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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. Key Points The bed load flux involves advective and diffusive parts Gradients in particle activity induce a diffusive flux Particle positions and velocities define a statistical ensemble

We report laboratory experiments on intense bed‐load driven by turbulent open‐channel flows. Using high‐speed cameras and a laser light sheet, we measured detailed profiles of granular velocity and concentration near the sidewall. The profiles provide new information on transport layer structure and its relation to the applied Shields stress. Contrary to expectations, we find that intense bed‐load layers respond to changes in flow conditions by adjusting their granular concentration at the base, slightly above the bed. Two mechanisms account for the resulting behavior: stresses generated by immersed granular collisions, and equilibration of the otherwise unstable shear layer by density stratification. Without parameter adjustment, the deduced constitutive relations capture the responses of velocity, concentration, and layer thickness to a ten‐fold increase in Shields stress. Key Points New measurements of velocity and concentration profiles Clarify the structure of bed‐load layers Allow derivation of physically based constitutive relation

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. Key Points Bed load particle activity systematically varies with scale Active particle configurations are preferentially selected by turbulence Particle velocities possess an exponential‐like distribution

Ongoing cryospheric change has modified sediment export from glacierized catchments substantially, with significant implications for ecosystems and downstream users, notably hydropower companies. Sediment is exported either as finer sediment in suspension or as coarser bedload with intermittent contact between sediment and the bed. To date, the difficulty in observing subglacial bedload transport limits the understanding of the physical processes associated with evacuating bedload compared with suspended load. We elucidate the factors controlling sediment export by inverting a physically-based numerical model of subglacial sediment production and sediment transport with suspended sediment and continuous bedload discharge records from an Alpine glacier. Comparable quantities of suspended sediment and bedload are exported, and model results suggest that both rely on the availability of sediment for transport. Yet, bedload export in subglacial channels also depends on particular hydraulic conditions, notably channel shape and hydraulic roughness. This makes exporting bedload-sized particles inefficient compared to fine-grained sediment. As a result, subglacial hydraulics should be explicitly considered when examining bedload export processes, and suspended and bedload transport should be considered separately. Inefficient bedload evacuation by melt water implies that glacial erosion may only continue when non-fluvial mechanisms evacuate sediment, such as sediment entrainment into the ice. Suspended and bedload sediment export from glaciers responds to different hydrogeomorphic processes. Understanding both processes is key for evaluating glaciers’ impacts on landscape evolution and sediment export as the climate warms.

Extreme precipitation is increasing the risk of dam breaks and formation occurring debris dams. Accurate prediction of dam‐break wave propagation is critical to disaster emergency management. Intense bed‐load transport by dam‐break floods can result in a dramatic change of topography, which in turn may affect flood propagation. However, only a very few studies have investigated the thin intense bed‐load layer under dam‐break floods. In this paper, a meshless two‐phase mathematical model is utilized to examine the water velocity, sediment velocity and volumetric fraction, and bed‐load transport flux as well as energy dissipation in bed‐load layer. The model is applied to simulate two‐ and three‐dimensional laboratory experiments of dam‐break wave over erodible beds. For the two‐dimensional experiment, the relative root mean square errors in computed water surface are all below 3.60% and those in profiles of bed‐load layer and static bed are mostly below 13.40%. For the three‐dimensional case, the relative error in computed highest water level is lower than 5.9%. Sediment stream‐wise velocity in bed‐load layer follows a power‐law vertical distribution while sediment volumetric fraction decreases linearly upwards. Accordingly, a formulation of the vertical distribution of bed‐load transport flux, contradictory to the parabolic law in existing studies, is proposed. Most of the water mechanical energy transferred to the sediment is dissipated due to the shear stress in the intense bed‐load layer while only a limit part is kept by the sediment grains. Energy dissipation due to sediment shear stress dominates the consumption of total mechanical energy in the two‐phase system. Key Points The thin intense bed‐load layer transported by dam‐break floods is numerically examined using a meshless two‐phase mathematical model A power‐law profile of sediment streamwise velocity and a linear one of sediment volumetric fraction in the thin bed‐load layer are proposed A new formulation of the sediment transport flux in the bed‐load layer is proposed

Recent studies have observed deviation from normal (Fickian) diffusion in sediment tracer dispersion that violates the assumption of statistical convergence to a Gaussian. Nikora et al. (2002) hypothesized that particle motion at short time scales is superdiffusive because of inertia, while long‐time subdiffusion results from heavy‐tailed rest durations between particle motions. Here we test this hypothesis with laboratory experiments that trace the motion of individual gravels under near‐threshold intermittent bed load transport (0.027 < τ* < 0.087). Particle behavior consists of two independent states: a mobile phase, in which indeed we find superdiffusive behavior, and an immobile phase, in which gravels distrained from the fluid remain stationary for long durations. Correlated grain motion can account for some but not all of the superdiffusive behavior for the mobile phase; invoking heterogeneity of grain size provides a plausible explanation for the rest. Grains that become immobile appear to stay at rest until the bed scours down to an elevation that exposes them to the flow. The return time distribution for bed scour is similar to the distribution of rest durations, and both have power law tails. Results provide a physical basis for scaling regimes of anomalous dispersion and the time scales that separate these regimes. Key Points Anomalous bed load sediment diffusion arises from grain‐scale interactions Particle inertia and grain heterogeneity explain superdiffusion at short times Heavy‐tailed particle waits are related to bed scour process

Given the challenges of re-creating complex bed load (BL) transport processes in rivers, models are preferred over gathering and examining field data. The highlight of the present research is to develop an approach to determine the ungauged bed load concentration (BLC u ) utilizing the measured suspended sediment concentration (SSC) and hydraulic variables of the last four decades for the Mahanadi River Basin. This technique employs shear stress and SSC equations for turbulent open channel flow. Besides, the predicted BLC u is correlated with SSC using a power relation to estimate BLC u on the river and tributaries. Eventually, different BL functions (BLF) efficiency is assessed across stations. The model predicted BLC u is comparable with the published data for sandy rivers and falls within ± 20%. Outliers in hydraulic and sedimentological statistics significantly influence estimating the BL fraction apart from higher relative ratios and catchment geology. The constants of power functions are physically linked to sediment transport configuration, mechanism, and inflow to the stream. The stream power-based BLF best predicts the BL transport, followed by shear stress and unit discharge approaches. The disparity in the estimation of BLC u results from station-specific physical factors, sampling data dispersion, and associated uncertainties.

Although several laboratory studies on the propagation of bed load pulses were carried out in the last decades, most studies neglect grain‐size‐specific aspects or use invasive measurement techniques. To remedy the situation, we present a novel, time‐efficient and non‐destructive laboratory technique to investigate grain‐size‐specific transport characteristics of bed load pulses. The method consists of a through‐water, high‐resolution image acquisition followed by the application of a supervised color classification algorithm (Gaussian Maximum Likelihood Classification). The analyzed bed load pulse consisted of five different grain size classes of dyed quartz sand and gravel, each having a unique color. The initial experimental bed was uni‐colored and contained the same size fractions as the augmented pulse. Quality assessment based on a confusion matrix approach and basic random sampling showed a high classification performance. By statistically analyzing the temporal and spatial color distribution of the experimental reach, characteristic parameters to describe the propagation behavior were determined. The bed load pulse presented in the application example initially showed strong deviations in the grain‐size‐specific advection and dispersion, and advection proved to be predominant in the transport process. Plain Language Summary In the present paper we introduce a novel laboratory method to investigate the grain‐size‐specific transport behavior of bed load pulses. A bedload pulse is a sudden increase in sand and gravel moving along the bottom of a river. We supplied a multi‐colored sediment input representing the downstream propagating pulse on a uni‐color experimental bed which contained the same size fractions than the pulse. Here, five different colors, each indicating a specific grain size fraction, were used as input. By automatically detecting the transported grains due to its color, we were able to analyze the grain‐size‐specific spreading and transport characteristics of the pulse. Therefore, we used high‐resolution photographs of our bed surface and applied a classification algorithm termed Gaussian Maximum Likelihood Classification. A quality assessment revealed a high classification accuracy of the used method. During the experiment, coarser fractions initially showed higher transport velocities. The transport process was dominated by a downstream shift (advection) of the plume rather than by the longitudinal spreading (dispersion). Key Points A novel image‐based, non‐destructive laboratory method to investigate grain‐size‐specific transport of bed load pulses is presented In addition to bed load pulse studies, the proposed method has high potential for other applications in sediment research The pulse in the application example evolves by a combination of advection and dispersion with a predominant advective component

We present a new random walk model for bed load sediment transport that explains the scale‐dependency generally observed in transport rates and captures the transient anomalous dispersion often seen in rivers. Particles alternate between mobile and resting phases, with a tempered stable probability distribution for both particle step length and resting time. Tempered fractional mobile‐immobile differential equations model the ensemble average of particle dynamics. The model is tested against data from three sediment dispersion experiments. Using tempering in both space and time, the new model is able to capture the full range of observed ensemble particle dynamics. The random walk model illuminates the physical meaning of all transport parameters in the mobile‐immobile equations and explains transitions between observed super‐diffusive, sub‐diffusive, and regular diffusive ensemble particle dynamics. By explicitly predicting the effects of spatial and temporal averaging on particle dynamics, this method can be used to link observations of fluvial sediment dynamics across scales. This approach is also generally applicable to a wide variety of geophysical and ecological dynamics, such as ecological dispersal, pathogen transmission in rivers, nutrient export from watersheds, and large‐scale geomorphodynamics associated with infrequent phenomena such as avalanches and turbidity currents. Key Points Quantify sediment transport across scales using a unified stochastic model Explain the variance scaling of bed‐load transport Test the random‐walk model using well‐known experiments

Spatial distributions of depth‐averaged water velocity, shear velocity, and apparent bed load velocity are mapped for the first time in a long reach of a wandering gravel bed river, lower Fraser River, British Columbia. Spatially intensive acoustic Doppler current profiler (aDcp) measurements were collected on the falling limbs of two freshets. Flow in the first year was near the threshold of motion, whereas in the second year discharge exceeded bankfull. Spatial distributions are interpolated from the point data using kriging. Joint density functions for shear velocity and flow depth throughout the reach are presented; marginal densities for shear velocity were near normally distributed but depth distributions were positively skewed by deep pools. The uncertainty of the spatial distributions is also assessed based on modeled temporal variability of the flow and bed load transport, measured aDcp error velocities, and calculated interpolation errors. The resulting maps are remarkably coherent, with maximum depth‐averaged velocity, shear velocity, and apparent bed load velocity following the thalweg. Largest values occur in channel bends at zones of flow convergence where the thalweg flow accelerates toward the bank. However, in the lower flow year the highest apparent bed load velocity was observed outside the thalweg in a deep pool downstream of a rapidly eroding cut bank. Erosion at this site was related to a flow confluence with relatively low shear but highly turbulent, strongly three‐dimensional separated flow.