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This paper is a study of the duality structure present in the semi geostrophic equations of meterology. We explore a new viewpoint, represented by Legendre transformations between alternative choices of independent variables, including space coordinates, geostrophic co ordinates 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 adoi:10.1098/rspa.1989.0074 fatcat:3bslarasmjg5hjpvr46q6dsfga