Siobhan L. Fathel

Experimental evidence of statistical ensemble behavior in bed load sediment transport

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 travel times. 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 five second 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 travel times 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 travel times. By taking into account the effects of experimental censorship associated with a finite sampling window, the relationship between streamwise hop distances and travel times, Lx∼Tpα, likely involves an exponent of α ∼ 2. These experimental results — an exponential distribution of travel times Tp and a Weibull distribution of hop distances Lx with shape parameter k 1.

Probability distributions of bed load particle velocities, accelerations, hop distances, and travel times informed by Jaynes's principle of maximum entropy

We describe the most likely forms of the probability distributions of bed load particle velocities, accelerations, hop distances, and travel times, in a manner that formally appeals to inferential statistics while honoring mechanical and kinematic constraints imposed by equilibrium transport conditions. The analysis is based on E. Jayness elaboration of the implications of the similarity between the Gibbs entropy in statistical mechanics and the Shannon entropy in information theory. By maximizing the information entropy of a distribution subject to known constraints on its moments, our choice of the form of the distribution is unbiased. The analysis suggests that particle velocities and travel times are exponentially distributed and that particle accelerations follow a Laplace distribution with zero mean. Particle hop distances, viewed alone, ought to be distributed exponentially. However, the covariance between hop distances and travel times precludes this result. Instead, the covariance structure suggests that hop distances follow a Weibull distribution. These distributions are consistent with high-resolution measurements obtained from high-speed imaging of bed load particle motions. The analysis brings us closer to choosing distributions based on our mechanical insight.

The elements and richness of particle diffusion during sediment transport at small timescales

The ideas of advection and diffusion of sediment particles are probabilistic constructs that emerge when the Master equation, a precise, probabilistic description of particle conservation, is approximated as a Fokker-Planck equation. The diffusive term approximates nonlocal transport. It “looks” upstream and downstream for variations in particle activity and velocities, whose effects modify the advective term. High-resolution measurements of bed load particle motions indicate that the mean squared displacement of tracer particles, when treated as a virtual plume, primarily reflects a nonlinear increase in the variance in hop distances with increasing travel time, manifest as apparent anomalous diffusion. In contrast, an ensemble calculation of the mean squared displacement involving paired coordinate positions independent of starting time indicates a transition from correlated random walks to normal (Fickian) diffusion. This normal behavior also is reflected in the particle velocity autocorrelation function. Spatial variations in particle entrainment produce a flux from sites of high entrainment toward sites of low entrainment. In the case of rain splash transport, this leads to topographic roughening, where differential rain splash beneath the canopy of a desert shrub contributes to the growth of a soil mound beneath the shrub. With uniform entrainment, rain splash transport, often described as a diffusive process, actually represents an advective particle flux that is proportional to the land-surface slope. Particle diffusion during both bed load and rain splash transport involves motions that mostly are patchy, intermittent and rarefied. The probabilistic framework of the Master equation reveals that continuous formulations of the flux and its divergence (the Exner equation) represent statistically expected behavior, analogous to Reynolds-averaged conditions. Key topics meriting clarification include the mechanical basis of particle diffusion, effects of rarefied conditions involving patchy, intermittent motions, and effects of rest times on diffusion of tracer particles and particle-borne substances. This article is protected by copyright. All rights reserved.