Erosion-deposition waves in shallow granular free-surface flows

Debris flows can spontaneously develop regular large-amplitude surge waves that are interspersed by periods in which the channel fill is completely stationary. These are important because each individual surge is much more destructive than a steady uniform flow with the same mass flux. In this paper small-scale experiments that exhibit similar behaviour are described. The flow consists of carborundum particles that flow down a rough inclined chute covered with a static erodible layer of the same grains. For inflow conditions close to the minimum depth required for steady uniform flows to exist, small disturbances are unstable, creating waves that rapidly coarsen and grow in size. As the waves become sufficiently large, the troughs between the wave crests drop below a critical thickness and come to rest. A series of steadily travelling waves develop which erode the static layer of particles in front of them and deposit grains behind them, to form a layer that is again stationary. This is, in turn, re-eroded and deposited by the next wave. We term these waves granular erosion-deposition waves. Although erosion and deposition problems are notoriously difficult, a simple model is developed which uses a depth-averaged version of the mu(I)-rheology and Pouliquen and Forterre's extended friction law. The viscous dissipation combines with dynamic, intermediate and static friction regimes to generate finite-length waves with static and mobile regions. The existence of stationary layers fundamentally distinguishes erosion-deposition waves from granular roll waves, which form in slightly deeper flows and are always completely mobilized. Numerical simulations show that the system of equations is able to model both erosion-deposition waves and granular roll waves. Moreover, the computed wave amplitude, wavespeed and coarsening dynamics are in good quantitative agreement with experiments.