The Contour-Advective Semi-Lagrangian Algorithm for the Shallow Water Equations

A new method for integrating shallow water equations, the contour-advective semi-Lagrangian (CASL) algorithm, is presented. This is the first implementation of a contour method to a system of equations for which exact potential vorticity invertibility does not exist. The new CASL method fuses the recent contour-advection technique with the traditional pseudospectral (PS) method. The potential vorticity field, which typically develops steep gradients and evolves into thin filaments, is discretized by level sets separated by contours that are advected in a fully Lagrangian way. The height and divergence fields, which are intrinsically broader in scale, are treated in an Eulerian way: they are discretized on an fixed grid and time stepped with a PS scheme. In fact, the CASL method is similar to the widely used semi-Lagrangian (SL) method in that material conservation of potential vorticity along particle trajectories is used to determine the potential vorticity at each time step from the previous one. The crucial difference is that, whereas in the CASL method the potential vorticity is merely advected, in the SL method the potential vorticity needs to be interpolated at each time step. This interpolation results in numerical diffusion in the SL method. By directly comparing the CASL, SL, and PS methods, it is demonstrated that the implicit diffusion associated with potential vorticity interpolation in the SL method and the explicit diffusion required for numerical stability in the PS method seriously degrade the solution accuracy compared with the CASL method. Moreover, it is shown that the CASL method is much more efficient than the SL and PS methods since, for a given solution accuracy, a much coarser grid can be used and hence much faster computations can be performed.