A Gegenbauer-based Shallow Water solver for a thick “ocean” over a rotating sphere

Gegenbauer Harmonics which are the eigenfunctions of the Linearized Shallow Water Equations in spherical coordinates for a thick layer of ocean are examined as alternative basis functions for global-scale spectral models. The performance of this basis is compared to that of the traditional Spherical Harmonics basis by testing the accuracy and stability with which the two bases simulate a single, analytic, wave mode of the Linearized Shallow Water Equations. For the linear equations our results show that for low initial wavenumbers the Spherical Harmonics are able to conserve the single wave mode with comparable accuracy to that of the proposed Gegenbauer Harmonics basis while at high initial wavenumbers the simulation with the Spherical Harmonics is significantly less accurate than the simulation with the Gegenbauer Harmonics. By considering a range of ocean thicknesses it is found that, for thin oceans, the Spherical Harmonics become numerically unstable after about 150 days, whereas the proposed Gegenbauer Harmonics remain stable even though they too are not the eigenfunctions of the Linearized Shallow Water Equations in thin oceans. This numerical instability of the Spherical Harmonics is independent of the wave's period and was not observed in thick oceans where the simulation remained stable for at least 200 days. Our results suggest that the numerical instability of the Spherical Harmonics originates at the poles. For the non-linear equations our results show that the Spherical Harmonics solutions are less accurate than the Gegenbauer Harmonics even in a thick ocean.