Stochastic-deterministic modeling of bed load transport in shallow water flow over erodible slope: Linear stability analysis and numerical simulation

In this article we propose a stochastic bed load transport formulation within the framework of the frictional shallow water equations in which the sediment transport rate results from the difference between the entrainment and deposition of particles. First we show that the Saint-Venant-Exner equations are linearly unstable in most cases for a uniform base flow down an inclined erodible bed for Shields numbers in excess of the threshold for incipient sediment motion allowing us to compute noise-induced pattern formation for Froude numbers below 2. The wavelength of the bed forms are selected naturally due to the absolute character of the bed instability and the existence of a maximum growth rate at a finite wavelength when the particle diffusivity coefficient and the water eddy viscosity are present as for Turing-like instability. A num e rical method is subsequently developed to analyze the performance of the model and theoretical results through three examples: the simulation of the fluctuations of the particle concentration using a stochastic Langevin equation, the deterministic simulation of anti-dunes formation over an erodible slope in full sediment-mobility conditions, and the computation of noise-induced pattern formation in hybrid stochasticdeterministic flows down a periodic flume. The full non-linear numerical simulations are in excellent agreement with the theoretical solutions. We conclude highlighting that the proposed depth-averaged formulation explains the developments of upstream migrating anti-dunes in straight flumes since the seminar experiments by Gilbert (1914). (C) 2015 Elsevier Ltd. All rights reserved.