Water Flowing over erodable surfacesleads to well-organized patterns of channels and characteristic scalings of the surfaces. Two non-linear conservation equations for water and sediment, together with appropriate constitutive equations, are sufficient to describe many of the main features of these surfaces. These features include characteristic profiles that vary significantly with boundary conditions; the emergence of channelized flows; the development of stable surfaces with ridges and valleys, which decline over time without tectonically-driven inputs; and characteristic surface scalings that change during the course of erosion. Initial instabilities caused by convergence of water and non-linearities in the sediment transport law stabilize with the convergence of water flows and convectively-driven diffusion. The initial processes are seeded
by noise generated in in the water flow over the surface and have their own distinctive scalings. The initial instabilities stabilize with the development of deeper water layers and convectively-driven diffusion.
Self-similar, separable, and stable solutions to the equations appear during "mature' phases of the process, with the associated surfaces being determined by a minimum-work principle.
These surfaces are characterized by well-defined scalings that range from the small scale of a river channel the large scale of whole fluvial landscapes. Initial instabilities caused by convergence of water and non-linearities in the sediment transport law "saturate" with the convergence of water flows and convectively-driven diffusion.
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