Hypothesis for a Channel Head Stability Criterion on the Basis of the Full Continuity Equation for Sediment Transport

The Smith and Bretherton instability criterion for perturbation growth allows incipient channelization to occur when the rate of change of sediment flux with contributing area overcomes the sediment flux-contributing area ratio. This criterion, a special case of the continuity equation for sediment transport, is based on the assumption that a small irregularity on the surface changes initially the contour curvature while leaves gradient and contributing area basically unaltered. This assumption might be valid for incipient channel growth, but does not apply to the case of fully formed hollows. In the present analysis we use the full continuity equation, including the perturbation induced change in area and slope, to derive a stability criterion for long-term equilibrium channel heads. The channel head is defined geometrically as the focus point of converging flow lines at the bottom of a hollow. It is demonstrated that stability at the channel head grows out of the competition between the rate of flow path convergence (dA/dL) and the degree of profile concavity (dS/dL). Analytical functions are derived to compute channel head-contributing area and -slope, flow path convergence and profile concavity as a function of perturbation depth, distance from the crest and the initial slope. In a numerical model these quantities point to the long-term equilibrium channel head position, which is shown to depend beside area and slope on the width to length ratio of hollows as well. Morphometrical measurements both in the field and on simulated topographies were used to test the hypothesis. We use this framework to interpret the contrasting sensitivity to erosion of concave mountain footslopes and convex hillslopes. It is shown that for geometrical reasons on mountain footslopes flowpath convergence plays a reduced role compared to profile concavity, which has the consequence that equilibrium channel head depth will be reduced, and streams will not incise. On the contrary the geometrical framework of convex hillslopes leads to higher sensitivity to fluvial incision that only can be counterbalanced by diffusion processes, such as creep and rainsplash.