TOWARDS AN ELEMENTARY THEORY OF DRAINAGE-BASIN EVOLUTION - II - A COMPUTATIONAL EVALUATION

Numerical solutions to a recently-introduced family of continuous mode ls provide realistic representations of the evolution of fluvial lands capes. The simplest subfamily of the models offers a characterization of the evolution of ''badlands'' as a process involving (1) a first, t ransient stage in which branching valleys emerge from unchanneled surf aces; (2) a second, ''equilibrium'' stage in which a fully-developed s urface with branching valleys and ridges declines in a stable, self-si milar mode; and (3) a final, dissipative stage in which regularities i n the landscape break down. In the transient stage of development, sma ll perturbations to the surface induce local variations in water flow, differential erosion, and the rapid emergence of a coherent, fine-sca le structure of channelized flow patterns. The small-scale features ev olve into larger scale features by a process in which small flows inte rsect and grow. Standard linearized analyses of the equations are inad equate for characterizing this process, which appears to be initially dominated by random effects and non-linear saturation. In the second s tage, the numerical solutions converge towards satisfaction of an opti mality principle by which the pat terns of ridges, valleys, and surfac e concavities minimize a function of the sediment Aux over the surface , subject to two constraints. This stage is in accordance with a theor etical analysis of the model presented in a previous paper, and the nu merical solutions are stable in accordance with this analysis. The opt imality principle is associated with both the emergence of separable s olutions to the conservation equations and a variety of regularities i n landscape form and evolution, including self-similar decline of form s and a ''law of height-proportional erosion''. The numerical solution s provide detailed insight into the co-evolution of landforms and flow s of water and sediment. The family of models provides an elementary t heory characterizing the evolution of drainage basin phenomena, and in particular (1) possesses interpretations in terms of various geomorph ological concepts and observations; (2) appears capable of explaining variations in geomorphic forms over a wide variety of environments; an d (3) unifies certain aspects of the continuous, discrete, and variati onal approaches to landscape modeling. (C) 1997 Elsevier Science Ltd.