Surface flows of granular materials: A modified picture for thick avalanches

Some time ago, Bouchaud et al. [J. Phys. I 4, 1383 (1994)] proposed a basic set of equations to describe surface flows. They assumed in particular that the rate of erosion (or accretion, depending on the slope) was proportional to the local amount R of rolling species. This is natural for thin avalanches, but not for thick avalanches. We discuss here the thick limit and assume that for $R\ensuremath{\gg}d$ (the grain diameter) the rates become independent of R. This leads to some different features: (i) filling of a silo, for which the steady-state slope is a (decreasing) function of the feeding rate; (ii) avalanches with a sink at the bottom end (``open cells''), for which the profile starts at a certain angle ${\ensuremath{\theta}}_{\mathrm{max}}$ and ends at the neutral angle ${\ensuremath{\theta}}_{n},$ where ${\ensuremath{\theta}}_{n}$ is the angle at which erosion balances accretion and is smaller than ${\ensuremath{\theta}}_{\mathrm{max}}$ $({\ensuremath{\theta}}_{n}={\ensuremath{\theta}}_{\mathrm{max}}\ensuremath{-}\ensuremath{\delta});$ and (iii) avalanches with a closed end (where the flow stops), for which the angle of repose is not ${\ensuremath{\theta}}_{n}$ but ${\ensuremath{\theta}}_{n}\ensuremath{-}\ensuremath{\delta}={\ensuremath{\theta}}_{\mathrm{max}}\ensuremath{-}2\ensuremath{\delta}.$ Each avalanche involves a cascade of successive regimes that are described analytically.