The international thermodynamic equation of seawater 2010:

Transcription

1 Manuals and Guides 56 Intergovernmental Oceanograhic Commission The international thermodynamic equation of seawater : Calculation and use of thermodynamic roerties

2 The Intergovernmental Oceanograhic Commission (IOC) of UNESCO celebrates its 5th anniversary in. Since taking the lead in coordinating the International Indian Ocean Exedition in 96, the IOC has worked to romote marine research, rotection of the ocean, and international cooeration. Today the Commission is also develoing marine services and caacity building, and is instrumental in monitoring the ocean through the Global Ocean Observing System (GOOS) and develoing marine-hazards warning systems in vulnerable regions. Recognized as the UN focal oint and mechanism for global cooeration in the study of the ocean, a key climate driver, IOC is a key layer in the study of climate change. Through romoting international cooeration, the IOC assists Member States in their decisions towards imroved management, sustainable develoment, and rotection of the marine environment.

3 Manuals and Guides 56 Intergovernmental Oceanograhic Commission The international thermodynamic equation of seawater : Calculation and use of thermodynamic roerties

4 The authors are resonsible for the choice and the resentation of the facts contained in this ublication and for the oinions exressed therein, which are not necessarily those of UNESCO, SCOR or IPSO and do not commit those Organizations. The hotograh on the front cover of a CTD and lowered DCP hovering just below the sea surface was taken south of Timor from the Southern Surveyor in ugust 3 by nn Gronell Thresher. For bibliograhic uroses, this document should be cited as follows: IOC, SCOR and IPSO, : The international thermodynamic equation of seawater : Calculation and use of thermodynamic roerties. Intergovernmental Oceanograhic Commission, Manuals and Guides No. 56, UNESCO (English), 96. Printed by UNESCO (IOC//MG/56) UNESCO/IOC et al.

5 iii Table of contents cknowledgements... vii Foreword. viii bstract.... Introduction.... Oceanograhic ractice Motivation for an udated thermodynamic descrition of seawater..3 SCOR/IPSO WG7 and the aroach taken guide to this TEOS- manual. 6.5 remark on units Recommendations.. 7. Basic Thermodynamic Proerties ITS-9 temerature Sea ressure Practical Salinity Reference Comosition and the Reference-Comosition Salinity Scale....5 bsolute Salinity..6 Gibbs function of seawater Secific volume Density Chemical otentials Entroy.... Internal energy..... Enthaly Helmholtz energy Osmotic coefficient Isothermal comressibility Isentroic and adiabatic comressibility....7 Sound seed....8 Thermal exansion coefficients Saline contraction coefficients. 3. Isobaric heat caacity.. 4. Isochoric heat caacity. 4. diabatic lase rate.. 5

6 iv 3. Derived Quantities Potential temerature Potential enthaly Conservative Temerature Potential density Density anomaly Potential density anomaly Secific volume anomaly Thermobaric coefficient Cabbeling coefficient Buoyancy frequency Neutral tangent lane Geostrohic, hydrostatic and thermal wind equations Neutral helicity Neutral Density Stability ratio Turner angle Proerty gradients along otential density surfaces Sloes of otential density surfaces and neutral tangent lanes comared Sloes of in situ density surfaces and secific volume anomaly surfaces Potential vorticity Vertical velocity through the sea surface Freshwater content and freshwater flux Heat transort Geootential Total energy Bernoulli function Dynamic height anomaly Montgomery geostrohic streamfunction Cunningham geostrohic streamfunction Geostrohic streamfunction in an aroximately neutral surface Pressure-integrated steric height Pressure to height conversion Freezing temerature Latent heat of melting Sublimation ressure Sublimation enthaly Vaour ressure Boiling temerature Latent heat of evaoration Relative humidity and fugacity Osmotic ressure Temerature of maximum density Conclusions... 6

7 v endix : Background and theory underlying the use of the Gibbs function of seawater ITS-9 temerature Sea ressure, gauge ressure and absolute ressure Reference Comosition and the Reference-Comosition Salinity Scale bsolute Salinity Satial variations in seawater comosition Gibbs function of seawater The fundamental thermodynamic relation The conservative and isobaric conservative roerties The otential roerty Proof that θ = θ ( S, η ) and = ( S ), θ Various isobaric derivatives of secific enthaly Differential relationshis between η, θ, and S The First Law of Thermodynamics dvective and diffusive heat fluxes θ θ.5 Derivation of the exressions for α, β, α and β Non-conservative roduction of entroy Non-conservative roduction of otential temerature Non-conservative roduction of Conservative Temerature Non-conservative roduction of density and of otential density... The reresentation of salinity in numerical ocean models The material derivatives of S *, S, S R and in a turbulent ocean The material derivatives of density and of locally-referenced otential density; the dianeutral velocity e The water-mass transformation equation Conservation equations written in otential density coordinates The vertical velocity through a general surface The material derivative of otential density The diaycnal velocity of layered ocean models (without rotation of the mixing tensor) The material derivative of orthobaric density The material derivative of Neutral Density Comutationally efficient 5-term exressions for the density of seawater in terms of and θ.... endix B: Derivation of the First Law of Thermodynamics.. 3 endix C: Publications describing the TEOS- thermodynamic descritions of seawater, ice and moist air endix D: Fundamental constants

8 vi endix E: lgorithm for calculating Practical Salinity E. Calculation of Practical Salinity in terms of K E. Calculation of Practical Salinity at oceanograhic temerature and ressure E.3 Calculation of conductivity ratio R for a given Practical Salinity E.4 Evaluating Practical Salinity using ITS-9 temeratures E.5 Towards SI-traceability of the measurement rocedure for Practical Salinity and bsolute Salinity endix F: Coefficients of the IPWS-95 Helmholtz function of fluid water (with extension down to 5K) endix G: Coefficients of the ure liquid water Gibbs function of IPWS endix H: Coefficients of the saline Gibbs function for seawater of IPWS endix I: Coefficients of the Gibbs function of ice Ih of IPWS endix J: Coefficients of the Helmholtz function of moist air of IPWS endix K: Coefficients of 5-term exressions for the density of seawater in terms of and of θ endix L: Recommended nomenclature, symbols and units in oceanograhy 56 endix M: Seawater-Ice-ir (SI) library of comuter software.. 6 endix N: Gibbs-SeaWater (GSW) Oceanograhic Toolbox endix O: Checking the Gibbs function of seawater against the original thermodynamic data.. 75,,,,,, endix P: Thermodynamic roerties based on g( S t ) h( S η ) h ( S θ ) and hˆ ( S,, ),, References Index

9 vii cknowledgements This TEOS Manual reviews and summarizes the work of the SCOR/IPSO Working Grou 7 on the Thermodynamics and Equation of State of Seawater. Dr John Gould and Professor Paola Malanotte Rizzoli layed ivotal roles in the establishment of the Working Grou and we have enjoyed rock solid scientific suort from the officers of SCOR, IPSO and IOC. TJMcD wishes to acknowledge fruitful discussions with Drs Jürgen Willebrand and Michael McIntyre regarding the contents of aendix B. We have benefited from extensive comments on drafts of this manual by Dr Stehen Griffies and Dr llyn Clarke. Dr Harry Bryden is thanked for valuable and timely advice on the treatment of salinity in ocean models. Louise Bell of CSIRO rovided much areciated advice on the layout of this document. TJMcD and DRJ wish to acknowledge artial financial suort from the Wealth from Oceans National Flagshi. This work contributes to the CSIRO Climate Change Research Program. This document is based on work artially suorted by the U.S. National Science Foundation to SCOR under Grant No. OCE 686. FJM wishes to acknowledge the Oceanograhic Section of the National Science Foundation and the National Oceanic and tmosheric dministration for suorting his work. This document has been written by the members of SCOR/IPSO Working Grou 7, Trevor J. McDougall, (chair), CSIRO, Hobart, ustralia Rainer Feistel, Leibniz Institut fuer Ostseeforschung, Warnemuende, Germany Daniel G. Wright +, Bedford Institute of Oceanograhy, Dartmouth, Canada Rich Pawlowicz, University of British Columbia, Vancouver, Canada Frank J. Millero, University of Miami, Florida, US David R. Jackett, CSIRO, Hobart, ustralia Brian. King, National Oceanograhy Centre, Southamton, UK Giles M. Marion, Desert Research Institute, Reno, US Steffen Seitz, Physikalisch Technische Bundesanstalt (PTB), Braunschweig, Germany Petra Sitzer, Physikalisch Technische Bundesanstalt (PTB), Braunschweig, Germany C T. rthur Chen, National Sun Yat Sen University, Taiwan, R.O.C. March + deceased, 8 th July.

10 viii Foreword This document describes the International Thermodynamic Equation Of Seawater (TEOS for short). This descrition of the thermodynamic roerties of seawater and of ice Ih has been adoted by the Intergovernmental Oceanograhic Commission at its 5 th ssembly in June 9 to relace EOS 8 as the official descrition of seawater and ice roerties in marine science. Fundamental to TEOS are the concets of bsolute Salinity and Reference Salinity. These variables are described in detail here, emhasising their relationshi to Practical Salinity. The science underinning TEOS has been described in a series of aers ublished in the refereed literature (see aendix C). The resent document may be called the TEOS Manual and acts as a guide to those ublished aers and concentrates on how the thermodynamic roerties obtained from TEOS are to be used in oceanograhy. In addition to the thermodynamic roerties of seawater, TEOS also describes the thermodynamic roerties of ice and of humid air, and these roerties are summarised in this document. The TEOS comuter software, this TEOS Manual and other documents may be obtained from In articular, there are two introductory articles about TEOS on this web site, namely What every oceanograher needs to know about TEOS (The TEOS Primer) (Pawlowicz, b) and Getting started with TEOS and the GSW Oceanograhic Toolbox (McDougall and Barker, ). succinct high level summary of the salient features of TEOS has also been ublished by the Intergovernmental Oceanograhic Commission as IOC et al. (b) [IOC, SCOR and IPSO, : The international thermodynamic equation of seawater : Summary for Policy Makers. Intergovernmental Oceanograhic Commission (Brochures Series)]. When referring to the use of TEOS, it is the resent document that should be referenced as IOC et al. (a) [IOC, SCOR and IPSO, : The international thermodynamic equation of seawater : Calculation and use of thermodynamic roerties. Intergovernmental Oceanograhic Commission, Manuals and Guides No. 56, UNESCO (English), 96.]. This version of the TEOS Manual includes corrections u to 7 th December.

11 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater bstract This document outlines how the thermodynamic roerties of seawater are evaluated using the International Thermodynamic Equation Of Seawater (TEOS ). This thermodynamic descrition of seawater is based on a Gibbs function formulation from which thermodynamic roerties such as entroy, secific volume, enthaly and otential enthaly are calculated directly. When determined from the Gibbs function, these quantities are fully consistent with each other. Entroy and enthaly are required for an accurate descrition of the advection and diffusion of heat in the ocean interior and for quantifying the ocean s role in exchanging heat with the atmoshere and with ice. The Gibbs function is a function of bsolute Salinity, temerature and ressure. In contrast to Practical Salinity, bsolute Salinity is exressed in SI units and it includes the influence of the small satial variations of seawater comosition in the global ocean. bsolute Salinity is the aroriate salinity variable for the accurate calculation of horizontal density gradients in the ocean. bsolute Salinity is also the aroriate salinity variable for the calculation of freshwater fluxes and for calculations involving the exchange of freshwater with the atmoshere and with ice. Potential functions are included for ice and for moist air, leading to accurate exressions for numerous thermodynamic roerties of ice and air including freezing temerature and latent heats of melting and of evaoration. This TEOS Manual describes how the thermodynamic roerties of seawater, ice and moist air are used in order to accurately reresent the transort of heat in the ocean and the exchange of heat with the atmoshere and with ice.

12 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater. Introduction. Oceanograhic ractice The Practical Salinity Scale, PSS 78, and the International Equation of State of Seawater (Unesco (98)) which exresses the density of seawater as a function of Practical Salinity, temerature and ressure, have served the oceanograhic community very well for thirty years. The Joint Panel on Oceanograhic Tables and Standards (JPOTS) (Unesco (983)) also romulgated the Millero, Perron and Desnoyers (973) algorithm for the secific heat caacity of seawater at constant ressure, the Chen and Millero (977) exression for the sound seed of seawater and the Millero and Leung (976) formula for the freezing oint temerature of seawater. Three other algorithms suorted under the ausices of JPOTS concerned the conversion between hydrostatic ressure and deth, the calculation of the adiabatic lase rate, and the calculation of otential temerature. The exressions for the adiabatic lase rate and for otential temerature could in rincile have been derived from the other algorithms of the EOS 8 set, but in fact they were based on the formulas of Bryden (973). We shall refer to all these algorithms jointly as EOS 8 for convenience because they reresent oceanograhic best ractice from the early 98s to 9.. Motivation for an udated thermodynamic descrition of seawater In recent years the following asects of the thermodynamics of seawater, ice and moist air have become aarent and suggest that it is timely to redefine the thermodynamic roerties of these substances. Several of the olynomial exressions of the International Equation of State of Seawater (EOS 8) are not totally consistent with each other as they do not exactly obey the thermodynamic Maxwell cross differentiation relations. The new aroach eliminates this roblem. Since the late 97s a more accurate and more broadly alicable thermodynamic descrition of ure water has been develoed by the International ssociation for the Proerties of Water and Steam, and has aeared as an IPWS Release (IPWS 95). lso since the late 97s some measurements of higher accuracy have been made of several roerties of seawater such as (i) heat caacity, (ii) sound seed and (iii) the temerature of maximum density. These can be incororated into a new thermodynamic descrition of seawater. The imact on seawater density of the variation of the comosition of seawater in the different ocean basins has become better understood. In order to further rogress this asect of seawater, a standard model of seawater comosition is needed to serve as a generally recognised reference for theoretical and chemical investigations. The increasing emhasis on the ocean as being an integral art of the global heat engine oints to the need for accurate exressions for the entroy, enthaly 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 atmoshere and ice (entroy, enthaly and internal energy were not available from EOS 8).

13 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater 3 The need for a thermodynamically consistent descrition of the interactions between seawater, ice and moist air; in articular, the need for accurate exressions for the latent heats of evaoration and freezing, both at the sea surface and in the atmoshere. The temerature scale has been revised from IPTS 68 to ITS 9 and revised IUPC (International Union of Pure and lied Chemistry) values have been adoted for the atomic weights of the elements (Wieser (6))..3 SCOR/IPSO WG7 and the aroach taken In 5 SCOR (Scientific Committee on Oceanic Research) and IPSO (International ssociation for the Physical Sciences of the Oceans) established Working Grou 7 on the Thermodynamics and Equation of State of Seawater (henceforth referred to as WG7). This grou has now develoed a collection of algorithms that incororate our best knowledge of seawater thermodynamics. The resent document summarizes the work of SCOR/IPSO Working Grou 7. To comute all thermodynamic roerties of seawater it is sufficient to know one of its so called thermodynamic otentials (Fofonoff 96, Feistel 993, lberty ). It was J.W. Gibbs (873) who discovered that an equation giving internal energy in terms of entroy and secific volume, or more generally any finite equation between internal energy, entroy and secific 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 roerties of the fluid (so far as reversible rocesses are concerned). The aroach taken by WG7 has been to develo a Gibbs function from which all the thermodynamic roerties of seawater can be derived by urely mathematical maniulations (such as differentiation). This aroach ensures that the various thermodynamic roerties are self consistent (in that they obey the Maxwell crossdifferentiation relations) and comlete (in that each of them can be derived from the given otential). The Gibbs function (or Gibbs otential) is a function of bsolute Salinity S (rather than of Practical Salinity S P ), temerature and ressure. bsolute Salinity is traditionally defined as the mass fraction of dissolved material in seawater. The use of bsolute Salinity as the salinity argument for the Gibbs function and for all other thermodynamic functions (such as density) is a major dearture from resent ractice (EOS 8). bsolute Salinity is referred over Practical Salinity because the thermodynamic roerties of seawater are directly influenced by the mass of dissolved constituents whereas Practical Salinity deends only on conductivity. Consider for examle exchanging a small amount of ure water with the same mass of silicate in an otherwise isolated seawater samle at constant temerature and ressure. Since silicate is redominantly non ionic, the conductivity (and therefore Practical Salinity S P ) is almost unchanged but the bsolute 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 samle, 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 rocesses actually cause comlications 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.5 and aendices.4,.5 and.. Here it is sufficient to oint out that the bsolute Salinity S which is the salinity argument of the TEOS Gibbs function is the version of absolute salinity that rovides the best estimate of the density of seawater; another name for S is Density Salinity.

14 4 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater The Gibbs function of seawater, ublished as Feistel (8), has been endorsed by the International ssociation for the Proerties of Water and Steam as the Release IPWS 8. This thermodynamic descrition of seawater roerties, together with the Gibbs function of ice Ih, IPWS 6, has been adoted by the Intergovernmental Oceanograhic Commission at its 5 th ssembly in June 9 to relace EOS 8 as the official descrition of seawater and ice roerties in marine science. The thermodynamic roerties of moist air have also recently been described using a Helmholtz function (Feistel et al. (a), IPWS ()) so allowing the equilibrium roerties at the air sea interface to be more accurately evaluated. The new aroach to the thermodynamic roerties of seawater, of ice Ih and of humid air is referred to collectively as the International Thermodynamic Equation Of Seawater, or TEOS for short. endix C lists the ublications which lie behind TEOS. notable difference of TEOS comared with EOS 8 is the adotion of bsolute 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 roerties of seawater. This recommendation deviates from the current ractice of working with Practical Salinity and tyically treating it as the best estimate of bsolute Salinity. This ractice is inaccurate and should be corrected. Note however that we strongly recommend that the salinity that is reorted to national data bases remain Practical Salinity as determined on the Practical Salinity Scale of 978 (suitably udated to ITS 9 temeratures as described in aendix E below). There are three very good reasons for continuing to store Practical Salinity rather than bsolute Salinity in such data reositories. First, Practical Salinity is an (almost) directly measured quantity whereas bsolute Salinity is generally a derived quantity. That is, we calculate Practical Salinity directly from measurements of conductivity, temerature and ressure, whereas to date we derive bsolute Salinity from a combination of these measurements lus other measurements and correlations that are not yet well established. Practical Salinity is referred over the actually measured in situ conductivity value because of its conservative nature with resect to changes of temerature or ressure, or dilution with ure water. Second, it is imerative that confusion is not created in national data bases where a change in the reorting of salinity may be mishandled at some stage and later be misinterreted as a real increase in the ocean s salinity. This second oint argues strongly for no change in resent ractice in the reorting of Practical Salinity S P in national data bases of oceanograhic data. Thirdly, the algorithms for determining the ʺbestʺ estimate of bsolute Salinity of seawater with non standard comosition are immature and will undoubtedly change in the future, so we cannot recommend storing bsolute Salinity in national data bases. Storage of a more robust intermediate value, the Reference Salinity, S R (defined as discussed in aendix.3 to give the best estimate of bsolute Salinity of Standard Seawater) would also introduce the ossibility of confusion in the stored salinity values without roviding any real advantage over storing Practical Salinity so we also avoid this ossibility. Values of Reference Salinity obtained from suitable observational techniques (for examle by direct measurement of the density of Standard Seawater) should be converted to corresonding numbers of Practical Salinity for storage, as described in sections.3.5. Note that the ractice of storing one tye of salinity in national data bases (Practical Salinity) but using a different tye of salinity in ublications (bsolute Salinity) is exactly analogous to our resent ractice with temerature; in situ temerature t is stored in data bases (since it is the measured quantity) but the temerature variable that is used in ublications is a calculated quantity, being either otential temerature θ or Conservative Temerature.

15 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater 5 In order to imrove the determination of bsolute Salinity we need to begin collecting and storing values of the salinity anomaly δ S= S SR based on measured values of density (such as can be measured with a vibrating tube densimeter, Kremling (97)). The 4 letter GF3 code (IOC (987)) DENS is currently defined for in situ measurements or comuted values from EOS 8. It is recommended that the density measurements made with a vibrating beam densimeter be reorted with the GF3 code DENS along with the laboratory temerature (TLB in C ) and laboratory ressure (PLB, the sea ressure in the laboratory, usually dbar). From this information and the Practical Salinity of the seawater samle, the absolute salinity anomaly δ S= S SR can be calculated using an inversion of the TEOS equation for density to determine S. For comleteness, it is advisable to also reort δ S under the new GF3 code DELS. The thermodynamic descrition of seawater and of ice Ih as defined in IPWS 8 and IPWS 6 has been adoted as the official descrition of seawater and of ice Ih by the Intergovernmental Oceanograhic Commission in June 9. These new international standards were adoted while recognizing that the techniques for estimating bsolute Salinity will likely imrove over the coming decades, and the algorithm for evaluating bsolute Salinity in terms of Practical Salinity, latitude, longitude and ressure will be udated from time to time, after relevant aroriately eer reviewed ublications have aeared, and that such an udated algorithm will aear on the web site. Users of this software should always state in their ublished work which version of the software was used to calculate bsolute Salinity. The more rominent advantages of TEOS comared with EOS 8 are The Gibbs function aroach allows the calculation of internal energy, entroy, enthaly, otential enthaly and the chemical otentials of seawater as well as the freezing temerature, and the latent heats of freezing and of evaoration. These quantities were not available from the International Equation of State 98 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 examle, the new TEOS temerature variable, Conservative Temerature,, is defined to be roortional to otential enthaly and is a very accurate measure of the heat content er unit mass of seawater; is two orders of magnitude more conservative than otential temerature θ. For the first time the influence of the satially varying comosition of seawater can systematically be taken into account through the use of bsolute Salinity. In the oen ocean, this has a non trivial effect on the horizontal density gradient comuted from the equation of state, and thereby on the ocean velocities and heat transorts calculated via the thermal wind relation. The thermodynamic quantities available from the new aroach are totally consistent with each other. The new salinity variable, bsolute Salinity, is measured in SI units. Moreover the treatment of freshwater fluxes in ocean models will be consistent with the use of bsolute Salinity, but is only aroximately so for Practical Salinity. The Reference Comosition of standard seawater suorts marine hysicochemical studies such as the solubility of sea salt constituents, the alkalinity, the H and the ocean acidification by rising concentrations of atmosheric CO.

16 6 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater.4 guide to this TEOS- manual The remainder of this manual begins by listing (in section ) the definitions of various thermodynamic quantities that follow directly from the Gibbs function of seawater by simle mathematical rocesses such as differentiation. These definitions are then followed in section 3 by the discussion of several derived quantities. The comuter software to evaluate these quantities is available from two searate libraries, the Seawater Ice ir (SI) library and the Gibbs SeaWater (GSW) Oceanograhic Toolbox, as described in aendices M and N. The functions in the SI library are generally available in basic SI units ( kg kg, kelvin and Pa), both for their inut arameters and for the oututs of the algorithms. Some additional routines are included in the SI library in terms of other commonly used units for the convenience of users. The SI library takes significantly more comuter time to evaluate most quantities (aroximately a factor of 65 more comuter time for many quantities, comaring otimized code in both cases) and rovides significantly more roerties than does the GSW Toolbox. The SI library uses the world wide standard for the thermodynamic descrition of ure water substance (IPWS 95). Since this is defined over extended ranges of temerature and ressure, the algorithms are long and their evaluation time consuming. The GSW Toolbox uses the Gibbs function of Feistel (3) (IPWS 9) to evaluate the roerties of ure water, and since this is valid only over the restricted ranges of temerature and ressure aroriate for the ocean, the algorithms are shorter and their execution is faster. The GSW Oceanograhic Toolbox is not as comrehensive as the SI library; for examle, the roerties of moist air are only available in the SI library. In addition, a comutationally efficient exression for density ρ in terms of Conservative Temerature (rather than in terms of in situ temerature) involving just 5 coefficients is also available and is described in aendix.3 and aendix K. The inut and outut arameters of the GSW Oceanograhic Toolbox are in units which oceanograhers will find more familiar than basic SI units. We exect that oceanograhers will mostly use this GSW Toolbox because of its greater simlicity and comutational efficiency, and because of the more familiar units comared with the SI 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 SI and GSW libraries, together with this TEOS Manual are available from the website Initially the SI library is being made available in Visual Basic and FORTRN while the GSW library is available mainly in MTLB. fter these descritions in sections and 3 of how to determine the thermodynamic quantities and various derived quantities, we end with some conclusions (section 4). dditional information on Practical Salinity, the Gibbs function, Reference Salinity, comosition anomalies, bsolute Salinity, and some fundamental thermodynamic roerties such as the First Law of Thermodynamics, the non conservative nature of many oceanograhic variables, a list of recommended symbols, and succinct lists of thermodynamic formulae are given in the aendices. Much of this work has aeared elsewhere in the ublished literature but is collected here in a condensed form for the usersʹ convenience. Two introductory articles about TEOS, namely What every oceanograher needs to know about TEOS (The TEOS Primer) (Pawlowicz, b), and Getting started with TEOS and the GSW Oceanograhic Toolbox (McDougall and Barker, ) are available from

17 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater 7.5 remark on units The most convenient variables and units in which to conduct thermodynamic investigations are bsolute Salinity S in units of kg kg, bsolute Temerature T (K), and bsolute Pressure P in Pa. These are the arameters and units used in the SI software library. Oceanograhic ractice to date has used non basic SI units for many variables, in articular, temerature is usually measured on the Celsius ( C ) scale, ressure is sea ressure quoted in decibars relative to the ressure of a standard atmoshere (.35 dbar), while salinity has had its own oceanograhy secific scale, the Practical Salinity Scale of 978. In the GSW Oceanograhic Toolbox we adot C for the temerature unit, ressure is sea ressure in dbar and bsolute Salinity S is exressed in units of g kg so that it takes numerical values close to those of Practical Salinity. doting these non basic SI units does not come without a enalty as there are many thermodynamic formulae that are more conveniently maniulated when exressed in SI units. s an examle, the freshwater fraction of seawater is written correctly as ( S ), but it is clear that in this instance bsolute Salinity must be exressed in kg kg not in gkg. There are also cases within the GSW Toolbox in which SI units are required and this may occasionally cause some confusion. common examle of this issue arises when a variable is differentiated or integrated with resect to ressure. Nevertheless, for many alications it is deemed imortant to remain close to resent oceanograhic ractice even though it means that one has to be vigilant to detect those exressions that need a variable to be exressed in the less familiar SI units..6 Recommendations In accordance with resolution XXV 7 of the Intergovernmental Oceanograhic Commission at its 5 th ssembly in June 9, and the several Releases and Guidelines of the International ssociation for the Proerties of Water and Steam, the TEOS thermodynamic descrition of seawater, of ice and of moist air is recommended for use by oceanograhers in lace of the International Equation Of State 98 (EOS 8). The software to imlement this change is available at the web site Under TEOS it is recognized that the comosition of seawater varies around the world ocean and that the thermodynamic roerties of seawater are more accurately reresented as functions of bsolute Salinity S than of Practical Salinity S P. It is useful to think of the transition from Practical Salinity to bsolute Salinity in two stes. In the first ste a seawater samle is effectively treated as though it is Standard Seawater and its Reference Salinity S R is calculated; Reference Salinity may be taken to be simly roortional to Practical Salinity. Reference Salinity has SI units (for examle, gkg ) and is the natural starting oint to consider the influence of any variation in comosition. In the second ste the bsolute Salinity nomaly is evaluated using one of several techniques, the easiest of which is via a comuter algorithm that effectively interolates between a satial atlas of these values. Then bsolute Salinity is estimated as the sum of Reference Salinity and bsolute Salinity nomaly. Of the four ossible versions of absolute salinity, the one that is used as the argument for the TEOS Gibbs function is designed to rovide accurate estimates of the density of seawater. It is recognized that our knowledge of how to estimate seawater comosition anomalies and their effect on thermodynamic roerties is limited. Nevertheless, we should not continue to ignore the influence of these comosition variations on seawater roerties and on ocean dynamics. s more knowledge is gained in this area over the coming decade or so, and after such knowledge has been duly ublished in the scientific

18 8 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater literature, any udated algorithm to evaluate the bsolute Salinity nomaly will be available (with its version number) from The storage of salinity in national data bases should continue to occur as Practical Salinity, as this will maintain continuity of this imortant time series. Oceanograhic databases label stored, rocessed or exorted arameters with the GF3 code PSL for Practical Salinity and SSL for salinity measured before 978 (IOC, 987). In order to avoid ossible confusion in data bases between different tyes of salinity it is very strongly recommended that under no circumstances should either Reference Salinity or bsolute Salinity be stored in national data bases. In order to accurately calculate the thermodynamic roerties of seawater, bsolute Salinity must be calculated by first calculating Reference Salinity and then adding on the bsolute Salinity nomaly. Because bsolute Salinity is the aroriate salinity variable for use with the equation of state, bsolute Salinity should be the salinity variable that is ublished in oceanograhic journals. The version number of the software, or the exact formula, that was used to convert Reference Salinity into bsolute Salinity should always be stated in ublications. Nevertheless, there may be some alications where the likely future changes in the algorithm that relates Reference Salinity to bsolute Salinity resents a concern, and for these alications it may be referable to ublish grahs and tables in Reference Salinity. For these studies or where it is clear that the effect of comositional variations are insignificant or not of interest, the Gibbs function may be called with S R rather than S, thus avoiding the need to calculate the bsolute Salinity nomaly. When this is done, it should be clearly stated that the salinity variable that is being grahed is Reference Salinity, not bsolute Salinity. The TEOS aroach of using thermodynamic otentials to describe the roerties of seawater, ice and moist air means that it is ossible to derive many more thermodynamic roerties than were available from EOS 8. The seawater roerties entroy, internal energy, enthaly and articularly otential enthaly were not available from EOS 8 but are central to accurately calculating the transort of heat in the ocean and hence the air sea heat flux in the couled climate system. Under EOS 8 the observed variables ( S P, t, ) were first used to calculate otential temerature θ and then water masses were analyzed on the SP θ diagram. Curved contours of otential density could also be drawn on this same SP θ diagram. Under TEOS, since density and otential density are now not functions of Practical Salinity S P but rather are functions of bsolute Salinity S, it is now not ossible to draw isolines of otential density on a SP θ diagram. Rather, because of the satial variations of seawater comosition, a given value of otential density defines an area on the SP θ diagram, not a curved line. Under TEOS, the observed variables ( S P, t, ), together with longitude and latitude, are used to first form bsolute Salinity S, and then Conservative Temerature is evaluated. Oceanograhic water masses are then analyzed on the S diagram, and otential density contours can also be drawn on this S diagram. Preformed Salinity S * is used internally in numerical ocean models where it is imortant that the salinity variable be conservative. When describing the use of TEOS, it is the resent document (the TEOS Manual) that should be referenced as IOC et al. () [IOC, SCOR and IPSO, : The international thermodynamic equation of seawater : Calculation and use of thermodynamic roerties. Intergovernmental Oceanograhic Commission, Manuals and Guides No. 56, UNESCO (English), 96 ]. Two introductory articles about TEOS, namely What every oceanograher needs to know about TEOS (The TEOS Primer) (Pawlowicz, b), and Getting started with TEOS and the GSW Oceanograhic Toolbox (McDougall and Barker, ) are available from

19 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater 9. Basic Thermodynamic Proerties. ITS 9 temerature In 99 the International Practical Temerature Scale 968 (IPTS 68) was relaced by the International Temerature Scale 99 (ITS 9). There are two main methods to convert between these two temerature scales; Rusby s (99) 8 th order fit valid over a wide range of temeratures, and Saunders (99).4 scaling widely used in the oceanograhic community. The two methods are formally indistinguishable in the oceanograhic temerature range because they differ by less than either the uncertainty in thermodynamic temerature (of order mk), or the ractical alication of the IPTS 68 and ITS 9 scales. The differences between the Saunders (99) and Rusby (99) formulae are less than mk throughout the temerature range C to 4 C and less than.3mk in the temerature range between C and C. Hence we recommend that the oceanograhic community continues to use the Saunders formula ( t ) ( t ) / C =.4 / C. (..) 68 9 One alication of this formula is in the udated comuter algorithm for the calculation of Practical Salinity (PSS 78) in terms of conductivity ratio. The algorithms for PSS 78 require t 68 as the temerature argument. In order to use these algorithms with t 9 data, t 68 may be calculated using (..). n extended discussion of the different temerature scales, their inherent uncertainty and the reasoning for our recommendation of (..) can be found in aendix... Sea ressure Sea ressure is defined to be the bsolute Pressure P less the bsolute Pressure of one standard atmoshere, P 35Pa; that is P P. (..) It is common oceanograhic ractice to exress sea ressure in decibars (dbar). nother common ressure variable that arises naturally in the calibration of sea board instruments gauge is gauge ressure which is bsolute Pressure less the bsolute Pressure of the atmoshere at the time of the instrument s calibration (erhas in the laboratory, or erhas at sea). Because atmosheric ressure changes in sace and time, sea ressure is referred as a thermodynamic variable as it is unambiguously related to bsolute Pressure. The seawater Gibbs function in the GSW Toolbox is exressed as a function of sea ressure (functionally equivalent to the use of bsolute Pressure P in the IPWS gauge Releases and in the SI library); that is, g is a function of, it is not a function of..3 Practical Salinity Practical Salinity S P is defined on the Practical Salinity Scale of 978 (Unesco (98, 983)) in terms of the conductivity ratio K 5 which is the electrical conductivity of the samle at temerature t 68 = 5 C and ressure equal to one standard atmoshere ( = dbar and absolute ressure P equal to 35 Pa), divided by the conductivity of a standard

20 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater otassium chloride (KCl) solution at the same temerature and ressure. The mass fraction of KCl (i.e., the mass of KCl er mass of solution) in the standard solution is When K 5 =, 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 su ; 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 aendix E along with the simle change that must be made to the Unesco (983) formulae so that the algorithm for Practical Salinity can be called with ITS 9 temerature as an inut arameter rather than the older t 68 temerature in which the PSS 78 algorithms were defined. The reader is also directed to the CDIC chater on Method for salinity (conductivity ratio) measurement which describes best ractice in measuring the conductivity ratio of seawater samles (Kawano (9)). Practical Salinity is defined only in the range < SP < 4. Practical Salinities below or above 4 comuted from conductivity, should be evaluated by the PSS 78 extensions of Hill et al. (986) and Poisson and Gadhoumi (993). Samles exceeding a Practical Salinity of 5 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 rocess. This is discussed further in aendix E. Data stored in national and international data bases should, as a matter of rincile, be measured values rather than derived quantities. Consistent with this, we recommend continuing to store the measured (in situ) temerature rather than the derived quantity, Conservative Temerature. Similarly we strongly recommend that Practical Salinity S P continue to be the salinity variable that is stored in such data bases since S P is closely related to the measured values of conductivity. This recommendation has the very imortant advantage that there is no change to the resent ractice and so there is less chance of transitional errors occurring in national and international data bases because of the adotion of bsolute Salinity in oceanograhy..4 Reference Comosition and the Reference Comosition Salinity Scale The reference comosition of seawater is defined by Millero et al. (8a) as the exact mole fractions given in Table D.3 of aendix D below. This comosition was introduced by Millero et al. (8a) as their best estimate of the comosition of Standard Seawater, being seawater from the surface waters of a certain region of the North tlantic. The exact location for the collection of bulk material for the rearation of Standard Seawater is not secified. Shis gathering this bulk material are given guidance notes by the Standard Seawater Service, requesting that water be gathered between longitudes 5 W and 4 W, in dee water, during daylight hours. Reference Comosition Salinity S R (or Reference Salinity for short) was designed by Millero et al. (8b) to be the best estimate of the mass fraction bsolute Salinity S of Standard Seawater. Indeendent of accuracy considerations, it rovides a recise measure of dissolved material in Standard Seawater and is the correct salinity argument to be used in the TEOS Gibbs function for Standard Seawater. For the range of salinities where Practical Salinities are defined (that is, in the range < S < 4) Millero et al. (8a) show that P SR upssp where u PS ( ) gkg. (.4.) In the range < SP < 4, this equation exresses the Reference Salinity of a seawater samle on the Reference Comosition Salinity Scale (Millero et al. (8a)). For ractical uroses, this relationshi can be taken to be an equality since the aroximate nature of

21 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater this relation only reflects the extent to which Practical Salinity, as determined from measurements of conductivity ratio, temerature and ressure, varies when a seawater samle is heated, cooled or subjected to a change in ressure but without exchange of mass with its surroundings. The Practical Salinity Scale of 978 was designed to satisfy this roerty as accurately as ossible within the constraints of the olynomial aroximations used to determine Chlorinity (and hence Practical Salinity) in terms of the measured conductivity ratio. From Eqn. (.4.), a seawater samle of Reference Comosition whose Practical Salinity S P is 35 has a Reference Salinity S R of g kg. Millero et al. (8a) estimate that the absolute uncertainty in this value is ±.7 g kg. The difference between the numerical values of Reference and Practical Salinities can be traced back to the original ractice of determining salinity by evaoration of water from seawater and weighing the remaining solid material. This rocess also evaorated some volatile comonents and most of the.65 4 g kg salinity difference is due to this effect. Measurements of the comosition of Standard Seawater at a Practical Salinity S P of 35 using mass sectrometry and/or ion chromatograhy are underway and may rovide udated estimates of both the value of the mass fraction of dissolved material in Standard Seawater and its uncertainty. ny udate of this value will not change the Reference Comosition Salinity Scale and so will not affect the calculation of Reference Salinity nor of bsolute Salinity as calculated from Reference Salinity lus the bsolute Salinity nomaly. Oceanograhic databases label stored, rocessed or exorted arameters with the GF3 code PSL for Practical Salinity and SSL for salinity measured before 978 (IOC, 987). In order to avoid ossible confusion in data bases between different tyes of salinity it is very strongly recommended that under no circumstances should either Reference Salinity or bsolute Salinity be stored in national data bases. Detailed information on Reference Comosition and Reference Salinity can be found in Millero et al. (8a). For the userʹs convenience a brief summary of information from Millero et al. (8a), including the recise definition of Reference Salinity is given in aendix.3 and in Table D3 of aendix D..5 bsolute Salinity bsolute Salinity is traditionally defined as the mass fraction of dissolved material in seawater. For seawater of Reference Comosition, Reference Salinity gives our current best estimate of bsolute Salinity. To deal with comosition anomalies in seawater, we need an extension of the Reference Comosition Salinity S R that rovides a useful measure of salinity over the full range of oceanograhic conditions and agrees recisely with Reference Salinity when the dissolved material has Reference Comosition. When comosition anomalies are resent, no single measure of dissolved material can fully reresent the influences on seawater roerties on all thermodynamic roerties, so it is clear that either additional information will be required or comromises 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., b) and maintains the high accuracy of PSS 78 necessary for oceanograhic alications. The introduction of ʺDensity Salinityʺ S addresses both of these issues; it is this tye of absolute salinity that in TEOS arlance is labeled S and called bsolute Salinity. In this section we exlain how S is defined and evaluated, but first we outline other choices that are available for the definition of absolute salinity in the resence of comosition variations in seawater. The most obvious definition of absolute salinity is the mass fraction of dissolved non HO material in a seawater samle at its temerature and ressure. This seemingly dens

22 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater simle definition is actually far more subtle than it first aears. Notably, there are questions about what constitutes water and what constitutes dissolved material. Perhas the most obvious examle of this issue occurs when CO is dissolved in water to roduce a mixture of CO, HCO3, HCO3, CO3, H +, OH and HO, with the relative roortions deending on dissociation constants that deend on temerature, ressure and H. Thus, the dissolution of a given mass of CO in ure water essentially transforms some of the water into dissolved material. change in the temerature and even an adiabatic change in ressure results in a change in absolute salinity defined in this way due to the deendence of chemical equilibria on temerature and ressure. Pawlowicz et al. () and Wright et al. (b) address this second issue by defining Solution bsolute Salinity (usually shortened to Solution Salinity ), S, as the mass fraction of dissolved non HO material after a seawater samle is brought to the constant temerature t = 5 C and the fixed sea ressure dbar (fixed bsolute Pressure of 35 Pa). add nother measure of absolute salinity is the dded Mass Salinity S which is S R lus the mass fraction of material that must be added to Standard Seawater to arrive at the concentrations of all the secies in the given seawater samle, after chemical equilibrium has been reached, and after the samle is brought to the constant temerature t = 5 C add and the fixed sea ressure of dbar. The estimation of absolute salinity S is not straightforward for seawater with anomalous comosition because while the final equilibrium state is known, one must iteratively determine the mass of anomalous solute rior to any chemical reactions with Reference Comosition seawater. Pawlowicz et al. () rovide an algorithm to achieve this, at least aroximately. This definition of add absolute salinity, S, is useful for laboratory studies of artificial seawater and it differs soln from S because of the chemical reactions that take lace between the several secies of the added material and the comonents of seawater that exist in Standard Seawater. dded Mass Salinity may be the most aroriate form of salinity for accurately accounting for the mass of salt discharged by rivers and hydrothermal vents into the ocean. Preformed bsolute Salinity (usually shortened to Preformed Salinity ), S *, is a different tye of absolute salinity which is secifically designed to be as close as ossible to being a conservative variable. That is, S * is designed to be insensitive to biogeochemical rocesses that affect the other tyes of salinity to varying degrees. Preformed Salinity S * is formed by first estimating the contribution of biogeochemical soln add rocesses to one of the salinity measures S, S, or S, and then subtracting this contribution from the aroriate salinity variable. In this way Preformed Salinity S * is designed to be a conservative salinity variable which is indeendent of the effects of the non conservative biogeochemical rocesses. S * will find a rominent role in ocean soln add modeling. The three tyes of absolute salinity S, S and S * are discussed in more detail in aendices.4 and., where aroximate relationshis between these dens variables and S S are resented, based on the work of Pawlowicz et al. () and Wright et al. (b). Note that for a samle of Standard Seawater, all of the five salinity soln add variables S R, S, S, S and S * and are equal. soln add There is no simle means to measure either S or S for the general case of the arbitrary addition of many comonents to Standard Seawater. Hence a more recise and easily determined measure of the amount of dissolved material in seawater is required dens and TEOS adots Density Salinity for this urose. Density Salinity S is defined as the value of the salinity argument of the TEOS exression for density which gives the samle s actual measured density at the temerature t = 5 C and at the sea ressure = dbar. When there is no risk of confusion, Density Salinity is also called dens bsolute Salinity with the label S, that is S S. Usually we do not have accurate measurements of density but rather we have measurements of Practical Salinity, soln

23 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater 3 temerature and ressure, and in this case, bsolute Salinity may be calculated using Practical Salinity and the comuter algorithm of McDougall, Jackett and Millero (a) which rovides an estimate of δ S= S SR. This comuter rogram was formed as follows. In a series of aers (Millero et al. (976a, 978,, 8b), McDougall et al. (a)), accurate measurements of the density of seawater samles, along with the Practical Salinity of those samles, gave estimates of δ S= S SR from most of the major basins of the world ocean. This was done by first calculating the Reference Density from the TEOS equation of state using the samle s Reference Salinity as the salinity argument (this calculation essentially assumes that the seawater samle has the comosition of Standard Seawater). The difference between the measured density and the Reference Density was then used to estimate the bsolute Salinity nomaly δ S= S SR (Millero et al. (8a)). The McDougall et al. (a) algorithm is based on the observed correlation between this S SR data and the silicate concentration of the seawater samles (Millero et al., 8a), with the silicate concentration being estimated by interolation of a global atlas (Gouretski and Koltermann (4)). The algorithm for bsolute Salinity takes the form ( φ λ ) S = S R + δs = S S P,,,, (.5.) Where φ is latitude (degrees North), λ is longitude (degrees east, ranging from E to 36 E) while is sea ressure. Heuristically the deendence of δ S= S SR on silicate can be thought of as reflecting the fact that silicate affects the density of a seawater samle without significantly affecting its conductivity or its Practical Salinity. In ractice this exlains about 6% of the effect and the remainder is due to the correlation of other comosition anomalies (such as nitrate) with silicate. In the McDougall et al. (a) algorithm the Baltic Sea is treated searately, following the work of Millero and Kremling (976) and Feistel et al. (c, d), because some rivers flowing into the Baltic are unusually high in calcium carbonate. Figure. sketch indicating how thermodynamic quantities such as density are calculated as functions of bsolute Salinity. bsolute Salinity is found by adding an estimate of the bsolute Salinity nomaly δ S to the Reference Salinity.

24 4 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater Since the density of seawater is rarely measured, we recommend the aroach illustrated in Figure as a ractical method to include the effects of comosition anomalies on estimates of bsolute Salinity and density. When comosition anomalies are not known, the algorithm of McDougall et al. (a) may be used to estimate bsolute Salinity in terms of Practical Salinity and the satial location of the measurement in the world oceans. The difference between bsolute Salinity and Reference Salinity, as estimated by the McDougall et al. (a) algorithm, is illustrated in Figure (a) at a ressure of dbar, and in a vertical section through the Pacific Ocean in Figure (b). Of the aroximately 8 samles of seawater from the world ocean that have been examined to date for δ S= S SR the standard error (square root of the mean squared value) of δ S= S SR is.7 g kg. That is, the tyical value of δ S= S SR of the 8 samles taken to date is.7 g kg. The standard error of the difference between the measured values of δ S= S SR and the values evaluated from the comuter algorithm of McDougall et al. (a) is.48 g kg. The maximum values of δ S= S SR of aroximately.5 g kg occur in the North Pacific. Figure (a). bsolute Salinity nomaly δ S at = dbar. Figure (b). vertical section of bsolute Salinity nomaly δ S along 8 o E in the Pacific Ocean. The thermodynamic descrition of seawater and of ice Ih as defined in IPWS-8 and IPWS-6 has been adoted as the official descrition of seawater and of ice Ih by the Intergovernmental Oceanograhic Commission in June 9. These thermodynamic descritions of seawater and ice were endorsed recognizing that the techniques for

25 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater 5 estimating bsolute Salinity will likely imrove over the coming decades. The algorithm for evaluating bsolute Salinity in terms of Practical Salinity, latitude, longitude and ressure, will likely be udated from time to time, after relevant aroriately eerreviewed ublications have aeared, and such an udated algorithm will aear on the web site. Users of this software should state in their ublished work which version of the software was used to calculate bsolute Salinity. The resent comuter software which evaluates bsolute Salinity S given the inut variables Practical Salinity S P, longitude λ, latitude φ and ressure is available at bsolute Salinity is also available as the inverse function of density S ( T, P, ρ ) in the SI library of comuter algorithms as the algorithm sea_sa_si (see aendix M) and in the GSW Toolbox as the algorithm gsw_s_from_rho..6 Gibbs function of seawater t is related to the secific enthaly h and = + where T = 73.5K is the Celsius zero oint. TEOS defines the Gibbs function of seawater as the sum of a ure water art and the saline art (IPWS 8) The Gibbs function of seawater g ( S,, ) entroy η, by g h ( T t) η W S (,, ) (, ) (,, ) g S t = g t + g S t. (.6.) S, The saline art of the Gibbs function, g is valid over the ranges < S < 4 g kg, 4 6. C < t < 4 C, and < < dbar, although its thermal and colligative roerties are valid u to t = 8 C and S = g kg at =. The ure water art of the Gibbs function, g W, can be obtained from the IPWS 95 Helmholtz function of ure water substance which is valid from the freezing temerature or from the sublimation temerature to 73 K. lternatively, the ure water art of the Gibbs function can be obtained from the IPWS 9 Gibbs function which is valid in the oceanograhic ranges of temerature and ressure, from less than the freezing temerature of seawater (at any ressure), u to 4 C (secifically from 4 (.65+( + P ).743 MPa ) C to 4 C), and in the ressure range < < dbar. For ractical uroses in oceanograhy it is exected that IPWS 9 will be used because it executes aroximately two orders of magnitude faster than the IPWS 95 code for ure water. However if one is concerned with temeratures between 4 C and 8 C W then one must use the IPWS 95 version of g (exressed in terms of absolute temerature (K) and absolute ressure (Pa)) rather than the IPWS 9 version. The thermodynamic roerties derived from the IPWS 95 (the Release roviding the Helmholtz function formulation for ure water) and IPWS 8 (the Release endorsing the Feistel (8) Gibbs function) combination are available from the SI software library, while that derived from the IPWS 9 (the Release endorsing the ure water art of Feistel (3)) and IPWS 8 combination are available from the GSW Oceanograhic Toolbox. The GSW Toolbox is restricted to the oceanograhic standard range in temerature and ressure, however the validity of results extends at = to bsolute Salinity u to mineral saturation concentrations (Marion et al. 9). Secific volume (which is the ressure derivative of the Gibbs function) is resently an extraolated quantity outside the Netunian range (i. e. the oceanograhic range) of temerature and bsolute Salinity at =, and exhibits errors there of u to 3%. We emhasize that models of seawater roerties that use a single salinity variable, S, as inut require aroximately fixed chemical comosition ratios (e.g., Na/Cl, Ca/Mg, Cl/HCO3, etc.). s seawater evaorates or freezes, eventually minerals such as CaCO3 will reciitate. Small anomalies are reasonably handled by using S as the inut variable (see section.5) but reciitation may cause large deviations from the nearly fixed ratios associated with

26 6 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater standard seawater. Under extreme conditions of reciitation, models of seawater based on the Millero et al. (8a) Reference Comosition will no longer be alicable. Figure 3 illustrates S t boundaries of validity (determined by the onset of reciitation) for 8 (CO = 385 μ atm ) and (CO = 55 μ atm ) (from Marion et al. (9)). Figure 3. The boundaries of validity of the Millero et al. (8a) comosition at = in Year 8 (solid lines) and otentially in Year (dashed lines). t high salinity, calcium carbonate saturates first and comes out of solution; thereafter the Reference Comosition of Standard Seawater of Millero et al. (8a) does not aly. The Gibbs function (.6.) contains four arbitrary constants that cannot be determined by any set of thermodynamic measurements. These arbitrary constants mean that the Gibbs function (.6.) is unknown and unknowable u to the arbitrary function of temerature and bsolute Salinity (where T is the Celsius zero oint, 73.5 K ) ( ) ( ) a+ a T+ t + a3+ a4 T+ t S (.6.) (see for examle Fofonoff (96) and Feistel and Hagen (995)). The first two coefficients a and a are arbitrary constants of the ure water Gibbs function g W ( t, ) while the second two coefficients a 3 and a 4 are arbitrary coefficients of the saline art of the Gibbs S function g ( S,, t ). Following generally acceted convention, the first two coefficients are chosen to make the entroy and internal energy of liquid water zero at the trile oint and W ( t ) η, = (.6.3) t t W ( ) u t, = (.6.4) t t as described in IPWS 95 and in more detail in Feistel et al. (8a) for the IPWS 95 Helmholtz function descrition of ure water substance. When the ure water Gibbs function g W ( t, ) of (.6.) is taken from the fitted Gibbs function of Feistel (3), the two arbitrary constants a and a are (in the aroriate non dimensional form) g and g of the table in aendix G below. These values of g and g are not identical to the values in Feistel (3) because the resent values have been taken from IPWS 9 and have been chosen to most accurately achieve the trile oint conditions (.6.3) and (.6.4) as discussed in Feistel et al. (8a).

27 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater 7 The remaining two arbitrary constants a 3 and a 4 of (.6.) are determined by ensuring that the secific enthaly h and secific entroy η of a samle of standard seawater with standard ocean roerties ( SSO, tso, SO ) = ( g kg, C, dbar) are both zero, that is that h S, t, = (.6.5) and ( ) SO SO SO ( S t ) η,, =. (.6.6) SO SO SO In more detail, these conditions are actually officially written as (Feistel (8), IPWS 8) and (,, ) (, ) (, ) S W W SO SO SO t t SO SO h S t = u t h t (.6.7) ( S, t, ) ( t, ) ( t, ) S W W SO SO SO t t SO SO η = η η. (.6.8) Written in this way, (.6.7) and (.6.8) use roerties of the ure water descrition (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 (.6.3) and (.6.4)), these constraints on the saline Gibbs function are written this way so that they are indeendent of any subsequent change in the arbitrary constants involved in the thermodynamic descrition of ure water. While the two slightly different thermodynamic descritions of ure water, namely IPWS 95 and IPWS 9, both achieve zero values of the internal energy and entroy at the trile oint of ure water, the values assigned to the enthaly and entroy of ure water at the temerature and W W ressure of the standard ocean, h ( tso, SO ) and η ( tso, SO ) on the right hand sides of W (.6.7) and (.6.8), are slightly different in the two cases. For examle h ( tso, SO ) is 3 3.3x Jkg from IPWS 9 (as described in the table of aendix G) comared with 8 the round off error of x Jkg when using IPWS 95 with double recision arithmetic. This issues is discussed in more detail in section 3.3. The olynomial form and the coefficients for the ure water Gibbs function g W ( t, ) from Feistel (3) and IPWS 9 are given in aendix G, while the combined olynomial and logarithmic form and the coefficients for the saline art of the Gibbs S function g ( S,, t ) (from Feistel (8) and IPWS 8) are reroduced in aendix H. SCOR/IPSO Working Grou 7 has indeendently checked that the Gibbs functions of Feistel (3) and of Feistel (8) do in fact fit the underlying data of various thermodynamic quantities to the accuracy quoted in those two fundamental aers. This checking was erformed by Giles M. Marion, and is summarized in aendix O. Further checking of these Gibbs functions has occurred in the rocess leading u to IPWS aroving these Gibbs function formulations as the Releases IPWS 8 and IPWS 9. Discussions of how well the Gibbs functions of Feistel (3) and Feistel (8) fit the underlying (laboratory) data of density, sound seed, secific heat caacity, temerature of maximum density etc may be found in those aers, along with comarisons with the corresonding algorithms of EOS 8. The IPWS 9 release discusses the accuracy to which the Feistel (3) Gibbs function fits the underlying thermodynamic otential of IPWS 95; in summary, for the variables density, thermal exansion coefficient and secific heat caacity, the rms misfit between IPWS 9 and IPWS 95, in the region of validity of IPWS 9, are a factor of between and less than the corresonding error in the laboratory data to which both thermodynamic otentials were fitted. Hence, in the oceanograhic range of arameters, IPWS 9 and IPWS 95 may be regarded as equally accurate thermodynamic descritions of ure liquid water. The Gibbs function g has units of Jkg in both the SI and GSW software libraries..7 Secific volume

28 8 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater The secific volume of seawater v is given by the ressure derivative of the Gibbs function at constant bsolute Salinity S and in situ temerature t, that is ( ) v = v S,, t = g = g P. (.7.) P S, T Notice that secific volume is a function of bsolute Salinity S rather than of Reference Salinity S R or Practical Salinity S P. The imortance of this oint is discussed in section.8. When derivatives are taken with resect to in situ temerature, or at constant in situ temerature, the symbol t is avoided as it can be confused with the same symbol for time. Rather, we use T in lace of t in the exressions for these derivatives. For many theoretical and modeling uroses in oceanograhy it is convenient to regard the indeendent temerature variable to be Conservative Temerature rather than in situ temerature t. We note here that the secific volume is equal to the ressure derivative of secific enthaly at fixed bsolute Salinity when any one of η, θ or is also held constant, as follows (from aendix.) h P = h P = h P = v. (.7.) S, η S, S, θ The use of P in these equations emhasizes that it must be in Pa not dbar. Secific 3 volume v has units of m kg in both the SI and GSW software libraries..8 Density The density of seawater ρ is the recirocal of the secific volume. It is given by the recirocal of the ressure derivative of the Gibbs function at constant bsolute Salinity S and in situ temerature t, that is ( S t ) ( gp ) ( g P S, T) ρ = ρ,, = =. (.8.) Notice that density is a function of bsolute Salinity S rather than of Reference Salinity S R or Practical Salinity S P. This is an extremely imortant oint because bsolute Salinity S in units of gkg is numerically greater than Practical Salinity by between.65 gkg and.95 gkg in the oen ocean so that if Practical Salinity were inadvertently used as the salinity argument for the density algorithm, a significant density 3 error of between. kg m 3 and.5 kg m would result. For many theoretical and modeling uroses in oceanograhy it is convenient to regard density to be a function of Conservative Temerature rather than of in situ temerature t. That is, it is convenient to form the following two functional forms of density, ( S ) ρ = ˆ ρ,,, (.8.) where is Conservative Temerature. We will adot the convention (see Table L. in aendix L) that when enthaly h, secific volume v or density ρ are taken to be functions of otential temerature they attract an over tilde as in v or ρ, and when they are taken to be functions of Conservative Temerature they attract a caret as in ˆv and ˆ ρ. With this convention, exressions involving artial derivatives such as (.7.) can be written more comactly as (from aendix.) ˆ h = h = h = v = ρ (.8.3) P P P since the other variables are taken to be constant during the artial differentiation. endix P lists exressions for many thermodynamic variables in terms of the thermodynamic otentials h = h ( S, η, ), h = h ( S, θ, ) and h = hˆ ( S,, ). (.8.4) Density ρ has units of 3 kg m in both the SI and GSW software libraries.

29 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater 9 Comutationally efficient exressions for ˆ ρ ( S,, ) and ρ ( S θ ),, involving 5 coefficients are available (McDougall et al. (b)) and are described in aendix.3 and aendix K. These exressions can be integrated with resect to ressure to rovide closed exressions for hˆ ( S,, ) and h ( S, θ, ) (see Eqn. (.3.6))..9 Chemical otentials s for any two comonent thermodynamic system, the Gibbs energy, G, of a seawater samle containing the mass of water m W and the mass of salt m S at temerature t and ressure can be written in the form (Landau and Lifshitz (959), lberty (), Feistel (8)) W S (,,, ) μ μ G m m t = m + m (.9.) W S W S where the chemical otentials of water in seawater defined by the artial derivatives μ W G =, and m W ms, T, W μ and of salt in seawater S mw, T, S μ are S G μ =. (.9.) m Identifying absolute salinity with the mass fraction of salt dissolved in seawater, S = ms / ( mw + ms) (Millero et al. (8a)), the secific Gibbs energy g is given by G W S W S W g( S,, t ) = = ( S) μ + Sμ = μ + S( μ μ ) (.9.3) m + m W S and is indeendent of the total mass of the samle. Note that this exression for g as the sum of a water art and a saline art is not the same as the ure water and the saline slit W in (.6.) ( μ is the chemical otential of water in seawater; it does not corresond to a W ure water samle as g does). This Gibbs energy g is used as the thermodynamic otential function (Gibbs function) for seawater. The above three equations can be used to W S write exressions for μ and μ in terms of the Gibbs function g as μ W ( ) + = = + + = mw ms g g S g g ( mw ms) g S mw S m, W S ms, T, T ms T, (.9.4) and for the chemical otential of salt in seawater, ( ) + S μ = = + + = + mw ms g g S g g ( mw ms) g ( S) ms S m, S S mw, T, T mw T, (.9.5) The relative chemical otential μ (commonly called the chemical otential of seawater ) follows from (.9.4) and (.9.5) as S W g μ = μ μ =, (.9.6) S T, and describes the change in the Gibbs energy of a arcel of seawater of fixed mass if a small amount of water is relaced by salt at constant temerature and ressure. lso, from the fundamental thermodynamic relation (Eqn. (.7.) in aendix.7) it follows that the chemical otential of seawater μ describes the change of enthaly dh if at constant ressure and entroy, a small mass fraction of water is relaced by salt, d S. Equations (.9.4) (.9.6) serve to define the three chemical otentials in terms of the Gibbs function g of seawater. Note that the weights of the sums that aear in Eqns. (.9.) (.9.5) are strictly the mass fractions of salt and of ure water in seawater, so that

30 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater for a seawater samle of anomalous comosition these mass fractions would be more soln dens accurately given in terms of S than by S S. In this regard, the Gibbs energy in Eqn. (.9.) should strictly be the weighted sum of the chemical otentials of all the constituents in seawater. However, ractically seaking, the vaour ressure, the latent heat and the freezing temerature are all rather weakly deendent on salinity, and hence the use of S in this section is recommended. The SI comuter software library (aendix M) redominantly uses basic SI units, so that S has units of kg kg S W and g, μ, μ and μ all have units of Jkg. In the GSW Oceanograhic Toolbox (aendix N) S has units of gkg S W while μ, μ and μ all have units of Jg. This adotion of oceanograhic (i.e. non basic SI) units for S means that secial care is needed in evaluating equations such as (.9.3) and (.9.5) where in the term ( S ) it is clear that S must have units of kg kg. The adotion of non basic SI units is common in oceanograhy, but often causes some difficulties such as this.. Entroy The secific entroy of seawater η is given by ( S t ) g g T η = η,, = =. (..) T S, When taking derivatives with resect to in situ temerature, the symbol T will be used for temerature in order that these derivatives not be confused with time derivatives. Entroy η has units of Jkg K in both the SI and GSW software libraries.. Internal energy The secific internal energy of seawater u is given by (where T is the Celsius zero oint, 73.5 K and P = 35Pa is the standard atmoshere ressure) g g u = u( S,, t ) = g+ ( T+ t) η ( + P) v = g ( T+ t) ( + P). (..) T P S, S, T This exression is an examle where the use of non basic SI units resents a roblem, because in the roduct ( + P ) v, ( + P ) = P must be in Pa if secific volume has its 3 regular units of m kg : hence here sea ressure must be exressed in Pa. lso, the ressure derivative in Eqn. (..) must be done with resect to ressure in Pa. Secific internal energy u has units of Jkg in both the SI and GSW software libraries.. Enthaly The secific enthaly of seawater h is given by Secific enthaly h has units of g h = h( S,, t ) = g+ ( T+ t) η = g ( T+ t). (..) T Jkg S, in both the SI and GSW software libraries.

31 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater.3 Helmholtz energy The secific Helmholtz energy of seawater f is given by g f = f ( S,, t ) = g ( + P) v = g ( + P). P S, T (.3.) This exression is another examle where the use of non basic SI units resents a roblem, because in the roduct ( + P ) v, must be in Pa if secific volume has its regular units of 3 m kg. The secific Helmholtz energy f has units of Jkg in both the SI and GSW comuter software libraries..4 Osmotic coefficient The osmotic coefficient of seawater φ is given by S g φ = φ( S ) ( ( )),, t = g S mswr T + t. S T, (.4.) The osmotic coefficient of seawater describes the change of the chemical otential of water er mole of added salt, exressed as multiles of the thermal energy, RT ( + t) (Millero and Leung (976), Feistel and Marion (7), Feistel (8)), W W (, t, ) ( S, t, ) m R( T t) μ = μ + + φ. (.4.) SW Here, R = Jmol K is the universal molar gas constant. The molality m SW is the number of dissolved moles of solutes (ions) of the Reference Comosition as defined by Millero et al. (8a), er kilogram of ure 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 age 59 of Feistel and Marion (7)). The freezing oint lowering equations (3.33., 3.33.) or the vaour ressure lowering can be comuted from the osmotic coefficient of seawater (see Millero and Leung (976), Bromley et al. (974))..5 Isothermal comressibility 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. t The isothermal and isohaline comressibility of seawater κ is defined by ρ κ κ ρ (,, ) v t t PP = S t = = v = P S, T P S, T gp g (.5.) where the second derivative of g is taken with resect to ressure (in Pa ) at constant S and t. The use of P in the ressure derivatives in Eqn. (.5.) serves to emhasize that these derivatives must be taken with resect to ressure in Pa not in dbar. The t isothermal comressibility of seawater κ roduced by both the SI and GSW comuter software libraries (aendices M and N) has units of Pa.

32 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater.6 Isentroic and isohaline comressibility When the entroy and bsolute Salinity are held constant while the ressure is changed, the isentroic and isohaline comressibility κ is obtained: ρ v ρ ρ κ = κ( S,, t ) = ρ = v = ρ = ρ P S, η P S, η P S, θ P S, (.6.) ( gtp gtt gpp ) =. gp gtt The isentroic and isohaline comressibility κ is sometimes called simly the isentroic comressibility (or sometimes the adiabatic comressibility ), on the unstated understanding that there is also no transfer of salt during the isentroic or adiabatic change in ressure. The isentroic and isohaline comressibility of seawater κ roduced by both the SI and GSW software libraries (aendices M and N) has units of Pa..7 Sound seed The seed of sound in seawater c is given by ( ) ρ ( ρκ) P TT ( TP TT PP) c = c S,, t = P = = g g g g g. (.7.) S, η Note that in these exressions in Eqn. (.7.), since sound seed is in ms and density 3 has units of kg m it follows that the ressure of the artial derivatives must be in Pa and the isentroic comressibility κ must have units of Pa. The sound seed c roduced by both the SI and the GSW software libraries (aendices M and N) has units of ms..8 Thermal exansion coefficients t The thermal exansion coefficient α with resect to in situ temerature t, is t t ρ v gtp α = α ( S,, t ) = = =. ρ T v T g S, S, P (.8.) θ The thermal exansion coefficient α with resect to otential temerature θ, is (see aendix.5) ( ) θ θ ρ v g gtt S,, TP r α = α ( S,, t, r) = = = θ, (.8.) ρ θ v θ g g S, S, where r is the reference ressure of the otential temerature. The g TT derivative in the numerator is evaluated at ( S, θ, r) whereas the other derivatives are all evaluated at ( S,, t ). The thermal exansion coefficient α with resect to Conservative Temerature, is (see aendix.5) ρ v g c TP α = α ( S,, t ) = = =. ρ v g T + g S, S, P P TT ( θ) TT (.8.3) Note that Conservative Temerature is defined only with resect to a reference ressure of dbar so that the θ in Eqn. (.8.3) is the otential temerature with r = dbar. ll the derivatives on the right hand side of Eqn. (.8.3) are evaluated at ( S,, t ). The constant c is defined in Eqn. (3.3.3) below.

33 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater 3.9 Saline contraction coefficients t The saline contraction coefficient β (sometimes also called the haline contraction coefficient) at constant in situ temerature t, is t t ρ v gsp β = β ( S,, t ) = = =. (.9.) ρ S v S g The saline contraction coefficient aendix.5) T, T, θ θ ρ v β = β ( S,, t, r) = = ρ S v S = P θ β at constant otential temerature θ, is (see θ, θ, (, θ, ) g g g S g g g g TP ST ST r TT SP P TT, (.9.) where r is the reference ressure of θ. One of the g ST derivatives in the numerator is evaluated at ( S, θ, r) whereas all the other derivatives are evaluated at ( S,, t ). The saline contraction coefficient β at constant Conservative Temerature, is (see aendix.5) β ρ v = β ( S,, t ) = = ρ S v S =,, ( θ) (, θ,) g g T + g S g g g g TP ST S TT SP P TT. (.9.3) Note that Conservative Temerature is defined only with resect to a reference ressure of dbar as indicated in this equation. The g S derivative in the numerator is evaluated at ( S, θ, ) whereas all the other derivatives are evaluated at ( S,, t ). In the SI comuter software (aendix M) all three saline contraction coefficients are roduced in units of kg kg while in the GSW Oceanograhic Toolbox (aendix N) all three saline contraction coefficients are roduced in units of kg g consistent with the referred oceanograhic unit for S in the GSW Toolbox being gkg.

34 4 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater. Isobaric heat caacity The secific isobaric heat caacity c is the rate of change of secific enthaly with temerature at constant bsolute Salinity S and ressure, so that h c = c S,, t = = T + t g. ( ) ( ) TT T S, (..) The isobaric heat caacity c varies over the S lane at = by aroximately 5%, as illustrated in Figure 4. Figure 4. Contours of isobaric secific heat caacity (in Jkg K ), Eqn. (..), at =. c of seawater The isobaric heat caacity comuter software libraries. c has units of Jkg K in both the SI and GSW. Isochoric heat caacity The secific isochoric heat caacity c v is the rate of change of secific internal energy u with temerature at constant bsolute Salinity S and secific volume, v, so that u cv = cv ( S,, t ) = = ( T + t)( gtt gpp gtp ) gpp. (..) T S, v Note that the isochoric and isobaric heat caacities are related by c t ( T + t)( α ) ( ρκ ) = c, and by cv = c κ. (..) t κ v t The isochoric heat caacity comuter software libraries. c v has units of Jkg K in both the SI and GSW

35 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater 5. The adiabatic lase rate The adiabatic lase rate Γ is the change of in situ temerature with ressure at constant entroy and bsolute Salinity, so that (McDougall and Feistel (3)) t t gtp h v Γ=Γ ( S,, t ) = = = = = = S, η S, TT S S, ( T + t) P P g η P η ρc θ ( T + θ) v ( T + θ) h ( T + θ) α ( T + θ) α c c P ρ c ρc ( S, θ,) = = = = S, S α t. (..) The adiabatic (and isohaline) lase rate is commonly (and incorrectly) exlained as being roortional to the work done on a fluid arcel as its volume changes in resonse to an increase in ressure. ccording to this exlanation the adiabatic lase rate would increase with both ressure and the fluid s comressibility, but this is not the case. Rather, the adiabatic lase rate is roortional to the thermal exansion coefficient and is indeendent of the fluid s comressibility. Indeed, the adiabatic lase rate changes sign at the temerature of maximum density whereas the comressibility and the work done by comression is always ositive. McDougall and Feistel (3) show that the adiabatic lase rate is indeendent of the increase in the internal energy that a arcel exeriences when it is comressed. Rather, the adiabatic lase rate reresents that change in temerature that is required to kee the entroy (and also θ and ) of a seawater arcel constant when its ressure is changed in an adiabatic and isohaline manner. The reference ressure of the otential temerature θ that aears in the last four exressions in Eqn. (..) is r = dbar. The adiabatic lase rate Γ outut of both the SI and the GSW comuter software libraries is in units of K Pa.

36 6 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater 3. Derived Quantities 3. Potential temerature The very useful concet of otential temerature was alied to the atmoshere originally by Helmholtz (888), first under the name of heat content, and later renamed otential temerature (Bezold (888)). These concets were transferred to oceanograhy by Helland Hansen (9). Potential temerature is the temerature that a fluid arcel would have if its ressure were changed to a fixed reference ressure r in an isentroic and isohaline manner. The hrase isentroic and isohaline is used reeatedly in this document. To these two qualifiers we should really also add without dissiation of mechanical energy. rocess that obeys all three restrictions is a thermodynamically reversible rocess. Note that one often (falsely) reads that the requirement of a reversible rocess is that the rocess occurs at constant entroy. However this statement is misleading because it is ossible for a fluid arcel to exchange some heat and some salt with its surroundings in just the right ratio so as to kee its entroy constant, but the rocesses is not reversible (see Eqn. (.7.)). Potential temerature referred to reference ressure r is often written as the ressure integral of the adiabatic lase rate (Fofonoff (96), (985)) P r ( r) ( [ ] ) θ = θ S,, t, = t+ Γ S, θ S,, t,, dp. (3..) P Note that this ressure integral needs to be done with resect to ressure exressed in Pa not dbar. The algorithm that is used with the TEOS Gibbs function aroach to seawater equates the secific entroies of two seawater arcels, one before and the other after the isentroic and isohaline ressure change. In this way, θ is evaluated using a Newton Rahson iterative solution technique to solve the following equation for θ η S, θ, = η S, t,, (3..) ( ) ( ) r or, in terms of the Gibbs function, g, ( θ ) ( ) g S,, = g S, t,. (3..3) T r T This relation is formally equivalent to Eqn. (3..). In the GSW Oceanograhic Toolbox θ 4 is found to machine recision ( C ) in two iterations of a modified Newton Rahson method, using a suitable initial value as described by McDougall et al. (b).. Note that the difference between the otential and in situ temeratures is not due to the work done in comressing a fluid arcel on going from one ressure 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 (3)). Rather, the difference between these temeratures is what is required to kee the entroy constant during the adiabatic and isohaline ressure change. The otential temerature θ outut of the SI software is in units of K while the outut from the GSW Toolbox is in C.

37 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater 7 3. Potential enthaly Potential enthaly h is the enthaly that a fluid arcel would have if its ressure were changed to a fixed reference ressure r in an isentroic and isohaline manner. Because heat fluxes into and out of the ocean occur mostly near the sea surface, the reference ressure for otential enthaly is always taken to be r = dbar (that is, at zero sea ressure). Potential enthaly can be exressed as the ressure integral of secific volume as (from McDougall (3) and see the discussion below Eqn. (.8.)) (,, ) = (, θ, ) = (, θ) = (,, ) (, θ[,,, ], ) h S t h S h S h S t v S S t dp P P (,, ) (, θ, ) P ( ) ˆ ( ) = h S,, t v S,, dp, P P = h( S,, t ) v( S, η, ) dp P = h S t v S dp P P (3..) and we emhasize that the ressure integrals here must be done with resect to ressure exressed in Pa rather than dbar. In terms of the Gibbs function, otential enthaly h is evaluated as h S, t, = h S, θ, = g S, θ, T + θ g S, θ,. (3..) ( ) ( ) ( ) ( ) ( ) T 3.3 Conservative Temerature Conservative Temerature is defined to be roortional to otential enthaly according to S,, t = S, θ = h S,, t c = h S, θ c (3.3.) ( ) ( ) ( ) ( ) where the value that is chosen for c is motivated in terms of otential enthaly evaluated at an bsolute Salinity of S = 35u = g kg and at θ = 5 C by SO PS (,5 C, ) (, C,) h SSO h SSO J kg K, (3.3.) (5 K) h S SO is zero according to the way the Gibbs function is defined in (.6.5). In fact we adot the exact definition for c to be the 5 digit value in (3.3.), so that J kg K. (3.3.3) noting that (, C,dbar) c and S SO differ from C and 5 C resectively by the round off amount of 5 C. When IPWS 9 (which is based on the aer of Feistel (3), see aendix G) is used for the ure water art of the Gibbs function, ( S SO, C,) differs from C 8 by 8.5 C and ( S SO,5 C,) differs from 5 C by C. Over the temerature range from C to 4 C the difference between Conservative Temerature 5 using IPWS 95 and IPWS 9 as the ure water art is no more than ±.5 C, a temerature difference that will be ignored. The value of c in (3.3.3) is very close to the average value of the secific heat caacity c at the sea surface of today s global ocean. This value of c also causes the average value of θ at the sea surface to be very close to zero. Since c is simly a constant of roortionality between otential enthaly and Conservative Temerature, it is totally When IPWS 95 is used for the ure water art of the Gibbs function, ( S SO, C,) (,5 C,)

38 8 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater arbitrary, and we see no reason why its value would need to change from (3.3.3) even when in future decades an imroved Gibbs function of seawater is agreed uon. endix.8 outlines why Conservative Temerature gets its name; it is aroximately two orders of magnitude more conservative comared with either otential temerature or entroy. The SI and GSW software libraries both include an algorithm for determining Conservative Temerature from values of bsolute Salinity S and otential temerature θ referenced to = dbar. These libraries also have an algorithm for evaluating otential temerature (referenced to dbar ) from S and. This inverse algorithm, ˆ θ ( S, ), has an initial seed based on a rational function aroximation and 4 finds otential temerature to machine recision ( C ) in one and a half iterations of a modified Newton Rahson technique (McDougall et al. (b)). 3.4 Potential density θ Potential density ρ is the density that a fluid arcel would have if its ressure were changed to a fixed reference ressure r in an isentroic and isohaline manner. Potential density referred to reference ressure r can be written as the ressure integral of the isentroic comressibility as P r ( r) ( ) ( [ ] ) ( [ ] ) θ ρ S,, t, = ρ S,, t + ρ S, θ S,, t,, κ S, θ S,, t,, dp. (3.4.) P The simler exression for otential density in terms of the Gibbs function is θ ( S ) ( [ ] ) t r S S t r r gp ( S [ S t r] r) ρ,,, = ρ, θ,,,, =, θ,,,,. (3.4.) Using the functional forms of Eqn. (.8.) and (.8.3) for in situ density, that is, either ρ = ρ ( S, θ, ) or ρ = ˆ ρ( S,, ), otential density with resect to reference ressure r (e. g. dbar) can be easily evaluated as θ ρ S,, t, = ρ S,, t, = ρ S, η, = ρ S, θ, = ˆ ρ S,,, (3.4.3) ( ) ( ) ( ) ( ) ( ) r r r r r where we note that the otential temerature θ in the enultimate exression is the otential temerature with resect to dbar. Once the reference ressure is fixed, otential density is a function only of bsolute Salinity and Conservative Temerature (or equivalently, of bsolute Salinity and otential temerature). Note that it is equally θ correct to label otential density as ρ or ρ η (or indeed as ρ ) because η, θ and are constant during the isentroic and isohaline ressure change from to r ; that is, these variables osses the otential roerty of aendix.9. Following the discussion after Eqn. (.8.) above, otential density may also be exressed in terms of the ressure derivative of the exressions h = h ( S, θ, ) and h = hˆ S,, for enthaly as (see also aendix P) ( ) ( S t ) = ( S t ) = h ( S = ) = hˆ ( S = ) θ ρ,,, r ρ,,, r P, θ, r P,, r. (3.4.4) 3.5 Density anomaly t Density anomaly σ is an old fashioned density measure that is now seldom used. It is the density evaluated at the in situ temerature but at zero sea ressure, minus 3 kg m, that is, t 3 3 σ ( S,, t ) = ρ( S,, t ) kgm = gp ( S,, t ) kgm. (3.5.) t θ σ was used as an aroximation to σ which avoided the comutational demand of t evaluating θ. Density anomaly σ is not rovided in the TEOS software libraries.

39 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater Potential density anomaly Potential density anomaly, θ θ σ or σ, is simly otential density minus kg m 3, θ ( S t r) = ( S t r) = ( S t r) = ρ ( S t r) θ [ ] σ,,, σ,,, ρ,,, kgm,,, kgm ( r r) P θ = g S, S, t,,, kg m. (3.6.) Note that it is equally correct to label otential density anomaly as σ or σ because both θ and are constant during the isentroic and isohaline ressure change from to r. 3.7 Secific volume anomaly The secific volume anomaly δ is defined as the difference between the secific volume and a given function of ressure. Traditionally δ has been defined as ( S,, t ) v( S,, t ) v( S,C, ) δ = (3.7.) SO (where the traditional value of Practical Salinity of 35 has been udated to an bsolute Salinity of SSO = 35u PS = g kg in the resent formulation). Note that the second term, v ( S SO, C, ), is a function only of ressure. In order to have a surface of constant secific volume anomaly more accurately aroximate neutral tangent lanes (see section 3.), it is advisable to relace the arguments S SO and C with more general values S and t that are carefully chosen (as say the median values of bsolute Salinity and temerature along the surface) so that the more general definition of secific volume anomaly is δ S,, t = v S,, t v S, t, = g S,, t g S, t,. (3.7.) ( ) ( ) ( ) P( ) P( ) The last terms in Eqns. (3.7.) and (3.7.) are simly functions of ressure and one has the freedom to choose any other function of ressure in its lace and still retain the dynamical roerties of secific volume anomaly. In articular, one can construct secific volume and enthaly to be functions of Conservative Temerature (rather than in situ temerature) as vˆ ( S,, ) and hˆ ( S,, ) and write a slightly different definition of secific volume anomaly as,, ˆ,, ˆ δ S = v S v S,, = h ˆ S,, hˆ S,,. (3.7.3) ( ) ( ) ( ) P( ) P( ) This is the form of secific volume anomaly adoted in the GSW Oceanograhic Toolbox where the default values of the reference values S and are S SO = g kg and C resectively. The same can also be done with otential temerature so that in terms of the secific volume v ( S, θ, ) and enthaly h ( S, θ, ) we can write another form of the secific volume anomaly as δ, θ,, θ,, θ,, θ,, θ,. (3.7.4) ( S ) = v( S ) v( S ) = hp( S ) hp( S ) These exressions exloit the fact that (see aendix.) h P = h P = h P = v. (3.7.5) S, η S, S, θ

40 3 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater 3.8 Thermobaric coefficient The thermobaric coefficient quantifies the rate of variation with ressure of the ratio of the thermal exansion coefficient and the saline contraction coefficient. With resect to otential temerature θ the thermobaric coefficient is (McDougall (987b)) θ θ ( ) α β θ θ θ θ θ θ α α β Tb = Tb ( S, t ) = β =. P P P S, θ θ β S, θ S, θ (3.8.) This exression for the thermobaric coefficient is most readily evaluated by differentiating an exression for density exressed as a function of otential temerature rather than in situ temerature, that is, with density exressed in the functional form ρ= ρ ( S, θ, ). With resect to Conservative Temerature the thermobaric coefficient is ( ) α β α α β Tb = Tb ( S, t ) = β =. P P P S, β S, S, (3.8.) This exression for the thermobaric coefficient is most readily evaluated by differentiating an exression for density exressed as a function of Conservative Temerature rather than in situ temerature, that is, with density exressed in the functional form ρ = ˆ ρ( S,, ). The thermobaric coefficient enters various quantities to do with the ath deendent nature of neutral trajectories and the ill defined nature of neutral surfaces (see (3.3.) (3.3.7)). The thermobaric dianeutral advection associated with the lateral mixing of heat Tb θ and salt along neutral tangent lanes is given by e = gn KTb nθ np or Tb e = gn KTb n np where nθ and n are the two dimensional gradients of either otential temerature or Conservative Temerature along the neutral tangent lane, n P is the corresonding eineutral gradient of absolute ressure and K is the eineutral diffusion coefficient. Note that the thermobaric dianeutral advection is roortional to the mesoscale eddy flux of heat along the neutral tangent lane, ck n, and is indeendent of the amount of small scale (dianeutral) turbulent mixing and hence is also indeendent of the dissiation of mechanical energy ε (Klocker and McDougall (a)). It is shown in aendix.4 below that while the eineutral diffusive fluxes K nθ and K n are different, the roduct of these fluxes with their θ resective thermobaric coefficients is the same, that is, Tb nθ = Tb n. Hence the Tb thermobaric dianeutral advection e is the same whether it is calculated as θ gn KTb nθ np or as gn KTb n np. Exressions for T θ b and T b in terms of enthaly in the functional forms h ( S, θ, ) and hˆ ( S ),, can be found in aendix P. Interestingly, for given magnitudes of the eineutral gradients of ressure and Tb Conservative Temerature, the dianeutral advection, e = gn KTb n np, of thermobaricity is maximized when these gradients are arallel, while neutral helicity is maximized when these gradients are erendicular, since neutral helicity is roortional to T b ( np n) k (see Eqn. (3.3.)). Tb This thermobaric vertical advection rocess, e, is absent from standard layered ocean models in which the vertical coordinate is a function only of S and (such as σ, otential density referenced to dbar). s described in aendix.7 below, the isoycnal diffusion of heat and salt in these layered models, caused by both arameterized 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 Tb velocity e through these surfaces (Klocker and McDougall (a)). In both the SI and GSW comuter software libraries the thermobaric arameter is outut in units of K Pa.

41 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater 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 lane. With resect to otential temerature θ the cabbeling coefficient is (McDougall (987b)) θ θ θ θ θ θ θ α α α α β Cb = Cb ( S, t ) = +. θ β β θ θ S S S, θ, θ, (3.9.) This exression for the cabbeling coefficient is most readily evaluated by differentiating an exression for density exressed as a function of otential temerature rather than in situ temerature, that is, with density exressed in the functional form ρ= ρ ( S, θ, ). With resect to Conservative Temerature the cabbeling coefficient is α α α α β Cb = Cb ( S, t ) = +. β S β S S,,, (3.9.) This exression for the cabbeling coefficient is most readily evaluated by differentiating an exression for density exressed as a function of Conservative Temerature rather than in situ temerature, that is, with density exressed in the functional form ρ = ˆ ρ( S,, ). The cabbeling dianeutral advection associated with the lateral mixing of heat and salt Cab along neutral tangent lanes is given by e = gn KC b n n (or less accurately by Cab e gn KC θ b nθ nθ ) where nθ and n are the two dimensional gradients of either otential temerature or Conservative Temerature along the neutral tangent lane and K is the eineutral diffusion coefficient. The cabbeling dianeutral advection is roortional to the mesoscale eddy flux of heat along the neutral tangent lane, K n, and is indeendent of the amount of small scale (dianeutral) turbulent mixing and hence is also indeendent of the dissiation of mechanical energy (Klocker and θ McDougall (a)). It is shown in aendix.4 that Cb nθ nθ Cb n n so that the estimate of the cabbeling dianeutral advection is different when calculated using otential temerature than when using Conservative Temerature. The estimate using otential temerature is slightly less accurate because of the non conservative nature of otential temerature. When the cabbeling and thermobaricity rocesses are analyzed by considering the mixing of two fluid arcels one finds that the density change is roortional to the square of the roerty ( and/or ) contrasts between the two fluid arcels (for the cabbeling case, see Eqn. (.9.4) in aendix.9). This leads to the thought that if an ocean front is slit u into a series of many smaller fronts then the effects of cabbeling and thermobaricity might be reduced by erhas the square of the number of such fronts. This is not the case. Rather, the total dianeutral transort across a frontal region deends on the roduct of the lateral flux of heat assing through the front and the contrast in temerature and/or ressure across the front, but is indeendent of the sharness of the front (Klocker and McDougall (a)). This can be understood by noting from above that Cab the dianeutral velocity due to cabbeling, e = gn KCb n n, is roortional to the scalar roduct of the eineutral flux of heat ck n and the eineutral temerature gradient. n Satially integrating this roduct over the area of the frontal region, one finds that the total dianeutral transort is roortional to the lateral heat flux times the difference in temerature across the frontal region (in the case of cabbeling) or the difference in ressure across the frontal region (in the case of thermobaricity). In both the SI and GSW software libraries the cabbeling arameter is outut in units of K. Exressions for C θ b and C b in terms of enthaly in the functional forms h ( S, θ, ) and hˆ ( S ),, can be found in aendix P.

42 3 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater 3. Buoyancy frequency The square of the buoyancy frequency (sometimes called the Brunt Väisälä frequency) is given in terms of the vertical gradients of density and ressure, or in terms of the vertical gradients of Conservative Temerature and bsolute Salinity (or in terms of the vertical gradients of otential temerature and bsolute Salinity) by (the g on the lefthand side is the gravitational acceleration, and x, y and z are the satial Cartesian coordinates) ( / ) = ρ ρ z + κ z = ρ ρz z g N P P c θ = αθ β S z z z x, y = α β S z θ x, y. N (3..) For two seawater arcels searated by a small distance Δ z in the vertical, an equally accurate method of calculating the buoyancy frequency is to bring both seawater arcels adiabatically and without exchange of matter to the average ressure and to calculate the difference in density of the two arcels after this change in ressure. In this way the otential density of the two seawater arcels are being comared at the same ressure. This common rocedure calculates the buoyancy frequency N according to g Δρ g Δρ N = =, (3..) ρ Δz ΔP where Δ ρ is the difference between the otential densities of the two seawater arcels with the reference ressure being the average of the two original ressures of the seawater arcels. The last art of Eqn. (3..) has used the hydrostatic relation Pz = gρ. 3. Neutral tangent lane The neutral lane is that lane in sace in which the local arcel of seawater can be moved an infinitesimal distance without being subject to a vertical buoyant restoring force; it is the lane of neutral or zero buoyancy. The normal vector to the neutral tangent lane n is given by g N n = ρ ρ + κ P = ρ ρ P/ c θ θ = α θ β S = α β S. ( ) (3..) s defined, n is not quite a unit normal vector, rather its vertical comonent is exactly k, θ θ that is, its vertical comonent is unity. It is clear that α θ β S is exactly equal to α β S. θ θ Interestingly, both α θ and β S are indeendent of the four arbitrary constants of the Gibbs function (see Eqn. (.6.)) while both α and β S contain an identical additional arbitrary term roortional to a3 S ; terms that exactly cancel in their difference, α β S, in Eqn. (3..). Exressing the two dimensional gradient of roerties in the neutral tangent lane by n, the roerty gradients in a neutral tangent lane obey ρ ρ + κ P = ρ ρ P/ c ( ) n n n n θ θ = α nθ β ns = α n β ns =. Here n is an examle of a rojected non orthogonal gradient (3..)

43 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater 33 τ τ r x r y r τ i + j + k, (3..3) that is widely used in oceanic and atmosheric theory and modelling. Horizontal distances are measured between the vertical lanes of constant latitude x and longitude y while the values of the roerty τ are evaluated on the r surface (e. g. an isoycnal surface, or in the case of n, a neutral tangent lane). This coordinate system is described by Sutcliffe (947), Bleck (978), McDougall (987b), McDougall (995) and Griffies (4). Note that rτ has no vertical comonent; it is not directed along the r surface, but rather it oints in exactly the horizontal direction. Finite difference versions of Eqn. (3..) such as α Δ β ΔS are also very accurate. Here α and β are the values of these coefficients evaluated at the average values of, S and of two arcels ( S,, ) and ( S,, ) on a neutral surface and Δ and Δ S are the roerty differences between the two arcels. The error involved with this finite amlitude version of Eqn. (3..), namely b ( ) T P P d, (3..4) is described in section and aendix (c) of Jackett and McDougall (997). n equally accurate finite amlitude version of Eqn. (3..) is to equate the otential densities of the =.5 +. two fluid arcels, each referenced to the average ressure ( ) 3. Geostrohic, hydrostatic and thermal wind equations The geostrohic aroximation to the horizontal momentum equations ((B9) below) equates the Coriolis term to the horizontal ressure gradient z P so that the geostrohic equation is z f k ρu = P or fv = k P. (3..) v k k u is the horizontal velocity where k is the vertical unit vector (ointing uwards) and f is the Coriolis arameter. The hydrostatic equation is an aroximation to the vertical momentum equation (see Eqn. (B9)), namely where u is the three dimensional velocity and = ( ) Pz ρ z = gρ. (3..) The use of P in these equations rather than serves to remind us that in order to retain the usual units for height, density and the gravitational acceleration, ressure in these dynamical equations must be exressed in Pa not dbar. The so called thermal wind equation is an equation for the vertical gradient of the horizontal velocity under the geostrohic aroximation. Vertically differentiating Eqn. (3..), using the hydrostatic equation Eqn. (3..) and ignoring the tiny term in ρ z (which is of Boussinesq magnitude), the thermal wind can be written as ( ) g N z = z z = zρ = n, ρ ρ gρ f v k P k k P (3..3) where z is the gradient oerator in the exactly horizontal direction along geootentials, and the last art of this equation relates the thermal wind to the ressure gradient in the neutral tangent lane, that is, effectively to the sloe of the neutral tangent lane (see McDougall (995)).

44 34 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater 3.3 Neutral helicity Neutral tangent lanes (which do exist) do not link u in sace to form a well defined n neutral surface unless the neutral helicity H is everywhere zero on the surface. Neutral helicity is defined as the scalar roduct of the vector α β S with its curl, n ( α β ) ( α β ) H S S (3.3.) and this is roortional to the thermobaric coefficient according to n H = β T P S b z β b b b ( ) = P T S k ( n n ) ( ) = g N T P k g N T P k a a T b of the equation of state (3.3.) where P z is simly the vertical gradient of ressure ( Pa m ) and n and are the two dimensional gradients of in the neural tangent lane and in the horizontal lane (actually the isobaric surface) resectively. The gradients a P and a are taken in an θ θ aroximately neutral surface. Since α θ β S and α β S are exactly equal, neutral helicity can be defined in Eqn. (3.3.) as the scalar roduct of this vector with its curl based on either formulation, so that (from the third line of Eqn. (3.3.), and θ bearing in mind that n and nθ are arallel vectors) we see that Tb nθ = Tb n, a 3 result that we use in section 3.8 and in aendix.4. Neutral helicity has units of m. Because of the non zero neutral helicity in the ocean, lateral motion following neutral tangent lanes has the character of helical motion. That is, if we ignore the effects of diaycnal mixing rocesses (as well as ignoring cabbeling and thermobaricity), the mean flow around ocean gyres still asses 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 exressed in Eqns. (.5.4) and (.5.6) in terms of the aroriate mean horizontal velocity and the difference between the sloe of the neutral tangent lane and the sloe of a well defined density surface. Neutral helicity is roortional to the comonent of the vertical shear of the geostrohic velocity ( v z, the thermal wind ) in the direction of the temerature gradient along the neutral tangent lane n, since, from Eqn. (3..3) and the third line of (3.3.) we find that n H = ρt fv (3.3.3) b z n. In the evolution equation of otential vorticity defined with resect to otential θ θ density ρ there is the baroclinic roduction term ρ ρ ρ (Straub (999)) and the first term in a Taylor series exansion for this baroclinic roduction term is roortional to neutral helicity and is given by (McDougall and Jackett (7)) ( ) ρ ρ ρ (3.3.4) θ P P n r P H where P r is the reference ressure of the otential density. Similarly, the curl in a otential density surface of the horizontal ressure gradient term in the horizontal, is given by (McDougall and Klocker ()) momentum equation, σ ( z ρ ) n ( ρ ) ( ) ρ r. σ z P k = H P P (3.3.5) z The fact that this curl is nonzero roves that a geostrohic streamfunction does not exist in a otential density surface. n Neutral helicity H also arises in the context of finding a closed exression for the mean velocity in the ocean. The comonent of the horizontal velocity in the direction

45 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater 35 along a contour of in a neutral tangent lane, namely the velocity comonent v k /, is given by (McDougall (995), Zika et al. (a, b)) n n n v n H vz k = +, φρft φ (3.3.6) so that the full exression for the horizontal velocity is n z b n H v z n n v φρ z ftb n φz n n n v = + k +. (3.3.7) Here φ z is the rate of siraling (radians er meter) in the vertical of the contours on neutral tangent lanes, and v is the velocity comonent across the contour on the neutral tangent lane (a velocity comonent that results from irreversible mixing rocesses). Neutral helicity arises in this context because it is roortional to the comonent of the thermal wind vector v z in the direction across the contour on the neutral tangent lane (see (3.3.3)). This equation (3.3.7) for the isoycnally averaged velocity v shows that in the absence of mixing rocesses (so that v = v z = ) and so long as (i) the eineutral contours siral in the vertical and (ii) n is not zero, then neutral n helicity H is required to be non zero in the ocean whenever the ocean is not motionless. Interestingly, for given magnitudes of the eineutral gradients of ressure and Conservative Temerature, neutral helicity is maximized when these gradients are erendicular since neutral helicity is roortional to T b ( np n) k (see Eqn. Tb (3.3.)), while the dianeutral advection of thermobaricity, e = gn KTb n np, is maximized when and are arallel (see section 3.8). n n P z 3.4 Neutral Density Neutral Density is the name given to a density variable that results from the comuter software described in Jackett and McDougall (997). Neutral Density is given the symbol n γ but it is not a thermodynamic variable as it is a function not only of salinity, temerature and ressure, but also of latitude and longitude. Because of the non zero n neutral helicity H in the ocean it is not ossible to form surfaces that are everywhere osculate with neutral tangent lanes (McDougall and Jackett (988)). Neutral Density surfaces minimize in some sense the global differences between the sloes of the neutral tangent lane and the Neutral Density surface. This sloe difference is given by ( β α ) nz az gn as a s = = (3.4.) where n z is the sloe of the neutral tangent lane, a z is the sloe of the aroximately neutral surface and a is the two dimensional gradient oerator in the aroximately neutral surface (of which a Neutral Density surface is one examle). The vertical velocity through an aroximately neutral surface due to lateral motion along a neutral tangent lane is the scalar roduct v s where v is the horizontal velocity (see Eqn. (.5.4)). Since Neutral Density is not a thermodynamic variable, it will not be described more fully in this manual.

46 36 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater 3.5 Stability ratio The stability ratio R ρ is the ratio of the vertical contribution from Conservative Temerature to that from bsolute Salinity to the static stability N of the water column. From (3..) above we find R ρ α z α θz =. (3.5.) β θ ( S ) β ( S ) z θ z 3.6 Turner angle The Turner angle Tu, named after J. Stewart Turner, is defined as the four quadrant arctangent (Ruddick (983) and McDougall et al. (988), articularly their Figure ) ( α z β ( ) α z β ( ) z z) θ θ θ θ αθ β ( S ) αθ β ( S ) Tu = tan + S, S ( z z ) tan +, z z (3.6.) 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 MTLB. The Turner angle Tu is quoted in degrees of rotation. Turner angles between 45 and 9 reresent the salt finger regime of double diffusive convection, with the strongest activity near 9. Turner angles between 45 and 9 reresent the diffusive regime of double diffusive convection, with the strongest activity near 9. Turner angles between 45 and 45 reresent regions where the stratification is stably stratified in both and S. Turner angles greater than 9 or less than 9 characterize a statically unstable water column in which N <. s 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 Oceanograhic Toolbox from the function gsw_turner_rsubrho_ct Proerty gradients along otential density surfaces The two dimensional gradient of a scalar ϕ along a otential density surface σϕ is related to the corresonding gradient in the neutral tangent lane nϕ by ϕ R [ z ρ r ] σϕ = nϕ + n (3.7.) z Rρ r (from McDougall (987a)), where r is the ratio of the sloe on the S diagram of an isoline of otential density with reference ressure r to the sloe of a otential density surface with reference ressure, and is defined by α ( S,, ) β ( S,, ) r =. (3.7.) α S,, β S,, ( ) ( ) r r Substituting ϕ = into (3.7.) gives the following relation between the (arallel) isoycnal and eineutral gradients of r Rρ σ = n = G n (3.7.3) Rρ r where the isoycnal temerature gradient ratio Rρ G (3.7.4) Rρ r

47 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater 37 has been defined as a shorthand exression for future use. This ratio G is available in the GSW Toolbox from the algorithm gsw_isoycnal_vs_nt_ct_ratio_ct5, while the ratio r of Eqn. (3.7.) is available there as gsw_isoycnal_sloe_ratio_ct5. Substituting ϕ = S into Eqn. (3.7.) gives the following relation between the (arallel) isoycnal and eineutral gradients of S = R = G S ρ σ ns ns R r r ρ. (3.7.5) 3.8 Sloes of otential density surfaces and neutral tangent lanes comared The two dimensional sloe of a surface is defined as the two dimensional gradient of height z of that surface. The two dimensional sloe of a surface is an exactly horizontal gradient vector; it has no vertical comonent. The sloe difference between the neutral tangent lane and a otential density surface with reference ressure r is given by (McDougall (988)) [ ] [ ] Rρ r n n R r n σ ρ σ nz σ z = = ( G ) = =. (3.8.) Rρ r z z z r Rρ z While otential density surfaces have been the most commonly used surfaces with which to searate isoycnal mixing rocesses from vertical mixing rocesses, many other tyes of density surface have been used. The list includes secific volume anomaly surfaces, atched otential density surfaces (Reid and Lynn (97)), Neutral Density surfaces (Jackett and McDougall (997)), orthobaric density surfaces (de Szoeke et al. ()) and some olynomial fits of Neutral Density as function of only salinity and either θ or (Eden and Willebrand (999), McDougall and Jackett (5b)). The most recent method for forming aroximately neutral surfaces is that of Klocker et al. (9a,b). This method is relatively comuter intensive but has the benefit that the remnant mis match between the final surface and the neutral tangent lane at each oint is due only to the neutral helicity of the data through which the surface asses. The relative skill of all these surfaces at aroximating the neutral tangent lane sloe at each oint has been summarized in the equations and histogram lots in the aers of McDougall (989, 995), McDougall and Jackett (5a, 5b), and Klocker et al. (9a,b). When lateral mixing with isoycnal diffusivity K is imosed along otential density surfaces rather than along neutral tangent lanes, a fictitious diaycnal diffusivity arises which is often labeled the Veronis effect after Veronis (975) (who considered the ill effects of exactly horizontal versus isoycnal mixing). This fictitious diaycnal diffusivity of density is equal to K times the square of the sloe error, Eqn. (3.8.) (Klocker et al. (9a)). 3.9 Sloes of in situ density surfaces and secific volume anomaly surfaces The vector sloe of an in situ density surface, ρ z, is defined to be the exactly horizontal vector = z z i + j + k, (3.9.) ρ z x ρ y ρ reresenting the di of the surface in both horizontal directions (note that height z is defined ositive uwards). This vector sloe can be related to the (very small) sloe of isobaric surface by ( g here is the gravitational acceleration) (McDougall (989))

48 38 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater g c ρ z z = ( nz z) +, (3.9.) N where c is the seed of sound and N is the buoyancy frequency. In the uer water column where the square of the buoyancy frequency is significantly larger than 5 g c 4.3x s, the in situ density surface has a similar sloe to the neutral tangent lane n z. In the dee ocean N is only about % of g c and so the surfaces of constant in situ density have a sloe of only % of the sloe of the neutral tangent lane. 5 t a ressure of about dbar where N s, the sloe of an in situ density surface is only about one fifth that of the neutral tangent lane. Neutrally buoyant floats in the ocean are usually metal cylinders that are much less comressible 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 aroximately occurs on surfaces of constant in situ density which at mid deth in the ocean are much closer to being isobaric surfaces than being locally referenced otential density surfaces. This is why these floats are sometimes described as isobaric floats. The sloe of a secific volume anomaly surface, z, can be exressed as g c g c δ z z = ( nz z) +, N N where c is the sound seed of the reference arcel ( S, ) δ (3.9.3) at ressure. This exression confirms that where the local seawater roerties are close to those of the reference arcel, the secific volume anomaly surface can closely aroximate the neutral tangent lane. The square bracket in Eqn. (3.9.3) is equal to ρgn δ z (from section 7 of McDougall (989) where δ is secific volume anomaly). 3. Potential vorticity Planetary otential vorticity is the Coriolis arameter f times the vertical gradient of a suitable variable. Potential density is sometimes used for that variable but using otential density (i) involves an inaccurate searation between lateral and diaycnal advection because otential density surfaces are not a good aroximation to neutral tangent lanes and (ii) incurs the non conservative baroclinic roduction term of Eqn. (3.3.4). Using aroximately neutral surfaces, ans, (such as Neutral Density surfaces) rovides an otimal searation between the effects of lateral and diaycnal mixing in the otential vorticity equation. In this case the otential vorticity variable is roortional to the recirocal of the thickness between a air of closely saced aroximately neutral surfaces. This lanetary otential vorticity variable is called Neutral Surface Potential Vorticity ( NSPV for short) and is related to fn by { ans b ( ) } n z a P a NSPV gρ f γ fn ex ρg N T P dl. (3..) The exonential exression was derived by McDougall (988) (his equation (47)) and is aroximate because the variation of the saline contraction coefficient β with ressure was neglected in comarison with the larger roortional change in the thermal exansion coefficient α with ressure. The integral in Eqn. (3..) is taken along an aroximately neutral surface from a location where NSPV is equal to fn. Interestingly the combination a P a P is simly the isobaric gradient of Conservative Temerature, P, which is almost the same as the horizontal gradient, z. more accurate version of this equation which does not ignore the variation of the saline contraction coefficient can be shown to be

49 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater { ( ) l ans } { ans P ρκ l} n z ex ( ) P P ( ) P P NSPV gρ f γ = fn g N ρα ρβ S d = fn ex g N ( ) d. 39 (3..) The exonential factor in Eqn. (3..) is aroximately the integrating factor b, defined n l l l l as b γ ρ ( ρ ρ ) where ρ ρ( β S α ), which allows satial integrals l n of ρb( β S α ) = b ρ γ to be aroximately indeendent of ath for vertical aths, that is, for aths in surfaces whose normal has zero vertical comonent. The gradient of fn is related to that of NSPV by (from Eqns. (3..) and (3..)) a ( fn ) ( NSPV) g N ρκ ρ g N T b ( P) ln ln = ( ). (3..3) a a P a P a The deficiencies of fn as a form of lanetary otential vorticity have not been widely areciated. Even in a lake, and also in the simle situation where temerature does not vary along a density surface ( = a ), the use of fn as lanetary otential vorticity is inaccurate since the right hand side of (3..3) is then aroximately Rρ ρ g N Tb P ap = Tb ap, (3..4) α Rρ and the mere fact that the density surface has a sloe (i. e. a P ) means that the contours of fn will not be arallel to contours of NSPV on the density surface. (In this situation (where = a ) the contours of NSPV along aroximately neutral surfaces coincide with those of isoycnal otential vorticity ( IPV ), the otential vorticity defined with resect to the vertical gradient of otential density by IPV = fgρ ρ z ). IPV is related to fn by (McDougall (988)) ( r) ( ) ( r) ( ) R r IPV gρ ρ β z ρ β = =, (3..5) fn N β Rρ β G G so that the ratio of NSPV to IPV lotted on an aroximately neutral surface is given by ( ) ( ) NSPV β Rρ = ex { g N ( ) ans P ρκ d }. l (3..6) IPV β r Rρ r You and McDougall (99) show that because of the highly differentiated nature of otential vorticity, isolines of IPV and NSPV do not coincide even at the reference ressure r of the otential density variable (see equations (4) (6) and Figure 4 of 3 that aer). NSPV, fn and IPV have the units s. The ratio IPV fn is available in the GSW Oceanograhic Toolbox as the function gsw_ipv_vs_fnsquared_ratio_ct5. 3. Vertical velocity through the sea surface There has been confusion regarding the exression that relates the net evaoration at the sea surface to the vertical velocity in the ocean through the sea surface. Since these exressions have often involved the salinity (through the factor ( S ) ) and so aear to be thermodynamic exressions, here we resent 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 atmosheric side of the interface (where ( E P) is the notional W vertical velocity of freshwater through the air sea interface with density ρ ; this density being that of ure water at the sea surface temerature and at atmosheric ressure). 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 ( S,,. t ) The vertical velocity through an arbitrary surface whose height is z = η ( xyt,, ) can be exressed as w V H η η t (where w is the vertical velocity through the geootential surface, see section 3.4, and

50 4 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater note that t is time in this context) and the mass flux associated with this dia surface vertical velocity comonent 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 (4), Warren (9)) w V η η t = ρ ρ W ( E P). (3..) H 3. Freshwater content and freshwater flux Oceanograhers traditionally call the ure 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 oceanograhic use of freshwater as being synonymous with ure water (i. e. S =, this ure water being in liquid, gaseous or solid ice forms). The freshwater content of seawater is ( S) = (. S /(g kg )). The first exression here clearly requires that bsolute Salinity is exressed in kg of sea salt er kg of solution. Note that the freshwater content is not based on Practical Salinity, that is, it is not (. SP ). The advective flux of mass er 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 S u. The advective flux of freshwater er unit area is the difference of these two mass fluxes, namely ρ ( S ) u. s outlined in section.5 and aendices.4 and., for water of anomalous comosition there are four tyes of absolute salinity that might be relevant to this discussion of dens freshwater fluxes; Density Salinity S S, Solution Salinity S, dded Mass Salinity add S, and Preformed Salinity S *. Since Preformed Salinity is designed to be a conservative variable with a zero flux air sea boundary condition, robably the best form of freshwater S =. S / (g kg ). content, at least in the context of an ocean model, is ( *) ( * ) soln 3.3 Heat transort flux of heat across the sea surface at a sea ressure of dbar is identical to the flux of otential enthaly which in turn is exactly equal to c times the flux of Conservative Temerature, where c is given by (3.3.3). By contrast, the same heat flux across the sea surface changes otential temerature θ in inverse roortion to c ( S, θ, ) and this heat caacity varies by 5% at the sea surface, deending mainly on salinity. The First Law of Thermodynamics, namely Eqn. (.3.) of aendix.3, can be aroximated as d R Q S ρc F F + ρε + hs ρ, S (3.3.) d t with an error in that is aroximately one ercent of the error incurred by treating either c θ or c ( S, θ, ) θ as the heat content of seawater (see McDougall (3) and aendices.3 and.8). Equation (3.3.) is exact at dbar while at great deth in the ocean the error with the aroximation (3.3.) is no larger than the neglect of the dissiation of mechanical energy term ρε in this equation (see aendix.). Because the left hand side of the First Law of Thermodynamics, Eqn. (3.3.), can be written as density times the material derivative of c it follows that can be treated as a conservative variable in the ocean and that c is transorted by advection and mixed by turbulent eineutral and dianeutral diffusion as though it is the heat content of seawater. For examle, the advective meridional flux of heat is the area integral of ρvh = ρvc (here v is the northward velocity). The error in comaring this advective

51 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater 4 meridional heat flux with the air sea heat flux is aroximately % of the error in so interreting the area integral of either ρvc θ or ρvc ( S, θ, ) θ. Similarly, turbulent diffusive fluxes of heat are accurately given by a turbulent diffusivity times the satial gradient of c but are less accurately aroximated by the same turbulent diffusivity times the satial gradient of cθ (see aendix.4 for a discussion of this oint). Warren (999, 6) has argued that because enthaly is unknown u to a linear function of salinity, it is only ossible 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 oinion seems to be widely held, but it is incorrect. Because enthaly is unknown and unknowable u to a linear function of S (i. e. u to the arbitrary function a+ a3s in terms of the constants defined in Eqn. (.6.)), the left hand side of Eqn. (3.3.) is unknowable to the extent Q S a3ρ ds d t. It is shown in aendix B that the terms F + h S ρ S on the right hand side of Eqn. (3.3.) are also unknowable to the same extent so that the effect of a 3 cancels from Eqn. (3.3.). Hence the fact that c is unknowable u to a linear function of S does not affect the usefulness of h or c as measures of heat content. Similarly, the difference between the meridional fluxes of c across two latitudes is equal to the areaintegrated air sea and geothermal heat fluxes between these latitudes (after allowing for any unsteady accumulation of c in the volume), irresective of whether there are nonzero fluxes of mass or salt across the sections. This owerful result follows directly from the fact that c is a conservative variable, obeying the simle conservation statement Eqn. (3.3.). This issue is discussed at greater length in section 6 of McDougall (3). 3.4 Geootential The geootential Φ is the gravitational otential energy er unit mass with resect to the height z =. llowing the gravitational acceleration to be a function of z, Φ is given by z Φ= g z dz. (3.4.) ( ) If the gravitational acceleration is taken to be constant Φ is simly 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 uward from this surface. In SI units Φ is measured in Jkg = m s. If the ocean is assumed to be in hydrostatic balance so that Pz = gρ (or g dz = v dp ) then the geootential Eqn. (3.4.) may be exressed as the vertical ressure integral of the secific volume in the water column, P P ( ) Φ = Φ v dp, (3.4.) where Φ is the value of the geootential at zero sea ressure, 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. (3.4.). 3.5 Total energy The total energy E is the sum of secific internal energy u, kinetic energy er unit mass.5u u ( where u is the three dimensional velocity vector) and the geootential Φ, E = u +Φ+ (3.5.) u u. Total energy E is not a function of only ( S ) quantity.,, t and so is not a thermodynamic

52 4 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater 3.6 Bernoulli function The Bernoulli function is the sum of secific enthaly h, kinetic energy er unit mass.5u u, and the geootential Φ, B = h +Φ + u u (3.6.). Using the exression (3..) that relates enthaly and otential enthaly, together with Eqn. (3.4.) for, Φ the Bernoulli function (3.6.) may be written as P P ( ) ( ) B = h +Φ + u u v vˆ S,, dp. (3.6.) The ressure integral term here is a version of the dynamic height anomaly (3.7.), this time for a secific volume anomaly defined with resect to the bsolute Salinity and Conservative Temerature (or equivalently, with resect to the bsolute Salinity and otential temerature) of the seawater arcel in question at ressure P. This ressure integral is equal to the Cunningham geostrohic streamfunction, Eqn. (3.9.). The Bernoulli function B is not a function of only ( S,, t ) and so is not a thermodynamic quantity. The Bernoulli function is dominated by the contribution of enthaly h to (3.6.) and by the contribution of otential enthaly h to (3.6.). The variation of kinetic energy or the geootential following a fluid arcel is tyically several thousand times less than the variation of enthaly or otential enthaly following the fluid motion. The definition of secific volume anomaly given in Eqn. (3.7.3) has been used by Saunders (995) to write (3.6.) as (with the dynamic height anomaly Ψ defined in (3.7.)) P (, C, ) ˆ(,, ) B = h +Φ + Ψ+ u u ˆ v SSO v S dp P (3.6.3) = h +Φ + Ψ+ u u hˆ S, C, + hˆ S, C, + hˆ S,, hˆ S,,. ˆ Note that h( S ) = c and ( ) ( ) ( ) ( ) ( ) SO SO hˆ S, C, =. SO 3.7 Dynamic height anomaly The dynamic height anomaly Ψ, given by the vertical integral P P ( [ ] [ ] ) Ψ = δ S, t, dp, (3.7.) is the geostrohic streamfunction for the flow at ressure P with resect to the flow at the sea surface and δ is the secific volume anomaly. Thus the two dimensional gradient of Ψ in the P ressure surface is simly related to the difference between the horizontal geostrohic velocity v at P and at the sea surface v according to k PΨ = fv fv. (3.7.) The definition Eqn. (3.7.) of dynamic height anomaly alies to all choices of the reference values S and t, θ or ˆ in the definition Eqns. ( ) of the secific volume anomaly δ. lso, δ in Eqn. (3.7.) can be relaced with secific volume v without affecting the isobaric gradient of the resulting streamfunction. That is, this substitution does not affect Eqn. (3.7.) because the additional term is a function only of ressure. Traditionally it was imortant to use secific volume anomaly in reference to secific volume as it was more accurate with comuter code which worked with singlerecision variables. Since comuters now regularly emloy double recision, this issue has been overcome and consequently either δ or v can be used in the integrand of Eqn. (3.7.), so making it either the dynamic height anomaly or the dynamic height. s in

53 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater 43 the case of Eqn. (3.4.), so also the dynamic height anomaly Eqn. (3.7.) has not assumed that the gravitational acceleration is constant and so Eqn. (3.7.) alies even when the gravitational acceleration is taken to vary in the vertical. The dynamic height anomaly Ψ should be quoted in units of m s. These are the units in which the GSW Toolbox (aendix N) oututs dynamic height anomaly in the function gsw_geo_strf_dyn_height. Note that the integration in Eqn. (3.7.) of secific volume anomaly with ressure in dbar would yield dynamic height anomaly in units of 3 m kg dbar, and the use of these units in Eqn. (3.7.) would not give the resultant horizontal gradient in the usual units, being the roduct of the Coriolis arameter (units of s ) and the velocity (units of m s ). This is the reason why the ressure integration is done with ressure in Pa and dynamic height anomaly is outut in m s. 3.8 Montgomery geostrohic streamfunction The Montgomery acceleration otential π defined by P π = P P δ δ S, t, dp (3.8.) ( ) ( [ ] [ ] ) P is the geostrohic streamfunction for the flow in the secific volume anomaly surface δ ( S,, t ) = δ relative to the flow at P = P (that is, at = dbar). Thus the twodimensional gradient of π in the δ secific volume anomaly surface is simly related to the difference between the horizontal geostrohic velocity v in the δ = δ surface and at the sea surface v according to = f f = k f v f v (3.8.) k v v or ( ) δ π δ π. The definition, Eqn. (3.8.), of the Montgomery geostrohic streamfunction alies to all choices of the reference values S and t in the definition, Eqn. (3.7.), of the secific volume anomaly δ. By carefully choosing these reference values the secific volume anomaly surface can be made to closely aroximate the neutral tangent lane (McDougall and Jackett (7)). It is not uncommon to read of authors using the Montgomery geostrohic streamfunction, Eqn. (3.8.), as a geostrohic streamfunction in surfaces other than secific volume anomaly surfaces. This incurs errors that should be recognized. For examle, the gradient of the Montgomery geostrohic streamfunction, Eqn. (3.8.), in a neutral tangent lane becomes (instead of Eqn. (3.8.) in the δ = δ surface) π = k fv fv + P P δ, (3.8.3) n ( ) ( ) where the last term reresents an error arising from using the Montgomery streamfunction in a surface other than the surface for which it was derived. Zhang and Hogg (99) subtracted an arbitrary ressure offset, ( P P ), from ( P P ) in the first term in Eqn. (3.8.), so defining the modified Montgomery streamfunction Z-H Z-H P ( ) ( [ ] [ ] ) π = P P δ δ S, t, dp. (3.8.4) The gradient of π in a neutral tangent lane becomes n Z-H P ( f f ) ( P P) π = k v v + (3.8.5) nδ, where the last term can be made significantly smaller than the corresonding term in Eqn. (3.8.3) by choosing the constant ressure P to be close to the average ressure on the surface. This term can be further minimized by suitably choosing the constant reference values S and in the definition, Eqn. (3.7.3), of secific volume anomaly δ so that this surface n

54 44 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater more closely aroximates the neutral tangent lane (McDougall (989)). imrovement is available because it can be shown that ( ) ( ) ( ) This ρ nδ = κ S,, κ S,, np Tb np. (3.8.6) The last term in Eqn. (3.8.5) is then aroximately (3.8.7) ( P P) ( ) ( ) nδ ρ T b n P P and hence suitable choices of P, S and can reduce the last term in Eqn. (3.8.5) that reresents the error in interreting the Montgomery geostrohic streamfunction, Eqn. (3.8.4), as the geostrohic streamfunction in a surface that is more neutral than a secific volume anomaly surface. The Montgomery geostrohic streamfunction should be quoted in units of m s. These are the units in which the GSW Toolbox (aendix N) oututs the Montgomery geostrohic streamfunction in the function gsw_geo_strf_montgomery. 3.9 Cunningham geostrohic streamfunction Cunningham () and lderson and Killworth (5), following Saunders (995) and Killworth (986), suggested that a suitable streamfunction on a density surface in a comressible ocean would be the difference between the Bernoulli function B and otential enthaly h. Since the kinetic energy er unit mass,.5u u, is a tiny comonent of the Bernoulli function, it was ignored and Cunningham () essentially roosed the streamfunction Π+Φ (see his equation ()), where h u u Π B Φ = h h + Φ Φ P ˆ P ( ) = hs (,, ) hs (,,) v S ( ), ( ), dp. (3.9.) The last line of this equation has used the hydrostatic equation Pz = gρ to exress Φ gz in terms of the vertical ressure integral of secific volume and the height of the sea surface where the geootential is Φ. The definition of otential enthaly, Eqn. (3..), is used to rewrite the last line of Eqn. (3.9.), showing that Cunningham s Π is also equal to P P ( ) ˆ( ) Π = vˆ S ( ), ( ), v S,, dp. (3.9.) In this form it aears very similar to the exression, Eqn. (3.7.), for dynamic height anomaly, the only difference being that in Eqn. (3.7.) the ressure indeendent values of bsolute Salinity and Conservative Temerature were S SO and C whereas here they are the local values on the surface, S and. While these local values of bsolute Salinity and Conservative Temerature are constant during the ressure integral in Eqn. (3.9.), they do vary with latitude and longitude along any density surface. The gradient of Π along the neutral tangent lane is { P } ( ) ρ T b P P ρ n z, n Π Φ (3.9.3) (from McDougall and Klocker ()) so that the error in nπ in using Π as the geostrohic streamfunction is aroximately ρ T ( ) b P P n. When using the Cunningham streamfunction Π in a otential density surface, the error in σ Π is aroximately ρ T b ( P P)( Pr P P) σ. The Cunningham geostrohic streamfunction should be quoted in units of m s and is available in the GSW Oceanograhic Toolbox (aendix N) as the function gsw_geo_strf_cunningham.

55 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater Geostrohic streamfunction in an aroximately neutral surface In order to evaluate a relatively accurate exression for the geostrohic streamfunction in an aroximately neutral surface (such as an ω surface of Klocker et al. (9a,b) or a Neutral Density surface of Jackett and McDougall (997)) a suitable reference seawater arcel ( S,, ) is selected from the aroximately neutral surface that one is considering, and the secific volume anomaly δ is defined as in (3.7.3) above. The aroximate geostrohic streamfunction is given by (from McDougall and Klocker ()) P n ϕ S,, P P δ S,, ρ T P P δ dp. (3.3.) ( ) = ( ) ( ) b ( )( ) P This exression is very accurate when the variation of conservative temerature with ressure along the aroximately neutral surface is either linear or quadratic. That is, in n these situations nϕ z P Φ = k ( f v f v ρ ) to a very good aroximation. In Eqn. (3.3.) ρ T 5 b is taken to be the constant value.7x K (Pa) m s. This McDougall Klocker geostrohic streamfunction is available from the GSW Oceanograhic Toolbox as the function gsw_geo_strf_mcd_klocker. 3.3 Pressure integrated steric height The deth integrated mass flux of the geostrohic Eulerian flow between two fixed ressure levels can also be reresented by a streamfunction. Using the hydrostatic relation Pz = gρ, and assuming the gravitational acceleration to be indeendent of height, the deth integrated mass flux ρv dz is given by g v dp and this motivates taking the ressure integral of the Dynamic Height nomaly Ψ (from Eqn. (3.7.)) to form the Pressure Integrated Steric Height PISH (also called Deth Integrated Steric Height DISH by Godfrey (989)), P P P ( ) δ ( [ ] [ ] ) PISH = Ψ = g Ψ dp = g S, t, dp dp P P P P P ( ) δ ( [ ] [ ] ) = g P P S, t, dp. (3.3.) The two dimensional gradient of Ψ is related to the deth integrated mass flux of the velocity difference with resect to the velocity at zero sea ressure, v, according to zp ( ) P k Ψ = f ρ v( z ) v dz = g f v( ) v dp. (3.3.) zp ( ) P The definition, Eqn. (3.3.), of PISH alies to all choices of the reference values S, S and t, θ or in the definitions, Eqns. ( ), of the secific volume anomaly. Since the velocity at deth in the ocean is generally much smaller than at the sea surface, it is customary to take the reference ressure to be some constant (dee) ressure P so that Eqn. (3.7.) becomes P P ( [ ], [ ], ) Ψ = δ S t dp (3.3.3) and PISH, reflecting the deth integrated horizontal mass transort from the sea surface to ressure P, relative to the flow at P, is

56 46 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater P P P ( ) δ ( [ ] [ ] ) PISH = Ψ = g Ψ dp = g S, t, dp dp P P P P P ( ) δ ( [ ], [ ], ) = g P P S t dp ( P P) = g δ S, t, d ( P P ). ( [ ] [ ] ) ( ) (3.3.4) The two dimensional gradient of Ψ is now related to the deth integrated mass flux of the velocity difference with resect to the velocity at P, v, according to zp ( ) P k Ψ = f ρ v( z ) v dz = g f v( ) v dp. (3.3.5) zp ( ) P The secific volume anomaly δ in Eqns. (3.3.), (3.3.3) and (3.3.4) can be relaced with secific volume v without affecting the isobaric gradient of the resulting streamfunction. That is, this substitution in Ψ does not affect Eqn. (3.3.) or Eqn. (3.3.5), as the additional term is a function only of ressure. With secific volume in lace of secific volume anomaly, Eqn. (3.3.4) becomes the deth integrated gravitational otential energy of the water column (lus a very small term that is resent because the atmosheric ressure is not zero, McDougall et al. (3)). PISH should be quoted in units of kg s so that its two dimensional gradient has the same units as the deth integrated flux of ρ v( z ) v times the Coriolis frequency. 3.3 Pressure to height conversion When vertically integrating the hydrostatic equation Pz = gρ in the context of an ocean model where bsolute Salinity S and Conservative Temerature (or otential temerature θ ) are iecewise constant in the vertical, the geootential (Eqn. (3.4.)) P P ( ) Φ = Φ v dp, (3.3.) can be evaluated as a series of exact differences. If there are a series of layers of index i i i searated by ressures and + i i (with + > ) then the integral can be exressed (making use of (3.7.5), namely h ˆ P = h S, P = v) as a sum over n layers of the differences in secific enthaly so that P P n ( ) ˆ i i i+ ( ) ˆ i i i ( ) Φ = Φ v dp = Φ h S,, h S,,. (3.3.) i = 3.33 Freezing temerature Freezing occurs at the temerature t f at which the chemical otential of water in seawater W Ih μ equals the chemical otential of ice μ. Thus, t f is found by solving the imlicit equation W Ih μ S, t, = μ t, (3.33.) ( ) ( ) f f or equivalently, in terms of the two Gibbs functions, Ih g ( S tf ) S g S ( S ) ( ) tf g tf Ih The Gibbs function for ice Ih, (, ),,,,, =,. (3.33.) g t is defined by IPWS 6 (IPWS (9a)) and Feistel and Wagner (6) and is summarized in aendix I below. In the secial case of zero salinity, the chemical otential of water in seawater reduces to the Gibbs function of

57 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater W W ure water, ( t ) g ( t ) μ,, =,. simle correlation function for the melting ressure as a function of temerature is available from IPWS (8b) and has been imlemented in the SI library. t the ocean surface, = dbar, from Eqn. (3.33.) the TEOS freezing oint of ure water is t f ( gkg, dbar) =. 59 C with an uncertainty of only μk, noting that the trile oint temerature of water is exactly 73.6 K by definition of the ITS 9 temerature scale. The freezing temerature of the standard ocean is t f ( S SO,dbar ) =.99 C with an uncertainty of mk. Note that Eqn. (3.33.) is valid for air free water/seawater. Dissolution of air in water lowers the freezing oint slightly; saturation with air lowers the freezing temeratures by about mk. To estimate the effects of small changes in the ressure or salinity on the freezing temerature, it is convenient to consider a ower series exansion of (3.33.). The result in the limit of an infinitesimal ressure change at fixed salinity gives the ressure coefficient of freezing oint lowering, as (Clausius Claeyron equation, Feistel et al. (a)), Ih S Ih T ST T t g S f g g = χ ( S, ) =. (3.33.3) S g S g g Its values, evaluated from TEOS, vary only weakly with salinity between χ ( gkg, dbar) =.749 mk/dbar for ure water and χ ( S SO,dbar ) =.7483 mk/dbar for the standard ocean. TEOS is consistent with the most accurate measurement of χ and its exerimental uncertainty of.5 mk/dbar (Feistel and Wagner (5), (6)). Since the value of χ always exceeds that of the adiabatic lase rate Γ, cold seawater may freeze and decomose into ice and brine during adiabatic ulift but this can never haen to a sinking arcel. In the limit of infinitesimal changes in bsolute Salinity at fixed ressure, we obtain the saline coefficient of freezing oint lowering, as (Raoult s law), t SgSS = χs ( S, ) =. S g S g g f Tyical numerical values are χ S ( gkg, dbar) Ih T ST T 47 (3.33.4) = 59. mk/(g kg ) for ure water and S ( S,dbar χ SO ) = 56.9 mk/(g kg ) for seawater. s a raw ractical estimate, Eqn. (3.33.4) can be exanded into owers of salinity, S using only the leading term of the TEOS saline Gibbs function, g RTS S ln S, which stems from Planck s ideal solution theory (Planck (888)). Here, RS = R MS = J kg K is the secific gas constant of sea salt, R is the universal molar gas constant, and M S = g mol is the molar mass of sea salt with Reference Comosition. The SI denominator of Eqn. (3.33.4) is roortional to the melting heat L, Eqn. (3.34.7). The convenient result obtained with these simlifications is tf RS ( ) T SI + tf 59 mk/(g kg ). (3.33.5) S L SI where we have used t f = C and L = 33 Jkg as aroximations that are aroriate for the standard ocean. This simle result is only weakly deendent on these choices and is in reasonable agreement with the exact values from Eqn. (3.33.4) and with Millero and Leung (976). The freezing temerature of seawater is always lower than that of ure water. When sea ice is formed, it often contains remnants of seawater included in brine ockets. t equilibrium, the salinity in these ockets deends only on temerature and ressure, rather than, for examle, on the ocket volume, and can be comuted in the functional form S ( ) t, as an imlicit solution of Eqn. (3.33.). Measured values for the brine salinity of ntarctic sea ice agree very well with those comuted of Eqn. (3.33.) u

58 48 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater to the saturation concentration of about gkg at surface ressure (Feistel et al. (b)). t high ressures, the validity of the Gibbs function of seawater, and therefore of the comuted freezing oint or brine salinity, too, is limited to only 5 gkg. We note that in the first aroximation, as inferred from Planck s theory of ideal solutions, the above roerties deend on the number of dissolved articles regardless of the article sizes, masses or charges. In other words, they deend mainly on the molar rather than on the mass density of the solute, in contrast to roerties such as the density of seawater and roerties derived from it. The roerties considered in the remainder of this section ( ) which share this attribute are referred to as the colligative roerties of seawater Latent heat of melting The melting rocess of ice in ure water can be conducted by sulying heat at constant ressure. If this is done slowly enough that equilibrium is maintained, then the temerature will also remain constant. The heat required er mass of molten ice is the WI latent heat, or enthaly, of melting, L. It is found as the difference between the secific enthaly of water, h W, and the secific enthaly of ice, h Ih, (Kirchhoff s law, Curry and Webster (999)): Here, t ( ) ( ) ( ) ( ) WI W Ih f L = h t, h t,. (3.34.) f is the freezing temerature of water, section The enthalies h and are available from IPWS 95 (IPWS (9b)) and IPWS 6 (IPWS (9a)), resectively. In the case of seawater, the melt water will additionally mix with the ambient brine, thus changing the salinity and the freezing temerature of the seawater. Consequently, the enthaly related to this hase transition will deend on the articular conditions under which the melting occurs. Here, we define the latent heat of melting as the enthaly increase er infinitesimal mass of molten ice of a comosite system consisting of ice and seawater, when the temerature is increased at constant ressure and at constant total masses of water and salt, in excess to the heat needed to warm u the seawater and ice hases individually (Feistel and Hagen (998), Feistel et al. (b)). Mass conservation of both water and salt during this thermodynamic rocess is essential to ensure the indeendence of the latent heat formula from the unknown absolute enthalies of salt and water that otherwise would accomany any mass exchange. SI, The enthaly of sea ice, h is additive with resect to its constituents ice, the mass fraction Ih Ih w, and seawater,, w : f h with the liquid mass fraction ( ) ( ) (,, ) (, ) SI Ih Ih Ih W Ih h h Ih, with h = w h S t + w h t. (3.34.) Uon warming, the mass of melt water changes the ice fraction S The related temerature derivative of Eqn. (3.34.) is. Ih Ih Ih Ih ( w ) ( w ) w ( h h) SI Ih Ih h h h S h w = T T S T T T Ih w and the brine salinity S, T,. (3.34.3) The rate of brine salinity change with temerature is given by the recirocal of Eqn. (3.33.4) and is related to the isobaric melting rate, w Ih / T, by the conservation of the Ih w S = const, in the form total salt, ( ) S S w = Ih T w T Ih. (3.34.4)

59 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater 49 Using this relation, Eqn. (3.34.3) takes the simlified form SI h T ( ) Ih Ih Ih SI w c w c L = + The coefficient in front of the melting rate, T, Ih w T SI h L ( S, ) = h S h S Ih. (3.34.5), (3.34.6) rovides the desired exression for isobaric melting enthaly, namely the difference between the artial secific enthalies of water in seawater and of ice. s is hysically required for any measurable thermodynamic quantity, the arbitrary absolute enthalies of ice, water and salt cancel in the formula (3.34.6), rovided that the reference state conditions for the ice and seawater formulations are chosen consistently (Feistel et al. (8a)). Note that because of h = g + ( T + t) η and Eqn. (3.33.), the latent heat can also be written in terms of entroies η rather than enthalies h, in the form SI η Ih L ( S, ) = ( T + tf) η S η. (3.34.7) S T, gain the result is indeendent of unknown (and unknowable) constants. The latent heat of melting deends only weakly on salinity and on ressure. t the SI WI surface ressure, the comuted value is L (,) = L ( ) = J kg for ure water, SI and L ( S SO, ) = J kg for the standard ocean, with a difference of about % due to the dissolved salt. t a ressure of dbar, these values reduce by.6% to SI WI L (,dbar) = L ( dbar) = J kg SI and L ( S SO,dbar ) = J kg. WI TEOS is consistent with the most accurate measurements of L and their exerimental uncertainties of J kg, or.6% (Feistel and Wagner (5), (6)) Sublimation ressure subl The sublimation ressure of ice P is defined as the absolute ressure P of water vaour in equilibrium with ice at a given temerature t, at or below the freezing V temerature. It is found by equating the chemical otential of water vaour μ with the chemical otential of ice μ Ih, so it is found by solving the imlicit equation μ ( tp ) μ ( tp ) V subl Ih subl or equivalently, in terms of the two Gibbs functions, Ih The Gibbs function for ice Ih, g (, ), =,, (3.35.) ( ) ( ) V subl Ih subl g t, P = g t, P. (3.35.) tp is defined by IPWS 6 and Feistel and Wagner (6) and is summarized in aendix I below. Note that here the absolute ressure P rather than the sea ressure is used because the sublimation ressure of ice at ambient conditions is much lower than the atmosheric ressure. The Gibbs function of vaour, g V (, ) tp, is available from the Helmholtz function of fluid water, as defined by IPWS 95; for details see for examle Feistel et al. (8a), (a), (b). The highest ossible sublimation ressure is found at the trile oint of water. The TEOS value of the maximum sublimation ressure (i.e., the trile oint subl ressure) comuted from Eqn. (3.35.) is P = P t = Pa and has an uncertainty of. Pa (IPWS 6, Feistel et al. (8a)). Reliable theoretical values for the sublimation ressure are available down to K (Feistel and Wagner (7)); a simle correlation function for the sublimation ressure down to 5 K is rovided by IPWS (8b) and is included as a function in the SI

60 5 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater V library. The IPWS 95 function μ required for Eqn. (3.35.) is only valid above 3 K. n extension to 5 K was develoed for TEOS (Feistel et al. (a)) and is available as the default otion in the SI library. In nature, vaour cannot reasonably be exected to exist below 5 K since it has extremely low density, even in the interstellar vacuum. For this reason, the ice of comets does not evaorate far from the sun. The lowest temeratures estimated for the terrestrial olar atmoshere do not go below 3 K. In the resence of air, ice is under higher total ressure than just its own sublimation va ressure. The artial ressure of vaour in humid air, P = xvp, is comuted from the total absolute ressure P and the mole fraction of vaour, x V. Similar to the absolute salinity S of seawater, the variable describes the mass fraction of dry air resent in humid air. Given, the mole fraction of vaour is comuted from xv =, (3.35.3) M / M ( ) W where M is the molar mass of dry air and M W is the molar mass of water. subl sat The sublimation ressure, P ( tp, ) = xv P, of ice in equilibrium with humid air is the sat artial ressure of vaour in saturated air. To comute x V from Eqn. (3.35.3), the required air fraction at saturation, = sat ( t, P), is found by equating the chemical V otential of water vaour in humid air μ W with the chemical otential of ice μ Ih, so that it is found by solving the imlicit equation ( tp) μ ( tp) V sat Ih or equivalently, in terms of the two Gibbs functions, μw,, =,, (3.35.4) (,, ) (,, ) (, ) V sat sat V sat Ih g t P g t P = g t P. (3.35.5) V The Gibbs function of humid air, g (,, ) tp, is defined by Feistel et al. (a). t t = C and atmosheric ressure, the sublimation ressure of ice has the value subl P ( C, 35 Pa) = Pa, comuted by solving Eqn. (3.35.4) for sat, then using (3.35.3) to determine the corresonding mole fraction and multilying the atmosheric ressure by this quantity. Similarly, at the freezing oint of the standard ocean the subl sublimation ressure is P (.99 C, 35 Pa) = Pa. The difference between observed or modelled artial vaour ressures and the sublimation ressure comuted from TEOS is an aroriate quantity for use in arameterizations of the mass flux between ice and the atmoshere Sublimation enthaly The sublimation rocess that occurs when ice is in contact with ure water vaour can be conducted by sulying heat at constant t and P, with t at or below the freezing temerature. The heat required er mass evaorated from the ice is the latent heat, or VI enthaly, of sublimation, L. It is found as the difference between the secific enthaly of water vaour, h V, and the secific enthaly of ice, h Ih : () ( ) ( ) VI V subl Ih subl L t = h t, P h t, P. (3.36.) subl Here, P () t is the sublimation ressure of ice at the temerature t, section The V Ih enthalies h and h are available from IPWS 95 and IPWS 6, resectively. Reliable values for the sublimation enthaly are theoretically available down to K from a simle correlation function (Feistel and Wagner (7)). t the trile oint of water, the TEOS VI sublimation enthaly is L (. C) = J kg with an uncertainty of J kg, or.3%.

61 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater 5 In the case when air is resent, the vaour resulting from the sublimation will add to sat the gas hase, thus increasing the mole fraction of vaour x V. If for examle the total sat ressure P is held constant, the artial ressure xv P will rise, and the ice must get subl sat warmer to maintain equilibrium at the modified sublimation ressure P = xv P. Consequently, the enthaly related to this hase transition will deend on the articular conditions under which the sublimation rocess 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 enthaly increase er infinitesimal mass of sublimated ice of a comosite system consisting of ice and humid air, when the temerature is increased at constant ressure and at constant total masses of water and dry air, in excess of the enthaly increase needed to warm u the ice and humid air hases individually (Feistel et al. (a)). Mass conservation of both total water and dry air during this thermodynamic rocess is essential to ensure the indeendence of the latent heat formula from the unknown absolute enthalies of air and water that otherwise would accomany any mass exchange. The enthaly of ice air, h I, is additive with resect to its constituents ice, h Ih, with the mass fraction w Ih, and humid air, V Ih, w : h with the gas fraction ( ) ( ) (,, ) (, ) I Ih V Ih Ih h = w h t + w h t. (3.36.) Uon warming, the mass of vaour roduced by sublimation reduces the ice fraction and increases the humidity, that is, decreases the relative dry air fraction of the gas hase. The related temerature derivative of Eqn. (3.36.) is Ih Ih Ih Ih V ( w ) ( w ) w ( h h ) I V V Ih Ih h h h h w = T T T T T, T, The air fraction change is related to the isobaric sublimation rate, Ih w = const, in the form conservation of the dry air, ( ) w = Ih T w T Using this relation, Eqn. (3.36.3) takes the simle form I h T ( ) Ih V Ih Ih I w = w c + w c L. T The coefficient in front of the sublimation rate, V I V h Ih (, ) = L h h Ih Ih / Ih w. (3.36.3) w T, by the. (3.36.4) T, Ih (3.36.5), (3.36.6) rovides the desired exression for isobaric sublimation enthaly, namely the difference between the artial secific enthalies of vaour in humid air and of ice. In the ideal gas aroximations for air and for vaour, the artial secific enthaly of vaour in humid V V V air, h h, equals the secific enthaly of vaour, h ( t ), as a function of only the temerature, indeendent of the ressure and of the resence of air (Feistel et al. (a)). In this case, Eqn. (3.36.6) coincides formally with Eqn. (3.36.), excet that the two are evaluated at the different ressures P and P subl, resectively. s is hysically required for any measurable thermodynamic quantity, the arbitrary absolute enthalies of ice, vaour and air cancel in the formula (3.36.6), rovided that the reference state conditions for the ice and humid air formulations are chosen consistently (Feistel et al. (8a), (a)). The latent heat of sublimation deends only weakly on the air fraction and on the ressure.

62 5 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater For saturated air over sea ice, the air fraction = can be comuted from the brine salinity, or from the sea surface salinity in the case of floating ice, section t the absolute surface ressure P SO = 35 Pa and the freezing oint t f =.99 C of the sat standard ocean, the TEOS value for saturated air with SO = ( tf, PSO ) = is I L ( SO, P SO ) = J kg subl. The related sublimation ressure is P ( tf, P SO) = Pa, see section Observational data show that the ambient air over the ocean surface is sub saturated in the climatological mean. Rather than being saturated, values for that corresond to a relative humidity of 75% 8% (see section 3.4) may be a more realistic estimate for the marine atmoshere (Dai (6)); these values reresent non equilibrium conditions that result in net evaoration as art of the global hydrological cycle. sat 3.37 Vaour ressure va The vaour ressure of seawater ( ) P S, t is defined as the absolute ressure P of water vaour in equilibrium with seawater at a given temerature t and salinity S. It is V found by equating the chemical otential of vaour μ with the chemical otential of W water in seawater μ so that it is found by solving the imlicit equation μ ( tp, ) μ ( S, tp, ) V va W va or equivalently, in terms of the two Gibbs functions, =, (3.37.) (, ) (,, ) S (,, ) V va va va g t P = g S t P S g S t P. (3.37.) Note that here we use the absolute ressure P rather than the sea ressure ; since the vaour ressure of water at ambient conditions is much lower than the atmosheric ressure, the corresonding sea ressure (P va 35 Pa) would be negative and near 5 Pa. The Gibbs functions of vaour and seawater, g V ( tp, ) and g( S, t, P ), are available from the Helmholtz function of fluid water, as defined by IPWS 95, and the Gibbs function of seawater, IPWS 8 or IPWS 9 (IPWS (9c)). In the case of ure water, S =, the solution of Eqn. (3.37.) is the so called saturation curve in the t P diagram of water, which connects the trile oint with the critical oint. The lowest ossible vaour ressure of ure liquid water is found at the trile oint of water. The TEOS value of this minimum vaour ressure, comuted va from Eqn. (3.37.), is P (,. C) = P t = Pa with an uncertainty of. Pa (IPWS 95, Feistel et al. (8a)). For comarison, the vaour ressure of the standard va ocean is P ( S SO, C) = Pa. t laboratory temerature the related values are va va P (, 5 C) = Pa and P ( S SO, 5 C) = 3.57 Pa. The relatively small vaour ressure lowering caused by the resence of dissolved salt can be comuted from the isothermal salinity derivative of Eqn. (3.37.) in the form (Raoult s law) va P SgSS =. (3.37.3) V S g S g g T P SP P s a raw ractical estimate, this equation can be exanded into owers of salinity, using S only the leading term of the TEOS saline Gibbs function, g RTS S ln S, which stems from Planck s ideal solution theory. Here, RS = R MS = J kg K is the secific gas constant of sea salt, R is the universal molar gas constant, and M S = g mol is the molar mass of sea salt with Reference Comosition. The secific volume of seawater, g, is neglected in comarison to that of vaour. The latter is aroximately V va considered as an ideal gas, g RT / ( MWP ), where M W = g mol is the molar mass of water. The convenient result obtained with these simlifications is

63 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater 53 P S va T M W va P.57 MS P va. (3.37.4) The vaour ressure of seawater is always lower than that of ure water. In the resence of air, seawater is under a higher ressure P than under its vaour ressure va va P. In this case, the vaour ressure of seawater P ( S, t, P ) is defined as the artial ressure of water vaour in humid air that is in equilibrium with seawater at a given ressure P, temerature t and salinity S. It is found by equating the chemical V otential of vaour in humid air μ V with the chemical otential of water in seawater W μ so that it is found by solving the imlicit equation cond for ( ) μ (, tp, ) μ ( S, tp, ) V cond W V = (3.37.5) S, t, P, or equivalently, in terms of the two Gibbs functions, (,, ) (,, ) (,, ) S (,, ) V cond cond V cond g tp g tp = g S tp S g S tp. (3.37.6) Since the vaour ressure is lowered in the resence of sea salt (Eqn. (3.37.4)), at vaour ressures above the condensation oint vaour condenses out of the air at the sea surface, even before the saturation oint (that is, relative humidity of %) is reached, to maintain local equilibrium with the seawater. The larger scale equilibration rocess may involve downward diffusion of water vaour to the sea surface rather than reciitation of dew or cond fog. From the calculated sub saturated air fraction of the condensation oint,, the cond V mole fraction of vaour x (3.53.), and in turn the vaour ressure va cond P ( S,, t P) = xv P are available from straightforward calculations. The Gibbs function V of humid air g is available from Feistel et al. (a) and is also lanned to be made available as the document IPWS (). va The TEOS value comuted from Eqn. (3.37.5) is P (, C, PSO) = Pa for ure water at surface air ressure; the vaour ressure of the standard ocean is va P ( S SO, C, P SO ) = 6.43 Pa. t laboratory temerature the related values are va va P (, 5 C, P SO ) = Pa and P ( S SO, 5 C, P SO ) = 34.3 Pa Boiling temerature boil The boiling temerature of water or seawater is defined as the temerature ( ) t S, P at which the vaour ressure (of section 3.37) equals a given ressure P. It is found by V equating the chemical otential of vaour μ with the chemical otential of water in W seawater μ so that it is found by solving the imlicit equation boil μ ( t P) μ ( S t P) V boil W boil, =,,, (3.38.) for t ( S, P ), or equivalently in terms of the two Gibbs functions, ( ) ( ) S ( ) V boil boil boil g t, P = g S, t, P S g S, t, P. (3.38.) boil The TEOS boiling temerature of ure water at atmosheric ressure is (, ) t P SO = C. This temerature is outside the validity range of u to 8 C of the TEOS Gibbs function for seawater Latent heat of evaoration The evaoration rocess of ure liquid water in contact with ure water vaour can be conducted by sulying heat at constant t and P. The heat required er mass evaorated from the liquid is the latent heat, or enthaly, of evaoration, L. It is found as the VW

64 54 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater difference between the secific enthaly of water vaour, liquid water, h W : va () V () (, ) (, ) VW V va W va h V, and the secific enthaly of L t = h t P h t P. (3.39.) Here, P t is the vaour ressure of water at the temerature t (section 3.37). The W enthalies h and h are available from IPWS 95. t the trile oint of water, the TEOS evaoration enthaly is L VW (. C) = 5 95 J kg. In the case of seawater in contact with air, the vaour resulting from the evaoration will add to the gas hase, thus increasing the mole fraction of vaour, while the liquid water loss will increase the brine salinity, and cause a change to the seawater enthaly. Consequently, the enthaly related to this hase transition will deend on the articular conditions under which the evaoration rocess occurs. Here, we define the latent heat of evaoration as the enthaly increase er infinitesimal mass of evaorated water of a comosite system consisting of seawater and humid air, when the temerature is increased at constant ressure and at constant total masses of water, salt and dry air, in excess of the enthaly increase needed to warm u the seawater and humid air hases individually (Feistel et al. (a)). Mass conservation during this thermodynamic rocess is essential to ensure the indeendence of the latent heat formula from the unknown absolute enthalies of air, salt and water that otherwise would accomany any mass exchange. S, The enthaly of sea air, h is additive with resect to its constituents, seawater, h, with the mass fraction w SW, and humid air, V, w SW : h with the gas fraction ( ) ( ) (,, ) (,, ) S SW V SW h = w h t + w h S t. (3.39.) Uon warming, the mass of water transferred from the liquid to the gas hase by evaoration reduces the seawater mass fraction w SW, increases the brine salinity S and increases the humidity, with a corresonding decrease in the dry air fraction of the gas hase. The related temerature derivative of Eqn. (3.39.) is SW SW ( w ) ( w ) S V V h h h = + T T T, T, S, T, ( ) SW h SW h S V w + w + w + h h T S T T SW / SW. (3.39.3) The isobaric evaoration rate w T is related to the air fraction change by the SW w = const, in the form conservation of the dry air, ( ) SW w = SW T w T and to the change of salinity by the conservation of the salt, SW. SW T S S w = T w Using these relations, Eqn. (3.39.3) takes the simlified form S h T ( ) SW V SW S w c w c L = + The coefficient in front of the evaoration rate,, (3.39.4) w SW SW w T S = const, in the form (3.39.5). (3.39.6) S V h h L (, S, t, ) = h h + S, S V T, T, (3.39.7)

65 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater 55 rovides the desired exression for isobaric evaoration enthaly, namely the difference between the artial secific enthalies of vaour in humid air (the first two terms) and of water in seawater (the last two terms). In the ideal gas aroximations for air and for V V vaour, the artial secific enthaly of vaour in humid air, h h, equals the V secific enthaly of vaour, h () t, as a function of only the temerature, indeendent of the ressure and of the resence of air (Feistel et al. (a)). s is hysically required for any measurable thermodynamic quantity, the arbitrary absolute enthalies of water, salt and air cancel in the formula (3.39.7), rovided that the reference state conditions for both the seawater and the humid air formulation are chosen consistently (Feistel et al. (8a), (a)). The latent heat of evaoration deends only weakly on salinity and on air fraction, and is an almost linear function of the temerature and of the ressure. Selected reresentative values for the air fraction at condensation, cond, and the latent heat of evaoration, S L, are given in Table Table 3.39.: Selected values for the equilibrium air fraction, comuted S from Eqn. (3.37.6), and the latent heat of evaoration, L, comuted from Eqn. (3.39.7), for different sea surface conditions. Note that the TEOS formulation for humid air is valid u to 5 MPa, i.e., almost 5 dbar sea ressure. Condition S g kg t C dbar cond % cond, S L J kg Pure water Brackish water Standard ocean Troical ocean High ressure In the derivation of Eqn. (3.39.7), the value of is indirectly assumed to be comuted from the equilibrium condition (3.37.6) between humid air and seawater, = cond. t cond this humidity the air is still sub saturated, > sat, but its vaour starts condensing at cond sat the sea surface. The values of and coincide only below the freezing oint of seawater, or at vanishing salinity, see also the following section 3.4. The evaoration rate, w SW / T, can be comuted from Eqn. (3.37.6), the equilibrium condition between humid air and seawater, at changing temerature and S constant ressure (Feistel et al. (a)). In contrast, the derivation of L using Eqns. (3.39.) (3.39.7) is a mere consideration of mass and enthaly balances; no equilibrium condition is actually involved. Hence, it is hysically evident that Eqn. (3.39.7) can also be alied to situations in which takes any given value different from cond, that is, it can be alied regardless of whether or not the humid air is actually at equilibrium with the sea surface. 3.4 Relative humidity and fugacity Parameterised formulas for the flux of water and heat through the ocean surface are usually exressed in terms of a given relative humidity of the air in contact with seawater. In this section we rovide the formulas for the relative humidity and the fugacity from the TEOS otential functions for seawater and humid air, and we exlain why the relative fugacity with resect to condensation rather than with resect to saturation should be used for oceanograhic flux estimates (Feistel et al. (a)). Near the saturation oint, the

66 56 TEOS- Manual: Calculation and use of the thermodynamic roerties 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. lso in this aroximation, relative humidity is only a roerty of fluid water at given temerature and ressure of the vaour hase, indeendent of the resence of air. The CCT definition of relative humidity is in terms of mole fraction: t given ressure and temerature, [the relative humidity is defined as] the ratio, exressed as a ercent, of the mole fraction of water vaour to the vaour mole fraction which the moist gas would have if it were saturated with resect to either water or ice at the same ressure and temerature. Consistent with CCT, IUPC defines relative humidity as the ratio, often exressed as a ercentage, of the artial ressure of water in the atmoshere at some observed temerature, to the saturation vaour ressure of ure water at this temerature (Calvert (99), IUPC (997)). This definition of the relative humidity takes the form xv RHCCT = (3.4.) sat xv with regard to the mole fraction of vaour xv ( ), Eqn. (3.35.3), and the saturated air sat cond fraction = ( t, P) = (, t, P) either from Eqn. (3.37.6) with resect to liquid water, at t above the freezing oint of ure water, or from Eqn. (3.35.5) with resect to ice, at t cond below the freezing oint of ure water. Here, ( S, t, P ) is the air fraction of humid air at equilibrium with seawater, Eqn. (3.37.5), which is subsaturated for S >. The WMO 3 definition of the relative humidity is (Pruacher and Klett (997), Jacobson (5)), r / RHWMO = = (3.4.) sat sat r / where r = ( ) / is the humidity ratio. If r is small, we can estimate xv rm / M W (from Eqn. (3.35.3)) and therefore RHWMO RHCCT, that is, we find aroximate consistency between Eqns. (3.4.) and (3.4.). Sometimes, esecially when considering hase or chemical equilibria, it is more convenient to use the fugacity (or activity) rather than artial ressure ratio (IUPC (997)). The fugacity of vaour in humid air is defined as V V, id V(,, ) V ex μ μ f T P = x P R. (3.4.3) WT Here, RW = R MW is the secific gas constant of water, μ V (, T, P) = g V g V is the V, id chemical otential of vaour in humid air, and μ (, T, P) is its ideal gas limit which is equal to the true chemical otential in the limit of very low ressure, T V, id V T V,id xvp μ ( T,, P) = g + c ( T' ) d T' + RWTln V T ' P. (3.4.4) V V V V T V,id The values of g, P and T of μ must be chosen consistently with the adjustable V V, id constants of g (Feistel et al. (a)). The ideal gas heat caacity of vaour c ( T ) is available from IPWS 95. In the ideal gas limit of infinite dilution, f V converges to the artial ressure of vaour (Glasstone (947)), P V ( ) va V lim f, T, P = x P = P. (3.4.5) CCT: Consultative Committee for Thermometry, IUPC: International Union of Pure and lied Chemistry, 3 WMO: World Meteorological Organisation,

67 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater 57 The saturation fugacity is defined by the equilibrium between liquid water (or ice) and V W μ, T, P = μ, T, P, that is, vaour in air, ( ) ( ) W where g (, T, P) (, T, P) μ (, T, P) μ W V,id sat sat sat fv = xv Pex, RWT (3.4.6) μ = is the chemical otential of liquid water (or the chemical otential Ih of ice, μ ). The relative fugacity ϕ of humid air is then defined, dividing Eqn. (3.4.3) by Eqn. (3.4.6) and making use of Eqn. (3.4.4), as W ( T,, P) μ (, T, P) V f V μ ϕ = = ex. sat fv RWT V V, id, (3.4.7) In the ideal gas limit, μ = μ and using (3.4.3) we see that the relative fugacity ϕ coincides with the relative humidity, Eqn. (3.4.). Taking Eqn. (3.4.7) at the condensation oint, = cond, Eqn. (3.37.5), it follows that the relative fugacity of humid air at equilibrium with seawater ( sea air for short) is ϕ S W ( S, T, P) μ (, T, P) S W f V μ = = ex. sat fv RWT (3.4.8) The chemical otential difference in the exonent is roortional to the osmotic coefficient of seawater, φ, which is comuted from the saline art of the Gibbs function as (Feistel and Marion (7), Feistel (8)), where S S g φ ( S, T, P) = g S, mswrt S TP, m SW is the molality of seawater (Millero et al. (8a)), S msw =. S M ( ) From the chemical otential of water in seawater, (3.4.) we infer for the relative fugacity of sea air the simle formula ϕ S S ( m M φ) SW W W S (3.4.9) (3.4.) µ = g S g, and Eqns. (3.4.8) = ex, (3.4.) which is identical to the activity a W of water in seawater. Similar to the ideal gas aroximation, the relative fugacity of sea air is indeendent of the resence or the S roerties of air. In Eqn. (3.4.), the relative fugacity ϕ exresses the fact that the vaour ressure of seawater is lower than that of ure water, i.e., that humid air in equilibrium with seawater above its freezing temerature is always sub saturated. s a raw ractical estimate, using a series exansion of Eqns. (3.4.) and (3.4.) with resect to salinity, we can obtain from the molality msw = S / MS + O( S ) and the osmotic coefficient φ = + O( S ) the linear relation S M W ϕ S, (3.4.) M S i.e., Raoult s law for the vaour ressure lowering of seawater, Eqn. (3.37.4). Below the freezing temerature of ure water at a given ressure, the saturation of vaour is defined by the chemical otential of ice rather than liquid water, i.e. by ( T, P) μ (, T, P) μ Ih V,id sat sat sat fv = xv Pex, RWT rather than Eqn. (3.4.6). Then, the relative fugacity of sea air is (3.4.3)

68 58 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater Ih ( S, T, P) μ ( T, P) S W S f V μ ϕ = = ex. sat (3.4.4) fv RWT When the temerature is lowered further to the freezing oint of seawater, the exonent of (3.4.4) vanishes and sea air is saturated, ϕ S =, for sea ice air at any lower temerature. Thermodynamic fluxes in non equilibrium states are driven by Onsager forces such as the gradient of μ / T (De Groot and Mazur (984)). t the sea surface, assuming the same temerature and ressure on both sides of the sea air interface, the dimensionless Onsager force XS (, S, T, P ) driving the transfer of water is the difference between the chemical otentials of water in humid air and in seawater, X S V V W (, T, P) μ ( S, T, P) μ μ =Δ = RWT RWT RWT This difference vanishes at the condensation oint, cond ( S T P ). (3.4.5) =,,, Eqn. (3.37.5), rather than at saturation. X S can also be exressed in terms of fugacities, Eqns. (3.4.7), (3.4.8) and (3.4.), in the form ϕ ( ) XS = ln = m S SWMWφ + ln ϕ( ). (3.4.6) ϕ S ( ) Rather than the relative humidity, Eqns. (3.4.), (3.4.), the sea air Onsager force X S, in conjunction with the formula (3.39.7), is relevant for the arameterization of nonequilibrium latent heat fluxes across the sea surface. In the secial case of limnological S =, which, in the ideal gas aroximation. ll roerties required for the calculation of the formula (3.4.6) are available from the TEOS thermodynamic otentials for seawater, ice, and humid air. alications, or below the freezing oint of seawater, it reduces to X lnϕ ( ) corresonds to the relative humidity, ln ( RH CCT ) 3.4 Osmotic ressure If ure water is searated from seawater by a semi ermeable membrane which allows water molecules to ass but not salt articles, water will enetrate into the seawater, thus diluting it and ossibly increasing its ressure, until the chemical otential of water in both boxes becomes the same (or the ure water reservoir is exhausted). In the usual model configuration, the two samles are thermally couled but may ossess different ressures; the resulting ressure difference required to maintain equilibrium is the osmotic ressure of seawater. n examle of a ractical alication is desalination by reverse osmosis; if the ressure on seawater in a vessel exceeds its osmotic ressure, freshwater can be squeezed out of solution through suitable membrane walls (Sherwood et al. (967)). The osmotic ressure of seawater is very imortant for marine organisms; it is considered resonsible for the small number of secies that can survive in brackish environments. The defining condition for the osmotic equilibrium is equality of the chemical W otentials of ure water at the ressure and of water in seawater at the ressure, W ( ) ( ) W g g t, = g S, t, S. S T, The solution of this imlicit relation for the osmotic ressure is osm ( ) W (3.4.) P S,, t = P P. (3.4.) The TEOS value for the osmotic ressure of the standard ocean is P S, C,dbar = Pa, comuted from Eqn. (3.4.). osm ( ) SO

69 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater Temerature of maximum density t about 4 C and atmosheric ressure, ure water has a density maximum below which the thermal exansion coefficient and the adiabatic lase rate change their signs (Röntgen (89), McDougall and Feistel (3)). t salinities higher than 3.8 g kg the temerature of maximum density t MD is below the freezing oint t f (Table 3.4.). The seasonal and satial interlay between density maximum and freezing oint is highly imortant for the stratification stability and the seasonal dee convection for brackish estuaries with ermanent vertical and lateral salinity gradients such as the Baltic Sea (Feistel et al. (8b), Leäranta and Myrberg (9), Reissmann et al. (9)). The temerature of maximum density t MD is comuted from the condition of vanishing thermal exansion coefficient, that is, from the solution of the imlicit equation for t MD ( S, ), gtp ( S, tmd, ) =. (3.4.) The temerature of maximum density is available in the GSW Oceanograhic Toolbox as function gsw_tems_maxdensity. This function also returns the otential temerature and the Conservative Temerature at this maximum density oint. Selected TEOS values comuted from Eqn. (3.4.) are given in Table Table 3.4.: Freezing temerature t f and temerature of maximum density t MD for air free brackish seawater with absolute salinities S between and 5 g kg, comuted at the surface ressure from TEOS. Values of t MD in arentheses are less than the freezing temerature. S t f t MD S t f g kg C C g kg C C g kg C (.36) (.48) (.54) t MD S t f t MD C

70 6 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater 4. Conclusions The International Thermodynamic Equation of Seawater (TEOS ) allows all the thermodynamic roerties of ure water, ice Ih, seawater and moist air to be evaluated in an internally self consistent manner. For the first time the effects of the small variations in seawater comosition around the world ocean can be included, esecially their effects on the density of seawater (which can be equivalent to ten times the recision of our Practical Salinity measurements at sea). Perhas the most aarent change in ractice comared with the International Equation of State of seawater (EOS 8) is the adotion of bsolute Salinity S instead of Practical Salinity S P (PSS 78) as the salinity argument for the thermodynamic roerties of seawater. Imortantly, Practical Salinity is retained as the salinity variable that is stored in data bases because Practical Salinity is virtually the measured variable (whereas bsolute Salinity is a calculated variable) and also so that national data bases do not become corruted with incorrectly labeled and stored salinity data. The adotion of bsolute Salinity as the argument for all the algorithms used to evaluate the thermodynamic roerties of seawater makes sense simly because the thermodynamic roerties of seawater deend on S rather than on S P ; seawater arcels that have the same values of temerature, ressure and of S P do not have the same density unless the arcels also share the same value of S. bsolute Salinity is measured in SI units and the calculation of the freshwater concentration and of freshwater fluxes follows naturally from bsolute Salinity, but not from Practical Salinity. bsolute Salinity is calculated in a two stage rocess. First Reference Salinity is calculated from measurements of Practical Salinity using Eqn. (.4.). Then the bsolute Salinity nomaly is estimated from the comuter algorithm of McDougall et al. (a) or by other means, and bsolute Salinity is formed as the sum of Reference Salinity and the bsolute Salinity nomaly. There are subtle issues in defining what is exactly meant by absolute salinity and at least four different definitions are ossible when comositional anomalies are resent. 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 bsolute Salinity nomaly reresents the state of the art as at, but this area of oceanograhy is relatively immature. It is likely that the accuracy of this algorithm will imrove as more seawater samles from around the world ocean have their density accurately measured. fter such future work is ublished and the results distilled into a revised algorithm for bsolute Salinity nomaly, such an algorithm will be served from Oceanograhers should ublish the version number of this software that is used to obtain thermodynamic roerties in their manuscrits. Because bsolute Salinity is the aroriate salinity variable for use with the equation of state, bsolute Salinity is the salinity variable that should be ublished in oceanograhic journals. The version number of the software that is used to convert Reference Salinity S R into bsolute Salinity S should always be stated in ublications. Nevertheless, there may be some alications where the likely future changes in the algorithm that relates Reference Salinity to bsolute Salinity resents a concern, and for these alications it may be referable to ublish grahs and tables in Reference Salinity. For these studies or where it is clear that the effect of comositional variations are insignificant or not of interest, the Gibbs function may be called with S rather than S, R

71 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater 6 thus avoiding the need to calculate the bsolute Salinity nomaly. When this is done, it should be clearly stated that the salinity variable that is being grahed is Reference Salinity, not bsolute Salinity. The treatment of salinity in ocean models is discussed in aendix.. The recommended aroach is to carry both Preformed Salinity S * and bsolute Salinity nomaly δ S as model variables so that Density Salinity can be calculated at each time ste of the model and used to accurately evaluate density. Potential temerature has been used in oceanograhy as though it is a conservative variable, and yet the secific heat of seawater varies by 5% at the sea surface, and otential temerature is not conserved when seawater arcels mix. The First Law of Thermodynamics can be very accurately regarded as the statement that otential enthaly 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 otential enthaly (i. e. the air sea flux of c ) means that otential enthaly can be treated as the heat content of seawater and fluxes of otential enthaly in the ocean can be treated as heat fluxes. Just as it is erfectly valid to talk of the flux of salinity anomaly, ( S constant), across an ocean section even when the mass flux across the section is non zero, so it is erfectly valid to treat the flux of c 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 temerature variable in ocean models is commonly regarded as being otential temerature, but since the non conservative source terms that are resent in the evolution equation for otential temerature are not included in models, it is aarent that the interior of ocean models already treat the rognostic temerature variable as Conservative Temerature. To comlete the transition to in ocean modeling, models should be initialized with rather than θ, the outut temerature must be comared to observed data rather than to θ data, and during the model run, any air sea fluxes that deend on the sea surface temerature (SST) must be calculated at each model time ste using θ = ˆ θ( S, ). The final ingredient needed for an ocean model is a comutationally efficient form of density in terms of Conservative Temerature, that is ρ = ˆ ρ( S,, ), such as that described in aendix.3 and aendix K of this TEOS Manual. Under EOS 8 the observed variables ( S P, t, ) were first used to calculate otential temerature θ and then water masses were analyzed on the SP θ diagram. Curved contours of otential density could also be drawn on this same SP θ diagram. Under TEOS, since density and otential density are now not functions of Practical Salinity S P but rather are functions of bsolute Salinity S, it is now not ossible to draw isolines of otential density on a SP θ diagram. Rather, because of the satial variations of seawater comosition, a given value of otential density defines an area on the SP θ diagram, not a curved line. Under TEOS, the observed variables ( S P, t, ), together with longitude and latitude, are used to first form bsolute Salinity S, and then Conservative Temerature is evaluated Oceanograhic water masses are then analyzed on the S diagram, and otential density contours can also be drawn on this S diagram. Preformed Salinity S * is used internally in numerical ocean models where it is imortant that the salinity variable be conservative. When describing the use of TEOS, it is the resent document (the TEOS Manual) that should be referenced as IOC et al. () [IOC, SCOR and IPSO, : The international thermodynamic equation of seawater : Calculation and use of thermodynamic roerties. Intergovernmental Oceanograhic Commission, Manuals and Guides No. 56, UNESCO (English), 96 ]. Two introductory articles about TEOS, namely What every oceanograher needs to know about TEOS (The TEOS Primer) (Pawlowicz, b), and Getting started with TEOS and the GSW Oceanograhic Toolbox (McDougall and Barker, ) are available from

72 6 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater PPENDIX : Background and theory underlying the use of the Gibbs function of seawater. ITS-9 temerature In order to understand the limitations of conversion between different temerature scales, it is helful to review the definitions of temerature and of the international scales on which it is reorted... Definition When considering temerature, the fundamental hysical quantity is thermodynamic temerature, symbol T. The unit for temerature is the kelvin. The name of the unit has a lowercase k. The symbol for the unit is uercase K. One kelvin is /73.6 of the thermodynamic temerature of the trile oint of water. ( recent evolution of the definition has been to secify the isotoic comosition of the water to be used as that of Vienna Standard Mean Ocean Water, VSMOW.) The Celsius temerature, symbol t, is defined by t C = T K 73.5, and C is the same size as K... ITS 9 temerature scale The definition of temerature scales is the resonsibility of the Consultative Committee for Thermometry (CCT) which reorts to the International Committee for Weights and Measures (often referred to as CIPM for its name in the French language). Over the last 4 years, two temerature scales have been used; the International Practical Temerature Scale 968 (IPTS 68), followed by the International Temerature Scale 99 (ITS 9). These are defined by Barber (969) and Preston Thomas (99). For information about the International Temerature Scales of 948 and 97 the reader is referred to Preston Thomas (99). In the oceanograhic range, temeratures are determined using a latinum resistance thermometer. The temerature scales are defined as functions of the ratio W, namely the ratio of the thermometer resistance at the temerature to be measured R() t to the resistance at a reference temerature R. In IPTS 68, R is R ( C, ) while in ITS 9 R is R (. C ). The details of these temerature scales and the differences between the two scales are therefore defined by the functions of W used to calculate T. For ITS 9, and in the range C < t 9 < C, t 9 is described by a olynomial with coefficients given by Table 4 of Preston Thomas (99). We note in assing that the conversions from W to T and from T to W are both defined by olynomials and these are not erfect inverses of one another. Preston Thomas oints out that the inverses are equivalent to within.3mk. In fact the inverses have a difference of.3 mk at 86 C, and a maximum error in the range C < t 9 < 4 C of.6 mk at 3 C. That the CCT allowed this discreancy between the two olynomials immediately rovides an indication of the absolute uncertainty in the determination, and indeed in the definition, of temerature.

73 TEOS- Manual: Calculation and Use of the Thermodynamic Proerties of Seawater 63 second uncertainty in the absolute realization of ITS 9 arises from what is referred to as sub range inconsistency. The olynomial referred to above describes the behaviour of an ideal thermometer. ny ractical thermometer has small deviations from this ideal behaviour. ITS 9 allows the deviations to be determined by measuring the resistance of the thermometer at u to five fixed oints: the trile oint of water and the freezing oints of tin, zinc, aluminium and silver, covering the range. C < t 9 < C. If not all of these oints are measured, then it is ermissible to estimate the deviation from as many of those oints as are measured. The melting oint of Gallium ( t 9 = C) and the trile oint of Mercury ( t 9 = C) may also be used if the thermometer is to oerate over a smaller temerature range. Hence the manner in which the thermometer may be used to interolate between the oints is not unique. Rather it deends on which fixed oints are measured, and there are several ossible outcomes, all equally valid within the definition. Sections 3.3. and of Preston Thomas (99) give recise details of the formulation of the deviation function. The difference between the deviation functions derived from different sets of fixed oints will deend on the thermometer, so it not ossible to state an uer bound on this non uniqueness. Common ractice in oceanograhic standards laboratories is to estimate the deviation function from measurements at the trile oint of water and the melting oint of Gallium ( t 9 = 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 9, in addition to any imerfections of measurement by any ractical thermometer (Rudtsch and Fischer (8), Feistel et al. (8a)). It is therefore not ossible to seek a unique and erfect conversion between IPTS 68 and ITS 9. Goldberg and Weir (99) and Mares and Kalova (8) have discussed the rocedures needed to convert measured thermohysical quantities (such as secific heat) from one temerature definition to another. When mechanical or electrical energy is used in a laboratory to heat a certain samle, this energy can be measured in electrical or mechanical units by aroriate instruments such as an amere meter, indeendent of any definition of a temerature scale. It is obvious from the fundamental thermodynamic relation (at constant bsolute Salinity), du = Tdη + Pd v, that the same energy difference Tdη results in different values for the entroy η, deending on the number read for T from a thermometer calibrated on the 99 comared with one calibrated on the 968 scale. similar deendence is found for numbers derived from entroy, for examle, for the heat caacity, c = Tη T S., Douglas (969) listed a systematic consideration of the quantitative relations between the measured values of various thermal roerties and the articular temerature scale used in the laboratory at the time the measurement was conducted. Conversion formulas to ITS 9 of readings on obsolete scales are rovided by Goldberg and Weir (99) and Weir and Goldberg (996). ny thermal exerimental data that entered the construction of the thermodynamic otentials that form TEOS were carefully converted by these rules, in addition to the conversion between the various older definitions of for examle calories and joules. This must be borne in mind when roerties comuted from TEOS are combined with historical measurements from the literature.

74 64 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater..3 Theoretical conversion between IPTS 68 and ITS 9 Having understood that the conversion between IPTS 68 and ITS 9 is not uniquely defined, we review the sources of uncertainty, or even flexibility, in the conversion between t 9 and t 68. Consider first why t 9 and t 68 temeratures differ: ) The fixed oints have new temerature definitions in ITS 9, due to imrovements in determining the absolute thermodynamic temeratures of the melting/freezing hysical states relative to the trile oint of water. ) For some given resistance ratio W the two scales have different algorithms for interolating between the fixed oints. Now consider why there is non uniqueness in the conversion: 3) In some range of ITS 9, the conversion of W to t 9 can be undertaken with a choice of coefficients that is made by the user (Preston Thomas (99) Sections to 3.3.3), referred to as sub range inconsistency. 4) The imact of the ITS 9 deviation function on the conversion is non linear. Therefore the size of the coefficients in the deviation function will affect the difference, t 9 t 68. The formal conversion is different for each actual thermometer that has been used to acquire data. The grou resonsible for develoing ITS 9 was well aware of the non uniqueness of the conversion. Table 6 of Preston Thomas (99) gives differences ( t9 t68 ) with a resolution of mk, because (a) the true thermodynamic temerature T was known to have uncertainties of order mk or larger in some ranges, (b) the sub range inconsistency of ITS 9 using the same calibration data gave an uncertainty of several tenths of mk. Therefore to attemt to define a generic conversion of ( t9 t68 ) with a resolution of say. mk would robably be meaningless and ossibly misleading as there isn t a unique generic conversion function...4 Practical conversion between IPTS 68 and ITS 9 Rusby (99) ublished an 8 th order olynomial that was a fit to Table 6 of Preston Thomas (99). This fit is valid in the range 73.5 K to K ( C to C). He reorts that the olynomial fits the table to within mk, commensurate with the nonuniqueness of IPTS 68. Rusby s 8 th order olynomial is in effect the official recommended conversion between IPTS 68 and ITS 9. This olynomial has been used to convert historical IPTS 68 data to ITS 9 for the rearation of the new thermodynamic roerties of seawater that are the main subject of this manual. s a convenient conversion valid in a narrower temerature range, Rusby (99) also roosed T T /K = -.5 T / K (..) ( ) ( ) in the range 6 K to 4 K ( 3 C to 7 C). Rusby (99) also exlicitly reminds readers (see his age 58) that comound quantities that involve temerature intervals such as heat caacity and thermal conductivity are affected by their deendence on the derivative dt ( 9 T68 )/ dt68. bout the same time that Rusby ublished his conversion from t 68 to t 9, Saunders (99) made a recommendation to oceanograhers that in the common oceanograhic temerature range C < t 68 < 4 C, conversion could be achieved using t / C = t / C.4. (..) ( ) ( ) 9 68

75 TEOS- Manual: Calculation and Use of the Thermodynamic Proerties of Seawater 65 The difference between Saunders (99) and Rusby (99) arises from the best sloe being.4 near C and.6 near C (recall that t 68 for the boiling oint of water was C while its t 9 is C). Thus Rusby (99) chose.5 over the wider range of C to C. In considering what is a reasonable conversion between the two temerature scales, we must recall that the uncertainty in conversion between measured resistance and either temerature scale is of order a few tenths of mk, and the uncertainty in the absolute thermodynamic temerature T is robably at least as large, and may be larger than mk in some arts of the oceanograhic range. For all ractical uroses data converted using Saunders.4 cannot be imroved uon; conversions using Rusby s (99) 8 th order fit are fully consistent with Saunders.4 in the oceanograhic temerature range within the limitations of the temerature scales...5 Recommendation regarding temerature conversion The ITS 9 scale was introduced to correct differences between true thermodynamic temerature T, and temeratures reorted in IPTS 68. There are remaining imerfections and residuals in T T9 (Rusby, ers. comm.), which may be as high as a coule of mk in the region of interest. This is being investigated by the Consultative Committee for Thermometry (CCT). t a meeting in (Rusby and White (3)) the CCT considered introducing a new temerature scale to incororate the known imerfections, referred to at that time as ITS XX. Further consideration by CCT WG has moved thinking away from the desirability of a new scale. The field of thermometry is undergoing raid advances at resent. Instead of a new temerature scale, the known limitations of the ITS 9 can be addressed in large art through the ITS 9 Technical nnex, and documentation from time to time of any known differences between thermodynamic temerature and ITS 9 (Rile et al. (8)). The two main conversions currently in use are Rusby s 8 th order fit valid over a wide range of temeratures, and Saunders.4 scaling widely used in the oceanograhic community. They are formally indistinguishable because they differ by less than both the uncertainty in thermodynamic temerature, and the uncertainty in the ractical alication of the IPTS 68 and ITS 9 scales. Nevertheless we note that Rusby (99) suggests a linear fit with sloe.5 in the range 3 C to 7 C, and that Saunders sloe.4 is a better fit in the range C to 4 C while Rusby s 8 th order fit is more robust for temeratures outside the oceanograhic range. The difference between Saunders (99) and Rusby (99) is less than mk everywhere in the range C to 4 C and less than.3mk in the range C to C. In conclusion, the algorithms for PSS 78 require t 68 as the temerature argument. In order to use these algorithms with t 9 data, t 68 may be calculated using Eqn. (..3) thus t / C =.4 t / C. (..3) ( ) ( ) 68 9

76 66 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater. Sea ressure, gauge ressure and absolute ressure Sea ressure is defined to be the bsolute Pressure P less the bsolute Pressure of one standard atmoshere, P 35Pa; that is P P. (..) lso, it is common oceanograhic ractice to exress sea ressure in decibars (dbar). nother common ressure variable that arises naturally in the calibration of sea board gauge instruments is gauge ressure which is bsolute Pressure less the bsolute Pressure of the atmoshere at the time of the instrument s calibration (erhas in the laboratory, or erhas at sea). Because atmosheric ressure changes in sace and time, sea ressure is referred as a thermodynamic variable as it is unambiguously related to bsolute Pressure. The seawater Gibbs function is naturally a function of sea ressure (or functionally equivalently, of bsolute Pressure P ); it is not a function of gauge ressure. Table.. Pressure unit conversion table Pascal (Pa) decibar (dbar) bar (bar) Technical atmoshere atmoshere (at) (atm) torr (Torr) oundforce er square inch (si) Pa N/m dbar 4 5 dyn/cm bar 6 dyn/cm at kgf/cm atm atm torr Torr si lbf/in Examle: Pa = N/m = 4 dbar = 5 bar =.97 6 at = atm, etc. The difference between sea ressure and gauge ressure is quite small and robably insignificant for many oceanograhic alications. Nevertheless it would be best ractice to ensure that the CTD ressure that is used in the seawater Gibbs function is calibrated on deck to read the atmosheric ressure as read from the shi s bridge barometer, less the absolute ressure of one standard atmoshere, P 35Pa. (When the CTD is lowered from the sea surface, the monitoring software may well dislay gauge ressure, indicating the distance from the surface.) Since there are a variety of different units used to exress atmosheric ressure, we resent a table (Table..) to assist in converting between these different units of ressure (see ISO (993)). Note that one decibar ( dbar) is exactly. bar, and that mmhg is very similar to torr with the actual relationshi being mmhg = torr. The torr is defined as exactly /76 of the bsolute Pressure of one standard atmoshere, so that one torr is exactly equal to ( 35/76) Pa.

77 TEOS- Manual: Calculation and Use of the Thermodynamic Proerties of Seawater 67.3 Reference Comosition and the Reference-Comosition Salinity Scale s mentioned in the main text, the Reference Comosition of seawater is defined by Millero et al. (8a) as the exact mole fractions given in Table D.3 of aendix D below. This comosition model was determined from the most accurate measurements available of the roerties of Standard Seawater, which is filtered seawater from the surface waters of the North tlantic as made available by the IPSO Standard Seawater Service. The Reference Comosition is erfectly consistent with charge balance of ocean waters and the most recent atomic weight estimates (Wieser (6)). For seawater with this reference comosition the Reference Comosition Salinity S R as defined below rovides our best estimate of the bsolute Salinity. The Reference Comosition includes all imortant comonents of seawater having mass fractions greater than about. gkg (i. e.. mg kg ) that can significantly affect either the conductivity or the density of seawater having a Practical Salinity of 35. The most significant ions not included are Li + (~.8 mg kg ) and Rb + (~. mg kg ). Dissolved gases N (~6 mg kg ) and O ( u to 8 mg kg in the ocean) are not included as neither have a significant effect on density or on conductivity. In addition, N remains within a few ercent of saturation at the measured temerature in almost all laboratory and in situ conditions. However, the dissolved gas CO (~.7 mg kg ), and the ion OH (~.8 mg kg ) are included in the Reference Comosition because of their imortant role in the equilibrium dynamics of the carbonate system. Changes in H which involve conversion of CO to and from ionic forms affect conductivity and density. Concentrations of the major nutrients Si(OH) 4, NO 3 3 and PO 4 are assumed to be negligible in Standard Seawater; their concentrations in the ocean range from 6 mg kg, mg kg, and. mg kg resectively. The Reference Comosition does not include organic matter. The comosition of Dissolved Organic Matter (DOM) is comlex and oorly known. DOM is tyically resent at concentrations of.5 mg kg in the ocean. Reference Comosition Salinity is defined to be conservative during mixing or evaoration that occurs without removal of sea salt from solution. Because of this roerty, the Reference Comosition Salinity of any seawater samle can be defined in terms of roducts determined from the mixture or searation of two recisely defined end members. Pure water and KCl normalized seawater are defined for this urose. Pure water is defined as Vienna Standard Mean Ocean Water, VSMOW, which is described in the Guideline of the International ssociation for the Proerties of Water and Steam (IPWS (5), BIPM (5)); it is taken as the zero reference value. KCl normalized seawater (or normalized seawater for short) is defined to corresond to a seawater samle with a Practical Salinity of 35. Thus, any seawater samle that has the same electrical conductivity as a solution of otassium chloride (KCl) in ure water with the KCl mass fraction of g kg when both are at the ITS 9 temerature t = C and one standard atmoshere ressure, P = 35 Pa is referred to as normalized seawater. Here, KCl refers to the normal isotoic abundances of otassium and chlorine as described by the International Union of Pure and lied Chemistry (Wieser (6)). s discussed below, any normalized seawater samle has a Reference Comosition Salinity of gkg. Since Reference Comosition Salinity is defined to be conservative during mixing, if a seawater samle of mass m and Reference Comosition Salinity S R is mixed with another seawater samle of mass m and Reference Comosition Salinity S R, the final Reference Comosition Salinity S R of this samle is S R ms = m + ms + m R R. (.3.)

78 68 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater Negative values of m and m, corresonding to the removal of seawater with the aroriate salinity are ermitted, so long as m( SR) + m( SR) >. In articular, if S R = (ure water) and m is the mass of ure water needed to normalize the seawater samle (that is, m is the mass needed to achieve S R = g kg ), then the original Reference Comosition Salinity of samle is given by S = [ + ( m / m )] g kg. (.3.) R The definitions and rocedures above allow one to determine the Reference Salinity of any seawater samle at the ITS 9 temerature t = C and one standard atmoshere ressure. To comlete the definition, we note that the Reference Comosition Salinity of a seawater samle at given temerature and ressure is equal to the Reference Comosition Salinity of the same samle at any other temerature and ressure rovided the transition rocess is conducted without exchange of matter, in articular, without evaoration, reciitation or degassing of substance from the solution. Note that this roerty is shared by Practical Salinity to the accuracy of the algorithms used to define this quantity in terms of the conductivity ratio R 5. We noted above that a Practical Salinity of 35 is associated with a Reference Salinity of gkg. This value was determined by Millero et al. (8a) using the reference comosition model, the most recent atomic weights (Wieser (6)) and the relation S = Cl / (g kg ) which was used in the original definition of Practical Salinity to convert between measured Chlorinity values and Practical Salinity. Since the relation between Practical Salinity and conductivity ratio was defined using the same conservation relation as satisfied by Reference Salinity, the Reference Salinity can be determined to the same accuracy as Practical Salinity wherever the latter is defined (that is, in the range < S < 4), as P SR upssp where u PS ( ) gkg. (.3.3) For ractical uroses, this relationshi can be taken to be an equality since the aroximate 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 recisely conservative during mixing and also during changes in temerature and ressure that occur without exchange of mass with the surroundings. The Reference Comosition Salinity Scale is defined such that a seawater samle whose Practical Salinity S P is 35 has a Reference Comosition Salinity S R of recisely g kg. Millero et al. (8a) estimate that the absolute uncertainty associated with using this value as an estimate of the bsolute Salinity of Reference Comosition Seawater is ±.7 g kg. Thus the numerical difference between the Reference Salinity exressed in gkg and Practical Salinity is about 4 times larger than this estimate of uncertainty. The difference is also large comared to our ability to measure Practical Salinity at sea (which can be as recise as ±. ). Understanding how this discreancy was introduced requires consideration of some historical details that influenced the definition of Practical Salinity. The details are resented in Millero et al. (8a) and in Millero () and are briefly reviewed below. There are two rimary sources of error that contribute to this discreancy. First, and most significant, in the original evaoration technique used by Sørensen in 9 (Forch et al. 9) to estimate salinity, some volatile comonents of the dissolved material were lost so the amount of dissolved material was underestimated. Second, the aroximate relation determined by Knudsen (9) to determine S ( ) from measurements of Cl ( ) was based on analysis of only nine samles (one from the Red Sea, one from the North tlantic, one from the North Sea and six from the Baltic Sea). Both the errors in estimating absolute Salinity by evaoration and the bias towards Baltic Sea conditions, where strong -

79 TEOS- Manual: Calculation and Use of the Thermodynamic Proerties of Seawater 69 comosition anomalies relative to North tlantic conditions are found, are reflected in Knudsenʹs formula, S K ( ) Cl ( ) = (.3.4) When the Practical Salinity Scale was decided uon in the late 97s it was known that this relation included significant errors, but it was decided to maintain numerical consistency with this acceted definition of salinity for tyical mid ocean conditions (Millero ()). To achieve this consistency while having salinity directly roortional to Chlorinity, the Joint Panel for Oceanograhic Tables and Standards (JPOTS) decided to determine the roortionality constant from Knudsenʹs formula at S K = 35 ( Cl = ), (Wooster et al., 969). This resulted in the conversion formula S ( ).8655 Cl( ) = (.3.5) being used in the definition of the ractical salinity scale as if it were an identity, thus introducing errors that have either been overlooked or acceted for the ast 3 years. We now break with this tradition in order to define a salinity scale based on a comosition model for Standard Seawater that was designed to give a much imroved estimate of the mass fraction salinity for Standard Seawater and for Reference Comosition Seawater. The introduction of this salinity scale rovides a more hysically meaningful measure of salinity and simlifies the task of systematically incororating the influence of satial variations of seawater comosition into the rocedure for estimating bsolute Salinity. Finally, we note that to define the Reference Comosition Salinity Scale we have introduced the quantity u PS in Eqn. (.3.3), defined by u PS ( ) gkg. This value was determined by the requirement that the Reference Comosition Salinity gives the best estimate of the mass fraction bsolute Salinity (that is, the mass fraction of non HO material) of Reference Comosition Seawater. However, the uncertainty in using S R to estimate the bsolute Salinity of Reference Comosition Seawater is at least.7 gkg at S = 35 (Millero et al. (8b)). Thus, although u PS is recisely secified in the definition of the Reference Comosition Salinity Scale, it must be noted that using the resulting definition of the Reference Salinity to estimate the bsolute Salinity of Reference Comosition Seawater does have a non zero uncertainty associated with it. This and related issues are discussed further in the next subsection..4 bsolute Salinity Millero et al. (8a) list the following six advantages of adoting Reference Salinity and bsolute Salinity S in reference to Practical Salinity S P.. The definition of Practical Salinity S P on the PSS 78 scale is searate from the system of SI units (BIPM (6)). Reference Salinity can be exressed in the unit (g kg ) as a measure of bsolute Salinity. doting bsolute Salinity and Reference Salinity will terminate the ongoing controversies in the oceanograhic literature about the use of PSU or PSS and make research aers more readable to the outside scientific community and consistent with SI.. The freshwater mass fraction of seawater is not (. S P ). Rather, it is (. S /( gkg )), where S is the bsolute Salinity, defined as the mass fraction of dissolved material in seawater. The values of S /( gkg ) and S P are known to differ by about.5%. There seems to be no good reason for continuing to ignore this known difference, for examle in ocean models. 3. PSS 78 is limited to the range < S P < 4. For a smooth crossover on one side to ure water, and on the other side to concentrated brines u to saturation, as for S R

80 7 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater examle encountered in sea ice at very low temeratures, salinities beyond these limits need to be defined. While this oses 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 examle in the determination of the Gibbs function of seawater, can only be comuted from a chemical comosition model, which is available for S R but not for S P. 5. For artificial seawater of Reference Comosition, S R has a fixed relation to Chlorinity, indeendent of conductivity, salinity, temerature, or ressure. 6. Stoichiometric anomalies can be secified accurately relative to Reference Comosition Seawater with its known comosition, but only uncertainly with resect to IPSO Standard Seawater with its unknown comosition. These variations in the comosition of seawater cause significant (a few ercent) variations in the horizontal density gradient. Regarding oint number, Practical Salinity S P is a dimensionless number of the order of 35 in the oen ocean; no units or their multiles are ermitted. There is however more freedom in choosing the reresentation of bsolute Salinity S since it is defined as the mass fraction of dissolved material in seawater. For examle, all the following quantities are equal (see ISO (993) and BIPM (6)), 34 g/kg = 34 mg/g =.34 kg/kg =.34 = 3.4 % = 34 m = 34 mg/kg. In articular, it is strictly correct to write the freshwater fraction of seawater as either (. S /( gkg )) or as ( S ) but it would be incorrect to write it as (. S ). Clearly it is essential to consider the units used for bsolute Salinity in any articular alication. If this is done, there should be no danger of confusion, but to maintain the numerical value of bsolute Salinity close to that of Practical Salinity S P we adot the first otion above, namely gkg as the referred unit for S, (as in S = g kg ). The Reference Salinity, S R, is defined to have the same units and follows the same conventions as S. Salinity S measured rior to PSS 78 available from the literature or from databases is usually reorted in or t (art er thousand) and is converted to the Reference Salinity, SR = ups S, by the numerical factor u PS from (.3.3). Regarding oint number 5, Chlorinity Cl is the concentration variable that was used in the laboratory exeriments for the fundamental determinations of the equation of state and other roerties, but has seldom been measured in the field since the definition of PSS 78 (Millero, ). Since the relation S = 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 Comosition Seawater. Thus, Chlorinity exressed in can be converted to Reference Comosition Salinity by the relation, SR = ucl Cl, with the numerical factor ucl = ups. These constants are recommended for the conversion of historical (re 9) data. The rimary source of error in using this relation will be the ossible resence of comosition anomalies in the historical data relative to Standard Seawater. Regarding oint number 6, the comosition 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 (975) and Millero () oint out that these satial variations in the relative comosition of seawater imact the relationshi between Practical Salinity (which is essentially a measure of the conductivity of seawater at a fixed temerature and ressure) and density. ll the thermohysical roerties of seawater as well as other multicomonent electrolyte solutions are directly related to the concentrations of the major comonents, not the salinity determined by conductivity; note that some of the variable nonelectrolytes (e.g.,

81 TEOS- Manual: Calculation and Use of the Thermodynamic Proerties of Seawater 7 Si(OH) 4, CO and dissolved organic material) do not have an areciable conductivity signal. It is for this reason that the new TEOS thermodynamic descrition of seawater (Millero et al. (8a), Millero ()) has the Gibbs function g of seawater exressed as a function of bsolute Salinity as g ( S,, t ) rather than as a function of Practical Salinity S P or of Reference Salinity, S R. The issue of the satial variation in the comosition of seawater is discussed more fully in aendix.5. Regarding oint number, we note that it is erhas debatable which of dens (. S /( gkg soln )), (. S /( gkg add )), (. S /( gkg )) or (. S * /( gkg )) is the most aroriate measure of the freshwater mass fraction. (These different versions of absolute salinity are defined in section.5 and also later in this aendix.) This is a minor oint comared with the resent use of S P in this context, and the choice of which of these exressions may deend on the use for the freshwater mass fraction. For examle, in the context of ocean modelling, if S * is the salinity variable that is treated as a conservative variable in an ocean model, then (. S * /( gkg )) is robably the most aroriate version of freshwater mass fraction. It should be noted that the quantity S aearing as an argument of the function g ( S,, t ) is the bsolute Salinity (the Density Salinity dens S S ) measured on the Reference Comosition Salinity Scale. This is imortant since the Gibbs function has been fitted to laboratory and field measurements with the bsolute Salinity values exressed on this scale. Thus, for examle, it is ossible that sometime in the future it will be determined that an imroved estimate of the mass fraction of dissolved material in Standard Seawater can be obtained by multilying S R by a factor slightly different from (uncertainties ermit values in the range ±.). We emhasize that since the Gibbs function is exressed in terms of the bsolute Salinity exressed on the Reference Comosition 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 roerties determined from the Gibbs function. In art for this reason, we recommend that the Reference Comosition Salinity continue to be measured on the scale defined by Millero et al. (8a) even if new results indicate that imroved estimates of the true mass fraction can be obtained using a modified scale. That is, we recommend that the value of u PS used in (.3.3) not be udated. If a more accurate mass fraction estimate is required for some urose in the future, such a revised estimate should definitely not be used as an argument of the TEOS Gibbs function. Finally, we note a second reason for recommending that the value assigned to u PS not be modified without very careful consideration. Working Grou 7 is recommending that the ractice of exressing salinity as Practical Salinity in ublications be hased out in favour of using bsolute Salinity for this urose. It is critically imortant that this new measure of salinity remain stable into the future. In articular, we note that any change in the value of u PS used in the determination of Reference Salinity would result in a change in reorted salinity values that would be unrelated to any real hysical change. For examle, a change in u PS from /35 to (35.654/35) x. for examle, would result in changes of the reorted salinity values of order.35 gkg which is more than ten times larger than the recision of modern salinometers. Thus changes associated with a series of imroved estimates of u PS (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 exressed as a function of bsolute Salinity measured on the Reference Comosition Salinity Scale as defined by Millero et al. (8a), we strongly recommend that the Reference Comosition Salinity continue to be exressed on this scale; no changes in the value of u PS should be introduced.

82 7 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater For seawater of Reference Comosition, Reference Salinity S R is the best available estimate of the mass fraction of non HO material in seawater. s discussed in sections.4 and.5, under TEOS S R was determined to rovide the best available estimate of the mass fraction of non HO material in Standard Seawater by Millero et al. (8a). Subsequently, Pawlowicz (a) has argued that the DIC content of the Reference Comosition is robably about 7 μmol kg low for SSW and also for the North tlantic surface water from which it was reared. 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 larger (Pawlowicz et al., ) but at.55 gkg it is still just below the uncertainty of.7 gkg assigned to the estimated bsolute Salinity by Millero et al. (8a). In fact, the largest uncertainties in Reference Salinity as a measure of the bsolute Salinity of SSW are associated with uncertainties in the mass fractions of other constituents such as sulhate, which may be as large as.5 gkg (Seitz et al., a). Nevertheless, it seems that the sulhate value of Reference Comosition Seawater lies within the 95% uncertainty range of the best laboratory determined estimates of SSW s sulhate concentration, so there is no justification for an udate of the Reference Comosition at this time. When the comosition of seawater differs from that of Standard Seawater, there are several ossible definitions of the absolute salinity of a seawater samle, as discussed in section.5. Concetually the simlest definition is the mass fraction of dissolved non HO material in a seawater samle at its temerature and ressure. One drawback of this definition is that because the equilibrium conditions between HO and several carbon comounds deends on temerature and ressure, this mass fraction would change as the temerature and ressure of the samle is changed, even without the addition or loss of any material from the samle. This drawback can be overcome by first bringing the samle to the constant temerature t = 5 C and the fixed sea ressure dbar, and when this is done, the resulting mass fraction of non HO material is called Solution bsolute soln Salinity (usually shortened to Solution Salinity ), S. nother measure of absolute add salinity is the dded Mass Salinity S which is S R lus the mass fraction of material that must be added to Standard Seawater to arrive at the concentrations of all the secies in the given seawater samle, after chemical equilibrium has been reached, and after the samle has been brought to t = 5 C and = dbar. nother form of absolute salinity, Preformed bsolute Salinity (usually shortened to Preformed Salinity ), S *, has been defined by Pawlowicz et al. () and Wright et al. (b). Preformed Salinity S * is designed to be as close as ossible to being a conservative variable. That is, S * is designed to be insensitive to biogeochemical rocesses that affect the other tyes of salinity to varying degrees. S * is formed by first estimating the contribution of biogeochemical rocesses to one of the salinity measures soln add S, S, or S, and then subtracting this contribution from the aroriate salinity variable. Because it is designed to be a conservative oceanograhic variable, S * will find a rominent role in ocean modeling. soln add There is still no simle means to measure either S or S for the general case of the arbitrary addition of many comonents to Standard Seawater. Hence a more recise and easily determined measure of the amount of dissolved material in seawater is required dens dens and TEOS adots Density Salinity S for this urose. Density Salinity S is defined as the value of the salinity argument of the TEOS exression for density which gives the samle s actual measured density at the temerature t = 5 C and at the sea ressure = dbar. When there is no risk of confusion, Density Salinity is also called dens bsolute Salinity with the label S, that is S S. There are two clear advantages of dens soln add S S over both S and S. First, it is ossible to measure the density of a seawater samle very accurately and in an SI traceable manner, and second, the use of

83 TEOS- Manual: Calculation and Use of the Thermodynamic Proerties of Seawater 73 dens S S yields the best available estimates of the density of seawater. This is imortant because in the field of hysical oceanograhy, it is density that needs to be known to the highest relative accuracy. Pawlowicz et al. () and Wright et al. (b) found that while the nature of the ocean s comosition variations changes from one ocean basin to another, the five different dens soln add salinity measures S R, S, S, S and S * are aroximately related by the following simle linear relationshis, (obtained by combining equations (55) (57) and (6) of Pawlowicz et al. ()) R dens R S S.35 δ S, (.4.) dens dens R. δ R S S S, (.4.) S soln dens SR.75 δsr, (.4.3) add dens R.78 δ R S S S. (.4.4) Eqn. (.4.) is simly the definition of the bsolute Salinity nomaly, dens dens δs δsr S SR. Note that in many TEOS ublications, the simler notation dens dens δ S is used for δ SR S SR, a salinity difference for which a global atlas is available (McDougall et al. (a)). In the context of ocean modelling, it is more convenient to cast these salinity differences with resect to the Preformed Salinity S as follows (using the above equations) R dens R S S.35 δ S, (.4.5) dens dens R S S.35 δ S, (.4.6) soln dens *. δ R S S S, (.4.7) add dens R S S.3 δ S. (.4.8) These relationshis are illustrated on the number line of salinity in Figure.4.. For SSW, dens soln add all five salinity variables S R, S, S, S and S * are equal. It should be noted that the simle relationshis of Eqns. (.4.) (.4.8) are derived from simle linear fits to model calculations that show more comlex variations. However, the variation about these relationshis is not larger than the tyical uncertainty of ocean measurements. These linear relationshis rovide a way by which the effects of anomalous seawater comosition may be addressed in ocean models (see aendix.). Figure.4.. Number line of salinity, illustrating the differences between various forms of salinity for seawater whose comosition differs from that of Standard Seawater. If measurements are available of the Total lkalinity, 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 aear on the left hand sides of

84 74 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater Eqns. (.4.) (.4.8). Pawlowicz et al. () have used a chemical model of conductivity and density to estimate how the many salinity differences introduced above deend on the measured roerties of seawater. The following equations corresond to Eqns. (.4.) (.4.4) above, and come from equations (5) (54) and (59) of Pawlowicz et al. (). These equations are written in terms of the values of the nitrate and silicate concentrations in the seawater samle (measured in mol kg ), the difference between the Total lkalinity ( T ) and Dissolved Inorganic Carbon ( DIC ) of the samle and the corresonding values of our best estimates of T and DIC in Standard Seawater, Δ T and Δ DIC, both measured in mol kg. For Standard Seawater our best estimates of T and DIC are.3 ( S P 35) mol kg and.8 ( S P 35) mol kg resectively (see Pawlowicz (a), Pawlowicz et al. () and the discussion of this asect of SSW versus RCSW in Wright et al. (b))). ( S* SR) ( 3 4) / (g kg ) = 8.ΔT 7.ΔDIC 43. NO +. Si(OH) (mol kg ), (.4.9) ( S SR) ( 3 4) dens / (g kg ) = 55.6Δ T + 4.7Δ DIC+38.9 NO Si(OH) (mol kg ), (.4.) ( S SR) ( 3 4) soln / (g kg ) = 7. Δ T + 47.Δ DIC NO Si(OH) (mol kg ), (.4.) ( S SR) ( 3 4) add / (g kg ) = 5.9 Δ T Δ DIC +6.NO + 6. Si(OH) (mol kg ). (.4.) The standard error of the model fits in Eqns. (.4.9) (.4.) are given by Pawlowicz et 4 3 al. () at less than kg m (in terms of density) which is equivalent to a factor of smaller than the accuracy to which Practical Salinity can be measured at sea. It is clear that if measurements of T, DIC, nitrate and silicate are available (and recognizing that these measurements will come with their own error bars), these exressions will likely give more accurate estimates of the salinity differences than the aroximate linear exressions resented in Eqns. (.4.) (.4.8). The coefficients in Eqn. (.4.) are reasonably similar to the corresonding exression of Brewer and Bradshaw (975) (as dens corrected by Millero et al. (976a)): when exressed as the salinity anomaly S SR rather than as the corresonding density anomaly ρ ρr, their exression corresonding to Eqn. (.4.) had the coefficients 7.4,.8, 3.9 and 59.9 comared with the coefficients 55.6, 4.7, 38.9 and 5.7 resectively in Eqn. (.4.). The salinity differences exressed with resect to Preformed Salinity S * which corresond to Eqns. (.4.5) (.4.8) can be found by linear combinations of Eqns. (.4.9) (.4.) as follows ( SR S* ) ( 3 4) / (g kg ) = 8.Δ T + 7.Δ DIC NO. Si(OH) (mol kg ), (.4.3) ( S S* ) ( 3 4) dens / (g kg ) = 73.7 Δ T +.8Δ DIC +8.9 NO Si(OH) (mol kg ), (.4.4) ( S S* ) ( 3 4) soln / (g kg ) = 5.3Δ T + 54.Δ DIC NO Si(OH) (mol kg ), (.4.5) ( S S* ) ( 3 4) add / (g kg ) = 44.Δ T +.Δ DIC+59.NO + 6. Si(OH) (mol kg ). (.4.6)

85 TEOS- Manual: Calculation and Use of the Thermodynamic Proerties of Seawater 75.5 Satial variations in seawater comosition When the oceanograhic data needed to evaluate Eqn. (.4.) for S SR is not available, the look u table method of McDougall et al. (a) is recommended to dens dens evaluate δs δsr S SR. The following aragrahs describe how this method was develoed. In a series of aers Millero et al. (976a, 978,, 8b) and McDougall et al. (a) have reorted on density measurements made in the laboratory on samles collected from around the world s oceans. Each samle has had its Practical Salinity measured in the laboratory as well as its density (measured with a vibrating tube densimeter at 5 C and atmosheric ressure). The Practical Salinity yields a Reference meas Salinity S R according to Eqn. (.3.3), while the density measurement ρ imlies an dens bsolute Salinity S S by using the equation of state and the equality ρ meas = ρ dens ( S ),5 C,dbar. The difference dens S SR between these two salinity measures is taken to be due to the comosition of the samle being different to that of Standard Seawater. In these aers Millero established that the salinity difference S SR could be estimated aroximately from knowledge of just the silicate concentration of the fluid samle. The reason for the exlaining ower 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.6) of the tyical variations in concentrations (g kg ) of the above secies and (c) being essentially non ionic; its resence has little effect on conductivity while having a direct effect on density. When the existing data on δ S, based on laboratory measurements of density, was regressed against the silicate concentration of the seawater samles, McDougall et al. (a) found the simle relation ( ) R 4 δ S /(gkg ) = ( S S )/(gkg ) = 98.4 Si(OH) /(molkg ). Global (.5.) This regression was done over all available density measurements from the world ocean, and the standard error in the fit was.54 g kg. The deendence of δ S on silicate concentration is observed to be different in each ocean basin, and this asect was exloited by McDougall et al. (a) to obtain a more accurate deendence of δ S on location in sace. For data in the Southern Ocean south of 3 o S the best simle fit was found to be δ S ( ) 4 /(gkg ) = Si(OH) /(mol kg ), Southern Ocean (.5.) and the associated standard error is.6 g kg. The data north of 3 o S in each of the Pacific, Indian and tlantic Oceans was treated searately. In each of these three regions the fit was constrained to match (.5.) at 3 o S and the sloe of the fit was allowed to vary linearly with latitude. The resulting fits were (for latitudes north of 3 o S, that is for λ 3 ) δs δs δs ( [ λ ])( ) 4 / (g kg ) = / 3 + Si(OH) / (mol kg ), Pacific (.5.3) ( [ λ ])( ) 4 / (g kg ) = / 3 + Si(OH) / (mol kg ), Indian (.5.4) ( [ λ ])( ) 4 / (g kg ) = / 3 + Si(OH) / (mol kg ). tlantic (.5.5) These relationshis between the bsolute Salinity nomaly δ S= S SR and silicate concentration have been used by McDougall, Jackett and Millero (a) in a comuter algorithm that uses an existing global data base of silicate (Gouretski and Koltermann (4)) and rovides an estimate of bsolute Salinity when given a seawater samle s dens

86 76 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater Practical Salinity as well as its satial location in the world ocean. This comuter algorithm accounts for the latest understanding of bsolute Salinity in the Baltic Sea, but it is silent on the influence of comositional variations in other marginal seas. The bsolute Salinity nomaly in the Baltic Sea has been quite variable over the ast few decades of observation (Feistel et al. (c)). The comuter algorithm of McDougall et al. (a) uses the relationshi found by Feistel et al. (c) that alies in the years 6 9, namely S S =.87g kg S S, (.5.6) ( ) R R SO where S SO = g kg is the standard ocean Reference Salinity that corresonds to the Practical Salinity of 35. In order to gauge the imortance of the satial variation of seawater comosition, the northward gradient of density at constant ressure is shown in Fig..5. for the data in a world ocean hydrograhic atlas deeer than m. The vertical axis in this figure is the magnitude of the difference between the northward density gradient at constant ressure dens when the TEOS algorithm for density is called with S S (as it should be) comared with calling the same TEOS density algorithm with S R as the salinity argument. Figure.5. shows that the thermal wind is misestimated by more than % for 58% of the data in the world ocean below a deth of m if the effects of the variable seawater comosition are ignored. Figure.5.. The northward density gradient at constant ressure (the horizontal axis) for data in the global ocean atlas of Gouretski and Koltermann (4) for > dbar. The vertical axis is the magnitude of the difference between evaluating the density gradient using S versus S R as the salinity argument in the TEOS exression for density. The imortance of the satial variations in seawater comosition illustrated in Fig..5. can be comared with the corresonding imrovement achieved by the TEOS Gibbs function for Standard Seawater comared with using EOS 8. This is done by ignoring satial variations in seawater comosition in both the evaluation of TEOS and in EOS8 by calling TEOS with S R and EOS 8 with S P. Figure.5. shows the magnitude of the imrovement in the thermal wind in the art of the ocean that is deeer than m through the adotion of TEOS but ignoring the influence of

87 TEOS- Manual: Calculation and Use of the Thermodynamic Proerties of Seawater 77 comositional variations. By comaring Figs..5. and.5. it is seen that the main benefit that TEOS delivers to the evaluation of the thermal wind is through the incororation of satial variations in seawater comosition; the greater accuracy of TEOS over EOS 8 for Standard Seawater is only 8% as large as the imrovement gained by the incororation of comositional variations into TEOS (i. e. the rms value of the vertical axis in Fig..5. is 8% of that of the vertical axis of Fig..5.). If the tlantic were excluded from this comarison, the relative imortance of comositional variations would be even larger. Figure.5.. The northward density gradient at constant ressure (the horizontal axis) for data in the global ocean atlas of Gouretski and Koltermann (4) for > 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 exression for density comared with using S P in the EOS 8 algorithm for density. The thermodynamic descrition of seawater and of ice Ih as defined in IPWS 8 and IPWS 6 has been adoted as the official descrition of seawater and of ice Ih by the Intergovernmental Oceanograhic Commission in June 9. The adotion of TEOS has recognized that this technique of estimating bsolute Salinity from readily measured quantities is erhas the least mature asect of the TEOS thermodynamic descrition of seawater. The resent comuter software, in both FORTRN and MTLB, which evaluates bsolute Salinity S given the inut variables Practical Salinity S P, longitude λ, latitude φ and sea ressure is available at It is exected, as new data (articularly density data) become available, that the determination of bsolute Salinity will imrove over the coming decades, and the algorithm for evaluating bsolute Salinity in terms of Practical Salinity, latitude, longitude and ressure, will be udated from time to time, after relevant aroriately eer reviewed ublications have aeared, and such an udated algorithm will aear on the web site. Users of this software should state in their ublished work which version of the software was used to calculate bsolute Salinity.

88 78 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater.6 Gibbs function of seawater The Gibbs function of seawater g ( S,, ) ure water W S g ( t, ) and the saline art of the Gibbs function (,, ) W S g ( S,, t ) g ( t, ) g ( S,, t ). t is defined as the sum of the Gibbs function for g S t so that = + (.6.) In this way at zero bsolute Salinity, the thermodynamic roerties of seawater are equal to those of ure water. This consistency is also maintained with resect to the Gibbs function for ice so that the roerties along the equilibrium curve can be accurately determined (such as the freezing temerature as a function of bsolute Salinity and ressure). The careful alignment of the thermodynamic otentials of ure water, ice Ih and seawater is described in Feistel et al. (8a). The internationally acceted thermodynamic descrition of the roerties of ure water (IPWS 95) is the official ure water basis uon which the Gibbs function of seawater is built according to (.6.). This g W ( t, ) Gibbs function of liquid water is valid over extended ranges of temerature and ressure from the freezing oint to the critical oint ( C < t < 374 C and 6 Pa < + P < MPa) however it is a comutationally exensive algorithm. Part of the reason for this comutational intensity is that the IPWS 95 formulation is in terms of a Helmholtz function which has the ressure as a function of temerature and density, so that an iterative rocedure is need to for the Gibbs function g W ( t, ) (see for examle, Feistel et al. (8a)) For ractical oceanograhic use in the oceanograhic ranges of temerature and ressure, from less than the freezing temerature of seawater (at any ressure), u to 4 C (secifically from.65 ( P ).743 MPa + + C to 4 C), and in the ressure 4 range < < dbar we also recommend the use of the ure water art of the Gibbs function of Feistel (3) which has been aroved by IPWS as the Sulementary Release, IPWS 9. The IPWS 9 release discusses the accuracy to which the Feistel (3) Gibbs function fits the underlying thermodynamic otential of IPWS 95; in summary, for the variables density, thermal exansion coefficient and secific heat caacity, the rms misfit between IPWS 9 and IPWS 95, in the region of validity of IPWS 9, are a factor of between and less than the corresonding error in the laboratory data to which IPWS 95 was fitted. Hence, in the oceanograhic range of arameters, IPWS 9 and IPWS 95 may be regarded as equally accurate thermodynamic descritions of ure liquid water. ll of the thermodynamic roerties of seawater that are described in this Manual are available as both FORTRN and MTLB imlementations. These imlementations are available for g W ( t, ) being IPWS 95 and IPWS 9, both being equally accurate relative to the laboratory determined known roerties, but with the comuter code based on IPWS 9 being aroximately a factor of 65 faster than that based on IPWS 95. Most of the exerimental seawater data that were already used for the construction of EOS 8 were exloited again for the IPWS 8 formulation after their careful adjustment to the new temerature and salinity scales and the imroved ure water reference IPWS 95. dditionally, IPWS 8 was significantly imroved (comared with EOS 8) by making use of theoretical relations such as the ideal solution law and the Debye Hückel limiting law, as well as by incororating additional accurate measurements such as the temeratures of maximum density, vaour ressures and mixing heats, and imlicitly by the enormous background data set which had entered the determination of IPWS 95 (Wagner and Pruß (), Feistel (3, 8)). For examle, Millero and Li (994) concluded that the ure water art of the EOS 8 sound seed formula of Chen and Millero (977) was resonsible for a deviation of.5 ms from Del Grosso s (974) formula for seawater at high ressures and temerature below 5 o C. Chen and Millero (977) only measured the differences in the sound seeds of seawater and ure water. The

89 TEOS- Manual: Calculation and Use of the Thermodynamic Proerties of Seawater 79 new Gibbs function in which we use IPWS 95 for the ure water art as well as sound seeds from Del Grosso (974), is erfectly consistent with Chen and Millero s (976) densities and Bradshaw and Schleicher s (97) thermal exansion data at high ressures. The accuracy of high ressure seawater densities has increased with the use of IPWS 95, directly as the ure water art, and indirectly by correcting earlier seawater measurements, making them ʺnewʺ seawater data. In this manner the known soundseed inconsistency of EOS 8 has been resolved in a natural manner..7 The fundamental thermodynamic relation The fundamental thermodynamic relation for a system comosed 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 arcel. The fundamental thermodynamic relation is ( ) dh vdp = T+ t dη + μds. (.7.) derivation of the fundamental thermodynamic relation can be found in Warren (6) (his equation (8)). The left hand side of Eqn. (.7.) is often written as du+ ( + P ) dv where ( + P ) is the absolute ressure. Here h is the secific enthaly (i.e. enthaly er unit mass of seawater), u is the secific internal energy, v = ρ is the secific volume, ( T + t) is the absolute temerature, η is the secific entroy and μ is the relative chemical otential. In fluid dynamics we usually deal with material derivatives, ddt, that is, derivatives defined following the fluid motion, ddt = t + u where u is the fluid velocity. In terms of this tye of derivative, and assuming local thermodynamic equilibrium (i. e. that local thermodynamic equilibrium is maintained during the temoral change), the fundamental thermodynamic relation is d h d P dη d = ( T + t) + μ S. (.7.) dt ρ dt dt dt Note that the constancy of entroy does not imly the absence of irreversible rocesses because, for examle, there can be irreversible changes of both salinity and enthaly at constant ressure in just the right ratio so as to have equal effects in Eqns. (.7.) or (.7.) so that the change of entroy in these equations is zero..8 The conservative and isobaric conservative roerties thermodynamic variable C is said to be conservative if its evolution equation (that is, its rognostic equation) has the form dc C ( ρc) + ( ρuc) = ρ = F. (.8.) t dt C For such a conservative roerty, in the absence of fluxes F at the boundary of a control volume, the total amount of C stuff is constant inside the control volume. The middle art of Eqn. (.8.) has used the continuity equation (which is the equation for the conservation of mass) ρ t + ρu =. (.8.) x, y, z In the secial case when the material derivative of a roerty is zero (that is, the middle art of Eqn. (.8.) is zero) the roerty is said to be materially conserved. The only quantity that can be regarded as % conservative in the ocean is mass [equivalent to taking C = and F C = in Eqn. (.8.)]. In fact, looking ahead to ( )

90 8 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater aendices. and., if we strictly interret ρu as the mass flux er unit area of ure seawater (i. e. of only ure water lus dissolved material) and secifically, that ρu excludes the flux of articulate matter, then the right hand side of the continuity equation (.8.) should be ρ S S, the non conservative source of mass due to biogeochemical rocesses. It can be shown that the influence of this source term ρ S S in the continuity equation on the evolution equation for bsolute Salinity is less imortant by the factor Sˆ ( Sˆ ) than the same source term that aears in this evolution equation for bsolute Salinity, Eqn. (..8). Hence the current ractice of assuming that the nonarticulate art of the ocean obeys the conservative form (.8.) of the continuity equation is confirmed even in the resence of biogeochemical rocesses. Two other variables, total energy E = u +.5u u + Φ (see Eqn. (B.5)) and Conservative Temerature (or equivalently, otential enthaly h ) are not comletely conservative, but the error in assuming them to be conservative is negligible (see aendix.). Other variables such as Reference Salinity S R, bsolute Salinity S, otential temerature θ, enthaly h, internal energy u, entroy η, density ρ, otential density θ ρ, secific volume anomaly δ and the Bernoulli function B = h +.5u u+φ (see Eqn. (B.7)) are not conservative variables. While both bsolute Salinity and Reference Salinity are conservative under the turbulent mixing rocess, both are affected in a non conservative way by the remineralization rocess. Because the dominant variations of the comosition of seawater are due to secies 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 aroximately 4 larger (in terms of its effects on density) than the error in assuming that is a conservative variable. Preformed Salinity S * is constructed so that it contains no signature of the biogeochemical rocesses that cause the satial variation of seawater comosition. In this way S * is secifically 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 resently limited by our knowledge of the biogeochemical rocesses (see aendices.4 and.5 and Pawlowicz et al. ()). 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 +.5u u + Φ, (iii) Conservative Temerature, and (iv) Preformed Salinity S *. different form of conservation attribute, namely isobaric conservation occurs when the total amount of the quantity is conserved when two fluid arcels are mixed at constant ressure without external inut of heat or matter. This isobaric conservative roerty is a very valuable attribute for an oceanograhic variable. ny conservative variable is also isobaric conservative, thus the four conservative variables listed above, namely mass, Conservative Temerature, Preformed Salinity S *, and total energy E are isobaric conservative. In addition, the Bernoulli function B and secific enthaly h are also isobaric conservative (see Eqn. (B.7) and the discussion thereafter). Some variables that are not isobaric conservative include otential temerature θ, θ internal energy u, entroy η, density ρ, otential density ρ, and secific volume anomaly δ. Enthaly h and Conservative Temerature are not exactly isobaric conservative because enthaly increases when the kinetic energy of fluid motion is dissiated by molecular viscosity inside the control volume and when there is a salinity source term due to the remineralization of articulate matter. However, these are tiny effects in the First Law of Thermodynamics (see aendix.) and traditionally we regard enthaly h as an isobaric conservative variable. Note that while h is isobaric conservative, it is not a conservative variable.

91 TEOS- Manual: Calculation and Use of the Thermodynamic Proerties of Seawater 8 endices.8 and. show that for all ractical uroses we can treat and as being conservative variables (and hence also isobaric conservative variables); doing so ignores the dissiation of mechanical energy ε and other terms of similar or smaller magnitude. Hence for all ractical uroses in oceanograhy we have mass and the following three other variables that are conservative and isobaric conservative ; () Conservative Temerature, (and otential enthaly () Preformed Salinity S *, and (3) total energy E. Here we comment briefly on the likely errors involved with assuming variables other than S * and to be conservative variables in ocean models. If one took bsolute Salinity S as an ocean model s salinity variable and treated it as being conservative, the salinity error would (after a long sin u time) be aroximately as large as the bsolute Salinity nomaly (as shown in Figure ), which is larger than.5 g kg in the North Pacific, 3 imlying density errors of. kg m. s a measure of the imortance of this tye of density error, we note that if the equation of state in an ocean model were called with S R instead of with S, the northward density gradient at fixed ressure (i. e. the thermal wind) would be misestimated by more than % for more than 58% of the data below a ressure of dbar in the world ocean. It is clearly desirable to not have this tye of systematic error in the dynamical equations of the ocean comonent of couled climate models. endix. discusses ractical ways of including the effects of the nonconservative remineralization source term in ocean models. The recommended otion 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 bsolute Salinity nomaly as described in section.. and Eqns. (..3) (..5). The errors incurred in ocean models by treating otential temerature θ as being conservative have not yet been thoroughly investigated, but McDougall (3) and Tailleux () have made a start on this toic. McDougall (3) found that tyical errors in θ are ±. C while in isolated regions such as where the fresh mazon water discharges into the ocean, the error can be as large as.4 C. The corresonding error in the meridional heat flux aears to be about.5 PW (or a relative error of.4%). The use of Conservative Temerature in ocean models reduces these errors by almost two orders of magnitude. If the ocean were in thermodynamic equilibrium, its temerature would be the same everywhere as would the chemical otentials of water and of each dissolved secies, while the entroy and the concentrations of each secies would be functions of ressure. Turbulent mixing acts in the comlementary direction, tending to make salinity and entroy constant but in the rocess causing gradients in temerature and the chemical otentials as functions of ressure. That is, turbulent mixing acts to maintain a nonequilibrium state. This difference between the roles of molecular versus turbulent mixing results from the symmetry breaking role of the gravity field; for examle, in a laboratory without gravity, turbulent and molecular mixing would have indistinguishable effects. S Note that the molecular flux of salt F is given by equation (58.) of Landau and S Lifshitz (959) and by Eqn. (B.3) below. F consists not only of the roduct of the usual molecular diffusivity and ρ S, but also contains two other terms that are roortional to the gradients of temerature and ressure resectively. 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 resence of turbulent mixing in the real ocean renders this rocess moot as turbulence tends to homogenize the ocean and maintains a relatively constant sea salt comosition. Note that the descrition conservation equation of a articular quantity is often used for the equation that describes how this quantity changes in resonse to the h ), h

92 8 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater divergence of various fluxes of the quantity and to non conservative source terms. For examle, it is usual to refer to the conservation equation for entroy or for otential temerature. 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 ractical uroses) conserved. For other variables we will refer to their evolution equation or their rognostic equation or their local balance equation..9 The otential roerty ny thermodynamic roerty of seawater that remains constant when a arcel of seawater is moved from one ressure to another adiabatically, without exchange of mass and without interior conversion between its turbulent kinetic and internal energies, is said to ossess the otential roerty, or in other words, to be a otential variable. Prime examles of otential variables are entroy η and all tyes of salinity. The constancy of entroy η can be seen from the First Law of Thermodynamics in Eqn. (B.9) below; with the right hand side of Eqn. (B.9) being zero, and with no change in bsolute Salinity, it follows that entroy is also constant. ny thermodynamic roerty that is a function of only bsolute Salinity and entroy also remains unchanged by this rocedure and is said to have the otential roerty. Thermodynamic roerties that osses the otential attribute include otential temerature θ, otential enthaly h, Conservative θ Temerature and otential density ρ (no matter what fixed reference ressure is chosen). Some thermodynamic roerties that do not osses the otential roerty are temerature t, enthaly h, internal energy u, secific volume v, density ρ, secific volume anomaly δ, total energy E and the Bernoulli function B. From Eqn. (B.7) we notice that in the absence of molecular fluxes and the source term of bsolute Salinity, the Bernoulli function B is constant following the fluid flow only if the ressure field is steady; in general this is not the case. The non otential nature of E is exlained in the discussion following Eqn. (B.7). Some authors have used the term quasi material to describe a variable that has the otential roerty. The name quasi material derives from the idea that the variable only changes as a result of irreversible mixing rocesses and does not change in resonse to adiabatic and isohaline changes in ressure. The word adiabatic is traditionally taken to mean a rocess during which there is no exchange of heat between the environment and the fluid arcel one is considering. With this definition of adiabatic it is still ossible for the entroy η, the otential temerature θ and the Conservative Temerature of a fluid arcel to change during an isohaline and adiabatic rocess. This is because the dissiation of mechanical energy ε causes increases in η, θ and (see the First Law of Thermodynamics, Eqns. (.3.3) (.3.5)). While the dissiation of mechanical energy is a small term whose influence is routinely neglected in the First Law of Thermodynamics in oceanograhy, it seems advisable to modify the meaning of the word adiabatic in oceanograhy so that our use of the word more accurately reflects the roerties we normally associate with an adiabatic rocess. ccordingly we roose that the word adiabatic in oceanograhy be taken to describe a rocess occurring without exchange of heat and also without the internal dissiation of mechanical energy. With this definition of adiabatic, a rocess that is both isohaline and adiabatic does imly that the entroy η, otential temerature θ and Conservative Temerature are all constant. Using this more restrictive definition of the word adiabatic we can restate the definition of a otential roerty as follows; any thermodynamic roerty of seawater that remains constant when a arcel of seawater is moved from one ressure to another

93 TEOS- Manual: Calculation and Use of the Thermodynamic Proerties of Seawater 83 adiabatically and without exchange of mass, is said to ossess the otential roerty, or in other words, to be a otential variable. In aendix.8 above we concluded that only mass, and the three variables E, S * and (aroximately) are conservative (and hence also isobaric conservative ). Since E does not osses the otential roerty, we now conclude that only mass and the two variables S * and osses all three highly desired roerties, namely that they are conservative, isobaric conservative and are otential variables. In the case of Conservative Temerature, its conservative (and therefore its isobaric conservative ) nature is aroximate: while is not a % conservative variable, it is aroximately two orders of magnitude closer to being a totally conservative variable than are either otential temerature or entroy. Similarly, Preformed Salinity S * is in rincile % conservative, but our ability to evaluate S * from hydrograhic observations is limited (for examle, by the aroximate relations (.4.) or (.4.9)). Table.9. The otential, conservative, isobaric conservative and the functional nature of various oceanograhic variables Variable otential? conservative? isobaric conservative? function of ( S,, t )? S * S x x SR, S P x x x t x x x θ x x η x x h x x, h 3 3 u x x x 4 B x x x 4 4 E x x ρ,v x x x θ ρ x x δ x x x ρ v x x x n γ x x x x The remineralization of organic matter changes SR less than it changes S. 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. (..3)). 4 Taking the effects of remineralization to be negligible. c ρ has been decided uon. 5 Once the reference sound seed function ( ) In Table.9. various oceanograhic variables are categorized according to whether they osses the otential roerty, whether they are conservative variables, whether they S t. are isobaric conservative variables, and whether they are functions of only ( ),,, x 5

94 84 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater Note that 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 ( S,, t ) but also of the comosition of seawater. Hence is the most ideal thermodynamic variable. If it were not for the non conservation of bsolute Salinity, it too would be an ideal thermodynamic variable, but in this sense, Preformed Salinity is suerior to bsolute Salinity. Conservative Temerature and Preformed Salinity S * are the only two variables in this table to be both otential and θ conservative. The last four rows of Table.9. are for otential density, ρ (see section 3.4), secific volume anomaly, δ (see section 3.7), orthobaric density, ρ v (see aendix n.8) and Neutral Density γ (see section 3.4 and aendix.9).. Proof that θ = θ ( S, η ) and = ( S θ ) Consider changes occurring at the sea surface, (secifically at = dbar) where the temerature is the same as the otential temerature referenced to dbar and the increment of ressure d is zero. Regarding secific enthaly h and chemical otential μ to be functions of entroy η (in lace of temerature t ), that is, considering the functional form of h and μ to be h = h ( S, η, ) and μ = μ ( S, η, ), it follows from the fundamental thermodynamic relation (Eqn. (.7.)) that hη ( S, η,) d η + hs ( S ) ( ) ( ), η, ds = T + θ d η + μ S, η, d S, (..) which shows that secific entroy η is simly a function of bsolute Salinity S and otential temerature θ, that is η = η ( S, θ ), with no searate deendence on ressure. It follows that θ = θ ( S, η ). Similarly, from the definition of otential enthaly and Conservative Temerature in Eqns. (3..) and (3.3.), at = dbar it can be seen that the fundamental thermodynamic relation (.7.) imlies, ( θ) η μ( θ ) c d = T + d + S,, d S. (..) This shows that Conservative Temerature is also simly a function of bsolute Salinity and otential temerature, = ( S, θ ), with no searate deendence on ressure. It then follows that may also be exressed as a function of only S and η. It follows that has the otential roerty.. Various isobaric derivatives of secific enthaly Because of the central role of enthaly in the transort and the conservation of heat in the ocean, the derivatives of secific enthaly at constant ressure are here derived with resect to bsolute Salinity and with resect to the three temerature like variables η, θ and as well as in situ temerature t. We begin by noting that the three standard derivatives of h = h ( S,, t ) when in situ temerature t is taken as the temerature like variable are and ( ) ( ) μ ( ) T, T h S = μ S,, t T + t S,, t, (..) S, ( ) ( ) η ( ) h T = c S,, t = T + t S,, t, (..) S, T T ( ) ( ) ( ) h P = v S,, t T + t v S,, t. (..3) T

95 TEOS- Manual: Calculation and Use of the Thermodynamic Proerties of Seawater 85 Now considering secific enthaly to be a function of entroy (rather than of temerature t ), that is, taking h = h ( S, η, ), the fundamental thermodynamic relation (.7.) becomes hη dη + hs ds = ( T + t) dη + μds while h P = v (..4) so that h η = T + t S, ( ) and h S = μ. η,, S, η (..5) Now taking secific enthaly to be a function of otential temerature (rather than of temerature t ), that is, taking h = h ( S, θ, ), the fundamental thermodynamic relation (.7.) becomes ( ) h θ dθ + h ds = T + t dη + μds while S h P = v (..6). S, θ To evaluate the h θ artial derivative, it is first written in terms of the derivative with resect to entroy as h h = η = η T + t, (..7) θ θ, S η θ S S, S ( ) where (..5) has been used. This equation can be evaluated at = when it becomes (the otential temerature used here is referenced to r = ) ( θ ) η ( θ) θ = = θ + S, = S h c S,, T. (..8) These two equations are used to arrive at the desired exression for h θ namely θ S, ( ) ( T ) + t θ ( T + θ ) h = c S,,. (..9) To evaluate the h S artial derivative, we first write secific enthaly in the functional form h = h ( S, η( S, θ), ) and then differentiate it, finding h S h. S h = + η η S (..) θ θ, η, S, The artial derivative of secific entroy η = gt (Eqn. (..)) with resect to bsolute Salinity, η S = g, S T is also equal to μt since chemical otential is defined by Eqn. (.9.6) as μ = g S. Since the artial derivative of entroy with resect to S in (..) is erformed at fixed otential temerature (rather than at fixed in situ temerature), this is equal to μt evaluated at =. Substituting both arts of (..5) into (..) we have the desired exression for h namely S ( ) ( ) ( ) S T θ, h = μ S,, t T + t μ S, θ,. (..) Notice that this exression contains some things that are evaluated at the general ressure and one evaluated at the reference ressure r =. Now considering secific enthaly to be a function of Conservative Temerature (rather than of temerature t ), that is, taking h = hˆ ( S,, ), the fundamental thermodynamic relation (.7.) becomes ( ) hˆ ˆ d + hs ds = T + t dη + μds while h ˆ P = v. (..) S, The artial derivative ĥ follows directly from this equation as t = this equation reduces to ( ) η ( ) hˆ = T + t = T + t η. (..3) S, S, S ( ) h ˆ c T θ η, S, = S = = + (..4) and combining these two equations gives the desired exression for ĥ namely

96 86 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater ( T + t) ( T + θ ) hˆ = c. S, (..5) To evaluate the h ˆS artial derivative we first write h in the functional form h = h( S, η ( S, ), ) and then differentiate it, finding (using both arts of Eqn. (..5)) h ˆ S = μ( S ) ( ),, t + T+ t ηs. (..6), The differential exression Eqn. (..) can be evaluated at = where the left hand side is simly c d so that from Eqn. (..) we find that η S so that the desired exression for S, ( S θ ) ( T ) = μ,,, +θ h ˆS is ( ) ( T + t) ( T + θ ) ( ) hˆ = μ S,, t μ S, θ,. (..7) (..8) The above boxed exressions for four different isobaric derivatives of secific enthaly are imortant as they are integral to forming the First Law of Thermodynamics in terms of otential temerature and in terms of Conservative Temerature.. Differential relationshis between η, θ, and S Evaluating the fundamental thermodynamic relation in the forms (..6) and (..) and using the four boxed equations in aendix., we find the relations ( ) ( ) ( T + t) ( T + θ ) ( ) ( T θ) ( ) ( ) ( ) ( ) T + t dη+ μ ds = c dθ + μ T + t μ ds T ( ) ( T θ) T+ t T+ t = c d + μ( ) μ( ) d S. + + The quantity ( ) whole equation is then multilied by ( T+ θ ) ( T+ t) obtaining ( θ) η ( ) θ ( θ) μ ( ) μ( ) (..) μ ds is now subtracted from each of these three exressions and the T + d = c d T + ds = c d d S. (..) T From this follows all the following artial derivatives between η, θ, and S, = c ( S θ ) c = μ( θ ) ( + θ) μ ( θ ) θ S,,, = ( T + θ ) c μ( S θ ) η S η S, S S T T S c θ S η,,,,, (..3) =,, c, (..4) = ( T + ) c ( S ) θ ( T θ) μ ( S θ ) c ( S θ ) θ θ, θ,, θ = +,,,,, (..5) S η T = c c ( S, θ, ), θ = μ( S θ ) ( T + θ) μ ( S θ ) c ( S θ ) S θ S,,,,,,, (..6) S T = c ( S,,) ( T + ), η μ ( θ ) η θ θ η =,,, (..7) S T S θ = c ( T + θ ), η μ( θ ) ( θ) S S = S,, T +. (..8) The three second order derivatives of ˆ η ( S, ) aendix P. The corresonding derivatives of ˆ θ ( S, ) are listed in Eqns. (P.4) and (P.5) of, namely ˆ θ, ˆS θ, ˆ θ, ˆS θ and θ can also be derived using Eqn. (P.3), obtaining ˆS S

97 TEOS- Manual: Calculation and Use of the Thermodynamic Proerties of Seawater 87 ˆ θ = ˆ = S, θs θ θ, ˆ θ = θθ ( ) 3 θ, θ ˆ θ S S θθ S = + 3 ( θ) ( θ), (..9a,b,c,d) and ˆ S S S θ S S θθ θs S = + θ θ θ θ, (..) θ in terms of the artial derivatives θ, S, θθ, θ S and SS which can be obtained S θ from the TEOS Gibbs function. by differentiating the olynomial ( ),.3 The First Law of Thermodynamics The law of the conservation of energy for thermodynamic equilibrium states was discovered in the 9 th century by Gibbs (873) and other early ioneers. It was formulated as a balance between internal energy, heat and work, similar to the fundamental equation (.7.), and referred to as the First Law of Thermodynamics (Thomson (85), Clausius (876), lberty ()). Under the weaker condition of a local thermodynamic equilibrium (Glansdorff and Prigogine (97)), the original thermodynamic concets can be suitably generalized to describe irreversible rocesses of fluid dynamics which are subject to molecular fluxes and macroscoic motion (Landau and Lifshitz (959), De Groot and Mazur (984)). In some circles the First Law of Thermodynamics is used to describe the evolution equation for total energy, being the sum of internal energy, otential energy and kinetic energy. Here we follow the more common ractice of regarding the First Law of Thermodynamics as the difference between the conservation equation of total energy and the evolution equation for kinetic energy lus otential energy, leaving what might loosely be termed the evolution equation of heat, Eqn. (.3.) (Landau and Lifshitz (959), McDougall (3), Griffies (4)). The First Law of Thermodynamics can therefore be written as (see Eqn. (B.9) and the other Eqns. (.3.3), (.3.4) and (.3.5) of this aendix; all of these equations are equally valid incarnations of the First Law of Thermodynamics) ρ dh dp R Q S = F F + hs, dt ρ dt ρε + ρ S (.3.) R Q where F is the sum of the boundary and radiative heat fluxes and F is the sum of all molecular diffusive fluxes of heat, being the normal molecular heat flux directed down the temerature gradient lus a term roortional to the molecular flux of salt (the Dufour Effect, see Eqn. (B.4) below). Lastly, ε is the rate of dissiation of mechanical energy er S unit mass, transformed into internal energy and h S ρ S is the rate of increase of enthaly due to the interior source term of bsolute Salinity caused by remineralization. The derivation of (.3.) is summarized in aendix B below, where we also discuss the related evolution equations for total energy and for the Bernoulli function. Following Fofonoff (96) we note that an imortant consequence of (.3.) is that when two finite sized arcels of seawater are mixed at constant ressure and under ideal conditions, the total amount of enthaly is conserved. To see this one combines (.3.) with the continuity equation ρ t + ( ρu ) = to find the following divergence form of the First Law of Thermodynamics, dp R Q S ( ρh) t + ( ρuh) = F F + ρε + hs ρ S. (.3.) dt One then integrates over the volume that encomasses both fluid arcels 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

98 88 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater ressure), mixes and contracts. The dissiation of mechanical energy by viscous friction and the source term due to the roduction of bsolute Salinity are also commonly ignored during such mixing rocesses but in fact these terms do cause a small increase in the enthaly of the mixture with resect to that of the two original arcels. art from these non conservative source terms, under these assumtions Eqn. (.3.) reduces to the statement that the volume integrated amount of ρ h is the same for the two initial fluid arcels as for the final mixed arcel, that is, the total amount of enthaly is unchanged. This result of non equilibrium thermodynamics is of the utmost imortance in oceanograhy. The fact that enthaly is conserved when fluid arcels mix at constant ressure is the central result uon which all of our understanding of heat fluxes and of heat content in the ocean rests. The imortance of this result cannot be overemhasized; it must form art of all our introductory courses on oceanograhy and climate dynamics. s imortant as this result is, it does not follow that enthaly is the best variable to reresent heat content in the ocean. Enthaly is a very oor reresentation of heat content in the ocean because it does not osses the otential roerty. It will be seen that otential enthaly h (referenced to zero sea ressure) is the best thermodynamic variable to reresent heat content in the ocean. The First Law of Thermodynamics (.3.) can be written (using Eqn. (.7.)) as an evolution equation for entroy as follows dη ds R Q S ρ ( T + t) + μ = F F + ρε + hs ρ S. (.3.3) dt dt The First Law of Thermodynamics (.3.) can also be written in terms of otential temerature θ (with resect to reference ressure r ) by substituting Eqns. (..9) and (..) into Eqn. (.3.) as (from Bacon and Fofonoff (996) and McDougall (3)) ( + ) ( + θ ) T t dθ ds ρ c( r) + μ( ) ( T + t) μt ( r) = T dt dt R Q F F + ρε + h ρ S S S, (.3.4) where T is the Celsius zero oint ( T is exactly 73.5 K), while in terms of Conservative Temerature, the First Law of Thermodynamics is (from McDougall (3), using Eqns. (..5) and (..8) above) ( ) ( θ) ( ) ( θ) T + t d T + t ds ρ c + μ( ) μ( ) = T + dt T + dt R Q F F + ρε + h ρ S S S, (.3.5) where c is the fixed constant defined by the exact 5 digit number in Eqn. (3.3.3). In aendices.6,.7 and.8 the non conservative roduction of entroy, otential temerature and Conservative Temerature are quantified, both as Taylor series exansions which identify the relevant non linear thermodynamic terms that cause the roduction of these variables, and also on the S diagram where variables are contoured which grahically illustrate the non conservation of these variables. In other words, aendices.6,.7 and.8 quantify the non ideal nature of the left hand sides of Eqns. (.3.3) (.3.5). That is, these aendices quantify the deviations of the lefthand sides of these equations from being roortional to ρ dη dt, ρ dθ dt and ρ d dt. quick ranking of these three variables, η, θ and, from the viewoint of the amount of their non conservation, can be gleaned by examining the range of the terms (at fixed ressure) that multily the material derivatives on the left hand sides of the above Eqns. (.3.3), (.3.4) and (.3.5). The ocean circulation may be viewed as a series of adiabatic and isohaline movements of seawater arcels interruted by a series of isolated turbulent mixing events. During any of the adiabatic and isohaline transort stages every

99 TEOS- Manual: Calculation and Use of the Thermodynamic Proerties of Seawater 89 otential roerty is constant, so each of the above variables, entroy, otential temerature and Conservative Temerature are % ideal during these adiabatic and isohaline advection stages. The turbulent mixing events occur at fixed ressure so the non conservative roduction of say entroy deends on the extent to which the coefficients ( T + t) and μ in Eqn. (.3.3) vary at fixed ressure. Similarly the nonconservative roduction of otential temerature deends on the extent to which the coefficients c ( r)( T + t) ( T + θ ) and μ( ) ( T + t) μt ( r) in Eqn. (.3.4) vary at fixed ressure, while the non conservative roduction of Conservative Temerature deends on the extent to which the coefficients ( T + t) ( T + θ ) and μ ( ) μ( )( T + t) ( T + θ) in Eqn. (.3.5) vary at fixed ressure. ccording to this way of looking at these equations we note that the material derivative of entroy aears in Eqn. (.3.3) multilied by the absolute temerature ( T + t) which varies by about 5% at the sea surface ( ( ) ), the term that multilies dθ dt in (.3.4) is dominated by the variations in the isobaric secific heat c ( S,, t r) which is mainly a function of S and which varies by 5% at the sea surface (see Figure 4), while the material derivative of Conservative Temerature d dt in Eqn. (.3.5) is multilied by the roduct of a constant heat caacity c and the factor ( T+ t) ( T+ θ ) which varies very little in the ocean, esecially when one realizes that it is only the variation of this ratio at each ressure level that is of concern. This factor is unity at the sea surface and is also very close to unity in the dee ocean. Figure.3.. Contours (in C ) of the difference θ between otential temerature θ and Conservative Temerature at the sea surface of the annually averaged atlas of Gouretski and Koltermann (4). Fortunately, Conservative Temerature is not only much more accurately conserved in the ocean than otential temerature but it is also relatively easy to use in oceanograhy. Because Conservative Temerature also ossesses the otential roerty, it is a very accurate reresentation of the heat content of seawater. The difference θ between otential temerature θ and Conservative Temerature at the sea surface is shown in Figure.3. (after McDougall, 3). If an ocean model is

100 9 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater written with otential temerature as the rognostic temerature variable rather than Conservative Temerature, and is run with the same constant value of the isobaric secific heat caacity ( c as given by Eqn. (3.3.3)), the neglect of the non conservative source terms that should aear in the rognostic equation for θ means that such an ocean model incurs errors in the model outut. These errors will deend on the nature of the surface boundary condition; for flux boundary conditions the errors are as shown in Figure.3.. This aendix has largely demonstrated the benefits of otential enthaly and Conservative Temerature from the viewoint of conservation equations, but the benefits can also be roven by the following arcel based argument. First, the air sea heat flux needs to be recognized as a flux of otential enthaly which is exactly c times the flux of Conservative Temerature. Second, the work of aendix.8 shows that while it is the in situ enthaly that is conserved when arcels mix, a negligible error is made when otential enthaly is assumed to be conserved during mixing at any deth. Third, note that the ocean circulation can be regarded as a series of adiabatic and isohaline movements during which is absolutely unchanged (because of its otential nature) followed by a series of turbulent mixing events during which is almost totally conserved. Hence it is clear that is the quantity that is advected and diffused in an almost conservative fashion and whose surface flux is exactly roortional to the air sea heat flux..4 dvective and diffusive heat fluxes In section 3.3 and aendices.8 and.3 the First Law of Thermodynamics is shown to be ractically equivalent to the conservation equation (..5) for Conservative Temerature. We have emhasized that this means that the advection of heat is very accurately given as the advection of c. In this way c can be regarded as the heat content er unit mass of seawater and the error involved with making this association is aroximately % of the error in assuming that either cθ or c ( S, θ, dbar ) θ is the heat content er unit mass of seawater (see also aendix. for a discussion of this oint). The conservative form (..5) imlies that the turbulent diffusive flux of heat should be directed down the mean gradient of Conservative Temerature rather than down the mean gradient of otential temerature. In this aendix we quantify the difference between these mean temerature gradients. Consider first the resective temerature gradients along the neutral tangent lane. From Eqn. (3..) we find that θ θ ( α β ) θ S ( α β ) = = (.4.) n n n, so that the eineutral gradients of θ and are related by the ratios of their resective thermal exansion and saline contraction coefficients, namely ( α β ) θ ( α θ β ) θ =. (.4.) n This roortionality factor between the arallel two dimensional vectors nθ and n is readily calculated and illustrated grahically. Before doing so we note two other equivalent exressions for this roortionality factor. The eineutral gradients of θ, and S are related by (using θ = ˆ θ( S, )) θ = ˆ θ + ˆ θ, (.4.3) n n S n S ns α β n and using the neutral relationshi ( ) n = we find

101 TEOS- Manual: Calculation and Use of the Thermodynamic Proerties of Seawater 9 ( ) ˆ ˆ nθ = θ + α β θs n. θ (.4.4) lso, in section 3.3 we found that Tb nθ = Tb n, so that we can write the equivalent exressions θ n n ( α β ) Tb θ α β θs ( α θ β θ ) T θ b = = = ˆ + ˆ, (.4.5) θ and it can be shown that α α ˆ θ θ ˆ ˆ β β = + α β θs θ, that is, θ β = β + α ˆ θ ˆ S θ. The artial derivatives ˆ θ and ˆS θ in the last art of Eqn. (.4.5) are both indeendent of ressure while α β is a function of ressure. This ratio, Eqn. (.4.5), of the eineutral gradients of θ and is shown in Figure.4. at =, indicating that the eineutral gradient of otential temerature is sometimes more that % different to that of Conservative Temerature. This ratio nθ n is only a weak function of ressure. This ratio, nθ n (i.e. Eqn. (.4.5)), is available in the GSW Oceanograhic Toolbox as function gsw_nt_t_vs_ct_ratio_ct5. Similarly to Eqn. (.4.3), the vertical gradients are related by θ = ˆ θ + ˆ θ, (.4.6) = and ( ) z z S S z and using the definition, Eqn. (3.5.), of the stability ratio we find that θz z ˆ R ˆ ρ S. = θ + α β θ (.4.7) For values of the stability ratio R ρ close to unity, the ratio θz z is close to the values of nθ n shown in Figure.4.. For other values of R ρ, Eqn. (.4.7) can be calculated and lotted. % at =, showing the ercentage difference between the eineutral gradients of θ and. The blue dots are from the ocean atlas of Gouretski and Koltermann (4) at =. Figure.4.. Contours of ( nθ n ) s noted in section 3.8 the dianeutral advection of thermobaricity is the same when quantified in terms of otential temerature as when done in terms of Conservative Temerature. The same is not true of the dianeutral velocity caused by cabbeling. The ratio of the cabbeling dianeutral velocity calculated using otential temerature to that using θ Conservative Temerature is given by ( Cb nθ nθ) ( Cb n n ) (see section 3.9) which can be exressed as

102 9 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater ( α β ) θ ( α θ β ) θ θ θ θ b nθ b b b b θ C b n b Cb Tb Cb ( θ α β θs ) C C C T C = = = ˆ + ˆ, C (.4.8) and this is contoured in Fig..4.. While the ratio of Eqn. (.4.8) is not exactly unity, it varies relatively little in the oceanograhic range, indicating that the dianeutral advection due to cabbeling estimated using θ or are within half a ercent of each other at =. θ Figure.4.. Contours of the ercentage difference of ( Cb nθ ) ( Cb n ) from unity at = dbar. θ θ.5 Derivation of the exressions for α, β, α and β This aendix derives the exressions in Eqns. (.8.) (.8.3) and (.9.) (.9.3) for θ the thermal exansion coefficients α and α θ and the haline contraction coefficients β and β. θ In order to derive Eqn. (.8.) for α we first need an exression for θ T. This S, is found by differentiating with resect to in situ temerature the entroy equality η( S,, t ) = η( S, θ[ S,, t, r], r) which defines otential temerature, obtaining θ ηt ( S,, t ) gtt ( S,, t ) = =. (.5.) T η S, θ, g S, θ, S, T ( ) ( ) r TT r This is then used to obtain the desired exression Eqn. (.8.) for ( ) ( ) θ α as follows ( θ ) ( ) θ v v θ gtp S,, t g S,, α = = = v θ S, v T S, T S, gp S,, t gtt S,, t TT r. (.5.) In order to derive Eqn. (.8.3) for α we first need an exression for t. This S, is found by differentiating with resect to in situ temerature the entroy equality η S,, t = ˆ η S, S,, t obtaining ( ) ( [ ]) T,,,,, ( ) ( θ) ( ) = ηt S t = T+ gtt S t c S, η S (.5.3) where the second art of this equation has used Eqn. (..4) for η. This is then used S to obtain the desired exression Eqn. (.8.3) for α as follows

103 TEOS- Manual: Calculation and Use of the Thermodynamic Proerties of Seawater 93 (,, ) c (,, ) ( θ ) (,, ) v v gtp S t α = = = v S, v T S, T S, gp S t T+ gtt S t θ. (.5.4) In order to derive Eqn. (.9.) for β we first need an exression for θ S,. This T is found by differentiating with resect to bsolute Salinity the entroy equality η S,, t = η S, θ S,, t,, which defines otential temerature, obtaining ( ) ( [ r] r) θ S T, η S ( S,, t ) ( S,, ) = θ ηs η θ S r ( T + θ ) μt ( S, θ, r) μt ( S, t, ) (, θ, r) ( ) ( θ ) ( θ ) = c S = gst S, t, gst S,, r gtt S,, r, (.5.5) where Eqns. (..5) and (..7) have been used with a general reference ressure r rather than with r =. By differentiating ρ= ρ( S, θ[ S, t,, r], ) with resect to bsolute Salinity it can be shown that (Gill (98), McDougall (987a)) θ ρ ρ θ θ β = S = S + α ρ ρ S θ, T, T,, (.5.6) and using Eqn. (.5.5) we arrive at the desired exression Eqn. (.9.) for β θ (,, ) ( ) (,, ) g ( ) ( ),, g, θ, r ( ) ( ) gsp S t gtp S t S T S t S T S = +. (.5.7) g S,, t g S,, t g S,, t P P TT Note that the terms in the natural logarithm of the square root of bsolute Salinity cancel from the two arts of the square brackets in Eqns. (.5.5) and (.5.7). In order to derive Eqn. (.9.3) for β we first need an exression for S,. T This is found by differentiating with resect to bsolute Salinity the entroy equality η S,, t = ˆ η S, S,, t obtaining (using Eqns. (..4) and (..8)) ( ) ( [ ]) θ β = η ηs ( S ),, t ˆ η S S S T, Differentiating ρ ˆ ρ( S, [ S, t, ], ) (,,) ( ) (,, ) = μ S θ T+ θ μt S t c ( θ ) ( θ) ( ) = gs S,, T gst S, t, + c. = with resect to bsolute Salinity leads to β ρ ρ = S = S + α ρ ρ S, T, T,, (.5.8) (.5.9) and using Eqn. (.5.8) we arrive at the desired exression (.9.3) for β namely (,, ) ( ) (,, ) g ( ) ( ) ( ),, g, θ, + θ ( ) ( ) gsp S t gtp S t S T S t S S T β gp S,, t gp S,, t gtt S,, t = +. (.5.) Note that the terms in the natural logarithm of the square root of bsolute Salinity cancel from the two arts of the square brackets in Eqns. (.5.8) and (.5.).

104 94 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater.6 Non-conservative roduction of entroy In this and the following three aendices (.6.9) the non conservative nature of several thermodynamic variables (entroy, otential temerature, Conservative Temerature and otential density) will be quantified by considering the mixing of airs of seawater arcels at fixed ressure. The mixing is taken to be comlete so that the end state is a seawater arcel that is homogeneous in bsolute Salinity and entroy. That is, we will be considering mixing to comletion by a turbulent mixing rocess. In aendix. the non conservative roduction of bsolute Salinity by the remineralization of articulate organic matter is considered. This rocess is not being considered in aendices.6.9. The non conservative roduction which is quantified in aendices.6.9 occurs in the absence of any variation in seawater comosition. Following Fofonoff (96), consider mixing two fluid arcels (arcels and ) that have initially different temeratures and salinities. The mixing rocess occurs at ressure. The mixing is assumed to haen to comletion so that in the final state bsolute Salinity, entroy and all the other roerties are uniform. ssuming that the mixing haens with a vanishingly small amount of dissiation of mechanical energy, the ε term can be droed from the First Law of Thermodynamics, (.3.), this equation becoming R ( ρh) + ( ρ h) = Q. t u F F at constant ressure (.6.) Note that this equation has the form (.8.) and so h is conserved during mixing at constant ressure, that is, h is isobaric conservative. In the case we are considering of mixing the two seawater arcels, the system is closed and there are no radiative, boundary or molecular heat fluxes coming through the outside boundary so the integral over sace and time of the right hand side of Eqn. (.6.) is zero. The surface integral of ( ρ u h) through the boundary is also zero. Hence it is aarent that the volume integral of ρ h is the same at the final state as it is at the initial state, that is, enthaly is conserved. Hence during the mixing rocess the mass, salt content and enthaly are conserved, that is m + m = m, (.6.) ms + ms = ms (.6.3), mh + mh = mh, (.6.4) while the non conservative nature of entroy means that it obeys the equation, m + m + m = m (.6.5) η η δη η. Here S, h and η are the values of bsolute Salinity, enthaly and entroy of the final mixed fluid and δη is the roduction of entroy, that is, the amount by which entroy is not conserved during the mixing rocess. Entroy η is now regarded as the functional form η = η ( S, h, ) and is exanded in a Taylor series of S and h about the values of S and h of the mixed fluid, retaining terms to second order in [ S S] =Δ S and in [ h h ] = Δ h. Then η and η are evaluated and (.6.4) and (.6.5) used to find mm δη = { η hh ( Δ h) + ηhs ΔhΔ S ( ) + ηss ΔS }. (.6.6) m Towards the end of this section the imlications of the roduction (.6.6) of entroy will be quantified, but for now we ask what constraints the Second Law of Thermodynamics might lace on the form of the Gibbs function g ( S,, t ) of seawater. The Second Law of Thermodynamics tells us that the entroy excess δη must not be negative for all ossible combinations of the differences in enthaly and salinity between the two fluid arcels. From (.6.6) this requirement imlies the following three inequalities, η hh <, (.6.7)

105 TEOS- Manual: Calculation and Use of the Thermodynamic Proerties of Seawater η <, SS hs < hh SS ( ) η η η, 95 (.6.8) (.6.9) where the last requirement reflects the need for the discriminant of the quadratic in (.6.6) to be negative. Since entroy 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 (.7.) we find that (where T is the absolute temerature, T = T + t) η ηh = = T (.6.) h η S S, η μ = =, S T h, (.6.) and from these relations the following exressions for the second order derivatives of η can be found, and η η hh Sh η T T = = = h c h S, S, ( μ T ) η μ = = = h S h c T S, ( μ ) ( μ ), T, (.6.) (.6.3) η T T h ηss = = S S h, h S T, S, T, (.6.4) μs T μ =. T c T T The last equation comes from regarding ηs as ηs = η ( [ ] ) S S, h S, t,,. The constraint (.6.7) that η hh < simly requires (from (.6.)) that the isobaric heat caacity c is ositive, or that g TT <. The constraint (.6.8) that η S, S < requires (from (.6.4)) that 3 T μ gs, S > c T (.6.5) T that is, the second derivative of the Gibbs function with resect to bsolute Salinity g SS must exceed some negative number. The constraint (.6.9) that ( ηhs) < ηhhηss requires that (substituting from (.6.), (.6.3) and (.6.4)) gss >, (.6.6) 3 Tc and since the isobaric heat caacity must be ositive, this requirement is that g S, S > and so is more demanding than (.6.5). We conclude that while there are the three requirements (.6.7) to (.6.9) on the functional form of entroy η = η ( S, h, ) in order to satisfy the constraint of the Second Law of Thermodynamics that entroy be roduced when water arcels mix, these three constraints are satisfied by the following two constraints on the form of the Gibbs function g ( S,, t ), g < (.6.7) and TT SS g >. (.6.8)

106 96 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater The Second Law of Thermodynamics does not imose any additional requirement on the cross derivatives g ST nor on any third order derivatives of the Gibbs function. The constraint (.6.8) can be understood by considering the molecular diffusion of salt which is known to be directed down the gradient of chemical otential μ ( S,, t ) (Landau and Lifshitz (959)). That is, the molecular flux of salt is roortional to μ. Exanding μ in terms of gradients of bsolute Salinity, of temerature, and of ressure, one finds that the first term is μ S S and in order to avoid an unstable exlosion of salt one must have μ S = g. SS > So the constraint (.6.8) amounts to the requirement that the molecular diffusivity of salt is ositive. The two constraints (.6.7) and (.6.8) on the Gibbs function are well known in the thermodynamics literature. Landau and Lifshitz (959) derive them on the basis of the contribution of molecular fluxes of heat and salt to the roduction of entroy (their equations 58.9 and 58.3). lternatively, Planck (935) as well as Landau and Lifshitz (98) in their 96 (this is 98 in editions before the 976 extension made by Lifshitz and Pitayevski) inferred such inequalities from thermodynamic stability considerations. It is leasing to obtain the same constraints on the seawater Gibbs function from the above Non Equilibrium Thermodynamics aroach of mixing fluid arcels since this aroach involves turbulent mixing which is the tye of mixing that dominates in the ocean; (molecular diffusion has the comlementary role of dissiating tracer variance). In addition to the Second Law requirements (.6.7) and (.6.8) there are other constraints which the seawater Gibbs function must obey. One is that the adiabatic (and isohaline) comressibility must be ositive for otherwise the fluid would exand in resonse to an increase in ressure which is an unstable situation. Taking g P > (since secific volume needs to be ositive) the requirement that the adiabatic (and isohaline) comressibility be ositive imoses the following two constraints (from (.6.)) g PP < (.6.9) and ( g ) TP < gpp gtt, (.6.) recognizing that g TT is negative ( g TP may, and does, take either sign). Equation (.6.) is more demanding of g PP than is (.6.9), requiring g PP to be less than a negative number rather than simly being less than zero. This last inequality can also be regarded t as a constraint on the thermal exansion coefficient α, imlying that its square must be less than gp gpp gtt or otherwise the relevant comressibility (κ ) would be negative and the sound seed comlex. The constraints on the seawater Gibbs function g ( S,, t ) that have been discussed above are summarized as g >, g S S >, g PP <, g TT <, and ( g ) TP < gpp gtt. (.6.) We return now to quantify the non conservative roduction of entroy in the ocean. When the mixing rocess occurs at =, the exression (.6.6) for the roduction of entroy can be exressed in terms of Conservative Temerature (since is simly roortional to h at = ) as follows (now entroy is taken to be the functional form η = ˆ η( S, )) mm δη = { ˆ η ( Δ ) + ˆ ηs ΔΔ S ( ) + ˆ ηss Δ S }. (.6.) m The maximum roduction occurs when arcels of equal mass are mixed so that mmm = and we adot this value in what follows. To illustrate the magnitude of 8 this non conservation of entroy we first scale entroy by a dimensional constant so that the resulting variable ( entroic temerature ) has the value 5 C at ( S, ) = ( SSO,5 C) and then is subtracted. The result is contoured in S sace in Figure.6..

107 TEOS- Manual: Calculation and Use of the Thermodynamic Proerties of Seawater 97 Figure.6.. Contours (in C ) of a variable which illustrates the non conservative roduction of entroy η in the ocean. The fact that the variable in Figure.6. is not zero over the whole S lane is because entroy is not a conservative variable. The non conservative roduction of entroy can be read off this figure by selecting two seawater samles and mixing along the straight line between these arcels and then reading off the roduction (in C ) of entroy from the figure. Taking the most extreme situation with one arcel at ( S ) (, = gkg, C) and the other at the warmest and saltiest corner of the figure, the non conservative roduction of η on mixing arcels of equal mass is aroximately.9 C. Since entroy can be exressed indeendently of ressure as a function of only bsolute Salinity and Conservative Temerature η = ˆ η( S, ), and since at any ressure in the ocean both S and may be considered conservative variables (see aendix.8 below), it is clear that the non conservative roduction given by (.6.) and illustrated in Figure.6. is equivalent to the slightly more accurate exression (.6.6) which alies at any ressure. The only discreancy between the roduction of entroy calculated from (.6.) and that from (.6.6) is due to the very small non conservative roduction of at ressures other than zero (as well as the fact that both exressions contain only the second order terms in an infinite Taylor series)..7 Non-conservative roduction of otential temerature When fluid arcels undergo irreversible and comlete mixing at constant ressure, the thermodynamic quantities that are conserved during the mixing rocess are mass, bsolute Salinity and enthaly. s in section.6 we again consider two arcels being mixed without external inut of heat or mass and the three equations that reresent the conservation of these quantities are again Eqns. (.6.) (.6.4). Potential temerature θ is not conserved during the mixing rocess and the roduction of otential temerature is given by m θ + m θ + mδθ = mθ (.7.). Enthaly in the functional form h h ( S θ ) =,, is exanded in a Taylor series of S and θ about the values S and θ of the mixed fluid, retaining terms to second order in

108 98 TEOS- Manual: Calculation and use of the thermodynamic roerties of seawater [ S S ] = Δ S and in [ ] θ θ = Δ θ. Then h and h are evaluated and Eqns. (.6.4) and (.7.) used to find mm h ( ) h S h S S S ( ) θθ θ δθ = θ θ S. m h Δ + Δ Δ + Δ θ h θ h (.7.) θ The maximum roduction occurs when arcels of equal mass are mixed so that mmm =. The heat caacity h 8 θ is not a strong function of θ but is a much stronger function of S so the first term in the curly brackets in Eqn. (.7.) is generally small comared with the second term. lso, the third term in Eqn. (.7.) which causes the so called dilution heating, is usually small comared with the second term. tyical value of h θ S is aroximately 5.4 Jkg K (gkg ) (see the deendence of isobaric heat caacity on S in Figure 4 of section.) so that an aroximate exression for the roduction of otential temerature δθ is ( ) δθ Δ Δ Δθ Since otential temerature θ = ˆ θ( S, ) can be exressed indeendently of ressure as a function of only bsolute Salinity and Conservative Temerature, and since during turbulent mixing both S and may be considered conservative variables (see section.8 below), it is clear that the non conservative roduction given by (.7.) can be aroximated by the corresonding roduction of otential temerature that would occur if the mixing had occurred at =, namely 4 4 h S S h θ θ 3.4x S / [g kg ]. (.7.3) δθ = mm Δ θ + Δθ Δ + Δ θ S SS ( ) S ( S ) θθ m θ θ θ, (.7.4) where the exact roortionality between otential enthaly and Conservative Temerature h c has been exloited. The maximum roduction occurs when arcels of equal mass are mixed so that mmm = and we adot this value in what follows. 8 Equations (.7.) or (.7.4) may be used to evaluate the non conservative roduction of otential temerature due to mixing a air of fluid arcels across a front at which there are known differences in salinity and temerature. The temerature difference θ is contoured in Figure.7. and can be used to illustrate Eqn. (.7.4). δθ can be read off this figure by selecting two seawater samles and mixing along the straight line between these arcels (along which both bsolute Salinity and Conservative Temerature are conserved) and then calculating the roduction (in C ) of θ from the contoured values of θ. Taking the most extreme situation with one arcel at ( S ) (, = g kg, C) and the other at the warmest and saltiest corner of Figure.7., the non conservative roduction of θ on mixing arcels of equal mass is aroximately.55 C. This is to be comared with the corresonding maximum roduction of entroy, as discussed above in connection with Figure.6., of aroximately.9 C. If Figure.7. were to be used to quantify the errors in oceanograhic ractice incurred by assuming that θ is a conservative variable, one might select roerty contrasts that were tyical of a rominent oceanic front and decide that because δθ is small at this one front, that the issue can be ignored (see for examle, Warren (6)). But the observed roerties in the ocean result from a large and indeterminate number of such rior mixing events and the non conservative roduction of θ accumulates during each of these mixing events, often in a sign definite fashion. How can we ossibly estimate the error that is made by treating otential temerature as a conservative variable during all of these unknowably many ast individual mixing events? This seemingly difficult issue is artially resolved by considering what is actually done in ocean models today. These models carry a temerature conservation equation that does not have non conservative source terms, so that the model s temerature variable is best interreted as being. This

109 TEOS- Manual: Calculation and Use of the Thermodynamic Proerties of Seawater 99 being the case, the temerature difference contoured in Figure.7. illustrates the error that is made by interreting the model temerature as being θ. That is, the values contoured in Figures.6. and.7. are reresentative of the error, exressed in C, associated with assuming that η and θ resectively are conservative variables. These contoured values of temerature difference encasulate the accumulated non conservative roduction that has occurred during all the many mixing rocesses that have lead to the ocean s resent state. The maximum such error for η is aroximately. C (from Figure.6.) while for θ the maximum error is aroximately.8 C (from Figure.7.). One ercent of the data at the sea surface of the world ocean have values of θ that lie outside a range that is.5 C wide (McDougall (3)), imlying that this is the magnitude of the error incurred by ocean models when they treat θ as a conservative quantity. To ut a temerature difference of.5 C in context, this is the tyical difference between in situ and otential temeratures for a ressure difference of 5 dbar, and it is aroximately times as large as the tyical differences between t 9 and t 68 in the ocean. From the curvature of the isolines on Figure.7. it is clear that the non conservative roduction of θ takes both ositive and negative signs. Figure.7.. Contours (in C ) of the difference between otential temerature and Conservative Temerature θ. This lot illustrates the nonconservative roduction of otential temerature θ in the ocean..8 Non-conservative roduction of Conservative Temerature When fluid arcels undergo irreversible and comlete mixing at constant ressure, the thermodynamic quantities that are conserved during the mixing rocess are mass, bsolute Salinity and enthaly. s in sections.6 and.7 we consider two arcels being mixed without external inut of heat or mass and the three equations that reresent the conservation of these quantities are again Eqns. (.6.) (.6.4). Potential enthaly