Dual Variables in Semigeostrophic Theory

This paper is a study of the duality structure present in the semigeostrophic equations of meterology. We explore a new viewpoint, represented by Legendre transformations between alternative choices of independent variables, including space coordinates, geostrophic coordinates and isentropic coordinates. Detailed examples are given, for both uniform and non-uniform potential vorticity, of the modelling of an atmospheric front at a given time in stable flow. In each case the front is a half-line of gradient discontinuity on a continuous convex surface, obtained as the self-intersection on a swallowtail surface, after the latter is convexified so that non-physical multivaluedness is removed.