The shallow-water equations in non-spherical geometry with latitudinal variation of gravity

The shallow-water equations in spherical geometry are well known. They are derived as a constant-density, constant-gravity specialization of the hydrostatic primitive equations for a thin layer of fluid, bounded below by topography and above by a free surface. It is shown herein that it is possible to derive an analogous set of shallow-water equations in non-spherical (but zonally symmetric) geometry using orthogonal curvilinear coordinates. This equation set is dynamically consistent, possessing conservation principles for mass, axial angular momentum, energy and potential vorticity. Furthermore, gravity is allowed to vary, as it does physically, as a function of latitude. This prepares the way for performing sensitivity tests, in an idealized framework, to assess the possible impact of latitudinal variation of gravity. Illustrative examples of models of gravity and specific non-spherical coordinate systems are given.