Radio‐tracking gravel particles in a large braided river in New Zealand: a field test of the stochastic theory of bed load transport proposed by Einstein

Abstract

Abstract:Hans A. Einstein initiated a probabilistic approach to modelling sediment transport in rivers. His formulae were based on theory and were stimulated by laboratory investigations. The theory assumes that bed load movement occurs in individual steps of rolling, sliding or saltation and rest periods. So far very few attempts have been made to measure stochastic elements in nature. For the first time this paper presents results of radio-tracing the travel path of individual particles in a large braided gravel bed river: the Waimakariri River of New Zealand. As proposed by Einstein, it was found that rest periods can be modelled by an exponential distribution, but particle step lengths are better represented by a gamma distribution. Einstein assumed an average travel distance of 100 grain-diameters for any bed load particle between consecutive points of deposition, but larger values of 6Ð7 m or 150 grain-diameters and 6Ð1 m or 120 graindiameters were measured for two test particle sizes. Together with other available large scale field data, a dependence of the mean step length on particle diameter relative to the D 50 of the bed surface was found. During small floods the time used for movement represents only 2Ð7% of the total time from erosion to deposition. The increase in percentage of time being used for transport means that it then has to be regarded in stochastic transport models. Tracing the flow path of bed load particles between erosion and deposition sites is a step towards explaining the interactions between sediment transport and river morphology.

Dynamics and mechanics of bed-load tracer particles

“…At the single-flood scale, tracer particle displacements have been shown to be well described by exponential, gamma, and power-law distributions (Phillips et al, 2013;Habersack, 2001;Hassan et al, 2013). Here we find that the majority of observations are well described by exponential distributions, with two exceptions that decay faster than exponentially (Fig.…”

“…Examining passive tracers in the field introduces an ambiguity; one measures particle displacement -i.e., the distance a particle travels between successive surveys of its position -but this displacement is composed of an unknown number of steps and rests. Displacement length distributions measured for individual floods, and at longer timescales over many floods, typically follow exponential or gamma-like distributions (Hassan et al, 1991;Schmidt and Ergenzinger, 1992;Habersack, 2001;Lamarre and Roy, 2008;Bradley and Tucker, 2012;Hassan et al, 2013;Phillips et al, 2013). We propose two simple limits for particle displacement during a flood: (1) the lower limit is that a particle executes a single step, with a characteristic length scale predicted by Eq.…”

Diffusion of bedload particles in open-channel flows: distribution of travel times and second-order statistics of particle trajectories

“…Available data suggest that particle step lengths at the intermediate scale (i.e., distances between long rests) are likely to follow exponential [7,8,13], gamma [12], or two-parameter gamma [11] distributions. In these references, it can also be found that the mean step lengths may vary, depending on travel conditions, from 100 to 150 particle diameters.…”

“…The information on the empirical distributions of rest periods is less extensive. The available data advocate that this distribution is exponential [7,8,11] or follows the power law when the deposited particles are buried by other particles [14,15,20]. The information on the step length and resting time distributions can be incorporated in a deterministic Lagrangian model of saltating grains [3], which in turn can provide a basis for extensive numerical simulations for the identification of the diffusive behavior of particles at different time scales.…”

Promise, performance and current limitations of a magnetic Bedload Movement Detector

“…These values are reasonable when compared to instantaneous field and laboratory velocities of individual particles, which range from 0·1 to 2 m/s (e.g. Meland and Norrman, 1966;Bridge and Dominic, 1984;Drake et al, 1988;Chacho et al, 1994;Habersack, 2001;Rennie and Millar, 2007).…”