A probabilistic description of the bed load sediment flux: 1. Theory

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117, F331, doi:1.129/212jf2352, 212 A probabilistic description of the bed load sediment flux: 1. Theory David Jon Furbish, 1 Peter K. Haff, 2 John C. Roseberry, 1 and Mark W. Schmeeckle 3 Received 25 January 212; revised 16 July 212; accepted 18 July 212; published 21 September 212. [1] We provide a probabilistic definition of the bed load sediment flux. In treating particle positions and motions as stochastic quantities, a flux form of the Master equation (a general expression of conservation) reveals that the volumetric flux involves an advective part equal to the product of an 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 a diffusivity that arises from the second moment of the probability density function of particle displacements. Gradients in the activity, instantaneous or time-averaged, therefore effect a particle flux. Time-averaged descriptions of the flux involve averaged products of the particle activity, the particle velocity and the diffusivity; the significance of these products depends on the scale of averaging. The flux form of the Exner equation looks like a Fokker-Planck equation (an advection-diffusion form of the Master equation). The entrainment form of the Exner equation similarly involves advective and diffusive terms, but because it is based on the joint probability density function of particle hop distances and associated travel times, this form involves a time derivative term that represents a lag effect associated with the exchange of particles between the static and active states. The formulation is consistent with experimental measurements and simulations of particle motions reported in companion papers. Citation: Furbish, D. J., P. K. Haff, J. C. Roseberry, and M. W. Schmeeckle (212), A probabilistic description of the bed load sediment flux: 1. Theory, J. Geophys. Res., 117, F331, doi:1.129/212jf Introduction [2] The bed load sediment flux, defined as the solid volume of bed load particles crossing a vertical surface per unit time per unit width, figures prominently in descriptions of sediment transport and the evolution of alluvial channels. Translating this definition of the flux into conceptually simple quantities that accurately characterize the collective motions of particles, however, is not necessarily straightforward, and quantitative definitions of the flux have several forms. We note at the outset that, when viewed at the particle scale, the instantaneous, vertically integrated flux q A (t) [L 2 t 1 ] associated with a surface A [L 2 ] is precisely 1 Department of Earth and Environmental Sciences and Department of Civil and Environmental Engineering, Vanderbilt University, Nashville, Tennessee, USA. 2 Division of Earth and Ocean Sciences, Nicholas School of the Environment, Duke University, Durham, North Carolina, USA. 3 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, 231 Vanderbilt Pl., Nashville, TN , USA. (david.j.furbish@vanderbilt.edu) 212. American Geophysical Union. All Rights Reserved /12/212JF2352 defined as the surface integral of surface-normal velocities of the solid fraction, namely q A ðtþ ¼ 1 Z u p nda; ð1þ b where u p [L t 1 ] is the discontinuous particle velocity field viewed at the surface A, n is the unit vector normal to A, and b [L] is the width of A, where A extends over the vertical domain of moving particles (Figure 1). This precise definition, however, is impractical. Except possibly using high-speed imaging of a small (observable) number of particles [Drake et al., 1988; Lajeunesse et al., 21; Roseberry et al., 212] at high resolution, the flux described by (1) is virtually impossible to measure, and we are far from possessing a theory of sediment transport that describes the velocity field u p as it responds to near-bed turbulence [Parker et al., 23]. Conventional descriptions of the flux therefore instead appeal to measures of collective particle behavior, specifically averaged quantities such as the average particle velocity and concentration, to replace the detailed information contained in the particle velocity field u p at the surface A. [3] With equilibrium (i.e., quasi-steady and uniform) bed and transport conditions, for example, the sediment flux normally is defined in flux form as the product of a mean particle velocity U p [L t 1 ] and a particle concentration, A F331 1of21

2 F331 F331 [1997a], Parker et al. [2], Seminara et al. [22], Wong et al. [27], Ganti et al. [21] and others. By this definition, with quasi-steady bed and transport conditions the flux is equal to the product of the volumetric rate of particle entrainment per unit streambed area, E [L t 1 ], and the mean particle hop distance, l [L], measured start to stop. That is, q x ¼ El: ð3þ Figure 1. Definition diagram for surface integral of surface-normal velocities u p = u p n of the discontinuous particle velocity field up at the surface A with width b. The surface A extends upward to a height necessary to include all bed load particles, and arrows are representative of vector components u p surrounding infinite sets of such vector components positioned over the solid fraction. The vector field u p at A is nonzero only over the domain consisting of intersections of (moving) particles with A. namely, the volume of particles in motion per unit streambed area [e.g., Bridge and Dominic, 1984; Wiberg and Smith, 1989; Seminara et al., 22; Parker et al., 23; Francalanci and Solari, 27; Wong et al., 27; Lajeunesse et al., 21], herein referred to as the bed load particle activity g [L]. That is, for one-dimensional transport in the x direction the flux q x [L 2 t 1 ]is q x ¼ U p g; with the caveat that U p and g represent macroscopic quantities averaged over stochastic fluctuations [Wong et al., 27]. Note that this is like the definition of advection associated with a continuous medium. As elaborated below, to describe the sediment flux as the product of a mean velocity and a concentration indeed assumes a continuum behavior where active (moving) particles are uniformly (albeit quasi-randomly) distributed. But as recently noted [Schmeeckle and Furbish, 27; Ancey, 21], the continuum assumption is rarely satisfied for sediment particles transported as bed load, particularly at low transport rates [Roseberry et al., 212], and the details of the averaging, whether involving ensemble, spatial or temporal averaging [Coleman and Nikora, 29], matter to the physical interpretation as well as the form of the definition of the flux. Ancey [21] notes in his review of several definitions of the flux that it remains unclear how the flux is actually related to the mean particle velocity and the particle concentration. [4] Another important definition of the bed load sediment flux is the entrainment form of this quantity, first introduced by Einstein [195] and recently elaborated by Wilcock ð2þ This is essentially a statement of conservation of particle volume where, assuming spatially uniform transport, rates of entrainment and deposition are steady, uniform and everywhere balanced. The value of this definition is highlighted in treating tracer particles [Ganti et al., 21], notably involving exchanges between the active and inactive layers of the streambed [Wong et al., 27]. What is unclear is how the ingredients of (3), notably the distribution of particle hop distances with mean l, translate to unsteady and nonuniform conditions [Lajeunesse et al., 21], and the extent to which this and other definitions overlap or match (2) [Ancey, 21]. On this point we note that any formulation of the flux must be consistent with (1). [5] At any instant the solid volume of bed load particles in motion per unit area of streambed, the particle activity g, can vary spatially due to short-lived near-bed turbulence excursions as well as longer-lived influences of bed form geometry on the mean flow [Drake et al., 1988; McLean et al., 1994; Nelson et al., 1995; Schmeeckle and Nelson, 23; Singh et al., 29; Roseberry et al., 212]. During their motions, particles respond to turbulent fluctuations and interact with the bed and with each other so that, at any instant within a given small area, some particles move faster and some move slower than the average within the area, and fluctuations in velocity are of the same order as the average velocity. Moreover, a hallmark of bed load particles is their propensity to alternate between states of motion and rest over a large range of timescales. These attributes mean that, in relation to the definition (2) above, the bed load particle flux involves an advective part, as is normally assumed, but more generally it also involves a diffusive part associated with variations in particle activity and velocity [Lisle et al., 1998; Schmeeckle and Furbish, 27; Furbish et al., 29a, 29b]. In relation to the definition (3) above, because the distribution of particle hop distances may be considered the marginal distribution of a joint probability density function of particle hop distances and associated travel times [Lajeunesse et al., 21], a more general form of this definition similarly involves a diffusive part as well as a time derivative term that represents a lag effect associated with the exchange of particles between the static and active states. [6] The purpose of this contribution is to clarify the points above, namely, how variations in particle activity and velocity influence the volumetric bed load flux, q = iq x + jq y [L 2 t 1 ], with components q x and q y parallel to the coordinates x [L] and y [L], selected here to coincide with the streamwise and cross-stream directions, respectively. Our analysis involves a probabilistic formulation wherein particle positions and motions are treated as stochastic quantities, leading to a kinematic description of q that illustrates how and why it involves both advective and diffusive terms, 2of21

3 F331 F331 Figure 2. Definition diagram of a particle of diameter D moving with a positive velocity parallel to x through a surface A positioned at x =, where x i denotes the distance that the nose of the particle is relative to x =, and ɛ denotes a small distance measured from the nose of the particle. borrowing key elements from closely related formulations [Furbish et al., 29a, 29b; Furbish and Haff, 21]. [7] It is straightforward using the probabilistic framework of the Master equation [e.g., Risken, 1984; Ebeling and Sokolov, 25] to formulate a statement of conservation of particle concentration c having the form c/ t = r q [Ancey, 21], and then by inspection extract from this statement a kinematic description of the flux q [e.g., Furbish et al., 29a], therein revealing that it has both advective and diffusive parts. More challenging, however, is to formulate a definition of q directly from a description of particle motions as Einstein [195] did for Brownian motions. This is particularly desirable inasmuch as a direct formulation more fully clarifies the geometrical and kinematic ingredients of the flux, including its relation to such quantities as particle hop distances and velocities [e.g., Drake et al., 1988; Wilcock, 1997b; Wong et al., 27; Ancey, 21], and variations in particle velocity and activity. [8] In section 2 we formulate a qualitative version of the one-dimensional flux q x, with the purpose of clarifying key geometrical and kinematic ingredients in the problem, notably particle size, shape and velocity, and spatial variations in particle concentration. We show how the definitions (1) and (2) are related. This provides the basis for illustrating that, in appealing to averaged particle quantities (specifically the mean particle velocity and concentration) to replace the detailed information contained in the discontinuous particle velocity field u p at the surface A, the resulting description of the flux must in general involve both advective and diffusive parts. In section 3 we provide a more formal, probabilistic description of the one-dimensional flux, and describe the implications of different definitions of the probability distribution of particle displacements versus hop distances. In the final part of this section we generalize to the twodimensional case. In section 4 we show the flux form of the Exner equation to illustrate how it is like a Fokker-Planck equation, and for comparison we obtain the entrainment form of the Exner equation to illustrate how this form involves a time derivative term (not contained in the Fokker- Planck equation) that represents a memory (or lag) effect associated with the exchange of particles between the static and active states. This result has implications for the use of the related entrainment formulation of conservation of tracer particles. In section 5 we elaborate how ensemble, spatial and temporal averaging matter in defining the flux, and we consider time averaging of the flux to suggest how persistent spatial variations in particle activity associated with bed topography influence the flux. For simplicity throughout, we consider transport of a single particle size, then briefly comment on the problem of generalizing the formulation to mixtures of sizes in section 6. [9] As noted by Ancey [21] and others, there is no unique way to define the solid volumetric flux. Nonetheless, an unambiguous, probabilistic definition exists. Beyond a definition of the bed load flux, moreover, the formulation highlights that the probability distribution of particle displacements, including details of how this distribution is defined, has a central role in describing particle motions across a range of scales. This is particularly significant in view of a growing interest in the possibility of non-fickian behavior in the transport of sediment and associated materials [e.g., Nikora et al., 22; Schumer et al., 29; Foufoula-Georgiou and Stark, 21; Bradley et al., 21; Ganti et al., 21; Voller and Paola, 21; Hill et al., 21; Ball, 212; Martin et al., 212], and in relation to connecting probabilistic descriptions of particle motions with treatments of fluid motion. [1] In companion papers [Roseberry et al., 212; Furbish et al., 212a, 212b] we present detailed measurements of bed load particle motions obtained from high-speed imaging in laboratory flume experiments. These measurements support key elements of the formulation described here. 2. Geometrical Ingredients of the One-Dimensional Flux [11] As an important reference point, here we present a discrete version of (1) to reveal details of particle shape and motion that figure into this deterministic definition of the flux. This provides the basis for illustrating that, in appealing to averaged particle quantities (specifically the mean particle velocity and concentration) to replace the detailed information embodied in (1), the resulting description of the flux must in general involve both advective and diffusive parts. We start with a rendering of the geometry and motion of a single particle. [12] Consider a particle of diameter D [L] that is moving parallel to x through a surface A positioned at x = (Figure 2). Let x i [L] denote the position of the nose of the particle relative to x =, and let V i (x i )[L 3 ] denote the volume of the particle that is to the right of x = as a function of x i. The particle volume discharge Q i (t) across A is (Appendix A) Q i ðtþ ¼S i ðx i Þu i ; where S i (x i )= V i / x i [L 2 ] is like a hypsometric function of the particle, equal to its cross-sectional area on the surface A at x =, and u i =dx i /dt [L t 1 ] is its velocity parallel to x. [13] Consider, then, a cloud of equal-sized particles which are moving with varying velocities parallel to x toward and ð4þ 3of21

4 F331 F331 to suspect that, at any instant, particles intersecting A with large (or small) cross-sectional area S i are any more (or less) likely to possess large (or small) velocity u i. In this case, q x ðtþ ¼ 1 b NS iðx i Þ u i ¼ S b u i ¼ gu i : ð8þ Figure 3. Definition diagram showing cloud of particles moving with varying velocities parallel to x toward and through a surface A positioned at x =, where x i denotes the distance that the nose of the ith particle is relative to x =. through a surface A of width b positioned at x = (Figure 3). Let N(t) denote the number of particles intersecting A at time t. If x i now denotes the distance between the nose of the ith particle and x =, then the instantaneous volumetric flux q x across A is X N q x ðtþ ¼ 1 Q i ðtþ ¼ 1 b b i¼1 X N i¼1 S i ðx i Þu i : This is a discrete version of (1). Namely, if the particle velocity (field) parallel to x is u p = u p n, and if H(u p ) is the Heaviside step function defined by H(u p ) = for u p < and H(u p ) = 1 for u p, then M A (t) =1 H(u p )H( u p ) denotes a mask projected onto A such that M A = 1 where u p and M A = elsewhere [Furbish et al., 29b], whence and q A ðtþ ¼q x ðtþ ¼ 1 b Z Z A M A da ¼ XN A u p nda ¼ 1 b i¼1 Z S i ðx i Þ A M A u p nda ¼ 1 b X N i¼1 ð5þ ð6þ S i ðx i Þu i ; which shows the relation between (1) and (5) with q A = q x. Note that N(t) is a stepped function of time as particles intersect and lose contact with A. At any instant, therefore, the derivative dn/dt strictly is either zero or undefined. Nonetheless, for sufficiently large N and rapidity of particles intersecting and losing contact with A, one can envision that N(t) begins to appear as a smooth function of time where brief fluctuations in q x become small relative to the magnitude of q x. [14] Letting an overbar denote an average over N particles, the last part of (5) may be written as q x ¼ð1=bÞNS i ðx i Þu i. For equal-sized particles, moreover, it is reasonable to assume that S i and u i are uncorrelated, as there is no reason ð7þ The product NS i ðx i Þ¼S½L 2 Š is the cross-sectional area of particles intersecting A, and the ratio S/b = g [L 2 L 1 = L 3 L 2 ] is equivalent to the particle activity, the volume of active particles per unit streambed area. Specifically, this is a local line averaged activity. Because Sdx is equal to the volume of active particles within a small spatial interval dx, Sdx/bdx = S/b = g is the volume of active particles within the small area bdx. Then, if it is assumed that u i is equal to the average velocity U p of all particles in the cloud in the vicinity of A, that is U p = u i, one may conclude that q x = gu p, which is the definition (2) of the flux normally assumed for quasisteady bed and transport conditions [e.g., Bridge and Dominic, 1984; Wiberg and Smith, 1989; Seminara et al., 22; Parker et al., 23; Francalanci and Solari, 27; Wong et al., 27; Lajeunesse et al., 21]. Two caveats, however, must accompany this assessment of averages. [15] First, envision a uniform cloud of equal-sized particles moving with varying velocities parallel to x toward and through a set of surfaces A located at various positions along x. By uniform we mean the following. For a specified width b, let n x (x, t) denote the number of particles per unit distance parallel to x, such that n x (x, t)dx is the number of particles whose noses are located within any small interval dx. Then, for a sufficiently large width b, assume that n x (x, t) varies negligibly with x. At any instant the number of particles N and the corresponding particle area S ¼ NS i intersecting each surface is the same, although the detailed configuration of S varies from surface to surface. Let U p denote the average particle velocity parallel to x, that is, the average of all particles in the cloud near any surface A rather than the average u i of particles intersecting a surface A. Because the cloud is uniform, each surface A samples at any instant the full distribution of possible velocities (for sufficiently large width b), in which case u i = U p for all surfaces. This is the situation for which Ancey [21] notes that an ensemble-like average over A, giving u i, is equivalent to a volume average over the particle cloud, giving U p.in contrast, envision a cloud of particles with average velocity U p whose concentration n x (x, t) at some instant decreases with increasing distance x. Now, both the number of particles N and the particle area S ¼ NS i intersecting each surface decrease with increasing x. Moreover, in this case the surface and volume averages are not equivalent, and u i > U p. Here is why. [16] Let a prime denote a fluctuation about an average. Then, at any instant S i ¼ S i þ S i and u i = U p + u i. In turn, q x ¼ð1=bÞNðS i U p þ S i u i þ U p S i þ S i u i Þ. Consider a plot of u i versus S i at an instant (Figure 4), which provides a perspective as viewed by an observer moving with the average velocity U p, although the conclusions below pertain equally to an Eulerian frame of reference. During a small interval of time dt, some points on this plot move to the right as the cross-sectional area S i increases for particles that are beginning to cross A, and some points move to the left as the 4of21

5 F331 F331 Figure 4. Plots of deviation in particle velocity u i = u i U p versus deviation in particle cross-sectional area S i ¼ S i S i as viewed by observer moving with the average velocity U p for (a) uniform particle cloud with u i ¼ and u i ¼ U p, and (b) particle cloud where the particle activity decreases with increasing distance x with u i > and u i > U p. cross-sectional area S i decreases for particles that have mostly crossed A. The rate of motion of the points to the right and left is proportional to the magnitude of the particle velocity u i, so motion is faster near the top and bottom and slower near the middle of the plot. Points at u i = along the S i axis do not move during dt. Some points vanish as particles leave A, and new points appear as particles arrive at and initially intersect A. Points arrive at the far left of the second and third quadrants where the small areas of intersection of arriving particles are less than the average intersection area, and move to the far right of the first and fourth quadrants as their fat middles exceed the average intersection area, and then move back to the far left of the second and third quadrants because the intersection area of their exiting tails is less than the average intersection area. [17] In the case of the uniform cloud of particles described above, the number of points and their scatter is similar across all surfaces A and on each surface over time, and u i = U p with S i u i ¼ U p S i ¼ S i u i ¼ (Figure 4a), so that q x ¼ ð1=bþns i ðx i Þu i ¼ gu i ¼ gu p as in (8) or (1). In the second case where the particle activity decreases with increasing x, this situation changes. At any instant the number of particles to the immediate left of A is greater than the number to the immediate right of A. The likelihood that a particle to the left or right of A will intersect A during a small interval of time dt, for a given magnitude of the velocity u i, increases with its proximity to A, and, for a given proximity to A, increases with the magnitude of its velocity u i. Of the particles that are at any given distance to the left of A, the faster ones (large positive u i ) are more likely than are slower ones to reach A. And, of the particles that are at any given distance to the right of A, the slower ones (large negative u i ) are more likely than are faster ones to reach A. Because of the greater number of particles to the left of A than to the right of A, the plot of u i versus S i becomes preferentially populated by faster moving particles and depleted of slower moving particles. The effect is to shift the surface-averaged velocity u i upward such that u i is finite (Figure 4b). That is, the surface A sees an average velocity u i > U p where u i ¼ U p þ u i with U p S i ¼ S i u i ¼ for the same particle surface area S ¼ NS i. In turn, the flux q x ¼ 1 b NðS iu p þ S i u i Þ¼gU p þ gu i ; in which an extra term involving velocity fluctuations about the mean appears in the definition of q x. The counterpart to this situation occurs when the particle activity g increases with distance x, in which case u i < U p. [18] The effect embodied in (9) can be readily visualized by considering the motion of a triangular cloud of particles which possess two velocities, 1 and 2, in equal proportions (Figure 5). The average velocity of all particles in the cloud is U p = 1.5. During a short interval of time dt the particles begin to segregate. At any position x in front of the crest of the cloud there is a greater proportion of fast particles, and at any position x behind the crest there is a greater proportion of slow particles. The average velocity of particles intersecting a surface A in the leading, fully segregated part of the cloud is 2, and the average velocity of particles intersecting a Figure 5. Triangular cloud of particles possessing two velocities, 1 and 2, in equal proportions. During a short interval of time dt the particles begin to segregate, whereas the cloud as a whole moves downstream with the average velocity U p. ð9þ 5of21

6 F331 F331 surface A in the trailing, fully segregated part is 1. The average velocity of particles intersecting a surface A at any x in front of the crest is greater than U p, and the average velocity of particles intersecting a surface A at any x behind the crest is less than U p. The cloud as a whole moves downstream with velocity U p. One must be careful, however, to limit this idea to small time dt, as it neglects time variations in particle velocities, including starting and stopping. [19] This effect of an activity gradient vanishes in the absence of fluctuating particle velocities (i.e., if u i ¼ ), and, as elaborated in the next section, this effect represents diffusion when particle motions are cast in probabilistic terms. We show in fact that whereas ð1=bþns i U p represents advection, the product 1=bÞNðS i u i is equivalent to a diffusive term that looks like (1/2) (kg)/ x, where k [L 2 t 1 ] is a particle diffusivity. [2] A second caveat that goes with the averaging above centers on particle size. As mentioned above, for equal-sized particles it may be assumed that S i and u i are uncorrelated. When considering a mixture of particle sizes, however, the covariance between S i and u i cannot be neglected inasmuch as some particle sizes preferentially move faster than other sizes. As briefly elaborated in section 6, this means that individual sizes must be treated separately. 3. Probabilistic Formulation of the One-Dimensional Flux [21] Here we present a more careful rendering of the collective behavior of particles to define the bed load sediment flux, wherein particle positions and motions are treated as stochastic quantities. The explicit functional notation used in this section, although bulky in places, figures importantly in the bookkeeping of the formulation. In functions such as f g (g; x, y, t) (defined below), random variables, g in this example, appear first within the parentheses (and as subscripts, which identify the probability density or distribution), followed by parametric quantities or independent variables after the semicolon. Here a parametric quantity means a key quantity that is not a random variable, and which can be treated mathematically as an independent variable. In a conditional function such as f r g (r g; x, dt), the quantity providing the conditioning, g in this case, is to be considered a parameter, so this function could just as well be written, for example, as f r;g (r;g, x, dt) Ensemble States of Particle Motions [22] Because the particle activity g varies stochastically over space and time at many scales, a particularly challenging part of defining the bed load sediment flux is taking this variability into account such that the local, instantaneous flux can be systematically related to spatially averaged or time-averaged expressions of the flux, and vice versa. We approach this by envisioning an ensemble of configurations of particle positions and velocities in a manner similar to (but not identical to) that outlined by Gibbs [192] for gas particle systems. As Kittel [1958, p. 8] notes The scheme introduced by Gibbs is to replace time averages over a single system by ensemble averages, which are averages at a fixed time over all systems in an ensemble. The problem of demonstrating the equivalence of the two types of averages is the subject of ergodic theory It may be argued, as Tolman [Tolman, 1938] has done, that the ensemble average really corresponds better to the actual situation than does the time average. We never know the initial conditions of the system, so we do not know exactly how to take the time average. The ensemble average describes our ignorance appropriately. In turn, the ergodic hypothesis suggests that (for gas systems) one may assume an ensemble average is the same as a time average over one realization, that is, a single system that evolves through time. Here we define the essentials of an ensemble appropriate to sediment particle motions. We use this as a starting point for our probabilistic formulation of the flux, and then return to it later to suggest how persistent time-averaged variations in particle activity associated with bed forms influence the flux. [23] Envision bed load particles moving over an area B [L 2 ] on a streambed that is subjected to steady macroscopic flow conditions, and momentarily assume for simplicity that the streambed is planar [e.g., Lajeunesse et al., 21; Roseberry et al., 212], albeit possibly involving small, stationary fluctuations in elevation [e.g., Wong et al., 27]. Over time, some particles stop and others start, some particles leave the area B across its boundaries and others arrive. We choose B to be sufficiently large that, during any small interval of time dt, any difference in the number of particles leaving B and the number arriving is negligibly small relative to the total number N a of active particles within B. Similarly, any difference in the number of particles that stop and start within B during dt is negligibly small relative to the total number N a of active particles. Then, N a may be considered the same from one instant to the next. We now envision all possible instantaneous configurations of the N a active particles as defined by their x, y positions within B at a fixed time, with the understanding that this set of configurations need not represent the same set of particles, only that N a is the same. This imagined set of possible configurations constitutes an ensemble of active particle positions, and, in the absence of any additional information, we initially assume that each configuration in the ensemble is equally probable (but see Roseberry et al. [212]). [24] Consider an elementary area db within B. If n xy (x, y, t) [L 2 ] denotes the number of active particles per unit area, then n xy (x, y, t)db is the number of particles within db and the associated activity g(x, y, t) =V p n xy (x, y, t) such that g may be considered a random variable. One can then envision that the ensemble of configurations of particle positions, each equally probable, yields for any area db a probability density function of the activity g, namely f g (g; x, y, t)[l 1 ], such that f g (g; x, y, t)dg is the probability that the activity within db at (x, y, t) falls between g and g +dg. The form of f g (g; x, y, t) and its parametric values (e.g., mean, variance) are specific to the sediment (size, shape) and the macroscopic flow conditions, including the turbulence structure. Equally important, the form of f g (g; x, y, t) varies with the size of db (Appendix B), which means that the magnitude of fluctuations in the bed load flux relative to mean conditions at a given position varies with scale. [25] To elaborate this important point, we momentarily focus on one-dimensional transport parallel to x. Let db = bdx. For a specified width b, ifn x (x, t) [L 1 ] denotes the number of active particles per unit distance parallel to x, then 6of21

7 F331 F331 Figure 6. Examples of the cumulative distribution F g (g; x, t) obtained from the probability density function f g (g; x, t) of the particle activity g as this varies with width b for b = 5D, 1D, 5D and 1,D with the same overall activity g = N a V p /B =.5 units [L]. The variance of F g (g; x, t) decreases with increasing b, as reflected by the increasing slope of F g (g; x, t) near g =.5. Individual values of g = S/b used to generate fg(g; x, t) are obtained numerically from 1, configurations of particles uniformly (albeit randomly) distributed over an area db = 1Db. n x (x, t)dx is the number of active particles within bdx. The local activity at position x is g(x, t) =V p n x (x, t)/b [L] where, in the limit of dx becomes g(x, t) =S/b, that is, the particle area S intersecting a surface A at x divided by the surface width b. For a specified area B and total number of particles N a with overall activity g = N a V p /B, envision a large number of configurations where, in each configuration, N a particles are randomly distributed over B. Each configuration gives a different activity g(x, t)=s/b calculated at one position x. Hence the ensemble of particle configurations, each equally probable, yields for any position x a probability density function of the activity g, namely f g (g; x, t)[l 1 ]. As the width b increases, the number of particles intersecting a surface at x on average increases. This means that for a given overall activity the form of f g (g; x, t) varies with b (Figure 6). Specifically, whereas the mean activity at x associated with this distribution is equal to the overall activity calculated by g = N a V p /B, the variance of f g (g; x, t) decreases with increasing b, which reflects on average smaller fluctuations in the number of particles intersecting the surface at x. Moreover, any actual realization of the activity at an instant in effect is a sample from f g (g; x, t), so the variability in such realizations from one instant to the next decreases with increasing b. We reconsider this point below and in Roseberry et al. [212]. [26] Returning to the two-dimensional case, each active particle in each possible configuration possesses an instantaneous velocity u p = iu p + jv p at time t. One can therefore associate with each particle at time t the small (pending) displacements r = u p dt [L] and s = v p dt [L] parallel to x and y, respectively, that occur during dt, that is, between t and t + dt. For each configuration there is a joint probability distribution of r and s associated with N a particles. But because within any elementary area db the number of active particles n xy (x, y, t)db, and thus the activity g(x, y, t), varies among configurations, there are likewise n xy db values of the pair r and s for each configuration. Furthermore, we must leave open the possibility, elaborated below, that the velocities u p, and therefore the displacements r and s, of the n xy db particles within db are correlated with the number of active particles n xy db. We now envision the ensemble as consisting of all possible instantaneous states defined by the joint occurrence of particle positions and displacements r and s, and we assume this ensemble defined over B yields for any area db a joint probability density function of the activity g and the displacements r and s, namely f g,r,s (g, r, s; x, y, dt) [L 3 ], where certain values of g, r and s, and their combinations, are more (or less) probable than are others. Like f g (g; x, y, t), the form of f g,r,s (g, r, s; x, y, dt) and its parametric values are specific to the sediment (size, shape) and the macroscopic flow conditions, including the turbulence structure. [27] Specifically, among the ensemble of possible configurations of particle positions and velocities, some configurations may be preferentially selected or excluded by the turbulence structure inasmuch as turbulent sweeps and bursts characteristically lead to patchy, fast-moving clouds of particles [Schmeeckle and Nelson, 23; Roseberry et al., 212], or because unusual configurations (e.g., all N a active particles are clustered within db) are excluded by the physics of coupled fluid-particle motions. Nonetheless, in the absence of a clear understanding of the influence of turbulence on the particle activity, we cannot suggest that any particular configuration of particle positions and velocities is not possible, and hence, the initial assumption that each configuration in the ensemble is equally probable is justified [Tolman, 1938]. This assumption, however, is not critical in that f g (g; x, y, t) orf g,r,s (g, r, s; x, y, dt) ultimately must be defined semi-empirically. Moreover, if the streambed and turbulence structure are homogeneous (in a probabilistic sense) over B, then it may be assumed that f g (g; x, y, t) and f g,r,s (g, r, s; x, y, dt) are the same for each elementary area db. And, because these probability densities vary smoothly with x, y position, their parametric values (e.g., mean, variance) also vary smoothly such that these values may be considered continuous fields, albeit uniform and steady in this initial example of a planar streambed. [28] If, in contrast, the streambed and turbulence structure vary over B, for example, due to the presence of bed forms, then one might expect concomitant, systematic variations in particle activity and motions. In this case the bed forms are to be considered part of the externally imposed macroscopic conditions, that is, as a bed condition that is compatible with the macroscopic flow and sediment properties. Then, we again may envision an ensemble of possible configurations of active particle positions and velocities, each configuration being equally probable. But here it is important to imagine, as Gibbs did, the set of configurations as being separate systems (realizations) with the same bed forms at a fixed time, not necessarily as a time series of one realization where the bed forms grow or migrate. As above, we assume this ensemble yields for any area db a probability density function of the activity, namely f g (g; x, y, t), and a joint probability density function of the activity g and the displacements r and s, namely f g,r,s (g, r, s; x, y, dt). Now the 7of21

8 F331 F331 Figure 7. Definition diagram for particle motions parallel to x coordinate, showing probability density function f r g (r g; x, dt) of displacement distances r during dt. forms of f g (g; x, y, t) and f g,r,s (g, r, s; x, y, dt) and their parametric values may vary with x, y position (and with time; see section 5), although it still may be that these values are continuous fields over B. [29] In the next three sections we consider for simplicity one-dimensional transport parallel to x, where our first objective is to obtain a probabilistic description of the sediment flux q x, and our second objective is to obtain the expected (ensemble-averaged) value of this flux. In this case the number density n xy (x, y, t), the activity g(x, y, t) = V p n xy (x, y, t), the density function f g (g; x, y, t) and the joint density function f g,r,s (g, r, s; x, y, dt) introduced above may be simplified to n x (x, t)[l 1 ], g(x, t) [L], f g (g; x, t)[l 1 ] and f g,r (g, r; x, dt) [L 2 ]. We also define the conditional probability density function f rjg ðrjg; x; dtþ ¼ f g;rðg; r; x; dtþ ; ð1þ f g ðg; x; tþ with units [L 1 ], where f g (g; x, t) may be considered the marginal distribution of f g,r (g, r; x, dt). That is, f r g (r g; x, dt) dr is the probability that a particle at x will move a distance between r and r +drduring dt given that, among all possible combinations of particle activity and displacements r, attention is restricted to the specific activity g(x, t) at time t. In turn we let F r g (r g; x, dt) denote the cumulative distribution function defined by F rjg ðrjg; x; dtþ ¼ Z r f rjg ðrjg; x; dtþdr; ð11þ where the lower limit of integration indicates that r may be positive or negative, a condition that we redefine below. That is, F r g (r g; x, dt) is the probability that a particle at x will move a distance less than or equal to r during dt, given the activity g(x, t) at time t Master Equation [3] To a good approximation most bed load particles move downstream. Nonetheless, there is value in considering the more general case of bidirectional motions. With reference to Figure 7, consider particle motions along a coordinate x, where it is convenient to treat motions in the positive and negative directions separately. For particles located at x = x at time t, let r denote a displacement in the positive x direction during dt, and let l denote a (positive) displacement in the negative x direction during dt. Further, let p(x, t) denote the probability that motion is in the positive x direction, and let q(x, t) denote the probability that motion is in the negative x direction. Thus, p(x, t)+q(x, t) = 1. Also note that a particle in motion during dt may also be in motion (or at rest) at either time t or time t +dt, or both. That is, r or l is the total displacement of an active particle for all motion that occurs over an interval less than or equal to dt. The displacements r and l therefore are not to be interpreted as hop distances measured start to stop, a point that we examine below. [31] Now, if F r g (r g; x, dt) denotes the probability that a particle starting at x (r = ) moves a distance less than or equal to r during dt, then R r g (r g; x, dt)=1 F r g (r g; x,dt) is the probability that a particle moves a distance greater than r during dt. By definition the conditional probability density of r is f r g (r g; x, dt) =df r g /dr = dr r g /dr[l 1 ]. In turn, if F l g (l g; x, dt) denotes the probability that a particle starting at x (l = ) moves a distance less than or equal to l during dt, then R l g (l g; x, dt) =1 F l g (l g; x, dt) is the probability that a particle moves a distance greater than l (in the negative x direction) during dt. The conditional probability density of l is f l g (l g; x, dt) =df l g /dl = dr l g /dl [L 1 ]. Note that because r and l are defined here as being positive displacements, the lower limit of integration in (11) defining F r g is now set to zero, and likewise for the (unwritten) companion definition of F l g. [32] If the location of a particle is specified by the position x of its nose, then over a specified area A of width b normal to x, let n x (x, t) [L 1 ] denote the number of active particles per unit distance parallel to x. Then, g(x, t)bdx = V p n x (x, t) dx denotes the associated volume of active particles at x at time t, and p(x, t)g(x, t)bdx is the volume of particles that moves in the positive x direction during dt. Moreover, the volume of particles passing position x in the positive x direction from x < x is p(x, t)g(x, t)r r g (x x g; x, dt)bdx, and the volume passing position x in the negative x direction from x > x is q(x, t)g(x, t)r l g (x x g; x, dt)bdx. The total volume of particles passing position x in the positive x direction during dt is V þ ðx; t þ dt; gþ ¼b Z x pðx ; tþgðx ; tþr rjg ðx x jg; x; dtþdx ; ð12þ and the total (negative) volume of particles passing position x in the negative x direction during dt is V ðx; t þ dt; gþ ¼ b x qðx ; tþgðx ; tþr ljg ðx xjg; x ; dtþdx : ð13þ The net volume of particles passing x in the positive x direction during dt is V(x, t + dt) = V + (x, t + dt) + V (x, t +dt), namely Vðx; t þ dt; gþ ¼b Z x b x pðx ; tþgðx ; tþr rjg ðx x jg; x ; dtþdx qðx ; tþgðx ; tþr ljg ðx xjg; x ; dtþdx : ð14þ This is a flux form of the Master equation [Risken, 1984; Ebeling and Sokolov, 25; Furbish et al., 29a, 29b], illustrating that the volume V(x, t + dt) passing x during 8of21

9 F331 F331 dt may be influenced by motions originating at positions both to the left and right of x. Note that nothing is assumed a priori regarding the forms of the conditional probability densities, f r g (r g; x, dt) and f l g (l g; x, dt), of the displacements r and l. Also note that the explicit appearance of the activity g as a parameter in the functional notation of the left side of (14) highlights that the particle volume V(x, t +dt; g) is conditional on the activity. This point is important in the idea of an ensemble average presented below Advection and Diffusion [33] The Master equation (14) may be recast in a more compact form involving advective and diffusive terms as follows. With r = x x (x < x) and l = x x (x > x), a change of variables in (14) gives Vðx; t þ dt; gþ ¼b b pðx r; tþgðx r; tþr rjg ðrjg; x r; dtþdr qðx þ l; tþgðx þ l; tþr ljg ðljg; x þ l; dtþdl: ð15þ Expanding the products p(x r, t)g(x r, t)r r g (r g; x r,dt) and q(x + l, t)g(x + l, t)r l g (l g; x + l,dt) as a Taylor series to first order then leads to Vðx; t þ dt; gþ ¼bpðx; tþgðx; tþ R rjg ðrjg; x; dtþdr bqðx; tþgðx; tþ R ljg ðljg; x; dtþdl b x pðx; tþgðx; tþ rr rjg ðrjg; x; dtþdr b x qðx; tþgðx; tþ lr ljg ðljg; x; dtþdl : ð16þ By definition the mean particle displacements during dt are (Appendix C) m r ðx; g; dtþ ¼ and m l ðx; g; dtþ ¼ rf rjg ðrjg; x; dtþdr ¼ lf ljg ðljg; x; dtþdl ¼ R rjg ðrjg; x; dtþdr ð17þ R ljg ðljg; x; dtþdl: ð18þ The second moments of these displacements about the local origin x are s 2 r ðx; g; dtþ ¼ r 2 f rjg ðrjg; x; dtþdr ¼ 2 and s 2 l ðx; g; dtþ ¼ l 2 f ljg ðljg; x; dtþdl ¼ 2 rr rjg ðrjg; x; dtþdr ð19þ lr ljg ðljg; x; dtþdl: ð2þ In turn, average velocities conditional to the activity g are defined by and 1 u r ðx; t; gþ ¼ lim dt dt 1 u l ðx; t; gþ ¼ lim dt dt rf rjg ðrjg; x; dtþdr lf ljg ðljg; x; dtþdl; ð21þ ð22þ and diffusivities are defined by [Risken, 1984; Ebeling and Sokolov, 25; Furbish et al., 29a, 29b] and 1 k r ðx; t; gþ ¼ lim dt dt 1 k l ðx; t; gþ ¼ lim dt dt r 2 f rjg ðrjg; x; dtþdr l 2 f ljg ðljg; x; dtþdl: ð23þ ð24þ [34] Substituting (17) through (2) into (16), dividing by dt, and taking the limit as dt thus gives the particle volume discharge, dvðx; tþ Q x ðx; t; gþ ¼ dt ¼ bgðx; tþ½pðx; tþu r ðx; t; gþ qðx; tþu l ðx; t; gþš b 1 f 2 x gðx; tþ½pðx; tþk rðx; t; gþþqðx; tþk l ðx; t; gþšg: ð25þ The first term on the right side of (25) is advective and the second is diffusive. The bracketed part of the first term is merely the weighted average particle velocity u [L t 1 ], namely, u(x, t; g)=p(x, t)u r (x, t; g) q(x, t)u l (x, t; g). That is, in the development above, for convenience we defined l as being a positive displacement in the negative x direction, so by this definition u l is positive. If for cosmetic reasons we now let u l carry the sign, then u(x, t; g) =p(x, t)u r (x, t; g) + q(x, t)u l (x, t; g). Similarly, the parenthetical part of the second term on the right side of (25) is a weighted diffusivity k [L 2 t 1 ], namely, k(x, t; g) =p(x, t)k r (x, t; g) + q(x, t)k l (x, t; g). With these definitions, dividing (25) by the width b gives the flux q x (x, t) [L 2 t 1 ], namely q x ðx; tþ ¼ug 1 2 x ðkgþ; ð26þ which suggests that spatial variations in g or k can effect a flux that is in addition to the advective flux. We consider the conditions under which the diffusive term in (26) may be important in section 5 below and in Furbish et al. [212a]. [35] The activity g(x, t) is treated above as being one of many possible instantaneous values of g at position x, whereas the velocity u and the diffusivity k are formally defined above as ensemble averages, that is, the (statistically) expected values of these quantities obtained from the ensemble of all possible configurations of particle positions and velocities, conditional to the activity g. The conditional probability densities f r g and f l g (as well as the related 9of21

10 F331 F331 Letting an overbar denote an ensemble average, dividing by b and by dt, and taking the limit as dt, this becomes Figure 8. Schematic diagram of three realizations of the joint probability density function f r,t (r, t), where a steep covariance relation (open circles) between r and t implies varying speeds due to varying displacements over a similar travel time, a weak covariance (gray circles) implies varying speeds due to similar displacements over varying travel times, and an intermediate covariance (black circles) implies relatively uniform speeds. functions R r g and R l g ) thus represent underlying (ensemble) populations and are smooth, continuous functions. In order to envision (26) as representing the local instantaneous flux, one must therefore imagine that u and k actually represent values obtained from an instantaneous sample drawn from the densities f r g and f l g. Over an elementary area bdx, this sample may involve few to many particles as determined by the instantaneous value of g and the width b, so the instantaneous distributions of displacements (drawn from f r g and f l g ) may look more like irregular histograms than like the smooth functions f r g and f l g, and the velocity u (26) is like the simple average u i in (8). We return to this point below. [36] Meanwhile, to complete the ensemble average over all values of the activity g we first substitute (17) through (2) into (16). Then, to simplify we redefine r to its original meaning as a displacement that is positive or negative, note that dl = dr, combine the integrals in (16), and use p + q =1 to give Vðx; t þ dt; gþ ¼bgðx; tþ rf rjg ðrjg; x; dtþdr b 1 Z gðx; tþ r 2 f rjg ðrjg; x; dtþdr ; ð27þ 2 x which, like (16), is the particle volume crossing x during dt associated with the activity g. In turn, multiplying (27) by the probability f g (g; x, t)dg weights this volume in proportion to the relative occurrence of g over the ensemble. Substituting (1) into (27), multiplying by f g (g; x, t)dg and integrating over the activity g thus gives Vðx; t þ dtþ ¼b b 1 2 x grf g;r ðg; r; x; dtþdr dg gr 2 f g;r ðg; r; x; dtþdr dg : ð28þ q x ðx; tþ ¼ug 1 ð 2 x kg Þ; ð29þ which is the ensemble-averaged flux. [37] A key point embodied in (29) is that the advective part involves the averaged product of the particle velocity and activity, and the diffusive term involves the averaged product of the diffusivity and activity. Indeed, experiments suggest that, at low transport rates, both the particle activity and the average velocity increase with increasing bed stress, where the activity increases faster than the velocity [Schmeeckle and Furbish, 27; Ancey et al., 28; Ancey, 21; Lajeunesse et al., 21; Roseberry et al., 212], clearly indicating that u and g are correlated. This figures importantly in considering how the ensemble average is related to time averaging, a topic that we address in section 5. Meanwhile we note that if u and g, and k and g, are independent, which may be the case at high transport rates (and is demonstrably correct in the case of rain splash transport treated as a stochastic advection-diffusion process [Furbish et al., 29a]), then (29) becomes q x ðx; tþ ¼u g 1 ð 2 x k g Þ: ð3þ [38] This formulation of particle advection and diffusion shares an important similarity with porous-media transport. Namely, in contrast to the advective-diffusive process in simple fluid-solvent systems, wherein the fluid velocity and the molecular diffusivity are independent [e.g., Furbish, 1997], the mean particle velocity u and the diffusivity k in (26) are highly correlated. For example, if particle velocities are distributed exponentially [Lajeunesse et al., 21; Roseberry et al., 212] with mean U p, then the diffusivity k = ts u 2 = tu p 2 [Taylor, 1922], where s u 2 is the variance of the particle velocities and t is the Lagrangian integral timescale obtained from the autocorrelation function of the particle velocities u. This relation highlights that the diffusive part of the flux in (26) fundamentally is associated with velocity fluctuations, and that this diffusive part vanishes in the absence of particle advection, entirely analogous to the relation between advection and mechanical dispersion in porous-media transport [Furbish et al., 212b] Hop Distances and Travel Times [39] In contrast to the small displacements r and l that occur during the small interval dt, as highlighted in the previous section, let l [L] denote a particle displacement measured start to stop that occurs over a travel time t [t]. Then let f l,t (l, t; x, t )[L 1 t 1 ] denote the joint probability density of l and t for particles whose motions start at position x at time t. With reference to Figure 8, a steep covariance relation between l and t implies varying speeds (defined by l/t) due to varying displacements over a similar travel time. A weak covariance implies varying speeds due to similar displacements over varying travel times. An intermediate covariance implies relatively uniform speeds. 1 of 21