Sedimentary bed evolution as a mean‐reverting random walk: Implications for tracer statistics

Abstract

Sediment tracers are increasingly employed to estimate bed load transport and landscape evolution rates. Tracer trajectories are dominated by periods of immobility ("waiting times") as they are buried and reexcavated 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 the particle feed rate and range of bed elevation fluctuations, respectively. Observations of bed evolution in flume experiments agree with model predictions; in particular, the model reproduces the 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.

Dynamics and mechanics of bed-load tracer particles

“…The stochastic erosion and deposition of the river-bed surface acts to bury and excavate particles, and recent laboratory results suggest that this produces heavy-tailed particle rest durations (Martin et al, 2012(Martin et al, , 2014. Although these rest durations cannot be measured from passive tracers in the field, our previous work used the dispersion of the tracer plume to infer similar behavior to laboratory experiments (Phillips et al, 2013).…”

Diffusion of bedload particles in open-channel flows: distribution of travel times and second-order statistics of particle trajectories

“…The information on the empirical distributions of rest periods is less extensive. The available data advocate that this distribution is exponential [7,8,11] or follows the power law when the deposited particles are buried by other particles [14,15,20]. The information on the step length and resting time distributions can be incorporated in a deterministic Lagrangian model of saltating grains [3], which in turn can provide a basis for extensive numerical simulations for the identification of the diffusive behavior of particles at different time scales.…”

Anomalous Diffusion of Sand Tracer Particles Under Waves

“…Continuing this process, it is apparent that a particle that jumps to a site x at time n = 0 subsequently executes a one‐dimensional random walk in the vertical direction (cf. Martin et al, ; Voepel et al, ), consisting of zero or more time steps with m ≥0, followed by one jump event. The probability of 2 n time steps occurring before the latter jump event is therefore equal to the first‐return probability for a one‐dimensional random walk (Klafter & Sokolov, ), …”

“…Some possible physical causes for heavy‐tailed jump length distributions include the influence of heterogeneous flow or topography at the scale of particle motion (Tucker & Bradley, ), or differing mobility within mixed‐size sediments (Ganti et al, ; Hill et al, ). Heavy‐tailed rest time distributions may be caused by particles being intermittently trapped or buried (Martin et al, ; Parker et al, ; Pelosi et al, , ; Voepel et al, ). Both types of anomalous diffusion behavior have been observed for sediment transport in streams (Bradley et al, ; Nikora et al, ), and the microscopic (particle‐scale) statistics that underlie the anomalous macroscopic behavior have also been the subject of experiments (Ancey et al, ; Ballio & Radice, ; Drake et al, ; Fathel et al, ; Habersack, ; Hassan et al, ; Heyman et al, ; Lajeunesse et al, ; Martin et al, ; Radice et al, ; Roseberry et al, ; Wilson & Hay, ) and theories (Ancey, ; Ancey et al, ; Fan et al, , ; Furbish & Schmeeckle, ).…”