Fractional dispersion in a sand bed river

Random walk models of fluvial bed load transport use probability distributions to describe the distance a grain travels during an episode of transport and the time it rests after deposition. These models typically employ probability distributions with finite first and second moments, reflecting an underlying assumption that all the factors that influence sediment transport tend to combine in such a way that the length of a step or the duration of a rest can be characterized by a mean value surrounded by a specific amount of variability. The observation that many transport systems exhibit apparent scale-dependent behavior and non-Fickian dispersion suggests that this assumption is not always valid. We revisit a nearly 50 year old tracer experiment in which the tracer plume exhibits the hallmarks of dispersive transport described by a step length distribution with a divergent second moment and no characteristic dispersive size. The governing equation of this type of random walk contains fractional-order derivatives. We use the data from the experiment to test two versions of a fractional-order model of dispersive fluvial bed load transport. The first version uses a heavy-tailed particle step length distribution with a divergent second moment to reproduce the anomalously high fraction of tracer mass observed in the downstream tail of the spatial distribution. The second version adds a feature that partitions mass into a detectable mobile phase and an undetectable, immobile phase. This two-phase transport model predicts other features observed in the data: a decrease in the amount of detected tracer mass over the course of the experiment and enhanced particle retention near the source. The fractional-order models match the observed plume shape and growth rates better than prior attempts with classical models.