A probabilistic description of the bed load sediment flux: 3. The particle velocity distribution and the diffusive flux

Abstract

[1] Particles transported as bed load within a specified streambed area possess at any instant a distribution of velocities. This distribution figures prominently in describing the rates of transport and dispersal of particles. High-speed imaging of sand particles transported as bed load over a planar bed reveals that the probability density functions of the streamwise and cross-stream particle velocities are exponential-like. For quasi-steady conditions the exponential-like density of streamwise velocities reflects a balance among three fluxes in momentum space: (1) an advection of streamwise momentum whose magnitude and sign vary with the momentum state; (2) a diffusion of momentum from higher to lower values of momentum density; and (3) a drift of momentum from regions in momentum space having high average rates of generation of kinetic energy toward regions having low rates of generation of kinetic energy. The probability density of cross-stream velocities similarly reflects a balance of fluxes of cross-stream momentum. Whereas the average net force acting on particles is zero under steady conditions, the mean, variance and asymmetry of the distribution of forces acting on particles vary with the momentum state of the particles. Numerical simulations of particle motions that are faithful to these statistical properties reproduce key empirical results, namely, the exponential-like velocity distribution and the nonlinear relation between hop distances and travel times. The simulations also illustrate how steady gradients in particle activity, the solid volume of particles in motion per unit streambed area, induce a diffusive flux as described in companion papers.Citation: Furbish, D. J., J. C. Roseberry, M. W. Schmeeckle (2012), A probabilistic description of the bed load sediment flux: 3. The particle velocity distribution and the diffusive flux,

A microstructural approach to bed load transport: mean behaviour and fluctuations of particle transport rates

“…We do not gloss over the importance of particle velocity, however, its calculation is more easily amenable to analysis when we focus on short timescales. On longer timescales, when particle paths are characterized by alternating mobile and resting phases, this analysis is more involved (Ganti et al 2010;Zhang et al 2012). In the former case, the central limit theorem applies and we expect that the arithmetic mean U n tends to a well-defined mean particle velocityū s .…”

“…In the former case, the central limit theorem applies and we expect that the arithmetic mean U n tends to a well-defined mean particle velocityū s . There are a number of models available for calculating the mean velocity of a single particle as a function of the flow velocity or shear stress (Wiberg & Smith 1985;Ancey et al 2002;Niño et al 2002;Chatanantavet et al 2013), as well as its statistical properties (Lajeunesse, Malverti & Charru 2010;Furbish, Roseberry & Schmeeckle 2012b). Therefore, for the sake of simplicity, we assume that the mean particle velocityū s (x, t;v, h) can be determined independently and is fully controlled by the water stream.…”

Numerical simulation of the dynamics of sedimentary river beds with a stochastic Exner equation

“…And a computation of the local solid discharge with [2,3] first requires to extend the model to non-uniform flow conditions. Last, with a small diffusive term, the kinetic approach of [18][19][20][21]42] cannot improve on standard Exner model: it is not a new equation and it still requires empirical fitting. Contrary to the latter interpretation of Exner equation starting with a kinetic framework and ending with a fully deterministic closure at the hydrodynamic scale, our model lets the coupled hydrodynamic depend stochastically (through the coupling) on the random particles activity.…”

“…In the series of papers [18][19][20][21]42], a kinetic approach is used in order to derive Exner equation, from the evolution equation of the probability density of particles possessing some "activity" γ at position x and time t, the time-derivative of that density supposedly being directly proportional to the time-derivative of the bed elevation. This yields a reinterpretation of Exner equation as a nonlinear advection-diffusion equation for the bed elevation, and consequently includes the possibility of a diffusive behaviour that is sometimes observed in experiments (depending on the flows and on the sampling-time of measures).…”

Swash zone boundary conditions and direct bed shear stress measurements over loose sediment beds

“…The probabilistic definition has been further applied to describe the bed load sediment flux, e.g. particle velocity distribution, particle motions (Furbish et al, 2012a;Furbish et al, 2012b;Furbish et al, 2012c;Roseberry et al, 2012).…”