Summary of accomplishments and research on forward-in-time differencing for shallow fluid flows on the sphere. Final technical report

This past year the authors have continued the development of their 3D grid point model. The general model is nonhydrostatic, is written in nonorthogonal terrain following coordinates, and consistently incorporates either Eulerian or semi-Lagrangian differencing. The non-hydrostatic formulation will be particularly useful when the grid size is small enough to resolve nonhydrostatic processes--for example: convective instabilities, and internal gravity wave dynamics in deep sheared atmospheres. However, the authors have also found that the nonhydrostatic formulation has numerical advantages on coarser grids, such as a better conditioning of the Laplacian matrix that must be inverted. In addition, use of the nonhydrostatic equations allows consistency as the mesh is refined, or when using nested grids. The authors have used this model as a vehicle to compare the advantages of Eulerian and semi-Lagrangian methods. They have incorporated more options into different versions of the basic model. They have implemented a fairly sophisticated subgrid scale turbulence parameterization in the basic model and performed a number of LES studies of planetary boundary layers to calibrate the model. In order to further improve the efficiency of the model, they have developed a new method for treating systems with a variety of time scales. Traditional strategies, including split explicit methods, semi-implicit methods, and mode splitting, all have some inaccuracies and/or stability problems. The new method, which they have named the method of averages (MOA), is based on three steps. To validate the method, they built a 2D shallow water model of an ocean basin