  The power spectrum S of linear transects of the earth's topography is often
observed to be a power-law function of wave number k with exponent close to -2:
S(k) is proportional to k^-2. In addition, river networks are fractal trees
that satisfy many power-law or fractal relationships between their morphologic
components. A model equation for the evolution of the earth's topography by
erosional processes which produces fractal topography and fractal river
networks is presented and its solutions compared in detail to real topography.
The model is the diffusion equation for sediment transport on hillslopes and
channels with the local diffusivity proportional to the square of the
discharge. The dependence of diffusivity on discharge follows from fundamental
equations of sediment transport. We study the model in two ways. In the first
analysis the diffusivity is parameterized as a function of relief and a Taylor
expansion procedure is carried out to obtain a differential equation for the
landform elevation which includes the spatially-variable diffusivity to first
order in the elevation. The solution to this equation is a self-affine or
fractal surface with linear transects that have power spectra S(k) is
proportional to k^-1.8, independent of the age of the topography, consistent
with observations. The hypsometry produced by the model is skewed such that
lowlands make up a larger area than highlands as in real topography. In the
second analysis we include river networks explicitly in a numerical simulation
by calculating... Abstract continued in paper.
