Spectral Transform Solutions to the Shallow Water Test Set

Abstract

Solutions to the test case suite proposed by Williamson et al. ( J. Comput. Phys. 102, 211 (1992)), for the shallow water equations in spherical geometry, are presented. The solutions have been generated using a conventional spectral transform technique combined with a semi-implicit time differencing scheme. For several of the test cases, closed-form solutions do not exist. For these cases, high-resolution numerical integrations of the spectral transform model are used to provide reference solutions against which alternative numerical schemes and lower resolution spectral transform solutions can be evaluated. The sensitivity of the high resolution numerical solutions, associated with temporal truncation, spatial truncation, and internal dissipation, are quantified in order to help bound their uncertainty. In almost all of the test cases, the spectral transform method proves to be a highly accurate solution technique. This is particularly the case at resolutions typically associated with atmospheric general circulation models used to simulate the atmosphere's climate. The most serious deficiency of the spectral transform method, in the context of the test cases, is the introduction of spurious minima and maxima into the solution (caused by Gibbs phenomenon), when sharp gradients exist. Although this behavior is not necessarily a problem for accurately simulating fluid flow, it can become a serious problem for atmospheric general circulation models if the spurious wave structures result in nonphysical states such as negative water vapor mixing ratio.