Flux atop an advancing slip face and the brink line curvature of barchan dunes

A two-dimensional argument by Bagnold for the flux over the brink line of a shape-invariantly moving dune is generalized to three dimensions. This is achieved by describing the slip face as the solution to an eikonal equation with an unusual Dirichlet boundary condition where part of the boundary is to be determined. With the assumption of potential flow the flux over a heap is obtained based on kinematics, by solving a Poisson equation and without making reference to the wind profile or sand flux laws. Matching it with the brink line flux can be used in the results of field observations by Sauermann et al. (Geomorphology 36:47–62, 2000) to explain one of the five measured shape parameters of a barchan, the brink line curvature, from the other four. More generally the brink line flux formula proposed here could serve as an evolution equation for the brink line position in a given height and flux profile, in the limit that the avalanching processes are much faster than the rest of the surface evolution.