Mechanism for global optimization of river networks from local erosion rules

We show that landscapes and their drainage networks evolve under erosion rules to obey a variational principle. Starting from hydrodynamics we first find by separation of variables a general relationship between erosion and a modified slope-area law. This law encompasses previous work and observations, and we confirm it by simulation. Secondly we show that this corresponds to a solution of a variational minimization model, generalized from Rodriguez-Iturbe et al. Thus a mechanism by which the local process of erosion acts to globally optimize a river network is produced.