  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.
