1

A probabilistic description of the bed loadsediment flux: 1. Theory

David Jon Furbish, Peter K. Haff, John C. Roseberry, and Mark W.1 2 1

Schmeeckle3

________ Department of Earth and Environmental Sciences and Department of Civil and Environmental1

Engineering, Vanderbilt University, Nashville, Tennessee, USA. Division of Earth and Ocean Sciences, Nicholas School of the Environment, Duke University,2

Durham, North Carolina, USA. School of Geographical Sciences, Arizona State University, Tempe, Arizona, USA.3

Abstract. We provide a probabilistic definition of the bed load sediment flux. In treating particle positions and motions as stochastic quantities, a flux form ofthe Master Equation reveals that the volumetric flux involves an advective partequal to the product of an average particle velocity and the particle activity (thesolid volume of particles in motion per unit streambed area), and a diffusive partinvolving the gradient of the product of the particle activity and a diffusivity thatarises from the second moment of the probability density function of particledisplacements. Gradients in the activity, instantaneous or time-averaged,therefore effect a particle flux. Time-averaged descriptions of the flux involveaveraged products of the particle activity, the particle velocity and the diffusivity,whose significance depends on the averaging timescale. The flux form of theExner equation looks like a Fokker-Planck equation. The entrainment form of theExner equation similarly involves advective and diffusive terms, but because it isbased on the joint probability density function of particle hop distances andassociated travel times, this form involves a time derivative term that represents alag effect associated with the exchange of particles between the static and activestates. The formulation highlights that the probability distribution of particledisplacements figures prominently in describing particle motions across a rangeof scales, notably bearing on the possibility of anomalous versus Fickian diffusivebehavior. The formulation is consistent with experimental measurements andsimulations of particle motions reported in companion papers.

1. IntroductionThe 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 andthe evolution of alluvial channels. Translating this definition of the flux into conceptually simplequantities that accurately characterize the collective motions of particles, however, is not necessarilystraightforward, and quantitative definitions of the flux have several forms. We note at the outset

Athat, when viewed at the particle scale, the instantaneous solid volume flux q [L t ] across a2 -1

surface A [L ] is precisely defined as the surface integral of surface-normal velocities of the solid2

2

fraction, namely

pwhere u [L t ] is the discontinuous particle velocity field viewed at the surface A, n is the unit-1

vector normal to A, and b [L] is the width of A (Figure 1). As a point of reference, because thevolumetric flux formally is a volume per unit area per unit time, the “flux” defined by (1) actuallyis a vertically integrated flux. This precise definition, however, is impractical. Except possiblyusing 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) isvirtually impossible to measure, and we are far from possessing a theory of sediment transport that

pdescribes the velocity field u as it responds to near-bed turbulence [Parker et al., 2003].Conventional descriptions of the flux therefore instead appeal to measures of collective particlebehavior, specifically averaged quantities such as the average particle velocity and concentration,

pto replace the detailed information contained in the particle velocity field u at the surface A.With equilibrium (i.e. quasi-steady and uniform) bed and transport conditions, for example, the

psediment flux normally is defined in “flux form” as the product of a mean particle velocity U [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

xbed load particle activity ( [L]. That is, for one-dimensional transport in the x direction the flux q[L t ] is2 -1

pwith the caveat that U and ( represent macroscopic quantities averaged over stochastic fluctuations[Wong et al., 2007]. Note that this is like the definition of advection associated with a continuousmedium. As elaborated below, to describe the sediment flux as the product of a mean velocity anda 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, whetherinvolving ensemble, spatial or temporal averaging [Coleman and Nikora, 2009], matter to thephysical interpretation as well as the form of the definition of the flux. Ancey [2010] notes in hisreview of several definitions of the flux that it remains unclear how the flux is actually related to themean particle velocity and the particle concentration.

Another important definition of the bed load sediment flux is the “entrainment form” of thisquantity, first introduced by Einstein [1950] and recently elaborated by Wilcock [1997], Parker etal. [2001], Seminara et al. [2002], Wong et al. [2007], Ganti et al. [2010] and others. By thisdefinition, with quasi-steady bed and transport conditions the flux is equal to the product of thevolumetric rate of particle entrainment per unit streambed area, E [L t ], and the mean particle hop-1

distance, [L], measured start to stop. That is,

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This is essentially a statement of conservation of particle volume where, assuming spatially uniformtransport, rates of entrainment and deposition are steady and everywhere balanced. The value of thisdefinition is highlighted in treating tracer particles [Ganti et al., 2010], notably involving exchangesbetween 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 , translate to

unsteady and nonuniform conditions [Lajeunesse et al., 2010], and the extent to which this and otherdefinitions overlap or match (2) [Ancey, 2010]. On this point we note that any formulation of theflux must be consistent with (1).

At any instant the solid volume of bed load particles in motion per unit area of streambed, theparticle activity (, can vary spatially due to short-lived near-bed turbulence excursions as well aslonger-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 witheach other so that, at any instant within a given small area, some particles move faster and somemove slower than the average within the area, and fluctuations in velocity are of the same order asthe average velocity. Moreover, a hallmark of bed load particles is their propensity to alternatebetween states of motion and rest over a large range of timescales. These attributes mean that, inrelation to the definition (2) above, the bed load particle flux involves an advective part, as isnormally assumed, but more generally it also involves a diffusive part associated with variations inparticle 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 hopdistances may be considered the marginal distribution of a joint probability density function ofparticle hop distances and associated travel times [Lajeunesse et al., 2010], a more general form ofthis definition similarly involves a diffusive part as well as a time derivative term that represents alag effect associated with the exchange of particles between the static and active states.

The purpose of this contribution is to clarify the points above, namely, how variations in particle

x yactivity and velocity influence the volumetric bed load flux, q = iq + jq [L t ], with components2 -1

x yq and q parallel to the coordinates x [L] and y [L], selected here to coincide with the averagesurface of the sediment-water interface viewed over a length scale much larger than that of sedimentbedforms such as ripples or dunes. Our analysis involves a probabilistic formulation whereinparticle positions and motions are treated as stochastic quantities, leading to a kinematic descriptionof q that illustrates how and why it involves both advective and diffusive terms, borrowing keyelements from closely related formulations [Furbish et al., 2009a, 2009b; Furbish and Haff, 2010].

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 chaving the form Mc/Mt = -L@q [Ancey, 2010], and then by inspection extract from this statement akinematic description of the flux q [e.g. Furbish et al., 2009a], therein revealing that it has bothadvective and diffusive parts. More challenging, however, is to formulate a definition of q directlyfrom a description of particle motions as Einstein [1905] did for Brownian motions. This isparticularly desirable inasmuch as a direct formulation more fully clarifies the geometrical andkinematic ingredients of the flux, including its relation to such quantities as particle hop distancesand velocities [e.g. Drake et al., 1988; Wilcock, 1997b; Wong et al., 2007; Ancey, 2010], andvariations in particle velocity and activity.

xIn section 2 we formulate a qualitative version of the one-dimensional flux q , with the purposeof clarifying key geometrical and kinematic ingredients in the problem, notably particle size, shapeand velocity, and spatial variations in particle concentration. We show how the definitions (1) and

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(2) are related. This provides the basis for illustrating that, in appealing to averaged particlequantities (specifically the mean particle velocity and concentration) to replace the detailed

pinformation contained in the discontinuous particle velocity field u at the surface A, the resultingdescription of the flux must in general involve both advective and diffusive parts. In section 3 weprovide a more formal, probabilistic description of the one-dimensional flux, and describe theimplications of different definitions of the probability distribution of particle displacements versushop distances. In the final part of this section we generalize to the two-dimensional case. In section4 we show the flux form of the Exner equation to illustrate how it is like a Fokker-Planck equation,and for comparison we obtain the entrainment form of the Exner equation to illustrate how this forminvolves 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 activestates. This result has implications for the use of the related entrainment formulation of conservationof tracer particles. In section 5 we elaborate how ensemble, spatial and temporal averaging matterin defining the flux, and we consider time averaging of the flux to suggest how persistent spatialvariations in particle activity associated with bed forms influence the flux. For simplicitythroughout, we consider transport of a single particle size, then briefly comment on the problem ofgeneralizing the formulation to mixtures of sizes in section 6.

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 loadflux, 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 motionsacross a range of scales. This is particularly significant in view of a growing interest in thepossibility of non-Fickian behavior in the transport of sediment and associated materials [e.g. Nikoraet al., 2002; Schumer et al., 2009; Foufoula-Georgiou and Stark, 2010; Bradley et al., 2010; Gantiet al., 2010; Voller and Paola, 2010; Hill et al., 2010; Martin et al., 2012], and in relation toconnecting probabilistic descriptions of particle motions with treatments of fluid motion.

In companion papers [Roseberry et al., 2012; Furbish et al., 2012a, 2012b] we present detailedmeasurements of bed load particle motions obtained from high-speed imaging in laboratory flumeexperiments. These measurements support key elements of the formulation described here. [Note:Although not yet accepted for publication, the companion papers Roseberry et al. [2012] andFurbish et al. [2012a, 2012b] are cited with a 2012 date for simplicity of reference.]

2. Geometrical Ingredients of the One-Dimensional Flux2.1. The Surface-Integral Flux

As an important reference point, here we present a discrete version of (1) to reveal details ofparticle shape and motion that figure into this deterministic definition of the flux. This provides thebasis for illustrating that, in appealing to averaged particle quantities (specifically the mean particlevelocity and concentration) to replace the detailed information embodied in (1), the resultingdescription of the flux must in general involve both advective and diffusive parts. We start with arendering of the geometry and motion of a single particle.

Consider a particle of diameter D [L] that is moving parallel to x through a surface A positioned

iat x = 0 (Figure 2). Let > [L] denote the position of the nose of the particle relative to x = 0, and let

i i iV (> ) [L ] denote the volume of the particle that is to the right of x = 0 as a function of > . The3

iparticle volume discharge Q (t) across A is (Appendix A)

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i i i iwhere S (> ) = MV /M> [L ] is like a hypsometric function of the particle, equal to its cross-sectional2

i iarea on the surface A at x = 0, and u = d> /dt [L t ] is its velocity parallel to x.-1

Consider, then, a cloud of equal-sized particles which are moving with varying velocities parallelto x toward and through a surface A of width b positioned at x = 0 (Figure 3). At time t a number

iN(t) of particles intersects A. If > now denotes the distance that the nose of the ith particle is relative

xto x = 0, then the instantaneous volumetric flux q across A is

p pThis is a discrete version of (1). Namely, if the particle velocity (field) parallel to x is u = u @n, and

p p p p pif H(u ) is the Heaviside step function defined by H(u ) = 0 for u < 0 and H(u ) = 1 for u $ 0, then

A p p A p AM (t) = 1 - H(u )H(-u ) denotes a “mask” projected onto A such that M = 1 where u � 0 and M =0 elsewhere [Furbish et al., 2009b], whence

and

A xwhich shows the relation between (1) and (5) with q = q . Note that N(t) is a stepped function oftime as particles intersect and lose contact with A. At any instant, therefore, the derivative dN/dtstrictly is either zero or undefined. Nonetheless, for sufficiently large N and rapidity of particlesintersecting and losing contact with A, one can envision that N(t) begins to appear as a “smooth”

x xfunction of time where brief fluctuations in q become small relative to the magnitude of q .

xLetting an overbar denote an average over N particles, the last part of (5) may be written as q

i i= (1/b)N . For equal-sized particles, moreover, it is reasonable to assume that S and u are

uncorrelated, as there is no reason to suspect that, at any instant, particles intersecting A with large

i i(or small) cross-sectional area S are any more (or less) likely to possess large (or small) velocity u .In this case,

The product N = S [L ] is the cross-sectional area of particles intersecting A, and the ratio S/b2

= ( [L L = L L ] is equivalent to the particle activity, the volume of active particles per unit2 -1 3 -2

streambed area. Specifically, this is a local “line” averaged activity. Because Sdx is equal to thevolume of active particles within a small interval dx, Sdx/bdx = S/b = ( is the volume of active

pparticles within the small area bdx. Then, if it is assumed that is equal to the average velocity U

p x pof all particles in the cloud in the vicinity of A, that is U = , one may conclude that q = (U ,

which is the definition (2) of the flux normally assumed for quasi-steady bed and transportconditions [e.g. Bridge and Dominic, 1984; Wiberg and Smith, 1989; Seminara et al., 2002; Parkeret al., 2003; Francalanci and Solari, 2007; Wong et al., 2007; Lajeunesse et al., 2010]. Twocaveats, however, must accompany this assessment of averages.

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First, envision a uniform cloud of equal-sized particles moving with varying velocities parallelto x toward and through a set of surfaces A located at various positions along x. By “uniform” we

xmean the following. For a specified width b, let n (x, t) denote the number of particles per unit

xdistance parallel to x, such that n (x, t)dx is the number of particles whose noses are located within

xany small interval dx. Then, for a sufficiently large width b, assume that n (x, t) varies negligibly

with x. At any instant the number of particles N and the corresponding particle area S = N

intersecting each surface is the same, although the detailed configuration of S varies from surface

pto surface. Let U 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 of particles intersecting a surface

A. Because the cloud is uniform, each surface A “samples” at any instant the full distribution of

ppossible velocities (for sufficiently large width b), in which case = U for all surfaces. This is the

situation for which Ancey [2010] notes that an ensemble-like average over A, giving , is equivalent

pto a “volume” average over the particle cloud, giving U . In contrast, envision a cloud of particles

p xwith average velocity U whose concentration n (x, t) at some instant decreases with increasing

distance x. Now, both the number of particles N and the particle area S = N intersecting each

surface decrease with increasing x. Moreover, in this case the surface and volume averages are not

pequivalent, and > U . Here is why.

i i i pLet a prime denote a fluctuation about an average. Then, at any instant S = + S N and u = U

i x i i+ u N. In turn, q = (1/b)N( ). Consider a plot of u N versus S N at an

instant (Figure 4), which provides a perspective as viewed by an observer moving with the average

pvelocity U , 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

iarea S N increases for particles that are beginning to cross A, and some points move to the left as the

icross-sectional area S N decreases for particles that have mostly crossed A. The rate of motion of the

ipoints to the right and left is proportional to the magnitude of the particle velocity u N, so motion is

i ifaster near the top and bottom and slower near the middle of the plot. Points at u N = 0 along the S Naxis do not move during dt. Some points vanish as particles leave A, and new points appear asparticles arrive at and initially intersect A. Points arrive at the far left of the second and thirdquadrants where the small areas of intersection of arriving particles are less than the averageintersection area, and move to the far right of the first and fourth quadrants as their fat middlesexceed the average intersection area, and then move back to the far left of the second and thirdquadrants because the intersection area of their exiting tails is less than the average intersection area.

In the case of the uniform cloud of particles described above, the number of points and their

pscatter is similar across all surfaces A and on each surface over time, and = U with

x p = 0 (Figure 4a), so that q = (1/b)N , = ( = (U 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 theimmediate right of A. The likelihood that a particle to the left or right of A will intersect A during

ia small interval of time dt, for a given magnitude of the velocity u N, increases with its proximity to

iA, and, for a given proximity to A, increases with the magnitude of its velocity u N. Of the particles

ithat are at any given distance to the left of A, the faster ones (large positive u N) are more likely thanare slower ones to reach A. And, of the particles that are at any given distance to the right of A, the

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islower ones (large negative u N) are more likely than are faster ones to reach A. Because of the

i igreater number of particles to the left of A than to the right of A, the plot of u N versus S N becomespreferentially populated by faster moving particles and depleted of slower moving particles. The

effect is to shift the surface-averaged velocity upward such that is finite (Figure 4b). That

p pis, the surface A “sees” an average velocity > U where = U + with = 0

for the same particle surface area S = N . In turn, the flux

in which an extra term involving velocity fluctuations about the mean appears in the definition of

xq . The counterpart to this situation occurs when the particle activity ( increases with distance x,

pin which case < U .

The effect embodied in (9) can be readily visualized by considering the motion of a triangularcloud of particles which possess two velocities, 1 and 2, in equal proportions (Figure 5). The

paverage velocity of all particles in the cloud is U = 1.5. During a short interval of time dt theparticles begin to segregate. At any position x in front of the crest of the cloud there is a greaterproportion of fast particles, and at any position x behind the crest there is a greater proportion ofslow 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 thetrailing (fully) segregated part is 1. The average velocity of particles intersecting a surface A at any

px in front of the crest is greater than U , and the average velocity of particles intersecting a surface

pA at any x behind the crest is less than U . The cloud as a whole moves downstream with velocity

pU . One must be careful, however, to limit this idea to small time dt, as it neglects time variationsin particle velocities, including starting and stopping.

This effect of an activity gradient vanishes in the absence of fluctuating particle velocities (i.e.,

if = 0), and, as elaborated below, it represents diffusion when particle motions are cast in

probabilistic terms. We show in fact that whereas (1/b)N represents advection, the product

(1/b)N( ) is equivalent to a diffusive term that looks like -(1/2)M(6()/Mx, where 6 [L t ] is a2 -1

diffusivity that derives from the second moment of the distribution of particle displacements.Meanwhile, we emphasize that the surface-integral definition (8) of the flux, which does not

distinguish between advection and diffusion, is precise so long as is exactly specified (and not

passumed to equal the overall average U ) in the presence of an activity gradient.A second caveat that goes with the averaging above centers on particle shape, and can be

illustrated with a simple example. Suppose that a single spherical particle intersects A at time t with

1 1 1 1 1distance > = D/2 so that S (> ) = (1/4)BD is at its largest value, and S = . Also suppose that u2

1 1= is a small value. Assuming Q = N , this correctly gives Q = S u . But suppose that a

2 2 2second, fast moving particle also intersects A where at time t the distance > is small so that S (> )

1 1 1 2 1 1 2 2n S (> ), whereas u n u . The actual discharge Q = S u + S u . The discharge using averaged

1 2 1 2 1 2quantities is calculated as Q = N = (S + S )(u + u )/2 . S u /2, which clearly is incorrect.

Although this effect of particle shape becomes proportionally less significant with an increasing

number of particles N such that any covariance vanishes (and, interestingly, is not present

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iwith non-rotating cubic particles with constant S ), this example points to the idea that the summedproduct in (5) generally does not equal the product of the averages for small N. Moreover, as

ielaborated in section 6, when considering a mixture of particle sizes, the covariance between S and

iu cannot be neglected inasmuch as some particle sizes preferentially move faster than other sizes.

2.2. The Geometry of Diffusion

x i i iWe emphasize that the precise definition (5) of the flux q (t) requires knowledge of > , S and ufor all N(t) particles intersecting A, and we now turn to the consequences of appealing to averagedquantities to replace this detailed information. We begin by providing an approximate geometricalinterpretation of the flux of spherical particles as described by (9).

Consider again a cloud of particles, each of diameter D, which are moving parallel to x with

uvarying velocities u. Let f (u) denote the probability density function of the velocities of particleswhose noses are located within any elementary area bdx at all positions x. We assume for simplicity

uthat f (u) is a positive function involving only downstream motions, and that its parametric values(mean, variance) are invariant along x. Note that in this formulation, whereas u represents the

ivelocity of a particle without reference to its proximity to a surface A, u pertains to the ith particleof N particles that intersect A.

Let J denote a small interval of time, and consider a particle whose nose is at position xN < x attime J = 0. In order for this particle to intersect a fixed surface A at position x at time J, it musttravel a distance greater than or equal to x - xN but less than or equal to x + D - xN during J. That is,the particle must possess a velocity between u = (x - xN)/J and u = (x + D - xN)/J. A particle whosenoses is at position xN such that x # xN # x + D at time J = 0 initially intersects A. In order for thisparticle to remain in contact with A during J (assuming only downstream motion), it must travel adistance greater than or equal to zero but less than or equal to x + D - xN. That is, it must possess avelocity between u = 0 and u = (x + D - xN)/J. The total number of particles N(x, J) intersecting afixed surface A at position x after a small interval of time J is therefore

xwhere, as above, n (x) is the number of active particles per unit distance parallel to x at J = 0.

pIf U denotes the average particle velocity, then based on recent experiments [Lajeunesse et al.,2010; Roseberry et al., 2012] we assume for illustration that

pHere we stress that U is equal to the average velocity of all particles in the cloud. Substituting (11)into (10) and integrating with respect to u,

We now assume that

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xwhich describes a linear variation in n about the position x. Substituting (13) into (12) andintegrating with respect to xN,

xIn the presence of a uniform particle gradient (Mn /Mx = 0), the total number of particles intersecting

xA is N(x, J) = n (x)D at time J. The second term on the right side of (14) is a correction due to the

xvariation in n over D. The last term in (14) describes the change in N(x, J) at x during J due to theoverall downstream (advective) motion of the particles. It follows that, locally at x, MN/MJ = -

p xDU Mn /Mx.

p pLet z denote a moving coordinate such that x = z - U J, where dz/dJ = U = -dx/dJ. Then, in a

p z x J = 0 z x J = 0reference frame moving with the average velocity U , n (z) = n (x)* , Mn /Mz = Mn /Mx* , and

where locally at z, MN/MJ = 0. Thus, at small time J, the position z “sees” a steady number of active

p zparticles intersecting A (which is moving with velocity U ) and a steady gradient in n .

iConsider the ith particle that intersects A at time J possessing the velocity u . This particle

ireached A with velocity u = u during J, starting from a position greater than or equal to xN = x - Ju

xbut less than or equal to xN = x + D - Ju. Because n (xN)dxN is the number of particles within the

u xinterval xN to xN + dxN initially (J = 0), f (u)n (xN)dxNdu is the number of particles within this intervalpossessing a velocity from u to u + du. Thus, the integral of this product from x - Ju to x + D - Juis the total number of particles intersecting A at time J possessing a velocity from uN = u to uN + duN

A i i= u + du. The probability density f (u ) of velocities u for particles intersecting A after a smallinterval J is therefore

Substituting (13) into (16) and integrating with respect to xN,

Substituting (11) into (17), multiplying by u and integrating from zero to infinity, the averagevelocity of particles intersecting A is

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x p xThus, with Mn /Mx < 0, the average is greater than the average U ; and with Mn /Mx > 0, the average

pis less than the average U .

p A i iIn turn, with u = U + uN, the probability density f (u N) of fluctuating velocities u N for particlesintersecting A after a small interval J is

pMultiplying (19) by uN and integrating from uN = -U to infinity, the average of the fluctuating

pvelocities of particles intersecting A in a reference frame moving with the average velocity U is

This is the “diffusive” velocity contribution to the surface-integral flux across A. Specifically,

multiplying (20) by S/b = N /b,

pHere we note that D = (1/2)V for spherical particles intersecting A (Appendix B). In turn the

p xparticle activity ( = V n /b, so

p xwhere 6 = JU is like a diffusivity. This demonstrates that q N, expressed in the form of a diffusive 2

flux involving the activity gradient, M(/Mx, is entirely consistent with a surface-integral definition

of this flux involving the average of the fluctuating velocities, . Also note that for the exponential

u p p udensity function (11), the variance F = U , so 6 = JU = JF , highlighting that the diffusive flux2 2 2 2

(22) fundamentally is associated with velocity fluctuations. Indeed, if J is identified as theLagrangian integral timescale obtained from the autocorrelation function of the particle velocitiesu, then this definition of 6 is equivalent to the classic definition provided by Taylor [1921] [seeFurbish et al., 2012b].

We emphasize that this idealized formulation is aimed at showing the consistency between thediffusion and surface-integral forms of the flux, rather than providing a general description of the

u p xflux. As such we have assumed that f (u) (and U ) are invariant along x, and that n initially varies

ulinearly in the vicinity of position x. The formulation therefore neglects any variations in f (u) (and

p xU ) along x, possible changes in Mn /Mx and particle velocities u during J, and the possibility thatparticle motions start or stop during J. Thus, J must be considered small, that is, smaller than theautocorrelation timescale of particle velocities. In addition, whereas the diffusivity 6 in (22) isdimensionally sound, its dependence on time J as written above is imprecise. In section 3 we turnto a more formal derivation of (22) that reveals the ingredients of 6. In companion papers[Roseberry et al., 2012; Furbish et al., 2012b] we describe the details of particle motions that setthe magnitude of the diffusivity 6 [Ball, 2012].

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3. Probabilistic Formulation of the One-Dimensional FluxHere 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. Theexplicit functional notation used in this section, although bulky in places, figures importantly in the

(bookkeeping of the formulation. In functions such as f ((; x, y, t) (defined below), randomvariables, ( in this example, appear first within the parentheses (and as subscripts, which identifythe probability density or distribution), followed by parametric quantities or independent variablesafter the semicolon. Here a “parametric quantity” means a key quantity that is not a randomvariable, and which can be treated mathematically as an independent variable. In a conditional

r*(function such as f (r*(; x, dt), the quantity providing the conditioning, ( in this case, is to be

r;(considered a parameter, so this function could just as well be written, for example, as f (r; (, x, dt).

3.1. Ensemble States of Particle MotionsBecause the particle activity ( varies stochastically over space and time at many scales, a

particularly challenging part of defining the bed load sediment flux is taking this variability intoaccount in a way where the local, instantaneous flux can be systematically related to spatiallyaveraged or time-averaged expressions of the flux, and vice versa. We approach this by envisioningan ensemble of configurations of particle positions and velocities in a manner similar to (but notidentical to) that outlined by Gibbs [1902] for gas particle systems. As Kittel [1958] notes, “Thescheme 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 demonstratingthe equivalence of the two types of averages is the subject of ergodic theory... It may be argued, asTolman [Tolman, 1938] has done, that the ensemble average really corresponds better to the actualsituation than does the time average. We never know the initial conditions of the system, so we donot know exactly how to take the time average. The ensemble average describes our ignoranceappropriately.” In turn, the ergodic hypothesis suggests that (for gas systems) one may assume anensemble average is the same as a time average over one realization, that is, a single system thatevolves through time. Here we define the essentials of an ensemble appropriate to sediment particlemotions. We use this as a starting point for our probabilistic formulation of the flux, and then returnto it later to suggest how persistent time-averaged variations in particle activity associated with bedforms influence the flux.

Envision bed load particles moving over an area B [L ] on a streambed that is subjected to steady2

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, stationaryfluctuations 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 besufficiently large that, during any small interval of time dt, any difference in the number of particles

aleaving B and the number arriving is negligibly small relative to the total number N of activeparticles within B. Similarly, any difference in the number of particles that stop and start within B

a aduring dt is negligibly small relative to the total number N of active particles. Then, N may beconsidered the same from one instant to the next. We now envision all possible instantaneous

aconfigurations of the N active particles as defined by their xy positions within B at a fixed time, withthe understanding that this set of configurations need not represent the same set of particles, only

athat N is the same. This imagined set of possible configurations constitutes an ensemble of activeparticle positions, and, in the absence of any additional information, we initially assume that eachconfiguration in the ensemble is equally probable (but see Roseberry et al. [2012]).

12

xyConsider an elementary area dB within B. If n (x, y, t) [L ] denotes the number of active-2

xyparticles per unit area, then n (x, y, t)dB is the number of particles within dB and the associated

p xyactivity ((x, y, t) = V n (x, y, t) such that ( may be considered a random variable. One can thenenvision that the ensemble of configurations of particle positions, each equally probable, yields for

( (any area dB a probability density function of the activity (, namely f ((; x, y, t) [L ], such that f ((;-1

x, y, t)d( is the probability that the activity within dB at (x, y, t) falls between ( and ( + d(. The

(form of f ((; x, y, t) and its parametric values (e.g. mean, variance) are specific to the sediment (size,shape) and the macroscopic flow conditions, including the turbulence structure. Equally important,

(the form of f ((; x, y, t) varies with the size of dB (Appendix C), which means that the magnitudeof fluctuations in the bed load flux relative to mean conditions at a given position varies with scale.

To elaborate this important point, we momentarily focus on one-dimensional transport parallel

xto x. Let dB = bdx. For a specified width b, if n (x, t) [L ] denotes the number of active particles-1

xper unit distance parallel to x, then n (x, t)dx is the number of active particles within bdx. The local

p xactivity at position x is ((x, t) = V n (x, t)/b [L] where, in the limit of dx 6 0 becomes ((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

a a parea B and total number of particles N with overall activity ( = N V /B, envision a large number of

aconfigurations where, in each configuration, N particles are randomly distributed over B. Eachconfiguration gives a different activity ((x, t) = S/b calculated at one position x. Hence the ensembleof particle configurations, each equally probable, yields for any position x a probability density

(function of the activity (, namely f ((; x, t) [L ]. As the width b increases, the number of particles-1

intersecting a surface at x on average increases. This means that for a given overall activity the form

(of f ((; x, t) varies with b (Figure 6). Specifically, whereas the mean activity at x associated with

a p (this distribution is equal to the overall activity calculated by ( = N V /B, the variance of f ((; x, t)decreases with increasing b, which reflects on average smaller fluctuations in the number of particlesintersecting the surface at x. Moreover, any actual realization of the activity at an instant in effect

(is a “sample” from f ((; x, t), so the variability in such realizations from one instant to the nextdecreases with increasing b. We reconsider this point below and in Roseberry et al. [2012].

Returning to the two-dimensional case, each active particle in each possible configuration

p p ppossesses an instantaneous velocity u = iu + jv at time t. One can therefore associate with each

p pparticle at time t the small (pending) displacements r [L] = u dt and s = v dt [L] parallel to x and y,respectively, that occur during dt, that is, between t and t + dt. For each configuration there is a joint

aprobability distribution of r and s associated with N particles. But because within any elementary

xyarea dB the number of active particles n (x, y, t)dB, and thus the activity ((x, y, t), varies among

xyconfigurations, there are likewise n dB values of the pair r and s for each configuration.

pFurthermore, we must leave open the possibility, elaborated below, that the velocities u , and

xytherefore the displacements r and s, of the n dB particles within dB are correlated with the number

xyof active particles n dB. We now envision the ensemble as consisting of all possible instantaneousstates defined by the joint occurrence of particle positions and displacements r and s, and we assumethis ensemble defined over B yields for any area dB a joint probability density function of the

(, r, sactivity ( and the displacements r and s, namely f ((, r, s; x, y, dt) [L ], where certain values of-3

((, r and s, and their combinations, are more (or less) probable than are others. Like f ((; x, y, t), the

(, r, sform of f ((, r, s; x, y, dt) and its parametric values are specific to the sediment (size, shape) andthe macroscopic flow conditions, including the turbulence structure.

Specifically, among the ensemble of possible configurations of particle positions and velocities,some configurations may be preferentially selected or excluded by the turbulence structure inasmuchas 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.

13

aall N active particles are clustered within dB) are excluded by the physics of coupled fluid-particlemotions. Nonetheless, we cannot claim the wisdom, given our current understanding of turbulenceover a mobile sediment boundary, to suggest that any particular configuration of particle positionsand velocities is not possible, and hence, the initial assumption that each configuration in the

(ensemble is equally probable is justified. This assumption, however, is not critical in that f ((; x,

(, r, sy, t) or f ((, r, s; x, y, dt) ultimately must be defined semi-empirically. Moreover, if the streambedand turbulence structure are homogeneous (in a probabilistic sense) over B, then it may be assumed

( (, r, sthat f ((; x, y, t) and f ((, r, s; x, y, dt) are the same for each elementary area dB. And, becausethese probability densities vary smoothly with xy position, their parametric values (e.g. mean,variance) also vary smoothly such that these values may be considered continuous fields, albeituniform and steady in this initial example of a planar streambed.

If, in contrast, the streambed and turbulence structure vary over B, for example, due to thepresence of bed forms, then one might expect concomitant, systematic variations in particle activityand motions. In this case the bed forms are to be considered part of the externally imposedmacroscopic conditions, that is, as a bed condition that is compatible with the macroscopic flow andsediment properties. Then, we again may envision an ensemble of possible configurations of activeparticle positions and velocities, each configuration being equally probably. But here it is importantto imagine, as Gibbs did, the set of configurations as being separate systems (realizations) with thesame bed forms at a fixed time, not necessarily as a time series of one realization where the bedforms grow or migrate. As above, we assume this ensemble yields for any area dB a probability

(density function of the activity, namely f ((; x, y, t), and a joint probability density function of the

(, r, s (activity ( and the displacements r and s, namely f ((, r, s; x, y, dt). Now the forms of f ((; x, y,

(, r, st) and f ((, r, s; x, y, dt) and their parametric values may vary with xy position (and with time; seesection 5), although it still may be that these values are continuous fields over B.

In the next three sections we consider for simplicity one-dimensional transport parallel to x,

xwhere our first objective is to obtain a probabilistic description of the sediment flux q , and oursecond objective is to obtain the expected (ensemble-averaged) value of this flux. In this case the

xy p xy (number density n (x, y, t), the activity ((x, y, t) = V n (x, y, t), the density function f ((; x, y, t) and

(, r, s xthe joint density function f ((, r, s; x, y, dt) introduced above may be simplified to n (x, t) [L ],-1

( (, r((x, t) [L], f ((; x, t) [L ] and f ((, r; x, dt) [L ]. We also define the conditional probability density-1 -2

function

( (, rwith units [L ], where f ((; x, t) may be considered the marginal distribution of f ((, r; x, dt). That-1

r*(is, f (r*(; x, dt)dr is the probability that a particle at x will move a distance between r and r + drduring dt given that, among all possible combinations of particle activity and displacements r,

r*(attention is restricted to the specific activity ((x, t) at time t. In turn we let F (r*(; x, dt) denotethe cumulative distribution function defined by

where the lower limit of integration indicates that r may be positive or negative, a condition that we

r*(redefine below. That is, F (r*(; x, dt) is the probability that a particle at x will move a distance lessthan or equal to r during dt, given the activity ((x, t) at time t.

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3.2. Master EquationTo 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, considerparticle motions along a coordinate x, where it is convenient to treat motions in the positive andnegative directions separately. For particles located at x = xN at time t, let r denote a displacementin the positive x direction during dt, and let l denote a (positive) displacement in the negative xdirection during dt. Further, let p(xN, t) denote the probability that motion is in the positive xdirection, and let q(xN, t) denote the probability that motion is in the negative x direction. Thus, p(xN,t) + q(xN, t) = 1. Also note that a particle in motion during dt may also be in motion (or at rest) ateither time t or time t + dt, or both. That is, r or l is the total displacement of an active particle forall motion that occurs over an interval less than or equal to dt. The displacements r and l thereforeare not to be interpreted as hop distances measured start to stop, a point that we examine below.

r*(Now, if F (r*(; xN, dt) denotes the probability that a particle starting at xN (r = 0) moves a

r*( r*(distance less than or equal to r during dt, then R (r*(; xN, dt) = 1 - F (r*(; xN, dt) is the probabilitythat a particle moves a distance greater than r during dt. By definition the conditional probability

r*( r*( r*( l*(density of r is f (r*(; xN, dt) = dF /dr = -dR /dr [L ]. In turn, if F (l*(; xN, dt) denotes the-1

probability that a particle starting at xN (l = 0) moves a distance less than or equal to l during dt, then

l*( l*(R (l*(; xN, dt) = 1 - F (l*(; xN, dt) is the probability that a particle moves a distance greater than

l*(l (in the negative x direction) during dt. The conditional probability density of l is f (l*(; xN, dt) =

l*( l*(dF /dl = -dR /dl [L ]. Note that because r and l are defined here as being positive displacements,-1

r*(the lower limit of integration in (24) defining F is now set to zero, and likewise for the (unwritten)

l*(companion definition of F .If the location of a particle is specified by the position x of its nose, then over a specified area

xA of width b normal to x, let n (x, t) [L ] denote the number of active particles per unit distance-1

p xparallel to x. Then, ((xN, t)bdxN = V n (xN, t)dxN denotes the associated volume of active particles atxN at time t, and p(xN, t)((xN, t)bdxN is the volume of particles that moves in the positive x directionduring dt. Moreover, the volume of particles passing position x in the positive x direction from xN

r*(< x is p(xN, t)((xN, t)R (x - xN*(; xN, dt)bdxN, and the volume passing position x in the negative x

l*(direction from xN > x is q(xN, t)((xN, t)R (xN - x*(; xN, dt)bdxN. The total volume of particles passingposition x in the positive x direction during dt is

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

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

-dt) + V (x, t + dt), namely

This is a flux form of the Master Equation [Risken, 1984; Ebeling and Sokolov, 2005; Furbish et al.,

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2009a, 2009b], illustrating that the volume V(x, t + dt) passing x during dt may be influenced bymotions originating at positions both to the left and right of x. Note that nothing is assumed a priori

r*( l*(regarding the forms of the conditional probability densities, f (r*(; xN, dt) and f (l*(, xN, dt), of thedisplacements r and l. Also note that the explicit appearance of the activity ( as a parameter in thefunctional notation of the left side of (27) highlights that the particle volume V(x, t + dt; () isconditional on the activity. This point is important in the idea of an ensemble average presentedbelow.

3.3. Advection and DiffusionThe Master Equation (27) may be recast in a more compact form involving advective and

diffusive terms as follows. With r = x - xN (xN < x) and l = xN - x (xN > x), a change of variables in(27) gives

r*( l*(Expanding the products p(x - r, t)((x - r, t)R (r*(; x - r, dt) and q(x + l, t)((x + l, t)R (l*(; x + l,dt) as Taylor series to first order then leads to

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

and

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

and

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

and

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

and

Substituting (30) through (33) into (29), dividing by dt, and taking the limit as dt 6 0 thus gives theparticle volume discharge,

The first term on the right side of (38) is advective and the second is diffusive. The bracketed partof the first term is merely the weighted average particle velocity u [L t ], namely, u(x, t; () = p(x,-1

r lt)u (x, t; () - q(x, t)u (x, t; (). That is, in the development above, for convenience we defined l as

lbeing a positive displacement in the negative x direction, so by this definition u is positive. If for

l r lcosmetic reasons we now let u carry the sign, then u(x, t; () = p(x, t)u (x, t; () + q(x, t)u (x, t; ().Similarly, the parenthetical part of the second term on the right side of (38) is a weighted diffusivity

r l6 [L t ], namely, 6(x, t; () = p(x, t)6 (x, t; () + q(x, t)6 (x, t; (). With these definitions, dividing2 -1

x(38) by the width b gives the flux q (x, t) [L t ], namely2 -1

which, consistent with the conclusions in section 2.2, suggests that spatial variations in ( or 6 caneffect a flux that is in addition to the advective flux. We consider the conditions under which thediffusive term in (39) may be important in section 5 below and in Furbish et al. [2012a].

The activity ((x, t) is treated above as being one of many possible instantaneous values of ( atposition x, whereas the velocity u and the diffusivity 6 are formally defined above as ensembleaverages, that is, the (statistically) expected values of these quantities obtained from the ensembleof all possible configurations of particle positions and velocities, conditional to the activity (. The

r*( l*( r*( l*(conditional probability densities f and f (as well as the related functions R and R ) thusrepresent underlying (ensemble) populations and are smooth, continuous functions. In order toenvision (39) as representing the local instantaneous flux, one must therefore imagine that u and 6

r*(actually represent values obtained from an instantaneous “sample” drawn from the densities f and

l*(f . Over an elementary area bdx, this sample may involve few to many particles as determined bythe instantaneous value of ( and the width b, so the instantaneous distributions of displacements

r*( l*( r*((drawn from f and f ) may look more like irregular histograms than like the smooth functions f

l*(and f , and the velocity u (39) is like the simple average in (8). We return to this point below.

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Meanwhile, to complete the ensemble average over all values of the activity ( we first substitute(30) through (33) into (29). Then, to simplify we redefine r to its original meaning as adisplacement that is positive or negative, note that dl = -dr, combine the integrals in (29), and usep + q = 1 to give

which, like (29), is the particle volume crossing x during dt associated with the activity (. In turn,

(multiplying (40) by the probability f ((; x, t)d( weights this volume in proportion to the relative

(occurrence of ( over the ensemble. Substituting (23) into (40), multiplying by f ((; x, t)d( andintegrating over the activity ( thus gives

Letting an overbar denote an ensemble average, dividing by b and by dt, and taking the limit as dt6 0, this becomes

which is the ensemble-averaged flux.A key point embodied in (42) 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 diffusivityand activity. Indeed, experiments suggest that, at low transport rates, both the particle activity andthe average velocity increase with increasing bed stress, where the activity increases faster than thevelocity [Schmeeckle and Furbish, 2007; Ancey et al., 2008; Ancey, 2010; Lajeunesse et al., 2010;Roseberry et al., 2012], clearly indicating that u and ( are correlated. This figures importantly inconsidering how the ensemble average is related to time averaging, a topic that we address in section5. Meanwhile we note that if u and (, and 6 and (, are independent, which may be the case at hightransport rates (and is demonstrably correct in the case of rain splash transport treated as a stochasticadvection-diffusion process [Furbish et al., 2009a]), then (42) becomes

Comparing this formulation with classic descriptions of transport involving simple fluids revealsseveral interesting points. For tracer molecules within a fluid the velocity in (42) maps to the

fadvective fluid velocity u (which is equal to the mean molecular velocity [Meyer, 1971; Furbish,1997]), and the activity in (42) maps to the tracer concentration c. In a simple fluid-solvent systemthese quantities are independent (i.e. their covariance is zero). Hence, the advective flux of tracers

fis the product of the advective velocity and the concentration, u c. Moreover, for isothermal

mconditions the molecular diffusivity 6 is a thermodynamic quantity that is independent of theconcentration and position, so the diffusivity is outside the differential in the diffusive term, which

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mthen looks like Fick’s law as normally written, namely -6 Mc/Mx. We further note that, whereas thefluid velocity and the molecular diffusivity are independent in simple fluid-solvent systems, themean particle velocity u and the diffusivity 6 in (39) are highly correlated. (Indeed, the approximate

pdescription (22) gives 6 = Ju with U = u.) That is, the diffusive part of the flux in (39) vanishes2

in the absence of particle advection, entirely analogous to the relation between advection andmechanical dispersion in porous-media transport [Furbish et al., 2012b].

3.4. Describing the Distribution of Particle DisplacementsTo keep the notation simple, here we omit the notation indicating a conditional dependence on

the activity (, but with the understanding that this dependence is implied. And, as in section 3.2,we let r denote a displacement in the positive x direction and l a (positive) displacement in thenegative x direction.

rWith reference to Figure (7), of the particles starting at xN, let P (r; xN, dt) [L ] denote the-1

proportion located at xN + r at time t + dt, relative to the proportion that moves beyond xN + r duringdt, namely

Integrating (44) from r = 0 to r then gives

from which it follows that the probability density of r is

lIn turn, of the particles starting at xN, let P (l; xN, dt) [L ] denote the proportion located at xN - l at-1

time t + dt, relative to the proportion that moves beyond xN - l during dt. By a development similarto that above one obtains

whence the probability density of l is

These relations merit further discussion.If r and l were considered displacements in time, specifically the “age” of an entity, rather than

r ldisplacements in position, as above, then the probabilities R (r; xN, dt) and R (l; xN, dt) are referred

rto as “reliability” or “survival” functions in reliability (or survival) theory, and the proportions P (r;

l rxN, dt) and P (l; xN, dt) are the associated “hazard” or “failure-rate” functions. The functions P (r;

lxN, dt) and P (l; xN, dt) must be non-negative and integrate to infinity over the domain [0, 4), butotherwise may have any form, monotonic (increasing or decreasing), nonmonotonic ordiscontinuous. These functions can be obtained directly from (44) for known distributions. But ofpotentially greater value is the idea of using (46) as a strategy for defining the probability density

r lfunction f (r; xN, dt) in terms of P (l; xN, dt), based on theoretical or empirical arguments for the form

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rof P (r; xN, dt). Indeed, this is the strategy used to select a suitable distribution in reliability/survivalanalysis, wherein several well known distributions arise, for example, the exponential, gamma,Weibull and Pareto distributions, possibly involving heavy-tailed behavior for certain distributions.

To briefly illustrate this point, we first emphasize that r and l denote displacements during dt;these displacements do not represent hop distances from start to stop. Particles may be in motion

rat either time t or t + dt, or both. Note that P (r; xN, dt)dr may be considered a conditional

rprobability. Namely, P (r; xN, dt)dr is the probability that during dt a particle will have adisplacement from r to r + dr, given that it achieves a displacement of at least r. That is, because

rall displacements r occur during dt, P (r; xN, dt)dr is equivalent to the probability that a particle has

p p p pa velocity within u to u + du such that during dt it experiences a displacement r = u dt to r + dr

p p p= (u + du )dt, given that its velocity is at least as fast as u = r/dt.Measurements of bed load particle motions using high-speed imaging suggest that, at low to

pmoderate transport rates, streamwise particle displacements r (or velocities u ) follow anexponential-like distribution [Lajeunesse et al., 2010; Roseberry et al., 2012]. As elaborated in

rRoseberry et al. [2012], an exponential-like density f (r; xN, dt) implies a constant failure rate,

r r r rnamely, P (r; xN, dt) = P = 1/: , where : is the mean displacement during dt. Namely, theprobability that a particle will experience a displacement within r to r + dr during dt is a fixed

rproportion, 1/: , of particles that experience displacements greater than r during dt. Or, the

p p pprobability that a particle possesses a velocity within u to u + du is a fixed proportion of particles

pmoving faster than u . In turn, Ganti et al. [2010] and Hill et al. [2010] point out that power-lawdistributions can arise from combinations of exponential distributions, and apply this idea totransport of mixed particle sizes, albeit involving distributions of travel distance (see below) rather

rthan the distribution of displacements, f (r; xN, dt).In contrast to a displacement r that occurs during dt, let 8 denote a particle displacement

8, Jmeasured start to stop that occurs over a travel time J [t], and let f (8, J) [L t ] denote the joint-1 -1

probability density of 8 and J. With reference to Figure 8, a steep covariance relation between 8and J implies varying speeds (defined by 8/J) due to varying displacements over a similar traveltime. A weak covariance implies varying speeds due to similar displacements over varying traveltimes. An intermediate covariance implies relatively uniform speeds.

8The marginal distribution f (8) [L ] of the displacements 8 is-1

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.,

82012]. By itself, f (8) contains no information regarding particle travel times or speeds. In turn the

Jmarginal distribution f (J) [t ] of the travel times J [e.g. Lajeunesse et al., 2010] is-1

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

JThe elements of (46) also suggest a strategy for clarifying the physical basis of the densities f (J)

8 J 8 Jand f (8). Namely, if we define P (J; xN) [t ] and P (8; xN) [L ] as in (44), then P is a temporal-1 -1

8“failure rate” function and P is a spatial “failure rate” function as normally defined inreliability/survival theory, where “failure” may be interpreted as particle disentrainment [Furbishand Haff, 2010]. Thus, a physical (probabilistic) understanding of how and why active bed loadparticles stop in relation to bed roughness and near-bed flow conditions is central to describing the

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probability densities of the hop distances 8 and the associated travel times J, beyond purely

J 8empirical descriptions. Moreover, descriptions of f (J) and f (8) must be mutually consistent,

8, Jinasmuch as these combine to form the joint density f (8, J).

J J 8 8 J 8Specifically, with P = P (J; xN) and P = P (8; xN), then the densities f (J) and f (8) depend onthe travel time J and the distance 8, and therefore on changing conditions following entrainment and

J J 8 8downstream of the initial position xN. If, however, P = P (xN) and P = P (xN), then these densities

J 8are independent of J and 8. For example, if for physical reasons P and P are constants that depend

J J 8 8 J Jonly on local conditions, namely P (xN) = 1/: and P (xN) = 1/: , then from (46), f (J) = (1/: )exp(-

J 8 8 8 JJ/: ) and f (8) = (1/: )exp(8/: ) are exponential densities with mean travel time : and mean hop

8 Jdistance : . (This example of constant P is analogous to a constant “failure rate” in reliability

J J Janalysis, where : would be interpreted as the mean longevity.) In contrast, if P = P (J; xN) = "J" -

J, for example, then f (J) = "J exp(-J ) is a standard Weibull distribution with shape factor "; or1 " - 1 "

J J m mif P (J; xN) = "/J, then f (J) = "J /J is a Pareto distribution with scale factor J . This idea of a" " + 1

8 Jdisentrainment rate function (P or P ) is conceptually similar to, but of a different form than, thedisentrainment rate function involving the distribution of particle hop distances as described byNakagawa and Tsujimoto [1980].

For completeness we note an additional definition of travel distances, where displacements 8 aremeasured over a specified time J, but involve rest times [e.g. Einstein, 1937; Hassan and Church,1991; Bradley et al., 2010; Hill et al., 2010]. Namely, if 8 is redefined to include multiple hops with

8rest times over an interval J, then the probability density of 8 may be denoted as f (8; J),emphasizing the significance of the interval J as a parameter. Connecting travel distances that

8, Jinvolve rest times to the density f (8, J) and associated particle speeds requires additionalinformation on rest times and/or numbers of hops [Hill et al., 2010].

We return below (section 4.2) to the joint probability density of hop distances and associatedtravel 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

8motions. Lajeunesse et al. [2010] present histograms representing the marginal distributions f (8)

Jand f (J) based on high-speed imaging of particle motions, and note that these possess well defined

8, Jmodes that are less than the means. Data concerning the joint density f (8, J) are also presentedin a companion paper [Roseberry et al., 2012].

3.5. The Two-Dimensional FluxAs in section 3.1 let r and s denote particle displacements parallel to x and y, respectively, and

r, slet f (r, s; dt) [L ] denote the joint probability density function of r and s. If u = iu + jv denotes the-2

xx yyensemble-average particle velocity, and if 6 denotes a diffusivity tensor with the elements 6 , 6

xy yx x y x yand 6 = 6 , then the component fluxes q and q of q = iq + jq are

and

xx yyHere u and v, and 6 and 6 , derive from the first and second moments, respectively, of the marginal

r s r, sdistributions, f (r; dt) and f (s; dt), of the joint density function f (r, s; dt) as described in section 3.2above. Also,

(51)

(52)

21

which is like a covariance (defined about the origin).Inasmuch as diffusive particle motions normal to the mean motion are centered about this mean

xx yymotion [Lajeunesse et al., 2010; Roseberry et al., 2012], the magnitudes of 6 and 6 vary with the

xydirection of the mean motion. Moreover, 6 is finite only when the mean motion is not parallel to

xx yythe x or y axis. As described above, the effect of the diffusive terms involving 6 and 6 is to

x ycontribute proportionally more (or fewer) particles to q and q relative to the contribution of those

xx yyparticles represented by u( and v(, depending on the sign of M(6 ()/Mx and M(6 ()/My. The effect

xy xy xyof the diffusive terms involving 6 is similar. For example, with finite 6 and negative M((6 )/My(due, say, to decreasing activity ( along y), proportionally more particles starting from positions at

S x Sy < y contribute to the flux q parallel to x across an elementary plane at y (Figure 9), relative to

S xythose particles represented by u( at y = y . Conversely, with positive M((6 )/My, proportional fewer

S x Sparticles starting from positions at y < y contribute to the flux q across an elementary plane at y .

xy ySimilar remarks pertain to the term involving M((6 )/Mx with respect to the flux q . The diffusivitiesin (51) and (52) are associated with the motion, not with any medium (as with heat conduction inan anisotropic solid). Moreover, like mechanical dispersion associated with flow in a porousmedium, the elements of 6 co-vary with u and v. This point is elaborated in companion papers[Roseberry et al., 2012; Furbish et al., 2012a, 2012b].

4. Exner Equation4.1. Flux Form

bLet 0(x, y, t) denote the local elevation of the streambed, and let c denote the volumetric particle

b x yconcentration of the bed. Then with c M0/Mt = -Mq /Mx - Mq /My, substitution of (51) and (52) gives

This formulation assumes that active particles effectively remain in contact with the bed, where 0is defined as the (local) average surface elevation of particles, including the (small) contribution tothe bed elevation associated with active particles. Note that (54) has the form of a Fokker-Planckequation, where the elements of 6 are inside both derivatives.

4.2. Entrainment FormHaving 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 theExner equation [e.g. Parker et al., 2000; Garcia, 2008; Ancey, 2010] for comparison with (54)above. Let E(x, y, t) [L t ] denote the volumetric rate of particle entrainment per unit streambed-1

area, and let D(x, y, t) [L t ] denote the volumetric rate of deposition per unit streambed area.-1

Assuming only downstream motions involving the streamwise hop distance 8 and the cross-streamhop distance R over the travel time J, we denote the joint probability density function of 8, R and

8, R, JJ as f (8, R, J; x, y, t), which depends on xy position and time t. By definition the rate ofdeposition is

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(54)

(55)

22

which explicitly incorporates the idea, neglected in previous formulations (Appendix E), thatparticles arriving at an xy position at time t started their hops 8 and R at many different times t - J,

8, R, Jas clearly reflected in histograms representing marginal distributions of f , that is, the

8 Jdistributions f (8) and f (J) [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 [Wonget al., 2007].

8, R, JStarting with (55) and assuming that f is not heavy-tailed [Roseberry et al., 2012], then it isstraightforward to show (Appendix F) that

where an overbar denotes an average. Specifically, and denote average hop distances and

denotes the associated average travel time. The averages and denote the second moments

of 8 and R, and denotes the averaged product of 8 and R. The terms in (56) involving spatial

derivatives are analogous to the terms in (49) involving spatial derivatives.In the case of a steady, uniform entrainment rate E with uniform and steady values of the average

b xhop distance and the travel time , then with the definition c M0/Mt = -Mq /Mx for one-dimensional

xtransport, it follows from (56) that the flux q = E , which is equivalent to the definition (3)

provided by Einstein [1950]. We further note that, contrary to the assertion of Lajeunesse et al.

[2010], the average hop distance is equal to the product of the ensemble average velocity and

the average travel time (Appendix G), namely . Equating the “flux” and “entrainment”

xforms of the flux thus gives q = ( = E = E . That is, under steady, uniform conditions the

activity ( = E , or = (/E, which has the interpretation of being the mean residence time of

particles within the nominal volume (B.The term on the right side of (56) 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 (56) with uniform and steady values of the average

hop distance and the travel time , namely

With a steady entrainment rate (ME/Mt = 0), the rate of change in the bed elevation 0 goes simply asthe divergence of the entrainment rate, ME/Mx. That is, there is a difference in the rates of depositionand entrainment at any position x because of a difference in the number of particles arriving at andleaving x. With a uniform entrainment rate (ME/Mx = 0), the (uniform) rate of change in the bedelevation goes as the rate of change in the entrainment rate, ME/Mt [L t ], modulated by the average-2

travel time. Thus, with small (which also implies small ), entrained particles quickly return to

the rest state (they “remember” to stop), and the difference D - E is small. But with increasing

(56)

(57)

23

average travel time , entrained particles increasingly “forget” to stop, so there is an increasing lag

between deposition and entrainment (or vice versa).Focusing on the difference D - E in (57), this formulation does not specify the style (rolling,

sliding, hopping) of particle motions; in fact, particles could be saltating high into the fluid column.

bSo in specifying that c M0/Mt = D - E [e.g. Parker et al., 2000; Garcia, 2008; Ancey, 2010], 0 iseffectively 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

b x xM(E )/Mt is like a source term. Specifically, writing c M0/Mt = -Mq /Mx, then it follows that M(q -

E )/Mx = S with S = M(E )/MJ. This indicates that the flux in excess of the steady, uniform flux E ,

xnamely q - E , increases (or decreases) downstream at the rate S with finite D - E.

We complete this section by noting the significance of the time derivative terms in the relatedentrainment formulation of conservation of tracer particles. Consider the simplified case of one-

Tdimensional transport parallel to x, and let f (x, t) denote the fraction of bed load particles that aretracers. The rate of deposition of tracers is

which incorporates the idea that tracer particles arriving at position x at time t started their hops 8

8, Jat many different times t - J. Again assuming that f is not heavy-tailed, (58) can be written as

Further assume for illustration a steady, uniform entrainment rate E with uniform and steady values

of the average hop distance and the travel time . Under these conditions,

T TIf all particles are tracers, f = 1, D = D and (60) reduces to the steady condition D - E = 0. Butotherwise, despite steady, uniform transport conditions (D - E = 0), the time derivatives in (60)cannot necessarily be neglected given that the spatiotemporal evolution of a non-uniform ensembleof tracers is an unsteady problem. Namely, if h denotes a nominal steady, uniform thickness of bed-

T T b Tsurface particles involved in transport, then D (x, t) - Ef (x, t) = c hMf /Mt and

The unsteady terms involving E account for the fact that tracers arriving at x at time t start fromdifferent positions upstream at different times, and, because they are entrained at different times, the

T T 1fraction f is changing at any specific starting position. For example, f (xN, t - J ) at position xN when

1 T 2tracer 1 is entrained at time t - J is different from f (xN, t - J ) when tracer 2 is entrained at the same

2position xN at time t - J , although both tracer particles arrive at position x downstream at time t

2because particle 2 has a shorter travel time J than does particle 1.

b RNote that c h/E = J is the mean residence time of particles within the thickness h, so upon

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(59)

(60)

(61)

24

dividing (61) by E and rearranging,

R R R Rwhere U = /(J + ) is a mean virtual velocity and 5 = /(J + ) is a virtual diffusivity. This

has the form of the advection-diffusion equation obtained by Ganti et al. [2010] assuming Fickian(normal) diffusion (their equation (11)), but differs in the explicit appearance of the mean residence

Rtime J and the mean travel time .

5. Averaged QuantitiesIn 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 f ((; x,

(, r, sy, t) of the activity (, and the joint probability density f ((, r, s; x, y, dt) of the activity ( and thedisplacements r and s, are associated with active particles within any elementary area dB. Focusing

r*(on displacements r, then likewise, the conditional density f (r*(; x, y, dt) is associated with activeparticles within dB = bdx. With small b, at any instant individual realizations drawn from the

( r*(densities f ((; x, y, t) and f (r*(; x, y, dt) at position y may be quite different from realizations from

( r*(f ((; x, y + )y, t) and f (r*(; x, y + )y, dt) at position y + )y. Upon lengthening b, the number ofactive particles within bdx generally increases, and the activity incorporates spatial variations that

(exist at scales smaller than b, so the probability density f obtained from the ensemble ofconfigurations 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 of y. Similarly, with increasing b, individual realizations (over bdx) of the conditional

r*(probability density of r approach the smooth function f (r*(; x, dt), independent of y.In effect, a lengthening of b is equivalent to sampling a greater number of possible states of

( r*(particle motions. However, b cannot be “too large” if the underlying forms of f and f changealong 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 fluxis inherently width-averaged over b. This also suggests that for equivalent macroscopic flowconditions, the magnitude of the fluctuations in the flux depend on the measurement width b. Wenow turn to time averaging of (39).

Of interest is the behavior of (39) when averaged over different characteristic timescales, andthe relation of this to ensemble averaging. Bed load transport rates vary over many timescales [e.g.see Table 1 in Gomez et al., 1989]; and for nominally steady flow conditions, the measurementinterval influences the calculated rate inasmuch as fluctuations in transport over durations shorterthan the measurement interval are averaged (and thus smoothed) in the calculation. To ourknowledge no systematic, simultaneous measurements of particle activity and velocities areavailable (beyond those reported in Roseberry et al. [2012], which are of short duration), so we lackan empirical basis for evaluating time averaging of these quantities. Nonetheless, we may surmisethe following in general terms.

Consider first a planar bed with steady (uniform) macroscopic flow conditions. With increasingaveraging period, one may assume that at any position x the bed experiences an increasingproportion of the set of possible (ensemble) configurations of particle activity and velocity, and witha sufficiently long averaging period the bed at x eventually experiences a fully representative set of

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25

possible configurations, in which case it is reasonable to assume that a (long) time average equalsthe ensemble average, as in (42). Moreover, for planar bed conditions the time-averaged product

of 6 and ( is independent of position, so the diffusive term vanishes and the flux .

However, it must be noted that only in the limit where the activity ( approaches a constant (e.g. for

sufficiently large b) does this become . Simultaneous measurements of particle activity

and velocities are presented in Roseberry et al. [2012].In contrast, consider three timescales associated with a homogeneous field of migrating bed

tforms. The first is a “short” turbulence timescale T , which we envision as being sufficiently longthat, at any position x, the bed experiences a representative sample of possible turbulencefluctuations specific to where x is located within the bed form field, but short enough that the localbed form morphology does not change significantly. The second is an intermediate bed form

btimescale T , which we envision as being comparable to the period required for migration of bed

t bforms (e.g. ripples or dunes) over one wavelength. (Note that T may be similar to T for small bed

fforms.) The third is a “long” bed-form field timescale T , which we envision as being long enoughthat, 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.

tWhen (39) is averaged over the turbulence timescale T ,

which looks like the ensemble average (42), and highlights that the flux retains its dependence on

ttime after averaging, as it varies over timescales longer than T . Moreover, whereas on a planar bedthe time-averaged flux is constant (uniform) and the diffusive term vanishes, within a field of activebed forms the flux varies with position x (otherwise bed forms would not form, grow or migrate),and the diffusive term in (63) may be nonzero due to persistent spatial variations in the averaged

product arising 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 reformulationof the stability analysis of Smith [1970] to include the diffusive flux suggests that this flux is asufficient, if not necessary, condition for selection of a preferred wavelength during initial ripplegrowth [Kahn and Furbish, 2010; Kahn, 2011].

bWhen (39) is averaged over the bed form timescale T , the result is the same as in (63) inasmuchas bed form geometries in a natural field are not identical. That is, in the idealization of identicalone-dimensional migrating bed forms, positions over a single wavelength in principle sample allpossible turbulence fluctuations, so a spatial average over one wavelength is the same as a time

baverage over one period T . In this idealization the diffusive term therefore vanishes. But withnaturally variable bed form geometries, the time-averaged diffusive term may be nonzero, and thetime-averaged flux retains its dependence on time, particularly given persistent interactions amongneighboring bedforms [Jerolmack and Mohrig, 2005]. In contrast, when (39) is averaged over the

ffield timescale T , the diffusive term in principle vanishes and the time-averaged flux becomes a

constant equal to , independent of time and position. Also, like the planar bed case, only in the

limit where the activity ( approaches a constant does this become .

For unsteady morphodynamic problems at the larger bar scale, analytical descriptions of theconstituents 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

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26

coupled motions of water and sediment leading to the growth of bars [Callander, 1969; Engelundand Skovgaard; 1973; Parker, 1976; Fredsøe, 1978; Blondeaux and Seminara, 1985; Nelson andSmith, 1989; Seminara and Tubino, 1989; Furbish, 1998]. These models, which start with theReynolds averaged momentum equations, in effect assume that local conditions coincide withensemble-averaged conditions, where bed forms change sufficiently slowly that the bed locallyexperiences a representative sample of the ensemble of particle activity and motions, and the localflux varies smoothly with xy position and time, consistent with the quasi-steady approximationapplied to turbulence conditions.

6. Discussion and ConclusionsThe 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 anydefinition that appeals to averaged quantities of particle motions (e.g. the mean particle velocity andactivity) to replace the detailed information embodied in (1) must be consistent with this definition.

With quasi-steady bed and transport conditions, the definition (2) of the one-dimensional flux

x pq parallel to x, involving the product of the average particle velocity U and the particle activity (,is consistent with (1) inasmuch as active particles are at any instant uniformly (albeit quasi-randomly) distributed over the streambed, and the flux-normal width b over which the particleactivity is calculated is sufficiently large to smooth over instantaneous small-scale variations in the

pactivity along b. In this situation the average velocity U of particles over the streambed is

equivalent to the average velocity of N particles that intersect a vertical surface of width b at any

x pposition x, and the flux q = (U = ( .

In contrast, in the presence of a particle activity gradient parallel to the mean particle motion,

M(/Mx, the average velocity of particles intersecting a surface at position x may be different from

pthe average velocity U of all particles in the vicinity of x. This occurs because the surface ispreferentially populated by faster moving particles and depleted of slower moving particles when

pM(/Mx < 0, or visa versa when M(/Mx > 0, even though the average velocity U is uniform over x. The

xflux calculated as q = ( is entirely consistent with the surface-integral definition (1). Moreover,

i p iupon writing the particle velocity u as the sum of the average U and a fluctuating part u N, namely

i p i x pu = U + u N, the flux looks like q = (U + ( . The term involving the averaged fluctuating

velocities is proportional to the activity gradient, M(/Mx, and it therefore may be interpreted as a“diffusive” flux. Thus, whereas the deterministic surface-integral definition of the flux (1) does notdistinguish between advection and diffusion, it can be formulated as consisting of these two parts.

A formal rendering of the collective behavior of active sediment particles, wherein particlepositions and motions are treated as stochastic quantities, yields a flux form of the Master Equation,namely (27). Assuming that particle motions do not involve heavy-tailed behavior, the formulationreveals that the volumetric flux involves an advective part equal to the product of the particleactivity and the ensemble-average particle velocity, and a diffusive part involving the gradient ofthe product of the particle activity and a diffusivity obtained as the time-derivative of the secondmoment of the probability density function of particle displacements. A key point in the formulationis that the average particle velocity and the diffusivity are correlated. Thus, the diffusive part of theflux vanishes, inasmuch as fluctuating particle velocities vanish, in the absence of overall advectiveparticle motions — which is entirely analogous to the relation between advection and mechanicaldispersion in porous-media transport. The effect of the diffusive flux therefore is to add to or

27

subtract from the advective flux in the presence of an activity gradient, as opposed to operatingindependently of the mean motion as in molecular diffusion.

Central to the formulation is the probability density function of particle displacements r that

r*(occur during a small interval of time dt, conditional on the particle activity (, namely f (r*(; x, dt).A useful way to think about the source of this smooth probability density is to envision an ensembleof states consisting of all possible configurations of particle positions and velocities. This beginswith selecting a streambed area B that is subjected to steady macroscopic flow conditions. This areamust be sufficiently large that, during any small interval of time dt, the total number of activeparticles 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 conditionsat a fixed time, rather than as a time series of one system where the bed and flow conditions evolve.

r*(Then, for any elementary area dB, this ensemble yields the smooth density function f (r*(; x, dt).In turn, the average displacement, as in (30), and the average particle velocity, as in (34), forexample, represent ensemble averages, where the “ensemble” consists of this set of systems at afixed time. (Note that this definition of an ensemble average is consistent with classic definitionsof such averages from statistical mechanics.) Particle activity data consistent with the idea of anensemble of particle configurations are presented in Roseberry et al. [2012].

Time-averaged descriptions of the flux involve averaged products of the particle activity, theparticle velocity and the diffusivity. The significance of the covariance parts of these productsdepends on the averaging timescale in relation to characteristic timescales of near-bed turbulenceand beform evolution. The covariances almost certainly underlie 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 bedform geometries, the flux,when averaged over a timescale nominally long enough to accommodate fluctuations associated withbedform evolution, nonetheless retains its dependence on time over longer timescales in the presenceof strong feedbacks between sediment transport and topography with interactions amongneighboring bedforms [Jerolmack and Mohrig, 2005].

The flux form of the Exner equation, (54), looks like a Fokker-Planck equation in whichgradients in the particle diffusivity, like gradients in the particle activity, can in principle contributeto changes in bed elevation. However, the significance of this idea requires clarification, theoreticalor 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,(56), similarly involves advective and diffusive terms, but also involves a time derivative term thatrepresents 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

xdistance and the mean travel time , then for one-dimensional transport the flux q = E , which

xis equivalent to the definition (3) provided by Einstein [1950]. For the unsteady case the flux q

cannot be expressed simply in terms of E, and 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

Rthe diffusivity contain the mean travel time and the mean residence time J within a nominal

Rthickness of active particles (neglecting burial and re-emergence). Inasmuch as n J , say, at low

R R R Rtransport rates, the virtual velocity U . /J and the virtual diffusivity 5 . /J .

The formulation of the sediment flux presented herein involves, for simplicity, a single particlediameter D, so definitions of the particle activity, velocity and diffusivity are specific to thissituation. In generalizing to a mixture of particle sizes, covariances between particle activity,

28

velocity and diffusivity become particularly important. For example, recall that in writing the last

x i ipart of (5) as q = (1/b)N , the covariance between S and u can be neglected for equal-sized

xparticles and this expression becomes (8), namely, q = (1/b)N = . Moreover, with uniform

p x p iactivity, = U , so q = (U as in (2). But with a mixture of sizes, the covariance between S and

iu cannot be neglected inasmuch as some particle sizes preferentially move faster than other sizes.

x p i In this case q = N , where ( = S /b is like an individual particle activity. Or, letting j denote

x jthe jth size fraction, we may write q = to denote the fractional flux [Wilcock and McArdell,

j1993; Wilcock, 1997a, 1997b], where ( is the activity of the jth fraction. The total flux is then the

sum over all j sizes, where each has advective and diffusive parts.

Consistent with the results of Lajeunesse et al. [2010], we show in a companion paper

r[Roseberry et al., 2012] that the probability density f (r; dt) of streamwise displacements of sand

u pparticles and the associated density of velocities f (u ) are exponential-like at low transport rates on

ra planar streambed. In relation to the “failure rate” function P (r; xN, dt) described in section 3.4,

r ran exponential function implies that P = P (xN, dt) is independent of the displacement r. This in turnmeans that the probability that a particle will experience a displacement within r to r + dr during dtis a fixed proportion of particles that experience displacements greater than r during dt. Or, the

p p pprobability that a particle possesses a velocity within u to u + du is a fixed proportion of particles

p 8 Jmoving faster than u . We also show that the failure rate functions P and P for the hop distance8 and the travel time J are not necessarily constants, but rather vary with 8 and J.

Appendix A: Particle Volume DischargeConsider a particle of diameter D [L] that is moving with a positive velocity parallel to x through

ia surface A positioned at x = 0 (Figure 2). Let > [L] denote the position of the nose of the particle

i irelative to x = 0, and let V (> ) [L ] denote the volume of the particle that is to the right of x = 0 as3

ia function of > . Also, let g [L] denote a small distance measured from the nose of the particle, where

i+0 # g # D. Then, at time t [t] the volume V (t) [L ] of the particle that is simultaneously to the leftg 3

i iof > - g and to the right of x = 0 for 0 # > # D is

i i i i iwhere H(> ) is the Heaviside step function defined by H(> ) = 0 for > < 0 and H(> ) = 1 for > $ 0.In this expression the Heaviside functions serve as off-on switches over the x domain. The second

i+ iterm on the right side of (A1) insures that V (t) is piecewise continuous without a jump at > = D,g

and the last two terms, as will be seen momentarily, insure that the derivative of (A1) is piecewise

icontinuous at > = 0 when g 6 0.

i i i+With > = > (t), the rate of change in V isg

(A1)

29

i i i i i iwhere *(> ) = dH/d> is the Dirac delta function. The derivative MV /M> = S (> ) [L ] is like a2

hypsometric function of the particle, and is equal to its cross-sectional area on the surface A at x =

i i0. The derivative d> /dt = u [L t ] is its velocity parallel to x. The terms involving the Dirac-1

function nominally represent instantaneous changes in the rates of gain and loss of volume when the

i iparticle arrives at and leaves A. Because these terms are non-zero only at > = 0 or > = D, the thirdand 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 g 6 0, the second term on the right side of

i+(A2) vanishes to give the particle volume discharge Q (t) across A in the positive x direction,namely

i-By symmetry the particle volume discharge Q (t) in the negative x direction has the same form as

i- i- i i i i ithe right side of (A3), namely Q (t) = dV /dt = S (> )u H(> )[1 - H(> - D)]. Moreover, at this pointg

i ithe product involving the Heaviside functions is redundant, as the surface area S (> ) is a piecewise

i i i i icontinuous function that is finite over 0 # > # D with S (> < 0) = S (> > D) = 0. Thus, in general the

iparticle volume discharge Q (t) across A is

which is (4) in the main text.

Appendix B: Averaged Particle Cross-Sectional Area and Volume

pHere we show that D = (1/2)V for spherical particles intersecting a surface A, although for

simplicity we start with cubic particles. Of the N active particles whose noses are between x = 0 and

xx = D, let n = n /N [L ] denote the proportion per unit distance parallel to x, and assume that any-1

instantaneous gradient in n, Mn/Mx, is uniform over a distance equal to the particle length D. Then

> i ithe probability density function f (> ) of the distance > is

> i > iNote that if Mn/Mx = 0, f (> ) reduces to a uniform density function, namely f (> ) = 1/D. Assume that

i i i i ithe cubic particles do not rotate while crossing through A. In this case V (> ) = D > with S (> ) =2

i iMV /M> = D , from which it immediately follows that = D .2 2

> i iUpon integrating (B1), the cumulative distribution function F (> ) of the distance > is

(A2)

(A3)

(A4)

(B1)

30

V i i V i > i i iThe cumulative distribution function F (V ) of the volume V is then F (V ) = F (> ) with > = V /D . 2

This yields

V i V iIn turn, the probability density function f (V ) = dF /dV is

and the average is

Thus, whereas = D is independent of the gradient, Mn/Mx, of active particles, depends 2

i i i ion this gradient. This occurs because S (> ) is symmetrical about D/2, whereas V (> ) is monotonic

i iover 0 # > # D. At lowest order = D /2 = V (D)/2. 3

The averages and for spherical particles are obtained in a similar manner, although

the derivation is far lengthier. Like cubic particles, the average = (B/12)D is independent 2

i pof the gradient Mn/Mx, and at lowest order the average = V (D)/2 = (1/2)V . Moreover, D

p= (B/12)D = (1/2)V .3

Appendix C: Probability Distribution of Activity in EnsembleaFor a streambed area B, let N 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

athe total number of configurations involving N particles distributed among m partitions is

eUsing the language of statistical mechanics, we may refer to each of these N configurations as a

1 2 3 m“macrostate.” In turn, if n , n , n , ... n denote the number of particles in each of the m partitions

eof an individual macrostate, then using Maxwell-Boltzmann counting there are n ways in which the

a eN particles may be rearranged amongst the m partitions, and we may refer to each of these narrangements as a “microstate.” The number of microstates in a given macrostate is

e eand the total number of microstates M over N macrostates is

(B2)

(B3)

(B4)

(B5)

(C1)

(C2)

31

aAs a point of reference, for N = 10 particles distributed among m = 2 partitions, there are a total of

e e a eN = 11 macrostates and a total of M = 1,024 microstates. For N = 10 and m = 5, there are N =

e a e1,001 macrostates and M = 9,765,625 microstates. And, for N = 10 and m = 10, there are N =

e a92,378 macrostates and M = 1 × 10 microstates. For N = 5 and m = 10, there are 2,00210

amacrostates and 100,000 microstates. And, for N = 5 and m = 20, there are 42,504 macrostates and3,200,000 microstates.

dB n dB eIf n denotes the number of particles within dB, then the proportion P (n ) of M microstates

dB dBhaving n particles within dB — that is, the probability distribution of n — is given by thebinomial distribution assuming each microstate is equally probable [Roseberry et al., 2012]. With

a alarge m relative to N , there is an increasing number of ways to partition N particles into m - 1 (or

dBm - 2, etc.) areas dB, so the likelihood of finding small numbers n within any dB increases, and the

n dB dBdistribution P (n ) is exponential-like, albeit decaying with increasing n faster than an exponential

a afunction. With decreasing m relative to N , there are fewer ways to partition N particles into an area

dB n dBdB having small n , and the distribution P (n ) takes an asymmetric form with finite mode. For

a n dBsmall m relative to N , the distribution P (n ) becomes Gaussian-like. As elaborated in Roseberryet al. [2012], this distribution forms the basis of the null hypothesis of spatial randomness in thepositions of active particles. We show that near-bed turbulence leads to decided patchiness inparticle positions, and that fluctuations in activity, and therefore in transport rates, are systematicallyrelated to the sampling area.

Appendix D: Means and Variances of Displacement DistancesThe definitions (30) through (33) are well known. Nonetheless, because these definitions are

not necessarily familiar, for completeness we show how they are obtained. For simplicity we omitthe conditional notation indicating a dependence on the activity (.

First note that by the product rule,

r rWith R (r) = 1 - F (r), substitution leads to

Integrating this from r = 0 to r = 4,

Evaluating the last integral then leads to

r rinsofar as the limit of rR (r) as r 6 4 is equal to zero. This is guaranteed if R (r) decays at least as

(C3)

(D1)

(D2)

(D3)

(D4)

32

rfast as a negative exponential function, in which case the product rR (r) looks like r/e , whose limit r

ris zero as r 6 4. Indeed, the absence of this condition being satisfied implies that f (r) is a heavy-tailed distribution without finite mean. In turn,

or

Integrating this from r = 0 to r = 4,

r rinsofar as the limit of r R (r) as r 6 4 is equal to zero. Again, this is guaranteed if R (r) decays at2

rleast as fast as a negative exponential function, in which case the product r R (r) looks like r /e ,2 2 r

whose limit is zero as r 6 4. The absence of this condition being satisfied implies that fr(r) is a

l lheavy-tailed distribution without finite variance. A similar development involving f (l) and R (l)

l lleads to comparable expressions for : and F .2

Appendix E: Formulation of the Deposition RateAssuming one-dimensional transport parallel to x, previous formulations [e.g. Parker et al.,

2000; Ganti et al., 2010] of the deposition rate D analogous to (55) have the form:

which neglects the idea that particles arriving at position x at time t started their hops 8 at manydifferent times t - J. As written, (E1) either assumes that particle hops 8 effectively occurinstantaneously, or that E and D are steady (in which case the appearance of time t is unnecessary)

8and f (8) is merely the uniformly distributed proportion of particles entrained at x - 8 which steadilyarrive at x.

Consider instead the following unsteady form of (E1):

8which allows for finite travel time J between entrainment and deposition. But now f (8; J) stillcannot be interpreted as a hop distance distribution as originally defined by Einstein [1950]. Rather,

8as described in section 3.4, f (8; J) is now akin to what Hill et al. [2010] and others refer to as adistribution of travel distances 8, where particles might experience multiple hops, with waitingtimes, during the specific interval J. This requires integration over all J, and therefore knowledge

8of how f evolves with J. It is instead far more straightforward and practical to consider the jointprobability density of hop distances 8 and associated travel times J and write

8, Jwhich is the one-dimensional version of (55). Note that D, E and f may each be unsteady and

(D5)

(D6)

(D7)

(E1)

(E2)

(E3)

33

nonuniform.The associated formulation of the rate of deposition of particle tracers is written as

TInasmuch as the tracer fraction f is assumed to vary with time, then as with (E1) above, it must beassumed that particle travel times are everywhere negligible.

Appendix F: Entrainment Form of Exner EquationWe expand the integrand in (55) as a Taylor series to first order about t and to second order

about x and y, namely

Substituting (F1) into (55), rearranging, and momentarily letting d7 = d8dRdJ, leads to

8, R, Jwhere the unwritten limits of integration match those of (55). The triple integral of f in the firstterm on the right side of (F2) by definition equals unity. Then, because the order of integration does

8 R Jnot matter, selectively integrating to obtain the marginal distributions f , f and f , and the joint

8, Rdistribution f ,

The first three integrals in (F3) equal the mean hop distances and and the mean travel time .

The fourth and sixth integrals equal the second moments and . The double integral equal the

averaged product . With these definitions, (F3) looks like (56).

(E4)

(F1)

(F2)

(F3)

34

Appendix G: Relation Between Flux DefinitionsAs described in section 3.1, consider a planar streambed area B large enough to sample steady,

homogeneous near-bed conditions of turbulence and transport. At any instant the number of activeparticles is approximately constant. That is, the rate of disentrainment within B equals the rate ofentrainment, and the rate at which particles leave B across its boundaries equals the rate at whichparticles enter B across its boundaries. Imagine recording particle motions within B for an interval

s sof time T [t] [e.g. Lajeunesse et al., 2010; Roseberry et al., 2012]. For T much longer than the

smean particle travel time, particle motions during T adequately represent the joint probability

8, Jdensity f (8,J) of hop distances 8 and travel times J without bias due to censorship of motions at

s 8 Jtimes t = 0 and t = T [Furbish et al., 1990]. The marginal distributions f (8) and f (J) possess means

and . And, at any instant the ensemble average particle velocity is .

iThe average velocity of the ith (individual) particle with travel time J is

s s sIn turn, letting N denote the number of particle motions during T , and assuming that N is large, theensemble average velocity

Thus, contrary to the assertion of Lajeunesse et al. [2010], the ensemble average hop distance

indeed is equal to the product of the ensemble averaged velocity and the mean travel time

Furbish et al. [2012a].

xThe quasi-steady (“equilibrium”) volumetric flux q on a planar bed, when written as anequivalence between its “flux” form and its “entrainment” form, is

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

That is, under steady conditions the activity ( = E , or

where, now, has the simple interpretation of being the mean residence time of particles within the

nominal volume (B. Thus, (G3), (G4) and (G5) show the relation between the two forms of the flux

xq .As a point of reference, when particles continue their motions indefinitely (that is, they do not

i sstart and stop), then experimentally J = T (the sample time) and (G2) becomes

(G1)

(G2)

(G3)

(G4)

(G5)

35

swhere now is the average displacement during T , and the average in (G6) is the same as the

average of an individual particle over long time.

NotationA surface area [L ].2

b width normal to flux [L].B streambed area [L ].2

c concentration.

bc volumetric particle concentration of bed.d7 differential equal to d8dRdJ [L t].2

D particle diameter [L]; volumetric particle deposition rate per unit streambed area [Lt ].-1

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

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

Af probability density function of velocities ui intersecting A [L t].-1

r lf , f probability density functions of displacements r and l [L ].-1

r*( l*(f , f conditional probability density functions of displacements r and l [L ].-1

(f probability density function of activity ( [L ].-1

(, rf joint probability density function of ( and r [L ].-2

(, r, sf joint probability density function of (, r and s [L ].-3

(F cumulative probability distribution function of (.

8f probability density function of 8 [L ].-1

Jf probability density function of J [t ].-1

8, Jf joint probability density function of 8 and J [L t ].-1 -1

8*Jf conditional probability density function of 8 [L ].-1

8, Jf joint probability density function of 8 and J [L-1 t-1].

8, R, Jf joint probability density function of 8, R and J [L t ].-2 -1

8, Rf joint probability density function of 8 and R [L ].-2

r lF , F cumulative probability distribution functions of r and l.

r*( l*(F , F cumulative conditional probability distribution functions of r and l.

Tf fraction of lead load particles that are tracers.

uf probability density function of particle velocity u [L t]-1

V if probability density function of volume V [L-3].

V iF cumulative probability distribution function of volume V .

> if probability density function of distance > [L-1].

> iF cumulative probability distribution function of distance > .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].

(G6)

36

L length scale [L].m number of partitions of B.

A p pM mask projected on surface A, equal to 1 - H(u )H(-u ).

eM total number of microstates in ensemble.n unit vector normal to A.n proportion of N particles per unit length [L ].-1

dBn number of particles within dB.

en number of microstates in a macrostate.

in number of particles in the ith partition of a macrostate.

x zn , n number of particles per unit length parallel to x and to z [L ]-1

xyn number of particles per unit area [L ].-2

N number of particles intersecting surface.

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

eN total number of configurations (macrostates) in ensemble.

sN number of particle motions.p probability that a particle moves in the positive x direction.

n e dBP proportion of N configurations having n particles within dB.

r l r r r l l lP , P proportions defined by P = f /(1 - F ) and P = f /(1 - F ) [L ].-1

8 J 8 8 8 J J JP , P proportions defined by P = f /(1 - F ) [L ] and P = f /(1 - F ) [t ].-1 -1

q probability that a particle moves in the negative x direction.q volumetric particle flux [L t ] and [L t ].-1 2 -1

Aq volumetric particle flux across surface A [L t ].2 -1

x yq , q volumetric particle flux components parallel to x and y [L t ].2 -1

x jq volumetric particle flux of jth size fraction [L t ].2 -1

Q volumetric particle discharge [L t ].3 -1

xQ volumetric particle discharge parallel to x [L t ].3 -1

iQ volume discharge of ith particle [L t ].3 -1

i+ i-Q , Q volume discharge of ith particle in positive and negative x direction [L t ].3 -1

r particle displacement parallel to x [L].

mr scale factor in Pareto distribution.

r l r r l lR , R functions defined by R = 1 - F and R = 1 - F .

r*( l*( r*( r*( l*( l*(R , R functions defined by R = 1 - F and R = 1 - F .s particle displacement parallel to y [L].

S cross-sectional area of particles intersecting A [L ]; source term, M(E )/Mt [L t ].2 -1

iS cross-sectional area of ith particle on surface A [L ].2

t time [t].

bT bed form timescale [t].

fT bed-form field timescale [t].

sT sampling interval [t].

tT turbulence timescale [t].u, v average particle velocity components parallel to x and y [L t ].-1

fu fluid velocity [L t ].-1

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

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

p pu , v particle velocity components parallel to x and y [L t-1].

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

37

r lu , u mean velocities associated with displacements r and l [L t ].-1

pU mean particle velocity [L t ].-1

RU mean virtual particle velocity, [L t ].-1

V volume of particles [L ].3

iV volume of ith particle to right of surface A [L ].3

+ -V , V volume of particles to right and left of surface A [L ].3

i+ i-V , V volume of ith particle to right and left of surface A [L ].g g 3

pV volume of particle [L ].3

w variable of integration associated with displacements r and l [L].x, y Cartesian coordinates in streamwise and cross-stream directions [L].

pz moving coordinate such that x = z - U J [L].

Sy specific position along y [L]." shape factor in Weibull and Pareto distributions.( particle activity [L].

p i ( individual particle activity defined by S /b [L].

j( particle activity of jth size fraction [L].* Dirac delta function [L ].-1

) increment.g small distance measured from front of particle [L].0 local elevation of streambed surface [L].6, 6 diffusivity, diffusivity tensor [L t ].2 -1

m6 molecular diffusivity [L t ].2 -1

r l6 , 6 diffusivities associated with displacements r and l [L t ].2 -1

xx yy xy6 , 6 , 6 elements of diffusivity 6 [L t ].2 -1

R5 virtual diffusivity, [L t ].2 -1

8 particle hop distance parallel to x [L].

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

8 J: , : mean hop distance [L] and mean travel time [t].

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

r lF , F second moments of particle displacements r and l during dt [L ].2 2 2

uF variance of particle velocities u [L t ].2 2 -2

J interval of time; particle travel time [t].

RJ mean residence time of particles in thickness h [t].R particle hop distance parallel to y [L].

Acknowledgments. We acknowledge support by the National Science Foundation (EAR-0744934), and appreciateAmelia Furbish’s insistence that we pay close attention to L’Hôpital’s rule.

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40

______________D. J. Furbish, Department of Earth and Environmental Sciences, Vanderbilt University, 2301 Vanderbilt Place, Station

B 35-1805, Nashville, Tennessee 37235, USA. (david.j.furbish@vanderbilt.edu)P. K. Haff, Earth and Ocean Sciences Division, Nicholas School of the Environment, Duke University, Durham, North

Carolina 27708, USA. (haff@duke.edu)J. C. Roseberry, Department of Earth and Environmental Sciences, Vanderbilt University, 2301 Vanderbilt Place, Station

B 35-1805, Nashville, Tennessee 37235, USA. (john.c.roseberry@vanderbilt.edu)M. W. Schmeeckle, School of Geographical Sciences, Arizona State University, Tempe, Arizona 85287, USA.

(schmeeckle@asu.edu)

Figure Captionsp pFigure 1. Definition diagram for surface integral of surface-normal velocities u = u @n of the

pdiscontinuous particle velocity field u at the surface A with width b. The surface A extendsupward to a height necessary to include all bed load particles, and arrows are representative of

pvector components u surrounding infinite sets of such vector components positioned over solid

pfraction. The vector field u at A is nonzero only over domain consisting of intersections of(moving) particles with A.

Figure 2. Definition diagram of a particle of diameter D moving with a positive velocity parallel

ito x through a surface A positioned at x = 0, where > denotes the distance that the nose of theparticle is relative to x = 0, and g denotes a small distance measured from the nose of theparticle.

Figure 3. Definition diagram showing cloud of particles moving with varying velocities parallel

ito x toward and through a surface A positioned at x = 0, where > denotes the distance that thenose of the ith particle is relative to x = 0.

i i pFigure 4. Plots of deviation in particle velocity u N = u - U versus deviation in particle cross-

i i psectional area S N = S - as viewed by observer moving with the average velocity U for (a)

buniform particle cloud with = 0 and = U , and (b) particle cloud where the particle activity

bdecreases with increasing distance x with > 0 and > U .

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

pmoves downstream with the average velocity U .

(Figure 6. Examples of the cumulative distribution F ((; x, t) obtained from the probability density

(function f ((; x, t) of the particle activity ( as this varies with width b for b = 50D, 100D, 500D

a p (and 1,000D with the same overall activity ( = N V /B = 0.05 units [L]. The variance of f ((; x,

(t) decreases with increasing b, as reflected by the increasing slope of F ((; x, t) near ( = 0.05.

(Individual values of ( = S/b used to generate F ((; x, t) are obtained numerically from 10,000configurations of particles uniformly (albeit randomly) distributed over an area dB = 10Db.

Figure 7. Definition diagram for particles motions parallel to x coordinate, showing probability

rdensity function f (r; xN, dt) of displacement distances r during dt.

r, JFigure 8. Schematic diagram of three realizations of the joint probability density function f (r, J),where a steep covariance relation (open circles) between r and J implies varying speeds due tovarying displacements over a similar travel time, a weak covariance (gray circles) impliesvarying speeds due to similar displacements over varying travel times, and an intermediatecovariance (black circles) implies relatively uniform speeds.

xyFigure 9. Schematic diagram showing effect of diffusive terms involving 6 , where, with negative

xyM((6 )/My due to decreasing activity ( along y, proportionally more particles starting from

S xpositions at y < y contribute (red arrows) to the flux q parallel to x across an elementary plane

41

S Sat y , relative to those particles represented by u( at y = y .

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