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

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.