## 1. Bed Load Particle Diffusion

[2] The idea of diffusion (or “dispersion”) of bed load particles is a central element of two compelling problems in the study of sediment transport. The first involves understanding the kinematics and mechanics of downstream and cross-stream diffusion of tracer particles in flume experiments or in natural channels at flood and longer timescales [e.g.,*Sayre and Hubbell*, 1965; *Drake et al.*, 1988; *Hassan and Church*, 1991; *Ferguson and Wathen*, 1998; *Nikora et al.*, 2002; *Ganti et al.*, 2010; *Martin et al.*, 2012]. The second involves understanding how bed load particle diffusion contributes to the local sediment flux under conditions of nonuniform transport [*Lisle et al.*, 1998; *Schmeeckle and Furbish*, 2007; *Furbish et al.*, 2012a, 2012b; *Ball*, 2012], notably in relation to spatial variations in particle activity associated with flow over bedforms. Both problems begin with the recognition that bed load particle motions, although deterministically governed in detail by coupled fluid-particle physics, nonetheless possess a distinctly probabilistic nature due to the stochastic (quasi-random) qualities of particle entrainment and disentrainment, and the inherent variability in particle velocities and displacements during transport [*Einstein*, 1937, 1950].

[3] The possibility that sediment particles exhibit anomalous rather than Fickian (normal) diffusion during transport [*Nikora et al.*, 2002; *Bradley et al.*, 2010; *Ganti et al.*, 2010; *Hill et al.*, 2010; *Martin et al.*, 2012] has far-reaching implications for how we conceptualize and calculate rates of transport and dispersal of particles and particle-borne substances [*Schumer et al.*, 2009; *Furbish et al.*, 2009a, 2009b; *Furbish and Haff*, 2010; *Furbish et al.*, 2012a], and raises a fundamental question. What is the physical basis for the appearance of anomalous diffusion in bed load particle motions? Inasmuch as these particle motions involve rolling, sliding and low hops with frequent interactions (collisions) with the bed [*Drake et al.*, 1988; *Lajeunesse et al.*, 2010; *Roseberry et al.*, 2012], thereby producing stochastic variations in particle velocities and displacements [*Einstein*, 1937, 1950; *Roseberry et al.*, 2012; *Furbish et al.*, 2012b], then these motions mimic random-walk behavior attributed to particles in other well known diffusive systems, abiotic and biotic [e.g.,*Einstein*, 1905; *Taylor*, 1922; *Viswanathan et al.*, 1996; *Metzler and Klafter*, 2000; *Okubo and Levin*, 2001; *Cantrell and Cosner*, 2003; *Trigger*, 2010; *Huang et al.*, 2011]. But individual bed load particle motions are brief, involving only a few to tens of collisions with the bed between start and stop [*Roseberry et al.*, 2012], far fewer than what particles in other (e.g., molecular) systems experience on similar timescales or over comparably scaled excursion distances. An understanding of bed load particle diffusion therefore requires a sharper description of the kinematics of particle motions than might be represented by simple random-walk models.

[4] Evidence for anomalous diffusion comes from measured displacements of tracer particles seeded in natural channels and flume experiments [*Nikora et al.*, 2002; *Bradley et al.*, 2010; *Hill et al.*, 2010; *Martin et al.*, 2012]. Specifically, letting **x**_{p} = (*x*_{p}, *y*_{p}) [L] denote the particle position with streamwise and cross-stream coordinates*x*_{p} [L] and *y*_{p}[L], then for Brownian-like (that is, normal or Fickian) diffusion the streamwise particle diffusivity*κ*_{x} [L^{2} t^{−1}] can be calculated from measurements using the Einstein-Smoluchowski equation [*Einstein*, 1905; *von Smoluchowski*, 1906],

where *τ* [t] is a time (lag) interval and (*τ*) = 〈*x*_{p}(*t* + *τ*) − *x*_{p}(*t*)〉 [L] is the expected (average) displacement associated with the time interval *τ.* The angle brackets in (1)denote an average over many starting times for a single particle, or an average over a specified group of particles, where in practice these two types of averaging can be combined. Assuming the expected cross-stream displacement (*τ*) = 〈*y*_{p}(*t* + *τ*) − *y*_{p}(*t*)〉 [L] is zero in the absence of net cross-stream transport, then for cross-stream motions, 2*κ*_{y}*τ* = 〈[*y*_{p}(*t* + *τ*) − *y*_{p}(*t*)]^{2}〉 with diffusivity *κ*_{y} [L^{2} t^{−1}]. More generally, letting *R*_{x} [L^{2}] denote the right side of (1), namely *R*_{x}(*τ*) = 〈[*x*_{p}(*t* + *τ*) − *x*_{p}(*t*) − (*τ*)]^{2}〉, then the idea of anomalous diffusion considers the scaling of the mean squared displacement *R*_{x}(*τ*) with the time interval *τ* as *R*_{x}(*τ*) ∼ *τ*^{σ}, where for normal (Fickian) diffusion the exponent *σ* = 1, for subdiffusion 0 < *σ* < 1, and for superdiffusion *σ* > 1 [e.g., *Metzler and Klafter*, 2000; *Nikora et al.*, 2002; *Schumer et al.*, 2009; *Trigger*, 2010]. The specific value *σ* = 2 represents true ballistic behavior underlying Brownian particle motion in molecular systems [e.g., *Huang et al.*, 2011; *Pusey*, 2011], and, as described below, this value represents a ballistic-like behavior in the case of bed load particle motions. Similar comments apply to the mean squared cross-stream displacement*R*_{y}(*τ*) = 〈[*y*_{p}(*t* + *τ*) − *y*_{p}(*t*)]^{2}〉 and the relation *R*_{y}(*τ*) ∼ *τ*^{σ}. To calculate *R*_{x}(*τ*) or *R*_{y}(*τ*) for an individual particle or for a group of particles observed at discrete intervals, the average for the interval *τ* is obtained over all paired observations separated by *τ*, where the number of paired observations necessarily diminishes with increasing *τ.* By definition, the value of *R*_{x}(*τ*) or *R*_{y}(*τ*) calculated for an individual particle approaches zero as *τ* approaches the particle travel time *T*_{p} [t] [*Roseberry et al.*, 2012].

[5] Bed load particle motions involve three timescales [*Nikora et al.*, 2002]: a short timescale characteristic of the interval between particle-bed collisions, analogous to the mean free time as defined for molecular systems; an intermediate timescale corresponding to the typical particle travel time (start to stop) involving multiple particle-bed collisions; and a long timescale spanning multiple particle hops and intervening rest periods. Plots of*R*_{x}(*τ*) for particle motions at the short timescale reflect a ballistic-like behavior, namely*σ* = 2 [*Roseberry et al.*, 2012], as in conventional diffusive systems. (*Huang et al.* [2011] report the first direct observation of true ballistic behavior underlying Brownian particle motion as suspected by A. Einstein. *Pusey* [2011] provides a lovely narrative of the historical context for this discovery and its implications.) Anomalous diffusion has been tentatively identified [*Nikora et al.*, 2002; *Martin et al.*, 2012] at intermediate and long timescales from plots of *R*_{x}(*τ*) and *R*_{y}(*τ*) for tracer particle motions from flume experiments and from a re-analysis of tracer motions in a field experiment (see*Nikora et al.* [2002] with reference to *Drake et al.* [1988]). The idea of subdiffusive behavior (*σ* < 1) of tracer particles involving multiple hops and rest times at long timescales is compelling [*Nikora et al.*, 2002; *Bradley et al.*, 2010; *Hill et al.*, 2010]. For intermediate timescales coinciding with particle travel times, however, plots of the mean squared displacement, *R*_{x}(*τ*) and *R*_{y}(*τ*), may ostensibly indicate non-Fickian behavior while actually reflecting effects of correlated random walks [*Viswanathan et al.*, 2005] associated with intrinsic periodicities in particle motions, not anomalous diffusion [*Roseberry et al.*, 2012]. Herein we provide the theoretical basis for this observed behavior, and we illustrate how the effective (Fickian) particle diffusivities *κ*_{x} and *κ*_{y}, specifically relevant to calculations of the bed load sediment flux [*Furbish et al.*, 2012a], obtain from G. I. Taylor's classic definition [*Taylor*, 1922] involving the particle velocity autocovariance, including its relation to the ensemble-averaged particle velocity as articulated by O. M. Phillips [*Phillips*, 1991].

[6] Using results of high-speed imaging of sand particles transported as bed load over a planar bed [*Schmeeckle and Furbish*, 2007; *Roseberry et al.*, 2012], our analysis reveals the ballistic-like behavior of particles at short timescales, behavior that*Nikora et al.* [2002]correctly anticipated but could not demonstrate with the data available to them, and it clarifies why the behavior of particles at the intermediate timescale corresponding to the typical particle travel time cannot represent superdiffusion. We present a proof-of-concept that Taylor's formulation yields a proper description of diffusion of bed load particles at low transport rates, consistent with Fickian diffusion. The analysis also points to the design of experimental measurements required to obtain precise estimates of the particle diffusivity and related quantities.