On multilayer shallow water systems

Abstract

The work is devoted to the derivation of a multilayer shallow water model. Considering flows with large friction coefficients, with significants water depth or with important wind effects, the horizontal velocity can hardly be approximated by a vertically constant velocity as in the classical shallow water system. In [2] multilayer shallow water model is proposed, by considering a constant profile of the horizontal velocity at each layer and by including mass and momentum exchange terms between the layers. In this work, starting from the 2D instationary incompressible Navier-Stokes equations and following the technique introduced in [1], we derive a model where the unknowns are the horizontal and vertical velocity as well as the pressure. The horizontal velocity averaging in each layer and the incompressible property lead to a mass equation involving the mass exchanges through the upper and lower interfaces of this layer. The system with that mass equation and the 2D Navier-Stokes problem is first time-discretised in each layer leading to a non linear problem, which is next written in a 2D mixte variational form. Afterwards, the proposed model is obtained by the projection of the 2D variational form on subspaces chosen by specifying the dependence on the vertical variable of the unknown functions. As a particular case we obtain a multilayer shallow water model with hydrostatic pressure, where the unknowns are the height of the fluid and the horizontal velocities at each layer, similarly to the model proposed in [2]. Althougth the definition of the momentum transference terms are different for these two models. Finally, we present some numerical tests.