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

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.

Flow partitioning and unstable divergence in fluviokarst evolution in central Kentucky

“…Likewise, the initiation of channelized surface drainage may be characterized by positive feedback (once initiated, channels collect more flow, increasing shear stress and enlarging or extending the channel). In some cases channel initiation has also been explicitly linked to dynamical instability (Smith and Bretherton, 1972;Loewenherz, 1991;Dietrich et al, 1992). In either case the selfreinforcement is finite due to limitations such as the extent or thickness of soluble rock in the case of karst depressions, or the production of runoff in the case of channels.…”

“…Over longer time scales, or with additional components added to the model, different or additional feedbacks are possible, which might affect model stability. This is unsurprising, as stability properties of hydrologic (and other earth surface) systems often change as temporal or spatial scales vary (Loewenherz, 1991;Phillips, 1999;Scheidegger, 1983;Smith and Bretherton, 1972). Our purpose in using the flow partitioning model is to determine whether flow competition dynamics alone are sufficient to account for karst/fluvial divergence.…”

Mathematical Analysis of a Model of River Channel Formation

“…Smith and Bretherton's study was later elaborated by LOEWENHERZ (1991), and LOE- WENHERZ-LAWRENCE (1994), who was particularly concerned with the issue of wavelength selection, something which also formed the principal concern of IZUMI and PARKER (1995,2000). Nonlinear studies of channel development and topographic evolution focussed on catchment scale problems, such as that of WILLGOOSE et al (1991), however such efforts were unable to compute the solution of the governing models directly, essentially because of the stiffness of the system.…”

Mathematical Models for Erosion and the Optimal Transportation of Sediment

“…The second group has produced remarkable simulations of evolving channel networks; see [52,53], [18], [48] and [35]. The third group has lead to an increasing understanding of the physical mechanisms that underlie erosion and channel formation; see [40], [42], [30], [36], [28], [27], [29], [43], [22,23,24,25,21], [44,45,46], [39], [50], [9], [15], [7], [41].…”

“…These equations improve the original model in [42] by including a pressure term (in addition to the gravitational and friction terms) that prevents water from accumulating in an unbounded manner in surface concavities; see [28]. They present a representation of the free water surface in a diffusion analogy approximation to the St. Venant equations; see [51].…”