The Matsuno baroclinic wave test case

Abstract

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.

Topography based local spherical Voronoi grid refinement on classical and moist shallow-water finite volume models

“…From Fig. 12c-d, comparing the black lines with the white line, we can conclude that the obtained EIGW is propagating in agreement with the analytical speed for both grids since the analytical solutions propagate zonally, the longitude-time Hovmöller diagrams are expected to be straight lines with slope k ω (Shamir et al, 2019). The same analysis holds for the RW Hovmöller diagrams depicted in Fig.…”

“…This test case has been proposed by Shamir et al (2019) and considers analytical wave solutions to the linearised shallowwater equations on the beta plane that approximate analytical solutions to the shallow-water equations on the sphere for low gravity waves speed and are taken as initial and reference solutions for this test case. It is recommended to analyse two types of waves: eastward inertia-gravity wave (EIGW, hereafter) and Rossby wave (RW, hereafter).…”

“…It is recommended to analyse two types of waves: eastward inertia-gravity wave (EIGW, hereafter) and Rossby wave (RW, hereafter). We follow Shamir et al (2019) analyzing both waves in this work. The mean atmosphere depth is set to H = 30m, the wavenumber is defined as k = 5 and the mode number is defined as n = 1.…”

“…In order to guarantee that waves pass through the whole Andes topography based locally refined region, we applied a threedimensional rotation of 21 • in the x − z plane in the VR7 grid. We integrate the shallow-water equations for 100 wave periods for each EIGW and RW, as it is suggested in Shamir et al (2019), for both rotated VR7 and UR7 grids to investigate any local refinement impact on the solution.…”

“…In the first framework, we will consider the non-linear shallow-water equations on the sphere and some standard tests proposed in the literature (Williamson et al, 1992;Galewsky et al, 2004) and a more recent test proposed by Shamir et al (2019). In the second framework, the moist shallow-water model proposed by Zerroukat and Allen (2015) is investigated.…”