A finite element exterior calculus framework for the rotating shallow-water equations

We describe discretisations of the shallow water equations on the sphere using the framework of finite element exterior calculus, which are extensions of the mimetic finite difference framework presented in Ringler, Thuburn, Klemp, and Skamarock (Journal of Computational Physics, 2010). The exterior calculus notation provides a guide to which finite element spaces should be used for which physical variables, and unifies a number of desirable properties. We present two formulations: a ``primal'' formulation in which the finite element spaces are defined on a single mesh, and a ``primal-dual'' formulation in which finite element spaces on a dual mesh are also used. Both formulations have velocity and layer depth as prognostic variables, but the exterior calculus framework leads to a conserved diagnostic potential vorticity. In both formulations we show how to construct discretisations that have mass-consistent (constant potential vorticity stays constant), stable and oscillation-free potential vorticity advection.