Dynamics of granular avalanches caused by local perturbations
  Surface flow of granular material is investigated within a continuum approach in two dimensions. The dynamics is described by a non-linear coupling between the two `states' of the granular material: a mobile layer and a static bed. Following previous studies, we use mass and momentum conservation to derive St-Venant like equations for the evolution of the thickness R of the mobile layer and the profile Z of the static bed. This approach allows the rheology in the flowing layer to be specified independently, and we consider in details the two following models: a constant plug flow and a linear velocity profile. We study and compare these models for non-stationary avalanches triggered by a localized amount of mobile grains on a static bed of constant slope. We solve analytically the non-linear dynamical equations by the method of characteristics. This enables us to investigate the temporal evolution of the avalanche size, amplitude and shape as a function of model parameters and initial conditions. In particular, we can compute their large time behavior as well as the condition for the formation of shocks. 
Thorsten Emig, Philippe Claudin, Jean-Philippe Bouchaud