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  - A.1.2 ITS-90 temperature scale
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The international thermodynamic

equation of seawater – 2010:

Calculation and use of thermodynamic properties

Manuals and Guides 56

Intergovernmental Oceanographic Commission

ManualsandGuides56

IntergovernmentalOceanographicCommission

Theinternationalthermodynamic

equationofseawater–2010:

Calculation and use of thermodynamic properties



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)

iii

IOC Manuals and Guides No. 56

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

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 …….………………. 33

3.13 Neutral helicity …………….…………………………………………………..….…. 34

3.14 Neutral Density ….…………………………………………………………..….…….. 35

3.15 Stability ratio …..………………………………………………………………...…. 36

3.16 Turner angle ….………………………………………………………………………. 36

3.17 Property gradients along potential density surfaces …………………………… 36

3.18 Slopes of potential density surfaces and neutral tangent planes compared ..… 37

3.19 Slopes of in situ density surfaces and specific volume anomaly surfaces …..… 37

3.20 Potential vorticity …………………………………….………………………………. 38

3.21 Vertical velocity through the sea surface …….…………………………………… 39

3.22 Freshwater content and freshwater flux …………………………………………. 40

3.23 Heat transport …………………….………………………………………………….. 40

3.24 Geopotential ………….……………………………………………………………….. 41

3.25 Total energy …………….…………………………………………………………….. 41

3.26 Bernoulli function ……….…………………………………………………………… 42

3.27 Dynamic height anomaly …………………………………………………………… 42

3.28 Montgomery geostrophic streamfunction ………….…………………………… 43

3.29 Cunningham geostrophic streamfunction ……….……………………………… 44

3.30 Geostrophic streamfunction in an approximately neutral surface ….………… 45

3.31 Pressure-integrated steric height ……..…………………………………………… 45

3.32 Pressure to height conversion …….………………………………………………… 46

3.33 Freezing temperature ……….…………………………………………………….….. 46

3.34 Latent heat of melting ….…………………………………………..………….…… 48

3.35 Sublimation pressure …………………………………………………………….… 49

3.36 Sublimation enthalpy …………………………………………………………….… 50

3.37 Vapour pressure ……………………………………………………………….…… 52

3.38 Boiling temperature ……….……………………………………….…………….… 53

3.39 Latent heat of evaporation ………………………………………………………… 53

3.40 Relative humidity and fugacity …………………………………………………… 55

3.41 Osmotic pressure …………………………………………………………………… 58

3.42 Temperature of maximum density …….….……………………………………… 59

4. Conclusions ………………………………………….……..……………………… 60

IOC Manuals and Guides No. 56

Appendix A: Background and theory underlying the use of the

Gibbs function of seawater ………………..……………...... 62

A.1 ITS-90 temperature …………………………………………………………………... 62

A.2 Sea pressure, gauge pressure and absolute pressure …………………………….... 66

A.3 Reference Composition and the Reference-Composition Salinity Scale …….…... 67

A.4 Absolute Salinity ……………………………………………….……………………... 69

A.5 Spatial variations in seawater composition ……………………………….………... 75

A.6 Gibbs function of seawater ………………………………….…………….……..…... 78

A.7 The fundamental thermodynamic relation ………………………………….……... 79

A.8 The “conservative” and “isobaric conservative” properties ……………….…..…. 79

A.9 The “potential” property ……………….……….………………………………...... 82

A.10 Proof that

()

A,S

θη

= and

(

)

A,S

Θ=Θ .…………………….…………………... 84

A.11 Various isobaric derivatives of specific enthalpy ……………...…..…………… 84

A.12 Differential relationships between ,,

and A

S …………...……...….………. 86

A.13 The First Law of Thermodynamics ………………….……………………...…...... 87

A.14 Advective and diffusive “heat” fluxes ………………….………………..……...... 90

A.15 Derivation of the expressions for ,,

θθ

βα

and

………………….……….. 92

A.16 Non-conservative production of entropy ………………………...……………...... 94

A.17 Non-conservative production of potential temperature ………………….……... 97

A.18 Non-conservative production of Conservative Temperature ………………….. 99

A.19 Non-conservative production of density and of potential density ………….. 102

A.20 Therepresentationofsalinityinnumericaloceanmodels………………........... 103

A.21 The material derivatives of *,S A,S R

S and

in a turbulent ocean ……….... 108

A.22 The material derivatives of density and of locally-referenced

potential density; the dianeutral velocity e

 …………….……............................ 112

A.23 The water-mass transformation equation …………….…….................................. 113

A.24 Conservation equations written in potential density coordinates ………..……. 114

A.25 The vertical velocity through a general surface …………….………………….... 116

A.26 The material derivative of potential density ………………………..….……..... 116

A.27 The diapycnal velocity of layered ocean models (without rotation

of the mixing tensor) ……………………………………………….….….……..... 117

A.28 The material derivative of orthobaric density ………………………..….……..... 118

A.29 The material derivative of Neutral Density …………………………….……..... 119

A.30 Computationally efficient 25-term expressions for the density

of seawater in terms of

and

………………………..…………………….. 120

Appendix B: Derivation of the First Law of Thermodynamics ….…….…… 123

Appendix C: Publications describing the TEOS-10 thermodynamic

descriptions of seawater, ice and moist air ……...………..…..…. 130

Appendix D: Fundamental constants …….…………………..……………………... 133

IOC Manuals and Guides No. 56

Appendix E: Algorithm for calculating Practical Salinity ……….…………….. 137

E.1 Calculation of Practical Salinity in terms of K15 ….……........................................... 137

E.2 Calculation of Practical Salinity at oceanographic temperature and pressure ..... 137

E.3 Calculation of conductivity ratio R for a given Practical Salinity .......................... 138

E.4 Evaluating Practical Salinity using ITS-90 temperatures ......................................... 139

E.5 Towards SI-traceability of the measurement procedure for Practical Salinity

and Absolute Salinity .................................................................................................. 139

Appendix F: Coefficients of the IAPWS-95 Helmholtz function of

fluid water (with extension down to 50K) …………...………...… 142

Appendix G: Coefficients of the pure liquid water Gibbs function

of IAPWS-09 ………………………...……………………….…...………. 145

Appendix H: Coefficients of the saline Gibbs function for seawater

of IAPWS-08 ………………………………………………..…….………. 146

Appendix I: Coefficients of the Gibbs function of ice Ih of IAPWS-06 …... 147

Appendix J: Coefficients of the Helmholtz function of moist air

of IAPWS-10 ……………………………..…….………………………… 149

Appendix K: Coefficients of 25-term expressions for the density

of seawater in terms of

and of

…….……………………....... 153

Appendix L: Recommended nomenclature, symbols and

units in oceanography ………………………………………………… 156

Appendix M: Seawater-Ice-Air (SIA) library of computer software ……….. 162

Appendix N: Gibbs-SeaWater (GSW) Oceanographic Toolbox ..................... 173

Appendix O: Checking the Gibbs function of seawater against the

original thermodynamic data ……………………….………….…… 175

Appendix P: Thermodynamic properties based on

(

)( )

,, , , , ,gS tp hS p



()

A,,hS p

 and

(

)

ˆ,,hS pΘ …………………….…………………... 178

References ……………………………..……………………………….………….…….… 182

Index ……………………………………..……………………………….…………….….… 191

vii

IOC Manuals and Guides No. 56

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.TJMcDandDRJwishtoacknowledgepartial

financialsupportfromtheWealthfromOceansNationalFlagship.Thisworkcontributes

totheCSIROClimateChangeResearchProgram.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),CSIRO,Hobart,Australia

RainerFeistel,Leibniz‐InstitutfuerOstseeforschung,Warnemuende,Germany

DanielG.Wright+,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,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.



viii

IOC Manuals and Guides No. 56

Foreword



ThisdocumentdescribestheInternationalThermodynamicEquationOfSeawater–2010

(TEOS‐10forshort).Thisdescriptionofthethermodynamicpropertiesofseawaterand

oficeIh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.

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.

WhenreferringtotheuseofTEOS‐10,itisthepresentdocument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.].



ThisversionoftheTEOS‐10Manualincludescorrectionsupto16thOctober2010.



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

IOC Manuals and Guides No. 56

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,potentialtemperature,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.

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

IOC Manuals and Guides No. 56

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

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).

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• 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freezing,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Ocean)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A

S(rather

thanofPracticalSalinityP

S),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P

S)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A

Swhichisthesalinityargumentofthe

TEOS‐10Gibbsfunctionistheversionofabsolutesalinitythatprovidesthebestestimate

ofthedensityofseawater;anothernameforA

Sis“DensitySalinity”.

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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P

S

innationaldatabasesofoceanographicdata.Thirdly,thealgorithmsfordeterminingthe

ʺbestʺ 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,R

S(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

publicationsisacalculatedquantity,beingeitherpotentialtemperature

or

ConservativeTemperatureΘ.

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InordertoimprovethedeterminationofAbsoluteSalinityweneedtobegincollecting

andstoringvaluesofthesalinityanomalyAAR

SSS

−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AAR

SSS

−canbecalculatedusingan

inversionoftheTEOS‐10equationfordensitytodetermineA.SForcompleteness,itis

advisabletoalsoreportA

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freezing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,anew

temperaturevariable,ConservativeTemperature,canbedefinedasbeing

proportionaltopotentialenthalpyandisavaluablemeasureofthe“heat”content

perunitmassofseawaterforuseinphysicaloceanographyandinclimatestudies,

asitisapproximatelytwoordersofmagnitudemoreconservativethanpotential

temperature.

• 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(1

kg kg−,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

intermsofConservativeTemperature(ratherthanin

termsofinsitutemperature)involvingjust25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availablemainlyinMATLAB.

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

usersʹconvenience.



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1.5 A remark on units

Themostconvenientvariablesandunitsinwhichtoconductthermodynamic

investigationsareAbsoluteSalinityA

Sinunitsofkgkg‐1,AbsoluteTemperatureT(K),

andAbsolutePressure

inPa.ThesearetheparametersandunitsusedintheSIA

softwarelibrary.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weadoptC°forthe

temperatureunit,pressureisseapressureindbarandAbsoluteSalinityA

S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

()

1S−,

butitisclearthatinthisinstanceAbsoluteSalinitymustbeexpressedin1

kg kg−notin

gkg .

−Therearealso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isrecommended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A

SthanofPracticalSalinityP

S.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R

Siscalculated;ReferenceSalinitymaybetakentobesimply

proportionaltoPracticalSalinity.ReferenceSalinityhasSIunits(forexample,1

gkg

−)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effectively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

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literature,anyupdatedalgorithmtoevaluatetheAbsoluteSalinityAnomalywillbe

available(withitsversionnumber)fromwww.TEOS‐10.org.

Thestorageofsalinityinnationaldatabasesshouldcontinuetooccuras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shouldbethesalinityvariablethatis

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R

SratherthanA

S,thusavoidingtheneedtocalculatetheAbsoluteSalinity

Anomaly.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

(

)

P,,Stpwerefirstusedtocalculatepotential

temperature

andthenwatermasseswereanalyzedontheP

−

diagram.Curved

contoursofpotentialdensitycouldalsobedrawnonthissameP

−

diagram.Under

TEOS‐10,sincedensityandpotentialdensityarenownotfunctionsofPracticalSalinityP

S

butratherarefunctionsofAbsoluteSalinityA

S,itisnownotpossibletodrawisolinesof

potentialdensityonaP

−diagram.Rather,becauseofthespatialvariationsof

seawatercomposition,agivenvalueofpotentialdensitydefinesanareaontheP

−

diagram,notacurvedline.UnderTEOS‐10,theobservedvariables

(

)

P,,Stp,together

withlongitudeandlatitude,areusedtofirstformAbsoluteSalinityA

SandConservative

Temperature.ΘOceanographicwatermassesarethenanalyzedontheA

S−Θdiagram,

andpotentialdensitycontourscanalsobedrawnonthisA

−

Θdiagram,while

PreformedSalinity*

Sisthenaturalsalinityvariabletobeusedinapplicationssuchas

numericalmodelling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]. 



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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

()

(

)

68 90

/ C = 1.00024 / C .tt°°(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68

tasthetemperatureargument.Inordertousethesealgorithmswith90

tdata,

t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

lesstheAbsolutePressureofone

standardatmosphere,0101 325Pa;P≡thatis

0.pPP

≡

−(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gauge

p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

intheIAPWS

ReleasesandintheSIAlibrary);thatis,

isafunctionofp,itisnotafunctionofgauge

p.



2.3PracticalSalinity



PracticalSalinityP

SisdefinedonthePracticalSalinityScaleof1978(Unesco(1981,1983))

intermsoftheconductivityratio15

Kwhichistheelectricalconductivityofthesampleat

temperature68

t=15°Candpressureequaltoonestandardatmosphere(p=0dbarand

absolutepressure

equalto101325Pa),dividedbytheconductivityofastandard

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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−

×.When15

K=1,thePracticalSalinityP

S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68

t

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P

242.S

<PracticalSalinitiesbelow2

orabove42computedfromconductivity,asmeasuredforexampleincoastallagoons,

shouldbeevaluatedbythePSS‐78extensionsofHilletal.(1986)andPoissonand

Gadhoumi(1993).SamplesexceedingaPracticalSalinityof50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.Thisisdiscussedfurther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,

potentialtemperature.SimilarlywestronglyrecommendthatPracticalSalinityP

S

continuetobethesalinityvariablethatisstoredinsuchdatabasessinceP

S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R

S(orReference

Salinityforshort)wasdesignedbyMilleroetal.(2008b)tobethebestestimateofthe

mass‐fractionAbsoluteSalinityA

S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

242S<<)Milleroetal.(2008a)showthat

RPSP

SuS≈where 1

PS (35.165 04 35) g kgu

−

≡.(2.4.1)

IntherangeP

242S<<,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P

Sis35hasaReferenceSalinityR

Sof1

35.165 04 g kg

−

.Milleroetal.(2008a)

estimatethattheabsoluteuncertaintyinthisvalueis1

0.007 g kg

−

±.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1

0.165 04 g kg

−

salinitydifferenceisduetothiseffect.

MeasurementsofthecompositionofStandardSeawaterataPracticalSalinityP

S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itis

verystronglyrecommendedthat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userʹs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R

S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.,2010b)andmaintainsthehighaccuracyofPSS‐78necessaryfor

oceanographicapplications.TheintroductionofʺDensitySalinityʺ dens

Saddressesbothof

theseissues;itisthistypeofabsolutesalinitythatinTEOS‐10parlanceislabeledA

Sand

calledAbsoluteSalinity.InthissectionweexplainhowA

S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.(2010b)addressthissecondissuebydefining“SolutionAbsolute

Salinity”(usuallyshortenedto“SolutionSalinity”),soln

S,asthemassfractionofdissolved

non‐H2Omaterialafteraseawatersampleisbroughttotheconstanttemperature25 Ct=°

andthefixedseapressure0dbar(fixedAbsolutePressureof101325Pa).

Anothermeasureofabsolutesalinityisthe“Added‐MassSalinity”add

SwhichisR

S

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25 Ct=°

andthefixedseapressureof0dbar.Theestimationofabsolutesalinityadd

S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,add

S,isusefulforlaboratorystudiesofartificialseawateranditdiffers

fromsoln

S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A

S,soln

S,oradd

S,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soln

S,add

Sand*

Sarediscussedinmore

detailinappendicesA.4andA.20,whereapproximaterelationshipsbetweenthese

variablesanddens

SS≡arepresented,basedontheworkofPawlowiczetal.(2010)and

Wrightetal.(2010b).NotethatforasampleofStandardSeawater,allofthefivesalinity

variablesR

S,A

S,soln

S,add

Sand*

Sandareequal.

Thereisnosimplemeanstomeasureeithersoln

Soradd

S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”dens

S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25 Ct

°andatthesea

pressurep=0dbar.Whenthereisnoriskofconfusion,“DensitySalinity”isalsocalled

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IOC Manuals and Guides No. 56

AbsoluteSalinitywiththelabelA

S,thatisdens

SS≡.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,JackettandMillero(2010a)

whichprovidesanestimateofAAR

SSS

−.Thiscomputerprogramwasformedas

follows.

Inaseriesofpapers(Milleroetal.(1976a,1978,2000,2008b),McDougalletal.(2010a)),

accuratemeasurementsofthedensityofseawatersamples,alongwiththePractical

Salinityofthosesamples,gaveestimatesofAAR

SSS

−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AAR

SSS

=−(Millero

etal.(2008a)).TheMcDougalletal.(2010a)algorithmisbasedontheobservedcorrelation

betweenthisAR

SS−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

(

)

ARAAP

,,, ,SSSSS p

δφλ

=+ = (2.5.1)

Where

islatitude(degreesNorth),

islongitude(degreeseast,rangingfrom0°Eto

360°E)whilepisseapressure.

HeuristicallythedependenceofAAR

SSS

−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.(2010a)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A

totheReferenceSalinity.



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IOC Manuals and Guides No. 56



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.(2010a)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.(2010a)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AAR

SSS

=−thestandarderror(squarerootofthemeansquared

value)ofAAR

SSS

=−is0.0107gkg‐1.Thatis,the“typical”valueofAAR

SSS

=−ofthe

811samplestakentodateis0.0107gkg‐1.Thestandarderrorofthedifferencebetweenthe

measuredvaluesofAAR

SSS

=−andthevaluesevaluatedfromthecomputeralgorithm

ofMcDougalletal.(2010a)is0.0048gkg‐1.ThemaximumvaluesofAAR

SSS

=−of

approximately0.025gkg‐1occurintheNorthPacific.



Figure2(a).AbsoluteSalinityAnomalyA

atp=2000dbar.



Figure2(b).AverticalsectionofAbsoluteSalinity

AnomalyA

along180oEinthePacificOcean.



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

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IOC Manuals and Guides No. 56

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A

Sgiventheinput

variablesPracticalSalinity P

S,longitude

,latitude

andpressureisavailableat

www.TEOS‐10.org.AbsoluteSalinityisalsoavailableastheinversefunctionofdensity

()

A,,STP

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.



2.6Gibbsfunctionofseawater



TheGibbsfunctionofseawater

(

)

A,,

Stpisrelatedtothespecificenthalpyhand

entropy,

by

()

hTt

=− + where0273.15KT

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)

()

(

)

(

)

,, , ,,

S tp g tp g S tp=+ .(2.6.1)

ThesalinepartoftheGibbsfunction,S,

isvalidovertheranges0<A

S<42gkg–1,

–6.0°C<t<40°C,and4

0< 10 dbarp<,althoughitsthermalandcolligativeproperties

arevaliduptot=80°CandA

S=120gkg–1atp=0.

Thepure‐waterpartoftheGibbsfunction,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 0.0743 MPa ) CpP −

−++× °

to40°C),andinthepressurerange4

0 < 10 dbarp<.

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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,A,Sasinputrequire

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A

Sastheinputvariable(seesection2.5)but

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IOC Manuals and Guides No. 56

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A

St−boundariesofvalidity(determinedbytheonsetofprecipitation)for2008

(pCO2=385atm

)and2100(pCO2=550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0

TistheCelsiuszeropoint,273.15K)

()

(

)

120 340 A

aaTt aaTtS

⎡⎤⎡⎤

+++++

⎣⎦⎣⎦

(2.6.2)

(seeforexampleFofonoff(1962)andFeistelandHagen(1995)).Thefirsttwocoefficients

aand2

aarearbitraryconstantsofthepurewaterGibbsfunction

()

tpwhilethe

secondtwocoefficients3

aand4

aarearbitrarycoefficientsofthesalinepartoftheGibbs

function

()

A,, .

Stp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

(

)

,0tp

(2.6.3)

and

(

)

,0utp

(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

()

tpof(2.6.1)istakenfromthefittedGibbsfunctionofFeistel(2003),thetwo

arbitraryconstants1

aand2

aare(intheappropriatenon‐dimensionalform)00

and10



ofthetableinappendixGbelow.Thesevaluesof00

and10

arenotidenticaltothe

valuesinFeistel(2003)becausethepresentvalueshavebeentakenfromIAPWS‐09and

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IOC Manuals and Guides No. 56

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3

aand4

aof(2.6.2)aredeterminedbyensuring

thatthespecificenthalpyhandspecificentropy

ofasampleofstandardseawaterwith

standard‐oceanproperties1

SO SO SO

( , , ) (35.165 04 g kg , 0 C, 0 dbar)Stp −

=°arebothzero,

thatisthat

()

SO SO SO

,, 0hS t p

(2.6.5)

and

()

SO SO SO

,, 0.Stp

(2.6.6)

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

(

)

(

)

(

)

SWW

SO SO SO t t SO SO

,, , ,hS t p u tp h t p=− (2.6.7)

and

(

)

(

)

(

)

SWW

SO SO SO t t SO SO

,, , ,Stp tp tp

ηηη

=− .(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,

(

)

SO SO

,htpand

(

)

SO SO

,tp

ontheright‐handsidesof

(2.6.7)and(2.6.8),areslightlydifferentinthetwocases.Forexample

()

SO SO

,htpis

3.3 10x−1

Jkg

−fromIAPWS‐09(asdescribedinthetableofappendixG)comparedwith

theround‐offerrorof8

210x−1

Jkg

−

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

()

tp

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

()

A,,

Stp(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

hasunitsof1

Jkg

−

inboththeSIAandGSWsoftwarelibraries.



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2.7Specificvolume



ThespecificvolumeofseawatervisgivenbythepressurederivativeoftheGibbs

functionatconstantAbsoluteSalinityA

Sandinsitutemperature,tthatis

()

,, .

PST

vvStpg gP===∂∂(2.7.1)

NoticethatspecificvolumeisafunctionofAbsoluteSalinityA

SratherthanofReference

SalinityR

SorPracticalSalinityP.S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potentialtemperature

or

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,

orΘisalsoheldconstant,asfollows(fromappendixA.11)

AA A

,, ,

SS S

hP hP hP v

ηθ

∂∂ =∂∂ =∂∂ = (2.7.2)

Theuseof

intheseequationsemphasizesthatitmustbeinPa notdbar. Specific

volumevhasunitsof31

mkg

−inboththeSIAandGSWsoftwarelibraries.



2.8Density



Thedensityofseawater

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A

S

andinsitutemperature,tthatis

()()

(

)

,, .

PST

Stp g gP

ρρ

−

===∂∂(2.8.1)

NoticethatdensityisafunctionofAbsoluteSalinityA

SratherthanofReferenceSalinity

SorPracticalSalinityP.SThisisanextremelyimportantpointbecauseAbsolute

SalinityA

Sinunitsof1

gkg

−isnumericallygreaterthanPracticalSalinitybybetween

0.1651

gkg

−and0.1951

gkg

−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3

0.12 kg m−and3

0.15 kg m

−

wouldresult.

Formanytheoreticalandmodelingpurposesinoceanographyitisconvenientto

regarddensitytobeafunctionofpotentialtemperature

orConservativeTemperature

Θratherthanofinsitutemperature.tThatis,itisconvenienttoformthefollowingtwo

functionalformsofdensity,

()

(

)

,, , , ,Sp Sp

ρρ θ ρ

==Θ

(2.8.2)

where

andΘarerespectivelypotentialtemperatureandConservativeTemperature,

bothreferencedtor0dbar.p=Wewilladopttheconvention(seeTableL.2inappendix

L)thatwhenenthalpy,hspecificvolumevordensity

aretakentobefunctionsof

potentialtemperaturetheyattractanover‐tildeasinv

or,

andwhentheyaretakento

befunctionsofConservativeTemperaturetheyattractacaretasinˆ

vandˆ.

Withthis

convention,expressionsinvolvingpartialderivativessuchas(2.7.2)canbewrittenmore

compactlyas(fromappendixA.11)

PPP

hhh v

−

== ==



(2.8.3)

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

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

IOC Manuals and Guides No. 56

()

A,,hhS p



,

(

)

A,,hhS p

=and

(

)

ˆ,, .

hhS p=Θ

(2.8.4)

Density

hasunitsof3

kg m−inboththeSIAandGSWsoftwarelibraries.

Computationallyefficientexpressionsfor

(

)

ˆ,,

Θand

(

)

A,,Sp

ρθ

involving25

coefficientsareavailable(McDougalletal.(2010b))andaredescribedinappendixA.30

andappendixK.Theseexpressionscanbeintegratedwithrespecttopressuretoprovide

closedexpressionsfor

(

)

ˆ,,

hS pΘand

(

)

A,,hS p

(seeEqn.(A.30.6)).



2.9Chemicalpotentials



Asforanytwo‐componentthermodynamicsystem,theGibbsenergy,,Gofaseawater

samplecontainingthemassofwaterW

mandthemassofsaltS

mattemperaturetand

pressurepcanbewrittenintheform(LandauandLifshitz(1959),Alberty(2001),Feistel

(2008))

()

WS W S

,,,Gm m tp m m

=+(2.9.1)

wherethechemicalpotentialsofwaterinseawaterW

andofsaltinseawaterS

are

definedbythepartialderivatives

W,,mTp

∂

=∂,and

S,,

mTp

∂

=∂(2.9.2)

Identifyingabsolutesalinitywiththemassfractionofsaltdissolvedinseawater,

()

AS WS

/Smmm=+(Milleroetal.(2008a)),thespecificGibbsenergy

isgivenby

() ()

()

WSW SW

AAAA

,, 1

gS tp S S S

μμμ μμ

==−+=+−

+(2.9.3)

andisindependentofthetotalmassofthesample.Notethatthisexpressionfor

asthe

sumofawaterpartandasalinepartisnotthesameasthepurewaterandthesalinesplit

in(2.6.1)(W

isthechemicalpotentialofwaterinseawater;itdoesnotcorrespondtoa

purewatersampleasW

does).ThisGibbsenergy

isusedasthethermodynamic

potentialfunction(Gibbsfunction)forseawater.Theabovethreeequationscanbeusedto

writeexpressionsforW

andS

intermsoftheGibbsfunction

as

() ()

WS A

WAWA

,, Tp Tp

mTp

mmg gS g

gm m gS

mSmS

⎡⎤

∂+ ∂∂ ∂

⎣⎦

==++=−

∂∂∂∂

(2.9.4)

andforthechemicalpotentialofsaltinseawater,

() () ()

WS A

SASA

Tp Tp

mTp

mmg gS g

gm m g S

mSmS

⎡⎤

∂+ ∂∂ ∂

⎣⎦

==++=+−

∂∂∂∂

(2.9.5)

Therelativechemicalpotential

(commonlycalledthe“chemicalpotentialofseawater”)

followsfrom(2.9.4)and(2.9.5)as

μμμ

∂

=− =

∂(2.9.6)

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

describesthechangeofenthalpydhifat

constantpressureandentropy,asmallmassfractionofwaterisreplacedbysalt,A

d.S

Equations(2.9.4)–(2.9.6)servetodefinethethreechemicalpotentialsintermsofthe

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

IOC Manuals and Guides No. 56

Gibbsfunction

ofseawater.NotethattheweightsofthesumsthatappearinEqns.

(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soln

Sthanbydens

SS≡.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A

Sinthissectionisrecommended.

TheSIAcomputersoftwarelibrary(appendixM)predominantlyusesbasicSIunits,so

thatA

Shasunitsof1

kg kg−andS

,,g

andW

allhaveunitsof1

Jkg .

−IntheGSW

OceanographicToolbox(appendixN)A

Shasunitsof1

gkg

−

whileS

andW

allhave

unitsof1

Jg .

−Thisadoptionofoceanographic(i.e.non‐basic‐SI)unitsforA

Smeansthat

specialcareisneededinevaluatingequationssuchas(2.9.3)and(2.9.5)whereintheterm

()

1S−itisclearthatA

Smusthaveunitsof1

kg kg

−

.Theadoptionofnon‐basic‐SIunits

iscommoninoceanography,butoftencausessomedifficultiessuchasthis.



2.10Entropy



Thespecificentropyofseawater

isgivenby

()

,, .

TSp

Stp g gT

ηη

==−=−∂∂(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

hasunitsof11

Jkg K

−−

inboththeSIAandGSWsoftwarelibraries.



2.11Internalenergy



Thespecificinternalenergyofseawateruisgivenby(where0

TistheCelsiuszeropoint,

273.15Kand0101 325PaP=isthestandardatmospherepressure)

()()() () ()

A000 0

,, .

Sp ST

uuStp gTt pPv gTt pP

∂∂

==++−+=−+−+

∂∂

(2.11.1)

Thisexpressionisanexamplewheretheuseofnon‐basicSIunitspresentsaproblem,

becauseintheproduct

()

pPv−+ ,

(

)

pP P

=mustbeinPaifspecificvolumehasits

regularunitsof31

mkg

−:‐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1

Jkg

−

inboththeSIAandGSWsoftware

libraries.



2.12Enthalpy



Thespecificenthalpyofseawaterhisgivenby

()() ()

A00

,, .

hhStp gTt gTtT

∂

==++=−+

∂(2.12.1)

Specificenthalpyhhasunitsof1

Jkg

−

inboththeSIAandGSWsoftwarelibraries.



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

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2.13Helmholtzenergy



ThespecificHelmholtzenergyofseawater

isgivenby

( ) () ()

A00

,, .

f fStp g pPv g pP P

∂

==−+=−+

∂(2.13.1)

Thisexpressionisanotherexamplewheretheuseofnon‐basicSIunitspresentsaproblem,

becauseintheproduct

()

pPv−+ ,pmustbeinPaifspecificvolumehasitsregularunitsof

mkg .

−ThespecificHelmholtzenergy

hasunitsof1

Jkg

−

inboththeSIAandGSW

computersoftwarelibraries.



2.14Osmoticcoefficient



Theosmoticcoefficientofseawater

isgivenby

() ()

()

AASW0

,, .

Stp g S mRT t

φφ

−

⎛⎞

∂

⎜⎟

==−− +

⎜⎟

∂

⎝⎠

(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,

()

RT t+(Millero

andLeung(1976),FeistelandMarion(2007),Feistel(2008)),

()

(

)

(

)

ASW0

0, , , ,tp S tp m RT t

μφ

=++.(2.14.2)

Here,R=8.31447211

Jmol K

−−

istheuniversalmolargasconstant.ThemolalitySW

m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t

isdefinedby

()

ST ST

Stp v

κκ ρ

−−

∂∂

===−=−

∂∂ (2.15.1)

wherethesecondderivativeof

istakenwithrespecttopressure(inPa )atconstantA

S

and.tTheuseof

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t

producedbyboththeSIAandGSWcomputer

softwarelibraries(appendicesMandN)hasunitsof1

Pa .

−





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

IOC Manuals and Guides No. 56

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

isobtained:

()

AAAA

1111

,,,,

SSSS

TP TT PP

PTT

Stp v

PPPP

ggg

ηηθ

ρρρ

κκ ρ ρ ρ

−−−−

∂∂∂∂

== =−= =

∂∂∂∂

−

(2.16.1)

Theisentropicandisohalinecompressibility

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

produced

byboththeSIAandGSWsoftwarelibraries(appendicesMandN)hasunitsof1

Pa .

−



2.17Soundspeed



Thespeedofsoundinseawatercisgivenby

() ()

(

)

,, .

PTT TP TTPP

ccStp P gg g gg

ρρκ

−

==∂∂== −

(2.17.1)

NotethatintheseexpressionsinEqn.(2.17.1),sincesoundspeedisin1

−anddensity

hasunitsof3

kg m−itfollowsthatthepressureofthepartialderivativesmustbeinPaand

theisentropiccompressibility

musthaveunitsof1

−

.Thesoundspeedcproduced

byboththeSIAandtheGSWsoftwarelibraries(appendicesMandN)hasunitsof1

−.



2.18Thermalexpansioncoefficients



Thethermalexpansioncoefficientt

withrespecttoinsitutemperature,tis

()

,, .

tt TP

Sp Sp

Stp TvT g

αα ρ

∂∂

==−==

∂∂ (2.18.1)

Thethermalexpansioncoefficient

withrespecttopotentialtemperature,

is(see

appendixA.15)

()

(

)

,, , ,

PTT

Sp Sp

gS p

Stpp vgg

θθ

αα ρθ θ

∂∂

==−==

∂∂ (2.18.2)

wherer

pisthereferencepressureofthepotentialtemperature.TheTT

derivativeinthe

numeratorisevaluatedat

()

,,Sp

whereastheotherderivativesareallevaluatedat

()

A,, .Stp

Thethermalexpansioncoefficient

withrespecttoConservativeTemperature,Θis

(seeappendixA.15)

() ()

,, .

PTT

Sp Sp

Stp vgTg

αα ρθ

ΘΘ ∂∂

==−==−

∂Θ ∂Θ + (2.18.3)

NotethatConservativeTemperature

isdefinedonlywithrespecttoareference

pressureof0dbarsothatthe

inEqn.(2.18.3)isthepotentialtemperaturewith

0dbar.

p=Allthederivativesontheright‐handsideofEqn.(2.18.3)areevaluatedat

()

A,, .StpTheconstant0

cisdefinedinEqn.(3.3.3)below.

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2.19Salinecontractioncoefficients



Thesalinecontractioncoefficientt

(sometimesalsocalledthehalinecontraction

coefficient)atconstantinsitutemperature,tis

()

,, .

Tp Tp

Stp SvS g

ββ ρ

∂∂

===−=−

∂∂ (2.19.1)

Thesalinecontractioncoefficient

atconstantpotentialtemperature,

is(see

appendixA.15)

()

AA A

,, ,

TP ST ST TT SP

PTT

Stpp SvS

gg gg

gSp

θθ

ββ ρ

∂∂

===−

∂∂

⎡⎤

−−

⎣⎦

(2.19.2)

wherer

pisthereferencepressureof.

OneoftheA

derivativesinthenumeratoris

evaluatedat

()

,,Sp

whereasalltheotherderivativesareevaluatedat

()

A,, .Stp

Thesalinecontractioncoefficient

atconstantConservativeTemperature,Θis(see

appendixA.15)

()

() ( )

AA A

,,0

TP S T S TT S P

PTT

Stp SvS

gg gg

gT S

ββ ρ

θθ

ΘΘ

−

∂∂

===−

∂∂

⎡⎤

−+ −

⎣⎦

(2.19.3)

NotethatConservativeTemperature

isdefinedonlywithrespecttoareference

pressureof0dbarasindicatedinthisequation.TheA

derivativeinthenumeratoris

evaluatedat

()

A,,0S

whereasalltheotherderivativesareevaluatedat

()

A,, .Stp

IntheSIAcomputersoftware(appendixM)allthreesalinecontractioncoefficientsare

producedinunitsof1

kg kg−whileintheGSWOceanographicToolbox(appendixN)all

threesalinecontractioncoefficientsareproducedinunitsof1

kg g

−

consistentwiththe

preferredoceanographicunitforA

SintheGSWToolboxbeing1

gkg .

−





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

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2.20Isobaricheatcapacity



Thespecificisobaricheatcapacityp

cistherateofchangeofspecificenthalpywith

temperatureatconstantAbsoluteSalinityA

Sandpressure,psothat

() ()

,, .

pp TT

ccStp Ttg

∂

===−+

∂(2.20.1)

Theisobaricheatcapacityp

cvariesovertheA

−

Θplaneatp=0byapproximately5%,

asillustratedinFigure4.



Figure4.Contoursofisobaricspecificheatcapacityp

cofseawater

(in 11

Jkg K

−−

),Eqn.(2.20.1),atp=0.



Theisobaricheatcapacityp

chasunitsof11

Jkg K

−

−inboththeSIAandGSW

computersoftwarelibraries.



2.21Isochoricheatcapacity



Thespecificisochoricheatcapacityv

cistherateofchangeofspecificinternalenergyu

withtemperatureatconstantAbsoluteSalinityA

Sandspecificvolume,,vsothat

() ()

()

,, .

v v TT PP TP PP

ccStp Ttgg g g

∂

===−+−

∂(2.21.1)

Notethattheisochoricandisobaricheatcapacitiesarerelatedby

()

vp t

ρκ

=− andby .

=(2.21.2)

Theisochoricheatcapacityv

chasunitsof11

Jkg K

−

−inboththeSIAandGSW

computersoftwarelibraries.



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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))

()

(

)

() () () ()

()

AA A

,, ,

00 00

00 0

,,0

TT p

SS Sp

pp p

Sp S

tt ghv

Stp PP g P c

TTTT

PcS

cc c

ηη ρ

θθθαθα

ρθ

∂∂ ∂ ∂

Γ=Γ = = = − = = =

∂∂ ∂∂∂

++++

∂∂

====

∂Θ ∂Θ∂

(2.22.1)

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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

and

)ofaseawaterparcel

constantwhenitspressureischangedinanadiabaticandisohalinemanner.Thereference

pressureofthepotentialtemperature

thatappearsinthelastfourexpressionsinEqn.

(2.22.1)isr0dbar.p=

TheadiabaticlapserateΓoutputofboththeSIAandtheGSWcomputersoftware

librariesisinunitsof1

K Pa−.



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

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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r

p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

mechanical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r

pisoftenwrittenasthepressure

integraloftheadiabaticlapserate(Fofonoff(1962),(1985))

()

[]

()

Ar AA

,, , , ,, , , .

S tpp t S S tpp p dP

θθ θ

′

′′

==+Γ

∫(3.1.1)

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,

isevaluatedusingaNewton‐

Raphsoniterativesolutiontechniquetosolvethefollowingequationfor



()

(

)

Ar A

,, ,, ,Sp Stp

ηθ η

=(3.1.2)

or,intermsoftheGibbsfunction,,



()

(

)

Ar A

,, ,, .

Sp gStp

−=−(3.1.3)

ThisrelationisformallyequivalenttoEqn.(3.1.1).IntheGSWOceanographicToolbox



isfoundtomachineprecision(14

10 C

−°∼)intwoiterationsofamodifiedNewton‐Raphson

method,usingasuitableinitialvalueasdescribedbyMcDougalletal.(2010b)..

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

outputoftheSIAsoftwareisin

unitsofKwhiletheoutputfromtheGSWToolboxisinC°.



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IOC Manuals and Guides No. 56

3.2Potentialenthalpy



Potentialenthalpy0

histheenthalpythatafluidparcelwouldhaveifitspressurewere

changedtoafixedreferencepressurer

p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r

p=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))

()()()()

[]

()

() ( )

AA AA AA

,, , ,0 , ,, , ,, , ,

,, , ,

,, , , ,

h S tp hS h S hS tp vS S tpp p dP

hS tp vS p dP

θθ θ

′′ ′

=== −

′′

=−

′′

=−

′′

=−Θ

∫







(3.2.1)

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0

his

evaluatedas

()( )

(

)

(

)

(

)

AA A0A

,, ,,0 ,,0 ,,0.

hStp hS gS T gS

θθθθ

== −+ (3.2.2)



3.3ConservativeTemperature



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

accordingto

()()

(

)

(

)

0000

AAA A

,, , ,, ,

Stp S hStpc hS c

θθ

Θ=Θ= =



(3.3.1)

wherethevaluethatischosenfor0

cismotivatedintermsofpotentialenthalpyevaluated

atanAbsoluteSalinityof1

SO PS

35 35.165 04 g kgSu

−

== andat25 C

°by

()()

SO SO 11

,25 C,0 ,0 C,0 3991.867 957 119 63 J kg K ,

(25 K)

hS hS

−

⎡⎤

°− °

⎣⎦

≈(3.3.2)

notingthat

()

SO ,0 C,0dbarhS °iszeroaccordingtothewaytheGibbsfunctionisdefined

in(2.6.5).Infactweadopttheexactdefinitionfor0

ctobethe15‐digitvaluein(3.3.2),so

that

011

3991.867 957 119 63 J kg K .

−

≡(3.3.3)

WhenIAPWS‐95isusedforthepurewaterpartoftheGibbsfunction,

()

SO ,0 C,0SΘ°and

()

SO ,25 C,0SΘ°differfrom0°Cand25°Crespectivelybytheround‐offamountof

510 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,

(

)

SO ,0 C,0SΘ°differsfrom0°C

by8

8.25 10 C

−

−× °and

()

SO ,25 C,0SΘ°differsfrom25°Cby6

9.3 10 C.

−

°Overthe

temperaturerangefrom0C°to40 C°thedifferencebetweenConservativeTemperature

usingIAPWS‐95andIAPWS‐09asthepurewaterpartisnomorethan5

1.5 10 C

−

±× °,a

temperaturedifferencethatwillbeignored.

Thevalueof0

cin(3.3.3)isveryclosetotheaveragevalueofthespecificheatcapacity

cattheseasurfaceoftoday’sglobalocean.Thisvalueof0

calsocausestheaverage

valueof

−Θattheseasurfacetobeveryclosetozero.Since0

cissimplyaconstantof

proportionalitybetweenpotentialenthalpyandConservativeTemperature,itistotally

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IOC Manuals and Guides No. 56

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.

AppendixA.18outlines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A

Sandpotential

temperature

referencedto0dbarp=.Theselibrariesalsohaveanalgorithmfor

evaluatingpotentialtemperature(referencedto0 dbar )fromA

SandΘ.Thisinverse

algorithm,

()

ˆ,S

Θ,hasaninitialseedbasedonarationalfunctionapproximationand

findspotentialtemperaturetomachineprecision(14

10 C

−°∼)inoneandahalfiterationsof

amodifiedNewton‐Raphsontechnique(McDougalletal.(2010b)).



3.4Potentialdensity



Potentialdensity

isthedensitythatafluidparcelwouldhaveifitspressurewere

changedtoafixedreferencepressurer

pinanisentropicandisohalinemanner.Potential

densityreferredtoreferencepressurer

pcanbewrittenasthepressureintegralofthe

isentropiccompressibilityas

()()

[]

()

[]

()

Ar A AA AA

,, , ,, , ,, , , , ,, , , .

Stpp Stp S Stpp p S Stpp pdP

ρρρθ κθ

′

′′′′

∫(3.4.1)

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

()

[

]

()

[

]

(

)

Ar AArr AArr

,, , , ,, , , , ,, , , .

Stpp S Stpp p g S Stpp p

ρρθ θ

−

== (3.4.2)

Usingeitherofthefunctionalforms(2.8.2)forinsitudensity,thatis,either

()

A,,Sp

ρρ θ

=or

()

ˆ,, ,Sp

ρρ

=Θpotentialdensitywithrespecttoreferencepressure

p(e.g.1000dbar)canbeeasilyevaluatedas

(

)

(

)

(

)

(

)

(

)

A r A r Ar Ar A r

,, , ,, , , , , , , , ,S tpp S tpp S p S p S p

ρρρηρθρ

====Θ



(3.4.3)

wherewenotethatthepotentialtemperature

inthethirdexpressionisthepotential

temperaturewithrespectto0dbar.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

or

Θ(orindeedas

)because

,

and

areconstantduring

theisentropicandisohalinepressurechangefromptor

p;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

()

A,,hhS p

=and

()

ˆ,,hhS p=Θ

forenthalpyas(seealsoappendixP)

()()( ) ( )

Ar Ar A r A r

,, , ,, , , , , , .

Stpp Stpp hS pp hS pp

ρρ θ

−

⎡

⎤

⎡⎤

====Θ=

⎣⎦

⎣

⎦

(3.4.4)



3.5Densityanomaly



Densityanomaly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 ,

−thatis,

()()

(

)

31 3

AA A

,, ,,0 1000kgm ,,0 1000kgm .

Stp St g St

σρ

−

−−

=− = − (3.5.1)

wasusedasanapproximationto

whichavoidedthecomputationaldemandof

evaluating

.Densityanomalyt

isnotprovidedintheTEOS‐10softwarelibraries.

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3.6Potentialdensityanomaly



Potentialdensityanomaly,

or,

issimplypotentialdensityminus1000kgm–3,

()()

(

)

()

[]

()

Ar Ar Ar

AA rr

,, , ,, , ,, , 1000kgm

,, , 1000kgm

,,,,, 1000kgm.

Stpp Stpp Stpp

Stpp

gS Stppp

θθ

σσρ

Θ−

−

==−

=−

(3.6.1)

Notethatitisequallycorrecttolabelpotentialdensityanomalyas

or

Θbecauseboth

andΘareconstantduringtheisentropicandisohalinepressurechangefromptor.p



3.7Specificvolumeanomaly



Thespecificvolumeanomaly

isdefinedasthedifferencebetweenthespecificvolume

andagivenfunctionofpressure.Traditionally

hasbeendefinedas

(

)( )

(

)

AASO

,, ,, ,0C,Stp vStp vS p

=−°(3.7.1)

(wherethetraditionalvalueofPracticalSalinityof35hasbeenupdatedtoanAbsolute

Salinityof1

SO 35 35.16504 g kg

−

== inthepresentformulation).Notethatthesecond

term,

()

SO ,0 C, ,vS 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

3.11),itisadvisabletoreplacetheargumentsSO

Sand0C°withmoregeneralvalues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

()()

(

)

()

(

)

AAA A A

,, ,, , , ,, , , .

Stp vStp vStp g Stp gStp

=−= −



(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

()

ˆ,,vS pΘand

(

)

ˆ,,hS pΘandwriteaslightlydifferentdefinitionof

specificvolumeanomalyas

()()

(

)

()

(

)

AAA A A

ˆˆ

,, ,, ,, ,, ,, .

SpvSpvSphSphSp

Θ= Θ− Θ= Θ− Θ

 

  (3.7.3)

Thesamecanalsobedonewithpotentialtemperaturesothatintermsofthespecific

volume

()

A,,vS p

andenthalpy

(

)

A,,hS p

wecanwriteanotherformofthespecific

volumeanomalyas

()()

(

)

()

(

)

AAA A A

,, ,, ,, ,, ,, .

S p vS pvS p hS phS p

δθ θ θ θ θ

=−= −









 (3.7.4)

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

AA A

,, ,

SS S

hP hP hP v

ηθ

∂∂ =∂∂ =∂∂ = (3.7.5)



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

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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

thethermobariccoefficientis(McDougall(1987b))

()

bbA

TTStp PP P

θθ θθθ

θθ θ

αβ ααβ

ββ

∂∂∂

== =−

∂∂ ∂

(3.8.1)

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

()

A,, .Sp

ρρ θ

=

WithrespecttoConservativeTemperature

thethermobariccoefficientis

()

bbA

TTStp PP P

αβ ααβ

ββ

ΘΘ ΘΘΘ

ΘΘ Θ

∂∂∂

== =−

∂∂ ∂

(3.8.2)

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

()

ˆ,, .Sp

ρρ

=Θ

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Tb 2

bnn

egNKT P

−

=− ∇ ⋅∇ or

Tb 2

bnn

egNKT P

−Θ

=− ∇Θ⋅∇ wheren

∇

andn

∇

Θarethetwo‐dimensionalgradientsof

eitherpotentialtemperatureorConservativeTemperaturealongtheneutraltangent

plane,n

∇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,

cK−∇Θandisindependentoftheamountofsmall‐scale(dianeutral)turbulentmixing

andhenceisalsoindependentofthedissipationofmechanicalenergy

(Klockerand

McDougall(2010a)).ItisshowninappendixA.14belowthatwhiletheepineutral

diffusivefluxesn

−∇andn

K−∇Θaredifferent,theproductofthesefluxeswiththeir

respectivethermobariccoefficientsisthesame,thatis,bb

∇

=∇ΘHencethe

thermobaricdianeutraladvectionTb

eisthesamewhetheritiscalculatedas

bnn

NKT P

−

−∇⋅∇oras2

NKT P

−Θ

−∇Θ⋅∇Expressionsfor

and

TΘintermsof

enthalpyinthefunctionalforms

(

)

A,,hS p

and

(

)

ˆ,,hS pΘcanbefoundinappendixP.

Interestingly,forgivenmagnitudesoftheepineutralgradientsofpressureand

ConservativeTemperature,thedianeutraladvection,Tb 2

bnn

egNKT P

−Θ

−∇Θ⋅∇,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

()

bnn

Θ∇×∇Θ⋅k(seeEqn.(3.13.2)).

Thisthermobaricverticaladvectionprocess,Tb

e,isabsentfromstandardlayered

oceanmodelsinwhichtheverticalcoordinateisafunctiononlyofA

SandΘ(suchas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Tb

ethroughthesesurfaces(KlockerandMcDougall(2010a)).

InboththeSIAandGSWcomputersoftwarelibrariesthethermobaricparameteris

outputinunitsof11

K Pa

−

−.

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

IOC Manuals and Guides No. 56

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

thecabbelingcoefficientis(McDougall(1987b))

()

bbA

,, ,

,2 .

Sp p p

CCStp SS

θθθθθ

θθ

ααααβ

θββ

⎛⎞

∂∂ ∂

==+ −

⎜⎟

∂∂ ∂

⎝⎠ (3.9.1)

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

()

A,, .Sp

ρρ θ

=

WithrespecttoConservativeTemperature

thecabbelingcoefficientis

()

bbA

,, ,

,2 .

Sp p p

CCStp SS

ααααβ

ββ

ΘΘΘΘΘ

ΘΘ

⎛⎞

∂∂ ∂

==+ −

⎜⎟

∂Θ ∂ ∂

⎝⎠ (3.9.2)

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

()

ˆ,, .Sp

ρρ

=Θ

Thecabbelingdianeutraladvectionassociatedwiththelateralmixingofheatandsalt

alongneutraltangentplanesisgivenbyCab 2

bnn

egNKC

−Θ

−∇Θ⋅∇Θ(orlessaccuratelyby

Cab 2

bnn

egNKC

−

≈− ∇ ⋅∇ )wheren

∇

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−∇Θandisindependentoftheamountofsmall‐scale(dianeutral)turbulentmixing

andhenceisalsoindependentofthedissipationofmechanicalenergy(Klockerand

McDougall(2010a)).ItisshowninappendixA.14thatbbnn n n

θθ

∇

⋅∇ ≠ ∇ Θ⋅∇ 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.19.4)inappendixA.19).Thisleadstothethoughtthatifanoceanfrontis

splitupintoaseriesofmanysmallerfrontsthentheeffectsofcabbelingand

thermobaricitymightbereducedbyperhapsthesquareof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,Cab 2

egNKC

−Θ

−∇Θ⋅∇Θisproportionaltothe

scalarproductoftheepineutralfluxofheat0

−

∇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2

K−.Expressionsfor

and

intermsofenthalpyinthefunctionalforms

()

A,,hS p

and

()

ˆ,,hS pΘcanbefoundinappendixP.

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

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3.10Buoyancyfrequency



Thesquareofthebuoyancyfrequency(sometimescalledtheBrunt‐Väisäläfrequency)2

N

isgivenintermsoftheverticalgradientsofdensityandpressure,orintermsofthe

verticalgradientsofpotentialtemperatureandAbsoluteSalinity(orintermsofthe

verticalgradientsofConservativeTemperatureandAbsoluteSalinity)by(the

onthe

left‐handsideisthegravitationalacceleration,andx,yandzarethespatialCartesian

coordinates)

(

)

12 1 1 2

zz zz

zxy

NPPc

θθ

ρρ κ ρ ρ

αθ β

αβ

−− −

ΘΘ

=− + =− −

=−∂∂

=Θ−∂∂

(3.10.1)

Fortwoseawaterparcelsseparatedbyasmalldistance

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

NzP

ρρ

ΘΘ

ΔΔ

=− =

ΔΔ

(3.10.2)

where

Δ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.ThelastpartofEqn.(3.10.2)hasusedthehydrostaticrelationz

=− .



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

(

)

12 1 1 2

NPPc

θθ

ρρκ ρ ρ

αθβ

αβ

−− −

ΘΘ

=− ∇ + ∇ =− ∇ −∇

=∇−∇

=∇Θ−∇

(3.11.1)

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A

θθ

αθβ

∇

−∇isexactlyequalto

A.S

αβ

ΘΘ

∇Θ − ∇ Interestingly,both

∇

andA

∇

areindependentofthefour

arbitraryconstantsoftheGibbsfunction(seeEqn.(2.6.2))whileboth

Θ∇Θ andA

Θ∇

containanidenticaladditionalarbitrarytermproportionalto3A

∇

;termsthatexactly

cancelintheirdifference,A,S

αβ

ΘΘ

∇Θ − ∇ inEqn.(3.11.1).

Expressingthetwo‐dimensionalgradientofpropertiesintheneutraltangentplaneby

∇thepropertygradientsinaneutraltangentplaneobey

(

)

112

nn nn

θθ

ρρκ ρ ρ

αθβ

αβ

−−

ΘΘ

−∇+∇ =− ∇−∇

=∇−∇

=∇Θ−∇

(3.11.2)

Heren

∇isanexampleofaprojectednon‐orthogonalgradient

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rxy

ττ

∂∂

∇≡ + +ijk(3.11.3)

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

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r

∇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A0S

αβ

ΘΘ

Θ− Δ ≈ arealsovery

accurate.Here

Θand

Θarethevaluesofthesecoefficientsevaluatedattheaverage

valuesofA

,SΘandpoftwoparcels

(

)

A11

,,SpΘand

(

)

A22

,,SpΘona“neutralsurface”

andΔΘ andA

SΔarethepropertydifferencesbetweenthetwoparcels.Theerror

involvedwiththisfiniteamplitudeversionofEqn.(3.11.2),namely

()

,TPPd

−−Θ

∫(3.11.4)

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

(

)

0.5 .ppp=+



3.12Geostrophic,hydrostaticand“thermalwind”equations



Thegeostrophicapproximationtothehorizontalmomentumequations((B9)below)

equatestheCoriolistermtothehorizontalpressuregradientz

∇

sothatthegeostrophic

equationis

×=−∇ku or 1.

=×∇vk (3.12.1)

whereuisthethreedimensionalvelocityand

(

)

−× ×vkkuisthehorizontalvelocity

wherekistheverticalunitvector(pointingupwards)and

istheCoriolisparameter.

Thehydrostaticequationisanapproximationtotheverticalmomentumequation(see

Eqn.(B9)),namely

−(3.12.2)

Theuseof

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),usingthehydrostaticequationEqn.(3.12.2)andignoringthetinyterminz



(whichisofBoussinesqmagnitude),thethermalwindcanbewrittenas

()

zzz z n

ρρρ

=×∇ =−×∇= ×∇vk k k(3.12.3)

wherez

∇isthegradientoperatorintheexactlyhorizontaldirectionalonggeopotentials,

andthelastpartofthisequationrelatesthe“thermalwind”tothepressuregradientinthe

neutraltangentplane,thatis,effectivelytotheslopeoftheneutraltangentplane(see

McDougall(1995)).



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3.13Neutralhelicity



Neutraltangentplanes(whichdoexist)donotlinkupinspacetoformawell‐defined

neutralsurfaceunlesstheneutralhelicityn

Hiseverywherezeroonthesurface.Neutral

helicityisdefinedasthescalarproductofthevectorA

αβ

ΘΘ

∇

Θ− ∇ withitscurl,

()

(

)

HS S

αβ αβ

ΘΘ ΘΘ

≡∇Θ−∇⋅∇×∇Θ−∇(3.13.1)

andthisisproportionaltothethermobariccoefficient

oftheequationofstate

accordingto

()

zpp

HTPS

PT S

gNT P

ΘΘ

−Θ

=∇⋅∇×∇Θ

=∇×∇Θ⋅

≈∇×∇Θ⋅

(3.13.2)

wherez

issimplytheverticalgradientofpressure(1

Pa m

−

)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

∇

anda

∇

Θaretakeninan

approximatelyneutralsurface.SinceA

θθ

αθβ

∇

−∇andA

αβ

ΘΘ

∇

Θ− ∇ 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n

∇areparallelvectors)weseethatbb

θθ

∇= ∇Θa

resultthatweuseinsection3.8andinappendixA.14.Neutralhelicityhasunitsof3

−

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isproportionaltothecomponentoftheverticalshearofthe

geostrophicvelocity(,

vthe“thermalwind”)inthedirectionofthetemperaturegradient

alongtheneutraltangentplane,

∇Θsince,fromEqn.(3.12.3)andthethirdlineof(3.13.2)

wefindthat

HTf

=⋅∇Θv(3.13.3)

Intheevolutionequationofpotentialvorticitydefinedwithrespecttopotential

density

thereisthebaroclinicproductionterm2p

ρρ ρ

−

∇

⋅∇ ×∇ (Straub(1999))and

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))

(

)

PPH

ρρ ρ

−∇⋅∇×∇≈ − (3.13.4)

wherer

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,

()

σρ

∇× ∇ isgivenby(McDougallandKlocker(2010))

()

1r.

zPHPPz

σρ

−

⎛⎞

∂

∇× ∇ ⋅ = − −

⎜⎟

∂

⎝⎠

k(3.13.5)

Thefactthatthiscurlisnonzeroprovesthatageostrophicstreamfunctiondoesnotexistin

apotentialdensitysurface.

Neutralhelicityn

Halsoarisesinthecontextoffindingaclosedexpressionforthe

meanvelocityintheocean.Thecomponentofthehorizontalvelocityinthedirection

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alongacontourofΘinaneutraltangentplane,namelythevelocitycomponent

⋅×∇Θ∇Θvk isgivenby(McDougall(1995),Zikaetal.(2010a,2010b))

φρ

⊥

∇Θ −

⋅× = +

∇Θ ∇Θ

vk (3.13.6)

sothatthefullexpressionforthehorizontalvelocityis

zn n

Hv v

φρ

⊥⊥

⎧⎫

∇

Θ∇Θ

−

⎪⎪

=+×+

⎨⎬

∇

Θ∇Θ

∇Θ

⎪⎪

⎩⎭

(3.13.7)

Herez

istherateofspiraling(radianspermeter)intheverticaloftheΘcontourson

neutraltangentplanes,andv⊥isthevelocitycomponentacrosstheΘcontouronthe

neutraltangentplane(avelocitycomponentthatresultsfromirreversiblemixing

processes).Neutralhelicityarisesinthiscontextbecauseitisproportionaltothe

componentofthethermalwindvectorz

vinthedirectionacrosstheΘcontouronthe

neutraltangentplane(see(3.13.3)).Thisequation(3.13.7)fortheisopycnally‐averaged

velocityvshowsthatintheabsenceofmixingprocesses(sothat0

⊥⊥

=)andsolong

as(i)theepineutral

contoursspiralintheverticaland(ii)n

∇

Θisnotzero,thenneutral

helicityn

Hisrequiredtobenon‐zerointheoceanwhenevertheoceanisnotmotionless.

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

(

)

bnn

∇

×∇ Θ ⋅k(seeEqn.

(3.13.2)),whilethedianeutraladvectionofthermobaricity,Tb 2

bnn

egNKT P

−Θ

−∇Θ⋅∇,is

maximizedwhenn

∇Θandn

∇areparallel(seesection3.8).



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

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n

H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

(

)

na a a

zzgN S

βα

−Θ Θ

=∇ −∇ = ∇ − ∇Θs(3.14.1)

wheren

∇istheslopeoftheneutraltangentplane,a

∇

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⋅vswherevisthehorizontalvelocity(seeEqn.(A.25.4)).

SinceNeutralDensityisnotathermodynamicvariable,itwillnotbedescribedmorefully

inthismanual.



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

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3.15Stabilityratio



ThestabilityratioR

istheratiooftheverticalcontributionfromConservative

TemperaturetothatfromAbsoluteSalinitytothestaticstability2

Nofthewatercolumn.

From(3.10.1)abovewefind

() ()

RSS

ρθ

ααθ

ββ

=≈

(3.15.1)



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)

(

)

(

)

()

() ()

()

tan ,

Tu S S

θθ θθ

αβ αβ

αθ β αθ β

−Θ Θ Θ Θ

−

=Θ+ Θ−

≈+ − (3.16.1)

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A.STurneranglesgreaterthan90°orlessthan−90°

characterizeastaticallyunstablewatercolumninwhich20.N

Asacheckonthe

calculationoftheTurnerangle,notethat

(

)

tan 45 .RTu

−+°

TheTurnerangleandthe

stabilityratioareavailableintheGSWOceanographicToolboxfromthefunction

gsw_Turner_Rsubrho_CT25.



3.17Propertygradientsalongpotentialdensitysurfaces



Thetwo‐dimensionalgradientofascalar

alongapotentialdensitysurface

∇is

relatedtothecorrespondinggradientintheneutraltangentplanen

∇

by

[

]

ϕϕ

−

∇=∇+ ∇Θ

Θ⎡⎤

−

⎣⎦

(3.17.1)

(fromMcDougall(1987a)),whereristheratiooftheslopeontheA

−

Θdiagramofan

isolineofpotentialdensitywithreferencepressurer

ptotheslopeofapotentialdensity

surfacewithreferencepressurep,andisdefinedby

(

)

(

)

()()

Ar Ar

,, ,,

Sp Sp

rSp Sp

αβ

ΘΘ

=ΘΘ

.(3.17.2)

Substituting

Θinto(3.17.1)givesthefollowingrelationbetweenthe(parallel)

isopycnalandepineutralgradientsof



rR G

⎡⎤

−

⎣⎦

∇Θ= ∇Θ= ∇Θ

⎡⎤

−

⎣⎦

(3.17.3)

wherethe“isopycnaltemperaturegradientratio”

GRr

⎡

⎤

−

⎣

⎦

≡

⎡

⎤

−

⎣

⎦

(3.17.4)

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hasbeendefinedasashorthandexpressionforfutureuse.ThisratioGΘisavailablein

theGSWToolboxfromthealgorithmgsw_isopycnal_vs_ntp_CT_ratio_CT25,whilethe

ratiorofEqn.(3.17.2)isavailablethereasgsw_isopycnal_slope_ratio_CT25.

SubstitutingA

=intoEqn.(3.17.1)givesthefollowingrelationbetweenthe(parallel)

isopycnalandepineutralgradientsofA

S

AAA

SSS

⎡⎤

−

⎣⎦

∇= ∇= ∇

⎡⎤

−

⎣⎦

(3.17.5)



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r

pisgivenby

(McDougall(1988))

[]

()

[

]

nnn

nzzz z

Rr Rr

zz G

Rr rR

ρ ρ

σσ

ρ ρ

−−

∇Θ ∇Θ ∇Θ−∇Θ ∇Θ

∇−∇ = = − = =

ΘΘΘ Θ

⎡⎤ ⎡⎤

−−

⎣⎦ ⎣⎦

(3.18.1)

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

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)(Klockeretal.

(2009a)).



3.19Slopesofinsitudensitysurfacesandspecificvolumeanomalysurfaces



Thevectorslopeofaninsitudensitysurface,,

∇

isdefinedtobetheexactlyhorizontal

vector

ρρρ

∂∂

∇= + +ijk(3.19.1)

representingthe“dip”ofthesurfaceinbothhorizontaldirections(notethatheight

is

definedpositiveupwards).Thisvectorslopecanberelatedtothe(verysmall)slopeof

isobaricsurfaceby(

hereisthegravitationalacceleration)(McDougall(1989))

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

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()

pnp

zz zz N

−

⎡

⎤

∇−∇ =∇−∇ +

⎢

⎥

⎣

⎦(3.19.2)

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

22 52

4.3 10

cxs

−−

≈,theinsitudensitysurfacehasasimilarslopetotheneutraltangent

plane.

∇Inthedeepocean2

Nisonlyabout1%of22

c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252

10 sN

−

≈,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”.

Theslopeofaspecificvolumeanomalysurface,,

∇

canbeexpressedas

()

22 22

pnp

gc gc

zz zz NN

−

⎡

⎤

∇−∇ =∇−∇ + −

⎢

⎥

⎣

⎦



(3.19.3)

wherec



isthesoundspeedofthereferenceparcel

(

)

A,S



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2

ρδ

−

∂

∂(fromsection7ofMcDougall

(1989)where

isspecificvolumeanomaly).



3.20Potentialvorticity



PlanetarypotentialvorticityistheCoriolisparameter

timestheverticalgradientofa

suitablevariable.Potentialdensityissometimesusedforthatvariablebutusingpotential

density(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.(3.13.4).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.ThisplanetarypotentialvorticityvariableiscalledNeutral‐Surface‐Potential‐

Vorticity(NSPV forshort)andisrelatedto2

Nby

(

)

{

}

1n 2 22

ans

exp .

zaPa

NSPV g f fN g N T P d

ργ ρ

−−Θ

≡− ≈ − ∇Θ−Θ ∇ ⋅

∫l(3.20.1)

TheexponentialexpressionwasderivedbyMcDougall(1988)(hisequation(47))andis

approximatebecausethevariationofthesalinecontractioncoefficient

Θwithpressure

wasneglectedincomparisonwiththelargerproportionalchangeinthethermalexpansion

coefficient

Θwithpressure.TheintegralinEqn.(3.20.1)istakenalonganapproximately

neutralsurfacefromalocationwhereNSPV isequalto2.

NInterestinglythe

combinationaPa

∇Θ−Θ∇ issimplytheisobaricgradientofConservativeTemperature,

∇Θwhichisalmostthesameasthehorizontalgradient,.

∇

ΘAmoreaccurateversion

ofthisequationwhichdoesnotignorethevariationofthesalinecontractioncoefficient

canbeshowntobe

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

IOC Manuals and Guides No. 56

(

)

{

}

{}

1n 2 2 2

ans

222

ans

exp ( ) ( )

exp ( ) .

zPPPP

NSPV g f fN g N S d

fN g N d

ργ ρα ρβ

ρκ

−−ΘΘ

−

≡− = − ∇ Θ− ∇ ⋅

=∇⋅

∫

l(3.20.2)

TheexponentialfactorinEqn.(3.20.2)isapproximatelytheintegratingfactorb,defined

asn()

lll

ρρρ

≡∇ ⋅∇ ∇ ⋅∇ whereA

()

ρρβ α

ΘΘ

∇

≡∇−∇Θ,whichallowsspatialintegrals

ofn

()

bS b

βα ργ

ΘΘ

∇− ∇Θ=∇≈∇tobeapproximatelyindependentofpathfor

“verticalpaths”,thatis,forpathsinsurfaceswhosenormalhaszeroverticalcomponent.

Thegradienta

∇of2

NisrelatedtothatofNSPV by(fromEqns.(3.20.2)and(3.20.1))

()

(

)

22222

ln ln ( ) .

aa P aPa

NNSPVgN gNT P

ρκ ρ

−−Θ

∇−∇ =−∇≈ ∇Θ−Θ∇(3.20.3)

Thedeficienciesof2

Nasaformofplanetarypotentialvorticityhavenotbeenwidely

appreciated.Eveninalake,andalsointhesimplesituationwheretemperaturedoesnot

varyalongadensitysurface(a

∇Θ=0),theuseof2

Nasplanetarypotentialvorticityis

inaccuratesincetheright‐handsideof(3.20.3)isthenapproximately

NT P T P

ρα

−Θ Θ

−

Θ∇ = ∇

⎡⎤

−

⎣⎦

,(3.20.4)

andthemerefactthatthedensitysurfacehasaslope(i.e.aP

∇

≠0)meansthatthe

contoursof2

NwillnotbeparalleltocontoursofNSPV onthedensitysurface.(Inthis

situation(wherea

∇Θ=0)thecontoursofNSPV alongapproximatelyneutralsurfaces

coincidewiththoseofisopycnal‐potential‐vorticity(

PV ),thepotentialvorticitydefined

withrespecttotheverticalgradientofpotentialdensityby1z

IPV fg

−

=− ).

PV isrelatedto2

Nby(McDougall(1988))

()

1rr

111

zRr

IPV g

fN N p p G G

ββ

ρρ

ββ

ΘΘ

−Θ

ΘΘΘΘ

⎡⎤

−

−⎣⎦

≡= = ≈

⎡⎤

−

⎣⎦

(3.20.5)

sothattheratioofNSPV to

PV plottedonanapproximatelyneutralsurfaceisgivenby

()

{}

ans

1exp ( ) .

NSPV

IPV pRr

βρκ

Θ−

⎡⎤

−

⎣⎦

=∇⋅

⎡⎤

−

⎣⎦

∫l(3.20.6)

YouandMcDougall(1991)showthatbecauseofthehighlydifferentiatednatureof

potentialvorticity,isolinesof

PV andNSPV donotcoincideevenatthereference

pressurer

pofthepotentialdensityvariable(seeequations(14)–(16)andFigure14of

thatpaper). NSPV ,2

Nand

PV havetheunits3

−

Theratio2

PV fN isavailablein

theGSWOceanographicToolboxasthefunctiongsw_IPV_vs_fNsquared_ratio_CT25.



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

(

)

1S−)andsoappearto

bethermodynamicexpressions,herewepresentthecorrectequationwhichwewillseeis

merelykinematics,notthermodynamics.Let

(

)

WEP

−betheverticalmassfluxthrough

theair‐seainterfaceontheatmosphericsideoftheinterface(where

(

)

−

isthenotional

verticalvelocityoffreshwaterthroughtheair‐seainterfacewithdensityW

;thisdensity

beingthatofpurewaterattheseasurfacetemperatureandatatmosphericpressure).The

samemassflux

(

)

WEP

−mustflowthroughtheair‐seainterfaceontheoceansideof

theinterfacewherethedensityis

(

)

A,,0.St

ρρ

=Theverticalvelocitythroughan

arbitrarysurfacewhoseheightis

(

)

xyt

=canbeexpressedasH

−⋅∇−∂∂V

(wherewistheverticalvelocitythroughthegeopotentialsurface,seesection3.24,and

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IOC Manuals and Guides No. 56

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,.



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))

(

)

1W .

wtEP

ηη ρρ

−

−⋅∇−∂∂= −V(3.21.1)



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.A0S=,thispurewaterbeinginliquid,gaseousorsolidiceforms).The

freshwatercontentofseawateris

(

)

(

)

110.001/(gkg).SS

−

−=− 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 .S−

Theadvectivefluxofmassperunitareaisu

whereuisthefluidvelocitythrough

thechosenareaelementwhiletheadvectivefluxofseasaltisA.Su

Theadvectivefluxof

freshwaterperunitareaisthedifferenceofthesetwomassfluxes,namely

()

1.Su

−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dens

SS≡,SolutionSalinitysoln

S,Added‐MassSalinity

add

S,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 1 0.001 / (g kg ) .SS

−

−=− 



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0

ctimesthefluxofConservative

TemperatureΘ,where0

cisgivenby(3.3.3).Bycontrast,thesameheatfluxacrossthe

seasurfacechangespotentialtemperature

ininverseproportionto

()

A,,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

0RQ

ρρερ

Θ≈−∇⋅ −∇⋅ + +

FF S(3.23.1)

withanerrorinΘthatisapproximatelyonepercentoftheerrorincurredbytreating

either0

or

()

A,,0

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mechanicalenergyterm

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0

itfollowsthat

canbetreatedas

aconservativevariableintheoceanandthat0

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

vh vc

ρρ

=Θ(herevisthenorthwardvelocity).Theerrorincomparingthisadvective

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meridional“heatflux”withtheair‐seaheatfluxisapproximately1%oftheerrorinso

interpretingtheareaintegralofeither0

or

(

)

A,,0

vc S

θθ

.Similarly,turbulent

diffusivefluxesof“heat”areaccuratelygivenbyaturbulentdiffusivitytimesthespatial

gradientof0

cΘbutarelessaccuratelyapproximatedbythesameturbulentdiffusivity

timesthespatialgradientof0

(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

linearfunctionofA

S(i.e.uptothearbitraryfunction13A

aaS

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

dd.aSt

ItisshowninappendixBthatthetermsA

−∇⋅ +FSontheright‐hand

sideofEqn.(3.23.1)arealsounknowabletothesameextentsothattheeffectof3

acancels

fromEqn.(3.23.1).Hencethefactthat0

isunknowableuptoalinearfunctionofA

S

doesnotaffecttheusefulnessof0

hor0

asmeasuresof“heatcontent”.Similarly,the

differencebetweenthemeridionalfluxesof0

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0

cΘ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0

cΘis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

=0.Allowingthegravitationalaccelerationtobeafunctionof

,Φisgivenby

()

zdz

′

Φ=∫(3.24.1)

Ifthegravitationalaccelerationistakentobeconstant

issimply.

z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

ismeasuredupwardfromthissurface.InSIunitsΦis

measuredin122

Jkg m s .

−−

=Iftheoceanisassumedtobeinhydrostaticbalancesothat

=− (or

dz vdP

′

−= )thenthegeopotentialEqn.(3.24.1)maybeexpressedasthe

verticalpressureintegralofthespecificvolumeinthewatercolumn,

()

Pvp dP

′

Φ=Φ−∫(3.24.2)

where0

Φ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

isthesumofspecificinternalenergy,ukineticenergyperunitmass

0.5 ⋅uu(whereuisthethree‐dimensionalvelocityvector)andthegeopotential,Φ

2.u

+Φ+ ⋅uu

(3.25.1)

Totalenergy

isnotafunctionofonly

(

)

A,,Stpandsoisnotathermodynamic

quantity.

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3.26Bernoullifunction



TheBernoullifunctionisthesumofspecificenthalpy,hkineticenergyperunitmass

0.5 ⋅uu,andthegeopotential,Φ

2.h=+Φ+ ⋅uu

(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

() ( )

2ˆ,, .

hvpvSpdP

′

′′

=+Φ+⋅− − Θ

∫

(3.26.2)

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

.Thispressure

integralisequaltotheCunninghamgeostrophicstreamfunction,Eqn.(3.29.2).

TheBernoullifunction

isnotafunctionofonly

(

)

A,,Stp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0

h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))

()()

()()()()

00 1SO A

00 1SO SO A A

ˆˆ

,0 C, , ,

ˆˆˆˆ

,0 C, ,0 C,0 , , , ,0 .

hvSpvSpdP

hhSphShSphS

′′′

=+Φ+Ψ+ ⋅− ° − Θ

=+Φ+Ψ+ ⋅− ° + ° + Θ − Θ

∫

(3.26.3)

Notethat

()

ˆ,,0 p

hS cΘ=Θand

()

ˆ,0 C,0 0hS °=

.



3.27Dynamicheightanomaly



ThedynamicheightanomalyΨ,givenbytheverticalintegral

[][]

()

A,, ,

PSptppdP

′

′′ ′

Ψ=−∫(3.27.1)

isthegeostrophicstreamfunctionfortheflowatpressure

withrespecttotheflowatthe

seasurfaceand

isthespecificvolumeanomaly.Thusthetwo‐dimensionalgradientof

Ψinthe

pressuresurfaceissimplyrelatedtothedifferencebetweenthehorizontal

geostrophicvelocityvat

andattheseasurface 0

vaccordingto

f×∇ Ψ = −kvv(3.27.2)

ThedefinitionEqn.(3.27.1)ofdynamicheightanomalyappliestoallchoicesofthe

referencevaluesA

and,t



orˆ

inthedefinitionEqns.(3.7.1–3.7.4)ofthespecific

volumeanomaly.

Also,

inEqn.(3.27.1)canbereplacedwithspecificvolumev

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specificvolumeanomalyinpreferenceto

specificvolume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

orv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

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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thevertical.

Thedynamicheightanomaly

shouldbequotedinunitsof22

m s

−.Thesearethe

unitsinwhichtheGSWToolbox(appendixN)outputsdynamicheightanomalyinthe

functiongsw_geo_strf_dyn_height.NotethattheintegrationinEqn.(3.27.1)ofspecific

volumeanomalywithpressureindbar wouldyielddynamicheightanomalyinunitsof

m kg dbar

−,andtheuseoftheseunitsinEqn.(3.27.2)wouldnotgivetheresultant

horizontalgradientintheusualunits,beingtheproductoftheCoriolisparameter(unitsof

s−)andthevelocity(unitsof1

m s

−

).Thisisthereasonwhythepressureintegrationis

donewithpressureinPa anddynamicheightanomalyisoutputin22

m s

−.



3.28Montgomerygeostrophicstreamfunction



TheMontgomery“accelerationpotential”

definedby

()

[][]

()

PSptppdP

πδδ

′

′′ ′

=− −

∫(3.28.1)

isthegeostrophicstreamfunctionfortheflowinthespecificvolumeanomalysurface

()

,,Stp

=relativetotheflowat0

(thatis,at0dbarp

).Thusthetwo‐

dimensionalgradientof

inthe1

specificvolumeanomalysurfaceissimplyrelatedto

thedifferencebetweenthehorizontalgeostrophicvelocityvinthe1

surfaceandat

theseasurface 0

vaccordingto

×∇ = −kvv

or

(

)

10.ff

δπ

∇=−×−kvv

(3.28.2)

Thedefinition,Eqn.(3.28.1),oftheMontgomerygeostrophicstreamfunctionappliestoall

choicesofthereferencevaluesA



andt



inthedefinition,Eqn.(3.7.2),ofthespecific

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1

surface)

(

)

(

)

00nn

ff PP

∇=−× − +−∇kvv ,(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−,from

()

P−inthefirstterminEqn.(3.28.1),sodefiningthemodifiedMontgomery

streamfunction

()

[][]

()

Z-H

A,, .

PSptppdP

πδδ

′

′′ ′

=− −

∫(3.28.4)

ThegradientofZ-H

inaneutraltangentplanebecomes

(

)

(

)

Z-H

ff PP

∇=−×−+−∇kvv (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

tobeclosetotheaveragepressureonthe

surface.

Thistermcanbefurtherminimizedbysuitablychoosingtheconstantreferencevalues



andΘ



inthedefinition,Eqn.(3.7.3),ofspecificvolumeanomaly



sothatthissurface

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morecloselyapproximatestheneutraltangentplane(McDougall(1989)).This

improvementisavailablebecauseitcanbeshownthat

()

(

)

(

)

,, ,, .

nnbn

Sp SpPT P

ρδ κ κ

⎡⎤

∇=− Θ− Θ ∇≈ Θ−Θ∇

⎢⎥

⎣⎦

 

  (3.28.6)

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

()

(

)

()

nbn

PTPP

δρ

−Θ

−∇ ≈ Θ−Θ∇ −



(3.28.7)

andhencesuitablechoicesof

,A



and



canreducethelastterminEqn.(3.28.5)that

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22

m s

−

.

ThesearetheunitsinwhichtheGSWToolbox(appendixN)outputstheMontgomery

geostrophicstreamfunctioninthefunctiongsw_geo_strf_Montgomery.



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

and

potentialenthalpy0.hSincethekineticenergyperunitmass,0.5 ⋅uu,isatiny

componentoftheBernoullifunction,itwasignoredandCunningham(2000)essentially

proposedthestreamfunction0

Π+Φ (seehisequation(12)),where

()

AA A

(,,) (,,0) (),(), .

hS p hS vS p p p dP

Π≡ − − ⋅−Φ

=− +Φ−Φ

′

′′ ′

=Θ−Θ− Θ

∫

(3.29.1)

Thelastlineofthisequationhasusedthehydrostaticequationz

=− toexpress

zΦ≈ intermsoftheverticalpressureintegralofspecificvolumeandtheheightofthe

seasurfacewherethegeopotentialis0.



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

isalsoequalto

()()

ˆˆ

(), (), ,, .

PvS p p p vS p dP

′

′′ ′ ′

Π=− Θ − Θ

∫(3.29.2)

Inthisformit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SO

Sand0C°whereasherethey

arethelocalvaluesonthesurface,A

Sand

.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.

Thegradientof

alongtheneutraltangentplaneis

{

}

()

0b0

nz n

PTPP

ρρ

−Θ

∇

Π≈ ∇ −∇Φ − − ∇Θ(3.29.3)

(fromMcDougallandKlocker(2010))sothattheerrorinn

∇

ΠinusingΠasthe

geostrophicstreamfunctionisapproximately

()

1b0

TPP

−Θ

−

−∇Θ

.Whenusingthe

CunninghamstreamfunctionΠinapotentialdensitysurface,theerrorin

∇Πis

approximately

()

(

)

1b0r0

22TPP PPP

−Θ

−−−−∇Θ

.TheCunninghamgeostrophic

streamfunctionshouldbequotedinunitsof22

m s

−

andisavailableintheGSW

OceanographicToolbox(appendixN)asthefunctiongsw_geo_strf_Cunningham.

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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(suchasan

‐surfaceofKlockeretal.(2009a,b)ora

NeutralDensitysurfaceofJackettandMcDougall(1997))asuitablereferenceseawater

parcel

()

A,,SpΘ







isselectedfromtheapproximatelyneutralsurfacethatoneisconsidering,

andthespecificvolumeanomaly



isdefinedasin(3.7.3)above.Theapproximate

geostrophicstreamfunctionisgivenby(fromMcDougallandKlocker(2010))

()

(

)

(

)

AAb

212

,, ,, .

Sp PPSp T PP dP

ϕδρ δ

−Θ ′

Θ= − Θ− Θ−Θ−−

∫











(3.30.1)

Thisexpressionisveryaccuratewhenthevariationofconservativetemperaturewith

pressurealongtheapproximatelyneutralsurfaceiseitherlinearorquadratic.Thatis,in

thesesituations

(

)

n100

∇≈∇−∇Φ=−×−kvvtoaverygoodapproximation.In

Eqn.(3.30.1)1

−Θ

istakentobetheconstantvalue15 1 2 2 2

2.7 10 K (Pa) m sx

−

−−−

.This

McDougall‐KlockergeostrophicstreamfunctionisavailablefromtheGSWOceanographic

Toolboxasthefunctiongsw_geo_strf_McD_Klocker.



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,

=− andassumingthegravitationalaccelerationtobeindependentof

height,thedepth‐integratedmassfluxdz

∫visgivenby1

−

−∫vandthismotivates

takingthepressureintegraloftheDynamicHeightAnomaly

(fromEqn.(3.27.1))to

formthePressure‐Integrated‐Steric‐Height

ISH (alsocalledDepth‐IntegratedSteric

Height

ISH byGodfrey(1989)),

()

[][]

()

[][]

()

000

=,,

,, .

PPP

ISH g p dP g S p t p p dP dP

g PP SptppdP

′′

−−

−

′′′′′ ′′′′′′

Ψ= Ψ =−

′′′′′

=− −

∫∫∫

∫

(3.31.1)

Thetwo‐dimensionalgradientof

′

isrelatedtothedepth‐integratedmassfluxofthe

velocitydifferencewithrespecttothevelocityatzeroseapressure,0,vaccordingto

()

zP P

pzP P

zdzgfpdP

−

′′′′′

⎡⎤ ⎡ ⎤

×∇ Ψ = − = −

⎣⎦ ⎣ ⎦

∫∫

kvvvv(3.31.2)

Thedefinition,Eqn.(3.31.1),of

ISH appliestoallchoicesofthereferencevaluesAA

,SS







and,t





orΘ



inthedefinitions,Eqns.(3.7.2–3.7.4),ofthespecificvolumeanomaly.

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

sothatEqn.(3.27.1)becomes

[][]

()

A,,

PSptppdP

′

′′ ′

Ψ= ∫(3.31.3)

and

ISH ,reflectingthedepth‐integratedhorizontalmasstransportfromtheseasurface

topressure1

,relativetotheflowat1

,is

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

IOC Manuals and Guides No. 56

()

[][]

()

[][]

()

[][]

()

111

1A0

=,,

,, .

PPP

ISH g p dP g S p t p p dP dP

gPPSptppdP

gSptppdPP

−−

′′

−

′

′′ ′′ ′ ′ ′ ′ ′′

Ψ= Ψ =

′′′′′

=−

′′′ ′

=−

∫∫∫

∫

(3.31.4)

Thetwo‐dimensionalgradientof

′

isnowrelatedtothedepth‐integratedmassfluxof

thevelocitydifferencewithrespecttothevelocityat 1

,1,vaccordingto

()

zP P

pzP P

zdzgfpdP

−

′

′′ ′ ′

⎡⎤ ⎡ ⎤

×∇ Ψ = − = −

⎣⎦ ⎣ ⎦

∫∫

kvvvv(3.31.5)

Thespecificvolumeanomaly

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)).

ISH shouldbequotedinunitsof2

kg s

−

sothatitstwo‐dimensionalgradienthasthe

sameunitsasthedepth‐integratedfluxof

(

)

′

⎡

⎤

−

⎣

⎦

vvtimestheCoriolisfrequency.



3.32Pressuretoheightconversion



Whenverticallyintegratingthehydrostaticequationz

−inthecontextofanocean

modelwhereAbsoluteSalinityA

SandConservativeTemperatureΘ(orpotential

temperature

)arepiecewiseconstantinthevertical,thegeopotential(Eqn.(3.24.2))

()

Pvp dP

′

Φ=Φ−∫(3.32.1)

canbeevaluatedasaseriesofexactdifferences.Ifthereareaseriesoflayersofindexi

separatedbypressuresi

pand1i

(with1ii

+>)thentheintegralcanbeexpressed

(makinguseof(3.7.5),namelyA,

hhv

=)asasumovernlayersofthedifferences

inspecificenthalpysothat

()

()()

001

ˆˆ

,, ,, .

Pniii iii

Pvp dP hS p hS p

−+

⎡

⎤

′′

Φ=Φ− =Φ− Θ − Θ

⎣

⎦

∑

∫(3.32.2)



3.33Freezingtemperature



Freezingoccursatthetemperaturef

tatwhichthechemicalpotentialofwaterinseawater

equalsthechemicalpotentialoficeIh

.Thus,f

tisfoundbysolvingtheimplicit

equation

()

(

)

WIh

Af f

,, ,Stp tp

μμ

=(3.33.1)

orequivalently,intermsofthetwoGibbsfunctions,

()

(

)

(

)

Af A Af f

,, ,, , .

Stp S Stp g tp−=(3.33.2)

TheGibbsfunctionforiceIh,

(

)

Ih ,,

tp 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

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IOC Manuals and Guides No. 56

purewater,

() ()

0, , , .tp g tp

=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

()

f0gkg , 0dbart−=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

()

fSO

,0dbartS =

‐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

withairlowersthefreezingtemperaturesbyabout2mK .

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)),

()

AIh

pSpp

pTSTT

Sg g

tSp

Sg g

−−

∂==−

∂−−

(3.33.3)

Itsvalues,evaluatedfromTEOS‐10,varyonlyweaklywithsalinitybetween

()

0gkg , 0dbar

−=–0.7429mK/dbar forpurewaterand

(

)

SO ,0dbar

=–0.7483

mK/dbar forthestandardocean.TEOS‐10isconsistentwiththemostaccurate

measurementofp

anditsexperimentaluncertaintyof0.0015mK/dbar (Feisteland

Wagner(2005),(2006)).Sincethevalueof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),

()

AIh

STSTT

tSp

SgSg g

∂==

∂−−

(3.33.4)

Typicalnumericalvaluesare

()

0gkg , 0dbar

−=–59.21

mK/(g kg )

−

forpurewaterand

()

SO ,0dbar

=–56.91

mK/(g kg )

−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,S

SA A

RTS S≈,which

stemsfromPlanck’sideal‐solutiontheory(Planck(1888)).Here,SS

RRM

=264.7599

Jkg–1K–1isthespecificgasconstantofseasalt,Ristheuniversalmolargasconstant,and

=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SI

L,Eqn.(3.34.7).The

convenientresultobtainedwiththesesimplificationsis

()

59 mK/(g kg )

tTt

−

∂≈− + ≈−

∂.(3.33.5)

wherewehaveusedf2Ct=− andSI

= 3301

Jkg

−

asapproximationsthatareappropriate

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

(

)

A,Stp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

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IOC Manuals and Guides No. 56

tothesaturationconcentrationofabout1101

gkg

−

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1

gkg

−.

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density

ofseawaterandpropertiesderivedfromit.Thepropertiesconsideredintheremainderof

thissection(3.33‐3.42)whichsharethisattributearereferredtoasthecolligative

propertiesofseawater.



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,WI

L.Itisfoundasthedifferencebetweenthespecific

enthalpyofwater,W,handthespecificenthalpyofice,Ih ,h(Kirchhoff’slaw,Curryand

Webster(1999)):

()

(

)

(

)

WI W Ih

,,.

phtphtp=− (3.34.1)

Here,

()

tpisthefreezingtemperatureofwater,section3.33.TheenthalpiesW

handIh

h

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to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,SI ,hisadditivewithrespecttoitsconstituentsice,Ih ,hwith

themassfractionIh ,wandseawater,,hwiththeliquidmassfraction

(

)

1:w−

()

(

)

(

)

SI Ih Ih Ih

1,, ,h w hS tp wh tp=− + .(3.34.2)

Uponwarming,themassofmeltwaterchangestheicefractionIh

wandthebrinesalinity

A.STherelatedtemperaturederivativeofEqn.(3.34.2)is

() () ()

SI Ih Ih

Ih Ih Ih Ih

Sp p

ppp

hh hShw

ww whh

TT STT T

∂∂ ∂∂∂∂

=− +− + + −

∂∂ ∂∂∂ ∂

.(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,Ih /p

−

∂∂,bytheconservationofthe

totalsalt,

()

1wS−=const,intheform

SSw

∂∂

−.(3.34.4)

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

()

SI Ih

Ih Ih Ih SI

1ppp

wc wc L

∂∂

=− + −

∂∂

.(3.34.5)

Thecoefficientinfrontofthemeltingrate,

()

SI Ih

LSp hS h

∂

=− −

∂,(3.34.6)

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

conditionsfortheiceandseawaterformulationsarechosenconsistently(Feisteletal.

(2008a)).Notethatbecauseof

(

)

hg Tt

=+ + andEqn.(3.33.2),thelatentheatcanalso

bewrittenintermsofentropies

ratherthanenthalpies,hintheform

()( )

SI Ih

A0fA

LSp T t S S

ηη

⎛⎞

∂

⎜⎟

=+×− −

⎜⎟

∂

⎝⎠

.(3.34.7)

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

surfacepressure,thecomputedvalueis

(

)

(

)

SI WI

0,0 0

LL==333426.5Jkg–1forpurewater,

and

()

SO ,0

LS =329928.5Jkg–1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 WI

0,1000dbar 1000dbar

LL==331528Jkg–1and

(

)

SO ,1000dbar

LS =328034Jkg–1.

TEOS‐10isconsistentwiththemostaccuratemeasurementsofWI

L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subl

isdefinedastheabsolutepressure

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V

withthe

chemicalpotentialoficeIh ,

soitisfoundbysolvingtheimplicitequation

()

(

)

Vsubl Ihsubl

,,,tP tP

μμ

=(3.35.1)

orequivalently,intermsofthetwoGibbsfunctions,

()

(

)

Vsubl Ihsubl

,,.gtP gtP=(3.35.2)

TheGibbsfunctionforiceIh,

(

)

Ih ,

tP isdefinedbyIAPWS‐06andFeistelandWagner

(2006)andissummarizedinappendixIbelow.Notethatheretheabsolutepressure



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,

(

)

tP ,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subl

=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

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library.TheIAPWS‐95function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,vap

xP=,iscomputedfromthe

totalabsolutepressure

andthemolefractionofvapour,V.

Similartotheabsolute

salinityA

Sofseawater,thevariable

describesthemassfractionofdryairpresentin

humidair.Given,

themolefractionofvapouriscomputedfrom

()

11 /

xAMM

−

=−− ,(3.35.3)

whereA

isthemolarmassofdryairandW

isthemolarmassofwater.

Thesublimationpressure,

(

)

subl sat

tP x P=,oficeinequilibriumwithhumidairisthe

partialpressureofvapourinsaturatedair.Tocomputesat

fromEqn.(3.35.3),the

requiredairfractionatsaturation,

(

)

sat ,,

AtP=isfoundbyequatingthechemical

potentialofwatervapourinhumidairAV

withthechemicalpotentialoficeIh ,

sothat

itisfoundbysolvingtheimplicitequation

()

AV sat Ih

W,, , ,

tP tP

μμ

=(3.35.4)

orequivalently,intermsofthetwoGibbsfunctions,

()

(

)

()

AV sat sat AV sat Ih

,, ,, ,

AtPAg AtPgtP−=.(3.35.5)

TheGibbsfunctionofhumidair,

()

AV ,,

AtP ,isdefinedbyFeisteletal.(2010a).

Att=0°Candatmosphericpressure,thesublimationpressureoficehasthevalue

subl

(0°C,101325Pa)=613.745Pa,computedbysolvingEqn.(3.35.4)forsat ,

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subl

(‐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,VI

L.Itisfoundasthedifferencebetweenthespecificenthalpy

ofwatervapour,V,handthespecificenthalpyofice,Ih :h

()

(

)

(

)

VI V subl Ih subl

,,.

Lt htP htP=− (3.36.1)

Here,

()

subl

tisthesublimationpressureoficeatthetemperature,tsection3.35.The

enthalpiesV

handIh

h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

sublimationenthalpyis

()

VI 0.01 C

L°=2834359Jkg–1withanuncertaintyof1000Jkg–1,or

0.03%.

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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sat

.Ifforexamplethetotal

pressure

isheldconstant,thepartialpressuresat

Pwillrise,andtheicemustget

warmertomaintainequilibriumatthemodifiedsublimationpressuresubl sat

xP=

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

latentheatformulafromtheunknownabsoluteenthalpiesofairandwaterthatotherwise

wouldaccompanyanymassexchange.

Theenthalpyoficeair,AI ,hisadditivewithrespecttoitsconstituentsice,Ih ,hwith

themassfractionIh ,wandhumidair,AV ,hwiththegasfraction

(

)

1:w−

()

(

)

(

)

AI Ih AV Ih Ih

1,, ,h w h Atp wh tp=− + .(3.36.2)

Uponwarming,themassofvapourproducedbysublimationreducestheicefractionIh

w

andincreasesthehumidity,thatis,decreasestherelativedry‐airfraction

ofthegas

phase.TherelatedtemperaturederivativeofEqn.(3.36.2)is

() () ()

AI AV AV Ih Ih

Ih Ih Ih Ih AV

pApTpp p

hh hAh w

ww whh

TT ATT T

∂∂ ∂∂∂ ∂

=− +− + + −

∂∂ ∂∂∂ ∂

.(3.36.3)

Theair‐fractionchangeisrelatedtotheisobaricsublimationrate,Ih /p

−

∂∂,bythe

conservationofthedryair,

()

1wA−=const,intheform

AAw

∂∂

−.(3.36.4)

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

()

AI Ih

Ih AV Ih Ih AI

ppp

wc wc L

∂∂

=− + −

∂∂

(3.36.5)

Thecoefficientinfrontofthesublimationrate,

()

AI AV Ih

LAp h A h

∂

=− −

∂,(3.36.6)

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,AV AV

hAh−,equalsthespecificenthalpyofvapour,

(

)

ht,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

andsubl ,P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.

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IOC Manuals and Guides No. 56

Forsaturatedairoverseaice,theairfractionsat

A=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SO

=101325Paandthefreezingpointf

t=‐1.919°Cofthe

standardocean,theTEOS‐10valueforsaturatedairwith

(

)

sat

SO f SO

AtP==0.99678is

()

SO SO

LAP=2833006Jkg–1.Therelatedsublimationpressureis

()

subl

fSO

tP =523.436

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.



3.37Vapourpressure



Thevapourpressureofseawater

(

)

vap

Stisdefinedastheabsolutepressure

of

watervapourinequilibriumwithseawateratagiventemperaturetandsalinityA.SItis

foundbyequatingthechemicalpotentialofvapourV

withthechemicalpotentialof

waterinseawater W

sothatitisfoundbysolvingtheimplicitequation

(

)

(

)

V vap W vap

,,,tP S tP

μμ

=,(3.37.1)

orequivalently,intermsofthetwoGibbsfunctions,

()

(

)

(

)

V vap vap vap

AAA

,,, ,,

gtP gStP Sg StP=− .(3.37.2)

Notethathereweusetheabsolutepressure

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,

(

)

tP andA

(,,),

StPare

availablefromtheHelmholtzfunctionoffluidwater,asdefinedbyIAPWS‐95,andthe

Gibbsfunctionofseawater,IAPWS‐08orIAPWS‐09(IAPWS(2009c)).

Inthecaseofpurewater,A0,S

thesolutionofEqn.(3.37.1)istheso‐called

saturationcurveinthetP−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vap

(0,0.01°C)=t

=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vap

(SO ,S0°C)=599.907Pa.Atlaboratorytemperaturetherelatedvaluesare

vap

(0,25°C)=3169.93Paandvap

(SO ,S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)

vap A

SP P

Sg g

∂=

∂−−

.(3.37.3)

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,S

SA A

RTS S≈,whichstems

fromPlanck’sideal‐solutiontheory.Here,SS

RRM

=264.7599Jkg–1K–1isthespecific

gasconstantofseasalt,Ristheuniversalmolargasconstant,andS

=31.40382gmol–1

isthemolarmassofseasaltwithReferenceComposition.Thespecificvolumeof

seawater,,

isneglectedincomparisontothatofvapour.Thelatterisapproximately

consideredasanidealgas,

(

)

Vvap

gRTMP≈,whereW

=18.015268gmol–1isthe

molarmassofwater.Theconvenientresultobtainedwiththesesimplificationsis

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IOC Manuals and Guides No. 56

vap

vap vap

0.57

∂≈− ≈− ×

∂.(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

thanunderitsvapour

pressurevap .PInthiscase,thevapourpressureofseawater

(

)

vap

A,,

StPisdefinedasthe

partialpressureofwatervapourinhumidairthatisinequilibriumwithseawaterata

givenpressure,

temperaturetandsalinityA.SItisfoundbyequatingthechemical

potentialofvapourinhumidairV

withthechemicalpotentialofwaterinseawater

sothatitisfoundbysolvingtheimplicitequation

()

Vcond W

AV A

,, ,,

tP S tP

μμ

=(3.37.5)

for

()

cond

A,,

StP,orequivalently,intermsofthetwoGibbsfunctions,

()

(

)

() ()

AV cond cond AV cond

AAA

,, ,, ,, ,,

A tP A g A tP gS tP S g S tP−=−.(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,cond

,the

molefractionofvapourcond

(3.53.2),andinturnthevapourpressure

()

vap cond

StP x P=areavailablefromstraightforwardcalculations.TheGibbsfunction

ofhumidairAV

isavailablefromFeisteletal.(2010a)andisalsoplannedtobemade

availableasthedocumentIAPWS(2010).

TheTEOS‐10valuecomputedfromEqn.(3.37.5)isvap

(0,0°C,PSO)=613.760Pafor

purewateratsurfaceairpressure;thevapourpressureofthestandardoceanis

vap

(SO ,S0°C,SO

)=602.403Pa.Atlaboratorytemperaturetherelatedvaluesare

vap

(0,25°C,SO

)=3183.73Paandvap

(SO ,S25°C,SO

)=3124.03Pa.



3.38Boilingtemperature



Theboilingtemperatureofwaterorseawaterisdefinedasthetemperature

()

boil

A,tSPat

whichthevapourpressure(ofsection3.37)equalsagivenpressure.

Itisfoundby

equatingthechemicalpotentialofvapourV

withthechemicalpotentialofwaterin

seawater W

sothatitisfoundbysolvingtheimplicitequation

()

(

)

Vboil W boil

,,,,tP StP

μμ

=(3.38.1)

forboil

(,)tSP,orequivalentlyintermsofthetwoGibbsfunctions,

(

)( )

(

)

V boil boil boil

AAA

,,, ,,.

tP gStPSgStP=− (3.38.2)

TheTEOS‐10boilingtemperatureofpurewateratatmosphericpressureis

()

boil

0,tP=

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.

Theheatrequiredpermassevaporated

fromtheliquidisthelatentheat,orenthalpy,ofevaporation,VW

L.Itisfoundasthe

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differencebetweenthespecificenthalpyofwatervapour,V,handthespecificenthalpyof

liquidwater,W:h

()

(

)

(

)

VW V vap W vap

Lt htP htP=−.(3.39.1)

Here,vap ()

tisthevapourpressureofwateratthetemperaturet(section3.37).The

enthalpiesV

handW

hareavailablefromIAPWS‐95.Atthetriplepointofwater,the

TEOS‐10evaporationenthalpyis

(

)

VW 0.01 C

L°=2500915Jkg–1.

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

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,SA ,hisadditivewithrespecttoitsconstituents,seawater,,h

withthemassfractionSW ,wandhumidair,AV ,hwiththegasfraction

(

)

1:w−

()

(

)

(

)

SA SW AV SW

1,, ,,hwhAtpwhStp=− + .(3.39.2)

Uponwarming,themassofwatertransferredfromtheliquidtothegasphaseby

evaporationreducestheseawatermassfractionSW ,wincreasesthebrinesalinityA

Sand

increasesthehumidity,withacorrespondingdecreaseinthedry‐airfraction

ofthegas

phase.TherelatedtemperaturederivativeofEqn.(3.39.2)is

() ()

()

SA AV AV

SW SW

SW SW AV

pApTp

Sp p

Tp p

hh hA

TT AT

hhS w

ww hh

TST T

∂∂ ∂∂

=− +−

∂∂ ∂∂

∂∂∂ ∂

++ +−

∂∂∂ ∂

(3.39.3)

TheisobaricevaporationrateSW /p

wT−∂ ∂ isrelatedtotheair‐fractionchangebythe

conservationofthedryair,

()

1wA−=const,intheform

AAw

∂∂

−,(3.39.4)

andtothechangeofsalinitybytheconservationofthesalt,SW

wS=const,intheform

SW .

SSw

∂∂

=−

∂∂

(3.39.5)

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

()

SA SW

SW AV SW SA

1ppp

wc wcL

∂∂

=− + −

∂∂

.(3.39.6)

Thecoefficientinfrontoftheevaporationrate,

()

SA AV

,,, ,

LAStp h A hS

∂∂

=− −+

∂∂

(3.39.7)

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IOC Manuals and Guides No. 56

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,AV AV

hAh−,equalsthe

specificenthalpyofvapour,

()

ht,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,cond ,Aandthelatent

heatofevaporation,SA

,aregiveninTable3.39.1.



Table3.39.1:Selectedvaluesfortheequilibriumairfraction,cond ,Acomputed

fromEqn.(3.37.6),andthelatentheatofevaporation,SA

,

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.



ConditionA

S

gkg–1

t

°C

p

dbar

cond



%



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 25 098.05933 2438971

Highpressure35.16504 0400 99.98943 2443759



InthederivationofEqn.(3.39.7),thevalueof

isindirectlyassumedtobecomputed

fromtheequilibriumcondition(3.37.6)betweenhumidairandseawater,

=cond .AAt

thishumiditytheairisstillsub‐saturated,cond sat ,

A>butitsvapourstartscondensingat

theseasurface.Thevaluesofcond

andsat

coincideonlybelowthefreezingpointof

seawater,oratvanishingsalinity,seealsothefollowingsection3.40.

Theevaporationrate,SW /p

wT−∂ ∂ ,canbecomputedfromEqn.(3.37.6),the

equilibriumconditionbetweenhumidairandseawater,atchangingtemperatureand

constantpressure(Feisteletal.(2010a)).Incontrast,thederivationofSA

L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

takesanygivenvaluedifferentfromcond ,Athatis,itcan

beappliedregardlessofwhetherornotthehumidairisactuallyatequilibriumwiththe

seasurface.



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

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

IOC Manuals and Guides No. 56

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

CCT sat

=(3.40.1)

withregardtothemolefractionofvapour

(

)

AEqn.(3.35.3),andthesaturatedair

fraction

() ( )

sat cond

,0,,

AtP A tP== 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,

(

)

cond

A,,

StPistheairfractionofhumid

airatequilibriumwithseawater,Eqn.(3.37.5),whichissubsaturatedforA0.S>

TheWMO3definitionoftherelativehumidityis(PruppacherandKlett(1997),

Jacobson(2005)),

WMO sat sat

1/ 1

RH rA

−

(3.40.2)

where

()

1/rAA=− isthehumidityratio.Ifrissmall,wecanestimateVAW

rM M≈

(fromEqn.(3.35.3))andthereforeWMO CCT

RH RH

≈

,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

()

VV, id

,, exp .fATP xP RT

μμ

⎧

⎫

−

⎪

⎨

⎬

⎪

⎩⎭

(3.40.3)

Here,WW

RRM=isthespecificgasconstantofwater,

(

)

VAVAV

TP g Ag

=− isthe

chemicalpotentialofvapourinhumidair,and

(

)

V, id ,,

isitsideal‐gaslimitwhichis

equaltothetruechemicalpotentialinthelimitofverylowpressure,

() ()

V, id V V,id V

,, 1 'd' ln

AT P g c T T R T

⎛⎞

=+ − +

⎜⎟

⎝⎠

∫.(3.40.4)

ThevaluesofV

,V

andV

TofV,id

mustbechosenconsistentlywiththeadjustable

constantsofAV

(Feisteletal.(2010a)).Theideal‐gasheatcapacityofvapour

()

V, id

cTis

availablefromIAPWS‐95.Intheideal‐gaslimitofinfinitedilution,V

convergestothe

partialpressureofvapour(Glasstone(1947)),

(

)

vap

lim , , .

PfATP xP P

→==(3.40.5)

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

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

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IOC Manuals and Guides No. 56

Thesaturationfugacityisdefinedbytheequilibriumbetweenliquidwater(orice)and

vapourinair,

()()

,, 0,,

TP TP

μμ

=,thatis,

()

(

)

WV,idsat

sat sat

0, , , ,

exp ,

TP A TP

fxP RT

μμ

⎧

⎫

−

⎪

⎨

⎬

⎪

⎩⎭

(3.40.6)

where

()

W0, ,

=isthechemicalpotentialofliquidwater(orthechemicalpotential

ofice,Ih

).Therelativefugacity

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

()()

sat

,, 0,,

exp .

AT P T P

μμ

⎧

⎫

−

⎪

⎨

⎬

⎪

⎩⎭

(3.40.7)

Intheideal‐gaslimit, VV, id

μμ

=andusing(3.40.3)weseethattherelativefugacity



coincideswiththerelativehumidity,Eqn.(3.40.1).

TakingEqn.(3.40.7)atthecondensationpoint,cond ,AA=Eqn.(3.37.5),itfollowsthat

therelativefugacityofhumidairatequilibriumwithseawater(“seaair”forshort)is

()()

SA V

sat

,, 0,,

exp .

STP TP

μμ

⎧

⎫

−

⎪

⎨

⎬

⎪

⎩⎭

(3.40.8)

Thechemicalpotentialdifferenceintheexponentisproportionaltotheosmoticcoefficient

ofseawater,,

whichiscomputedfromthesalinepartoftheGibbsfunctionas(Feistel

andMarion(2007),Feistel(2008)),

()

SW A ,

,, ,

STP g S

mRT S

⎡

⎤

∂

⎢

⎥

=− − ∂

⎢

⎥

⎣

⎦

(3.40.9)

whereSW

misthemolalityofseawater(Milleroetal.(2008a)),

()

mSM

=−(3.40.10)

Fromthechemicalpotentialofwaterinseawater,A

gSg=− ,andEqns.(3.40.8)‐

(3.40.10)weinferfortherelativefugacityofseaairthesimpleformula

(

)

SW W

exp ,mM

=− (3.40.11)

whichisidenticaltotheactivityW

a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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2

SW A S A

/()mSMOS=+andthe

osmoticcoefficient

()

1OS

=+ thelinearrelation

SA W

≈− (3.40.12)

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

()

(

)

Ih V,id sat

sat sat

,,,

exp ,

TP A TP

fxP RT

μμ

⎧⎫

−

⎪⎪

=⎨⎬

⎪⎪

⎩⎭

(3.40.13)

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

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

IOC Manuals and Guides No. 56

()()

WIh

SA V

sat

,, ,

exp .

STP TP

μμ

⎧

⎫

−

⎪

⎨

⎬

⎪

⎩⎭

(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,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/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

()

SA A

,,,XASTPdrivingthetransferofwateristhedifferencebetweenthechemical

potentialsofwaterinhumidairandinseawater,

(

)

(

)

AV A

WW W

,, ,, .

TP S TP

XRT RT RT

μμ

⎛⎞

=Δ = −

⎜⎟

⎝⎠ (3.40.15)

Thisdifferencevanishesatthecondensationpoint,

(

)

cond

A,, ,

ASTP=Eqn.(3.37.5),

ratherthanatsaturation. SA

Xcanalsobeexpressedintermsoffugacities,Eqns.(3.40.7),

(3.40.8)and(3.40.11),intheform

(

)

() ()

SA SW W

ln ln .

XmMA

ϕφϕ

==+(3.40.16)

Ratherthantherelativehumidity,Eqns.(3.40.1),(3.40.2),thesea‐airOnsagerforceSA ,X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

()

SA lnXA

=,which

correspondstotherelativehumidity,

(

)

CCT

ln RH ,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thepressureW

pandofwaterinseawateratthepressure,p

()

,,, .

gtp gStp SS

∂

=−

∂(3.41.1)

Thesolutionofthisimplicitrelationfortheosmoticpressureis

(

)

osm W

A,, .

Stp P P=− (3.41.2)

TheTEOS‐10valuefortheosmoticpressureofthestandardoceanis

()

osm

SO ,0 C,0dbarPS°=

2354684Pa,computedfromEqn.(3.41.1).



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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MD

tisbelowthefreezingpointf

t(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MD

tiscomputedfromtheconditionof

vanishingthermalexpansioncoefficient,thatis,fromthesolutionoftheimplicitequation

forMD A

(,)tSp,

(

)

AMD

,, 0.

gSt p

(3.42.1)

ThetemperatureofmaximumdensityisavailableintheGSWOceanographicToolboxas

functiongsw_temps_maxdensity.Thisfunctionalsoreturnsthepotentialtemperature

andtheConservativeTemperatureatthismaximumdensitypoint.SelectedTEOS‐10

valuescomputedfromEqn.(3.42.1)aregiveninTable3.42.1.



Table3.42.1:Freezingtemperaturef

tandtemperatureofmaximumdensityMD

t

forair‐freebrackishseawaterwithabsolutesalinitiesA

Sbetween0

and1

25 g kg−,computedatthesurfacepressurefromTEOS‐10.

ValuesofMD

tinparenthesesarelessthanthefreezingtemperature.



g kg–1

°C

t

°C

g kg–1

°C

t

°C

g kg–1

°C

t

°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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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.IceIhisthenaturallyabundantformofice,having

hexagonalcrystals.Forthefirsttimetheeffectsofthesmallvariationsinseawater

compositionaroundtheworldoceancanbe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changecomparedtotheInternationalEquationofStateof

seawater(EOS‐80)istheadoptionofAbsoluteSalinityA

SinsteadofPracticalSalinityP

S

(PSS‐78)asthesalinityargumentforthethermodynamicpropertiesofseawater.

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A

SratherthanonP

S;seawaterparcels

thathavethesamevaluesoftemperature,pressureandofP

Sdonothavethesame

densityunlesstheparcelsalsosharethesamevalueofA

S.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inatwo‐stageprocess.FirstReferenceSalinityis

calculatedfrommeasurementsofPracticalSalinityusingEqn.(2.4.1).ThentheAbsolute

SalinityAnomalyisestimatedfromthecomputeralgorithmofMcDougalletal.(2010a)or

byothermeans,andAbsoluteSalinityisformed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forAbsoluteSalinity

Anomaly,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.

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thatisusedtoconvert

ReferenceSalinityR

SintoAbsoluteSalinityA

S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.

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IOC Manuals and Guides No. 56

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R

SratherthanA

S,

thusavoidingtheneedtocalculatetheAbsoluteSalinityAnomaly.Whenthisisdone,it

shouldbeclearlystatedthatthesalinityvariablethatisbeinggraphedisReference

Salinity,notAbsoluteSalinity.

ThetreatmentofsalinityinoceanmodelsisdiscussedinappendixA.20.The

recommendedapproachistocarrybothPreformedSalinity*

SandAbsoluteSalinity

AnomalyA

asmodelvariablessothatDensitySalinitycanbecalculatedateachtime

stepofthemodelandusedtoaccuratelyevaluatedensity.

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

isaconservativevariable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0

)

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,A

(constant)S

−

,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

cΘacrossanoceansectionasthe“heatflux”acrossthesectionevenwhenthefluxesof

massandofsaltacrossthesectionarenon‐zero.

Thetemperaturevariableinoceanmodelsiscommonlyregardedasbeingpotential

temperature,butsincethenon‐conservativesourcetermsthatarepresentintheevolution

equationforpotentialtemperaturearenotincludedinmodels,itisapparentthatthe

interiorofoceanmodelsalreadytreattheprognostictemperaturevariableasConservative

Temperature.ΘTocompletethetransitionto

inoceanmodeling,modelsshouldbe

initializedwithΘratherthan

,theoutputtemperaturemustbecomparedtoobserved

Θdataratherthanto

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

()

ˆ,.

θθ

=ΘThefinalingredientneededforanoceanmodelisacomputationally

efficientformofdensityintermsofConservativeTemperature,thatis

()

ˆ,, ,

ρρ

=Θ

suchasthatdescribedinappendixA.30andappendixKofthisTEOS‐10Manual.

UnderEOS‐80theobservedvariables

(

)

P,,Stpwerefirstusedtocalculatepotential

temperature

andthenwatermasseswereanalyzedontheP

−

diagram.Curved

contoursofpotentialdensitycouldalsobedrawnonthissameP

−

diagram.Under

TEOS‐10,sincedensityandpotentialdensityarenownotfunctionsofPracticalSalinityP

S

butratherarefunctionsofAbsoluteSalinityA

S,itisnownotpossibletodrawisolinesof

potentialdensityonaP

−diagram.Rather,becauseofthespatialvariationsof

seawatercomposition,agivenvalueofpotentialdensitydefinesanareaontheP

−

diagram,notacurvedline.UnderTEOS‐10,theobservedvariables

(

)

P,,Stp,together

withlongitudeandlatitude,areusedtofirstformAbsoluteSalinityA

SandConservative

Temperature.ΘOceanographicwatermassesarethenanalyzedontheA

S−Θdiagram,

andpotentialdensitycontourscanalsobedrawnonthisA

−

Θdiagram,while

PreformedSalinity*

Sisthenaturalsalinityvariabletobeusedinapplicationssuchas

numericalmodelling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]. 

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

IOC Manuals and Guides No. 56

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C K 273.15,tT°= − 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

()

Rttothe

resistanceatareferencetemperature0.RInIPTS‐68,0

Ris

(

)

0C,R°whileinITS‐900

Ris

()

0.01 C .R°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<90

t<968.71°C,90

t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Ware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<90

t<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<90

t<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(90

t=29.7646°C)andthe

triplepointofMercury(90

t=‐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(90

t=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),dd d,uT Pv

+thatthesameenergydifference

resultsindifferentvaluesfortheentropy,

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,

A,.

=

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.



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

IOC Manuals and Guides No. 56

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90

tand68.t

Considerfirstwhy90

tand68

t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Wto90

t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,90 68.tt−

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

()

90 68

tt−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

(

)

90 68

tt−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

(

)

(

)

90 68 68

/K = - 0.00025 / K - 273.15TT T−(A.1.1)

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

()

90 68 68

/.dT T dT−AboutthesametimethatRusbypublishedhisconversionfrom68

tto

90 ,tSaunders(1990)madearecommendationtooceanographersthatinthecommon

oceanographictemperaturerange‐2°C<68

t<40°C,conversioncouldbeachievedusing

()

(

)

90 68

/ C = / C 1.00024.tt°° (A.1.2)

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IOC Manuals and Guides No. 56

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68

tfortheboilingpointofwaterwas

100°Cwhileits90

t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90

−

(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68

tasthetemperatureargument.In

ordertousethesealgorithmswith90

tdata,68

tmaybecalculatedusingEqn.(A.1.3)thus

()

(

)

68 90

/ C = 1.00024 / C .tt°°(A.1.3)



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

IOC Manuals and Guides No. 56

A.2 Sea pressure, gauge pressure and absolute pressure

SeapressurepisdefinedtobetheAbsolutePressure

lesstheAbsolutePressureofone

standardatmosphere,0101 325Pa;P≡thatis

0.pPP

≡

−(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gauge

p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isnaturallyafunctionofseapressurep

(orfunctionallyequivalently,ofAbsolutePressure

);itisnotafunctionofgauge

pressure.



TableA.2.1Pressureunitconversiontable





Pascal

(Pa)



decibar

(dbar)



bar

(bar)



Technical

atmosphere

(at)



atmosphere

(atm)



torr

(Torr)

pound‐

forceper

square

inch

(psi)

1Pa≡1N/m210−410−510.197×10−69.8692×10−67.5006×10−3145.04×10−6

1dbar104≡105dyn/cm20.10.1019798.692×10−375.0061.45037744

1bar10000010≡106dyn/cm21.01970.98692750.0614.5037744

1at98066.59.806650.980665≡1kgf/cm20.967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1.3332×10−31.3595×10−31.3158×10−3≡1Torr19.337×10−3

1psi6894.7570.6894868.948×10−370.307×10−368.046×10−351.715≡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,0101 325Pa.P

≡

(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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IOC Manuals and Guides No. 56

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R

S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1

gkg

−

(i.e.1.01

mg kg

−

)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1

mg kg

−

)andRb+(~0.121

mg kg

−

).

Dissolvedgases2

N(~161

mg kg−)and2

O(upto81

mg kg

−

intheocean)arenotincluded

asneitherhaveasignificanteffectondensityoronconductivity.Inaddition,2

Nremains

withinafewpercentofsaturationatthemeasuredtemperatureinalmostalllaboratory

andinsituconditions.However,thedissolvedgas2

CO (~0.71

mg kg

−

),andtheionOH

−



(~0.081

mg kg−)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2

CO toandfromionicformsaffectconductivityanddensity.

Concentrationsofthemajornutrients4

Si(OH) ,3

−

and3

−

areassumedtobe

negligibleinStandardSeawater;theirconcentrationsintheoceanrangefrom0‐16

mg kg−,0‐21

mg kg

−

,and0‐0.21

mg kg

−

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1

mg kg

−



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,

=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1

gkg .

−

SinceReference‐CompositionSalinityisdefinedtobeconservativeduringmixing,ifa

seawatersampleofmass1

mandReference‐CompositionSalinityR1

Sismixedwith

anotherseawatersampleofmass2

mandReference‐CompositionSalinityR2,Sthefinal

Reference‐CompositionSalinityR12

Softhissampleis

1R1 2R2

R12

mS mS

Smm

=+.(A.3.1)

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

IOC Manuals and Guides No. 56

Negativevaluesof1

mand2,mcorrespondingtotheremovalofseawaterwiththe

appropriatesalinityarepermitted,solongas

(

)

(

)

1R12R2

110mSmS

−

+−>.Inparticular,if

R2 0S=(purewater)and2

misthemassofpurewaterneededtonormalizetheseawater

sample(thatis,2

misthemassneededtoachieveR12

S=35.16504gkg−1),thentheoriginal

Reference‐CompositionSalinityofsample1isgivenby

-1

R1 2 1

[1 ( / )] 35.16504 g kgSmm=+ × .(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15.R

WenotedabovethataPracticalSalinityof35isassociatedwithaReferenceSalinityof

35.165041

gkg .

−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 /1

(g kg )

−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

242S<<),as

RPSP

SuS≈where 1

PS (35.165 04 35) g kgu

−

≡.(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P

Sis35hasaReference‐CompositionSalinityR

Sofprecisely

35.165 04 g kg−.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1

0.007 g kg−

±.ThusthenumericaldifferencebetweentheReferenceSalinity

expressedin1

gkg

−andPracticalSalinityisabout24timeslargerthanthisestimateof

uncertainty.Thedifference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

TEOS-10 Manual: Calculation and Use of the Thermodynamic Properties of Seawater

IOC Manuals and Guides No. 56

compositionanomaliesrelativetoNorthAtlanticconditionsarefound,arereflectedin

Knudsenʹsformula,

()

(

)

K‰0.031.805 ‰.SCl=+ (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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KnudsenʹsformulaatK

S=35‰(Cl =

19.3740‰),(Woosteretal.,1969).Thisresultedintheconversionformula

()

(

)

‰ 1.80655 ‰SCl=(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PS

uinEqn.(A.3.3),definedby1

PS (35.165 04 35) g kgu−

≡.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R

S

toestimatetheAbsoluteSalinityofReferenceCompositionSeawaterisatleast0.007

gkg

−atS=35(Milleroetal.(2008b)).Thus,althoughPS

u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R

S

andAbsoluteSalinityA

SinpreferencetoPracticalSalinity P.S



1. ThedefinitionofPracticalSalinityP

S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 )

−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P

S).Rather,itis

(1–0.001A

S/( 1

gkg

−)),whereA

SistheAbsoluteSalinity,definedasthemass

fractionofdissolvedmaterialinseawater.ThevaluesofA

S/( 1

gkg

−)andP

S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<P

S<42.Forasmoothcrossoverononesideto

purewater,andontheothersidetoconcentratedbrinesuptosaturation,asfor

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IOC Manuals and Guides No. 56

exampleencounteredinseaiceatverylowtemperatures,salinitiesbeyondthese

limitsneedtobedefined.WhilethisposesachallengeforP,SitistrivialforR.S

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R

Sbut

notforP.S

5. ForartificialseawaterofReferenceComposition,R

S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P

S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A

S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A

S/( 1

gkg

−

))oras(1–A

S)butitwouldbeincorrecttowriteitas(1–0.001A

S).

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P

Sweadoptthefirst

optionabove,namely1

gkg

−asthepreferredunitforA,S(asinA

S=35.16504gkg−1).

TheReferenceSalinity,R,Sisdefinedtohavethesameunitsandfollowsthesame

conventionsasA.S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,RPS

‰,SuS=bythenumericalfactorPS

u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1.806 55SCl

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, RCl

,SuCl=withthenumericalfactorCl PS

1.806 55 .uu

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.,

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IOC Manuals and Guides No. 56

Si (OH) ,2

CO anddissolvedorganicmaterial)donothaveanappreciableconductivity

signal.ItisforthisreasonthatthenewTEOS‐10thermodynamicdescriptionofseawater

(Milleroetal.(2008a),Millero(2010))hastheGibbsfunction

ofseawaterexpressedasa

functionofAbsoluteSalinityas

(

)

A,,

StpratherthanasafunctionofPracticalSalinity

SorofReferenceSalinity,R.STheissueofthespatialvariationinthecompositionof

seawaterisdiscussedmorefullyinappendixA.5.

Regardingpointnumber2,wenotethatitisperhapsdebatablewhichof

(1–0.001dens

S/( 1

gkg

−)),(1–0.001soln

S/( 1

gkg

−

)),(1–0.001add

S/( 1

gkg

−))or

(1–0.001*

S/( 1

gkg

−

))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P

Sinthiscontext,and

thechoiceofwhichofthese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/( 1

gkg

−))is

probablythemostappropriateversionoffreshwatermassfraction.

ItshouldbenotedthatthequantityA

Sappearingasanargumentofthefunction

()

A,,

StpistheAbsoluteSalinity(the“DensitySalinity”dens

SS≡)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R

S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PS

u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PS

unot

bemodifiedwithoutverycarefulconsideration.WorkingGroup127isrecommending

thatthepracticeofexpressingsalinityasPracticalSalinityinpublicationsbephasedoutin

favourofusingAbsoluteSalinityforthispurpose.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PS

u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PS

u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1

gkg

−

whichismorethan

tentimeslargerthantheprecisionofmodernsalinometers.Thuschangesassociatedwith

aseriesofimprovedestimatesofPS

u(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),westronglyrecommend

thattheReference‐CompositionSalinitycontinuetobeexpressedonthisscale;nochanges

inthevalueofPS

ushouldbeintroduced.

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IOC Manuals and Guides No. 56

ForseawaterofReferenceComposition,ReferenceSalinityR

S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R

S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(2010)hasarguedthattheDICcontentoftheReference

Compositionisprobablyabout1171

mol kg

−

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.,

2010)butat0.00551

gkg

−itisstilljustbelowtheuncertaintyof0.0071

gkg

−assignedto

theestimated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1

gkg

−(Seitzetal.,2010a).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,sothereisno

justificationforanupdateoftheReferenceCompositionatthistime.

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‐

HOmaterialinaseawatersampleatitstemperatureandpressure”.Onedrawbackof

thisdefinitionisthatbecausetheequilibriumconditionsbetween2

HO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25 Ct

°andthefixedseapressure0dbar,andwhen

thisisdone,theresultingmass‐fractionofnon‐2

HOmaterialiscalled“SolutionAbsolute

Salinity”(usuallyshortenedto“SolutionSalinity”),soln

S.Anothermeasureofabsolute

salinityisthe“Added‐MassSalinity”add

SwhichisR

S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25 Ct=°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.(2010)andWrightetal.

(2010b).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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,soln

S,oradd

S,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.

Thereisstillnosimplemeanstomeasureeithersoln

Soradd

S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”dens

Sforthispurpose.“DensitySalinity”dens

S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25 Ct

°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A

S,thatisdens

SS≡.Therearetwoclearadvantagesof

dens

SS≡overbothsoln

Sandadd

S.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

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IOC Manuals and Guides No. 56

dens

SS≡yieldsthebestavailableestimatesofthedensityofseawater.Thisisimportant

becauseinthefieldofphysicaloceanography,itisdensitythatneedstobeknowntothe

highestrelativeaccuracy.

Pawlowiczetal.(2010)andWrightetal.(2010b)foundthatwhilethenatureofthe

ocean’scompositionvariationschangesfromoneoceanbasintoanother,thefivedifferent

salinitymeasuresR

S,dens

S,soln

S,add

Sand*

Sareapproximatelyrelatedbythefollowing

simplelinearrelationships,(obtainedbycombiningequations(55)–(57)and(62)of

Pawlowiczetal.(2010))

 dens

0.35SS S

∗−≈− ,(A.4.1)

dens dens

AR R

1.0SS S

−≡ ,(A.4.2)

soln dens

AR R

1.75 δSSS−≈ ,(A.4.3)

add dens

AR R

0.78SS S

−≈ .(A.4.4)

Eqn.(A.4.2)issimplythedefinitionoftheAbsoluteSalinityAnomaly,

dens dens

ARAR

SSSS

δδ

≡≡−.NotethatinmanyTEOS‐10publications,thesimplernotation

isusedfordens dens

RAR

SSS

≡−,asalinitydifferenceforwhichaglobalatlasis

available(McDougalletal.(2010a)).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)

dens

0.35SS S

∗

−≈ ,(A.4.5)

dens dens

1.35SS S

∗

−≈ ,(A.4.6)

 soln dens

A* R

2.1SS S

−≈ ,(A.4.7)

add dens

1.13SS S

∗

−≈ .(A.4.8)

TheserelationshipsareillustratedonthenumberlineofsalinityinFigureA.4.1.ForSSW,

allfivesalinityvariablesR

S,dens

S,soln

S,add

Sand*

Sareequal.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.

Theselinearrelationshipsprovide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variousformsofsalinity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foursalinitydifferencesthatappearontheleft‐handsidesof

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

IOC Manuals and Guides No. 56

Eqns.(A.4.1)–(A.4.8).Pawlowiczetal.(2010)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.(2010).

Theseequationsarewrittenintermsofthevaluesofthenitrateandsilicateconcentrations

intheseawatersample(measuredin1

mol kg

−

),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,TA

andDICΔ,both

measuredin1

mol kg

−

.ForStandardSeawaterourbestestimatesofTAandDICare

0.0023 ( 35)S1

mol kg−andP

0.00208 ( 35)S1

mol kg

−

respectively(seePawlowicz(2010),

Pawlowiczetal.(2010)andthediscussionofthisaspectofSSWversusRCSWinWrightet

al.(2010b))).

()

1 1

*R 3 4

/ (g kg ) 18.1 TA 7.1 DIC 43.0 NO 0.1 Si(OH) (mol kg )SS −−−

−=−Δ−Δ−+ ,(A.4.9)

() ( )

dens 1 1

AR 3 4

/ (g kg ) 55.6 TA 4.7 DIC +38.9 NO 50.7 Si(OH) (mol kg )SS −−−

−=Δ+Δ + ,(A.4.10)

() ( )

soln 1 1

AR 3 4

/ (g kg ) 7.2 TA 47.0 DIC+ 36.5 NO 96.0 Si(OH) (mol kg )SS −−−

−=Δ+Δ + ,(A.4.11)

() ( )

add 1 1

AR 3 4

/ (g kg ) 25.9 TA 4.9 DIC +16.1NO 60.2 Si(OH) (mol kg )SS −−−

−=Δ+Δ + .(A.4.12)

ThestandarderrorofthemodelfitsinEqns.(A.4.9)–(A.4.11)aregivenbyPawlowiczet

al.(2010)atlessthan43

10 kg m

−−

(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dens

SS−

ratherthanasthecorrespondingdensityanomaly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

()

1 1

R* 3 4

/ (g kg ) 18.1 TA 7.1 DIC 43.0 NO 0.1 Si(OH) (mol kg )SS −−−

−=Δ+Δ+− ,(A.4.13)

() ( )

dens 1 1

A* 3 4

/ (g kg ) 73.7 TA 11.8 DIC +81.9 NO 50.6 Si(OH) (mol kg )SS −−−

−=Δ+Δ + ,(A.4.14)

() ( )

soln 1 1

A* 3 4

/ (g kg ) 25.3 TA 54.1 DIC +79.5 NO 95.9 Si(OH) (mol kg )SS −−−

−=Δ+Δ + ,(A.4.15)

() ( )

add 1 1

A* 3 4

/ (g kg ) 44.0 TA 12.0 DIC+59.1NO 60.1 Si(OH) (mol kg )SS −−−

−=Δ+Δ + .(A.4.16)



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A.5 Spatial variations in seawater composition

WhentheoceanographicdataneededtoevaluateEqn.(A.4.10)fordens

SS−isnot

available,thelook‐uptablemethodofMcDougalletal.(2010a)isrecommendedto

evaluatedens dens

AR AR

SS SS

δδ

≡≡−.Thefollowingparagraphsdescribehowthismethod

wasdeveloped.

InaseriesofpapersMilleroetal.(1976a,1978,2000,2008b)andMcDougalletal.

(2010a)havereportedondensitymeasurementsmadeinthelaboratoryonsamples

collectedfromaroundtheworld’soceans.Eachsamplehas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R

SaccordingtoEqn.(A.3.3),whilethedensitymeasurementmeas

impliesan

AbsoluteSalinitydens

SS≡byusingtheequationofstateandtheequality

()

meas dens

A,25 C,0dbarS

ρρ

=°.Thedifferencedens

SS−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AR

SS−

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1

(g kg )

−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dataonA

,basedonlaboratorymeasurementsofdensity,was

regressedagainstthesilicateconcentrationoftheseawatersamples,McDougalletal.

(2010a)foundthesimplerelation

(

)

11 1

AAR 4

/(gkg ) ( )/(gkg ) 98.24 Si(OH) /(mol kg )SSS

−− −

=− = .Global(A.5.1)

Thisregressionwasdoneoverallavailabledensitymeasurementsfromtheworldocean,

andthestandarderrorinthefitwas0.0054gkg‐1.

ThedependenceofA

onsilicateconcentrationisobservedtobedifferentineach

oceanbasin,andthisaspectwasexploitedbyMcDougalletal.(2010a)toobtainamore

accuratedependenceofA

onlocationinspace.FordataintheSouthernOceansouthof

30oSthebestsimplefitwasfoundtobe

()

/(gkg ) 74.884 Si(OH) /(mol kg )S

−−

=,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30

≥− °)

[

]

()

(

)

1 1

/ (g kg ) 74.884 1 0.3622 / 30 1 Si(OH) / (mol kg )S

δλ

− −

=+ °+ ,Pacific(A.5.3)

[

]

()

(

)

1 1

/ (g kg ) 74.884 1 0.3861 / 30 1 Si(OH) / (mol kg )S

δλ

− −

=+ °+ ,Indian(A.5.4)

[

]

()

(

)

1 1

/ (g kg ) 74.884 1 1.0028 / 30 1 Si(OH) / (mol kg ) .S

δλ

− −

=+ °+ Atlantic(A.5.5)

TheserelationshipsbetweentheAbsoluteSalinityAnomalyAAR

SSS

−andsilicate

concentrationhavebeenusedbyMcDougall,JackettandMillero(2010a)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

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IOC Manuals and Guides No. 56

PracticalSalinityaswellasitsspatiallocationintheworldocean.Thiscomputer

algorithm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.(2010a)

usestherelationshipfoundbyFeisteletal.(2010c)thatappliesintheyears2006‐2009,

namely

()

AR RSO

0.087 g kg 1SS SS

−

−= ×− ,(A.5.6)

whereSO

S=35.16504gkg–1isthestandard‐oceanReferenceSalinitythatcorrespondsto

thePracticalSalinityof35.

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dens

SS≡(asitshouldbe)

comparedwithcallingthesameTEOS‐10densityalgorithmwithR

S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.



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

 1000p>dbar.Theverticalaxisisthemagnitudeofthedifference

betweenevaluatingthedensitygradientusingA

SversusR

S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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R

SandEOS‐80withP

S.FigureA.5.2showsthe

magnitudeoftheimprovementinthe“thermalwind”inthepartoftheoceanthatis

deeperthan1000 m throughtheadoptionofTEOS‐10butignoringtheinfluenceof

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IOC Manuals and Guides No. 56

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18%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18%ofthatoftheverticalaxisofFig.A.5.1).Ifthe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

 1000 dbarp>.Theverticalaxisisthemagnitudeofthedifference

betweenevaluatingthedensitygradientusingR

Sasthesalinity

argumentintheTEOS‐10expressionfordensitycomparedwithusingP

S

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A

SgiventheinputvariablesPracticalSalinity P

S,longitude

,latitude

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

()

A,,

StpisdefinedasthesumoftheGibbsfunctionfor

purewater

()

tpandthesalinepartoftheGibbsfunction

(

)

A,,

Stpsothat

()

(

)

(

)

,, , ,, .

S tp g tp g S tp=+ (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

()

tp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needtofortheGibbsfunction

()

tp(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 0.0743 MPa CpP −

⎡⎤

−++× °

⎣⎦

to40°C),andinthepressure

range4

0 < 10 dbarp<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.

All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

()

tp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hadenteredthedeterminationof

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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

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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(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ʺnewʺ 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

(

)

dd d dhvP Tt S

ημ

−=++ .(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

()

ddupPv++ 

where

()

pP+istheabsolutepressure.Herehisthespecificenthalpy(i.e.enthalpyper

unitmassofseawater),uisthespecificinternalenergy,1

−

=isthespecificvolume,

()

Tt+istheabsolutetemperature,

isthespecificentropyand

istherelative

chemicalpotential.Influiddynamicsweusuallydealwithmaterialderivatives,ddt,that

is,derivativesdefinedfollowingthefluidmotion,ddtt

∂∂ + ⋅∇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

()

d1d d d

dd d d

hP S

tt tt

−=++ (A.7.2)

Notethattheconstancyofentropy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

() ( )

ρρρ

+∇⋅ = = −∇⋅uF(A.8.1)

Forsucha“conservative”property,intheabsenceoffluxesC

F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)

(

)

,, 0.

xyz

ρρ

∂∂ +∇⋅ =u(A.8.2)

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1C=andC

F0inEqn.(A.8.1)].Infact,lookingaheadto

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appendicesA.20andA.21,ifwestrictlyinterpret

uasthemassfluxperunitareaof

pureseawater(i.e.ofonlypurewaterplusdissolvedmaterial)andspecifically,that

u

excludesthefluxofparticulatematter,thentheright‐handsideofthecontinuityequation

(A.8.2)shouldbeA

S,thenon‐conservativesourceofmassduetobiogeochemical

processes.ItcanbeshownthattheinfluenceofthissourcetermA

Sinthecontinuity

equationontheevolutionequationforAbsoluteSalinityislessimportantbythefactor

()

ˆˆ

1SS−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0.5u

+⋅+Φuu

(seeEqn.(B.15))and

ConservativeTemperatureΘ(orequivalently,potentialenthalpy0

h)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R,SAbsoluteSalinityA,Spotential

temperature,

enthalpy,hinternalenergy,uentropy,

density,

potentialdensity

specificvolumeanomaly

andtheBernoullifunction0.5h

+⋅+Φuu

(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the

remineralization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R

Stobeaconservativevariable.

However,wenotethattheerrorinvolvedwithassumingthatR

S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andA.5andPawlowiczet

al.(2010)).

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

0.5u=+ ⋅+Φuu

,(iii)ConservativeTemperature

,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

,PreformedSalinity*

S,andtotalenergy



are“isobaricconservative”.Inaddition,theBernoullifunction

andspecificenthalpyh

arealso“isobaricconservative”(seeEqn.(B.17)andthediscussionthereafter).

Somevariablesthatarenot“isobaricconservative”includepotentialtemperature,



internalenergy,uentropy,

density,

potentialdensity,

andspecificvolume

anomaly.

EnthalpyhandConservativeTemperature

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.

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AppendicesA.18andA.21showthatforallpracticalpurposeswecantreatΘand0

h

asbeing“conservative”variables(andhencealso“isobaricconservative”variables);doing

soignoresthedissipationofmechanicalenergy

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,

(andpotentialenthalpy0

h),

(2)PreformedSalinity*

S,and

(3)totalenergy

.



Herewecommentbrieflyonthelikelyerrorsinvolvedwithassumingvariablesother

than*

SandΘtobeconservativevariablesinoceanmodels.IfonetookAbsoluteSalinity

S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1

0.025 g kg

−

intheNorthPacific,

implyingdensityerrorsof3

0.020 kg m

−

.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R

S

insteadofwithA

S,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remineralization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anAbsoluteSalinityAnomaly

asdescribedinsectionA.20.1andEqns.(A.20.3)–(A.20.5).

Theerrorsincurredinoceanmodelsbytreatingpotentialtemperature

asbeing

conservativehavenotyetbeenthoroughlyinvestigated,butMcDougall(2003)and

Tailleux(2010)havemadeastartonthistopic.McDougall(2003)foundthattypicalerrors

in

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

inoceanmodelsreducestheseerrorsbyalmost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

theentropyandtheconcentrationsofeachspecieswouldbefunctionsofpressure.

Turbulentmixingactsinthecomplementarydirection,tendingtomakesalinityand

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S

Fisgivenbyequation(58.11)ofLandauand

Lifshitz(1959)andbyEqn.(B.23)below. S

Fconsistsnotonlyoftheproductoftheusual

moleculardiffusivityandA

−∇ ,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

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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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

andalltypesofsalinity.Theconstancyof

entropy

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havethe“potential”property.Thermodynamicpropertiesthatpossesthe“potential”

attributeincludepotentialtemperature,

potentialenthalpy0,hConservative

TemperatureΘandpotentialdensity

(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,

specific

volumeanomaly,

totalenergy

andtheBernoullifunction.

FromEqn.(B.17)we

noticethatintheabsenceofmolecularfluxesandthesourcetermofAbsoluteSalinity,the

Bernoullifunction

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

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

,thepotentialtemperature

andtheConservativeTemperature

ofafluidparceltochangeduringanisohaline

andadiabaticprocess.Thisisbecausethedissipationofmechanicalenergy

causes

increasesin

,

andΘ(seetheFirstLawofThermodynamics,Eqns.(A.13.3)‐(A.13.5)).

Whilethedissipationofmechanical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

mechanicalenergy.Withthisdefinitionof“adiabatic”,aprocessthatisbothisohalineand

adiabaticdoesimplythattheentropy

,potentialtemperature

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

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“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

,*

S

andΘ(approximately)are“conservative”(andhencealso“isobaricconservative”).Since

doesnotpossesthe“potential”property,wenowconcludethatonlymassandthetwo

variables*

Sand

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

isnota100%conservativevariable,itis

approximatelytwoordersofmagnitudeclosertobeingatotallyconservativevariable

thanareeitherpotentialtemperatureorentropy.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”? “conservative”? “isobaric conservative”? function of

()

A,,Stp?

S x

S x 1 x 1

,SS

x 1 x 1 x

t x x x

x x

h x x 2

,hΘ 3 3

u x x x

x x 4 x

x 4 4 x

x x x

x x

x x x

x x x 5

x x x x



1TheremineralizationoforganicmatterchangesR

SlessthanitchangesA.S

2Taking

andtheeffectsofremineralizationtobenegligible.

3Taking

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

(

)

0,cp

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”variables,andwhethertheyarefunctionsofonly

()

A,,Stp.

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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

()

A,,Stpbutalsoof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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,

(seesection

3.4),specificvolumeanomaly,

(seesection3.7),orthobaricdensity,v

(seeappendix

A.28)andNeutralDensityn

(seesection3.14andappendixA.29).



A.10 Proof that

()

A,S

θη

= and

(

)

A,S

Θ=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

tobefunctionsofentropy

(inplaceoftemperaturet),thatis,consideringthe

functionalformofhand

tobe

(

)

A,,hhS p



and

(

)

A,, ,Sp

μμ η



itfollowsfromthe

fundamentalthermodynamicrelation(Eqn.(A.7.1))that

() ()

(

)

(

)

AAA0AA

,,0d ,,0 d d ,,0d ,

hS h S S T S S

ηη η θημη

+=++



(A.10.1)

whichshowsthatspecificentropy

issimplyafunctionofAbsoluteSalinityA

Sand

potentialtemperature,

thatis

(

)

A,S

ηθ

=,withnoseparatedependenceonpressure.

Itfollowsthat

(

)

A,.S

θη

=

Similarly,fromthedefinitionofpotentialenthalpyandConservativeTemperaturein

Eqns.(3.2.1)and(3.3.1),at0p=dbaritcanbeseenthatthefundamentalthermodynamic

relation(A.7.1)implies

()

(

)

0AA

dd,,0d.

cT SS

θη μ θ

Θ= + +(A.10.2)

ThisshowsthatConservativeTemperatureisalsosimplyafunctionofAbsoluteSalinity

andpotentialtemperature,

()

A,S

Θ=Θ ,withnoseparatedependenceonpressure.Itthen

followsthatΘmayalsobeexpressedasafunctionofonlyA

Sand.

Itfollowsthat



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

andΘaswellasinsitutemperature.t

Webeginbynotingthatthethreestandardderivativesof

()

A,,hhStp=whenin

situtemperature tistakenasthe“temperature‐like”variableare

()

(

)

(

)

AA0A

,,, ,, ,

hS Stp T t Stp

μμ

∂∂ = − + (A.11.1)

()

(

)

(

)

AA0A

,,, ,, ,

hT c Stp T t Stp

∂∂ = = + (A.11.2)

and

()

(

)

(

)

AA0A

,,, ,, .

h P vS tp T tv S tp∂∂ = − + (A.11.3)

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Nowconsideringspecificenthalpytobeafunctionofentropy(ratherthanof

temperaturet),thatis,taking

(

)

A,, ,hhS p



thefundamentalthermodynamicrelation

(A.7.1)becomes

()

AA0 A

dd dd

hhSTt S

ηημ

+=++



while

A,,

hP v

∂

∂=



(A.11.4)

sothat

()

,Sp

hTt

∂∂ = +



and A,.

∂

∂=



(A.11.5)

Nowtakingspecificenthalpytobeafunctionofpotentialtemperature(ratherthanof

temperaturet),thatis,taking

(

)

A,, ,hhS p

=thefundamentalthermodynamicrelation

(A.7.1)becomes

()

AA0 A

dd dd

hhSTt S

θθημ

+=++

 while

A,.

hP v

∂

∂=

(A.11.6)

Toevaluatetheh

partialderivative,itisfirstwrittenintermsofthederivativewith

respecttoentropyas

()

AA0

,,,

Sp Sp

hh Tt

θθηθ

ηη

==+



(A.11.7)

where(A.11.5)hasbeenused.Thisequationcanbeevaluatedat0p

whenitbecomes

(thepotentialtemperatureusedhereisreferencedtor0p

)

()

(

)

AA0

,0 ,,0 .

hcS T

θθ

ηθ

===+

(A.11.8)

Thesetwoequationsareusedtoarriveatthedesiredexpressionforh

namely

()

(

)

()

,,0 .

hcST

(A.11.9)

ToevaluatetheA

partialderivative,wefirstwritespecificenthalpyinthefunctional

form

()

,,,hhS S p

ηθ



andthendifferentiateit,finding

AA A

,,,

SS S

ppSp

hhh

θηη



(A.11.10)

ThepartialderivativeofspecificentropyT

−(Eqn.(2.10.1))withrespecttoAbsolute

Salinity,AA

SST

=− isalsoequaltoT

−

sincechemicalpotentialisdefinedbyEqn.

(2.9.6)asA

=.SincethepartialderivativeofentropywithrespecttoA

Sin(A.11.10)is

performedatfixedpotentialtemperature(ratherthanatfixedinsitutemperature),thisis

equaltoT

−evaluatedat0.p=Substitutingbothpartsof(A.11.5)into(A.11.10)we

havethedesiredexpressionforA

namely

()

(

)

(

)

AA0A

,,, , ,0.

hStpTtS

μμθ

=−+

(A.11.11)

Noticethatthisexpressioncontainssomethingsthatareevaluatedatthegeneralpressure

pandoneevaluatedatthereferencepressurer0.p



NowconsideringspecificenthalpytobeafunctionofConservativeTemperature

(ratherthanoftemperaturet),thatis,taking

(

)

ˆ,, ,hhS p

=Θthefundamental

thermodynamicrelation(A.7.1)becomes

()

AA0 A

ˆˆ

dd dd

hhSTt S

ημ

ΘΘ+ = + + while

ˆ.

hP v

∂

∂=(A.11.12)

Thepartialderivativeˆ

hΘfollowsdirectlyfromthisequationas

() ()

ˆ.

Sp S

hTt Tt

ηη

ΘΘΘ

=+ =+ (A.11.13)

At0p=thisequationreducesto

()

ˆ,

hcT

θη

ΘΘ

===+ (A.11.14)

andcombiningthesetwoequationsgivesthedesiredexpressionforˆ

namely

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

IOC Manuals and Guides No. 56

(

)

()

ˆ.

=+(A.11.15)

ToevaluatetheA

ˆS

hpartialderivativewefirstwritehinthefunctionalform

()

,,,hhS S p

=Θ



andthendifferentiateit,finding(usingbothpartsofEqn.(A.11.5))

()()

ˆ,, .

hStpTt

μη

Θ=++(A.11.16)

ThedifferentialexpressionEqn.(A.11.12)canbeevaluatedat0p

wheretheleft‐hand

sideissimply0d

cΘsothatfromEqn.(A.11.12)wefindthat

(

)

()

,,0,

μθ

ηθ

Θ=− +(A.11.17)

sothatthedesiredexpressionforA

ˆS

his

()

(

)

()

ˆ,, , ,0.

hStp S

μμθ

=−

+(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.



A.12 Differential relationships between ,,

and A

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

() () ()

()

() ( ) ( ) ()

()

() () ()

()

0A 0A

dd 0d 0d

d0d.

Tt pS c p Tt S

Tt Tt

cp S

ημ θ μ μ

μμ

θθ

+⎡⎤

++ = + −+

⎣⎦

⎡⎤

=Θ+−

⎢⎥

⎣⎦

(A.12.1)

Thequantity

()

dpS

isnowsubtractedfromeachofthesethreeexpressionsandthe

wholeequationisthenmultipliedby

(

)

(

)

TTt

+obtaining

() ()

(

)

(

)

(

)

00AA

d0d 0dd0d.

pTp

TcT Sc S

θη θ θμ μ

+= −+ =Θ− (A.12.2)

Fromthisfollowsallthefollowingpartialderivativesbetween,,

andA,S



()

A,,0 ,

ScS c

θθ

Θ= 

(

)

(

)

(

)

A0A

,,0 ,,0 ,

STp

STSc

θμθ θμ θ

⎡⎤

Θ= −+

⎣⎦

(A.12.3)



()

STc

Θ=+ 

(

)

A,,0 ,

μθ

Θ= (A.12.4)



()

(

)

A0A

,,0,

STcS

θθθ

=+ 

(

)

(

)

(

)

A0A A

,,0 ,,0,

STp

TScS

θθμθθ

=+ (A.12.5)

()

A,,0,

SccS

θθ

Θ=

(

)

(

)

(

)( )

AA0A A

,,0 ,,0 ,,0,

STp

STScS

θμθ θμθ θ

Θ⎡⎤

=− − +

⎣⎦

(A.12.6)



()()

AA0

,,0 ,

ScS T

θθ

=+

(

)

AA,,0,

ημθ

=− (A.12.7)



()

ScT

Θ=+

(

)

(

)

AA0

,,0 .

SST

μθ θ

Θ=− + (A.12.8)

Thethreesecondorderderivativesof

(

)

ˆ,S

arelistedinEqns.(P.14)and(P.15)of

appendixP.Thecorrespondingderivativesof

()

ˆ,S

,namelyˆ

,A

ˆS

,ˆ

ΘΘ ,A

ˆS

Θand

ˆSS

canalsobederivedusingEqn.(P.13),obtaining

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IOC Manuals and Guides No. 56

Θ=Θ

, A

ˆS

=−Θ



,

()

θθ

ΘΘ

=− Θ



,

() ()

A23

ˆSS

θθ

ΘΘ

=− +

ΘΘ





,(A.12.9a,b,c,d)

and AA A A A

ˆ2

SS S S S

θθ θ θ

⎛⎞

ΘΘΘΘ

=− + −

⎜⎟

ΘΘΘΘΘ

⎝⎠







,(A.12.10)

intermsofthepartialderivatives

,A

,

,A

andAA

whichcanbeobtained

bydifferentiatingthepolynomial

(

)

A,S

fromtheTEOS‐10Gibbsfunction.



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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)).

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kineticenergypluspotentialenergy,leavingwhatmight

looselybetermedtheevolutionequationof“heat”,Eqn.(A.13.1)(LandauandLifshitz

(1959),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)

d1d ,

hP h

ρρερ

⎛⎞

−=−∇⋅−∇⋅++

⎜⎟

⎝⎠

FF S(A.13.1)

whereR

FisthesumoftheboundaryandradiativeheatfluxesandQ

F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,

istherateofdissipationofmechanicalenergyper

unitmass,transformedintointernalenergyandA

Sistherateofincreaseof

enthalpyduetotheinteriorsourcetermofAbsoluteSalinitycausedbyremineralization.

Thederivationof(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(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(A.13.1)

withthecontinuityequation

(

)

ρρ

∂∂+∇⋅ =utofindthefollowingdivergenceformof

theFirstLawofThermodynamics,

() ()

ht h h

ρρ ρερ

∂∂+∇⋅ −=−∇⋅−∇⋅++uFFS(A.13.2)

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

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pressure),mixesandcontracts.Thedissipationofmechanical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thesetermsdocauseasmallincreaseinthe

enthalpyofthemixturewithrespecttothatofthetwooriginalparcels.Apartfromthese

non‐conservativesourceterms,undertheseassumptionsEqn.(A.13.2)reducestothe

statementthatthevolumeintegratedamountof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0

h(referencedtozeroseapressure)isthebestthermodynamic

variabletorepresent“heatcontent”intheocean.

TheFirstLawofThermodynamics(A.13.1)canbewritten(usingEqn.(A.7.2))asan

evolutionequationforentropyasfollows

()

dd .

Tt h

ρμ ρερ

⎛⎞

++ =−∇⋅−∇⋅++

⎜⎟

⎝⎠

FF S(A.13.3)

TheFirstLawofThermodynamics(A.13.1)canalsobewrittenintermsofpotential

temperature

(withrespecttoreferencepressurer

p)bysubstitutingEqns.(A.11.9)and

(A.11.11)intoEqn.(A.13.1)as(fromBaconandFofonoff(1996)andMcDougall(2003))

()

() () ( ) ( )

r0r

Tt S

cp p T t p

Tt t

ρμμ

ρε ρ

⎛⎞

+⎡⎤

+−+ =

⎜⎟

⎣⎦

⎜⎟

⎝⎠

−∇⋅ −∇⋅ + +FF S

(A.13.4)

where0

TistheCelsiuszeropoint(0

T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)

()

() () ()

()

Tt Tt S

Tt T t

ρμμ

θθ

ρε ρ

⎛⎞

⎡⎤

Θ+− =

⎜⎟

⎢⎥

⎜⎟

⎢⎥

⎣⎦

⎝⎠

−∇⋅ −∇⋅ + +FF S

(A.13.5)

where0

c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A

−

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ddt

,ddt

andddt

Θ.

Aquickrankingofthesethreevariables,,



and,

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

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“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

()

Tt+and

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

()

(

)( )

r0 0p

cpT t T

++and

(

)

(

)

(

)

0rT

pTt p

μμ

⎡

⎤

−+

⎣

⎦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

(

)( )

TtT

++and

() ()( )( )

0pTtT

μθ

⎡⎤

−++

⎣⎦

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

()

Tt+whichvariesbyabout15%attheseasurface(

(

)

273.15 40 273.15 1.146+≈),the

termthatmultipliesddt

in(A.13.4)isdominatedbythevariationsintheisobaric

specificheat

()

cStpwhichismainlyafunctionofA

Sandwhichvariesby5%atthe

seasurface(seeFigure4),whilethematerialderivativeofConservativeTemperature

ddtΘinEqn.(A.13.5)ismultipliedbytheproductofaconstant“heatcapacity”0

cand

thefactor

()

(

)

TtT

++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andisalsoveryclosetounityinthedeepocean.



FigureA.13.1.Contours(inC°)ofthedifference

−

Θbetweenpotential

temperature

andConservativeTemperature

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betweenpotentialtemperature

andConservativeTemperatureΘat

theseasurfaceisshowninFigureA.13.1(afterMcDougall,2003).Ifanoceanmodelis

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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(0

casgivenbyEqn.(3.3.3)),theneglectofthenon‐conservativesource

termsthatshouldappearintheprognosticequationfor

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0

c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

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.



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0.

Inthisway0

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0

or

(

)

A,,0dbar

isthe

“heatcontent”perunitmassofseawater(seealsoappendixA.21foradiscussionofthis

point).

Theconservativeform(A.21.15)impliesthattheturbulentdiffusive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difference

betweenthesemeantemperaturegradients.

Considerfirsttherespectivetemperaturegradientsalongtheneutraltangentplane.

FromEqn.(3.11.2)wefindthat

()

(

)

nn n

θθ

αβ θ αβ

ΘΘ

∇=∇ = ∇Θ(A.14.1)

sothattheepineutralgradientsof

and

arerelatedbytheratiosoftheirrespective

thermalexpansionandsalinecontractioncoefficients,namely

(

)

()

θθ

αβ

θαβ

ΘΘ

∇= ∇Θ(A.14.2)

Thisproportionalityfactorbetweentheparalleltwo‐dimensionalvectorsn

∇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

,

andA

Sarerelatedby(using

()

ˆ,S

θθ

Θ)

ˆˆ

nnSn

θθ θ

∇=∇Θ+ ∇ (A.14.3)

andusingtheneutralrelationship

(

)

Ann

αβ

ΘΘ

∇

=∇Θwefind

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()

ˆˆ

nSn

θθ αβθ

ΘΘ

Θ⎡⎤

∇= + ∇Θ

⎣⎦ (A.14.4)

Also,insection3.13wefoundthat,

bn bn

∇

=∇Θsothatwecanwritetheequivalent

expressions

()

ˆˆ

nbS

θθ

αβ

αβθ

αβ

ΘΘ ΘΘΘ

∇⎡⎤

===+

⎣⎦

∇Θ (A.14.5)

anditcanbeshownthatˆ

αθ

=and

(

)

ˆˆ

βαβθθ

ΘΘΘ

⎡⎤

⎣⎦ ,thatis,

ˆˆ

βαθθ

ΘΘ

=+ Thepartialderivativesˆ

andA

ˆS

inthelastpartofEqn.(A.14.5)

arebothindependentofpressurewhile

Θisafunctionofpressure.Thisratio,Eqn.

(A.14.5),oftheepineutralgradientsof

and

isshowninFigureA.14.1at0p=,

indicatingthattheepineutralgradientofpotentialtemperatureissometimesmorethat1%

differenttothatofConservativeTemperature.Thisrationn

∇

∇Θisonlyaweak

functionofpressure.Thisratio,nn

∇

∇Θ(i.e.Eqn.(A.14.5)),isavailableintheGSW

OceanographicToolboxasfunctiongsw_ntp_pt_vs_CT_ratio_CT25.

SimilarlytoEqn.(A.14.3),theverticalgradientsarerelatedby

ˆˆ

zzS

θθ θ

=Θ+ (A.14.6)

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

ˆˆ

αβθ

−ΘΘ

Θ⎡⎤

=+⎣⎦

Θ(A.14.7)

Forvaluesofthestabilityratio

closetounity,theratiozz

isclosetothevaluesof

∇∇ΘshowninFigureA.14.1.ForothervaluesofR

,Eqn.(A.14.7)canbe

calculatedandplotted.



FigureA.14.1.Contoursof

()

1100%

∇∇Θ−× at0p

,showingthepercentage

differencebetweentheepineutralgradientsof

andΘ.Thebluedots

arefromtheoceanatlasofGouretskiandKoltermann(2004)at0p=.



Asnotedinsection3.8thedianeutraladvectionofthermobaricityisthesamewhen

quantifiedintermsofpotentialtemperatureaswhendoneintermsofConservative

Temperature.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

(

)

(

)

bbnn n n

θθθ

∇

⋅∇ ∇ Θ⋅∇ Θ (seesection3.9)which

canbeexpressedas

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()

bbbbb

bbbb

ˆˆ

CCCTC

CCTC

θθθθ

θθ

αβ

θθαβθ

αβ

ΘΘ ΘΘΘ

ΘΘΘ

⎛⎞

∇⎡⎤

===+

⎜⎟ ⎣⎦

∇Θ ⎝⎠ (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

or

arewithinhalfapercentofeachotherat0p=.



FigureA.14.2.Contoursofthepercentagedifferenceof

(

)()

bbnn

θθ

∇

∇Θ 

fromunityat0p

dbar.



A.15 Derivation of the expressions for ,,

θθ

βα

and

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

and

andthehalinecontractioncoefficients



and.

Θ

InordertoderiveEqn.(2.18.2)for

wefirstneedanexpressionforA,.

∂∂ This

isfoundbydifferentiatingwithrespecttoinsitutemperaturetheentropyequality

()

[

]

()

AAArr

,, , ,, , ,Stp S Stpp p

ηηθ

=whichdefinespotentialtemperature,obtaining

(

)

()

(

)

()

Ar Ar

,, ,, .

,, ,,

TTT

Stp g Stp

TSpgSp

ηθ θ

∂==

∂(A.15.1)

ThisisthenusedtoobtainthedesiredexpressionEqn.(2.18.2)for

asfollows

()

AAA

AAr

,,,

,, , ,

11 .

,, ,,

TP TT

PTT

Sp Sp Sp

Stpg S p

v v T T g S tp g S tp

αθ

−

⎛⎞

∂∂∂

== =

⎜⎟

∂∂∂

⎝⎠

(A.15.2)

InordertoderiveEqn.(2.18.3)for

wefirstneedanexpressionforA,.

t∂Θ ∂ This

isfoundbydifferentiatingwithrespecttoinsitutemperaturetheentropyequality

()

[

]

()

AAA

,, , ,,

Stp S Stp

ηη

=Θ obtaining

() ()( )

A0A

,, ,, ,

TTTp

Sp S

Stp T g Stpc

ηθ

∂Θ ∂Θ

==−+

∂∂ (A.15.3)

wherethesecondpartofthisequationhasusedEqn.(A.12.4)for

ΘThisisthenused

toobtainthedesiredexpressionEqn.(2.18.3)for

asfollows

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IOC Manuals and Guides No. 56

()

()( )

AAA

A0 A

,,,

11 .

,, ,,

PTT

Sp Sp Sp

gStp

vvTT gStpTgStp

αθ

−

Θ⎛⎞

∂∂∂Θ

== =−

⎜⎟

∂Θ ∂ ∂ +

⎝⎠

(A.15.4)

InordertoderiveEqn.(2.19.2)for

wefirstneedanexpressionforA,.

∂∂ This

isfoundbydifferentiatingwithrespecttoAbsoluteSalinitytheentropyequality

()

[

]

()

AAArr

,, , ,, , ,Stp S Stpp p

ηηθ

=whichdefinespotentialtemperature,obtaining

()( )

()

()()

()()()

AAr

Ar A

AArAr

,, , ,

,, ,,

,, ,, ,, ,

ST ST TT

Stp S p

TSp Stp

cS p

Stp g S p g S p

θθη η θ

θμθ μ

θθ

∂⎡⎤

=−

⎣⎦

∂

+⎡⎤

=−

⎣⎦

⎡⎤

=−

⎣⎦

(A.15.5)

whereEqns.(A.12.5)and(A.12.7)havebeenusedwithageneralreferencepressurer

p

ratherthanwithr0.p=Bydifferentiating

[

]

(

)

AA r

,,,,,SStppp

ρρ θ

=withrespectto

AbsoluteSalinityitcanbeshownthat(Gill(1982),McDougall(1987a))

AAA

,,,

11 ,

pTpTp

SSS

θθ

ρρθ

βα

ρρ

∂∂∂

== +

∂∂∂

(A.15.6)

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



()

()()( )

()()

AAA Ar

AAA

,, ,, , ,

,, .

,, ,, ,,

TP S T S T

PPTT

gStp Stp S p

gStp

g S tp g S tpg S tp

θθ

⎡

⎤

−

⎣

⎦

=− + (A.15.7)

Notethattheterms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).

InordertoderiveEqn.(2.19.3)for

wefirstneedanexpressionforA,.

S∂Θ ∂ 

ThisisfoundbydifferentiatingwithrespecttoAbsoluteSalinitytheentropyequality

()

[

]

()

AAA

,, , ,,Stp S Stp

ηη

=Θ obtaining(usingEqns.(A.12.4)and(A.12.8))

()

()()()

A0A

A0 A

,,0 ,,

,,0 ,, .

SSTp

Stp

STStpc

STgStpc

ηηη

μθ θμ

θθ

∂Θ ⎡⎤

=Θ −

⎣⎦

∂

⎡⎤

=−+

⎣⎦

⎡⎤

=−+

⎣⎦

(A.15.8)

Differentiating

[]

(

)

ˆ,,,,SStpp

ρρ

=Θ withrespecttoAbsoluteSalinityleadsto

AAA

,,,

11 ,

pTpTp

SSS

ρρ

βα

ρρ

ΘΘ

∂∂∂Θ

==+

∂∂∂

(A.15.9)

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

Θnamely

()

()()()()

()()

AAAA0

AAA

,, ,, , ,0

,, .

,, ,, ,,

TP S T S

PPTT

gStp Stp S T

gStp

g S tp g S tpg S tp

θθ

⎡

⎤

−+

⎣

⎦

=− + (A.15.10)

Notethattheterms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).



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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potentialdensity)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mechanicalenergy,the

term

canbedroppedfromtheFirstLawofThermodynamics,(A.13.1),thisequationbecoming

() ( )

ρρ

+∇⋅ = −∇⋅ −∇⋅uFF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

()

u

throughtheboundaryisalsozero.Henceitisapparentthatthevolumeintegralof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

12 ,mmm

=(A.16.2)

1A1 2A2 A

,mS mS mS+=(A.16.3)

11 2 2 ,mh mh mh+=(A.16.4)

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

11 2 2 .mm mm

ηδηη

++=(A.16.5)

HereA,Shand

arethevaluesofAbsoluteSalinity,enthalpyandentropyofthefinal

mixedfluidand

istheproductionofentropy,thatis,theamountbywhichentropyis

notconservedduringthemixingprocess.Entropy

isnowregardedasthefunctional

form

()

A,,Shp

ηη

=andisexpandedinaTaylorseriesofA

Sandhaboutthevaluesof

Sandhofthemixedfluid,retainingtermstosecondorderin

[

]

A2 A1 A

SS S

−

=Δ andin

[]

21 .hh h−=ΔThen1

and2

areevaluatedand(A.16.4)and(A.16.5)usedtofind

() ( )

{

}

AAA

1AA

222.

hh hS S S

mm hhS S

δη η η η

=− Δ + Δ Δ + Δ

  (A.16.6)

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

(

)

A,,

Stpofseawater.

TheSecondLawofThermodynamicstellsusthattheentropyexcess

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,

 0,

(A.16.7)

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AA 0,

(A.16.8)

()

AAA

hS hh S S

ηηη



(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,

theconstraintswouldseemtobethreedifferentconstraintsonvariousthirdderivative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,0

TT t=+)



hSp

−

∂

(A.16.10)

∂

==−

∂

(A.16.11)

andfromtheserelationsthefollowingexpressionsforthesecondorderderivativesof



canbefound,

212

hh p

Sp Sp

−−

∂∂ −

== =

∂

(A.16.12)

(

)

Sh pT

pSp

hS h c T

ημ

∂−

∂⎛⎞

== =−

⎜⎟

∂∂ ∂ ⎝⎠

(A.16.13)

and

(

)

(

)

,,,

hp Tp S p

ShS

TcT

μμ

∂− ∂−

∂∂

== −

∂∂∂

∂

⎡⎤

⎛⎞

=− − ⎢⎥

⎜⎟

⎝⎠

⎣⎦



(A.16.14)

ThelastequationcomesfromregardingA

as

[

]

(

)

,,,,.

ShStpp

ηη

 

Theconstraint(A.16.7)that0

simplyrequires(from(A.16.12))thattheisobaric

heatcapacityp

cispositive,orthat0.

Theconstraint(A.16.8)thatAA 0,



requires(from(A.16.14))that

SS pT

gcT

⎡

⎤

⎛⎞

>−

⎢

⎥

⎜⎟

⎝⎠

⎣

⎦(A.16.15)

thatis,thesecondderivativeoftheGibbsfunctionwithrespecttoAbsoluteSalinityAA

g

mustexceedsomenegativenumber.Theconstraint(A.16.9)that

()

AAA

hS hh S S

ηηη





requiresthat(substitutingfrom(A.16.12),(A.16.13)and(A.16.14))

30,

Tc >(A.16.16)

andsincetheisobaricheatcapacitymustbepositive,thisrequirementisthatAA 0,

g>

andsoismoredemandingthan(A.16.15).

Weconcludethatwhiletherearethethreerequirements(A.16.7)to(A.16.9)onthe

functionalformofentropy

()

A,,Shp

ηη

=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

()

A,,

Stp,

(A.16.17)

and

AA 0.

g>(A.16.18)

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TheSecondLawofThermodynamicsdoesnotimposeanyadditionalrequirementonthe

crossderivativesA

noronanythirdorderderivativesoftheGibbsfunction.

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

()

A,,Stp



(LandauandLifshitz(1959)).Thatis,themolecularfluxofsaltisproportionalto.

−∇ 

Expanding

−∇ intermsofgradientsofAbsoluteSalinity,oftemperature,andof

pressure,onefindsthatthefirsttermisAASS

−

∇andinordertoavoidanunstable

explosionofsaltonemusthaveAAA

SSS

>So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

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0

g>(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))

(A.16.19)

and

()

TP PP TT

gg<(A.16.20)

recognizingthatTT

isnegative(TP

may,anddoes,takeeithersign).Equation(A.16.20)

ismoredemandingof

thanis(A.16.19),requiring

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t

,implyingthatitssquaremustbe

lessthan2

PP TT

−orotherwisetherelevantcompressibility(

)wouldbenegativeand

thesoundspeedcomplex.

TheconstraintsontheseawaterGibbsfunction

(

)

A,,

Stp

thathavebeendiscussed

abovearesummarizedas

g> AA 0

g>, 0,

0,

and

()

TP PP TT

gg<(A.16.21)

Wereturnnowtoquantifythenon‐conservativeproductionofentropyintheocean.

Whenthemixingprocessoccursat0,p

theexpression(A.16.6)fortheproductionof

entropycanbeexpressedintermsofConservativeTemperature

(sinceΘissimply

proportionaltohat0p=)asfollows(nowentropyistakentobethefunctionalform

()

ˆ,S

ηη

=Θ)

() ( )

{

}

AAA

1AA

22ˆˆ ˆ

SSS

mm SS

δη η η η

ΘΘ Θ

=− ΔΘ + ΔΘΔ + Δ (A.16.22)

Themaximumproductionoccurswhenparcelsofequalmassaremixedsothat

mmm

−=andweadoptthisvalueinwhatfollows.Toillustratethemagnitudeof

this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

()( )

ASO

,,25CSSΘ= ° andthenΘissubtracted.TheresultiscontouredinA

S−Θspace

inFigureA.16.1.

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FigureA.16.1.Contours(inC°)ofavariablewhichillustratesthenon‐conservative

productionofentropy

intheocean.



ThefactthatthevariableinFigureA.16.1isnotzerooverthewholeA

S−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

()

A,0gkg,0CS−

Θ= ° andtheotheratthewarmestandsaltiestcornerofthefigure,the

non‐conservativeproductionof

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

(

)

ˆ,S

ηη

Θ,andsinceatanypressure

intheoceanbothA

SandΘmay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equivalenttotheslightlymoreaccurateexpression(A.16.6)which

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).



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).Potentialtemperature

isnotconservedduringthemixingprocessandtheproductionofpotential

temperatureisgivenby

11 2 2 .mmm m

θδθ θ

++=(A.17.1)

Enthalpyinthefunctionalform

(

)

A,,hhS p

=isexpandedinaTaylorseriesofA

Sand

aboutthevaluesA

Sand

ofthemixedfluid,retainingtermstosecondorderin

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[]

A2 A1 A

SS S−=Δandin

[]

21 .

θθ

−=ΔThen1

hand2

hareevaluatedandEqns.(A.16.4)

and(A.17.1)usedtofind

() ()

AAA

1AA

222.

SSS

mm SS

hh h

θθ

θθ θ

δθ θ θ

⎧⎫

⎪⎪

=Δ+ΔΔ+Δ

⎨⎬

⎪⎪

⎩⎭





 

(A.17.2)

Themaximumproductionoccurswhenparcelsofequalmassaremixedsothat

.mmm

−=The“heatcapacity”h

isnotastrongfunctionof

butisamuch

strongerfunctionofA

S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A

isapproximately–5.411 11

Jkg K (gkg )

−

−−−

(seethedependenceof

isobaricheatcapacityonA

SinFigure4ofsection2.20)sothatanapproximateexpression

fortheproductionofpotentialtemperature

is

()

1AA

43.4 10 / [g kg ] .

hSh x S

θθ

−−

≈Δ ≈− Δ

 (A.17.3)

Sincepotentialtemperature

(

)

ˆ,S

θθ

Θcanbeexpressedindependentlyofpressure

asafunctionofonlyAbsoluteSalinityandConservativeTemperature,andsinceduring

turbulentmixingbothA

SandΘmaybeconsidered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0p=,namely

() ()

AAA

1AA

222SSS

mm SS

θθ

θθ θ

δθ θ θ

ΘΘ

⎧⎫

=Δ+ΔΔ+Δ

⎨⎬

ΘΘ Θ

⎩⎭

,(A.17.4)

wheretheexactproportionalitybetweenpotentialenthalpyandConservative

Temperature00

hc≡Θhasbeenexploited.Themaximumproductionoccurswhenparcels

ofequalmassaremixedsothat2

mmm

−

andweadoptthisvalueinwhatfollows.

Equations(A.17.2)or(A.17.4)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iscontouredinFigureA.17.1andcanbeusedtoillustrateEqn.(A.17.4).

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

fromthe

contouredvaluesof

−Θ.Takingthemostextremesituationwithoneparcelat

()

A,0gkg,0CS−

Θ= ° andtheotheratthewarmestandsaltiestcornerofFigureA.17.1,

thenon‐conservativeproductionof

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

isaconservativevariable,onemightselectproperty

contraststhatweretypicalofaprominentoceanicfrontanddecidethatbecause

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

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

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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

.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

and

respectivelyareconservativevariables.These

contouredvaluesoftemperaturedifferenceencapsulatetheaccumulatednon‐conservative

productionthathasoccurredduringallthemanymixingprocessesthathaveleadtothe

ocean’spresentstate.Themaximumsucherrorfor

isapproximately‐1.0C°(from

FigureA.16.1)whilefor

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

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

asaconservative

quantity.FromthecurvatureoftheisolinesonFigureA.17.1itisclearthatthenon‐

conservativeproductionofpotentialtemperaturetakesbothpositiveandnegativesigns.



FigureA.17.1.Contours(inC°)ofthedifferencebetweenpotentialtemperature

andConservativeTemperature

−

Θ.Thisplotillustratesthenon‐

conservativeproductionofpotentialtemperature

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duringthemixingprocess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againEqns.(A.16.2)–(A.16.4).Potentialenthalpy

handConservativeTemperature

arenotexactlyconservedduringthemixingprocess

andtheproductionofΘisgivenby

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11 2 2 .mmmm

Θ+ Θ+ Θ= Θ(A.18.1)

Enthalpyinthefunctionalform

(

)

ˆ,,hhS p=Θ

isexpandedinaTaylorseriesofA

Sand



aboutthevaluesA

SandΘofthemixedfluid,retainingtermstosecondorderin

[]

A2 A1 A

SS S−=Δ

andin

[]

21 .Θ−Θ =ΔΘThen1

hand2

hareevaluatedandEqns.(A.16.4)

and(A.18.1)areusedtofind

() ()

AAA

1AA

ˆˆ

ˆˆ ˆ

SSS

mm SS

mhh h

ΘΘ

ΘΘ Θ

⎧⎫

⎪⎪

Θ= ΔΘ + ΔΘΔ + Δ

⎨⎬

⎪⎪

⎩⎭

(A.18.2)

Inordertoevaluatethesepartialderivatives,wefirstwriteenthalpyintermsofpotential

enthalpy(i.e.0

cΘ)usingEqn.(3.2.1),as

() ( )

ˆˆ

,, ,, .

hhSpc vSpdP

′

=Θ=Θ+ Θ

∫(A.18.3)

Thisisdifferentiatedwithrespectto

giving

ˆ.

Sp P

hhc dP

αρ

ΘΘ

′

==+

∫(A.18.4)

Theright‐handsideofEqn.(A.18.4)scalesas

(

)

cPP

−

+− whichismorethan0

c

byonlyabout0

0.0015 p

cfor

()

P−of7

410×Pa(4,000dbar).Hence,toaverygood

approximation,ˆ

hΘinEqn.(A.18.2)maybetakentobesimply0

c.Itisinterestingto

examinewhythisapproximationissoaccuratewhenthedifferencebetweenenthalpy,,h

andpotentialenthalpy,0,hasgivenbyEqns.(3.2.1)and(A.18.3),scalesas1

−whichis

aslargeastypicalvaluesofpotentialenthalpy.ThereasonisthattheintegralinEqns.

(3.2.1)or(A.18.3)isdominatedbytheintegralofthemeanvalueof1,

−socausinga

significantoffsetbetweenhand0

hasafunctionofpressurebutnotaffectingthepartial

derivativeˆ

hΘwhichistakenatfixedpressure.Eventhedependenceofdensityon

pressurealonedoesnotaffectˆ.hΘ

Thesecondorderderivativesofˆ

hareneededinEqn.(A.18.2),andthesecanbe

estimatedintermsofthestrengthofcabbelingasfollows.Equation(A.18.4)is

differentiatedwithrespecttoConservativeTemperature,giving

(

)

ˆˆ,

h v dP dP

αρ

ΘΘ ΘΘ Θ

′

∫∫ (A.18.5)

sothatwemaywriteEqn.(A.18.2)approximatelyas(assuming12

)

(

)() ( )

{

}

AAA

0ˆˆ ˆ

8SSS

PP vvSvS

ΘΘ Θ

−

Θ≈ ΔΘ + ΔΘΔ + Δ (A.18.6)

ThedominantterminEqn.(A.18.6)isusuallytheterminˆ

Θwhichisapproximately

2ˆ

−ΘΘ

−,andfromEqn.(A.19.4)belowweseethat

isapproximatelyproportionalto

thenon‐conservativeproductionofdensityatfixedpressure,oftenreferredtolooselyas

“cabbeling”(McDougall,1987b),thatis,

()

(

)()

(

)

000

00220

ˆ.

ppp

PP PP PP

ccc

δρ

ρρ

ΘΘ

−−−

Θ ≈ ΔΘ ≈ − ΔΘ ≈ (A.18.7)

TheproductionofΘcausesanincreaseintemperatureandaconsequentdecreasein

densityof

αδ

−Θ.Theratioofthischangeindensity(usingEqn.(A.18.6))tothat

causedbycabbeling(fromEqn.(A.19.4))is

(

)

−− whichisabout0.0015fora

valueof

()

P−of40MPa.Henceitisclearthatcabbelinghasamuchlargereffecton

densitythandoesthenon‐conservationof.

Nevertheless,fromEqn.(A.18.6)wesee

thatthenon‐conservativeproductionof

isapproximatelyproportionaltotheproductof

seapressureandthestrengthofcabbeling.

ThefirstterminthebracketinEqn.(A.18.6)isusuallyaboutafactoroftenlargerthan

theothertwoterms(McDougall(1987b)),sotheproductionofConservativeTemperature

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101

ΘasaratioofthecontrastinConservativeTemperature21

Θ=Θ −Θmaybe

approximatedas(since21

ρρα

−−Θ

ΘΘ ΘΘ Θ

≈− ≈ )

() ()()

03.3 10 dbar K .

PP p

Θ−

−ΔΘ

Θ≈≈×ΔΘ

ΔΘ (A.18.8)

where

Θhasbeentakentobe52

1.1 10 K

−

×(McDougall,1987b).

AttheseasurfaceConservativeTemperature

istotallyconserved(0

Θ= ).The

non‐conservativeproductionofConservativeTemperature,,

increaseslinearlywith

pressure(seeEqn.(A.18.6))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



decreasesproportionallytothesquareofthetemperatureandsalinitycontrasts.

McDougall(2003)concludedthattheproduction

betweenextremeseawaterparcelsat

eachpressureislargestat600dbar,andthemagnitudeofthenon‐conservative

productionofConservativeTemperature,,

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600p

dbar.Thisconservativequantitywas

constructedbytakingtheconservativepropertyenthalpyhatthispressureandadding

thelinearfunctionofA

Swhichmakestheresultequaltozeroat

(

)

A0, 0 CS

Θ= ° andat

()

A35.165 04 g kg , 0 C .S−

=Θ=°Thisquantityisthenscaledsothatitbecomes25 C°at

()

A35.165 04 g kg , 25 C .S−

=Θ=°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



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

410 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.McDougall(2003)concludesthatthe

maximumnon‐conservationofΘintherealoceanisafactorofapproximatelyahundred

lessthanthemaximumnon‐conservativeproductionofpotentialtemperature.



FigureA.18.1.Contours(inC°)ofavariablethatisusedtoillustratethenon‐

conservativeproductionofConservativeTemperatureΘat

600p

dbar.Thecloudofpointsshowwheremostofthe

oceanicdataresideat600p

dbar.Thethreepointsthatare

forcedtobezeroareshownwithblackdots.

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FromthecurvatureoftheisolinesonFigureA.18.1itisclearthatthenon‐conservative

productionofConservativeTemperatureat600p

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ˆ0hΘΘ >,AA

ˆ0

h>andAAA

ˆˆˆ

()

SSS

hhh

ΘΘΘ

<

everywhere,anditcanbeshownthattheserequirementsareinfactmet.

FromEqns.(A.18.8)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

()()

()

10 dbar K / [g kg ] .pS

δθ

−

−−

Θ≈− ΔΘ Δ (A.18.9)

Takingatypicalratiooftemperaturedifferencestosalinitydifferencesinthedeepocean

tobe1

10 K / [g kg ]

−,Eqn.(A.18.9)becomes

(

)

10 dbar .p

δδθ

−

Θ≈− Atapressureof4000

dbarthisratiois0.4

δθ

Θ≈−implyingthatConservativeTemperatureisalmostasnon‐

conservativeasispotentialtemperature.Whilethisisthecase,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

and.

ΘThe

largestnon‐conservativeincrementofConservativeTemperature

seemstooccurata

pressureofabout600dbar(McDougall(2003))andthisvalueof

isapproximatelytwo

ordersofmagnitudelessthanthemaximumvalueof

whichoccursattheseasurface.

ThematerialinappendicesA.16‐A.18hascloselyfollowedthepaperofMcDougall

(2003).



A.19 Non-conservative production of density and of potential density

Forthepurposeofcalculatingthenon‐conservativeproductionofdensitywetakeboth

AbsoluteSalinityA

SandConservativeTemperature

tobe100%conservative(see

appendixA.18above).Densityiswritteninthefunctionalform

(

)

ˆ,,Sp

ρρ

=Θ(A.19.1)

andthesamemixingprocessbetweentwofluidparcelsisconsideredasintheprevious

appendices.MassandAbsoluteSalinityareconservedduringtheturbulentmixing

process(Eqns.(A.16.2)and(A.16.3))asisConservativeTemperature,thatis

11 22 ,mm mΘ+ Θ= Θ(A.19.2)

whilethenon‐conservativenatureofdensitymeansthatitobeystheequation,

11 2 2 .mm mm

ρδρρ

++=(A.19.3)

DensityisexpandedinaTaylorseriesofA

Sand

aboutthevaluesofA

SandΘofthe

mixedfluid,retainingtermstosecondorderin

[

]

A2 A1 A

SS S

−

=Δ andin

[]

21 .Θ−Θ =ΔΘ

Then1

and2

areevaluatedand(A.19.3)isusedtofind

() ( )

{

}

AAA

1AA

22ˆˆ ˆ

SSS

mm SS

δρ ρ ρ ρ

ΘΘ Θ

=− ΔΘ + ΔΘΔ + Δ (A.19.4)

Thenon‐conservativeproductionofdensity

ofEqn.(A.19.4)isillustratedinFigure

A.19.1formixingat0p=dbar.Thatis,thisfigureshowstheproduction

of

potentialdensity

(notethatthesymbol

couldequallywellbeusedforpotential

density).Thequantitycontouredonthisfigureisformedasfollows.Firstthelinear

functionofA

Sisfoundthatisequalto

at

(

)

A0, 0 CS

Θ= ° andat

()

A35.165 04 g kg , 0 C .S−

=Θ=°ThislinearfunctionofA

Sissubtractedfrom

andthe

resultisscaledtoequal25 C°at

(

)

A35.165 04 g kg , 25 C .S−

Θ= ° Thevariablethatis

contouredinFigureA.19.1isthedifferencebetweenthisscaledlinearcombinationof



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103

andA

S,andConservativeTemperature.Thisfigureallowsthenon‐conservativenature

ofdensitytobeunderstoodintemperatureunits.Themixingofextremefluidparcelson

FigureA.19.1causesthesameincreaseindensityasacoolingofapproximately10 C.°

FromFigureA.18.1itisseenthatthe(tiny)non‐conservativenatureofΘisafactorof

approximately4000smallerthanthis.

FigureA.19.1.Contours(inC°)ofavariablethatisusedtoillustratethenon‐

conservativeproductionofpotentialdensity.

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derivativesforuseinthehydrostatic

relationshipandfrequently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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.4.5)and(A.4.6)

betweenthesalinityvariablesR*

,SSanddens

SS=thatwerediscussedinsectionA.4.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restatingEqns.(A.4.5)and(A.4.6),namely

dens

R1R

SSrS

∗

−≈ ,(A.20.1)

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dens dens

A1R

(1 )SS rS

∗

−≈+ ,(A.20.2)

where1

rwillbetakentobetheconstant0.35basedontheworkofPawlowiczetal.(2010),

andinthissectiontheseapproximaterelationswillbetakentobeexact.TheAbsolute

SalinityAnomalydens dens

RA R

SS S

≡−isthesalinitydifferencethatcanbedirectlymeasured

fromseawatersamplesusingavibratingbeamdensimeterandknowledgeofthesample’s

PracticalSalinity,andagloballook‐uptableexistsforthisquantity(McDougalletal.

(2010a)).Becausethisparticularsalinitydifferenceisbasedondirectmeasurements,itis

thenaturalmeasureoftheanomalouscompositionofseawatertouseindevelopingthe

followingoptionsfornumericalmodelingofcompositionanomalies.



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)),

()

ˆˆ

dˆ.

dnn

hK S D

⎛⎞

∂

=∇⋅ ∇ +

⎜⎟

∂

⎝⎠

(A.20.3)

AsexplainedinappendixA.21,theover‐tildeof*

S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.20.3)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),while

theright‐handsideofthisequationisthestandardnotationindicatingthat*

Sisbeing

diffusedalongneutraltangentplaneswiththediffusivityKandintheverticaldirection

withthediapycnaldiffusivityD(andhhereistheaveragethicknessbetweentwoclosely

spacedneutraltangentplanes).

Inordertoevaluatedensityduringtherunningofanoceanmodel,DensitySalinity

mustbeevaluated.ThiscanbedonefromEqn.(A.20.2)asthesumofthemodel’ssalinity

variable*

Sanddens

(1 )rS

+.Thiscouldbedonebysimplyaddingtothemodel’s

salinityvariable1

(1 )r

timesthefixedspatialmapof

()

dens

RobsS

asobservedtoday(and

asisavailablefromthecomputeralgorithmofMcDougalletal.(2010a)).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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Absolute

SalinityAnomaliesdens

R(, , )Sxyp

whilenotpermittingthemtodeveloplargedifferences

fromtheobservedvaluesdens

R(obs)S

,werecommendcarryinganevolutionequationfor

dens

sothatitbecomesanextramodelvariablewhichevolvesaccordingto

() ()

dens dens

dens 1 dens dens

1RRR

d(obs) .

dnn

hK S D S S

δδ

δτδδ

−

⎛⎞

∂

=∇⋅ ∇ + + −

⎜⎟

∂

⎝⎠ (A.20.4)

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Herethemodelvariabledens

wouldbeinitializedbasedonobservations,dens

R(obs)S

,

andadvectedanddiffusedlikeanyothertracer,butinaddition,thereisanon‐

conservativesourceterm

()

1 dens dens

(obs)SS

τδ δ

−−whichservestorestorethemodel

variabledens

towardstheobservedvaluewitharestoringtime

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ofEqn.

(A.20.4)providesawayforward.Anindicationofhowthisapproachmightbeimproved

inthefuturecanbegleanedfromlookingatEqn.(A.4.14)fordens

∗

−(takenfrom

Pawlowiczetal.(2010)).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includedbiogeochemicalprocessesonthemodelʹs

densityanditscirculation.

Insummary,theapproachsuggestedherecarriestheevolutionEqns.(A.20.3)and

(A.20.4)for*

Sanddens

,whiledens

Siscalculatedbythemodelateachtimestep

accordingto

dens dens

A1R

ˆˆ

(1 )SS rS

∗

=++ ,(A.20.5)

withourbestpresentestimateof1

(1 )r

being1.35 .Themodelisinitializedwithvalues

ofPreformedSalinityusingEqn.(A.20.1)(namelydens

R1 R

ˆˆ

SSrS

∗=− )basedon

observationsofReferenceSalinityandontheglobaldatabaseof

()

dens

RobsS

from

McDougalletal.(2010a).



A.20.2IncludingasourcetermintheevolutionequationforAbsoluteSalinity

Anequivalentprocedureistocarrythefollowingevolutionequation(A.20.6)forAbsolute

Salinity,whichmorespecifically,iscalledDensitySalinity,dens

.SS≡Oninspectionof

Eqn.(A.20.2),dens dens

A1R

(1 )SS rS

∗

−≈+ ,andrecognizingthatS

∗

isaconservativevariable,

itisclearthatthenon‐conservativeproductionofdens

Smustoccurattherate1

(1 )r+

timestherateatwhichthesamenon‐conservativeprocessesaffecttheAbsoluteSalinity

Anomalydens

.Since(fromEqn.(A.20.4))thenon‐conservativesourcetermfordens

is

()

1 dens dens

(obs)SS

τδ δ

−−,wefindthattheevolutionequationforDensitySalinitytobe

()

dens dens

dens 1 dens dens

1A1RR

dens

dens A

ˆˆ

dˆ1(obs)

ˆˆ

hK S D r S S

hK S D z

τδ δ

−

⎛⎞

∂

=∇⋅ ∇ + ++ −

⎜⎟

∂

⎝⎠

⎛⎞

∂

=∇⋅ ∇ + +

⎜⎟

∂

⎝⎠

(A.20.6)

Alternatively,thisequationcanbederivedbysummingEqn.(A.20.3)plus1

(1 )r+times

Eqn.(A.20.4).Herethenon‐conservativesourcetermintheevolutionequationfor

DensitySalinityhasbeengiventhelabelA

ˆS

Sforlateruse.

Inthisapproachtheevolutionequation(A.20.4)fordens

isalsocarriedandthe

model’ssalinityvariable,dens

S,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DensitySalinityusingEqn.(A.4.2)(namelydens dens

ARR

ˆˆ

SSS

=+ )basedon

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observationsofReferenceSalinityandontheglobaldatabaseof

()

dens

RobsS

from

McDougalletal.(2010a).

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.20.6)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DensitySalinity.Suchair‐seafluxesdonotaffectPreformedSalinity.



A.20.3IncludingasourcetermintheevolutionequationforReferenceSalinity

Anequivalentprocedureistocarrythefollowingevolutionequation(A.20.7)for

ReferenceSalinity.OninspectionofEqn.(A.20.1),dens

R1R

SSrS

∗

−≈ ,andrecognizing

thatS∗isaconservativevariable,itisclearthatthenon‐conservativeproductionofR

S

mustoccurattherate1

rtimestherateatwhichthesamenon‐conservativeprocesses

affecttheAbsoluteSalinityAnomalydens

.Since(fromEqn.(A.20.4))thenon‐

conservativesourcetermfordens

is

(

)

1 dens dens

(obs)SS

τδ δ

−−,theevolutionequationfor

ReferenceSalinityis

()

1 dens dens

1R1RR

ˆˆ

dˆ(obs)

ˆˆ

hK S D r S S

hK S D zr

τδ δ

−

⎛⎞

∂

=∇⋅ ∇ + + −

⎜⎟

∂

⎝⎠

⎛⎞

∂

=∇⋅ ∇ + +

⎜⎟

∂+

⎝⎠

,(A.20.7)

wherethenon‐conservativesourcetermis11

(1 )rr

timesthecorrespondingsourceterm

intheevolutionequationforDensitySalinityinEqn.(A.20.6).

InthisapproachtheevolutionEqns.(A.20.7)and(A.20.4)forR

Sanddens

are

carriedbytheoceanmodel,whiledens

Siscalculatedbythemodelateachtimestep

accordingtoEqn.(A.4.2),namely

dens dens

ARR

ˆˆ

SSS

=+ .(A.20.8)

Thisapproach,likethatofsectionA.20.2shouldgiveidenticalresultstothatdescribed

insectionA.20.1usingPreformedSalinityexceptforthemorecomplicatedair‐seaflux

boundaryconditionforReferenceSalinitythanforPreformedSalinity.Itdoesseemthat

theconservativenatureofEqn.(A.20.3)forPreformedSalinityisasignificantadvantage,

andsothisapproachislikelytobepreferredbyoceanmodelers.



A.20.4Discussionoftheconsequencesifremineralizationisignored

IfanoceanmodeldoesnotcarrytheevolutionequationforAbsoluteSalinityAnomaly

(Eqn.(A.20.4))andthemodel’ssalinityevolutionequationdoesnotcontainthe

appropriatenon‐conservativesourceterm,istherethenanypreferenceto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.20.3).Inthisapproachthereisstillachoiceofhowtoinitialize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.20.3)containsnonon‐conservativesourceterms.Inthisapproachtheequationofstate

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107

willbecalledwith*

Sratherthandens

S,andthesesalinitiesdifferbyapproximately

dens

(1 )rS

+.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

dens

SandwithR

S.ThefigureshowsthatwhenusingR

S,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*

SasthesalinityargumentratherthanR

S,theerrorswouldbelarger

intheratio1

(1 ) 1.35r+≈ .Thatis,for58%ofthedataintheworldoceandeeperthan1000

m,the“thermalwind”relationwouldbemisestimatedby2.7%

≈

if*

Sisusedinplaceof

dens

Sasthesalinityargumenttotheequationofstate.Also,thesepercentageerrorsin

“thermalwind”arelargerinthePacificOcean.

Anotherchoiceofthesalinitydatatoinitializethemodelisdens

S.Thischoicehasthe

advantagethatforaninitialperiodoftimeafterinitializationtheequationofstateiscalled

withthecorrectsalinityvariable.Howeveratlatertimes,theneglectofthenon‐

conservativesourceterminEqn.(A.20.6)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DensitySalinitytobegin

tobeimportant.Thisisnottosuggestthattherelaxationtimescale

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

non‐conservativesourcetermA

ˆS

SinEqn.(A.20.6)tobecomesignificant.

AthirdchoiceistoinitializethemodelwithReferenceSalinity,R

S.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

thatrepresenttheeffectsofbiogeochemistryonconductivityandR

S.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studiessincewehave

routinelyinitializedmodelswithobservationsofPracticalSalinityandhavetreateditas

thoughitwereaconservativevariableandhaveuseditasthesalinityargumentforthe

equationofstate.

Inprinciple,thereareothercombinationsofoptionswheretheevolutionequationfor

AbsoluteSalinityAnomaly(A.20.4)iscarriedbyanoceanmodel,butsomeoptionother

thanthosediscussedinsubsectionsA.20.1–A.20.3ispursued.Wedonotconsiderthese

variousoptionshere,sinceifonegoestothetroubleofcarryingtheevolutionequationfor

AbsoluteSalinityAnomaly,thenoneshouldbesufficientlycarefultoimplementoneof

theoptionsdiscussedinsubsectionsA.20.1–A.20.3.



A.20.5Discussionoftheoptionsforincludingremineralization

TheapproachesofsubsectionsA.20.1–A.20.3ofthisappendixcaneachaccountforthe

non‐conservativeeffectsofremineralizationif1

risaconstantandsolongasthe

appropriateboundaryconditionsareimposed.Theadvantageofusing*

Sisthatitobeys

astandardconservativeevolutionequation(A.20.3)withnosourcetermontheright‐hand

side.Since*

Sisdesignedtobeaconservativesalinityvariable,itwouldappeartobethe

mostappropriatesalinityvariableforhavingasanaxisofthetraditional“S

−diagram”,

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whichwouldthenbecomethe*

S−Θdiagram.Similarly,*

Swouldalsobethebestchoice

forthesalinityvariableinaninversestudy.

IfanoceanmodelweretoberunwithoutcarryingtheevolutionequationforAbsolute

SalinityAnomaly(A.20.4)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Density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.

Itshouldbenotedthateachofthemodellingapproachesdescribedinsubsections

A.20.1–A.20.3arerelatedtotoday’sestimateofthe

()

dens

RobsS

field.Thisfieldwill

changenotonlyastheobservationaldatabaseimprovesbutalsoastheoceancomposition

evolveswithtime.



A.21 The material derivatives of *,S A,S R

S and

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))

() ( )

ρρρ

+∇⋅ = = −∇⋅uF(A.21.1)

ThereareseveraldifferentcontributionstothemolecularfluxofsaltS

F,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

(

)

ρρ

+∇⋅ =u(A.21.2)

TemporallyaveragingthisequationinCartesiancoordinates(i.e.atfixed,,

yz)gives

(

)

ρρ

+∇⋅ =u(A.21.3)

whichwechoosetowriteinthefollowingform,afterdivisionbyaconstantdensity0



(usuallytakentobe3

1035 kg m−,seeGriffies(2004))

(

)

ρρ

+∇⋅ =u

where 0.

≡uu

(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)

()()

()

00 00

*11

** * *

SS S S

ρρ ρ

ρρ

ρρ ρρ

∂′′ ′′

+∇⋅ = + ⋅∇ = − ∇⋅ − ∇⋅

∂

uu Fu

 (A.21.5)

HerethePreformedSalinityhasbeendensity‐weightedaveraged,thatis,**

≡,

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processes,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

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109

() ()()

()

000000

*** *

ˆˆ

ˆˆˆ ˆ

ˆˆ

ˆ.

SSeS Se

hK S D z

ρρρρρρ

∂∂

+∇ ⋅ + = + ⋅∇ +

∂∂

⎛⎞

∂

=∇⋅ ∇ +

⎜⎟

∂

⎝⎠





(A.21.6)

Heretheover‐caretmeansthatthevariable(e.g.*

S)hasbeenaveragedinathickness‐and‐

density‐weightedmannerbetweenapairof“neutralsurfaces”asmalldistancehapartin

thevertical,ˆ

v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.

Thedensityvalue

isthedensitywhoseaverage

heightistheheightatwhichtheequationisevaluated.Theissuesofaveraging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

()

** *

ˆˆ ˆ

ˆˆ

ˆ.

nnn

SS S

Se hKS D

tz z

⎛⎞

∂∂ ∂

+⋅∇ + =∇⋅ ∇ +

⎜⎟

∂∂ ∂

⎝⎠

v(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ˆ

v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A.SNotethatA

SistheconvenientgenericsymbolforDensitySalinitydens

S;

unlessthereisroomforconfusionwiththeothermeasuresofabsolutesalinity,soln

Sand

add

S,itprovesconvenienttousethesimplersymbolA

Sratherthandens

Sandtousethe

descriptionAbsoluteSalinityratherthanDensitySalinity.

AbsoluteSalinityobeystheinstantaneousevolutionequation(basedonEqn.(A.8.1))

() ( )

ρρρ ρ

+∇⋅ = = −∇⋅ +uFS(A.21.8)

ThesourcetermA

SisdescribedinappendixA.20(seeeqn.(A.20.6)).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

()()

()

00 0

AA A

SS S

ρρ ρ

ρρ

ρρ ρ

∂

+∇⋅ = + ⋅∇

∂

′′ ′′

=− ∇⋅ − ∇⋅ +



(A.21.9)

whichcorrespondstoEqn.(A.21.5)above.Whenaveragedoverthemesoscalethe

prognosticequationforAbsoluteSalinitybecomes

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() ()()

()

000000

AAA A

ˆˆ

ˆˆˆ ˆ

ˆˆ

ˆ,

SSeS Se

hK S D z

ρρρρρρ

∂∂

+∇ ⋅ + = + ⋅∇ +

∂∂

⎛⎞

∂

=∇⋅ ∇ + +

⎜⎟

∂

⎝⎠







(A.21.10)

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

()

AA A

ˆˆ ˆ

ˆˆ

ˆ.

nnn

n z

SS S

Se hKS D

tz z

⎛⎞

∂∂ ∂

+⋅∇ + =∇⋅ ∇ + +

⎜⎟

∂∂ ∂

⎝⎠

vS(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ˆ

vandthetemporallyaveraged

dianeutralvelocitye

ofdensitycoordinates.Apartfromthenon‐conservativesource

termA

ˆS

S,theright‐handsideisthedivergenceoftheturbulentfluxesofAbsolute

Salinity.

ThecorrespondingturbulentevolutionequationforReferenceSalinityis

()

RR R1

ˆˆ ˆ

ˆˆ

ˆ.

nnn

n z

SS Sr

Se hKS D

tz zr

⎛⎞

∂∂ ∂

+⋅∇ + =∇⋅ ∇ + +

⎜⎟

∂∂ ∂+

⎝⎠

vS(A.21.12)

Thenon‐conservativesourcetermhereisjustifiedinsubsectionA.20.3ofappendixA.20.

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1

rbe

takentobetheconstant10.35.r=HenceweseethatReferenceSalinityisaffectedby

biogeochemicalprocessesatabout0.26(0.35/ 1.35

≈

)ofthecorrespondinginfluenceof

biogeochemistryonDensitySalinity.

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,

(

)

()

() ()

()

0RQ

tTt

ρρερ

θμμ ρ

Θ=−∇⋅−∇⋅++

⎡⎤

−−−∇⋅+

⎢⎥

⎣⎦

(A.21.13)

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Q

−

∇⋅FandsaltS

−∇⋅ F

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)arenolargerthanthe

dissipationterm

whichisalsojustifiablyneglectedinoceanography(McDougall

(2003)).ThesourcetermA

S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,A,

hthatappears

inEqn.(A.21.13)isabout1

65 J g

−

−(i.e.65

−

111

Jkg (gkg )

−

−−

)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A

SfromtheNorthAtlantictotheNorthPacificissufficienttocausea

changeinAbsoluteSalinityof0.0251

gkg

−

,sothemaximumcontributiontoanerrorin



TEOS-10 Manual: Calculation and Use of the Thermodynamic Properties of Seawater

IOC Manuals and Guides No. 56

111

fromthesourceterm

()

(

)

A00

hTTt

ρθ

+SinEqn.(A.21.13),whenintegratedoverthe

wholeocean,isapproximately

(

)

(

)

01 1 1

( ) 65 J g 0.025 g kg 0.4 mK

c−− −

≈.Theothertermin

SinEqn.(A.21.13)ismultipliedbythesquarebracketwhichfromequation(27)of

McDougall(2003)isequalto

()

(

)

TTt

++timesapproximately1

Θ−

−,sothatthis

squarebracketisapproximately301

−

(i.e.30 111

Jkg (gkg )

−

−−

)atapressurepof4000

dbar(40MPa)sothecontributionofthistermislessthanhalfthatoftheterminA

Sin

thefirstlineofEqn.(A.21.13).ThisconfirmsthatthepresenceofthetwotermsinA

S

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mechanical

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Q

−

∇⋅Fandthetermduetotheboundary

andradiativeheatfluxes,

()

(

)

.TTt

−+∇⋅ +FAttheseasurfacethepotential

temperature

andinsitutemperaturetareequalsothatthistermissimplyR

−∇⋅Fso

thattherearenoapproximationswithtreatingtheair‐seasensible,latentandradiative

heatfluxesasbeingfluxesof0.

cΘThereisanissueattheseafloorwheretheboundary

heatflux(thegeothermalheatflux)affectsConservativeTemperaturethroughthe“heat

capacity”

()()

00p

TtcT

++ratherthansimply0.

cThatis,theinputofacertain

amountofgeothermalheatwillcausealocalchangein

asthoughtheseawaterhadthe

“specificheatcapacity”

()

(

)

00p

TtcT

++ratherthan0.

c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

()()

TTt

++.

WeconcludethatitissufficientlyaccuratetoassumethatConservativeTemperature

isinfactconservativeandthattheinstantaneousconservationequationis

() ( )

00 0 RQ

pp p

cc c

ρρρ

Θ + ∇⋅ Θ = = −∇⋅ − ∇⋅uFF(A.21.14)

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))

()( )

bound

ˆˆ

ˆˆˆ

ˆ.

nnnz

ehKDF

∂Θ ∂Θ

+ ⋅∇Θ+ = ∇ ⋅ ∇Θ + Θ−

∂∂

v(A.21.15)

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

ound

Fwhichisthesumoftheseboundaryheatfluxes

dividedby0

Anyconservativevariable,,Cobeysaconservationequationidentical

informtoEqns.(A.21.7)and(A.21.15),withˆ

Csimplyreplacing*

Sorˆ

Θ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

asbeing

conservativehavenotyetbeenthoroughlyinvestigated,butMcDougall(2003)and

Tailleux(2010)havemadeastartonthistopic.McDougall(2003)foundthattypicalerrors

in

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twoordersof

magnitude.

Itispossibletoderiveanevolutionequationforpotentialtemperaturewhich

resemblesEqn.(A.21.15)butwhichcontainsadditionalnon‐conservativesourcetermson

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

IOC Manuals and Guides No. 56

theright‐handsideoftheequation.However,thereseemslittlepointindoingsohere,as

itismuchmoreconvenienttoinsteadadopttheConservativeTemperaturevariable.Note

thattheconsequencesfordynamicaloceanographyofignoringthenon‐conservative

sourcetermsinthepotentialtemperatureevolutionequationareaslargeasignoringthe

variationsinseawatercomposition;a

rangeof0.2 C°correspondstoadensityrangeof

0.04 kg m−whichistwiceaslargeasthedensityerrorduetoignoringthemaximum

valueofAR

SS−of1

0.025 g kg−.

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ˆ

vtotheEulerian‐mean

horizontal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

()( )

* bound

dˆˆˆ ˆˆ

ˆ,

dtzznn z

zwhKDF

Θ=Θ + ⋅∇Θ+ Θ = ∇ ⋅ ∇Θ + Θ −v(A.21.16)

wheretheverticalvelocity*

wisrelatedtoe

by

*ˆ.

wz ze=+⋅∇+v(A.21.17)



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.

()

ˆ,,Sp

ρρ

=Θ)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

ˆˆ

dddd

ˆ,

dddd

tttt

ρβακ

−ΘΘ

=−+

(A.22.1)

whereˆ

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),dP dt

,butitismorerelevantandmoreinterestingtoconsider

thematerialderivativeofthelogarithmofthelocally‐referencedpotentialdensity,ˆ,

since

thisvariableislocallyconstantintheneutraltangentplane.Thematerialderivativeofˆl



isgivenby

11 A

ˆˆ

ddddd

ˆˆ .

ddddd

lPS

ttttt

ρρ

ρρκβα

−− ΘΘ

=−= −(A.22.2)

SubstitutingfromEqns.(A.21.11)and(A.21.15)above,andnotingthatboththetemporal

andthelateralgradientsofˆl

vanishalongtheneutraltangentplane(thatis,

ˆˆ

αβ

ΘΘ

∇Θ− ∇ =0andA

ˆˆ

tnn

αβ

ΘΘ

Θ− =),thematerialderivativeofˆl

amounts

tothefollowingequationforthedianeutralvelocitye

(notethattheboundaryheatflux

ound

Falsoneedstobeincludedforfluidvolumesthatabuttheseasurface)

()

(

)

(

)

() ( )

ˆˆ ˆ ˆ

ˆˆ .

znnnn

zzz

eShhKhhKS

DDS

αβ α β

αβ β

Θ Θ Θ− Θ−

ΘΘ Θ

Θ− = ∇⋅ ∇Θ− ∇⋅ ∇

+Θ− −



S(A.22.3)

Theleft‐handsideisequalto12

eg N

−

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

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113

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

and

)

()

(

)

(

)

bb A

ˆˆ ˆ ˆ ˆ .

nn nn z

eg N K C T p D DS

αβ β

−Θ Θ ΘΘΘ

=− ∇Θ⋅∇Θ+ ∇Θ⋅∇ + Θ − −

S(A.22.4)

Thecabbelingnonlinearity(the

CΘterm)alwayscauses“densification”,thatis,italways

causesanegativedianeutralvelocity,e

,whilethethermobaricnonlinearity(the

TΘterm)

cancauseeitherdiapycnalupwellingordownwelling.

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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,

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flavours,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straight‐forwardtoincludetheseprocessesinthese

conservationequations(seeforexampleMcDougall(1994,1997b)).



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ˆ

Θ,thuseliminatinge

andobtainingthefollowingequationforthe

temporalandspatialevolutionofˆ

Θalongtheneutraltangentplane(McDougall(1984))

() ( )

()

1bb

23 A

ˆˆˆˆˆˆˆ

ˆˆ

ˆ,

ˆ1

n n n z nn nn

hK KgN C T p

DgN dR

βα

−Θ Θ

Θ−

∂Θ + ⋅∇ Θ = ∇ ⋅ ∇ Θ + Θ ∇ Θ⋅∇ Θ + ∇ Θ⋅∇

∂

+Θ+

Θ−

(A.23.1)

where

isthestabilityratioofthewatercolumn,A

ˆˆ

αβ

ΘΘ

=Θ Theterminvolving

hasbeenwrittenasproportionaltothecurvatureoftheA

ˆˆ

−

Θdiagramofavertical

cast;thistermcanalsobewrittenas

(

)

ˆˆ ˆ ˆ .

zz z

zzz

DgN S S

Θ−

Θ−Θ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affectedbydiapycnalturbulentmixingonlytotheextentthatthe

verticalA

ˆˆ

S−Θdiagramisnotlocallystraight,and(iii)arenotinfluencedbythevertical

variationof

sincez

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

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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

() ( )

()

AAAbb

23 A

ˆˆˆˆˆˆˆ

ˆˆ

ˆ,

ˆ1

nnn nnnn

SShKSKgNSC Tp

DgN dR

−Θ Θ

Θ−

∂+ ⋅∇ = ∇ ⋅ ∇ + ∇ Θ⋅∇ Θ + ∇ Θ⋅∇

∂

+Θ+

Θ−

(A.23.2)

anditeasytoshowthat

Θtimestheright‐handsideofEqn.(A.23.1)isequalto

Θtimes

theright‐handsideofEqn.(A.23.2).

Toconstructthewater‐masstransformationequationofaconservativetracer,Cthe

meandianeutralvelocitye

iseliminatedfromtheˆ

Cconservationequation(A.24.1)using

Eqn.(A.22.4)giving(fromMcDougall(1984))

() ( )

()

1bb

223 2

A22

ˆˆˆˆˆˆˆ

ˆˆ ˆ ˆ

ˆˆˆ

ˆˆ ˆ

n n n z nn nn

zzz

CChKCKgNCC Tp

dC C dS

DS D gN CgN

dS S d

αβ

−Θ Θ

Θ− −Θ

∂+ ⋅∇ = ∇ ⋅ ∇ + ∇Θ⋅∇Θ+ ∇Θ⋅∇

∂

++Θ+

(A.23.3)

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

ˆˆ

SC−andtheA

ˆˆ

−

Θcurves.Thetermsinvolving

canalsobewrittenas

()

() ()

223 A

A22

AAA AAA

ˆˆ ˆ

ˆˆ

ˆˆ ˆ

ˆˆ ˆ ˆ ˆ ˆ ˆˆ ˆˆ ˆ

zz z zz z z

zz z z z zz

dC C dS

DS D gN

dS S d

SC S C S DC gN S S S

Θ−

+Θ=

−+Θ−Θ

(A.23.4)



A.24 Conservation equations written in potential density coordinates

Thematerialderivativeofa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),

()()

* d

ˆˆˆ ˆ

ˆˆˆˆˆ

ˆˆˆ

ˆˆ

znzz

nn z

CCC C

Cw CeC CeC

tzt t

hK C DC

∂∂∂ ∂

+⋅∇+ = +⋅∇+ = +⋅∇ +

∂∂∂ ∂

=∇⋅ ∇ +

vvv



(A.24.1)

where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

ound

F

(e.g.,theseasurface)wouldneedtobeaddedastheextraterm

ound

F−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))

() ()

rrA

ˆˆ

ppS

σσ

αβ

ΘΘ

Θ− =and

(

)

(

)

rrA

ˆˆ

ppS

σσ

αβ

ΘΘ

∇

Θ− ∇ =0,(A.24.2)

where

()

Θand

()

Θareshorthandnotationsfor

(

)

ˆˆ

,,Sp

ΘΘand

()

ˆˆ

,,Sp

ΘΘ

respectively,andr

pisthereferencepressureofthepotentialdensity.UsingEqns.(3.17.1)

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115

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))

()()

()

ˆˆ

ˆˆˆ

ˆˆ

ehKD

GhK

GG K

σσσ

ΘΘ

∂Θ ∂Θ

+ ⋅∇Θ+ = ∇ ⋅ ∇Θ + Θ

∂∂

⎡⎤

−∇⋅ − ∇Θ

⎣⎦

⎛⎞

∇Θ⋅∇Θ

⎡⎤

−−

⎜⎟

⎣⎦

⎜⎟

⎝⎠

v

(A.24.3)

wherethe“isopycnaltemperaturegradientratio”G

isdefinedas(fromEqn.(3.17.4))

1GrR Rr

ρρ

Θ⎡⎤⎡⎤

=− −

⎣⎦⎣⎦

andrisdefinedinEqn.(3.17.2)astheratioof/

ΘΘ

atthein

situpressureptothatevaluatedatthereferencepressurer.pThecorresponding

equationforAbsoluteSalinityis

()()

AAA

ˆˆ ˆ

ˆˆˆ

ˆˆ

Se hKS DS

GhK S

SG GK

σσσ

ΘΘ

∂∂

+⋅∇ + = ∇⋅ ∇ + +

∂∂

⎛⎞

⎡⎤

−∇⋅ − ∇

⎜⎟

⎢⎥

⎜⎟

⎣⎦

⎝⎠

⎛⎞

∇Θ⋅∇

⎡⎤

−−

⎜⎟

⎣⎦

⎜⎟

⎝⎠

vS

(A.24.4)

Thetermsinthesecondandthird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.

MultiplyingEqn.(A.24.4)by

(

)

Θandsubtracting

(

)

ΘtimesEqn.(A.24.3)we

findthecorrespondingexpressionforthediapycnalvelocityd

(followingMcDougall

(1991))

()

(

)

()

(

)

()

()( )

()

() ()

()

d11

rAr

rA r r

1ˆˆ

ˆˆ

1ˆ

ˆˆ ˆ

nn nn

nn n

ephKSphK

pDS pD p

pr hK p Kr

pGGK

pGKC T

σσ

σσ σσ

βα

βαβ

αα

ββ

ΘΘ

ΘΘΘ

ΘΘ

ΘΘΘ

ΘΘΘΘ

∂= ∇⋅∇− ∇⋅∇Θ

∂

+−Θ+

+−∇⋅∇Θ+ ∇⋅∇Θ

⎛⎞ ∇Θ⋅∇Θ

⎡⎤

−−

⎜⎟

⎣⎦

⎜⎟ Θ

⎝⎠

⎡⎤

−−∇Θ⋅∇Θ+∇

⎢⎥

⎣⎦



()

npΘ⋅∇

(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1Gr

=theselastthreelinesdonotreducetozerobut

rathertobˆ

TK p

Θ∇Θ⋅∇ showingthatthethermobariceffectremains.

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bewrittenas

()

br ˆˆ

CpK

σσ

∇

Θ⋅∇ Θ(c.f.thelastlineofEqn.(A.27.2)below).

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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

()

AA A

ˆˆ ˆ

ˆˆ

ˆ,

ann

a z

SS S

Se hKS D

tz z

⎛⎞

∂∂ ∂

+⋅∇ + =∇⋅ ∇ + +

⎜⎟

∂∂ ∂

⎝⎠

vS(A.25.1)

and

()()

ˆˆ

ˆˆˆ

ˆ.

annz

ehKD

∂Θ ∂Θ

+⋅∇Θ+ =∇⋅ ∇Θ+ Θ

∂∂

v(A.25.2)

Cross‐multiplyingtheseequationsby

(

)

ˆˆ

,,Sp

ββ

ΘΘ

=Θand

()

ˆˆ

,,Sp

αα

ΘΘ

=Θand

subtractinggivesthefollowingequationfortheverticalvelocitythroughthe

approximatelyneutralsurface,

()

() ( )

()

ˆˆ ˆ

ˆˆ

ˆ.

ann nn

zzz

egNKC T p

gN D DS gN

gN S gN tt

αβ β

βα β α

−Θ Θ

−Θ Θ −Θ

−Θ Θ −Θ Θ

=− ∇Θ⋅∇Θ+ ∇Θ⋅∇

+Θ− −

⎡

⎤

∂∂Θ

⎡⎤

+⋅∇−∇Θ+ −

⎢

⎥

⎣⎦

∂∂

⎢

⎥

⎣

⎦



S(A.25.3)

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)

()

ˆˆ

ˆˆˆˆ

aa na

gN S z z

βα ρ

−Θ Θ ∇

⎡⎤

⋅∇−∇Θ=−⋅ =⋅∇−∇=⋅

⎣⎦

vvvvs(A.25.4)

and

ˆˆ

tatt

lna

Nzz

βα ρ

−Θ Θ

⎡⎤

∂∂Θ

−=−=−

⎢⎥

∂∂

⎢⎥

⎣⎦

(A.25.5)

whereˆl

isthe(thickness‐weighted)locally‐referencedpotentialdensity.

CombiningtheseresultswithEqn.(A.22.4)wehavetherathersimplekinematicresult

that

ˆ,

att

ee zz=+⋅+ −vs

 (A.25.6)

showingthattheverticalvelocitythroughageneral“a”surface,,

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,ˆ,

⋅vs

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,

z−



A.26 The material derivative of potential density

Thematerialderivativeofthenaturallogarithmofpotentialdensityis

()

Θtimesthe

materialderivativeEqn.(A.21.11)ofAbsolutesalinityminus

(

)

Θ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

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117

respecttopotentialdensitysurfaces,onefindsthatthetemporalandisopycnalgradient

termscancelleavingonlytheterminthemeandiapycnalvelocityd

asfollows

()

d11

rAr

rA r r

1ˆˆ

ˆˆ ,

nn nn

e p hK S p hK

pDS pD p

ρβα

βαβ

ΘΘΘ

∂= ∇⋅∇ − ∇⋅∇Θ

∂

+−Θ+



(A.26.1)

wherethefollowingexactexpressionfortheverticalgradientofpotentialdensityhasbeen

used,

() ()

rA r

1ˆˆ

ˆzz

pS p

ρβα

ΘΘΘ

∂=−Θ

∂(A.26.2)

Equation(A.26.1)canbewrittenmoreinformativelyas(followingMcDougall,1991)

()

() () ()

{}

()

[]

()

rrArA

ˆˆ

1ˆ

ˆˆ

ˆˆˆˆ

ˆˆ ˆ ,

zS z S

nn nn

Dp pS pS

pr hK

pKC T p

ρρ

αα β

ΘΘ

ΘΘ Θ

ΘΘΘ

⎛⎞

∂∂

⎜⎟

∂∂

⎝⎠

+Θ+Θ−

+−∇⋅∇Θ

+ ∇ Θ⋅∇ Θ + ∇ Θ⋅∇

S

(A.26.3)

whererisdefinedinEqn.(3.17.2)astheratioof/

Θattheinsitupressureptothat

evaluatedatthereferencepressurer.pIftheequationofstatewerelinear,onlythefirst

twotermswouldbepresentontherightofEqn.(A.26.3).



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r

p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

ratherhavemixedlaterallyalongthepotentialdensitycoordinatedirection.The

diapycnalvelocityd_model

inthisclassofmodelobeystheequation(c.f.Eqn.(A.26.1)

above)

()

d_model 11

rAr

rA r r

1ˆˆ

ˆˆ ,

ephKSphK

pDS pD p

σσ

σσ σσ

ρβα

βαβ

ΘΘΘ

∂

∇⋅ ∇ − ∇⋅ ∇Θ

∂

+−Θ+



(A.27.1)

where

∇isthegradientoperatoralongthepotentialdensitycoordinate,K

isthelateral

diffusivityalongthelayersandh

isthethicknessbetweenapairofpotentialdensity

surfacesinthevertical.Thisequationcanberewrittenas

()

() () ()

{}

()

d_model

rrArA

ˆˆ

1ˆ

ˆˆ

ˆˆˆˆ

ˆˆ

zS z S

Dp pS pS

KC p

σσσ

ρρ

αα β

ΘΘ

ΘΘ Θ

⎛⎞

∂∂

⎜⎟

∂∂

⎝⎠

+Θ+Θ−

+∇Θ⋅∇Θ

S

(A.27.2)

Thetermsintheverticalturbulentdiffusivity

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

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IOC Manuals and Guides No. 56

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

()

[]

()

rˆ

1nn

pr h hK

Θ−

−

∇⋅ ∇Θ

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fromEqn.(A.27.2);thatis,

thetermproportionaltobˆ

KT p

Θ∇Θ⋅∇ inEqn.(A.26.3)isabsentfromEqn.(A.27.2),as

firstremarkedonbyIudiconeetal.(2008).Thethermobaricdiapycnaladvectionis

probablysignificantintheSouthernOcean(KlockerandMcDougall(2010a))andthis

wouldarguefortherotationofthelateralmixingtensorinlayeredmodelstothelocal

directionoftheneutraltangentplane,asisdoneinheight‐coordinateoceanmodels.Also

missingfromlayeredoceanmodelsisthemeanverticaladvectionˆ

⋅

vsduetothehelical

natureofneutraltrajectoriesintheocean(seesection3.13,Eqn.(A.25.4)andKlockerand

McDougall(2010b)).



A.28 The material derivative of orthobaric density

Orthobaricdensity

()

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v

canbeexpressedwithrespecttoorthobaricdensitysurfacesas

ˆ,

vvv

vee

tzz

ρρ

∂∂∂

+⋅∇ + =

∂∂∂

v

(A.28.1)

wherethetemporallyaveragedverticalvelocitythroughthev

surfaceisgivenby(from

McDougallandJackett(2005a))

()

(

)

Aˆ

egN S p pp

αβ ψ

−Θ Θ

=Θ−+−+⋅∇v



(A.28.2)

where(fromdeSzoekeetal.(2000))

() ()

223 2 2

0b0

12 ,,gN c c gT N p

ρρ

−− Θ−

⎡

⎤

−≈ Δ≈− Θ−Θ

⎣

⎦(A.28.3)

andcΔisthedifferencebetweenthereferencesoundspeedfunction

()

0,cp

andthe

soundspeedofseawaterwhichcanbeexpressedinthefunctionalform

()

,, .cp

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

(

)

0,.p

ΘHereA

is

shorthandforthematerialderivativeofA

Sandisexpressedintermsofmixingprocesses

bytheright‐handsideofEqn.(A.21.11);

issimilarlyshorthandforthematerial

derivativeofˆ

Θandisgivenbytheright‐handsideofEqn.(A.21.15).

ThefirsttermontherightofEqn.(A.28.2)representstheeffectsofirreversiblemixing

processesontheflowthroughorthobaricdensitysurfaces,andthiscontributionto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.



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119

A.29 The material derivative of Neutral Density

NeutralDensity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n

canbeexpressedas

nn n

ˆ,

γγ

∂+⋅∇ + =

∂v

(A.29.1)

wherethetemporallyaveragedverticalvelocitythroughthen

surfaceisgivenby(from

McDougallandJackett(2005b))

()

Aref

ref ref

ref 1

ref ref

ref

ref ref

ref

() () ˆ

() ()

()

ˆˆ

1ˆ

()

() ()

2( 1)

() ()

ppS

eppS

ppp

ppS

γγ

γγγ

αβ

ψαβ

ΘΘ

−

ΘΘ

Θ−

≈+⋅

Θ−

+− +⋅∇

⎛⎞

−

+− ⋅∇ − ⋅∇Θ

⎜⎟

Θ−Θ

⎝⎠

Θ−

+− Θ−

+−⋅







(A.29.2)

HereA

isshorthandforthematerialderivativeofA

Sfollowingtheappropriatemean

velocityandisexpressedintermsofmixingprocessesbytheright‐handsideofEqn.

(A.21.11),Θ

issimilarlyshorthandforthematerialderivativeofˆ

andisgivenbyEqn.

(A.21.15),and

()

−isdefinedby

()

2ref

2 2 ref ref ref

ref b b

()

() ()

NgT gTPP

ψρ

ΘΘ

−Θ−Θ

−=

⎡⎤

+Θ−Θ+ −Θ

⎣⎦

(A.29.3)

Here2

ref

Nisthesquareofthebuoyancyfrequencyofthepre‐labelledreferencedataset.

Equation(A.29.3)showsthat

()

−

isnonzerototheextentthatthereisawatermass

contrastref

()Θ−Θ 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

reasonablevaluesofref

()Θ−Θ andref

()pp−thedenominatorinEqn.(A.29.3)iscloseto

ref

Nand

()

−issmall.Intheseexpressionsthethermalexpansioncoefficient

()

Θ

andsalinecontractioncoefficient

(

)

Θ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,

()

Θand

()

Θareshorthandfor

(

)

A,,Sp

ΘΘand

()

A,, .Sp

ΘΘ

ThefirstterminEqn.(A.29.2)isexpectedasNeutralDensitychangesinresponseto

theirreversiblemixingprocessesΘ

andS

.ThenextterminEqn.(A.29.2),ref

ˆ,⋅

vs isalso

expected;itisthemeanverticalmotionthroughthe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

arisingfromtheseemingly

innocuousslidingmotionalongthesloping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

throughthen

surface.Notethatas

(

)

ref

Θ−Θ tendstozero,

()

−also

tendstozerosothatthethirdlineofEqn.(A.29.2)iswell‐behavedandbecomes

proportionalto1ref ref

() .

ppp

−−⋅∇Θv

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

IOC Manuals and Guides No. 56

A.30 Computationally efficient 25-term expressions for the density of seawater

in terms of Θ and

Oceanmodelstodatehavetreatedtheirsalinityandtemperaturevariablesasbeing

PracticalSalinityP

Sandpotentialtemperature

.AsthefullimplicationsofTEOS‐10are

incorporatedintooceanmodelstheywillneedtocarryPreformedSalinity*

Sand

ConservativeTemperatureΘasconservativevariables(asdiscussedinappendicesA.20

andA.21),andacomputationallyefficientexpressionfordensityintermsofAbsolute

SalinityA

SandConservativeTemperature

willbeneeded.

FollowingtheworkofMcDougalletal.(2003)andJackettetal.(2006),theTEOS‐10

density

hasbeenapproximatedbyarationalfunctionofthesameformasinthose

papers.Thefittedexpressionistheratiooftwopolynomialsof

(

)

A,,SpΘ

25 25

num denom

ρρ

≈= .(A.30.1)

Thedensitydatahasbeenfittedina“funnel”ofdatapointsin

(

)

A,,Stpspacewhichis

describedinmoredetailinMcDougalletal.(2010b).The“funnel”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5500dbar,themaximumtemperatureofthefitted

datais12 C°andtheminimumAbsoluteSalinityis1

PS 30 g kgu

−

.Thatis,thefithasbeen

performedoveraregionofparameterspacewhichincludeswaterthatisapproximately

10 C°warmerand1

5gkg

−fresherinthedeepoceanthanexistsinthepresentocean(see

Figure1ofJackettetal.(2006)).TableK.1ofappendixKcontainsthe25coefficientsofthe

expression(A.30.1)fordensityintermsof

(

)

A,,SpΘ.

AsoutlinedinappendixK,this25‐termrational‐functionexpressionfor

yieldsthe

thermalexpansionandhalinecontractioncoefficients,

and

,thatareessentiallyas

accurateasthosederivedfromthefullTEOS‐10Gibbsfunctionfordatainthe

“oceanographicfunnel”.Thesamecannotbeclaimedforthesoundspeedderivedby

differentiatingEqn.(A.30.1)withrespecttopressure;thissoundspeedhasanrmserrorin

the“funnel”ofalmost1

0.25 m s−whereasTEOS‐10fitstheavailablesoundspeeddata

withanrmserrorofonly1

0.035 m s

−

.

Indynamicaloceanographyitisthethermalexpansionandhalinecontraction

coefficients

Θand

Θwhicharethemostimportantaspectsoftheequationofstate

sincethe“thermalwind”isproportionaltoApp

αβ

ΘΘ

∇Θ− ∇ andtheverticalstatic

stabilityisgivenintermsofthebuoyancyfrequencyNby12

()

gN S

αβ

−ΘΘ

=Θ− .

Hencefordynamicaloceanographywemaytakethe25‐termrationalfunctionexpression

fordensity,Eqn.(A.30.1),asessentiallyreflectingthefullaccuracyofTEOS‐10.Thisis

confirmedinFig.A.30.1wheretheerrorinusingthe25‐termexpressionfordensityto

calculatetheisobaricnorthwarddensitygradientisshown.Theverticalaxisonthisfigure

isthemagnitudeofthedifferenceinthenorthwardisobaricdensitygradientintheworld

oceanbelow1000 m whenevaluatedusingEqn.(A.30.1)versususingthefullTEOS‐10

Gibbsfunction.Thescalesoftheaxesofthisfigurehavebeenchosentobethesameas

thoseofFig.A.5.1ofappendixA.5sothatthesmallnessoftheerrorsassociatedwithusing

the25‐termdensityexpressioncanbeappreciated.TheerrorsrepresentedinFig.A.30.1

representperhapshalfoftheremaininguncertaintyinourknowledgeofseawater

properties,andbycomparingFigs.A.30.1andA.5.1itisclearthatthemuchmore

importantissueistoproperlyrepresenttheeffectsofseawatercompositiononseawater

density.ThermsvalueoftheverticalaxisinFig.A.30.1is11.4%ofthatofFig.A.5.1.

McDougalletal.(2010b)havealsoprovideda25‐termrational‐functionexpressionfor

densityintermsof

(

)

A,,Sp

.The25coefficientscanbefoundinTableK.2ofappendix

K.Asanapproximationtodensity,this25‐termrationalfunctionisapproximatelyas

accurateastheonedescribedaboveintermsof

(

)

A,,SpΘ.Itmustbeemphasizedthough

TEOS-10 Manual: Calculation and Use of the Thermodynamic Properties of Seawater

IOC Manuals and Guides No. 56

121

thatanoceanmodelthattreatedpotentialtemperatureasaconservativevariablewould

makeerrorsinitstreatmentofheatfluxes,asdescribedinappendicesA.13,A.14andA.17,

andasillustratedinFiguresA.13.1,A.14.1andA.17.1.



FigureA.30.1.Thenorthwarddensitygradientatconstantpressure(thehorizontal

axis)fordataintheworldoceanatlasofGouretskiandKoltermann

(2004)for1000 dbarp>.Theverticalaxisisthemagnitudeofthe

differencebetweenevaluatingthedensitygradientusingthe25‐term

expressionEqn.(A.30.1)insteadofusingthefullTEOS‐10expression,

usingAbsoluteSalinityA

Sasthesalinityargumentinbothcases.



AppendixPdescribeshowanexpressionfortheenthalpyofseawaterintermsof

ConservativeTemperature,specificallythefunctionalform

(

)

ˆ,,hS pΘ,togetherwithan

expressionforentropyintheform

(

)

ˆ,S

,canbeusedasanalternativethermodynamic

potentialtotheGibbsfunction

(

)

A,,

Stp.Theneedforthefunctionalform

(

)

ˆ,,hS pΘ

alsoarisesinsection3.32andinEqns.(3.26.3)and(3.29.1).The25‐termexpression,Eqn.

(A.30.1),for

()

25 25

ˆ,,Sp

ρρ

=Θcanbeusedtofindaclosedexpressionfor

(

)

ˆ,,hS pΘby

integratingthereciprocalof

()

ˆ,,Sp

Θwithrespecttopressure(inPa ),since

ˆP

−

== (seeEqn.(2.8.3)).

The25‐termexpressionforspecificvolume,Eqn.(A.30.1),isfirstwrittenexplicitlyas

theratiooftwopolynomialsinseapressurep(indbar )as

25 01 2 3

25 2

01 2

ˆˆ2

aapapap

vbbpbp

++ +

== ++ ,(A.30.2)

wherethecoefficients0

ato3

aand0

bto2

barethefollowingfunctionsofA

SandΘ

() ()

1.5 1.5

234 3 2

0 14 15 16 17 18 A 19 A 20 A 21 A 22 A

1acccccScScScScS=+ Θ+ Θ+ Θ+ Θ+ + Θ+ Θ+ + Θ,

123

ac=,

224

ac=Θ,

325

ac=Θ,

()

012 3 4 5A6A 7A

bcccccScScS=+Θ+Θ+Θ+ + Θ+ ,

()

18910A

0.5bcccS=+Θ+,

21112

bc c=+Θ,

andthenumberedcoefficients1

cto25

caresoidentifiedinTableK.1(notethat13 1c=).

ItisnotdifficulttorearrangeEqn.(A.30.2)intotheform

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

IOC Manuals and Guides No. 56

()

25 25 31 32

A22

2012

ˆˆ ,, ,

ab aaNMp

vvS p p

bbbpbp

⎛⎞ +

=Θ=−++

⎜⎟ ++

⎝⎠ (A.30.3)

whereNand

aregivenby

301 20

abb ab

Na b

=+ − and 1

330 21

ab ab ab

Ma bb

=+ − − .(A.30.4)

ThepressureintegralofthelastterminEqn.(A.30.3)iswellknown(seeforexample

section2.103ofGradshteynandRyzhik(1980))andisdependentonthesignofthe

discriminantofthedenominator.Inourcaseitcanbeshownthat2

102

bbb>overthe

domainofthe“funnel”andalsothatboth0

band1

barepositive,while2

bisnegativeand

boundedawayfromzero.Theindefiniteintegral,withrespecttoseapressuremeasured

inPa ,ofthelastterminEqn.(A.30.3)is(with *4

10NN=and*4

M=)

***

21102

221

01 2

222

01 2 21 02 2 1 1 02

ln 2 ln

bp b b bb

NMp M NbMb

dP b b p b p

bbpbp b b bb b p b b bb

∫+− −

+−

′=+++

++ −++−

,(A.30.5)

Theenthalpy

()

ˆ,,

hS pΘisthedefiniteintegralofEqn.(A.30.3)from0

to

,plus0

,

beingthevalueofenthalpyat0

(i.e.at0dbarp

).Hencethefullexpressionfor

()

ˆ,,

hS pΘis(with2

1102

bbbb=− − and2

1102

Bb b bb=+ − )

()

25 0 4 4 2

31 32

12 2

200 2

ˆ,, 10 10

ln 1 ln 1 .

pab a

hS p c p p

NM BA

Mbb bb

pp p

bbb BA ABbp

⎛⎞

Θ=Θ+ − +

⎜⎟

⎝⎠

−⎛⎞

−

⎛⎞

++++ +

⎜⎟

−+

⎝⎠ ⎝⎠

(A.30.6)

Thefactorof4

10 thatappearshereandin*

Nand*

effectivelyservestoconvertthe

unitsoftheintegrationvariablefromdbar toPa sothat

()

ˆ,,

hS pΘhasunitsof1

Jkg .

−



IntheseequationsA

Sisin1

gkg

−,

inC°andpisindbar. Theargumentsofthetwo

naturallogarithmsinEqn.(A.30.6)arealwaysgreaterthan1,andinfacttheyarebetween

1and1.2evenforpaslargeas4

10 dbar (notethatboth2

band

arenegative).Also,

whentheenthalpydifferenceatthesamevaluesofA

Sand

butatdifferentpressures

(seeEqn.(3.32.2))isevaluatedusingEqn.(A.30.6),theexpressioncanalsobearrangedto

containonlytwologarithmterms.

FollowingYoung(2010),thedifferencebetweenspecificenthalpyand0

cΘmaybe

called“dynamicenthalpy”andcanbereadilycalculatedfromEqn.(A.30.6),recognizing

thatthisequationisbasedonthecomputationallyefficient25‐termexpressionfordensity

ofMcDougalletal.(2010b)ratherthanbeingevaluatedfromthefullTEOS‐10Gibbs

function.Similarly,thepartialderivativesof

()

ˆ,,

hS pΘwithrespecttoAbsolute

SalinityA

SandwithrespecttoConservativeTemperature

canbecalculatedeitherby

algebraicdifferentiationofEqn.(A.30.6)orbyfirstalgebraicallydifferentiatingEqn.

(A.30.1)andthennumericallyintegratingthisexpressionwithrespecttopressure(this

secondprocedureismotivatedbytakingtheappropriateA

Sor

derivativesofEqn.

(3.2.1);seeEqns.(A.18.4)and(A.18.5)).

Anexpression

()

A,,hS p

forenthalpyasafunctionofpotentialtemperature

can

befoundinasimilarmannertothatoutlinedabove,butwiththecoefficientsofthe25‐

termrational‐functionexpressionfordensitynowbeingtakenfromTableK.2,andwith

thefirsttermbeingexpressedastheexactpolynomialexpressionfor

(

)

A,,0hS

insteadof

as0

cΘ.

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123

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Q

anditsvolumeVbeingchanged.The

infinitesimalchangeintheinternalenergyoftheparceliswrittenas

()

ddUQpPV

=−+ 

where

()

0dpP V−+ 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,

andQ

perunitvolume,,q

as

d1d .

ddd

hPq

ttt

ρρ

⎛⎞

−=

⎜⎟

⎝⎠

forpurewater(B.1)

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mechanicalenergyinto“heat”,

whichcontributes

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,R

F,andminusthedivergenceofthemolecularfluxofheat

kT=− ∇F(whereT

kisthemoleculardiffusivityofheat),sothattheFirstLawof

Thermodynamicsforpurewateris

()

d1d .

ddd

hPq kT

ttt

ρε

⎛⎞

−==−∇⋅+∇⋅∇+

⎜⎟

⎝⎠ F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))

[

]

(

)

dd d,

HVP Q Tt MS

δμ μ

−=+−+ (B.3)

where

isthemassofthefluidparcel.Whenwrittenintermsofthespecificenthalpy

,handQ

perunitvolume,q

,thisequationbecomes(usingA

dd S

=−∇⋅F)

[]

()

d1d

ddd T

hPq Tt

ttt

ρμμ

⎛⎞

−=−−+∇⋅

⎜⎟

⎝⎠ F.(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mechanicalenergyandthemolecular,radiativeand

boundaryfluxesof“heat”andsalt?Ifthe“heating”termdqt

in(B.4)werethesameas

inthepurewatercase(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dqt



in(B.4)isnotthesameasthatinthepurewatercase,(B.2).

Substitutingtheexpressionfordqt

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.24))for

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

IOC Manuals and Guides No. 56

themolecularfluxQ

F).ThetwoversionsoftheFirstLawofThermodynamicsare

differentby

[]

()

[

]

Tt b k

μμ μ

⎡

⎤

⎛⎞

−−+ ∇⋅+∇⋅ +

⎢

⎥

⎜⎟

⎢

⎥

⎝⎠

⎣

⎦

FF.(B.5)

Thisinconsistencymeansthattheratherpoorlydefined“rateofheating”dqt

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andintheabsenceofnewinspirationwhichwehave

notfoundintheliterature,wetentativelyconcludethatanyattempttoderivetheFirst

LawofThermodynamicsviathisrouteinvolvingthelooselydefined“rateofheating”

dqt

isdoomedtofailure.

SincethereappearstobenowayofderivingtheFirstLawofThermodynamicsthat

involvesthe“heating”termdqt

,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.



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

dd ,

xyz

tt=∂ ∂ + ⋅∇u

() ()

d1d d d d d

ddd d d d

hPu v S

pP T t

ttt t tt

−=++=++ (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

()

huPvu pPv=+ =+ + where

visthespecificvolume.



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125

Gravitationalpotentialenergy



Ifthegravitationalaccelerationistakentobeconstantthegravitationalpotentialenergy

perunitmasswithrespecttotheheight

=0issimply.

z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0

zisdefinedby

()

zdz

′

Φ=∫(B.7)

AtafixedlocationinspaceΦisindependentoftimewhileitsspatialgradientisgivenby

∇Φ = kwherekistheunitvectorpointingupwardsintheverticaldirection.The

evolutionequationforΦisthenreadilyconstructedas

() ( )

ρρρρ

Φ+∇⋅ Φ = =u(B.8)

Wherewistheverticalcomponentofthethree‐dimensionalvelocity,thatis.w=⋅uk

(Clearlyinthissection

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.



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socalleddynamicviscosityvisc

vandthevelocity

divergence∇⋅u(followingGill(1982)),soarrivingat

(

)

visc

dfPgv

ρρρρ

+×=−∇− +∇⋅ ∇

uku k u

(B.9)

Where

istheCoriolisfrequency,visc

vistheviscosityand

∇

uistwicethesymmetrized

velocityshear,

(

)

ij ji

ux u x∇=∂ ∂ +∂ ∂u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

()

[]

()

[]

()

visc 1

dd ,

tPgwv

ρρ

ρρ ρε

⋅+∇⋅ ⋅

=⋅=−⋅∇−+∇⋅∇⋅−

uu u uu

uu u uu (B.10)

wherethedissipationofmechanicalenergy

isthepositivedefinitequantity

(

)

visc

2.v

≡∇⋅∇uu(B.11)



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

IOC Manuals and Guides No. 56

Evolutionequationforthesumofkineticandgravitationalpotentialenergies



Theevolutionequationfortotalmechanicalenergy0.5

⋅

+Φuu isfoundbyaddingEqns.

(B8)and(B10)giving

()()

()

[]

()

visc 1

dd .

tPv

ρρ

ρρε

⎡⎤ ⎡⎤

⋅+Φ +∇⋅ ⋅+Φ

⎣⎦ ⎣⎦

= ⋅ +Φ = − ⋅∇ +∇⋅ ∇ ⋅ −

uu u uu

uu u uu (B.12)

Noticethattheterm

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

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

andtheAbsolutesalinityA

S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

(

)

dd ddut pP vt=− + sothattheinternal

energychangesonlyasaresultoftheworkdoneincompressingthefluidparcel.Realizing

that1

−

=andusingthecontinuityEqn.(A.8.1)intheformdd 0,t

∇⋅ =uddutcan

beexpressedinthissituationofnomolecular,radiativeorboundaryfluxesas

()

dd .ut pP

−

=− + ∇⋅uAddingthisequationtotheinviscid,non‐dissipativeversionof

(B.12)gives

() ( )

[

]

(

)

ttpP

ρρρ

+∇⋅ = = −∇⋅ +uu

EEE

,nomolecularfluxes(B.13)

wherethetotalenergy

u=+ ⋅+Φuu

(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.14).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

⋅uσsothatthereneedstobetheadditionalterm

(

)

−

∇⋅ ⋅u

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

visc

=− ∇uσsothattheextratermis

[]

(

)

visc 1

2.v

∇⋅ ∇ ⋅uu Alsoheat

fluxesattheoceanboundariesandbyradiationR

FandmoleculardiffusionQ

F

necessitatetheadditionaltermsRQ

−∇⋅ − ∇ ⋅FF.Atthisstagewehavenotspecifiedthe

formofthemoleculardiffusivefluxofheatQ

Fintermsofgradientsoftemperatureand

AbsoluteSalinity;thisisdonebelowineq(B.24).Thenon‐conservativeproductionof

AbsoluteSalinitybytheremineralizationofsinkingparticulatematter,A

S,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

() ( )

[

]

(

)

[]

()

visc 1

tpP

ρρρ

ρρ

+∇⋅ = = −∇⋅ + −∇⋅ −∇⋅

+∇⋅ ∇ ⋅ +

uuFF

uu S

EEE

(B.15)

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IOC Manuals and Guides No. 56

127

where

()

A0ST

hTt

=− + (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).

Ifitwerenotfortheremineralizationsourceterm,A

S,theright‐handsideof

the

conservationequation(B.15)wouldbethedivergenceofaflux,ensuringthattotal

energy

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

(

)

ρρ

+⋅+Φuu

whilethe

divergencepartoftheleft‐handsideisbasedonadifferentquantity,namelytheBernoulli

function 1

2.h=+ ⋅+Φuu

Thisformofthetotalenergyequationis

() ( )

[

]

(

)

RQ visc

tvh

ρρ ρ ρ

+∇⋅ =−∇⋅ −∇⋅ +∇⋅ ∇ ⋅ +uFF uuS

(B.16)

Inanoceanmodellingcontext,itisratherstrangetocontemplatetheenergyvariablethat

isadvectedthroughthefaceofamodelgrid,

,tobedifferenttotheenergyvariablethat

ischangedinthegridcell,

.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1

+⋅+Φuu

sothattheprognosticequation

fortheBernoullifunctionis

() ( )

[

]

(

)

RQ visc

dd .

ttP v h

ρρρ ρ ρ

+∇⋅ = = −∇⋅ −∇⋅ +∇⋅ ∇ ⋅ +uFFuu

BBB

S(B.17)

ThesourcetermA

SofAbsoluteSalinitycausedbytheremineralizationofparticulate

matteraffectsenthalpyattherate

(

)

A0ST

hTt

=− + 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

)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

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assumption,andintheabsenceofA

Sthe

Bernoullifunction

possessesthe“potential”property.Thenegativeaspectofthis



evolutionequation(B.17)isthatinthemoregeneralsituationwheretheflowisunsteady,

thepresenceofthet

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

is“isobaricconservative”butisnota

“conservative”variablenordoesitpossesthe“potential”property.

Notingthatthetotalenergy

isrelatedtotheBernoullifunctionby

()

=−+

andcontinuingtotakethewholeoceantobeinasteadystateand

withA0

=S,sothat

hasthe“potential”property,itisclearthat

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

P=everywhere,then

remainsconstantwhiletheparcel’stotalenergy

changes

bythedifferenceinthequantity

(

)

−+ betweenthetwolocations.Hencewe

concludethateveninasteadyocean

doesnotpossesthe“potential”property.

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

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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

() ( )

(

)

dd .

uuutpP h

ρρρ ρερ

+∇⋅ = = − + ∇⋅ −∇⋅ −∇⋅ + +uuFFS(B.18)

Usingthecontinuityequationintheformddvt

∇⋅uandthefundamental

thermodynamicrelation(A.7.2),thisequationcanbewrittenas

() ()

d1d d d d d

dd d d d d

hP u v S

pP T t

tt t t tt

ρρ ρμ

ρε ρ

⎛⎞

⎛⎞⎛ ⎞

−=++=++

⎜⎟

⎜⎟⎜ ⎟

⎝⎠⎝ ⎠

⎝⎠

=−∇⋅ −∇⋅ + +FF S

,(B.19)

whichistheFirstLawofThermodynamics.Thecorrespondingevolutionequationfor

AbsoluteSalinityis(Eqn.(A.21.8))

() ( )

SSS

ρρ ρ ρ

= +∇⋅ = −∇⋅ +uFS(B.20)

whereS

FisthemolecularfluxofsaltandA

S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terms

andA

Sontheright‐handsideandtheleft‐handsideisnot



timesthe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A

S.Second,itisratherunclearhowonewould

otherwisearriveatthemolecularfluxes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”dqt

anddidnotleadus

totheFirstLaw.Forcompleteness,themolecularfluxesQ

FandS

Farenowwrittenin

termsofthegradientsofAbsoluteSalinity,temperatureandpressure.

LandauandLifshitz(1959)(theirsection58)showthatthemolecularfluxesQ

Fand

Faregivenintermsofthechemicalpotential

andthegradientsoftemperatureandof

chemicalpotentialby



TEOS-10 Manual: Calculation and Use of the Thermodynamic Properties of Seawater

IOC Manuals and Guides No. 56

129

S,abT

=− ∇ − ∇F(B.21)

and

() ,bT t T

μγ μ

=− + ∇ − ∇ +FF(B.22)

where,aband

arethreeindependentcoefficients.Notethesymmetrybetweensome

ofthecross‐diffusiontermsinthatthesamecoefficientbappearsinbothequations.

WhenwrittenintermsofthegradientsofAbsoluteSalinity,temperatureandpressure

theseexpressionsforthemolecularfluxesQ

FandS

Fbecome

SSt

kS bT P

ρμ β

ρμμ

⎛⎞

=− ∇ − + ∇ + ∇

⎜⎟

⎝⎠

F(B.23)

(usingtherelationtp

ρμ

=− thatfollowsfromthedefinitionofbotht

and

interms

oftheGibbsfunction)and

()

bkT

μρ

⎛⎞

=+ −∇

⎜⎟

⎝⎠

FF(B.24)

Theseexpressionsinvolvethepuremoleculardiffusivitiesoftemperatureandsalinity(T

k

andS

k)andthesingleparameterbthatappearsinpartofboththecross‐diffusionofsalt

downthetemperaturegradientandpartofthecross‐diffusionof“heat”downthe

gradientofAbsoluteSalinity.Theotherparametersintheseequationsfollowdirectly

fromtheGibbsfunctionofseawater.Thelasttermin(B.23)represents“barodiffusion”as

itcausesafluxofsaltdownthegradientofpressure.Themiddletermin(B.23)isafluxof

saltduetothegradientofinsitutemperatureandiscalledtheSoreteffectwhilethefirst

termin(B.24)isafluxof“heat”causedbythemolecularfluxofsalt,S

F,andthis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(seeEqn.(2.6.2)).SinceT

contains

thearbitraryadditionalconstant4,athefactthatS

Fcontainsnoarbitraryconstants

impliesthatthecross‐diffusioncoefficientbinEqns.(B.21)–(B.24)isarbitrarytothe

extentA

−.

Themolecularfluxof“heat”Q

FisunknowabletotheextentS

aF(since

is

arbitrarytotheextent

()

340

aaTt++andthepresenceofbinEqn.(B.24)cancelsthe

influenceof4

a).ThismeansthattheQ

−

∇⋅FtermontherightoftheFirstLawEqn.

(B.19)isunknowabletotheextentS

3.a

−

∇⋅FTheleft‐handsideofEqn.(B.19)is

unknowabletotheextent3A

ddaSt

(sincespecificenthalpyhcontainsthearbitrary

component13A

aaS+).ThelastterminEqn.(B.19)containsthearbitrarytermA

S

(sinceA

hisarbitrarybytheamount3

a).Thesethreearbitrary,unknowable

contributionstotheFirstLawofThermodynamicsEqn.(B.19)sumto3

a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.

RegardingEqns.(B.21)–(B.24),itisnotedthatstrictlyspeakingthegradientofthe

chemicalpotential

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saltʺ.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.

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

IOC Manuals and Guides No. 56

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inthetext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,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,Paris.Available

fromhttp://www.TEOS‐10.org[thepresentdocument,calledtheTEOS‐10manual]

TEOS-10 Manual: Calculation and Use of the Thermodynamic Properties of Seawater

IOC Manuals and Guides No. 56

131

Tertiarystandarddocuments



McDougall,T.J.,D.R.JackettandF.J.Millero,2010a:AnalgorithmforestimatingAbsolute

Salinityintheglobalocean.submittedtoOceanScience,apreliminaryversionisavailable

atOceanSci.Discuss.,6,215‐242.http://www.ocean‐sci‐discuss.net/6/215/2009/osd‐6‐215‐

2009‐print.pdfandthecomputersoftwareisavailablefromhttp://www.TEOS‐10.org



Backgroundpaperstothedeclaredstandards



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.ocean‐

sci.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.ocean‐sci.net/4/275/2008/os‐4‐275‐2008.html

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A

STP

−

−models,OceanSci.,5,

285‐291.www.ocean‐sci.net/5/285/2009/

Millero,F.J.,2000.Effectofchangesinthecompositionofseawateronthedensity‐salinity

relationship.Deep‐SeaRes.I47,1583‐1590.

Millero,F.J.,2010:Historyoftheequationofstateofseawater.Oceanography,23,18‐33.

Millero,F.J.,F.Huang,N.Williams,J.WatersandR.Woosley,2009:Theeffectofcomposition

onthedensityofSouthPacificOceanwaters,Mar.Chem.,114,56‐62.

Millero,F.J.,J.Waters,R.Woosley,F.Huang,andM.Chanson,2008b:Theeffectof

compositiononthedensityofIndianOceanwaters,Deep‐SeaRes.I,55,460‐470.

Pawlowicz,R.,2010: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.,D.G.WrightandF.J.Millero,2010:Theeffectsofbiogeochemicalprocesseson

oceanicconductivity/salinity/densityrelationshipsandthecharacterizationofreal

seawater.OceanScienceDiscussions,7,773–836.

http://www.ocean‐sci‐discuss.net/7/773/2010/osd‐7‐773‐2010‐print.pdf

Seitz,S.,R.Feistel,D.G.Wright,S.Weinreben,P.SpitzerandP.deBievre,2010b:Metrological

TraceabilityofOceanographicSalinityMeasurementResults.OceanScienceDiscussions,7,

1303–1346.http://www.ocean‐sci‐discuss.net/7/1303/2010/osd‐7‐1303‐2010‐print.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,2010b:Absolute

Salinity,“DensitySalinity”andtheReference‐CompositionSalinityScale:presentand

futureuseintheseawaterstandardTEOS‐10.OceanSci.Discuss.,7,1559‐1625.

http://www.ocean‐sci‐discuss.net/7/1559/2010/osd‐7‐1559‐2010‐print.pdf



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

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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.ocean‐sci.net/6/633/2010/os‐6‐633‐2010.pdfandhttp://www.ocean‐

sci.net/6/633/2010/os‐6‐633‐2010‐supplement.pdf

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,2010a: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.ocean‐sci.net/6/695/2010/os‐6‐695‐2010.pdfandhttp://www.ocean‐

sci.net/6/695/2010/os‐6‐695‐2010‐supplement.pdf

McDougallT.J.,D.R.Jackett,P.M.Barker,C.Roberts‐Thomson,R.FeistelandR.W.Hallberg,

2010b:Acomputationallyefficient25‐termexpressionforthedensityofseawaterin

termsofConservativeTemperature,andrelatedpropertiesofseawater.submittedto

OceanScienceDiscussions.Computersoftwareisavailablefromhttp://www.TEOS‐10.org

McDougall,T.J.,D.R.JackettandF.J.Millero,2010a:AnalgorithmforestimatingAbsolute

Salinityintheglobalocean.submittedtoOceanScience,apreliminaryversionisavailable

atOceanSci.Discuss.,6,215‐242.



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.TheGSW

MATLABOceanographicToolboxcontainsmanyhelpfiles,includingonecalled“Getting

startedwiththeGibbsSeaWater(GSW)OceanographicToolboxofTEOS‐10”whichserves

asasuccinctintroductiontotheuseofTEOS‐10inphysicaloceanography.



NotethatseveralofthepaperslistedinthisappendixareappearinginOceanScienceinthe

specialissue“ThermophysicalPropertiesofSeawater”.



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133

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.Theexactvalueoftheisobaric“heatcapacity”0

cisgiveninTable

D.5.



TableD.1.FundamentalconstantsfromCODATA2006(Mohretal.(2008))andISO(1993).

SymbolValueUncertaintyUnitComment

R8.3144720.000015Jmol–1K–1molargasconstant

101325exactPanormalpressure

T273.15exact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

18.0152680.000002gmol–1molarmass

t3.9781210.04°Cmaximumdensity,temperature

999.974950.00084kgm–3maximumdensityat0



999.84310.001kgm–3densityat0

Tand0

,00

1/v

=

(

)

∂∂6.774876×10–20.06×10–2kgm–3K–1

(

)

∂∂ at0

Tand0



T273.16exactKtriplepointtemperature

611.6570.01Patriplepointpressure

999.7930.01kgm–3triplepointdensity

0exactJkg–1K–1triplepointentropy

u0exactJkg–1triplepointinternalenergy

T273.1525190.000002Kfreezingpointat0



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

IOC Manuals and Guides No. 56

TableD.3.Theseasaltcompositiondefinitionforreferencesalinityofthestandardocean.

Thestandardoceanisat25°Cand101325Pa.X–molefractions,Z–valences,

W–massfractions(Milleroetal.2008a).MolarmassesMfromWieser(2006)

withtheiruncertaintiesgiveninthebrackets.Themassfractionsarewith

respecttothemassofsolutionratherthanthemassofpurewaterinsolution.



Solute j Zj Mj

g mol–1

10–7 Xj × Zj

10–7 Wj

Na+ +1 22.989 769 28(2) 4188071 4188071

0.3065958

Mg2+ +2 24.305 0(6) 471678 943356

0.0365055

Ca2+ +2 40.078(4) 91823 183646

0.0117186

K+ +1 39.098 3(1) 91159 91159

0.0113495

Sr2+ +2 87.62(1) 810 1620

0.0002260

Cl– –1 35.453(2) 4874839 –4874839

0.5503396

SO4

2– –2 96.062 6(50) 252152 –504304

0.0771319

HCO3

– –1 61.016 84(96) 15340 –15340

0.0029805

Br– –1 79.904(1) 7520 –7520

0.0019134

CO3

2– –2 60.008 9(10) 2134 –4268

0.0004078

B(OH)4

– –1 78.840 4(70) 900 –900

0.0002259

F– –1 18.998 403 2(5) 610 –610

0.0000369

OH– –1 17.007 33(7) 71 –71

0.0000038

B(OH)3 0 61.833 0(70) 2807 0

0.0005527

CO2 0 44.009 5(9) 86 0

0.0000121

Sum 1 000 000 0 0 1.0

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135

TableD.4.SelectedpropertiesoftheKCl‐normalisedreferenceseawater

(Milleroetal.2008a),andproposalsofWG127(2006).



Symbol Value Uncertainty Unit Comment

31.403 8218 0.001 g mol–1

referencesalinitymolarmass

XM=

∑

1.245 2898 exacta -

referencesalinityvalencefactor

XZ=

∑

N 6.022 141 79 × 1023 3 × 1016 1

mol

−

Avogadroconstant

N 1.917 6461 × 1022 6 × 1017 g–1 referencesalinityparticlenumber

SAS

/NNM

u 1.004 715… exacta g kg–1 unitconversionfactor,

≡

35.165 04 g kg–1 / 35

SSO 35.165 04 exacta g kg–1 standardoceanreferencesalinity,

35 PS

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

p 0 exactPa standardoceansurfaceseapressure

p = PSO – P0

hSO 0 exactJ kg–1 standardoceansurfaceenthalpy

hSO = ut

SO 0 exactJ kg–1 K–1 standardoceansurfaceentropy

SO =

Su 40.188 617… exacta g kg–1 unit‐relatedscalingconstant,

40 PS

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

a 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.



Symbol Value Uncertainty Unit Comment

c 3991.867 957 119 63 exact J kg–1 K–1 See Eqn. (3.3.3)



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

IOC Manuals and Guides No. 56



TableD.6.ChemicalcompositionofdryairwithafixedCO2level.

MolefractionsarefromPicardetal.(2008)exceptfor2

N

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.



Gas Mole

fraction

Mass

fraction

Molar mass

g mol–1

N2 0.780 847 9 0.755 184 73 28.013 4(3)

O2 0.209 390 0 0.231 318 60 31.998 8(4)

Ar 0.009 332 0 0.012 870 36 39.948 (1)

CO2 0.000 400 0 0.000 607 75 44.009 5(9)

Ne 0.000 018 2 0.000 012 68 20.179 7(6)

He 0.000 005 2 0.000 000 72 4.002 602(2)

CH4 0.000 001 5 0.000 000 83 16.042 46(81)

Kr 0.000 001 1 0.000 003 18 83.798 (2)

H2 0.000 000 5 0.000 000 03 2.015 88(10)

N2O 0.000 000 3 0.000 000 46 44.012 8(4)

CO 0.000 000 2 0.000 000 19 28.010 1(9)

Xe 0.000 000 1 0.000 000 45 131.293 (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)

7.292 1150 10 sx

−

Ω= ,(D.1)

(Groten(2004))andtheCoriolisparameterfis(inradianspersecond)

2 sin 1.458 423 00 10 sin sfx

φφ

−

=Ω = ,(D.2)

where

islatitude(

hasoppositesignsinthetwohemispheres).



GravitationalAcceleration

Thegravitationalacceleration

intheoceancanbetakentobethefollowingfunctionof

latitude

andseapressurep,orheight

relativetothegeoid,

()

(

)

()

232627

32 54 7

(m s ) 9.780 327 1 5.3024 10 sin 5.8 10 sin 2 1 2.26 10 (m)

9.780 327 1 5.2792 10 sin 2.32 10 sin 1 2.26 10 (m)

9.780 327 1 5.2792 10 sin 2.32 10 sin 1 2.22 10 (dbar) .

gxxxz

xx xz

xx xp

φφ

−−−−

−− −

=+ − −

=+ + −

≈+ + +

(D.3)

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variationof

with

andpintheoceaninEqn.(D.3)isderivedinMcDougalletal.(2010b).The

height

abovethegeoidisnegativeintheocean.Notethat

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0p

,2

9.8062 m s ,g

−

=whichisavaluecommonlyused

inoceanmodels.Thevalueof

averagedovertheearth’ssurfaceis2

9.7976 m s ,g

−

=

whilethevalueaveragedoverthesurfaceoftheoceanis2

9.7963 m sg

−

=(Griffies(2004)).

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137

AppendixE:

AlgorithmforcalculatingPracticalSalinity



E.1 Calculation of Practical Salinity in terms of K15

PracticalsalinityP

SisdefinedonthePracticalSalinityScaleof1978(Unesco(1981,1983))

intermsoftheconductivityratio15

Kwhichistheelectricalconductivityofthesampleat

temperature68

t=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15

K=1,thePracticalSalinityP

S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15

Kisnotunity,P

Sand15

Karerelatedby

(Unesco,1981,1983)thePSS‐78equation

()

P15

SaK

=∑where

(

)

()

P68

,15C,0

35, 15 C,0

CS t

KCt

=°

==° (E.1.1)

andthecoefficientsi

aaregiveninthefollowingtable.Notethatthesumofthesixi

a

coefficientsisprecisely35,whilethesumofthesixi

bcoefficientsispreciselyzero.

Equation(E.1.1)isvalidintherangeP

242.S

<



i i

a i

b i

c i

d i

0 0.0080 0.0005 6.766097 x 10-1

1 - 0.1692 - 0.0056 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 14.0941 - 0.0375 - 6.9698 x 10-7 4.215 x 10-1 3.989 x10-15

4 - 7.0261 0.0636 1.0031 x 10-9 - 3.107 x 10-3

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2C 35Ct−° ≤ ≤ °

and0 10 000dbar.

p≤≤ Measurementsofsalinityinthefieldgenerallymeasurethe

conductivityratioR

()

(

)

()

(

)

()

P68 P68 P68 68

68 P 68 68 68

,, ,, ,,0 35,,0

35, 15 C,0 , ,0 35, ,0 35, 15 C,0

CS t p CS t p CS t C t

RCt CSt Ct Ct

=° =° (E.2.1)

whichhasbeenexpressedin(E.2.1)astheproductofthreefactors,whicharelabeled

andt

rasfollows

()

P68

,, .

35, 15 C,0 ptt

CS t p

RRRr

=° (E.2.2)

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IOC Manuals and Guides No. 56

Thelastfactort

rhasbeenfittedtoexperimentaldataasthepolynomialintemperature

(68

t)

()

rct

∑(E.2.3)

andthefactorp

Rhasbeenfittedtoexperimentaldataasafunctionof,p68

tandRas

()() ()

1 68 2 68 3 4 68

1/C /C /C

dt dt Rd dt

⎡

⎤

+°+°++°

⎣

⎦

∑

(E.2.4)

ThusforanysamplemeasurementofRitispossibletoevaluatet

randp

Randhence

calculate

RRr

=(E.2.5)

Atatemperatureof68 15 C,t=°t

Rissimply15

KandPracticalSalinityP

Scanbe

determinedform(E.1.1).Fortemperaturesotherthan68 15 Ct

°,PracticalSalinityP

Sis

givenbythefollowingfunctionoft

Rwith0.0162,k



()

(

)

()

/C 15 .

1/C15

it it

SaR bR

°−

⎡⎤

+°−

⎣⎦

∑∑

(E.2.6)

Equations(E.1.1)and(E.2.6)arevalidonlyintherangeP

242.S

<Outsidethis

rangeP

Scanbedeterminedbydilutionwithpurewaterorevaporationofaseawater

sample.PracticalSalinityP

ScanalsobeestimatedfromtheextensionsofthePractical

SalinityScaleproposedbyHilletal.(1986)forP

02S

<andbyPoissonandGadhoumi

(1993)forP

42 50.S<< ThevaluesofPracticalSalinityP

Sestimatedinthismannermay

thenbeusedinEqn.(2.4.1),namelyRPSP

SuS

≈

toestimateReferenceSalinityR.S

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90

t

temperature,itisnecessaryfirsttoconvertthetemperatureto68

tusing68 90

1.00024tt=

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t

R.Becauset

risa

functiononlyoftemperature,atthisstagebotht

Randt

r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)andisalsoavailableinthe

GSWOceanographicToolboxasthefunctiongsw_cndr_from_SP.Notethatthisiterative

inverseprocedureisdoneintermsof68

t;thecodeaccepts90

tastheinputand

immediatelyconvertsthistoa68

ttemperaturebeforeperformingtheaboveiterative

procedure.TheiterationisstoppedwhenthePracticalSalinitycorrespondingtothe

outputconductivityratiodiffersfromtheinputPracticalSalinitybylessthan10

10−.



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139

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15

KandthedefiningmassfractionofKCl.Suppose

PracticalSalinityP

Sweretobeevaluatedusingthepolynomial(E.1.1)butusing15 90

−



insteadof15

K,where15 90

K−isdefined

(

)

()

P90

15 90

,15C,0

35, 15 C,0

CS t

KCt

−

=°

==° (E.4.1)

Themagnitudeofthedifference15 90 15

−

canbecalculatedandisfoundtobeless

than6.8x10‐7everywhereintherangeP

242.S

<Furthercalculationshowsthat

P15

41SK∂∂ <everywhereinthevalidrangeofPracticalSalinity,sothattheconsequence

ofusing15 90

K−in(E.1.1)insteadof15

K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90 15 C,t=°

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15

KratioinEqn.

(E.1.1)canequivalentlyrefertoaratioofconductivitiesat90 14.996 C.t

°

Thefactthattheconductivityratiot

Risratherweaklydependentonthetemperature

atwhichtheratioisdeterminedisimportantforthepracticaluseofbenchsalinometers.

Itisimportantthatsamplesandseawaterstandardsshouldberunatthesame

temperature,stableatorder1mK.Thisisachievedbytheuseofalargewaterbathinthe

instrument.However,itisnotcriticaltoknowthestablebathtemperaturetoanybetter

than10or20mK.

Theratios,

andt

r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68 90

1.00024tt=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

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

IOC Manuals and Guides No. 56

andpressure,fromwhichsalinity,densityandotherpropertiesarecomputedinturnby

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.(2010b)).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newReference‐CompositionSalinityScale(Milleroetal.

(2008a)),computethesalinityvaluefromameasuredconductivityratiowithrespectto

the15

KconductivityratioofIAPSOStandardSeawater(SSW,CulkinandRidout(1998)

and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15

K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

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

IOC Manuals and Guides No. 56

141

byarbitraryindependentlaboratoriesisneitherknownnorevengranted.Aslowdriftof

artefact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15

K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fortheoceanobservationsystem(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.(2010b)).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.



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

IOC Manuals and Guides No. 56

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

()

(

)

(

)

flu V,id res

,, ,fT f T RT

ρϕτδ

=+ , (F.1)

where

()

V,id ,fT

istheideal‐gaspart,(F.2),W

R=461.51805Jkg–1K–1isthespecificgas

constantofwaterusedinIAPWS‐95,and

(

)

res ,

τδ

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,W

RRM==461.52364Jkg–1K–1,where

=18.015268gmol–1isthemolarmassofwater(IAPWS(2005)).

Theideal‐gaspart,

()

V,id ,,fT

ofthespecificHelmholtzenergyforwatervapour

is(fromIAPWS(2009b),WagnerandPruß(2002),Feisteletal.(2010a))

() () ()

V,id 0 ex

,,.fT RT

ϕτδ ϕ τ

⎡

⎤

⎣

⎦ (F.2)

Notethattheterm

()

hasbeenaddedbyFeisteletal.(2010a)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

()

τδ

wasobtainedfromanequationforthespecificisobaricheat

capacityofvapourandreads

()

(

)

00000

12 3

,ln ln ln1

nn n n e

ϕτδ δ τ τ

−

=+++ + −

∑. (F.3)

The“reduceddensity”c

ρρ

=and“reducedtemperature”c/TT

arespecifiedby

c322 kg m

−

=,c647.096 K.T

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

18.320 446 483 749 693n=−

and26.683 210 527 593 226.n=

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

(

)

ex 0.

ϕτ

Therangecanbeextendedto

includetheregion50–130Kwiththefollowingcorrectionfunction

()

addedto(F.2)

inthistemperaturerange,

() ()

223

13 99

τττ

ϕτ τε

τεε

εε

⎛⎞

=×− − + − + +

⎜⎟

⎝⎠

, for50 K ≤ T ≤ 130 K, (F.4)

whereE

T=130K,E=0.278296458178592,andcE

/TT

.At,



()

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.

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143

deviationof4

610

−

×inPW

.cR 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),butithasnotyetbeenendorsedbyIAPWS.

Theresidualpartof(F.1)hastheform

()

()()

()

751

res

54 56

52 55

exp

ii ii i

ii i

dt dt c

dt b

iiiiii

ϕδτδτδ

ταδεβτγ δψ

=+ −

+−−−−+Δ

∑∑

(F.5)

withtheabbreviations

21−+=Δ

δθ

, i

δτθ

11 −+−= ,and

() ()

(

)

exp 1 1 .

ψδτ

=−−− −(F.6)

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1

nand2

nareslightlydifferent

fromthoseofTEOS‐10givenhere(Feisteletal.(2008a)).



i 0

n 0

1 –8.32044648374969

2 6.68321052759323

3 3.00632

4 0.012436 1.28728967

5 0.97315 3.53734222

6 1.2795 7.74073708

7 0.96956 9.24437796

8 0.24873 27.5075105



TableF.2.Coefficientsoftheresidualpart(F.5).



i ci d

i t

i n

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

22 1 15 1 –6.2639586912454× 10–10

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23 2 1 7 –0.10793600908932

24 2 2 1 0.017611491008752

25 2 2 9 0.22132295167546

26 2 2 10 –0.40247669763528

27 2 3 10 0.58083399985759

28 2 4 3 4.9969146990806× 10–3

29 2 4 7 –0.031358700712549

30 2 4 10 –0.74315929710341

31 2 5 10 0.4780732991548

32 2 6 6 0.020527940895948

33 2 6 10 –0.13636435110343

34 2 7 10 0.014180634400617

35 2 9 1 8.3326504880713× 10–3

36 2 9 2 –0.029052336009585

37 2 9 3 0.038615085574206

38 2 9 4 –0.020393486513704

39 2 9 8 –1.6554050063734× 10–3

40 2 10 6 1.9955571979541× 10–3

41 2 10 9 1.5870308324157× 10–4

42 2 12 8 –1.638856834253× 10–5

43 3 3 16 0.043613615723811

44 3 4 22 0.034994005463765

45 3 4 23 –0.076788197844621

46 3 5 23 0.022446277332006

47 4 14 10 –6.2689710414685× 10–5

48 6 3 50 –5.5711118565645× 10–10

49 6 6 44 –0.19905718354408

50 6 6 46 0.31777497330738

51 6 6 50 –0.11841182425981



TableF.3.Coefficientsoftheresidualpart(F.5).



i di t

i n

i 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 b

i B

i n

i C

i D

i A

i 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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145

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

()

tpisthefollowingfunctionoftheindependentvariablesITS‐

90Celsiustemperature,u

tt y=×,andseapressure,u

pp z

×

(, )

Wjk

tp g g yz

=∑∑ ,(G.1)

withthereducedtemperatureu

ytt

andthereduced(dimensionless)pressure

pp=.Theunit‐relatedconstantsuu

,tpandu

aregiveninTableD4ofappendixD

(e.g.84

10 Pa 10 dbar

p== ).Coefficientsnotcontainedinthetablebelowhavethevalue

=0.Twoofthese41parameters(00

and10

)arearbitraryandarecomputedfrom

thereference‐stateconditionsofvanishingspecificentropy,,

andspecificinternal

energy,,uofliquidH2Oatthetriplepoint,

()

,0,Tp

=and

(

)

,0.uT p

(G.2)

Notethatthevaluesof00

and10

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 k gjk j k gjk

0 0

0.101 342 743 139 674 × 103 3 2

0.499 360 390 819 152 × 103

0 1

0.100 015 695 367 145 × 106 3 3

–0.239 545 330 654 412 × 103

0 2

–0.254 457 654 203 630 × 104 3 4

0.488 012 518 593 872 × 102

0 3

0.284 517 778 446 287 × 103 3 5

–0.166 307 106 208 905 × 10

0 4

–0.333 146 754 253 611 × 102 4 0

–0.148 185 936 433 658 × 103

0 5

0.420 263 108 803 084 × 10 4 1

0.397 968 445 406 972 × 103

0 6 –0.546 428 511 471 039 4 2 –0.301 815 380 621 876 × 103

1 0

0.590 578 347 909 402 × 10 4 3

0.152 196 371 733 841 × 103

1 1

–0.270 983 805 184 062 × 103 4 4

–0.263 748 377 232 802 × 102

1 2

0.776 153 611 613 101 × 103 5 0

0.580 259 125 842 571 × 102

1 3

–0.196 512 550 881 220 × 103 5 1

–0.194 618 310 617 595 × 103

1 4

0.289 796 526 294 175 × 102 5 2

0.120 520 654 902 025 × 103

1 5

–0.213 290 083 518 327 × 10 5 3

–0.552 723 052 340 152 × 102

2 0

–0.123 577 859 330 390 × 105 5 4

0.648 190 668 077 221 × 10

2 1

0.145 503 645 404 680 × 104 6 0

–0.189 843 846 514 172 × 102

2 2

–0.756 558 385 769 359 × 103 6 1

0.635 113 936 641 785 × 102

2 3

0.273 479 662 323 528 × 103 6 2

–0.222 897 317 140 459 × 102

2 4

–0.555 604 063 817 218 × 102 6 3

0.817 060 541 818 112 × 10

2 5

0.434 420 671 917 197 × 10 7 0

0.305 081 646 487 967 × 10

3 0

0.736 741 204 151 612 × 103 7 1

–0.963 108 119 393 062 × 10

3 1

–0.672 507 783 145 070 × 103

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AppendixH:Coefficientsofthesaline

GibbsfunctionforseawaterofIAPWS‐08



Non‐zerocoefficientsijk

ofthesalinespecificGibbsenergy

(

)

A,,

Stp

asafunctionof

theindependentvariablesabsolutesalinity,Au

²SSx

×,ITS‐90Celsiustemperature,

tt y=×,andseapressure,u

pp z=×:

Au1

(,,) ln ijk

jk ijk

jk i

Stp g g x x gx yz

⎧⎫

⎨⎬

⎩⎭

∑∑

. (H.1)

Theunit‐relatedconstantsuu

St pandu

aregiveninTableD4ofappendixD(e.g.

10 Pa 10 dbar

p== ).Coefficientswithk>0areadoptedfromFeistel(2003).Pure‐

watercoefficientswithi=0donotoccurinthesalinecontribution.Thecoefficients200



and210

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 gijk i j k gijk i j k gijk

1 0 0 5812.81456626732 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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147

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,

(

)

Ih ,.

tp 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(0

pPP

−)

() ( )( )( )( )

()

00t t

,Relnln2ln

kk k k k kk

gtp g sT T rt t t t t t t

gp g P

rp r P

τττττ

⎡

⎤

=− ⋅+ − −++ +− −

⎢

⎥

⎣

⎦

⎛⎞

=⋅

⎜⎟

⎝⎠

⎛⎞

=⋅

⎜⎟

⎝⎠

∑

(I.1)

withthereducedtemperature

(

)

TtT

=+ andt

Tandt

aregiveninTableI.1.Ifthe

seapressurepisexpressedindbar thent

mustalsobegivenintheseunitsas

t0.061 1657 dbarP=.Therealconstants00

to04

and0

,thecomplexconstants1

t,1

r,

t,and20

rto22

r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 t

611.657 Pa

Numerical triple-point pressure num

P 611.654 771 007 894 Pa

Normal pressure 0

101325 Pa

Triple-point temperature t

T 273.16 K



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

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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 – 0.632 020 233 335 886 × 106 J kg–1

g01 0.655 022 213 658 955 J kg–1

g02 – 0.189 369 929 326 131 × 10-7 J kg–1

g03 0.339 746 123 271 053 × 10-14 J kg–1

g04 – 0.556 464 869 058 991 × 10-21 J kg–1

s0 (absolute) 0.189 13 × 103 J kg–1 K–1

s0 (IAPWS-95) – 0.332 733 756 492 168 × 104 J kg–1 K–1

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 J kg–1 K–1

t2 0.337 315 741 065 416 0.335 449 415 919 309

r20 – 0.725 974 574 329 220 × 102 – 0.781 008 427 112 870 × 102 J kg–1 K–1

r21 – 0.557 107 698 030 123 × 10-4 0.464 578 634 580 806 × 10-4 J kg–1 K–1

r22 0.234 801 409 215 913 × 10-10 – 0.285 651 142 904 972 × 10-10 J kg–1 K–1



Thenumericaltriplepointpressurenum

P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

()

ismeantastheprincipalvalue,i.e.itevaluatestoimaginarypartsinthe

interval

(

)

Im ln z

⎡⎤

−< ≤+

⎣⎦ .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

<double>),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0

usedintheTEOS‐10algorithms.Inthe

ʹabsoluteʹversion,0

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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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149

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)[an

IAPWSGuideline,inpreparation])isrepresentedintermsofaHelmholtzfunctionwhich

expressesthespecificHelmholtzenergyasafunctionofdry‐airmassfraction,

absolute

temperatureTandhumid‐airmassdensity,,

andtakestheform

()()

(

)

(

)

(

)

AV VV AAmix

,, 1 , , ,,fAT AfT AfT fAT

ρρ ρ

=− + + . (J.1)

ThevapourpartisgivenbytheIAPWS‐95Helmholtzfunctionforfluidwater(IAPWS

(2009b)),

()

(

)

VV fluV

,,,fT f T

ρρ

≡ (J.2)

iscomputedatthevapourdensity,

(

)

=− ,andisdefinedinEqn.(F.1)ofappendix

F.Thedry‐airpart,

()

,,fT

iscomputedatthedry‐airdensity,A,

=andis

definedbyEqn.(J.3).Theair‐watercross‐overpart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 Symbol Value Unit Reference

Molar gas constant L

R 8.314 51 J mol–1 K–1 Lemmon et al. (2000)

Molar gas constant R 8.314 472 J mol–1 K–1 IAPWS (2005)

Molar mass of dry air MA 28.958 6 g mol–1 Lemmon et al. (2000)

Molar mass of dry air MA 28.965 46 g mol–1 IAPWS (2010)

Molar mass of water MW 18.015 268 g mol–1 IAPWS (2005)

Celsius zero point T0 273.15 K Preston-Thomas (1990)

Normal pressure 0

101 325 Pa ISO(1993)



ThespecificHelmholtzenergyfordryairis(Lemmonetal.(2000)),

()

() ()

AA id res

,,,

fT M

ατδ α τδ

⎡

⎤

⎣

⎦. (J.3)

ThevaluestobeusedformolarmassA

ofdryair,andforthemolargasconstantL

R

aregiveninTableJ.1.Thefunction

(

)

id ,

τδ

istheideal‐gaspart,

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

IOC Manuals and Guides No. 56

()

() ()

id 0 4 0 1.5 0 0 0

67 8 11

000 0

91210 13

,ln lnln1exp

ln 1 exp ln 2 / 3 exp

nnnn n

nnn n

τδ δ τ τ τ τ

ττ

−

⎡

⎤

=+ + + + −−

⎣

⎦

⎡⎤⎡ ⎤

+−−+ +

⎣⎦⎣ ⎦

∑ (J.4)

and

()

res ,

τδ

istheresidualpart,

()

10 19

res

111

,exp.

kk kk k

ij ij l

ατδ δτ δτ δ

=+ −

∑∑ (J.5)

The“reducedvariables”inEqns.(J.3)‐(J.5)are*

A/TT

=withthereducingtemperature

A132.6312 KT=,andA*

ρρ

=withthereducingdensity*3

10.4477 mol dm

−

=×.

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(0

nand0

n)listedinTableJ.2arearbitraryandarecomputed

herefromthereference‐stateconditionsofvanishingspecificentropy,A,

andspecific

enthalpy,A,hofdryairatthetemperature0

Tandthenormalpressure0,

asgivenin

TableJ.1,

(

)

,0,TP

(J.6)

(

)

,0.hTP

(J.7)

TheHelmholtzfunctionmix

inEqn.(J.1)describesthewater‐airinteractionandis

definedby

()

() () () () ()

mix AW AAW AWW

AW A W

,, 2 .

AART A

fAT BT C T C T

MM M M

ρρ

⎧⎫

⎡

⎤

−−

⎪⎪

=++

⎨⎬

⎢

⎥

⎪⎪

⎣

⎦

⎩⎭

(J.8)

Thevaluesusedforthemolargasconstant,Rthemolarmassofdryair,A,

andthe

molarmassofwater,W,MaregiveninTableJ.1.

Thesecondcross‐virialcoefficient,

(

)

AW ,BTisgivenbyHarveyandHuang(2007)as

() * .

BT b c

=∑ (J.9)

ThecoefficientsofEqn.(J.9)aregiveninTableJ.4.

Thethirdcross‐virialcoefficients

(

)

AAW

CTand

(

)

AWW

CTaredefinedinHylandand

Wexler(1983),intheform

()

AAW

CTca

−

=∑ (J.10)

and

()

AWW

*exp i

CT c b

−

⎧

⎫

=−

⎨

⎬

⎩⎭

∑. (J.11)

Thecoefficientsi

aandi

bofEqns.(J.10)and(J.11)aregiveninTableJ.4.



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151

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and0

n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 0

n i 0

1 0.605 719 400 000 000 × 10–7 8 0.791 309 509 000 000

2 –0.210 274 769 000 000 × 10–4 9 0.212 236 768 000 000

3 –0.158 860 716 000 000 × 10–3 10 –0.197 938 904 000 000

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 j

k l

k n

1 1 0 0 0.118 160 747 229

2 1 0.33 0 0.713 116 392 079

3 1 1.01 0

–0.161 824 192 067 × 10

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



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TableJ.4.Coefficientsofthecross‐virialcoefficients

(

)

AW ,

T

()

AAW

CT and

(

)

AWW ,CT Eqns.(J.9)‐(J.11).Thereducingfactorsare63 1

*10mmolb−−

=

and66 2

*10mmol,c−−

=the“reducedtemperature”is

()

/100KT

=.



i ai b

i c

i d

0 0.482 737 × 10–3 –0.107 288 76 × 102

1 0.105 678 × 10–2 0.347 802 00 × 102 0.665 687 × 102 –0.237

2 –0.656 394 × 10–2 –0.383 383 00 × 102 –0.238 834 × 103 –1.048

3 0.294 442 × 10–1 0.334 060 00 × 102 –0.176 755 × 103 –3.183

4 –0.319 317 × 10–1



Theequationofstate,Eqn.(J.1),isvalidforhumidairwithinthetemperatureand

pressurerange

193K ≤T≤473Kand10nPa≤

≤5MPa.(J.12)

Thepressureiscomputedfrom2AV

.Pf

=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TP−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

cantakeanyvaluebetween0and1providedthatthepartial

vapourpressure,vap

xP=,(V

isthemolefractionofvapour,Eqn.(3.35.3))doesnot

exceeditssaturationvalue,i.e.,

01A≤≤ and

(

)

sat ,.

TP A≤(J.13)

Theexactvalueoftheairfraction

(

)

sat ,

TP 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sat

ofhumid

aircanbeestimatedbythecorrelationfunctionforeitherthevapourpressure,

()

liq ,

Tof

purewater(IAPWS(2007)),orforthesublimationpressure,

(

)

subl ,

Tofice(IAPWS

(2008b)),toobtain

()

sat sat sat

,/1/,ATP PP PP M M

⎡

⎤

=− − −

⎣

⎦fromEqn.(3.35.3)asa

practicallysufficientapproximation.



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153

AppendixK:Coefficientsof25‐termexpressions

forthedensityofseawaterintermsofΘandof





TheTEOS‐10Gibbsfunctionofseawater

(

)

A,,

Stp

iswrittenasapolynomialinterms

ofinsitutemperaturet,whileforoceanmodels,densityneedstobeexpressedasa

computationallyefficientexpressionintermsofeitherConservativeTemperatureΘor

potentialtemperature

(referencedtor0p

dbar).McDougalletal.(2010b)havefitted

theTEOS‐10valuesofdensity

toA,S

andpina“funnel”ofdatapointsin

(

)

A,,Stpspace.Thefittedexpressionisintheformofarationalfunction,beingtheratio

oftwopolynomialsof

(

)

A,,SpΘ

25 25

num denom

ρρ

=.(K.1)

The“funnel”ofdatapointsin

(

)

A,,StpspaceisdescribedinmoredetailinMcDougallet

al.(2010b);attheseasurfaceitcoversthefullrangeoftemperatureandsalinitywhilefor

pressuregreaterthan5500dbar,themaximumtemperatureofthefitteddatais12 C°and

theminimumAbsoluteSalinityis1

PS 30 g kgu

−

.Themaximumpressureofthe“funnel”

is8000dbar .TableK.1containsthe25coefficientsoftheexpression(K.1)fordensityin

termsof

(

)

A,,SpΘ.Thecoefficients112

−

inthistablehaveunitsof3

kg m−andthe

coefficients13 25

cc−aredimensionless,andthenormalizingvaluesofA,SΘandpare

takentobe1

1g kg

−,1Kand1dbarrespectively.

Thermserrorofthis25‐termapproximationtotheTEOS‐10densityislessthan

0.0015 kg m−overthe“funnel”;thiscanbecomparedwiththermsuncertaintyof

0.004 kg m−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,

αρ

Θ∂

=− ∂Θ ,(K.2)

ofthe25‐termequationofstateisdifferentfromthesamethermalexpansioncoefficient

evaluatedfromTEOS‐10withanrmserrorinthe“funnel”oflessthan61

0.3 10 Kx−−

,

comparedwiththermserrorofthethermalexpansioncoefficientofthelaboratorydata

towhichtheFeistel(2008)Gibbsfunctionwasfittedof61

0.6 10 Kx

−

−(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.1)ismoreaccuratethanthethermalexpansion

coefficient.Hencewemayconsiderthe25‐termrationalfunctionexpressionfordensity,

Eqn.(K.1),tobeequallyasaccurateasthefullTEOS‐10expressionsfordensity,the

thermalexpansioncoefficientandthesalinecontractioncoefficientfordatathatreside

insidethe“oceanographicfunnel”.

Thesoundspeedevaluatedfromthe25‐termrationalfunctionEqn.(K.1),hasanrms

erroroverthe“funnel”ofalmost0.251

−

whichisapproximatelyfivetimestherms

erroroftheunderlyingsoundspeeddatathatwasincorporatedintotheFeistel(2008)

Gibbsfunction(seerows7to9ofTableO.1ofappendixO).Hence,the25‐term

expressionfordensityisnotaparticularlyaccurateexpressionforthesoundspeedin

seawater,eveninthe“funnel”.Butfordynamicaloceanographywhere

Θand

Θare

theaspectsoftheequationofstatethat,togetherwithspatialgradientsofA

Sand



driveoceancurrentsandaffectthecalculationofthebuoyancyfrequency,wemaytake

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

IOC Manuals and Guides No. 56

the25‐termrational‐functionexpressionfordensity,Eqn.(K.1),asessentiallyreflecting

thefullaccuracyofTEOS‐10.



num

Coefficients 25

denom

Coefficients

c 9.998 438 029 070 821 4 x 102 13

c 1.0

c Θ 7.118 809 067 894 091 0 x 100 14

7.054 768 189 607 157 6 x 10-3

c 2

Θ -1.945 992 251 337 968 7 x 10-2 15

-1.175 369 560 585 864 7 x 10-5

c 3

Θ 6.174 840 445 587 464 1 x 10-4 16

5.921 980 948 827 490 3 x 10-7

c A

S 2.892 573 154 127 765 3 x 100 17

3.488 790 222 801 251 9 x 10-10

c A

SΘ 2.147 149 549 326 832 4 x 10-3 18

S 2.077 771 608 561 845 8 x 10-3

()

S 1.945 753 175 118 305 9 x 10-3 19

-2.221 085 729 372 299 8 x 10-8

c p 1.193 068 181 853 174 8 x 10-2 20

-3.662 814 106 789 528 2 x 10-10

c 2

pΘ 2.696 914 801 183 075 8 x 10-7 21

()

1.5

S 3.468 821 075 791 734 0 x 10-6

c A

pS 5.935 568 592 503 565 3 x 10-6 22

()

1.5 2

8.019 054 152 807 065 5 x 10-10

c 2

p -2.594 338 980 742 903 9 x 10-8 23

cp 6.831 462 955 412 332 4 x 10-6

c 22

pΘ -7.273 411 171 282 270 7 x 10-12 24

c23

-8.529 479 483 448 544 6 x 10-17

-9.227 532 514 503 807 0 x 10-18

TABLEK.1Coefficients of the polynomials

(

)

num A ,,

Θ and

(

)

denom A ,,

Θ that

define the 25-term rational-function Eqn. (K.1) for density.



ThesameprocedurehasbeenusedbyMcDougalletal.(2010b)tofitarational

functionoftheformofEqn.(K.1)butwherethepolynomialsinthenumeratorand

denominatorarefunctionsof

(

)

A,,Sp

ratherthanof

(

)

A,,SpΘ.Thisformofthe25‐

termrationalfunctionexpressionfordensityisofapproximatelythesameaccuracyas

thatdescribedabove,andthe25coefficientsofthisexpressionaregiveninTableK.2.

Thecoefficients112

−

inthistablehaveunitsof3

kg m

−

andthecoefficients13 25

cc−are

dimensionless,andthenormalizingvaluesofA,S

andparetakentobe1

1g kg

−,1K

and1dbarrespectively.



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

IOC Manuals and Guides No. 56

155

num

Coefficients 25

denom

Coefficients

c 9.998 427 704 040 868 8 x 102 13

c 1.0

7.353 990 725 780 200 0 x 100 14

7.288 277 317 994 539 7 x 10-3

c 2

-5.272 502 484 658 053 7 x 10-2 15

-4.427 042 357 570 579 5 x 10-5

c 3

5.105 140 542 790 050 1 x 10-4 16

4.821 816 757 416 573 2 x 10-7

c A

S 2.837 207 495 416 299 4 x 100 17

1.966 643 777 649 954 1 x 10-10

c A

-5.746 287 373 866 898 5 x 10-3 18

S 2.019 220 131 573 115 6 x 10-3

()

S 2.016 582 840 401 100 5 x 10-3 19

-7.838 666 741 074 767 1 x 10-6

c p 1.150 668 012 876 069 5 x 10-2 20

-2.749 397 117 121 584 4 x 10-10

c 2

1.202 602 702 900 458 1 x 10-7 21

()

1.5

S 4.661 419 029 016 429 3 x 10-6

c A

pS 5.536 190 936 504 846 6 x 10-6 22

()

1.5 2

1.518 271 263 728 829 5 x 10-9

c 2

p -2.756 315 640 465 192 8 x 10-8 23

cp 6.414 629 356 742 288 6 x 10-6

c 22

-5.883 476 945 993 336 4 x 10-12 24

c23

-9.536 284 588 639 736 0 x 10-17

-9.623 745 548 627 732 0 x 10-18



TABLEK.2Coefficients of the polynomials

(

)

num A ,,

and

(

)

denom A ,,

that

define the 25-term rational-function Eqn. (K.1) for density.



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

IOC Manuals and Guides No. 56

AppendixL:Recommendednomenclature,

symbolsandunitsinoceanography



L.1 Recommended nomenclature

ThestrictSIunitsofAbsoluteSalinity,temperatureandpressureare1

kg kg−,Absolute

TemperatureinKandAbsolutePressure

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A

Sin1

gkg ,

−

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 WG127 is recommending that Chlorinity be

defined in terms of a mass fraction as

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. Hence WG127 recommends that mass

fraction units are used for Chlorinity.

Standard Ocean

Reference Salinity SO

S g kg–1 35.165 04 g kg–1 being exactly PS

35 u,

corresponding to the standard ocean Practical

Salinity of 35.

Freezing temperatures ff

,,tθ

ºC In situ, potential and conservative values, each as a

function of A

S and p.

Absolute pressure P Pa 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 software

in the GSW Toolbox.

p dbar Equal to 0

−

and usually expressed in dbar not

Pa.

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IOC Manuals and Guides No. 56

157

Gauge pressure. Gauge

pressure (also called

applied pressure) is

sometimes reported from

ship-born instruments.

gauge

p 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 gauge ,p

as p is the argument to the seawater Gibbs

function.

Reference pressure r

p dbar The value of the sea pressure p to which potential

temperature and/or potential density are

referenced.

One standard atmosphere 0

Pa exactly 101 325 Pa (= 10.1325 dbar)

Isopycnal slope ratio r 1

(

)

(

)

() ()

αβ

ΘΘ

Stability ratio R

(

)

(

)

RS S

θθ

αβ αθβ

ΘΘ

=Θ ≈

Isopycnal temperature

gradient ratio

GΘ 1 1GrR Rr

ρρ

⎡

⎤⎡ ⎤

−−

⎣

⎦⎣ ⎦

; n

∇

Θ= ∇Θ

Practical Salinity P

S 1 Defined in the range P

242S

< by PSS-78 based

on measured conductivity ratios.

Reference Salinity R

S g kg-1 Reference-Composition Salinity (or Reference

Salinity for short) is the Absolute Salinity of

seawater samples that have Reference

Composition. At P

S = 35, R

S is exactly PS P

while in the range P

242S

< RPSP

.SuS≈

Absolute Salinity

(This is the salinity

argument to all the

GSW Toolbox functions.)

dens

SS=g kg-1 AR A PSP A

SS S uS S

=+ ≈ +

Absolute Salinity is the sum of R

S on the Millero

et al. (2008a) Reference-Salinity Scale and the

Absolute Salinity Anomaly. The full symbol for

S is dens

S 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 dens

SS= is

“Density Salinity”.

Absolute Salinity

Anomaly A

g kg-1 AAR

SSS

=−

, the difference between Absolute

Salinity, dens

,SS= and Reference-Composition

Salinity. An algorithm to evaluate A

available (McDougall et al. (2010a)). In terms of

the full nomenclature of Pawlowicz et al. (2010),

Wright et al. (2010b) and appendix A.4 herein, the

Absolute Salinity Anomaly A

is dens

“Preformed Absolute

Salinity”,

often shortened to

“Preformed Salinity”

S g kg-1 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).

“Solution Absolute

Salinity”, often shortened

to “Solution Salinity”

soln

S g kg-1 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).

“Added-Mass Salinity” add

S g kg-1 add

−

is the estimated mass fraction of non-

H2O 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.

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

IOC Manuals and Guides No. 56

Temperature t ºC

Absolute Temperature T K

(

)

(

)

/ K / K / C 273.15 / CTT t t

≡

+°= +°

temperature derivatives T K When a quantity is differentiated with respect to in

situ temperature, the symbol T is used in order to

distinguish this variable from time.

Celsius zero point 0

T K 0273.15 KT

≡

Potential temperature

ºC Defined implicitly by Eqn. (3.1.3)

Conservative Temperature Θ ºC Defined in Eqn. (3.3.1) as exactly potential

enthalpy divided by 0

A constant “specific

heat”, for use with

Conservative Temperature

c J kg–1 K–1 011

3991.867 957 119 63 J kg K .

c−−

≡ This 15-digit

number is defined to be the exact value of 0

Combined standard

uncertainty

uc Varies

Enthalpy H J

Specific enthalpy h J kg–1

(

)

0.hu pPv=+ +

Here p and 0

must be in Pa not dbar.

Specific potential

enthalpy

h0 J kg–1 Specific enthalpy referenced to zero sea pressure,

[

]

(

)

AA r r

,,,,0,0hhS Stpp p

Specific isobaric heat

capacity

c J kg–1 K–1

pSp

chT=∂ ∂

Internal energy U J

Specific internal energy u J kg–1

Specific isochoric heat

capacity v

c J kg–1 K–1

vSv

cuT=∂ ∂

Gibbs function

(Gibbs energy)

G J

Specific Gibbs function

(Gibbs energy)

g J kg–1

Specific Helmholtz

Energy

f J kg–1

Unit conversion factor for

salinities PS

u g kg–1 11

PS (35.16504 35) g kg 1.004 715... g kgu

−

≡≈

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)).

Entropy Σ J K–1

Specific entropy

J kg–1 K–1 In many other publications the symbol s is used for

specific entropy.

Density

kg m–3

Density anomaly t

kg m–3

(

)

A,,0St

– 1000 kg m–3

Potential density anomaly

referenced to a sea

pressure of 1000 dbar

kg m–3

[

]

(

)

AA rr

,,,,,SStppp

ρθ

– 1000 kg m-3 where

r1000 dbarp

Potential density anomaly

referenced to a sea

pressure of 4000 dbar

kg m–3

[

]

(

)

AA rr

,,,,,SStppp

ρθ

– 1000 kg m-3 where

r4000 dbarp

Thermal expansion

coefficient with respect to

in situ temperature

K–1 AA

Sp Sp

vvT T

ρρ

−−

∂∂ =− ∂ ∂

Thermal expansion

coefficient with respect to

potential temperature

K–1 AA

Sp Sp

θρρθ

−−

∂∂ =− ∂ ∂

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159

Thermal expansion

coefficient with respect to

Conservative Temperature

Θ K–1 AA

Sp Sp

ρρ

−−

∂∂Θ =− ∂ ∂Θ

Saline contraction

coefficient at constant in

situ temperature

kg g–1

A, A,

Tp Tp

vvS S

ρρ

−−

−∂∂ = ∂∂

Note that the units for t

are consistent with SA

being in g kg-1.

Saline contraction

coefficient at constant

potential temperature

kg g–1

A, A,

vvS S

ρρ

−−

−∂∂ = ∂∂

Note that the units for

are consistent with SA

being in g kg-1.

Saline contraction

coefficient at constant

Conservative Temperature

Θ kg g–1

A, A,

vvS S

ρρ

−−

ΘΘ

−∂∂ =+∂∂

Note that the units for

are consistent with SA

being in g kg-1.

Isothermal

compressibility

Pa–1

Isentropic and isohaline

compressibility

Pa–1

Chemical potential of

water in seawater

J g–1

Chemical potential of sea

salt in seawater

J g–1

Relative chemical

potential of (sea salt and

water in) seawater

J g–1

(

)

A,tp

∂∂ = −

Dissipation rate of kinetic

energy per unit mass

J kg–1 s–1

= m2 s–3

Adiabatic lapse rate Γ K Pa–1

AAA

,,,SSS

ttt

PPP

∂∂∂

Γ= = =

∂∂∂

Sound speed c m s–1

Specific volume v m3 kg–1 1

−

Specific volume anomaly

m3 kg–1

Thermobaric coefficient

based on

KPa

−

(

)

b,S

θθθθ

βαβ

=∂ ∂

Thermobaric coefficient

based on Θ

TΘ 11

KPa

−

(

)

b,S

βαβ

ΘΘΘΘ

=∂ ∂

Cabbeling coefficient

based on

−

bAA

,, ,

Sp p p

CSS

θθ

θθ θ θ

αα

ββ

αθ α β

⎛⎞

=∂ ∂ + ∂ ∂ − ∂ ∂

⎜⎟

⎝⎠

Cabbeling coefficient

based on Θ

CΘ 2

−

bAA

,, ,

Sp p p

CSS

αα

ββ

αα β

ΘΘ

ΘΘ Θ Θ

ΘΘ

⎛⎞

=∂ ∂Θ + ∂ ∂ − ∂ ∂

⎜⎟

⎝⎠

Buoyancy frequency N 1

−

(

)

(

)

Ng S g S

θθ

αβ αθβ

ΘΘ

=Θ− = −

Neutral helicity n

H 3

−

Defined by Eqns. (3.13.1) and (3.13.2)

Neutral Density n

kg m–3 A density variable whose iso-surfaces are designed

to be approximately neutral, i. e.

A.S

γγ

αβ

ΘΘ

∇Θ≈ ∇

Neutral-Surface-Potential-

Vorticity

NSPV 3

−

SPV g f

−

=− where f is the Coriolis

parameter.

Dynamic height anomaly ′

Φ 22

−

31 22

Pa m kg m s

−

Montgomery geostrophic

streamfunction

−

31 22

Pa m kg m s

−

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

IOC Manuals and Guides No. 56

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

−

1.458 42 10 sin sx

−

−, where

is latitude

Molality

mol kg–1

()

==−

∑ where S

is the

mole-weighted average atomic weight of the

elements of sea salt,

S0.031 403 821 8 kg molM−

=…

Ionic strength

mol kg–1

()

0.622 644 9

622.644 9 mol kg

31.403 821 8 1

ImZ mz

−

≈−

∑

Osmotic coefficient

()

(

)

(

)

()

SW 0

0, , , ,

tp S tp

STp mRT t

−

where the molar gas constant,

R = 8.314 472 J mol–1 K–1. See also Eqn. (3.40.9)

for an equivalent definition of .



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161

L.2 Suggested Symbols when variables are functions of ,

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μmol/kgandnotcm3dm–3,ml/lorμmol/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

()

A,,Stp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

()

A,,hhS p



,

()

A,,hhS p

=and

()

ˆ,,

hhS p=ΘbytheexpressionsinappendixP.



TableL.2.SuggestedSymbolswhenvariablesarefunctionsof,

andΘ

quantity function of symbol for this

functional form

enthalpy,h

specificvolume,v

density,



entropy,



(

)

A,,Stp

(

)

A,,hhStp=

(

)

A,,vvStp=

(

)

A,,Stp

ρρ

(

)

A,,Stp

ηη

enthalpy,h

specificvolume,v

density,



potentialtemperature,



(

)

A,,Sp

(

)

A,,hhS p



(

)

A,,vvS p



(

)

A,,Sp

ρρ η



(

)

A,S

θη



enthalpy,h

specificvolume,v

density,



entropy,



(

)

A,,Sp

(

)

A,,hhS p

=

(

)

A,,vvS p

=

(

)

A,,Sp

ρρ θ

=

(

)

A,S

ηθ

=

enthalpy,h

specificvolume,v

density,



entropy,



(

)

A,,SpΘ

(

)

ˆ,,

hhS p=Θ

(

)

ˆ,,

vvS p=Θ

(

)

ˆ,,

ρρ

=Θ

(

)

ˆ,

ηη

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

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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A

Sinunitsof1

kg kg−(sothatfor

examplestandardoceanReferenceSalinityisinputtoSIAfunctionsas0.03516504

kg kg−)ratherthan35.165 04 (1

gkg

−

),insitutemperatureisinputasAbsolute

TemperatureTinK,andpressureisinputasAbsolutePressure

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AAP

(,,,)SSS P

=thatcalculatesAbsoluteSalinity(inkgkg‐1)when

givenPracticalSalinityP

S(whichisunitlessandtakesnumberslike35not0.035)as

itssalinityinputvariable,alongwithlocationintheformoflongitude

(°E)latitude

(°N)andAbsolutePressure

(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

()

PPA

,,, .SSS P

φλ

=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.

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

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163

• 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<

<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

2(, )PfT

=for

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 2(, )PfT

=.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≤ A

S≤0.12kgkg‐1,262K≤T≤353K,and

100Pa≤P≤ 8

10 Pa.

Sincethismanualfocusesonseawater,wereferthereadertoFeisteletal.(2010b)and

Wrightetal.(2010a)fordetailsontheiceandaircomponentsoftheSIAlibrary.

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

IOC Manuals and Guides No. 56

However,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.(2010a);werefertheinterestedreadertoFeistel

etal.(2010b))andWrightetal.(2010a))fordetailedinformation.Wrightetal.(2010a)

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.

DuetothefactthateachleveloftheSIAlibrarybuildsonlowerlevelsandthefact

that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.(2010a))andthesoftwareisservedfromthewww.TEOS‐10.org

website.



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165

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.(2010a)wheredetailedinformationontheuseofeachfunctionis

provided.

Level 0 routines

Constants_0

Public Parameter Values

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

Constants_0 (Cont'd)

Parameter Values (cont'd)

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

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

Maths_0

Uses

constants_0

Public Routines

get_cubicroots

matrix_solve

(S2) Convert_0

Uses

constants_0

Public Routines

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)

Uses

constants_0

Public Routines

chk_iapws95_table6

chk_iapws95_table7

flu_f_si

(S4) Ice_1 (IAPWS06)

Uses

constants_0

Public Routines

chk_iapws06_table6

ice_g_si

(S5) Sal_1 (IAPWS08)

Uses

constants_0

Public Routines

sal_g_term_si

(S6) Air_1

Uses

constants_0

Public Routines

air_baw_m3mol

air_caaw_m6mol2

air_caww_m6mol2

dry_f_si

dry_init_clear

dry_init_Lemmon2000

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Level 2 routines

(S7) Flu_2

Uses

constants_0, flu_1

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

((S8) Ice_2

Uses

constants_0, ice_1

Public Routines

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

(S9) Sal_2

Uses

constants_0, sal_1

Public Routines

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

(S10) Air_2

Uses

constants_0, flu_1, air_1

Public Routines

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

Level 3 routines

(S11) Flu_3a

Uses

constants_0, convert_0,

maths_0, flu_1

Public Routines

get_it_ctrl_density

liq_density_si

liq_g_si

set_it_ctrl_density

vap_density_si

vap_g_si

(S12) Sea_3a

Uses

constants_0, sal_1, sal_2,

flu_3a (convert_0, maths_0,

flu_1)

Public Routines

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

(S13) Air_3a

Uses

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

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(S14) Flu_3b

Uses

constants_0, flu_2, flu_3a

(convert_0, maths_0,

flu_1)

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

(S15) Sea_3b

Uses

constants_0, sal_2, flu_3a,

sea_3a (convert_0, maths_0,

flu_1, sal_1)

Public Routines

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

(S16) Air_3b

Uses

constants_0, convert_0,

air_1, air_2, air_3a (maths_0,

flu_1)

Public Routines

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

Uses

constants_0, sea_3a, sea_3b

(convert_0, maths_0, flu_1,

sal_1, sal_2, flu_3a)

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

(S18) Air_3c

Uses

constants_0, convert_0,

air_2, air_3a, air_3b

(maths_0, air_1, flu_1)

Public Routines

air_h_si

air_potdensity_si

air_potenthalpy_si

air_pottemp_si

air_temperature_si

set_it_ctrl_air_pottemp

(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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Level 4 routines

(S20) Liq_Vap_4

Uses

constants_0, maths_0, flu_1,

flu_2, flu_3a (Convert_0)

Public Routines

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

(S21) Ice_Vap_4

Uses

constants_0, maths_0, flu_1,

flu_2, ice_1, ice_2

Public Routines

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

(S22) Sea_Vap_4

Uses

constants_0, maths_0, flu_1,

sal_1, sal_2, flu_3a, sea_3a,

flu_3b (convert_0, flu_2)

Public Routines

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

(S23) Ice_Liq_4

Uses

constants_0, maths_0, flu_1,

ice_1, flu_2, ice_2

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

(S24) Sea_Liq_4

Uses

constants_0, flu_1, sal_1, flu_2,

sal_2, flu_3a (convert_0,

maths_0)

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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(S27) Liq_Ice_Air_4

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)

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

(S28) Sea_Ice_Vap_4

Uses

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

(S29) Liq_Air_4a

Uses

constants_0, convert_0,

maths_0, flu_1, air_1, flu_2,

air_2, flu_3a, air_3a, air_3b,

liq_vap_4

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

(S30) Ice_Air_4a

Uses

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

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

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(S31) Liq_Air_4b

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)

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

(S32) Ice_Air_4b

Uses

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

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

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)

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

(S34) Ice_Air_4c

Uses

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

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

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

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Level 5 routines

(S35) Flu_IF97_5

Uses

constants_0

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

(S36) Ice_Flu_5

Uses

constants_0

Public Routines

fit_ice_liq_pressure_si

fit_ice_liq_temperature_si

fit_ice_vap_pressure_si

(S37) Sea_5a

Uses

constants_0, sea_3a,

sea_3b, sea_3c (convert_0,

maths_0, flu_1, sal_1, sal_2,

flu_3a)

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

(S38) Air_5

Uses

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

air_lapserate_moist

_c100m

(S39) Liq_F03_5

Uses

constants_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

(S40) OS2008_5

Uses

flu_1, flu_2,

flu_3a, ice_1, liq_vap_4,

sal_1, sal_2 (constants_0,

convert_0, maths_0)

Public Routines

chk_os2008_table

(S41) GSW_Library_5

Uses

constants_0, maths_0,

liq_f03_5, flu_1, flu_3a,

sal_1, sal_2, sea_3a,

sea_3b, sea_5a (convert_0)

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

(S42) Convert_5

Uses

constants_0,

convert_0

Public Routines

cnv_pressure

cnv_salinity

cnv_temperature

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

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173

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

()

1S−onlywhenAbsoluteSalinityA

S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A

Sin1

gkg

−(sothatforexample,standardoceanreferencesalinitySO

S

is35.165041

gkg

−[not0.035165041

kg kg

−

]),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oceanographic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paperbyMcDougalletal.(2010b)describeshowmanyofthese

functionsarecalculated.

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25‐termcomputationallyefficient

expressionfordensityasafunctionofAbsoluteSalinity,ConservativeTemperatureand

pressure(seeappendixA.30andappendixK).Bothformsgivevaluesofdensity,the

thermalexpansioncoefficientetc.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25‐termversionfordensity.

Version1oftheGSWToolboxwasreleasedinJanuary2009andversion2.0in

October2010.Thisversion2.0oftheGSWToolboxisavailabletodateonlyinMATLAB

(version1isalsoavailableinFORTRAN)fromthewebsiteatwww.TEOS‐10.org.Aquick

introductiontoTEOS‐10isavailableontheTEOS‐10website(www.TEOS‐10.org)asthe

shortdocumentcalled“GettingstartedwiththeGibbsSeaWater(GSW)Oceanographic

ToolboxofTEOS‐10”.SomeofthefunctionnamesintheGSWOceanographicToolbox

aregiveninthefollowingtable.

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

IOC Manuals and Guides No. 56

TableN.1.Aselectionoffunctionsinversion2.0oftheGSWOceanographicToolbox.

Thermodynamic

Property

Functionname

(inMATLAB)

Remarks

AbsoluteSalinity A

Sgsw_SA_from_SP theMcDougall,JackettandMillero(2010a)

algorithmforA

Susingalook‐uptableforA



PreformedSalinity*

Sgsw_Sstar_from_SP PreformedSalinity*

SfromPracticalSalinityP

S

Conservative

TemperatureΘgsw_CT_from_t ConservativeTemperatureΘ,from

()

A,,Stp

Gibbsfunction

andits

1stand2ndderivatives gsw_gibbs thesumoftheIAPWS‐09andIAPWS‐08Gibbs

functions,andthederivativesofthissum

specificvolumevgsw_specvol

(

)

A,,vS tpspecificvolumeusinggsw_gibbs

density

gsw_rho

(

)

A,,Stp

insitudensityusinggsw_gibbs

potentialdensity

Θgsw_pot_rho

(

)

,, ,Stpp

Θusinggsw_gibbs

density

,and

potentialdensity

Θ gsw_rho_CT25

(

)

ˆ,,Sp

Θ,insitudensityusingthe25‐term

expressionfordensityintermsofConservative

Temperature(CTforshort),Θ.Potentialdensity

withrespecttopressurer

p,usingthis25‐term

expression,isobtainedbysimplycalling

gsw_rho_CT25withthispressure,obtaining

()

ˆ,,

ρρ

Θ=Θ.

specificentropy

gsw_entropy

(

)

A,,Stp

usinggsw_gibbs

specificenthalpyh gsw_enthalpy

(

)

A,,hS tpusinggsw_gibbs

firstorderderivativesof

enthalpywithrespectto

ConservativeTemperature

gsw_enthalpy_first_derivati

ves

andA

ˆS

husinggsw_gibbs(seeEqns.(A.11.15)

and(A.11.18))

secondorderderivativesof

enthalpywithrespectto

ConservativeTemperature

gsw_enthalpy_second_deri

vatives A

ˆˆ

ΘΘ Θ andAA

ˆSS

husinggsw_gibbs

soundspeedcgsw_sound_speed

(

)

A,,cS tpusinggsw_gibbs

Conservative

TemperatureΘgsw_CT_from_pt

(

)

A,S

Θ,founddirectlyfromgsw_gibbs.

Here

ispotentialtemperaturewithr

p=0.

potentialtemperature

gsw_pt_from_t

(

)

,, ,Stpp

foundbyusinggsw_gibbsandby

equatingtwovaluesofentropy

potentialtemperature

gsw_pt0_from_t

(

)

A,,Stp

,acomputationallyfasterversionof

gsw_pt_from_twhenr0p

dbar.

potentialtemperature

gsw_pt_from_CT

(

)

A,S

,foundbyNewton_Raphsoniteration,

beingtheinversefunctionofgsw_CT_from_pt

thermalexpansion

coefficientwithrespectto

Conservative

Temperature

Θ

gsw_alpha_wrt_CT

()

A,,Stp

Θusinggsw_gibbs

salinecontraction

coefficientatconstant

Conservative

TemperatureΘ

gsw_beta_const_CT

()

A,,Stp

Θusinggsw_gibbs

density,thermal

expansionandsaline

contractioncoefficient

gsw_rho_alpha_beta_CT25

(

)

ˆ,,Sp

Θ,

()

ˆ,,

ΘΘand

()

ˆ,,

ΘΘ

usingthe25‐termexpressionfordensityinterms

ofConservativeTemperatureΘ

dynamicheightanomalygsw_geo_strf_dyn_height geostrophicstreamfunctioninanisobaricsurface

McDougall‐Klocker

geostrophicstreamfunctiongsw_geo_strf_McD_Klocker geostrophicstreamfunctioninanapproximately

neutralsurface,seeEqn.(3.30.1)

Montgomerygeostrophic

streamfunction gsw_geo_strf_Montgomery geostrophicstreamfunctioninaspecificvolume

anomalysurface,seeEqn.(3.28.1)

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

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175

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

,isobaricspecificheatcapacityp

c,

thermalexpansioncoefficientt

,soundspeed,cspecificvolume,vfreezing

temperaturef

tmixingheat.hΔ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

r.m.s F03 - expt.datum

⎡⎤

=⎢⎥

⎣⎦

∑(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P

Scolumnsweremislabeledas

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

IOC Manuals and Guides No. 56

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P

S=20.062,t=25°C,andp=588.0barsshouldhave

been0.9643933

kg m

−

,not0.9643833

kg m

−

.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1

gkg

−

(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,

mixingheathΔ,freezingpointdepressionf

twatervapourpressurevap ,pandthe

Debye‐Hückellimitinglaw.

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.

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

IOC Manuals and Guides No. 56

177

Table O.1. Summary of data used in the regression to determine the coefficients of the Feistel

(2003) Gibbs potential.

Quantity

Source

()

AgkgS−

t /°C

P/MPa

Points

Required

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

PBB80 5-42 0-30 0 345 4 ppm 4.0 ppm 4.2 ppm

PG93 34-50 15-30 0 81 10 ppm 12.0i ppm 11.9 ppm

cp BDSW70 10-50 0-40 0 25 2 J/(kg K) 1.45ii J/(kg K) 1.43 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)

C78 10-30 -6-1 0.7-33 31 0.6x10-6 K-1 0.73x10-6 K-1 0.74x10-6 K-1

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

vS BS70 30-40 -2-30 1-100 221 4 ppm 2.6 ppm 2.6 ppm

t DK74 4-40 -2-0 0 32 2 J/kg 1.8 J/kg 1.9 J/kg

h B68 0-33 25 0 24 4 J 2.4 J 2.4 J

h MHH73 1-41 0-30 0 95 0.4 J/kg 0.5 J/kg 0.5 J/kg

i TheoriginalvalueinTable9ofFeistel(2003)of11.3ppmreferstothespecificvolume.

ii 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.

Quantity

Source

()

AgkgS−

t /°C

P/MPa

Points

Resulting

r.m.s.

Verifying

r.m.s.

cp BDCW67 11-117 2-80 0 221 3.46 J/(kg K) 3.46 J/(kg K)

c 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)

h B68 0-97 25 0 33 0.75 J/kg 0.75 J/kg

h C70 35-36 2-25 0 19 7.2 J/kg 7.1 J/kg

h MHH73 1-35 0-30 0 120 3.3 J/kg 3.3 J/kg

t DK74 4-40 -0.2-(-2.2) 0 32 1.6 mK 1.6 mK

t FM07 5-109 -0.3-(-6.9) 0 22 1.2 mK 1.0 mK

pvap R54 18-40 25 0 13 2.8 J/kg 2.8 J/kg

oil

t BSRSR74 6-70 60-80 0 32 9.1 J/kg 9.3 J/kg

gLL F08 35 -5-95 0 21 0.091 J/kg 0.092 J/kg

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

IOC Manuals and Guides No. 56

AppendixP:Thermodynamicproperties

basedon

(

)

(

)

(

)

AA A

,, , , , , , ,gS tp hS p hS p

ηθ



and

(

)

ˆ,,hS pΘ



ThethermodynamicpotentialonwhichTEOS‐10isbasedistheGibbsfunctionof

seawater.BeingaGibbsfunction,

(

)

A,,

Stp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

()

A,, .hhS p



Itprovestheoreticallyconvenienttoconsidertheadditionalfunctional

forms

()

A,,hhS p

=and

()

ˆ,,hhS p=Θ

forenthalpy.Thesetwofunctionalformsdonot

constituteacompletethermodynamicdescriptionofseawaterbutwhensupplementedby

theexpressions

()

A,S

ηθ

=and

(

)

ˆ,S

ηη

Θforentropy,theydoformcomplete

thermodynamicpotentials.Intheexpressions

(

)

A,,hhS p

=and

()

A,S

ηθ

=itis

possibletochooseanyfixedreferencepressurer

pforthedefinitionofpotential

temperature,

.Howeverthereisnoadvantagetochoosingthereferencepressuretobe

differentfromr0p

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(theresultsinthistablefor

(

)

A,,hS p



mostlycome

fromFeistel(2008)andFeisteletal.(2010b)).Notethatthereferencepressurer

pthat

appearsinthelastthreecolumnsofthe

rowofTableP.1isthereferencepressureof

potentialdensity,notof

,whereasintheGibbsfunctioncolumn,thisgeneralreference

pressuremustalsobeusedtoevaluate

.

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))

AA A

PPS PS PS

PP P P P

PS P S

hh vv

Tvvv

θθθ θθ θθ

ρρ

ρρρ

=− =− =−+



 





 

 (P.1)

AA A

ˆˆ ˆ

ˆˆˆ

ˆˆ ,

ˆˆˆ ˆˆˆ

ˆˆˆ

PPS PS PS

PP P P P

PPSP

hh vv

Tvvv

hhh

ρρ

ρρρ

ΘΘΘ ΘΘ ΘΘ

=− =− =−+ (P.2)

AAA AAA

AAA

PS PS S S S S

PP P

PS P PS P

SSS

hh vv

hh h vv v

Cvvv v v

hh h h

θθ

θθθ θ θ θθ θ θ

θθ θ θ

ρρ

ρρ ρ

ρρρρ ρ

⎛⎞ ⎛⎞

=− + =− +

⎜⎟ ⎜⎟

⎜⎟

⎜⎟ ⎝⎠

⎝⎠

⎛⎞

=− + −

⎜⎟

⎝⎠



  

 

     



 

  

(P.3)

AAA AAA

AAA

ˆˆ

ˆˆ ˆ ˆˆ

ˆˆ ˆ

ˆˆˆ ˆ ˆ ˆˆˆˆ ˆ

ˆˆ

ˆˆ ˆ

ˆˆˆ ˆ ˆ

PS PSS S SS

PP P

PPSP PS P

SSS

hh vv

hh h vv v

Cvvvv v

hhh h h

ρρ

ρρ ρ

ρρρ ρ ρ

ΘΘ

ΘΘΘ Θ Θ ΘΘ Θ Θ

ΘΘ Θ Θ

⎛⎞ ⎛⎞

⎜⎟

=− + =− +

⎜⎟

⎜⎟ ⎝⎠

⎝⎠

⎛⎞

=− + −⎜⎟

⎜⎟

⎝⎠

(P.4)

HerefollowssomeinterestingexpressionsthatcanbegleanedfromTableP.1.

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

IOC Manuals and Guides No. 56

179

() ()

0ˆ

00 ,

chc

θθ

ΘΘ

==Θ=−



()

+= 

(

)

()

Tt hh

Tch

+∂

+∂(P.5)

() ()

ˆˆ

ˆ,

ˆ0

Sp S p

ηθη

θθ α

θηα

ΘΘ ΘΘ

∂∂

====− =− = =

∂Θ ∂Θ Θ

(P.6)

() ()

ˆˆ

μθη

=− = − + (P.7)

()

AA A

010

AA A

ˆˆ

ˆ.

SpS S

SS S c

ηθη

θθ

θη

−

ΘΘ

ΘΘ Θ

Θ+

∂

∂∂

= = =− =− = = −

∂∂ ∂



(P.8)

SeealsoEqn.(A.12.6)foranalternativeexpressionforA

ˆS

.Equation(P.8)canalsobe

writtenas

()

ˆ.

−

∂+ =

∂(P.9)

NowweconsiderhowallthetermsinthelastcolumnofTableP.1maybeevaluated

intermsoftheexpression

()

ˆ,,hS pΘofEqn.(A.30.6);thisbeingtheexpressionfor

specificenthalpythatfollowsfromthe25‐termexpressionfordensityasafunctionof

()

A,,SpΘasdescribedinEqn.(K.1)andTableK.1.Thesuperscript“25”isaddedto

()

ˆ,,hS pΘtoemphasizethatthisisanapproximateexpressionforspecificenthalpy.

Thefirststepistoevaluate

exactlyfromthefollowingimplicitexpressionforΘin

termsoftheGibbsfunctionat0p=(seeEqn.(2.12.1)),asdiscussedinsection3.3,

()

(

)

(

)

(

)

AA0A

,,0 ,,0 ,,0

chSt gSt TgSt

θθθθ

Θ= = = = − + = .(P.10)

Next,weremindourselvesthatweknowthefunctionalformsof

(

)

A,S

,

()

A,S

and

()

A,,0S

μθ

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))

() ( )

,,,0

SgSt

ηθ θ

=− =

,

(

)

(

)

,,0 , ,0

SgSt

μθ θ

,(P.11a,b)

andEqn.(P.10)isrepeatedhereemphasizingthefunctionalformoftheleft‐handside,

()( )

(

)

(

)

AA 0A

,,,0 ,,0

cS gSt T gSt

θθθθ

Θ==−+ =

.(P.12)

Thepartialderivativeswithrespectto

andwithrespectto

,bothatconstantA

S

andp,andthepartialderivativeswithrespecttoA

S,arerelatedby

Sp Sp

θθ

∂∂

∂Θ ∂

,and A

AA ,

∂∂ ∂

=−

∂∂ ∂



.(P.13a,b)

Useoftheseexpressions,actingonentropyyields(with0p

everywhere,andusing

Eqn.(P.7)[orEqn.(A.12.8b)]andEqn.(P.8))

()

ˆp

Θ=≡



,

()

ˆp

ΘΘ =−Θ+

,

(

)

()

,,0

ˆS

μθ

ηθ

=− +



,(P.14a,b,c)

()

ˆSp

=Θ+



,and

()

(

)

()

,,0

ˆS

μθ

ηθ

=− −

+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

()

ˆ,,hS pΘandtheexactexpressions(P.14)

and(P.15)whicharewrittenintermsofpotentialtemperature

whichisfoundfromthe

exactimplicitequation(P.10).Thiscompletesthediscussionofhow

()

ˆ,,hS pΘcanbe

usedasanalternativethermodynamicpotentialofseawater.Thepartialderivativesof

entropyinEqns.(P.14)and(P.15)areavailableintheGSWToolboxfromthefunctions

gsw_entropy_first_derivativesandgsw_entropy_second_derivatives.



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

IOC Manuals and Guides No. 56

Table P.1. Expressions for various thermodynamic variables based on four different thermodynamic potentials

Expressionsbasedon

()

A,,

Stp

Expressionsbasedon

(

)

A,,hS p



Expressionsbasedon

()

A,,hS p

and

(

)

A,S



Expressionsbasedon

(

)

ˆ,,hS pΘand

(

)

ˆ,S

t t

(

)

Tt h



()

Tt h



(

)

0ˆˆ

Tt h

(

)( )

Ar A

,, ,,

SpgStp

=

[thisisanimplicitequationfor

]

(

)

(

)

00Th



()

(

)

;0Th

θη



(

)

0ˆ

(

)( )

(

)

A0A

,,0 ,,0

gS T g S

θθθ

−+

Θ=

(

)

hcΘ=



(

)

hcΘ= 

(

)

hcΘΘ=

()

A,,

gS tp=

=−



gh h

=−



ˆˆ

gh h

=−

(

)

A,S

ηθ

=

(

)

ˆ,S

ηη

()

hgTtg=− +

(

)

A,,hhS p



(

)

A,,hhS p

=

(

)

ˆ,,hhS p=Θ

vg=

vh=



vh= ˆ

vh=

()

−

(

)

−



(

)

−

=

(

)

ˆP

−

= A



ηη

=−



ˆˆ

ηη

=−

(

)( )

00TP

ugTtg pPg=− + − +

(

)

uh pPh=− +



(

)

uh pPh=− +



(

)

ˆˆ

uh pPh=− +

()

gpPg=−+

(

)

hh pPh

=− − +





(

)

hh pPh

θθ

ηη

=− − +

 



(

)

ˆˆ ˆ

ˆˆ

hh pPh

ηη

ΘΘ

=− − +

()

0pTT

cTtg=− + p

chh

ηη



(

)

ch hh

θθθθθθθ

ηη η

=−



 

(

)

ˆˆˆ

ˆˆ ˆ

ch h h

ηη η

ΘΘΘΘΘΘΘ

=−

()()()

A0A

,,0 ,,0

hgS T gS

θθθ

=−+

(

)

00hh=



(

)

00hh=

(

)

ˆ0p

hh c

=Θ

()

gS p

ρθ

−

⎡⎤

=⎣⎦

()

−

⎡

⎤

⎣

⎦



()

−

⎡

⎤

⎣

⎦



()

ˆP

ρρ

−

⎡

⎤

⎣

⎦

−

=−

PPP P

hh h h

−−

=− +



 



()

11tP

PPP P

hh h hh

θθ

θθ θθ θ

−−

=− − −



 



()

ˆˆ

ˆˆ ˆ

ˆˆ

PPP P

hh h

−−

ΘΘ

ΘΘ ΘΘ Θ

=− − −

PPP P

gg g

−−

=− + 1

−

=−



−

=− 1

ˆˆ

−

=−

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

IOC Manuals and Guides No. 56

181

Table P.1. (cont’d) Expressions for various thermodynamic variables based on four different thermodynamic potentials

Expressionsbasedon

()

A,,

Stp

Expressionsbasedon

(

)

A,,hS p



Expressionsbasedon

()

A,,hS p

and

(

)

A,S



Expressionsbasedon

(

)

ˆ,,hS pΘand

(

)

ˆ,S

()

TT TP TT PP

cgg g gg=−

ch h=−



ch h=−



ˆˆ

ch h=−

Γ TP TT

=−

Γ=



Γ=  ˆˆ

Γ=

tTP

= 1

−



()

hh hh

θθ θθ θ

−

=− −





()

1ˆ

ˆˆ

−Θ

ΘΘ ΘΘ Θ

=− −

()

PTT

()

ηη

−



 1

−

=

ˆˆ

hh c

−Θ

=−

()

αθ

Θ=− +

()

hh h

Θ−

=



()

hh h

Θ−

=

 1

ˆˆ

Θ−

−

=− A

PPS

hh h

−

=−







()

PPS

hh hh

θθθ

θθθ θθθ

ηη

−

=−

−

+−



()

ˆˆ

PPS

ηη

−

ΘΘΘ

−Θ

ΘΘΘ ΘΘΘ

=−

−

+−

()

PPS

TP ST ST

PTT

gSp

−

=−

⎡⎤

−

⎣⎦

()

PPS

hh h

ηη

−

=−





 A

−

=− A

ˆˆ ˆˆ

PPS PP

hh hh

θη

βη

−−

=− +

()( )

,,0

PPS

TP S T S

PTT

gT S

θθ

Θ−

−

=−

⎡

⎤

−+

⎣

⎦

()

PPS

hh h

Θ−

−

=−







(

)

()

PPS PP

hh hh h

Θ− −

=− +



 

 A

ˆˆ

Θ−

=−

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

IOC Manuals and Guides No. 56

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