Introduction

The effect of the various discretizations in space and time may be studied systematically in the context of a linearized model (Grotjhan and O’Brien, 1976). With such a model, it is possible to determine analytical solutions both for the system of equations under consideration and for the system obtained after discretization and therefore, by straightforward comparison, to appraise the effect of the chosen numerical schemes. The shallow water barotropic model is used as a study tool, as it has as solutions the two types of waves described by the primitive equation models: slow waves associated with advection terms and fast inertia-gravity waves associated with the Coriolis terms and the adaptation terms (pressure force in the equations of motion and divergence in the continuity equation). Moreover, it can be shown that a primitive equation model with n levels may be considered, once linearized, as the superposition of n shallow water barotropic models of decreasing equivalent heights.

The linearized barotropic model

5.2.1 The equations for the perturbations