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Keywords:

  • high-order discontinuous Galerkin method;
  • shallow water equations;
  • geophysical flows;
  • boundary discretization;
  • Rattray island

Abstract

In this paper some preliminary results concerning the application of the high-order discontinuous Galerkin (DG) method for the resolution of realistic problems of tidal flows around shallow water islands are presented. In particular, tidal flows are computed around the Rattray island located in the Great Barrier Reef. This island is a standard benchmark problem well documented in the literature providing useful in situ measurements for validation of the model. Realistic elements of the simulation are a tidal flow forcing, a variable bathymetry and a non-trivial coastline.

The computation of tidal flows in shallow water around an island is very similar to the simulation of the Euler equations around bluff bodies in quasi-steady flows. The main difference lies in the high irregularity of islands' shapes and in the fact that, in the framework of large-scale ocean models, the number of elements to represent an island is drastically limited compared with classical engineering computations. We observe that the high-order DG method applied to shallow water flows around bluff bodies with poor linear boundary representations produces oscillations and spurious eddies. Surprisingly those eddies may have the right size and intensity but may be generated by numerical diffusion and are not always mathematically relevant. Although not interested in solving accurately the boundary layers of an island, we show that a high-order boundary representation is mandatory to avoid non-physical eddies and spurious oscillations. It is then possible to parametrize accurately the subgrid-scale processes to introduce the correct amount of diffusion in the model. The DG results around the Rattray island are eventually compared with current measurements and reveal good agreement. Copyright © 2008 John Wiley & Sons, Ltd.