This paper presents the combined isentropic and spherical geostrophic coordinate version of semigeostrophic theory. This is accomplished by first proposing a spherical coordinate generalization of the geostrophic momentum approximation and discussing its associated conservation principles for absolute angular momentum, total energy, potential vorticity and potential pseudodensity. We then show how the use of the spherical geostrophic coordinates allows the equations of the geostrophic momentum approximation to be written in a canonical form that makes ageostrophic advection implicit. This leads to a simple equation for the prediction of the potential pseudodensity. The potential pseudodensity can then be inverted to obtain the associated wind and mass fields. In this way the more general semigeostrophic theory retains the same simple mathematical structure as quasi-geostrophic theory—a single predictive equation which does not explicitly contain ageostrophic advection and an invertibility principle. The combined use of isentropic and spherical geostrophic coordinates is crucial to retaining this simplicity.

In order to demonstrate how the theory applies to problems of barotropic–baroclinic instability and Rossby–Haurwitz wave dispersion, we derive the semigeostrophic generalization of the Charney–Stern theorem and compare the semigeostrophic Rossby–Haurwitz wave frequencies with those of Laplace's tidal equations. The agreement between these frequencies is generally better than 0.5%. Thus, the theory appears to encompass a wide range of meteorological phenomena including both planetary-scale and synoptic-scale waves, along with their finer scale aspects such as fronts and jets.