To Meander or Not to Meander: That is the Question. Rod-Chain Models May Hold the Answer.

Meandering is a common phenomenon in rivers and other fluid and quasi-fluid systems. Although these systems differ widely in their scales and characteristics in two and three dimensions, a commonality between them appears to be that decelerating systems are unstable, while accelerating systems are stable. Despite this simple feature of meandering systems, no universal formulation for the meander instability currently exists.

Toward that end, we take a "bottom-up" approach and consider a rod-chain model, starting with a simple rod that is hinged in its mass center, with forces acting upon its ends in the direction of its orientation. If these forces result in tension in the rod, the system is stable, while if they result in compression, the system is unstable to small perturbations and pivots on its hinge. The simple formulation of this instability is well-characterized.

A slightly more complex system with two rods is then analyzed. Because meandering usually manifests itself in lateral displacement, we consider both situations in which the rods are either hinged in their centers or are allowed to translate in the lateral direction. In both cases, we incorporate lateral resistance force modeled by spring/damper systems and derive a threshold for the decelerating force resulting in meandering of the chain.

Multi-rod systems and their continuum limits are far more complex. In this case we will take a new approach and utilizes an intricate Lyapunov method combined with the novel nonlinear energy method to rigorously justify the stability/instability criterion and capture convergent estimates of critical thresholds.

An important difference between the rod-chain model and fluids is that the latter accommodates shear between elements. Thus, the next step is to connect rods with dashpots/springs to allow shear and resist sharp bending, and to take it to the limit to characterize fluid instability. However, this transition remains a grand mathematical challenge, especially in refining the rod-chain model to join it to well-known fluid instabilities formulated in a "top-down" approach. Once accomplished, it may show that the rod-chain model responds to the same meander instability as fluid systems and our analysis will lead to a more complete understanding of meandering instability in rivers and a variety of other systems.