Final report for symmetric truncations of the shallow water equations and modelling of coherent structures in a boundary layer

In this final report the authors will briefly describe three achievements in the funding period, related to the original goals of the research, and attach some representative papers for each achievement. The original goal of the project was to implement a numerical scheme to numerically solve the shallow water equations (in periodic boundary conditions and on the sphere), in such a way as to preserve not just the enstrophy, but all other invariants consistent with the level of the numerical truncation. The theoretical framework for this work was set up successfully and simple numerical runs were carried out. However because the scheme was set up in Lagrangian coordinates, and even though preserved the invariants in the desired way, it was inefficient. Therefore this idea was put aside. The second goal achieved arose indirectly from the work on the original project however. The Hamiltonian formalism used in the constructions of the original methods, were used successfully in treating a problem in boundary layer theory. This problem is of general fluid dynamical interest and therefore applies to the atmospheric boundary layer as well. The third goal achieved is much closer in spirit to the original project. Unfortunately, at the time the proposal for renewal was written, this was still not developed well, and could not be included in the renewal request. However, now the project to be described is nearly complete. This project led ultimately to an interesting method for solving the shallow water equations on the sphere. This method was implemented, and led to successful results, and in their opinion, an improvement over the shallow water solver supplied by NCAR