[1] High-speed imaging of coarse sand particles transported as bed load over a planar bed reveals that the particle activity, the solid volume of particles in motion per unit streambed area, fluctuates as particles respond to near-bed fluid turbulence while simultaneously interacting with the bed. The relative magnitude of these fluctuations systematically varies with the size of the sampling area. The particle activity within a specified sampling area is distributed in a manner that is consistent with the existence of an ensemble of configurations of particle positions wherein certain configurations are preferentially selected or excluded by the turbulence structure, manifest as patchiness of active particles. The particle activity increases with increasing bed stress far faster than does the average particle velocity, so changes in the transport rate with changing stress are dominated by changes in the activity, not velocity. The probability density functions of the streamwise and cross-stream particle velocities are exponential-like and lack heavy tails. Plots of the mean squared particle displacement versus time may ostensibly indicate non-Fickian diffusive behavior while actually reflecting effects of correlated random walks associated with intrinsic periodicities in particle motions, not anomalous diffusion. The probability density functions of the particle hop distance (start-to-stop) and the associated travel time are gamma-like, which provides the empirical basis for showing that particle disentrainment rates vary with hop distance and travel time.

[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.

[1] Rain splash transport of sediment on a sloping surface arises from a downslope drift of grains displaced ballistically by raindrop impacts. We use high-speed imaging of drop impacts on dry sand to describe the drop-to-grain momentum transfer as this varies with drop size and grain size and to clarify ingredients of downslope grain drift. The ''splash'' of many grains involves ejection of surface grains accelerated by grain-to-grain collisions ahead of the radially spreading front of a drop as it deforms into a saucer shape during impact. For a given sand size, splash distances are similar for different drop sizes, but the number of displaced grains increases with drop size in proportion to the momentum of the drop not infiltrated within the first millisecond of impact. We present a theoretical formulation for grain ejection which assumes that the proportion of ejected grains within any small azimuthal angular interval dq about the center of impact is proportional to the momentum density of the spreading drop within dq and that the momentum of ejected grains at angle q is, on average, proportional to the momentum of the spreading drop at q. This formulation, consistent with observed splash distances, suggests that downslope grain transport involves an asymmetry in both quantity and distance: more grains move downslope than upslope with increasing surface slope, and, on average, grains move farther downslope. This latter effect is primarily due to the radial variation in the surface-parallel momentum of the spreading drop. Surface-parallel transport increases approximately linearly with slope.

A high‐resolution data set obtained from high‐speed imaging of coarse sand particles transported as bed load allows us to confidently describe the forms and qualities of the ensemble distributions of particle velocities, accelerations, hop distances, and traveltimes. Autocorrelation functions of frame‐averaged values (and the decay of these functions) support the idea that the forms of these distributions become time invariant within the 5 s imaging interval. Distributions of streamwise and cross‐stream particle velocities are exponential, consistent with previous experiments and theory. Importantly, streamwise particle velocities possess a “light” tail, where the largest velocities are limited by near‐bed fluid velocities. Distributions of streamwise and cross‐stream particle accelerations are Laplace in form and are centered on zero, consistent with equilibrium transport conditions. The majority of particle hops, measured start to stop, involve short displacements, and streamwise hop distances possess a Weibull distribution. In contrast to previous work, the distribution of traveltimes is exponential, consistent with a fixed temporal disentrainment rate. The Weibull distribution of hop distances is consistent with a decreasing spatial disentrainment rate and is related to the exponential distribution of traveltimes. By taking into account the effects of experimental censorship associated with a finite sampling window, the relationship between streamwise hop distances and traveltimes, Lx∼Tpα, likely involves an exponent of α ∼ 2. These experimental results—an exponential distribution of traveltimes Tp and a Weibull distribution of hop distances Lx with shape parameter k < 1—are consistent with a nonlinear relationship between these quantities with α > 1.

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