Physical basis for anomalous sediment dispersion

Anomalous sediment dispersion, i.e. sedimentary particle spreading that diverges from normal Brownian diffusion, has been observed in a variety of settings. There is currently some debate regarding the dispersion of bedload particles in a river. Some observations show departures from Brownian motion in the macroscopic statistics of dispersion; however, experimental observation of microscopic grain motion has found only Brownian behavior. Nikora and colleagues (WRR, 2002) hypothesized that bedload transport should be superdiffusive at short timescales due to particle inertia as grains undergo nearly ballistic trajectories, and should transition to subdiffusion at long timescales because grains spend most of their time at rest. Here we use small-scale experiments and numerical modeling to show that this hypothesis is broadly correct. We identify the physical processes underlying anomalous bedload dispersion and the characteristic timescales associated with super- and sub-diffusive behavior. Flume experiments were set up to photographically track (30 frames / second) individual painted sediment grains along a well-sorted gravel bed (D50 = 0.71 mm) under steady water discharge. Streamwise mobile particle positions determined from image analysis disperse approximately as x~t^0.7 for timescales up to several seconds, and this superdiffusion was found to be robust over an order of magnitude of boundary shear stresses in experiments. Streamwise particle velocity showed correlations over similar timescales, and this property was also independent of boundary shear stress. Data indicate that short-range velocity correlation from particle momentum produces superdiffusion, consistent with hard-sphere collision models that also exhibit superdiffusion. Motivated by such models, we construct a simple 1d numerical model of inertial particles whose momentum is gradually dissipated by fluid drag and bed friction. This model confirms that short-range velocity correlations of the type observed in our experiments lead to sediment superdiffusion that is in general agreement with observations. Sediment grains were also tracked at long timescales (hours). Most of the time, grains are at rest on the bed, thus their long time motions are dominated by waiting times on the bed. Tracking 243 grains through time, waiting times were found to exhibit a heavy-tailed distribution, with cumulative waiting time probability decreasing approximately as p~t^-1 at long times (>1 min). These heavy-tailed waiting time dynamics are likely related to fluctuations in bed topography arising from formation and breakup of granular structures. While the limitations imposed by the length of the flume make it impossible to track sediment dispersion over long times, incorporation of heavy-tailed waiting times into the 1d numerical model produces robust subdiffusive particle dispersion at long timescales.