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Keywords:

  • Fokker-Planck equation;
  • advection;
  • bed load sediment;
  • diffusion

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Geometrical Ingredients of the One-Dimensional Flux
  5. 3. Probabilistic Formulation of the One-Dimensional Flux
  6. 4. Exner Equation
  7. 5. Averaged Quantities
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Particle Volume Discharge
  11. Appendix B:: Probability Distribution of Activity in Ensemble
  12. Appendix C:: Means and Variances of Displacement Distances
  13. Appendix D:: Entrainment Form of Exner Equation
  14. Appendix E:: Relation Between Definitions of the Flux
  15. Acknowledgments
  16. References

[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.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Geometrical Ingredients of the One-Dimensional Flux
  5. 3. Probabilistic Formulation of the One-Dimensional Flux
  6. 4. Exner Equation
  7. 5. Averaged Quantities
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Particle Volume Discharge
  11. Appendix B:: Probability Distribution of Activity in Ensemble
  12. Appendix C:: Means and Variances of Displacement Distances
  13. Appendix D:: Entrainment Form of Exner Equation
  14. Appendix E:: Relation Between Definitions of the Flux
  15. Acknowledgments
  16. References

[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 qA(t) [L2 t−1] associated with a surface A [L2] is precisely defined as the surface integral of surface-normal velocities of the solid fraction, namely

  • display math

where up [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., 2010; Roseberry et al., 2012] 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 upas it responds to near-bed turbulence [Parker et al., 2003]. 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 up at the surface A.

image

Figure 1. Definition diagram for surface integral of surface-normal velocitiesup = upn 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 up surrounding infinite sets of such vector components positioned over the solid fraction. The vector field up at A is nonzero only over the domain consisting of intersections of (moving) particles with A.

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[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 velocityUp [L t−1] and a particle concentration, namely, the 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; Wong et al., 2007; Lajeunesse et al., 2010], herein referred to as the bed load particle activity γ[L]. That is, for one-dimensional transport in thex direction the flux qx [L2 t−1] is

  • display math

with the caveat that Up and γ represent macroscopic quantities averaged over stochastic fluctuations [Wong et al., 2007]. 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, 2007; Ancey, 2010], the continuum assumption is rarely satisfied for sediment particles transported as bed load, particularly at low transport rates [Roseberry et al., 2012], and the details of the averaging, whether involving ensemble, spatial or temporal averaging [Coleman and Nikora, 2009], matter to the physical interpretation as well as the form of the definition of the flux. Ancey [2010] 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 [1950] and recently elaborated by Wilcock [1997a], Parker et al. [2000], Seminara et al. [2002], Wong et al. [2007], Ganti et al. [2010]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, inline image [L], measured start to stop. That is,

  • display math

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., 2010], notably involving exchanges between the active and inactive layers of the streambed [Wong et al., 2007]. What is unclear is how the ingredients of (3), notably the distribution of particle hop distances with mean inline image, translate to unsteady and nonuniform conditions [Lajeunesse et al., 2010], and the extent to which this and other definitions overlap or match (2) [Ancey, 2010]. 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 γ, 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, 2003; Singh et al., 2009; Roseberry et al., 2012]. 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, 2007; Furbish et al., 2009a, 2009b]. 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., 2010], 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 = iqx + jqy [L2 t−1], with components qx and qy 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 ofq that illustrates how and why it involves both advective and diffusive terms, borrowing key elements from closely related formulations [Furbish et al., 2009a, 2009b; Furbish and Haff, 2010].

[7] It is straightforward using the probabilistic framework of the Master equation [e.g., Risken, 1984; Ebeling and Sokolov, 2005] to formulate a statement of conservation of particle concentration c having the form ∂c/t = −∇ · q [Ancey, 2010], and then by inspection extract from this statement a kinematic description of the flux q [e.g., Furbish et al., 2009a], 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 [1905] 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., 2007; Ancey, 2010], and variations in particle velocity and activity.

[8] In section 2we formulate a qualitative version of the one-dimensional fluxqx, 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 up at the surface A, the resulting description of the flux must in general involve both advective and diffusive parts. In section 3we 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 two-dimensional case. Insection 4we 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. Insection 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 [2010]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., 2002; Schumer et al., 2009; Foufoula-Georgiou and Stark, 2010; Bradley et al., 2010; Ganti et al., 2010; Voller and Paola, 2010; Hill et al., 2010; Ball, 2012; Martin et al., 2012], and in relation to connecting probabilistic descriptions of particle motions with treatments of fluid motion.

[10] In companion papers [Roseberry et al., 2012; Furbish et al., 2012a, 2012b] 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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Geometrical Ingredients of the One-Dimensional Flux
  5. 3. Probabilistic Formulation of the One-Dimensional Flux
  6. 4. Exner Equation
  7. 5. Averaged Quantities
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Particle Volume Discharge
  11. Appendix B:: Probability Distribution of Activity in Ensemble
  12. Appendix C:: Means and Variances of Displacement Distances
  13. Appendix D:: Entrainment Form of Exner Equation
  14. Appendix E:: Relation Between Definitions of the Flux
  15. Acknowledgments
  16. References

[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 = 0 (Figure 2). Let ξi [L] denote the position of the nose of the particle relative to x = 0, and let Vi(ξi) [L3] denote the volume of the particle that is to the right of x = 0 as a function of ξi. The particle volume discharge Qi(t) across A is (Appendix A)

  • display math

where Si(ξi) = ∂Vi/∂ξi [L2] is like a hypsometric function of the particle, equal to its cross-sectional area on the surfaceA at x = 0, and ui = dξi/dt [L t−1] is its velocity parallel to x.

image

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 = 0, where ξi denotes the distance that the nose of the particle is relative to x = 0, and ε denotes a small distance measured from the nose of the particle.

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[13] Consider, then, a cloud of equal-sized particles which are moving with varying velocities parallel tox toward and through a surface A of width b positioned at x = 0 (Figure 3). Let N(t) denote the number of particles intersecting A at time t. If ξi now denotes the distance between the nose of the ith particle and x = 0, then the instantaneous volumetric flux qx across A is

  • display math

This is a discrete version of (1). Namely, if the particle velocity (field) parallel to x is up = upn, and if H(up) is the Heaviside step function defined by H(up) = 0 for up < 0 and H(up) = 1 for up ≥ 0, then MA(t) = 1 − H(up)H(−up) denotes a “mask” projected onto A such that MA = 1 where up ≠ 0 and MA = 0 elsewhere [Furbish et al., 2009b], whence

  • display math

and

  • display math

which shows the relation between (1) and (5) with qA = qx. 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 qx become small relative to the magnitude of qx.

image

Figure 3. Definition diagram showing cloud of particles moving with varying velocities parallel to x toward and through a surface A positioned at x = 0, where ξi denotes the distance that the nose of the ith particle is relative to x = 0.

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[14] Letting an overbar denote an average over N particles, the last part of (5) may be written as inline image. For equal-sized particles, moreover, it is reasonable to assume thatSi and ui are uncorrelated, as there is no reason to suspect that, at any instant, particles intersecting Awith large (or small) cross-sectional areaSi are any more (or less) likely to possess large (or small) velocity ui. In this case,

  • display math

The product inline imageis the cross-sectional area of particles intersectingA, and the ratio S/b = γ [L2 L−1 = L3 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 = γ is the volume of active particles within the small area bdx. Then, if it is assumed that inline image is equal to the average velocity Up of all particles in the cloud in the vicinity of A, that is Up =  inline image, one may conclude that qx = γUp, which is the definition (2)of the flux normally assumed for quasi-steady bed and transport conditions [e.g.,Bridge and Dominic, 1984; Wiberg and Smith, 1989; Seminara et al., 2002; Parker et al., 2003; Francalanci and Solari, 2007; Wong et al., 2007; Lajeunesse et al., 2010]. Two caveats, however, must accompany this assessment of averages.

[15] First, envision a uniform cloud of equal-sized particles moving with varying velocities parallel tox 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 nx(x, t) denote the number of particles per unit distance parallel to x, such that nx(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 nx(x, t) varies negligibly with x. At any instant the number of particles N and the corresponding particle area inline image intersecting each surface is the same, although the detailed configuration of S varies from surface to surface. Let Up 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 inline image 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 inline image = Up for all surfaces. This is the situation for which Ancey [2010]notes that an ensemble-like average overA, giving inline image, is equivalent to a “volume” average over the particle cloud, giving Up. In contrast, envision a cloud of particles with average velocity Up whose concentration nx(x, t) at some instant decreases with increasing distance x. Now, both the number of particles N and the particle area inline image intersecting each surface decrease with increasing x. Moreover, in this case the surface and volume averages are not equivalent, and inline image > Up. Here is why.

[16] Let a prime denote a fluctuation about an average. Then, at any instant inline image and ui = Up + ui′. In turn, inline image. Consider a plot of ui versus Si at an instant (Figure 4), which provides a perspective as viewed by an observer moving with the average velocity Up, 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 areaSi increases for particles that are beginning to cross A, and some points move to the left as the cross-sectional areaSi 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 ui, so motion is faster near the top and bottom and slower near the middle of the plot. Points at ui = 0 along the Si 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.

image

Figure 4. Plots of deviation in particle velocity ui′ = ui − Upversus deviation in particle cross-sectional area inline image as viewed by observer moving with the average velocity Up for (a) uniform particle cloud with inline image and inline image, and (b) particle cloud where the particle activity decreases with increasing distance x with inline image and inline image.

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[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 inline image = Up with inline image (Figure 4a), so that inline image 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 ui, increases with its proximity to A, and, for a given proximity to A, increases with the magnitude of its velocity ui. Of the particles that are at any given distance to the left of A, the faster ones (large positive ui) 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 ui) 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 ui versus Sibecomes preferentially populated by faster moving particles and depleted of slower moving particles. The effect is to shift the surface-averaged velocity inline image upward such that inline image is finite (Figure 4b). That is, the surface A “sees” an average velocity inline image > Up where inline image with inline image for the same particle surface area inline image. In turn, the flux

  • display math

in which an extra term involving velocity fluctuations about the mean appears in the definition of qx. The counterpart to this situation occurs when the particle activity γ increases with distance x, in which case inline image < Up.

[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 Up = 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 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 Up, and the average velocity of particles intersecting a surface A at any x behind the crest is less than Up. The cloud as a whole moves downstream with velocity Up. 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.

image

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 Up.

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[19] This effect of an activity gradient vanishes in the absence of fluctuating particle velocities (i.e., if inline image), 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 inline image represents advection, the product inline image is equivalent to a diffusive term that looks like −(1/2) ∂(κγ)/∂x, where κ [L2 t−1] is a particle diffusivity.

[20] A second caveat that goes with the averaging above centers on particle size. As mentioned above, for equal-sized particles it may be assumed thatSi and ui are uncorrelated. When considering a mixture of particle sizes, however, the covariance between Si and ui 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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Geometrical Ingredients of the One-Dimensional Flux
  5. 3. Probabilistic Formulation of the One-Dimensional Flux
  6. 4. Exner Equation
  7. 5. Averaged Quantities
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Particle Volume Discharge
  11. Appendix B:: Probability Distribution of Activity in Ensemble
  12. Appendix C:: Means and Variances of Displacement Distances
  13. Appendix D:: Entrainment Form of Exner Equation
  14. Appendix E:: Relation Between Definitions of the Flux
  15. Acknowledgments
  16. References

[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 fγ(γ; x, y, t) (defined below), random variables, γ 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 fr|γ(r|γ; x, dt), the quantity providing the conditioning, γ in this case, is to be considered a parameter, so this function could just as well be written, for example, as fr;γ(r;γ, x, dt).

3.1. Ensemble States of Particle Motions

[22] Because the particle activity γvaries 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, andvice 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 [1902] 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 [L2] 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., 2010; Roseberry et al., 2012], albeit possibly involving small, stationary fluctuations in elevation [e.g., Wong et al., 2007]. 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 Na 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 Na of active particles. Then, Na may be considered the same from one instant to the next. We now envision all possible instantaneous configurations of the Na 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 Na 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. [2012]).

[24] Consider an elementary area dB within B. If nxy(x, y, t) [L−2] denotes the number of active particles per unit area, then nxy(x, y, t)dB is the number of particles within dB and the associated activity γ(x, y, t) = Vpnxy(x, y, t) such that γ 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 γ, namely fγ(γ; x, y, t) [L−1], such that fγ(γ; x, y, t)dγ is the probability that the activity within dB at (x, y, t) falls between γ and γ + dγ. The form of fγ(γ; 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 fγ(γ; 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 tox. Let dB = bdx. For a specified width b, if nx(x, t) [L−1] denotes the number of active particles per unit distance parallel to x, then nx(x, t)dx is the number of active particles within bdx. The local activity at position x is γ(x, t) = Vpnx(x, t)/b [L] where, in the limit of dx [RIGHTWARDS ARROW] 0 becomes γ(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 Na with overall activity γ = NaVp/B, envision a large number of configurations where, in each configuration, Na particles are randomly distributed over B. Each configuration gives a different activity γ(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 γ, namely fγ(γ; 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 fγ(γ; 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 γ = NaVp/B, the variance of fγ(γ; 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 fγ(γ; 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. [2012].

image

Figure 6. Examples of the cumulative distribution Fγ(γ; x, t) obtained from the probability density function fγ(γ; x, t) of the particle activity γ as this varies with width b for b = 50D, 100D, 500D and 1,000D with the same overall activity γ = NaVp/B = 0.05 units [L]. The variance of Fγ(γ; x, t) decreases with increasing b, as reflected by the increasing slope of Fγ(γ; x, t) near γ = 0.05. Individual values of γ = S/b used to generate (γ; x, t) are obtained numerically from 10,000 configurations of particles uniformly (albeit randomly) distributed over an area dB = 10Db.

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[26] Returning to the two-dimensional case, each active particle in each possible configuration possesses an instantaneous velocityup = iup + jvp at time t. One can therefore associate with each particle at time t the small (pending) displacements r = updt [L] and s = vpdt [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 Na particles. But because within any elementary area dB the number of active particles nxy(x, y, t)dB, and thus the activity γ(x, y, t), varies among configurations, there are likewise nxydB values of the pair r and s for each configuration. Furthermore, we must leave open the possibility, elaborated below, that the velocities up, and therefore the displacements r and s, of the nxydB particles within dB are correlated with the number of active particles nxydB. 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 γ and the displacements r and s, namely fγ,r,s(γ, r, s; x, y, dt) [L−3], where certain values of γ, r and s, and their combinations, are more (or less) probable than are others. Like fγ(γ; x, y, t), the form of fγ,r,s(γ, 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, 2003; Roseberry et al., 2012], or because “unusual” configurations (e.g., all Na 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 fγ(γ; x, y, t) or fγ,r,s(γ, r, s; x, y, dt) ultimately must be defined semi-empirically. Moreover, if the streambed and turbulence structure are homogeneous (in a probabilistic sense) overB, then it may be assumed that fγ(γ; x, y, t) and fγ,r,s(γ, 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 fγ(γ; x, y, t), and a joint probability density function of the activity γ and the displacements r and s, namely fγ,r,s(γ, r, s; x, y, dt). Now the forms of fγ(γ; x, y, t) and fγ,r,s(γ, 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 tox, where our first objective is to obtain a probabilistic description of the sediment flux qx, and our second objective is to obtain the expected (ensemble-averaged) value of this flux. In this case the number densitynxy(x, y, t), the activity γ(x, y, t) = Vpnxy(x, y, t), the density function fγ(γ; x, y, t) and the joint density function fγ,r,s(γ, r, s; x, y, dt) introduced above may be simplified to nx(x, t) [L−1], γ(x, t) [L], fγ(γ; x, t) [L−1] and fγ,r(γ, r; x, dt) [L−2]. We also define the conditional probability density function

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with units [L−1], where fγ(γ; x, t) may be considered the marginal distribution of fγ,r(γ, r; x, dt). That is, fr|γ(r|γ; x, dt)dr is the probability that a particle at x will move a distance between r and r + dr during dt given that, among all possible combinations of particle activity and displacements r, attention is restricted to the specific activity γ(x, t) at time t. In turn we let Fr|γ(r|γ; x, dt) denote the cumulative distribution function defined by

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where the lower limit of integration indicates that r may be positive or negative, a condition that we redefine below. That is, Fr|γ(r|γ; 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 γ(x, t) at time t.

3.2. Master Equation

[30] 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.

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Figure 7. Definition diagram for particle motions parallel to x coordinate, showing probability density function fr|γ(r|γx′, dt) of displacement distances r during dt.

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[31] Now, if Fr|γ(r|γ; x′, dt) denotes the probability that a particle starting at x′ (r = 0) moves a distance less than or equal to r during dt, then Rr|γ(r|γx′, dt) = 1 − Fr|γ(r|γx′, dt) is the probability that a particle moves a distance greater than r during dt. By definition the conditional probability density of r is fr|γ(r|γx′, dt) = dFr|γ/dr = −dRr|γ/dr[L−1]. In turn, if Fl|γ(l|γx′, dt) denotes the probability that a particle starting at x′ (l = 0) moves a distance less than or equal to l during dt, then Rl|γ(l|γx′, dt) = 1 − Fl|γ(l|γx′, 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 fl|γ(l|γx′, dt) = dFl|γ/dl = − dRl|γ/dl [L−1]. Note that because r and l are defined here as being positive displacements, the lower limit of integration in (11) defining Fr|γ is now set to zero, and likewise for the (unwritten) companion definition of Fl|γ.

[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 nx(x, t) [L−1] denote the number of active particles per unit distance parallel to x. Then, γ(x′, t)bdx′ = Vpnx(x′, t)dx′ denotes the associated volume of active particles at x′ at time t, and p(x′, t)γ(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)γ(x′, t)Rr|γ(x − x′|γx′, dt)bdx′, and the volume passing position x in the negative x direction from x′ > x is q(x′, t)γ(x′, t)Rl|γ(x′ − x|γx′, dt)bdx′. The total volume of particles passing position x in the positive x direction during dt is

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and the total (negative) volume of particles passing position x in the negative x direction during dt is

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The net volume of particles passing x in the positive x direction during dt is V(xt + dt) = V+(xt + dt) + V(xt + dt), namely

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This is a flux form of the Master equation [Risken, 1984; Ebeling and Sokolov, 2005; Furbish et al., 2009a, 2009b], illustrating that the volume V(x, t + dt) passing x during 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, fr|γ(r|γx′, dt) and fl|γ(l|γx′, dt), of the displacements r and l. Also note that the explicit appearance of the activity γ as a parameter in the functional notation of the left side of (14) highlights that the particle volume V(x, t + dt; γ) is conditional on the activity. This point is important in the idea of an ensemble average presented below.

3.3. 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 = xx′ (x′ < x) and l = x′ − x (x′ > x), a change of variables in (14) gives

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Expanding the products p(x − r, t)γ(xr, t)Rr|γ(r|γ; x − r, dt) and q(x + lt)γ(x + lt)Rl|γ(l|γx + l, dt) as a Taylor series to first order then leads to

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By definition the mean particle displacements during dt are (Appendix C)

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and

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The second moments of these displacements about the local origin x are

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and

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In turn, average velocities conditional to the activity γ are defined by

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and

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and diffusivities are defined by [Risken, 1984; Ebeling and Sokolov, 2005; Furbish et al., 2009a, 2009b]

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and

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[34] Substituting (17) through (20) into (16), dividing by dt, and taking the limit as dt [RIGHTWARDS ARROW] 0 thus gives the particle volume discharge,

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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(xtγ) = p(xt)ur(xtγ) − q(xt)ul(xtγ). 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 ul is positive. If for cosmetic reasons we now let ul carry the sign, then u(xtγ) = p(xt)ur(xtγ) + q(xt)ul(xtγ). Similarly, the parenthetical part of the second term on the right side of (25) is a weighted diffusivity κ [L2 t−1], namely, κ(xtγ) = p(xt)κr(xtγ) + q(xt)κl(xtγ). With these definitions, dividing (25) by the width b gives the flux qx(x, t) [L2 t−1], namely

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which suggests that spatial variations in γ or κ 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. [2012a].

[35] The activity γ(x, t) is treated above as being one of many possible instantaneous values of γ at position x, whereas the velocity u and the diffusivity κ 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 γ. The conditional probability densities fr|γ and fl|γ (as well as the related functions Rr|γ and Rl|γ) 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 κ actually represent values obtained from an instantaneous “sample” drawn from the densities fr|γ and fl|γ. Over an elementary area bdx, this sample may involve few to many particles as determined by the instantaneous value of γ and the width b, so the instantaneous distributions of displacements (drawn from fr|γ and fl|γ) may look more like irregular histograms than like the smooth functions fr|γ and fl|γ, and the velocity u (26) is like the simple average inline image in (8). We return to this point below.

[36] Meanwhile, to complete the ensemble average over all values of the activity γ we first substitute (17) through (20) 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

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which, like (16), is the particle volume crossing x during dt associated with the activity γ. In turn, multiplying (27) by the probability fγ(γ; x, t)dγ weights this volume in proportion to the relative occurrence of γ over the ensemble. Substituting (10) into (27), multiplying by fγ(γ; x, t)dγ and integrating over the activity γ thus gives

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Letting an overbar denote an ensemble average, dividing by b and by dt, and taking the limit as dt [RIGHTWARDS ARROW] 0, this becomes

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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, 2007; Ancey et al., 2008; Ancey, 2010; Lajeunesse et al., 2010; Roseberry et al., 2012], clearly indicating that u and γ 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 γ, and κ and γ, 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., 2009a]), then (29) becomes

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[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 κ in (26) are highly correlated. For example, if particle velocities are distributed exponentially [Lajeunesse et al., 2010; Roseberry et al., 2012] with mean Up, then the diffusivity κ = τσu2 = τUp2 [Taylor, 1922], where σu2 is the variance of the particle velocities and τ 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., 2012b].

3.4. 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 λ [L] denote a particle displacement measured start to stop that occurs over a travel time τ [t]. Then let fλ,τ(λτx′, t′) [L−1 t−1] denote the joint probability density of λ and τ for particles whose motions start at position x′ at time t′. With reference to Figure 8, a steep covariance relation between λ and τ implies varying speeds (defined by λ/τ) 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.

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Figure 8. Schematic diagram of three realizations of the joint probability density function fr,τ(rτ), where a steep covariance relation (open circles) between r and τ 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.

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[40] The marginal distribution fλ(λ; x′, t′) [L−1] of the displacements λ is

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which defines a distribution of hop distances, start to stop, without rest times [e.g., Einstein, 1950; Wong et al., 2007; Bradley et al., 2010; Ganti et al., 2010; Lajeunesse et al., 2010; Roseberry et al., 2012]. By itself, fλ(λx′, t′) contains no information regarding particle travel times or speeds. In turn the marginal distribution fτ(τx′, t′) [t−1] of the travel times τ [e.g., Lajeunesse et al., 2010] is

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which similarly, by itself, contains no information regarding particle hop distances or speeds.

[41] With reference to Figure 9, of the particles starting at x′ at time t′, let Pλ(λx′, t′) [L−1] denote the proportion that moves to x′ + λ, relative to the proportion that moves beyond x′ + λ, namely

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Integrating (33) from λ = 0 to λ then gives

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from which it follows that the probability density of λ is

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In turn, of the particles starting at x′ at time t′, let Pτ(τx′, t′) [t−1] denote the proportion that ceases motion at time t′ + τ relative to the proportion that continues moving beyond t′ + τ. By a development similar to that above one obtains

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whence the probability density of τ is

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image

Figure 9. Definition diagram of particle hop distance λ and travel time τ for particles whose motions begin at position x′ or at time t′, showing probability density function fλ(λx′, t′) or fτ(τx′, t′).

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[42] The elements of (35) and (37) suggest a strategy for clarifying the physical basis of the densities fτ(τx′, t′) and fλ(λx′, t′). Namely, Pτ(τx′, t′) is a temporal “failure rate” function and Pλ(λx′; t′) is a spatial “failure rate” function as normally defined in reliability/survival theory, where “failure” may be interpreted as particle disentrainment [Furbish and Haff, 2010]. Thus, a physical (probabilistic) understanding of how and why active bed load particles stop in relation to bed roughness and near-bed flow conditions is central to describing the probability densities of the hop distancesλ and the associated travel times τ, beyond purely empirical descriptions. Moreover, descriptions of fτ(τx′, t′) and fλ(λx′, t′) must be mutually consistent, in as much as these combine to form the joint density fλ,τ(λτx′, t′).

[43] Specifically, with Pτ = Pτ(τx′, t′) and Pλ = Pλ(λx′, t′), then the densities fτ(τx′, t′) and fλ(λx′, t′) depend on the travel time τ and the distance λ, and therefore on changing conditions following entrainment at time t′ and downstream of the initial position x′. If, however, Pτ = Pτ(x′, t′) and Pλ = Pλ(x′, t′), then these densities are independent of τ and λ. For example, if for physical reasons Pτ and Pλ are constants that depend only on local conditions, namely Pτ(x′, t′) = 1/μτ and Pλ(x′, t′) = 1/μλ, then from (37) and (35), fτ(τx′, t′) = (1/μτ)exp(−τ/μτ) and fλ(λx′, t′) = (1/μλ)exp(λ/μλ) are exponential densities with mean travel time μτ and mean hop distance μλ. (This example of constant Pτ is analogous to a constant “failure rate” in reliability analysis, where μτ would be interpreted as the mean longevity.) In contrast, if Pτ = Pτ(τx′, t′) = ατα−1, for example, then fτ(τx′, t′) = ατα−1exp(−τα) is a standard Weibull distribution with shape factor α; or if Pτ(τx′, t′) = α/τ, then fτ(τx′, t′) = ατmα/τα+1 is a Pareto distribution with scale factor τm. This idea of a disentrainment rate function (Pλ or Pτ) is conceptually similar to, but of a different form than, the disentrainment rate function involving the distribution of particle hop distances as described by Nakagawa and Tsujimoto [1980].

[44] For completeness we note an additional definition of travel distances, where displacements λ are measured over a specified time τ, but involve rest times [e.g., Einstein, 1937; Hassan and Church, 1991; Bradley et al., 2010; Hill et al., 2010]. Namely, if λ is redefined to include multiple hops with rest times over an interval τ, then the probability density of λ may be denoted as fλ(λ; τ), emphasizing the significance of the interval τ as a parameter. Connecting travel distances that involve rest times to the density fλ(λ; τ) and associated particle speeds requires additional information on rest times and/or numbers of hops [Hill et al., 2010].

[45] We return below (section 4.2) to the joint probability density of hop distances and associated travel times in considering the entrainment form of the Exner equation [e.g., Parker et al., 2000; Garcia, 2008; Ancey, 2010], where we generalize this density to include cross-stream particle motions.Lajeunesse et al. [2010] present histograms representing the marginal distributions fλ(λ) and fτ(τ) based on high-speed imaging of particle motions, and note that these possess well defined modes that are less than the means. Data concerning the joint densityfλ,τ(λ; τ) are also presented in a companion paper [Roseberry et al., 2012].

3.5. The Two-Dimensional Flux

[46] As in section 3.1 let r and s denote particle displacements parallel to x and y, respectively, and let fr,s(r, s; dt) [L−2] denote the joint probability density function of r and s. If u = iu + jvdenotes the ensemble-average particle velocity, and ifκ denotes a diffusivity tensor with the elements κxx, κyy and κxy = κyx, then the component fluxes qx and qy of q = iqx + jqy are

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and

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Here u and v, and κxx and κyy, derive from the first and second moments, respectively, of the marginal distributions, fr(r; dt) and fs(s; dt), of the joint density function fr,s(r, s; dt) as described in section 3.2 above. Also,

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which is like a covariance.

[47] Inasmuch as diffusive particle motions normal to the mean motion are centered about this mean motion [Lajeunesse et al., 2010; Roseberry et al., 2012], the magnitudes of κxx and κyy vary with the direction of the mean motion. Moreover, κxy is finite only when the mean motion is not parallel to the x or y axis. As described above, the effect of the diffusive terms involving κxx and κyy is to contribute proportionally more (or fewer) particles to qx and qy relative to the contribution of those particles represented by and , depending on the sign of ∂(κxxγ)/∂x and ∂(κyyγ)/∂y. The effect of the diffusive terms involving κxy is similar. For example, with finite κxy and negative ∂(κxyγ)/∂y (due, say, to decreasing activity γ along y), proportionally more particles starting from positions at y < yS contribute to the flux qx parallel to x across an elementary plane at yS (Figure 10), relative to those particles represented by at y = yS. Conversely, with positive ∂(κxyγ)/∂y, proportional fewer particles starting from positions at y < yS contribute to the flux qx across an elementary plane at yS. Similar remarks pertain to the term involving ∂(κxyγ)/∂x with respect to the flux qy. The diffusivities in (38) and (39) are associated with the motion, not with any medium (as with heat conduction in an anisotropic solid). Moreover, like mechanical dispersion associated with flow in a porous medium, the elements of κco-vary withu and v. This point is elaborated in companion papers [Roseberry et al., 2012; Furbish et al., 2012a, 2012b].

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Figure 10. Schematic diagram showing effect of diffusive terms involving κxy, where, with negative ∂(γκxy)/∂y due to decreasing activity γ along y, proportionally more particles starting from positions at y < yS contribute (grey arrows) to the flux qx parallel to x across an elementary plane at yS, relative to those particles represented by uγ at y = yS.

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4. Exner Equation

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Geometrical Ingredients of the One-Dimensional Flux
  5. 3. Probabilistic Formulation of the One-Dimensional Flux
  6. 4. Exner Equation
  7. 5. Averaged Quantities
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Particle Volume Discharge
  11. Appendix B:: Probability Distribution of Activity in Ensemble
  12. Appendix C:: Means and Variances of Displacement Distances
  13. Appendix D:: Entrainment Form of Exner Equation
  14. Appendix E:: Relation Between Definitions of the Flux
  15. Acknowledgments
  16. References

4.1. Flux Form

[48] Let η(x, y, t) denote the local elevation of the streambed, and let cb denote the volumetric particle concentration of the bed. Then with cbη/∂t = − ∂qx/∂x − ∂qy/∂y, that is, neglecting motion associated with uplift or subsidence, substitution of (38) and (39) gives

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This formulation assumes that active particles effectively remain in contact with the bed, where η is defined as the (local) average surface elevation of particles, including the (small) contribution to the bed elevation associated with active particles. Note that (41)has the form of a Fokker-Planck equation, where the elements ofκ are inside both derivatives.

4.2. Entrainment Form

[49] Having introduced the joint probability density function of particle hop distances and associated travel times in section 3.4 above, here we generalize this idea to obtain the entrainment form of the Exner equation [e.g., Parker et al., 2000; Garcia, 2008; Ancey, 2010] for comparison with (41) above. Let E(x, y, t) [L t−1] denote the volumetric rate of particle entrainment per unit streambed area, and let D(x, y, t) [L t−1] denote the volumetric rate of deposition per unit streambed area. Assuming only downstream motions involving the streamwise hop distance λand the cross-stream hop distanceψ over the travel time τ, we denote the joint probability density function of λ, ψ and τ as fλ,ψ,τ(λψτxyt), which depends on x, y position and time t. By definition the rate of deposition is

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which explicitly incorporates the idea that particles arriving at an x, y position at time t started their hops λ and ψ at many different times tτ, as clearly reflected in histograms representing marginal distributions of fλ,ψ,τ, that is, the distributions fλ(λ) and fτ(τ) [Lajeunesse et al., 2010; Roseberry et al., 2012]. For simplicity, however, we are neglecting a possible dependence of the hop distance on the entrainment rate [Wong et al., 2007].

[50] Starting with (42) and assuming that fλ,ψ,τis not heavy-tailed [Roseberry et al., 2012], then it is straightforward to show (Appendix D) that

  • display math

where an overbar denotes an average. Specifically, inline image and inline image denote average hop distances and inline image denotes the associated average travel time. The averages inline image and inline image denote the second moments of λ and ψ, and inline image denotes the averaged product of λ and ψ. The terms in (43) involving spatial derivatives are analogous to the terms in (41) involving spatial derivatives.

[51] In the case of a steady, uniform entrainment rate E with uniform and steady values of the average hop distance inline image and the travel time inline image, then with the definition cbη/∂t = −∂qx/∂xfor one-dimensional transport, it follows from(43) that the flux inline image, which is equivalent to the definition (3) provided by Einstein [1950]. We further note that the average hop distance inline image is equal to the product of the ensemble average velocity ū and the average travel time inline image (Appendix E), namely inline image. Equating the “flux” and “entrainment” forms of the flux thus gives inline image. That is, under steady, uniform conditions the activity inline image, or inline image, which has the interpretation of being the mean residence time of particles within the nominal volume γB.

[52] The term on the right side of (43)involving the time derivative represent a “memory” associated with the exchange of particles between the static (rest) and active states. To illustrate this point, consider a simplified one-dimensional version of(43) with uniform and steady values of the average hop distance inline image and the travel time inline image, namely

  • display math

With a steady entrainment rate (∂E/∂t = 0), the rate of change in the bed elevation η goes simply as the divergence of the entrainment rate, ∂E/∂x. That is, there is a difference in the rates of deposition and entrainment at any position x because of a difference in the number of particles arriving at and leaving x. With a uniform entrainment rate (∂E/∂x = 0), the (uniform) rate of change in the bed elevation goes as the rate of change in the entrainment rate, ∂E/∂t [L t−2], modulated by the average travel time. Thus, with small inline image (which also implies small inline image), entrained particles quickly return to the rest state (they “remember” to stop), and the difference DE is small. But with increasing average travel time inline image, entrained particles increasingly “forget” to stop, so there is an increasing lag between deposition and entrainment (or vice versa).

[53] Focusing on the difference DE in (44), this formulation does not specify the style (rolling, sliding, hopping) of particle motions; in fact, particles could be saltating high into the fluid column. So in specifying that cbη/∂t = DE [e.g., Parker et al., 2000; Garcia, 2008; Ancey, 2010], η is effectively defined as the (local) average surface elevation of particles at rest, neglecting the (small) contribution to the bed elevation associated with active particles in contact with the bed. In this case inline image is like a source term. Specifically, writing cbη/∂t = −∂qx/∂x, then it follows that inline image with inline image. This indicates that the flux in excess of the steady, uniform flux inline image, namely inline image, increases (or decreases) downstream at the rate S with finite DE.

[54] We complete this section by noting the significance of the time derivative terms in the related entrainment formulation of conservation of tracer particles. Consider the simplified case of one-dimensional transport parallel tox, and let fT(x, t) denote the fraction of bed load particles that are tracers. The rate of deposition of tracers is

  • display math

which incorporates the idea that tracer particles arriving at position x at time t started their hops λ at many different times tτ. Again assuming that fλ,τis not heavy-tailed,(45) can be written as

  • display math

Further assume for illustration a steady, uniform entrainment rate E with uniform and steady values of the average hop distance inline image and the travel time inline image. Under these conditions,

  • display math

If all particles are tracers, fT = 1, DT = D and (47) reduces to the steady condition DE = 0. But otherwise, despite steady, uniform transport conditions (DE = 0), the time derivatives in (47)cannot necessarily be neglected given that the spatiotemporal evolution of a non-uniform ensemble of tracers is an unsteady problem. Namely, ifhdenotes a nominal steady, uniform thickness of bed-surface particles involved in transport, thenDT(xt) − E fT(xt) = cbh ∂fT/∂t and

  • display math

The unsteady term involving E accounts for the fact that tracers arriving at x at time t start from different positions upstream at different times, and, because they are entrained at different times, the fraction fT is changing at any specific starting position. For example, fT(x′, tτ1) at position x′ when tracer 1 is entrained at time tτ1 is different from fT(x′, tτ2) when tracer 2 is entrained at the same position x′ at time tτ2, although both tracer particles arrive at position x downstream at time t because particle 2 has a shorter travel time τ2 than does particle 1.

[55] Note that cbh/E = τR is the mean residence time of particles within the thickness h, so upon dividing (48) by E and rearranging,

  • display math

where inline image is a mean virtual velocity and inline imageis a virtual diffusivity. This has the form of the advection-diffusion equation obtained byGanti et al. [2010] assuming Fickian (normal) diffusion (their equation (11)), but differs in the explicit appearance of the mean residence time τR and the mean travel time inline image.

5. Averaged Quantities

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Geometrical Ingredients of the One-Dimensional Flux
  5. 3. Probabilistic Formulation of the One-Dimensional Flux
  6. 4. Exner Equation
  7. 5. Averaged Quantities
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Particle Volume Discharge
  11. Appendix B:: Probability Distribution of Activity in Ensemble
  12. Appendix C:: Means and Variances of Displacement Distances
  13. Appendix D:: Entrainment Form of Exner Equation
  14. Appendix E:: Relation Between Definitions of the Flux
  15. Acknowledgments
  16. References

[56] In the formulation above the width b is not explicitly specified, as the flux is considered a “per unit width” quantity. But this deserves further consideration, returning to the idea of an ensemble of configurations of particle positions and velocities. In section 3.1, the probability density fγ(γxyt) of the activity γ, and the joint probability density fγ,r,s(γrsxy, dt) of the activity γ and the displacements r and s, are associated with active particles within any elementary area dB. Focusing on displacements r, then likewise, the conditional density fr|γ(r|γxy, dt) is associated with active particles within dB = bdx. With small b, at any instant individual realizations drawn from the densities fγ(γxyt) and fr|γ(r|γxy, dt) at position y may be quite different from realizations from fγ(γxy + Δy, t) and fr|γ(r|γxy + Δy, dt) at position y + Δy. Upon lengthening b, the number of active particles within bdx generally increases, and the activity incorporates spatial variations that exist at scales smaller than b, so the probability density fγ obtained from the ensemble of configurations defined for bdx, centered about the same average, possesses a smaller variance (Figure 6). With sufficiently large b, the activity tends to a constant, the ensemble-averaged activity, independent ofy. Similarly, with increasing b, individual realizations (over bdx) of the conditional probability density of r approach the smooth function fr|γ(r|γx, dt), independent of y.

[57] In effect, a lengthening of b is equivalent to sampling a greater number of possible states of particle motions. However, b cannot be “too large” if the underlying forms of fγ and fr|γ change along y, say, in relation to changing near-bed turbulence structure in the mean. For the “right”b, a reasonable description of the instantaneous flux is given by (39), where the realization of the flux is inherently width-averaged overb. This also suggests that for equivalent macroscopic flow conditions, the magnitude of the fluctuations in the flux depend on the measurement width b. We now turn to time averaging of (26).

[58] Of interest is the behavior of (26) when averaged over different characteristic timescales, and the relation of this to ensemble averaging. Bed load transport rates vary over many timescales [e.g., see Gomez et al., 1989, Table 1]; and for nominally steady flow conditions, the measurement interval influences the calculated rate inasmuch as fluctuations in transport over durations shorter than the measurement interval are averaged in the calculation. To our knowledge no systematic, simultaneous measurements of particle activity and velocities are available (beyond those reported in Roseberry et al. [2012], which are of short duration), so we lack an empirical basis for evaluating time averaging of these quantities. Nonetheless, we may surmise the following in general terms.

[59] Consider first a planar bed with steady (uniform) macroscopic flow conditions. With increasing averaging period, one may assume that at any position x the bed experiences an increasing proportion of the set of possible (ensemble) configurations of particle activity and velocity, and with a sufficiently long averaging period the bed at x eventually experiences a fully representative set of possible configurations, in which case it is reasonable to assume that a (long) time average equals the ensemble average, as in (29). Moreover, for planar bed conditions the time-averaged product ofκ and γ is independent of position, so the diffusive term vanishes and the flux inline image. However, it must be noted that only in the limit where the activity γ approaches a constant (e.g., for sufficiently large b) does this become inline image.

[60] In contrast, consider three timescales associated with a homogeneous field of migrating bed forms. The first is a “short” turbulence timescale Tt, which we envision as being sufficiently long that, at any position x, the bed experiences a representative sample of possible turbulence fluctuations specific to where x is located within the bed form field, but short enough that the local bed form morphology does not change significantly. The second is an intermediate bed form timescale Tb, which we envision as being comparable to the period required for migration of bed forms (e.g., ripples or dunes) over one wavelength. (Note that Tt may be similar to Tbfor small bed forms.) The third is a “long” bed-form field timescaleTf, which we envision as being long enough that, at any position x, the bed experiences a fully representative sample of all possible positions (heights, proximity to crests, etc.) on bed forms within the migrating field.

[61] When (26) is averaged over the turbulence timescale Tt,

  • display math

which looks like the ensemble average (29), and highlights that the flux retains its dependence on time after averaging, as it varies over timescales longer than Tt. Moreover, whereas on a planar bed the time-averaged flux is constant (uniform) and the diffusive term vanishes, within a field of active bed forms the flux varies with positionx (otherwise bed forms would not form, grow or migrate), and the diffusive term in (50) may be nonzero due to persistent spatial variations in the averaged product inline imagearising from the influence of bed form topography on the near-bed turbulence [e.g.,McLean et al., 1994; Nelson et al., 1995; Jerolmack and Mohrig, 2005]. Indeed, a reformulation of the stability analysis of Smith [1970] to include the diffusive flux suggests that this flux is a sufficient, if not necessary, condition for selection of a preferred wavelength during initial ripple growth [Kahn and Furbish, 2010; Kahn, 2011].

[62] When (26) is averaged over the bed form timescale Tb, the result is the same as in (50)inasmuch as bed form geometries in a natural field are not identical. That is, in the idealization of identical one-dimensional migrating bed forms, positions over a single wavelength in principle sample all possible turbulence fluctuations, so a spatial average over one wavelength is the same as a time average over one periodTb. In this idealization the diffusive term therefore vanishes. But with naturally variable bed form geometries, the time-averaged diffusive term may be nonzero, and the time-averaged flux retains its dependence on time, particularly given persistent interactions among neighboring bed forms [Jerolmack and Mohrig, 2005]. In contrast, when (26) is averaged over the field timescale Tf, the diffusive term in principle vanishes and the time-averaged flux becomes a constant equal to inline image, independent of time and position. Also, like the planar bed case, only in the limit where the activity γ approaches a constant does this become overline inline image.

[63] For unsteady morphodynamic problems at the larger bar scale, analytical descriptions of the constituents of sediment transport and conservation normally treat these as continuous two-dimensional fields. An example is the class of models aimed at describing the instability of the coupled motions of water and sediment leading to the growth of bars [Callander, 1969; Engelund and Skovgaard, 1973; Parker, 1976; Fredsøe, 1978; Blondeaux and Seminara, 1985;Nelson and Smith, 1989; Seminara and Tubino, 1989; Furbish, 1998]. These models, which start with the Reynolds averaged momentum equations, in effect assume that local conditions coincide with ensemble-averaged conditions, where bed forms change sufficiently slowly that the bed locally experiences a representative sample of the ensemble of particle activity and motions, and the local flux varies smoothly withx, yposition and time, consistent with the quasi-steady approximation applied to turbulence conditions.

6. Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Geometrical Ingredients of the One-Dimensional Flux
  5. 3. Probabilistic Formulation of the One-Dimensional Flux
  6. 4. Exner Equation
  7. 5. Averaged Quantities
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Particle Volume Discharge
  11. Appendix B:: Probability Distribution of Activity in Ensemble
  12. Appendix C:: Means and Variances of Displacement Distances
  13. Appendix D:: Entrainment Form of Exner Equation
  14. Appendix E:: Relation Between Definitions of the Flux
  15. Acknowledgments
  16. References

[64] The surface-integral definition(1) of the instantaneous flux of bed load sediment, although impractical as a guide for direct measurements of the flux, nonetheless is precise. Thus any definition that appeals to averaged quantities of particle motions (e.g., the mean particle velocity and activity) to replace the detailed information embodied in (1) must be consistent with this definition.

[65] With quasi-steady bed and transport conditions, the definition(2)of the one-dimensional fluxqx parallel to x, involving the product of the average particle velocity Up and the particle activity γ, is consistent with (1)inasmuch as active particles are at any instant uniformly (albeit quasi-randomly) distributed over the streambed, and the flux-normal widthbover which the particle activity is calculated is sufficiently large to smooth over instantaneous small-scale variations in the activity alongb. In this situation the average velocity Up of particles over the streambed is equivalent to the average velocity inline image of N particles that intersect a vertical surface of width b at any position x, and the flux inline image.

[66] In contrast, in the presence of a particle activity gradient parallel to the mean particle motion, ∂γ/∂x, the average velocity inline image of particles intersecting a surface at position x may be different from the average velocity Up of all particles in the vicinity of x. This occurs because the surface is preferentially populated by faster moving particles and depleted of slower moving particles when ∂γ/∂x < 0, or vice versa when ∂γ/∂x > 0. Moreover, upon writing the particle velocity ui as the sum of the average Up and a fluctuating part ui, namely ui = Up + ui, the flux looks like inline image. The term involving the averaged fluctuating velocities is proportional to the activity gradient, ∂γ/∂x, and it therefore may be interpreted as a “diffusive” flux. Thus, whereas the deterministic surface-integral definition of the flux(1) does not distinguish between advection and diffusion, it can be formulated as consisting of these two parts.

[67] A formal rendering of the collective behavior of active sediment particles, wherein particle positions and motions are treated as stochastic quantities, yields a flux form of the Master equation, namely (14). Assuming that particle motions do not involve heavy-tailed behavior, the formulation reveals that the volumetric flux involves an advective part equal to the product of the particle activity and the ensemble-average particle velocity, and a diffusive part involving the gradient of the product of the particle activity and a diffusivity. A key point in the formulation is that the average particle velocity and the diffusivity are correlated. Thus, the diffusive part of the flux vanishes, inasmuch as fluctuating particle velocities vanish, in the absence of overall advective particle motions — which is entirely analogous to the relation between advection and mechanical dispersion in porous-media transport. The effect of the diffusive flux therefore is to add to or subtract from the advective flux in the presence of an activity gradient, as opposed to operating independently of the mean motion as in molecular diffusion.

[68] Central to the formulation is the probability density function of particle displacements r that occur during a small interval of time dt, conditional on the particle activity γ, namely fr|γ(r|γx, dt). A useful way to think about the source of this smooth probability density is to envision an ensemble of states consisting of all possible configurations of particle positions and velocities. This begins with selecting a streambed area B that is subjected to steady macroscopic flow conditions. This area must be sufficiently large that, during any small interval of time dt, the total number of active particles within B remains effectively steady. We then imagine, as Gibbs [1902] did, the set of states (particle positions and velocities) as being separate systems with the same bed and flow conditions at a fixed time, rather than as a time series of one system where the bed and flow conditions evolve. Then, for any elementary area dB, this ensemble yields the smooth density function fr|γ(r|γx, dt).

[69] Time-averaged descriptions of the flux involve averaged products of the particle activity, the particle velocity and the diffusivity. The significance of the covariance parts of these products depends on the averaging timescale in relation to characteristic timescales of near-bed turbulence and bed form evolution. The covariances likely contribute to fluctuations in transport rates [e.g.,Gomez et al., 1989] inasmuch as the particle activity and velocity, and the velocity and diffusivity, are strongly correlated. And, it may be that with naturally variable bed form geometries, the flux, when averaged over a timescale nominally long enough to accommodate fluctuations associated with bed form evolution, nonetheless retains its dependence on time over longer timescales in the presence of strong feedbacks between sediment transport and topography with interactions among neighboring bed forms [Jerolmack and Mohrig, 2005].

[70] The flux form of the Exner equation, (41), looks like a Fokker-Planck equation in which gradients in the particle diffusivity, like gradients in the particle activity, can in principle contribute to changes in bed elevation. However, the significance of this idea requires clarification, theoretical or experimental, aimed at showing how the diffusivity varies with the particle activity and velocity [Furbish et al., 2012b] in relation to bed topography. The entrainment form of the Exner equation, (43), similarly involves advective and diffusive terms, but also involves a time derivative term that represents a lag effect associated with the exchange of particles between the static and active states. In the case of a steady, uniform entrainment rate E with uniform and steady values of the mean hop distance inline image and the mean travel time inline image, then for one-dimensional transport the flux inline image, which is equivalent to the definition (3) provided by Einstein [1950]. For the unsteady case the flux qx cannot be expressed simply in terms of E, inline image and inline image due to lag effects. As applied to tracer particles under the conditions of a steady, uniform entrainment rate E, the virtual tracer particle velocity and the diffusivity contain the mean travel time inline image and the mean residence time τRwithin a nominal thickness of active particles (neglecting burial and re-emergence). Inasmuch as inline image, say, at low transport rates, the virtual velocity inline image and the virtual diffusivity inline image.

[71] The formulation of the sediment flux presented herein involves, for simplicity, a single particle diameter D, so definitions of the particle activity, velocity and diffusivity are specific to this situation. In generalizing to a mixture of particle sizes, covariances between particle activity, velocity and diffusivity become particularly important. For example, recall that in writing the last part of (5) as inline image, the covariance between Si and uican be neglected for equal-sized particles and this expression becomes(8), namely, inline image. Moreover, with uniform activity, inline image, so qx = γUp as in (2). But with a mixture of sizes, the covariance between Si and ui cannot be neglected inasmuch as some particle sizes preferentially move faster than other sizes. In this case inline image, where γp = Si/b is like an individual particle activity. Or, letting j denote the jth size fraction, we may write inline image to denote the fractional flux [Wilcock and McArdell, 1993; Wilcock, 1997a, 1997b], where γj is the activity of the jth fraction. The total flux is then the sum over all j sizes, where each inline image has advective and diffusive parts.

7. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Geometrical Ingredients of the One-Dimensional Flux
  5. 3. Probabilistic Formulation of the One-Dimensional Flux
  6. 4. Exner Equation
  7. 5. Averaged Quantities
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Particle Volume Discharge
  11. Appendix B:: Probability Distribution of Activity in Ensemble
  12. Appendix C:: Means and Variances of Displacement Distances
  13. Appendix D:: Entrainment Form of Exner Equation
  14. Appendix E:: Relation Between Definitions of the Flux
  15. Acknowledgments
  16. References

[72] Under quasi-steady, uniform transport conditions, the one-dimensional flux is equal to the product of the particle activity and the average particle velocity inasmuch as active particles are at any instant uniformly (albeit quasi-randomly) distributed, and the flux-normal width over which the activity is calculated is large enough to smooth over instantaneous small-scale variations in the activity normal to the flux. The analysis yields the entrainment form of the flux, equal to the product of the volumetric entrainment rate and the average particle hop distance, in the special situation of quasi-steady, uniform transport conditions. For unsteady transport, the entrainment form of the flux cannot be expressed simply in terms of the entrainment rate, the average hop distance and the average travel time.

[73] A description of particle motions within the framework provided by the Master equation reveals that the flux generally consists of an advective part involving the product of the particle activity and the average particle velocity, and a diffusive part involving the gradient of the product of the particle activity and a diffusivity. The average particle velocity and the particle diffusivity are by definition highly correlated. The diffusive part of the flux vanishes, inasmuch as fluctuating particle velocities vanish, in the absence of advective particle motions. Covariances in the particle activity and velocity, and the activity and diffusivity, contribute to fluctuations in the volumetric flux.

[74] We may envision an ensemble of particle states consisting of all possible configurations of particle positions and velocities compatible with steady macroscopic flow conditions. This description, similar in concept to the definition of an ensemble of particle states in classic statistical mechanics, provides a framework for describing how the probability distribution of particle activity, and therefore the volumetric flux, varies with sampling area.

[75] The flux form of the Exner equation looks like a Fokker-Planck equation with advective and diffusive terms, highlighting the possible effects of particle diffusion on bed form growth under nonuniform transport conditions. The entrainment form of the Exner equation similarly involves advective and diffusive terms, and the unsteady term involves a lag effect associated with the exchange of particles between the static and active states. Under conditions of steady, uniform entrainment, the virtual velocity and diffusivity of tracer particles are inversely proportional to the sum of the average travel time and the average residence time of particles within the active layer thickness.

Appendix A:: Particle Volume Discharge

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Geometrical Ingredients of the One-Dimensional Flux
  5. 3. Probabilistic Formulation of the One-Dimensional Flux
  6. 4. Exner Equation
  7. 5. Averaged Quantities
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Particle Volume Discharge
  11. Appendix B:: Probability Distribution of Activity in Ensemble
  12. Appendix C:: Means and Variances of Displacement Distances
  13. Appendix D:: Entrainment Form of Exner Equation
  14. Appendix E:: Relation Between Definitions of the Flux
  15. Acknowledgments
  16. References

[76] Consider a particle of diameter D [L] that is moving with a positive velocity parallel to x through a surface A positioned at x = 0 (Figure 2). Let ξi [L] denote the position of the nose of the particle relative to x = 0, and let Vi(ξi) [L3] denote the volume of the particle that is to the right of x = 0 as a function of ξi. Also, let ε [L] denote a small distance measured from the nose of the particle, where 0 ≤ εD. Then, at time t [t] the volume Vi+ε(t) [L3] of the particle that is simultaneously to the left of ξiε and to the right of x = 0 for 0 ≤ ξiD is

  • display math

where H(ξi) is the Heaviside step function defined by H(ξi) = 0 for ξi < 0 and H(ξi) = 1 for ξi≥ 0. In this expression the Heaviside functions serve as off-on switches over thex domain. The second term on the right side of (A1) insures that Vi+ε(t) is piecewise continuous without a jump at ξi = D, and the last two terms, as will be seen momentarily, insure that the derivative of (A1) is piecewise continuous at ξi = 0 when ε [RIGHTWARDS ARROW] 0.

[77] With ξi = ξi(t), the rate of change in Vi+ε is

  • display math

where δ(ξi) = dH/dξi is the Dirac delta function. The derivative ∂Vi/∂ξi = Si(ξi) [L2] is like a hypsometric function of the particle, and is equal to its cross-sectional area on the surfaceA at x = 0. The derivative dξi/dt = ui [L t−1] is its velocity parallel to x. The terms involving the Dirac function nominally represent instantaneous changes in the rates of gain and loss of volume when the particle arrives at and leaves A.Because these terms are non-zero only atξi = 0 or ξi = D, the third and fourth terms, the fifth and seventh terms, and the sixth and eighth terms on the right side of (A2), respectively, cancel each other. Then, upon letting ε [RIGHTWARDS ARROW] 0, the second term on the right side of (A2) vanishes to give the particle volume discharge Qi+(t) across A in the positive x direction, namely

  • display math

By symmetry the particle volume discharge Qi(t) in the negative x direction has the same form as the right side of (A3), namely Qi(t) = dViε/dt = Si(ξi)uiH(ξi)[1 − H(ξi − D)]. Moreover, at this point the product involving the Heaviside functions is redundant, as the surface area Si(ξi) is a piecewise continuous function that is finite over 0 ≤ ξiD with Si(ξi < 0) = Si(ξi > D) = 0. Thus, in general the particle volume discharge Qi(t) across A is

  • display math

which is (4) in the main text.

Appendix B:: Probability Distribution of Activity in Ensemble

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Geometrical Ingredients of the One-Dimensional Flux
  5. 3. Probabilistic Formulation of the One-Dimensional Flux
  6. 4. Exner Equation
  7. 5. Averaged Quantities
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Particle Volume Discharge
  11. Appendix B:: Probability Distribution of Activity in Ensemble
  12. Appendix C:: Means and Variances of Displacement Distances
  13. Appendix D:: Entrainment Form of Exner Equation
  14. Appendix E:: Relation Between Definitions of the Flux
  15. Acknowledgments
  16. References

[78] For a streambed area B, let Na denote the total number of active particles in each possible configuration of the ensemble. If m denotes the number of partitions of B of area dB = B/m, then the total number of configurations involving Na particles distributed among m partitions is

  • display math

Using the language of statistical mechanics, we may refer to each of these Ne configurations as a “macrostate.” In turn, if n1n2n3, …, nm denote the number of particles in each of the mpartitions of an individual macrostate, then using Maxwell-Boltzmann counting there arene ways in which the Na particles may be rearranged amongst the m partitions, and we may refer to each of these ne arrangements as a “microstate.” The number of microstates in a given macrostate is

  • display math

and the total number of microstates Me over Ne macrostates is

  • display math

As a point of reference, for Na = 10 particles distributed among m = 2 partitions, there are a total of Ne = 11 macrostates and a total of Me = 1,024 microstates. For Na = 10 and m = 5, there are Ne = 1,001 macrostates and Me = 9,765,625 microstates. And, for Na = 10 and m = 10, there are Ne = 92,378 macrostates and Me = 1 × 1010 microstates. For Na = 5 and m = 10, there are 2,002 macrostates and 100,000 microstates. And, for Na = 5 and m = 20, there are 42,504 macrostates and 3,200,000 microstates.

[79] If ndB denotes the number of particles within dB, then the proportion Pn(ndB) of Me microstates having ndB particles within dB — that is, the probability distribution of ndB — is given by the binomial distribution assuming each microstate is equally probable [Roseberry et al., 2012]. With large m relative to Na, there is an increasing number of ways to partition Na particles into m − 1 (or m − 2, etc.) areas dB, so the likelihood of finding small numbers ndB within any dB increases, and the distribution Pn(ndB) is exponential-like, albeit decaying with increasingndB faster than an exponential function. With decreasing m relative to Na, there are fewer ways to partition Na particles into an area dB having small ndB, and the distribution Pn(ndB) takes an asymmetric form with finite mode. For small m relative to Na, the distribution Pn(ndB) becomes Gaussian-like. As elaborated inRoseberry et al. [2012], this distribution forms the basis of the null hypothesis of spatial randomness in the positions of active particles. We show that near-bed turbulence leads to decided patchiness in particle positions, and that fluctuations in activity, and therefore in transport rates, are systematically related to the sampling area.

Appendix C:: Means and Variances of Displacement Distances

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Geometrical Ingredients of the One-Dimensional Flux
  5. 3. Probabilistic Formulation of the One-Dimensional Flux
  6. 4. Exner Equation
  7. 5. Averaged Quantities
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Particle Volume Discharge
  11. Appendix B:: Probability Distribution of Activity in Ensemble
  12. Appendix C:: Means and Variances of Displacement Distances
  13. Appendix D:: Entrainment Form of Exner Equation
  14. Appendix E:: Relation Between Definitions of the Flux
  15. Acknowledgments
  16. References

[80] The definitions (17) through (20) are well known. Nonetheless, because these definitions are not necessarily familiar, for completeness we show how they are obtained. For simplicity we omit the conditional notation indicating a dependence on the activity γ.

[81] First note that by the product rule,

  • display math

With Rr(r) = 1 − Fr(r), substitution leads to

  • display math

Integrating this from r = 0 to r = ∞,

  • display math

Evaluating the last integral then leads to

  • display math

insofar as the limit of rRr(r) as r [RIGHTWARDS ARROW] ∞ is equal to zero. This is guaranteed if Rr(r) decays at least as fast as a negative exponential function, in which case the product rRr(r) looks like r/er, whose limit is zero as r [RIGHTWARDS ARROW] ∞. Indeed, the absence of this condition being satisfied implies that fr(r) is a heavy-tailed distribution without finite mean. In turn,

  • display math

or

  • display math

Integrating this from r = 0 to r = ∞,

  • display math

insofar as the limit of r2Rr(r) as r [RIGHTWARDS ARROW] ∞ is equal to zero. Again, this is guaranteed if Rr(r) decays at least as fast as a negative exponential function, in which case the product r2Rr(r) looks like r2/er, whose limit is zero as r [RIGHTWARDS ARROW] ∞. The absence of this condition being satisfied implies that fr(r) is a heavy-tailed distribution without finite variance. A similar development involvingfl(l) and Rl(l) leads to comparable expressions for μl and σl2.

Appendix D:: Entrainment Form of Exner Equation

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Geometrical Ingredients of the One-Dimensional Flux
  5. 3. Probabilistic Formulation of the One-Dimensional Flux
  6. 4. Exner Equation
  7. 5. Averaged Quantities
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Particle Volume Discharge
  11. Appendix B:: Probability Distribution of Activity in Ensemble
  12. Appendix C:: Means and Variances of Displacement Distances
  13. Appendix D:: Entrainment Form of Exner Equation
  14. Appendix E:: Relation Between Definitions of the Flux
  15. Acknowledgments
  16. References

[82] We expand the integrand in (42) as a Taylor series to first order about t and to second order about x and y, namely

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Substituting (D1) into (42), rearranging, and momentarily letting dΛ = dλ dψ dτ, leads to

  • display math

where the unwritten limits of integration match those of (42). The triple integral of fλ,ψ,τ in the first term on the right side of (D2) by definition equals unity. Then, because the order of integration does not matter, selectively integrating to obtain the marginal distributions fλ, fψ and fτ, and the joint distribution fλ,ψ,

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The first three integrals in (D3) equal the mean hop distances inline image and inline image and the mean travel time inline image. The fourth and sixth integrals equal the second moments inline image and inline image. The double integral equal the averaged product inline image. With these definitions, (E3) looks like (43).

Appendix E:: Relation Between Definitions of the Flux

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Geometrical Ingredients of the One-Dimensional Flux
  5. 3. Probabilistic Formulation of the One-Dimensional Flux
  6. 4. Exner Equation
  7. 5. Averaged Quantities
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Particle Volume Discharge
  11. Appendix B:: Probability Distribution of Activity in Ensemble
  12. Appendix C:: Means and Variances of Displacement Distances
  13. Appendix D:: Entrainment Form of Exner Equation
  14. Appendix E:: Relation Between Definitions of the Flux
  15. Acknowledgments
  16. References

[83] As described in section 3.1, consider a planar streambed area Blarge enough to sample steady, homogeneous near-bed conditions of turbulence and transport. At any instant the number of active particles is approximately constant. That is, the rate of disentrainment withinB equals the rate of entrainment, and the rate at which particles leave B across its boundaries equals the rate at which particles enter B across its boundaries. Imagine recording particle motions within B for an interval of time Ts [t] [e.g., Lajeunesse et al., 2010; Roseberry et al., 2012]. For Ts much longer than the mean particle travel time, particle motions during Ts adequately represent the joint probability density fλ,τ(λ, τ) of hop distances λ and travel times τ without bias due to censorship of motions at times t = 0 and t = Ts [Furbish et al., 1990]. The marginal distributions fλ(λ) and fτ(τ) possess means inline image and inline image. And, at any instant the ensemble average particle velocity is inline image.

[84] The average velocity of the ith (individual) particle with travel time τi is

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

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Thus, contrary to the assertion of Lajeunesse et al. [2010], the ensemble average hop distance inline image indeed is equal to the product of the ensemble averaged velocity ū and the mean travel time inline image [Furbish et al., 2012b].

[85] The quasi-steady (“equilibrium”) volumetric fluxqx on a planar bed, when written as an equivalence between its “flux” form and its “entrainment” form, is

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where γ is the particle activity (the volume of active particles in motion per unit streambed area) and E is the entrainment rate (the volumetric rate at which particles become active per unit streambed area). So evidently,

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That is, under steady conditions the activity inline image, or

  • display math

where, now, inline image has the simple interpretation of being the mean residence time of particles within the nominal volume γB. Thus, (E3), (E4) and (E5) show the relation between the two forms of the flux qx.

Notation
A

surface area [L2].

b

width normal to flux [L].

B

streambed area [L2].

c

concentration.

cb

volumetric particle concentration of bed.

differential equal to dλ dψ dτ [L2 t].

D

particle diameter [L]; volumetric particle deposition rate per unit streambed area [L t−1].

DT

volumetric tracer deposition rate per unit streambed area [L t−1].

E

volumetric particle entrainment per unit streambed area [L t−1].

fr, fl

probability density functions of displacements r and l [L−1].

fr|γ, fl|γ

conditional probability density functions of displacements r and l [L−1].

fγ

probability density function of activity γ [L−1].

fγ,r

joint probability density function of γ and r [L−2].

fγ,r,s

joint probability density function of γ, r and s [L−3].

Fγ

cumulative probability distribution function of γ.

fλ

probability density function of λ [L−1].

fτ

probability density function of τ [t−1].

fλ,τ

joint probability density function of λ and τ [L−1 t−1].

fλ|τ

conditional probability density function of λ [L−1].

fλ,ψ,τ

joint probability density function of λ, ψ and τ [L−2 t−1].

fλ,ψ

joint probability density function of λ and ψ [L−2].

Fr, Fl

cumulative probability distribution functions of r and l.

Fr|γ, Fl|γ

cumulative conditional probability distribution functions of r and l.

fT

fraction of bed load particles that are tracers.

h

effective thickness of active bed load particles [L].

H

Heaviside step function.

i

designation of the ith of N particles.

j

designation of the jth particle-size fraction.

l

particle displacement in negative x direction [L].

L

length scale [L].

m

number of partitions of B.

MA

mask projected on surface A, equal to 1 − H(up)H(−up).

Me

total number of microstates in ensemble.

n

unit vector normal to A.

n

proportion of N particles per unit length [L−1].

ndB

number of particles within dB.

ne

number of microstates in a macrostate.

ni

number of particles in the ith partition of a macrostate.

nx, nz

number of particles per unit length parallel to x and to z [L−1].

nxy

number of particles per unit area [L−2].

N

number of particles intersecting surface.

Na

total number of active particles in each configuration (macrostate) of ensemble.

Ne

total number of configurations (macrostates) in ensemble.

Ns

number of particle motions.

p

probability that a particle moves in the positive x direction.

Pn

proportion of Ne configurations having ndB particles within dB.

Pλ, Pτ

proportions defined by Pλ = fλ/(1 − Fλ) [L−1] and Pτ = fτ/(1 − Fτ) [t−1].

q

probability that a particle moves in the negative x direction.

q

volumetric particle flux [L t−1] and [L2 t−1].

qA

volumetric particle flux across surface A [L2 t−1].

qx, qy

volumetric particle flux components parallel to x and y [L2 t−1].

qxj

volumetric particle flux of jth size fraction [L2 t−1].

Q

volumetric particle discharge [L3 t−1].

Qx

volumetric particle discharge parallel to x [L3 t−1].

Qi

volume discharge of ith particle [L3 t−1].

Qi+, Qi

volume discharge of ith particle in positive and negative x direction [L3 t−1].

r

particle displacement parallel to x [L].

rm

scale factor in Pareto distribution.

Rr, Rl

functions defined by Rr = 1 − Fr and Rl = 1 − Fl.

Rλ, Rτ

functions defined by Rλ = 1 − Fλ and Rτ = 1 − Fτ.

Rr|γ, Rl|γ

functions defined by Rr|γ = 1 − Fr|γ and Rl|γ = 1 − Fl|γ.

s

particle displacement parallel to y [L].

S

cross-sectional area of particles intersectingA [L2]; source term, inline image [L t−1].

Si

cross-sectional area ofith particle on surface A [L2].

t

time [t].

Tb

bed form timescale [t].

Tf

bed-form field timescale [t].

Ts

sampling interval [t].

Tt

turbulence timescale [t].

u, v

average particle velocity components parallel to x and y [L t−1].

ui

velocity component parallel to x of ith particle [L t−1].

up

particle velocity component normal to surface A [L t−1].

up, vp

particle velocity components parallel to x and y [L t−1].

up

particle velocity field at surface A [L t−1].

ur, ul

mean velocities associated with displacements r and l [L t−1].

Up

mean particle velocity [L t−1].

UR

mean virtual particle velocity, inline image [L t−1].

V

volume of particles [L3].

Vi

volume of ith particle to right of surface A [L3].

V+, V

volume of particles to right and left of surface A [L3].

Vi+εViε

volume of ith particle to right and left of surface A [L3].

Vp

volume of particle [L3].

w

variable of integration associated with displacements r and l [L].

x, y

Cartesian coordinates in streamwise and cross-stream directions [L].

yS

specific position along y [L].

α

shape factor in Weibull and Pareto distributions.

γ

particle activity [L].

γp

individual particle activity defined by Si/b [L].

γj

particle activity of jth size fraction [L].

δ

Dirac delta function [L−1].

Δ

increment.

ε

small distance measured from front of particle [L].

η

local elevation of streambed surface [L].

κ, κ

diffusivity, diffusivity tensor [L2 t−1].

κr, κl

diffusivities associated with displacements r and l [L2 t−1].

κxx, κyy, κxy

elements of diffusivity κ [L2 t−1].

KR

virtual diffusivity, inline image [L2 t−1].

λ

particle hop distance parallel to x [L].

μr, μl

first moments (means) of particle displacements r and l during dt [L].

μλ, μτ

mean hop distance [L] and mean travel time [t].

ξi

distance of ith particle to right of surface at x = 0 [L].

σr2, σl2

second moments of particle displacements r and l during dt [L2].

σu2

variance of particle velocities u [L2 t−2].

τ

interval of time; particle travel time [t].

τR

mean residence time of particles in thickness h [t].

ψ

particle hop distance parallel to y [L].

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Geometrical Ingredients of the One-Dimensional Flux
  5. 3. Probabilistic Formulation of the One-Dimensional Flux
  6. 4. Exner Equation
  7. 5. Averaged Quantities
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Particle Volume Discharge
  11. Appendix B:: Probability Distribution of Activity in Ensemble
  12. Appendix C:: Means and Variances of Displacement Distances
  13. Appendix D:: Entrainment Form of Exner Equation
  14. Appendix E:: Relation Between Definitions of the Flux
  15. Acknowledgments
  16. References

[86] We appreciate comments by the Editor (Alex Densmore) and Associate Editor (Emmanuel Gabet) and reviews provided by Efi Foufoula-Georgiou and Chris Paola. We acknowledge support by the National Science Foundation (EAR-0744934), and appreciate Amelia Furbish's insistence that we pay close attention to L'Hôpital's rule.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Geometrical Ingredients of the One-Dimensional Flux
  5. 3. Probabilistic Formulation of the One-Dimensional Flux
  6. 4. Exner Equation
  7. 5. Averaged Quantities
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A:: Particle Volume Discharge
  11. Appendix B:: Probability Distribution of Activity in Ensemble
  12. Appendix C:: Means and Variances of Displacement Distances
  13. Appendix D:: Entrainment Form of Exner Equation
  14. Appendix E:: Relation Between Definitions of the Flux
  15. Acknowledgments
  16. References