1LAGA, University of Paris 13, Villetaneuse, France
2Department of Atmospheric Science, Colorado State University, Fort Collins, USA
Received: 13 Sep 2016 – Discussion started: 04 Oct 2016
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 for 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.
Revised: 28 Dec 2016 – Accepted: 04 Jan 2017 – Published: 17 Feb 2017
Eldred, C. and Randall, D.: Total energy and potential enstrophy conserving schemes for the shallow water equations using Hamiltonian methods – Part 1: Derivation and properties, Geosci. Model Dev., 10, 791-810, doi:10.5194/gmd-10-791-2017, 2017.