Nonlinear Evolution Equations and the Braiding of Weakly Transporting Flows Over Gravel Beds

The flow of a river that transports sediment in the form of gravel as bedload is investigated for the case when the transport is small. The linear stability of such flows is discussed and used to formulate some strongly nonlinear investigations describing the interaction of bar instabilities that are known to occur. The key spatial scales in the asymptotic limit of small transport are identified, and highly nonlinear evolution equations derived for each case. A generalized KDV equation is found to govern the nonlinear evolution at small wavenumbers, while at O(1) wavenumbers an infinite set of "triad-like" amplitude equations describes the flow. The interactions demonstrate the natural tendency of rivers of width significantly higher than the critical width at which instability first occurs to form complex patterns that may be associated with braided rivers. The weak transport limit used in our anaysis makes our work directly relevant to rivers experiencing flood conditions where the onset of a flood causes transport to begin. The results shown suggest that in the highly nonlinear stages, bars take the form of slabs tilted in the flow direction with steep edges. In addition, it is found that there is no equilibrium state. These findings are consistent with observations.