Landscape Evolution

Abstract

Chapter 6 describes a model for the evolution of hillslope topography by means of fluvial erosion. Coupled equations for the hillslope elevation and depth of water flow represent conservation of sediment and water mass, respectively. A uniform downhill slope is unstable to the Smith–Bretherton rilling instability, and this is regularised at large wave number by a singular term involving the ratio of water depth to hillslope elevation. The same theory is then used to describe the form and evolution of finite depth channels. These can be described by a nonlinear diffusion equation coupled with an integral constraint, the combination of which appears to give a well-posed problem for the channel depth; in addition, the channel profile selects its own width automatically. The issue of combining channel dynamics with longer term hillslope erosion is then discussed.