skip to main content

Title: Analytical Solution for Anomalous Diffusion of Bedload Tracers Gradually Undergoing Burial

Accounting for the burial of tracer particles during bedload transport is an important component in the formulation of tracer dispersal in rivers. Herein we propose a modified active layer formulation, which accounts for the effect of burial and admits analytical solutions, enabling insightful exploration of the phenomenon of superdiffusion of bedload tracers at the intermediate timescale. This phenomenon has been observed in recent numerical results using the 2‐D Exner‐Based Master Equation. By assuming that tracers in the active layer can exchange with nontracer particles in the substrate layer to preserve mass, and that tracers entering the substrate layer get permanently trapped during the timescale of analysis, we are able to deduce governing equations for the tracer concentration in both layers. The active layer tracer concentration is shown to be governed by an advection‐diffusion equation with a sink term, and the increase of tracers in the substrate layer is driven by a corresponding source term. The solution for the variance of tracer population is analytically determined and can be approximated by the sum of a diffusion‐induced scaling (t1) and an advection‐induced scaling (t3) terms at the intermediate timescale, which explains the phenomenon of superdiffusion. The proposed formulation is shown to be able to capture the key characteristics of tracer transport as inferred by comparison with available results of numerical simulations.

more » « less
Award ID(s):
1209402 1242458
Author(s) / Creator(s):
 ;  ;  ;  ;  ;  
Publisher / Repository:
DOI PREFIX: 10.1029
Date Published:
Journal Name:
Journal of Geophysical Research: Earth Surface
Page Range / eLocation ID:
p. 21-37
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Abstract

    To date, there is no consensus on the probability distribution of particle velocities during bedload transport, with some studies suggesting an exponential‐like distribution while others a Gaussian‐like distribution. Yet, the form of this distribution is key for the determination of sediment flux and the dispersion characteristics of tracers in rivers. Combining theoretical analysis of the Fokker‐Planck equation for particle motions, numerical simulations of the corresponding Langevin equation, and measurements of motion in high‐speed imagery from particle‐tracking experiments, we examine the statistics of bedload particle trajectories, revealing a two‐regime distance‐time (LTp) scaling for the particle hops (measured from start to stop). We show that particles of short hop distances scale asL~giving rise to the Weibull‐like front of the hop distance distribution, while particles of long hop distances transition to a different scaling regime ofL~Tpleading to the exponential‐like tail of the hop distance distribution. By demonstrating that the predominance of mostly long hop particles results in a Gaussian‐like velocity distribution, while a mixture of both short and long hop distance particles leads to an exponential‐like velocity distribution, we argue that the form of the probability distribution of particle velocities can depend on the physical environment within which particle transport occurs, explaining and unifying disparate views on particle velocity statistics reported in the literature.

    more » « less
  2. Abstract

    Predicting the transport of bedload tracer particles is a problem of significant theoretical and practical interest. Yet, little understanding exists for transport in rivers in the presence of bedforms, which may trap grains and thereby influence travel distance. In a series of flume experiments with a sandy gravel bed in a large experimental flume, bed elevation and tracer travel distances were measured at high resolution for a range of discharges. As discharge increased, bedform height increased and bedform length decreased, increasing bedform steepness. For all tracer sizes and flow conditions, bedforms act as primary controls on the tracer travel distances. Bedform trapping increases linearly with the ratio of bedform height to tracer grain size, with 50% trapping efficiency for a ratio of two and 90% trapping efficiency for a ratio of four. A theoretical model based on the extended active layer formulation for sediment transport is able to capture much of the distribution of measured travel distances for all tracer sizes and discharges, providing a first connection between tracer transport theory and bedform trapping and indicating normal diffusion of tracers at relatively small timescales. Variable bedform geometry can influence trap efficiency for individual bedforms and the theoretical model can help identify “preferential trapping” conditions. The distribution of tracer travel distances for a mixture of grain sizes and variable discharge, as expected in natural rivers, displays heavy tail characteristics.

    more » « less
  3. Abstract

    Modeling transport, erosion, and deposition of nonuniform sediment over temporal intervals that are short compared to those characterizing channel bed aggradation and degradation remains an open problem due to the complex quantification of the sediment fluxes between the bed material load and the alluvial deposit. Parker, Paola, and Leclair in 2000 proposed a morphodynamic (PPL) framework to overcome this problem. This framework is used here to model the dispersal of a patch of gravel tracers in three different settings, a laboratory flume, a mountain creek, and a braided river. To simplify the problem, (a) the bed slope, bedload transport rate, and bed configuration are assumed to be constant in space and time (equilibrium), (b) sediment entrainment and deposition are modeled with a constant step length formulation, and (c) the PPL framework is implemented in a one‐dimensional (laterally averaged) model. Model validation against laboratory experiments suggests that, as the transport capacity of the flow increases, the maximum elevation‐specific density of sediment entrainment may migrate downward in the deposit. The comparison between model results and field data shows that the equilibrium solution can reasonably capture tracer dispersal. The equilibrium model can also reproduce subdiffusion and superdiffusion of a patch of tracers in the streamwise direction, depending on the magnitude of the short‐term bed level changes. Finally, the average tracer elevation in a cross‐section decreases in time because particles that are buried deep in the deposit are only rarely reentrained into bedload transport.

    more » « less
  4. Abstract

    Our ability to test probabilistic models linking clast movements to bedload diffusion is most limited by basic field data, because measuring transport statistics during natural floods is difficult. We embedded accelerometers and gyroscopes into artificial cobbles, and measured transport during 28 daily snowmelt floods in Halfmoon Creek, Colorado, USA. The tracers captured ≈6 orders of temporal magnitude of rest durations in one data set for the first time. Motions and rests suggest a scaling transition around ≈12.5 min from subdiffusion to superdiffusion with increasing shear stress and timescale. We interpret that diurnal hydrograph cyclicity may cause another diffusion scaling break at ≈12 h. Shear stress controls and scaling uncertainties may explain differences in diffusion exponents found in several field data sets, suggesting that gravel superdiffusion scaling may be relatively universal over minutes to seasons. Methodologically, “smartrocks” can quantify field transport probabilities previously only possible in laboratory experiments.

    more » « less
  5. The cytoskeleton–a composite network of biopolymers, molecular motors, and associated binding proteins–is a paradigmatic example of active matter. Particle transport through the cytoskeleton can range from anomalous and heterogeneous subdiffusion to superdiffusion and advection. Yet, recapitulating and understanding these properties–ubiquitous to the cytoskeleton and other out-of-equilibrium soft matter systems–remains challenging. Here, we combine light sheet microscopy with differential dynamic microscopy and single-particle tracking to elucidate anomalous and advective transport in actomyosin-microtubule composites. We show that particles exhibit multi-mode transport that transitions from pronounced subdiffusion to superdiffusion at tunable crossover timescales. Surprisingly, while higher actomyosin content increases the range of timescales over which transport is superdiffusive, it also markedly increases the degree of subdiffusion at short timescales and generally slows transport. Corresponding displacement distributions display unique combinations of non-Gaussianity, asymmetry, and non-zero modes, indicative of directed advection coupled with caged diffusion and hopping. At larger spatiotemporal scales, particles in active composites exhibit superdiffusive dynamics with scaling exponents that are robust to changing actomyosin fractions, in contrast to normal, yet faster, diffusion in networks without actomyosin. Our specific results shed important new light on the interplay between non-equilibrium processes, crowding and heterogeneity in active cytoskeletal systems. More generally, our approach is broadly applicable to active matter systems to elucidate transport and dynamics across scales. 
    more » « less