Linear Stability of Finite-Difference Approximations on a Uniform Latitude-Longitude Grid with Fourier Filtering

Abstract

The linear stability condition for explicit, second-order, centered difference approximations to the shallow-water equations on a uniform latitude-longitude spherical grid is determined. The grid has points at the poles and on the equator. Two modifications involving Fourier filtering in the longitudinal direction are considered to permit a longer stable time step. In the first, the prognostic variables are filtered, while in the second the zonal pressure gradient in the zonal momentum equation and the zonal divergence in the continuity equation are filtered. Both modifications permit approximately the same time step when the wavenumber cutoff is the same. Although filtering can he performed at all latitudes, the most efficient procedure involves filtering only near the poles. To allow a reasonable time step, all but approximately the six longest waves must be removed at the first row of points next to the pole. This filtering next to the pole determines a time step such that additional filtering is necessary at only a few latitudes, usually poleward of 80°. The stability condition is also found for second- and fourth-order approximations on a shifted mesh in which the first grid points are a half-grid interval away from the pole. Only filtering of the prognostic variables is considered. As in the previous case, to allow a reasonable time step, only the longest waves can be. retained next to the pole. This filtering determines a time step such that filtering is not required at all latitudes. When such filtering is performed, the second- and fourth-order schemes have approximately the same stable time step. The modifications to the free oscillations of the model caused by the filtering are also discussed.