Initiation of Channel Head Bifurcation by Overland Flow

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 Ω are computed. Because the channel head radius R_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 R_c, bifurcation cannot be initiated. As R_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 R_c for the onset of bifurcation.