A probabilistic description of the bed load
sediment flux: 1. Theory
David Jon Furbish,1 Peter K. Haff, 2 John C. Roseberry,1 and Mark W.
Schmeeckle3
________
1
Department of Earth and Environmental Sciences and Department of Civil and Environmental
Engineering, Vanderbilt University, Nashville, Tennessee, USA.
2
Division of Earth and Ocean Sciences, Nicholas School of the Environment, Duke University,
Durham, North Carolina, USA.
3
School of Geographical Sciences, Arizona State University, Tempe, Arizona, USA.
Abstract. We provide a probabilistic definition of the bed load sediment flux.
In treating particle positions and motions as stochastic quantities, a flux form of
the Master Equation reveals that the volumetric flux involves an advective part
equal to the product of an average particle velocity and the particle activity (the
solid volume of particles in motion per unit streambed area), and a diffusive part
involving the gradient of the product of the particle activity and a diffusivity that
arises from the second moment of the probability density function of particle
displacements. Gradients in the activity, instantaneous or time-averaged,
therefore effect a particle flux. Time-averaged descriptions of the flux involve
averaged products of the particle activity, the particle velocity and the diffusivity,
whose significance depends on the averaging timescale. The flux form of the
Exner equation looks like a Fokker-Planck equation. The entrainment form of the
Exner equation similarly involves advective and diffusive terms, but because it is
based on the joint probability density function of particle hop distances and
associated travel times, this form involves a time derivative term that represents a
lag effect associated with the exchange of particles between the static and active
states. The formulation highlights that the probability distribution of particle
displacements figures prominently in describing particle motions across a range
of scales, notably bearing on the possibility of anomalous versus Fickian diffusive
behavior. The formulation is consistent with experimental measurements and
simulations of particle motions reported in companion papers.
1. Introduction
The bed load sediment flux, defined as the solid volume of bed load particles crossing a vertical
surface per unit time per unit width, figures prominently in descriptions of sediment transport and
the evolution of alluvial channels. Translating this definition of the flux into conceptually simple
quantities that accurately characterize the collective motions of particles, however, is not necessarily
straightforward, and quantitative definitions of the flux have several forms. We note at the outset
that, when viewed at the particle scale, the instantaneous solid volume flux qA [L2 t-1] across a
surface A [L2] is precisely defined as the surface integral of surface-normal velocities of the solid
1
fraction, namely
(1)
where up [L t-1] is the discontinuous particle velocity field viewed at the surface A, n is the unit
vector normal to A, and b [L] is the width of A (Figure 1). As a point of reference, because the
volumetric flux formally is a volume per unit area per unit time, the “flux” defined by (1) actually
is a vertically integrated flux. This precise definition, however, is impractical. Except possibly
using high-speed imaging of a small (observable) number of particles [Drake et al., 1988;
Lajeunesse et al., 2010; Roseberry et al., 2012] at high resolution, the flux described by (1) is
virtually impossible to measure, and we are far from possessing a theory of sediment transport that
describes the velocity field up as it responds to near-bed turbulence [Parker et al., 2003].
Conventional descriptions of the flux therefore instead appeal to measures of collective particle
behavior, specifically averaged quantities such as the average particle velocity and concentration,
to replace the detailed information contained in the particle velocity field up at the surface A.
With equilibrium (i.e. quasi-steady and uniform) bed and transport conditions, for example, the
sediment flux normally is defined in “flux form” as the product of a mean particle velocity Up [L t-1]
and a particle concentration, namely, the volume of particles in motion per unit streambed area [e.g.
Bridge and Dominic, 1984; Wiberg and Smith, 1989; Seminara et al., 2002; Parker et al., 2003;
Francalanci and Solari, 2007; Wong et al., 2007; Lajeunesse et al., 2010], herein referred to as the
bed load particle activity ( [L]. That is, for one-dimensional transport in the x direction the flux qx
[L2 t-1] is
(2)
with the caveat that Up and ( represent macroscopic quantities averaged over stochastic fluctuations
[Wong et al., 2007]. Note that this is like the definition of advection associated with a continuous
medium. As elaborated below, to describe the sediment flux as the product of a mean velocity and
a concentration indeed assumes a continuum behavior where active (moving) particles are uniformly
(albeit quasi-randomly) distributed. But as recently noted [Schmeeckle and Furbish, 2007; Ancey,
2010], the continuum assumption is rarely satisfied for sediment particles transported as bed load,
particularly at low transport rates [Roseberry et al., 2012], and the details of the averaging, whether
involving ensemble, spatial or temporal averaging [Coleman and Nikora, 2009], matter to the
physical interpretation as well as the form of the definition of the flux. Ancey [2010] notes in his
review of several definitions of the flux that it remains unclear how the flux is actually related to the
mean particle velocity and the particle concentration.
Another important definition of the bed load sediment flux is the “entrainment form” of this
quantity, first introduced by Einstein [1950] and recently elaborated by Wilcock [1997], Parker et
al. [2001], Seminara et al. [2002], Wong et al. [2007], Ganti et al. [2010] and others. By this
definition, with quasi-steady bed and transport conditions the flux is equal to the product of the
volumetric rate of particle entrainment per unit streambed area, E [L t-1], and the mean particle hop
distance, [L], measured start to stop. That is,
(3)
2
This is essentially a statement of conservation of particle volume where, assuming spatially uniform
transport, rates of entrainment and deposition are steady and everywhere balanced. The value of this
definition is highlighted in treating tracer particles [Ganti et al., 2010], notably involving exchanges
between the active and inactive layers of the streambed [Wong et al., 2007]. What is unclear is how
the ingredients of (3), notably the distribution of particle hop distances with mean , translate to
unsteady and nonuniform conditions [Lajeunesse et al., 2010], and the extent to which this and other
definitions overlap or match (2) [Ancey, 2010]. On this point we note that any formulation of the
flux must be consistent with (1).
At any instant the solid volume of bed load particles in motion per unit area of streambed, the
particle activity (, can vary spatially due to short-lived near-bed turbulence excursions as well as
longer-lived influences of bed form geometry on the mean flow [Drake et al., 1988; McLean et al.,
1994; Nelson et al., 1995; Schmeeckle and Nelson, 2003; Singh et al., 2009; Roseberry et al., 2012].
During their motions, particles respond to turbulent fluctuations and interact with the bed and with
each other so that, at any instant within a given small area, some particles move faster and some
move slower than the average within the area, and fluctuations in velocity are of the same order as
the average velocity. Moreover, a hallmark of bed load particles is their propensity to alternate
between states of motion and rest over a large range of timescales. These attributes mean that, in
relation to the definition (2) above, the bed load particle flux involves an advective part, as is
normally assumed, but more generally it also involves a diffusive part associated with variations in
particle activity and velocity [Lisle et al., 1998; Schmeeckle and Furbish, 2007; Furbish et al.,
2009a, 2009b]. In relation to the definition (3) above, because the distribution of particle hop
distances may be considered the marginal distribution of a joint probability density function of
particle hop distances and associated travel times [Lajeunesse et al., 2010], a more general form of
this definition similarly involves a diffusive part as well as a time derivative term that represents a
lag effect associated with the exchange of particles between the static and active states.
The purpose of this contribution is to clarify the points above, namely, how variations in particle
activity and velocity influence the volumetric bed load flux, q = iqx + jqy [L2 t-1], with components
qx and qy parallel to the coordinates x [L] and y [L], selected here to coincide with the average
surface of the sediment-water interface viewed over a length scale much larger than that of sediment
bedforms such as ripples or dunes. Our analysis involves a probabilistic formulation wherein
particle positions and motions are treated as stochastic quantities, leading to a kinematic description
of q that illustrates how and why it involves both advective and diffusive terms, borrowing key
elements 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 c
having the form Mc/Mt = -L@q [Ancey, 2010], and then by inspection extract from this statement a
kinematic description of the flux q [e.g. Furbish et al., 2009a], therein revealing that it has both
advective and diffusive parts. More challenging, however, is to formulate a definition of q directly
from a description of particle motions as Einstein [1905] did for Brownian motions. This is
particularly desirable inasmuch as a direct formulation more fully clarifies the geometrical and
kinematic ingredients of the flux, including its relation to such quantities as particle hop distances
and velocities [e.g. Drake et al., 1988; Wilcock, 1997b; Wong et al., 2007; Ancey, 2010], and
variations in particle velocity and activity.
In section 2 we formulate a qualitative version of the one-dimensional flux qx, with the purpose
of clarifying key geometrical and kinematic ingredients in the problem, notably particle size, shape
and velocity, and spatial variations in particle concentration. We show how the definitions (1) and
3
(2) are related. This provides the basis for illustrating that, in appealing to averaged particle
quantities (specifically the mean particle velocity and concentration) to replace the detailed
information contained in the discontinuous particle velocity field up at the surface A, the resulting
description of the flux must in general involve both advective and diffusive parts. In section 3 we
provide a more formal, probabilistic description of the one-dimensional flux, and describe the
implications of different definitions of the probability distribution of particle displacements versus
hop distances. In the final part of this section we generalize to the two-dimensional case. In section
4 we show the flux form of the Exner equation to illustrate how it is like a Fokker-Planck equation,
and for comparison we obtain the entrainment form of the Exner equation to illustrate how this form
involves a time derivative term (not contained in the Fokker-Planck equation) that represents a
“memory” (or lag) effect associated with the exchange of particles between the static and active
states. This result has implications for the use of the related entrainment formulation of conservation
of tracer particles. In section 5 we elaborate how ensemble, spatial and temporal averaging matter
in defining the flux, and we consider time averaging of the flux to suggest how persistent spatial
variations in particle activity associated with bed forms influence the flux. For simplicity
throughout, we consider transport of a single particle size, then briefly comment on the problem of
generalizing the formulation to mixtures of sizes in section 6.
As noted by Ancey [2010] and others, there is no unique way to define the solid volumetric flux.
Nonetheless, an unambiguous, probabilistic definition exists. Beyond a definition of the bed load
flux, moreover, the formulation highlights that the probability distribution of particle displacements,
including details of how this distribution is defined, has a central role in describing particle motions
across a range of scales. This is particularly significant in view of a growing interest in the
possibility of non-Fickian behavior in the transport of sediment and associated materials [e.g. Nikora
et al., 2002; Schumer et al., 2009; Foufoula-Georgiou and Stark, 2010; Bradley et al., 2010; Ganti
et al., 2010; Voller and Paola, 2010; Hill et al., 2010; Martin et al., 2012], and in relation to
connecting probabilistic descriptions of particle motions with treatments of fluid motion.
In companion papers [Roseberry et al., 2012; Furbish et al., 2012a, 2012b] we present detailed
measurements of bed load particle motions obtained from high-speed imaging in laboratory flume
experiments. These measurements support key elements of the formulation described here. [Note:
Although not yet accepted for publication, the companion papers Roseberry et al. [2012] and
Furbish et al. [2012a, 2012b] are cited with a 2012 date for simplicity of reference.]
2. Geometrical Ingredients of the One-Dimensional Flux
2.1. The Surface-Integral Flux
As an important reference point, here we present a discrete version of (1) to reveal details of
particle shape and motion that figure into this deterministic definition of the flux. This provides the
basis for illustrating that, in appealing to averaged particle quantities (specifically the mean particle
velocity and concentration) to replace the detailed information embodied in (1), the resulting
description of the flux must in general involve both advective and diffusive parts. We start with a
rendering of the geometry and motion of a single particle.
Consider a particle of diameter D [L] that is moving parallel to x through a surface A positioned
at x = 0 (Figure 2). Let >i [L] denote the position of the nose of the particle relative to x = 0, and let
Vi(>i) [L3] denote the volume of the particle that is to the right of x = 0 as a function of >i. The
particle volume discharge Qi(t) across A is (Appendix A)
(4)
4
where Si(>i) = MVi/M>i [L2] is like a hypsometric function of the particle, equal to its cross-sectional
area on the surface A at x = 0, and ui = d>i/dt [L t-1] is its velocity parallel to x.
Consider, then, a cloud of equal-sized particles which are moving with varying velocities parallel
to x toward and through a surface A of width b positioned at x = 0 (Figure 3). At time t a number
N(t) of particles intersects A. If >i now denotes the distance that the nose of the ith particle is relative
to x = 0, then the instantaneous volumetric flux qx across A is
(5)
This is a discrete version of (1). Namely, if the particle velocity (field) parallel to x is up = up@n, and
if H(up) is the Heaviside step function defined by H(up) = 0 for up < 0 and H(up) = 1 for up $ 0, then
MA(t) = 1 - H(up)H(-up) denotes a “mask” projected onto A such that MA = 1 where up … 0 and MA =
0 elsewhere [Furbish et al., 2009b], whence
(6)
and
(7)
which shows the relation between (1) and (5) with qA = qx. Note that N(t) is a stepped function of
time as particles intersect and lose contact with A. At any instant, therefore, the derivative dN/dt
strictly is either zero or undefined. Nonetheless, for sufficiently large N and rapidity of particles
intersecting and losing contact with A, one can envision that N(t) begins to appear as a “smooth”
function of time where brief fluctuations in qx become small relative to the magnitude of qx.
Letting an overbar denote an average over N particles, the last part of (5) may be written as qx
= (1/b)N
. For equal-sized particles, moreover, it is reasonable to assume that Si and ui are
uncorrelated, as there is no reason to suspect that, at any instant, particles intersecting A with large
(or small) cross-sectional area Si are any more (or less) likely to possess large (or small) velocity ui.
In this case,
(8)
The product N
= S [L2] is the cross-sectional area of particles intersecting A, and the ratio S/b
= ( [L2 L-1 = L3 L-2] is equivalent to the particle activity, the volume of active particles per unit
streambed area. Specifically, this is a local “line” averaged activity. Because Sdx is equal to the
volume of active particles within a small interval dx, Sdx/bdx = S/b = ( is the volume of active
particles within the small area bdx. Then, if it is assumed that is equal to the average velocity Up
of all particles in the cloud in the vicinity of A, that is Up =
, one may conclude that qx = (Up,
which is the definition (2) of the flux normally assumed for quasi-steady bed and transport
conditions [e.g. Bridge and Dominic, 1984; Wiberg and Smith, 1989; Seminara et al., 2002; Parker
et al., 2003; Francalanci and Solari, 2007; Wong et al., 2007; Lajeunesse et al., 2010]. Two
caveats, however, must accompany this assessment of averages.
5
First, envision a uniform cloud of equal-sized particles moving with varying velocities parallel
to x toward and through a set of surfaces A located at various positions along x. By “uniform” we
mean the following. For a specified width b, let nx(x, t) denote the number of particles per unit
distance parallel to x, such that nx(x, t)dx is the number of particles whose noses are located within
any small interval dx. Then, for a sufficiently large width b, assume that nx(x, t) varies negligibly
with x. At any instant the number of particles N and the corresponding particle area S = N
intersecting each surface is the same, although the detailed configuration of S varies from surface
to surface. Let Up denote the average particle velocity parallel to x, that is, the average of all
particles in the cloud near any surface A rather than the average of particles intersecting a surface
A. Because the cloud is uniform, each surface A “samples” at any instant the full distribution of
possible velocities (for sufficiently large width b), in which case = Up for all surfaces. This is the
situation for which Ancey [2010] notes that an ensemble-like average over A, giving
, is equivalent
to a “volume” average over the particle cloud, giving Up. In contrast, envision a cloud of particles
with average velocity Up whose concentration nx(x, t) at some instant decreases with increasing
distance x. Now, both the number of particles N and the particle area S = N intersecting each
surface decrease with increasing x. Moreover, in this case the surface and volume averages are not
equivalent, and > Up. Here is why.
Let a prime denote a fluctuation about an average. Then, at any instant Si =
+ uiN. In turn, qx = (1/b)N(
+ SiN and ui = Up
). Consider a plot of uiN versus SiN at an
instant (Figure 4), which provides a perspective as viewed by an observer moving with the average
velocity Up, although the conclusions below pertain equally to an Eulerian frame of reference.
During a small interval of time dt, some points on this plot move to the right as the cross-sectional
area SiN increases for particles that are beginning to cross A, and some points move to the left as the
cross-sectional area SiN decreases for particles that have mostly crossed A. The rate of motion of the
points to the right and left is proportional to the magnitude of the particle velocity uiN, so motion is
faster near the top and bottom and slower near the middle of the plot. Points at uiN = 0 along the SiN
axis do not move during dt. Some points vanish as particles leave A, and new points appear as
particles arrive at and initially intersect A. Points arrive at the far left of the second and third
quadrants where the small areas of intersection of arriving particles are less than the average
intersection area, and move to the far right of the first and fourth quadrants as their fat middles
exceed the average intersection area, and then move back to the far left of the second and third
quadrants because the intersection area of their exiting tails is less than the average intersection area.
In the case of the uniform cloud of particles described above, the number of points and their
scatter is similar across all surfaces A and on each surface over time, and
= Up with
= 0 (Figure 4a), so that qx = (1/b)N
,=(
= (Up as in (8) or (1).
In the second case where the particle activity decreases with increasing x, this situation changes.
At any instant the number of particles to the immediate left of A is greater than the number to the
immediate right of A. The likelihood that a particle to the left or right of A will intersect A during
a small interval of time dt, for a given magnitude of the velocity uiN, increases with its proximity to
A, and, for a given proximity to A, increases with the magnitude of its velocity uiN. Of the particles
that are at any given distance to the left of A, the faster ones (large positive uiN) are more likely than
are slower ones to reach A. And, of the particles that are at any given distance to the right of A, the
6
slower ones (large negative uiN) are more likely than are faster ones to reach A. Because of the
greater number of particles to the left of A than to the right of A, the plot of uiN versus SiN becomes
preferentially 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
is, the surface A “sees” an average velocity
> Up where
= Up +
with
=0
for the same particle surface area S = N . In turn, the flux
(9)
in which an extra term involving velocity fluctuations about the mean appears in the definition of
qx. The counterpart to this situation occurs when the particle activity ( increases with distance x,
in which case < Up.
The effect embodied in (9) can be readily visualized by considering the motion of a triangular
cloud of particles which possess two velocities, 1 and 2, in equal proportions (Figure 5). The
average velocity of all particles in the cloud is Up = 1.5. During a short interval of time dt the
particles begin to segregate. At any position x in front of the crest of the cloud there is a greater
proportion of fast particles, and at any position x behind the crest there is a greater proportion of
slow particles. The average velocity of particles intersecting a surface A in the leading (fully)
segregated part of the cloud is 2, and the average velocity of particles intersecting a surface A in the
trailing (fully) segregated part is 1. The average velocity of particles intersecting a surface A at any
x in front of the crest is greater than Up, and the average velocity of particles intersecting a surface
A at any x behind the crest is less than Up. The cloud as a whole moves downstream with velocity
Up. One must be careful, however, to limit this idea to small time dt, as it neglects time variations
in particle velocities, including starting and stopping.
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
(1/b)N(
represents advection, the product
) is equivalent to a diffusive term that looks like -(1/2)M(6()/Mx, where 6 [L2 t-1] is a
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
assumed to equal the overall average Up) 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
distance >1 = D/2 so that S1(>1) = (1/4)BD2 is at its largest value, and S1 = . Also suppose that u1
=
is a small value. Assuming Q = N
, this correctly gives Q = S1u1. But suppose that a
second, fast moving particle also intersects A where at time t the distance >2 is small so that S2(>2)
n S1(>1), whereas u1 n u2. The actual discharge Q = S1u1 + S2u2. The discharge using averaged
quantities is calculated as Q = N
= (S1 + S2)(u1 + u2)/2 . S1u2/2, 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
7
with non-rotating cubic particles with constant Si), this example points to the idea that the summed
product in (5) generally does not equal the product of the averages for small N. Moreover, as
elaborated in section 6, when considering a mixture of particle sizes, the covariance between Si and
ui cannot be neglected inasmuch as some particle sizes preferentially move faster than other sizes.
2.2. The Geometry of Diffusion
We emphasize that the precise definition (5) of the flux qx(t) requires knowledge of >i, Si and ui
for all N(t) particles intersecting A, and we now turn to the consequences of appealing to averaged
quantities to replace this detailed information. We begin by providing an approximate geometrical
interpretation 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
varying velocities u. Let fu(u) denote the probability density function of the velocities of particles
whose noses are located within any elementary area bdx at all positions x. We assume for simplicity
that fu(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
velocity of a particle without reference to its proximity to a surface A, ui pertains to the ith particle
of N particles that intersect A.
Let J denote a small interval of time, and consider a particle whose nose is at position xN < x at
time J = 0. In order for this particle to intersect a fixed surface A at position x at time J, it must
travel 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 whose
noses is at position xN such that x # xN # x + D at time J = 0 initially intersects A. In order for this
particle to remain in contact with A during J (assuming only downstream motion), it must travel a
distance greater than or equal to zero but less than or equal to x + D - xN. That is, it must possess a
velocity between u = 0 and u = (x + D - xN)/J. The total number of particles N(x, J) intersecting a
fixed surface A at position x after a small interval of time J is therefore
(10)
where, as above, nx(x) is the number of active particles per unit distance parallel to x at J = 0.
If Up denotes the average particle velocity, then based on recent experiments [Lajeunesse et al.,
2010; Roseberry et al., 2012] we assume for illustration that
(11)
Here we stress that Up is equal to the average velocity of all particles in the cloud. Substituting (11)
into (10) and integrating with respect to u,
(12)
We now assume that
8
(13)
which describes a linear variation in nx about the position x. Substituting (13) into (12) and
integrating with respect to xN,
(14)
In the presence of a uniform particle gradient (Mnx/Mx = 0), the total number of particles intersecting
A is N(x, J) = nx(x)D at time J. The second term on the right side of (14) is a correction due to the
variation in nx over D. The last term in (14) describes the change in N(x, J) at x during J due to the
overall downstream (advective) motion of the particles. It follows that, locally at x, MN/MJ = DUpMnx/Mx.
Let z denote a moving coordinate such that x = z - UpJ, where dz/dJ = Up = -dx/dJ. Then, in a
reference frame moving with the average velocity Up, nz(z) = nx(x)*J = 0, Mnz/Mz = Mnx/Mx*J = 0, and
(15)
where locally at z, MN/MJ = 0. Thus, at small time J, the position z “sees” a steady number of active
particles intersecting A (which is moving with velocity Up) and a steady gradient in nz.
Consider the ith particle that intersects A at time J possessing the velocity ui. This particle
reached A with velocity ui = u during J, starting from a position greater than or equal to xN = x - Ju
but less than or equal to xN = x + D - Ju. Because nx(xN)dxN is the number of particles within the
interval xN to xN + dxN initially (J = 0), fu(u)nx(xN)dxNdu is the number of particles within this interval
possessing a velocity from u to u + du. Thus, the integral of this product from x - Ju to x + D - Ju
is the total number of particles intersecting A at time J possessing a velocity from uN = u to uN + duN
= u + du. The probability density fA(ui) of velocities ui for particles intersecting A after a small
interval J is therefore
(16)
Substituting (13) into (16) and integrating with respect to xN,
(17)
Substituting (11) into (17), multiplying by u and integrating from zero to infinity, the average
velocity of particles intersecting A is
(18)
9
Thus, with Mnx/Mx < 0, the average
is greater than the average Up; and with Mnx/Mx > 0, the average
is less than the average Up.
In turn, with u = Up + uN, the probability density fA(uiN) of fluctuating velocities uiN for particles
intersecting A after a small interval J is
(19)
Multiplying (19) by uN and integrating from uN = -Up to infinity, the average of the fluctuating
velocities of particles intersecting A in a reference frame moving with the average velocity Up is
(20)
This is the “diffusive” velocity contribution to the surface-integral flux across A. Specifically,
multiplying (20) by S/b = N /b,
(21)
Here we note that
D = (1/2)Vp for spherical particles intersecting A (Appendix B). In turn the
particle activity ( = Vpnx/b, so
(22)
where 6 = JUp 2 is like a diffusivity. This demonstrates that qxN, expressed in the form of a diffusive
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
density function (11), the variance Fu2 = Up2, so 6 = JUp 2 = JFu2, highlighting that the diffusive flux
(22) fundamentally is associated with velocity fluctuations. Indeed, if J is identified as the
Lagrangian integral timescale obtained from the autocorrelation function of the particle velocities
u, then this definition of 6 is equivalent to the classic definition provided by Taylor [1921] [see
Furbish et al., 2012b].
We emphasize that this idealized formulation is aimed at showing the consistency between the
diffusion and surface-integral forms of the flux, rather than providing a general description of the
flux. As such we have assumed that fu(u) (and Up) are invariant along x, and that nx initially varies
linearly in the vicinity of position x. The formulation therefore neglects any variations in fu(u) (and
Up) along x, possible changes in Mnx/Mx and particle velocities u during J, and the possibility that
particle motions start or stop during J. Thus, J must be considered small, that is, smaller than the
autocorrelation timescale of particle velocities. In addition, whereas the diffusivity 6 in (22) is
dimensionally sound, its dependence on time J as written above is imprecise. In section 3 we turn
to 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 set
the magnitude of the diffusivity 6 [Ball, 2012].
10
3. Probabilistic Formulation of the One-Dimensional Flux
Here we present a more careful rendering of the collective behavior of particles to define the bed
load sediment flux, wherein particle positions and motions are treated as stochastic quantities. The
explicit functional notation used in this section, although bulky in places, figures importantly in the
bookkeeping of the formulation. In functions such as f(((; x, y, t) (defined below), random
variables, ( in this example, appear first within the parentheses (and as subscripts, which identify
the probability density or distribution), followed by parametric quantities or independent variables
after the semicolon. Here a “parametric quantity” means a key quantity that is not a random
variable, and which can be treated mathematically as an independent variable. In a conditional
function such as fr*((r*(; x, dt), the quantity providing the conditioning, ( in this case, is to be
considered a parameter, so this function could just as well be written, for example, as fr;((r; (, x, dt).
3.1. Ensemble States of Particle Motions
Because 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 into
account in a way where the local, instantaneous flux can be systematically related to spatially
averaged or time-averaged expressions of the flux, and vice versa. We approach this by envisioning
an ensemble of configurations of particle positions and velocities in a manner similar to (but not
identical to) that outlined by Gibbs [1902] for gas particle systems. As Kittel [1958] notes, “The
scheme introduced by Gibbs is to replace time averages over a single system by ensemble averages,
which are averages at a fixed time over all systems in an ensemble. The problem of demonstrating
the equivalence of the two types of averages is the subject of ergodic theory... It may be argued, as
Tolman [Tolman, 1938] has done, that the ensemble average really corresponds better to the actual
situation than does the time average. We never know the initial conditions of the system, so we do
not know exactly how to take the time average. The ensemble average describes our ignorance
appropriately.” In turn, the ergodic hypothesis suggests that (for gas systems) one may assume an
ensemble average is the same as a time average over one realization, that is, a single system that
evolves through time. Here we define the essentials of an ensemble appropriate to sediment particle
motions. We use this as a starting point for our probabilistic formulation of the flux, and then return
to it later to suggest how persistent time-averaged variations in particle activity associated with bed
forms influence the flux.
Envision bed load particles moving over an area B [L2] on a streambed that is subjected to steady
macroscopic flow conditions, and momentarily assume for simplicity that the streambed is planar
[e.g. Lajeunesse et al., 2010; Roseberry et al., 2012], albeit possibly involving small, stationary
fluctuations in elevation [e.g. Wong et al., 2007]. Over time, some particles stop and others start,
some particles leave the area B across its boundaries and others arrive. We choose B to be
sufficiently large that, during any small interval of time dt, any difference in the number of particles
leaving B and the number arriving is negligibly small relative to the total number Na of active
particles within B. Similarly, any difference in the number of particles that stop and start within B
during dt is negligibly small relative to the total number Na of active particles. Then, Na may be
considered the same from one instant to the next. We now envision all possible instantaneous
configurations of the Na active particles as defined by their xy positions within B at a fixed time, with
the understanding that this set of configurations need not represent the same set of particles, only
that Na is the same. This imagined set of possible configurations constitutes an ensemble of active
particle positions, and, in the absence of any additional information, we initially assume that each
configuration in the ensemble is equally probable (but see Roseberry et al. [2012]).
11
Consider an elementary area dB within B. If nxy(x, y, t) [L-2] denotes the number of active
particles per unit area, then nxy(x, y, t)dB is the number of particles within dB and the associated
activity ((x, y, t) = Vpnxy(x, y, t) such that ( may be considered a random variable. One can then
envision that the ensemble of configurations of particle positions, each equally probable, yields for
any area dB a probability density function of the activity (, namely f(((; x, y, t) [L-1], such that f(((;
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 magnitude
of 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
to x. Let dB = bdx. For a specified width b, if nx(x, t) [L-1] denotes the number of active particles
per unit distance parallel to x, then nx(x, t)dx is the number of active particles within bdx. The local
activity at position x is ((x, t) = Vpnx(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
area B and total number of particles Na with overall activity ( = NaVp/B, envision a large number of
configurations where, in each configuration, Na particles are randomly distributed over B. Each
configuration gives a different activity ((x, t) = S/b calculated at one position x. Hence the ensemble
of particle configurations, each equally probable, yields for any position x a probability density
function of the activity (, namely f(((; x, t) [L-1]. As the width b increases, the number of particles
intersecting a surface at x on average increases. This means that for a given overall activity the form
of f(((; x, t) varies with b (Figure 6). Specifically, whereas the mean activity at x associated with
this distribution is equal to the overall activity calculated by ( = NaVp/B, the variance of f(((; x, t)
decreases with increasing b, which reflects on average smaller fluctuations in the number of particles
intersecting the surface at x. Moreover, any actual realization of the activity at an instant in effect
is a “sample” from f(((; x, t), so the variability in such realizations from one instant to the next
decreases with increasing b. We reconsider this point below and in Roseberry et al. [2012].
Returning to the two-dimensional case, each active particle in each possible configuration
possesses an instantaneous velocity up = iup + jvp at time t. One can therefore associate with each
particle at time t the small (pending) displacements r [L] = updt and s = vpdt [L] parallel to x and y,
respectively, that occur during dt, that is, between t and t + dt. For each configuration there is a joint
probability distribution of r and s associated with Na particles. But because within any elementary
area dB the number of active particles nxy(x, y, t)dB, and thus the activity ((x, y, t), varies among
configurations, there are likewise nxydB values of the pair r and s for each configuration.
Furthermore, we must leave open the possibility, elaborated below, that the velocities up, and
therefore the displacements r and s, of the nxydB particles within dB are correlated with the number
of active particles nxydB. We now envision the ensemble as consisting of all possible instantaneous
states defined by the joint occurrence of particle positions and displacements r and s, and we assume
this ensemble defined over B yields for any area dB a joint probability density function of the
activity ( and the displacements r and s, namely f(, r, s((, r, s; x, y, dt) [L-3], where certain values of
(, r and s, and their combinations, are more (or less) probable than are others. Like f(((; x, y, t), the
form of f(, r, s((, r, s; x, y, dt) and its parametric values are specific to the sediment (size, shape) and
the macroscopic flow conditions, including the turbulence structure.
Specifically, among the ensemble of possible configurations of particle positions and velocities,
some configurations may be preferentially selected or excluded by the turbulence structure inasmuch
as turbulent sweeps and bursts characteristically lead to patchy, fast-moving clouds of particles
[Schmeeckle and Nelson, 2003; Roseberry et al., 2012], or because “unusual” configurations (e.g.
12
all Na active particles are clustered within dB) are excluded by the physics of coupled fluid-particle
motions. Nonetheless, we cannot claim the wisdom, given our current understanding of turbulence
over a mobile sediment boundary, to suggest that any particular configuration of particle positions
and velocities is not possible, and hence, the initial assumption that each configuration in the
ensemble is equally probable is justified. This assumption, however, is not critical in that f(((; x,
y, t) or f(, r, s((, r, s; x, y, dt) ultimately must be defined semi-empirically. Moreover, if the streambed
and turbulence structure are homogeneous (in a probabilistic sense) over B, then it may be assumed
that f(((; x, y, t) and f(, r, s((, r, s; x, y, dt) are the same for each elementary area dB. And, because
these 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, albeit
uniform 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 the
presence of bed forms, then one might expect concomitant, systematic variations in particle activity
and motions. In this case the bed forms are to be considered part of the externally imposed
macroscopic conditions, that is, as a bed condition that is compatible with the macroscopic flow and
sediment properties. Then, we again may envision an ensemble of possible configurations of active
particle positions and velocities, each configuration being equally probably. But here it is important
to imagine, as Gibbs did, the set of configurations as being separate systems (realizations) with the
same bed forms at a fixed time, not necessarily as a time series of one realization where the bed
forms grow or migrate. As above, we assume this ensemble yields for any area dB a probability
density function of the activity, namely f(((; x, y, t), and a joint probability density function of the
activity ( and the displacements r and s, namely f(, r, s((, r, s; x, y, dt). Now the forms of f(((; x, y,
t) and f(, r, s((, r, s; x, y, dt) and their parametric values may vary with xy position (and with time; see
section 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,
where our first objective is to obtain a probabilistic description of the sediment flux qx, and our
second objective is to obtain the expected (ensemble-averaged) value of this flux. In this case the
number density nxy(x, y, t), the activity ((x, y, t) = Vpnxy(x, y, t), the density function f(((; x, y, t) and
the joint density function f(, r, s((, r, s; x, y, dt) introduced above may be simplified to nx(x, t) [L-1],
((x, t) [L], f(((; x, t) [L-1] and f(, r((, r; x, dt) [L-2]. We also define the conditional probability density
function
(23)
with units [L-1], where f(((; x, t) may be considered the marginal distribution of f(, r((, r; x, dt). That
is, fr*((r*(; x, dt)dr is the probability that a particle at x will move a distance between r and r + dr
during dt given that, among all possible combinations of particle activity and displacements r,
attention is restricted to the specific activity ((x, t) at time t. In turn we let Fr*((r*(; x, dt) denote
the cumulative distribution function defined by
(24)
where the lower limit of integration indicates that r may be positive or negative, a condition that we
redefine below. That is, Fr*((r*(; x, dt) is the probability that a particle at x will move a distance less
than or equal to r during dt, given the activity ((x, t) at time t.
13
3.2. Master Equation
To a good approximation most bed load particles move downstream. Nonetheless, there is value
in considering the more general case of bidirectional motions. With reference to Figure 7, consider
particle motions along a coordinate x, where it is convenient to treat motions in the positive and
negative directions separately. For particles located at x = xN at time t, let r denote a displacement
in the positive x direction during dt, and let l denote a (positive) displacement in the negative x
direction during dt. Further, let p(xN, t) denote the probability that motion is in the positive x
direction, 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) at
either time t or time t + dt, or both. That is, r or l is the total displacement of an active particle for
all motion that occurs over an interval less than or equal to dt. The displacements r and l therefore
are not to be interpreted as hop distances measured start to stop, a point that we examine below.
Now, if Fr*((r*(; xN, dt) denotes the probability that a particle starting at xN (r = 0) moves a
distance less than or equal to r during dt, then Rr*((r*(; xN, dt) = 1 - Fr*((r*(; xN, dt) is the probability
that a particle moves a distance greater than r during dt. By definition the conditional probability
density of r is fr*((r*(; xN, dt) = dFr*(/dr = -dRr*(/dr [L-1]. In turn, if Fl*((l*(; xN, dt) denotes the
probability that a particle starting at xN (l = 0) moves a distance less than or equal to l during dt, then
Rl*((l*(; xN, dt) = 1 - Fl*((l*(; xN, dt) is the probability that a particle moves a distance greater than
l (in the negative x direction) during dt. The conditional probability density of l is fl*((l*(; xN, dt) =
dFl*(/dl = -dRl*(/dl [L-1]. Note that because r and l are defined here as being positive displacements,
the lower limit of integration in (24) defining Fr*( is now set to zero, and likewise for the (unwritten)
companion definition of Fl*(.
If the location of a particle is specified by the position x of its nose, then over a specified area
A of width b normal to x, let nx(x, t) [L-1] denote the number of active particles per unit distance
parallel to x. Then, ((xN, t)bdxN = Vpnx(xN, t)dxN denotes the associated volume of active particles at
xN at time t, and p(xN, t)((xN, t)bdxN is the volume of particles that moves in the positive x direction
during dt. Moreover, the volume of particles passing position x in the positive x direction from xN
< x is p(xN, t)((xN, t)Rr*((x - xN*(; xN, dt)bdxN, and the volume passing position x in the negative x
direction from xN > x is q(xN, t)((xN, t)Rl*((xN - x*(; xN, dt)bdxN. The total volume of particles passing
position x in the positive x direction during dt is
(25)
and the total (negative) volume of particles passing position x in the negative x direction during dt
is
(26)
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
(27)
This is a flux form of the Master Equation [Risken, 1984; Ebeling and Sokolov, 2005; Furbish et al.,
14
2009a, 2009b], illustrating that the volume V(x, t + dt) passing x during dt may be influenced by
motions originating at positions both to the left and right of x. Note that nothing is assumed a priori
regarding the forms of the conditional probability densities, fr*((r*(; xN, dt) and fl*((l*(, xN, dt), of the
displacements r and l. Also note that the explicit appearance of the activity ( as a parameter in the
functional notation of the left side of (27) highlights that the particle volume V(x, t + dt; () is
conditional on the activity. This point is important in the idea of an ensemble average presented
below.
3.3. Advection and Diffusion
The 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
(28)
Expanding the products p(x - r, t)((x - r, t)Rr*((r*(; x - r, dt) and q(x + l, t)((x + l, t)Rl*((l*(; x + l,
dt) as Taylor series to first order then leads to
(29)
By definition the mean particle displacements during dt are (Appendix D)
(30)
and
(31)
The second moments of these displacements about the local origin x are
(32)
and
(33)
In turn, average velocities conditional to the activity ( are defined by
(34)
and
15
(35)
and diffusivities are defined by [Risken, 1984; Ebeling and Sokolov, 2005; Furbish et al., 2009a,
2009b]
(36)
and
(37)
Substituting (30) through (33) into (29), dividing by dt, and taking the limit as dt 6 0 thus gives the
particle volume discharge,
(38)
The first term on the right side of (38) is advective and the second is diffusive. The bracketed part
of the first term is merely the weighted average particle velocity u [L t-1], namely, u(x, t; () = p(x,
t)ur(x, t; () - q(x, t)ul(x, t; (). That is, in the development above, for convenience we defined l as
being a positive displacement in the negative x direction, so by this definition ul is positive. If for
cosmetic reasons we now let ul carry the sign, then u(x, t; () = p(x, t)ur(x, t; () + q(x, t)ul(x, t; ().
Similarly, the parenthetical part of the second term on the right side of (38) is a weighted diffusivity
6 [L2 t-1], namely, 6(x, t; () = p(x, t)6r(x, t; () + q(x, t)6l(x, t; (). With these definitions, dividing
(38) by the width b gives the flux qx(x, t) [L2 t-1], namely
(39)
which, consistent with the conclusions in section 2.2, suggests that spatial variations in ( or 6 can
effect a flux that is in addition to the advective flux. We consider the conditions under which the
diffusive 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 ( at
position x, whereas the velocity u and the diffusivity 6 are formally defined above as ensemble
averages, that is, the (statistically) expected values of these quantities obtained from the ensemble
of all possible configurations of particle positions and velocities, conditional to the activity (. The
conditional probability densities fr*( and fl*( (as well as the related functions Rr*( and Rl*() thus
represent underlying (ensemble) populations and are smooth, continuous functions. In order to
envision (39) as representing the local instantaneous flux, one must therefore imagine that u and 6
actually represent values obtained from an instantaneous “sample” drawn from the densities fr*( and
fl*(. Over an elementary area bdx, this sample may involve few to many particles as determined by
the instantaneous value of ( and the width b, so the instantaneous distributions of displacements
(drawn from fr*( and fl*() may look more like irregular histograms than like the smooth functions fr*(
and fl*(, and the velocity u (39) is like the simple average in (8). We return to this point below.
16
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 a
displacement that is positive or negative, note that dl = -dr, combine the integrals in (29), and use
p + q = 1 to give
(40)
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( and
integrating over the activity ( thus gives
(41)
Letting an overbar denote an ensemble average, dividing by b and by dt, and taking the limit as dt
6 0, this becomes
(42)
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 diffusivity
and activity. Indeed, experiments suggest that, at low transport rates, both the particle activity and
the average velocity increase with increasing bed stress, where the activity increases faster than the
velocity [Schmeeckle and Furbish, 2007; Ancey et al., 2008; Ancey, 2010; Lajeunesse et al., 2010;
Roseberry et al., 2012], clearly indicating that u and ( are correlated. This figures importantly in
considering how the ensemble average is related to time averaging, a topic that we address in section
5. Meanwhile we note that if u and (, and 6 and (, are independent, which may be the case at high
transport rates (and is demonstrably correct in the case of rain splash transport treated as a stochastic
advection-diffusion process [Furbish et al., 2009a]), then (42) becomes
(43)
Comparing this formulation with classic descriptions of transport involving simple fluids reveals
several interesting points. For tracer molecules within a fluid the velocity in (42) maps to the
advective fluid velocity uf (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 system
these quantities are independent (i.e. their covariance is zero). Hence, the advective flux of tracers
is the product of the advective velocity and the concentration, ufc. Moreover, for isothermal
conditions the molecular diffusivity 6m is a thermodynamic quantity that is independent of the
concentration and position, so the diffusivity is outside the differential in the diffusive term, which
17
then looks like Fick’s law as normally written, namely -6mMc/Mx. We further note that, whereas the
fluid velocity and the molecular diffusivity are independent in simple fluid-solvent systems, the
mean particle velocity u and the diffusivity 6 in (39) are highly correlated. (Indeed, the approximate
description (22) gives 6 = Ju2 with Up = u.) That is, the diffusive part of the flux in (39) vanishes
in the absence of particle advection, entirely analogous to the relation between advection and
mechanical dispersion in porous-media transport [Furbish et al., 2012b].
3.4. Describing the Distribution of Particle Displacements
To 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 the
negative x direction.
With reference to Figure (7), of the particles starting at xN, let Pr(r; xN, dt) [L-1] denote the
proportion located at xN + r at time t + dt, relative to the proportion that moves beyond xN + r during
dt, namely
(44)
Integrating (44) from r = 0 to r then gives
(45)
from which it follows that the probability density of r is
(46)
In turn, of the particles starting at xN, let Pl(l; xN, dt) [L-1] denote the proportion located at xN - l at
time t + dt, relative to the proportion that moves beyond xN - l during dt. By a development similar
to that above one obtains
(47)
whence the probability density of l is
(48)
These relations merit further discussion.
If r and l were considered displacements in time, specifically the “age” of an entity, rather than
displacements in position, as above, then the probabilities Rr(r; xN, dt) and Rl(l; xN, dt) are referred
to as “reliability” or “survival” functions in reliability (or survival) theory, and the proportions Pr(r;
xN, dt) and Pl(l; xN, dt) are the associated “hazard” or “failure-rate” functions. The functions Pr(r;
xN, dt) and Pl(l; xN, dt) must be non-negative and integrate to infinity over the domain [0, 4), but
otherwise may have any form, monotonic (increasing or decreasing), nonmonotonic or
discontinuous. These functions can be obtained directly from (44) for known distributions. But of
potentially greater value is the idea of using (46) as a strategy for defining the probability density
function fr(r; xN, dt) in terms of Pl(l; xN, dt), based on theoretical or empirical arguments for the form
18
of Pr(r; xN, dt). Indeed, this is the strategy used to select a suitable distribution in reliability/survival
analysis, 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
at either time t or t + dt, or both. Note that Pr(r; xN, dt)dr may be considered a conditional
probability. Namely, Pr(r; xN, dt)dr is the probability that during dt a particle will have a
displacement from r to r + dr, given that it achieves a displacement of at least r. That is, because
all displacements r occur during dt, Pr(r; xN, dt)dr is equivalent to the probability that a particle has
a velocity within up to up + dup such that during dt it experiences a displacement r = updt to r + dr
= (up + dup)dt, given that its velocity is at least as fast as up = r/dt.
Measurements of bed load particle motions using high-speed imaging suggest that, at low to
moderate transport rates, streamwise particle displacements r (or velocities up) follow an
exponential-like distribution [Lajeunesse et al., 2010; Roseberry et al., 2012]. As elaborated in
Roseberry et al. [2012], an exponential-like density fr(r; xN, dt) implies a constant failure rate,
namely, Pr(r; xN, dt) = Pr = 1/:r, where :r is the mean displacement during dt. Namely, the
probability that a particle will experience a displacement within r to r + dr during dt is a fixed
proportion, 1/:r, of particles that experience displacements greater than r during dt. Or, the
probability that a particle possesses a velocity within up to up + dup is a fixed proportion of particles
moving faster than up. In turn, Ganti et al. [2010] and Hill et al. [2010] point out that power-law
distributions can arise from combinations of exponential distributions, and apply this idea to
transport of mixed particle sizes, albeit involving distributions of travel distance (see below) rather
than the distribution of displacements, fr(r; xN, dt).
In contrast to a displacement r that occurs during dt, let 8 denote a particle displacement
measured start to stop that occurs over a travel time J [t], and let f8, J(8, J) [L-1 t-1] denote the joint
probability density of 8 and J. With reference to Figure 8, a steep covariance relation between 8
and J implies varying speeds (defined by 8/J) due to varying displacements over a similar travel
time. A weak covariance implies varying speeds due to similar displacements over varying travel
times. An intermediate covariance implies relatively uniform speeds.
The marginal distribution f8(8) [L-1] of the displacements 8 is
(49)
which defines a distribution of hop distances, start to stop, without rest times [e.g. Einstein, 1950;
Wong et al., 2007; Bradley et al., 2010; Ganti et al., 2010; Lajeunesse et al., 2010; Roseberry et al.,
2012]. By itself, f8(8) contains no information regarding particle travel times or speeds. In turn the
marginal distribution fJ(J) [t-1] of the travel times J [e.g. Lajeunesse et al., 2010] is
(50)
which similarly, by itself, contains no information regarding particle hop distances or speeds.
The elements of (46) also suggest a strategy for clarifying the physical basis of the densities fJ(J)
and f8(8). Namely, if we define PJ(J; xN) [t-1] and P8(8; xN) [L-1] as in (44), then PJ is a temporal
“failure rate” function and P8 is a spatial “failure rate” function as normally defined in
reliability/survival theory, where “failure” may be interpreted as particle disentrainment [Furbish
and Haff, 2010]. Thus, a physical (probabilistic) understanding of how and why active bed load
particles stop in relation to bed roughness and near-bed flow conditions is central to describing the
19
probability densities of the hop distances 8 and the associated travel times J, beyond purely
empirical descriptions. Moreover, descriptions of fJ(J) and f8(8) must be mutually consistent,
inasmuch as these combine to form the joint density f8, J(8, J).
Specifically, with PJ = PJ(J; xN) and P8 = P8(8; xN), then the densities fJ(J) and f8(8) depend on
the travel time J and the distance 8, and therefore on changing conditions following entrainment and
downstream of the initial position xN. If, however, PJ = PJ(xN) and P8 = P8(xN), then these densities
are independent of J and 8. For example, if for physical reasons PJ and P8 are constants that depend
only on local conditions, namely PJ(xN) = 1/:J and P8(xN) = 1/:8, then from (46), fJ(J) = (1/:J)exp(J/:J) and f8(8) = (1/:8)exp(8/:8) are exponential densities with mean travel time :J and mean hop
distance :8. (This example of constant PJ is analogous to a constant “failure rate” in reliability
analysis, where :J would be interpreted as the mean longevity.) In contrast, if PJ = PJ(J; xN) = "J" 1
, for example, then fJ(J) = "J" - 1exp(-J") is a standard Weibull distribution with shape factor "; or
if PJ(J; xN) = "/J, then fJ(J) = "Jm"/J" + 1 is a Pareto distribution with scale factor Jm. This idea of a
disentrainment rate function (P8 or PJ) is conceptually similar to, but of a different form than, the
disentrainment rate function involving the distribution of particle hop distances as described by
Nakagawa and Tsujimoto [1980].
For completeness we note an additional definition of travel distances, where displacements 8 are
measured 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
rest times over an interval J, then the probability density of 8 may be denoted as f8(8; J),
emphasizing the significance of the interval J as a parameter. Connecting travel distances that
involve rest times to the density f8, J(8, J) and associated particle speeds requires additional
information 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 associated
travel times in considering the entrainment form of the Exner equation [e.g. Parker et al., 2000;
Garcia, 2008; Ancey, 2010], where we generalize this density to include cross-stream particle
motions. Lajeunesse et al. [2010] present histograms representing the marginal distributions f8(8)
and fJ(J) based on high-speed imaging of particle motions, and note that these possess well defined
modes that are less than the means. Data concerning the joint density f8, J(8, J) are also presented
in a companion paper [Roseberry et al., 2012].
3.5. The Two-Dimensional Flux
As in section 3.1 let r and s denote particle displacements parallel to x and y, respectively, and
let fr, s(r, s; dt) [L-2] denote the joint probability density function of r and s. If u = iu + jv denotes the
ensemble-average particle velocity, and if 6 denotes a diffusivity tensor with the elements 6xx, 6yy
and 6xy = 6yx, then the component fluxes qx and qy of q = iqx + jqy are
(51)
and
(52)
Here u and v, and 6xx and 6yy, derive from the first and second moments, respectively, of the marginal
distributions, fr(r; dt) and fs(s; dt), of the joint density function fr, s(r, s; dt) as described in section 3.2
above. Also,
20
(53)
which is like a covariance (defined about the origin).
Inasmuch as diffusive particle motions normal to the mean motion are centered about this mean
motion [Lajeunesse et al., 2010; Roseberry et al., 2012], the magnitudes of 6xx and 6yy vary with the
direction of the mean motion. Moreover, 6xy is finite only when the mean motion is not parallel to
the x or y axis. As described above, the effect of the diffusive terms involving 6xx and 6yy is to
contribute proportionally more (or fewer) particles to qx and qy relative to the contribution of those
particles represented by u( and v(, depending on the sign of M(6xx()/Mx and M(6yy()/My. The effect
of the diffusive terms involving 6xy is similar. For example, with finite 6xy and negative M((6xy)/My
(due, say, to decreasing activity ( along y), proportionally more particles starting from positions at
y < yS contribute to the flux qx parallel to x across an elementary plane at yS (Figure 9), relative to
those particles represented by u( at y = yS. Conversely, with positive M((6xy)/My, proportional fewer
particles starting from positions at y < yS contribute to the flux qx across an elementary plane at yS.
Similar remarks pertain to the term involving M((6xy)/Mx with respect to the flux qy. The diffusivities
in (51) and (52) are associated with the motion, not with any medium (as with heat conduction in
an anisotropic solid). Moreover, like mechanical dispersion associated with flow in a porous
medium, the elements of 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 Equation
4.1. Flux Form
Let 0(x, y, t) denote the local elevation of the streambed, and let cb denote the volumetric particle
concentration of the bed. Then with cbM0/Mt = -Mqx/Mx - Mqy/My, substitution of (51) and (52) gives
(54)
This formulation assumes that active particles effectively remain in contact with the bed, where 0
is defined as the (local) average surface elevation of particles, including the (small) contribution to
the bed elevation associated with active particles. Note that (54) has the form of a Fokker-Planck
equation, where the elements of 6 are inside both derivatives.
4.2. Entrainment Form
Having introduced the joint probability density function of particle hop distances and associated
travel times in section 3.4 above, here we generalize this idea to obtain the entrainment form of the
Exner equation [e.g. Parker et al., 2000; Garcia, 2008; Ancey, 2010] for comparison with (54)
above. Let E(x, y, t) [L t-1] denote the volumetric rate of particle entrainment per unit streambed
area, and let D(x, y, t) [L t-1] denote the volumetric rate of deposition per unit streambed area.
Assuming only downstream motions involving the streamwise hop distance 8 and the cross-stream
hop distance R over the travel time J, we denote the joint probability density function of 8, R and
J as f8, R, J(8, R, J; x, y, t), which depends on xy position and time t. By definition the rate of
deposition is
(55)
21
which explicitly incorporates the idea, neglected in previous formulations (Appendix E), that
particles arriving at an xy position at time t started their hops 8 and R at many different times t - J,
as clearly reflected in histograms representing marginal distributions of f8, R, J, that is, the
distributions f8(8) and fJ(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 [Wong
et al., 2007].
Starting with (55) and assuming that f8, R, J is not heavy-tailed [Roseberry et al., 2012], then it is
straightforward to show (Appendix F) that
(56)
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
hop distance and the travel time , then with the definition cbM0/Mt = -Mqx/Mx for one-dimensional
transport, it follows from (56) that the flux qx = 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”
forms of the flux thus gives qx = ( = 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
(57)
With a steady entrainment rate (ME/Mt = 0), the rate of change in the bed elevation 0 goes simply as
the divergence of the entrainment rate, ME/Mx. That is, there is a difference in the rates of deposition
and entrainment at any position x because of a difference in the number of particles arriving at and
leaving x. With a uniform entrainment rate (ME/Mx = 0), the (uniform) rate of change in the bed
elevation goes as the rate of change in the entrainment rate, ME/Mt [L t-2], modulated by the average
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
22
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.
So in specifying that cbM0/Mt = D - E [e.g. Parker et al., 2000; Garcia, 2008; Ancey, 2010], 0 is
effectively defined as the (local) average surface elevation of particles at rest, neglecting the (small)
contribution to the bed elevation associated with active particles in contact with the bed. In this case
M(E )/Mt is like a source term. Specifically, writing cbM0/Mt = -Mqx/Mx, then it follows that M(qx E )/Mx = S with S = M(E )/MJ. This indicates that the flux in excess of the steady, uniform flux E ,
namely qx - 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 related
entrainment formulation of conservation of tracer particles. Consider the simplified case of onedimensional transport parallel to x, and let fT(x, t) denote the fraction of bed load particles that are
tracers. The rate of deposition of tracers is
(58)
which incorporates the idea that tracer particles arriving at position x at time t started their hops 8
at many different times t - J. Again assuming that f8, J is not heavy-tailed, (58) can be written as
(59)
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,
(60)
If all particles are tracers, fT = 1, DT = D and (60) reduces to the steady condition D - E = 0. But
otherwise, 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 ensemble
of tracers is an unsteady problem. Namely, if h denotes a nominal steady, uniform thickness of bedsurface particles involved in transport, then DT(x, t) - EfT(x, t) = cbhMfT/Mt and
(61)
The unsteady terms involving E account for the fact that tracers arriving at x at time t start from
different positions upstream at different times, and, because they are entrained at different times, the
fraction fT is changing at any specific starting position. For example, fT(xN, t - J1) at position xN when
tracer 1 is entrained at time t - J1 is different from fT(xN, t - J2) when tracer 2 is entrained at the same
position xN at time t - J2, although both tracer particles arrive at position x downstream at time t
because particle 2 has a shorter travel time J2 than does particle 1.
Note that cbh/E = JR is the mean residence time of particles within the thickness h, so upon
23
dividing (61) by E and rearranging,
(62)
where UR = /(JR + ) is a mean virtual velocity and 5R = /(JR + ) 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
time JR and the mean travel time .
5. Averaged Quantities
In the formulation above the width b is not explicitly specified, as the flux is considered a “per
unit width” quantity. But this deserves further consideration, returning to the idea of an ensemble
of configurations of particle positions and velocities. In section 3.1, the probability density f(((; x,
y, t) of the activity (, and the joint probability density f(, r, s((, r, s; x, y, dt) of the activity ( and the
displacements r and s, are associated with active particles within any elementary area dB. Focusing
on displacements r, then likewise, the conditional density fr*((r*(; x, y, dt) is associated with active
particles within dB = bdx. With small b, at any instant individual realizations drawn from the
densities f(((; x, y, t) and fr*((r*(; x, y, dt) at position y may be quite different from realizations from
f(((; x, y + )y, t) and fr*((r*(; x, y + )y, dt) at position y + )y. Upon lengthening b, the number of
active particles within bdx generally increases, and the activity incorporates spatial variations that
exist at scales smaller than b, so the probability density f( obtained from the ensemble of
configurations defined for bdx, centered about the same average, possesses a smaller variance
(Figure 6). With sufficiently large b, the activity tends to a constant, the ensemble-averaged activity,
independent of y. Similarly, with increasing b, individual realizations (over bdx) of the conditional
probability density of r approach the smooth function fr*((r*(; x, dt), independent of y.
In effect, a lengthening of b is equivalent to sampling a greater number of possible states of
particle motions. However, b cannot be “too large” if the underlying forms of f( and fr*( change
along y, say, in relation to changing near-bed turbulence structure in the mean. For the “right” b,
a reasonable description of the instantaneous flux is given by (39), where the realization of the flux
is inherently width-averaged over b. This also suggests that for equivalent macroscopic flow
conditions, the magnitude of the fluctuations in the flux depend on the measurement width b. We
now turn to time averaging of (39).
Of interest is the behavior of (39) when averaged over different characteristic timescales, and
the relation of this to ensemble averaging. Bed load transport rates vary over many timescales [e.g.
see Table 1 in Gomez et al., 1989]; and for nominally steady flow conditions, the measurement
interval influences the calculated rate inasmuch as fluctuations in transport over durations shorter
than the measurement interval are averaged (and thus smoothed) in the calculation. To our
knowledge no systematic, simultaneous measurements of particle activity and velocities are
available (beyond those reported in Roseberry et al. [2012], which are of short duration), so we lack
an empirical basis for evaluating time averaging of these quantities. Nonetheless, we may surmise
the following in general terms.
Consider first a planar bed with steady (uniform) macroscopic flow conditions. With increasing
averaging period, one may assume that at any position x the bed experiences an increasing
proportion of the set of possible (ensemble) configurations of particle activity and velocity, and with
a sufficiently long averaging period the bed at x eventually experiences a fully representative set of
24
possible configurations, in which case it is reasonable to assume that a (long) time average equals
the 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
forms. The first is a “short” turbulence timescale Tt, which we envision as being sufficiently long
that, at any position x, the bed experiences a representative sample of possible turbulence
fluctuations specific to where x is located within the bed form field, but short enough that the local
bed form morphology does not change significantly. The second is an intermediate bed form
timescale Tb, which we envision as being comparable to the period required for migration of bed
forms (e.g. ripples or dunes) over one wavelength. (Note that Tt may be similar to Tb for small bed
forms.) The third is a “long” bed-form field timescale Tf, which we envision as being long enough
that, at any position x, the bed experiences a fully representative sample of all possible positions
(heights, proximity to crests, etc.) on bed forms within the migrating field.
When (39) is averaged over the turbulence timescale Tt,
(63)
which looks like the ensemble average (42), and highlights that the flux retains its dependence on
time after averaging, as it varies over timescales longer than Tt. Moreover, whereas on a planar bed
the time-averaged flux is constant (uniform) and the diffusive term vanishes, within a field of active
bed forms the flux varies with 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 reformulation
of the stability analysis of Smith [1970] to include the diffusive flux suggests that this flux is a
sufficient, if not necessary, condition for selection of a preferred wavelength during initial ripple
growth [Kahn and Furbish, 2010; Kahn, 2011].
When (39) is averaged over the bed form timescale Tb, the result is the same as in (63) inasmuch
as bed form geometries in a natural field are not identical. That is, in the idealization of identical
one-dimensional migrating bed forms, positions over a single wavelength in principle sample all
possible turbulence fluctuations, so a spatial average over one wavelength is the same as a time
average over one period Tb. In this idealization the diffusive term therefore vanishes. But with
naturally variable bed form geometries, the time-averaged diffusive term may be nonzero, and the
time-averaged flux retains its dependence on time, particularly given persistent interactions among
neighboring bedforms [Jerolmack and Mohrig, 2005]. In contrast, when (39) is averaged over the
field timescale Tf, 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 the
constituents of sediment transport and conservation normally treat these as continuous twodimensional fields. An example is the class of models aimed at describing the instability of the
25
coupled motions of water and sediment leading to the growth of bars [Callander, 1969; Engelund
and Skovgaard; 1973; Parker, 1976; Fredsøe, 1978; Blondeaux and Seminara, 1985; Nelson and
Smith, 1989; Seminara and Tubino, 1989; Furbish, 1998]. These models, which start with the
Reynolds averaged momentum equations, in effect assume that local conditions coincide with
ensemble-averaged conditions, where bed forms change sufficiently slowly that the bed locally
experiences a representative sample of the ensemble of particle activity and motions, and the local
flux varies smoothly with xy position and time, consistent with the quasi-steady approximation
applied to turbulence conditions.
6. Discussion and Conclusions
The surface-integral definition (1) of the instantaneous flux of bed load sediment, although
impractical as a guide for direct measurements of the flux, nonetheless is precise. Thus any
definition that appeals to averaged quantities of particle motions (e.g. the mean particle velocity and
activity) to replace the detailed information embodied in (1) must be consistent with this definition.
With quasi-steady bed and transport conditions, the definition (2) of the one-dimensional flux
qx parallel to x, involving the product of the average particle velocity Up and the particle activity (,
is consistent with (1) inasmuch as active particles are at any instant uniformly (albeit quasirandomly) distributed over the streambed, and the flux-normal width b over which the particle
activity is calculated is sufficiently large to smooth over instantaneous small-scale variations in the
activity along b. In this situation the average velocity Up of particles over the streambed is
equivalent to the average velocity of N particles that intersect a vertical surface of width b at any
position x, and the flux qx = (Up = ( .
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
the average velocity Up of all particles in the vicinity of x. This occurs because the surface is
preferentially populated by faster moving particles and depleted of slower moving particles when
M(/Mx < 0, or visa versa when M(/Mx > 0, even though the average velocity Up is uniform over x. The
flux calculated as qx = ( is entirely consistent with the surface-integral definition (1). Moreover,
upon writing the particle velocity ui as the sum of the average Up and a fluctuating part uiN, namely
ui = Up + uiN, the flux looks like qx = (Up + ( . 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 not
distinguish 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 particle
positions 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 formulation
reveals that the volumetric flux involves an advective part equal to the product of the particle
activity and the ensemble-average particle velocity, and a diffusive part involving the gradient of
the product of the particle activity and a diffusivity obtained as the time-derivative of the second
moment of the probability density function of particle displacements. A key point in the formulation
is that the average particle velocity and the diffusivity are correlated. Thus, the diffusive part of the
flux vanishes, inasmuch as fluctuating particle velocities vanish, in the absence of overall advective
particle motions — which is entirely analogous to the relation between advection and mechanical
dispersion in porous-media transport. The effect of the diffusive flux therefore is to add to or
26
subtract from the advective flux in the presence of an activity gradient, as opposed to operating
independently of the mean motion as in molecular diffusion.
Central to the formulation is the probability density function of particle displacements r that
occur during a small interval of time dt, conditional on the particle activity (, namely fr*((r*(; x, dt).
A useful way to think about the source of this smooth probability density is to envision an ensemble
of states consisting of all possible configurations of particle positions and velocities. This begins
with selecting a streambed area B that is subjected to steady macroscopic flow conditions. This area
must be sufficiently large that, during any small interval of time dt, the total number of active
particles within B remains effectively steady. We then imagine, as Gibbs [1902] did, the set of states
(particle positions and velocities) as being separate systems with the same bed and flow conditions
at a fixed time, rather than as a time series of one system where the bed and flow conditions evolve.
Then, for any elementary area dB, this ensemble yields the smooth density function fr*((r*(; x, dt).
In turn, the average displacement, as in (30), and the average particle velocity, as in (34), for
example, represent ensemble averages, where the “ensemble” consists of this set of systems at a
fixed time. (Note that this definition of an ensemble average is consistent with classic definitions
of such averages from statistical mechanics.) Particle activity data consistent with the idea of an
ensemble of particle configurations are presented in Roseberry et al. [2012].
Time-averaged descriptions of the flux involve averaged products of the particle activity, the
particle velocity and the diffusivity. The significance of the covariance parts of these products
depends on the averaging timescale in relation to characteristic timescales of near-bed turbulence
and 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 with
bedform evolution, nonetheless retains its dependence on time over longer timescales in the presence
of strong feedbacks between sediment transport and topography with interactions among
neighboring bedforms [Jerolmack and Mohrig, 2005].
The flux form of the Exner equation, (54), looks like a Fokker-Planck equation in which
gradients in the particle diffusivity, like gradients in the particle activity, can in principle contribute
to changes in bed elevation. However, the significance of this idea requires clarification, theoretical
or experimental, aimed at showing how the diffusivity varies with the particle activity and velocity
[Furbish et al., 2012b] in relation to bed topography. The entrainment form of the Exner equation,
(56), similarly involves advective and diffusive terms, but also involves a time derivative term that
represents a lag effect associated with the exchange of particles between the static and active states.
In the case of a steady, uniform entrainment rate E with uniform and steady values of the mean hop
distance and the mean travel time , then for one-dimensional transport the flux qx = E , which
is equivalent to the definition (3) provided by Einstein [1950]. For the unsteady case the flux qx
cannot be expressed simply in terms of E, 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
the diffusivity contain the mean travel time and the mean residence time JR within a nominal
thickness of active particles (neglecting burial and re-emergence). Inasmuch as n JR, say, at low
transport rates, the virtual velocity UR . /JR and the virtual diffusivity 5R . /JR.
The formulation of the sediment flux presented herein involves, for simplicity, a single particle
diameter D, so definitions of the particle activity, velocity and diffusivity are specific to this
situation. In generalizing to a mixture of particle sizes, covariances between particle activity,
27
velocity and diffusivity become particularly important. For example, recall that in writing the last
part of (5) as qx = (1/b)N
, the covariance between Si and ui can be neglected for equal-sized
particles and this expression becomes (8), namely, qx = (1/b)N
activity,
=
. Moreover, with uniform
= Up, so qx = (Up as in (2). But with a mixture of sizes, the covariance between Si and
ui cannot be neglected inasmuch as some particle sizes preferentially move faster than other sizes.
In this case qx = N
, where (p = Si /b is like an individual particle activity. Or, letting j denote
the jth size fraction, we may write qx j =
to denote the fractional flux [Wilcock and McArdell,
1993; Wilcock, 1997a, 1997b], where (j 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
[Roseberry et al., 2012] that the probability density fr(r; dt) of streamwise displacements of sand
particles and the associated density of velocities fu(up) are exponential-like at low transport rates on
a planar streambed. In relation to the “failure rate” function Pr(r; xN, dt) described in section 3.4,
an exponential function implies that Pr = Pr(xN, dt) is independent of the displacement r. This in turn
means that the probability that a particle will experience a displacement within r to r + dr during dt
is a fixed proportion of particles that experience displacements greater than r during dt. Or, the
probability that a particle possesses a velocity within up to up + dup is a fixed proportion of particles
moving faster than up. We also show that the failure rate functions P8 and PJ for the hop distance
8 and the travel time J are not necessarily constants, but rather vary with 8 and J.
Appendix A: Particle Volume Discharge
Consider a particle of diameter D [L] that is moving with a positive velocity parallel to x through
a surface A positioned at x = 0 (Figure 2). Let >i [L] denote the position of the nose of the particle
relative to x = 0, and let Vi(>i) [L3] denote the volume of the particle that is to the right of x = 0 as
a function of >i. Also, let g [L] denote a small distance measured from the nose of the particle, where
0 # g # D. Then, at time t [t] the volume Vi+g(t) [L3] of the particle that is simultaneously to the left
of >i - g and to the right of x = 0 for 0 # >i # D is
(A1)
where H(>i) is the Heaviside step function defined by H(>i) = 0 for >i < 0 and H(>i) = 1 for >i $ 0.
In this expression the Heaviside functions serve as off-on switches over the x domain. The second
term on the right side of (A1) insures that Vi+g(t) is piecewise continuous without a jump at >i = D,
and the last two terms, as will be seen momentarily, insure that the derivative of (A1) is piecewise
continuous at >i = 0 when g 6 0.
With >i = >i(t), the rate of change in Vi+g is
28
where *(>i) = dH/d>i is the Dirac delta function. The derivative MVi/M>i = Si(>i) [L2] is like a
hypsometric function of the particle, and is equal to its cross-sectional area on the surface A at x =
0. The derivative d>i/dt = ui [L t-1] is its velocity parallel to x. The terms involving the Dirac
(A2)
function nominally represent instantaneous changes in the rates of gain and loss of volume when the
particle arrives at and leaves A. Because these terms are non-zero only at >i = 0 or >i = D, the third
and fourth terms, the fifth and seventh terms, and the sixth and eighth terms on the right side of
(A2), respectively, cancel each other. Then, upon letting g 6 0, the second term on the right side of
(A2) vanishes to give the particle volume discharge Qi+(t) across A in the positive x direction,
namely
(A3)
By symmetry the particle volume discharge Qi-(t) in the negative x direction has the same form as
the right side of (A3), namely Qi-(t) = dVi-g/dt = Si(>i)uiH(>i)[1 - H(>i - D)]. Moreover, at this point
the product involving the Heaviside functions is redundant, as the surface area Si(>i) is a piecewise
continuous function that is finite over 0 # >i # D with Si(>i < 0) = Si(>i > D) = 0. Thus, in general the
particle volume discharge Qi(t) across A is
(A4)
which is (4) in the main text.
Appendix B: Averaged Particle Cross-Sectional Area and Volume
Here we show that
D = (1/2)Vp 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
x = D, let n = nx/N [L-1] denote the proportion per unit distance parallel to x, and assume that any
instantaneous gradient in n, Mn/Mx, is uniform over a distance equal to the particle length D. Then
the probability density function f>(>i) of the distance >i is
(B1)
Note that if Mn/Mx = 0, f>(>i) reduces to a uniform density function, namely f>(>i) = 1/D. Assume that
the cubic particles do not rotate while crossing through A. In this case Vi(>i) = D2>i with Si(>i) =
MVi/M>i = D2, from which it immediately follows that = D2.
Upon integrating (B1), the cumulative distribution function F>(>i) of the distance >i is
29
(B2)
The cumulative distribution function FV(Vi) of the volume Vi is then FV(Vi) = F>(>i) with >i = Vi/D 2.
This yields
(B3)
In turn, the probability density function fV(Vi) = dFV/dVi is
(B4)
and the average
is
(B5)
Thus, whereas
= D 2 is independent of the gradient, Mn/Mx, of active particles,
depends
on this gradient. This occurs because Si(>i) is symmetrical about D/2, whereas Vi(>i) is monotonic
over 0 # >i # D. At lowest order
= D 3/2 = Vi(D)/2.
The averages
and
for spherical particles are obtained in a similar manner, although
the derivation is far lengthier. Like cubic particles, the average
of the gradient Mn/Mx, and at lowest order the average
= (B/12)D 2 is independent
= Vi(D)/2 = (1/2)Vp. Moreover,
D
= (B/12)D3 = (1/2)Vp.
Appendix C: Probability Distribution of Activity in Ensemble
For a streambed area B, let Na denote the total number of active particles in each possible
configuration of the ensemble. If m denotes the number of partitions of B of area dB = B/m, then
the total number of configurations involving Na particles distributed among m partitions is
(C1)
Using the language of statistical mechanics, we may refer to each of these Ne configurations as a
“macrostate.” In turn, if n1, n2, n3, ... nm denote the number of particles in each of the m partitions
of an individual macrostate, then using Maxwell-Boltzmann counting there are ne ways in which the
Na particles may be rearranged amongst the m partitions, and we may refer to each of these ne
arrangements as a “microstate.” The number of microstates in a given macrostate is
(C2)
and the total number of microstates Me over Ne macrostates is
30
(C3)
As a point of reference, for Na = 10 particles distributed among m = 2 partitions, there are a total of
Ne = 11 macrostates and a total of Me = 1,024 microstates. For Na = 10 and m = 5, there are Ne =
1,001 macrostates and Me = 9,765,625 microstates. And, for Na = 10 and m = 10, there are Ne =
92,378 macrostates and Me = 1 × 1010 microstates. For Na = 5 and m = 10, there are 2,002
macrostates and 100,000 microstates. And, for Na = 5 and m = 20, there are 42,504 macrostates and
3,200,000 microstates.
If ndB denotes the number of particles within dB, then the proportion Pn(ndB) of Me microstates
having ndB particles within dB — that is, the probability distribution of ndB — is given by the
binomial distribution assuming each microstate is equally probable [Roseberry et al., 2012]. With
large m relative to Na, there is an increasing number of ways to partition Na particles into m - 1 (or
m - 2, etc.) areas dB, so the likelihood of finding small numbers ndB within any dB increases, and the
distribution Pn(ndB) is exponential-like, albeit decaying with increasing ndB faster than an exponential
function. With decreasing m relative to Na, there are fewer ways to partition Na particles into an area
dB having small ndB, and the distribution Pn(ndB) takes an asymmetric form with finite mode. For
small m relative to Na, the distribution Pn(ndB) becomes Gaussian-like. As elaborated in Roseberry
et al. [2012], this distribution forms the basis of the null hypothesis of spatial randomness in the
positions of active particles. We show that near-bed turbulence leads to decided patchiness in
particle positions, and that fluctuations in activity, and therefore in transport rates, are systematically
related to the sampling area.
Appendix D: Means and Variances of Displacement Distances
The 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 omit
the conditional notation indicating a dependence on the activity (.
First note that by the product rule,
(D1)
With Rr(r) = 1 - Fr(r), substitution leads to
(D2)
Integrating this from r = 0 to r = 4,
(D3)
Evaluating the last integral then leads to
(D4)
insofar as the limit of rRr(r) as r 6 4 is equal to zero. This is guaranteed if Rr(r) decays at least as
31
fast as a negative exponential function, in which case the product rRr(r) looks like r/e r, whose limit
is zero as r 6 4. Indeed, the absence of this condition being satisfied implies that fr(r) is a heavytailed distribution without finite mean. In turn,
(D5)
or
(D6)
Integrating this from r = 0 to r = 4,
(D7)
insofar as the limit of r2Rr(r) as r 6 4 is equal to zero. Again, this is guaranteed if Rr(r) decays at
least as fast as a negative exponential function, in which case the product r2Rr(r) looks like r2/e r,
whose limit is zero as r 6 4. The absence of this condition being satisfied implies that fr(r) is a
heavy-tailed distribution without finite variance. A similar development involving fl(l) and Rl(l)
leads to comparable expressions for :l and Fl2.
Appendix E: Formulation of the Deposition Rate
Assuming 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:
(E1)
which neglects the idea that particles arriving at position x at time t started their hops 8 at many
different times t - J. As written, (E1) either assumes that particle hops 8 effectively occur
instantaneously, or that E and D are steady (in which case the appearance of time t is unnecessary)
and f8(8) is merely the uniformly distributed proportion of particles entrained at x - 8 which steadily
arrive at x.
Consider instead the following unsteady form of (E1):
(E2)
which allows for finite travel time J between entrainment and deposition. But now f8(8; J) still
cannot be interpreted as a hop distance distribution as originally defined by Einstein [1950]. Rather,
as described in section 3.4, f8(8; J) is now akin to what Hill et al. [2010] and others refer to as a
distribution of travel distances 8, where particles might experience multiple hops, with waiting
times, during the specific interval J. This requires integration over all J, and therefore knowledge
of how f8 evolves with J. It is instead far more straightforward and practical to consider the joint
probability density of hop distances 8 and associated travel times J and write
(E3)
which is the one-dimensional version of (55). Note that D, E and f8, J may each be unsteady and
32
nonuniform.
The associated formulation of the rate of deposition of particle tracers is written as
(E4)
Inasmuch as the tracer fraction fT is assumed to vary with time, then as with (E1) above, it must be
assumed that particle travel times are everywhere negligible.
Appendix F: Entrainment Form of Exner Equation
We expand the integrand in (55) as a Taylor series to first order about t and to second order
about x and y, namely
(F1)
Substituting (F1) into (55), rearranging, and momentarily letting d7 = d8dRdJ, leads to
(F2)
where the unwritten limits of integration match those of (55). The triple integral of f8, R, J in the first
term on the right side of (F2) by definition equals unity. Then, because the order of integration does
not matter, selectively integrating to obtain the marginal distributions f8, fR and fJ, and the joint
distribution f8, R,
(F3)
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).
33
Appendix G: Relation Between Flux Definitions
As 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 active
particles is approximately constant. That is, the rate of disentrainment within B equals the rate of
entrainment, and the rate at which particles leave B across its boundaries equals the rate at which
particles enter B across its boundaries. Imagine recording particle motions within B for an interval
of time Ts [t] [e.g. Lajeunesse et al., 2010; Roseberry et al., 2012]. For Ts much longer than the
mean particle travel time, particle motions during Ts adequately represent the joint probability
density f8, J(8,J) of hop distances 8 and travel times J without bias due to censorship of motions at
times t = 0 and t = Ts [Furbish et al., 1990]. The marginal distributions f8(8) and fJ(J) possess means
and . And, at any instant the ensemble average particle velocity is .
The average velocity of the ith (individual) particle with travel time Ji is
(G1)
In turn, letting Ns denote the number of particle motions during Ts, and assuming that Ns is large, the
ensemble average velocity
(G2)
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].
The quasi-steady (“equilibrium”) volumetric flux qx on a planar bed, when written as an
equivalence between its “flux” form and its “entrainment” form, is
(G3)
where ( is the particle activity (the volume of active particles in motion per unit streambed area) and
E is the entrainment rate (the volumetric rate at which particles become active per unit streambed
area). So evidently,
(G4)
That is, under steady conditions the activity ( = E , or
(G5)
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
qx.
As a point of reference, when particles continue their motions indefinitely (that is, they do not
start and stop), then experimentally Ji = Ts (the sample time) and (G2) becomes
34
(G6)
where now
is the average displacement during Ts, and the average in (G6) is the same as the
average of an individual particle over long time.
Notation
A
b
B
c
cb
d7
D
DT
E
fA
fr , f l
fr*(, fl*(
f(
f(, r
f(, r, s
F(
f8
fJ
f8, J
f8*J
f8, J
f8, R, J
f8, R
Fr, Fl
Fr*(, Fl*(
fT
fu
fV
FV
f>
F>
h
H
i
j
l
surface area [L2].
width normal to flux [L].
streambed area [L2].
concentration.
volumetric particle concentration of bed.
differential equal to d8dRdJ [L2 t].
particle diameter [L]; volumetric particle deposition rate per unit streambed area [L
t-1].
volumetric tracer deposition rate per unit streambed area [L t-1].
volumetric particle entrainment per unit streambed area [L t-1].
probability density function of velocities ui intersecting A [L-1 t].
probability density functions of displacements r and l [L-1].
conditional probability density functions of displacements r and l [L-1].
probability density function of activity ( [L-1].
joint probability density function of ( and r [L-2].
joint probability density function of (, r and s [L-3].
cumulative probability distribution function of (.
probability density function of 8 [L-1].
probability density function of J [t-1].
joint probability density function of 8 and J [L-1 t-1].
conditional probability density function of 8 [L-1].
joint probability density function of 8 and J [L-1 t-1].
joint probability density function of 8, R and J [L-2 t-1].
joint probability density function of 8 and R [L-2].
cumulative probability distribution functions of r and l.
cumulative conditional probability distribution functions of r and l.
fraction of lead load particles that are tracers.
probability density function of particle velocity u [L-1 t]
probability density function of volume Vi [L-3].
cumulative probability distribution function of volume Vi.
probability density function of distance >i [L-1].
cumulative probability distribution function of distance >i.
effective thickness of active bed load particles [L].
Heaviside step function.
designation of the ith of N particles.
designation of the jth particle-size fraction.
particle displacement in negative x direction [L].
35
L
m
MA
Me
n
n
ndB
ne
ni
nx, nz
nxy
N
Na
Ne
Ns
p
Pn
Pr, Pl
P8, PJ
q
q
qA
qx, qy
qx j
Q
Qx
Qi
Qi+, Qir
rm
Rr, Rl
Rr*(, Rl*(
s
S
Si
t
Tb
Tf
Ts
Tt
u, v
uf
ui
up
up, vp
up
length scale [L].
number of partitions of B.
mask projected on surface A, equal to 1 - H(up)H(-up).
total number of microstates in ensemble.
unit vector normal to A.
proportion of N particles per unit length [L-1].
number of particles within dB.
number of microstates in a macrostate.
number of particles in the ith partition of a macrostate.
number of particles per unit length parallel to x and to z [L-1]
number of particles per unit area [L-2].
number of particles intersecting surface.
total number of active particles in each configuration (macrostate) of ensemble.
total number of configurations (macrostates) in ensemble.
number of particle motions.
probability that a particle moves in the positive x direction.
proportion of Ne configurations having ndB particles within dB.
proportions defined by Pr = fr/(1 - Fr) and Pl = fl/(1 - Fl) [L-1].
proportions defined by P8 = f8/(1 - F8) [L-1] and PJ = fJ/(1 - FJ) [t-1].
probability that a particle moves in the negative x direction.
volumetric particle flux [L t-1] and [L2 t-1].
volumetric particle flux across surface A [L2 t-1].
volumetric particle flux components parallel to x and y [L2 t-1].
volumetric particle flux of jth size fraction [L2 t-1].
volumetric particle discharge [L3 t-1].
volumetric particle discharge parallel to x [L3 t-1].
volume discharge of ith particle [L3 t-1].
volume discharge of ith particle in positive and negative x direction [L3 t-1].
particle displacement parallel to x [L].
scale factor in Pareto distribution.
functions defined by Rr = 1 - Fr and Rl = 1 - Fl.
functions defined by Rr*( = 1 - Fr*( and Rl*( = 1 - Fl*(.
particle displacement parallel to y [L].
cross-sectional area of particles intersecting A [L2]; source term, M(E )/Mt [L t-1].
cross-sectional area of ith particle on surface A [L2].
time [t].
bed form timescale [t].
bed-form field timescale [t].
sampling interval [t].
turbulence timescale [t].
average particle velocity components parallel to x and y [L t-1].
fluid velocity [L t-1].
velocity component parallel to x of ith particle [L t-1].
particle velocity component normal to surface A [L t-1].
particle velocity components parallel to x and y [L t-1].
particle velocity field at surface A [L t-1].
36
ur, ul
Up
UR
mean velocities associated with displacements r and l [L t-1].
mean particle velocity [L t-1].
mean virtual particle velocity,
[L t-1].
V
Vi
V+, VVi+g, Vi-g
Vp
w
x, y
z
yS
"
(
(p
(j
*
)
g
0
6, 6
6m
6r , 6l
6xx, 6yy, 6xy
5R
volume of particles [L3].
volume of ith particle to right of surface A [L3].
volume of particles to right and left of surface A [L3].
volume of ith particle to right and left of surface A [L3].
volume of particle [L3].
variable of integration associated with displacements r and l [L].
Cartesian coordinates in streamwise and cross-stream directions [L].
moving coordinate such that x = z - UpJ [L].
specific position along y [L].
shape factor in Weibull and Pareto distributions.
particle activity [L].
individual particle activity defined by Si /b [L].
particle activity of jth size fraction [L].
Dirac delta function [L-1].
increment.
small distance measured from front of particle [L].
local elevation of streambed surface [L].
diffusivity, diffusivity tensor [L2 t-1].
molecular diffusivity [L2 t-1].
diffusivities associated with displacements r and l [L2 t-1].
elements of diffusivity 6 [L2 t-1].
virtual diffusivity,
[L2 t-1].
8
:r, :l
:8 , : J
>i
Fr 2, F l 2
Fu 2
J
JR
R
particle hop distance parallel to x [L].
first moments (means) of particle displacements r and l during dt [L].
mean hop distance [L] and mean travel time [t].
distance of ith particle to right of surface at x = 0 [L].
second moments of particle displacements r and l during dt [L2].
variance of particle velocities u [L2 t-2].
interval of time; particle travel time [t].
mean residence time of particles in thickness h [t].
particle hop distance parallel to y [L].
Acknowledgments. We acknowledge support by the National Science Foundation (EAR-0744934), and appreciate
Amelia Furbish’s insistence that we pay close attention to L’Hôpital’s rule.
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39
______________
D. J. Furbish, Department of Earth and Environmental Sciences, Vanderbilt University, 2301 Vanderbilt Place, Station
B 35-1805, Nashville, Tennessee 37235, USA. ([email protected])
P. K. Haff, Earth and Ocean Sciences Division, Nicholas School of the Environment, Duke University, Durham, North
Carolina 27708, USA. ([email protected])
J. C. Roseberry, Department of Earth and Environmental Sciences, Vanderbilt University, 2301 Vanderbilt Place, Station
B 35-1805, Nashville, Tennessee 37235, USA. ([email protected])
M. W. Schmeeckle, School of Geographical Sciences, Arizona State University, Tempe, Arizona 85287, USA.
([email protected])
Figure Captions
Figure 1. Definition diagram for surface integral of surface-normal velocities up = up@n of the
discontinuous particle velocity field up at the surface A with width b. The surface A extends
upward to a height necessary to include all bed load particles, and arrows are representative of
vector components up surrounding infinite sets of such vector components positioned over solid
fraction. The vector field up 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
to x through a surface A positioned at x = 0, where >i denotes the distance that the nose of the
particle is relative to x = 0, and g denotes a small distance measured from the nose of the
particle.
Figure 3. Definition diagram showing cloud of particles moving with varying velocities parallel
to x toward and through a surface A positioned at x = 0, where >i denotes the distance that the
nose of the ith particle is relative to x = 0.
Figure 4. Plots of deviation in particle velocity uiN = ui - Up versus deviation in particle crosssectional area SiN = Si - as viewed by observer moving with the average velocity Up for (a)
uniform particle cloud with
= 0 and
decreases with increasing distance x with
= Ub, and (b) particle cloud where the particle activity
> 0 and
> Ub.
Figure 5. Triangular cloud of particles possessing two velocities, 1 and 2, in equal proportions.
During a short interval of time dt the particles begin to segregate, whereas the cloud as a whole
moves downstream with the average velocity Up.
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
and 1,000D with the same overall activity ( = NaVp/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 10,000
configurations of particles uniformly (albeit randomly) distributed over an area dB = 10Db.
Figure 7. Definition diagram for particles motions parallel to x coordinate, showing probability
density function fr(r; xN, dt) of displacement distances r during dt.
Figure 8. Schematic diagram of three realizations of the joint probability density function fr, J(r, J),
where a steep covariance relation (open circles) between r and J implies varying speeds due to
varying displacements over a similar travel time, a weak covariance (gray circles) implies
varying speeds due to similar displacements over varying travel times, and an intermediate
covariance (black circles) implies relatively uniform speeds.
Figure 9. Schematic diagram showing effect of diffusive terms involving 6xy, where, with negative
M((6xy)/My due to decreasing activity ( along y, proportionally more particles starting from
positions at y < yS contribute (red arrows) to the flux qx parallel to x across an elementary plane
40
at yS, relative to those particles represented by u( at y = yS.
41
Figure 1.
Figure 2.
Figure 3.
Figure 4.
Figure 5.
Figure 6.
Figure 7.
Figure 8.
Figure 9.