Stability and the initiation of channelized surface drainage: A reassessment of the short wavelength limit

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Abstract
Instability leading to channel initiation occurs when advective processes dominate diffusive processes in the transport of surface material. Smith and Bretherton's (1972) linear treatment of this stability/instability phenomenon is here reexamined. Slopes of finite length are considered and a turning point analysis is used to construct a full asymptotic solution for the short wavelength limit. The analysis shows that the initial longitudinal profiles of marginally unstable surface incisions are deepest at a point corresponding to the convex/concave boundary in the base state profile and may actually extend upslope some distance from that point. The extent of the upslope incision is dependent upon the diffusion coefficient for the surface material and the wave number of the disturbance. In the original stability model, the growth rate of the short wavelength instability increases with decreasing wavelength, so that the shortest wavelengths are the most unstable. By introducing a surface material transport function which accounts for microscale nonlocal transfer, the shortest wavelengths are damped and the unstable growth rates are largest at moderate values of the wave number. The lengthscale set by nonlocal material transport represents the spatial scale at which the continuum description underlying the stability model breaks down. The neutral stability diagram for slope length‐wave number space illustrates that when the very shortest wavelengths are damped, concave slope segments may exhibit stability, at least while the system is within a linear regime. The most promising field application of the linear model lies in the characterization of shallow surface incisions, such as rills, which are in initial stages of development.

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