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 atmospheric and oceanic 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 small speeds of gravity waves such as those of the baroclinic modes in the atmosphere and ocean. 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 large speeds of gravity waves 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 amplitude) for both planetary and inertia gravity waves, and confirms the accuracy of the simulation by employing Hovmöller diagrams and temporal and spatial spectra. The proposed test case is successfully applied to a standard finite-difference, equatorial, non-linear, shallow water model in spherical coordinates, which demonstrates that Matsuno’s wave-solutions can be accurately simulated for at least 10 wave-periods, which for oceanic planetary waves is nearly 1300 days. In order to facilitate the use of the proposed test case, we provide Matlab, Python and Fortran codes for computing the analytic solutions at any time on arbitrary latitude-longitude grids.