A GLOBAL SHALLOW-WATER NUMERICAL-MODEL BASED ON THE SEMI-LAGRANGIAN ADVECTION OF POTENTIAL VORTICITY

A global shallow-water primitive equation model based on the semi-Lagr angian advection of potential vorticity is presented. A modification o f the basic advection scheme needed to stabilize Rossby waves is intro duced. The divergence and continuity equations are the remaining gover ning equations. A two-time-level semi-Lagrangian semi-implicit numeric al scheme that avoids forward extrapolation of non-linear terms is use d. This leads to a set of non-linear implicit equations to be solved a t each time-step. The wind field is expressed in terms of a streamfunc tion and velocity potential, and a spatial discretization based on sec ond-order finite differences on an unstaggered grid is used. The impli cit equations are solved simultaneously using a non-linear multigrid m ethod. The model is integrated for periods of up to 50 days at various resolutions, using a variety of initial conditions including real dat a. Comparisons with an existing semi-Lagrangian finite-difference shal low-water model and an Eulerian spectral shallow-water model are carri ed out. The new model is found to integrate stably and efficiently, an d 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 o f highly non-linear flow (where the true solution is unknown) is prese nted in which it gives results that stand notably apart.