Self-similarity and multifractality of fluvial erosion topography. 1. Mathematical conditions and physical origin

Abstract

It is suggested that the scaling laws satisfied by fluvial erosion topography and river networks reflect a basic self-similarity or multifractality property of the topographic surface within river basins. By analyzing the symmetries of fluvial topography, we conclude that this self-similarity or multifractality condition should be expressed in a particular way in terms of the topographic increments within subbasins. We then analyze whether self-similar or multifractal topographies can be stationary or transient solutions of dynamic evolution models of the type δh/δt = U - f{β, τ}, where U is the uplift rate, f is the fluvial erosion rate, β is a vector of erodibility parameters, and τ is hydraulic shear stress. The hydraulic stress on a channel bed is assumed to satisfy τ proportional to AmS(n), where A is contributing area, S is slope, and m and n are parameters. We allow U to vary randomly in time and β to vary randomly in space and determine conditions on these random functions as well as the parameters m and n under which the topography may remain in a self-similar or multifractal state. Simulation shows that self-similar states are attractive also for non-self-similar boundary and initial conditions.