Meandering river channels evolve as a result of fluid mechanic and sedimentary processes. Their evolution can be described by differential equations that dictate how channel curvature gives rise to local perturbations in fluid velocity, prompting preferential erosion and sediment deposition–which is exactly the process of meandering. Here we extend standard meander theory to provide the first correct explanation of a perplexing behavior observed in flumes, where a fixed inlet leads to the long-term decay of all meanders. Not only is a continuous perturbation (on the time scale of tectonic plate movement for most terrestrial rivers) required for sustained meandering, but the frequency of that driving determines whether the meanders grow or decay. We use scaling arguments to predict the rate of river migration and the shapes of meanders.
We use semi-discrete theory to stabilize our nonlinear meandering river simulation–the first demonstrably accurate, converged solutions for the meander equations. Apart from making gorgeous shapes, we find that the nonlinear behavior close to the upstream boundary and in cases of a “clamped” inlet agrees well with our linearized theory.
One of the ongoing mysteries of meandering river research is why, from 1945 until present, efforts to reproduce meandering rivers in laboratory settings have been largely unsuccessful. Our simulations of previous experimental work are in good heuristic agreement with work that has been done, but also show that the conditions in those experiments were not consistent with what theory predicts will lead to sustained meandering in flumes. Between the linear theory, the scaling analysis, and the numerical simulation, we determine the appropriate length for experimental flumes, the appropriate duration of experimental runs, and the necessary properties of sediment.
This colloquium will give a introduction to many of the most powerful tools in applied mathematics: linear theory, scaling analysis, semi-discrete analysis, and numerical methods for nonlinear equations.