A subordinated advection model for uniform bed load transport from local to regional scales

Abstract

Sediment tracers moving as bed load can exhibit anomalous dispersion behavior deviating from Fickian diffusion. The presence of heavy-tailed resting time distributions and thin-tailed step length distributions motivate adoption of fractional-derivative models (FDMs) to describe sediment dispersion, but these models require many parameters that are difficult to quantify. Here we propose a considerably simplified FDM for anomalous transport of uniformly sized grains along straight channels, the subordinated advection equation (SAE), which is based on the concept of time subordination. Unlike previous FDM models with time index between 0 and 1, our SAE model adopts a value of between 1 and 2. This describes random velocities deviating significantly from the mean velocity and models both long resting periods and relatively fast displacements. We show that the model quantifies the dynamics of four bed load transport experiments recorded in the literature. In addition to , SAE model parameters-velocity and capacity coefficient-are related to the mean and variance of particle velocities, respectively. Successful application of the SAE model also implies a universal probability density for the heavy-tailed waiting time distribution (with finite mean) and a relatively lighter tailed step length distribution for uniform bed load transport from local to regional scales.

Exploring a semimechanistic episodic Langevin model for bed load transport: Emergence of normal and anomalous advection and diffusion regimes

“…However, following the seminal work of Nikora et al . [], studies of anomalous diffusion have recently blossomed from flume experiments [e.g., Martin et al ., ; Ballio et al ., ; Campagnol et al, ], field investigations [e.g., Hassan et al ., ; Phillips et al ., ; Haschenburger , ], and numerical simulations [e.g., Bradley et al ., ; Bialik et al ., ; Zhang et al ., ; Pelosi et al ., ].…”

“…Generally, the number of step lengths extracted from experiments is often relatively small, preventing a robust analysis on the tails of the distribution [ Martin et al ., ]; besides, large step lengths might also be missed due to limitations in the sample size window [ Phillips et al ., ], implying that the tails of the PDFs of from the previous studies might not be so clear to interpret. Similar to step lengths , the PDFs of resting times have not been consistently quantified by several studies [see Zhang et al ., for a review]. For example, both exponential distributions [ Einstein , ; Yang and Sayre , ; Schmidt and Ergenzinger , ; Habersack , ; Wu and Yang , ; Valyrakis et al ., ; Furbish et al ., ], and heavy‐tailed distributions (power‐law, Martin et al .…”

Influence of bed clusters and size gradation on operational time distribution for non‐uniform bed‐load transport

“…Zhang et al () proposed the following subordinated advection model for uniform bed‐load transport from local to regional scales: where P ( x , t ) [ ML −1 ] is the sediment concentration probability density, V [ LT −1 ] is the mean particle velocity, β ( x ) [ T γ−1 ] is the space‐dependent capacity coefficient, and γ [dimensionless] (1 < γ < 2) is the index which defines the order of the time‐fractional Caputo derivative ( Miller and Ross , ). Subordination is a mathematical method where physical time is randomized to represent the operational time experienced by each individual target (Baeumer et al, ), the sediment particles in this study.…”

“…Subordination is a mathematical method where physical time is randomized to represent the operational time experienced by each individual target (Baeumer et al, ), the sediment particles in this study. The relationship between the OTD and model can be simply described as this: the time index γ in model affects the overall trend of the OTD, while the capacity coefficient β scales the OTD (Zhang et al, ). After parameters ( γ , β , and v ) in model are fitted by using the measured sediment data (such as sediment transport rates), the underlying OTD is defined.…”

“…As reviewed by Metzler and Klafter (, ) and explicated by many other researchers (see for example, the review by Zhang et al, ), the FDMs are superior to the Fick's law based models in quantifying anomalous diffusion with heavier tails than exponential PDFs for particle displacement and/or resting periods. The simplest FDM is the subordinated advection model, proposed recently by Zhang et al () (see further discussion in the section on Stochastic Model), which was found to be more efficient than other FDMs in capturing the complex dynamics of uniform sand transport under steady‐state flow conditions. The core of the subordinated advection model is defining the time that bed‐load sediment has spent in motion by quantifying the operational time distribution (OTD), therefore, providing critical information on the resting and displacement of bed‐load sediment.…”