The Matsuno baroclinic wave test case

Abstract

The analytic wave solutions obtained by Matsuno (1966) in his seminal work on equatorial waves provide a simple and informative way of assessing the performance of atmospheric models by measuring the accuracy with which they simulate these waves. These solutions approximate the solutions of the shallow-water equations on the sphere for low gravity-wave speeds such as those of the baroclinic modes in the atmosphere. This is in contrast to the solutions of the non-divergent barotropic vorticity equation, used in the Rossby–Haurwitz test case, which are only accurate for high gravity-wave speeds such as those of the barotropic mode. The proposed test case assigns specific values to the wave parameters (gravity-wave speed, zonal wave number, meridional wave mode and wave amplitude) for both planetary and inertia-gravity waves, and suggests simple assessment criteria suitable for zonally propagating wave solutions. The test is successfully applied to a spherical shallow-water model in an equatorial channel and to a global-scale model. By adding a small perturbation to the initial fields it is demonstrated that the chosen initial waves remain stable for at least 100 wave periods. The proposed test case can also be used as a resolution convergence test.