In this work we study the modeling of one-dimensional avalanche
flows made of a moving layer over a static base, where the
interface between the two can be time dependent. We propose a general
model, obtained by looking for an approximate solution with constant
velocity profile to the incompressible Euler equations. This model
has an energy dissipation equation that is consistent with the depth
integrated energy equation of the Euler system. It has physically
relevant steady state solutions, and, for constant slope, it gives
a particular exact solution to the incompressible hydrostatic
Euler equations. Then, we propose a simplified model, for which
the energy conservation holds only up to third-order terms.
Its associated
eigenvalues depend on the mass exchange velocity between the
static and moving layers. We show that a simplification used in
some previously proposed models gives a non-consistent energy
equation. Our models do not use, nor provide, any equation for the
moving interface, thus other arguments have to be used in order to
close the system. With special assumptions, and in particular
small velocity, we can nevertheless obtain an equation for the
evolution of the interface. Furthermore, the unknown parameters of
the model proposed by Bouchaud, Cates, Ravi and Edwards (BCRE
model) can be derived. For the quasi-stationary case
we compare and discuss the equation for the moving interface with
Khakhar's model.

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