)SAref
z

)

+ v̂ ⋅s ref

( )(
)(p )
⎛
⎞
( p− p )
+ (ψ −1) ⎜ v̂ ⋅∇ p −
v̂ ⋅∇ Θ ⎟ ( p )
(Θ̂ − Θ )
⎝
⎠
(α ( p) Θ − β ( p)S )
+ 2(ψ −1)
−1

+ ψ γ −1 pt + v̂ ⋅∇γ p
γ

ref

γ

γ

γ

(

z

)

(α

ref

γ

ref

Θ

Θ

( p)Θ ref
z

Θ

A

Θ

− β ( p)SAref
z

ref

z

−1

  

(A.29.2)  

)

+ ψ γ −1 v̂ ⋅s ref .
Here   SA    is   shorthand   for   the   material   derivative   of   Ŝ A    following   the   appropriate   mean  
velocity   and   is   expressed   in   terms   of   mixing   processes   by   the   right-­‐‑hand   side   of   Eqn.  

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    is   similarly   shorthand   for   the   material   derivative   of   Θ̂    and   is   given   by   Eqn.  
(A.21.11),   Θ
(A.21.15),  and   ψ γ − 1   is  defined  by    

( )
(ψ −1)
γ

=

ˆ Θref )
− 12 ρ g 2TbΘ (Θ−
.   
ˆ Θref ) + 1 gT Θ ( P − P ref ) Θref ⎤
+ ρ g 2TbΘ (Θ−
b
z ⎦
2

2
⎡ N ref
⎣

(A.29.3)  

2
   is   the   square   of   the   buoyancy   frequency   of   the   pre-­‐‑labelled   reference   data   set.      
Here   N ref
Equation   (A.29.3)   shows   that   ψ γ − 1    is   nonzero   to   the   extent   that   there   is   a   water   mass  
contrast   (Θ− Θref )   between  the  seawater  parcel  that  is  being  labeled  and  the  data  on  the  pre-­‐‑
labeled   reference   data   set   that   communicates   neutrally   with   the   seawater   sample.      For  
ˆ Θref )   and   ( p − p ref )   the  denominator  in  Eqn.  (A.29.3)  is  close  to   N 2   
reasonable  values  of   (Θ−
ref
and   ψ γ − 1   is  small.    In  these  expressions  the  thermal  expansion  coefficient   α Θ ( p )   and  saline  
contraction   coefficient   β Θ ( p )    are   evaluated   at   the   average   of   the   properties   of   the   parcel  
being  labeled  and  the  parcel  in  the  reference  data  set  to  which  it  is  neutrally  related,  that  is,  
α Θ ( p )   and   β Θ ( p )   are  shorthand  for   α Θ ( SA , Θ, p )   and   β Θ ( SA , Θ, p ).       
The   first   term   in   Eqn.   (A.29.2)   is   expected  as   Neutral   Density   changes   in   response   to   the  
    and   S .      The   next   term   in   Eqn.   (A.29.2),   vˆ ⋅ sref ,    is   also  
irreversible   mixing   processes   Θ
A
expected;   it   is   the   mean   vertical   motion   through   the   γ n    surface   due   to   the   helical   motion   of  
neutral  trajectories  in  the  reference  data  set,  caused  in  turn  by  the  non-­‐‑zero  neutral  helicity  of  
the  reference  data  set.    The  remaining  terms  in  the  last  four  lines  of  Eqn.  (A.29.2)  arise  because  
of  the  non-­‐‑quasi-­‐‑material  (non-­‐‑potential)  nature  of  Neutral  Density.    The  second  line  of  Eqn.  
(A.29.2)   represents   the   contribution   to   eγ    arising   from   the   seemingly   innocuous   sliding  
motion   along   the   sloping   γ n    surface   and   from   the   vertical   heaving   of   these   surfaces.      The  
lateral  gradients  of  properties  in  the  reference  data  set  also  affect  the  mean  flow   eγ   through  
the   γ n    surface.      Note   that   as   Θ̂− Θref    tends   to   zero,   ψ γ − 1    also   tends   to   zero   so   that   the  
third  line  of  Eqn.  (A.29.2)  is  well-­‐‑behaved  and  becomes  proportional  to   pz−1 ( p − p ref ) vˆ ⋅ ∇γ Θref .       
  
  

(

(

)

)

(

)

(

)

A.30 Computationally efficient 75-term expression for the specific volume of seawater
in terms of Θ
Ocean  models  that  pre-­‐‑date  TEOS-­‐‑10  have  treated  their  salinity  and  temperature  variables  as  
being   Practical   Salinity   SP    and   potential   temperature   θ .      Ocean   models   that   are   TEOS-­‐‑10  
compatible  calculate  Absolute  Salinity   S A   and  Conservative  Temperature   Θ   (as  discussed  in  
appendices  A.20  and  A.21),  and  they  use  a  computationally  efficient  expression  for  calculating  
specific   volume   (or   density)   in   terms   of   Absolute   Salinity   S A ,   Conservative   Temperature   Θ   
and  pressure   p .      
Earlier   versions   of   the   GSW   Oceanographic   Toolbox   and   of   this   TEOS-­‐‑10   Manual   have  
included  25-­‐‑term  and  48-­‐‑term  rational  functions  for  specific  volume  in  terms  of   S A ,   Θ   and   p .    
These   GSW   functions   (and   the   corresponding   earlier   versions   of   the   TEOS-­‐‑10   Manual)   are  
archived  and  are  still  available  from  the  TEOS-­‐‑10  web  site.    When  implementing  TEOS-­‐‑10  in  
numerical   ocean   models   it   became   clear   that   the   rational   function   form   for   specific   volume  
was  not  very  computationally  efficient  and  Roquet  et  al.  (2015)  showed  that  a  straightforward  
polynomial  (as  opposed  to  a  rational  function)  is  a  better  form  for  ocean  modelling.    Following  
the   publication   of   this   paper,   the   GSW   Oceanographic   Toolbox   has   adopted   this   polynomial  
form,   v̂ 75 SA ,Θ, p ,   as   an   accurate   alternative   to   first   calculating   in   situ   temperature   from  
Conservative  Temperature  and  then  using  the  full  Gibbs  function  to  evaluate  specific  volume.      
The   75-­‐‑term   polynomial   by   Roquet   et   al.   (2015)   is   expressed   in   terms   of   the   following  
three  dimensionless  salinity,  temperature  and  pressure  variables,    

(

)

s ≡

SA + 24 g kg −1
Θ
p
  ,                 τ ≡
              and             π ≡
  ,  
Θu
pu
SA u

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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

131

in  terms  of  the  unit-­‐‑related  scaling  constants    

SA u ≡ 40× 35.16504g kg −1 / 35 ,               Θ u ≡ 40°C             and           pu ≡ 104 dbar .    

(A.30.2)  

Their  polynomial  expression  for  the  specific  volume  of  seawater  is    

v̂(SA ,Θ, p) = vu ∑ vijk s i τ jπ k ,  

(A.30.3)  

i, j,k

where   vu ≡ 1 m 3kg −1    and   the   non-­‐‑zero   dimensionless   constants   vijk    are   given   in   Table   K.1   of  
appendix   K.      The   specific   volume   data   was   fitted   in   a   “funnel”   of   data   points   in   ( SA , Θ, p )   
space  (McDougall  et  al.  (2003))  which  extends  to  a  pressure  of  8000   dbar .    At  the  sea  surface  
the  “funnel”  covers  the  full  range  of  temperature  and  salinity  while  for  pressures  greater  than  
6500   dbar   the   maximum   temperature   of   the   fitted   data   is   10°C    and   the   minimum   Absolute  
Salinity   is   30 g kg −1 .      That   is,   the   fit   has   been   performed   over   a   region   of   parameter   space  
which   includes   water   that   is   approximately   8°C    warmer   and   5 g kg −1    fresher   in   the   deep  
ocean  than  the  seawater  which  exists  in  the  present  ocean.      
As   outlined   in   appendix   K,   this   75-­‐‑term   polynomial   expression   for   v    yields   the   thermal  
expansion  and  saline  contraction  coefficients,   α Θ   and   β Θ ,  that  are  essentially  as  accurate  as  
those  derived  from  the  full  TEOS-­‐‑10  Gibbs  function  for  data  in  the  “oceanographic  funnel”.    In  
dynamical   oceanography   it   is   these   thermal   expansion   and   haline   contraction   coefficients  
which   are   the   most   important   aspects   of   the   equation   of   state   since   the   “thermal   wind”   is  
proportional   to   α Θ∇ p Θ − β Θ∇ p SA    and   the   vertical   static   stability   is   given   in   terms   of   the  
buoyancy  frequency   N   by   g −1 N 2 = α Θ Θ z − β Θ (SA ) z .    Hence  for  dynamical  oceanography  we  
may   take   Roquet   et   al.’s   (2015)   75-­‐‑term   polynomial   expression   for   specific   volume   as  
essentially  reflecting  the  full  accuracy  of  TEOS-­‐‑10.      
Appendix   P   describes   how   an   expression   for   the   enthalpy   of   seawater   in   terms   of  
Conservative   Temperature,   specifically   the   functional   form   hˆ ( S A , Θ, p ) ,   together   with   an  
expression   for   entropy   in   the   form   ηˆ ( SA , Θ) ,   can   be   used   as   an   alternative   thermodynamic  
potential  to  the  Gibbs  function   g ( SA , t , p ) .    The  need  for  the  functional  form   hˆ ( S A , Θ, p )   also  
arises  in  section  3.32  and  in  Eqns.  (3.26.3)  and  (3.29.1).    The  75-­‐‑term  expression,  Eqn.  (A.30.3)  
for   v 75 = v̂ 75 SA ,Θ, p    can   be   used   to   find   a   closed   expression   for   hˆ ( S A , Θ, p )    by   integrating  
v̂ 75 SA ,Θ, p    with   respect   to   pressure   (in   Pa ),   since   hˆP = v = ρ −1   (see   Eqn.   (2.8.3)).      Specific  
enthalpy  calculated  from   v̂ 75 SA ,Θ, p   is  available  in  the  GSW  Oceanographic  Toolbox  as  the  
function   gsw_enthalpy(SA,CT,p).      Using   gsw_enthalpy   to   evaluate   hˆ ( S A , Θ, p )    is   5   times  
faster   than   first   evaluating   the   in   situ   temperature   t    (from   gsw_t_from_CT(SA,CT,p))   and  
then   calculating   enthalpy   from   the   full   Gibbs   function   expression   h ( SA , t, p )    using  
gsw_enthalpy_t_exact(SA,t,p).      (These   last   two   function   calls   have   also   been   combined   into  
the  one  function,  gsw_enthalpy_CT_exact(SA,CT,p).)    
Also,  the  enthalpy  difference  at  the  same  values  of   S A   and   Θ   but  at  different  pressures  
(see  Eqn.  (3.32.5))  is  available  as  the  function  gsw_enthalpy_diff(SA,CT,p_shallow,p_deep).      
Following  Young  (2010),  the  difference  between   h   and   c0p Θ   is  called  “dynamic  enthalpy”  
and   can   be   found   using   the   function   gsw_dynamic_enthalpy(SA,CT,p)   in   the   GSW  
Oceanographic  Toolbox.      
  
  

(

)

(

)

(

)

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Appendix  B:  
Derivation  of  the  First  Law  of  Thermodynamics    

  
  
  

Motivation    

  
For   a   pure   fluid   in   which   there   is   no   dissolved   material   (such   as   pure   water   with   zero  
Absolute   Salinity)   the   derivation   of   the   First   Law   of   Thermodynamics   usually   starts   with   a  
discussion   of   how   the   internal   energy   U    of   a   fixed   mass   of   fluid   is   changed   under   the  
influence   of   it   being   “heated”   by   the   amount   δ Q    and   its   volume   V    being   changed.      The  
infinitesimal   change   in   the   internal   energy   of   the   parcel   is   written   as   dU = δ Q − ( p + P0 ) dV   
where   − ( p + P0 ) dV    is   the   mechanical   work   done   on   the   fluid   by   the   pressure   at   the   moving  
boundaries   of   the   fluid   parcel.      This  relationship  can   be   written   in   terms   of   the   specific   (i.   e.  
per  unit  mass)  enthalpy   h,   the  density   ρ ,   and   δ Q   per  unit  volume,   δ q,   as    

⎛ dh 1 dP ⎞
δq
for  pure  water  (B.1)  
.   
−
⎟ =
dt
⎝ dt ρ dt ⎠
It  is  recognized  that  the  right-­‐‑hand  side  of  (B.1)  is  not  the  divergence  of  a  “heat”  flux,  and  the  
term   that   causes   this   complication   is   the   dissipation   of   kinetic   energy   into   “heat”,   which  
contributes   ρε   to  the  right-­‐‑hand  side  of  (B.1).    Apart  from  this  familiar  dissipation  term,  the  
right-­‐‑hand  side  is  minus  the  divergence  of  the  sum  of  the  boundary  and  radiative  heat  fluxes,  
FR ,  and  minus  the  divergence  of  the  molecular  flux  of  heat   FQ = − ρc p k T ∇T   (where   k T   is  the  
molecular   diffusivity   of   temperature),   so   that   the   First   Law   of   Thermodynamics   for   pure  
water  is    

ρ⎜

⎛ dh 1 dP ⎞
δq
=
ρ⎜
−
= − ∇ ⋅F R − ∇ ⋅FQ + ρε .   
⎟
dt
⎝ dt ρ dt ⎠

for  pure  water  (B.2)  

Now  consider  seawater  in  which  the  Absolute  Salinity  and  its  gradients  are  non-­‐‑zero.    The  
same  traditional  discussion  of  the  First  Law  of  Thermodynamics  involving  the  “heating”,  the  
application   of   compression   work   and   the   change   of   salinity   to   a   fluid   parcel   shows   that   the  
change  of  enthalpy  of  the  fluid  parcel  is  given  by  (see  equations  6b  and  17b  of  Warren  (2006))    

dH − VdP = δ Q + ( µ − [T0 + t ] µT ) M dS A ,   

(B.3)  

where   M   is  the  mass  of  the  fluid  parcel.    When  written  in  terms  of  the  specific  enthalpy   h,   
and   δ Q   per  unit  volume,   δ q ,  this  equation  becomes  (using   ρ dSA dt = −∇ ⋅ F S )    

⎛ dh 1 dp ⎞
δq
−
− ( µ − [T0 + t ] µT ) ∇ ⋅ FS .  
⎟ =
t
ρ
t
t
d
d
d
⎝
⎠

ρ⎜

(B.4)  

Does  this  help  with  the  task  of  constructing  an  expression  for  the  right-­‐‑hand  side  of  (B.4)  
in  terms  of  the  dissipation  of  kinetic  energy  and  the  molecular,  radiative  and  boundary  fluxes  
of   “heat”   and   salt?      If   the   “heating”   term   δ q dt    in   Eqn.   (B.4)   were   the   same   as   in   the   pure  
water   case   Eqn.   (B.2)   then   we   would   have   successfully   derived   the   First   Law   of  
Thermodynamics  in  a  saline  ocean  via  this  route.    However,  we  will  now  show  that   δ q dt   in  
Eqn.  (B.4)  is  not  the  same  as  that  in  the  pure  water  case,  Eqn.  (B.2).      
Substituting  the  expression  for   δ q dt   from  (B.2)  into  the  right-­‐‑hand  side  of  (B.4)  we  find  
that  the  right-­‐‑hand  side  is  not  the  same  as  the  First  Law  of  Thermodynamics  (B.19)  which  we  
derive  below  (this  comparison  involves  using  the  correct  expression  (B.30))  for  the  molecular  
flux   FQ ).    The  two  versions  of  the  First  Law  of  Thermodynamics  are  different  by    

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133

⎡ B′ µ S
⎤
(B.5)  
+ ∇ ⋅ ⎢ S A FS ⎥ .  
⎢⎣ ρ k ⎡⎣T0 + t ⎤⎦ ⎥⎦
This   inconsistency   means   that   the   rather   poorly   defined   “rate   of   heating”   δ q dt    must   be  
different  in  the  saline  case  than  in  the  pure  water  situation  by  this  amount.    We  know  of  no  
way  of  justifying  this  difference,  so  we  conclude  that  any  attempt  to  derive  the  First  Law  of  
Thermodynamics   via   this   route   involving   the   loosely   defined   “rate   of   heating”   δ q dt    is  
doomed  to  failure.    This  is  not  to  say  that  Eqn.  (B.4)  is  incorrect.    Rather,  the  point  is  that  it  is  
not  useful,  since   δ q dt   cannot  be  deduced  directly  by  physical  reasoning  (for  example,  how  
would  one  guess  how  the  Dufour  effect  contributes  to   δ q dt ?)      
Since   there   appears   to   be   no   way   of   deriving   the   First   Law   of   Thermodynamics   that  
involves  the  “heating”  term   δ q dt ,  we  follow  Landau  and  Lifshitz  (1959)  and  derive  the  First  
Law   via   the   following   circuitous   route.      Rather   than   attempting   to   guess   the   form   of   the  
molecular   forcing   terms   in   this   equation   directly,   we   first   construct   a   conservation   equation  
for  the  total  energy,  being  the  sum  of  the  kinetic,  gravitational  potential  and  internal  energies.    
It   is   in   this   equation   that   we   insert   the   molecular   fluxes   of   heat   and   momentum   and   the  
radiative  and  boundary  fluxes  of  heat.    We  know  that  the  evolution  equation  for  total  energy  
must   have   the   conservative   form,   and   so   we   insist   that   the   forcing   terms   in   this   equation  
appear  as  the  divergence  of  fluxes.      
Having  formed  the  conservation  equation  for  total  energy,  the  known  evolution  equations  
for   two   of   the   types   of   energy,   namely   the   kinetic   and   gravitational   potential   energies,   are  
subtracted,   leaving   a   prognostic   equation   for   the   internal   energy,   that   is,   the   First   Law   of  
Thermodynamics.      
We  start  by  developing  the  evolution  equations  for  gravitational  potential  energy  and  for  
kinetic   energy   (via   the   momentum   equation).      The   sum   of   these   two   evolution   equations   is  
noted.      We   then   step   back   a   little   and   consider   the   simplified   situation   where   there   are   no  
molecular  fluxes  of  heat  and  salt  and  no  effects  of  viscosity  and  no  radiative  or  boundary  heat  
fluxes.      In   this   “adiabatic”   limit   we   are   able   to   develop   the   conservation   equation   for   total  
energy,   being   the   sum   of   internal   energy,   kinetic   energy   and   gravitational   potential   energy.    
To   this   equation   we   introduce   the   molecular,   radiative   and   boundary   flux   divergences.    
Finally   the   First   Law   of   Thermodynamics   is   found   by   subtracting   from   this   total   energy  
equation   the   conservation   statement   for   the   sum   of   the   kinetic   and   gravitational   potential  
energies.      
  
  

(

FS ⋅∇ µ − ⎡⎣T0 + t ⎤⎦ µT

)

The  fundamental  thermodynamic  relation    

  
Recall   the   fundamental   thermodynamic   relation   (A.7.1)   repeated   here   in   the   form   (A.7.2)   in  
terms   of   material   derivatives   following   the   instantaneous   motion   of   a   fluid   parcel  
d dt = ∂ ∂t
+ u ⋅ ∇ ,     
x, y, z

dh 1 dP
du
dv
dη
dS
−
=
+ ( p + P0 )
= (T0 + t )
+ µ A .   
dt ρ dt
dt
dt
dt
dt

(B.6)  

The  use  of  the  same  symbol   t   for  time  and  for  in  situ  temperature  in  °C  is  noted  but  should  
not  cause  confusion.    The  middle  expression  in  (B.6)  uses  the  fact  that  specific  enthalpy   h   and  
specific   internal   energy   u    are   related   by   h = u + Pv = u + ( p + P0 ) v    where   v    is   the   specific  
volume.      

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Gravitational  potential  energy    

  
If  the  gravitational  acceleration  is  taken  to  be  constant  the  gravitational  potential  energy  per  
unit   mass   with   respect   to   the   height   z    =   0   is   simply   gz .       Allowing   the   gravitational  
acceleration  to  be  a  function  of  height  means  that  the  gravitational  potential  energy  per  unit  
mass   Φ   with  respect  to  some  fixed  height   z0   is  defined  by      
z

Φ=

∫ g ( z′) dz′.   

(B.7)  

z0

At   a   fixed   location   in   space   Φ    is   independent   of   time   while   its   spatial   gradient   is   given   by  
∇Φ = g k   where   k   is  the  unit  vector  pointing  upwards  in  the  vertical  direction.    The  evolution  
equation  for   Φ   is  then  readily  constructed  as    

dΦ
(B.8)  
= ρ gw ,   
dt
where   w    is   the   vertical   component   of   the   three-­‐‑dimensional   velocity,   that   is   w = u ⋅ k .     
(Clearly  in  this  section   g   is  the  gravitational  acceleration,  not  the  Gibbs  function).    Note  that  
this  local  balance  equation  for  gravitational  potential  energy  is  not  in  the  form  (A.8.1)  required  
of   a   conservative   variable   since   the   right-­‐‑hand   side   of   (B.8)   is   not   minus   the   divergence   of   a  
flux.      
  
  

( ρΦ )t + ∇ ⋅ ( ρΦu)

= ρ

Momentum  evolution  equation    

  
The   momentum   evolution   equation   is   derived   in   many   textbooks   including   Landau   and  
Lifshitz   (1959),   Batchelor   (1970),   Gill   (1982)   and   Griffies   (2004).      The   molecular   viscosity  
appears   in   the   exact   momentum   evolution   equation   in   the   rather   complicated   expressions  
appearing   in   equations   (3.3.11)   and   (3.3.12)   of   Batchelor   (1970).      We   ignore   the   term   that  
depends   on   the   product   of   the   kinematic   viscosity   v visc    and   the   velocity   divergence   ∇ ⋅ u   
(following  Gill  (1982)),  so  arriving  at    

ρ

(

)

du
 ,   
+ f k × ρ u = − ∇P − ρ gk + ∇ ⋅ ρ v visc ∇u
dt

(B.9)  

    is   twice   the  
where   f    is   the   Coriolis   frequency,   v visc    is   the   kinematic   viscosity   and   ∇u

symmetrized  velocity  shear,   ∇u = ∂ui ∂x j + ∂u j ∂xi .     Under  the  same  assumption  as  above  
of   ignoring   the   velocity   divergence,   the   pressure   P    that   enters   (B.9)   can   be   shown   to   be  
equivalent  to  the  equilibrium  pressure  that  is  rightly  the  pressure  argument  of  the  equation  of  
state   (Batchelor   (1970)).      The   centripetal   acceleration   associated   with   the   coordinate   system  
being   on   a   rotating   planet   can   be   taken   into   account   by   an   addition   to   the   gravitational  
acceleration  in  (B.9)  (Griffies  (2004)).      
  
  

(

)

Kinetic  energy  evolution  equation    

  
The  kinetic  energy  evolution  equation  is  found  by  taking  the  scalar  product  of  Eqn.  (B.9)  with  
u   giving    

( ρ 12 u ⋅ u )t

+ ∇ ⋅ ( ρ u 12 [u ⋅ u ])
= ρ d ( 12 u ⋅ u ) dt

(

)

= − u ⋅ ∇P − ρ gw + ∇ ⋅ ρ v visc∇ 12 [u ⋅ u ] − ρε ,

  

(B.10)  

where  the  dissipation  of  kinetic  energy   ε   is  the  positive  definite  quantity    

ε ≡
  
  

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1
2

(

)

 ⋅∇u
 .   
v visc ∇u

(B.11)  

TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

135

Evolution  equation  for  the  sum  of  kinetic  and  gravitational  potential  energies    

  
The  evolution  equation  for  total  mechanical  energy   12 u ⋅ u + Φ   is  found  by  adding  Eqns.  (B8)  
and  (B10)  giving    

( ρ ⎡⎣

1
2

u ⋅ u + Φ ⎤⎦

)

t

(

+ ∇ ⋅ ρ u ⎡⎣ 12 u ⋅ u + Φ ⎤⎦

)

(

)

= ρ d ( 12 u ⋅ u + Φ ) dt = − u ⋅ ∇P + ∇ ⋅ ρ v visc∇ 12 [u ⋅ u] − ρε .

  

(B.12)  

Notice   that   the   term   ρ gw    which   has   the   role   of   exchanging   energy   between   the   kinetic   and  
gravitational  potential  forms  has  cancelled  when  these  two  evolution  equations  were  added.      
  
  

Conservation  equation  for  total  energy   E   in  the  absence  of  molecular  fluxes    

  
In  the  absence  of  molecular  or  other  irreversible  processes  (such  as  radiation  of  heat),  and  in  the  
absence   of   the   non-­‐‑conservative   source   term   for   Absolute   Salinity   that   is   associated   with  
remineralization,  both  the  specific  entropy   η   and  the  Absolute  salinity   S A   of  each  fluid  parcel  is  
constant  following  the  fluid  motion  so  that  the  right-­‐‑hand  side  of  (B.6)  is  zero  and  the  material  
derivative  of  internal  energy  satisfies   du dt = − ( p + P0 ) dv dt   so  that  the  internal  energy  changes  
only   as   a   result   of   the   work   done   in   compressing   the   fluid   parcel.      Realizing   that   v = ρ −1    and  
using  the  continuity  Eqn.  (A.8.1)  in  the  form   dρ dt + ρ∇⋅ u = 0,    du dt   can  be  expressed  in  this  
situation   of   no   molecular,   radiative   or   boundary   fluxes   as   du dt = − ρ −1 ( p + P0 ) ∇ ⋅ u .       Adding  
this  equation  to  the  inviscid,  non-­‐‑dissipative  version  of  (B.12)  gives    

( ρE )t

+ ∇ ⋅ ( ρ uE

)

= − ∇ ⋅ ([ p + P0 ] u ) ,  

= ρ dE dt

no  molecular  fluxes  (B.13)  

where  the  total  energy    

E = u + 12 u ⋅ u + Φ   

(B.14)  

is  defined  as  the  sum  of  the  internal,  kinetic  and  gravitational  potential  energies.      
  
  

Conservation  equation  for  total  energy  in  the  presence  of  molecular  fluxes  and  
remineralization    

  
Now,   following   section   49   Landau   and   Lifshitz   (1959)   we   need   to   consider   how   molecular  
fluxes  of  heat  and  salt  and  the  radiation  of  heat  will  alter  the  simplified  conservation  equation  
of  total  energy  (B.13).    The  molecular  viscosity  gives  rise  to  a  stress  in  the  fluid  represented  by  
the   tensor   σ ,    and   the   interior   flux   of   energy   due   to   this   stress   tensor   is   u ⋅ σ    so   that   there  
needs   to   be   the   additional   term   −∇ ⋅ ( u ⋅ σ )    added   to   the   right-­‐‑hand   side   of   the   total   energy  
conservation   equation.      Consistent   with   Eqn.   (B.9)   above   we   take   the   stress   tensor   to   be  
    so   that   the   extra   term   is   ∇ ⋅ ρ v visc∇ 1 u ⋅ u .       Also   heat   fluxes   at   the   ocean  
σ = − ρ v visc ∇u
]
2[
R
boundaries  and  by  radiation   F   and  molecular  diffusion   FQ   necessitate  the  additional  terms  
−∇ ⋅ FR − ∇ ⋅ FQ .    At  this  stage  we  have  not  specified  the  form  of  the  molecular  diffusive  flux  of  
heat   FQ   in  terms  of  gradients  of  temperature  and  Absolute  Salinity;  this  is  done  below  in  Eqn.  
(B.24).      The   non-­‐‑conservative   production   of   Absolute   Salinity   by   the   remineralization   of  
S
sinking  particulate  matter,   ρ S A ,  introduces  a  source  of  energy  because  the  specific  internal  
energy   and   the   specific   enthalpy   of   seasalt   are   not   the   same   as   for   pure   water.      The   total  
energy  conservation  equation  in  the  presence  of  molecular,  radiative  and  boundary  fluxes,  as  
well  as  the  interior  source  of  salinity  is    

(

( ρE )t + ∇ ⋅ ( ρuE ) = ρ dE

)

(

)

dt = − ∇ ⋅ ⎡⎣ p + P0 ⎤⎦ u − ∇ ⋅F R − ∇ ⋅FQ

(

)

S
+ ∇ ⋅ ρ v visc∇ 12 ⎡⎣ u ⋅ u ⎤⎦ + hS ρ S A .

  

(B.15)  

A

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where   hSA = µ − (T0 + t ) µT   (see  Eqn.  (A.11.1))  is  the  partial  derivative  of  specific  enthalpy  with  
respect  to  Absolute  Salinity  at  fixed  temperature  and  pressure.    This  last  term  in  Eqn.  (B.15)  is  
more  readily  justified  in  Eqn.  (B.17)  below,  which  is  a  rearranged  form  of  Eqn.  (B.15).      
S
If  it  were  not  for  the  remineralization  source  term,   hS ρ S A ,  the  right-­‐‑hand  side  of  the   E   
A
conservation  equation  (B.15)  would  be  the  divergence  of  a  flux,  ensuring  that  total  energy   E   
would   be   both   a   “conservative”   variable   and   an   “isobaric   conservative”   variable   (see  
appendix  A.8  for  the  definition  of  these  characteristics).        
  
  

Two  alternative  forms  of  the  conservation  equation  for  total  energy      

  
Another  way  of  expressing  the  total  energy  equation  (B.15)  is  to  write  it  in  a  quasi-­‐‑divergence  
form,  with  the  temporal  derivative  being  of   ρE = ρ ( u + 12 u ⋅ u + Φ )   while  the  divergence  part  
of   the   left-­‐‑hand   side   is   based   on   a   different   quantity,   namely   the   Bernoulli  
function B = h + 12 u ⋅ u +Φ .     This  form  of  the  total  energy  equation  is    

( ρE )t

(

)

(

)

S
+ ∇ ⋅ ρ uB = − ∇ ⋅F R − ∇ ⋅FQ + ∇ ⋅ ρ v visc∇ 12 ⎡⎣ u ⋅ u ⎤⎦ + hS ρ S A .   

(B.16)  

A

In  an  ocean  modelling  context,  it  is  rather  strange  to  contemplate  the  energy  variable  that  is  
advected   through   the   face   of   a   model   grid,   B ,   to   be   different   to   the   energy   variable   that   is  
changed   in   the   grid   cell,   E .      Hence   this   form   of   the   total   energy   equation   has   not   proved  
popular.      
A  third  way  of  expressing  the  total  energy  equation  (B.15)  is  to  write  the  left-­‐‑hand  side  in  
terms  of  only  the  Bernoulli  function   B = h + 12 u ⋅ u +Φ   so  that  the  prognostic  equation  for  the    
Bernoulli  function  is    

( ρB )t + ∇ ⋅ ( ρuB ) = ρ dB

(

)

S
dt = Pt − ∇ ⋅F R − ∇ ⋅FQ + ∇ ⋅ ρ v visc∇ 12 ⎡⎣ u ⋅ u ⎤⎦ + hS ρ S A .   (B.17)  
A

The   source   term   ρ S    of   Absolute   Salinity   caused   by   the   remineralization   of   particulate  
matter   affects   enthalpy   at   the   rate   hSA = µ − (T0 + t ) µT    and   can   be   thought   of   as   replacing  
some   seasalt   in   place   of   water   molecules,   occurring   at   fixed   pressure   and   temperature,   as  
might  occur  through  two  syringes  in  the  interior  of  a  seawater  parcel,  one  supplying  pure  salt  
and  the  other  extracting  pure  water,  at  the  same  temperature  and  pressure.    The  influence  of  
the  salinity  increment  caused  by  this  source  term  on  enthalpy  (and  therefore  on  the  Bernoulli  
function   B )  is  similar  to  the  way  an  increment  of  Absolute  Salinity  enters  Eqn.  (B.3).    When  
the  flow  is  steady,  and  in  particular,  when  the  pressure  field  is  time  invariant  at  every  point  in  
space,   this   Bernoulli   form   of   the   total   energy   equation   has   the   desirable   property   that   B    is  
conserved   following   the   fluid   motion   in   the   absence   of   radiative,   boundary   and   molecular  
fluxes  and  in  the  absence  of  non-­‐‑conservative  salinity  production.    Subject  to  this  steady-­‐‑state  
S
assumption,  and  in  the  absence  of   ρ S A   the  Bernoulli  function   B   possesses  the  “potential”  
property.    The  negative  aspect  of  this   B   evolution  equation  (B.17)  is  that  in  the  more  general  
situation   where   the   flow   is   unsteady,   the   presence   of   the   Pt    term   means   that   the   Bernoulli  
function   does   not   behave   as   a   conservative   variable   because   the   right-­‐‑hand   side   of   (B.17)   is  
not  the  divergence  of  a  flux.    In  this  general  non-­‐‑steady  situation   B   is  “isobaric  conservative”  
but  is  not  a  “conservative”  variable  nor  does  it  posses  the  “potential”  property.      
Noting  that  the  total  energy   E   is  related  to  the  Bernoulli  function  by   E = B − ( p + P0 ) ρ   
S
and  even  if  we  take  the  whole  ocean  to  be  in  a  steady  state  and  with   ρ S A = 0 ,  so  that   B   has  
the   “potential”   property,   it   is   clear   that   E    does   not   have   the   “potential”   property   in   this  
situation.    That  is,  if  a  seawater  parcel  moves  from  say  2000  dbar  to  0  dbar  without  exchange  
of   material   or   heat   with   its   surroundings   and   with   Pt = 0    everywhere,   then   B    remains  
constant   while   the   parcel’s   total   energy   E    changes   by   the   difference   in   the   quantity  
− ( p + P0 ) ρ   between  the  two  locations.    Hence  we  conclude  that  even  in  a  steady  ocean   E   
does  not  possess  the  “potential”  property.    This  means  that  total  energy   E   is  useless  as  far  as  
being  a  marker  of  fluid  flow.      
SA

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(

137

)

When   the   viscous   production   term   ∇ ⋅ ρ v visc∇ 12 ⎡⎣ u ⋅ u ⎤⎦    in   the   above   equations   is  
integrated  over  the  ocean  volume,  the  contribution  from  the  sea  surface  is  the  power  input  by  
the  wind  stress   τ ,  namely  the  area  integral  of   τ ⋅ usurf   where   usurf   is  the  surface  velocity  of  
the  ocean.      
  
  

Obtaining  the  First  Law  of  Thermodynamics  by  subtraction    

  
The   evolution   equation   (B.12)   for   the   sum   of   kinetic   and   gravitational   potential   energies   is  
now  subtracted  from  the  total  energy  conservation  equation  (B.15)  giving    

( ρu )t + ∇ ⋅ ( ρuu ) = ρ du dt = − ( p + P0 ) ∇ ⋅ u

− ∇ ⋅F R − ∇ ⋅FQ + ρε + hS ρ S A .   
S

A

(B.18)  

Using   the   continuity   equation   in   the   form   ρ dv dt = ∇⋅ u    and   the   fundamental  
thermodynamic  relation  (A.7.2),  this  equation  can  be  written  as    

⎛
dS ⎞
⎛ dh 1 dP ⎞
⎛ du
dv ⎞
dη
= ρ⎜
ρ⎜
−
+ ( p + P0 ) ⎟ = ρ ⎜ (T0 + t )
+ µ A⎟
⎟
dt
dt ⎠
dt ⎠
⎝ dt
⎝ dt ρ dt ⎠
⎝

    

(B.19)  

= − ∇ ⋅F − ∇ ⋅F + ρε + hS ρ S ,
R

SA

Q

A

which   is   the   First   Law   of   Thermodynamics.      The   corresponding   evolution   equation   for  
Absolute  Salinity  is  (Eqn.  (A.21.8))    

ρ

(

)

(

)

dSA
S
= ρ SA + ∇ ⋅ ρ u SA = − ∇ ⋅FS + ρ S A ,   
t
dt

(B.20)  

where   FS   is  the  molecular  flux  of  salt  and   ρ S A   is  the  non-­‐‑conservative  source  of  Absolute  
Salinity   due   to   the   remineralization   of   particulate   matter.      For   many   purposes   in  
oceanography  the  exact  dependence  of  the  molecular  fluxes  of  heat  and  salt  on  the  gradients  
of  Absolute  Salinity,  temperature  and  pressure  is  unimportant,  nevertheless,  Eqns.  (B.23)  and  
(B.24)  below  list  these  molecular  fluxes  in  terms  of  the  spatial  gradients  of  these  quantities.      
At  first  sight  Eqn.  (B.19)  has  little  to  recommend  it;  there  are  two  non-­‐‑conservative  source  
S
terms   ρε    and   hS ρ S A    on   the   right-­‐‑hand   side   and   the   left-­‐‑hand   side   is   not   ρ    times   the  
A
material  derivative  of  any  quantity  as  is  required  of  a  conservation  equation  of  a  conservative  
variable.    Equation  (B.19)  corresponds  to  equation  (57.6)  of  Landau  and  Lifshitz  (1959)  and  is  
repeated  at  Eqns.  (A.13.1)  and  (A.13.3)  above.      
The   approach   used   here   to   develop   the   First   Law   of   Thermodynamics   seems   rather  
convoluted   in   that   the   conservation   equation   for   total   energy   is   first   formed,   and   then   the  
evolution  equations  for  kinetic  and  gravitational  potential  energies  are  subtracted.    Moreover,  
the   molecular,   radiative   and   boundary   fluxes   were   included   into   the   total   energy  
conservation   equation   as   separate   deliberate   flux   divergences,   rather   than   coming   from   an  
underlying   basic   conservation   equation.      This   is   the   approach   of   Landau   and   Lifshitz   (1959)  
and   it   is   adopted   for   the   following   reasons.      First   this   approach   ensures   that   the   molecular,  
radiative   and   boundary   fluxes   do   enter   the   total   energy   conservation   equation   (B.15)   as   the  
divergence  of  fluxes  so  that  the  total  energy  is  guaranteed  to  be  a  conservative  variable  (apart  
from   the   salinity   source   term).      This   is   essential;   total   energy   can   only   be   allowed   to  
spontaneously   appear   or   disappear   when   there   is   a   bona   fide   interior   source   term   such   as  
S
hS ρ S A .    Second,  it  is  rather  unclear  how  one  would  otherwise  arrive  at  the  molecular  fluxes  
A
of   heat   and   salt   on   the   right-­‐‑hand   side   of   the   First   Law   of   Thermodynamics   since   the   direct  
approach  which  was  attempted  at  the  beginning  of  this  appendix  involved  the  poorly  defined  
“rate  of  heating”   δ q dt   and  did  not  lead  us  to  the  First  Law.    For  completeness,  the  molecular  
fluxes   FQ   and   FS   are  now  written  in  terms  of  the  gradients  of  Absolute  Salinity,  temperature  
and  pressure.      
  
S

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The  molecular  fluxes  of  heat  and  salt    
  

The  molecular  fluxes  of  salt  and  heat,   FS   and   FQ ,  are  now  written  in  the  general  matrix  
form  in  terms  of  the  thermodynamic  Onsager  “forces”   ∇ − µ T   and   ∇ 1 T   as  (see  de  Groot  
and  Mazur  (1984))    

(

)

( )

(
)
( )
B∇ ( − µ T ) + C ∇ (1 T ) ,  

FS = A∇ − µ T + B∇ 1 T ,  

(B.21)  

FQ =

(B.22)  

where   A, B    and   C    are   three   independent   coefficients.      The   equality   of   the   off-­‐‑diagonal  
diffusion  coefficients,   B ,  results  from  the  Onsager  (1931a,b)  reciprocity  relation.    When  these  
fluxes  are  substituted  into  the  First  Law  of  Thermodynamics  Eqn.  (B.19)  and  this  is  written  as  
an  evolution  equation  for  entropy,  we  find    
⎛ −µ ⎞
1
dη
∇ ⋅FS   ,  
ρ
= ρη + ∇ ⋅ ρ uη = − ∇ ⋅FQ − ⎜
(B.23)  
⎟
t
dt
T
T
⎠
⎝

( )

(

)

where   we   have   ignored   the   radiative   flux   divergence,   the   dissipation   of   turbulent   kinetic  
energy   and   the   non-­‐‑conservative   production   of   Absolute   Salinity   due   to   biogeochemistry.    
The   right-­‐‑hand   side   of   this   equation   is   now   massaged   into   the   divergence   of   a   flux   plus   a  
remainder  term    

ρ

dη
dt

( )t

= ρη

(

+ ∇ ⋅ ρ uη

)

⎛1
µ ⎞
= − ∇ ⋅ ⎜ F Q − FS ⎟
T ⎠
⎝T
⎛ 1⎞
⎛ −µ ⎞
+ F ⋅∇ ⎜ ⎟ + FS ⋅∇ ⎜
.
⎝T⎠
⎝ T ⎟⎠

      

(B.24)  

Q

The   second   line   of   this   equation   contains   the   non-­‐‑conservative   source   terms.      If   these   terms  
were   not   present   then   specific   entropy   would   be   a   conservative   thermodynamic   variable.    
Now   we   will   investigate   what   is   required   for   this   second   line   of   Eqn.   (B.24)   to   be   always  
positive;   that   is,   what   requirement   does   this   positivity   constraint   place   on   A, B    and   C    of  
Eqns.  (B.21)  and  (B.22)?    Substituting  Eqns.  (B.21)  and  (B.22)  into  the  second  line  of  Eqn.  (B.24)  
we  have    
⎛ 1⎞
⎛ −µ ⎞
⎛ 1⎞ ⎛ 1⎞
⎛ 1 ⎞ ⎛ −µ ⎞
⎛ −µ ⎞ ⎛ −µ ⎞
= C∇ ⎜ ⎟ ⋅∇ ⎜ ⎟ + 2B∇ ⎜ ⎟ ⋅∇ ⎜
+ A∇ ⎜
⋅∇
FQ ⋅∇ ⎜ ⎟ + FS ⋅∇ ⎜
.    (B.25)  
⎟
⎟
⎝T⎠
⎝ T ⎠
⎝T⎠ ⎝T⎠
⎝T⎠ ⎝ T ⎠
⎝ T ⎟⎠ ⎜⎝ T ⎟⎠
For   the   right-­‐‑hand   side   of   this   equation   to   always   be   positive   requires   A > 0 ,   C > 0    and   a  
condition  that   B   not  be  too  large.    In  terms  of  the  directions  of  the  spatial  gradients   ∇ 1 T   
and   ∇ − µ T ,   the   B    term   is   largest   in   magnitude   when   these   gradients   are   parallel   or  
antiparallel.    Hence  the  right-­‐‑hand  side  of  this  equation  may  be  considered  a  simple  quadratic,  
and   the   requirement   that   we   seek   is   that   there   are   no   real   solutions   of   the   right-­‐‑hand   side  
being   zero,   requiring   that   the   discriminant   of   the   quadratic   be   negative.      That   is   4B 2 − 4 AC   
must   be   negative.      So   the   three   constraints   are   A > 0 ,   C > 0    and   AC > B 2 ,   which   can   be  
reduced  to  simply  two  constraints  such  as   A > 0   and   C > B 2 A .      
The  part  of  the  salt  flux  of  Eqn.  (B.21)  that  is  proportional  to   −∇SA   is  traditionally  written  
as   − ρ k S ∇SA   implying  that   A = ρ k S T µ S .    The  molecular  fluxes  of  salt  and  heat,   FS   and   FQ ,  
A
can  now  be  written  in  terms  of  the  gradients  of  Absolute  Salinity,  temperature  and  pressure  as    

(

( )

)

⎛
⎞
⎛ ρk ST ⎛ µ ⎞
µ
B⎞
+
FS = − ρ k S ⎜ ∇SA + P ∇P⎟ − ⎜
⎟ ∇T ,  
⎜⎝
⎟⎠
⎜⎝ µ S ⎜⎝ T ⎟⎠ T T 2 ⎟⎠
µS
A
A
FQ = −

Bµ S
Bµ S
1 ⎛
B2 ⎞
S
T
A
A
∇T
+
C
−
F
=
−
ρ
c
k
∇T
+
FS ,  
⎜
⎟
p
A⎠
T2 ⎝
ρk ST
ρk ST

(B.26)  
(B.27)  

where  the  fact  that   C > B 2 A   has  been  used  to  write  the  regular  diffusion  of  heat  down  the  
temperature   gradient   as   − ρ c p k T ∇T    where   k T    is   the   positive   molecular   diffusivity   of  
temperature.      These   expressions   involve   the   (strictly   positive)   molecular   diffusivities   of  

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139

temperature  and  salinity  ( k T   and   k S )  and  the  single  cross-­‐‑diffusion  parameter   B .    The  other  
parameters  in  these  equations  follow  directly  from  the  Gibbs  function  of  seawater.      
Sometimes  a  “reduced  heat  flux”  is  introduced  by  reducing  the  molecular  flux  of  heat  by  
FS = µ − T µT FS ,  being  the  flux  of  enthalpy  due  to  the  molecular  flux  of  salt.    This  
∂h ∂SA
T,p
prompts  the  introduction  of  a  revised  cross-­‐‑diffusion  coefficient  defined  by    

(

)

B′ ≡ B +

ρk ST 3 ⎛ µ ⎞
,  
µ S ⎜⎝ T ⎟⎠ T

(B.28)  

A

and  in  terms  of  this  cross-­‐‑diffusion  coefficient  Eqns.  (B.26)  and  (B.27)  can  be  written  as    

⎛
⎞
µ
B′
FS = − ρ k S ⎜ ∇SA + P ∇P⎟ − 2 ∇T ,  
⎜⎝
⎟⎠
µS
T
A

(B.29)  

and    

(

)

FQ − µ − T µT FS = − ρ c p k T ∇T +
=

B′ µ S

A

ρk ST

FS

⎞
B′ µ S ⎛
µ
A
− ρ c p K ∇T −
⎜ ∇SA + P ∇P⎟ ,
⎟⎠
T ⎜⎝
µS
A

  

(B.30)  

T

( )

where   K T   is  a  revised  molecular  diffusivity  of  temperature,   ρ c p K T = ρ c p k T + B′ 2 AT 2 .      
The   term   in   (B.29)   that   is   proportional   to   the   pressure   gradient   ∇P    represents  
“barodiffusion”  as  it  causes  a  flux  of  salt  down  the  gradient  of  pressure.    In  an  undisturbed  
ocean   that   is   in   vertical   diffusive   equilibrium,   the   barodiffusion   term   would   cause   Absolute  
Salinity  to  increase  with  depth  in  the  ocean  at  the  rate  of   ~ 3g kg −1   per  1000m.    The  turbulent  
nature  of  the  ocean  means  that  this  molecular  diffusive  balance  does  not  occur.    The  last  term  
in  (B.29)  is  a  flux  of  salt  due  to  the  gradient  of  in  situ  temperature  and  is  called  the  Soret  effect,  
while  the  last  term  in  the  second  line  of  Eqn.  (B.30)  is  called  the  Dufour  effect.      
The  molecular  flux  of  salt  is  independent  of  the  four  arbitrary  constants  (Fofonoff  (1962))  
that   appear   in   the   Gibbs   function   of   seawater   (Eqn.   (2.6.2)).      This   implies   that   the   cross-­‐‑
diffusion  coefficient   B   in  Eqns.  (B.21)–(B.22)  is  arbitrary  to  the  extent   a3ρ k S T µ S (since   µ   is  
A
arbitrary   to   the   extent   a3 + a4 (T0 + t ) ).      From   Eqn.   (B.27)   we   find   that   the   molecular   flux   of  
“heat”   FQ   is  unknowable  to  the  extent   a3FS .    This  means  that  the   − ∇ ⋅ FQ   term  on  the  right  
of  the  First  Law  Eqn.  (B.19)  is  unknowable  to  the  extent   − a3∇ ⋅ FS .     The  left-­‐‑hand  side  of  Eqn.  
(B.19)   is   unknowable   to   the   extent   a3 ρ dSA dt    (since   specific   enthalpy   h    contains   the  
arbitrary   component   a1 + a3SA ).      The   last   term   in   Eqn.   (B.19)   contains   the   arbitrary   term  
S
a3ρ S A    (since   hS    is   arbitrary   by   the   amount   a3 ).      These   three   arbitrary,   unknowable  
A
contributions  to  the  First  Law  of  Thermodynamics  Eqn.  (B.19)  sum  to   a3   times  the  evolution  
equation  (B.20)  for  Absolute  Salinity.    This  allows  these  arbitrary  terms  to  be  subtracted  from  
Eqn.   (B.19),   confirming   that   the   four   arbitrary   unknowable   constants   of   Eqn.   (2.6.2)   have   no  
measureable   consequences   on   the   First   Law   of   Thermodynamics.      The   cross-­‐‑diffusion  
coefficient   B′   of  Eqns.  (B.28)  –  (B.30)  does  not  contain  any  arbitrary  constants.      
Regarding   Eqns.   (B.21)–(B.30),   it   is   noted   that   strictly   speaking   the   gradient   of   the  
chemical   potential   µ    must   be   replaced   by   the   gradients   of   the   chemical   potentials   of   the  
individual   constituents   of   sea   salt,   and   the   diffusion   coefficients   in   front   of   these   many  
gradients   are   different   for   each   constituent,   since   there   is   no   uniform   molecular   diffusion   of  
the  mixture  "ʺsea  salt"ʺ.    When  additional  processes  act  to  keep  the  composition  approximately  
fixed,   the   use   of   only   one   chemical   potential   for   sea   salt   is   permitted   in   non-­‐‑equilibrium  
situations.    These  processes  are  mainly  ion  relaxation  by  Coulomb  forces,  which  in  the  form  of  
ambipolar   diffusion   prevent   any   local   electrical   charge   separation,   and   secondly,   turbulent  
mixing   which   has   the   same   transport   coefficient   for   each   species   and   whose   fluxes   are  
proportional  to  the  concentration  gradients  of  “potential”  quantities  (see  appendix  A.9)  rather  
than  to  the  gradients  of  the  individual  chemical  potentials.      

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Appendix  C:  

Publications  describing  the  TEOS-­‐‑10  thermodynamic  
descriptions  of  seawater,  ice  and  moist  air    
  

Primary  standard  documents    
  

Harvey,  A.  H.  and  P.  H.  Huang,  2007:  First-­‐‑Principles  Calculation  of  the  Air–Water  Second  Virial  
Coefficient.    Int.  J.  Thermophys.,  28,  556–565.      
Hyland,   R.   W.   and   A.   Wexler,   1983:   Formulations   for   the   thermodynamic   properties   of   dry   air  
from  173.15  to  473.15  K,  and  of  saturated  moist  air  from  173.15  to  372.15  K,  at  pressures  up  to  
5Mpa.    ASHRAE  Transact.  89,  520–535.      
IAPWS,   2008a:   Release   on   the   IAPWS   Formulation   2008   for   the   Thermodynamic   Properties   of  
Seawater.   The   International   Association   for   the   Properties   of   Water   and   Steam.   Berlin,  
Germany,  September  2008,  available  from  http://www.iapws.org. This  Release  is  referred  to  in  
the  text  as  IAPWS-­‐‑08.      
IAPWS,   2009a:   Revised   Release   on   the   Equation   of   State   2006   for   H2O   Ice   Ih.   The   International  
Association  for  the  Properties  of  Water  and  Steam.  Doorwerth,  The  Netherlands,  September  
2009,   available   from   http://www.iapws.org. This   revised   Release   is   referred   to   in   the   text   as  
IAPWS-­‐‑06.      
IAPWS,   2009b:   Revised   Release   on   the   IAPWS   Formulation   1995   for   the   Thermodynamic  
Properties   of   Ordinary   Water   Substance   for   General   and   Scientific   Use.   The   International  
Association  for  the  Properties  of  Water  and  Steam.  Doorwerth,  The  Netherlands,  September  
2009,   available   from   http://www.iapws.org. This   revised   Release   is   referred   to   in   the   text   as  
IAPWS-­‐‑95.      
IAPWS,   2009c:   Supplementary   Release   on   a   Computationally   Efficient   Thermodynamic  
Formulation   for   Liquid   Water   for   Oceanographic   Use.   The   International   Association   for   the  
Properties  of  Water  and  Steam.  Doorwerth,  The  Netherlands,  September  2009,  available  from  
http://www.iapws.org. This  Release  is  referred  to  as  IAPWS-­‐‑09.      
IAPWS,  2010:  Guideline  on  an  Equation  of  State  for  Humid  Air  in  Contact  with  Seawater  and  Ice,  
Consistent  with  the  IAPWS  Formulation  2008  for  the  Thermodynamic  Properties  of  Seawater.  
The  International  Association  for  the  Properties  of  Water  and  Steam.  Niagara  Falls,  Canada,  
July   2010,   available   from   http://www.iapws.org. This   Guideline   is   referred   to   in   the   text   as  
IAPWS-­‐‑10.        
Lemmon,   E.   W.,   R.   T.   Jacobsen,   S.   G.   Penoncello   and   D.   G.   Friend,   2000:   Thermodynamic  
properties  of  air  and  mixtures  of  nitrogen,  argon  and  oxygen  from  60  to  2000  K  at  pressures  to  
2000  MPa.    J.  Phys.  Chem.  Ref.  Data,  29,  331–362.      
Millero,   F.   J.,   R.   Feistel,   D.   G.   Wright,   and   T.   J.   McDougall,   2008a:   The   composition   of   Standard  
Seawater   and   the   definition   of   the   Reference-­‐‑Composition   Salinity   Scale,   Deep-­‐‑Sea   Res.   I,   55,  
50-­‐‑72.      
  

Secondary  standard  documents    

  
IOC,   SCOR   and   IAPSO,   2010a:   The   international   thermodynamic   equation   of   seawater   –   2010:  
Calculation   and   use   of   thermodynamic   properties.      Intergovernmental   Oceanographic  
Commission,  Manuals  and  Guides  No.  56,  UNESCO  (English),  196  pp,  Paris.    Available  from  
www.TEOS-10.org [the  present  document,  called  the  TEOS-­‐‑10  manual]      
McDougall,   T.   J.,   D.   R.   Jackett,   F.   J.   Millero,   R.   Pawlowicz   and   P.   M.   Barker,   2012:   A   global  
algorithm   for   estimating   Absolute   Salinity.      Ocean   Science,   8,   1123-­‐‑1134. http://www.oceansci.net/8/1123/2012/os-8-1123-2012.pdf The   computer   software   is   available   from www.TEOS10.org

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Background  papers  to  the  declared  standards    
  

141

Feistel,  R.,  2003:  A  new  extended  Gibbs  thermodynamic  potential  of  seawater.    Progr.  Oceanogr.,  58,  
43-­‐‑114.      
Feistel,  R.,  2008:  A  Gibbs  function  for  seawater  thermodynamics  for  −6  to  80  °C  and  salinity  up  to  
120  g  kg–1.    Deep-­‐‑Sea  Res.  I,  55,  1639-­‐‑1671.      
Feistel,  R.  and  W.  Wagner,  2006:  A  New  Equation  of  State  for  H2O  Ice  Ih.    J.  Phys.  Chem.  Ref.  Data,  
35,  2,  1021-­‐‑1047.      
Feistel,  R.,  S.  Weinreben,  H.  Wolf,  S.  Seitz,  P.  Spitzer,  B.  Adel,  G.  Nausch,  B.  Schneider  and  D.  G.  
Wright,  2010c:  Density  and  Absolute  Salinity  of  the  Baltic  Sea  2006–2009.    Ocean  Science,  6,  3–
24. http://www.ocean-sci.net/6/3/2010/os-6-3-2010.pdf   
Feistel,   R.,   D.   G.   Wright,   H.-­‐‑J.   Kretzschmar,   E.   Hagen,   S.   Herrmann   and   R.   Span,   2010a:  
Thermodynamic   properties   of   sea   air.   Ocean   Science,   6,   91–141.      http://www.oceansci.net/6/91/2010/os-6-91-2010.pdf   
Feistel,  R.,  D.  G.  Wright,  K.  Miyagawa,  A.  H.  Harvey,  J.  Hruby,  D.  R.  Jackett,  T.  J.  McDougall  and  
W.   Wagner,   2008:   Mutually   consistent   thermodynamic   potentials   for   fluid   water,   ice   and  
seawater:   a   new   standard   for   oceanography.   Ocean   Science,   4,   275-­‐‑291.   http://www.oceansci.net/4/275/2008/os-4-275-2008.pdf   
IOC,   SCOR   and   IAPSO,   2010b:   The   international   thermodynamic   equation   of   seawater   –   2010:   A  
Summary   for   Policy   Makers.      Intergovernmental   Oceanographic   Commission   (Brochures  
Series).    Available  from  www.TEOS-10.org   
McDougall,   T.   J.,   2003:   Potential   enthalpy:   A   conservative   oceanic   variable   for   evaluating   heat  
content  and  heat  fluxes.  Journal  of  Physical  Oceanography,  33,  945-­‐‑963.      
Marion,   G.   M.,   F.   J.   Millero,   and   R.   Feistel,   2009:   Precipitation   of   solid   phase   calcium   carbonates  
and   their   effect   on   application   of   seawater   SA − T − P    models,   Ocean   Sci.,   5,   285-­‐‑291.  
http://www.ocean-sci.net/5/285/2009/os-5-285-2009.pdf   
Millero,   F.   J.,   2000.   Effect   of   changes   in   the   composition   of   seawater   on   the   density-­‐‑salinity  
relationship.    Deep-­‐‑Sea  Res.  I  47,  1583-­‐‑1590.      
Pawlowicz,   R.,   2010a:   A   model   for   predicting   changes   in   the   electrical   conductivity,   Practical  
Salinity,  and  Absolute  Salinity  of  seawater  due  to  variations  in  relative  chemical  composition.  
Ocean  Science,  6,  361–378.    http://www.ocean-sci.net/6/361/2010/os-6-361-2010.pdf   
Pawlowicz,   R.,   T.   McDougall,   R.   Feistel   and   R.   Tailleux,   2012:   An   historical   perspective   on   the  
development   of   the   Thermodynamic   Equation   of   Seawater   –   2010:      Ocean   Sci.,   8,   161-­‐‑174.    
http://www.ocean-sci.net/8/161/2012/os-8-161-2012.pdf   
Pawlowicz,   R.,   D.   G.   Wright   and   F.   J.   Millero,   2011:   The   effects   of   biogeochemical   processes   on  
oceanic   conductivity/salinity/density   relationships   and   the   characterization   of   real   seawater.  
Ocean  Science,  7,  363–387.    Available  from  http://www.ocean-sci.net/7/363/2011/os-7-363-2011.pdf
Seitz,   S.,   R.   Feistel,   D.G.   Wright,   S.   Weinreben,   P.   Spitzer   and   P.   de   Bievre,   2011:   Metrological  
Traceability   of   Oceanographic   Salinity   Measurement   Results.      Ocean   Science,   7,   45–62.    
http://www.ocean-sci.net/7/45/2011/os-7-45-2011.pdf   
Wagner,  W.  and  Pruß,  A.,  2002:  The  IAPWS  formulation  1995  for  the  thermodynamic  properties  of  
ordinary  water  substance  for  general  and  scientific  use.    J.  Phys.  Chem.  Ref.  Data,  31,  387-­‐‑535.    
Wright,  D.  G.,  R.  Pawlowicz,  T.  J.  McDougall,  R.  Feistel  and  G.  M.  Marion,  2011:  Absolute  Salinity,  
“Density   Salinity”   and   the   Reference-­‐‑Composition   Salinity   Scale:   present   and   future   use   in  
the   seawater   standard   TEOS-­‐‑10.      Ocean   Sci.,   7,   1-­‐‑26.      http://www.ocean-sci.net/7/1/2011/os-7-12011.pdf   
  

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Papers  describing  computer  software    
  

Feistel,  R.,  D.  G.  Wright,  D.  R.  Jackett,  K.  Miyagawa,  J.  H.  Reissmann,  W.  Wagner,  U.  Overhoff,  C.  
Guder,   A.   Feistel   and   G.   M.   Marion,   2010b:   Numerical   implementation   and   oceanographic  
application  of  the  thermodynamic  potentials  of  liquid  water,  water  vapour,  ice,  seawater  and  
humid   air   -­‐‑   Part   1:   Background   and   equations.      Ocean   Science,   6,   633-­‐‑677.      http://www.oceansci.net/6/633/2010/os-6-633-2010.pdf
and   http://www.ocean-sci.net/6/633/2010/os-6-633-2010supplement.pdf   
McDougall   T.   J.   and   P.   M.   Barker,   2011:   Getting   started   with   TEOS-­‐‑10   and   the   Gibbs   Seawater  
(GSW)   Oceanographic   Toolbox,   28pp.,   SCOR/IAPSO   WG127,   ISBN   978-­‐‑0-­‐‑646-­‐‑55621-­‐‑5,  
available  from  www.TEOS-10.org   
Roquet,  F.,  G.  Madec,  T.  J.  McDougall  and  P.  M.  Barker,  2015:  Accurate  polynomial  expressions  for  
the  density  and  specific  volume  of  seawater  using  the  TEOS-­‐‑10  standard.    Ocean  Modelling,  90,  
29-­‐‑43,  http://dx.doi.org/10.1016/j.ocemod.2015.04.002     
McDougall,   T.   J.,   D.   R.   Jackett,   F.   J.   Millero,   R.   Pawlowicz   and   P.   M.   Barker,   2012:   A   global  
algorithm   for   estimating   Absolute   Salinity.      Ocean   Science,   8,   1123-­‐‑1134. http://www.oceansci.net/8/1123/2012/os-8-1123-2012.pdf The   computer   software   is   available   from www.TEOS10.org
Wright,  D.  G.,  R.  Feistel,  J.  H.  Reissmann,  K.  Miyagawa,  D.  R.  Jackett,  W.  Wagner,  U.  Overhoff,  C.  
Guder,   A.   Feistel   and   G.   M.   Marion,   2010:   Numerical   implementation   and   oceanographic  
application  of  the  thermodynamic  potentials  of  liquid  water,  water  vapour,  ice,  seawater  and  
humid   air   -­‐‑   Part   2:   The   library   routines.      Ocean   Science,   6,   695-­‐‑718.      http://www.oceansci.net/6/695/2010/os-6-695-2010.pdf
and   http://www.ocean-sci.net/6/695/2010/os-6-695-2010supplement.pdf   
  

TEOS-­‐‑10  web  site    
  

SCOR/IAPSO   Working   Group   127   has   created   the   web   site   www.TEOS-10.org which   serves  
many   of   the   TEOS-­‐‑10   papers,   this   TEOS-­‐‑10   manual   as   well   as   the   SIA   (Seawater   Ice   Air)   and  
GSW   (Gibbs   SeaWater)   libraries   of   oceanographic   computer   software.      Each   function   in   the  
GSW   MATLAB   Oceanographic   Toolbox   contains   a   help   file   which   describes   the   derivation,  
attributes  and  use  of  each  function.      
In  addition,  the  www.TEOS-10.org web  site  has  two  documents  entitled    
• “Getting   started   with   TEOS-­‐‑10   and   the   Gibbs   Seawater   (GSW)   Oceanographic  
Toolbox”  (McDougall  and  Barker,  2011)  and    
• “What   every   oceanographer   needs   to   know   about   TEOS-­‐‑10   (The   TEOS-­‐‑10   Primer)”    
    (Pawlowicz,  2010b).      
Together   these   documents   serve   as   a   succinct   introduction   to   the   use   of   TEOS-­‐‑10   in   physical  
oceanography.      

  
  
  
Note  that  several  of  the  papers  listed  in  this  appendix  have  appeared  in  Ocean  Science  in  the  special  
issue  “Thermophysical  Properties  of  Seawater”,  see http://www.ocean-sci.net/special_issue14.html          
  
  
  
  
Note   that   when   referring   to   the   use   of   TEOS-­‐‑10,   it   is   the   present   document   that   should   be  
referenced  as  IOC  et  al.  (2010),  with  the  full  citation  being    
IOC,  SCOR  and  IAPSO,  2010:  The  international  thermodynamic  equation  of  seawater  –  2010:  
Calculation   and   use   of   thermodynamic   properties.      Intergovernmental   Oceanographic  
Commission,  Manuals  and  Guides  No.  56,  UNESCO  (English),  196  pp.      
  
  

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143

Appendix  D:  Fundamental  constants    
  
  
  
Following   the   recommendation   of   IAPWS   (2005),   the   values   of   the   fundamental   constants  
were  taken  from  CODATA  2006  (Mohr  et  al.  (2008)),  as  listed  in  Table  D.1.    Selected  properties  
of   pure   water   were   taken   from   IAPWS   (1996,   1997,   2005,   2006)   as   listed   in   Table   D.2.      The  
chemical  Reference  Composition  of  seawater  from  Millero  et  al.  (2008a)  is  given  in  Table  D.3.    
Selected  seawater  constants  derived  from  the  Reference  Composition  are  listed  in  Table  D.4.    
0
The  exact  value  of  the  isobaric  “heat  capacity”   c p   is  given  in  Table  D.5.      
  
  
  
Table  D.1.  Fundamental  constants  from  CODATA  2006  (Mohr  et  al.  (2008))  and  ISO  (1993).    
Symbol  
R   
P0   

Value  

Uncertainty  

Unit  

8.314  472  
101  325  

0.000  015  
exact  

J  mol   K   

T0   

273.15  

exact  

–1

Pa  

Comment  
–1

molar  gas  constant  
normal  pressure  

K  
Celsius  zero  point  
  
  
  
Table  D.2.  Selected  properties  of  liquid  water  from  IAPWS  (1996,  1997,  2005,  2006)    
                                        and  Feistel  (2003).  
  
Symbol  
Value  
Uncertainty  
Unit  
Comment  
M W   
18.015  268  
0.000  002  
molar  mass  
g  mol–1  

tMD   

3.978  121  

0.04  

°C  

ρ MD   

999.974  95  

0.000  84  

kg  m–3  

maximum  density  at   P0     

ρ 0   

999.8431  

0.001  

kg  m–3  

density  at   T0   and   P0 ,   ρ 0 = 1 / v 0     

(∂ρ

0

/ ∂T

)

P

   6.774  876  ×  10–2  

0.06  ×  10–2  

maximum  density  ,  temperature  

kg  m–3  K–1   ( ∂ρ / ∂T ) P at   T0   and   P0     

Tt   
Pt   

273.16  

exact  

K  

triple  point  temperature  

611.657  

0.01  

Pa  

triple  point  pressure  

ρ t   

999.793  

0.01  

kg  m–3  

triple  point  density  

ηt   

0  

exact  

J  kg–1  K–1  

triple  point  entropy  

ut   

0  

exact  

J  kg–1  

Tf0   

273.152  519  

0.000  002  

K  

triple  point  internal  energy  
freezing  point  at   P0     

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Table  D.3.    The  sea  salt  composition  definition  for  seawater  of  Reference-­‐‑Composition    
at   25°C   and   101325   Pa.         Xj   –   mole   fractions,   Zj   –   valences,   Wj   –   mass   fractions  
(Millero   et   al.   2008a).      The   molar   masses   Mj   are   from   Wieser   (2006)   with   their  
uncertainties  in  the  last  one  or  two  digits  given  in  the  brackets.    The  mass  fractions  
Mj  are  the  mass  of  a  particular  solute  as  a  fraction  of  the  total  mass  of  solute.    The  
mole   fractions   Xj   in   this   table   are   extracted   from   Table   4   of   Millero   et   al.   (2008a)  
which  is  the  official  definition  of  Reference-­‐‑Composition  seawater.      
  
Solute j

Zj
+1
+2
+2

Na+
2+

Mg
Ca2+
K+

Mj
g mol–1
22.989 769 28(2)
24.305 0(6)
40.078(4)

Xj
10–7

Xj Zj
10–7

4188071

4188071

0.3065958

471678
91823
91159

943356
183646
91159

0.0365055
0.0117186

810

1620

4874839

–4874839

252152

–504304

15340
7520

–15340
–7520

2134

–4268

900
610

–900
–610

0.0002259

Wj

+1
+2

39.098 3(1)
87.62(1)

–1
–2

35.453(2)
96.062 6(50)

–1

61.016 84(96)

–1
–2

79.904(1)
60.008 9(10)

–1

78.840 4(70)

–1
–1

18.998 403 2(5)

–

17.007 33(7)

71

–71

0.0000038

B(OH)3
CO2

0
0

61.833 0(70)
44.009 5(9)

2807
86

0
0

0.0005527
0.0000121

10 000 000

0

1.0

Sr

2+

Cl–
SO4

2–

HCO3
Br–
CO3

–

2–

B(OH)4
F–
OH

–

Sum

IOC Manuals and Guides No. 56

0.0113495
0.0002260
0.5503396
0.0771319
0.0029805
0.0019134
0.0004078
0.0000369

TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

Table  D.4.      Selected  properties  of  KCl-­‐‑normalised  Reference  Seawater,    
                                            from  Millero  et  al.  (2008a).    
  
Symbol
Value
Uncertainty
Unit
Comment
Reference  Salinity  molar  mass    

MS

31.403 8218

0.001

g mol

MS = ∑ X j M j

–1

j

Reference  Salinity  valence  factor    

Z2

exact

1.245 2898

a

Z 2 = ∑ X j Z 2j

-

j

a

Avogadro  constant    

NA

6.022 141 79 × 1023

3 × 1016

mol −1

NS

1.917 6461 × 1022

6 × 1017

g–1

uPS

1.004 715…

exacta

g kg–1

unit  conversion  factor,    
uPS ≡ 35.165 04 g kg–1 / 35

SSO

35.165 04

exacta

g kg–1

Standard  Ocean  Reference  Salinity,    
35 uPS

TSO

273.15

exact  

K

Standard  Ocean  temperature    
TSO = T0

tSO

0

exact  

°C

Standard  Ocean  temperature    
tSO = TSO – T0

PSO

101 325

exact  

Pa

Standard  Ocean  surface  pressure    
PSO = P0

pSO

0

exact  

Pa

Standard  Ocean  surface  sea  pressure  
pSO = PSO – P0

hSO

0

exact  

J kg–1

Standard  Ocean  surface  enthalpy    
hSO = ut

ηSO

0

exact  

J kg–1 K–1

Standard  Ocean  surface  entropy    
ηSO = ηt

Su

40.188 617…

exacta

g kg–1

unit-­‐‑related  scaling  constant,    
40 uPS

tu

40

exact  

°C

unit-­‐‑related  scaling  constant  

pu

108

exact  

Pa

unit-­‐‑related  scaling  constant  

gu

1

exact  

J kg–1

unit-­‐‑related  scaling  constant  

Reference  Salinity  particle  number    

NS = N A / M S

by  definition  of  Reference  Salinity  and  Reference  Composition

Table  D.5.  The  exact  definition  of  the  isobaric  “heat  capacity”  that  relates    
                                        potential  enthalpy  to  Conservative  Temperature   Θ .       
Symbol

c 0p

Value

3991.867 957 119 63

Uncertainty

Unit

exact

–1

J kg K

Comment
–1

See Eqn. (3.3.3)

  

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146

  

TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

Table  D.6.    Chemical  composition  of  dry  air  with  a  fixed  CO2  level.      
Mole   fractions   are   from   Picard   et   al.   (2008)   except   for   N 2   
which   was   adjusted   by   subtracting   all   other   mole   fractions  
from   1   (Picard   et   al.   (2008)).      Uncertainties   of   the   molar  
masses  (Wieser  (2006))  are  given  in  brackets.      
Mole
fraction
N2
0.780 847 9
O2
0.209 390 0
Ar
0.009 332 0
CO2 0.000 400 0
Ne 0.000 018 2
He 0.000 005 2
CH4 0.000 001 5
Kr
0.000 001 1
H2
0.000 000 5
N2O 0.000 000 3
CO 0.000 000 2
Xe 0.000 000 1
Gas

Mass
fraction
0.755 184 73
0.231 318 60
0.012 870 36
0.000 607 75
0.000 012 68
0.000 000 72
0.000 000 83
0.000 003 18
0.000 000 03
0.000 000 46
0.000 000 19
0.000 000 45

Molar mass
g mol–1
28.013 4(3)
31.998 8(4)
39.948 (1)
44.009 5(9)
20.179 7(6)
4.002 602(2)
16.042 46(81)
83.798 (2)
2.015 88(10)
44.012 8(4)
28.010 1(9)
131.293 (6)

Air 1.000 000 0 0.999 999 98 28.965 46(33)
  
  
  
  
Coriolis  Parameter    
The  rotation  rate  of  the  earth   Ω   is  (in  radians  per  second)    

Ω = 7.292 1150 × 10−5 s −1 ,  

(D.1)  

(Groten  (2004))  and  the  Coriolis  parameter  f  is  (in  radians  per  second)    

f = 2Ω sin φ = 1.458 423 00×10−4 sin φ s −1 ,  

(D.2)  

where   φ   is  latitude  ( φ   has  opposite  signs  in  the  two  hemispheres).          
  
  
  
Gravitational  Acceleration  
The   gravitational   acceleration   g    in   the   ocean   can   be   taken   to   be   the   following   function   of  
latitude   φ   and  sea  pressure   p ,  or  height   z   relative  to  the  geoid,    

(
= 9.780 327 (1 + 5.2792 × 10
≈ 9.780 327 (1 + 5.2792 × 10

)(
sin φ ) (1 − 2.26 × 10
sin φ ) (1 + 2.22 × 10

)
z (m) )

g (m s −2 ) = 9.780 327 1 + 5.3024 × 10−3 sin 2 φ − 5.8 × 10−6 sin 2 2φ 1 − 2.26 × 10−7 z (m)
−3

sin 2 φ + 2.32 × 10−5

−3

sin 2 φ + 2.32 × 10−5

4

−7

4

−7

      (D.3)  

)

p (dbar) .

The   dependence   on   latitude   in   Eqn.   (D.3)   is   from   Moritz   (2000)   and   is   the   gravitational  
acceleration  on  the  surface  of  an  ellipsoid  which  approximates  the  geoid.    The  height   z   above  
the   geoid   is   negative   in   the   ocean.         Note   that   g    increases   with   depth   in   the   ocean   at   about  
71.85%  of  the  rate  at  which  it  decreases  with  height  in  the  atmosphere.      
At  a  latitude  of   45° N   and  at   p = 0 ,   g = 9.8062 m s−2 ,   which  is  a  value  commonly  used  in  
ocean  models.    The  value  of   g   averaged  over  the  earth’s  surface  is   g = 9.7976 m s−2 ,   while  the  
value  averaged  over  the  surface  of  the  ocean  is   g = 9.7963 m s−2   (Griffies  (2004)).      

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147

Appendix  E:    
Algorithm  for  calculating  Practical  Salinity    

E.1 Calculation of Practical Salinity in terms of K15
Practical   Salinity   S P    is   defined   on   the   Practical   Salinity   Scale   of   1978   (UNESCO   (1981,  
1983))   in   terms   of   the   conductivity   ratio   K15    which   is   the   electrical   conductivity   of   the  
sample  at  temperature   t68   =  15  °C  and  pressure  equal  to  one  standard  atmosphere  ( p   =  0  
dbar   and   Absolute   Pressure   P   equal   to   101   325   Pa),   divided   by   the   conductivity   of   a  
standard   potassium   chloride   (KCl)   solution   at   the   same   temperature   and   pressure.      The  
mass   fraction   of   KCl   in   the   standard   solution   is   32.4356x10-­‐‑3   (mass   of   KCl   per   mass   of  
solution).    When   K15   =  1,  the  Practical  Salinity   S P   is  by  definition  35.    Note  that  Practical  
Salinity  is  a  unit-­‐‑less  quantity.    Though  sometimes  convenient,  it  is  technically  incorrect  to  
quote  Practical  Salinity  in  “psu”;  rather  it  should  be  quoted  as  a  certain  Practical  Salinity  
“on  the  Practical  Salinity  Scale  PSS-­‐‑78”.    When   K15   is  not  unity,   S P   and   K15   are  related  by  
(UNESCO,  1981,  1983)  the  PSS-­‐‑78  equation    

SP =

5

∑ ai ( K15 )

i 2

i =0

      where         K15 =

C ( SP , t68 = 15°C,0)
C ( 35, t68 = 15°C,0)

,   

(E.1.1)  

and   the   coefficients   ai    are   given   in   the   following   table.      Note   that   the   sum   of   the   six   ai   
coefficients   is   precisely   35,   while   the   sum   of   the   six   bi    coefficients   is   precisely   zero.    
Equation  (E.1.1)  is  valid  in  the  range   2 < SP < 42.       
  

i

ai

bi

ci

di

ei

0
1

0.0080
- 0.1692

0.0005
- 0.0056

6.766097 x 10-1
2.00564 x 10-2

3.426 x 10-2

2.070 x 10-5

2

25.3851

- 0.0066

1.104259 x 10-4

4.464 x 10-4

- 6.370 x10-10

3
4

14.0941
- 7.0261

- 0.0375
0.0636

- 6.9698 x 10-7
1.0031 x 10-9

4.215 x 10-1
- 3.107 x 10-3

3.989 x10-15

5

2.7081

- 0.0144

E.2 Calculation of Practical Salinity at oceanographic temperature and pressure
The   following   formulae   from   UNESCO   (1983)   are   valid   over   the   range   −2 °C ≤ t ≤ 35 °C   
and   0 ≤ p ≤ 10 000dbar.       Measurements   of   salinity   in   the   field   generally   measure   the  
conductivity  ratio   R     

R=

C ( SP , t68 , p )

C ( 35, t68 = 15 °C,0)

=

C ( SP , t68 , p ) C ( SP , t68 ,0)

C (35, t68 ,0)

C ( SP , t68 ,0) C (35, t68 ,0 ) C (35, t68 = 15 °C,0 )

  

(E.2.1)  

which   has   been   expressed   in   (E.2.1)   as   the   product   of   three   factors,   which   are   labeled  
Rp , Rt   and   rt   as  follows    

R=

C ( SP , t68 , p )

C ( 35, t68 = 15 °C,0)

= R p Rt rt .   

(E.2.2)  

The  last  factor   rt   has  been  fitted  to  experimental  data  as  the  following  polynomial  in   t68     

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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

rt =

4

∑ ci (t68 / °C)   
i

(E.2.3)  

i =0

and  the  factor   R p   has  been  fitted  to  experimental  data  as  a  function  of   p,    t68   and   R   as    
3

∑ ei pi

(E.2.4)  
.   
2
1 + d1 ( t68 / °C ) + d 2 (t68 / °C ) + R ⎡⎣d 3 + d 4 (t68 / °C )⎤⎦
For  any  measurement  of   R   it  is  possible  to  evaluate   rt   and   R p   and  hence  calculate    
i =1

Rp = 1 +

Rt =

R
.   
R p rt

(E.2.5)  

At   a   temperature   of   t68 =15 °C,    Rt    is   simply   K15    and   Practical   Salinity   S P    can   be  
determined   form   (E.1.1).      For   temperatures   other   than   t68 =15 °C ,   Practical   Salinity   S P    is  
given  by  the  following  function  of   Rt   with   k = 0.0162,     
5

SP = ∑ ai ( Rt )

i2

+

i =0

5
(t68 / °C − 15)
i2
∑ bi ( Rt ) .   
⎡⎣1 + k (t68 / °C − 15)⎤⎦ i = 0

(E.2.6)  

Equations   (E.1.1)   and   (E.2.6)   are   valid   only   in   the   range   2 < SP < 42.       Outside   this  
range   S P    can   be   determined   by   dilution   with   pure   water   or   evaporation   of   a   seawater  
sample.      Practical   Salinity   S P    can   also   be   estimated   using   the   extension   of   the   Practical  
Salinity   Scale   proposed   by   Hill   et   al.   (1986)   for   0 < SP < 2 .      The   GSW   Oceanographic  
Toolbox   incorporates   a   modified   form   of   the   extension   of   Hill   et   al.   (1986)   for   0 < SP < 2 .    
The  modification  ensures  that  the  algorithm  is  exactly  PSS-­‐‑78  for   SP ≥ 2   and  is  continuous  
at   SP = 2 .    The  values  of  Practical  Salinity   S P   estimated  in  this  manner  may  then  be  used  
in  Eqn.  (2.4.1),  namely   SR ≈ uPS SP   to  estimate  Reference  Salinity   SR .       
When  using  a  laboratory  salinometer  to  evaluate  Practical  Salinity,  use  is  made  of  Eqn.  
(E.2.6)   since   the   salinometer   returns   Rt    and   the   instrument’s   bath   temperature   is   known  
(and  is  easily  converted  from  a  measured  temperature  on  the  ITS-­‐‑90  scale  to   t68 ).      
The  temperatures  in  Eqns.  (E.2.1)  to  (E.2.6)  are  all  on  the  IPTS-­‐‑68  scale.    The  functions  
and   coefficients   have   not   been   refitted   to   ITS-­‐‑90   temperatures.      Therefore   in   order   to  
calculate   Practical   Salinity   from   conductivity   ratio   at   a   measured   pressure   and   t90   
temperature,  it  is  necessary  first  to  convert  the  temperature  to   t68   using   t68 = 1.00024 t90   as  
described  Eqn.  (A.1.3)  of  appendix  A.1.    This  is  done  as  the  first  line  of  the  computer  code  
described   in   the   GSW   Oceanographic   Toolbox   (appendix   N).      Further   remarks   on   the  
implications   of   the   different   temperature   scales   on   the   definition   and   calculation   of  
Practical  Salinity  can  be  found  in  appendix  E.4  below.      
  
  

E.3 Calculation of conductivity ratio R for a given Practical Salinity
When   Practical   Salinity   is   known   and   one   wants   to   deduce   the   conductivity   ratio   R   
associated  with  this  value  of  Practical  Salinity  at  a  given  temperature,  a  Newton-­‐‑Raphson  
iterative  inversion  of  Eqn.  (E.2.6)  is  first  performed  to  evaluate   Rt .    Because   rt   is  a  function  
only   of   temperature,   at   this   stage   both   Rt    and   rt    are   known   so   that   Eqn.   (E.2.4)   can   be  
written   as   a   quadratic   in   R    with   known   coefficients   which   is   solved   to   yield   R .       This  
procedure   is   outlined   in   more   detail   in   UNESCO   (1983).      Computer   software   to   perform  
this   procedure   is   available   in   the   GSW   Oceanographic   Toolbox   as   the   functions  
gsw_R_from_SP   and   gsw_C_from_SP   which   return   conductivity   ratio   and   conductivity  
−1
(in   mS ( cm ) )  respectively.    Note  that  this  iterative  inverse  procedure  is  done  in  terms  of  
t68 ;  the  code  accepts   t90   as  the  input  and  immediately  converts  this  to  a   t68   temperature  
before  performing  the  above  iterative  procedure.      

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149

E.4 Evaluating Practical Salinity using ITS-90 temperatures
We  first  consider  the  consequence  of  the  change  from  IPTS-­‐‑68  to  ITS-­‐‑90  for  the  definition  
of  Practical  Salinity  as  a  function  of   K15   and  the  defining  mass  fraction  of  KCl.    Suppose  
Practical   Salinity   S P    were   to   be   evaluated   using   the   polynomial   (E.1.1)   but   using   K15−90   
instead  of   K15 ,  where   K15−90   is  defined    

K15−90 =

C ( SP , t90 = 15 °C,0)
C ( 35, t90 = 15 °C,0)

.   

(E.4.1)  

The  magnitude  of  the  difference   K15−90 − K15   can  be  calculated  and  is  found  to  be  less  than  
6.8x10-­‐‑7   everywhere   in   the   range   2 < SP < 42.       Further   calculation   shows   that  
∂SP ∂K15 < 41  everywhere  in  the  valid  range  of  Practical  Salinity,  so  that  the  consequence  
of   using   K15−90    in   (E.1.1)   instead   of   K15    incurs   a   change   in   Practical   Salinity   of   less   than  
3x10-­‐‑5.      This   is   nearly   two   orders   of   magnitude   below   the   measurement   accuracy   of   a  
sample,  and  an  order  of  magnitude  smaller  than  the  error  caused  by  the  uncertainty  in  the  
definition  of  the  mass  fraction  of  KCl.    If  all  the  original  measurements  that  form  the  basis  
of   the   Practical   Salinity   Scale   were   converted   to   ITS-­‐‑90,   and   the   analysis   repeated   to  
determine   the   appropriate   mass   fraction   to   give   the   required   conductivity   at   t90 =15 °C,   
the  same  mass  fraction  32.4356x10-­‐‑3  would  be  derived.      
Not   withstanding   the   insensitivity   of   this   conductivity   ratio   to   such   a   small  
temperature   difference,   following   Millero   et   al.   (2008a)   the   definition   of   Practical   Salinity  
can   be   restated   with   reference   to   the   ITS-­‐‑90   scale   by   noting   that   the   K15    ratio   in   Eqn.  
(E.1.1)  can  equivalently  refer  to  a  ratio  of  conductivities  at   t90 =14.996 °C.       
The  fact  that  the  conductivity  ratio   Rt   is  rather  weakly  dependent  on  the  temperature  
at   which   the   ratio   is   determined   is   important   for   the   use   of   bench   salinometers.      It   is  
important  that  samples  and  seawater  standards  be  run  at  the  same  temperature,  stable  at  
order  1  mK,  which  is  achieved  by  the  use  of  a  large  water  bath.    However,  it  is  not  critical  
to  know  the  stable  bath  temperature  to  any  better  than  10  or  20  mK.      
The   ratios   Rp , Rt    and   rt    that   underlie   the   temperature-­‐‑dependent   expression   (E.2.6)  
for   Practical   Salinity   are   more   sensitive   to   the   difference   between   IPTS-­‐‑68   and   ITS-­‐‑90  
temperatures   and   this   is   the   reason   why   we   recommend   retaining   the   original   computer  
algorithms  for  these  ratios,  and  to  simply  convert  the  input  temperature  (which  these  days  
is   on   the   ITS-­‐‑90   temperature   scale)   in   to   the   corresponding   IPTS-­‐‑68   temperature   using  
t68 = 1.00024 t90    as   the   first   operation   in   the   software.      Thereafter   the   software   proceeds  
according  to  (E.2.1)  –  (E.2.6).      
  
  

E.5 Towards SI-traceability of the measurement procedure for Practical Salinity
and Absolute Salinity
The  observation  of  climate  change  taking  place  in  the  world  ocean  on  a  global  scale  over  
decades   or   centuries   requires   measurement   techniques   that   permit   the   highest   accuracy  
currently   available,   long-­‐‑term   stability   and   world-­‐‑wide   comparability   of   the   measured  
values.    The  highest  reliability  for  this  purpose  can  be  ensured  only  by  traceability  of  these  
measurement   results   to   the   primary   standards   of   the   International   System   of   Units   (SI),  
supported  by  the  National  Metrological  Institutes  such  as  the  NIST  (National  Institute  of  
Standards  and  Technology)  in  the  US,  the  NPL  (National  Physical  Laboratory)  in  the  UK,  
or  the  PTB  (Physikalisch-­‐‑Technische  Bundesanstalt)  in  Germany.    
In   order   to   compute   the   thermodynamic   properties   of   a   seawater   sample   with  
standard   composition,   three   independent   parameters   must   be   measured.      Since   the  
introduction   of   the   Practical   Salinity   Scale   of   1978   as   an   international   standard   for  
oceanography,  these  three  properties  have  been  electrolytic  conductivity,  temperature  and  
pressure,   from   which   salinity,   density   and   other   properties   are   computed   in   turn   by  

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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

standard   algorithms.      The   traceability   of   temperature   and   pressure   measurement   results,  
for  example  by  CTD  sensors,  is  ensured  due  to  established  calibration  procedures  carried  
out  by  the  manufacturer  or  other  laboratories  and  will  not  be  considered  here  any  further.    
The   observation   of   the   ocean’s   salinity   is   a   more   complicated   task   (Millero   et   al.  
(2008a)).  Even  though  over  the  last  century  different  and  permanently  improved  methods  
were   developed   and   introduced   in   oceanography,   traceability   of   salinity   measurement  
results  to  SI  units  has  not  yet  been  achieved  (Seitz  et  al.  2008  and  Seitz  et  al.  (2011)).    This  
implies   the   risk   that   readings   taken   today   may   possess   an   enlarged   uncertainty   when  
being  compared  with  observations  taken  a  hundred  years  from  now,  a  circumstance  that  
will  reduce  the  accuracy  of  long-­‐‑term  trend  analyses  performed  in  the  future.      
A  quantity,  quite  generally,  is  a  “property  of  a  phenomenon,  body  or  substance,  where  
the   property   has   a   magnitude   that   can   be   expressed   as   a   number”   (ISO/IEC,   2007).      The  
process  to  obtain  this  number  is  called  measurement.    The  value  of  the  indicated  number  
(the   quantity   value)   is   determined   by   a   calibration   of   the   measuring   system   with   a  
reference  having  a  known  quantity  value  of  the  same  kind.    In  turn,  the  quantity  value  of  
the   reference   is   assigned   in   a   superior   measurement   procedure,   which   is   likewise  
calibrated   with   a   reference   and   so   on.      This   calibration   hierarchy   ends   in   a   primary  
reference  procedure  used  to  assign  a  quantity  value  and  a  unit  to  a  primary  standard  for  
that   kind   of   quantity.      Thus,   the   unit   of   a   measured   quantity   value   expresses   its   link   (its  
metrological   traceability)   to   the   quantity   value   of   the   corresponding   primary   standard.    
Obviously,  quantity  values  measured  at  different  times  or  locations,  by  different  persons  
with  different  devices  or  methods  can  be  compared  with  each  other  only  if  they  are  linked  
to  the  same  reference  standard,  whose  corresponding  quantity  value  must  be  reproducible  
with  a  high  degree  of  reliability.      
Concerning   comparability   of   measured   quantity   values   a   second   aspect   is   of  
importance.      The   quantity   value   of   a   primary   standard   can   only   be   realised   with   an  
inevitable   uncertainty.      The   same   holds   for   every   measurement   and   calibration.      A  
measurement  result  therefore  always  has  to  indicate  the  measured  quantity  value  and  its  
uncertainty.      Obviously,   the   latter   increases   with   every   calibration   step   down   the  
calibration   hierarchy.      Measured   quantity   values   can   evidently   only   be   assumed  
equivalent   if   their   difference   is   smaller   than   their   measurement   uncertainty  
(compatibility).      On   the   other   hand   they   can   only   be   assumed   reliably   different,   if   the  
difference  is  larger  than  the  uncertainty.      
To   ensure   comparability   in   practice,   the   International   System   of   Units   (SI)   was  
established.      National   Metrological   Institutes   (NMIs)   have   developed   primary   reference  
procedures  to  realise  the  SI  units  in  the  form  of  primary  standards.    Extensive  (ongoing)  
efforts   are   made   to   link   these   units   to   fundamental   and   physical   constants   in   order   to  
achieve   the   highest   degree   of   reproducibility.      Moreover,   the   NMIs   periodically   conduct  
international  comparison  measurements  under  the  umbrella  of  the  International  Bureau  of  
Weights   and   Measures,   in   order   to   ensure   the   compatibility   of   the   quantity   values   of  
national  standards.      
PSS-­‐‑78,  and  similarly  the  Reference-­‐‑Composition  Salinity  Scale  (Millero  et  al.  (2008a)),  
compute   the   salinity   value   from   a   measured   conductivity   ratio   with   respect   to   the   K15   
conductivity  ratio  of  IAPSO  Standard  Seawater  (SSW,  Culkin  and  Ridout  (1998),  Bacon  et  
al.   (2007)),   which   plays   the   role   of   a   primary   standard.      The   production   procedure   if  
IAPSO  Standard  Seawater,  and  in  particular  the  adjustment  of  its  conductivity  to  that  of  a  
potassium  chloride  (KCl)  solution  of  definite  purity  and  the  corresponding  assignment  of  
the   K15    ratio,   can   be   seen   as   a   primary   reference   procedure.      However   both   of   these  
solutions   are   artefacts   lying   outside   the   SI   system;   they   are   not   subject   to   regular  
international   inter-­‐‑comparisons;   their   sufficiently   precise   replicability   by   arbitrary  
independent   laboratories   is   neither   known   nor   even   granted.      A   slow   drift   of   artefact  

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properties  cannot  rigorously  be  excluded,  similar  in  principle  to  the  “evaporation”  of  mass  
from  the  kilogram  prototype  stored  in  Paris.    It  is  impossible  to  foresee  effects  that  might  
affect  the  conductivity  of  SSW  solution  one  day.    Thus,  with  respect  to  decadal  or  century  
time   scales,   there   is   an   uncertainty   of   its   K15    ratio,   which   a   priori   can   not   be   quantified  
and  puts  long  term  comparability  of  salinity  measurement  results  at  risk.      
This  fundamental  problem,  which  is  related  to  any  artificial  reference  standard,  can,  at  
least   in   principle,   be   avoided   if   the   conductivity   of   seawater   is   measured   traceable   to  
primary  SI  standards  (“absolute”  conductivity)  rather  than  relying  on  a  conductivity  ratio.    
Unfortunately   the   related   uncertainty   of   absolute   conductivity   measurements   with  
present-­‐‑day   state-­‐‑of-­‐‑the-­‐‑art   technology   is   one   order   of   magnitude   larger   than   that   of   the  
relative  measurements  presently  used  in  oceanography  (Seitz  et  al.  (2008)).      
A   way   out   of   this   practical   dilemma   is   the   measurement   of   a   different   seawater  
quantity  that  is  traceable  to  SI  standards  and  possesses  the  demanded  small  uncertainty,  
and   from   which   the   salinity   can   be   computed   via   an   empirical   relation   that   is   very  
precisely  known  (Seitz  et  al.  (2011)).    Among  the  potential  candidates  for  this  purpose  are  
the  sound  speed,  the  refractive  index,  chemical  analysis  (e.g.  by  mass  spectroscopy)  of  the  
sea-­‐‑salt   constituents,   in   particular   chlorine,   and   direct   density   measurements.      The   latter  
has  three  important  advantages,  i)  SI-­‐‑traceable  density  measurements  of  seawater  can  be  
carried  out  with  a  relative  uncertainty  of  1  ppm  (Wolf  (2008)),  which  perfectly  meets  the  
needs  of  ocean  observation,  ii)  a  relation  exists  between  density  and  the  Absolute  Salinity  
of   seawater   is   available   with   a   relative   uncertainty   of   4   ppm   in   the   form   of   the   TEOS-­‐‑10  
Gibbs  function,  iii)  the  measurand,  density,  is  of  immediate  relevance  for  oceanography,  in  
contrast  to  other  options.      
It   is   important   to   note   that   the   actual   measuring   procedure   for   a   quantity   value   is  
irrelevant   for   its   traceability.      To   measure   the   weight   of   a   person,   a   mass   balance   can   be  
used,  a  spring  or  a  magnetic  coil;  it  is  the  quantity  value  that  is  traceable,  not  the  method  
to  achieve  this  value.    The  method  in  use  is  not  intrinsically  important  except  in  so  far  as  it  
is   responsible   for   the   uncertainty   of   the   quantity   value.      Hence,   we   may   measure   the  
density   of   seawater   with   a   CTD   conductivity   sensor,   provided   this   sensor   is   properly  
calibrated  with  respect  to  an  SI-­‐‑traceable  density  reference  standard.    In  practice,  this  will  
mean   that   the   sensor   calibration   in   oceanographic   labs   must   be   done   with   standard  
seawater   samples   of   certified   density   rather   than  certified  Practical  Salinity.    The  density  
value   returned   from   the   CTD   reading   at   sea   is   then   converted   into   an   Absolute   Salinity  
value  by  means  of  the  equation  of  state  of  seawater,  and  eventually  into  a  Practical  Salinity  
number   for   storage   in   data   centres.      The   latter   step   may   include   some   modification  
regarding   local   sea   salt   composition   anomalies.      Storing   a   salinity   value   rather   than   the  
related   density   reading   has   the   advantage   of   conservativity   with   respect   to   dilution   or  
changes  of  temperature  or  pressure.      
This   conceptual   proposal   of   WG127   is   still   immature   and   needs   to   be   worked   out   in  
more   detail   in   the   following   years.      Although   it   may   imply   only   minor   changes   in   the  
practical  use  of  a  CTD  or  similar  devices,  the  new  concept  is  very  promising  regarding  the  
long-­‐‑term  reliability  of  observations  made  in  the  near  future  for  climatic  trend  analyses  to  
be  performed  by  the  coming  generations.    An  immediate  consequence  of  this  proposal  is  
to  have  the  density  (at  a  given  temperature  and  pressure)  of  several  samples  of  each  batch  
of  IAPSO  Standard  Seawater  measured  when  they  are  produced  and  have  these  densities  
made  available  as  reference  values  for  each  batch.      
  

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Appendix  F:    

Coefficients  of  the  IAPWS-­‐‑95  Helmholtz  function  
of  fluid  water  (with  extension  down  to  50  K)    

  
  
The  specific  Helmholtz  energy  for  fluid  (gaseous  and  liquid)  water  is  given  by  the  revised  
IAPWS  Release,  IAPWS  (2009b),  which  is  based  mainly  on  the  work  of  Wagner  and  Pruß  
(2002).      This   revised   release   is   still   referred   to   as   IAPWS-­‐‑95.      The   specific   Helmholtz  
energy  of  IAPWS-­‐‑95  is  defined  by    

f flu (T , ρ ) = f V,id (T , ρ ) + RWT ϕ res (τ , δ ) ,

(F.1)

where   f
(T , ρ )   is  the  ideal-­‐‑gas  part,  (F.2),   RW   =  461.518  05  J  kg–1  K–1  is  the  specific  gas  
constant   of   water   used   in   IAPWS-­‐‑95,   and   ϕ res (τ , δ )    is   the   dimensionless   residual   part  
consisting  of  56  terms,  available  from  (F.5)  and  Tables  F.2  -­‐‑  F.4.    Note  that  the  gas  constant  
used   here   differs   from   the   most   recent   value,   RW = R M W =   461.523   64   J   kg–1   K–1,   where  
M W   =  18.015  268  g  mol–1  is  the  molar  mass  of  water  (IAPWS  (2005)).      
The  ideal-­‐‑gas  part,   f V,id (T , ρ ) ,   of  the  specific  Helmholtz  energy  for  water  vapour  
is  (from  IAPWS  (2009b),  Wagner  and  Pruß  (2002),  Feistel  et  al.  (2010a))    
V,id

f V,id (T , ρ ) = RWT ⎡⎣ϕ 0 (τ , δ ) + ϕ ex (τ )⎤⎦ .

(F.2)

Note  that  the  term   ϕ ex (τ )   has  been  added  by  Feistel  et  al.  (2010a)  (see  IAPWS-­‐‑12)  in  order  
to   extend   the   formulation   to   extraterrestrial   applications,   and   because   sublimation  
pressure   values   are   now   available   down   to   50   K   from   Feistel   and   Wagner   (2007)   and  
IAPWS   (2008b);   an   extreme   range   where   no   related   experiments   have   been   performed.    
This   term   is   additional   to   the   specific   Helmholtz   energy   of   IAPWS   (2009b)   and   Wagner  
and   Pruß   (2002).      The   function   ϕ 0 (τ , δ )    was   obtained   from   an   equation   for   the   specific  
isobaric  heat  capacity  of  vapour  and  reads  
8

(

)

ϕ 0 (τ , δ ) = ln δ + n10 + n20τ + n30 lnτ + ∑ ni0 ln 1 − e−γ i τ .
i =4

0

(F.3)

The   “reduced   density”   δ = ρ / ρc    and   “reduced   temperature”   τ = Tc / T    are   specified   by  
ρc = 322 kg m−3 ,   Tc = 647.096 K.     The  coefficients  of  (F.3)  are  available  from  Table  F.1.    The  
IAPWS-­‐‑95  reference  state  conditions  define  the  internal  energy  and  the  entropy  of  liquid  
water  to  be  zero  at  the  triple  point.    A  highly  accurate  numerical  implementation  of  these  
conditions   gave   the   following   values   rounded   to   16   digits   for   the   adjustable   coefficients  
n1o = −8.320 446 483 749 693    and   n2o = 6.683 210 527 593 226.    These   are   the   values   used   in  
TEOS-­‐‑10  (IAPWS  (2009b),  Feistel  et  al.  (2008a)).      
The   temperature   T    is   measured   on   the   ITS-­‐‑90   scale.      The   range   of   validity   is   130   –  
2000   K   without  the  extension  (F.4),  that  is  with   ϕ ex (τ ) = 0.     The  range  can  be  extended  to  
include  the  region  50  –  130  K  with  the  following  correction  function   ϕ ex (τ )   added  to  (F.2)  
in  this  temperature  range,    
⎛ 1
3
τ 9 9τ
τ2 ⎞
for              50 K ≤ T ≤ 130 K,
(F.4)
ϕ ex (τ ) = E × ⎜ − − 2 (τ + ε ) ln − + 2 + 3 ⎟ ,
2
τ
ε
2
ε
ε
2
ε
2
ε
⎝
⎠
where   TE   =  130   K  ,   E   =  0.278  296  458  178  592,  and   ε = Tc / TE .    At   τ = ε ,    ϕ ex (τ )   is  zero,  as  
well  as  its  first,  second,  third  and  fourth  temperature  derivatives.    This  correction  has  been  
determined  such  that  when  applied  to  the  formula  used  in  IAPWS-­‐‑95,  it  results  in  a  fit  to  
the  heat  capacity  data  of  Woolley  (1980)  between  50  and  130  K  with  an  r.m.s.  deviation  of  

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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

153

6 ×10−4    in   cP RW .       This   extension   formula   has   been   developed   particularly   for  

implementation   in   TEOS-­‐‑10   (Feistel   et   al.   (2010a)),   it   is   consistent   with   the   correlation  
function   given   in   IAPWS   (2008b),   and   it   is   expected   to   be   endorsed   as   the   IAPWS  
Guideline  IAPWS-­‐‑12.          
The  residual  part  of  (F.1)  has  the  form    

ϕ res =

7

∑ niδ diτ ti +
i =1

+

54

∑ niδ

i =52

∑ niδ d τ t exp ( −δ c )
51

i

i

i

i =8

(

)

56

(F.5)

τ exp −αi (δ − ε i ) − βi (τ − γ i ) + ∑ ni Δ δψ

di ti

2

2

bi

i =55

with  the  abbreviations    
1

(

)

2
2
Δ = θ 2 + Bi δ − 1 i ,       θ = 1 − τ + Ai δ − 1 βi ,      and      ψ = exp −Ci (δ − 1) − Di (τ − 1) .    (F.6)  

2a

The  coefficients  of  (F.5)  are  available  from  Tables  F.2  –  F.4.      
  
  
Table  F.1.    Coefficients  appearing  in  Eqn.  (F.3).    Note  that  the  originally  published  values  
(Wagner   and   Pruß   (2002))   of   the   adjustable   coefficients   n1o    and   n2o    are   slightly   different  
from  those  of  TEOS-­‐‑10  given  here  (Feistel  et  al.  (2008a)).      
  
i
ni0
γ i0
1
2
3
4
5
6
7
8

–8.32044648374969
6.68321052759323
3.00632
0.012436
0.97315
1.2795
0.96956
0.24873

1.28728967
3.53734222
7.74073708
9.24437796
27.5075105

  
  
Table  F.2.    Coefficients  of  the  residual  part  (F.5).      
  
i
ci
di
ti
ni
1
0
1 –0.5
0.012533547935523
2
0
1 0.875
7.8957634722828
3
0
1
1
–8.7803203303561
4
0
2
0.5
0.31802509345418
5
0
2 0.75 –0.26145533859358
6
0
3 0.375 –7.8199751687981× 10–3
7
0
4
1
8.8089493102134× 10–3
8
1
1
4
–0.66856572307965
9
1
1
6
0.20433810950965
10
1
1
12
–6.6212605039687× 10–5
11
1
2
1
–0.19232721156002
12
1
2
5
–0.25709043003438
13
1
3
4
0.16074868486251
14
1
4
2
–0.040092828925807
15
1
4
13
3.9343422603254× 10–7
16
1
5
9
–7.5941377088144× 10–6
17
1
7
3
5.6250979351888× 10–4
18
1
9
4
–1.5608652257135× 10–5
19
1
10
11
1.1537996422951× 10–9
20
1
11
4
3.6582165144204× 10–7
21
1
13
13
–1.3251180074668× 10–12

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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51

1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
4
6
6
6
6

15
1
2
2
2
3
4
4
4
5
6
6
7
9
9
9
9
9
10
10
12
3
4
4
5
14
3
6
6
6

1
7
1
9
10
10
3
7
10
10
6
10
10
1
2
3
4
8
6
9
8
16
22
23
23
10
50
44
46
50

–6.2639586912454× 10–10
–0.10793600908932
0.017611491008752
0.22132295167546
–0.40247669763528
0.58083399985759
4.9969146990806× 10–3
–0.031358700712549
–0.74315929710341
0.4780732991548
0.020527940895948
–0.13636435110343
0.014180634400617
8.3326504880713× 10–3
–0.029052336009585
0.038615085574206
–0.020393486513704
–1.6554050063734× 10–3
1.9955571979541× 10–3
1.5870308324157× 10–4
–1.638856834253× 10–5
0.043613615723811
0.034994005463765
–0.076788197844621
0.022446277332006
–6.2689710414685× 10–5
–5.5711118565645× 10–10
–0.19905718354408
0.31777497330738
–0.11841182425981

  
  
Table  F.3.    Coefficients  of  the  residual  part  (F.5).      
  
i
d i ti
ni
αi
βi
γi
εi
52 3 0 –31.306260323435 20 150 1.21 1
53 3 1
31.546140237781 20 150 1.21 1
54 3 4 –2521.3154341695 20 250 1.25 1
  
  
Table  F.4.    Coefficients  of  the  residual  part  (F.5).      
  
i
ai
bi
Bi
ni
Ci
Di
Ai
βi
55 3.5 0.85 0.2 –0.14874640856724 28 700 0.32 0.3
56 3.5 0.95 0.2
0.31806110878444 32 800 0.32 0.3
  
Equation   (F.1)   is   valid   between   50   and   1273   K   and   for   pressures   up   to   1000   MPa   in   the  
stable  single-­‐‑phase  region  of  fluid  water.    Uncertainty  estimates  are  available  from  IAPWS  
(2009b)  and  Wagner  and  Pruß  (2002).      
  
  

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155

Appendix  G:  Coefficients  of  the  pure  liquid    
water  Gibbs  function  of  IAPWS-­‐‑09    

  
  
The   pure   liquid   water   part   of   the   Gibbs   function   of   Feistel   (2003)   has   been   approved   by  
IAPWS   (IAPWS   (2009c))   as   an   alternative   thermodynamic   description   of   pure   water   to  
IAPWS-­‐‑95   in   the   oceanographic   ranges   of   temperature   and   pressure.      The   pure   water  
specific  Gibbs  energy   g W ( t , p )   is  the  following  function  of  the  independent  variables  ITS-­‐‑
90  Celsius  temperature,   t = tu × y ,  and  sea  pressure,   p = pu × z     
7

6

gW (t , p ) = g u ∑∑ g jk y j z k ,  

(G.1)  

j =0 k =0

with   the   reduced   temperature   y = t tu    and   the   reduced   (dimensionless)   pressure  
z = p / pu .    The  unit-­‐‑related  constants   tu , pu   and   g u   are  given  in  Table  D4  of  appendix  D  
(e.  g.   pu = 108 Pa = 104 dbar ).    Coefficients  not  contained  in  the  table  below  have  the  value  
g jk   =  0.    Two  of  these  41  parameters  ( g00   and   g10 )  are  arbitrary  and  are  computed  from  
the   reference-­‐‑state   conditions   of   vanishing   specific   entropy,   η ,    and   specific   internal  
energy,   u,   of  liquid  H2O  at  the  triple  point,    

η (Tt , pt ) = 0,                       and                 u (Tt , pt ) = 0.   

(G.2)  

Note  that  the  values  of   g 00   and   g10   in  the  table  below  are  taken  from  Feistel  et  al.  (2008a)  
and   IAPWS   (2009),   and   are   not   identical   to   the   values   in   Feistel   (2003).      The   modified  
values  have  been  chosen  to  most  accurately  achieve  the  triple-­‐‑point  conditions  (G.2)  (see  
Feistel  et  al.  (2008a)  for  a  discussion  of  this  point).      

j
0
0
0
0
0
0
0
1
1
1
1
1
1
2
2
2
2
2
2
3
3

k
0
1
2
3
4
5
6
0
1
2
3
4
5
0
1
2
3
4
5
0
1

gjk
0.101 342 743 139 674 × 10
0.100 015 695 367 145 × 106
–0.254 457 654 203 630 × 104
0.284 517 778 446 287 × 103
–0.333 146 754 253 611 × 102
0.420 263 108 803 084 × 10
–0.546 428 511 471 039
3

0.590 578 347 909 402 × 10
–0.270 983 805 184 062 × 103
0.776 153 611 613 101 × 103
–0.196 512 550 881 220 × 103
0.289 796 526 294 175 × 102
–0.213 290 083 518 327 × 10
–0.123 577 859 330 390 × 105
0.145 503 645 404 680 × 104
–0.756 558 385 769 359 × 103
0.273 479 662 323 528 × 103
–0.555 604 063 817 218 × 102
0.434 420 671 917 197 × 10
0.736 741 204 151 612 × 103
–0.672 507 783 145 070 × 103

j

k

3
3
3
3
4
4
4
4
4
5
5
5
5
5
6
6
6
6
7
7

2
3
4
5
0
1
2
3
4
0
1
2
3
4
0
1
2
3
0
1

gjk
0.499 360 390 819 152 × 103
–0.239 545 330 654 412 × 103
0.488 012 518 593 872 × 102
–0.166 307 106 208 905 × 10
–0.148 185 936 433 658 × 103
0.397 968 445 406 972 × 103
–0.301 815 380 621 876 × 103
0.152 196 371 733 841 × 103
–0.263 748 377 232 802 × 102
0.580 259 125 842 571 × 102
–0.194 618 310 617 595 × 103
0.120 520 654 902 025 × 103
–0.552 723 052 340 152 × 102
0.648 190 668 077 221 × 10
–0.189 843 846 514 172 × 102
0.635 113 936 641 785 × 102
–0.222 897 317 140 459 × 102
0.817 060 541 818 112 × 10
0.305 081 646 487 967 × 10
–0.963 108 119 393 062 × 10

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Appendix  H:  Coefficients  of  the  saline    
Gibbs  function  for  seawater  of  IAPWS-­‐‑08    
  
  
  
Non-­‐‑zero  coefficients   gijk   of  the  saline  specific  Gibbs  energy   g S ( S A , t , p )   as  a  function  of  
the   independent   variables   Absolute   Salinity,   SA = Su × x ² ,   ITS-­‐‑90   Celsius   temperature,  
t = tu × y ,  and  sea  pressure,   p = pu × z :  

⎧
⎫
g S ( SA , t, p) = g u ∑ ⎨ g1 jk x 2 ln x + ∑ gijk x i ⎬ y j z k .    
j ,k ⎩
i >1
⎭

(H.1)  

The   unit-­‐‑related   constants   Su , tu , pu    and   g u    are   given   in   Table   D4   of   appendix   D   (e.   g.  
pu = 108 Pa = 104 dbar ).         Coefficients   with   k    >   0   are   adopted   from   Feistel   (2003).      Pure-­‐‑
water  coefficients  with   i   =  0  do  not  occur  in  the  saline  contribution.    The  coefficients   g200   
and   g210   were  determined  to  exactly  achieve  Eqns.  (2.6.7)  and  (2.6.8)  when  the  pure  water  
Gibbs  function  was  that  of  IAPWS-­‐‑95.      
  
  

i

j

k

i

j

k

gijk

i

j

k

gijk

1

0

0

5812.81456626732

gijk

2

5

0

–21.6603240875311

3

2

2

–54.1917262517112

1

1

0

851.226734946706

4

5

0

2.49697009569508

2

3

2

–204.889641964903

2

0

0

1416.27648484197

2

6

0

2.13016970847183

2

4

2

74.7261411387560

3

0

0

–2432.14662381794

2

0

1

–3310.49154044839

2

0

3

–96.5324320107458

4

0

0

2025.80115603697

3

0

1

199.459603073901

3

0

3

68.0444942726459

5

0

0

–1091.66841042967

4

0

1

–54.7919133532887

4

0

3

–30.1755111971161

6

0

0

374.601237877840

5

0

1

36.0284195611086

2

1

3

124.687671116248

7

0

0

–48.5891069025409

2

1

1

729.116529735046

3

1

3

–29.4830643494290

2

1

0

168.072408311545

3

1

1

–175.292041186547

2

2

3

–178.314556207638

3

1

0

–493.407510141682

4

1

1

–22.6683558512829

3

2

3

25.6398487389914

4

1

0

543.835333000098

2

2

1

–860.764303783977

2

3

3

113.561697840594

5

1

0

–196.028306689776

3

2

1

383.058066002476

2

4

3

–36.4872919001588

6

1

0

36.7571622995805

2

3

1

694.244814133268

2

0

4

15.8408172766824

2

2

0

880.031352997204

3

3

1

–460.319931801257

3

0

4

–3.41251932441282

3

2

0

–43.0664675978042

2

4

1

–297.728741987187

2

1

4

–31.6569643860730

4

2

0

–68.5572509204491

3

4

1

234.565187611355

2

2

4

44.2040358308000

2

3

0

–225.267649263401

2

0

2

384.794152978599

2

3

4

–11.1282734326413

3

3

0

–10.0227370861875

3

0

2

–52.2940909281335

2

0

5

–2.62480156590992

4

3

0

49.3667694856254

4

0

2

–4.08193978912261

2

1

5

7.04658803315449

2

4

0

91.4260447751259

2

1

2

–343.956902961561

2

2

5

–7.92001547211682

3

4

0

0.875600661808945

3

1

2

83.1923927801819

4

4

0

–17.1397577419788

2

2

2

337.409530269367

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157

Appendix  I:  Coefficients  of  the  Gibbs  function    
of  ice  Ih  of  IAPWS-­‐‑06    

  
  
  
The  Gibbs  energy  of  ice  Ih,  the  naturally  abundant  form  of  ice,  having  hexagonal  crystals,  
is  a  function  of  temperature  (ITS-­‐‑90)  and  sea  pressure,   g Ih ( t , p ) .     This  Gibbs  function  has  
been  derived  by  Feistel  and  Wagner  (2006)  and  was  adopted  as  an  IAPWS  Release  in  2006  
and  revised  in  2009  (IAPWS  (2009a)),  here  referred  to  as  IAPWS-­‐‑06.    This  equation  of  state  
for   ice   Ih   is   given   by   Eqn.   (I.1)   as   a   function   of   temperature,   with   two   of   its   coefficients  
being  polynomial  functions  of  sea  pressure   p   ( p = P − P0 )    
2
⎡
τ2 ⎤
g Ih ( t , p ) = g0 − s0Tt ⋅ τ + Tt Re ∑ rk ⎢( tk − τ ) ln ( tk − τ ) + (tk + τ ) ln (tk + τ ) − 2tk ln tk − ⎥
tk ⎦
k =1 ⎣
4
⎛ p⎞
g0 ( p ) = ∑ g0k ⋅ ⎜ ⎟
k =0
⎝ Pt ⎠

k

   (I.1)  

k

2
⎛ p⎞
r2 ( p ) = ∑ r2 k ⋅ ⎜ ⎟ ,
k =0
⎝ Pt ⎠
with  the  reduced  temperature   τ = (T0 + t ) Tt   and   Tt   and   Pt   are  given  in  Table  I.1.    If  the  
sea   pressure   p    is   expressed   in   dbar    then   Pt    must   also   be   given   in   these   units   as  
Pt = 0.0611657 dbar .    The  real  constants   g00   to   g04   and   s0 ,  the  complex  constants   t1 ,   r1 ,  
t2 ,  and   r20   to   r22   are  listed  in  Table  I.2.      
  
  
TABLE  I.1      Special  constants  and  values  used  in  the  ice  Ih  Gibbs  function.      
  

Quantity

Symbol

Value

Unit

Experimental triple-point pressure

Pt

611.657

Pa

Numerical triple-point pressure

num

611.654 771 007 894

Pa

101325

Pa

273.16

K

Normal pressure
Triple-point temperature

Pt
P0
Tt

  

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TABLE  I.2    Coefficients  of  the  equation  of  state  (Gibbs  potential  function)  of  ice  Ih    
                                          as  given    by  Eqn.  (I.1).      
  
  
Coefficient
Real part
Imaginary part
Unit
g00
g01
g02

– 0.632 020 233 335 886 × 106
0.655 022 213 658 955

J kg

– 0.189 369 929 326 131 × 10-7

g03

0.339 746 123 271 053 × 10-14

g04

– 0.556 464 869 058 991 × 10-21

s0 (absolute)
s0 (IAPWS-95)

J kg

J kg
J kg

– 0.332 733 756 492 168 × 104

t1

0.368 017 112 855 051 × 10-1

0.510 878 114 959 572 ×10-1

r1

0.447 050 716 285 388 × 102

0.656 876 847 463 481 × 102

t2

0.337 315 741 065 416

0.335 449 415 919 309

r20

– 0.725 974 574 329 220 × 10

r21

– 0.557 107 698 030 123 × 10-4

r22

0.234 801 409 215 913 × 10-10

–1
–1
–1

–1
J kg
–1 –1
J kg K
–1 –1
J kg K

0.189 13 × 103

2

–1

– 0.781 008 427 112 870 × 102
0.464 578 634 580 806 × 10-4
– 0.285 651 142 904 972 × 10-10

J kg

–1 –1
K

–1 –1
K
–1 –1
J kg K
–1 –1
J kg K
J kg

  
  
The   numerical   triple   point   pressure   Ptnum    listed   in   Table   I.1   was   derived   in   Feistel   et   al.  
(2008a)   as   the   Absolute   Pressure   at   which   the   three   phases   of   water   were   in  
thermodynamic   equilibrium   at   the   triple   point   temperature,   using   the   mathematical  
descriptions   of   the   three   phases   as   given   by   IAPWS-­‐‑95   and   IAPWS-­‐‑06.      The   complex  
logarithm   ln ( z )   is  meant  as  the  principal  value,  i.e.  it  evaluates  to  imaginary  parts  in  the  
interval   −π < Im ⎡⎣ln ( z )⎤⎦ ≤ + π .      The   complex   notation   used   here   has   no   direct   physical  
basis   but   serves   for   convenience   of   analytical   partial   derivatives   and   for   compactness   of  
the  resulting  formulae,  especially  in  program  code.    Complex  data  types  are  supported  by  
scientific   computer   languages   like   Fortran   (as   COMPLEX*16)   or   C++   (as   complex  
),  thus  allowing  an  immediate  implementation  of  the  formulae  given,  without  the  
need   for   prior   conversion   to   much   more   complicated   real   functions,   or   for   experience   in  
complex  calculus.    
The  residual  entropy  coefficient  s0  is  given  in  Table  I.2  in  the  form  of  two  alternative  
values.   Its   ‘IAPWS-­‐‑95’   version   is   required   for   phase   equilibria   studies   between   ice   and  
fluid  water  and  seawater.    This  is  the  value  of   s0   used  in  the  TEOS-­‐‑10  algorithms.    In  the  
'ʹabsolute'ʹ  version,   s0   is  the  statistical  non-­‐‑zero  entropy  ice  possesses  at  the  zero  point  (0  K)  
resulting   from   the   multiplicity   of   its   energetically   equivalent   crystal   configurations   (for  
details,  see  Feistel  and  Wagner  (2005)).      
The   value   of   g 00    listed   in   table   I.2   is   the   value   in   the   revised   IAPWS-­‐‑2006   Ice   Ih  
Release  (IAPWS  (2009a))  which  improves  the  numerical  consistency  (Feistel  et  al.  (2008a))  
with  the  IAPWS-­‐‑1995  Release  for  the  fluid  phase  of  water.      
  
  
  

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159

Appendix  J:    
Coefficients  of  the  Helmholtz  function  of    
moist  air  of  IAPWS-­‐‑10    

The  equation  of  state  of  humid  air  described  here  (Feistel   et  al.  (2010a),  IAPWS  (2010))  is  
represented   in   terms   of   a   Helmholtz   function   which   expresses   the   specific   Helmholtz  
energy   as   a   function   of   dry-­‐‑air   mass   fraction   A,    absolute   temperature   T    and   humid-­‐‑air  
mass  density,   ρ ,   and  takes  the  form    

(

)

(

)

f AV ( A, T , ρ ) = (1 − A) f V T , ρ V + Af A T , ρ A + f mix ( A, T , ρ ) .

(J.1)

The   vapour   part   is   given   by   the   IAPWS-­‐‑95   Helmholtz   function   for   fluid   water   (IAPWS  
(2009b)),    

(

)

(

)

f V T , ρ V ≡ f flu T , ρ V ,

(J.2)

is  computed  at  the  vapour  density,   ρ V = (1 − A) ρ ,  and  is  defined  in  Eqn.  (F.1)  of  appendix  
F.      The   dry-­‐‑air   part, f A T , ρ A ,    is   computed   at   the   dry-­‐‑air   density,   ρ A = Aρ ,    and   is  
defined  by  Eqn.  (J.3).    The  air-­‐‑water  cross-­‐‑over  part   f mix   is  defined  by  Eqn.  (J.8).      
  
  
Table  J.1.    Special  constants  and  values  used  in  this  appendix.    Note  that  the    
molar  gas  constant  used  here  differs  from  the  most  recent  value  (IAPWS  
(2005)),   and   the   molar   mass   of   dry   air   used   here   differs   from   the   most  
recent  value  (Picard  et  al.  (2008)),  Table  D6.      
  

(

Quantity
Molar gas constant
Molar gas constant
Molar mass of dry air
Molar mass of dry air
Molar mass of water
Celsius zero point
Normal pressure

)

Symbol

Value

Unit

RL

8.314 51

J mol–1 K–1

R
MA
MA
MW
T0

8.314 472
28.958 6
28.965 46
18.015 268
273.15

P0

101 325

–1

J mol K
g mol–1
g mol–1
g mol–1
K

Reference

–1

Lemmon et al. (2000)
IAPWS (2005)
Lemmon et al. (2000)
IAPWS (2010)
IAPWS (2005)
Preston-Thomas (1990)

Pa

ISO(1993)

  
The  specific  Helmholtz  energy  for  dry  air  is  (Lemmon  et  al.  (2000)),    

(

)

f A T,ρA =

R LT id
⎡α (τ , δ ) + α res (τ , δ )⎤ .
⎦
MA ⎣

(J.3)

The  values  to  be  used  for  molar  mass   M A   of  dry  air,  and  for  the  molar  gas  constant   R L   
are  given  in  Table  J.1.    The  function   α id (τ , δ )   is  the  ideal-­‐‑gas  part,    

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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

(

5

)⎦

0
α id (τ , δ ) = ln δ + ∑ ni0τ i −4 + n60τ 1.5 + n70 ln τ + n80 ln ⎡1 − exp −n11
τ ⎤

⎣

i =1

+

n90 ln ⎡1 − exp
⎣

(

0
τ
−n12

and   α res (τ , δ )   is  the  residual  part,    

α res (τ , δ ) =

10

∑ nkδ ikτ jk +
k =1

)

( )

(J.4)

0
0
⎤ + n10
ln ⎡ 2 / 3 + exp n13
τ ⎤
⎦
⎣
⎦

19

∑ nkδ i τ j
k

k =11

k

(

)

exp −δ lk .

(J.5)

The  “reduced  variables”  in  Eqns.  (J.3)  -­‐‑   (J.5)  are   τ = TA* / T   with  the  reducing  temperature  
TA* = 132.6312 K ,   and   δ = ρ A / ρ A*    with   the   reducing   density   ρ A* = 10.4477 mol dm−3 × M A .    
M A   is  given  in  Table  J.1.    The  coefficients  of  Eqns.  (J.4)  and  (J.6)  are  given  in  Tables  J.2  and  
J.3.      
Two  of  the  parameters  ( n40   and   n50 )  listed  in  Table  J.2  are  arbitrary  and  are  computed  
here   from   the   reference-­‐‑state   conditions   of   vanishing   specific   entropy,   η A ,    and   specific  
enthalpy,   h A ,    of   dry   air   at   the   temperature   T0    and   the   normal   pressure   P0 ,    as   given   in  
Table  J.1,    

η A (T0 , P0 ) = 0,

(J.6)

h A (T0 , P0 ) = 0.

(J.7)

The   Helmholtz   function   f mix in   Eqn.   (J.1)   describes   the   water-­‐‑air   interaction   and   is  
defined  by    

f mix ( A, T , ρ ) =

A (1 − A) ρ RT ⎧⎪ AW
(1 − A) C AWW T ⎤ ⎫⎪ .
3 ⎡ A AAW
C
(T ) +
( )⎥ ⎬
⎨ 2 B (T ) + ρ ⎢
M AM W
2 ⎣ MA
MW
⎪⎩
⎦ ⎪⎭

(J.8)

The   values   used   for   the   molar   gas   constant   R,    the   molar   mass   of   dry   air,   M A ,    and   the  
molar  mass  of  water,   M W ,   are  given  in  Table  J.1.      
The  second  cross-­‐‑virial  coefficient,   B AW (T ) ,   is  given  by  Harvey  and  Huang  (2007)  as    
3

B AW (T ) = b * ∑ ciτ di .

(J.9)

i =1

The  coefficients  of  Eqn.  (J.9)  are  given  in  Table  J.4.      
The  third  cross-­‐‑virial  coefficients   C AAW (T )   and   C AWW (T )   are  defined  in  Hyland  and  
Wexler  (1983),  in  the  form    
4

C AAW (T ) = c * ∑ aiτ −i ,

(J.10)

i =0

and    

⎧3
⎫
C AWW (T ) = − c *exp ⎨∑ biτ −i ⎬ .
⎩ i =0
⎭
The  coefficients   ai   and   bi   of  Eqns.  (J.10)  and  (J.11)  are  given  in  Table  J.4.      
  

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(J.11)

TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

Table  J.2.    Dimensionless  coefficients  and  exponents  for  the  ideal-­‐‑gas  part,    
Eqn.  (J.4),  for  dry  air  (Lemmon  et  al.  (2000)).    In  TEOS-­‐‑10,  the  coefficients  
n 40   and   n50   are  re-­‐‑adjusted  to  the  reference  state  conditions,  Eqns.  (J.6,  J.7),  
and  deviate  from  the  originally  published  values  of  Lemmon  et  al.  (2000).      
  

i
1

  
  
  
  

i

ni0
0.605 719 400 000 000 × 10–7

ni0

8

0.791 309 509 000 000

9

0.212 236 768 000 000

10

–0.197 938 904 000 000

2

–0.210 274 769 000 000 × 10

–4

3

–0.158 860 716 000 000 × 10

–3

4

0.974 502 517 439 480 × 10

11

0.253 636 500 000 000 × 102

5

0.100 986 147 428 912 × 102

12

0.169 074 100 000 000 × 102

6

–0.195 363 420 000 000 × 10–3

13

0.873 127 900 000 000 × 102

7

0.249 088 803 200 000 × 10

Table   J.3.      Coefficients   and   exponents   for   the   residual   part,   Eqn.   (J.5),    
                                          for  dry  air  (Lemmon  et  al.  (2000)).      

  

k

ik

jk

lk

nk

1

1

0

0

0.118 160 747 229

2

1

0.33

0

0.713 116 392 079

3

1

1.01

0

4

2

0

0

0.714 140 178 971 × 10–1

5

3

0

0

–0.865 421 396 646 × 10–1

6

3

0.15

0

0.134 211 176 704

7

4

0

0

0.112 626 704 218 × 10–1

8

4

0.2

0

–0.420 533 228 842 × 10–1

9

4

0.35

0

0.349 008 431 982 × 10–1

10

6

1.35

0

0.164 957 183 186 × 10–3

11

1

1.6

1

–0.101 365 037 912

12

3

0.8

1

–0.173 813 690 970

13

5

0.95

1

–0.472 103 183 731 × 10–1

14

6

1.25

1

–0.122 523 554 253 × 10–1

15

1

3.6

2

–0.146 629 609 713

16

3

6

2

–0.316 055 879 821 × 10–1

17

11

3.25

2

0.233 594 806 142 × 10–3

18

1

3.5

3

0.148 287 891 978 × 10–1

19

3

15

3

–0.938 782 884 667 × 10–2

–0.161 824 192 067 × 10

  
  

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162

  

TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

Table  J.4.    Coefficients  of  the  cross-­‐‑virial  coefficients   B AW (T ) ,    C AAW (T ) and    
C AWW (T ) , Eqns.  (J.9)  -­‐‑  (J.11).    The  reducing  factors  are   b* = 10−6 m3 mol −1   
and   c* = 10−6 m6 mol −2 ,   the  “reduced  temperature”  is   τ = T / (100 K ) .      
i
0
1
2
3
4

ai
0.482 737 × 10–3
0.105 678 × 10–2
–0.656 394 × 10–2
0.294 442 × 10–1
–0.319 317 × 10–1

bi
–0.107 288 76 × 102
0.347 802 00 × 102
–0.383 383 00 × 102
0.334 060 00 × 102

ci

di

0.665 687 × 102
–0.238 834 × 103
–0.176 755 × 103

–0.237
–1.048
–3.183

  
The   equation   of   state,   Eqn.   (J.1),   is   valid   for   humid   air   within   the   temperature   and  
pressure  range    
(J.12)  
193  K ≤   T   ≤  473  K        and        10  nPa  ≤   P   ≤  5  MPa.  
The  pressure  is  computed  from   P = ρ 2 f ρAV .     All  validity  regions  of  the  formulas  combined  
in  Eqn.  (J.1),  including  the  Helmholtz  functions  of  water  vapour  and  of  dry  air,  as  well  as  
the  cross-­‐‑virial  coefficients,  overlap  only  in  this  range.    The  separate  ranges  of  validity  of  
the  individual  components  are  wider;  some  of  them  significantly  wider.    Therefore,  Eqn.  
(J.1)   will   provide   reasonable   results   outside   of   the   T − P    range   given   above   under   the  
condition   that   a   certain   component   dominates   numerically   in   Eqn.   (J.1)   and   is   evaluated  
within  its  particular  range  of  validity.      
The   air   fraction   A    can   take   any   value   between   0   and   1   provided   that   the   partial  
vapour   pressure,   P vap = xV P ,   ( xV    is   the   mole   fraction   of   vapour,   Eqn.   (3.35.3))   does   not  
exceed  its  saturation  value,  i.e.,      

0 ≤ A ≤ 1 and

Asat (T , P ) ≤ A .   

(J.13)  

The   exact   value   of   the   air   fraction   Asat (T , P )    of   saturated   humid   air   is   given   by   equal  
chemical  potentials  of  water  vapour  in  humid  air  and  of  either  liquid  water,  Eqn.  (3.37.5),  
if  the  temperature  is  above  the  freezing  point,  or  of  ice,  Eqn.  (3.35.4),  if  the  temperature  is  
below  the  freezing  point.    At  low  density,  the  saturation  vapour  pressure   Psat   of  humid  air  
can   be   estimated   by   the   correlation   function   for   either   the   vapour   pressure,   P liq (T ) ,    of  
pure   water   (IAPWS   (2007)),   or   for   the   sublimation   pressure,   P subl (T ) ,    of   ice   (IAPWS  
(2008b)),   to   obtain   Asat (T , P ) = P − Psat / ⎡ P − Psat (1 − M W / M A )⎤ ,    from   Eqn.   (3.35.3)   as   a  
⎣
⎦
practically  sufficient  approximation.      
  
  

(

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)

TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

163

Appendix  K:  Coefficients  of  the  75-­‐‑term  expression  
      for  the  specific  volume  of  seawater  in  terms  of  Θ    

  
  
  
The  TEOS-­‐‑10  Gibbs  function  of  seawater   g ( SA , t , p )   is  written  as  a  polynomial  in  terms  of  
in   situ   temperature   t ,   while   for   ocean   models,   specific   volume   (or   density)   needs   to   be  
expressed  as  a  computationally  efficient  expression  in  terms  of  Conservative  Temperature  
Θ .      Roquet   et   al.   (2015)   have   published   such   a   computationally   efficient   polynomial   for  
specific  volume.    Their   non-­‐‑dimensional  (root)  salinity   s ,  temperature   τ ,  and  pressure  
π ,  variables  are    

s ≡

SA + 24 g kg −1
Θ
p
  ,                 τ ≡
              and             π ≡
  ,  
Θu
pu
SA u

(K.1)  

in  terms  of  the  unit-­‐‑related  scaling  constants    

SA u ≡ 40× 35.16504g kg −1 / 35 ,               Θ u ≡ 40°C             and           pu ≡ 104 dbar .    

(K.2)  

Their  polynomial  expression  for  the  specific  volume  of  seawater  is    

v̂(SA ,Θ, p) = vu ∑ vijk s i τ jπ k ,  

(K.3)  

i, j,k

where   vu ≡ 1 m 3kg −1   and  the  non-­‐‑zero  dimensionless  constants   vijk   are  given  in  Table  K.1.      
Roquet  et  al.  (2015)  fitted  the  TEOS-­‐‑10  values  of  specific  volume   v   to   SA , Θ   and   p   in  a  
“funnel”   of   data   points   in   ( SA , Θ, p )    space.      This   is   the   same   “funnel”   of   data   points   as  
used   in   McDougall   et   al.   (2003);   at   the   sea   surface   it   covers   the   full   range   of   temperature  
and   salinity   while   for   pressure   greater   than   6500   dbar,   the   maximum   temperature   of   the  
fitted   data   is   10°C    and   the   minimum   Absolute   Salinity   is   30 g kg −1 .      The   maximum  
pressure   of   the   “funnel”   is   8000   dbar .      Table   K.1   contains   the   75   coefficients   of   the  
expression  (K.3)  for  specific  volume  in  terms  of   ( SA , Θ, p ) .      
The  rms  error  of  this  75-­‐‑term  approximation  to  the  full  Gibbs  function-­‐‑derived  TEOS-­‐‑
10  specific  volume  over  the  “funnel”  is   0.2x10−9 m 3kg −1 ;  this  can  be  compared  with  the  rms  
uncertainty  of   4x10−9 m 3kg −1   of  the  underlying  laboratory  density  data  to  which  the  TEOS-­‐‑
10  Gibbs  function  was  fitted  (see  the  first  two  rows  of  Table  O.1  of  appendix  O).    Similarly,  
the  appropriate  thermal  expansion  coefficient,    

αΘ =

1 ∂v
v ∂Θ S

= −
A, p

1 ∂ρ
ρ ∂Θ S

,  

(K.4)  

A, p

of   the   75-­‐‑term   equation   of   state   is   different   from   the   same   thermal   expansion   coefficient  
evaluated  from  the  full  Gibbs  function-­‐‑derived  TEOS-­‐‑10  with  an  rms  error  in  the  “funnel”  
of   0.03x10−6 K −1 ;   this   can   be   compared   with   the   rms   error   of   the   thermal   expansion  
coefficient   of   the   laboratory   data   to   which   the   Feistel   (2008)   Gibbs   function   was   fitted   of  
0.73 x10−6 K −1    (see   row   six   of   Table   O.1   of   appendix   O).      In   terms   of   the   evaluation   of  
density   gradients,   the   haline   contraction   coefficient   evaluated   from   Eqn.   (K.3)   is   many  
times  more  accurate  than  the  thermal  expansion  coefficient.    Hence  we  may  consider  the  
75-­‐‑term  polynomial  expression  for  specific  volume,  Eqn.  (K.3),  to  be  equally  as  accurate  as  
the  full  TEOS-­‐‑10  expressions  for  specific  volume,  for  the  thermal  expansion  coefficient  and  
for  the  saline  contraction  coefficient  for  data  that  reside  inside  the  “oceanographic  funnel”.    
The   sound   speed   evaluated   from   the   75-­‐‑term   polynomial   of   Eqn.   (K.3)   has   an   rms  
error   over   the   “funnel”   of   0.025 m s −1    which   is   a   little   less   than   the   rms   error   of   the  

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164

TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

underlying  sound  speed  data  that  was  incorporated  into  the  Feistel  (2008)  Gibbs  function,  
being   0.035 m s −1   (see  rows  7  to  9  of  Table  O.1  of  appendix  O).    Hence,  especially  for  the  
purposes  of  dynamical  oceanography  where   α Θ   and   β Θ   are  the  aspects  of  the  equation  of  
state  that,  together  with  spatial  gradients  of   S A   and   Θ ,  drive  ocean  currents  and  affect  the  
calculation   of   the   buoyancy   frequency,   we   may   take   the   75-­‐‑term   expression   for   specific  
volume,  Eqn.  (K.3),  as  essentially  reflecting  the  full  accuracy  of  TEOS-­‐‑10.      
The   use   of   Eqn.   (K.3)   to   evaluate   v̂ SA ,Θ, p    or   ρˆ ( SA , Θ, p )    from  
gsw_specvol(SA,CT,p)   or   gsw_rho(SA,CT,p)   is   approximately   five   times   faster   than   first  
evaluating  the  in  situ  temperature   t   (from  gsw_t_from_CT(SA,CT,p))  and  then  calculating  
in   situ   specific   volume   or   density   from   the   full   Gibbs   function   expression   v SA ,t, p    or  
ρ ( SA , t, p )    via   gsw_specvol_t_exact(SA,t,p)   or   gsw_rho_t_exact(SA,t,p).      (These   two  
function   calls   have   been   combined   into   gsw_specvol_CT_exact(SA,CT,P)   and  
gsw_rho_CT_exact(SA,CT,P).)      
  
  
Table  K.1.    Coefficients  of  the  75-­‐‑term  polynomial  of  Roquet  et  al.  (2015).      
  

(

)

(

i

j

k

vijk

i

j

k

vijk

i

j

k

vijk

0

0

0

0

5

0

0

2

0

0

1

5

0

2

0

2

2

0

0

0

6

0

3

0

2

3

0

0

0

0

1

4

0

2

4

0

0

1

0

1

0

1

2

5

0

0

2

0

1

1

1

2

6

0

0

3

0

1

2

1

2

0

1

0

4

0

1

1

1

0

5

0

1

2

1

0

0

1

1

3

1

0

1

1

1

4

1

0

2

1

1

5

1

0

3

1

1

0

2

0

4

1

1

1

2

0

0

2

1

2

2

0

1

2

1

3

2

0

2

2

1

4

2

0

3

2

1

0

3

0

0

3

1

1

3

0

1

3

1

2

3

0

2

3

1

3

3

0

0

4

1

0

4

0

1

4

1

1

4

0

0

5

1

2

4

0

0

0

2

-8.0539615540e-7
-3.3052758900e-7
2.0543094268e-7
-6.0799143809e-5
2.4262468747e-5
-3.4792460974e-5
3.7470777305e-5
-1.7322218612e-5
3.0927427253e-6
1.8505765429e-5
-9.5677088156e-6
1.1100834765e-5
-9.8447117844e-6
2.5909225260e-6
-1.1716606853e-5
-2.3678308361e-7
2.9283346295e-6
-4.8826139200e-7
7.9279656173e-6
-3.4558773655e-6
3.1655306078e-7
-3.4102187482e-6
1.2956717783e-6
5.0736766814e-7
9.9856169219e-6

1

1

1.0769995862e-3
-3.1038981976e-4
6.6928067038e-4
-8.5047933937e-4
5.8086069943e-4
-2.1092370507e-4
3.1932457305e-5
-1.5649734675e-5
3.5009599764e-5
-4.3592678561e-5
3.4532461828e-5
-1.1959409788e-5
1.3864594581e-6
2.7762106484e-5
-3.7435842344e-5
3.5907822760e-5
-1.8698584187e-5
3.8595339244e-6
-1.6521159259e-5
2.4141479483e-5
-1.4353633048e-5
2.2863324556e-6
6.9111322702e-6
-8.7595873154e-6
4.3703680598e-6

-5.8484432984e-7
-4.8122251597e-6
4.9263106998e-6
-1.7811974727e-6
-1.1736386731e-6
-5.5699154557e-6
5.4620748834e-6
-1.3544185627e-6
2.1305028740e-6
3.9137387080e-7
-6.5731104067e-7
-4.6132540037e-7
7.7618888092e-9
-6.3352916514e-8
-1.1309361437e-6
3.6310188515e-7
1.6746303780e-8
-3.6527006553e-7
-2.7295696237e-7
2.8695905159e-7
1.0531153080e-7
-1.1147125423e-7
3.1454099902e-7
-1.2647261286e-8
1.9613503930e-9

  

IOC Manuals and Guides No. 56

3

1

2

0

2

2

1

2

2

2

2

2

0

3

2

1

3

2

0

4

2

0

0

3

1

0

3

2

0

3

0

1

3

1

1

3

0

2

3

0

0

4

1

0

4

0

1

4

0

0

5

0

0

6

)

TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

  
  
  

165

Appendix  L:  Recommended  nomenclature,  
symbols  and  units  in  oceanography    

L.1 Recommended nomenclature
The   strict   SI   units   of   Absolute   Salinity,   temperature   and   pressure   are   kg kg −1,   Absolute  
Temperature   in   K    and   Absolute   Pressure   P    in   Pa.      These   are   the   units   predominantly  
adopted   in   the   SIA   computer   software   for   the   input   and   output   variables.      If  
oceanographers  were  to  adopt  this  practice  of  using  strictly  SI  quantities  it  would  simplify  
many  thermodynamic  expressions  at  the  cost  of  using  unfamiliar  units.      
The  GSW  Oceanographic  Toolbox  (appendix  N)  adopts  as  far  as  possible  the  currently  
used  oceanographic  units,  so  that  the  input  variables  for  all  the  computer  algorithms  are  
Absolute   Salinity   in   S A    in   g kg −1,    temperature   in   ° C    and   pressure   as   sea   pressure   in  
dbar.    The  outputs  of  the  functions  are  also  generally  consistent  with  this  choice  of  units,  
but  some  variables  are  more  naturally  expressed  in  SI  units.      
It   seems   impractical   to   recommend   that   the   field   of   oceanography   fully   adopt   strict  
basic   SI   units.      It   is   however   very   valuable   to   have   the   field   adopt   uniform   symbols   and  
units,   and   in   the   interests   of   achieving   this   uniformity   we   recommend   the   following  
symbols   and   units.      These   are   the   symbols   and   units   we   have   adopted   in   the   GSW  
Oceanographic  Toolbox.      
  
  

Table  L.1.  Recommended  Symbols  and  Units  in  Oceanography    
Quantity

Symbol Units

Comments

Chlorinity

Cl

g kg–1

Chlorinity is defined as the following mass
fraction; it is 0.328 523 4 times the ratio of the
mass of pure silver required to precipitate all
dissolved chloride, bromide and iodide in seawater
to the mass of seawater.

Standard Ocean
Reference Salinity

SSO

g kg–1

freezing temperatures

t f , Θf

ºC

Absolute Pressure

P

Pa

35.165 04 g kg–1 being exactly 35 uPS ,
corresponding to the standard ocean Practical
Salinity of 35.
in situ and conservative values, each as a function
of S A and p.
When Absolute Pressure is used it should always
be in Pa, not in Mpa nor in dbar.

sea pressure. Sea pressure
is the pressure argument
to all the
GSW Toolbox functions.

p

dbar

Equal to P − P0 and usually expressed in dbar not
Pa.

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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

gauge pressure. Gauge
pressure (also called
applied pressure) is
sometimes reported from
ship-born instruments.

p gauge

dbar

Equal to the Absolute Pressure P minus the local
atmospheric pressure at the time of the instrument
calibration, and expressed in dbar not Pa. Sea
pressure p is preferred over gauge pressure p gauge ,
as p is the argument to the seawater Gibbs
function.

reference pressure

pr

dbar

The value of the sea pressure p to which potential
temperature and/or potential density are
referenced.

one standard atmosphere
isopycnal slope ratio

P0
r

Pa
1

exactly 101 325 Pa (= 10.1325 dbar)

Stability Ratio

Rρ

1

Rρ = α ΘΘz β Θ ( SA )z ≈ α θ θ z β θ ( SA )z .

isopycnal temperature
gradient ratio
Practical Salinity

GΘ

1

G Θ = r ⎡⎣ Rρ −1⎤⎦ ⎡⎣ Rρ − r ⎤⎦ ; ∇σ Θ = G Θ∇n Θ

SP

1

Defined in the range 2 < SP < 42 by PSS-78 based
on measured conductivity ratios.

Reference Salinity

SR

g kg-1

Reference-Composition Salinity (or Reference
Salinity for short) is the Absolute Salinity of
seawater samples that have Reference
Composition. At S P = 35, S R is exactly uPS SP .
while in the range 2 < SP < 42 SR ≈ uPS SP .

Absolute Salinity
(This is the salinity
argument of all the
GSW Toolbox functions.)

-1
SA = SAdens g kg

Absolute Salinity
Anomaly

δ SA

g kg-1

“Preformed Absolute
Salinity”,

S*

g kg-1

“Solution Absolute
Salinity”, often shortened
to “Solution Salinity”

S Asoln

g kg-1

“Added-Mass Salinity”

SAadd

g kg-1

often shortened to
“Preformed Salinity”

IOC Manuals and Guides No. 56

r =

α Θ ( p) β Θ ( p)
α Θ ( pr ) β Θ ( pr )

SA = SR + δ SA ≈ uPS SP + δ SA
Absolute Salinity is the sum of S R on the Millero
et al. (2008a) Reference-Salinity Scale and the
Absolute Salinity Anomaly. The full symbol for
S A is SAdens as it is the type of absolute salinity
which delivers the best estimate of density when
used as the salinity argument of the TEOS-10
Gibbs function. Another name for SA = SAdens is
“Density Salinity”.
δ SA = SA − SR , the difference between Absolute
Salinity, SA = SAdens , and Reference-Composition
Salinity. In terms of the full nomenclature of
Pawlowicz et al. (2010), Wright et al. (2010b) and
appendix A.4 herein, the Absolute Salinity
Anomaly δ SA is δ SRdens .
Preformed Absolute Salinity S* is a salinity
variable that is designed to be as conservative as
possible, by removing the estimated
biogeochemical influences on the seawater
composition from other forms of salinity (see
Pawlowicz et al. (2010), Wright et al. (2010b) and
appendix A.4 herein).
The mass fraction of non-H2O constituents in
seawater after it has been brought to chemical
equilibrium at t = 25°C and p = 0 dbar (see
Pawlowicz et al. (2010), Wright et al. (2010b) and
appendix A.4 herein).
SAadd − SR is the estimated mass fraction of nonH2O constituents needed as ingredients to be added
to Standard Seawater which when mixed and
brought to chemical equilibrium at t = 25°C and
p = 0 dbar results in the observed seawater
composition.

TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

167

temperature
Absolute Temperature

t
T

ºC
K

temperature derivatives

T

K

Celsius zero point

T0
θ
Θ

K

T0 ≡ 273.15 K

ºC
ºC

Defined implicitly by Eqn. (3.1.3)
Defined in Eqn. (3.3.1) as exactly potential
enthalpy divided by c 0p .

c0p ≡ 3991.867 957 119 63 J kg −1 K −1 . This 15-digit
number is defined to be the exact value of c 0p .
c 0p is the ratio of potential enthalpy h0 to Θ .

potential temperature
Conservative Temperature

T / K ≡ T0 / K + t / (°C) = 273.15 + t / (°C)
When a quantity is differentiated with respect to in
situ temperature, the symbol T is used in order to
distinguish this variable from time.

the “specific heat”, for use
with Conservative
Temperature

c 0p

J kg–1 K–1

combined standard
uncertainty
enthalpy
specific enthalpy

uc

Varies

H
h

J
J kg–1

specific potential enthalpy

h0

J kg–1
–1

h = u + ( p + P0 ) v .
Here p and P0 must be in Pa not dbar.

specific enthalpy referenced to zero sea pressure,
–1

specific isobaric heat
capacity

cp

J kg K

internal energy
specific internal energy
specific isochoric heat
capacity

U
u

cv

J
J kg–1
J kg–1 K–1

Gibbs function
(Gibbs energy)
specific Gibbs function
(Gibbs energy)
specific Helmholtz energy
unit conversion factor for
salinities

G

J

g

J kg–1

f

J kg–1
g kg–1

entropy
specific entropy

Σ

density
density anomaly
potential density anomaly
referenced to a sea
pressure of 2000 dbar
potential density anomaly
referenced to a sea
pressure of 4000 dbar
thermal expansion
coefficient with respect to
in situ temperature
thermal expansion
coefficient with respect to
potential temperature θ

uPS

η

ρ

σt
σ2

J K–1
J kg–1 K–1
kg m–3
kg m–3
kg m–3

h0 = h ( SA ,θ [ SA , t , p, pr = 0], pr = 0 )

c p = ∂h ∂T

SA , p

cv = ∂u ∂T

SA , v

uPS ≡ (35.16504 35) g kg−1 ≈ 1.004 715... g kg−1
The first part of this expression is exact. This
conversion factor is an important and invariant
constant of the 2008 Reference-Salinity Scale
(Millero et al. (2008a)).

In many other publications the symbol s is used for
specific entropy.
–3
ρ ( SA , t,0) – 1000 kg m

-3
ρ ( SA , θ [ SA , t , p, pr ], pr ) – 1000 kg m where

pr = 2000 dbar

σ4

kg m–3

-3
ρ ( SA , θ [ SA , t , p, pr ], pr ) – 1000 kg m where

pr = 4000 dbar

αt

K–1

v −1∂v / ∂T

SA , p

= − ρ −1∂ρ / ∂T

SA , p

αθ

K–1

v −1∂v / ∂θ

SA , p

= − ρ −1∂ρ / ∂θ

SA , p

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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

thermal expansion
coefficient with respect to
Conservative Temperature

αΘ

K–1

v −1∂v / ∂Θ

saline contraction
coefficient at constant in
situ temperature

βt

kg g–1

− v −1∂v / ∂SA

saline contraction
coefficient at constant
potential temperature

βθ

saline contraction
coefficient at constant
Conservative Temperature

βΘ

isothermal compressibility

κt

Pa–1

isentropic and isohaline
compressibility
chemical potential of
water in seawater
chemical potential of sea
salt in seawater
relative chemical potential
of (sea salt and water in)
seawater
dissipation rate of kinetic
energy per unit mass

κ

Pa–1

µW

J g–1

µS
µ

J g–1

ε

J kg–1 s–1
= m2 s–3

adiabatic lapse rate

Γ

K Pa–1

SA , p

= − ρ −1∂ρ / ∂Θ

SA , p

Θ

= ρ −1∂ρ / ∂SA

T,p

Note that the units for β t are consistent with SA
being in g kg-1.
kg g–1

− v −1∂v / ∂SA

θ,p

Note that the units for β θ are consistent with SA
being in g kg-1.
kg g–1

− v −1∂v / ∂SA

Θ, p

= + ρ −1∂ρ / ∂SA

Note that the units for β
being in g kg-1.

J g–1

(∂g

Γ=

Θ

Θ, p

are consistent with SA

∂SA )t , p = µ S − µ W

∂t
∂P S

=
A ,θ

∂t
∂P S

=
A ,Θ

∂t
∂P S

=
A ,η

∂v
∂η

=

(T0 +θ ) ∂v
c0p

SA , p

∂Θ S

A, p

–1

c
v

ms
m3 kg–1

specific volume anomaly
thermobaric coefficient
based on θ
thermobaric coefficient
based on Θ
cabbeling coefficient
based on θ
cabbeling coefficient
based on Θ
buoyancy frequency

δ

m3 kg–1

Tbθ

K −1Pa −1

Tbθ = β θ ∂ α θ β θ

TbΘ

K −1Pa −1

TbΘ = β Θ ∂ α Θ β Θ

Neutral Density

= ρ −1∂ρ / ∂SA

θ,p

sound speed
specific volume

neutral helicity

T,p

θ

Cb

C bΘ
N

H
γn

K
K
s

n

−2
−2

−1

m

−3

v = ρ −1

(

) ∂P

SA , θ

(

) ∂P

SA , Θ

Cbθ = ∂α θ ∂θ
CbΘ = ∂α Θ ∂Θ

(

SA , p

SA , p

2

θ
+ 2 α θ ∂α θ ∂SA
β

⎛ θ⎞
− ⎜ α ⎟ ∂β θ ∂SA
θ , p ⎝ βθ ⎠
θ, p

Θ
+ 2 α Θ ∂α Θ ∂SA

⎛ Θ⎞
− ⎜ α Θ ⎟ ∂β Θ ∂SA
Θ, p
Θ, p
⎝β ⎠

β

)

(

N = g α Θz − β SA z = g α θ θ z − β θ SA z
2

Θ

Θ

2

)

defined by Eqns. (3.13.1) and (3.13.2)

kg m–3

a density variable whose iso-surfaces are designed
to be approximately neutral, i. e.
α Θ∇γ Θ ≈ β Θ∇γ SA .

Neutral-Surface-PotentialVorticity

NSPV

s −3

NSPV = − g ρ −1 f γ zn where f is the Coriolis
parameter.

dynamic height anomaly

Ψ

m 2 s −2

Pa m3 kg −1 = m2 s−2

Montgomery geostrophic
streamfunction

ΨM

m 2 s −2

Pa m3 kg −1 = m2 s−2

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169

PISH (Pressure-Integrated
Steric Height)

Ψ′

kg s-2

streamfunction for f times the depth-integrated
relative mass flux, see Eqns. (3.31.1) – (3.31.5).

Coriolis parameter

f

s −1

1.458 42 x 10−4 sin φ s−1 , where φ is latitude

molar mass of Reference
Seawater

MS

g mol−1

molality of seasalt in
Reference Seawater

mSW

mol kg–1

M S is the mole-weighted average atomic weight
of the constituents of Reference Seawater,
M S = 31.403 821 8... g mol−1 , from Millero et al.
(2008a).
mSW = ∑ i mi =

SA
1
.
M S 1− SA

(

)

mi is the molality

of constituent i in Reference Seawater.

valence factor of
Reference Seawater

Z2

1

ionic strength of
Reference Seawater

I

mol kg–1

Z 2 = ∑ i X i Z i2 ≡ 1.245 289 8 where Z i is the
charge of seawater constituent i which is present
at the mole fraction X i in Reference Seawater
(from Millero et al. (2008a)).

I =

1
2

mSW Z 2 =

1
2

∑ i mi Zi2

= 0.622 644 9 mSW
≈

SA
0.622 644 9
mol kg −1
.
0.031 403 821 8
1− SA

(

)

mi is the molality of constituent i in Reference
Seawater.

osmotic coefficient

φ

1

φ ( SA , T , p ) =

g ( 0, t , p ) − µ W ( SA , t , p )

mSW R (T0 + t )
where the molar gas constant,
R = 8.314 472 J mol–1 K–1. See also Eqns. (2.14.1)
and (3.40.9) for an equivalent definition of φ .

  
  

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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

L.2 Recommended symbols when variables are functions of η, θ and Θ
Note  that  whether  using  standard  notation  or  variants  from  it,  all  variables  should  be  
explicitly  defined  in  publications  when  first  used.    Standard  notation  should  be  considered  
as  an  additional  aid  to  improve  readability,  not  as  a  replacement  for  explicit  definitions.      
Note  that  oxygen  should  be  reported  in  µμmol/kg  and  not  cm3dm–3,  ml/l  or  µμmol/l  (this  
reflects  a  desire  for  consistency  with  reporting  of  other  quantities  and  will  avoid  problems  
associated  with  conversion  between  moles  and  ml  using  the  gas  equations).      
When   thermodynamic   variables   are   taken   to   be   functions   of   variables   other   than   the  
standard   combination   ( SA , t, p )    it   is   convenient   to   indicate   this   by   a   marking   on   the  
variable.      This   greatly   simplifies   the   nomenclature   for   partial   derivatives.      Table   L.2   lists  
the   suggested   markings   on   the   variables   that   arise   commonly   in   this   context.  
   The  

thermodynamic   variables   are   related   to   the   thermodynamic   potentials   h = h SA ,η , p ,  
h = h SA ,θ , p   and   h = hˆ ( SA , Θ, p )   by  the  expressions  in  appendix  P.      
  
  

(

(

)

Table L.2. Recommended symbols when variables are functions of η, θ , Θ and h
  

quantity
enthalpy,   h     
specific  volume,   v   

function of

( SA , t , p )

( SA ,η, p )

density,   ρ       
Conservative  Temperature,   Θ     
enthalpy,   h     
specific  volume,   v   

( SA , θ , p )

density,   ρ       
entropy,   η     
enthalpy,   h     
specific  volume,   v   

( SA , Θ, p )

density,   ρ       
entropy,   η   

IOC Manuals and Guides No. 56

(
)
(
)
(
)
( )
h = h ( SA ,θ , p )
v = v ( SA ,θ , p )
ρ = ρ ( SA ,θ , p )
η = η ( SA ,θ )
h = hˆ ( SA , Θ, p )

v = vˆ ( SA , Θ, p )

η = ηˆ ( SA , Θ)

entropy,   η     

specific  volume,   v   


h = h S A ,η , p

v = v S A ,η , p

ρ = ρ S A ,η , p

Θ = Θ S A ,η

ρ = ρˆ ( SA , Θ, p )

density,   ρ       

Conservative  Temperature,   Θ   

v = v ( SA , t , p )

η = η ( SA , t , p )

entropy,   η     
specific  volume,   v   

h = h ( SA , t , p )

ρ = ρ ( SA , t , p )

density,   ρ       
enthalpy,   h     

symbol for this
functional form

( SA , h, p )


Θ = Θ SA ,h, p

v = v SA ,h, p

ρ = ρ SA ,h, p

η = η SA ,h, p

(

(
(
(

)

)
)
)

)

TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

171

Appendix  M:    
Seawater-­‐‑Ice-­‐‑Air  (SIA)  library  of  computer  software    

  
  
  
This  software  library,  the  Seawater-­‐‑Ice-­‐‑Air  library  (the  SIA  library  for  short),  contains  the  
TEOS-­‐‑10   subroutines   for   evaluating   a   wide   range   of   thermodynamic   properties   of   pure  
water   (using   IAPWS-­‐‑95),   seawater   (using   IAPWS-­‐‑08   for   the   saline   part),   ice   Ih   (using  
IAPWS-­‐‑06)  and  for  moist  air  (using  Feistel  et  al.  (2010a),  IAPWS  (2010)).    It  is  divided  into  
six   levels   (levels   0   through   5)   with   each   successive   level   building   on   the   functional  
capabilities  introduced  at  lower  levels.    Briefly,    
• level   0   defines   fundamental   constants,   sets   options   used   throughout   the   library   and  
provides  routines  to  convert  between  Practical  Salinity  and  Absolute  Salinity    
• level   1   defines   a   complete   set   of   independent   and   consistent   elements   that   are   based  
on   previous   work   and   form   the   essential   building   blocks   for   the   rest   of   the   library  
routines    
• level  2  provides  access  to  a  set  of  properties  for  individual  mediums  (liquid  or  vapour  
water,  ice,  seawater  and  dry  or  humid  air)  that  can  be  calculated  from  the  level  0  and  1  
routines  without  additional  approximations    
• level   3   introduces   additional   functions   that   require   numerical   solution   of   equations.    
Most  importantly,  it  is  at  this  level  that  the  density  of  pure  fluid  water  is  determined  
from   temperature   and   pressure   information.      This   permits   the   definition   of   Gibbs  
functions   for   pure   water   and   seawater   that   make   use   of   the   IAPWS-­‐‑95   Helmholtz  
function  as  a  fundamental  building  block    
• level   4   deals   with   a   fairly   broad   (but   not   exhaustive)   selection   of   equilibrium  
properties  involving  fluid  water,  seawater,  ice  and  air;  and    
• level  5  includes  a  set  of  routines  that  build  on  the  SIA  routines  but  violate  principals  
adhered   to   throughout   levels   0   though   4.      In   particular,   non-­‐‑basic   SI   units   are  
permitted  at  this  level  as  discussed  below.      
  
As  a  general  rule,  the  inputs  and  the  outputs  of  the  algorithms  in  the  SIA  library  are  in  
basic   SI   units.      Hence   the   salinity   is   Absolute   Salinity   S A    in   units   of   kg kg −1   (so   that   for  
example   standard   ocean   Reference   Salinity   is   input   to   SIA   functions   as   0.035   165   04  
( kg kg −1)   rather   than   35.165 04    ( g kg −1 ),   in   situ   temperature   is   input   as   Absolute  
Temperature   T    in   K,   and   pressure   is   input   as   Absolute   Pressure   P    in   Pa.      Use   of   these  
basic  SI  units  simplifies  the  calculation  of  theoretical  expressions  in  thermodynamics.    The  
only  exceptions  to  this  rule  for  the  units  of  the  inputs  and  outputs  in  the  SIA  library  are  as  
follows.    
• The   function   SA = SA ( SP ,φ, λ, P)    that   calculates   Absolute   Salinity   (in   kg   kg-­‐‑1)   when  
given  Practical  Salinity   S P   (which  is  unitless  and  takes  numbers  like  35  not  0.035)  as  
its  salinity  input  variable,  along  with  location  in  the  form  of  longitude   λ   (°E)  latitude  
φ   (°N)  and  Absolute  Pressure   P   (Pa).    Location  is  required  in  this  routine  to  account  
for  the  influence  of  composition  anomalies  through  a  lookup  table  adopted  from  the  
GSW  Oceanographic  Toolbox.      
• The   inverse   function   SP = SP ( SA , φ , λ , P ) .    This   and   the   previous   routine   are   found   at  
level   0   since   Absolute   Salinity   is   required   as   an   input   to   many   of   the   higher   level  
library  routines.      

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•

•

TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

General   purpose   routines   that   allow   for   conversions   between   a   broad   range   of  
pressure,   temperature   and   salinity   units   that   are   in   common   usage   are   provided   at  
level  5.    The  numerical  input  value  and  its  unit  are  provided  by  the  user  and  results  
are  returned  in  a  specified  output  unit.    
Algorithms  are  included  at  level  5  that  use  non-­‐‑basic  SI  units  as  input  and  as  output.    
Most  noteworthy  is  the  GSW  library  module  that  uses  the  SIA  routines  to  mimic  many  
of  the  routines  in  the  GSW  Oceanographic  Toolbox.    These  routines  use  IAPWS-­‐‑09  for  
pure  water  in  place  of  IAPWS-­‐‑95  to  provide  improved  computational  efficiency.    They  
have   been   used   to   provide   independent   checks   on   the   corresponding   routines   in   the  
GSW  Oceanographic  Toolbox.          

  

Because  the  IAPWS-­‐‑95  description  of  pure  water  substance  (both  liquid  and  vapour)  is  
the   world-­‐‑wide   standard   for   pure   water   substance,   the   SIA   library   is   the   official  
description  of  seawater,  although  it  should  be  noted  that  the  computer  software  does  not  
come  with  any  warranty.    Rather  it  is  the  underlying  papers  as  listed  in  appendix  C  that  
are  the  officially  warranted  descriptions.      
The   SIA   library   benefits   from   the   full   range   of   applicability   of   the   IAPWS-­‐‑95  
Helmholtz  function  for  pure  water,  0  kg  m-­‐‑3  <   ρ   <  1200  kg  m-­‐‑3,  130  K  <  T  <  1273  K,  plus  an  
extension   down   to   50   K   introduced   by   Feistel   et   al.   (2010a).      It   does   however   have   two  
disadvantages  as  far  as  the  field  of  oceanography  is  concerned.    First,  because  IAPWS-­‐‑95  is  
valid   over   very   wide   ranges   of   temperature   and   pressure,   it   is   necessarily   an   extensive  
series   of   polynomials   and   exponentials   which   is   not   as   fast   computationally   as   the  
equation  of  state  EOS-­‐‑80  with  which  oceanographers  are  familiar.    Second,  the  IAPWS-­‐‑95  
thermodynamic   potential   is   a   Helmholtz   function   which   expresses   thermodynamic  
properties   in   terms   of   density   and   temperature   rather   than   pressure   and   temperature   as  
normally  used  in  oceanography.    Since  IAPWS-­‐‑95  describes  not  only  liquid  water  but  also  
water  vapour,  this  Helmholtz  form  of  the  thermodynamic  potential  is  natural.    Although  
the   library   also   includes   a   Gibbs   function   formulation   with   temperature   and   pressure   as  
independent   variables,   the   core   routines   implement   this   formulation   by   first   solving  
P = ρ 2 f ρ (T , ρ )    for   ρ    and   then   using   IAPWS-­‐‑95,   which   is   a   computationally   expensive  
procedure.        
In   the   GSW   Oceanographic   Toolbox   (appendix   N)   we   present   an   alternative  
thermodynamic  description  of  seawater  properties  based  on  the  IAPWS-­‐‑09  description  of  
the   pure   liquid   water   part   as   a   Gibbs   function.      The   GSW   formulation   is   limited   to   the  
Neptunian   range   (i.   e.   the   oceanographic   range)   of   temperature   and   pressure   and   deals  
only  with  liquid  water,  but  it  is  far  more  computationally  efficient  since  the  limited  range  
of   validity   allows   equivalent   accuracy   with   fewer   terms   and   the   Gibbs   function  
formulation  avoids  the  need  to  invert  the  relation P = ρ 2 f ρ (T , ρ ) .    This  formulation  is  also  
implemented  at  level  5  of  the  SIA  library  as  a  cross-­‐‑check  on  the  GSW  routines  and  for  the  
convenience   of   SIA   library   users   working   on   applications   requiring   increased  
computational  efficiency.    Note  however  that  some  of  routines  in  the  SIA  implementation  
of  the  GSW  routines  are  not  as  fully  optimized  as  the  corresponding  routines  in  the  GSW  
Oceanographic  Toolbox.      
The   presence   of   dissolved   salts   in   seawater   reduces   the   range   of   applicability   of   the  
SIA  and  GSW  seawater  routines  in  comparison  with  the  IAPWS-­‐‑95  range  of  applicability  
for  pure  fluid  water,  whether  or  not  the  IAPWS-­‐‑09  Gibbs  formulation  is  used  to  represent  
pure  water  properties.    This  is  because  the  range  of  applicability  of  the  saline  component  
of   the   Gibbs   function   is   limited   to   0   kg   kg-­‐‑1   ≤   S A    ≤   0.12   kg   kg-­‐‑1,      262   K   ≤   T   ≤   353   K,   and    
100  Pa  ≤  P  ≤   108   Pa.  
Since  this  manual  focuses  on  seawater,  we  refer  the  reader  to  Feistel  et  al.  (2010b)  and  
Wright  et  al.  (2010)  for  details  on  the  ice  and  air  components  of  the  SIA  library.    However,  

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173

below,  we  discuss  a  few  features  of  the  library  that  relate  to  these  additional  components.    
First,  we  note  that  the  thermodynamic  potentials  of  pure  water,  ice,  the  saline  part  of  the  
seawater  Gibbs  function  and  the  Gibbs  function  of  moist  air  have  been  carefully  adjusted  
to   make   them   fully   compatible   with   each   other   (Feistel   et   al.   (2008a)).      Only   by   so   doing  
can  the  equilibrium  properties  of  coincident  phases  be  accurately  evaluated  (for  example,  
the   freezing   temperature   of   pure   water   and   of   seawater).      Many   functions   involving  
equilibrium   properties   of   water,   vapour,   ice,   seawater   and   dry   or   humid   air   are  
implemented   in   level   4   of   the   SIA   library.      To   provide   an   indication   of   the   range   of  
functions   available,   we   have   listed   the   routine   names   in   Table   M.1   below.      This   table  
comes  from  Table  3.1  of  Wright  et  al.  (2010);  we  refer  the  interested  reader  to  Feistel  et  al.  
(2010b))  and  Wright  et  al.  (2010))  for  detailed  information.    Wright  et  al.   (2010)  provide  a  
supplementary  table  that  is  cross-­‐‑referenced  to  their  Table  3.1  to  give  details  on  the  usage  
of  each  routine  and  each  table  in  their  supplement  references  in  turn  the  relevant  sections  
of  Feistel  et  al.  (2010b)  for  additional  background  information.    
Because  each  level  of  the  SIA  library  builds  on  lower  levels  and  since  multiple  phases  
may   be   involved   in   the   equilibrium   calculations,   the   determination   of   the   ranges   of  
validity  of  the  routines  in  the  SIA  library  can  become  rather  involved.    To  deal  with  this  
issue,  a  procedure  has  been  implemented  in  the  library  to  return  an  error  code  for  function  
evaluations   that   depend   on   results   from   any   of   the   basic   building   block   routines   from  
outside  of  their  individual  ranges  of  validity.    Numerical  check  values  are  provided  with  
each  of  the  routines  in  the  library  and  auxiliary  routines  are  provided  to  assist  users  in  the  
validation  of  local  implementations.      
Further   details   of   the   SIA   software   library   are   provided   in   the   papers   Feistel   et   al.  
(2010b))   and   Wright   et   al.   (2010))   and   the   software   is   served   from   the   www.TEOS-10.org
web  site.      
  
  
  

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Table  M.1.    The  SIA  library  contents.  Module  names  are  in  bold  and  user-­‐‑accessible    

routines   are   in   plain   type.      Each   of   the   Public   Routines   can   be   accessed   by  
users.      The   underlined   routines   are   thermodynamic   potential   functions  
including   first   and   second   derivatives.      The   bracketed   numbers   preceding  
most   module   names   give   the   related   table   numbers   in   the   supplement   to  
Wright  et  al.  (2010)  where  detailed  information  on  the  use  of  each  function  is  
provided.      

Level 0 routines
Constants_0

Constants_0 (Cont'd)

Maths_0

(S2) Convert_0

Public Parameter Values

Parameter Values (cont'd)

Uses

Uses

celsius_temperature_si
check_limits
cp_chempot_si
cp_density_si
cp_pressure_si
cp_temperature_si
dry_air_dmax
dry_air_dmin
dry_air_tmax
dry_air_tmin
errorreturn
flu_dmax
flu_dmin
flu_tmax
flu_tmin
gas_constant_air_si
gas_constant_air_L2000
gas_constant_molar_si
gas_constant_molar_L2000
gas_constant_h2O_si
gas_constant_h2O_iapws95
ice_pmax
ice_pmin
ice_tmax
ice_tmin
isextension2010
isok

mix_air_dmax
mix_air_dmin
mix_air_tmax
mix_air_tmin
molar_mass_air_si
molar_mass_air_l2000
molar_mass_h2o_si
molar_mass_seasalt_si
pi
sal_pmax
sal_pmin
sal_smax
sal_smin
sal_tmax
sal_tmin
sealevel_pressure_si
so_salinity_si
so_temperature_si
so_pressure_si
tp_density_ice_iapws95_si
tp_density_liq_iapws95_si
tp_density_vap_iapws95_si
tp_enthalpy_ice_si
tp_enthalpy_vap_si
tp_pressure_exp_si
tp_pressure_iapws95_si
tp_temperature_si

constants_0

constants_0

Public Routines

Public Routines

get_cubicroots
matrix_solve

air_massfraction_air_si
air_massfraction_vap_si
air_molar_mass_si
air_molfraction_air_si
air_molfraction_vap_si
asal_from_psal
psal_from_asal

Level 1 routines
(S3) Flu_1 (IAPWS95)

(S4) Ice_1 (IAPWS06)

(S5) Sal_1 (IAPWS08)

(S6) Air_1

Uses

Uses

Uses

Uses

constants_0

constants_0

constants_0

constants_0

Public Routines

Public Routines

Public Routines

Public Routines

chk_iapws95_table6
chk_iapws95_table7
flu_f_si

chk_iapws06_table6
ice_g_si

sal_g_term_si

air_baw_m3mol
air_caaw_m6mol2
air_caww_m6mol2
dry_f_si
dry_init_clear
dry_init_Lemmon2000

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175

Level 2 routines
(S7) Flu_2

((S8) Ice_2

(S9) Sal_2

(S10) Air_2

Uses

Uses

Uses

Uses

constants_0, flu_1

constants_0, ice_1

constants_0, sal_1

constants_0, flu_1, air_1

Public Routines

Public Routines

Public Routines

Public Routines

flu_cp_si
flu_cv_si
flu_enthalpy_si
flu_entropy_si
flu_expansion_si
flu_gibbs_energy_si
flu_internal_energy_si
flu_kappa_s_si
flu_kappa_t_si
flu_lapserate_si
flu_pressure_si
flu_soundspeed_si

ice_chempot_si
ice_cp_si
ice_density_si
ice_enthalpy_si
ice_entropy_si
ice_expansion_si
ice_helmholtz_energy_si
ice_internal_energy_si
ice_kappa_s_si
ice_kappa_t_si
ice_lapserate_si
ice_p_coefficient_si
ice_specific_volume_si

sal_act_coeff_si
sal_act_potential_si
sal_activity_w_si
sal_chem_coeff_si
sal_chempot_h2o_si
sal_chempot_rel_si
sal_dilution_si
sal_g_si
sal_mixenthalpy_si
sal_mixentropy_si
sal_mixvolume_si
sal_molality_si
sal_osm_coeff_si
sal_saltenthalpy_si
sal_saltentropy_si
sal_saltvolume_si

air_f_si
air_f_cp_si
air_f_cv_si
air_f_enthalpy_si
air_f_entropy_si
air_f_expansion_si
air_f_gibbs_energy_si
air_f_internal_energy_si
air_f_kappa_s_si
air_f_kappa_t_si
air_f_lapserate_si
air_f_mix_si
air_f_pressure_si
air_f_soundspeed_si
chk_iapws10_table

(S11) Flu_3a

(S12) Sea_3a

(S13) Air_3a

Uses

Uses

Uses

constants_0, convert_0,
maths_0, flu_1

constants_0, sal_1, sal_2,
flu_3a (convert_0, maths_0,
flu_1)

constants_0, convert_0,
maths_0, air_1, air_2 (flu_1)

Public Routines

air_density_si
air_g_si
get_it_ctrl_airdensity
set_it_ctrl_airdensity

Level 3 routines

Public Routines
get_it_ctrl_density
liq_density_si
liq_g_si
set_it_ctrl_density
vap_density_si
vap_g_si

chk_iapws08_table8a
chk_iapws08_table8b
chk_iapws08_table8c
sea_chempot_h2o_si
sea_chempot_rel_si
sea_cp_si
sea_density_si
sea_enthalpy_si
sea_entropy_si
sea_g_si
sea_g_contraction_t_si
sea_g_expansion_si
sea_gibbs_energy_si
sea_internal_energy_si
sea_kappa_s_si
sea_kappa_t_si
sea_lapserate_si
sea_osm_coeff_si
sea_soundspeed_si
sea_temp_maxdensity_si

Public Routines

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(S14) Flu_3b

(S15) Sea_3b

(S16) Air_3b

Uses

Uses

Uses

constants_0, flu_2, flu_3a
(convert_0, maths_0,
flu_1)

constants_0, sal_2, flu_3a,
sea_3a (convert_0, maths_0,
flu_1, sal_1)

constants_0, convert_0,
air_1, air_2, air_3a (maths_0,
flu_1)

Public Routines

Public Routines

Public Routines

liq_cp_si
liq_cv_si
liq_enthalpy_si
liq_entropy_si
liq_expansion_si
liq_gibbs_energy_si
liq_internal_energy_si
liq_kappa_s_si
liq_kappa_t_si
liq_lapserate_si
liq_soundspeed_si
vap_cp_si
vap_cv_si
vap_enthalpy_si
vap_entropy_si
vap_expansion_si
vap_gibbs_energy_si
vap_internal_energy_si
vap_kappa_s_si
vap_kappa_t_si
vap_lapserate_si
vap_soundspeed_si

sea_h_si
sea_h_contraction_h_si
sea_h_contraction_t_si
sea_h_contraction_theta_si
sea_h_expansion_h_si
sea_h_expansion_t_si
sea_h_expansion_theta_si
sea_potdensity_si
sea_potenthalpy_si
sea_pottemp_si
sea_temperature_si
set_it_ctrl_pottemp

air_g_chempot_vap_si
air_g_compressibility
factor_si
air_g_contraction_si
air_g_cp_si
air_g_cv_si
air_g_density_si
air_g_enthalpy_si
air_g_entropy_si
air_g_expansion_si
air_g_gibbs_energy_si
air_g_internal_energy_si
air_g_kappa_s_si
air_g_kappa_t_si
air_g_lapserate_si
air_g_soundspeed_si
chk_lemmon_etal_2000

(S17) Sea_3c

(S18) Air_3c

Uses

Uses

constants_0, sea_3a, sea_3b
(convert_0, maths_0, flu_1,
sal_1, sal_2, flu_3a)

constants_0, convert_0,
air_2, air_3a, air_3b
(maths_0, air_1, flu_1)

Public Routines
sea_eta_contraction_h_si
sea_eta_contraction_t_si
sea_eta_contraction_theta_si
sea_eta_density_si
sea_eta_entropy_si
sea_eta_expansion_h_si
sea_eta_expansion_t_si
sea_eta_expansion_theta_si
sea_eta_potdensity_si
sea_eta_pottemp_si
sea_eta_temperature_si
set_it_ctrl_entropy_si

(S19) Sea_3d

Uses
constants_0, sal_2, flu_3a
(convert_0, maths_0, flu_1,
sal_1)

Public Routines
sea_sa_si
set_it_ctrl_salinity

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Public Routines
air_h_si
air_potdensity_si
air_potenthalpy_si
air_pottemp_si
air_temperature_si
set_it_ctrl_air_pottemp

TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

Level 4 routines
(S20) Liq_Vap_4

(S21) Ice_Vap_4

(S22) Sea_Vap_4

Uses

Uses

Uses

constants_0, maths_0, flu_1,
flu_2, flu_3a (Convert_0)

constants_0, maths_0, flu_1,
flu_2, ice_1, ice_2

Public Routines

Public Routines

constants_0, maths_0, flu_1,
sal_1, sal_2, flu_3a, sea_3a,
flu_3b (convert_0, flu_2)

chk_iapws95_table8
liq_vap_boilingtemperature_si
liq_vap_chempot_si
liq_vap_density_liq_si
liq_vap_density_vap_si
liq_vap_enthalpy_evap_si
liq_vap_enthalpy_liq_si
liq_vap_enthalpy_vap_si
liq_vap_entropy_evap_si
liq_vap_entropy_liq_si
liq_vap_entropy_vap_si
liq_vap_pressure_liq_si
liq_vap_pressure_vap_si
liq_vap_temperature_si
liq_vap_vapourpressure_si
liq_vap_volume_evap_si
set_liq_vap_eq_at_p
set_liq_vap_eq_at_t
set_it_ctrl_liq_vap

ice_vap_chempot_si
ice_vap_density_ice_si
ice_vap_density_vap_si
ice_vap_enthalpy_ice_si
ice_vap_enthalpy_subl_si
ice_vap_enthalpy_vap_si
ice_vap_entropy_ice_si
ice_vap_entropy_subl_si
ice_vap_entropy_vap_si
ice_vap_pressure_vap_si
ice_vap_sublimationpressure_si
ice_vap_sublimationtemp_si
ice_vap_temperature_si
ice_vap_volume_subl_si
set_ice_vap_eq_at_p
set_ice_vap_eq_at_t
set_it_ctrl_ice_vap

Public Routines

(S23) Ice_Liq_4

(S24) Sea_Liq_4

Uses

Uses

constants_0, maths_0, flu_1,
ice_1, flu_2, ice_2

constants_0, flu_1, sal_1, flu_2,
sal_2, flu_3a (convert_0,
maths_0)

Public Routines
ice_liq_chempot_si
ice_liq_density_ice_si
ice_liq_density_liq_si
ice_liq_enthalpy_ice_si
ice_liq_enthalpy_liq_si
ice_liq_enthalpy_melt_si
ice_liq_entropy_ice_si
ice_liq_entropy_liq_si
ice_liq_entropy_melt_si
ice_liq_meltingpressure_si
ice_liq_meltingtemperature_si
ice_liq_pressure_liq_si
ice_liq_temperature_si
ice_liq_volume_melt_si
set_ice_liq_eq_at_p
set_ice_liq_eq_at_t
set_it_ctrl_ice_liq

sea_vap_boilingtemperature_si
sea_vap_brinefraction_seavap_si
sea_vap_brinesalinity_si
sea_vap_cp_seavap_si
sea_vap_density_sea_si
sea_vap_density_seavap_si
sea_vap_density_vap_si
sea_vap_enthalpy_evap_si
sea_vap_enthalpy_sea_si
sea_vap_enthalpy_seavap_si
sea_vap_enthalpy_vap_si
sea_vap_entropy_sea_si
sea_vap_entropy_seavap_si
sea_vap_entropy_vap_si
sea_vap_expansion_seavap_si
sea_vap_g_si
sea_vap_kappa_t_seavap_si
sea_vap_pressure_si
sea_vap_salinity_si
sea_vap_temperature_si
sea_vap_vapourpressure_si
sea_vap_volume_evap_si
set_it_ctrl_sea_vap
set_sea_vap_eq_at_s_p
set_sea_vap_eq_at_s_t
set_sea_vap_eq_at_t_p

Public Routines
sea_liq_osmoticpressure_si
set_sea_liq_eq_at_s_t_p
set_it_ctrl_sea_liq

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(S25) Sea_Ice_4

Uses
constants_0, convert_0,
maths_0, flu_1, ice_1, sal_1,
ice_2, sal_2, flu_3a, sea_3a,
flu_3b (flu_2)

Public Routines
sea_ice_brinefraction_seaice_si
sea_ice_brinesalinity_si
sea_ice_cp_seaice_si
sea_ice_density_ice_si
sea_ice_density_sea_si
sea_ice_density_seaice_si
sea_ice_dtfdp_si
sea_ice_dtfds_si
sea_ice_enthalpy_ice_si
sea_ice_enthalpy_melt_si
sea_ice_enthalpy_sea_si
sea_ice_enthalpy_seaice_si
sea_ice_entropy_ice_si
sea_ice_entropy_sea_si
sea_ice_entropy_seaice_si
sea_ice_expansion_seaice_si
sea_ice_freezingtemperature_si
sea_ice_g_si
sea_ice_kappa_t_seaice_si
sea_ice_meltingpressure_si
sea_ice_pressure_si
sea_ice_salinity_si
sea_ice_temperature_si
sea_ice_volume_melt_si
set_it_ctrl_sea_ice
set_sea_ice_eq_at_s_p
set_sea_ice_eq_at_s_t
set_sea_ice_eq_at_t_p

(S26) Sea_Air_4

Uses
constants_0, convert_0,
maths_0, flu_1, sal_1, air_1,
flu_2, sal_2, air_2, flu_3a,
sea_3a, air_3a, air_3b,
liq_vap_4, liq_air_4a

Public Routines
sea_air_chempot_evap_si
sea_air_condense_temp_si
sea_air_density_air_si
sea_air_density_vap_si
sea_air_enthalpy_evap_si
sea_air_entropy_air_si
sea_air_massfraction_air_si
sea_air_vapourpressure_si
set_it_ctrl_sea_air
set_sea_air_eq_at_s_a_p
set_sea_air_eq_at_s_t_p

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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

(S27) Liq_Ice_Air_4

(S28) Sea_Ice_Vap_4

Uses

Uses

constants_0, convert_0,
maths_0, flu_1, ice_1, air_1,
flu_2, ice_2, air_2, air_3b,
ice_liq_4 (air_3a)

constants_0, maths_0, flu_1,
ice_1, sal_1, sal_2

Public Routines
sea_ice_vap_density_vap_si
sea_ice_vap_pressure_si
sea_ice_vap_salinity_si
sea_ice_vap_temperature_si
set_it_ctrl_sea_ice_vap
set_sea_ice_vap_eq_at_p
set_sea_ice_vap_eq_at_s
set_sea_ice_vap_eq_at_t

Public Routines
liq_ice_air_airfraction_si
liq_ice_air_density_si
liq_ice_air_dryairfraction_si
liq_ice_air_enthalpy_si
liq_ice_air_entropy_si
liq_ice_air_ifl_si
liq_ice_air_iml_si
liq_ice_air_liquidfraction_si
liq_ice_air_pressure_si
liq_ice_air_solidfraction_si
liq_ice_air_temperature_si
liq_ice_air_vapourfraction_si
set_liq_ice_air_eq_at_a
set_liq_ice_air_eq_at_p
set_liq_ice_air_eq_at_t
set_liq_ice_air_eq_at
_wa_eta_wt
set_liq_ice_air_eq_at
_wa_wl_wi
set_it_ctrl_liq_ice_air

(S29) Liq_Air_4a

(S30) Ice_Air_4a

Uses

Uses

constants_0, convert_0,
maths_0, flu_1, air_1, flu_2,
air_2, flu_3a, air_3a, air_3b,
liq_vap_4

constants_0, convert_0,
maths_0, air_1, ice_1, ice_2,
air_2, air_3a, air_3b, ice_vap_4
(flu_1, flu_2)

Public Routines

Public Routines

liq_air_a_from_rh_cct_si
liq_air_a_from_rh_wmo_si
liq_air_condensationpressure_si
liq_air_density_air_si
liq_air_density_liq_si
liq_air_density_vap_si
liq_air_dewpoint_si
liq_air_enthalpy_evap_si
liq_air_entropy_air_si
liq_air_icl_si
liq_air_ict_si
liq_air_massfraction_air_si
liq_air_pressure_si
liq_air_rh_cct_from_a_si
liq_air_rh_wmo_from_a_si
liq_air_temperature_si
set_it_ctrl_liq_air
set_liq_air_eq_at_a_eta
set_liq_air_eq_at_a_p
set_liq_air_eq_at_a_t
set_liq_air_eq_at_t_p

ice_air_a_from_rh_cct_si
ice_air_a_from_rh_wmo_si
ice_air_condensationpressure_si
ice_air_density_air_si
ice_air_density_ice_si
ice_air_density_vap_si
ice_air_enthalpy_subl_si
ice_air_frostpoint_si
ice_air_icl_si
ice_air_ict_si
ice_air_massfraction_air_si
ice_air_pressure_si
ice_air_rh_cct_from_a_si
ice_air_rh_wmo_from_a_si
ice_air_sublimationpressure_si
ice_air_temperature_si
set_ice_air_eq_at_a_eta
set_ice_air_eq_at_a_p
set_ice_air_eq_at_a_t
set_ice_air_eq_at_t_p
set_it_ctrl_ice_air

IOC Manuals and Guides No. 56

179

180

TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

(S31) Liq_Air_4b

(S32) Ice_Air_4b

Uses

Uses

constants_0, flu_3a, air_3a,
liq_air_4a (convert_0, maths_0,
flu_1, air_1, flu_2, air_2, air_3b,
liq_vap_4)

constants_0, convert_0, ice_1,
air_3a, ice_air_4a (maths_0,
flu_1, air_1, flu_2, ice_2, air_2,
air_3b, ice_vap_4)

Public Routines

Public Routines

liq_air_g_si
liq_air_g_cp_si
liq_air_g_density_si
liq_air_g_enthalpy_si
liq_air_g_entropy_si
liq_air_g_expansion_si
liq_air_g_kappa_t_si
liq_air_g_lapserate_si
liq_air_liquidfraction_si
liq_air_vapourfraction_si

ice_air_g_si
ice_air_g_cp_si
ice_air_g_density_si
ice_air_g_enthalpy_si
ice_air_g_entropy_si
ice_air_g_expansion_si
ice_air_g_kappa_t_si
ice_air_g_lapserate_si
ice_air_solidfraction_si
ice_air_vapourfraction_si

(S33) Liq_Air_4c

(S34) Ice_Air_4c

Uses

Uses

constants_0, air_3a, ice_liq_4,
liq_air_4a, liq_air_4b
(convert_0, maths_0, flu_1,
ice_1, air_1, flu_2, ice_2 air_2,
flu_3a, air_3b, liq_vap_4)

constants_0, convert_0,
ice_liq_4, ice_air_4b (maths_0,
flu_1, ice_1, air_1, flu_2, ice_2,
air_2, air_3a, air_3b, ice_air_4a,
ice_vap_4)

Public Routines

Public Routines

liq_air_h_si
liq_air_h_cp_si
liq_air_h_density_si
liq_air_h_kappa_s_si
liq_air_h_lapserate_si
liq_air_h_temperature_si
liq_air_potdensity_si
liq_air_potenthalpy_si
liq_air_pottemp_si
set_it_ctrl_liq_air_pottemp

ice_air_h_si
ice_air_h_cp_si
ice_air_h_density_si
ice_air_h_kappa_s_si
ice_air_h_lapserate_si
ice_air_h_temperature_si
ice_air_potdensity_si
ice_air_potenthalpy_si
ice_air_pottemp_si
set_it_ctrl_ice_air_pottemp

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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

181

Level 5 routines
(S35) Flu_IF97_5

(S36) Ice_Flu_5

(S37) Sea_5a

(S38) Air_5

Uses

Uses

Uses

Uses

constants_0

constants_0

Public Routines

Public Routines

chk_iapws97_table
fit_liq_density_if97_si
fit_liq_g_if97_si
fit_vap_density_if97_si
fit_vap_g_if97_si

fit_ice_liq_pressure_si
fit_ice_liq_temperature_si
fit_ice_vap_pressure_si

constants_0, sea_3a,
sea_3b, sea_3c (convert_0,
maths_0, flu_1, sal_1, sal_2,
flu_3a)

constants_0,
air_3b, liq_air_4a
(convert_0,
maths_0, flu_1,
flu_2, flu_3a, air_1,
air_2, air_3a,
liq_vap_4)

Public Routines
sea_alpha_ct_si
sea_alpha_pt0_si
sea_alpha_t_si
sea_beta_ct_si
sea_beta_pt0_si
sea_beta_t_si
sea_cabb_ct_si
sea_cabb_pt0_si
sea_ctmp_from_ptmp0_si
sea_ptmp0_from_ctmp_si
sea_thrmb_ct_si
sea_thrmb_pt0_si

Public Routines
air_lapserate_moist
_c100m

(S39) Liq_F03_5

(S40) OS2008_5

(S41) GSW_Library_5

(S42) Convert_5

Uses

Uses

Uses

Uses

constants_0

flu_1, flu_2,
flu_3a, ice_1, liq_vap_4,
sal_1, sal_2 (constants_0,
convert_0, maths_0)

constants_0, maths_0,
liq_f03_5, flu_1, flu_3a,
sal_1, sal_2, sea_3a,
sea_3b, sea_5a (convert_0)

constants_0,
convert_0

Public Routines
chk_iapws09_table6
fit_liq_cp_f03_si
fit_liq_density_f03_si
fit_liq_expansion_f03_si
fit_liq_g_f03_si
fit_liq_kappa_t_f03_si
fit_liq_soundspeed_f03_si

Public Routines
chk_os2008_table

Public Routines
gsw_alpha_ct
gsw_alpha_pt0
gsw_alpha_t
gsw_asal_from_psal
gsw_beta_ct
gsw_beta_pt0
gsw_beta_t
gsw_cabb_ct
gsw_cabb_pt0
gsw_cp
gsw_ctmp_from_ptmp0
gsw_dens
gsw_enthalpy
gsw_entropy
gsw_g
gsw_kappa
gsw_kappa_t
gsw_pden
gsw_psal_from_asal
gsw_ptmp
gsw_ptmp0_from_ctmp
gsw_specvol
gsw_svel
gsw_thrmb_ct
gsw_thrmb_pt0

Public Routines
cnv_pressure
cnv_salinity
cnv_temperature

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182

TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

Appendix  N:    
Gibbs-­‐‑SeaWater  (GSW)  Oceanographic  Toolbox    

  
  
  
This   Gibbs-­‐‑SeaWater   (GSW)   Oceanographic   Toolbox   (the   “GSW   Toolbox”   for   short),  
contains   the   TEOS-­‐‑10   subroutines   for   evaluating   the   thermodynamic   properties   of   pure  
water   (using   IAPWS-­‐‑09)   and   seawater   (using   IAPWS-­‐‑08   for   the   saline   part).      The   GSW  
Oceanographic  Toolbox  does  not  provide  properties  of  ice  or  of  moist  air  (these  properties  
can   be   found   in   the   SIA   library).      This   GSW   Oceanographic   Toolbox   does   not   adhere   to  
strict  basic-­‐‑SI  units  but  rather  oceanographic  units  are  adopted.    While  it  is  comfortable  for  
oceanographers  to  adopt  these  familiar  non-­‐‑basic  SI  units,  doing  so  comes  at  a  price,  since  
many   of   the   thermodynamic   expressions   demand   that   variables   be   expressed   in   basic-­‐‑SI  
units.      The   simplest   example   is   the   pure   water   fraction   (the   so-­‐‑called   “freshwater  
fraction”)  which  is   (1 − SA )   only  when  Absolute  Salinity   S A   is  in  basic-­‐‑SI  units.    The  price  
that  one  pays  for  adopting  comfortable  units  is  that  one  must  be  vigilant  when  evaluating  
thermodynamic   expressions;   there   are   traps   for   the   unwary,   particularly   concerning   the  
units  of  Absolute  Salinity  and  of  pressure.      
This   GSW   Oceanographic   Toolbox   has   inputs   in   “oceanographic”   units,   namely  
Absolute   Salinity   S A    in   g kg −1    (so   that   for   example,   Standard   Ocean   Reference   Salinity  
SSO   is  35.165  04   g kg −1   [not  0.035  165  04   kg kg −1]),  in  situ  temperature   t   in  °C  and  pressure  
as  sea  pressure   p   in  dbar.      
The  GSW  Oceanographic  Toolbox  is  designed  as  a  successor  to  the  Seawater  library  of  
MATLAB  routines  which  has  been  widely  used  by  oceanographers  in  the  past  fifteen  years;  
see   http://www.cmar.csiro.au/datacentre/ext_docs/seawater.htm. Many   of   the   non-­‐‑
thermodynamic  subroutines  of  the  Seawater  library  have  been  retained  or  updated  in  the  
GSW  Toolbox  (for  example,  a  function  to  calculate  the  square  of  the  buoyancy  frequency,  
and  functions  to  calculate  a  selection  of  different  geostrophic  streamfunctions).      
The   thermodynamic   variables   density   and   enthalpy,   and   several   thermodynamic  
variables   derived   from   density   and   enthalpy,   are   available   in   the   GSW   Toolbox   in   two  
forms.    One  form  uses  the  full  TEOS-­‐‑10  Gibbs  function  (being  the  sum  of  IAPWS-­‐‑09  and  
IAPWS-­‐‑08)   while   the   other   form   is   based   on   a   75-­‐‑term   computationally   efficient  
expression   for   specific   volume   as   a   function   of   Absolute   Salinity,   Conservative  
Temperature  and  pressure  (see  appendix  A.30  and  appendix  K).    Both  forms  give  values  
of   density   and   the   thermal   expansion   coefficient   within   the   accuracy   of   laboratory-­‐‑
determined  values  for  these  quantities,  so  that  for  oceanographic  purposes  the  two  forms  
can  be  regarded  as  equally  accurate.    Certainly,  the  present  uncertainty  in  accounting  for  
the  spatial  variations  in  seawater  composition  has  a  larger  impact  on  density  etc.  than  the  
small   difference   incurred   by   using   the   computationally   efficient   75-­‐‑term   version   for  
specific  volume.      
Version   1   of   the   GSW   Toolbox   was   released   in   January   2009,   version   2.0   in   October  
2010   and   version   3.0   in   May   2011.      The   GSW   Toolbox   is   available   in   MATLAB,   FORTRAN  
and   C   from   the   web   site   at   www.TEOS-10.org. A   quick   introduction   to   TEOS-­‐‑10   is  
available   on   the   TEOS-­‐‑10   web   site   as   the   short   document   called   “Getting   started   with  
TEOS-­‐‑10  and  the  GSW  Oceanographic  Toolbox”.    The  next  four  pages  list  all  the  functions  
in   version   3.05   of   the   GSW   Oceanographic   Toolbox   and   this   is   followed   by   Table   N.1  
which  describes  some  of  the  GSW  functions  in  more  detail.      
  

IOC Manuals and Guides No. 56

Practical Salinity from conductivity, C (incl. for SP < 2)
conductivity, C, from Practical Salinity (incl. for SP < 2)
Practical Salinity from conductivity ratio, R (incl. for SP < 2)
conductivity ratio, R, from Practical Salinity (incl. for SP < 2)
Practical Salinity from a laboratory salinometer (incl. for SP < 2)
Practical Salinity from Knudsen Salinity

Absolute Salinity from Practical Salinity
Preformed Salinity from Practical Salinity
Conservative Temperature from in-situ temperature

gsw_deltaSA_from_SP
gsw_SA_Sstar_from_SP
gsw_SR_from_SP
gsw_SP_from_SR
gsw_SP_from_SA
gsw_Sstar_from_SA
gsw_SA_from_Sstar
gsw_SP_from_Sstar
gsw_pt_from_CT
gsw_t_from_CT
gsw_CT_from_pt
gsw_pot_enthalpy_from_pt
gsw_pt_from_t
gsw_pt0_from_t
gsw_t_from_pt0
gsw_t90_from_t48
gsw_t90_from_t68
gsw_z_from_p
gsw_p_from_z
gsw_z_from_depth
gsw_depth_from_z
gsw_Abs_Pressure_from_p
gsw_p_from_Abs_Pressure
gsw_entropy_from_CT
gsw_CT_from_entropy
gsw_entropy_from_pt
gsw_pt_from_entropy
gsw_entropy_from_t
gsw_t_from_entropy
gsw_adiabatic_lapse_rate_from_CT
gsw_adiabatic_lapse_rate_from_t
gsw_molality_from_SA
gsw_ionic_strength_from_SA

Absolute Salinity Anomaly from Practical Salinity
Absolute Salinity & Preformed Salinity from Practical Salinity
Reference Salinity from Practical Salinity
Practical Salinity from Reference Salinity
Practical Salinity from Absolute Salinity
Preformed Salinity from Absolute Salinity
Absolute Salinity from Preformed Salinity
Practical Salinity from Preformed Salinity
potential temperature from Conservative Temperature
in-situ temperature from Conservative Temperature
Conservative Temperature from potential temperature
potential enthalpy from potential temperature
potential temperature
potential temperature with reference pressure of 0 dbar
in-situ temperature from potential temperature with p_ref of 0 dbar
ITS-90 temperature from IPTS-48 temperature
ITS-90 temperature from IPTS-68 temperature
height from pressure
pressure from height
height from depth
depth from height
Absolute Pressure, P, from sea pressure, p
sea pressure, p, from Absolute Pressure, P
entropy from Conservative Temperature
Conservative Temperature from entropy
entropy from potential temperature
potential temperature from entropy
entropy from in-situ temperature
in-situ temperature from entropy
adiabatic lapse rate from Conservative Temperature
adiabatic lapse rate from in-situ temperature
molality of seawater
ionic strength of seawater

other conversions between temperatures, salinities, entropy, pressure and height

gsw_SA_CT_plot
function to plot Absolute Salinity – Conservative Temperature
			 profiles on the SA-CT diagram, including the freezing line
			 and selected potential density contours

Absolute Salinity – Conservative Temperature plotting function

gsw_SA_from_SP
gsw_Sstar_from_SP
gsw_CT_from_t

Absolute Salinity (SA), Preformed Salinity (Sstar) and Conservative Temperature (CT)

gsw_SP_from_C
gsw_C_from_SP
gsw_SP_from_R
gsw_R_from_SP
gsw_SP_salinometer
gsw_SP_from_SK

Practical Salinity (SP), PSS-78
gsw_specvol
gsw_alpha
gsw_beta
gsw_alpha_on_beta
gsw_specvol_alpha_beta
gsw_specvol_first_derivatives
gsw_specvol_second_derivatives
gsw_specvol_first_derivatives_wrt_enthalpy
gsw_specvol_second_derivatives_wrt_enthalpy
gsw_specvol_anom
gsw_specvol_anom_standard
gsw_rho
gsw_rho_alpha_beta
gsw_rho_first_derivatives
gsw_rho_second_derivatives
gsw_rho_first_derivatives_wrt_enthalpy
gsw_rho_second_derivatives_wrt_enthalpy
gsw_sigma0
gsw_sigma1
gsw_sigma2
gsw_sigma3
gsw_sigma4
gsw_cabbeling
gsw_thermobaric
gsw_enthalpy
gsw_enthalpy_diff
gsw_dynamic_enthalpy
gsw_enthalpy_first_derivatives
gsw_enthalpy_second_derivatives
gsw_sound_speed
gsw_kappa
gsw_internal_energy
gsw_internal_energy_first_derivatives
gsw_internal_energy_second_derivatives
gsw_CT_from_enthalpy
gsw_SA_from_rho
gsw_CT_from_rho
gsw_CT_maxdensity

GSW version 3.05.5

specific volume
thermal expansion coefficient with respect to CT
saline contraction coefficient at constant CT
alpha divided by beta
specific volume, thermal expansion and saline contraction coefficients
first derivatives of specific volume
second derivatives of specific volume
first derivatives of specific volume with respect to enthalpy
second derivatives of specific volume with respect to enthalpy
specific volume anomaly
specific volume anomaly realtive to SSO & 0°C
in-situ density and potential density
in-situ density, thermal expansion and saline contraction coefficients
first derivatives of density
second derivatives of density
first derivatives of density with respect to enthalpy
second derivatives of density with respect to enthalpy
sigma0 with reference pressure of 0 dbar
sigma1 with reference pressure of 1000 dbar
sigma2 with reference pressure of 2000 dbar
sigma3 with reference pressure of 3000 dbar
sigma4 with reference pressure of 4000 dbar
cabbeling coefficient
thermobaric coefficient
enthalpy
difference of enthalpy between two pressures
dynamic enthalpy
first derivatives of enthalpy
second derivatives of enthalpy
sound speed
isentropic compressibility
internal energy
first derivatives of internal energy
second derivatives of internal energy
Conservative Temperature from enthalpy
Absolute Salinity from density
Conservative Temperature from density
Conservative Temperature of maximum density of seawater

specific volume, density and enthalpy

Gibbs SeaWater (GSW) Oceanographic Toolbox of TEOS –10

Page 1.

gsw_CT_freezing
gsw_CT_freezing_poly
gsw_t_freezing
gsw_t_freezing_poly
gsw_pot_enthalpy_ice_freezing
gsw_pot_enthalpy_ice_freezing_poly
gsw_SA_freezing_from_CT
gsw_SA_freezing_from_CT_poly
gsw_SA_freezing_from_t
gsw_SA_freezing_from_t_poly
gsw_pressure_freezing_CT
gsw_CT_freezing_first_derivatives
gsw_CT_freezing_first_derivatives_poly
gsw_t_freezing_first_derivatives
gsw_t_freezing_first_derivatives_poly
gsw_pot_enthalpy_ice_freezing_first_derivatives
gsw_pot_enthalpy_ice_freezing_first_derivatives_poly
gsw_latentheat_melting

Conservative Temperature freezing temp of seawater
Conservative Temperature freezing temp of seawater (poly)
in-situ freezing temperature of seawater
in-situ freezing temperature of seawater (poly)
potential enthalpy of ice at which seawater freezes
potential enthalpy of ice at which seawater freezes (poly)
SA of seawater at the freezing temp (for given CT)
SA of seawater at the freezing temp (for given CT) (poly)
SA of seawater at the freezing temp (for given t)
SA of seawater at the freezing temp (for given t) (poly)
pressure of seawater at the freezing temp (for given CT)
first derivatives of CT freezing temp of seawater
first derivatives of CT freezing temp of seawater (poly)
first derivatives of in-situ freezing temp of seawater
first derivatives of in-situ freezing temp of seawater (poly)
first derivatives of potential enthalpy of ice at freezing
first derivatives of potential enthalpy of ice at freezing (poly)
latent heat of melting of ice into seawater

seawater and ice properties at freezing temperatures

gsw_latentheat_evap_CT
latent heat of evaporation of water from seawater (isobaric
			 evaporation enthalpy) with CT as input temperature
gsw_latentheat_evap_t
latent heat of evaporation of water from seawater (isobaric
			 evaporation enthalpy) with in-situ temperature, t, as input

isobaric evaporation enthalpy

gsw_geo_strf_dyn_height
dynamic height anomaly
gsw_geo_strf_dyn_height_pc
dynamic height anomaly for piecewise constant profiles
gsw_geo_strf_isopycnal
approximate isopycnal geostrophic streamfunction
gsw_geo_strf_isopycnal_pc
approximate isopycnal geostrophic streamfunction for
			 piecewise constant profiles
gsw_geo_strf_Cunningham
Cunningham geostrophic streamfunction
gsw_geo_strf_Montgomery
Montgomery geostrophic streamfunction
gsw_geo_strf_steric_height
dynamic height anomaly divided by 9.7963 m s-2
gsw_geo_strf_PISH
pressure integrated steric height
gsw_travel_time
acoustic travel time
gsw_geostrophic_velocity
geostrophic velocity

geostrophic streamfunctions, acoustic travel time and geostrophic velocity

gsw_Turner_Rsubrho
Turner angle & Rsubrho
gsw_Nsquared
buoyancy (Brunt-Väisäla) frequency squared (N2)
gsw_Nsquared_min
minimum buoyancy frequency squared (N2)
gsw_stabilise_SA_const_t
minimally adjust SA to produce a stable water column,
			 keeping in-situ temperature constant
gsw_stabilise_SA_CT
minimally adjusts SA & CT to produce a stable water column
gsw_mlp
mixed-layer pressure
gsw_Nsquared_lowerlimit
specified profile of minimum buoyancy frequency squared
gsw_IPV_vs_fNsquared_ratio
ratio of isopycnal potential vorticity to f times N2

vertical stability

gsw_specvol_ice
gsw_alpha_wrt_t_ice
gsw_rho_ice
gsw_pressure_coefficient_ice
gsw_sound_speed_ice
gsw_kappa_ice
gsw_kappa_const_t_ice
gsw_internal_energy_ice
gsw_enthalpy_ice
gsw_entropy_ice
gsw_cp_ice
gsw_chem_potential_water_ice
gsw_Helmholtz_energy_ice
gsw_adiabatic_lapse_rate_ice
gsw_pt0_from_t_ice
gsw_pt_from_t_ice
gsw_t_from_pt0_ice
gsw_t_from_rho_ice
gsw_pot_enthalpy_from_pt_ice
gsw_pt_from_pot_enthalpy_ice
gsw_pot_enthalpy_from_pt_ice_poly
gsw_pt_from_pot_enthalpy_ice_poly
gsw_pot_enthalpy_from_specvol_ice
gsw_specvol_from_pot_enthalpy_ice
gsw_pot_enthalpy_from_specvol_ice_poly
gsw_specvol_from_pot_enthalpy_ice_poly

thermodynamic properties of ice Ih

gsw_melting_seaice_SA_CT_ratio
gsw_melting_seaice_SA_CT_ratio_poly
gsw_melting_seaice_equilibrium_SA_CT_ratio
gsw_melting_seaice_equilibrium_SA_CT_ratio_poly
gsw_seaice_fraction_to_freeze_seawater
gsw_melting_seaice_into_seawater

GSW version 3.05.5

SA to CT ratio when ice melts into seawater
SA to CT ratio when ice melts into seawater (poly)
SA to CT ratio when ice melts, near equilibrium
SA to CT ratio when ice melts, near equilibrium (poly)
ice mass fraction to freeze seawater
SA and CT when ice melts in seawater
ratios of SA, CT and P changes during frazil ice formation
ratios of SA, CT and P changes during frazil ice formation (poly)
SA, CT & ice mass fraction from bulk SA & bulk enthalpy
SA, CT & ice fraction from bulk SA & bulk potential enthalpy
SA, CT & ice fraction from bulk SA & bulk potential enthalpy (poly)

specific volume of ice
thermal expansion coefficient of ice with respect to in-situ temp
in-situ density of ice
pressure coefficient of ice
sound speed of ice (compression waves)
isentropic compressibility of ice
isothermal compressibility of ice
internal energy of ice
enthalpy of ice
entropy of ice
isobaric heat capacity of ice
chemical potential of water in ice
Helmholtz energy of ice
adiabatic lapse rate of ice
potential temperature of ice with reference pressure of 0 dbar
potential temperature of ice
in-situ temp from potential temp of ice with p_ref of 0 dbar
in-situ temp from density of ice
potential enthalpy from potential temperature of ice
potential temperature from potential enthalpy of ice
potential enthalpy from potential temperature of ice (poly)
potential temperature from potential enthalpy of ice (poly)
potential enthalpy from specific volume of ice
specific volume from potential enthalpy of ice
potential enthalpy from specific volume of ice (poly)
specific volume from potential enthalpy of ice (poly)

SA to CT ratio when sea ice melts into seawater
SA to CT ratio when sea ice melts into seawater (poly)
SA to CT ratio when sea ice melts, near equilibrium
SA to CT ratio when sea ice melts, near equilibrium (poly)
sea ice mass fraction to freeze seawater
SA and CT when sea ice melts into seawater

thermodynamic interaction between sea ice and seawater

gsw_melting_ice_SA_CT_ratio
gsw_melting_ice_SA_CT_ratio_poly
gsw_melting_ice_equilibrium_SA_CT_ratio
gsw_melting_ice_equilibrium_SA_CT_ratio_poly
gsw_ice_fraction_to_freeze_seawater
gsw_melting_ice_into_seawater
gsw_frazil_ratios_adiabatic
gsw_frazil_ratios_adiabatic_poly
gsw_frazil_properties
gsw_frazil_properties_potential
gsw_frazil_properties_potential_poly

thermodynamic interaction between ice and seawater

Gibbs SeaWater (GSW) Oceanographic Toolbox of TEOS –10

Page 2.

spiciness with reference pressure of 0 dbar
spiciness with reference pressure of 1000 dbar
spiciness with reference pressure of 2000 dbar

Celsius zero point; 273.15 K
one standard atmosphere; 101 325 Pa
Standard Ocean Reference Salinity; 35.165 04 g/kg
unit conversion factor for salinities; (35.165 04/35) g/kg
the “specific heat” for use with CT; 3991.867 957 119 63 (J/kg)/K
conductivity of SSW at SP=35, t_68=15, p=0; 42.9140 mS/cm
ratio of SP to Chlorinity; 1.80655 (g/kg) -1
valence factor of sea salt; 1.2452898
mole-weighted atomic weight of sea salt; 31.4038218... g/mol

Coriolis parameter
gravitational acceleration
spherical earth distance between points in the ocean

first derivatives of Conservative Temperature
second derivatives of Conservative Temperature
first derivatives of entropy
second derivatives of entropy
first derivatives of potential temperature
second derivatives of potential temperature

ratio of the slopes of isopycnals on the SA-CT diagram for
p & p_ref
ratio of the gradient of CT in a potential density surface to
that in the neutral tangent plane
ratio of gradients of pt & CT in a neutral tangent plane

gsw_SA_from_rho_t_exact
gsw_deltaSA_from_rho_t_exact
gsw_rho_t_exact

Absolute Salinity from density
Absolute Salinity Anomaly from density
in-situ density

laboratory functions, for use with densimeter measurements

gsw_T0
gsw_P0
gsw_SSO
gsw_uPS
gsw_cp0
gsw_C3515
gsw_SonCl
gsw_valence_factor
gsw_atomic_weight

TEOS–10 constants

gsw_f
gsw_grav
gsw_distance

planet Earth properties

gsw_CT_first_derivatives
gsw_CT_second_derivatives
gsw_entropy_first_derivatives
gsw_entropy_second_derivatives
gsw_pt_first_derivatives
gsw_pt_second_derivatives

derivatives of entropy, CT and pt

gsw_ntp_pt_vs_CT_ratio

gsw_isopycnal_vs_ntp_CT_ratio

gsw_isopycnal_slope_ratio

neutral versus isopycnal slopes and ratios

gsw_spiciness0
gsw_spiciness1
gsw_spiciness2

spiciness

GSW version 3.05.5

gsw_specvol_CT_exact
specific volume
gsw_alpha_CT_exact
thermal expansion coefficient with respect to CT
gsw_beta_CT_exact
saline contraction coefficient at constant CT
gsw_alpha_on_beta_CT_exact
alpha divided by beta
gsw_specvol_alpha_beta_CT_exact
specific volume, thermal expansion and saline
			 contraction coefficients
gsw_specvol_first_derivatives_CT_exact
first derivatives of specific volume
gsw_specvol_second_derivatives_CT_exact
second derivatives of specific volume
gsw_specvol_first_derivatives_wrt_enthalpy_CT_exact
first derivatives of specific volume with respect
			 to enthalpy
gsw_specvol_second_derivatives_wrt_enthalpy_CT_exact
second derivatives of specific volume with respect
			 to enthalpy
gsw_specvol_anom_CT_exact
specific volume anomaly
gsw_specvol_anom_standard_CT_exact
specific volume anomaly realtive to SSO & 0°C
gsw_rho_CT_exact
in-situ density and potential density
gsw_rho_alpha_beta_CT_exact
in-situ density, thermal expansion and saline
			 contraction coefficients
gsw_rho_first_derivatives_CT_exact
first derivatives of density
gsw_rho_second_derivatives_CT_exact
second derivatives of density
gsw_rho_first_derivatives_wrt_enthalpy_CT_exact
first derivatives of density with respect to enthalpy
gsw_rho_second_derivatives_wrt_enthalpy_CT_exact
second derivatives of density with respect to enthalpy
gsw_sigma0_CT_exact
sigma0 with reference pressure of 0 dbar
gsw_sigma1_CT_exact
sigma1 with reference pressure of 1000 dbar
gsw_sigma2_CT_exact
sigma2 with reference pressure of 2000 dbar
gsw_sigma3_CT_exact
sigma3 with reference pressure of 3000 dbar
gsw_sigma4_CT_exact
sigma4 with reference pressure of 4000 dbar
gsw_cabbeling_CT_exact
cabbeling coefficient
gsw_thermobaric_CT_exact
thermobaric coefficient
gsw_enthalpy_CT_exact
enthalpy
gsw_enthalpy_diff_CT_exact
difference of enthalpy between two pressures
gsw_dynamic_enthalpy_CT_exact
dynamic enthalpy
gsw_enthalpy_first_derivatives_CT_exact
first derivatives of enthalpy
gsw_enthalpy_second_derivatives_CT_exact
second derivatives of enthalpy
gsw_sound_speed_CT_exact
sound speed
gsw_kappa_CT_exact
isentropic compressibility
gsw_internal_energy_CT_exact
internal energy
gsw_internal_energy_first_derivatives_CT_exact
first derivatives of internal energy
gsw_internal_energy_second_derivatives_CT_exact
second derivatives of internal energy
gsw_CT_from_enthalpy_exact
Conservative Temperature from enthalpy
gsw_SA_from_rho_CT_exact
Absolute Salinity from density
gsw_CT_from_rho_exact
Conservative Temperature from density
gsw_CT_maxdensity_exact
Conservative Temperature of maximum density
			 of seawater

specific volume, density and enthalpy in terms of CT, based on the exact Gibbs function

Gibbs SeaWater (GSW) Oceanographic Toolbox of TEOS –10

Page 3.

argon solubility from SA and CT
argon solubility from SP and pt
helium solubility from SA and CT
helium solubility from SP and pt
krypton solubility from SA and CT
krypton solubility from SP and pt
nitrogen solubility from SA and CT
nitrogen solubility from SP and pt
neon solubility from SA and CT
neon solubility from SP and pt
oxygen solubility from SA and CT
oxygen solubility from SP and pt

gsw_specvol_t_exact
specific volume
gsw_alpha_wrt_CT_t_exact
thermal expansion coefficient with respect to Conservative
			 Temperature
gsw_alpha_wrt_pt_t_exact
thermal expansion coefficient with respect to potential temperature
gsw_alpha_wrt_t_exact
thermal expansion coefficient with respect to in-situ temperature
gsw_beta_const_CT_t_exact
saline contraction coefficient at constant Conservative Temperature
gsw_beta_const_pt_t_exact
saline contraction coefficient at constant potential temperature
gsw_beta_const_t_exact
saline contraction coefficient at constant in-situ temperature
gsw_specvol_anom_standard_t_exact
specific volume anomaly realtive to SSO & 0°C
gsw_rho_t_exact
in-situ density
gsw_pot_rho_t_exact
potential density
gsw_sigma0_pt0_exact
sigma0 from pt0 with reference pressure of 0 dbar
gsw_enthalpy_t_exact
enthalpy
gsw_dynamic_enthalpy_t_exact
dynamic enthalpy
gsw_CT_first_derivatives_wrt_t_exact
first derivatives of Conservative Temperature with respect to t
gsw_enthalpy_first_derivatives_wrt_t_exact first derivatives of enthalpy with respect to t
gsw_sound_speed_t_exact
sound speed
gsw_kappa_t_exact
isentropic compressibility
gsw_kappa_const_t_exact
isothermal compressibility
gsw_internal_energy_t_exact
internal energy
gsw_SA_from_rho_t_exact
Absolute Salinity from density
gsw_t_from_rho_exact
in-situ temperature from density
gsw_t_maxdensity_exact
in-situ temperature of maximum density of seawater
gsw_cp_t_exact
isobaric heat capacity
gsw_isochoric_heat_cap_t_exact
isochoric heat capacity
gsw_chem_potential_relative_t_exact
relative chemical potential
gsw_chem_potential_water_t_exact
chemical potential of water in seawater
gsw_chem_potential_salt_t_exact
chemical potential of salt in seawater
gsw_t_deriv_chem_potential_water_t_exact temperature derivative of chemical potential of water
gsw_dilution_coefficient_t_exact
dilution coefficient of seawater
gsw_Helmholtz_energy_t_exact
Helmholtz energy
gsw_osmotic_coefficient_t_exact
osmotic coefficient of seawater
gsw_osmotic_pressure_t_exact
osmotic pressure of seawater

basic thermodynamic properties in terms of in-situ t, based on the exact Gibbs function

gsw_Arsol
gsw_Arsol_SP_pt
gsw_Hesol
gsw_Hesol_SP_pt
gsw_Krsol
gsw_Krsol_SP_pt
gsw_N2sol
gsw_N2sol_SP_pt
gsw_Nesol
gsw_Nesol_SP_pt
gsw_O2sol
gsw_O2sol_SP_pt

dissolved gasses

GSW version 3.05.5

the TEOS-10 Gibbs function of seawater and its derivatives
the TEOS-10 Gibbs function of ice and its derivatives
Absolute Salinity Anomaly Ratio (excluding the Baltic Sea)
ratio of Absolute to Preformed Salinity, minus 1
Absolute Salinity Anomaly atlas value (excluding the Baltic Sea)
Calculates Absolute Salinity in the Baltic Sea
Calculates Practical Salinity in the Baltic Sea
“oceanographic funnel” check for the 75-term equation
entropy minus the terms that are a function of only SA
entropy_part evaluated at 0 dbar
linearly interpolates the reference cast
linearly interpolates (SA,CT,p) to the desired p
Reiniger & Ross (1968) interpolation of (SA,CT,p) to the desired p
gibbs(0,2,0,SA,t,0)
part of gibbs_ice(1,0,t,p)
part of gibbs_ice(1,0,pt0,0)
specvol(35.16504,0,p)
enthalpy(35.16504,0,p)
Hill ratio at a Practical Salinity of 2

www.TEOS–10.org

The GSW Toolbox is available from

gsw_front_page
gsw_check_functions
gsw_demo
gsw_ver
gsw_licence

documentation set
front page to the GSW Oceanographic Toolbox
checks that all the GSW functions work correctly
demonstrates many GSW functions and features
displays the GSW version number
creative commons licence for the GSW Oceanographic Toolbox

The GSW data set:
gsw_data_v3_0
This file contains:
		
(1) the global data set of Absolute Salinity Anomaly Ratio,
		
(2) the global data set of Absolute Salinity Anomaly Ref.,
		
(3) a reference cast (for the isopycnal streamfunction),
		
(4) two reference casts that are used by gsw_demo
		
(5) three vertical profiles of (SP, t, p) at known long & lat, plus the
			
outputs of all the GSW functions for these 3 profiles,
			
and the required accuracy of all these outputs.

gsw_gibbs
gsw_gibbs_ice
gsw_SAAR
gsw_Fdelta
gsw_deltaSA_atlas
gsw_SA_from_SP_Baltic
gsw_SP_from_SA_Baltic
gsw_infunnel
gsw_entropy_part
gsw_entropy_part_zerop
gsw_interp_ref_cast
gsw_linear_interp_SA_CT
gsw_rr68_interp_SA_CT
gsw_gibbs_pt0_pt0
gsw_gibbs_ice_part_t
gsw_gibbs_ice_pt0
gsw_specvol_SSO_0
gsw_enthalpy_SSO_0
gsw_Hill_ratio_at_SP2

The GSW functions call the following library functions:

Library functions of the GSW toolbox (internal functions; not intended to be called by users)

Gibbs SeaWater (GSW) Oceanographic Toolbox of TEOS –10

Page 4.

TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

187

Table  N.1.  A  selection  of  functions  in  version  3.05  of  the  GSW  Oceanographic  Toolbox.        
Thermodynamic  
Property  

Function  name  
(in  MATLAB)  

Remarks    

Absolute  Salinity S A   

gsw_SA_from_SP    

the  McDougall  et  al.  (2012)  algorithm  for   S A   using  
a  look-­‐‑up  table    

Preformed  Salinity   S*   

gsw_Sstar_from_SP      

Preformed  Salinity   S*   from  Practical  Salinity   S P     

gsw_CT_from_t      

Conservative  Temperature   Θ ,  from  

Conservative  
Temperature   Θ   
Gibbs  function   g   and  its  
1st  and  2nd  derivatives    
specific  volume   v   
density   ρ   

gsw_gibbs    
gsw_specvol_t_exact      
gsw_rho_t_exact      

( SA , t, p )     

the  sum  of  the  IAPWS-­‐‑09  and  IAPWS-­‐‑08  Gibbs  
functions,  and  the  derivatives  of  this  sum        
v SA , t, p   specific  volume  using  gsw_gibbs    

(
)
ρ ( SA , t, p ) in  situ  density  using  gsw_gibbs      

potential  density   ρ   

gsw_pot_rho_t_exact      

ρ Θ ( SA , t , p, pr )   using  gsw_gibbs    

density   ρ ,  and    
potential  density   ρ Θ     

gsw_rho(SA,CT,p)    

ρˆ SA , Θ, p ,  in  situ  density  using  the  75-­‐‑term  
expression  for  density  in  terms  of   Θ .    Potential  
density  with  respect  to  pressure   pr   is  obtained  by  
calling  gsw_rho  with  this  pressure,  obtaining  
ρ Θ = ρˆ ( SA , Θ, pr ) .      

Θ

specific  entropy   η   

gsw_entropy_from_t  
gsw_entropy_from_CT    

specific  enthalpy   h     

gsw_enthalpy    
gsw_enthalpy_t_exact    

first  order  derivatives  of  

hˆ ( S A , Θ, p )   

second  order  derivatives  of  

hˆ ( S A , Θ, p )   
sound  speed   c   

(

)

specific  entropy   η   using  gsw_gibbs  with  the  input  
temperature  either  being  in  situ  temperature  or  
Conservative  Temperature    

(

)

h SA , Θ, p   from  using  the  75-­‐‑term  expression  
for  density,  or  from   h SA , t, p   using  gsw_gibbs    

(

)

gsw_enthalpy_first_derivative
s    

ĥΘ   and   ĥS   using  the  75-­‐‑term  expression  for  
A
density  in  terms  of   Θ .        
ˆ
gsw_enthalpy_second_derivati hˆ , hˆ
ΘΘ
ΘSA   and   hSA SA   using  the  75-­‐‑term  
ves    
expression  for  density  in  terms  of   Θ .        

(

)

gsw_sound_speed    
gsw_sound_speed_CT_exact    
gsw_sound_speed_t_exact  

c SA , Θ, p   from  using  the  75-­‐‑term  expression  for  
density,  or  from  the  full  TEOS-­‐‑10  Gibbs  function  as  
either   c SA , Θ, p   or   c SA , t, p     

Conservative  
Temperature   Θ   

gsw_CT_from_pt      

Θ ( SA ,θ ) ,  found  directly  from  gsw_gibbs  .      
Here   θ   is  potential  temperature  with   pr   =  0.      

potential  temperature   θ   

gsw_pt_from_t      

θ ( SA , t, p, pr )   found  by  using  gsw_gibbs    and  by  
equating  two  values  of  entropy    

potential  temperature   θ   

gsw_pt0_from_t      

θ ( SA , t, p ) ,  a  computationally  faster  version  of  
gsw_pt_from_t    when   pr = 0   dbar.        

potential  temperature   θ   

gsw_pt_from_CT      

θ ( SA , Θ) ,  found  by  Newton_Raphson  iteration,  
being  the  inverse  function  of  gsw_CT_from_pt      

thermal  expansion  
coefficient  with  respect  to  
Θ ,   α Θ   

gsw_alpha    
gsw_alpha_wrt_CT_t_exact      

α Θ ( SA , Θ, p )   using  the  75-­‐‑term  expression  for  
Θ
density,  or   α ( S A , t , p )   from  gsw_gibbs    

saline  contraction  
coefficient  at  constant   Θ     

gsw_beta    
gsw_beta_const_CT_t_exact      

density,  thermal  
expansion  and  saline  
contraction  coefficient  

gsw_rho_alpha_beta    
  

β Θ ( SA , Θ, p )   using  the  75-­‐‑term  expression  for  
Θ
density,  or   β ( S A , t , p )   from  gsw_gibbs  
ρˆ ( SA , Θ, p ) ,   αˆ Θ ( SA , Θ, p )   and   βˆ Θ ( SA , Θ, p )   

dynamic  height  anomaly  

gsw_geo_strf_dyn_height    

(

)

(

)

using  the  75-­‐‑term  expression  for  density  in  terms  
of   Θ     

geostrophic  streamfunction  in  an  isobaric  surface    

approximate  isopycnal    
gsw_geo_strf_isopycnal  
geostrophic  streamfunction  

geostrophic  streamfunction  in  an  approximately  
neutral  surface,  see  Eqn.  (3.30.1)    

Montgomery  geostrophic  
streamfunction    

geostrophic  streamfunction  in  a  specific  volume  
anomaly  surface,  see  Eqn.  (3.28.1)    

gsw_geo_strf_Montgomery    

IOC Manuals and Guides No. 56

188

TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

Appendix  O:    
Checking  the  Gibbs  function  of  seawater  
against  the  original  thermodynamic  data    

  
  
  
One   of   the   tasks   undertaken   by   SCOR/IAPSO   Working   Group   127   was   to   verify   the  
accuracy   of   the   Feistel   (2003)   and   Feistel   (2008)   Gibbs   functions   against   the   underlying  
laboratory  data  to  which  these  Gibbs  functions  were  fitted.    This  checking  was  performed  
by  Giles  M.  Marion,  and  is  reported  here.      
  
Verification  of  the  Feistel  (2003)  Gibbs  function    
Table  9  of  Feistel  (2003)  included  a  root  mean  square  (r.m.s.)  estimate  of  the  fit  of  the  
Gibbs   function   to   the   original   experimental   data.      In   Table   O.1   here,   this   estimate   is   the  
column  labeled  “Resulting  r.m.s.”.    All  the  data  in  Table  O.1  are  from  Feistel  (2003)  except  
for   the   last   column,   where   Giles   M.   Marion   has   estimated   an   independent   “Verifying  
r.m.s.”.      
The   seawater   properties   that   were   used   to   develop   the   Feistel   (2003)   Gibbs   function  
(see   Column   1   of   Table   O.1)   included   density   ρ ,   isobaric   specific   heat   capacity   c p ,  
thermal  expansion  coefficient   α t ,  sound  speed   c,   specific  volume   v,   freezing  temperature  
tf   mixing  heat   Δh .     This  dataset  included  1834  observations.    Column  2  of  Table  O.1  are  
the   data   sources   that   are   listed   in   the   references.      The   r.m.s.   values   were   calculated   with  
the  equation:  
0.5

⎡1
2⎤
r.m.s = ⎢ ∑ ( F03 - expt.datum ) ⎥   
⎣n n
⎦

(O.1)  

where  F03  refers  to  output  of  the  FORTRAN  code  that  implements  Feistel’s  (2003)  Gibbs  
function.    In  many  cases,  the  experimental  data  had  to  be  adjusted  to  bring  this  data  into  
conformity   with   recent   definitions   of   temperature   and   the   thermal   properties   of   pure  
water  (see  Feistel  (2003)  for  the  specifics  of  the  datasets  used  and  the  internal  assumptions  
involved  in  developing  the  Gibbs  function).      
Comparisons   of   the   “Resulting”   (Feistel)   and   “Verifying”   (Marion)   columns   in   Table  
O.1   show   that   they   are   in   excellent   agreement.      The   small   differences   between   the   two  
r.m.s.  columns  are  likely  due  to  (1)  the  number  of  digits  used  in  the  calculations,  (2)  small  
variations   in   the   exact   equations   used   for   the   calculations,   or   (3)   small   errors   in   model  
inputs.    In  any  case,  these  small  differences  are  insignificant.      
There   were   two   typographical   errors   in   the   original   Table   9   of   Feistel   (2003)   in   the  
“Resulting  r.m.s.”  column.    The  original  value  for  the  PG93  dataset  was  listed  as  11.3  ppm,  
which   is   slightly   different   from   the   verifying   value   of   11.9   ppm.      A   subsequent   check  
indicated   that   this   value   should   have   been   listed   as   12.0   ppm,   which   is   in   excellent  
agreement  with  the  value  of  11.9  ppm.    Similarly,  the  original  “Resulting  r.m.s.”  value  for  
the  BDSW70  dataset  was  listed  as  0.54  J/(kg  K),  which  is  significantly  at  variance  from  the  
verifying   estimate   of   1.43   J/(kg   K).      A   subsequent   check   indicated   that   this   value   should  
have   been   listed   as   1.45   J/(kg   K),   which   is   in   excellent   agreement   with   the   independent  
estimate  of  1.43  J/(Kg  K).      
There   were   three   minor   errors   between   the   original   literature   data   and   the   Feistel  
(2003)  compilation  of  this  data.    In  the  BS70  dataset,  two   S P   columns  were  mislabeled  as  

IOC Manuals and Guides No. 56

TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

189

30.504   and   30.502,   where   the   correct   order   should   have   been   30.502   and   30.504.      In   the  
CM76  dataset,  the  correct  value  at   S P   =  20.062,   t   =  25  °C,  and  p  =  588.0  bars  should  have  
been   0.964393   kg m −3 ,   not   0.964383   kg m −3 .      These   minor   errors   are   insignificant.      The  
independent   comparisons   in   Table   O.1   verify   the   accuracy   of   the   Feistel   (2003)   Gibbs  
function.      
  
Verification  of  the  Feistel  (2008)  saline  part  of  the  Gibbs  function  of  seawater  
The   saline   Gibbs   function   Feistel   (2008)   was   designed   to   increase   the   temperature  
range  up  to  80  °C  and  the  salinity  range  up  to  120   g kg −1   (Feistel,  2008).    Table  7  of  Feistel  
(2008)   included   a   root   mean   square   (r.m.s.)   estimate   of   the   model   fit   to   the   original  
experimental   data   (see   the   column   “Resulting   r.m.s.”   in   the   attached  Table   O.2).      All   the  
data  in  this  table  are  from  the  Feistel  (2008)  paper  except  for  the  last  column,  where  Giles  
M.  Marion  has  estimated  an  independent  “Verifying  r.m.s.”.      
The  new  seawater  salinity  databases  that  were  used  to  develop  the  Feistel  (2008)  Gibbs  
function   (see   Column   1   of   Table  O.2)   included   isobaric   specific   heat   capacity   c p ,   mixing  
heat   Δh ,  freezing  point  depression   tf   water  vapour  pressure   p vap ,   and  the  Debye-­‐‑Hückel  
limiting  law   g LL .     This  salinity  dataset  included  602  observations.    Column  2  of  Table  O.2  
are  the  data  sources  that  are  listed  in  the  references.    In  many  cases,  the  experimental  data  
had   to   be   adjusted   to   bring   this   data   into   conformity   with   recent   definitions   of  
temperature   and   the   thermal   properties   of   pure   water   (see   Feistel   (2008)   for   the   specifics  
on  the  datasets  used  and  the  internal  assumptions  involved  in  model  development).      
Comparisons   of   the   “Resulting”   (Feistel)   and   “Verifying”   (Marion)   “r.m.s.”   columns  
show  that  they  are  in  excellent  agreement.    The  most  likely  explanation  for  the  few  small  
differences   is   the   number   of   digits   used   in   the   calculations.      In   general,   the   greater   the  
number  of  digits  used  in  these  calculations,  the  more  accurate  the  calculations.      
This   independent   check   reveals   that   there   are   no   significant   differences   between   the  
Feistel   and   Marion   estimations   of   r.m.s.   values   for   these   comparisons   (Table   O.2),   which  
verifies  the  accuracy  of  the  Feistel  (2008).      
  
Verification  of  the  Pure  Water  part  of  the  Feistel  (2003)  Gibbs  function  
The  pure  water  part  of  the  Feistel  (2003)  Gibbs  function  was  itself  a  fit  to  the  IAPWS-­‐‑95  
Helmholtz  function  of  pure  water  substance.    The  accuracy  of  this  fit  to  IAPWS-­‐‑95  in  the  
oceanographic   ranges   of   temperature   and   pressure   has   been   checked   independently   by  
two   members   of   the   SCOR/IAPSO   Working   Group   127   (Dan   G.   Wright   and   David   R.  
Jackett).    The  accuracy  of  this  pure  water  part  of  the  Feistel  (2003)  Gibbs  function  has  also  
been   checked   by   an   evaluation   committee   of   IAPWS   in   the   process   of   approving   the  
Feistel  (2003)  Gibbs  function  as  an  IAPWS  Release  (IAPWS-­‐‑09).    In  IAPWS-­‐‑09  it  is  shown  
that  the  pure  water  part  of  the  Feistel  (2003)  Gibbs  function  fits  the  IAPWS-­‐‑95  properties  
more  precisely  than  the  uncertainty  of  the  data  that  underlies  IAPWS-­‐‑95.    Hence  we  can  be  
totally   comfortable   with   the   use   of   the   Feistel   (2003)   Gibbs   function   to   represent   the  
properties  of  pure  liquid  water  in  the  oceanographic  ranges  of  temperature  and  pressure.      

IOC Manuals and Guides No. 56

190

TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

Table O.1. Summary of data used in the regression to determine the coefficients of the Feistel
(2003) Gibbs potential.
Quantity Source

−1

t /°C

P/MPa

#
Required
Points r.m.s.

Resulting
r.m.s.

Verifying
r.m.s.

MGW76c

0.5-40

0-40

0

122

4 ppm

4.1 ppm

4.2 ppm

ρ

PBB80

5-42

0-30

0

345

4 ppm

4.0 ppm

4.2 ppm

PG93

34-50

15-30

0

81

i

10 ppm

12.0 ppm

11.9 ppm

ii

cp

BDSW70

10-50

0-40

0

25

2 J/(kg K)

cp

MPD73

1-40

5-35

0

48

0.5 J/(kg K) 0.52 J/(kg K) 0.45 J/(kg K)

α

C78

10-30

-6-1

0.7-33

31

c

D74(I-III)

29-43

0-35

0-2

92

5 cm/s

1.7 cm/s

1.6 cm/s

c

D74(IVa-d) 29-43

0-30

0.1-5

32

5 cm/s

1.2 cm/s

1.2 cm/s

c

D74(V-VI) 33-37

0-5

0-100

128

5 cm/s

3.5 cm/s

3.5 cm/s

v

CM76

5-40

0-40

0-100

558

10 ppm

11.0 ppm

11.2 ppm

BS70

30-40

-2-30

1-100

221

4 ppm

2.6 ppm

2.6 ppm

tf

DK74

4-40

-2-0

0

32

2 J/kg

1.8 J/kg

1.9 J/kg

Δh

B68

0-33

25

0

24

4J

2.4 J

2.4 J

Δh

MHH73

1-41

0-30

0

95

0.4 J/kg

0.5 J/kg

0.5 J/kg

v

ii

(g kg )

ρ
ρ

i

SA

t

S

-6

0.6x10 K

-1

1.45 J/(kg K) 1.43 J/(kg K)

-6

0.73x10 K

-1

-6

0.74x10 K

The  original  value  in  Table  9  of  Feistel  (2003)  of  11.3  ppm  refers  to  the  specific  volume.  
The  original  value  in  Table  9  of  Feistel  (2003)  was  0.54  J/(kg  K),  which  apparently  was  a  
typographical  error.  

Table O.2. Summary of extra datasets used in the regression to determine the coefficients of
the Feistel (2008) Gibbs potential.
t /°C

P/MPa Points

Resulting
r.m.s.

Verifying
r.m.s.

11-117

2-80

0

221

3.46 J/(kg K)

3.46 J/(kg K)

MPD73

1-40

5-35

0

48

0.57 J/(kg K)

0.57 J/(kg K)

cp

MP05

1-35

10-40

0

41

1.30 J/(kg K)

1.30 J/(kg K)

Δh

B68

0-97

25

0

33

0.75 J/kg

0.75 J/kg

Δh

C70

35-36

2-25

0

19

7.2 J/kg

7.1 J/kg

Δh

MHH73

1-35

0-30

0

120

3.3 J/kg

3.3 J/kg

tf

DK74

4-40

-0.2-(-2.2)

0

32

1.6 mK

1.6 mK

tf

FM07

5-109

-0.3-(-6.9)

0

22

1.2 mK

1.0 mK

vap

R54

18-40

25

0

13

2.8 J/kg

2.8 J/kg

BSRSR74

6-70

60-80

0

32

9.1 J/kg

9.3 J/kg

F08

35

-5-95

0

21

0.091 J/kg

0.092 J/kg

Source

SA

cp

BDCW67

c Sp

Quantity

p

tboil
g

LL

(g kg )
−1

IOC Manuals and Guides No. 56

-1

TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

191

Appendix  P:  Thermodynamic  properties    

based  on   g ( SA ,t, p ), h ( SA ,η, p ), h ( SA ,θ , p )   and   ĥ( SA ,Θ, p )     

  
  
The   thermodynamic   potential   on   which   TEOS-­‐‑10   is   based   is   the   Gibbs   function   of  
seawater.    Being  a  Gibbs  function,   g ( SA , t , p )   is  naturally  a  function  of  Absolute  Salinity,  in  
situ   temperature   and   pressure.      There   are   other   choices   for   a   thermodynamic   potential.    
One  such  choice  is  enthalpy  h  as  a  function  of  Absolute  Salinity,  entropy  and  pressure,  and  

we  give  this  functional  form  for  enthalpy  a  boomerang  over  the  h  so  that   h = h SA ,η , p .     It  
proves  theoretically  convenient  to  consider  the  additional  functional  forms   h = h SA ,θ , p   
and   h = hˆ ( SA , Θ, p )   for  enthalpy.    These  two  functional  forms  do  not  constitute  a  complete  
thermodynamic   description   of   seawater   but   when   supplemented   by   the   expressions  
η = η SA ,θ    and   η = ηˆ ( SA , Θ)    for   entropy,   they   do   form   complete   thermodynamic  
potentials.    In  the  expressions   h = h SA ,θ , p   and   η = η SA ,θ   it  is  possible  to  choose  any  
fixed  reference  pressure   pr   for  the  definition  of  potential  temperature,   θ .    However  there  
is  no  advantage  to  choosing  the  reference  pressure  to  be  different  from   pr = 0   and  it  is  this  
value  that  is  taken  in  Table  P.1  and  throughout  this  appendix.    Table  P.1  lists  expressions  
for   some   common   thermodynamic   quantities   in   terms   of   these   potential   functions.      Note  
that  the  reference  pressure   pr   that  appears  in  the  last  three  columns  of  the   ρ θ   row  of  Table  
P.1  is  the  reference  pressure  of  potential  density,  not  of   θ ,  whereas  in  the  Gibbs  function  
column,  this  general  reference  pressure  must  also  be  used  to  evaluate   θ .      
In   addition   to   Table   P.1   we   have   the   following   expressions   for   the   thermobaric   and  
cabbeling  coefficients  (of  Eqns.  (3.8.1)  –  (3.9.2))    

(

(

)

(

   Tbθ

=

TbΘ =

Cbθ =

hPPθ
ĥP

A

A

ĥP

(P.1)  

A

(P.2)  

2
2


hPθ hPθ SA ⎛ hPθ ⎞ hPSA SA
vθθ
vθ vθ SA ⎛ vθ ⎞ vSA SA
−2
+⎜  ⎟
=
−2
+⎜
⎟
⎜⎝ vS ⎟⎠
⎜⎝ hPS ⎟⎠
v
vS v
v
hPS hP
hP
A
A
A
A

ρθθ
ρ
+2 θ
ρ
ρ
S

ρθ S
ρ

2

A

⎛ ρ ⎞ ρ S S
A A
−⎜ θ ⎟
,
⎜⎝ ρ S ⎟⎠
ρ
A

⎛ hˆ
hˆ
hˆ hˆPΘSA
= PΘΘ − 2 PΘ
+ ⎜ PΘ
⎜ hˆPS
hˆP
hˆPSA hˆP
⎝ A

2

⎞ hˆPS S
vˆ
vˆ vˆΘSA ⎛ vˆΘ
A A
⎟
= ΘΘ − 2 Θ
+⎜
ˆ
⎜ vˆS
⎟ hP
vˆ
vˆSA vˆ
⎝ A
⎠

⎛ ρˆ
ρˆ
ρˆ ρˆ ΘSA
= − ΘΘ + 2 Θ
− ⎜ Θ
⎜ ρˆ
ρˆ
ρˆ SA ρˆ
⎝ SA

)

)


ρ Pθ
ρθ ρ PSA
hPθ hPPSA
v Pθ
vθ v PSA
− 
=
−
=
−
+
,   
ρ
ρ S
ρ
v
vS
v
hPS hP

A

CbΘ

(

hˆPPΘ
hˆ hˆPPSA
vˆ
vˆ vˆPSA
ρˆ
ρˆ ρˆ PSA
− PΘ
= PΘ − Θ
= − PΘ + Θ
,   
vˆ
vˆSA vˆ
ρˆ
ρˆ SA ρˆ
hˆP
hˆPSA hˆP

hPθθ

= −

)

)

(

  

(P.3)  

2

⎞ vˆSA SA
⎟
⎟
vˆ
⎠

2

   (P.4)  

⎞ ρˆ SA SA
.
⎟
⎟
ˆ
ρ
⎠

Here  follows  some  interesting  expressions  that  can  be  gleaned  from  Table  P.1.      

c 0p
η̂Θ2
(T + t ) = hˆΘ = ∂h
0 

,                (T0 + θ ) =
c p 0 = hθ 0 = c pΘθ = −
,                0
η̂ΘΘ
ηˆΘ
(T0 + θ ) c0p ∂h0

()

()

,   

(P.5)  

SA , p

  

IOC Manuals and Guides No. 56

  

gT ( SA ,θ , pr ) = gT ( SA , t, p )   

g ( SA ,θ ,0 ) − (T0 + θ ) gT ( SA ,θ ,0 )

θ

Θ

µ = gSA

µ = g − SA g S

u = g − (T0 + t ) gT − ( p + P0 ) g P

f = g − ( p + P0 ) g P

c p = − (T0 + t ) gTT

h 0 = g ( SA ,θ ,0 ) − (T0 + θ ) gT ( SA ,θ ,0 )

µ

µW

u

f

cp

h0

κ

t

κ =−

IOC Manuals and Guides No. 56

t

g P−1g PP

ρ θ = ⎡⎣ gP ( SA ,θ , pr ) ⎤⎦

−1

ρ = ( gP )

ρ

ρθ

v = gP

v

A

h = g − (T0 + t ) gT

h

(

)

)


 −1 
 −1 hη2P
κ = − hP hPP + hP 
hηη
t

(

( )

()

−1
ρθ = ⎡⎣ hP ( pr ) ⎤⎦

(

η

η = − gT

η

h = h S A ,η , p

v = hP
 −1
ρ = hP

µ = hS
A



W
µ = h − η hη − SA hS
A


u = h − p + P0 hP



f = h − η hη − p + P0 hP
 
c p = hη hηη

h0 = h 0



g = h − η hη

−1

)

()

(

A

(

)

A

)

()
−1
ρθ = ⎡⎣ hP ( pr ) ⎤⎦
κ t = − hP−1 hPP − hP−1

)

(

hθ2Pηθ
hθ ηθθ − hθθ ηθ

h = h 0

0

c p = hθ ηθ2 ηθ hθθ − hθ ηθθ

(

f = h − η hθ ηθ − p + P0 hP

u = h − p + P0 hP

(

A

)

)

µW = h − η hθ ηθ − SA hS − η S hθ ηθ

A

−1

)

µ = hS − η S hθ ηθ

( )

ρ = hP

v = hP

h = h SA ,θ , p

(

η = η ( SA ,θ )

g = h − η hθ ηθ

θ;

(

(T0 + t ) = hθ ηθ
(T0 +θ ) = hθ (0) ηθ

)

Θ = h 0 c0p

()

()

(

Expressions  based  on    
h S ,θ , p   and   η S ,θ
A
A


Θ = h 0 c0p

)

)

)

g = g ( SA , t , p )

c 0p

(

(

(

Expressions  based  on    

h S A ,η , p

T0 + t = hη

T0 + θ = hη 0

g

Θ =

[this  is  an  implicit  equation  for   θ ]  

t

g ( SA , t , p )

Expressions  based  on    

)

()

Θ = ĥ 0 c0p

A

(

κ t = − hP hPP − hP

Θ ΘΘ

(hˆ ηˆ

−1

)

− hˆΘΘηˆΘ

ρ θ = ρ Θ = ⎡ hˆP ( pr )⎤
⎣
⎦
2
hˆΘPηˆΘ
ˆ −1 ˆ
ˆ −1

h0 = hˆ (0) = c0p Θ

c p = hˆΘηˆΘ2 ηˆΘhˆΘΘ − hˆΘηˆΘΘ

f = hˆ − ηˆ hˆΘ ηˆΘ − ( p + P0 ) hˆP

u = hˆ − ( p + P0 ) hˆP

A

)

µW = ĥ− η̂ ĥΘ η̂Θ − SA ĥS − η̂S ĥΘ η̂Θ

(

−1

µ = hˆSA − ηˆSA hˆΘ ηˆΘ

( )
ρ = hˆP

v = hˆP

h = hˆ ( SA , Θ, p )

η = ηˆ ( SA , Θ)

g = hˆ − ηˆ hˆΘ ηˆΘ

Θ;

(T0 + t ) = hˆΘ ηˆΘ
(T0 +θ ) = c0p ηˆΘ

Expressions  based  on    
ˆ
h ( SA , Θ, p )   and   ηˆ ( SA , Θ)

Table P.1. Expressions for various thermodynamic variables based on four different thermodynamic potentials

TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

t

192

)

  

βΘ

β

θ

βt

α

Θ

αθ

c0p

gTP gTT ( S A , θ , pr )
gP
gTT

β t = − gP−1 gPSA

g P gTT

−1
gTP ⎡ g SAT − (T0 + θ ) g SA ( SA ,θ ,0 ) ⎤
⎣
⎦
+
g P gTT

β Θ = − g P−1 g PSA

+

)

gTP ⎡⎣ g SAT − g SAT ( SA ,θ , pr )⎤⎦

β θ = − g P−1 g PSA

Θ

g
α = − TP
g P (T0 + θ ) gTT

αθ =

gTP
gP

αt =

2
gTP
gTT

− gTT gPP

αt

2
TP

(g

+

g P−1

Γ = − gTP gTT

c = g P gTT

κ =−

g P−1 g PP

Γ

c

κ

g ( SA , t , p )

Expressions  based  on    

A

A

()
()
()
()


 −1  hSAη 0
+ hP hPη 
hηη 0
 
β Θ = − hP−1 hPS
A

 −1  hSA 0
+ hP hPη 
hη 0

 
β θ = − hP−1 hPS


 −1  hSAη
+ hP hPη 
hηη

 
β t = − hP−1 hPS

()

 −1 
c0p
α = hP hPη 
hη 0
Θ

()


Γ = hPη

 −1 hPη
t
α = hP 
hηη

 −1 hPη
θ
α = hP 
hηη 0
θ θθ

( h η

)

)

hS ( 0 )
β Θ = − hP−1 hPS + hP−1hPθ  A
A
hθ ( 0 )

A

− hθθ ηθ
θ θθ

( h η

A

− hS θ ηθ
θ SAθ

( h η

A

β θ = − hP−1 hPS

+ hP−1hPθ

β t = − hP−1 hPS

()

c0p
−1 

α = hP hPθ 
hθ 0
Θ

)

)

− hθθ ηθ

ηθ2

α θ = hP−1hPθ

α t = − hP−1hPθ

Γ = hPθ ηθ

− hPP

(

c = hP


− hPP

)


c = hP

(

κ = − hP−1 hPP

)

 
κ = − hP−1 hPP

(

Expressions  based  on    
h SA ,θ , p   and   η SA ,θ

Expressions  based  on    

h S A ,η , p

−hˆPP

Θ ΘΘ

Θ SA Θ

(hˆ ηˆ
(hˆ ηˆ

)
)
ˆ

− hˆΘΘηˆΘ

− hˆSAΘηˆΘ

)

IOC Manuals and Guides No. 56

β Θ = − hˆP−1 hˆPSA

ηS Θ
β θ = − hˆP−1 hˆPSA + hˆP−1hˆPΘ A
ηˆΘΘ

+ hˆP−1hˆPΘ

β t = − hˆP−1 hˆPSA

α Θ = hˆP−1 hˆPΘ

ηˆΘ2
0
c pηˆΘΘ

− hˆΘΘηˆΘ

ηˆΘ2
Θ ΘΘ

(hˆ ηˆ

α θ = − hˆP−1hˆPΘ

α t = − hˆP−1hˆPΘ

Γ = hˆPΘ ηˆΘ

c = hˆP

κ = − hˆP−1 hˆPP

hˆ ( SA , Θ, p )   and   ηˆ ( SA , Θ)

Expressions  based  on    

Table P.1. (cont’d) Expressions for various thermodynamic variables based on four different thermodynamic potentials

TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater 193

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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

∂θ
∂Θ S

(

A

Θ, p

c 0p ηˆSA

ηˆΘ

(P.6)  

()

A

µ (0) = −
∂θ
∂SA

)2

c0p η̂ΘΘ
c0p
T0 + θ η̂ΘΘ
∂θ
1
αΘ
=
= θˆΘ =  = −
=
−
=
=
,   
∂Θ S
Θθ
cp 0
η̂Θ2
c0p
αθ
,p

= − (T0 + θ ) ηˆSA ,   

(P.7)  

(

)

2

−1
Θ
c0p η̂S Θ
T0 + θ η̂S Θ
SA
0 ∂η̂Θ
A
A
ˆ
= θS = −  = −
= cp
= −
.   
2
0
A
∂S
Θ
η̂
c
A
θ
Θ
p
Θ

∂θ
=
∂SA

(P.8)  

Θ

See  Eqn.  (A.12.6)  for  an  alternative  expression  for   θˆS .    Eqn.  (P.8)  can  also  be  written  as    
A

(

∂ (T0 + θ )

−1

) ∂S

A

Θ

= ηˆSAΘ c0p .   

(P.9)  

Now  we  consider  how  all  the  terms  in  the  last  column  of  Table  P.1  may  be  evaluated  
in   terms   of   ĥ75 SA ,Θ, p    of   Eqn.   (A.30.6);   this   being   the   expression   for   specific   enthalpy  
that  follows  from  the  75-­‐‑term  expression  for  specific  volume  as  a  function  of   ( SA , Θ, p )   as  
described   in   Eqn.   (K.1)   and   Table   K.1.      The   first   step   is   to   evaluate   θ    exactly   from   the  
following   implicit   expression   for   Θ    in   terms   of   the   Gibbs   function   at   p = 0    (see   Eqn.  
(2.12.1)),  as  discussed  in  section  3.3,    

(

)

c0p Θ = h ( SA , t =θ ,0) = g ( SA , t =θ ,0) − (T0 + θ ) gT ( SA , t = θ ,0) .  

(

)

(

(P.10)  

)

 S ,θ    and  
Next,   we   remind   ourselves   that   we   know   the   functional   forms   of   η SA ,θ ,   Θ
A
µ SA ,θ , 0    in   terms   of   the   coefficients   of   the   Gibbs   function   of   seawater   as   the   exact  
polynomial  and  logarithm  terms  given  by  (from  Eqns.  (2.10.1)  and  (2.9.6))    

(

)

η ( SA ,θ ) = − gT ( SA ,t = θ , 0 ) ,                 µ ( SA ,θ , 0 ) = g S ( SA ,t = θ , 0 ) ,  

(P.11a,b)  

A

and  Eqn.  (P.10)  is  repeated  here  emphasizing  the  functional  form  of  the  left-­‐‑hand  side,    
 S ,θ = g S ,t = θ ,0 − T + θ g S ,t = θ ,0 .  
c0 Θ
(P.12)  

(

p

)

A

(

) (

A

) T(

0

)

A

The  partial  derivatives  with  respect  to   Θ   and  with  respect  to   θ ,  both  at  constant   S A   
and   p ,  and  the  partial  derivatives  with  respect  to   S A ,  are  related  by    

Θ
1 ∂
∂
∂
∂
∂
S
= 
=
− A
,        and        
.  
(P.13a,b)  
∂Θ
∂S
∂S
Θ ∂θ
Θ ∂θ
θ

SA , p

θ

A θ,p

A Θ, p

SA , p

SA , p

Use   of   these   expressions,   acting   on   entropy   yields   (with   p = 0    everywhere,   and   using  
Eqn.  (P.7)  [or  Eqn.  (A.12.8b)]  and  Eqn.  (P.8))    
c0p
c0p
µ SA , θ , 0
η
1
, η̂ΘΘ = − 
,           η̂S = −
,  
(P.14a,b,c)  
η̂Θ =  θ ≡
2
A
Θθ T + θ
Θθ
T0 + θ
T0 + θ
0

(

η̂S

AΘ


Θ
S
= A
Θ
θ

)

c0p

(T0 +θ )2

(

,          and         η̂S

A SA

= −

(
(

)

µ S

( SA ,θ , 0)
−
(T0 +θ )

A

(Θ )

)

)

2

c0p

SA

(T0 +θ )2


Θ
θ

,  

(P.15a,b)  

in  terms  of  the  partial  derivatives  of  the  exact  polynomial  expressions  (P.11b)  and  (P.12).      
All   of   the   thermodynamic   variables   of   the   last   column   of   Table   P.1   can   now   be  
evaluated   using   the   partial   derivatives   of   ĥ75 SA ,Θ, p    and   the   exact   expressions   (P.14)  
and  (P.15)  which  are  written  in  terms  of   θ   which  is  found  from  the  exact  implicit  equation  
(P.10).      This   completes   the   discussion   of   how   ĥ75 SA ,Θ, p    can   be   used   as   an   alternative  
thermodynamic   potential   of   seawater.      The   partial   derivatives   of   entropy   in   Eqns.   (P.14)  
and   (P.15)   are   available   from   the   functions   gsw_entropy_first_derivatives   and  
gsw_entropy_second_derivatives.      The   Second   Law   constraint   on   ηˆ ( SA , Θ )    of   entropy  
2
production   for   turbulent   mixing   is   guaranteed   if   ηˆΘΘ < 0 ,   ηˆSA Θ < ηˆSA SAηˆΘΘ    and  
ηˆSA SA < 0 .    From  Eqns.  (P.14)  and  (P.15)  we  find  that  these  three  constraints  are  satisfied  iff  
    and   µ    are   positive,   and   these   two   constraints   are   the   same   as   those   of   Eqns.  
both   Θ
SA
θ
(A.16.17)  and  (A.16.18),  namely   gTT < 0   and   g SA SA > 0 ,  and  so  are  guaranteed  to  hold.        

(

)

(

)

(

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195

  
  

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Index    

  
  
  
Absolute  Pressure,  9,  73,  165  
Absolute  Salinity   SA = SAdens ,  11-­‐‑15,  76-­‐‑85,    
        112-­‐‑121,  166    
Absolute  Salinity  Anomaly,  13,  78-­‐‑85,  166    
absolute  temperature,  69    
Added-­‐‑mass  Salinity   SAadd ,  12,  79-­‐‑81,  166    
adiabatic  lapse  rate,  25    
  
Bernoulli  function,  47,  136    
boiling  temperature,  60    
Boussinesq  approximation,  117    
buoyancy  frequency,  32    
  
cabbeling  coefficient,  31,  100,  122  168,  191    
chemical  potentials,  19,  168,  192    
Chlorinity,  11,  74-­‐‑76,  165    
composition  variation,  3,  11-­‐‑15,  82-­‐‑85,    
        112-­‐‑116    
“conservative”  property,  87-­‐‑90    
Conservative  Temperature   Θ ,  4-­‐‑5,  8,  18,    
          22,  27,  106-­‐‑111,  117-­‐‑121,  167  
Coriolis  parameter,  148,  169    
Cunningham  streamfunction,  50,    
  
density,  18,  129-­‐‑133,  165-­‐‑167    
density,  75-­‐‑term  expression,  129-­‐‑131,    
        163-­‐‑164    
Density  Salinity   SAdens = SA ,  11-­‐‑15,  76-­‐‑85,    
        112-­‐‑120,  166    
dianeutral  advection,  121-­‐‑123    
dianeutral  velocity,  121-­‐‑123    
dynamic  height  anomaly,  48    
  
enthalpy,  18-­‐‑19,  20,  87-­‐‑94,  132-­‐‑133,  134,  139,    
        167,  191-­‐‑194      
enthalpy  as  thermodynamic  potential,  18,    
        19,  132-­‐‑133,  191-­‐‑194    
entropy,  20,  26,  87,  92,  95,  97,  102-­‐‑106,  167,    
        191-­‐‑194    
EOS-­‐‑80,  2,  85,  Fig.  A.5.2    
  
First  Law  of  Thermodynamics,  95-­‐‑98,    
        132-­‐‑139    
freshwater  content,  46    
freshwater  flux,  46    
freezing  temperature,  53    
fundamental  thermodynamic  relation,  87    
fugacity,  62    

geostrophic  streamfunctions,  42-­‐‑51    
Gibbs  function  of  ice  Ih,  7,  157    
Gibbs  function  of  pure  water,  15-­‐‑17,  86,    
        152-­‐‑155,  155    
Gibbs  function  of  seawater,  5,  15-­‐‑17,  86,    
        156    
gravitational  acceleration,  146    
GSW  Oceanographic  Toolbox,  183-­‐‑186    
  
haline  contraction  coefficients,  23,  32,    
        101-­‐‑102,  168  
heat  transport,  5,  27,  46-­‐‑47,  95-­‐‑100,  117-­‐‑121,    
        132-­‐‑139  
heat  diffusion,  5,  27,  46-­‐‑47,  95-­‐‑100,  117-­‐‑121,    
        132-­‐‑139    
Helmholtz  energy,  21    
Helmholtz  function  of  fluid  water,  152-­‐‑154    
Helmholtz  function  of  moist  air,  159-­‐‑162    
  
IAPSO,  3    
IAPWS,  4,    
IAPWS-­‐‑95,  15,  140,  152-­‐‑154    
IAPWS-­‐‑06,  140,  157-­‐‑158    
IAPWS-­‐‑08,  15,  140,  156    
IAPWS-­‐‑09,  15,  140,  155    
IAPWS-­‐‑10,  140,  159-­‐‑162    
internal  energy,  20,  87,  132-­‐‑139    
IPTS-­‐‑68  temperature,  3,  9,  69-­‐‑72,  147-­‐‑149    
IOC,  7,  68    
ionic  strength,  169    
isentropic  and  adiabatic    
        compressibility,  22,  32,  168    
isochoric  heat  capacity,  24,  167    
isobaric  heat  capacity,  24,  25,  92-­‐‑94,  95,    
        97,  167,  194  
isopycnal-­‐‑potential-­‐‑vorticity,  45    
isothermal  compressibility,  21,  24    
“isobaric-­‐‑conservative”  property,  87-­‐‑90    
ITS-­‐‑90  temperature,  3,  9,  69-­‐‑72,  147-­‐‑151    
  
latent  heat  of  evaporation,  60    
latent  heat  of  melting,  55    
  
material  derivatives,  117-­‐‑123    
molality,  21,  64,  169    
Montgomery  streamfunction,  49-­‐‑50,  168    
  

IOC Manuals and Guides No. 56

206

TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

Neutral  Density,  39,  127    
neutral  helicity,  30,  35-­‐‑39,  44,  130,  168    
Neutral-­‐‑Surface-­‐‑Potential-­‐‑Vorticity,  42-­‐‑45,    
        168    
neutral  tangent  plane,  29,  30,  31,  32-­‐‑39,    
        42-­‐‑44,  99,  121-­‐‑123    
nomenclature,  165-­‐‑170    
  
orthobaric  density,  128-­‐‑129    
osmotic  coefficient,  21,  65,  172    
osmotic  pressure,  65    
  
Preformed  Salinity   S* ,  12,  79-­‐‑81,  112-­‐‑118    
potential  density,  28    
potential  enthalpy   h 0 ,  27,  167    
“potential”  property,  90-­‐‑92    
potential  temperature   θ ,  26,  95,  167,    
        191-­‐‑194    
potential  vorticity,  38,  42-­‐‑45,  168    
Practical  Salinity   S P ,  9,  75,  147-­‐‑151    
pressure,  9,  73,  165-­‐‑166    
  
Absolute  Pressure,  9,  73,  165    
  
gauge  pressure,  9,  73,  166    
  
sea  pressure,  9,  73,  165    
Pressure-­‐‑Integrated  Steric  Height,  51    
  
recommended  nomenclature,  165-­‐‑170    
recommended  symbols,  165-­‐‑170    
Reference  Composition  (RC),  10-­‐‑11,  74-­‐‑81    
Reference-­‐‑Composition  Salinity   S R ,  10,    
        11,  74-­‐‑85,  112-­‐‑116    
Reference-­‐‑Composition  Salinity  Scale,  10,    
        11,  74-­‐‑85    
Reference  Density,  13    
Reference  Salinity   S R ,  10-­‐‑11,  74-­‐‑85,    
        112-­‐‑116    
relative  humidity,  62-­‐‑65    
  
saline  contraction  coefficients,  23,  32,    
        99-­‐‑100,  101-­‐‑102,  121-­‐‑129,  168  
salinity    
  
Absolute  Salinity   SA = SAdens ,  11-­‐‑15,    
                      76-­‐‑85,  112-­‐‑118,  166    
  
Added-­‐‑mass  Salinity   SAadd ,  12,  78-­‐‑81,  
                    166      
  
Density  Salinity   SAdens = SA ,  11-­‐‑15,    
                      76-­‐‑85,  112-­‐‑118,  166    
  
Practical  Salinity   S P ,  9,  76-­‐‑85,  112-­‐‑118    
  
Preformed  Salinity   S* ,  12,  78-­‐‑81,    
                    112-­‐‑118,  166    
  
Reference  Salinity   S R ,  10-­‐‑11,  74-­‐‑85,    
                    112-­‐‑118,  166    
  
Solution  Salinity   S Asoln ,  12,  78-­‐‑81,  166    
  

IOC Manuals and Guides No. 56

salinity  in  ocean  models,  112-­‐‑119    
SCOR,  3    
sea  pressure,  73,  165    
SIA  software  library,  171-­‐‑181    
SI-­‐‑traceability  of  salinity,  149-­‐‑151    
slopes  of  surfaces,  40-­‐‑42    
Solution  Salinity   S Asoln ,  12,  78-­‐‑81,  166    
sound  speed,  22  
specific  volume,  18,  29,  47-­‐‑53,  111-­‐‑112,    
        168  
specific  volume  anomaly,  29,  47-­‐‑53    
stability  ratio,  39,  40-­‐‑41,  122-­‐‑124,  166    
Standard  Seawater  (SSW),  10-­‐‑15,  74-­‐‑81    
sublimation  enthalpy,  57    
sublimation  pressure,  56    
  
75-­‐‑term  expression  for  specific  volume,    
        130-­‐‑133,  163-­‐‑164    
TEOS-­‐‑10,  4-­‐‑8,  67-­‐‑68    
TEOS-­‐‑10  web  site,  67,  142    
Temperature  
  
absolute  temperature   T ,  69    
  
Celsius  temperature   t ,  69    
  
Conservative  Temperature   Θ ,  5,  7-­‐‑8,    
                    18,  22,  23  27,  106-­‐‑111,  117-­‐‑121,  167,    
                    191-­‐‑194    
  
in  situ  temperature,  26    
  
potential  temperature   θ ,  26,  87,    
                    106-­‐‑111,  167,  191-­‐‑194      
temperature  of  maximum  density,  65-­‐‑66    
thermal  expansion  coefficients,  22,  25,  32,    
        99-­‐‑102,  121-­‐‑124,  167-­‐‑168    
“thermal  wind”,  34-­‐‑35,  84-­‐‑85,  131    
thermodynamic  potentials,  3,  132-­‐‑133,    
        191-­‐‑194    
thermobaric  coefficient,  30,  36,  44,  99,  168  
total  energy,  47,  88-­‐‑91,  135-­‐‑137    
Turner  angle,  39    
  
vapour  pressure,  59    
  
water-­‐‑mass  transformation,  123-­‐‑125    
WG127,  3    
  
  
  
  
  
  
  
  
  

TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater

207

    

Changes  made  to  this  TEOS-­‐‑10  manual,  since  the  13th  April  2010  
version  which  was  printed  by  IOC.      
17th  April  2010  

Page  102,  line  6,  an  error  in  the  inequality  fixed,  and  changed  to  be  in  terms  of   ĥ .    

4th  July  2010  

Page  39,  Eqn.  (3.20.4).    An  error  fixed  in  this  equation;  an  extra  factor  of   Rρ .    

20th  Aug  2010  

Throughout   the   document,   changed   h p    to   hP    and   so   help   to   clarify   when   pressure  
must  be  expressed  in  Pa  rather  than  dbar.      

3rd  Sept  2010  

Extensive  changes  to  page  122  (the  description  of  enthalpy  obtained  from  the  25-­‐‑term  
expression  for  density),  page  174  (the  list  of  GSW  Toolbox  function  names)  and  page  
179   (the   description   of   how   the   25-­‐‑term   expression   for   density,   along   with  
knowledge   of   the   exact   Gibbs   function   at   p   =   0   dbar,   can   be   used   as   the   full  
thermodynamic  potential  of  seawater).      

14th  Sept  2010  

Corrected  a  typo  on  the  left-­‐‑hand  side  of  Eqn.  (A.11.16);   hS   was  replaced  by   hˆS .       
A
A

22nd  Sept  2010  

Page  100.    Changes  in  Eqns.  (A.18.5)  –  (A.18.7).        

  

Page  122.    Simplified  Eqn.  (A.30.6).        

7th  Feb  2011  

Changes   to   appendices   A.5   and   A.20   concerning   the   calculation   of   the   Absolute  
Salinity   Anomaly   from   the   look-­‐‑up   table   method   of   McDougall   et   al.   (2012),   and   the  
use  of  this  changed  method  in  ocean  modelling.      

20th  March  2011   Changes  to  appendix  A.30  and  appendix  K,  replacing  the  25-­‐‑term  rational  function  for  
ρˆ ( SA , Θ, p )   with  a  48-­‐‑term  version.      
10th  May  2013  

Material  added  to  sections  3.11,  3.12,  3.13,  3.20  and  appendices  A.22  and  A.23.      

4th  May  2015  

Changes  to  appendix  A.30  and  appendix  K,  replacing  the  48-­‐‑term  rational  function  for  
v̂ SA ,Θ, p   with  a  75-­‐‑term  polynomial.      

31st  Oct  2015  

Change  to  Eqn.  (3.31.1).      

2nd  Nov  2015  

Change  in  the  definition  of   b   just  below  Eqn.  (3.20.14).      

(

)

  
  
  
  
In   this   TEOS-­‐‑10   Manual   the   text   is   10.5   pt   Palatino   linotype   at   exactly   14   pt   vertical  
spacing.    The  references  are  10  pt  Palatino  linotype  at  exactly  13  pt  vertical  spacing.      The  
text  is  both  right  and  left  justified.    Left  margin  is  3.17cm  and  the  right  margin  is  3.10cm  
respectively.      The   top   and   bottom   margins   are   both   2.05cm.      The   header   and   footer   are  
both   70%   grey.      The   MATHTYPE   size   settings   are   10.5,   7,   6,   16,   9,   75%,   100%   and   150%  
respectively.      
  
  
  
  
  

IOC Manuals and Guides No. 56

Intergovernmental Oceanographic Commission (IOC)
United Nations Educational, Scientific and Cultural Organization
1, rue Miollis
75732 Paris Cedex 15, France
Tel: +33 1 45 68 10 10
Fax: +33 1 45 68 58 12
Website: http://ioc.unesco.org



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