Preservation of the Discrete Geostrophic Equilibrium in Shallow Water Flows
We are interested in the numerical simulation of large scale phenomena in geophysical flows. In these cases, Coriolis forces play an important role and the circulations are often perturbations of the so-called geostrophic equilibrium. Hence, it is essential to design a numerical strategy that preserves a discrete version of this equilibrium. In this article we work on the shallow water equations in a finite volume framework and we propose a first step in this direction by introducing an auxiliary pressure that is in geostrophic equilibrium with the velocity field and that is computed thanks to the solution of an elliptic problem. Then the complete solution is obtained by working on the deviating part of the pressure. Some numerical examples illustrate the improvement through comparisons with classical discretizations.
KeywordsGeostrophic Adjustment Shallow Water Flows Finite Volume Method Well-balanced Scheme
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- 4.F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources, Birkaüser (2004).Google Scholar
- 5.F. Bouchut, J. Le Sommer, V. Zeitlin, Frontal geostrophic adjustment and nonlinear wave phenomena in one dimensional rotating shallow water. Part 2: high-resolution numerical simulations, J. Fluid Mech., 513, 35–63 (2004).Google Scholar
- 6.M.J. Castro, J.A. Lopez, C. Pares, Finite Volume Simulation of the Geostrophic Adjustment in a Rotating Shallow-Water System SIAM J. on Scientific Computing, 31, 444–477 (2008).Google Scholar
- 11.J. Pedlosky, Geophysical Fluid Dynamics, Springer, 2nd edition (1990).Google Scholar
- 12.G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation, Cambridge University Press (2006).Google Scholar