A fractal derivative model to quantify bed-load transport along a heterogeneous sand bed

Bed-load transport is a complex process exhibiting anomalous dynamics, which cannot be efficiently described using the traditional advection-diffusion equation. This study aims at developing and testing a Hausdorff fractal derivative model to characterize scale-dependent, anomalous dynamics of bed-load transport through a heterogeneous gravel-bed. Applications show that the Hausdorff fractal derivative model generally matches the bed sediment snapshots measured in flume experiments with both continuous and instantaneous sediment sources. The order of the Hausdorff fractal derivative is a scale-dependent indicator varying with bed heterogeneity and particle size. For example, bed armoring and size selective transport can cause the fast downstream motion of fine sediment and the enhanced trapping for coarse materials, which can be conveniently quantified by selecting the corresponding order of the Hausdorff fractal derivative in the new model proposed by this study. Further comparison with the fractional derivative model (containing a nonlocal operator to capture long-term memory embedded in both motion and resting of sediment particles) shows that both models can capture anomalous bed-load dynamics, while the Hausdorff fractal derivative model is more attractive due to its local operator and convenient numerical solution.