A probabilistic description of the bed load sediment flux: 4. Fickian diffusion at low transport rates

High-speed imaging of coarse sand particles transported as bed load reveals how particle motions possess intrinsic periodicities associated with their start-and-stop behavior. The dominant harmonics in these motions have a primary influence on the rate at which the mean squared particle displacement R(τ) - a measure conventionally used to assess the possibility of anomalous diffusion - increases with the time interval τ. Over a timescale corresponding to the typical travel time of particles, calculations of R(τ) may ostensibly indicate non-Fickian behavior while actually reflecting the effects of periodicities in particle motions, not anomalous diffusion. We provide the theoretical basis for this observed behavior, and we illustrate how the effective (Fickian) particle diffusivity obtains from G. I. Taylor's classic definition involving the particle velocity autocovariance, including its relation to the ensemble-averaged particle velocity as articulated by O. M. Phillips. Cross-stream diffusivities are an order of magnitude smaller than streamwise diffusivities.