Total energy and potential enstrophy conserving schemes for the shallow water equations using Hamiltonian methods  Part 1: Derivation and properties
Abstract
The shallow water equations provide a useful analogue of the fully compressible Euler equations since they have similar characteristics: conservation laws, inertiagravity 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) Cgrid total energy and potential enstrophy conserving scheme, and the 2007 Salmon (S07) Zgrid 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 nonorthogonal 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.
 Publication:

Geoscientific Model Development
 Pub Date:
 February 2017
 DOI:
 10.5194/gmd107912017
 arXiv:
 arXiv:1609.03797
 Bibcode:
 2017GMD....10..791E
 Keywords:

 Mathematics  Numerical Analysis