It is shown that the solution of the semi-geostrophic equations 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 geostrophic balance are identified as energy minimizers and a procedure for finding the minimizers is constructed, which is a form of potential vorticity inversion. This defines a generalization 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 Lagrangian conservation 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 geostrophic motion, with additional terms coming from the curvature of the sphere and extra ‘curvature’ resulting from the variable Coriolis parameter in the generalized coordinate transformation. These terms are estimated, and are found to be very small for normal data. The estimate is verified in computations, confirming the accuracy of the local $f$-plane approximation usually made with semi-geostrophic theory.