Deep circulations under simple classes of stratification

Abstract

Deep circulations, where the motion field is vertically aligned over one or more scale heights, are considered within the framework of the Primitive Equations in three dimensions. Important in both tropospheric and stratospheric applications, such motions are admitted by special classes of stratification, where two families of thermodynamic surfaces are not completely independent, as is true under generally baroclinic conditions. Under barotropic and equivalent barotropic stratifications, the number of degrees of freedom are reduced by one, so that the attending circulations are governed by a two-dimensional reduction of the full primitive equations.

The reduced two-dimensional equations governing these motions are derived from the three-dimensional primitive equations in spherical geometry. In so doing, a mapping is established between the full primitive equations and a reduced two-dimensional system, under these special classes of stratification. Very little approximation is required in the derivation of this system, particularly within the framework of isentropic coordinates. Although not strictly equal to the shallow water equations, the reduced set is of the same general form and therefore is amenable to the same methods of solution. Moreover, while the shallow water equations serve as a heuristic analogue of atmospheric behavior, the equivalent barotropic system derived here explicitly represents atmospheric variables. The equivalent depth, an arbitrary parameter in the shallow water equations, emerges implicitly in the derivation of this system, which obeys a conservation principle analogous to Ertel's theorem for three-dimensional flow.

Owing to its correspondence to the full primitive equations, the equivalent barotropic system provides a physical basis for carrying out two-dimensional integrations (e.g., detailed transport calculations of very high resolution) to investigate deep atmospheric motions. The reduced system also provides a natural framework for investigating the behavior of total column abundances of atmospheric constituents, e.g., total ozone, measured by satellite. Although the immediate application of this system is to study barotropic circulations, because temperature variations along a lower bounding material surface are fully accounted for, the reduced system may be attractive for investigating a wider class of motions, e.g., baroclinic disturbances where the heat flux is confined to the surface.