A numerical study of an operator splitting method for rotating flows with large ageostrophic initial data

We propose an operator splitting method which is especially suitable for lo ng-time integration of geophysical equations characterized by the presence of multiple-time scales and weak-operator splitting. The method is illustra ted on the classical rotating shallow-water equations on a periodic domain with large ageostrophic (unprepared) initial data. The asymptotic splitting decomposes the solution into a first part which solves the quasigeostrophi c equation; a second one which is the "slow" ageostrophic component of the how; and a corrector. The particular decomposition we use ensures that the corrector is small for large rotation. By considering only the "slow" ageos trophic and quasigeostrophic components a numerical approximation to the sh allow-water equations is derived that effectively removes the time-step res trictions caused by the presence of fast waves. The splitting is exact in t he asymptotic limit of large rotation and includes the nonlinearity of the equations. Numerical examples are included. These examples demonstrate a si gnificant reduction in the computational cost over direct numerical approxi mations of the shallow-water equations. We conclude with an outline of a ge neral operator splitting method for more general primitive geophysical equa tions.