Using the shallow water equations, we develop a numerical framework on a spherical geodesic grid that conserves domain-integrated mass, potential vorticity, potential enstrophy, and total energy. The numerical scheme is equally applicable to hexagonal grids on a plane and to spherical geodesic grids. We compare this new numerical scheme to its predecessor and show that the new scheme does considerably better in conserving potential enstrophy and energy. Furthermore, in a simulation of

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... c turbulence, the new numerical scheme produces energy and enstrophy spectra with slopes of approximately and , respectively, where is the total wave number. 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.