Vorticity-Divergence Semi-Lagrangian Shallow-Water Model of the Sphere Based on Compact Finite Differences

The semi-Lagrangian representation of advection allows circumventing of the CFL restriction on time steps, which is especially severe for finite-difference models on the regular latitude–longitude grid. The distinct features of the presented semi-Lagrangian model are the use of vorticity and divergence as prognostic variables in conjunction with the fourth-order compact finite differences on the unstaggered regular latitude–longitude grid. The key point of this approach is the solution of the Poisson equations on the sphere, which is necessary for reconstructing the velocity field from vorticity and divergence. The accurate and efficient direct solver for this problem is described. The results of the standard test set for shallow-water equations on the sphere demonstrate the accuracy and computational efficiency of the model with the time steps several times greater than in the Eulerian model.