Numerical simulation of the dynamics of sedimentary river beds with a stochastic Exner equation

Abstract

Abstract. At the scale of a river reach, the dynamics of the river bed is typically modelled by Exner equation (conservation of the solid mass) with an empirical solid flux of transported sediments, which is a simple deterministic algebraic formula function of i) the sediment physical characteristics (size and mass) and of ii) the averaged hydrodynamical description of the ambient water flow. This model has proved useful, in particular through numerical simulations, for hydraulic engineering purposes (like estimating the mass of sediments that is drained through an open dam). Though, the model is also coarse. And its applicability at various space and time scales remains a question of considerable interest for sedimentologists. In particular, physical experiments from the grain scale to the laboratory scale reveal important fluctuations of the solid flux in given hydrodynamical conditions. This work is a preliminary study of the coupling of a stochastic Exner equation with a hydrodynamical model for large scales. (Stochastic models with a probabilistic solid flux are currently being investigated, but most often only from the viewpoint of theoretical physics at the grain scale.) We introduce a new stochastic Exner model and discuss it using numerical simulations in an appropriate test case.Résumé. A l'échelle d'un bras de rivière, la dynamique du lit de la rivière est généralement modélisée par l'équation d'Exner (conservation de la masse solide) avec un flux solide empirique pour le transport de sédiments, flux défini par une formule algébrique et déterministe dépendant i) des caractéristiques physiques des sédiments (taille et masse) et ii) de la description hydraulique moyenne de l'écoulement local. Ce modèle s'est avéré utile, notammentà travers des simulations numériques, pour des applications en ingénierie hydraulique (par exemple, estimation de la masse de sédiments mobilisée lors d'une vidange de barrage). Néanmoins, le modèle estégalement grossier. Et son utilisation pour deséchelles de temps et d'espace variées reste une question d'un intérêt considérable pour les sédimentologues. En particulier, les expériences physiques allant de l'échelle du grainà celle du laboratoire révèlent d'importantes fluctuations du flux solide sous des conditions hydrodynamiques données.Ce travail est uneétude préliminaire du couplage d'uneéquation d'Exner stochastique avec un modèle hydrodynamique pour les grandeséchelles. (Des modèles stochastiques prenant en compte un flux solide probabiliste sont actuellementétudiés, mais le plus souvent seulement du point de vue de la physique théoriqueà l'échelle du grain.) On introduit un nouveau modèle d'Exner stochastique possible puis on le discuteà l'aide de simulations numériquesà travers un cas test approprié.

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

“…We have also illustrated with examples that the error created in using the advection‐diffusion equation for computing mean particle activity is small except in regions marked by sharp variations in the particle activity or activity gradients. Similar observations were made with the stochastic Burgers equation, a nonlinear equation that is close to the advection‐diffusion equation and in which particle activity is replaced by velocity: the solution to which the scheme converges depends on the mesh size [ Hairer and Voss , ] and velocity discontinuities are smoothed out even in the absence of diffusive terms [ Audusse et al , ]. This peculiar behavior of stochastic advection seems to be a common feature of stochastic partial differential equations (as the topic is quite recent, we may miss the broader view).…”

Numerical scheme for a water flow-driven forward stratigraphic model

“…It bears strong analogies with but remains differ from that of Birnir et al [6]. Let us also mention a few remote works that rest upon a similar coupling philosophy but the scales of which are much smaller than ours: Delestre et al [9] on rainwater overland-flows, Cordier et al [8] on bedload transport, and Audusse et al [2] on sedimentary river beds.…”

“…Nevertheless, we observe that the preconditioner choice leads to important changes in the computational time. In our case, it is better to use ILU (2) or ILU (3). For stronger preconditioning methods, the cost of each iteration becomes too important to gain in efficiency.…”

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

“…Indeed, in real-world scenarios, water flow is rarely uniform as a result of the complex interplay between bed morphology, hydrodynamics, and sediment transport. As a consequence, stochastic models must be coupled with governing equations for the water phase (e.g., the shallow water equations), a task that raises numerous theoretical and computational problems [Bohorquez and Ancey, 2015;Audusse et al, 2015]. One of these problems is that existing stochastic models introduce a number of parameters (e.g., the particle diffusivity, the entrainment and deposition rates) without specifying how they depend on water flow.…”