Comparison of numerical methods for modelling ocean circulation in basins with irregular coasts

Comprehension of global oceanic currents and, ultimately, of climate variability requires the use of computer modelling. Although much effort has been spent on the accuracy of traditional finite difference (FD) models used in ocean modelling, there are still concerns, especially since these models have a crude representation of the geometry of oceanic basins. Such a crude representation may influence the accuracy of modelling boundary currents, or unrealisticly represent the impinging of eddies or the propagation of Kelvin waves along the coastline. This motivated the use of alternative modelling techniques applied on completely irregular geometries such as finite element (FE) and spectral element (SE) methods. In this thesis, we want to investigate the accuracy and cost-effectiveness of these three numerical methods in irregular domains and to understand to which extent the unstructured grid FE and SE methods constitute an improvement over the more traditional FD methods. To accomplish this, we limit ourselves to modelling the shallow water equations in presence of irregular coastlines with no bottom topography.In the first part of the thesis, we compare the performances of FD methods on Cartesian grids with FE and SE methods in various geometries for linear and non-linear applications. We argue that the SE method is to a certain extent superior to FD methods. In a second part, we study the influence of step-like walls on vorticity budgets for wind-driven shallow water FD models. We show that vorticity budgets can be very sensitive to the FD formulation. This has certain implications for using vorticity budgets as a diagnostic tool in FD models. In the final part, we use a SE shallow water model for investigating the "inertial runaway problem" in irregular domains for the single-gyre Munk problem. Ideally, one would like the statistical equilibrium observed at large Reynolds number to be insensitive to model choices that are not well founded, e.g., the precise value of the viscous coefficient, and choice of dynamic boundary condition. Simple models of geophysical flows are indeed very sensitive to these choices. For example, flows typically converge to unrealisticly strong circulations, particularly under free-slip boundary conditions, even at rather modest Reynolds numbers. This is referred to as the "inertial runaway problem". We show that the addition of irregular coastlines to the canonical problem helps to slow considerably the circulation, but does not prevent runway