A probabilistic description of the bed load sediment flux: 3. The particle velocity distribution and the diffusive flux

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117,, doi: /2012jf002355, 2012 A probabilistic description of the bed load sediment flux: 3. The particle velocity distribution and the diffusive flux David Jon Furbish, 1 John C. Roseberry, 1 and Mark W. Schmeeckle 2 Received 25 January 2012; revised 16 July 2012; accepted 18 July 2012; published 21 September [1] Particles transported as bed load within a specified streambed area possess at any instant a distribution of velocities. This distribution figures prominently in describing the rates of transport and dispersal of particles. High-speed imaging of sand particles transported as bed load over a planar bed reveals that the probability density functions of the streamwise and cross-stream particle velocities are exponential-like. For quasi-steady conditions the exponential-like density of streamwise velocities reflects a balance among three fluxes in momentum space: (1) an advection of streamwise momentum whose magnitude and sign vary with the momentum state; (2) a diffusion of momentum from higher to lower values of momentum density; and (3) a drift of momentum from regions in momentum space having high average rates of generation of kinetic energy toward regions having low rates of generation of kinetic energy. The probability density of cross-stream velocities similarly reflects a balance of fluxes of cross-stream momentum. Whereas the average net force acting on particles is zero under steady conditions, the mean, variance and asymmetry of the distribution of forces acting on particles vary with the momentum state of the particles. Numerical simulations of particle motions that are faithful to these statistical properties reproduce key empirical results, namely, the exponential-like velocity distribution and the nonlinear relation between hop distances and travel times. The simulations also illustrate how steady gradients in particle activity, the solid volume of particles in motion per unit streambed area, induce a diffusive flux as described in companion papers. Citation: Furbish, D. J., J. C. Roseberry, M. W. Schmeeckle (2012), A probabilistic description of the bed load sediment flux: 3. The particle velocity distribution and the diffusive flux, J. Geophys. Res., 117,, doi: /2012jf Introduction [2] Particles transported as bed load within a specified streambed area B [L 2 ] possess at any instant a distribution of velocities. This distribution of particle velocities figures prominently in describing the rate of sediment transport and the rate of dispersal of particles during transport. 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) [e.g., Bridge and Dominic, 1984; Wiberg and Smith, 1989; Seminara et al., 2002; Parker et al., 2003; Francalanci and Solari, 2007; 1 Department of Earth and Environmental Sciences and Department of Civil and Environmental Engineering, Vanderbilt University, Nashville, Tennessee, USA. 2 School of Geographical Sciences and Urban Planning, Arizona State University, Tempe, Arizona, USA. Corresponding author: D. J. Furbish, Department of Earth and Environmental Sciences, Vanderbilt University, 2301 Vanderbilt Pl., Nashville, TN , USA. (david.j.furbish@vanderbilt.edu) American Geophysical Union. All Rights Reserved /12/2012JF Wong et al., 2007; Lajeunesse et al., 2010], and a diffusive part involving the gradient of the product of the particle activity and a diffusivity that arises from the time derivative of the second moment of the probability distribution of particle displacements occurring during a small interval dt [t]. This diffusive contribution to the flux thus may be important under conditions of nonuniform transport, notably in relation to flow over bed forms. Moreover, the nature of the fluctuations in particle velocities in part reflected by the second moment of the particle velocity distribution bears on how we conceptualize particle diffusion, including tracer particle motions [Roseberry et al., 2012; Ball, 2012; Furbish et al., 2012b]. [3] In independent experiments [Lajeunesse et al., 2010; Roseberry et al., 2012], high-speed imaging of sand particles transported as bed load over a planar bed reveals that the probability density function f up (u p )[L 1 t] of streamwise particle velocities u p [L t 1 ] is exponential-like (Figure 1), and the probability density f vp (v p )[L 1 t] of cross-stream velocities v p [L t 1 ] is peaked at v p = 0, decaying approximately exponentially with jv p j (Figure 2). Specifically, Lajeunesse et al. [2010, Figure 6] plotted one example of the exponential-like distribution of streamwise velocities u p, but implied in their description that this distribution 1of15

2 Figure 1. Discrete probability density of streamwise velocities u p, and semi-log plot (inset) of probability density versus u p with straight-line fit to data illustrating exponentiallike form of distribution; note decline in density below the line at u p 24 cm s 1. consistently appears across their experimental runs involving several particle sizes and bed stresses. Roseberry et al. [2012] described exponential-like distributions from four runs involving one particle size and different bed stresses, of which data from R3B are presented here (Figure 1). Of particular note is the consistent form of the streamwise velocity distribution, which may possess a light tail reflecting that particle velocities are limited by near-bed fluid velocities [Roseberry et al., 2012]. Similarly, the peaked, symmetrical distribution of cross-stream velocities v p (Figure 2) is consistent among these separate experiments. This consistency, notably given the small bed area B and the short time intervals involved in the sampling, suggests that the forms of these distributions represent a persistent behavior of bed load particles transported under macroscopically steady conditions. [4] The consistent appearance of an exponential-like distribution of particle velocities must reflect a systematic behavior in the exchanges of momentum during fluidparticle and particle-bed interactions during steady transport. Herein we describe what these velocity distributions reveal about how particle momenta vary as particles accelerate and decelerate between entrainment and disentrainment in response to fluid drag and interactions with the bed. Our analysis begins with a rendering of the Master equation, a general expression of conservation, to describe how particles collectively change from one momentum state to another [e.g., Trigger, 2010]. Assuming the probability density function of particle accelerations at any given momentum state is not heavy-tailed, as elaborated below, the formulation leads to a Fokker-Planck equation describing how the distribution of particle momentum changes in response to advective and diffusive fluxes in momentum space. We then show that, for steady conditions, the exponential-like density of streamwise velocities reflects a balance among three fluxes in momentum space: (1) an advection of streamwise momentum, the magnitude and sign of which vary with the momentum state; (2) a diffusion of momentum from higher to lower values of momentum density; and (3) a drift of momentum associated with variations over momentum space in the average rate of change of particle kinetic energy. The probability density of cross-stream velocities similarly reflects a balance of fluxes of cross-stream momentum. [5] The analysis reveals that the form (mean, variance, asymmetry) of the distribution of forces acting on particles varies with the momentum state of the particles. This information provides the basis for simulating motions of bed load particles, including entrainment and disentrainment, in a way that is faithful to the kinematics of these motions. The algorithm reproduces key empirical results reported in Roseberry et al. [2012], including the exponential-like velocity distributions (Figures 1 and 2), the gamma-like distributions of the particle hop distances and travel times, and the nonlinear, heteroscedastic relation between hop distances and travel times. Moreover, the algorithm allows us to illustrate how gradients in particle activity, the solid volume of particles in motion per unit streambed area, induce a diffusive particle flux that is in addition to the advective flux consisting of the product of the particle activity and the mean particle velocity [Furbish et al., 2012a]. The simulations naturally reproduce essential features of the idea of a flux saturation, wherein a certain downstream distance, the saturation length, is required for the flux to reach an equilibrium value compatible with the imposed flow and bed-stress conditions [e.g., Bagnold, 1941; Parker, 1975; Nakagawa and Tsujimoto, 1980; Fourrière et al., 2010; Nelson et al., 2011]. These results connect the probabilistic description of the bed load sediment flux developed in the first paper in this series [Furbish et al., 2012a] with the measurements of particle motions presented in the second paper [Roseberry et al., 2012]. 2. Laboratory Experiments and Particle Velocity Measurements [6] As described in Roseberry et al. [2012], our analysis concerns the results of high-speed imaging of sand particles transported as bed load over a planar bed. The experiments Figure 2. Discrete probability density of cross-stream velocities v p, and semi-log plot (inset) of probability density versus v p with straight-line fit to data illustrating exponential-like form of one-sided distribution; note misfit near v p =0. 2of15

3 were conducted with an 8.5 m 0.3 m recirculating flume in the River Dynamics Laboratory at Arizona State University. The sand particles are approximately uniform in size with a nominal diameter of 0.05 cm. (The sand is filter-grade such that all particle diameters are between cm and cm.) Flow conditions involved four values of bed stress, although here we focus on two of these (R2B and R3B). The average particle activity in the lower stress experiment (R2B) was about 0.19 particles per cm 2, and the average activity in the higher stress experiment (R3B) was about 2.8 particles per cm 2. High-speed imaging of particle motions covered a 7.6 cm 6.1 cm domain of the bed at a 1,280 1,024 pixel resolution using a rate of 250 frames per second over a duration of 0.4 s. [7] For the runs R2B and R3B we mapped frame-byframe the streamwise position x p [L] and the cross-stream position y p [L] of each particle that experienced motion, which yields a Lagrangian (x p, y p, t) field of particle motions. Some particles were in motion over only a few frames, whereas some particles remained in motion over the entire 100 frames. We then calculated the streamwise and cross-stream displacements Dr = x p (t + Dt) x p (t) [L] and Ds = y p (t + Dt) y p (t) [L] between successive frames. Dividing these displacements by the interval Dt [t] between frames then gives estimates of the instantaneous particle velocity components, namely, u p Dr/Dt [L t 1 ] and v p Ds/Dt [L t 1 ]. Run R2B involved measurements of 21 particles and run R3B involved measurements of 311 particles yielding a total of 874 and 12,944 estimates, respectively, of the paired velocity components. These paired velocity components involved numerous instants with v p =0 and finite u p, and fewer instants with u p = 0 and finite v p. Although particles mostly moved downstream, some particles occasionally moved upstream (u p < 0). We considered a particle with u p = v p = 0 to be at rest, even if for only one frame interval. Conversely, a particle is considered to be active if either u p or v p is finite. 3. Distributions of Particle Velocities [8] The particle velocities described above collectively represent a large sample of all possible particle velocities likely to occur under similar macroscopic flow conditions. Thus, we assume that when pooled these data represent a sample of an underlying quasi-steady distribution of velocities that exists at any instant within a specified streambed area B sufficiently large to sample all possible particle velocities from one instant to the next [Furbish et al., 2012a; Roseberry et al., 2012]. [9] As described in Roseberry et al. [2012], the distribution of pooled streamwise velocities is exponential-like (Figure 1), consistent with the experimental results of Lajeunesse et al. [2010] involving coarser sand particles, but includes small negative velocities. Thus, neglecting negative velocities, f up 1 u p ¼ e up=up u p 0; ð1þ u p where u p is the mean streamwise particle velocity. For crossstream velocities, the distribution is symmetrical about v p =0, and although it is Gaussian-like, this distribution actually is too peaked to be Gaussian. A semi-log plot (Figure 2) reveals that the distribution is more like a two-sided exponential distribution, f vp v p 1 ¼ v p vp e j j= vp j j ; where v p is the mean cross-stream velocity magnitude, although this is not an ideal fit either, particularly near the origin ( v p = 0). The cross-stream velocity data presented by Lajeunesse et al. [2010] similarly display a sharply peaked mode at v p = 0. As described in Roseberry et al. [2012], the data are equally well fit by a plot of ( log f vp ) versus v p at large v p, consistent with a Gaussian distribution. In either case, the data do not exhibit heavy tails. 4. Particle Momentum [10] The exponential-like distributions, f up (u p ) and f vp (v p ), of the streamwise and cross-stream particle velocities, u p and v p (Figures 1 and 2), come from a sample consisting of a large number of particles at many instants during their motions and at all stages in these motions just after entrainment and just before disentrainment, and throughout the intervening period, whether of relatively short or long duration. This sample thus consists of a large set of instantaneous particle configurations and velocities, and we may therefore assume that f up (u p ) and f vp (v p ) are representative of the set of possible velocity states that is, the ensemble of such states consistent with the macroscopic flow conditions [Furbish et al., 2012a]. With equal-sized particles, multiplying the velocities u p and v p by the particle mass m gives the streamwise and cross-stream particle momenta p x = mu p [M L t 1 ] and p y = mv p [M L t 1 ], and we then define the probability density functions of p x and p y, namely f px ( p x )[M 1 L 1 t] and f py (p y )[M 1 L 1 t], which we also may assume are exponential-like. Note that, by focusing on streamwise and cross-stream motions separately, we are effectively treating p x and p y each as a scalar quantity. [11] In time series of the streamwise velocity of individual particles [see Roseberry et al., 2012], positive fluctuations in u p (and thus p x ) represent accelerations due to fluid drag and release of particles from pockets or from behind stationary particles, particle-particle impacts, and hopping as particles lose contact with the bed. Negative fluctuations in u p represent decelerations due to decreasing fluid drag, particles bumping into other bed particles, momentary increases in particle-particle friction, and changes in the mode of motion, say, from rolling to sliding. In time series of the cross-stream velocity of individual particles, both positive and negative fluctuations in v p (and thus p y ) represent accelerations due to fluid drag as well as decelerations due to particle-bed interactions. [12] The densities f px ( p x ) and f py ( p y ) may evolve with time as particles change from one momentum state to another. Thus, in general f px = f px ( p x, t) and f py = f py ( p y, t). Let q x = p x (t +dt) p x (t) [MLt 1 ] and q y = p y (t +dt) p y (t) [M L t 1 ] denote small changes or displacements in particle momentum during a small interval of time dt, and let f qx (q x ; p x, t) [M 1 L 1 t] and f qy (q y ; p y, t) [M 1 L 1 t] denote the probability density functions of the momentum displacements q x and q y. We now start with the Master equation, the most general expression of conservation ð2þ 3of15

4 possible, to describe how the densities f px (p x ) and f py (p y ) may change with time. Namely, according to the Master equation [Risken, 1984; Ebeling and Sokolov, 2005; Furbish et al., 2009a, 2009b], the change in f px (p x, t) atp x during dt is f px ðp x ; t þ dtþ f px ðp x ; tþ ¼ Z f px Z f px ðp x ; tþf qx ðp x p x ; p x ; tþdp x ðp x ; tþf qx ðp x p x ; p x ; tþdp x : Similarly, the change in f py (p y, t) atp y during dt is Z f py p y ; t þ dt fpy p y ; t ¼ f py p y ; t f qy p y p y ; p y ; t dp y Z p y ; t fqy p y p y ; p y ; t dp y : f py The first integrals in (3) and (4) describe the arrival of particles to the momentum states p x and p y at time t +dtfrom other momentum states p x and p y at time t. The second integrals describe the departure of particles with momentum p x and p y at time t to other possible states p x and p y at time t +dt. Note that the momentum coordinates p x and p y are physically the same as p x and p y. The primes are merely a convenient way to distinguish two distinct momentum states of the particles, one at time t and one at time t +dt. [13] We now initially focus on (3) involving streamwise particle motions, with the understanding that the following manipulations are identical for (4) involving cross-stream motions. We assume that the probability density of momentum displacements f qx (q x ; p x, t) is not heavy-tailed (as described below). Then, as presented in detail by [Risken, 1984; Ebeling and Sokolov, 2005; Furbish et al., 2009a, 2009b], a change of variables q x = p x p x followed by a Taylor expansion of the integrands to second order about p x leads to f px ðp x ; t þ dtþ f px ðp x ; tþ ¼ Z f px ðp x ; tþ q x f qx ðq x ; p x ; tþdq x p x þ 1 2 Z f px ðp x ; tþ q 2 x 2 f q x ðq x ; p x ; tþdq x : p 2 x Dividing (5) by dt and taking the limit as dt 0thenleads to a Fokker-Planck equation [Risken, 1984; Ebeling and Sokolov, 2005], f px t ¼ 1 k 1x f px þ p x 2 2 p 2 x ð3þ ð4þ ð5þ k 2x f px ; ð6þ where the first transition moment deriving from f qx is k 1x Z 1 dt 0 ðp x ; tþ ¼ lim dt and the second transition moment is k 2x Z 1 dt 0 ðp x ; tþ ¼ lim dt q x f qx ðq x ; p x ; tþdq x ð7þ q 2 x f q x ðq x ; p x ; tþdq x : ð8þ In the language of statistical transport theory, the first transition moment k 1x [M L t 2 ]islikea drift speed, the average rate of change in the streamwise particle momentum, and the second transition moment k 2x [M 2 L 2 t 3 ]islikea diffusivity, the rate of change in the second moment of the momentum displacements. [14] In physical terms these transition moments have the following interpretations. Among a total of N active particles, consider those in the momentum state p x = mu p at time t. The moment k 1x is equivalent to the (instantaneous) average streamwise force component acting on these particles, or the particle mass times the average streamwise acceleration of the particles. As elaborated below, whereas with steady conditions the average net force acting on all N particles is zero, the average force k 1x may vary with the momentum state p x. Turning to the moment k 2x, again consider those particles in the momentum state p x = mu p. At time t the particles are experiencing both positive and negative accelerations. Because q x = p x (t +dt) p x (t) =m[u p (t +dt) u p (t)] = mq u, where q u = u p (t +dt) u p (t) [Lt 1 ] denotes a velocity displacement during dt, (8) may be rewritten as m 2 k 2x ðp x ; tþ ¼ lim dt 0 dt Z q 2 u f q u ðq u ; p x ; tþdq u : ð9þ Here, f qu (q u ; p x, t) is the probability density function of the velocity displacements q u. Now k 2x =2mĖ x, where, letting angle brackets denoting the average in (9), Ė x = lim dt 0 (1/dt)(m/2) q u 2 [M L 2 t 3 ] is the average rate of change in streamwise kinetic energy of particles with momentum p x, that is, the rate of change in energy in the fluctuating motions of particles with velocity u p. [15] By conservation of probability f px = t = Q px = p x, so from the Fokker-Planck equation (6) the flux of probability Q px [t 1 ] from one part of the domain of the probability density function f px toward another part of this domain is Q px ¼ k 1x f px 1 k 2x f px ; ð10þ 2 p x with advective and diffusive terms. Further, with steady conditions ( f px / t = 0) and Q px = 0 at all momentum states p x, k 1x f px 1 d k 2x f px ¼ 0; ð11þ 2 dp x where now f px = f px (p x ) and f qx = f qx (q x ; p x ) are independent of time so that the moments k 1x and k 2x are independent of time. An analogous expression is obtained with respect to crossstream motions involving the density f py and the transition moments k 1y and k 2y, all independent of time. Namely, k 1y f py 1 d k 2y f py ¼ 0: ð12þ 2 dp y [16] Particles start and stop, so the time-averaged streamwise acceleration of any individual particle that is, the average over all instants of a particle s motion is zero. This also means that the time-averaged streamwise force component on the particle is zero. At any instant the probability that a particle possesses a momentum within the small interval p x to p x +dp x is f px ( p x )dp x. The number of 4of15

5 particles within this interval is Nf px (p x )dp x, and the sum of the streamwise forces acting on these particles is Nk 1x ( p x ) f px ( p x )dp x. The net streamwise force on all N particles must equal zero, so Z k 1x ðp x Þf px ðp x Þdp x ¼ 0; ð13þ which also means that the average net streamwise force on the particles is zero. Similarly, the net cross-stream force on all N particles must equal zero, so Z k 1y p y fpy p y dpy ¼ 0: ð14þ Momentarily assuming that all particle motions are downstream, the lower limit of integration in (13) becomes zero. In turn, integrating (11) from zero to infinity leads to Z 0 Z k 1x f px dp x 1 d k 2x f px dpx 2 0 dp x ¼ k 2x ð0þf px ð0þ k 2x ð Þf px ð Þ ¼ 0; ð15þ which indicates that, inasmuch as the product k 2x ( ) f px ( )is zero in the limit of p x, either k 2x (0) or f px (0), or both, must equal zero. However, particles can move upstream, so k 2x (0) does not necessarily vanish at p x = 0, a point to which we return below. [17] The possibility exists that the average forces k 1x and k 1y depend on the local momentum states p x and p y. Consider the particles at any instant within a small interval p x to p x + dp x (or within p y to p y + dp y ). If the particles within this interval are preferentially accelerating, then k 1x > 0 (or k 1y > 0); and if the particles are preferentially decelerating, then k 1x < 0 (or k 1y < 0). However, unless k 1x varies with p x such that it is positive over some domain of p x and negative over the rest of p x, (13) cannot be satisfied. Moreover, with k 1x positive and negative over p x, to satisfy (11) requires a compensation by the diffusive term wherein the product k 2x f px is non-monotonic. Similarly, k 1y must be positive over some domain of p y and negative over the rest of p y to satisfy (14); and because symmetry about p y =0 requires that k 1y =0atp y = 0, this in turn requires that k 1y be antisymmetric about p y =0. [18] Expanding (11), and expanding (12), k 1x f px k 2x 2 k 1y f py k 2y 2 df px f p x dk 2x ¼ 0; dp x 2 dp x df py f p y dk 2y ¼ 0: dp y 2 dp y ð16þ ð17þ The first terms in (16) and (17) describe an advection of momentum at the rates k 1x and k 1y. The second terms describe a diffusion of momentum from higher to lower values of momentum density in association with the gradients df px =dp x and df py =dp y. The third terms in (16) and (17) describe a drift of momentum at the rates dk 2x =dp x and dk 2y =dp y from regions within the domains of p x and p y having high values of the transition moments k 2x and k 2y toward regions having low values of these moments [e.g., Monin and Yaglom, 1965; Legg and Raupach, 1982; Thomson, 1984; Visser, 1997; Yamazaki, 2005]. With reference to the description of (9) above, this is equivalent to saying that the drift of momentum described by the third terms in (16) and (17) is from regions in momentum space having high average rates of change in kinetic energy toward regions having low rates of change in kinetic energy. The units of dk 2x =dp x = 2mdĖ x =dp x and dk 2y =dp y =2mdĖ y =dp y,likek 1x and k 1y,are those of a force [M L t 2 ]. But unlike the real forces k 1x and k 1y,dk 2x =dp x and dk 2y =dp y represent pseudo-forces associated with gradients in the rates of change in kinetic energy. [19] Assuming only downstream motions, integrating (16) with respect to p x, f px ðp x Þ ¼ f px0e R px 0 2k1x=k2x ð 1=k2x Þdk2x=dpx and integrating (17) with respect to p y, f py p y ½ Šdpx ; ð18þ R py ¼ Ce ½ 2k1y=k2y ð 1=k2y Þdk2y=dpyŠdpy ; ð19þ where f px 0 is the value of f px at p x = 0, and C is a constant determined by normalization. These expressions illustrate that the essence of defining the probability densities f px and f py resides in understanding the transition moments k 1x and k 2x, and k 1y and k 2y, notably how these vary with the momentum states p x and p y. In the absence of a theory for these moments, here we appeal to our experimental results from runs R2B and R3B to constrain their behavior. 5. Measurements [20] Recall that the data set for run R3B is an order-ofmagnitude larger than the data set for run R2B (section 2). Certain calculated (averaged) values described below therefore exhibit larger statistical uncertainty for R2B than for R3B. Nonetheless, the results for R2B and R3B are consistent, so in order to limit the number of figures we focus on the results from R3B. [21] In order to work directly with the particle velocity data, we multiply the integral in (18) by m/m (which is equivalent to dividing k 1x by m, k 2x by m 2 and p x by m) and consider the probability density of velocities u p, namely f up up ¼ fup0 er 0 ½2a 1x=a 2x ð1=a 2x Þda 2x=du p Šdu p ; ð20þ u p where f up 0 is the value of f up at u p =0,a 1x (u p )=k 1x =m [L t 2 ] is the average particle acceleration and a 2x (u p )=k 2x =m 2 = (2/m)Ė x [L 2 t 3 ] is the rate of change in the second moment of the velocity displacements. Both a 1x and a 2x may depend on the local velocity state u p. We can then estimate a 1x and a 2x as follows. As above, let q u = u p (t + Dt) u p (t) =q x =m (Figure 3) and q u 2 = [u p (t + Dt) u p (t)] 2 = (q x =m) 2 (Figure 4). In turn, we calculate a 1x and a 2x from the averages of q u and q u 2 for each 1 cm s 1 increment of u p, namely 1 X n a 1x u p ndt i¼1 1 X n ðq u Þ i and a 2x u p ndt i¼1 q 2 u i : ð21þ where n is the total number of measurements within the increment of u p. Note that for simplicity of estimation we 5of15

6 Figure 3. Plot of velocity displacements q u versus streamwise velocity u p, with local average (gray circles) and one standard error bars based on one cm s 1 increments of u p.points with u p > 0 falling below the solid line represent instants where particles with positive (downstream) velocities assume negative (upstream) velocities during Dt (Figure 1), and points with u p < 0 falling above the solid line represent instants where particles with negative velocities assume positive velocities during Dt, where the total number of points below the line with u p > 0 equals the number with u p < 0, as represented in Figure 1. report values of a 1x and a 2x in the figures described below as the products a 1x Dt and a 2x Dt, where Dt = 1/250 = sec in the experiments. [22] With streamwise motions, data for q u, q u 2 (and thus for a 1x and a 2x ) become sparse for u p greater than about 15 cm s 1. Figure 4. Plot of squared displacements q 2 u versus streamwise velocity u p, with local average (gray circles) and one standard error bars based on 1 cm s 1 increments of u p. Figure 5. Plot of product of average acceleration a 1x and time interval Dt versus streamwise velocity u p, with one standard error bars based on 1 cm s 1 increments of u p ; solid line fit by eye, vertical dashed line is at u p = 4.6 cm s 1. With cross-stream motions, data become sparse for v p greater than about 5 cm s 1. We thus focus on the data-rich domains of u p and v p in the curve fitting described below. Moreover, eye-fitted relations suffice for obtaining firstorder estimates of the coefficients. [23] The average acceleration a 1x is positive over the domain 0 < u p u p and negative over u p u p (Figure 5). In these experimental runs the particles therefore are preferentially accelerating (a 1x > 0) for u p u p and preferentially decelerating (a 1x < 0) for u p u p. Histograms of q u (Figure 6) also reveal a systematic variation in the asymmetry of particle accelerations. At small u p, particles experience larger positive accelerations than negative accelerations. Near the average velocity (u p = 4.6 cm s 1 ), positive and negative accelerations are approximately symmetrical about an average of zero (a 1x 0). And, at large u p, particles experience larger negative accelerations than positive accelerations. In turn, estimated values of a 2x increase approximately linearly with particle velocity u p (Figure 7), with a slope da 2x =du p = b 1/Dt [L t 2 ] and an intercept of zero, consistent with the result obtained from (15), that k 2x (0) = m 2 a 2x (0) = 0. [24] Using these estimates of a 1x, a 2x and da 2x =du p from Figures 5 and 7, we can immediately calculate the bracketed integrand in (20) to reveal that it is approximately constant over u p (Figure 8). This result is consistent with an exponential density function f up (u p ) (Figure 1). Namely, if this integrand equals a constant 1/A [L 1 t], then A [L t 1 ] may be interpreted as the average particle velocity. Indeed, a value of 1/A = 1/4.6 = 0.22 provides a good visual fit to the data in Figure 8. Alternatively, inasmuch as f up is an exponential density, we may rewrite the integrand in (20) as 2a 1x da 2x du p þ a 2x A ¼ 0: ð22þ If on empirical grounds we assume to first order that a 1x = a 1x0 + au p over the domain 0 < u p <20cms 1 (Figure 5), 6of15

7 Figure 6. Example histograms (proportion) of velocity displacements q u illustrating systematic variation in asymmetry of displacements with increasing streamwise velocity u p. 7of15

8 Figure 7. Plot of product of diffusivity a 2x and time interval Dt versus streamwise velocity u p, with one standard error bars based on 1 cm s 1 increments of u p ; solid line has slope of bdt =1. where a [t 1 ] is like a rate constant, and if we further assume that a 2x = bu p as above, then substituting these expressions into (22), 2a 1x0 þ 2au p b þ b A u p ¼ 0: ð23þ This is satisfied if 2a 1x0 + b and if A = b/2a. In turn, the average velocity u p = A = b/2a. Our first-order estimates indeed give bdt 1 and a 1x 0 Dt 0.5 bdt/2. Moreover, adt 0.1 so u p 5cms cm s 1, as above. Figure 9. Plot of product a 2x f up Dt versus streamwise velocity u p illustrating that this product is non-monotonic as required by the acceleration a 1x being both positive and negative over the u p domain; vertical dashed line is at u p = 4.6 cm s 1. [25] In addition, upon dividing (13) by m 2 and replacing f px (p x ) with f up (u p ), numerical integration over the positive domain of the data gives a value close to zero, namely This is only about 5% of the positive contribution of the integration over the domain 0 u p u p, consistent with the expectation that the net streamwise force acting on the particles is zero. Moreover, with the average force k 1x = ma 1x being both positive and negative over p x = mu p, the product k 2x f px in (11) indeed is non-monotonic, increasing over p x p x (u p u p ) and decreasing over p x p x (u p u p ) (Figure 9). This last point further implies that much of the total fluctuating kinetic energy is generated about the average momentum state p x. [26] The exponential function (1) is thus consistent with the idea that a steady momentum density f px over p x reflects a balance among three fluxes: (1) an advection of momentum at the rate k 1x, positive and decreasing from p x =0top x p x, and negative and increasing over p x p x ; (2) a diffusion of momentum from higher to lower values of momentum density in association with the gradient df px =dp x ; and (3) a drift of momentum from higher to lower values of the diffusivity k 2x, equivalent to a drift from regions within the domain of p x having high average rates of change in kinetic energy toward regions having low rates of change in kinetic energy. [27] With respect to cross-stream motions, the analogous form of (20) is f vp v p ¼ Ce R vp ½ ð Šdvp : ð24þ 2a1y=a2y 1=a2y Þda2y=dvp Figure 8. Plot of integrand 1/A =2a 1x =a 2x (1/a 2x )da 2x =du p versus streamwise velocity u p illustrating that this integrand is approximately constant over positive u p ; solid line is at 1/A = 1/4.6 = 0.22 (cm 1 s). As above we calculate a 1y and a 2y from the averages of q v (Figure 10) and q 2 v (Figure 11) for increments of v p. The average acceleration a 1y indeed is antisymmetric across v p = 0 (Figure 10). That is, a 1y is zero near v p = 0, increasingly positive with increasingly negative v p, and 8of15

9 Figure 10. Plot of velocity displacements q v versus crossstream velocity v p, with local average (gray circles) and one standard error bars based on 0.5 and 1 cm s 1 increments of v p. increasingly negative with increasingly positive v p. This reflects an overall advection of cross-stream momentum toward v p = 0. When transformed to a one-sided function, a 1y is zero at v p = 0, and increasingly negative with increasing v p (Figure 12). The diffusivity a 2y is finite at v p = 0, and increases approximately linearly with v p (Figure 13). Histograms of velocity displacements q v reveal that cross-stream accelerations are approximately symmetrical about the local average. [28] Using estimates of a 1y, a 2y and da 2y /dv p from Figures 12 and 13, we calculate the bracketed integrand in Figure 12. Plot of product of average acceleration a 1y and time interval Dt versus cross-stream velocity v p, with one standard error bars based on 0.5 and 1 cm s 1 increments of v p. (24) to reveal that it varies approximately linearly over v p except near the origin ( v p = 0) (Figure 14). On purely empirical grounds, this result suggests that the integral in (24) involves v p 2 away from the origin, which is consistent with a Gaussian-like distribution [Roseberry et al., 2012] for large v p. 6. Numerical Simulations [29] Numerical simulations involving particles that undergo random walks over the domain of u p are consistent Figure 11. Plot of squared displacements q 2 v versus crossstream velocity v p, with local average (gray circles) and one standard error bars based on 0.5 and 1 cm s 1 increments of v p. Figure 13. Plot of product of diffusivity a 2y and time interval Dt versus cross-stream velocity v p, with one standard error bars based on 0.5 and 1 cm s 1 increments of v p. 9of15

10 Figure 14. Plot of integrand 2a 1y /a 2y (1/a 2y )da 2y /d v p versus cross-stream velocity v p illustrating that this integrand varies approximately linearly with v p except near v p = 0; gray circles are based on averaging over 0.2 cm s 1 increments of v p. with the analysis above. Starting with the definition of the velocity displacement q u = u p (t + Dt) u p (t), the form of the random walk is u p ðt þ DtÞ ¼ u p ðþþe t qu u p ðþ t þ Squ u p ðþ t ; ð25þ where E qu is the expected (average) velocity displacement and S qu is a random fluctuation, each of which depends on the velocity state u p (t). Note that E qu a 1x Dt, and that S qu is selected so as to be faithful to the variance of the distribution f qu (q u ; u p, t). Thus, the simulations of particle transitions between velocity states based on (25) are faithful to the measured statistics of these transitions. [30] In these simulations we assume that f qu (q u ; u p, t) can be approximated by a normal distribution based on the histograms of q u in Figure (6). We then select a velocity displacement q u for each particle at each time step from this normal distribution with mean (that is, an expected value E qu ) equal to a 1x Dt =(a 1x0 + au p )Dt (Figure 5) and second moment a 2x Dt = bu p Dt (Figure 7) based on the numerical fits to the data in these figures. (Note that this neglects the asymmetry (Figure 6) of the displacements q u.) Upon substituting this expression for E qu into (25) and rearranging, [31] In the experiments some particles at any instant may possess negative velocities u p (Figure 1). For simplicity in our random-walk simulations we reflect displacements q u across u p = 0. These reflections therefore are analogous to momentary (unrecorded) negative excursions, and to stopping and starting of particles. This means that during any time step disentrainment is numerically balanced by entrainment, consistent with steady transport conditions. Moreover, on physical grounds a 2x cannot be unbounded, as particles cannot accelerate to velocities greater than instantaneous near-bed fluid velocities, so for simplicity we randomly reset u p for the few particles that experience displacements to u p >10u p. We then iteratively calculate the velocities of a large number of particles. Starting with N particles whose initial velocities are selected from a peaked distribution f up centered near zero (with mean much less than u p ), the simulated version of f up based on the N particles relaxes in less than the equivalent of 0.1 sec real time to a steady exponential-like form with mean equal to u p (Figure 15). We stress that these random-walk simulations are not conditioned a priori to give an exponential form with specified mean. The random-walk algorithm does not know the form of f up nor the value of u p, only how a 1x and a 2x vary with u p. [32] Numerical simulations involving particles that undergo random walks over the domain v p, positive and negative, similarly are consistent with the exponential-like form of f vp. In these simulations we select a displacement q v for each particle at each time step from a normal distribution. On empirical grounds we specify the mean as a 1y Dt = 0.04 v p 2 (Figure 12) and the second moment as a 2y Dt = 0.3+ v p ) (Figure 13) based on numerical fits to the data in these figures. (Note that the approximate linear fit for a 2y Dt underestimates measured values at large v p. Also, as above, this neglects any asymmetry of the displacements q v.) This random-walk algorithm is effectively the same as reflecting u p ðt þ DtÞ ¼ fu p ðþþa t 1x0 Dt þ S qu u p ðþ t ; ð26þ which has the form of an autoregressive model [e.g., Box and Jenkins, 1976] wherein f = 1 + adt represents a firstorder autoregressive coefficient. However, (26) is not a simple autoregressive model, notably with the appearance of the term a 1x0 Dt and the dependence of S qu on the velocity u p. Thus, in treating the particle velocity u p as a Markov-like process, attention must be paid to how the distribution f qu (q u ; u p, t) varies with the particle velocity state u p. Figure 15. Plot of probability density function f up of streamwise velocities u p showing results of random-walk simulation (circles) involving 10,000 particles versus theoretical distribution (line) with mean u p = 4.6 cm s of 15

11 where F x [M L t 2 ] is the net streamwise force acting on a particle, fluid drag plus particle-to-bed friction. This means that the probability distribution of the impulses I = F x Dt [M L t 1 ] acting on particles in motion during any small interval Dt has the same form as the distribution of velocity displacements q u, whose mean, variance and asymmetry depend on u p (Figures 3 7). In the limit of Dt 0, the form of the distribution of instantaneous accelerations of particles in motion matches that of the net streamwise forces acting on them. Thus, simulations of particle motions that do not involve directly solving Newton s laws for the coupled motions of fluid and particles nonetheless require faithfulness to the dependence of the average acceleration a 1x = k 1x =m and the diffusivity a 2x = k 2x =m 2 on the velocity (or momentum) state. This is essential for the simulations of particle motions described in the next section. Figure 16. Plot of probability density function f vp of crossstream velocities v p showing results of random-walk simulation (circles) involving 10,000 particles versus theoretical distribution (line) with mean jv p j = 1.8 cm s 1. across u p = 0 in the one-sided (positive u p ) simulations above. Zero crossings in these simulations may be interpreted as representing cross-stream fluctuations in the particle velocity v p when the streamwise velocity u p is finite, and disentrainment or entrainment when the crossings coincide with u p 0. These simulations lead to an exponential-like distribution consistent with (2), although the mean v p is slightly overestimated (Figure 16). [33] For convenience these simulations assume that the displacements q u and q v during each time step are normally distributed and uncorrelated from one time step to the next. More realistically, the probability density of q u is asymmetric at small and large u p (Figure 6), reflecting that large positive excursions are more likely than are negative excursions at small particle velocities, and that large positive excursions are unlikely at large particle velocities, limited by near-bed fluid velocities. This asymmetry would have the effect of exporting momentum from small u p toward the mean, and likewise from large u p toward the mean. The effect of any autocorrelation in q u or q v is to alter the numerical relaxation rather than influence the emergent forms of the distributions f up and f vp and their mean values. Whereas the particle velocities u p and v p are autocorrelated with a correlation timescale of less than 0.1 sec [Furbish et al., 2012b], the differencing of these velocities to give q u and q v generally weakens the autocorrelation of the resulting series (i.e. q u and q v ) relative to the original series (u p and v p ) [e.g., Box and Jenkins, 1976]. Thus the assumption that the displacements q u and q v are uncorrelated from one time step to the next is reasonable. [34] These simulations suggest another interesting point. According to Newton s second law, q u 1 m F xðþdt t ¼ 1 m It ðþ; ð27þ 7. The Diffusive Flux [35] The formulation presented in the first paper of this series [Furbish et al., 2012a] suggests that the streamwise volumetric sediment flux q s [L 2 t 1 ]is q s ¼ ug 1 ð 2 x kg Þ; ð28þ where u [L t 1 ] is the average velocity of active particles, k [L 2 t 1 ] is the diffusivity, and g [L] is the particle activity, the volume of particles in motion per unit streambed area. Within the present context, u(t) represents the average of an instantaneous sample drawn from f up, and the time average u equals the ensemble average u p if at all times the activity g is finite [Furbish et al., 2012a; Roseberry et al., 2012]. Values of u, k and g may be viewed as representing timeaveraged quantities, or they may be considered instantaneous quantities if measured over a bed area B (or width b [L]) sufficiently large to sample all possible velocities from one instant to the next. Of particular interest is the magnitude of the diffusive part of (28) relative to the advective part, given that experiments designed to measure the diffusive flux are not yet available, whereas analyses of measurements based on the assumption that transport is described by the advective part of (28) are numerous. Here we note that the random-walk algorithm described in the previous section provides the essential ingredients for simulating bed load particle motions, including entrainment and disentrainment, in order to evaluate the magnitude of the diffusive part of (28). Although not based on the equations of motion describing the coupled behavior of particles and fluid, the algorithm nonetheless is faithful to the kinematics of particle motions. [36] The numerical analysis described next considers particle motions parallel to x and consists of two parts. The first part involves simulating particle hop distances l and associated travel times t for comparison with the joint probability density of l and t reported in Roseberry et al. [2012]. This part is aimed at providing additional evidence that the random-walk algorithm yields results that are consistent with experiments as a lead in to the second part of the analysis. The second part involves simulating conditions that yield a diffusive flux over x as represented in (28). These numerical 11 of 15

12 Figure 17. Plot of hop distance l versus travel time t showing the relation l t 5/3 (solid line); black circles are obtained from random-walk simulation involving 279 motions of 100 particles, and white circles are experimental data from runs R2B and R3B involving 176 particle motions as reported in Roseberry et al. [2012]. analyses therefore connect the theory leading to (28) developed in the first paper in this series [Furbish et al., 2012a] with the measurements of particle motions presented in the second paper [Roseberry et al., 2012]. [37] As in section 6 above we select a change in velocity q u for each of N particles at each time step from a normal distribution with mean equal to a 1x Dt =(a 1x0 + au p )Dt and second moment a 2x Dt = bu p Dt. (Values of a 1x0 Dt = 0.5, adt = 0.1 and bdt = 1 with Dt = sec are the same as those used in the simulations above, based on the results of run R3B.) This change in velocity q u is added to the velocity in the previous time step, and the particle is displaced accordingly. If the velocity falls below zero, the particle is treated as being disentrained. Simultaneously a new active particle is started ( entrained ) elsewhere in the computational domain of x (described below) so that N remains constant, consistent with steady transport conditions. At the beginning of a simulation and with the entrainment of each new particle, particle positions along x are randomly selected as described below. The simulations are spun up to a steady state condition, which involves a few tenths of a second in real time. The simulations also involve a large number of particles N. This is done for numerical reasons rather than being meant to represent large transport rates. That is, results from a single run with large numbers are effectively equivalent to averaging over several simulations with small numbers. Large N also ensures that numerical fluctuations in computed quantities such as the activity and flux are small relative to the overall magnitudes of these quantities. We then record the particle activity and flux at specified positions x, and we calculate particle hop distances and travel times. As mentioned above, values of u, k and g obtained from the simulations may be viewed as representing timeaveraged quantities, or they may be considered instantaneous quantities if measured over a bed area B (or width b) sufficiently large to sample all possible velocities from one instant to the next [Furbish et al., 2012a; Roseberry et al., 2012]. [38] The simulations qualitatively reproduce the gammalike distributions of the particle hop distance l and the associated travel time t reported by Roseberry et al. [2012, Figure 14], and they closely reproduce the nonlinear, heteroscedastic relation between l and t, namely l t 5/3 (Figure 17) [see also Roseberry et al., 2012, Figure 13]. Note that the data in Figure 17 represent a sample from the joint probability distribution of l and t. Here we again emphasize that the simulations are not conditioned a priori to give this relationship. That is, the random-walk algorithm does not know how l and t are related, only how a 1x and a 2x vary with u p. These results, together with those described in section 6 above, indicate that the simulated kinematics of particle motions, start to stop, are consistent with the experiments. In addition, the mean hop distance l and the mean travel time t calculated from the simulations are not subject to experimental censorship wherein the motions of particles that enter or leave the video sampling area are spatially censored, or where particles that are in motion when the video starts or stops are temporally censored. Thus the simulations yield the expected result that u p = l/t [Furbish et al., 2012a, Appendix G]. That is, the ensemble average hop distance l is equal to the product of the ensemble average velocity u p and the mean travel time t. [39] The parametric quantities used in these simulations are based on experiments involving a single particle activity and associated mean velocity (run R3B). The simulations thus neglect the possibility that the mean velocity u and diffusivity k vary with the activity g. The simulations therefore mimic conditions that are equivalent to assuming that the volumetric flux q s is q s ¼ ug k 2 g x : ð29þ Thus, with the diffusivity k removed from the derivative between (28) and (29), the simulations described next are focused on effects of variations in the activity g. [40] To initially illustrate the diffusive behavior described by (29), we set up a special situation. Let E [L t 1 ] denote the volumetric rate of particle entrainment per unit streambed area, and let D [L t 1 ] denote the volumetric rate of disentrainment per unit streambed area. Particles are randomly positioned (entrained) within a domain of length X [L] equal to 10 cm. As above, a particle is entrained when another is disentrained, so N is fixed. The particle activity along x is then determined by the balance between entrainment, disentrainment and downstream motion. (Because the width of the computational domain is arbitrary, specific values of activity are unimportant.) Specifically, under steady conditions conservation within the domain X requires that dq s =dx E + D = 0, and conservation downstream of X requires that dq s =dx + D = 0 (with E = 0). In turn, the probability of disentrainment within any small interval dx 12 of 15