A simple mathematical model of channel bifurcation

It is commonly observed that, in the development processes of channels, a channel head splits into two or more than two branches. By repeating such channel bifurcation, simple patterns of channels, which tend to appear in the initial stages of development, evolve into complex patterns of channel networks. In this study, we propose a simple mathematical model of channel bifurcation, in which a channel head is modeled by an opening on a flat plane composed of erodible bed material. A linear stability analysis is performed with the use of the depth-averaged momentum equations of flow and the Exner equation for beds subject to erosion. The analysis shows that the opening becomes unstable when the Froude critical depth divided by the bottom friction coefficient becomes sufficiently small compared with the radius of the opening. The implication is that channel bifurcation tends to take place as the discharge attracted by the channel head decreases.