Mean-Reverting Random Walk Model for Sedimentary Bed Evolution, Bed Forms, and Tracer Waiting Times

Abstract:

Sediment tracers move intermittently with trajectories dominated by periods of immobility (“waiting times”). Particle waiting times are governed by the process of burial and re-excavation in the stochastically evolving river bed. Here, we model bed evolution as a random walk with mean-reverting tendency (Ornstein-Uhlenbeck process) originating from the restoring effect of erosion and deposition. The Ornstein-Uhlenbeck model contains two parameters, a and b, related to particle feed rate and range of bed elevation fluctuations, respectively. Observations of bed evolution in idealized flume experiments of glass spheres moving in a two-dimensional channel under constant shear stress agree with model predictions; in particular, the model reproduces the observed asymptotic t^-1 tail in the tracer waiting time exceedance probability distribution. This waiting time distribution is similar to that inferred for tracers in natural gravel streams and avalanching rice piles, indicating applicability of the Ornstein-Uhlenbeck mean-reverting model to many disordered transport systems with tracer burial and excavation. We explore possible application of our model for understanding steady-state evolution of bed forms.