Spatial Discretization of the Shallow Water Equations in Spherical Geometry Using Osherʹs Scheme

The shallow water equations in spherical geometry provide a first prototype for developing and testing numerical algorithms for atmospheric circulation models. Since the seventies these models have often been solved with spectral methods. Increasing demands on grid resolution combined with massive parallelism and local grid refinement seem to offer significantly better perspectives for gridpoint methods. In this paper we study the use of Osherʹs finite-volume scheme for the spatial discretization of the shallow water equations on the rotating sphere. This finite volume scheme of upwind type is well suited for solving a hyperbolic system of equations. Special attention is paid to the pole problem. To that end Osherʹs scheme is applied on the common (reduced) latitude-longitude grid and on a stereographic grid. The latter is most appropriate in the polar region as in stereographic coordinates the pole singularity does not exist. The latitude-longitude grid is preferred on lower latitudes. Therefore, across the sphere we apply Osherʹs scheme on a combined grid connecting the two grids at high latitude. We will show that this provides an attractive spatial discretization for explicit integration methods, as it can greatly reduce the time step limitation incurred by the pole singularity when using a latitude-longitude grid only. When time step limitation plays no significant role, the standard (reduced) latitude-longitude grid is advocated provided that the grid is kept sufficiently fine in the polar region to resolve flow over the poles.