Fractional advection‐dispersion equations for modeling transport at the Earth surface

Abstract

[1] Characterizing the collective behavior of particle transport on the Earth surface is a key ingredient in describing landscape evolution. We seek equations that capture essential features of transport of an ensemble of particles on hillslopes, valleys, river channels, or river networks, such as mass conservation, superdiffusive spreading in flow fields with large velocity variation, or retardation due to particle trapping. Development of stochastic partial differential equations such as the advection-dispersion equation (ADE) begins with assumptions about the random behavior of a single particle: possible velocities it may experience in a flow field and the length of time it may be immobilized. When assumptions underlying the ADE are relaxed, a fractional ADE (fADE) can arise, with a non-integer-order derivative on time or space terms. Fractional ADEs are nonlocal; they describe transport affected by hydraulic conditions at a distance. Space fractional ADEs arise when velocity variations are heavy tailed and describe particle motion that accounts for variation in the flow field over the entire system. Time fractional ADEs arise as a result of power law particle residence time distributions and describe particle motion with memory in time. Here we present a phenomenological discussion of how particle transport behavior may be parsimoniously described by a fADE, consistent with evidence of superdiffusive and subdiffusive behavior in natural and experimental systems.

Does the flow of information in a landscape have direction?

“…[7] Up to this point, research on the application of equation (5), or its equivalent (6), in modeling landscapes has examined the role of the locality factor a, with a focus on understanding how this value is related to the statistics of physically observed features in the landscape [Schumer et al, 2009;Foufoula-Georgiou et al, 2010;Voller and Paola, 2010]. Here we study instead the non-locality direction coefficient À1 ≤ b ≤ 1.…”

“…and the landscapes themselves are known to exhibit variability over a range of scales (that is manifested in the fractal structure of landscapes [Trucotte, 1992]). These observations have driven an emerging view that transport in landscapes may be better treated with non-local models [e.g., Schumer et al, 2009;Foufoula-Georgiou et al, 2010;Voller and Paola, 2010]. 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.…”

“…[6] It is worth noting at this point that there are a number of alternative definitions for a fractional derivative-see Podlubny [1998] for a general treatment of the alternatives or Voller and Paola [2010, Appendix] for a discussion in a geophysical setting. Here, we choose the Caputo derivative for two reasons: (i) it is not singular at the origin and (ii) Caputo derivatives of a constant are zero.…”

“…[5] A standard approach for formally introducing nonlocality into the Exner transport model-appropriate when the heterogeneity length scales on the fluvial surface are powerlaw distributed-is to use fractional calculus [Podlubny, 1998]. In this way, recalling that in our simple model the erosional and depositional components form a single contiguous domain, a non-local expression for the unit discharge can be written as a weighted average of the left-and right-handed Caputo derivatives, i.e.,…”

“…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.…”