The scaling of fluvial landscapes

The analysis of a family of physically based landscape models leads to the analysis of two stochastic processes that seem to determine the shape and s tructure of river basins. The partial differential equation determine the s caling invariances of the landscape through these processes. The models bri dge the gap between the stochastic and deterministic approach to landscape evolution because they produce noise by sediment divergences seeded by inst abilities in the water flow. The first process is a channelization process corresponding to Brownian motion of the initial slopes. It is driven by whi te noise and characterized by the spatial roughness coefficient of 0.5. The second process, driven by colored noise, is a maturation process where the landscape moves closer to a mature landscape determined by separable solut ions. This process is characterized by the spatial roughness coefficient of 0.75 and is analogous to an interface driven through random media with que nched noise. The values of the two scaling exponents, which are interpreted as reflecting universal, but distinct, physical mechanisms involving diffu sion driven by noise, correspond well with field measurements from areas fo r which the advective sediment transport processes of our models are applic able. Various other scaling laws, such as Hack's law and the law of exceede nce probabilities, are shown to result from the two scalings, and Horton's laws for a river network are derived from the first one. (C) 2001 Elsevier Science Ltd. All rights reserved.