Abstract
The existing theories of channel formation by surface sheet flow assume self-preserving base state slope profiles which migrate upstream without changing their shapes because formal stability analysis becomes inpossible if base state slope profiles are in unsteady states. In this study, the theories of Izumi and Parker2), and Izumi and Fujii3) are extended to include the case of channelization on slopes with arbitrary shapes with the use of the "frozen time approach" and the "momentary stability concept". In the frozen time approach, time derivative terms of base state equations are dropped under the assumption that the growth of perturbations is sufficiently faster than the evolution of base states as a first approximation. The stability of unsteady base states is further studied in terms of the momentary stability concept in which base states are assumed to become unstable for perturbations if the growth of perturbations is faster than the evolution of base states.
The analysis shows that, as long as their curvatures lie in appropriate ranges, slopes become unstable to evolve into slopes incised by channels, the spacing of which is on the order of one thousand times the depth of surface sheet flow. In the erosion-dominant case such as chennelization on slopes composed of strongly cohesive soils, the spacing between incipient channels are slightly decreased with increasing slope curvatures. In the range of sufficiently large curvatures, slopes are momentarily stable as the evolution rate of base states surpasses the growth rate of perturbations. In the case that both erosion and deposition are significant such as channelization on slopes composed of weakly cohesive soils, slopes with small curvatures are found to be unstable though clear dominant channel spacing does not appear. In the range of sufficiently large curvatures, slopes tend to be momentarily stable as well as in the erosion-dominant case.
