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

From Batchelor (1967), Kamenkovich (1977), Gill (1982), Gregg (1984), and Davis (1994), the first law of thermodynamics may be written as

where ε is the internal energy, h is the specific enthalpy, defined by h ≡ ε + (p₀ + p)/ρ, ρ is in situ density, p is the excess of the real pressure over the fixed atmospheric reference pressure, p₀ = 0.101 325 MPa (Feistel and Hagen 1995), d/dt ≡ ∂/∂t + u · ∇ is the material derivative following the instantaneous fluid velocity, F_Q is the flux of heat by all manner of molecular fluxes and by radiation, and ρε_M is the rate of dissipation of kinetic energy (W m⁻³) into thermal energy. As explained by Landau and Lifshitz [1959; see their Eqs. (57.6) and (58.12)], Fofonoff (1962), and Davis (1994), F_Q includes the cross-diffusion of heat by the gradient of salinity (the Dufour effect) as well as the heat of transfer due to the flux of salt. A “reduced heat flux,” F_q, can be defined that does not include the heat of transfer, so that

F_QF_qh_SF_SF_qμT₀Tμ_TF_S(3)

where h_S = μ − (T₀ + T)μ_T is the partial derivative of specific enthalpy with respect to salinity at constant in situ temperature and pressure, F_S is the flux of salt, T₀ = 273.15 K is the temperature offset between kelvins and degrees Celsius (see Feistel and Hagen 1995), T is in degrees Celsius, μ is the relative chemical potential of salt in seawater (i.e., μ is the difference between the partial chemical potential of salt μ_S and the partial chemical potential of water μ_W in seawater), and μ_T is its derivative with respect to in situ temperature and both μ and μ_T are evaluated at (S, T, p).

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, ε_M, 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 ε_MN⁻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 ε_M/C_p.

The other term on the right-hand sides of these instantaneous conservation equations (1) and (2) is the divergence of a molecular heat flux, −∇·F_Q. 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 F_Q 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 F_Q 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) F_Q is equal to the reduced heat flux F_q. 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).

Because the right-hand sides of (1) and (2) are in the form of the divergence of a flux, namely −∇·F_Q, the key to finding a new variable whose conservation represents the first law of thermodynamics is to find one such that the left-hand side of (1) or (2) is ρ times the material derivative of that variable, for if that were possible, the first law of thermodynamics could be written in the standard conservation form, being the same form as the salt conservation equation,

where F_S is the flux of salt by all manner of molecular processes.

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 second law of thermodynamics defines the specific entropy, σ, whose total derivative obeys the Gibbs relation

dhρdpT₀TdσμdS.(5)

This relation is sometimes called the fundamental thermodynamic relation and it can also be regarded as the mathematical definition of entropy. Consider the movement, without exchange of heat or salt, of a fluid parcel from its in situ pressure p to a fixed reference pressure p_r. Neither salinity nor entropy change during this motion, so that it is apparent from (5) that

hp_S,σρ,(6)

so that the enthalpy at the reference pressure, which we call potential enthalpy, h⁰(S, θ, p_r), is related to in situ enthalpy, h, by

Here we have chosen to regard enthalpy and in situ density ρ as functions of potential temperature θ rather than of in situ temperature. Note that for a fixed reference pressure, h⁰ is a function of only S and θ.

The total derivative of (7) is taken, finding that

where α̃ = −ρ⁻¹∂ρ/∂θ|_S,p and β̃ = ρ⁻¹∂ρ/∂S|_θ,p. The typical value of the left-hand side of this equation is C_pdθ/dt and a typical value for the last two terms is [α̃(p − p_r)/ρ]dθ/dt. The ratio of the last two terms to the dominant term in (8) is then approximately α̃(p − p_r)/ρC_p, and for a pressure difference of 4000 dbar (4 × 10⁷ Pa) this ratio is typically 0.0015, implying that the right-hand side of (8) is almost the material derivative of potential enthalpy. Were it not for these two small terms in (8), potential enthalpy would be the conservative “heatlike” variable that we seek whose conservation equation would be exactly the first law of thermodynamics and (2) would become ρdh⁰/dt = −∇·F_Q + ρε_M.

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

Regarding enthalpy as a function of potential temperature, that is as h(S, θ, p), the first law of thermodynamics [(2)], takes the form

From the Gibbs relation [(5)], we have

where the second expression has used the fact that at constant S and p, both h and σ can be regarded simply as functions of θ. The second part of (10) can be evaluated not only at p but also at the reference pressure where the left-hand side is the heat capacity at that pressure, C_p(p_r) [which is shorthand for C_p(S, θ, p_r)], so that

Again from the Gibbs relation, we have

and regarding h to be the functional form h[S, σ(S, θ), p] leads to

The last part of this equation has used the identity (obtained for example from the definition of the Gibbs function) that ∂σ/∂S|_T,p = −∂μ/∂T|_S,p. Notice that in (13) μ and (T₀ + T) are evaluated in situ while μ_T(p_r) is evaluated at the reference pressure.

Substituting (11) and (13) into (9) gives the first law of thermodynamics expressed in terms of changes of potential temperature and salinity as

This equation was derived by Bacon and Fofonoff (1996) by a slightly different route. We now derive the corresponding result in terms of conservative temperature.

5. The first law in terms of Θ

Regarding enthalpy now as a function of conservative temperature, that is as h(S, Θ, p), the first law of thermodynamics takes the form

From the Gibbs relation, we find that

and (17) can be evaluated not only at p but also at the reference pressure where the left-hand side is C⁰_p, so that

Regarding h to be the functional form h[S, σ(S, Θ), p] leads to

From the Gibbs relation, we have

and when this is evaluated at the reference pressure, it becomes

so that (19) becomes

Substituting (18) and (22) into (16) gives the first law of thermodynamics expressed in terms of changes of conservative temperature and salinity as

At the sea surface T = θ and (23) reduces to ρC⁰_pdΘ/dt = −∇·F_Q + ρε_M. Regarding enthalpy and density to be functions of Θ, potential enthalpy is given by

(where here and henceforth the reference pressure is taken to be zero) and the first law of thermodynamics can be written as

where α = −ρ⁻¹∂ρ/∂Θ|_S,p and β = ρ⁻¹∂ρ/∂S|_Θ,p are the thermal expansion and haline contraction coefficients defined with respect to conservative temperature. The coefficients of dΘ/dt and of dS/dt in (23) and (25) can be equated to find the following two exact relations for T in terms of θ, similar to the traditional relationship for θ as T plus the pressure integral of the lapse rate,

6. Potential enthalpy as heat content

The key finding in this paper amounts to proving that in comparing (16) to (9), h_S|_Θ,p ≪ h_S|_θ,p and that the heat capacity defined with respect to Θ, namely h_Θ|_S,p, varies much less from the constant value C⁰_p than does the heat capacity defined with respect to θ, namely, h_θ|_S,p. For example, even at a pressure as large as 4 × 10⁷ Pa (4000 dbar), h_Θ|_S,p is at most 1.0015 C⁰_p, while h_θ|_S,p varies by more than 5% [see (11) and Fig. 1]. Moreover, h_θ|_S,p suffers this 5% variation in the upper ocean where the spatial contrasts of θ are much larger than at depth so that the variation in h_θ|_S,p, can do more damage than the variation in h_Θ|_S,p, which occurs only at depth where the temperature gradients are small.

This paper argues that (23) or (25) can be approximated as

Taking a mean dianeutral advection velocity of 10⁻⁷ m s⁻¹ and Θ_z of 2 × 10⁻³ K m⁻¹ in the deep ocean, a typical value of dΘ/dt is 2 × 10⁻¹⁰ K s⁻¹ and the terms that have been neglected in going from (25) to (28) are smaller than this by three orders of magnitude. These neglected terms amount to no more that the dissipation of kinetic energy in (28), assuming ε_M = 10⁻⁹ W kg⁻¹.

The air–sea flux of heat appears in (28) as F_Q 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 h_Θ|_S,p which, at a pressure of 4 × 10⁷ Pa (4000 dbar), is about 1.0015 C⁰_p, which can be taken to be C⁰_p with high accuracy.

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 F_Q is the same as the reduced heat flux F_q through the boundaries. This is convenient since h_S = μ − (T₀ + T)μ_T 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.

Thus far I have considered only instantaneous conservation equations. Here the issue of averaging is addressed. First, (28) is written as the instantaneous conservation equation for Θ:

ρ_tρuF_Θ(29)

where F_Θ ≡ F_Q/C⁰_p is the molecular and boundary flux of Θ, and the dissipation of kinetic energy term ρε_M has been dropped. McDougall et al. (2002) have argued that the most sensible way of averaging (29) in z coordinates results in the form

where the last term on the right is the turbulent flux term and Θ^ρ is the density-weighted average value of Θ. The key point in that paper is that the “velocity” variable that is carried by ocean models is actually proportional to the average mass flux per unit area, so that in (30), ũ = ρu/ρ₀, and ρ₀ is the constant value of density that is used in the horizontal pressure gradient term in the horizontal momentum equations. Hence when evaluating the flux of h⁰ through a section of an ocean model, one should form the area integral of ρ₀C⁰_p times the product of the model's velocity and temperature, ũ Θ^ρ. This contrasts with the common practice of including an extra factor of in situ density in the area integral, which actually introduces a Boussinesq error into the calculation, since McDougall et al. (2002) show that these models are actually free of the Boussinesq approximation error in steady state if the model variables are interpreted according to (30).

7. C_p(S, θ, p_r)θ as heat content

In a recent paper, Bacon and Fofonoff (1996) advocated the use of C_p(S, θ, p_r) θ as heat content but here it is proven that this is actually less accurate than simply using C⁰_pθ. In arguing that h⁰ is an almost conservative oceanic “heat” variable, the present work approximates the first factor (T₀ + T)/(T₀ + θ) in (23) by unity and also ignores the square bracket in (23). This is equivalent to neglecting the two pressure integral terms in (25). When considering the first law of thermodynamics in the form (14), Bacon and Fofonoff (1996) also took (T₀ + T)/(T₀ + θ) to be unity, but they justified this choice by introducing a “surface equivalent heat flux” and claiming that the thermodynamic balance in (14) could be “brought to the surface” where p = 0. This justification is incorrect because the conservation laws must be obeyed by seawater at the pressure at which the physical processes, such as mixing, occur. While we agree that the approximation (T₀ + T)/(T₀ + θ) ≈ 1 in (14) is a very good approximation, and in advocating h⁰ and Θ we make an approximation of the same magnitude, we stress that this is indeed an approximation.

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)μ_T(p_r)]dS/dt in (14) could be ignored so that the material derivative of heat was taken to be C_p(p_r)dθ/dt. While it is true that the ignored term is much smaller than C_p(p_r)dθ/dt, it will be shown here that it is inconsistent to ignore this term if C_p(p_r) 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 C_p(S, θ, p_r) dθ/dt was the integral of C_p(S, θ, p_r)θ times the mass flux per unit area over the bounding area of the ocean volume. This oversight falsely assumes that d[C_p(S, θ, p_r)θ]/dt is the total derivative of “heat” rather than what they had arrived at, namely C_p(S, θ, p_r)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).

To examine the nonconservative production of C_p[S, θ, p_r]θ the material derivative

dh⁰dth⁰_θdθdth⁰_SdSdt(31)

is rewritten as

If one interprets C_p(S, θ, p_r)θ = h⁰_θθ as “heat content” when evaluating the meridional “heat flux,” then the right-hand side of (32) has been assumed to be zero. As explained above, Bacon and Fofonoff (1996) were aware that the last term in (32) was being neglected but, due to an oversight, were apparently not aware that they had also discarded the θdh⁰_θ/dt term.

The difference θ − Θ is equivalent to the difference between C⁰_pθ and h⁰, and using (31),

where the right-hand side terms make C⁰_pθ different to h⁰. We find in appendix B that the dominant nonlinearity in the function h⁰(S, θ) that causes θ to be nonconservative is the term in 2h⁰_Sθ and this has equal contributions from the variations of the two partial derivatives on the right-hand side of (33). That is, the variation of h⁰_S with θ is just as important as the variation of h⁰_θ = C_p(p_r) with S in causing the nonconservation of θ. Hence it is inconsistent to ignore the same term, −h⁰_SdS/dt, in (32) when examining how well C_p(S, θ, p_r)θ approximates h⁰.

We conclude that past attempts to justify C_p(p_r)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., C_p(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 C_p(p_r)θ and we show below that C_p(S, T, p)θ is less accurate than both C_p(p_r)θ and C⁰_pθ.

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, C_p(p_r)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 C⁰_p, also when S is close to 35 psu (see Fig. 2a). The production of θ on mixing any two fluid parcels can be deduced from this diagram. For example, the mixing of equal masses of the two parcels (S = 0, Θ = 0) and (S = 40, Θ = 40) means that the mixed fluid is at (S = 20, Θ = 20). We can read off Fig. 2a that θ − Θ is zero for one parent water mass and is about 0.22°C for the other, so the average θ of these two parcels is 20.11°C. However, at (S = 20, Θ = 20) Fig. 2a has θ − Θ about −0.44°C so the mixture actually has θ = 19.56°C which is cooler than the average θ by 0.55°C. In order to correct for this production of θ, one must abandon θ and adopt Θ, and Fig. 2a shows that the maximum difference between these temperatures is almost 2°C in the very fresh and warm region of the diagram near (S = 0, Θ = 40).

The error in taking C_p(S, θ, p_r)θ to be h⁰ is shown on the full S − Θ plane in Fig. 2b. This error is expressed in temperature units as [C_p(p_r)θ − h⁰]/C⁰_p = C_p(p_r)θ/C⁰_p − Θ. Whereas the maximum variation of θ − Θ is about −2°C (see Fig. 2a), the maximum variation in C_p(p_r)θ/C⁰_p − Θ is only 0.22°C. The maximum amount of nonconservative production on the S − Θ plane is about a factor of 5 less for C_p(p_r)θ/C⁰_p than for θ, being about −0.1°C compared with −0.55°C. However when one considers only data from the real ocean, which is mostly clustered near 35 psu, C_p(p_r)θ/C⁰_p is no better than θ as can be seen from Fig. 3. This is confirmed by comparing the root-mean-square value of θ − Θ for the whole of the global ocean atlas of Koltermann et al. (2003), namely 0.018°C, with the corresponding root-mean-square value of C_p(p_r)θ/C⁰_p − Θ, which is 0.019°C. Also, in the next section it will be shown that the use of C_p(p_r)θ as heat content to calculate the meridional heat flux is no more accurate than simply using the meridional flux of potential temperature with a fixed value of specific heat.

Having already compared the production of θ with that of C_p(S, θ, p_r)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/C⁰_p is the air–sea flux of Θ [see (29)]. This is exactly how today's ocean models relate the air–sea heat flux to the flux of the model's temperature variable, and since the model's temperature obeys a standard conservation equation, the most obvious interpretation of the model's temperature is as conservative temperature Θ. Given the values of Θ and S at each location in the model, it is possible to calculate the value of potential temperature at every point. The magnitude of the errors in existing ocean models is illustrated in Fig. 7 where the temperature difference, θ − Θ, is shown at the sea surface, calculated from the Koltermann et al. (2003) atlas. For the annually averaged data, values of θ − Θ as large as 0.09°C are seen in the North Atlantic while the −0.06°C contour is evident in the eastern equatorial Pacific. These patterns of θ − Θ represent the errors in today's ocean models due to the neglect of the nonconservative production of θ. Larger values of θ − Θ occur in the Mediterranean Sea (up to 0.2°C) and larger negative values occur where warm freshwater from rivers enter the ocean (values as low as −1.2°C; see Fig. 2a at S = 0, Θ = 25°C). These are the largest errors in the SST that are currently incurred by the neglect of the nonconservative production terms in the θ evolution equation when an ocean model is driven by specified air–sea fluxes. These errors reduce to no more than 1 mK when the model's temperature variable is interpreted as conservative temperature.

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 ρ₀C⁰_pυ̃(θ − Θ), where υ̃ is the model's northward velocity [see (30)]. This difference in heat flux is shown by the solid line in Figs. 9b and 9c for data from the model of Hirst et al. (2000) and has a maximum value of 0.0046 PW or about 0.4% of the maximum heat flux across any latitude circle. This change in meridional heat flux implies a corresponding difference in the air–sea heat flux (of about 0.2 W m⁻² in the vicinity of 20°S), which is expected to be very similar to the error in the air–sea heat flux in present models that are run with a prescribed SST pattern. This heat flux error is approximately 10% of the change in surface insolation expected under a doubling of greenhouse gases in the atmosphere. The full line with dots in Fig. 9b shows the error in the meridional heat flux if it is calculated using C_p(p_r)θ as heat content rather than the accurate heat content h⁰ = C⁰_pΘ. While the error in using C_p(p_r)θ as heat content has a different dependence on latitude, the typical error in the meridional heat flux is very similar to that using C⁰_pθ. The dashed line in Fig. 9b shows the error in the meridional heat flux when the in situ heat capacity is used to define the heat content as C_p(S, T, p)θ. This choice, dating back to Bryan (1962), has larger errors than when simply using a fixed heat capacity (compare with the solid line in Fig. 9b).

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 C_pθ as an approximation to the flux of internal energy, where C_p was evaluated at zero pressure and at the salinity of the fluid parcel as the average heat capacity between the temperatures zero and θ. In this way, C_pθ is actually equal to h⁰(S, θ) − h⁰(S, 0) (D. Jackett 2002, personal communication) and since h⁰(S, 0) varies by only 125 J kg⁻¹, equivalent to 0.031°C, [see Fig. 2a of this paper or Table A4 of Feistel and Hagen (1995)] over the full range of salinity, Warren's C_pθ is very nearly potential enthalpy. Figure 9c confirms that while the meridional flux of C_pθ is not a particularly accurate expression for the flux of internal energy, it is quite an accurate approximation for the flux of h⁰. Warren (1999) showed that the meridional flux of C_pθ was a very good approximation to the flux of the Bernoulli function; a result that is consistent with the next section of this paper where it is found that the Bernoulli function and potential enthalpy are the same up to about 0.003°C in temperature units, that is, B = h⁰ ± 0.003 C⁰_p.

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

Adding the first law of thermodynamics [(2)] to the conservation statements for kinetic energy, (1/2)u · u, and for the geopotential, Φ = gz, a conservation equation is found for the Bernoulli function, B ≡ h + Φ + (1/2)u · u, namely (see Batchelor 1967 or Gill 1982)

The last term here is negligible in the ocean interior, being many orders of magnitude smaller than even the tiny term ρε_M in (2). Hence apart from the unsteady pressure term, (34) is in the form of a clean conservation equation [like (4)]. If it were not for the p_t term the Bernoulli function would be the quantity whose conservation statement would resemble the first law of thermodynamics, with the right-hand side being (minus) the divergence of the molecular flux of heat, −∇·F_Q. There is a sense in which both (2) and (34) are conservation equations for total energy; the difference being that the kinetic energy equation has been used to reexpress the dissipation of mechanical energy, ρε_M, in (2) to obtain (34). In the same sense, one could call both (2) and (34) the first law of thermodynamics. However, we follow accepted practice in the literature and call (2) the first law of thermodynamics and (34) the conservation of total energy [see, e.g., sections 1–5 and 1–10 of Haltiner and Williams (1980)].

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.

The principal difficulty with using B as an oceanographic energy-like variable is not however due to the rather small dependence of B on pressure, but rather is due to it not being a locally determined quantity: in addition to B being a function of the locally measured properties S, T, and p, it also contains dynamical information in the geopotential function (as well as being dependent on the magnitude of the three-dimensional velocity vector). While both p and Φ are known when one is running a prognostic ocean model, Φ is not a locally observed quantity in ocean data. On using the hydrostatic equation to express Φ in terms of the height of the sea surface where the geopotential is Φ⁰, B can be written as [using (7) and ignoring the tiny kinetic energy]

In order to calculate B from observed data one needs to both (a) have knowledge of Φ⁰ and (b) perform a vertical pressure integral all the way to the sea surface. Hence B is not a locally determined quantity. The geopotential at the sea surface, Φ⁰, requires satellite altimeter data or the performance of an inverse model. The fact that the Bernoulli function cannot be determined from local thermodynamic properties means that it is unsuitable for use as a water-mass property.

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.

The first law of thermodynamics can be cast in terms of conservation equations for potential temperature θ and for conservative temperature Θ as [from (14) and (23), ignoring the dissipation of kinetic energy]

The ratio of the absolute temperatures, (T₀ + θ)/(T₀ + T), that multiplies the divergence of the molecular flux of heat in (37) varies from 1.0 by only 0.15% in the ocean so that the first term on the right of (37) is very close to being the divergence of the molecular heat flux (divided by the constant, C⁰_p). The corresponding term in (36) is divided by the heat capacity, C_p(p_r), which varies by 5% in the ocean and so it is much less accurate to regard this term as the divergence of the molecular flux of heat. Similar remarks can be made for the terms in these equations that multiply the divergence of the molecular flux of salt. The result is that it is more than a hundred times more accurate to regard the right-hand side of (37) as −∇·F_Q/C⁰_p than to do so in (36) and this is the root cause of the more conservative nature of Θ than of θ.

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 C⁰_pθ or C_p(p_r)θ as heat content (Fig. 9b). That is, the remaining error in the meridional heat flux from using h⁰ is estimated to be less than 5 × 10⁻⁵ PW.

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,” C⁰_p (=3989.244 952 928 15 J kg⁻¹ K⁻¹). Since ocean models (i) have their temperature obeying a standard conservation statement and (ii) have the heat capacity at the sea surface being constant, it is apparent that the temperature variable in these ocean models is actually conservative temperature' Θ rather than potential temperature θ. The error in interpreting Θ as the temperature variable in ocean models is likely to be no more than 1% of the error in θ, that is 1% of approximately ±0.1°C, namely, ±10⁻³°C. The typical temperature difference θ − Θ of ±0.1°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 1500 dbar. Since ocean models have been careful to deal with potential temperature rather than in situ temperature, it would also make sense to convert ocean models to Θ rather than θ.

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 C⁰_p (=3989.244 952 928 15 J kg⁻¹ K⁻¹). If the temperature in an ocean model were to really be θ then (i) additional nonconservative production terms would be needed in the temperature equation, and (ii), the heat capacity that is used at the sea surface to relate the air–sea heat flux to the surface flux of potential temperature would have to vary in space and time by up to 5% because heat capacity is a function of θ and S. Interestingly, the heat capacity for cool freshwater would need to be 5% larger than C⁰_p even though the temperature error θ − Θ is very small for such cool fresh seawater (Figs. 1 and 2a).

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.