Towards an elementary theory of drainage basin evolution: II. A computational evaluation

Abstract

Numerical solutions to a recently-introduced family of continuous models provide realistic representations of the evolution of fluvial landscapes. The simplest subfamily of the models offers a characterization of the evolution of “badlands” as a process involving (1) a first, transient stage in which branching valleys emerge from unchanneled surfaces; (2) a second, “equilibrium” stage in which a fully-developed surface with branching valleys and ridges declines in a stable, self-similar mode; and (3) a final, dissipative stage in which regularities in the landscape break down. In the transient stage of development, small perturbations to the surface induce local variations in water flow, differential erosion, and the rapid emergence of a coherent, fine-scale structure of channelized flow patterns. The small-scale features evolve into larger scale features by a process in which small flows intersect and grow. Standard linearized analyses of the equations are inadequate for characterizing this process, which appears to be initially dominated by random effects and non-linear saturation. In the second stage, the numerical solutions converge towards satisfaction of an optimality principle by which the patterns of ridges, valleys, and surface concavities minimize a function of the sediment flux over the surface, subject to two constraints. This stage is in accordance with a theoretical analysis of the model presented in a previous paper, and the numerical solutions are stable in accordance with this analysis. The optimality principle is associated with both the emergence of separable solutions to the conservation equations and a variety of regularities in landscape form and evolution, including self-similar decline of forms and a “law of height-proportional erosion”. The numerical solutions provide detailed insight into the co-evolution of landforms and flows of water and sediment. The family of models provides an elementary theory characterizing the evolution of drainage basin phenomena, and in particular (1) possesses interpretations in terms of various geomorphological concepts and observations; (2) appears capable of explaining variations in geomorphic forms over a wide variety of environments; and (3) unifies certain aspects of the continuous, discrete, and variational approaches to landscape modeling.