Equations of Motion Using Thermodynamic Coordinates

Abstract

The forms of the primitive equations of motion and continuity are obtained when an arbitrary thermodynamic state variable=mrestricted only to be vertically monotonic=mis used as the vertical coordinate. Natural generalizations of the Montgomery and Exner functions suggest themselves. For a multicomponent fluid 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 coefficient that can be completely described given the coordinate variable and the equation of state. In the vorticity balance this term produces a contribution identified with the baroclinicity vector. Only when the coordinate variable is a function only of pressure and in situ specific volume does the coefficient of salinity gradient vanish and the baroclinicity vector disappear. This coefficient is explicitly calculated and displayed for potential specific volume as thermodynamic coordinate, and for patched potential specific volume, where different reference pressures are used in various pressure subranges. Except within a few hundred decibars of the reference pressures, the salinity-gradient coefficient is not negligible and ought to be taken into account in ocean circulation models.