A global shallow‐water model using an expanded spherical cube: Gnomonic versus conformal coordinates

A model using shallow-water equations with an Arakawa-type scheme for momentum terms is tested on a quasi-uniform geometry on the sphere, derived by a spherical expansion of the inscribed cube based on the gnomonic projection. Thereby, a quasi-homogeneous distribution of grid points is achieved, and a global finite-difference model is designed which does not require Fourier filtering or suffer from the burden of redundant computational points at high latitudes. Difficulties resulting from the directional discontinuity of the coordinate lines crossing the edges of the expanded cube are almost completely eliminated by using the Arakawa B-grid, so that only scalar points are placed along the edges. An alternative approach is developed based on numerical orthogonalization of the grid whereby, inter alia, the directional discontinuity at the edges is avoided at the cost of some accumulation of points in the vicinity of the vertices of the cube. In the customary Rossby-Haurwitz wave-4 tests, both approaches are shown to converge to a visually indistinguishable solution as the resolution is increased. However, with the orthogonalized, conformal grid, convergence towards the asymptotic solution was substantially faster.