A Duality Principle in Semigeostrophic Theory

Abstract

The semigeostrophic equations are a filtered approximation to the primitive equations that have been applied with considerable success to the study of atmospheric frontogenesis and, increasingly, to the study of the influence of orography on the balanced component of flow. In the simplest Boussinesq and f-plane form of these equations the requirement that the instantaneous solutions should not be symmetrically unstable can be expressed geometrically as the need for a certain potential related to the geopotential field to be a convex function. Conversely, any convex function can be associated directly with a stable solution of the semigeostrophic equations. This property of convexity is sufficient to allow a natural mapping from any instantaneous solution to a conjugate solution in which the roles of cyclonic and anticyclonic regions are reversed. In terms of this surprising duality,a solution possessing a mature surface front is associated with a conjugate solution whose flow is blocked or separated by orography, allowing some general conclusions concerning the appearance of surface fronts to be translated directly to corresponding statements concerning orographic blocking, and vice versa. It is noted that instantaneous semigeostrophic solutions can be defined by a minimum energy principle, which also provides a physicgl interpretation of the convex potentials we introduce. Due to this variational principle it is possible to show that the typical onset of frontogenesis or of orographic separation from initially smooth data can be viewed in terms of the geometrical singularities of elementary catastrophe theory.