  A few years ago, Bouchaud al. introduced a phenomenological model to describe
surface flows of granular materials [J. Phys. Fr. I, 4, 1383 (1994)]. According
to this model, one can distinguish between a static phase and a rolling phase
that are able to exchange grains through an erosion/accretion mechanism.
Boutreux et al. [Phys. Rev. E, 58, 4692 (1998)] proposed a modification of the
exchange term in order to describe thicker flows where saturation effects are
present. However, these approaches assumed that the downhill convection
velocity of the grains is constant inside the rolling phase, a hypothesis that
is not verified experimentally. In this article, we therefore modify the above
models by introducing a velocity profile in the flow, and study the physical
consequences of this modification in the simple situation of an avalanche in an
open cell. We present a complete analytical description of the avalanche in the
case of a linear velocity profile, and generalize the results for a power-law
dependency. We show, in particular, that the amplitude of the avalanche is
strongly affected by the velocity profile.
