Scaling and river networks: A Landau theory for erosion

We propose a coarse-grained theory for the formation of a river network in the form of a Langevin equation for the erosion of the landscape coupled to a conservation law for the surface water flow. We claim that this is the universal form for large-scale behavior. We show by simulations of a discrete model that represents the same dynamics that the slope-area law, the basin size distribution law, and Horton’s laws agree with real rivers. We discuss the relationship to optimal channel networks and to self-organized criticality.