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 structure of river basins. The partial differential equation determine the scaling invariances of the landscape through these processes. The models bridge the gap between the stochastic and deterministic approach to landscape evolution because they produce noise by sediment divergences seeded by instabilities in the water flow. The first process is a channelization process corresponding to Brownian motion of the initial slopes. It is driven by white 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 solutions. This process is characterized by the spatial roughness coefficient of 0¢ 75 and is analogous to an interface driven through random media with quenched noise. The values of the two scaling exponents, which are interpreted as reflecting universal, but distinct, physical mechanisms involving diffusion driven by noise, correspond well with field measurements from areas for which the advective sediment transport processes of our models are applicable. Various other scaling laws, such as Hack's law and the Law of Exceedence Probabilities, are shown to result from the two scalings, and Horton's Laws for a river network are derived from the first one.

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A family of models governing the evolution of surface drainage systems is introduced. The models involve an equation for water flow based on the continuity equation and special cases of the Navier-Stokes equations, and an erosion equation that is based on the conservation of sediment and sediment transport laws. These models embody the concept of a free water surface and include the equations of Smith and Bretherton as special cases. Numerical solutions, based on a 2-step MacCormack predictor-corrector finite difference scheme, were computed for a variety of special cases of the family of models. The special cases have the form a = V. V(z + h) nh513 -