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

Abstract

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. Citation:Ancey, C., P. Bohorquez, and J. Heyman (2015), Stochastic interpretation of the advection-diffusion equation and its relevance to bed load transport, p particle volume (m 3 ).Gaussian noise increment (s −1∕2 ). b space and time Gaussian noise increment (m −1∕2 s −1∕2 ) .

Stochastic-deterministic modeling of bed load transport in shallow water flow over erodible slope: Linear stability analysis and numerical simulation

“…For one-space variable problems, the simplest morphodynamic model comprises the shallow-water (Saint-Venant) equations for the conservation of mass and momentum of the water phase and the Exner equation for the continuity equation of the bed [37]: ∂h ∂t + ∂hv ∂x = 0, (1) ∂hv ∂t (2) ( (3) in which h(x, t ) = y s − y b denotes the flow depth, y b (x, t) and y s (x, t) are the positions of the bed and free surfaces,v is the depth-averaged velocity, x is the downstream position, t is time, ϱ is the water density, τ b is the bottom shear stress, ζ b is the bed porosity,q s is the average bed load transport rate (see (7) below), and D and E represent the deposition and entrainment rates, respectively. The bed slope is defined as tan θ = −∂ x y b .…”

“…We can also deduce the mean sediment transport rate and make insightful links between the stochastic erosion-deposition formulation (5) and the Exner equation (3). Taking the ensemble average of (5) and using the Itô convention, we end up with the governing equa-…”

“…For instance, we can relate the γ and b averages using (4) [5]. Higher-order moments are more difficult to express explicitly [3]. The last step prior to numerical solutions concerns the closure equations, which specify the dependence of the parameters τ b , ν,ū s , λ and σ involved in the Saint-Venant-Exner equations.…”

“…Next, Section 4 is devoted to the description of the hybrid finite-differences/finitevolume method and the numerical simulations. Accuracy and performance are evaluated by comparing numerical simulations with available theoretical solutions [3,4,43]. We also study the evolution of infinitesimal disturbances on a uniform base flow down an inclined plane, which leads to pattern formation and anti-dunes.…”

Why and How Do We Study Sediment Transport? Focus on Coastal Zones and Ongoing Methods

“…New types of models are developed, tested and evaluated on simple configurations to improve knowledge of basic mechanisms at a fine scale, e.g., direct numerical simulation of bedform evolution and/or sediment transport [298,299], emerging methods of smoothed particle hydrodynamics for fluid-flow interaction [194]. In-depth theoretical analysis is also ongoing on bedload (e.g., [300,301]), on bedforms [302], on the transition from bedload to suspended load [303,304], on the granular flow rheology in bedload transport [305], amongst others. These works, not yet adapted to the study of natural environments, precede the models evolution and move forward our knowledge of the involved mechanisms.…”