Bernard Di Martino

Resolution to a three‐dimensional physical oceanographic problem using the non‐linear Galerkin method

This paper presents the numerical results concerning the adaptation of the non-linear Galerkin method to three-dimensional geophysical fluid equations. This method was developed by Marion and Temam to solve the Navier-Stokes two-dimensional equations. It allows a substantial decrease in calculation costs due to the application of an appropriate treatment to each mode based on its position in the spectrum. The large scales involved in the study of geophysical flow require that the earths rotational effects and the existence of a high degree of stratification be taken into account. These phenomena play an important role in the distribution of the energy spectrum. It is shown here that the non-linear Galerkin method is very well-suited to the treatment of these phenomena. First, the method for the particular situation of a rigid-lid with a flat bottom is validated, for which the functional basis used is particularly well-adapted. Then the more general case of a domain exhibiting variable bathymetry is presented, which necessitates the use of the transformation a, thus providing a study domain with a cylindrical configuration.

Simulation of the spread of a viscous fluid using a bidimensional shallow water model

In this paper we propose a numerical method to solve the Cauchy problem based on the viscous shallow water equations in an horizontally moving domain. More precisely, we are interested in a flooding and drying model, used to modelize the overflow of a river or the intrusion of a tsunami on ground. We use a nonconservative form of the two-dimensional shallow water equations, in eight velocity formulation and we build a numerical approximation, based on the Arbitrary lagrangian eulerian formulation, in order to compute the solution in the moving domain.