  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.
