A global shallow‐water numerical model based on the semi‐lagrangian advection of potential vorticity

Abstract

Abstract A global shallow‐water primitive equation model based on the semi‐Lagrangian advection of potential vorticity is presented. A modification of the basic advection scheme needed to stabilize Rossby waves is introduced. the divergence and continuity equations are the remaining governing equations. A two‐time‐level semi‐Lagrangian semi‐implicit numerical scheme that avoids forward extrapolation of non‐linear terms is used. This leads to a set of non‐linear implicit equations to be solved at each time‐step. the wind field is expressed in terms of a streamfunction and velocity potential, and a spatial discretization based on second‐order finite differences on an unstaggered grid is used. the implicit equations are solved simultaneously using a non‐linear multigrid method. The model is integrated for periods of up to 50 days at various resolutions, using a variety of initial conditions including real data. Comparisons with an existing semi‐Lagrangian finite‐difference shallow‐water model and an Eulerian spectral shallow‐water model are carried out. the new model is found to integrate stably and efficiently, and to require no noise suppressors other than the inherent diffusivity associated with the interpolations. the model gives results that are, in general, very close to those of the comparison models, but a case of highly non‐linear flow (where the true solution is unknown) is presented in which it gives results that stand notably apart.