A shallow‐water semi‐geostrophic model on a sphere

Abstract

AbstractThe f‐plane semi‐geostrophic equations have provided much insight into atmospheric motions and we wish to see if more can be gained by removing the f‐plane restriction. As a first step we begin by considering the shallow‐water semi‐geostrophic equations on a sphere and describe a numerical solution procedure for them based on a semi‐Lagrangian predictor–corrector method. It is demonstrated, numerically, that the method appears to converge to a true solution of the semi‐geostrophic equations for fixed resolution, and this suggests that there is a unique solution to the continuous equations. The results from several idealized and real‐data test problems are presented and show that the scheme is stable and accurate. The results are compared with those obtained from a shallow‐water semi‐Lagrangian primitive‐equation model. The simulations are broadly similar but the semi‐geostrophic solution maintains stronger anticyclones than the primitive‐equation model. This is in agreement with results expected after considering f‐plane baroclinic wave simulations with the effects of spherical geometry taken into account.