Joris Heyman

Statistics of bedload transport over steep slopes: Separation of time scales and collective motion

Steep slope streams show large fluctuations of sediment discharge across several time scales. These fluctuations may be inherent to the internal dynamics of the sediment transport process. A probabilistic framework thus seems appropriate to analyze such a process. In this paper, we present an experimental study of bedload transport over a steep slope flume for small to moderate Shields numbers. The sampling technique allows the acquisition of high-resolution time series of the solid discharge. The resolved time scales range from 10− 1 s up to 105 s. We show that two distinct time scales can be observed in the probability density function for the waiting time between moving particles. We make the point that the separation of time scales is related to collective dynamics. Proper statistics of a Markov process including collective entrainment are derived. The separation of time scales is recovered theoretically for low entrainment rates.

Stochastic interpretation of the advection‐diffusion equation and its relevance to bed load transport

The advection diffusion equation is one of the most widespread equations in physics. It arises quite often in the context of sediment transport, e.g., for describing time and space variations in the particle activity (the solid volume of particles in motion per unit streambed area). Phenomenological laws are usually sufficient to derive this equation and interpret its terms. Stochastic models can also be used to derive it with the significant advantage that they provide information on the statistical properties of particle activity. These models are quite useful when sediment transport exhibits large fluctuations (typically at low transport rates), making the measurement of mean values difficult. Among these stochastic models, the most common approach consists of random walk models. For instance, they have been used to model the random displacement of tracers in river streams. Here we explore an alternative approach, which involves monitoring the evolution of the number of particles moving within an array of cells of finite length. Birth-death Markov processes are well-suited to this objective. While the topic has been explored in detail for diffusion-reaction systems, the treatment of advection has received no attention. We therefore look into the possibility of deriving the advection diffusion equation (with a source term) within the framework of birth-death Markov processes. We show that in the continuum limit (when the cell size becomes vanishingly small), we can derive an advection diffusion equation for particle activity. Yet, while this derivation is formally valid in the continuum limit, it runs into difficulty in the practical applications involving cells or meshes of finite length. Indeed, within our stochastic framework, particle advection produces nonlocal effects, which are more or less significant depending on the cell size and particles velocity. Albeit nonlocal, these effects look like (local) diffusion and add to the intrinsic particle diffusion (dispersal due to velocity fluctuations), with the important consequence that local measurements depend on both the intrinsic properties of particles displacement and the dimensions of the measurement system.

Bed load transport over a broad range of timescales: Determination of three regimes of fluctuations

This paper describes the relationship between the statistics of bed load transport flux and the timescale over which it is sampled. A stochastic formulation is developed for the probability distribution function of bed load transport flux, based on the Ancey et al. (2008) theory. An analytical solution for the variance of bed load transport flux over differing sampling timescales is presented. The solution demonstrates that the timescale dependence of the variance of bed load transport flux reduces to a three-regime relation demarcated by an intermittency timescale (t(I)) and a memory timescale (t(c)). As the sampling timescale increases, this variance passes through an intermittent stage ( t(c)). We propose a dimensionless number (Ra) to represent the relative strength of fluctuation, which provides a common ground for comparison of fluctuation strength among different experiments, as well as different sampling timescales for each experiment. Our analysis indicates that correlated motion and the discrete nature of bed load particles are responsible for this three-regime behavior. We use the data from three experiments with high temporal resolution of bed load transport flux to validate the proposed three-regime behavior. The theoretical solution for the variance agrees well with all three sets of experimental data. Our findings contribute to the understanding of the observed fluctuations of bed load transport flux over monosize/multiple-size grain beds, to the characterization of an inherent connection between short-term measurements and long-term statistics, and to the design of appropriate sampling strategies for bed load transport flux.

Entrainment, motion, and deposition of coarse particles transported by water over a sloping mobile bed

In gravel bed rivers, bed load transport exhibits considerable variability in time and space. Recently, stochastic bed load transport theories have been developed to address the mechanisms and effects of bed load transport fluctuations. Stochastic models involve parameters such as particle diffusivity, entrainment, and deposition rates. The lack of hard information on how these parameters vary with flow conditions is a clear impediment to their application to real-world scenarios. In this paper, we determined the closure equations for the above parameters from laboratory experiments. We focused on shallow supercritical flow on a sloping mobile bed in straight channels, a setting that was representative of flow conditions in mountain rivers. Experiments were run at low sediment transport rates under steady nonuniform flow conditions (i.e., the water discharge was kept constant, but bed forms developed and migrated upstream, making flow nonuniform). Using image processing, we reconstructed particle paths to deduce the particle velocity and its probability distribution, particle diffusivity, and rates of deposition and entrainment. We found that on average, particle acceleration, velocity, and deposition rate were responsive to local flow conditions, whereas entrainment rate depended strongly on local bed activity. Particle diffusivity varied linearly with the depth-averaged flow velocity. The empirical probability distribution of particle velocity was well approximated by a Gaussian distribution when all particle positions were considered together. In contrast, the particles located in close vicinity to the bed had exponentially distributed velocities. Our experimental results provide closure equations for stochastic or deterministic bed load transport models.