Bedload Sediment Flux Research Articles

Check dam impact on sediment loads: example of the Guerbe River in the Swiss Alps – a catchment scale experiment

Abstract. The construction of check dams is a common practice around the world where the aim is to reduce the damage by flooding events through mountain streams. However, quantifying the effectiveness of such engineering structures has remained very challenging and requires well-selected case studies, since the outcome of such an evaluation depends on site-specific geometric, geologic and climatic conditions. Conventionally, the check dams' effectiveness has been estimated using information about how the bedload sediment flux in the stream changes after the check dams are constructed. A permanent lowering of the bedload flux not only points to a success in reducing the probability of sediment transport occurrence but also implies that the sediment input through the system is likely to decrease. Here, we applied a method for data acquisition and two different equations (Meyer-Peter–Müller and Recking approach) to estimate and compare the sediment transport in a mountain stream in Switzerland under engineered and non-engineered conditions. Whereas the first equation is derived from a classical approach that is based on flume experiment data with a slope of less than 0.02 m m−1, the second equation (Recking) has been derived based on a bedload field dataset comprising active mountain streams under steeper conditions. We selected the Guerbe (Gürbe) River situated in the Swiss Alps as a case study, which has been engineered since the end of the 19th century. This has resulted in more than 110 check dams along a ca. 5 km reach where sediment has continuously been supplied from adjacent hillslopes, primarily by landsliding. We measured the riverbed grain size, topographic gradients and river widths within selected segments along this reach. Additionally, a gauging station downstream of the reach engineered with check dams yielded information to calibrate the hydroclimatic situation for the study reach, thus offering ideal conditions for our catchment-scale experiment. Using the acquired data and the dataset about historical runoff covering the time interval between 2009 and 2021 and considering the current engineered conditions, we estimated a mean annual volume of transported bedload which ranges from 900 to 6000 m3 yr−1. We then envisaged possible channel geometries before the check dams were constructed. We inferred (1) higher energy gradients which we averaged over the length of several check dams and which we considered a proxy for the steeper river slope under natural conditions; (2) channel widths that are smaller than those measured today, thereby anticipating that the channel was more confined in the past; and (3) larger grain size percentiles, which we consider to be similar to the values measured from preserved landslides in the region. Using such potential non-engineered scenarios as constraints, the two equations both point towards a larger sediment flux compared to the engineered state, although the results of these equations differed significantly in magnitude. Whereas the Recking approach returned estimates where the bedload sediment flux is ca. 10 times larger in comparison with the current situation, the use of the Meyer-Peter–Müller equation predicts an increase of ca. 100 times in bedload fluxes for a state without check dams. These results suggest that the check dams in the Guerbe River are highly efficient not only in regulating sediment transport by decreasing the probability of high sediment flux occurrence during torrential conditions but also in stabilizing the channel bed by avoiding incision. The most likely consequence is a stabilization of the terrain around such structures by reducing the activation of landslides.

Lateral sediment connectivity by curve number and a proposed approach to soil erodibility at the watershed scale

The flux of suspended sediment and bedload in a watershed is closely linked to the resistance of soil, known as erodibility. However, the relation between the resistance generated by soil to sediment transport from the hillslope to the river flow is still a challenge. Typically, the bedload flux is related to the shear stress of the main channel. However, the link between lateral sediment flux from hillslopes to suspended sediment flux of the main channel and the relation between bedload and suspended sediment flux to soil erodibility need to be developed. For three Brazilian watersheds, suspended sediment and bedload transport, soils, and hydrological data were obtained to determine watershed soil erodibility. The landscape connectivity expressed by hydrological-sediment connectivity was applied to determine the lateral sediment transport to the primary watershed channel. The flow resistance and lateral suspended sediment discharge were represented by the curve number, which summarized the resistance from the soil-group complex leading to soil erodibility. The curve number addressed the lateral flow and suspended sediment transport to represent the lateral connectivity on the landscape. Sediment transport (sand and suspended sediment) from hillslopes was added to the bedload transport along the longitudinal axis of the watershed, and the watershed soil erodibility (KW) was, thus, obtained for each of the three evaluated watersheds.

EFFECTS OF WAVE SKEWNESS AND ASYMMETRY ON THE EVOLUTION OF FIRE ISLAND, NEW YORK

Bedload transport of sediment by waves and currents is one of the key physical processes that affect the evolution of coasts, nearshore areas, and the engineering practices there. Wave skewness and asymmetry, both of which increase as waves shoal, result in a net bedload sediment flux over a wave cycle. The impacts of this mechanism on large-scale coastal and shoreline change are investigated in this study, using field observations and Coupled Ocean Atmosphere Wave Sediment Transport (COAWST), a hydrodynamic process-based numerical modeling system (Warner et al., 2010). The study site is Fire Island, New York, located at the Atlantic Coast of the USA, with a focus on the persistent shoreline shape, at the western half of this 50-km-long barrier island, that has been hypothesized to be linked to the sand deposits at the shoreface.

Numerical investigation of scour around the monopile using CFD-DEM coupling method

The risks posed by local scour around offshore monopile foundations are substantial and threaten the structural safety of these installations. It is therefore essential to study the mechanisms of local scour to mitigate these risks. Numerical simulation is an effective tool to help us understand the mechanisms of local scour. However, the accurate modeling of the local scour process remains to be challenging due to the complexity of sediment transportation, in which the continuity threatment of sediment phase is typically used, the discrete behavior of the sediment is hard to be described. Therefore, in this paper, in order to conduct the deepin investigation into the mechanism of the local scour around monopile, a three-dimensional CFD-DEM model was established by coupling the computational fluid dynamic method (CFD) and discrete element method (DEM). To increase the efficiency and accuracy of the CFD-DEM model, the coarse grain method (CGM) was used to decrease the number of required DEM particles in the simulation, and the angle of repose was used to calibrate the contact parameters between DEM particles. During the validation process of the numerical model, it was found that the development of the scour depth and the morphology of scour pit obtained by the CFD-DEM model showed good agreement with the previously published results. The CFD-DEM simulation revealed that seepage-induced vertical drag force plays a crucial role in the initial stage of local scour. Two sediment transportation processes, "Pile toe erosion-Slope avalanche" and "Push-Accumulation-Wash", were observed and the relationship between bedload sediment flux and the Shields number was quantified. In the end of this paper, the critical evaluation of the CFD-DEM method was conducted.

Grain shape effects in bed load sediment transport.

Bed load sediment transport, in which wind or water flowing over a bed of sediment causes grains to roll or hop along the bed, is a critically important mechanism in contexts ranging from river restoration1 to planetary exploration2. Despite its widespread occurrence, predictions of bed load sediment flux are notoriously imprecise3,4. Many studies have focused on grain size variability5 as a source of uncertainty, but few have investigated the role of grain shape, even though shape has long been suspected to influence transport rates6. Here we show that grain shape can modify bed load transport rates by an amount comparable to the scatter in many sediment transport datasets4,7,8. We develop a theory that accounts for grain shape effects on fluid drag and granular friction and predicts that the onset and efficiency of transport depend on the coefficients of drag and bulk friction of the transported grains. Laboratory experiments confirm these predictions and reveal that the effect of grain shape on sediment transport can be difficult to intuit from the appearance of grains. We propose a shape-corrected sediment transport law that collapses our experimental measurements. Our results enable greater accuracy in predictions of sediment transport and help reconcile theories developed for spherical particles with the behaviour of natural sediment grains.

A Novel Sediment Transport Model (STM) Accounting Phase Lag Effect. A Resonance Condition

The classical Exner model coupled with a bed-load sediment flux formula is widely used to describe the morphodynamics of coastal environments. However, the main drawbacks of this model are (i) Lack of robustness, (ii) Lack of differentiation between sediment and fluid velocities, and (iii) Generation of instabilities when the interactions between sediment and fluid flow become more important. Moreover, Exner's model does not allow us to know with which characteristic velocity the bottom is moving. This set of drawbacks weakens the effectiveness of most sediment transport models proposed in the literature, particularly the Exner model. In this work, we reformulate the bed-load equation and we propose a new averaged sediment transport model for application in coastal or estuarine environments. The proposed model incorporates phase shift effects into the bed-load equation. The bedform's characteristic velocity, sediment, and fluid velocity are differentiated. We developed a new first-order, well-balanced, positivity-preserving, path-preserving, and central wind (WBPP-PCCU) scheme to solve the proposed hyperbolic sediment transport model (HSTM). We used the Averaging Essentially Non-Oscillatory (AENO) reconstruction coupled with the third-order Runge-Kutta Semi-Implicit (SI-RK3) method to achieve second-order accuracy. The balance and positivity of the water depth properties were proven. In this work, a resonance condition is proposed. The model facilitates the application of several other schemes such as Roe, HLLC, HLLEM, PVM (polynomial viscosity matrix), RVM (rational viscosity matrix), which require the diagonalization of the Jacobian matrix. The accuracy, robustness, positivity preservation, and equilibrium properties of the resulting model are evaluated using a series of carefully selected test cases. The proposed model provides an excellent ability to simulate sediment transport in a wide range of coastal environments.

Probabilistic description of bedload fluxes from the aggregate dynamics of individual grains

Abstract. We formulate the bedload sediment flux probability distribution from the Lagrangian dynamics of individual grains. Individual particles obey Langevin equations wherein the stochastic forces driving particle motions are switched on and off by particle entrainment and deposition. The flux is calculated as the rate of many such particles crossing a control surface within a specified observation time. Flux distributions inherit observation time dependence from the on–off motions of particles. At the longest observation times, distributions converge to sharp peaks around classically expected values, but at short times, fluctuations are erratic. We relate this scale dependence of bedload transport rates to the movement characteristics of individual sediment grains. This work provides a statistical mechanics description for the fluctuations and observation-scale dependence of sediment transport rates.

Spatial Trends and Drivers of Bedload and Suspended Sediment Fluxes in Global Rivers

AbstractBedload is notoriously challenging to measure and model; its dynamics, therefore, remains largely unknown in most fluvial systems worldwide. We present results from a global scale bedload flux model as part of the WBMsed modeling framework that well predict the distribution of water discharge, suspended sediment and bedload. The sensitivity of bedload predictions to river slope, particle size, discharge, river width, and suspended sediment were analyzed, showing the model to be most responsive to spatial dynamics in river discharge and slope. The relationship between bedload and total sediment flux is analyzed globally, and for representative longitudinal river profiles (Amazon, Mississippi, and Lena Rivers). The results show that while the proportion of bedload decreases from headwaters to the coasts, there is considerable variability between basins and along river corridors. The topographic and hydrological longitudinal profiles of rivers are shown to be the key drivers of bedload trends, with fluctuations in slope controlling its more local dynamics. Estimates of water and sediment fluxes to global oceans from 2,067 largest river outlets (draining 67% of the global continental area) are provided. Estimated water discharge at 30,579 km3/y corresponds well to past estimates; however, sediment flux is higher. Total global particulate load of 17.8 Gt/y is delivered to global oceans, 14.8 Gt/y as washload, 1.1 Gt/y as bedload, and 2.6 Gt/y as suspended bed material. The largest 25 rivers are predicted to transport more than half of the total sediment flux to global oceans.

Observations of Bedform Migration and Bedload Sediment Transport in Combined Wave‐Current Flows

AbstractBedload transport is an important mechanism for sediment flux in the nearshore. Yet few studies examine the relationship between bedform evolution and net sediment transport. Our work contributes concurrent observations of bedform mobility and bedload transport in response to wave dominant, current dominant, and combined wave‐current flows in the nearshore. Bedload sediment flux from migrating bedforms during combined wave‐current conditions accounted for at least 20% more bedload transport when compared with wave dominant flows and at least 80% more than current‐dominant flows. Bedforms were observed to transport the most sediment during periods with strong currents, with high‐energy skewed waves, and while bedform orientation and transport direction were aligned. Regardless of flow type, bedform migration rates were directly proportional to the total kinetic energy contained in the flow field. Eleven bedload transport models formulated to be used in combined flows (both shear and energetics based) were compared with sediment flux estimated from measured bedform migration. An energetics based sediment transport model was most representative for our data.

Quantification of bedform dynamics and bedload sediment flux in sandy braided rivers from airborne and satellite imagery

AbstractImages from specially‐commissioned aeroplane sorties (manned aerial vehicle, MAV), repeat unmanned aerial vehicle (UAV) surveys, and Planet CubeSat satellites are used to quantify dune and bar dynamics in the sandy braided South Saskatchewan River, Canada. Structure‐from‐Motion (SfM) techniques and application of a depth‐brightness model are used to produce a series of Digital Surface Models (DSMs) at low and near‐bankfull flows. A number of technical and image processing challenges are described that arise from the application of SfM in dry and submerged environments. A model for best practice is presented and analysis suggests a depth‐brightness model approach can represent the different scales of bedforms present in sandy braided rivers with low‐turbidity and shallow (< 2 m deep) water.The aerial imagery is used to quantify the spatial distribution of unit bar and dune migration rate in an 18 km reach and three ~1 km long reaches respectively. Dune and unit bar migration rates are highly variable in response to local variations in planform morphology. Sediment transport rates for dunes and unit bars, obtained by integrating migration rates (from UAV) with the volume of sediment moved (from DSMs using MAV imagery) show near‐equivalence in sediment flux. Hence, reach‐based sediment transport rate estimates can be derived from unit bar data alone. Moreover, it is shown that reasonable estimates of sediment transport rate can be made using just unit bar migration rates as measured from 2D imagery, including from satellite images, so long as informed assumptions are made regarding average bar shape and height. With recent availability of frequent, repeat satellite imagery, and the ease of undertaking repeat MAV and UAV surveys, for the first time, it may be possible to provide global estimates of bedload sediment flux for large or inaccessible low‐turbidity rivers that currently have sparse information on bedload sediment transport rates. © 2018 The Authors. Earth Surface Processes and Landforms published by John Wiley & Sons Ltd.

Measuring Bedload Sediment Transport with an Acoustic Doppler Velocity Profiler

Acoustic Doppler velocity profilers (ADVP) measure the velocity simultaneously in a linear array of bins. They have been successfully used in the past to measure three-dimensional turbulent flow and the dynamics of suspended sediment. The capability of ADVP systems to measure bedload sediment flux remains uncertain. The main outstanding question relates to the physical meaning of the velocity measured in the region where bedload sediment transport occurs. The main hypothesis of the paper, that the ADVP measures the velocity of the moving bedload particles, is validated in laboratory experiments that range from weak to intense bedload transport. First, a detailed analysis of the raw return signals recorded by the ADVP reveals a clear footprint of the bedload sediment particles, demonstrating that these are the main scattering sources. Second, time-averaged and temporal fluctuations of bedload transport derived from high-speed videography are in good agreement with ADVP estimates. Third, ADVP based estimates of bedload velocity and thickness of the bedload layer comply with semi-theoretical expressions based on previous results. An ADVP configuration optimized for bedload measurements is found to perform only marginally better than the standard configuration for flow measurements, indicating that the standard ADVP configuration can be used for sediment flux investigations. Data treatment procedures are developed that identify the immobile-bed surface, the layers of rolling/sliding and saltating bedload particles, and the thickness of the bedload layer. Combining ADVP measurements of the bedload velocity with measurements of particle concentration provided by existing technology would provide the sediment flux.

An approach to assess the impact of landscape connectivity and effective catchment area upon bedload sediment flux in Saco Creek Watershed, Semiarid Brazil

Sediment flux in semi-arid drainage networks is influenced by landscape connectivity, wherein natural landforms and hydro infrastructure networks such as low order dams impede the downstream transfer of sediment. This study analyzes how drainage impediments affect bedload transport within Saco Creek watershed, in semi-arid northeast Brazil. Geomorphology and land use maps are constructed to assess changes to short term landscape connectivity. Bedload conveyance is appraised for precipitation/runoff events of variable magnitude-frequency relationships. From this, ‘effective’ rainfall events are determined. Analysis of the type and location/pattern of drainage impediments is used to assess effective catchment area under three distinct runoff scenarios. In the low magnitude event scenario, only 33.8% of the watershed area contributed effectively to sediment flux. In moderate and high magnitude events this value increased to 61.6% and 70.8% respectively. The relatively small variation from moderate to high magnitude events reflects the operation of dams which store bedload sediments. Findings from this study can be used to show how landscape connectivity analysis can support dam management in semiarid agrarian landscapes.

Video observations of bed form morphodynamics in a meander bend

A new optical remote sensing technique for estimating water depth from an oblique camera view is described. The water surface and the bed were imaged simultaneously to create time-dependent maps of the water surface velocities and the bed elevations that can be used to validate numerical models at high spatial and temporal resolution. The technique was applied in a sandy meander bend at the University of Minnesota Saint Anthony Falls Laboratory Outdoor StreamLab. The root mean square differences between optical estimates of the bed and in situ observations ranged between 0.01 and 0.03 m. Mean bed form wavelength was 0.73 m and mean crest height was 0.07 m, but both varied with distance around the meander bend. Bed form classification varied with distance downstream, and sinuosity of bed forms varied with local radius of curvature. Bed form roughness scaled similarly to other natural riverine environments although wavelength and height magnitude and variability were larger than predicted by empirical formulations for straight reaches. Bed form translation rate varied between 1 and 5 mm s−1. Estimates of velocity from particle image velocimetry (PIV) on the water surface were ∼10% higher than in situ observations collected ∼0.05 m below the water surface. Using the PIV observations to drive simple equations for bed load sediment flux, we explained up to 72% of the observed variance in downstream sediment flux. The new methodology described here provides nonintrusive, high spatial and temporal resolution measurements of both the bed and the flow.

A physical model for seismic noise generation by turbulent flow in rivers

AbstractPrevious studies suggest that the seismic noise induced by rivers may be used to infer river transport properties, and previous theoretical work showed that bedload sediment flux can be inverted from seismic data. However, the lack of a theoretical framework relating water flow to seismic noise prevents these studies from providing accurate bedload fluxes and quantitative information on flow processes. Here we propose a forward model of seismic noise caused by turbulent flow. In agreement with previous observations, modeled turbulent flow‐induced noise operates at lower frequencies than bedload‐induced noise. Moreover, the differences in the spectral signatures of turbulent flow‐induced and bedload‐induced forces at the riverbed are significant enough that these two processes can be characterized independently using seismic records acquired at various distances from the river. In cases with isolated turbulent flow noise, we suggest that riverbed stress can be inverted. Finally, we validate our model by comparing predictions to previously reported observations. We show that our model captures the spectral peak located around 6–7 Hz and previously attributed to water flow at Hance Rapids in the Colorado River (United States); we also show that turbulent flow causes a significant part of the seismic noise recorded at the Trisuli River in Nepal, which reveals that the hysteresis curve previously reported there does not solely include bedload, but is also largely influenced by turbulent flow‐induced noise. We expect the framework presented here to be useful to invert realistic bedload fluxes by enabling the removal of the turbulent flow contribution from seismic data.

Non-additive increases in sediment stability are generated by macroinvertebrate species interactions in laboratory streams.

Previous studies have shown that biological structures such as plant roots can have large impacts on landscape morphodynamics, and that physical models that do not incorporate biology can generate qualitatively incorrect predictions of sediment transport. However, work to date has focused almost entirely on the impacts of single, usually dominant, species. Here we ask whether multiple, coexisting species of hydropsychid caddisfly larvae have different impacts on sediment mobility compared to single-species systems due to competitive interactions and niche differences. We manipulated the presence of two common species of net-spinning caddisfly (Ceratopsyche oslari, Arctopsyche californica) in laboratory mesocosms and measured how their silk filtration nets influence the critical shear stress required to initiate sediment grain motion when they were in monoculture versus polyculture. We found that critical shear stress increases non-additively in polycultures where species were allowed to interact. Critical shear stress was 26% higher in multi-species assemblages compared to the average single-species monoculture, and 21% greater than levels of stability achieved by the species having the largest impact on sediment motion in monoculture. Supplementary behavioral experiments suggest the non-additive increase in critical shear stress may have occurred as competition among species led to shifts in the spatial distribution of the two populations and complementary habitat use. To explore the implications of these results for field conditions, we used results from the laboratory study to parameterize a common model of sediment transport. We then used this model to estimate potential bed movement in a natural stream for which we had measurements of channel geometry, grain size, and daily discharge. Although this extrapolation is speculative, it illustrates that multi-species impacts could be sufficiently large to reduce bedload sediment flux over annual time scales in streams where multiple species of caddisfly are present.

Morphodynamics of river bed variation with variable bedload step length

Abstract. Here we consider the 1-D morphodynamics of an erodible bed subject to bedload transport. Fluvial bed elevation variation is typically modeled by the Exner equation, which, in its classical form, expresses mass conservation in terms of the divergence of the bedload sediment flux. An entrainment form of the Exner equation can be written as an alternative description of the same bedload processes, by introducing the notions of an entrainment rate into bedload and of a particle step length, and assuming a certain probability distribution for the step length. This entrainment form implies some degree of nonlocality, which is absent from the standard flux form, so that these two expressions, which are different ways to look at same conservation principle (i.e., sediment continuity), may no longer become equivalent in cases when channel complexity and flow conditions allow for long particle saltation steps (including, but not limited to the case where particle step length has a heavy tailed distribution) or when the domain of interest is not long compared to the step length (e.g., laboratory scales, or saltation over relatively smooth surfaces). We perform a systematic analysis of the effects of the nonlocality in the entrainment form of the Exner equation on transient aggradational/degradational bed profiles by using the flux form as a benchmark. As expected, the two forms converge to the same results as the step length converges to zero, in which case nonlocality is negligible. As step length increases relative to domain length, the mode of aggradation changes from an upward-concave form to a rotational, and then eventually a downward-concave form. Corresponding behavior is found for the case of degradation. These results may explain anomalously flat, aggradational, long profiles that have been observed in some short laboratory flume experiments.

New approach to understanding sediment transport

Sediment transport is an important factor in the evolution of river channels. Bed load sediment flux, defined as the volume of particles crossing a vertical surface per unit time per unit width, is a common way to describe sediment transport. However, quantifying bed load sediment flux in terms of conceptually simple quantities is challenging, as the motion of particles in turbulent water is very complex. In a set of four related papers, Furbish et al. developed a new probabilistic definition of bed load transport and verified their approach using laboratory experiments and model simulations. Their approach, which treats particle positions and motions as stochastic quantities, shows how the particle activity and velocity distributions are key elements in describing sediment flux. Their experiments, using high‐speed imaging of sand particle motions in a laboratory tank, allowed them to directly observe particle behavior and test their theory. Model simulations also verified their approach. (Journal of Geophysical Research‐Earth Surface, doi:10.1029/2012JF002352, 2012, doi:10.1029/2012JF002353, 2012, doi:10.1029/2012JF002355, 2012, and doi:10.1029/2012JF002356, 2012)

A probabilistic description of the bed load sediment flux: 3. The particle velocity distribution and the diffusive flux

Particles transported as bed load within a specified streambed area possess at any instant a distribution of velocities. This distribution figures prominently in describing the rates of transport and dispersal of particles. High‐speed imaging of sand particles transported as bed load over a planar bed reveals that the probability density functions of the streamwise and cross‐stream particle velocities are exponential‐like. For quasi‐steady conditions the exponential‐like density of streamwise velocities reflects a balance among three fluxes in momentum space: (1) an advection of streamwise momentum whose magnitude and sign vary with the momentum state; (2) a diffusion of momentum from higher to lower values of momentum density; and (3) a drift of momentum from regions in momentum space having high average rates of generation of kinetic energy toward regions having low rates of generation of kinetic energy. The probability density of cross‐stream velocities similarly reflects a balance of fluxes of cross‐stream momentum. Whereas the average net force acting on particles is zero under steady conditions, the mean, variance and asymmetry of the distribution of forces acting on particles vary with the momentum state of the particles. Numerical simulations of particle motions that are faithful to these statistical properties reproduce key empirical results, namely, the exponential‐like velocity distribution and the nonlinear relation between hop distances and travel times. The simulations also illustrate how steady gradients in particle activity, the solid volume of particles in motion per unit streambed area, induce a diffusive flux as described in companion papers.

A probabilistic description of the bed load sediment flux: 2. Particle activity and motions

High‐speed imaging of coarse sand particles transported as bed load over a planar bed reveals that the particle activity, the solid volume of particles in motion per unit streambed area, fluctuates as particles respond to near‐bed fluid turbulence while simultaneously interacting with the bed. The relative magnitude of these fluctuations systematically varies with the size of the sampling area. The particle activity within a specified sampling area is distributed in a manner that is consistent with the existence of an ensemble of configurations of particle positions wherein certain configurations are preferentially selected or excluded by the turbulence structure, manifest as patchiness of active particles. The particle activity increases with increasing bed stress far faster than does the average particle velocity, so changes in the transport rate with changing stress are dominated by changes in the activity, not velocity. The probability density functions of the streamwise and cross‐stream particle velocities are exponential‐like and lack heavy tails. Plots of the mean squared particle displacement versus time may ostensibly indicate non‐Fickian diffusive behavior while actually reflecting effects of correlated random walks associated with intrinsic periodicities in particle motions, not anomalous diffusion. The probability density functions of the particle hop distance (start‐to‐stop) and the associated travel time are gamma‐like, which provides the empirical basis for showing that particle disentrainment rates vary with hop distance and travel time.

A probabilistic description of the bed load sediment flux: 1. Theory

We provide a probabilistic definition of the bed load sediment flux. In treating particle positions and motions as stochastic quantities, a flux form of the Master equation (a general expression of conservation) reveals that the volumetric flux involves an advective part equal to the product of an average particle velocity and the particle activity (the solid volume of particles in motion per unit streambed area), and a diffusive part involving the gradient of the product of the particle activity and a diffusivity that arises from the second moment of the probability density function of particle displacements. Gradients in the activity, instantaneous or time‐averaged, therefore effect a particle flux. Time‐averaged descriptions of the flux involve averaged products of the particle activity, the particle velocity and the diffusivity; the significance of these products depends on the scale of averaging. The flux form of the Exner equation looks like a Fokker‐Planck equation (an advection‐diffusion form of the Master equation). The entrainment form of the Exner equation similarly involves advective and diffusive terms, but because it is based on the joint probability density function of particle hop distances and associated travel times, this form involves a time derivative term that represents a lag effect associated with the exchange of particles between the static and active states. The formulation is consistent with experimental measurements and simulations of particle motions reported in companion papers.