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Using the shallow water equations, a numerical framework on a spherical geodesic grid that conserves domain- integrated mass, potential vorticity, potential enstrophy, and total energy is developed. The numerical scheme is equally applicable to hexagonal grids on a plane and to spherical geodesic grids. This new numerical scheme is compared to its predecessor and it is shown that the new scheme does considerably better in conserving potential enstrophy and energy. Furthermore, in a simulation of geostrophic turbulence, the new numerical scheme produces energy and enstrophy spectra with slopes of approximately K23 and K 21, respectively, where K is the total wavenumber. These slopes are in agreement with theoretical predictions. This work also exhibits a discrete momentum equation that is compatible with the Z-grid vorticity-divergence equation.

A Potential Enstrophy and Energy Conserving Numerical Scheme for Solution of the Shallow-Water Equations on a Geodesic Grid