Total energy and potential enstrophy conserving schemes for the shallow water equations using Hamiltonian methods: Derivation and Properties (Part 1)

TL;DR

This paper extends energy and enstrophy conserving schemes for shallow water equations to arbitrary polygonal grids using Hamiltonian and Discrete Exterior Calculus methods, enhancing their applicability beyond traditional grid restrictions.

Contribution

It develops generalized schemes for shallow water equations that conserve key physical quantities on complex grids, combining Hamiltonian and Discrete Exterior Calculus techniques.

Findings

Schemes conserve total energy and potential enstrophy on non-orthogonal polygonal grids.

Extension to arbitrary orthogonal spherical polygonal grids achieved.

Framework sets the stage for accurate, structure-preserving numerical simulations.

Abstract

The shallow water equations provide a useful analogue of the fully compressible Euler equations since they have similar characteristics: conservation laws, inertia-gravity and Rossby waves and a (quasi-) balanced state. In order to obtain realistic simulation results, it is desirable that numerical models have discrete analogues of these properties. Two prototypical examples of such schemes are the 1981 Arakawa and Lamb (AL81) C-grid total energy and potential enstrophy conserving scheme, and the 2007 Salmon (S07) Z-grid total energy and potential enstrophy conserving scheme. Unfortunately, the AL81 scheme is restricted to logically square, orthogonal grids; and the S07 scheme is restricted to uniform square grids. The current work extends the AL81 scheme to arbitrary non-orthogonal polygonal grids and the S07 scheme to arbitrary orthogonal spherical polygonal grids in a manner that allows both total energy and potential enstrophy conservation, by combining Hamiltonian methods (work done by Salmon, Gassmann, Dubos and others) and Discrete Exterior Calculus (Thuburn, Cotter, Dubos, Ringler, Skamarock, Klemp and others). Detailed results of the schemes applied to standard test cases are deferred to Part 2 of this series of papers.