Surface flow of granular material is investigated within a continuum approach in two dimensions. The dynamics is described by the nonlinear coupling between a mobile layer and an erodible bed of static grains. 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 bed. This approach allows the rheology in the flowing layer to be specified independently, and we consider in detail the two following models: a constant plug flow and a linear velocity profile. We study and compare these models for nonstationary avalanches triggered by a localized amount of mobile grains on a bed of initially constant slope. We solve analytically the nonlinear 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.
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