A probabilistic description of the bed load sediment flux: 1. Theory

Abstract

[1] We provide a probabilistic definition of the bed load sediment flux. In treating particle positions and motions as stochastic quantities, a flux form of the Master equation (a general expression of conservation) reveals that the volumetric flux involves an advective part equal to the product of an average particle velocity and the particle activity (the solid volume of particles in motion per unit streambed area), and a diffusive part involving the gradient of the product of the particle activity and a diffusivity that arises from the second moment of the probability density function of particle displacements. Gradients in the activity, instantaneous or time-averaged, therefore effect a particle flux. Time-averaged descriptions of the flux involve averaged products of the particle activity, the particle velocity and the diffusivity; the significance of these products depends on the scale of averaging. The flux form of the Exner equation looks like a Fokker-Planck equation (an advection-diffusion form of the Master equation). The entrainment form of the Exner equation similarly involves advective and diffusive terms, but because it is based on the joint probability density function of particle hop distances and associated travel times, this form involves a time derivative term that represents a lag effect associated with the exchange of particles between the static and active states. The formulation is consistent with experimental measurements and simulations of particle motions reported in companion papers.

Sediment connectivity: a framework for understanding sediment transfer at multiple scales

“…Sediment flux is defined as the solid volume of sediment particles crossing a surface per unit time, per unit width. From a transport-distance perspective, sediment flux at a point is the integral of the detached material from upslope that moves at least the distance between the point of detachment and the point of measurement (Einstein, 1950;Wainwright et al, 2001;Parsons et al, 2004;Furbish et al, 2012). Sediment-detachment and sediment-transport mechanisms can be ordered along a spectrum depending on the degree of hydrological control.…”

Monitoring Riverbank Erosion in Mountain Catchments Using Terrestrial Laser Scanning

“…Soil erosion has several on-site and off-site impacts on the environment: (1) loss of fertile soil with important consequences on agriculture [1]; (2) silting of reservoirs that reduces the storage capacity and interferes with dam operations [2][3][4]; (3) migration of pollution in which sediment transport is considered a means of transport for contaminants [5,6]; (4) increase of flood risk [7] and debris flow events [8]; and (5) geomorphic evolution of river beds [9] with possible impacts on the surrounding structures. At the basin scale, sediment production is the result of the complex interaction between different geomorphic processes: splash erosion, sheet erosion, rill erosion, gully erosion, bank erosion as well as mass movements [10].…”

Scales of collective entrainment and intermittent transport in collision-driven bed load

“…As expected, the waiting times between emigration events seem to be a function of the driving frequency. When the former are non-dimensionalized by the latter, the variance among 30 the experiments is significantly reduced (figure 6B). Moderate dispersion remains among the different experiments, however, indicating that driving rate is not the only factor controlling the waiting times.…”

“…In particular, at low driving rates we see waiting and averaging times that are significantly longer than expected, suggesting that the first-order kinematic avalanching model described above is incomplete. One timescale that has not been considered is the relaxation time of avalanches, which for our experiments would be the deposition timescale of mobile clusters 30 following an entrainment event. This timescale may not be independent of driving rate, and it becomes impossible to measure when avalanches overlap in time.…”