Modelling River Channel Formation

Abstract

A coupled model describing the evolution of the topographic elevation and the depth of the overland water film is here studied when considering the overland flow of water over an erodible sediment. We complete the previous modelling of the problems by SMITH and BRETHERTON (1972) and FOWLER et al. (2007), obtaining a model which involves a degenerate nonlinear parabolic equation (satisfied on the interior of the support of the solution) with a super-linear source term and a prescribed constant mass. The degeneracy of the equation causes the channel width to be self-selecting. We propose here a global formulation of the problem, formulated in the whole space, beyond the support of the solution. An important feature of the model proposed here is that despite of the presence of the superlinear forcing term at the equation, a solution to it can not blow up thanks to the mass constraint.