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.

Bedload transport refers to conditions when grains bounce and skid along the riverbed

Particle-based, stochastic approaches have been developed from which mean values, probability distributions, and the dependence of measured values on averaging scales can all be obtained

The original stochastic description of bedload displacement is due to Einstein, who calculated particle trajectories as a random sequence of rests interrupted by instantaneous steps

The velocities of moving grains fluctuate due to hydrodynamic forcing and particle–bed collisions

Motion–rest alternation and the fluctuating movement velocities of individual particles both lend variability to the sediment flux

The sediment flux has been described as a stochastic process using both renewal theory and population modeling approaches. These methods rely on additional Eulerian characteristics of particle transport, which apply to an ensemble of particles occupying a volume or crossing a surface.
Renewal models introduce inter-arrival time distributions

In this paper, our objective is to formulate the stochastic sediment flux directly from the Lagrangian dynamics of individual grains rather than by introducing additional Eulerian quantities, such as volumetric entrainment and deposition rates or inter-arrival time distributions. To achieve this, in Sect.

The starting point for our analysis is an idealized one-dimensional domain populated with sediment particles on the surface of a sedimentary bed.
Particles are set in motion by the flow and move downstream until they deposit, and the cycle repeats. The downstream coordinate is

From these assumptions, we propose an equation of motion for the individual sediment grain including two features. First, particles should alternate between motion and rest, similar to the earlier motion–rest models summarized by Eq. (

Panel

These equations represent Newtonian dynamics that are

The transport process described by Eq. (

The formulation of Eq. (

The time evolution of Eq. (

A master equation for the phase space density can be formulated by noting that the combined process

To understand the structure of Eq. (

To calculate the sediment transport rate we will use the probability distribution of position for bedload particles, defined as

Fortunately, bedload experiments often show Gaussian velocities for moving particles

When motions are intermittent as in Eq. (

We can actually obtain the overdamped approximation for the phase space equation (Eq.

We integrate Eq. (

Equation (

To phrase the probability distribution of the sediment flux in terms of these particle dynamics, we apply a method very similar to the one developed by

Panel

From this initial configuration, the flux is calculated as the average rate of particles crossing to the right of the control surface at

The probability density

The average over initial conditions and possible trajectories of the indicator for the

Inserting Eq. (

Equation (

Equation (

The overdamped master equation (Eq.

The overdamped master equation (Eq.

Panel

The moments of particle position from Eq. (

The formalism in Sect.

This result indicates a nuanced observation-scale dependence in the sediment flux. We can better understand Eq. (

Panel

The limiting form of

Figure

Figure

In this paper, we have formulated a mechanistic description of the bedload sediment flux using a detailed stochastic model of individual particle displacements. The resulting sediment flux distribution shows Poissonian fluctuations that depend on observation scale. Our displacement model applies over a wider range of timescales than earlier formulations because it includes both Newtonian velocities and motion–rest alternation. In appropriate simplified limits, the displacement model Eq. (

We solved the displacement model analytically to obtain the displacement probability distribution. This derivation relied on the “overdamped” approximation that particles accelerate rapidly following entrainment. This approximation is only possible when the velocities of moving grains are Gaussian. We then formulated the stochastic sediment flux using the resulting particle displacement statistics. The obtained flux distribution mimics earlier renewal theory descriptions of the bedload flux

Our description of individual particle displacements in Sect.

A simplified element of our approach is the representation of entrainment and deposition by instantaneous alternation between motion and rest states

The sediment flux probability distribution derived in Sect.

A vast set of processes generates transport rate fluctuations in real channels. At the shortest timescales, fluctuations arise from the intermittent arrivals of individual grains, as we have described here. Over longer timescales, activity waves

Phrasing the transport rate in terms of the Lagrangian dynamics of individual grains produces a flux distribution that adjusts with the observation time. According to Eq. (

Both

We have formulated the bedload flux probability distribution from the statistical mechanics of individual grains in transport. This formulation produces Poissonian flux distributions having scale-dependent rates, meaning transport rate fluctuations are relatively narrow, and transport characteristics shift with the timescales over which they are observed. In laboratory experiments, sediment transport fluctuations are typically wider than Poissonian. Notably, we can assert that the Poisson flux distribution derived in this paper originates exclusively from the independence of individual grains: the Poisson form is completely indifferent to the forces driving particles downstream so long as these forces do not introduce correlations between particles. In the future, it will be necessary to refine the statistical mechanics formulation presented here to produce wider transport fluctuations. We expect that introducing any component in Eq. (

Because the joint process

In the limit of vanishing

The position probability distribution can be obtained from Eq. (

Now we take Fourier transforms over space and Laplace transforms over time of the overdamped master equation (Eq.

The Laplace transform of Eq. (

The behavior of Eq. (

Python scripts used for the Monte Carlo simulation of Eq. (

All of the data presented in the paper is freely available at

All authors (JKP, MAH, and RMLF) contributed equally to ideation and paper preparation. JKP performed all calculations and constructed all figures.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We would like to thank the two anonymous reviewers for their constructive reviews of the paper.

This paper was edited by Jens Turowski and reviewed by two anonymous referees.