Drainage basin evolution is modeled as the time development of an initial surface subject to conservation of sediment and water and a transport law qs = F(S, q) connecting the sediment flux qs with the local slope S and the discharge of surface water q. Two models are presented. The first is appropriate to a smooth surface on which no discrete channels have formed, and the second is appropriate to a family of V‐shaped valleys, each containing a separate stream of negligible width. For the first model, some solutions are presented that describe the evolution of a long ridge for which the profile is independent of one spatial coordinate. The stability of such surfaces is then discussed. It is shown that, if F/q < ∂F/∂q, disturbances of small amplitude and small lateral scale will grow rapidly and presumably will lead to the formation of closely spaced channels directed down the slope, whereas, if F/q ≥ ∂F/∂q, such channels will tend to disappear. For a surface eroding without change of shape, convex portions are stable, and concave segments are unstable. The second model is less well developed, since conservation principles alone are insufficient to determine its evolution without some additional postulate describing the sideways migration of individual streams. It is shown how, if each stream moves so that the sediment fluxes entering from its two side slopes remain equal, a system of similar parallel valleys is unstable, and neighbors will tend to coalesce on a time scale comparable to that for erosion through a layer as thick as a valley is deep.

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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