TEOS 10 Manual
TEOS-10_Manual
TEOS-10_Manual
User Manual:
Open the PDF directly: View PDF 
.
Page Count: 218
| Download | |
| Open PDF In Browser | View PDF |
Manuals and Guides 56
Intergovernmental Oceanographic Commission
The international thermodynamic
equation of seawater – 2010:
Calculation and use of thermodynamic properties
The Intergovernmental Oceanographic Commission (IOC) of UNESCO celebrates
its 50th anniversary in 2010. Since taking the lead in coordinating the International
Indian Ocean Expedition in 1960, the IOC has worked to promote marine research,
protection of the ocean, and international cooperation. Today the Commission is also
developing marine services and capacity building, and is instrumental in monitoring
the ocean through the Global Ocean Observing System (GOOS) and developing
marine-hazards warning systems in vulnerable regions. Recognized as the UN
focal point and mechanism for global cooperation in the study of the ocean, a key
climate driver, IOC is a key player in the study of climate change. Through promoting
international cooperation, the IOC assists Member States in their decisions towards
improved management, sustainable development, and protection of the marine
environment.
ManualsȱandȱGuidesȱ56ȱ
IntergovernmentalȱOceanographicȱCommissionȱ
Theȱinternationalȱthermodynamicȱȱ
equationȱofȱseawaterȱ–ȱ2010:ȱȱ
Calculation and use of thermodynamic properties
ȱ
The authors are responsible for the choice and the presentation of the facts contained in this publication
and for the opinions expressed therein, which are not necessarily those of UNESCO, SCOR or IAPSO and
do not commit those Organizations.
The photograph on the front cover of a CTD and lowered ADCP hovering just below the sea surface was
taken south of Timor from the Southern Surveyor in August 2003 by Ann Gronell Thresher.
For bibliographic purposes, this document should be cited as follows:
IOC, SCOR and IAPSO, 2010: The international thermodynamic equation of seawater – 2010: Calculation and
use of thermodynamic properties. Intergovernmental Oceanographic Commission, Manuals and Guides No.
56, UNESCO (English), 196 pp.
Printed by UNESCO
(IOC/2010/MG/56 Rev.)
© UNESCO/IOC et al. 2010
iii
Table of contents
Acknowledgements ……………………………………………………………………... vii
Foreword ……………………………………………………………………………….…… viii
Abstract ………………………………………………………………………………...……… 1
1. Introduction ………………………………………………………………….…..… 2
1.1 Oceanographic practice 1978 -‐‑ 2009 ……………………………………………………. 2
1.2 Motivation for an updated thermodynamic description of seawater ………….… 2
1.3 SCOR/IAPSO WG127 and the approach taken ………………....……………….… 3
1.4 A guide to this TEOS-‐‑10 manual ………………………………………………………. 6
1.5 A remark on units …………………………………………………………………..…… 7
1.6 Recommendations …………………………………………………………………..…… 7
2. Basic Thermodynamic Properties ……………….………………..…. 9
2.1 ITS-‐‑90 temperature ………………………………...…..…………………………..……. 9
2.2 Sea pressure ………………………………….…………………..……….……………. 9
2.3 Practical Salinity …………………………..…………………………………………..… 9
2.4 Reference Composition and the Reference-‐‑Composition Salinity Scale …..……. 10
2.5 Absolute Salinity ………………………………………………………………………. 11
2.6 Gibbs function of seawater .…….………………………………………………… 15
2.7 Specific volume ………….…….……………………….…….……………………… 18
2.8 Density ………...…………………...……………………………………………….… 18
2.9 Chemical potentials ……..………………………………..………………………… 19
2.10 Entropy …………………………………………………….............................……… 20
2.11 Internal energy …………………………………………………..............….……… 20
2.12 Enthalpy ………..………………………………………………………….…...…… 20
2.13 Helmholtz energy ….…………………………………………………….....……… 21
2.14 Osmotic coefficient ….………………………………………………….….........… 21
2.15 Isothermal compressibility ..…….………………………………………………... 21
2.16 Isentropic and adiabatic compressibility …..…………….……………………… 22
2.17 Sound speed ……………………….……………………………………………..… 22
2.18 Thermal expansion coefficients ……...………………………………………….... 22
2.19 Saline contraction coefficients ……………………………………………….…… 23
2.20 Isobaric heat capacity ………..…………………………………………………… 24
2.21 Isochoric heat capacity ……….…………………………………………………… 24
2.22 Adiabatic lapse rate ………..……………………………………………………… 25
IOC Manuals and Guides No. 56
iv
3. Derived Quantities …………………………………..……….………………. 26
3.1 Potential temperature …………………………………………………………………. 26
3.2 Potential enthalpy ………………………………………....………………………… 27
3.3 Conservative Temperature ……………….………………….………………………. 27
3.4 Potential density ……………………………………………….……………………… 28
3.5 Density anomaly …………………….………………………….…………………… 28
3.6 Potential density anomaly ……….…………………………….…………………… 29
3.7 Specific volume anomaly ………………………………………….…………………. 29
3.8 Thermobaric coefficient ………….……………………………………………………. 30
3.9 Cabbeling coefficient ………….…………………………………………………....…. 31
3.10 Buoyancy frequency ……….…………………………………………….……..….… 32
3.11 Neutral tangent plane …….……………………………………………………..…. 32
3.12 Geostrophic, hydrostatic and “thermal wind” equations …….………………. 34
3.13 Neutral helicity …………….…………………………………………………..….…. 35
3.14 Neutral Density ….…………………………………………………………..….…….. 39
3.15 Stability ratio …..………………………………………………………………...…. 39
3.16 Turner angle ….………………………………………………………………………. 39
3.17 Property gradients along potential density surfaces …………………………… 40
3.18 Slopes of potential density surfaces and neutral tangent planes compared ..… 40
3.19 Slopes of in situ density surfaces and specific volume anomaly surfaces …..… 41
3.20 Planetary potential vorticity …………………………………….......………………. 42
3.21 Vertical velocity through the sea surface …….…………………………………… 45
3.22 Freshwater content and freshwater flux …………………………………………. 46
3.23 Heat transport …………………….………………………………………………….. 46
3.24 Geopotential ………….……………………………………………………………….. 47
3.25 Total energy …………….…………………………………………………………….. 47
3.26 Bernoulli function ……….…………………………………………………………… 47
3.27 Dynamic height anomaly …………………………………………………………… 48
3.28 Montgomery geostrophic streamfunction ………….…………………………… 49
3.29 Cunningham geostrophic streamfunction ……….……………………………… 50
3.30 Geostrophic streamfunction in an approximately neutral surface ….………… 51
3.31 Pressure-‐‑integrated steric height ……..…………………………………………… 51
3.32 Pressure to height conversion …….………………………………………………… 52
3.33 Freezing temperature ……….…………………………………………………….….. 53
3.34 Latent heat of melting ….…………………………………………..………….…… 55
3.35 Sublimation pressure …………………………………………………………….… 56
3.36 Sublimation enthalpy …………………………………………………………….… 57
3.37 Vapour pressure ……………………………………………………………….…… 59
3.38 Boiling temperature ……….……………………………………….…………….… 60
3.39 Latent heat of evaporation ………………………………………………………… 60
3.40 Relative humidity and fugacity …………………………………………………… 62
3.41 Osmotic pressure …………………………………………………………………… 65
3.42 Temperature of maximum density …….….……………………………………… 65
4. Conclusions ………………………………………….……..……………………… 67
IOC Manuals and Guides No. 56
v
Appendix A: Background and theory underlying the use of the
Gibbs function of seawater ………………..……………...... 69
A.1 ITS-‐‑90 temperature …………………………………………………………………... 69
A.2 Sea pressure, gauge pressure and Absolute Pressure …………….………….... 73
A.3 Reference Composition and the Reference-‐‑Composition Salinity Scale …….…... 74
A.4 Absolute Salinity ……………………………………………….……………………... 76
A.5 Spatial variations in seawater composition ……………………………….………... 82
A.6 Gibbs function of seawater ………………………………….…………….……..…... 86
A.7 The fundamental thermodynamic relation ………………………………….……... 87
A.8 The “conservative” and “isobaric conservative” properties ……………….…..…. 87
A.9 The “potential” property ……………….……….………………………………...... 90
A.10 Proof that θ = θ ( SA ,η ) and Θ= Θ ( SA ,θ ) .…………………….…………………... 92
A.11 Various isobaric derivatives of specific enthalpy ……………...…..…………… 92
A.12 Differential relationships between η, θ , Θ and S A …………...……...….………. 95
A.13 The First Law of Thermodynamics ………………….……………………...…...... 95
A.14 Advective and diffusive “heat” fluxes ………………….………………..……...... 99
A.15 Derivation of the expressions for α θ , β θ , α Θ and β Θ ………………….…….. 101
A.16 Non-‐‑conservative production of entropy ………………………...……………..... 102
A.17 Non-‐‑conservative production of potential temperature ………………….…... 106
A.18 Non-‐‑conservative production of Conservative Temperature ……………….. 108
A.19 Non-‐‑conservative production of specific volume …………………………..…... 111
A.20 The representation of salinity in numerical ocean models ………………........... 112
A.21 The material derivatives of S* , SA , S R and Θ in a turbulent ocean ……….... 117
A.22 The material derivatives of density and of locally-‐‑referenced
potential density; the dianeutral velocity e …………….……............................ 121
A.23 The water-‐‑mass transformation equation …………….…….................................. 123
A.24 Conservation equations written in potential density coordinates ………..……. 125
A.25 The vertical velocity through a general surface …………….………………….... 126
A.26 The material derivative of potential density ………………………..….……..... 127
A.27 The diapycnal velocity of layered ocean models (without rotation
of the mixing tensor) ……………………………………………….….….……..... 128
A.28 The material derivative of orthobaric density ………………………..….……..... 128
A.29 The material derivative of Neutral Density …………………………….……..... 129
A.30 Computationally efficient 75-‐‑term expression for the specific volume
of seawater in terms of Θ …………………….....………..…………………….. 130
Appendix B: Derivation of the First Law of Thermodynamics ….…….…… 132
Appendix C: Publications describing the TEOS-‐‑10 thermodynamic
descriptions of seawater, ice and moist air ……...………..…..…. 140
Appendix D: Fundamental constants …….…………………..……………………... 143
IOC Manuals and Guides No. 56
vi
Appendix E: Algorithm for calculating Practical Salinity ……….…………….. 147
E.1 Calculation of Practical Salinity in terms of K15 ….……........................................... 147
E.2 Calculation of Practical Salinity at oceanographic temperature and pressure ..... 147
E.3 Calculation of conductivity ratio R for a given Practical Salinity .......................... 148
E.4 Evaluating Practical Salinity using ITS-‐‑90 temperatures ......................................... 149
E.5 Towards SI-‐‑traceability of the measurement procedure for Practical Salinity
and Absolute Salinity .................................................................................................. 149
Appendix F: Coefficients of the IAPWS-‐‑95 Helmholtz function of
fluid water (with extension down to 50K) …………...………...… 152
Appendix G: Coefficients of the pure liquid water Gibbs function
of IAPWS-‐‑09 ………………………...……………………….…...………. 155
Appendix H: Coefficients of the saline Gibbs function for seawater
of IAPWS-‐‑08 ………………………………………………..…….………. 156
Appendix I: Coefficients of the Gibbs function of ice Ih of IAPWS-‐‑06 …... 157
Appendix J: Coefficients of the Helmholtz function of moist air
of IAPWS-‐‑10 ……………………………..…….………………………… 159
Appendix K: Coefficients of the 75-‐‑term expression for the specific
volume of seawater in terms of Θ ……………….….………….... 163
Appendix L: Recommended nomenclature, symbols and
units in oceanography ………………………………………………… 165
Appendix M: Seawater-‐‑Ice-‐‑Air (SIA) library of computer software ……….. 171
Appendix N: Gibbs-‐‑SeaWater (GSW) Oceanographic Toolbox ..................... 182
Appendix O: Checking the Gibbs function of seawater against the
original thermodynamic data ……………………….………….…… 188
Appendix P: Thermodynamic properties based on g SA ,t, p , h SA ,η , p ,
(
)
(
)
(
) (
)
h SA ,θ , p and ĥ SA ,Θ, p …………………….…………………... 191
References ……………………………..……………………………….………….…….… 195
Index ……………………………………..……………………………….…………….….… 205
Changes made to the TEOS-‐‑10 Manual since 13th April 2010 .…………......… 207
IOC Manuals and Guides No. 56
vii
Acknowledgements
This TEOS-‐‑10 Manual reviews and summarizes the work of the SCOR/IAPSO Working
Group 127 on the Thermodynamics and Equation of State of Seawater. Dr John Gould and
Professor Paola Malanotte-‐‑Rizzoli played pivotal roles in the establishment of the
Working Group and we have enjoyed rock-‐‑solid scientific support from the officers of
SCOR, IAPSO and IOC. TJMcD wishes to acknowledge fruitful discussions with Drs
Jürgen Willebrand and Michael McIntyre regarding the contents of appendix B. We have
benefited from extensive comments on drafts of this manual by Dr Stephen Griffies and Dr
Allyn Clarke. Dr Harry Bryden is thanked for valuable and timely advice on the
treatment of salinity in ocean models. Louise Bell of CSIRO provided much appreciated
advice on the layout of this document. Dr Paul Barker is thanked for carefully proof-‐‑
reading this TEOS-‐‑10 Manual, for writing most of the GSW Oceanographic Toolbox
functions and for designing the TEOS-‐‑10 web site, www.TEOS-10.org. This document is
based on work partially supported by the U.S. National Science Foundation to SCOR
under Grant No. OCE-‐‑0608600. FJM wishes to acknowledge the Oceanographic Section of
the National Science Foundation and the National Oceanic and Atmospheric
Administration for supporting his work.
This document has been written by the members of SCOR/IAPSO Working Group 127,
Trevor J. McDougall, (chair), University of New South Wales, Sydney, Australia
Rainer Feistel, Leibniz-‐‑Institut fuer Ostseeforschung, Warnemuende, Germany
Daniel G. Wright+, formerly of Bedford Institute of Oceanography, Dartmouth, Canada
Rich Pawlowicz, University of British Columbia, Vancouver, Canada
Frank J. Millero, University of Miami, Florida, USA
David R. Jackett++, formerly of CSIRO, Hobart, Australia
Brian A. King, National Oceanography Centre, Southampton, UK
Giles M. Marion, Desert Research Institute, Reno, USA
Steffen Seitz, Physikalisch-‐‑Technische Bundesanstalt (PTB), Braunschweig, Germany
Petra Spitzer, Physikalisch-‐‑Technische Bundesanstalt (PTB), Braunschweig, Germany
C-‐‑T. Arthur Chen, National Sun Yat-‐‑Sen University, Taiwan, R.O.C.
March 2010
+ deceased, 8th July 2010
++ deceased, 31st March 2012
IOC Manuals and Guides No. 56
viii
Foreword
This document describes the International Thermodynamic Equation Of Seawater – 2010
(TEOS-‐‑10 for short). TEOS-‐‑10 defines the thermodynamic properties of seawater and of
ice Ih and has been adopted by the Intergovernmental Oceanographic Commission at its
25th Assembly in June 2009, replacing EOS-‐‑80 as the official description of seawater and
ice properties in marine science.
Fundamental to TEOS-‐‑10 are the concepts of Absolute Salinity and Reference
Salinity. These variables are described in detail here, emphasising their relationship to
Practical Salinity.
The science underpinning TEOS-‐‑10 has been described in a series of papers
published in the refereed literature (see appendix C). The present document may be
called the “TEOS-‐‑10 Manual” and acts as a guide to those published papers and
concentrates on how the thermodynamic properties obtained from TEOS-‐‑10 are to be
used in oceanography.
In addition to the thermodynamic properties of seawater, TEOS-‐‑10 also describes the
thermodynamic properties of ice and of humid air, and these properties are summarised
in this document. The TEOS-‐‑10 computer software, this TEOS-‐‑10 Manual and other
documents may be obtained from www.TEOS-10.org. In particular, there are two
introductory articles about TEOS-‐‑10 on this web site, namely “What every
oceanographer needs to know about TEOS-‐‑10 (The TEOS-‐‑10 Primer)” (Pawlowicz,
2010b) and “Getting started with TEOS-‐‑10 and the Gibbs Seawater (GSW)
Oceanographic Toolbox” (McDougall and Barker, 2011). An historical account of how
TEOS-‐‑10 was developed has appeared in Pawlowicz et al. (2012).
When referring to the use of TEOS-‐‑10, it is the present document that should be
referenced as IOC et al. (2010a) [IOC, SCOR and IAPSO, 2010: The international
thermodynamic equation of seawater – 2010: Calculation and use of thermodynamic properties.
Intergovernmental Oceanographic Commission, Manuals and Guides No. 56, UNESCO
(English), 196 pp.].
This version of the TEOS-‐‑10 Manual includes corrections up to 20th December 2016.
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
1
Abstract
This document outlines how the thermodynamic properties of seawater are evaluated
using the International Thermodynamic Equation Of Seawater – 2010 (TEOS-‐‑10). This
thermodynamic description of seawater is based on a Gibbs function formulation from
which thermodynamic properties such as entropy, specific volume, enthalpy and
potential enthalpy are calculated directly. When determined from the Gibbs function,
these quantities are fully consistent with each other. Entropy and enthalpy are required
for an accurate description of the advection and diffusion of heat in the ocean interior and
for quantifying the ocean’s role in exchanging heat with the atmosphere and with ice. The
Gibbs function is a function of Absolute Salinity, temperature and pressure. In contrast to
Practical Salinity, Absolute Salinity is expressed in SI units and it includes the influence of
the small spatial variations of seawater composition in the global ocean. Absolute Salinity
is the appropriate salinity variable for the accurate calculation of horizontal density
gradients in the ocean. Absolute Salinity is also the appropriate salinity variable for the
calculation of freshwater fluxes and for calculations involving the exchange of freshwater
with the atmosphere and with ice. Potential functions are included for ice and for moist
air, leading to accurate expressions for numerous thermodynamic properties of ice and air
including freezing temperature and latent heats of melting and of evaporation. This
TEOS-‐‑10 Manual describes how the thermodynamic properties of seawater, ice and moist
air are used in order to accurately represent the transport of heat in the ocean and the
exchange of heat with the atmosphere and with ice.
IOC Manuals and Guides No. 56
2
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
1. Introduction
1.1 Oceanographic practice 1978 -‐‑ 2009
The Practical Salinity Scale, PSS-‐‑78, and the International Equation of State of Seawater
(UNESCO (1981)) which expresses the density of seawater as a function of Practical
Salinity, temperature and pressure, have served the oceanographic community very well
for thirty years. The Joint Panel on Oceanographic Tables and Standards (JPOTS)
(UNESCO (1983)) also promulgated the Millero, Perron and Desnoyers (1973) algorithm
for the specific heat capacity of seawater at constant pressure, the Chen and Millero (1977)
expression for the sound speed of seawater and the Millero and Leung (1976) formula for
the freezing point temperature of seawater. Three other algorithms supported under the
auspices of JPOTS concerned the conversion between hydrostatic pressure and depth, the
calculation of the adiabatic lapse rate, and the calculation of potential temperature. The
expressions for the adiabatic lapse rate and for potential temperature could in principle
have been derived from the other algorithms of the EOS-‐‑80 set, but in fact they were based
on the formulas of Bryden (1973). We shall refer to all these algorithms jointly as ‘EOS-‐‑80’
for convenience because they represent oceanographic best practice from the early 1980s to
2009.
1.2 Motivation for an updated thermodynamic description of seawater
In recent years the following aspects of the thermodynamics of seawater, ice and moist air
have become apparent and suggest that it is timely to redefine the thermodynamic
properties of these substances.
• Several of the polynomial expressions of the International Equation of State of
Seawater (EOS-‐‑80) are not totally consistent with each other as they do not exactly
obey the thermodynamic Maxwell cross-‐‑differentiation relations. The new
approach eliminates this problem.
• Since the late 1970s a more accurate and more broadly applicable thermodynamic
description of pure water has been developed by the International Association for
the Properties of Water and Steam, and has appeared as an IAPWS Release (IAPWS-‐‑
95). Also since the late 1970s some measurements of higher accuracy have been
made of several properties of seawater such as (i) heat capacity, (ii) sound speed and
(iii) the temperature of maximum density. These can be incorporated into a new
thermodynamic description of seawater.
• The impact on seawater density of the variation of the composition of seawater in
the different ocean basins has become better understood. In order to further
progress this aspect of seawater, a standard model of seawater composition is
needed to serve as a generally recognised reference for theoretical and chemical
investigations.
• The increasing emphasis on the ocean as being an integral part of the global heat
engine points to the need for accurate expressions for the entropy, enthalpy and
internal energy of seawater so that heat fluxes can be more accurately determined in
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
•
•
3
the ocean and across the interfaces between the ocean and the atmosphere and ice
(entropy, enthalpy and internal energy were not available from EOS-‐‑80).
The need for a thermodynamically consistent description of the interactions between
seawater, ice and moist air; in particular, the need for accurate expressions for the
latent heats of evaporation and melting, both at the sea surface and in the
atmosphere.
The temperature scale has been revised from IPTS-‐‑68 to ITS-‐‑90 and revised IUPAC
(International Union of Pure and Applied Chemistry) values have been adopted for
the atomic weights of the elements (Wieser (2006)).
1.3 SCOR/IAPSO WG127 and the approach taken
In 2005 SCOR (Scientific Committee on Oceanic Research) and IAPSO (International
Association for the Physical Sciences of the Oceans) established Working Group 127 on the
“Thermodynamics and Equation of State of Seawater” (henceforth referred to as WG127).
This group has now developed a collection of algorithms that incorporate our best
knowledge of seawater thermodynamics. The present document summarizes the work of
SCOR/IAPSO Working Group 127.
To compute all thermodynamic properties of seawater it is sufficient to know one of its
so-‐‑called thermodynamic potentials (Fofonoff 1962, Feistel 1993, Alberty 2001). It was J.W.
Gibbs (1873) who discovered that “an equation giving internal energy in terms of entropy and
specific volume, or more generally any finite equation between internal energy, entropy and specific
volume, for a definite quantity of any fluid, may be considered as the fundamental thermodynamic
equation of that fluid, as from it… may be derived all the thermodynamic properties of the fluid (so
far as reversible processes are concerned).”
The approach taken by WG127 has been to develop a Gibbs function from which all
the thermodynamic properties of seawater can be derived by purely mathematical
manipulations (such as differentiation). This approach ensures that the various
thermodynamic properties are self-‐‑consistent (in that they obey the Maxwell cross-‐‑
differentiation relations) and complete (in that each of them can be derived from the given
potential).
The Gibbs function (or Gibbs potential) is a function of Absolute Salinity S A (rather
than of Practical Salinity S P ), temperature and pressure. Absolute Salinity is traditionally
defined as the mass fraction of dissolved material in seawater. The use of Absolute
Salinity as the salinity argument for the Gibbs function and for all other thermodynamic
functions (such as density) is a major departure from present practice (EOS-‐‑80). Absolute
Salinity is preferred over Practical Salinity because the thermodynamic properties of
seawater are directly influenced by the mass of dissolved constituents whereas Practical
Salinity depends only on conductivity. Consider for example exchanging a small amount
of pure water with the same mass of silicate in an otherwise isolated seawater sample at
constant temperature and pressure. Since silicate is predominantly non-‐‑ionic, the
conductivity (and therefore Practical Salinity S P ) is almost unchanged but the Absolute
Salinity is increased, as is the density. Similarly, if a small mass of say NaCl is added and
the same mass of silicate is taken out of a seawater sample, the mass fraction absolute
salinity will not have changed (and so the density should be almost unchanged) but the
Practical Salinity will have increased.
The variations in the relative concentrations of seawater constituents caused by
biogeochemical processes actually cause complications in even defining what exactly is
meant by “absolute salinity”. These issues have not been well studied to date, but what is
known is summarized in section 2.5 and appendices A.4, A.5 and A.20. Here it is
sufficient to point out that the Absolute Salinity S A that is the salinity argument of the
IOC Manuals and Guides No. 56
4
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
TEOS-‐‑10 Gibbs function is the version of absolute salinity that provides the best estimate
of the density of seawater; another name for S A is “Density Salinity”.
The Gibbs function of seawater, published as Feistel (2008), has been endorsed by the
International Association for the Properties of Water and Steam as the Release IAPWS-‐‑08.
This thermodynamic description of seawater properties, together with the Gibbs function
of ice Ih, IAPWS-‐‑06, has been adopted by the Intergovernmental Oceanographic
Commission at its 25th Assembly in June 2009 to replace EOS-‐‑80 as the official description
of seawater and ice properties in marine science. The thermodynamic properties of moist
air have also recently been described using a Helmholtz function (Feistel et al. (2010a),
IAPWS (2010)) so allowing the equilibrium properties at the air-‐‑sea interface to be more
accurately evaluated. The new approach to the thermodynamic properties of seawater, of
ice Ih and of humid air is referred to collectively as the “International Thermodynamic
Equation Of Seawater – 2010”, or “TEOS-‐‑10” for short. Appendix C lists the publications
which lie behind TEOS-‐‑10.
A notable difference of TEOS-‐‑10 compared with EOS-‐‑80 is the adoption of Absolute
Salinity to be used in journals to describe the salinity of seawater and to be used as the
salinity argument in algorithms that give the various thermodynamic properties of
seawater. This recommendation deviates from the current practice of working with
Practical Salinity and typically treating it as the best estimate of Absolute Salinity. This
practice is inaccurate and should be corrected. Note however that we strongly
recommend that the salinity that is reported to national data bases remain Practical
Salinity as determined on the Practical Salinity Scale of 1978 (suitably updated to ITS-‐‑90
temperatures as described in appendix E below).
There are three very good reasons for continuing to store Practical Salinity rather than
Absolute Salinity in such data repositories. First, Practical Salinity is an (almost) directly
measured quantity whereas Absolute Salinity is generally a derived quantity. That is, we
calculate Practical Salinity directly from measurements of conductivity, temperature and
pressure, whereas to date we derive Absolute Salinity from a combination of these
measurements plus other measurements and correlations that are not yet well established.
Practical Salinity is preferred over the actually measured in situ conductivity value
because of its conservative nature with respect to changes of temperature or pressure, or
dilution with pure water. Second, it is imperative that confusion is not created in national
data bases where a change in the reporting of salinity may be mishandled at some stage
and later be misinterpreted as a real increase in the ocean’s salinity. This second point
argues strongly for no change in present practice in the reporting of Practical Salinity S P
in national data bases of oceanographic data. Thirdly, the algorithms for determining the
"ʺbest"ʺ estimate of Absolute Salinity of seawater with non-‐‑standard composition are
immature and will undoubtedly change in the future, so we cannot recommend storing
Absolute Salinity in national data bases. Storage of a more robust intermediate value, the
Reference Salinity, S R (defined as discussed in appendix A.3 to give the best estimate of
Absolute Salinity of Standard Seawater) would also introduce the possibility of confusion
in the stored salinity values without providing any real advantage over storing Practical
Salinity so we also avoid this possibility. Values of Reference Salinity obtained from
suitable observational techniques (for example by direct measurement of the density of
Standard Seawater) should be converted to corresponding numbers of Practical Salinity
for storage, as described in sections 2.3 -‐‑ 2.5.
Note that the practice of storing one type of salinity in national data bases (Practical
Salinity) but using a different type of salinity in publications (Absolute Salinity) is exactly
analogous to our present practice with temperature; in situ temperature t is stored in data
bases (since it is the measured quantity) but the temperature variable that is used in
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
5
publications is a calculated quantity, being potential temperature θ under EOS-‐‑80 and is
now Conservative Temperature Θ under TEOS-‐‑10.
In order to improve the determination of Absolute Salinity we need to begin collecting
and storing values of the salinity anomaly δ SA = SA − SR based on measured values of
density (such as can be measured with a vibrating tube densimeter, Kremling (1971)). The
4-‐‑letter GF3 code (IOC (1987)) DENS is currently defined for in situ measurements or
computed values from EOS-‐‑80. It is recommended that the density measurements made
with a vibrating beam densimeter be reported with the GF3 code DENS along with the
laboratory temperature (TLAB in ° C ) and laboratory pressure (PLAB, the sea pressure in
the laboratory, usually 0 dbar). From this information and the Practical Salinity of the
seawater sample, the Absolute Salinity Anomaly δ SA = SA − SR can be calculated using
an inversion of the TEOS-‐‑10 equation for density to determine SA . For completeness, it is
advisable to also report δ SA under the new GF3 code DELS.
The thermodynamic description of seawater and of ice Ih as defined in IAPWS-‐‑08 and
IAPWS-‐‑06 has been adopted as the official description of seawater and of ice Ih by the
Intergovernmental Oceanographic Commission in June 2009. These new international
standards were adopted while recognizing that the techniques for estimating Absolute
Salinity will likely improve over the coming decades, and the algorithm for evaluating
Absolute Salinity in terms of Practical Salinity, latitude, longitude and pressure will be
updated from time to time, after relevant appropriately peer-‐‑reviewed publications have
appeared, and that such an updated algorithm will appear on the www.TEOS-10.org web
site. Users of this software should always state in their published work which version of
the software was used to calculate Absolute Salinity.
The more prominent advantages of TEOS-‐‑10 compared with EOS-‐‑80 are
•
The Gibbs function approach allows the calculation of internal energy, entropy,
enthalpy, potential enthalpy and the chemical potentials of seawater as well as the
freezing temperature, and the latent heats of melting and of evaporation. These
quantities were not available from the International Equation of State 1980 but are
essential for the accurate accounting of “heat” in the ocean and for the consistent
and accurate treatment of air-‐‑sea and ice-‐‑sea heat fluxes. For example, the new
TEOS-‐‑10 temperature variable, Conservative Temperature, Θ , is defined to be
proportional to potential enthalpy and is a very accurate measure of the “heat”
content per unit mass of seawater; Θ is two orders of magnitude more conservative
than potential temperature θ .
•
For the first time the influence of the spatially varying composition of seawater can
systematically be taken into account through the use of Absolute Salinity. In the
open ocean, this has a non-‐‑trivial effect on the horizontal density gradient computed
from the equation of state, and thereby on the ocean velocities and heat transports
calculated via the “thermal wind” relation.
•
The thermodynamic quantities available from the new approach are totally
consistent with each other.
•
The new salinity variable, Absolute Salinity, is measured in SI units. Moreover the
treatment of freshwater fluxes in ocean models will be consistent with the use of
Absolute Salinity, but is only approximately so for Practical Salinity.
•
The Reference Composition of standard seawater supports marine physicochemical
studies such as the solubility of sea salt constituents, the alkalinity, the pH and the
ocean acidification by rising concentrations of atmospheric CO2.
IOC Manuals and Guides No. 56
6
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
1.4 A guide to this TEOS-‐‑10 manual
The remainder of this manual begins by listing (in section 2) the definitions of various
thermodynamic quantities that follow directly from the Gibbs function of seawater by
simple mathematical processes such as differentiation. These definitions are then
followed in section 3 by the discussion of several derived quantities. The computer
software to evaluate these quantities is available from two separate libraries, the Seawater-‐‑
Ice-‐‑Air (SIA) library and the Gibbs-‐‑SeaWater (GSW) Oceanographic Toolbox, as described
in appendices M and N. The functions in the SIA library are generally available in basic-‐‑SI
units ( kg kg −1 , kelvin and Pa), both for their input parameters and for the outputs of the
algorithms. Some additional routines are included in the SIA library in terms of other
commonly used units for the convenience of users. The SIA library takes significantly
more computer time to evaluate most quantities (approximately a factor of 65 more
computer time for many quantities, comparing optimized code in both cases) and
provides significantly more properties than does the GSW Toolbox. The SIA library uses
the world-‐‑wide standard for the thermodynamic description of pure water substance
(IAPWS-‐‑95). Since this is defined over extended ranges of temperature and pressure, the
algorithms are long and their evaluation time-‐‑consuming. The GSW Toolbox uses the
Gibbs function of Feistel (2003) (IAPWS-‐‑09) to evaluate the properties of pure water, and
since this is valid only over the restricted ranges of temperature and pressure appropriate
for the ocean, the algorithms are shorter and their execution is faster. The GSW
Oceanographic Toolbox is not as comprehensive as the SIA library; for example, the
properties of moist air are only available in the SIA library. In addition, a computationally
efficient expression for density specific volume in terms of Conservative Temperature
(rather than in terms of in situ temperature) involving just 75 coefficients is also available
and is described in appendix A.30 and appendix K.
The input and output parameters of the GSW Oceanographic Toolbox are in units
which oceanographers will find more familiar than basic SI units. We expect that
oceanographers will mostly use this GSW Toolbox because of its greater simplicity and
computational efficiency, and because of the more familiar units compared with the SIA
library. The name GSW (Gibbs-‐‑SeaWater) has been chosen to be similar to, but different
from the existing “sw” (Sea Water) library which is already in wide circulation. Both the
SIA and GSW libraries, together with this TEOS-‐‑10 Manual are available from the website
www.TEOS-10.org. Initially the SIA library is being made available in Visual Basic and
FORTRAN while the GSW library is in MATLAB with some functions in FORTRAN and C.
After these descriptions in sections 2 and 3 of how to determine the thermodynamic
quantities and various derived quantities, we end with some conclusions (section 4).
Additional information on Practical Salinity, the Gibbs function, Reference Salinity,
composition anomalies, Absolute Salinity, and some fundamental thermodynamic
properties such as the First Law of Thermodynamics, the non-‐‑conservative nature of many
oceanographic variables, a list of recommended symbols, and succinct lists of
thermodynamic formulae are given in the appendices. Much of this work has appeared
elsewhere in the published literature but is collected here in a condensed form for the
users'ʹ convenience.
Two introductory articles about TEOS-‐‑10, namely “What every oceanographer needs
to know about TEOS-‐‑10 (The TEOS-‐‑10 Primer)” (Pawlowicz, 2010b), and “Getting
started with TEOS-‐‑10 and the Gibbs Seawater (GSW) Oceanographic Toolbox”
(McDougall and Barker, 2011) are available from www.TEOS-10.org. An introductory
article, Pawlowicz et al. (2012), describes the multi-‐‑year scientific puzzles with which we
wrestled during the development of TEOS-‐‑10.
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
7
1.5 A remark on units
The SIA software library of TEOS-‐‑10 is written in terms of Absolute Salinity S A in units of
kg kg-‐‑1, Absolute Temperature T in K, and Absolute Pressure P in Pa, however
oceanographic practice to date has used non-‐‑basic-‐‑SI units for many variables, in
particular, temperature is usually measured on the Celsius ( °C ) scale, pressure is sea
pressure quoted in decibars relative to the pressure of a standard atmosphere (10.1325
dbar), while salinity has had its own oceanography-‐‑specific scale, the Practical Salinity
Scale of 1978. In the GSW Oceanographic Toolbox of TEOS-‐‑10 we adopt °C for the
temperature unit, pressure is sea pressure in dbar and Absolute Salinity S A is expressed
in units of g kg−1 so that it takes numerical values close to those of Practical Salinity.
Adopting these non-‐‑basic-‐‑SI units does not come without a penalty as there are many
thermodynamic formulae that are more conveniently manipulated when expressed in SI
units. As an example, the freshwater fraction of seawater is written correctly as (1 − SA ) ,
but it is clear that in this instance Absolute Salinity must be expressed in kg kg −1 not in
g kg −1. Thus there are cases within the GSW Toolbox in which SI units are required and
this may occasionally cause some confusion. A common example of this issue arises when
a variable is differentiated or integrated with respect to pressure. Nevertheless, for many
applications it is deemed important to remain close to present oceanographic practice even
though it means that one has to be vigilant to detect those expressions that need a variable
to be expressed in the less-‐‑familiar SI units.
1.6 Recommendations
In accordance with resolution XXV-‐‑7 of the Intergovernmental Oceanographic
Commission at its 25th Assembly in June 2009, and the several Releases and Guidelines of
the International Association for the Properties of Water and Steam, the TEOS-‐‑10
thermodynamic description of seawater, of ice and of moist air has been adopted for use
by oceanographers in place of the International Equation Of State – 1980 (EOS-‐‑80). The
software to implement this change is available at the web site www.TEOS-10.org.
Under TEOS-‐‑10 it is recognized that the composition of seawater varies around the
world ocean and that the thermodynamic properties of seawater are more accurately
represented as functions of Absolute Salinity S A than of Practical Salinity S P . It is useful
to think of the transition from Practical Salinity to Absolute Salinity in two steps. In the
first step a seawater sample is effectively treated as though it is Standard Seawater and its
Reference Salinity S R is calculated; Reference Salinity may be taken to be simply
proportional to Practical Salinity. Reference Salinity has SI units (for example, g kg −1 ) and
is the natural starting point to consider the influence of any variation in composition. In
the second step the Absolute Salinity Anomaly is evaluated using one of several
techniques, the easiest of which is via a computer algorithm that essentially interpolates
between a spatial atlas of these values. Then Absolute Salinity is estimated as the sum of
Reference Salinity and Absolute Salinity Anomaly. Of the four possible versions of
absolute salinity, the one that is used as the argument for the TEOS-‐‑10 Gibbs function is
designed to provide accurate estimates of the density of seawater.
It is recognized that our knowledge of how to estimate seawater composition
anomalies and their effect on thermodynamic properties is limited. Nevertheless, we
should not continue to ignore the influence of these composition variations on seawater
properties and on ocean dynamics. As more knowledge is gained in this area over the
coming decade or so, and after such knowledge has been duly published in the scientific
literature, any updated algorithm to evaluate the Absolute Salinity Anomaly will be made
available (with its version number) from www.TEOS-10.org.
IOC Manuals and Guides No. 56
8
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
The salinity that is stored in national data bases should continue to be Practical
Salinity, as this will maintain continuity of this important time series. Oceanographic
databases label stored, processed or exported parameters with the GF3 code PSAL for
Practical Salinity and SSAL for salinity measured before 1978 (IOC, 1987). In order to
avoid possible confusion in data bases between different types of salinity it is very
strongly recommended that under no circumstances should either Reference Salinity or
Absolute Salinity be stored in national data bases.
In order to accurately calculate the thermodynamic properties of seawater, Absolute
Salinity must be calculated by first calculating Reference Salinity and then adding on the
Absolute Salinity Anomaly. Because Absolute Salinity is the appropriate salinity variable
for use with the equation of state, Absolute Salinity is the salinity variable that should be
published in oceanographic journals. The version number of the software, or the exact
formula, that was used to convert Reference Salinity into Absolute Salinity should always
be stated in publications. Nevertheless, there may be some applications where the likely
future changes in the algorithm that relates Reference Salinity to Absolute Salinity
presents a concern, and for these applications it may be preferable to publish graphs and
tables in Reference Salinity. For these studies or where it is clear that the effect of
compositional variations are insignificant or not of interest, the Gibbs function may be
called with S R rather than S A . When this is done, it should be clearly stated that the
salinity variable that is being graphed is Reference Salinity, not Absolute Salinity.
The TEOS-‐‑10 approach of using thermodynamic potentials to describe the properties
of seawater, ice and moist air means that it is possible to derive many more
thermodynamic properties than were available from EOS-‐‑80. The seawater properties
entropy, internal energy, enthalpy and particularly potential enthalpy were not available
from EOS-‐‑80 but are central to accurately calculating the transport of “heat” in the ocean
and hence the air-‐‑sea heat flux in the coupled climate system.
Under EOS-‐‑80 the observed variables ( SP , t , p ) were first used to calculate potential
temperature θ and then water masses were analyzed on the SP − θ diagram. Curved
contours of potential density ρ θ could also be drawn on this same SP − θ diagram.
Under TEOS-‐‑10, since density and potential density are now not functions of Practical
Salinity SP but rather are functions of Absolute Salinity S A , it is no longer possible to
draw isolines of potential density on a SP − θ diagram. Rather, because of the spatial
variations of seawater composition, a given value of potential density defines an area on
the SP − θ diagram, not a curved line.
Under TEOS-‐‑10, the observed variables ( SP , t , p ) , together with longitude and
latitude, are first used to form Absolute Salinity S A , and then Conservative Temperature
Θ is evaluated. Oceanographic water masses are then analyzed on the SA − Θ diagram,
and potential density ρ Θ contours can also be drawn on this SA − Θ diagram. The
computationally-‐‑efficient 75-‐‑term expression for the specific volume of seawater (of
appendix K) is a convenient and accurate equation of state for observational and
theoretical studies and for ocean modelling. Preformed Salinity S* is used internally in
numerical ocean models where it is important that the salinity variable be conservative.
When describing the use of TEOS-‐‑10, it is the present document (the TEOS-‐‑10
Manual) that should be referenced as IOC et al. (2010) [IOC, SCOR and IAPSO, 2010: The
international thermodynamic equation of seawater – 2010: Calculation and use of thermodynamic
properties. Intergovernmental Oceanographic Commission, Manuals and Guides No. 56,
UNESCO (English), 196 pp]. Two introductory articles about TEOS-‐‑10, namely “Getting
started with TEOS-‐‑10 and the Gibbs Seawater (GSW) Oceanographic Toolbox”
(McDougall and Barker, 2011), and “What every oceanographer needs to know about
TEOS-‐‑10:-‐‑ The TEOS-‐‑10 Primer” (Pawlowicz, 2010b), are available from www.TEOS10.org.
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
9
2. Basic Thermodynamic Properties
2.1 ITS-‐‑90 temperature
In 1990 the International Practical Temperature Scale 1968 (IPTS-‐‑68) was replaced by the
International Temperature Scale 1990 (ITS-‐‑90). There are two main methods to convert
between these two temperature scales; Rusby’s (1991) 8th order fit valid over a wide range
of temperatures, and Saunders’ (1990) 1.00024 scaling widely used in the oceanographic
community. The two methods are formally indistinguishable in the oceanographic
temperature range because they differ by less than either the uncertainty in
thermodynamic temperature (of order 1 mK), or the practical application of the IPTS-‐‑68
and ITS-‐‑90 scales. The differences between the Saunders (1990) and Rusby (1991)
formulae are less than 1 mK throughout the temperature range -‐‑2 °C to 40 °C and less than
0.03mK in the temperature range between -‐‑2 °C and 10 °C. Hence we recommend that the
oceanographic community continues to use the Saunders formula
(t68 /°C)
= 1.00024 (t90 /°C) .
(2.1.1)
One application of this formula is in the updated computer algorithm for the calculation of
Practical Salinity (PSS-‐‑78) in terms of conductivity ratio. The algorithms for PSS-‐‑78
require t68 as the temperature argument. In order to use these algorithms with t90 data,
t68 may be calculated using (2.1.1).
An extended discussion of the different temperature scales, their inherent uncertainty
and the reasoning for our recommendation of (2.1.1) can be found in appendix A.1.
2.2 Sea pressure
Sea pressure p is defined to be the Absolute Pressure P less the Absolute Pressure of one
standard atmosphere, P0 ≡ 101 325Pa; that is
p ≡ P − P0 .
(2.2.1)
It is common oceanographic practice to express sea pressure in decibars (dbar). Another
common pressure variable that arises naturally in the calibration of sea-‐‑board instruments
is gauge pressure p gauge which is Absolute Pressure less the Absolute Pressure of the
atmosphere at the time of the instrument’s calibration (perhaps in the laboratory, or
perhaps at sea). Because atmospheric pressure changes in space and time, sea pressure p
is preferred as a thermodynamic variable as it is unambiguously related to Absolute
Pressure. The seawater Gibbs function in the GSW Toolbox is expressed as a function of
sea pressure p (functionally equivalent to the use of Absolute Pressure P in the IAPWS
Releases and in the SIA library); that is, g is a function of p , it is not a function of p gauge .
2.3 Practical Salinity
Practical Salinity S P is defined on the Practical Salinity Scale of 1978 (UNESCO (1981,
1983)) in terms of the conductivity ratio K15 which is the electrical conductivity of the
sample at temperature t68 = 15 °C and pressure equal to one standard atmosphere ( p = 0
dbar and Absolute Pressure P equal to 101 325 Pa), divided by the conductivity of a
IOC Manuals and Guides No. 56
10
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
standard potassium chloride (KCl) solution at the same temperature and pressure. The
mass fraction of KCl (i.e., the mass of KCl per mass of solution) in the standard solution is
32.4356 × 10−3 . When K15 = 1, the Practical Salinity S P is by definition 35. Note that
Practical Salinity is a unit-‐‑less quantity. Though sometimes convenient, it is technically
incorrect to quote Practical Salinity in “psu”; rather it should be quoted as a certain
Practical Salinity “on the Practical Salinity Scale PSS-‐‑78”. The formula for evaluating
Practical Salinity can be found in appendix E along with the simple change that must be
made to the UNESCO (1983) formulae so that the algorithm for Practical Salinity can be
called with ITS-‐‑90 temperature as an input parameter rather than the older t68
temperature in which the PSS-‐‑78 algorithms were defined. The reader is also directed to
the CDIAC chapter on “Method for salinity (conductivity ratio) measurement” which
describes best practice in measuring the conductivity ratio of seawater samples (Kawano
(2009)).
Practical Salinity is defined only in the range 2 < SP < 42. Practical Salinities below 2
can be evaluated from conductivity using the PSS-‐‑78 extension of Hill et al. (1986). We
have modified this Hill et al. (1986) extension to make the result a continuous function of is
arguments; this function is available as gsw_SP_from_C in the GSW Oceanographic
Toolbox. Samples exceeding a Practical Salinity of 42 must be diluted to the valid salinity
range and the measured value should be adjusted based on the added water mass and the
conservation of sea salt during the dilution process (as discussed in appendix E).
Data stored in national and international data bases should, as a matter of principle, be
measured values rather than derived quantities. Consistent with this, we recommend
continuing to store the measured (in situ) temperature rather than the derived quantity,
Conservative Temperature. Similarly we strongly recommend that Practical Salinity S P
continue to be the salinity variable that is stored in such data bases since S P is closely
related to the measured values of conductivity. This recommendation has the very
important advantage that there is no change to the present practice and so there is less
chance of transitional errors occurring in national and international data bases because of
the adoption of Absolute Salinity in oceanography.
2.4 Reference Composition and the Reference-‐‑Composition Salinity Scale
The reference composition of seawater is defined by Millero et al. (2008a) as the exact mole
fractions given in Table D.3 of appendix D below. This composition was introduced by
Millero et al. (2008a) as their best estimate of the composition of Standard Seawater, being
seawater from the surface waters of a certain region of the North Atlantic. The exact
location for the collection of bulk material for the preparation of Standard Seawater is not
specified. Ships gathering this bulk material are given guidance notes by the Standard
Seawater Service, requesting that water be gathered between longitudes 50°W and 40°W,
in deep water, during daylight hours. Reference-‐‑Composition Salinity S R (or Reference
Salinity for short) was designed by Millero et al. (2008a) to be the best estimate of the mass-‐‑
fraction Absolute Salinity SA of Standard Seawater. Independent of accuracy
considerations, it provides a precise measure of dissolved material in Standard Seawater
and is the correct salinity argument to be used in the TEOS-‐‑10 Gibbs function for Standard
Seawater.
For the range of salinities where Practical Salinities are defined (that is, in the range
2 < SP < 42 ) Millero et al. (2008a) show that
SR ≈ uPS SP where uPS ≡ (35.165 04 35) g kg −1 .
(2.4.1)
In the range 2 < SP < 42 , this equation expresses the Reference Salinity of a seawater sample
on the Reference-‐‑Composition Salinity Scale (Millero et al. (2008a)). For practical
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
11
purposes, this relationship can be taken to be an equality since the approximate nature of
this relation only reflects the extent to which Practical Salinity, as determined from
measurements of conductivity ratio, temperature and pressure, varies when a seawater
sample is heated, cooled or subjected to a change in pressure but without exchange of
mass with its surroundings. The Practical Salinity Scale of 1978 was designed to satisfy
this property as accurately as possible within the constraints of the polynomial
approximations used to determine Chlorinity (and hence Practical Salinity) in terms of the
measured conductivity ratio.
From Eqn. (2.4.1), a seawater sample of Reference Composition whose Practical
Salinity S P is 35 has a Reference Salinity S R of 35.165 04 g kg−1 . Millero et al. (2008a)
estimate that the absolute uncertainty in this value is ± 0.007 g kg −1 . The difference
between the numerical values of Reference and Practical Salinities can be traced back to
the original practice of determining salinity by evaporation of water from seawater and
weighing the remaining solid material. This process also evaporated some volatile
components and most of the 0.165 04 g kg−1 salinity difference is due to this effect.
Measurements of the composition of Standard Seawater at a Practical Salinity S P of 35
using mass spectrometry and/or ion chromatography are underway and may provide
updated estimates of both the value of the mass fraction of dissolved material in Standard
Seawater and its uncertainty. Any update of this value will not change the Reference-‐‑
Composition Salinity Scale and so will not affect the calculation of Reference Salinity nor
of Absolute Salinity as calculated from Reference Salinity plus the Absolute Salinity
Anomaly.
Oceanographic databases label stored, processed or exported parameters with the GF3
code PSAL for Practical Salinity and SSAL for salinity measured before 1978 (IOC, 1987).
In order to avoid possible confusion in data bases between different types of salinity,
under no circumstances should either Reference Salinity or Absolute Salinity be stored in
national data bases.
Detailed information on Reference Composition and Reference Salinity can be found
in Millero et al. (2008a). For the user'ʹs convenience a brief summary of information from
Millero et al. (2008a), including the precise definition of Reference Salinity is given in
appendix A.3 and in Table D3 of appendix D.
2.5 Absolute Salinity
Absolute Salinity is traditionally defined as the mass fraction of dissolved material in
seawater. For seawater of Reference Composition, Reference Salinity gives our current
best estimate of Absolute Salinity. To deal with composition anomalies in seawater, we
need an extension of the Reference-‐‑Composition Salinity S R that provides a useful
measure of salinity over the full range of oceanographic conditions and agrees precisely
with Reference Salinity when the dissolved material has Reference Composition. When
composition anomalies are present, no single measure of dissolved material can fully
represent the influences on seawater properties on all thermodynamic properties, so it is
clear that either additional information will be required or compromises will have to be
made. In addition, we would like to introduce a measure of salinity that is traceable to the
SI (Seitz et al., 2011) and maintains the high accuracy of PSS-‐‑78 necessary for
oceanographic applications. The introduction of "ʺDensity Salinity"ʺ SAdens addresses both of
these issues; it is this type of absolute salinity that in TEOS-‐‑10 parlance is labeled S A and
called Absolute Salinity. In this section we explain how S A is defined and evaluated, but
first we outline other choices that are available for the definition of absolute salinity in the
presence of composition variations in seawater.
IOC Manuals and Guides No. 56
12
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
The most obvious definition of absolute salinity is “the mass fraction of dissolved non-‐‑
H2O material in a seawater sample at its temperature and pressure”. This seemingly
simple definition is actually far more subtle than it first appears. Notably, there are
questions about what constitutes water and what constitutes dissolved material. Perhaps
the most obvious example of this issue occurs when CO2 is dissolved in water to produce a
mixture of CO2, H2CO3, HCO3-‐‑, CO32-‐‑, H+, OH-‐‑ and H2O, with the relative proportions
depending on dissociation constants that depend on temperature, pressure and pH. Thus,
the dissolution of a given mass of CO2 in pure water essentially transforms some of the
water into dissolved material. A change in the temperature and even an adiabatic change
in pressure results in a change in absolute salinity defined in this way due to the
dependence of chemical equilibria on temperature and pressure. Pawlowicz et al. (2010)
and Wright et al. (2011) address this second issue by defining “Solution Absolute Salinity”
(usually shortened to “Solution Salinity”), S Asoln , as the mass fraction of dissolved non-‐‑H2O
material after a seawater sample is brought to the constant temperature t = 25°C and the
fixed sea pressure 0 dbar (fixed Absolute Pressure of 101 325 Pa).
Another measure of absolute salinity is the “Added-‐‑Mass Salinity” SAadd which is S R
plus the mass fraction of material that must be added to Standard Seawater to arrive at the
concentrations of all the species in the given seawater sample, after chemical equilibrium
has been reached, and after the sample is brought to the constant temperature t = 25°C
and the fixed sea pressure of 0 dbar. The estimation of absolute salinity SAadd is not
straightforward for seawater with anomalous composition because while the final
equilibrium state is known, one must iteratively determine the mass of anomalous solute
prior to any chemical reactions with Reference-‐‑Composition seawater. Pawlowicz et al.
(2010) provide an algorithm to achieve this, at least approximately. This definition of
absolute salinity, SAadd , is useful for laboratory studies of artificial seawater and it differs
from S Asoln because of the chemical reactions that take place between the several species of
the added material and the components of seawater that exist in Standard Seawater.
Added-‐‑Mass Salinity may be the most appropriate form of salinity for accurately
accounting for the mass of salt discharged by rivers and hydrothermal vents into the
ocean.
“Preformed Absolute Salinity” (usually shortened to “Preformed Salinity”), S* , is a
different type of absolute salinity which is specifically designed to be as close as possible
to being a conservative variable. That is, S* is designed to be insensitive to
biogeochemical processes that affect the other types of salinity to varying degrees.
Preformed Salinity S* is formed by first estimating the contribution of biogeochemical
processes to one of the salinity measures S A , S Asoln , or SAadd , and then subtracting this
contribution from the appropriate salinity variable. In this way Preformed Salinity S* is
designed to be a conservative salinity variable which is independent of the effects of the
non-‐‑conservative biogeochemical processes. S* will find a prominent role in ocean
modeling. The three types of absolute salinity S Asoln , SAadd and S* are discussed in more
detail in appendices A.4 and A.20, where approximate relationships between these
variables and SA ≡ SAdens are presented, based on the work of Pawlowicz et al. (2010) and
Wright et al. (2011). Note that for a sample of Standard Seawater, all of the five salinity
variables S R , S A , S Asoln , SAadd and S* and are equal.
There is no simple means to measure either S Asoln or SAadd for the general case of the
arbitrary addition of many components to Standard Seawater. Hence a more precise and
easily determined measure of the amount of dissolved material in seawater is required
and TEOS-‐‑10 adopts “Density Salinity” for this purpose. “Density Salinity” SAdens is
defined as the value of the salinity argument of the TEOS-‐‑10 expression for density which
gives the sample’s actual measured density at the temperature t = 25°C and at the sea
pressure p = 0 dbar. When there is no risk of confusion, “Density Salinity” is also called
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
13
Absolute Salinity with the label S A , that is SA ≡ SAdens . Usually we do not have accurate
measurements of density but rather we have measurements of Practical Salinity,
temperature and pressure, and in this case, Absolute Salinity may be calculated using
Practical Salinity and the computer algorithm of McDougall, et al. (2012) which provides
an estimate of δ SA = SA − SR . This computer program was formed as follows.
In a series of papers (Millero et al. (1976a, 1978, 2000, 2008b), McDougall et al. (2012)),
accurate measurements of the density of seawater samples, along with the Practical
Salinity of those samples, gave estimates of δ SA = SA − SR from most of the major basins of
the world ocean. This was done by first calculating the “Reference Density” from the
TEOS-‐‑10 equation of state using the sample’s Reference Salinity as the salinity argument
(this calculation essentially assumes that the seawater sample has the composition of
Standard Seawater). The difference between the measured density and the “Reference
Density” was then used to estimate the Absolute Salinity Anomaly δ SA = SA − SR (Millero
et al. (2008a)). The McDougall et al. (2012) algorithm is based on the observed correlation
between this SA − SR data and the silicate concentration of the seawater samples (Millero
et al. , 2008a), with the silicate concentration being estimated by interpolation of a global
atlas (Gouretski and Koltermann (2004)).
The algorithm for Absolute Salinity takes the form
SA = SR + δ SA = SA ( SP , φ , λ , p ) ,
(2.5.1)
Where φ is latitude (degrees North), λ is longitude (degrees east, ranging from 0°E to
360°E) while p is sea pressure.
Heuristically the dependence of δ SA = SA − SR on silicate can be thought of as
reflecting the fact that silicate affects the density of a seawater sample without
significantly affecting its conductivity or its Practical Salinity. In practice this explains
about 60% of the effect and the remainder is due to the correlation of other composition
anomalies (such as nitrate) with silicate. In the McDougall et al. (2012) algorithm the Baltic
Sea is treated separately, following the work of Millero and Kremling (1976) and Feistel et
al. (2010c, 2010d), because some rivers flowing into the Baltic are unusually high in
calcium carbonate.
Figure 1. A sketch indicating how thermodynamic quantities
such as density are calculated as functions of Absolute Salinity.
Absolute Salinity is found by adding an estimate of the
Absolute Salinity Anomaly δ SA to the Reference Salinity.
IOC Manuals and Guides No. 56
14
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Since the density of seawater is rarely measured, we recommend the approach
illustrated in Figure 1 as a practical method to include the effects of composition
anomalies on estimates of Absolute Salinity and density. When composition anomalies
are not known, the algorithm of McDougall et al. (2012) may be used to estimate Absolute
Salinity in terms of Practical Salinity and the spatial location of the measurement in the
world oceans.
The difference between Absolute Salinity and Reference Salinity, as estimated by the
McDougall et al. (2012) algorithm, is illustrated in Figure 2 (a) at a pressure of 2000 dbar,
and in a vertical section through the Pacific Ocean in Figure 2 (b).
Of the approximately 800 samples of seawater from the world ocean that have been
examined to date for δ SA = SA − SR the standard error (square root of the mean squared
value) of δ SA = SA − SR is 0.0107 g kg-‐‑1. That is, the “typical” value of δ SA = SA − SR of the
811 samples taken to date is 0.0107 g kg-‐‑1. The standard error of the difference between the
measured values of δ SA = SA − SR and the values evaluated from the computer algorithm
of McDougall et al. (2012) is 0.0048 g kg-‐‑1. The maximum values of δ SA = SA − SR of
approximately 0.025 g kg-‐‑1 occur in the North Pacific.
Figure 2 (a). Absolute Salinity Anomaly δ SA at p = 2000 dbar.
Figure 2 (b). A vertical section of Absolute Salinity
Anomaly δ SA along 180oE in the Pacific Ocean.
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
15
The thermodynamic description of seawater and of ice Ih as defined in IAPWS-‐‑08 and
IAPWS-‐‑06 has been adopted as the official description of seawater and of ice Ih by the
Intergovernmental Oceanographic Commission in June 2009. These thermodynamic
descriptions of seawater and ice were endorsed recognizing that the techniques for
estimating Absolute Salinity will likely improve over the coming decades. The algorithm
for evaluating Absolute Salinity in terms of Practical Salinity, latitude, longitude and
pressure, will likely be updated from time to time, after relevant appropriately peer-‐‑
reviewed publications have appeared, and such an updated algorithm will appear on the
www.TEOS-10.org web site. Users of this software should state in their published work
which version of the software was used to calculate Absolute Salinity.
The present computer software which evaluates Absolute Salinity S A given the input
variables Practical Salinity S P , longitude λ , latitude φ and pressure is available at
www.TEOS-10.org. Absolute Salinity is also available as the inverse function of density
SA (T , P, ρ ) in the SIA library of computer algorithms as the algorithm sea_sa_si (see
appendix M) and in the GSW Toolbox as the algorithm gsw_SA_from_rho_t_exact.
2.6 Gibbs function of seawater
The Gibbs function of seawater g ( SA , t , p ) is related to the specific enthalpy h and
entropy η , by g = h − (T0 + t )η where T0 = 273.15K is the Celsius zero point. TEOS-‐‑10
defines the Gibbs function of seawater as the sum of a pure water part and the saline part
(IAPWS-‐‑08)
g ( SA , t , p ) = g W ( t , p ) + g S ( S A , t , p ) .
(2.6.1)
The saline part of the Gibbs function, g S , is valid over the ranges 0 < S A < 42 g kg–1,
–6.0 °C < t < 40 °C, and 0 < p < 104 dbar , although its thermal and colligative properties
are valid up to t = 80 °C and S A = 120 g kg–1 at p = 0.
The pure-‐‑water part of the Gibbs function, g W , can be obtained from the IAPWS-‐‑95
Helmholtz function of pure-‐‑water substance which is valid from the freezing temperature
or from the sublimation temperature to 1273 K. Alternatively, the pure-‐‑water part of the
Gibbs function can be obtained from the IAPWS-‐‑09 Gibbs function which is valid in the
oceanographic ranges of temperature and pressure, from less than the freezing
temperature of seawater (at any pressure), up to 40 °C (specifically from
− (2.65 + ( p + P0 ) × 0.0743 MPa −1 ) °C to 40 °C), and in the pressure range 0 < p < 104 dbar .
For practical purposes in oceanography it is expected that IAPWS-‐‑09 will be used because
it executes approximately two orders of magnitude faster than the IAPWS-‐‑95 code for
pure water. However if one is concerned with temperatures between 40 °C and 80 °C
then one must use the IAPWS-‐‑95 version of g W (expressed in terms of absolute
temperature (K) and Absolute Pressure (Pa)) rather than the IAPWS-‐‑09 version.
The thermodynamic properties derived from the IAPWS-‐‑95 (the Release providing the
Helmholtz function formulation for pure water) and IAPWS-‐‑08 (the Release endorsing the
Feistel (2008) Gibbs function) combination are available from the SIA software library,
while that derived from the IAPWS-‐‑09 (the Release endorsing the pure water part of
Feistel (2003)) and IAPWS-‐‑08 combination are available from the GSW Oceanographic
Toolbox. The GSW Toolbox is restricted to the oceanographic standard range in
temperature and pressure, however the validity of results extends at p = 0 to Absolute
Salinity up to mineral saturation concentrations (Marion et al. 2009). Specific volume
(which is the pressure derivative of the Gibbs function) is presently an extrapolated
quantity outside the Neptunian range (i. e. the oceanographic range) of temperature and
Absolute Salinity at p = 0, and exhibits errors there of up to 3%. We emphasize that
models of seawater properties that use a single salinity variable, SA , as input require
IOC Manuals and Guides No. 56
16
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
approximately fixed chemical composition ratios (e.g., Na/Cl, Ca/Mg, Cl/HCO3, etc.). As
seawater evaporates or freezes, eventually minerals such as CaCO3 will precipitate. Small
anomalies are reasonably handled by using S A as the input variable (see section 2.5) but
precipitation may cause large deviations from the nearly fixed ratios associated with
standard seawater. Under extreme conditions of precipitation, models of seawater based
on the Millero et al. (2008a) Reference Composition will no longer be applicable. Figure 3
illustrates SA − t boundaries of validity (determined by the onset of precipitation) for 2008
(pCO2 = 385 µ atm ) and 2100 (pCO2 = 550 µ atm ) (from Marion et al. (2009)).
Figure 3. The boundaries of validity of the Millero et al. (2008a)
composition at p = 0 in Year 2008 (solid lines) and potentially
in Year 2100 (dashed lines). At high salinity, calcium carbonate
saturates first and comes out of solution; thereafter the
Reference Composition of Standard Seawater of Millero et al.
(2008a) does not apply.
The Gibbs function (2.6.1) contains four arbitrary constants that cannot be determined
by any set of thermodynamic measurements. These arbitrary constants mean that the
Gibbs function (2.6.1) is unknown and unknowable up to the arbitrary function of
temperature and Absolute Salinity (where T0 is the Celsius zero point, 273.15 K )
⎡⎣a1 + a2 (T0 + t )⎤⎦ + ⎡⎣a3 + a4 (T0 + t )⎤⎦ SA
(2.6.2)
(see for example Fofonoff (1962) and Feistel and Hagen (1995)). The first two coefficients
a1 and a2 are arbitrary constants of the pure water Gibbs function g W ( t , p ) while the
second two coefficients a3 and a4 are arbitrary coefficients of the saline part of the Gibbs
function g S ( SA , t , p ) . Following generally accepted convention, the first two coefficients
are chosen to make the entropy and internal energy of liquid water zero at the triple point
and
η W ( tt , pt ) = 0
(2.6.3)
u W ( tt , pt ) = 0
(2.6.4)
as described in IAPWS-‐‑95 and in more detail in Feistel et al. (2008a) for the IAPWS-‐‑95
Helmholtz function description of pure water substance. When the pure-‐‑water Gibbs
function g W ( t , p ) of (2.6.1) is taken from the fitted Gibbs function of Feistel (2003), the two
arbitrary constants a1 and a2 are (in the appropriate non-‐‑dimensional form) g 00 and g10
of the table in appendix G below. These values of g 00 and g10 are not identical to the
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
17
values in Feistel (2003) because the present values have been taken from IAPWS-‐‑09 and
have been chosen to most accurately achieve the triple-‐‑point conditions (2.6.3) and (2.6.4)
as discussed in Feistel et al. (2008a).
The remaining two arbitrary constants a3 and a4 of (2.6.2) are determined by ensuring
that the specific enthalpy h and specific entropy η of a sample of standard seawater with
standard-‐‑ocean properties ( SSO , tSO , pSO ) = (35.165 04 g kg −1, 0 °C, 0 dbar) are both zero,
that is that
(2.6.5)
h ( SSO , tSO , pSO ) = 0
and
η ( SSO , tSO , pSO ) = 0.
(2.6.6)
In more detail, these conditions are actually officially written as (Feistel (2008), IAPWS-‐‑08)
and
hS ( SSO , tSO , pSO ) = u W (tt , pt ) − h W ( tSO , pSO )
(2.6.7)
η S ( SSO , tSO , pSO ) = η W (tt , pt ) − η W ( tSO , pSO ) .
(2.6.8)
Written in this way, (2.6.7) and (2.6.8) use properties of the pure water description (the
right-‐‑hand sides) to constrain the arbitrary constants in the saline Gibbs function. While
the first terms on the right-‐‑hand sides of these equations are zero (see (2.6.3) and (2.6.4)),
these constraints on the saline Gibbs function are written this way so that they are
independent of any subsequent change in the arbitrary constants involved in the
thermodynamic description of pure water. While the two slightly different
thermodynamic descriptions of pure water, namely IAPWS-‐‑95 and IAPWS-‐‑09, both
achieve zero values of the internal energy and entropy at the triple point of pure water,
the values assigned to the enthalpy and entropy of pure water at the temperature and
pressure of the standard ocean, h W ( tSO , pSO ) and η W ( tSO , pSO ) on the right-‐‑hand sides of
(2.6.7) and (2.6.8), are slightly different in the two cases. For example h W ( tSO , pSO ) is
3.3x10−3 J kg −1 from IAPWS-‐‑09 (as described in the table of appendix G) compared with
the round-‐‑off error of 2 x10−8 J kg −1 when using IAPWS-‐‑95 with double-‐‑precision
arithmetic. This issues is discussed in more detail in section 3.3.
The polynomial form and the coefficients for the pure water Gibbs function g W ( t , p )
from Feistel (2003) and IAPWS-‐‑09 are given in appendix G, while the combined
polynomial and logarithmic form and the coefficients for the saline part of the Gibbs
function g S ( S A , t , p ) (from Feistel (2008) and IAPWS-‐‑08) are reproduced in appendix H.
SCOR/IAPSO Working Group 127 has independently checked that the Gibbs functions
of Feistel (2003) and of Feistel (2008) do in fact fit the underlying data of various
thermodynamic quantities to the accuracy quoted in those two fundamental papers. This
checking was performed by Giles M. Marion, and is summarized in appendix O. Further
checking of these Gibbs functions has occurred in the process leading up to IAPWS
approving these Gibbs function formulations as the Releases IAPWS-‐‑08 and IAPWS-‐‑09.
Discussions of how well the Gibbs functions of Feistel (2003) and Feistel (2008) fit the
underlying (laboratory) data of density, sound speed, specific heat capacity, temperature
of maximum density etc may be found in those papers, along with comparisons with the
corresponding algorithms of EOS-‐‑80. The IAPWS-‐‑09 release discusses the accuracy to
which the Feistel (2003) Gibbs function fits the underlying thermodynamic potential of
IAPWS-‐‑95; in summary, for the variables density, thermal expansion coefficient and
specific heat capacity, the rms misfit between IAPWS-‐‑09 and IAPWS-‐‑95, in the region of
validity of IAPWS-‐‑09, are a factor of between 20 and 100 less than the corresponding error
in the laboratory data to which both thermodynamic potentials were fitted. Hence, in the
oceanographic range of parameters, IAPWS-‐‑09 and IAPWS-‐‑95 may be regarded as equally
accurate thermodynamic descriptions of pure liquid water.
The Gibbs function g has units of J kg −1 in both the SIA and GSW software libraries.
IOC Manuals and Guides No. 56
18
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
2.7 Specific volume
The specific volume of seawater v is given by the pressure derivative of the Gibbs
function at constant Absolute Salinity S A and in situ temperature t , that is
v = v ( SA , t, p ) = g P = ∂g ∂P S
A ,T
.
(2.7.1)
Notice that specific volume is a function of Absolute Salinity S A rather than of Reference
Salinity S R or Practical Salinity S P . The importance of this point is discussed in section
2.8. When derivatives are taken with respect to in situ temperature, or at constant in situ
temperature, the symbol t is avoided as it can be confused with the same symbol for time.
Rather, we use T in place of t in the expressions for these derivatives.
For many theoretical and modeling purposes in oceanography it is convenient to
regard the independent temperature variable to be Conservative Temperature Θ rather
than in situ temperature t . We note here that the specific volume is equal to the pressure
derivative of specific enthalpy at fixed Absolute Salinity when any one of η, θ or Θ is also
held constant, as follows (from appendix A.11)
∂h ∂P S
A ,η
= ∂h ∂P S
A,Θ
= ∂h ∂P S
A ,θ
= v .
(2.7.2)
The use of P in these equations emphasizes that it must be in Pa not dbar. Specific
volume v has units of m3 kg −1 in both the SIA and GSW software libraries.
2.8 Density
The density of seawater ρ is the reciprocal of the specific volume. It is given by the
reciprocal of the pressure derivative of the Gibbs function at constant Absolute Salinity S A
and in situ temperature t , that is
ρ = ρ ( SA , t , p ) = ( g P )
−1
(
= ∂g ∂P S
A ,T
)
−1
.
(2.8.1)
Notice that density is a function of Absolute Salinity S A rather than of Reference Salinity
S R or Practical Salinity S P . This is an extremely important point because Absolute
Salinity S A in units of g kg −1 is numerically greater than Practical Salinity by between
0.165 g kg −1 and 0.195 g kg −1 in the open ocean so that if Practical Salinity were
inadvertently used as the salinity argument for the density algorithm, a significant density
error of between 0.12 kg m−3 and 0.15 kg m−3 would result.
For many theoretical and modeling purposes in oceanography it is convenient to
regard density to be a function of Conservative Temperature Θ rather than of in situ
temperature t . That is, it is convenient to form the following functional form of density,
ρ = ρˆ ( SA , Θ, p ) ,
(2.8.2)
where Θ is Conservative Temperature. We will adopt the convention (see Table L.2 in
appendix L) that when enthalpy h, specific volume v or density ρ are taken to be
functions of potential temperature they attract an over-‐‑tilde as in v or ρ , and when they
are taken to be functions of Conservative Temperature they attract a caret as in v̂ and ρˆ .
With this convention, expressions involving partial derivatives such as (2.7.2) can be
written more compactly as (from appendix A.11)
(2.8.3)
hP = hP = ĥP = v = ρ −1
since the other variables are taken to be constant during the partial differentiation.
Appendix P lists expressions for many thermodynamic variables in terms of the
thermodynamic potentials
h = h SA ,η , p , h = h SA ,θ , p and h = hˆ ( SA , Θ, p ) .
(2.8.4)
(
IOC Manuals and Guides No. 56
)
(
)
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
19
Density ρ has units of kg m −3 in both the SIA and GSW software libraries.
The computationally efficient expression for v̂ SA ,Θ, p involving 75 coefficients
(Roquet et al. (2015)) is described in appendix A.30 and appendix K and is available in the
GSW computer software library as the function gsw_specvol(SA,CT,p). Note that
potential density with respect to reference pressure p_ref is calculated using the same
function, as gsw_rho(SA,CT,p_ref). Note that v̂ SA ,Θ, p can be integrated with respect to
pressure to provide a closed expression for hˆ ( S A , Θ, p ) (see Eqns. (2.8.3) and (3.2.1)) which
is available as the function gsw_enthalpy(SA,CT,p).
(
(
)
)
2.9 Chemical potentials
As for any two-‐‑component thermodynamic system, the Gibbs energy, G, of a seawater
sample containing mass of water mW and mass of salt mS at temperature t and pressure
p can be written in the form (Landau and Lifshitz (1959), Alberty (2001), Feistel (2008))
G ( mW , mS , t , p ) = mW µ W + mS µ S
(2.9.1)
where the chemical potentials of water in seawater µ W and of salt in seawater µ S are
defined by the partial derivatives
µW =
∂G
∂mW
, and µ S =
mS , T , p
∂G
∂mS
.
(2.9.2)
mW ,T , p
Identifying absolute salinity with the mass fraction of salt dissolved in seawater,
SA = mS / ( mW + mS ) (Millero et al. (2008a)), the specific Gibbs energy g is given by
(
)
G
(2.9.3)
= (1 − SA ) µ W + SA µ S = µ W + SA µ S − µ W .
mW + mS
Note that this expression for g as the sum of a water part and a saline part is not the same
as the pure water and the saline split in Eqn. (2.6.1) ( µ W is the chemical potential of water
in seawater; it does not correspond to a pure water sample as g W does in Eqn. (2.6.1)).
This Gibbs energy g is used as the thermodynamic potential function (Gibbs function) for
seawater. The above three equations can be used to write expressions for µ W and µ S in
terms of the Gibbs function g of seawater as
g ( SA , t , p ) =
∂ ⎡( mW + mS ) g ⎦⎤
µW = ⎣
∂mW
= g + ( mW + mS )
mS ,T , p
∂g
∂SA
T,p
∂SA
∂mW
T,p
∂SA
∂mS
= g − SA
mS
∂g
∂SA
(2.9.4)
T,p
and for the chemical potential of salt in seawater,
∂ ⎡( mW + mS ) g ⎦⎤
µS = ⎣
∂mS
= g + ( mW + mS )
mW ,T , p
∂g
∂SA
= g + (1 − S A )
mW
∂g
∂SA
(2.9.5)
T,p
The relative chemical potential µ (commonly called the “chemical potential of seawater”)
follows from (2.9.4) and (2.9.5) as
µ = µS − µ W =
∂g
∂SA
,
(2.9.6)
T,p
and describes the change in the Gibbs energy of a parcel of seawater of fixed mass if a
small amount of water is replaced by salt at constant temperature and pressure. Also,
from the fundamental thermodynamic relation (Eqn. (A.7.1) in appendix A.7) it follows
that the chemical potential of seawater µ describes the change of enthalpy dh if at
constant pressure and entropy, a small mass fraction of water is replaced by salt, dSA .
Equations (2.9.4) – (2.9.6) serve to define the three chemical potentials in terms of the
Gibbs function g of seawater. Note that the weights of the sums that appear in Eqns.
IOC Manuals and Guides No. 56
20
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
(2.9.1) – (2.9.5) are strictly the mass fractions of salt and of pure water in seawater, so that
for a seawater sample of anomalous composition these mass fractions would be more
accurately given in terms of S Asoln than by SA ≡ SAdens . In this regard, the Gibbs energy in
Eqn. (2.9.1) should strictly be the weighted sum of the chemical potentials of all the
constituents in seawater. However, practically speaking, the vapour pressure, the latent
heat and the freezing temperature are all rather weakly dependent on salinity, and hence
the use of S A in this section is recommended.
Note that both µ and µ S have singularities at SA = 0 g kg −1 while µ W is well-‐‑
behaved there.
The SIA computer software library (appendix M) predominantly uses basic SI units, so
that S A has units of kg kg −1 and g , µ , µ S and µ W all have units of J kg−1. In the GSW
Oceanographic Toolbox (appendix N) S A has units of g kg −1 while µ , µ S and µ W all have
units of J g −1. This adoption of oceanographic (i.e. non-‐‑basic-‐‑SI) units for S A means that
special care is needed in evaluating equations such as (2.9.3) and (2.9.5) where in the term
(1 − SA ) it is clear that SA must have units of kg kg −1. The adoption of non-‐‑basic-‐‑SI units
is common in oceanography, but often causes some difficulties such as this. To be specific,
the use of oceanographic units for Absolute Salinity (such as in the GSW Oceanographic
Toolbox) means that the above equations (2.9.4) – (2.9.5) are evaluated as
µW =
S ∂g
g
− A
R ∂SA
R
, and µ S =
T ,p
⎛
S ⎞ ∂g
g
+ ⎜1 − A ⎟
R ⎠ ∂SA
R ⎝
(2.9.7)
T ,p
where the constant R is defined as R = 1000 g kg −1 , while Eqn. (2.9.6) is unchanged.
2.10 Entropy
The specific entropy of seawater η is given by
η = η ( SA , t, p ) = − gT = − ∂g ∂T S
A, p
.
(2.10.1)
When taking derivatives with respect to in situ temperature, the symbol T will be used for
temperature in order that these derivatives not be confused with time derivatives.
Entropy η has units of J kg −1 K −1 in both the SIA and GSW software libraries.
2.11 Internal energy
The specific internal energy of seawater u is given by (where T0 is the Celsius zero point,
273.15 K and P0 = 101 325Pa is the standard atmosphere pressure)
∂g
∂g
. (2.11.1)
u = u ( SA , t, p ) = g + (T0 + t )η − ( p + P0 ) v = g − (T0 + t )
− ( p + P0 )
∂T SA , p
∂P SA , T
This expression is an example where the use of non-‐‑basic SI units presents a problem,
because in the product − ( p + P0 ) v , ( p + P0 ) = P must be in Pa if specific volume has its
regular units of m3 kg −1 :-‐‑ hence here sea pressure p must be expressed in Pa . Also, the
pressure derivative in Eqn. (2.11.1) must be done with respect to pressure in Pa .
Specific internal energy u has units of J kg −1 in both the SIA and GSW software
libraries.
2.12 Enthalpy
The specific enthalpy of seawater h is given by
h = h ( SA , t, p ) = g + (T0 + t )η = g − (T0 + t )
IOC Manuals and Guides No. 56
∂g
∂T
.
SA , p
(2.12.1)
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
21
Specific enthalpy h has units of J kg −1 in both the SIA and GSW software libraries. Also,
note that potential enthalpy is defined in section 3.2 below, and dynamic enthalpy is
defined as enthalpy minus potential enthalpy (Young, 2010).
2.13 Helmholtz energy
The specific Helmholtz energy of seawater f is given by
f = f ( SA , t, p ) = g − ( p + P0 ) v = g − ( p + P0 )
∂g
.
∂P SA , T
(2.13.1)
This expression is another example where the use of non-‐‑basic SI units presents a problem,
because in the product − ( p + P0 ) v , p must be in Pa if specific volume has its regular units of
m3 kg −1. The specific Helmholtz energy f has units of J kg −1 in both the SIA and GSW
computer software libraries.
2.14 Osmotic coefficient
The osmotic coefficient of seawater φ is given by
⎛
φ = φ ( SA , t , p ) = − ⎜ g ( SA , t , p ) − g ( 0, t , p ) − S A
⎜
⎝
∂g
∂SA
⎞
−1
⎟ ( mSW R (T0 + t ) ) .
⎟
T, p ⎠
(2.14.1)
The osmotic coefficient of seawater describes the change of the chemical potential of water
per mole of added salt, expressed as multiples of the thermal energy, R (T0 + t ) (Millero
and Leung (1976), Feistel and Marion (2007), Feistel (2008)),
µ W ( 0, t, p ) = µ W ( SA , t , p ) + mSW R (T0 + t ) φ .
(2.14.2)
Here, R = 8.314 472 J mol −1 K −1 is the universal molar gas constant. The molality mSW is
the number of dissolved moles of solutes (ions) of the Reference Composition as defined
by Millero et al. (2008a), per kilogram of pure water. Note that the molality of seawater
may take different values if neutral molecules of salt rather than ions are counted (see the
discussion on page 519 of Feistel and Marion (2007)). The freezing-‐‑point lowering
equations (3.33.1, 3.33.2) or the vapour-‐‑pressure lowering can be computed from the
osmotic coefficient of seawater (see Millero and Leung (1976), Bromley et al. (1974)).
2.15 Isothermal compressibility
The thermodynamic quantities defined so far are all based on the Gibbs function itself and
its first derivatives. The remaining quantities discussed in this section all involve higher
order derivatives.
The isothermal and isohaline compressibility of seawater κ t is defined by
κ t = κ t ( SA , t, p ) = ρ −1
∂ρ
∂v
g
= − v −1
= − PP
∂P SA , T
∂P SA , T
gP
(2.15.1)
where the second derivative of g is taken with respect to pressure (in Pa ) at constant S A
and t. The use of P in the pressure derivatives in Eqn. (2.15.1) serves to emphasize that
these derivatives must be taken with respect to pressure in Pa not in dbar . The
isothermal compressibility of seawater κ t produced by both the SIA and GSW computer
software libraries (appendices M and N) has units of Pa −1.
IOC Manuals and Guides No. 56
22
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
2.16 Isentropic and isohaline compressibility
When the entropy and Absolute Salinity are held constant while the pressure is changed,
the isentropic and isohaline compressibility κ is obtained:
κ = κ ( SA , t , p ) = ρ −1
(g
=
∂ρ
∂P
2
TP
= − v −1
SA ,η
− gTT g PP
g P gTT
∂v
∂P
).
= ρ −1
SA ,η
∂ρ
∂P
= ρ −1
SA , θ
∂ρ
∂P
SA , Θ
(2.16.1)
The isentropic and isohaline compressibility κ is sometimes called simply the isentropic
compressibility (or sometimes the “adiabatic compressibility”), on the unstated
understanding that there is also no transfer of salt during the isentropic or adiabatic
change in pressure. The isentropic and isohaline compressibility of seawater κ produced
by both the SIA and GSW software libraries (appendices M and N) has units of Pa −1.
2.17 Sound speed
The speed of sound in seawater c is given by
(
)
0.5
c = c ( SA , t , p ) = ∂P ∂ρ S
= ( ρκ )
(
−0.5
)
0.5
2
.
(2.17.1)
= g P gTT ⎡⎣ gTP
− gTT g PP ⎤⎦
A ,η
Note that in these expressions in Eqn. (2.17.1), since sound speed is in m s−1 and density
has units of kg m −3 it follows that the pressure of the partial derivatives must be in Pa and
the isentropic compressibility κ must have units of Pa −1 . The sound speed c produced
by both the SIA and the GSW software libraries (appendices M and N) has units of m s−1 .
2.18 Thermal expansion coefficients
The thermal expansion coefficient α t with respect to in situ temperature t , is
α t = α t ( SA , t , p ) = −
1 ∂ρ
ρ ∂T
=
SA , p
1 ∂v
v ∂T
=
SA , p
gTP
.
gP
(2.18.1)
The thermal expansion coefficient α θ with respect to potential temperature θ , is (see
appendix A.15)
α θ = α θ ( SA , t, p, pr ) = −
1 ∂ρ
ρ ∂θ
=
SA , p
1 ∂v
v ∂θ
=
SA , p
gTP gTT ( SA ,θ , pr )
,
gP
gTT
(2.18.2)
where pr is the reference pressure of the potential temperature. The gTT derivative in the
numerator is evaluated at ( SA ,θ , pr ) whereas the other derivatives are all evaluated at
( SA , t, p ) .
The thermal expansion coefficient α Θ with respect to Conservative Temperature Θ , is
(see appendix A.15)
α Θ = α Θ ( SA , t , p ) = −
c0p
1 ∂ρ
1 ∂v
g
=
= − TP
.
ρ ∂Θ SA , p v ∂Θ SA , p
g P (T0 + θ ) gTT
(2.18.3)
Note that Conservative Temperature Θ is defined only with respect to a reference
pressure of 0 dbar so that the θ in Eqn. (2.18.3) is the potential temperature with
pr = 0 dbar. All the derivatives on the right-‐‑hand side of Eqn. (2.18.3) are evaluated at
( SA , t, p ) . The constant c0p is defined in Eqn. (3.3.3) below.
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
23
2.19 Saline contraction coefficients
The saline contraction coefficient β t (sometimes also called the haline contraction
coefficient) at constant in situ temperature t , is
β t = β t ( SA , t , p ) =
1 ∂ρ
ρ ∂SA
= −
T, p
1 ∂v
v ∂SA
= −
T, p
gS
AP
gP
.
(2.19.1)
The saline contraction coefficient β θ at constant potential temperature θ , is (see
appendix A.15)
β θ = β θ ( S A , t , p , pr ) =
=
1 ∂ρ
ρ ∂SA
=−
θ, p
1 ∂v
v ∂SA
θ, p
gTP ⎡⎣ g SAT − g SAT ( SA , θ , pr )⎤⎦ − gTT g SA P
g P gTT
(2.19.2)
,
where pr is the reference pressure of θ . One of the gS T derivatives in the numerator is
A
evaluated at ( SA ,θ , pr ) whereas all the other derivatives are evaluated at ( SA , t, p ) .
The saline contraction coefficient β Θ at constant Conservative Temperature Θ , is (see
appendix A.15)
β Θ = β Θ ( SA , t , p ) =
1 ∂ρ
ρ ∂SA
=−
Θ, p
1 ∂v
v ∂SA
Θ, p
gTP ⎡ g SAT − (T0 + θ ) g SA ( SA ,θ ,0 )⎤ − gTT g SA P
⎣
⎦
=
.
g P gTT
−1
(2.19.3)
Note that Conservative Temperature Θ is defined only with respect to a reference
pressure of 0 dbar as indicated in this equation. The gS derivative in the numerator is
A
evaluated at ( SA ,θ , 0) whereas all the other derivatives are evaluated at ( SA , t, p ) .
In the SIA computer software (appendix M) all three saline contraction coefficients are
produced in units of kg kg −1 while in the GSW Oceanographic Toolbox (appendix N) all
three saline contraction coefficients are produced in units of kg g−1 consistent with the
preferred oceanographic unit for S A being g kg −1.
IOC Manuals and Guides No. 56
24
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
2.20 Isobaric heat capacity
The specific isobaric heat capacity c p is the rate of change of specific enthalpy with
temperature at constant Absolute Salinity S A and pressure p, so that
c p = c p ( SA , t , p ) =
∂h
∂T
= − (T0 + t ) gTT .
(2.20.1)
SA , p
The isobaric heat capacity c p varies over the SA − Θ plane at p = 0 by approximately 5%,
as illustrated in Figure 4.
Figure 4. Contours of isobaric specific heat capacity c p of seawater
(in J kg −1 K −1 ), Eqn. (2.20.1), at p = 0.
The isobaric heat capacity c p has units of J kg −1 K −1 in both the SIA and GSW
computer software libraries.
2.21 Isochoric heat capacity
The specific isochoric heat capacity cv is the rate of change of specific internal energy u
with temperature at constant Absolute Salinity S A and specific volume, v, so that
cv = cv ( SA , t, p ) =
∂u
∂T
(
2
= − (T0 + t ) gTT g PP − gTP
SA , v
)
g PP .
(2.21.1)
Note that the isochoric and isobaric heat capacities are related by
cv = c p −
(T0 + t ) (α t )
(ρ κ )
t
2
, and by cv = c p
κ
.
κt
(2.21.2)
The isochoric heat capacity cv has units of J kg −1 K −1 in both the SIA and GSW
computer software libraries.
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
25
2.22 The adiabatic lapse rate
The adiabatic lapse rate Γ is the change of in situ temperature with pressure at constant
entropy and Absolute Salinity, so that (McDougall and Feistel (2003))
∂t
Γ =
∂P S
=
A
∂t
=
∂P S
,η
(T0 + θ )α θ
ρ c p ( SA ,θ ,0 )
A
η
g
∂v
= − P = − TP =
ηT
gTT
∂η
,Θ
=
(T0 + θ )
c0p
∂v
∂Θ S
=
A, p
SA
∂2 h
=
∂η ∂P
,p
(T0 + θ )
c0p
∂2 h
∂Θ ∂P
=
SA
=
SA
(T0 + t )α t
ρ cp
(T0 + θ )α Θ .
(2.22.1)
ρ c0p
The adiabatic (and isohaline) lapse rate is commonly (and incorrectly) explained as being
proportional to the work done on a fluid parcel as its volume changes in response to an
increase in pressure. According to this explanation the adiabatic lapse rate would increase
with both pressure and the fluid’s compressibility, but this is not the case. Rather, the
adiabatic lapse rate is proportional to the thermal expansion coefficient and is
independent of the fluid’s compressibility. Indeed, the adiabatic lapse rate changes sign at
the temperature of maximum density whereas the compressibility κ and the work done
by compression is always positive. McDougall and Feistel (2003) show that the adiabatic
lapse rate is independent of the increase in the internal energy that a parcel experiences
when it is compressed. Rather, the adiabatic lapse rate represents that change in
temperature that is required to keep the entropy (and also θ and Θ ) of a seawater parcel
constant when its pressure is changed in an adiabatic and isohaline manner. The reference
pressure of the potential temperature θ that appears in the last four expressions in Eqn.
(2.22.1) is pr = 0 dbar.
The adiabatic lapse rate Γ in the GSW computer software library is evaluated via the
functions gsw_adiabatic_lapse_rate_from_t and gsw_adiabatic_lapse_rate_from_CT
(depending on whether the input temperature is in situ temperature or Conservative
Temperature). In both cases the expression used is − gTP gTT = −η P ηT (see the top line
of Eqn. (2.22.1)) calculated directly from the Gibbs function of seawater g SA ,t, p
(IAPWS-‐‑08 and IAPWS-‐‑09). This is consistent with the exact use of η = η SA ,t, p
throughout the GSW Toolbox to convert between in situ temperature and potential
temperature. An alternative option for calculating Γ would be to use the 75-‐‑term
expression for specific volume in the expressions in the second line of Eqn. (2.22.1). This
option is not adopted as it would mean that the small errors in the thermal expansion
coefficient α Θ would cause an rms error in the adiabatic lapse rate Γ of 4.7x10−12 K Pa −1 .
This error, while small, would then conflict with the exact relationships that have been
chosen to relate in situ temperature, potential temperature, Conservative Temperature,
entropy and the adiabatic lapse rate.
The adiabatic lapse rate Γ output of both the SIA and the GSW computer software
libraries is in units of K Pa −1 .
(
(
)
)
IOC Manuals and Guides No. 56
26
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
3. Derived Quantities
3.1 Potential temperature
The very useful concept of potential temperature was applied to the atmosphere originally
by Helmholtz (1888), first under the name of ‘heat content’, and later renamed ‘potential
temperature’ (Bezold (1888)). These concepts were transferred to oceanography by
Helland-‐‑Hansen (1912). Potential temperature is the temperature that a fluid parcel
would have if its pressure were changed to a fixed reference pressure pr in an isentropic
and isohaline manner. The phrase “isentropic and isohaline” is used repeatedly in this
document. To these two qualifiers we should really also add “without dissipation of
kinetic energy”. A process that obeys all three restrictions is a thermodynamically
reversible process. Note that one often (falsely) reads that the requirement of a reversible
process is that the process occurs at constant entropy. However this statement is
misleading because it is possible for a fluid parcel to exchange some heat and some salt
with its surroundings in just the right ratio so as to keep its entropy constant, but the
processes is not reversible (see Eqn. (A.7.1)).
Potential temperature referred to reference pressure pr is often written as the pressure
integral of the adiabatic lapse rate (Fofonoff (1962), (1985))
Pr
θ = θ ( SA , t, p, pr ) = t + ∫ Γ ( SA ,θ [SA , t, p, p′], p′) dP′.
(3.1.1)
P
Note that this pressure integral needs to be done with respect to pressure expressed in Pa
not dbar .
The algorithm that is used with the TEOS-‐‑10 Gibbs function approach to seawater
equates the specific entropies of two seawater parcels, one before and the other after the
isentropic and isohaline pressure change. In this way, θ is evaluated using a Newton-‐‑
Raphson iterative solution technique to solve the following equation for θ
(3.1.2)
η ( SA ,θ , pr ) = η ( SA , t, p ) ,
or, in terms of the Gibbs function, g ,
− gT ( SA ,θ , pr ) = − gT ( SA , t, p ) .
(3.1.3)
This relation is formally equivalent to Eqn. (3.1.1). In the GSW Oceanographic Toolbox θ
is found to machine precision ( 10−14 °C ) in two iterations of a modified Newton-‐‑Raphson
method (McDougall and Wotherspoon (2014)), using a suitable initial value.
Note that the difference between the potential and in situ temperatures is not due to
the work done in compressing a fluid parcel on going from one pressure to another:-‐‑ the
sign of this work is often in the wrong sense and the magnitude is often wrong by a few
orders of magnitude (McDougall and Feistel (2003)). Rather, the difference between these
temperatures is what is required to keep the entropy constant during the adiabatic and
isohaline pressure change. The potential temperature θ output of the SIA software is in
units of K while the output from the GSW Toolbox is in °C .
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
27
3.2 Potential enthalpy
Potential enthalpy h 0 is the enthalpy that a fluid parcel would have if its pressure were
changed to a fixed reference pressure pr in an isentropic and isohaline manner. Because
heat fluxes into and out of the ocean occur mostly near the sea surface, the reference
pressure for potential enthalpy is always taken to be pr = 0 dbar (that is, at zero sea
pressure). Potential enthalpy can be expressed as the pressure integral of specific volume
as (from McDougall (2003) and see the discussion below Eqn. (2.8.2))
(
)
(
)
(
)
(
)
(
)
(
)
(
)
P
(
)
h0 SA ,t, p = h SA ,θ ,0 = h 0 SA ,θ = h SA ,t, p − ∫ v SA ,θ ⎡⎣ SA ,t, p, p′ ⎤⎦ , p′ dP′
P0
(
)
P
= h SA ,t, p − ∫ v SA ,η , p′ dP′
P0
P
(
)
(
)
(3.2.1)
= h SA ,t, p − ∫ v SA ,θ , p′ dP′
P0
P
= h SA ,t, p − ∫ v̂ SA ,Θ, p′ dP′ ,
P0
and we emphasize that the pressure integrals here must be done with respect to pressure
expressed in Pa rather than dbar. In terms of the Gibbs function, potential enthalpy h 0 is
evaluated as
h 0 ( SA , t , p ) = h ( SA , θ , 0 ) = g ( SA ,θ , 0 ) − (T0 + θ ) gT ( SA ,θ , 0 ) .
(3.2.2)
Also, note that dynamic enthalpy is defined as enthalpy minus potential enthalpy (Young,
2010) and is available as the function gsw_dynamic_enthalpy in the GSW Toolbox.
3.3 Conservative Temperature
Conservative Temperature Θ is defined to be proportional to potential enthalpy,
(
)
(
)
(
)
(
S ,θ = h0 S ,t, p c0 = h 0 S ,θ
Θ SA ,t, p = Θ
A
A
p
A
)
c0p
(3.3.1)
where the value that is chosen for c 0p is motivated in terms of potential enthalpy evaluated
at an Absolute Salinity of SSO = 35 uPS = 35.165 04 g kg−1 and at θ = 25 °C by
⎡⎣ h ( SSO , 25 °C, 0 ) − h ( SSO , 0 °C, 0 )⎤⎦
≈ 3991.867 957 119 63 J kg −1 K −1 ,
(25 K)
(3.3.2)
noting that h ( SSO , 0 °C, 0dbar ) is zero according to the way the Gibbs function is defined
in (2.6.5). We adopt the exact definition for c 0p to be the 15-‐‑digit value in (3.3.2), so that
c0p ≡ 3991.867 957 119 63 J kg −1 K −1 .
(3.3.3)
When IAPWS-‐‑95 is used for the pure water part of the Gibbs function, Θ ( SSO ,0 °C,0) and
Θ ( SSO ,25 °C,0) differ from 0 °C and 25 °C respectively by the round-‐‑off amount of
5 × 10−12 °C. When IAPWS-‐‑09 (which is based on the paper of Feistel (2003), see appendix
G) is used for the pure water part of the Gibbs function, Θ ( SSO ,0 °C,0) differs from 0 °C
by −8.25 × 10−8 °C and Θ ( SSO ,25 °C,0) differs from 25 °C by 9.3 × 10−6 °C. Over the
temperature range from 0 ° C to 40 ° C the difference between Conservative Temperature
using IAPWS-‐‑95 and IAPWS-‐‑09 as the pure water part is no more than ± 1.5 × 10−5 °C , a
temperature difference that will be ignored.
The value of c 0p in (3.3.3) is very close to the average value of the specific heat capacity
c p at the sea surface of today’s global ocean. This value of c 0p also causes the average
value of θ − Θ at the sea surface to be very close to zero. Since c 0p is simply a constant of
proportionality between potential enthalpy and Conservative Temperature, it is totally
IOC Manuals and Guides No. 56
28
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
arbitrary, and we see no reason why its value would need to change from (3.3.3) even
when in future decades an improved Gibbs function of seawater is agreed upon.
McDougall (2003), Graham and McDougall (2013) and appendix A.18 outline why
Conservative Temperature gets its name; it is approximately two orders of magnitude
more conservative compared with either potential temperature or entropy.
The SIA and GSW software libraries both include an algorithm for determining
Conservative Temperature Θ from values of Absolute Salinity S A and potential
temperature θ referenced to p = 0 dbar . These libraries also have an algorithm for
evaluating potential temperature (referenced to 0 dbar ) from S A and Θ . This inverse
algorithm, θˆ ( SA , Θ ) , has an initial seed based on a rational function approximation and
finds potential temperature to machine precision ( 10−14 °C ) in one and a half iterations of
a modified Newton-‐‑Raphson technique.
3.4 Potential density
Potential density ρ θ is the density that a fluid parcel would have if its pressure were
changed to a fixed reference pressure pr in an isentropic and isohaline manner. Potential
density referred to reference pressure pr can be written as the pressure integral of the
isentropic compressibility κ as
Pr
ρ θ ( SA , t, p, pr ) = ρ ( SA , t, p ) + ∫ ρ ( SA ,θ [SA , t, p, p′], p′) κ ( SA ,θ [SA , t, p, p′], p′) dP′. (3.4.1)
P
The simpler expression for potential density in terms of the Gibbs function is
ρ θ ( SA , t, p, pr ) = ρ ( SA ,θ [SA , t, p, pr ], pr ) = g P−1 ( SA ,θ [SA , t, p, pr ], pr ).
(3.4.2)
Using the functional forms of Eqn. (2.8.2) and (2.8.3) for in situ density, that is, either
ρ = ρ SA ,θ , p or ρ = ρˆ ( SA , Θ, p ) , potential density with respect to reference pressure pr
(e. g. 1000 dbar) can be easily evaluated as
ρθ SA ,t, p, pr = ρ Θ SA ,t, p, pr = ρ SA ,η , pr = ρ SA ,θ , pr = ρ̂ SA ,Θ, pr ,
(3.4.3)
(
(
)
)
(
)
(
)
(
)
(
)
where we note that the potential temperature θ in the penultimate expression is the
potential temperature with respect to 0 dbar. Once the reference pressure is fixed,
potential density is a function only of Absolute Salinity and Conservative Temperature (or
equivalently, of Absolute Salinity and potential temperature). Note that it is equally
correct to label potential density as ρ θ or ρ Θ (or indeed as ρ η ) because η , θ and Θ are
constant during the isentropic and isohaline pressure change from p to pr ; that is, these
variables posses the “potential” property of appendix A.9.
Following the discussion after Eqn. (2.8.2) above, potential density may also be
expressed in terms of the pressure derivative of the expressions h = h SA ,θ , p and
h = hˆ ( SA , Θ, p ) for enthalpy as (see also appendix P)
(
ρθ ( SA ,t, p, pr ) = ρ Θ ( SA ,t, p, pr ) = ⎡⎣ hP ( SA ,θ , p = pr ) ⎤⎦
−1
(
)
−1
= ⎡⎣ ĥP SA ,Θ, p = pr ⎤⎦ .
)
(3.4.4)
3.5 Density anomaly
Density anomaly σ t is an old-‐‑fashioned density measure that is now seldom used. It is
the density evaluated at the in situ temperature but at zero sea pressure, minus 1000
kg m−3 , that is,
σ t ( SA , t, p ) = ρ ( SA , t,0 ) − 1000 kg m −3 = g P−1 ( SA , t,0 ) − 1000 kg m −3.
θ
(3.5.1)
σ was used as an approximation to σ which avoided the computational demand of
evaluating θ . Density anomaly σ t is not provided in the TEOS-‐‑10 software libraries.
t
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
29
3.6 Potential density anomaly
Potential density anomaly, σ θ or σ Θ , is simply potential density minus 1000 kg m–3,
σ θ ( SA , t, p, pr ) = σ Θ ( SA , t, p, pr ) = ρ θ ( SA , t , p, pr ) − 1000 kg m −3
= ρ Θ ( SA , t , p, pr ) − 1000 kg m −3
(3.6.1)
= g P−1 ( SA ,θ [SA , t , p, pr ], pr ) − 1000 kg m −3.
Note that it is equally correct to label potential density anomaly as σ θ or σ Θ because both
θ and Θ are constant during the isentropic and isohaline pressure change from p to pr .
3.7 Specific volume anomaly
The specific volume anomaly δ is defined as the difference between the specific volume
and a given function of pressure. Traditionally δ has been defined as
δ ( SA , t, p ) = v ( SA , t, p ) − v ( SSO ,0°C, p )
(3.7.1)
(where the traditional value of Practical Salinity of 35 has been updated to an Absolute
Salinity of SSO = 35 uPS = 35.16504 g kg −1 in the present formulation). Note that the second
term, v ( SSO ,0°C, p ) , is a function only of pressure. In order to have a surface of constant
specific volume anomaly more accurately approximate neutral tangent planes (see section
S
and
with more general values
3.11), it is advisable to replace the arguments
0
°
C
S
SO
A
and t that are carefully chosen (as say the median values of Absolute Salinity and
temperature along the surface) so that the more general definition of specific volume
anomaly is
δ SA ,t, p = v SA ,t, p − v SA , t , p = g P SA ,t, p − g P SA , t , p .
(3.7.2)
(
)
(
(
)
)
(
(
)
)
The last terms in Eqns. (3.7.1) and (3.7.2) are simply functions of pressure and one has
the freedom to choose any other function of pressure in its place and still retain the
dynamical properties of specific volume anomaly. In particular, one can construct specific
volume and enthalpy to be functions of Conservative Temperature (rather than in situ
temperature) as vˆ ( SA , Θ, p ) and hˆ ( SA , Θ, p ) and write a slightly different definition of
specific volume anomaly as
(
)
(
)
δ ( SA ,Θ, p ) = v̂ ( SA ,Θ, p ) − v̂ SA , Θ,
p = ĥP ( SA ,Θ, p ) − ĥP SA , Θ,
p .
(3.7.3)
This is the form of specific volume anomaly adopted in the GSW Oceanographic Toolbox
are S = 35.165 04 g kg −1 and
where the default values of the reference values SA and Θ
SO
0°C respectively. The same can also be done with potential temperature so that in terms
of the specific volume v SA ,θ , p and enthalpy h SA ,θ , p we can write another form of
the specific volume anomaly as
(
(
)
)
(
(
)
)
(
)
(
)
v SA ,θ , p − v SA ,θ , p = hP SA ,θ , p − hP SA ,θ , p .
(3.7.4)
These expressions exploit the fact that (see appendix A.11)
∂h ∂P S
A ,η
= ∂h ∂P S
A,Θ
= ∂h ∂P S
A ,θ
= v .
(3.7.5)
IOC Manuals and Guides No. 56
30
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
3.8 Thermobaric coefficient
The thermobaric coefficient quantifies the rate of variation with pressure of the ratio of the
thermal expansion coefficient and the saline contraction coefficient. With respect to
potential temperature θ the thermobaric coefficient is (McDougall (1987b))
Tbθ = Tbθ ( SA , t , p ) = β θ
(
∂ αθ βθ
)
∂P
∂α θ
∂P
=
S A ,θ
−
S A ,θ
α θ ∂β θ
β θ ∂P
.
(3.8.1)
S A ,θ
This expression for the thermobaric coefficient is most readily evaluated by differentiating
an expression for density expressed as a function of potential temperature rather than in
situ temperature, that is, with density expressed in the functional form ρ = ρ SA ,θ , p .
With respect to Conservative Temperature Θ the thermobaric coefficient is
(
TbΘ
=
TbΘ
( SA , t , p ) = β
Θ
(
∂ αΘ βΘ
)
∂P
=
SA , Θ
∂α Θ
∂P
−
SA , Θ
α Θ ∂β Θ
β Θ ∂P
)
.
(3.8.2)
SA , Θ
This expression for the thermobaric coefficient is most readily evaluated by differentiating
an expression for density expressed as a function of Conservative Temperature rather than
in situ temperature, that is, with density expressed in the functional form ρ = ρˆ ( SA , Θ, p ) .
The thermobaric coefficient enters various quantities to do with the path-‐‑dependent
nature of neutral trajectories and the ill-‐‑defined nature of neutral surfaces (see (3.13.1) –
(3.13.7)). The thermobaric dianeutral advection associated with the lateral mixing of heat
and salt along neutral tangent planes is given by eTb = − gN −2 K Tbθ ∇nθ ⋅ ∇n P or
eTb = − gN −2 K TbΘ∇n Θ⋅ ∇n P where ∇nθ and ∇n Θ are the two-‐‑dimensional gradients of
either potential temperature or Conservative Temperature along the neutral tangent
plane, ∇n P is the corresponding epineutral gradient of Absolute Pressure and K is the
epineutral diffusion coefficient. Note that the thermobaric dianeutral advection is
proportional to the mesoscale eddy flux of “heat” along the neutral tangent plane,
− c0p K ∇n Θ , and is independent of the amount of small-‐‑scale (dianeutral) turbulent mixing
and hence is also independent of the dissipation of kinetic energy ε (Klocker and
McDougall (2010a)). It is shown in appendix A.14 below that while the epineutral
diffusive fluxes − K ∇nθ and − K ∇n Θ are different, the product of these fluxes with their
respective thermobaric coefficients is the same, that is, Tbθ ∇nθ = TbΘ∇n Θ . Hence the
thermobaric dianeutral advection e Tb is the same whether it is calculated as
− gN −2 K Tbθ ∇nθ ⋅ ∇n P or as − gN −2 K TbΘ∇n Θ⋅ ∇n P. Expressions for Tbθ and TbΘ in terms of
enthalpy in the functional forms h SA ,θ , p and hˆ ( SA , Θ, p ) can be found in appendix P.
Interestingly, for given magnitudes of the epineutral gradients of pressure and
Conservative Temperature, the dianeutral advection, eTb = − gN −2 K TbΘ∇n Θ⋅ ∇n P , of
thermobaricity is maximized when these gradients are parallel, while neutral helicity is
maximized when these gradients are perpendicular, since neutral helicity is proportional
to TbΘ ( ∇n P × ∇n Θ ) ⋅ k (see Eqn. (3.13.2)).
This thermobaric vertical advection process, e Tb , is absent from standard layered
ocean models in which the vertical coordinate is a function only of S A and Θ (such as σ 2 ,
potential density referenced to 2000 dbar). As described in appendix A.27 below, the
isopycnal diffusion of heat and salt in these layered models, caused by both parameterized
diffusion along the coordinate and by eddy-‐‑resolved motions, does give rise to the
cabbeling advection through the coordinate surfaces but does not allow the thermobaric
velocity e Tb through these surfaces (Klocker and McDougall (2010a)).
In both the SIA and GSW computer software libraries the thermobaric parameter is
output in units of K −1 Pa −1 .
(
IOC Manuals and Guides No. 56
)
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
31
3.9 Cabbeling coefficient
The cabbeling coefficient quantifies the rate at which dianeutral advection occurs as a
result of mixing of heat and salt along the neutral tangent plane. With respect to potential
temperature θ the cabbeling coefficient is (McDougall (1987b))
∂α θ
Cb = Cb ( SA , t , p ) =
∂θ
θ
θ
SA , p
α θ ∂α θ
+2 θ
β ∂SA
θ, p
⎛ αθ
− ⎜⎜ θ
⎝β
2
⎞ ∂β θ
⎟⎟
⎠ ∂SA
.
(3.9.1)
θ, p
This expression for the cabbeling coefficient is most readily evaluated by differentiating an
expression for density expressed as a function of potential temperature rather than in situ
temperature, that is, with density expressed in the functional form ρ = ρ SA ,θ , p .
With respect to Conservative Temperature Θ the cabbeling coefficient is
(
CbΘ
=
CbΘ
∂α Θ
( SA , t , p ) =
∂Θ
SA
α Θ ∂α Θ
+2 Θ
β ∂SA
,p
)
2
Θ, p
⎛ α Θ ⎞ ∂β Θ
− ⎜⎜ Θ ⎟⎟
⎝ β ⎠ ∂SA
.
(3.9.2)
Θ, p
This expression for the cabbeling coefficient is most readily evaluated by differentiating an
expression for density expressed as a function of Conservative Temperature rather than in
situ temperature, that is, with density expressed in the functional form ρ = ρˆ ( SA , Θ, p ) .
The cabbeling dianeutral advection associated with the lateral mixing of heat and salt
along neutral tangent planes is given by eCab = − gN −2 K CbΘ∇n Θ⋅ ∇n Θ (or less accurately by
eCab ≈ − gN −2 K Cbθ ∇nθ ⋅ ∇nθ ) where ∇nθ and ∇n Θ are the two-‐‑dimensional gradients of
either potential temperature or Conservative Temperature along the neutral tangent plane
and K is the epineutral diffusion coefficient. The cabbeling dianeutral advection is
proportional to the mesoscale eddy flux of “heat” along the neutral tangent plane,
− K ∇n Θ , and is independent of the amount of small-‐‑scale (dianeutral) turbulent mixing
and hence is also independent of the dissipation of kinetic energy (Klocker and McDougall
(2010a)). It is shown in appendix A.14 that Cbθ ∇nθ ⋅ ∇nθ ≠ CbΘ∇n Θ⋅ ∇n Θ so that the
estimate of the cabbeling dianeutral advection is different when calculated using potential
temperature than when using Conservative Temperature. The estimate using potential
temperature is slightly less accurate because of the non-‐‑conservative nature of potential
temperature.
When the cabbeling and thermobaricity processes are analyzed by considering the
mixing of two fluid parcels one finds that the density change is proportional to the square
of the property ( Θ and/or p ) contrasts between the two fluid parcels (for the cabbeling
case, see Eqn. (A.19.2) in appendix A.19). This leads to the thought that if an ocean front is
split up into a series of many less intense fronts then the effects of cabbeling and
thermobaricity might be reduced in proportion to the number of such fronts. This is not
the case. Rather, the total dianeutral transport across a frontal region depends on the
product of the lateral flux of heat passing through the front and the contrast in
temperature and/or pressure across the front, but is independent of the sharpness of the
front (Klocker and McDougall (2010a)). This can be understood by noting from above that
the dianeutral velocity due to cabbeling, eCab = − gN −2 K CbΘ∇n Θ⋅ ∇n Θ, is proportional to the
scalar product of the epineutral flux of heat − c0p K ∇n Θ and the epineutral temperature
gradient ∇n Θ . Spatially integrating this product over the area of the frontal region, one
finds that the total dianeutral transport is proportional to the lateral heat flux times the
difference in temperature across the frontal region (in the case of cabbeling) or the
difference in pressure across the frontal region (in the case of thermobaricity).
In both the SIA and GSW software libraries the cabbeling parameter is output in units
of K −2 . Expressions for Cbθ and C bΘ in terms of enthalpy in the functional forms
h SA ,θ , p and hˆ ( SA , Θ, p ) can be found in appendix P.
(
)
IOC Manuals and Guides No. 56
32
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
3.10 Buoyancy frequency
The square of the buoyancy frequency (sometimes called the Brunt-‐‑Väisälä frequency) N 2
is given in terms of the vertical gradients of density and pressure, or in terms of the
vertical gradients of Conservative Temperature and Absolute Salinity (or in terms of the
vertical gradients of potential temperature and Absolute Salinity) by (the g on the left-‐‑
hand side is the gravitational acceleration, and x, y and z are the spatial Cartesian
coordinates)
(
g −1N 2 = − ρ −1ρ z + κ Pz = − ρ −1 ρ z − Pz / c 2
= α θ θ z − β θ ∂SA ∂z x , y
)
(3.10.1)
= α ΘΘ z − β Θ ∂SA ∂z x , y .
For two seawater parcels separated by a small distance Δz in the vertical, an equally
accurate method of calculating the buoyancy frequency is to bring both seawater parcels
adiabatically and without exchange of matter to the average pressure and to calculate the
difference in density of the two parcels after this change in pressure. In this way the
potential density of the two seawater parcels are being compared at the same pressure.
This common procedure calculates the buoyancy frequency N according to
(
N 2 = g α Θ Θ z − β Θ SA z
)
≈−
g Δρ Θ
g 2 Δρ Θ
, or N 2 = g 2 ρ β Θ SAP − α ΘΘP ≈
, (3.10.2)
ρ Δz
ΔP
(
)
where Δρ Θ is the difference between the potential densities of the two seawater parcels
with the reference pressure being the average of the two original pressures of the seawater
parcels. Eqn. (3.10.2b) has made use of the hydrostatic relation Pz = − g ρ .
3.11 Neutral tangent plane
The neutral plane is that plane in space in which the local parcel of seawater can be moved
an infinitesimal distance without being subject to a vertical buoyant restoring force; it is
the plane of neutral-‐‑ or zero-‐‑ buoyancy. The normal vector to the neutral tangent plane n
is given by
(
g −1 N 2 n = − ρ −1∇ρ + κ∇P = − ρ −1 ∇ρ − ∇P / c 2
θ
θ
= α ∇θ − β ∇SA
Θ
)
(3.11.1)
Θ
= α ∇Θ − β ∇SA .
As defined, n is not quite a unit normal vector, rather its vertical component is exactly k ,
that is, its vertical component is unity. It is clear that α θ ∇θ − β θ ∇SA is exactly equal to
α Θ∇Θ − β Θ ∇SA . Interestingly, both α θ ∇θ and β θ ∇SA are independent of the four
arbitrary constants of the Gibbs function (see Eqn. (2.6.2)) while both α Θ∇Θ and β Θ ∇SA
contain an identical additional arbitrary term proportional to a3 ∇SA ; terms that exactly
cancel in their difference, α Θ∇Θ − β Θ ∇SA , in Eqn. (3.11.1).
Expressing the two-‐‑dimensional gradient of properties in the neutral tangent plane by
∇n , the property gradients in a neutral tangent plane obey
(
)
− ρ −1∇ n ρ + κ∇ n P = − ρ −1 ∇ n ρ − ∇ n P / c 2 = α θ ∇ nθ − β θ ∇ n SA
= α Θ∇ n Θ − β Θ ∇ n SA
(3.11.2)
= 0.
Here ∇n is an example of a projected gradient
∇rτ ≡
IOC Manuals and Guides No. 56
∂τ
∂x r
i +
∂τ
∂y r
j + 0 k ,
(3.11.3)
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
33
that is widely used in oceanic and atmospheric theory and modelling. Horizontal
distances are measured between the vertical planes of constant latitude x and longitude y
while the values of the property τ are evaluated on the r surface (e. g. an isopycnal
surface, or in the case of ∇n , a neutral tangent plane). This coordinate system is described
by Sutcliffe (1947), Bleck (1978), McDougall (1987b), McDougall (1995) and Griffies (2004).
Note that ∇rτ has no vertical component; it is not directed along the r surface, but rather
it points in exactly the horizontal direction.
Finite difference versions of Eqn. (3.11.2) such as α ΘΔΘ − β Θ ΔSA ≈ 0 are also very
accurate. Here α Θ and β Θ are the values of these coefficients evaluated at the average
values of Θ, SA and p of two parcels SA1 , Θ1 , p1 and SA2 , Θ2 , p2 on a “neutral surface”
and ΔΘ and ΔSA are the property differences between the two parcels. The error
involved with this finite amplitude version of Eqn. (3.11.2), namely
(
)
− TbΘ ∫ ( P − P ) d Θ ,
2
(
)
(3.11.4)
1
is described in section 2 and appendix A(c) of Jackett and McDougall (1997). An equally
accurate finite amplitude version of Eqn. (3.11.2) is to equate the potential densities of the
two fluid parcels, each referenced to the average pressure p = 0.5 ( p1 + p2 ) .
The reason why oceanographers take the strong lateral mixing of mesoscale eddies to
be directed along the neutral tangent plane is because of the smallness of the observed
dissipation of kinetic energy ε in the ocean interior. If the lateral diffusivity
K ≈ 102 − 103 m 2 s −1 of mesoscale dispersion and subsequent molecular diffusion were to
occur along a surface that differed in slope from the neutral tangent plane by an angle
whose tangent was s, then the individual fluid parcels would be transported above and
below the neutral tangent plane and would need to subsequently sink or rise in order to
attain a vertical position of neutral buoyancy.
Figure 5. Sketch of the consequences of the adiabatic movement followed by
release of fluid parcels along a plane that is different to a neutral tangent plane.
This vertical motion would either (i) involve no small-‐‑scale turbulent mixing, in which
case the combined process is equivalent to epineutral mixing, or (ii), the sinking and rising
parcels would mix with and entrain the surrounding ocean in a plume-‐‑like fashion (see
Figure 5), so suffering irreversible diffusion. In this second case, the dissipation of kinetic
energy associated with the diapycnal mixing would be observed. But in fact the
dissipation of kinetic energy in the main thermocline is consistent with a diapycnal
diffusivity of only 10−5 m 2 s −1 . This small value of the diapycnal (vertical) diffusivity has
been confirmed by purposely released tracer experiments.
When lateral diffusion with diffusivity K is taken to occur along a surface other than
a neutral tangent plane, some dianeutral diffusion occurs, and the amount of this
dianeutral diffusion is the same as achieved by a vertical diffusivity of s 2 K where s 2 is
the square of the vector slope ∇r z − ∇n z between the mixing direction and the neutral
tangent plane. This result is proven as follows.
IOC Manuals and Guides No. 56
34
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
The lateral flux of Neutral Density along the direction of mixing, the r surface is
− K∇ rγ = − K γ z ( ∇r z − ∇n z ) ,
(3.11.5)
and the component of this lateral flux across the neutral tangent plane is
− K∇ rγ ⋅ ( ∇r z − ∇n z ) = − K γ z ( ∇r z − ∇n z ) .
2
(3.11.6)
Dividing by minus the vertical gradient of Neutral Density, −γ z , shows that this flux is the
same as that caused by the positive fictitious vertical diffusivity of density
2
∇r z − ∇n z K = s 2 K .
Hence if all of this observed diapycnal diffusivity (based on the observed dissipation
of turbulent kinetic energy ε ) were due to mesoscale eddies mixing along a direction
different to neutral tangent planes, the (tangent of the) angle between this mesoscale
mixing direction and the neutral tangent plane, s, would satisfy 10−5 m 2 s −1 = s 2 K . Using
K ≈ 103 m 2 s −1 gives the maximum value of s to be 10−4 . Since we believe that bona fide
interior diapycnal mixing processes (such as breaking internal gravity waves) are
responsible for the bulk of the observed diapycnal diffusivity, we conclude that the
angular difference s between the direction of mesoscale eddy mixing and the neutral
tangent plane must be substantially less than 10−4 ; say 2x10−5 for argument’s sake.
(
)
3.12 Geostrophic, hydrostatic and “thermal wind” equations
The geostrophic approximation to the horizontal momentum equations (Eqn. (B9) below)
equates the Coriolis term to the horizontal pressure gradient ∇ z P so that the geostrophic
equation is
(3.12.1)
f k × ρ u = −∇z P or fv = ρ1 k × ∇ z P = g k × ∇p z ,
where u is the three dimensional velocity and v = − k × ( k × u ) is the horizontal velocity
where k is the vertical unit vector (pointing upwards) and f is the Coriolis parameter.
The last part of the above equation has used ∇ z P = − Pz ∇ p z from Eqn. (3.12.4b) below and
the hydrostatic approximation, which is the following approximation to the vertical
momentum equation (B9),
(3.12.2)
Pz = − g ρ .
The use of P in these equations rather than p serves to remind us that in order to retain
the usual units for height, density and the gravitational acceleration, pressure in these
dynamical equations must be expressed in Pa not dbar.
The so called “thermal wind” equation is an equation for the vertical gradient of the
horizontal velocity under the geostrophic approximation. Vertically differentiating Eqn.
(3.12.1) and using the hydrostatic equation Eqn. (3.12.2), the thermal wind can be written
f vz =
( ) k ×∇ P +
1
ρ z
z
1
ρ
k ×∇ z ( Pz ) = −
g
ρ
k ×∇ p ρ =
N2
gρ
k ×∇ n P,
(3.12.3)
where ∇ p is the projected lateral gradient operator in the isobaric surface (see Eqn.
(3.11.3)). The last part of this equation relates the “thermal wind”, f v z , to the pressure
gradient in the neutral tangent plane (McDougall, 1995). Note that the Boussinesq
approximation has not been made to derive any part of Eqn. (3.12.3). Under the
Boussinesq approximation, ∇ p ρ is approximated by ∇ z ρ , and the last term in Eqn.
(3.12.3) is approximated as − N 2 k ×∇ n z . The derivation of Eqn. (3.12.3) proceeds as
follows. To go from the second part of Eqn. (3.12.3) to the third part use is made of
∇ p ρ = ∇ z ρ + ρ z ∇ p z and ∇ p P = 0 = ∇ z P + Pz ∇ p z .
(3.12.4a,b)
To go from the third part of Eqn. (3.12.3) to the final part, use is made of Eqn. (3.12.4a) and
∇ n ρ = ∇ z ρ + ρ z ∇ n z , which, when combined gives
(
)
∇ p ρ = ∇ n ρ − ρ z ∇ n z − ∇ p z .
IOC Manuals and Guides No. 56
(3.12.5)
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
35
Now Eqn. (3.12.4b) is used together with ∇ n P = ∇ z P + Pz ∇ n z to find
(
)
∇ n P = Pz ∇ n z − ∇ p z ,
(3.12.6)
and this is substituted into Eqn. (3.12.5) to find
∇ p ρ = ∇ n ρ − ρ z ∇ n P Pz .
(3.12.7)
Now along a neutral tangent plane we know that ∇ n ρ = ρκ ∇ n P ( κ is the isentropic and
isohaline compressibility of seawater) and substituting this into Eqn. (3.12.7) leads to the
2
final expression of Eqn. (3.12.3), namely Ng ρ k × ∇ n P (recognizing that the buoyancy
frequency is defined by N 2 = g κ Pz − ρ1 ρ z ).
The rotation of the horizontal velocity vector with height can be determined as
follows. Let the angle of the horizontal velocity v with respect to due east (measured
counter-‐‑clockwise) be ϕ so that v = v cosϕ , sin ϕ . Vertically differentiating this
2
equation and taking the cross product with v leads to v × v z = k ϕ z v which shows that
the rate of spiraling of the horizontal velocity vector in the vertical ϕ z is proportional to
the amount by which this velocity is not parallel to the direction of the “thermal wind”
shear v z . The last relation can be rewritten as
(
)
(
ϕz v
2
)
= k ⋅ v × v z = uvz − vuz = − v ⋅k × v z = − v ⋅∇ × v ,
(3.12.8)
which demonstrates that the rotation of the horizontal velocity with height is proportional
to the helicity of the horizontal velocity, v ⋅∇ × v .
Now, substituting Eqn. (3.12.3) for the “thermal wind” v z , into Eqn. (3.12.8) we find
ϕz v
2
= − v ⋅k × v z =
N2
fg ρ
( )
Under the usual Boussinesq approximation − g ρ
neutral tangent plane, ∇n z , so that we have
ϕz v
2
≈ −
N2
f
v ⋅∇n P .
−1
(3.12.9)
∇n P is set equal to the slope of the
v ⋅∇n z ,
(3.12.10)
and since the vertical velocity through geopotentials, w , is given by the simple
geometrical relationship w = zt + v ⋅∇n z + e (where e is the dianeutral velocity, that is,
n
the vertical velocity through the neutral tangent plane), we have
ϕz v
2
≈ −
N2
f
( w − e − z ),
(3.12.11)
t n
showing that the rotation of the horizontal velocity vector with height is not simply
proportional to the vertical velocity of the flow but rather only to the sliding motion along
the neutral tangent plane, v ⋅∇n z .
3.13 Neutral helicity
The neutral tangent plane was defined in section 3.11 as the plane in which parcels can be
moved in an adiabatic and isohaline manner without experiencing a vertical buoyant
force. The normal n to the neutral tangent plane is given by Eqn. (3.11.1) and it is natural
to think that all these little tangent planes would link up and form a well-‐‑defined surface,
but this is not actually the case in the ocean. In order to understand why the ocean
chooses to be so ornery we need to understand what property the normal n to a surface
must fulfill in order that the surface exists.
In general, for a surface to exist in x, y, z space there must be a function φ x, y, z
that is constant on the surface and whose gradient ∇φ is in the direction of the normal to
the surface, n . That is, there must be an integrating factor b x, y, z such that ∇φ = bn .
Assuming now that the surface does exist, consider a line integral of bn along a closed
curved path in the surface. Since the line element of the integration path is everywhere
normal to n , the closed line integral is zero, and by Stokes’s theorem, the area integral of
(
)
(
(
)
)
IOC Manuals and Guides No. 56
36
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
( )
∇ × bn must be zero over the area enclosed by the closed curved path. Since the area
element of integration dA is in the direction n , it is clear that ∇ × bn ⋅ dA is proportional
to ∇ × bn ⋅ n . The only way that this area integral can be guaranteed to be zero for all
such closed paths is if the integrand is zero everywhere on the surface, that is, if
∇ × bn ⋅ n = ∇b × n ⋅ n + b ∇ × n ⋅ n = 0 , that is, if n ⋅∇ × n = 0 at all locations on the
surface.
For the case in hand, the normal to the neutral tangent plane is in the direction
α Θ∇Θ − β Θ ∇SA and we define the neutral helicity H n as the scalar product of
α Θ∇Θ − β Θ ∇SA with its curl,
( )
( )
( )
(
)
(
)
(
)
(
)
H n ≡ α Θ∇Θ − β Θ ∇SA ⋅ ∇ × α Θ∇Θ − β Θ ∇SA .
(3.13.1)
Neutral tangent planes (which do exist) do not link up in space to form a well-‐‑defined
neutral surface unless the neutral helicity H n is everywhere zero on the surface.
Recognizing that both the thermal expansion coefficient and the saline contraction
coefficient are functions of SA ,Θ, p , neutral helicity H n may be expressed as the
following four expressions, all of which are proportional to the thermobaric coefficient TbΘ
of the equation of state,
(
Hn =
=
)
β Θ TbΘ ∇P ⋅ ∇SA × ∇Θ
(
)
Pz β Θ TbΘ ∇ p SA × ∇ p Θ ⋅ k
= g −1 N 2TbΘ ( ∇n P × ∇n Θ ) ⋅ k
(3.13.2)
≈ g −1 N 2TbΘ ( ∇a P × ∇a Θ) ⋅ k
where Pz is simply the vertical gradient of pressure ( Pa m−1 ) and ∇n Θ and ∇ p Θ are the
two-‐‑dimensional gradients of Θ in the neural tangent plane and in the horizontal plane
(actually the isobaric surface) respectively. The gradients ∇a P and ∇a Θ are taken in an
approximately neutral surface.
Since α θ ∇θ − β θ ∇SA and α Θ∇Θ − β Θ ∇SA are exactly equal, neutral helicity can be
defined in Eqn. (3.13.1) as the scalar product of this vector with its curl based on either
formulation, so that (from the third line of Eqn. (3.13.2), and bearing in mind that ∇n Θ
and ∇nθ are parallel vectors) we see that Tbθ ∇nθ = TbΘ∇n Θ, a result that we use in section
3.8 and in appendix A.14. Neutral helicity has units of m −3 .
Interestingly, for given magnitudes of the epineutral gradients of pressure and
Conservative Temperature, neutral helicity is maximized when these gradients are
perpendicular since neutral helicity is proportional to TbΘ ( ∇n P × ∇n Θ ) ⋅ k (see Eqn.
(3.13.2)), while the dianeutral advection of thermobaricity, eTb = − gN −2 K TbΘ∇n Θ⋅ ∇n P , is
maximized when ∇n Θ and ∇n P are parallel (see section 3.8).
Because of the non-‐‑zero neutral helicity in the ocean, lateral motion following neutral
tangent planes has the character of helical motion. That is, if we ignore the effects of
diapycnal mixing processes (as well as ignoring cabbeling and thermobaricity), the mean
flow around ocean gyres still passes through any well-‐‑defined “density” surface because
of the helical nature of neutral trajectories, caused in turn by the non-‐‑zero neutral helicity.
This dia-‐‑surface flow is expressed in Eqns. (A.25.4) and (A.25.6) in terms of the
appropriate mean horizontal velocity and the difference between the slope of the neutral
tangent plane and the slope of a well-‐‑defined “density” surface.
Neutral helicity in the world ocean is observed to be small in some sense. One way of
visualizing this smallness of H n is to examine all the hydrographic data in SA ,Θ, p
space. When this is done for an entire ocean basin (for example, the whole of the
combined North and South Atlantic oceans), and the data is spun in this three-‐‑
dimensional SA ,Θ, p space, it is clear that the ocean hydrography lies close to a single
surface in this SA ,Θ, p space. We will now show that if all the SA ,Θ, p data from the
ocean lie exactly on a single surface f SA ,Θ, p = 0 in SA ,Θ, p space, then this requires
(
(
(
)
)
IOC Manuals and Guides No. 56
(
)
(
)
(
)
)
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
(
)
37
∇SA × ∇Θ ⋅∇P = 0 everywhere in physical x, y, z space. That is, we will prove that the
“skinny” nature of the ocean hydrography in SA ,Θ, p space is a direct indication of the
smallness of neutral helicity H n .
Taking the spatial gradient of f SA ,Θ, p = 0 in physical x, y, z space we have
∇f = 0 since f is zero at every point in physical x, y, z space. Expanding ∇f in terms
of the spatial gradients ∇SA , ∇Θ , and ∇P , and taking the scalar product with ∇SA × ∇Θ
gives
(
fP
SA ,Θ
(
)
(
)
)
(
)
∇SA × ∇Θ ⋅∇P = 0 .
(3.13.3)
In the general case of f P ≠ 0 , the result ∇SA × ∇Θ ⋅∇P = 0 is proven. In the special case
f P = 0 , f is independent of P so that there is a simpler equation for the surface f ,
being f SA ,Θ = 0 , which is the equation for a single line on the SA ,Θ diagram; a single
“water-‐‑mass” for the whole world ocean. In this case, changes in SA are locally
proportional to those of Θ so that ∇SA × ∇Θ = 0 which guarantees ∇SA × ∇Θ ⋅∇P = 0 .
Hence we have proven that the “skinniness” of the ocean hydrography in SA ,Θ, p space
is a direct indication of the smallness of neutral helicity H n .
The “skinny” nature of the North and South Atlantic hydrography is illustrated in
Figure 6, which shows all the hydrographic data on the SA − Θ diagram at a pressure of
500 dbar . This cut at constant pressure through the hydrographic data in three-‐‑
dimensional SA ,Θ, p space, and similar cuts at different fixed pressures, show that the
data from the whole physical x, y, z volume of the North and South Atlantic lie close to a
single surface in the three-‐‑dimensional SA ,Θ, p space. Figure 6 also illustrates the
method of formation of one of Reid and Lynn’s (1971) “isopycnals” and how the potential
density anomaly with respect to the sea surface, σ Θ , of 27.3 kg m −3 is matched to σ 1 of
31.938 kg m −3 in the Southern Ocean but to a different σ Θ of 27.44 kg m −3 in the North
Atlantic.
(
)
(
)
(
(
)
(
)
(
)
)
Figure 6. Hydrographic data from the ocean atlas of Gouretski and
Koltermann (2004) for the North and South Atlantic at a pressure of 500 dbar.
The colour of the data points indicates the latitude, from blue in the south
through green at the equator to red in the north.
Neutral helicity is proportional to the component of the vertical shear of the
geostrophic velocity ( v z , the “thermal wind”) in the direction of the temperature gradient
along the neutral tangent plane ∇n Θ , since, from Eqn. (3.12.3) and the third line of (3.13.2)
we find that
H n = ρTbΘ fv z ⋅ ∇n Θ.
(3.13.4)
In the evolution equation of potential vorticity defined with respect to potential
density ρ θ there is the baroclinic production term ρ −2∇ρ θ ⋅∇ρ × ∇P (Straub (1999)) and
IOC Manuals and Guides No. 56
38
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
the first term in a Taylor series expansion for this baroclinic production term is
proportional to neutral helicity and is given by (McDougall and Jackett (2007))
ρ −2∇ρ θ ⋅∇ρ × ∇P ≈ ( Pr − P ) H n
(3.13.5)
where Pr is the reference pressure of the potential density. Similarly, the curl in a
potential density surface of the horizontal pressure gradient term in the horizontal
momentum equation, ∇σ × ρ1 ∇z p , is given by (McDougall and Klocker (2010))
(
)
−1
⎛ ∂ρ Θ ⎞
(3.13.6)
∇σ × ρ ∇ z P ⋅ k = H ( Pr − P ) ⎜ −
⎟ .
⎝ ∂z ⎠
The fact that this curl is nonzero proves that a geostrophic streamfunction does not exist in
a potential density surface.
The absolute velocity vector in the ocean can be written as a closed expression
involving neutral helicity, and this expression is derived as follows. First the Eulerian-‐‑
mean horizontal velocity is related directly to mixing processes by invoking the water-‐‑
mass transformation equation (A.23.1), so that
(
1
(
)
n
)
(
v ⋅∇ nΘ̂ = γ z ∇ n ⋅ γ z−1 K∇ nΘ̂ + KgN −2Θ̂ z CbΘ∇ nΘ̂ ⋅∇ nΘ̂ + TbΘ∇ nΘ̂ ⋅∇ n P
+ Dβ Θ gN −2Θ̂3z
2
d ŜA
− Ψz ⋅∇ nΘ̂ − Θ̂t ,
)
(3.13.7)
n
d Θ̂ 2
where the thickness-‐‑weighted mean velocity of density-‐‑coordinate averaging, v̂ , has been
written as v̂ = v + Ψz , that is, as the sum of the Eulerian-‐‑mean horizontal velocity v and
the quasi-‐‑Stokes eddy-‐‑induced horizontal velocity Ψz (McDougall and McIntosh (2001)).
The quasi-‐‑Stokes vector streamfunction Ψ is usually expressed in terms of an imposed
lateral diffusivity and the slope of the locally-‐‑referenced potential density surface (Gent et
al., (1995)). More generally, at least in a steady state when Θ̂t is zero, the right-‐‑hand side
n
of Eqn. (3.13.7) is due only to mixing processes and once the form of the lateral and
vertical diffusivities are known, these terms are known in terms of the ocean’s
hydrography. Eqn. (3.13.7) is written more compactly as
v ⋅ τ = v ⊥ where τ ≡ ∇ nΘ̂ ∇ nΘ̂ ,
(3.13.8)
and v ⊥ is interpreted as being due to mixing processes.
Following Needler (1985) and McDougall (1995) the mean horizontal velocity v is
split into components along and across the contours of Θ̂ on the neutral tangent plane,
v = v τ × k + v ⊥ τ ,
(3.13.9)
where v = v ⋅ τ × k . Note that if τ points northwards then τ × k points eastward. The
expression v ⋅ τ = v ⊥ of Eqn. (3.13.8) is now vertically differentiated to obtain
v ⋅ τ z = − v z ⋅ τ + vz⊥
= −
N2
fg ρ
k × ∇n P ⋅ τ + vz⊥ ,
(3.13.10)
2
where we have used the “thermal wind” equation (3.12.3), v z = Nfg ρ k × ∇n P . We will now
show that the left-‐‑hand side of this equation is − φ z v where φ z is the rate of rotation of the
direction of the unit vector τ with respect to height (in radians per metre). By expressing
the two-‐‑dimensional unit vector τ in terms of the angle φ (measured counter-‐‑clockwise)
of τ with respect to due east so that τ = cos φ , sin φ , we see that τ × k = sin φ , − cos φ ,
τ z = − φ z τ × k and k ⋅ τ × τ z = φ z . Interestingly, φ z is also equal to minus the helicity of τ
(and to minus the helicity of τ × k ), that is, φ z = − τ ⋅∇ × τ = − τ × k ⋅∇ × τ × k , where
the helicity of a vector is defined to be the scalar product of the vector with its curl. From
the velocity decomposition (3.13.9) and the equation τ z = − φ z τ × k we see that the left-‐‑
hand side of Eqn. (3.13.10), v ⋅ τ z , is − φ z v , hence v can be expressed as
(
)
(
(
v =
)
(
v⊥
vz⊥
N 2 k ⋅∇n P × τ
Hn
− z or v =
−
,
fg ρ
φz
φz
φz
φ z ρ f TbΘ ∇ nΘ̂
IOC Manuals and Guides No. 56
)
)
(3.13.11)
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
39
where we have used the definition of neutral helicity H n , Eqn. (3.13.2). The expression for
both horizontal components of the Eulerian-‐‑mean horizontal velocity vector v is
⎧⎪ N 2 k ⋅∇ P × τ
v ⊥ ⎫⎪
n
v = ⎨
− z ⎬ τ × k + v ⊥ τ ,
(3.13.12)
φz
φ z ⎪⎭
⎪⎩ fg ρ
and the horizontal velocity due to solely the two mixing terms can be expressed as
vz⊥
(v ⊥ )2 ⎛ τ × k ⎞
1 ⊥
, whose magnitude is
−
τ × k + v⊥ τ =
v τ
φz
φ z ⎜⎝ v ⊥ ⎟⎠ z
φz
( )
z
=
(v τ )
⊥
φ
. (3.13.13)
Equation (3.13.12) for the Eulerian-‐‑mean horizontal velocity v shows that in the
absence of mixing processes (so that v ⊥ = vz⊥ = 0 ) and so long as (i) the epineutral Θ̂
contours do spiral in the vertical and (ii) ∇ nΘ̂ is not zero, then neutral helicity H n (which
is proportional to k ⋅∇n P × τ ) is required to be non-‐‑zero in the ocean whenever the ocean is
not motionless. Neutral helicity arises in this context because it is proportional to the
component of the thermal wind vector v z in the direction across the Θ̂ contour on the
neutral tangent plane (see Eqn. (3.13.4)). This derivation of the expression for the mean
absolute horizontal velocity vector v is based on McDougall (1995) and Zika et al. (2010a).
3.14 Neutral Density
Neutral Density is the name given to a density variable that results from the computer
software described in Jackett and McDougall (1997). Neutral Density is given the symbol
γ n but it is not a thermodynamic variable as it is a function not only of salinity,
temperature and pressure, but also of latitude and longitude. Because of the non-‐‑zero
neutral helicity H n in the ocean it is not possible to form surfaces that are everywhere
osculate with neutral tangent planes (McDougall and Jackett (1988)). Neutral Density
surfaces minimize in some sense the global differences between the slopes of the neutral
tangent plane and the Neutral Density surface. This slope difference is given by
(
)
s = ∇ n z − ∇ a z = − gN −2 α Θ∇ aΘ − β Θ∇ a SA ,
(3.14.1)
where ∇n z is the slope of the neutral tangent plane, ∇a z is the slope of the approximately
neutral surface and ∇a is the two-‐‑dimensional gradient operator in the approximately
neutral surface (of which a Neutral Density surface is one example). The vertical velocity
through an approximately neutral surface due to lateral motion along a neutral tangent
plane is the scalar product v ⋅ s where v is the horizontal velocity (see Eqn. (A.25.4)).
3.15 Stability ratio
The stability ratio Rρ is the ratio of the vertical contribution from Conservative
Temperature to that from Absolute Salinity to the static stability N 2 of the water column.
From (3.10.1) above we find
Rρ =
α ΘΘ z
β Θ ( SA )z
.
(3.15.1)
Rρ is available in the GSW Oceanographic Toolbox as the function gsw_Turner_Rsubrho.
3.16 Turner angle
The Turner angle Tu , named after J. Stewart Turner, is defined as the four-‐‑quadrant
arctangent (Ruddick (1983) and McDougall et al. (1988), particularly their Figure 1)
(
( )z
( )z ) ,
Tu = tan −1 α ΘΘ z + β Θ SA , α ΘΘ z − β Θ SA
(3.16.1)
IOC Manuals and Guides No. 56
40
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
where the first of the two arguments of the arctangent function is the “y”-‐‑argument and
the second one the “x”-‐‑argument, this being the common order of these arguments in
Fortran and MATLAB. The Turner angle Tu is quoted in degrees of rotation. Turner
angles between 45° and 90° represent the “salt-‐‑finger” regime of double-‐‑diffusive
convection, with the strongest activity near 90°. Turner angles between − 45° and − 90°
represent the “diffusive” regime of double-‐‑diffusive convection, with the strongest activity
near −90°. Turner angles between − 45° and 45° represent regions where the stratification
is stably stratified in both Θ and SA . Turner angles greater than 90° or less than − 90°
characterize a statically unstable water column in which N 2 < 0. As a check on the
calculation of the Turner angle, note that Rρ = − tan (Tu + 45° ) . The Turner angle and the
stability ratio are available in the GSW Oceanographic Toolbox from the function
gsw_Turner_Rsubrho.
3.17 Property gradients along potential density surfaces
The two-‐‑dimensional gradient of a scalar ϕ along a potential density surface ∇σ ϕ is
related to the corresponding gradient in the neutral tangent plane ∇nϕ by
∇σ ϕ = ∇nϕ +
ϕ z Rρ [r − 1]
(3.17.1)
∇n Θ
Θz ⎡⎣ Rρ − r ⎤⎦
(from McDougall (1987a)), where r is the ratio of the slope on the SA − Θ diagram of an
isoline of potential density with reference pressure pr to the slope of a potential density
surface with reference pressure p , and is defined by
α Θ ( SA , Θ, p ) β Θ ( SA , Θ, p )
.
(3.17.2)
r = Θ
α ( SA , Θ, pr ) β Θ ( SA , Θ, pr )
Substituting ϕ = Θ into (3.17.1) gives the following relation between the (parallel)
isopycnal and epineutral gradients of Θ
r ⎡⎣ Rρ − 1⎤⎦
∇n Θ = G Θ∇n Θ
⎡⎣ Rρ − r ⎤⎦
where the “isopycnal temperature gradient ratio”
∇σ Θ =
(3.17.3)
⎡⎣ Rρ − 1⎤⎦
⎡⎣ Rρ r − 1⎤⎦
(3.17.4)
⎡ Rρ − 1⎤⎦
GΘ
∇σ SA = ⎣
∇n SA =
∇n SA .
r
⎡⎣ Rρ − r ⎤⎦
(3.17.5)
GΘ ≡
has been defined as a shorthand expression for future use. This ratio G Θ is available in
the GSW Toolbox from the algorithm gsw_isopycnal_vs_ntp_CT_ratio, while the ratio r
of Eqn. (3.17.2) is available there as gsw_isopycnal_slope_ratio. Substituting ϕ = SA into
Eqn. (3.17.1) gives the following relation between the (parallel) isopycnal and epineutral
gradients of S A
3.18 Slopes of potential density surfaces and neutral tangent planes compared
The two-‐‑dimensional slope of a surface is defined as the two-‐‑dimensional gradient of
height z of that surface. The two-‐‑dimensional slope of a surface is an exactly horizontal
gradient vector; it has no vertical component. The slope difference between the neutral
tangent plane and a potential density surface with reference pressure pr is given by
(McDougall (1988))
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
∇n z − ∇σ z =
Rρ [1 − r ] ∇n Θ
=
⎣⎡ Rρ − r ⎦⎤ Θz
(1 − G ) ∇ΘΘ
Θ
n
=
z
41
Rρ [1 − r ] ∇σ Θ
∇n Θ − ∇σ Θ
=
. (3.18.1)
Θz
r ⎣⎡ Rρ − 1⎦⎤ Θz
While potential density surfaces have been the most commonly used surfaces with
which to separate “isopycnal” mixing processes from vertical mixing processes, many
other types of density surface have been used. The list includes specific volume anomaly
surfaces, patched potential density surfaces (Reid and Lynn (1971)), Neutral Density
surfaces (Jackett and McDougall (1997)), orthobaric density surfaces (de Szoeke et al.
(2000)) and some polynomial fits of Neutral Density as function of only salinity and either
θ or Θ (Eden and Willebrand (1999), McDougall and Jackett (2005b)). The most recent
method for forming approximately neutral surfaces is that of Klocker et al. (2009a,b). This
method is relatively computer intensive but has the benefit that the remnant mis-‐‑match
between the final surface and the neutral tangent plane at each point is due only to the
neutral helicity of the data through which the surface passes. The relative skill of all these
surfaces at approximating the neutral tangent plane slope at each point has been
summarized in the equations and histogram plots in the papers of McDougall (1989, 1995),
McDougall and Jackett (2005a, 2005b), and Klocker et al. (2009a,b).
When lateral mixing with isopycnal diffusivity K is imposed along potential density
surfaces rather than along neutral tangent planes, a fictitious diapycnal diffusivity arises
which is often labeled the “Veronis effect” after Veronis (1975) (who considered the ill
effects of exactly horizontal versus isopycnal mixing). This fictitious diapycnal diffusivity
of density is equal to K times the square of the slope error, Eqn. (3.18.1).
3.19 Slopes of in situ density surfaces and specific volume anomaly surfaces
The vector slope of an in situ density surface, ∇ρ z , is defined to be the horizontal vector
∇ρ z =
∂z
∂x ρ
i +
∂z
∂y ρ
j + 0 k ,
(3.19.1)
representing the “dip” of the surface in both horizontal directions (note that height z is
defined positive upwards). The difference between this vector slope and the (very small)
slope of an isobaric surface ∇ p z can be related to the slope of the neutral tangent plane
with respect to the isobaric surface, ∇ n z − ∇ p z , by ( g is the gravitational acceleration)
(McDougall (1989))
−1
⎡
g 2 c2 ⎤
(3.19.2)
∇ρ z − ∇ p z = ∇n z − ∇ p z ⎢1 +
⎥ ,
N2 ⎦
⎣
where c is the speed of sound and N is the buoyancy frequency. In the upper water
column where the square of the buoyancy frequency is significantly larger than
g 2 c 2 ≈ 4.3x10−5 s −2 , the in situ density surface has a similar slope to the neutral tangent
plane ∇n z . In the deep ocean N 2 is only about 1% of g 2 c 2 and so the surfaces of
constant in situ density have a slope of only 1% of the slope of the neutral tangent plane.
At a pressure of about 1000 dbar where N 2 ≈ 10−5 s−2 , the slope of an in situ density surface
is only about one fifth that of the neutral tangent plane. Neutrally buoyant floats in the
ocean are usually metal cylinders that are much less compressible than seawater. These
floats have a constant mass and an almost constant volume. Hence these floats have an
almost constant in situ density and their motion approximately occurs on surfaces of
constant in situ density which at mid depth in the ocean are much closer to being isobaric
surfaces than being locally-‐‑referenced potential density surfaces. This is why these floats
are sometimes described as “isobaric floats”, and is the reason why a “compressee” is
sometimes added to a float so that its compressibility approximates that of seawater.
The slope of a specific volume anomaly surface, ∇δ z , can be expressed as
(
)
IOC Manuals and Guides No. 56
42
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
−1
⎡
g 2 c2
g 2 c 2 ⎤
(3.19.3)
∇δ z − ∇ p z = ∇ n z − ∇ p z ⎢1 +
−
⎥ ,
N2
N 2 ⎥⎦
⎢⎣
where c is the sound speed of the reference parcel SA , Θ
at pressure p. This expression
confirms that where the local seawater properties are close to those of the reference parcel,
the specific volume anomaly surface can closely approximate the neutral tangent plane.
The square bracket in Eqn. (3.19.3) is equal to ρ gN −2 ∂δ ∂z (from section 7 of McDougall
(1989) where δ is specific volume anomaly).
(
)
( )
3.20 Planetary potential vorticity
Planetary potential vorticity is the Coriolis parameter f times the vertical gradient of a
suitable variable. Potential density is often used for that variable but its use (i) involves an
inaccurate separation between lateral and diapycnal advection because potential density
surfaces are not a good approximation to neutral tangent planes and (ii) incurs the non-‐‑
conservative baroclinic production term of Eqn. (3.13.5). Using approximately neutral
surfaces, “ans”, (such as Neutral Density surfaces) provides an optimal separation
between the effects of lateral and diapycnal mixing in the potential vorticity equation. In
this case the potential vorticity variable is proportional to the reciprocal of the thickness
between a pair of closely spaced approximately neutral surfaces.
The evolution equation for planetary potential vorticity is derived by first taking the
epineutral “divergence” ∇n ⋅ of the geostrophic relationship from Eqn. (3.12.1), namely
fv = g k × ∇p z . The projected “divergences” of a two-‐‑dimensional vector a in the
neutral tangent plane and in an isobaric surface, are ∇n ⋅a = ∇z ⋅a + a z ⋅∇n z and
∇ p ⋅a = ∇z ⋅a + a z ⋅∇ p z from which we find (using Eqn. (3.12.6), ∇ n z − ∇ p z = ∇ n P Pz )
∇n ⋅a = ∇p ⋅a + a z ⋅∇n P Pz .
(3.20.1)
Applying this relationship to the two-‐‑dimensional vector fv = g k × ∇p z we have
(
∇n ⋅ ( fv ) = g ∇p ⋅ k × ∇p z
) + fv
z
⋅∇n P Pz = 0 .
(3.20.2)
The first part of this expression can be seen to be zero by simply calculating its
components, and the second part is zero because the thermal wind vector v z points in the
direction k × ∇n P (see Eqn. (3.12.3)). It can be shown that ∇r ⋅ fv is zero in any surface
r = r ρ , P which is defined as a function of in situ density and pressure.
Eqn. (3.20.2), namely ∇n ⋅ fv = 0 , can be interpreted as the divergence form of the
evolution equation of planetary potential vorticity since
(
( )
)
( )
⎛ qv⎞
∇n ⋅ ( fv ) = ∇n ⋅ ⎜ ⎟ = 0 ,
⎝γz ⎠
(3.20.3)
where q = f γ z is the planetary potential vorticity, being the Coriolis parameter times the
vertical gradient of Neutral Density. This instantaneous equation can be averaged in a
thickness-‐‑weighted sense in density coordinates yielding
⎛ q̂ v̂ ⎞
⎛ v ′′q′′ ⎞
−1
∇n ⋅ ⎜
= − ∇n ⋅ ⎜
⎟
⎟ = ∇n ⋅ γ z K∇ nq̂ ,
γ
γ
⎝ z⎠
⎝ z ⎠
(
)
(3.20.4)
where the double-‐‑primed quantities are deviations of the instantaneous values from the
thickness-‐‑weighted mean quantities. Here the epineutral eddy flux of planetary potential
vorticity per unit area has been taken to be down the epineutral gradient of q̂ with the
epineutral diffusivity K . The thickness-‐‑weighted mean planetary potential vorticity is
⎛ q⎞
q̂ ≡ γ z ⎜ ⎟ = f γ z ,
⎝γ z⎠ γ
IOC Manuals and Guides No. 56
(3.20.5)
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
43
and the averaging in the above equations is consistent with the difference between the
thickness-‐‑weighted mean velocity and the velocity averaged on the Neutral Density
surface, v̂ − v (the bolus velocity), being v̂ − v = K ∇n ln q̂ , since Eqn. (3.20.4) can be
written as ∇n ⋅ f v̂ = ∇n ⋅ γ z−1 K∇ nq̂ while the epineutral temporal average of Eqn. (3.20.3)
is ∇n ⋅ f v = 0 .
The divergence form of the mean planetary potential vorticity evolution equation,
Eqn. (3.20.4), is quite different to that of a normal conservative variable such as Absolute
Salinity or Conservative Temperature in that (i) neither the vertical diffusivity nor the
dianeutral velocity makes an appearance, and (ii) there is no temporal tendency term in
the equation.
The mean planetary potential vorticity equation (3.20.4) may be put into the advective
form by subtracting q̂ times the mean continuity equation,
( )
(
( )
()
)
⎛ 1 ⎞
⎛ v̂ ⎞
ez
= 0 ,
⎜
⎟ + ∇n ⋅ ⎜ ⎟ +
γ z
⎝ γ z ⎠
⎝ γ z n ⎠ t
from Eqn. (3.20.4), yielding ( γ z−1 times)
q̂t
n
(
(3.20.6)
)
+ v̂ ⋅ ∇nq̂ = γ z ∇n ⋅ γ z−1 K∇ nq̂ + q̂ez ,
or
(
(3.20.7)
)
dq̂
(3.20.8)
= γ z ∇n ⋅ γ z−1 K∇ nq̂ + ( q̂e )z .
dt
In this form, it is clear that planetary potential vorticity behaves like a conservative
variable as far as epineutral mixing is concerned, but it is quite unlike a normal
conservative variable as far as vertical mixing is concerned.
If q̂ were a normal conservative variable the last term in Eqn. (3.20.8) would be
( Dq̂z )z where D is the vertical diffusivity. The term that actually appears in Eqn. (3.20.8),
( q̂e )z , is different to ( Dq̂z )z by ( q̂e − Dq̂z )z = f ( eγ z − Dγ zz )z . Equation (A.22.4) for the
mean dianeutral velocity e can be expressed as e ≈ Dz + D γ zz γ z if the following three
aspects of the non-‐‑linear equation of state are ignored; (1) cabbeling and thermobaricity,
(2) the vertical variation of the thermal expansion coefficient and the saline contraction
coefficient, and (3) the vertical variation of the integrating factor b ( x, y, z ) of Eqns. (3.20.10)
-‐‑ (3.20.15) below. Even when ignoring these three different implications of the nonlinear
equation of state, the evolution equations (3.20.7) and (3.20.8) of q̂ are unlike normal
conservation equations because of the extra term
q̂t
n
+ v̂ ⋅ ∇nq̂ + eq̂z =
( q̂e − Dq̂z )z
= f ( eγ z − Dγ zz )z ≈ f ( Dzγ z )z =
( Dz q̂ )z
(3.20.9)
on their right-‐‑hand sides. This presence of this additional term can result in “unmixing”
of q̂ in the vertical. Consider a situation where both q̂ and Θ̂ are locally linear functions
of ŜA down a vertical water column, so that the ŜA − q̂ and ŜA − Θ̂ diagrams are both
locally straight lines, exhibiting no curvature. Imposing a large amount of vertical mixing
at one height (e. g. a delta function of D ) will not change the ŜA − Θ̂ diagram because of
the zero ŜA − Θ̂ curvature (see the water-‐‑mass transformation equation (A.23.1)).
However, the additional term Dz q̂ z of Eqn. (3.20.9) means that there will be a change in
q̂ of Dz q̂ z = q̂Dzz + q̂z Dz ≈ q̂Dzz along the neural tangent plane (that is, in Eqn. (3.20.7)).
This is q̂ times a negative anomaly at the central height of the extra vertical diffusion, and
is q̂ times a positive anomaly on the flanking heights above and below the central height.
In this way, a delta function of extra vertical diffusion induces structure in the initially
straight ŜA − q̂ line which is a telltale sign of “unmixing”.
This planetary potential vorticity variable, q̂ = f γ z , is often mapped on Neutral
Density surfaces to give insight into the mean circulation of the ocean on density surfaces.
The reasoning is that if the influence of dianeutral advection (the last term in Eqn. (3.20.7))
is small, and the epineutral mixing of q̂ is also small, then in a steady ocean v̂ ⋅ ∇nq̂ = 0
(
)
(
)
IOC Manuals and Guides No. 56
44
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
and the thickness-‐‑weighted mean flow on density surfaces v̂ will be along contours of
thickness-‐‑weighted planetary potential vorticity q̂ = f γ z .
Because the square of the buoyancy frequency, N 2 , accurately represents the vertical
static stability of a water column, there is a strong urge to regard fN 2 as the appropriate
planetary potential vorticity variable, and to map its contours on Neutral Density surfaces.
This urge must be resisted, as spatial maps of fN 2 are significantly different to those of
q̂ = f γ z . To see why this is the case the relationship between the epineutral gradients of
q̂ and fN 2 will be derived.
For the present purposes Neutral Helicity will be assumed sufficiently small that the
existence of neutral surfaces is a good approximation, and we seek the integrating factor
b = b x, y, z which allows the construction of Neutral Density surfaces ( γ surfaces)
according to
⎞
⎛ ∇ρ
∇γ
= b β Θ∇SA − α Θ∇Θ = b ⎜
− κ ∇P⎟ .
(3.20.10)
γ
⎠
⎝ ρ
(
)
(
)
Taking the curl of this equation gives
∇ρ ⎞
∇b ⎛
= − ∇κ × ∇P .
× ⎜ κ ∇P −
ρ ⎟⎠
b ⎝
(3.20.11)
The bracket on the left-‐‑hand side is normal to the neutral tangent plane and points in the
direction n = − ∇ n z + k and is g −1 N 2 −∇ n z + k . Taking the component of Eqn. (3.20.11)
in the direction of the normal to the neutral tangent plane, n , we find
(
(∇ nκ
0 = ∇κ × ∇P ⋅ n =
)
+ κ z n ) × ( ∇ n P + Pz n ) ⋅ n
(
)
= ∇ nκ × ∇ n P ⋅ n = ∇ nκ × ∇ n P ⋅k = κ S ∇ n SA + κ Θ∇ nΘ × ∇ n P ⋅k
A
(3.20.12)
= TbΘ∇ n P × ∇ nΘ ⋅k = g N −2 H n ,
which simply says that the neutral helicity H n must be zero in order for the dianeutral
component of Eqn. (3.20.11) to hold, that is, ∇ n P × ∇ nΘ ⋅k must be zero. Here the
have been used.
equalities κ S = β PΘ and κ Θ = − α Θ
P
A
Writing ∇b as ∇ nb + bz n , Eqn. (3.20.11) becomes
(
)
g −1 N 2 ∇n ln b × ( −∇ n z + k ) = − Pz ∇p κ × −∇ p z + k ,
(
)
(
)
(3.20.13)
where ∇P = Pz −∇ p z + k has been used on the right-‐‑hand side, −∇ p z + k being the
normal to the isobaric surface. Concentrating on the horizontal components of this
equation, g −1 N 2 ∇n ln b = − Pz ∇p κ , and using the hydrostatic equation Pz = − g ρ gives
(
∇n ln b = ρ g 2 N −2 ∇p κ = − ρ g 2 N −2 α ΘP ∇p Θ − β PΘ∇p SA
(
)
)
= − ρ g 2 TbΘ N −2 ∇n Θ − gN −2 α ΘP Θ z − β PΘ SA z ∇n P.
(3.20.14)
The second line of this equation has used the geometric relationship
∇p Θ = ∇ nΘ − Θ z ∇ n P Pz and the corresponding equation for Absolute Salinity.
defined
by
Eqn.
(3.20.10),
that
is
The
integrating
factor
b
b ≡ −gN −2γ −1∇γ ⋅ n (n ⋅ n) = −gN −2γ −1∇γ ⋅ n (1 + ∇n z ⋅∇n z) , allows spatial integrals of
b ( β Θ∇SA − α Θ∇Θ) ≈ ∇ ln γ to be approximately independent of path for “vertical paths”,
that is, for paths in planes whose normal has zero vertical component.
By analogy with fN 2 , Neutral Surface Potential Vorticity ( NSPV ) is defined as −gγ −1
times q̂ = f γ z , so that NSPV = b fN 2 (from the vertical component of Eqn. (3.20.10)), so
that the ratio of NSPV to fN 2 is found by spatially integrating Eqn. (3.20.14) to be
NSPV
fN 2
=b =−
( )z = exp ⎧⎪−
g ln γ
N2
IOC Manuals and Guides No. 56
⎫⎪
⎧⎪
⎫⎪
ρ TbΘ
α ΘP Θ z − β PΘ SA z
∇
P
⋅
dl
⎨ ∫ans −2 2 ∇n Θ ⋅ dl ⎬ exp ⎨− ∫ans
⎬ . (3.20.15)
n
g N
g −1 N 2
⎩⎪
⎭⎪
⎩⎪
⎭⎪
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
45
The integral here is taken along an approximately neutral surface (such as a Neutral
Density surface) from a location where NSPV is equal to fN 2 .
The deficiencies of fN 2 as a form of planetary potential vorticity have not been widely
appreciated. Even in a lake, the use of fN 2 as planetary potential vorticity is inaccurate
since the right-‐‑hand side of (3.20.14) is then
− ρ g 2 N −2 α ΘP ∇p Θ = ρ g 2 N −2 α ΘP Θ z ∇Θ P Pz = −
α ΘP
∇Θ P ,
(3.20.16)
αΘ
where the geometrical relationship ∇p Θ = − Θ z ∇Θ P Pz has been used along with the
hydrostatic equation. The mere fact that the Conservative Temperature surfaces in a lake
have a slope (i. e. ∇Θ P ≠ 0 ) means that the spatial variation of contours of fN 2 will not be
the same as that of the contours of NSPV on a Θ surface in a lake.
In the situation where there is no gradient of Conservative Temperature along a
Neutral Density surface ( ∇γ Θ = 0 ) the contours of NSPV along the Neutral Density
surface coincide with those of isopycnal-‐‑potential-‐‑vorticity ( IPV ), the potential vorticity
defined with respect to the vertical gradient of potential density by IPV = − fg ρ −1ρ zΘ .
IPV is related to fN 2 by (McDougall (1988))
β Θ ( pr )
IPV
− g ρ −1ρ zΘ
≡
=
fN 2
N2
β Θ ( p)
⎡⎣ Rρ r − 1⎤⎦
β Θ ( pr ) 1
1
=
≈ Θ ,
Θ
Θ
⎡⎣ Rρ − 1⎤⎦
G
β ( p) G
(3.20.17)
so that the ratio of NSPV to IPV , evaluated on an approximately neutral surface, is
( )
( )
Θ
⎧⎪
⎫⎪
⎧⎪
⎫⎪
ρTΘ
α ΘΘ − β Θ S
NSPV β p ⎡⎣ Rρ −1⎤⎦
= Θ
exp ⎨− ∫ans −2 b 2 ∇n Θ ⋅ dl ⎬ exp ⎨− ∫ans P z −1 2P A z ∇n P ⋅ dl ⎬ . (3.20.18)
IPV
β pr ⎡ Rρ ⎤
g N
g N
⎪⎩
⎪⎭
⎪⎩
⎪⎭
⎢ −1⎥
⎢⎣ r
⎥⎦
You and McDougall (1991) show that because of the highly differentiated nature of
potential vorticity, isolines of IPV and NSPV do not coincide even at the reference
pressure pr of the potential density variable (see equations (14) – (16) and Figure 14 of
that paper). NSPV , fN 2 and IPV have the units s −3 . The ratio IPV fN 2 , evaluated
according to the middle expression in Eqn. (3.20.17), is available in the GSW
Oceanographic Toolbox as the function gsw_IPV_vs_fNsquared_ratio.
3.21 Vertical velocity through the sea surface
There has been confusion regarding the expression that relates the net evaporation at the
sea surface to the vertical velocity in the ocean through the sea surface. Since these
expressions have often involved the salinity (through the factor (1 − SA ) ) and so appear to
be thermodynamic expressions, here we present the correct equation which we will see is
merely kinematics, not thermodynamics. Let ρ W ( E − P ) be the vertical mass flux through
the air-‐‑sea interface on the atmospheric side of the interface (where ( E − P ) is the notional
vertical velocity of freshwater through the air-‐‑sea interface with density ρ W ; this density
being that of pure water at the sea surface temperature and at atmospheric pressure). The
same mass flux ρ W ( E − P ) must flow through the air-‐‑sea interface on the ocean side of
the interface where the density is ρ = ρ ( SA , t,0) . The vertical velocity through an
arbitrary surface whose height is z = η ( x, y, t ) can be expressed as w − v ⋅∇η − ∂η ∂t
(where w is the vertical velocity through the geopotential surface, see section 3.24, and
note that t is time in this context) and the mass flux associated with this dia-‐‑surface
vertical velocity component is this vertical velocity times the density of the seawater, ρ .
By equating the two mass fluxes on either side of the air-‐‑sea interface we arrive at the
vertical ocean velocity through the air-‐‑sea interface as (Griffies (2004), Warren (2009))
(
)
w − v ⋅∇η − ∂η ∂t = ρ −1ρ W E − P .
(3.21.1)
IOC Manuals and Guides No. 56
46
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
3.22 Freshwater content and freshwater flux
Oceanographers traditionally call the pure water fraction of seawater the “freshwater
fraction” or the “freshwater content”. This can cause confusion because in some science
circles “freshwater” is used to describe water of low but non-‐‑zero salinity. Nevertheless,
here we retain the oceanographic use of “freshwater” as being synonymous with pure
water (i. e. SA = 0 , this pure water being in liquid, gaseous or solid ice forms). The
freshwater content of seawater is (1 − SA ) = 1 − 0.001SA / (g kg −1 ) . The first expression
here clearly requires that Absolute Salinity is expressed in kg of sea salt per kg of solution.
Note that the freshwater content is not based on Practical Salinity, that is, it is not
(1− 0.001 SP ) .
The advective flux of mass per unit area is ρ u where u is the fluid velocity through
the chosen area element while the advective flux of sea salt is ρ SAu . The advective flux of
freshwater per unit area is the difference of these two mass fluxes, namely ρ (1 − SA ) u . As
outlined in section 2.5 and appendices A.4 and A.20, for water of anomalous composition
there are four types of absolute salinity that might be relevant to this discussion of
freshwater fluxes; Density Salinity SAdens ≡ SA , Solution Salinity S Asoln , Added-‐‑Mass Salinity
SAadd , and Preformed Salinity S* . Since Preformed Salinity is designed to be a conservative
variable with a zero flux air-‐‑sea boundary condition, probably the best form of freshwater
content, at least in the context of an ocean model, is (1 − S* ) = 1 − 0.001S* / (g kg −1 ) .
(
)
(
)
3.23 Heat transport
A flux of heat across the sea surface at a sea pressure of 0 dbar is identical to the flux of
potential enthalpy which in turn is exactly equal to c 0p times the flux of Conservative
Temperature Θ , where c 0p is given by (3.3.3). By contrast, the same heat flux across the
sea surface changes potential temperature θ in inverse proportion to c p ( SA ,θ , 0 ) and this
heat capacity varies by 5% at the sea surface, depending mainly on salinity.
The First Law of Thermodynamics, namely Eqn. (A.13.1) of appendix A.13, can be
approximated as
dΘ
S
ρ c0p
≈ − ∇ ⋅F R − ∇ ⋅FQ + ρε + hS ρ S A,
(3.23.1)
A
dt
with an error in Θ that is approximately one percent of the error incurred by treating
either c0p θ or c p ( SA ,θ , 0 ) θ as the “heat content” of seawater (see McDougall (2003) and
appendices A.13 and A.18). Equation (3.23.1) is exact at 0 dbar while at great depth in the
ocean the error with the approximation (3.23.1) is no larger than the neglect of the
dissipation of kinetic energy term ρε in this equation (see appendix A.21).
Because the left-‐‑hand side of the First Law of Thermodynamics, Eqn. (3.23.1), can be
written as density times the material derivative of c0p Θ it follows that Θ can be treated as
a conservative variable in the ocean and that c0p Θ is transported by advection and mixed
by turbulent epineutral and dianeutral diffusion as though it is the “heat content” of
seawater. For example, the advective meridional flux of “heat” is the area integral of
ρ vh0 = ρ vc0p Θ (here v is the northward velocity). The error in comparing this advective
meridional “heat flux” with the air-‐‑sea heat flux is approximately 1% of the error in so
interpreting the area integral of either ρ vc0pθ or ρ v c p ( S A , θ ,0 ) θ . Similarly, turbulent
diffusive fluxes of “heat” are accurately given by a turbulent diffusivity times the spatial
gradient of c0p Θ but are less accurately approximated by the same turbulent diffusivity
times the spatial gradient of c0pθ (see appendix A.14 for a discussion of this point).
Warren (1999, 2006) has argued that because enthalpy is unknown up to a linear
function of salinity, it is only possible to talk of a flux of “heat” through an ocean section if
the fluxes of mass and salt through the ocean section are both zero. This opinion seems to
be widely held, but it is incorrect. Because enthalpy is unknown and unknowable up to a
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
47
linear function of S A (i. e. up to the arbitrary function a1 + a3SA in terms of the constants
defined in Eqn. (2.6.2)), the left-‐‑hand side of Eqn. (3.23.1) is unknowable to the extent
S
a3ρ dSA dt . It is shown in appendix B that the terms −∇ ⋅FQ + hSA ρ S A on the right-‐‑hand
side of Eqn. (3.23.1) are also unknowable to the same extent so that the effect of a3 cancels
from Eqn. (3.23.1). Hence the fact that c0p Θ is unknowable up to a linear function of S A
does not affect the usefulness of h 0 or c0p Θ as measures of “heat content”. Similarly, the
difference between the meridional fluxes of c0p Θ across two latitudes is equal to the area-‐‑
integrated air-‐‑sea and geothermal heat fluxes between these latitudes (after allowing for
any unsteady accumulation of c0p Θ in the volume), irrespective of whether there are non-‐‑
zero fluxes of mass or salt across the sections. This powerful result follows directly from
the fact that c0p Θ is taken to be a conservative variable, obeying the simple conservation
statement Eqn. (3.23.1). This issue is discussed at greater length in section 6 of McDougall
(2003).
3.24 Geopotential
The geopotential Φ is the gravitational potential energy per unit mass with respect to the
height z = 0. Allowing the gravitational acceleration to be a function of z , Φ is given by
z
( )
Φ = ∫ g z ′ dz ′ .
0
(3.24.1)
If the gravitational acceleration is taken to be constant Φ is simply gz . Note that height
and Φ are negative quantities in the ocean since the sea surface (or the geoid) is taken as
the reference height and z is measured upward from this surface. In SI units Φ is
measured in J kg−1 = m2 s−2 . If the ocean is assumed to be in hydrostatic balance so that
Pz = − g ρ (or − g dz′ = v dP′ ) then the geopotential Eqn. (3.24.1) may be expressed as the
vertical pressure integral of the specific volume in the water column,
P
( )
Φ = Φ0 − ∫ v p′ dP′ ,
P0
(3.24.2)
where Φ0 is the value of the geopotential at zero sea pressure, that is, the gravitational
acceleration times the height of the free surface above the geoid. Note that the
gravitational acceleration has not been assumed to be constant in Eqn. (3.24.2).
3.25 Total energy
The total energy E is the sum of specific internal energy u, kinetic energy per unit mass
0.5 u ⋅ u ( where u is the three-‐‑dimensional velocity vector) and the geopotential Φ ,
E = u + Φ + 12 u ⋅ u .
(3.25.1)
Total energy E is not a function of only ( SA , t, p ) and so is not a thermodynamic
quantity.
3.26 Bernoulli function
The Bernoulli function is the sum of specific enthalpy h, kinetic energy per unit mass
0.5 u ⋅ u , and the geopotential Φ ,
B = h +Φ + 12 u ⋅ u.
(3.26.1)
Using the expression (3.2.1) that relates enthalpy and potential enthalpy, together with
Eqn. (3.24.2) for Φ , the Bernoulli function (3.26.1) may be written as
IOC Manuals and Guides No. 56
48
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
P
B = h0 + Φ 0 + 12 u ⋅ u − ∫ vˆ ( p′ ) − vˆ ( S A , Θ, p ′ ) dP′ .
(3.26.2)
P0
The pressure integral term here is a version of the dynamic height anomaly (3.27.1), this
time for a specific volume anomaly defined with respect to the Absolute Salinity and
Conservative Temperature (or equivalently, with respect to the Absolute Salinity and
potential temperature) of the seawater parcel in question at pressure P . This pressure
integral is equal to the Cunningham geostrophic streamfunction, Eqn. (3.29.2).
The Bernoulli function B is not a function of only ( SA , t, p ) and so is not a
thermodynamic quantity.
The Bernoulli function is dominated by the contribution of enthalpy h to (3.26.1) and
by the contribution of potential enthalpy h 0 to (3.26.2). The variation of kinetic energy or
the geopotential following a fluid parcel is typically several thousand times less than the
variation of enthalpy or potential enthalpy following the fluid motion.
The definition of specific volume anomaly given in Eqn. (3.7.3) has been used by
Saunders (1995) to write (3.26.2) as (with the dynamic height anomaly Ψ defined in
(3.27.1))
(
)
B = h0 + Φ0 + 12 u ⋅ u + Ψ + ∫ v̂ ( SA ,Θ, p′ ) − v̂ SSO ,0°C, p′ dP′
P
P0
(
) (
) (
u ⋅ u + Ψ − ĥ ( SSO ,0°C, p ) + ĥ ( SA ,Θ, p ) − c0pΘ ,
)
(
)
= h0 + Φ0 + 12 u ⋅ u + Ψ − ĥ SSO ,0°C, p + ĥ SSO ,0°C,0 + ĥ SA ,Θ, p − ĥ SA ,Θ,0 (3.26.3)
= h +Φ +
0
0
1
2
where the last line has used hˆ ( SSO ,0°C,0 ) = 0 and hˆ ( SA , Θ,0) = c0p Θ . The sum of the last
two terms in this equation, namely ĥ SA ,Θ, p − c0pΘ , is dynamic enthalpy.
(
)
3.27 Dynamic height anomaly
The dynamic height anomaly Ψ with respect to the sea surface is given by
P
(
)
Ψ = − ∫ δˆ SA ⎡⎣ p′ ⎤⎦ ,Θ ⎡⎣ p′ ⎤⎦ , p′ dP′, where δˆ ( SA , Θ, p ) = vˆ ( SA , Θ, p ) − vˆ ( S SO ,0°C, p ) . (3.27.1)
P0
This is the geostrophic streamfunction for the flow at pressure P with respect to the flow
at the sea surface and δˆ is the specific volume anomaly. Thus the two-‐‑dimensional
gradient of Ψ in the P pressure surface is simply related to the difference between the
horizontal geostrophic velocity v at P and at the sea surface v 0 according to
k × ∇P Ψ = fv − fv0 .
(3.27.2)
Dynamic height anomaly is also commonly called the “geopotential anomaly”. The
specific volume anomaly, δˆ in the vertical integral in Eqn. (3.27.1) can be replaced with
specific volume v̂ without affecting the isobaric gradient of the resulting streamfunction.
That is, this substitution does not affect Eqn. (3.27.2) because the additional term is a
function only of pressure. Traditionally it was important to use δˆ in preference to v̂ as it
was more accurate with computer code which worked with single-‐‑precision variables.
Since computers now regularly employ double-‐‑precision, this issue has been overcome
and consequently either δˆ or v̂ can be used in the integrand of Eqn. (3.27.1), so making it
either the “dynamic height anomaly” or the “dynamic height”. As in the case of Eqn.
(3.24.2), so also the dynamic height anomaly Eqn. (3.27.1) has not assumed that the
gravitational acceleration is constant and so Eqn. (3.27.2) applies even when the
gravitational acceleration is taken to vary in both the vertical and in the horizontal.
The dynamic height anomaly Ψ should be quoted in units of m2 s−2 . These are the
units in which the GSW Toolbox (appendix N) outputs dynamic height anomaly in the
function gsw_geo_strf_dyn_height(SA,CT,p,p_ref). When the last argument of this
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
49
function, p_ref, is other than zero, the function returns the dynamic height anomaly with
respect to a (deep) reference pressure p_ref, rather than with respect to P0 (i.e. zero dbar
sea pressure) as in Eqn. (3.27.1). In this case the lateral isobaric gradient of the
streamfunction represents the geostrophic velocity difference relative to the (deep) pref
pressure surface, that is,
(3.27.3)
k × ∇ P Ψ = fv − fv ref .
Note that the integration in Eqn. (3.27.1) of specific volume anomaly with pressure must
be done with pressure in Pa (not dbar ) in order to have the resultant isobaric gradient,
∇ P Ψ , in the usual units, being the product of the Coriolis parameter (units of s −1 ) and the
velocity (units of m s −1 ). The GSW function gsw_steric_height(SA,CT,p,p_ref) returns Ψ
divided by the constant gravitational acceleration g0 = 9.7963 ms−2 . Hence steric height
remains proportional to an exact geostrophic streamfunction but the spatial variation of
the gravitational acceleration ensures that it cannot be exactly equal to the height of an
isobaric surface above a geopotential surface.
3.28 Montgomery geostrophic streamfunction
The Montgomery “acceleration potential” (Montgomery, 1937) Ψ M defined by
(
)
P
(
)
Ψ M = P − P0 δˆ − ∫ δˆ SA ⎡⎣ p′ ⎤⎦ ,Θ ⎡⎣ p′ ⎤⎦ , p′ dP′ =
P0
( P − P0 )δˆ + Ψ
(3.28.1)
is the geostrophic streamfunction for the flow in the specific volume anomaly surface
δˆ ( SA , Θ, p ) = δˆ1 relative to the flow at P = P0 (that is, at p = 0 dbar ). Thus the two-‐‑
dimensional gradient of Ψ M in the δˆ specific volume anomaly surface is simply related
1
to the difference between the horizontal geostrophic velocity v in the δˆ = δˆ1 surface and
at the sea surface v 0 according to
k × ∇δˆ ΨM = fv − fv0 or ∇δˆ ΨM = − k × ( fv − fv0 ).
1
(3.28.2)
1
The definition, Eqn. (3.28.1), of the Montgomery geostrophic streamfunction applies to all
in the definition, Eqn. (3.7.3), of the specific
choices of the reference values SA and Θ
volume anomaly. By carefully choosing these reference values the specific volume
anomaly surface can be made to closely approximate the neutral tangent plane
(McDougall and Jackett (2007)).
It is not uncommon to read of authors using the Montgomery geostrophic
streamfunction, Eqn. (3.28.1), as a geostrophic streamfunction in surfaces other than
specific volume anomaly surfaces. This incurs errors that should be recognized. For
example, the gradient of the Montgomery geostrophic streamfunction, Eqn. (3.28.1), in a
neutral tangent plane becomes (instead of Eqn. (3.28.2) in the δˆ = δˆ1 surface)
∇ n Ψ M = − k × ( fv − fv 0 ) + ( P − P0 ) ∇ nδˆ ,
(3.28.3)
where the last term represents an error arising from using the Montgomery
streamfunction in a surface other than the surface for which it was derived.
Zhang and Hogg (1992) subtracted an arbitrary pressure offset, P − P0 , from
( P − P0 ) in the first term in Eqn. (3.28.1), so defining the modified Montgomery
streamfunction
(
(
)
Ψ Z-H = P − P δˆ − ∫ δˆ ( S A [ p ′] , Θ [ p ′] , p ′ ) dP′ .
P
)
(3.28.4)
P0
The gradient of Ψ Z-H in a neutral tangent plane becomes
(
)
∇n Ψ Z-H = − k × ( fv − fv 0 ) + P − P ∇ nδˆ ,
(3.28.5)
where the last term can be made significantly smaller than the corresponding term in Eqn.
(3.28.3) by choosing the constant pressure P to be close to the average pressure on the
IOC Manuals and Guides No. 56
50
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
surface. This term can be further minimized by suitably choosing the constant reference
in the definition, Eqn. (3.7.3), of specific volume anomaly δ so that this
values SA and Θ
surface more closely approximates the neutral tangent plane (McDougall (1989)). This
improvement is available because it can be shown that
(
)
( )
∇ P.
ρ ∇ nδ = − ⎡⎢κˆ ( SA ,Θ, p ) − κˆ SA , Θ,
p ⎤ ∇ n P ≈ TbΘ Θ − Θ
n
⎣
⎦⎥
(3.28.6)
The last term in Eqn. (3.28.5) is then approximately
( P − P ) ∇ nδ
( )
∇ ( P − P )2
(3.28.7)
ρ −1TbΘ Θ − Θ
n
can reduce the last term in Eqn. (3.28.5) that
and hence suitable choices of P , SA and Θ
represents the error in interpreting the Montgomery geostrophic streamfunction, Eqn.
(3.28.4), as the geostrophic streamfunction in a surface that is more neutral than a specific
volume anomaly surface.
The Montgomery geostrophic streamfunction should be quoted in units of m2 s−2 .
These are the units in which the GSW Toolbox outputs the Montgomery geostrophic
streamfunction in the function gsw_geo_strf_Montgomery(SA,CT,p,p_ref). When the last
argument of this function, p_ref, is other than zero, the function returns the Montgomery
geostrophic streamfunction with respect to a (deep) reference sea pressure p_ref, rather
than with respect to p = 0 dbar (i.e. P = P0 ) as in Eqn. (3.28.1).
≈
1
2
3.29 Cunningham geostrophic streamfunction
Cunningham (2000) and Alderson and Killworth (2005), following Saunders (1995) and
Killworth (1986), suggested that a suitable streamfunction on a density surface in a
compressible ocean would be the difference between the Bernoulli function B and
potential enthalpy h 0 . Since the kinetic energy per unit mass, 12 u ⋅ u , is a tiny component
of the Bernoulli function, it was ignored and Cunningham (2000) essentially proposed the
streamfunction Π + Φ 0 (see his equation (12)), where
Π ≡ B − h0 − 12 u ⋅ u − Φ 0
= h − h0 + Φ − Φ 0
(3.29.1)
P
= hˆ( SA , Θ, p) − hˆ( SA , Θ,0) − ∫ vˆ ( SA ( p′), Θ( p′), p′ ) dP′.
P0
The last line of this equation has used the hydrostatic equation Pz = − g ρ to express
Φ ≈ gz in terms of the vertical pressure integral of specific volume and the height of the
sea surface where the geopotential is Φ 0 . The difference between enthalpy and potential
enthalpy h − h0 in this equation has been named “dynamic enthalpy” by Young (2010).
The definition of potential enthalpy, Eqn. (3.2.1), is used to rewrite the last line of Eqn.
(3.29.1), showing that Cunningham’s Π is also equal to
P
(
) (
)
Π = − ∫ v̂ SA ( p′ ), Θ( p′ ), p′ − v̂ SA , Θ, p′ dP′
P0
(
)
(
)
= Ψ − ĥ SSO ,0°C, p + ĥ SA ,Θ, p −
c0pΘ
(3.29.2)
.
The first line of this equations appears very similar to the expression, Eqn. (3.27.1), for
dynamic height anomaly, the only difference being that in Eqn. (3.27.1) the pressure-‐‑
independent values of Absolute Salinity and Conservative Temperature were SSO and
0°C whereas here they are the local values on the surface, SA and Θ . While these local
values of Absolute Salinity and Conservative Temperature are constant during the
pressure integral in Eqn. (3.29.2), they do vary with latitude and longitude along any
“density” surface.
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
51
The gradient of Π along the neutral tangent plane is
∇n Π ≈
{
1
ρ
}
∇z P −∇Φ0 − 12 ρ −1TbΘ ( P − P0 ) ∇nΘ ,
2
(3.29.3)
(from McDougall and Klocker (2010)) so that the error in ∇n Π in using Π as the
2
geostrophic streamfunction is approximately − 12 ρ −1TbΘ ( P − P0 ) ∇n Θ . When using the
Cunningham streamfunction Π in a potential density surface, the error in ∇σ Π is
approximately − 12 ρ −1TbΘ ( P − P0 )( 2 Pr − P − P0 ) ∇σ Θ . The Cunningham geostrophic
streamfunction should be quoted in units of m2 s−2 and is available in the GSW
Oceanographic Toolbox as the function gsw_geo_strf_Cunningham(SA,CT,p,p_ref).
When the last argument of this function, p_ref, is other than zero, the function returns the
Cunningham geostrophic streamfunction with respect to a (deep) reference sea pressure
p_ref, rather than with respect to p = 0 dbar (i.e. P = P0 ) as in Eqn. (3.29.1).
3.30 Geostrophic streamfunction in an approximately neutral surface
In order to evaluate a relatively accurate expression for the geostrophic streamfunction in
an approximately neutral surface a suitable reference seawater parcel SA , Θ,
p is selected
from the approximately neutral surface that one is considering, and the specific volume
anomaly δ is defined as in (3.7.3) above. The approximate geostrophic streamfunction ϕ n
is given by (from McDougall and Klocker (2010))
(
ϕn =
1
2
( P − P )δ ( S ,Θ, p) −
A
1
12
(
)(
P − P
ρ −1TbΘ Θ − Θ
)
2
P
−
)
∫ δ dP′ .
(3.30.1)
P0
This expression is more accurate than the Montgomery and Cunningham geostrophic
streamfunctions when used in potential density surfaces, in the ω -‐‑surfaces of Klocker et
al. (2009a,b) and in the Neutral Density surfaces of Jackett and McDougall (1997). That is,
in these surfaces ∇nϕ n ≈ ρ1 ∇ z P −∇Φ 0 = − k × ( fv − fv 0 ) to a very good approximation. In
Eqn. (3.30.1) ρ −1TbΘ is taken to be the constant value 2.7 x10−15 K −1 (Pa) −2 m2s−2 . This
approximate isopycnal geostrophic streamfunction of McDougall and Klocker (2010) is
available as the function gsw_geo_strf_isopycnal in the GSW Toolbox. When the last
argument of this function, p_ref, is other than zero, the function returns the isopycnal
geostrophic streamfunction with respect to a (deep) reference sea pressure p_ref, rather
than with respect to the sea surface at p = 0 dbar (i.e. P = P0 ) as in Eqn. (3.30.1).
3.31 Pressure-‐‑integrated steric height
The depth-‐‑integrated mass flux of the geostrophic Eulerian flow between two fixed
pressure levels can also be represented by a streamfunction. Using the hydrostatic
relation Pz = − g ρ , and assuming the gravitational acceleration to be independent of
height, the depth-‐‑integrated mass flux ∫ ρ v dz is given by − g −1 ∫ v dP and this motivates
taking the pressure integral of the Dynamic Height Anomaly Ψ (from Eqn. (3.27.1)) to
form the Pressure-‐‑Integrated-‐‑Steric-‐‑Height PISH (also called Depth-‐‑Integrated Steric
Height DISH by Godfrey (1989)),
P
( )
P P′′
(
)
PISH = Ψ ′ = − g −1 ∫ Ψ p ′′ dP′′ = g −1 ∫ ∫ δˆ SA ⎡⎣ p ′ ⎤⎦ , Θ ⎡⎣ p ′ ⎤⎦ , p ′ dP′ dP′′
P0
P
(
) (
P0 P0
)
(3.31.1)
= g −1 ∫ P − P′ δˆ SA ⎡⎣ p ′ ⎤⎦ , Θ ⎡⎣ p ′ ⎤⎦ , p ′ dP′.
P0
The two-‐‑dimensional gradient of Ψ′ is related to the depth-‐‑integrated mass flux of the
velocity difference with respect to the velocity at zero sea pressure, v 0 , according to
IOC Manuals and Guides No. 56
52
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
z ( P0 )
P
z( P )
P0
k × ∇ p Ψ′ = f ∫ ρ ⎡⎣ v ( z′) − v0 ⎤⎦ dz′ = g −1 f ∫ ⎡⎣ v ( p′) − v0 ⎤⎦ dP′.
(3.31.2)
S
The definition, Eqn. (3.31.1), of
applies to all choices of the reference values
PISH
A , S A
in the definitions, Eqns. (3.7.2 – 3.7.4), of the specific volume anomaly.
and t , θ or Θ
Since the velocity at depth in the ocean is generally much smaller than at the sea
surface, it is customary to take the reference pressure to be some constant (deep) pressure
P1 so that Eqn. (3.27.1) becomes
(
P1
)
Ψ = ∫ δˆ SA ⎡⎣ p ′ ⎤⎦ ,Θ ⎡⎣ p ′ ⎤⎦ , p ′ dP′ ,
P
(3.31.3)
and PISH , reflecting the depth-‐‑integrated horizontal mass transport from the sea surface
to pressure P1 , relative to the flow at P1 , is
P1
(
P1 P1
( )
)
PISH = Ψ ′ = g −1 ∫ Ψ p ′′ dP′′ = g −1 ∫ ∫ δˆ SA ⎡⎣ p ′ ⎤⎦ ,Θ ⎡⎣ p ′ ⎤⎦ , p ′ dP′ dP′′
P0
P1
P0 P′′
) (
(
)
= g −1 ∫ P′ − P0 δˆ SA ⎡⎣ p ′ ⎤⎦ ,Θ ⎡⎣ p ′ ⎤⎦ , p ′ dP′
P0
=
1
2
g
−1
( P1 − P0 )2
∫
0
(
(3.31.4)
)
2
δˆ SA ⎡⎣ p ′ ⎤⎦ ,Θ ⎡⎣ p ′ ⎤⎦ , p ′ d ⎛ ( P′ − P0 ) ⎞ .
⎝
⎠
The two-‐‑dimensional gradient of Ψ′ is now related to the depth-‐‑integrated mass flux of
the velocity difference with respect to the velocity at P1 , v1, according to
z ( P0 )
P1
z ( P1 )
P0
k × ∇ p Ψ′ = f ∫ ρ ⎡⎣ v ( z′) − v1 ⎤⎦ dz′ = g −1 f ∫ ⎡⎣ v ( p′) − v1 ⎤⎦ dP′.
(3.31.5)
The specific volume anomaly δˆ in Eqns. (3.31.1), (3.31.3) and (3.31.4) can be replaced with
specific volume v without affecting the isobaric gradient of the resulting streamfunction.
That is, this substitution in Ψ′ does not affect Eqn. (3.31.2) or Eqn. (3.31.5), as the
additional term is a function only of pressure. With specific volume in place of specific
volume anomaly, Eqn. (3.31.4) becomes the depth-‐‑integrated gravitational potential
energy of the water column (plus a very small term that is present because the
atmospheric pressure is not zero, McDougall et al. (2003)).
PISH should be quoted in units of kg s−2 so that its two-‐‑dimensional gradient has the
same units as the depth-‐‑integrated flux of ρ ⎡⎣ v ( z′) − v1 ⎤⎦ times the Coriolis frequency.
3.32 Pressure to height conversion
The vertical integral of the hydrostatic equation ( g = − v Pz ) can be written as
0
0
∫ g ( z ′ ) dz ′ = Φ − ∫ v ( p′ ) dP′ = − ∫ v̂ ( SSO ,0°C, p′ ) dP′ + Ψ + Φ
z
P
P
0
P0
P0
(
)
(3.32.1)
= − ĥ SSO ,0°C, p + Ψ + Φ ,
0
where the dynamic height anomaly Ψ is expressed in terms of the specific volume
anomaly δˆ = vˆ ( SA , Θ, p ) − vˆ ( SSO ,0°C, p ) by
P
Ψ = − ∫ δˆ ( p′ ) dP′ ,
(3.32.2)
P0
where P0 = 101 325Pa is the standard atmosphere pressure. Writing the gravitational
acceleration of Eqn. (D.3) as g = g (φ , z ) = g (φ ,0) (1 − γ z ) , the left-‐‑hand side of Eqn.
(3.32.1) becomes g (φ ,0 ) z − 12 γ z 2 , and using the 75-‐‑term expression for the specific
enthalpy of Standard Seawater, Eqn. (3.32.1) becomes
(
(
)
)
( )(
)
ĥ75 SSO , 0°C, p − Ψ − Φ0 + g φ ,0 z − 12 γ z 2 = 0 .
IOC Manuals and Guides No. 56
(3.32.3)
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
53
This is the equation that is solved for height z in the GSW function gsw_z_from_p. It is
traditional to ignore Ψ + Φ0 when converting between pressure and height, and this can
be done by simply calling this function with only two arguments, as in
gsw_z_from_p(p,lat). Ignoring Ψ + Φ0 makes a difference to z of up to 4m at 5000
dbar. Note that height z is negative in the ocean. When the code is called with three
arguments, the third argument is taken to be Ψ + Φ0 and this is used in the solution of
Eqn. (3.32.3). Dynamic height anomaly Ψ can be evaluated using the GSW function
gsw_geo_strf_dyn_height. The GSW function gsw_p_from_z is the exact inverse
function of gsw_z_from_p; these functions yield outputs that are consistent with each
other to machine precision.
When vertically integrating the hydrostatic equation Pz = − g ρ in the context of an
ocean model where Absolute Salinity S A and Conservative Temperature Θ are
piecewise constant in the vertical, the geopotential (Eqn. (3.24.2))
z
( )
P
( )
Φ = ∫ g z ′ dz ′ = Φ0 − ∫ v p′ dP′ ,
0
P0
(3.32.4)
can be evaluated as a series of exact differences. If there are a series of layers of index i
separated by pressures p i and p i +1 (with pi +1 > pi ) then the integral can be expressed
(making use of (3.7.5), namely hP
= hˆP = v ) as a sum over n layers of the differences
SA , Θ
in specific enthalpy so that
P
( )
n
(
) (
)
Φ = Φ0 − ∫ v p′ dP′ = Φ0 − ∑ ⎡ ĥ SAi ,Θ i , p i+1 − ĥ SAi ,Θ i , p i ⎤ .
⎦
i =1 ⎣
P
0
(3.32.5)
The difference in enthalpy at two different pressures for given values of S A and Θ is
available in the GSW Oceanographic Toolbox via the function gsw_enthalpy_diff. The
summation of a series of such differences in enthalpy occurs in the GSW functions to
evaluate two geostrophic streamfunctions from piecewise-‐‑constant vertical property
profiles, gsw_geo_strf_dyn_height_pc and gsw_geo_strf_isopycnal_pc.
3.33 Freezing temperature
Freezing occurs at the temperature tf at which the chemical potential of water in seawater
µ W equals the chemical potential of ice µ Ih . Thus, tf is found by solving the implicit
equation
µ W ( SA , tf , p ) = µ Ih ( tf , p )
(3.33.1)
or equivalently, in terms of the two Gibbs functions,
g ( SA , tf , p ) − SA gSA ( SA , tf , p ) = g Ih ( tf , p ) .
(3.33.2)
The Gibbs function for ice Ih, g ( t , p ) , is defined by IAPWS-‐‑06 (IAPWS (2009a)) and
Feistel and Wagner (2006) and is summarized in appendix I below. In the special case of
zero salinity, the chemical potential of water in seawater reduces to the Gibbs function of
pure water, µ W ( 0, t , p ) = g W ( t , p ) . A simple correlation function for the melting pressure
as a function of temperature is available from IAPWS (2008b) and has been implemented
in the SIA library.
At the ocean surface, p = 0 dbar, from Eqn. (3.33.1) the TEOS-‐‑10 freezing point of pure
water is tf 0g kg −1 , 0dbar = 0.002 519 °C with an uncertainty of only 2 µK , noting that the
triple point temperature of water is exactly 273.16 K by definition of the ITS-‐‑90
temperature scale. The freezing temperature of the standard ocean is tf ( SSO , 0dbar ) =
-‐‑1.919 °C with an uncertainty of 2 mK. Note that Eqn. (3.33.1) is valid for air-‐‑free
water/seawater. Dissolution of air in water lowers the freezing point slightly; saturation
Ih
(
)
IOC Manuals and Guides No. 56
54
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
with air lowers the freezing temperatures by about 2.4 mK for pure water and about 1.9
mK for seawater with an Absolute Salinity of SA = SSO = 35.16504 g kg −1 .
To estimate the effects of small changes in the pressure or salinity on the freezing
temperature, it is convenient to consider a power series expansion of (3.33.1). The result in
the limit of an infinitesimal pressure change at fixed salinity gives the pressure coefficient
of freezing point lowering, as (Clausius-‐‑Clapeyron equation, Feistel et al. (2010a)),
g P − SA g PS − g PIh
∂tf
A
= χ P SA , p = −
.
∂P S
gT − SA g S T − gTIh
(
)
(3.33.3)
A
A
Its values, evaluated from TEOS-‐‑10, vary only weakly with salinity between
χ p 0g kg −1, 0dbar = –0.7429 mK/dbar for pure water and χ p ( SSO , 0dbar ) = –0.7483
mK/dbar for the standard ocean. TEOS-‐‑10 is consistent with the most accurate
measurement of χ p and its experimental uncertainty of 0.0015 mK/dbar (Feistel and
Wagner (2005), (2006)). Since the value of χ p always exceeds that of the adiabatic lapse
rate Γ , cold seawater may freeze and decompose into ice and brine during adiabatic
uplift but this can never happen to a sinking parcel.
In the limit of infinitesimal changes in Absolute Salinity at fixed pressure, we obtain
the saline coefficient of freezing point lowering, as (Raoult’s law),
(
)
∂tf
∂SA
= χ S ( SA , p ) =
p
(
S A g SA SA
gT − SA g SAT − gTIh
.
(3.33.4)
)
Typical numerical values are χ S 0g kg −1 , 0dbar = –59.2 mK/(g kg −1 ) for pure water and
χ S ( SSO , 0dbar ) = –56.9 mK/(g kg −1 ) for seawater.
As a raw practical estimate, Eqn. (3.33.4) can be expanded into powers of salinity,
using only the leading term of the TEOS-‐‑10 saline Gibbs function, g S ≈ RSTSA ln SA , which
stems from Planck’s ideal-‐‑solution theory (Planck (1888)). Here, RS = R M S = 264.7599
J kg–1 K–1 is the specific gas constant of sea salt, R is the universal molar gas constant, and
M S = 31.403 82 g mol–1 is the molar mass of sea salt with Reference Composition. The
denominator of Eqn. (3.33.4) is proportional to the melting heat LSIp , Eqn. (3.34.7). The
convenient result obtained with these simplifications is
∂tf
∂SA
≈ −
p
RS
2
T + tf ) ≈ − 59 mK/(g kg −1 ) .
SI ( 0
Lp
(3.33.5)
where we have used tf = −2 oC and LSIp = 330 J kg −1 as appropriate approximations for the
standard ocean. This simple result is only weakly dependent on these choices and is in
reasonable agreement with the exact values from Eqn. (3.33.4) and with Millero and Leung
(1976). The freezing temperature of seawater is always lower than that of pure water.
When sea-‐‑ice is formed, it often contains remnants of seawater included in brine
pockets. At equilibrium, the salinity in these pockets depends only on temperature and
pressure, rather than, for example, on the pocket volume, and can be computed in the
functional form SA ( t, p ) as an implicit solution of Eqn. (3.33.1). Measured values for the
brine salinity of Antarctic sea ice agree very well with those computed of Eqn. (3.33.1) up
to the saturation concentration of about 110 g kg −1 at surface pressure (Feistel et al.
(2010b)). At high pressures, the validity of the Gibbs function of seawater, and therefore
of the computed freezing point or brine salinity, too, is limited to only 50 g kg −1 .
We note that in the first approximation, as inferred from Planck’s theory of ideal
solutions, the above properties depend on the number of dissolved particles regardless of
the particle sizes, masses or charges. In other words, they depend mainly on the molar
rather than on the mass density of the solute, in contrast to properties such as the specific
volume of seawater and properties derived from it. The properties considered in this and
the following sections (sections 3.33 – 3.42) which share this attribute are referred to as the
colligative properties of seawater.
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
55
3.34 Latent heat of melting
The melting process of ice in pure water can be conducted by supplying heat at constant
pressure. If this is done slowly enough that equilibrium is maintained, then the
temperature will also remain constant. The heat required per mass of molten ice is the
latent heat, or enthalpy, of melting, LWI
p . It is found as the difference between the specific
enthalpy of water, h W , and the specific enthalpy of ice, hIh , (Kirchhoff’s law, Curry and
Webster (1999)):
W
Ih
LWI
p ( p ) = h (tf , p ) − h (tf , p ) .
(3.34.1)
Here, tf ( p ) is the freezing temperature of water, section 3.33. The enthalpies h W and h Ih
are available from IAPWS-‐‑95 (IAPWS (2009b)) and IAPWS-‐‑06 (IAPWS (2009a)),
respectively.
In the case of seawater, the melt water will additionally mix with the ambient brine,
thus changing the salinity and the freezing temperature of the seawater. Consequently,
the enthalpy related to this phase transition will depend on the particular conditions
under which the melting occurs.
Here, we define the latent heat of melting as the enthalpy increase per infinitesimal
mass of molten ice of a composite system consisting of ice and seawater, when the
temperature is increased at constant pressure and at constant total masses of water and
salt, in excess of the heat needed to warm up the seawater and ice phases individually
(Feistel and Hagen (1998), Feistel et al. (2010b)). Mass conservation of both water and salt
during this thermodynamic process is essential to ensure the independence of the latent
heat formula from the unknown absolute enthalpies of salt and water that otherwise
would accompany any mass exchange.
The enthalpy of sea ice, hSI , is additive with respect to its constituents ice, hIh , with
the mass fraction wIh , and seawater, h, with the liquid mass fraction 1 − wIh :
(
(
)
hSI = 1 − wIh h ( SA , t, p ) + wIh h Ih (t , p ) .
)
(3.34.2)
Upon warming, the mass of melt water changes the ice fraction w Ih and the brine salinity
SA . The related temperature derivative of Eqn. (3.34.2) is
∂hSI
∂T
(
= 1 − wIh
p
)
∂h
∂T
(
+ 1 − wIh
SA , p
)
∂h
∂SA
T,p
∂SA
∂T
+ w Ih
p
∂h Ih
∂T
(
+ h Ih − h
p
)
∂w Ih
.
∂T p
(3.34.3)
The rate of brine salinity change with temperature is given by the reciprocal of Eqn.
(3.33.4) and is related to the isobaric melting rate, −∂wIh / ∂T , by the conservation of the
p
total salt, 1 − wIh SA = const, in the form
(
)
∂SA
∂T
=
p
SA ∂wIh
.
1 − wIh ∂T p
(3.34.4)
Using this relation, Eqn. (3.34.3) takes the simplified form
∂hSI
∂T
(
)
SI
= 1 − wIh c p + w Ih c Ih
p − Lp
p
∂w Ih
.
∂T p
(3.34.5)
The coefficient in front of the melting rate,
LSIp ( SA , p ) = h − S A
∂h
∂SA
− h Ih ,
(3.34.6)
T,p
provides the desired expression for isobaric melting enthalpy, namely the difference
between the partial specific enthalpies of water in seawater and of ice. As is physically
required for any measurable thermodynamic quantity, the arbitrary absolute enthalpies of
ice, water and salt cancel in the formula (3.34.6), provided that the reference state
IOC Manuals and Guides No. 56
56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
conditions for the ice and seawater formulations are chosen consistently (Feistel et al.
(2008a)). Note that because of h = g + (T0 + t )η and Eqn. (3.33.2), the latent heat can also
be written in terms of entropies η rather than enthalpies h, in the form
⎛
⎞
∂η
(3.34.7)
− η Ih ⎟ .
LSIp ( SA , p ) = (T0 + tf ) × ⎜η − SA
⎜
⎟
∂SA T , p
⎝
⎠
Again the result is independent of unknown (and unknowable) constants.
The latent heat of melting depends only weakly on salinity and on pressure. At the
–1
surface pressure, the computed value is LSIp (0,0) = LWI
p (0) = 333 426.5 J kg for pure water,
SI
–1
and Lp ( SSO ,0) = 329 928.5 J kg for the standard ocean, with a difference of about 1% due
to the dissolved salt. At a pressure of 1000 dbar, these values reduce by 0.6% to
SI
–1
–1
LSIp (0,1000dbar ) = LWI
p ( 1000dbar ) = 331 528 J kg and Lp ( SSO ,1000dbar ) = 328 034 J kg .
WI
TEOS-‐‑10 is consistent with the most accurate measurements of Lp and their experimental
uncertainties of 200 J kg–1, or 0.06% (Feistel and Wagner (2005), (2006)).
3.35 Sublimation pressure
The sublimation pressure of ice Psubl is defined as the Absolute Pressure P of water
vapour in equilibrium with ice at a given temperature t, at or below the freezing
temperature. It is found by equating the chemical potential of water vapour µ V with the
chemical potential of ice µ Ih , so it is found by solving the implicit equation
(
)
(
)
µ V t, Psubl = µ Ih t, Psubl ,
(3.35.1)
or equivalently, in terms of the two Gibbs functions,
(
)
(
)
g V t , Psubl = g Ih t , Psubl .
(3.35.2)
The Gibbs function for ice Ih, g Ih ( t , P ) is defined by IAPWS-‐‑06 and Feistel and Wagner
(2006) and is summarized in appendix I below. Note that here the Absolute Pressure P
rather than the sea pressure p is used because the sublimation pressure of ice at ambient
conditions is much lower than the atmospheric pressure.
The Gibbs function of vapour, g V ( t , P ) , is available from the Helmholtz function of
fluid water, as defined by IAPWS-‐‑95; for details see for example Feistel et al. (2008a),
(2010a), (2010b). The highest possible sublimation pressure is found at the triple point of
water. The TEOS-‐‑10 value of the maximum sublimation pressure (i.e., the triple point
pressure) computed from Eqn. (3.35.1) is Psubl = Pt = 611.655 Pa and has an uncertainty of
0.01 Pa (IAPWS-‐‑06, Feistel et al. (2008a)).
Reliable theoretical values for the sublimation pressure are available down to 20 K
(Feistel and Wagner (2007)); a simple correlation function for the sublimation pressure
down to 50 K is provided by IAPWS (2008b) and is included as a function in the SIA
library. The IAPWS-‐‑95 function µ V required for Eqn. (3.35.1) is only valid above 130 K.
An extension to 50 K was developed for TEOS-‐‑10 (Feistel et al. (2010a)) and is available as
the default option in the SIA library. In nature, vapour cannot reasonably be expected to
exist below 50 K since it has extremely low density, even in the interstellar vacuum. For
this reason, the ice of comets does not evaporate far from the sun. The lowest
temperatures estimated for the terrestrial polar atmosphere do not go below 130 K.
In the presence of air, ice is under higher total pressure than just its own sublimation
pressure. The partial pressure of vapour in humid air, P vap = xV P , is computed from the
total Absolute Pressure P and the mole fraction of vapour, xV . Similar to the Absolute
Salinity S A of seawater, the variable A describes the mass fraction of dry air present in
humid air. Given A, the mole fraction of vapour is computed from
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
xV =
1− A
,
1 − A (1 − M W / M A )
57
(3.35.3)
where M A is the molar mass of dry air and M W is the molar mass of water.
The sublimation pressure, P subl ( t , P ) = xVsat P , of ice in equilibrium with humid air is the
partial pressure of vapour in saturated air. To compute xVsat from Eqn. (3.35.3), the
required air fraction at saturation, A = Asat ( t , P ) , is found by equating the chemical
AV
potential of water vapour in humid air µW
with the chemical potential of ice µ Ih , so that
it is found by solving the implicit equation
(
)
AV
µW
Asat , t, P = µ Ih (t, P ) ,
or equivalently, in terms of the two Gibbs functions,
(
)
(
(3.35.4)
)
g AV Asat , t, P − Asat g AAV Asat , t , P = g Ih (t , P ) .
(3.35.5)
The Gibbs function of humid air, g AV ( A, t , P ) , is defined by Feistel et al. (2010a).
At t = 0 °C and atmospheric pressure, the sublimation pressure of ice has the value
Psubl (0 °C, 101 325 Pa) = 613.745 Pa, computed by solving Eqn. (3.35.4) for Asat , then using
(3.35.3) to determine the corresponding mole fraction and multiplying the atmospheric
pressure by this quantity. Similarly, at the freezing point of the standard ocean the
sublimation pressure is Psubl (-‐‑1.919 °C, 101 325 Pa) = 523.436 Pa.
The difference between observed or modelled partial vapour pressures and the
sublimation pressure computed from TEOS-‐‑10 is an appropriate quantity for use in
parameterizations of the mass flux between ice and the atmosphere.
3.36 Sublimation enthalpy
The sublimation process that occurs when ice is in contact with pure water vapour can be
conducted by supplying heat at constant t and P, with t at or below the freezing
temperature. The heat required per mass evaporated from the ice is the latent heat, or
enthalpy, of sublimation, LVI
p . It is found as the difference between the specific enthalpy
of water vapour, h V , and the specific enthalpy of ice, h Ih :
(
)
(
)
V
LVI
t , Psubl − h Ih t , Psubl .
p (t ) = h
(3.36.1)
Here, P subl ( t ) is the sublimation pressure of ice at the temperature t , section 3.35. The
enthalpies h V and h Ih are available from IAPWS-‐‑95 and IAPWS-‐‑06, respectively. Reliable
values for the sublimation enthalpy are theoretically available down to 20 K from a simple
correlation function (Feistel and Wagner (2007)). At the triple point of water, the TEOS-‐‑10
–1
–1
sublimation enthalpy is LVI
p ( 0.01°C) = 2 834 359 J kg with an uncertainty of 1000 J kg , or
0.03%.
In the case when air is present, the vapour resulting from the sublimation will add to
the gas phase, thus increasing the mole fraction of vapour xVsat . If for example the total
pressure P is held constant, the partial pressure xVsat P will rise, and the ice must get
warmer to maintain equilibrium at the modified sublimation pressure Psubl = xVsat P.
Consequently, the enthalpy related to this phase transition will depend on the particular
conditions under which the sublimation process occurs. These effects are small under
ambient conditions but may be relevant at higher air densities.
Here, we define the latent heat of sublimation as the enthalpy increase per
infinitesimal mass of sublimated ice of a composite system consisting of ice and humid air,
when the temperature is increased at constant pressure and at constant total masses of
water and dry air, in excess of the enthalpy increase needed to warm up the ice and humid
air phases individually (Feistel et al. (2010a)). Mass conservation of both total water and
dry air during this thermodynamic process is essential to ensure the independence of the
IOC Manuals and Guides No. 56
58
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
latent heat formula from the unknown absolute enthalpies of air and water that otherwise
would accompany any mass exchange.
The enthalpy of ice air, h AI , is additive with respect to its constituents ice, hIh , with
the mass fraction wIh , and humid air, h AV , with the gas fraction 1 − wIh :
h
AI
(
= 1− w
Ih
)h
AV
( A, t, p )
+ w h
Ih Ih
(
(t, p ) .
)
(3.36.2)
Upon warming, the mass of vapour produced by sublimation reduces the ice fraction w Ih
and increases the humidity, that is, decreases the relative dry-‐‑air fraction A of the gas
phase. The related temperature derivative of Eqn. (3.36.2) is
∂h AI
∂T
(
= 1 − wIh
p
)
∂h AV
∂T
(
+ 1 − wIh
)
A, p
∂h AV
∂A
T,p
∂A
∂T
+ wIh
p
∂h Ih
∂T
(
+ h Ih − h AV
p
)
∂w Ih
. (3.36.3)
∂T p
The air-‐‑fraction change is related to the isobaric sublimation rate, −∂wIh / ∂T , by the
p
conservation of the dry air, 1 − w Ih A = const, in the form
(
)
∂A
∂T
=
p
A ∂w Ih
.
1 − w Ih ∂T p
(3.36.4)
Using this relation, Eqn. (3.36.3) takes the simple form
∂h AI
∂T
(
)
Ih Ih
AI
= 1 − wIh c AV
p + w c p − Lp
p
∂wIh
.
∂T p
(3.36.5)
The coefficient in front of the sublimation rate,
AV
LAI
−A
p ( A, p ) = h
∂h AV
∂A
− h Ih ,
(3.36.6)
T,p
provides the desired expression for isobaric sublimation enthalpy, namely the difference
between the partial specific enthalpies of vapour in humid air and of ice. In the ideal-‐‑gas
approximations for air and for vapour, the partial specific enthalpy of vapour in humid
air, h AV − AhAAV , equals the specific enthalpy of vapour, h V ( t ) , as a function of only the
temperature, independent of the pressure and of the presence of air (Feistel et al. (2010a)).
In this case, Eqn. (3.36.6) coincides formally with Eqn. (3.36.1), except that the two are
evaluated at the different pressures P and Psubl , respectively. As is physically required
for any measurable thermodynamic quantity, the arbitrary absolute enthalpies of ice,
vapour and air cancel in the formula (3.36.6), provided that the reference state conditions
for the ice and humid air formulations are chosen consistently (Feistel et al. (2008a),
(2010a)). The latent heat of sublimation depends only weakly on the air fraction and on
the pressure.
For saturated air over sea ice, the air fraction A = Asat can be computed from the brine
salinity, or from the sea surface salinity in the case of floating ice, section 3.38. At the
absolute surface pressure PSO = 101325 Pa and the freezing point tf = -‐‑1.919 °C of the
standard ocean, the TEOS-‐‑10 value for saturated air with ASO = Asat ( tf , PSO ) = 0.996 78 is
subl
–1
LAI
( tf , PSO ) = 523.436
p ( ASO , PSO ) = 2 833 006 J kg . The related sublimation pressure is P
Pa, see section 3.35.
Observational data show that the ambient air over the ocean surface is sub-‐‑saturated
in the climatological mean. Rather than being saturated, values for A that correspond to a
relative humidity of 75% – 82% (see section 3.40) may be a more realistic estimate for the
marine atmosphere (Dai (2006)); these values represent non-‐‑equilibrium conditions that
result in net evaporation as part of the global hydrological cycle.
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
59
3.37 Vapour pressure
The vapour pressure of seawater P vap ( S A , t ) is defined as the Absolute Pressure P of
water vapour in equilibrium with seawater at a given temperature t and salinity SA . It is
found by equating the chemical potential of vapour µ V with the chemical potential of
water in seawater µ W so that it is found by solving the implicit equation
(
)
(
)
µ V t, P vap = µ W SA , t, P vap ,
or equivalently, in terms of the two Gibbs functions,
(
) (
)
(3.37.1)
(
)
g V t, P vap = g SA , t , P vap − SA g SA SA , t , P vap .
(3.37.2)
Note that here we use the Absolute Pressure P rather than the sea pressure p ; since the
vapour pressure of water at ambient conditions is much lower than the atmospheric
pressure, the corresponding sea pressure (Pvap – 101325 Pa) would be negative and near
-‐‑105 Pa. The Gibbs functions of vapour and seawater, g V ( t , P ) and g ( SA , t, P), are
available from the Helmholtz function of fluid water, as defined by IAPWS-‐‑95, and the
Gibbs function of seawater, IAPWS-‐‑08.
In the case of pure water, SA = 0, the solution of Eqn. (3.37.1) is the so-‐‑called
saturation curve in the t − P diagram of water, which connects the triple point with the
critical point. The lowest possible vapour pressure of pure liquid water is found at the
triple point of water. The TEOS-‐‑10 value of this minimum vapour pressure, computed
from Eqn. (3.37.1), is P vap (0, 0.01 °C) = Pt = 611.655 Pa with an uncertainty of 0.01 Pa
(IAPWS-‐‑95, Feistel et al. (2008a)). For comparison, the vapour pressure of the standard
ocean is P vap ( SSO , 0 °C) = 599.907 Pa. At laboratory temperature the related values are
P vap (0, 25 °C) = 3169.93 Pa and P vap ( SSO , 25 °C) = 3110.57 Pa.
The relatively small vapour pressure lowering caused by the presence of dissolved salt
can be computed from the isothermal salinity derivative of Eqn. (3.37.1) in the form
(Raoult’s law)
S A g SA SA
∂P vap
.
(3.37.3)
=
∂SA T
g P − SA g SA P − g PV
As a raw practical estimate, this equation can be expanded into powers of salinity, using
only the leading term of the TEOS-‐‑10 saline Gibbs function, g S ≈ RSTSA ln SA , which stems
from Planck’s ideal-‐‑solution theory. Here, RS = R M S = 264.7599 J kg–1 K–1 is the specific
gas constant of sea salt, R is the universal molar gas constant, and M S = 31.403 82 g mol–1
is the molar mass of sea salt with Reference Composition. The specific volume of
seawater, g p , is neglected in comparison to that of vapour. The latter is approximately
considered as an ideal gas, g Vp ≈ RT / M W P vap , where M W = 18.015 268 g mol–1 is the
molar mass of water. The convenient result obtained with these simplifications is
(
∂P vap
∂SA
≈ −
T
)
M W vap
P
≈ − 0.57 × P vap .
MS
(3.37.4)
The vapour pressure of seawater is always lower than that of pure water.
In the presence of air, seawater is under a higher pressure P than under its vapour
pressure P vap . In this case, the vapour pressure of seawater P vap ( SA , t , P ) is defined as the
partial pressure of water vapour in humid air that is in equilibrium with seawater at a
given pressure P, temperature t and salinity SA . It is found by equating the chemical
V
potential of vapour in humid air µAV
with the chemical potential of water in seawater
W
so that it is found by solving the implicit equation
µ
(
)
V
µAV
Acond , t, P = µ W ( SA , t , P )
(3.37.5)
for Acond ( S A , t , P ) , or equivalently, in terms of the two Gibbs functions,
IOC Manuals and Guides No. 56
60
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
(
)
(
)
g AV Acond , t, P − Acond g AAV Acond , t , P = g ( SA , t , P ) − SA g SA ( SA , t , P ) .
(3.37.6)
Since the vapour pressure is lowered in the presence of sea salt (Eqn. (3.37.4)), at vapour
pressures above the condensation point vapour condenses out of the air at the sea surface,
even before the saturation point (that is, relative humidity of 100%) is reached, to maintain
local equilibrium with the seawater. The larger scale equilibration process may involve
downward diffusion of water vapour to the sea surface rather than precipitation of dew or
fog. From the calculated sub-‐‑saturated air fraction of the condensation point, Acond , the
mole fraction of vapour xVcond (3.35.3), and in turn the vapour pressure
P vap ( SA , t , P ) = xVcond P are available from straightforward calculations. The Gibbs function
of humid air g AV is available from Feistel et al. (2010a) and also as the IAPWS Guideline,
IAPWS-‐‑10 (IAPWS (2010)).
The TEOS-‐‑10 value computed from Eqn. (3.37.5) is P vap (0, 0 °C, PSO) = 613.760 Pa for
pure water at surface air pressure; the vapour pressure of the standard ocean is
P vap ( SSO , 0 °C, PSO ) = 602.403 Pa. At laboratory temperature the related values are
P vap (0, 25 °C, PSO ) = 3183.73 Pa and P vap ( SSO , 25 °C, PSO ) = 3124.03 Pa.
3.38 Boiling temperature
The boiling temperature of water or seawater is defined as the temperature t boil ( S A , P ) at
which the vapour pressure (of section 3.37) equals a given pressure P. It is found by
equating the chemical potential of vapour µ V with the chemical potential of water in
seawater µ W so that it is found by solving the implicit equation
(
)
(
)
µ V t boil , P = µ W SA , t boil , P ,
(3.38.1)
for t boil ( SA , P) , or equivalently in terms of the two Gibbs functions,
(
g V t boil , P
)
(
)
(
)
= g SA , t boil , P − SA g SA SA , t boil , P .
(3.38.2)
The TEOS-‐‑10 boiling temperature of pure water at atmospheric pressure is t boil ( 0, PSO ) =
99.974 °C. This temperature is outside the validity range of up to 80 °C of the TEOS-‐‑10
Gibbs function for seawater.
3.39 Latent heat of evaporation
The evaporation process of pure liquid water in contact with pure water vapour can be
conducted by supplying heat at constant t and P. The heat required per mass evaporated
from the liquid is the latent heat, or enthalpy, of evaporation, LVW
p . It is found as the
difference between the specific enthalpy of water vapour, h V , and the specific enthalpy of
liquid water, h W :
(
)
(
)
V
LVW
t , P vap − h W t , P vap .
p (t ) = h
(3.39.1)
Here, P (t ) is the vapour pressure of water at the temperature t (section 3.37). The
enthalpies h V and h W are available from IAPWS-‐‑95. At the triple point of water, the
–1
TEOS-‐‑10 evaporation enthalpy is LVW
p ( 0.01°C ) = 2 500 915 J kg .
In the case of seawater in contact with air, the vapour resulting from the evaporation
will add to the gas phase, thus increasing the mole fraction of vapour, while the liquid
water loss will increase the brine salinity, and cause a change to the seawater enthalpy.
Consequently, the enthalpy related to this phase transition will depend on the particular
conditions under which the evaporation process occurs.
Here, we define the latent heat of evaporation as the enthalpy increase per
infinitesimal mass of evaporated water of a composite system consisting of seawater and
humid air, when the temperature is increased at constant pressure and at constant total
vap
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
61
masses of water, salt and dry air, in excess of the enthalpy increase needed to warm up the
seawater and humid air phases individually (Feistel et al. (2010a)). Mass conservation
during this thermodynamic process is essential to ensure the independence of the latent
heat formula from the unknown absolute enthalpies of air, salt and water that otherwise
would accompany any mass exchange.
The enthalpy of sea air, hSA , is additive with respect to its constituents, seawater, h,
with the mass fraction wSW , and humid air, h AV , with the gas fraction 1 − wSW :
(
(
)
hSA = 1 − wSW h AV ( A, t, p ) + wSW h ( SA , t , p ) .
)
(3.39.2)
Upon warming, the mass of water transferred from the liquid to the gas phase by
evaporation reduces the seawater mass fraction wSW , increases the brine salinity S A and
increases the humidity, with a corresponding decrease in the dry-‐‑air fraction A of the gas
phase. The related temperature derivative of Eqn. (3.39.2) is
∂hSA
∂T
(
= 1 − wSW
p
+w
SW
) ∂∂hT
(
AV
∂h
∂T
+ 1 − wSW
A, p
+w
SW
SA , p
∂h
∂SA
) ∂h∂A
T,p
AV
∂SA
∂T
T,p
∂A
∂T
p
(
+ h−h
p
AV
)
∂wSW
.
∂T p
(3.39.3)
The isobaric evaporation rate −∂wSW / ∂T is related to the air-‐‑fraction change by the
p
conservation of the dry air, 1 − wSW A = const, in the form
(
)
∂A
∂T
=
p
A ∂wSW
,
1 − wSW ∂T p
(3.39.4)
and to the change of salinity by the conservation of the salt, wSW SA = const, in the form
∂SA
∂T
= −
p
SA ∂wSW
.
wSW ∂T p
(3.39.5)
Using these relations, Eqn. (3.39.3) takes the simplified form
∂hSA
∂T
(
)
= 1 − wSW c AV
+ wSW c p − LSA
p
p
p
∂wSW
.
∂T p
(3.39.6)
The coefficient in front of the evaporation rate,
AV
LSA
−A
p ( A, S A , t , p ) = h
∂h AV
∂A
− h + SA
T,p
∂h
∂SA
,
(3.39.7)
T,p
provides the desired expression for isobaric evaporation enthalpy, namely the difference
between the partial specific enthalpies of vapour in humid air (the first two terms) and of
water in seawater (the last two terms). In the ideal-‐‑gas approximations for air and for
vapour, the partial specific enthalpy of vapour in humid air, h AV − AhAAV , equals the
specific enthalpy of vapour, h V ( t ) , as a function of only the temperature, independent of
the pressure and of the presence of air (Feistel et al. (2010a)). As is physically required for
any measurable thermodynamic quantity, the arbitrary absolute enthalpies of water, salt
and air cancel in the formula (3.39.7), provided that the reference state conditions for both
the seawater and the humid-‐‑air formulation are chosen consistently (Feistel et al. (2008a),
(2010a)). The latent heat of evaporation depends only weakly on salinity and on air
fraction, and is an almost linear function of the temperature and of the pressure.
Selected representative values for the air fraction at condensation, Acond , and the latent
heat of evaporation, LSA
p , are given in Table 3.39.1.
In the derivation of Eqn. (3.39.7), the value of A is indirectly assumed to be computed
from the equilibrium condition (3.37.6) between humid air and seawater, A = Acond . At
this humidity the air is still sub-‐‑saturated, Acond > Asat , but its vapour starts condensing at
IOC Manuals and Guides No. 56
62
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
the sea surface. The values of Acond and Asat coincide only below the freezing point of
seawater, or at vanishing salinity, see also the following section 3.40.
The evaporation rate, − ∂wSW / ∂T , can be computed from Eqn. (3.37.6), the
p
equilibrium condition between humid air and seawater, at changing temperature and
constant pressure (Feistel et al. (2010a)). In contrast, the derivation of LSA
p using Eqns.
(3.39.2) -‐‑ (3.39.7) is a mere consideration of mass and enthalpy balances; no equilibrium
condition is actually involved. Hence, it is physically evident that Eqn. (3.39.7) can also be
applied to situations in which A takes any given value different from Acond , that is, it can
be applied regardless of whether or not the humid air is actually at equilibrium with the
sea surface.
Table 3.39.1: Selected values for the equilibrium air fraction, Acond , computed
from Eqn. (3.37.6), and the latent heat of evaporation, LSA
p ,
computed from Eqn. (3.39.7), for different sea-‐‑surface conditions.
Note that the TEOS-‐‑10 formulation for humid-‐‑air is valid up to 5
MPa, i.e., almost 500 dbar sea pressure.
p
t
LSA
SA
Acond
p
Condition
–1
g kg
°C dbar
%
J kg–1
Pure water
0
0
0 99.622 31 2 499 032
Brackish water
10
0
0 99.624 27 2 499 009
Standard ocean 35.165 04 0
0 99.629 31 2 498 510
Tropical ocean 35.165 04 25
0 98.059 33 2 438 971
High pressure 35.165 04 0 400 99.989 43 2 443 759
3.40 Relative humidity and fugacity
Parameterised formulas for the flux of water and heat through the ocean surface are
usually expressed in terms of a given relative humidity of the air in contact with seawater.
In this section we provide the formulas for the relative humidity and the fugacity from the
TEOS-‐‑10 potential functions for seawater and humid air, and we explain why the relative
fugacity with respect to condensation rather than with respect to saturation should be
used for oceanographic flux estimates (Feistel et al. (2010a)). Near the saturation point, the
two flux formulas may even exhibit different signs (different flux directions) since
condensation occurs at the sea surface at sub-‐‑saturated values of relative humidity.
Relative humidity is not uniquely defined in the literature, but the common definitions
give the same results in the ideal-‐‑gas limit of humid air. Also in this approximation,
relative humidity is only a property of fluid water at given temperature and pressure of
the vapour phase, independent of the presence of air.
The CCT1 definition of relative humidity is in terms of mole fraction: “At given
pressure and temperature, [the relative humidity is defined as] the ratio, expressed as a
percent, of the mole fraction of water vapour to the vapour mole fraction which the moist
gas would have if it were saturated with respect to either water or ice at the same pressure
and temperature.” Consistent with CCT, IUPAC2 defines relative humidity “as the ratio,
often expressed as a percentage, of the partial pressure of water in the atmosphere at some
observed temperature, to the saturation vapour pressure of pure water at this
temperature” (Calvert (1990), IUPAC (1997)). This definition of the relative humidity
takes the form
1 CCT: Consultative Committee for Thermometry, www.bipm.org/en/committees/cc/cct/
2 IUPAC: International Union of Pure and Applied Chemistry, www.iupac.org
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
RH CCT =
xV
xVsat
63
(3.40.1)
with regard to the mole fraction of vapour xV ( A) , Eqn. (3.35.3), and the saturated air
fraction A = Asat ( t , P ) = Acond ( 0, t , P ) either from Eqn. (3.37.6) with respect to liquid water,
at t above the freezing point of pure water, or from Eqn. (3.35.5) with respect to ice, at t
below the freezing point of pure water. Here, Acond ( S A , t , P ) is the air fraction of humid
air at equilibrium with seawater, Eqn. (3.37.5), which is subsaturated for SA > 0.
The WMO3 definition of the relative humidity is (Pruppacher and Klett (1997),
Jacobson (2005)),
1 / A −1
(3.40.2)
r
1 / Asat − 1
where r = (1 − A) / A is the humidity ratio. If r is small, we can estimate xV ≈ rM A / M W
(from Eqn. (3.35.3)) and therefore RH WMO ≈ RH CCT , that is, we find approximate
consistency between Eqns. (3.40.1) and (3.40.2).
Sometimes, especially when considering phase or chemical equilibria, it is more
convenient to use the fugacity (or activity) rather than partial pressure ratio (IUPAC
(1997)). The fugacity of vapour in humid air is defined as
RH WMO =
r
sat
=
V
V, id ⎫
⎪⎧ µ − µ
⎪
(3.40.3)
f V ( A, T , P ) = xV P exp ⎨
⎬ .
⎪⎩ RWT ⎪⎭
Here, RW = R M W is the specific gas constant of water, µ V ( A, T , P ) = g AV − Ag AAV is the
chemical potential of vapour in humid air, and µ V, id ( A, T , P ) is its ideal-‐‑gas limit which is
equal to the true chemical potential in the limit of very low pressure,
T
x P
T
V, id
V
(3.40.4)
µ
( A, T , P ) = g0 + ∫ ⎛⎜1 − ⎞⎟ c V,id
(T ' ) dT ' + RWT ln VV .
p
T'⎠
P
V⎝
0
T
0
The values of g0V ,
AV
P0V and
T0V of
µ V,id must be chosen consistently with the adjustable
id
constants of g (Feistel et al. (2010a)). The ideal-‐‑gas heat capacity of vapour c V,
(T ) is
p
available from IAPWS-‐‑95. In the ideal-‐‑gas limit of infinite dilution, f V converges to the
partial pressure of vapour (Glasstone (1947)),
lim f V ( A, T , P ) = xV P = P vap .
(3.40.5)
P→0
The saturation fugacity is defined by the equilibrium between liquid water (or ice) and
vapour in air, µ V ( A, T , P ) = µ W ( 0, T , P ) , that is,
f Vsat
=
(
)
⎧ µ W ( 0, T , P ) − µ V,id Asat , T , P ⎫
⎪
⎬ ,
RWT
⎪⎩
⎪⎭
⎪
xVsat P exp ⎨
(3.40.6)
where µ W = g ( 0, T , P ) is the chemical potential of liquid water (or the chemical potential
of ice, µ Ih ). The relative fugacity ϕ of humid air is then defined, dividing Eqn. (3.40.3) by
Eqn. (3.40.6) and making use of Eqn. (3.40.4), as
⎧⎪ µ V ( A, T , P ) − µ W ( 0, T , P ) ⎫⎪
fV
ϕ = sat = exp ⎨
⎬ .
RWT
fV
⎪⎩
⎪⎭
(3.40.7)
In the ideal-‐‑gas limit, µ V = µ V, id , and using (3.40.3) we see that the relative fugacity ϕ
coincides with the relative humidity, Eqn. (3.40.1).
Taking Eqn. (3.40.7) at the condensation point, A = Acond , Eqn. (3.37.5), it follows that
the relative fugacity of humid air at equilibrium with seawater (“sea air” for short) is
3 WMO: World Meteorological Organisation, www.wmo.int
IOC Manuals and Guides No. 56
64
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
ϕ SA =
⎧⎪ µ W ( SA , T , P ) − µ W ( 0, T , P ) ⎫⎪
f VSA
=
exp
⎨
⎬ .
RWT
f Vsat
⎩⎪
⎭⎪
(3.40.8)
The chemical potential difference in the exponent is proportional to the osmotic coefficient
of seawater, φ , which is computed from the saline part of the Gibbs function as (Feistel
and Marion (2007), Feistel (2008)),
φ ( SA , T , P ) = −
⎡
∂g
⎢ g ( SA , T , P ) − g ( 0, T , P ) − SA
∂SA
mSW RT ⎢
⎣
1
⎤
⎥ ,
⎥
T,P ⎦
where mSW is the molality of seawater (Millero et al. (2008a)),
SA
mSW =
.
(1 − SA ) M S
(3.40.9)
(3.40.10)
From the chemical potential of water in seawater, µW = g − SA g S , and Eqns. (3.40.8) -‐‑
A
(3.40.10) we infer for the relative fugacity of sea air the simple formula
ϕ SA = exp ( − mSW M W φ ) ,
(3.40.11)
which is identical to the activity aW of water in seawater. Similar to the ideal gas
approximation, the relative fugacity of sea air is independent of the presence or the
properties of air. In Eqn. (3.40.11), the relative fugacity ϕ SA ≤ 1 expresses the fact that the
vapour pressure of seawater is lower than that of pure water, i.e., that humid air in
equilibrium with seawater above its freezing temperature is always sub-‐‑saturated.
As a raw practical estimate, using a series expansion of Eqns. (3.40.10) and (3.40.11)
with respect to salinity, we can obtain from the molality mSW = SA / M S + O( SA2 ) and the
osmotic coefficient φ = 1 + O ( SA ) the linear relation
MW
(3.40.12)
SA ,
MS
i.e., Raoult’s law for the vapour-‐‑pressure lowering of seawater, Eqn. (3.37.4).
Below the freezing temperature of pure water at a given pressure, the saturation of
vapour is defined by the chemical potential of ice rather than liquid water, i.e. by
ϕ SA ≈ 1 −
f Vsat
=
(
)
⎧ µ Ih (T , P ) − µ V,id Asat , T , P ⎫
⎪
⎬,
RWT
⎪⎩
⎪⎭
⎪
xVsat P exp ⎨
(3.40.13)
rather than Eqn. (3.40.6). Then, the relative fugacity of sea air is
ϕ
SA
⎧⎪ µ W ( SA , T , P ) − µ Ih (T , P ) ⎫⎪
f VSA
= sat = exp ⎨
⎬ .
RWT
fV
⎪⎩
⎪⎭
(3.40.14)
When the temperature is lowered further to the freezing point of seawater, the exponent of
(3.40.14) vanishes and sea air is saturated, ϕ SA = 1, for sea-‐‑ice air at any lower temperature.
Thermodynamic fluxes in non-‐‑equilibrium states are driven by Onsager “forces” such
as the gradient of − µ / T (de Groot and Mazur (1984)). At the sea surface, assuming the
same temperature and pressure on both sides of the sea-‐‑air interface, the dimensionless
Onsager force X SA ( A, SA , T , P ) driving the transfer of water is the difference between the
chemical potentials of water in humid air and in seawater,
V
µAV
A, T , P )
µ W ( SA , T , P )
⎛ µ ⎞
(
(3.40.15)
X SA = Δ ⎜
.
−
⎟ =
RWT
RWT
⎝ RWT ⎠
This difference vanishes at the condensation point, A = Acond ( SA , T , P ) , Eqn. (3.37.5),
rather than at saturation. X SA can also be expressed in terms of fugacities, Eqns. (3.40.7),
(3.40.8) and (3.40.11), in the form
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
X SA = ln
ϕ ( A)
ϕ
SA
( SA )
= mSW M Wφ + ln ϕ ( A) .
65
(3.40.16)
Rather than the relative humidity, Eqns. (3.40.1), (3.40.2), the sea-‐‑air Onsager force X SA , in
conjunction with the formula (3.39.7), is relevant for the parameterization of non-‐‑
equilibrium latent heat fluxes across the sea surface. In the special case of limnological
applications, or below the freezing point of seawater, it reduces to X SA = ln ϕ ( A) , which
corresponds to the relative humidity, ln ( RH CCT ) , in the ideal-‐‑gas approximation. All
properties required for the calculation of the formula (3.40.16) are available from the
TEOS-‐‑10 thermodynamic potentials for seawater, ice, and humid air.
3.41 Osmotic pressure
If pure water is separated from seawater by a semi-‐‑permeable membrane which allows
water molecules to pass but not salt particles, water will penetrate into the seawater, thus
diluting it and possibly increasing its pressure, until the chemical potential of water in
both boxes becomes the same (or the pure water reservoir is exhausted). In the usual
model configuration, the two samples are thermally coupled but may possess different
pressures; the resulting pressure difference required to maintain equilibrium is the
osmotic pressure of seawater. An example of a practical application is desalination by
reverse osmosis; if the pressure on seawater in a vessel exceeds its osmotic pressure,
freshwater can be “squeezed” out of solution through suitable membrane walls
(Sherwood et al. (1967)). The osmotic pressure of seawater is very important for marine
organisms; it is considered responsible for the small number of species that can survive in
brackish environments.
The defining condition for the osmotic equilibrium is equality of the chemical
potentials of pure water at pressure p W and of water in seawater at the pressure p,
(
)
g 0, t , p W = g ( SA , t , p ) − S A
∂g
∂SA
.
(3.41.1)
T, p
The solution of this implicit relation for p (given values of S A , t and p W ) leads to the
osmotic pressure p osm
posm = p − p W .
(3.41.2)
An example of the TEOS-‐‑10 value for the osmotic pressure of standard seawater is
posm SA = SSO , t = 0 °C, p W = 0dbar = 235.4684 dbar . Osmotic pressure may be calculated
using the gsw_osmotic_pressure_t_exact(SA,t,pw) function of the GSW Oceanographic
Toolbox.
(
)
3.42 Temperature of maximum density
At about 4 °C and atmospheric pressure, pure water has a density maximum below which
the thermal expansion coefficient and the adiabatic lapse rate change their signs (Röntgen
(1892), McDougall and Feistel (2003)). At salinities higher than 23.8 g kg–1 the temperature
of maximum density tMD is below the freezing point tf (Table 3.42.1). The seasonal and
spatial interplay between density maximum and freezing point is highly important for the
stratification stability and the seasonal deep convection for brackish estuaries with
permanent vertical and lateral salinity gradients such as the Baltic Sea (Feistel et al.
(2008b), Leppäranta and Myrberg (2009), Reissmann et al. (2009)).
The temperature of maximum density tMD is computed from the condition of
vanishing thermal expansion coefficient, that is, from the solution of the implicit equation
for tMD ( SA , p) ,
IOC Manuals and Guides No. 56
66
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
gTP ( SA , tMD , p ) = 0.
(3.42.1)
The temperature of maximum density is available in the GSW Oceanographic Toolbox as
the function gsw_t_maxdensity_exact. Selected TEOS-‐‑10 values computed from Eqn.
(3.42.1) are given in Table 3.42.1.
Table 3.42.1: Freezing temperature tf and temperature of maximum density tMD
for air-‐‑free brackish seawater with absolute salinities S A between 0
and 25 g kg −1, computed at the surface pressure from TEOS-‐‑10.
Values of tMD in parentheses are less than the freezing temperature.
tf
tf
tf
SA
tMD
SA
tMD SA
tMD
g kg–1
°C
g kg–1
°C
g kg–1
°C
°C
°C
°C
0
+0.003
3.978
8.5
–0.456 2.128 17
–0.912
0.250
0.5
–0.026
3.868
9
–0.483 2.019 17.5
–0.939
0.139
1
–0.054
3.758
9.5
–0.509 1.909 18
–0.966
0.027
1.5
–0.081
3.649
10
–0.536 1.800 18.5
–0.994 –0.085
2
–0.108
3.541
10.5
–0.563 1.690 19
–1.021 –0.196
2.5
–0.135
3.432
11
–0.590 1.580 19.5
–1.048 –0.308
3
–0.162
3.324
11.5
–0.616 1.470 20
–1.075 –0.420
3.5
–0.189
3.215
12
–0.643 1.360 20.5
–1.102 –0.532
4
–0.216
3.107
12.5
–0.670 1.249 21
–1.130 –0.644
4.5
–0.243
2.999
13
–0.697 1.139 21.5
–1.157 –0.756
5
–0.269
2.890
13.5
–0.724 1.028 22
–1.184 –0.868
5.5
–0.296
2.782
14
–0.750 0.917 22.5
–1.212 –0.980
6
–0.323
2.673
14.5
–0.777 0.807 23
–1.239 –1.092
6.5
–0.349
2.564
15
–0.804 0.696 23.5
–1.267 –1.204
7
–0.376
2.456
15.5
–0.831 0.584 24
–1.294 (–1.316)
7.5
–0.403
2.347
16
–0.858 0.473 24.5
–1.322 (–1.428)
8
–0.429
2.238
16.5
–0.885 0.362 25
–1.349 (–1.540)
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
67
4. Conclusions
The International Thermodynamic Equation Of Seawater – 2010 (TEOS-‐‑10) allows all the
thermodynamic properties of pure water, ice Ih, seawater and moist air to be evaluated in
an internally self-‐‑consistent manner. For the first time the effects of the small variations in
seawater composition around the world ocean are included, especially their effects on the
density of seawater (which can be equivalent to ten times the precision of our Practical
Salinity measurements at sea).
Perhaps the most apparent changes compared with the International Equation of State
of seawater (EOS-‐‑80) are (i) the adoption of Absolute Salinity S A instead of Practical
Salinity S P (PSS-‐‑78) as the salinity argument for the thermodynamic properties of
seawater, and (ii) the use of Conservative Temperature Θ in place of potential
temperature θ . Importantly, Practical Salinity is retained as the salinity variable that is
stored in data bases because Practical Salinity is virtually the measured variable (whereas
Absolute Salinity is a calculated variable) and also so that national data bases do not
become corrupted with incorrectly labeled and stored salinity data.
The adoption of Absolute Salinity as the argument for all the algorithms used to
evaluate the thermodynamic properties of seawater makes sense simply because the
thermodynamic properties of seawater depend on S A rather than on S P ; seawater parcels
that have the same values of temperature, pressure and of S P do not have the same
density unless the parcels also share the same value of S A . Absolute Salinity is measured
in SI units and the calculation of the freshwater concentration and of freshwater fluxes
follows naturally from Absolute Salinity, but not from Practical Salinity.
Absolute Salinity is calculated from the computer algorithm of McDougall et al. (2012)
or by other means, as the sum of Reference Salinity and the Absolute Salinity Anomaly.
There are subtle issues in defining what is exactly meant by “absolute salinity” and at least
four different definitions are possible when compositional anomalies are present. We
have chosen the definition that yields the most accurate estimates of seawater density
since the ocean circulation is sensitive to rather small gradients of density. The algorithm
that estimates Absolute Salinity Anomaly represents the state of the art as at 2010, but this
area of oceanography is relatively immature. It is likely that the accuracy of this algorithm
will improve as more seawater samples from around the world ocean have their density
accurately measured. After such future work is published and the results distilled into a
revised algorithm, such an algorithm will be served from www.TEOS-10.org.
Oceanographers should publish the version number of this software that is used to obtain
thermodynamic properties in their manuscripts.
For these reasons the TEOS-‐‑10 salinity variable to appear in publications is Absolute
Salinity S A . The version number of the software that is used to convert Reference Salinity
S R into Absolute Salinity SA should always be stated in publications. Nevertheless, there
may be some applications where the likely future changes in the algorithm that relates
Reference Salinity to Absolute Salinity presents a concern, and for these applications it
may be preferable to publish graphs and tables in Reference Salinity. For these studies or
where it is clear that the effect of compositional variations are insignificant or not of
interest, the Gibbs function may be called with S R rather than S A , thus avoiding the need
to calculate the Absolute Salinity Anomaly. When this is done, it should be clearly stated
that Reference Salinity is being used, not Absolute Salinity.
IOC Manuals and Guides No. 56
68
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
The recommended treatment of salinity in ocean models is to carry evolution
equations for both Preformed Salinity S* and another variable, F δ , which is related to the
Absolute Salinity Anomaly, so that Absolute Salinity can be calculated at each time step of
the model and used to accurately evaluate density (as discussed in appendix A.20).
Potential temperature has been used in oceanography as though it is a conservative
variable, and yet the specific heat of seawater varies by 5% at the sea surface, and potential
temperature is not conserved when seawater parcels mix. The First Law of
Thermodynamics can be very accurately regarded as the statement that potential enthalpy
h0 and Conservative Temperature Θ are conservative variables in the ocean. This,
together with the knowledge that the air-‐‑sea heat flux is exactly the air-‐‑sea flux of
potential enthalpy (i. e. the air-‐‑sea flux of c0p Θ ) means that potential enthalpy can be
treated as the “heat content” of seawater, and fluxes of potential enthalpy in the ocean can
be treated as “heat fluxes”. Just as it is perfectly valid to talk of the flux of salinity
anomaly ( SA − constant) across an ocean section even when the mass flux across the
section is non-‐‑zero, so it is perfectly valid to treat the flux of c0p Θ across an ocean section
as the “heat flux” even when the fluxes of mass and of salt across the section are non-‐‑zero.
The temperature variable in ocean models has been taken to be potential temperature
θ , but to date the non-‐‑conservative source terms that are present in the evolution
equation of potential temperature have not been included. To be TEOS-‐‑10 compatible,
ocean models need to treat their temperature variable as Conservative Temperature Θ .
Ocean models should be initialized with Θ rather than θ , the output temperature must
be compared to observed Θ data rather than to θ data, and during the model run, any
air-‐‑sea fluxes that depend on the sea-‐‑surface temperature (SST) must be calculated at each
model time step using θ = θˆ ( SA , Θ ) .
Under EOS-‐‑80 the observed variables ( SP , t , p ) were first used to calculate potential
temperature θ and then water masses were analyzed on the SP − θ diagram. Curved
contours of potential density ρ θ could also be drawn on this same SP − θ diagram.
Under TEOS-‐‑10, since density and potential density are now not functions of Practical
Salinity SP but rather are functions of Absolute Salinity S A , it is no longer possible to
draw isolines of potential density on a SP − θ diagram. Rather, because of the spatial
variations of seawater composition, a given value of potential density defines an area on
the SP − θ diagram, not a curved line.
Under TEOS-‐‑10, the observed variables ( SP , t , p ) , together with longitude and
latitude, are first used to form Absolute Salinity S A , and then Conservative Temperature
Θ is evaluated. Oceanographic water masses are then analyzed on the SA − Θ diagram,
and potential density ρ Θ contours can also be drawn on this SA − Θ diagram. The
computationally-‐‑efficient 75-‐‑term expression for the specific volume of seawater (of
appendix K) is a convenient and accurate equation of state for observational and
theoretical studies and for ocean modelling. Preformed Salinity S* is used internally in
numerical ocean models where it is important that the salinity variable be conservative.
Appendix L lists the recommended nomenclature, symbols and units of
thermodynamic quantities for use by oceanographers.
When describing the use of TEOS-‐‑10, it is the present document (the TEOS-‐‑10
Manual) that should be referenced as IOC et al. (2010) [IOC, SCOR and IAPSO, 2010: The
international thermodynamic equation of seawater – 2010: Calculation and use of thermodynamic
properties. Intergovernmental Oceanographic Commission, Manuals and Guides No. 56,
UNESCO (English), 196 pp]. Two introductory articles about TEOS-‐‑10, namely “Getting
started with TEOS-‐‑10 and the Gibbs Seawater (GSW) Oceanographic Toolbox”
(McDougall and Barker, 2011), and “What every oceanographer needs to know about
TEOS-‐‑10:-‐‑ The TEOS-‐‑10 Primer” (Pawlowicz, 2010b), are available from
www.TEOS-10.org.
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
69
APPENDIX A:
Background and theory underlying
the use of the Gibbs function of seawater
A.1 ITS-90 temperature
In order to understand the limitations of conversion between different temperature scales, it is
helpful to review the definitions of temperature and of the international scales on which it is
reported.
A.1.1 Definition
When considering temperature, the fundamental physical quantity is thermodynamic
temperature, symbol T. The unit for temperature is the kelvin. The name of the unit has a
lowercase k. The symbol for the unit is uppercase K. One kelvin is 1/273.16 of the
thermodynamic temperature of the triple point of water. (A recent evolution of the definition
has been to specify the isotopic composition of the water to be used as that of Vienna
Standard Mean Ocean Water, VSMOW.) The Celsius temperature, symbol t , is defined by
t °C = T K − 273.15, and 1 °C is the same size as 1 K.
A.1.2 ITS-‐‑90 temperature scale
The definition of temperature scales is the responsibility of the Consultative Committee for
Thermometry (CCT) which reports to the International Committee for Weights and Measures
(often referred to as CIPM for its name in the French language). Over the last 40 years, two
temperature scales have been used; the International Practical Temperature Scale 1968 (IPTS-‐‑
68), followed by the International Temperature Scale 1990 (ITS-‐‑90). These are defined by
Barber (1969) and Preston-‐‑Thomas (1990). For information about the International
Temperature Scales of 1948 and 1927 the reader is referred to Preston-‐‑Thomas (1990).
In the oceanographic range, temperatures are determined using a platinum resistance
thermometer. The temperature scales are defined as functions of the ratio W , namely the
ratio of the thermometer resistance at the temperature to be measured R (t ) to the resistance at
a reference temperature R0 . In IPTS-‐‑68, R0 is R ( 0°C ) , while in ITS-‐‑90 R0 is R ( 0.01°C ) . The
details of these temperature scales and the differences between the two scales are therefore
defined by the functions of W used to calculate T . For ITS-‐‑90, and in the range 0 °C < t90 <
968.71 °C, t90 is described by a polynomial with 10 coefficients given by Table 4 of Preston-‐‑
Thomas (1990).
We note in passing that the conversions from W to T and from T to W are both defined
by polynomials and these are not perfect inverses of one another. Preston-‐‑Thomas points out
that the inverses are equivalent to within 0.13mK. In fact the inverses have a difference of 0.13
mK at 861°C, and a maximum error in the range 0 °C < t90 < 40 °C of 0.06 mK at 31 °C. That
the CCT allowed this discrepancy between the two polynomials immediately provides an
indication of the absolute uncertainty in the determination, and indeed in the definition, of
temperature.
IOC Manuals and Guides No. 56
70
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
A second uncertainty in the absolute realization of ITS-‐‑90 arises from what is referred to
as sub-‐‑range inconsistency. The polynomial referred to above describes the behaviour of an
‘ideal’ thermometer. Any practical thermometer has small deviations from this ideal
behaviour. ITS-‐‑90 allows the deviations to be determined by measuring the resistance of the
thermometer at up to five fixed points: the triple point of water and the freezing points of tin,
zinc, aluminium and silver, covering the range 0.01 °C < t90 < 961.78 °C. If not all of these
points are measured, then it is permissible to estimate the deviation from as many of those
points as are measured. The melting point of Gallium ( t90 = 29.7646 °C) and the triple point of
Mercury ( t90 = -‐‑ 38.8344 °C) may also be used if the thermometer is to operate over a smaller
temperature range. Hence the manner in which the thermometer may be used to interpolate
between the points is not unique. Rather it depends on which fixed points are measured, and
there are several possible outcomes, all equally valid within the definition. Sections 3.3.2 and
3.3.3 of Preston-‐‑Thomas (1990) give precise details of the formulation of the deviation
function. The difference between the deviation functions derived from different sets of fixed
points will depend on the thermometer, so it not possible to state an upper bound on this non-‐‑
uniqueness. Common practice in oceanographic standards laboratories is to estimate the
deviation function from measurements at the triple point of water and the melting point of
Gallium ( t90 = 29.7646 °C). This allows a linear deviation function to be determined, but no
higher order terms.
In summary, there is non-‐‑uniqueness in the definition of ITS-‐‑90, in addition to any
imperfections of measurement by any practical thermometer (Rudtsch and Fischer (2008),
Feistel et al. (2008a)). It is therefore not possible to seek a unique and perfect conversion
between IPTS-‐‑68 and ITS-‐‑90.
Goldberg and Weir (1992) and Mares and Kalova (2008) have discussed the procedures
needed to convert measured thermophysical quantities (such as specific heat) from one
temperature definition to another. When mechanical or electrical energy is used in a
laboratory to heat a certain sample, this energy can be measured in electrical or mechanical
units by appropriate instruments such as an ampere meter, independent of any definition of a
temperature scale. It is obvious from the fundamental thermodynamic relation (at constant
Absolute Salinity), du = Tdη + Pdv, that the same energy difference Tdη results in different
values for the entropy η , depending on the number read for T from a thermometer
calibrated on the 1990 compared with one calibrated on the 1968 scale. A similar dependence
is found for numbers derived from entropy, for example, for the heat capacity,
c p = T ηT
.
SA , p
Douglas (1969) listed a systematic consideration of the quantitative relations between the
measured values of various thermal properties and the particular temperature scale used in
the laboratory at the time the measurement was conducted. Conversion formulas to ITS-‐‑90 of
readings on obsolete scales are provided by Goldberg and Weir (1992) and Weir and Goldberg
(1996).
Any thermal experimental data that entered the construction of the thermodynamic
potentials that form TEOS-‐‑10 were carefully converted by these rules, in addition to the
conversion between the various older definitions of for example calories and joules. This
must be borne in mind when properties computed from TEOS-‐‑10 are combined with historical
measurements from the literature.
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
71
A.1.3 Theoretical conversion between IPTS-‐‑68 and ITS-‐‑90
Having understood that the conversion between IPTS-‐‑68 and ITS-‐‑90 is not uniquely defined,
we review the sources of uncertainty, or even flexibility, in the conversion between t90 and
t68 .
Consider first why t90 and t68 temperatures differ:
1) The fixed points have new temperature definitions in ITS-‐‑90, due to improvements in
determining the absolute thermodynamic temperatures of the melting/freezing physical
states relative to the triple point of water.
2) For some given resistance ratio W the two scales have different algorithms for interpolating
between the fixed points.
Now consider why there is non-‐‑uniqueness in the conversion:
3) In some range of ITS-‐‑90, the conversion of W to t90 can be undertaken with a choice of
coefficients that is made by the user (Preston-‐‑Thomas (1990) Sections 3.3.2.1 to 3.3.3),
referred to as sub-‐‑range inconsistency.
4) The impact of the ITS-‐‑90 deviation function on the conversion is non-‐‑linear. Therefore the
size of the coefficients in the deviation function will affect the difference, t90 − t68 . The
formal conversion is different for each actual thermometer that has been used to acquire
data.
The group responsible for developing ITS-‐‑90 was well aware of the non-‐‑uniqueness of
the conversion. Table 6 of Preston-‐‑Thomas (1990) gives differences (t90 − t68 ) with a resolution
of 1 mK, because
(a) the true thermodynamic temperature T was known to have uncertainties of order
1 mK or larger in some ranges,
(b) the sub-‐‑range inconsistency of ITS-‐‑90 using the same calibration data gave an
uncertainty of several tenths of 1 mK.
Therefore to attempt to define a generic conversion of (t90 − t68 ) with a resolution of say
0.1 mK would probably be meaningless and possibly misleading as there isn’t a unique
generic conversion function.
A.1.4 Practical conversion between IPTS-‐‑68 and ITS-‐‑90
Rusby (1991) published an 8th order polynomial that was a fit to Table 6 of Preston-‐‑Thomas
(1990). This fit is valid in the range 73.15 K to 903.89 K (-‐‑200 °C to 630.74 °C). He reports that
the polynomial fits the table to within 1 mK, commensurate with the non-‐‑uniqueness of IPTS-‐‑
68.
Rusby’s 8th order polynomial is in effect the ‘official recommended’ conversion between
IPTS-‐‑68 and ITS-‐‑90. This polynomial has been used to convert historical IPTS-‐‑68 data to ITS-‐‑
90 for the preparation of the new thermodynamic properties of seawater that are the main
subject of this manual.
As a convenient conversion valid in a narrower temperature range, Rusby (1991) also
proposed
(A.1.1)
(T90 − T68 ) /K = -0.00025 (T68 / K - 273.15)
in the range 260 K to 400 K (-‐‑13 °C to 127 °C). Rusby (1991) also explicitly reminds readers
(see his page 1158) that compound quantities that involve temperature intervals such as heat
capacity and thermal conductivity are affected by their dependence on the derivative
d (T90 − T68 ) /dT68 . About the same time that Rusby published his conversion from t68 to t90 ,
Saunders (1990) made a recommendation to oceanographers that in the common
oceanographic temperature range -‐‑2 °C < t68 < 40 °C, conversion could be achieved using
(t90 /°C) = (t68 /°C) 1.00024.
(A.1.2)
IOC Manuals and Guides No. 56
72
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
The difference between Saunders (1990) and Rusby (1991) arises from the best slope being
1.00024 near 0 °C and 1.00026 near 100 °C (recall that t68 for the boiling point of water was
100 °C while its t90 is 99.974 °C). Thus Rusby (1991) chose 1.00025 over the wider range of 0
°C to 100 °C.
In considering what is a ‘reasonable’ conversion between the two temperature scales, we
must recall that the uncertainty in conversion between measured resistance and either
temperature scale is of order a few tenths of mK, and the uncertainty in the absolute
thermodynamic temperature T is probably at least as large, and may be larger than 1 mK in
some parts of the oceanographic range. For all practical purposes data converted using
Saunders’ 1.00024 cannot be improved upon; conversions using Rusby’s (1991) 8th order fit are
fully consistent with Saunders’ 1.00024 in the oceanographic temperature range within the
limitations of the temperature scales.
A.1.5 Recommendation regarding temperature conversion
The ITS-‐‑90 scale was introduced to correct differences between true thermodynamic
temperature T , and temperatures reported in IPTS-‐‑68.
There are remaining imperfections and residuals in T −T90 (Rusby, pers. comm.), which
may be as high as a couple of mK in the region of interest. This is being investigated by the
Consultative Committee for Thermometry (CCT). At a meeting in 2000 (Rusby and White
(2003)) the CCT considered introducing a new temperature scale to incorporate the known
imperfections, referred to at that time as ITS-‐‑XX. Further consideration by CCT WG1 has
moved thinking away from the desirability of a new scale. The field of thermometry is
undergoing rapid advances at present. Instead of a new temperature scale, the known
limitations of the ITS-‐‑90 can be addressed in large part through the ITS-‐‑90 Technical Annex,
and documentation from time to time of any known differences between thermodynamic
temperature and ITS-‐‑90 (Ripple et al. (2008)).
The two main conversions currently in use are Rusby’s 8th order fit valid over a wide
range of temperatures, and Saunders’ 1.00024 scaling widely used in the oceanographic
community. They are formally indistinguishable because they differ by less than both the
uncertainty in thermodynamic temperature, and the uncertainty in the practical application of
the IPTS-‐‑68 and ITS-‐‑90 scales. Nevertheless we note that Rusby (1991) suggests a linear fit
with slope 1.00025 in the range -‐‑13 °C to 127 °C, and that Saunders’ slope 1.00024 is a better fit
in the range -‐‑2 °C to 40 °C while Rusby’s 8th order fit is more robust for temperatures outside
the oceanographic range. The difference between Saunders (1990) and Rusby (1991) is less
than 1 mK everywhere in the range -‐‑2 °C to 40 °C and less than 0.03mK in the range -‐‑2 °C to
10 °C.
In conclusion, the algorithms for PSS-‐‑78 require t68 as the temperature argument. In
order to use these algorithms with t90 data, t68 may be calculated using Eqn. (A.1.3) thus
(t68 /°C)
IOC Manuals and Guides No. 56
= 1.00024 (t90 /°C) .
(A.1.3)
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
73
A.2 Sea pressure, gauge pressure and Absolute Pressure
Sea pressure p is defined to be the Absolute Pressure P less the Absolute Pressure of one
standard atmosphere, P0 ≡ 101 325Pa; that is
p ≡ P − P0 .
(A.2.1)
Also, it is common oceanographic practice to express sea pressure in decibars (dbar). Another
common pressure variable that arises naturally in the calibration of sea-‐‑board instruments is
gauge pressure p gauge which is Absolute Pressure less the Absolute Pressure of the
atmosphere at the time of the instrument’s calibration (perhaps in the laboratory, or perhaps
at sea). Because atmospheric pressure changes in space and time, sea pressure p is preferred
over p gauge as a thermodynamic variable as it is unambiguously related to Absolute Pressure.
The seawater Gibbs function is naturally a function of sea pressure p (or functionally
equivalently, of Absolute Pressure P ); it is not a function of gauge pressure.
Table A.2.1 Pressure unit conversion table
1 Pa
1 dbar
Pascal
decibar
bar
(Pa)
(dbar)
(bar)
≡ 1 N/m2
10−4
10−5
104
≡ 105 dyn/cm2
0.1
≡ 106 dyn/cm2
Technical
atmosphere atmosphere
(at)
(atm)
torr
(Torr)
pound-‐‑
force per
square inch
(psi)
10.197×10−6 9.8692×10−6 7.5006×10−3 145.04×10−6
0.101 97
98.692×10−3
75.006
1.450 377 44
1.0197
0.986 92
750.06
14.503 7744
1 bar
100 000
10
1 at
98 066.5
9.806 65
0.980 665
≡ 1 kgf/cm2
0.967 841
735.56
14.223
1 atm
101 325
10.1325
1.013 25
1.0332
≡ 1 atm
760
14.696
1 torr
133.322
1.3332×10−2
≡ 1 Torr
19.337×10−3
1.3332×10−3 1.3595×10−3 1.3158×10−3
1 psi 6 894.757
0.689 48
68.948×10−3 70.307×10−3 68.046×10−3
51.715
≡ 1 lbf/in2
Example: 1 Pa = 1 N/m2 = 10−4 dbar = 10−5 bar = 10.197×10−6 at = 9.8692×10−6 atm, etc.
The difference between sea pressure and gauge pressure is quite small and probably
insignificant for many oceanographic applications. Nevertheless it would be best practice to
ensure that the CTD pressure that is used in the seawater Gibbs function is calibrated on deck
to read the atmospheric pressure as read from the ship’s bridge barometer, less the Absolute
Pressure of one standard atmosphere, P0 ≡ 101 325Pa. (When the CTD is lowered from the
sea surface, the monitoring software may well display gauge pressure, indicating the distance
from the surface.)
Since there are a variety of different units used to express atmospheric pressure, we
present a table (Table A.2.1) to assist in converting between these different units of pressure
(see ISO (1993)). Note that one decibar (1 dbar) is exactly 0.1 bar, and that 1 mmHg is very
similar to 1 torr with the actual relationship being 1 mmHg = 1.000 000 142 466 321... torr. The
torr is defined as exactly 1/760 of the Absolute Pressure of one standard atmosphere, so that
one torr is exactly equal to (101 325/760) Pa.
IOC Manuals and Guides No. 56
74
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
A.3 Reference Composition and the Reference-Composition Salinity Scale
As mentioned in the main text, the Reference Composition of seawater is defined by Millero et
al. (2008a) as the exact mole fractions given in Table D.3 of appendix D below. This
composition model was determined from the most accurate measurements available of the
properties of Standard Seawater, which is filtered seawater from the surface waters of the
North Atlantic as made available by the IAPSO Standard Seawater Service. The Reference
Composition is perfectly consistent with charge balance of ocean waters and the most recent
atomic weight estimates (Wieser (2006)). For seawater with this reference composition the
Reference-‐‑Composition Salinity S R as defined below provides our best estimate of the
Absolute Salinity.
The Reference Composition includes all important components of seawater having mass
fractions greater than about 0.001 g kg −1 (i. e. 1.0 mg kg −1 ) that can significantly affect either
the conductivity or the density of seawater having a Practical Salinity of 35. The most
significant ions not included are Li + (~0.18 mg kg −1 ) and Rb + (~0.12 mg kg −1 ). Dissolved
gases N 2 (~16 mg kg −1 ) and O2 ( up to 8 mg kg −1 in the ocean) are not included as neither
have a significant effect on density or on conductivity. In addition, N 2 remains within a few
percent of saturation at the measured temperature in almost all laboratory and in situ
conditions. However, the dissolved gas CO2 (~0.7 mg kg −1 ), and the ion OH − (~0.08 mg kg −1 )
are included in the Reference Composition because of their important role in the equilibrium
dynamics of the carbonate system. Changes in pH which involve conversion of CO2 to and
from ionic forms affect conductivity and density. Concentrations of the major nutrients
Si(OH)4 , NO3− and PO34− are assumed to be negligible in Standard Seawater; their
concentrations in the ocean range from 0-‐‑16 mg kg −1 , 0-‐‑2 mg kg −1 , and 0-‐‑0.2 mg kg −1
respectively. The Reference Composition does not include organic matter. The composition
of Dissolved Organic Matter (DOM) is complex and poorly known. DOM is typically present
at concentrations of 0.5-‐‑2 mg kg −1 in the ocean.
Reference-‐‑Composition Salinity is defined to be conservative during mixing or
evaporation that occurs without removal of sea salt from solution. Because of this property,
the Reference-‐‑Composition Salinity of any seawater sample can be defined in terms of
products determined from the mixture or separation of two precisely defined end members.
Pure water and KCl-‐‑normalized seawater are defined for this purpose. Pure water is defined as
Vienna Standard Mean Ocean Water, VSMOW, which is described in the 2001 Guideline of
the International Association for the Properties of Water and Steam (IAPWS (2005), BIPM
(2005)); it is taken as the zero reference value. KCl-‐‑normalized seawater (or normalized seawater
for short) is defined to correspond to a seawater sample with a Practical Salinity of 35. Thus,
any seawater sample that has the same electrical conductivity as a solution of potassium
chloride (KCl) in pure water with the KCl mass fraction of 32.4356 g kg-‐‑1 when both are at the
ITS-‐‑90 temperature t = 14.996 °C and one standard atmosphere pressure, P = 101325 Pa is
referred to as normalized seawater. Here, KCl refers to the normal isotopic abundances of
potassium and chlorine as described by the International Union of Pure and Applied
Chemistry (Wieser (2006)). As discussed below, any normalized seawater sample has a
Reference-‐‑Composition Salinity of 35.165 04 g kg −1.
Since Reference-‐‑Composition Salinity is defined to be conservative during mixing, if a
seawater sample of mass m1 and Reference-‐‑Composition Salinity SR1 is mixed with another
seawater sample of mass m2 and Reference-‐‑Composition Salinity S R 2 , the final Reference-‐‑
Composition Salinity SR12 of this sample is
m1SR1 + m2 SR2
.
(A.3.1)
m1 + m2
Negative values of m1 and m2 , corresponding to the removal of seawater with the
appropriate salinity are permitted, so long as m1 (1 − SR1 ) + m2 (1 − SR2 ) > 0 . In particular, if
SR2 = 0 (pure water) and m2 is the mass of pure water needed to normalize the seawater
SR12 =
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
75
sample (that is, m2 is the mass needed to achieve SR12 = 35.165 04 g kg−1), then the original
Reference-‐‑Composition Salinity of sample 1 is given by
SR1 = [1 + (m2 / m1 )] × 35.16504 g kg-1 .
(A.3.2)
The definitions and procedures above allow one to determine the Reference Salinity of
any seawater sample at the ITS-‐‑90 temperature t = 14.996 °C and one standard atmosphere
pressure. To complete the definition, we note that the Reference-‐‑Composition Salinity of a
seawater sample at given temperature and pressure is equal to the Reference-‐‑Composition
Salinity of the same sample at any other temperature and pressure provided the transition
process is conducted without exchange of matter, in particular, without evaporation,
precipitation or degassing of substance from the solution. Note that this property is shared by
Practical Salinity to the accuracy of the algorithms used to define this quantity in terms of the
conductivity ratio R15.
We noted above that a Practical Salinity of 35 is associated with a Reference Salinity of
35.165 04 g kg −1. This value was determined by Millero et al. (2008a) using the reference
composition model, the most recent atomic weights (Wieser (2006)) and the relation S = 1.806
55 Cl / (g kg −1 ) which was used in the original definition of Practical Salinity to convert
between measured Chlorinity values and Practical Salinity. Since the relation between
Practical Salinity and conductivity ratio was defined using the same conservation relation as
satisfied by Reference Salinity, the Reference Salinity can be determined to the same accuracy
as Practical Salinity wherever the latter is defined (that is, in the range 2< S P < 42 ), as
SR ≈ uPS SP where uPS ≡ (35.165 04 35) g kg −1 .
(A.3.3)
For practical purposes, this relationship can be taken to be an equality since the approximate
nature of this relation only reflects the accuracy of the algorithms used in the definition of
Practical Salinity. This follows from the fact that the Practical Salinity, like Reference Salinity,
is intended to be precisely conservative during mixing and also during changes in
temperature and pressure that occur without exchange of mass with the surroundings.
The Reference-‐‑Composition Salinity Scale is defined such that a seawater sample whose
Practical Salinity S P is 35 has a Reference-‐‑Composition Salinity S R of precisely
35.165 04 g kg−1 . Millero et al. (2008a) estimate that the absolute uncertainty associated with
using this value as an estimate of the Absolute Salinity of Reference-‐‑Composition Seawater is
± 0.007 g kg−1 . Thus the numerical difference between the Reference Salinity expressed in
g kg −1 and Practical Salinity is about 24 times larger than this estimate of uncertainty. The
difference, 0.165 04 , is also large compared to our ability to measure Practical Salinity at sea
(which can be as precise as ± 0.002 ). Understanding how this discrepancy was introduced
requires consideration of some historical details that influenced the definition of Practical
Salinity. The details are presented in Millero et al. (2008a) and in Millero (2010) and are briefly
reviewed below.
There are two primary sources of error that contribute to this discrepancy. First, and most
significant, in the original evaporation technique used by Sørensen in 1900 (Forch et al. 1902)
to estimate salinity, some volatile components of the dissolved material were lost so the
amount of dissolved material was underestimated. Second, the approximate relation
determined by Knudsen (1901) to determine S ( ‰ ) from measurements of Cl ( ‰ ) was based
on analysis of only nine samples (one from the Red Sea, one from the North Atlantic, one from
the North Sea and six from the Baltic Sea). Both the errors in estimating absolute Salinity by
evaporation and the bias towards Baltic Sea conditions, where strong composition anomalies
relative to North Atlantic conditions are found, are reflected in Knudsen'ʹs formula,
SK ( ‰ ) = 0.03 + 1.805 Cl ( ‰ ) .
(A.3.4)
When the Practical Salinity Scale was decided upon in the late 1970s it was known that
this relation included significant errors, but it was decided to maintain numerical consistency
IOC Manuals and Guides No. 56
76
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
with this accepted definition of salinity for typical mid-‐‑ocean conditions (Millero (2010)). To
achieve this consistency while having salinity directly proportional to Chlorinity, the Joint
Panel for Oceanographic Tables and Standards (JPOTS) decided to determine the
proportionality constant from Knudsen'ʹs formula at S K = 35 ‰ ( Cl = 19.3740 ‰ ), (Wooster et
al., 1969). This resulted in the conversion formula
S ( ‰ ) = 1.80655 Cl ( ‰ )
(A.3.5)
being used in the definition of the practical salinity scale as if it were an identity, thus
introducing errors that have either been overlooked or accepted for the past 30 years. We
now break with this tradition in order to define a salinity scale based on a composition model
for Standard Seawater that was designed to give a much improved estimate of the mass-‐‑
fraction salinity for Standard Seawater and for Reference-‐‑Composition Seawater. The
introduction of this salinity scale provides a more physically meaningful measure of salinity
and simplifies the task of systematically incorporating the influence of spatial variations of
seawater composition into the procedure for estimating Absolute Salinity.
Finally, we note that to define the Reference-‐‑Composition Salinity Scale we have
introduced the quantity uPS in Eqn. (A.3.3), defined by uPS ≡ (35.165 04 35) g kg −1 . This value
was determined by the requirement that the Reference-‐‑Composition Salinity gives the best
estimate of the mass-‐‑fraction Absolute Salinity (that is, the mass-‐‑fraction of non-‐‑H2O material)
of Reference-‐‑Composition Seawater. However, the uncertainty in using S R to estimate the
Absolute Salinity of Reference-‐‑Composition Seawater is at least 0.007 g kg −1 at S = 35
(Millero et al. (2008b)). Thus, although uPS is precisely specified in the definition of the
Reference-‐‑Composition Salinity Scale, it must be noted that using the resulting definition of
the Reference Salinity to estimate the Absolute Salinity of Reference-‐‑Composition Seawater
does have a non-‐‑zero uncertainty associated with it. This and related issues are discussed
further in the next subsection.
A.4 Absolute Salinity
Millero et al. (2008a) list the following six advantages of adopting Reference Salinity S R and
Absolute Salinity S A in preference to Practical Salinity S P .
1. The definition of Practical Salinity S P on the PSS-‐‑78 scale is separate from the system
of SI units (BIPM (2006)). Reference Salinity can be expressed in the unit
(g kg −1 ) as a measure of Absolute Salinity. Adopting Absolute Salinity and Reference
Salinity will terminate the ongoing controversies in the oceanographic literature about
the use of “PSU” or “PSS” and make research papers more readable to the outside
scientific community and consistent with SI.
2. The freshwater mass fraction of seawater is not (1 – 0.001 S P ). Rather, it is
(1 – 0.001 S A /( g kg −1 )), where S A is the Absolute Salinity, defined as the mass fraction
of dissolved material in seawater. The values of S A /( g kg −1 ) and S P are known to
differ by about 0.5%. There seems to be no good reason for continuing to ignore this
known difference, for example in ocean models.
3. PSS-‐‑78 is limited to the range 2 < S P < 42. For a smooth crossover on one side to pure
water, and on the other side to concentrated brines up to saturation, as for example
encountered in sea ice at very low temperatures, salinities beyond these limits need to
be defined. While this poses a challenge for S P , it is trivial for SR .
4. The theoretical Debye-‐‑Hückel limiting laws of seawater behavior at low salinities,
used for example in the determination of the Gibbs function of seawater, can only be
computed from a chemical composition model, which is available for S R but not for
S P .
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
77
5. For artificial seawater of Reference Composition, S R has a fixed relation to Chlorinity,
independent of conductivity, salinity, temperature, or pressure.
6. Stoichiometric anomalies can be specified accurately relative to Reference-‐‑
Composition Seawater with its known composition, but only uncertainly with respect
to IAPSO Standard Seawater with its unknown composition. These variations in the
composition of seawater cause significant (a few percent) variations in the horizontal
density gradient.
Regarding point number 2, Practical Salinity S P is a dimensionless number of the order of
35 in the open ocean; no units or their multiples are permitted. There is however more
freedom in choosing the representation of Absolute Salinity S A since it is defined as the mass
fraction of dissolved material in seawater. For example, all the following quantities are equal
(see ISO (1993) and BIPM (2006)),
34 g/kg = 34 mg/g = 0.034 kg/kg = 0.034 = 3.4 % = 34 000 ppm = 34 000 mg/kg.
In particular, it is strictly correct to write the freshwater fraction of seawater as either
(1 – 0.001 S A /( g kg −1 )) or as (1 – S A ) but it would be incorrect to write it as (1 – 0.001 S A ).
Clearly it is essential to consider the units used for Absolute Salinity in any particular
application. If this is done, there should be no danger of confusion, but to maintain the
numerical value of Absolute Salinity close to that of Practical Salinity S P we adopt the first
option above, namely g kg −1 as the preferred unit for SA , (as in S A = 35.165 04 g kg−1). The
Reference Salinity, SR , is defined to have the same units and follows the same conventions as
SA . Salinity “S‰” measured prior to PSS-‐‑78 available from the literature or from databases is
usually reported in ‰ or ppt (part per thousand) and is converted to the Reference Salinity,
S R = uPS S‰, by the numerical factor uPS from (A.3.3).
Regarding point number 5, Chlorinity Cl is the concentration variable that was used in
the laboratory experiments for the fundamental determinations of the equation of state and
other properties, but has seldom been measured in the field since the definition of PSS-‐‑78
(Millero, 2010). Since the relation SP = 1.806 55 Cl for Standard Seawater was used in the
definition of Practical Salinity this may be taken as an exact relation for Standard Seawater
and it is also our best estimate for Reference-‐‑Composition Seawater. Thus, Chlorinity
expressed in ‰ can be converted to Reference-‐‑Composition Salinity by the
relation, SR = uCl Cl , with the numerical factor uCl = 1.806 55 uPS. These constants are
recommended for the conversion of historical (pre 1900) data. The primary source of error in
using this relation will be the possible presence of composition anomalies in the historical data
relative to Standard Seawater.
Regarding point number 6, the composition of dissolved material in seawater is not
constant but varies a little from one ocean basin to another, and the variation is even stronger
in estuaries, semi-‐‑enclosed or even enclosed seas. Brewer and Bradshaw (1975) and Millero
(2000) point out that these spatial variations in the relative composition of seawater impact the
relationship between Practical Salinity (which is essentially a measure of the conductivity of
seawater at a fixed temperature and pressure) and density. All the thermophysical properties
of seawater as well as other multicomponent electrolyte solutions are directly related to the
concentrations of the major components, not the salinity determined by conductivity; note
that some of the variable nonelectrolytes (e.g., Si(OH)4 , CO2 and dissolved organic material)
do not have an appreciable conductivity signal. It is for this reason that the TEOS-‐‑10
thermodynamic description of seawater has the Gibbs function g of seawater expressed as a
function of Absolute Salinity as g ( SA , t , p ) rather than as a function of Practical Salinity S P or
of Reference Salinity, SR . The issue of the spatial variation in the composition of seawater is
discussed more fully in appendix A.5.
IOC Manuals and Guides No. 56
78
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Regarding point number 2, we note that it is debatable which of (1 – 0.001 SAdens /( g kg −1 )),
(1 – 0.001 S Asoln /( g kg −1 )), (1 – 0.001 SAadd /( g kg −1 )) or (1 – 0.001 S* /( g kg −1 )) is the most
appropriate measure of the freshwater mass fraction. (These different versions of absolute
salinity are defined in section 2.5 and also later in this appendix.) This is a minor point
compared with the present use of (1 – 0.001 SP ) in this context, and the choice of which of the
above expressions may depend on the use for the freshwater mass fraction. For example, in
the context of ocean modelling, if S* is the salinity variable that is treated as a conservative
variable in an ocean model, then (1 – 0.001 S* /( g kg −1 )) is probably the most appropriate
version of freshwater mass fraction.
It should be noted that the quantity S A appearing as an argument of the function
g ( SA , t, p ) is the Absolute Salinity (the “Density Salinity” SA ≡ SAdens ) measured on the
Reference-‐‑Composition Salinity Scale. This is important since the Gibbs function has been
fitted to laboratory and field measurements with the Absolute Salinity values expressed on
this scale. Thus, for example, it is possible that sometime in the future it will be determined
that an improved estimate of the mass fraction of dissolved material in Standard Seawater can
be obtained by multiplying S R by a factor slightly different from 1 (uncertainties permit
values in the range 1 ± 0.002). We emphasize that since the Gibbs function is expressed in
terms of the Absolute Salinity expressed on the Reference-‐‑Composition Salinity Scale, use of
any other scale (even one that gives more accurate estimates of the true mass fraction of
dissolved substances in Standard Seawater) will reduce the accuracy of the thermodynamic
properties determined from the Gibbs function. In part for this reason, we recommend that
the Reference-‐‑Composition Salinity continue to be measured on the scale defined by Millero et
al. (2008a) even if new results indicate that improved estimates of the true mass fraction can be
obtained using a modified scale. That is, we recommend that the value of uPS used in (A.3.3)
not be updated. If a more accurate mass fraction estimate is required for some purpose in the
future, such a revised estimate should definitely not be used as an argument of the
TEOS-‐‑10 Gibbs function.
Finally, we note a second reason for recommending that the value assigned to uPS not be
modified without very careful consideration. Under TEOS-‐‑10, Absolute Salinity replaces
Practical Salinity as the salinity variable in publications, and it is critically important that this
new measure of salinity remain stable into the future. In particular, we note that any change
in the value of uPS used in the determination of Reference Salinity would result in a change in
reported salinity values that would be unrelated to any real physical change. For example, a
change in uPS from 35.16504/35 to (35.16504/35) x 1.001 for example, would result in changes
of the reported salinity values of order 0.035 g kg −1 which is more than ten times the precision
of modern salinometers. Thus changes associated with a series of improved estimates of uPS
(as a measure of the mass fraction of dissolved salts in Standard Seawater) could cause very
serious confusion for researchers who monitor salinity as an indicator of climate change.
Based on this concern, and the fact that the Gibbs function is expressed as a function of
Absolute Salinity measured on the Reference-‐‑Composition Salinity Scale as defined by Millero
et al. (2008a), no changes in the value of uPS should be introduced.
For seawater of Reference Composition, Reference Salinity S R is the best available
estimate of the mass-‐‑fraction of non-‐‑H2O material in seawater. As discussed in sections 2.4
and 2.5, under TEOS-‐‑10 S R was determined to provide the best available estimate of the
mass-‐‑fraction of non-‐‑H2O material in Standard Seawater by Millero et al. (2008a).
Subsequently, Pawlowicz (2010a) has argued that the DIC content of the Reference
Composition is probably about 117 µ mol kg −1 low for SSW and also for the North Atlantic
surface water from which it was prepared. This difference in DIC causes a negligible effect on
both conductivity and density, and hence on Reference Salinity and Density Salinity. The
influence on Solution Salinity is nearly a factor of 10 larger (Pawlowicz et al., 2011) but at
0.0055 g kg −1 it is still just below the uncertainty of 0.007 g kg −1 assigned to the estimated
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
79
Absolute Salinity by Millero et al. (2008a). In fact, the largest uncertainties in Reference
Salinity as a measure of the Absolute Salinity of SSW are associated with uncertainties in the
mass fractions of other constituents such as sulphate, which may be as large as 0.05 g kg −1
(Seitz et al., 2010). Nevertheless, it seems that the sulphate value of Reference-‐‑Composition
Seawater lies within the 95% uncertainty range of the best laboratory-‐‑determined estimates of
SSW’s sulphate concentration.
When the composition of seawater differs from that of Standard Seawater, there are
several possible definitions of the absolute salinity of a seawater sample, as discussed in
section 2.5. Conceptually the simplest definition is “the mass fraction of dissolved non-‐‑ H2O
material in a seawater sample at its temperature and pressure”. One drawback of this
definition is that because the equilibrium conditions between H2O and several carbon
compounds depends on temperature and pressure, this mass-‐‑fraction would change as the
temperature and pressure of the sample is changed, even without the addition or loss of any
material from the sample. This drawback can be overcome by first bringing the sample to the
constant temperature t = 25°C and the fixed sea pressure 0 dbar, and when this is done, the
resulting mass-‐‑fraction of non-‐‑ H2O material is called “Solution Absolute Salinity” (usually
shortened to “Solution Salinity”), S Asoln . Another measure of absolute salinity is the “Added-‐‑
Mass Salinity” SAadd which is S R plus the mass fraction of material that must be added to
Standard Seawater to arrive at the concentrations of all the species in the given seawater
sample, after chemical equilibrium has been reached, and after the sample has been brought
to t = 25°C and p = 0 dbar.
Another form of absolute salinity, “Preformed Absolute Salinity” (usually shortened to
“Preformed Salinity”), S* , has been defined by Pawlowicz et al. (2011) and Wright et al.
(2011). Preformed Salinity S* is designed to be as close as possible to being a conservative
variable. That is, S* is designed to be insensitive to the biogeochemical processes that affect
the other types of salinity to varying degrees. S* is formed by first estimating the contribution
of biogeochemical processes to one of the salinity measures S A , S Asoln , or SAadd , and then
subtracting this contribution from the appropriate salinity variable. Because it is designed to
be a conservative oceanographic variable, S* will find a prominent role in ocean modeling.
Since S* is designed to be a conservative salinity variable, it would appear to also be the
best choice for the salinity variable in inverse models. An argument can also be made that S*
should be the salinity variable that is used as an axis of the traditional “ S − θ diagram”, which
would then become the S* −Θ diagram. However, this argument is resisted because potential
density contours cannot be drawn on the S* −Θ diagram because density is a function of
Absolute Salinity, not of Preformed Salinity.
There are no simple methods available to measure either S Asoln or SAadd for the general case
of the arbitrary addition of many components to Standard Seawater. Hence a more precise
and easily determined measure of the amount of dissolved material in seawater is required,
and TEOS-‐‑10 adopts “Density Salinity” SAdens for this purpose. “Density Salinity” SAdens is
defined as the value of the salinity argument of the TEOS-‐‑10 expression for density which
gives the sample’s actual measured density at the temperature t = 25°C and at the sea
pressure p = 0 dbar. When there is no risk of confusion, “Density Salinity” is also called
Absolute Salinity with the label S A , that is SA ≡ SAdens . There are two clear advantages of
SA ≡ SAdens over both S Asoln and SAadd . First, it is possible to measure the density of a seawater
sample very accurately and in an SI-‐‑traceable manner, and second, the use of SA ≡ SAdens yields
the best available estimates of the density of seawater. This is important because amongst
various thermodynamic properties in the field of physical oceanography, it is density that
needs to be known to the highest relative accuracy.
Pawlowicz et al. (2011) and Wright et al. (2011) found that while the nature of the ocean’s
composition variations changes from one ocean basin to another, the five different salinity
measures S R , SAdens , S Asoln , SAadd and S* are approximately related by the following simple
IOC Manuals and Guides No. 56
80
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
linear relationships, (obtained by combining equations (55) – (57) and (62) of Pawlowicz et al.
(2011))
S∗ − SR ≈ − 0.35 δ SA ,
Eqn.
(A.4.2)
is
(A.4.1)
SAdens − SR ≡ 1.0 δ SA ,
(A.4.2)
SAsoln − SR ≈ 1.75 δ SA ,
(A.4.3)
SAadd − SR ≈ 0.78 δ SA .
(A.4.4)
simply
the
definition
of
the
Absolute
Salinity
Anomaly,
δ SA ≡ δ SRdens ≡ SAdens − SR . Note that here and in many TEOS-‐‑10 publications, the simpler
notation δ SA is used for δ SRdens ≡ SAdens − SR , a salinity difference that can now be estimated
from a global atlas (McDougall et al. (2012)).
In the context of ocean modelling, it is more convenient to cast these salinity differences
with respect to the Preformed Salinity S∗ as follows (using the above equations)
SR − S∗ ≈ 0.35 δ SA ,
(A.4.5)
SAdens − S∗ ≈ 1.35 δ SA ,
(A.4.6)
SAsoln − S* ≈ 2.1 δ SA ,
(A.4.7)
SAadd − S∗ ≈ 1.13 δ SA .
(A.4.8)
For SSW, all five salinity variables S R , SA ≡ SAdens , S Asoln , SAadd and S* are equal. The
relationships (A.4.1), (A.4.2), (A.4.5) and (A.4.6) are illustrated on the number line of salinity
in Figure A.4.1. It should be noted that the simple relationships of Eqns. (A.4.1) – (A.4.8) are
derived from simple linear fits to model calculations that show more complex variations.
However, the variation about these relationships is not larger than the typical uncertainty of
ocean measurements. Eqn. (A.4.6) provides a way by which the effects of anomalous seawater
composition may be addressed in ocean models (see appendix A.20).
Figure A.4.1. Number line of salinity, illustrating the differences between
Preformed Salinity S* , Reference Salinity S R , and Absolute
Salinity S A for seawater whose composition differs from that
of Standard Seawater.
If measurements are available of the Total Alkalinity, Dissolved Inorganic Carbon, and the
nitrate and silicate concentrations, but not of density anomalies, then alternative formulae are
available for the salinity differences that appear on the left-‐‑hand sides of Eqns. (A.4.1) –
(A.4.8). Pawlowicz et al. (2011) have used a chemical model of conductivity and density to
estimate how the many salinity differences introduced above depend on the measured
properties of seawater. The following equations correspond to Eqns. (A.4.1) – (A.4.4) above,
and come from equations (51) – (54) and (59) of Pawlowicz et al. (2011). These equations are
written in terms of the values of the nitrate and silicate concentrations in the seawater sample
(measured in mol kg −1 ), the difference between the Total Alkalinity ( TA ) and Dissolved
Inorganic Carbon ( DIC ) of the sample and the corresponding values of our best estimates of
TA and DIC in Standard Seawater, ΔTA and ΔDIC , both measured in mol kg −1 . For
Standard Seawater our best estimates of TA and DIC are 0.0023 ( SP 35) mol kg −1 and
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
81
0.00208 ( SP 35) mol kg −1 respectively (see Pawlowicz (2010a), Pawlowicz et al. (2011) and the
discussion of this aspect of SSW versus RCSW in Wright et al. (2011))).
(
)
(mol kg −1 ) , (A.4.9)
)
(
)
(mol kg −1 ) , (A.4.10)
)
(
)
(mol kg−1 ) , (A.4.11)
)
(
)
(mol kg−1 ) . (A.4.12)
( S* − SR ) / (g kg−1 )
(S
dens
A
(S
− SR / (g kg −1 ) = 55.6 ΔTA + 4.7 ΔDIC+38.9 NO3− + 50.7 Si(OH)4
soln
A
(S
= −18.1 ΔTA − 7.1 ΔDIC − 43.0 NO3− + 0.1 Si(OH)4
add
A
− SR / (g kg −1 ) = 7.2 ΔTA + 47.0 ΔDIC+36.5 NO3− + 96.0 Si(OH)4
− SR / (g kg −1 ) = 25.9 ΔTA + 4.9 ΔDIC+16.1NO3− + 60.2 Si(OH) 4
The standard error of the model fits in Eqns. (A.4.9) – (A.4.11) are given by Pawlowicz et al.
(2011) at less than 10−4 kg m −3 (in terms of density) which is equivalent to a factor of 20
smaller than the accuracy to which Practical Salinity can be measured at sea. It is clear that if
measurements of TA, DIC, nitrate and silicate are available (and recognizing that these
measurements will come with their own error bars), these expressions will likely give more
accurate estimates of the salinity differences than the approximate linear expressions
presented in Eqns. (A.4.1) – (A.4.8). The coefficients in Eqn. (A.4.10) are reasonably similar to
the corresponding expression of Brewer and Bradshaw (1975) (as corrected by Millero et al.
(1976a)):-‐‑ when expressed as the salinity anomaly SAdens − SR rather than as the corresponding
density anomaly ρ − ρ R , their expression corresponding to Eqn. (A.4.10) had the coefficients
71.4, -‐‑12.8, 31.9 and 59.9 compared with the coefficients 55.6, 4.7, 38.9 and 50.7 respectively in
Eqn. (A.4.10).
The salinity differences expressed with respect to Preformed Salinity S* which correspond
to Eqns. (A.4.5) – (A.4.8) can be found by linear combinations of Eqns. (A.4.9) – (A.4.12) as
follows
( SR − S* ) / (g kg−1 )
(S
(S
(
= 18.1 ΔTA + 7.1 ΔDIC + 43.0 NO3− − 0.1 Si(OH)4
)
(mol kg −1 ) , (A.4.13)
)
(
)
(mol kg −1 ) , (A.4.14)
)
(
)
(mol kg−1 ) , (A.4.15)
)
(
)
(mol kg−1 ) . (A.4.16)
dens
A
− S* / (g kg −1 ) = 73.7 ΔTA + 11.8 ΔDIC+81.9 NO3− + 50.6 Si(OH)4
soln
A
− S* / (g kg −1 ) = 25.3 ΔTA + 54.1 ΔDIC+79.5 NO3− + 95.9 Si(OH)4
(S
add
A
− S* / (g kg −1 ) = 44.0 ΔTA + 12.0 ΔDIC+59.1NO3− + 60.1 Si(OH)4
IOC Manuals and Guides No. 56
82
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
A.5 Spatial variations in seawater composition
When the oceanographic data needed to evaluate Eqn. (A.4.10) for SAdens − SR ≡ SA − SR is
not available, the look-‐‑up table method of McDougall et al. (2012) is recommended to evaluate
δ SA ≡ δ SRdens ≡ SA − SR . The following describes how this method was developed.
In a series of papers Millero et al. (1976a, 1978, 2000, 2008b) and McDougall et al. (2012)
have reported on density measurements made in the laboratory on samples collected from
around the world’s oceans. Each sample had its Practical Salinity measured in the laboratory
as well as its density (measured with a vibrating tube densimeter at 25 °C and atmospheric
pressure). The Practical Salinity yields a Reference Salinity S R according to Eqn. (A.3.3),
while the density measurement ρ meas implies an Absolute Salinity SA ≡ SAdens by using the
equation of state and the equality ρ meas = ρ SAdens , 25 °C, 0dbar . The difference SAdens − SR
between these two salinity measures is taken to be due to the composition of the sample being
different to that of Standard Seawater. In these papers Millero established that the salinity
difference SA − SR could be estimated approximately from knowledge of just the silicate
concentration of the fluid sample. The reason for the explaining power of silicate alone is
thought to be that (a) it is itself substantially correlated with other relevant variables (e.g. total
alkalinity, nitrate concentration, DIC [often called total carbon dioxide]), (b) it accounts for a
substantial fraction (about 0.6) of the typical variations in concentrations (g kg −1 ) of the above
species and (c) being essentially non-‐‑ionic; its presence has little effect on conductivity while
having a direct effect on density.
When the existing δ SA data, based on laboratory measurements of density, was regressed
against the silicate concentration of the seawater samples, McDougall et al. (2012) found the
simple relation
(
)
(
)
δ SA / (g kg −1 ) = ( SA − SR ) / (g kg −1 ) = 98.24 Si(OH)4 / (mol kg −1 ) .
Global (A.5.1)
This regression was done over all available density measurements from the world ocean, and
the standard error of the fit was 0.0054 g kg −1.
The dependence of δ SA on silicate concentration is observed to be different in each ocean
basin, and this aspect was exploited by McDougall et al. (2012) to obtain a more accurate
dependence of δ SA on location in space. For data in the Southern Ocean south of 30oS the
best simple fit was found to be
(
)
δ SA / (g kg −1 ) = 74.884 Si(OH)4 / (mol kg −1 ) ,
Southern Ocean (A.5.2)
and the associated standard error is 0.0026 g kg −1 .
The data north of 30oS in each of the Pacific, Indian and Atlantic Oceans was treated
separately. In each of these three regions the fit was constrained to match (A.5.2) at 30oS and
the slope of the fit was allowed to vary linearly with latitude. The resulting fits were (for
latitudes north of 30oS, that is for λ ≥ − 30° )
(
)
) = 74.884 (1+ 0.3861[λ / 30°+1]) (Si(OH) / (mol kg ) ) ,
) = 74.884 (1 +1.0028 [λ / 30°+1]) (Si(OH) / (mol kg ) ) .
δ SA / (g kg −1 ) = 74.884 (1+ 0.3622 [λ / 30°+1]) Si(OH)4 / (mol kg −1 ) ,
Pacific (A.5.3)
δ SA / (g kg −1
−1
4
Indian (A.5.4)
δ SA / (g kg −1
−1
4
Atlantic (A.5.5)
These relationships between the Absolute Salinity Anomaly δ SA = SA − SR and silicate
concentration have been used by McDougall et al. (2012) in a computer algorithm that uses an
existing global data base of silicate (Gouretski and Koltermann (2004)) and provides an
estimate of Absolute Salinity when given a seawater sample’s Practical Salinity as well as its
spatial location in the world ocean.
Since version 3.0, this computer algorithm works as follows. The values of both the
Reference Salinity and the Absolute Salinity Anomaly, calculated from the global Gouretski
and Koltermann (2004) hydrographic atlas using Eqns. (A.5.2) – (A.5.5), were used to form the
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
83
ratio Rδ ≡ δ SAatlas S Ratlas of these atlas values of Absolute Salinity Anomaly and Reference
Salinity. These values of the Absolute Salinity Anomaly Ratio, Rδ , were stored as a function
of latitude, longitude and pressure on a regular 4° × 4° grid in latitude and longitude. These
values of Rδ are interpolated onto the latitude, longitude and pressure of an oceanographic
observation (the details of the interpolation method can be found in McDougall et al. (2012))
and the Absolute Salinity Anomaly δ SA of an oceanographic observation is calculated from
δ SA = Rδ SR where Rδ ≡ δ SAatlas S Ratlas ,
(A.5.6)
where S R is the Reference Salinity of the oceanographic observation. For the bulk of the
ocean this expression for δ SA is almost the same as simply setting δ SA equal to δ SAatlas , but
the use of Eqn. (A.5.6) is preferable in situations where the sample’s Reference Salinity is
small, such as in rivers, in estuaries and after a rain shower at the sea surface in the open
ocean. In these situations the influence of the ocean’s biogeochemical processes on δ SA
should approach zero as S R approaches zero, and this is achieved by Eqn. (A.5.6).
Where the nutrient and carbon chemistry data are available to evaluate Eqn. (A.4.10), the
results obtained are similar although not identical to those obtained from Eqn. (A.5.6) using
the McDougall et al. (2012) algorithm.
The relationships between the three salinity variables SA , S* and S R are found as follows.
First we note the relationships between these salinities (from Eqns. (A.4.2), (A.4.1) and (A.4.6))
SA = SR + δ SA ,
(A.5.7)
S* = SR − r1δ SA ,
(A.5.8)
SA = S* + (1+ r1 )δ SA .
(A.5.9)
Substituting Eqn. (A.5.6) into these equations gives the following simple linear relationships
between the three different salinities,
(
)
(1 − r Rδ ) ,
(1 + R ) = S 1+ F where F
( )
(1 − r R )
SA = SR 1 + Rδ ,
(A.5.10)
S* = SR
(A.5.11)
1
δ
SA = S*
δ
δ
δ
*
1
=
[1+ r1 ] Rδ
(1 − r Rδ )
.
(A.5.12)
1
These three equations are used in the six functions in the GSW Oceanographic Toolbox that
relate one salinity variable to another, where r1 is taken to be 0.35 while Rδ is obtained from
the look-‐‑up table of McDougall et al. (2012).
This approach has so far assumed that the Absolute Salinity Anomaly in fresh water is
zero. This is usually a good assumption for rainwater, but is often not true of water in rivers.
For example, the river water flowing into the Baltic has an Absolute Salinity Anomaly of
approximately 0.087 g kg −1 . When one has knowledge of the Absolute Salinity Anomaly due
to river water inflow, this can be incorporated as follows
δ SA = Rδ SR + δ SAriver ,
leading to (using Eqn. (A.5.7))
(
SA = SR 1 + Rδ
) + δS
river
A
(A.5.13)
.
(A.5.14)
In turn, an estimate for δ SAriver might be constructed in the vicinity of a particular river from
prior knowledge of the Absolute Salinity Anomaly at the river mouth δ SAriver_mouth (this is
actually the Absolute Salinity Anomaly appropriate for river water extrapolated to S R = 0 ) by
a formula such as (drawing inspiration from the formula for the Baltic, see below)
(
)
δ SAriver = δ SAriver_mouth 1 − S R S Ratlas .
(A.5.15)
IOC Manuals and Guides No. 56
84
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
The computer algorithm of McDougall et al. (2012) accounts for the latest understanding
of Absolute Salinity in the Baltic Sea, but it is silent on the influence of compositional
variations in other marginal seas. The Absolute Salinity Anomaly in the Baltic Sea has been
quite variable over the past few decades of observation (Feistel et al. (2010c)). The computer
algorithm of McDougall et al. (2012) uses the relationship found by Feistel et al. (2010c) that
applies in the years 2006-‐‑2009, namely
SA − SR = δ SA = 0.087 g kg −1 × (1 − SR SSO ) ,
Baltic (A.5.16)
where SSO = 35.165 04 g kg–1 is the standard-‐‑ocean Reference Salinity that corresponds to the
Practical Salinity of 35. The Absolute Salinity Anomaly in the Baltic Sea is not due to
biogeochemical activity, but rather is due to the rivers bringing material of anomalous
composition into the Baltic. Hence Absolute Salinity in the Baltic is a conservative variable
and Preformed Salinity is defined to be equal to Absolute Salinity in the Baltic. That is, in the
Baltic S* = SA , implying that r1 = − 1 and F δ = 0 (see Eqns. (A.5.7) – (A.5.9) and (A.5.12)).
In order to gauge the importance of the spatial variation of seawater composition, the
northward gradient of density at constant pressure is shown in Fig. A.5.1 for the data in a
world ocean hydrographic atlas deeper than 1000m. The vertical axis in this figure is the
magnitude of the difference between the northward density gradient at constant pressure
when the TEOS-‐‑10 algorithm for density is called with SA ≡ SAdens (as it should be) compared
with calling the same TEOS-‐‑10 density algorithm with SR as the salinity argument. Figure
A.5.1 shows that the “thermal wind” is misestimated by more than 2% for 58% of the data in
the world ocean below a depth of 1000m if the effects of the variable seawater composition are
ignored. When this same comparison is done for only the North Pacific, it is found that 60%
of the data deeper than 1000m has “thermal wind” misestimated by more than 10% if S R is
used in place of S A .
Figure A.5.1. The northward density gradient at constant pressure (the horizontal axis)
for data in the global ocean atlas of Gouretski and Koltermann (2004) for
p > 1000 dbar. The vertical axis is the magnitude of the difference
between evaluating the density gradient using S A versus S R as the
salinity argument in the TEOS-‐‑10 expression for density.
The importance of the spatial variations in seawater composition illustrated in Fig. A.5.1
can be compared with the corresponding improvement achieved by the TEOS-‐‑10 Gibbs
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
85
function for Standard Seawater compared with using EOS-‐‑80. This is done by ignoring spatial
variations in seawater composition in both the evaluation of TEOS-‐‑10 and in EOS80 by calling
TEOS-‐‑10 with S R and EOS-‐‑80 with SP . Figure A.5.2 shows the magnitude of the
improvement in the “thermal wind” in the part of the ocean that is deeper than 1000m
through the adoption of TEOS-‐‑10 but ignoring the influence of compositional variations. By
comparing Figs. A.5.1 and A.5.2 it is seen that the main benefit that TEOS-‐‑10 delivers to the
evaluation of the “thermal wind” is through the incorporation of spatial variations in seawater
composition; the greater accuracy of TEOS-‐‑10 over EOS-‐‑80 for Standard Seawater is only 17%
as large as the improvement gained by the incorporation of compositional variations into
TEOS-‐‑10 (i. e. the rms value of the vertical axis in Fig. A.5.2 is 17% of that of the vertical axis of
Fig. A.5.1). If the North Atlantic were excluded from this comparison, the relative importance
of compositional variations would be even larger.
Figure A.5.2. The northward density gradient at constant pressure (the horizontal axis)
for data in the global ocean atlas of Gouretski and Koltermann (2004) for
p > 1000 dbar . The vertical axis is the magnitude of the difference
between evaluating the density gradient using S R as the salinity
argument in the TEOS-‐‑10 expression for density compared with using SP
in the EOS-‐‑80 algorithm for density.
The thermodynamic description of seawater and of ice Ih as defined in IAPWS-‐‑08 and
IAPWS-‐‑06 has been adopted as the official description of seawater and of ice Ih by the
Intergovernmental Oceanographic Commission in June 2009. The adoption of TEOS-‐‑10 has
recognized that this technique of estimating Absolute Salinity from readily measured
quantities is perhaps the least mature aspect of the TEOS-‐‑10 thermodynamic description of
seawater. The present computer software, in both FORTRAN and MATLAB, which evaluates
Absolute Salinity S A given the input variables Practical Salinity S P , longitude λ , latitude φ
and sea pressure p is available at www.TEOS-10.org. It is expected, as new data (particularly
density data) become available, that the determination of Absolute Salinity will improve over
the coming decades, and the algorithm for evaluating Absolute Salinity in terms of Practical
Salinity, latitude, longitude and pressure, will be updated from time to time, after relevant
appropriately peer-‐‑reviewed publications have appeared, and such an updated algorithm will
appear on the www.TEOS-10.org web site. Users of this software should state in their
published work which version of the software was used to calculate Absolute Salinity.
IOC Manuals and Guides No. 56
86
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
A.6 Gibbs function of seawater
The Gibbs function of seawater g ( SA , t , p ) is defined as the sum of the Gibbs function for pure
water g W ( t , p ) and the saline part of the Gibbs function g S ( S A , t , p ) so that
g ( SA , t , p ) = g W ( t , p ) + g S ( S A , t , p ) .
(A.6.1)
In this way at zero Absolute Salinity, the thermodynamic properties of seawater are equal to
those of pure water. This consistency is also maintained with respect to the Gibbs function for
ice so that the properties along the equilibrium curve can be accurately determined (such as
the freezing temperature as a function of Absolute Salinity and pressure). The careful
alignment of the thermodynamic potentials of pure water, ice Ih and seawater is described in
Feistel et al. (2008a).
The internationally accepted thermodynamic description of the properties of pure water
(IAPWS-‐‑95) is the official pure-‐‑water basis upon which the Gibbs function of seawater is built
according to (A.6.1). This g W ( t , p ) Gibbs function of liquid water is valid over extended
ranges of temperature and pressure from the freezing point to the critical point (–22 °C < t <
374 °C and 600 Pa < p + P0 < 1000 MPa) however it is a computationally expensive algorithm.
Part of the reason for this computational intensity is that the IAPWS-‐‑95 formulation is in
terms of a Helmholtz function which has the pressure as a function of temperature and
density, so that an iterative procedure is needed to form the Gibbs function g W ( t , p ) (see for
example, Feistel et al. (2008a)).
For practical oceanographic use in the oceanographic ranges of temperature and pressure,
from less than the freezing temperature of seawater (at any pressure), up to 40 °C (specifically
from − ⎡2.65 + ( p + P0 ) × 0.0743 MPa −1 ⎤ °C to 40 °C), and in the pressure range 0 < p < 104 dbar
⎣
⎦
we also recommend the use of the pure water part of the Gibbs function of Feistel (2003)
which has been approved by IAPWS as the Supplementary Release, IAPWS-‐‑09. The IAPWS-‐‑
09 release discusses the accuracy to which the Feistel (2003) Gibbs function fits the underlying
thermodynamic potential of IAPWS-‐‑95; in summary, for the variables density, thermal
expansion coefficient and specific heat capacity, the rms misfit between IAPWS-‐‑09 and
IAPWS-‐‑95, in the region of validity of IAPWS-‐‑09, are a factor of between 20 and 100 less than
the corresponding error in the laboratory data to which IAPWS-‐‑95 was fitted. Hence, in the
oceanographic range of parameters, IAPWS-‐‑09 and IAPWS-‐‑95 may be regarded as equally
accurate thermodynamic descriptions of pure liquid water.
Many of the thermodynamic properties of seawater that are described in this Manual are
available as both FORTRAN and MATLAB implementations. These implementations are
available for g W ( t , p ) being IAPWS-‐‑95 and IAPWS-‐‑09, both being equally accurate relative to
the laboratory-‐‑determined known properties, but with the computer code based on IAPWS-‐‑09
being approximately a factor of 65 faster than that based on IAPWS-‐‑95.
Most of the experimental seawater data that were already used for the construction of
EOS-‐‑80 were exploited again for the IAPWS-‐‑08 formulation after their careful adjustment to
the new temperature and salinity scales and the improved pure-‐‑water reference IAPWS-‐‑95.
Additionally, IAPWS-‐‑08 was significantly improved (compared with EOS-‐‑80) by making use
of theoretical relations such as the ideal-‐‑solution law and the Debye-‐‑Hückel limiting law, as
well as by incorporating additional accurate measurements such as the temperatures of
maximum density, vapour pressures and mixing heats, and implicitly by the enormous
background data set which underlies IAPWS-‐‑95 (Wagner and Pruß (2002), Feistel (2003,
2008)). For example, Millero and Li (1994) concluded that the pure-‐‑water part of the EOS-‐‑80
sound-‐‑speed formula of Chen and Millero (1977) was responsible for a deviation of 0.5 m s−1
from Del Grosso’s (1974) formula for seawater at high pressures and temperature below 5 oC.
Chen and Millero (1977) only measured the differences in the sound speeds of seawater and
pure water. The new Gibbs function in which we use IAPWS-‐‑95 for the pure-‐‑water part as
well as sound speeds from Del Grosso (1974), is perfectly consistent with Chen and Millero’s
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
87
(1976) densities and Bradshaw and Schleicher’s (1970) thermal expansion data at high
pressures. The accuracy of high-‐‑pressure seawater densities has increased with the use of
IAPWS-‐‑95, directly as the pure-‐‑water part, and indirectly by correcting earlier seawater
measurements, making them "ʺnew"ʺ seawater data. In this manner the known sound-‐‑speed
inconsistency of EOS-‐‑80 has been resolved in a natural manner.
A.7 The fundamental thermodynamic relation
The fundamental thermodynamic relation for a system composed of a solvent (water) and a
solute (sea salt) relates the total differentials of thermodynamic quantities for the case where
the transitions between equilibrium states are reversible. This restriction is satisfied for
infinitesimally small changes of an infinitesimally small seawater parcel. The fundamental
thermodynamic relation is
dh − v dP = (T0 + t ) dη + µ dSA .
(A.7.1)
A derivation of the fundamental thermodynamic relation can be found in Warren (2006) (his
equation (8)). The left-‐‑hand side of Eqn. (A.7.1) is often written as du + ( p + P0 ) dv where
( p + P0 ) = P is the Absolute Pressure. Here h is the specific enthalpy (i.e. enthalpy per unit
mass of seawater), u is the specific internal energy, v = ρ −1 is the specific volume, (T0 + t ) = T
is the absolute temperature, η is the specific entropy and µ is the relative chemical potential.
In fluid dynamics we usually deal with material derivatives, d dt , that is, derivatives defined
following the fluid motion, d dt = ∂ ∂t + u ⋅ ∇ where u is the fluid velocity. In terms of this
type of derivative, and assuming local thermodynamic equilibrium (i. e. that local
thermodynamic equilibrium is maintained during the temporal change), the fundamental
thermodynamic relation is
dh 1 dP
dη
dS
−
= (T0 + t )
+ µ A .
dt ρ dt
dt
dt
(A.7.2)
Note that the constancy of entropy in a given situation does not imply the absence of
irreversible processes because, for example, there can be irreversible changes of both salinity
and enthalpy at constant pressure in just the right ratio so as to have equal effects in Eqns.
(A.7.1) or (A.7.2) so that the change of entropy in these equations is zero.
A.8 The “conservative” and “isobaric conservative” properties
A thermodynamic variable C is said to be “conservative” if its evolution equation (that is, its
prognostic equation) has the form
dC
(A.8.1)
= − ∇ ⋅ FC .
dt
For such a “conservative” property, in the absence of fluxes FC at the boundary of a control
volume, the total amount of C-‐‑stuff is constant inside the control volume. The middle part of
Eqn. (A.8.1) has used the continuity equation (which is the equation for the conservation of
mass)
(A.8.2)
∂ρ ∂t x , y , z + ∇ ⋅ ( ρ u ) = 0 .
( ρ C )t + ∇ ⋅ ( ρ uC )
= ρ
In the special case when the material derivative of a property is zero (that is, the middle part
of Eqn. (A.8.1) is zero) the property is said to be “materially conserved”.
The only quantity that can be regarded as 100% conservative in the ocean is mass
[equivalent to taking C = 1 and FC = 0 in Eqn. (A.8.1)]. In fact, looking ahead to appendices
A.20 and A.21, if we strictly interpret ρ u as the mass flux per unit area of pure seawater (i. e.
of only pure water plus dissolved material) and specifically, that ρ u excludes the flux of
IOC Manuals and Guides No. 56
88
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
particulate matter, then the right-‐‑hand side of the continuity equation (A.8.2) should be ρ S A ,
the non-‐‑conservative source of mass due to biogeochemical processes. It can be shown that
S
the influence of this source term ρ S A in the continuity equation on the evolution equation
for Absolute Salinity is less important by the factor SˆA 1 − SˆA than the same source term
that appears in this evolution equation for Absolute Salinity, Eqn. (A.21.8). Hence the current
practice of assuming that the non-‐‑particulate part of the ocean obeys the conservative form
(A.8.2) of the continuity equation is confirmed even in the presence of biogeochemical
processes.
Two other variables, total energy E = u + 0.5 u ⋅ u + Φ (see Eqn. (B.15)) and Conservative
Temperature Θ (or equivalently, potential enthalpy h 0 ) are not completely conservative, but
the error in assuming them to be conservative is negligible (see appendix A.21). Other
variables such as Reference Salinity SR , Absolute Salinity SA , potential temperature θ ,
enthalpy h, internal energy u, entropy η , density ρ , potential density ρ θ , specific volume
anomaly δ and the Bernoulli function B = h + 0.5 u ⋅ u +Φ (see Eqn. (B.17)) are not
conservative variables.
While both Absolute Salinity and Reference Salinity are conservative under the turbulent
mixing process, both are affected in a non-‐‑conservative way by biogeochemical process.
Because the dominant variations of the composition of seawater are due to species which do
not have a strong signature in conductivity, in some situations it may be sufficiently accurate
to take Reference Salinity S R to be a conservative variable. However, we note that the error
involved with assuming that S R is a conservative variable is a factor of approximately 40
larger (in terms of its effects on density) than the error in assuming that Θ is a conservative
variable. Preformed Salinity S* is constructed so that it contains no signature of the
biogeochemical processes that cause the spatial variation of seawater composition. In this
way S* is specifically designed to be a conservative oceanic salinity variable. Having said
that, the accuracy with which we can construct Preformed Salinity S* from ocean
observations is presently limited by our knowledge of the biogeochemical processes (see
appendices A.4 -‐‑ A.5 and Pawlowicz et al. (2011)).
Summarizing this discussion thus far, the quantities that can be considered conservative
in the ocean are (in descending order of accuracy) (i) mass, (ii) total energy
E = u + 0.5 u ⋅ u + Φ , (iii) Conservative Temperature Θ , and (iv) Preformed Salinity S* .
A different form of “conservation” attribute, namely “isobaric conservation” occurs when
the total amount of the quantity is conserved when two fluid parcels are mixed at constant
pressure without external input of heat or matter. This “isobaric conservative” property is a
very valuable attribute for an oceanographic variable. Any “conservative” variable is also
“isobaric conservative”, thus the four conservative variables listed above, namely mass,
Conservative Temperature Θ , Preformed Salinity S* , and total energy E are almost (but not
exactly) “isobaric conservative”. In addition, the Bernoulli function B and specific enthalpy
h are also almost exactly “isobaric conservative” (see Eqn. (B.17) and the discussion
thereafter).
Some variables that are not “isobaric conservative” include potential temperature θ ,
internal energy u, entropy η , density ρ , potential density ρ θ , and specific volume anomaly
δ . Enthalpy h and Conservative Temperature Θ are not exactly “isobaric conservative”
because enthalpy increases when the kinetic energy of fluid motion is dissipated by molecular
viscosity inside the control volume and when there is a salinity source term due to the
remineralization of particulate matter. However, these are tiny effects in the First Law of
Thermodynamics (see appendix A.21) and traditionally we regard enthalpy h as an “isobaric
conservative” variable. Note that while h is “isobaric conservative”, it is not a “conservative”
variable.
Appendices A.18 and A.21 show that for all practical purposes we can treat Θ and h 0 as
being “conservative” variables (and hence also “isobaric conservative” variables); doing so
S
(
IOC Manuals and Guides No. 56
)
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
89
ignores the dissipation of kinetic energy ε and other terms of similar or smaller magnitude.
Hence for all practical purposes in oceanography we have mass and the following three other
variables that are “conservative” and “isobaric conservative”;
(1) Conservative Temperature Θ, (and potential enthalpy h 0 ),
(2) Preformed Salinity S* , and
(3) total energy E .
Here we comment briefly on the likely errors involved with treating variables other than
S* and Θ to be conservative variables in ocean models. If one took Absolute Salinity SA as
an ocean model’s salinity variable and treated it as being conservative, the salinity error
would (after a long spin-‐‑up time) be approximately as large as the Absolute Salinity Anomaly
(as shown in Figure 2), which is larger than 0.025 g kg −1 in the North Pacific, implying density
errors of 0.020 kg m−3 . As a measure of the importance of this type of density error, we note
that if the equation of state in an ocean model were called with S R instead of with S A , the
northward density gradient at fixed pressure (i. e. the thermal wind) would be misestimated
by more than 2% for more than 58% of the data below a pressure of 1000 dbar in the world
ocean. It is clearly desirable to not have this type of systematic error in the dynamical
equations of the ocean component of coupled climate models. Appendix A.20 discusses
practical ways of including the effects of the non-‐‑conservative biogeochemical source term in
ocean models. The recommended option is that ocean models carry Preformed Salinity S* as
the model’s conservative salinity model variable, and that they also carry an evolution
equation for as defined in Eqn. (A.5.12), as described in section A.20.1 and Eqns. (A.20.4) –
(A.20.6).
The errors incurred in ocean models by treating potential temperature θ as being
conservative have not yet been thoroughly investigated, but McDougall (2003), Tailleux (2010)
and Graham and McDougall (2013) have made a start on this topic. McDougall (2003) found
that typical errors in θ are ± 0.1°C while in isolated regions such as where the fresh Amazon
water discharges into the ocean, the error can be as large as 1.4 °C . The corresponding error
in the meridional heat flux appears to be about 0.005 PW (or a relative error of 0.4%). The use
of Conservative Temperature Θ in ocean models reduces these errors by approximately two
orders of magnitude.
If the ocean were in thermodynamic equilibrium, its temperature would be the same
everywhere as would the chemical potentials of water and of each dissolved species, while its
entropy and the concentrations of each species would be functions of pressure. Turbulent
mixing acts in the complementary direction, tending to homogenize the concentration of each
species and to make entropy constant, but in the process causing gradients in temperature and
the chemical potentials as functions of pressure. That is, turbulent mixing acts to maintain a
non-‐‑equilibrium state. This difference between the roles of molecular versus turbulent mixing
results from the symmetry breaking role of the gravity field; for example, in a laboratory
without gravity, turbulent and molecular mixing would have indistinguishable effects.
Note that the molecular flux of salt FS is given by equation (58.11) of Landau and Lifshitz
(1959) and by Eqn. (B.23) below. FS consists not only of the product of the usual molecular
diffusivity and −ρ∇SA , but also contains two other terms that are proportional to the
gradients of temperature and pressure respectively. It is these terms that cause the
equilibrium vertical gradients of the dissolved solutes in a non-‐‑turbulent ocean to be different
and non-‐‑zero; the last term being called the baro-‐‑diffusion effect. The presence of turbulent
mixing in the real ocean renders this process moot as turbulence tends to homogenize the
ocean and maintains a relatively constant sea-‐‑salt composition.
Note that the description “conservation equation” of a particular quantity is often used for
the equation that describes how this quantity changes in response to the divergence of various
fluxes of the quantity and to non-‐‑conservative “source” terms. For example, it is usual to refer
to the “conservation equation” for entropy or for “potential temperature”. Since these
IOC Manuals and Guides No. 56
90
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
variables are not conservative variables it seems unnatural to refer to their evolution
equations as “conservation equations”. Hence here we will use the term “conservation
equation” only for a variable that is (for all practical purposes) conserved. For other variables
we will refer to their “evolution equation” or their “prognostic equation” or their “local
balance equation”.
A.9 The “potential” property
Any thermodynamic property of seawater that remains constant when a parcel of seawater is
moved from one pressure to another adiabatically, without exchange of mass and without
interior conversion between its turbulent kinetic and internal energies, is said to possess the
“potential” property, or in other words, to be a “potential” variable. Prime examples of
“potential” variables are entropy η and all types of salinity. The constancy of entropy η can
be seen from the First Law of Thermodynamics in Eqn. (B.19) below; with the right-‐‑hand side
of Eqn. (B.19) being zero, and with no change in Absolute Salinity, it follows that entropy is
also constant. Any thermodynamic property that is a function of only Absolute Salinity and
entropy also remains unchanged by this procedure and is said to possess the “potential”
property. Thermodynamic properties that posses the “potential” attribute include potential
temperature θ , potential enthalpy h0 , Conservative Temperature Θ and potential density ρ θ
(no matter what fixed reference pressure is chosen). Some thermodynamic properties that do
not posses the potential property are temperature t , enthalpy h, internal energy u, specific
volume v, density ρ , specific volume anomaly δ , total energy E and the Bernoulli function
B . From Eqn. (B.17) we notice that in the absence of molecular fluxes and the source term of
Absolute Salinity, the Bernoulli function B is constant following the fluid flow only if the
pressure field is steady; in general this is not the case. The non-‐‑potential nature of E is
explained in the discussion following Eqn. (B.17).
Some authors have used the term “quasi-‐‑material” to describe a variable that has the
“potential” property. The name “quasi-‐‑material” derives from the idea that the variable only
changes as a result of irreversible mixing processes and does not change in response to
adiabatic and isohaline changes in pressure.
The word “adiabatic” is traditionally taken to mean a process during which there is no
exchange of heat between the environment and the fluid parcel one is considering. With this
definition of “adiabatic” it is still possible for the entropy η , the potential temperature θ and
the Conservative Temperature Θ of a fluid parcel to change during an isohaline and adiabatic
process. This is because the dissipation of kinetic energy ε causes increases in η , θ and Θ
(see the First Law of Thermodynamics, Eqns. (A.13.3) -‐‑ (A.13.5)). While the dissipation of
kinetic energy is a small term whose influence is routinely neglected in the First Law of
Thermodynamics in oceanography, it seems advisable to modify the meaning of the word
“adiabatic” in oceanography so that our use of the word more accurately reflects the
properties we normally associate with an adiabatic process. Accordingly we propose that the
word “adiabatic” in oceanography be taken to describe a process occurring without exchange
of heat and also without the internal dissipation of kinetic energy. With this definition of
“adiabatic”, a process that is both isohaline and adiabatic does imply that the entropy η ,
potential temperature θ and Conservative Temperature Θ are all constant.
Using this more restrictive definition of the word “adiabatic” we can restate the definition
of a “potential” property as follows; any thermodynamic property of seawater that remains
constant when a parcel of seawater is moved from one pressure to another “adiabatically” and
without exchange of mass, is said to possess the “potential” property, or in other words, to be
a “potential” variable.
In appendix A.8 above we concluded that only mass, and the three variables E , S* and
Θ (approximately) are “conservative” (and hence also “isobaric conservative”). Since E
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
91
does not posses the “potential” property, we now conclude that only mass and the two
variables S* and Θ posses all three highly desired properties, namely that they are
“conservative”, “isobaric conservative” and are “potential” variables. In the case of
Conservative Temperature Θ, its “conservative” (and therefore its “isobaric conservative”)
nature is approximate:-‐‑ while Θ is not a 100% conservative variable, it is approximately two
orders of magnitude closer to being a totally conservative variable than is potential
temperature. Similarly, Preformed Salinity S* is in principle 100% conservative, but our
ability to evaluate S* from hydrographic observations is limited (for example, by the
approximate relations (A.4.1) or (A.4.9)).
Table A.9.1 The “potential”, “conservative”, “isobaric conservative” and
the functional nature, of various oceanographic variables
Variable
“potential”?
S*
SA
SR , SP
t
x
θ
η
h
Θ, h 0
u
B
E
ρ,v
ρθ
δ
ρv
γn
x
“conservative”? “isobaric conservative”?
x
x
x
x
x
x
1
1
x
x
x
x
x
x
x
x
3 Taking
1
x
x
x
4
4
x
x
x
x
x
x
3
4
x
x
x
x
x
1 The remineralization of organic matter changes
2 Taking
x
1
2
3
x
x
x
x
function of ( SA , t, p ) ?
x
x
5
x
S R less than it changes SA .
ε and the effects of remineralization to be negligible.
ε and other terms of similar size to be negligible (see the discussion
following Eqn. (A.21.13)).
4 Taking the effects of remineralization to be negligible.
5 Once the reference sound speed function
c0 ( p, ρ ) has been decided upon.
In Table A.9.1 various oceanographic variables are categorized according to whether they
posses the “potential” property, whether they are “conservative” variables, whether they are
“isobaric conservative”, and whether they are functions of only ( SA , t, p ) . Note that Θ is the
only variable that achieve four “ticks” in this table, while Preformed Salinity S* has ticks in
the first three columns, but not in the last column since it is a function not only of ( SA , t, p )
(since it also depends on the composition of seawater). Hence Θ is the most “ideal”
thermodynamic variable. If it were not for the non-‐‑conservation of Absolute Salinity, it too
IOC Manuals and Guides No. 56
92
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
would be an “ideal” thermodynamic variable, but in this sense, Preformed Salinity is superior
to Absolute Salinity. Conservative Temperature Θ and Preformed Salinity S* are the only
two variables in this table to be both “potential” and “conservative”. The last four rows of
Table A.9.1 are for potential density, ρ θ (see section 3.4), specific volume anomaly, δ (see
section 3.7), orthobaric density, ρ v (see appendix A.28) and Neutral Density γ n (see section
3.14 and appendix A.29).
A.10 Proof that θ = θ ( SA ,η ) and Θ=Θ ( SA ,θ )
Consider changes occurring at the sea surface, (specifically at p = 0 dbar) where the
temperature is the same as the potential temperature referenced to 0 dbar and the increment
of pressure dp is zero. Regarding specific enthalpy h and chemical potential µ to be
functions of entropy η (in place of temperature t ), that is, considering the functional form of
h and µ to be h = h SA ,η , p and µ = µ SA ,η , p , it follows from the fundamental
thermodynamic relation (Eqn. (A.7.1)) that
hη SA ,η ,0 dη + hS SA ,η ,0 dSA = T0 + θ dη + µ SA ,η ,0 dSA ,
(A.10.1)
(
(
)
)
A
(
(
)
)
(
)
(
)
which shows that specific entropy η is simply a function of Absolute Salinity S A and
potential temperature θ , that is η = η SA ,θ , with no separate dependence on pressure. It
follows that θ = θ ( SA ,η ) .
Similarly, from the definition of potential enthalpy and Conservative Temperature in
Eqns. (3.2.1) and (3.3.1), at p = 0 dbar it can be seen that the fundamental thermodynamic
relation (A.7.1) implies
(
(
)
)
(
)
c0p dΘ = T0 + θ dη + µ SA ,θ ,0 dSA .
(A.10.2)
This shows that Conservative Temperature is also simply a function of Absolute Salinity and
potential temperature, Θ = Θ SA ,θ , with no separate dependence on pressure. It then follows
that Θ may also be expressed as a function of only S A and η . It follows that Θ has the
“potential” property.
(
)
A.11 Various isobaric derivatives of specific enthalpy
Because of the central role of enthalpy in the transport and the conservation of “heat” in the
ocean, the derivatives of specific enthalpy at constant pressure are here derived with respect
to Absolute Salinity and with respect to the three “temperature-‐‑like” variables η, θ and Θ as
well as in situ temperature t.
We begin by noting that the three standard derivatives of h = h ( SA , t, p ) when in situ
temperature t is taken as the “temperature-‐‑like” variable are
∂h ∂SA T , p = µ ( SA , t , p ) − (T0 + t ) µT ( SA , t , p ) ,
∂h ∂T
SA , p
∂h ∂P S
A ,T
(A.11.1)
= c p ( SA , t, p ) = (T0 + t ) ηT ( SA , t, p ) ,
(A.11.2)
= v ( SA , t , p ) − (T0 + t ) v T ( SA , t , p ) .
(A.11.3)
Now considering specific enthalpy to be a function of entropy (rather than of temperature
t ), that is, taking h = h SA ,η , p , the fundamental thermodynamic relation (A.7.1) becomes
hη dη + hS dSA = T0 + t dη + µ dSA while ∂ h ∂P
(A.11.4)
= v ,
(
)
(
A
so that
∂ h ∂η
SA , p
)
= T0 + t and ∂ h ∂SA
IOC Manuals and Guides No. 56
(
)
SA ,η
η, p
= µ .
(A.11.5)
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
93
Now taking specific enthalpy to be a function of potential temperature (rather than of in
situ temperature t ), that is, taking h = h SA ,θ , p , the fundamental thermodynamic relation
(A.7.1) becomes
(
(
)
)
hθ dθ + hS dSA = T0 + t dη + µ dSA while ∂ h ∂P
A
SA ,θ
= v.
(A.11.6)
To evaluate the hθ partial derivative, it is first written in terms of the derivative with respect
to entropy as
(A.11.7)
= hη
ηθ
= T0 + t ηθ ,
hθ
SA , p
(
SA , p
SA , p
)
SA
where (A.11.5) has been used. This equation can be evaluated at p = 0 when it becomes (the
potential temperature used here is referenced to pr = 0 )
hθ
SA , p=0
(
) (
)
= c p SA ,θ ,0 = T0 + θ ηθ
SA
.
(A.11.8)
These two equations are used to arrive at the desired expression for hθ namely
hθ
SA , p
(
= c p SA ,θ ,0
T +t
) ((T0 +θ))
(
) (
)
= − T0 + t gTT SA ,θ ,0 .
(A.11.9)
0
hS partial derivative, we first write specific enthalpy in the functional form
To evaluate the
A
h = h SA ,η SA ,θ , p and then differentiate it, finding
= hS
+ hη
ηS .
hS
(A.11.10)
(
(
) )
A θ, p
A η, p
A θ
SA , p
The partial derivative of specific entropy η = − gT (Eqn. (2.10.1)) with respect to Absolute
Salinity, ηS = − gS T , is also equal to − µT since chemical potential is defined by Eqn. (2.9.6)
A
A
as µ = gS . Since the partial derivative of entropy with respect to S A in (A.11.10) is
A
performed at fixed potential temperature (rather than at fixed in situ temperature), this is
equal to − µT evaluated at p = 0. Substituting both parts of (A.11.5) into (A.11.10) we have
the desired expression for hS namely
A
hS
A θ, p
(
) ( ) (
)
( SA ,t, p ) − (T0 + t ) gTS ( SA ,θ ,0).
= µ SA ,t, p − T0 + t µT SA ,θ ,0
= gS
A
(A.11.11)
A
Notice that this expression contains some things that are evaluated at the general pressure p
and one that is evaluated at the reference pressure pr = 0.
Now considering specific enthalpy to be a function of Conservative Temperature (rather
than of in situ temperature t ), that is, taking h = hˆ ( S A , Θ, p ) , the fundamental thermodynamic
relation (A.7.1) becomes
hˆΘ dΘ + hˆSA dSA = (T0 + t ) dη + µ dSA while ∂hˆ ∂P
SA , Θ
= v .
(A.11.12)
The partial derivative ĥΘ follows directly from this equation as
hˆΘ
= (T0 + t )ηΘ S
SA , p
A,p
= (T0 + t )ηΘ S .
A
(A.11.13)
At p = 0 this equation reduces to
hˆΘ
SA , p = 0
= c0p = (T0 + θ )ηΘ S ,
(A.11.14)
A
and combining these two equations gives the desired expression for ĥΘ namely
hˆΘ
SA , p
=
(T0 + t ) c0 .
(T0 + θ ) p
(A.11.15)
To evaluate the hˆS partial derivative we first write h in the functional form
A
h = h SA ,η SA ,Θ , p and then differentiate it, finding (using both parts of Eqn. (A.11.5))
(
(
) )
IOC Manuals and Guides No. 56
94
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
hˆSA
Θ, p
= µ ( SA , t, p ) + (T0 + t )ηSA .
(A.11.16)
Θ
The differential expression Eqn. (A.11.12) can be evaluated at p = 0 where the left-‐‑hand side
is simply c0p dΘ so that from Eqn. (A.11.12) we find that
ηSA
Θ
= −
µ ( SA ,θ ,0 )
,
(T0 + θ )
(A.11.17)
so that the desired expression for hˆS is
A
hˆSA
Θ, p
(T0 + t ) µ S ,θ ,0
(
)
(T0 + θ ) A
T0 + t )
(
g SA ( S A , t , p ) −
g ( S ,θ ,0 ).
(T0 + θ ) SA A
= µ ( SA , t , p ) −
=
(A.11.18)
The above boxed expressions for four different isobaric derivatives of specific enthalpy are
important as they are integral to forming the First Law of Thermodynamics in terms of
potential temperature and in terms of Conservative Temperature. The partial derivatives ĥΘ
and hˆS of Eqns. (A.11.15) and (A.11.18) can be calculated using the GSW Oceanographic
A
Toolbox function gsw_enthalpy_first_derivatives_CT_exact.
The second order partial derivatives ĥΘΘ , hˆS Θ and hˆS S can be written in terms of the
A
A A
seawater Gibbs function as (these second order partial derivatives can be evaluated using the
GSW Oceanographic Toolbox function gsw_enthalpy_second_derivatives_CT_exact.)
hΘΘ
SA , p
= hˆΘΘ =
(c )
2
0
p
(T0 + θ )
2
⎛ (T0 + t )
1
1
−
⎜⎜
,
,0
T
g
S
g
S
θ
θ
+
) TT ( A )
TT ( A , t , p )
⎝( 0
g S AT ( S A , t , p )
⎛ (T0 + t ) g SAT ( SA ,θ ,0 )
−
⎜⎜
gTT ( SA , t , p )
(T0 + θ ) ⎝ (T0 + θ ) gTT ( SA ,θ ,0 )
c0p
hˆSA Θ =
⎞
⎟⎟
⎠
⎞
⎟⎟ ,
⎠
c 0p g SA ( SA ,θ ,0 ) ⎛ (T0 + t )
⎞
1
1
−
−
⎜⎜
⎟,
2
gTT ( SA , t , p ) ⎟⎠
(T0 + θ )
⎝ (T0 + θ ) gTT ( SA ,θ ,0 )
hˆSA SA = g SA SA ( SA , t , p ) −
−
(A.11.20)
(T0 + t ) g
( S ,θ ,0)
(T0 + θ ) SA SA A
(T0 + t ) ⎡⎣ g SAT ( SA ,θ ,0)⎤⎦ − ⎡⎣ g SAT ( SA , t, p )⎤⎦
gTT ( S A , t , p )
(T0 + θ ) gTT ( SA ,θ ,0 )
g S AT ( S A , t , p ) ⎞
2 g SA ( SA , θ ,0 ) ⎛ (T0 + t ) g SAT ( SA , θ ,0 )
−
⎜⎜
⎟
gTT ( SA , t , p ) ⎟⎠
(T0 + θ ) ⎝ (T0 + θ ) gTT ( SA ,θ ,0)
2
+
(A.11.19)
2
(A.11.21)
⎡ g S ( SA , θ ,0 )⎤⎦ ⎛ (T0 + t )
⎞
1
1
−
+ ⎣ A
⎜
⎟.
2
gTT ( SA , t , p ) ⎠
(T0 + θ )
⎝ (T0 + θ ) gTT ( S A , θ ,0 )
2
The first order partial derivatives ĥΘ and hˆS evaluated from the 75-‐‑term expression for
A
specific volume, v̂ SA ,Θ, p , are available in the GSW Oceanographic Toolbox from the
function gsw_enthalpy_first_derivatives, while the second order partial derivatives ĥΘΘ ,
hˆSA Θ and hˆSA SA are evaluated from the same 75-‐‑term expression for specific volume by the
function gsw_enthalpy_second_derivatives.
(
)
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
95
A.12 Differential relationships between η , θ , Θ and SA
Evaluating the fundamental thermodynamic relation in the forms (A.11.6) and (A.11.12) and
using the four boxed equations in appendix A.11, we find the relations
(T0 + t ) dη + µ ( p ) dSA
=
=
(T0 + t ) c 0 dθ + ⎡ µ p − T + t µ 0 ⎤ dS
( )
⎣ ( ) ( 0 ) T ( )⎦ A
(T0 + θ ) p
(T0 + t ) c0 dΘ + ⎡ µ p − (T0 + t ) µ 0 ⎤ dS .
( )⎥ A
⎢ ( )
(T0 + θ ) p
(T0 + θ )
⎢⎣
⎥⎦
(A.12.1)
The quantity µ ( p ) dSA is now subtracted from each of these three expressions and the whole
equation is then multiplied by (T0 + θ ) (T0 + t ) obtaining
(T0 +θ ) dη
= c p (0) dθ − (T0 + θ ) µT (0) dSA = c0p dΘ − µ (0) dSA .
(A.12.2)
From this follows all the following partial derivatives between η, θ , Θ and SA ,
Θθ
Θη
θη
SA
= c p ( SA ,θ ,0) c0p , ΘSA = ⎡⎣ µ ( SA ,θ ,0) − (T0 + θ ) µT ( SA ,θ ,0)⎤⎦ c0p ,
θ
(A.12.3)
SA
= (T0 + θ ) c0p , ΘSA = µ ( SA ,θ ,0) c0p ,
(A.12.4)
SA
= (T0 + θ ) c p ( SA ,θ ,0) , θ SA = (T0 + θ ) µT ( SA ,θ ,0) c p ( SA ,θ ,0) ,
(A.12.5)
η
η
θΘ S = c0p c p ( SA ,θ ,0) , θSA = − ⎡⎣ µ ( SA ,θ ,0) − (T0 + θ ) µT ( SA ,θ ,0)⎤⎦ c p ( SA ,θ ,0) , (A.12.6)
A
Θ
ηθ
SA
= c p ( SA ,θ ,0) (T0 + θ ) , ηSA = − µT ( SA ,θ ,0) ,
(A.12.7)
ηΘ
SA
= c0p (T0 + θ ) , ηSA = − µ ( SA ,θ ,0) (T0 + θ ) .
(A.12.8)
θ
Θ
The three second order derivatives of ηˆ ( SA , Θ) are listed in Eqns. (P.14) and (P.15) of
appendix P. The corresponding derivatives of θˆ ( SA , Θ ) , namely θˆΘ , θˆS , θˆΘΘ , θˆS Θ and
A
A
θˆS S can also be derived using Eqn. (P.13), obtaining
A A
Θ
Θ
Θ
Θ
Θ
1
S
θ SA
SA θθ
+
θˆΘ = , θˆS = − A , θˆΘΘ = − θθ 3 , θˆS Θ = −
,
A
A
Θθ
Θθ
2
3
Θ
Θ
Θ
θ
θ
θ
( )
( )
( )
(A.12.9a,b,c,d)
⎞2
⎛Θ
Θ
Θ
Θ
S
θ
S
S
A
A
(A.12.10)
and θˆS S
− ⎜ A ⎟ θθ ,
A A
⎜
⎟
Θθ Θθ
θ
⎝ Θθ ⎠ Θθ
, Θ
, Θ
, Θ
in terms of the partial derivatives Θ
SA
θ SA and Θ SA SA which can be obtained by
θ
θθ
( S ,θ ) from the TEOS-‐‑10 Gibbs function.
differentiating the polynomial Θ
A
Θ
S S
= − A A + 2
Θ
A.13 The First Law of Thermodynamics
The law of the conservation of energy for thermodynamic equilibrium states was discovered
in the 19th century by Gibbs (1873) and other early pioneers. It was formulated as a balance
between internal energy, heat and work, similar to the fundamental thermodynamic equation
(A.7.1), and referred to as the First Law of Thermodynamics (Thomson (1851), Clausius (1876),
Alberty (2001)). Under the weaker condition of a local thermodynamic equilibrium
(Glansdorff and Prigogine (1971)), the original thermodynamic concepts can be suitably
generalized to describe irreversible processes of fluid dynamics which are subject to molecular
fluxes and macroscopic motion (Landau and Lifshitz (1959), de Groot and Mazur (1984)).
IOC Manuals and Guides No. 56
96
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
In some circles the “First Law of Thermodynamics” is used to describe the evolution
equation for total energy, being the sum of internal energy, potential energy and kinetic
energy. Here we follow the more common practice of regarding the First Law of
Thermodynamics as the difference between the conservation equation of total energy and the
evolution equation for the sum of kinetic energy and potential energy, leaving what might
loosely be termed the evolution equation of “heat”, Eqn. (A.13.1) (Landau and Lifshitz (1959),
de Groot and Mazur (1984), McDougall (2003), Griffies (2004)).
The First Law of Thermodynamics can therefore be written as (see Eqn. (B.19) and the
other Eqns. (A.13.3), (A.13.4) and (A.13.5) of this appendix; all of these equations are equally
valid incarnations of the First Law of Thermodynamics)
⎛ dh 1 dP ⎞
S
= − ∇ ⋅F R − ∇ ⋅FQ + ρε + hS ρ S A,
ρ⎜
−
⎟
A
⎝ dt ρ dt ⎠
(A.13.1)
where FR is the sum of the boundary and radiative heat fluxes and FQ is the sum of all
molecular diffusive fluxes of heat, being the normal molecular heat flux directed down the
temperature gradient plus a term proportional to the molecular flux of salt (the Dufour effect,
see Eqn. (B.24) below). Lastly, ε is the rate of dissipation of kinetic energy per unit mass and
S
hS ρ S A is the rate of increase of enthalpy due to the interior source term of Absolute Salinity
A
caused by remineralization. The derivation of Eqn. (A.13.1) is summarized in appendix B
below, where we also discuss the related evolution equations for total energy and for the
Bernoulli function.
Following Fofonoff (1962) we note that an important consequence of Eqn. (A.13.1) is that
when two finite sized parcels of seawater are mixed at constant pressure and under ideal
conditions, the total amount of enthalpy is conserved. To see this one combines Eqn. (A.13.1)
with the continuity equation ∂ρ ∂t + ∇ ⋅ ( ρ u ) = 0 to find the following divergence form of the
First Law of Thermodynamics,
dP
S
(A.13.2)
∂ ρ h ∂t + ∇ ⋅ ρ uh −
= − ∇ ⋅F R − ∇ ⋅FQ + ρε + hS ρ S A .
A
dt
One then integrates over the volume that encompasses both fluid parcels while assuming
there to be no radiative, boundary or molecular fluxes across the boundary of the control
volume. This control volume may change with time as the fluid moves (at constant pressure),
mixes and contracts. The dissipation of kinetic energy by viscous friction and the source term
due to the production of Absolute Salinity are also commonly ignored during such mixing
processes but in fact the dissipation term does cause a small increase in the enthalpy of the
mixture with respect to that of the two original parcels (see Appendix A.21). Apart from these
non-‐‑conservative source terms, under these assumptions Eqn. (A.13.2) reduces to the
statement that the volume integrated amount of ρ h is the same for the two initial fluid
parcels as for the final mixed parcel, that is, the total amount of enthalpy is unchanged.
This result of non-‐‑equilibrium thermodynamics is of the utmost importance in
oceanography. The fact that enthalpy is conserved when fluid parcels mix at constant
pressure is the central result upon which all of our understanding of “heat fluxes” and of
“heat content” in the ocean rests. The importance of this result cannot be overemphasized; it
must form part of all our introductory courses on oceanography and climate dynamics. As
important as this result is, it does not follow that enthalpy is the best variable to represent
“heat content” in the ocean. Enthalpy is a very poor representation of “heat content” in the
ocean because it does not posses the “potential” property. It will be seen that potential
enthalpy h 0 (referenced to zero sea pressure) is the best thermodynamic variable to represent
“heat content” in the ocean.
The First Law of Thermodynamics, Eqn. (A.13.1), can be written (using Eqn. (A.7.2)) as an
evolution equation for entropy as follows
( )
(
IOC Manuals and Guides No. 56
)
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
⎛
dS ⎞
dη
S
ρ ⎜ (T0 + t )
+ µ A ⎟ = − ∇ ⋅F R − ∇ ⋅FQ + ρε + hS ρ S A .
A
dt
dt ⎠
⎝
97
(A.13.3)
The First Law of Thermodynamics (A.13.1) can also be written in terms of potential
temperature θ (with respect to reference pressure pr ) by using Eqns. (A.11.9) and (A.11.11) in
Eqn. (A.13.1) as (from Bacon and Fofonoff (1996) and McDougall (2003))
⎛ (T + t )
dS ⎞
dθ
ρ⎜ 0
c p ( pr )
+ ⎡⎣ µ ( p ) − (T0 + t ) µT ( pr ) ⎤⎦ A ⎟ =
dt
dt ⎠
⎝ (T0 + θ )
(A.13.4)
− ∇ ⋅F R − ∇ ⋅FQ + ρε + hS ρ S A ,
S
A
where T0 is the Celsius zero point ( T0 is exactly 273.15 K), while in terms of Conservative
Temperature Θ , the First Law of Thermodynamics is (from McDougall (2003), using Eqns.
(A.11.15) and (A.11.18) above)
⎛ (T + t ) d Θ ⎡
(T + t ) µ (0)⎤⎥ d SA ⎞⎟ =
ρ⎜ 0
c0p
+ ⎢µ ( p) − 0
⎜⎝ (T0 + θ ) d t ⎢
(T0 + θ ) ⎥⎦ d t ⎟⎠
⎣
(A.13.5)
− ∇ ⋅F R − ∇ ⋅FQ + ρε + hS ρ S A ,
S
A
where c 0p is the fixed constant defined by the exact 15-‐‑digit number in Eqn. (3.3.3).
In appendices A.16, A.17 and A.18 the non-‐‑conservative production of entropy, potential
temperature and Conservative Temperature are quantified, both as Taylor series expansions
which identify the relevant non-‐‑linear thermodynamic terms that cause the production of
these variables, and also on the SA − Θ diagram where variables are contoured which
graphically illustrate the non-‐‑conservation of these variables. In other words, appendices
A.16, A.17 and A.18 quantify the non-‐‑ideal nature of the left-‐‑hand sides of Eqns. (A.13.3) -‐‑
(A.13.5). That is, these appendices quantify the deviations of the left-‐‑hand sides of these
equations from being proportional to ρ dη dt , ρ dθ dt and ρ dΘ dt .
A quick ranking of these three variables, η , θ and Θ , from the viewpoint of the amount
of their non-‐‑conservation, can be gleaned by examining the range of the terms (at fixed
pressure) that multiply the material derivatives on the left-‐‑hand sides of the above Eqns.
(A.13.3), (A.13.4) and (A.13.5). The ocean circulation may be viewed as a series of adiabatic
and isohaline movements of seawater parcels interrupted by a series of isolated turbulent
mixing events. During any of the adiabatic and isohaline transport stages every “potential”
property is constant, so each of the above variables, entropy, potential temperature and
Conservative Temperature are 100% ideal during these adiabatic and isohaline advection
stages. The turbulent mixing events occur at fixed pressure so the non-‐‑conservative
production of say entropy depends on the extent to which the coefficients (T0 + t ) and µ in
Eqn. (A.13.3) vary at fixed pressure. Similarly the non-‐‑conservative production of potential
temperature depends on the extent to which the coefficients c p ( pr )(T0 + t ) (T0 + θ ) and
⎡⎣ µ ( p ) − (T0 + t ) µT ( pr )⎤⎦ in Eqn. (A.13.4) vary at fixed pressure, while the non-‐‑conservative
production of Conservative Temperature depends on the extent to which the coefficients
(T0 + t ) (T0 + θ ) and ⎡⎣ µ ( p ) − µ (0)(T0 + t ) (T0 + θ )⎤⎦ in Eqn. (A.13.5) vary at fixed pressure.
According to this way of looking at these equations we note that the material derivative of
entropy appears in Eqn. (A.13.3) multiplied by the absolute temperature (T0 + t ) which varies
by about 15% at the sea surface ( ( 273.15 + 40) 273.15 ≈ 1.146 ), the term that multiplies dθ dt
in (A.13.4) is dominated by the variations in c p ( S A , t , pr ) which is mainly a function of S A and
which varies by 5% at the sea surface (see Figure 4), while the material derivative of
Conservative Temperature dΘ dt in Eqn. (A.13.5) is multiplied by the product of a constant
“heat capacity” c 0p and the factor (T0 + t ) (T0 + θ ) which varies very little in the ocean,
especially when one realizes that it is only the variation of this ratio at each pressure level that
is of concern. This factor is unity at the sea surface.
IOC Manuals and Guides No. 56
98
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Graham and McDougall (2013) have derived the evolution equations for potential
temperature, Conservative Temperature and specific entropy in a turbulent ocean,
demonstrating that the non-‐‑conservative source terms of potential temperature are two orders
of magnitude larger than those for Conservative Temperature. They also showed that the
dissipation of kinetic energy is an order of magnitude larger than the non-‐‑conservative source
terms in the evolution equation for Conservative Temperature.
Figure A.13.1. The difference θ − Θ (in °C ) between potential temperature θ
and Conservative Temperature Θ at the sea surface of the
annually-‐‑averaged atlas of Gouretski and Koltermann (2004).
Fortunately, Conservative Temperature is not only much more accurately conserved in
the ocean than potential temperature but it is also relatively easy to use in oceanography.
Because Conservative Temperature also possesses the “potential” property, it is a very
accurate representation of the “heat content” of seawater. The difference θ − Θ between
potential temperature θ and Conservative Temperature Θ at the sea surface is shown in
Figure A.13.1 (after McDougall, 2003). If an ocean model is written with potential
temperature as the prognostic temperature variable rather than Conservative Temperature,
and is run with the same constant value of the isobaric specific heat capacity ( c 0p as given by
Eqn. (3.3.3)), the neglect of the non-‐‑conservative source terms that should appear in the
prognostic equation for θ means that such an ocean model incurs errors in the model output.
These errors will depend on the nature of the surface boundary condition; for flux boundary
conditions the errors are as shown in Figure A.13.1.
This appendix has largely demonstrated the benefits of potential enthalpy and
Conservative Temperature from the viewpoint of conservation equations, but the benefits can
also be proven by the following parcel-‐‑based argument. First, the air-‐‑sea heat flux needs to be
recognized as a flux of potential enthalpy which is exactly c 0p times the flux of Conservative
Temperature. Second, the work of appendix A.18 shows that while it is the in situ enthalpy
that is conserved when parcels mix, a negligible error is made when potential enthalpy is
assumed to be conserved during mixing at any depth. Third, note that the ocean circulation
can be regarded as a series of adiabatic and isohaline movements during which Θ is
absolutely unchanged (because of its “potential” nature) followed by a series of turbulent
mixing events during which Θ is almost totally conserved. Hence it is clear that Θ is the
quantity that is advected and diffused in an almost conservative fashion and whose surface
flux is exactly proportional to the air-‐‑sea heat flux.
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
99
A.14 Advective and diffusive “heat” fluxes
In section 3.23 and appendices A.8 and A.13 the First Law of Thermodynamics is shown to be
practically equivalent to the conservation equation (A.21.15) for Conservative Temperature Θ.
We have emphasized that this means that the advection of “heat” is very accurately given as
the advection of c0p Θ. In this way c0p Θ can be regarded as the “heat content” per unit mass of
seawater and the error involved with making this association is approximately 1% of the error
in assuming that either c0pθ or c p ( SA ,θ , 0dbar )θ is the “heat content” per unit mass of
seawater (see also appendix A.21 for a discussion of this point).
The turbulent flux of a “potential” property can be thought of as the exchange of parcels
of equal mass but contrasting values of the “potential” property, and the turbulent flux can be
parameterized as being down the gradient of the “potential” property. The conservative form
of Eqn. (A.21.15) implies that the turbulent flux of heat should be directed down the mean
gradient of Conservative Temperature rather than down the mean gradient of potential
temperature. In this appendix we quantify the ratio of the mean gradients of potential
temperature and Conservative Temperature.
Consider first the respective temperature gradients along the neutral tangent plane. From
Eqn. (3.11.2) we find that
(α
θ
)
(
)
β θ ∇nθ = ∇n SA = α Θ β Θ ∇n Θ ,
(A.14.1)
so that the epineutral gradients of θ and Θ are related by the ratios of their respective
thermal expansion and saline contraction coefficients, namely
(α
(α θ
Θ
∇nθ =
βΘ
β
θ
) ∇ Θ.
)
(A.14.2)
n
This proportionality factor between the parallel two-‐‑dimensional vectors ∇nθ and ∇n Θ is
readily calculated and illustrated graphically. Before doing so we note two other equivalent
expressions for this proportionality factor.
The epineutral gradients of θ , Θ and S A are related by (using θ = θˆ ( S A , Θ ) )
∇nθ = θˆΘ ∇n Θ + θˆSA ∇n SA ,
and using the neutral relationship ∇n SA =
(
(α
Θ
)
(A.14.3)
β Θ ∇n Θ we find
)
∇nθ = θˆΘ + ⎡⎣α Θ β Θ ⎤⎦ θˆSA ∇n Θ.
(A.14.4)
Also, in section 3.13 we found that Tbθ ∇nθ = TbΘ∇n Θ, so that we find the expressions
∇nθ
∇n Θ
(α
(α θ
Θ
=
)
βθ )
βΘ
=
TbΘ
= θˆΘ + ⎡⎣α Θ β Θ ⎤⎦ θˆSA ,
θ
Tb
(
(A.14.5)
)
and it can be shown that α Θ α θ = θˆΘ and β θ β Θ = 1 + ⎡α Θ β Θ ⎤ θˆS θˆΘ , that is,
⎣
⎦ A
β θ = β Θ + α Θ θˆSA θˆΘ . The ratios α Θ α θ and β θ β Θ have been plotted by Graham and
McDougall (2013); interestingly α Θ α θ is approximately a linear function of S A while
β θ β Θ is approximately a function of only Θ . The partial derivatives θˆΘ and θˆSA in the last
part of Eqn. (A.14.5) are both independent of pressure while α Θ β Θ is a function of pressure.
The ratio, Eqn. (A.14.5), of the epineutral gradients of θ and Θ is shown in Figure A.14.1 at
p = 0 , indicating that the epineutral gradient of potential temperature is sometimes more that
1% different to that of Conservative Temperature. This ratio ∇nθ ∇n Θ is only a weak
function of pressure. This ratio, ∇nθ ∇n Θ (i.e. Eqn. (A.14.5)), is available in the GSW
Oceanographic Toolbox as function gsw_ntp_pt_vs_CT_ratio.
Similarly to Eqn. (A.14.3), the vertical gradients are related by
θ z = θˆΘ Θz + θˆSA SAz ,
(A.14.6)
and using the definition, Eqn. (3.15.1), of the stability ratio we find that
IOC Manuals and Guides No. 56
100
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
θ
z
(A.14.7)
= θˆΘ + Rρ−1 ⎡⎣α Θ β Θ ⎤⎦ θˆSA .
Θz
For values of the stability ratio Rρ close to unity, the ratio θ z Θz is close to the values of
∇nθ ∇n Θ shown in Figure A.14.1.
Figure A.14.1. Contours of ∇nθ ∇n Θ − 1 × 100% at p = 0 , showing the percentage
difference between the epineutral gradients of θ and Θ . The red dots
are from the ocean atlas of Gouretski and Koltermann (2004) at p = 0 .
(
)
As noted in section 3.8 the dianeutral advection of thermobaricity is the same when
quantified in terms of θ as when done in terms of Θ . The same is not true of the dianeutral
velocity caused by cabbeling. The ratio of the cabbeling dianeutral velocity calculated using
potential temperature to that using Conservative Temperature is given by
Cbθ ∇nθ ⋅ ∇nθ CbΘ∇n Θ ⋅ ∇n Θ (see section 3.9) which can be expressed as
(
)(
Cbθ ∇nθ
2
CbΘ ∇n Θ
2
Cθ
= bΘ
Cb
)
(α
(α
Θ
θ
βΘ
β
θ
)
)
2
2
2
=
Cbθ ⎛ TbΘ ⎞
Cθ
= bΘ θˆΘ + ⎡⎣α Θ β Θ ⎤⎦ θˆSA
Θ ⎜ θ ⎟
Cb ⎝ Tb ⎠
Cb
(
) ,
2
(A.14.8)
and this is contoured in Fig. A.14.2. While the ratio of Eqn. (A.14.8) is not exactly unity, it
varies relatively little in the oceanographic range, indicating that the dianeutral advection due
to cabbeling estimated using θ or Θ are within half a percent of each other at p = 0 .
(
Figure A.14.2. Contours of the percentage difference of Cbθ ∇nθ
from unity at p = 0 dbar.
IOC Manuals and Guides No. 56
2
) (C
Θ
b
∇n Θ
2
)
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
101
A.15 Derivation of the expressions for α θ , β θ , α Θ and β Θ
This appendix derives the expressions in Eqns. (2.18.2) – (2.18.3) and (2.19.2) – (2.19.3) for the
thermal expansion coefficients α θ and α Θ and the haline contraction coefficients β θ and β Θ .
In order to derive Eqn. (2.18.2) for α θ we first need an expression for ∂θ ∂T
. This is
SA , p
found by differentiating with respect to in situ temperature the entropy equality
η ( SA , t, p ) = η ( SA ,θ [SA , t , p, pr ], pr ) which defines potential temperature, obtaining
∂θ
∂T
=
SA , p
g ( S , t, p )
ηT ( SA , t , p )
(T +θ ) c p ( SA , t, p ) .
= TT A
= 0
gTT ( SA ,θ , pr )
ηT ( SA ,θ , pr )
(T0 + t ) c p ( SA ,θ , pr )
(A.15.1)
This is then used to obtain the desired expression Eqn. (2.18.2) for α θ as follows
1 ∂v
α =
v ∂θ
θ
1 ∂v
=
v ∂T
SA , p
⎛ ∂θ
⎜
⎜
SA , p ⎝ ∂T
⎞
⎟
⎟
SA , p ⎠
−1
=
gTP ( SA , t, p ) gTT ( SA ,θ , pr )
g P ( SA , t , p ) gTT ( SA , t , p )
.
(A.15.2)
In order to derive Eqn. (2.18.3) for α Θ we first need an expression for ∂Θ ∂T
. This is
SA , p
found by differentiating with respect to in situ temperature the entropy equality
η ( SA , t , p ) = ηˆ ( SA , Θ [ SA , t , p ]) obtaining
g ( S , t, p )
(T + θ ) c p ( SA , t , p ) ,
∂Θ
∂Θ
= ηT ( SA , t , p )
= − (T0 + θ ) TT A0
= 0
∂T SA , p
∂η S
cp
c0p
(T0 + t )
(A.15.3)
A
where the second part of this equation has used Eqn. (A.12.4) for Θη
obtain the desired expression Eqn. (2.18.3) for α Θ as follows
1 ∂v
1 ∂v
=
α =
v ∂Θ SA , p
v ∂T
Θ
SA
. This is then used to
−1
⎛ ∂Θ
⎞
c0p
gTP ( SA , t, p )
.
⎜
⎟ = −
⎜
⎟
g P ( SA , t , p ) (T0 + θ ) gTT ( SA , t , p )
SA , p ⎝ ∂T SA , p ⎠
(A.15.4)
In order to derive Eqn. (2.19.2) for β θ we first need an expression for ∂θ ∂SA
. This is
T, p
found by differentiating with respect to Absolute Salinity at fixed in situ temperature and
pressure the entropy equality η ( SA , t , p ) = η ( SA ,θ [SA , t , p, pr ], pr ) which defines potential
temperature, obtaining
∂θ
∂SA
= θη
T, p
=
=
SA
⎡ηS ( SA , t, p ) − ηS ( SA ,θ , pr )⎤
A
⎣ A
⎦
(T0 + θ ) ⎡ µ S ,θ , p − µ S , t, p ⎤
)⎦
T ( A
r)
T ( A
c p ( S A , θ , pr ) ⎣
⎡ g S T ( SA , t , p ) − g S T ( SA , θ , pr )⎤ gTT ( SA , θ , pr ) ,
A
⎣ A
⎦
(A.15.5)
where Eqns. (A.12.5) and (A.12.7) have been used with a general reference pressure pr rather
than with pr = 0. By differentiating ρ = ρ SA ,θ ⎡⎣ SA ,t, p, pr ⎤⎦ , p with respect to Absolute
Salinity it can be shown that (Gill (1982), McDougall (1987a))
(
βθ =
1 ∂ρ
ρ ∂SA
=
θ, p
1 ∂ρ
ρ ∂SA
)
+ αθ
T, p
∂θ
∂SA
,
(A.15.6)
T, p
and using Eqn. (A.15.5) we arrive at the desired expression Eqn. (2.19.2) for β θ
βθ = −
( SA , t , p )
g P ( SA , t , p )
gS
AP
+
gTP ( SA , t , p ) ⎡⎣ g SAT ( SA , t , p ) − g SAT ( S A ,θ , pr )⎤⎦
g P ( S A , t , p ) gTT ( S A , t , p )
.
(A.15.7)
Note that the terms in the Gibbs function in the natural logarithm of the square root of
Absolute Salinity cancel from the two parts of the square brackets in Eqns. (A.15.5) and
(A.15.7).
IOC Manuals and Guides No. 56
102
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
In order to derive Eqn. (2.19.3) for β Θ we first need an expression for ∂Θ ∂SA
. This is
T, p
found by differentiating with respect to Absolute Salinity at fixed in situ temperature and
pressure the entropy equality η ( SA , t , p ) = ηˆ ( SA , Θ [ SA , t , p ]) obtaining (using Eqns. (A.12.4)
and (A.12.8))
∂Θ
∂SA
= Θη
T, p
SA
⎡η ( S , t , p ) − ηˆ
⎤
SA
Θ⎦
⎣ SA A
= ⎡⎣ µ ( SA ,θ ,0 ) − (T0 + θ ) µT ( SA , t , p )⎤⎦ c 0p
(A.15.8)
= ⎡⎣ g SA ( SA , θ ,0 ) − (T0 + θ ) g SAT ( SA , t , p )⎤⎦ c 0p .
Differentiating ρ = ρˆ ( SA , Θ [ SA , t , p ], p ) with respect to Absolute Salinity leads to
βΘ =
1 ∂ρ
ρ ∂SA
=
Θ, p
1 ∂ρ
ρ ∂SA
+ αΘ
T, p
∂Θ
∂SA
,
(A.15.9)
T, p
and using Eqn. (A.15.8) we arrive at the desired expression (2.19.3) for β Θ namely
βΘ = −
( SA , t , p )
g P ( SA , t , p )
gS
AP
+
gTP ( SA , t , p ) ⎡⎣ g SAT ( SA , t , p ) − g SA ( SA , θ ,0 ) (T0 + θ )⎤⎦
g P ( SA , t , p ) gTT ( SA , t , p )
. (A.15.10)
Note that the terms in the Gibbs function in the natural logarithm of the square root of
Absolute Salinity cancel from the two parts of the square brackets in Eqns. (A.15.8) and
(A.15.10).
A.16 Non-conservative production of entropy
In this and the following three appendices (A.16 – A.19) the non-‐‑conservative nature of
several thermodynamic variables (entropy, potential temperature, Conservative Temperature
and specific volume) will be quantified by considering the mixing of pairs of seawater parcels
at fixed pressure. The mixing is taken to be complete so that the end state is a seawater parcel
that is homogeneous in Absolute Salinity and entropy. That is, we will be considering mixing
to completion by a turbulent mixing process. In appendix A.20 the non-‐‑conservative
production of Absolute Salinity by the remineralization of particulate organic matter is
considered. This process is not being considered in appendices A.16 – A.19. The non-‐‑
conservative production which is quantified in appendices A.16 – A.19 occurs in the absence
of any variation in seawater composition.
Following Fofonoff (1962), consider mixing two fluid parcels (parcels 1 and 2) that have
initially different temperatures and salinities. The mixing process occurs at pressure p. The
mixing is assumed to happen to completion so that in the final state Absolute Salinity, entropy
and all the other properties are uniform. Assuming that the mixing happens with a
vanishingly small amount of dissipation of kinetic energy, the ε term can be dropped from
the First Law of Thermodynamics, (A.13.1), this equation becoming
( ρ h )t
+ ∇⋅ ( ρ u h ) = − ∇⋅ FR − ∇⋅ FQ .
at constant pressure (A.16.1)
Note that this equation has the form (A.8.1) and so h is conserved during mixing at constant
pressure, that is, h is “isobaric conservative”. In the case we are considering of mixing the
two seawater parcels, the system is closed and there are no radiative, boundary or molecular
heat fluxes coming through the outside boundary so the integral over space and time of the
right-‐‑hand side of Eqn. (A.16.1) is zero. The surface integral of ( ρ u h ) through the boundary
is also zero. Hence it is apparent that the volume integral of ρ h is the same at the final state
as it is at the initial state, that is, enthalpy is conserved. Hence during the mixing process the
mass, salt content and enthalpy are conserved, that is
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
103
m1 + m2 = m ,
(A.16.2)
m1 SA1 + m2 SA2 = m SA ,
(A.16.3)
m1 h1 + m2 h2 = mh ,
(A.16.4)
while the non-‐‑conservative nature of entropy means that it obeys the equation,
m1 η1 + m2 η2 + m δη = mη .
(A.16.5)
Here SA , h and η are the values of Absolute Salinity, enthalpy and entropy of the final mixed
fluid and δη is the production of entropy, that is, the amount by which entropy is not
conserved during the mixing process. Entropy η is now regarded as the functional form
η = η SA ,h, p and is expanded in a Taylor series of SA and h about the values of SA and h
of the mixed fluid, retaining terms to second order in [SA2 − SA1 ] = ΔSA and in [h2 − h1 ] = Δh .
Then η1 and η2 are evaluated and (A.16.4) and (A.16.5) used to find
(
)
δη = −
1
2
m1 m2
m
2
{
2
ηhh ( Δh ) + 2 ηhS Δh ΔSA + ηS
A
A SA
( ΔSA )2 } .
(A.16.6)
Graham and McDougall (2013) have derived the following evolution equation for entropy
in a turbulent ocean, involving the epineutral diffusivity K and the vertical turbulent
diffusivity D (see appendix A.21 for the meaning of the symbols in this thickness-‐‑weighted
averaged equation)
(
) ( ) + (T ε+ t )
dη̂
∂η̂
∂η̂
=
+ v̂ ⋅∇ nη̂ + e
= γ z ∇ n ⋅ γ z−1 K∇ nη̂ + Dη̂ z
dt
∂t n
∂z
⎛ ĥ
+ K ⎜ η̂Θ ΘΘ
ĥΘ
⎝
z
0
⎛ ĥ
⎞
⎞
ΘSA
− η̂ΘΘ ⎟ ∇ nΘ̂ ⋅∇ nΘ̂ + 2K ⎜ η̂Θ
− η̂ΘS ⎟ ∇ nΘ̂ ⋅∇ n ŜA
A⎟
⎜⎝
ĥΘ
⎠
⎠
⎛ ĥ
⎞
S S
+ K ⎜ η̂Θ A A − η̂S S ⎟ ∇ n ŜA ⋅∇ n ŜA
A A⎟
⎜⎝
ĥΘ
⎠
(A.16.7)
⎛ ĥ
⎞
⎛ ĥ
⎞
ΘSA
+ D ⎜ η̂Θ ΘΘ − η̂ΘΘ ⎟ Θ̂ 2z + 2D ⎜ η̂Θ
− η̂ΘS ⎟ Θ̂ z ŜA
A⎟
z
⎜⎝
ĥΘ
ĥΘ
⎝
⎠
⎠
⎛ ĥ
⎞
S S
+ D ⎜ η̂Θ A A − η̂S S ⎟ ŜA
A A⎟
z
⎜⎝
ĥΘ
⎠
( )
2
.
Towards the end of this section the implications of the production (A.16.6) of entropy will
be quantified, but for now we ask what constraints the Second Law of Thermodynamics
might place on the form of the Gibbs function g ( SA , t , p ) of seawater. The Second Law of
Thermodynamics tells us that the entropy excess δη must not be negative for all possible
combinations of the differences in enthalpy and salinity between the two fluid parcels. From
(A.16.6) this requirement implies the following three inequalities,
(A.16.8)
ηhh < 0 , ηS S < 0 ,
(η )
hSA
A A
2
< ηhh ηS
A SA
,
(A.16.9)
where the last requirement reflects the need for the discriminant of the quadratic in (A.16.6) to
be negative. Since entropy is already a first derivative of the Gibbs function, these constraints
would seem to be three different constraints on various third derivatives of the Gibbs
function. In fact, we will see that they amount to only two rather well-‐‑known constraints on
second order derivatives of the Gibbs function.
From the fundamental thermodynamic relation (A.7.1) we find that (where T is the
absolute temperature, T = T0 + t )
IOC Manuals and Guides No. 56
104
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
∂η
ηh =
∂h
1
T
=
SA , p
∂η
ηS =
A
∂SA
(A.16.10)
µ
,
T
= −
h, p
(A.16.11)
and from these relations the following expressions for the second order derivatives of η can
be found,
∂2 η
ηhh =
∂h2
ηS
ηS
Ah
A SA
=
=
∂T −1
∂h
=
SA , p
∂2 η
∂h∂SA
∂2 η
∂SA2
h, p
(
p
(
SA , p
∂ −µ T
=
=
=
∂h
∂ −µ T
∂SA
−T −2
,
cp
)
= −
SA , p
)
−
T,p
(
1 ⎛ µ⎞
,
c p ⎜⎝ T ⎟⎠ T
∂ −µ T
∂h
(A.16.12)
)
SA , p
∂h
∂SA
(A.16.13)
T,p
(A.16.14)
⎡
⎤
⎞
⎛
T
µ
= − A −
⎢
⎥ .
T
c p ⎢⎣⎜⎝ T ⎟⎠ T ⎥⎦
The last equation comes from regarding ηS as ηS = ηS SA ,h ⎡⎣ SA ,t, p ⎤⎦ , p .
A
A
A
The constraint (A.16.8) that ηhh < 0 simply requires (from (A.16.12)) that the isobaric heat
capacity c p is positive, or that gTT < 0 . The constraint (A.16.8) that ηS S < 0 , requires (from
A A
(A.16.14)) that
µS
2
2
(
)
2
⎡⎛ µ ⎞ ⎤
(A.16.15)
g SA SA
⎢⎜ ⎟ ⎥ ,
⎣ ⎝ T ⎠T ⎦
that is, the second derivative of the Gibbs function with respect to Absolute Salinity gS S
A A
must exceed some negative number. The constraint (A.16.9) that (ηhS )2 < ηhh ηS S requires
A
A A
that (substituting from (A.16.12), (A.16.13) and (A.16.14))
g SA SA
(A.16.16)
> 0 ,
T 3c p
T3
> −
cp
and since the isobaric heat capacity must be positive, this requirement is that gS S > 0 , and
A A
so is more demanding than (A.16.15).
We conclude that while there are the three requirements (A.16.8) to (A.16.9) on the
functional form of entropy η = η SA ,h, p in order to satisfy the constraint of the Second Law
of Thermodynamics that entropy be produced when water parcels mix, these three constraints
are satisfied by the following two constraints on the form of the Gibbs function g ( SA , t , p ) ,
(A.16.17)
gTT < 0
(
and
)
gSASA > 0.
(A.16.18)
The Second Law of Thermodynamics does not impose any additional requirement on the
cross derivatives gS T nor on any third order derivatives of the Gibbs function.
A
The constraint (A.16.18) can be understood by considering the molecular diffusion of salt,
which is known to be directed down the gradient of chemical potential µ ( SA , t, p ) (see Eqn.
(B.21)). That is, the molecular flux of salt is proportional to −∇µ . Expanding −∇µ in terms of
gradients of Absolute Salinity, of temperature, and of pressure, one finds that the first term is
−µSA ∇SA and in order to avoid an unstable explosion of salt one must have µSA = gSASA > 0.
Hence the constraint (A.16.18) amounts to the requirement that the molecular diffusivity of
salt is positive.
The two constraints (A.16.17) and (A.16.18) on the Gibbs function are well known in the
thermodynamics literature. Landau and Lifshitz (1959) derive them on the basis of the
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
105
contribution of molecular fluxes of heat and salt to the production of entropy (their equations
58.9 and 58.13). Alternatively, Planck (1935) as well as Landau and Lifshitz (1980) in their §96
(this is §98 in editions before the 1976 extension made by Lifshitz and Pitayevski) inferred
such inequalities from thermodynamic stability considerations. It is pleasing to obtain the
same constraints on the seawater Gibbs function from the above Non-‐‑Equilibrium
Thermodynamics approach of mixing fluid parcels since this approach involves turbulent
mixing which is the type of mixing that dominates in the ocean; molecular diffusion has the
complementary role of dissipating tracer variance.
In addition to the Second Law requirements (A.16.17) and (A.16.18) there are other
constraints which the seawater Gibbs function must obey. One is that the adiabatic (and
isohaline) compressibility must be positive for otherwise the fluid would expand in response
to an increase in pressure which is an unstable situation. Taking gP > 0 (since specific volume
needs to be positive) the requirement that the adiabatic (and isohaline) compressibility be
positive imposes the following two constraints (from (2.16.1))
g PP < 0
(A.16.19)
and
( gTP )2
< gPP gTT ,
(A.16.20)
recognizing that gTT is negative ( gTP may, and does, take either sign). Equation (A.16.20) is
more demanding of gPP than is (A.16.19), requiring gPP to be less than a negative number
rather than simply being less than zero. This last inequality can also be regarded as a
constraint on the thermal expansion coefficient α t , implying that its square must be less than
g P−2 g PP gTT or otherwise the relevant compressibility ( κ ) would be negative and the sound
speed complex.
The constraints on the seawater Gibbs function g ( SA , t, p ) that have been discussed above
are summarized as
g p > 0, gSASA > 0 , gPP < 0 , gTT < 0 , and ( gTP ) < g PP gTT .
2
(A.16.21)
We return now to quantify the non-‐‑conservative production of entropy in the ocean.
When the mixing process occurs at p = 0, the expression (A.16.6) for the production of
entropy can be expressed in terms of Conservative Temperature Θ (since Θ is simply
proportional to h at p = 0 ) as follows (now entropy is taken to be the functional form
η = ηˆ ( SA , Θ) )
{
}
m1 m2
2
2
(A.16.22)
ηˆΘΘ ( ΔΘ) + 2ηˆΘSA ΔΘΔSA + ηˆSASA ( ΔSA ) .
m2
The maximum production occurs when parcels of equal mass are mixed so that
1
m m m −2 = 81 and we adopt this value in what follows. To illustrate the magnitude of this
2 1 2
non-‐‑conservation of entropy we first scale entropy by a dimensional constant so that the
resulting variable (“entropic temperature”) has the value 25 °C at ( SA , Θ) = ( SSO ,25 °C ) and
then Θ is subtracted. The result is contoured in SA − Θ space in Figure A.16.1.
The fact that the variable in Figure A.16.1 is not zero over the whole SA − Θ plane is
because entropy is not a conservative variable. The non-‐‑conservative production of entropy
can be read off this figure by selecting two seawater samples and mixing along the straight
line between these parcels and then reading off the production (in °C ) of entropy from the
figure. Taking the most extreme situation with one parcel at ( SA , Θ) = 0 g kg −1,0 °C and the
other at the warmest and saltiest corner of the figure, the production of η on mixing parcels
of equal mass is approximately 0.9 °C .
Since entropy can be expressed independently of pressure as a function of only Absolute
Salinity and Conservative Temperature η = ηˆ ( SA , Θ) , and since at any pressure in the ocean
both S A and Θ may quite accurately be considered conservative variables (see appendix A.18
below), it is clear that the non-‐‑conservative production given by (A.16.22) and illustrated in
Figure A.16.1 is very nearly equivalent to the slightly more accurate expression (A.16.6) which
δη = −
1
2
(
)
IOC Manuals and Guides No. 56
106
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
applies at any pressure. The only discrepancy between the production of entropy calculated
from (A.16.22) and that from (A.16.6) is due to the very small non-‐‑conservative production of
Θ at pressures other than zero (as well as the fact that both expressions contain only the
second order terms in an infinite Taylor series).
Figure A.16.1. Contours (in °C ) of a variable which illustrates the non-‐‑conservative
production of entropy η in the ocean.
A.17 Non-conservative production of potential temperature
When fluid parcels undergo irreversible and complete mixing at constant pressure, the
thermodynamic quantities that are conserved during the mixing process are mass, Absolute
Salinity and enthalpy. As in section A.16 we again consider two parcels being mixed without
external input of heat or mass and the three equations that represent the conservation of these
quantities are again Eqns. (A.16.2) – (A.16.4). The production of potential temperature during
the mixing process is given by
m1 θ1 + m2 θ2 + m δθ = mθ .
(
(A.17.1)
)
Enthalpy in the functional form h = h SA ,θ , p is expanded in a Taylor series of S A and θ
about the values S A and θ of the mixed fluid, retaining terms to second order in
[SA2 − SA1 ] = ΔSA and in [θ2 −θ1 ] = Δθ . Then h1 and h2 are evaluated and Eqns. (A.16.4) and
(A.17.1) used to find
⎫
hθ S
hS S
m m ⎧⎪ h
2
2⎪
δθ = 12 1 2 2 ⎨ θθ Δθ + 2 A Δθ ΔSA + A A ΔSA ⎬ .
(A.17.2)
hθ
hθ
m ⎪ hθ
⎪
⎩
⎭
The maximum production occurs when parcels of equal mass are mixed so that
1
m m m −2 = 81 . The “heat capacity” hθ is not a strong function of θ but is a stronger
2 1 2
function of SA so the first term in the curly brackets in Eqn. (A.17.2) is generally small
compared with the second term. Also, the third term in Eqn. (A.17.2) which causes the so-‐‑
called “dilution heating”, is usually small compared with the second term. A typical value of
hθ S is approximately –5.4 J kg−1 K −1 (g kg −1 ) −1 (see the dependence of isobaric heat capacity
A
on S A in Figure 4 of section 2.20) so that an approximate expression for the production of
potential temperature δθ is
( )
δθ
≈
Δθ
1
h ΔSA
4 θ SA
(
(
)
)
hθ ≈ − 3.4x10−4 ΔSA / [g kg −1] .
(A.17.3)
The same form of the non-‐‑conservative production terms in Eqn. (A.17.2) also appears in the
following turbulent evolution equation for potential temperature, in both the epineutral and
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
107
vertical diffusion terms (Graham and McDougall, 2013). (See appendix A.21 for an
explanation of the symbols that appear in this thickness-‐‑weighted averaged equation.)
∂θˆ
∂θˆ
dθˆ
=
+ v̂ ⋅∇ nθˆ + e
= γ z ∇ n ⋅ γ z−1 K∇ nθˆ + Dθˆz + ε hθ
z
dt
∂t
∂z
(
) ( )
n
⎛ h
⎞
hθ S
hS S
+ K ⎜ θθ ∇ nθˆ ⋅∇ nθˆ + 2 A ∇ nθˆ ⋅∇ n ŜA + A A ∇ n ŜA ⋅∇ n ŜA ⎟ (A.17.4)
⎜⎝ hθ
⎟⎠
hθ
hθ
⎛ h
2⎞
hθ S
hS S
+ D ⎜ θθ θˆz2 + 2 A θˆz ŜA + A A ŜA ⎟ .
z
z
⎜⎝ hθ
⎟⎠
hθ
hθ
Since potential temperature θ = θˆ ( S , Θ ) can be expressed independently of pressure as a
( )
A
function of only Absolute Salinity and Conservative Temperature, and since during turbulent
mixing both S A and Θ may be considered approximately conservative variables (see section
A.18 below), it is clear that the non-‐‑conservative production given by (A.17.2) can be
approximated by the corresponding production of potential temperature that would occur if
the mixing had occurred at p = 0 , namely
δθ ≈
1
2
m1 m2 ⎧⎪ Θ
θθ
Δθ
⎨
2
Θ
m ⎪ θ
⎩
( )
2
Θ
Θ
θS
S S
+ 2 A Δθ ΔSA + A A ΔSA
Θθ
Θθ
(
⎫
)2 ⎪⎬ ,
⎪⎭
(A.17.5)
where the exact proportionality between potential enthalpy and Conservative Temperature
h 0 ≡ c0p Θ has been exploited. The maximum production occurs when parcels of equal mass
are mixed so that 12 m1 m2 m −2 = 81 and we adopt this value in what follows.
Equations (A.17.2) or (A.17.5) may be used to evaluate the non-‐‑conservative production of
potential temperature due to mixing a pair of fluid parcels across a front at which there are
known differences in salinity and temperature. The temperature difference θ − Θ is
contoured in Figure A.17.1 and can be used to illustrate Eqn. (A.17.5). δθ can be read off this
figure by selecting two seawater samples and mixing along the straight line between these
parcels (along which both Absolute Salinity and Conservative Temperature are conserved)
and then calculating the production (in °C ) of θ from the contoured values of θ − Θ . Taking
the most extreme situation with one parcel at ( SA , Θ) = 0 g kg −1 ,0 °C and the other at the
warmest and saltiest corner of Figure A.17.1, the non-‐‑conservative production of θ on mixing
parcels of equal mass is approximately -‐‑0.55 °C . This is to be compared with the
corresponding maximum production of entropy, as discussed above in connection with
Figure A.16.1, of approximately 0.9 °C .
If Figure A.17.1 were to be used to quantify the errors in oceanographic practice incurred
by assuming that θ is a conservative variable, one might select property contrasts that were
typical of a prominent oceanic front and decide that because δθ is small at this one front, that
the issue can be ignored (see for example, Warren (2006)). But the observed properties in the
ocean result from a large and indeterminate number of such prior mixing events and the non-‐‑
conservative production of θ accumulates during each of these mixing events, often in a sign-‐‑
definite fashion. How can we possibly estimate the error that is made by treating potential
temperature as a conservative variable during all of these unknowably many past individual
mixing events? This seemingly difficult issue is partially resolved by considering what is
actually done in ocean models today. These models carry a temperature conservation
equation that does not have non-‐‑conservative source terms, so that the model’s temperature
variable is best interpreted as being Θ . This being the case, the temperature difference
contoured in Figure A.17.1 illustrates the error that is made by interpreting the model
temperature as being θ . That is, the values contoured in Figures A.16.1 and A.17.1 are
representative of the error, expressed in °C , associated with assuming that η and θ
respectively are conservative variables. These contoured values of temperature difference
encapsulate the accumulated non-‐‑conservative production that has occurred during all the
(
)
IOC Manuals and Guides No. 56
108
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
many mixing processes that have lead to the ocean’s present state. The maximum such error
for η is approximately -‐‑1.0 °C (from Figure A.16.1) while for θ the maximum error is
approximately -‐‑1.8 °C (from Figure A.17.1). One percent of the data at the sea surface of the
world ocean have values of θ − Θ that lie outside a range that is 0.25 °C wide (McDougall
(2003)), implying that this is the magnitude of the error incurred by ocean models when they
treat θ as a conservative quantity. To put a temperature difference of 0.25 °C in context, this
is the typical difference between in situ and potential temperatures for a pressure difference of
2500 dbar, and it is approximately 100 times as large as the typical differences between t90
and t68 in the ocean. From the curvature of the isolines on Figure A.17.1 it is clear that the
non-‐‑conservative production of θ takes both positive and negative signs.
Figure A.17.1. Contours (in °C ) of the difference between potential temperature
and Conservative Temperature θ − Θ . This plot illustrates the non-‐‑
conservative production of potential temperature θ in the ocean.
A.18 Non-conservative production of Conservative Temperature
When fluid parcels undergo irreversible and complete mixing at constant pressure, the
thermodynamic quantities that are conserved are mass, Absolute Salinity and enthalpy. As in
sections A.16 and A.17 we consider two parcels being mixed without external input of heat or
mass, and the three equations that represent the conservation of these quantities are Eqns.
(A.16.2) – (A.16.4). Neither potential enthalpy h0 nor Conservative Temperature Θ are
exactly conserved during the mixing process and the production of Θ is given by
(A.18.1)
m1 Θ1 + m2 Θ2 + m δΘ = m Θ.
ˆ
Enthalpy in the functional form h = h ( SA , Θ, p ) is expanded in a Taylor series of S A and Θ
about the values S A and Θ of the mixed fluid, retaining terms to second order in
[SA2 − SA1 ] = ΔSA and in [Θ2 −Θ1 ] = ΔΘ. Then h1 and h2 are evaluated and Eqns. (A.16.4) and
(A.18.1) are used to find
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
δΘ =
1
2
⎧⎪ hˆ
⎫
hˆΘSA
hˆSA SA
2
2⎪
ΘΘ
⎨ ˆ ( ΔΘ) + 2 ˆ ΔΘΔSA + ˆ ( ΔSA ) ⎬ .
hΘ
hΘ
⎩⎪ hΘ
⎭⎪
m1 m2
m2
109
(A.18.2)
Graham and McDougall (2013) have shown that the same form of the non-‐‑conservative
production terms in Eqn. (A.18.2) also appears in the following turbulent evolution equation
for Conservative Temperature, in both the epineutral and vertical diffusion terms (see
appendix A.21 for an explanation of the symbols that appear in this thickness-‐‑weighted
averaged equation),
(
) (
)
∂Θ̂
dΘ̂
∂Θ̂
=
+ v̂ ⋅∇ nΘ̂ + e
= γ z ∇ n ⋅ γ z−1 K∇ nΘ̂ + DΘ̂ z + ε ĥΘ
z
∂t
dt
∂z
n
⎛ ĥ
⎞
ĥS S
ĥΘS
+ K ⎜ ΘΘ ∇ nΘ̂ ⋅∇ nΘ̂ + 2 A ∇ nΘ̂ ⋅∇ n ŜA + A A ∇ n ŜA ⋅∇ n ŜA ⎟ (A.18.3)
⎜⎝ ĥΘ
⎟⎠
ĥΘ
ĥΘ
⎛ ĥ
ĥΘS
ĥS S
+ D ⎜ ΘΘ Θ̂ 2z + 2 A Θ̂ z ŜA + A A ŜA
z
z
⎜⎝ ĥΘ
ĥΘ
ĥΘ
2⎞
( ) ⎟⎟⎠ .
The mesoscale epineutral turbulent fluxes in Eqn. (A.18.3) have been expressed in terms of the
epineutral diffusivity K . In terms of the turbulent mesoscale fluxes themselves rather than
ˆ ⋅∇ Θ
ˆ and K ∇ Sˆ ⋅ ∇ Sˆ are minus the scalar
their parameterized versions, the terms K∇n Θ
n
n A
n A
product of the epineutral flux of Θ and S A with their respective epineutral gradients, while
ˆ ⋅ ∇ Sˆ is the sum of minus the scalar product of the epineutral flux of Θ and ∇ Sˆ ,
2 K ∇n Θ
n A
n A
ˆ .
and minus the scalar product of the epineutral flux of S A and ∇ n Θ
In order to evaluate the partial derivatives in Eqns. (A.18.2) and (A.18.3), we first write
enthalpy in terms of potential enthalpy (i. e. c0p Θ ) using Eqn. (3.2.1), as
h = hˆ ( SA , Θ, p ) = c0p Θ +
∫P vˆ ( SA , Θ, p′) dP′.
P
(A.18.4)
0
This is differentiated with respect to Θ giving
hΘ S
A, p
= hˆΘ = c0p +
P
∫P α
Θ
ρ dP′.
(A.18.5)
0
The right-‐‑hand side of Eqn. (A.18.5) scales as c0p + ρ −1 ( P − P0 )α Θ , which is more than c 0p by
only about 0.0015 c0p for ( P − P0 ) of 4 × 107 Pa (4,000 dbar). Hence, to a very good
approximation, ĥΘ in Eqns. (A.18.2) and (A.18.3) may be taken to be simply c 0p . It is
interesting to examine why this approximation is so accurate when the difference between
enthalpy, h, and potential enthalpy, h0 , as given by Eqns. (3.2.1) and (A.18.4), scales as ρ −1P
which is as large as typical values of potential enthalpy. The reason is that the integral in
Eqns. (3.2.1) or (A.18.4) is dominated by the integral of the mean value of ρ −1 , so causing a
significant offset between h and h 0 as a function of pressure but not affecting the partial
derivative ĥΘ which is taken at fixed pressure. Even the dependence of density on pressure
alone does not affect hˆΘ .
The second order derivatives of ĥ are needed in Eqns. (A.18.2) and (A.18.3), and these can
be estimated by differentiating Eqn. (A.18.4) or (A.18.5), so that, for example,
hˆΘΘ =
P
∫P vˆΘΘ
0
dP′ =
∫P (α
P
0
Θ
ρ
)
Θ
dP′ ,
(A.18.6)
so that we may write Eqn. (A.18.2) approximately as (assuming m1 = m2 )
δΘ ≈
( P − P0 )
8c0p
{ vˆ
ΘΘ
( ΔΘ)2 + 2 vˆSAΘ ΔΘ ΔSA
+ vˆSA SA ( ΔSA )
2
} .
(A.18.7)
This equation is approximate because the variation of v̂ΘΘ , v̂ΘS and v̂S S with pressure has
A
A A
been ignored. The dominant term in Eqn. (A.18.7) is usually the term in v̂ΘΘ and from Eqn.
(A.19.2) below we see that δΘ is approximately proportional to the non-‐‑conservative
IOC Manuals and Guides No. 56
110
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
destruction of specific volume at fixed pressure caused by the “cabbeling” non-‐‑linearities in
the equation of state (McDougall, 1987b), so that
( P − P0 ) vˆ ΔΘ 2 ≈ − ( P − P0 ) δ v .
(A.18.8)
δΘ ≈
)
ΘΘ (
c0p
8c0p
The production of Θ causes an increase in Conservative Temperature and a consequent
decrease in density of − ρα ΘδΘ . The ratio of this change in density (using Eqn. (A.18.7)) to
that caused by cabbeling (from Eqn. (A.19.2) and using δρ ≈ − ρ 2 δ v ) is − ( P − P0 ) α Θ ρ c0p
which is about 0.0015 for a value of ( P − P0 ) of 40 MPa. Hence it is clear that cabbeling has a
much larger effect on density than does the non-‐‑conservation of Θ . Nevertheless, it is
interesting to note from Eqn. (A.18.7) that the non-‐‑conservative production of Θ is
approximately proportional to the product of sea pressure and the strength of cabbeling.
The first term in the bracket in Eqn. (A.18.7) is usually about a factor of ten larger than the
other two terms (McDougall (1987b)), so the production of Conservative Temperature δΘ as a
ratio of the contrast in Conservative Temperature ΔΘ = Θ2 −Θ1 may be approximated as
(since v̂ΘΘ ≈ − ρ −2 ρˆ ΘΘ ≈ ρ −1αΘΘ )
δΘ
ΔΘ
≈
( P − P0 )αΘΘ ΔΘ
8ρ c0p
≈ 3.3 × 10−9 ( p dbar )( ΔΘ K ) .
(A.18.9)
where α ΘΘ has been taken to be 1.1 × 10−5 K −2 (McDougall, 1987b).
At the sea surface Conservative Temperature Θ is totally conserved ( δΘ = 0 ). The
expression for the non-‐‑conservative production of Conservative Temperature, δΘ , increases
almost linearly with pressure (see Eqn. (A.18.7)) but at larger pressure the range of
temperature and salinity in the ocean decreases, and from the above equations it is clear that
the magnitude of δΘ is proportional to the square of the temperature and salinity contrasts.
McDougall (2003) concluded that the production δΘ between extreme seawater parcels at
each pressure is largest at 600 dbar. The magnitude of the non-‐‑conservative production of
Conservative Temperature, δΘ , is illustrated in Figure A.18.1 for data at this pressure. The
quantity contoured on this figure is the difference between Θ and the following totally
conservative quantity at p = 600 dbar. This conservative quantity was constructed by taking
the conservative property enthalpy h at this pressure and adding the linear function of S A
which makes the result equal to zero at
( SA = 0, Θ= 0 °C) and at
SA = 35.165 04 g kg −1, Θ= 0 °C . This quantity is then scaled so that it becomes 25 °C at
SA = 35.165 04 g kg −1, Θ = 25 °C . In this manner the quantity that is contoured in Figure
A.18.1 has units of °C and represents the amount by which Conservative Temperature Θ is
not a totally conservative variable at a pressure of 600 dbar. The maximum amount of
production by mixing seawater parcels at the boundaries of Figure A.18.1 is about 4 × 10−3 °C
although the range of values encountered in the real ocean at this pressure is actually quite
small, as indicated in Figure A.18.1.
From the curvature of the isolines on Figure A.18.1 it is clear that the non-‐‑conservative
production of Conservative Temperature at p = 600 dbar is positive, so that an ocean model
that ignores this production of Conservative Temperature will slightly underestimate Θ .
From Eqn. (A.18.2) one sees the non-‐‑conservative production of Conservative Temperature is
always positive if hˆΘΘ > 0 , hˆS S > 0 and (hˆΘS )2 < hˆΘΘ hˆS S , and Graham and McDougall
A A
A
A A
(2013) have shown that these requirements are met everywhere in the full TEOS-‐‑10 ranges of
salinity, temperature and pressure for the full TEOS-‐‑10 Gibbs function and this is also the case
when using the 75-‐‑term polynomial for specific volume in its range of applicability.
From Eqns. (A.18.9) and (A.17.3) we can write the ratio of the production of Conservative
Temperature to the production of potential temperature when two seawater parcels mix as the
approximate expression
(
(
)
)
δΘ
≈ −10−5 ( p dbar )( ΔΘ K ) ΔSA / [g kg −1 ]
δθ
(
IOC Manuals and Guides No. 56
)
−1
.
(A.18.10)
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
111
Taking a typical ratio of temperature differences to salinity differences in the deep ocean to be
5K / [g kg −1 ] , Eqn. (A.18.10) becomes δΘ δθ ≈ − 5 x10−5 ( p dbar ). At a pressure of 4000 dbar
this ratio is δΘ δθ ≈ − 0.2 implying that Conservative Temperature is a factor of five more
conservative than potential temperature at these great depths. Note also that the temperature
and salinity contrasts in the deep ocean are small, so the non-‐‑conservation of both types of
temperature amount to very small temperature increments of both δθ and δΘ. The largest
non-‐‑conservative increment of Conservative Temperature δΘ seems to occur at a pressure of
about 600 dbar (McDougall (2003)) and this value of δΘ is approximately two orders of
magnitude less than the maximum value of δθ which occurs at the sea surface. The material
in appendices A.16 -‐‑ A.18 has closely followed McDougall (2003).
Figure A.18.1. Contours (in °C ) of a variable that is used to illustrate the non-‐‑
conservative production of Conservative Temperature Θ at p = 600 dbar.
The cloud of points show where most of the oceanic data reside at p = 600
dbar. The three points that are forced to be zero are shown with black dots.
A.19 Non-conservative production of specific volume
Following Graham and McDougall (2013) specific volume is expressed as a function of
Absolute Salinity S A , specific enthalpy h and pressure as v = v SA ,h, p and the same mixing
process between two fluid parcels is considered as in the previous appendices. Mass, salt and
enthalpy are conserved during the turbulent mixing process (Eqns. (A.16.2) -‐‑ (A.16.4)) while
the non-‐‑conservative nature of specific volume means that it obeys the equation,
(
)
m1 v1 + m2 v2 + m δ v = mv .
(A.19.1)
Specific volume is expanded in a Taylor series of S A and h about the values of S A and h of
the mixed fluid at pressure p , retaining terms to second order in [SA2 − SA1 ] = ΔSA and in
[h2 − h1 ] = Δh . Then v1 and v2 are evaluated and (A.19.1) is used to find
δv = −
1
2
≈ −
1
2
m1 m2
2
m
m1 m2
{
{
( )
2
vhh Δh + 2 vhS Δh ΔSA + vS
( )
A
2
A SA
( ΔSA )2 }
(
)}
(A.19.2)
2
v̂ΘΘ ΔΘ + 2 v̂ΘS ΔΘ ΔSA + v̂S S ΔSA .
A
A A
m2
The non-‐‑conservative destruction of specific volume of Eqn. (A.19.2) is illustrated in
Figure A.19.1 for mixing at p = 0 dbar. The quantity contoured on this figure is formed as
follows. First the linear function of S A is found that is equal to specific volume at
IOC Manuals and Guides No. 56
112
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
( SA = 0, Θ= 0 °C)
(
)
and at SA = 35.165 04 g kg −1, Θ= 0 °C . This linear function of S A is
subtracted from v and the result is scaled to equal 25 °C at SA = 35.165 04 g kg −1 , Θ= 25 °C .
The variable that is contoured in Figure A.19.1 is the difference between this scaled linear
combination of v and S A , and Conservative Temperature. This figure allows the non-‐‑
conservative nature of specific volume to be understood in temperature units. The mixing of
extreme fluid parcels on Figure A.19.1 causes the same decrease in specific volume as a
cooling of approximately 10 °C , which is approximately 4000 times larger than the
corresponding non-‐‑conservative production of Θ at 600dbar (from Figure A.18.1).
From Eqn. (A.19.2) it follows that specific volume is always destroyed by turbulent
mixing processes if vhh > 0 , vS S > 0 and ( vhS )2 < vS S vhh , and Graham and McDougall
A
A A
A A
(2013) have shown that these conditions are satisfied over the full TEOS-‐‑10 ranges of salinity,
temperature and pressure by the full TEOS-‐‑10 Gibbs function and this is also true of the 75-‐‑
term expression for specific volume of Appendix K. Note that in contrast to the case of
specific volume, the non-‐‑conservation of density is not sign-‐‑definite. That is, while turbulent
mixing always destroys specific volume, it does not always produce density ρ = v −1 .
The fact that turbulent mixing at constant pressure always destroys specific volume also
implies that internal energy is always produced by this turbulent mixing at constant pressure
(see the First Law of Thermodynamics, Eqn. (B.19)).
(
)
Figure A.19.1. Contours (in °C ) of a variable that is used to illustrate the non-‐‑
conservative production of specific volume at p = 0 dbar. The
three points that are forced to be zero are shown with black dots.
A.20 The representation of salinity in numerical ocean models
Ocean models need to evaluate salinity at every time step as a necessary prelude to using the
equation of state to determine density and its partial derivatives for use in the hydrostatic
relationship and in neutral mixing algorithms. The current practice in numerical models is to
treat salinity as a perfectly conserved quantity in the interior of the ocean; salinity changes at
the surface and at coastal boundaries due to evaporation, precipitation, brine rejection, ice
melt and river runoff, and satisfies an advection-‐‑diffusion equation away from these
boundaries. The inclusion of composition anomalies necessitates several changes to this
approach. These changes can be divided into two broad categories. First, in addition to fresh
water inputs and brine rejection, all sources of dissolved material entering through the surface
and coastal boundaries of the model should be considered as possible sources of composition
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
113
anomalies. Second, within the interior of the model, changes due to the growth, decay and
remineralization of biological material must be considered. Here, we focus on this second
issue. While the ultimate resolution of these issues will involve biogeochemical models, in
this appendix we discuss some practical ways forward based on the approximate relations
(A.5.7) -‐‑ (A.5.12) between the salinity variables SR , S* and SA = SAdens . At the time of writing,
the suggested approaches here have not been tested, so it must be acknowledged that the
treatment of seawater composition anomalies in ocean models is currently a work in progress.
We begin by repeating Eqns. (A.5.11) and (A.5.12), namely
(A.20.1)
SA =
(A.20.2)
where
Rδ ≡
(
)
S (1+ F δ ) ,
S* = SR 1 − r1Rδ ,
δ SAatlas
S Ratlas
*
[1+ r1 ] Rδ
and F δ =
(1 − r Rδ )
.
(A.20.3)
1
δ
Recall that the Absolute Salinity Anomaly Ratio, R ≡ δ SAatlas S Ratlas , is the ratio of the atlas
values of Absolute Salinity Anomaly and Reference Salinity. The stored values of Rδ are
interpolated onto the latitude, longitude and pressure of an oceanographic observation.
Rδ is bounded between zero 0.001 in the global ocean. With r1 taken to be 0.35 we note the
following approximate expression F δ = SA S* − 1 ≈ 1.35 Rδ .
A.20.1 Using Preformed Salinity S* as the conservative salinity variable
Because Preformed Absolute Salinity S* (henceforth referred to by the shortened name,
Preformed Salinity) is designed to be a conservative salinity variable, blind to the effects of
biogeochemical processes, its evolution equation will be in the conservative form (A.8.1).
When this type of conservation equation is averaged in the appropriate manner (see appendix
A.21) the conservation equation for Preformed Salinity becomes (from Eqn. (A.21.7)),
⎛ ∂ Ŝ ⎞
dŜ*
= γ z ∇ n ⋅ γ z−1 K∇ n Ŝ* + ⎜ D * ⎟ .
dt
⎝ ∂z ⎠ z
(
)
(A.20.4)
As explained in appendix A.21, the over-‐‑tilde of Ŝ* indicates that this variable is the
thickness-‐‑weighted average Preformed Salinity, having been averaged between a pair of
closely spaced neutral tangent planes. The material derivative on the left-‐‑hand side of Eqn.
(A.20.4) is with respect to the sum of the Eulerian and quasi-‐‑Stokes velocities of height
coordinates (equivalent to the description in appendix A.21 in terms of the thickness-‐‑weighted
average horizontal velocity and the mean dianeutral velocity). The right-‐‑hand side of this
equation is the standard notation indicating that Ŝ* is diffused along neutral tangent planes
with the diffusivity K and in the vertical direction with the diapycnal diffusivity D (and γ z−1 is
proportional to the average thickness between two closely spaced neutral tangent planes).
In order to evaluate density during the running of an ocean model, Absolute Salinity
SA = SAdens must be evaluated. This can be done from Eqn. (A.20.2) as the product of the
model’s salinity variable Ŝ* and (1 + F δ ) . This could be done by simply multiplying the
model’s salinity by the fixed spatial map of (1 + F δ ) as observed today (using r1 = 0.35 and
the value of Rδ obtained from the computer algorithm of McDougall et al. (2012)). However
experience has shown that even a smooth field of density errors can result in significant
anomalies in diagnostic model calculations, primarily due to the misalignment of the density
errors and the model bottom topography. Indeed, even if the correct mean density could
somehow be determined, approximations associated with the specification of the model
bottom topography would result in significant errors in bottom pressure torques that can
degrade the model solution. One way to minimize such errors is to allow some dynamical
adjustment of the specified density field so that, for example, density contours tend to align
IOC Manuals and Guides No. 56
114
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
with bottom depth contours where the flow is constrained to follow bottom topography. This
simple idea is the key to the success of the robust diagnostic approach (Sarmiento and Bryan
(1982)). To allow dynamical adjustment of the salinity difference S A − S* while not permitting
SA − S* to drift too far from the observed values, we recommend carrying an evolution
equation for F δ so that it becomes an extra model variable which evolves according to
⎛ ∂F δ ⎞
dF δ
= γ z ∇ n ⋅ γ z−1 K∇ n F δ + ⎜ D
+ τ −1 F δ obs − F δ .
dt
∂z ⎟⎠
⎝
z
(
)
(
)
(A.20.5)
Here the model variable F δ would be initialized based on observations, F δ obs (using Eqn.
(A.20.3) with r1 = 0.35 and the interpolated values of Rδ from McDougall et al. (2012)), and
advected and diffused like any other tracer, but in addition, there is a non-‐‑conservative source
term τ −1 F δ obs − F δ which serves to restore the model variable F δ towards the observed
value with a restoring time τ that can be chosen to suit particular modeling needs. It should
be at least 30 days to permit significant adjustment, but it might prove appropriate to allow a
much longer adjustment period (up to several years) if drift from observations is sufficiently
slow. The lower bound is based on a very rough estimate of the time required for the density
field to be aligned with topography by advective processes. The upper bound is set by the
requirement to have the restoring time relatively short compared to vertical and basin-‐‑scale
horizontal redistribution times.
Ideally one would like the non-‐‑conservative source term to reflect the actual physical and
chemical processes responsible for remineralization in the ocean interior, but until our
knowledge of these processes improves such that this is possible, the approach based on Eqn.
(A.20.5) provides a way forward. An indication of how an approach based on modeled
biogeochemical processes might be implemented in the future can be gleaned from looking at
Eqn. (A.4.14) for S A − S∗ . If a biogeochemical model produced estimates of the quantities on
the right-‐‑hand side of this equation, it could be immediately integrated into an ocean model to
diagnose the effects of the biogeochemical processes on the model'ʹs density and its circulation.
In summary, the approach suggested here carries the evolution Eqns. (A.20.4) and (A.20.5)
for Ŝ* and F δ , while Ŝ A is calculated by the model at each time step according to
(
)
(
)
SˆA = Sˆ* 1+ F δ .
(A.20.6)
The model is initialized with values of Preformed Salinity using Eqn. (A.20.1) based on
observations of Reference Salinity and on the interpolated global database of Rδ from
McDougall et al. (2012) using r1 = 0.35 . This approach applies to the open ocean, but the
Baltic Sea is to be treated differently. As described in appendix A.5, the observed Absolute
Salinity Anomaly δ SA in the Baltic Sea is not primarily due to non-‐‑conservative
biogeochemical source terms but rather is due to rivers delivering water to the Baltic with
much larger Absolute Salinity than would be expected from the Practical Salinity of the river
discharge. In the Baltic Sea, SA = S∗ , r1 = − 1 and F δ = 0 (as discussed in appendix A.5) so
that in the Baltic region of an ocean model the equation of state should be called with the
model’s salinity variable, Preformed Salinity S∗ . The discharges (mass fluxes) of river water
and of Absolute Salinity should both appear as source terms at the edges of the Baltic Sea.
A.20.2 Including a source term in the evolution equation for Absolute Salinity
An alternative procedure would be to carry an evolution equation for Absolute Salinity rather
than for Preformed Salinity in an ocean model. Using Eqns. (A.20.4) -‐‑ (A.20.6), the following
evolution equation for Absolute Salinity can be constructed,
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
⎛ ∂ Ŝ ⎞
∂ Ŝ
dŜA
Ŝ
= γ z ∇ n ⋅ γ z−1 K∇ n ŜA + ⎜ D A ⎟ − 2K∇ n Ŝ* ⋅∇ n F δ − 2DFzδ * + * F δ obs − F δ
dt
∂z
τ
⎝ ∂z ⎠ z
(
(
= γ z∇ n ⋅ γ
)
−1
K∇ n ŜA
z
)
(
⎛ ∂ Ŝ ⎞
S
+ ⎜ D A ⎟ + Sˆ A .
⎝ ∂z ⎠ z
)
115
(A.20.7)
Here the non-‐‑conservative source term in the evolution equation for Absolute Salinity has
S
been given the label Sˆ A for later use. If the ocean model resolves mesoscale eddies then the
δ
term − 2 K ∇n Sˆ* ⋅ ∇ n F in Eqn. (A.20.7) becomes the scalar product of ∇ n F δ and the
epineutral flux of S* plus the scalar product of ∇n Sˆ* and the epineutral flux of F δ . In this
approach the evolution equation (A.20.5) for F δ is also carried and the model’s salinity
variable, Ŝ A , is used directly as the argument of the equation of state and other
thermodynamic functions in the model. The model would be initialized with values of
Absolute Salinity using Eqn. (A.5.10) (namely SA = SR 1 + Rδ ) based on observations of
Reference Salinity and on the global data base of Rδ from McDougall et al. (2012). The
production terms involving Ŝ* in Eqn. (A.20.7) would need to be evaluated in terms of the
model’s salinity variable Ŝ A using Eqn. (A.20.6).
This approach should give identical results to that described in section A.20.1 using
Preformed Salinity. One disadvantage of having Density Salinity as the model’s salinity
variable is that its evolution equation (A.20.7) is not in the conservative form so that, for
example, it is not possible to perform easy global budgets of salinity to test for the numerical
integrity of the model code. Another disadvantage is that the air-‐‑sea flux of carbon dioxide
and other gases may need to be taken into account as the surface boundary condition of
Absolute Salinity. Such air-‐‑sea fluxes do not affect Preformed Salinity. But perhaps the
largest disadvantage of this approach is the difficulty in evaluating the non-‐‑conservative
terms − 2K ∇n Sˆ* ⋅ ∇n F δ − 2DFzδ ∂Sˆ* ∂z in Eqn. (A.20.7), especially when meso-‐‑scale eddies are
present, as discussed above.
A.20.3 Discussion of the consequences if remineralization is ignored
If an ocean model does not carry the evolution equation for F δ (Eqn. (A.20.5)) and the
model’s salinity evolution equation does not contain the appropriate non-‐‑conservative source
term, is there then any preference for initializing and interpreting the model’s salinity variable
as either Preformed Salinity, Absolute Salinity or Reference Salinity? That is, the simplest
method of dealing with these salinity issues is to continue the general approach that has been
taken for the past several decades of simply taking one type of salinity in the model, and that
salinity is taken to be conservative. Under this approximation the salinity that is used in the
equation of state to calculate density in the model is the same as the salinity that obeys a
normal conservation equation of the form Eqn. (A.20.4). In this approach there is still a choice
of how to initialize and to interpret the salinity in a model, and here we discuss the relative
virtues of these options.
If the model is initialized with a data set of estimated Preformed Salinity S* , then S*
should evolve correctly, since S* is a conservative variable and its evolution equation Eqn.
(A.20.4) contains no non-‐‑conservative source terms. In this approach the equation of state will
be called with Ŝ* rather than ŜA , and these salinities differ by approximately (1 + r1 ) δ SA .
The likely errors with this approach can be estimated using the simple example of Figure
A.5.1. The vertical axis in this figure is the difference between the northward density gradient
at constant pressure when the equation of state is called with ŜA and with Ŝ R . The figure
shows that when using Ŝ R , for all the data in the world ocean below a depth of 1000 m , 58%
of this data is in error by more than 2%. If this graph were re-‐‑done with Ŝ* as the salinity
argument rather than Ŝ R , the errors would be larger by the ratio (1 + r1 ) ≈ 1.35 . That is, for
58% of the data in the world ocean deeper than 1000 m , the “thermal wind” relation would be
(
)
IOC Manuals and Guides No. 56
116
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
misestimated by ≈ 2.7% if Ŝ* is used in place of ŜA as the salinity argument of the equation
of state. Also, these percentage errors in “thermal wind” are much larger in the North Pacific.
Another choice of the salinity data to initialize the model is ŜA . An advantage of this
choice is that initially the equation of state is called with the correct salinity variable.
However at later times, the neglect of the non-‐‑conservative source terms in Eqn. (A.20.7)
means that the model’s salinity variable will depart from reality and errors will creep in due
to the lack of these legitimate non-‐‑conservative source terms. How long might it be
acceptable to integrate such a model before the errors approached those described in the
previous paragraph? One could imagine that in the upper ocean the influence of these
different salinity variables is dwarfed by other physics such as air sea interaction and active
gyral motions. If one considered a depth of 1000m as being a depth where the influence of the
different salinities would be both apparent and would make a significant impact on the
thermal wind equation, then one might guess that it would take several decades for the
neglect of the non-‐‑conservative source terms in the evolution equation for Absolute Salinity to
begin to be important. This is not to suggest that the relaxation time scale τ should be chosen
to be as long as this, rather this is an estimate of how long it would take for the neglect of the
S
non-‐‑conservative source term Sˆ A in Eqn. (A.20.7) to become significant.
A third choice is to initialize the model with Reference Salinity, Ŝ R . This choice incurs the
errors displayed in Figure A.5.1 right from the start of any numerical simulation. Thereafter,
on some unknown timescale, further errors will arise because the conservation equation for
Reference Salinity is missing the legitimate non-‐‑conservative source terms representing the
effects of biogeochemistry on conductivity and Ŝ R . Hence this choice is the least desired of
the three considered in this subsection. Note that this choice is basically the approach that has
been used to date in ocean modeling since we have routinely initialized models with
observations of Practical Salinity and have treated it as a conservative variable and have used
it as the salinity argument for the equation of state.
To summarize, the approaches of both subsections A.20.1 and A.20.2 of this appendix can
each account for the non-‐‑conservative effects of remineralization if r1 is a constant and so long
as the appropriate boundary conditions are imposed. The advantage of using Ŝ* is that it
obeys a standard conservative evolution equation (A.20.4) with no source term on the right-‐‑
hand side. If an ocean model were to be run without carrying the evolution equation for F δ
and hence without the ability to incorporate the appropriate non-‐‑conservative source terms in
either Eqns. (A.20.6) or (A.20.7), then the model must resort to carrying only one salinity
variable, and this salinity variable must be treated as a conservative variable in the ocean
model. In this circumstance, we advise that the ocean’s salinity variable be interpreted as
Absolute Salinity, and initialized as such. In this way, the errors in the thermal wind equation
will develop only slowly over a time scale of several decades or more in the deep ocean.
The use of an existing climatology for F δ and the introduction of a rather arbitrary
relaxation time τ are less than desirable features of this way of treating salinity in ocean
models. An alternative strategy is available in an ocean model that includes biogeochemical
processes and carries evolution equations for Total Alkalinity (TA), Dissolved Inorganic
Carbon (DIC) as well as nitrate and silicate concentrations. Having these quantities available
during the running of an ocean model allows the use of the following equation (this is Eqn.
(A.4.14), from Pawlowicz et al., 2011) to evaluate Absolute Salinity
( SA − S* ) / (g kg−1 ) = (73.7 ΔTA +11.8ΔDIC+81.9 NO3− +50.6 Si(OH)4 )
(molkg −1 ) .
(A.20.8)
Under this approach, Preformed Salinity would be carried as the model’s conservative
prognostic salinity variable as in Eqn. (A.20.4), and the above equation for SA − S* in terms of
the biogeochemical variables would be used to evaluate Absolute Salinity for use in the
model’s expression for specific volume.
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
117
A.21 The material derivatives of S* , S A , SR and Θ in a turbulent ocean
Preformed Salinity S* is designed to be a conservative variable which obeys the following
instantaneous conservation equation (based on Eqn. (A.8.1))
dS*
(A.21.1)
= − ∇ ⋅ FS .
dt
There are several different contributions to the molecular flux of salt FS , expressions for
which can be seen at equation (58.11) of Landau and Lifshitz (1959) and in Eqn. (B.23) below.
For completeness, we repeat the continuity equation (A.8.2) here as
( ρ S* )t + ∇ ⋅ ( ρ uS* )
= ρ
ρt + ∇ ⋅ ( ρ u ) = 0.
(A.21.2)
Temporally averaging this equation in Cartesian coordinates (i. e. at fixed x, y , z ) gives
( )
ρt + ∇⋅ ρ u = 0,
(A.21.3)
which we choose to write in the following form, after division by a constant density ρ 0
(usually taken to be 1035 kg m −3 , see Griffies (2004))
( ρ ρ0 )t + ∇ ⋅ u = 0 where u ≡ ρu
ρ0 .
(A.21.4)
This velocity u is actually proportional to the average mass flux of seawater per unit area.
The conservation equation for Preformed Salinity (A.21.1) is now averaged in the
corresponding manner obtaining (McDougall et al. 2002)
⎛ ρ ⎞
⎛ ρ ρ⎞
⎜⎝ ρ0 S* ⎟⎠ + ∇ ⋅ ⎜⎝ S* u ⎟⎠ =
t
ρ
ρ0
∂S*
∂t
ρ
ρ
+ u ⋅∇S* = −
1
ρ0
∇ ⋅FS −
1
ρ0
(
)
∇ ⋅ ρ S*′′u′′ .
(A.21.5)
ρ
Here the Preformed Salinity has been density-‐‑weighted averaged, that is, S* ≡ ρ S* ρ , and
the double primed quantities are deviations of the instantaneous quantity from its density-‐‑
weighted average value. Since the turbulent fluxes are many orders of magnitude larger than
molecular fluxes in the ocean, the molecular flux of salt is henceforth ignored.
The averaging process involved in Eqn. (A.21.5) has not invoked the traditional
Boussinesq approximation. The above averaging process is best viewed as an average over
many small-‐‑scale mixing processes over several hours, but not over mesoscale time and space
scales. This later averaging over the energetic mesoscale eddies is not always necessary,
depending on the scale of the piece of ocean or ocean model that is under investigation. The
two-‐‑stage averaging process, without invoking the Boussinesq approximation, over first
small-‐‑scale mixing processes (several meters) followed by averaging over the mesoscale (of
order 100 km) has been performed by Greatbatch and McDougall (2003), yielding the
prognostic equation for Preformed Salinity
h−1
(
ρ!
ρ0
hŜ*
)
t n
+ h−1∇ n ⋅
(
ρ!
ρ0
) (
hv̂Ŝ* +
ρ!
ρ0
e! Ŝ*
)
=
z
ρ!
ρ0
∂ Ŝ*
∂t
= γ! z ∇ n ⋅
+
ρ!
ρ0
v̂ ⋅∇ n Ŝ* +
n
(
γ! z−1 K∇ n Ŝ*
)
ρ!
ρ0
e!
∂ Ŝ*
∂z
⎛ ∂ Ŝ ⎞
+ ⎜D *⎟ .
⎝ ∂z ⎠ z
(A.21.6)
Here the over-‐‑caret means that the variable (e.g. Ŝ* ) has been averaged in a thickness-‐‑and-‐‑
density-‐‑weighted manner between a pair of “neutral surfaces” a small distance apart in the
vertical, v̂ is the thickness-‐‑and-‐‑density-‐‑weighted horizontal velocity, e is the dianeutral
velocity (the vertical velocity that penetrates through the neutral tangent plane) and e is the
temporal average of e on the “neutral surface” (that is, e is not thickness-‐‑weighted). The
turbulent fluxes are parameterized by the epineutral diffusivity K and the dianeutral (or
vertical) diffusivity D . γ! z is the vertical gradient of a suitable compressibility-‐‑corrected
density such as Neutral Density or locally-‐‑referenced potential density, and the averaging
involved in forming γ! z is done to preserve the average thickness between closely-‐‑spaced
neutral tangent planes; that is, the averaging is performed on γ z−1 . The issues of averaging
IOC Manuals and Guides No. 56
118
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
involved in Eqns. (A.21.5) and (A.21.6) are subtle, and are not central to our purpose in this
thermodynamic manual. Hence we proceed with the more standard Boussinesq approach,
but retain the over-‐‑carets to remind the reader of the thickness-‐‑weighted nature of the
variables.
Having derived this evolution equation (A.21.6) for Preformed Salinity without invoking
the Boussinesq approximation, we now follow common practice and invoke this
approximation, finding the simpler expression
∂ Ŝ*
∂t
+ v̂ ⋅∇ n Ŝ* + e!
n
⎛ ∂ Ŝ ⎞
∂ Ŝ*
= γ! z ∇ n ⋅ γ! z−1 K∇ n Ŝ* + ⎜ D * ⎟ .
∂z
⎝ ∂z ⎠ z
(
)
(A.21.7)
The left-‐‑hand side is the material derivative of the thickness-‐‑weighted Preformed Salinity
with respect to the thickness-‐‑weighted horizontal velocity v̂ and the temporally averaged
dianeutral velocity e of density coordinates. The right-‐‑hand side is the divergence of the
turbulent fluxes of Preformed Salinity; the fact that the lateral diffusion term is the divergence
of a flux can be seen when it is transformed to Cartesian coordinates. The same conservation
statement Eqn. (A.21.7) can be derived without making the Boussinesq approximation by a
simple reinterpretation of the vertical coordinate as being pressure, and this interpretation is
now becoming common in ocean modelling (see Bleck (1978), Huang et al. (2001), de Szoeke
and Samelson (2002), Losch et al. (2004) and Griffies (2004)).
We now proceed to develop the corresponding evolution equation for Absolute Salinity
SA . Note that SA is the convenient generic symbol for Density Salinity SAdens ; unless there is
room for confusion with the other measures of absolute salinity, S Asoln and SAadd , it proves
convenient to use the simpler symbol S A rather than SAdens and to use the description Absolute
Salinity rather than Density Salinity.
Absolute Salinity obeys the instantaneous evolution equation (based on Eqn. (A.8.1))
( ρ SA )t + ∇ ⋅ ( ρuSA )
= ρ
dSA
S
= − ∇ ⋅FS + ρ S A .
dt
(A.21.8)
The source term S A is described in appendix A.20 (see eqn. (A.20.7)). This non-‐‑conservative
source term is due to biogeochemical processes, for example, the remineralization of
biological material; the turning of particulate matter into dissolved seasalt. When this
equation is density-‐‑weighted averaged, we find
S
ρ⎞
⎛ ρ ⎞
⎛ ρ
⎜⎝ ρ0 SA ⎟⎠ + ∇ ⋅ ⎜⎝ SA u ⎟⎠ =
t
= −
ρ
ρ0
1
ρ0
∂SA
∂t
ρ
∇ ⋅FS −
+ u ⋅∇SA
1
ρ0
ρ
(
)
∇ ⋅ ρ SA′′ u′′ +
ρ
ρ0
S
SA
ρ
(A.21.9)
,
which corresponds to Eqn. (A.21.5) above. When averaged over the mesoscale the prognostic
equation for Absolute Salinity becomes
h−1
(
ρ!
ρ0
hŜA
)
t n
+ h−1∇ n ⋅
(
ρ!
ρ0
)(
hv̂ŜA +
ρ!
ρ0
e! ŜA
)=
z
ρ!
ρ0
∂ ŜA
∂t
= γ! z ∇ n ⋅
+
ρ!
ρ0
v̂ ⋅∇ n ŜA +
n
(
γ! z−1 K∇ n ŜA
)
ρ!
ρ0
e!
∂ ŜA
∂z
⎛ ∂ Ŝ ⎞
+⎜D A⎟ +
⎝ ∂z ⎠ z
(A.21.10)
ρ!
ρ0
ˆ SA
S
,
and when the Boussinesq approximation is made we find the simpler expression
∂ ŜA
∂t
+ v̂ ⋅∇ n ŜA + e!
n
⎛ ∂ Ŝ ⎞
∂ ŜA
S
= γ! z ∇ n ⋅ γ! z−1 K∇ n ŜA + ⎜ D A ⎟ + Sˆ A .
∂z
⎝ ∂z ⎠ z
(
)
(A.21.11)
The left-‐‑hand side is the material derivative of the thickness-‐‑weighted Absolute Salinity with
respect to the thickness-‐‑weighted horizontal velocity v̂ and the temporally averaged
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
119
dianeutral velocity e of density coordinates. Apart from the non-‐‑conservative source term
S
Sˆ A , the right-‐‑hand side is the divergence of the turbulent fluxes of Absolute Salinity.
The corresponding turbulent evolution equation for Reference Salinity can be shown to be
⎛ ∂ Ŝ ⎞
∂ Ŝ
r1
∂ Ŝ R
S
(A.21.12)
Sˆ A .
+ v̂ ⋅∇ n Ŝ R + e! R = γ! z ∇ n ⋅ γ! z−1 K∇ n Ŝ R + ⎜ D R ⎟ +
∂t
∂z
1 + r1
⎝ ∂z ⎠
(
)
n
z
(
)
As discussed in appendices A.4 and A.20, given our rather elementary knowledge of the way
variations in seawater composition affect conductivity, we recommend that r1 be taken to be
the constant r1 = 0.35. Hence we see that Reference Salinity is affected by biogeochemical
processes at the fraction 0.35/ 1.35 ( ≈ 0.26 ) of the corresponding influence of biogeochemistry
on Absolute Salinity S A .
We turn now to consider the material derivative of Conservative Temperature in a
turbulent ocean. From Eqns. (A.13.5) and (A.21.8) the instantaneous material derivative of Θ
is, without approximation,
ρ c0p
(T0 + θ ) − ∇ ⋅F R − ∇ ⋅FQ + ρε + h ρ S S
)
S
(T0 + t ) (
⎡ (T0 + θ )
⎤
S
S
− ⎢
µ ( p ) − µ ( 0 ) ⎥ ( −∇ ⋅F + ρ S ) .
⎢⎣ (T0 + t )
⎥⎦
dΘ
=
dt
A
A
(A.21.13)
A
The fact that the right-‐‑hand side of Eqn. (A.21.13) is not the divergence of a flux means that Θ
is not a 100% conservative variable. However, the finite-‐‑amplitude analysis of mixing pairs of
seawater parcels in appendix A.18 has shown that the non-‐‑constant coefficients of the
divergences of the molecular fluxes of heat − ∇ ⋅ FQ and salt −∇ ⋅ FS appearing on the right-‐‑
hand side of Eqn. (A.21.13) are of no practical consequence as they cause an error in
Conservative Temperature of no more than 1.2 mK (see Figure A.18.1). These non-‐‑ideal
terms on the right-‐‑hand side of Eqn. (A.21.13) in a turbulent ocean have been shown to be an
order of magnitude less than the dissipation term ρε which is also justifiably neglected in
S
oceanography (Graham and McDougall, 2013). The source term ρ S A was not considered in
the mixing of seawater parcels in appendix A.18, and we now show that these terms also
make negligible contributions to Eqn. (A.21.13).
The partial derivative of enthalpy with respect to Absolute Salinity, hS , that appears in
A
Eqn. (A.21.13) is about − 65 J g −1 (i.e. − 65 J kg−1 (g kg −1 ) −1 ) at a temperature of 10 °C . This
value can be deduced from Figure A.17.1 and also from Figure 30(c) and Table 12 of Feistel
(2003), albeit for the Gibbs function of seawater that immediately predated the TEOS-‐‑10 saline
Gibbs function of Feistel (2008) and IAPWS (2008). The spatial integral of the source term
S
ρ S A from the North Atlantic to the North Pacific is sufficient to cause a change in Absolute -‐‑
Salinity of 0.025 g kg −1 , so the maximum contribution to an error in Θ from the source term
S
hS ρ S A T0 + θ T0 + t in Eqn. (A.21.13), when integrated over the whole ocean, is
A
S
approximately (c0p ) −1 65 J g −1 0.025 g kg −1 ≈ 0.4 mK . The other term in ρ S A in Eqn.
(A.21.13) is multiplied by the square bracket which from equation (27) of McDougall (2003) is
equal to (T0 + θ ) (T0 + t ) times approximately − pβ Θ ρ −1 , so that this square bracket is
approximately 30 J g −1 (i.e. 30 J kg−1 (g kg −1 ) −1 ) at a pressure p of 4000 dbar (40 MPa) so the
S
contribution of this term is less than half that of the term in ρ S A in the first line of Eqn.
S
(A.21.13). This confirms that the presence of the two terms in ρ S A in the First Law of
Thermodynamics has less impact than even the non-‐‑ideal nature of the molecular flux
divergence terms in Eqn. (A.21.13) and the dissipation of kinetic energy in this equation.
Hence with negligible error, the right-‐‑hand side of Eqn. (A.21.13) may be regarded as the
sum of the ideal molecular flux of heat term − ∇ ⋅ FQ and the term due to the boundary and
radiative heat fluxes, − (T0 + θ ) ∇ ⋅ F R (T0 + t ) . At the sea surface the potential temperature
θ and in situ temperature t are equal so that this term is simply − ∇ ⋅ FR so that there are no
approximations with treating the air-‐‑sea sensible, latent and radiative heat fluxes as being
(
)(
)
(
)(
)
IOC Manuals and Guides No. 56
120
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
fluxes of c0p Θ. There is an issue at the sea floor where the boundary heat flux (the geothermal
heat flux) affects Conservative Temperature through the “heat capacity” (T0 + t ) c0p (T0 + θ )
rather than simply c 0p . That is, the input of a certain amount of geothermal heat flux will
cause a local change in Θ as though the seawater had the “specific heat capacity”
(T0 + t ) c0p (T0 + θ ) rather than c 0p . These two specific heat capacities differ from each other
by no more than 0.15% at a pressure of 4000 dbar. If this small percentage change in the
effective “specific heat capacity” was ever considered important, it could be corrected by
artificially multiplying the geothermal heat flux at the sea floor by T0 + θ ⎡⎣T0 + t ⎤⎦ c0p , so
becoming the geothermal flux of Conservative Temperature.
Graham and McDougall (2013) have derived the evolution equation for Conservative
Temperature for a turbulent ocean while retaining the non-‐‑conservative source terms (see
Eqn. (A.18.3) above), and have used these terms to quantify the extent of the non-‐‑conservation
of Θ in a realistic ocean model. This work has confirmed that Conservative Temperature is
two orders of magnitude more conservative in the ocean than is potential temperature, and
has also shown that the neglect of the dissipation of kinetic energy ε is more than an order of
magnitude more important than the neglect of the non-‐‑ideal nature of the Conservative
Temperature variable (that is, the neglect of the last two lines of Eqn. (A.18.3)). We conclude
that for the purpose of accounting for the transport of “heat” in the ocean it is sufficiently
accurate to assume that Conservative Temperature is in fact conservative and that its
instantaneous conservation equation is
(
)
(
)
dΘ
(A.21.14)
= − ∇ ⋅ FR − ∇ ⋅ FQ .
dt
Now we perform the same two-‐‑stage averaging procedure as outlined above in the case of
Preformed Salinity. The Boussinesq form of the mesoscale-‐‑averaged equation is (analogous to
Eqn. (A.21.7))
c0p ( ρ Θ)t + c0p ∇ ⋅ ( ρ Θu ) = ρ c0p
(
) (
)
∂Θ̂
∂Θ̂
+ v̂ ⋅∇ nΘ̂ + e!
= γ! z ∇ n ⋅ γ! z−1 K∇ nΘ̂ + DΘ̂ z − F bound .
z
∂t
∂z
(A.21.15)
n
As in the case of the S* equation (A.21.7), the molecular flux of heat has been ignored in
comparison with the turbulent fluxes of Conservative Temperature. The air-‐‑sea fluxes of
sensible and latent heat, the radiative and the geothermal heat fluxes remain in Eqn. (A.21.15)
in the vertical heat flux F bound which is the sum of these boundary heat fluxes divided by
ρ0c0p . Any conservative variable, C , obeys a conservation equation identical in form to Eqns.
(A.21.7) and (A.21.15), with Ĉ simply replacing Ŝ* or Θ̂ in these equations, and of course
with the boundary flux being the boundary flux of property C .
The errors incurred in ocean models by treating potential temperature θ as being
conservative have not yet been thoroughly investigated, but McDougall (2003), Tailleux (2010)
and Graham and McDougall (2013) have made a start on this topic. McDougall (2003) found
that typical errors in θ are ± 0.1°C while in isolated regions such as where the fresh Amazon
water discharges into the ocean, the error can be as large as 1.4 °C . The corresponding error
in the meridional heat flux appears to be about 0.005 PW (or a relative error of 0.4%). The use
of Conservative Temperature Θ in ocean models reduces the non-‐‑conservative source terms
associated with the use of potential temperature by two orders of magnitude (Graham and
McDougall, 2013). Note that the consequences for dynamical oceanography of ignoring the
non-‐‑conservative source terms in the potential temperature evolution equation are larger than
ignoring the variations in seawater composition; a θ range of 0.2 °C corresponds to a density
range of 0.04 kg m−3 which is twice as large as the density error due to ignoring the maximum
value of SA − SR of 0.025 g kg −1 .
The evolution equations of Preformed Salinity (A.21.7) and Conservative Temperature
(A.21.15) are the underpinning conservation equations for these variables in ocean models.
An important issue for ocean models is how to relate v̂ to the Eulerian-‐‑mean horizontal
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
121
velocity v . This area of research involves temporal-‐‑residual-‐‑mean theory and the quasi-‐‑
Stokes streamfunction (Gent and McWilliams (1990), Gent et al. (1995), McDougall and
McIntosh (2001) and Griffies (2004)). We will not discuss this topic here. Suffice it to say that
the mean advection can be expressed in Cartesian coordinates, with for example, Eqn.
(A.21.15) becoming
(
) (
)
dΘ̂
= Θ̂ t + v̂ ⋅∇ z Θ̂ + w*Θ̂ z = γ! z ∇ n ⋅ γ! z−1 K∇ nΘ̂ + DΘ̂ z − F bound ,
z
z
dt
(A.21.16)
where the vertical velocity w* is related to e by
w* = zt + v̂ ⋅∇ n z + e .
(A.21.17)
n
A.22 The material derivatives of density and of locally-referenced
potential density; the dianeutral velocity e
Regarding density to be a function of Conservative Temperature (i. e. ρ = ρˆ ( SA , Θ, p ) ) and
taking the material derivative of the natural logarithm of density following the mesoscale-‐‑
thickness-‐‑weighted-‐‑averaged mean flow (as in Eqns. (A.21.15) or (A.21.16)), we have
ˆ
dρˆ
dSˆ
dΘ
dP
(A.22.1)
= β Θ A − αΘ
+ κ
,
dt
dt
dt
dt
where ρ̂ is the thickness-‐‑weighted average value of density. One can continue to consider
the material derivative of in situ density, and in so doing, one carries along the last term in
Eqn. (A.22.1), κ dP dt , but it is more relevant and more interesting to consider the material
derivative of the logarithm of the locally-‐‑referenced potential density, ρˆ l , since this variable is
locally constant in the neutral tangent plane. The material derivative of ρˆ l is given by
ˆ
dρˆ l
dρˆ
dP
dSˆ
dΘ
(A.22.2)
ρˆ −1
= ρˆ −1
−κ
= β Θ A − αΘ
.
dt
dt
dt
dt
dt
Substituting from Eqns. (A.21.11) and (A.21.15) above, and noting that both the temporal and
the lateral gradients of ρˆ l vanish along the neutral tangent plane (that is,
ˆ − β Θ∇ Sˆ = 0 and α Θ Θ
ˆ − β Θ Sˆ
α Θ∇n Θ
= 0 ), the material derivative of ρˆ l amounts to
t
At
n A
n
n
the following equation for the dianeutral velocity e (note that the boundary heat flux F bound
also needs to be included for fluid volumes that abut the sea surface)
ρˆ −1
(
e α ΘΘ̂ z − β Θ ŜA
) = α γ ∇ ⋅ (γ
Θ
z
z
n
(
−1
z K∇ nΘ̂
)
) − β γ ∇ ⋅ (γ
Θ
(
+ α Θ DΘ̂ z − β Θ DŜA
z
)
z z
z
n
−1
z K∇ n Ŝ A
S
− β Θ Sˆ A .
)
(A.22.3)
The left-‐‑hand side is equal to e g −1 N 2 and the first two terms on the right hand side would
sum to zero if the equation of state were linear. This equation can be rewritten as the
following equation for the temporally averaged vertical velocity through the neutral tangent
plane at a given longitude and latitude (from McDougall (1987b), and see Eqns. (3.8.2) and
(3.9.2) for the definitions of C bΘ and TbΘ )
(
)
(
)
(
e g −1 N 2 = − K CbΘ∇ nΘ̂ ⋅∇ nΘ̂ + TbΘ∇ nΘ̂ ⋅∇ n P + α Θ DΘ̂ z − β Θ DŜA
z
)
z z
S
− β Θ Sˆ A . (A.22.4)
The cabbeling nonlinearity (the C bΘ term) always causes “densification”, that is, it always
causes a negative dianeutral velocity, e , while the thermobaric nonlinearity (the TbΘ term)
can cause either diapycnal upwelling or downwelling. The vertical turbulent diffusion terms
can be re-‐‑expressed in terms of DN 2 so that Eqn. (A.22.4) becomes
IOC Manuals and Guides No. 56
122
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
(
)
S
e N 2 = − gK CbΘ∇ nΘ̂ ⋅∇ nΘ̂ + TbΘ∇ nΘ̂ ⋅∇ n P − g β Θ Sˆ A
+
(
)
Rρ
1
ADN 2 − DN 2
z
A
Rρ −1
(
)
⎡ α Θ β Θ 1 ⎤
⎢ zΘ − zΘ
⎥,
β Rρ ⎥⎦
⎢⎣ α
(A.22.5)
where the area A = A( z ) of the density surfaces is included (Klocker and McDougall (2010a)).
This is the complete equation relating upwelling e to diffusion D in the sense of the “abyssal
recipes” of Munk (1966) and Munk and Wunsch (1998). In this context, the Osborn (1980)
relation DN 2 = Γε ≈ 0.2 ε can be used in the second line of Eqn. (A.22.5) to relate upwelling to
the dissipation of turbulent kinetic energy, ε (Klocker and McDougall (2010a)).
The thermobaric and cabbeling dianeutral advection processes are illustrated in Figure
A.22.1. Water parcels A and B are brought together in an adiabatic and isohaline manner
until they meet at location D. During this adiabatic advection process their values of
Absolute Salinity and Conservative Temperature are constant, and since they meet at the
pressure at D, they must have the same value of potential density with respect to the pressure
at D (see this potential isopycnal on panel (b) of the figure). Also, during this adiabatic and
isohaline motion, both parcels A and B fall off the neutral trajectory that links the original
positions of the parcels. This vertical motion occurs because these parcels have a different
compressibility to the water on the neutral trajectory (because they have different
temperatures and salinities to the corresponding parcels on the neutral trajectory). Once
parcels A and B mix intimately, the density of the mixed parcel is greater than that of the
original parcels and so the combined parcel sinks vertically from location D to location E.
This sinking is due to cabbeling, that is, it is due to the potential density surfaces being curved
on the SA − Θ diagram.
Figure A.22.1. Sketch of the dianeutral advection processes, thermobaricity and cabbeling.
To summarize this appendix A.22; we have found that the material derivative of in situ
density Eqn. (A.22.1), when adjusted for the dynamically passive compressibility term,
becomes the material derivative of locally-‐‑referenced potential density Eqn. (A.22.2) which
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
123
can be interpreted as an expression Eqn. (A.22.4) for e , the temporally-‐‑averaged vertical
velocity through the local neutral tangent plane. This dianeutral velocity e is not a separate
mixing process, but rather is a direct result of mixing processes such as (i) small-‐‑scale
turbulent mixing as parameterized by the diffusivity D, and (ii) lateral turbulent mixing of
heat and salt along the neutral tangent plane (as parameterized by the lateral turbulent
diffusivity K ) acting in conjunction with the cabbeling and thermobaric nonlinearities of the
equation of state. Note that a common diapycnal mixing mechanism, double-‐‑diffusive
convection (which actually comes in two separate flavors, a salt-‐‑fingering type and a
“diffusive” type of double-‐‑diffusive convection) is omitted from the conservation equations
(A.21.11) and (A.21.15) and also from the mean dianeutral velocity equation (A.22.4). It is
however straightforward to include these processes in these conservation equations (see for
example McDougall (1984, 1987b)).
A.23 The water-mass transformation equation
It is instructive to substitute Eqn. (A.22.4) for e into the expression (A.21.15) for the material
derivative of Θ̂ , thus eliminating e and obtaining the following equation for the temporal
and spatial evolution of Θ̂ along the neutral tangent plane (McDougall (1984))
(
)
(
∂Θ̂
+ v̂ ⋅∇ nΘ̂ = γ! z ∇ n ⋅ γ! z−1 K∇ nΘ̂ + KgN −2Θ̂ z CbΘ∇ nΘ̂ ⋅∇ nΘ̂ + TbΘ∇ nΘ̂ ⋅∇ n P
∂t
)
n
Θ
+ Dβ gN
−2
Θ̂ 3z
d 2 ŜA
d Θ̂ 2
(A.23.1)
β Θ Rρ
S
Sˆ A ,
+ Θ
α Rρ −1
(
)
ˆ β Θ Sˆ . The term involving D
where Rρ is the stability ratio of the water column, Rρ = α ΘΘ
z
Az
ˆ diagram of a vertical cast; this
has been written as proportional to the curvature of the SˆA − Θ
ˆ Sˆ − Sˆ Θ
ˆ
term can also be written as Dβ Θ gN −2 Θ
z A zz
A z zz . The form of Eqn. (A.23.1) illustrates
that when analyzed in density coordinates, Conservative Temperature (and Absolute Salinity)
(i) are affected not only by the expected lateral diffusion process along density surfaces but
also by the nonlinear dianeutral advection processes, cabbeling and thermobaricity, (ii) are
ˆ diagram is
affected by diapycnal turbulent mixing only to the extent that the vertical SˆA − Θ
not locally straight, and (iii) are not influenced by the vertical variation of D since Dz does
not appear in this equation.
Equations (A.21.11) and (A.21.15) are the fundamental conservation equations of salinity
and Conservative Temperature in a turbulent ocean, and the pair of equations (A.22.4) and
(A.23.1) are simply derived as linear combinations of Eqns. (A.21.11) and (A.21.15). The
“density” conservation equation (A.22.4) and the “water-‐‑mass transformation” equation
(A.23.1) are in some sense the “normal modes” of Eqns. (A.21.11) and (A.21.15). That is, Eqn.
(A.22.4) expresses how mixing processes contribute to the mean vertical velocity e through
the neutral tangent plane, while (A.23.1) expresses how the tracer called “Conservative
Temperature measured along the neutral direction” is affected by mixing processes; this
equation does not contain e .
For completeness, the water-‐‑mass conservation equation for Absolute Salinity that
corresponds to Eqn. (A.23.1) is
(
(
)
)
(
∂ ŜA
+ v̂ ⋅∇ n ŜA = γ! z ∇ n ⋅ γ! z−1 K∇ n ŜA + K gN −2 ŜA CbΘ∇ nΘ̂ ⋅∇ nΘ̂ + TbΘ∇ nΘ̂ ⋅∇ n P
z
∂t
n
Θ
+ Dα gN
−2
Θ̂ 3z
d 2 ŜA
d Θ̂ 2
+
Rρ
( R −1)
)
(A.23.2)
S
Sˆ A ,
ρ
IOC Manuals and Guides No. 56
124
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
and it easy to show that α Θ times the right-‐‑hand side of Eqn. (A.23.1) is equal to β Θ times the
right-‐‑hand side of Eqn. (A.23.2).
The water-‐‑mass transformation rates of Absolute Salinity and of Conservative
Temperature are illustrated in Figure A.23.1 for an ocean in steady-‐‑state. In this situation, the
water-‐‑mass transformation rates in terms of SA and Θ (from Eqns. (A.23.1) and (A.23.2)) are
v̂ ⋅∇ n ŜA and v̂ ⋅∇ nΘ̂ respectively, and these are illustrated as a vector in the figure, directed
along the neutral direction. By contrast, the material derivative of SA and Θ (from Eqns.
(A.21.11) and (A.21.15), also shown in the figure) include contributions from the mean
dianeutral velocity e . The contribution to the material derivatives from purely horizontal
advection along the local isobaric surface is also sketched in the figure. The advantage of the
water-‐‑mass transformation approach using the neutral framework, namely v̂ ⋅∇ n ŜA and
v̂ ⋅∇ nΘ̂ is that it can be observed in the ocean due to spatial (or corresponding temporal)
changes along neutral density surfaces. In contrast, one seldom has a reliable estimate of the
dianeutral advection e at any particular location in the ocean and so the material derivatives
dSA dt and dΘ dt are not observable quantities. Moreover, in contrast to the isobaric
gradients, the epineutral gradients of “water-‐‑mass conversion”, v̂ ⋅∇ n ŜA and v̂ ⋅∇ nΘ̂ are not
affected by the passive vertical motion of a water column caused by adiabatic vertical heaving
motion.
Figure A.23.1. Sketch of the water-‐‑mass transformation, compared with the material
derivative of Absolute Salinity and Conservative Temperature. Two vertical
casts at different horizontal locations are sketched in the figure.
To construct the water-‐‑mass transformation equation of a conservative tracer C , the mean
dianeutral velocity e is eliminated from the Ĉ conservation equation (A.24.1) using Eqn.
(A.22.4) giving (from McDougall (1984))
(
)
(
∂Ĉ
+ v̂ ⋅∇ nĈ = γ! z ∇ n ⋅ γ! z−1 K∇ nĈ + K gN −2 Ĉz CbΘ∇ nΘ̂ ⋅∇ nΘ̂ + TbΘ∇ nΘ̂ ⋅∇ n P
∂t
n
( )
+ D ŜA
2
z
d 2Ĉ
dŜA2
+
Ĉz
ŜA
z
Θ
Dα gN
−2
Θ̂ 3z
d 2 ŜA
d Θ̂ 2
−2
)
(A.23.3)
Θ
S
+ Ĉz gN β Sˆ A .
This equation shows that vertical turbulent mixing processes affect the tracer on neutral
tangent planes according to the curvatures of vertical casts as displayed on both the SˆA − Cˆ
ˆ curves. The terms involving D can also be written as
and the SˆA − Θ
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
( )
D SˆA z
2
2ˆ
d 2Cˆ
Cˆ z
Θ
−2 ˆ 3 d S A
+
α
Θ
=
D
gN
z
ˆ2
dSˆA2
SˆA z
dΘ
(
D SˆA z Cˆ zz − SˆA zz Cˆ z
)
Θ
SˆA z + DCˆ zα gN
−2
(Θˆ Sˆ
ˆ
− SˆA z Θ
zz
z A zz
)
125
(A.23.4)
SˆA z .
A.24 Conservation equations written in potential density coordinates
The material derivative of a conservative quantity C can be expressed with respect to the
Cartesian reference frame, the neutral tangent plane, or a potential density reference frame so
that the conservation equation of a conservative variable can be written as (see Eqn. (A.21.16),
∂Ĉ
∂Ĉ
∂Ĉ
+ v̂ ⋅∇ z Ĉ + w*
=
∂t
∂z
∂t
z
+ v̂ ⋅∇ nĈ + e! Ĉz =
n
∂Ĉ
∂t
+ v̂ ⋅∇σ Ĉ + e! d Ĉz
σ
(
) ( )
(A.24.1)
= γ! z ∇ n ⋅ γ! z−1 K∇ nĈ + DĈz ,
z
where e d is the mean vertical component of the total transport velocity that moves through
the potential density surface. Any flux of C across the ocean boundaries F bound (e.g., the sea
surface) would need to be added as the extra term − Fzbound on the last line of Eqn. (A.24.1).
Notice that the lateral diffusion occurs along the neutral tangent plane. In this section we
consider what terms are neglected if this lateral mixing term is instead regarded as diffusion
occurring along potential density surfaces.
The temporal and lateral gradients of Absolute Salinity and Conservative Temperature in
a potential density surface are related by (McDougall (1991))
ˆ
α Θ ( pr ) Θ
t
σ
− β Θ ( pr ) SˆAt
σ
ˆ − β Θ ( p ) ∇ Sˆ = 0 ,
= 0 and α Θ ( pr ) ∇σ Θ
r
σ A
(
(A.24.2)
)
(
)
ˆ , p and β Θ Sˆ , Θ
ˆ , p
where α Θ ( pr ) and β Θ ( pr ) are shorthand notations for α Θ SˆA , Θ
r
A
r
respectively, and pr is the reference pressure of the potential density. Using Eqns. (3.17.1) to
(3.17.5) which relate the gradients of properties in a potential density surface to those in a
neutral tangent plane, the following form of the conservation equation (A.21.15) for
Conservative Temperature can be derived (see equation (26) of McDougall (1991))
∂Θ̂
∂t
+ v̂ ⋅∇σ Θ̂ + e! d
σ
(
) (
∂Θ̂
= σ! z ∇σ ⋅ σ! z−1 K∇σ Θ̂ + DΘ̂ z
∂z
(
)
z
)
⎛
∇ Θ̂ ⋅∇ nΘ̂ ⎞
− h−1∇ n ⋅ ⎡G Θ −1⎤ hK∇ nΘ̂ − ⎜ G Θ ⎡G Θ −1⎤ K n
⎟ ,
⎣
⎦
⎣
⎦
Θ̂ z
⎝
⎠z
(A.24.3)
where the “isopycnal temperature gradient ratio” G Θ is defined as (from Eqn. (3.17.4))
G Θ = ∇σ Θ ∇ n Θ = r ⎡⎣ Rρ −1⎤⎦ ⎡⎣ Rρ − r ⎤⎦ and r is defined in Eqn. (3.17.2) as the ratio of α Θ / β Θ
at the in situ pressure p to that evaluated at the reference pressure pr . σ z−1 is the averaged
value of the reciprocal of the vertical gradient of potential density, while σ z is simply the
reciprocal of σ z−1 . The corresponding equation for Absolute Salinity is
∂ ŜA
∂t
+ v̂ ⋅∇σ ŜA + e! d
σ
(
) (
∂ ŜA
= σ! z ∇σ ⋅ σ! z−1 K∇σ ŜA + DŜA
z
∂z
)
z
S
+ Sˆ A
⎛ ⎡ GΘ
⎞
⎛ GΘ
⎤
∇ Θ̂ ⋅∇ n ŜA ⎞
⎡G Θ −1⎤ K n
− h−1∇ n ⋅ ⎜ ⎢
−1⎥ hK∇ n ŜA ⎟ − ⎜
⎟ .
⎣
⎦
Θ̂ z
⎝ r
⎠z
⎥⎦
⎝ ⎢⎣ r
⎠
(A.24.4)
The terms in the second lines of Eqns. (A.24.3) and (A.24.4) arise because in the first line of
these equations, the lateral diffusion is written as being along potential density surfaces rather
than along neutral tangent planes. As explained in McDougall (1991), these terms are non
zero even at the reference pressure of the potential density variable.
IOC Manuals and Guides No. 56
126
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Multiplying Eqn. (A.24.4) by β Θ ( pr ) and subtracting α Θ ( pr ) times Eqn. (A.24.3) the
corresponding expression for the diapycnal velocity e d is (following McDougall (1991))
e! d
(
)
(
1 ∂ ρ̂ Θ
= β Θ pr σ! z ∇σ ⋅ σ! z−1 K∇σ ŜA − α Θ pr σ! z ∇σ ⋅ σ! z−1 K∇σ Θ̂
ρ̂ Θ ∂z
( )
( )(
+ β Θ pr DŜA
) −α
z z
Θ
( )
( pr )( DΘ̂ z )z + β Θ ( pr ) Sˆ S
)
A
)
(
) ( ) Gr K∇nr ⋅∇nΘ̂
⎛ α Θ ( p )⎞
∇ Θ̂ ⋅∇ nΘ̂
− β Θ ( pr ) ⎜ Θ r ⎟ G Θ ⎡G Θ −1⎤ K n
⎣
⎦
Θ̂ z
⎝ β ( pr ) ⎠ z
⎤
β Θ ( p ) ⎡ GΘ
− Θ r ⎢
− 1⎥ K ( CbΘ∇ nΘ̂ ⋅∇ nΘ̂ + TbΘ∇ nΘ̂ ⋅∇ n P ) .
β ( p ) ⎢⎣ r
⎥⎦
( )(
+ α Θ pr r −1 γ z ∇ n ⋅ γ z−1 K∇ nΘ̂ + α Θ pr
Θ
(A.24.5)
All the terms in the last three lines of this equation occur because the first line has lateral
mixing along potential density surfaces rather than along neutral tangent planes. Even at the
reference pressure where G Θ = r =1 these last three lines do not reduce to zero but rather to
ˆ ⋅ ∇ P showing that the thermobaric effect remains.
TbΘ K ∇n Θ
n
In summary, this section has written down the expressions for the material derivatives of
Conservative Temperature, Absolute Salinity and potential density in a form where one can
identify the many rather nasty terms that are neglected if one assumes that the ocean mixes
laterally along potential density surfaces instead of the physically correct neutral tangent
planes. It is noted in passing that the first line of the right-‐‑hand side of Eqn. (A.24.5) can also
ˆ ⋅∇ Θ
ˆ
be written as CbΘ ( pr ) K ∇σ Θ
σ (c.f. the last line of Eqn. (A.27.2) below).
A.25 The vertical velocity through a general surface
Consider a general surface which we identify with the label “a” (for example, this could stand
for “approximately neutral surface”). The material derivative on the left-‐‑hand sides of the
conservation equations (A.21.11) and (A.21.15) for Absolute Salinity and Conservative
Temperature are now written with respect to this general “a” coordinate as
∂ ŜA
∂t
+ v̂ ⋅∇ a ŜA + e! a
a
⎛ ∂ Ŝ ⎞
∂ ŜA
S
= γ! z ∇ n ⋅ γ! z−1 K∇ n ŜA + ⎜ D A ⎟ + Sˆ A ,
∂z
∂z
⎝
⎠z
(
)
(A.25.1)
and
(
) ( )
(Sˆ , Θˆ , p) and
∂Θ̂
∂Θ̂
+ v̂ ⋅∇ aΘ̂ + e! a
= γ! z ∇ n ⋅ γ! z−1 K∇ nΘ̂ + DΘ̂ z .
z
∂t
∂z
a
(A.25.2)
(
)
ˆ , p and
Cross-‐‑multiplying these equations by β Θ = β Θ A
α Θ = α Θ SˆA , Θ
subtracting gives the following equation for the vertical velocity through the approximately
neutral surface,
(
e a = − g N −2 K CbΘ∇ nΘ̂ ⋅∇ nΘ̂ + TbΘ∇ nΘ̂ ⋅∇ n P
(
)
(
)
)
S
+ g N −2 ⎛ α Θ DΘ̂ z − β Θ DŜA ⎞ − g N −2β Θ Sˆ A
⎝
z z⎠
z
(A.25.3)
⎡ ∂ Ŝ
∂Θ̂ ⎤
⎥.
+ g N −2 v̂ ⋅ ⎡ β Θ∇ a ŜA − α Θ∇ aΘ̂ ⎤ + g N −2 ⎢ β Θ A − α Θ
⎣
⎦
∂t
∂t ⎥
⎢⎣
a ⎦
a
The terms in the third line of this equation represent the deviation of the “a” coordinate from
neutrality and these terms can be shown to be (from Klocker and McDougall (2010b) and
from Eqn. (3.14.1) above, assuming the surfaces are not vertical)
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
l
ˆ ⎤ = − vˆ ⋅ ∇a ρˆ = vˆ ⋅ (∇ z − ∇ z ) = vˆ ⋅ s
g N −2 vˆ ⋅ ⎡ β Θ∇a SˆA − α Θ∇a Θ
n
a
⎣
⎦
ρˆ zl
127
(A.25.4)
and
ρˆtl
⎡
ˆ ⎤
∂Sˆ
∂Θ
(A.25.5)
g N −2 ⎢ β Θ A − α Θ
⎥ = − la = zt n − zt a
ˆz
∂t a
∂t a ⎥
ρ
⎢⎣
⎦
l
where ρˆ is the (thickness-‐‑weighted) locally-‐‑referenced potential density.
Combining these results with Eqn. (A.22.4) we have the simple kinematic result that
e a = e + v̂ ⋅s + zt − zt ,
n
(A.25.6)
a
showing that the vertical velocity through a general “a” surface, e a , is that through the
neutral tangent plane e plus that due to the “a” surface having a different slope in space to
the neutral tangent plane, vˆ ⋅ s, plus that due to the “a” surface moving vertically in time (at
fixed latitude and longitude) at a different rate than the neutral tangent plane, zt n − zt a .
A.26 The material derivative of potential density
The material derivative of the natural logarithm of potential density is β Θ ( pr ) times the
material derivative Eqn. (A.21.11) of Absolute Salinity minus α Θ ( pr ) times the material
derivative Eqn. (A.21.15) of Conservative Temperature. Using the relationships Eqn. (A.24.2)
that relate the gradients of Absolute Salinity and Conservative Temperature in potential
density surfaces, and taking the material derivative of potential density with respect to
potential density surfaces, one finds that the temporal and isopycnal gradient terms cancel
leaving only the term in the mean diapycnal velocity e d as follows
e! d
(
)
( )
(
)
( r)
)
1 ∂ ρ̂ Θ
= β Θ pr γ! z ∇ n ⋅ γ! z−1 K∇ n ŜA − α Θ pr γ! z ∇ n ⋅ γ! z−1 K∇ nΘ̂
ρ̂ Θ ∂z
SA
Θ
Θ
Θ
ˆ
+ β p DŜ
− α p DΘ̂ + β p S ,
( )
( r )(
Az
)
( r )(
z
z z
(A.26.1)
where the exact expression for the vertical gradient of potential density has been used,
1 ∂ρˆ Θ
ˆ .
(A.26.2)
= β Θ ( pr ) SˆA z − α Θ ( pr ) Θ
z
ρˆ Θ ∂z
Equation (A.26.1) can be written more informatively as (following McDougall, 1991)
e! d
⎛ D ∂ ρ̂ Θ ⎞
1 ∂ ρ̂ Θ
Θ
ˆ SA
=
⎜ Θ ∂z ⎟ + β pr S
ρ̂ Θ ∂z
ρ̂
⎝
⎠z
( )
{
( )
( )
( )
+ D α ΘΘ pr Θ̂ 2z + 2α SΘ pr Θ̂ z ŜA − β SΘ pr ŜA2
+α
Θ
( )
A
(
z
)
pr ⎡⎣ r −1⎤⎦ γ! z ∇ n ⋅ γ! z−1 K∇ nΘ̂ +
A
β Θ ( pr )
β
Θ
( p)
(
z
}
(A.26.3)
)
K CbΘ∇ nΘ̂ ⋅∇ nΘ̂ + TbΘ∇ nΘ̂ ⋅∇ n P ,
where r is defined in Eqn. (3.17.2) as the ratio of α Θ / β Θ at the in situ pressure p to that
evaluated at the reference pressure pr . If the equation of state were linear, only the first two
terms would be present on the right of Eqn. (A.26.3).
IOC Manuals and Guides No. 56
128
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
A.27 The diapycnal velocity of layered ocean models (without rotation of the mixing
tensor)
Layered models of the ocean circulation have a potential density variable (usually with a
reference pressure pr of 2000 dbar) as their vertical coordinate. To date these models have
not rotated the direction of lateral mixing to align with the neutral tangent plane but have
mixed laterally along the potential density coordinate direction. The diapycnal velocity
e d_model in this class of model obeys the equation (c.f. Eqn. (A.26.1) above)
e! d_model
(
) ( ) (
( )( )
( )
)
1 ∂ ρ̂ Θ
= β Θ pr σ! z ∇σ ⋅ σ! z−1 K σ ∇σ ŜA − α Θ pr σ! z ∇σ ⋅ σ! z−1 K σ ∇σ Θ̂
ρ̂ Θ ∂z
SA
Θ
Θ
Θ
ˆ
+ β p DŜ
− α p DΘ̂ + β p S ,
( )
( r )(
Az
)
r
z
z z
(A.27.1)
r
where ∇σ is the gradient operator along the potential density coordinate, K σ is the lateral
diffusivity along the layers, σ z−1 is the averaged value of the reciprocal of the vertical gradient
of potential density, while σ z is simply the reciprocal of σ z−1 . This equation can be written as
e d_model
⎛ D ∂ ρ̂ Θ ⎞
1 ∂ ρ̂ Θ
Θ
ˆ SA
=
⎜ Θ ∂z ⎟ + β pr S
ρ̂ Θ ∂z
⎝ ρ̂
⎠z
( )
{
( )
( )
K σ CbΘ ( pr ) ∇σ Θ̂ ⋅∇σ Θ̂ .
( )
}
+ D α ΘΘ pr Θ̂ 2z + 2α SΘ pr Θ̂ z ŜA − β SΘ pr ŜA2
+
A
z
A
z
(A.27.2)
The terms in the vertical turbulent diffusivity D are identical to those in the correct equation
(A.26.3) while the diapycnal velocity due to cabbeling is quite similar to that in the correct
expression Eqn. (A.26.3); the difference mostly being that the cabbeling coefficient is here
evaluated at the reference pressure instead of at the in situ pressure, and that the lateral
temperature gradient is here evaluated along the potential density surface rather than along
the neutral tangent plane (these gradients are proportional to each other via the relation
(3.17.3)). Another difference is that the term α Θ pr ⎡⎣ r −1⎤⎦ γ z ∇ n ⋅ γ z−1 K∇ nΘ̂ in Eqn. (A.26.3) is
missing from Eqn. (A.27.2). This type of difference is to be expected since the direction of the
lateral mixing is different.
Notice the absence of the thermobaric diapycnal advection in Eqn. (A.27.2); that is, the
ˆ ⋅ ∇ P in Eqn. (A.26.3) does not appear in Eqn. (A.27.2); this was
term proportional to K TbΘ∇n Θ
n
first pointed out by Iudicone et al. (2008). The thermobaric diapycnal advection is significant
in the Southern Ocean (Klocker and McDougall (2010a)) and its omission from layered ocean
models amounts to a non-‐‑trivial inherent limitation of this type of ocean model. Also missing
from layered ocean models is the mean vertical advection vˆ ⋅ s due to the helical nature of
neutral trajectories in the ocean (see section 3.13, Eqn. (A.25.4) and Klocker and McDougall
(2010b)), whereas this physical process occurs naturally in z-‐‑coordinate ocean models.
( )
(
)
A.28 The material derivative of orthobaric density
Orthobaric density ρ v ( p, ρ ) has been defined by de Szoeke et al. (2000) as a pressure
corrected form of in situ density. The construction of orthobaric density requires the
isentropic compressibility to be approximated as a function of pressure and in situ density.
While orthobaric density has the advantage of being a thermodynamic variable, orthobaric
density surfaces are often not particularly good approximations to neutral tangent planes (see
McDougall and Jackett (2005a) and Klocker et al. (2009a,b)). The material derivative of ρ v can
be expressed with respect to orthobaric density surfaces as
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
∂ ρv
∂t
+ v̂ ⋅∇ ρ ρv + e
ρv
v
ρv
∂ ρv
ρ ∂ρ
= e v v ,
∂z
∂z
129
(A.28.1)
where the temporally averaged vertical velocity through the ρ v surface is given by (from
McDougall and Jackett (2005a))
e
ρv
(
) ( )
− β Θ S + ψ −1 ⎛ p
= gN −2 α ΘΘ
A
⎝ t
ρv
+ v̂ ⋅∇ ρ p ⎞ pz ,
⎠
v
(A.28.2)
where (from de Szoeke et al. (2000))
(ψ −1)
ˆ Θ ( p, ρ )⎤ ,
≈ 2 g 2 N −2c0−3Δc ≈ − ρ g 2TbΘ N −2 ⎡Θ−
0
⎣
⎦
(A.28.3)
and Δc is the difference between the reference sound speed function c0 ( p, ρ ) and the sound
speed of seawater which can be expressed in the functional form c ( p, ρ , Θ) . This difference in
the sound speed is equivalent to the difference between the actual Conservative Temperature
of a water parcel and the reference value Θ0 ( p, ρ ) . Here SA is shorthand for the material
derivative of Ŝ A and is expressed in terms of mixing processes by the right-‐‑hand side of Eqn.
is similarly shorthand for the material derivative of Θ̂ and is given by the right-‐‑
(A.21.11); Θ
hand side of Eqn. (A.21.15).
The first term on the right of Eqn. (A.28.2) represents the effects of irreversible mixing
ρ
processes on the flow through orthobaric density surfaces, and this contribution to e v is
exactly the same as the flow through neutral tangent planes, e (Eqn. (A.22.4)). The second
term in Eqn. (A.28.2) arises from the non-‐‑quasi-‐‑material (non-‐‑potential) nature of orthobaric
density. This vertical advection arises from the seemingly innocuous sliding motion along the
sloping orthobaric density surface and from the vertical heaving of these surfaces.
A.29 The material derivative of Neutral Density
Neutral Density γ n is not a thermodynamic function since it depends on latitude and
longitude. The Neutral Density algorithm finds the data point in a pre-‐‑labeled reference data
set that has the same potential density as the data point that is being labeled; the reference
pressure of this potential density is the average of the pressures of the two parcels. The
material derivative of γ n can be expressed as
∂γ n
∂t
+ v̂ ⋅∇γ γ n + eγ γ zn = eγ γ zn ,
(A.29.1)
γ
where the temporally averaged vertical velocity through the γ n surface is given by (from
McDougall and Jackett (2005b))
e
γ
≈
(
(α
α
Θ
Θ
− β Θ ( p) S
( p) Θ
A
( p)Θ ref
z
−β
Θ
)
( p)SAref
z
)
+ v̂ ⋅s ref
( )(
)(p )
⎛
⎞
( p− p )
+ (ψ −1) ⎜ v̂ ⋅∇ p −
v̂ ⋅∇ Θ ⎟ ( p )
(Θ̂ − Θ )
⎝
⎠
(α ( p) Θ − β ( p)S )
+ 2(ψ −1)
−1
+ ψ γ −1 pt + v̂ ⋅∇γ p
γ
ref
γ
γ
γ
(
z
)
(α
ref
γ
ref
Θ
Θ
( p)Θ ref
z
Θ
A
Θ
− β ( p)SAref
z
ref
z
−1
(A.29.2)
)
+ ψ γ −1 v̂ ⋅s ref .
Here SA is shorthand for the material derivative of Ŝ A following the appropriate mean
velocity and is expressed in terms of mixing processes by the right-‐‑hand side of Eqn.
IOC Manuals and Guides No. 56
130
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
is similarly shorthand for the material derivative of Θ̂ and is given by Eqn.
(A.21.11), Θ
(A.21.15), and ψ γ − 1 is defined by
( )
(ψ −1)
γ
=
ˆ Θref )
− 12 ρ g 2TbΘ (Θ−
.
ˆ Θref ) + 1 gT Θ ( P − P ref ) Θref ⎤
+ ρ g 2TbΘ (Θ−
b
z ⎦
2
2
⎡ N ref
⎣
(A.29.3)
2
is the square of the buoyancy frequency of the pre-‐‑labelled reference data set.
Here N ref
Equation (A.29.3) shows that ψ γ − 1 is nonzero to the extent that there is a water mass
contrast (Θ− Θref ) between the seawater parcel that is being labeled and the data on the pre-‐‑
labeled reference data set that communicates neutrally with the seawater sample. For
ˆ Θref ) and ( p − p ref ) the denominator in Eqn. (A.29.3) is close to N 2
reasonable values of (Θ−
ref
and ψ γ − 1 is small. In these expressions the thermal expansion coefficient α Θ ( p ) and saline
contraction coefficient β Θ ( p ) are evaluated at the average of the properties of the parcel
being labeled and the parcel in the reference data set to which it is neutrally related, that is,
α Θ ( p ) and β Θ ( p ) are shorthand for α Θ ( SA , Θ, p ) and β Θ ( SA , Θ, p ).
The first term in Eqn. (A.29.2) is expected as Neutral Density changes in response to the
and S . The next term in Eqn. (A.29.2), vˆ ⋅ sref , is also
irreversible mixing processes Θ
A
expected; it is the mean vertical motion through the γ n surface due to the helical motion of
neutral trajectories in the reference data set, caused in turn by the non-‐‑zero neutral helicity of
the reference data set. The remaining terms in the last four lines of Eqn. (A.29.2) arise because
of the non-‐‑quasi-‐‑material (non-‐‑potential) nature of Neutral Density. The second line of Eqn.
(A.29.2) represents the contribution to eγ arising from the seemingly innocuous sliding
motion along the sloping γ n surface and from the vertical heaving of these surfaces. The
lateral gradients of properties in the reference data set also affect the mean flow eγ through
the γ n surface. Note that as Θ̂− Θref tends to zero, ψ γ − 1 also tends to zero so that the
third line of Eqn. (A.29.2) is well-‐‑behaved and becomes proportional to pz−1 ( p − p ref ) vˆ ⋅ ∇γ Θref .
(
(
)
)
(
)
(
)
A.30 Computationally efficient 75-term expression for the specific volume of seawater
in terms of Θ
Ocean models that pre-‐‑date TEOS-‐‑10 have treated their salinity and temperature variables as
being Practical Salinity SP and potential temperature θ . Ocean models that are TEOS-‐‑10
compatible calculate Absolute Salinity S A and Conservative Temperature Θ (as discussed in
appendices A.20 and A.21), and they use a computationally efficient expression for calculating
specific volume (or density) in terms of Absolute Salinity S A , Conservative Temperature Θ
and pressure p .
Earlier versions of the GSW Oceanographic Toolbox and of this TEOS-‐‑10 Manual have
included 25-‐‑term and 48-‐‑term rational functions for specific volume in terms of S A , Θ and p .
These GSW functions (and the corresponding earlier versions of the TEOS-‐‑10 Manual) are
archived and are still available from the TEOS-‐‑10 web site. When implementing TEOS-‐‑10 in
numerical ocean models it became clear that the rational function form for specific volume
was not very computationally efficient and Roquet et al. (2015) showed that a straightforward
polynomial (as opposed to a rational function) is a better form for ocean modelling. Following
the publication of this paper, the GSW Oceanographic Toolbox has adopted this polynomial
form, v̂ 75 SA ,Θ, p , as an accurate alternative to first calculating in situ temperature from
Conservative Temperature and then using the full Gibbs function to evaluate specific volume.
The 75-‐‑term polynomial by Roquet et al. (2015) is expressed in terms of the following
three dimensionless salinity, temperature and pressure variables,
(
)
s ≡
SA + 24 g kg −1
Θ
p
, τ ≡
and π ≡
,
Θu
pu
SA u
IOC Manuals and Guides No. 56
(A.30.1)
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
131
in terms of the unit-‐‑related scaling constants
SA u ≡ 40× 35.16504g kg −1 / 35 , Θ u ≡ 40°C and pu ≡ 104 dbar .
(A.30.2)
Their polynomial expression for the specific volume of seawater is
v̂(SA ,Θ, p) = vu ∑ vijk s i τ jπ k ,
(A.30.3)
i, j,k
where vu ≡ 1 m 3kg −1 and the non-‐‑zero dimensionless constants vijk are given in Table K.1 of
appendix K. The specific volume data was fitted in a “funnel” of data points in ( SA , Θ, p )
space (McDougall et al. (2003)) which extends to a pressure of 8000 dbar . At the sea surface
the “funnel” covers the full range of temperature and salinity while for pressures greater than
6500 dbar the maximum temperature of the fitted data is 10°C and the minimum Absolute
Salinity is 30 g kg −1 . That is, the fit has been performed over a region of parameter space
which includes water that is approximately 8°C warmer and 5 g kg −1 fresher in the deep
ocean than the seawater which exists in the present ocean.
As outlined in appendix K, this 75-‐‑term polynomial expression for v yields the thermal
expansion and saline contraction coefficients, α Θ and β Θ , that are essentially as accurate as
those derived from the full TEOS-‐‑10 Gibbs function for data in the “oceanographic funnel”. In
dynamical oceanography it is these thermal expansion and haline contraction coefficients
which are the most important aspects of the equation of state since the “thermal wind” is
proportional to α Θ∇ p Θ − β Θ∇ p SA and the vertical static stability is given in terms of the
buoyancy frequency N by g −1 N 2 = α Θ Θ z − β Θ (SA ) z . Hence for dynamical oceanography we
may take Roquet et al.’s (2015) 75-‐‑term polynomial expression for specific volume as
essentially reflecting the full accuracy of TEOS-‐‑10.
Appendix P describes how an expression for the enthalpy of seawater in terms of
Conservative Temperature, specifically the functional form hˆ ( S A , Θ, p ) , together with an
expression for entropy in the form ηˆ ( SA , Θ) , can be used as an alternative thermodynamic
potential to the Gibbs function g ( SA , t , p ) . The need for the functional form hˆ ( S A , Θ, p ) also
arises in section 3.32 and in Eqns. (3.26.3) and (3.29.1). The 75-‐‑term expression, Eqn. (A.30.3)
for v 75 = v̂ 75 SA ,Θ, p can be used to find a closed expression for hˆ ( S A , Θ, p ) by integrating
v̂ 75 SA ,Θ, p with respect to pressure (in Pa ), since hˆP = v = ρ −1 (see Eqn. (2.8.3)). Specific
enthalpy calculated from v̂ 75 SA ,Θ, p is available in the GSW Oceanographic Toolbox as the
function gsw_enthalpy(SA,CT,p). Using gsw_enthalpy to evaluate hˆ ( S A , Θ, p ) is 5 times
faster than first evaluating the in situ temperature t (from gsw_t_from_CT(SA,CT,p)) and
then calculating enthalpy from the full Gibbs function expression h ( SA , t, p ) using
gsw_enthalpy_t_exact(SA,t,p). (These last two function calls have also been combined into
the one function, gsw_enthalpy_CT_exact(SA,CT,p).)
Also, the enthalpy difference at the same values of S A and Θ but at different pressures
(see Eqn. (3.32.5)) is available as the function gsw_enthalpy_diff(SA,CT,p_shallow,p_deep).
Following Young (2010), the difference between h and c0p Θ is called “dynamic enthalpy”
and can be found using the function gsw_dynamic_enthalpy(SA,CT,p) in the GSW
Oceanographic Toolbox.
(
)
(
)
(
)
IOC Manuals and Guides No. 56
132
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Appendix B:
Derivation of the First Law of Thermodynamics
Motivation
For a pure fluid in which there is no dissolved material (such as pure water with zero
Absolute Salinity) the derivation of the First Law of Thermodynamics usually starts with a
discussion of how the internal energy U of a fixed mass of fluid is changed under the
influence of it being “heated” by the amount δ Q and its volume V being changed. The
infinitesimal change in the internal energy of the parcel is written as dU = δ Q − ( p + P0 ) dV
where − ( p + P0 ) dV is the mechanical work done on the fluid by the pressure at the moving
boundaries of the fluid parcel. This relationship can be written in terms of the specific (i. e.
per unit mass) enthalpy h, the density ρ , and δ Q per unit volume, δ q, as
⎛ dh 1 dP ⎞
δq
for pure water (B.1)
.
−
⎟ =
dt
⎝ dt ρ dt ⎠
It is recognized that the right-‐‑hand side of (B.1) is not the divergence of a “heat” flux, and the
term that causes this complication is the dissipation of kinetic energy into “heat”, which
contributes ρε to the right-‐‑hand side of (B.1). Apart from this familiar dissipation term, the
right-‐‑hand side is minus the divergence of the sum of the boundary and radiative heat fluxes,
FR , and minus the divergence of the molecular flux of heat FQ = − ρc p k T ∇T (where k T is the
molecular diffusivity of temperature), so that the First Law of Thermodynamics for pure
water is
ρ⎜
⎛ dh 1 dP ⎞
δq
=
ρ⎜
−
= − ∇ ⋅F R − ∇ ⋅FQ + ρε .
⎟
dt
⎝ dt ρ dt ⎠
for pure water (B.2)
Now consider seawater in which the Absolute Salinity and its gradients are non-‐‑zero. The
same traditional discussion of the First Law of Thermodynamics involving the “heating”, the
application of compression work and the change of salinity to a fluid parcel shows that the
change of enthalpy of the fluid parcel is given by (see equations 6b and 17b of Warren (2006))
dH − VdP = δ Q + ( µ − [T0 + t ] µT ) M dS A ,
(B.3)
where M is the mass of the fluid parcel. When written in terms of the specific enthalpy h,
and δ Q per unit volume, δ q , this equation becomes (using ρ dSA dt = −∇ ⋅ F S )
⎛ dh 1 dp ⎞
δq
−
− ( µ − [T0 + t ] µT ) ∇ ⋅ FS .
⎟ =
t
ρ
t
t
d
d
d
⎝
⎠
ρ⎜
(B.4)
Does this help with the task of constructing an expression for the right-‐‑hand side of (B.4)
in terms of the dissipation of kinetic energy and the molecular, radiative and boundary fluxes
of “heat” and salt? If the “heating” term δ q dt in Eqn. (B.4) were the same as in the pure
water case Eqn. (B.2) then we would have successfully derived the First Law of
Thermodynamics in a saline ocean via this route. However, we will now show that δ q dt in
Eqn. (B.4) is not the same as that in the pure water case, Eqn. (B.2).
Substituting the expression for δ q dt from (B.2) into the right-‐‑hand side of (B.4) we find
that the right-‐‑hand side is not the same as the First Law of Thermodynamics (B.19) which we
derive below (this comparison involves using the correct expression (B.30)) for the molecular
flux FQ ). The two versions of the First Law of Thermodynamics are different by
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
133
⎡ B′ µ S
⎤
(B.5)
+ ∇ ⋅ ⎢ S A FS ⎥ .
⎢⎣ ρ k ⎡⎣T0 + t ⎤⎦ ⎥⎦
This inconsistency means that the rather poorly defined “rate of heating” δ q dt must be
different in the saline case than in the pure water situation by this amount. We know of no
way of justifying this difference, so we conclude that any attempt to derive the First Law of
Thermodynamics via this route involving the loosely defined “rate of heating” δ q dt is
doomed to failure. This is not to say that Eqn. (B.4) is incorrect. Rather, the point is that it is
not useful, since δ q dt cannot be deduced directly by physical reasoning (for example, how
would one guess how the Dufour effect contributes to δ q dt ?)
Since there appears to be no way of deriving the First Law of Thermodynamics that
involves the “heating” term δ q dt , we follow Landau and Lifshitz (1959) and derive the First
Law via the following circuitous route. Rather than attempting to guess the form of the
molecular forcing terms in this equation directly, we first construct a conservation equation
for the total energy, being the sum of the kinetic, gravitational potential and internal energies.
It is in this equation that we insert the molecular fluxes of heat and momentum and the
radiative and boundary fluxes of heat. We know that the evolution equation for total energy
must have the conservative form, and so we insist that the forcing terms in this equation
appear as the divergence of fluxes.
Having formed the conservation equation for total energy, the known evolution equations
for two of the types of energy, namely the kinetic and gravitational potential energies, are
subtracted, leaving a prognostic equation for the internal energy, that is, the First Law of
Thermodynamics.
We start by developing the evolution equations for gravitational potential energy and for
kinetic energy (via the momentum equation). The sum of these two evolution equations is
noted. We then step back a little and consider the simplified situation where there are no
molecular fluxes of heat and salt and no effects of viscosity and no radiative or boundary heat
fluxes. In this “adiabatic” limit we are able to develop the conservation equation for total
energy, being the sum of internal energy, kinetic energy and gravitational potential energy.
To this equation we introduce the molecular, radiative and boundary flux divergences.
Finally the First Law of Thermodynamics is found by subtracting from this total energy
equation the conservation statement for the sum of the kinetic and gravitational potential
energies.
(
FS ⋅∇ µ − ⎡⎣T0 + t ⎤⎦ µT
)
The fundamental thermodynamic relation
Recall the fundamental thermodynamic relation (A.7.1) repeated here in the form (A.7.2) in
terms of material derivatives following the instantaneous motion of a fluid parcel
d dt = ∂ ∂t
+ u ⋅ ∇ ,
x, y, z
dh 1 dP
du
dv
dη
dS
−
=
+ ( p + P0 )
= (T0 + t )
+ µ A .
dt ρ dt
dt
dt
dt
dt
(B.6)
The use of the same symbol t for time and for in situ temperature in °C is noted but should
not cause confusion. The middle expression in (B.6) uses the fact that specific enthalpy h and
specific internal energy u are related by h = u + Pv = u + ( p + P0 ) v where v is the specific
volume.
IOC Manuals and Guides No. 56
134
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Gravitational potential energy
If the gravitational acceleration is taken to be constant the gravitational potential energy per
unit mass with respect to the height z = 0 is simply gz . Allowing the gravitational
acceleration to be a function of height means that the gravitational potential energy per unit
mass Φ with respect to some fixed height z0 is defined by
z
Φ=
∫ g ( z′) dz′.
(B.7)
z0
At a fixed location in space Φ is independent of time while its spatial gradient is given by
∇Φ = g k where k is the unit vector pointing upwards in the vertical direction. The evolution
equation for Φ is then readily constructed as
dΦ
(B.8)
= ρ gw ,
dt
where w is the vertical component of the three-‐‑dimensional velocity, that is w = u ⋅ k .
(Clearly in this section g is the gravitational acceleration, not the Gibbs function). Note that
this local balance equation for gravitational potential energy is not in the form (A.8.1) required
of a conservative variable since the right-‐‑hand side of (B.8) is not minus the divergence of a
flux.
( ρΦ )t + ∇ ⋅ ( ρΦu)
= ρ
Momentum evolution equation
The momentum evolution equation is derived in many textbooks including Landau and
Lifshitz (1959), Batchelor (1970), Gill (1982) and Griffies (2004). The molecular viscosity
appears in the exact momentum evolution equation in the rather complicated expressions
appearing in equations (3.3.11) and (3.3.12) of Batchelor (1970). We ignore the term that
depends on the product of the kinematic viscosity v visc and the velocity divergence ∇ ⋅ u
(following Gill (1982)), so arriving at
ρ
(
)
du
,
+ f k × ρ u = − ∇P − ρ gk + ∇ ⋅ ρ v visc ∇u
dt
(B.9)
is twice the
where f is the Coriolis frequency, v visc is the kinematic viscosity and ∇u
symmetrized velocity shear, ∇u = ∂ui ∂x j + ∂u j ∂xi . Under the same assumption as above
of ignoring the velocity divergence, the pressure P that enters (B.9) can be shown to be
equivalent to the equilibrium pressure that is rightly the pressure argument of the equation of
state (Batchelor (1970)). The centripetal acceleration associated with the coordinate system
being on a rotating planet can be taken into account by an addition to the gravitational
acceleration in (B.9) (Griffies (2004)).
(
)
Kinetic energy evolution equation
The kinetic energy evolution equation is found by taking the scalar product of Eqn. (B.9) with
u giving
( ρ 12 u ⋅ u )t
+ ∇ ⋅ ( ρ u 12 [u ⋅ u ])
= ρ d ( 12 u ⋅ u ) dt
(
)
= − u ⋅ ∇P − ρ gw + ∇ ⋅ ρ v visc∇ 12 [u ⋅ u ] − ρε ,
(B.10)
where the dissipation of kinetic energy ε is the positive definite quantity
ε ≡
IOC Manuals and Guides No. 56
1
2
(
)
⋅∇u
.
v visc ∇u
(B.11)
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
135
Evolution equation for the sum of kinetic and gravitational potential energies
The evolution equation for total mechanical energy 12 u ⋅ u + Φ is found by adding Eqns. (B8)
and (B10) giving
( ρ ⎡⎣
1
2
u ⋅ u + Φ ⎤⎦
)
t
(
+ ∇ ⋅ ρ u ⎡⎣ 12 u ⋅ u + Φ ⎤⎦
)
(
)
= ρ d ( 12 u ⋅ u + Φ ) dt = − u ⋅ ∇P + ∇ ⋅ ρ v visc∇ 12 [u ⋅ u] − ρε .
(B.12)
Notice that the term ρ gw which has the role of exchanging energy between the kinetic and
gravitational potential forms has cancelled when these two evolution equations were added.
Conservation equation for total energy E in the absence of molecular fluxes
In the absence of molecular or other irreversible processes (such as radiation of heat), and in the
absence of the non-‐‑conservative source term for Absolute Salinity that is associated with
remineralization, both the specific entropy η and the Absolute salinity S A of each fluid parcel is
constant following the fluid motion so that the right-‐‑hand side of (B.6) is zero and the material
derivative of internal energy satisfies du dt = − ( p + P0 ) dv dt so that the internal energy changes
only as a result of the work done in compressing the fluid parcel. Realizing that v = ρ −1 and
using the continuity Eqn. (A.8.1) in the form dρ dt + ρ∇⋅ u = 0, du dt can be expressed in this
situation of no molecular, radiative or boundary fluxes as du dt = − ρ −1 ( p + P0 ) ∇ ⋅ u . Adding
this equation to the inviscid, non-‐‑dissipative version of (B.12) gives
( ρE )t
+ ∇ ⋅ ( ρ uE
)
= − ∇ ⋅ ([ p + P0 ] u ) ,
= ρ dE dt
no molecular fluxes (B.13)
where the total energy
E = u + 12 u ⋅ u + Φ
(B.14)
is defined as the sum of the internal, kinetic and gravitational potential energies.
Conservation equation for total energy in the presence of molecular fluxes and
remineralization
Now, following section 49 Landau and Lifshitz (1959) we need to consider how molecular
fluxes of heat and salt and the radiation of heat will alter the simplified conservation equation
of total energy (B.13). The molecular viscosity gives rise to a stress in the fluid represented by
the tensor σ , and the interior flux of energy due to this stress tensor is u ⋅ σ so that there
needs to be the additional term −∇ ⋅ ( u ⋅ σ ) added to the right-‐‑hand side of the total energy
conservation equation. Consistent with Eqn. (B.9) above we take the stress tensor to be
so that the extra term is ∇ ⋅ ρ v visc∇ 1 u ⋅ u . Also heat fluxes at the ocean
σ = − ρ v visc ∇u
]
2[
R
boundaries and by radiation F and molecular diffusion FQ necessitate the additional terms
−∇ ⋅ FR − ∇ ⋅ FQ . At this stage we have not specified the form of the molecular diffusive flux of
heat FQ in terms of gradients of temperature and Absolute Salinity; this is done below in Eqn.
(B.24). The non-‐‑conservative production of Absolute Salinity by the remineralization of
S
sinking particulate matter, ρ S A , introduces a source of energy because the specific internal
energy and the specific enthalpy of seasalt are not the same as for pure water. The total
energy conservation equation in the presence of molecular, radiative and boundary fluxes, as
well as the interior source of salinity is
(
( ρE )t + ∇ ⋅ ( ρuE ) = ρ dE
)
(
)
dt = − ∇ ⋅ ⎡⎣ p + P0 ⎤⎦ u − ∇ ⋅F R − ∇ ⋅FQ
(
)
S
+ ∇ ⋅ ρ v visc∇ 12 ⎡⎣ u ⋅ u ⎤⎦ + hS ρ S A .
(B.15)
A
IOC Manuals and Guides No. 56
136
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
where hSA = µ − (T0 + t ) µT (see Eqn. (A.11.1)) is the partial derivative of specific enthalpy with
respect to Absolute Salinity at fixed temperature and pressure. This last term in Eqn. (B.15) is
more readily justified in Eqn. (B.17) below, which is a rearranged form of Eqn. (B.15).
S
If it were not for the remineralization source term, hS ρ S A , the right-‐‑hand side of the E
A
conservation equation (B.15) would be the divergence of a flux, ensuring that total energy E
would be both a “conservative” variable and an “isobaric conservative” variable (see
appendix A.8 for the definition of these characteristics).
Two alternative forms of the conservation equation for total energy
Another way of expressing the total energy equation (B.15) is to write it in a quasi-‐‑divergence
form, with the temporal derivative being of ρE = ρ ( u + 12 u ⋅ u + Φ ) while the divergence part
of the left-‐‑hand side is based on a different quantity, namely the Bernoulli
function B = h + 12 u ⋅ u +Φ . This form of the total energy equation is
( ρE )t
(
)
(
)
S
+ ∇ ⋅ ρ uB = − ∇ ⋅F R − ∇ ⋅FQ + ∇ ⋅ ρ v visc∇ 12 ⎡⎣ u ⋅ u ⎤⎦ + hS ρ S A .
(B.16)
A
In an ocean modelling context, it is rather strange to contemplate the energy variable that is
advected through the face of a model grid, B , to be different to the energy variable that is
changed in the grid cell, E . Hence this form of the total energy equation has not proved
popular.
A third way of expressing the total energy equation (B.15) is to write the left-‐‑hand side in
terms of only the Bernoulli function B = h + 12 u ⋅ u +Φ so that the prognostic equation for the
Bernoulli function is
( ρB )t + ∇ ⋅ ( ρuB ) = ρ dB
(
)
S
dt = Pt − ∇ ⋅F R − ∇ ⋅FQ + ∇ ⋅ ρ v visc∇ 12 ⎡⎣ u ⋅ u ⎤⎦ + hS ρ S A . (B.17)
A
The source term ρ S of Absolute Salinity caused by the remineralization of particulate
matter affects enthalpy at the rate hSA = µ − (T0 + t ) µT and can be thought of as replacing
some seasalt in place of water molecules, occurring at fixed pressure and temperature, as
might occur through two syringes in the interior of a seawater parcel, one supplying pure salt
and the other extracting pure water, at the same temperature and pressure. The influence of
the salinity increment caused by this source term on enthalpy (and therefore on the Bernoulli
function B ) is similar to the way an increment of Absolute Salinity enters Eqn. (B.3). When
the flow is steady, and in particular, when the pressure field is time invariant at every point in
space, this Bernoulli form of the total energy equation has the desirable property that B is
conserved following the fluid motion in the absence of radiative, boundary and molecular
fluxes and in the absence of non-‐‑conservative salinity production. Subject to this steady-‐‑state
S
assumption, and in the absence of ρ S A the Bernoulli function B possesses the “potential”
property. The negative aspect of this B evolution equation (B.17) is that in the more general
situation where the flow is unsteady, the presence of the Pt term means that the Bernoulli
function does not behave as a conservative variable because the right-‐‑hand side of (B.17) is
not the divergence of a flux. In this general non-‐‑steady situation B is “isobaric conservative”
but is not a “conservative” variable nor does it posses the “potential” property.
Noting that the total energy E is related to the Bernoulli function by E = B − ( p + P0 ) ρ
S
and even if we take the whole ocean to be in a steady state and with ρ S A = 0 , so that B has
the “potential” property, it is clear that E does not have the “potential” property in this
situation. That is, if a seawater parcel moves from say 2000 dbar to 0 dbar without exchange
of material or heat with its surroundings and with Pt = 0 everywhere, then B remains
constant while the parcel’s total energy E changes by the difference in the quantity
− ( p + P0 ) ρ between the two locations. Hence we conclude that even in a steady ocean E
does not possess the “potential” property. This means that total energy E is useless as far as
being a marker of fluid flow.
SA
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
(
137
)
When the viscous production term ∇ ⋅ ρ v visc∇ 12 ⎡⎣ u ⋅ u ⎤⎦ in the above equations is
integrated over the ocean volume, the contribution from the sea surface is the power input by
the wind stress τ , namely the area integral of τ ⋅ usurf where usurf is the surface velocity of
the ocean.
Obtaining the First Law of Thermodynamics by subtraction
The evolution equation (B.12) for the sum of kinetic and gravitational potential energies is
now subtracted from the total energy conservation equation (B.15) giving
( ρu )t + ∇ ⋅ ( ρuu ) = ρ du dt = − ( p + P0 ) ∇ ⋅ u
− ∇ ⋅F R − ∇ ⋅FQ + ρε + hS ρ S A .
S
A
(B.18)
Using the continuity equation in the form ρ dv dt = ∇⋅ u and the fundamental
thermodynamic relation (A.7.2), this equation can be written as
⎛
dS ⎞
⎛ dh 1 dP ⎞
⎛ du
dv ⎞
dη
= ρ⎜
ρ⎜
−
+ ( p + P0 ) ⎟ = ρ ⎜ (T0 + t )
+ µ A⎟
⎟
dt
dt ⎠
dt ⎠
⎝ dt
⎝ dt ρ dt ⎠
⎝
(B.19)
= − ∇ ⋅F − ∇ ⋅F + ρε + hS ρ S ,
R
SA
Q
A
which is the First Law of Thermodynamics. The corresponding evolution equation for
Absolute Salinity is (Eqn. (A.21.8))
ρ
(
)
(
)
dSA
S
= ρ SA + ∇ ⋅ ρ u SA = − ∇ ⋅FS + ρ S A ,
t
dt
(B.20)
where FS is the molecular flux of salt and ρ S A is the non-‐‑conservative source of Absolute
Salinity due to the remineralization of particulate matter. For many purposes in
oceanography the exact dependence of the molecular fluxes of heat and salt on the gradients
of Absolute Salinity, temperature and pressure is unimportant, nevertheless, Eqns. (B.23) and
(B.24) below list these molecular fluxes in terms of the spatial gradients of these quantities.
At first sight Eqn. (B.19) has little to recommend it; there are two non-‐‑conservative source
S
terms ρε and hS ρ S A on the right-‐‑hand side and the left-‐‑hand side is not ρ times the
A
material derivative of any quantity as is required of a conservation equation of a conservative
variable. Equation (B.19) corresponds to equation (57.6) of Landau and Lifshitz (1959) and is
repeated at Eqns. (A.13.1) and (A.13.3) above.
The approach used here to develop the First Law of Thermodynamics seems rather
convoluted in that the conservation equation for total energy is first formed, and then the
evolution equations for kinetic and gravitational potential energies are subtracted. Moreover,
the molecular, radiative and boundary fluxes were included into the total energy
conservation equation as separate deliberate flux divergences, rather than coming from an
underlying basic conservation equation. This is the approach of Landau and Lifshitz (1959)
and it is adopted for the following reasons. First this approach ensures that the molecular,
radiative and boundary fluxes do enter the total energy conservation equation (B.15) as the
divergence of fluxes so that the total energy is guaranteed to be a conservative variable (apart
from the salinity source term). This is essential; total energy can only be allowed to
spontaneously appear or disappear when there is a bona fide interior source term such as
S
hS ρ S A . Second, it is rather unclear how one would otherwise arrive at the molecular fluxes
A
of heat and salt on the right-‐‑hand side of the First Law of Thermodynamics since the direct
approach which was attempted at the beginning of this appendix involved the poorly defined
“rate of heating” δ q dt and did not lead us to the First Law. For completeness, the molecular
fluxes FQ and FS are now written in terms of the gradients of Absolute Salinity, temperature
and pressure.
S
IOC Manuals and Guides No. 56
138
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
The molecular fluxes of heat and salt
The molecular fluxes of salt and heat, FS and FQ , are now written in the general matrix
form in terms of the thermodynamic Onsager “forces” ∇ − µ T and ∇ 1 T as (see de Groot
and Mazur (1984))
(
)
( )
(
)
( )
B∇ ( − µ T ) + C ∇ (1 T ) ,
FS = A∇ − µ T + B∇ 1 T ,
(B.21)
FQ =
(B.22)
where A, B and C are three independent coefficients. The equality of the off-‐‑diagonal
diffusion coefficients, B , results from the Onsager (1931a,b) reciprocity relation. When these
fluxes are substituted into the First Law of Thermodynamics Eqn. (B.19) and this is written as
an evolution equation for entropy, we find
⎛ −µ ⎞
1
dη
∇ ⋅FS ,
ρ
= ρη + ∇ ⋅ ρ uη = − ∇ ⋅FQ − ⎜
(B.23)
⎟
t
dt
T
T
⎠
⎝
( )
(
)
where we have ignored the radiative flux divergence, the dissipation of turbulent kinetic
energy and the non-‐‑conservative production of Absolute Salinity due to biogeochemistry.
The right-‐‑hand side of this equation is now massaged into the divergence of a flux plus a
remainder term
ρ
dη
dt
( )t
= ρη
(
+ ∇ ⋅ ρ uη
)
⎛1
µ ⎞
= − ∇ ⋅ ⎜ F Q − FS ⎟
T ⎠
⎝T
⎛ 1⎞
⎛ −µ ⎞
+ F ⋅∇ ⎜ ⎟ + FS ⋅∇ ⎜
.
⎝T⎠
⎝ T ⎟⎠
(B.24)
Q
The second line of this equation contains the non-‐‑conservative source terms. If these terms
were not present then specific entropy would be a conservative thermodynamic variable.
Now we will investigate what is required for this second line of Eqn. (B.24) to be always
positive; that is, what requirement does this positivity constraint place on A, B and C of
Eqns. (B.21) and (B.22)? Substituting Eqns. (B.21) and (B.22) into the second line of Eqn. (B.24)
we have
⎛ 1⎞
⎛ −µ ⎞
⎛ 1⎞ ⎛ 1⎞
⎛ 1 ⎞ ⎛ −µ ⎞
⎛ −µ ⎞ ⎛ −µ ⎞
= C∇ ⎜ ⎟ ⋅∇ ⎜ ⎟ + 2B∇ ⎜ ⎟ ⋅∇ ⎜
+ A∇ ⎜
⋅∇
FQ ⋅∇ ⎜ ⎟ + FS ⋅∇ ⎜
. (B.25)
⎟
⎟
⎝T⎠
⎝ T ⎠
⎝T⎠ ⎝T⎠
⎝T⎠ ⎝ T ⎠
⎝ T ⎟⎠ ⎜⎝ T ⎟⎠
For the right-‐‑hand side of this equation to always be positive requires A > 0 , C > 0 and a
condition that B not be too large. In terms of the directions of the spatial gradients ∇ 1 T
and ∇ − µ T , the B term is largest in magnitude when these gradients are parallel or
antiparallel. Hence the right-‐‑hand side of this equation may be considered a simple quadratic,
and the requirement that we seek is that there are no real solutions of the right-‐‑hand side
being zero, requiring that the discriminant of the quadratic be negative. That is 4B 2 − 4 AC
must be negative. So the three constraints are A > 0 , C > 0 and AC > B 2 , which can be
reduced to simply two constraints such as A > 0 and C > B 2 A .
The part of the salt flux of Eqn. (B.21) that is proportional to −∇SA is traditionally written
as − ρ k S ∇SA implying that A = ρ k S T µ S . The molecular fluxes of salt and heat, FS and FQ ,
A
can now be written in terms of the gradients of Absolute Salinity, temperature and pressure as
(
( )
)
⎛
⎞
⎛ ρk ST ⎛ µ ⎞
µ
B⎞
+
FS = − ρ k S ⎜ ∇SA + P ∇P⎟ − ⎜
⎟ ∇T ,
⎜⎝
⎟⎠
⎜⎝ µ S ⎜⎝ T ⎟⎠ T T 2 ⎟⎠
µS
A
A
FQ = −
Bµ S
Bµ S
1 ⎛
B2 ⎞
S
T
A
A
∇T
+
C
−
F
=
−
ρ
c
k
∇T
+
FS ,
⎜
⎟
p
A⎠
T2 ⎝
ρk ST
ρk ST
(B.26)
(B.27)
where the fact that C > B 2 A has been used to write the regular diffusion of heat down the
temperature gradient as − ρ c p k T ∇T where k T is the positive molecular diffusivity of
temperature. These expressions involve the (strictly positive) molecular diffusivities of
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
139
temperature and salinity ( k T and k S ) and the single cross-‐‑diffusion parameter B . The other
parameters in these equations follow directly from the Gibbs function of seawater.
Sometimes a “reduced heat flux” is introduced by reducing the molecular flux of heat by
FS = µ − T µT FS , being the flux of enthalpy due to the molecular flux of salt. This
∂h ∂SA
T,p
prompts the introduction of a revised cross-‐‑diffusion coefficient defined by
(
)
B′ ≡ B +
ρk ST 3 ⎛ µ ⎞
,
µ S ⎜⎝ T ⎟⎠ T
(B.28)
A
and in terms of this cross-‐‑diffusion coefficient Eqns. (B.26) and (B.27) can be written as
⎛
⎞
µ
B′
FS = − ρ k S ⎜ ∇SA + P ∇P⎟ − 2 ∇T ,
⎜⎝
⎟⎠
µS
T
A
(B.29)
and
(
)
FQ − µ − T µT FS = − ρ c p k T ∇T +
=
B′ µ S
A
ρk ST
FS
⎞
B′ µ S ⎛
µ
A
− ρ c p K ∇T −
⎜ ∇SA + P ∇P⎟ ,
⎟⎠
T ⎜⎝
µS
A
(B.30)
T
( )
where K T is a revised molecular diffusivity of temperature, ρ c p K T = ρ c p k T + B′ 2 AT 2 .
The term in (B.29) that is proportional to the pressure gradient ∇P represents
“barodiffusion” as it causes a flux of salt down the gradient of pressure. In an undisturbed
ocean that is in vertical diffusive equilibrium, the barodiffusion term would cause Absolute
Salinity to increase with depth in the ocean at the rate of ~ 3g kg −1 per 1000m. The turbulent
nature of the ocean means that this molecular diffusive balance does not occur. The last term
in (B.29) is a flux of salt due to the gradient of in situ temperature and is called the Soret effect,
while the last term in the second line of Eqn. (B.30) is called the Dufour effect.
The molecular flux of salt is independent of the four arbitrary constants (Fofonoff (1962))
that appear in the Gibbs function of seawater (Eqn. (2.6.2)). This implies that the cross-‐‑
diffusion coefficient B in Eqns. (B.21)–(B.22) is arbitrary to the extent a3ρ k S T µ S (since µ is
A
arbitrary to the extent a3 + a4 (T0 + t ) ). From Eqn. (B.27) we find that the molecular flux of
“heat” FQ is unknowable to the extent a3FS . This means that the − ∇ ⋅ FQ term on the right
of the First Law Eqn. (B.19) is unknowable to the extent − a3∇ ⋅ FS . The left-‐‑hand side of Eqn.
(B.19) is unknowable to the extent a3 ρ dSA dt (since specific enthalpy h contains the
arbitrary component a1 + a3SA ). The last term in Eqn. (B.19) contains the arbitrary term
S
a3ρ S A (since hS is arbitrary by the amount a3 ). These three arbitrary, unknowable
A
contributions to the First Law of Thermodynamics Eqn. (B.19) sum to a3 times the evolution
equation (B.20) for Absolute Salinity. This allows these arbitrary terms to be subtracted from
Eqn. (B.19), confirming that the four arbitrary unknowable constants of Eqn. (2.6.2) have no
measureable consequences on the First Law of Thermodynamics. The cross-‐‑diffusion
coefficient B′ of Eqns. (B.28) – (B.30) does not contain any arbitrary constants.
Regarding Eqns. (B.21)–(B.30), it is noted that strictly speaking the gradient of the
chemical potential µ must be replaced by the gradients of the chemical potentials of the
individual constituents of sea salt, and the diffusion coefficients in front of these many
gradients are different for each constituent, since there is no uniform molecular diffusion of
the mixture "ʺsea salt"ʺ. When additional processes act to keep the composition approximately
fixed, the use of only one chemical potential for sea salt is permitted in non-‐‑equilibrium
situations. These processes are mainly ion relaxation by Coulomb forces, which in the form of
ambipolar diffusion prevent any local electrical charge separation, and secondly, turbulent
mixing which has the same transport coefficient for each species and whose fluxes are
proportional to the concentration gradients of “potential” quantities (see appendix A.9) rather
than to the gradients of the individual chemical potentials.
IOC Manuals and Guides No. 56
140
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Appendix C:
Publications describing the TEOS-‐‑10 thermodynamic
descriptions of seawater, ice and moist air
Primary standard documents
Harvey, A. H. and P. H. Huang, 2007: First-‐‑Principles Calculation of the Air–Water Second Virial
Coefficient. Int. J. Thermophys., 28, 556–565.
Hyland, R. W. and A. Wexler, 1983: Formulations for the thermodynamic properties of dry air
from 173.15 to 473.15 K, and of saturated moist air from 173.15 to 372.15 K, at pressures up to
5Mpa. ASHRAE Transact. 89, 520–535.
IAPWS, 2008a: Release on the IAPWS Formulation 2008 for the Thermodynamic Properties of
Seawater. The International Association for the Properties of Water and Steam. Berlin,
Germany, September 2008, available from http://www.iapws.org. This Release is referred to in
the text as IAPWS-‐‑08.
IAPWS, 2009a: Revised Release on the Equation of State 2006 for H2O Ice Ih. The International
Association for the Properties of Water and Steam. Doorwerth, The Netherlands, September
2009, available from http://www.iapws.org. This revised Release is referred to in the text as
IAPWS-‐‑06.
IAPWS, 2009b: Revised Release on the IAPWS Formulation 1995 for the Thermodynamic
Properties of Ordinary Water Substance for General and Scientific Use. The International
Association for the Properties of Water and Steam. Doorwerth, The Netherlands, September
2009, available from http://www.iapws.org. This revised Release is referred to in the text as
IAPWS-‐‑95.
IAPWS, 2009c: Supplementary Release on a Computationally Efficient Thermodynamic
Formulation for Liquid Water for Oceanographic Use. The International Association for the
Properties of Water and Steam. Doorwerth, The Netherlands, September 2009, available from
http://www.iapws.org. This Release is referred to as IAPWS-‐‑09.
IAPWS, 2010: Guideline on an Equation of State for Humid Air in Contact with Seawater and Ice,
Consistent with the IAPWS Formulation 2008 for the Thermodynamic Properties of Seawater.
The International Association for the Properties of Water and Steam. Niagara Falls, Canada,
July 2010, available from http://www.iapws.org. This Guideline is referred to in the text as
IAPWS-‐‑10.
Lemmon, E. W., R. T. Jacobsen, S. G. Penoncello and D. G. Friend, 2000: Thermodynamic
properties of air and mixtures of nitrogen, argon and oxygen from 60 to 2000 K at pressures to
2000 MPa. J. Phys. Chem. Ref. Data, 29, 331–362.
Millero, F. J., R. Feistel, D. G. Wright, and T. J. McDougall, 2008a: The composition of Standard
Seawater and the definition of the Reference-‐‑Composition Salinity Scale, Deep-‐‑Sea Res. I, 55,
50-‐‑72.
Secondary standard documents
IOC, SCOR and IAPSO, 2010a: The international thermodynamic equation of seawater – 2010:
Calculation and use of thermodynamic properties. Intergovernmental Oceanographic
Commission, Manuals and Guides No. 56, UNESCO (English), 196 pp, Paris. Available from
www.TEOS-10.org [the present document, called the TEOS-‐‑10 manual]
McDougall, T. J., D. R. Jackett, F. J. Millero, R. Pawlowicz and P. M. Barker, 2012: A global
algorithm for estimating Absolute Salinity. Ocean Science, 8, 1123-‐‑1134. http://www.oceansci.net/8/1123/2012/os-8-1123-2012.pdf The computer software is available from www.TEOS10.org
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Background papers to the declared standards
141
Feistel, R., 2003: A new extended Gibbs thermodynamic potential of seawater. Progr. Oceanogr., 58,
43-‐‑114.
Feistel, R., 2008: A Gibbs function for seawater thermodynamics for −6 to 80 °C and salinity up to
120 g kg–1. Deep-‐‑Sea Res. I, 55, 1639-‐‑1671.
Feistel, R. and W. Wagner, 2006: A New Equation of State for H2O Ice Ih. J. Phys. Chem. Ref. Data,
35, 2, 1021-‐‑1047.
Feistel, R., S. Weinreben, H. Wolf, S. Seitz, P. Spitzer, B. Adel, G. Nausch, B. Schneider and D. G.
Wright, 2010c: Density and Absolute Salinity of the Baltic Sea 2006–2009. Ocean Science, 6, 3–
24. http://www.ocean-sci.net/6/3/2010/os-6-3-2010.pdf
Feistel, R., D. G. Wright, H.-‐‑J. Kretzschmar, E. Hagen, S. Herrmann and R. Span, 2010a:
Thermodynamic properties of sea air. Ocean Science, 6, 91–141. http://www.oceansci.net/6/91/2010/os-6-91-2010.pdf
Feistel, R., D. G. Wright, K. Miyagawa, A. H. Harvey, J. Hruby, D. R. Jackett, T. J. McDougall and
W. Wagner, 2008: Mutually consistent thermodynamic potentials for fluid water, ice and
seawater: a new standard for oceanography. Ocean Science, 4, 275-‐‑291. http://www.oceansci.net/4/275/2008/os-4-275-2008.pdf
IOC, SCOR and IAPSO, 2010b: The international thermodynamic equation of seawater – 2010: A
Summary for Policy Makers. Intergovernmental Oceanographic Commission (Brochures
Series). Available from www.TEOS-10.org
McDougall, T. J., 2003: Potential enthalpy: A conservative oceanic variable for evaluating heat
content and heat fluxes. Journal of Physical Oceanography, 33, 945-‐‑963.
Marion, G. M., F. J. Millero, and R. Feistel, 2009: Precipitation of solid phase calcium carbonates
and their effect on application of seawater SA − T − P models, Ocean Sci., 5, 285-‐‑291.
http://www.ocean-sci.net/5/285/2009/os-5-285-2009.pdf
Millero, F. J., 2000. Effect of changes in the composition of seawater on the density-‐‑salinity
relationship. Deep-‐‑Sea Res. I 47, 1583-‐‑1590.
Pawlowicz, R., 2010a: A model for predicting changes in the electrical conductivity, Practical
Salinity, and Absolute Salinity of seawater due to variations in relative chemical composition.
Ocean Science, 6, 361–378. http://www.ocean-sci.net/6/361/2010/os-6-361-2010.pdf
Pawlowicz, R., T. McDougall, R. Feistel and R. Tailleux, 2012: An historical perspective on the
development of the Thermodynamic Equation of Seawater – 2010: Ocean Sci., 8, 161-‐‑174.
http://www.ocean-sci.net/8/161/2012/os-8-161-2012.pdf
Pawlowicz, R., D. G. Wright and F. J. Millero, 2011: The effects of biogeochemical processes on
oceanic conductivity/salinity/density relationships and the characterization of real seawater.
Ocean Science, 7, 363–387. Available from http://www.ocean-sci.net/7/363/2011/os-7-363-2011.pdf
Seitz, S., R. Feistel, D.G. Wright, S. Weinreben, P. Spitzer and P. de Bievre, 2011: Metrological
Traceability of Oceanographic Salinity Measurement Results. Ocean Science, 7, 45–62.
http://www.ocean-sci.net/7/45/2011/os-7-45-2011.pdf
Wagner, W. and Pruß, A., 2002: The IAPWS formulation 1995 for the thermodynamic properties of
ordinary water substance for general and scientific use. J. Phys. Chem. Ref. Data, 31, 387-‐‑535.
Wright, D. G., R. Pawlowicz, T. J. McDougall, R. Feistel and G. M. Marion, 2011: Absolute Salinity,
“Density Salinity” and the Reference-‐‑Composition Salinity Scale: present and future use in
the seawater standard TEOS-‐‑10. Ocean Sci., 7, 1-‐‑26. http://www.ocean-sci.net/7/1/2011/os-7-12011.pdf
IOC Manuals and Guides No. 56
142
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Papers describing computer software
Feistel, R., D. G. Wright, D. R. Jackett, K. Miyagawa, J. H. Reissmann, W. Wagner, U. Overhoff, C.
Guder, A. Feistel and G. M. Marion, 2010b: Numerical implementation and oceanographic
application of the thermodynamic potentials of liquid water, water vapour, ice, seawater and
humid air -‐‑ Part 1: Background and equations. Ocean Science, 6, 633-‐‑677. http://www.oceansci.net/6/633/2010/os-6-633-2010.pdf
and http://www.ocean-sci.net/6/633/2010/os-6-633-2010supplement.pdf
McDougall T. J. and P. M. Barker, 2011: Getting started with TEOS-‐‑10 and the Gibbs Seawater
(GSW) Oceanographic Toolbox, 28pp., SCOR/IAPSO WG127, ISBN 978-‐‑0-‐‑646-‐‑55621-‐‑5,
available from www.TEOS-10.org
Roquet, F., G. Madec, T. J. McDougall and P. M. Barker, 2015: Accurate polynomial expressions for
the density and specific volume of seawater using the TEOS-‐‑10 standard. Ocean Modelling, 90,
29-‐‑43, http://dx.doi.org/10.1016/j.ocemod.2015.04.002
McDougall, T. J., D. R. Jackett, F. J. Millero, R. Pawlowicz and P. M. Barker, 2012: A global
algorithm for estimating Absolute Salinity. Ocean Science, 8, 1123-‐‑1134. http://www.oceansci.net/8/1123/2012/os-8-1123-2012.pdf The computer software is available from www.TEOS10.org
Wright, D. G., R. Feistel, J. H. Reissmann, K. Miyagawa, D. R. Jackett, W. Wagner, U. Overhoff, C.
Guder, A. Feistel and G. M. Marion, 2010: Numerical implementation and oceanographic
application of the thermodynamic potentials of liquid water, water vapour, ice, seawater and
humid air -‐‑ Part 2: The library routines. Ocean Science, 6, 695-‐‑718. http://www.oceansci.net/6/695/2010/os-6-695-2010.pdf
and http://www.ocean-sci.net/6/695/2010/os-6-695-2010supplement.pdf
TEOS-‐‑10 web site
SCOR/IAPSO Working Group 127 has created the web site www.TEOS-10.org which serves
many of the TEOS-‐‑10 papers, this TEOS-‐‑10 manual as well as the SIA (Seawater Ice Air) and
GSW (Gibbs SeaWater) libraries of oceanographic computer software. Each function in the
GSW MATLAB Oceanographic Toolbox contains a help file which describes the derivation,
attributes and use of each function.
In addition, the www.TEOS-10.org web site has two documents entitled
• “Getting started with TEOS-‐‑10 and the Gibbs Seawater (GSW) Oceanographic
Toolbox” (McDougall and Barker, 2011) and
• “What every oceanographer needs to know about TEOS-‐‑10 (The TEOS-‐‑10 Primer)”
(Pawlowicz, 2010b).
Together these documents serve as a succinct introduction to the use of TEOS-‐‑10 in physical
oceanography.
Note that several of the papers listed in this appendix have appeared in Ocean Science in the special
issue “Thermophysical Properties of Seawater”, see http://www.ocean-sci.net/special_issue14.html
Note that when referring to the use of TEOS-‐‑10, it is the present document that should be
referenced as IOC et al. (2010), with the full citation being
IOC, SCOR and IAPSO, 2010: The international thermodynamic equation of seawater – 2010:
Calculation and use of thermodynamic properties. Intergovernmental Oceanographic
Commission, Manuals and Guides No. 56, UNESCO (English), 196 pp.
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
143
Appendix D: Fundamental constants
Following the recommendation of IAPWS (2005), the values of the fundamental constants
were taken from CODATA 2006 (Mohr et al. (2008)), as listed in Table D.1. Selected properties
of pure water were taken from IAPWS (1996, 1997, 2005, 2006) as listed in Table D.2. The
chemical Reference Composition of seawater from Millero et al. (2008a) is given in Table D.3.
Selected seawater constants derived from the Reference Composition are listed in Table D.4.
0
The exact value of the isobaric “heat capacity” c p is given in Table D.5.
Table D.1. Fundamental constants from CODATA 2006 (Mohr et al. (2008)) and ISO (1993).
Symbol
R
P0
Value
Uncertainty
Unit
8.314 472
101 325
0.000 015
exact
J mol K
T0
273.15
exact
–1
Pa
Comment
–1
molar gas constant
normal pressure
K
Celsius zero point
Table D.2. Selected properties of liquid water from IAPWS (1996, 1997, 2005, 2006)
and Feistel (2003).
Symbol
Value
Uncertainty
Unit
Comment
M W
18.015 268
0.000 002
molar mass
g mol–1
tMD
3.978 121
0.04
°C
ρ MD
999.974 95
0.000 84
kg m–3
maximum density at P0
ρ 0
999.8431
0.001
kg m–3
density at T0 and P0 , ρ 0 = 1 / v 0
(∂ρ
0
/ ∂T
)
P
6.774 876 × 10–2
0.06 × 10–2
maximum density , temperature
kg m–3 K–1 ( ∂ρ / ∂T ) P at T0 and P0
Tt
Pt
273.16
exact
K
triple point temperature
611.657
0.01
Pa
triple point pressure
ρ t
999.793
0.01
kg m–3
triple point density
ηt
0
exact
J kg–1 K–1
triple point entropy
ut
0
exact
J kg–1
Tf0
273.152 519
0.000 002
K
triple point internal energy
freezing point at P0
IOC Manuals and Guides No. 56
144
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Table D.3. The sea salt composition definition for seawater of Reference-‐‑Composition
at 25°C and 101325 Pa. Xj – mole fractions, Zj – valences, Wj – mass fractions
(Millero et al. 2008a). The molar masses Mj are from Wieser (2006) with their
uncertainties in the last one or two digits given in the brackets. The mass fractions
Mj are the mass of a particular solute as a fraction of the total mass of solute. The
mole fractions Xj in this table are extracted from Table 4 of Millero et al. (2008a)
which is the official definition of Reference-‐‑Composition seawater.
Solute j
Zj
+1
+2
+2
Na+
2+
Mg
Ca2+
K+
Mj
g mol–1
22.989 769 28(2)
24.305 0(6)
40.078(4)
Xj
10–7
Xj Zj
10–7
4188071
4188071
0.3065958
471678
91823
91159
943356
183646
91159
0.0365055
0.0117186
810
1620
4874839
–4874839
252152
–504304
15340
7520
–15340
–7520
2134
–4268
900
610
–900
–610
0.0002259
Wj
+1
+2
39.098 3(1)
87.62(1)
–1
–2
35.453(2)
96.062 6(50)
–1
61.016 84(96)
–1
–2
79.904(1)
60.008 9(10)
–1
78.840 4(70)
–1
–1
18.998 403 2(5)
–
17.007 33(7)
71
–71
0.0000038
B(OH)3
CO2
0
0
61.833 0(70)
44.009 5(9)
2807
86
0
0
0.0005527
0.0000121
10 000 000
0
1.0
Sr
2+
Cl–
SO4
2–
HCO3
Br–
CO3
–
2–
B(OH)4
F–
OH
–
Sum
IOC Manuals and Guides No. 56
0.0113495
0.0002260
0.5503396
0.0771319
0.0029805
0.0019134
0.0004078
0.0000369
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Table D.4. Selected properties of KCl-‐‑normalised Reference Seawater,
from Millero et al. (2008a).
Symbol
Value
Uncertainty
Unit
Comment
Reference Salinity molar mass
MS
31.403 8218
0.001
g mol
MS = ∑ X j M j
–1
j
Reference Salinity valence factor
Z2
exact
1.245 2898
a
Z 2 = ∑ X j Z 2j
-
j
a
Avogadro constant
NA
6.022 141 79 × 1023
3 × 1016
mol −1
NS
1.917 6461 × 1022
6 × 1017
g–1
uPS
1.004 715…
exacta
g kg–1
unit conversion factor,
uPS ≡ 35.165 04 g kg–1 / 35
SSO
35.165 04
exacta
g kg–1
Standard Ocean Reference Salinity,
35 uPS
TSO
273.15
exact
K
Standard Ocean temperature
TSO = T0
tSO
0
exact
°C
Standard Ocean temperature
tSO = TSO – T0
PSO
101 325
exact
Pa
Standard Ocean surface pressure
PSO = P0
pSO
0
exact
Pa
Standard Ocean surface sea pressure
pSO = PSO – P0
hSO
0
exact
J kg–1
Standard Ocean surface enthalpy
hSO = ut
ηSO
0
exact
J kg–1 K–1
Standard Ocean surface entropy
ηSO = ηt
Su
40.188 617…
exacta
g kg–1
unit-‐‑related scaling constant,
40 uPS
tu
40
exact
°C
unit-‐‑related scaling constant
pu
108
exact
Pa
unit-‐‑related scaling constant
gu
1
exact
J kg–1
unit-‐‑related scaling constant
Reference Salinity particle number
NS = N A / M S
by definition of Reference Salinity and Reference Composition
Table D.5. The exact definition of the isobaric “heat capacity” that relates
potential enthalpy to Conservative Temperature Θ .
Symbol
c 0p
Value
3991.867 957 119 63
Uncertainty
Unit
exact
–1
J kg K
Comment
–1
See Eqn. (3.3.3)
IOC Manuals and Guides No. 56
145
146
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Table D.6. Chemical composition of dry air with a fixed CO2 level.
Mole fractions are from Picard et al. (2008) except for N 2
which was adjusted by subtracting all other mole fractions
from 1 (Picard et al. (2008)). Uncertainties of the molar
masses (Wieser (2006)) are given in brackets.
Mole
fraction
N2
0.780 847 9
O2
0.209 390 0
Ar
0.009 332 0
CO2 0.000 400 0
Ne 0.000 018 2
He 0.000 005 2
CH4 0.000 001 5
Kr
0.000 001 1
H2
0.000 000 5
N2O 0.000 000 3
CO 0.000 000 2
Xe 0.000 000 1
Gas
Mass
fraction
0.755 184 73
0.231 318 60
0.012 870 36
0.000 607 75
0.000 012 68
0.000 000 72
0.000 000 83
0.000 003 18
0.000 000 03
0.000 000 46
0.000 000 19
0.000 000 45
Molar mass
g mol–1
28.013 4(3)
31.998 8(4)
39.948 (1)
44.009 5(9)
20.179 7(6)
4.002 602(2)
16.042 46(81)
83.798 (2)
2.015 88(10)
44.012 8(4)
28.010 1(9)
131.293 (6)
Air 1.000 000 0 0.999 999 98 28.965 46(33)
Coriolis Parameter
The rotation rate of the earth Ω is (in radians per second)
Ω = 7.292 1150 × 10−5 s −1 ,
(D.1)
(Groten (2004)) and the Coriolis parameter f is (in radians per second)
f = 2Ω sin φ = 1.458 423 00×10−4 sin φ s −1 ,
(D.2)
where φ is latitude ( φ has opposite signs in the two hemispheres).
Gravitational Acceleration
The gravitational acceleration g in the ocean can be taken to be the following function of
latitude φ and sea pressure p , or height z relative to the geoid,
(
= 9.780 327 (1 + 5.2792 × 10
≈ 9.780 327 (1 + 5.2792 × 10
)(
sin φ ) (1 − 2.26 × 10
sin φ ) (1 + 2.22 × 10
)
z (m) )
g (m s −2 ) = 9.780 327 1 + 5.3024 × 10−3 sin 2 φ − 5.8 × 10−6 sin 2 2φ 1 − 2.26 × 10−7 z (m)
−3
sin 2 φ + 2.32 × 10−5
−3
sin 2 φ + 2.32 × 10−5
4
−7
4
−7
(D.3)
)
p (dbar) .
The dependence on latitude in Eqn. (D.3) is from Moritz (2000) and is the gravitational
acceleration on the surface of an ellipsoid which approximates the geoid. The height z above
the geoid is negative in the ocean. Note that g increases with depth in the ocean at about
71.85% of the rate at which it decreases with height in the atmosphere.
At a latitude of 45° N and at p = 0 , g = 9.8062 m s−2 , which is a value commonly used in
ocean models. The value of g averaged over the earth’s surface is g = 9.7976 m s−2 , while the
value averaged over the surface of the ocean is g = 9.7963 m s−2 (Griffies (2004)).
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
147
Appendix E:
Algorithm for calculating Practical Salinity
E.1 Calculation of Practical Salinity in terms of K15
Practical Salinity S P is defined on the Practical Salinity Scale of 1978 (UNESCO (1981,
1983)) in terms of the conductivity ratio K15 which is the electrical conductivity of the
sample at temperature t68 = 15 °C and pressure equal to one standard atmosphere ( p = 0
dbar and Absolute Pressure P equal to 101 325 Pa), divided by the conductivity of a
standard potassium chloride (KCl) solution at the same temperature and pressure. The
mass fraction of KCl in the standard solution is 32.4356x10-‐‑3 (mass of KCl per mass of
solution). When K15 = 1, the Practical Salinity S P is by definition 35. Note that Practical
Salinity is a unit-‐‑less quantity. Though sometimes convenient, it is technically incorrect to
quote Practical Salinity in “psu”; rather it should be quoted as a certain Practical Salinity
“on the Practical Salinity Scale PSS-‐‑78”. When K15 is not unity, S P and K15 are related by
(UNESCO, 1981, 1983) the PSS-‐‑78 equation
SP =
5
∑ ai ( K15 )
i 2
i =0
where K15 =
C ( SP , t68 = 15°C,0)
C ( 35, t68 = 15°C,0)
,
(E.1.1)
and the coefficients ai are given in the following table. Note that the sum of the six ai
coefficients is precisely 35, while the sum of the six bi coefficients is precisely zero.
Equation (E.1.1) is valid in the range 2 < SP < 42.
i
ai
bi
ci
di
ei
0
1
0.0080
- 0.1692
0.0005
- 0.0056
6.766097 x 10-1
2.00564 x 10-2
3.426 x 10-2
2.070 x 10-5
2
25.3851
- 0.0066
1.104259 x 10-4
4.464 x 10-4
- 6.370 x10-10
3
4
14.0941
- 7.0261
- 0.0375
0.0636
- 6.9698 x 10-7
1.0031 x 10-9
4.215 x 10-1
- 3.107 x 10-3
3.989 x10-15
5
2.7081
- 0.0144
E.2 Calculation of Practical Salinity at oceanographic temperature and pressure
The following formulae from UNESCO (1983) are valid over the range −2 °C ≤ t ≤ 35 °C
and 0 ≤ p ≤ 10 000dbar. Measurements of salinity in the field generally measure the
conductivity ratio R
R=
C ( SP , t68 , p )
C ( 35, t68 = 15 °C,0)
=
C ( SP , t68 , p ) C ( SP , t68 ,0)
C (35, t68 ,0)
C ( SP , t68 ,0) C (35, t68 ,0 ) C (35, t68 = 15 °C,0 )
(E.2.1)
which has been expressed in (E.2.1) as the product of three factors, which are labeled
Rp , Rt and rt as follows
R=
C ( SP , t68 , p )
C ( 35, t68 = 15 °C,0)
= R p Rt rt .
(E.2.2)
The last factor rt has been fitted to experimental data as the following polynomial in t68
IOC Manuals and Guides No. 56
148
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
rt =
4
∑ ci (t68 / °C)
i
(E.2.3)
i =0
and the factor R p has been fitted to experimental data as a function of p, t68 and R as
3
∑ ei pi
(E.2.4)
.
2
1 + d1 ( t68 / °C ) + d 2 (t68 / °C ) + R ⎡⎣d 3 + d 4 (t68 / °C )⎤⎦
For any measurement of R it is possible to evaluate rt and R p and hence calculate
i =1
Rp = 1 +
Rt =
R
.
R p rt
(E.2.5)
At a temperature of t68 =15 °C, Rt is simply K15 and Practical Salinity S P can be
determined form (E.1.1). For temperatures other than t68 =15 °C , Practical Salinity S P is
given by the following function of Rt with k = 0.0162,
5
SP = ∑ ai ( Rt )
i2
+
i =0
5
(t68 / °C − 15)
i2
∑ bi ( Rt ) .
⎡⎣1 + k (t68 / °C − 15)⎤⎦ i = 0
(E.2.6)
Equations (E.1.1) and (E.2.6) are valid only in the range 2 < SP < 42. Outside this
range S P can be determined by dilution with pure water or evaporation of a seawater
sample. Practical Salinity S P can also be estimated using the extension of the Practical
Salinity Scale proposed by Hill et al. (1986) for 0 < SP < 2 . The GSW Oceanographic
Toolbox incorporates a modified form of the extension of Hill et al. (1986) for 0 < SP < 2 .
The modification ensures that the algorithm is exactly PSS-‐‑78 for SP ≥ 2 and is continuous
at SP = 2 . The values of Practical Salinity S P estimated in this manner may then be used
in Eqn. (2.4.1), namely SR ≈ uPS SP to estimate Reference Salinity SR .
When using a laboratory salinometer to evaluate Practical Salinity, use is made of Eqn.
(E.2.6) since the salinometer returns Rt and the instrument’s bath temperature is known
(and is easily converted from a measured temperature on the ITS-‐‑90 scale to t68 ).
The temperatures in Eqns. (E.2.1) to (E.2.6) are all on the IPTS-‐‑68 scale. The functions
and coefficients have not been refitted to ITS-‐‑90 temperatures. Therefore in order to
calculate Practical Salinity from conductivity ratio at a measured pressure and t90
temperature, it is necessary first to convert the temperature to t68 using t68 = 1.00024 t90 as
described Eqn. (A.1.3) of appendix A.1. This is done as the first line of the computer code
described in the GSW Oceanographic Toolbox (appendix N). Further remarks on the
implications of the different temperature scales on the definition and calculation of
Practical Salinity can be found in appendix E.4 below.
E.3 Calculation of conductivity ratio R for a given Practical Salinity
When Practical Salinity is known and one wants to deduce the conductivity ratio R
associated with this value of Practical Salinity at a given temperature, a Newton-‐‑Raphson
iterative inversion of Eqn. (E.2.6) is first performed to evaluate Rt . Because rt is a function
only of temperature, at this stage both Rt and rt are known so that Eqn. (E.2.4) can be
written as a quadratic in R with known coefficients which is solved to yield R . This
procedure is outlined in more detail in UNESCO (1983). Computer software to perform
this procedure is available in the GSW Oceanographic Toolbox as the functions
gsw_R_from_SP and gsw_C_from_SP which return conductivity ratio and conductivity
−1
(in mS ( cm ) ) respectively. Note that this iterative inverse procedure is done in terms of
t68 ; the code accepts t90 as the input and immediately converts this to a t68 temperature
before performing the above iterative procedure.
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
149
E.4 Evaluating Practical Salinity using ITS-90 temperatures
We first consider the consequence of the change from IPTS-‐‑68 to ITS-‐‑90 for the definition
of Practical Salinity as a function of K15 and the defining mass fraction of KCl. Suppose
Practical Salinity S P were to be evaluated using the polynomial (E.1.1) but using K15−90
instead of K15 , where K15−90 is defined
K15−90 =
C ( SP , t90 = 15 °C,0)
C ( 35, t90 = 15 °C,0)
.
(E.4.1)
The magnitude of the difference K15−90 − K15 can be calculated and is found to be less than
6.8x10-‐‑7 everywhere in the range 2 < SP < 42. Further calculation shows that
∂SP ∂K15 < 41 everywhere in the valid range of Practical Salinity, so that the consequence
of using K15−90 in (E.1.1) instead of K15 incurs a change in Practical Salinity of less than
3x10-‐‑5. This is nearly two orders of magnitude below the measurement accuracy of a
sample, and an order of magnitude smaller than the error caused by the uncertainty in the
definition of the mass fraction of KCl. If all the original measurements that form the basis
of the Practical Salinity Scale were converted to ITS-‐‑90, and the analysis repeated to
determine the appropriate mass fraction to give the required conductivity at t90 =15 °C,
the same mass fraction 32.4356x10-‐‑3 would be derived.
Not withstanding the insensitivity of this conductivity ratio to such a small
temperature difference, following Millero et al. (2008a) the definition of Practical Salinity
can be restated with reference to the ITS-‐‑90 scale by noting that the K15 ratio in Eqn.
(E.1.1) can equivalently refer to a ratio of conductivities at t90 =14.996 °C.
The fact that the conductivity ratio Rt is rather weakly dependent on the temperature
at which the ratio is determined is important for the use of bench salinometers. It is
important that samples and seawater standards be run at the same temperature, stable at
order 1 mK, which is achieved by the use of a large water bath. However, it is not critical
to know the stable bath temperature to any better than 10 or 20 mK.
The ratios Rp , Rt and rt that underlie the temperature-‐‑dependent expression (E.2.6)
for Practical Salinity are more sensitive to the difference between IPTS-‐‑68 and ITS-‐‑90
temperatures and this is the reason why we recommend retaining the original computer
algorithms for these ratios, and to simply convert the input temperature (which these days
is on the ITS-‐‑90 temperature scale) in to the corresponding IPTS-‐‑68 temperature using
t68 = 1.00024 t90 as the first operation in the software. Thereafter the software proceeds
according to (E.2.1) – (E.2.6).
E.5 Towards SI-traceability of the measurement procedure for Practical Salinity
and Absolute Salinity
The observation of climate change taking place in the world ocean on a global scale over
decades or centuries requires measurement techniques that permit the highest accuracy
currently available, long-‐‑term stability and world-‐‑wide comparability of the measured
values. The highest reliability for this purpose can be ensured only by traceability of these
measurement results to the primary standards of the International System of Units (SI),
supported by the National Metrological Institutes such as the NIST (National Institute of
Standards and Technology) in the US, the NPL (National Physical Laboratory) in the UK,
or the PTB (Physikalisch-‐‑Technische Bundesanstalt) in Germany.
In order to compute the thermodynamic properties of a seawater sample with
standard composition, three independent parameters must be measured. Since the
introduction of the Practical Salinity Scale of 1978 as an international standard for
oceanography, these three properties have been electrolytic conductivity, temperature and
pressure, from which salinity, density and other properties are computed in turn by
IOC Manuals and Guides No. 56
150
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
standard algorithms. The traceability of temperature and pressure measurement results,
for example by CTD sensors, is ensured due to established calibration procedures carried
out by the manufacturer or other laboratories and will not be considered here any further.
The observation of the ocean’s salinity is a more complicated task (Millero et al.
(2008a)). Even though over the last century different and permanently improved methods
were developed and introduced in oceanography, traceability of salinity measurement
results to SI units has not yet been achieved (Seitz et al. 2008 and Seitz et al. (2011)). This
implies the risk that readings taken today may possess an enlarged uncertainty when
being compared with observations taken a hundred years from now, a circumstance that
will reduce the accuracy of long-‐‑term trend analyses performed in the future.
A quantity, quite generally, is a “property of a phenomenon, body or substance, where
the property has a magnitude that can be expressed as a number” (ISO/IEC, 2007). The
process to obtain this number is called measurement. The value of the indicated number
(the quantity value) is determined by a calibration of the measuring system with a
reference having a known quantity value of the same kind. In turn, the quantity value of
the reference is assigned in a superior measurement procedure, which is likewise
calibrated with a reference and so on. This calibration hierarchy ends in a primary
reference procedure used to assign a quantity value and a unit to a primary standard for
that kind of quantity. Thus, the unit of a measured quantity value expresses its link (its
metrological traceability) to the quantity value of the corresponding primary standard.
Obviously, quantity values measured at different times or locations, by different persons
with different devices or methods can be compared with each other only if they are linked
to the same reference standard, whose corresponding quantity value must be reproducible
with a high degree of reliability.
Concerning comparability of measured quantity values a second aspect is of
importance. The quantity value of a primary standard can only be realised with an
inevitable uncertainty. The same holds for every measurement and calibration. A
measurement result therefore always has to indicate the measured quantity value and its
uncertainty. Obviously, the latter increases with every calibration step down the
calibration hierarchy. Measured quantity values can evidently only be assumed
equivalent if their difference is smaller than their measurement uncertainty
(compatibility). On the other hand they can only be assumed reliably different, if the
difference is larger than the uncertainty.
To ensure comparability in practice, the International System of Units (SI) was
established. National Metrological Institutes (NMIs) have developed primary reference
procedures to realise the SI units in the form of primary standards. Extensive (ongoing)
efforts are made to link these units to fundamental and physical constants in order to
achieve the highest degree of reproducibility. Moreover, the NMIs periodically conduct
international comparison measurements under the umbrella of the International Bureau of
Weights and Measures, in order to ensure the compatibility of the quantity values of
national standards.
PSS-‐‑78, and similarly the Reference-‐‑Composition Salinity Scale (Millero et al. (2008a)),
compute the salinity value from a measured conductivity ratio with respect to the K15
conductivity ratio of IAPSO Standard Seawater (SSW, Culkin and Ridout (1998), Bacon et
al. (2007)), which plays the role of a primary standard. The production procedure if
IAPSO Standard Seawater, and in particular the adjustment of its conductivity to that of a
potassium chloride (KCl) solution of definite purity and the corresponding assignment of
the K15 ratio, can be seen as a primary reference procedure. However both of these
solutions are artefacts lying outside the SI system; they are not subject to regular
international inter-‐‑comparisons; their sufficiently precise replicability by arbitrary
independent laboratories is neither known nor even granted. A slow drift of artefact
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
151
properties cannot rigorously be excluded, similar in principle to the “evaporation” of mass
from the kilogram prototype stored in Paris. It is impossible to foresee effects that might
affect the conductivity of SSW solution one day. Thus, with respect to decadal or century
time scales, there is an uncertainty of its K15 ratio, which a priori can not be quantified
and puts long term comparability of salinity measurement results at risk.
This fundamental problem, which is related to any artificial reference standard, can, at
least in principle, be avoided if the conductivity of seawater is measured traceable to
primary SI standards (“absolute” conductivity) rather than relying on a conductivity ratio.
Unfortunately the related uncertainty of absolute conductivity measurements with
present-‐‑day state-‐‑of-‐‑the-‐‑art technology is one order of magnitude larger than that of the
relative measurements presently used in oceanography (Seitz et al. (2008)).
A way out of this practical dilemma is the measurement of a different seawater
quantity that is traceable to SI standards and possesses the demanded small uncertainty,
and from which the salinity can be computed via an empirical relation that is very
precisely known (Seitz et al. (2011)). Among the potential candidates for this purpose are
the sound speed, the refractive index, chemical analysis (e.g. by mass spectroscopy) of the
sea-‐‑salt constituents, in particular chlorine, and direct density measurements. The latter
has three important advantages, i) SI-‐‑traceable density measurements of seawater can be
carried out with a relative uncertainty of 1 ppm (Wolf (2008)), which perfectly meets the
needs of ocean observation, ii) a relation exists between density and the Absolute Salinity
of seawater is available with a relative uncertainty of 4 ppm in the form of the TEOS-‐‑10
Gibbs function, iii) the measurand, density, is of immediate relevance for oceanography, in
contrast to other options.
It is important to note that the actual measuring procedure for a quantity value is
irrelevant for its traceability. To measure the weight of a person, a mass balance can be
used, a spring or a magnetic coil; it is the quantity value that is traceable, not the method
to achieve this value. The method in use is not intrinsically important except in so far as it
is responsible for the uncertainty of the quantity value. Hence, we may measure the
density of seawater with a CTD conductivity sensor, provided this sensor is properly
calibrated with respect to an SI-‐‑traceable density reference standard. In practice, this will
mean that the sensor calibration in oceanographic labs must be done with standard
seawater samples of certified density rather than certified Practical Salinity. The density
value returned from the CTD reading at sea is then converted into an Absolute Salinity
value by means of the equation of state of seawater, and eventually into a Practical Salinity
number for storage in data centres. The latter step may include some modification
regarding local sea salt composition anomalies. Storing a salinity value rather than the
related density reading has the advantage of conservativity with respect to dilution or
changes of temperature or pressure.
This conceptual proposal of WG127 is still immature and needs to be worked out in
more detail in the following years. Although it may imply only minor changes in the
practical use of a CTD or similar devices, the new concept is very promising regarding the
long-‐‑term reliability of observations made in the near future for climatic trend analyses to
be performed by the coming generations. An immediate consequence of this proposal is
to have the density (at a given temperature and pressure) of several samples of each batch
of IAPSO Standard Seawater measured when they are produced and have these densities
made available as reference values for each batch.
IOC Manuals and Guides No. 56
152
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Appendix F:
Coefficients of the IAPWS-‐‑95 Helmholtz function
of fluid water (with extension down to 50 K)
The specific Helmholtz energy for fluid (gaseous and liquid) water is given by the revised
IAPWS Release, IAPWS (2009b), which is based mainly on the work of Wagner and Pruß
(2002). This revised release is still referred to as IAPWS-‐‑95. The specific Helmholtz
energy of IAPWS-‐‑95 is defined by
f flu (T , ρ ) = f V,id (T , ρ ) + RWT ϕ res (τ , δ ) ,
(F.1)
where f
(T , ρ ) is the ideal-‐‑gas part, (F.2), RW = 461.518 05 J kg–1 K–1 is the specific gas
constant of water used in IAPWS-‐‑95, and ϕ res (τ , δ ) is the dimensionless residual part
consisting of 56 terms, available from (F.5) and Tables F.2 -‐‑ F.4. Note that the gas constant
used here differs from the most recent value, RW = R M W = 461.523 64 J kg–1 K–1, where
M W = 18.015 268 g mol–1 is the molar mass of water (IAPWS (2005)).
The ideal-‐‑gas part, f V,id (T , ρ ) , of the specific Helmholtz energy for water vapour
is (from IAPWS (2009b), Wagner and Pruß (2002), Feistel et al. (2010a))
V,id
f V,id (T , ρ ) = RWT ⎡⎣ϕ 0 (τ , δ ) + ϕ ex (τ )⎤⎦ .
(F.2)
Note that the term ϕ ex (τ ) has been added by Feistel et al. (2010a) (see IAPWS-‐‑12) in order
to extend the formulation to extraterrestrial applications, and because sublimation
pressure values are now available down to 50 K from Feistel and Wagner (2007) and
IAPWS (2008b); an extreme range where no related experiments have been performed.
This term is additional to the specific Helmholtz energy of IAPWS (2009b) and Wagner
and Pruß (2002). The function ϕ 0 (τ , δ ) was obtained from an equation for the specific
isobaric heat capacity of vapour and reads
8
(
)
ϕ 0 (τ , δ ) = ln δ + n10 + n20τ + n30 lnτ + ∑ ni0 ln 1 − e−γ i τ .
i =4
0
(F.3)
The “reduced density” δ = ρ / ρc and “reduced temperature” τ = Tc / T are specified by
ρc = 322 kg m−3 , Tc = 647.096 K. The coefficients of (F.3) are available from Table F.1. The
IAPWS-‐‑95 reference state conditions define the internal energy and the entropy of liquid
water to be zero at the triple point. A highly accurate numerical implementation of these
conditions gave the following values rounded to 16 digits for the adjustable coefficients
n1o = −8.320 446 483 749 693 and n2o = 6.683 210 527 593 226. These are the values used in
TEOS-‐‑10 (IAPWS (2009b), Feistel et al. (2008a)).
The temperature T is measured on the ITS-‐‑90 scale. The range of validity is 130 –
2000 K without the extension (F.4), that is with ϕ ex (τ ) = 0. The range can be extended to
include the region 50 – 130 K with the following correction function ϕ ex (τ ) added to (F.2)
in this temperature range,
⎛ 1
3
τ 9 9τ
τ2 ⎞
for 50 K ≤ T ≤ 130 K,
(F.4)
ϕ ex (τ ) = E × ⎜ − − 2 (τ + ε ) ln − + 2 + 3 ⎟ ,
2
τ
ε
2
ε
ε
2
ε
2
ε
⎝
⎠
where TE = 130 K , E = 0.278 296 458 178 592, and ε = Tc / TE . At τ = ε , ϕ ex (τ ) is zero, as
well as its first, second, third and fourth temperature derivatives. This correction has been
determined such that when applied to the formula used in IAPWS-‐‑95, it results in a fit to
the heat capacity data of Woolley (1980) between 50 and 130 K with an r.m.s. deviation of
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
153
6 ×10−4 in cP RW . This extension formula has been developed particularly for
implementation in TEOS-‐‑10 (Feistel et al. (2010a)), it is consistent with the correlation
function given in IAPWS (2008b), and it is expected to be endorsed as the IAPWS
Guideline IAPWS-‐‑12.
The residual part of (F.1) has the form
ϕ res =
7
∑ niδ diτ ti +
i =1
+
54
∑ niδ
i =52
∑ niδ d τ t exp ( −δ c )
51
i
i
i
i =8
(
)
56
(F.5)
τ exp −αi (δ − ε i ) − βi (τ − γ i ) + ∑ ni Δ δψ
di ti
2
2
bi
i =55
with the abbreviations
1
(
)
2
2
Δ = θ 2 + Bi δ − 1 i , θ = 1 − τ + Ai δ − 1 βi , and ψ = exp −Ci (δ − 1) − Di (τ − 1) . (F.6)
2a
The coefficients of (F.5) are available from Tables F.2 – F.4.
Table F.1. Coefficients appearing in Eqn. (F.3). Note that the originally published values
(Wagner and Pruß (2002)) of the adjustable coefficients n1o and n2o are slightly different
from those of TEOS-‐‑10 given here (Feistel et al. (2008a)).
i
ni0
γ i0
1
2
3
4
5
6
7
8
–8.32044648374969
6.68321052759323
3.00632
0.012436
0.97315
1.2795
0.96956
0.24873
1.28728967
3.53734222
7.74073708
9.24437796
27.5075105
Table F.2. Coefficients of the residual part (F.5).
i
ci
di
ti
ni
1
0
1 –0.5
0.012533547935523
2
0
1 0.875
7.8957634722828
3
0
1
1
–8.7803203303561
4
0
2
0.5
0.31802509345418
5
0
2 0.75 –0.26145533859358
6
0
3 0.375 –7.8199751687981× 10–3
7
0
4
1
8.8089493102134× 10–3
8
1
1
4
–0.66856572307965
9
1
1
6
0.20433810950965
10
1
1
12
–6.6212605039687× 10–5
11
1
2
1
–0.19232721156002
12
1
2
5
–0.25709043003438
13
1
3
4
0.16074868486251
14
1
4
2
–0.040092828925807
15
1
4
13
3.9343422603254× 10–7
16
1
5
9
–7.5941377088144× 10–6
17
1
7
3
5.6250979351888× 10–4
18
1
9
4
–1.5608652257135× 10–5
19
1
10
11
1.1537996422951× 10–9
20
1
11
4
3.6582165144204× 10–7
21
1
13
13
–1.3251180074668× 10–12
IOC Manuals and Guides No. 56
154
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
4
6
6
6
6
15
1
2
2
2
3
4
4
4
5
6
6
7
9
9
9
9
9
10
10
12
3
4
4
5
14
3
6
6
6
1
7
1
9
10
10
3
7
10
10
6
10
10
1
2
3
4
8
6
9
8
16
22
23
23
10
50
44
46
50
–6.2639586912454× 10–10
–0.10793600908932
0.017611491008752
0.22132295167546
–0.40247669763528
0.58083399985759
4.9969146990806× 10–3
–0.031358700712549
–0.74315929710341
0.4780732991548
0.020527940895948
–0.13636435110343
0.014180634400617
8.3326504880713× 10–3
–0.029052336009585
0.038615085574206
–0.020393486513704
–1.6554050063734× 10–3
1.9955571979541× 10–3
1.5870308324157× 10–4
–1.638856834253× 10–5
0.043613615723811
0.034994005463765
–0.076788197844621
0.022446277332006
–6.2689710414685× 10–5
–5.5711118565645× 10–10
–0.19905718354408
0.31777497330738
–0.11841182425981
Table F.3. Coefficients of the residual part (F.5).
i
d i ti
ni
αi
βi
γi
εi
52 3 0 –31.306260323435 20 150 1.21 1
53 3 1
31.546140237781 20 150 1.21 1
54 3 4 –2521.3154341695 20 250 1.25 1
Table F.4. Coefficients of the residual part (F.5).
i
ai
bi
Bi
ni
Ci
Di
Ai
βi
55 3.5 0.85 0.2 –0.14874640856724 28 700 0.32 0.3
56 3.5 0.95 0.2
0.31806110878444 32 800 0.32 0.3
Equation (F.1) is valid between 50 and 1273 K and for pressures up to 1000 MPa in the
stable single-‐‑phase region of fluid water. Uncertainty estimates are available from IAPWS
(2009b) and Wagner and Pruß (2002).
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
155
Appendix G: Coefficients of the pure liquid
water Gibbs function of IAPWS-‐‑09
The pure liquid water part of the Gibbs function of Feistel (2003) has been approved by
IAPWS (IAPWS (2009c)) as an alternative thermodynamic description of pure water to
IAPWS-‐‑95 in the oceanographic ranges of temperature and pressure. The pure water
specific Gibbs energy g W ( t , p ) is the following function of the independent variables ITS-‐‑
90 Celsius temperature, t = tu × y , and sea pressure, p = pu × z
7
6
gW (t , p ) = g u ∑∑ g jk y j z k ,
(G.1)
j =0 k =0
with the reduced temperature y = t tu and the reduced (dimensionless) pressure
z = p / pu . The unit-‐‑related constants tu , pu and g u are given in Table D4 of appendix D
(e. g. pu = 108 Pa = 104 dbar ). Coefficients not contained in the table below have the value
g jk = 0. Two of these 41 parameters ( g00 and g10 ) are arbitrary and are computed from
the reference-‐‑state conditions of vanishing specific entropy, η , and specific internal
energy, u, of liquid H2O at the triple point,
η (Tt , pt ) = 0, and u (Tt , pt ) = 0.
(G.2)
Note that the values of g 00 and g10 in the table below are taken from Feistel et al. (2008a)
and IAPWS (2009), and are not identical to the values in Feistel (2003). The modified
values have been chosen to most accurately achieve the triple-‐‑point conditions (G.2) (see
Feistel et al. (2008a) for a discussion of this point).
j
0
0
0
0
0
0
0
1
1
1
1
1
1
2
2
2
2
2
2
3
3
k
0
1
2
3
4
5
6
0
1
2
3
4
5
0
1
2
3
4
5
0
1
gjk
0.101 342 743 139 674 × 10
0.100 015 695 367 145 × 106
–0.254 457 654 203 630 × 104
0.284 517 778 446 287 × 103
–0.333 146 754 253 611 × 102
0.420 263 108 803 084 × 10
–0.546 428 511 471 039
3
0.590 578 347 909 402 × 10
–0.270 983 805 184 062 × 103
0.776 153 611 613 101 × 103
–0.196 512 550 881 220 × 103
0.289 796 526 294 175 × 102
–0.213 290 083 518 327 × 10
–0.123 577 859 330 390 × 105
0.145 503 645 404 680 × 104
–0.756 558 385 769 359 × 103
0.273 479 662 323 528 × 103
–0.555 604 063 817 218 × 102
0.434 420 671 917 197 × 10
0.736 741 204 151 612 × 103
–0.672 507 783 145 070 × 103
j
k
3
3
3
3
4
4
4
4
4
5
5
5
5
5
6
6
6
6
7
7
2
3
4
5
0
1
2
3
4
0
1
2
3
4
0
1
2
3
0
1
gjk
0.499 360 390 819 152 × 103
–0.239 545 330 654 412 × 103
0.488 012 518 593 872 × 102
–0.166 307 106 208 905 × 10
–0.148 185 936 433 658 × 103
0.397 968 445 406 972 × 103
–0.301 815 380 621 876 × 103
0.152 196 371 733 841 × 103
–0.263 748 377 232 802 × 102
0.580 259 125 842 571 × 102
–0.194 618 310 617 595 × 103
0.120 520 654 902 025 × 103
–0.552 723 052 340 152 × 102
0.648 190 668 077 221 × 10
–0.189 843 846 514 172 × 102
0.635 113 936 641 785 × 102
–0.222 897 317 140 459 × 102
0.817 060 541 818 112 × 10
0.305 081 646 487 967 × 10
–0.963 108 119 393 062 × 10
IOC Manuals and Guides No. 56
156
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Appendix H: Coefficients of the saline
Gibbs function for seawater of IAPWS-‐‑08
Non-‐‑zero coefficients gijk of the saline specific Gibbs energy g S ( S A , t , p ) as a function of
the independent variables Absolute Salinity, SA = Su × x ² , ITS-‐‑90 Celsius temperature,
t = tu × y , and sea pressure, p = pu × z :
⎧
⎫
g S ( SA , t, p) = g u ∑ ⎨ g1 jk x 2 ln x + ∑ gijk x i ⎬ y j z k .
j ,k ⎩
i >1
⎭
(H.1)
The unit-‐‑related constants Su , tu , pu and g u are given in Table D4 of appendix D (e. g.
pu = 108 Pa = 104 dbar ). Coefficients with k > 0 are adopted from Feistel (2003). Pure-‐‑
water coefficients with i = 0 do not occur in the saline contribution. The coefficients g200
and g210 were determined to exactly achieve Eqns. (2.6.7) and (2.6.8) when the pure water
Gibbs function was that of IAPWS-‐‑95.
i
j
k
i
j
k
gijk
i
j
k
gijk
1
0
0
5812.81456626732
gijk
2
5
0
–21.6603240875311
3
2
2
–54.1917262517112
1
1
0
851.226734946706
4
5
0
2.49697009569508
2
3
2
–204.889641964903
2
0
0
1416.27648484197
2
6
0
2.13016970847183
2
4
2
74.7261411387560
3
0
0
–2432.14662381794
2
0
1
–3310.49154044839
2
0
3
–96.5324320107458
4
0
0
2025.80115603697
3
0
1
199.459603073901
3
0
3
68.0444942726459
5
0
0
–1091.66841042967
4
0
1
–54.7919133532887
4
0
3
–30.1755111971161
6
0
0
374.601237877840
5
0
1
36.0284195611086
2
1
3
124.687671116248
7
0
0
–48.5891069025409
2
1
1
729.116529735046
3
1
3
–29.4830643494290
2
1
0
168.072408311545
3
1
1
–175.292041186547
2
2
3
–178.314556207638
3
1
0
–493.407510141682
4
1
1
–22.6683558512829
3
2
3
25.6398487389914
4
1
0
543.835333000098
2
2
1
–860.764303783977
2
3
3
113.561697840594
5
1
0
–196.028306689776
3
2
1
383.058066002476
2
4
3
–36.4872919001588
6
1
0
36.7571622995805
2
3
1
694.244814133268
2
0
4
15.8408172766824
2
2
0
880.031352997204
3
3
1
–460.319931801257
3
0
4
–3.41251932441282
3
2
0
–43.0664675978042
2
4
1
–297.728741987187
2
1
4
–31.6569643860730
4
2
0
–68.5572509204491
3
4
1
234.565187611355
2
2
4
44.2040358308000
2
3
0
–225.267649263401
2
0
2
384.794152978599
2
3
4
–11.1282734326413
3
3
0
–10.0227370861875
3
0
2
–52.2940909281335
2
0
5
–2.62480156590992
4
3
0
49.3667694856254
4
0
2
–4.08193978912261
2
1
5
7.04658803315449
2
4
0
91.4260447751259
2
1
2
–343.956902961561
2
2
5
–7.92001547211682
3
4
0
0.875600661808945
3
1
2
83.1923927801819
4
4
0
–17.1397577419788
2
2
2
337.409530269367
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
157
Appendix I: Coefficients of the Gibbs function
of ice Ih of IAPWS-‐‑06
The Gibbs energy of ice Ih, the naturally abundant form of ice, having hexagonal crystals,
is a function of temperature (ITS-‐‑90) and sea pressure, g Ih ( t , p ) . This Gibbs function has
been derived by Feistel and Wagner (2006) and was adopted as an IAPWS Release in 2006
and revised in 2009 (IAPWS (2009a)), here referred to as IAPWS-‐‑06. This equation of state
for ice Ih is given by Eqn. (I.1) as a function of temperature, with two of its coefficients
being polynomial functions of sea pressure p ( p = P − P0 )
2
⎡
τ2 ⎤
g Ih ( t , p ) = g0 − s0Tt ⋅ τ + Tt Re ∑ rk ⎢( tk − τ ) ln ( tk − τ ) + (tk + τ ) ln (tk + τ ) − 2tk ln tk − ⎥
tk ⎦
k =1 ⎣
4
⎛ p⎞
g0 ( p ) = ∑ g0k ⋅ ⎜ ⎟
k =0
⎝ Pt ⎠
k
(I.1)
k
2
⎛ p⎞
r2 ( p ) = ∑ r2 k ⋅ ⎜ ⎟ ,
k =0
⎝ Pt ⎠
with the reduced temperature τ = (T0 + t ) Tt and Tt and Pt are given in Table I.1. If the
sea pressure p is expressed in dbar then Pt must also be given in these units as
Pt = 0.0611657 dbar . The real constants g00 to g04 and s0 , the complex constants t1 , r1 ,
t2 , and r20 to r22 are listed in Table I.2.
TABLE I.1 Special constants and values used in the ice Ih Gibbs function.
Quantity
Symbol
Value
Unit
Experimental triple-point pressure
Pt
611.657
Pa
Numerical triple-point pressure
num
611.654 771 007 894
Pa
101325
Pa
273.16
K
Normal pressure
Triple-point temperature
Pt
P0
Tt
IOC Manuals and Guides No. 56
158
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
TABLE I.2 Coefficients of the equation of state (Gibbs potential function) of ice Ih
as given by Eqn. (I.1).
Coefficient
Real part
Imaginary part
Unit
g00
g01
g02
– 0.632 020 233 335 886 × 106
0.655 022 213 658 955
J kg
– 0.189 369 929 326 131 × 10-7
g03
0.339 746 123 271 053 × 10-14
g04
– 0.556 464 869 058 991 × 10-21
s0 (absolute)
s0 (IAPWS-95)
J kg
J kg
J kg
– 0.332 733 756 492 168 × 104
t1
0.368 017 112 855 051 × 10-1
0.510 878 114 959 572 ×10-1
r1
0.447 050 716 285 388 × 102
0.656 876 847 463 481 × 102
t2
0.337 315 741 065 416
0.335 449 415 919 309
r20
– 0.725 974 574 329 220 × 10
r21
– 0.557 107 698 030 123 × 10-4
r22
0.234 801 409 215 913 × 10-10
–1
–1
–1
–1
J kg
–1 –1
J kg K
–1 –1
J kg K
0.189 13 × 103
2
–1
– 0.781 008 427 112 870 × 102
0.464 578 634 580 806 × 10-4
– 0.285 651 142 904 972 × 10-10
J kg
–1 –1
K
–1 –1
K
–1 –1
J kg K
–1 –1
J kg K
J kg
The numerical triple point pressure Ptnum listed in Table I.1 was derived in Feistel et al.
(2008a) as the Absolute Pressure at which the three phases of water were in
thermodynamic equilibrium at the triple point temperature, using the mathematical
descriptions of the three phases as given by IAPWS-‐‑95 and IAPWS-‐‑06. The complex
logarithm ln ( z ) is meant as the principal value, i.e. it evaluates to imaginary parts in the
interval −π < Im ⎡⎣ln ( z )⎤⎦ ≤ + π . The complex notation used here has no direct physical
basis but serves for convenience of analytical partial derivatives and for compactness of
the resulting formulae, especially in program code. Complex data types are supported by
scientific computer languages like Fortran (as COMPLEX*16) or C++ (as complex
), thus allowing an immediate implementation of the formulae given, without the
need for prior conversion to much more complicated real functions, or for experience in
complex calculus.
The residual entropy coefficient s0 is given in Table I.2 in the form of two alternative
values. Its ‘IAPWS-‐‑95’ version is required for phase equilibria studies between ice and
fluid water and seawater. This is the value of s0 used in the TEOS-‐‑10 algorithms. In the
'ʹabsolute'ʹ version, s0 is the statistical non-‐‑zero entropy ice possesses at the zero point (0 K)
resulting from the multiplicity of its energetically equivalent crystal configurations (for
details, see Feistel and Wagner (2005)).
The value of g 00 listed in table I.2 is the value in the revised IAPWS-‐‑2006 Ice Ih
Release (IAPWS (2009a)) which improves the numerical consistency (Feistel et al. (2008a))
with the IAPWS-‐‑1995 Release for the fluid phase of water.
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
159
Appendix J:
Coefficients of the Helmholtz function of
moist air of IAPWS-‐‑10
The equation of state of humid air described here (Feistel et al. (2010a), IAPWS (2010)) is
represented in terms of a Helmholtz function which expresses the specific Helmholtz
energy as a function of dry-‐‑air mass fraction A, absolute temperature T and humid-‐‑air
mass density, ρ , and takes the form
(
)
(
)
f AV ( A, T , ρ ) = (1 − A) f V T , ρ V + Af A T , ρ A + f mix ( A, T , ρ ) .
(J.1)
The vapour part is given by the IAPWS-‐‑95 Helmholtz function for fluid water (IAPWS
(2009b)),
(
)
(
)
f V T , ρ V ≡ f flu T , ρ V ,
(J.2)
is computed at the vapour density, ρ V = (1 − A) ρ , and is defined in Eqn. (F.1) of appendix
F. The dry-‐‑air part, f A T , ρ A , is computed at the dry-‐‑air density, ρ A = Aρ , and is
defined by Eqn. (J.3). The air-‐‑water cross-‐‑over part f mix is defined by Eqn. (J.8).
Table J.1. Special constants and values used in this appendix. Note that the
molar gas constant used here differs from the most recent value (IAPWS
(2005)), and the molar mass of dry air used here differs from the most
recent value (Picard et al. (2008)), Table D6.
(
Quantity
Molar gas constant
Molar gas constant
Molar mass of dry air
Molar mass of dry air
Molar mass of water
Celsius zero point
Normal pressure
)
Symbol
Value
Unit
RL
8.314 51
J mol–1 K–1
R
MA
MA
MW
T0
8.314 472
28.958 6
28.965 46
18.015 268
273.15
P0
101 325
–1
J mol K
g mol–1
g mol–1
g mol–1
K
Reference
–1
Lemmon et al. (2000)
IAPWS (2005)
Lemmon et al. (2000)
IAPWS (2010)
IAPWS (2005)
Preston-Thomas (1990)
Pa
ISO(1993)
The specific Helmholtz energy for dry air is (Lemmon et al. (2000)),
(
)
f A T,ρA =
R LT id
⎡α (τ , δ ) + α res (τ , δ )⎤ .
⎦
MA ⎣
(J.3)
The values to be used for molar mass M A of dry air, and for the molar gas constant R L
are given in Table J.1. The function α id (τ , δ ) is the ideal-‐‑gas part,
IOC Manuals and Guides No. 56
160
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
(
5
)⎦
0
α id (τ , δ ) = ln δ + ∑ ni0τ i −4 + n60τ 1.5 + n70 ln τ + n80 ln ⎡1 − exp −n11
τ ⎤
⎣
i =1
+
n90 ln ⎡1 − exp
⎣
(
0
τ
−n12
and α res (τ , δ ) is the residual part,
α res (τ , δ ) =
10
∑ nkδ ikτ jk +
k =1
)
( )
(J.4)
0
0
⎤ + n10
ln ⎡ 2 / 3 + exp n13
τ ⎤
⎦
⎣
⎦
19
∑ nkδ i τ j
k
k =11
k
(
)
exp −δ lk .
(J.5)
The “reduced variables” in Eqns. (J.3) -‐‑ (J.5) are τ = TA* / T with the reducing temperature
TA* = 132.6312 K , and δ = ρ A / ρ A* with the reducing density ρ A* = 10.4477 mol dm−3 × M A .
M A is given in Table J.1. The coefficients of Eqns. (J.4) and (J.6) are given in Tables J.2 and
J.3.
Two of the parameters ( n40 and n50 ) listed in Table J.2 are arbitrary and are computed
here from the reference-‐‑state conditions of vanishing specific entropy, η A , and specific
enthalpy, h A , of dry air at the temperature T0 and the normal pressure P0 , as given in
Table J.1,
η A (T0 , P0 ) = 0,
(J.6)
h A (T0 , P0 ) = 0.
(J.7)
The Helmholtz function f mix in Eqn. (J.1) describes the water-‐‑air interaction and is
defined by
f mix ( A, T , ρ ) =
A (1 − A) ρ RT ⎧⎪ AW
(1 − A) C AWW T ⎤ ⎫⎪ .
3 ⎡ A AAW
C
(T ) +
( )⎥ ⎬
⎨ 2 B (T ) + ρ ⎢
M AM W
2 ⎣ MA
MW
⎪⎩
⎦ ⎪⎭
(J.8)
The values used for the molar gas constant R, the molar mass of dry air, M A , and the
molar mass of water, M W , are given in Table J.1.
The second cross-‐‑virial coefficient, B AW (T ) , is given by Harvey and Huang (2007) as
3
B AW (T ) = b * ∑ ciτ di .
(J.9)
i =1
The coefficients of Eqn. (J.9) are given in Table J.4.
The third cross-‐‑virial coefficients C AAW (T ) and C AWW (T ) are defined in Hyland and
Wexler (1983), in the form
4
C AAW (T ) = c * ∑ aiτ −i ,
(J.10)
i =0
and
⎧3
⎫
C AWW (T ) = − c *exp ⎨∑ biτ −i ⎬ .
⎩ i =0
⎭
The coefficients ai and bi of Eqns. (J.10) and (J.11) are given in Table J.4.
IOC Manuals and Guides No. 56
(J.11)
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Table J.2. Dimensionless coefficients and exponents for the ideal-‐‑gas part,
Eqn. (J.4), for dry air (Lemmon et al. (2000)). In TEOS-‐‑10, the coefficients
n 40 and n50 are re-‐‑adjusted to the reference state conditions, Eqns. (J.6, J.7),
and deviate from the originally published values of Lemmon et al. (2000).
i
1
i
ni0
0.605 719 400 000 000 × 10–7
ni0
8
0.791 309 509 000 000
9
0.212 236 768 000 000
10
–0.197 938 904 000 000
2
–0.210 274 769 000 000 × 10
–4
3
–0.158 860 716 000 000 × 10
–3
4
0.974 502 517 439 480 × 10
11
0.253 636 500 000 000 × 102
5
0.100 986 147 428 912 × 102
12
0.169 074 100 000 000 × 102
6
–0.195 363 420 000 000 × 10–3
13
0.873 127 900 000 000 × 102
7
0.249 088 803 200 000 × 10
Table J.3. Coefficients and exponents for the residual part, Eqn. (J.5),
for dry air (Lemmon et al. (2000)).
k
ik
jk
lk
nk
1
1
0
0
0.118 160 747 229
2
1
0.33
0
0.713 116 392 079
3
1
1.01
0
4
2
0
0
0.714 140 178 971 × 10–1
5
3
0
0
–0.865 421 396 646 × 10–1
6
3
0.15
0
0.134 211 176 704
7
4
0
0
0.112 626 704 218 × 10–1
8
4
0.2
0
–0.420 533 228 842 × 10–1
9
4
0.35
0
0.349 008 431 982 × 10–1
10
6
1.35
0
0.164 957 183 186 × 10–3
11
1
1.6
1
–0.101 365 037 912
12
3
0.8
1
–0.173 813 690 970
13
5
0.95
1
–0.472 103 183 731 × 10–1
14
6
1.25
1
–0.122 523 554 253 × 10–1
15
1
3.6
2
–0.146 629 609 713
16
3
6
2
–0.316 055 879 821 × 10–1
17
11
3.25
2
0.233 594 806 142 × 10–3
18
1
3.5
3
0.148 287 891 978 × 10–1
19
3
15
3
–0.938 782 884 667 × 10–2
–0.161 824 192 067 × 10
IOC Manuals and Guides No. 56
161
162
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Table J.4. Coefficients of the cross-‐‑virial coefficients B AW (T ) , C AAW (T ) and
C AWW (T ) , Eqns. (J.9) -‐‑ (J.11). The reducing factors are b* = 10−6 m3 mol −1
and c* = 10−6 m6 mol −2 , the “reduced temperature” is τ = T / (100 K ) .
i
0
1
2
3
4
ai
0.482 737 × 10–3
0.105 678 × 10–2
–0.656 394 × 10–2
0.294 442 × 10–1
–0.319 317 × 10–1
bi
–0.107 288 76 × 102
0.347 802 00 × 102
–0.383 383 00 × 102
0.334 060 00 × 102
ci
di
0.665 687 × 102
–0.238 834 × 103
–0.176 755 × 103
–0.237
–1.048
–3.183
The equation of state, Eqn. (J.1), is valid for humid air within the temperature and
pressure range
(J.12)
193 K ≤ T ≤ 473 K and 10 nPa ≤ P ≤ 5 MPa.
The pressure is computed from P = ρ 2 f ρAV . All validity regions of the formulas combined
in Eqn. (J.1), including the Helmholtz functions of water vapour and of dry air, as well as
the cross-‐‑virial coefficients, overlap only in this range. The separate ranges of validity of
the individual components are wider; some of them significantly wider. Therefore, Eqn.
(J.1) will provide reasonable results outside of the T − P range given above under the
condition that a certain component dominates numerically in Eqn. (J.1) and is evaluated
within its particular range of validity.
The air fraction A can take any value between 0 and 1 provided that the partial
vapour pressure, P vap = xV P , ( xV is the mole fraction of vapour, Eqn. (3.35.3)) does not
exceed its saturation value, i.e.,
0 ≤ A ≤ 1 and
Asat (T , P ) ≤ A .
(J.13)
The exact value of the air fraction Asat (T , P ) of saturated humid air is given by equal
chemical potentials of water vapour in humid air and of either liquid water, Eqn. (3.37.5),
if the temperature is above the freezing point, or of ice, Eqn. (3.35.4), if the temperature is
below the freezing point. At low density, the saturation vapour pressure Psat of humid air
can be estimated by the correlation function for either the vapour pressure, P liq (T ) , of
pure water (IAPWS (2007)), or for the sublimation pressure, P subl (T ) , of ice (IAPWS
(2008b)), to obtain Asat (T , P ) = P − Psat / ⎡ P − Psat (1 − M W / M A )⎤ , from Eqn. (3.35.3) as a
⎣
⎦
practically sufficient approximation.
(
IOC Manuals and Guides No. 56
)
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
163
Appendix K: Coefficients of the 75-‐‑term expression
for the specific volume of seawater in terms of Θ
The TEOS-‐‑10 Gibbs function of seawater g ( SA , t , p ) is written as a polynomial in terms of
in situ temperature t , while for ocean models, specific volume (or density) needs to be
expressed as a computationally efficient expression in terms of Conservative Temperature
Θ . Roquet et al. (2015) have published such a computationally efficient polynomial for
specific volume. Their non-‐‑dimensional (root) salinity s , temperature τ , and pressure
π , variables are
s ≡
SA + 24 g kg −1
Θ
p
, τ ≡
and π ≡
,
Θu
pu
SA u
(K.1)
in terms of the unit-‐‑related scaling constants
SA u ≡ 40× 35.16504g kg −1 / 35 , Θ u ≡ 40°C and pu ≡ 104 dbar .
(K.2)
Their polynomial expression for the specific volume of seawater is
v̂(SA ,Θ, p) = vu ∑ vijk s i τ jπ k ,
(K.3)
i, j,k
where vu ≡ 1 m 3kg −1 and the non-‐‑zero dimensionless constants vijk are given in Table K.1.
Roquet et al. (2015) fitted the TEOS-‐‑10 values of specific volume v to SA , Θ and p in a
“funnel” of data points in ( SA , Θ, p ) space. This is the same “funnel” of data points as
used in McDougall et al. (2003); at the sea surface it covers the full range of temperature
and salinity while for pressure greater than 6500 dbar, the maximum temperature of the
fitted data is 10°C and the minimum Absolute Salinity is 30 g kg −1 . The maximum
pressure of the “funnel” is 8000 dbar . Table K.1 contains the 75 coefficients of the
expression (K.3) for specific volume in terms of ( SA , Θ, p ) .
The rms error of this 75-‐‑term approximation to the full Gibbs function-‐‑derived TEOS-‐‑
10 specific volume over the “funnel” is 0.2x10−9 m 3kg −1 ; this can be compared with the rms
uncertainty of 4x10−9 m 3kg −1 of the underlying laboratory density data to which the TEOS-‐‑
10 Gibbs function was fitted (see the first two rows of Table O.1 of appendix O). Similarly,
the appropriate thermal expansion coefficient,
αΘ =
1 ∂v
v ∂Θ S
= −
A, p
1 ∂ρ
ρ ∂Θ S
,
(K.4)
A, p
of the 75-‐‑term equation of state is different from the same thermal expansion coefficient
evaluated from the full Gibbs function-‐‑derived TEOS-‐‑10 with an rms error in the “funnel”
of 0.03x10−6 K −1 ; this can be compared with the rms error of the thermal expansion
coefficient of the laboratory data to which the Feistel (2008) Gibbs function was fitted of
0.73 x10−6 K −1 (see row six of Table O.1 of appendix O). In terms of the evaluation of
density gradients, the haline contraction coefficient evaluated from Eqn. (K.3) is many
times more accurate than the thermal expansion coefficient. Hence we may consider the
75-‐‑term polynomial expression for specific volume, Eqn. (K.3), to be equally as accurate as
the full TEOS-‐‑10 expressions for specific volume, for the thermal expansion coefficient and
for the saline contraction coefficient for data that reside inside the “oceanographic funnel”.
The sound speed evaluated from the 75-‐‑term polynomial of Eqn. (K.3) has an rms
error over the “funnel” of 0.025 m s −1 which is a little less than the rms error of the
IOC Manuals and Guides No. 56
164
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
underlying sound speed data that was incorporated into the Feistel (2008) Gibbs function,
being 0.035 m s −1 (see rows 7 to 9 of Table O.1 of appendix O). Hence, especially for the
purposes of dynamical oceanography where α Θ and β Θ are the aspects of the equation of
state that, together with spatial gradients of S A and Θ , drive ocean currents and affect the
calculation of the buoyancy frequency, we may take the 75-‐‑term expression for specific
volume, Eqn. (K.3), as essentially reflecting the full accuracy of TEOS-‐‑10.
The use of Eqn. (K.3) to evaluate v̂ SA ,Θ, p or ρˆ ( SA , Θ, p ) from
gsw_specvol(SA,CT,p) or gsw_rho(SA,CT,p) is approximately five times faster than first
evaluating the in situ temperature t (from gsw_t_from_CT(SA,CT,p)) and then calculating
in situ specific volume or density from the full Gibbs function expression v SA ,t, p or
ρ ( SA , t, p ) via gsw_specvol_t_exact(SA,t,p) or gsw_rho_t_exact(SA,t,p). (These two
function calls have been combined into gsw_specvol_CT_exact(SA,CT,P) and
gsw_rho_CT_exact(SA,CT,P).)
Table K.1. Coefficients of the 75-‐‑term polynomial of Roquet et al. (2015).
(
)
(
i
j
k
vijk
i
j
k
vijk
i
j
k
vijk
0
0
0
0
5
0
0
2
0
0
1
5
0
2
0
2
2
0
0
0
6
0
3
0
2
3
0
0
0
0
1
4
0
2
4
0
0
1
0
1
0
1
2
5
0
0
2
0
1
1
1
2
6
0
0
3
0
1
2
1
2
0
1
0
4
0
1
1
1
0
5
0
1
2
1
0
0
1
1
3
1
0
1
1
1
4
1
0
2
1
1
5
1
0
3
1
1
0
2
0
4
1
1
1
2
0
0
2
1
2
2
0
1
2
1
3
2
0
2
2
1
4
2
0
3
2
1
0
3
0
0
3
1
1
3
0
1
3
1
2
3
0
2
3
1
3
3
0
0
4
1
0
4
0
1
4
1
1
4
0
0
5
1
2
4
0
0
0
2
-8.0539615540e-7
-3.3052758900e-7
2.0543094268e-7
-6.0799143809e-5
2.4262468747e-5
-3.4792460974e-5
3.7470777305e-5
-1.7322218612e-5
3.0927427253e-6
1.8505765429e-5
-9.5677088156e-6
1.1100834765e-5
-9.8447117844e-6
2.5909225260e-6
-1.1716606853e-5
-2.3678308361e-7
2.9283346295e-6
-4.8826139200e-7
7.9279656173e-6
-3.4558773655e-6
3.1655306078e-7
-3.4102187482e-6
1.2956717783e-6
5.0736766814e-7
9.9856169219e-6
1
1
1.0769995862e-3
-3.1038981976e-4
6.6928067038e-4
-8.5047933937e-4
5.8086069943e-4
-2.1092370507e-4
3.1932457305e-5
-1.5649734675e-5
3.5009599764e-5
-4.3592678561e-5
3.4532461828e-5
-1.1959409788e-5
1.3864594581e-6
2.7762106484e-5
-3.7435842344e-5
3.5907822760e-5
-1.8698584187e-5
3.8595339244e-6
-1.6521159259e-5
2.4141479483e-5
-1.4353633048e-5
2.2863324556e-6
6.9111322702e-6
-8.7595873154e-6
4.3703680598e-6
-5.8484432984e-7
-4.8122251597e-6
4.9263106998e-6
-1.7811974727e-6
-1.1736386731e-6
-5.5699154557e-6
5.4620748834e-6
-1.3544185627e-6
2.1305028740e-6
3.9137387080e-7
-6.5731104067e-7
-4.6132540037e-7
7.7618888092e-9
-6.3352916514e-8
-1.1309361437e-6
3.6310188515e-7
1.6746303780e-8
-3.6527006553e-7
-2.7295696237e-7
2.8695905159e-7
1.0531153080e-7
-1.1147125423e-7
3.1454099902e-7
-1.2647261286e-8
1.9613503930e-9
IOC Manuals and Guides No. 56
3
1
2
0
2
2
1
2
2
2
2
2
0
3
2
1
3
2
0
4
2
0
0
3
1
0
3
2
0
3
0
1
3
1
1
3
0
2
3
0
0
4
1
0
4
0
1
4
0
0
5
0
0
6
)
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
165
Appendix L: Recommended nomenclature,
symbols and units in oceanography
L.1 Recommended nomenclature
The strict SI units of Absolute Salinity, temperature and pressure are kg kg −1, Absolute
Temperature in K and Absolute Pressure P in Pa. These are the units predominantly
adopted in the SIA computer software for the input and output variables. If
oceanographers were to adopt this practice of using strictly SI quantities it would simplify
many thermodynamic expressions at the cost of using unfamiliar units.
The GSW Oceanographic Toolbox (appendix N) adopts as far as possible the currently
used oceanographic units, so that the input variables for all the computer algorithms are
Absolute Salinity in S A in g kg −1, temperature in ° C and pressure as sea pressure in
dbar. The outputs of the functions are also generally consistent with this choice of units,
but some variables are more naturally expressed in SI units.
It seems impractical to recommend that the field of oceanography fully adopt strict
basic SI units. It is however very valuable to have the field adopt uniform symbols and
units, and in the interests of achieving this uniformity we recommend the following
symbols and units. These are the symbols and units we have adopted in the GSW
Oceanographic Toolbox.
Table L.1. Recommended Symbols and Units in Oceanography
Quantity
Symbol Units
Comments
Chlorinity
Cl
g kg–1
Chlorinity is defined as the following mass
fraction; it is 0.328 523 4 times the ratio of the
mass of pure silver required to precipitate all
dissolved chloride, bromide and iodide in seawater
to the mass of seawater.
Standard Ocean
Reference Salinity
SSO
g kg–1
freezing temperatures
t f , Θf
ºC
Absolute Pressure
P
Pa
35.165 04 g kg–1 being exactly 35 uPS ,
corresponding to the standard ocean Practical
Salinity of 35.
in situ and conservative values, each as a function
of S A and p.
When Absolute Pressure is used it should always
be in Pa, not in Mpa nor in dbar.
sea pressure. Sea pressure
is the pressure argument
to all the
GSW Toolbox functions.
p
dbar
Equal to P − P0 and usually expressed in dbar not
Pa.
IOC Manuals and Guides No. 56
166
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
gauge pressure. Gauge
pressure (also called
applied pressure) is
sometimes reported from
ship-born instruments.
p gauge
dbar
Equal to the Absolute Pressure P minus the local
atmospheric pressure at the time of the instrument
calibration, and expressed in dbar not Pa. Sea
pressure p is preferred over gauge pressure p gauge ,
as p is the argument to the seawater Gibbs
function.
reference pressure
pr
dbar
The value of the sea pressure p to which potential
temperature and/or potential density are
referenced.
one standard atmosphere
isopycnal slope ratio
P0
r
Pa
1
exactly 101 325 Pa (= 10.1325 dbar)
Stability Ratio
Rρ
1
Rρ = α ΘΘz β Θ ( SA )z ≈ α θ θ z β θ ( SA )z .
isopycnal temperature
gradient ratio
Practical Salinity
GΘ
1
G Θ = r ⎡⎣ Rρ −1⎤⎦ ⎡⎣ Rρ − r ⎤⎦ ; ∇σ Θ = G Θ∇n Θ
SP
1
Defined in the range 2 < SP < 42 by PSS-78 based
on measured conductivity ratios.
Reference Salinity
SR
g kg-1
Reference-Composition Salinity (or Reference
Salinity for short) is the Absolute Salinity of
seawater samples that have Reference
Composition. At S P = 35, S R is exactly uPS SP .
while in the range 2 < SP < 42 SR ≈ uPS SP .
Absolute Salinity
(This is the salinity
argument of all the
GSW Toolbox functions.)
-1
SA = SAdens g kg
Absolute Salinity
Anomaly
δ SA
g kg-1
“Preformed Absolute
Salinity”,
S*
g kg-1
“Solution Absolute
Salinity”, often shortened
to “Solution Salinity”
S Asoln
g kg-1
“Added-Mass Salinity”
SAadd
g kg-1
often shortened to
“Preformed Salinity”
IOC Manuals and Guides No. 56
r =
α Θ ( p) β Θ ( p)
α Θ ( pr ) β Θ ( pr )
SA = SR + δ SA ≈ uPS SP + δ SA
Absolute Salinity is the sum of S R on the Millero
et al. (2008a) Reference-Salinity Scale and the
Absolute Salinity Anomaly. The full symbol for
S A is SAdens as it is the type of absolute salinity
which delivers the best estimate of density when
used as the salinity argument of the TEOS-10
Gibbs function. Another name for SA = SAdens is
“Density Salinity”.
δ SA = SA − SR , the difference between Absolute
Salinity, SA = SAdens , and Reference-Composition
Salinity. In terms of the full nomenclature of
Pawlowicz et al. (2010), Wright et al. (2010b) and
appendix A.4 herein, the Absolute Salinity
Anomaly δ SA is δ SRdens .
Preformed Absolute Salinity S* is a salinity
variable that is designed to be as conservative as
possible, by removing the estimated
biogeochemical influences on the seawater
composition from other forms of salinity (see
Pawlowicz et al. (2010), Wright et al. (2010b) and
appendix A.4 herein).
The mass fraction of non-H2O constituents in
seawater after it has been brought to chemical
equilibrium at t = 25°C and p = 0 dbar (see
Pawlowicz et al. (2010), Wright et al. (2010b) and
appendix A.4 herein).
SAadd − SR is the estimated mass fraction of nonH2O constituents needed as ingredients to be added
to Standard Seawater which when mixed and
brought to chemical equilibrium at t = 25°C and
p = 0 dbar results in the observed seawater
composition.
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
167
temperature
Absolute Temperature
t
T
ºC
K
temperature derivatives
T
K
Celsius zero point
T0
θ
Θ
K
T0 ≡ 273.15 K
ºC
ºC
Defined implicitly by Eqn. (3.1.3)
Defined in Eqn. (3.3.1) as exactly potential
enthalpy divided by c 0p .
c0p ≡ 3991.867 957 119 63 J kg −1 K −1 . This 15-digit
number is defined to be the exact value of c 0p .
c 0p is the ratio of potential enthalpy h0 to Θ .
potential temperature
Conservative Temperature
T / K ≡ T0 / K + t / (°C) = 273.15 + t / (°C)
When a quantity is differentiated with respect to in
situ temperature, the symbol T is used in order to
distinguish this variable from time.
the “specific heat”, for use
with Conservative
Temperature
c 0p
J kg–1 K–1
combined standard
uncertainty
enthalpy
specific enthalpy
uc
Varies
H
h
J
J kg–1
specific potential enthalpy
h0
J kg–1
–1
h = u + ( p + P0 ) v .
Here p and P0 must be in Pa not dbar.
specific enthalpy referenced to zero sea pressure,
–1
specific isobaric heat
capacity
cp
J kg K
internal energy
specific internal energy
specific isochoric heat
capacity
U
u
cv
J
J kg–1
J kg–1 K–1
Gibbs function
(Gibbs energy)
specific Gibbs function
(Gibbs energy)
specific Helmholtz energy
unit conversion factor for
salinities
G
J
g
J kg–1
f
J kg–1
g kg–1
entropy
specific entropy
Σ
density
density anomaly
potential density anomaly
referenced to a sea
pressure of 2000 dbar
potential density anomaly
referenced to a sea
pressure of 4000 dbar
thermal expansion
coefficient with respect to
in situ temperature
thermal expansion
coefficient with respect to
potential temperature θ
uPS
η
ρ
σt
σ2
J K–1
J kg–1 K–1
kg m–3
kg m–3
kg m–3
h0 = h ( SA ,θ [ SA , t , p, pr = 0], pr = 0 )
c p = ∂h ∂T
SA , p
cv = ∂u ∂T
SA , v
uPS ≡ (35.16504 35) g kg−1 ≈ 1.004 715... g kg−1
The first part of this expression is exact. This
conversion factor is an important and invariant
constant of the 2008 Reference-Salinity Scale
(Millero et al. (2008a)).
In many other publications the symbol s is used for
specific entropy.
–3
ρ ( SA , t,0) – 1000 kg m
-3
ρ ( SA , θ [ SA , t , p, pr ], pr ) – 1000 kg m where
pr = 2000 dbar
σ4
kg m–3
-3
ρ ( SA , θ [ SA , t , p, pr ], pr ) – 1000 kg m where
pr = 4000 dbar
αt
K–1
v −1∂v / ∂T
SA , p
= − ρ −1∂ρ / ∂T
SA , p
αθ
K–1
v −1∂v / ∂θ
SA , p
= − ρ −1∂ρ / ∂θ
SA , p
IOC Manuals and Guides No. 56
168
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
thermal expansion
coefficient with respect to
Conservative Temperature
αΘ
K–1
v −1∂v / ∂Θ
saline contraction
coefficient at constant in
situ temperature
βt
kg g–1
− v −1∂v / ∂SA
saline contraction
coefficient at constant
potential temperature
βθ
saline contraction
coefficient at constant
Conservative Temperature
βΘ
isothermal compressibility
κt
Pa–1
isentropic and isohaline
compressibility
chemical potential of
water in seawater
chemical potential of sea
salt in seawater
relative chemical potential
of (sea salt and water in)
seawater
dissipation rate of kinetic
energy per unit mass
κ
Pa–1
µW
J g–1
µS
µ
J g–1
ε
J kg–1 s–1
= m2 s–3
adiabatic lapse rate
Γ
K Pa–1
SA , p
= − ρ −1∂ρ / ∂Θ
SA , p
Θ
= ρ −1∂ρ / ∂SA
T,p
Note that the units for β t are consistent with SA
being in g kg-1.
kg g–1
− v −1∂v / ∂SA
θ,p
Note that the units for β θ are consistent with SA
being in g kg-1.
kg g–1
− v −1∂v / ∂SA
Θ, p
= + ρ −1∂ρ / ∂SA
Note that the units for β
being in g kg-1.
J g–1
(∂g
Γ=
Θ
Θ, p
are consistent with SA
∂SA )t , p = µ S − µ W
∂t
∂P S
=
A ,θ
∂t
∂P S
=
A ,Θ
∂t
∂P S
=
A ,η
∂v
∂η
=
(T0 +θ ) ∂v
c0p
SA , p
∂Θ S
A, p
–1
c
v
ms
m3 kg–1
specific volume anomaly
thermobaric coefficient
based on θ
thermobaric coefficient
based on Θ
cabbeling coefficient
based on θ
cabbeling coefficient
based on Θ
buoyancy frequency
δ
m3 kg–1
Tbθ
K −1Pa −1
Tbθ = β θ ∂ α θ β θ
TbΘ
K −1Pa −1
TbΘ = β Θ ∂ α Θ β Θ
Neutral Density
= ρ −1∂ρ / ∂SA
θ,p
sound speed
specific volume
neutral helicity
T,p
θ
Cb
C bΘ
N
H
γn
K
K
s
n
−2
−2
−1
m
−3
v = ρ −1
(
) ∂P
SA , θ
(
) ∂P
SA , Θ
Cbθ = ∂α θ ∂θ
CbΘ = ∂α Θ ∂Θ
(
SA , p
SA , p
2
θ
+ 2 α θ ∂α θ ∂SA
β
⎛ θ⎞
− ⎜ α ⎟ ∂β θ ∂SA
θ , p ⎝ βθ ⎠
θ, p
Θ
+ 2 α Θ ∂α Θ ∂SA
⎛ Θ⎞
− ⎜ α Θ ⎟ ∂β Θ ∂SA
Θ, p
Θ, p
⎝β ⎠
β
)
(
N = g α Θz − β SA z = g α θ θ z − β θ SA z
2
Θ
Θ
2
)
defined by Eqns. (3.13.1) and (3.13.2)
kg m–3
a density variable whose iso-surfaces are designed
to be approximately neutral, i. e.
α Θ∇γ Θ ≈ β Θ∇γ SA .
Neutral-Surface-PotentialVorticity
NSPV
s −3
NSPV = − g ρ −1 f γ zn where f is the Coriolis
parameter.
dynamic height anomaly
Ψ
m 2 s −2
Pa m3 kg −1 = m2 s−2
Montgomery geostrophic
streamfunction
ΨM
m 2 s −2
Pa m3 kg −1 = m2 s−2
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
169
PISH (Pressure-Integrated
Steric Height)
Ψ′
kg s-2
streamfunction for f times the depth-integrated
relative mass flux, see Eqns. (3.31.1) – (3.31.5).
Coriolis parameter
f
s −1
1.458 42 x 10−4 sin φ s−1 , where φ is latitude
molar mass of Reference
Seawater
MS
g mol−1
molality of seasalt in
Reference Seawater
mSW
mol kg–1
M S is the mole-weighted average atomic weight
of the constituents of Reference Seawater,
M S = 31.403 821 8... g mol−1 , from Millero et al.
(2008a).
mSW = ∑ i mi =
SA
1
.
M S 1− SA
(
)
mi is the molality
of constituent i in Reference Seawater.
valence factor of
Reference Seawater
Z2
1
ionic strength of
Reference Seawater
I
mol kg–1
Z 2 = ∑ i X i Z i2 ≡ 1.245 289 8 where Z i is the
charge of seawater constituent i which is present
at the mole fraction X i in Reference Seawater
(from Millero et al. (2008a)).
I =
1
2
mSW Z 2 =
1
2
∑ i mi Zi2
= 0.622 644 9 mSW
≈
SA
0.622 644 9
mol kg −1
.
0.031 403 821 8
1− SA
(
)
mi is the molality of constituent i in Reference
Seawater.
osmotic coefficient
φ
1
φ ( SA , T , p ) =
g ( 0, t , p ) − µ W ( SA , t , p )
mSW R (T0 + t )
where the molar gas constant,
R = 8.314 472 J mol–1 K–1. See also Eqns. (2.14.1)
and (3.40.9) for an equivalent definition of φ .
IOC Manuals and Guides No. 56
170
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
L.2 Recommended symbols when variables are functions of η, θ and Θ
Note that whether using standard notation or variants from it, all variables should be
explicitly defined in publications when first used. Standard notation should be considered
as an additional aid to improve readability, not as a replacement for explicit definitions.
Note that oxygen should be reported in µμmol/kg and not cm3dm–3, ml/l or µμmol/l (this
reflects a desire for consistency with reporting of other quantities and will avoid problems
associated with conversion between moles and ml using the gas equations).
When thermodynamic variables are taken to be functions of variables other than the
standard combination ( SA , t, p ) it is convenient to indicate this by a marking on the
variable. This greatly simplifies the nomenclature for partial derivatives. Table L.2 lists
the suggested markings on the variables that arise commonly in this context.
The
thermodynamic variables are related to the thermodynamic potentials h = h SA ,η , p ,
h = h SA ,θ , p and h = hˆ ( SA , Θ, p ) by the expressions in appendix P.
(
(
)
Table L.2. Recommended symbols when variables are functions of η, θ , Θ and h
quantity
enthalpy, h
specific volume, v
function of
( SA , t , p )
( SA ,η, p )
density, ρ
Conservative Temperature, Θ
enthalpy, h
specific volume, v
( SA , θ , p )
density, ρ
entropy, η
enthalpy, h
specific volume, v
( SA , Θ, p )
density, ρ
entropy, η
IOC Manuals and Guides No. 56
(
)
(
)
(
)
( )
h = h ( SA ,θ , p )
v = v ( SA ,θ , p )
ρ = ρ ( SA ,θ , p )
η = η ( SA ,θ )
h = hˆ ( SA , Θ, p )
v = vˆ ( SA , Θ, p )
η = ηˆ ( SA , Θ)
entropy, η
specific volume, v
h = h S A ,η , p
v = v S A ,η , p
ρ = ρ S A ,η , p
Θ = Θ S A ,η
ρ = ρˆ ( SA , Θ, p )
density, ρ
Conservative Temperature, Θ
v = v ( SA , t , p )
η = η ( SA , t , p )
entropy, η
specific volume, v
h = h ( SA , t , p )
ρ = ρ ( SA , t , p )
density, ρ
enthalpy, h
symbol for this
functional form
( SA , h, p )
Θ = Θ SA ,h, p
v = v SA ,h, p
ρ = ρ SA ,h, p
η = η SA ,h, p
(
(
(
(
)
)
)
)
)
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
171
Appendix M:
Seawater-‐‑Ice-‐‑Air (SIA) library of computer software
This software library, the Seawater-‐‑Ice-‐‑Air library (the SIA library for short), contains the
TEOS-‐‑10 subroutines for evaluating a wide range of thermodynamic properties of pure
water (using IAPWS-‐‑95), seawater (using IAPWS-‐‑08 for the saline part), ice Ih (using
IAPWS-‐‑06) and for moist air (using Feistel et al. (2010a), IAPWS (2010)). It is divided into
six levels (levels 0 through 5) with each successive level building on the functional
capabilities introduced at lower levels. Briefly,
• level 0 defines fundamental constants, sets options used throughout the library and
provides routines to convert between Practical Salinity and Absolute Salinity
• level 1 defines a complete set of independent and consistent elements that are based
on previous work and form the essential building blocks for the rest of the library
routines
• level 2 provides access to a set of properties for individual mediums (liquid or vapour
water, ice, seawater and dry or humid air) that can be calculated from the level 0 and 1
routines without additional approximations
• level 3 introduces additional functions that require numerical solution of equations.
Most importantly, it is at this level that the density of pure fluid water is determined
from temperature and pressure information. This permits the definition of Gibbs
functions for pure water and seawater that make use of the IAPWS-‐‑95 Helmholtz
function as a fundamental building block
• level 4 deals with a fairly broad (but not exhaustive) selection of equilibrium
properties involving fluid water, seawater, ice and air; and
• level 5 includes a set of routines that build on the SIA routines but violate principals
adhered to throughout levels 0 though 4. In particular, non-‐‑basic SI units are
permitted at this level as discussed below.
As a general rule, the inputs and the outputs of the algorithms in the SIA library are in
basic SI units. Hence the salinity is Absolute Salinity S A in units of kg kg −1 (so that for
example standard ocean Reference Salinity is input to SIA functions as 0.035 165 04
( kg kg −1) rather than 35.165 04 ( g kg −1 ), in situ temperature is input as Absolute
Temperature T in K, and pressure is input as Absolute Pressure P in Pa. Use of these
basic SI units simplifies the calculation of theoretical expressions in thermodynamics. The
only exceptions to this rule for the units of the inputs and outputs in the SIA library are as
follows.
• The function SA = SA ( SP ,φ, λ, P) that calculates Absolute Salinity (in kg kg-‐‑1) when
given Practical Salinity S P (which is unitless and takes numbers like 35 not 0.035) as
its salinity input variable, along with location in the form of longitude λ (°E) latitude
φ (°N) and Absolute Pressure P (Pa). Location is required in this routine to account
for the influence of composition anomalies through a lookup table adopted from the
GSW Oceanographic Toolbox.
• The inverse function SP = SP ( SA , φ , λ , P ) . This and the previous routine are found at
level 0 since Absolute Salinity is required as an input to many of the higher level
library routines.
IOC Manuals and Guides No. 56
172
•
•
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
General purpose routines that allow for conversions between a broad range of
pressure, temperature and salinity units that are in common usage are provided at
level 5. The numerical input value and its unit are provided by the user and results
are returned in a specified output unit.
Algorithms are included at level 5 that use non-‐‑basic SI units as input and as output.
Most noteworthy is the GSW library module that uses the SIA routines to mimic many
of the routines in the GSW Oceanographic Toolbox. These routines use IAPWS-‐‑09 for
pure water in place of IAPWS-‐‑95 to provide improved computational efficiency. They
have been used to provide independent checks on the corresponding routines in the
GSW Oceanographic Toolbox.
Because the IAPWS-‐‑95 description of pure water substance (both liquid and vapour) is
the world-‐‑wide standard for pure water substance, the SIA library is the official
description of seawater, although it should be noted that the computer software does not
come with any warranty. Rather it is the underlying papers as listed in appendix C that
are the officially warranted descriptions.
The SIA library benefits from the full range of applicability of the IAPWS-‐‑95
Helmholtz function for pure water, 0 kg m-‐‑3 < ρ < 1200 kg m-‐‑3, 130 K < T < 1273 K, plus an
extension down to 50 K introduced by Feistel et al. (2010a). It does however have two
disadvantages as far as the field of oceanography is concerned. First, because IAPWS-‐‑95 is
valid over very wide ranges of temperature and pressure, it is necessarily an extensive
series of polynomials and exponentials which is not as fast computationally as the
equation of state EOS-‐‑80 with which oceanographers are familiar. Second, the IAPWS-‐‑95
thermodynamic potential is a Helmholtz function which expresses thermodynamic
properties in terms of density and temperature rather than pressure and temperature as
normally used in oceanography. Since IAPWS-‐‑95 describes not only liquid water but also
water vapour, this Helmholtz form of the thermodynamic potential is natural. Although
the library also includes a Gibbs function formulation with temperature and pressure as
independent variables, the core routines implement this formulation by first solving
P = ρ 2 f ρ (T , ρ ) for ρ and then using IAPWS-‐‑95, which is a computationally expensive
procedure.
In the GSW Oceanographic Toolbox (appendix N) we present an alternative
thermodynamic description of seawater properties based on the IAPWS-‐‑09 description of
the pure liquid water part as a Gibbs function. The GSW formulation is limited to the
Neptunian range (i. e. the oceanographic range) of temperature and pressure and deals
only with liquid water, but it is far more computationally efficient since the limited range
of validity allows equivalent accuracy with fewer terms and the Gibbs function
formulation avoids the need to invert the relation P = ρ 2 f ρ (T , ρ ) . This formulation is also
implemented at level 5 of the SIA library as a cross-‐‑check on the GSW routines and for the
convenience of SIA library users working on applications requiring increased
computational efficiency. Note however that some of routines in the SIA implementation
of the GSW routines are not as fully optimized as the corresponding routines in the GSW
Oceanographic Toolbox.
The presence of dissolved salts in seawater reduces the range of applicability of the
SIA and GSW seawater routines in comparison with the IAPWS-‐‑95 range of applicability
for pure fluid water, whether or not the IAPWS-‐‑09 Gibbs formulation is used to represent
pure water properties. This is because the range of applicability of the saline component
of the Gibbs function is limited to 0 kg kg-‐‑1 ≤ S A ≤ 0.12 kg kg-‐‑1, 262 K ≤ T ≤ 353 K, and
100 Pa ≤ P ≤ 108 Pa.
Since this manual focuses on seawater, we refer the reader to Feistel et al. (2010b) and
Wright et al. (2010) for details on the ice and air components of the SIA library. However,
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
173
below, we discuss a few features of the library that relate to these additional components.
First, we note that the thermodynamic potentials of pure water, ice, the saline part of the
seawater Gibbs function and the Gibbs function of moist air have been carefully adjusted
to make them fully compatible with each other (Feistel et al. (2008a)). Only by so doing
can the equilibrium properties of coincident phases be accurately evaluated (for example,
the freezing temperature of pure water and of seawater). Many functions involving
equilibrium properties of water, vapour, ice, seawater and dry or humid air are
implemented in level 4 of the SIA library. To provide an indication of the range of
functions available, we have listed the routine names in Table M.1 below. This table
comes from Table 3.1 of Wright et al. (2010); we refer the interested reader to Feistel et al.
(2010b)) and Wright et al. (2010)) for detailed information. Wright et al. (2010) provide a
supplementary table that is cross-‐‑referenced to their Table 3.1 to give details on the usage
of each routine and each table in their supplement references in turn the relevant sections
of Feistel et al. (2010b) for additional background information.
Because each level of the SIA library builds on lower levels and since multiple phases
may be involved in the equilibrium calculations, the determination of the ranges of
validity of the routines in the SIA library can become rather involved. To deal with this
issue, a procedure has been implemented in the library to return an error code for function
evaluations that depend on results from any of the basic building block routines from
outside of their individual ranges of validity. Numerical check values are provided with
each of the routines in the library and auxiliary routines are provided to assist users in the
validation of local implementations.
Further details of the SIA software library are provided in the papers Feistel et al.
(2010b)) and Wright et al. (2010)) and the software is served from the www.TEOS-10.org
web site.
IOC Manuals and Guides No. 56
174
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Table M.1. The SIA library contents. Module names are in bold and user-‐‑accessible
routines are in plain type. Each of the Public Routines can be accessed by
users. The underlined routines are thermodynamic potential functions
including first and second derivatives. The bracketed numbers preceding
most module names give the related table numbers in the supplement to
Wright et al. (2010) where detailed information on the use of each function is
provided.
Level 0 routines
Constants_0
Constants_0 (Cont'd)
Maths_0
(S2) Convert_0
Public Parameter Values
Parameter Values (cont'd)
Uses
Uses
celsius_temperature_si
check_limits
cp_chempot_si
cp_density_si
cp_pressure_si
cp_temperature_si
dry_air_dmax
dry_air_dmin
dry_air_tmax
dry_air_tmin
errorreturn
flu_dmax
flu_dmin
flu_tmax
flu_tmin
gas_constant_air_si
gas_constant_air_L2000
gas_constant_molar_si
gas_constant_molar_L2000
gas_constant_h2O_si
gas_constant_h2O_iapws95
ice_pmax
ice_pmin
ice_tmax
ice_tmin
isextension2010
isok
mix_air_dmax
mix_air_dmin
mix_air_tmax
mix_air_tmin
molar_mass_air_si
molar_mass_air_l2000
molar_mass_h2o_si
molar_mass_seasalt_si
pi
sal_pmax
sal_pmin
sal_smax
sal_smin
sal_tmax
sal_tmin
sealevel_pressure_si
so_salinity_si
so_temperature_si
so_pressure_si
tp_density_ice_iapws95_si
tp_density_liq_iapws95_si
tp_density_vap_iapws95_si
tp_enthalpy_ice_si
tp_enthalpy_vap_si
tp_pressure_exp_si
tp_pressure_iapws95_si
tp_temperature_si
constants_0
constants_0
Public Routines
Public Routines
get_cubicroots
matrix_solve
air_massfraction_air_si
air_massfraction_vap_si
air_molar_mass_si
air_molfraction_air_si
air_molfraction_vap_si
asal_from_psal
psal_from_asal
Level 1 routines
(S3) Flu_1 (IAPWS95)
(S4) Ice_1 (IAPWS06)
(S5) Sal_1 (IAPWS08)
(S6) Air_1
Uses
Uses
Uses
Uses
constants_0
constants_0
constants_0
constants_0
Public Routines
Public Routines
Public Routines
Public Routines
chk_iapws95_table6
chk_iapws95_table7
flu_f_si
chk_iapws06_table6
ice_g_si
sal_g_term_si
air_baw_m3mol
air_caaw_m6mol2
air_caww_m6mol2
dry_f_si
dry_init_clear
dry_init_Lemmon2000
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
175
Level 2 routines
(S7) Flu_2
((S8) Ice_2
(S9) Sal_2
(S10) Air_2
Uses
Uses
Uses
Uses
constants_0, flu_1
constants_0, ice_1
constants_0, sal_1
constants_0, flu_1, air_1
Public Routines
Public Routines
Public Routines
Public Routines
flu_cp_si
flu_cv_si
flu_enthalpy_si
flu_entropy_si
flu_expansion_si
flu_gibbs_energy_si
flu_internal_energy_si
flu_kappa_s_si
flu_kappa_t_si
flu_lapserate_si
flu_pressure_si
flu_soundspeed_si
ice_chempot_si
ice_cp_si
ice_density_si
ice_enthalpy_si
ice_entropy_si
ice_expansion_si
ice_helmholtz_energy_si
ice_internal_energy_si
ice_kappa_s_si
ice_kappa_t_si
ice_lapserate_si
ice_p_coefficient_si
ice_specific_volume_si
sal_act_coeff_si
sal_act_potential_si
sal_activity_w_si
sal_chem_coeff_si
sal_chempot_h2o_si
sal_chempot_rel_si
sal_dilution_si
sal_g_si
sal_mixenthalpy_si
sal_mixentropy_si
sal_mixvolume_si
sal_molality_si
sal_osm_coeff_si
sal_saltenthalpy_si
sal_saltentropy_si
sal_saltvolume_si
air_f_si
air_f_cp_si
air_f_cv_si
air_f_enthalpy_si
air_f_entropy_si
air_f_expansion_si
air_f_gibbs_energy_si
air_f_internal_energy_si
air_f_kappa_s_si
air_f_kappa_t_si
air_f_lapserate_si
air_f_mix_si
air_f_pressure_si
air_f_soundspeed_si
chk_iapws10_table
(S11) Flu_3a
(S12) Sea_3a
(S13) Air_3a
Uses
Uses
Uses
constants_0, convert_0,
maths_0, flu_1
constants_0, sal_1, sal_2,
flu_3a (convert_0, maths_0,
flu_1)
constants_0, convert_0,
maths_0, air_1, air_2 (flu_1)
Public Routines
air_density_si
air_g_si
get_it_ctrl_airdensity
set_it_ctrl_airdensity
Level 3 routines
Public Routines
get_it_ctrl_density
liq_density_si
liq_g_si
set_it_ctrl_density
vap_density_si
vap_g_si
chk_iapws08_table8a
chk_iapws08_table8b
chk_iapws08_table8c
sea_chempot_h2o_si
sea_chempot_rel_si
sea_cp_si
sea_density_si
sea_enthalpy_si
sea_entropy_si
sea_g_si
sea_g_contraction_t_si
sea_g_expansion_si
sea_gibbs_energy_si
sea_internal_energy_si
sea_kappa_s_si
sea_kappa_t_si
sea_lapserate_si
sea_osm_coeff_si
sea_soundspeed_si
sea_temp_maxdensity_si
Public Routines
IOC Manuals and Guides No. 56
176
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
(S14) Flu_3b
(S15) Sea_3b
(S16) Air_3b
Uses
Uses
Uses
constants_0, flu_2, flu_3a
(convert_0, maths_0,
flu_1)
constants_0, sal_2, flu_3a,
sea_3a (convert_0, maths_0,
flu_1, sal_1)
constants_0, convert_0,
air_1, air_2, air_3a (maths_0,
flu_1)
Public Routines
Public Routines
Public Routines
liq_cp_si
liq_cv_si
liq_enthalpy_si
liq_entropy_si
liq_expansion_si
liq_gibbs_energy_si
liq_internal_energy_si
liq_kappa_s_si
liq_kappa_t_si
liq_lapserate_si
liq_soundspeed_si
vap_cp_si
vap_cv_si
vap_enthalpy_si
vap_entropy_si
vap_expansion_si
vap_gibbs_energy_si
vap_internal_energy_si
vap_kappa_s_si
vap_kappa_t_si
vap_lapserate_si
vap_soundspeed_si
sea_h_si
sea_h_contraction_h_si
sea_h_contraction_t_si
sea_h_contraction_theta_si
sea_h_expansion_h_si
sea_h_expansion_t_si
sea_h_expansion_theta_si
sea_potdensity_si
sea_potenthalpy_si
sea_pottemp_si
sea_temperature_si
set_it_ctrl_pottemp
air_g_chempot_vap_si
air_g_compressibility
factor_si
air_g_contraction_si
air_g_cp_si
air_g_cv_si
air_g_density_si
air_g_enthalpy_si
air_g_entropy_si
air_g_expansion_si
air_g_gibbs_energy_si
air_g_internal_energy_si
air_g_kappa_s_si
air_g_kappa_t_si
air_g_lapserate_si
air_g_soundspeed_si
chk_lemmon_etal_2000
(S17) Sea_3c
(S18) Air_3c
Uses
Uses
constants_0, sea_3a, sea_3b
(convert_0, maths_0, flu_1,
sal_1, sal_2, flu_3a)
constants_0, convert_0,
air_2, air_3a, air_3b
(maths_0, air_1, flu_1)
Public Routines
sea_eta_contraction_h_si
sea_eta_contraction_t_si
sea_eta_contraction_theta_si
sea_eta_density_si
sea_eta_entropy_si
sea_eta_expansion_h_si
sea_eta_expansion_t_si
sea_eta_expansion_theta_si
sea_eta_potdensity_si
sea_eta_pottemp_si
sea_eta_temperature_si
set_it_ctrl_entropy_si
(S19) Sea_3d
Uses
constants_0, sal_2, flu_3a
(convert_0, maths_0, flu_1,
sal_1)
Public Routines
sea_sa_si
set_it_ctrl_salinity
IOC Manuals and Guides No. 56
Public Routines
air_h_si
air_potdensity_si
air_potenthalpy_si
air_pottemp_si
air_temperature_si
set_it_ctrl_air_pottemp
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Level 4 routines
(S20) Liq_Vap_4
(S21) Ice_Vap_4
(S22) Sea_Vap_4
Uses
Uses
Uses
constants_0, maths_0, flu_1,
flu_2, flu_3a (Convert_0)
constants_0, maths_0, flu_1,
flu_2, ice_1, ice_2
Public Routines
Public Routines
constants_0, maths_0, flu_1,
sal_1, sal_2, flu_3a, sea_3a,
flu_3b (convert_0, flu_2)
chk_iapws95_table8
liq_vap_boilingtemperature_si
liq_vap_chempot_si
liq_vap_density_liq_si
liq_vap_density_vap_si
liq_vap_enthalpy_evap_si
liq_vap_enthalpy_liq_si
liq_vap_enthalpy_vap_si
liq_vap_entropy_evap_si
liq_vap_entropy_liq_si
liq_vap_entropy_vap_si
liq_vap_pressure_liq_si
liq_vap_pressure_vap_si
liq_vap_temperature_si
liq_vap_vapourpressure_si
liq_vap_volume_evap_si
set_liq_vap_eq_at_p
set_liq_vap_eq_at_t
set_it_ctrl_liq_vap
ice_vap_chempot_si
ice_vap_density_ice_si
ice_vap_density_vap_si
ice_vap_enthalpy_ice_si
ice_vap_enthalpy_subl_si
ice_vap_enthalpy_vap_si
ice_vap_entropy_ice_si
ice_vap_entropy_subl_si
ice_vap_entropy_vap_si
ice_vap_pressure_vap_si
ice_vap_sublimationpressure_si
ice_vap_sublimationtemp_si
ice_vap_temperature_si
ice_vap_volume_subl_si
set_ice_vap_eq_at_p
set_ice_vap_eq_at_t
set_it_ctrl_ice_vap
Public Routines
(S23) Ice_Liq_4
(S24) Sea_Liq_4
Uses
Uses
constants_0, maths_0, flu_1,
ice_1, flu_2, ice_2
constants_0, flu_1, sal_1, flu_2,
sal_2, flu_3a (convert_0,
maths_0)
Public Routines
ice_liq_chempot_si
ice_liq_density_ice_si
ice_liq_density_liq_si
ice_liq_enthalpy_ice_si
ice_liq_enthalpy_liq_si
ice_liq_enthalpy_melt_si
ice_liq_entropy_ice_si
ice_liq_entropy_liq_si
ice_liq_entropy_melt_si
ice_liq_meltingpressure_si
ice_liq_meltingtemperature_si
ice_liq_pressure_liq_si
ice_liq_temperature_si
ice_liq_volume_melt_si
set_ice_liq_eq_at_p
set_ice_liq_eq_at_t
set_it_ctrl_ice_liq
sea_vap_boilingtemperature_si
sea_vap_brinefraction_seavap_si
sea_vap_brinesalinity_si
sea_vap_cp_seavap_si
sea_vap_density_sea_si
sea_vap_density_seavap_si
sea_vap_density_vap_si
sea_vap_enthalpy_evap_si
sea_vap_enthalpy_sea_si
sea_vap_enthalpy_seavap_si
sea_vap_enthalpy_vap_si
sea_vap_entropy_sea_si
sea_vap_entropy_seavap_si
sea_vap_entropy_vap_si
sea_vap_expansion_seavap_si
sea_vap_g_si
sea_vap_kappa_t_seavap_si
sea_vap_pressure_si
sea_vap_salinity_si
sea_vap_temperature_si
sea_vap_vapourpressure_si
sea_vap_volume_evap_si
set_it_ctrl_sea_vap
set_sea_vap_eq_at_s_p
set_sea_vap_eq_at_s_t
set_sea_vap_eq_at_t_p
Public Routines
sea_liq_osmoticpressure_si
set_sea_liq_eq_at_s_t_p
set_it_ctrl_sea_liq
IOC Manuals and Guides No. 56
177
178
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
(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
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
(S27) Liq_Ice_Air_4
(S28) Sea_Ice_Vap_4
Uses
Uses
constants_0, convert_0,
maths_0, flu_1, ice_1, air_1,
flu_2, ice_2, air_2, air_3b,
ice_liq_4 (air_3a)
constants_0, maths_0, flu_1,
ice_1, sal_1, sal_2
Public Routines
sea_ice_vap_density_vap_si
sea_ice_vap_pressure_si
sea_ice_vap_salinity_si
sea_ice_vap_temperature_si
set_it_ctrl_sea_ice_vap
set_sea_ice_vap_eq_at_p
set_sea_ice_vap_eq_at_s
set_sea_ice_vap_eq_at_t
Public Routines
liq_ice_air_airfraction_si
liq_ice_air_density_si
liq_ice_air_dryairfraction_si
liq_ice_air_enthalpy_si
liq_ice_air_entropy_si
liq_ice_air_ifl_si
liq_ice_air_iml_si
liq_ice_air_liquidfraction_si
liq_ice_air_pressure_si
liq_ice_air_solidfraction_si
liq_ice_air_temperature_si
liq_ice_air_vapourfraction_si
set_liq_ice_air_eq_at_a
set_liq_ice_air_eq_at_p
set_liq_ice_air_eq_at_t
set_liq_ice_air_eq_at
_wa_eta_wt
set_liq_ice_air_eq_at
_wa_wl_wi
set_it_ctrl_liq_ice_air
(S29) Liq_Air_4a
(S30) Ice_Air_4a
Uses
Uses
constants_0, convert_0,
maths_0, flu_1, air_1, flu_2,
air_2, flu_3a, air_3a, air_3b,
liq_vap_4
constants_0, convert_0,
maths_0, air_1, ice_1, ice_2,
air_2, air_3a, air_3b, ice_vap_4
(flu_1, flu_2)
Public Routines
Public Routines
liq_air_a_from_rh_cct_si
liq_air_a_from_rh_wmo_si
liq_air_condensationpressure_si
liq_air_density_air_si
liq_air_density_liq_si
liq_air_density_vap_si
liq_air_dewpoint_si
liq_air_enthalpy_evap_si
liq_air_entropy_air_si
liq_air_icl_si
liq_air_ict_si
liq_air_massfraction_air_si
liq_air_pressure_si
liq_air_rh_cct_from_a_si
liq_air_rh_wmo_from_a_si
liq_air_temperature_si
set_it_ctrl_liq_air
set_liq_air_eq_at_a_eta
set_liq_air_eq_at_a_p
set_liq_air_eq_at_a_t
set_liq_air_eq_at_t_p
ice_air_a_from_rh_cct_si
ice_air_a_from_rh_wmo_si
ice_air_condensationpressure_si
ice_air_density_air_si
ice_air_density_ice_si
ice_air_density_vap_si
ice_air_enthalpy_subl_si
ice_air_frostpoint_si
ice_air_icl_si
ice_air_ict_si
ice_air_massfraction_air_si
ice_air_pressure_si
ice_air_rh_cct_from_a_si
ice_air_rh_wmo_from_a_si
ice_air_sublimationpressure_si
ice_air_temperature_si
set_ice_air_eq_at_a_eta
set_ice_air_eq_at_a_p
set_ice_air_eq_at_a_t
set_ice_air_eq_at_t_p
set_it_ctrl_ice_air
IOC Manuals and Guides No. 56
179
180
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
(S31) Liq_Air_4b
(S32) Ice_Air_4b
Uses
Uses
constants_0, flu_3a, air_3a,
liq_air_4a (convert_0, maths_0,
flu_1, air_1, flu_2, air_2, air_3b,
liq_vap_4)
constants_0, convert_0, ice_1,
air_3a, ice_air_4a (maths_0,
flu_1, air_1, flu_2, ice_2, air_2,
air_3b, ice_vap_4)
Public Routines
Public Routines
liq_air_g_si
liq_air_g_cp_si
liq_air_g_density_si
liq_air_g_enthalpy_si
liq_air_g_entropy_si
liq_air_g_expansion_si
liq_air_g_kappa_t_si
liq_air_g_lapserate_si
liq_air_liquidfraction_si
liq_air_vapourfraction_si
ice_air_g_si
ice_air_g_cp_si
ice_air_g_density_si
ice_air_g_enthalpy_si
ice_air_g_entropy_si
ice_air_g_expansion_si
ice_air_g_kappa_t_si
ice_air_g_lapserate_si
ice_air_solidfraction_si
ice_air_vapourfraction_si
(S33) Liq_Air_4c
(S34) Ice_Air_4c
Uses
Uses
constants_0, air_3a, ice_liq_4,
liq_air_4a, liq_air_4b
(convert_0, maths_0, flu_1,
ice_1, air_1, flu_2, ice_2 air_2,
flu_3a, air_3b, liq_vap_4)
constants_0, convert_0,
ice_liq_4, ice_air_4b (maths_0,
flu_1, ice_1, air_1, flu_2, ice_2,
air_2, air_3a, air_3b, ice_air_4a,
ice_vap_4)
Public Routines
Public Routines
liq_air_h_si
liq_air_h_cp_si
liq_air_h_density_si
liq_air_h_kappa_s_si
liq_air_h_lapserate_si
liq_air_h_temperature_si
liq_air_potdensity_si
liq_air_potenthalpy_si
liq_air_pottemp_si
set_it_ctrl_liq_air_pottemp
ice_air_h_si
ice_air_h_cp_si
ice_air_h_density_si
ice_air_h_kappa_s_si
ice_air_h_lapserate_si
ice_air_h_temperature_si
ice_air_potdensity_si
ice_air_potenthalpy_si
ice_air_pottemp_si
set_it_ctrl_ice_air_pottemp
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
181
Level 5 routines
(S35) Flu_IF97_5
(S36) Ice_Flu_5
(S37) Sea_5a
(S38) Air_5
Uses
Uses
Uses
Uses
constants_0
constants_0
Public Routines
Public Routines
chk_iapws97_table
fit_liq_density_if97_si
fit_liq_g_if97_si
fit_vap_density_if97_si
fit_vap_g_if97_si
fit_ice_liq_pressure_si
fit_ice_liq_temperature_si
fit_ice_vap_pressure_si
constants_0, sea_3a,
sea_3b, sea_3c (convert_0,
maths_0, flu_1, sal_1, sal_2,
flu_3a)
constants_0,
air_3b, liq_air_4a
(convert_0,
maths_0, flu_1,
flu_2, flu_3a, air_1,
air_2, air_3a,
liq_vap_4)
Public Routines
sea_alpha_ct_si
sea_alpha_pt0_si
sea_alpha_t_si
sea_beta_ct_si
sea_beta_pt0_si
sea_beta_t_si
sea_cabb_ct_si
sea_cabb_pt0_si
sea_ctmp_from_ptmp0_si
sea_ptmp0_from_ctmp_si
sea_thrmb_ct_si
sea_thrmb_pt0_si
Public Routines
air_lapserate_moist
_c100m
(S39) Liq_F03_5
(S40) OS2008_5
(S41) GSW_Library_5
(S42) Convert_5
Uses
Uses
Uses
Uses
constants_0
flu_1, flu_2,
flu_3a, ice_1, liq_vap_4,
sal_1, sal_2 (constants_0,
convert_0, maths_0)
constants_0, maths_0,
liq_f03_5, flu_1, flu_3a,
sal_1, sal_2, sea_3a,
sea_3b, sea_5a (convert_0)
constants_0,
convert_0
Public Routines
chk_iapws09_table6
fit_liq_cp_f03_si
fit_liq_density_f03_si
fit_liq_expansion_f03_si
fit_liq_g_f03_si
fit_liq_kappa_t_f03_si
fit_liq_soundspeed_f03_si
Public Routines
chk_os2008_table
Public Routines
gsw_alpha_ct
gsw_alpha_pt0
gsw_alpha_t
gsw_asal_from_psal
gsw_beta_ct
gsw_beta_pt0
gsw_beta_t
gsw_cabb_ct
gsw_cabb_pt0
gsw_cp
gsw_ctmp_from_ptmp0
gsw_dens
gsw_enthalpy
gsw_entropy
gsw_g
gsw_kappa
gsw_kappa_t
gsw_pden
gsw_psal_from_asal
gsw_ptmp
gsw_ptmp0_from_ctmp
gsw_specvol
gsw_svel
gsw_thrmb_ct
gsw_thrmb_pt0
Public Routines
cnv_pressure
cnv_salinity
cnv_temperature
IOC Manuals and Guides No. 56
182
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Appendix N:
Gibbs-‐‑SeaWater (GSW) Oceanographic Toolbox
This Gibbs-‐‑SeaWater (GSW) Oceanographic Toolbox (the “GSW Toolbox” for short),
contains the TEOS-‐‑10 subroutines for evaluating the thermodynamic properties of pure
water (using IAPWS-‐‑09) and seawater (using IAPWS-‐‑08 for the saline part). The GSW
Oceanographic Toolbox does not provide properties of ice or of moist air (these properties
can be found in the SIA library). This GSW Oceanographic Toolbox does not adhere to
strict basic-‐‑SI units but rather oceanographic units are adopted. While it is comfortable for
oceanographers to adopt these familiar non-‐‑basic SI units, doing so comes at a price, since
many of the thermodynamic expressions demand that variables be expressed in basic-‐‑SI
units. The simplest example is the pure water fraction (the so-‐‑called “freshwater
fraction”) which is (1 − SA ) only when Absolute Salinity S A is in basic-‐‑SI units. The price
that one pays for adopting comfortable units is that one must be vigilant when evaluating
thermodynamic expressions; there are traps for the unwary, particularly concerning the
units of Absolute Salinity and of pressure.
This GSW Oceanographic Toolbox has inputs in “oceanographic” units, namely
Absolute Salinity S A in g kg −1 (so that for example, Standard Ocean Reference Salinity
SSO is 35.165 04 g kg −1 [not 0.035 165 04 kg kg −1]), in situ temperature t in °C and pressure
as sea pressure p in dbar.
The GSW Oceanographic Toolbox is designed as a successor to the Seawater library of
MATLAB routines which has been widely used by oceanographers in the past fifteen years;
see http://www.cmar.csiro.au/datacentre/ext_docs/seawater.htm. Many of the non-‐‑
thermodynamic subroutines of the Seawater library have been retained or updated in the
GSW Toolbox (for example, a function to calculate the square of the buoyancy frequency,
and functions to calculate a selection of different geostrophic streamfunctions).
The thermodynamic variables density and enthalpy, and several thermodynamic
variables derived from density and enthalpy, are available in the GSW Toolbox in two
forms. One form uses the full TEOS-‐‑10 Gibbs function (being the sum of IAPWS-‐‑09 and
IAPWS-‐‑08) while the other form is based on a 75-‐‑term computationally efficient
expression for specific volume as a function of Absolute Salinity, Conservative
Temperature and pressure (see appendix A.30 and appendix K). Both forms give values
of density and the thermal expansion coefficient within the accuracy of laboratory-‐‑
determined values for these quantities, so that for oceanographic purposes the two forms
can be regarded as equally accurate. Certainly, the present uncertainty in accounting for
the spatial variations in seawater composition has a larger impact on density etc. than the
small difference incurred by using the computationally efficient 75-‐‑term version for
specific volume.
Version 1 of the GSW Toolbox was released in January 2009, version 2.0 in October
2010 and version 3.0 in May 2011. The GSW Toolbox is available in MATLAB, FORTRAN
and C from the web site at www.TEOS-10.org. A quick introduction to TEOS-‐‑10 is
available on the TEOS-‐‑10 web site as the short document called “Getting started with
TEOS-‐‑10 and the GSW Oceanographic Toolbox”. The next four pages list all the functions
in version 3.05 of the GSW Oceanographic Toolbox and this is followed by Table N.1
which describes some of the GSW functions in more detail.
IOC Manuals and Guides No. 56
Practical Salinity from conductivity, C (incl. for SP < 2)
conductivity, C, from Practical Salinity (incl. for SP < 2)
Practical Salinity from conductivity ratio, R (incl. for SP < 2)
conductivity ratio, R, from Practical Salinity (incl. for SP < 2)
Practical Salinity from a laboratory salinometer (incl. for SP < 2)
Practical Salinity from Knudsen Salinity
Absolute Salinity from Practical Salinity
Preformed Salinity from Practical Salinity
Conservative Temperature from in-situ temperature
gsw_deltaSA_from_SP
gsw_SA_Sstar_from_SP
gsw_SR_from_SP
gsw_SP_from_SR
gsw_SP_from_SA
gsw_Sstar_from_SA
gsw_SA_from_Sstar
gsw_SP_from_Sstar
gsw_pt_from_CT
gsw_t_from_CT
gsw_CT_from_pt
gsw_pot_enthalpy_from_pt
gsw_pt_from_t
gsw_pt0_from_t
gsw_t_from_pt0
gsw_t90_from_t48
gsw_t90_from_t68
gsw_z_from_p
gsw_p_from_z
gsw_z_from_depth
gsw_depth_from_z
gsw_Abs_Pressure_from_p
gsw_p_from_Abs_Pressure
gsw_entropy_from_CT
gsw_CT_from_entropy
gsw_entropy_from_pt
gsw_pt_from_entropy
gsw_entropy_from_t
gsw_t_from_entropy
gsw_adiabatic_lapse_rate_from_CT
gsw_adiabatic_lapse_rate_from_t
gsw_molality_from_SA
gsw_ionic_strength_from_SA
Absolute Salinity Anomaly from Practical Salinity
Absolute Salinity & Preformed Salinity from Practical Salinity
Reference Salinity from Practical Salinity
Practical Salinity from Reference Salinity
Practical Salinity from Absolute Salinity
Preformed Salinity from Absolute Salinity
Absolute Salinity from Preformed Salinity
Practical Salinity from Preformed Salinity
potential temperature from Conservative Temperature
in-situ temperature from Conservative Temperature
Conservative Temperature from potential temperature
potential enthalpy from potential temperature
potential temperature
potential temperature with reference pressure of 0 dbar
in-situ temperature from potential temperature with p_ref of 0 dbar
ITS-90 temperature from IPTS-48 temperature
ITS-90 temperature from IPTS-68 temperature
height from pressure
pressure from height
height from depth
depth from height
Absolute Pressure, P, from sea pressure, p
sea pressure, p, from Absolute Pressure, P
entropy from Conservative Temperature
Conservative Temperature from entropy
entropy from potential temperature
potential temperature from entropy
entropy from in-situ temperature
in-situ temperature from entropy
adiabatic lapse rate from Conservative Temperature
adiabatic lapse rate from in-situ temperature
molality of seawater
ionic strength of seawater
other conversions between temperatures, salinities, entropy, pressure and height
gsw_SA_CT_plot
function to plot Absolute Salinity – Conservative Temperature
profiles on the SA-CT diagram, including the freezing line
and selected potential density contours
Absolute Salinity – Conservative Temperature plotting function
gsw_SA_from_SP
gsw_Sstar_from_SP
gsw_CT_from_t
Absolute Salinity (SA), Preformed Salinity (Sstar) and Conservative Temperature (CT)
gsw_SP_from_C
gsw_C_from_SP
gsw_SP_from_R
gsw_R_from_SP
gsw_SP_salinometer
gsw_SP_from_SK
Practical Salinity (SP), PSS-78
gsw_specvol
gsw_alpha
gsw_beta
gsw_alpha_on_beta
gsw_specvol_alpha_beta
gsw_specvol_first_derivatives
gsw_specvol_second_derivatives
gsw_specvol_first_derivatives_wrt_enthalpy
gsw_specvol_second_derivatives_wrt_enthalpy
gsw_specvol_anom
gsw_specvol_anom_standard
gsw_rho
gsw_rho_alpha_beta
gsw_rho_first_derivatives
gsw_rho_second_derivatives
gsw_rho_first_derivatives_wrt_enthalpy
gsw_rho_second_derivatives_wrt_enthalpy
gsw_sigma0
gsw_sigma1
gsw_sigma2
gsw_sigma3
gsw_sigma4
gsw_cabbeling
gsw_thermobaric
gsw_enthalpy
gsw_enthalpy_diff
gsw_dynamic_enthalpy
gsw_enthalpy_first_derivatives
gsw_enthalpy_second_derivatives
gsw_sound_speed
gsw_kappa
gsw_internal_energy
gsw_internal_energy_first_derivatives
gsw_internal_energy_second_derivatives
gsw_CT_from_enthalpy
gsw_SA_from_rho
gsw_CT_from_rho
gsw_CT_maxdensity
GSW version 3.05.5
specific volume
thermal expansion coefficient with respect to CT
saline contraction coefficient at constant CT
alpha divided by beta
specific volume, thermal expansion and saline contraction coefficients
first derivatives of specific volume
second derivatives of specific volume
first derivatives of specific volume with respect to enthalpy
second derivatives of specific volume with respect to enthalpy
specific volume anomaly
specific volume anomaly realtive to SSO & 0°C
in-situ density and potential density
in-situ density, thermal expansion and saline contraction coefficients
first derivatives of density
second derivatives of density
first derivatives of density with respect to enthalpy
second derivatives of density with respect to enthalpy
sigma0 with reference pressure of 0 dbar
sigma1 with reference pressure of 1000 dbar
sigma2 with reference pressure of 2000 dbar
sigma3 with reference pressure of 3000 dbar
sigma4 with reference pressure of 4000 dbar
cabbeling coefficient
thermobaric coefficient
enthalpy
difference of enthalpy between two pressures
dynamic enthalpy
first derivatives of enthalpy
second derivatives of enthalpy
sound speed
isentropic compressibility
internal energy
first derivatives of internal energy
second derivatives of internal energy
Conservative Temperature from enthalpy
Absolute Salinity from density
Conservative Temperature from density
Conservative Temperature of maximum density of seawater
specific volume, density and enthalpy
Gibbs SeaWater (GSW) Oceanographic Toolbox of TEOS –10
Page 1.
gsw_CT_freezing
gsw_CT_freezing_poly
gsw_t_freezing
gsw_t_freezing_poly
gsw_pot_enthalpy_ice_freezing
gsw_pot_enthalpy_ice_freezing_poly
gsw_SA_freezing_from_CT
gsw_SA_freezing_from_CT_poly
gsw_SA_freezing_from_t
gsw_SA_freezing_from_t_poly
gsw_pressure_freezing_CT
gsw_CT_freezing_first_derivatives
gsw_CT_freezing_first_derivatives_poly
gsw_t_freezing_first_derivatives
gsw_t_freezing_first_derivatives_poly
gsw_pot_enthalpy_ice_freezing_first_derivatives
gsw_pot_enthalpy_ice_freezing_first_derivatives_poly
gsw_latentheat_melting
Conservative Temperature freezing temp of seawater
Conservative Temperature freezing temp of seawater (poly)
in-situ freezing temperature of seawater
in-situ freezing temperature of seawater (poly)
potential enthalpy of ice at which seawater freezes
potential enthalpy of ice at which seawater freezes (poly)
SA of seawater at the freezing temp (for given CT)
SA of seawater at the freezing temp (for given CT) (poly)
SA of seawater at the freezing temp (for given t)
SA of seawater at the freezing temp (for given t) (poly)
pressure of seawater at the freezing temp (for given CT)
first derivatives of CT freezing temp of seawater
first derivatives of CT freezing temp of seawater (poly)
first derivatives of in-situ freezing temp of seawater
first derivatives of in-situ freezing temp of seawater (poly)
first derivatives of potential enthalpy of ice at freezing
first derivatives of potential enthalpy of ice at freezing (poly)
latent heat of melting of ice into seawater
seawater and ice properties at freezing temperatures
gsw_latentheat_evap_CT
latent heat of evaporation of water from seawater (isobaric
evaporation enthalpy) with CT as input temperature
gsw_latentheat_evap_t
latent heat of evaporation of water from seawater (isobaric
evaporation enthalpy) with in-situ temperature, t, as input
isobaric evaporation enthalpy
gsw_geo_strf_dyn_height
dynamic height anomaly
gsw_geo_strf_dyn_height_pc
dynamic height anomaly for piecewise constant profiles
gsw_geo_strf_isopycnal
approximate isopycnal geostrophic streamfunction
gsw_geo_strf_isopycnal_pc
approximate isopycnal geostrophic streamfunction for
piecewise constant profiles
gsw_geo_strf_Cunningham
Cunningham geostrophic streamfunction
gsw_geo_strf_Montgomery
Montgomery geostrophic streamfunction
gsw_geo_strf_steric_height
dynamic height anomaly divided by 9.7963 m s-2
gsw_geo_strf_PISH
pressure integrated steric height
gsw_travel_time
acoustic travel time
gsw_geostrophic_velocity
geostrophic velocity
geostrophic streamfunctions, acoustic travel time and geostrophic velocity
gsw_Turner_Rsubrho
Turner angle & Rsubrho
gsw_Nsquared
buoyancy (Brunt-Väisäla) frequency squared (N2)
gsw_Nsquared_min
minimum buoyancy frequency squared (N2)
gsw_stabilise_SA_const_t
minimally adjust SA to produce a stable water column,
keeping in-situ temperature constant
gsw_stabilise_SA_CT
minimally adjusts SA & CT to produce a stable water column
gsw_mlp
mixed-layer pressure
gsw_Nsquared_lowerlimit
specified profile of minimum buoyancy frequency squared
gsw_IPV_vs_fNsquared_ratio
ratio of isopycnal potential vorticity to f times N2
vertical stability
gsw_specvol_ice
gsw_alpha_wrt_t_ice
gsw_rho_ice
gsw_pressure_coefficient_ice
gsw_sound_speed_ice
gsw_kappa_ice
gsw_kappa_const_t_ice
gsw_internal_energy_ice
gsw_enthalpy_ice
gsw_entropy_ice
gsw_cp_ice
gsw_chem_potential_water_ice
gsw_Helmholtz_energy_ice
gsw_adiabatic_lapse_rate_ice
gsw_pt0_from_t_ice
gsw_pt_from_t_ice
gsw_t_from_pt0_ice
gsw_t_from_rho_ice
gsw_pot_enthalpy_from_pt_ice
gsw_pt_from_pot_enthalpy_ice
gsw_pot_enthalpy_from_pt_ice_poly
gsw_pt_from_pot_enthalpy_ice_poly
gsw_pot_enthalpy_from_specvol_ice
gsw_specvol_from_pot_enthalpy_ice
gsw_pot_enthalpy_from_specvol_ice_poly
gsw_specvol_from_pot_enthalpy_ice_poly
thermodynamic properties of ice Ih
gsw_melting_seaice_SA_CT_ratio
gsw_melting_seaice_SA_CT_ratio_poly
gsw_melting_seaice_equilibrium_SA_CT_ratio
gsw_melting_seaice_equilibrium_SA_CT_ratio_poly
gsw_seaice_fraction_to_freeze_seawater
gsw_melting_seaice_into_seawater
GSW version 3.05.5
SA to CT ratio when ice melts into seawater
SA to CT ratio when ice melts into seawater (poly)
SA to CT ratio when ice melts, near equilibrium
SA to CT ratio when ice melts, near equilibrium (poly)
ice mass fraction to freeze seawater
SA and CT when ice melts in seawater
ratios of SA, CT and P changes during frazil ice formation
ratios of SA, CT and P changes during frazil ice formation (poly)
SA, CT & ice mass fraction from bulk SA & bulk enthalpy
SA, CT & ice fraction from bulk SA & bulk potential enthalpy
SA, CT & ice fraction from bulk SA & bulk potential enthalpy (poly)
specific volume of ice
thermal expansion coefficient of ice with respect to in-situ temp
in-situ density of ice
pressure coefficient of ice
sound speed of ice (compression waves)
isentropic compressibility of ice
isothermal compressibility of ice
internal energy of ice
enthalpy of ice
entropy of ice
isobaric heat capacity of ice
chemical potential of water in ice
Helmholtz energy of ice
adiabatic lapse rate of ice
potential temperature of ice with reference pressure of 0 dbar
potential temperature of ice
in-situ temp from potential temp of ice with p_ref of 0 dbar
in-situ temp from density of ice
potential enthalpy from potential temperature of ice
potential temperature from potential enthalpy of ice
potential enthalpy from potential temperature of ice (poly)
potential temperature from potential enthalpy of ice (poly)
potential enthalpy from specific volume of ice
specific volume from potential enthalpy of ice
potential enthalpy from specific volume of ice (poly)
specific volume from potential enthalpy of ice (poly)
SA to CT ratio when sea ice melts into seawater
SA to CT ratio when sea ice melts into seawater (poly)
SA to CT ratio when sea ice melts, near equilibrium
SA to CT ratio when sea ice melts, near equilibrium (poly)
sea ice mass fraction to freeze seawater
SA and CT when sea ice melts into seawater
thermodynamic interaction between sea ice and seawater
gsw_melting_ice_SA_CT_ratio
gsw_melting_ice_SA_CT_ratio_poly
gsw_melting_ice_equilibrium_SA_CT_ratio
gsw_melting_ice_equilibrium_SA_CT_ratio_poly
gsw_ice_fraction_to_freeze_seawater
gsw_melting_ice_into_seawater
gsw_frazil_ratios_adiabatic
gsw_frazil_ratios_adiabatic_poly
gsw_frazil_properties
gsw_frazil_properties_potential
gsw_frazil_properties_potential_poly
thermodynamic interaction between ice and seawater
Gibbs SeaWater (GSW) Oceanographic Toolbox of TEOS –10
Page 2.
spiciness with reference pressure of 0 dbar
spiciness with reference pressure of 1000 dbar
spiciness with reference pressure of 2000 dbar
Celsius zero point; 273.15 K
one standard atmosphere; 101 325 Pa
Standard Ocean Reference Salinity; 35.165 04 g/kg
unit conversion factor for salinities; (35.165 04/35) g/kg
the “specific heat” for use with CT; 3991.867 957 119 63 (J/kg)/K
conductivity of SSW at SP=35, t_68=15, p=0; 42.9140 mS/cm
ratio of SP to Chlorinity; 1.80655 (g/kg) -1
valence factor of sea salt; 1.2452898
mole-weighted atomic weight of sea salt; 31.4038218... g/mol
Coriolis parameter
gravitational acceleration
spherical earth distance between points in the ocean
first derivatives of Conservative Temperature
second derivatives of Conservative Temperature
first derivatives of entropy
second derivatives of entropy
first derivatives of potential temperature
second derivatives of potential temperature
ratio of the slopes of isopycnals on the SA-CT diagram for
p & p_ref
ratio of the gradient of CT in a potential density surface to
that in the neutral tangent plane
ratio of gradients of pt & CT in a neutral tangent plane
gsw_SA_from_rho_t_exact
gsw_deltaSA_from_rho_t_exact
gsw_rho_t_exact
Absolute Salinity from density
Absolute Salinity Anomaly from density
in-situ density
laboratory functions, for use with densimeter measurements
gsw_T0
gsw_P0
gsw_SSO
gsw_uPS
gsw_cp0
gsw_C3515
gsw_SonCl
gsw_valence_factor
gsw_atomic_weight
TEOS–10 constants
gsw_f
gsw_grav
gsw_distance
planet Earth properties
gsw_CT_first_derivatives
gsw_CT_second_derivatives
gsw_entropy_first_derivatives
gsw_entropy_second_derivatives
gsw_pt_first_derivatives
gsw_pt_second_derivatives
derivatives of entropy, CT and pt
gsw_ntp_pt_vs_CT_ratio
gsw_isopycnal_vs_ntp_CT_ratio
gsw_isopycnal_slope_ratio
neutral versus isopycnal slopes and ratios
gsw_spiciness0
gsw_spiciness1
gsw_spiciness2
spiciness
GSW version 3.05.5
gsw_specvol_CT_exact
specific volume
gsw_alpha_CT_exact
thermal expansion coefficient with respect to CT
gsw_beta_CT_exact
saline contraction coefficient at constant CT
gsw_alpha_on_beta_CT_exact
alpha divided by beta
gsw_specvol_alpha_beta_CT_exact
specific volume, thermal expansion and saline
contraction coefficients
gsw_specvol_first_derivatives_CT_exact
first derivatives of specific volume
gsw_specvol_second_derivatives_CT_exact
second derivatives of specific volume
gsw_specvol_first_derivatives_wrt_enthalpy_CT_exact
first derivatives of specific volume with respect
to enthalpy
gsw_specvol_second_derivatives_wrt_enthalpy_CT_exact
second derivatives of specific volume with respect
to enthalpy
gsw_specvol_anom_CT_exact
specific volume anomaly
gsw_specvol_anom_standard_CT_exact
specific volume anomaly realtive to SSO & 0°C
gsw_rho_CT_exact
in-situ density and potential density
gsw_rho_alpha_beta_CT_exact
in-situ density, thermal expansion and saline
contraction coefficients
gsw_rho_first_derivatives_CT_exact
first derivatives of density
gsw_rho_second_derivatives_CT_exact
second derivatives of density
gsw_rho_first_derivatives_wrt_enthalpy_CT_exact
first derivatives of density with respect to enthalpy
gsw_rho_second_derivatives_wrt_enthalpy_CT_exact
second derivatives of density with respect to enthalpy
gsw_sigma0_CT_exact
sigma0 with reference pressure of 0 dbar
gsw_sigma1_CT_exact
sigma1 with reference pressure of 1000 dbar
gsw_sigma2_CT_exact
sigma2 with reference pressure of 2000 dbar
gsw_sigma3_CT_exact
sigma3 with reference pressure of 3000 dbar
gsw_sigma4_CT_exact
sigma4 with reference pressure of 4000 dbar
gsw_cabbeling_CT_exact
cabbeling coefficient
gsw_thermobaric_CT_exact
thermobaric coefficient
gsw_enthalpy_CT_exact
enthalpy
gsw_enthalpy_diff_CT_exact
difference of enthalpy between two pressures
gsw_dynamic_enthalpy_CT_exact
dynamic enthalpy
gsw_enthalpy_first_derivatives_CT_exact
first derivatives of enthalpy
gsw_enthalpy_second_derivatives_CT_exact
second derivatives of enthalpy
gsw_sound_speed_CT_exact
sound speed
gsw_kappa_CT_exact
isentropic compressibility
gsw_internal_energy_CT_exact
internal energy
gsw_internal_energy_first_derivatives_CT_exact
first derivatives of internal energy
gsw_internal_energy_second_derivatives_CT_exact
second derivatives of internal energy
gsw_CT_from_enthalpy_exact
Conservative Temperature from enthalpy
gsw_SA_from_rho_CT_exact
Absolute Salinity from density
gsw_CT_from_rho_exact
Conservative Temperature from density
gsw_CT_maxdensity_exact
Conservative Temperature of maximum density
of seawater
specific volume, density and enthalpy in terms of CT, based on the exact Gibbs function
Gibbs SeaWater (GSW) Oceanographic Toolbox of TEOS –10
Page 3.
argon solubility from SA and CT
argon solubility from SP and pt
helium solubility from SA and CT
helium solubility from SP and pt
krypton solubility from SA and CT
krypton solubility from SP and pt
nitrogen solubility from SA and CT
nitrogen solubility from SP and pt
neon solubility from SA and CT
neon solubility from SP and pt
oxygen solubility from SA and CT
oxygen solubility from SP and pt
gsw_specvol_t_exact
specific volume
gsw_alpha_wrt_CT_t_exact
thermal expansion coefficient with respect to Conservative
Temperature
gsw_alpha_wrt_pt_t_exact
thermal expansion coefficient with respect to potential temperature
gsw_alpha_wrt_t_exact
thermal expansion coefficient with respect to in-situ temperature
gsw_beta_const_CT_t_exact
saline contraction coefficient at constant Conservative Temperature
gsw_beta_const_pt_t_exact
saline contraction coefficient at constant potential temperature
gsw_beta_const_t_exact
saline contraction coefficient at constant in-situ temperature
gsw_specvol_anom_standard_t_exact
specific volume anomaly realtive to SSO & 0°C
gsw_rho_t_exact
in-situ density
gsw_pot_rho_t_exact
potential density
gsw_sigma0_pt0_exact
sigma0 from pt0 with reference pressure of 0 dbar
gsw_enthalpy_t_exact
enthalpy
gsw_dynamic_enthalpy_t_exact
dynamic enthalpy
gsw_CT_first_derivatives_wrt_t_exact
first derivatives of Conservative Temperature with respect to t
gsw_enthalpy_first_derivatives_wrt_t_exact first derivatives of enthalpy with respect to t
gsw_sound_speed_t_exact
sound speed
gsw_kappa_t_exact
isentropic compressibility
gsw_kappa_const_t_exact
isothermal compressibility
gsw_internal_energy_t_exact
internal energy
gsw_SA_from_rho_t_exact
Absolute Salinity from density
gsw_t_from_rho_exact
in-situ temperature from density
gsw_t_maxdensity_exact
in-situ temperature of maximum density of seawater
gsw_cp_t_exact
isobaric heat capacity
gsw_isochoric_heat_cap_t_exact
isochoric heat capacity
gsw_chem_potential_relative_t_exact
relative chemical potential
gsw_chem_potential_water_t_exact
chemical potential of water in seawater
gsw_chem_potential_salt_t_exact
chemical potential of salt in seawater
gsw_t_deriv_chem_potential_water_t_exact temperature derivative of chemical potential of water
gsw_dilution_coefficient_t_exact
dilution coefficient of seawater
gsw_Helmholtz_energy_t_exact
Helmholtz energy
gsw_osmotic_coefficient_t_exact
osmotic coefficient of seawater
gsw_osmotic_pressure_t_exact
osmotic pressure of seawater
basic thermodynamic properties in terms of in-situ t, based on the exact Gibbs function
gsw_Arsol
gsw_Arsol_SP_pt
gsw_Hesol
gsw_Hesol_SP_pt
gsw_Krsol
gsw_Krsol_SP_pt
gsw_N2sol
gsw_N2sol_SP_pt
gsw_Nesol
gsw_Nesol_SP_pt
gsw_O2sol
gsw_O2sol_SP_pt
dissolved gasses
GSW version 3.05.5
the TEOS-10 Gibbs function of seawater and its derivatives
the TEOS-10 Gibbs function of ice and its derivatives
Absolute Salinity Anomaly Ratio (excluding the Baltic Sea)
ratio of Absolute to Preformed Salinity, minus 1
Absolute Salinity Anomaly atlas value (excluding the Baltic Sea)
Calculates Absolute Salinity in the Baltic Sea
Calculates Practical Salinity in the Baltic Sea
“oceanographic funnel” check for the 75-term equation
entropy minus the terms that are a function of only SA
entropy_part evaluated at 0 dbar
linearly interpolates the reference cast
linearly interpolates (SA,CT,p) to the desired p
Reiniger & Ross (1968) interpolation of (SA,CT,p) to the desired p
gibbs(0,2,0,SA,t,0)
part of gibbs_ice(1,0,t,p)
part of gibbs_ice(1,0,pt0,0)
specvol(35.16504,0,p)
enthalpy(35.16504,0,p)
Hill ratio at a Practical Salinity of 2
www.TEOS–10.org
The GSW Toolbox is available from
gsw_front_page
gsw_check_functions
gsw_demo
gsw_ver
gsw_licence
documentation set
front page to the GSW Oceanographic Toolbox
checks that all the GSW functions work correctly
demonstrates many GSW functions and features
displays the GSW version number
creative commons licence for the GSW Oceanographic Toolbox
The GSW data set:
gsw_data_v3_0
This file contains:
(1) the global data set of Absolute Salinity Anomaly Ratio,
(2) the global data set of Absolute Salinity Anomaly Ref.,
(3) a reference cast (for the isopycnal streamfunction),
(4) two reference casts that are used by gsw_demo
(5) three vertical profiles of (SP, t, p) at known long & lat, plus the
outputs of all the GSW functions for these 3 profiles,
and the required accuracy of all these outputs.
gsw_gibbs
gsw_gibbs_ice
gsw_SAAR
gsw_Fdelta
gsw_deltaSA_atlas
gsw_SA_from_SP_Baltic
gsw_SP_from_SA_Baltic
gsw_infunnel
gsw_entropy_part
gsw_entropy_part_zerop
gsw_interp_ref_cast
gsw_linear_interp_SA_CT
gsw_rr68_interp_SA_CT
gsw_gibbs_pt0_pt0
gsw_gibbs_ice_part_t
gsw_gibbs_ice_pt0
gsw_specvol_SSO_0
gsw_enthalpy_SSO_0
gsw_Hill_ratio_at_SP2
The GSW functions call the following library functions:
Library functions of the GSW toolbox (internal functions; not intended to be called by users)
Gibbs SeaWater (GSW) Oceanographic Toolbox of TEOS –10
Page 4.
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
187
Table N.1. A selection of functions in version 3.05 of the GSW Oceanographic Toolbox.
Thermodynamic
Property
Function name
(in MATLAB)
Remarks
Absolute Salinity S A
gsw_SA_from_SP
the McDougall et al. (2012) algorithm for S A using
a look-‐‑up table
Preformed Salinity S*
gsw_Sstar_from_SP
Preformed Salinity S* from Practical Salinity S P
gsw_CT_from_t
Conservative Temperature Θ , from
Conservative
Temperature Θ
Gibbs function g and its
1st and 2nd derivatives
specific volume v
density ρ
gsw_gibbs
gsw_specvol_t_exact
gsw_rho_t_exact
( SA , t, p )
the sum of the IAPWS-‐‑09 and IAPWS-‐‑08 Gibbs
functions, and the derivatives of this sum
v SA , t, p specific volume using gsw_gibbs
(
)
ρ ( SA , t, p ) in situ density using gsw_gibbs
potential density ρ
gsw_pot_rho_t_exact
ρ Θ ( SA , t , p, pr ) using gsw_gibbs
density ρ , and
potential density ρ Θ
gsw_rho(SA,CT,p)
ρˆ SA , Θ, p , in situ density using the 75-‐‑term
expression for density in terms of Θ . Potential
density with respect to pressure pr is obtained by
calling gsw_rho with this pressure, obtaining
ρ Θ = ρˆ ( SA , Θ, pr ) .
Θ
specific entropy η
gsw_entropy_from_t
gsw_entropy_from_CT
specific enthalpy h
gsw_enthalpy
gsw_enthalpy_t_exact
first order derivatives of
hˆ ( S A , Θ, p )
second order derivatives of
hˆ ( S A , Θ, p )
sound speed c
(
)
specific entropy η using gsw_gibbs with the input
temperature either being in situ temperature or
Conservative Temperature
(
)
h SA , Θ, p from using the 75-‐‑term expression
for density, or from h SA , t, p using gsw_gibbs
(
)
gsw_enthalpy_first_derivative
s
ĥΘ and ĥS using the 75-‐‑term expression for
A
density in terms of Θ .
ˆ
gsw_enthalpy_second_derivati hˆ , hˆ
ΘΘ
ΘSA and hSA SA using the 75-‐‑term
ves
expression for density in terms of Θ .
(
)
gsw_sound_speed
gsw_sound_speed_CT_exact
gsw_sound_speed_t_exact
c SA , Θ, p from using the 75-‐‑term expression for
density, or from the full TEOS-‐‑10 Gibbs function as
either c SA , Θ, p or c SA , t, p
Conservative
Temperature Θ
gsw_CT_from_pt
Θ ( SA ,θ ) , found directly from gsw_gibbs .
Here θ is potential temperature with pr = 0.
potential temperature θ
gsw_pt_from_t
θ ( SA , t, p, pr ) found by using gsw_gibbs and by
equating two values of entropy
potential temperature θ
gsw_pt0_from_t
θ ( SA , t, p ) , a computationally faster version of
gsw_pt_from_t when pr = 0 dbar.
potential temperature θ
gsw_pt_from_CT
θ ( SA , Θ) , found by Newton_Raphson iteration,
being the inverse function of gsw_CT_from_pt
thermal expansion
coefficient with respect to
Θ , α Θ
gsw_alpha
gsw_alpha_wrt_CT_t_exact
α Θ ( SA , Θ, p ) using the 75-‐‑term expression for
Θ
density, or α ( S A , t , p ) from gsw_gibbs
saline contraction
coefficient at constant Θ
gsw_beta
gsw_beta_const_CT_t_exact
density, thermal
expansion and saline
contraction coefficient
gsw_rho_alpha_beta
β Θ ( SA , Θ, p ) using the 75-‐‑term expression for
Θ
density, or β ( S A , t , p ) from gsw_gibbs
ρˆ ( SA , Θ, p ) , αˆ Θ ( SA , Θ, p ) and βˆ Θ ( SA , Θ, p )
dynamic height anomaly
gsw_geo_strf_dyn_height
(
)
(
)
using the 75-‐‑term expression for density in terms
of Θ
geostrophic streamfunction in an isobaric surface
approximate isopycnal
gsw_geo_strf_isopycnal
geostrophic streamfunction
geostrophic streamfunction in an approximately
neutral surface, see Eqn. (3.30.1)
Montgomery geostrophic
streamfunction
geostrophic streamfunction in a specific volume
anomaly surface, see Eqn. (3.28.1)
gsw_geo_strf_Montgomery
IOC Manuals and Guides No. 56
188
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Appendix O:
Checking the Gibbs function of seawater
against the original thermodynamic data
One of the tasks undertaken by SCOR/IAPSO Working Group 127 was to verify the
accuracy of the Feistel (2003) and Feistel (2008) Gibbs functions against the underlying
laboratory data to which these Gibbs functions were fitted. This checking was performed
by Giles M. Marion, and is reported here.
Verification of the Feistel (2003) Gibbs function
Table 9 of Feistel (2003) included a root mean square (r.m.s.) estimate of the fit of the
Gibbs function to the original experimental data. In Table O.1 here, this estimate is the
column labeled “Resulting r.m.s.”. All the data in Table O.1 are from Feistel (2003) except
for the last column, where Giles M. Marion has estimated an independent “Verifying
r.m.s.”.
The seawater properties that were used to develop the Feistel (2003) Gibbs function
(see Column 1 of Table O.1) included density ρ , isobaric specific heat capacity c p ,
thermal expansion coefficient α t , sound speed c, specific volume v, freezing temperature
tf mixing heat Δh . This dataset included 1834 observations. Column 2 of Table O.1 are
the data sources that are listed in the references. The r.m.s. values were calculated with
the equation:
0.5
⎡1
2⎤
r.m.s = ⎢ ∑ ( F03 - expt.datum ) ⎥
⎣n n
⎦
(O.1)
where F03 refers to output of the FORTRAN code that implements Feistel’s (2003) Gibbs
function. In many cases, the experimental data had to be adjusted to bring this data into
conformity with recent definitions of temperature and the thermal properties of pure
water (see Feistel (2003) for the specifics of the datasets used and the internal assumptions
involved in developing the Gibbs function).
Comparisons of the “Resulting” (Feistel) and “Verifying” (Marion) columns in Table
O.1 show that they are in excellent agreement. The small differences between the two
r.m.s. columns are likely due to (1) the number of digits used in the calculations, (2) small
variations in the exact equations used for the calculations, or (3) small errors in model
inputs. In any case, these small differences are insignificant.
There were two typographical errors in the original Table 9 of Feistel (2003) in the
“Resulting r.m.s.” column. The original value for the PG93 dataset was listed as 11.3 ppm,
which is slightly different from the verifying value of 11.9 ppm. A subsequent check
indicated that this value should have been listed as 12.0 ppm, which is in excellent
agreement with the value of 11.9 ppm. Similarly, the original “Resulting r.m.s.” value for
the BDSW70 dataset was listed as 0.54 J/(kg K), which is significantly at variance from the
verifying estimate of 1.43 J/(kg K). A subsequent check indicated that this value should
have been listed as 1.45 J/(kg K), which is in excellent agreement with the independent
estimate of 1.43 J/(Kg K).
There were three minor errors between the original literature data and the Feistel
(2003) compilation of this data. In the BS70 dataset, two S P columns were mislabeled as
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
189
30.504 and 30.502, where the correct order should have been 30.502 and 30.504. In the
CM76 dataset, the correct value at S P = 20.062, t = 25 °C, and p = 588.0 bars should have
been 0.964393 kg m −3 , not 0.964383 kg m −3 . These minor errors are insignificant. The
independent comparisons in Table O.1 verify the accuracy of the Feistel (2003) Gibbs
function.
Verification of the Feistel (2008) saline part of the Gibbs function of seawater
The saline Gibbs function Feistel (2008) was designed to increase the temperature
range up to 80 °C and the salinity range up to 120 g kg −1 (Feistel, 2008). Table 7 of Feistel
(2008) included a root mean square (r.m.s.) estimate of the model fit to the original
experimental data (see the column “Resulting r.m.s.” in the attached Table O.2). All the
data in this table are from the Feistel (2008) paper except for the last column, where Giles
M. Marion has estimated an independent “Verifying r.m.s.”.
The new seawater salinity databases that were used to develop the Feistel (2008) Gibbs
function (see Column 1 of Table O.2) included isobaric specific heat capacity c p , mixing
heat Δh , freezing point depression tf water vapour pressure p vap , and the Debye-‐‑Hückel
limiting law g LL . This salinity dataset included 602 observations. Column 2 of Table O.2
are the data sources that are listed in the references. In many cases, the experimental data
had to be adjusted to bring this data into conformity with recent definitions of
temperature and the thermal properties of pure water (see Feistel (2008) for the specifics
on the datasets used and the internal assumptions involved in model development).
Comparisons of the “Resulting” (Feistel) and “Verifying” (Marion) “r.m.s.” columns
show that they are in excellent agreement. The most likely explanation for the few small
differences is the number of digits used in the calculations. In general, the greater the
number of digits used in these calculations, the more accurate the calculations.
This independent check reveals that there are no significant differences between the
Feistel and Marion estimations of r.m.s. values for these comparisons (Table O.2), which
verifies the accuracy of the Feistel (2008).
Verification of the Pure Water part of the Feistel (2003) Gibbs function
The pure water part of the Feistel (2003) Gibbs function was itself a fit to the IAPWS-‐‑95
Helmholtz function of pure water substance. The accuracy of this fit to IAPWS-‐‑95 in the
oceanographic ranges of temperature and pressure has been checked independently by
two members of the SCOR/IAPSO Working Group 127 (Dan G. Wright and David R.
Jackett). The accuracy of this pure water part of the Feistel (2003) Gibbs function has also
been checked by an evaluation committee of IAPWS in the process of approving the
Feistel (2003) Gibbs function as an IAPWS Release (IAPWS-‐‑09). In IAPWS-‐‑09 it is shown
that the pure water part of the Feistel (2003) Gibbs function fits the IAPWS-‐‑95 properties
more precisely than the uncertainty of the data that underlies IAPWS-‐‑95. Hence we can be
totally comfortable with the use of the Feistel (2003) Gibbs function to represent the
properties of pure liquid water in the oceanographic ranges of temperature and pressure.
IOC Manuals and Guides No. 56
190
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Table O.1. Summary of data used in the regression to determine the coefficients of the Feistel
(2003) Gibbs potential.
Quantity Source
−1
t /°C
P/MPa
#
Required
Points r.m.s.
Resulting
r.m.s.
Verifying
r.m.s.
MGW76c
0.5-40
0-40
0
122
4 ppm
4.1 ppm
4.2 ppm
ρ
PBB80
5-42
0-30
0
345
4 ppm
4.0 ppm
4.2 ppm
PG93
34-50
15-30
0
81
i
10 ppm
12.0 ppm
11.9 ppm
ii
cp
BDSW70
10-50
0-40
0
25
2 J/(kg K)
cp
MPD73
1-40
5-35
0
48
0.5 J/(kg K) 0.52 J/(kg K) 0.45 J/(kg K)
α
C78
10-30
-6-1
0.7-33
31
c
D74(I-III)
29-43
0-35
0-2
92
5 cm/s
1.7 cm/s
1.6 cm/s
c
D74(IVa-d) 29-43
0-30
0.1-5
32
5 cm/s
1.2 cm/s
1.2 cm/s
c
D74(V-VI) 33-37
0-5
0-100
128
5 cm/s
3.5 cm/s
3.5 cm/s
v
CM76
5-40
0-40
0-100
558
10 ppm
11.0 ppm
11.2 ppm
BS70
30-40
-2-30
1-100
221
4 ppm
2.6 ppm
2.6 ppm
tf
DK74
4-40
-2-0
0
32
2 J/kg
1.8 J/kg
1.9 J/kg
Δh
B68
0-33
25
0
24
4J
2.4 J
2.4 J
Δh
MHH73
1-41
0-30
0
95
0.4 J/kg
0.5 J/kg
0.5 J/kg
v
ii
(g kg )
ρ
ρ
i
SA
t
S
-6
0.6x10 K
-1
1.45 J/(kg K) 1.43 J/(kg K)
-6
0.73x10 K
-1
-6
0.74x10 K
The original value in Table 9 of Feistel (2003) of 11.3 ppm refers to the specific volume.
The original value in Table 9 of Feistel (2003) was 0.54 J/(kg K), which apparently was a
typographical error.
Table O.2. Summary of extra datasets used in the regression to determine the coefficients of
the Feistel (2008) Gibbs potential.
t /°C
P/MPa Points
Resulting
r.m.s.
Verifying
r.m.s.
11-117
2-80
0
221
3.46 J/(kg K)
3.46 J/(kg K)
MPD73
1-40
5-35
0
48
0.57 J/(kg K)
0.57 J/(kg K)
cp
MP05
1-35
10-40
0
41
1.30 J/(kg K)
1.30 J/(kg K)
Δh
B68
0-97
25
0
33
0.75 J/kg
0.75 J/kg
Δh
C70
35-36
2-25
0
19
7.2 J/kg
7.1 J/kg
Δh
MHH73
1-35
0-30
0
120
3.3 J/kg
3.3 J/kg
tf
DK74
4-40
-0.2-(-2.2)
0
32
1.6 mK
1.6 mK
tf
FM07
5-109
-0.3-(-6.9)
0
22
1.2 mK
1.0 mK
vap
R54
18-40
25
0
13
2.8 J/kg
2.8 J/kg
BSRSR74
6-70
60-80
0
32
9.1 J/kg
9.3 J/kg
F08
35
-5-95
0
21
0.091 J/kg
0.092 J/kg
Source
SA
cp
BDCW67
c Sp
Quantity
p
tboil
g
LL
(g kg )
−1
IOC Manuals and Guides No. 56
-1
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
191
Appendix P: Thermodynamic properties
based on g ( SA ,t, p ), h ( SA ,η, p ), h ( SA ,θ , p ) and ĥ( SA ,Θ, p )
The thermodynamic potential on which TEOS-‐‑10 is based is the Gibbs function of
seawater. Being a Gibbs function, g ( SA , t , p ) is naturally a function of Absolute Salinity, in
situ temperature and pressure. There are other choices for a thermodynamic potential.
One such choice is enthalpy h as a function of Absolute Salinity, entropy and pressure, and
we give this functional form for enthalpy a boomerang over the h so that h = h SA ,η , p . It
proves theoretically convenient to consider the additional functional forms h = h SA ,θ , p
and h = hˆ ( SA , Θ, p ) for enthalpy. These two functional forms do not constitute a complete
thermodynamic description of seawater but when supplemented by the expressions
η = η SA ,θ and η = ηˆ ( SA , Θ) for entropy, they do form complete thermodynamic
potentials. In the expressions h = h SA ,θ , p and η = η SA ,θ it is possible to choose any
fixed reference pressure pr for the definition of potential temperature, θ . However there
is no advantage to choosing the reference pressure to be different from pr = 0 and it is this
value that is taken in Table P.1 and throughout this appendix. Table P.1 lists expressions
for some common thermodynamic quantities in terms of these potential functions. Note
that the reference pressure pr that appears in the last three columns of the ρ θ row of Table
P.1 is the reference pressure of potential density, not of θ , whereas in the Gibbs function
column, this general reference pressure must also be used to evaluate θ .
In addition to Table P.1 we have the following expressions for the thermobaric and
cabbeling coefficients (of Eqns. (3.8.1) – (3.9.2))
(
(
)
(
Tbθ
=
TbΘ =
Cbθ =
hPPθ
ĥP
A
A
ĥP
(P.1)
A
(P.2)
2
2
hPθ hPθ SA ⎛ hPθ ⎞ hPSA SA
vθθ
vθ vθ SA ⎛ vθ ⎞ vSA SA
−2
+⎜ ⎟
=
−2
+⎜
⎟
⎜⎝ vS ⎟⎠
⎜⎝ hPS ⎟⎠
v
vS v
v
hPS hP
hP
A
A
A
A
ρθθ
ρ
+2 θ
ρ
ρ
S
ρθ S
ρ
2
A
⎛ ρ ⎞ ρ S S
A A
−⎜ θ ⎟
,
⎜⎝ ρ S ⎟⎠
ρ
A
⎛ hˆ
hˆ
hˆ hˆPΘSA
= PΘΘ − 2 PΘ
+ ⎜ PΘ
⎜ hˆPS
hˆP
hˆPSA hˆP
⎝ A
2
⎞ hˆPS S
vˆ
vˆ vˆΘSA ⎛ vˆΘ
A A
⎟
= ΘΘ − 2 Θ
+⎜
ˆ
⎜ vˆS
⎟ hP
vˆ
vˆSA vˆ
⎝ A
⎠
⎛ ρˆ
ρˆ
ρˆ ρˆ ΘSA
= − ΘΘ + 2 Θ
− ⎜ Θ
⎜ ρˆ
ρˆ
ρˆ SA ρˆ
⎝ SA
)
)
ρ Pθ
ρθ ρ PSA
hPθ hPPSA
v Pθ
vθ v PSA
−
=
−
=
−
+
,
ρ
ρ S
ρ
v
vS
v
hPS hP
A
CbΘ
(
hˆPPΘ
hˆ hˆPPSA
vˆ
vˆ vˆPSA
ρˆ
ρˆ ρˆ PSA
− PΘ
= PΘ − Θ
= − PΘ + Θ
,
vˆ
vˆSA vˆ
ρˆ
ρˆ SA ρˆ
hˆP
hˆPSA hˆP
hPθθ
= −
)
)
(
(P.3)
2
⎞ vˆSA SA
⎟
⎟
vˆ
⎠
2
(P.4)
⎞ ρˆ SA SA
.
⎟
⎟
ˆ
ρ
⎠
Here follows some interesting expressions that can be gleaned from Table P.1.
c 0p
η̂Θ2
(T + t ) = hˆΘ = ∂h
0
, (T0 + θ ) =
c p 0 = hθ 0 = c pΘθ = −
, 0
η̂ΘΘ
ηˆΘ
(T0 + θ ) c0p ∂h0
()
()
,
(P.5)
SA , p
IOC Manuals and Guides No. 56
gT ( SA ,θ , pr ) = gT ( SA , t, p )
g ( SA ,θ ,0 ) − (T0 + θ ) gT ( SA ,θ ,0 )
θ
Θ
µ = gSA
µ = g − SA g S
u = g − (T0 + t ) gT − ( p + P0 ) g P
f = g − ( p + P0 ) g P
c p = − (T0 + t ) gTT
h 0 = g ( SA ,θ ,0 ) − (T0 + θ ) gT ( SA ,θ ,0 )
µ
µW
u
f
cp
h0
κ
t
κ =−
IOC Manuals and Guides No. 56
t
g P−1g PP
ρ θ = ⎡⎣ gP ( SA ,θ , pr ) ⎤⎦
−1
ρ = ( gP )
ρ
ρθ
v = gP
v
A
h = g − (T0 + t ) gT
h
(
)
)
−1
−1 hη2P
κ = − hP hPP + hP
hηη
t
(
( )
()
−1
ρθ = ⎡⎣ hP ( pr ) ⎤⎦
(
η
η = − gT
η
h = h S A ,η , p
v = hP
−1
ρ = hP
µ = hS
A
W
µ = h − η hη − SA hS
A
u = h − p + P0 hP
f = h − η hη − p + P0 hP
c p = hη hηη
h0 = h 0
g = h − η hη
−1
)
()
(
A
(
)
A
)
()
−1
ρθ = ⎡⎣ hP ( pr ) ⎤⎦
κ t = − hP−1 hPP − hP−1
)
(
hθ2Pηθ
hθ ηθθ − hθθ ηθ
h = h 0
0
c p = hθ ηθ2 ηθ hθθ − hθ ηθθ
(
f = h − η hθ ηθ − p + P0 hP
u = h − p + P0 hP
(
A
)
)
µW = h − η hθ ηθ − SA hS − η S hθ ηθ
A
−1
)
µ = hS − η S hθ ηθ
( )
ρ = hP
v = hP
h = h SA ,θ , p
(
η = η ( SA ,θ )
g = h − η hθ ηθ
θ;
(
(T0 + t ) = hθ ηθ
(T0 +θ ) = hθ (0) ηθ
)
Θ = h 0 c0p
()
()
(
Expressions based on
h S ,θ , p and η S ,θ
A
A
Θ = h 0 c0p
)
)
)
g = g ( SA , t , p )
c 0p
(
(
(
Expressions based on
h S A ,η , p
T0 + t = hη
T0 + θ = hη 0
g
Θ =
[this is an implicit equation for θ ]
t
g ( SA , t , p )
Expressions based on
)
()
Θ = ĥ 0 c0p
A
(
κ t = − hP hPP − hP
Θ ΘΘ
(hˆ ηˆ
−1
)
− hˆΘΘηˆΘ
ρ θ = ρ Θ = ⎡ hˆP ( pr )⎤
⎣
⎦
2
hˆΘPηˆΘ
ˆ −1 ˆ
ˆ −1
h0 = hˆ (0) = c0p Θ
c p = hˆΘηˆΘ2 ηˆΘhˆΘΘ − hˆΘηˆΘΘ
f = hˆ − ηˆ hˆΘ ηˆΘ − ( p + P0 ) hˆP
u = hˆ − ( p + P0 ) hˆP
A
)
µW = ĥ− η̂ ĥΘ η̂Θ − SA ĥS − η̂S ĥΘ η̂Θ
(
−1
µ = hˆSA − ηˆSA hˆΘ ηˆΘ
( )
ρ = hˆP
v = hˆP
h = hˆ ( SA , Θ, p )
η = ηˆ ( SA , Θ)
g = hˆ − ηˆ hˆΘ ηˆΘ
Θ;
(T0 + t ) = hˆΘ ηˆΘ
(T0 +θ ) = c0p ηˆΘ
Expressions based on
ˆ
h ( SA , Θ, p ) and ηˆ ( SA , Θ)
Table P.1. Expressions for various thermodynamic variables based on four different thermodynamic potentials
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
t
192
)
βΘ
β
θ
βt
α
Θ
αθ
c0p
gTP gTT ( S A , θ , pr )
gP
gTT
β t = − gP−1 gPSA
g P gTT
−1
gTP ⎡ g SAT − (T0 + θ ) g SA ( SA ,θ ,0 ) ⎤
⎣
⎦
+
g P gTT
β Θ = − g P−1 g PSA
+
)
gTP ⎡⎣ g SAT − g SAT ( SA ,θ , pr )⎤⎦
β θ = − g P−1 g PSA
Θ
g
α = − TP
g P (T0 + θ ) gTT
αθ =
gTP
gP
αt =
2
gTP
gTT
− gTT gPP
αt
2
TP
(g
+
g P−1
Γ = − gTP gTT
c = g P gTT
κ =−
g P−1 g PP
Γ
c
κ
g ( SA , t , p )
Expressions based on
A
A
()
()
()
()
−1 hSAη 0
+ hP hPη
hηη 0
β Θ = − hP−1 hPS
A
−1 hSA 0
+ hP hPη
hη 0
β θ = − hP−1 hPS
−1 hSAη
+ hP hPη
hηη
β t = − hP−1 hPS
()
−1
c0p
α = hP hPη
hη 0
Θ
()
Γ = hPη
−1 hPη
t
α = hP
hηη
−1 hPη
θ
α = hP
hηη 0
θ θθ
( h η
)
)
hS ( 0 )
β Θ = − hP−1 hPS + hP−1hPθ A
A
hθ ( 0 )
A
− hθθ ηθ
θ θθ
( h η
A
− hS θ ηθ
θ SAθ
( h η
A
β θ = − hP−1 hPS
+ hP−1hPθ
β t = − hP−1 hPS
()
c0p
−1
α = hP hPθ
hθ 0
Θ
)
)
− hθθ ηθ
ηθ2
α θ = hP−1hPθ
α t = − hP−1hPθ
Γ = hPθ ηθ
− hPP
(
c = hP
− hPP
)
c = hP
(
κ = − hP−1 hPP
)
κ = − hP−1 hPP
(
Expressions based on
h SA ,θ , p and η SA ,θ
Expressions based on
h S A ,η , p
−hˆPP
Θ ΘΘ
Θ SA Θ
(hˆ ηˆ
(hˆ ηˆ
)
)
ˆ
− hˆΘΘηˆΘ
− hˆSAΘηˆΘ
)
IOC Manuals and Guides No. 56
β Θ = − hˆP−1 hˆPSA
ηS Θ
β θ = − hˆP−1 hˆPSA + hˆP−1hˆPΘ A
ηˆΘΘ
+ hˆP−1hˆPΘ
β t = − hˆP−1 hˆPSA
α Θ = hˆP−1 hˆPΘ
ηˆΘ2
0
c pηˆΘΘ
− hˆΘΘηˆΘ
ηˆΘ2
Θ ΘΘ
(hˆ ηˆ
α θ = − hˆP−1hˆPΘ
α t = − hˆP−1hˆPΘ
Γ = hˆPΘ ηˆΘ
c = hˆP
κ = − hˆP−1 hˆPP
hˆ ( SA , Θ, p ) and ηˆ ( SA , Θ)
Expressions based on
Table P.1. (cont’d) Expressions for various thermodynamic variables based on four different thermodynamic potentials
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater 193
194
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
∂θ
∂Θ S
(
A
Θ, p
c 0p ηˆSA
ηˆΘ
(P.6)
()
A
µ (0) = −
∂θ
∂SA
)2
c0p η̂ΘΘ
c0p
T0 + θ η̂ΘΘ
∂θ
1
αΘ
=
= θˆΘ = = −
=
−
=
=
,
∂Θ S
Θθ
cp 0
η̂Θ2
c0p
αθ
,p
= − (T0 + θ ) ηˆSA ,
(P.7)
(
)
2
−1
Θ
c0p η̂S Θ
T0 + θ η̂S Θ
SA
0 ∂η̂Θ
A
A
ˆ
= θS = − = −
= cp
= −
.
2
0
A
∂S
Θ
η̂
c
A
θ
Θ
p
Θ
∂θ
=
∂SA
(P.8)
Θ
See Eqn. (A.12.6) for an alternative expression for θˆS . Eqn. (P.8) can also be written as
A
(
∂ (T0 + θ )
−1
) ∂S
A
Θ
= ηˆSAΘ c0p .
(P.9)
Now we consider how all the terms in the last column of Table P.1 may be evaluated
in terms of ĥ75 SA ,Θ, p of Eqn. (A.30.6); this being the expression for specific enthalpy
that follows from the 75-‐‑term expression for specific volume as a function of ( SA , Θ, p ) as
described in Eqn. (K.1) and Table K.1. The first step is to evaluate θ exactly from the
following implicit expression for Θ in terms of the Gibbs function at p = 0 (see Eqn.
(2.12.1)), as discussed in section 3.3,
(
)
c0p Θ = h ( SA , t =θ ,0) = g ( SA , t =θ ,0) − (T0 + θ ) gT ( SA , t = θ ,0) .
(
)
(
(P.10)
)
S ,θ and
Next, we remind ourselves that we know the functional forms of η SA ,θ , Θ
A
µ SA ,θ , 0 in terms of the coefficients of the Gibbs function of seawater as the exact
polynomial and logarithm terms given by (from Eqns. (2.10.1) and (2.9.6))
(
)
η ( SA ,θ ) = − gT ( SA ,t = θ , 0 ) , µ ( SA ,θ , 0 ) = g S ( SA ,t = θ , 0 ) ,
(P.11a,b)
A
and Eqn. (P.10) is repeated here emphasizing the functional form of the left-‐‑hand side,
S ,θ = g S ,t = θ ,0 − T + θ g S ,t = θ ,0 .
c0 Θ
(P.12)
(
p
)
A
(
) (
A
) T(
0
)
A
The partial derivatives with respect to Θ and with respect to θ , both at constant S A
and p , and the partial derivatives with respect to S A , are related by
Θ
1 ∂
∂
∂
∂
∂
S
=
=
− A
, and
.
(P.13a,b)
∂Θ
∂S
∂S
Θ ∂θ
Θ ∂θ
θ
SA , p
θ
A θ,p
A Θ, p
SA , p
SA , p
Use of these expressions, acting on entropy yields (with p = 0 everywhere, and using
Eqn. (P.7) [or Eqn. (A.12.8b)] and Eqn. (P.8))
c0p
c0p
µ SA , θ , 0
η
1
, η̂ΘΘ = −
, η̂S = −
,
(P.14a,b,c)
η̂Θ = θ ≡
2
A
Θθ T + θ
Θθ
T0 + θ
T0 + θ
0
(
η̂S
AΘ
Θ
S
= A
Θ
θ
)
c0p
(T0 +θ )2
(
, and η̂S
A SA
= −
(
(
)
µ S
( SA ,θ , 0)
−
(T0 +θ )
A
(Θ )
)
)
2
c0p
SA
(T0 +θ )2
Θ
θ
,
(P.15a,b)
in terms of the partial derivatives of the exact polynomial expressions (P.11b) and (P.12).
All of the thermodynamic variables of the last column of Table P.1 can now be
evaluated using the partial derivatives of ĥ75 SA ,Θ, p and the exact expressions (P.14)
and (P.15) which are written in terms of θ which is found from the exact implicit equation
(P.10). This completes the discussion of how ĥ75 SA ,Θ, p can be used as an alternative
thermodynamic potential of seawater. The partial derivatives of entropy in Eqns. (P.14)
and (P.15) are available from the functions gsw_entropy_first_derivatives and
gsw_entropy_second_derivatives. The Second Law constraint on ηˆ ( SA , Θ ) of entropy
2
production for turbulent mixing is guaranteed if ηˆΘΘ < 0 , ηˆSA Θ < ηˆSA SAηˆΘΘ and
ηˆSA SA < 0 . From Eqns. (P.14) and (P.15) we find that these three constraints are satisfied iff
and µ are positive, and these two constraints are the same as those of Eqns.
both Θ
SA
θ
(A.16.17) and (A.16.18), namely gTT < 0 and g SA SA > 0 , and so are guaranteed to hold.
(
)
(
)
(
IOC Manuals and Guides No. 56
)
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
195
References
Alberty, R. A., 2001: Use of Legendre transforms in chemical thermodynamics. Pure Appl.
Chem., 73, 1349–1380.
Alderson, S. G. and P. D. Killworth, 2005: A preoperational scheme for calculating sea surface
height by Bernoulli inverse of Agro float data in the North Atlantic. J. Atmos. Oceanic
Tech., 22, 1416-‐‑1422.
Bacon, S., F. Culkin, N. Higgs and P. Ridout, 2007: IAPSO Standard Seawater: Definition of the
uncertainty in the calibration procedure, and stability of recent batches. J Atmos. Oceanic
Technol. 24, 1785-‐‑1799.
Bacon, S. and N Fofonoff, 1996: Oceanic heat flux calculation. J Atmos. Oceanic Technol. 13,
1327-‐‑1329.
Barber, C. R., 1969: The International Temperature Scale of 1968. Metrologia, 5, 35-‐‑44.
Batchelor, G. K., 1970: An Introduction to Fluid Dynamics. Cambridge University Press, 615pp.
Bezold, W.v., 1888: Zur Thermodynamik der Atmosphaere. Potentielle Temperatur. Verticaler
Temperaturgradient. Zusammengesetzte Convection. Sitzungsberichte der Koeniglich
Preussischen Akademie der Wissenschaften zu Berlin, 46, 1189-‐‑1206.
http://bibliothek.bbaw.de/bibliothekdigital/digitalequellen/schriften/anzeige/index_html?band=10-sitz/1888-2&seite:int=530
BIPM, 2005: Clarification of the definition of the kelvin, unit of thermodynamic temperature
RECOMMENDATION 2 (CI-‐‑2005) The International Committee for Weights and
Measures (CIPM), http://www.bipm.org/cc/CIPM/Allowed/94/CIPM-Recom2CI-2005-EN.pdf
BIPM, 2006: The International System of Units (SI), 8th edition. Bureau International des Poids et
Mesures, Organisation Intergouvernementale de la Convention du Mètre, Sèvres, France.
http://www.bipm.fr/utils/common/pdf/si_brochure_8_en.pdf
Bleck, R., 1978: Finite difference equations in generalized vertical coordinates, I, Total energy
conservation. Contrib. Atmos. Phys., 51, 360-‐‑372.
Bradshaw, A. and K. E. Schleicher, 1970: Direct measurements of thermal expansion of sea
water under pressure. Deep-‐‑Sea Res., 17, 691-‐‑706.
Brewer, P. G., and A. Bradshaw, 1975. The effect of non-‐‑ideal composition of seawater on
salinity and density. J. Mar. Res. 33, 157-‐‑175.
Bromley, L. A., 1968: Relative enthalpies of sea salt solutions at 25 °C. Journal of Chemical and
Engineering Data, 13, 399-‐‑402.
Bromley, L. A., V. A. Desaussure, J. C. Clipp, and J. S. Wright, 1967: Heat capacities of sea
water solutions at salinities of 1 to 12% and temperatures of 2° to 80°. Journal of Chemical
and Engineering Data, 12, 202-‐‑206.
Bromley, L. A., A. E. Diamond, E. Salami, and D. G. Wilkins, 1970: Heat capacities and
enthalpies of sea salt solutions to 200 °C. Journal of Chemical and Engineering Data, 15, 246-‐‑
253.
Bromley, L. A., D. Singh, P. Ray, S. Sridhar, and S. M. Read, 1974: Thermodynamic properties
of sea salt solutions. AIChE Journal, 20, 326-‐‑335.
Bryden, H. L., 1973: New polynomials for thermal expansion, adiabatic temperature gradient
and potential temperature of seawater. Deep-‐‑Sea Res., 20, 401-‐‑408.
Caldwell, D. R., 1978: The maximum density points of pure and saline water. Deep-‐‑Sea
Research, 25, 175-‐‑181.
Calvert, J. G., 1990: Glossary of atmospheric chemistry terms (Recommendations 1990), Pure &
App. Chem., 62, 2167-‐‑2219.
Chen, C.-‐‑T. and F. J. Millero, 1976: The specific volume of seawater at high pressures. Deep-‐‑Sea
Res. 23, 595-‐‑612.
IOC Manuals and Guides No. 56
196
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Chen, C.-‐‑T. and F. J. Millero, 1977: Sound speed of seawater at high pressures. J. Acoust. Soc.
Am., 62, 1129-‐‑1135.
Clausius, R., 1876: Die Mechanische Wärmetheorie. Friedrich Vieweg und Sohn,
Braunschweig.
Culkin, F. and P. S. Ridout, 1998: Stability of IAPSO Standard Seawater, J. Atmos. Oceanic.
Technol., 15, 1072–1075.
Cunningham, S. A., 2000: Circulation and volume flux of the North Atlantic using synoptic
hydrographic data in a Bernoulli inverse. J. Marine Res., 58, 1-‐‑35.
Curry, J. A. and P. J. Webster, 1999: Thermodynamics of atmospheres and oceans. Academic Press,
San Diego, USA.
Dai, A., 2006: Recent Climatology, Variability, and Trends in Global Surface Humidity. J.
Climate, 19, 3589-‐‑3605.
de Groot, S. R. and P. Mazur, 1984: Non-‐‑equilibrium thermodynamics. Dover Publications, Inc.,
New York.
Del Grosso, V. A., 1974: New equation for the speed of sound in natural waters (with
comparison to other equations). J. Acoust. Soc. Am., 56, 1084-‐‑1091.
de Szoeke, R. A., 2000: Equations of Motion Using Thermodynamic Coordinates. Journal of
Physical Oceanography, 30, 2814-‐‑2829.
de Szoeke, R. A. and R. M. Samelson, 2002: The duality between the Boussinesq and non-‐‑
Boussinesq hydrostatic equations of motion. Journal of Physical Oceanography, 32, 2194-‐‑2203.
de Szoeke, R. A., S. R. Springer and D. M. Oxilia, 2000: Orthobaric density: A thermodynamic
variable for ocean circulation studies. J. Phys. Oceanogr., 30, 2830-‐‑2852.
Douglas, T. B., 1969: Conversion of existing calorimetrically determined thermodynamic
properties to the bases of the International Practical Scale of 1968. Jour. of Res. NBS -‐‑ A:
Physics and Chemistry, 73A, 451–470.
Eden, C. and J. Willebrand, 1999: Neutral density revisited. Deep-‐‑Sea Res. II, 46, 33-‐‑54.
Feistel, R., 1993: Equilibrium thermodynamics of seawater revisited, Progr. Oceanogr., 31, 101-‐‑
179.
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 E. Hagen, 1995: On the GIBBS thermodynamic potential of seawater. Prog. in
Oceanogr., 36, 249-‐‑327.
Feistel, R. and E. Hagen, 1998: A Gibbs thermodynamic potential of sea ice, Cold Regions Sci.
Technol., 28, 83-‐‑142, 1998.
Feistel, R. and G. M. Marion, 2007: A Gibbs-‐‑Pitzer function of high-‐‑salinity seawater
thermodynamics. Progress in Oceanography, 74, 515-‐‑539.
Feistel, R., G. M. Marion, R. Pawlowicz and D. G. Wright, 2010d: Thermophysical property
anomalies of Baltic seawater. Ocean Science, 6, 949-‐‑981. http://www.oceansci.net/6/949/2010/os-6-949-2010.pdf
Feistel, R., G. Nausch, N. Wasmund (Eds), 2008b: State and Evolution of the Baltic Sea, 1952 -‐‑
2005. A Detailed 50-‐‑Year Survey of Meteorology and Climate, Physics, Chemistry,
Biology, and Marine Environment. John Wiley & Sons, Inc., Hoboken.
Feistel, R. and W. Wagner, 2005. High-‐‑pressure thermodynamic Gibbs functions of ice and sea
ice. Journal of Marine Research, 63, 95-‐‑139.
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. and W. Wagner, 2007: Sublimation pressure and sublimation enthalpy of H2O ice Ih
between 0 and 273.16 K, Geochim. Cosmochim. Acta, 71, 36–45, 2007.
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, D. R. Jackett, K. Miyagawa, J. H. Reissmann, W. Wagner, U. Overhoff,
C. Guder, A. Feistel and G. M. Marion, 2010b: Numerical implementation and
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
197
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.oceansci.net/6/633/2010/os-6-633-2010-supplement.pdf
Feistel, R., D. G. Wright, H.-‐‑J. Kretzschmar, E. Hagen, S. Herrmann and R. Span, 2010a:
Thermodynamic properties of sea air. Ocean Science, 6, 91–141. http://www.oceansci.net/6/91/2010/os-6-91-2010.pdf
Feistel, R., D. G. Wright, K. Miyagawa, A. H. Harvey, J. Hruby, D. R. Jackett, T. J. McDougall
and W. Wagner, 2008a: 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.pdf
Fofonoff, N. P., 1962: Physical properties of seawater. The Sea, Vol. 1, M N Hill, Ed., Wiley-‐‑
Interscience, 3-‐‑30.
Fofonoff, N. P., 1985: Physical properties of seawater: A new salinity scale and equation of
state for seawater. J. Geophys. Res., 90, 3322–3342.
Gent, P. R. and J. C. McWilliams, 1990: Isopycnal mixing in ocean circulation models. Journal of
Physical Oceanography, 20, 150-‐‑155.
Gent, P. R., J. Willebrand, T. J. McDougall and J. C. McWilliams, 1995: Parameterizing eddy-‐‑
induced tracer transports in ocean circulation models. Journal of Physical Oceanography, 25,
463-‐‑474.
Gibbs, J. W., 1873: Graphical methods in the thermodynamics of fluids, Trans. Connecticut
Acad. Arts and Sci., 2, 309-‐‑342.
Gill, A. E., 1982: Atmosphere-‐‑Ocean Dynamics. Academic Press, 662pp.
Glandorff P., and I. Prigogine, 1971: Thermodynamic theory of structure, stability and fluctuations.
Wiley-‐‑Interscience, 306pp.
Glasstone, S., 1947: Thermodynamics for Chemists, D. van Nostrand Company, Inc., Princeton,
New Jersey, USA.
Godfrey, J. S., 1989: A Sverdrup model of the depth-‐‑integrated flow for the world ocean
allowing for island circulations. Geophysical and Astrophysical Fluid Dynamics. 45, 89-‐‑119.
Goldberg, R.N. and R. D. Weir, 1992: Conversion of temperatures and thermodynamic
properties to the basis of the International Temperature Scale of 1990. Pure & Applied
Chemistry, 64, 1545-‐‑1562.
Gouretski, V. V. and K. P. Koltermann, 2004: WOCE global hydrographic climatology. Berichte
des Bundesamtes für Seeschifffahrt und Hydrographie Tech. Rep. 35/2004, 49 pp.
Gradshteyn, I. S. and I. M. Ryzhik, 1980: Tables of Integrals, Series and Products. Academic Press,
1160 pp.
Graham, F. S. and T. J. McDougall, 2013: Quantifying the non-‐‑conservative production of
Conservative Temperature, potential temperature and entropy. Journal of Physical
Oceanography, 43, 838-‐‑862.
Greatbatch, R. J. and T. J. McDougall, 2003: The non-‐‑Boussinesq Temporal-‐‑Residual-‐‑Mean.
Journal of Physical Oceanography, 33, 1231-‐‑1239.
Griffies, S. M., 2004: Fundamentals of Ocean Climate Models. Princeton, NJ: Princeton University
Press, 518 pp + xxxiv.
Groten, E., 2004: Fundamental Parameters and Current (2004) Best Estimates of the Parameters
of Common Relevance to Astronomy, Geodesy, and Geodynamics. Journal of Geodesy, 77,
724-‐‑797.
Harvey, A. H. and P. H. Huang, 2007: First-‐‑Principles Calculation of the Air–Water Second
Virial Coefficient. Int. J. Thermophys., 28, 556–565.
Helland-‐‑Hansen, B., 1912: The Ocean Waters. Intern. Rev. Hydrobiol. Hydrog., Hydrog. Supp.
Ser. 1 No. 2, Leipzig.
Helmholtz, H.v., 1888: Ueber atmosphaerische Bewegungen. Sitzungsberichte der Koeniglich
Preussischen Akademie der Wissenschaften zu Berlin, 46, 647–663.
http://bibliothek.bbaw.de/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=10sitz/1888-1&seite:int=665
IOC Manuals and Guides No. 56
198
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Hill, K. D., T. M. Dauphinee and D. J. Woods, 1986: The extension of the Practical Salinity Scale
1978 to low salinities, IEEE J. Oceanic Eng., 11, 109–112.
Huang, R. X., X. Jin and X. Zhang, 2001: An oceanic general circulation model in pressure
coordinates. Advances in Atmospheric Physics, 18, 1-‐‑22.
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, 2005: Guideline on the Use of Fundamental Physical Constants and Basic Constants of
Water. The International Association for the Properties of Water and Steam. Gaithersburg,
Maryland, USA, September 2001. Revised July 2005, available from http://www.iapws.org
IAPWS, 2007: Revised Release on the IAPWS Industrial Formulation 1997 for the
Thermodynamic Properties of Water and Steam, Lucerne, Switzerland, August 2007,
available from www.iapws.org
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 www.iapws.org. This Release is referred to in
the text as IAPWS-‐‑08.
IAPWS, 2008b: Revised Release on the Pressure along the Melting and Sublimation Curves of
Ordinary Water Substance. The International Association of the Properties of Water and
Steam, Berlin, Germany, September 2008, available from http://www.iapws.org.
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 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 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.
IAPWS, 2012: Guideline on a Low-‐‑Temperature Extension of the IAPWS-‐‑95 Formulation for
Water Vapor. The International Association for the Properties of Water and Steam.
Boulder Colorado, USA, October 2012, available from http://www.iapws.org.
This
Guideline is referred to in the text as IAPWS-‐‑12.
IOC, 1987: GF3 -‐‑ A General Formatting System for Geo-‐‑Referenced Data. Vol. 2, Technical
Description of the GF3 Format and Code Tables. Intergovernmental Oceanographic
Commission, Manuals and Guides 17, UNESCO.
IOC, SCOR and IAPSO, 2010a: The international thermodynamic equation of seawater – 2010:
Calculation and use of thermodynamic properties. Intergovernmental Oceanographic
Commission, Manuals and Guides No. 56, UNESCO (English), 196 pp. Available from
www.TEOS-10.org [the present document, called the TEOS-‐‑10 manual]
IOC, SCOR and IAPSO, 2010b: The international thermodynamic equation of seawater – 2010: A
Summary for Policy Makers. Intergovernmental Oceanographic Commission (Brochures
Series). Available from http://www.TEOS-10.org
ISO, 1993: ISO Standards Handbook: Quantities and Units. International Organization for
Standardization, Geneva, Switzerland.
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
199
ISO/IEC, 2007: International Vocabulary of Metrology (VIM) – Basic and General Concepts and
Associated Terms, ISO/IEC Guide 99-‐‑12. available from http://www.bipm.org/en/publications/
Iudicone, D., G. Madec, and T. J. McDougall, 2008: Water-‐‑mass transformations in a neutral
density framework and the key role of light penetration. Journal of Physical Oceanography,
38, 1357-‐‑1376.
IUPAC, 1997: Compendium of Chemical Terminology, 2nd ed. (the "ʺGold Book"ʺ). Compiled
by A. D. McNaught and A. Wilkinson. Blackwell Scientific Publications, Oxford 1997.
XML on-‐‑line corrected version: http://goldbook.iupac.org (2006-‐‑) created by M. Nic, J. Jirat,
B. Kosata; updates compiled by A. Jenkins. ISBN 0-‐‑9678550-‐‑9-‐‑8. doi:10.1351/goldbook.
Jackett, D. R. and T. J. McDougall, 1997: A neutral density variable for the world’s oceans.
Journal of Physical Oceanography, 27, 237-‐‑263.
Jackett, D. R., T. J. McDougall, R. Feistel, D. G. Wright and S. M. Griffies, 2006: Algorithms for
density, potential temperature, Conservative Temperature and freezing temperature of
seawater. Journal of Atmospheric and Oceanic Technology, 23, 1709-‐‑1728.
Jacobson, M. Z., 2005: Fundamentals of Atmospheric Modeling, 2nd Edition, University Press,
Cambridge, UK.
Kawano, T., 2009: Method for salinity (conductivity ratio) measurement. Global Ship-‐‑based Repeat
Hydrography Manual, IOCCP Report No. 14, ICPO Publication Series no. 134, 13pp,
available from http://www.go-ship.org/HydroMan.html
Killworth, P. D., 1986: A Bernoulli inverse method for determining the ocean circulation.
Journal of Physical Oceanography, 16, 2031-‐‑2051.
Klocker, A. and T. J. McDougall, 2010a: Influence of the nonlinear equation of state on global
estimates of dianeutral advection and diffusion. Journal of Physical Oceanography, 40, 1690-‐‑
1709.
Klocker, A. and T. J. McDougall, 2010b: Quantifying the consequences of the ill-‐‑defined nature
of neutral surfaces. Journal of Physical Oceanography, 40, 1866-‐‑1880.
Klocker, A., T. J. McDougall and D. R. Jackett, 2009a: A new method for forming
approximately neutral surfaces. Ocean Sci., 5, 155-‐‑172.
http://www.ocean-sci.net/5/155/2009/os-5-155-2009.pdf
Klocker, A., T. J. McDougall and D. R. Jackett, 2009b: Corrigendum to "ʺA new method for
forming approximately neutral surfaces”, published in Ocean Science, 5, 155–172, 2009,
Ocean Sci., 5, 191-‐‑191. http://www.ocean-sci.net/5/191/2009/os-5-191-2009.pdf
Knudsen, M., 1901: Hydrographische Tabellen, G. E. C. Gad, Copenhagen; L. Friedrichsen & Co.,
Hamburg; Williams & Norgate, London; and Buchdruckerei Bianco Luno, 63pp.
Kremling, K., 1971: New Method for measuring density of seawater, Nature, 229, 109–110.
Landau, L. D. and E. M. Lifshitz, 1959: Fluid Mechanics. Pergamon, 536 pp.
Landau, L. D. and E. M. Lifshitz, 1980: Statistical Physics. Reed Educational and Professional
Publishing Ltd. 544 pp.
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.
Leppäranta, M., K. Myrberg, 2009: Physical Oceanography of the Baltic Sea. Springer Verlag
Heidelberg, Germany.
Losch, M., A. Adcroft and J.-‐‑M. Campin, 2004: How sensitive are coarse general circulation
models to fundamental approximations in the equations of motion? Journal of Physical
Oceanography, 34, 306-‐‑319.
Mares, R. and J. Kalova, 2008: Thermophysical properties converted from data and equations
based on old temperature scales. In: Water, Steam, and Aqueous Solutions: Advances in
Science and Technology for Power Generation. Proceedings of the 15th International
Conference on the Properties of Water and Steam. September 7-‐‑11, 2008, Berlin/Germany.
ISBN 978-‐‑3-‐‑931384-‐‑64-‐‑7.
Marion, G. M., F. J. Millero, and R. Feistel, 2009: Precipitation of solid phase calcium
carbonates and their effect on application of seawater SA − T − P models, Ocean Sci., 5,
285-‐‑291. http://www.ocean-sci.net/5/285/2009/os-5-285-2009.pdf
IOC Manuals and Guides No. 56
200
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
McDougall, T. J., 1984: The relative roles of diapycnal and isopycnal mixing on subsurface
water mass conversion. Journal of Physical Oceanography, 14, 1577-‐‑1589.
McDougall, T. J., 1987a: Neutral surfaces. Journal of Physical Oceanography, 17, 1950-‐‑1964.
McDougall, T. J., 1987b: Thermobaricity, cabbeling, and water-‐‑mass conversion. J. Geophys.
Res., 92, 5448-‐‑5464.
McDougall, T. J., 1988: Neutral-‐‑surface potential vorticity. Progress in Oceanography, 20, 185-‐‑
221.
McDougall, T. J., 1989: Streamfunctions for the lateral velocity vector in a compressible ocean.
J. Marine Research, 47, 267-‐‑284.
McDougall, T. J., 1991: Parameterizing mixing in inverse models in Dynamics of Oceanic Internal
Gravity Waves, edited by P. Müller and D. Henderson, Proceedings of the sixth ’Aha
Huliko’a Hawaiian Winter Workshop, University of Hawaii at Manoa, 355-‐‑386.
McDougall, T. J., 1995: The influence of ocean mixing on the absolute velocity vector. Journal of
Physical Oceanography, 25, 705-‐‑725.
McDougall, T. J., 2003: Potential enthalpy: A conservative oceanic variable for evaluating heat
content and heat fluxes. Journal of Physical Oceanography, 33, 945-‐‑963.
McDougall T. J. and P. M. Barker, 2011: Getting started with TEOS-‐‑10 and the Gibbs Seawater
(GSW) Oceanographic Toolbox, 28pp., SCOR/IAPSO WG127, ISBN 978-‐‑0-‐‑646-‐‑55621-‐‑5,
available from www.TEOS-10.org
McDougall, T. J., J. A. Church and D. R. Jackett, 2003: Does the nonlinearity of the equation of
state impose an upper bound on the buoyancy frequency? Journal of Marine Research, 61,
745-‐‑764.
McDougall, T. J. and R. Feistel, 2003: What causes the adiabatic lapse rate? Deep-‐‑Sea Research I,
50, 1523-‐‑1535.
McDougall, T. J., R. J. Greatbatch and Y. Lu, 2002: On conservation equations in oceanography:
How accurate are Boussinesq ocean models? Journal of Physical Oceanography, 32, 1574-‐‑
1584.
McDougall, T. J. and D. R. Jackett, 1988: On the helical nature of neutral trajectories in the
ocean. Progress in Oceanography, 20, 153-‐‑183.
McDougall, T. J. and D. R. Jackett, 2005a: An assessment of orthobaric density in the global
ocean. Journal of Physical Oceanography, 35, 2054–2075.
McDougall, T. J. and D. R. Jackett, 2005b: The material derivative of neutral density. Journal of
Marine Research, 63, 159-‐‑185.
McDougall, T. J. and D. R. Jackett, 2007: The thinness of the ocean in S − Θ − p space and the
implications for mean diapycnal advection. Journal of Physical Oceanography, 37, 1714-‐‑
1732.
McDougall, T. J., D. R. Jackett, F. J. Millero, R. Pawlowicz and P. M. Barker, 2012: A global
algorithm for estimating Absolute Salinity. Ocean Science, 8, 1123-‐‑1134. http://www.oceansci.net/8/1123/2012/os-8-1123-2012.pdf The computer software is available from www.TEOS10.org
McDougall, T. J., D. R. Jackett, D. G. Wright and R. Feistel, 2003: Accurate and computationally
efficient algorithms for potential temperature and density of seawater. J. Atm. Ocean.
Technol., 20, 730-‐‑741.
McDougall, T. J. and A. Klocker, 2010: An approximate geostrophic streamfunction for use in
density surfaces. Ocean Modelling, 32, 105-‐‑117.
McDougall, T. J. and P. C. McIntosh, 2001: The temporal-‐‑residual-‐‑mean velocity. Part II:
Isopycnal interpretation and the tracer and momentum equations. Journal of Physical
Oceanography, 31, 1222-‐‑1246.
McDougall, T. J., S. A. Thorpe and C. H. Gibson, 1988: Small-‐‑scale turbulence and mixing in
the ocean: A glossary, in Small-‐‑scale turbulence and mixing in the ocean, edited by J. C. J.
Nihoul and B. M. Jamart, Elsevier, Amsterdam. 3-‐‑9.
McDougall T. J. and S. J. Wotherspoon, 2014: A simple modification of Newton’s method to
achieve convergence of order 1+ 2 . Applied Mathematics Letters, 29, 20-‐‑25.
http://dx.doi.org/10.1016/j.aml.2013.10.008
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
201
Millero, F. J., 1974: Seawater as a multicomponent electrolyte solution. The Sea, Vol. 5, E. D.
Goldberg, Ed., Wiley-‐‑Interscience, 3-‐‑80.
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., 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.
Millero, F. J., Forsht, D., Means, D., Giekes, J., Kenyon, K., 1978: The density of North Pacific
Ocean waters. J. Geophys. Res., 83, 2359-‐‑2364.
Millero, F. J., Gonzalez, A., Brewer, P.G., Bradshaw, A., 1976a: The density of North Atlantic
and North Pacific deep waters. Earth and Planetary Sci. Letters, 32, 468-‐‑472.
Millero, F. J., A. Gonzalez, A. and G. K. Ward, 1976c: The density of seawater solutions at one
atmosphere as a function of temperature and salinity. J. Mar. Res., 34, 61-‐‑93.
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., Lawson, D., Gonzalez, A., 1976b: The density of artificial river and estuarine
waters. J. Geophys. Res., 81, 1177-‐‑1179.
Millero, F. J. and W. H. Leung, 1976: The thermodynamics of seawater at one atmosphere, Am.
J. Sci., 276, 1035– 1077.
Millero, F. J. and X. Li, 1994: Comments on “On equations for the speed of sound in seawater”
[J. Acoust. Soc. Am. 94, 255-‐‑275 (1993)]. J. Acoust. Soc. Am. 95, 2757-‐‑2759.
Millero, F. J. and K. Kremling, 1976: The densities of Baltic Sea waters. Deep-‐‑Sea Research, 23,
1129-‐‑1138.
Millero, F. J., G. Perron and J. F. Desnoyers, 1973: Heat capacity of seawater solutions from 5 to
35 °C and .05 to 22 %0 Chlorinity. J. Geophys. Res., 78, 4499-‐‑4506.
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.
Mohr, P.J. and B. N. Taylor, 2005: CODATA recommended values of the fundamental physical
constants: 2002. Rev. Mod. Phys., 77, 1-‐‑107.
Mohr, P. J., B. N. Taylor and D. B. Newell, 2008: CODATA recommended values of the
fundamental physical constants: 2006. Rev. Mod. Phys., 80, 633-‐‑730; also published in J.
Phys. Chem. Ref. Data, 37, 1187-‐‑1284 (2008).
Montgomery, R. B., 1937. A suggested method for representing gradient flow in isentropic
surfaces. Bull. Amer. Meteor. Soc. 18, 210–212.
Moritz, H., 2000: Geodetic reference system 1980. Journal of Geodesy, 74, 128-‐‑133.
Munk, W. H., 1966: Abyssal recipes. Deep-‐‑Sea Res., 13, 707–730.
Munk, W. H. and C. Wunsch, 1998: Abyssal recipes II: Energetics of tidal and wind mixing.
Deep-‐‑Sea Res. I, 45, 1977–2010.
Needler, G. T., 1985: The absolute velocity as a function of conserved measurable quantities.
Progress in Oceanography, 14, 421-‐‑429.
Onsager, L. 1931a: Reciprocal relations in irreversible processes. I. Physical Review, 37, 405-‐‑426.
Onsager, L. 1931b: Reciprocal relations in irreversible processes. II. Physical Review, 38, 2265-‐‑
2279.
Osborn, T. R., 1980: Estimates of the local rate of vertical diffusion from dissipation
measurements. J. Phys. Oceanogr., 10, 83–89.
Pawlowicz, R., 2010a: A model for predicting changes in the electrical conductivity, Practical
Salinity, and Absolute Salinity of seawater due to variations in relative chemical
composition. Ocean Science, 6, 361–378.
http://www.ocean-sci.net/6/361/2010/os-6-361-2010.pdf
Pawlowicz, R., 2010b: What every oceanographer needs to know about TEOS-‐‑10 (The TEOS-‐‑10
Primer), unpublished manuscript, available from www.TEOS-10.org
Pawlowicz, R., T. McDougall, R. Feistel and R. Tailleux, 2012: An historical perspective on the
development of the Thermodynamic Equation of Seawater – 2010: Ocean Sci., 8, 161-‐‑174.
http://www.ocean-sci.net/8/161/2012/os-8-161-2012.pdf
IOC Manuals and Guides No. 56
202
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Pawlowicz, R., D. G. Wright and F. J. Millero, 2011: The effects of biogeochemical processes on
oceanic conductivity/salinity/density relationships and the characterization of real
seawater. Ocean Science, 7, 363–387. http://www.ocean-sci.net/7/363/2011/os-7-363-2011.pdf
Picard, A., R. S. Davis, M. Gläser and K. Fujii, 2008: Revised formula for the density of moist
air (CIPM-‐‑2007), Metrologia, 45, 149-‐‑155.
Planck, M., 1888: Das chemische Gleichgewicht in verdünnten Lösungen. Annalen der Physik
und Chemie, Neue Folge, 34, 139-‐‑154. Available from
http://www.archive.org/stream/annalenderphysi53unkngoog#page/n151/mode/1up
Planck, M. 1935: Bemerkungen über Quantitätsparameter, Intensitätsparameter und stabiles
Gleichgewicht. Physica II, 1029-‐‑1032. Reprinted in Ebeling, W., Hoffmann, D. (Eds.), 2008,
Über Thermodynamische Gleichgewichte, von Max Planck. Verlag Harry Deutsch, Frankfurt
am Main, 236 pp.
Preston-‐‑Thomas, H., 1990: The international temperature scale of 1990 (ITS-‐‑90), Metrologia 27,
3-‐‑10.
Pruppacher, H. R. and J. D. Klett, 1997: Microphysics of Clouds and Precipitation. Second Revised
and Enlarged Edition with an Introduction to Cloud Chemistry and Cloud Electricity,
Kluwer Academic Publishers, Dordrecht/Boston/London.
Reid, J. L. and R. J. Lynn, 1971: On the influence of the Norwegian-‐‑Greenland and Weddell
seas upon the bottom waters of the Indian and Pacific oceans. Deep-‐‑Sea Res., 18, 1063-‐‑1088.
Robinson, R. A., 1954: The vapour pressure and osmotic equivalence of sea water. Journal of
the Marine Biological Association of the U.K. 33, 449-‐‑455.
Roquet, F., G. Madec, T. J. McDougall and P. M. Barker, 2015: Accurate polynomial
expressions for the density and specific volume of seawater using the TEOS-‐‑10 standard.
Ocean Modelling, 90, 29-‐‑43, http://dx.doi.org/10.1016/j.ocemod.2015.04.002
Reissmann, J. H., H. Burchard, R. Feistel, E. Hagen, H. U. Lass, V. Mohrholz, G. Nausch, L.
Umlauf , G. Wieczorek, 2009: State-‐‑of-‐‑the-‐‑art review on vertical mixing in the Baltic Sea
and consequences for eutrophication. Progress in Oceanography, 82, 47-‐‑80.
Ripple D., B. Fellmuth, J. Fischer, G. Machin, P. Steur, O. Tamura and D. R. White, 2008: Report
from the Mise en Pratique Task Group: the next international temperature scale and the
mise en pratique for the definition of the kelvin. CCT/08-‐‑17/rev. 8pp. Available from
http://www.bipm.org/cc/CCT/Allowed/24/D17_rev_MePK_Report_2008c.pdf
Röntgen, W. C., 1892: Ueber die Constitution des flüssigen Wassers, Annalen der Physik, 45, 91-‐‑
97.
Ruddick, B., 1983: A practical indicator of the stability of the water column to double-‐‑diffusive
activity. Deep-‐‑Sea Res., 30, 1105–1107.
Rudtsch, S. and J. Fischer, 2008: Temperature measurements according to the International
Temperature Scale of 1990 and its associated uncertainties. Accred. Qual. Assur., 13, 607–
609.
Rusby, R. L., 1991: The conversion of thermal reference values to the ITS-‐‑90. J. Chem.
Thermodynamics, 23, 1153-‐‑1161.
Rusby, R. L. and R. White, 2003: CCT Workshop: Toward the ITS-‐‑XX: Summary of the
proceedings, CCT/03-‐‑01.
Sarmiento, J. L. and K. Bryan, 1982: An ocean transport model for the North Atlantic. Journal of
Geophysical Research, 87(C1), 394-‐‑408.
Sarmiento, J. L. and N. Gruber, 2006: Ocean Biogeochemical Dynamics, Princeton University
Press, Princeton, 503pp.
Saunders, P. M, 1981: Practical conversion of pressure to depth. Journal of Physical
Oceanography, 11, 573-‐‑574.
Saunders, P. 1990: The International Temperature Scale of 1990, ITS-‐‑90. WOCE Newsletter 10,
IOS, Wormley, UK.
Saunders, P. M., 1995: The Bernoulli function and flux of energy in the ocean. J. Geophys. Res.,
100, 22,647 -‐‑ 22,648.
Saunders, P. M. and N. P. Fofonoff, 1976: Conversion of pressure to depth in the ocean. Deep-‐‑
Sea Research, 23, 109-‐‑111.
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
203
Seitz, S., R. Feistel, D.G. Wright, S. Weinreben, P. Spitzer and P. de Bievre, 2011: Metrological
Traceability of Oceanographic Salinity Measurement Results. Ocean Science, 7, 45–62.
http://www.ocean-sci.net/7/45/2011/os-7-45-2011.pdf
Seitz, S., P. Spitzer and R. J. C. Brown, 2008: Consistency of practical salinity measurements
traceable to primary conductivity standards: Euromet project 918, Accred. Qual. Assur. 13,
601–605.
Seitz, S., P. Spitzer and R. J. C. Brown, 2010: CCQM-‐‑P111 study on traceable determination of
practical salinity and mass fraction of major seawater components. Accred. Qual. Assur.
15, 9–17.
Sherwood, T. K., P. L. T. Brian and R. E. Fisher, 1967: Desalination by reverse osmosis, Ind.
Eng. Chem. Fundamen., 6, 2-‐‑12.
Straub, D. N., 1999: On thermobaric production of potential vorticity in the ocean. Tellus A, 51,
314-‐‑325.
Sutcliffe, R. C. A., 1947: A contribution to the problem of development. Quart. J. Roy. Meteor.
Soc., 73, 370-‐‑383.
Tailleux, R., 2010: Identifying and quantifying nonconservative energy production/destruction
terms in hydrostatic Boussinesq primitive equation models. Ocean Modelling, 34, 125-‐‑136.
Thomson, W., 1851: On the dynamical theory of heat; with numerical results deduced from
Mr. Joule'ʹs equivalent of a thermal unit and M. Regnault'ʹs observations on steam. Trans.
Roy. Soc. Edinburgh, 3, 48-‐‑52.
UNESCO, 1981: The Practical Salinity Scale 1978 and the International Equation of State of
Seawater 1980. UNESCO technical papers in marine science 36, 25pp.
UNESCO, 1983: Algorithms for computation of fundamental properties of seawater. UNESCO
technical papers in marine science 44, 53pp.
Veronis, G., 1975: The role of models in tracer studies, in: Numerical Models of Ocean Circulation,
pp. 133–146, National Academy of Science.
Wagner, W. and A. Pruß, 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.
Warren, B. A., 1999: Approximating the energy transport across oceanic sections. J. Geophys.,
Res., 104, 7915-‐‑7919.
Warren, B. A., 2006: The first law of thermodynamics in a salty ocean. Prog. in Oceanogr., 70,
149-‐‑167.
Warren, B. A., 2009: Note on the vertical velocity and diffusive salt flux induced by
evaporation and precipitation. Journal of Physical Oceanography, 39, 2680–2682.
Weir, R. D. and R. N. Goldberg, 1996: On the conversion of thermodynamic properties to the
bases of the International Temperature Scale of 1990. J. Chem. Thermodyn., 28, 261-‐‑276.
Wieser, M. E., 2006: Atomic weights of the elements 2005 (IUPAC Technical Report). Pure
Appl. Chem. 78, 2051-‐‑2066. www.iupac.org/publications/pac/78/11/2051/pdf/
Wolf, H., 2008: Determination of water density: limitations at the uncertainty level of 1 x 10–6,
Accred. Qual. Assur. 13, 587–591.
Woolley, H.W., 1980: Thermodynamic properties for H2O in the ideal gas state, in: Straub, J.,
and Scheffler, K. (Eds.), Water and Steam, Proceedings of the 9th International Conference
on the Properties of Steam, München, 1979. Pergamon Press, Oxford-‐‑New York-‐‑Toronto-‐‑
Sydney-‐‑Paris-‐‑Frankfurt, pp.166–175.
Wooster, W. S., A. J. Lee and G. Dietrich, 1969: Redefinition of salinity, Limnology.
Oceanography, 14, 437-‐‑438.
Wright, D. G., R. Feistel, J. H. Reissmann, K. Miyagawa, D. R. Jackett, W. Wagner, U. Overhoff,
C. Guder, A. Feistel and G. M. Marion, 2010: Numerical implementation and
oceanographic application of the thermodynamic potentials of liquid water, water
vapour, ice, seawater and humid air -‐‑ Part 2: The library routines. Ocean Science, 6, 695-‐‑
718. http://www.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
Wright, D. G., R. Pawlowicz, T. J. McDougall, R. Feistel and G. M. Marion, 2011: Absolute
Salinity, “Density Salinity” and the Reference-‐‑Composition Salinity Scale: present and
IOC Manuals and Guides No. 56
204
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
future use in the seawater standard TEOS-‐‑10. Ocean Sci., 7, 1-‐‑26. http://www.oceansci.net/7/1/2011/os-7-1-2011.pdf
You, Y. and T. J. McDougall, 1990: Neutral surfaces and potential vorticity in the World'ʹs
Oceans. Journal of Geophysical Research, 95, 13235-‐‑13261.
Young, W. R., 2010: Dynamic enthalpy, Conservative Temperature, and the seawater
Boussinesq approximation. Journal of Physical Oceanography, 40, 394–400.
Zhang, H-‐‑M and N. G. Hogg, 1992: Circulation and water mass balance in the Brazil Basin. J.
Marine Research, 50, 385-‐‑420.
Zika, J. D., T. J. McDougall and B. M. Sloyan, 2010a: A tracer-‐‑contour inverse method for
estimating ocean circulation and mixing. J. Physical Oceanography, 40, 26–47.
Zika, J. D., T. J. McDougall and B. M. Sloyan, 2010b: Weak mixing in the eastern North
Atlantic: An application of the tracer-‐‑contour inverse method. J. Physical Oceanography,
40, 1881-‐‑1893.
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
205
Index
Absolute Pressure, 9, 73, 165
Absolute Salinity SA = SAdens , 11-‐‑15, 76-‐‑85,
112-‐‑121, 166
Absolute Salinity Anomaly, 13, 78-‐‑85, 166
absolute temperature, 69
Added-‐‑mass Salinity SAadd , 12, 79-‐‑81, 166
adiabatic lapse rate, 25
Bernoulli function, 47, 136
boiling temperature, 60
Boussinesq approximation, 117
buoyancy frequency, 32
cabbeling coefficient, 31, 100, 122 168, 191
chemical potentials, 19, 168, 192
Chlorinity, 11, 74-‐‑76, 165
composition variation, 3, 11-‐‑15, 82-‐‑85,
112-‐‑116
“conservative” property, 87-‐‑90
Conservative Temperature Θ , 4-‐‑5, 8, 18,
22, 27, 106-‐‑111, 117-‐‑121, 167
Coriolis parameter, 148, 169
Cunningham streamfunction, 50,
density, 18, 129-‐‑133, 165-‐‑167
density, 75-‐‑term expression, 129-‐‑131,
163-‐‑164
Density Salinity SAdens = SA , 11-‐‑15, 76-‐‑85,
112-‐‑120, 166
dianeutral advection, 121-‐‑123
dianeutral velocity, 121-‐‑123
dynamic height anomaly, 48
enthalpy, 18-‐‑19, 20, 87-‐‑94, 132-‐‑133, 134, 139,
167, 191-‐‑194
enthalpy as thermodynamic potential, 18,
19, 132-‐‑133, 191-‐‑194
entropy, 20, 26, 87, 92, 95, 97, 102-‐‑106, 167,
191-‐‑194
EOS-‐‑80, 2, 85, Fig. A.5.2
First Law of Thermodynamics, 95-‐‑98,
132-‐‑139
freshwater content, 46
freshwater flux, 46
freezing temperature, 53
fundamental thermodynamic relation, 87
fugacity, 62
geostrophic streamfunctions, 42-‐‑51
Gibbs function of ice Ih, 7, 157
Gibbs function of pure water, 15-‐‑17, 86,
152-‐‑155, 155
Gibbs function of seawater, 5, 15-‐‑17, 86,
156
gravitational acceleration, 146
GSW Oceanographic Toolbox, 183-‐‑186
haline contraction coefficients, 23, 32,
101-‐‑102, 168
heat transport, 5, 27, 46-‐‑47, 95-‐‑100, 117-‐‑121,
132-‐‑139
heat diffusion, 5, 27, 46-‐‑47, 95-‐‑100, 117-‐‑121,
132-‐‑139
Helmholtz energy, 21
Helmholtz function of fluid water, 152-‐‑154
Helmholtz function of moist air, 159-‐‑162
IAPSO, 3
IAPWS, 4,
IAPWS-‐‑95, 15, 140, 152-‐‑154
IAPWS-‐‑06, 140, 157-‐‑158
IAPWS-‐‑08, 15, 140, 156
IAPWS-‐‑09, 15, 140, 155
IAPWS-‐‑10, 140, 159-‐‑162
internal energy, 20, 87, 132-‐‑139
IPTS-‐‑68 temperature, 3, 9, 69-‐‑72, 147-‐‑149
IOC, 7, 68
ionic strength, 169
isentropic and adiabatic
compressibility, 22, 32, 168
isochoric heat capacity, 24, 167
isobaric heat capacity, 24, 25, 92-‐‑94, 95,
97, 167, 194
isopycnal-‐‑potential-‐‑vorticity, 45
isothermal compressibility, 21, 24
“isobaric-‐‑conservative” property, 87-‐‑90
ITS-‐‑90 temperature, 3, 9, 69-‐‑72, 147-‐‑151
latent heat of evaporation, 60
latent heat of melting, 55
material derivatives, 117-‐‑123
molality, 21, 64, 169
Montgomery streamfunction, 49-‐‑50, 168
IOC Manuals and Guides No. 56
206
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Neutral Density, 39, 127
neutral helicity, 30, 35-‐‑39, 44, 130, 168
Neutral-‐‑Surface-‐‑Potential-‐‑Vorticity, 42-‐‑45,
168
neutral tangent plane, 29, 30, 31, 32-‐‑39,
42-‐‑44, 99, 121-‐‑123
nomenclature, 165-‐‑170
orthobaric density, 128-‐‑129
osmotic coefficient, 21, 65, 172
osmotic pressure, 65
Preformed Salinity S* , 12, 79-‐‑81, 112-‐‑118
potential density, 28
potential enthalpy h 0 , 27, 167
“potential” property, 90-‐‑92
potential temperature θ , 26, 95, 167,
191-‐‑194
potential vorticity, 38, 42-‐‑45, 168
Practical Salinity S P , 9, 75, 147-‐‑151
pressure, 9, 73, 165-‐‑166
Absolute Pressure, 9, 73, 165
gauge pressure, 9, 73, 166
sea pressure, 9, 73, 165
Pressure-‐‑Integrated Steric Height, 51
recommended nomenclature, 165-‐‑170
recommended symbols, 165-‐‑170
Reference Composition (RC), 10-‐‑11, 74-‐‑81
Reference-‐‑Composition Salinity S R , 10,
11, 74-‐‑85, 112-‐‑116
Reference-‐‑Composition Salinity Scale, 10,
11, 74-‐‑85
Reference Density, 13
Reference Salinity S R , 10-‐‑11, 74-‐‑85,
112-‐‑116
relative humidity, 62-‐‑65
saline contraction coefficients, 23, 32,
99-‐‑100, 101-‐‑102, 121-‐‑129, 168
salinity
Absolute Salinity SA = SAdens , 11-‐‑15,
76-‐‑85, 112-‐‑118, 166
Added-‐‑mass Salinity SAadd , 12, 78-‐‑81,
166
Density Salinity SAdens = SA , 11-‐‑15,
76-‐‑85, 112-‐‑118, 166
Practical Salinity S P , 9, 76-‐‑85, 112-‐‑118
Preformed Salinity S* , 12, 78-‐‑81,
112-‐‑118, 166
Reference Salinity S R , 10-‐‑11, 74-‐‑85,
112-‐‑118, 166
Solution Salinity S Asoln , 12, 78-‐‑81, 166
IOC Manuals and Guides No. 56
salinity in ocean models, 112-‐‑119
SCOR, 3
sea pressure, 73, 165
SIA software library, 171-‐‑181
SI-‐‑traceability of salinity, 149-‐‑151
slopes of surfaces, 40-‐‑42
Solution Salinity S Asoln , 12, 78-‐‑81, 166
sound speed, 22
specific volume, 18, 29, 47-‐‑53, 111-‐‑112,
168
specific volume anomaly, 29, 47-‐‑53
stability ratio, 39, 40-‐‑41, 122-‐‑124, 166
Standard Seawater (SSW), 10-‐‑15, 74-‐‑81
sublimation enthalpy, 57
sublimation pressure, 56
75-‐‑term expression for specific volume,
130-‐‑133, 163-‐‑164
TEOS-‐‑10, 4-‐‑8, 67-‐‑68
TEOS-‐‑10 web site, 67, 142
Temperature
absolute temperature T , 69
Celsius temperature t , 69
Conservative Temperature Θ , 5, 7-‐‑8,
18, 22, 23 27, 106-‐‑111, 117-‐‑121, 167,
191-‐‑194
in situ temperature, 26
potential temperature θ , 26, 87,
106-‐‑111, 167, 191-‐‑194
temperature of maximum density, 65-‐‑66
thermal expansion coefficients, 22, 25, 32,
99-‐‑102, 121-‐‑124, 167-‐‑168
“thermal wind”, 34-‐‑35, 84-‐‑85, 131
thermodynamic potentials, 3, 132-‐‑133,
191-‐‑194
thermobaric coefficient, 30, 36, 44, 99, 168
total energy, 47, 88-‐‑91, 135-‐‑137
Turner angle, 39
vapour pressure, 59
water-‐‑mass transformation, 123-‐‑125
WG127, 3
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
207
Changes made to this TEOS-‐‑10 manual, since the 13th April 2010
version which was printed by IOC.
17th April 2010
Page 102, line 6, an error in the inequality fixed, and changed to be in terms of ĥ .
4th July 2010
Page 39, Eqn. (3.20.4). An error fixed in this equation; an extra factor of Rρ .
20th Aug 2010
Throughout the document, changed h p to hP and so help to clarify when pressure
must be expressed in Pa rather than dbar.
3rd Sept 2010
Extensive changes to page 122 (the description of enthalpy obtained from the 25-‐‑term
expression for density), page 174 (the list of GSW Toolbox function names) and page
179 (the description of how the 25-‐‑term expression for density, along with
knowledge of the exact Gibbs function at p = 0 dbar, can be used as the full
thermodynamic potential of seawater).
14th Sept 2010
Corrected a typo on the left-‐‑hand side of Eqn. (A.11.16); hS was replaced by hˆS .
A
A
22nd Sept 2010
Page 100. Changes in Eqns. (A.18.5) – (A.18.7).
Page 122. Simplified Eqn. (A.30.6).
7th Feb 2011
Changes to appendices A.5 and A.20 concerning the calculation of the Absolute
Salinity Anomaly from the look-‐‑up table method of McDougall et al. (2012), and the
use of this changed method in ocean modelling.
20th March 2011 Changes to appendix A.30 and appendix K, replacing the 25-‐‑term rational function for
ρˆ ( SA , Θ, p ) with a 48-‐‑term version.
10th May 2013
Material added to sections 3.11, 3.12, 3.13, 3.20 and appendices A.22 and A.23.
4th May 2015
Changes to appendix A.30 and appendix K, replacing the 48-‐‑term rational function for
v̂ SA ,Θ, p with a 75-‐‑term polynomial.
31st Oct 2015
Change to Eqn. (3.31.1).
2nd Nov 2015
Change in the definition of b just below Eqn. (3.20.14).
(
)
In this TEOS-‐‑10 Manual the text is 10.5 pt Palatino linotype at exactly 14 pt vertical
spacing. The references are 10 pt Palatino linotype at exactly 13 pt vertical spacing. The
text is both right and left justified. Left margin is 3.17cm and the right margin is 3.10cm
respectively. The top and bottom margins are both 2.05cm. The header and footer are
both 70% grey. The MATHTYPE size settings are 10.5, 7, 6, 16, 9, 75%, 100% and 150%
respectively.
IOC Manuals and Guides No. 56
Intergovernmental Oceanographic Commission (IOC)
United Nations Educational, Scientific and Cultural Organization
1, rue Miollis
75732 Paris Cedex 15, France
Tel: +33 1 45 68 10 10
Fax: +33 1 45 68 58 12
Website: http://ioc.unesco.org
Source Exif Data:
File Type : PDF File Type Extension : pdf MIME Type : application/pdf PDF Version : 1.7 Linearized : Yes Create Date : 2016:12:19 21:00:05Z Creator : Word Modify Date : 2016:12:21 12:01:23+11:00 Has XFA : No XMP Toolkit : Adobe XMP Core 5.4-c005 78.147326, 2012/08/23-13:03:03 Creator Tool : Word Metadata Date : 2016:12:21 12:01:23+11:00 Keywords : Producer : Mac OS X 10.10.5 Quartz PDFContext Format : application/pdf Title : Microsoft Word - Thermodynamic_Manual_04Oct16_1st_part.doc Document ID : uuid:fc930e88-1f23-4a43-bb80-64a94a8ca2a3 Instance ID : uuid:0b116101-a8f9-452a-8b27-072577653c67 Page Count : 218EXIF Metadata provided by EXIF.tools