A two-layer shallow water model for bedload sediment transport: convergence to Saint-Venant-Exner model

Abstract

A two-layer shallow water type model is proposed to describe bedload sediment transport. The upper layer is lled by water and the lower one by sediment. The key point falls on the de nition of the friction laws between the two layers, which are a generalization of those introduced in Fern andez-Nieto et al. (ESAIM: M2AN, 51:115- 145, 2017). This de nition allows to apply properly the two-layer shallow water model for the case of intense and slow bedload sediment transport. Moreover, we prove that the two-layer model converges to a Saint-Venant-Exner system (SVE) including gravitational e ects when the ratio between the hydrodynamic and morphodynamic time scales is small. The SVE with gravitational e ects is a degenerated nonlinear parabolic system. This means that its numerical approximation is very expensive from a computational point of view, see for example T. Morales de Luna et al. (J. Sci. Comp., 48(1): 258{273, 2011). In this work, gravitational e ects are introduced into the two-layer system without such extra computational cost. Finally, we also consider a generalization of the model that includes a non-hydrostatic pressure correction for the uid layer and the boundary condition at the sediment surface. Numerical tests show that the model provides promising results and behave well in low transport rate regimes as well as in many other situations.