A family of mixed finite element pairs with optimal geostrophic balance

We introduce a family of mixed finite element pairs for use on geodesic grids and with adaptive mesh refinement for numerical weather prediction and ocean modelling. We prove that when these finite element pairs are applied to the linear rotating shallow water equations, the geostrophically balanced states are exactly steady, which means that the numerical schemes do not introduce any spurious inertia-gravity waves; this makes these finite element pairs in some sense optimal for numerical weather prediction and ocean modelling applications. We further prove that these finite element pairs satisfy an inf-sup condition which means that they are free of spurious pressure modes which would pollute the numerical solution over the timescales required for large-scale geophysical applications. We then discuss the extension to incompressible Euler-Boussinesq equations with rotation, and show that for the linearised equations the balanced states are again exactly steady on arbitrary unstructured meshes. We also show that the discrete pressure Poisson equation resulting from these discretisations satisfies an optimal stencil property. All these properties make the discretisations in this family excellent candidates for numerical weather prediction and large-scale ocean modelling applications when unstructured grids are required.