Linear stability analysis of channel inception: downstream-driven theory

Abstract

A linear stability analysis of incipient channellization on hillslopes is performed using the shallow-water equations and a description of the erosion of a cohesive bed. The base state consists of a laterally uniform Froude-subcritical sheet flow down a smooth, downward-concave hillslope profile. The downstream boundary condition consists of the imposition of a Froude number of unity. The process of channellization is thus driven from the downstream end. The flow and bed profiles describe a base state that migrates at constant, slow speed in the upstream direction due to bed erosion. Transverse perturbations corresponding to a succession of parallel incipient channels are introduced. It is found that these perturbations grow in time, so describing incipient channellization, only when the characteristic spacing between incipient channels is on the order of 6–100 times the Froude-critical depth divided by the resistance coefficient. The characteristic wavelength associated with maximum perturbation growth rate is found to scale as 10 times the Froude-critical depth divided by the resistance coefficient. Evaluating the friction coefficient as on the order of 0.01, an estimate of incipient channel spacing on the order of 1000 times the Froude-critical depth is obtained. The analysis reveals that downstream-driven channellization becomes more difficult as (a) the critical shear stress required to erode the bed becomes so large that it approaches the Froude-critical shear stress reached at the downstream boundary and (b) the Froude number of the subcritical equilibrium flow attained far upstream approaches unity. Alternative mechanisms must be invoked to explain channellization on slopes high enough to maintain Froude-supercritical sheet flow.