Abstract

A new algorithm is presented for the solution of the shallow water equations on quasi-uniform spherical grids. It combines a mimetic spatial discretization with a Crank-Nicolson time scheme for fast waves and an accurate and conservative forward-in-time advection scheme for mass and potential vorticity. The algorithm is tested on two families of grids: hexagonal-icosahedral Voronoi grids, and modified equiangular cube-sphere grids. For the cubed-sphere case, a key ingredient is the development of a suitable discrete Hodge star operator for the non-orthogonal grid.

Results of several test cases will be presented. The results confirm a number of desirable properties for which the scheme was designed: exact mass conservation, very good available energy and potential enstrophy conservation, vanishing curl of grad, steady geostrophic modes, and accurate PV advection. The scheme is stable for large wave Courant numbers and for advective Courant numbers up to about one.

The accuracy of the scheme appears to be limited by the accuracy of the various mimetic spatial operators. On the hexagonal grid there is no evidence for damaging effects of computational Rossby modes, despite attempts to force them explicitly.