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 coor- dinate, 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, ˙ ϵ D/Dt, plays a role analogous to vertical temperature. Geostrophic vertical shear is proportional velocity, w ϵ Dz/Dt, in conventional geometrical co- solely to the gradient of the height of potential tem- ordinates. The former is zero (if 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 pro- cesses (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
᭧ 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, water prop- ordinate. Some simple examples are considered in sec- erties (e.g., potential temperature, nutrient concentra- tion 3. tions, passive scalars) are mapped on potential density The thermobaric effect disappears if a linear model surfaces (conventional or patched); sections of water of the equation of state for seawater is assumed. In this properties are compared with sections of potential den- case potential density is a useful generalization of po- sity; sometimes potential density is used as vertical co- tential temperature, with its attendant dynamical prop- ordinate in water property sections. An important water erties (Spiegel and Veronis 1960). Effectively, in situ property that is often mapped or displayed in sections density in the equations of motion may be replaced by or maps is potential vorticity, which is itself inversely potential density. However, the linear model is not an adequate approximation to the seawater equation of state over the full pressure range encountered in the ocean (Fofonoff 1985; Feistel 1993). The equations of motion 1 Additional irreversible processes, such as cabbeling, may con- have nevertheless been often rendered in terms of po- tribute to potential density change (McDougall and Garrett 1992). tential density as vertical coordinate assuming implicitly 2816 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30 a linear equation of state (Robinson 1965; de Szoeke p and Bennett 1993). These equations may only be used H(p, , S) ϭ ͵ ␣(pЈ, , S) dpЈ, (2.4) over limited pressure ranges. Some versions of numer- P (pЈ, , S)␣ץ ical models of ocean circulation based on potential den- p ⌸(p, , S) ϭ dpЈ, (2.5) ץ ͵ sity as the coordinate appear to have overlooked this important quali®cation (Bleck and Smith 1990) and are P (pЈ, , S)␣ץ therefore potentially de®cient in their quantitative rep- p (p, , S) ϭ dpЈ. (2.6) Sץ ͵ resentation of ¯ows at pressures or depths far from the reference pressure. These models are being revised and P corrected to take account of thermobaric effects (Sun et We also de®ne al. 1999). M ϭ H ϩ⌽, (2.7) McDougall (1987a; McDougall and Jackett 1988; where ⌽ϭgz is the geopotential. If were speci®c Jackett and McDougall 1997) developed the idea of neu- entropy, H, ⌸, and would be, within arbitrary additive tral density to overcome some of the dif®culties with terms, the speci®c enthalpy, the temperature, and the potential density. Neutral density, being a function of chemical pseudopotential [the last of these is the term longitude and latitude as well as pressure, temperature, used by Davis (1994)]. When is potential temperature, and salinity, is not a thermodynamic state variable and M is called the Montgomery function or potential, and so not a member of the class considered here. Its relation ⌸ is called the Exner function (Eliassen and Klein- to some of the results established in section 3 will be schmidt 1957).3 We shall apply these names to M and brie¯y discussed. ⌸ in the general case. We call the chemical potential analogue. From (2.6) 2. A generalized thermodynamic coordinate ␣ץ ץ ϭ , (2.6)Ј Sץ pץ -Let us consider the general case of replacing the ver tical coordinate z in the primitive equations of motion ,S p, (in which hydrostatic balance is assumed) by , a ther- which is a kind of haline contraction coef®cient. Its modynamic variable. It is necessary to assume only that relation to the usual haline coef®cient, when ϭ po- z does not change sign. Possible examples of tential density, is exposed in the appendix and calculatedץ/ץ include potential speci®c volume, in situ speci®c vol- in section 4. ume, potential temperature, entropy, etc. The Montgomery and Exner functions can assume In a binary mixture (as we shall take seawater to be) very disparate guises depending on the choice made of the thermodynamic state of any parcel, and hence the . Some examples can be found in section 3. Sun et al. (1999) presented a speci®c formulation for the choice value of the variable , is uniquely speci®ed by three of ϭ potential density. What these authors call Mont- independent variables such as in situ temperature T, gomery function differs from the present M. pressure p, and salinity S: The transformation of the three-dimensional gradient ١p is rather special since p is itself a thermodynamic ϭ (p, T, S). (2.1) variable. The ®rst task is to transform the hydrostatic relation The equation of state relates these three variables to pץ :␣ speci®c volume ␣ ϭϪg. (2.8) zץ ␣ ϭ ␣(p, T, S). (2.2) This becomes zץ pץ Eliminating T between these two equations, one obtains2 ␣ ϭϪg . (2.9) ץ ץ ␣ ϭ ␣(p, ,S). (2.3) Here the derivatives are taken with the horizontal Three thermodynamic functions may be de®ned relative to an arbitrary reference pressure P (usually chosen to be zero for seawater): 3 The example of an ideal gas, with ϭ potential temperature, 1Ϫ1/ furnishes the classical Exner function: ⌸ϭcp(p/P) , where cp is speci®c heat at constant pressure, and is the ratio of speci®c heats
(ϭ1.4 for dry air). The Montgomery function is then M ϭ cpT ϩ gz ϭ⌸ ϩ gz. In the atmosphere, this quantity is the streamfunction 2 More generally, T and S may be eliminated in favor of and of the geostrophic gradient wind in isentropic surfaces (Montgomery (independent thermodynamic coordinates). Then would appear in- 1937). This useful property, among others, does not generalize to the stead of S in what follows. ocean, except for special choices of . NOVEMBER 2000 DE SZOEKE 2817
coordinates x, y held constant. We will encounter two components only, F is the horizontal friction force, and usages of partial derivatives throughout this paper: as (2.14) has been used in (2.15). This equation differs a vertical spatial derivative, as in (2.9), and as a ther- from Hsu and Arakawa's (1990) only by the inclusion
.modynamic derivative, as in (2.5), for example. In the of the ١IS term. The extra term is crucial, however latter case, where there is risk of confusion, as in the It prevents the extension to the ocean of the inference equations following, we shall emphasize thermodynam- (valid for the dry atmosphere, for which ϭ 0) from
,()p,S, indicating that p and S (2.15) that, in the absence of friction forces (F ϭ 0ץ/Hץ) ic usage by writing - shall indicate, as M is a potential for geocentric accelerations. The conץ/ץ ,are held constant, etc. Otherwise in (2.9), that x, y are to be held constant. Thus tinuity equation becomes ץץ Sץ Hץ pץ Hץ Hץ Hץ (١I ´(pu) ϩ (p Ç ) ϭ 0, (2.16 ϭϩ ϩ . (2.10) p ϩ ץ tץ ץS ,pץ ץ p ,Sץ p,S ץ ץ Substituting from (2.4)±(2.6), this may be written where Ϫp is called the thickness. Equation (2.9) shows S that this is literally the weight of material per unit ץ pץ Hץ t isץ/ץ ,(ϭ⌸ϩ␣ ϩ . (2.10)Ј interval in the vertical, per unit area. In (2.16 ץ ץ ץ the time derivative with held ®xed, that is, within . is taken with x, y, t ®xedץ/ץ ;Finally, from (2.9) and (2.7) we have surfaces of constant -S The substantial derivative is invariant under the transץ Mץ ϭ⌸ϩ . (2.11) formation to coordinates and is given by ץ ץ ץץ D (ϩ Ç , (2.17 ١ ´ This is an alternative formulation of the hydrostatic bal- ϭϩu ץ t Iץ ance, in terms of M rather than p. Dt In the horizontal components of the momentum bal- ance the force per unit mass due to pressure gradients where is p. This is transformed to gradients with held ١H D␣ ®xed by using the identity Ç ϵ (2.18) Dt pץ z ١ (p Ϫ I , (2.12 ١ p ϭ ١ plays the role of vertical pseudovelocity, and is speci®edץ ץ/zץHI diagnostically by a combination of the reversible and y)| is the gradient operator along sur- irreversible processes that transport or are sources of ץ ,xץ) ١I ϵ where faces of constant . From (2.4) (see below). These equations must be supplemented by an equa- ,H tion for salinity concentrationץ (IS. (2.13 ١ p ϩ ١␣ ١IIH ϭ ,S pץ DS (S) (١Ip and using SÇ ϵϭq , (2.19␣ Multiplying (2.12) by ␣, eliminating (2.10), (2.6), (2.7), one obtains Dt
١IM Ϫ ١IS. (2.14) and of course, the hydrostatic equation in the form ١Hp ϭ␣ (2.11). , ١IS terms in (2.11) andץ/SץThe occurrence of (2.14) arises from the dependence of the thermodynamic A word on the schematic way in which this system coordinate on S, apart from p and ␣. But for the of equations is to be solved may be in order. The prog- nostic equation (2.16) is solved for p , given u and Ç terms, the latter dependence permits the replacement of [the latter speci®ed diagnostically by (2.18)]. Then p is )M Ϫ⌸k. It is important to noteץ ,١I) ١p ϩ gk by␣ that equations like (2.11) and the resulting identity obtained by integrating with respect to , given a bound- (2.14) occur, with appropriately de®ned M, ⌸, and , ary condition like whatever choice is made for . p ϭ 0at ϭ . The horizontal momentum balance and continuity sfc equations can be obtained in terms of x, y, ,tas in- Simultaneously, Eq. (2.19) is solved for S. Thus enough dependent variables. This transformation has been often information is available to calculate the right side of done, for example, by Hsu and Arakawa (1990), so that (2.11), ⌸ and being speci®ed from substituting p and we may content ourselves with stating the results. The S with (2.5) and (2.6). Hence (2.11) is integrated from horizontal momentum balance becomes bot to sfc to furnish M. Last, prognostic equation (2.15) Du is solved for u. More details may be found by consulting S ϩ F. (2.15) the literature on isopycnic/isentropic coordinate models M ϩ ١ ١ ϭϪf k ϫ u Ϫ Dt II (Bleck and Smith 1990; Hsu and Arakawa 1990; Higdon Here u is the velocity vector composed of the horizontal and Bennett 1996; Higdon and de Szoeke 1997). 2818 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30 uץ a. Pseudovorticity, thermal wind, and potential S)/p ١ Ϫ ١ ⌸ϩS ١)Ϫf k ϫϭ p I I I ץ vorticity
(p, (2.27/(␣ ˜١ p ϫ ˜١) We will obtain counterparts of the familiar vorticity ϭ k ϫ balance with respect to x, y, coordinates. First, we de®ne the relative pseudovorticity, where the second equality follows from (2.24). Equation (2.23) is a prognostic equation for a quantity u ϭ (Ϫ , u, x Ϫ uy), (2.20) similar to Ertel's potential vorticity. Transforming the ˜ ϵϫ١ third component of back to partial derivatives with :) is the gradient operator with respect respect to level coordinatesץ ,١I) ϭ ˜١ where to x, y, , and u is formally taken to have corresponding zuץz Ϫ zyץϫ (´´´), f ϩ ( x Ϫ uy)| ϭ f ϩ ( xϪ uy)|z ϩ zx ˜١ components u, , 0. Taking the pseudocurl of the three-dimensional momentum balance posed by (z), (2.28 ١ ϭ ( f k ϩ Ј)´(k Ϫ the horizontal equations (2.15), and the hydrostatic bal- I
(١Iz is the slope of an isopleth. Dividing (2.28 ance (2.11), and using (2.16), one obtains equations for where Ϫ1 the components of , by p ϭϪg␣ z, we see that
D u f ϩ ( xyϪ u )| (١. (2.29´(u Ϫ f ϭϪgϪ1␣( f k ϩ Ј ˜١ ´ϪϪ Dt p p p p
⌸Ϫy F2 y SϪ Sy The right side of this, involving only conventional gra- ϭϩ (2.21) Ϫ1 pp dients, would be (apart from the constant factor Ϫg ) Ertel's potential vorticity satz de®ned relative to , were
D u the latter a function only of p and ␣ (see below). By ,( Ϫ f observing that, from (2.6 ˜١ ´ Ϫ Dt p p p ץ ץ ץ Ϫ⌸x ϩ F1 x SϪ Sx d ϭ dp ϩ d ϩ dS, Sץ ץ pץ (ϭϪ (2.22 pp the last term on the right of (2.23) may be written (be- D f ϩ xyϪ u ١I ϭ 0) as cause Dt p ץ 11 (S ´ k. (2.30 ١ IIp ϫ ١ S ´ k ϭ ١ ١II ϫ pץf k ϩ F2x Ϫ F1yxyyx S Ϫ S pp Ç ϩϩ , (2.23) ˜١ ´ ϭ ppp -١I␣ ´ k/p;itex ١Ip ϫ By (2.24), this term is also where k ϭ (0, 0, 1). Only the hydrostatic approximation presses the effect of a remnant of the speci®c volume .has been made in obtaining this set of equations. It may gradient, p١IS, that persists in surfaces of constant S)p, may be comparedץ/␣ץ) p),S ϭץ/ץ) be established from (2.4)±(2.6) that The coef®cient S)p, ϭϪ␣bЈ, where bЈ is the haline contractionץ/␣ץ) to .2.24) coef®cient of seawater) .␣ ˜١ p ˜⌸ ϫk ϭϫ١ ˜١ S ˜ ϫϩ١ ˜١ The expression on the right is the baroclinicity vector (Gill 1982), written in terms of thermodynamic coor- b. Buoyancy frequency; gain factor dinates; that on the left is the combination of terms appearing in Eqs. (2.21)±(2.23). Buoyancy frequency n is de®ned by ␣ץ These equations may be compared with the equations g 2 2 for the conventional relative vorticity, n ϭϪg ⌫, (2.31) zץ ␣
Јϭ(Ϫ z, uz,( x Ϫ uy)|z), (2.25) 2 where the adiabatic compressibility, ␣⌫ ϭ ␣/c (c is which may be found in Pedlosky (1987). [The z sub- sound speed), is given by script is written after the staff in the third component ␣ץ 1 of (2.25) to emphasize that this variable is held constant ⌫ϭ ϭϪ␣Ϫ2 . (2.32) 2 p entropy,Sץduring the x, y differentiations.] The ®rst two compo- c nents of and Ј are related by Now regarding ␣ as a function of p, ,S, Ϫ1 (p ϭϪg ␣uz, (2.26ץ/uץu/p ϭ Sץ ␣ץ ץ ␣ץ pץ ␣ץ ␣ץ ϭϩϩ (2.33) zץ Sץ zץ ץ zץ pץ zץ using (2.9). Neglecting all but the last terms on the left of (2.21), (2.22), and the friction terms on the right, one ,S p,S p, ,␣/z ϭϪgץ/pץ obtains the thermal wind relation for geostrophic shear, so that, using the hydrostatic relation NOVEMBER 2000 DE SZOEKE 2819
c. Vertical pseudovelocityץ ⌸ץ g␣␣ץg22 n2 ϭϪ ϩ ϩ ,(z From (2.3ץ pץ ␣pcץ22␣ [],S ,S Çץ ץ ץ (S Ç ϭ pÇ ϩ ␣Ç ϩ S, (2.40ץ ץ g Sץ ␣ץ pץ (ϩ . (2.34 zץ pץ ␣ ,S where SÇ is given by (2.19), and 2 We write the ®rst term as Ϫg ⌬⌫, where ␣⌬⌫ is the ␣2 compressibility anomaly associated with , de®ned by ␣Ç ϩ pÇ ϭ q(␣), (2.41) c2 Ϫ2 Ϫ2 p),S. (2.35) q (␣) being the irreversible sources of speci®c volumeץ/␣ץ) ␣ ϭ c ϩ ⌫⌬ If were entropy, potential temperature, or potential (Davis 1994). A little manipulation puts (2.40) into the speci®c volume (whatever the reference pressure), this form ץ anomaly would vanish. On the other hand, if were some function of p and ␣ only, then ϵ 0, and the Ç ϭϪ␣2(⌬⌫pÇ ϩ q ␣) Ϫ q(S). (2.42) p ,S ץthird term of (2.34) disappears, while Given the irreversible sources, q (␣) and q (S), this relation (p) ϭ ␣Ϫ2 / (2.36ץ/␣ץ)Ϫ␣Ϫ2 p ␣ diagnostically speci®esÇ . We callÇ the vertical pseu- is a function of p and ␣, which we may denote dovelocity because it is the kinematic rate of change of Ϫ1 1/cc22 (p, ␣); ␣⌫ ϭ ␣/ may be called the virtual com- the coordinate ; the quantity Ϫg pÇ is the instan- 000 Ϫ2 Ϫ1 pressibility. Hence taneous mass ¯ux (in kg m s , positive when upward) across surfaces. Unless ⌬⌫ ϭ 0, pÇ makes a contri- Ϫ2 Ϫ2 ⌬⌫ ϭ c Ϫ c0 . (2.37) bution toÇ . Because, from (2.17),
(١I)p ϩ Ç p, (2.43 ´ t ϩ uץ) By de®ning pÇ ϭ Eq. (2.42) may be written ()p,S, (2.38ץ/␣ץ) p),S ϭץ/⌸ץ) ϭ ١I)p ´ t ϩ uץ)(pÇ ϭ ( Ϫ 1 .zץ/ץ we may write the second term of (2.34) as g␣Ϫ1 ϩ p Ϫ1(q (␣) Ϫ q (S)), (2.44) This is the apparent stability, the contribution of gra- p dient to stability. As a proportion of true stability n 2,it where z ϭ 1/z ϭϪg/␣pץ/ץ may be written, after setting ϵ (Ϫ1 Ϫ Ϫ1 S )Ϫ1 ϭ (1 ϩ Ϫ1␣ 2⌬⌫p )Ϫ1. [from (2.9)], p (2.45) S The ®rst equality of (2.45) de®nes a modi®ed gain factorץ gץ ⌫⌬gg22 ϵϭ1 ϩϪ . (2.39) 22 2 2 . The second equality is used in obtaining (2.44) and zץ p ,S ␣nץϪ␣ pn n 2 follows from the elimination of Sz ϭϪgS/␣p and n We shall call this ratio the buoyancy gain factor. The by employing the two equalities of (2.39). If ⌬⌫ ϭ 0,
condition for validity of the transformation of z to , then ϭ 1, whatever the value of ; conversely, if p z be the same sign everywhere, positive say, ϭ 0, then ϭ . The modi®ed gain factor multipliesץ/ץ that requires that Ͼ 0 everywhere. the irreversible buoyancy source terms in (2.44). If were potential speci®c volume (⌬⌫ ϭ 0), the If the modi®ed gain factor is close to 1, it attenuates gain factor would differ from 1 by an amount pro- the apparent vertical motion of surfaces of constant . p and the vertical salinity gradient. This From (2.4), (2.6), (2.7) it is easy to verify thatץ/ץ portional to extra term can reverse the sign of , reversing the ap- gϪ1 ( u ´ )p ( u ´ )z gϪ1( u ´ )M ١I t ϩץ ١I Ϫ t ϩץ ١I ϭ t ϩץ ␣ parent stability, examples of which Lynn and Reid Ϫ Ϫ1 (١I)S. (2.46 ´ t ϩ uץ)showed. Such extreme situations emphasize the ϩ g (1968) importance of the terms in Eqs. (2.11) and (2.15). The ®rst term on the right gives the rate of change of The gain factor may be written in terms of R ϭ the height of a surface of constant ; variations in height a /b S , the stability ratio (a is the thermal expansion Јz Ј z Ј up to several hundred meters may be expected. The coef®cient), as corresponding variations in gϪ1M are very similar to R Ϫ variations in dynamic height of an surface with respect ϭ . (2.39)Ј to reference pressure P; these are bounded in order of R Ϫ 1 magnitude by O(1 m). The variations contributed by the McDougall (1987a) showed how this ratio relates po- salinity term are of similar or lesser magnitude. Hence, to better than 1%, one may approximate (2.46) by tential temperature and salinity gradients on potential Ϫ1 (١I)z. (2.47 ´ t ϩ uץ) ١I)p ഡ ´ t ϩ uץ) density surfaces and on neutral tangent planes. Ϫ␣g 2820 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30
3. Special examples preted as potential speci®c volume. In section 4 we will see the inadequacy of this substitution. A narrower class of thermodynamic variables is made up of those that can be thought of as functions of pres- sure and in situ speci®c volume only, namely, b. Pressure ϭ (p, ␣). (3.1) Suppose we take If such a variable is used instead of one of the more c2 ϭ p Ϫ 1 (3.4) general class depending also on S, Eq. (2.6) implies that ␣ ϵ 0. This greatly simpli®es the forms of the hydro- static equation (2.11), the horizontal momentum balance as the vertical coordinate, where c1 is a constant. (This 2 (2.15), and the pseudovorticity equations (2.21)±(2.23). does not have speci®c volume units, althoughϪc1 / In particular (2.23) furnishes Ertel's potential vorticity would have.) This relation is easily inverted, theorem (Pedlosky 1987), which states that if Ç ϵ ␣ ϭ c2(p Ϫ )Ϫ1, (3.5) D /Dt is negligible (i.e., is nearly conservative), 1 along with friction F, then the potential vorticity given and we may obtain, from (2.4), (2.5), and (2.38), by (2.29) is conserved. Only in constant surfaces of variables of type (3.1) is the associated Montgomery p Ϫ ␣p function the streamfunction of the geostrophic current. H ϭ c2 ln ϭϪc2 ln Ϫ 1 11 c2 When is chosen to have units of speci®c volume 1 (which is trivial to arrange), we call a variable of the 22 2 Ϫcc11␣c 1 type (3.1) an orthobaric speci®c volume (meaning pres- ⌸ϭ Ϫ ϭϪ␣ Ϫ . (3.6) p Ϫ ␣p Ϫ c2 sure-corrected speci®c volume). In a sequel paper (de 1 22 Szoeke et al. 2000) we shall describe an empirically c1 ␣ ϭϭ 22 derived orthobaric speci®c volume designed to mini- (p Ϫ ) c1 mize the compressibility anomaly ⌬⌫ [Eq. (2.35)]. Ϫ1 Ϫ2 p) ϭ c1 , so the function c 0( p, ␣) de®nedץ/ ␣ץ) Also after Eq. (2.36) must coincide with the constant c1. a. In situ speci®c volume Equation (2.39) gives The paradigmatic example of an orthobaric speci®c g2 11 volume variable is in situ speci®c volume ␣ itself. Then ϭ 1 ϩϪ. (3.7) 22 2 Eqs. (2.4)±(2.6) give, for ϭ ϭ ␣, nc c1 H ϭ ␣p, ⌸ϭp, ϭ 0. (3.2) The choice of pressure as coordinate, ϭ p, is recovered → in the limit c1 0. [This choice of coordinate is amply Equation (2.36) shows that the choice of c 0 that gives discussed by Haltiner and Williams (1980)]. In this lim- ϭ ␣ is ϱ. Also, from (2.38), ϭ 1, and from (2.39), it, H → 0, ⌸ → Ϫ␣. Hence from (2.7), M → gz so that 4 the gain factor is (2.11) becomes g2 zץ ( ϭ 1 ϩ . (3.3 22 g ϭϪ␣, (3.8) pץ nc Ϫ1 For n ϭ 1 cph, c ϭ 1500 m s , this parameter is O(10). which is the expected form of the hydrostatic relation. This is quite large. It means that at this strati®cation, To lowest order, (3.8) allows the approximate replace- the compression by the hydrostatic pressure has about Ϫ1 ment of p by Ϫg␣ 0 z in the oceans, where ␣ ϭ ␣ 0 ϩ 10 times the effect on in situ density of thermal and ␦. A more careful calculation gives the geopotential haline strati®cation. Such a large gain factor attests that anomaly, or dynamic height, of one pressure surface, p, ␣ is certainly not a conservative variable. In terms of with respect to another, P, namely (2.42), the effect of compression on␣Ç is paramount. Nevertheless, (3.2) has often been adopted as a basis P for a coordinate framework in oceanic models (Rob- g⌬z ϭ⌬⌽ϭ͵ ␦ dpЈ (3.8)Ј inson 1965; Bleck and Smith 1990; Oberhuber 1993; p de Szoeke and Bennett 1993), although with ␣ inter- (Sverdrup et al. 1942). Equation (2.42) for the pseu- dovelocity gives
cDp2 4 1 2 Ϫ2(␣) This factor is the ratio of vertical gradient of in situ speci®c Ç ϭ 1 Ϫϩc1␣ q , (3.9) volume (multiplied by Ϫg/␣)ton2. It is also the ratio of the difference cDt2 between the slopes of neutral tangent planes and isobaric surfaces to the difference between the slopes of in situ density surfaces and which approaches Dp/Dt in the limit. This equation be- isobaric surface (McDougall 1989). comes merely an identity. On the other hand, because NOVEMBER 2000 DE SZOEKE 2821
-p can be quite large at remote pressures. The stanץ/ץ p ϭ 1, (2.16) becomes a diagnostic equation, which is used to calculateÇ from the divergence of motion in dard for comparison is the haline contraction coef®cient, pressure surfaces. Equation (2.41) is prognostic and ␣bЈϭ0.8 ϫ 10Ϫ6 m3 kgϪ1 psuϪ1, as indicated in the p isץ/ץ must be integrated to furnish ␣. In this way, the role of discussion of the term (2.30) above. Unless → these two equations is reversed in the limit c1 0. The substantially smaller than ␣bЈ, say within the bounds limits of and do not exist, which is of no conse- Ϯ0.1␣bЈ, the use of ␣P has dubious dynamical merit, quence except to af®rm that p is not conservative. as a substantial force due to salinity gradients,
.S), appears in the momentum balance, Eqsץ ,١IS)These two extreme examples (c 0 ϭϱand c 0 ϭ 0) serve to emphasize that sensible, useful coordinate (2.11), (2.15), and the baroclinicity vector in the po- choices may result, even when c 0 is not particularly tential vorticity balance, Eq. (2.23), is substantial. From p for P ϭ 0 may be largerץ/⌸ץclose to true sound speed c(p, ␣,S), though such co- Fig. 1 we see that ϭ ordinates will not be conservative (Ç ± 0). than 3 at great pressure; it is larger than 1.5 for P Ͻ 4ЊC, p Ͼ 2 hbar. It is in any case signi®cantly larger than 1, which is what it is for the paradigmatic example 4. Potential density (3.2). Replacing ⌸ by p in the equations of motion In this section we shall consider the use of potential incurs a large error in the calculation of currents from density as an independent variable. It seems that this is pressure gradients. The difference of from 1 comes what is being done whenever water properties are dis- about because of the dependence of the seawater thermal played and interpreted on potential density surfaces. To expansion coef®cient on pressure [McDougall 1987b; complete the dynamical picture one should try to un- see Eq. (A5) in the appendix]. A variable very similar derstand the force balance on such surfaces. The frame- to has been plotted by McDougall (1987a; see Fig. work for doing this has been erected in section 2. Part 4 of that paper.) p, the latterץ/ץ of what is required is the determination of the ther- There is a strong link between and modynamic functions ⌸, , etc. The salinity gradient scaled by ␣bЈ, where bЈ is the haline contraction co- on coordinate surfaces, multiplied by chemical potential ef®cient, so that the right panels of Fig. 1 closely re- analogue , appears as a force in the momentum bal- semble one another. This relation is derived as Eq. (A7) ance. It prevents the generalized Montgomery function in the appendix. Except within a few hundred decibars p)(␣bЈ)Ϫ1 is larger thanץ/ץ) ,from being a geostrophic streamfunction; that is, the of the reference pressure slope of potential density surfaces alone is not propor- 0.1 for potential temperatures lower than 10ЊC (the dot- p panel of Fig. 1), and even largerץ/ץ tional to the thermal wind shear, but is augmented by ted line on the the baroclinicity vector components lying in the poten- than 1 (dashed line) for pressures greater than 3000 dbar, p. and potential temperatures lower than 2ЊC. This meansץ/ץ tial density surface. The latter are proportional to Potential vorticity is altered by the baroclinicity com- that the explicit dependence in the equations of motion ponent normal to potential density surfaces. The chem- on the salinity gradient is reduced by the use of ␣ 0 as ical potential analogue differs signi®cantly from zero coordinate to smaller than one-tenth what it would oth- only because of the dependence of seawater compress- erwise be only within a few hundred decibars of the sea ibility on temperature (the thermobaric effect). surface. It is scarcely reduced more than 50% at pres- We shall calculate also the buoyancy gain factor sures greater than 1.5 hbar. Similar statements could be for potential density, which is the ratio of the vertical made about the use of ␣ 2 (Fig. 2). The coef®cient ϭ p hews a little closer to 1 than in Fig. 1. Henceץ/⌸ץ gradient of potential density to the true stability. This .p| for P ϭ 2 hbar is not as large as in Figץ/ץ|is an environmental parameter, rather than a thermo- (␣bЈ)Ϫ1 dynamic state variable (like ), and depends on the 1 for P ϭ 0. Nonetheless this coef®cient well exceeds vertical spatial relationship of thermodynamic phases in 0.1 (dotted lines in Fig. 2) except within a few hundred the ocean. It is essential that it be positive, and for decibars of 2 hbar, and is O(1) beyond 1 hbar from the preference close to 1. But for the thermobaric effect, it reference pressure. would indeed be very nearly 1. The buoyancy gain factor 2, corresponding to the The commonest practical example of is potential choice of ␣ 2 as vertical coordinate and calculated from density anomaly P ϭ P Ϫ 1000 referenced to pressure Eq. (2.39), is shown in Fig. 3 for north±south hydro- P. (Common choices of P are 0, 1, 2, 4 hbar [1 hectobar graphic sections in the Atlantic and Paci®c Oceans (Tsu- ϭ 100 bar].) It is slightly more convenient to argue in chiya et al. 1992, 1994). This parameter is close to 1 Ϫ1 terms of potential speci®c volume, ␣P ϵ P .Inthe throughout both ocean sections, especially in the Paci®c, appendix we show how to calculate the ⌸ and var- if layers close to the bottom and the surface are ex- 2 z and n are small and dif®cultץ/2 ␣ץ iables of (2.5), (2.6) for ϭ ␣P. In Figs. 1 and 2 we cluded, where both show these variables and their pressure derivatives for to evaluate reliably. [The 0.4 cph (ϭ7.0 ϫ 10Ϫ4 rad sϪ1) reference pressure P ϭ 0, 2 hbar, as functions of pressure buoyancy frequency contour is marked on Fig. 3 as a
- conץ/Sץ p, and potential temperature P. Their dependence on guide.] This closeness to 1 indicates that the salinity S is rather slight. By de®nition (2.6), and tributions to the momentum and vorticity balances may p are small near the reference pressure P. However, be small. An important exception occurs in the deepץ/ץ 2822 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30
p (top panels); Exner function, ⌸, and its pressureץ/ץ ,FIG. 1. Chemical potential analogue, , and its pressure derivative p (bottom panels) corresponding to potential speci®c volume ␣0 referenced to 0 dbar; for 0 (potential temperatureץ/⌸ץ ,derivative referenced to 0 dbar) ϭϪ1 (dashed), 0, 2, ...,20ЊC; S ϭ 35.0 psu.
Ϫ1 -South Atlantic where 2 values of 0.6 and smaller occur. of u S ϵϪf k ϫ ١I S vectors calculated from sa Ϫ3 This feature is clearly associated with waters below the linity gradients on the 2 ϭ 35.1 kg m surface, deep salinity-maximum core layer, which show very superimposed on the topography of that surface, both strong vertical salinity gradients. It was precisely of obtained from the National Oceanographic Data Cen- z ter global dataset (Boyer and Levitus 1994). Quiteץ/0 ␣ץ these waters that Lynn and Reid (1968) noted that (in present notation) became negative. A section show- large values, occasionally exceeding 5 cm sϪ1 , are ing 0 (de®ned in terms of ␣ 0) would exhibit negative found in the Gulf Stream, Kuroshio Extension, and values in these waters. You and McDougall (1990) dis- Antarctic Circumpolar Current; also around the south- played maps of 0/ 0 (called by those authors) on eastern edges of the subtropical gyres. The chosen several ``neutral surfaces'' in the global ocean, though isopycnal of Fig. 4 is found between the sea surface none as deep as the salinity maximum layer in the South and 800 m, quite far from the reference pressure of Atlantic. 2000 dbar. This is quite a severe test of the smallness
It is incorrect to conclude, even when 2 is close of ١I S, therefore, though not an unreasonable one ١I S ´ k [Eq. (2.24)], of from the viewpoint of representing the wind-driven ١I ϫ to 1, that component S force in the mo- gyres of the upper pycnocline on 2 surfaces. In any ˜the baroclinicity vector, or the ١ mentum balance, is negligible. Figure 4 shows a map case, in ocean models with 2 (or similar) as the ver- NOVEMBER 2000 DE SZOEKE 2823
FIG. 2. As in Fig. 1 but for ␣2 (potential speci®c volume referenced to 2000 dbar); for 2 values (potential temperature referenced to 2000 dbar) given in Fig. 1. tical coordinate, it is necessary to retain the onal velocities (McDougall 1988) are smaller by a sig-
- S) terms in the horizontal momentum bal- ni®cant factorÐprobably because the 0 ϭ 27.25 sur ץ ,١I S) ance and the hydrostatic balance. face does not stray as far from its reference pressure as
We also calculated, though we do not show it, the does the 2 ϭ 35.1 surface. Ϫ1 ١IS ´ k, which ١I ϫ  baroclinicity remnant (2.30) as differs from the meridional component of uS by a. Patched potential density f Ϫ1 ´ u . It is a noisier calculation than u , involving ١I S S To overcome some of the disadvantages of potential  two gradients and their relative orientation. Its magni- Ϫ1 density it is common to use several potential speci®c tude is of order 1 cm s , except in boundary currents volumes referenced to central pressures in particular re- and where the density surface shoals to intersect the sea Ϫ1 stricted pressure ranges (Reid and Lynn 1971). For ex- surface, where it reaches magnitudes of order 5 cm s . ample, consider the following variable, which is called You and McDougall (1990) compute and display a map patched potential speci®c volume: of an analogous quantity on the 0 ϭ 27.25 potential density surface (as well as others). The topography of ␣Јϭ␣ ,0Յ p Ͻ 1 hbar, this surface is shallower than the ϭ 35.1 surface 0 2 ϭ ␣ , 1 hbar Յ p Ͻ 3 hbar, (4.1) used in Fig. 4. You and McDougall's maps appear noisy 2 too, though the magnitudes of their equivalent meridi- ϭ ␣4, 3 hbar Յ p Ͻ p bottom; 2824 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30
FIG. 3. The buoyancy gain factor 2 corresponding to ␣2 for north±south sections in Atlantic and Paci®c Oceans. Buoyancy frequency contour of 0.4 cph is shown dashed. NOVEMBER 2000 DE SZOEKE 2825
Ϫ3 FIG. 4. The geostrophic velocity uS driven by salinity gradients on the 2 ϭ 35.1 kg m potential density surface, superimposed on the topography (in decibars) of that surface. No vectors are plotted where the pressure on the surface is less than 100 dbar.
(١ Ϫ⌫١p ϭϪaЈ١ ϩ bЈ١S, (4.2 -where ␣ 0, ␣ 2, ␣ 4 are potential speci®c volumes refer enced to 0, 2, 4 hbar. This variable, though discontin- uous, certainly is a thermodynamic state variable and where aЈ and bЈ are thermal and haline expansion co- ®ts into the framework of section 2. Figure 5 shows the ef®cients. This vector is not irrotational in general for ⌸ and variables, and their pressure derivatives, cor- a multicomponent ¯uid (nor is any scalar multiple of responding to ␣Ј. The ⌸ variable is quite close to p it), and so does not possess a potential, which would -p otherwise be the obvious choice for ␥. Jackett and Mcץ/ץ over the whole pressure range. By construction is close to zero near the several reference pressures. Dougall (1997) describe how the form of ␥(p, T, S, x, y) Nonetheless it is still comparable to bЈ␣ over much of is determined empirically by ®tting surfaces to the glob- the pressure range. Except within a few hundred deci- al hydrographic dataset. They emphasize the explicit p| exceeds dependence of ␥ on longitude, x, and latitude, yÐaץ/ץ|bars of the reference pressure, (bЈ␣)Ϫ1 0.1 (dotted lines, Fig. 5). Reid and Lynn (1971) con- dependence that formally disquali®es it as a thermo- structed patched surfaces by joining surfaces of partic- dynamic state variable under the meaning of section 2. Although the ideas of section 2 may be extended to ular ␣ 0, ␣ 2, ␣ 4 values that seemed to match well across the transitions between the pressure ranges. The matches cover neutral density also, at least in principle, by in- were not perfect, so there were discontinuities among troducing local de®nitions (varying in x, y) of the func- the surfaces at the transitions. The properties of these tions H, ⌸, and , the utility of such an approach does surfaces will be examined in a sequel paper (de Szoeke not appear to be high. Instead, one may note that, upon et al. 2000). Suf®ce it to say that the neglect of terms regarding ␥ as a function of x, y, z, t, the hydrostatic in (2.11), (2.15), or (2.21)±(2.23) is not well justi®ed balance becomes on the evidence of Fig. 5. 0 ϭ ␣p␥ ϩ gz␥ , (4.3) and the horizontal pressure gradient may be replaced in b. Neutral density the momentum equations by using the identity (z. (4.4 p ϩ g١ ١␣ p ϭ ١␣ Jackett and McDougall (1997) have expounded the concept of neutral density surfaces on which a variable H I I
١I is the gradient along neutral surfaces, ␥ ϭ is constant, and so constructed that spatial gradients (Here ␥ of ␥ are nearly aligned with the dianeutral vector, de- const.) When the pseudocurl of the momentum balance ®ned by is taken, one obtains, in contrast to (2.21)±(2.23), 2826 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30
FIG. 5. Like Fig. 1 but for patched potential speci®c volume; for 0 given in Fig. 1.
D ␥␥␥ u ␣ F2␥ the left. The third equation is prognostic for a potential (u Ϫ f ϭϪ␣ ϩ p Ϫ (4.5 ˜١ ´ϪϪ Dt p p pyy p p vorticity satz based on ␥. In these three equations one ␥␥ ␥ ␥ ␥ readily discerns the transformed contributions of the
-١p. If neutral density is near ϫ ␣D u␥␥␥ ␣F1␥ baroclinity vector Ϫ١ , Ϫ f ϭ ␣xxϪ p ϩ (4.6) ly conserved ˜١ ´ Ϫ Dt p␥␥ p p ␥ p ␥ p ␥ D␥ Dfkf ϩ xyϪ u ϩ ␥Ç ϵ ഠ 0, (4.8) Ç Dt␥ ˜١ ´ ϭ Dt p␥␥ p and friction is negligible, only the baroclinicity vector Ϫ␣ p ϩ ␣ p contributes to potential vorticity change. For the latter ϩ xy yx p to make exactly no contribution it is necessary that ␥ ␥ be a function of p and ␣ alone. However, McDougall F Ϫ F (1988, 1995) makes an argument for the approximate ϩ 2x 1y . (4.7) p conservation of neutral-density potential vorticity, ␥ which depends on the neglect of the baroclinicity term. The ®rst two of these furnish the thermal wind equations On the other hand, You and McDougall (1990) com- upon neglecting friction and all but the third terms on puted and displayed estimates of the baroclinicity term NOVEMBER 2000 DE SZOEKE 2827 for several potential density surfaces in the middepth cient, , multiplying the salinity gradient in the mo- North and South Atlantic. mentum balance generates the baroclinicity vector, Eden and Willebrand (1999) calculated variables sim- which is not negligible for either potential speci®c vol- ilar to Jackett and McDougall's (1997) neutral density ume or patched potential speci®c volume. The several by objectively minimizing gradients of the dianeutral components of the baroclinicity vector occur in the gen- vector (4.2). In particular, they obtained an approximate eralized thermal wind balance for geostrophic shear and neutral density variable that is constrained to be a func- in the equation for potential vorticity. As the baroclin- tion of and S. The latter property ensures that this icity vector is a cross-product between salinity gradient variable, like potential density, is quasi-material, and a and pressure gradient, the accurate estimation of its thermodynamic function, so that it falls into the class components places a premium on accuracy in these two of variables considered here. Its associated Montgomery gradients. This sensitivity ought to be considered when function is not an acceleration potential (geostrophic using potential speci®c volume or any similar thermo- streamfunction). dynamic variable as vertical coordinate in ocean cir- culation models. A simpler, more restricted class of thermodynamic 5. Summary variables exists that are functions only of in situ speci®c It was shown how to write the equations of motion, volume and pressure. These have the property that they continuity, and vorticity in terms of , a quite general generate no salinity gradient terms in the momentum thermodynamic variable (except that it must be mono- balance ( ϵ 0), nor any contribution from the baro- tonic in the vertical), as the vertical coordinate. The clinicity vector in the vorticity balance. (It was of course hydrostatic relation becomes for the latter reason that Ertel (1942) selected them in de®ning his potential vorticity satz.) On the other hand ,S ץM Ϫ ץ⌸ϭ remnant effects of compressibility contribute to the rate while the horizontal pressure gradient force per unit of change of such variables, which is interpreted as a mass becomes cross-coordinate pseudovelocity. In a sequel paper we shall take up the question of whether a form of density .S M Ϫ ١ ١ p ϭ ١␣ H I I corrected only for pressureÐhence a member of this Once is speci®ed as a thermodynamic variable, the classÐcan be constructed that is nearly conservative, forms of M and ⌸, the generalized Montgomery and within acceptable limits, and unambiguously stable. Exner functions, and the chemical potential analogue It is impossible to de®ne a thermodynamic variable are determined as a matter of course from thermody- for seawater that is quasi-conservative and on whose namic manipulations with the equation of state. The surfaces a geostrophic streamfunction can be de®ned ( variable comes about only if depends on salinity S ϵ 0). However, variables that are either quasi-conser- independently of pressure p and speci®c volume ␣. vative or possess geostrophic streamfunctions can be More than that, is signi®cantly nonzero only because readily de®ned. In either case, it is useful to have the seawater compressibility depends on pressure (McDou- equations of motion, continuity, and thermodynamics gall 1987b). given in terms of the designated variable as independent In this paper we have calculated ⌸ and and their coordinate. pressure derivatives when is potential speci®c volume An index of the quality of a thermodynamic variable (the reciprocal of potential density) referenced to var- as coordinate was devised. We called this the buoyancy ious pressures. The former, ⌸, is often tacitly assumed gain factor, , because it gives the ratio of apparent to be approximately the same as pressure, while the stability to the true stability (buoyancy frequency latter, , is overlooked. It was shown that this was not squared). The positiveness of this factor is essential to tenable except within a few hundred decibars of the the monotoneness of the coordinate. Closeness of the reference pressure. Indeed, as a standard of comparison buoyancy gain factor to 1 is considered desirable in a for assessing the importance of , one should compare thermodynamic coordinate, but is by no means essential. Ϫ1 p),S to bЈ, the haline contraction coef®cient: (Pressure is a respectable choice of coordinate for whichץ/ץ) ␣ the ratio of these terms differs signi®cantly from zero, the gain factor is not close to 1.) In fact for 2 as co- indeed is comparable to 1, except near the reference ordinate (potential density referenced to 2000 dbar), the pressure (Figs. 1 and 2). gain factor is quite close to 1 throughout most of the We also calculated ⌸ and for patched potential oceans (Fig. 3). This closeness is still no guarantee that speci®c volume, for which a different central reference a geostrophic streamfunction is available (i.e., that is pressure is used in various pressure ranges. While it negligible) on surfaces of that variable. Ϫ1 p),S is againץ/ץ) ␣ ,appears super®cially that ⌸ ഠ p comparable to bЈ except near the reference pressures Acknowledgments. I thank S. Springer and A. Bennett (Fig. 5). for useful discussions and D. Oxilia for assistance with In examining the vorticity balance it was shown that the calculations. T. McDougall and an anonymous re- the term formed by the chemical-potential-like coef®- viewer provided valuable criticism of an earlier draft. 2828 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30
(p)aЈ(p)␣␣ץ -This research was supported by National Science Foun dation Grants 9319892, 9402891, by the Jet Propulsion ϭϭ , (A.5) (P)aЈ(P)␣␣ץ Laboratory under the TOPEX/Poseidon Announcement P p,S of Opportunity, Contract 958127, and by NASA Grant ␣ץ ץ NAGS-4947. ϭϭ␣(P)bЈ(P) Ϫ ␣(p)bЈ(p), (A.6) Sץp ,Sp, ץ ␣PP␣
APPENDIX where the common dependences on P, S have been suppressed. Because ␣bЈ depends only weakly on p, Eq. Calculation of the H, ⌸, and Functions (A6) shows that
ץ It is shown how to calculate the H, ⌸, and functions (␣bЈ)Ϫ1 Ϫ 1. (A.7) of (2.4)±(2.6) for ϭ ␣P, where ␣P is potential speci®c ഡ p ,Sץ volume, referenced to pressure P. It is convenient to ␣p
-p)␣P,S from 0, deץ/ץ) express (2.4)±(2.6) in differential form: The variation of from 1, and pends on the variation of the thermal expansion coef- -H ®cient aЈ with pressure, as (A5) shows. If no such varץ ϭ ␣, (A.1a) iation occurs (no thermobaric effect), then the integra- p ,Sץ ␣P tion of (A.1), (A.2) gives ϭ 0. In particular, this happens when an equation of state is assumed that is ␣ץ ⌸ץ ϭ , (A.1b) linear in its variables. P p,S Hence, Eqs. (A.1) for H, ⌸, and can be integrated␣ץp ,S ץ ␣ P forward in p by using, for example, a high-order Runge± Kutta scheme. Initial conditions are given by (A.2). The ␣ץ ץ ϭ , (A.1c) variables will depend, as well as on p, on the initial Sץp ,Sp, ץ ␣ ␣ PP parameters P and S. subject to REFERENCES H ϭ⌸ϭ ϭ 0atp ϭ P. (A.2) Bleck, R., 1974: Short-range prediction in isentropic coordinates with The right sides of (A.1) must be calculated from the ®ltered and un®ltered numerical models. Mon. Wea. Rev., 102, equation of state, ␣ ϭ ␣(p, T, S), and the potential spe- 813±829. ci®c volume ␣ ϭ ␣(P, , S) with respect to reference , and L. T. Smith, 1990: A wind-driven isopycnic model of the P P North and equatorial Atlantic Ocean. 1. Model development and pressure P as a function of potential temperature supporting experiments. J. Geophys. Res., 95, 3273±3285. P(p, T, S) referenced to P. By sedulously transforming Boyer, T., and S. Levitus, 1994: Quality control and processing of variables, one obtains historical oceanographic temperature, salinity, and oxygen data. National Oceanic and Atmospheric Administration Tech. Rep. Ϫ1 NESDIS 81, U.S. Department of Commerce, 64 pp. ץ ␣ץ ␣ץ ␣ץ ϭ PP Davis, R. E., 1994: Diapycnal mixing in the ocean: Equations for .T large-scale budgets. J. Phys. Oceanogr., 24, 777±800ץ ץ Tץ ␣ץ PPp,S p,S[] S p,S de Szoeke, R. A., and A. F. Bennett, 1993: Microstructure ¯uxes across density surfaces. J. Phys. Oceanogr., 23, 2254±2264. S. R. Springer, and D. M. Oxilia, 2000: Orthobaric density: A , ␣ץ ␣ץ ϭ P and (A.3) . thermodynamic variable for ocean circulation studies. J. Physץ ץ PPp,S S Oceanogr., 30, 2830±2852. Eden, C., and J. Willebrand, 1999: Neutral density revisited. Deep- Sea Res. II, 46, 33±54. P ␣ץ ץPP␣ץ ␣ץ ␣ץ ␣ץ ϭϪ ϩ Ekman, V. W., 1934: Review of ``Das Bodenwasser und die Glied- .S erung der Atlantischen Tiefsee'' by G. WuÈst. J. Cons. perm. intץS p,T ץ SץPP p,S␣ץS p,T ץS p, ץ ␣P []P Explor. Mer, 9, 102±104. .Eliassen, A., and E. Kleinschmidt, 1957: Dynamic meteorology ␣ץ ␣ץ ␣ץ ϭϪ P . (A.4) Handbuch der Physik, S. FluÈgge, Ed., Vol. 48, Springer-Verlag, 1±154. S ץ P p,S␣ץS p, ץ PP Ertel, H., 1942: Ein neuer hydrodynamischer Wirbelsatz. Meteor. Z., The expressions after the ®rst equalities of (A.3), (A.4) 59, 277±281. Feistel, R., 1993: Equilibrium thermodynamics of seawater revisited. are written in terms of the usual thermal expansion and Progress in Oceanography, Vol. 31, Pergamon, 101±179. Ϫ1 T)p,S, b ϭ Fofonoff, N. P., 1985: Physical properties of seawater: A new salinityץ/␣ץ) ␣ haline contraction coef®cients a ϭ Ϫ1 Ϫ1 scale and equation of state for seawater. J. Geophys. Res., 90 /␣ץ) ␣S)p,T. The second equalities use aЈϭץ/␣ץ) ␣Ϫ .S) , where ␣ is thought of as (C2), 3332±3342ץ/␣ץ) ) , bЈϭϪ␣Ϫ1ץ P p,S p,P Gill, A. E., 1982: Atmosphere±Ocean Dynamics. Academic Press, a function of p, P, S (Gill 1982; Jackett and McDougall 662 pp. 1995). Haltiner, G. J., and R. T. Williams, 1980: Numerical Prediction and Hence Dynamic Meteorology. 2d ed. Wiley, 477 pp. NOVEMBER 2000 DE SZOEKE 2829
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