A General Method for Conserving Energy and Potential Enstrophy in Shallow-Water Models

Rick Salmon Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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Abstract

The shallow-water equations may be posed in the form df /dt = {F, H, Z}, where H is the energy, Z is the potential enstrophy, and the Nambu bracket {F, H, Z} is completely antisymmetric in its three arguments. This makes it very easy to construct numerical models that conserve analogs of the energy and potential enstrophy; one need only discretize the Nambu bracket in such a way that the antisymmetry property is maintained. Using this strategy, this paper derives explicit finite-difference approximations to the shallow-water equations that conserve mass, circulation, energy, and potential enstrophy on a regular square grid and on an unstructured triangular mesh. The latter includes the regular hexagonal grid as a special case.

Corresponding author address: Rick Salmon, Dept. 0213, 9500 Gilman Drive, UCSD, La Jolla, CA 92093-0213. Email: rsalmon@ucsd.edu

Abstract

The shallow-water equations may be posed in the form df /dt = {F, H, Z}, where H is the energy, Z is the potential enstrophy, and the Nambu bracket {F, H, Z} is completely antisymmetric in its three arguments. This makes it very easy to construct numerical models that conserve analogs of the energy and potential enstrophy; one need only discretize the Nambu bracket in such a way that the antisymmetry property is maintained. Using this strategy, this paper derives explicit finite-difference approximations to the shallow-water equations that conserve mass, circulation, energy, and potential enstrophy on a regular square grid and on an unstructured triangular mesh. The latter includes the regular hexagonal grid as a special case.

Corresponding author address: Rick Salmon, Dept. 0213, 9500 Gilman Drive, UCSD, La Jolla, CA 92093-0213. Email: rsalmon@ucsd.edu

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