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Manuals and Guides 56
Intergovernmental Oceanographic Commission
The international thermodynamic
equation of seawater – 2010:
Calculation and use of thermodynamic properties
Manuals and Guides 56
Intergovernmental Oceanographic Commission
The international thermodynamic
equation of seawater – 2010:
Calculation and use of thermodynamic properties
The authors are responsible for the choice and the presentation of the facts contained in this publication
and for the opinions expressed therein, which are not necessarily those of UNESCO, SCOR or IAPSO and
do not commit those Organizations.
The photograph on the front cover of a CTD and lowered ADCP hovering just below the sea surface was
taken south of Timor from the Southern Surveyor in August 2003 by Ann Gronell Thresher.
For bibliographic purposes, this document should be cited as follows:
IOC, SCOR and IAPSO, 2010: The international thermodynamic equation of seawater – 2010: Calculation and
use of thermodynamic properties. Intergovernmental Oceanographic Commission, Manuals and Guides No.
56, UNESCO (English), 196 pp.
Printed by UNESCO
(IOC/2010/MG/56)
© 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
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3. Derived Quantities …………………………………..……….………………. 26
3.1 Potential temperature …………………………………………………………………. 26
3.2 Potential enthalpy ………………………………………....………………………… 27
3.3 Conservative Temperature ……………….………………….………………………. 27
3.4 Potential density ……………………………………………….……………………… 28
3.5 Density anomaly …………………….………………………….…………………… 28
3.6 Potential density anomaly ………….…………………………….…………………… 29
3.7 Specific volume anomaly ………………………………………….…………………. 29
3.8 Thermobaric coefficient ………….……………………………………………………. 30
3.9 Cabbeling coefficient ………….…………………………………………………....…. 31
3.10 Buoyancy frequency ……….…………………………………………….……..….… 32
3.11 Neutral tangent plane …….……………………………………………………..…. 32
3.12 Geostrophic, hydrostatic and “thermal wind” equations …….………………. 33
3.13 Neutral helicity …………….…………………………………………………..….…. 34
3.14 Neutral Density ….…………………………………………………………..….…….. 35
3.15 Stability ratio …..………………………………………………………………...…. 36
3.16 Turner angle ….………………………………………………………………………. 36
3.17 Property gradients along potential density surfaces …………………………… 36
3.18 Slopes of potential density surfaces and neutral tangent planes compared ..… 37
3.19 Slopes of in situ density surfaces and specific volume anomaly surfaces …..… 37
3.20 Potential vorticity …………………………………….………………………………. 38
3.21 Vertical velocity through the sea surface …….…………………………………… 39
3.22 Freshwater content and freshwater flux …………………………………………. 40
3.23 Heat transport …………………….………………………………………………….. 40
3.24 Geopotential ………….……………………………………………………………….. 41
3.25 Total energy …………….…………………………………………………………….. 41
3.26 Bernoulli function ……….…………………………………………………………… 42
3.27 Dynamic height anomaly …………………………………………………………… 42
3.28 Montgomery geostrophic streamfunction ………….…………………………… 43
3.29 Cunningham geostrophic streamfunction ……….……………………………… 44
3.30 Geostrophic streamfunction in an approximately neutral surface ….………… 45
3.31 Pressure-integrated steric height ……..…………………………………………… 45
3.32 Pressure to height conversion …….………………………………………………… 46
3.33 Freezing temperature ……….…………………………………………………….….. 46
3.34 Latent heat of melting ….…………………………………………..………….…… 48
3.35 Sublimation pressure …………………………………………………………….… 49
3.36 Sublimation enthalpy …………………………………………………………….… 50
3.37 Vapour pressure ……………………………………………………………….…… 52
3.38 Boiling temperature ……….……………………………………….…………….… 53
3.39 Latent heat of evaporation ………………………………………………………… 53
3.40 Relative humidity and fugacity …………………………………………………… 55
3.41 Osmotic pressure …………………………………………………………………… 58
3.42 Temperature of maximum density …….….……………………………………… 59
4. Conclusions ………………………………………….……..……………………… 60
IOC Manuals and Guides No. 56
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Appendix A: Background and theory underlying the use of the
Gibbs function of seawater ………………..……………...... 62
A.1 ITS-90 temperature …………………………………………………………………... 62
A.2 Sea pressure, gauge pressure and absolute pressure …………………………….... 66
A.3 Reference Composition and the Reference-Composition Salinity Scale …….…... 67
A.4 Absolute Salinity ……………………………………………….……………………... 69
A.5 Spatial variations in seawater composition ……………………………….………... 75
A.6 Gibbs function of seawater ………………………………….…………….……..…... 78
A.7 The fundamental thermodynamic relation ………………………………….……... 79
A.8 The “conservative” and “isobaric conservative” properties ……………….…..…. 79
A.9 The “potential” property ……………….……….………………………………...... 82
A.10 Proof that θ = θ ( SA ,η ) and Θ = Θ ( SA , θ ) .…………………….…………………... 84
A.11 Various isobaric derivatives of specific enthalpy ……………...…..…………… 84
A.12 Differential relationships between η , θ , Θ and S A …………...……...….………. 86
A.13 The First Law of Thermodynamics ………………….……………………...…...... 87
A.14 Advective and diffusive “heat” fluxes ………………….………………..……...... 90
A.15 Derivation of the expressions for α θ , β θ , α Θ and β Θ ………………….……….. 92
A.16 Non-conservative production of entropy ………………………...……………...... 94
A.17 Non-conservative production of potential temperature ………………….……... 97
A.18 Non-conservative production of Conservative Temperature ………………….. 99
A.19 Non-conservative production of density and of potential density ………….. 102
A.20 The representation of salinity in numerical ocean models ………………........... 103
A.21 The material derivatives of S* , S A , S R and Θ in a turbulent ocean ……….... 108
A.22 The material derivatives of density and of locally-referenced
potential density; the dianeutral velocity e …………….……............................ 112
A.23 The water-mass transformation equation …………….…….................................. 113
A.24 Conservation equations written in potential density coordinates ………..……. 114
A.25 The vertical velocity through a general surface …………….………………….... 116
A.26 The material derivative of potential density ………………………..….……..... 116
A.27 The diapycnal velocity of layered ocean models (without rotation
of the mixing tensor) ……………………………………………….….….……..... 117
A.28 The material derivative of orthobaric density ………………………..….……..... 118
A.29 The material derivative of Neutral Density …………………………….……..... 119
A.30 Computationally efficient 25-term expressions for the density
of seawater in terms of Θ and θ ………………………..…………………….. 120
Appendix B: Derivation of the First Law of Thermodynamics ….…….…… 123
Appendix C: Publications describing the TEOS-10 thermodynamic
descriptions of seawater, ice and moist air ……...………..…..…. 130
Appendix D: Fundamental constants …….…………………..……………………... 133
IOC Manuals and Guides No. 56
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Appendix E: Algorithm for calculating Practical Salinity ……….…………….. 137
E.1 Calculation of Practical Salinity in terms of K15 ….……........................................... 137
E.2 Calculation of Practical Salinity at oceanographic temperature and pressure ..... 137
E.3 Calculation of conductivity ratio R for a given Practical Salinity .......................... 138
E.4 Evaluating Practical Salinity using ITS-90 temperatures ......................................... 139
E.5 Towards SI-traceability of the measurement procedure for Practical Salinity
and Absolute Salinity .................................................................................................. 139
Appendix F: Coefficients of the IAPWS-95 Helmholtz function of
fluid water (with extension down to 50K) …………...………...… 142
Appendix G: Coefficients of the pure liquid water Gibbs function
of IAPWS-09 ………………………...……………………….…...………. 145
Appendix H: Coefficients of the saline Gibbs function for seawater
of IAPWS-08 ………………………………………………..…….………. 146
Appendix I: Coefficients of the Gibbs function of ice Ih of IAPWS-06 …... 147
Appendix J: Coefficients of the Helmholtz function of moist air
of IAPWS-10 ……………………………..…….………………………… 149
Appendix K: Coefficients of 25-term expressions for the density
of seawater in terms of Θ and of θ …….……………………....... 153
Appendix L: Recommended nomenclature, symbols and
units in oceanography ………………………………………………… 156
Appendix M: Seawater-Ice-Air (SIA) library of computer software ……….. 162
Appendix N: Gibbs-SeaWater (GSW) Oceanographic Toolbox ..................... 173
Appendix O: Checking the Gibbs function of seawater against the
original thermodynamic data ……………………….………….…… 175
Appendix P: Thermodynamic properties based on g ( SA , t , p ) , h ( SA ,η , p ) ,
h ( SA , θ , p ) and hˆ ( SA , Θ, p ) …………………….…………………... 178
References ……………………………..……………………………….………….…….… 182
Index
……………………………………..……………………………….…………….….… 191
IOC Manuals and Guides No. 56
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Acknowledgements
This TEOS‐10 Manual reviews and summarizes the work of the SCOR/IAPSO Working
Group 127 on the Thermodynamics and Equation of State of Seawater. Dr John Gould and
Professor Paola Malanotte‐Rizzoli played pivotal roles in the establishment of the
Working Group and we have enjoyed rock‐solid scientific support from the officers of
SCOR, IAPSO and IOC. TJMcD wishes to acknowledge fruitful discussions with Drs
Jürgen Willebrand and Michael McIntyre regarding the contents of appendix B. We have
benefited from extensive comments on drafts of this manual by Dr Stephen Griffies and Dr
Allyn Clarke. Dr Harry Bryden is thanked for valuable and timely advice on the
treatment of salinity in ocean models. Louise Bell of CSIRO provided much appreciated
advice on the layout of this document. TJMcD and DRJ wish to acknowledge partial
financial support from the Wealth from Oceans National Flagship. This work contributes
to the CSIRO Climate Change Research Program. This document is based on work
partially supported by the U.S. National Science Foundation to SCOR under Grant No.
OCE‐0608600. FJM wishes to acknowledge the Oceanographic Section of the National
Science Foundation and the National Oceanic and Atmospheric Administration for
supporting his work.
This document has been written by the members of SCOR/IAPSO Working Group 127,
Trevor J. McDougall, (chair), CSIRO, Hobart, Australia
Rainer Feistel, Leibniz‐Institut fuer Ostseeforschung, Warnemuende, Germany
Daniel G. Wright+, Bedford Institute of Oceanography, Dartmouth, Canada
Rich Pawlowicz, University of British Columbia, Vancouver, Canada
Frank J. Millero, University of Miami, Florida, USA
David R. Jackett, CSIRO, Hobart, Australia
Brian A. King, National Oceanography Centre, Southampton, UK
Giles M. Marion, Desert Research Institute, Reno, USA
Steffen Seitz, Physikalisch‐Technische Bundesanstalt (PTB), Braunschweig, Germany
Petra Spitzer, Physikalisch‐Technische Bundesanstalt (PTB), Braunschweig, Germany
C‐T. Arthur Chen, National Sun Yat‐Sen University, Taiwan, R.O.C.
March 2010
+
deceased, 8th July 2010.
IOC Manuals and Guides No. 56
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Foreword
This document describes the International Thermodynamic Equation Of Seawater – 2010
(TEOS‐10 for short). This description of the thermodynamic properties of seawater and
of ice Ih has been adopted by the Intergovernmental Oceanographic Commission at its
25th Assembly in June 2009 to replace EOS‐80 as the official description of seawater and
ice properties in marine science.
Fundamental to TEOS‐10 are the concepts of Absolute Salinity and Reference
Salinity. These variables are described in detail here, emphasising their relationship to
Practical Salinity.
The science underpinning TEOS‐10 has been described in a series of papers
published in the refereed literature (see appendix C). The present document may be
called the “TEOS‐10 Manual” and acts as a guide to those published papers and
concentrates on how the thermodynamic properties obtained from TEOS‐10 are to be
used in oceanography.
In addition to the thermodynamic properties of seawater, TEOS‐10 also describes the
thermodynamic properties of ice and of humid air, and these properties are summarised
in this document. The TEOS‐10 computer software, this TEOS‐10 Manual and other
documents may be obtained from www.TEOS‐10.org.
When referring to the use of TEOS‐10, it is the present document that should be
referenced as IOC et al. (2010) [IOC, SCOR and IAPSO, 2010: The international
thermodynamic equation of seawater – 2010: Calculation and use of thermodynamic properties.
Intergovernmental Oceanographic Commission, Manuals and Guides No. 56, UNESCO
(English), 196 pp.].
This version of the TEOS‐10 Manual includes corrections up to 16th October 2010.
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, potential temperature, enthalpy and
potential enthalpy are calculated directly. When determined from the Gibbs function,
these quantities are fully consistent with each other. Entropy and enthalpy are required
for an accurate description of the advection and diffusion of heat in the ocean interior and
for quantifying the ocean’s role in exchanging heat with the atmosphere and with ice. The
Gibbs function is a function of Absolute Salinity, temperature and pressure. In contrast to
Practical Salinity, Absolute Salinity is expressed in SI units and it includes the influence of
the small spatial variations of seawater composition in the global ocean. Absolute Salinity
is the appropriate salinity variable for the accurate calculation of horizontal density
gradients in the ocean. Absolute Salinity is also the appropriate salinity variable for the
calculation of freshwater fluxes and for calculations involving the exchange of freshwater
with the atmosphere and with ice. Potential functions are included for ice and for moist
air, leading to accurate expressions for numerous thermodynamic properties of ice and air
including freezing temperature and latent heats of melting and of evaporation. This
TEOS‐10 Manual describes how the thermodynamic properties of seawater, ice and moist
air are used in order to accurately represent the transport of heat in the ocean and the
exchange of heat with the atmosphere and with ice.
IOC Manuals and Guides No. 56
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
1. Introduction
1.1 Oceanographic practice 1978 - 2009
The Practical Salinity Scale, PSS‐78, and the International Equation of State of Seawater
(Unesco (1981)) which expresses the density of seawater as a function of Practical Salinity,
temperature and pressure, have served the oceanographic community very well for thirty
years. The Joint Panel on Oceanographic Tables and Standards (JPOTS) (Unesco (1983))
also promulgated the Millero, Perron and Desnoyers (1973) algorithm for the specific heat
capacity of seawater at constant pressure, the Chen and Millero (1977) expression for the
sound speed of seawater and the Millero and Leung (1976) formula for the freezing point
temperature of seawater. Three other algorithms supported under the auspices of JPOTS
concerned the conversion between hydrostatic pressure and depth, the calculation of the
adiabatic lapse rate, and the calculation of potential temperature. The expressions for the
adiabatic lapse rate and for potential temperature could in principle have been derived
from the other algorithms of the EOS‐80 set, but in fact they were based on the formulas of
Bryden (1973). We shall refer to all these algorithms jointly as ‘EOS‐80’ for convenience
because they represent oceanographic best practice from the early 1980s to 2009.
1.2 Motivation for an updated thermodynamic description of seawater
In recent years the following aspects of the thermodynamics of seawater, ice and moist air
have become apparent and suggest that it is timely to redefine the thermodynamic
properties of these substances.
• Several of the polynomial expressions of the International Equation of State of
Seawater (EOS‐80) are not totally consistent with each other as they do not exactly
obey the thermodynamic Maxwell cross‐differentiation relations.
The new
approach eliminates this problem.
• Since the late 1970s a more accurate and more broadly applicable thermodynamic
description of pure water has been developed by the International Association for
the Properties of Water and Steam, and has appeared as an IAPWS Release (IAPWS‐
95). Also since the late 1970s some measurements of higher accuracy have been
made of several properties of seawater such as (i) heat capacity, (ii) sound speed and
(iii) the temperature of maximum density. These can be incorporated into a new
thermodynamic description of seawater.
• The impact on seawater density of the variation of the composition of seawater in
the different ocean basins has become better understood. In order to further
progress this aspect of seawater, a standard model of seawater composition is
needed to serve as a generally recognised reference for theoretical and chemical
investigations.
• The increasing emphasis on the ocean as being an integral part of the global heat
engine points to the need for accurate expressions for the entropy, enthalpy and
internal energy of seawater so that heat fluxes can be more accurately determined in
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).
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
•
•
3
The need for a thermodynamically consistent description of the interactions between
seawater, ice and moist air; in particular, the need for accurate expressions for the
latent heats of evaporation and freezing, both at the sea surface and in the
atmosphere.
The temperature scale has been revised from IPTS‐68 to ITS‐90 and revised IUPAC
(International Union of Pure and Applied Chemistry) values have been adopted for
the atomic weights of the elements (Wieser (2006)).
1.3 SCOR/IAPSO WG127 and the approach taken
In 2005 SCOR (Scientific Committee on Oceanic Research) and IAPSO (International
Association for the Physical Sciences of the Ocean) established Working Group 127 on the
“Thermodynamics and Equation of State of Seawater” (henceforth referred to as WG127).
This group has now developed a collection of algorithms that incorporate our best
knowledge of seawater thermodynamics. The present document summarizes the work of
SCOR/IAPSO Working Group 127.
To compute all thermodynamic properties of seawater it is sufficient to know one of its
so‐called thermodynamic potentials (Fofonoff 1962, Feistel 1993, Alberty 2001). It was J.W.
Gibbs (1873) who discovered that “an equation giving internal energy in terms of entropy and
specific volume, or more generally any finite equation between internal energy, entropy and specific
volume, for a definite quantity of any fluid, may be considered as the fundamental thermodynamic
equation of that fluid, as from it… may be derived all the thermodynamic properties of the fluid (so
far as reversible processes are concerned).”
The approach taken by WG127 has been to develop a Gibbs function from which all
the thermodynamic properties of seawater can be derived by purely mathematical
manipulations (such as differentiation).
This approach ensures that the various
thermodynamic properties are self‐consistent (in that they obey the Maxwell cross‐
differentiation relations) and complete (in that each of them can be derived from the given
potential).
The Gibbs function (or Gibbs potential) is a function of Absolute Salinity 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 which is the salinity argument of the
TEOS‐10 Gibbs function is the version of absolute salinity that provides the best estimate
of the density of seawater; another name for S A is “Density Salinity”.
IOC Manuals and Guides No. 56
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
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
ʺbestʺ estimate of Absolute Salinity of seawater with non‐standard composition are
immature and will undoubtedly change in the future, so we cannot recommend storing
Absolute Salinity in national data bases. Storage of a more robust intermediate value, the
Reference Salinity, 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
publications is a calculated quantity, being either potential temperature θ or
Conservative Temperature Θ .
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
5
In order to improve the determination of Absolute Salinity we need to begin collecting
and storing values of the salinity anomaly δ S A = SA − S R 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 S A = SA − S R can be calculated using an
inversion of the TEOS‐10 equation for density to determine S A . For completeness, it is
advisable to also report δ S A under the new GF3 code DELS.
The thermodynamic description of seawater and of ice Ih as defined in IAPWS‐08 and
IAPWS‐06 has been adopted as the official description of seawater and of ice Ih by the
Intergovernmental Oceanographic Commission in June 2009. These new international
standards were adopted while recognizing that the techniques for estimating Absolute
Salinity will likely improve over the coming decades, and the algorithm for evaluating
Absolute Salinity in terms of Practical Salinity, latitude, longitude and pressure will be
updated from time to time, after relevant appropriately peer‐reviewed publications have
appeared, and that such an updated algorithm will appear on the www.TEOS‐10.org web
site. Users of this software should always state in their published work which version of
the software was used to calculate Absolute Salinity.
The more prominent advantages of TEOS‐10 compared with EOS‐80 are
•
The Gibbs function approach allows the calculation of internal energy, entropy,
enthalpy, potential enthalpy and the chemical potentials of seawater as well as the
freezing temperature, and the latent heats of freezing and of evaporation. These
quantities were not available from the International Equation of State 1980 but are
essential for the accurate accounting of “heat” in the ocean and for the consistent
and accurate treatment of air‐sea and ice‐sea heat fluxes. For example, a new
temperature variable, Conservative Temperature, can be defined as being
proportional to potential enthalpy and is a valuable measure of the “heat” content
per unit mass of seawater for use in physical oceanography and in climate studies,
as it is approximately two orders of magnitude more conservative than potential
temperature.
•
For the first time the influence of the spatially varying composition of seawater can
systematically be taken into account through the use of Absolute Salinity. In the
open ocean, this has a non‐trivial effect on the horizontal density gradient computed
from the equation of state, and thereby on the ocean velocities and heat transports
calculated via the “thermal wind” relation.
•
The thermodynamic quantities available from the new approach are totally
consistent with each other.
•
The new salinity variable, Absolute Salinity, is measured in SI units. Moreover the
treatment of freshwater fluxes in ocean models will be consistent with the use of
Absolute Salinity, but is only approximately so for Practical Salinity.
•
The Reference Composition of standard seawater supports marine physicochemical
studies such as the solubility of sea salt constituents, the alkalinity, the pH and the
ocean acidification by rising concentrations of atmospheric CO2.
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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 ρ in terms of Conservative Temperature (rather than in
terms of in situ temperature) involving just 25 coefficients is also available and is described
in appendix A.30 and appendix K.
The input and output parameters of the GSW Oceanographic Toolbox are in units
which oceanographers will find more familiar than basic SI units. We expect that
oceanographers will mostly use this GSW Toolbox because of its greater simplicity and
computational efficiency, and because of the more familiar units compared with the SIA
library. The name GSW (Gibbs‐SeaWater) has been chosen to be similar to, but different
from the existing “sw” (Sea Water) library which is already in wide circulation. Both the
SIA and GSW libraries, together with this TEOS‐10 Manual are available from the website
www.TEOS‐10.org. Initially the SIA library is being made available in Visual Basic and
FORTRAN while the GSW library is available mainly in MATLAB.
After these descriptions in sections 2 and 3 of how to determine the thermodynamic
quantities and various derived quantities, we end with some conclusions (section 4).
Additional information on Practical Salinity, the Gibbs function, Reference Salinity,
composition anomalies, Absolute Salinity, and some fundamental thermodynamic
properties such as the First Law of Thermodynamics, the non‐conservative nature of many
oceanographic variables, a list of recommended symbols, and succinct lists of
thermodynamic formulae are given in the appendices. Much of this work has appeared
elsewhere in the published literature but is collected here in a condensed form for the
usersʹ convenience.
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1.5 A remark on units
The most convenient variables and units in which to conduct thermodynamic
investigations are Absolute Salinity S A in units of kg kg‐1, Absolute Temperature T (K),
and Absolute Pressure P in Pa. These are the parameters and units used in the SIA
software library. Oceanographic practice to date has used non‐basic‐SI units for many
variables, in particular, temperature is usually measured on the Celsius ( °C ) scale,
pressure is sea pressure quoted in decibars relative to the pressure of a standard
atmosphere (10.1325 dbar), while salinity has had its own oceanography‐specific scale, the
Practical Salinity Scale of 1978. In the GSW Oceanographic Toolbox we adopt °C for the
temperature unit, pressure is sea pressure in dbar and Absolute Salinity 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. There are also cases within the GSW Toolbox in which SI units are required and
this may occasionally cause some confusion. A common example of this issue arises when
a variable is differentiated or integrated with respect to pressure. Nevertheless, for many
applications it is deemed important to remain close to present oceanographic practice even
though it means that one has to be vigilant to detect those expressions that need a variable
to be expressed in the less‐familiar SI units.
1.6 Recommendations
In accordance with resolution XXV‐7 of the Intergovernmental Oceanographic
Commission at its 25th Assembly in June 2009, and the several Releases and Guidelines of
the International Association for the Properties of Water and Steam, the TEOS‐10
thermodynamic description of seawater, of ice and of moist air is recommended for use by
oceanographers in place of the International Equation Of State – 1980 (EOS‐80). The
software to implement this change is available at the web site www.TEOS‐10.org.
Under TEOS‐10 it is recognized that the composition of seawater varies around the
world ocean and that the thermodynamic properties of seawater are more accurately
represented as functions of Absolute Salinity 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 effectively interpolates
between a spatial atlas of these values. Then Absolute Salinity is estimated as the sum of
Reference Salinity and Absolute Salinity Anomaly. Of the four possible versions of
absolute salinity, the one that is used as the argument for the TEOS‐10 Gibbs function is
designed to provide accurate estimates of the density of seawater.
It is recognized that our knowledge of how to estimate seawater composition
anomalies and their effect on thermodynamic properties is limited. Nevertheless, we
should not continue to ignore the influence of these composition variations on seawater
properties and on ocean dynamics. As more knowledge is gained in this area over the
coming decade or so, and after such knowledge has been duly published in the scientific
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literature, any updated algorithm to evaluate the Absolute Salinity Anomaly will be
available (with its version number) from www.TEOS‐10.org.
The storage of salinity in national data bases should continue to occur as Practical
Salinity, as this will maintain continuity of this important time series. Oceanographic
databases label stored, processed or exported parameters with the GF3 code PSAL for
Practical Salinity and SSAL for salinity measured before 1978 (IOC, 1987). In order to
avoid possible confusion in data bases between different types of salinity it is very
strongly recommended that under no circumstances should either Reference Salinity or
Absolute Salinity be stored in national data bases.
In order to accurately calculate the thermodynamic properties of seawater, Absolute
Salinity must be calculated by first calculating Reference Salinity and then adding on the
Absolute Salinity Anomaly. Because Absolute Salinity is the appropriate salinity variable
for use with the equation of state, Absolute Salinity should be the salinity variable that is
published in oceanographic journals. The version number of the software, or the exact
formula, that was used to convert Reference Salinity into Absolute Salinity should always
be stated in publications. Nevertheless, there may be some applications where the likely
future changes in the algorithm that relates Reference Salinity to Absolute Salinity
presents a concern, and for these applications it may be preferable to publish graphs and
tables in Reference Salinity. For these studies or where it is clear that the effect of
compositional variations are insignificant or not of interest, the Gibbs function may be
called with 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 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 S P − θ diagram. Curved
contours of potential density could also be drawn on this same S P − 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 SA , it is now not 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 used to first form Absolute Salinity SA and Conservative
Temperature Θ. Oceanographic water masses are then analyzed on the SA − Θ diagram,
and potential density contours can also be drawn on this SA − Θ diagram, while
Preformed Salinity S* is the natural salinity variable to be used in applications such as
numerical modelling where it is important that the salinity variable be conservative.
When describing the use of TEOS‐10, it is the present document (the TEOS‐10 Manual)
that should be referenced as IOC et al. (2010) [IOC, SCOR and IAPSO, 2010: The
international thermodynamic equation of seawater – 2010: Calculation and use of thermodynamic
properties. Intergovernmental Oceanographic Commission, Manuals and Guides No. 56,
UNESCO (English), 196 pp].
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2. Basic Thermodynamic Properties
2.1 ITS‐90 temperature
In 1990 the International Practical Temperature Scale 1968 (IPTS‐68) was replaced by the
International Temperature Scale 1990 (ITS‐90). There are two main methods to convert
between these two temperature scales; Rusby’s (1991) 8th order fit valid over a wide range
of temperatures, and Saunders’ (1990) 1.00024 scaling widely used in the oceanographic
community. The two methods are formally indistinguishable in the oceanographic
temperature range because they differ by less than either the uncertainty in
thermodynamic temperature (of order 1 mK), or the practical application of the IPTS‐68
and ITS‐90 scales. The differences between the Saunders (1990) and Rusby (1991)
formulae are less than 1 mK throughout the temperature range ‐2 °C to 40 °C and less than
0.03mK in the temperature range between ‐2 °C and 10 °C. Hence we recommend that the
oceanographic community continues to use the Saunders formula
( 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 ≡ 101 325 Pa; 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 SP 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
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potassium chloride (KCl) solution at the same temperature and pressure. The mass
fraction of KCl (i.e., the mass of KCl per mass of solution) in the standard solution is
32.4356 × 10−3 . When K15 = 1, the Practical Salinity SP 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
or above 42 computed from conductivity, as measured for example in coastal lagoons,
should be evaluated by the PSS‐78 extensions of Hill et al. (1986) and Poisson and
Gadhoumi (1993). Samples exceeding a Practical Salinity of 50 must be diluted to the valid
salinity range and the measured value should be adjusted based on the added water mass
and the conservation of sea salt during the dilution process. This is discussed further in
appendix E.
Data stored in national and international data bases should, as a matter of principle, be
measured values rather than derived quantities. Consistent with this, we recommend
continuing to store the measured (in situ) temperature rather than the derived quantity,
potential temperature. Similarly we strongly recommend that Practical Salinity SP
continue to be the salinity variable that is stored in such data bases since SP 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. (2008b) 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
S R ≈ uPS S P
where
uPS ≡ (35.165 04 35) g kg −1 .
(2.4.1)
In the range 2 < SP < 42 , this equation expresses the Reference Salinity of a seawater sample
on the Reference‐Composition Salinity Scale (Millero et al. (2008a)). For practical
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purposes, this relationship can be taken to be an equality since the approximate nature of
this relation only reflects the extent to which Practical Salinity, as determined from
measurements of conductivity ratio, temperature and pressure, varies when a seawater
sample is heated, cooled or subjected to a change in pressure but without exchange of
mass with its surroundings. The Practical Salinity Scale of 1978 was designed to satisfy
this property as accurately as possible within the constraints of the polynomial
approximations used to determine Chlorinity (and hence Practical Salinity) in terms of the
measured conductivity ratio.
From Eqn. (2.4.1), a seawater sample of Reference Composition whose Practical
Salinity SP 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 SP of 35
using mass spectrometry and/or ion chromatography are underway and may provide
updated estimates of both the value of the mass fraction of dissolved material in Standard
Seawater and its uncertainty. Any update of this value will not change the Reference‐
Composition Salinity Scale and so will not affect the calculation of Reference Salinity nor
of Absolute Salinity as calculated from Reference Salinity plus the Absolute Salinity
Anomaly.
Oceanographic databases label stored, processed or exported parameters with the GF3
code PSAL for Practical Salinity and SSAL for salinity measured before 1978 (IOC, 1987).
In order to avoid possible confusion in data bases between different types of salinity it is
very strongly recommended that under no circumstances should either Reference Salinity
or Absolute Salinity be stored in national data bases.
Detailed information on Reference Composition and Reference Salinity can be found
in Millero et al. (2008a). For the userʹs convenience a brief summary of information from
Millero et al. (2008a), including the precise definition of Reference Salinity is given in
appendix A.3 and in Table D3 of appendix D.
2.5 Absolute Salinity
Absolute Salinity is traditionally defined as the mass fraction of dissolved material in
seawater. For seawater of Reference Composition, Reference Salinity gives our current
best estimate of Absolute Salinity. To deal with composition anomalies in seawater, we
need an extension of the Reference‐Composition Salinity SR that provides a useful
measure of salinity over the full range of oceanographic conditions and agrees precisely
with Reference Salinity when the dissolved material has Reference Composition. When
composition anomalies are present, no single measure of dissolved material can fully
represent the influences on seawater properties on all thermodynamic properties, so it is
clear that either additional information will be required or compromises will have to be
made. In addition, we would like to introduce a measure of salinity that is traceable to the
SI (Seitz et al., 2010b) and maintains the high accuracy of PSS‐78 necessary for
oceanographic applications. The introduction of ʺDensity Salinityʺ SAdens addresses both of
these issues; it is this type of absolute salinity that in TEOS‐10 parlance is labeled SA and
called Absolute Salinity. In this section we explain how SA is defined and evaluated, but
first we outline other choices that are available for the definition of absolute salinity in the
presence of composition variations in seawater.
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The most obvious definition of absolute salinity is “the mass fraction of dissolved non‐
H2O material in a seawater sample at its temperature and pressure”. This seemingly
simple definition is actually far more subtle than it first appears. Notably, there are
questions about what constitutes water and what constitutes dissolved material. Perhaps
the most obvious example of this issue occurs when CO2 is dissolved in water to produce a
mixture of CO2, H2CO3, HCO3‐, CO32‐, H+, OH‐ and H2O, with the relative proportions
depending on dissociation constants that depend on temperature, pressure and pH. Thus,
the dissolution of a given mass of CO2 in pure water essentially transforms some of the
water into dissolved material. A change in the temperature and even an adiabatic change
in pressure results in a change in absolute salinity defined in this way due to the
dependence of chemical equilibria on temperature and pressure. Pawlowicz et al. (2010)
and Wright et al. (2010b) address this second issue by defining “Solution Absolute
Salinity” (usually shortened to “Solution Salinity”), SAsoln , 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 SAsoln 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 SA , SAsoln , 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 SAsoln , 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. (2010b). Note that for a sample of Standard Seawater, all of the five salinity
variables SR , SA , SAsoln , SAadd and S* and are equal.
There is no simple means to measure either SAsoln or SAadd for the general case of the
arbitrary addition of many components to Standard Seawater. Hence a more precise and
easily determined measure of the amount of dissolved material in seawater is required
and TEOS‐10 adopts “Density Salinity” for this purpose. “Density Salinity” SAdens is
defined as the value of the salinity argument of the TEOS‐10 expression for density which
gives the sample’s actual measured density at the temperature t = 25°C and at the sea
pressure p = 0 dbar. When there is no risk of confusion, “Density Salinity” is also called
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Absolute Salinity with the label SA , 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, Jackett and Millero (2010a)
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. (2010a)),
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. (2010a) 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. (2010a) algorithm the
Baltic Sea is treated separately, following the work of Millero and Kremling (1976) and
Feistel et al. (2010c, 2010d), because some rivers flowing into the Baltic are unusually high
in calcium carbonate.
Figure 1. A sketch indicating how thermodynamic quantities
such as density are calculated as functions of Absolute Salinity.
Absolute Salinity is found by adding an estimate of the
Absolute Salinity Anomaly δ SA to the Reference Salinity.
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Since the density of seawater is rarely measured, we recommend the approach
illustrated in Figure 1 as a practical method to include the effects of composition
anomalies on estimates of Absolute Salinity and density. When composition anomalies
are not known, the algorithm of McDougall et al. (2010a) may be used to estimate Absolute
Salinity in terms of Practical Salinity and the spatial location of the measurement in the
world oceans.
The difference between Absolute Salinity and Reference Salinity, as estimated by the
McDougall et al. (2010a) algorithm, is illustrated in Figure 2 (a) at a pressure of 2000 dbar,
and in a vertical section through the Pacific Ocean in Figure 2 (b).
Of the approximately 800 samples of seawater from the world ocean that have been
examined to date for 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. (2010a) 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.
The thermodynamic description of seawater and of ice Ih as defined in IAPWS-08 and
IAPWS-06 has been adopted as the official description of seawater and of ice Ih by the
Intergovernmental Oceanographic Commission in June 2009. These thermodynamic
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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 peerreviewed 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 SA given the input
variables Practical Salinity SP , 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.
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 = 273.15 K 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 ( SA , t, p ) .
(2.6.1)
The saline part of the Gibbs function, g S , is valid over the ranges 0 < SA < 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 SA = 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
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 SA as the input variable (see section 2.5) but
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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 ( S A , 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 ( t t , pt ) = 0
(2.6.3)
u W ( t t , 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) g00 and g10
of the table in appendix G below. These values of g00 and g10 are not identical to the
values in Feistel (2003) because the present values have been taken from IAPWS‐09 and
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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
h ( SSO , tSO , pSO ) = 0
(2.6.5)
and
η ( SSO , tSO , pSO ) = 0.
(2.6.6)
In more detail, these conditions are actually officially written as (Feistel (2008), IAPWS‐08)
and
hS ( SSO , tSO , pSO ) = u W ( tt , pt ) − h W ( tSO , pSO )
(2.6.7)
η S ( SSO , tSO , pSO ) = η W ( tt , pt ) − η W ( tSO , pSO ) .
(2.6.8)
Written in this way, (2.6.7) and (2.6.8) use properties of the pure water description (the
right‐hand sides) to constrain the arbitrary constants in the saline Gibbs function. While
the first terms on the right‐hand sides of these equations are zero (see (2.6.3) and (2.6.4)),
these constraints on the saline Gibbs function are written this way so that they are
independent of any subsequent change in the arbitrary constants involved in the
thermodynamic description of pure water.
While the two slightly different
thermodynamic descriptions of pure water, namely IAPWS‐95 and IAPWS‐09, both
achieve zero values of the internal energy and entropy at the triple point of pure water,
the values assigned to the enthalpy and entropy of pure water at the temperature and
pressure of the standard ocean, h W ( tSO , pSO ) and η W ( tSO , pSO ) on the right‐hand sides of
(2.6.7) and (2.6.8), are slightly different in the two cases. For example h W ( tSO , pSO ) is
3.3x10−3 J kg −1 from IAPWS‐09 (as described in the table of appendix G) compared with
the round‐off error of 2 x10−8 J kg −1 when using IAPWS‐95 with double‐precision
arithmetic. This issues is discussed in more detail in section 3.3.
The polynomial form and the coefficients for the pure water Gibbs function g W ( t , p )
from Feistel (2003) and IAPWS‐09 are given in appendix G, while the combined
polynomial and logarithmic form and the coefficients for the saline part of the Gibbs
function g S ( S A , t , p ) (from Feistel (2008) and IAPWS‐08) are reproduced in appendix H.
SCOR/IAPSO Working Group 127 has independently checked that the Gibbs functions
of Feistel (2003) and of Feistel (2008) do in fact fit the underlying data of various
thermodynamic quantities to the accuracy quoted in those two fundamental papers. This
checking was performed by Giles M. Marion, and is summarized in appendix O. Further
checking of these Gibbs functions has occurred in the process leading up to IAPWS
approving these Gibbs function formulations as the Releases IAPWS‐08 and IAPWS‐09.
Discussions of how well the Gibbs functions of Feistel (2003) and Feistel (2008) fit the
underlying (laboratory) data of density, sound speed, specific heat capacity, temperature
of maximum density etc may be found in those papers, along with comparisons with the
corresponding algorithms of EOS‐80. The IAPWS‐09 release discusses the accuracy to
which the Feistel (2003) Gibbs function fits the underlying thermodynamic potential of
IAPWS‐95; in summary, for the variables density, thermal expansion coefficient and
specific heat capacity, the rms misfit between IAPWS‐09 and IAPWS‐95, in the region of
validity of IAPWS‐09, are a factor of between 20 and 100 less than the corresponding error
in the laboratory data to which both thermodynamic potentials were fitted. Hence, in the
oceanographic range of parameters, IAPWS‐09 and IAPWS‐95 may be regarded as equally
accurate thermodynamic descriptions of pure liquid water.
The Gibbs function g has units of J kg −1 in both the SIA and GSW software libraries.
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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 potential temperature θ or
Conservative Temperature Θ rather than in situ temperature t . We note here that the
specific volume is equal to the pressure derivative of specific enthalpy at fixed Absolute
Salinity when any one of 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 SA rather than of Reference Salinity
S R or Practical Salinity S P . This is an extremely important point because Absolute
Salinity SA 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 potential temperature θ or Conservative Temperature
Θ rather than of in situ temperature t . That is, it is convenient to form the following two
functional forms of density,
ρ = ρ ( SA ,θ , p ) = ρˆ ( SA , Θ, p ) ,
(2.8.2)
where θ and Θ are respectively potential temperature and Conservative Temperature,
both referenced to pr = 0 dbar. 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)
hP = hP = hˆP = v = ρ −1
(2.8.3)
since the other variables are taken to be constant during the partial differentiation.
Appendix P lists expressions for many thermodynamic variables in terms of the
thermodynamic potentials
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h = h ( SA ,η , p ) , h = h ( SA ,θ , p ) and h = hˆ ( SA , Θ, p ) .
19
(2.8.4)
Density ρ has units of kg m −3 in both the SIA and GSW software libraries.
Computationally efficient expressions for ρˆ ( SA , Θ, p ) and ρ ( SA ,θ , p ) involving 25
coefficients are available (McDougall et al. (2010b)) and are described in appendix A.30
and appendix K. These expressions can be integrated with respect to pressure to provide
closed expressions for hˆ ( SA , Θ, p ) and h ( SA ,θ , p ) (see Eqn. (A.30.6)).
2.9 Chemical potentials
As for any two‐component thermodynamic system, the Gibbs energy, G , of a seawater
sample containing the mass of water mW and the 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,
S A = mS / ( mW + mS ) (Millero et al. (2008a)), the specific Gibbs energy g is given by
g ( SA , t, p ) =
(
G
= (1 − SA ) μ W + SA μ S = μ W + SA μ S − μ W
mW + mS
)
(2.9.3)
and is independent of the total mass of the sample. 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 (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). 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 as
∂ ⎡( 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 − SA )
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
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Gibbs function g of seawater. Note that the weights of the sums that appear in Eqns.
(2.9.1) – (2.9.5) are strictly the mass fractions of salt and of pure water in seawater, so that
for a seawater sample of anomalous composition these mass fractions would be more
accurately given in terms of S Asoln than by S A ≡ 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.
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.
2.10 Entropy
The specific entropy of seawater η is given by
η = η ( SA , t , p ) = − gT = − ∂g ∂T
SA , 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 = 101 325 Pa is the standard atmosphere pressure)
∂g
∂g
u = u ( SA , t , p ) = g + (T0 + t )η − ( p + P0 ) v = g − (T0 + t )
− ( p + P0 )
.
(2.11.1)
∂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 ( S A , t , p ) = g + (T0 + t )η = g − (T0 + t )
∂g
∂T
.
SA , p
Specific enthalpy h has units of J kg −1 in both the SIA and GSW software libraries.
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21
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
.
(2.13.1)
SA , T
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 S − SA
⎜
⎝
∂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 ( S A , t , p ) = ρ −1
∂ρ
∂P
= − v −1
SA , T
∂v
∂P
= −
SA , T
g PP
gP
(2.15.1)
where the second derivative of g is taken with respect to pressure (in Pa ) at constant S A
and t. The use of P in the pressure derivatives in Eqn. (2.15.1) serves to emphasize that
these derivatives must be taken with respect to pressure in Pa not in dbar . The
isothermal compressibility of seawater κ t produced by both the SIA and GSW computer
software libraries (appendices M and N) has units of Pa −1.
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2.16 Isentropic and isohaline compressibility
When the entropy and Absolute Salinity are held constant while the pressure is changed,
the isentropic and isohaline compressibility κ is obtained:
κ = κ ( SA , t , p ) = ρ −1
=
(g
∂ρ
∂P
2
TP
= − v −1
SA ,η
− gTT g PP
g P gTT
∂v
∂P
).
= ρ −1
SA ,η
∂ρ
∂P
= ρ −1
SA , θ
∂ρ
∂P
SA , Θ
(2.16.1)
The isentropic and isohaline compressibility κ is sometimes called simply the isentropic
compressibility (or sometimes the “adiabatic compressibility”), on the unstated
understanding that there is also no transfer of salt during the isentropic or adiabatic
change in pressure. The isentropic and isohaline compressibility of seawater κ produced
by both the SIA and GSW software libraries (appendices M and N) has units of Pa −1.
2.17 Sound speed
The speed of sound in seawater c is given by
c = c ( SA , t , p ) =
∂P ∂ρ
SA ,η
( ρκ )−1
=
= g P gTT
(g
2
TP
)
− gTT g PP .
(2.17.1)
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 ( S A , 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 ) = −
c 0p
g
1 ∂ρ
1 ∂v
=
= − TP
.
g P (T0 + θ ) gTT
ρ ∂Θ SA , p v ∂Θ SA , p
(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 c 0p is defined in Eqn. (3.3.3) below.
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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 g SAT derivatives in the numerator is
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 g SA derivative in the numerator is
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 in the GSW Toolbox being g kg −1.
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2.20 Isobaric heat capacity
The specific isobaric heat capacity c p is the rate of change of specific enthalpy with
temperature at constant Absolute Salinity S A and pressure p, so that
c p = c p ( SA , t , p ) =
∂h
∂T
= − (T0 + t ) gTT .
(2.20.1)
SA , p
The isobaric heat capacity c p varies over the S A − Θ plane at p = 0 by approximately 5%,
as illustrated in Figure 4.
Figure 4. Contours of isobaric specific heat capacity c p of seawater
(in J kg −1 K −1 ), Eqn. (2.20.1), at p = 0.
The isobaric heat capacity c p has units of J kg −1 K −1 in both the SIA and GSW
computer software libraries.
2.21 Isochoric heat capacity
The specific isochoric heat capacity cv is the rate of change of specific internal energy u
with temperature at constant Absolute Salinity S A and specific volume, v, so that
cv = cv ( SA , t , p ) =
∂u
∂T
(
2
= − (T0 + t ) gTT g PP − gTP
SA , v
)
g PP .
(2.21.1)
Note that the isochoric and isobaric heat capacities are related by
cv = c p −
(T0 + t ) (α t )
(ρ κ )
t
2
,
and by
cv = c p
κ
.
κt
(2.21.2)
The isochoric heat capacity cv has units of J kg −1 K −1 in both the SIA and GSW
computer software libraries.
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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))
Γ = Γ ( SA , t, p ) =
∂t
∂P
=
SA ,η
∂t
∂P
= −
SA , Θ
gTP
∂ 2h
=
gTT
∂η∂P
=
SA
∂v
∂η
=
SA , p
(T0 + t ) α t
ρ cp
(2.22.1)
=
(T0 + θ )
c 0p
∂v
∂Θ S
=
A, p
(T0 + θ )
c 0p
∂ 2h
∂Θ∂P
=
SA
(T0 + θ ) α Θ
ρ c 0p
=
(T0
+ θ )αθ
ρ c p ( SA ,θ ,0 )
.
The adiabatic (and isohaline) lapse rate is commonly (and incorrectly) explained as being
proportional to the work done on a fluid parcel as its volume changes in response to an
increase in pressure. According to this explanation the adiabatic lapse rate would increase
with both pressure and the fluid’s compressibility, but this is not the case. Rather, the
adiabatic lapse rate is proportional to the thermal expansion coefficient and is
independent of the fluid’s compressibility. Indeed, the adiabatic lapse rate changes sign at
the temperature of maximum density whereas the compressibility and the work done by
compression is always positive. McDougall and Feistel (2003) show that the adiabatic
lapse rate is independent of the increase in the internal energy that a parcel experiences
when it is compressed. Rather, the adiabatic lapse rate represents that change in
temperature that is required to keep the entropy (and also 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 output of both the SIA and the GSW computer software
libraries is in units of K Pa −1 .
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3. Derived Quantities
3.1 Potential temperature
The very useful concept of potential temperature was applied to the atmosphere originally
by Helmholtz (1888), first under the name of ‘heat content’, and later renamed ‘potential
temperature’ (Bezold (1888)). These concepts were transferred to oceanography by
Helland‐Hansen (1912). Potential temperature is the temperature that a fluid parcel
would have if its pressure were changed to a fixed reference pressure 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
mechanical energy”. A process that obeys all three restrictions is a thermodynamically
reversible process. Note that one often (falsely) reads that the requirement of a reversible
process is that the process occurs at constant entropy. However this statement is
misleading because it is possible for a fluid parcel to exchange some heat and some salt
with its surroundings in just the right ratio so as to keep its entropy constant, but the
processes is not reversible (see Eqn. (A.7.1)).
Potential temperature referred to reference pressure 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
η ( S A , θ , pr ) = η ( S A , t , p ) ,
(3.1.2)
or, in terms of the Gibbs function, g ,
− gT ( S A , θ , p r ) = − gT ( S A , 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, using a suitable initial value as described by McDougall et al. (2010b). .
Note that the difference between the potential and in situ temperatures is not due to
the work done in compressing a fluid parcel on going from one pressure to another:‐ the
sign of this work is often in the wrong sense and the magnitude is often wrong by a few
orders of magnitude (McDougall and Feistel (2003)). Rather, the difference between these
temperatures is what is required to keep the entropy constant during the adiabatic and
isohaline pressure change. The potential temperature θ output of the SIA software is in
units of K while the output from the GSW Toolbox is in °C .
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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))
h 0 ( SA , t , p ) = h ( SA ,θ ,0 ) = h 0 ( SA ,θ ) = h ( S A , t , p ) − ∫ v ( S A ,θ [ SA , t , p, p′] , p ′) dP′
P
P0
P
= h ( SA , t , p ) − ∫ v ( SA ,η , p′) dP′
P0
P
= h ( SA , t , p ) − ∫ v ( SA , θ , p ′) dP′
(3.2.1)
P0
P
= h ( SA , t , p ) − ∫ vˆ ( 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)
3.3 Conservative Temperature
Conservative Temperature Θ is defined to be proportional to potential enthalpy
according to
Θ ( S A , t , p ) = Θ ( SA ,θ ) = h 0 ( SA , t , p ) c 0p = h 0 ( SA ,θ ) c 0p
(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). In fact we adopt the exact definition for c 0p to be the 15‐digit value in (3.3.2), so
that
c 0p ≡ 3991.867 957 119 63 J kg −1 K −1 .
(3.3.3)
When IAPWS‐95 is used for the pure water part of the Gibbs function, Θ ( SSO ,0 °C,0 ) and
Θ ( SSO , 25 °C,0 ) differ from 0 °C and 25 °C respectively by the round‐off amount of
5 × 10−12 °C. When IAPWS‐09 (which is based on the paper of Feistel (2003), see appendix
G) is used for the pure water part of the Gibbs function, Θ ( SSO ,0 °C,0) differs from 0 °C
by −8.25 × 10−8 °C and Θ ( SSO , 25 °C,0 ) differs from 25 °C by 9.3 × 10−6 °C. Over the
temperature range from 0 ° C to 40 ° C the difference between Conservative Temperature
using IAPWS‐95 and IAPWS‐09 as the pure water part is no more than ± 1.5 × 10−5 °C , a
temperature difference that will be ignored.
The value of c 0p in (3.3.3) is very close to the average value of the specific heat capacity
c p at the sea surface of today’s global ocean. This value of c 0p also causes the average
value of θ − Θ at the sea surface to be very close to zero. Since c 0p is simply a constant of
proportionality between potential enthalpy and Conservative Temperature, it is totally
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arbitrary, and we see no reason why its value would need to change from (3.3.3) even
when in future decades an improved Gibbs function of seawater is agreed upon.
Appendix A.18 outlines why Conservative Temperature gets its name; it is
approximately two orders of magnitude more conservative compared with either
potential temperature or entropy.
The SIA and GSW software libraries both include an algorithm for determining
Conservative Temperature Θ from values of Absolute Salinity 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 ( S A , 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 (McDougall et al. (2010b)).
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 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 [ S A , t , p, pr ] , pr ) = g P−1 ( SA ,θ [ SA , t, p, pr ] , pr ) .
(3.4.2)
Using either of the functional forms (2.8.2) 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
ρ θ ( SA , t , p, pr ) = ρ Θ ( SA , t , p, pr ) = ρ ( SA ,η , pr ) = ρ ( SA ,θ , pr ) = ρˆ ( SA , Θ, pr ) ,
(3.4.3)
where we note that the potential temperature θ in the third 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 ρη ) 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)
ρ θ ( SA , t , p, pr ) = ρ Θ ( SA , t, p, pr ) = ⎣⎡ hP ( SA ,θ , p = pr ) ⎦⎤
−1
−1
= ⎡ hˆP ( S A , Θ, p = pr ) ⎤ .
⎣
⎦
(3.4.4)
3.5 Density anomaly
Density anomaly σ t is an old‐fashioned density measure that is now seldom used. It is
the density evaluated at the in situ temperature but at zero sea pressure, minus 1000
kg m −3 , that is,
σ t ( SA , t, p ) = ρ ( SA , t,0 ) − 1000 kg m −3 = g P−1 ( SA , t ,0 ) − 1000 kg m −3 .
θ
(3.5.1)
σ was used as an approximation to σ which avoided the computational demand of
evaluating θ . Density anomaly σ t is not provided in the TEOS‐10 software libraries.
t
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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
3.11), it is advisable to replace the arguments SSO and 0°C with more general values S 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 ( S A , 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 ) = vˆ ( SA , Θ, p ) − vˆ SA , Θ, p = hˆP ( SA , Θ, p ) − hˆP SA , Θ, p .
(3.7.3)
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
(
)
(
)
δ ( SA ,θ , p ) = 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)
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3.8 Thermobaric coefficient
The thermobaric coefficient quantifies the rate of variation with pressure of the ratio of the
thermal expansion coefficient and the saline contraction coefficient. With respect to
potential temperature θ the thermobaric coefficient is (McDougall (1987b))
Tb = Tb ( SA , t p ) = β
θ
θ
θ
(
∂ αθ β θ
)
∂P
∂α θ
∂P
=
SA , θ
−
SA , θ
α θ ∂β θ
β θ ∂P
.
(3.8.1)
SA ,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 ( S A ,θ , p ) .
With respect to Conservative Temperature Θ the thermobaric coefficient is
TbΘ
=
TbΘ
( SA , t p ) = β
Θ
(
∂ αΘ β Θ
∂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,
− c 0p K ∇n Θ , and is independent of the amount of small‐scale (dianeutral) turbulent mixing
and hence is also independent of the dissipation of mechanical 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 eTb 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, eTb , 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 eTb through these surfaces (Klocker and McDougall (2010a)).
In both the SIA and GSW computer software libraries the thermobaric parameter is
output in units of K −1 Pa −1 .
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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 ( S A , t p ) =
∂θ
θ
θ
SA , p
α θ ∂α θ
+2 θ
β ∂SA
2
θ, p
⎛ α θ ⎞ ∂β θ
−⎜ θ⎟
⎝ β ⎠ ∂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 mechanical 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.19.4) in appendix A.19). This leads to the thought that if an ocean front is
split up into a series of many smaller fronts then the effects of cabbeling and
thermobaricity might be reduced by perhaps the square of the number of such fronts. This
is not the case. Rather, the total dianeutral transport across a frontal region depends on
the product of the lateral flux of heat passing through the front and the contrast in
temperature and/or pressure across the front, but is independent of the sharpness of the
front (Klocker and McDougall (2010a)). This can be understood by noting from above that
the dianeutral velocity due to cabbeling, eCab = − gN −2 K CbΘ∇n Θ ⋅ ∇n Θ, is proportional to the
scalar product of the epineutral flux of heat − c 0p 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 CbΘ in terms of enthalpy in the functional forms
h ( SA ,θ , p ) and hˆ ( SA , Θ, p ) can be found in appendix P.
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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 potential temperature and Absolute Salinity (or in terms of the
vertical gradients of Conservative 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 −1 N 2 = − ρ −1ρ z + κ Pz = − ρ −1 ρ z − Pz / c 2
)
= α θ θ z − β θ ∂S A ∂z x , y
(3.10.1)
= α Θ Θ 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
N2 = −
g Δρ Θ
g 2 Δρ Θ
,
=
ρ Δz
ΔP
(3.10.2)
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. The last part of Eqn. (3.10.2) has used 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 ∇S A
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
θ
= α ∇nθ − β ∇n SA
= α Θ∇ n Θ − β Θ ∇ n S A
= 0.
Here ∇n is an example of a projected non‐orthogonal gradient
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)
θ
(3.11.2)
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
∇ rτ ≡
∂τ
∂x r
i +
∂τ
∂y r
j + 0k,
33
(3.11.3)
that is widely used in oceanic and atmospheric theory and modelling. Horizontal
distances are measured between the vertical planes of constant latitude x and longitude y
while the values of the property 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 Θ ΔΘ − β Θ ΔS A ≈ 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 S A1 , Θ1 , p1 and S A2 , 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
(
)
(
)
2
− TbΘ ∫ ( P − P ) d Θ ,
(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 ) .
3.12 Geostrophic, hydrostatic and “thermal wind” equations
The geostrophic approximation to the horizontal momentum equations ((B9) below)
equates the Coriolis term to the horizontal pressure gradient ∇ z P so that the geostrophic
equation is
f k × ρ u = − ∇z P
fv =
or
1
ρ
k ×∇ z P .
(3.12.1)
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 hydrostatic equation is an approximation to the vertical momentum equation (see
Eqn. (B9)), namely
Pz = − g ρ .
(3.12.2)
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), using the hydrostatic equation Eqn. (3.12.2) and ignoring the tiny term in ρ z
(which is of Boussinesq magnitude), the thermal wind can be written as
f vz =
1
ρ
k ×∇ z ( Pz ) = −
g
ρ
k ×∇ z ρ =
N2
gρ
k ×∇n P,
(3.12.3)
where ∇ z is the gradient operator in the exactly horizontal direction along geopotentials,
and the last part of this equation relates the “thermal wind” to the pressure gradient in the
neutral tangent plane, that is, effectively to the slope of the neutral tangent plane (see
McDougall (1995)).
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
3.13 Neutral helicity
Neutral tangent planes (which do exist) do not link up in space to form a well‐defined
neutral surface unless the neutral helicity H n is everywhere zero on the surface. Neutral
helicity is defined as the scalar product of the vector 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)
and this is proportional to the thermobaric coefficient TbΘ of the equation of state
according to
H n = β Θ TbΘ ∇P ⋅ ∇SA × ∇Θ
(
)
Pz β Θ TbΘ ∇ p SA × ∇ p Θ ⋅ k
=
(3.13.2)
= g −1 N 2TbΘ ( ∇n P × ∇n Θ) ⋅ k
≈ 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 .
Because of the non‐zero neutral helicity in the ocean, lateral motion following neutral
tangent planes has the character of helical motion. That is, if we ignore the effects of
diapycnal mixing processes (as well as ignoring cabbeling and thermobaricity), the mean
flow around ocean gyres still passes through any well‐defined “density” surface because
of the helical nature of neutral trajectories, caused in turn by the non‐zero neutral helicity.
This dia‐surface flow is expressed in Eqns. (A.25.4) and (A.25.6) in terms of the
appropriate mean horizontal velocity and the difference between the slope of the neutral
tangent plane and the slope of a well‐defined “density” surface.
Neutral helicity is proportional to the component of the vertical shear of the
geostrophic velocity ( v 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 = ρ TbΘ fv z ⋅ ∇n Θ .
(3.13.3)
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
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
(3.13.4)
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 ⎞
∇σ × ρ ∇ z P ⋅ k = H ( Pr − P ) ⎜ −
⎟ .
⎝ ∂z ⎠
(
1
)
n
(3.13.5)
The fact that this curl is nonzero proves that a geostrophic streamfunction does not exist in
a potential density surface.
Neutral helicity H n also arises in the context of finding a closed expression for the
mean velocity in the ocean. The component of the horizontal velocity in the direction
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
35
along a contour of Θ in a neutral tangent plane, namely the velocity component
v ⋅ k × ∇n Θ / ∇n Θ , is given by (McDougall (1995), Zika et al. (2010a, 2010b))
v⋅k ×
∇n Θ
∇n Θ
=
−H n
v z⊥
+
,
φz
φz ρ f TbΘ ∇n Θ
(3.13.6)
so that the full expression for the horizontal velocity is
⎧⎪
∇n Θ
v z⊥ ⎫⎪
−H n
v = ⎨
+
⎬k ×
Θ
φz ⎪⎭
∇n Θ
⎩⎪ φz ρ f Tb ∇n Θ
+ v⊥
∇n Θ
.
∇n Θ
(3.13.7)
Here φz is the rate of spiraling (radians per meter) in the vertical of the Θ contours on
neutral tangent planes, and v ⊥ is the velocity component across the Θ contour on the
neutral tangent plane (a velocity component that results from irreversible mixing
processes). Neutral helicity arises in this context because it is proportional to the
component of the thermal wind vector v z in the direction across the Θ contour on the
neutral tangent plane (see (3.13.3)). This equation (3.13.7) for the isopycnally‐averaged
velocity v shows that in the absence of mixing processes (so that v ⊥ = vz⊥ = 0 ) and so long
as (i) the epineutral Θ contours spiral in the vertical and (ii) ∇n Θ is not zero, then neutral
helicity H n is required to be non‐zero in the ocean whenever the ocean is not motionless.
Interestingly, for given magnitudes of the epineutral gradients of pressure and
Conservative Temperature, neutral helicity is maximized when these gradients are
perpendicular since neutral helicity is proportional to 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).
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 β Θ∇a SA − α Θ∇a Θ
)
(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)).
Since Neutral Density is not a thermodynamic variable, it will not be described more fully
in this manual.
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
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
α ΘΘz
Rρ =
β Θ ( SA ) z
≈
αθθz
β θ ( SA ) z
.
(3.15.1)
3.16 Turner angle
The Turner angle Tu , named after J. Stewart Turner, is defined as the four‐quadrant
arctangent (Ruddick (1983) and McDougall et al. (1988), particularly their Figure 1)
(
(α θ θ
Tu = tan −1 α Θ Θ z + β Θ ( SA ) z , α ΘΘ z − β Θ ( SA ) z
≈ tan −1
z
+ β θ ( SA ) z , α θ θ z − β θ ( SA ) z
)
)
(3.16.1)
where the first of the two arguments of the arctangent function is the “y”‐argument and
the second one the “x”‐argument, this being the common order of these arguments in
Fortran and MATLAB. The Turner angle Tu is quoted in degrees of rotation. Turner
angles between 45° and 90° represent the “salt‐finger” regime of double‐diffusive
convection, with the strongest activity near 90°. Turner angles between −45° and −90°
represent the “diffusive” regime of double‐diffusive convection, with the strongest activity
near −90°. Turner angles between −45° and 45° represent regions where the stratification is
stably stratified in both Θ and S A . 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_CT25.
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]
∇n Θ
(3.17.1)
Θ z ⎡⎣ Rρ − r ⎤⎦
(from McDougall (1987a)), where r is the ratio of the slope on the S A − 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 ( S , Θ, p ) β Θ ( SA , Θ, p )
r = Θ A
.
(3.17.2)
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 =
GΘ ≡
IOC Manuals and Guides No. 56
⎡⎣ Rρ − 1⎤⎦
⎡⎣ Rρ r − 1⎤⎦
(3.17.3)
(3.17.4)
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
37
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_CT25, while the
ratio r of Eqn. (3.17.2) is available there as gsw_isopycnal_slope_ratio_CT25.
Substituting ϕ = SA into Eqn. (3.17.1) gives the following relation between the (parallel)
isopycnal and epineutral gradients of S A
⎡ Rρ − 1⎤⎦
GΘ
∇σ SA = ⎣
∇n SA =
∇n SA .
r
⎡⎣ Rρ − r ⎤⎦
(3.17.5)
3.18 Slopes of potential density surfaces and neutral tangent planes compared
The two‐dimensional slope of a surface is defined as the two‐dimensional gradient of
height z of that surface. The two‐dimensional slope of a surface is an exactly horizontal
gradient vector; it has no vertical component. The slope difference between the neutral
tangent plane and a potential density surface with reference pressure pr is given by
(McDougall (1988))
∇n z − ∇σ z =
Rρ [1 − r ] ∇n Θ
=
⎣⎡ Rρ − r ⎦⎤ Θ z
(1 − G ) ∇ΘΘ
Θ
n
=
z
Rρ [1 − r ] ∇σ Θ
∇n Θ − ∇σ Θ
=
. (3.18.1)
Θz
r ⎡⎣ Rρ − 1⎤⎦ Θ z
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) (Klocker et al.
(2009a)).
3.19 Slopes of in situ density surfaces and specific volume anomaly surfaces
The vector slope of an in situ density surface, ∇ ρ z , is defined to be the exactly horizontal
vector
∇ρ z =
∂z
∂x ρ
i +
∂z
∂y ρ
j + 0k ,
(3.19.1)
representing the “dip” of the surface in both horizontal directions (note that height z is
defined positive upwards). This vector slope can be related to the (very small) slope of
isobaric surface by ( g here is the gravitational acceleration) (McDougall (1989))
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
∇ρ z − ∇ p z =
(
−1
⎡
g 2 c2 ⎤
∇n z − ∇ p z ⎢1 +
⎥ ,
N2 ⎦
⎣
)
(3.19.2)
where c is the speed of sound and N is the buoyancy frequency. In the upper water
column where the square of the buoyancy frequency is significantly larger than
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”.
The slope of a specific volume anomaly surface, ∇δ z , can be expressed as
∇δ z − ∇ p z =
(
−1
⎡
g 2 c2
g2 c2 ⎤
∇n z − ∇ p z ⎢1 +
−
⎥ ,
N2
N2 ⎦
⎣
)
(
(3.19.3)
)
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 Potential vorticity
Planetary potential vorticity is the Coriolis parameter f times the vertical gradient of a
suitable variable. Potential density is sometimes used for that variable but using potential
density (i) involves an inaccurate separation between lateral and diapycnal advection
because potential density surfaces are not a good approximation to neutral tangent planes
and (ii) incurs the non‐conservative baroclinic production term of Eqn. (3.13.4). Using
approximately neutral surfaces, “ans”, (such as Neutral Density surfaces) provides an
optimal separation between the effects of lateral and diapycnal mixing in the potential
vorticity equation. In this case the potential vorticity variable is proportional to the
reciprocal of the thickness between a pair of closely spaced approximately neutral
surfaces. This planetary potential vorticity variable is called Neutral‐Surface‐Potential‐
Vorticity ( NSPV for short) and is related to fN 2 by
{
}
NSPV ≡ − g ρ −1 f γ zn ≈ fN 2 exp − ∫ans ρ g 2 N −2 TbΘ ( ∇a Θ − ΘP ∇a P ) ⋅ dl .
(3.20.1)
The exponential expression was derived by McDougall (1988) (his equation (47)) and is
approximate because the variation of the saline contraction coefficient β Θ with pressure
was neglected in comparison with the larger proportional change in the thermal expansion
coefficient α Θ with pressure. The integral in Eqn. (3.20.1) is taken along an approximately
neutral surface from a location where NSPV is equal to fN 2 . Interestingly the
combination ∇a Θ − ΘP ∇a P is simply the isobaric gradient of Conservative Temperature,
∇ P Θ, which is almost the same as the horizontal gradient, ∇ z Θ . A more accurate version
of this equation which does not ignore the variation of the saline contraction coefficient
can be shown to be
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
{
) }
(
NSPV ≡ − g ρ −1 f γ zn = fN 2 exp − ∫ans g 2 N −2 ( ρα Θ ) P ∇ P Θ − ( ρβ Θ ) P ∇ P S A ⋅ dl
{
}
= fN 2 exp ∫ans g 2 N −2∇ P ( ρκ ) ⋅ dl .
39
(3.20.2)
The exponential factor in Eqn. (3.20.2) is approximately the integrating factor b , defined
as b ≡ ∇γ n ⋅ ∇ρ l (∇ρ l ⋅ ∇ρ l ) where ∇ρ l ≡ ρ ( β Θ∇SA − α Θ∇Θ) , which allows spatial integrals
of ρ b( β Θ∇SA − α Θ∇Θ) = b∇ρ l ≈ ∇γ n to be approximately independent of path for
“vertical paths”, that is, for paths in surfaces whose normal has zero vertical component.
The gradient ∇a of fN 2 is related to that of NSPV by (from Eqns. (3.20.2) and (3.20.1))
(
)
∇a ln fN 2 − ∇a ( ln NSPV ) = − g 2 N −2∇ P ( ρκ ) ≈ ρ g 2 N −2TbΘ ( ∇a Θ − ΘP ∇a P ) .
(3.20.3)
The deficiencies of fN 2 as a form of planetary potential vorticity have not been widely
appreciated. Even in a lake, and also in the simple situation where temperature does not
vary along a density surface ( ∇a Θ = 0 ), the use of fN 2 as planetary potential vorticity is
inaccurate since the right‐hand side of (3.20.3) is then approximately
Rρ
− ρ g 2 N −2TbΘΘP ∇a P = Θ
TbΘ∇a P ,
(3.20.4)
α ⎡⎣ Rρ − 1⎤⎦
and the mere fact that the density surface has a slope (i. e. ∇a P ≠ 0 ) means that the
contours of fN 2 will not be parallel to contours of NSPV on the density surface. (In this
situation (where ∇a Θ = 0 ) the contours of NSPV along approximately neutral surfaces
coincide with those of isopycnal‐potential‐vorticity ( 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 )
− g ρ −1ρ zΘ
IPV
≡
=
fN 2
N2
β Θ ( p)
⎡⎣ Rρ r − 1⎤⎦
β Θ ( pr ) 1
1
=
≈ Θ ,
Θ
Θ
⎡⎣ Rρ − 1⎤⎦
G
β ( p) G
(3.20.5)
so that the ratio of NSPV to IPV plotted on an approximately neutral surface is given by
NSPV
IPV
=
β Θ ( p)
β Θ ( pr )
⎡⎣ Rρ − 1⎤⎦
exp ∫ans g 2 N −2∇ P ( ρκ ) ⋅ dl .
⎡⎣ Rρ r − 1⎤⎦
{
}
(3.20.6)
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 is available in
the GSW Oceanographic Toolbox as the function gsw_IPV_vs_fNsquared_ratio_CT25.
3.21 Vertical velocity through the sea surface
There has been confusion regarding the expression that relates the net evaporation at the
sea surface to the vertical velocity in the ocean through the sea surface. Since these
expressions have often involved the salinity (through the factor (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 − VH ⋅ ∇η − ∂η ∂t
(where w is the vertical velocity through the geopotential surface, see section 3.24, and
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
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 − VH ⋅ ∇η − ∂η ∂t = ρ −1ρ W ( E − P ) .
(3.21.1)
3.22 Freshwater content and freshwater flux
Oceanographers traditionally call the pure water fraction of seawater the “freshwater
fraction” or the “freshwater content”. This can cause confusion because in some science
circles “freshwater” is used to describe water of low but non‐zero salinity. Nevertheless,
here we retain the oceanographic use of “freshwater” as being synonymous with pure
water (i. e. 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 SA u . 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 S Adens ≡ SA , Solution Salinity S Asoln , Added‐Mass Salinity
S Aadd , 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
≈ − ∇ ⋅ F R − ∇ ⋅ FQ + ρε + hSA ρ S SA,
ρ c 0p
(3.23.1)
dt
with an error in Θ that is approximately one percent of the error incurred by treating
either c 0p 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 mechanical 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 c 0p Θ it follows that Θ can be treated as
a conservative variable in the ocean and that c 0p 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 vh 0 = ρ vc 0p Θ (here v is the northward velocity). The error in comparing this advective
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41
meridional “heat flux” with the air‐sea heat flux is approximately 1% of the error in so
interpreting the area integral of either ρ vc 0pθ or ρ v c p ( SA ,θ ,0 ) θ . Similarly, turbulent
diffusive fluxes of “heat” are accurately given by a turbulent diffusivity times the spatial
gradient of c 0p Θ but are less accurately approximated by the same turbulent diffusivity
times the spatial gradient of c 0pθ (see appendix A.14 for a discussion of this point).
Warren (1999, 2006) has argued that because enthalpy is unknown up to a linear
function of salinity, it is only possible to talk of a flux of “heat” through an ocean section if
the fluxes of mass and salt through the ocean section are both zero. This opinion seems to
be widely held, but it is incorrect. Because enthalpy is unknown and unknowable up to a
linear function of 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
a3ρ dSA dt . It is shown in appendix B that the terms −∇ ⋅ FQ + hSA ρ S SA 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 c 0p Θ is unknowable up to a linear function of S A
does not affect the usefulness of h 0 or c 0p Θ as measures of “heat content”. Similarly, the
difference between the meridional fluxes of c 0p 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 c 0p 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 c 0p Θ is a conservative variable, obeying the simple conservation statement
Eqn. (3.23.1). This issue is discussed at greater length in section 6 of McDougall (2003).
3.24 Geopotential
The geopotential Φ is the gravitational potential energy per unit mass with respect to the
height z = 0. Allowing the gravitational acceleration to be a function of z , Φ is given by
z
Φ = ∫ g ( z ′) dz ′ .
(3.24.1)
0
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 = m 2 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 = Φ 0 − ∫ v ( p′) dP′ ,
(3.24.2)
P0
where Φ 0 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 .
Total energy E
quantity.
is not a function of only
( SA , t , p )
(3.25.1)
and so is not a thermodynamic
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3.26 Bernoulli function
The Bernoulli function is the sum of specific enthalpy h, kinetic energy per unit mass
0.5 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
P
B = h 0 + Φ 0 + 12 u ⋅ u − ∫ v ( p′) − vˆ ( SA , 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))
P
B = h 0 + Φ 0 + Ψ + 12 u ⋅ u − ∫ vˆ ( SSO ,0°C, p′) − vˆ ( SA , Θ, p′) dP′
P0
(3.26.3)
= h + Φ + Ψ + u ⋅ u − hˆ ( SSO ,0°C, p ) + hˆ ( SSO ,0°C,0 ) + hˆ ( S A , Θ, p ) − hˆ ( S A , Θ,0 ) .
0
0
1
2
Note that hˆ ( SA , Θ,0 ) = c 0p Θ and hˆ ( SSO ,0°C,0 ) = 0 .
3.27 Dynamic height anomaly
The dynamic height anomaly Ψ , given by the vertical integral
Ψ = − ∫ δ ( SA [ p′] , t [ p′] , p′) dP′,
P
(3.27.1)
P0
is the geostrophic streamfunction for the flow at pressure P with respect to the flow at the
sea surface and δ 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 − fv 0 .
(3.27.2)
The definition Eqn. (3.27.1) of dynamic height anomaly applies to all choices of the
reference values S A and t , θ or Θ̂ in the definition Eqns. (3.7.1 – 3.7.4) of the specific
volume anomaly δ . Also, δ in Eqn. (3.27.1) can be replaced with specific volume v
without affecting the isobaric gradient of the resulting streamfunction. That is, this
substitution does not affect Eqn. (3.27.2) because the additional term is a function only of
pressure. Traditionally it was important to use specific volume anomaly in preference to
specific volume as it was more accurate with computer code which worked with single‐
precision variables. Since computers now regularly employ double‐precision, this issue
has been overcome and consequently either δ or v can be used in the integrand of Eqn.
(3.27.1), so making it either the “dynamic height anomaly” or the “dynamic height”. As in
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43
the case of Eqn. (3.24.2), so also the dynamic height anomaly Eqn. (3.27.1) has not assumed
that the gravitational acceleration is constant and so Eqn. (3.27.2) applies even when the
gravitational acceleration is taken to vary in the vertical.
The dynamic height anomaly 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. Note that the integration in Eqn. (3.27.1) of specific
volume anomaly with pressure in dbar would yield dynamic height anomaly in units of
m3 kg −1dbar , and the use of these units in Eqn. (3.27.2) would not give the resultant
horizontal gradient in the usual units, being the product of the Coriolis parameter (units of
s−1 ) and the velocity (units of m s −1 ). This is the reason why the pressure integration is
done with pressure in Pa and dynamic height anomaly is output in m2 s−2 .
3.28 Montgomery geostrophic streamfunction
The Montgomery “acceleration potential” π defined by
π = ( P − P0 ) δ − ∫ δ ( SA [ p′] , t [ p′] , p′) dP′
P
(3.28.1)
P0
is the geostrophic streamfunction for the flow in the specific volume anomaly surface
δ ( SA , t , p ) = δ1 relative to the flow at P = P0 (that is, at p = 0 dbar ). Thus the two‐
dimensional gradient of π in the δ1 specific volume anomaly surface is simply related to
the difference between the horizontal geostrophic velocity v in the δ = δ1 surface and at
the sea surface v 0 according to
k × ∇δ1π = fv − fv 0
or
∇δ1π = − k × ( fv − fv 0 ) .
(3.28.2)
The definition, Eqn. (3.28.1), of the Montgomery geostrophic streamfunction applies to all
choices of the reference values S A and t in the definition, Eqn. (3.7.2), of the specific
volume anomaly δ . 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 δ = δ1 surface)
∇nπ = − k × ( fv − fv 0 ) + ( P − P0 ) ∇nδ ,
(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
π Z-H = ( P − P ) δ − ∫ δ ( SA [ p′] , t [ p′] , p′) dP′ .
P
(3.28.4)
P0
The gradient of π Z-H in a neutral tangent plane becomes
∇nπ Z-H = − k × ( fv − fv 0 ) + ( P − P ) ∇ nδ ,
(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
surface.
This term can be further minimized by suitably choosing the constant reference values
S A and Θ in the definition, Eqn. (3.7.3), of specific volume anomaly δ so that this surface
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more closely approximates the neutral tangent plane (McDougall (1989)).
improvement is available because it can be shown that
(
)⎦⎥
(
)
ρ ∇nδ = − ⎡κ ( SA , Θ, p ) − κ SA , Θ, p ⎤ ∇n P ≈ TbΘ Θ − Θ ∇n P .
⎣⎢
This
(3.28.6)
The last term in Eqn. (3.28.5) is then approximately
( P − P ) ∇ nδ
≈
1
2
(
)
ρ −1TbΘ Θ − Θ ∇n ( P − P )
2
(3.28.7)
and hence suitable choices of P , S A and Θ can reduce the last term in Eqn. (3.28.5) that
represents the error in interpreting the Montgomery geostrophic streamfunction, Eqn.
(3.28.4), as the geostrophic streamfunction in a surface that is more neutral than a specific
volume anomaly surface.
The Montgomery geostrophic streamfunction should be quoted in units of m2 s −2 .
These are the units in which the GSW Toolbox (appendix N) outputs the Montgomery
geostrophic streamfunction in the function gsw_geo_strf_Montgomery.
3.29 Cunningham geostrophic streamfunction
Cunningham (2000) and Alderson and Killworth (2005), following Saunders (1995) and
Killworth (1986), suggested that a suitable streamfunction on a density surface in a
compressible ocean would be the difference between the Bernoulli function B and
potential enthalpy h 0 . Since the kinetic energy per unit mass, 0.5 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 − h 0 − 12 u ⋅ u − Φ 0
= h − h0 + Φ − Φ 0
(3.29.1)
P
= h ( S A , Θ, p ) − h ( 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 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 = − ∫ vˆ ( SA ( p′), Θ( p ′), p′) − vˆ ( S A , Θ, p′) dP′.
(3.29.2)
P0
In this form it appears very similar to the expression, Eqn. (3.27.1), for dynamic height
anomaly, the only difference being that in Eqn. (3.27.1) the pressure‐independent values
of Absolute Salinity and Conservative Temperature were SSO and 0°C whereas here they
are the local values on the surface, S A 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.
The gradient of Π along the neutral tangent plane is
∇n Π ≈
{
1
ρ
}
∇ z P − ∇Φ 0 − 21 ρ −1TbΘ ( P − P0 ) ∇n Θ ,
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 (appendix N) as the function gsw_geo_strf_Cunningham.
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3.30 Geostrophic streamfunction in an approximately neutral surface
In order to evaluate a relatively accurate expression for the geostrophic streamfunction in
an approximately neutral surface (such as an ω ‐surface of Klocker et al. (2009a,b) or a
Neutral Density surface of Jackett and McDougall (1997)) 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 is given by (from McDougall and Klocker (2010))
(
)
ϕ n ( SA , Θ, p ) =
1
2
(
)
P − P δ ( SA , Θ, p ) −
1
12
(
)(
ρ −1TbΘ Θ − Θ P − P
)
2
P
− ∫ δ dP′ .
(3.30.1)
P0
This expression is very accurate when the variation of conservative temperature with
pressure along the approximately neutral surface is either linear or quadratic. That is, in
these situations ∇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
McDougall‐Klocker geostrophic streamfunction is available from the GSW Oceanographic
Toolbox as the function gsw_geo_strf_McD_Klocker.
3.31 Pressure‐integrated steric height
The depth‐integrated mass flux of the geostrophic Eulerian flow between two fixed
pressure levels can also be represented by a streamfunction. Using the hydrostatic
relation 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′′
PISH = Ψ ′ = g −1 ∫ Ψ ( p ′′) dP′′ = − g −1 ∫ ∫ δ ( SA [ p′] , t [ p′] , p′) dP′ dP′′
P
P0
= −g
−1
P0 P0
(3.31.1)
∫ ( P − P′) δ ( SA [ p ′] , t [ p′] , p ′) dP′.
P
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
z ( P0 )
P
z( P )
P0
k × ∇ p Ψ′ = f ∫ ρ ⎡⎣ v ( z ′) − v 0 ⎤⎦ dz ′ = g −1 f ∫ ⎡⎣ v ( p′) − v 0 ⎤⎦ dP′.
(3.31.2)
The definition, Eqn. (3.31.1), of PISH applies to all choices of the reference values SA , SA
and t , θ or Θ in the definitions, Eqns. (3.7.2 – 3.7.4), of the specific volume anomaly.
Since the velocity at depth in the ocean is generally much smaller than at the sea
surface, it is customary to take the reference pressure to be some constant (deep) pressure
P1 so that Eqn. (3.27.1) becomes
P1
Ψ = ∫ δ ( S A [ p′] , t [ p′] , p′) dP′
(3.31.3)
P
and PISH , reflecting the depth‐integrated horizontal mass transport from the sea surface
to pressure P1 , relative to the flow at P1 , is
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P1
P1 P1
P0
P0 P′′
PISH = Ψ′ = g −1 ∫ Ψ ( p′′) dP′′ = g −1 ∫ ∫ δ ( SA [ p ′] , t [ p′] , p′) dP′ dP′′
P1
= g −1 ∫ ( P′ − P0 ) δ ( SA [ p ′] , t [ p′] , p′) dP′
(3.31.4)
P0
=
1
2
g
−1
( P1 − P0 )2
∫
0
(
)
δ ( SA [ p′] , t [ p′] , p′) d ( P′ − P0 ) .
2
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 δ 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
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 (or potential
temperature θ ) are piecewise constant in the vertical, the geopotential (Eqn. (3.24.2))
P
Φ = Φ 0 − ∫ v ( p′) dP′ ,
(3.32.1)
P0
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 p i +1 > p i ) then the integral can be expressed
(making use of (3.7.5), namely hP S , Θ = hˆP = v ) as a sum over n layers of the differences
A
in specific enthalpy so that
n −1
P
(
)
(
)
Φ = Φ 0 − ∫ v ( p′) dP′ = Φ 0 − ∑ ⎡ hˆ SAi , Θi , p i +1 − hˆ SAi , Θi , p i ⎤ .
⎦
i =1 ⎣
P
(3.32.2)
0
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 g SA ( SA , tf , p ) = g Ih ( tf , p ) .
(3.33.2)
The Gibbs function for ice Ih, g Ih ( 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
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47
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
with air lowers the freezing temperatures by about 2 mK .
To estimate the effects of small changes in the pressure or salinity on the freezing
temperature, it is convenient to consider a power series expansion of (3.33.1). The result in
the limit of an infinitesimal pressure change at fixed salinity gives the pressure coefficient
of freezing point lowering, as (Clausius‐Clapeyron equation, Feistel et al. (2010a)),
(
)
∂tf
∂p
= χ p ( SA , p ) = −
SA
g p − SA g SA p − g Ih
p
gT − SA g SAT − gTIh
.
(3.33.3)
Its values, evaluated from TEOS‐10, vary only weakly with salinity between
χ p 0 g 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 0 g 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 C and LSIp = 330 J kg −1 as approximations that are appropriate
for the standard ocean. This simple result is only weakly dependent on these choices and
is in reasonable agreement with the exact values from Eqn. (3.33.4) and with Millero and
Leung (1976). The freezing temperature of seawater is always lower than that of pure
water.
When sea‐ice is formed, it often contains remnants of seawater included in brine
pockets. At equilibrium, the salinity in these pockets depends only on temperature and
pressure, rather than, for example, on the pocket volume, and can be computed in the
functional form S A ( 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
IOC Manuals and Guides No. 56
48
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
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 density
of seawater and properties derived from it. The properties considered in the remainder of
this section (3.33‐3.42) which share this attribute are referred to as the colligative
properties of seawater.
3.34 Latent heat of melting
The melting process of ice in pure water can be conducted by supplying heat at constant
pressure. If this is done slowly enough that equilibrium is maintained, then the
temperature will also remain constant. The heat required per mass of molten ice is the
latent heat, or enthalpy, of melting, LWI
p . It is found as the difference between the specific
enthalpy of water, h W , and the specific enthalpy of ice, h Ih , (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 to the heat needed to warm up the seawater and ice phases individually
(Feistel and Hagen (1998), Feistel et al. (2010b)). Mass conservation of both water and salt
during this thermodynamic process is essential to ensure the independence of the latent
heat formula from the unknown absolute enthalpies of salt and water that otherwise
would accompany any mass exchange.
The enthalpy of sea ice, hSI , is additive with respect to its constituents ice, h Ih , with
the mass fraction w Ih , and seawater, h, with the liquid mass fraction 1 − wIh :
(
(
)
)
hSI = 1 − wIh h ( SA , t , p ) + w Ih h Ih ( t , p ) .
(3.34.2)
Upon warming, the mass of melt water changes the ice fraction wIh and the brine salinity
S A . 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
+ wIh
p
∂h Ih
∂T
(
+ h Ih − h
p
)
∂w Ih
∂T
.
(3.34.3)
p
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
IOC Manuals and Guides No. 56
=
p
S A ∂wIh
1 − wIh ∂T
.
p
(3.34.4)
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
49
Using this relation, Eqn. (3.34.3) takes the simplified form
∂hSI
∂T
(
)
SI
= 1 − wIh c p + wIh c Ih
p − Lp
p
∂wIh
∂T
.
(3.34.5)
p
The coefficient in front of the melting rate,
LSIp ( SA , p ) = h − SA
∂h
∂S A
− 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
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
LSIp ( SA , p ) = (T0 + tf ) × ⎜η − SA
⎜
∂SA
⎝
T,p
⎞
− η Ih ⎟ .
⎟
⎠
(3.34.7)
Again the result is independent of unknown (and unknowable) constants.
The latent heat of melting depends only weakly on salinity and on pressure. At the
–1
surface pressure, the computed value is LSIp ( 0,0 ) = LWI
p ( 0 ) = 333 426.5 J kg for pure water,
SI
and Lp ( SSO ,0 ) = 329 928.5 J kg–1 for the standard ocean, with a difference of about 1% due
to the dissolved salt. At a pressure of 1000 dbar, these values reduce by 0.6% to
–1 and LSI S
–1
LSIp ( 0,1000dbar ) = LWI
p ( 1000dbar ) = 331 528 J kg
p ( SO ,1000dbar ) = 328 034 J kg .
WI
TEOS‐10 is consistent with the most accurate measurements of L p 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 P subl 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
(
)
(
)
(3.35.1)
)
(3.35.2)
μ V t , P subl = μ Ih t , P subl ,
or equivalently, in terms of the two Gibbs functions,
(
)
(
g V t , P subl = g Ih t , P subl .
The Gibbs function for ice Ih, g ( 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 P subl = 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
Ih
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50
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
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
xV =
1− A
,
1 − A (1 − M W / M A )
(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 ) ,
(3.35.4)
or equivalently, in terms of the two Gibbs functions,
(
)
(
)
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
subl
P (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 P subl (‐1.919 °C, 101 325 Pa) = 523.436 Pa.
The difference between observed or modelled partial vapour pressures and the
sublimation pressure computed from TEOS‐10 is an appropriate quantity for use in
parameterizations of the mass flux between ice and the atmosphere.
3.36 Sublimation enthalpy
The sublimation process that occurs when ice is in contact with pure water vapour can be
conducted by supplying heat at constant t and P, with t at or below the freezing
temperature. The heat required per mass evaporated from the ice is the latent heat, or
enthalpy, of sublimation, LVI
p . It is found as the difference between the specific enthalpy
V
of water vapour, h , and the specific enthalpy of ice, h Ih :
(
)
(
)
V
LVI
t , Psubl − h Ih t , Psubl .
p (t ) = h
(3.36.1)
Here, P ( 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%.
subl
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
51
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 P subl = 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
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, h Ih , with
the mass fraction w Ih , and humid air, h AV , with the gas fraction 1 − w Ih :
(
(
)
)
h AI = 1 − wIh h AV ( A, t , p ) + wIh h Ih ( t, p ) .
(3.36.2)
Upon warming, the mass of vapour produced by sublimation reduces the ice fraction wIh
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
AV
) ∂∂hT
(
+ 1 − wIh
A, p
AV
) ∂h∂A
T ,p
∂A
∂T
+ wIh
p
∂h Ih
∂T
(
+ h Ih − h AV
p
Ih
) ∂∂wT
. (3.36.3)
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 ∂wIh
1 − w Ih ∂T
.
(3.36.4)
p
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.
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
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.
3.37 Vapour pressure
The vapour pressure of seawater P vap ( SA , t ) is defined as the absolute pressure P of
water vapour in equilibrium with seawater at a given temperature t and salinity S A . 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 ,
(3.37.1)
or equivalently, in terms of the two Gibbs functions,
(
) (
)
(
)
g V t , P vap = g SA , t , P vap − SA g SA S A , 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 or IAPWS‐09 (IAPWS (2009c)).
In the case of pure water, S A = 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
(
IOC Manuals and Guides No. 56
)
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
∂P vap
∂SA
≈ −
T
M W vap
≈ − 0.57 × P vap .
P
MS
53
(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 S A . It is found by equating the chemical
V
potential of vapour in humid air μAV
with the chemical potential of water in seawater
W
μ so that it is found by solving the implicit equation
(
)
V
μAV
Acond , t , P = μ W ( S A , t , P )
(3.37.5)
for Acond ( SA , t , P ) , or equivalently, in terms of the two Gibbs functions,
(
)
(
)
g AV Acond , t , P − Acond g AAV Acond , t , P = g ( S A , 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 (3.53.2), 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 is also planned to be made
available as the document 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 ( SA , 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 ,
for t
boil
(3.38.1)
( 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 ( 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.
boil
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
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
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 vap (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
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 :
h
(
SA
SW
= 1− w
)h
(
AV
( A, t, p )
SW
+w
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
SW
+w
AV
) ∂∂hT
∂h
∂T
(
+ 1 − wSW
A, p
SW
+w
SA , p
∂h
∂SA
AV
) ∂h∂A
T,p
∂SA
∂T
T,p
∂A
∂T
p
(
+ h−h
AV
)
p
∂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
,
(3.39.4)
p
and to the change of salinity by the conservation of the salt, wSW S A = 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
.
(3.39.6)
p
The coefficient in front of the evaporation rate,
AV
LSA
−A
p ( A, S A , t , p ) = h
IOC Manuals and Guides No. 56
∂h AV
∂A
− h + SA
T,p
∂h
∂SA
,
T,p
(3.39.7)
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
55
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.
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.
Condition
Pure water
Brackish water
Standard ocean
Tropical ocean
High pressure
SA
g kg–1
0
10
35.165 04
35.165 04
35.165 04
t
°C
0
0
0
25
0
p
dbar
0
0
0
0
400
Acond
%
99.622 31
99.624 27
99.629 31
98.059 33
99.989 43
LSA
p
J kg–1
2 499 032
2 499 009
2 498 510
2 438 971
2 443 759
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
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.
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
IOC Manuals and Guides No. 56
56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
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 CCT 1 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, IUPAC 2 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
xV
RH CCT = sat
(3.40.1)
xV
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 ( SA , t , P ) is the air fraction of humid
air at equilibrium with seawater, Eqn. (3.37.5), which is subsaturated for SA > 0.
The WMO 3 definition of the relative humidity is (Pruppacher and Klett (1997),
Jacobson (2005)),
r
1/ A −1
RH WMO = sat =
(3.40.2)
1 / Asat − 1
r
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
⎧⎪ μ V − μ V, id ⎫⎪
f V ( A, T , P ) = xV P exp ⎨
⎬.
⎩⎪ RWT ⎭⎪
(3.40.3)
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,
μ V, id ( A, T , P ) = g0V +
T
T ⎞ V,id
x P
⎛
∫ ⎜⎝1 − T ' ⎟⎠ c p (T ') dT ' + RWT ln PV0V .
(3.40.4)
T0V
The values of g0V , P0V and T0V of μ V,id must be chosen consistently with the adjustable
id
constants of g AV (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 .
P →0
1
(3.40.5)
CCT: Consultative Committee for Thermometry, www.bipm.org/en/committees/cc/cct/
IUPAC: International Union of Pure and Applied Chemistry, www.iupac.org
3 WMO: World Meteorological Organisation, www.wmo.int
2
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
57
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
ϕ =
V
W
fV
⎪⎧ μ ( A, T , P ) − μ ( 0, T , P ) ⎪⎫
exp
=
⎨
⎬.
RWT
f Vsat
⎪⎩
⎪⎭
(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
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 ) = −
1
mSW RT
⎡
∂g S
⎢ g S − SA
∂SA
⎢⎣
⎤
⎥,
T ,P ⎥
⎦
(3.40.9)
where mSW is the molality of seawater (Millero et al. (2008a)),
mSW =
SA
(1 − SA ) M S
.
(3.40.10)
From the chemical potential of water in seawater, µ W = g − SA g SA , and Eqns. (3.40.8) ‐
(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 ( S A ) the linear relation
M
ϕ SA ≈ 1 − W SA ,
(3.40.12)
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
f Vsat
=
(
)
⎧ μ Ih (T , P ) − μ V,id Asat , T , P ⎫
⎪
⎬,
R
T
W
⎪⎩
⎪⎭
⎪
xVsat P exp ⎨
(3.40.13)
rather than Eqn. (3.40.6). Then, the relative fugacity of sea air is
IOC Manuals and Guides No. 56
58
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
ϕ SA =
⎧⎪ μ W ( S A , T , P ) − μ Ih (T , P ) ⎫⎪
f VSA
exp
=
⎨
⎬.
RWT
f Vsat
⎩⎪
⎭⎪
(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
⎛ μ ⎞
( A, T , P ) − μ W ( SA , T , P ) .
=
X SA = Δ ⎜
⎟
RWT
RWT
⎝ RWT ⎠
(3.40.15)
This difference vanishes at the condensation point, A = Acond ( S A , 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
ϕ ( A)
= mSW M Wφ + ln ϕ ( A) .
(3.40.16)
X SA = ln SA
ϕ ( SA )
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 the pressure p W and of water in seawater at the pressure p,
(
)
g W t , p W = g ( SA , t, p ) − SA
∂g
∂SA
.
(3.41.1)
T, p
The solution of this implicit relation for the osmotic pressure is
P osm ( SA , t , p ) = P − P W .
The TEOS‐10 value for the osmotic pressure of the
P osm ( SSO ,0 °C,0dbar ) = 2 354 684 Pa, computed from Eqn. (3.41.1).
IOC Manuals and Guides No. 56
(3.41.2)
standard
ocean
is
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
59
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 ) ,
gTP ( S A , tMD , p ) = 0.
(3.42.1)
The temperature of maximum density is available in the GSW Oceanographic Toolbox as
function gsw_temps_maxdensity. This function also returns the potential temperature
and the Conservative Temperature at this maximum density point. Selected TEOS‐10
values computed from Eqn. (3.42.1) are given in Table 3.42.1.
Table 3.42.1: Freezing temperature 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.
SA
g kg–1
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
8
tf
°C
+0.003
–0.026
–0.054
–0.081
–0.108
–0.135
–0.162
–0.189
–0.216
–0.243
–0.269
–0.296
–0.323
–0.349
–0.376
–0.403
–0.429
tMD
°C
3.978
3.868
3.758
3.649
3.541
3.432
3.324
3.215
3.107
2.999
2.890
2.782
2.673
2.564
2.456
2.347
2.238
SA
g kg–1
8.5
9
9.5
10
10.5
11
11.5
12
12.5
13
13.5
14
14.5
15
15.5
16
16.5
tf
°C
–0.456
–0.483
–0.509
–0.536
–0.563
–0.590
–0.616
–0.643
–0.670
–0.697
–0.724
–0.750
–0.777
–0.804
–0.831
–0.858
–0.885
tMD
°C
2.128
2.019
1.909
1.800
1.690
1.580
1.470
1.360
1.249
1.139
1.028
0.917
0.807
0.696
0.584
0.473
0.362
SA
g kg–1
17
17.5
18
18.5
19
19.5
20
20.5
21
21.5
22
22.5
23
23.5
24
24.5
25
tf
°C
–0.912
–0.939
–0.966
–0.994
–1.021
–1.048
–1.075
–1.102
–1.130
–1.157
–1.184
–1.212
–1.239
–1.267
–1.294
–1.322
–1.349
tMD
°C
0.250
0.139
0.027
–0.085
–0.196
–0.308
–0.420
–0.532
–0.644
–0.756
–0.868
–0.980
–1.092
–1.204
(–1.316)
(–1.428)
(–1.540)
IOC Manuals and Guides No. 56
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
4. Conclusions
The International Thermodynamic Equation of Seawater – 2010 (TEOS‐10) allows all the
thermodynamic properties of pure water, ice Ih, seawater and moist air to be evaluated in
an internally self‐consistent manner. Ice Ih is the naturally abundant form of ice, having
hexagonal crystals. For the first time the effects of the small variations in seawater
composition around the world ocean can be included, especially their effects on the density
of seawater (which can be equivalent to ten times the precision of our Practical Salinity
measurements at sea).
Perhaps the most apparent change compared to the International Equation of State of
seawater (EOS‐80) is the adoption of Absolute Salinity S A instead of Practical Salinity S P
(PSS‐78) as the salinity argument for the thermodynamic properties of seawater.
Importantly, Practical Salinity is retained as the salinity variable that is stored in data
bases because Practical Salinity is virtually the measured variable (whereas Absolute
Salinity is a calculated variable) and also so that national data bases do not become
corrupted with incorrectly labeled and stored salinity data.
The adoption of Absolute Salinity as the argument for all the algorithms used to
evaluate the thermodynamic properties of seawater makes sense simply because the
thermodynamic properties of seawater depend on 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 in a two‐stage process. First Reference Salinity is
calculated from measurements of Practical Salinity using Eqn. (2.4.1). Then the Absolute
Salinity Anomaly is estimated from the computer algorithm of McDougall et al. (2010a) or
by other means, and Absolute Salinity is formed as the sum of Reference Salinity and the
Absolute Salinity Anomaly. There are subtle issues in defining what is exactly meant by
“absolute salinity” and at least four different definitions are possible when compositional
anomalies are present. We have chosen the definition that yields the most accurate
estimates of seawater density since the ocean circulation is sensitive to rather small
gradients of density. The algorithm that estimates Absolute Salinity Anomaly represents
the state of the art as at 2010, but this area of oceanography is relatively immature. It is
likely that the accuracy of this algorithm will improve as more seawater samples from
around the world ocean have their density accurately measured. After such future work
is published and the results distilled into a revised algorithm for Absolute Salinity
Anomaly, such an algorithm will be served from www.TEOS‐10.org. Oceanographers
should publish the version number of this software that is used to obtain thermodynamic
properties in their manuscripts.
Because Absolute Salinity is the appropriate salinity variable for use with the equation
of state, Absolute Salinity is the salinity variable that should be published in
oceanographic journals. The version number of the software that is used to convert
Reference Salinity S R into Absolute Salinity S A 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.
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
61
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 the salinity variable that is being graphed is Reference
Salinity, not Absolute Salinity.
The treatment of salinity in ocean models is discussed in appendix A.20. The
recommended approach is to carry both Preformed Salinity S* and Absolute Salinity
Anomaly δ S A as model variables so that Density Salinity can be calculated at each time
step of the model and used to accurately evaluate density.
Potential temperature has been used in oceanography as though it is a conservative
variable, and yet the specific heat of seawater varies by 5% at the sea surface, and potential
temperature is not conserved when seawater parcels mix.
The First Law of
Thermodynamics can be very accurately regarded as the statement that potential enthalpy
is a conservative variable in the ocean. This, together with the knowledge that the air‐sea
heat flux is exactly the air‐sea flux of potential enthalpy (i. e. the air‐sea flux of c 0p 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, ( S A − 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
c 0p Θ across an ocean section as the “heat flux” across the section even when the fluxes of
mass and of salt across the section are non‐zero.
The temperature variable in ocean models is commonly regarded as being potential
temperature, but since the non‐conservative source terms that are present in the evolution
equation for potential temperature are not included in models, it is apparent that the
interior of ocean models already treat the prognostic temperature variable as Conservative
Temperature Θ . To complete the transition to Θ in ocean modeling, 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 ) . The final ingredient needed for an ocean model is a computationally
efficient form of density in terms of Conservative Temperature, that is ρ = ρˆ ( SA , Θ, p ) ,
such as that described in appendix A.30 and appendix K of this TEOS‐10 Manual.
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 S P − θ diagram. Curved
contours of potential density could also be drawn on this same S P − 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 now not possible to draw isolines of
potential density on a S P − θ diagram. Rather, because of the spatial variations of
seawater composition, a given value of potential density defines an area on the S P − θ
diagram, not a curved line. Under TEOS‐10, the observed variables ( SP , t , p ) , together
with longitude and latitude, are used to first form Absolute Salinity S A and Conservative
Temperature Θ. Oceanographic water masses are then analyzed on the S A − Θ diagram,
and potential density contours can also be drawn on this S A − Θ diagram, while
Preformed Salinity S* is the natural salinity variable to be used in applications such as
numerical modelling where it is important that the salinity variable be conservative.
When describing the use of TEOS‐10, it is the present document (the TEOS‐10 Manual)
that should be referenced as IOC et al. (2010) [IOC, SCOR and IAPSO, 2010: The
international thermodynamic equation of seawater – 2010: Calculation and use of thermodynamic
properties. Intergovernmental Oceanographic Commission, Manuals and Guides No. 56,
UNESCO (English), 196 pp].
IOC Manuals and Guides No. 56
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
APPENDIX A:
Background and theory underlying
the use of the Gibbs function of seawater
A.1 ITS-90 temperature
In order to understand the limitations of conversion between different temperature scales,
it is helpful to review the definitions of temperature and of the international scales on
which it is reported.
A.1.1 Definition
When considering temperature, the fundamental physical quantity is thermodynamic
temperature, symbol T. The unit for temperature is the kelvin. The name of the unit has a
lowercase k. The symbol for the unit is uppercase K. One kelvin is 1/273.16 of the
thermodynamic temperature of the triple point of water. (A recent evolution of the
definition has been to specify the isotopic composition of the water to be used as that of
Vienna Standard Mean Ocean Water, VSMOW.) The Celsius temperature, symbol t , is
defined by t °C = T K − 273.15, and 1 °C is the same size as 1 K.
A.1.2 ITS‐90 temperature scale
The definition of temperature scales is the responsibility of the Consultative Committee
for Thermometry (CCT) which reports to the International Committee for Weights and
Measures (often referred to as CIPM for its name in the French language). Over the last 40
years, two temperature scales have been used; the International Practical Temperature
Scale 1968 (IPTS‐68), followed by the International Temperature Scale 1990 (ITS‐90). These
are defined by Barber (1969) and Preston‐Thomas (1990). For information about the
International Temperature Scales of 1948 and 1927 the reader is referred to Preston‐
Thomas (1990).
In the oceanographic range, temperatures are determined using a platinum resistance
thermometer. The temperature scales are defined as functions of the ratio W , namely the
ratio of the thermometer resistance at the temperature to be measured R ( t ) to the
resistance at a reference temperature R0 . In IPTS‐68, R0 is R ( 0°C ) , while in ITS‐90 R0 is
R ( 0.01°C ) . The details of these temperature scales and the differences between the two
scales are therefore defined by the functions of W used to calculate T . For ITS‐90, and in
the range 0 °C < t90 < 968.71 °C, t90 is described by a polynomial with 10 coefficients given
by Table 4 of Preston‐Thomas (1990).
We note in passing that the conversions from W to T and from T to W are both
defined by polynomials and these are not perfect inverses of one another. Preston‐
Thomas points out that the inverses are equivalent to within 0.13mK. In fact the inverses
have a difference of 0.13 mK at 861°C, and a maximum error in the range 0 °C < t90 < 40 °C
of 0.06 mK at 31 °C. That the CCT allowed this discrepancy between the two polynomials
immediately provides an indication of the absolute uncertainty in the determination, and
indeed in the definition, of temperature.
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A second uncertainty in the absolute realization of ITS‐90 arises from what is referred
to as sub‐range inconsistency. The polynomial referred to above describes the behaviour
of an ‘ideal’ thermometer. Any practical thermometer has small deviations from this ideal
behaviour. ITS‐90 allows the deviations to be determined by measuring the resistance of
the thermometer at up to five fixed points: the triple point of water and the freezing points
of tin, zinc, aluminium and silver, covering the range 0.01 °C < t90 < 961.78 °C. If not all of
these points are measured, then it is permissible to estimate the deviation from as many of
those points as are measured. The melting point of Gallium ( t90 = 29.7646 °C) and the
triple point of Mercury ( t90 = ‐ 38.8344 °C) may also be used if the thermometer is to
operate over a smaller temperature range. Hence the manner in which the thermometer
may be used to interpolate between the points is not unique. Rather it depends on which
fixed points are measured, and there are several possible outcomes, all equally valid
within the definition. Sections 3.3.2 and 3.3.3 of Preston‐Thomas (1990) give precise
details of the formulation of the deviation function. The difference between the deviation
functions derived from different sets of fixed points will depend on the thermometer, so it
not possible to state an upper bound on this non‐uniqueness. Common practice in
oceanographic standards laboratories is to estimate the deviation function from
measurements at the triple point of water and the melting point of Gallium ( t90 = 29.7646
°C). This allows a linear deviation function to be determined, but no higher order terms.
In summary, there is non‐uniqueness in the definition of ITS‐90, in addition to any
imperfections of measurement by any practical thermometer (Rudtsch and Fischer (2008),
Feistel et al. (2008a)). It is therefore not possible to seek a unique and perfect conversion
between IPTS‐68 and ITS‐90.
Goldberg and Weir (1992) and Mares and Kalova (2008) have discussed the
procedures needed to convert measured thermophysical quantities (such as specific heat)
from one temperature definition to another. When mechanical or electrical energy is used
in a laboratory to heat a certain sample, this energy can be measured in electrical or
mechanical units by appropriate instruments such as an ampere meter, independent of
any definition of a temperature scale. It is obvious from the fundamental thermodynamic
relation (at constant Absolute Salinity), du = Tdη + Pdv, that the same energy difference
Tdη results in different values for the entropy η , depending on the number read for T
from a thermometer calibrated on the 1990 compared with one calibrated on the 1968
scale. A similar dependence is found for numbers derived from entropy, for example, for
the heat capacity,
c p = T ηT S , p .
A
Douglas (1969) listed a systematic consideration of the quantitative relations between the
measured values of various thermal properties and the particular temperature scale used
in the laboratory at the time the measurement was conducted. Conversion formulas to
ITS‐90 of readings on obsolete scales are provided by Goldberg and Weir (1992) and Weir
and Goldberg (1996).
Any thermal experimental data that entered the construction of the thermodynamic
potentials that form TEOS‐10 were carefully converted by these rules, in addition to the
conversion between the various older definitions of for example calories and joules. This
must be borne in mind when properties computed from TEOS‐10 are combined with
historical measurements from the literature.
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A.1.3 Theoretical conversion between IPTS‐68 and ITS‐90
Having understood that the conversion between IPTS‐68 and ITS‐90 is not uniquely
defined, we review the sources of uncertainty, or even flexibility, in the conversion
between t90 and t68 .
Consider first why t90 and t68 temperatures differ:
1) The fixed points have new temperature definitions in ITS‐90, due to improvements in
determining the absolute thermodynamic temperatures of the melting/freezing physical
states relative to the triple point of water.
2) For some given resistance ratio W the two scales have different algorithms for
interpolating between the fixed points.
Now consider why there is non‐uniqueness in the conversion:
3) In some range of ITS‐90, the conversion of W to t90 can be undertaken with a choice of
coefficients that is made by the user (Preston‐Thomas (1990) Sections 3.3.2.1 to 3.3.3),
referred to as sub‐range inconsistency.
4) The impact of the ITS‐90 deviation function on the conversion is non‐linear. Therefore
the size of the coefficients in the deviation function will affect the difference, t90 − t68 .
The formal conversion is different for each actual thermometer that has been used to
acquire data.
The group responsible for developing ITS‐90 was well aware of the non‐uniqueness
of the conversion. Table 6 of Preston‐Thomas (1990) gives differences ( t90 − t68 ) with a
resolution of 1 mK, because
(a) the true thermodynamic temperature T was known to have uncertainties of order
1 mK or larger in some ranges,
(b) the sub‐range inconsistency of ITS‐90 using the same calibration data gave an
uncertainty of several tenths of 1 mK.
Therefore to attempt to define a generic conversion of ( t90 − t68 ) with a resolution of say
0.1 mK would probably be meaningless and possibly misleading as there isn’t a unique
generic conversion function.
A.1.4 Practical conversion between IPTS‐68 and ITS‐90
Rusby (1991) published an 8th order polynomial that was a fit to Table 6 of Preston‐
Thomas (1990). This fit is valid in the range 73.15 K to 903.89 K (‐200 °C to 630.74 °C). He
reports that the polynomial fits the table to within 1 mK, commensurate with the non‐
uniqueness of IPTS‐68.
Rusby’s 8th order polynomial is in effect the ‘official recommended’ conversion
between IPTS‐68 and ITS‐90. This polynomial has been used to convert historical IPTS‐68
data to ITS‐90 for the preparation of the new thermodynamic properties of seawater that
are the main subject of this manual.
As a convenient conversion valid in a narrower temperature range, Rusby (1991) also
proposed
(A.1.1)
(T90 − T68 ) /K = -0.00025 (T68 / K - 273.15)
in the range 260 K to 400 K (‐13 °C to 127 °C). Rusby (1991) also explicitly reminds readers
(see his page 1158) that compound quantities that involve temperature intervals such as
heat capacity and thermal conductivity are affected by their dependence on the derivative
d (T90 − T68 ) /dT68 . About the same time that Rusby published his conversion from t68 to
t90 , Saunders (1990) made a recommendation to oceanographers that in the common
oceanographic temperature range ‐2 °C < t68 < 40 °C, conversion could be achieved using
( t90 /°C )
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= ( t68 /°C ) 1.00024.
(A.1.2)
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65
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 )
= 1.00024 ( t90 /°C ) .
(A.1.3)
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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 ≡ 101 325 Pa; 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 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
pound‐
force per
square
inch
(Torr)
(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
1.3332×10−3 1.3595×10−3 1.3158×10−3
≡ 1 Torr
19.337×10−3
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 ≡ 101 325 Pa. (When the CTD is
lowered from the sea surface, the monitoring software may well display gauge pressure,
indicating the distance from the surface.)
Since there are a variety of different units used to express atmospheric pressure, we
present a table (Table A.2.1) to assist in converting between these different units of
pressure (see ISO (1993)). Note that one decibar (1 dbar) is exactly 0.1 bar, and that 1
mmHg is very similar to 1 torr with the actual relationship being 1 mmHg = 1.000 000 142
466 321... torr. The torr is defined as exactly 1/760 of the Absolute Pressure of one
standard atmosphere, so that one torr is exactly equal to (101 325/760) Pa.
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A.3 Reference Composition and the Reference-Composition Salinity Scale
As mentioned in the main text, the Reference Composition of seawater is defined by
Millero et al. (2008a) as the exact mole fractions given in Table D.3 of appendix D below.
This composition model was determined from the most accurate measurements available
of the properties of Standard Seawater, which is filtered seawater from the surface waters
of the North Atlantic as made available by the IAPSO Standard Seawater Service. The
Reference Composition is perfectly consistent with charge balance of ocean waters and the
most recent atomic weight estimates (Wieser (2006)). For seawater with this reference
composition the Reference‐Composition Salinity S R as defined below provides our best
estimate of the Absolute Salinity.
The Reference Composition includes all important components of seawater having
mass fractions greater than about 0.001 g kg −1 (i. e. 1.0 mg kg −1 ) that can significantly
affect either the conductivity or the density of seawater having a Practical Salinity of 35.
The most significant ions not included are Li + (~0.18 mg kg −1 ) and Rb+ (~0.12 mg kg −1 ).
Dissolved gases N 2 (~16 mg kg −1 ) and O2 ( up to 8 mg kg −1 in the ocean) are not included
as neither have a significant effect on density or on conductivity. In addition, N 2 remains
within a few percent of saturation at the measured temperature in almost all laboratory
and in situ conditions. However, the dissolved gas CO2 (~0.7 mg kg −1 ), and the ion OH −
(~0.08 mg kg −1 ) are included in the Reference Composition because of their important role
in the equilibrium dynamics of the carbonate system. Changes in pH which involve
conversion of CO2 to and from ionic forms affect conductivity and density.
Concentrations of the major nutrients Si(OH) 4 , NO3− and PO34− are assumed to be
negligible in Standard Seawater; their concentrations in the ocean range from 0‐16
mg kg −1 , 0‐2 mg kg −1 , and 0‐0.2 mg kg −1 respectively. The Reference Composition does
not include organic matter. The composition of Dissolved Organic Matter (DOM) is
complex and poorly known. DOM is typically present at concentrations of 0.5‐2 mg kg −1
in the ocean.
Reference‐Composition Salinity is defined to be conservative during mixing or
evaporation that occurs without removal of sea salt from solution. Because of this
property, the Reference‐Composition Salinity of any seawater sample can be defined in
terms of products determined from the mixture or separation of two precisely defined end
members. Pure water and KCl‐normalized seawater are defined for this purpose. Pure water
is defined as Vienna Standard Mean Ocean Water, VSMOW, which is described in the
2001 Guideline of the International Association for the Properties of Water and Steam
(IAPWS (2005), BIPM (2005)); it is taken as the zero reference value. KCl‐normalized
seawater (or normalized seawater for short) is defined to correspond to a seawater sample
with a Practical Salinity of 35. Thus, any seawater sample that has the same electrical
conductivity as a solution of potassium chloride (KCl) in pure water with the KCl mass
fraction of 32.4356 g kg‐1 when both are at the ITS‐90 temperature t = 14.996 °C and one
standard atmosphere pressure, P = 101325 Pa is referred to as normalized seawater.
Here, KCl refers to the normal isotopic abundances of potassium and chlorine as described
by the International Union of Pure and Applied Chemistry (Wieser (2006)). As discussed
below, any normalized seawater sample has a Reference‐Composition Salinity of
35.165 04 g kg −1.
Since Reference‐Composition Salinity is defined to be conservative during mixing, if a
seawater sample of mass m1 and Reference‐Composition Salinity S R1 is mixed with
another seawater sample of mass m2 and Reference‐Composition Salinity S R 2 , the final
Reference‐Composition Salinity S R12 of this sample is
S R12 =
m1S R1 + m2 S R 2
.
m1 + m2
(A.3.1)
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Negative values of m1 and m2 , corresponding to the removal of seawater with the
appropriate salinity are permitted, so long as m1 (1 − S R1 ) + m2 (1 − S R 2 ) > 0 . In particular, if
S R 2 = 0 (pure water) and m2 is the mass of pure water needed to normalize the seawater
sample (that is, m2 is the mass needed to achieve S R12 = 35.165 04 g kg−1), then the original
Reference‐Composition Salinity of sample 1 is given by
S R1 = [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
S R ≈ uPS S P
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 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
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composition anomalies relative to North Atlantic conditions are found, are reflected in
Knudsenʹs formula,
S K ( ‰ ) = 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 with this accepted definition of salinity for typical mid‐ocean conditions
(Millero (2010)). To achieve this consistency while having salinity directly proportional to
Chlorinity, the Joint Panel for Oceanographic Tables and Standards (JPOTS) decided to
determine the proportionality constant from Knudsenʹs formula at 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
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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 S R .
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 .
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 S A , (as in S A = 35.165 04 g kg−1).
The Reference Salinity, S R , is defined to have the same units and follows the same
conventions as S A . 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 S = 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, S R = 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.,
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Si (OH) 4 , CO2 and dissolved organic material) do not have an appreciable conductivity
signal. It is for this reason that the new TEOS‐10 thermodynamic description of seawater
(Millero et al. (2008a), Millero (2010)) has the Gibbs function 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, S R . The issue of the spatial variation in the composition of
seawater is discussed more fully in appendix A.5.
Regarding point number 2, we note that it is perhaps debatable which of
(1 – 0.001 SAdens /( g kg −1 )), (1 – 0.001 S Asoln /( g kg −1 )), (1 – 0.001 S Aadd /( 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 S P in this context, and
the choice of which of these expressions may depend on the use for the freshwater mass
fraction. For example, in the context of ocean modelling, if S* is the salinity variable that
is treated as a conservative variable in an ocean model, then (1 – 0.001 S* /( 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. Working Group 127 is recommending
that the practice of expressing salinity as Practical Salinity in publications be phased out in
favour of using Absolute Salinity for this purpose. It is critically important that this new
measure of salinity remain stable into the future. In particular, we note that any change in
the value of 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 larger than 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), we strongly recommend
that the Reference‐Composition Salinity continue to be expressed on this scale; no changes
in the value of uPS should be introduced.
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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 (2010) 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.,
2010) but at 0.0055 g kg −1 it is still just below the uncertainty of 0.007 g kg −1 assigned to
the estimated Absolute Salinity by Millero et al. (2008a). In fact, the largest uncertainties in
Reference Salinity as a measure of the Absolute Salinity of SSW are associated with
uncertainties in the mass fractions of other constituents such as sulphate, which may be as
large as 0.05 g kg −1 (Seitz et al., 2010a). Nevertheless, it seems that the sulphate value of
Reference‐Composition Seawater lies within the 95% uncertainty range of the best
laboratory‐determined estimates of SSW’s sulphate concentration, so there is no
justification for an update of the Reference Composition at this time.
When the composition of seawater differs from that of Standard Seawater, there are
several possible definitions of the absolute salinity of a seawater sample, as discussed in
section 2.5. Conceptually the simplest definition is “the mass fraction of dissolved non‐
H 2O material in a seawater sample at its temperature and pressure”. One drawback of
this definition is that because the equilibrium conditions between H 2O 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‐ H 2O material is called “Solution Absolute
Salinity” (usually shortened to “Solution Salinity”), S Asoln . Another measure of absolute
salinity is the “Added‐Mass Salinity” S Aadd 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. (2010) and Wright et al.
(2010b). Preformed Salinity S* is designed to be as close as possible to being a
conservative variable. That is, S* is designed to be insensitive to biogeochemical
processes that affect the other types of salinity to varying degrees. S* is formed by first
estimating the contribution of biogeochemical processes to one of the salinity measures
SA , SAsoln , 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.
There is still no simple means to measure either SAsoln 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 SA , that is SA ≡ SAdens . There are two clear advantages of
SA ≡ SAdens over both SAsoln 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
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SA ≡ SAdens yields the best available estimates of the density of seawater. This is important
because in the field of physical oceanography, it is density that needs to be known to the
highest relative accuracy.
Pawlowicz et al. (2010) and Wright et al. (2010b) found that while the nature of the
ocean’s composition variations changes from one ocean basin to another, the five different
salinity measures S R , SAdens , SAsoln , SAadd and S* are approximately related by the following
simple linear relationships, (obtained by combining equations (55) – (57) and (62) of
Pawlowicz et al. (2010))
S∗ − SR ≈ − 0.35 δ SRdens ,
(A.4.1)
SAdens − SR ≡ 1.0 δ SRdens ,
(A.4.2)
SAsoln − S R ≈ 1.75 δSdens
R ,
(A.4.3)
SAadd − S R ≈ 0.78 δ S Rdens .
(A.4.4)
Eqn. (A.4.2) is simply the definition of the Absolute Salinity Anomaly,
δ SA ≡ δ SRdens ≡ SAdens − SR . Note that in many TEOS‐10 publications, the simpler notation
δ SA is used for δ S Rdens ≡ SAdens − SR , a salinity difference for which a global atlas is
available (McDougall et al. (2010a)). In the context of ocean modelling, it is more
convenient to cast these salinity differences with respect to the Preformed Salinity S∗ as
follows (using the above equations)
S R − S∗ ≈ 0.35 δ SRdens ,
(A.4.5)
SAdens − S∗ ≈ 1.35 δ SRdens ,
(A.4.6)
SAsoln − S* ≈ 2.1 δ SRdens ,
(A.4.7)
SAadd − S∗ ≈ 1.13 δ S Rdens .
(A.4.8)
These relationships are illustrated on the number line of salinity in Figure A.4.1. For SSW,
all five salinity variables S R , SAdens , SAsoln , SAadd and S* are equal. It should be noted that
the simple relationships of Eqns. (A.4.1) – (A.4.8) are derived from simple linear fits to
model calculations that show more complex variations. However, the variation about
these relationships is not larger than the typical uncertainty of ocean measurements.
These linear relationships provide a way by which the effects of anomalous seawater
composition may be addressed in ocean models (see appendix A.20).
Figure A.4.1.
Number line of salinity, illustrating the differences
between various forms of salinity for seawater whose
composition differs from that of Standard Seawater.
If measurements are available of the Total Alkalinity, Dissolved Inorganic Carbon, and
the nitrate and silicate concentrations, but not of density anomalies, then alternative
formulae are available for the four salinity differences that appear on the left‐hand sides of
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Eqns. (A.4.1) – (A.4.8). Pawlowicz et al. (2010) have used a chemical model of conductivity
and density to estimate how the many salinity differences introduced above depend on
the measured properties of seawater. The following equations correspond to Eqns. (A.4.1)
– (A.4.4) above, and come from equations (51) – (54) and (59) of Pawlowicz et al. (2010).
These equations are written in terms of the values of the nitrate and silicate concentrations
in the seawater sample (measured in 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 ( S P 35) mol kg−1 and 0.00208 ( SP 35) mol kg −1 respectively (see Pawlowicz (2010),
Pawlowicz et al. (2010) and the discussion of this aspect of SSW versus RCSW in Wright et
al. (2010b))).
(
)
(mol kg −1 ) ,
)
(
)
(mol kg −1 ) , (A.4.10)
)
(
)
(mol kg −1 ) , (A.4.11)
)
(
)
(mol kg −1 ) .
( S* − SR ) / (g kg −1 )
(S
dens
A
(S
− S R / (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
− S R / (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
(A.4.9)
(A.4.12)
The standard error of the model fits in Eqns. (A.4.9) – (A.4.11) are given by Pawlowicz et
al. (2010) at less than 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 ) ,
)
(
)
(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
(A.4.13)
− S* / (g kg −1 ) = 44.0 ΔTA + 12.0 ΔDIC +59.1NO3− + 60.1 Si(OH)4
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A.5 Spatial variations in seawater composition
When the oceanographic data needed to evaluate Eqn. (A.4.10) for SAdens − SR is not
available, the look‐up table method of McDougall et al. (2010a) is recommended to
evaluate δ SA ≡ δ SRdens ≡ SAdens − SR . The following paragraphs describe how this method
was developed.
In a series of papers Millero et al. (1976a, 1978, 2000, 2008b) and McDougall et al.
(2010a) have reported on density measurements made in the laboratory on samples
collected from around the world’s oceans. Each sample has had its Practical Salinity
measured in the laboratory as well as its density (measured with a vibrating tube
densimeter at 25 °C and atmospheric pressure). The Practical Salinity yields a Reference
Salinity 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 data on δ SA , based on laboratory measurements of density, was
regressed against the silicate concentration of the seawater samples, McDougall et al.
(2010a) 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 in 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. (2010a) 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
4
Indian (A.5.4)
δ SA / (g kg −1
4
−1
−1
Atlantic (A.5.5)
These relationships between the Absolute Salinity Anomaly δ SA = SA − SR and silicate
concentration have been used by McDougall, Jackett and Millero (2010a) in a computer
algorithm that uses an existing global data base of silicate (Gouretski and Koltermann
(2004)) and provides an estimate of Absolute Salinity when given a seawater sample’s
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Practical Salinity as well as its spatial location in the world ocean. This computer
algorithm accounts for the latest understanding of Absolute Salinity in the Baltic Sea, but it
is silent on the influence of compositional variations in other marginal seas. The Absolute
Salinity Anomaly in the Baltic Sea has been quite variable over the past few decades of
observation (Feistel et al. (2010c)). The computer algorithm of McDougall et al. (2010a)
uses the relationship found by Feistel et al. (2010c) that applies in the years 2006‐2009,
namely
S A − S R = 0.087 g kg −1 × (1 − S R SSO ) ,
(A.5.6)
where SSO = 35.165 04 g kg–1 is the standard‐ocean Reference Salinity that corresponds to
the Practical Salinity of 35.
In order to gauge the importance of the spatial variation of seawater composition, the
northward gradient of density at constant pressure is shown in Fig. A.5.1 for the data in a
world ocean hydrographic atlas deeper than 1000m. The vertical axis in this figure is the
magnitude of the difference between the northward density gradient at constant pressure
when the TEOS‐10 algorithm for density is called with SA ≡ SAdens (as it should be)
compared with calling the same TEOS‐10 density algorithm with S R as the salinity
argument. Figure A.5.1 shows that the “thermal wind” is misestimated by more than 2%
for 58% of the data in the world ocean below a depth of 1000m if the effects of the variable
seawater composition are ignored.
Figure A.5.1. The northward density gradient at constant pressure (the horizontal axis)
for data in the global ocean atlas of Gouretski and Koltermann (2004) for
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 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 1000 m through the adoption of TEOS‐10 but ignoring the influence of
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compositional variations. By comparing Figs. A.5.1 and A.5.2 it is seen that the main
benefit that TEOS‐10 delivers to the evaluation of the “thermal wind” is through the
incorporation of spatial variations in seawater composition; the greater accuracy of TEOS‐
10 over EOS‐80 for Standard Seawater is only 18% as large as the improvement gained by
the incorporation of compositional variations into TEOS‐10 (i. e. the rms value of the
vertical axis in Fig. A.5.2 is 18% of that of the vertical axis of Fig. A.5.1). If the Atlantic
were excluded from this comparison, the relative importance of compositional variations
would be even larger.
Figure A.5.2. The northward density gradient at constant pressure (the horizontal axis)
for data in the global ocean atlas of Gouretski and Koltermann (2004) for
p > 1000 dbar . The vertical axis is the magnitude of the difference
between evaluating the density gradient using S R as the salinity
argument in the TEOS‐10 expression for density compared with using SP
in the EOS‐80 algorithm for density.
The thermodynamic description of seawater and of ice Ih as defined in IAPWS‐08 and
IAPWS‐06 has been adopted as the official description of seawater and of ice Ih by the
Intergovernmental Oceanographic Commission in June 2009. The adoption of TEOS‐10
has recognized that this technique of estimating Absolute Salinity from readily measured
quantities is perhaps the least mature aspect of the TEOS‐10 thermodynamic description of
seawater. The present computer software, in both FORTRAN and MATLAB, which evaluates
Absolute Salinity S A given the input variables Practical Salinity S P , longitude λ , latitude
φ and sea pressure p is available at www.TEOS‐10.org. It is expected, as new data
(particularly density data) become available, that the determination of Absolute Salinity
will improve over the coming decades, and the algorithm for evaluating Absolute Salinity
in terms of Practical Salinity, latitude, longitude and pressure, will be updated from time
to time, after relevant appropriately peer‐reviewed publications have appeared, and such
an updated algorithm will appear on the www.TEOS‐10.org web site. Users of this
software should state in their published work which version of the software was used to
calculate Absolute Salinity.
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A.6 Gibbs function of seawater
The Gibbs function of seawater g ( SA , t , p ) is defined as the sum of the Gibbs function for
pure water g W ( t , p ) and the saline part of the Gibbs function g S ( S A , t , p ) so that
g ( SA , t, p ) = g W ( t, p ) + g S ( SA , 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 need to for 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.
All of the thermodynamic properties of seawater that are described in this Manual are
available as both FORTRAN and MATLAB implementations. These implementations are
available for 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 had entered the determination of
IAPWS‐95 (Wagner and Pruß (2002), Feistel (2003, 2008)). For example, Millero and Li
(1994) concluded that the pure‐water part of the EOS‐80 sound‐speed formula of Chen and
Millero (1977) was responsible for a deviation of 0.5 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
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new Gibbs function in which we use IAPWS‐95 for the pure‐water part as well as sound
speeds from Del Grosso (1974), is perfectly consistent with Chen and Millero’s (1976)
densities and Bradshaw and Schleicher’s (1970) thermal expansion data at high pressures.
The accuracy of high‐pressure seawater densities has increased with the use of IAPWS‐95,
directly as the pure‐water part, and indirectly by correcting earlier seawater
measurements, making them ʺnewʺ seawater data. In this manner the known sound‐
speed inconsistency of EOS‐80 has been resolved in a natural manner.
A.7 The fundamental thermodynamic relation
The fundamental thermodynamic relation for a system composed of a solvent (water) and
a solute (sea salt) relates the total differentials of thermodynamic quantities for the case
where the transitions between equilibrium states are reversible. This restriction is satisfied
for infinitesimally small changes of an infinitesimally small seawater parcel. The
fundamental thermodynamic relation is
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 ) 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 ) 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
dt
dt
ρ dt
(A.7.2)
Note that the constancy of entropy does not imply the absence of irreversible processes
because, for example, there can be irreversible changes of both salinity and enthalpy at
constant pressure in just the right ratio so as to have equal effects in Eqns. (A.7.1) or (A.7.2)
so that the change of entropy in these equations is zero.
A.8 The “conservative” and “isobaric conservative” properties
A thermodynamic variable C is said to be “conservative” if its evolution equation (that is,
its prognostic equation) has the form
dC
= − ∇ ⋅ FC .
(A.8.1)
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)
∂ρ ∂t x , y , z + ∇ ⋅ ( ρ u ) = 0 .
(A.8.2)
( 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
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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 particulate matter, then the right‐hand side of the continuity equation
(A.8.2) should be ρ S SA , the non‐conservative source of mass due to biogeochemical
processes. It can be shown that the influence of this source term ρ S SA 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 S R , 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 the
remineralization process. Because the dominant variations of the composition of seawater
are due to species which do not have a strong signature in conductivity, in some situations
it may be sufficiently accurate to take Reference Salinity 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 and A.5 and Pawlowicz et
al. (2010)).
Summarizing this discussion thus far, the quantities that can be considered
conservative in the ocean are (in descending order of accuracy) (i) mass, (ii) total energy
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 “isobaric conservative”. In addition, the Bernoulli function B and specific enthalpy h
are also “isobaric conservative” (see Eqn. (B.17) and the discussion thereafter).
Some variables that are not “isobaric conservative” include potential temperature 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.
(
)
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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 ignores the dissipation of mechanical 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 assuming 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 SA , the northward density gradient at fixed pressure (i. e. the thermal
wind) would be misestimated by more than 2% for more than 58% of the data below a
pressure of 1000 dbar in the world ocean. It is clearly desirable to not have this type of
systematic error in the dynamical equations of the ocean component of coupled climate
models. Appendix A.20 discusses practical ways of including the effects of the non‐
conservative remineralization source term in ocean models. The recommended option is
that ocean models carry Preformed Salinity S* as the model’s conservative salinity model
variable, and that they also carry an evolution equation for an Absolute Salinity Anomaly
as described in section A.20.1 and Eqns. (A.20.3) – (A.20.5).
The errors incurred in ocean models by treating potential temperature θ as being
conservative have not yet been thoroughly investigated, but McDougall (2003) and
Tailleux (2010) have made a start on this topic. McDougall (2003) found that typical errors
in 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 almost two
orders of magnitude.
If the ocean were in thermodynamic equilibrium, its temperature would be the same
everywhere as would the chemical potentials of water and of each dissolved species, while
the entropy and the concentrations of each species would be functions of pressure.
Turbulent mixing acts in the complementary direction, tending to make salinity and
entropy constant but in the process causing gradients in temperature and the chemical
potentials as functions of pressure. That is, turbulent mixing acts to maintain a non‐
equilibrium state. This difference between the roles of molecular versus turbulent mixing
results from the symmetry breaking role of the gravity field; for example, in a laboratory
without gravity, turbulent and molecular mixing would have indistinguishable effects.
Note that the molecular flux of salt 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
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divergence of various fluxes of the quantity and to non‐conservative “source” terms. For
example, it is usual to refer to the “conservation equation” for entropy or for “potential
temperature”. Since these variables are not conservative variables it seems unnatural to
refer to their evolution equations as “conservation equations”. Hence here we will use the
term “conservation equation” only for a variable that is (for all practical purposes)
conserved. For other variables we will refer to their “evolution equation” or their
“prognostic equation” or their “local balance equation”.
A.9 The “potential” property
Any thermodynamic property of seawater that remains constant when a parcel of
seawater is moved from one pressure to another adiabatically, without exchange of mass
and without interior conversion between its turbulent kinetic and internal energies, is said
to possess the “potential” property, or in other words, to be a “potential” variable. Prime
examples of “potential” variables are entropy 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 have the “potential” property. Thermodynamic properties that posses the “potential”
attribute include potential temperature θ , potential enthalpy h 0 , 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 mechanical 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 mechanical energy is a small term whose influence is routinely
neglected in the First Law of Thermodynamics in oceanography, it seems advisable to
modify the meaning of the word “adiabatic” in oceanography so that our use of the word
more accurately reflects the properties we normally associate with an adiabatic process.
Accordingly we propose that the word “adiabatic” in oceanography be taken to describe a
process occurring without exchange of heat and also without the internal dissipation of
mechanical energy. With this definition of “adiabatic”, a process that is both isohaline and
adiabatic does imply that the entropy 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
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“adiabatically” and without exchange of mass, is said to possess the “potential” property,
or in other words, to be a “potential” variable.
In appendix A.8 above we concluded that only mass, and the three variables E , S*
and Θ (approximately) are “conservative” (and hence also “isobaric conservative”). Since
E 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 are either potential temperature or entropy. Similarly, Preformed Salinity S* is in
principle 100% conservative, but our ability to evaluate S* from hydrographic
observations is limited (for example, by the approximate relations (A.4.1) or (A.4.9)).
Table A.9.1 The “potential”, “conservative”, “isobaric conservative” and
the functional nature of various oceanographic variables
Variable
“potential”?
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
1
1
x
x
3
x
4
4
x
x
x
x
x
x
2
3
x
x
x
x
x
x
x
x
x
function of ( SA , t , p ) ?
4
x
x
x
x
x
x
x
5
x
The remineralization of organic matter changes S R less than it changes S A .
Taking ε and the effects of remineralization to be negligible.
3 Taking 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 c
0 ( p, ρ ) has been decided upon.
1
2
In Table A.9.1 various oceanographic variables are categorized according to whether they
posses the “potential” property, whether they are “conservative” variables, whether they
are “isobaric conservative” variables, and whether they are functions of only ( SA , t , p ) .
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Note that Θ is the only variable that achieve four “ticks” in this table, while Preformed
Salinity S* has ticks in the first three columns, but not in the last column since it is a
function not only of ( SA , t, p ) but also of the composition of seawater. Hence Θ is the
most “ideal” thermodynamic variable. If it were not for the non‐conservation of Absolute
Salinity, it too would be an “ideal” thermodynamic variable, but in this sense, Preformed
Salinity is superior to Absolute Salinity. Conservative Temperature Θ and Preformed
Salinity S* are the only two variables in this table to be both “potential” and
“conservative”. The last four rows of Table A.9.1 are for potential density, 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 ( S A ,η ,0 ) dη + hSA ( SA ,η ,0 ) dSA = (T0 + θ ) dη + μ ( SA ,η ,0 ) dSA ,
(A.10.1)
which shows that specific entropy η is simply a function of Absolute Salinity SA 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
c 0p dΘ =
(T0 + θ ) dη
+ μ ( S A , 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 ( S A , 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
and
∂h ∂P S
A ,T
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(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)
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85
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 + hSA dSA = (T0 + t ) dη + μ dSA while ∂h ∂P
= v,
(A.11.4)
SA ,η
so that
∂h ∂η
SA , p
= (T0 + t )
∂h ∂SA
and
η, p
= μ.
(A.11.5)
Now taking specific enthalpy to be a function of potential temperature (rather than of
temperature t ), that is, taking h = h ( SA ,θ , p ) , the fundamental thermodynamic relation
(A.7.1) becomes
hθ dθ + hSA dSA = (T0 + t ) dη + μ dSA
∂h ∂P
while
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
hθ
= ηθ S hη
= ηθ S (T0 + t ) ,
(A.11.7)
SA , p
A
SA , p
A
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 ) = ηθ
SA
(T0 + θ ) .
(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 )
(T0 + t ) .
(T0 + θ )
(A.11.9)
To evaluate
the hSA partial derivative, we first write specific enthalpy in the functional
form h = h ( S A ,η ( SA , θ ) , p ) and then differentiate it, finding
hSA
= hSA
+ hη
η SA .
(A.11.10)
θ, p
η, p
θ
SA , p
The partial derivative of specific entropy η = − gT (Eqn. (2.10.1)) with respect to Absolute
Salinity, ηSA = − g SAT , is also equal to − μT since chemical potential is defined by Eqn.
(2.9.6) as μ = g SA . Since the partial derivative of entropy with respect to SA in (A.11.10) is
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 hSA namely
hSA
θ, p
= μ ( SA , t, p ) − (T0 + t ) μT ( SA ,θ ,0 ) .
(A.11.11)
Notice that this expression contains some things that are evaluated at the general pressure
p and one evaluated at the reference pressure pr = 0.
Now considering specific enthalpy to be a function of Conservative Temperature
(rather than of temperature t ), that is, taking h = hˆ ( SA , Θ, p ) , the fundamental
thermodynamic relation (A.7.1) becomes
hˆΘ dΘ + hˆSA dS A = (T0 + t ) dη + μ dSA
∂hˆ ∂P
while
SA , Θ
= v.
(A.11.12)
The partial derivative ĥΘ follows directly from this equation as
hˆΘ
SA , p
= (T0 + t )ηΘ S
A,p
= (T0 + t )ηΘ S .
(A.11.13)
A
At p = 0 this equation reduces to
hˆΘ
SA , p =0
= c 0p = (T0 + θ )ηΘ S ,
(A.11.14)
A
and combining these two equations gives the desired expression for ĥΘ namely
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hˆΘ
SA , p
(T0 + t ) c0 .
(T0 + θ ) p
=
(A.11.15)
To evaluate the hˆSA partial derivative we first write h in the functional form
h = h ( SA ,η ( SA , Θ ) , p ) and then differentiate it, finding (using both parts of Eqn. (A.11.5))
hˆSA
Θ, p
= μ ( SA , t , p ) + (T0 + t )ηSA
Θ
.
(A.11.16)
The differential expression Eqn. (A.11.12) can be evaluated at p = 0 where the left‐hand
side is simply c 0p dΘ so that from Eqn. (A.11.12) we find that
η SA
Θ
= −
μ ( SA ,θ ,0)
,
(T0 + θ )
so that the desired expression for hˆSA is
hˆSA
Θ, p
= μ ( SA , t , p ) −
(T0 + t ) μ S ,θ ,0 .
(
)
(T0 + θ ) A
(A.11.17)
(A.11.18)
The above boxed expressions for four different isobaric derivatives of specific enthalpy are
important as they are integral to forming the First Law of Thermodynamics in terms of
potential temperature and in terms of Conservative Temperature.
A.12 Differential relationships between η , θ , Θ and S A
Evaluating the fundamental thermodynamic relation in the forms (A.11.6) and (A.11.12)
and using the four boxed equations in appendix A.11, we find the relations
(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 )
(T0 + θ ) p
⎣⎢
⎦⎥
(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 = c 0p dΘ − μ ( 0 ) dSA .
(A.12.2)
From this follows all the following partial derivatives between η , θ , Θ and S A ,
Θθ
Θη
θη
SA
SA
SA
= c p ( SA ,θ ,0) c 0p ,
ΘSA = ⎡⎣ μ ( SA ,θ ,0 ) − (T0 + θ ) μT ( SA ,θ ,0) ⎤⎦ c 0p ,
θ
(A.12.3)
= (T0 + θ ) c 0p ,
ΘSA = μ ( SA , θ ,0 ) c 0p ,
(A.12.4)
= (T0 + θ ) c p ( SA , θ ,0 ) ,
θ SA = (T0 + θ ) μT ( SA ,θ ,0 ) c p ( S A ,θ ,0 ) ,
(A.12.5)
θ Θ S = c 0p c p ( SA ,θ ,0 ) , θ SA
A
ηθ
SA
= c p ( SA ,θ ,0) (T0 + θ ) ,
ηΘ S = c0p (T0 + θ ) ,
A
η
η
Θ
= − ⎡⎣ μ ( SA ,θ ,0 ) − (T0 + θ ) μT ( SA ,θ ,0 ) ⎤⎦ c p ( SA ,θ ,0 ) , (A.12.6)
ηSA = − μT ( SA ,θ ,0 ) ,
θ
η SA
Θ
= − μ ( SA ,θ ,0 ) (T0 + θ ) .
(A.12.7)
(A.12.8)
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 ( S A , Θ ) , namely θˆΘ , θˆSA , θˆΘΘ , θˆSA Θ and
θˆSA SA can also be derived using Eqn. (P.13), obtaining
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TEOS-10 Manual: Calculation and Use of the Thermodynamic Properties of Seawater
Θ
S
1
Θ
θ SA
Θ
θˆΘ = , θˆSA = − A , θˆΘΘ = − θθ 3 , θˆSAΘ = −
Θθ
Θθ
( Θ )
( Θ
θ
θ
)
2
+
Θ
Θ
SA θθ
( )
Θ
θ
3
,
87
(A.12.9a,b,c,d)
2
⎞ Θ
⎛Θ
Θ
Θ
SA SA
SA Θθ SA
SA
θθ
+
−
(A.12.10)
2
⎜
⎟
⎟ Θ
,
⎜
Θ
Θ
Θ
Θ
θ
θ
θ
θ
θ
⎝
⎠
, Θ
, Θ
, Θ
in terms of the partial derivatives Θ
θ
θθ
SA
θ SA and Θ SA SA which can be obtained
( S ,θ ) from the TEOS‐10 Gibbs function.
by differentiating the polynomial Θ
A
and
θˆSA SA = −
A.13 The First Law of Thermodynamics
The law of the conservation of energy for thermodynamic equilibrium states was
discovered in the 19th century by Gibbs (1873) and other early pioneers. It was formulated
as a balance between internal energy, heat and work, similar to the fundamental equation
(A.7.1), and referred to as the First Law of Thermodynamics (Thomson (1851), Clausius
(1876), Alberty (2001)). Under the weaker condition of a local thermodynamic equilibrium
(Glansdorff and Prigogine (1971)), the original thermodynamic concepts can be suitably
generalized to describe irreversible processes of fluid dynamics which are subject to
molecular fluxes and macroscopic motion (Landau and Lifshitz (1959), De Groot and
Mazur (1984)).
In some circles the “First Law of Thermodynamics” is used to describe the evolution
equation for total energy, being the sum of internal energy, potential energy and kinetic
energy. Here we follow the more common practice of regarding the First Law of
Thermodynamics as the difference between the conservation equation of total energy and
the evolution equation for kinetic energy plus potential energy, leaving what might
loosely be termed the evolution equation of “heat”, Eqn. (A.13.1) (Landau and Lifshitz
(1959), McDougall (2003), Griffies (2004)).
The First Law of Thermodynamics can therefore be written as (see Eqn. (B.19) and the
other Eqns. (A.13.3), (A.13.4) and (A.13.5) of this appendix; all of these equations are
equally valid incarnations of the First Law of Thermodynamics)
⎛ dh 1 dP ⎞
SA
R
Q
−
(A.13.1)
⎟ = − ∇ ⋅ F − ∇ ⋅ F + ρε + hSA ρ S ,
ρ dt ⎠
⎝ dt
where F R 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 mechanical energy per
unit mass, transformed into internal energy and hSA ρ S SA is the rate of increase of
enthalpy due to the interior source term of Absolute Salinity caused by remineralization.
The derivation of (A.13.1) is summarized in appendix B below, where we also discuss the
related evolution equations for total energy and for the Bernoulli function.
Following Fofonoff (1962) we note that an important consequence of (A.13.1) is that
when two finite sized parcels of seawater are mixed at constant pressure and under ideal
conditions, the total amount of enthalpy is conserved. To see this one combines (A.13.1)
with the continuity equation ∂ρ ∂t + ∇ ⋅ ( ρ u ) = 0 to find the following divergence form of
the First Law of Thermodynamics,
dP
∂ ( ρ h ) ∂t + ∇ ⋅ ( ρ uh ) −
= − ∇ ⋅ F R − ∇ ⋅ FQ + ρε + hSA ρ S SA .
(A.13.2)
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
ρ⎜
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
pressure), mixes and contracts. The dissipation of mechanical energy by viscous friction
and the source term due to the production of Absolute Salinity are also commonly ignored
during such mixing processes but in fact these terms do cause a small increase in the
enthalpy of the mixture with respect to that of the two original parcels. Apart from these
non‐conservative source terms, under these assumptions Eqn. (A.13.2) reduces to the
statement that the volume integrated amount of 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 (A.13.1) can be written (using Eqn. (A.7.2)) as an
evolution equation for entropy as follows
dη
dS ⎞
⎛
(A.13.3)
+ μ A ⎟ = − ∇ ⋅ F R − ∇ ⋅ FQ + ρε + hSA ρ S SA .
dt
dt ⎠
⎝
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 substituting Eqns. (A.11.9) and
(A.11.11) into Eqn. (A.13.1) as (from Bacon and Fofonoff (1996) and McDougall (2003))
ρ ⎜ ( T0 + t )
⎛ (T0 + t )
dθ
dS
c p
+ ⎡ μ p − T + t ) μT ( pr ) ⎤⎦ A
⎜ (T + θ ) p ( r ) d t ⎣ ( ) ( 0
dt
⎝ 0
ρ⎜
⎞
⎟⎟ =
⎠
− ∇ ⋅ F R − ∇ ⋅ FQ + ρε + hSA ρ S
(A.13.4)
SA
,
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)
⎛ (T0 + t ) 0 d Θ ⎡
(T + t ) μ 0 ⎤ d SA
cp
+ ⎢μ ( p) − 0
( )⎥
⎜ (T0 + θ )
t
T
+
d
θ
(
)
⎢
0
⎣
⎦⎥ d t
⎝
ρ⎜
⎞
⎟ =
⎟
⎠
(A.13.5)
− ∇ ⋅ F − ∇ ⋅ F + ρε + hSA ρ S
R
c 0p
Q
SA
,
where
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 S A − Θ diagram where variables are
contoured which graphically illustrate the non‐conservation of these variables. In other
words, appendices A.16, A.17 and A.18 quantify the non‐ideal nature of the left‐hand sides
of Eqns. (A.13.3) ‐ (A.13.5). That is, these appendices quantify the deviations of the left‐
hand sides of these equations from being proportional to 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
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89
“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 the isobaric
specific heat c p ( SA , 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 and is also very close to unity in the deep ocean.
Figure A.13.1. Contours (in °C ) of the difference θ − Θ 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
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
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.
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 c 0p Θ. In this way c 0p 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 c 0pθ 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 conservative form (A.21.15) implies that the turbulent diffusive flux of heat should
be directed down the mean gradient of Conservative Temperature rather than down the
mean gradient of potential temperature. In this appendix we quantify the difference
between these mean temperature gradients.
Consider first the respective temperature gradients along the neutral tangent plane.
From Eqn. (3.11.2) we find that
(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 β Θ ∇n Θ we find
IOC Manuals and Guides No. 56
(A.14.3)
TEOS-10 Manual: Calculation and Use of the Thermodynamic Properties of Seawater
)
(
∇nθ = θˆΘ + ⎡⎣α Θ β Θ ⎤⎦ θˆSA ∇n Θ .
91
(A.14.4)
Also, in section 3.13 we found that Tbθ ∇nθ = TbΘ∇n Θ, so that we can write the equivalent
expressions
∇nθ
∇n Θ
(α
(α θ
Θ
=
)
θ
β )
βΘ
TbΘ
= θˆΘ + ⎡⎣α Θ β Θ ⎤⎦ θˆSA ,
Tbθ
=
(A.14.5)
)
(
and it can be shown that α Θ α θ = θˆΘ and β θ β Θ = 1 + ⎡⎣α Θ β Θ ⎤⎦ θˆSA θˆΘ , that is,
β θ = β Θ + α Θ θˆSA θˆΘ . 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. This 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_CT25.
Similarly to Eqn. (A.14.3), the vertical gradients are related by
θ z = θˆΘ Θ z + θˆSA SA z ,
(A.14.6)
and using the definition, Eqn. (3.15.1), of the stability ratio we find that
θz
Θz
= θˆΘ + Rρ−1 ⎡⎣α Θ β Θ ⎤⎦ θˆSA .
(A.14.7)
For values of the stability ratio Rρ close to unity, the ratio θ z Θ z is close to the values of
∇nθ ∇n Θ shown in Figure A.14.1. For other values of Rρ , Eqn. (A.14.7) can be
calculated and plotted.
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 blue 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 potential temperature as when done in terms of Conservative
Temperature. The same is not true of the dianeutral velocity caused by cabbeling. The ratio
of the cabbeling dianeutral velocity calculated using potential temperature to that using
Conservative Temperature is given by Cbθ ∇nθ ⋅ ∇nθ
CbΘ∇n Θ ⋅ ∇n Θ (see section 3.9) which
can be expressed as
(
)(
)
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Cbθ ∇nθ
2
CbΘ ∇n Θ
2
(
(
Θ
Θ
Cθ α β
= bΘ
Cb α θ 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.
2
) (C
Θ
b
∇n Θ
2
)
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 S , p . This
A
is 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
ηT ( SA , t, p )
g ( S , t, p )
.
= TT A
ηT ( SA ,θ , pr )
gTT ( 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 ( S A , t , p )
.
(A.15.2)
In order to derive Eqn. (2.18.3) for α Θ we first need an expression for ∂Θ ∂t S , p . This
A
is found by differentiating with respect to in situ temperature the entropy equality
η ( SA , t , p ) = ηˆ ( SA , Θ [ SA , t , p ]) obtaining
∂Θ
∂Θ
= ηT ( SA , t , p )
= − (T0 + θ ) gTT ( SA , t , p ) c 0p ,
∂T SA , p
∂η S
(A.15.3)
A
where the second part of this equation has used Eqn. (A.12.4) for Θη
to obtain the desired expression Eqn. (2.18.3) for α Θ as follows
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SA
. This is then used
TEOS-10 Manual: Calculation and Use of the Thermodynamic Properties of Seawater
1 ∂v
1 ∂v
α =
=
v ∂Θ SA , p
v ∂T
Θ
93
−1
⎛ ∂Θ
⎞
c 0p
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 T , p . This
is found by differentiating with respect to Absolute Salinity 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
T ( A
r)
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 [ S A , 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 ( S A , t , p ) ⎡⎣ g SAT ( S A , t , p ) − g SAT ( SA ,θ , pr ) ⎤⎦
g P ( SA , t , p ) gTT ( SA , t , p )
.
(A.15.7)
Note that the terms in the natural logarithm of the square root of Absolute Salinity cancel
from the two parts of the square brackets in Eqns. (A.15.5) and (A.15.7).
In order to derive Eqn. (2.19.3) for β Θ we first need an expression for ∂Θ ∂SA T , p .
This is found by differentiating with respect to Absolute Salinity 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 ( S A ,θ ,0 ) − (T0 + θ ) μT ( SA , t , p ) ⎤⎦ c 0p
(A.15.8)
= ⎡⎣ g SA ( S A , θ ,0 ) − (T0 + θ ) g SAT ( SA , t , p ) ⎤⎦ c 0p .
Differentiating ρ = ρˆ ( SA , Θ [ S A , 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 natural logarithm of the square root of Absolute Salinity cancel
from the two parts of the square brackets in Eqns. (A.15.8) and (A.15.10).
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A.16 Non-conservative production of entropy
In this and the following three appendices (A.16 – A.19) the non‐conservative nature of
several thermodynamic variables (entropy, potential temperature, Conservative
Temperature and potential density) will be quantified by considering the mixing of pairs
of seawater parcels at fixed pressure. The mixing is taken to be complete so that the end
state is a seawater parcel that is homogeneous in Absolute Salinity and entropy. That is,
we will be considering mixing to completion by a turbulent mixing process. In appendix
A.20 the non‐conservative production of Absolute Salinity by the remineralization of
particulate organic matter is considered. This process is not being considered in
appendices A.16 – A.19. The non‐conservative production which is quantified in
appendices A.16 – A.19 occurs in the absence of any variation in seawater composition.
Following Fofonoff (1962), consider mixing two fluid parcels (parcels 1 and 2) that
have initially different temperatures and salinities. The mixing process occurs at pressure
p. The mixing is assumed to happen to completion so that in the final state Absolute
Salinity, entropy and all the other properties are uniform. Assuming that the mixing
happens with a vanishingly small amount of dissipation of mechanical energy, the ε term
can be dropped from the First Law of Thermodynamics, (A.13.1), this equation becoming
( ρ h )t
+ ∇ ⋅ ( ρ u h ) = − ∇ ⋅ F R − ∇ ⋅ 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
m1 + m2 = m ,
(A.16.2)
m1 SA1 + m2 SA2 = m SA ,
(A.16.3)
m1 h1 + m2 h2 = m h ,
(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 S A , 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 S A and h about the values of
S A and h of the mixed fluid, retaining terms to second order in [ S A2 − SA1 ] = ΔSA and in
[h2 − h1 ] = Δh . Then η1 and η2 are evaluated and (A.16.4) and (A.16.5) used to find
{
}
m1 m2
2
2
ηhh ( Δh ) + 2ηhSA Δh ΔSA + ηSA SA ( ΔSA ) .
(A.16.6)
2
m
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,
ηhh < 0 ,
(A.16.7)
δη = −
1
2
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η SA SA < 0 ,
(η )
2
hSA
95
(A.16.8)
< ηhh ηSA 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,
the constraints would seem to be three different constraints on various third derivative of
the Gibbs function. In fact, we will see that they amount to only two rather well‐known
constraints on second order derivatives of the Gibbs function.
From the fundamental thermodynamic relation (A.7.1) we find that (where T is the
absolute temperature, T = T0 + t )
∂η
ηh =
= T −1
(A.16.10)
∂h SA , p
η SA =
∂η
∂SA
= −
h, p
μ
T
,
(A.16.11)
and from these relations the following expressions for the second order derivatives of η
can be found,
ηhh =
η SA h =
∂ 2η
∂h 2
=
SA , p
∂ 2η
∂h ∂SA
=
∂ 2η
∂SA2
=
=
SA , p
∂h
∂SA
= −
SA , p
∂ (− μ T )
h, p
−T −2
,
cp
∂ ( −μ T )
p
and
η SA SA =
∂ T −1
∂h
−
T, p
(A.16.12)
1 ⎛μ⎞
,
c p ⎜⎝ T ⎟⎠T
∂ (− μ T )
∂h
SA , p
∂h
∂SA
(A.16.13)
T, p
(A.16.14)
T 2 ⎡⎛ μ ⎞ ⎤
= −
−
⎢
⎥ .
T
c p ⎣⎜⎝ T ⎟⎠T ⎦
The last equation comes from regarding ηSA as ηSA = ηSA ( SA , h [ SA , t , p ] , p ) .
The constraint (A.16.7) 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 ηSA SA < 0 ,
requires (from (A.16.14)) that
2
μ SA
g SA SA
> −
T3
cp
2
⎡⎛ μ ⎞ ⎤
⎢⎜ ⎟ ⎥ ,
⎣⎝ T ⎠T ⎦
(A.16.15)
that is, the second derivative of the Gibbs function with respect to Absolute Salinity g SA SA
2
must exceed some negative number. The constraint (A.16.9) that ηhSA < ηhh ηSA SA
requires that (substituting from (A.16.12), (A.16.13) and (A.16.14))
g SA SA
(A.16.16)
> 0 ,
T 3c p
(
)
and since the isobaric heat capacity must be positive, this requirement is that g SA SA > 0 ,
and so is more demanding than (A.16.15).
We conclude that while there are the three requirements (A.16.7) to (A.16.9) on the
functional form of entropy 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 ) ,
gTT < 0
(A.16.17)
and
g SA SA > 0.
(A.16.18)
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The Second Law of Thermodynamics does not impose any additional requirement on the
cross derivatives g SAT nor on any third order derivatives of the Gibbs function.
The constraint (A.16.18) can be understood by considering the molecular diffusion of
salt which is known to be directed down the gradient of chemical potential μ ( SA , t , p )
(Landau and Lifshitz (1959)). 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 ∇S A and in order to avoid an unstable
explosion of salt one must have μSA = g SA SA > 0. So the constraint (A.16.18) amounts to
the requirement that the molecular diffusivity of salt is positive.
The two constraints (A.16.17) and (A.16.18) on the Gibbs function are well known in
the thermodynamics literature. Landau and Lifshitz (1959) derive them on the basis of the
contribution of molecular fluxes of heat and salt to the production of entropy (their
equations 58.9 and 58.13). Alternatively, Planck (1935) as well as Landau and Lifshitz
(1980) in their §96 (this is §98 in editions before the 1976 extension made by Lifshitz and
Pitayevski) inferred such inequalities from thermodynamic stability considerations. It is
pleasing to obtain the same constraints on the seawater Gibbs function from the above
Non‐Equilibrium Thermodynamics approach of mixing fluid parcels since this approach
involves turbulent mixing which is the type of mixing that dominates in the ocean;
(molecular diffusion has the complementary role of dissipating tracer variance).
In addition to the Second Law requirements (A.16.17) and (A.16.18) there are other
constraints which the seawater Gibbs function must obey. One is that the adiabatic (and
isohaline) compressibility must be positive for otherwise the fluid would expand in
response to an increase in pressure which is an unstable situation. Taking g P > 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
< g PP gTT ,
(A.16.20)
recognizing that gTT is negative ( gTP may, and does, take either sign). Equation (A.16.20)
is more demanding of g PP than is (A.16.19), requiring g PP 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, g SA SA > 0 , g PP < 0 , gTT < 0 , and
( gTP )2
< g PP gTT .
(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 ) )
m m
2
2
δη = − 12 1 2 2 ηˆΘΘ ( ΔΘ) + 2ηˆΘSA ΔΘ ΔSA + ηˆSA SA ( ΔSA ) .
(A.16.22)
m
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
2 1 2
this non‐conservation of entropy we first scale entropy by a dimensional constant so that
the resulting variable (“entropic temperature”) has the value 25 °C
at
( SA , Θ) = ( SSO , 25 °C ) and then Θ is subtracted. The result is contoured in SA − Θ space
in Figure A.16.1.
{
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Figure A.16.1. Contours (in °C ) of a variable which illustrates the non‐conservative
production of entropy η in the ocean.
The fact that the variable in Figure A.16.1 is not zero over the whole S A − 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
non‐conservative 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 be considered conservative variables (see appendix A.18
below), it is clear that the non‐conservative production given by (A.16.22) and illustrated
in Figure A.16.1 is equivalent to the slightly more accurate expression (A.16.6) which
applies at any pressure. The only discrepancy between the production of entropy
calculated from (A.16.22) and that from (A.16.6) is due to the very small non‐conservative
production of Θ at pressures other than zero (as well as the fact that both expressions
contain only the second order terms in an infinite Taylor series).
(
)
A.17 Non-conservative production of potential temperature
When fluid parcels undergo irreversible and complete mixing at constant pressure, the
thermodynamic quantities that are conserved during the mixing process are mass,
Absolute Salinity and enthalpy. As in section A.16 we again consider two parcels being
mixed without external input of heat or mass and the three equations that represent the
conservation of these quantities are again Eqns. (A.16.2) – (A.16.4). Potential temperature
θ is not conserved during the mixing process and the production of potential
temperature is given by
m1 θ1 + m2 θ 2 + m δθ = m θ .
(A.17.1)
Enthalpy in the functional form h = h ( S ,θ , p ) is expanded in a Taylor series of S and
A
A
θ about the values SA and θ of the mixed fluid, retaining terms to second order in
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[ 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
δθ =
1
2
m1 m2
m2
⎧⎪
⎨
⎪⎩
hθ SA
hSA SA
hθθ
2
2⎫
⎪
θ
2
θ
Δ
+
Δ
Δ
+
Δ
S
S
(
)
(
)
⎬.
A
A
hθ
hθ
hθ
⎪⎭
(A.17.2)
The maximum production occurs when parcels of equal mass are mixed so that
1
m m m −2 = 18 . The “heat capacity” hθ is not a strong function of θ but is a much
2 1 2
stronger function of S A 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 SA is approximately –5.4 J kg −1 K −1 (g kg −1 ) −1 (see the dependence of
isobaric heat capacity 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
4 θ S A ΔS A
(
)
hθ ≈ − 3.4 x10−4 ΔSA / [g kg −1 ] .
(A.17.3)
Since potential temperature θ = θˆ ( S A , Θ ) can be expressed independently of pressure
as a function of only Absolute Salinity and Conservative Temperature, and since during
turbulent mixing both S A and Θ may be considered conservative variables (see section
A.18 below), it is clear that the non‐conservative production given by (A.17.2) can be
approximated by the corresponding production of potential temperature that would occur
if the mixing had occurred at p = 0 , namely
δθ =
1
2
Θ
Θ
⎫
m1 m2 ⎧ Θθθ
( Δθ )2 + 2 θ SA Δθ ΔSA + SA SA ( ΔSA )2 ⎬ ,
2 ⎨
Θθ
Θθ
m ⎩ Θθ
⎭
(A.17.4)
where the exact proportionality between potential enthalpy and Conservative
Temperature h 0 ≡ c 0p 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.17.4) may be used to evaluate the non‐conservative
production of potential temperature due to mixing a pair of fluid parcels across a front at
which there are known differences in salinity and temperature. The temperature
difference θ − Θ is contoured in Figure A.17.1 and can be used to illustrate Eqn. (A.17.4).
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
(
)
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99
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 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. From the curvature of the isolines on Figure A.17.1 it is clear that the non‐
conservative production of potential temperature takes both positive and negative signs.
Figure A.17.1. Contours (in °C ) of the difference between potential temperature
and Conservative Temperature θ − 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 during the mixing process are mass,
Absolute Salinity and enthalpy. As in sections A.16 and A.17 we consider two parcels
being mixed without external input of heat or mass and the three equations that represent
the conservation of these quantities are again Eqns. (A.16.2) – (A.16.4). Potential enthalpy
h 0 and Conservative Temperature Θ are not exactly conserved during the mixing process
and the production of Θ is given by
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m1 Θ1 + m2 Θ2 + m δΘ = m Θ .
(A.18.1)
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
δΘ =
1
2
⎧⎪ hˆ
⎫
hˆΘSA
hˆSA SA
2
2⎪
ΘΘ
⎨ ˆ ( ΔΘ ) + 2 ˆ ΔΘΔSA + ˆ ( ΔSA ) ⎬ .
hΘ
hΘ
⎩⎪ hΘ
⎭⎪
m1 m2
m2
(A.18.2)
In order to evaluate these partial derivatives, we first write enthalpy in terms of potential
enthalpy (i. e. c 0p Θ ) using Eqn. (3.2.1), as
h = hˆ ( SA , Θ, p ) = c 0p Θ +
∫P vˆ ( SA , Θ, p′) dP′.
P
(A.18.3)
0
This is differentiated with respect to Θ giving
hΘ S
A, p
= hˆΘ = c 0p +
P
∫P α
Θ
ρ dP′ .
(A.18.4)
0
The right‐hand side of Eqn. (A.18.4) scales as c 0p + ρ −1 ( P − P0 ) α Θ , which is more than c 0p
by only about 0.0015 c 0p for ( P − P0 ) of 4 × 107 Pa (4,000 dbar). Hence, to a very good
approximation, ĥΘ in Eqn. (A.18.2) 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, h 0 , as given by Eqns. (3.2.1) and (A.18.3), 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.18.3) 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 Eqn. (A.18.2), and these can be
estimated in terms of the strength of cabbeling as follows. Equation (A.18.4) is
differentiated with respect to Conservative Temperature, giving
P
hˆΘΘ =
∫P vˆΘΘ
dP′ =
0
∫P (α
P
0
Θ
ρ
)
Θ
dP′ ,
(A.18.5)
so that we may write Eqn. (A.18.2) approximately as (assuming m1 = m2 )
δΘ ≈
( P − P0 )
8c 0p
{ vˆ
ΘΘ
( ΔΘ)2 + 2 vˆSAΘ ΔΘ ΔSA
+ vˆSA SA ( ΔSA )
2
}.
(A.18.6)
The dominant term in Eqn. (A.18.6) is usually the term in v̂ΘΘ which is approximately
− ρ −2 ρˆ ΘΘ , and from Eqn. (A.19.4) below we see that δΘ is approximately proportional to
the non‐conservative production of density at fixed pressure, often referred to loosely as
“cabbeling” (McDougall, 1987b), that is,
δΘ ≈
( P − P0 ) vˆ
8c 0p
ΘΘ
( ΔΘ)2
≈ −
( P − P0 )
8c 0p
( P − P0 ) δρ Θ .
ρˆ ΘΘ
2
ΔΘ ) ≈
2 (
ρ
ρ 2 c 0p
(A.18.7)
The production of Θ causes an increase in temperature and a consequent decrease in
density of − ρα ΘδΘ . The ratio of this change in density (using Eqn. (A.18.6)) to that
caused by cabbeling (from Eqn. (A.19.4)) is − ( P − P0 ) α Θ ρ c 0p 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, from Eqn. (A.18.6) we see
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.18.6) is usually about a factor of ten larger than
the other two terms (McDougall (1987b)), so the production of Conservative Temperature
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δΘ as a ratio of the contrast in Conservative Temperature ΔΘ = Θ2 − Θ1 may be
approximated as (since v̂ΘΘ ≈ − ρ −2 ρˆ ΘΘ ≈ ρ −1α ΘΘ )
δΘ
ΔΘ
≈
( P − P0 ) αΘΘ ΔΘ
8ρ c 0p
≈ 3.3 × 10−9 ( p dbar )( ΔΘ K ) .
(A.18.8)
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
non‐conservative production of Conservative Temperature, δΘ , increases linearly with
pressure (see Eqn. (A.18.6)) but at larger pressure the range of temperature and salinity in
the ocean decreases, and from the above equations it is clear that the magnitude of δΘ
decreases proportionally 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, and 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. McDougall (2003) concludes that the
maximum non‐conservation of Θ in the real ocean is a factor of approximately a hundred
less than the maximum non‐conservative production of potential temperature θ .
(
(
)
)
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.
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From the curvature of the isolines on Figure A.18.1 it is clear that the non‐conservative
production of Conservative Temperature at 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 , hˆSA SA > 0 and ( hˆΘSA ) 2 < hˆΘΘ hˆSA SA
everywhere, and it can be shown that these requirements are in fact met.
From Eqns. (A.18.8) and (A.17.3) we can write the ratio of the production of
Conservative Temperature to the production of potential temperature when two seawater
parcels mix as the approximate expression
δΘ
≈ − 10−5 ( p dbar )( ΔΘ K ) ΔSA / [g kg −1 ]
δθ
(
)
−1
.
(A.18.9)
Taking a typical ratio of temperature differences to salinity differences in the deep ocean
to be 10 K / [g kg −1 ] , Eqn. (A.18.9) becomes δΘ δθ ≈ − 10−4 ( p dbar ) . At a pressure of 4000
dbar this ratio is δΘ δθ ≈ − 0.4 implying that Conservative Temperature is almost as non‐
conservative as is potential temperature. While this is the case, the temperature and
salinity contrasts in the deep ocean are small, so the non‐conservation of both types of
temperature amount to very small temperature increments of both 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 the paper of McDougall
(2003).
A.19 Non-conservative production of density and of potential density
For the purpose of calculating the non‐conservative production of density we take both
Absolute Salinity S A and Conservative Temperature Θ to be 100% conservative (see
appendix A.18 above). Density is written in the functional form
ρ = ρˆ ( SA , Θ, p )
(A.19.1)
and the same mixing process between two fluid parcels is considered as in the previous
appendices. Mass and Absolute Salinity are conserved during the turbulent mixing
process (Eqns. (A.16.2) and (A.16.3)) as is Conservative Temperature, that is
m1 Θ1 + m2 Θ2 = m Θ ,
(A.19.2)
while the non‐conservative nature of density means that it obeys the equation,
m1 ρ1 + m2 ρ 2 + m δρ = m ρ .
(A.19.3)
Density is expanded in a Taylor series of S A and Θ about the values of S A and Θ of the
mixed fluid, retaining terms to second order in [ S A2 − SA1 ] = ΔSA and in [Θ2 − Θ1 ] = ΔΘ .
Then ρ1 and ρ 2 are evaluated and (A.19.3) is used to find
{
}
m1 m2
2
2
ρˆ ΘΘ ( ΔΘ) + 2 ρˆ SA Θ ΔΘ ΔSA + ρˆ SA SA ( ΔSA ) .
(A.19.4)
m2
The non‐conservative production of density δρ of Eqn. (A.19.4) is illustrated in Figure
A.19.1 for mixing at p = 0 dbar. That is, this figure shows the production δρ θ of
potential density ρ θ (note that the symbol ρ Θ could equally well be used for potential
density). The quantity contoured on this figure is formed as follows. First the linear
function of S A is found that is equal to ρ θ at ( 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 ρ θ 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 ρ θ
δρ = −
1
2
(
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)
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and S A , and Conservative Temperature. This figure allows the non‐conservative nature
of density to be understood in temperature units. The mixing of extreme fluid parcels on
Figure A.19.1 causes the same increase in density as a cooling of approximately 10 °C.
From Figure A.18.1 it is seen that the (tiny) non‐conservative nature of Θ is a factor of
approximately 4000 smaller than this.
Figure A.19.1. Contours (in °C ) of a variable that is used to illustrate the non‐
conservative production of potential density ρ θ . The three
points that are forced to be zero are shown with black dots.
A.20 The representation of salinity in numerical ocean models
Ocean models need to evaluate salinity at every time step as a necessary prelude to using
the equation of state to determine density and its derivatives for use in the hydrostatic
relationship and frequently in neutral mixing algorithms. The current practice in
numerical models is to treat salinity as a perfectly conserved quantity in the interior of the
ocean; salinity changes at the surface and at coastal boundaries due to evaporation,
precipitation, brine rejection, ice melt and river runoff and satisfies an advection‐diffusion
equation away from these boundaries.
The inclusion of composition anomalies
necessitates several changes to this approach. These changes can be divided into two
broad categories. First, in addition to fresh water inputs and brine rejection, all sources of
dissolved material entering through the surface and coastal boundaries of the model
should be considered as possible sources of composition anomalies. Second, within the
interior of the model, changes due to the growth, decay and remineralization of biological
material must be considered. Here, we focus on this second issue. While the ultimate
resolution of these issues will involve biogeochemical models, in this appendix we discuss
some practical ways forward based on the approximate relations (A.4.5) and (A.4.6)
between the salinity variables S R , S* and SA = SAdens that were discussed in section A.4. At
the time of writing, the suggested approaches here have not been tested, so it must be
acknowledged that the treatment of seawater composition anomalies in ocean models is
currently a work in progress.
We begin by restating Eqns. (A.4.5) and (A.4.6), namely
S R − S∗ ≈ r1 δ S Rdens ,
(A.20.1)
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S Adens − S∗ ≈ (1 + r1 ) δ S Rdens ,
(A.20.2)
where r1 will be taken to be the constant 0.35 based on the work of Pawlowicz et al. (2010),
and in this section these approximate relations will be taken to be exact. The Absolute
Salinity Anomaly δ S Rdens ≡ SAdens − S R is the salinity difference that can be directly measured
from seawater samples using a vibrating beam densimeter and knowledge of the sample’s
Practical Salinity, and a global look‐up table exists for this quantity (McDougall et al.
(2010a)). Because this particular salinity difference is based on direct measurements, it is
the natural measure of the anomalous composition of seawater to use in developing the
following options for numerical modeling of composition anomalies.
A.20.1 Using Preformed Salinity S* as the conservative salinity variable
Because Preformed Absolute Salinity S* (henceforth referred to by the shortened name,
Preformed Salinity) is designed to be a conservative salinity variable, blind to the effects of
biogeochemical processes, its evolution equation will be in the conservative form (A.8.1).
When this type of conservation equation is averaged in the appropriate manner (see
appendix A.21) the conservation equation for Preformed Salinity becomes (from Eqn.
(A.21.7)),
⎛ ∂Sˆ ⎞
dSˆ*
= h1 ∇n ⋅ hK ∇n Sˆ* + ⎜ D * ⎟ .
(A.20.3)
⎜ ∂z ⎟
dt
⎝
⎠z
(
)
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.20.3) is with respect to the sum of the Eulerian and quasi‐Stokes velocities of
height coordinates (equivalent to the description in appendix A.21 in terms of the
thickness‐weighted average horizontal velocity and the mean dianeutral velocity), while
the right‐hand side of this equation is the standard notation indicating that Ŝ* is being
diffused along neutral tangent planes with the diffusivity K and in the vertical direction
with the diapycnal diffusivity D (and h here is the average thickness between two closely
spaced neutral tangent planes).
In order to evaluate density during the running of an ocean model, Density Salinity
must be evaluated. This can be done from Eqn. (A.20.2) as the sum of the model’s salinity
variable Ŝ* and (1 + r1 ) δ S Rdens . This could be done by simply adding to the model’s
salinity variable (1 + r1 ) times the fixed spatial map of δ S Rdens ( obs ) as observed today (and
as is available from the computer algorithm of McDougall et al. (2010a)). However
experience has shown that even a smooth field of density errors can result in significant
anomalies in diagnostic model calculations, primarily due to the misalignment of the
density errors and the model bottom topography. Indeed, even if the correct mean
density could somehow be determined, approximations associated with the specification
of the model bottom topography would result in significant errors in bottom pressure
torques that can degrade the model solution. One way to minimize such errors is to allow
some dynamical adjustment of the specified density field so that, for example, density
contours tend to align with bottom depth contours where the flow is constrained to follow
bottom topography. This simple idea is the key to the success of the robust diagnostic
approach (Sarmiento and Bryan (1982)). To allow dynamical adjustment of the Absolute
Salinity Anomalies δ S Rdens ( x, y , p ) while not permitting them to develop large differences
from the observed values δ S Rdens (obs) , we recommend carrying an evolution equation for
δ SRdens so that it becomes an extra model variable which evolves according to
dδ S Rdens
=
dt
1
h
⎛ ∂δ S Rdens ⎞
dens
dens
−1
∇n ⋅ hK ∇nδ S Rdens + ⎜ D
⎟ + τ δ S R (obs) − δ S R .
z
∂
⎝
⎠z
(
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)
(
)
(A.20.4)
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105
Here the model variable δ S Rdens would be initialized based on observations, δ S Rdens (obs) ,
and advected and diffused like any other tracer, but in addition, there is a non‐
conservative source term τ −1 δ S Rdens (obs) − δ S Rdens which serves to restore the model
variable δ S Rdens 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 of Eqn.
(A.20.4) provides a way forward. An indication of how this approach might be improved
in the future can be gleaned from looking at Eqn. (A.4.14) for S Adens − S∗ (taken from
Pawlowicz et al. (2010)). If a biogeochemical model produced estimates of the quantities
on the right‐hand side of this equation, it could be immediately integrated into an ocean
model to diagnose the effects of the included biogeochemical processes on the modelʹs
density and its circulation.
In summary, the approach suggested here carries the evolution Eqns. (A.20.3) and
(A.20.4) for Ŝ* and δ S Rdens , while ŜAdens is calculated by the model at each time step
according to
SˆAdens = Sˆ∗ + (1 + r1 ) δ S Rdens ,
(A.20.5)
(
)
with our best present estimate of (1 + r1 ) being 1.35 . The model is initialized with values
of Preformed Salinity using Eqn. (A.20.1) (namely Sˆ∗ = SˆR − r1 δ SRdens ) based on
observations of Reference Salinity and on the global data base of δ S Rdens ( obs ) from
McDougall et al. (2010a).
A.20.2 Including a source term in the evolution equation for Absolute Salinity
An equivalent procedure is to carry the following evolution equation (A.20.6) for Absolute
Salinity, which more specifically, is called Density Salinity, S A ≡ SAdens . On inspection of
Eqn. (A.20.2), S Adens − S∗ ≈ (1 + r1 ) δ S Rdens , and recognizing that S∗ is a conservative variable,
it is clear that the non‐conservative production of SAdens must occur at the rate (1 + r1 )
times the rate at which the same non‐conservative processes affect the Absolute Salinity
Anomaly δ S Rdens . Since (from Eqn. (A.20.4)) the non‐conservative source term for δ S Rdens is
τ −1 δ SRdens (obs) − δ SRdens , we find that the evolution equation for Density Salinity to be
(
)
dSˆAdens
=
dt
⎛ ∂Sˆ dens ⎞
∇n ⋅ hK ∇n SˆAdens + ⎜ D A ⎟ + (1 + r1 )τ −1 δ S Rdens (obs) − δ S Rdens
⎜
∂z ⎟⎠ z
⎝
(A.20.6)
dens ⎞
ˆ
⎛
∂
S
= h1 ∇n ⋅ hK ∇n SˆAdens + ⎜ D A ⎟ + Sˆ SA
⎜
∂z ⎟⎠ z
⎝
Alternatively, this equation can be derived by summing Eqn. (A.20.3) plus (1 + r1 ) times
Eqn. (A.20.4). Here the non‐conservative source term in the evolution equation for
Density Salinity has been given the label Sˆ SA for later use.
In this approach the evolution equation (A.20.4) for δ S Rdens is also carried and the
model’s salinity variable, ŜAdens , is used directly as the argument of the equation of state
and other thermodynamic functions in the model. The model would be initialized with
values of Density Salinity using Eqn. (A.4.2) (namely SˆAdens = SˆR + δ SRdens ) based on
1
h
(
)
(
)
(
)
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
observations of Reference Salinity and on the global data base of δ S Rdens ( obs ) from
McDougall et al. (2010a).
This approach should give identical results to that described in section A.20.1 using
Preformed Salinity. One disadvantage of having Density Salinity as the model’s salinity
variable is that its evolution equation (A.20.6) is not in the conservative form so that, for
example, it is not possible to perform easy global budgets of salinity to test for the
numerical integrity of the model code. Another disadvantage is that the air‐sea flux of
carbon dioxide and other gases may need to be taken into account as the surface boundary
condition of Density Salinity. Such air‐sea fluxes do not affect Preformed Salinity.
A.20.3 Including a source term in the evolution equation for Reference Salinity
An equivalent procedure is to carry the following evolution equation (A.20.7) for
Reference Salinity. On inspection of Eqn. (A.20.1), S R − S∗ ≈ r1 δ S Rdens , and recognizing
that S∗ is a conservative variable, it is clear that the non‐conservative production of S R
must occur at the rate r1 times the rate at which the same non‐conservative processes
affect the Absolute Salinity Anomaly δ S Rdens . Since (from Eqn. (A.20.4)) the non‐
conservative source term for δ S Rdens is τ −1 δ S Rdens (obs) − δ S Rdens , the evolution equation for
Reference Salinity is
(
dSˆR
=
dt
)
ˆ ⎞
−1
dens
dens
R
⎟⎟ + r1 τ δ S R (obs) − δ S R
⎝
⎠z
,
(A.20.7)
⎛ ∂SˆR ⎞
r1
SA
1
ˆ
ˆ
= h ∇n ⋅ hK ∇n S R + ⎜ D
+
S
⎜ ∂z ⎟⎟
(1 + r1 )
⎝
⎠z
where the non‐conservative source term is r1 (1 + r1 ) times the corresponding source term
in the evolution equation for Density Salinity in Eqn. (A.20.6).
In this approach the evolution Eqns. (A.20.7) and (A.20.4) for Ŝ R and δ S Rdens are
carried by the ocean model, while ŜAdens is calculated by the model at each time step
according to Eqn. (A.4.2), namely
1
h
⎛
(
) + ⎜⎜ D ∂∂Sz
(
)
∇n ⋅ hK ∇n SˆR
SˆAdens = SˆR + δ S Rdens .
(
)
(A.20.8)
This approach, like that of section A.20.2 should give identical results to that described
in section A.20.1 using Preformed Salinity except for the more complicated air‐sea flux
boundary condition for Reference Salinity than for Preformed Salinity. It does seem that
the conservative nature of Eqn. (A.20.3) for Preformed Salinity is a significant advantage,
and so this approach is likely to be preferred by ocean modelers.
A.20.4 Discussion of the consequences if remineralization is ignored
If an ocean model does not carry the evolution equation for Absolute Salinity Anomaly
(Eqn. (A.20.4)) and the model’s salinity evolution equation does not contain the
appropriate non‐conservative source term, is there then any preference to initializing and
interpreting the model’s salinity variable as either Preformed Salinity, Absolute Salinity or
Reference Salinity? That is, the simplest method of dealing with these salinity issues is to
continue the general approach that has been taken for the past several decades of simply
taking one type of salinity in the model and that salinity is taken to be conservative.
Under this approximation the salinity that is used in the equation of state to calculate
density in the model is the same as the salinity that obeys a normal conservation equation
of the form Eqn. (A.20.3). In this approach there is still a choice of how to initialize the
salinity in a model, and here we discuss the relative virtues of these options.
If the model is initialized with a data set of estimated Preformed Salinity S* , then S*
should evolve correctly, since S* is a conservative variable and its evolution equation Eqn.
(A.20.3) contains no non‐conservative source terms. In this approach the equation of state
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107
will be called with Ŝ* rather than ŜAdens , and these salinities differ by approximately
(1 + r1 ) δ S Rdens . 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
ŜAdens 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
in 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 misestimated by ≈ 2.7% if Ŝ* is used in place of
ŜAdens as the salinity argument to the equation of state. Also, these percentage errors in
“thermal wind” are larger in the Pacific Ocean.
Another choice of the salinity data to initialize the model is ŜAdens . This choice has the
advantage that for an initial period of time after initialization the equation of state is called
with the correct salinity variable. However at later times, the neglect of the non‐
conservative source term in Eqn. (A.20.6) means that the model’s salinity variable will
depart from reality and errors will creep in due to the lack of these legitimate non‐
conservative source terms. How long might it be acceptable to integrate such a model
before the errors approached those described in the previous paragraph? One could
imagine that in the upper ocean the influence of these different salinity variables is
dwarfed by other physics such as air sea interaction and active gyral motions. If one
considered a depth of 1000m as being a depth where the influence of the different
salinities would be both apparent and would make a significant impact on the thermal
wind equation, then one might guess that it would take several decades for the neglect of
the non‐conservative source terms in the evolution equation for Density Salinity to begin
to be important. This is not to suggest that the relaxation time scale 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
non‐conservative source term Sˆ SA in Eqn. (A.20.6) 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
that represent 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 studies since we have
routinely initialized models with observations of Practical Salinity and have treated it as
though it were a conservative variable and have used it as the salinity argument for the
equation of state.
In principle, there are other combinations of options where the evolution equation for
Absolute Salinity Anomaly (A.20.4) is carried by an ocean model, but some option other
than those discussed in subsections A.20.1 – A.20.3 is pursued. We do not consider these
various options here, since if one goes to the trouble of carrying the evolution equation for
Absolute Salinity Anomaly, then one should be sufficiently careful to implement one of
the options discussed in subsections A.20.1 – A.20.3.
A.20.5 Discussion of the options for including remineralization
The approaches of subsections A.20.1 – A.20.3 of this appendix can each account for the
non‐conservative effects of remineralization if 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.20.3) with no source term on the right‐hand
side. Since S* is designed to be a conservative salinity variable, it would appear to be the
most appropriate salinity variable for having as an axis of the traditional “ S − θ diagram”,
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which would then become the S* − Θ diagram. Similarly, Ŝ* would also be the best choice
for the salinity variable in an inverse study.
If an ocean model were to be run without carrying the evolution equation for Absolute
Salinity Anomaly (A.20.4) and hence without the ability to incorporate the appropriate
non‐conservative source terms in either Eqns. (A.20.6) or (A.20.7), then the model must
resort to carrying only one salinity variable, and this salinity variable must be treated as a
conservative variable in the ocean model. In this circumstance, we advise that the ocean’s
salinity variable be interpreted as Density Salinity, and initialized as such. In this way, the
errors in the thermal wind equation will develop only slowly over a time scale of several
decades or more in the deep ocean.
It should be noted that each of the modelling approaches described in subsections
A.20.1 – A.20.3 are related to today’s estimate of the δ S Rdens ( obs ) field. This field will
change not only as the observational data base improves but also as the ocean composition
evolves with time.
A.21 The material derivatives of S* , SA , S R 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))
( ρ S* )t + ∇ ⋅ ( ρ uS* )
= ρ
dS*
= − ∇ ⋅ FS .
dt
(A.21.1)
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 t + ∇ ⋅ ( ρ u ) = 0.
(A.21.2)
Temporally averaging this equation in Cartesian coordinates (i. e. at fixed x, y , z ) gives
( )
ρ t + ∇ ⋅ 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
ρ 0 )t + ∇ ⋅ u = 0 where u ≡ ρ u ρ 0 .
(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
ρ0
S*
ρ
) + ∇ ⋅ (S ) =
ρ
t
*
u
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 processes, without invoking the Boussinesq
approximation, over first small‐scale mixing processes (several meters) followed by
averaging over the mesoscale (of order 100 km) has been performed by Greatbatch and
McDougall (2003), yielding the prognostic equation for Preformed Salinity
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(
ρ
ρ0
Sˆ*
)
t n
+ ∇n ⋅
(
ρ
ρ0
) (
Sˆ* vˆ +
ρ
ρ0
)
e Sˆ*
ρ
ρ0
=
z
=
1
h
∂Sˆ*
∂t
+
ρ
ρ0
ρ
ρ0
vˆ ⋅ ∇n Sˆ* +
e
n
∂Sˆ*
∂z
(A.21.6)
⎛ ∂Sˆ ⎞
∇n ⋅ hK ∇n Sˆ* + ⎜ D * ⎟ .
⎜ ∂z ⎟
⎝
⎠z
(
109
)
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 h 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. The density value ρ is the density whose average
height is the height at which the equation is evaluated. The issues of averaging involved
in Eqns. (A.21.5) and (A.21.6) are subtle, and are not central to our purpose in this
thermodynamic manual. Hence we proceed with the more standard Boussinesq
approach, but retain the over‐carets to remind the reader of the thickness‐weighted nature
of the variables.
Having derived this evolution equation (A.21.6) for Preformed Salinity without
invoking the Boussinesq approximation, we now follow common practice and invoke this
approximation, finding the simpler expression
∂Sˆ*
∂t
+ vˆ ⋅ ∇n Sˆ* + e
n
∂Sˆ*
=
∂z
1
h
⎛ ∂Sˆ ⎞
∇n ⋅ hK ∇n Sˆ* + ⎜ D * ⎟ .
⎜ ∂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 S A . Note that S A 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
S Aadd , 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 + ∇ ⋅ ( ρ uSA )
= ρ
dS A
= − ∇ ⋅ FS + ρ S
dt
SA
.
(A.21.8)
The source term S SA is described in appendix A.20 (see eqn. (A.20.6)). This non‐
conservative source term is due to biogeochemical processes, for example, the
remineralization of biological material; the turning of particulate matter into dissolved
seasalt. When this equation is density‐weighted averaged, we find
(
ρ
ρ0
SA
ρ
)
t
(
ρ
+ ∇ ⋅ SA u
)
=
ρ
ρ0
= −
∂ SA
∂t
1
ρ0
ρ
+ u ⋅ ∇ SA
S
∇⋅F −
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
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(
ρ
ρ0
SˆA
)
t n
+ ∇n ⋅
(
ρ
ρ0
) (
SˆA vˆ +
ρ
ρ0
e SˆA
)
=
z
=
ρ
ρ0
1
h
∂SˆA
∂t
+
ρ
ρ0
vˆ ⋅ ∇n SˆA +
ρ
ρ0
n
⎛ ∂Sˆ
∇n ⋅ hK ∇n SˆA + ⎜ D A
⎜ ∂z
⎝
(
)
e
∂SˆA
∂z
⎞
⎟⎟ +
⎠z
(A.21.10)
ρ
ρ0
Sˆ SA ,
and when the Boussinesq approximation is made we find the simpler expression
∂SˆA
∂t
+ vˆ ⋅ ∇n SˆA + e
n
∂SˆA
=
∂z
1
h
⎛ ∂Sˆ
∇n ⋅ hK ∇n SˆA + ⎜ D A
⎜ ∂z
⎝
(
)
⎞
S
⎟⎟ + Sˆ A .
⎠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
dianeutral velocity e of density coordinates. Apart from the non‐conservative source
term Sˆ SA , the right‐hand side is the divergence of the turbulent fluxes of Absolute
Salinity.
The corresponding turbulent evolution equation for Reference Salinity is
∂SˆR
∂t
+ vˆ ⋅ ∇n SˆR + e
n
∂SˆR
=
∂z
1
h
⎛ ∂Sˆ ⎞
r1
Sˆ SA .
∇n ⋅ hK ∇n SˆR + ⎜ D R ⎟ +
⎜ ∂z ⎟
+
r
1
(
)
1
⎝
⎠z
(
)
(A.21.12)
The non‐conservative source term here is justified in subsection A.20.3 of appendix A.20.
As discussed in appendices A.4 and A.20, given our rather elementary knowledge of the
way variations in seawater composition affect conductivity, we recommend that r1 be
taken to be the constant r1 = 0.35. Hence we see that Reference Salinity is affected by
biogeochemical processes at about 0.26 ( ≈ 0.35 / 1.35 ) of the corresponding influence of
biogeochemistry on Density Salinity.
We turn now to consider the material derivative of Conservative Temperature in a
turbulent ocean. From Eqns. (A.13.5) and (A.21.8) the instantaneous material derivative of
Θ is, without approximation,
ρ c 0p
dΘ
=
dt
(T0 + θ ) − ∇ ⋅ FR − ∇ ⋅ FQ + ρε + h ρ S SA
(
)
SA
(T0 + t )
⎡ (T + θ )
⎤
μ ( p ) − μ ( 0 ) ⎥ ( −∇ ⋅ FS + ρ S SA ) .
− ⎢ 0
⎢⎣ (T0 + t )
⎥⎦
(A.21.13)
The fact that the right‐hand side of Eqn. (A.21.13) is not the divergence of a flux means
that Θ is not a 100% conservative variable. However, the finite‐amplitude analysis of
mixing pairs of seawater parcels in appendix A.18 has shown that the non constant
coefficients of the divergences of the molecular fluxes of heat − ∇ ⋅ 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) are no larger than the
dissipation term ρε which is also justifiably neglected in oceanography (McDougall
(2003)). The source term ρ S SA 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, hSA , that appears
in 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 ρ S SA 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
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111
from the source term hSA ρ S SA (T0 + θ ) (T0 + t ) in Eqn. (A.21.13), when integrated over the
whole ocean, is approximately ( c 0p ) −1 65 J g −1 0.025 g kg −1 ≈ 0.4 mK . The other term in
ρ S SA 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 − p β 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 contribution of this term is less than half that of the term in ρ S SA in
the first line of Eqn. (A.21.13). This confirms that the presence of the two terms in ρ S SA
in the First Law of Thermodynamics has less impact than even the non‐ideal nature of the
molecular flux divergence terms in Eqn. (A.21.13) and the dissipation of mechanical
energy in this equation.
Hence with negligible error, the right‐hand side of Eqn. (A.21.13) may be regarded as
the sum of the ideal molecular flux of heat term − ∇ ⋅ 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 − ∇ ⋅ F R so
that there are no approximations with treating the air‐sea sensible, latent and radiative
heat fluxes as being fluxes of c 0p 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 ) c 0p (T0 + θ ) rather than simply c 0p . That is, the input of a certain
amount of geothermal heat will cause a local change in Θ as though the seawater had the
“specific heat capacity” (T0 + t ) c 0p (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 ) .
We conclude that it is sufficiently accurate to assume that Conservative Temperature
is in fact conservative and that the instantaneous conservation equation is
(
)(
c 0p ( ρ Θ)t + c0p ∇ ⋅ ( ρ Θu ) = ρ c 0p
)
dΘ
= − ∇ ⋅ F R − ∇ ⋅ FQ .
dt
(A.21.14)
Now we perform the same two‐stage averaging procedure as outlined above in the case of
Preformed Salinity. The Boussinesq form of the mesoscale‐averaged equation is
(analogous to Eqn. (A.21.7))
ˆ
ˆ
∂Θ
ˆ + e ∂Θ =
+ vˆ ⋅ ∇n Θ
∂t n
∂z
1
h
(
) (
ˆ + DΘ
ˆ − F bound
∇n ⋅ hK ∇n Θ
z
).
z
(A.21.15)
As in the case of the S* equation (A.21.7), the molecular flux of heat has been ignored in
comparison with the turbulent fluxes of Conservative Temperature. The air‐sea fluxes of
sensible and latent heat, the radiative and the geothermal heat fluxes remain in Eqn.
(A.21.15) in the vertical heat flux F bound which is the sum of these boundary heat fluxes
divided by ρ 0 c 0p . 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) and
Tailleux (2010) 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 two orders of
magnitude.
It is possible to derive an evolution equation for potential temperature which
resembles Eqn. (A.21.15) but which contains additional non‐conservative source terms on
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the right‐hand side of the equation. However, there seems little point in doing so here, as
it is much more convenient to instead adopt the Conservative Temperature variable. Note
that the consequences for dynamical oceanography of ignoring the non‐conservative
source terms in the potential temperature evolution equation are as large as ignoring the
variations in seawater composition; a 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 − S R 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 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
ˆ
dΘ
ˆ
=Θ
t
dt
z
ˆ + w*Θ
ˆ =
+ vˆ ⋅ ∇ z Θ
z
1
h
(
) (
ˆ + DΘ
ˆ − F bound
∇n ⋅ hK ∇n Θ
z
),
z
(A.21.16)
where the vertical velocity w* is related to e by
w* = zt n + vˆ ⋅ ∇n z + e .
(A.21.17)
A.22 The material derivatives of density and of locally-referenced
potential density; the dianeutral velocity e
Regarding density to be a function of Conservative Temperature (i. e. 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
ρˆ −1
ˆ
dρˆ
dSˆ
dΘ
dP
= β Θ A − αΘ
+ κ
,
dt
dt
dt
dt
(A.22.1)
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
ρˆ −1
= ρˆ −1
− κ
= β Θ A − αΘ
(A.22.2)
.
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
n A
t
At
n
n
to 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)
(
ˆ − β Θ Sˆ
e α ΘΘ
z
Az
)
(
)
(
ˆ − β Θ h −1∇ ⋅ hK ∇ Sˆ
= α Θ h −1∇n ⋅ hK ∇n Θ
n
n A
(
ˆ
+ α Θ DΘ
z
−1
2
)
z
(
− β Θ DSˆA z
)
z
− β Θ Sˆ SA .
)
(A.22.3)
The left‐hand side is equal to e g N and the first two terms on the right hand side
would sum to zero if the equation of state were linear. This equation can be rewritten as
the following equation for the temporally averaged vertical velocity through the neutral
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113
tangent plane at a given longitude and latitude (from McDougall (1987b), and see Eqns.
(3.8.2) and (3.9.2) for the definitions of CbΘ and TbΘ )
(
)
(
Θ
ˆ ⋅∇ Θ
ˆ + T Θ∇ Θ
ˆ
ˆ
e g −1 N 2 = − K CbΘ∇n Θ
DΘ
b
n
n ⋅ ∇n p + α
z
)
z
(
− β Θ DSˆA z
)
z
− β Θ Sˆ SA . (A.22.4)
The cabbeling nonlinearity (the CbΘ 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.
To summarize this appendix A.22; we have found that the material derivative of in
situ density Eqn. (A.22.1), when adjusted for the dynamically passive compressibility
term, becomes the material derivative of locally‐referenced potential density Eqn. (A.22.2)
which can be interpreted as an expression Eqn. (A.22.4) for e, the temporally‐averaged
vertical velocity through the local neutral tangent plane. This dianeutral velocity e is not
a separate mixing process, but rather is a direct result of mixing processes such as (i)
small‐scale turbulent mixing as parameterized by the diffusivity 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 flavours, a salt‐
fingering type and a “diffusive” type of double‐diffusive convection) is omitted from the
conservation equations (A.21.11) and (A.21.15) and also from the mean dianeutral velocity
equation (A.22.4). It is however straight‐forward to include these processes in these
conservation equations (see for example McDougall (1994, 1997b)).
A.23 The water-mass transformation equation
It is instructive to substitute Eqn. (A.22.4) for e into the expression (A.21.15) for the
material derivative of Θ̂ , thus eliminating e and obtaining the following equation for the
temporal and spatial evolution of Θ̂ along the neutral tangent plane (McDougall (1984))
ˆ
∂Θ
ˆ =
+ vˆ ⋅ ∇n Θ
∂t n
1
h
(
)
(
Θ
ˆ + KgN −2 Θ
ˆ C Θ∇ Θ
ˆ
ˆ
ˆ
∇n ⋅ hK ∇n Θ
b n ⋅ ∇ n Θ + Tb ∇ n Θ ⋅ ∇ n p
z
2ˆ
Θ
Rρ
ˆ 3 d SA + β
Sˆ SA ,
+ D β gN Θ
z
Θ
2
ˆ
α Rρ − 1
dΘ
Θ
−2
(
)
(A.23.1)
)
ˆ β Θ Sˆ . The term involving
where Rρ is the stability ratio of the water column, Rρ = α Θ Θ
z
Az
ˆ diagram of a vertical
D has been written as proportional to the curvature of the SˆA − Θ
ˆ Sˆ − Sˆ Θ
ˆ
cast; this term can also be written as D β Θ gN −2 Θ
A z zz . The form of Eqn. (A.23.1)
z A zz
illustrates that when analyzed in density coordinates, Conservative Temperature (and
Absolute Salinity) (i) are affected not only by the expected lateral diffusion process along
density surfaces but also by the nonlinear dianeutral advection processes, cabbeling and
thermobaricity, (ii) are affected by diapycnal turbulent mixing only to the extent that the
ˆ diagram is not locally straight, and (iii) are not influenced by the vertical
vertical SˆA − Θ
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
(
)
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how the tracer called “Conservative Temperature measured along the neutral direction” is
affected by mixing processes; this equation does not contain e .
For completeness, the water‐mass conservation equation for Absolute Salinity that
corresponds to Eqn. (A.23.1) is
∂SˆA
+ vˆ ⋅ ∇n SˆA =
∂t n
1
h
(
)
(
ˆ ⋅∇ Θ
ˆ + T Θ∇ Θ
ˆ
∇n ⋅ hK ∇n SˆA + K gN −2 SˆA z CbΘ∇n Θ
b
n
n ⋅ ∇n p
2ˆ
ˆ 3 d SA +
+ Dα gN Θ
z
ˆ2
dΘ
Θ
−2
)
(A.23.2)
Rρ
( Rρ −1)
Sˆ SA ,
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).
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))
∂Cˆ
+ vˆ ⋅ ∇n Cˆ =
∂t n
1
h
(
)
(
ˆ ⋅∇ Θ
ˆ + T Θ∇ Θ
ˆ
∇n ⋅ hK ∇n Cˆ + K gN −2 Cˆ z CbΘ∇n Θ
b
n
n ⋅ ∇n p
( )
+ D SˆA z
2
)
(A.23.3)
2ˆ
d 2Cˆ
Cˆ
ˆ 3 d SA + Cˆ gN −2 β Θ Sˆ SA .
+ z Dα Θ gN −2 Θ
z
z
2
ˆ2
dSˆA
SˆA z
dΘ
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
ˆ curves. The terms involving D can also be written as
SˆA − Cˆ and the SˆA − Θ
( )
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
)
(
ˆ Sˆ
ˆ ˆ
SˆA z + DCˆ zα Θ gN −2 Θ
z A zz − S A z Θ zz
)
(A.23.4)
SˆA z .
A.24 Conservation equations written in potential density coordinates
The material derivative of a quantity C can be expressed with respect to the Cartesian
reference frame, the neutral tangent plane, or a potential density reference frame so that
the conservation equation of a conservative variable can be written as (see Eqn. (A.21.16),
∂Cˆ
∂t
z
∂Cˆ
∂Cˆ
+ vˆ ⋅ ∇ z Cˆ + w*
=
∂z
∂t
n
∂Cˆ
+ vˆ ⋅ ∇ nCˆ + e Cˆ z =
+ vˆ ⋅ ∇σ Cˆ + e dCˆ z
∂t σ
=
1
h
(
) (
∇n ⋅ hK ∇nCˆ + DCˆ z
),
(A.24.1)
z
d
where e 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)
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115
to (3.17.5) which relate the gradients of properties in a potential density surface to those in
a neutral tangent plane, the following form of the conservation equation (A.21.15) for
Conservative Temperature can be derived (see equation (26) of McDougall (1991))
ˆ
ˆ
∂Θ
ˆ + e d ∂Θ =
+ vˆ ⋅ ∇σ Θ
∂t σ
∂z
1
hσ
(
) (
ˆ + DΘ
ˆ
∇σ ⋅ hσ K ∇σ Θ
z
)
z
)
(
ˆ
− h1 ∇n ⋅ ⎡⎣G Θ − 1⎤⎦ hK ∇n Θ
ˆ ⋅∇ Θ
ˆ⎞
⎛
∇ Θ
n
− ⎜ G Θ ⎡⎣G Θ − 1⎤⎦ K n
⎟⎟ ,
⎜
ˆ
Θ
z
⎝
⎠z
(A.24.3)
where the “isopycnal temperature gradient ratio” G Θ is defined as (from Eqn. (3.17.4))
G 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 . The corresponding
equation for Absolute Salinity is
∂SˆA
∂t
+ vˆ ⋅ ∇σ SˆA + e d
σ
∂SˆA
=
∂z
1
hσ
(
) (
∇σ ⋅ hσ K ∇σ SˆA + DSˆA z
)
z
⎛ ⎡ GΘ
⎞
⎤
− h1 ∇n ⋅ ⎜ ⎢
− 1⎥ hK ∇n SˆA ⎟
⎜ r
⎟
⎦
⎝⎣
⎠
Θ
ˆ
⎛G
∇ Θ ⋅ ∇n SˆA
⎡G Θ − 1⎤ K n
−⎜
⎣
⎦
⎜
ˆ
Θ
z
⎝ r
+ Sˆ SA
(A.24.4)
⎞
⎟⎟ .
⎠z
The terms in the second and third lines of Eqns. (A.24.3) and (A.24.4) arise because in the
first line of these equations, the lateral diffusion is written as being along potential density
surfaces rather than along neutral tangent planes. As explained in McDougall (1991),
these terms are non zero even at the reference pressure of the potential density variable.
Multiplying Eqn. (A.24.4) by β Θ ( pr ) and subtracting α Θ ( pr ) times Eqn. (A.24.3) we
find the corresponding expression for the diapycnal velocity e d (following McDougall
(1991))
e d
1 ∂ρˆ Θ
ˆ
= β Θ ( pr ) 1σ ∇σ ⋅ hσ K ∇σ SˆA − α Θ ( pr ) 1σ ∇σ ⋅ hσ K ∇σ Θ
h
h
ˆ
ρ Θ ∂z
+ βΘ
+ αΘ
(
)
(
)
( p ) ( DSˆ ) − α ( p ) ( DΘˆ ) + β ( p ) Sˆ
G
( p )( r −1) ∇ ⋅ ( hK ∇ Θˆ ) + α ( p ) K ∇ r ⋅ ∇ Θˆ
r
r
r
Θ
Az
z
1
h
Θ
r
z
z
Θ
n
n
r
SA
Θ
r
n
n
(A.24.5)
ˆ ⋅∇ Θ
ˆ
⎛ αΘ ( p ) ⎞
∇ Θ
n
− β Θ ( pr ) ⎜ Θ r ⎟ G Θ ⎣⎡G Θ − 1⎦⎤ K n
ˆ
⎜ β (p )⎟
Θ
r ⎠z
z
⎝
−
⎤
β Θ ( pr ) ⎡ G Θ
ˆ ⋅∇ Θ
ˆ + T Θ∇ Θ
ˆ
− 1⎥ K CbΘ∇n Θ
⎢
n
n ⋅ ∇n p .
b
Θ
β ( p) ⎣ r
⎦
(
)
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
ˆ ⋅ ∇ p showing that the thermobaric effect remains.
rather to 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.
ˆ ⋅∇ Θ
ˆ (c.f. the last line of Eqn. (A.27.2) below).
(A.24.5) can also be written as CbΘ ( pr ) K ∇σ Θ
σ
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A.25 The vertical velocity through a general surface
Consider a general surface which we identify with the label “a” (for example, this could
stand for “approximately neutral surface”). The material derivative on the left‐hand sides
of the conservation equations (A.21.11) and (A.21.15) for Absolute Salinity and
Conservative Temperature are now written with respect to this general “a” coordinate as
⎛ ∂Sˆ ⎞
∂SˆA
∂Sˆ
(A.25.1)
+ vˆ ⋅ ∇a SˆA + e a A = h1 ∇n ⋅ hK ∇n SˆA + ⎜ D A ⎟ + Sˆ SA ,
⎜ ∂z ⎟
∂t a
∂z
⎝
⎠z
and
ˆ
ˆ
∂Θ
ˆ + e a ∂Θ = 1 ∇ ⋅ hK ∇ Θ
ˆ + DΘ
ˆ
+ vˆ ⋅ ∇a Θ
(A.25.2)
.
n
z
h n
z
∂t a
∂z
(
)
(
) (
(
)
)
(
)
ˆ , p and α Θ = α Θ Sˆ , Θ
ˆ , p and
Cross‐multiplying these equations by β Θ = β Θ SˆA , Θ
A
subtracting gives the following equation for the vertical velocity through the
approximately neutral surface,
(
ˆ ⋅∇ Θ
ˆ + T Θ∇ Θ
ˆ
e a = − g N −2 K CbΘ∇n Θ
n
b
n ⋅ ∇n p
)
( ( ) − β ( DSˆ ) ) − g N
ˆ
+ g N −2 α Θ DΘ
z
Θ
z
Az
z
−2
β Θ Sˆ SA
(A.25.3)
⎡
ˆ
ˆ ⎤
ˆ ⎤ + g N −2 ⎢ β Θ ∂SA − α Θ ∂Θ ⎥ .
+ g N −2 vˆ ⋅ ⎡ β Θ∇a SˆA − α Θ∇a Θ
⎣
⎦
∂t a
∂t 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)
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
(A.25.4)
and
⎡
∂Sˆ
g N −2 ⎢ β Θ A
∂t
⎢⎣
ρˆ tl
ˆ ⎤
∂Θ
−α
⎥ = − la = zt
∂t a ⎥
ρˆ z
⎦
Θ
a
n
− zt
a
(A.25.5)
where ρˆ is the (thickness‐weighted) locally‐referenced potential density.
Combining these results with Eqn. (A.22.4) we have the rather simple kinematic result
that
l
e a = e + vˆ ⋅ s + zt
n
− zt a ,
(A.25.6)
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
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117
respect to potential density surfaces, one finds that the temporal and isopycnal gradient
terms cancel leaving only the term in the mean diapycnal velocity e d as follows
e d
1 ∂ρˆ Θ
ˆ
= β Θ ( pr ) h1 ∇n ⋅ hK ∇n SˆA − α Θ ( pr ) h1 ∇n ⋅ hK ∇n Θ
Θ
ρˆ ∂z
(
+β
Θ
)
( pr ) ( DSˆA z ) z
− α
Θ
(
( pr ) ( DΘˆ z ) z
+ β
Θ
)
(A.26.1)
( pr ) S
ˆ SA
,
where the following 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 ∂ρˆ Θ ⎞
1 ∂ρˆ Θ
S
Θ
= ⎜ Θ
⎟ + β ( pr ) Sˆ A
Θ
ˆ
ˆ
∂
∂
z
z
ρ
⎝ρ
⎠z
ˆ 2 + 2α Θ ( p ) Θ
ˆ Sˆ − β Θ ( p ) Sˆ 2
+ D α ΘΘ ( pr ) Θ
z
SA
z Az
SA
r
r
Az
{
(
ˆ
+ α Θ ( pr ) [ r − 1] h1 ∇n ⋅ hK ∇n Θ
)
}
(A.26.3)
β Θ ( pr )
ˆ ⋅∇ Θ
ˆ + T Θ∇ Θ
ˆ
K CbΘ∇n Θ
b
n
n ⋅ ∇n p ,
Θ
β ( 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).
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
rather 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 ∂ρˆ Θ
ˆ
= β Θ ( pr ) 1σ ∇σ ⋅ hσ K σ ∇σ SˆA − α Θ ( pr ) 1σ ∇σ ⋅ hσ K σ ∇σ Θ
Θ
h
h
ˆ
∂
z
ρ
(
+β
Θ
( pr ) (
DSˆA z
)
)
z
− α
Θ
( pr ) (
(
ˆ
DΘ
z
)
+ β ( pr ) Sˆ SA ,
)
(A.27.1)
Θ
z
where ∇σ is the gradient operator along the potential density coordinate, K σ is the lateral
diffusivity along the layers and hσ is the thickness between a pair of potential density
surfaces in the vertical. This equation can be rewritten as
e d_model
⎛ D ∂ρˆ Θ ⎞
1 ∂ρˆ Θ
S
Θ
=
⎜ Θ
⎟ + β ( pr ) Sˆ A
Θ
ρˆ ∂z
⎝ ρˆ ∂z ⎠ z
ˆ 2 + 2α Θ ( p ) Θ
ˆ Sˆ − β Θ ( p ) Sˆ 2
+ D α ΘΘ ( pr ) Θ
z
SA
z Az
SA
r
r
Az
{
}
(A.27.2)
ˆ ⋅∇ Θ
ˆ .
+ K σ CbΘ ( pr ) ∇σ Θ
σ
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
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
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
ˆ
via the relation (3.17.3)). Another difference is that the term α Θ ( pr ) [ r − 1] h −1∇n ⋅ hK ∇n Θ
in Eqn. (A.26.3) is missing from Eqn. (A.27.2). This type of difference is to be expected
since the direction of the lateral mixing is different.
Notice the absence of the thermobaric diapycnal advection from Eqn. (A.27.2); that is,
ˆ ⋅ ∇ p in Eqn. (A.26.3) is absent from Eqn. (A.27.2), as
the term proportional to K TbΘ∇n Θ
n
first remarked on by Iudicone et al. (2008). The thermobaric diapycnal advection is
probably significant in the Southern Ocean (Klocker and McDougall (2010a)) and this
would argue for the rotation of the lateral mixing tensor in layered models to the local
direction of the neutral tangent plane, as is done in height‐coordinate ocean models. Also
missing from layered ocean models is the mean vertical advection 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)).
(
)
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
∂ρ v
∂t
ρv
ρ
+ vˆ ⋅ ∇ ρv ρ v + e v
∂ρ v
∂z
ρ
= e v
∂ρ v
,
∂z
(A.28.1)
where the temporally averaged vertical velocity through the ρ v surface is given by (from
McDougall and Jackett (2005a))
(
)
(
ρ
− β Θ S + (ψ − 1) p
e v = gN −2 α Θ Θ
t
A
ρv
+ vˆ ⋅ ∇ ρv p
)
pz ,
(A.28.2)
where (from de Szoeke et al. (2000))
(ψ −1)
ˆ − Θ ( p, ρ ) ⎤ ,
≈ 2 g 2 N −2 c0−3Δc ≈ − 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
is similarly shorthand for the material
by the right‐hand side of Eqn. (A.21.11); Θ
derivative of Θ̂ and is given by the right‐hand side of Eqn. (A.21.15).
The first term on the right of Eqn. (A.28.2) represents the effects of irreversible mixing
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.
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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
+ vˆ ⋅ ∇γ γ n + e γ γ zn = e γ γ zn ,
(A.29.1)
γ
where the temporally averaged vertical velocity through the γ n surface is given by (from
McDougall and Jackett (2005b))
e
γ
≈
(α
Θ
− β Θ ( p ) S
( p) Θ
A
)
+ vˆ ⋅ s ref
(
)
+ (ψ γ − 1) ( p γ + vˆ ⋅ ∇γ p ) ( p )
α
Θ
−β
( p )Θref
z
Θ
t
( p ) SArefz
−1
z
⎛
⎞
( p − p ref )
−1
+ ψ γ − 1 ⎜ vˆ ⋅ ∇γ p ref −
vˆ ⋅ ∇γ Θref ⎟ ( pz )
ref
ˆ
(Θ − Θ )
⎝
⎠
Θ
Θ
α ( p ) Θ − β ( p ) SA
+ 2(ψ γ − 1) Θ
Θ
ref
α ( p )Θref
z − β ( p ) SA z
(
)
(
)
(
(
)
(A.29.2)
)
+ ψ γ − 1 vˆ ⋅ 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.
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 γ −1)
=
2
⎡ N ref
⎣
ˆ − Θref )
− 12 ρ g 2TbΘ ( Θ
.
ˆ − Θref ) + 1 gT Θ ( P − P ref ) Θref ⎤
+ ρ g 2TbΘ ( Θ
z ⎦
b
2
(A.29.3)
2
Here N ref
is the square of the buoyancy frequency of the pre‐labelled reference data set.
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
reasonable values of ( Θ
2
γ
N ref and ψ − 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
and S . The next term in Eqn. (A.29.2), vˆ ⋅ sref , is also
the irreversible mixing processes Θ
expected; it is the mean vertical motion through the γ n surface due to the helical motion
of neutral trajectories in the reference data set, caused in turn by the non‐zero neutral
helicity of the reference data set. The remaining terms in the last four lines of Eqn. (A.29.2)
arise because of the non‐quasi‐material (non‐potential) nature of Neutral Density. The
second line of Eqn. (A.29.2) represents the contribution to e γ arising from the seemingly
innocuous sliding motion along the sloping γ n surface and from the vertical heaving of
these surfaces. The lateral gradients of properties in the reference data set also affect the
mean flow e γ through the γ n surface. Note that as Θ̂ − Θref tends to zero, ψ γ − 1 also
tends to zero so that the third line of Eqn. (A.29.2) is well‐behaved and becomes
proportional to pz−1 ( p − p ref ) vˆ ⋅ ∇γ Θref .
(
(
)
)
(
)
(
)
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A.30 Computationally efficient 25-term expressions for the density of seawater
in terms of Θ and θ
Ocean models to date have treated their salinity and temperature variables as being
Practical Salinity SP and potential temperature θ . As the full implications of TEOS‐10 are
incorporated into ocean models they will need to carry Preformed Salinity S* and
Conservative Temperature Θ as conservative variables (as discussed in appendices A.20
and A.21), and a computationally efficient expression for density in terms of Absolute
Salinity S A and Conservative Temperature Θ will be needed.
Following the work of McDougall et al. (2003) and Jackett et al. (2006), the TEOS‐10
density ρ has been approximated by a rational function of the same form as in those
papers. The fitted expression is the ratio of two polynomials of ( SA , Θ, p )
ρ 25
ρ 25
ρ ≈ ρ 25 = Pnum Pdenom .
(A.30.1)
The density data has been fitted in a “funnel” of data points in ( SA , t , p ) space which is
described in more detail in McDougall et al. (2010b). The “funnel” extends to a pressure of
8000 dbar . At the sea surface the “funnel” covers the full range of temperature and
salinity while for pressures greater than 5500 dbar, the maximum temperature of the fitted
data is 12°C and the minimum Absolute Salinity is uPS 30 g kg −1 . That is, the fit has been
performed over a region of parameter space which includes water that is approximately
10°C warmer and 5 g kg −1 fresher in the deep ocean than exists in the present ocean (see
Figure 1 of Jackett et al. (2006)). Table K.1 of appendix K contains the 25 coefficients of the
expression (A.30.1) for density in terms of ( SA , Θ, p ) .
As outlined in appendix K, this 25‐term rational‐function expression for ρ yields the
thermal expansion and haline 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”. The same cannot be claimed for the sound speed derived by
differentiating Eqn. (A.30.1) with respect to pressure; this sound speed has an rms error in
the “funnel” of almost 0.25 m s −1 whereas TEOS‐10 fits the available sound speed data
with an rms error of only 0.035 m s −1 .
In dynamical oceanography it is the thermal expansion and haline contraction
coefficients α Θ and β Θ 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 the 25‐term rational function expression
for density, Eqn. (A.30.1), as essentially reflecting the full accuracy of TEOS‐10. This is
confirmed in Fig. A.30.1 where the error in using the 25‐term expression for density to
calculate the isobaric northward density gradient is shown. The vertical axis on this figure
is the magnitude of the difference in the northward isobaric density gradient in the world
ocean below 1000 m when evaluated using Eqn. (A.30.1) versus using the full TEOS‐10
Gibbs function. The scales of the axes of this figure have been chosen to be the same as
those of Fig. A.5.1 of appendix A.5 so that the smallness of the errors associated with using
the 25‐term density expression can be appreciated. The errors represented in Fig. A.30.1
represent perhaps half of the remaining uncertainty in our knowledge of seawater
properties, and by comparing Figs. A.30.1 and A.5.1 it is clear that the much more
important issue is to properly represent the effects of seawater composition on seawater
density. The rms value of the vertical axis in Fig. A.30.1 is 11.4% of that of Fig. A.5.1.
McDougall et al. (2010b) have also provided a 25‐term rational‐function expression for
density in terms of ( SA , θ , p ) . The 25 coefficients can be found in Table K.2 of appendix
K. As an approximation to density, this 25‐term rational function is approximately as
accurate as the one described above in terms of ( SA , Θ, p ) . It must be emphasized though
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121
that an ocean model that treated potential temperature as a conservative variable would
make errors in its treatment of heat fluxes, as described in appendices A.13, A.14 and A.17,
and as illustrated in Figures A.13.1, A.14.1 and A.17.1.
Figure A.30.1. The northward density gradient at constant pressure (the horizontal
axis) for data in the world ocean atlas of Gouretski and Koltermann
(2004) for p > 1000 dbar . The vertical axis is the magnitude of the
difference between evaluating the density gradient using the 25‐term
expression Eqn. (A.30.1) instead of using the full TEOS‐10 expression,
using Absolute Salinity S A as the salinity argument in both cases.
Appendix P describes how an expression for the enthalpy of seawater in terms of
Conservative Temperature, specifically the functional form hˆ ( SA , 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 ( SA , Θ, p )
also arises in section 3.32 and in Eqns. (3.26.3) and (3.29.1). The 25‐term expression, Eqn.
(A.30.1), for ρ 25 = ρˆ 25 ( SA , Θ, p ) can be used to find a closed expression for hˆ ( S A , Θ, p ) by
integrating the reciprocal of ρˆ 25 ( SA , Θ, p ) with respect to pressure (in Pa ), since
hˆP = v = ρ −1 (see Eqn. (2.8.3)).
The 25‐term expression for specific volume, Eqn. (A.30.1), is first written explicitly as
the ratio of two polynomials in sea pressure p (in dbar ) as
vˆ 25 =
a + a p + a2 p 2 + a3 p 3
1
= 0 1
,
25
b0 + 2b1 p + b2 p 2
ρˆ
(A.30.2)
where the coefficients a0 to a3 and b0 to b2 are the following functions of S A and Θ
a0 = 1 + c14 Θ + c15 Θ 2 + c16 Θ3 + c17 Θ 4 + c18 SA + c19 SA Θ + c20 S A Θ3 + c21 ( SA )
1.5
+ c22 ( SA ) Θ 2 ,
1.5
a1 = c23 ,
a2 = c24 Θ3 ,
a3 = c25 Θ ,
b0 = c1 + c2 Θ + c3Θ 2 + c4 Θ3 + c5 SA + c6 SA Θ + c7 ( SA ) ,
2
(
)
b1 = 0.5 c8 + c9 Θ2 + c10 SA ,
2
b2 = c11 + c12 Θ ,
and the numbered coefficients c1 to c25 are so identified in Table K.1 (note that c13 = 1 ).
It is not difficult to rearrange Eqn. (A.30.2) into the form
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⎛a
2a b ⎞
a
N + Mp
vˆ 25 = vˆ 25 ( SA , Θ, p ) = ⎜ 2 − 32 1 ⎟ + 3 p +
,
b2
b2 ⎠
b0 + 2b1 p + b2 p 2
⎝ b2
where N and M are given by
4a3b12 a3b0 2a2b1
2a3b0b1
a2b0
.
−
and
.
N = a0 +
M
a
=
+
−
−
1
b2
b2
b2
b22
b22
(A.30.3)
(A.30.4)
The pressure integral of the last term in Eqn. (A.30.3) is well known (see for example
section 2.103 of Gradshteyn and Ryzhik (1980)) and is dependent on the sign of the
discriminant of the denominator. In our case it can be shown that b12 > b0b2 over the
domain of the “funnel” and also that both b0 and b1 are positive, while b2 is negative and
bounded away from zero. The indefinite integral, with respect to sea pressure measured
in Pa , of the last term in Eqn. (A.30.3) is (with N * = 104 N and M * = 104 M )
b2 p + b1 − b12 − b0b2
N + Mp
M*
N *b2 − M *b1
2
′
dP
b
b
p
b
p
ln
2
ln
, (A.30.5)
=
+
+
+
∫
0
1
2
2b2
b0 + 2b1 p + b2 p 2
b2 p + b1 + b12 − b0b2
2b2 b12 − b0b2
The enthalpy hˆ 25 ( SA , Θ, p ) is the definite integral of Eqn. (A.30.3) from P0 to P , plus c 0p Θ ,
being the value of enthalpy at P0 (i. e. at p = 0 dbar ). Hence the full expression for
hˆ 25 ( SA , Θ, p ) is (with A = b1 − b12 − b0b2 and B = b1 + b12 − b0b2 )
⎛a
2a b ⎞
a
hˆ 25 ( SA , Θ, p ) = c 0p Θ + 104 ⎜ 2 − 32 1 ⎟ p + 104 3 p 2
2b2
b2 ⎠
⎝ b2
(A.30.6)
b
N* − 1 M*
*
⎛
⎛
⎞
2b
M
b
b2
b ( B − A) ⎞
+
ln ⎜ 1 + 1 p + 2 p 2 ⎟ +
ln ⎜ 1 + p 2
⎟.
2b2 ⎝
b0
b0 ⎠
A ( B + b2 p ) ⎠
( B − A)
⎝
The factor of 104 that appears here and in N * and M * effectively serves to convert the
units of the integration variable from dbar to Pa so that hˆ 25 ( SA , Θ, p ) has units of J kg −1.
In these equations S A is in g kg −1 , Θ in °C and p is in dbar. The arguments of the two
natural logarithms in Eqn. (A.30.6) are always greater than 1, and in fact they are between
1 and 1.2 even for p as large as 104 dbar (note that both b2 and A are negative). Also,
when the enthalpy difference at the same values of S A and Θ but at different pressures
(see Eqn. (3.32.2)) is evaluated using Eqn. (A.30.6), the expression can also be arranged to
contain only two logarithm terms.
Following Young (2010), the difference between specific enthalpy and c 0p Θ may be
called “dynamic enthalpy” and can be readily calculated from Eqn. (A.30.6), recognizing
that this equation is based on the computationally efficient 25‐term expression for density
of McDougall et al. (2010b) rather than being evaluated from the full TEOS‐10 Gibbs
function. Similarly, the partial derivatives of hˆ 25 ( SA , Θ, p ) with respect to Absolute
Salinity S A and with respect to Conservative Temperature Θ can be calculated either by
algebraic differentiation of Eqn. (A.30.6) or by first algebraically differentiating Eqn.
(A.30.1) and then numerically integrating this expression with respect to pressure (this
second procedure is motivated by taking the appropriate S A or Θ derivatives of Eqn.
(3.2.1); see Eqns. (A.18.4) and (A.18.5)).
An expression h 25 ( SA ,θ , p ) for enthalpy as a function of potential temperature θ can
be found in a similar manner to that outlined above, but with the coefficients of the 25‐
term rational‐function expression for density now being taken from Table K.2, and with
the first term being expressed as the exact polynomial expression for h ( SA ,θ , 0 ) instead of
as c 0p Θ .
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Appendix B:
Derivation of the First Law of Thermodynamics
Motivation
For a pure fluid in which there is no dissolved material (such as pure water with zero
Absolute Salinity) the derivation of the First Law of Thermodynamics usually starts with a
discussion of how the internal energy U of a fixed mass of fluid is changed under the
influence of it being “heated” by the amount δ Q and its volume V being changed. The
infinitesimal change in the internal energy of the parcel is written as dU = δ Q − ( p + P0 ) dV
where − ( p + P0 ) dV is the mechanical work done on the fluid by the pressure at the
moving boundaries of the fluid parcel. This relationship can be written in terms of the
specific (i. e. per unit mass) enthalpy h, the density ρ , and δ Q per unit volume, δ q, as
⎛ dh 1 dP ⎞
δq
−
.
⎟ =
ρ dt ⎠
dt
⎝ dt
ρ⎜
for pure water (B.1)
It is recognized that the right‐hand side of (B.1) is not the divergence of a “heat” flux, and
the term that causes this complication is the dissipation of mechanical energy into “heat”,
which contributes 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, F R , and minus the divergence of the molecular flux of heat
Fq = − k T ∇T (where k T is the molecular diffusivity of heat), so that the First Law of
Thermodynamics for pure water is
⎛ dh 1 dP ⎞
δq
−
= − ∇ ⋅ F R + ∇ ⋅ k T ∇T + ρε .
⎟ =
ρ dt ⎠
dt
⎝ dt
(
ρ⎜
)
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 dSA ,
(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
t
t
d
ρ
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 mechanical energy and the molecular, radiative and
boundary fluxes of “heat” and salt? If the “heating” term δ q dt in (B.4) were the same as
in the pure water case (B.2) then we would have successfully derived the First Law of
Thermodynamics in a saline ocean via this route. However, we will now show that δ q dt
in (B.4) is not the same as that in the pure water case, (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.24)) for
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the molecular flux FQ ).
different by
The two versions of the First Law of Thermodynamics are
⎡⎛
[T0 + t ] μSA
− ( μ − [T0 + t ] μT ) ∇ ⋅ FS + ∇ ⋅ ⎢⎜ μ + b
⎜
ρ kS
⎣⎢⎝
⎞ S⎤
⎟⎟ F ⎥ .
⎠ ⎦⎥
(B.5)
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 and in the absence of new inspiration which we have
not found in the literature, we tentatively conclude that any attempt to derive the First
Law of Thermodynamics via this route involving the loosely defined “rate of heating”
δ q dt is doomed to failure.
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.
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 x , y , z + u ⋅ ∇ ,
dh 1 dP
du
dv
dη
dS
−
=
+ ( p + P0 )
= (T0 + t )
+μ A .
ρ dt
dt
dt
dt
dt
dt
(B.6)
The use of the same symbol t for time and for in situ temperature in °C is noted but
should not cause confusion. The middle expression in (B.6) uses the fact that specific
enthalpy h and specific internal energy u are related by h = u + Pv = u + ( p + P0 ) v where
v is the specific volume.
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Gravitational potential energy
If the gravitational acceleration is taken to be constant the gravitational potential energy
per unit mass with respect to the height z = 0 is simply gz . Allowing the gravitational
acceleration to be a function of height means that the gravitational potential energy per
unit mass Φ with respect to some fixed height z0 is defined by
z
Φ=
∫ g ( z′) dz′.
(B.7)
z0
At a fixed location in space Φ is independent of time while its spatial gradient is given by
∇Φ = g k where k is the unit vector pointing upwards in the vertical direction. The
evolution equation for Φ is then readily constructed as
( ρΦ ) t + ∇ ⋅ ( ρΦ u )
= ρ
dΦ
= ρ gw ,
dt
(B.8)
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.
Momentum evolution equation
The momentum evolution equation is derived in many textbooks including Landau and
Lifshitz (1959), Batchelor (1970), Gill (1982) and Griffies (2004). The molecular viscosity
appears in the exact momentum evolution equation in the rather complicated expressions
appearing in equations (3.3.11) and (3.3.12) of Batchelor (1970). We ignore the term that
depends on the product of the so called dynamic viscosity v visc and the velocity
divergence ∇ ⋅ u (following Gill (1982)), so arriving at
(
)
du
pu ,
+ f k × ρ u = − ∇P − ρ gk + ∇ ⋅ ρ v visc ∇
(B.9)
dt
pu is twice the symmetrized
Where f is the Coriolis frequency, v visc is the viscosity and ∇
p
velocity shear, ∇u = ∂u ∂x + ∂u ∂x . Under the same assumption as above of ignoring
ρ
(
i
j
j
i
)
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 mechanical energy ε is the positive definite quantity
ε ≡
1
2
(
)
pu ⋅ ∇
pu .
v visc ∇
(B.11)
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Evolution equation for the sum of kinetic and gravitational potential energies
The evolution equation for total mechanical energy 0.5 u ⋅ u + Φ is found by adding Eqns.
(B8) and (B10) giving
( ρ ⎡⎣
1
2
u ⋅ u + Φ ⎤⎦
)
(
+ ∇ ⋅ ρ u ⎡⎣ 12 u ⋅ u + Φ ⎤⎦
t
)
(
)
= 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
)
= ρ dE dt
= − ∇ ⋅ ([ p + P0 ] u ) ,
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.14). The molecular viscosity gives rise to a stress in the fluid
represented by the tensor σ , and the interior flux of energy due to this stress tensor is
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
pu so that the extra term is ∇ ⋅ ρ v visc ∇ 1 [u ⋅ u] . Also heat
stress tensor to be σ = − ρ v visc ∇
2
fluxes at the ocean boundaries and by radiation F R and molecular diffusion FQ
necessitate the additional terms −∇ ⋅ F R − ∇ ⋅ 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 eq (B.24). The non‐conservative production of
Absolute Salinity by the remineralization of sinking particulate matter, ρ S SA , 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 + ∇ ⋅ ( ρ uE )
)
= ρ dE dt = − ∇ ⋅ ([ p + P0 ] u ) − ∇ ⋅ F R − ∇ ⋅ FQ
(
)
+ ∇ ⋅ ρ v visc ∇ 12 [u ⋅ u ] + hSA ρ S
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SA
.
(B.15)
TEOS-10 Manual: Calculation and Use of the Thermodynamic Properties of Seawater
127
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).
If it were not for the remineralization source term, hSA ρ S SA , the right‐hand side of
the E 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
+ ∇ ⋅ ( ρ uB
)
(
)
= − ∇ ⋅ F R − ∇ ⋅ FQ + ∇ ⋅ ρ v visc ∇ 12 [u ⋅ u] + hSA ρ S
SA
.
(B.16)
In an ocean modelling context, it is rather strange to contemplate the energy variable that
is advected through the face of a model grid, 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
(
)
+ ∇ ⋅ ( ρ uB ) = ρ dB dt = Pt − ∇ ⋅ F R − ∇ ⋅ FQ + ∇ ⋅ ρ v visc ∇ 12 [u ⋅ u ] + hSA ρ S
SA
. (B.17)
The source term ρ S SA 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 assumption, and in the absence of ρ S SA 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 and continuing to take the whole ocean to be in a steady state and
with ρ S SA = 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 posses the “potential” property.
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Obtaining the First Law of Thermodynamics by subtraction
The evolution equation (B.12) for the sum of kinetic and gravitational potential energies is
now subtracted from the total energy conservation equation (B.15) giving
( ρ u )t + ∇ ⋅ ( ρ u u ) = ρ du
dt = − ( p + P0 ) ∇ ⋅ u − ∇ ⋅ F R − ∇ ⋅ FQ + ρε + hSA ρ S
Using the continuity equation in the form ρ dv dt = ∇ ⋅ u
thermodynamic relation (A.7.2), this equation can be written as
SA
(B.18)
and the fundamental
⎛ dh 1 dP ⎞
dv ⎞
dη
dS ⎞
⎛ du
⎛
−
+ ( p + P0 ) ⎟ = ρ ⎜ (T0 + t )
+μ A⎟
⎟ = ρ⎜
ρ dt ⎠
dt ⎠
dt
dt ⎠
⎝ dt
⎝
⎝ dt
ρ⎜
= − ∇ ⋅ F − ∇ ⋅ F + ρε + hSA ρ S
R
.
Q
,
(B.19)
SA
which is the First Law of Thermodynamics. The corresponding evolution equation for
Absolute Salinity is (Eqn. (A.21.8))
ρ
dS A
= ( ρ SA )t + ∇ ⋅ ( ρ u SA ) = − ∇ ⋅ FS + ρ S
dt
SA
,
(B.20)
where FS is the molecular flux of salt and ρ S SA is the non‐conservative source of
Absolute Salinity due to the remineralization of particulate matter. For many purposes in
oceanography the exact dependence of the molecular fluxes of heat and salt on the
gradients of Absolute Salinity, temperature and pressure is unimportant, nevertheless,
Eqns. (B.23) and (B.24) below list these molecular fluxes in terms of the spatial gradients of
these quantities.
At first sight Eqn. (B.19) has little to recommend it; there are two non‐conservative
source terms ρε and hSA ρ S SA on the right‐hand side and the left‐hand side is not ρ
times the material derivative of any quantity as is required of a conservation equation of a
conservative variable. Equation (B.19) corresponds to equation (57.6) of Landau and
Lifshitz (1959) and is repeated at Eqns. (A.13.1) and (A.13.3) above.
The approach used here to develop the First Law of Thermodynamics seems rather
convoluted in that the conservation equation for total energy is first formed, and then the
evolution equations for kinetic and gravitational potential energies are subtracted.
Moreover, the molecular, radiative and boundary fluxes were included into the total
energy conservation equation as separate deliberate flux divergences, rather than coming
from an underlying basic conservation equation. This is the approach of Landau and
Lifshitz (1959) and it is adopted for the following reasons. First this approach ensures that
the molecular, radiative and boundary fluxes do enter the total energy conservation
equation (B.15) as the divergence of fluxes so that the total energy is guaranteed to be a
conservative variable (apart from the salinity source term). This is essential; total energy
can only be allowed to spontaneously appear or disappear when there is a bona fide
interior source term such as hSA ρ S SA . Second, it is rather unclear how one would
otherwise arrive at the molecular fluxes of heat and salt on the right‐hand side of the First
Law of Thermodynamics since the direct approach which was attempted at the beginning
of this appendix involved the poorly defined “rate of heating” 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.
Landau and Lifshitz (1959) (their section 58) show that the molecular fluxes FQ and
FS are given in terms of the chemical potential μ and the gradients of temperature and of
chemical potential by
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FS = − a ∇μ − b ∇T ,
(B.21)
FQ = − b (T0 + t ) ∇μ − γ ∇T + μ FS ,
(B.22)
and
where a, b and γ are three independent coefficients. Note the symmetry between some
of the cross‐diffusion terms in that the same coefficient b appears in both equations.
When written in terms of the gradients of Absolute Salinity, temperature and pressure
these expressions for the molecular fluxes FQ and FS become
⎛ ρ k S μT
⎞
kSβ t
+ b ⎟ ∇T +
∇P ,
FS = − ρ k S ∇SA − ⎜
⎜ μS
⎟
μ SA
A
⎝
⎠
(B.23)
(using the relation β t = − ρ μ p that follows from the definition of both β t and μ in terms
of the Gibbs function) and
⎛
(T0 + t ) μSA ⎞ S T
FQ = ⎜ μ + b
(B.24)
⎟⎟ F − k ∇T .
⎜
ρ kS
⎝
⎠
These expressions involve the pure molecular diffusivities of temperature and salinity ( k T
and k S ) and the single parameter b that appears in part of both the cross‐diffusion of salt
down the temperature gradient and part of the cross‐diffusion of “heat” down the
gradient of Absolute Salinity. The other parameters in these equations follow directly
from the Gibbs function of seawater. The last term in (B.23) represents “barodiffusion” as
it causes a flux of salt down the gradient of pressure. The middle term in (B.23) is a flux of
salt due to the gradient of in situ temperature and is called the Soret effect while the first
term in (B.24) is a flux of “heat” caused by the molecular flux of salt, FS , and this is called
the Dufour effect.
The molecular flux of salt is independent of the four arbitrary constants (Fofonoff
(1962)) that appear in the Gibbs function of seawater (see Eqn. (2.6.2)). Since μT contains
the arbitrary additional constant a4 , the fact that FS contains no arbitrary constants
implies that the cross‐diffusion coefficient b in Eqns. (B.21)–(B.24) is arbitrary to the
extent − a4 ρ k S μSA .
The molecular flux of “heat” FQ is unknowable to the extent a3FS (since μ is
arbitrary to the extent a3 + a4 (T0 + t ) and the presence of b in Eqn. (B.24) cancels the
influence of a4 ). 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 a3ρ S SA
(since hSA is arbitrary by the amount a3 ). These three arbitrary, unknowable
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.
Regarding Eqns. (B.21)–(B.24), 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 saltʺ. When additional processes act to keep the composition
approximately fixed, the use of only one chemical potential for sea salt is permitted in
non‐equilibrium situations. These processes are mainly ion relaxation by Coulomb forces,
which in the form of ambipolar diffusion prevent any local electrical charge separation,
and secondly, turbulent mixing which has the same transport coefficient for each species
and whose fluxes are proportional to the concentration gradients of “potential” quantities
(see appendix A.9) rather than to the gradients of the individual chemical potentials.
IOC Manuals and Guides No. 56
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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 in the text as IAPWS‐09.
IAPWS, 2010: Guideline on an Equation of State for Humid Air in Contact with Seawater and
Ice, Consistent with the IAPWS Formulation 2008 for the Thermodynamic Properties of
Seawater. The International Association for the Properties of Water and Steam. Niagara
Falls, Canada, July 2010, available from http://www.iapws.org. This Guideline is referred
to in the text as IAPWS‐10.
Lemmon, E. W., R. T. Jacobsen, S. G. Penoncello and D. G. Friend, 2000: Thermodynamic
properties of air and mixtures of nitrogen, argon and oxygen from 60 to 2000 K at
pressures to 2000 MPa. J. Phys. Chem. Ref. Data, 29, 331–362.
Millero, F. J., R. Feistel, D. G. Wright, and T. J. McDougall, 2008a: The composition of Standard
Seawater and the definition of the Reference‐Composition Salinity Scale, Deep‐Sea Res. I,
55, 50‐72.
Secondary standard documents
IOC, SCOR and IAPSO, 2010: The international thermodynamic equation of seawater – 2010:
Calculation and use of thermodynamic properties.
Intergovernmental Oceanographic
Commission, Manuals and Guides No. 56, UNESCO (English), 196 pp, Paris. Available
from http://www.TEOS‐10.org [the present document, called the TEOS‐10 manual]
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and Use of the Thermodynamic Properties of Seawater
131
Tertiary standard documents
McDougall, T. J., D. R. Jackett and F. J. Millero, 2010a: An algorithm for estimating Absolute
Salinity in the global ocean. submitted to Ocean Science, a preliminary version is available
at Ocean Sci. Discuss., 6, 215‐242. http://www.ocean‐sci‐discuss.net/6/215/2009/osd‐6‐215‐
2009‐print.pdf and the computer software is available from http://www.TEOS‐10.org
Background papers to the declared standards
Feistel, R., 2003: A new extended Gibbs thermodynamic potential of seawater. Progr.
Oceanogr., 58, 43‐114.
Feistel, R., 2008: A Gibbs function for seawater thermodynamics for −6 to 80 °C and salinity up
to 120 g kg–1. Deep‐Sea Res. I, 55, 1639‐1671.
Feistel, R. and W. Wagner, 2006: A New Equation of State for H2O Ice Ih. J. Phys. Chem. Ref.
Data, 35, 2, 1021‐1047.
Feistel, R., S. Weinreben, H. Wolf, S. Seitz, P. Spitzer, B. Adel, G. Nausch, B. Schneider and D.
G. Wright, 2010c: Density and Absolute Salinity of the Baltic Sea 2006–2009. Ocean
Science, 6, 3–24. http://www.ocean‐sci.net/6/3/2010/os‐6‐3‐2010.pdf
Feistel, R., D. G. Wright, H.‐J. Kretzschmar, E. Hagen, S. Herrmann and R. Span, 2010a:
Thermodynamic properties of sea air. Ocean Science, 6, 91–141. http://www.ocean‐
sci.net/6/91/2010/os‐6‐91‐2010.pdf
Feistel, R., D. G. Wright, K. Miyagawa, A. H. Harvey, J. Hruby, D. R. Jackett, T. J. McDougall
and W. Wagner, 2008: Mutually consistent thermodynamic potentials for fluid water, ice
and seawater: a new standard for oceanography. Ocean Science, 4, 275‐291.
http://www.ocean‐sci.net/4/275/2008/os‐4‐275‐2008.html
McDougall, T. J., 2003: Potential enthalpy: A conservative oceanic variable for evaluating heat
content and heat fluxes. Journal of Physical Oceanography, 33, 945‐963.
Marion, G. M., F. J. Millero, and R. Feistel, 2009: Precipitation of solid phase calcium
carbonates and their effect on application of seawater S A − T − P models, Ocean Sci., 5,
285‐291. www.ocean‐sci.net/5/285/2009/
Millero, F. J., 2000. Effect of changes in the composition of seawater on the density‐salinity
relationship. Deep‐Sea Res. I 47, 1583‐1590.
Millero, F. J., 2010: History of the equation of state of seawater. Oceanography, 23, 18‐33.
Millero, F. J., F. Huang, N. Williams, J. Waters and R. Woosley, 2009: The effect of composition
on the density of South Pacific Ocean waters, Mar. Chem., 114, 56‐62.
Millero, F. J., J. Waters, R. Woosley, F. Huang, and M. Chanson, 2008b: The effect of
composition on the density of Indian Ocean waters, Deep‐Sea Res. I, 55, 460‐470.
Pawlowicz, R., 2010: A model for predicting changes in the electrical conductivity, Practical
Salinity, and Absolute Salinity of seawater due to variations in relative chemical
composition. Ocean Science, 6, 361–378. http://www.ocean‐sci.net/6/361/2010/os‐6‐361‐
2010.pdf
Pawlowicz, R., D. G. Wright and F. J. Millero, 2010: The effects of biogeochemical processes on
oceanic conductivity/salinity/density relationships and the characterization of real
seawater. Ocean Science Discussions, 7, 773–836.
http://www.ocean‐sci‐discuss.net/7/773/2010/osd‐7‐773‐2010‐print.pdf
Seitz, S., R. Feistel, D.G. Wright, S. Weinreben, P. Spitzer and P. de Bievre, 2010b: Metrological
Traceability of Oceanographic Salinity Measurement Results. Ocean Science Discussions, 7,
1303–1346. http://www.ocean‐sci‐discuss.net/7/1303/2010/osd‐7‐1303‐2010‐print.pdf
Wagner, W. and Pruß, A., 2002: The IAPWS formulation 1995 for the thermodynamic
properties of ordinary water substance for general and scientific use. J. Phys. Chem. Ref.
Data, 31, 387‐535.
Wright, D. G., R. Pawlowicz, T. J. McDougall, R. Feistel and G. M. Marion, 2010b: Absolute
Salinity, “Density Salinity” and the Reference‐Composition Salinity Scale: present and
future use in the seawater standard TEOS‐10. Ocean Sci. Discuss., 7, 1559‐1625.
http://www.ocean‐sci‐discuss.net/7/1559/2010/osd‐7‐1559‐2010‐print.pdf
IOC Manuals and Guides No. 56
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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.ocean‐sci.net/6/633/2010/os‐6‐633‐2010.pdf and http://www.ocean‐
sci.net/6/633/2010/os‐6‐633‐2010‐supplement.pdf
Wright, D. G., R. Feistel, J. H. Reissmann, K. Miyagawa, D. R. Jackett, W. Wagner, U. Overhoff,
C. Guder, A. Feistel and G. M. Marion, 2010a: Numerical implementation and
oceanographic application of the thermodynamic potentials of liquid water, water
vapour, ice, seawater and humid air ‐ Part 2: The library routines. Ocean Science, 6, 695‐
718. http://www.ocean‐sci.net/6/695/2010/os‐6‐695‐2010.pdf and http://www.ocean‐
sci.net/6/695/2010/os‐6‐695‐2010‐supplement.pdf
McDougall T. J., D. R. Jackett, P. M. Barker, C. Roberts‐Thomson, R. Feistel and R. W. Hallberg,
2010b: A computationally efficient 25‐term expression for the density of seawater in
terms of Conservative Temperature, and related properties of seawater. submitted to
Ocean Science Discussions. Computer software is available from http://www.TEOS‐10.org
McDougall, T. J., D. R. Jackett and F. J. Millero, 2010a: An algorithm for estimating Absolute
Salinity in the global ocean. submitted to Ocean Science, a preliminary version is available
at Ocean Sci. Discuss., 6, 215‐242.
TEOS‐10 web site
SCOR/IAPSO Working Group 127 has created the web site www.TEOS‐10.org which serves
many of the TEOS‐10 papers, this TEOS‐10 manual as well as the SIA (Seawater Ice Air)
and GSW (Gibbs SeaWater) libraries of oceanographic computer software. The GSW
MATLAB Oceanographic Toolbox contains many help files, including one called “Getting
started with the Gibbs SeaWater (GSW) Oceanographic Toolbox of TEOS‐10” which serves
as a succinct introduction to the use of TEOS‐10 in physical oceanography.
Note that several of the papers listed in this appendix are appearing in Ocean Science in the
special issue “Thermophysical Properties of Seawater”.
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and Use of the Thermodynamic Properties of Seawater
133
Appendix D: Fundamental constants
Following the recommendation of IAPWS (2005), the values of the fundamental constants
were taken from CODATA 2006 (Mohr et al. (2008)), as listed in Table D.1. Selected
properties of pure water were taken from IAPWS (1996, 1997, 2005, 2006) as listed in Table
D.2. The chemical Reference Composition of seawater from Millero et al. (2008a) is given
in Table D.3. Selected seawater constants derived from the Reference Composition are
0
listed in Table D.4. 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
Pa
molar gas constant
normal pressure
T0
273.15
exact
K
Celsius zero point
–1
Comment
–1
Table D.2. Selected properties of liquid water from IAPWS (1996, 1997, 2005, 2006)
and Feistel (2003).
Symbol
MW
Value
Uncertainty
Unit
18.015 268
0.000 002
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
ρ0
999.8431
0.001
kg m–3
density at T0 and P0 , ρ 0 = 1 / v 0
6.774 876 × 10–2
0.06 × 10–2
kg m–3 K–1
( ∂ρ
0
/ ∂T
)
P
Comment
molar mass
maximum density , temperature
( ∂ρ / ∂T ) P at T0 and P0
Tt
Pt
273.16
exact
K
triple point temperature
611.657
0.01
Pa
triple point pressure
ρt
999.793
0.01
kg m–3
triple point density
ηt
0
exact
J kg–1 K–1
triple point entropy
ut
0
exact
J kg–1
Tf0
273.152 519
0.000 002
K
triple point internal energy
freezing point at P0
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Table D.3. The sea salt composition definition for reference salinity of the standard ocean.
The standard ocean is at 25°C and 101325 Pa. X – mole fractions, Z – valences,
W – mass fractions (Millero et al. 2008a). Molar masses M from Wieser (2006)
with their uncertainties given in the brackets. The mass fractions are with
respect to the mass of solution rather than the mass of pure water in solution.
Solute j
Mj
Zj
–1
Na+
Mg2+
Ca2+
K+
Sr2+
+1
+2
+2
+1
+2
g mol
22.989 769 28(2)
24.305 0(6)
40.078(4)
39.098 3(1)
87.62(1)
Cl–
–1
–2
35.453(2)
96.062 6(50)
–1
–1
–2
61.016 84(96)
79.904(1)
60.008 9(10)
B(OH)4
F–
OH–
–1
78.840 4(70)
–1
–1
18.998 403 2(5)
17.007 33(7)
B(OH)3
CO2
0
0
61.833 0(70)
44.009 5(9)
2–
SO4
–
HCO3
Br–
CO32–
–
Sum
IOC Manuals and Guides No. 56
Xj × Zj
10–7
Xj
10–7
Wj
4188071
471678
91823
91159
810
4188071
943356
183646
91159
1620
0.3065958
0.0365055
0.0117186
0.0113495
0.0002260
4874839
–4874839
252152
–504304
0.5503396
0.0771319
15340
7520
–15340
–7520
2134
–4268
900
610
71
–900
–610
–71
0.0002259
2807
86
0
0
0.0005527
0.0000121
1 000 000 0
0
1.0
0.0029805
0.0019134
0.0004078
0.0000369
0.0000038
TEOS-10 Manual: Calculation and Use of the Thermodynamic Properties of Seawater
135
Table D.4. Selected properties of the KCl‐normalised reference seawater
(Millero et al. 2008a), and proposals of WG127 (2006).
Symbol
Value
Uncertainty
Unit
Comment
reference salinity molar mass
31.403 8218
MS
0.001
–1
MS = ∑ X j M j
g mol
j
reference salinity valence factor
a
1.245 2898
Z2
Z 2 = ∑ X j Z 2j
-
exact
j
a
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
ηSO
0
exact
J kg–1 K–1
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
Avogadro constant
reference salinity particle number
NS = N A / M S
standard ocean surface enthalpy
hSO = ut
standard ocean surface entropy
ηSO = ηt
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
Value
c 0p
3991.867 957 119 63
Uncertainty
Unit
exact
–1
J kg K
Comment
–1
See Eqn. (3.3.3)
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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
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
0.999 999 98
28.965 46(33)
Gas
1.000 000 0
Coriolis Parameter
The rotation rate of the earth Ω is (in radians per second)
Ω = 7.292 1150 x 10−5 s−1 ,
(D.1)
(Groten (2004)) and the Coriolis parameter f is (in radians per second)
f = 2 Ω sin φ = 1.458 423 00 x 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 x10
≈ 9.780 327 (1 + 5.2792 x10
)(
sin φ )(1 − 2.26 x10
sin φ )(1 + 2.22 x10
)
z (m) )
g (m s−2 ) = 9.780 327 1 + 5.3024 x10−3 sin 2 φ − 5.8 x10−6 sin 2 2φ 1 − 2.26 x10−7 z (m)
−3
sin φ + 2.32 x10
−3
2
−5
sin 2 φ + 2.32 x10−5
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 variation of
g with z and p in the ocean in Eqn. (D.3) is derived in McDougall et al. (2010b). 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
137
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 )
i2
where
i =0
K15 =
C ( S P , 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 < S P < 42.
i
0
1
2
3
4
5
ai
0.0080
- 0.1692
25.3851
14.0941
- 7.0261
2.7081
bi
0.0005
- 0.0056
- 0.0066
- 0.0375
0.0636
- 0.0144
ci
di
ei
6.766097 x 10-1
2.00564 x 10-2
1.104259 x 10-4
- 6.9698 x 10-7
1.0031 x 10-9
3.426 x 10-2
4.464 x 10-4
4.215 x 10-1
- 3.107 x 10-3
2.070 x 10-5
- 6.370 x10-10
3.989 x10-15
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 ( S P , t68 , p )
C ( 35, t68 = 15 °C,0 )
=
C ( S P , t68 , p ) C ( SP , t68 ,0 )
C ( 35, t68 ,0 )
C ( S P , 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
R p , Rt and rt as follows
R=
C ( S P , t68 , p )
C ( 35, t68 = 15 °C,0 )
= R p Rt rt .
(E.2.2)
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
The last factor rt has been fitted to experimental data as the polynomial in temperature
( t68 )
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
Rp = 1 +
∑ ei pi
i =1
1 + d1 ( t68 / °C ) + d 2 ( t68 / °C ) + R ⎡⎣ d 3 + d 4 ( t68 / °C ) ⎤⎦
2
.
(E.2.4)
Thus for any sample measurement of R it is possible to evaluate rt and R p and hence
calculate
R
Rt =
.
(E.2.5)
R p rt
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,
SP =
5
∑ ai ( Rt )
i 2
i =0
+
5
( t68 / °C − 15)
i 2
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 < S P < 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 from the extensions of the Practical
Salinity Scale proposed by Hill et al. (1986) for 0 < S P < 2 and by Poisson and Gadhoumi
(1993) for 42 < S P < 50. The values of Practical Salinity S P estimated in this manner may
then be used in Eqn. (2.4.1), namely S R ≈ uPS S P to estimate Reference Salinity S R .
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) and is also available in the
GSW Oceanographic Toolbox as the function gsw_cndr_from_SP. Note that this iterative
inverse procedure is done in terms of t68 ; the code accepts t90 as the input and
immediately converts this to a t68 temperature before performing the above iterative
procedure. The iteration is stopped when the Practical Salinity corresponding to the
output conductivity ratio differs from the input Practical Salinity by less than 10−10 .
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139
E.4 Evaluating Practical Salinity using ITS-90 temperatures
We first consider the consequence of the change from IPTS‐68 to ITS‐90 for the definition
of Practical Salinity as a function of 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 ( S P , 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 < S P < 42. Further calculation shows that
∂S P ∂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 practical use of bench salinometers.
It is important that samples and seawater standards should be run at the same
temperature, stable at order 1 mK. This is achieved by the use of a large water bath in the
instrument. However, it is not critical to know the stable bath temperature to any better
than 10 or 20 mK.
The ratios R p , 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
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140
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
and pressure, from which salinity, density and other properties are computed in turn by
standard algorithms. The traceability of temperature and pressure measurement results,
for example by CTD sensors, is ensured due to established calibration procedures carried
out by the manufacturer or other laboratories and will not be considered here any further.
The observation of the ocean’s salinity is a more complicated task (Millero et al.
(2008a)). Even though over the last century different and permanently improved methods
were developed and introduced in oceanography, traceability of salinity measurement
results to SI units has not yet been achieved (Seitz et al. 2008 and Seitz et al. (2010b)). This
implies the risk that readings taken today may possess an enlarged uncertainty when
being compared with observations taken a hundred years from now, a circumstance that
will reduce the accuracy of long‐term trend analyses performed in the future.
A quantity, quite generally, is a “property of a phenomenon, body or substance,
where the property has a magnitude that can be expressed as a number” (ISO/IEC, 2007).
The process to obtain this number is called measurement. The value of the indicated
number (the quantity value) is determined by a calibration of the measuring system with
a reference having a known quantity value of the same kind. In turn, the quantity value
of the reference is assigned in a superior measurement procedure, which is likewise
calibrated with a reference and so on. This calibration hierarchy ends in a primary
reference procedure used to assign a quantity value and a unit to a primary standard for
that kind of quantity. Thus, the unit of a measured quantity value expresses its link (its
metrological traceability) to the quantity value of the corresponding primary standard.
Obviously, quantity values measured at different times or locations, by different persons
with different devices or methods can be compared with each other only if they are
linked to the same reference standard, whose corresponding quantity value must be
reproducible with a high degree of reliability.
Concerning comparability of measured quantity values a second aspect is of
importance. The quantity value of a primary standard can only be realised with an
inevitable uncertainty. The same holds for every measurement and calibration. A
measurement result therefore always has to indicate the measured quantity value and its
uncertainty. Obviously, the latter increases with every calibration step down the
calibration hierarchy. Measured quantity values can evidently only be assumed
equivalent if their difference is smaller than their measurement uncertainty
(compatibility). On the other hand they can only be assumed reliably different, if the
difference is larger than the uncertainty.
To ensure comparability in practice, the International System of Units (SI) was
established. National Metrological Institutes (NMIs) have developed primary reference
procedures to realise the SI units in the form of primary standards. Extensive (ongoing)
efforts are made to link these units to fundamental and physical constants in order to
achieve the highest degree of reproducibility. Moreover, the NMIs periodically conduct
international comparison measurements under the umbrella of the International Bureau
of Weights and Measures, in order to ensure the compatibility of the quantity values of
national standards.
PSS‐78, and similarly the new Reference‐Composition Salinity Scale (Millero et al.
(2008a)), compute the salinity value from a measured conductivity ratio with respect to
the K15 conductivity ratio of IAPSO Standard Seawater (SSW, Culkin and Ridout (1998)
and Bacon et al. (2007)), which plays the role of a primary standard. The production
procedure if IAPSO Standard Seawater, and in particular the adjustment of its
conductivity to that of a potassium chloride (KCl) solution of definite purity and the
corresponding assignment of the 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
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
141
by arbitrary independent laboratories is neither known nor even granted. A slow drift of
artefact properties cannot rigorously be excluded, similar in principle to the
“evaporation” of mass from the kilogram prototype stored in Paris. It is impossible to
foresee effects that might affect the conductivity of SSW solution one day. Thus, with
respect to decadal or century time scales, there is an uncertainty of its 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 for the ocean observation system (Seitz et al.
(2008)).
A way out of this practical dilemma is the measurement of a different seawater
quantity that is traceable to SI standards and possesses the demanded small uncertainty,
and from which the salinity can be computed via an empirical relation that is very
precisely known (Seitz et al. (2010b)). Among the potential candidates for this purpose
are the sound speed, the refractive index, chemical analysis (e.g. by mass spectroscopy) of
the sea‐salt constituents, in particular chlorine, and direct density measurements. The
latter has three important advantages, i) SI‐traceable density measurements of seawater
can be carried out with a relative uncertainty of 1 ppm (Wolf (2008)), which perfectly
meets the needs of ocean observation, ii) a relation exists between density and the
Absolute Salinity of seawater is available with a relative uncertainty of 4 ppm in the form
of the TEOS‐10 Gibbs function, iii) the measurand, density, is of immediate relevance for
oceanography, in contrast to other options.
It is important to note that the actual measuring procedure for a quantity value is
irrelevant for its traceability. To measure the weight of a person, a mass balance can be
used, a spring or a magnetic coil; it is the quantity value that is traceable, not the method
to achieve this value. The method in use is not intrinsically important except in so far as
it is responsible for the uncertainty of the quantity value. Hence, we may measure the
density of seawater with a CTD conductivity sensor, provided this sensor is properly
calibrated with respect to an SI‐traceable density reference standard. In practice, this will
mean that the sensor calibration in oceanographic labs must be done with standard
seawater samples of certified density rather than certified Practical Salinity. The density
value returned from the CTD reading at sea is then converted into an Absolute Salinity
value by means of the equation of state of seawater, and eventually into a Practical
Salinity number for storage in data centres. The latter step may include some
modification regarding local sea salt composition anomalies. Storing a salinity value
rather than the related density reading has the advantage of conservativity with respect
to dilution or changes of temperature or pressure.
This conceptual proposal of WG127 is still immature and needs to be worked out in
more detail in the following years. Although it may imply only minor changes in the
practical use of a CTD or similar devices, the new concept is very promising regarding
the long‐term reliability of observations made in the near future for climatic trend
analyses to be performed by the coming generations. An immediate consequence of this
proposal is to have the density (at a given temperature and pressure) of several samples
of each batch of IAPSO Standard Seawater measured when they are produced and have
these densities made available as reference values for each batch.
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Appendix F:
Coefficients of the IAPWS‐95 Helmholtz function
of fluid water (with extension down to 50 K)
The specific Helmholtz energy for fluid (gaseous and liquid) water is given by the revised
IAPWS Release, IAPWS (2009b), which is based mainly on the work of Wagner and Pruß
(2002). This revised release is still referred to as IAPWS‐95. The specific Helmholtz
energy of IAPWS‐95 is defined by
f flu (T , ρ ) = f V,id (T , ρ ) + RWT ϕ res (τ , δ ) ,
(F.1)
where f
(T , ρ ) is the ideal‐gas part, (F.2), RW = 461.518 05 J kg–1 K–1 is the specific gas
constant of water used in IAPWS‐95, and ϕ res (τ , δ ) is the dimensionless residual part
consisting of 56 terms, available from (F.5) and Tables F.2 ‐ F.4. Note that the gas constant
used here differs from the most recent value, RW = R M W = 461.523 64 J kg–1 K–1, where
M W = 18.015 268 g mol–1 is the molar mass of water (IAPWS (2005)).
The ideal‐gas part, f V,id (T , ρ ) , of the specific Helmholtz energy for water vapour
is (from IAPWS (2009b), Wagner and Pruß (2002), Feistel et al. (2010a))
V,id
f V,id (T , ρ ) = RWT ⎡⎣ϕ 0 (τ , δ ) + ϕ ex (τ ) ⎤⎦ .
(F.2)
Note that the term ϕ ex (τ ) has been added by Feistel et al. (2010a) 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 ln τ + ∑ ni0 ln 1 − e −γ i τ .
i =4
0
(F.3)
The “reduced density” δ = ρ / ρ c and “reduced temperature” τ = Tc / T are specified by
ρ c = 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
n1 = −8.320 446 483 749 693 and n2 = 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 ⎟,
ε 2ε 2ε
2ε ⎠
⎝ 2τ 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.
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
143
deviation of 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), but it has not yet been endorsed by IAPWS.
The residual part of (F.1) has the form
ϕ res =
7
∑ niδ d τ t
i
i
+
i =1
+
54
∑ niδ
i =52
51
∑ niδ d τ t exp ( −δ c )
i
i
i
i =8
(
)
56
(F.5)
τ exp −αi (δ − ε i ) − βi (τ − γ i ) + ∑ ni Δ δψ
d i ti
2
2
bi
i =55
with the abbreviations
Δ = θ 2 + Bi δ − 1
2 ai
)
(
1
,
θ = 1 − τ + Ai δ − 1 β , and ψ = exp −Ci (δ − 1)2 − Di (τ − 1)2 . (F.6)
i
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 n1 and n2 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
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
ci
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
di
1
1
1
2
2
3
4
1
1
1
2
2
3
4
4
5
7
9
10
11
13
15
ti
–0.5
0.875
1
0.5
0.75
0.375
1
4
6
12
1
5
4
2
13
9
3
4
11
4
13
1
ni
0.012533547935523
7.8957634722828
–8.7803203303561
0.31802509345418
–0.26145533859358
–7.8199751687981× 10–3
8.8089493102134× 10–3
–0.66856572307965
0.20433810950965
–6.6212605039687× 10–5
–0.19232721156002
–0.25709043003438
0.16074868486251
–0.040092828925807
3.9343422603254× 10–7
–7.5941377088144× 10–6
5.6250979351888× 10–4
–1.5608652257135× 10–5
1.1537996422951× 10–9
3.6582165144204× 10–7
–1.3251180074668× 10–12
–6.2639586912454× 10–10
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
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
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
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
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
–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
52
53
54
di
3
3
3
ti
0
1
4
ni
–31.306260323435
31.546140237781
–2521.3154341695
αi
20
20
20
βi
150
150
250
γi
1.21
1.21
1.25
εi
1
1
1
Table F.4. Coefficients of the residual part (F.5).
i
55
56
ai
3.5
3.5
bi
0.85
0.95
Bi
0.2
0.2
ni
–0.14874640856724
0.31806110878444
Ci
28
32
Di
700
800
Ai
0.32
0.32
βi
0.3
0.3
Equation (F.1) is valid between 50 and 1273 K and for pressures up to 1000 MPa in the
stable single‐phase region of fluid water. Uncertainty estimates are available from
IAPWS (2009b) and Wagner and Pruß (2002).
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145
Appendix G: Coefficients of the pure liquid
water Gibbs function of IAPWS‐09
The pure liquid water part of the Gibbs function of Feistel (2003) has been approved by
IAPWS (IAPWS (2009c)) as an alternative thermodynamic description of pure water to
IAPWS‐95 in the oceanographic ranges of temperature and pressure. The pure water
specific Gibbs energy 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 g00 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
3
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
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
146
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, S A = 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 g 200
and g 210 were determined to exactly achieve Eqns. (2.6.7) and (2.6.8) when the pure water
Gibbs function was that of IAPWS‐95.
i
j
k
gijk
i
j
k
gijk
i
j
k
gijk
1
1
2
3
4
5
6
7
2
3
4
5
6
2
3
4
2
3
4
2
3
4
0
1
0
0
0
0
0
0
1
1
1
1
1
2
2
2
3
3
3
4
4
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5812.81456626732
851.226734946706
1416.27648484197
–2432.14662381794
2025.80115603697
–1091.66841042967
374.601237877840
–48.5891069025409
168.072408311545
–493.407510141682
543.835333000098
–196.028306689776
36.7571622995805
880.031352997204
–43.0664675978042
–68.5572509204491
–225.267649263401
–10.0227370861875
49.3667694856254
91.4260447751259
0.875600661808945
–17.1397577419788
2
4
2
2
3
4
5
2
3
4
2
3
2
3
2
3
2
3
4
2
3
2
5
5
6
0
0
0
0
1
1
1
2
2
3
3
4
4
0
0
0
1
1
2
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
–21.6603240875311
2.49697009569508
2.13016970847183
–3310.49154044839
199.459603073901
–54.7919133532887
36.0284195611086
729.116529735046
–175.292041186547
–22.6683558512829
–860.764303783977
383.058066002476
694.244814133268
–460.319931801257
–297.728741987187
234.565187611355
384.794152978599
–52.2940909281335
–4.08193978912261
–343.956902961561
83.1923927801819
337.409530269367
3
2
2
2
3
4
2
3
2
3
2
2
2
3
2
2
2
2
2
2
2
3
4
0
0
0
1
1
2
2
3
4
0
0
1
2
3
0
1
2
2
2
2
3
3
3
3
3
3
3
3
3
4
4
4
4
4
5
5
5
–54.1917262517112
–204.889641964903
74.7261411387560
–96.5324320107458
68.0444942726459
–30.1755111971161
124.687671116248
–29.4830643494290
–178.314556207638
25.6398487389914
113.561697840594
–36.4872919001588
15.8408172766824
–3.41251932441282
–31.6569643860730
44.2040358308000
–11.1282734326413
–2.62480156590992
7.04658803315449
–7.92001547211682
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
147
Appendix I: Coefficients of the Gibbs function
of ice Ih of IAPWS‐06
The Gibbs energy of ice Ih, the naturally abundant form of ice, having hexagonal crystals,
is a function of temperature (ITS‐90) and sea pressure, 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 ⎣
⎛ p⎞
g0 ( p ) = ∑ g0 k ⋅ ⎜ ⎟
k =0
⎝ Pt ⎠
4
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 = 0.061 1657 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
Pt
P0
611.654 771 007 894
Pa
101325
Pa
Tt
273.16
K
Normal pressure
Triple-point temperature
IOC Manuals and Guides No. 56
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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
g00
g01
g02
Real part
Imaginary part
– 0.632 020 233 335 886 × 106
J kg
– 0.189 369 929 326 131 × 10-7
0.339 746 123 271 053 × 10-14
g04
– 0.556 464 869 058 991 × 10-21
s0 (IAPWS-95)
J kg
0.655 022 213 658 955
g03
s0 (absolute)
J kg
J kg
0.189 13 × 103
– 0.332 733 756 492 168 × 104
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
2
r20
– 0.725 974 574 329 220 × 10
r21
– 0.557 107 698 030 123 × 10-4
0.234 801 409 215 913 × 10-10
–1
–1
–1
–1
–1
J kg
–1 –1
J kg K
–1 –1
J kg K
t1
r22
Unit
– 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
ʹabsoluteʹ 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 g00 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
149
Appendix J:
Coefficients of the Helmholtz function of
moist air of IAPWS‐10
The equation of state of humid air described here (Feistel et al. (2010a), IAPWS (2010) [an
IAPWS Guideline, in preparation]) is represented in terms of a Helmholtz function which
expresses the specific Helmholtz energy as a function of dry‐air mass fraction 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
R
MA
MA
MW
T0
8.314 51
J mol–1 K–1
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
150
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
)
( )
(J.4)
0
0
⎤ + n10
τ ⎤
ln ⎡ 2 / 3 + exp n13
⎦
⎣
⎦
and α res (τ , δ ) is the residual part,
α res (τ , δ ) =
10
∑ nkδ ik τ
jk
+
k =1
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
151
Table J.2. Dimensionless coefficients and exponents for the ideal‐gas part,
Eqn. (J.4), for dry air (Lemmon et al. (2000)). In TEOS‐10, the coefficients
n40 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
i
ni0
ni0
1
0.605 719 400 000 000 × 10–7
8
0.791 309 509 000 000
2
–4
9
0.212 236 768 000 000
–3
–0.197 938 904 000 000
–0.210 274 769 000 000 × 10
3
–0.158 860 716 000 000 × 10
10
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
152
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
193 K ≤ T ≤ 473 K and 10 nPa ≤ P ≤ 5 MPa.
(J.12)
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 P sat 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 − P sat / ⎡⎣ 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
153
Appendix K: Coefficients of 25‐term expressions
for the density of seawater in terms of Θ and 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, density needs to be expressed as a
computationally efficient expression in terms of either Conservative Temperature Θ or
potential temperature θ (referenced to pr = 0 dbar). McDougall et al. (2010b) have fitted
the TEOS‐10 values of density ρ to SA , Θ and p in a “funnel” of data points in
( SA , t , p ) space. The fitted expression is in the form of a rational function, being the ratio
of two polynomials of ( SA , Θ, p )
ρ 25
ρ 25
ρ = Pnum Pdenom .
(K.1)
The “funnel” of data points in ( SA , t , p ) space is described in more detail in McDougall et
al. (2010b); at the sea surface it covers the full range of temperature and salinity while for
pressure greater than 5500 dbar, the maximum temperature of the fitted data is 12°C and
the minimum Absolute Salinity is uPS 30 g kg −1 . The maximum pressure of the “funnel”
is 8000 dbar . Table K.1 contains the 25 coefficients of the expression (K.1) for density in
terms of ( SA , Θ, p ) . The coefficients c1 − c12 in this table have units of kg m −3 and the
coefficients c13 − c25 are dimensionless, and the normalizing values of SA , Θ and p are
taken to be 1 g kg −1 , 1 K and 1 dbar respectively.
The rms error of this 25‐term approximation to the TEOS‐10 density is less than
0.0015 kg m −3 over the “funnel”; this can be compared with the rms uncertainty of
0.004 kg m −3 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,
1 ∂ρ
αΘ = −
,
(K.2)
ρ ∂Θ SA , p
of the 25‐term equation of state is different from the same thermal expansion coefficient
evaluated from TEOS‐10 with an rms error in the “funnel” of less than 0.3 x10−6 K −1 ,
compared with the rms error of the thermal expansion coefficient of the laboratory data
to which the Feistel (2008) Gibbs function was fitted of 0.6 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.1) is more accurate than the thermal expansion
coefficient. Hence we may consider the 25‐term rational function expression for density,
Eqn. (K.1), to be equally as accurate as the full TEOS‐10 expressions for density, the
thermal expansion coefficient and the saline contraction coefficient for data that reside
inside the “oceanographic funnel”.
The sound speed evaluated from the 25‐term rational function Eqn. (K.1), has an rms
error over the “funnel” of almost 0.25 m s −1 which is approximately five times the rms
error of the underlying sound speed data that was incorporated into the Feistel (2008)
Gibbs function (see rows 7 to 9 of Table O.1 of appendix O). Hence, the 25‐term
expression for density is not a particularly accurate expression for the sound speed in
seawater, even in the “funnel”. But for dynamical oceanography where α Θ and β Θ are
the aspects of the equation of state that, together with spatial gradients of SA and Θ
drive ocean currents and affect the calculation of the buoyancy frequency, we may take
IOC Manuals and Guides No. 56
154
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
the 25‐term rational‐function expression for density, Eqn. (K.1), as essentially reflecting
the full accuracy of TEOS‐10.
ρ 25
Pnum
c1
ρ 25
Pdenom
Coefficients
9.998 438 029 070 821 4 x 102
c13
Coefficients
1.0
c2
Θ
7.118 809 067 894 091 0 x 100
c14
Θ
7.054 768 189 607 157 6 x 10-3
c3
Θ2
-1.945 992 251 337 968 7 x 10-2
c15
Θ2
-1.175 369 560 585 864 7 x 10-5
c4
Θ3
6.174 840 445 587 464 1 x 10-4
c16
Θ3
5.921 980 948 827 490 3 x 10-7
c5
SA
2.892 573 154 127 765 3 x 100
c17
Θ4
3.488 790 222 801 251 9 x 10-10
c6
SA Θ
2.147 149 549 326 832 4 x 10-3
c18
SA
2.077 771 608 561 845 8 x 10-3
c7
( SA )2
1.945 753 175 118 305 9 x 10-3
c19
SA Θ
-2.221 085 729 372 299 8 x 10-8
c8
p
1.193 068 181 853 174 8 x 10-2
c20
S A Θ3
-3.662 814 106 789 528 2 x 10-10
c9
p Θ2
2.696 914 801 183 075 8 x 10-7
c21
( SA )1.5
3.468 821 075 791 734 0 x 10-6
c10
p SA
5.935 568 592 503 565 3 x 10-6
c22
( SA )1.5 Θ2
8.019 054 152 807 065 5 x 10-10
c11
p2
-2.594 338 980 742 903 9 x 10-8
c23
p
6.831 462 955 412 332 4 x 10-6
c12
p 2 Θ2
-7.273 411 171 282 270 7 x 10-12
c24
p 2 Θ3
-8.529 479 483 448 544 6 x 10-17
c25
p3 Θ
-9.227 532 514 503 807 0 x 10-18
ρ 25
ρ 25
TABLE K.1 Coefficients of the polynomials Pnum ( SA , Θ, p ) and Pdenom
( SA , Θ, p ) that
define the 25-term rational-function Eqn. (K.1) for density.
The same procedure has been used by McDougall et al. (2010b) to fit a rational
function of the form of Eqn. (K.1) but where the polynomials in the numerator and
denominator are functions of ( SA , θ , p ) rather than of ( SA , Θ, p ) . This form of the 25‐
term rational function expression for density is of approximately the same accuracy as
that described above, and the 25 coefficients of this expression are given in Table K.2.
The coefficients c1 − c12 in this table have units of kg m −3 and the coefficients c13 − c25 are
dimensionless, and the normalizing values of S A , θ and p are taken to be 1 g kg −1 , 1 K
and 1 dbar respectively.
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
ρ 25
Pnum
c1
ρ 25
Pdenom
Coefficients
9.998 427 704 040 868 8 x 102
c13
155
Coefficients
1.0
c2
θ
7.353 990 725 780 200 0 x 100
c14
θ
7.288 277 317 994 539 7 x 10-3
c3
θ2
-5.272 502 484 658 053 7 x 10-2
c15
θ2
-4.427 042 357 570 579 5 x 10-5
c4
θ3
5.105 140 542 790 050 1 x 10-4
c16
θ3
4.821 816 757 416 573 2 x 10-7
c5
SA
2.837 207 495 416 299 4 x 100
c17
θ4
1.966 643 777 649 954 1 x 10-10
c6
S Aθ
-5.746 287 373 866 898 5 x 10-3
c18
SA
2.019 220 131 573 115 6 x 10-3
c7
( SA )2
2.016 582 840 401 100 5 x 10-3
c19
S Aθ
-7.838 666 741 074 767 1 x 10-6
c8
p
1.150 668 012 876 069 5 x 10-2
c20
SAθ 3
-2.749 397 117 121 584 4 x 10-10
c9
pθ 2
1.202 602 702 900 458 1 x 10-7
c21
( SA )1.5
4.661 419 029 016 429 3 x 10-6
c10
p SA
5.536 190 936 504 846 6 x 10-6
c22
( SA )1.5 θ 2
1.518 271 263 728 829 5 x 10-9
c11
p2
-2.756 315 640 465 192 8 x 10-8
c23
p
6.414 629 356 742 288 6 x 10-6
c12
p 2θ 2
-5.883 476 945 993 336 4 x 10-12
c24
p2 θ 3
-9.536 284 588 639 736 0 x 10-17
c25
p3 θ
-9.623 745 548 627 732 0 x 10-18
ρ 25
ρ 25
TABLE K.2 Coefficients of the polynomials Pnum ( SA ,θ , p ) and Pdenom
( SA ,θ , p ) that
define the 25-term rational-function Eqn. (K.1) for density.
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
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
Chlorinity
Cl
g kg–1
WG127 is recommending that Chlorinity be
defined in terms of a mass fraction as
0.328 523 4 times the ratio of the mass of pure
silver required to precipitate all dissolved chloride,
bromide and iodide in seawater to the mass of
seawater. Hence WG127 recommends that mass
fraction units are used for Chlorinity.
Standard Ocean
Reference Salinity
SSO
g kg–1
Freezing temperatures
t f , θ f,
ºC
35.165 04 g kg–1 being exactly 35 uPS ,
corresponding to the standard ocean Practical
Salinity of 35.
In situ, potential and conservative values, each as a
function of S A and p.
Θf
Comments
Absolute pressure
P
Pa
When absolute pressure is used it should always be
in Pa, not in Mpa nor in dbar.
Sea pressure. Sea
pressure is the pressure
argument to all software
in the GSW Toolbox.
p
dbar
Equal to P − P 0 and usually expressed in dbar not
Pa.
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157
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
One standard atmosphere
Isopycnal slope ratio
P0
r
Pa
1
The value of the sea pressure p to which potential
temperature and/or potential density are
referenced.
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
Θ
G Θ = r ⎡⎣ Rρ − 1⎤⎦ ⎡⎣ Rρ − r ⎤⎦ ; ∇σ Θ = G ∇n Θ
SP
1
Defined in the range 2 < S P < 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 < S P < 42 SR ≈ uPS SP .
-1
Absolute Salinity
SA = SAdens g kg
(This is the salinity
argument to all the
GSW Toolbox functions.)
Absolute Salinity
Anomaly
δ SA
g kg-1
“Preformed Absolute
Salinity”,
S*
g kg-1
“Solution Absolute
Salinity”, often shortened
to “Solution Salinity”
SAsoln
g kg-1
“Added-Mass Salinity”
SAadd
g kg-1
often shortened to
“Preformed Salinity”
r =
α Θ ( p) β Θ ( p)
α Θ ( p r ) β Θ ( pr )
SA = SR + δ SA ≈ uPS SP + δ SA
Absolute Salinity is the sum of SR on the Millero
et al. (2008a) Reference-Salinity Scale and the
Absolute Salinity Anomaly. The full symbol for
SA 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. An algorithm to evaluate δ SA is
available (McDougall et al. (2010a)). In terms of
the full nomenclature of Pawlowicz et al. (2010),
Wright et al. (2010b) and appendix A.4 herein, the
Absolute Salinity Anomaly 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 − S R 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.
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Temperature
Absolute Temperature
t
T
ºC
K
temperature derivatives
T
K
Celsius zero point
T0
K
Potential temperature
Conservative Temperature
θ
ºC
ºC
A constant “specific
heat”, for use with
Conservative Temperature
Combined standard
uncertainty
Enthalpy
Specific enthalpy
c 0p
J kg–1 K–1
uc
Varies
H
h
J
J kg–1
Θ
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.
T0 ≡ 273.15 K
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 .
h = u + ( p + P0 ) v .
Here p and P0 must be in Pa not dbar.
Specific potential
enthalpy
h0
Specific isobaric heat
capacity
cp
J kg–1 K–1
c p = ∂h ∂T
SA , p
Internal energy
Specific internal energy
Specific isochoric heat
capacity
U
u
cv
J
J kg–1
J kg–1 K–1
cv = ∂u ∂T
SA , v
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
uPS
g kg–1
Entropy
Specific entropy
Σ
Density
Density anomaly
Potential density anomaly
referenced to a sea
pressure of 1000 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
J kg–1
Specific enthalpy referenced to zero sea pressure,
h0 = h ( SA ,θ [ SA , t , p, pr = 0] , pr = 0 )
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)).
J K–1
η
J kg–1 K–1
ρ
kg m–3
kg m–3
σt
σ1
kg m–3
In many other publications the symbol s is used for
specific entropy.
–3
ρ ( SA , t ,0 ) – 1000 kg m
ρ ( SA , θ [ SA , t, p, pr ] , pr ) – 1000 kg m-3 where
pr = 1000 dbar
σ4
–3
kg m
ρ ( SA , θ [ SA , t, p, pr ] , pr ) – 1000 kg m-3 where
pr = 4000 dbar
αt
K–1
v −1∂v / ∂T
αθ
K–1
v −1∂v / ∂θ
IOC Manuals and Guides No. 56
SA , p
= − ρ −1∂ρ / ∂T
SA , p
SA , p
= − ρ −1∂ρ / ∂θ
SA , p
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Thermal expansion
coefficient with respect to
Conservative Temperature
Θ
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
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
κt
Pa–1
κ
Pa–1
μW
J g–1
μS
J g–1
μ
J g–1
ε
J kg–1 s–1
= m2 s–3
Adiabatic lapse rate
Γ
K Pa–1
Sound speed
Specific volume
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
= − ρ −1∂ρ / ∂Θ
SA , p
= ρ −1∂ρ / ∂SA
T,p
159
SA , p
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
= ρ −1∂ρ / ∂SA
θ,p
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.
( ∂g
∂S A ) t , p = μ S − μ W
Γ=
∂t
∂P
=
SA , θ
∂t
∂P
=
SA , Θ
Θ
∂t
∂P
Θ, p
are consistent with SA
SA ,η
–1
θ
Cb
CbΘ
N
Neutral helicity
H
Neutral Density
γ
Neutral-Surface-PotentialVorticity
K
−2
K
−2
s
n
−1
m
−3
v = ρ −1
(
)
∂P
(
)
∂P
Cbθ = ∂α θ ∂θ
CbΘ = ∂α Θ ∂Θ
(
SA , p
SA , p
SA ,θ
SA , Θ
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 ∂S A
⎛ Θ⎞
− ⎜ α Θ ⎟ ∂β Θ ∂ S A
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 .
NSPV
s −3
NSPV = − g ρ −1 f γ zn where f is the Coriolis
Dynamic height anomaly
Φ′
m2 s−2
Pa m3 kg −1 = m2 s −2
Montgomery geostrophic
streamfunction
π
m2 s−2
Pa m3 kg −1 = m2 s −2
n
parameter.
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
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
Molality
mSW
mol kg–1
mSW =
∑i mi =
1
SA
where M S is the
M S (1 − SA )
mole-weighted average atomic weight of the
elements of sea salt,
M S = 0.031 403 821 8… kg mol −1
Ionic strength
I
mol kg–1
I =
φ
1
mSW Z 2 =
1
2
∑i mi zi2
= 0.622 644 9 mSW
≈
Osmotic coefficient
1
2
622.644 9
SA
mol kg −1
31.403 821 8
(1 − SA )
φ ( SA , T , p ) =
g ( 0, t , p ) − μ W ( S A , t , p )
mSW R (T0 + t )
where the molar gas constant,
R = 8.314 472 J mol–1 K–1. See also Eqn. (3.40.9)
for an equivalent definition of φ.
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161
L.2 Suggested 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 μmol/kg and not cm3dm–3, ml/l or μmol/l
(this reflects a desire for consistency with reporting of other quantities and will avoid
problems associated with conversion between moles and ml using the gas equations).
When thermodynamic variables are taken to be functions of variables other than the
standard combination ( 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. Suggested Symbols when variables are functions of η , θ and Θ
quantity
enthalpy, h
function of
( SA , t, p )
symbol for this
functional form
h = h ( SA , t, p )
specific volume, v
v = v ( SA , t , p )
density, ρ
ρ = ρ ( SA , t , p )
entropy, η
η = η ( SA , t , p )
enthalpy, h
( SA ,η , p )
h = h ( SA ,η , p )
specific volume, v
v = v ( SA ,η , p )
density, ρ
ρ = ρ ( SA ,η , p )
potential temperature, θ
θ = θ ( SA ,η )
enthalpy, h
( SA ,θ , p )
h = h ( SA ,θ , p )
specific volume, v
v = v ( SA ,θ , p )
density, ρ
ρ = ρ ( SA ,θ , p )
entropy, η
η = η ( SA ,θ )
enthalpy, h
( SA , Θ, p )
h = hˆ ( SA , Θ, p )
specific volume, v
v = vˆ ( SA , Θ, p )
density, ρ
ρ = ρˆ ( SA , Θ, p )
entropy, η
η = ηˆ ( SA , Θ )
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
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 ( S P , 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 S P = S P ( S A , φ , λ , P ) . This and the previous routine are found at
level 0 since Absolute Salinity is required as an input to many of the higher level
library routines.
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•
•
163
General purpose routines that allow for conversions between a broad range of
pressure, temperature and salinity units that are in common usage are provided at
level 5. The numerical input value and its unit are provided by the user and results
are returned in a specified output unit.
Algorithms are included at level 5 that use non‐basic SI units as input and as output.
Most noteworthy is the GSW library module that uses the SIA routines to mimic
many of the routines in the GSW Oceanographic Toolbox. These routines use
IAPWS‐09 for pure water in place of IAPWS‐95 to provide improved computational
efficiency.
They have been used to provide independent checks on the
corresponding routines in the GSW Oceanographic Toolbox.
Because the IAPWS‐95 description of pure water substance (both liquid and vapour)
is the world‐wide standard for pure water substance, the SIA library is the official
description of seawater, although it should be noted that the computer software does not
come with any warranty. Rather it is the underlying papers as listed in appendix C that
are the officially warranted descriptions.
The SIA library benefits from the full range of applicability of the IAPWS‐95
Helmholtz function for pure water, 0 kg m‐3 < 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. (2010a) for details on the ice and air components of the SIA library.
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
However, below, we discuss a few features of the library that relate to these additional
components. First, we note that the thermodynamic potentials of pure water, ice, the
saline part of the seawater Gibbs function and the Gibbs function of moist air have been
carefully adjusted to make them fully compatible with each other (Feistel et al. (2008a)).
Only by so doing can the equilibrium properties of coincident phases be accurately
evaluated (for example, the freezing temperature of pure water and of seawater). Many
functions involving equilibrium properties of water, vapour, ice, seawater and dry or
humid air are implemented in level 4 of the SIA library. To provide an indication of the
range of functions available, we have listed the routine names in Table M.1 below. This
table comes from Table 3.1 of Wright et al. (2010a); we refer the interested reader to Feistel
et al. (2010b)) and Wright et al. (2010a)) for detailed information. Wright et al. (2010a)
provide a supplementary table that is cross‐referenced to their Table 3.1 to give details on
the usage of each routine and each table in their supplement references in turn the
relevant sections of Feistel et al. (2010b) for additional background information.
Due to the fact that each level of the SIA library builds on lower levels and the fact
that multiple phases may be involved in the equilibrium calculations, the determination
of the ranges of validity of the routines in the SIA library can become rather involved. To
deal with this issue, a procedure has been implemented in the library to return an error
code for function evaluations that depend on results from any of the basic building block
routines from outside of their individual ranges of validity. Numerical check values are
provided with each of the routines in the library and auxiliary routines are provided to
assist users in the validation of local implementations.
Further details of the SIA software library are provided in the papers Feistel et al.
(2010b)) and Wright et al. (2010a)) and the software is served from the www.TEOS‐10.org
web site.
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
165
Table M.1. The SIA library contents. Module names are in bold and user‐accessible
routines are in plain type. Each of the Public Routines can be accessed by
users. The underlined routines are thermodynamic potential functions
including first and second derivatives. The bracketed numbers preceding
most module names give the related table numbers in the supplement to
Wright et al. (2010a) where detailed information on the use of each function is
provided.
Level 0 routines
Constants_0
Constants_0 (Cont'd)
Maths_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
(S3) Flu_1 (IAPWS95)
Uses
constants_0
(S2) Convert_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
(S4) Ice_1 (IAPWS06)
(S5) Sal_1 (IAPWS08)
(S6) Air_1
Uses
Uses
Uses
constants_0
constants_0
constants_0
Level 1 routines
Public Routines
Public Routines
Public Routines
Public Routines
chk_iapws95_table6
chk_iapws95_table7
flu_f_si
chk_iapws06_table6
ice_g_si
sal_g_term_si
air_baw_m3mol
air_caaw_m6mol2
air_caww_m6mol2
dry_f_si
dry_init_clear
dry_init_Lemmon2000
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Level 2 routines
(S7) Flu_2
((S8) Ice_2
(S9) Sal_2
Uses
Uses
Uses
Uses
constants_0, flu_1
constants_0, ice_1
constants_0, sal_1
constants_0, flu_1, air_1
(S10) Air_2
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
(S12) Sea_3a
(S13) Air_3a
Level 3 routines
(S11) Flu_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
Public Routines
get_it_ctrl_density
liq_density_si
liq_g_si
set_it_ctrl_density
vap_density_si
vap_g_si
IOC Manuals and Guides No. 56
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
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
(S14) Flu_3b
(S15) Sea_3b
167
(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
Public Routines
air_h_si
air_potdensity_si
air_potenthalpy_si
air_pottemp_si
air_temperature_si
set_it_ctrl_air_pottemp
(S19) Sea_3d
Uses
constants_0, sal_2, flu_3a
(convert_0, maths_0, flu_1,
sal_1)
Public Routines
sea_sa_si
set_it_ctrl_salinity
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Level 4 routines
(S20) Liq_Vap_4
(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
constants_0, maths_0, flu_1,
sal_1, sal_2, flu_3a, sea_3a,
flu_3b (convert_0, flu_2)
Public Routines
Public Routines
chk_iapws95_table8
liq_vap_boilingtemperature_si
liq_vap_chempot_si
liq_vap_density_liq_si
liq_vap_density_vap_si
liq_vap_enthalpy_evap_si
liq_vap_enthalpy_liq_si
liq_vap_enthalpy_vap_si
liq_vap_entropy_evap_si
liq_vap_entropy_liq_si
liq_vap_entropy_vap_si
liq_vap_pressure_liq_si
liq_vap_pressure_vap_si
liq_vap_temperature_si
liq_vap_vapourpressure_si
liq_vap_volume_evap_si
set_liq_vap_eq_at_p
set_liq_vap_eq_at_t
set_it_ctrl_liq_vap
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
(S23) Ice_Liq_4
Public Routines
sea_vap_boilingtemperature_si
sea_vap_brinefraction_seavap_si
sea_vap_brinesalinity_si
sea_vap_cp_seavap_si
sea_vap_density_sea_si
sea_vap_density_seavap_si
sea_vap_density_vap_si
sea_vap_enthalpy_evap_si
sea_vap_enthalpy_sea_si
sea_vap_enthalpy_seavap_si
sea_vap_enthalpy_vap_si
sea_vap_entropy_sea_si
sea_vap_entropy_seavap_si
sea_vap_entropy_vap_si
sea_vap_expansion_seavap_si
sea_vap_g_si
sea_vap_kappa_t_seavap_si
sea_vap_pressure_si
sea_vap_salinity_si
sea_vap_temperature_si
sea_vap_vapourpressure_si
sea_vap_volume_evap_si
set_it_ctrl_sea_vap
set_sea_vap_eq_at_s_p
set_sea_vap_eq_at_s_t
set_sea_vap_eq_at_t_p
(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
IOC Manuals and Guides No. 56
Public Routines
sea_liq_osmoticpressure_si
set_sea_liq_eq_at_s_t_p
set_it_ctrl_sea_liq
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(S25) Sea_Ice_4
Uses
constants_0, convert_0,
maths_0, flu_1, ice_1, sal_1,
ice_2, sal_2, flu_3a, sea_3a,
flu_3b (flu_2)
Public Routines
sea_ice_brinefraction_seaice_si
sea_ice_brinesalinity_si
sea_ice_cp_seaice_si
sea_ice_density_ice_si
sea_ice_density_sea_si
sea_ice_density_seaice_si
sea_ice_dtfdp_si
sea_ice_dtfds_si
sea_ice_enthalpy_ice_si
sea_ice_enthalpy_melt_si
sea_ice_enthalpy_sea_si
sea_ice_enthalpy_seaice_si
sea_ice_entropy_ice_si
sea_ice_entropy_sea_si
sea_ice_entropy_seaice_si
sea_ice_expansion_seaice_si
sea_ice_freezingtemperature_si
sea_ice_g_si
sea_ice_kappa_t_seaice_si
sea_ice_meltingpressure_si
sea_ice_pressure_si
sea_ice_salinity_si
sea_ice_temperature_si
sea_ice_volume_melt_si
set_it_ctrl_sea_ice
set_sea_ice_eq_at_s_p
set_sea_ice_eq_at_s_t
set_sea_ice_eq_at_t_p
(S26) Sea_Air_4
Uses
constants_0, convert_0,
maths_0, flu_1, sal_1, air_1,
flu_2, sal_2, air_2, flu_3a,
sea_3a, air_3a, air_3b,
liq_vap_4, liq_air_4a
Public Routines
sea_air_chempot_evap_si
sea_air_condense_temp_si
sea_air_density_air_si
sea_air_density_vap_si
sea_air_enthalpy_evap_si
sea_air_entropy_air_si
sea_air_massfraction_air_si
sea_air_vapourpressure_si
set_it_ctrl_sea_air
set_sea_air_eq_at_s_a_p
set_sea_air_eq_at_s_t_p
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(S28) Sea_Ice_Vap_4
(S27) Liq_Ice_Air_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
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(S31) Liq_Air_4b
171
(S32) Ice_Air_4b
Uses
Uses
constants_0, flu_3a, air_3a,
liq_air_4a (convert_0,
maths_0, flu_1, air_1, flu_2,
air_2, air_3b, liq_vap_4)
constants_0, convert_0, ice_1,
air_3a, ice_air_4a (maths_0,
flu_1, air_1, flu_2, ice_2, air_2,
air_3b, ice_vap_4)
Public Routines
Public Routines
liq_air_g_si
liq_air_g_cp_si
liq_air_g_density_si
liq_air_g_enthalpy_si
liq_air_g_entropy_si
liq_air_g_expansion_si
liq_air_g_kappa_t_si
liq_air_g_lapserate_si
liq_air_liquidfraction_si
liq_air_vapourfraction_si
ice_air_g_si
ice_air_g_cp_si
ice_air_g_density_si
ice_air_g_enthalpy_si
ice_air_g_entropy_si
ice_air_g_expansion_si
ice_air_g_kappa_t_si
ice_air_g_lapserate_si
ice_air_solidfraction_si
ice_air_vapourfraction_si
(S33) Liq_Air_4c
(S34) Ice_Air_4c
Uses
Uses
constants_0, air_3a, ice_liq_4,
liq_air_4a, liq_air_4b
(convert_0, maths_0, flu_1,
ice_1, air_1, flu_2, ice_2 air_2,
flu_3a, air_3b, liq_vap_4)
constants_0, convert_0,
ice_liq_4, ice_air_4b (maths_0,
flu_1, ice_1, air_1, flu_2, ice_2,
air_2, air_3a, air_3b,
ice_air_4a, ice_vap_4)
Public Routines
Public Routines
liq_air_h_si
liq_air_h_cp_si
liq_air_h_density_si
liq_air_h_kappa_s_si
liq_air_h_lapserate_si
liq_air_h_temperature_si
liq_air_potdensity_si
liq_air_potenthalpy_si
liq_air_pottemp_si
set_it_ctrl_liq_air_pottemp
ice_air_h_si
ice_air_h_cp_si
ice_air_h_density_si
ice_air_h_kappa_s_si
ice_air_h_lapserate_si
ice_air_h_temperature_si
ice_air_potdensity_si
ice_air_potenthalpy_si
ice_air_pottemp_si
set_it_ctrl_ice_air_pottemp
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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
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
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
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
(S41) GSW_Library_5
Public Routines
air_lapserate_moist
_c100m
(S42) Convert_5
(S39) Liq_F03_5
(S40) OS2008_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
IOC Manuals and Guides No. 56
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
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
173
Appendix N:
Gibbs‐SeaWater (GSW) Oceanographic Toolbox
This Gibbs‐SeaWater (GSW) Oceanographic Toolbox (the “GSW Toolbox” for short),
contains the TEOS‐10 subroutines for evaluating the thermodynamic properties of pure
water (using IAPWS‐09) and seawater (using IAPWS‐08 for the saline part). The GSW
Oceanographic Toolbox does not provide properties of ice or of moist air (these
properties can be found in the SIA library). This GSW Oceanographic Toolbox does not
adhere to strict basic‐SI units but rather oceanographic units are adopted. While it is
comfortable for oceanographers to adopt these familiar non‐basic SI units, doing so
comes at a price, since many of the thermodynamic expressions demand that variables be
expressed in basic‐SI units. The simplest example is the pure water fraction (the so‐called
“freshwater fraction”) which is (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 oceanographic MATLAB routines which has been widely used by oceanographers in the
past fifteen years; see http://www.cmar.csiro.au/datacentre/ext_docs/seawater.htm Many
of the non‐thermodynamic subroutines of the Seawater library have been retained or
updated in the GSW Toolbox (for example, a function to calculate the square of the
buoyancy frequency, and functions to calculate a selection of different geostrophic
streamfunctions). The paper by McDougall et al. (2010b) describes how many of these
functions are calculated.
The thermodynamic variables density and enthalpy, and several thermodynamic
variables derived from density and enthalpy, are available in the GSW Toolbox in two
forms. One form uses the full TEOS‐10 Gibbs function (being the sum of IAPWS‐09 and
IAPWS‐08) while the other form is based on a 25‐term computationally efficient
expression for density as a function of Absolute Salinity, Conservative Temperature and
pressure (see appendix A.30 and appendix K). Both forms give values of density, the
thermal expansion coefficient etc. within the accuracy of laboratory‐determined values
for these quantities, so that for oceanographic purposes the two forms can be regarded as
equally accurate. Certainly, the present uncertainty in accounting for the spatial
variations in seawater composition has a larger impact on density etc. than the small
difference incurred by using the computationally efficient 25‐term version for density.
Version 1 of the GSW Toolbox was released in January 2009 and version 2.0 in
October 2010. This version 2.0 of the GSW Toolbox is available to date only in MATLAB
(version 1 is also available in FORTRAN) from the web site at www.TEOS‐10.org. A quick
introduction to TEOS‐10 is available on the TEOS‐10 web site (www.TEOS‐10.org) as the
short document called “Getting started with the Gibbs SeaWater (GSW) Oceanographic
Toolbox of TEOS‐10”. Some of the function names in the GSW Oceanographic Toolbox
are given in the following table.
IOC Manuals and Guides No. 56
174
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Table N.1. A selection of functions in version 2.0 of the GSW Oceanographic Toolbox.
Thermodynamic
Property
Function name
(in MATLAB)
Remarks
Absolute Salinity S A
gsw_SA_from_SP
the McDougall, Jackett and Millero (2010a)
algorithm for S A using a look‐up table for
Preformed Salinity S*
gsw_Sstar_from_SP
Preformed Salinity S* from Practical Salinity S P
Conservative
Temperature Θ
Gibbs function g and its
1st and 2nd derivatives
specific volume v
density
ρ
gsw_gibbs
gsw_specvol
gsw_rho
Θ
potential density
ρ
density ρ , and
potential density
ρΘ
specific entropy
gsw_CT_from_t
gsw_pot_rho
gsw_rho_CT25
η
gsw_entropy
δ SA
Conservative Temperature Θ , from ( S A , 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
ρ Θ ( SA , t, p, pr ) using gsw_gibbs
ρˆ ( SA , Θ, p ) , in situ density using the 25‐term
expression for density in terms of Conservative
Temperature (CT for short), Θ . Potential density
with respect to pressure pr , using this 25‐term
expression, is obtained by simply calling
gsw_rho_CT25 with this pressure, obtaining
ρ Θ = ρˆ ( SA , Θ, pr ) .
η ( SA , t , p ) using gsw_gibbs
h ( SA , t , p ) using gsw_gibbs
specific enthalpy h
gsw_enthalpy
first order derivatives of
enthalpy with respect to
Conservative Temperature
gsw_enthalpy_first_derivati
ĥΘ and hˆSA using gsw_gibbs (see Eqns. (A.11.15)
ves
and (A.11.18))
second order derivatives of
enthalpy with respect to
Conservative Temperature
gsw_enthalpy_second_deri
vatives
hˆΘΘ , hˆΘSA and hˆSA SA using gsw_gibbs
sound speed c
gsw_sound_speed
c ( SA , t , p ) using gsw_gibbs
Conservative
Temperature Θ
gsw_CT_from_pt
Θ ( S A , 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 ( S A , Θ ) , found by Newton_Raphson iteration,
being the inverse function of gsw_CT_from_pt
gsw_alpha_wrt_CT
α Θ ( SA , t , p ) using gsw_gibbs
gsw_beta_const_CT
β Θ ( SA , t , p ) using gsw_gibbs
gsw_rho_alpha_beta_CT25
ρˆ ( SA , Θ, p ) , αˆ Θ ( SA , Θ, p ) and βˆ Θ ( SA , Θ, p )
thermal expansion
coefficient with respect to
Conservative
Θ
Temperature α
saline contraction
coefficient at constant
Conservative
Temperature Θ
density, thermal
expansion and saline
contraction coefficient
using the 25‐term expression for density in terms
of Conservative Temperature Θ
dynamic height anomaly
gsw_geo_strf_dyn_height
geostrophic streamfunction in an isobaric surface
McDougall‐Klocker
geostrophic streamfunction
gsw_geo_strf_McD_Klocker
geostrophic streamfunction in an approximately
neutral surface, see Eqn. (3.30.1)
Montgomery geostrophic
streamfunction
gsw_geo_strf_Montgomery
geostrophic streamfunction in a specific volume
anomaly surface, see Eqn. (3.28.1)
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
175
Appendix O:
Checking the Gibbs function of seawater
against the original thermodynamic data
One of the tasks undertaken by SCOR/IAPSO Working Group 127 was to verify the
accuracy of the Feistel (2003) and Feistel (2008) Gibbs functions against the underlying
laboratory data to which these Gibbs functions were fitted. This checking was performed
by Giles M. Marion, and is reported here.
Verification of the Feistel (2003) Gibbs function
Table 9 of Feistel (2003) included a root mean square (r.m.s.) estimate of the fit of the
Gibbs function to the original experimental data. In Table O.1 here, this estimate is the
column labeled “Resulting r.m.s.”. All the data in Table O.1 are from Feistel (2003) except
for the last column, where Giles M. Marion has estimated an independent “Verifying
r.m.s.”.
The seawater properties that were used to develop the Feistel (2003) Gibbs function
(see Column 1 of Table O.1) included density 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:
⎡1
2⎤
r.m.s = ⎢ ∑ ( F03 - expt.datum ) ⎥
n
⎣ n
⎦
0.5
(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
176
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
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
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
177
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
0.6x10-6 K-1 0.73x10-6 K-1 0.74x10-6 K-1
c
D74(I-III)
29-43
0-35
0-2
92
5 cm/s
1.7 cm/s
1.6 cm/s
c
D74(IVa-d) 29-43
0-30
0.1-5
32
5 cm/s
1.2 cm/s
1.2 cm/s
c
D74(V-VI) 33-37
0-5
0-100
128
5 cm/s
3.5 cm/s
3.5 cm/s
v
CM76
5-40
0-40
0-100
558
10 ppm
11.0 ppm
11.2 ppm
v
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
t
S
ii
( g kg )
ρ
ρ
i
SA
1.45 J/(kg K) 1.43 J/(kg 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
178
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Appendix P: Thermodynamic properties
based on g ( SA , t, p ) , h ( SA ,η, p ) , h ( SA ,θ , p ) and hˆ ( 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 (the results in this table for h ( S A ,η , p ) mostly come
from Feistel (2008) and Feistel et al. (2010b)). 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 =
ρ
ρ ρ PSA
hPPθ
h hPPSA
v
v vPSA
− Pθ
= Pθ − θ
= − Pθ + θ
,
ˆ
ρ
ρ SA ρ
v
v SA v
hPSA hP
hP
(P.1)
TbΘ =
ρˆ
ρˆ ρˆ PSA
hˆPPΘ
hˆ hˆPPSA
vˆ
vˆ vˆPSA
,
− PΘ
= PΘ − Θ
= − PΘ + Θ
ρˆ
ρˆ SA ρˆ
vˆ
vˆSA vˆ
hˆP
hˆPSA hˆP
(P.2)
h
h hPθ SA ⎛ hPθ
+⎜
Cb = Pθθ − 2 Pθ
⎜ hPS
hPSA hP
hˆP
⎝ A
θ
= −
CbΘ
ρθθ
ρ
+2 θ
ρ
ρ SA
ρθ SA ⎛ ρθ
−⎜
⎜ρ
ρ
⎝ SA
⎛ hˆ
hˆ
hˆ hˆPΘSA
= PΘΘ − 2 PΘ
+ ⎜ PΘ
⎜ hˆPS
hˆP
hˆPSA hˆP
⎝ A
= −
ρˆ ΘΘ
ρˆ
+2 Θ
ˆ
ρ
ρˆ SA
2
⎞ hPS S
⎛ v
v
v vθ SA
A A
= θθ − 2 θ
+⎜ θ
⎟
⎜ vS
⎟ hP
v
v SA v
⎝ A
⎠
2
⎞ v SA S A
⎟
⎟
v
⎠
2
⎞ ρ SA SA
,
⎟
⎟
ρ
⎠
2
⎞ hˆPS S
vˆ
vˆ vˆΘSA ⎛ vˆΘ
A A
⎟
= ΘΘ − 2 Θ
+⎜
⎜ vˆS
⎟ hˆP
vˆ
vˆSA vˆ
⎝ A
⎠
⎛ ρˆ
ρˆ ΘSA
− ⎜ Θ
⎜ ρˆ
ρˆ
⎝ SA
2
⎞ vˆSA SA
⎟
⎟
vˆ
⎠
2
⎞ ρˆ SA SA
.
⎟
⎟
ρˆ
⎠
Here follows some interesting expressions that can be gleaned from Table P.1.
IOC Manuals and Guides No. 56
(P.3)
(P.4)
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
c p ( 0 ) = hθ ( 0 ) = c 0p Θθ = −
ηˆΘ2
,
ηˆΘΘ
(T0 + θ ) =
(T0 + t )
(T0 + θ )
c0p
,
ηˆΘ
hˆΘ
∂h
= 0
c0p
∂h
=
179
,
(P.5)
SA , p
c 0p ηˆΘΘ
(T0 + θ ) ηˆΘΘ = c0p = α Θ ,
1
∂θ
∂θ
=
= θˆΘ =
= −
=
−
∂Θ SA , p ∂Θ SA
c p ( 0)
Θθ
ηˆΘ2
αθ
c0p
2
μ ( 0) = −
∂θ
∂SA
∂θ
=
∂SA
Θ, p
c 0p ηˆSA
(P.6)
= − (T0 + θ ) ηˆSA ,
ηˆΘ
−1
Θ SA
c 0p ηˆSA Θ
0 ∂ηˆΘ
ˆ
= θ SA = −
= −
=
c
p
∂SA
Θθ
ηˆΘ2
Θ
(P.7)
= −
(T0 + θ )2 ηˆSAΘ
c 0p
Θ
.
(P.8)
See also Eqn. (A.12.6) for an alternative expression for θˆSA . Equation (P.8) can also be
written as
∂ (T0 + θ )
−1
∂S A
=
Θ
ηˆSA Θ
c 0p
.
(P.9)
Now we consider how all the terms in the last column of Table P.1 may be evaluated
in terms of the expression hˆ 25 ( SA , Θ, p ) of Eqn. (A.30.6); this being the expression for
specific enthalpy that follows from the 25‐term expression for density as a function of
( SA , Θ, p ) as described in Eqn. (K.1) and Table K.1. The superscript “25” is added to
hˆ 25 ( SA , Θ, p ) to emphasize that this is an approximate expression for specific enthalpy.
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,
c 0p Θ = h ( S A , t = θ ,0 ) = g ( SA , t = θ ,0 ) − (T0 + θ ) gT ( SA , t = θ ,0 ) .
(P.10)
Next, we remind ourselves that we know the functional forms of η ( SA ,θ ) , Θ ( S A ,θ ) and
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 ( S A , θ ) = − gT ( S A , t = θ , 0 ) ,
μ ( S A , θ , 0 ) = g SA ( S A , t = θ , 0 ) ,
(P.11a,b)
and Eqn. (P.10) is repeated here emphasizing the functional form of the left‐hand side,
c 0p Θ ( SA , θ ) = g ( SA , t = θ ,0 ) − (T0 + θ ) gT ( S A , t = θ ,0 ) .
(P.12)
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
ΘSA ∂
∂
1 ∂
∂
∂
=
, and
.
(P.13a,b)
=
−
∂SA Θ, p
∂SA θ , p
∂Θ SA , p
Θθ ∂θ S , p
Θθ ∂θ SA , p
A
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))
c 0p
c 0p
μ ( SA , θ , 0)
η
1
, ηˆΘΘ = −
, ηˆSA = −
,
(P.14a,b,c)
ηˆΘ = θ ≡
2
Θθ
Θθ (T0 + θ )
(T0 + θ )
(T0 + θ )
ηˆSA Θ =
Θ SA
Θθ
c 0p
(T0 + θ )2
,
and
ηˆSA SA
(
Θ SA
μS ( SA ,θ , 0 )
= − A
−
Θθ
(T0 + θ )
)
2
c 0p
(T0 + θ )2
,
(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 hˆ 25 ( SA , Θ, p ) and the exact expressions (P.14)
and (P.15) which are written in terms of potential temperature θ which is found from the
exact implicit equation (P.10). This completes the discussion of how hˆ 25 ( 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 in the GSW Toolbox from the functions
gsw_entropy_first_derivatives and gsw_entropy_second_derivatives.
IOC Manuals and Guides No. 56
180
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
Table P.1. Expressions for various thermodynamic variables based on four different thermodynamic potentials
Expressions based on
g ( SA , t, p )
Expressions based on
h ( S A ,η , p )
Expressions based on
h ( SA ,θ , p ) and η ( SA ,θ )
Expressions based on
ˆ
h ( SA , Θ, p ) and ηˆ ( SA , Θ )
(T0 + t ) = hθ
(T0 + t ) = hˆΘ ηˆΘ
(T0 + θ ) = c0p ηˆΘ
t
t
(T0 + t ) = hη
θ
gT ( S A , θ , p r ) = gT ( S A , t , p )
(T0 + θ ) = hη ( 0)
[this is an implicit equation for θ ]
Θ
Θ =
g ( SA ,θ ,0 ) − (T0 + θ ) gT ( SA ,θ ,0 )
c 0p
θ;
ηθ
(T0 + θ ) = hθ ( 0) ηθ
Θ = h ( 0 ) c 0p
Θ = h ( 0 ) c 0p
Θ;
Θ = hˆ ( 0 ) c 0p
g
g = g ( SA , t, p )
g = h −η hη
g = h − η hθ ηθ
g = hˆ − ηˆ hˆΘ ηˆΘ
η
η = − gT
η
η = η ( SA ,θ )
η = ηˆ ( SA , Θ)
h
h = g − (T0 + t ) gT
h = h ( SA ,η , p )
h = h ( SA ,θ , p )
h = hˆ ( SA , Θ, p )
v
v = gP
v = hP
v = hP
v = hˆP
ρ
ρ = ( gP )
μ
μ = g SA
μ = hSA
μ = hSA − ηSA hθ ηθ
μ = hˆSA − ηˆSA hˆΘ ηˆΘ
u
u = g − (T0 + t ) gT − ( p + P0 ) g P
u = h − ( p + P0 ) hP
u = h − ( p + P0 ) hP
u = hˆ − ( p + P0 ) hˆP
f
f = g − ( p + P0 ) g P
f = h −η hη − ( p + P0 ) hP
f = h − η hθ ηθ − ( p + P0 ) hP
f = hˆ − ηˆ hˆΘ ηˆΘ − ( p + P0 ) hˆP
cp
c p = − (T0 + t ) gTT
c p = hη hηη
h0
h 0 = g ( SA ,θ ,0 ) − (T0 + θ ) gT ( SA ,θ ,0 )
ρθ
ρ θ = ⎡⎣ g P ( SA ,θ , pr ) ⎤⎦
κ
t
κ
( )
−1
ρ = hP
κ = − g P−1 g PP + g P−1
IOC Manuals and Guides No. 56
ρ θ = ⎡⎣ hP ( pr ) ⎤⎦
κ =
t
2
gTP
gTT
ρ = hP
− hP−1hPP
+
ρ = hˆP
(
)
h0 = h ( 0)
−1
2
−1 hη P
hP
κ = − hP−1 hPP
( )
−1
c p = hθηθ2 ηθ hθθ − hθηθθ
h0 = h ( 0)
−1
κ t = − g P−1 g PP
( )
−1
hηη
ρ θ = ⎡⎣ hP ( pr ) ⎤⎦
κ t = − hP−1 hPP − hP−1
(
c p = hˆΘηˆΘ2 ηˆΘhˆΘΘ − hˆΘηˆΘΘ
)
h 0 = hˆ ( 0 ) = c 0p Θ
−1
hθ2Pηθ
( hθηθθ − hθθηθ )
κ = − hP−1 hPP
−1
ρ θ = ρ Θ = ⎡⎣ hˆP ( pr ) ⎤⎦
κ t = − hˆP−1 hˆPP − hˆP−1
(
−1
hˆΘ2 PηˆΘ
hˆΘηˆΘΘ − hˆΘΘηˆΘ
κ = − hˆP−1 hˆPP
)
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
181
Table P.1. (cont’d) Expressions for various thermodynamic variables based on four different thermodynamic potentials
Expressions based on
g ( SA , t, p )
c
c = g P gTT
(g
2
TP
− gTT g PP
Γ
Γ = − gTP gTT
αt
αt =
Expressions based on
h ( S A ,η , p )
)
gTP
gP
αθ =
αΘ
αΘ = −
c 0p
gTP
g P (T0 + θ ) gTT
β t = − g P−1 g PSA
βt
−hPP
α θ = hP−1
hPη
hηη
hPη
hηη ( 0 )
α Θ = hP−1hPη
βθ
β θ = − g P−1 g PSA
+
βΘ
c 0p
hη ( 0 )
hSAη
hηη
β θ = − hP−1 hPSA
gTP ⎡⎣ g SAT − g SAT ( SA , θ , pr ) ⎤⎦
g P gTT
β Θ = − g P−1 g PSA
+ hP−1hPη
hSAη ( 0 )
gTP ⎡ g SAT − (T0 + θ ) g SA ( SA ,θ ,0 ) ⎤
⎣
⎦
+
g P gTT
+ hP−1hPη
c 0p
Θ ΘΘ
− hˆΘΘηˆΘ
)
ηˆΘ2
0
c pηˆΘΘ
α Θ = hˆP−1 hˆPΘ
hθ ( 0 )
( hθη θ − h θηθ )
( hθηθθ − hθθηθ )
SA
ηˆΘ2
( hˆ ηˆ
α θ = − hˆP−1hˆPΘ
α θ = hP−1hPθ
+ hP−1hPθ
α t = − hˆP−1hˆPΘ
SA
β t = − hˆP−1 hˆPSA
+ hˆP−1hˆPΘ
( hˆ ηˆ
( hˆ ηˆ
)
)
Θ SA Θ
− hˆSA ΘηˆΘ
Θ ΘΘ
− hˆΘΘηˆΘ
ηS Θ
β θ = − hˆP−1 hˆPSA + hˆP−1hˆPΘ A
ηˆΘΘ
ˆ
β θ = − hP−1 hPSA
hηη ( 0 )
β Θ = − hP−1 hPSA
−1
( hθηθθ − hθθηθ )
α Θ = hP−1hPθ
−hˆPP
Γ = hˆPΘ ηˆΘ
ηθ2
α t = − hP−1hPθ
Expressions based on
ˆ
h ( SA , Θ, p ) and ηˆ ( SA , Θ )
c = hˆP
−hPP
β t = − hP−1 hPSA
β t = − hP−1 hPSA
+ hP−1 hPη
c = hP
Γ = hPθ ηθ
Γ = hPη
α t = hP−1
gTP gTT ( SA ,θ , pr )
gP
gTT
αθ
c = hP
Expressions based on
h ( SA ,θ , p ) and η ( SA ,θ )
hSA ( 0 )
β Θ = − hP−1 hPSA + hP−1hPθ
hSA( 0 )
hθ ( 0 )
β Θ = − hˆP−1 hˆPSA
hη ( 0 )
IOC Manuals and Guides No. 56
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TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
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Index
absolute pressure, 9, 66, 156
Absolute Salinity S A = SAdens , 11‐15, 69‐77,
103‐110, 157
Absolute Salinity Anomaly, 13, 72‐77, 157
absolute temperature, 62
Added‐mass Salinity S Aadd , 12, 72‐74, 157
adiabatic lapse rate, 25
Gibbs function of ice Ih, 7, 147
Gibbs function of pure water, 15‐17, 78,
142‐144, 145
Gibbs function of seawater, 5, 15‐17, 78,
146
gravitational acceleration, 136
GSW Oceanographic Toolbox, 173‐174
Bernoulli function, 42
boiling temperature, 53
Boussinesq approximation, 109
buoyancy frequency, 32
haline contraction coefficients, 23, 32,
90‐91, 93, 112‐114, 159, 181
heat transport, 5, 27, 40‐41, 79‐84, 90‐92,
99‐102, 108‐112
heat diffusion, 5, 27, 40‐41, 79‐84, 90‐92,
99‐102, 108‐112
Helmholtz energy, 21
Helmholtz function of fluid water, 142
Helmholtz function of moist air, 149‐152
cabbeling coefficient, 31, 91, 159, 178
chemical potentials, 19, 159, 180
Chlorinity, 11, 68‐70, 156
composition variation, 3, 11‐14, 72‐77,
103‐111
“conservative” property, 79‐84
Conservative Temperature Θ , 3, 6, 18,
22, 27, 97‐102, 110‐112, 158, 178‐181
Coriolis parameter, 136, 160
Cunningham streamfunction, 44,
density, 18, 120, 152‐155
density, 25‐term expression, 120, 152‐155
Density Salinity S Adens = SA , 11‐15, 69‐77,
103‐110, 157
dianeutral advection, 112
dianeutral velocity, 112
dynamic height anomaly, 42
enthalpy, 18‐19, 20, 84‐86, 87, 100,
121‐122, 124, 158, 178‐181
enthalpy as thermodynamic potential, 18,
19, 121‐122, 178‐181
entropy, 20, 26, 79, 84, 88, 94, 158, 178‐181
EOS‐80, vii, 2, 77
First Law of Thermodynamics, 87‐90,
123‐129
freshwater content, 40
freshwater flux, 40
freezing temperature, 46
fundamental thermodynamic relation, 79
fugacity, 55
geostrophic streamfunctions, 43‐45
IAPSO, 3
IAPWS, 4,
IAPWS‐95, 130, 142‐144
IAPWS‐06, 130, 147‐148
IAPWS‐08, 130, 146
IAPWS‐09, 130, 145
IAPWS‐10, 130, 149‐152
internal energy, 20, 79,
IPTS‐68 temperature, 3, 9, 62‐65, 137‐139
IOC, 7, 61
ionic strength, 160
isentropic and adiabatic
compressibility, 22, 32, 159, 180
isochoric heat capacity, 24, 158
isobaric heat capacity, 24, 25, 84‐85, 86,
88, 95, 158, 179, 180
isopycnal‐potential‐vorticity, 39
isothermal compressibility, 21, 24
“isobaric‐conservative” property, 79‐84
ITS‐90 temperature, 3, 9, 62‐65, 137‐139,
146
latent heat of evaporation, 53
latent heat of melting, 48
material derivatives, 108‐114
molality, 21, 57, 160
Montgomery streamfunction, 43‐44, 159
Neutral Density, 35, 119
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neutral helicity, 30, 34‐35, 37, 119, 159
Neutral‐Surface‐Potential‐Vorticity, 38,
39, 159
neutral tangent plane, 29, 30, 31, 32‐33,
34‐39, 43, 90, 104, 109, 112‐118
nomenclature, 156‐161
orthobaric density, 118
osmotic coefficient, 21, 160
osmotic pressure, 58
Preformed Salinity S* , 12, 72‐74, 103‐109
potential density, 28
potential enthalpy h 0 , 27, 158
“potential” property, 82‐84
potential temperature θ , 26, 86, 158,
178‐181
potential vorticity, 34, 38‐39, 159
Practical Salinity S P , 9, 68, 137‐141
pressure, 9, 66, 156, 157
absolute pressure, 9, 66, 156
gauge pressure, 9, 66, 157
sea pressure, 9, 66, 156
Pressure‐Integrated Steric Height, 45
recommended nomenclature, 156‐161
recommended symbols, 156‐161
Reference Composition (RC), 10‐11, 67‐74
Reference‐Composition Salinity S R , 10,
11, 67‐77, 103‐110
Reference‐Composition Salinity Scale, 10,
11, 67‐77
Reference Density, 13
Reference Salinity S R , 10‐11, 67‐77,
103‐110
relative humidity, 55
saline contraction coefficients, 23, 32,
90‐91, 93, 112‐114, 159, 181
salinity
Absolute Salinity S A = SAdens , 11‐15,
69‐77, 103‐110, 157
Added‐mass Salinity S Aadd , 12, 72‐74,
157
Density Salinity S Adens = SA , 11‐15,
69‐77, 103‐110, 157
Practical Salinity S P , 9, 69‐77, 103‐110
Preformed Salinity S* , 12, 72‐74,
103‐109
Reference Salinity S R , 10‐11, 67‐77,
103‐110
Solution Salinity S Asoln , 12, 72‐74, 157
IOC Manuals and Guides No. 56
salinity in ocean models, 103‐110
SCOR, 3
sea pressure, 9, 66, 156
SIA software library, 162‐172
SI‐traceability of salinity, 139‐141
slopes of surfaces, 37‐38
Solution Salinity S Asoln , 12, 72‐74, 157
sound speed, 22, 159, 181
specific volume, 18, 29, 41, 42, 44, 46, 85,
159, 161, 180
specific volume anomaly, 29, 42, 43, 45
stability ratio, 36, 37, 39, 91, 113, 157
Standard Seawater (SSW), 10‐15, 67‐74
sublimation enthalpy, 50
sublimation pressure, 49
25‐term expression for density, 120‐122,
153‐155
TEOS‐10, 4‐8, 60‐61
TEOS‐10 web site, 60, 132
Temperature
absolute temperature T , 62
Celsius temperature t , 62
Conservative Temperature Θ , 3, 6,
18, 22, 27, 97‐102, 110‐112, 158,
178‐181
in situ temperature, 26
potential temperature θ , 26, 86, 158,
178‐181
temperature of maximum density, 59
thermal expansion coefficients, 22, 25, 32,
90‐91, 92, 112‐114, 158‐159, 181
“thermal wind”, 33, 76‐77, 120‐121
thermodynamic potentials, 3, 121‐122,
178‐181
thermobaric coefficient, 30, 34, 90, 159, 178
total energy, 41, 80‐83, 126‐127
Turner angle, 36
vapour pressure, 52
water‐mass transformation, 113‐114
WG127, 3
TEOS-10 Manual: Calculation and Use of the Thermodynamic Properties of Seawater
193
IOC Manuals and Guides
No.
Title
1 rev. 2
Guide to IGOSS Data Archives and Exchange (BATHY and TESAC). 1993. 27
pp. (English, French, Spanish, Russian)
2
International Catalogue of Ocean Data Station. 1976. (Out of stock)
3 rev. 3
Guide to Operational Procedures for the Collection and Exchange of JCOMM
Oceanographic Data. Third Revised Edition, 1999. 38 pp. (English, French,
Spanish, Russian)
4
Guide to Oceanographic and Marine Meteorological Instruments and
Observing Practices. 1975. 54 pp. (English)
5 rev. 2
Guide for Establishing a National Oceanographic Data Centre. Second Revised
Edition, 2008. 27 pp. (English) (Electronic only)
6 rev.
Wave Reporting Procedures for Tide Observers in the Tsunami Warning
System. 1968. 30 pp. (English)
7
Guide to Operational Procedures for the IGOSS Pilot Project on Marine
Pollution (Petroleum) Monitoring. 1976. 50 pp. (French, Spanish)
8
(Superseded by IOC Manuals and Guides No. 16)
9 rev.
Manual on International Oceanographic Data Exchange. (Fifth Edition). 1991. 82
pp. (French, Spanish, Russian)
9 Annex I
(Superseded by IOC Manuals and Guides No. 17)
9 Annex II
Guide for Responsible National Oceanographic Data Centres. 1982. 29 pp.
(English, French, Spanish, Russian)
10
(Superseded by IOC Manuals and Guides No. 16)
11
The Determination of Petroleum Hydrocarbons in Sediments. 1982. 38 pp.
(French, Spanish, Russian)
12
Chemical Methods for Use in Marine Environment Monitoring. 1983. 53 pp.
(English)
13
Manual for Monitoring Oil and Dissolved/Dispersed Petroleum Hydrocarbons
in Marine Waters and on Beaches. 1984. 35 pp. (English, French, Spanish,
Russian)
14
Manual on Sea‐Level Measurements and Interpretation. (English, French,
Spanish, Russian)
Vol. I: Basic Procedure. 1985. 83 pp. (English)
Vol. II: Emerging Technologies. 1994. 72 pp. (English)
Vol. III: Reappraisals and Recommendations as of the year 2000. 2002. 55 pp.
(English)
Vol. IV: An Update to 2006. 2006. 78 pp. (English)
IOC Manuals and Guides No. 56
194
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
No.
Title
15
Operational Procedures for Sampling the Sea‐Surface Microlayer. 1985. 15 pp.
(English)
16
Marine Environmental Data Information Referral Catalogue. Third Edition.
1993. 157 pp. (Composite English/French/Spanish/Russian)
17
GF3: A General Formatting System for Geo‐referenced Data
Vol. 1: Introductory Guide to the GF3 Formatting System. 1993. 35 pp. (English,
French, Spanish, Russian)
Vol. 2: Technical Description of the GF3 Format and Code Tables. 1987. 111 pp.
(English, French, Spanish, Russian)
Vol. 3: Standard Subsets of GF3. 1996. 67 pp. (English)
Vol. 4: User Guide to the GF3‐Proc Software. 1989. 23 pp. (English, French,
Spanish, Russian)
Vol. 5: Reference Manual for the GF3‐Proc Software. 1992. 67 pp. (English,
French, Spanish, Russian)
Vol. 6: Quick Reference Sheets for GF3 and GF3‐Proc. 1989. 22 pp. (English,
French, Spanish, Russian)
18
User Guide for the Exchange of Measured Wave Data. 1987. 81 pp. (English,
French, Spanish, Russian)
19
Guide to IGOSS Specialized Oceanographic Centres (SOCs). 1988. 17 pp.
(English, French, Spanish, Russian)
20
Guide to Drifting Data Buoys. 1988. 71 pp. (English, French, Spanish, Russian)
21
(Superseded by IOC Manuals and Guides No. 25)
22
GTSPP Real‐time Quality Control Manual. 1990. 122 pp. (English)
23
Marine Information Centre Development: An Introductory Manual. 1991. 32 pp.
(English, French, Spanish, Russian)
24
Guide to Satellite Remote Sensing of the Marine Environment. 1992. 178 pp.
(English)
25
Standard and Reference Materials for Marine Science. Revised Edition. 1993.
577 pp. (English)
26
Manual of Quality Control Procedures for Validation of Oceanographic Data.
1993. 436 pp. (English)
27
Chlorinated Biphenyls in Open Ocean Waters: Sampling, Extraction, Clean‐up
and Instrumental Determination. 1993. 36 pp. (English)
28
Nutrient Analysis in Tropical Marine Waters. 1993. 24 pp. (English)
29
Protocols for the Joint Global Ocean Flux Study (JGOFS) Core Measurements.
1994. 178 pp . (English)
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and Use of the Thermodynamic Properties of Seawater
No.
30
195
Title
MIM Publication Series:
Vol. 1: Report on Diagnostic Procedures and a Definition of Minimum
Requirements for Providing Information Services on a National and/or Regional
Level. 1994. 6 pp. (English)
Vol. 2: Information Networking: The Development of National or Regional
Scientific Information Exchange. 1994. 22 pp. (English)
Vol. 3: Standard Directory Record Structure for Organizations, Individuals and
their Research Interests. 1994. 33 pp. (English)
31
HAB Publication Series:
Vol. 1: Amnesic Shellfish Poisoning. 1995. 18 pp. (English)
32
Oceanographic Survey Techniques and Living Resources Assessment Methods.
1996. 34 pp. (English)
33
Manual on Harmful Marine Microalgae. 1995. (English) [superseded by a sale
publication in 2003, 92‐3‐103871‐0. UNESCO Publishing]
34
Environmental Design and Analysis in Marine Environmental Sampling. 1996. 86
pp. (English)
35
IUGG/IOC Time Project. Numerical Method of Tsunami Simulation with the Leap‐
Frog Scheme. 1997. 122 pp. (English)
36
Methodological Guide to Integrated Coastal Zone Management. 1997. 47 pp.
(French, English)
37
Post‐Tsunami Survey Field Guide. First Edition. 1998. 61 pp. (English, French,
Spanish, Russian)
38
Guidelines for Vulnerability Mapping of Coastal Zones in the Indian Ocean. 2000.
40 pp. (French, English)
39
Manual on Aquatic Cyanobacteria – A photo guide and a synopsis of their
toxicology. 2006. 106 pp. (English)
40
Guidelines for the Study of Shoreline Change in the Western Indian Ocean Region.
2000. 73 pp. (English)
41
Potentially Harmful Marine Microalgae of the Western Indian Ocean
Microalgues potentiellement nuisibles de lʹocéan Indien occidental. 2001. 104 pp.
(English/French)
42
Des outils et des hommes pour une gestion intégrée des zones côtières ‐ Guide
méthodologique, vol.II. Steps and Tools Towards Integrated Coastal Area
Management – Methodological Guide, Vol. II. 2001. 64 pp. (French, English;
Spanish)
43
Black Sea Data Management Guide (Cancelled)
44
Submarine Groundwater Discharge in Coastal Areas – Management implications,
measurements and effects. 2004. 35 pp. (English)
IOC Manuals and Guides No. 56
196
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
No.
Title
45
A Reference Guide on the Use of Indicators for Integrated Coastal Management.
2003. 127 pp. (English). ICAM Dossier No. 1
46
A Handbook for Measuring the Progress and Outcomes of Integrated Coastal and
Ocean Management. 2006. iv + 215 pp. (English). ICAM Dossier No. 2
47
TsunamiTeacher – An information and resource toolkit building capacity to
respond to tsunamis and mitigate their effects. 2006. DVD (English, Bahasa
Indonesia, Bangladesh Bangla, French, Spanish, and Thai)
48
Visions for a Sea Change. Report of the first international workshop on marine
spatial planning. 2007. 83 pp. (English). ICAM Dossier No. 4
49
Tsunami preparedness. Information guide for disaster planners. 2008. (English,
French, Spanish)
50
Hazard Awareness and Risk Mitigation in Integrated Coastal Area Management.
2009. 141 pp. (English). ICAM Dossier No. 5
51
IOC Strategic Plan for Oceanographic Data and Information Management (2008–
2011). 2008. 46 pp. (English)
52
Tsunami risk assessment and mitigation for the Indian Ocean; knowing your
tsunami risk – and what to do about it. 2009. 82 pp. (English)
53
Marine Spatial Planning. A Step‐by‐step Approach. 2009. 96 pp. (English). ICAM
Dossier No. 6
54
Ocean Data Standards Series:
Vol. 1: Recommendation to Adopt ISO 3166‐1 and 3166‐3 Country Codes as the
Standard for Identifying Countries in Oceanographic Data Exchange. 2010. 13 pp.
(English)
55
Microscopic and Molecular Methods for Quantitative Phytoplankton Analysis.
2010. 114 pp. (English)
56
The International Thermodynamic Equation of Seawater—2010: Calculation and
Use of Thermodynamic Properties. 2010. 196 pp. (English)
+ User’s Guide to the International Thermodynamic Equation of Seawater–2010
(abridged edition). 2010. (Arabic, Chinese, English, French, Russian, Spanish)
IOC Manuals and Guides No. 56
TEOS-10 Manual: Calculation and Use of the Thermodynamic Properties of Seawater
197
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 .
5th July 2010
Page 123‐124. Fixed an error in Eqn. (B.4) and changed Eqn. (B.5).
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. These changes were quite extensive.
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); hSA was replaced by hˆSA .
22nd Sept 2010
Page 100. Changes in Eqns. (A.18.5) – (A.18.7).
Page 122. Simplified Eqn. (A.30.6).
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
198
TEOS-10 Manual: Calculation and use of the thermodynamic properties of seawater
IOC Manuals and Guides No. 56
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.
Intergovernmental Oceanographic Commission (IOC)
United Nations Educational, Scientific and Cultural Organization
1, rue Miollis
75732 Paris Cedex 15, France
Tel: +33 1 45 68 10 10
Fax: +33 1 45 68 58 12
Website: http://ioc.unesco.org
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