Self-organization and scaling relationships of evolving river networks

The power spectra S of linear transects of Earth's topography is often observed to be a power law function of wave number k with exponent close to -2: S(k) ∝ k^-2. In addition, river networks are fractal trees that satisfy several power law relationships between their morphologic components. A model equation for the evolution of Earth's topography by transport-limited erosional processes which produces fractal topography and fractal river networks is presented, and its solutions are compared in detail to real topography. The model is the diffusion equation for sediment transport on hillslopes and channels with the diffusivity constant on hillslopes and proportional to the three-halves power of discharge in channels. The dependence of diffusivity on discharge is consistent with sediment rating curves. We study the model in two ways. In the first analysis the diffusivity is parameterized as a function of elevation, 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) ∝ k^-1.8, independent of the age of the topography, consistent with observations of real topography. The hypsometry produced by the model equation is skewed such that lowlands make up a larger fraction of the total area than highlands as observed in real topography. In the second analysis we include river networks explicitly in a numerical simulation by calculating the discharge at every point. We characterize the morphology of real river basins with five independent scaling relations between six morphometric variables. Scaling exponents are calculated for seven river networks from a variety of tectonic environments using high-quality digital elevation models. River networks formed in the model match the observed scaling laws and satisfy Tokunaga side-branching statistics.