Adaptive Grid Refinement for Numerical Weather Prediction.

This dissertation describes the application of an adaptive solution technique to the dynamical equations used in numerical weather models. The adaptive technique employed is that of Berger and Oliger. It uses a finite difference method to integrate the dynamical equations first on a coarse grid and then on finer grids. The location of the fine grids is determined using a Richardson-type estimate of the truncation error in the coarse grid solution. By correctly coupling the integrations on the various grids, periodically re-estimating the error and recreating the finer grids, approximately uniformly accurate solutions are economically produced. Two horizontally refining adaptive models, based on different sets of equations, are developed. The first, based upon the hydrostatic "primitive" equations of meteorology, is used to solve for the advection of a barotropic cyclone and to simulate the development of a baroclinic disturbance which results from the perturbation of an unstable jet. These integrations demonstrate the feasibility of using multiple, rotated, overlapping fine grids. Direct computations of the truncation error confirm the accuracy of the Richardson -type truncation error estimates. The primitive equations do not form a well-posed Initial Boundary Value Problem (IBVP). The second adaptive model, based upon a non-hydrostatic set of equations which do form a well-posed IBVP, is developed and then tested by simulating a developing baroclinic disturbance. The well-posedness of the equations, the necessity for less filtering in the finite difference model and the ability to extend integrations to non-hydrostatic motions are significant reasons for using the new set of dynamical equations in place of the hydrostatic primitive equations. Incorporating vertical refinement into an adaptive model is investigated. The ill-posedness of the primitive equations is a direct result of the hydrostatic approximation and may lead to instabilities in a vertically refining model. This is not a problem with the second set of equations. A more immediate and unsolved problem is that of vertically interpolating the thermodynamic variables of the hydrostatic approximation and the near geostrophic balance present in large scale flows. The atmosphere is very nearly in hydrostatic balance, thus even for the nonhydrostatic model the interpolation problem remains.