# Comparison of free-surface and rigid-lid finite element models of barotropic instabilities

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## Abstract

The main goal of this work is to appraise the finite element method in the way it represents barotropic instabilities. To that end, three different formulations are employed. The free-surface formulation solves the primitive shallow-water equations and is of predominant use for ocean modeling. The vorticity–stream function and velocity–pressure formulations resort to the rigid-lid approximation and are presented because theoretical results are based on the same approximation. The growth rates for all three formulations are compared for hyperbolic tangent and piecewise linear shear flows. Structured and unstructured meshes are utilized. The investigation is also extended to time scales that allow for instability meanders to unfold, permitting the formation of eddies. We find that all three finite element formulations accurately represent barotropic instablities. In particular, convergence of growth rates toward theoretical ones is observed in all cases. It is also shown that the use of unstructured meshes allows for decreasing the computational cost while achieving greater accuracy. Overall, we find that the finite element method for free-surface models is effective at representing barotropic instabilities when it is combined with an appropriate advection scheme and, most importantly, adapted meshes.

## Keywords

Finite element method Unstructured meshes Barotropic instabilities Free-surface flow## Notes

### Acknowledgements

Laurent White is a research fellow and Eric Deleersnijder is a research associate with the Belgian National Fund for Scientific Research (FNRS). The present study was carried out within the scope of the project “A second-generation model of the ocean system,” which is funded by the Communauté Française de Belgique, as Actions de Recherche Concertées, under contract ARC 04/09-316. This work is a contribution to the construction of SLIM, the Second-Generation Louvain-la-Neuve Ice-ocean Model (http://www.climate.be/SLIM). The authors are indebted to Benoit Cushman-Roisin for the comments he provided during the first stages of the preparation of this paper, and they would also like to thank the two anonymous reviewers for useful suggestions that helped improve this manuscript.

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