A thermodynamic potential is derived for seawater as a function of Conservative Temperature, Absolute Salinity and pressure. From this thermodynamic potential, all the equilibrium thermodynamic properties of seawater can be found, just as all these thermodynamic properties can be found from the TEOS-10 (the International Thermodynamic Equation of Seawater – 2010; IOC et al., 2010) Gibbs function (which is a function of in situ temperature, Absolute Salinity, and pressure). Present oceanographic practice in the Gibbs SeaWater Oceanographic Toolbox uses a polynomial expression for specific volume (and enthalpy) in terms of Conservative Temperature (as well as of Absolute Salinity and pressure), whereas the relationship between in situ temperature and Conservative Temperature is based on the Gibbs function. This mixed practice introduces (numerically small) inconsistencies and superfluous conversions between variables. The proposed thermodynamic potential of seawater, being expressed as an explicit function of Conservative Temperature, overcomes these small numerical inconsistencies, and in addition, the new approach allows for greater computational efficiency in the evaluation of sea surface temperature from Conservative Temperature. It is also shown that when using Conservative Temperature, the thermodynamic information in enthalpy is independent of that contained in entropy. This contrasts with the cases where either in situ temperature or potential temperature is used. In these cases, a single thermodynamic potential serves the important purpose of avoiding having to impose a separate consistency requirement between the functional forms of enthalpy and entropy.

The TEOS-10 (the International Thermodynamic Equation of Seawater – 2010; IOC et al., 2010) Gibbs function of seawater is a thermodynamic potential whose arguments are Absolute Salinity, in situ temperature, and pressure. The adoption in 2010 of TEOS-10 as the official description of the thermodynamic properties of seawater came with the recommendation that the observed variables Practical Salinity

The Absolute Salinity variable of TEOS-10 is defined on the Reference-Composition Salinity Scale of Millero et al. (2008) as an approximation to the mass fraction of dissolved material in seawater. As described in Pawlowicz (2010, 2011) and Wright et al. (2011), while the Gibbs function of a multi-component solution such as seawater should depend on the concentrations of all its constituents, Absolute Salinity on the Reference-Composition Salinity Scale is defined so that its use yields accurate values of the specific volume of seawater.

This paper was motivated by the following question: “is it possible to define a thermodynamic potential in terms of Conservative Temperature rather than, for example, in terms of in situ temperature, as is the case for the TEOS-10 Gibbs function of seawater (Feistel, 2008; IAWPS, 2008)?”. Progress had already been made towards answering this question in Appendix P of the TEOS-10 Manual (IOC et al., 2010) where it was shown that if expressions were available for both the enthalpy and the entropy of seawater as functions of Absolute Salinity, Conservative Temperature, and pressure, then all the thermodynamic properties of seawater could be derived.

While in situ temperature is an observed variable, its dependence on pressure (even for adiabatic variations of pressure at constant salinity) and its non-conservative nature under turbulent mixing processes have led to the adoption of Conservative Temperature in order to approximate the “heat content” per unit mass of seawater. It is Conservative Temperature that is now used as the temperature axis of “salinity–temperature” diagrams and as the model's temperature variable in ocean models (McDougall et al., 2021) because it is approximately conserved under mixing processes: the amount of non-conservation is typically 2 orders of magnitude less than that of potential temperature. In order to facilitate the use of Conservative Temperature in oceanography, Roquet et al. (2015) provided a 75-term polynomial for specific volume,

The First Law of Thermodynamics (see sections 49, 57, and 58 of Landau and Lifshitz (1959) and Appendix B of IOC et al., 2010),

Table of symbols.

Continued.

Equation (1) illustrates how the work performed by the environment on the fluid parcel due to its change in volume at pressure

Clausius (1876) considered the cyclic reversible exchange of heat between a control volume and the environment and inferred that there must be a state variable, which he named entropy,

We note that there is a fundamental difference in the language and symbols used in thermodynamics versus in fluid dynamics. As we have noted, the FTR, Eq. (2), applies only to reversible processes, and yet the FTR has been combined with the First Law of Thermodynamics, Eq. (1), to arrive at Eqs. (3) and (4) which are written in typical fluid dynamics form using material derivatives. There is a disconnect here, a disconnect that is common in the literature and is the source of much confusion. In fluid dynamics we do not require mixing processes to occur only for an instant and then to have these process switch off while the fluid slowly comes to thermodynamic equilibrium (as would be required to technically obey the thermodynamic restrictions associated with the FTR which we have used). Rather, in fluid dynamics we imagine the mixing processes and the dissipation of turbulent kinetic energy to occur continuously. Moreover, a state of thermodynamic equilibrium has spatially uniform fields of in situ temperature and chemical potential, and such a state is not what we observe or expect in the ocean which is mixed by turbulent mixing processes (see the discussion of this point on the last page of Appendix B of IOC et al., 2010). Hence it is clear that the restrictions associated with use of the FTR are not fulfilled when we combine it with the First Law and write the result using fluid dynamic notation and interpretation as though it might apply to the real ocean. We conclude that there are small thermodynamic inconsistencies involved with combining the FTR and the First Law into the forms of Eqs. (3) and (4). This same inconsistency is common to all advanced thermodynamics textbooks and is rarely discussed; a rare mention of the issue appears on the last page of Sect. 49 of Landau and Lifshitz (1959). Importantly, we point out below (in the paragraph that contains our Eq. 6) that in physical oceanography we do not need to use the evolution of entropy as it appears in Eqs. (3) and (4), but rather we exploit the fact that entropy is a function only of state variables and so can be expressed in the functional form

A test of the conservative nature (or otherwise) of an oceanographic variable is to consider the turbulent mixing of two seawater parcels. If the total amount of the variable in the final mixed product is the sum of the amounts in the two original parcels, then the variable is conservative. This is rigorously true for enthalpy in an isobaric mixing process (apart from the dissipation of turbulent kinetic energy which needs to be budgeted separately) and is close to being true of Conservative Temperature (McDougall, 2003; Graham and McDougall, 2013).

Temporarily setting aside the reservations we have outlined above with the evolution expressions for entropy in the forms Eqs. (3) and (4), it is customary to note from Eq. (4) that entropy is not a conservative variable because of the three terms

To understand and quantify the non-conservative production of entropy when turbulent mixing occurs between different seawater parcels, a different approach is required because the production terms

The

In practice the FTR is used extensively in the construction of the thermodynamic potentials that describes seawater, so that all the thermodynamic variables are related to each other using equations that apply for reversible processes. Because each of these thermodynamic variables are state variables, the use of the FTR is justified; its use essentially finds a route through parameter space caused by a series of reversible processes, even though there are many other ways of traversing between two

Two important characteristics of oceanographic variables are (i) whether they are “potential” variables and (ii) whether they are conservative variables: these characteristics are discussed at length in Sects. A.8 and A.9 of IOC et al. (2010). For example, Absolute Salinity is a potential variable since if the salt flux divergence,

Since Conservative Temperature

The fundamental thermodynamic relationship of Eq. (2) can be regarded as an expression for the total derivative of enthalpy when it is expressed as a function of

Importantly, all thermodynamic potentials obey the three general criteria which characterize axiomatic systems (Feistel, 2008, 2018). That is, thermodynamic potentials must exhibit consistency (that is, they exclude the possibility of deducing two different mathematical expressions for the same property), independence (that is, they prevent any derived function from being deducible from another one), and completeness (that is, they provide an equation for every equilibrium thermodynamic bulk property). For an arbitrary given thermodynamic property equation, the validity of these criteria is not trivially fulfilled and needs to be demonstrated in order to regard that equation a thermodynamic potential. The new thermodynamic potentials of this paper do obey these three essential criteria.

In this paper we derive a new thermodynamic potential of seawater,

In Sect. 2 we compare two ways of defining the properties of seawater. In one way we claim to have knowledge of both enthalpy and entropy as functions of in situ temperature; that is, we claim to know both

In Sect. 3 we find a new thermodynamic potential

The FTR, Eq. (2), in its original form,

The Gibbs function

Each of the thermodynamic potentials

The discussion of the derivation, definition, and use of the Gibbs function can be approached via a slightly different line of reasoning. We introduce this alternative line of reasoning because it resonates with the same line of reasoning that we use to derive/justify the thermodynamic potential

The last step in this alternative narrative that leads to the Gibbs function is to note that it is more convenient to combine the knowledge contained in

Comparing the traditional with the alternative reasoning surrounding the Gibbs function, we see that via the traditional approach, in order to deduce at the FTR from knowledge of the Gibbs function one needs to know both (i) how the Gibbs function is found from the observed data, namely, the differential expression Eq. (7) and (ii) the definition of the Gibbs function in terms of enthalpy and entropy,

Now we consider the case of Conservative Temperature

Expressions for various thermodynamic variables based on different thermodynamic potentials.

There are two useful features that follow directly from the definition of Conservative Temperature as being proportional to potential enthalpy referenced to

Note that in this

In the next section we introduce the new thermodynamic potential

Since Conservative Temperature

In this paper we adopt a similar integration of entropy and specific volume but now with respect to Conservative Temperature (rather than in situ temperature) to define the new thermodynamic potential of seawater

In summary, we are using polynomial fits to entropy and enthalpy (or equivalently, specific volume), as functions of Conservative Temperature, knowing from Appendix P of IOC et al. (2010) that these fits in the forms

In Sect. 2 we suggested that internal energy expressed as a function of Absolute Salinity, entropy, and specific volume is the most natural thermodynamic potential, but since mixing processes in the ocean occur at constant pressure rather than at constant volume, a more useful thermodynamic potential for seawater is enthalpy in the functional form

In Sect. 2 we introduced an alternate route to deriving the Gibbs function, using knowledge/observations of

Similarly, we showed in the

We conclude that the new thermodynamic potential

Having argued that the two thermodynamic potentials,

In summary, we have used the TEOS-10 Gibbs function of seawater to relate the different temperature variables and to evaluate both specific volume and entropy, which were then fitted with polynomials in the three independent variables

In order to construct an accurate polynomial expression for the thermodynamic potential of seawater

The specific entropy of a perfect gas can be expressed in terms of the Celsius potential temperature

Panels

One wonders how accurate a correspondingly simple logarithm expression would be for the entropy of seawater, defined by either

While the fit to entropy is better in Fig. 1b than in Fig. 1a, neither is particularly accurate for our purposes. For example, in determining potential temperature

The Second Law of Thermodynamics requires that entropy must be produced when mixing occurs, and the approximation

The accuracy of the approximate expression

In order to obtain an expression for

When the thermodynamic potential

When calculating Conservative Temperature

When adopting the approach of the present paper, the conversion from in situ temperature

Having converted observations of in situ temperature into Conservative Temperature, other calculations are more computationally efficient when using the enthalpy and entropy combination of

Similar gains in computational efficiency occur when evaluating potential density at a variety of reference pressures when using

We have written algorithms to evaluate all of the thermodynamic quantities of seawater using only one or both of

While in situ temperature is relatively simple to measure in the ocean, it is neither a potential property, nor is it a conservative property, and these deficiencies of in situ temperature have led to the adoption of Conservative Temperature

When the Roquet et al. (2015) 75-term polynomial for specific volume,

In this paper we have provided an accurate expression for entropy as a function of Conservative Temperature,

Converting from observed values of in situ temperature to Conservative Temperature takes a similar amount of computer time using the

In the

Moreover, we have been able to combine the expressions for specific volume and for entropy into a single thermodynamic potential function,

The fact that we have been able to form the new thermodynamic potential,

The thermodynamic independence of the information contained in

Equation (14) (or Eq. 13) above is the proposed definition of the thermodynamic potential of seawater defined with respect to Conservative Temperature, but it is not the only possible functional form, and here we present other possibilities. Equations (13) and (14) resemble the integral definition of the Gibbs function, Eq. (12), and now we follow an analogy with the

In the case of

We conclude that both Eqs. (14) and (A1) are straightforward to use as thermodynamic potentials in terms of

We note that the functional form of Eq. (A2) also works as a thermodynamic potential when potential temperature

Of the other functional forms we have used above, namely Eqs. (14), (A1), and (A3), the only other viable form we have found as a function of

If one did want to express all the thermodynamic variables in terms of

The polynomial-based expression for specific entropy as a function of Absolute Salinity and Conservative Temperature is given by Eq. (21) as the sum of the two dominant logarithm terms plus an eighth-order polynomial in the two dimensionless variables

Equations (15) and (16) for entropy

Considering changes occurring at constant Absolute Salinity and pressure, the FTR in the forms Eqs. (8) and (11) shows that in situ temperature

The expression for specific volume in terms of the Gibbs function is very neat and compact, being

In terms of the Gibbs function, the adiabatic lapse rate (the rate of change of in situ temperature during an adiabatic and isohaline change in pressure; see McDougall and Feistel, 2003) is

The relative chemical potential

The adiabatic and isohaline compressibility has the following compact expression in terms of

These expressions for the various thermodynamic variables are summarized in Table 2.

When we take the second-order cross-derivatives of the thermodynamic potential

Note that the equality between

It can be shown by coordinate transformation that each of Eqs. (C21)–(C23) contains exactly the same information as Eqs. (C14)–(C16). That is, each of the equations Eqs. (C21)–(C23) can be found by transforming the corresponding equation in Eqs. (C14)–(C16) from

The fundamental thermodynamic relationship (FTR) can be deduced from knowledge of the total differential of the Gibbs function

We write the total differential of

Because of the definition of Conservative Temperature,

On acceptance of this paper for publication, 24 of the Gibbs SeaWater Oceanographic Toolbox (GSW) subroutines in MATLAB were replaced by subroutines of the same name in the GSW code on the TEOS-10 website (

TJM discovered the forms, Eqs. (14), (A1)–(A3), (A8) and (A11), of the thermodynamic potential of seawater, PMB wrote and tested the 24 new computer subroutines that were needed to implement the ideas of this paper in the GSW computer software of TEOS-10, and RF ensured that the thermodynamic reasoning in the paper was precise, while FR performed the fit of entropy to Absolute Salinity and Conservative Temperature. All authors contributed to writing the manuscript.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

We thank William Dewar, Stephen Griffies, and Remi Tailleux for many helpful suggestions which improved the paper. This paper contributes to the tasks of the IAPSO/SCOR/IAPWS Joint Committee on the Properties of Seawater.

This research has been supported by the Australian Research Council (grant no. FL150100090).

This paper was edited by Mehmet Ilicak and reviewed by Stephen M. Griffies and one anonymous referee.