A Hermite-based Shallow Water solver for a thin “ocean” over a rotating sphere

Present-day, spectral or spectral transform, Shallow Water solvers in spherical coordinates employ the Spherical Harmonics functions as their basis functions since these functions are the eigenfunctions of the Laplacian on a sphere. Recent theoretical advances made in the study of the Shallow Water Equations have demonstrated that the eigenfunctions of the Shallow Water operator in a thin layer of fluid over a rotating sphere are accurately approximated by Hermite functions. This finding is employed here to examine the performance of Hermite functions as basis functions of global scale spectral solvers. Initialization of a linear Shallow Water spectral solver by an exact, zonally propagating, wave solution of the Linearized Shallow Water Equations enables an assessment of the accuracy of the numerical calculations at subsequent times. Our results show that the Hermite functions solver simulates the exact analytical solution after 100 days with no distortion of the initial wave field and with a less than 1% error in the wave's propagation speed while a Spherical Harmonics solver yields numerical solutions that severely distort the exact solution after 4 days only. By defining appropriate quantitative measures and error criteria the single eigenmode simulations are also examined for their potential application as test cases for global-scale models.