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We describe a compatible finite element discretisation for the shallow waterequations on the rotating sphere, concentrating on integrating consistentupwind stabilisation into the framework. Although the prognostic variables arevelocity and layer depth, the discretisation has a diagnostic potentialvorticity that satisfies a stable upwinded advection equation through aTaylor-Galerkin scheme; this provides a mechanism for dissipating enstrophy atthe gridscale whilst retaining optimal order consistency. We also use upwinddiscontinuous Galerkin schemes for the transport of layer depth. Thesetransport schemes are incorporated into a semi-implicit formulation that isfacilitated by a hybridisation method for solving the resulting mixed Helmholtzequation. We illustrate our discretisation with some standard rotating spheretest problems.

Higher-order compatible finite element schemes for the nonlinear rotating shallow water equations on the sphere