Equations of Motion Using Thermodynamic Coordinates

2814 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30 Equations of Motion Using Thermodynamic Coordinates ROLAND A. DE SZOEKE College of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, Oregon (Manuscript received 21 October 1998, in ®nal form 30 December 1999) ABSTRACT The forms of the primitive equations of motion and continuity are obtained when an arbitrary thermodynamic state variableÐrestricted only to be vertically monotonicÐis used as the vertical coordinate. Natural general- izations of the Montgomery and Exner functions suggest themselves. For a multicomponent ¯uid like seawater the dependence of the coordinate on salinity, coupled with the thermobaric effect, generates contributions to the momentum balance from the salinity gradient, multiplied by a thermodynamic coef®cient that can be com- pletely described given the coordinate variable and the equation of state. In the vorticity balance this term produces a contribution identi®ed with the baroclinicity vector. Only when the coordinate variable is a function only of pressure and in situ speci®c volume does the coef®cient of salinity gradient vanish and the baroclinicity vector disappear. This coef®cient is explicitly calculated and displayed for potential speci®c volume as thermodynamic coordinate, and for patched potential speci®c volume, where different reference pressures are used in various pressure subranges. Except within a few hundred decibars of the reference pressures, the salinity-gradient coef®cient is not negligible and ought to be taken into account in ocean circulation models. 1. Introduction perature surfaces is a potential for the acceleration ®eld, Potential temperature, the temperature a ¯uid parcel when friction is negligible. Because the gradient of this would have if removed adiabatically and reversibly from function gives the geostrophic ¯ow when it balances ambient pressure to a reference pressure, is a valuable Coriolis force, it is also called the geostrophic stream- concept in the study of the atmosphere and oceans. In function. Potential vorticity (absolute vorticity normal a single-component ¯uid (like dry air or pure water), it to potential temperature surfaces, scaled by their spac- is an alias of speci®c entropy, one of the key variables ing) is quasi-conservative (Ertel 1942). of thermodynamics. This means that as a ¯uid parcel The dual thermodynamic and mechanical signi®cance moves about, its potential temperature is changed only of potential temperature has commended it as a very by irreversible processes, such as molecular diffusion, useful variable for both descriptive (PalmeÂn and Newton diabatic heating, and turbulent mixing (Davis 1994). We 1969) and theoretical analysis of atmospheric motions. may call this the quasi-conservative, or quasi-material, As an example of the latter, potential temperature has property. (The use of this term is not meant to imply been used as the independent variable in place of the that irreversible processes are negligible.) Potential tem- geometrical vertical coordinate (Eliassen and Klein- perature in a single-component ¯uid is also the sole schmidt 1957). Numerical models of atmospheric cir- determinant of the buoyant force due to variations of culation, called isentropic-coordinate models, have been density. In particular, its vertical gradient is a measure developed on the basis of such a coordinate (Bleck 1974; of the local gravitational stability of the spatial arrange- Haltiner and Williams 1980; Hsu and Arakawa 1990). ment of the density ®eld. Important circulation theorems In such models, the rate of change of potential tem- can also be established on surfaces of constant potential perature, u˙ [ Du/Dt, plays a role analogous to vertical temperature. Geostrophic vertical shear is proportional velocity, w [ Dz/Dt, in conventional geometrical co- solely to the gradient of the height of potential tem- ordinates. The former is zero (if u is perfectly conser- perature surfaces (this is the thermal wind relation). A vative), or perhaps small (if quasi-conservative). In the function can be found whose gradient on potential tem- quasi-conservative case, it is calculated diagnostically as the result of subgrid-scale dientropic mixing processes (de Szoeke and Bennett 1993). This clean sep- aration of dientropic mixing from along-isentropic mix- Corresponding author address: Dr. Roland A. de Szoeke, College of Oceanic and Atmospheric Sciences, Oregon State University, 104 ing within the isentropic-coordinate framework is a con- Oceanography Admin. Building, Corvallis, OR 97331-5503. siderable advantage of its use. E-mail: [email protected] In a multicomponent ¯uid like seawater, potential q 2000 American Meteorological Society NOVEMBER 2000 DE SZOEKE 2815 temperature, notwithstanding its utility in descriptive proportional to the spacing of potential density surfaces physical oceanography, lacks the preponderant advan- (Keffer 1985; Talley 1988). tages it possesses in a dry atmosphere. Its dynamical Montgomery (1938) thought that large-scale turbu- role in determining the buoyancy force must be shared lent motions in the ocean, constrained to lie in potential with salinity. For this reason the idea of potential den- density surfaces approximately, would effect horizontal sity, the density of a water parcel moved from ambient mixing largely along such surfaces. This idea, some- pressure to a reference pressure, appears to have been times framed in terms of neutral trajectories or neutral developed as a generalization of potential temperature tangent planes (McDougall 1987a; Eden and Willebrand (WuÈst 1933; Montgomery 1937, 1938). Potential den- 1999) rather than potential density, has persisted in sity is indeed a quasi-conservative scalar, like potential physical oceanography ever since. In order to begin to temperature.1 However its dynamical properties are se- quantify this notion, it is useful to consider the balances verely lacking: buoyant force does not depend on po- represented by the equations of motion, continuity, ther- tential density alone (except in pressure ranges quite modynamics, and so on, from the viewpoint of surfaces close to the reference pressure), but also on salinity. of constant potential density, neutral density [a variable One consequence is that inversions of potential density devised by Jackett and McDougall (1997)], or any other can occur even when the local stability is positive (Ek- such variable. man 1934; Lynn and Reid 1968). Geostrophic shear For all these reasons it seems valuable to set down does not depend solely on the slope of potential density the primitive equations of motion in terms of potential surfaces, as will be shown in section 2. Not only does density as vertical coordinate. More than this, it would the proportionality vary with pressure (McDougall and be useful to have the equations of motion in terms of Jackett 1988), but there are extra terms in the geo- an arbitrary thermodynamic state variable, any function strophic balance if salinity surfaces do not coincide with of pressure, temperature, and salinity, of which potential potential density surfaces. No acceleration potential or density is one example. This is accomplished in section geostrophic streamfunction exists on potential density 2. The general theoretical framework presented there surfaces (McDougall 1989). Nor is potential vorticity permits the making of some interesting links to other de®ned in relation to potential density surfaces strictly special types of variables. In particular a natural gen- conserved, or even quasi-conserved: in addition to ir- eralization of the Montgomery function emerges, as reversible sources due to diffusion or friction there is a does a thermodynamic coef®cientÐanalogous to chem- reversible source due to a remnant of the baroclinicity ical potential (which is what it would be if speci®c vector (Pedlosky 1987), again associated with salinity entropy were the coordinate)Ðthat multiplies the salin- and potential density surfaces intersecting one another. ity gradient on coordinate surfaces and produces a term These complications involving the use of potential that enters the momentum balance as an additional force. density arise when potential density surfaces and salinity This chemical-potential analogue is de®ned as a matter surfaces do not coincide, and because seawater com- of course when the thermodynamical coordinate-vari- pressibility varies with temperature (Ekman 1934; Reid able is speci®ed. It is displayed in particular for the and Lynn 1971), called the thermobaric effect (Mc- choice of potential density (several reference levels are Dougall 1987b). Reid and Lynn (1971) proposed patch- considered, along with patched potential density) as ver- ing together surfaces of potential density de®ned over tical coordinate (section 4). In these cases, the chemical- narrower pressure ranges (of order 1000 dbar) and ref- potential analogue depends strongly on pressure and erenced to a central pressure in that range. This usually potential temperature and is hardly negligible. The overcomes the dif®culty of potential density inversions chemical-potential-analogue coef®cient vanishes com- occurring despite stable strati®cation. pletely for any member of a special family of thermo- Potential density surfaces (or patched potential den- dynamic variables that can be expressed as functions of sity surfaces) are frequently used in descriptive physical pressure and density alone, when such is used as co- oceanography (Reid 1981). For example,

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