We investigate the entrainment, deposition and motion of coarse spherical particles within a turbulent shallow water stream down a steep slope. This is an idealization of bed-load transport in mountain streams. Earlier investigations have described this kind of sediment transport using empirical correlations or concepts borrowed from continuum mechanics. The intermittent character of particle transport at low-water discharges led us to consider it as a random process. Sediment transport in this regime results from the imbalance between entrainment and deposition of particles rather than from momentum balance between water and particles. We develop a birth-death immigration-emigration Markov process to describe the particle exchanges between the bed and the water stream. A key feature of the model is its long autocorrelation times and wide, frequent fluctuations in the solid discharge, a phenomenon never previously explained despite its ubiquity in both nature and laboratory experiments. We present experimental data obtained using a nearly two-dimensional channel and glass beads as a substitute for sediment. Entrainment, trajectories, and deposition were monitored using a high-speed digital camera. The empirical probability distributions of the solid discharge and deposition frequency were properly described by the theoretical model. Experiments confirmed the existence of wide and frequent fluctuations of the solid discharge, and revealed the existence of long autocorrelation time, but theory overestimates the autocorrelation times by a factor of around three. Particle velocity was weakly dependent on the fluid velocity contrary to the predictions of the theoretical model, which performs well when a single particle is moving. For our experiments, the dependence of the solid discharge on the fluid velocity is entirely controlled by the number of moving particles rather than by their velocity. We also noted significant changes in the behaviour of particle transport when the bed slope or the water discharge was increased. The more vigorous the stream was, the more continuous the solid discharge became. Moreover, although 90 % of the energy supplied by gravity to the stream is dissipated by turbulence for slopes lower than 10 %, particles dissipate more and more energy when the bed slope is increased, but surprisingly, the dissipation rate is nearly independent of fluid velocity. A movie is available with the online version of the paper.

The objective of this review is to examine how the concept of plasticity is used in geophysical fluid dynamics. Rapid mass movements such as snow avalanches or debris flows involve slurries of solid particles (ice, boulder, clay, etc.) within an interstitial fluid (air, water). The bulk behavior of these materials has often been modeled as plastic materials, i.e., a plastic material yields and starts to flow once its stress state has significantly departed from equilibrium. Two plastic theories are of common use in fluid dynamics: Coulomb plasticity and viscoplasticity. These theories have little in common, since ideal Coulomb materials are two-phase materials for which pore pressure and friction play the key role in the bulk dynamics, whereas viscoplastic materials (e.g., Bingham fluids) typically behave as single-phase fluids on the macroscopic scale and exhibit a viscous behavior after yielding. Determining the rheological behavior of geophysical materials remains difficult because they encompass coarse, irregular particles over a very wide range of size. Consequently, the true nature of plastic behavior for geophysical flows is still vigorously debated. In this review, we first set out the continuum-mechanics principles used for describing plastic behavior. The notion of yield surface rather than yield stress is emphasized in order to better understand how tensorial constitutive equations can be derived from experimental data. The notion of single-phase or two-phase behaviors on the macroscopic scale is then examined using a microstructural analysis on idealized suspensions of spheres within a Newtonian fluid; for these suspensions, the single-phase approximation is valid only at very high or low Stokes numbers. Within this framework, the bulk stress tensor can also be constructed, which makes it possible to give a physical interpretation to yield stress. Most of the time, depending on the bulk properties (especially, particle size) and flow features, bulk behavior is either Coulomb-like or viscoplastic in simple-shear experiments. The consequences of the rheological properties on the flow features are also examined. Some remarkable properties of the governing equations describing thin layers flowing down inclined surfaces are discussed. Finally, the question of parameter fitting is tackled: since rheological properties cannot be measured directly in most cases, they must be evaluated from field data. As an example, we show that the Coulomb model successfully captures the main traits of avalanche motion, but statistical analysis demonstrates that the probability distribution of the friction coefficient is not universal.

This paper concerns a model of bed load transport, which describes the advection and dispersion of coarse particles carried by a turbulent water stream. The challenge is to develop a microstructural approach that, on the one hand, yields a parsimonious description of particle transport at the microscopic scale and, on the other hand, leads to averaged equations at the macroscopic scale that can be consistently interpreted in light of the continuum equations used in hydraulics. The cornerstone of the theory is the proper determination of the particle flux fluctuations. Apart from turbulence-induced noise, fluctuations in the particle transport rate are generated by particle exchanges with the bed consisting of particle entrainment and deposition. At the particle scale, the evolution of the number of moving particles can be described probabilistically using a coupled set of reaction-diffusion master equations. Theoretically, this is interesting but impractical, as solving the governing equations is fraught with difficulty. Using the Poisson representation, we show that these multivariate master equations can be converted into Fokker-Planck equations without any simplifying approximations. Thus, in the continuum limit, we end up with a Langevin-like stochastic partial differential equation that governs the time and space variations of the probability density function for the number of moving particles. For steady-state flow conditions and a fixed control volume, the probability distributions of the number of moving particles and the particle flux can be calculated analytically. Taking the average of the microscopic governing equations leads to an average mass conservation equation, which takes the form of the classic Exner equation under certain conditions carefully addressed in the paper. Analysis also highlights the specific part played by a process we refer to as collective entrainment, i.e. a nonlinear feedback process in particle entrainment. In the absence of collective entrainment the fluctuations in the number of moving particles are Poissonian, which implies that at the macroscopic scale they act as white noise that mediates bed evolution. In contrast, when collective entrainment occurs, large non-Poissonian fluctuations arise, with the important consequence that the evolution at the macroscopic scale may depart significantly that predicted by the averaged Exner equation. Comparison with experimental data gives satisfactory results for steady-state flows.

Stratification patterns are formed when a bidisperse mixture of large rough grains and smaller more mobile particles is poured between parallel plates to form a heap. At low flow rates discrete avalanches flow down the free surface and are brought to rest by the propagation of shock waves. Experiments performed in this paper show that the larger particles are segregated to the top of the avalanche, where the velocity is greatest, and are transported to the flow front. Here the particles are overrun but may rise to the free surface again by size segregation to create a recirculating coarse-grained front. Once the front is established composite images show that there is a steady regime in which any additional large grains that reach the front are deposited. This flow is therefore analogous to finger formation in geophysical mass flows, where the larger less mobile particles are shouldered aside to spontaneously form static lateral levees rather than being removed by basal deposition in two dimensions. At the heart of all these phenomena is a dynamic feedback between the bulk flow and the evolving particle-size distribution within the avalanche. A fully coupled theory for such segregation-mobility feedback effects is beyond the scope of this paper. However, it is shown how to derive a simplified uncoupled travellingwave solution for the avalanche motion and reconstruct the bulk two-dimensional flow field using assumed velocity profiles through the avalanche depth. This allows a simple hyperbolic segregation theory to be used to construct exact solutions for the particle concentration and for the recirculation within the bulk flow. Depending on the material composition and the strength of the segregation and deposition, there are three types of solution. The coarse-particle front grows in length if more large particles arrive than can be deposited. If there are fewer large grains and if the segregation is strong enough, a breaking size-segregation wave forms at a unique position behind the front. It consists of two expansion fans, two shocks and a central 'eye' of constant concentration that are arranged in a 'lens-like' structure. Coarse grains just behind the front are recirculated, while those reaching the head are overrun and deposited. Upstream of the wave, the size distribution resembles a small-particle 'sandwich' with a raft of rapidly flowing large particles on top and a coarse deposited layer at the bottom, consistent with the experimental observations made here. If the segregation is weak, the central eye degenerates, and all the large particles are deposited without recirculation. †

We experimentally study particle scale dynamics during segregation of a bidisperse mixture under oscillatory shear. Large and small particles show an underlying asymmetry that is dependent on the local particle concentration, with small particles segregating faster in regions of many large particles and large particles segregating slower in regions of many small particles. We quantify the asymmetry on bulk and particle scales, and capture it theoretically. This gives new physical insight into segregation and reveals a similarity with sedimentation, traffic flow, and particle diffusion.