A particle-mesh method for the shallow water equations near geostrophic balance

In this paper we outline several new particle-mesh methods that conserve potential vorticity (PV) in geophysical flows. The new methods are based on an operator splitting of the shallow water equations into an Eulerian gravity wave part and a Lagrangian advection part, each of which exactly preserves PV. Some of these methods are tested and compared with standard pseudospectral methods, based on qualitative features of the solutions obtained (i.e.~dispersion or smoothness of PV contours), as well as quantitative measures of energy and generalized enstrophy preservation, and increase in mean divergence level. Based on these comparisons we suggest several directions for further research