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Keywords:

  • bed load sediment;
  • diffusion;
  • velocity autocorrelation

Abstract

  1. Top of page
  2. Abstract
  3. 1. Bed Load Particle Diffusion
  4. 2. Experimental Measurements
  5. 3. Particle Motions and the Mean Squared Displacement
  6. 4. Particle Diffusivity
  7. 5. Discussion and Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[1] High-speed imaging of coarse sand particles transported as bed load reveals how particle motions possess intrinsic periodicities associated with their start-and-stop behavior. The dominant harmonics in these motions have a primary influence on the rate at which the mean squared particle displacementR(τ) — a measure conventionally used to assess the possibility of anomalous diffusion — increases with the time interval τ. Over a timescale corresponding to the typical travel time of particles, calculations of R(τ) may ostensibly indicate non-Fickian behavior while actually reflecting the effects of periodicities in particle motions, not anomalous diffusion. We provide the theoretical basis for this observed behavior, and we illustrate how the effective (Fickian) particle diffusivity obtains from G. I. Taylor's classic definition involving the particle velocity autocovariance, including its relation to the ensemble-averaged particle velocity as articulated by O. M. Phillips. Cross-stream diffusivities are an order of magnitude smaller than streamwise diffusivities.

1. Bed Load Particle Diffusion

  1. Top of page
  2. Abstract
  3. 1. Bed Load Particle Diffusion
  4. 2. Experimental Measurements
  5. 3. Particle Motions and the Mean Squared Displacement
  6. 4. Particle Diffusivity
  7. 5. Discussion and Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[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 xp = (xp, yp) [L] denote the particle position with streamwise and cross-stream coordinatesxp [L] and yp[L], then for Brownian-like (that is, normal or Fickian) diffusion the streamwise particle diffusivityκx [L2 t−1] can be calculated from measurements using the Einstein-Smoluchowski equation [Einstein, 1905; von Smoluchowski, 1906],

  • display math

where τ [t] is a time (lag) interval and inline image (τ) = 〈xp(t + τ) − xp(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 inline image (τ) = 〈yp(t + τ) − yp(t)〉 [L] is zero in the absence of net cross-stream transport, then for cross-stream motions, 2κyτ = 〈[yp(t + τ) − yp(t)]2〉 with diffusivity κy [L2 t−1]. More generally, letting Rx [L2] denote the right side of (1), namely Rx(τ) = 〈[xp(t + τ) − xp(t) − inline image(τ)]2〉, then the idea of anomalous diffusion considers the scaling of the mean squared displacement Rx(τ) with the time interval τ as Rx(τ) ∼ τσ, 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 displacementRy(τ) = 〈[yp(t + τ) − yp(t)]2〉 and the relation Ry(τ) ∼ τσ. To calculate Rx(τ) or Ry(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 Rx(τ) or Ry(τ) calculated for an individual particle approaches zero as τ approaches the particle travel time Tp [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 ofRx(τ) for particle motions at the short timescale reflect a ballistic-like behavior, namelyσ = 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 Rx(τ) and Ry(τ) for tracer particle motions from flume experiments and from a re-analysis of tracer motions in a field experiment (seeNikora 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, Rx(τ) and Ry(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 κx and κy, 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 thatNikora 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.

2. Experimental Measurements

  1. Top of page
  2. Abstract
  3. 1. Bed Load Particle Diffusion
  4. 2. Experimental Measurements
  5. 3. Particle Motions and the Mean Squared Displacement
  6. 4. Particle Diffusivity
  7. 5. Discussion and Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[7] Our analysis highlights results of high-speed imaging of sand particles transported as bed load over a planar bed. As described inRoseberry et al. [2012], the experiments were conducted using an 8.5 m × 0.3 m recirculating flume in the River Dynamics Laboratory at Arizona State University. For several flow conditions (Table 1), fluid velocities were measured with an acoustic Doppler velocimeter at a position one cm above the bed surface, from which bed shear stresses were calculated using the logarithmic law of the wall and a value of the roughness length z0 [L] equal to D50/30. The Froude number varied from 0.30 to 0.35, and the Shields number varied from 0.034 to 0.063. Bed material consisted of relatively uniform coarse sand with an average diameter D50of 0.05 cm. (The sand is filter-grade such that all particle diameters are between 0.045 cm and 0.055 cm.) The bed was smoothed before each experiment. High-speed imaging at 250 frames per second over a 7.57 cm (streamwise) by 6.05 cm (cross-stream) bed-surface domain with 1,280 × 1,024 pixel resolution provided the basis for tracking particle motions. A small Plexiglas “sled” window was placed on the water surface so that the camera had a clear view of the bed surface through the water column without effects of image distortion by light refraction with water-surface undulations. Flow depths were sufficiently large that the window did not interfere with the flow at the bed surface in the area filmed.

Table 1. Experimental Conditions
Run -Depth (cm)Velocity at 1 cm (cm s−1)Sampling Window Size (pixels)Run Time (sec)Sampling Interval (sec)Mean Activity (number cm−2)Mean Particle Velocity (cm s1)
R1A8.525.4500 × 50018.30.0120.04473.45
R2A11.528.2300 × 30019.60.0121.004.33
R3A12.531.0100 × 10013.20.0044.125.02
R5A14.534.850 × 503.50.00421.35.46
R2B11.528.21280 × 10240.40.0040.1903.43
R3B12.531.01280 × 10240.40.0042.834.57

[8] The image series involved a duration of 19.65 s (4,912 frames) for each of four stress conditions. We then performed two sets of measurements. In the first set, designated as A, we used runs R1, R2, R3 and R5 to track all active particles within a specified window at one of two sampling intervals over varying time durations (Table 1). In the second set of measurements, designated as B, we used runs R2 and R3 to track virtually all active particles over the full 1,280 × 1,024 pixel domain using a frame interval of 0.004 sec over a shorter duration (0.4 sec). The four series in set A provide a description of particle activities and velocities, and fluctuations in these quantities, over durations much longer than the average particle hop time. The two series in set B, although of shorter duration, provide a detailed description of particle motions over the full image domain at a finer resolution than that provided in set A.

[9] For set A, we used ImageJ (an open source code available from the National Institutes of Health) to mark the centroid of each active particle as it moved within successive frames, recording the centroid pixel coordinates. These were converted to streamwise and cross-stream coordinate positions,xp [L] and yp[L]. For set B, images were imported into ArcGIS 9.3 and spatial coordinates were edited as a point shapefile. All particles that visibly moved over the entire duration of video R2B were tracked, giving 870 unique spatial coordinates from 20 particles. In video R3B, the spatial coordinates of approximately 95% of all particles in motion were tracked, giving over 13,000 spatial positions from 311 particles. The particles not tracked in R3B were those whose identities were too difficult to maintain through the video or that exited the field of view early, or entered the field of view late. In our images, the diameter of an individual particle is represented by about 10 pixels. ImageJ allows the user to quickly mark a particle within successive images to track its coordinate position, but locks onto the nearest pixel. ArcGIS allows the user to mark the particle coordinate position at a sub-pixel resolution.

[10] For both sets of measurements (A and B), we calculated the streamwise and cross-stream particle displacements Δxp = xp(t + Δt) − xp(t) [L] and Δyp = yp(t + Δt) − yp(t) [L] between frames, and from these we estimated the “instantaneous” particle velocity components up = Δxpt [L t−1] and vp = Δypt [L t−1], where Δt [t] is the selected sampling interval (0.012 sec for R1A, R2A; 0.004 sec for R3A, R5A, R2B, R3B). These paired velocity components involved numerous instants with vp = 0 and finite up, and fewer instants with up = 0 and finite vp. Although particles mostly moved downstream, some particles occasionally moved upstream (up < 0). We considered a particle with up = vp = 0 to be at rest, even if for only one frame interval. Conversely, a particle is considered to be active if either up or vp is finite.

[11] These experiments indicate that particle motions consist of rolling, sliding and low hops that involve frequent interactions with particles on the bed [Drake et al., 1988; Lajeunesse et al., 2010; Roseberry et al., 2012]. Particles are accelerated to their highest velocities by sweeping fluid motions rather than being carried upward into high momentum flow, and most of the total hop distance of a particle (start to stop) occurs during periods of high velocity rather than during prolonged periods of low velocity [Roseberry et al., 2012]. The particle activity, the solid volume of particles in motion per unit streambed area, fluctuates as particles respond to near-bed fluid turbulence while simultaneously interacting with the bed, where the magnitude of the fluctuations in activity relative to the overall level of activity depends on the size of the sampling area. The activity increases with increasing bed stress faster than does the average particle velocity. Moreover, the probability density functions, image [L−1 t] and image [L−1t], of the streamwise and cross-stream particle velocities,up [L t−1] and vp [L t−1], are exponential-like, consistent with the experimental results ofLajeunesse et al. [2010], whereas the probability density functions of the streamwise particle hop distance Lx [L] and the associated travel time Tp[t] are gamma-like. In turn, the hop distance varies with travel time asLxTp5/3 [Roseberry et al., 2012; Furbish et al., 2012b].

3. Particle Motions and the Mean Squared Displacement

  1. Top of page
  2. Abstract
  3. 1. Bed Load Particle Diffusion
  4. 2. Experimental Measurements
  5. 3. Particle Motions and the Mean Squared Displacement
  6. 4. Particle Diffusivity
  7. 5. Discussion and Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

3.1. The Effect of Periodicities in Particle Motions

[12] The motion of a particle, start to stop, over a hop distance Lx during the travel time Tp by definition starts and ends with zero velocity (up(0) = up(Tp) = 0) with finite peak velocity in between. So regardless of how the particle velocity up(t) varies in detail during the interval 0 ≤ tTp, the velocity signal up(t) must possess at its most basic level a fundamental harmonic with period T = 2Tp [t] (although variations on this assertion, elaborated below, are possible). Treating this harmonic as a sinusoid, it is straightforward to show that integration of the velocity signal up(t) yields a displacement signal xp(t) composed of the sum of two parts, a mean motion equal to (Lx/Tp)t, and a fluctuating motion possessing a fundamental harmonic with period T = Tp, normally the dominant harmonic [Roseberry et al., 2012; Ball, 2012] (but see section 3.2 below) (Figure 1). Moreover, the travel time Tp of a particle influences the amplitude of its fundamental velocity harmonic and, in turn, the amplitude of the harmonic of the fluctuating part of the displacement signal xp(t). Namely, particles with long travel times on average are more likely to be accelerated to large peak velocities than are particles with short travel times [Roseberry et al., 2012]. The corollary is that particles with short travel times on average are limited to relatively small peak velocities. For illustration let xp(t) = (Lx/Tp)t + a sin (2πt/Tp), where a [L] is the amplitude of the dominant (and in this case, the fundamental) harmonic with period T = Tp. If Up [L t−1] is the amplitude of the underlying harmonic of the velocity signal up(t) with period T = 2Tp, then UpLx/Tp and the amplitude a is directly proportional to the product UpTpLx.

image

Figure 1. Plot of four examples from R2B and R3B of (a) streamwise particle displacement xp versus time t and (b) deviation in particle displacement about the average motion, xp − (Lx/Tp)t, versus time t for particles that gradually accelerate from rest to their peak velocities then gradually decelerate before returning to rest. In this case the displacement about the average motion is approximately sinusoidal with a principal (dominant) harmonic whose period T is equal to the travel time, namely T = Tp, with Tp = 0.056 sec (dark gray), Tp = 0.224 sec (black), Tp = 0.232 sec (white) and Tp = 0.272 sec (light gray).

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[13] Consider the contribution to the mean squared displacement Rx(τ) due to the periodic part of the motion of a particle. We start by assuming that the fundamental, or dominant, harmonic of the displacement signal xp(t) of a particle is given by xp(t) = (Lx/Tp)t + asin(ωt), where ω = 2π/T is the angular frequency and the period T is not necessarily equal to the travel time Tp. The expected displacement inline image(τ) associated with an interval τ is

  • display math

Substituting the harmonic expression above for xp(t) into (2) and evaluating the integral then gives

  • display math

where it becomes clear that inline image(τ) is not the same as the average displacement of the particle given by (Lx/Tp)t (with t = τ). Using the right side of the Einstein-Smoluchowskiequation (1), the mean squared displacement Rx(τ) is

  • display math

Substituting (3) and the harmonic expression for xp(t) into (4) then leads to

  • display math

in which the function F is

  • display math

where it is clear that Rx(τ) is not the same as the average of the squared deviations given by 〈[xp(t) − (Lx/Tp)t]2〉 (with t = τ) [Ball, 2012]. Thus, as the amplitude a goes to zero, the expected displacement inline image(τ) becomes linear in τ and the mean squared displacement Rx(τ) vanishes, so any contribution to the mean squared displacement of a particle is entirely due to its Brownian-like (non-periodic) part. Whereas the expected displacement inline image(τ) depends on the hop distance Lx, the mean squared displacement Rx(τ) of an individual particle does not. Moreover, expanding terms in (5) in powers of τ/T, then at lowest order for small times τ,

  • display math

which has the appearance of ballistic behavior, namely Rx(τ) ∼ τ2.

[14] In turn, when calculated for a group of particles the expected displacement inline image(τ) is an average over all particles. Thus, whereas the sinusoidal motion of an individual particle gives positive and negative deviations about the (individual) expected displacement, this motion may involve mostly positive or mostly negative deviations about the expected displacement of the group, depending on the average particle velocity and hop distance relative to the group averaged velocity and hop distance. This is a fundamentally distinguishing feature in physical interpretations of the expected displacement inline image(τ) and the mean squared displacement Rx(τ) for bed load particle motions versus particle motions that continue indefinitely, as in molecular systems or as envisioned by Taylor [1922] for particles suspended in a turbulent flow. Because bed load particles start and stop, the expected values inline image(τ) and Rx(τ) of individual particles do not in any ergodic sense possess an asymptotic long-time equality with the expected values of the group of particles. For example, each particle suspended in a homogeneous turbulent flow eventually experiences (in a probabilistic sense) the full suite of possible turbulent motions, so after an interval longer than the decorrelation timescale (see below), plots of the expected values inline image(τ) and Rx(τ) versus τ for individual particles converge and equal the expected values calculated for the group. In contrast, each bed load particle does not remain in motion long enough to experience the full suite of possible interactions with the fluid and bed, so the expected values inline image(τ) and Rx(τ) versus τfor each particle is unlike any other, and unlike values calculated for the group. Similar comments pertain to the mean squared cross-stream displacementRy(τ).

3.2. Experimental Results

[15] Imaging reveals that streamwise particle motions, start to stop, typically involve one of three types of net displacement. Based on 157 motions from R2B and R3B, most particles (63%) gradually accelerate from rest to a peak velocity then gradually decelerate before returning to rest. In this case the displacement about the average motion, xp − (Lx/Tp)t, is approximately sinusoidal with a principal (dominant) harmonic whose period is equal to the travel time, namely T = Tp (Figure 1). Some particles (27%) undergo a brief interval of rapid acceleration from rest to a peak velocity followed by gradual deceleration before deposition, or conversely, a gradual acceleration followed by a brief interval of rapid deceleration in returning to rest. In this case the displacement about the average motion appears as a dominant harmonic whose period T ≈ 2Tp (Figure 2). Finally, a few particles (10%) accelerate then decelerate more than once during a full hop such that the displacement about the average motion involves a (dominant) harmonic whose period is a fraction m of the travel time, namely TmTp (Figure 3).

image

Figure 2. Plot of four examples from R2B and R3B of (a) streamwise particle displacement xp versus time t and (b) deviation in particle displacement about the average motion, xp − (Lx/Tp)t, versus time t for particles that undergo a brief interval of rapid acceleration from rest to a peak velocity followed by gradual deceleration before deposition, or conversely, a gradual acceleration followed by a brief interval of rapid deceleration in returning to rest. In this case the displacement about the average motion appears as a dominant harmonic whose period T ≈ 2Tp, with Tp = 0.040 sec (white), Tp = 0.088 sec (light gray), Tp = 0.140 sec (dark gray) and Tp = 0.204 sec (black).

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image

Figure 3. Plot of four examples (different colors) from R2B and R3B of (a) streamwise particle displacement xp versus time t and (b) deviation in particle displacement about the average motion, xp − (Lx/Tp)t, versus time t for particles that accelerate then decelerate more than once during a full hop. In this case the displacement about the average motion involves a (dominant) harmonic whose period T is a fraction m of the travel time, namely TmTp, with Tp = 0.048 sec (dark gray), Tp = 0.048 sec (white), Tp = 0.188 sec (light gray) and Tp > 0.400 sec (black). Note that the dark gray circles are mostly hidden in the left panel.

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[16] Cross-stream motions similarly involve net displacements that possess periodic structure, but are less systematic than streamwise displacements (Figure 4). That is, dominant harmonics are not systematically related to the travel time Tp. Cross-stream motions are more erratic than streamwise motions.

image

Figure 4. Plot of four examples (different colors) from R2B and R3B of cross-stream particle displacementsyp versus time t for particles motions illustrated in Figures (a) 1 and (b) 2.

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[17] The effect of brief, rapid accelerations due to particle-fluid and particle-bed interactions, including collisions with particles on the bed, is to add high-frequency “noise” to the periodic part of the motion of a particle, that is, to randomize this periodic motion. Thus one may consider the motion of a particle as consisting of a correlated random walk — albeit a brief walk — wherein the motion is correlated from one instant to the next and involves a few to tens of collisions during an individual hop.

[18] The close fits between plots of (5) and (7) and empirically calculated values of Rx(τ) for individual particles from R2B and R3B (Figure 5) reveal the primary influence of the dominant harmonic in the streamwise motion of a particle on the mean squared displacement Rx(τ). (Similar fits were obtained for virtually all recorded particle motions completing full hops, start to stop.) Namely, with reference to the scaling relation Rx(τ) ∼ τσ, calculated values of Rx(τ) typically exhibit a ballistic-like behavior withσ ≈ 2 for τ ≤ 0.01 sec [Roseberry et al., 2012; Ball, 2012]. This represents for the specific conditions of our experiments the characteristic interval between particle-bed collisions, analogous to the mean-free path, a behavior thatNikora et al. [2002] correctly anticipated (but could not demonstrate with the data available to them). Moreover, the effect of the intrinsically periodic motion of a particle is to give Rx(τ) the appearance of non-Fickian (superdiffusive) behavior withσ > 1 for 0.01 ≤ τT/2, that is, before the slope of Rx(τ) begins to decline near τT/2. By the definition of the mean squared displacement Rx(τ) for the motion of an individual bed load particle, Rx(τ) returns to zero as τ approaches the travel time Tp — which is fundamentally different from the result that obtains for a particle whose motion continues indefinitely (see section 4 below).

image

Figure 5. Plots of calculated (circles) and theoretical (solid lines) values of the mean squared displacement Rx(τ) versus time interval τ for particle motions illustrated in (a) Figure 1 with T = Tp, (b) Figure 2 with T = 2Tp, and (c) Figure 3 with T = 0.5Tp (black), T = 0.4Tp (light gray), T = 0.7Tp (dark gray) and T = 0.5Tp (white); dashed lines are given by equation (7) and possess slopes of σ = 2.

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[19] When Rx(τ) is calculated for all streamwise particle motions in each of our experiments (Figure 6), the ballistic-like behavior observed for individual particles over the domainτ ≤ 0.01 sec persists for measurements involving a sampling interval Δt = 0.004 sec, and the appearance of anomalous diffusion (σ > 1) extends over the domain 0.01 ≤ τ ≤ 0.1 sec. (We note that the small window size of R3A sampled only short particle motions, wherein hop distances slightly larger than the average hop distance are excluded. The slope of Rx(τ) therefore declines at τ less than 0.1 sec. The window size of R5A is smaller than that of R3A, so we have not plotted the R5A data.) However, this apparent superdiffusive behavior (σ> 1) merely represents the collective effect of the correlated (sinusoidal) random walks of particles that are increasingly (but not completely) randomized by particle-fluid and particle-bed interactions over a timescale corresponding to the typical travel time of particles. Moreover, short motions are more akin to Brownian-like motions than are long duration motions, so short motions contribute more to the random part ofRx(τ), just as do higher harmonics within longer duration motions. With increasing τ, fewer particle motions are involved in the calculation of Rx(τ), specifically, only those whose travel time Tpτ. Values of Rx(τ) at the largest values of τ are based on particles with the largest travel times Tp. Thus, Rx(τ) tends to decline at large τ (but does not necessarily return to zero as with individual particles).

image

Figure 6. Plot of mean squared displacement Rx(τ) versus time interval τfor all particle motions within experiment R1A (black), R2A (light gray circles), R3A (white circles), R2B (light gray triangles) and R3B (white triangles). Eye-fit estimates of the slope ofRx(τ) over the domain 0.01 ≤ τ ≤ 0.1 sec vary from about 1.5 to 1.8, and dashed line possesses slope of σ = 2.

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[20] For particles completing full hops in R2B and R3B, estimates of the amplitude a systematically increase with the hop distance Lx, and, the hop distance Lx systematically increases with the (individual) mean velocity calculated as Lx/Tp (Figure 7). This reinforces the point made above, that particles with long travel times (or hop distances) on average are more likely to be accelerated to large peak velocities Up than are particles with short travel times. In turn, with LxTp5/3 [Roseberry et al., 2012], then UpLx/TpTp2/3 [Ball, 2012]. Thus, whereas the hop distance on average increases at a growing rate with increasing travel time, the mean velocity increases less rapidly with increasing travel time.

image

Figure 7. Plot of (a) amplitude a versus hop distance Lx and (b) hop distance Lx versus mean velocity Lx/Tp for particles completing full hops in R2B (white) and R3B (black); not shown is one datum in Figure 7a with coordinates (L, a) = 2.31,0.74.

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[21] When Ry(τ) is calculated for all cross-stream motions in each experiment, no ballistic-like behavior is apparent at smallτ, and the slope σ (i.e., the exponent in Rx(τ) ∼ τσ) over the domain 0.01 ≤ τ ≤ 0.1 varies from about 1 to 1.8 (Figure 8). The magnitudes of cross-stream particle velocities typically are much smaller than streamwise velocities [Roseberry et al., 2012], and our measurements of small cross-stream displacements are less precise.

image

Figure 8. Plot of mean squared displacement Ry(τ) versus time interval τfor all particle motions within experiment R1A (black), R2A (light gray circles), R3A (white circles), R2B (light gray triangles) and R3B (white triangles). Eye-fit estimates of the slope ofRy(τ) over the domain 0.01 ≤ τ ≤ 0.1 sec vary from about 1 to 1.8, where lower dashed line has slope of 1 and upper dashed line has slope of 2.

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4. Particle Diffusivity

  1. Top of page
  2. Abstract
  3. 1. Bed Load Particle Diffusion
  4. 2. Experimental Measurements
  5. 3. Particle Motions and the Mean Squared Displacement
  6. 4. Particle Diffusivity
  7. 5. Discussion and Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

4.1. Definition of the Diffusivity

[22] The motion of a bed load particle over its travel time Tpis continuous, albeit involving quasi-random (high-frequency) fluctuations in the velocity associated with fluid accelerations and particle-bed collisions [Lajeunesse et al., 2010; Roseberry et al., 2012; Furbish et al., 2012b]. An appropriate description of the particle diffusivities κx and κy therefore can be obtained from the classic definition provided by Taylor [1922]. It is important, however, to be explicit about the averaging involved in this definition, inasmuch as G. I. Taylor envisioned particle motions that continue indefinitely, as opposed to the start-and-stop motions of bed load particles.

[23] As described in Furbish et al. [2012a] and Roseberry et al. [2012], consider a planar streambed area B [L2] large enough to fully sample steady, homogeneous near-bed conditions of turbulence and transport. At any instant the numberN of active particles is approximately constant. That is, the rate of disentrainment within B equals the rate of entrainment, and the rate at which particles leave B across its boundaries equals the rate at which particles enter B across its boundaries. Imagine recording particle motions within B for an interval of time Ts [t] [e.g., Lajeunesse et al., 2010; Roseberry et al., 2012]. For Ts much longer than the mean particle travel time (also see below), particle motions during Ts adequately represent the joint probability density image Ly, Tp) [L−2 t−1] of hop distances Lx and Ly, and travel times Tp [Furbish et al., 2012a] without bias due to experimental censorship of motions at times t = 0 and t = Ts [Furbish et al., 1990], wherein longer duration motions are incompletely sampled. The marginal probability densities image [L−1], image [L−1] and image [t−1] possess the means inline image [L], inline image [L] and inline image [t]. Moreover, at any instant, the velocities of active particles within B possess the probability densities image [L−1 t] and image [L−1 t] with means inline image [L t−1] and inline image [L t−1] and variances image [L2 t−2] and image [L2 t−2], where it may be assumed that these represent ensemble averaged quantities [Furbish et al., 2012a, 2012b; Roseberry et al., 2012].

[24] The average streamwise velocity of the ith active particle with travel time Tpi is

  • display math

In turn, letting Ns denote the number of particle motions during Ts (note that NsN), and assuming that Ns is large, the ensemble average velocity

  • display math

Thus, the ensemble average hop distance inline image is equal to the product of the ensemble averaged velocity inline image and the mean travel time inline image. (But note that inline image/ inline image ≠ 〈Lxi/Tpi〉.)

[25] As a point of reference, when particles continue their motions indefinitely (that is, they do not start and stop), then experimentally Tpi = Ts (the sample time) and (9) becomes

  • display math

where now inline image is the average displacement during Ts, and the average in (10) is the same as the average of an individual particle over long time.

[26] Letting upi = upiinline image, then the autocovariance Cx(τ) [L2 t−2] of the streamwise particle velocities is

  • display math

where

  • display math

is the number of particle motions with travel time Tpτ. When τ = 0, Nτ = Ns, and Cx(0) = image Moreover, when particles continue their motions indefinitely,

  • display math

Taylor [1922] demonstrated that for continuous particle motions,

  • display math

Inasmuch as the integral in (14) converges as τ [RIGHTWARDS ARROW] ∞, then the streamwise particle diffusivity is

  • display math

in which τL [t] is the Lagrangian integral timescale defined by

  • display math

where Ax(τ) = image is the autocorrelation of the streamwise particle velocities. Note that in this development we are envisioning TsτL. By a similar development the cross-stream diffusivity is

  • display math

where image the variance of the cross-stream velocitiesvp.

4.2. Experimental Results

[27] The autocovariance Cx(τ) decays to zero by about τ ≈ 0.1 sec for our experiments (Figure 9). In turn, the integral in (15) converges, where numerically computed values level off at τ ≈ 0.1 to 0.15 sec (Figure 10), giving estimates of κx from about 0.3 cm2 s−1 to 0.8 cm2 s−1 over the range of experimental conditions. Inasmuch as variations in particle velocity are of the same order as the mean velocity inline image, [Roseberry et al., 2012; Furbish et al., 2012b], then as suggested by Phillips [1991],

  • display math

where δinline imageτL [L] is a characteristic distance of motion over which the autocovariance Cx(τ) is significant, and is similar in magnitude to the mean hop distance. Moreover, there is clear evidence that particle velocities uppossess an exponential-like density image with ensemble average inline image [Lajeunesse et al., 2010; Roseberry et al., 2012; Furbish et al., 2012b], in which case image = inline image2, which reinforces the point in (18), that κxinline imageδ. In the language of transport in porous media flows, δis the so-called “dispersivity.” In addition, estimates of the Lagrangian integral timescale suggest thatτL ≤ 0.1 sec, similar to the mean travel time estimated for particles in R2B and R3B [Roseberry et al., 2012]. We may therefore assume that κx = k inline image2 inline image = k inline image, where k is a dimensionless factor of order unity.

image

Figure 9. Plot of autocorrelation Ax(τ) = image of streamwise particle velocities up calculated for R1A (black), R2A (light gray circles), R3A (white circles), R5A (dark gray circles), R2B (light gray triangles) and R3B (white triangles).

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image

Figure 10. Plot of integral of the autocovariance Cx(τ) of streamwise particle velocities up calculated for R1A (black), R2A (light gray circles), R3A (white circles), R5A (dark gray circles), R2B (light gray triangles) and R3B (white triangles).

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[28] For cross-stream motions the autocovarianceCy(τ) decays to zero by about τ ≈ 0.1 sec (Figure 11). The integral in (17) converges, where numerically computed values level off at τ ≈ 0.1 sec (Figure 12), giving estimates of κy from about 0.05 cm2 s−1 to 0.1 cm2 s−1 over the range of experimental conditions, approximately an order of magnitude smaller than values of κx.

image

Figure 11. Plot of autocorrelation Ay(τ) = image cross-stream particle velocitiesvp calculated for R1A (black), R2A (light gray circles), R3A (white circles), R5A (dark gray circles) and R2B (light gray triangles).

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image

Figure 12. Plot of integral of the autocovariance Cy(τ) of cross-stream particle velocitiesvp calculated for R1A (black), R2A (light gray circles), R3A (white circles), R5A (dark gray circles) and R2B (light gray triangles).

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[29] We emphasize that calculated values of Cx(τ) and Cy(τ) at a given interval τ are based on all velocity signals for which Tpτ. Thus, for small τ these values are based on the velocity signals of most particles in each experiment, and for large τ these values are based on fewer signals with longer travel times. Uncertainty in the estimates of Cx(τ) and Cy(τ) (or Ax(τ) and Ay(τ)) therefore increases with increasing τ. Estimates at ττL cannot be interpreted as being significantly different from zero.

5. Discussion and Conclusions

  1. Top of page
  2. Abstract
  3. 1. Bed Load Particle Diffusion
  4. 2. Experimental Measurements
  5. 3. Particle Motions and the Mean Squared Displacement
  6. 4. Particle Diffusivity
  7. 5. Discussion and Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[30] As pointed out in Roseberry et al. [2012], problems of diffusion in molecular systems typically involve processes in which an individual particle experiences, say, 106–1010 collisions per second. Motions continue indefinitely, and any anomalous diffusive behavior (as characterized, for example, by hard sphere theory [e.g., Alder and Wainwright, 1967, 1970; Paul and Pusey, 1981]) emerges rapidly and persists. In ecological systems, hundreds to thousands of “collisions” (meaning changes in direction) of a “particle” — such as an albatross or a honey bee — can occur during an individual Lévy flight [Viswanathan et al., 1996; Reynolds et al., 2007]. In contrast, the sediment particle motions described herein involve a few to tens of collisions with the bed during one particle hop. The (apparent) superdiffusive behavior manifest in plots of the mean squared displacement Rx(τ) over a timescale 0.01 ≤ τ≤ 0.1 sec actually reflects the effects of periodicities that are inherent in streamwise particle motions, not (scale-invariant) superdiffusion — a behavior that cannot in any case persist at longer timescales.

[31] The idea that bed load particle motions exhibit a ballistic-like behavior at small time intervalsτis not the same as ballistic behavior as (conventionally) defined for molecular systems in which particles travel unimpeded within a vacuum between collisions. For example, because the mean-free time and associated mean-free path of air molecules are small at Earth-surface pressure and temperature conditions, molecular motions can be approximated as straight lines with constant velocity between collisions. (In detail these motions are parabolic in Earth's gravitational field, decidedly so at the rarefied conditions of the outer atmosphere.) The motion of a Brownian particle in the inertial (ballistic) regime is highly correlated before collisions with surrounding molecules randomize its motion. In contrast, bed load particle motions are strongly coupled with fluid motions, and particles rarely move faster than the fluid. Insofar as bed load particles travel at approximately constant velocity between collisions with the bed — leading toRx(τ) ∼ τ2— then although the particles possess inertia, like Brownian particles, this constant-velocity behavior is a result of the particle-fluid coupling, not ballistic behavior. The magnitudes of cross-stream particle velocities typically are much smaller than streamwise velocities [Roseberry et al., 2012]. The absence of ballistic-like behavior in plots ofRy(τ) at small τlikely reflects that small cross-stream particle motions are influenced proportionally more by interactions with the bed, and possibly less precision in our measurements of these cross-stream motions.

[32] At any instant, and from one instant to the next, the N active particles within the streambed area B represent all possible stages of motions over each possible particle hop represented by the underlying distribution of hop distances, image and image It is therefore appropriate to calculate Rx(τ), Ry(τ), Cx(τ) and Cy(τ) based on all paired observations of xp, yp, up and vp separated by the interval τ (as opposed to setting the initial time t of each motion to zero and calculating averages only across the number of motions for each time τ) in order to ensure that these quantities represent ensemble averages. For example, a value of Rx(τ) or Cx(τ) calculated for small τ includes those particles near the end of long hops as well as those near the beginning of short hops and long hops, in proportion to the likely occurrence of the various stages of motion of the N active particles sampled from all hops at any instant.

[33] That the autocovariances Cx(τ) and Cy(τ) decay to zero over a short interval τindicates a Fickian-like diffusive behavior [Taylor, 1922; Garrett, 2006; Ferrari, 2007] in both streamwise and cross-stream particle motions. In contrast, ifCx(τ), for example, possesses a long tail, then (15) does not converge rapidly and the diffusivity increases, albeit slowly, at times larger than τL [Garrett, 2006]. But here it is important to reemphasize that bed load particle motions do not continue indefinitely, so in fact a finite Cx(τ) for ττLis not physically meaningful in characterizing diffusive behavior. More basically, a Fickian-like behavior is anticipated from the exponential-like particle velocity distributions ofup and vp, possibly involving “light” tails [Roseberry et al., 2012; Furbish et al., 2012b], where the diffusivity arises from the time derivative of the second moment of the underlying probability density function of particle displacements occurring during a small interval dt [Furbish et al., 2012a]. The rapid decay of Cx(τ) and Cy(τ) over τ provides a clearer measure of diffusive behavior than does the form of Rx(τ) or Ry(τ), which is intrinsically sensitive to effects of periodicities in particle motions that start and stop with an average travel time inline imageτL.

[34] As described in companion papers [Furbish et al., 2012a, 2012b], the volumetric bed load sediment flux involves an advective part equal to the product of the average particle velocity and the particle activity (the solid volume of particles in motion per unit streambed area), and a diffusive part involving the gradient of the product of the particle activity and the diffusivity. This diffusive contribution to the flux may be important under conditions of nonuniform transport, and Taylor's formulation of the diffusivity, as described above, yields a proper description of the diffusion of bed load particles at low transport rates, consistent with Fickian diffusion. The problem of diffusion (dispersion) of tracer particles involving the effects of multiple hops and rest times [Sayre and Hubbell, 1965; Drake et al., 1988; Hassan and Church, 1991; Ferguson and Wathen, 1998; Nikora et al., 2002; Martin et al., 2012] requires a different formalism [Bradley et al., 2010; Ganti et al., 2010; Hill et al., 2010] that describes the effects of rest times.

[35] The analysis also points to the design of experimental measurements required to obtain precise estimates of the particle diffusivity and related quantities. Here are lessons we have learned. To confidently estimate the displacement and velocity statistics described above, including the mean hop distance and travel time (and their distributions), the sampling window size and time interval are critical. The window must be large enough, and the sampling time must be long enough (TsτL), to obtain a sufficient count of hops representing the full range of hop distances occurring in the near-bed conditions of turbulence. Our runs R2B and R3B, like R1A and R2A, had sufficiently large windows, but suffered from short sampling times. Runs R3A and R5A, like R1A and R2A, involved sufficient sampling times, but suffered from small windows. Moreover, to see ballistic-like behavior requires a sampling interval shorter than the typical interval between particle-bed collisions. Experiments involving a wider range of flow conditions and particle sizes are required to clarify the relation between particle velocities and diffusivities as suggested insection 4.2.

Notation
a

amplitude of sinusoidal particle displacement [L].

Ax, Ay

autocorrelation of streamwise and cross-stream particle velocities.

B

streambed area [L2].

Cx, Cy

autocovariance of streamwise and cross-stream particle velocities [L2 t−2].

D

particle diameter [L].

image, image

probability density functions of streamwise and cross-stream particle hop distances [L−1].

image

probability density function of particle travel times [t−1].

image

joint probability density function of particle hop distances and travel times [L−1 t−1].

image, image

probability density functions of streamwise and cross-stream particle velocities [L−1 t].

F

function defined by (6).

k

dimensionless factor of order unity.

Lx, Ly

streamwise and cross-stream particle hop distances [L].

m

fraction of travel time.

N

number of active particles within the streambed area B.

Ns

number of particle motions during the sampling interval Ts.

Nτ

number of particle motions with travel time Tpτ.

Rx, Ry

mean squared streamwise and cross-stream particle displacements [L2].

t

time [t].

T

period of sinusoidal particle displacement [t].

Tp

particle travel time [t].

Ts

sampling interval [t].

up

streamwise particle velocity [L t−1].

up

deviation in streamwise particle velocity about the average [L t−1].

Up

amplitude of sinusoidal particle velocity [L t−1].

vp

cross-stream particle velocity [L t−1].

xp

streamwise particle position or displacement [L].

xp

particle position or displacement vector [L].

yp

cross-stream particle position or displacement [L].

z0

roughness length [L].

δ

characteristic distance of particle motion during τL [L].

κx, κy

streamwise and cross-stream particle diffusivities [L2 t−1].

σ

exponent in scaling relation R(τ) ∼ τσ.

image, image

variance of streamwise and cross-stream particle velocities [L2 t−2].

τ

time (lag) interval [t].

τL

Lagrangian integral timescale [t].

ω

angular frequency equal to 2π/T [t−1].

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Bed Load Particle Diffusion
  4. 2. Experimental Measurements
  5. 3. Particle Motions and the Mean Squared Displacement
  6. 4. Particle Diffusivity
  7. 5. Discussion and Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[36] We are grateful to Peter Haff for critical discussions and insight. We appreciate comments by the Editor (Alex Densmore) and Associate Editor (Emmanuel Gabet) and reviews provided by Michael Church, Raleigh Martin and James Pizzuto. We acknowledge support by the National Science Foundation (EAR-0744934). All data described herein are available to the community.

References

  1. Top of page
  2. Abstract
  3. 1. Bed Load Particle Diffusion
  4. 2. Experimental Measurements
  5. 3. Particle Motions and the Mean Squared Displacement
  6. 4. Particle Diffusivity
  7. 5. Discussion and Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Bed Load Particle Diffusion
  4. 2. Experimental Measurements
  5. 3. Particle Motions and the Mean Squared Displacement
  6. 4. Particle Diffusivity
  7. 5. Discussion and Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information
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