Energy-enstrophy conserving compatible finite element schemes for the rotating shallow water equations with slip boundary conditions

TL;DR

This paper introduces a new compatible finite element scheme for the rotating shallow water equations that conserves energy and enstrophy even with slip boundary conditions, extending previous boundary-free formulations.

Contribution

It develops an energy-enstrophy conserving discretisation that handles slip boundary conditions, requiring boundary vorticity variables, applicable to arbitrary meshes and boundaries.

Findings

Conserves energy and enstrophy on bounded domains.

Works with arbitrary meshes and boundary shapes.

Validated with numerical simulations on a rotating hemisphere.

Abstract

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.