Approximation of the hydrostatic Navier-Stokes system for density stratified flows by a multilayer model: Kinetic interpretation and numerical solution

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 such as temperature or salinity. The derivation of the multilayer model is obtained by a Galerkin-type vertical discretization of the Navier-Stokes system with piecewise constant basis functions. In contrast with classical multilayer models in the literature that assume 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 robust finite volume scheme for its numerical approximation. Several numerical experiments are presented, including simulations of wind-driven stratified flows.