Self-organized complexity in geomorphology: Observations and models

The focus of this paper is to relate fundamental statistical properties of landforms and drainage networks to models that have been developed in statistical physics. Relevant properties and models are reviewed and a general overview is presented. Landforms and drainage networks are clearly complex, but well-defined scaling laws are found. Coastlines, topography contours, and lakes are classic self-similar fractals. The height of topography along a linear track is well approximated as a Brownian walk, a self-affine fractal. This type of behavior has also been found in surface physics, for example the surface roughness of a fracture. An applicable model is the Langevin equation, the heat equation with a stochastic white-noise driver. This model also reproduces the statistics of sediment deposition. Drainage networks were one of the original examples of self-similar fractal trees. An important advance in quantifying the structure of drainage networks is the application of the Tokunaga fractal side-branching statistics. A classic problem in statistical physics is the diffusion-limited aggregation. The resulting tree like structures have been shown to also satisfy the Tokunaga statistics. A modified version of the diffusion-limited aggregation model reproduces the statistics of drainage networks. It is concluded that the models developed in statistical physics have direct applicability to the fundamental problems in geomorphology.