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 numerical 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 stochastic-deterministic 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).