Numerical integration of a linear barotropic model using three methods of treating meteorological and gravitational modes separately

Summary

The analytical solution of a linear barotropic model is derived, including details of the quasi-geostrophic initialization procedure. The prognostic equations are integrated using three different methods of treating the meteorological and gravitational modes separately. These are a semi-Eulerian, semi-implicit (EI) technique, a semi-Lagrangian, semi-implicit (LI) procedure, and a split-explicit (SE) method. The stability criteria and phase speeds are derived for each of the three techniques.

The following theoretical conclusions are derived. Of course, in actual numerical integrations particularly those using more complex models, the results are not so unequivocal.

The stability of the EI procedure is governed by the CFL criterion for the meteorological mode. Gravity waves have no effect on the timestep but move more slowly than the analytical waves. The LI method is unconditionally stable with respect to both meteorological and gravitational modes. There is thus no timestep restriction. However, the gravity waves have the same reduced phase speed as in the EI technique. The SE procedure has CFL timestep criteria for both the meteorological and gravitational calculations. However, its gravity wave phase speeds are relatively accurate. Moreover, it is the only one of the three methods that handles the nearly-compensating pressure gradient and Coriolis forces together. From the point of computational efficiency, the LI technique is probably the best.