On bed particle diffusion in gravel bed flows under weak bed load transport



[1] We introduce a new conceptual model for longitudinal and transverse diffusion of moving bed particles under weak bed load transport. For both rolling/sliding and saltating modes the model suggests that the particle motion is diffusive and comprises at least three ranges of temporal and spatial scales with different diffusion regimes: (1) the local range (ballistic diffusion), (2) the intermediate range (normal or anomalous diffusion), and (3) the global range (subdiffusion). The local range corresponds to ballistic particle trajectories between two successive collisions with the static bed particles. The intermediate range corresponds to particle trajectories between two successive periods of rest. These trajectories consist of many local trajectories and may include tens or hundreds of collisions with the bed. The global range of scales corresponds to particle trajectories consisting of many intermediate trajectories, just as intermediate trajectories consist of many local trajectories. Our data from the Balmoral Canal (the intermediate range) and Drake et al.'s [1988] data from the Duck Creek (the global range) provide strong support for this conceptual model and identify anomalous diffusion regimes for the intermediate range (superdiffusion) and the global range (subdiffusion).

1. Introduction

[2] Sediment transport in gravel bed flows is probably one of the most intriguing (and difficult) problems among a wide range of processes observed in rivers. This problem combines various physical phenomena, many of which still await clarification and better understanding. In this paper we consider one of probably the least studied sediment transport phenomena, bed particle diffusion, whose understanding is still in its infancy. Indeed, although from time to time terms “diffusion” and “dispersion” are used in research papers they often describe processes or parameters of only marginal relation to the true diffusive properties of bed particle motion. The existing knowledge and research intuition suggest that bed particle motion should be diffusive, at least on the plane parallel to the bed, with x axes along the flow and y axes across the flow. Mathematically, diffusion processes are described by the scaling growth in time of the central moments of particle coordinates, i.e., equation image and equation image, where X(t) and Y(t) are longitudinal (along the flow) and transverse (across the flow) coordinates of bed particles, X′(t) = X(t) − equation image(t), Y′(t) = Y(t) − equation image(t), q is the moment order, and overbar denotes ensemble averaging. For the normal (Fickian) diffusion of bed particles we have for even moments γx(q) ≡ γy(q) ≡ γ ≡ 0.5, while all odd moments are equal to zero. This diffusion regime is consistent with theoretical considerations of Einstein [1937, 1942] and his followers in stochastic studies of bed particle motion [e.g., Yang and Sayre, 1971; Stelczer, 1981; Sun and Donahue, 2000]. Indeed, it is usually assumed that probability distributions for both length steps and rest periods are exponential (or close to exponential), and thus the Central Limit Theorem applies, leading to γx(q) ≡ γy(q) ≡ γ ≡ 0.5, [e.g., Yang and Sayre, 1971; Bouchaud and Georges, 1990; Weeks et al., 1996]. The normal particle diffusion is often used, implicitly or explicitly, also in experimental studies, as, for instance, that by Drake et al. [1988]. However, because of experimental difficulties, there have been no systematic studies, to our knowledge, convincingly confirming this assumption for a wide range of scales.

[3] In fact, there may potentially be several diffusion regimes in bed particle motion: ballistic [γx(q) ≡ γy(q) ≡ γ ≡ 1], superdiffusive [(γx, γy) > 0.5], normal [γx = γy = 0.5], or subdiffusive [(γx, γy) < 0.5]. Diffusion with γ ≠ 0.5 is known as anomalous diffusion [e.g., Havlin and Ben-Avraham, 1987; Bouchaud and Georges, 1990; Metzler and Klafter, 2000]. In turn, following Castiglione et al. [1999], the anomalous diffusion can be classified as weak [γ(q) ≡ γ = CONST ≠ 0.5] or strong [γ = γ(q) ≠ 0.5]. Also, scaling properties for anomalous diffusion may be isotropic [γx(q) = γy(q) ≠ 0.5] or anisotropic [γx(q) ≠ γy(q) ≠ 0.5]. It is important to note that the diffusion exponents may directly relate to parameters of probability distributions of particle motion characteristics such as length steps and/or rest periods [Bouchaud and Georges, 1990; Weeks and Swinney, 1998; Weeks et al., 1996], as well as to statistical properties of the bed surface [Havlin and Ben-Avraham, 1987; Wang, 1994]. Sometimes diffusion properties may be different for different ranges of scales, depending on diffusion-generating mechanisms.

[4] In general, the normal diffusion (e.g., Brownian motion) results from the Central Limit Theorem of probability theory. In case of anomalous diffusion the usual form of the Central Limit Theorem fails because of the presence of either “broad” distributions (with diverging first or second moment) and/or of “long-range” correlations [Bouchaud and Georges, 1990]. These statistical mechanisms may be induced by external fields (e.g., long-range correlations in the velocity field of a turbulent flow may potentially generate superdiffusive motion of bed particles), or they may be induced by the dynamics itself (e.g., trapping of particles on the bed may generate “broad” distributions of rest periods leading to subdiffusion). The theoretical identification of diffusion regimes and scaling exponents for moving bed particles is difficult (if possible at all) and should be supplemented with experimental studies [e.g., Habersack, 2001] and computer simulations [e.g., McEwan et al., 2001]. In this paper we first suggest a conceptual model of bed particle diffusion, and, second, examine field measurements which provide convincing support for this model.

2. Concept

[5] A bed load particle may either be at rest on the bed or be in one of the two modes of motion: (1) sliding/rolling mode or (2) saltation (hopping/bouncing) mode [Graf, 1996, p. 151; Raudkivi, 1998, p. 135]. There is also experimental evidence for a transitional regime when these two modes may coexist. Recently, Nikora et al. [2001] suggested a conceptual model for diffusion of saltating particles which we use here as a basis for a more general conceptual model covering both modes of particle motion, i.e., sliding/rolling and saltation. Considering saltating particles, Nikora et al. [2001] identified three ranges (local, intermediate, and global) of spatial and temporal scales with different diffusion regimes.

[6] The local range corresponds to ballistic particle trajectories between two successive collisions with the bed [γx(q) ≡ γy(q) ≡ γ ≡ 1]. These trajectories are a result of ballistic motion; that is, they are smooth (nonfractal) and with no pauses or sudden changes in directions. A classical example of “ballistic” diffusion is the famous Taylor's turbulent diffusion at small times (much less than the Lagrangian timescale) when equation image with γ = 1 [Taylor, 1921].

[7] The intermediate range corresponds to particle trajectories between two successive rests or periods of repose. The (intermediate) trajectories from this range consist of many local trajectories and may include tens or hundreds of collisions with the bed. The streamwise component of the straight line connecting the ends of the intermediate particle trajectory is equivalent to the “quick length step” in Einstein's [1937, 1942] theory of bed load. Recall that one of the key assumptions of his theory was that “bed load movement is to be considered as the motion of bed particles in quick steps with comparatively long intermediate periods of rest.” [Einstein, 1942, p. 563]. In the intermediate range the diffusion may be, in principle, either slow ((γx, γy) < 0.5), normal (γx = γy = 0.5), super ((γx, γy) > 0.5), or mixed (e.g., γx > 0.5 and γy < 0.5), depending on what factors dominate. For instance, the bed topography and near-bed turbulence may have opposite effects on bed particle diffusion. A “fractal” bed may slow down diffusion processes (γ < 0.5 [e.g., Havlin and Ben-Avraham, 1987; Bouchaud and Georges, 1990; Metzler and Klafter, 2000]), while turbulence may enhance them (γ > 0.5 [e.g., Monin and Yaglom, 1975; Weeks and Swinney, 1998]), or they can mutually cancel their effects (γ = 0.5). A classical example of superdiffusion in turbulence is the “4/3” Richardson's law with the diffusion exponent γ = 3/2 [Monin and Yaglom, 1975]. The range of scales of the “4/3” Richardson's law corresponds to the Kolmogorov's “−5/3” inertial subrange. It is very plausible that the eddies from the inertial subrange influence velocities of moving bed particles. Thus the “superdiffusive” behavior of the inertial subrange may well be imposed on the bed particle motion leading to an increased scaling diffusion exponent (γ > 0.5).

[8] The global range of scales corresponds to particle trajectories consisting of many intermediate trajectories, just as intermediate trajectories consist of many local trajectories. The particle behavior in the global range of scales is most likely subdiffusive (γ < 0.5), as a result of potentially infinite rest periods which may lead to the diverging first moment of the rest period distribution. Indeed, it has been shown that the distribution of particle length steps is close to exponential [e.g., Stelczer, 1981], while the distribution of rest periods T, which is much less studied, is more likely to be “broad”, i.e., P(T) ∝ Tv with v < 2. The combination of these distributions for the length steps and rest periods produces subdiffusive behavior with γ < 0.5. A comprehensive treatment of such a statistical scenario is given by Bouchaud and Georges [1990, pp. 141–147] and Metzler and Klafter [2000, pp. 18–25]. There are also some other statistical scenarios which may be applicable for the bed particle diffusion in the global range and which also suggest subdiffusive regime [e.g., Weeks et al., 1996; Weeks and Swinney, 1998; Carreras et al., 1999; Metzler and Klafter, 2000, pp. 25–30].

[9] The three-range conceptual model for saltating bed particles described above was well-supported by computer simulations [Nikora et al., 2001] which revealed anomalous diffusion regimes with anisotropic diffusion exponents (i.e., γx ≠ γy). The computer simulations also suggest that the ratio of the travelling particle diameter to the prevailing diameter of static bed particles (or the height of bed roughness) is one of the key parameters controlling particle diffusion. We call this ratio the relative particle size. In this paper we extend the described concept to also cover sliding/rolling mode which is the dominating mode in bed particle motion at low transport capacity conditions [Graf, 1996, p. 151; Raudkivi, 1998, p. 135].

[10] We can reasonably assume that the three ranges of scales with different scaling behavior described above should also exist for the case of sliding/rolling particles. Such a particle may change direction appreciably only if it collides with a similar or larger static bed particle. We define such collisions as significant. The particle displacements between these significant collisions, which are reasonably straight, may be identified as local trajectories analogous to the local trajectories of saltating particles. This consideration suggests that the boundary between the local and intermediate ranges of scales should depend on characteristic time and length scales between these significant collisions rather than on the mean time and distances between all collisions. Therefore the relative particle size is an important parameter when one considers particle diffusion. The definitions of the intermediate and global ranges of scales for sliding/rolling particles are identical to those presented above for saltating particles. It is also expected that the diffusion regimes for sliding/rolling particles should be similar to those for saltating particles. Thus we suggest here that both sliding/rolling and saltating particle motions occur within at least three ranges of scales, which we conceptually define as local, intermediate, and global.

[11] As a first approximation, we can present the qth central moments of the particle coordinates as a function of eight variables, i.e., equation image, equation image = fq[u*, g, d, D, v, t, ρ, (ρs − ρ)] where u* is the shear velocity; g is gravity acceleration; d is diameter of a travelling particle; D is a prevailing diameter of static bed particles (or some other roughness length characteristic of the surface); v is fluid viscosity; ρ and ρs are densities of water and solid particles, respectively; and t is the particle travelling time including rest periods. For the case of the second moment we have equation image, equation image = f2[u*, g, d, D, v, t, ρ, (ρs − ρ)], which leads to the following simplified relationship:

equation image

Making a reasonable assumption of incomplete self-similarity [Barenblatt, 1996] with respect to (tu*/d), neglecting viscous effects (when u*d/v ≥ 100 [Raudkivi, 1998, p. 35]), and assuming dD we can replace (1) with relationship (2):

equation image

where u*2/gd and ρ/(ρs − ρ) are combined together to represent the normalized bed shear stress known as the Shields parameter [e.g., Raudkivi, 1998]. The diffusion scaling exponents γx and γy in (2) have to be defined from experiments. In general, they may depend on the relative particle size d/D and the mode of particle motion. The latter may be quantified using the argument of f2b, i.e., the normalized bed shear stress ρu*2/[(ρs − ρ)gd]. We believe that relationship (2) is a convenient starting point for data presentation and comparison of diffusion regimes. Figure 1 shows a schematic trajectory of a bed particle for large t, which illustrates the scale considerations presented above. In summary, in Figure 1 we suggest that the motion of sliding/rolling and saltating particles in unidirectional water flows is diffusive and comprises three ranges (local, intermediate, and global) of spatial and temporal scales with different scaling behavior and diffusion properties. Indeed, the local trajectories for both modes are smooth (nonfractal), and one would thus expect that the local range diffusion is ballistic; that is, the variances of particle coordinates increase in time as equation image. In the intermediate range of scales the diffusion may be either slow, normal, or super, while in the global range of scales it is most likely slower than normal (Fickian) diffusion. The boundaries between the local, intermediate, and global ranges depend on motion mode and dimensionless numbers in (1) and should be defined from experiments. In section 4.2 we provide support for this model using our own field video records of a mobile bed as well as similar data from Drake et al. [1988].

Figure 1.

Conceptual representation of a bed particle trajectory consisting of three distinct ranges of scales: (1) local; (2) intermediate; and (3) global. We hypothesize the ballistic diffusion regime for the local range (γ = 1), the normal (γ = 0.5) or anomalous (γ ≠ 0.5) diffusion regime for the intermediate range, and the subdiffusion regime for the global range (γ < 0.5). The exact identification of diffusion regimes should be made from experiments, as, for instance, in Figure 8.

3. Field Measurements and Data Analysis

[12] Video movies of a mobile bed used in our analysis were recorded in May 1998 in the Balmoral Irrigation Canal (North Canterbury, New Zealand). The video recording was a part of a series of experiments designed to study turbulence in open-channel flows over fixed and mobile beds [Nikora and Goring, 1999, 2000a, 2000b]. The cross-sectional shape of the channel is close to trapezoidal with top width of 6.2–7.0 m, bottom width of 3.5–4.5 m, and depth of ≈1 m. The channel banks were covered by crushed rock with d50 = 30 to 40 mm, while the central flat part of the channel consisted of greywacke gravel with d50 = 13.4 mm (median particle size d50 is from Wolman “count” method). The experimental design included measurements of instantaneous velocity vectors, suspended sediment concentration, bed particle size distribution, bed topography, as well as standard hydrometric measurements of flow rate, bed and water surface slopes, and channel cross sections. To minimize sidewall effects all velocity measurements and bed video recording were made at the central part of the channel. The bed load activity and motion of individual particles on the bed were observed visually and recorded using an underwater video camera. The bed was essentially flat (at the macroscopic scale); that is, no obvious bed forms were observed during experiments. Particles moved intermittently, mainly in the form of rolling and sliding, and, to a small degree, in saltation mode. These visual observations agree well with a quantitative criterion for bed load (i.e., sliding and rolling) suggested by Raudkivi [1998, p. 135]. According to this criterion the bed particles move in rolling/sliding mode if the ratio of the particle fall velocity to the shear velocity is in the range from 2 to 6. The estimates of this criterion for our case study are in the range from 2.2 to 5.9. Similar behavior of bed particles in field experiments was reported by Drake et al. [1988]. The normalized bed shear stress (Shields parameter) τ* = ρu*2/(ρs − ρ)gd for d50 = 13.4 mm was τ* ≈ 0.02 which is slightly below the assumed critical value equation image (equation image ≈ 0.03–0.06 [Graf, 1996]). However, bed load transport still occurred, dominated by smaller particles. Thus the measurements were conducted during weak bed load transport. The main hydraulic parameters for the experiments used in this paper are shown in Table 1. More details about experimental conditions and instrumentation are given by Nikora and Goring [2000a]. Values in Table 1 of the roughness Reynolds number (Re* ≫ 100), depth to width ratio (7.57), and nonuniformity parameter β (−0.997, see Table 1 for definition) show that the flow investigated may be characterized as quasi-uniform, two-dimensional (2-D) flow with fully rough bed, at least in the central part of the channel [Nikora and Goring, 2000a]. Thus the obtained video records provide data for the simplest case of bed particle diffusion in uniform, steady, 2-D flow in an irrigation canal which may be viewed as a large “flume.” In addition to having nearly the same advantages as laboratory flumes, this canal contains flows with high Reynolds numbers, comparable to those in natural geophysical flows.

Table 1. Background Conditions of the Field Measurements
ParametersMean Valuesa
  • a

    Values are obtained from repeated measurements.

  • b

    Velocity is as described by Graf and Altinakar [1998].

  • c

    The shear velocity is obtained by extrapolating the Reynolds stress toward the bed [Nikora and Goring, 2000a].

  • d

    Particle sizes, σb, and Lx are determined at fixed-bed conditions when the flow rate in the canal was significantly reduced; σb, and Lx were measured and calculated as by Nikora et al. [1998].

  • e

    Particle sizes are determined from Helley-Smith sampler using sieving method.

Global parameters at experimental cross section
Flow discharge Q, m3/s5.90
Cross-section averaged velocity , Ua, m/s1.04
Cross-section mean depth , Ha, m0.84
Hydraulic radius R, m0.75
Flow width W at the water surface, m6.36
Width to depth ratio, W/Ha7.57
Bed slope Sb0.00132
Water surface slope Sw0.00128
Global shear velocitybu*g = [gRSwRUa(dUa/dx)]0.5, cm/s9.7
Nonuniformity parameterb β = −gRSw/u*g2−0.997
Local flow parameters at measuring location
Flow depth H, m1.06
Vertically averaged velocity Uv, m/s 1.23
Reynolds number Re = UvH/v1.30 × 106
Froude number Fr = Uv/equation image0.38
Shear velocitycu*, cm/s6.70
Local fixed-bed parameters at measuring locationd
Bed particle size d16, d50, d84; mm7.4, 13.4, 22.1
Standard deviation of bed elevations σb mm6.9
Longitudinal length-scale of bed elevations Lx, mm41
Relative roughness d50H0.0126
Normalized shear stress equation image0.021
Bed particle Reynolds number Re* = equation image898
Local mobile-bed parameters at measuring locatione
Bed particle size d16, d50, d84; mm2.4, 6.4, 14.1
Normalized shear stress equation image0.043
Bed particle Reynolds number Re* = * = equation image429

[13] The image analysis of video records in our study was similar to those described by Papanicolaou et al. [1999] and Keshavarzy and Ball [1999]. The video movies were first transformed into digital form. Then, special semiautomatic procedures were developed which allowed us to obtain coordinates X(t) and Y(t) of individual particle trajectories within a spatial window 20 × 23 cm2, with the sampling interval Δt = 0.04 s. At each i time step, the longest Ai and the shortest Ci axes of the projection of a moving particle on the x-y plane were also measured. The “true” longest A and shortest C sizes of measured particles were determined as A = max[Ai] and C = min[Ci]. In our analysis we also use B = (A + C)/2, which is close to the “true” intermediate particle size [Nikora et al., 1998]. In total, 159 complete particle trajectories were quantified and analyzed. These trajectories represent the intermediate range of scales as each trajectory is from entrainment point to trapping point. Figure 2 shows the cumulative size distributions of the intermediate particle size B for static particles on the bed (Wolman “count” method [Wolman, 1954]), moving particles collected by Helley-Smith sampler (sieving “weight” method), and moving particles measured from the video records. As one can see from Figure 2, the video sample may be considered as quite representative. We used the obtained data set to analyze particle shape, velocities, and diffusion.

Figure 2.

Cumulative size distribution for static bed particles (using Wolman's “count” method [Wolman, 1954] at significantly reduced flow rate), moving bed particles (from Helley-Smith sampler using sieving “weight” method), and particles sampled from the video records analogous to Wolman's “count” method.

4. Results

4.1. Bed Particle Shape and Velocity

4.1.1. Particle shape

[14] In our analysis we use the ratio A/C as a particle shape index. At A/C ≈ 1 particles may be identified as “spherical,” while at A/C > 1 they have elongated shape. Figure 3 shows that the index A/C of moving particles depended on their size. Indeed, the relations between A and C are clearly different for particles with A < 15 mm and A > 15 mm (Figure 3a). Moreover, for A < 15 mm the ratio A/C is not constant and increases as A/C = e0.065A, from approximately 1.2 to 2.7. For particles with A > 15 mm the ratio A/C is approximately the same, A/C ≈ 2.7 (Figure 3b). Thus our “diffusion” data set combines particles with different shape, from smaller nearly “spherical” particles to larger elongated particles. The transition value A = 15 mm corresponds to B ≈ (A + C)/2 = 10.3 mm. It should be noted that although we use word spherical for characterizing small particles, in reality their surface was quite rough, similar to larger particles. The observed range of A/C is typical for gravel bed streams [Nikora et al., 1998].

Figure 3.

Relationships between (a) particle sizes A (longest) and C (shortest); and (b) shape index A/C versus particle size A.

4.1.2. Particle velocities

[15] Although the bed particle velocity is an important parameter in most of bed load theories [e.g., Stelczer, 1981; Sun and Donahue, 2000], information available in the literature is rather limited and mainly relates to the mean particle velocity. Our data set provides some additional information on the mean velocity of rolling/sliding particles as well as on velocity statistics. Figure 4 shows mean particle velocities, variation coefficient equation image, skewness and kurtosis coefficients for both longitudinal U and transverse V velocity components versus the particle size B = (A + C)/2 (σ is standard deviation of either U or V, and equation image is the mean velocity modulus). In spite of the difference in shape between smaller (B < 10.3 mm) and larger (B > 10.3 mm) particles their velocity statistics appear to be insensitive to this difference as well as to the particle size (Figure 4). This observation permits averaging of velocity statistics among all particles studied, and these values are presented in Table 2. The corresponding parameters of the near-bed fluid velocities, at approximately 1 cm above the mean bed level, are also shown in Table 2 for comparison (see Nikora and Goring [2000a] for details). Reasonably assuming, as did Stelczer [1981], that the mean particle velocity equation image is equal to the difference between the near-bed fluid velocity equation image and the critical bottom velocity uc, i.e., equation image = equation imageuc (overbar defines time averaging), we can obtain from Table 2 that equation imagec ≈ 37 cm/s. This value agrees very well with laboratory measurements for similar bed particles summarized by Stelczer, [1981, pp. 167, 228]. The skewness and kurtosis coefficients for particle and fluid velocities appeared to be similar, suggesting that the near-bed turbulence may influence particle velocity fluctuations. At the same time, the variation coefficients for gravel particles exceed those for the fluid velocities by more than 2 times, emphasising the importance of friction effects between moving particles and static bed (e.g., collisions). The correlation coefficient between the longitudinal and transverse components of the bed particle velocity was statistically negligible.

Figure 4.

Particle velocity statistics versus particle size.

Table 2. Bed Particle Velocity Statistics
ParameterMean, mm/sσ/|equation image|SkewnessKurtosis
  • a

    |V| = (U2+V2)0.5; σ is standard deviation for the velocities U or V of individual particles; for all values written as a fraction, the denominator shows the parameter standard deviation obtained for the whole set of measured particles.

  • b

    Similar definitions also apply for the fluid parameters [Nikora and Goring, 2000a].

Particle Parametersa
Longitudinal U83.2/40.60.73/0.170.77/0.610.48/1.78
Transverse V−1.75/28.90.52/0.170.10/0.850.81/1.65
Fluid Parametersb
Longitudinal u4500.300.540.52
Transverse v00.220.050.83

4.2. Bed Particle Diffusion

[16] Our data set from the Balmoral Canal provides some useful information about particle diffusion within the intermediate range of scales only. The local and global ranges of scales are not covered by our data because of measurement limitations. Indeed, Figures 5 and 6 suggest that the upper bound of the local range in our experiment was comparable with the sampling interval Δt = 0.04 s; that is, the local range was not properly resolved. In terms of spatial scales it is equivalent to 3–7 mm, which is of the order of the particle size. The global range of scales was also not covered by our data as all measured trajectories are only from entrainment point to trapping point. There is only one study [Drake et al., 1988] known to the writers, which provides appropriate information covering particle diffusion within the global range, and we use their data in our analysis.

Figure 5.

Particle trajectories (a) Y(t) and (b) X(t). Note that trajectories for B > d50 are less dispersed than trajectories for B < d50 (d50 = 13.4 mm for static bed particles).

Figure 6.

Change in time of the second-order moments of particle positions equation image, equation image = f(t).

4.2.1. Intermediate range

[17] The bed particle trajectories were measured with different starting points (xoj, yoj) within the measurement window, where subscript j denotes the jth particle from the particle ensemble. Therefore the first step in the analysis was to shift measured trajectories in the x-y plane using simple transforms Xj = xjxoj and Yj = yjyoj, so all trajectories start from the same “point of release.” The “clouds” of these shifted intermediate trajectories, shown in Figure 5, resemble those for well-established diffusion phenomena. Also, separate consideration for small (B < d50 = 13.4 mm) and large (B > d50 = 13.4 mm) particles suggest that smaller particles should have more intensive diffusion compared with larger particles. This qualitative observation is confirmed in Figure 6 where variances for particle coordinates equation imageequation image = f(t) are shown separately for particles with B < 13.4 mm and B > 13.4 mm. Note that the size d50 = 13.4 mm (Table 1) is selected to represent a characteristic height in static bed topography where moving particles travel, i.e., we assume that in relationship (1), D = d50. Thus we subdivide our data set into two subsets: (1) travelling particles are less than prevailing static particles, d < D = d50 and (2) travelling particles are larger than prevailing static particles, d >D = d50. Figure 6 reveals scaling behavior equation image and equation imagecovering more than a decade on the time axis for both small and large particles. Following our conceptual model, we identify this range of scales as the intermediate range. The exponent γy for this range of scales appears to be approximately the same for small and large particles, i.e., γy ≈ 0.83. In contrast, the exponent γx for small and large particles has appreciably different values, i.e., γx ≈ 0.87 for B > 13.4 mm and γx ≈ 0.77 for B < 13.4 mm. It seems that for large particles the diffusion is probably isotropic, i.e., γx ≈ γy, while for the case of small particles it may be anisotropic as γx < γy. Validity of the observed difference between γx and γy for small particles may be argued as it is about 7–8% only. However, the deviation of γx and γy from the “normal” value of 0.5 and from the “ballistic” value of 1.0 are hardly questionable. Such a deviation means that bed particle diffusion in the intermediate range may be qualified as anomalous superdiffusion since the inequalities 0.5 < γx, γy < 1.0 are statistically significant.

[18] The higher moments, skewness SK(t) = equation image and kurtosis >KU(t) = equation image coefficients, are shown in Figure 7 (Z stands for X or Y). The empirical functions SK(t) and KU(t) in Figure 7 significantly deviate from Gaussian values at small t and tend to Gaussian zeros at large t. The time tG required for SK(t) and KU(t) to approach Gaussian values is approximately 0.2 to 0.3 s for the Y coordinate and 0.4 to 0.5 s for the X coordinate. Such behavior of SK(t) and KU(t) implies that the initial stage in particle motion is highly intermittent and skewed. The duration of this initial “intermittent” period is approximately 2 times larger for the longitudinal direction compared with the transverse direction. This ratio is close to the ratio of the longitudinal turbulence scale to the transverse scale (i.e., ≈2 [Nikora and Goring, 1999]), suggesting that it may be controlled by turbulence. Note that there is no clear difference in high-order moments behavior for small and large bed particles. Nonscaling behavior of SK(t) and KU(t) and their near-Gaussian values at t > tG suggest that particle diffusion in the intermediate range of scales may be fairly identified as weak (anomalous) superdiffusion, i.e., equation image and equation image, where γx,y(q) ≡ γx,y > 0.5.

Figure 7.

Change in time of the normalized third- and fourth-order moments of particle positions. Skewness is SK(t) = equation image and kurtosis is KU(t) = equation image, where Z stands for X or Y.

4.2.2. Global range

[19] In their paper, Drake et al. [1988] presented a plot [Drake et al., 1988, Figure 9b, p. 209] showing how the standard deviations of the particle “cloud” in the longitudinal and transverse directions change in time. The measurements were conducted in the Duck Creek, by monitoring 125 orange-painted particles of 4–8 mm in size. This size range represented the majority of static bed particles. In other words, the relative size d/D of moving particles was approximately 1 (see relationships (1) and (2)). The mean water depth and the shear velocity were 50 cm and 8.86 cm/s, respectively. Drake et al. [1988] plotted their data using standard (linear) coordinates. Then, they approximated the data points “by eye” following relationships equation image and equation image to get particle diffusivities Kx and Ky. Thus it was implicitly assumed that particle diffusion is Fickian (normal) with γx = γy = 0.5. However, when we replotted their data in the log-log coordinates, we found that the authors' assumption about the diffusion type is too strong. Figure 8 shows Drake et al.'s [1988] data in the log-log coordinates and with normalization on d (d = 6 mm) and u* (u* = 8.86 cm/s), suggested by (1) and (2). For the range ≈ 200 < (tu*/d) < ≈ 2000, which is supported by the majority of tracer particles, we have: γx ≈ 0.33 and γy ≈ 0.19, i.e., γx ≠ γy < 0.5 but not γx = γy = 0.5 as was originally assumed by Drake et al. [1988]. In our estimates we do not use data for (tu*/d) > ≈2000 as a number of particles for this range was reduced (some particles were buried, some moved out from the sampling window, etc.). Thus the reanalysis of Drake et al.'s [1988] data strongly supports our conceptual model and suggests that bed particle diffusion within the global range of scales is not only slower than normal (γx, γy < 0.5) but also anisotropic (γx ≠ γy). Figure 8 also shows our Balmoral Canal data obtained for the range from B = 9 to 18 mm which, if normalized on the median size, is equivalent to the range from 4 to 8 mm of Drake et al. [1988]. For normalization of equation image, equation image, and t we used d = 13.4 mm and u* = 6.7 cm/s (Table 1). Thus the relative size of moving particles in both cases was approximately 1. The diffusion exponents for particles from this size range (i.e., 9 to 18 mm) are γx ≈ γy ≈ 0.78, similar to those described above for B < 13.4 mm and B > 13.4 mm. Figure 8 suggests that the boundary between the intermediate range (Balmoral Canal data) and the global range (Duck Creek data) should be approximately (tu*/d) ≈ 15. The position of this boundary should probably also depend on the Shields (or particle mobility) parameter ρu*2/[(ρs − ρ)gd] and the relative size of moving particles. Although we consider Figure 8 as a quite successful attempt to quantify the conceptual plot in Figure 1, we still believe that many more experiments are required to identify diffusion regimes and their bounds accurately.

Figure 8.

Change in time of the normalized second-order moments of particle positions covering both the intermediate (Balmoral Canal data) and global (Duck Creek data) ranges of scales. Note that this plot specifies the diffusion regimes for the intermediate range as superdiffusion (γ > 0.5) and for the global range as subdiffusion (γ < 0.5). This plot may be viewed as an experimental validation of the conceptual model in Figure 1.

5. Conclusions

[20] In this paper we have introduced a new conceptual model for diffusion of moving bed particles. For both modes, sliding/rolling and saltation, the model suggests that the particle motion is diffusive and comprises at least three ranges of temporal and spatial scales with different diffusion regimes: (1) the local range (ballistic diffusion), (2) the intermediate range (normal or anomalous diffusion), and (3) the global range (subdiffusion). Our data from the Balmoral Canal and Drake et al.'s [1988] data from the Duck Creek provide strong support for this conceptual model and identify anomalous diffusion regimes for the intermediate range (superdiffusion) and the global range (subdiffusion).

[21] The next step should be development of a physically based model explaining such anomalous particle diffusion. There are several approaches currently available which may be used to model anomalous (super- and sub-) diffusion of bed particles. Among them are generalized (fractional) diffusion equations, Levy walks/flights models, fractional Fokker-Plank equations, fractional Brownian motion, and generalized Langevin equations [e.g., Bouchaud and Georges, 1990; Wang, 1994; Metzler and Klafter, 2000]. In future studies, these theoretical approaches should be strongly linked with experiments covering various transport conditions. We believe that studying diffusion regimes of bed particle motion will provide valuable information about their physics which eventually should help in building advanced bed load models.


[22] The research was conducted under the contracts CO1X0024 from the Foundation for Research Science and Technology (New Zealand) and NIW001 from the Marsden Fund administered by the Royal Society of New Zealand. Helmut Habersack thanks the Austrian Science Foundation (Schroedinger-scholarship J1687-GEO) and NIWA for funding a research stay in New Zealand. Ian McEwan was supported by a Technology Foresight Award from the Royal Academy of Engineering (UK) during the conduct of this work. The authors are grateful to J. Walsh, M. Duncan, D.M. Hicks, and S. Brown for assistance with field measurements and data analysis, and A. Huber for software support. The Associate Editor, two anonymous reviewers, and R. Spigel provided thorough reviews and helpful criticisms and suggestions which we gratefully incorporated into the final manuscript.