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Upstream migrating antidunes develop in streams under supercritical flow conditions. These bedforms are often organised in trains (i.e. sequences of well-developed antidunes). In shallow flows over coarse sediments, the dynamics of a single antidune can be significantly influenced by the surrounding bedforms. The drivers and behaviour of these interactions remain unclear. We conducted an experimental study to investigate the morphodynamics of antidune trains in narrow streams with a particular focus on mechanisms that could systematically influence bedform geometry as antidunes migrate upstream. The incipient antidunes near the flume outlet (which is a bed discontinuity) often exhibited migration periods notably longer than the typical value observed further upstream. Far from the boundaries, antidune sequences were mostly affected by transient downstream travelling disturbances and we did not observe any systematic change of antidune shape and dynamics along their migration path. Such insights into potential sources of non-uniformity are helpful for interpreting the differences between the outcomes of experimental and numerical simulations.

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 rivers. 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 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 particle 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 particle displacement and the dimensions of the measurement system.

The paper reviews our recent attempts at modelling bed load transport in mountain rivers. This is a longstanding issue that has attracted considerable attention over the last century. While a number of field and laboratory studies have been instrumental in getting the big picture, there is a clear lack of efficient methods for predicting bed evolution and particle flux. Most approaches to bed load transport have emphasized the existence of a oneto-one relationship between the particle flux and water discharge, but this result conflicts with the spread of data, which often spans over several orders of magnitude. A possible interpretation lies in the significance of the fluctuations of the particle flux together with the propagation of bed forms. We have therefore developed a theoretical model based on birth-death Markov processes to describe the random exchanges between the stream and bed, which then allows us to derive a governing equation for the particle flux fluctuations. We end up with the probability distribution function of the sediment transport rate. A striking feature is the existence of large fluctuations even under steady flow conditions. Numerical simulations have been carried out to compute the flow features, for the moment with no sediment transport. These simulations have shown that the kinematic wave approximation (which leads to a significant simplification of the Saint-Venant equation into a nonlinear advection equation) performs well for a wide range of water discharges. Remarkably, it has been found that under steady flow conditions, the local flow variables (wetted section and water discharge, or flow depth and mean velocity) exhibit a Froude similarity, i.e. regardless of the water discharge, the Froude number remains fairly constant at a given place of the river. Future work will consider the inclusion of a stochastic sediment transport equation in the Saint-Venant equations.

The advection-diffusion equation 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). Stochastic models can also be used to derive this equation, with the significant advantage that they provide information on the statistical properties of particle activity. Stochastic models are quite useful when sediment transport exhibits large fluctuations (typically at low transport rates), making the measurement of mean values difficult. We develop an approach based on birth-death Markov processes, which involves monitoring the evolution of the number of particles moving within an array of cells of finite length. While the topic has been explored in detail for diffusion-reaction systems, the treatment of advection has received little attention. We show that particle advection produces nonlocal effects, which are more or less significant depending on the cell size and particle 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 particle displacement and the dimensions of the measurement system.

As a result of interactions between streamflow andsediment, riverbeds are seldom smooth, but depending onhydraulic and sediment transport conditions, they exhibitbedforms, which grow and migrate over time (upstreamor downstream). Morphology migration may explainbedload transport pulses observed in field surveys andlaboratory experiments, even under steady supplyconditions (Gomez et al., 1989; Dhont and Ancey, 2018).Experimental investigations into the coupling betweenstream and bedform commonly focus on downstreammigrating forms (dunes or bars). Guala et al. (2014)documented the importance of considering the scale-dependent relationship between wave geometry andmigration celerity for estimating the sediment fluxes dueto downstream migrating dunes. Taking inspiration from their work, we investigated thecase of upstream migrating bedforms, called antidunes,on steep slopes. Antidunes of variable wavelengths havebeen observed in gravel-bed streams (see Video Clip S2in Froude et al., 2017). Well-controlled flumeexperiments are useful for assessing how bedformgeometry and celerity are interrelated.

Describing bedload transport as a stochastic process is an idea that emerged in the 1930s with the pioneering work of Einstein. For a long time, the stochastic approach attracted marginal attention, but the situation has radically changed over the last decade with the recent advances in the theory of bedload transport and high-resolution measurements of sediment transport rates. This paper outlines the main achievements in our approach to bedload transport in mountain streams under low to moderate flow conditions.

We consider one-dimensional bedforms (e.g., anti-dunes, dunes, ripples), and investigate their stability and stochastic dynamics by combining numerical stochastic simulations and linear stability analysis. We focus on anti-dune development in supercritical flows on sloping gravel beds. We also discuss how theory can be extended to ripples and dunes in subcritical flows on sand beds. We developed a model consisting of the classic Saint-Venant-Exner (SVE) equations and a stochastic advection-diffusion equation for particle activity (the solid volume of particles in motion per unit streambed area). The model is solved numerically using a weighted essentially nonoscillatory finitevolume method. It is applied to two case studies: (i) particle activity fluctuation over fixed plane beds with exact analytical solutions and (ii) the nonlinear simulation of the ensemble-averaged SVE for uniform background flow. Simulations are in excellent agreement with theoretical solutions. Also, they successfully capture the antidune regime and wavelength observed experimentally. A third set of simulations shows that the stochastic approach performs better than deterministic ones (referred to as non-equilibrium sediment transport equations in the hydraulic literature) at capturing not only the average sediment transport rate, but also its standard deviation and higher-order moments. Finally, a benchmark analysis of different bedload transport models provides evidence that those including particle diffusion perform better than classic models.