Meandering river dynamics: finite-domain theory and driven boundary conditions

Abstract:

The evolution of meandering river channels results from interactions amongst turbulent water flow, sediment transport, and channel geometry. Most current physics-based models derive from the meander-morphodynamics equations introduced by Ikeda et al. (1981). Corresponding linear theories, which serve as our theoretical understanding of meander behavior, have depended on the assumption of an infinite domain. We believe that infinite-domain theory cannot fully describe the behavior of experimental systems, which are finite in flume length, and which do not generally make use of periodic boundary conditions that would justify such an approximation. Here we consider any solution to the meander-morphodynamics equations as a series of eigenvalues and eigenvectors, which can capture the initial and boundary conditions of an experimental system. Rigorous mathematical consideration of the equations demonstrates that for a solution to be unique, it is necessary to specify three boundary conditions. In particular, whenever the upstream boundary is ‘clamped’ (i.e., a fixed injection point), theory predicts that a straight channel will propagate downstream. We believe this describes behavior that has been observed in most experimental work and may be the reason that meandering channels are so notoriously difficult to reproduce in a laboratory setting: mathematical theory shows that a self-maintained dynamic meandering channel cannot persist without a continuous upstream perturbation. This result implies that the formative conditions of natural meandering rivers also involves upstream driving. The numerical algorithms presented in this work are fully implicit and yield 2D solutions to the meander-morphodynamics equations with second order spatial and temporal convergence. We explore the characteristics of various upstream boundary conditions, and especially driven boundary conditions. Consideration of the growth patterns for spatially growing waves provides some insight for the design of experimental systems exhibiting self-sustaining river meanders.