Approximation of the hydrostatic Navier-Stokes system for density stratified flows by a multilayer model

We present a multilayer Saint-Venant system for the numerical simulation of free surface density-stratified flows over variable topography. The proposed model formally approximates the hydrostatic Navier-Stokes equations with a density that varies depending on the spatial and temporal distribution of a transported quantity (temperature, salinity, ...). The derivation is obtained by finite element Galerkin type discretization of the Navier-Stokes sytem with zeros order polynomials. In contrast with the classical multilayer approach that assumes immiscible fluids, we allow here for mass exchange between layers. We show that the multilayer system admits a kinetic interpretation, and we use this result to formulate a stable finite volume scheme for its numerical approximation. Several numerical experiments are presented including simulations of wind-driven stratified flows and upwelling processes.