VOLUME 33 JOURNAL OF PHYSICAL OCEANOGRAPHY MAY 2003

Potential Enthalpy: A Conservative Oceanic Variable for Evaluating Heat Content and Heat Fluxes

TREVOR J. MCDOUGALL* Antarctic CRC, University of Tasmania, Hobart, Tasmania, Australia

(Manuscript received 21 December 2001, in ®nal form 7 October 2002)

ABSTRACT Potential temperature is used in oceanography as though it is a conservative variable like salinity; however, turbulent mixing processes conserve enthalpy and usually destroy potential temperature. This negative production of potential temperature is similar in magnitude to the well-known production of entropy that always occurs during mixing processes. Here it is shown that potential enthalpyÐthe enthalpy that a water parcel would have if raised adiabatically and without exchange of salt to the sea surfaceÐis more conservative than potential temperature by two orders of magnitude. Furthermore, it is shown that a ¯ux of potential enthalpy can be called ``the heat ¯ux'' even though potential enthalpy is unde®ned up to a linear function of salinity. The exchange of heat across the sea surface is identically the ¯ux of potential enthalpy. This same ¯ux is not proportional to the ¯ux of potential temperature because of variations in heat capacity of up to 5%. The geothermal heat ¯ux across the ocean ¯oor is also approximately the ¯ux of potential enthalpy with an error of no more that 0.15%. These results prove that potential enthalpy is the quantity whose advection and diffusion is equivalent to advection and diffusion of ``heat'' in the ocean. That is, it is proven that to very high accuracy, the ®rst law of thermo- dynamics in the ocean is the conservation equation of potential enthalpy. It is shown that potential enthalpy is to be preferred over the Bernoulli function. A new temperature variable called ``conservative temperature'' is advanced that is simply proportional to potential enthalpy. It is shown that present ocean models contain typical errors of 0.1ЊC and maximum errors of 1.4ЊC in their temperature because of the neglect of the nonconservative production of potential temperature. The meridional ¯ux of heat through oceanic sections found using this conservative approach is different by up to 0.4% from that calculated by the approach used in present ocean models in which the nonconservative nature of potential temperature is ignored and the speci®c heat at the sea surface is assumed to be constant. An alternative approach that has been recommended and is often used with observed section data, namely, calculating the meridional heat ¯ux using the speci®c heat (at zero pressure) and potential temperature, rests on an incorrect theoretical foundation, and this estimate of heat ¯ux is actually less accurate than simply using the ¯ux of potential temperature with a constant heat capacity.

1. Introduction the variation of speci®c heat with salinity and (ii) the dependence of the total differential of enthalpy on var- The quest in this work is to derive a variable that is iations of salinity. conservative, independent of adiabatic changes in pres- Fofonoff (1962) pointed out that when ¯uid parcels sure, and whose conservation equation is the oceanic mix at constant pressure, the thermodynamic variable version of the ®rst law of thermodynamics. That is, we that is conserved is enthalpy, and he showed this implied seek a variable whose advection and diffusion can be that potential temperature is not a conservative variable. interpreted as the advection and diffusion of ``heat.'' In It is natural then to consider enthalpy as a candidate other words, we seek to answer the question, ``what is conservative variable for embodying the meaning of the heat'' in the ocean? The variable that is currently used ®rst law of thermodynamics. However, this attempt is for this purpose in ocean models is potential temperature thwarted by the strong dependence of enthalpy on pres- referenced to the sea surface, , but it does not accu- ␪ sure. For example, an increase in pressure of 107 Pa rately represent the conservation of heat because of (i) (1000 dbar), without exchange of heat or salt, causes a change in enthalpy that is equivalent to about 2.5ЊC. * Additional af®liation: CSIRO Marine Research, Hobart, Tas- We show in this paper that in contrast to enthalpy, po- mania, Australia. tential enthalpy does have the desired properties to em- body the meaning of the ®rst law. Present treatment of oceanic heat ¯uxes is clearly Corresponding author address: Dr. Trevor J. McDougall, Antarctic CRC, CSIRO Division of Marine Research, GPO Box 1538, Hobart, inconsistent. Ocean models treat potential temperature Tasmania 7001, Australia. as a conservative variable and calculate the heat ¯ux E-mail: [email protected] across oceanic sections using a constant value of heat

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Unauthenticated | Downloaded 10/01/21 07:16 AM UTC 946 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33 capacity. By contrast, heat ¯ux through sections of ob- is the ¯ux of heat by all manner of molecular ¯uxes served data is often calculated using a variable speci®c and by radiation, and ␳␧M is the rate of dissipation of heat multiplying the ¯ux of potential temperature per kinetic energy (W mϪ3) into thermal energy. As ex- unit area (Bryan 1962; Macdonald et al. 1994; Saunders plained by Landau and Lifshitz [1959; see their Eqs. 1995; Bacon and Fofonoff 1996). Here it is shown that (57.6) and (58.12)], Fofonoff (1962), and Davis (1994), the theoretical justi®cation of this second approach is FQ includes the cross-diffusion of heat by the gradient ¯awed on three counts. While the errors involved are of salinity (the Dufour effect) as well as the heat of small, it is clearly less than satisfactory to have con- transfer due to the ¯ux of salt. A ``reduced heat ¯ux,''

¯icting practices in the observational and modeling parts Fq, can be de®ned that does not include the heat of of physical oceanography, particularly as an accurate transfer, so that and convenient solution can be found. F ϭ F ϩ h F ϭ F ϩ [␮ Ϫ (T ϩ T)␮ ]F , (3) Warren (1999) has claimed that because internal en- QqSSq 0 TS ergy is unknown up to a linear function of salinity, it where hS ϭ ␮ Ϫ (T 0 ϩ T)␮T is the partial derivative is inappropriate to talk of a ¯ux of heat across an ocean of speci®c enthalpy with respect to salinity at constant section unless there are zero ¯uxes of mass and of salt in situ temperature and pressure, FS is the ¯ux of salt, across the section. Here it is shown that this pessimism T 0 ϭ 273.15 K is the temperature offset between kelvins is unfounded; it is perfectly valid to talk of potential and degrees Celsius (see Feistel and Hagen 1995), T is enthalpy, h 0, as the ``heat content'' and to regard the in degrees Celsius, ␮ is the relative chemical potential ¯ux of h 0 as the ``heat ¯ux.'' Moreover, h 0 is shown to of salt in seawater (i.e., ␮ is the difference between the be more conserved than is ␪ by more than two orders partial chemical potential of salt ␮S and the partial chem- 0 of magnitude. This paper proves that the ¯uxes of h ical potential of water ␮W in seawater), and ␮T is its across oceanic sections can be accurately compared with derivative with respect to in situ temperature and both the air±sea heat ¯ux, irrespective of whether the ¯uxes ␮ and ␮T are evaluated at (S, T, p). of mass and of salt are zero across these ocean sections. In words, the ®rst law of thermodynamics [(1)] says This has implications for best oceanographic practice that the internal energy of a ¯uid parcel can change due for the analysis of ocean observations and for the in- to (i) the work done when the parcel's volume is changed terpretation of ``temperature'' in models. at pressure (p 0 ϩ p), (ii) the divergence of the ¯ux of The ®rst law of thermodynamics is compared with heat, and (iii) the dissipation of turbulent kinetic energy the equation for the conservation of total energy (the into heat. The effect of the dissipation of kinetic energy Bernoulli equation). It is shown that while the Bernoulli in these equations is very small and is always ignored. Ϫ9 function and potential enthalpy differ by only about 3 For example, a typical dissipation rate, ␧M,of10 W ϫ 10Ϫ3ЊC (when expressed in temperature units), the kgϪ1 causes a warming of only 10Ϫ3 K (100 yr)Ϫ1. An- Bernoulli function cannot be considered a water-mass other way of quantifying the unimportance of this term property as it varies with the adiabatic vertical heaving is to compare it to the magnitude of diapycnal mixing. Ϫ2 of wave motions. A larger drawback of the Bernoulli The turbulent diapycnal diffusivity scales as 0.2 ␧MN function is that it cannot be determined from the local (Osborn 1980) and the diapycnal mixing of potential thermodynamic coordinates S, T, p. For these reasons temperature that this diffusivity causes is typically more the Bernoulli function is not an attractive variable com- than one thousand times larger than that caused by the 0 pared with h . dissipation of kinetic energy ␧M/Cp. The other term on the right-hand sides of these in- stantaneous conservation equations (1) and (2) is the 2. The ®rst law of thermodynamics FQ. When´divergence of a molecular heat ¯ux, Ϫ١ From Batchelor (1967), Kamenkovich (1977), Gill these conservation equations are averaged over all man- (1982), Gregg (1984), and Davis (1994), the ®rst law ner of turbulent motions, this term will also be quite of thermodynamics may be written as negligible compared with the turbulent heat ¯uxes ex- cept at the ocean's boundaries; the air±sea heat ¯ux d␧ 1 d␳ occurs as the average of F at the sea surface and the F ϩ ␳␧ , or (1) Q ´ ␳ Ϫ (p ϩ p) ϭϪ١ []dt 0 ␳2 dt QM geothermal heat ¯ux that the ocean receives from the solid earth also appears in the conservation equations

dh 1 dp through the average of FQ at the sea¯oor. We note in FQMϩ ␳␧ , (2) passing that at both the sea surface and at the ocean ´ ␳ ϪϭϪ١ dt ␳ dt ΂΃ ¯oor the ¯ux of salt is zero and so the heat of transfer where ␧ is the internal energy, h is the speci®c enthalpy, due to the ¯ux of salt is also zero and so from (3) FQ de®ned by h ϵ␧ϩ(p 0 ϩ p)/␳, ␳ is in situ density, p is equal to the reduced heat ¯ux Fq. Note also that in is the excess of the real pressure over the ®xed atmo- hot smokers, the ¯ux of salt (and heat) is advective in spheric reference pressure, p 0 ϭ 0.101 325 MPa (Feistel nature and so will be captured by the advection terms .(is the material on the left-hand side of (1) and (2 ١ ´ t ϩ uץ/ץand Hagen 1995), d/dt ϵ derivative following the instantaneous ¯uid velocity, FQ Because the right-hand sides of (1) and (2) are in the

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FQ, the parcel adiabatically (and without exchange of salt) to´form of the divergence of a ¯ux, namely Ϫ١ key to ®nding a new variable whose conservation rep- the sea surface pressure, its enthalpy is evaluated there resents the ®rst law of thermodynamics is to ®nd one and called potential enthalpy. During the adiabatic pres- such that the left-hand side of (1) or (2) is ␳ times the sure excursion the potential enthalpy of ¯uid parcels are material derivative of that variable, for if that were pos- unchanged and one wonders how much damage is done sible, the ®rst law of thermodynamics could be written by forcing the ¯uid parcels to migrate to zero pressure in the standard conservation form, being the same form before allowing them to mix rather than simply mixing as the salt conservation equation, in situ as they do in practice. This thinking was the dS motivation for examining potential enthalpy as a can- .F , (4) didate heat content ´ ␳uS) ϭϪ١)´ ١ ␳ ϭ (␳S)tSϩ dt The second law of thermodynamics de®nes the spe- where FS is the ¯ux of salt by all manner of molecular ci®c entropy, ␴, whose total derivative obeys the Gibbs processes. relation Physicists sometime caution against using ``heat'' as dh Ϫ (1/␳)dp ϭ (T ϩ T)d␴ ϩ ␮dS. (5) a noun because the ®rst law of thermodynamics is con- 0 cerned with changes in internal energy that are related This relation is sometimes called the fundamental ther- to not only heat ¯uxes but also to the doing of work. modynamic relation and it can also be regarded as the At the beginning of their book, Bohren and Albrecht mathematical de®nition of entropy. Consider the move- (1998) devote several pages discussing some examples ment, without exchange of heat or salt, of a ¯uid parcel in which the word ``heat'' is used imprecisely. Later in from its in situ pressure p to a ®xed reference pressure their book (section 3) they point out that the word ``en- pr. Neither salinity nor entropy change during this mo- thalpy'' can often be accurately used in place of ``heat tion, so that it is apparent from (5) that (p | ϭ 1/␳, (6ץ/hץ content per unit mass.'' In the present paper it will be shown that with negligible error, a new oceanic heat- S,␴ like variable called ``potential enthalpy,'' obeys a clean so that the enthalpy at the reference pressure, which we 0 conservation equation of the form (4) with the right- call potential enthalpy, h (S, ␪,pr), is related to in situ hand side being (minus) the divergence of the molecular enthalpy, h, by ¯ux of heat. That is, it will be shown that the left-hand p side of (2) is equal to ␳ times the material derivative of h0(S, ␪, p ) ϭ h(S, ␪, p) Ϫ 1/␳(S, ␪, pЈ) dpЈ. (7) r ͵ potential enthalpy plus a negligible error term. This pr means that the conservation equation of potential en- Here we have chosen to regard enthalpy and in situ thalpy in the ocean is equivalent to the ®rst law of density ␳ as functions of potential temperature ␪ rather thermodynamics. Given this, calling potential enthalpy than of in situ temperature. Note that for a ®xed ref- ``heat content'' can cause no harm or imprecise thinking erence pressure, h 0 is a function of only S and ␪. in oceanography. Potential enthalpy and ``heat content'' The total derivative of (7) is taken, ®nding that are effectively alternative names for the same thing be- cause potential enthalpy is the variable whose advection dh 1 dp dh0 d␪␣p Ä (S, ␪, pЈ) Ϫϭϩ dpЈ and diffusion throughout the ocean can be accurately dt ␳ dt dt dt ͵ ␳(S, ␪, pЈ) compared with the boundary ¯uxes of heat. Just as the pr advection and diffusion of a passive conservative tracer dS p ␤˜ (S, ␪, pЈ) Ϫ dpЈ, (8) in the ocean can be accurately compared with the bound- dt ͵ ␳(S, ␪, pЈ) ary ¯uxes of the passive tracer, this same association of pr Ϫ1 Ä Ϫ1 S | ␪,p. Theץ/␳ץ ␪ | S,p and ␤ ϭ ␳ץ/␳ץ the boundary ¯uxes and the tracer content justi®es the where ␣Ä ϭϪ␳ association of the word ``heat content'' with the new typical value of the left-hand side of this equation is variable, potential enthalpy. Cpd␪/dt and a typical value for the last two terms is [(␣Ä p Ϫ pr)/␳]d␪/dt. The ratio of the last two terms to the dominant term in (8) is then approximately␣Ä (p Ϫ 3. Potential enthalpy pr)/␳Cp, and for a pressure difference of 4000 dbar (4 It follows from the form of (2) that when mixing ϫ 107 Pa) this ratio is typically 0.0015, implying that occurs at constant pressure, enthalpy is conserved [this the right-hand side of (8) is almost the material deriv- is more obvious when (2) is written in divergence form ative of potential enthalpy. Were it not for these two using the continuity equation]. As an example, mixing small terms in (8), potential enthalpy would be the con- between ¯uid parcels at the sea surface where the pres- servative ``heatlike'' variable that we seek whose con- sure is constant (p ϭ 0 and the total pressure is p 0) servation equation would be exactly the ®rst law of conserves the enthalpy evaluated at that (zero) pressure. thermodynamics and (2) would become ␳dh0/dt ϭ

.FQ ϩ ␳␧M´Just as the concept of potential temperature is well es- Ϫ١ tablished in oceanography, consider now the ``poten- The rest of this paper will quantify the error made tial'' concept applied to enthalpy. After bringing a ¯uid by ignoring the last two terms in (8) and treating po-

Unauthenticated | Downloaded 10/01/21 07:16 AM UTC 948 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33 tential enthalpy as a conservative variable. It will be This equation was derived by Bacon and Fofonoff proven that the error in so doing is negligible, being no (1996) by a slightly different route. We now derive the larger than the neglect of the dissipation of kinetic en- corresponding result in terms of conservative temper- ergy into heat. It will be deduced that the temperature ature. 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 5. The ®rst law in terms of ⌰ treat potential temperature as a conservative variable. It is convenient to de®ne a temperature-like variable, ⌰, which is proportional to potential enthalpy, as

4. The ®rst law in terms of ␪ 00 ⌰ϵh /C p, Regarding enthalpy as a function of potential tem- where perature, that is as h(S, ␪,p), the ®rst law of thermo- 0 dynamics [(2)], takes the form C p ϵ h(S ϭ 35, ␪ ϭ 25, 0)/25, (15)

hd hdS and we call ⌰ the ``conservative temperature.'' This Ϫ1 Ϫ1 0 ץ ␪ ץ FQMϩ ␳␧ . (9) value ofC p (ϭ3989.244 952 928 15 J kg K ) de®ned ´ ␳ ϩϭϪ١ Sdt␪,p in (15) [using the algorithms of Feistel and Hagenץ ␪ΗΗS,p dtץ΂΃ (1995)] was chosen so as to minimize the difference From the Gibbs relation [(5)], we have 0 0 between C p␪ and potential enthalpy h when averaged .␴ over all the data at the sea surface of today's oceanץ hץ hץ (T T)or (T T) , (10) ϭ 00ϩ ϭ ϩ That is, with this constant value of heat capacity, the ␪ Sץ␪ S,pץ␴ΗΗΗS,pץ average value of ␪ Ϫ⌰at the sea surface in today's 0 where the second expression has used the fact that at ocean is almost zero. Also,C p is very close to being constant S and p, both h and ␴ can be regarded simply the spatially averaged value of heat capacity at the sea as functions of ␪. The second part of (10) can be eval- surface of today's ocean. Algorithms for potential en- uated not only at p but also at the reference pressure thalpy and conservative temperature in terms of salinity where the left-hand side is the heat capacity at that and potential temperature are given in appendix A. pressure, Cp (pr) [which is shorthand for Cp(S, ␪,pr)], Regarding enthalpy now as a function of conservative so that temperature, that is as h(S, ⌰, p), the ®rst law of ther- modynamics takes the form (h (T0 ϩ Tץ ϭ Cpr(p ). (11) hdSץ⌰hdץ (␪ΗS,p (T0 ϩ ␪ץ (F ϩ ␳␧ . (16 ´ ␳ ϩϭϪ١ dt Sdt QM p,⌰ ץ ΗΗS,p⌰ץAgain from the Gibbs relation, we have ΂΃ h From the Gibbs relation, we ®nd thatץ ϭ ␮, (12) ␴ץ hץ SΗ␴,pץ ϭ (T ϩ T) (17) , ⌰ץ 0⌰ץ and regarding h to be the functional form h[S, ␴(S, ␪), ΗΗS,p S p] leads to and (17) can be evaluated not only at p but also at the ␴ reference pressure where the left-hand side isC 0 , so thatץ hץ hץ hץ ϭϩ p (h (T ϩ Tץ S ␪ץ ␴ S,pץS ␴,pץ SΗΗΗΗ␪,pץ 0 0 ϭ C p. (18) (ΗS,p (T0 ϩ ␪⌰ץ (ϭ ␮ Ϫ(T ϩ T)␮ (p ). (13 0 Tr The last part of this equation has used the identity (ob- Regarding h to be the functional form h[S, ␴(S, ⌰), p] tained for example from the de®nition of the Gibbs func- leads to (T | . Notice that in (13ץ/␮ץS | ϭϪץ/␴ץ tion) that ␴ץ hץ hץ hץ T,p S,p ␮ and (T 0 ϩ T) are evaluated in situ while ␮T(pr)is ϭϩ ⌰ Sץ ␴ S,pץS ␴,pץ SΗΗΗΗ⌰,pץ .evaluated at the reference pressure Substituting (11) and (13) into (9) gives the ®rst law ␴ץ of thermodynamics expressed in terms of changes of ϭ ␮ ϩ (T ϩ T) . (19) potential temperature and salinity as 0 ⌰SΗץ

(T0 ϩ T) d␪ dS From the Gibbs relation, we have ␳ Cpr(p ) ϩ [␮(p) Ϫ (T0 ϩ T)␮Tr(p )] ␴␮ץ Ά·(T0 ϩ ␪) dt dt ϭϪ , (20) (SΗh,p (T0 ϩ Tץ (FQMϩ ␳␧ . (14 ´ ϭϪ١

Unauthenticated | Downloaded 10/01/21 07:16 AM UTC MAY 2003 MCDOUGALL 949 and when this is evaluated at the reference pressure, it becomes ( ␴␮(pץ ϭϪ r , (21) (SΗ⌰ (T0 ϩ ␪ץ so that (19) becomes

(h (T0 ϩ Tץ ϭ ␮ Ϫ ␮(pr). (22) (SΗ⌰,p (T0 ϩ ␪ץ Substituting (18) and (22) into (16) gives the ®rst law of thermodynamics expressed in terms of changes of conservative temperature and salinity as

(T ϩ T) d⌰ (T ϩ T) dS 000 ␳ C prϩ ␮(p) Ϫ ␮(p ) Ά·(T00ϩ ␪) dt [](T ϩ ␪) dt FIG. 1. Heat capacity (at the sea surface) minus the constant value (FQMϩ ␳␧ . (23 ´ ϭϪ١ 0 Ϫ1 Ϫ1 C p contoured on the S±⌰ diagram (J kg ЊC ). Heat capacity is 0 At the sea surface T ϭ ␪ and (23) reduces to ␳C pd⌰/ de®ned here with respect to potential temperature so that it is Cp(S, 0 -␪ | S. If heat capacity is de®ned with respect to conserץ/ hץF ϩ ␳␧ . Regarding enthalpy and density ␪, 0) ϵ´dt ϭϪ١ Q M 0 S, then it is exactly the constant value | ⌰ץ/ hץ vative temperature as to be functions of ⌰, potential enthalpy is given by 0 C p. p 00 h ϭ C p⌰ϭh(S, ⌰, p) Ϫ 1/␳(S, ⌰, pЈ) dpЈ (24) ͵ h␪ | S,p. For example, even at a pressure as large as 4 ϫ 0 7 0 10 Pa (4000 dbar), h | S,p is at most 1.0015C p , while (where here and henceforth the reference pressure is ⌰ h | S,p varies by more than 5% [see (11) and Fig. 1]. taken to be zero) and the ®rst law of thermodynamics ␪ Moreover, h␪ | S,p suffers this 5% variation in the upper can be written as ocean where the spatial contrasts of ␪ are much larger than at depth so that the variation in h | , can do more d⌰ d⌰ p ␣(S, ⌰, pЈ) ␪ S,p 0 damage than the variation in h | , which occurs only ␳ Cp ϩ dpЈ ⌰ S,p dt dt ͵ ␳(S, ⌰, pЈ) at depth where the temperature gradients are small. [ 0 This paper argues that (23) or (25) can be approxi- dS p ␤(S, ⌰, pЈ) mated as (FQMϩ ␳␧ , (25 ´ Ϫ dpЈϭϪ١ dt ͵ ␳(S, ⌰, pЈ) 0000 (⌰C ␳u)´ ١ ␳uh ) ϭ (C ␳⌰) ϩ)´ ١ ␳h )tptpϩ) [ 0 where Ϫ1 / | and Ϫ1 / S | are the (F ϩ ␳␧ . (28 ´ p ϭϪ١,⌰ ץ ␳ץ S,p ␤ ϭ ␳ ⌰ץ ␳ץ ϭϪ␳ ␣ thermal expansion and haline contraction coef®cients QM de®ned with respect to conservative temperature. The Taking a mean dianeutral advection velocity of 10Ϫ7 m Ϫ1 Ϫ3 Ϫ1 coef®cients of d⌰/dt and of dS/dt in (23) and (25) can s and ⌰z of 2 ϫ 10 Km in the deep ocean, a be equated to ®nd the following two exact relations for typical value of d⌰/dt is 2 ϫ 10Ϫ10 KsϪ1 and the terms T in terms of ␪, similar to the traditional relationship that have been neglected in going from (25) to (28) are for ␪ as T plus the pressure integral of the lapse rate, smaller than this by three orders of magnitude. These p neglected terms amount to no more that the dissipation (T Ϫ ␪) ␣(S, ⌰, pЈ) Ϫ9 Ϫ1 0 of kinetic energy in (28), assuming ␧M ϭ 10 Wkg . C p ϭ dpЈ, (26) (T0 ϩ ␪) ͵ ␳(S, ⌰, pЈ) The air±sea ¯ux of heat appears in (28) as F and 0 Q and since this ¯ux occurs at zero pressure, there is no error at all in equating the air±sea ¯ux with the ¯ux of po- p tential enthalpy [because the last two terms on the left- (T0 ϩ T) ␤(S, ⌰, pЈ) ␮(p) Ϫ ␮(pr) ϭϪ dpЈ. (27) hand side of (25) are zero at the sea surface]. The geo- (T0 ϩ ␪) ͵ ␳(S, ⌰, pЈ) []0 thermal heat ¯ux occurs at great depth and the local increase in ⌰ caused by the divergence of the geother- mal heat ¯ux should be evaluated using the speci®c heat 6. Potential enthalpy as heat content 7 h⌰ | S,p which, at a pressure of 4 ϫ 10 Pa (4000 dbar), 00 The key ®nding in this paper amounts to proving that is about 1.0015CCpp , which can be taken to be with in comparing (16) to (9), hS | ⌰,p K hS | ␪,p and that the high accuracy. heat capacity de®ned with respect to ⌰, namely h⌰ | S,p, This association of the air±sea and geothermal heat 0 0 varies much less from the constant valueC p than does ¯uxes with the ¯ux of h is particularly clear since there the heat capacity de®ned with respect to ␪, namely, is no ¯ux of salt across either the sea surface or the

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sea¯oor, so that from (3) the total boundary heat ¯ux For these reasons it is clear that h 0 and ⌰ are the

FQ is the same as the reduced heat ¯ux Fq through the oceanic thermodynamic quantities whose conservation boundaries. This is convenient since hS ϭ ␮ Ϫ (T 0 ϩ represents the ®rst law of thermodynamics. Further- 0 T)␮T is only known up to a constant, re¯ecting the fact more, it is legitimate to call h ``the heat content per that enthalpy itself is unknown up to a linear function unit mass'' and to call the ¯ux of h 0 ``the heat ¯ux,'' of salinity. bearing in mind that this nomenclature assumes the par- We come now to the question of whether it is possible ticular linear function of S that Feistel and Hagen (1995) to regard h 0 as heat content and the ¯ux of h 0 as heat adopted, just as the corresponding ¯ux of potential tem- ¯ux. Warren (1999) argued that because enthalpy is un- perature is dependent on the temperature scale on which known up to a linear function of salinity, it is only the potential temperature is measured. possible to talk of a ¯ux of heat through an ocean section Thus far I have considered only instantaneous con- if the ¯uxes of both mass and salt through that section servation equations. Here the issue of averaging is ad- are zero. Technically this is true, but only in the same dressed. First, (28) is written as the instantaneous con- narrow sense that it is not possible to talk of the ¯ux servation equation for ⌰: of ␪ through an ocean section because there is always (F , (29´␳u⌰) ϭϪ١)´١ ␳⌰) ϩ) the question of adding or subtracting a constant offset t ⌰ 0 to the temperature scale. Once we de®ne what scale where F⌰ ϵ FQ/C p is the molecular and boundary ¯ux (kelvins or degrees Celsius) is being used to measure of ⌰, and the dissipation of kinetic energy term ␳␧M has ␪, the issue is resolved and one can legitimately talk of been dropped. McDougall et al. (2002) have argued that a ¯ux of ␪ even though the mass ¯ux may be nonzero. the most sensible way of averaging (29) in z coordinates A similar argument can now be applied to potential results in the form enthalpy. In de®ning the Gibbs function of sea water, ␳␳ ( ⌰ uÄ)´ ١ ␳⌰ /␳ ) ϩ) Feistel and Hagen (1995) made arbitrary choices for 0 t

four constants, and two of these choices amount to mak- 1 ␳ (30) ,( ⌰K١)´ ١ F⌰ ϩ ´ ١ ing a speci®c choice for the unknowable linear function ϭϪ ␳ of salinity in the de®nition of h 0. The key thing to realize 0 is that for any arbitrary choice of this linear function where the last term on the right is the turbulent ¯ux ␳ of salinity, the conservation equation, (28), of h 0 is un- term and⌰ is the density-weighted average value of changed, and also, such arbitrary choices do not affect ⌰. The key point in that paper is that the ``velocity'' the air±sea and geothermal heat ¯uxes. Hence h 0 is the variable that is carried by ocean models is actually pro- correct property with which to track the advection and portional to the average mass ¯ux per unit area, so that

diffusion of heat in the ocean, irrespective of the ar- in (30), uÄ ϭ ␳u/␳ 0, and ␳ 0 is the constant value of bitrary function of salinity that is contained in the def- density that is used in the horizontal pressure gradient inition of h 0. term in the horizontal momentum equations. Hence For example, the difference between the meridional when evaluating the ¯ux of h 0 through a section of an ¯ux of h 0 across two latitudes is equal to the area-in- ocean model, one should form the area integral of tegrated air±sea and geothermal heat ¯uxes between ␳ C 0 times the product of the model's velocity and tem- 0 p ␳ those latitudes (after also allowing for any unsteady ac- perature,uÄ ⌰ . This contrasts with the common practice cumulation of h 0 in the volume), irrespective of whether of including an extra factor of in situ density in the area there are nonzero ¯uxes of mass or of salt across either integral, which actually introduces a Boussinesq error or both meridional sections. This powerful result fol- into the calculation, since McDougall et al. (2002) show lows directly from the fact that h 0 obeys a standard that these models are actually free of the Boussinesq conservation equation, (28), no matter what linear func- approximation error in steady state if the model vari- tion of salinity is chosen in the de®nition of h 0.Asa ables are interpreted according to (30). consequence, it does make perfect sense to talk of the meridional ¯ux of heat (i.e., the ¯ux of h 0) in the Indian 7. C (S, ␪,p)␪ as heat content and South Paci®c Oceans separately, just as it makes p r sense to discuss the meridional ¯uxes of mass, fresh- In a recent paper, Bacon and Fofonoff (1996) advo-

water, tritium, salt, and salinity anomaly (S Ϫ 35) cated the use of Cp(S, ␪,pr) ␪ as heat content but here through these individual ocean sections. Just as it is it is proven that this is actually less accurate than simply 0 0 valid and oftentimes advantageous to carry equations in using C p␪. In arguing that h is an almost conservative inverse models for salinity anomaly rather than the full oceanic ``heat'' variable, the present work approximates

salinity (McDougall 1991; Sloyan and Rintoul 2000; the ®rst factor (T 0 ϩ T)/(T 0 ϩ ␪) in (23) by unity and Ganachaud and Wunsch 2000), so it is valid to use equa- also ignores the square bracket in (23). This is equiv- tions for the anomaly of conservative temperature, (⌰ alent to neglecting the two pressure integral terms in

Ϫ⌰0). Doing so often has the effect of decreasing the (25). When considering the ®rst law of thermodynamics in¯uence of a relatively uncertain velocity ®eld on the in the form (14), Bacon and Fofonoff (1996) also took

heat budget. (T 0 ϩ T)/(T 0 ϩ ␪) to be unity, but they justi®ed this

Unauthenticated | Downloaded 10/01/21 07:16 AM UTC MAY 2003 MCDOUGALL 951 choice by introducing a ``surface equivalent heat ¯ux'' heat content have been ¯awed on theoretical grounds, and claiming that the thermodynamic balance in (14) and since we show below that this approach is no more could be ``brought to the surface'' where p ϭ 0. This accurate than simply using a constant heat capacity, it justi®cation is incorrect because the conservation laws should be abandoned. Prior to the Bacon and Fofonoff must be obeyed by seawater at the pressure at which (1996) paper various authors had used the in situ value the physical processes, such as mixing, occur. While we of heat capacity together with potential temperature [i.e., agree that the approximation (T 0 ϩ T)/(T 0 ϩ ␪) ഠ 1in Cp(S, T, p)␪] as heat content (Bryan 1962; Macdonald (14) is a very good approximation, and in advocating et al. 1994; Saunders 1995) but there is even less the- 0 h and ⌰ we make an approximation of the same mag- oretical justi®cation for this choice than for Cp(pr)␪ and nitude, we stress that this is indeed an approximation. we show below that Cp(S, T, p)␪ is less accurate than 0 Another step that Bacon and Fofonoff (1996) took in both Cp(pr)␪ and C p␪. their treatment of the ®rst law of thermodynamics was The production of ␪ and ⌰ on mixing between ¯uid to assume that [␮(p) Ϫ (T 0 ϩ T)␮T(pr)]dS/dt in (14) parcels is considered in appendix B and appendix C. could be ignored so that the material derivative of heat Figure C1 illustrates the result of mixing ¯uid parcels was taken to be Cp(pr)d␪/dt. While it is true that the with extreme property contrasts that are widely sepa- ignored term is much smaller than Cp(pr)d␪/dt, it will rated in space (at a series of ®xed pressures) and one be shown here that it is inconsistent to ignore this term wonders about the relevance of this procedure to the if Cp(pr) is allowed to vary. The third error in Bacon real ocean. The importance of these mixing arguments and Fofonoff (1996) was to state [their Eq. (8)] that the depends on the heat ¯ux that travels by these paths, so volume integral of the advective part of Cp(S, ␪,pr) d␪/ that, for example, if most of the oceanic heat transport dt was the integral of Cp(S, ␪,pr)␪ times the mass ¯ux entered the ocean where the ocean is very warm and per unit area over the bounding area of the ocean vol- salty and exited the ocean where it was very cool and ume. This oversight falsely assumes that d[Cp(S, ␪, very fresh, then the production of potential temperature pr)␪]/dt is the total derivative of ``heat'' rather than what of Ϫ0.4ЊC would be a realistic estimate for the bulk of they had arrived at, namely Cp(S, ␪,pr)d␪/dt. One can- the ocean. [In a similar manner, the total amount of not move the heat capacity inside the derivative when cabbeling (McDougall 1987) that occurs along a neutral the heat capacity is allowed to vary as in Bacon and density surface depends on the ¯ux of heat being trans- Fofonoff (1996). ported down the temperature contrast on that surface

To examine the nonconservative production of Cp[S, even though the individual mixing events occur between ␪,pr]␪ the material derivative parcels with very small ␪ and S contrasts.] Because mixing involves both epineutral and dianeutral mixing, dh00/dt ϭ hd␪/dt ϩ hdS 0/dt (31) ␪ S and because the heat ¯ux achieved by the various mixing is rewritten as paths is rather complex, the mixing arguments that lie behind the plots in Fig. C1 do not obviously provide a d[C (S, ␪, p )␪]/dt Ϫ dh000/dt ϭ ␪dh /dt Ϫ hdS/dt. (32) pr ␪ S realistic estimate of the importance of the nonconser- 0 If one interprets Cp(S, ␪,pr)␪ ϭ h␪␪ as ``heat content'' vative production of ␪ or of ⌰ in the ocean. The im- when evaluating the meridional ``heat ¯ux,'' then the portant message that is gleaned from Fig. C1 is that the right-hand side of (32) has been assumed to be zero. nonconservative production of ⌰ is at least one hundred As explained above, Bacon and Fofonoff (1996) were times smaller that the production of ␪. A realistic as- aware that the last term in (32) was being neglected but, sessment of the errors inherent in present oceanic prac- due to an oversight, were apparently not aware that they tice can then be found by examining the temperature 0 had also discarded the ␪dh␪ /dt term. difference ␪ Ϫ⌰as described later in this paper, and The difference ␪ Ϫ⌰is equivalent to the difference the error remaining in the use of ⌰ is taken to be less 0 0 between C p␪ and h , and using (31), than 1% of the temperature difference, ␪ Ϫ⌰. Appendix D considers internal energy and potential internal energy d(C 00000␪)/dt Ϫ dh /dt ϭϪ(h Ϫ C )d␪/dt Ϫ hdS/dt, (33) p ␪ pS as candidates for ``heat content'' but it is shown that 0 where the right-hand side terms make C p␪ different to they are not as suitable as potential enthalpy. h 0. We ®nd in appendix B that the dominant nonlinearity in the function h 0(S, ␪) that causes ␪ to be nonconser- 0 8. Quantifying the errors in ␪,Cp(pr)␪, and ⌰ vative is the term in 2hS ␪ and this has equal contributions from the variations of the two partial derivatives on the The nonconservative nature of potential temperature 0 right-hand side of (33). That is, the variation ofhS with can be illustrated on a variant of the usual S Ϫ ␪ dia- 0 0 ␪ is just as important as the variation of h␪ ϭ Cp(pr) gram. Since both h and ⌰ are conserved when mixing with S in causing the nonconservation of ␪. Hence it is occurs at p ϭ 0, it follows that any variation of the 0 inconsistent to ignore the same term, ϪhSdS/dt, in (32) difference, ␪ Ϫ⌰,onaS Ϫ⌰diagram must be due when examining how well Cp(S, ␪,pr)␪ approximates to the production of ␪ when mixing occurs at p ϭ 0. h 0. Enthalpy, h, is evaluated using the Gibbs function of

We conclude that past attempts to justify Cp(pr)␪ as Feistel and Hagen (1995). The arbitrary linear function

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0 FIG. 3. Contours of (a) ␪ Ϫ⌰and of (b) Cp(S, ␪, 0)␪/C p Ϫ⌰in a smaller range of salinity than in Figs. 1 and 4. Panel (a) illustrates 0 the error in regarding C p␪ as heat content; panel (b) illustrates the

error in regarding Cp(S, ␪, 0)␪ as heat content, in both cases measured in temperature units. The background cloud of points illustrate where there is data from somewhere in the World Ocean.

Fig. 2a has ␪ Ϫ⌰about Ϫ0.44ЊC so the mixture actually has ␪ ϭ 19.56ЊC which is cooler than the average ␪ by 0.55ЊC. In order to correct for this production of ␪, one must abandon ␪ and adopt ⌰, and Fig. 2a shows that the maximum difference between these temperatures is almost 2ЊC in the very fresh and warm region of the diagram near (S ϭ 0, ⌰ϭ40). 0 The error in taking Cp(S, ␪,pr)␪ to be h is shown on the full S Ϫ⌰plane in Fig. 2b. This error is expressed 0 0 in temperature units as [Cp(pr)␪ Ϫ h ]/C p ϭ Cp(pr)␪/ 0 FIG. 2. (a) Contours of the difference ␪ Ϫ⌰between potential C p Ϫ⌰. Whereas the maximum variation of ␪ Ϫ⌰is temperature ␪ and conservative temperature ⌰. (b) Contours of Cp(S, about Ϫ2ЊC (see Fig. 2a), the maximum variation in 0 0 ␪, 0)␪/C p Ϫ⌰, which is the error in regarding Cp(S, ␪, 0)␪ as ``heat Cp(pr)␪/C p Ϫ⌰is only 0.22ЊC. The maximum amount content,'' measured in temperature units. of nonconservative production on the S Ϫ⌰plane is 0 about a factor of 5 less for Cp(pr)␪/C p than for ␪, being about Ϫ0.1ЊC compared with Ϫ0.55ЊC. However when of S that is inherent in any de®nition of h was chosen one considers only data from the real ocean, which is 0 by Feistel and Hagen (1995) so that h is zero at (S, T, mostly clustered near 35 psu, Cp(pr)␪/C p is no better p) of (0, 0, 0) and (35, 0, 0). Our de®nition, (15), of ⌰ than ␪ as can be seen from Fig. 3. This is con®rmed by means that it can be regarded as a function of S and ␪, comparing the root-mean-square value of ␪ Ϫ⌰for the ⌰ϭ⌰(S, ␪), and ensures that ⌰ϭ␪ at the three points whole of the global ocean atlas of Koltermann et al. (0, 0), (35, 0), and (35, 25) on the S Ϫ ␪ plane. (2003), namely 0.018ЊC, with the corresponding root- 0 The temperature difference, ␪ Ϫ⌰, is quite small mean-square value of Cp(pr)␪/C p Ϫ⌰, which is when the temperature is close to zero and, because of 0.019ЊC. Also, in the next section it will be shown that 0 our choice ofC p , also when S is close to 35 psu (see the use of Cp(pr)␪ as heat content to calculate the me- Fig. 2a). The production of ␪ on mixing any two ¯uid ridional heat ¯ux is no more accurate than simply using parcels can be deduced from this diagram. For example, the meridional ¯ux of potential temperature with a ®xed the mixing of equal masses of the two parcels (S ϭ 0, value of speci®c heat. ⌰ϭ0) and (S ϭ 40, ⌰ϭ40) means that the mixed Having already compared the production of ␪ with

¯uid is at (S ϭ 20, ⌰ϭ20). We can read off Fig. 2a that of Cp(S, ␪,pr)␪, here we brie¯y document the non- that ␪ Ϫ⌰is zero for one parent water mass and is conservative production of other thermodynamic quan- about 0.22ЊC for the other, so the average ␪ of these tities. In each case the quantity concerned is multiplied two parcels is 20.11ЊC. However, at (S ϭ 20, ⌰ϭ20) by a positive constant and then a linear function of S

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FIG. 4. Contours (ЊC) of a variable that is used to illustrate the FIG. 5. Contours (ЊC) of a variable that is used to illustrate the nonconservative production of conservative temperature ⌰ at a pres- nonconservative production of entropy ␴. The three points that are sure of 600 dbar. The three points that are forced to be zero are shown forced to be zero are shown with black dots. with black dots and the cloud of points near S ϭ 35 psu show where data from the World Ocean at 600 dbar are clustered. than 1% of the error in ␪. With the bulk of the ocean having a ␪ error less than 0.1ЊC (from Fig. 3a) the max- and ⌰ is subtracted so that the resulting quantity is zero imum error in ⌰ is estimated at less than 10Ϫ3ЊC. at the (S, ⌰) points (0, 0), (35, 0), and (35, 25) while The corresponding result for entropy is shown in Fig. the coef®cient of ⌰ in the ®nal expression is arranged 5. Here the temperature-like variable that is derived to be is Ϯ1. In this way the variable that is plotted in from entropy, ␴, is simply proportional to ␴ with the Figs. 4, 5 and 6 (like those in Figs. 2 and 3) have proportionality constant chosen so that the resulting contours measured in temperature units. Because these ``entropic temperature'' is 25ЊC at (35, 25). From this plots are simply a scaled version of the original variable ®gure we deduce that entropy is produced at approxi- plus a linear combination of S and ⌰, they can be used mately three times the rate at which ␪ is produced. to determine the nonconservative production of the orig- The cabbeling nonlinearity of the equation of state inal variable, measured in temperature units. can also be compared with the above nonlinear pro- First the nonconservation of potential enthalpy, h 0, ductions by taking the appropriate linear combination is illustrated for mixing of ¯uid parcels at 600 dbar of potential density (referenced to the sea surface), S which, from Fig. C1b, is the pressure at which the great- and ⌰ that is also zero at (0, 0) and (35, 0) and (35, est production of h 0 occurs. Enthalpy evaluated at 600 25). From Fig. 6 we conclude that nonlinear productions dbar is conserved during mixing at this pressure and the linear function of enthalpy, S and ⌰ that is zero at (0, 0) and (35, 0) and (35, 25) is contoured in Fig. 4. The maximum value of the production of ⌰ when mixing at 600 dbar can be deduced from the contours in this ®gure, namely about 4 ϫ 10Ϫ3 ЊC. However, this requires mixing across the full scale of the axes in this ®gure, 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 ®gure. The actual maximum value of ␦⌰ at 600 dbar is almost an order of magnitude less than this value at 6.3 ϫ 10Ϫ4 ЊC (from Fig. C1b). [The vertical axis in Fig. 4 should really be proportional to the conservative variable h(S, ⌰, 6 MPa), but when this is done, the changes are im- perceptible, just as Fig. 2a can be drawn with ␪ as the vertical axis which causes only a small but perceptible change to the ®gure.] Because the nonconservative pro- FIG. 6. Contours (ЊC) of a variable that is used to illustrate the duction of ⌰ is less than 1% of the nonconservative nonconservative production of potential density ␳␪. The three points production of ␪, we conclude that the error in ⌰ is less that are forced to be zero are shown with black dots.

Unauthenticated | Downloaded 10/01/21 07:16 AM UTC 954 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33 larger than 14ЊC are possible for mixtures of some pairs to accurately represent heat content, while for fresh- of water parcels. This can be compared with the max- water, ␪ is less than ⌰. We have also examined the imum nonlinear production of ␪ of about Ϫ0.55ЊC. This variation of ␪ Ϫ⌰at the sea surface throughout the suggests that the nonlinear production of density by the year and the range of ␪ Ϫ⌰is shown in Fig. 7b. One cabbeling process is roughly 25 times as large as the percent of the values have a seasonal range of ␪ Ϫ⌰ effect on density of neglecting the production of ␪. This that exceeds 0.16ЊC. A temperature difference of is con®rmed by comparing the range of ␪ Ϫ⌰in Fig. 0.25ЊC is not completely negligible in the oceanÐit is 2a (2ЊC) with the range of the variable of Fig. 6 (27.5ЊC), the same as the difference ␪ Ϫ T between potential indicating that ␪ is about 14 times more conservative and in situ temperatures for a pressure excursion of than is potential density. about 4000 dbar. Another way of looking at these errors In appendix E it is shown that the use of potential is the plots in Fig. 8 of the root-mean-square and range enthalpy gives rise to a new expression for the available (maximum minus minimum) of ␪ Ϫ⌰as a function potential energy in the ocean and in particular, clearly of pressure in the World Ocean. This shows that the associates the difference between available potential en- range of ␪ Ϫ⌰is almost 0.4ЊC over the upper 1000 ergy and the available gravitational potential energy as m of the water column, and is actually as large as 1.4ЊC being due to the thermobaric nature of the equation of near the surface. state of seawater. The difference in the meridional heat ¯uxes under the two different interpretations of model temperature is 0 9. Errors in present ocean models calculated by taking the area integral of ␳ 0C p␷Ä (␪ Ϫ⌰), where␷Ä is the model's northward velocity [see (30)]. Consider an ocean model exchanging heat with the This difference in heat ¯ux is shown by the solid line atmosphere at the rate Q(x, y, t). We have established in Figs. 9b and 9c for data from the model of Hirst et that this heat enters or leaves the ocean as a ¯ux of al. (2000) and has a maximum value of 0.0046 PW or 0 potential enthalpy, so that Q/C p is the air±sea ¯ux of ⌰ about 0.4% of the maximum heat ¯ux across any latitude [see (29)]. This is exactly how today's ocean models circle. This change in meridional heat ¯ux implies a relate the air±sea heat ¯ux to the ¯ux of the model's corresponding difference in the air±sea heat ¯ux (of temperature variable, and since the model's temperature about 0.2 W mϪ2 in the vicinity of 20ЊS), which is obeys a standard conservation equation, the most ob- expected to be very similar to the error in the air±sea vious interpretation of the model's temperature is as heat ¯ux in present models that are run with a prescribed conservative temperature ⌰. Given the values of ⌰ and SST pattern. This heat ¯ux error is approximately 10% S at each location in the model, it is possible to calculate of the change in surface insolation expected under a the value of potential temperature at every point. The doubling of greenhouse gases in the atmosphere. The magnitude of the errors in existing ocean models is il- full line with dots in Fig. 9b shows the error in the lustrated in Fig. 7 where the temperature difference, ␪ meridional heat ¯ux if it is calculated using Cp(pr)␪ as Ϫ⌰, is shown at the sea surface, calculated from the heat content rather than the accurate heat content h 0 ϭ Koltermann et al. (2003) atlas. For the annually aver- C 0⌰. While the error in using C (p )␪ as heat content aged data, values of as large as 0.09 C are seen p p r ␪ Ϫ⌰ Њ has a different dependence on latitude, the typical error in the North Atlantic while the Ϫ0.06ЊC contour is ev- in the meridional heat ¯ux is very similar to that using ident in the eastern equatorial Paci®c. These patterns of C 0␪. The dashed line in Fig. 9b shows the error in the ␪ Ϫ⌰represent the errors in today's ocean models due p meridional heat ¯ux when the in situ heat capacity is to the neglect of the nonconservative production of ␪. used to de®ne the heat content as C (S, T, p)␪. This Larger values of ␪ Ϫ⌰occur in the Mediterranean Sea p (up to 0.2ЊC) and larger negative values occur where choice, dating back to Bryan (1962), has larger errors warm freshwater from rivers enter the ocean (values as than when simply using a ®xed heat capacity (compare low as Ϫ1.2ЊC; see Fig. 2a at S ϭ 0, ⌰ϭ25ЊC). These with the solid line in Fig. 9b). are the largest errors in the SST that are currently in- Warren (1999) chose to examine the meridional ¯ux curred by the neglect of the nonconservative production of internal energy, ␧, and implied that this is the quantity terms in the ␪ evolution equation when an ocean model that should be compared with the air±sea heat ¯ux. For is driven by speci®ed air±sea ¯uxes. These errors reduce the same model data of Hirst et al. (2000) the difference to no more than 1 mK when the model's temperature between the meridional ¯ux of ␧ and of h 0 is shown as variable is interpreted as conservative temperature. the solid line with dots in Fig. 9c. It is seen that the One handy way of expressing the error involved with meridional ¯ux of ␧ is no closer to being regarded as using potential temperature is to note that 0.5% of the the meridional heat ¯ux than is the ¯ux of ␪ using a annual-mean SST values in the ocean atlas have ®xed heat capacity. The reason for this is the second ␪ Ϫ⌰ϽϪ0.15ЊC and 0.5% have ␪ Ϫ⌰Ͼ0.10ЊC. term on the left-hand side of (1), which also means that That is, 1% of the annual-mean SST data lie outside internal energy does not have the ``potential'' property. an error range of 0.25ЊC. In salty water potential tem- Warren then derived the meridional ¯ux of Cp ␪ as an perature tends to be larger than it should be if it were approximation to the ¯ux of internal energy, where

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FIG. 7. (a) The difference, ␪ Ϫ⌰(ЊC), between potential temperature ␪ and conservative temperature ⌰ 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 ¯ux ®eld. (b) The range (max Ϫ min value) of ␪ Ϫ⌰(ЊC) at the sea surface during the 12 months of the year.

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FIG. 8. Plots of (a) the root-mean-square and (b) range (max Ϫ min) values of ␪ Ϫ⌰as a function of pressure for all data in the World Ocean.

FIG. 9. (a) The meridional heat ¯ux borne by the resolved-scale velocity ®eld in the oceanic 0 0 component of the coupled model of Hirst et al. (2000). This calculation uses h ϭ⌰C p as heat 0 content. (b) The error in the meridional heat ¯ux when using C p␪ as heat content is shown by the

solid line and the error in heat ¯ux when using Cp(pr)␪ as heat content is shown by the full line

with dots. The heat ¯ux error when using Cp(S, T, p)␪ is shown by the dashed line. (c) The solid line is the same as (b), namely the error in using potential temperature with a ®xed heat capacity. The meridional ¯ux of internal energy ␧ is different to the ¯ux of h0 by the full line with dots 0 0 while Warren's (1999) suggestion of using Cp ␪ ϭ [h (S, ␪) Ϫ h (S, 0)] is quite accurate with the error in the meridional heat ¯ux shown by the dashed line.

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Cp was evaluated at zero pressure and at the salinity of servation statement would resemble the ®rst law of ther- the ¯uid parcel as the average heat capacity between modynamics, with the right-hand side being (minus) the

FQ. There´the temperatures zero and ␪. In this way, Cp ␪ is actually divergence of the molecular ¯ux of heat, Ϫ١ equal to h 0(S, ␪) Ϫ h 0(S, 0) (D. Jackett 2002, personal is a sense in which both (2) and (34) are conservation communication) and since h 0(S, 0) varies by only 125 equations for total energy; the difference being that the JkgϪ1, equivalent to 0.031ЊC, [see Fig. 2a of this paper kinetic energy equation has been used to reexpress the or Table A4 of Feistel and Hagen (1995)] over the full dissipation of mechanical energy, ␳␧M, in (2) to obtain range of salinity, Warren's Cp ␪ is very nearly potential (34). In the same sense, one could call both (2) and (34) enthalpy. Figure 9c con®rms that while the meridional the ®rst law of thermodynamics. However, we follow

¯ux of Cp ␪ is not a particularly accurate expression for accepted practice in the literature and call (2) the ®rst the ¯ux of internal energy, it is quite an accurate ap- law of thermodynamics and (34) the conservation of proximation for the ¯ux of h 0. Warren (1999) showed total energy [see, e.g., sections 1±5 and 1±10 of Haltiner that the meridional ¯ux of Cp ␪ was a very good ap- and Williams (1980)]. proximation to the ¯ux of the Bernoulli function; a result Continuing to ignore the last term in (34) we see that that is consistent with the next section of this paper B is totally conserved when ¯uid parcels mix at constant where it is found that the Bernoulli function and po- pressure. In this regard B is superior to h 0 because po- tential enthalpy are the same up to about 0.003ЊCin tential enthalpy is not 100% conserved when mixing 0 0 temperature units, that is, B ϭ h Ϯ 0.003C p . happens in the subsurface ocean, and as a result ⌰ is It is concluded that present ocean models contain typ- in error by up to 1 mK. The range of pressure variation ical errors of Ϯ0.1ЊC due to the neglect of the noncon- at ®xed depth (due to the movement of mesoscale ed- servative production of ␪ although the error is as large dies) is typically 104 Pa (1 dbar) which is equivalent to as 1.4ЊC in isolated regions such as where the warm a change in enthalpy of 10 J kgϪ1, which in turn is fresh Amazon water discharges into the ocean. The cor- equivalent to a temperature change of 2.5 mK. An adi- responding typical error in the meridional heat ¯ux is abatic and isohaline change in pressure will cause a 0.005 PW (or 0.4%). To eliminate these errors one must change in the Bernoulli function of this magnitude, (i) interpret the model's temperature variable as ⌰ rather whereas potential enthalpy is totally independent of such than as ␪, (ii) carry the equation of state as ␳ ϭ ␳(S, pressure variations. In this regard h 0 and ⌰ are superior ⌰, p) (the above discussion has assumed that the chang- to B. es arising from having this different equation of state It is possible to imagine an ocean model carrying the are small, but this remains to be con®rmed), and (iii) Bernoulli function as its ``temperature'' variable. The calculate ␪ using the inverse function ␪(S, ⌰) when SST temporal change of pressure would need to be added as is required (e.g., in order to calculate air±sea ¯uxes with a forcing term in the model's B conservation equation, bulk formulas). These issues will be explored in a sub- as in (34). An ocean model would know both p and ⌽ sequent paper. While errors of 0.4% in the meridional at each time step so it would be possible to calculate heat ¯ux are much smaller than our ability to determine enthalpy from h ϭ B Ϫ⌽Ϫ(1/2)u ´ u and to use this these heat ¯uxes from observations, errors of Ϯ0.1ЊC as an argument of an equation of state in the functional in sea surface temperature do not seem to be totally form ␳(S, h, p). In this way the small error of 1 mK that trivial. is inherent in conserving ⌰ could be avoided. [Another way of avoiding this tiny error would be to carry the small source terms in the ⌰ equation, i.e., to carry the 10. The total energy, or Bernoulli equation two pressure integral terms in (25).] While implementing Adding the ®rst law of thermodynamics [(2)] to the the B conservation equation (34) in an ocean model conservation statements for kinetic energy, (1/2)u ´ u, would avoid any approximations in the total energy bud- and for the geopotential, ⌽ϭgz, a conservation equa- get, what would be lost is the notion that the model tion is found for the Bernoulli function, B ϵ h ϩ⌽ϩ variable B is a property of a water mass. Rather, B varies (1/2)u ´ u, namely (see Batchelor 1967 or Gill 1982) with pressure to the extent of 2.5 mK. This temperature increment happens to be the stated accuracy of modern

␳uB) CTD instruments and is larger than the maximum error)´ ١ ␳B)t ϩ) (1 mK) in using conservative temperature ⌰. -p 1 The principal dif®culty with using B as an oceanoץ (u ´ u . (34 ␮١ ´ F ϩϩ١ ´ ϭϪ١ t []΂΃2 graphic energy-like variable is not however due to theץ Q rather small dependence of B on pressure, but rather is The last term here is negligible in the ocean interior, due to it not being a locally determined quantity: in being many orders of magnitude smaller than even the addition to B being a function of the locally measured tiny term ␳␧M in (2). Hence apart from the unsteady properties S, T, and p, it also contains dynamical in- pressure term, (34) is in the form of a clean conservation formation in the geopotential function (as well as being equation [like (4)]. If it were not for the pt term the dependent on the magnitude of the three-dimensional Bernoulli function would be the quantity whose con- velocity vector). While both p and ⌽ are known when

Unauthenticated | Downloaded 10/01/21 07:16 AM UTC 958 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33 one is running a prognostic ocean model, ⌽ is not a property as it varies with the adiabatic heaving of wave locally observed quantity in ocean data. On using the motions; and (ii) unlike S and ⌰, B cannot be determined hydrostatic equation to express ⌽ in terms of the height from the local thermodynamic properties. of the sea surface where the geopotential is ⌽0, B can be written as [using (7) and ignoring the tiny kinetic energy] 11. Summary B ϭ h ϩ⌽ The aim of this work has been to develop a variable whose conservation statement is equivalent to the ®rst p 11 ϭ h00ϩ⌽ Ϫ Ϫ dpЈ. law of thermodynamics so that this variable can be ac- ͵ ␳[S(pЈ), ␪(pЈ), pЈ] ␳(S, ␪, pЈ) curately called ``heat content.'' This quest led to the 0 (35) thermodynamic quantity, potential enthalpy, which is the enthalpy that a ¯uid parcel would have if its pressure was changed, in an adiabatic and isohaline fashion, to In order to calculate B from observed data one needs the pressure of the sea surface. With an error that is to both (a) have knowledge of ⌽0 and (b) perform a more than two orders of magnitude less than present vertical pressure integral all the way to the sea surface. practice, the ¯ux of potential enthalpy is the correct ¯ux Hence B is not a locally determined quantity. The geo- of ``heat'' that can therefore be accurately compared potential at the sea surface, ⌽ 0, requires satellite altim- with air±sea and geothermal boundary ¯uxes of heat. eter data or the performance of an inverse model. The The ®rst law of thermodynamics can be cast in terms fact that the Bernoulli function cannot be determined of conservation equations for potential temperature ␪ from local thermodynamic properties means that it is and for conservative temperature ⌰ as [from (14) and unsuitable for use as a water-mass property. (23), ignoring the dissipation of kinetic energy] Potential enthalpy is by far the dominant contribution FQ ´ ١(to B, and when expressed in terms of ⌰, the oceanic d␪ (T0 ϩ ␪ 0 ␳ ϭϪ range of h is about 30ЊC. Hence the dynamical infor- dt (T0 ϩ T)Cpr(p ) mation that is contained in B, namely, B Ϫ h 0, being no more than 10 m2 sϪ2, is a factor of 10 000 less than FS ´ ١ (T0 ϩ ␪) 0 the dominant thermodynamic contribution, h , as found ϩ ␮(p) Ϫ (T0 ϩ ␪)␮Tr(p ) (T ϩ T) C (p ) by Cunningham (2000). Moreover, at the magnitude of []0 pr this dynamical information, B is not conserved at lead- (36) ing order because of the unsteady pressure term in (34). F ´ ١(d⌰ (T ϩ ␪ 0 That is, once the thermodynamic contribution, h ,is ␳ ϭϪ 0 Q 0 subtracted from B, the advection of the remainder is the dt (T0 ϩ T)C p same magnitude as the unsteady pressure term which is F ´ ١ (usually ignored. For dynamical information, the Mont- (T ϩ ␪ ϩ 0 ␮(p) Ϫ ␮(p )S . (37) gomery potential [or other suitable geostrophic stream- (T ϩ T) r C 0 []0 p function; see Montgomery 1937; McDougall 1989, his

Eq. (43)] has the advantage over B that it is not dom- The ratio of the absolute temperatures, (T 0 ϩ ␪)/(T 0 ϩ inated by a heat balance that is a factor of 10 000 larger T), that multiplies the divergence of the molecular ¯ux than the information contained in the geostrophic of heat in (37) varies from 1.0 by only 0.15% in the streamfunction. Here it is noted in passing that atmo- ocean so that the ®rst term on the right of (37) is very spheric scientists use the term Montgomery stream- close to being the divergence of the molecular heat ¯ux 0 function for h ϩ⌽whereas oceanographers use the term (divided by the constant,C p ). The corresponding term Montgomery streamfunction for the geostrophic stream- in (36) is divided by the heat capacity, Cp(pr), which function appropriate to any surface of interest, such as varies by 5% in the ocean and so it is much less accurate the streamfunction originally proposed by Montgomery to regard this term as the divergence of the molecular (1937) for geostrophic ¯ow in a steric anomaly surface. ¯ux of heat. Similar remarks can be made for the terms It is concluded that there is more information to be in these equations that multiply the divergence of the had by considering the potential enthalpy balance and molecular ¯ux of salt. The result is that it is more than the geostrophic streamfunction separately than by com- a hundred times more accurate to regard the right-hand 0 FQ/C p than to do so in (36) and this´bining these two pieces of information together into the side of (37) as Ϫ١ one Bernoulli equation. The present work supports the is the root cause of the more conservative nature of ⌰ argument of Bacon and Fofonoff (1996) that the ``use than of ␪. of the Bernoulli function is an unnecessary con¯ation This paper has largely proved the bene®ts of potential of mechanical and nonmechanical energy, given that enthalpy h 0 from the viewpoint of conservation equa- they evolve practically independently.'' The major tions, but the bene®ts can also be understood from the drawbacks with using the Bernoulli function are that (i) following parcel arguments. First, the air±sea heat ¯ux unlike S and ⌰, B cannot be considered a water-mass needs to be recognized as a ¯ux of h 0. Second, the work

Unauthenticated | Downloaded 10/01/21 07:16 AM UTC MAY 2003 MCDOUGALL 959 of appendixes B and C shows that while it is the in situ temperature error ␪ Ϫ⌰is very small for such cool enthalpy that is conserved when parcels mix, a negli- fresh seawater (Figs. 1 and 2a). gible error is made when h 0 is assumed to be conserved After submitting this manuscript for publication I during mixing at any depth. Third, note that the ocean have become aware that the Goddard Institute for Space circulation can be regarded as a series of adiabatic and Studies (GISS) ocean model already carries potential isohaline movements during which h 0 is absolutely un- enthalpy as its heatlike variable (Russell et al. 1995). changed followed by a series of turbulent mixing events The present paper can then be regarded as supplying during which h 0 is almost totally conserved. Hence it the theoretical motivation for converting such ocean is clear that h 0 is the quantity that is advected and dif- models from using potential temperature to potential fused in an almost conservative fashion and whose sur- enthalpy. face ¯ux is the air±sea heat ¯ux. The fact that enthalpy is only known up to a linear The small error involved with calling potential en- function of salinity does not diminish the usefulness of thalpy ``heat content'' has been shown to be no larger potential enthalpy as heat content nor the ¯ux of h 0 as than the effect of the dissipation of kinetic energy in heat ¯ux. It is proven that the meridional ¯ux of h 0 does the ®rst law of thermodynamics and so is utterly neg- represent a valid ¯ux of heat even when the meridional ligible. Without an exact total differential to represent ¯uxes of mass and of salt are nonzero. We have also the conservation of ``heat'' it is not possible to neatly shown here that the Bernoulli function and potential illustrate the errors involved with calling potential en- enthalpy differ by only about 3 ϫ 10Ϫ3ЊC (when ex- thalpy ``heat content,'' but the error in the meridional pressed in temperature units). Nevertheless, for study heat ¯ux is likely to amount to less than 1% of the error of heat budgets h 0 is more useful than the Bernoulli 0 0 involved when using either C p␪ or Cp(pr)␪ as heat con- function because in contrast to B, h , and ⌰ have the tent (Fig. 9b). That is, the remaining error in the me- distinct advantage of being locally determined ther- ridional heat ¯ux from using h 0 is estimated to be less modynamic quantities that are totally invariant under than 5 ϫ 10Ϫ5 PW. adiabatic and isohaline changes of pressure. Hence, h 0 It is convenient to de®ne a new temperature variable, and ⌰ are properties of water masses while B is not. called ``conservative temperature,'' ⌰, which is simply proportional to potential enthalpy with the proportion- Acknowledgments. I thank Dr. David Jackett for cod- 0 ality constant being the ®xed ``heat capacity,'' C p ing the thermodynamic algorithms based on the Gibbs (ϭ3989.244 952 928 15 J kgϪ1 KϪ1). Since ocean mod- function of Feistel and Hagen (1995) and for preparing els (i) have their temperature obeying a standard con- all the ®gures. Dr. Rainer Feistel kindly provided an servation statement and (ii) have the heat capacity at electronic version of the Gibbs function algorithm, and the sea surface being constant, it is apparent that the he, Dr. Bruce Warren, Dr. Stephen Grif®es, and Pro- temperature variable in these ocean models is actually fessor JuÈrgen Willebrand are thanked for their comments conservative temperature' ⌰ rather than potential tem- on a draft of this paper. Dr. Siobhan O'Farrell kindly perature ␪. The error in interpreting ⌰ as the temperature provided the data of Hirst et al. (2000) that is used in variable in ocean models is likely to be no more than Fig. 9. This work contributes to the CSIRO Climate 1% of the error in ␪, that is 1% of approximately Change Research Program. Ϯ0.1ЊC, namely, Ϯ10Ϫ3ЊC. The typical temperature dif- ference ␪ Ϫ⌰of Ϯ0.1ЊC is not completely negligible APPENDIX A in the oceanÐit is the same as the difference ␪ Ϫ T between potential and in situ temperatures for a pressure Algorithm for Conservative Temperature excursion of about 1500 dbar. Since ocean models have been careful to deal with potential temperature rather Enthalpy, h(S, T, p), is evaluated by differentiating than in situ temperature, it would also make sense to the Gibbs function, G(S, T, p), of Feistel and Hagen convert ocean models to ⌰ rather than ␪. (1995) according to h ϭ G Ϫ (T 0 ϩ T)GT. Potential The realization that ocean models carry ⌰ rather than enthalpy h 0 is enthalpy evaluated at the reference pres- ␪ means that the heat capacity of seawater in these model sure of zero and at the potential temperature; that is, codes should not be user-speci®ed but should be hard- h 0(S, ␪) ϭ h(S, ␪, 0). Following Feistel and Hagen 0 Ϫ1 Ϫ1 wired to beC p (ϭ3989.244 952 928 15 J kg K ). If (1995), the polynomial for potential enthalpy is written the temperature in an ocean model were to really be ␪ in terms of the scaled salinity and potential temperature then (i) additional nonconservative production terms variables, s ϭ S/40 and ␶ ϭ ␪/40. The coef®cients of would be needed in the temperature equation, and (ii), the polynomial h 0(s, ␶) are given in Table A1. Here, S the heat capacity that is used at the sea surface to relate is salinity in psu, ␪ is potential temperature in degrees the air±sea heat ¯ux to the surface ¯ux of potential Celsius (on the ITS-90 temperature scale) and h 0 is in temperature would have to vary in space and time by joules per kilogram. up to 5% because heat capacity is a function of ␪ and The conservative temperature ⌰ is de®ned as in (15) 0 00 S. Interestingly, the heat capacity for cool freshwater to be ⌰ϵh /CCpp , where is 3989.244 952 928 15 J 0 Ϫ1 Ϫ1 would need to be 5% larger thanC p even though the kg K . Check values for ⌰ are ⌰(S ϭ 20 psu, ␪ ϭ

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TABLE A1. Terms and coef®cients of the polynomial for potential The maximum production occurs when parcels of equal 0 enthalpy, h (s, ␶). Ϫ2 mass are mixed so that (1/2)m1m2m ϭ 1/8. The heat 0 0 h (s,␶) terms Coef®cients capacity, Cp(S, ␪, 0) ϭ h␪, is not a strong function of ␶ 1.686 946 314 599 218 ϫ 105 ␪ but is a much stronger function of S, so the ®rst term ␶ 2 Ϫ2.956 216 410 340 123 ϫ 103 in the curly brackets in (B5) is small compared with the 3 3 0 2 ␶ 2.918 123 904 030 847 4 ϫ 10 second term. Also, the third term in (B5),hSS (⌬S) , ␶ 4 Ϫ1.504 175 120 993 792 5 ϫ 103 which causes the so-called dilution heating, is small ␶ 5 2.521 066 860 081 389 5 ϫ 102 6 1 compared with the second term. A typical value of ␶ 5.919 813 629 534 450 4 ϫ 10 0 Ϫ1 Ϫ1 Ϫ1 s 4.332 174 782 958 425 ϫ 102 h␪S is Ϫ5.4Jkg K (psu) (Feistel and Hagen 1995) s␶ Ϫ1.222 348 964 386 558 5 ϫ 104 so that an approximate expression for the production of s␶ 2 3.392 152 985 502 741 6 ϫ 103 potential temperature is s␶ 3 Ϫ1.099 733 971 639 728 5 ϫ 103 s␶ 4 Ϫ5.905 091 691 243 234 ϫ 101 1 ⌬S 0 Ϫ4 5 1 ␦␪ ϭ h ⌬␪ ഠ Ϫ3.4 ϫ 10 ⌬␪⌬S. (B6) s␶ 1.261 732 848 665 46 ϫ 10 4 ␪S h0 s1.5 7.731 194 944 352 569 ϫ 102 ␪ s1.5␶ 1.785 072 635 133 068 2 ϫ 103 1.5 2 2 s ␶ Ϫ7.861 964 038 192 168 ϫ 10 APPENDIX C s1.5␶ 3 2.121 051 924 469 151 4 ϫ 102 s1.5␶ 4 3.312 672 232 980 99 ϫ 101 s2 Ϫ2.211 656 537 527 129 3 ϫ 103 The Nonconservative Production of ⌰ s2.5 1.382 873 680 456 869 8 ϫ 103 s3 Ϫ4.611 526 139 838 933 ϫ 102 The quantities that are conserved when two ¯uid par- cels are mixed at a general pressure p are mass, salt, and enthalpy h, while potential enthalpy h 0 will not be

20ЊC) ϭ 20.446 377 553 919 2ЊC, ⌰(0, 0) ϭ 0ЊC, ⌰(35, conserved (unless p ϭ pr). The equations for the three 0) ϭ 0ЊC, and ⌰(35, 25) ϭ 25ЊC. conserved quantities are (B1), (B2), and mh ϩ mh ϭ mh, (C1) APPENDIX B 11 22 while the nonconservative nature of potential enthalpy The Nonconservative Production of ␪ means that it obeys the equation

00 00 When ¯uid parcels under go irreversible and complete mh11ϩ mh 22ϩ m␦h ϭ mh , (C2) mixing at constant pressure, the thermodynamic quan- 0 0 tity that is conserved during the mixing process is en- where ␦h is the nonconservative production of h . En- thalpy is now expressed in the functional form, h ϭ thalpy, as can be deduced from (2). In addition, mass 0 and salt are also conserved. In this appendix we consider h(S, h , p), and expanded as a Taylor series of S and h 0 at ®xed pressure, p, about the properties of the mixed only mixing at the sea surface where p ϭ 0 and enthalpy, ¯uid, retaining terms to second order in (S S ) h, is potential enthalpy, h 0. When a ¯uid parcel of mass 2 Ϫ 1 ϭ S and in (hh00) h 0. Then h and h are evaluated m is mixed with another of mass m , the mass m, sa- ⌬ 21Ϫ ϭ⌬ 1 2 1 2 and (C1) and (C2) used to ®nd linity S, and potential enthalpy h 0 of the mixed ¯uid h ץ 22hץ h 1 mmץ :obey these simple equations ␦h0 ϭ 12 (⌬h02) ϩ 2 ⌬h0⌬S m ϩ m ϭ m, (B1) 02000 Sץ hץ hץ hץ] h 2 mץ 12 mS mS mS, (B2) 2hץ 11ϩ 22ϭ ϩ (⌬S)2 . (C3) SץSץ (mh000mh mh , (B3 11ϩ 22ϭ ] while the nonconservative nature of potential temper- In order to evaluate these partial derivatives, (24) is ature means that it obeys differentiated to ®nd p ␣ hץ (m11␪ ϩ m 22␪ ϩ m␦␪ ϭ m␪, (B4 0 Ϫ1 ϭ 1 ϩ (C p) dpЈ. (C4) h0 ͵ ␳ץ where ␪ is the potential temperature of the mixed ¯uid ΗS,p pr and ␦␪ is the ``production'' of potential temperature. Following Fofonoff (1962), h 0 is expanded in a Taylor The right-hand side of (C4) scales as 1 ϩ ␣(p Ϫ pr)/ 0 series of S and ␪ about the values S and ␪ of the mixed ␳C p, which is more than unity by only about 0.0015 [for (p Ϫ p )of4ϫ 107 Pa (4000 dbar)]. Hence, to a ¯uid, retaining terms to second order in (S 2 Ϫ S1) ϭ r 00 very good approximation, we may regard the left-hand ⌬S and in (␪ 2 Ϫ ␪1) ϭ⌬␪. Thenhh12 and are evaluated and (B3) and (B4) used to ®nd side of (C3) as simply the production of potential en- thalpy, ␦h 0. It is interesting to examine why this ap- 1 mm ␦␪ h0 ϭ 12[h02(⌬␪) ϩ 2h 0⌬␪⌬S ϩ h 02(⌬S)]. proximation is so accurate when the difference between ␪␪␪␪2 m2 SSS enthalpy, h, and potential enthalpy, h 0, as given by (24),

(B5) scales as (p Ϫ pr)/␳, which is as large as typical values

Unauthenticated | Downloaded 10/01/21 07:16 AM UTC MAY 2003 MCDOUGALL 961 of enthalpy itself. The reason is that the integral in (24) is dominated by the integral of the mean value of 1/␳, so causing a signi®cant offset between h and h 0 but not h 0, which is taken atץ/hץ affecting the partial derivative ®xed pressure. Even the dependence of density on pres- .h 0ץ/hץ sure alone does not affect As an example of the second order derivatives of h in (C3) we differentiate (C4), giving

␣ 2h pץ ϭ (C 0)Ϫ2 dpЈ; (C5) h p ͵ ␳ץh00ץ S,p p ΂΃ Η r ⌰ hence we may write (C3) approximately as 1 ␦h0 ϭ ␳Ϫ12(p Ϫ p )[␣ (⌬⌰) ϩ 2␣ ⌬⌰⌬S Ϫ ␤ (⌬S)], 2 8 r ⌰ SS (C6) FIG. C1. The largest amount of nonconservative production of where the integral in (C5) has been approximated as (a) potential temperature ␪ and of (b) conservative temperature proportional to the pressure difference and it is recog- ⌰ for pairs of water parcels drawn from the ocean at each pressure. nized that the thermal expansion coef®cient is a much stronger function of ⌰ and S than is density. Also in amining every possible combination of ¯uid parcels (C6), m1 ϭ m 2 has been assumed. Equation (C6) shows that the nonconservative pro- and storing the largest values of these quantities. The duction of potential enthalpy is proportional to the non- approximate values of ␦␪ and ␦⌰ were then calculated conservative production of density called cabbeling from (B6) and (C8) and are shown as the dashed lines 2 in Figs. C1a and C1b. For the same pair of parcels (McDougall 1987), (1/8)␳[␣Ä ␪(⌬␪) ϩ 2␣Ä S⌬␪⌬S Ϫ 2 Ä 2 that produced the largest values of ⌬␪⌬S and of (⌬␪) ␤S(⌬S) ], where for this purpose we do not distinguish between the two slightly different forms of the thermal the accurate values of ␦␪ and ␦⌰ were also calculated expansion coef®cient [in fact the bracket here is exactly and are shown as the full lines in Fig. C1. These the same as in (C6) even though the individual terms accurate values were determined by mixing the salin- are slightly different]. The production of h 0 causes a ity and the enthalpy of the two ¯uid parcels linearly 0 0 and then deducing, by Newton±Raphson iteration, the temperature change of ␦h /C p , which causes a change 0 0 in situ temperature of the mixed ¯uid from Feistel and in density of ␳␣␦h /C p . The ratio of this increase in density to that caused by cabbeling is ␣(p Ϫ p )/␳C 0 Hagen's (1995) expression for h(S, T, p). From this r p in situ temperature, ␪,h0 , ␦␪, ␦h 0, and ␦⌰ were cal- which is about 0.0015 for (p Ϫ pr) of 4000 dbar. Hence it is clear that cabbeling has a much larger effect on culated. The fact that the largest negative value of ␦␪ density than does the nonconservation of ⌰. in Figure B1a is only 1/10 of the Ϫ0.55ЊC identi®ed McDougall (1987) has shown that the ®rst term in the above re¯ects the fact that the atlas does not contain bracket in (C6) is usually about a factor of 10 larger than fresh meltwater near the poles. the other two terms, so we may approximate ␦h0 as The largest production of conservative temperature is seen to occur at a pressure of 600 dbar and is about 6.3 1 ϫ 10Ϫ4ЊC whereas the largest production of potential ␦h0 ϭ ␳Ϫ12(p Ϫ p )␣ (⌬⌰) 8 r ⌰ temperature is about Ϫ3 ϫ 10Ϫ2ЊC and this occurs at the sea surface. If we append to the atlas the missing 1 ഠ ␳Ϫ12(p Ϫ p )␣Ä (⌬␪) , (C7) cool fresh meltwater near the sea surface, the maximum 8 r ␪ value of ␦⌰ is unchanged but the extreme value of ␦␪ becomes Ϫ0.4ЊC. It is clear then that ⌰ is a factor of which gives the production of conservative temperature, about 600 more conservative than is ␪. It is for this ␦⌰ϭ␦h 0/,asC 0 p reason that we claim that ⌰ better represents ``heat'' Ϫ13 2 ␦⌰ ഠ 3.3 ϫ 10 (p Ϫ pr)(⌬␪), (C8) than does ␪ by a factor of more than two orders of magnitude. Ϫ5 Ϫ2 where ␣Ä ␪ has been taken to be 1.1 ϫ 10 K (Mc- Dougall 1987) and (p Ϫ pr) is in pascals. APPENDIX D In order to better compare the production of ␪ and ⌰ in today's ocean we have searched the annually A Discussion of Potential Internal Energy averaged oceanic atlas of Koltermann et al. (2003) in the following way. At each standard pressure the larg- Internal energy, ␧, does not posses the ``potential'' 2 /(p | S,␪ is not zero but is (p 0 ϩ pץ/␧ץ est values of ⌬␪⌬S and of (⌬␪) were found by ex- property in that

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(␳ 2c 2), which when integrated over a pressure range of where the pressure is constant, the left-hand side of 4000 dbar becomes a change of 360 J kgϪ1, equivalent (8) becomes exactly ␳dh 0 /dt, whereas this is not quite to a temperature change of approximately 0.1ЊC. We equal to ␳d␧ 0 /dt, there being the additional tiny term would not adopt a potential temperature algorithm that p 0 (␣d␪/dt Ϫ ␤dS/dt). had this type of error so it is clear that ␧ is not the heat- like variable we seek. However, potential internal en- APPENDIX E 0 0 ergy, ␧ ϭ h Ϫ p 0/␳(S, ␪, 0), does posses the ``po- tential'' property so it is totally invariant under adiabatic Available Potential Energy and isohaline changes in pressure. A similar analysis to The available gravitational potential energy of the 0 that in appendix C shows that the production of ␧ is ocean can be written (Reid et al. 1981), given by APEgrav 1 0 Ϫ12 p ␦␧ϭ ␳ (p ϩ p0)[␣⌰(⌬⌰) ϩ 2␣S ⌬⌰⌬S b 8 ϭ gpϪ1 ͵͵ ͵ {␳Ϫ1[S(p), ␪(p), p] 2 A 0 Ϫ ␤S(⌬S) ], (D1) Ϫ ␳Ϫ1[S˜˜˜(p), ␪˜ (p), p]} dp dA, where the reference pressure has been taken to be p r ϭ (E1) 0. Since p 0 is small compared with typical oceanic pres- sures, it is clear that the nonconservative production of where the area integral is over the whole ocean area, 0 0 ˜˜ ␧ is almost the same as that of h [cf. with the corre- pb(x, y) is the pressure at the ocean ¯oor andS˜ and ␪˜ sponding expression, (C6) for h 0]. are the pro®les after the whole ocean has been leveled Potential internal energy, ␧ 0 , can be written as the in an adiabatic and isohaline fashion so that neutral sum of internal energy, ␧, and the pressure integral density surfaces coincide with geopotential surfaces. 2 2 of Ϫ( p 0 ϩ p)/(␳ c ) and from this relationship, the APEgrav does not represent the total available potential left-hand side of (1) can be written as the material energy (APE) because of exchanges between gravita- derivative of ␧ 0 plus several other terms, the largest tional and internal energy during the leveling process. of which is smaller than d␧ 0 /dt by the factor, ␣( p ϩ The APE is the volume integral of the difference in p 0 )/␳C p , which is very similar to the ratio found for enthalpy between the two states, namely the terms that are additional to the material derivative pb of h 0 [see the discussion following (8)]. Hence we APE ϭ gϪ1 ͵͵ ͵ {h[S(p), ␪(p), p] conclude that for all practical purposes, potential in- A 0 0 ternal energy, ␧ , may be used instead of potential Ϫ h[S˜˜˜(p), ␪˜ (p), p]} dp dA. enthalpy, h 0 , as the variable whose conservation state- ment is the ®rst law of thermodynamics in the ocean. (E2) However, h 0 is preferred because at the sea surface Our equation (7) relating h to h 0 is now used, obtaining

pb APE ϭ gϪ10͵͵ ͵ {h [S(p), ␪(p)] Ϫ h 0[S˜˜˜(p), ␪˜ (p)]} dp dA A 0

ppb ϩ gϪ1 ͵͵ ͵ ͵ {␳Ϫ1[S(p), ␪(p), pЈ] Ϫ ␳Ϫ1[S˜˜˜(p), ␪˜ (p), pЈ]} dpЈ dp dA. (E3) A 00

The ®rst term here is identically zero because during sure; in other words, because of the thermobaric nature the adiabatic and isohaline rearrangement each ¯uid par- of the equation of state of seawater. This same conclu- cel retains it potential enthalpy so that the mass-weight- sion was found by Reid et al. (1981) by using a Taylor ed volume integral of h 0 is unchanged. This leaves the expansion of enthalpy in (E2). second part of (E3), which is a new expression for APE; it involves only a double integral of speci®c volume REFERENCES with no other dependence on enthalpy. By comparing (E1) and (E3) it is clear that APE is only different to Bacon, S., and N. Fofonoff, 1996: Oceanic heat ¯ux calculation. J. Atmos. Oceanic Technol., 13, 1327±1329. APEgrav because the thermal expansion coef®cient and Batchelor, G. K., 1967: An Introduction to Fluid Dynamics. Cam- the haline contraction coef®cient are functions of pres- bridge University Press, 615 pp.

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