Water erosion as a fractal growth process

The time evolution of river patterns and earth's relief is simulated on lattice by modeling the process of water erosion. Starting from a randomly perturbed surface, the river pattern and earth's relief develop simultaneously. The river pattern becomes stationary after all lakes have vanished. In the stationary state the river pattern shows some fractal properties such as a power-law size distribution of the drainage basin area and Horton's laws. The fractalities are shown to be not exactly self-similar but self-affine. A mean-field theory for the river pattern is discussed.