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

Abstract

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 earth's 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 σ, thus providing a study domain with a cylindrical configuration. Copyright © 1999 John Wiley & Sons, Ltd.

The exponential behavior of a stochastic Cahn-Hilliard-Navier-Stokes model with multiplicative noise

“…The long-time behavior of flows is a very interesting and important problem in the theory of fluid dynamic. As the vast literature [27,15,3,21,23,25,20,19,11,10] shows, the problem has been receiving very much attention over the last three decades. Another interesting question is to analyze the effects produced on a deterministic system by some stochastic or random disturbances appearing in the model.…”