Alternate Function Bases for Global Scale Spectral General Circulation Models

Solutions of the Shallow Water Equations (SWE) on a rotating sphere (AKA Laplace Tidal Equations) were recently derived for baroclinic and barotropic oceans. In baroclinic oceans where the speed of gravity waves is of the order of a few meters per second the solutions are very well approximated by Hermite Functions (i.e. Hermite Polynomials multiplied by a Gaussian “envelope”) while in barotropic oceans where the speed of gravity waves is of the order of 100 m/s the eigenfunctions are Gegenbauer Functions i.e. Gengenbauer Polynomials multiplieג by an “envelope” which is a high power of cos(latitude). These two sets of functions approximate the solutions of the eigenvalue problem associated with zonally propagating wave solutions of the SWE and therefore they provide alternate bases to the Spherical Harmonics basis in spectral general circulation models. Our results in simulating exact solutions of the SWE demonstrate that in barotropic oceans at high wavenumbers numerical simulations by the Gegenbauer Functions are significantly more accurate than simulations by the Spherical Harmonic. In baroclinic oceans numerical simulations by the Hermite Harmonics are far more accurate than simulations by the Spherical Harmonics and time interval over which the latter simulations can be carried out is much shorter than by the former. Similar advantages of the new bases prevail in simulations of the nonlinear equations even though the base functions are not eigensolutions of the nonlinear SWE. Numerical simulations by the Gegenbauer Harmonics are more stable than those by the Spherical Harmonics even in baroclinic oceans where neither set of eigenfunctions is a solution of the associated eigenvalue problem. The poor performance of the Spherical Harmonics basis in baroclinic oceans is attributed to the fact that higher modes have increased resolution mainly near the poles while regular solutions decay with latitude over a scale proportional to the radius of deformation.