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

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Potential Enthalpy: a Conservative Oceanic Variable for Evaluating Heat Content and Heat Fluxes 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 0.18C and maximum errors of 1.48C 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- u 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 2.58C. * 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 q 2003 American Meteorological Society 945 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 r«M is the rate of dissipation of heat multiplying the ¯ux of potential temperature per kinetic energy (W m23) 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 5 F 1 h F 5 F 1 [m 2 (T 1 T)m ]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 5 m 2 (T 0 1 T)mT 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 5 273.15 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, m 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., m is the difference between the be more conserved than is u by more than two orders partial chemical potential of salt mS and the partial chem- 0 of magnitude. This paper proves that the ¯uxes of h ical potential of water mW in seawater), and mT 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 m and mT 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 1 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. 29 function and potential enthalpy differ by only about 3 For example, a typical dissipation rate, «M,of10 W 3 10238C (when expressed in temperature units), the kg21 causes a warming of only 1023 K (100 yr)21. 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. 22 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 divergence of a molecular heat ¯ux, 2=´FQ. When 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 dr occurs as the average of F at the sea surface and the r 2 (p 1 p) 52= ´ F 1 r« , or (1) Q []dt 0 r2 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.
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