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Geoscientific Model Development An interactive open-access journal of the European Geosciences Union
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Volume 10, issue 2 | Copyright
Geosci. Model Dev., 10, 791-810, 2017
https://doi.org/10.5194/gmd-10-791-2017
© Author(s) 2017. This work is distributed under
the Creative Commons Attribution 3.0 License.

Development and technical paper 17 Feb 2017

Development and technical paper | 17 Feb 2017

Total energy and potential enstrophy conserving schemes for the shallow water equations using Hamiltonian methods – Part 1: Derivation and properties

Christopher Eldred1 and David Randall2 Christopher Eldred and David Randall
  • 1LAGA, University of Paris 13, Villetaneuse, France
  • 2Department of Atmospheric Science, Colorado State University, Fort Collins, USA

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.

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This paper represents research done on improving our ability to make future predictions about weather and climate, through the use of computer models. Specifically, we are aiming to improve the ability of such simulations to represent fundamental physical processes such as conservation laws. We found that it was possible to obtain a computer model with better conservation properties by using a specific set of mathematical tools (called Hamiltonian methods).
This paper represents research done on improving our ability to make future predictions about...
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