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

Transcription

1 VOLUME 33 JOURNAL OF PHYSICAL OCEANOGRAPHY MAY 23 Potential Enthaly: A Conservative Oceanic Variable for Evaluating Heat Content and Heat Fluxes TREVOR J. MCDOUGALL* Antarctic CRC, University of Tasmania, Hobart, Tasmania, Australia (Manuscrit received 21 December 21, in final form 7 October 22) ABSTRACT Potential temerature is used in oceanograhy as though it is a conservative variable like salinity; however, turbulent mixing rocesses conserve enthaly and usually destroy otential temerature. This negative roduction of otential temerature is similar in magnitude to the well-known roduction of entroy that always occurs during mixing rocesses. Here it is shown that otential enthaly the enthaly that a water arcel would have if raised adiabatically and without exchange of salt to the sea surface is more conservative than otential temerature by two orders of magnitude. Furthermore, it is shown that a flux of otential enthaly can be called the heat flux even though otential enthaly is undefined u to a linear function of salinity. The exchange of heat across the sea surface is identically the flux of otential enthaly. This same flux is not roortional to the flux of otential temerature because of variations in heat caacity of u to 5%. The geothermal heat flux across the ocean floor is also aroximately the flux of otential enthaly with an error of no more that.15%. These results rove that otential enthaly is the quantity whose advection and diffusion is equivalent to advection and diffusion of heat in the ocean. That is, it is roven that to very high accuracy, the first law of thermodynamics in the ocean is the conservation equation of otential enthaly. It is shown that otential enthaly is to be referred over the Bernoulli function. A new temerature variable called conservative temerature is advanced that is simly roortional to otential enthaly. It is shown that resent ocean models contain tyical errors of.1c and maximum errors of 1.4C in their temerature because of the neglect of the nonconservative roduction of otential temerature. The meridional flux of heat through oceanic sections found using this conservative aroach is different by u to.4% from that calculated by the aroach used in resent ocean models in which the nonconservative nature of otential temerature is ignored and the secific heat at the sea surface is assumed to be constant. An alternative aroach that has been recommended and is often used with observed section data, namely, calculating the meridional heat flux using the secific heat (at zero ressure) and otential temerature, rests on an incorrect theoretical foundation, and this estimate of heat flux is actually less accurate than simly using the flux of otential temerature with a constant heat caacity. 1. Introduction The quest in this work is to derive a variable that is conservative, indeendent of adiabatic changes in ressure, 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 interreted 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 urose in ocean models is otential temerature referenced to the sea surface,, but it does not accurately reresent the conservation of heat because of (i) * Additional affiliation: CSIRO Marine Research, Hobart, Tasmania, Australia. Corresonding author address: Dr. Trevor J. McDougall, Antarctic CRC, CSIRO Division of Marine Research, GPO Box 1538, Hobart, Tasmania 71, Australia. trevor.mcdougall@csiro.au the variation of secific heat with salinity and (ii) the deendence of the total differential of enthaly on variations of salinity. Fofonoff (1962) ointed out that when fluid arcels mix at constant ressure, the thermodynamic variable that is conserved is enthaly, and he showed this imlied that otential temerature is not a conservative variable. It is natural then to consider enthaly as a candidate conservative variable for embodying the meaning of the first law of thermodynamics. However, this attemt is thwarted by the strong deendence of enthaly on ressure. For examle, an increase in ressure of 1 7 Pa (1 dbar), without exchange of heat or salt, causes a change in enthaly that is equivalent to about 2.5C. We show in this aer that in contrast to enthaly, otential enthaly does have the desired roerties to embody the meaning of the first law. Present treatment of oceanic heat fluxes is clearly inconsistent. Ocean models treat otential temerature as a conservative variable and calculate the heat flux across oceanic sections using a constant value of heat 23 American Meteorological Society 945

2 946 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33 caacity. By contrast, heat flux through sections of observed data is often calculated using a variable secific heat multilying the flux of otential temerature er 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 aroach is flawed on three counts. While the errors involved are small, it is clearly less than satisfactory to have conflicting ractices in the observational and modeling arts of hysical oceanograhy, articularly as an accurate and convenient solution can be found. Warren (1999) has claimed that because internal energy is unknown u to a linear function of salinity, it is inaroriate 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 essimism is unfounded; it is erfectly valid to talk of otential enthaly, h, as the heat content and to regard the flux of h as the heat flux. Moreover, h is shown to be more conserved than is by more than two orders of magnitude. This aer roves that the fluxes of h across oceanic sections can be accurately comared with the air sea heat flux, irresective of whether the fluxes of mass and of salt are zero across these ocean sections. This has imlications for best oceanograhic ractice for the analysis of ocean observations and for the interretation of temerature in models. The first law of thermodynamics is comared with the equation for the conservation of total energy (the Bernoulli equation). It is shown that while the Bernoulli function and otential enthaly differ by only about C (when exressed in temerature units), the Bernoulli function cannot be considered a water-mass roerty 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,. For these reasons the Bernoulli function is not an attractive variable comared with h. 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 [ ] d 1 d ( ) FQ M, or (1) 2 dt dt dh 1 d FQ M, (2) dt dt where is the internal energy, h is the secific enthaly, defined by h ( )/, is in situ density, is the excess of the real ressure over the fixed atmosheric reference ressure, 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 M is the rate of dissiation of kinetic energy (W m 3 ) into thermal energy. As exlained 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 FQ Fq hsfs Fq [ (T T) T]F S, (3) where h S (T T) T is the artial derivative of secific enthaly with resect to salinity at constant in situ temerature and ressure, F S is the flux of salt, T K is the temerature offset between kelvins and degrees Celsius (see Feistel and Hagen 1995), T is in degrees Celsius, is the relative chemical otential of salt in seawater (i.e., is the difference between the artial chemical otential of salt S and the artial chemical otential of water W in seawater), and T is its derivative with resect to in situ temerature and both and T are evaluated at (S, T, ). In words, the first law of thermodynamics [(1)] says that the internal energy of a fluid arcel can change due to (i) the work done when the arcel s volume is changed at ressure ( ), (ii) the divergence of the flux of heat, and (iii) the dissiation of turbulent kinetic energy into heat. The effect of the dissiation of kinetic energy in these equations is very small and is always ignored. For examle, a tyical dissiation rate, M,of1 9 W kg 1 causes a warming of only 1 3 K (1 yr) 1. Another way of quantifying the unimortance of this term is to comare it to the magnitude of diaycnal mixing. The turbulent diaycnal diffusivity scales as.2 M N 2 (Osborn 198) and the diaycnal mixing of otential temerature that this diffusivity causes is tyically more than one thousand times larger than that caused by the dissiation of kinetic energy M /C. 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 comared with the turbulent heat fluxes excet 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 aears in the conservation equations through the average of F Q at the seafloor. We note in assing 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 catured 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

3 MAY 23 MCDOUGALL 947 form of the divergence of a flux, namely F Q, the key to finding a new variable whose conservation reresents the first law of thermodynamics is to find one such that the left-hand side of (1) or (2) is times the material derivative of that variable, for if that were ossible, the first law of thermodynamics could be written in the standard conservation form, being the same form as the salt conservation equation, ds (S) t (us) F S, (4) dt where F S is the flux of salt by all manner of molecular rocesses. 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 ages discussing some examles in which the word heat is used imrecisely. Later in their book (section 3) they oint out that the word enthaly can often be accurately used in lace of heat content er unit mass. In the resent aer it will be shown that with negligible error, a new oceanic heatlike variable called otential enthaly, obeys a clean conservation equation of the form (4) with the righthand 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 times the material derivative of otential enthaly lus a negligible error term. This means that the conservation equation of otential enthaly in the ocean is equivalent to the first law of thermodynamics. Given this, calling otential enthaly heat content can cause no harm or imrecise thinking in oceanograhy. Potential enthaly and heat content are effectively alternative names for the same thing because otential enthaly is the variable whose advection and diffusion throughout the ocean can be accurately comared with the boundary fluxes of heat. Just as the advection and diffusion of a assive conservative tracer in the ocean can be accurately comared with the boundary fluxes of the assive tracer, this same association of the boundary fluxes and the tracer content justifies the association of the word heat content with the new variable, otential enthaly. 3. Potential enthaly It follows from the form of (2) that when mixing occurs at constant ressure, enthaly is conserved [this is more obvious when (2) is written in divergence form using the continuity equation]. As an examle, mixing between fluid arcels at the sea surface where the ressure is constant ( and the total ressure is ) conserves the enthaly evaluated at that (zero) ressure. Just as the concet of otential temerature is well established in oceanograhy, consider now the otential concet alied to enthaly. After bringing a fluid arcel adiabatically (and without exchange of salt) to the sea surface ressure, its enthaly is evaluated there and called otential enthaly. During the adiabatic ressure excursion the otential enthaly of fluid arcels are unchanged and one wonders how much damage is done by forcing the fluid arcels to migrate to zero ressure before allowing them to mix rather than simly mixing in situ as they do in ractice. This thinking was the motivation for examining otential enthaly as a candidate heat content. The second law of thermodynamics defines the secific entroy,, whose total derivative obeys the Gibbs relation dh (1/)d (T T)d ds. (5) This relation is sometimes called the fundamental thermodynamic relation and it can also be regarded as the mathematical definition of entroy. Consider the movement, without exchange of heat or salt, of a fluid arcel from its in situ ressure to a fixed reference ressure r. Neither salinity nor entroy change during this motion, so that it is aarent from (5) that h/ S, 1/, (6) so that the enthaly at the reference ressure, which we call otential enthaly, h (S,, r ), is related to in situ enthaly, h, by h (S,, r) h(s,, ) 1/(S,, ) d. (7) r Here we have chosen to regard enthaly and in situ density as functions of otential temerature rather than of in situ temerature. Note that for a fixed reference ressure, h is a function of only S and. The total derivative of (7) is taken, finding that dh 1 d dh d (S,, ) d dt dt dt dt (S,, ) ds (S,, ) d, (8) dt (S,, ) r where 1 / S, and 1 /S,. The tyical value of the left-hand side of this equation is C d/dt and a tyical value for the last two terms is [ ( r )/]d/dt. The ratio of the last two terms to the dominant term in (8) is then aroximately ( r )/C, and for a ressure difference of 4 dbar (4 1 7 Pa) this ratio is tyically.15, imlying that the right-hand side of (8) is almost the material derivative of otential enthaly. Were it not for these two small terms in (8), otential enthaly would be the conservative heatlike variable that we seek whose conservation equation would be exactly the first law of thermodynamics and (2) would become dh /dt F Q M. The rest of this aer will quantify the error made by ignoring the last two terms in (8) and treating o- r

4 948 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33 tential enthaly as a conservative variable. It will be roven that the error in so doing is negligible, being no larger than the neglect of the dissiation of kinetic energy into heat. It will be deduced that the temerature error in ocean models that conserve otential enthaly are no more than 1 mk, which is a factor of more than 1 less than the errors in resent ocean models that treat otential temerature as a conservative variable. 4. The first law in terms of Regarding enthaly as a function of otential temerature, that is as h(s,, ), the first law of thermodynamics [(2)], takes the form h d h ds FQ M. (9) dt S dt S,, From the Gibbs relation [(5)], we have h h (T T) or (T T), (1) S, S, S where the second exression has used the fact that at constant S and, both h and can be regarded simly as functions of. The second art of (1) can be evaluated not only at but also at the reference ressure where the left-hand side is the heat caacity at that ressure, C ( r ) [which is shorthand for C (S,, r )], so that h (T T) C ( r ). (11) (T ) S, Again from the Gibbs relation, we have h, (12) S, and regarding h to be the functional form h[s, (S, ), ] leads to h h h S S S,, S, (T T) ( ). (13) T r The last art of this equation has used the identity (obtained for examle from the definition of the Gibbs function) that /S T, /T S,. Notice that in (13) and (T T) are evaluated in situ while T ( r )is evaluated at the reference ressure. Substituting (11) and (13) into (9) gives the first law of thermodynamics exressed in terms of changes of otential temerature and salinity as (T T) d ds C ( r) [() (T T) T( r)] (T ) dt dt FQ M. (14) This equation was derived by Bacon and Fofonoff (1996) by a slightly different route. We now derive the corresonding result in terms of conservative temerature. 5. The first law in terms of It is convenient to define a temerature-like variable,, which is roortional to otential enthaly, as where h /C, C h(s 35, 25, )/25, (15) and we call the conservative temerature. This value of C ( J kg 1 K 1 ) defined in (15) [using the algorithms of Feistel and Hagen (1995)] was chosen so as to minimize the difference between C and otential enthaly h when averaged over all the data at the sea surface of today s ocean. That is, with this constant value of heat caacity, the average value of at the sea surface in today s ocean is almost zero. Also, C is very close to being the satially averaged value of heat caacity at the sea surface of today s ocean. Algorithms for otential enthaly and conservative temerature in terms of salinity and otential temerature are given in aendix A. Regarding enthaly now as a function of conservative temerature, that is as h(s,, ), the first law of thermodynamics takes the form h d h ds FQ M. (16) dt S dt S,, From the Gibbs relation, we find that h (T T) (17), S, and (17) can be evaluated not only at but also at the reference ressure where the left-hand side is, so that S C h (T T) C. (18) (T ) S, Regarding h to be the functional form h[s, (S, ), ] leads to h h h S S S,, S, (T T). (19) S From the Gibbs relation, we have, (2) S (T T) h,

5 MAY 23 MCDOUGALL 949 and when this is evaluated at the reference ressure, it becomes ( r ), (21) S (T ) so that (19) becomes h (T T) ( r ). (22) S (T ), Substituting (18) and (22) into (16) gives the first law of thermodynamics exressed in terms of changes of conservative temerature and salinity as [ ] (T T) d (T T) ds C () ( r) (T ) dt (T ) dt FQ M. (23) At the sea surface T and (23) reduces to C d/ dt F Q M. Regarding enthaly and density to be functions of, otential enthaly is given by h C h(s,, ) 1/(S,, ) d (24) (where here and henceforth the reference ressure is taken to be zero) and the first law of thermodynamics can be written as [ d d (S,, ) C d dt dt (S,, ) Q ] ds (S,, ) d F M, (25) dt (S,, ) where 1 / S, and 1 /S, are the thermal exansion and haline contraction coefficients defined with resect to conservative temerature. The coefficients of d/dt and of ds/dt in (23) and (25) can be equated to find the following two exact relations for T in terms of, similar to the traditional relationshi for as T lus the ressure integral of the lase rate, and [ ] (T ) (S,, ) (T ) (S,, ) C d, (26) (T T) (S,, ) r (T ) (S,, ) () ( ) d. (27) 6. Potential enthaly as heat content The key finding in this aer amounts to roving that in comaring (16) to (9), h S, K h S, and that the heat caacity defined with resect to, namely h S,, varies much less from the constant value C than does the heat caacity defined with resect to, namely, FIG. 1. Heat caacity (at the sea surface) minus the constant value C contoured on the S diagram (J kg 1 C 1 ). Heat caacity is defined here with resect to otential temerature so that it is C (S,, ) h / S. If heat caacity is defined with resect to conservative temerature as h / S, then it is exactly the constant value C. h S,. For examle, even at a ressure as large as Pa (4 dbar), h S, is at most 1.15 C, while h S, varies by more than 5% [see (11) and Fig. 1]. Moreover, h S, suffers this 5% variation in the uer ocean where the satial contrasts of are much larger than at deth so that the variation in h S,, can do more damage than the variation in h S,, which occurs only at deth where the temerature gradients are small. This aer argues that (23) or (25) can be aroximated as (h ) t (uh ) (C ) t (Cu) FQ M. (28) Taking a mean dianeutral advection velocity of 1 7 m s 1 and z of Km 1 in the dee ocean, a tyical value of d/dt is Ks 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 dissiation of kinetic energy in (28), assuming M 1 9 Wkg 1. The air sea flux of heat aears in (28) as F Q and since this flux occurs at zero ressure, there is no error at all in equating the air sea flux with the flux of otential enthaly [because the last two terms on the lefthand side of (25) are zero at the sea surface]. The geothermal heat flux occurs at great deth and the local increase in caused by the divergence of the geothermal heat flux should be evaluated using the secific heat h S, which, at a ressure of Pa (4 dbar), is about 1.15 C, which can be taken to be C with high accuracy. This association of the air sea and geothermal heat fluxes with the flux of h is articularly clear since there is no flux of salt across either the sea surface or the

6 95 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33 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 (T T) T is only known u to a constant, reflecting the fact that enthaly itself is unknown u to a linear function of salinity. We come now to the question of whether it is ossible to regard h as heat content and the flux of h as heat flux. Warren (1999) argued that because enthaly is unknown u to a linear function of salinity, it is only ossible to talk of a flux of heat through an ocean section if the fluxes of both mass and salt through that section are zero. Technically this is true, but only in the same narrow sense that it is not ossible to talk of the flux of through an ocean section because there is always the question of adding or subtracting a constant offset to the temerature scale. Once we define what scale (kelvins or degrees Celsius) is being used to measure, the issue is resolved and one can legitimately talk of a flux of even though the mass flux may be nonzero. A similar argument can now be alied to otential enthaly. 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 secific choice for the unknowable linear function of salinity in the definition of h. The key thing to realize is that for any arbitrary choice of this linear function of salinity, the conservation equation, (28), of h is unchanged, and also, such arbitrary choices do not affect the air sea and geothermal heat fluxes. Hence h is the correct roerty with which to track the advection and diffusion of heat in the ocean, irresective of the arbitrary function of salinity that is contained in the definition of h. For examle, the difference between the meridional flux of h 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 h in the volume), irresective of whether there are nonzero fluxes of mass or of salt across either or both meridional sections. This owerful result follows directly from the fact that h obeys a standard conservation equation, (28), no matter what linear function of salinity is chosen in the definition of h.asa consequence, it does make erfect sense to talk of the meridional flux of heat (i.e., the flux of h ) in the Indian and South Pacific Oceans searately, 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 2; Ganachaud and Wunsch 2), so it is valid to use equations for the anomaly of conservative temerature, ( ). 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 h and are the oceanic thermodynamic quantities whose conservation reresents the first law of thermodynamics. Furthermore, it is legitimate to call h the heat content er unit mass and to call the flux of h the heat flux, bearing in mind that this nomenclature assumes the articular linear function of S that Feistel and Hagen (1995) adoted, just as the corresonding flux of otential temerature is deendent on the temerature scale on which the otential temerature 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 : () t (u) F, (29) where F F Q / C is the molecular and boundary flux of, and the dissiation of kinetic energy term M has been droed. McDougall et al. (22) have argued that the most sensible way of averaging (29) in z coordinates results in the form ( / ) t (ũ ) 1 F (K ), (3) where the last term on the right is the turbulent flux term and is the density-weighted average value of. The key oint in that aer is that the velocity variable that is carried by ocean models is actually roortional to the average mass flux er unit area, so that in (3), ũ u/, and is the constant value of density that is used in the horizontal ressure gradient term in the horizontal momentum equations. Hence when evaluating the flux of h through a section of an ocean model, one should form the area integral of C times the roduct of the model s velocity and tem- erature, ũ. This contrasts with the common ractice 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. (22) show that these models are actually free of the Boussinesq aroximation error in steady state if the model variables are interreted according to (3). 7. C (S,, r ) as heat content In a recent aer, Bacon and Fofonoff (1996) advocated the use of C (S,, r ) as heat content but here it is roven that this is actually less accurate than simly using C. In arguing that h is an almost conservative oceanic heat variable, the resent work aroximates the first factor (T T)/(T ) in (23) by unity and also ignores the square bracket in (23). This is equivalent to neglecting the two ressure integral terms in (25). When considering the first law of thermodynamics in the form (14), Bacon and Fofonoff (1996) also took (T T)/(T ) to be unity, but they justified this

7 MAY 23 MCDOUGALL 951 choice by introducing a surface equivalent heat flux and claiming that the thermodynamic balance in (14) could be brought to the surface where. This justification is incorrect because the conservation laws must be obeyed by seawater at the ressure at which the hysical rocesses, such as mixing, occur. While we agree that the aroximation (T T)/(T ) 1in (14) is a very good aroximation, and in advocating h and we make an aroximation of the same magnitude, we stress that this is indeed an aroximation. Another ste that Bacon and Fofonoff (1996) took in their treatment of the first law of thermodynamics was to assume that [() (T T) T ( r )]ds/dt in (14) could be ignored so that the material derivative of heat was taken to be C ( r )d/dt. While it is true that the ignored term is much smaller than C ( r )d/dt, it will be shown here that it is inconsistent to ignore this term if C ( 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 art of C (S,, r ) d/ dt was the integral of C (S,, r ) times the mass flux er unit area over the bounding area of the ocean volume. This oversight falsely assumes that d[c (S,, r )]/dt is the total derivative of heat rather than what they had arrived at, namely C (S,, r )d/dt. One cannot move the heat caacity inside the derivative when the heat caacity is allowed to vary as in Bacon and Fofonoff (1996). To examine the nonconservative roduction of C [S,, r ] the material derivative dh /dt hd/dt hds/dt S (31) is rewritten as d[c (S,, r)]/dt dh /dt dh /dt hds/dt. S (32) If one interrets C (S,, r ) h as heat content when evaluating the meridional heat flux, then the right-hand side of (32) has been assumed to be zero. As exlained above, Bacon and Fofonoff (1996) were aware that the last term in (32) was being neglected but, due to an oversight, were aarently not aware that they had also discarded the d h /dt term. The difference is equivalent to the difference between C and h, and using (31), d(c )/dt dh /dt (h C )d/dt hds/dt, S (33) where the right-hand side terms make C different to h. We find in aendix B that the dominant nonlinearity in the function h (S, ) that causes to be nonconservative is the term in 2h S and this has equal contributions from the variations of the two artial derivatives on the right-hand side of (33). That is, the variation of h S with is just as imortant as the variation of h C ( r ) with S in causing the nonconservation of. Hence it is inconsistent to ignore the same term, h S ds/dt, in (32) when examining how well C (S,, r ) aroximates h. We conclude that ast attemts to justify C ( r ) as heat content have been flawed on theoretical grounds, and since we show below that this aroach is no more accurate than simly using a constant heat caacity, it should be abandoned. Prior to the Bacon and Fofonoff (1996) aer various authors had used the in situ value of heat caacity together with otential temerature [i.e., C (S, T, )] as heat content (Bryan 1962; Macdonald et al. 1994; Saunders 1995) but there is even less theoretical justification for this choice than for C ( r ) and we show below that C (S, T, ) is less accurate than both C ( r ) and C. The roduction of and on mixing between fluid arcels is considered in aendix B and aendix C. Figure C1 illustrates the result of mixing fluid arcels with extreme roerty contrasts that are widely searated in sace (at a series of fixed ressures) and one wonders about the relevance of this rocedure to the real ocean. The imortance of these mixing arguments deends on the heat flux that travels by these aths, so that, for examle, if most of the oceanic heat transort 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 roduction of otential temerature of.4c 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 deends on the flux of heat being transorted down the temerature contrast on that surface even though the individual mixing events occur between arcels with very small and S contrasts.] Because mixing involves both eineutral and dianeutral mixing, and because the heat flux achieved by the various mixing aths is rather comlex, the mixing arguments that lie behind the lots in Fig. C1 do not obviously rovide a realistic estimate of the imortance of the nonconservative roduction of or of in the ocean. The imortant message that is gleaned from Fig. C1 is that the nonconservative roduction of is at least one hundred times smaller that the roduction of. A realistic assessment of the errors inherent in resent oceanic ractice can then be found by examining the temerature difference as described later in this aer, and the error remaining in the use of is taken to be less than 1% of the temerature difference,. Aendix D considers internal energy and otential internal energy as candidates for heat content but it is shown that they are not as suitable as otential enthaly. 8. Quantifying the errors in, C ( r ), and The nonconservative nature of otential temerature can be illustrated on a variant of the usual S diagram. Since both h and are conserved when mixing occurs at, it follows that any variation of the difference,,onas diagram must be due to the roduction of when mixing occurs at. Enthaly, h, is evaluated using the Gibbs function of Feistel and Hagen (1995). The arbitrary linear function

8 952 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33 FIG. 3. Contours of (a) and of (b) C (S,, )/ C in a smaller range of salinity than in Figs. 1 and 4. Panel (a) illustrates the error in regarding C as heat content; anel (b) illustrates the error in regarding C (S,, ) as heat content, in both cases measured in temerature units. The background cloud of oints illustrate where there is data from somewhere in the World Ocean. FIG. 2. (a) Contours of the difference between otential temerature and conservative temerature. (b) Contours of C (S,, )/ C, which is the error in regarding C (S,, ) as heat content, measured in temerature units. 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, ) of (,, ) and (35,, ). Our definition, (15), of means that it can be regarded as a function of S and, (S, ), and ensures that at the three oints (, ), (35, ), and (35, 25) on the S lane. The temerature difference,, is quite small when the temerature is close to zero and, because of our choice of C, also when S is close to 35 su (see Fig. 2a). The roduction of on mixing any two fluid arcels can be deduced from this diagram. For examle, the mixing of equal masses of the two arcels (S, ) and (S 4, 4) means that the mixed fluid is at (S 2, 2). We can read off Fig. 2a that is zero for one arent water mass and is about.22c for the other, so the average of these two arcels is 2.11C. However, at (S 2, 2) Fig. 2a has about.44c so the mixture actually has 19.56C which is cooler than the average by.55c. In order to correct for this roduction of, one must abandon and adot, and Fig. 2a shows that the maximum difference between these temeratures is almost 2C in the very fresh and warm region of the diagram near (S, 4). The error in taking C (S,, r ) to be h is shown on the full S lane in Fig. 2b. This error is exressed in temerature units as [C ( r ) h ]/ C C ( r )/ C. Whereas the maximum variation of is about 2C (see Fig. 2a), the maximum variation in C ( r )/ C is only.22c. The maximum amount of nonconservative roduction on the S lane is about a factor of 5 less for C ( r )/ C than for, being about.1c comared with.55c. However when one considers only data from the real ocean, which is mostly clustered near 35 su, C ( r )/ C is no better than as can be seen from Fig. 3. This is confirmed by comaring the root-mean-square value of for the whole of the global ocean atlas of Koltermann et al. (23), namely.18c, with the corresonding root- mean-square value of C ( r )/ C, which is.19c. Also, in the next section it will be shown that the use of C ( r ) as heat content to calculate the meridional heat flux is no more accurate than simly using the meridional flux of otential temerature with a fixed value of secific heat. Having already comared the roduction of with that of C (S,, r ), here we briefly document the nonconservative roduction of other thermodynamic quantities. In each case the quantity concerned is multilied by a ositive constant and then a linear function of S

9 MAY 23 MCDOUGALL 953 FIG. 4. Contours (C) of a variable that is used to illustrate the nonconservative roduction of conservative temerature at a ressure of 6 dbar. The three oints that are forced to be zero are shown with black dots and the cloud of oints near S 35 su show where data from the World Ocean at 6 dbar are clustered. and is subtracted so that the resulting quantity is zero at the (S, ) oints (, ), (35, ), and (35, 25) while the coefficient of in the final exression is arranged to be is 1. In this way the variable that is lotted in Figs. 4, 5 and 6 (like those in Figs. 2 and 3) have contours measured in temerature units. Because these lots are simly a scaled version of the original variable lus a linear combination of S and, they can be used to determine the nonconservative roduction of the original variable, measured in temerature units. First the nonconservation of otential enthaly, h, is illustrated for mixing of fluid arcels at 6 dbar which, from Fig. C1b, is the ressure at which the greatest roduction of h occurs. Enthaly evaluated at 6 dbar is conserved during mixing at this ressure and the linear function of enthaly, S and that is zero at (, ) and (35, ) and (35, 25) is contoured in Fig. 4. The maximum value of the roduction of when mixing at 6 dbar can be deduced from the contours in this figure, namely about C. However, this requires mixing across the full scale of the axes in this figure, but the range of temerature and salinity in the ocean at 6 dbar is much smaller as is illustrated by the cloud of data oints from the whole of the Koltermann et al. (23) global atlas, suerimosed on this figure. The actual maximum value of at 6 dbar is almost an order of magnitude less than this value at C (from Fig. C1b). [The vertical axis in Fig. 4 should really be roortional to the conservative variable h(s,, 6 MPa), but when this is done, the changes are imercetible, just as Fig. 2a can be drawn with as the vertical axis which causes only a small but ercetible change to the figure.] Because the nonconservative roduction of is less than 1% of the nonconservative roduction of, we conclude that the error in is less FIG. 5. Contours (C) of a variable that is used to illustrate the nonconservative roduction of entroy. The three oints that are forced to be zero are shown with black dots. than 1% of the error in. With the bulk of the ocean having a error less than.1c (from Fig. 3a) the maximum error in is estimated at less than 1 3 C. The corresonding result for entroy is shown in Fig. 5. Here the temerature-like variable that is derived from entroy,, is simly roortional to with the roortionality constant chosen so that the resulting entroic temerature is 25C at (35, 25). From this figure we deduce that entroy is roduced at aroximately three times the rate at which is roduced. The cabbeling nonlinearity of the equation of state can also be comared with the above nonlinear roductions by taking the aroriate linear combination of otential density (referenced to the sea surface), S and that is also zero at (, ) and (35, ) and (35, 25). From Fig. 6 we conclude that nonlinear roductions FIG. 6. Contours (C) of a variable that is used to illustrate the nonconservative roduction of otential density. The three oints that are forced to be zero are shown with black dots.

10 954 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33 larger than 14C are ossible for mixtures of some airs of water arcels. This can be comared with the maximum nonlinear roduction of of about.55c. This suggests that the nonlinear roduction of density by the cabbeling rocess is roughly 25 times as large as the effect on density of neglecting the roduction of. This is confirmed by comaring the range of in Fig. 2a (2C) with the range of the variable of Fig. 6 (27.5C), indicating that is about 14 times more conservative than is otential density. In aendix E it is shown that the use of otential enthaly gives rise to a new exression for the available otential energy in the ocean and in articular, clearly associates the difference between available otential energy and the available gravitational otential energy as being due to the thermobaric nature of the equation of state of seawater. 9. Errors in resent ocean models Consider an ocean model exchanging heat with the atmoshere at the rate Q(x, y, t). We have established that this heat enters or leaves the ocean as a flux of otential enthaly, so that Q/ C is the air sea flux of [see (29)]. This is exactly how today s ocean models relate the air sea heat flux to the flux of the model s temerature variable, and since the model s temerature obeys a standard conservation equation, the most obvious interretation of the model s temerature is as conservative temerature. Given the values of and S at each location in the model, it is ossible to calculate the value of otential temerature at every oint. The magnitude of the errors in existing ocean models is illustrated in Fig. 7 where the temerature difference,, is shown at the sea surface, calculated from the Koltermann et al. (23) atlas. For the annually averaged data, values of as large as.9c are seen in the North Atlantic while the.6c contour is evident in the eastern equatorial Pacific. These atterns of reresent the errors in today s ocean models due to the neglect of the nonconservative roduction of. Larger values of occur in the Mediterranean Sea (u to.2c) and larger negative values occur where warm freshwater from rivers enter the ocean (values as low as 1.2C; see Fig. 2a at S, 25C). These are the largest errors in the SST that are currently incurred by the neglect of the nonconservative roduction terms in the evolution equation when an ocean model is driven by secified air sea fluxes. These errors reduce to no more than 1 mk when the model s temerature variable is interreted as conservative temerature. One handy way of exressing the error involved with using otential temerature is to note that.5% of the annual-mean SST values in the ocean atlas have.15c and.5% have.1c. That is, 1% of the annual-mean SST data lie outside an error range of.25c. In salty water otential temerature tends to be larger than it should be if it were to accurately reresent heat content, while for freshwater, is less than. We have also examined the variation of at the sea surface throughout the year and the range of is shown in Fig. 7b. One ercent of the values have a seasonal range of that exceeds.16c. A temerature difference of.25c is not comletely negligible in the ocean it is the same as the difference T between otential and in situ temeratures for a ressure excursion of about 4 dbar. Another way of looking at these errors is the lots in Fig. 8 of the root-mean-square and range (maximum minus minimum) of as a function of ressure in the World Ocean. This shows that the range of is almost.4c over the uer 1 m of the water column, and is actually as large as 1.4C near the surface. The difference in the meridional heat fluxes under the two different interretations of model temerature is calculated by taking the area integral of C ( ), where is the model s northward velocity [see (3)]. 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. (2) and has a maximum value of.46 PW or about.4% of the maximum heat flux across any latitude circle. This change in meridional heat flux imlies a corresonding difference in the air sea heat flux (of about.2 W m 2 in the vicinity of 2S), which is exected to be very similar to the error in the air sea heat flux in resent models that are run with a rescribed SST attern. This heat flux error is aroximately 1% of the change in surface insolation exected under a doubling of greenhouse gases in the atmoshere. The full line with dots in Fig. 9b shows the error in the meridional heat flux if it is calculated using C ( r ) as heat content rather than the accurate heat content h. While the error in using C ( r ) as heat content C has a different deendence on latitude, the tyical error in the meridional heat flux is very similar to that using C. The dashed line in Fig. 9b shows the error in the meridional heat flux when the in situ heat caacity is used to define the heat content as C (S, T, ). This choice, dating back to Bryan (1962), has larger errors than when simly using a fixed heat caacity (comare with the solid line in Fig. 9b). Warren (1999) chose to examine the meridional flux of internal energy,, and imlied that this is the quantity that should be comared with the air sea heat flux. For the same model data of Hirst et al. (2) the difference between the meridional flux of and of h is shown as the solid line with dots in Fig. 9c. It is seen that the meridional flux of is no closer to being regarded as the meridional heat flux than is the flux of using a fixed heat caacity. 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 otential roerty. Warren then derived the meridional flux of C as an aroximation to the flux of internal energy, where

11 MAY 23 MCDOUGALL 955 FIG. 7. (a) The difference, (C), between otential temerature and conservative temerature at the sea surface for annually averaged data. These differences illustrate the errors in SST in resent ocean models when forced with a given heat flux field. (b) The range (max min value) of (C) at the sea surface during the 12 months of the year.

12 956 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33 FIG. 8. Plots of (a) the root-mean-square and (b) range (max min) values of as a function of ressure for all data in the World Ocean. FIG. 9. (a) The meridional heat flux borne by the resolved-scale velocity field in the oceanic comonent of the couled model of Hirst et al. (2). This calculation uses h C as heat content. (b) The error in the meridional heat flux when using C as heat content is shown by the solid line and the error in heat flux when using C ( r ) as heat content is shown by the full line with dots. The heat flux error when using C (S, T, ) is shown by the dashed line. (c) The solid line is the same as (b), namely the error in using otential temerature with a fixed heat caacity. The meridional flux of internal energy is different to the flux of h by the full line with dots while Warren s (1999) suggestion of using C [h (S, ) h (S, )] is quite accurate with the error in the meridional heat flux shown by the dashed line.

13 MAY 23 MCDOUGALL 957 C was evaluated at zero ressure and at the salinity of the fluid arcel as the average heat caacity between the temeratures zero and. In this way, C is actually equal to h (S, ) h (S, ) (D. Jackett 22, ersonal communication) and since h (S, ) varies by only 125 Jkg 1, equivalent to.31c, [see Fig. 2a of this aer or Table A4 of Feistel and Hagen (1995)] over the full range of salinity, Warren s C is very nearly otential enthaly. Figure 9c confirms that while the meridional flux of C is not a articularly accurate exression for the flux of internal energy, it is quite an accurate aroximation for the flux of h. Warren (1999) showed that the meridional flux of C was a very good aroximation to the flux of the Bernoulli function; a result that is consistent with the next section of this aer where it is found that the Bernoulli function and otential enthaly are the same u to about.3c in temerature units, that is, B h.3 C. It is concluded that resent ocean models contain tyical errors of.1c due to the neglect of the nonconservative roduction of although the error is as large as 1.4C in isolated regions such as where the warm fresh Amazon water discharges into the ocean. The corresonding tyical error in the meridional heat flux is.5 PW (or.4%). To eliminate these errors one must (i) interret the model s temerature variable as rather than as, (ii) carry the equation of state as (S,, ) (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 using the inverse function (S, ) when SST is required (e.g., in order to calculate air sea fluxes with bulk formulas). These issues will be exlored in a subsequent aer. While errors of.4% in the meridional heat flux are much smaller than our ability to determine these heat fluxes from observations, errors of.1c in sea surface temerature do not seem to be totally trivial. 1. 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 geootential, gz, a conservation equation is found for the Bernoulli function, B h (1/2)u u, namely (see Batchelor 1967 or Gill 1982) (B) (ub) t [ ] 1 FQ u u. (34) t 2 The last term here is negligible in the ocean interior, being many orders of magnitude smaller than even the tiny term M in (2). Hence aart from the unsteady ressure term, (34) is in the form of a clean conservation equation [like (4)]. If it were not for the 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 reexress the dissiation of mechanical energy, 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 acceted ractice 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 1 of Haltiner and Williams (198)]. Continuing to ignore the last term in (34) we see that B is totally conserved when fluid arcels mix at constant ressure. In this regard B is suerior to h because otential enthaly is not 1% conserved when mixing haens in the subsurface ocean, and as a result is in error by u to 1 mk. The range of ressure variation at fixed deth (due to the movement of mesoscale eddies) is tyically 1 4 Pa (1 dbar) which is equivalent to a change in enthaly of 1 J kg 1, which in turn is equivalent to a temerature change of 2.5 mk. An adiabatic and isohaline change in ressure will cause a change in the Bernoulli function of this magnitude, whereas otential enthaly is totally indeendent of such ressure variations. In this regard h and are suerior to B. It is ossible to imagine an ocean model carrying the Bernoulli function as its temerature variable. The temoral change of ressure 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 and at each time ste so it would be ossible to calculate enthaly from h B (1/2)u u and to use this as an argument of an equation of state in the functional form (S, h, ). In this way the small error of 1 mk that 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 two ressure integral terms in (25).] While imlementing the B conservation equation (34) in an ocean model would avoid any aroximations in the total energy budget, what would be lost is the notion that the model variable B is a roerty of a water mass. Rather, B varies with ressure to the extent of 2.5 mk. This temerature increment haens to be the stated accuracy of modern CTD instruments and is larger than the maximum error (1 mk) in using conservative temerature. The rincial difficulty with using B as an oceanograhic energy-like variable is not however due to the rather small deendence of B on ressure, but rather is due to it not being a locally determined quantity: in addition to B being a function of the locally measured roerties S, T, and, it also contains dynamical information in the geootential function (as well as being deendent on the magnitude of the three-dimensional velocity vector). While both and are known when