Initiation of Channel Head Bifurcation by Overland Flow

Abstract

Channel head bifurcation is a key factor for generating complexity of channel networks. Here we investigate incipient channel head bifurcation using linear stability analysis. Channel heads are simplified as circular hollows, toward which surface sheet flow accelerates in the radial direction. Sinusoidal perturbations in the angular direction with different angular wave numbers k are imposed on the bed, and their growth rates truenormalΩ~ are computed. Because the channel head radius trueR~c is extending over time, the base state (circular hollow in the absence of perturbations) also evolves continuously. With the use of the momentary stability concept, the evolving base state is defined as momentarily unstable to the imposed perturbation if the disturbance is growing faster than the evolution of the base state. It was found that in the range of sufficiently small trueR~c, bifurcation cannot be initiated. As trueR~c increases, bifurcation starts to be possible with k≈ 3–5. A higher k implies bifurcation with a narrower channel junction angle (θ = 2π/k). The average junction angle of the Colorado High Plains for the smallest drainage area is about 85° with a standard deviation of 35° (Sólyom and Tucker, 2007). Our predicted angles (75°–120°) agree qualitatively with the observed angles. Finally, we propose a simple criterion to compute the threshold trueR~c for the onset of bifurcation.

Analytic Theory of Equilibrium Fluvial Landscapes: The Integration of Hillslopes and Channels

“…In relation to transport‐limited erosion (e.g., Howard, ), the results of Smith and Bretherton () show that a channel‐cutting instability relating to nonlinearity in the transport of sediment underlies the process of channelization, with the boundary between stable and unstable flows forming an emergent CSB. An analogous instability for the case of detachment‐limited erosion (e.g., Howard, ) is described by Izumi and Parker (, ) and Pornprommin et al (), although the results of their stability analyses have been questioned (e.g., Smith, ). The analyses of Loewenherz () and Loewenherz‐Lawrence () indicate the necessity of stabilizing mechanisms that are capable of controlling the channel‐cutting instability and that lead to well‐defined channel networks.…”

Laboratory Experiment of Channel Head Bifurcation by Radial Overland Flow

“…To investigate the threshold condition, we adopted the recent linear stability analysis [3]. We calculated the threshold channel head radius for channelization (Rth) and compare it with the experimental average radius (R avg ) for every 1 second interval.…”

“…Thus, both studies showed that channel head bifurcation should be initiated by hydrophysical mechanisms (processes due to water forces such as erosion), not as a consequence of random topology. Pornprommin et al [3] performed a linear stability analysis using the flow equations in the polar coordination system to investigate incipient channel head bifurcation. Circular holes were assumed to be channel heads, toward which radial overland flow accelerates.…”

“…Then, channel bifurcation is defined as the holes lose their circular shapes when they are unstable to the imposed perturbations. They estimated that channel head does not bifurcate if channel head radius is not larger than approximately 10 times the Froude-critical depth of overland flow divided by the friction coefficient [3].…”

“…In this paper, we conducted a laboratory experiment of channel inception by overland flow in a circular chamber following the concept of channel bifurcation proposed by [3]. Our two main objectives in this study were; (i) to examine the early stage of channel network evolution (channel inception); and (ii) to compare the experimental results with the theoretical study [3].…”