Generalized semi-geostrophic theory on a sphere

Abstract

It is shown that the solution of the semi-geostrophic equatios for shallow water flow can be found and analysed in spherical geometry by methods similar to those used in the existing f-plane solutions. Stable states in geophtrophic balance are identified as energy minimisers and a procedure for finding the minimisers is contructed, which is a form of potential vorticity inversion. This defines a generalisation of the geostrophic coordinate transformation used in the f-plane theory. The procedure is demonstrated in computations. The evolution equations take a simple form in the transformed coordinates, though, as expected from previous work in the literature, they cannot be expressed exactly as geostrophic motion. The associated potential vorticity does not obey a langrangian conservtaion law, but it does obey a flux conservation law, with an associated circulation theorem. The divergence of the flow in the transformed coordinates is primarily that naturally associated with geostraphic motion, with additional terms coming from the curvature of the sphere and extra 'curvature' resulting from the variable Coriolis parameter in the generalised coordinate transformation. The estimate is verified in computations, confirming the accuracy of the local f-plane approximation usually made with semi-geostrophic theory.

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