Accuracy and consistency in finite element ocean modeling

by White, Laurent

The intrinsic flexibility of unstructured meshes is compelling for numerical ocean modeling. Complex topographic features, such as coastlines, islands and narrow straits, can faithfully be represented by locally increasing the mesh resolution and because there is no constraint on the mesh topology. In that respect, the finite element method is particularly promising. Not only does it allow for naturally handling unstructured meshes but it also offers additional flexibility in the choice of interpolation and is sustained by a rich and rigorous mathematical framework. This doctoral research was carried out under the auspices of the SLIM (Second-generation Louvain-la-Neuve Ice-ocean Model) project, the objective of which is to develop an ocean general circulation model using the finite element method. This PhD dissertation deals with one-, two- and three-dimensional finite element ocean modeling. We chiefly focus on the accurate representation of some selected oceanic processes and we devote much effort toward using a consistent finite element method to solve the underlying equations. We first concentrate on the finite element solution to a one-dimensional benchmark for the propagation of Poincaré waves with particular emphasis on the discontinuous Galerkin method and a physical justification for computing the numerical fluxes. We then compare three finite element formulations (vorticity - streamfunction, velocity - pressure and free-surface) for the solution to geophysical fluid flow instabilities problems. The prominent -- and remaining -- part of this work deals with three-dimensional ocean modeling on moving meshes. It covers the selection of the right elements for the vertical velocity and tracers through achieving strict tracer conservation and local consistency between the elevation, continuity and tracer equations. The ensuing three-dimensional model is successfully validated against a realistic tidal flow around a shallow-water island. New physical insights are proposed as to the physical processes encountered in such flows.