Energy-enstrophy conserving compatible finite element schemes for the rotating shallow water equations with slip boundary conditions
This is an incremental extension of an existing method to handle more realistic boundary conditions, important for geophysical fluid dynamics simulations.
The paper extends an energy-enstrophy conserving discretisation for the rotating shallow water equations to domains with slip boundary conditions, removing the previous restriction to boundary-free domains. Numerical tests on a rotating hemisphere confirm conservation properties.
We describe an energy-enstrophy conserving discretisation for the rotating shallow water equations with slip boundary conditions. This relaxes the assumption of boundary-free domains (periodic solutions or the surface of a sphere, for example) in the energy-enstrophy conserving formulation of McRae and Cotter (2014). This discretisation requires extra prognostic vorticity variables on the boundary in addition to the prognostic velocity and layer depth variables. The energy-enstrophy conservation properties hold for any appropriate set of compatible finite element spaces defined on arbitrary meshes with arbitrary boundaries. We demonstrate the conservation properties of the scheme with numerical solutions on a rotating hemisphere.