Shallow water flow on the sphere

The vector spherical harmonic spectral transform method is used to compute nonlinear time-dependent shallow water flows on the sphere. Several computational experiments are performed. The first consists of a Gaussian geopotential dome on a steady sphere that collapses while producing a wave that travels from the dome and reconstructs at its antipole. The reconstruct is not identical because the flow is nonlinear. The flow continues to travel between the poles and remains smooth until day 7 when it develops a sharp spicule at its original position. Near corners in the velocity appear to develop at earlier times. The method is quite stable without either implicit or explicit smoothing either in space or time. The two-thirds rule is not required, and the solutions converge quadratically corresponding to the temporal error. The second experiment starts with a Gaussian vortex dome on a steady sphere. The vortex dome does not collapse but rather is sustained by a geopotential low that develops early in the flow. The third experiment is the same as the first but on a rotating sphere. The resulting flows are considerably different as demonstrated by an apparent lack of singularities.