A Vector Derivation of the Semigeostrophic Potential Vorticity Equation

Abstract

The semigeostrophic potential vorticity equation is derived in a vector-based notation for the shallow-water and primitive equations, as models for atmospheric flow. The derivation proceeds from knowledge of the functional form of potential vorticity and starts directly from a vector form of the governing equations. The method makes use of highly involved vector identities and provides for a clearer picture of the nature of the final result, as compared to a component-based derivation. It is, however, limited by the need to know a priori the appropriate functional vector form of the dynamically relevant quantity, such as potential vorticity.