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Orthobaric Density: a Thermodynamic Variable for Ocean Circulation Studies

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Orthobaric Density: a Thermodynamic Variable for Ocean Circulation Studies 2830 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30 Orthobaric Density: A Thermodynamic Variable for Ocean Circulation Studies ROLAND A. DE SZOEKE,SCOTT R. SPRINGER, AND DAVID M. OXILIA College of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, Oregon (Manuscript received 21 October 1998, in ®nal form 30 December 1999) ABSTRACT A new density variable, empirically corrected for pressure, is constructed. This is done by ®rst ®tting com- pressibility (or sound speed) computed from global ocean datasets to an empirical function of pressure and in situ density (or speci®c volume). Then, by replacing true compressibility by this best-®t virtual compressibility in the thermodynamic density equation, an exact integral of a Pfaf®an differential form can be found; this is called orthobaric density. The compressibility anomaly (true minus best-®t) is not neglected, but used to develop a gain factor c on the irreversible processes that contribute to the density equation and drive diapycnal motion. The complement of the gain factor, c 2 1, multiplies the reversible motion of orthobaric density surfaces to make a second contribution to diapycnal motion. The gain factor is a diagnostic of the materiality of orthobaric density: gain factors of 1 would indicate that orthobaric density surfaces are as material as potential density surfaces. Calculations of the gain factor for extensive north±south ocean sections in the Atlantic and Paci®c show that it generally lies between 0.8 and 1.2. Orthobaric density in the ocean possesses advantages over potential density that commend its use as a vertical coordinate for both descriptive and modeling purposes. A geostrophic streamfunction exists for the momentum equations transformed to orthobaric density coordinates so that the gradients of orthobaric density surfaces give precisely the geostrophic shear. A form of Ertel's potential vorticity can be de®ned whose evolution equation contains no contribution from the baroclinicity vector. Orthobaric density surfaces are invariant to the choice of reference pressure. All of these are properties that potential density lacks. In the continuous limit, patched potential density surfaces, which are formed by joining segments of locally referenced potential density surfaces in various pressure ranges and are extensively used in descriptive physical oceanography, become a particular form of orthobaric density surfaces. The method of selecting the segments is equivalent to choosing the virtual compressibility function. This correspondence is a useful aid in interpreting patched potential density surfaces. In particular, there is a material ¯ow across such surfaces that is analogous to the reversible ¯ow across orthobaric isopycnals. 1. Introduction that is determined by the speci®cation of the coordinate variable. As this coef®cient is the chemical potential In a companion paper (de Szoeke 2000) the equations when entropy is used as coordinate, it is termed the of motion were set down in a form in which an arbitrary chemical potential analogue. If conventional potential thermodynamical variable was used as vertical coordi- density is used as vertical coordinate, these salinity- nate instead of depth. The variable was expressed as a gradient terms are by no means negligible in pressure 1 function of pressure, speci®c volume, and salinity ; the ranges more than a few hundred decibars from the ref- only restriction on it was that it be monotonic with erence pressure. respect to depth. It was shown that, in general, using Potential density of seawater was originally put for- such a variable as a coordinate introduces into the mo- ward as a generalization of the notion of potential tem- mentum equations terms explicitly involving salinity perature for a single-component ¯uid like dry air (WuÈst gradients, multiplied by a thermodynamic coef®cient 1933; Montgomery 1938). In the case of dry air, po- tential temperature possesses dual thermodynamic and dynamic properties that make it uniquely useful. It is an alias of entropy, so it can change only because of 1 The third variable, salinity, could be replaced by any variable independent of the other two. The ensuing discussion would still irreversible processes, such as molecular diffusion, tur- follow, with appropriate modi®cations. bulent mixing, diabatic heating, cabbeling, etc.; we may call such a variable quasi-conservative or quasi-mate- rial. It is also the sole determinant in the ¯uid of the Corresponding author address: Dr. Roland A. de Szoeke, College of Oceanic and Atmospheric Sciences, Oregon State University, 104 buoyant force due to density variations. Potential den- Oceanography Admin. Building, Corvallis, OR 97331-5503. sity of seawater, on the other hand, though it may be E-mail: [email protected] quasi-conservative, is severely lacking in its dynamical q 2000 American Meteorological Society NOVEMBER 2000 DE SZOEKE ET AL. 2831 properties. It may exhibit inversions even in regions faces (Reid and Lynn 1971), will be examined (section where the ocean is stably strati®ed (Ekman 1934; Lynn 4). An interesting result will be established; namely, that and Reid 1968). It is not the sole determinant of buoyant in the limit of arbitrarily closely spaced locally refer- force, for salinity gradients on potential density surfaces enced potential density segments, a form of orthobaric may still generate signi®cant contributions to the mo- density is obtained. The speci®cation of the matching mentum balance (de Szoeke 2000). This behavior is a of potential densities at the interfaces between their consequence of the thermobaric character of seawater, pressure ranges is equivalent to selecting the ``best-®t,'' namely, that the thermal expansion coef®cient of sea- or virtual, compressibility. Just as regionally differen- water depends strongly on pressure (Ekman 1934; Mc- tiated methods of patching potential density are em- Dougall 1987a). ployed (Reid 1994), geographically variable extensions McDougall (1987b; also Jackett and McDougall of the orthobaric density idea may likewise be contem- 1997) introduced the notion of neutral density to over- plated. The properties of the latter may be used to gain come some of the dif®culties associated with potential insight into the former. density, including the problem of monotonicity when true strati®cation is stable. However, neutral density is 2. Toward a quasi-conservative thermodynamic not a thermodynamic function; it depends not only on variable the thermodynamic state of a water sample, character- ized by the triplet of pressure, temperature, and salinity, A principal aim of this paper is to determine a variable but also on the geographical location where it was col- n 5 n(p, a), (2.1) lectedÐa dependence determined by reference to a global hydrographic dataset. This hybrid character a function only of pressure p and speci®c volume a 5 makes it dif®cult to work with, both theoretically and r21, that is nearly quasi-conservative2; that is, its sur- practically. faces are nearly normal to the dianeutral vector. We start In this paper we shall consider thermodynamic var- from the density equation, averaged over small-scale iables that are pycnotropicÐfunctions of pressure, p, processes, and speci®c volume, a, only. Such a variable, when used Dr Dp as vertical coordinate, produces no extra gradient terms, 2Gr 5q. (2.2) besides the Montgomery function gradient, in the mo- Dt Dt mentum balance equations (de Szoeke 2000). This pyc- Here q speci®es the sources and sinks of density due to notropic property is what commends such variables for (i) irreversible molecular ¯uxes of heat and salt, which use in de®ning Ertel's (1942) potential vorticity. How- are usually negligible; (ii) recti®cation due to nonli- ever, in general, pycnotropic variables lack the quasi- nearities in the equation of state, which produce cab- conservative property that potential density possesses: beling and thermobaric effects; and (iii) small-scale tur- that the variable changes only because of irreversible bulent eddy transport divergences (McDougall 1987a; processes that transport or create heat and salt. McDougall and Garrett 1992; Davis 1994). The second The aim of this paper is to construct a variable, n, term on the left represents the reversible adiabatic com- which we call empirical orthobaric speci®c volume, that pression in which Gr is the coef®cient of adiabatic com- is a function of p and a, monotonic in depth, and ap- pressibility,3 and may be considered a function of p, a,S: proximately quasi-conservative. By ``approximate,'' we 1/c 2(p, ,S), (2.3) shall mean that adiabatic changes in n due to com- Gr 5 a pressibility are as small as can be arranged for such a c being the speed of sound. Suppose we write variable. We shall construct this variable by empirically G 5G 1DG, (2.4a) ®tting adiabatic compressibility (or sound speed) to a r 0 function of p and a, and then using this best-®t com- where pressibility to construct an integral of the Pfaf®an dif- 1 ferential form for density. G5 , (2.4b) 0 2 The signi®cance of orthobaric speci®c volume (or its c0 (p, a) reciprocal, orthobaric density) is twofold. It may be used 11 as an alternative to potential density, conventional or DG 5 2 , (2.4c) 22 patched, for purposes of descriptive display of ocean- c (p, a, S) c0 (p, a) ographic data. Because of its attractive dynamical and 2 andc0 (p, a) shall be suitably chosen. We call G 0 the thermodynamic properties, it may be useful for theo- retical manipulations and for numerical modeling of ocean circulation. Its dual usefulness for description and modeling derives from the same attributes. 2 Exact conservation of any such variable is impossible (not only because of irreversible processes). The estimation of the degree of The relation of orthobaric density surfaces to patched ``nearness'' to conservation is part of the aim of this paper. potential density surfaces, the latter constructed by join- 3 Usually aGr is called the compressibility.
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