The formation of river channels

We consider a deterministic model of landscape evolution through the mechanism of overland flow over an erodible substrate, using the St. Venant equations of hydraulics together with the Exner equation for hillslope erosion. A novelty in the model is the allowance for a nonzero bedload layer thickness, which is necessary to distinguish between transport limited and detachment limited sediment removal. It has long been known that transport limited uniform flow is unstable when the hillslope topography is geomorphologically concave (i.e., the center of curvature is above ground). In this paper, we show how finite amplitude development of the consequent channel flow leads to an evolution equation for its depth h of the form h(t) = h(3/2) + (h(3/2))(YY), where Y is the cross-stream space variable. We show that solutions of compact support exist but that, despite appearances, blow up does not occur because of an associated integral constraint, and the channel equation admits a unique and apparently globally stable steady state. The consequences for hillslope evolution models are discussed.