Exner‐Based Master Equation for transport and dispersion of river pebble tracers: Derivation, asymptotic forms, and quantification of nonlocal vertical dispersion

Pelosi, Anna; Parker, Gary; Schumer, R.; Ma, Hsi-Pin

doi:10.1002/2014jf003130

Cited by 37 publications

(56 citation statements)

References 25 publications

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“…Two master equations for tracer pebble transport and dispersion are described in Pelosi et al . []. These equations were derived from the Exner equation of sediment mass conservation within a probabilistic framework as defined by Parker et al .…”

Section: Exner‐based Master Equations For Conservation Of Tracer Pebblesmentioning

confidence: 99%

“…[] and reformulated by Pelosi et al . [] in terms of continuous‐time random walks and the associated concept of a master equation (ME), which allowed them to describe the system evolution probabilistically.…”

Section: Introductionmentioning

confidence: 99%

“…The Exner‐Based Master Equation (EBME) capable of describing this coevolution of streamwise and vertical dispersion was applied by Pelosi et al . [] to a hypothetical problem for vertical dispersion alone, using plausible forms for the probability densities. Here we use an EBME to model two‐dimensional tracer plume spread and advection, using input parameters estimated from a laboratory experiment.…”

Section: Introductionmentioning

confidence: 99%

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The cause of advective slowdown of tracer pebbles in rivers: Implementation of Exner‐Based Master Equation for coevolving streamwise and vertical dispersion

et al. 2016

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Tracer pebbles are often used to study bed load transport processes in gravel bed rivers. Models have been proposed for their downstream dispersion, and also for vertical dispersion, but not for the combined effects of downstream and vertical movement. Here we use the Exner‐Based Master Equation to characterize the transient coevolution of streamwise and vertical advection‐diffusion of tracer pebbles under equilibrium transport conditions (no net aggradation or degradation). The coevolution of streamwise and vertical dispersion gives rise to behavior that can differ markedly from that associated with purely streamwise processes with no vertical exchange. One example is streamwise advective slowdown. Particles that are advected downward into zones where the probability of reentrainment becomes asymptotically small are essentially trapped and can no longer participate in streamwise advection. As a result, the mean streamwise velocity of the tracer plume declines in time. Qualitative and quantitative comparisons with two field experiments show encouraging agreement despite the simplified boundary conditions in the model.

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Section: Exner‐based Master Equations For Conservation Of Tracer Pebblesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The cause of advective slowdown of tracer pebbles in rivers: Implementation of Exner‐Based Master Equation for coevolving streamwise and vertical dispersion

et al. 2016

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“…This likely influences the dispersion of bedload tracers (Martin et al, 2012;Houssais and Lajeunesse, 2012;Pelosi et al, 2014). For the sake of simplicity, however, we restrict our analysis to a bed of uniform particles of size d s .…”

Section: Entrainment Of Tracersmentioning

confidence: 99%

“…Some particles may then be temporary buried within the bed, inducing streamwise 5 dispersion (Crickmore and Lean, 1962;Pelosi et al, 2014). Here, we neglect this mechanism and restrict our analysis to steady and uniform sediment transport.…”

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confidence: 99%

Advection and dispersion of bedload tracers

Lajeunesse¹,

Devauchelle²,

James³

2017

Preprint

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We use the erosion-deposition model introduced by Charru et al. (2004) to simulate numerically the evolution of a plume of bedload tracers entrained by a steady flow. In this model, the propagation of the plume results from the stochastic exchange of particles between the bed and the bedload layer. We find a transition between two asymptotic regimes. At early time, the tracers, initially at rest, are progressively set into motion by the flow. During this entrainment regime, the plume, strongly skewed in 5 the direction of propagation, continuously accelerates while spreading non-linearly. With time, the skewness of the plume eventually reaches a maximum value before decreasing. This marks the transition to an advection-diffusion regime in which the plume becomes increasingly symmetrical, spreads linearly, and advances at constant velocity. We derive analytically the expressions of the position, the variance and the skewness of the plume, and investigate their asymptotic regimes. Our model assumes steady state. In the field, however, bedload transport is intermittent. We show that the asymptotic regimes become 10 insensitive to this intermittency when expressed in terms of the distance traveled by the plume. If this finding applies to the field, it might provide an estimate for the average bedload transport rate.

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

Ancey

Bohorquez

Heyman

2015

J. Geophys. Res. Earth Surf.

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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 ) .

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Exner‐Based Master Equation for transport and dispersion of river pebble tracers: Derivation, asymptotic forms, and quantification of nonlocal vertical dispersion

Cited by 37 publications

References 25 publications

The cause of advective slowdown of tracer pebbles in rivers: Implementation of Exner‐Based Master Equation for coevolving streamwise and vertical dispersion

The cause of advective slowdown of tracer pebbles in rivers: Implementation of Exner‐Based Master Equation for coevolving streamwise and vertical dispersion

Advection and dispersion of bedload tracers

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

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