1. Introduction
The quest in this work is to derive a variable that is conservative, independent of adiabatic changes in pressure, and whose conservation equation is the oceanic version of the first law of thermodynamics. That is, we seek a variable whose advection and diffusion can be interpreted as the advection and diffusion of “heat.” In other words, we seek to answer the question, “what is heat” in the ocean? The variable that is currently used for this purpose in ocean models is potential temperature referenced to the sea surface, θ, but it does not accurately represent the conservation of heat because of (i) the variation of specific heat with salinity and (ii) the dependence of the total differential of enthalpy on variations of salinity.
Fofonoff (1962) pointed out that when fluid parcels mix at constant pressure, the thermodynamic variable that is conserved is enthalpy, and he showed this implied that potential temperature is not a conservative variable. It is natural then to consider enthalpy as a candidate conservative variable for embodying the meaning of the first law of thermodynamics. However, this attempt is thwarted by the strong dependence of enthalpy on pressure. For example, an increase in pressure of 107 Pa (1000 dbar), without exchange of heat or salt, causes a change in enthalpy that is equivalent to about 2.5°C. We show in this paper that in contrast to enthalpy, potential enthalpy does have the desired properties to embody the meaning of the first law.
Present treatment of oceanic heat fluxes is clearly inconsistent. Ocean models treat potential temperature as a conservative variable and calculate the heat flux across oceanic sections using a constant value of heat capacity. By contrast, heat flux through sections of observed data is often calculated using a variable specific heat multiplying the flux of potential temperature per unit area (Bryan 1962; Macdonald et al. 1994; Saunders 1995; Bacon and Fofonoff 1996). Here it is shown that the theoretical justification of this second approach is flawed on three counts. While the errors involved are small, it is clearly less than satisfactory to have conflicting practices in the observational and modeling parts of physical oceanography, particularly as an accurate and convenient solution can be found.
Warren (1999) has claimed that because internal energy is unknown up to a linear function of salinity, it is inappropriate to talk of a flux of heat across an ocean section unless there are zero fluxes of mass and of salt across the section. Here it is shown that this pessimism is unfounded; it is perfectly valid to talk of potential enthalpy, h0, as the “heat content” and to regard the flux of h0 as the “heat flux.” Moreover, h0 is shown to be more conserved than is θ by more than two orders of magnitude. This paper proves that the fluxes of h0 across oceanic sections can be accurately compared with the air–sea heat flux, irrespective of whether the fluxes of mass and of salt are zero across these ocean sections. This has implications for best oceanographic practice for the analysis of ocean observations and for the interpretation of “temperature” in models.
The first law of thermodynamics is compared with the equation for the conservation of total energy (the Bernoulli equation). It is shown that while the Bernoulli function and potential enthalpy differ by only about 3 × 10−3°C (when expressed in temperature units), the Bernoulli function cannot be considered a water-mass property as it varies with the adiabatic vertical heaving of wave motions. A larger drawback of the Bernoulli function is that it cannot be determined from the local thermodynamic coordinates S, T, p. For these reasons the Bernoulli function is not an attractive variable compared with h0.
2. The first law of thermodynamics

In words, the first law of thermodynamics [(1)] says that the internal energy of a fluid parcel can change due to (i) the work done when the parcel's volume is changed at pressure (p0 + p), (ii) the divergence of the flux of heat, and (iii) the dissipation of turbulent kinetic energy into heat. The effect of the dissipation of kinetic energy in these equations is very small and is always ignored. For example, a typical dissipation rate, εM, of 10−9 W kg−1 causes a warming of only 10−3 K (100 yr)−1. Another way of quantifying the unimportance of this term is to compare it to the magnitude of diapycnal mixing. The turbulent diapycnal diffusivity scales as 0.2 εMN−2 (Osborn 1980) and the diapycnal mixing of potential temperature that this diffusivity causes is typically more than one thousand times larger than that caused by the dissipation of kinetic energy εM/Cp.
The other term on the right-hand sides of these instantaneous conservation equations (1) and (2) is the divergence of a molecular heat flux, −∇·FQ. When these conservation equations are averaged over all manner of turbulent motions, this term will also be quite negligible compared with the turbulent heat fluxes except at the ocean's boundaries; the air–sea heat flux occurs as the average of FQ at the sea surface and the geothermal heat flux that the ocean receives from the solid earth also appears in the conservation equations through the average of FQ at the seafloor. We note in passing that at both the sea surface and at the ocean floor the flux of salt is zero and so the heat of transfer due to the flux of salt is also zero and so from (3) FQ is equal to the reduced heat flux Fq. Note also that in hot smokers, the flux of salt (and heat) is advective in nature and so will be captured by the advection terms on the left-hand side of (1) and (2).

Physicists sometime caution against using “heat” as a noun because the first law of thermodynamics is concerned with changes in internal energy that are related to not only heat fluxes but also to the doing of work. At the beginning of their book, Bohren and Albrecht (1998) devote several pages discussing some examples in which the word “heat” is used imprecisely. Later in their book (section 3) they point out that the word “enthalpy” can often be accurately used in place of “heat content per unit mass.” In the present paper it will be shown that with negligible error, a new oceanic heat-like variable called “potential enthalpy,” obeys a clean conservation equation of the form (4) with the right-hand side being (minus) the divergence of the molecular flux of heat. That is, it will be shown that the left-hand side of (2) is equal to ρ times the material derivative of potential enthalpy plus a negligible error term. This means that the conservation equation of potential enthalpy in the ocean is equivalent to the first law of thermodynamics. Given this, calling potential enthalpy “heat content” can cause no harm or imprecise thinking in oceanography. Potential enthalpy and “heat content” are effectively alternative names for the same thing because potential enthalpy is the variable whose advection and diffusion throughout the ocean can be accurately compared with the boundary fluxes of heat. Just as the advection and diffusion of a passive conservative tracer in the ocean can be accurately compared with the boundary fluxes of the passive tracer, this same association of the boundary fluxes and the tracer content justifies the association of the word “heat content” with the new variable, potential enthalpy.
3. Potential enthalpy
It follows from the form of (2) that when mixing occurs at constant pressure, enthalpy is conserved [this is more obvious when (2) is written in divergence form using the continuity equation]. As an example, mixing between fluid parcels at the sea surface where the pressure is constant (p = 0 and the total pressure is p0) conserves the enthalpy evaluated at that (zero) pressure. Just as the concept of potential temperature is well established in oceanography, consider now the “potential” concept applied to enthalpy. After bringing a fluid parcel adiabatically (and without exchange of salt) to the sea surface pressure, its enthalpy is evaluated there and called potential enthalpy. During the adiabatic pressure excursion the potential enthalpy of fluid parcels are unchanged and one wonders how much damage is done by forcing the fluid parcels to migrate to zero pressure before allowing them to mix rather than simply mixing in situ as they do in practice. This thinking was the motivation for examining potential enthalpy as a candidate heat content.


The rest of this paper will quantify the error made by ignoring the last two terms in (8) and treating potential enthalpy as a conservative variable. It will be proven that the error in so doing is negligible, being no larger than the neglect of the dissipation of kinetic energy into heat. It will be deduced that the temperature error in ocean models that conserve potential enthalpy are no more than 1 mK, which is a factor of more than 100 less than the errors in present ocean models that treat potential temperature as a conservative variable.
4. The first law in terms of θ






5. The first law in terms of Θ











6. Potential enthalpy as heat content
The key finding in this paper amounts to proving that in comparing (16) to (9), hS|Θ,p ≪ hS|θ,p and that the heat capacity defined with respect to Θ, namely hΘ|S,p, varies much less from the constant value

The air–sea flux of heat appears in (28) as FQ and since this flux occurs at zero pressure, there is no error at all in equating the air–sea flux with the flux of potential enthalpy [because the last two terms on the left-hand side of (25) are zero at the sea surface]. The geothermal heat flux occurs at great depth and the local increase in Θ caused by the divergence of the geothermal heat flux should be evaluated using the specific heat hΘ|S,p which, at a pressure of 4 × 107 Pa (4000 dbar), is about 1.0015
This association of the air–sea and geothermal heat fluxes with the flux of h0 is particularly clear since there is no flux of salt across either the sea surface or the seafloor, so that from (3) the total boundary heat flux FQ is the same as the reduced heat flux Fq through the boundaries. This is convenient since hS = μ − (T0 + T)μT is only known up to a constant, reflecting the fact that enthalpy itself is unknown up to a linear function of salinity.
We come now to the question of whether it is possible to regard h0 as heat content and the flux of h0 as heat flux. Warren (1999) argued that because enthalpy is unknown up to a linear function of salinity, it is only possible to talk of a flux of heat through an ocean section if the fluxes of both mass and salt through that section are zero. Technically this is true, but only in the same narrow sense that it is not possible to talk of the flux of θ through an ocean section because there is always the question of adding or subtracting a constant offset to the temperature scale. Once we define what scale (kelvins or degrees Celsius) is being used to measure θ, the issue is resolved and one can legitimately talk of a flux of θ even though the mass flux may be nonzero.
A similar argument can now be applied to potential enthalpy. In defining the Gibbs function of sea water, Feistel and Hagen (1995) made arbitrary choices for four constants, and two of these choices amount to making a specific choice for the unknowable linear function of salinity in the definition of h0. The key thing to realize is that for any arbitrary choice of this linear function of salinity, the conservation equation, (28), of h0 is unchanged, and also, such arbitrary choices do not affect the air–sea and geothermal heat fluxes. Hence h0 is the correct property with which to track the advection and diffusion of heat in the ocean, irrespective of the arbitrary function of salinity that is contained in the definition of h0.
For example, the difference between the meridional flux of h0 across two latitudes is equal to the area-integrated air–sea and geothermal heat fluxes between those latitudes (after also allowing for any unsteady accumulation of h0 in the volume), irrespective of whether there are nonzero fluxes of mass or of salt across either or both meridional sections. This powerful result follows directly from the fact that h0 obeys a standard conservation equation, (28), no matter what linear function of salinity is chosen in the definition of h0. As a consequence, it does make perfect sense to talk of the meridional flux of heat (i.e., the flux of h0) in the Indian and South Pacific Oceans separately, just as it makes sense to discuss the meridional fluxes of mass, freshwater, tritium, salt, and salinity anomaly (S − 35) through these individual ocean sections. Just as it is valid and oftentimes advantageous to carry equations in inverse models for salinity anomaly rather than the full salinity (McDougall 1991; Sloyan and Rintoul 2000; Ganachaud and Wunsch 2000), so it is valid to use equations for the anomaly of conservative temperature, (Θ − Θ0). Doing so often has the effect of decreasing the influence of a relatively uncertain velocity field on the heat budget.
For these reasons it is clear that h0 and Θ are the oceanic thermodynamic quantities whose conservation represents the first law of thermodynamics. Furthermore, it is legitimate to call h0 “the heat content per unit mass” and to call the flux of h0 “the heat flux,” bearing in mind that this nomenclature assumes the particular linear function of S that Feistel and Hagen (1995) adopted, just as the corresponding flux of potential temperature is dependent on the temperature scale on which the potential temperature is measured.

7. Cp(S, θ, pr)θ as heat content
In a recent paper, Bacon and Fofonoff (1996) advocated the use of Cp(S, θ, pr) θ as heat content but here it is proven that this is actually less accurate than simply using
Another step that Bacon and Fofonoff (1996) took in their treatment of the first law of thermodynamics was to assume that [μ(p) − (T0 + T)μT(pr)]dS/dt in (14) could be ignored so that the material derivative of heat was taken to be Cp(pr)dθ/dt. While it is true that the ignored term is much smaller than Cp(pr)dθ/dt, it will be shown here that it is inconsistent to ignore this term if Cp(pr) is allowed to vary. The third error in Bacon and Fofonoff (1996) was to state [their Eq. (8)] that the volume integral of the advective part of Cp(S, θ, pr) dθ/dt was the integral of Cp(S, θ, pr)θ times the mass flux per unit area over the bounding area of the ocean volume. This oversight falsely assumes that d[Cp(S, θ, pr)θ]/dt is the total derivative of “heat” rather than what they had arrived at, namely Cp(S, θ, pr)dθ/dt. One cannot move the heat capacity inside the derivative when the heat capacity is allowed to vary as in Bacon and Fofonoff (1996).


We conclude that past attempts to justify Cp(pr)θ as heat content have been flawed on theoretical grounds, and since we show below that this approach is no more accurate than simply using a constant heat capacity, it should be abandoned. Prior to the Bacon and Fofonoff (1996) paper various authors had used the in situ value of heat capacity together with potential temperature [i.e., Cp(S, T, p)θ] as heat content (Bryan 1962; Macdonald et al. 1994; Saunders 1995) but there is even less theoretical justification for this choice than for Cp(pr)θ and we show below that Cp(S, T, p)θ is less accurate than both Cp(pr)θ and
The production of θ and Θ on mixing between fluid parcels is considered in appendix B and appendix C. Figure C1 illustrates the result of mixing fluid parcels with extreme property contrasts that are widely separated in space (at a series of fixed pressures) and one wonders about the relevance of this procedure to the real ocean. The importance of these mixing arguments depends on the heat flux that travels by these paths, so that, for example, if most of the oceanic heat transport entered the ocean where the ocean is very warm and salty and exited the ocean where it was very cool and very fresh, then the production of potential temperature of −0.4°C would be a realistic estimate for the bulk of the ocean. [In a similar manner, the total amount of cabbeling (McDougall 1987) that occurs along a neutral density surface depends on the flux of heat being transported down the temperature contrast on that surface even though the individual mixing events occur between parcels with very small θ and S contrasts.] Because mixing involves both epineutral and dianeutral mixing, and because the heat flux achieved by the various mixing paths is rather complex, the mixing arguments that lie behind the plots in Fig. C1 do not obviously provide a realistic estimate of the importance of the nonconservative production of θ or of Θ in the ocean. The important message that is gleaned from Fig. C1 is that the nonconservative production of Θ is at least one hundred times smaller that the production of θ. A realistic assessment of the errors inherent in present oceanic practice can then be found by examining the temperature difference θ − Θ as described later in this paper, and the error remaining in the use of Θ is taken to be less than 1% of the temperature difference, θ − Θ. Appendix D considers internal energy and potential internal energy as candidates for “heat content” but it is shown that they are not as suitable as potential enthalpy.
8. Quantifying the errors in θ, Cp(pr)θ, and Θ
The nonconservative nature of potential temperature can be illustrated on a variant of the usual S − θ diagram. Since both h0 and Θ are conserved when mixing occurs at p = 0, it follows that any variation of the difference, θ − Θ, on a S − Θ diagram must be due to the production of θ when mixing occurs at p = 0. Enthalpy, h, is evaluated using the Gibbs function of Feistel and Hagen (1995). The arbitrary linear function of S that is inherent in any definition of h was chosen by Feistel and Hagen (1995) so that h is zero at (S, T, p) of (0, 0, 0) and (35, 0, 0). Our definition, (15), of Θ means that it can be regarded as a function of S and θ, Θ = Θ(S, θ), and ensures that Θ = θ at the three points (0, 0), (35, 0), and (35, 25) on the S − θ plane.
The temperature difference, θ − Θ, is quite small when the temperature is close to zero and, because of our choice of
The error in taking Cp(S, θ, pr)θ to be h0 is shown on the full S − Θ plane in Fig. 2b. This error is expressed in temperature units as [Cp(pr)θ − h0]/
Having already compared the production of θ with that of Cp(S, θ, pr)θ, here we briefly document the nonconservative production of other thermodynamic quantities. In each case the quantity concerned is multiplied by a positive constant and then a linear function of S and Θ is subtracted so that the resulting quantity is zero at the (S, Θ) points (0, 0), (35, 0), and (35, 25) while the coefficient of Θ in the final expression is arranged to be is ±1. In this way the variable that is plotted in Figs. 4, 5 and 6 (like those in Figs. 2 and 3) have contours measured in temperature units. Because these plots are simply a scaled version of the original variable plus a linear combination of S and Θ, they can be used to determine the nonconservative production of the original variable, measured in temperature units.
First the nonconservation of potential enthalpy, h0, is illustrated for mixing of fluid parcels at 600 dbar which, from Fig. C1b, is the pressure at which the greatest production of h0 occurs. Enthalpy evaluated at 600 dbar is conserved during mixing at this pressure and the linear function of enthalpy, S and Θ that is zero at (0, 0) and (35, 0) and (35, 25) is contoured in Fig. 4. The maximum value of the production of Θ when mixing at 600 dbar can be deduced from the contours in this figure, namely about 4 × 10−3 °C. However, this requires mixing across the full scale of the axes in this figure, but the range of temperature and salinity in the ocean at 600 dbar is much smaller as is illustrated by the cloud of data points from the whole of the Koltermann et al. (2003) global atlas, superimposed on this figure. The actual maximum value of δΘ at 600 dbar is almost an order of magnitude less than this value at 6.3 × 10−4 °C (from Fig. C1b). [The vertical axis in Fig. 4 should really be proportional to the conservative variable h(S, Θ, 6 MPa), but when this is done, the changes are imperceptible, just as Fig. 2a can be drawn with θ as the vertical axis which causes only a small but perceptible change to the figure.] Because the nonconservative production of Θ is less than 1% of the nonconservative production of θ, we conclude that the error in Θ is less than 1% of the error in θ. With the bulk of the ocean having a θ error less than 0.1°C (from Fig. 3a) the maximum error in Θ is estimated at less than 10−3°C.
The corresponding result for entropy is shown in Fig. 5. Here the temperature-like variable that is derived from entropy, σ, is simply proportional to σ with the proportionality constant chosen so that the resulting “entropic temperature” is 25°C at (35, 25). From this figure we deduce that entropy is produced at approximately three times the rate at which θ is produced.
The cabbeling nonlinearity of the equation of state can also be compared with the above nonlinear productions by taking the appropriate linear combination of potential density (referenced to the sea surface), S and Θ that is also zero at (0, 0) and (35, 0) and (35, 25). From Fig. 6 we conclude that nonlinear productions larger than 14°C are possible for mixtures of some pairs of water parcels. This can be compared with the maximum nonlinear production of θ of about −0.55°C. This suggests that the nonlinear production of density by the cabbeling process is roughly 25 times as large as the effect on density of neglecting the production of θ. This is confirmed by comparing the range of θ − Θ in Fig. 2a (2°C) with the range of the variable of Fig. 6 (27.5°C), indicating that θ is about 14 times more conservative than is potential density.
In appendix E it is shown that the use of potential enthalpy gives rise to a new expression for the available potential energy in the ocean and in particular, clearly associates the difference between available potential energy and the available gravitational potential energy as being due to the thermobaric nature of the equation of state of seawater.
9. Errors in present ocean models
Consider an ocean model exchanging heat with the atmosphere at the rate Q(x, y, t). We have established that this heat enters or leaves the ocean as a flux of potential enthalpy, so that Q/
One handy way of expressing the error involved with using potential temperature is to note that 0.5% of the annual-mean SST values in the ocean atlas have θ − Θ < −0.15°C and 0.5% have θ − Θ > 0.10°C. That is, 1% of the annual-mean SST data lie outside an error range of 0.25°C. In salty water potential temperature tends to be larger than it should be if it were to accurately represent heat content, while for freshwater, θ is less than Θ. We have also examined the variation of θ − Θ at the sea surface throughout the year and the range of θ − Θ is shown in Fig. 7b. One percent of the values have a seasonal range of θ − Θ that exceeds 0.16°C. A temperature difference of 0.25°C is not completely negligible in the ocean—it is the same as the difference θ − T between potential and in situ temperatures for a pressure excursion of about 4000 dbar. Another way of looking at these errors is the plots in Fig. 8 of the root-mean-square and range (maximum minus minimum) of θ − Θ as a function of pressure in the World Ocean. This shows that the range of θ − Θ is almost 0.4°C over the upper 1000 m of the water column, and is actually as large as 1.4°C near the surface.
The difference in the meridional heat fluxes under the two different interpretations of model temperature is calculated by taking the area integral of ρ0
Warren (1999) chose to examine the meridional flux of internal energy, ε, and implied that this is the quantity that should be compared with the air–sea heat flux. For the same model data of Hirst et al. (2000) the difference between the meridional flux of ε and of h0 is shown as the solid line with dots in Fig. 9c. It is seen that the meridional flux of ε is no closer to being regarded as the meridional heat flux than is the flux of θ using a fixed heat capacity. The reason for this is the second term on the left-hand side of (1), which also means that internal energy does not have the “potential” property. Warren then derived the meridional flux of
It is concluded that present ocean models contain typical errors of ±0.1°C due to the neglect of the nonconservative production of θ although the error is as large as 1.4°C in isolated regions such as where the warm fresh Amazon water discharges into the ocean. The corresponding typical error in the meridional heat flux is 0.005 PW (or 0.4%). To eliminate these errors one must (i) interpret the model's temperature variable as Θ rather than as θ, (ii) carry the equation of state as ρ = ρ(S, Θ, p) (the above discussion has assumed that the changes arising from having this different equation of state are small, but this remains to be confirmed), and (iii) calculate θ using the inverse function θ(S, Θ) when SST is required (e.g., in order to calculate air–sea fluxes with bulk formulas). These issues will be explored in a subsequent paper. While errors of 0.4% in the meridional heat flux are much smaller than our ability to determine these heat fluxes from observations, errors of ±0.1°C in sea surface temperature do not seem to be totally trivial.
10. The total energy, or Bernoulli equation

Continuing to ignore the last term in (34) we see that B is totally conserved when fluid parcels mix at constant pressure. In this regard B is superior to h0 because potential enthalpy is not 100% conserved when mixing happens in the subsurface ocean, and as a result Θ is in error by up to 1 mK. The range of pressure variation at fixed depth (due to the movement of mesoscale eddies) is typically 104 Pa (1 dbar) which is equivalent to a change in enthalpy of 10 J kg−1, which in turn is equivalent to a temperature change of 2.5 mK. An adiabatic and isohaline change in pressure will cause a change in the Bernoulli function of this magnitude, whereas potential enthalpy is totally independent of such pressure variations. In this regard h0 and Θ are superior to B.
It is possible to imagine an ocean model carrying the Bernoulli function as its “temperature” variable. The temporal change of pressure would need to be added as a forcing term in the model's B conservation equation, as in (34). An ocean model would know both p and Φ at each time step so it would be possible to calculate enthalpy from h = B − Φ − (1/2)u · u and to use this as an argument of an equation of state in the functional form ρ(S, h, p). In this way the small error of 1 mK that is inherent in conserving Θ could be avoided. [Another way of avoiding this tiny error would be to carry the small source terms in the Θ equation, i.e., to carry the two pressure integral terms in (25).] While implementing the B conservation equation (34) in an ocean model would avoid any approximations in the total energy budget, what would be lost is the notion that the model variable B is a property of a water mass. Rather, B varies with pressure to the extent of 2.5 mK. This temperature increment happens to be the stated accuracy of modern CTD instruments and is larger than the maximum error (1 mK) in using conservative temperature Θ.

Potential enthalpy is by far the dominant contribution to B, and when expressed in terms of Θ, the oceanic range of h0 is about 30°C. Hence the dynamical information that is contained in B, namely, B − h0, being no more than 10 m2 s−2, is a factor of 10 000 less than the dominant thermodynamic contribution, h0, as found by Cunningham (2000). Moreover, at the magnitude of this dynamical information, B is not conserved at leading order because of the unsteady pressure term in (34). That is, once the thermodynamic contribution, h0, is subtracted from B, the advection of the remainder is the same magnitude as the unsteady pressure term which is usually ignored. For dynamical information, the Montgomery potential [or other suitable geostrophic streamfunction; see Montgomery 1937; McDougall 1989, his Eq. (43)] has the advantage over B that it is not dominated by a heat balance that is a factor of 10 000 larger than the information contained in the geostrophic streamfunction. Here it is noted in passing that atmospheric scientists use the term Montgomery streamfunction for h + Φ whereas oceanographers use the term Montgomery streamfunction for the geostrophic streamfunction appropriate to any surface of interest, such as the streamfunction originally proposed by Montgomery (1937) for geostrophic flow in a steric anomaly surface.
It is concluded that there is more information to be had by considering the potential enthalpy balance and the geostrophic streamfunction separately than by combining these two pieces of information together into the one Bernoulli equation. The present work supports the argument of Bacon and Fofonoff (1996) that the “use of the Bernoulli function is an unnecessary conflation of mechanical and nonmechanical energy, given that they evolve practically independently.” The major drawbacks with using the Bernoulli function are that (i) unlike S and Θ, B cannot be considered a water-mass property as it varies with the adiabatic heaving of wave motions; and (ii) unlike S and Θ, B cannot be determined from the local thermodynamic properties.
11. Summary
The aim of this work has been to develop a variable whose conservation statement is equivalent to the first law of thermodynamics so that this variable can be accurately called “heat content.” This quest led to the thermodynamic quantity, potential enthalpy, which is the enthalpy that a fluid parcel would have if its pressure was changed, in an adiabatic and isohaline fashion, to the pressure of the sea surface. With an error that is more than two orders of magnitude less than present practice, the flux of potential enthalpy is the correct flux of “heat” that can therefore be accurately compared with air–sea and geothermal boundary fluxes of heat.

This paper has largely proved the benefits of potential enthalpy h0 from the viewpoint of conservation equations, but the benefits can also be understood from the following parcel arguments. First, the air–sea heat flux needs to be recognized as a flux of h0. Second, the work of appendixes B and C shows that while it is the in situ enthalpy that is conserved when parcels mix, a negligible error is made when h0 is assumed to be conserved during mixing at any depth. Third, note that the ocean circulation can be regarded as a series of adiabatic and isohaline movements during which h0 is absolutely unchanged followed by a series of turbulent mixing events during which h0 is almost totally conserved. Hence it is clear that h0 is the quantity that is advected and diffused in an almost conservative fashion and whose surface flux is the air–sea heat flux.
The small error involved with calling potential enthalpy “heat content” has been shown to be no larger than the effect of the dissipation of kinetic energy in the first law of thermodynamics and so is utterly negligible. Without an exact total differential to represent the conservation of “heat” it is not possible to neatly illustrate the errors involved with calling potential enthalpy “heat content,” but the error in the meridional heat flux is likely to amount to less than 1% of the error involved when using either
It is convenient to define a new temperature variable, called “conservative temperature,” Θ, which is simply proportional to potential enthalpy with the proportionality constant being the fixed “heat capacity,”
The realization that ocean models carry Θ rather than θ means that the heat capacity of seawater in these model codes should not be user-specified but should be hard-wired to be
After submitting this manuscript for publication I have become aware that the Goddard Institute for Space Studies (GISS) ocean model already carries potential enthalpy as its heatlike variable (Russell et al. 1995). The present paper can then be regarded as supplying the theoretical motivation for converting such ocean models from using potential temperature to potential enthalpy.
The fact that enthalpy is only known up to a linear function of salinity does not diminish the usefulness of potential enthalpy as heat content nor the flux of h0 as heat flux. It is proven that the meridional flux of h0 does represent a valid flux of heat even when the meridional fluxes of mass and of salt are nonzero. We have also shown here that the Bernoulli function and potential enthalpy differ by only about 3 × 10−3°C (when expressed in temperature units). Nevertheless, for study of heat budgets h0 is more useful than the Bernoulli function because in contrast to B, h0, and Θ have the distinct advantage of being locally determined thermodynamic quantities that are totally invariant under adiabatic and isohaline changes of pressure. Hence, h0 and Θ are properties of water masses while B is not.
I thank Dr. David Jackett for coding the thermodynamic algorithms based on the Gibbs function of Feistel and Hagen (1995) and for preparing all the figures. Dr. Rainer Feistel kindly provided an electronic version of the Gibbs function algorithm, and he, Dr. Bruce Warren, Dr. Stephen Griffies, and Professor Jürgen Willebrand are thanked for their comments on a draft of this paper. Dr. Siobhan O'Farrell kindly provided the data of Hirst et al. (2000) that is used in Fig. 9. This work contributes to the CSIRO Climate Change Research Program.
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APPENDIX A
Algorithm for Conservative Temperature
Enthalpy, h(S, T, p), is evaluated by differentiating the Gibbs function, G(S, T, p), of Feistel and Hagen (1995) according to h = G − (T0 + T)GT. Potential enthalpy h0 is enthalpy evaluated at the reference pressure of zero and at the potential temperature; that is, h0(S, θ) = h(S, θ, 0). Following Feistel and Hagen (1995), the polynomial for potential enthalpy is written in terms of the scaled salinity and potential temperature variables, s = S/40 and τ = θ/40. The coefficients of the polynomial h0(s, τ) are given in Table A1. Here, S is salinity in psu, θ is potential temperature in degrees Celsius (on the ITS-90 temperature scale) and h0 is in joules per kilogram.
The conservative temperature Θ is defined as in (15) to be Θ ≡ h0/
APPENDIX B
The Nonconservative Production of θ



APPENDIX C
The Nonconservative Production of Θ




Equation (C6) shows that the nonconservative production of potential enthalpy is proportional to the nonconservative production of density called cabbeling (McDougall 1987), (1/8)ρ[

In order to better compare the production of θ and Θ in today's ocean we have searched the annually averaged oceanic atlas of Koltermann et al. (2003) in the following way. At each standard pressure the largest values of ΔθΔS and of (Δθ)2 were found by examining every possible combination of fluid parcels and storing the largest values of these quantities. The approximate values of δθ and δΘ were then calculated from (B6) and (C8) and are shown as the dashed lines in Figs. C1a and C1b. For the same pair of parcels that produced the largest values of ΔθΔS and of (Δθ)2 the accurate values of δθ and δΘ were also calculated and are shown as the full lines in Fig. C1. These accurate values were determined by mixing the salinity and the enthalpy of the two fluid parcels linearly and then deducing, by Newton–Raphson iteration, the in situ temperature of the mixed fluid from Feistel and Hagen's (1995) expression for h(S, T, p). From this in situ temperature, θ, h0, δθ, δh0, and δΘ were calculated. The fact that the largest negative value of δθ in Figure B1a is only 1/10 of the −0.55°C identified above reflects the fact that the atlas does not contain fresh meltwater near the poles.
The largest production of conservative temperature is seen to occur at a pressure of 600 dbar and is about 6.3 × 10−4°C whereas the largest production of potential temperature is about −3 × 10−2°C and this occurs at the sea surface. If we append to the atlas the missing cool fresh meltwater near the sea surface, the maximum value of δΘ is unchanged but the extreme value of δθ becomes −0.4°C. It is clear then that Θ is a factor of about 600 more conservative than is θ. It is for this reason that we claim that Θ better represents “heat” than does θ by a factor of more than two orders of magnitude.
APPENDIX D
A Discussion of Potential Internal Energy

Potential internal energy, ε0, can be written as the sum of internal energy, ε, and the pressure integral of −(p0 + p)/(ρ2c2) and from this relationship, the left-hand side of (1) can be written as the material derivative of ε0 plus several other terms, the largest of which is smaller than dε0/dt by the factor, α(p + p0)/ρCp, which is very similar to the ratio found for the terms that are additional to the material derivative of h0 [see the discussion following (8)]. Hence we conclude that for all practical purposes, potential internal energy, ε0, may be used instead of potential enthalpy, h0, as the variable whose conservation statement is the first law of thermodynamics in the ocean. However, h0 is preferred because at the sea surface where the pressure is constant, the left-hand side of (8) becomes exactly ρdh0/dt, whereas this is not quite equal to ρdε0/dt, there being the additional tiny term p0(αdθ/dt − βdS/dt).
APPENDIX E
Available Potential Energy






Heat capacity (at the sea surface) minus the constant value
Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<0945:PEACOV>2.0.CO;2

(a) Contours of the difference θ − Θ between potential temperature θ and conservative temperature Θ. (b) Contours of Cp(S, θ, 0)θ/
Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<0945:PEACOV>2.0.CO;2

Contours of (a) θ − Θ and of (b) Cp(S, θ, 0)θ/
Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<0945:PEACOV>2.0.CO;2

Contours (°C) of a variable that is used to illustrate the nonconservative production of conservative temperature Θ at a pressure of 600 dbar. The three points that are forced to be zero are shown with black dots and the cloud of points near S = 35 psu show where data from the World Ocean at 600 dbar are clustered
Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<0945:PEACOV>2.0.CO;2

Contours (°C) of a variable that is used to illustrate the nonconservative production of entropy σ. The three points that are forced to be zero are shown with black dots
Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<0945:PEACOV>2.0.CO;2

Contours (°C) of a variable that is used to illustrate the nonconservative production of potential density ρθ. The three points that are forced to be zero are shown with black dots
Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<0945:PEACOV>2.0.CO;2

(a) The difference, θ − Θ (°C), between potential temperature θ and conservative temperature Θ at the sea surface for annually averaged data. These differences illustrate the errors in SST in present ocean models when forced with a given heat flux field. (b) The range (max − min value) of θ − Θ (°C) at the sea surface during the 12 months of the year
Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<0945:PEACOV>2.0.CO;2

Plots of (a) the root-mean-square and (b) range (max − min) values of θ − Θ as a function of pressure for all data in the World Ocean
Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<0945:PEACOV>2.0.CO;2

(a) The meridional heat flux borne by the resolved-scale velocity field in the oceanic component of the coupled model of Hirst et al. (2000). This calculation uses h0 =
Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<0945:PEACOV>2.0.CO;2

Fig. C1. The largest amount of nonconservative production of (a) potential temperature θ and of (b) conservative temperature Θ for pairs of water parcels drawn from the ocean at each pressure.
Citation: Journal of Physical Oceanography 33, 5; 10.1175/1520-0485(2003)033<0945:PEACOV>2.0.CO;2
Table A1. Terms and coefficients of the polynomial for potential enthalpy, h0 (s, τ)
