A nonlocal theory of sediment transport on hillslopes

Foufoula‐Georgiou, Efi; Ganti, Vamsi; Dietrich, W. E.

doi:10.1029/2009jf001280

Cited by 101 publications

(176 citation statements)

References 64 publications

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“…In this respect preliminary analyses are generally positive. Modeling of erosional systems with fully biased up-stream non-locality explains many observed features in erosional hillslope profiles [Foufoula-Georgiou et al, 2010]. Likewise a fully biased downstream model has been shown to explain anomalies in fluvial profiles formed in experimental depositional basins [Voller and Paola, 2010].…”

Section: Resultsmentioning

confidence: 90%

“…[12] The central contribution of this work is to show that a direct consequence of the emerging non-local models of landscape dynamics based on the fractional calculus [Schumer et al, 2009;Foufoula-Georgiou et al, 2010;Voller and Paola, 2010] is a strict directional dependence that is reversed between erosional and depositional systems. As such, experimental or field measurement of the difference in directional control between depositional and erosional systems would be a decisive step toward confirming the physical validity of current non-local sediment transport models.…”

Section: Resultsmentioning

confidence: 99%

“…Within this approach, diffusive flux is estimated as a power-law weighted sum of slopes across a region that could extend both up-and downstream of the point of calculation. Recent results have shown that such diffusive models can provide a better match to the observed convexities of erosional hill-slopes [Foufoula-Georgiou et al, 2010] and experimental depositional fluvial profiles [Voller and Paola, 2010]. In this light, we explore the consequences of introducing a non-local treatment in our example erosion-depositional diffusive landscape transport model.…”

Section: A Non-local Modelmentioning

confidence: 95%

“…Here we study instead the non-locality direction coefficient À1 ≤ b ≤ 1. In particular, on using the basic properties of Caputo derivatives [Foufoula-Georgiou et al, 2010;Podlubny, 1998]-that the fractional derivative of a positive power is (4a) and (4b)). The solid line is thy fluvial profile in the uplift/erosional domain, the broken line is the profile in the subsiding/depositional domain.…”

Section: The Influence Of the Direction Of Non-local Informationmentioning

confidence: 99%

“…Specific examples include the up-and downstream migration of meanders (depending on the channel aspect ratio and meander wave number) [Seminara, 2006;Zolezzi and Seminara, 2001;Zolezzi et al, 2005], upstream migration of erosional fronts [Tucker and Slingerland, 1994], and upstream-propagating waves of deposition [Hoyal and Sheets, 2009]. In contrast, the spatial convolution integrals at the core of the emerging non-local models of landscape dynamics [Schumer et al, 2009;Foufoula-Georgiou et al, 2010;Voller and Paola, 2010] imply that conditions at a point may depend intrinsically on conditions elsewhere in the landscape, without the need for a wave-like propagation. Here, by considering a simple realization of a source-to-sink sediment transport model, we show that, in this non-local framework, physically plausible solutions for fluvial long profiles require a binary partitioning in the direction of influence between erosional and depositional landscapes.…”

Section: Introductionmentioning

confidence: 99%

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Does the flow of information in a landscape have direction?

Voller

Ganti

Paola

et al. 2012

Geophysical Research Letters

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There is an emerging viewpoint that the sediment flux at a given point on the landscape may be influenced by landscape properties in a region extending away from the point of interest. Using a general sediment transport model that incorporates this non‐local nature via fractional derivatives, we find a strong asymmetry in the direction in which the non‐locality affects a given point: For erosional landscapes, physically plausible elevation profiles are obtained only when the spatial influence is restricted to the region upstream of the point of interest. By contrast, in depositional landscapes, the non‐local model is guaranteed to produce physically plausible topographic profiles only when the spatial influence is restricted to the region downstream of the point of interest. These results suggest that information flows downstream in an erosional landscape and upstream in depositional landscapes.

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Section: Resultsmentioning

confidence: 90%

Section: Resultsmentioning

confidence: 99%

Section: A Non-local Modelmentioning

confidence: 95%

Section: The Influence Of the Direction Of Non-local Informationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Does the flow of information in a landscape have direction?

Voller

Ganti

Paola

et al. 2012

Geophysical Research Letters

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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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A direct simulation demonstrating the role of spacial heterogeneity in determining anomalous diffusive transport

Voller

2015

Water Resources Research

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Typically the spreading length-scale in a diffusion process increases in time as ' $ t n ; n5 1 2 . In the presence of spatial heterogeneities, however, so-called anomalous diffusion can occur, here the time exponent n 6 ¼ 1 2 . The objective of this paper is to present a numerical simulation that directly demonstrates the link between spacial heterogeneity and anomalous diffusion. The simulation is of the infiltration of a fluid into a horizontal unit area Hele-Shaw cell in which the permeability at specified locations, laid out as a Sierpinski fractal carpet pattern, can differ from the nominal value K 5 1. When there is no permeability contrast, the fluid infiltration has a diffusion-like behavior, i.e., the filled plan-form area increases in time as F5At n ; n5 1 2 . When there is a permeability contrast, however, although the evolution of infiltration still follows a power law, anomalous behavior is observed; subdiffusive (n < 1 2 ) when K < 1 and superdiffusive (n > 1 2 ) when K > 1. These anomalous behaviors persist, even when the permeability contrast is only imposed over the largest sized fractal pattern element. But, if the pattern over which the permeability contrast is imposed has a subdomain length scale (obtained by filling in the largest sized element in the Sierpinski carpet with the smaller sized elements), normal diffusion n5 1 2 behavior is recovered. In the special case that the imposed permeability in the pattern elements is K ( 1, an approximate model, directly relating the coefficient and exponent in the infiltration power law to the porosity and fractal dimension of the carpet pattern, is derived and validated.

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A nonlocal theory of sediment transport on hillslopes

Cited by 101 publications

References 64 publications

Does the flow of information in a landscape have direction?

Does the flow of information in a landscape have direction?

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

A direct simulation demonstrating the role of spacial heterogeneity in determining anomalous diffusive transport

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