ESurfEarth Surface DynamicsESurfEarth Surf. Dynam.2196-632XCopernicus PublicationsGöttingen, Germany10.5194/esurf-6-1089-2018Determining the scales of collective entrainment in collision-driven bed loadScales of collective entrainment in bed loadLeeDylan B.JerolmackDouglassediment@sas.upenn.eduhttps://orcid.org/0000-0003-4358-6999Earth and Environmental Science, University of Pennsylvania, 240 S 33rd St, Philadelphia, PA 19104, USAMechanical Engineering and Applied Mechanics, University of Pennsylvania, 220 S 33rd St, Philadelphia, PA 19104, USADouglas Jerolmack (sediment@sas.upenn.edu)22November2018641089109930January20186February201822June20184July2018This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://esurf.copernicus.org/articles/6/1089/2018/esurf-6-1089-2018.htmlThe full text article is available as a PDF file from https://esurf.copernicus.org/articles/6/1089/2018/esurf-6-1089-2018.pdf
Fluvial bed-load
transport is notoriously unpredictable, especially near the threshold of
motion where stochastic fluctuations in sediment flux are large. Laboratory
and field observations suggest that particles are entrained collectively, but
this behavior is not well resolved. Collective entrainment introduces new
length scales and timescales of correlation into probabilistic formulations
of bed-load flux. We perform a series of experiments to directly quantify
spatially clustered movement of particles (i.e., collective motion), using a
steep-slope 2-D flume in which centimeter-scale marbles are fed at varying
rates into a shallow and turbulent water flow. We observe that entrainment
results exclusively from particle collisions and is generally collective,
while particles deposit independently of each other. The size distribution of
collective motion events is roughly exponential and constant across sediment
feed rates. The primary effect of changing feed rate is simply to change the
entrainment frequency, although the relation between these two diverges from
the expected linear form in the slowly driven limit. The total displacement
of all particles entrained in a collision event is proportional to the
kinetic energy deposited in the bed by the impactor. The first-order picture
that emerges is similar to generic avalanching dynamics in sandpiles:
“avalanches” (collective entrainment events) of a characteristic size relax
with a characteristic timescale regardless of feed rate, but the frequency of
avalanches increases in proportion to the feed rate. The transition from
intermittent to continuous bed-load transport then results from the
progressive merger of entrainment avalanches with increasing transport rate.
As most bed-load transport occurs in the intermittent regime, the length
scale of collective entrainment should be considered a fundamental addition
to any probabilistic bed-load framework.
Introduction
Bed load, the motion of particles along a stream bed by rolling, hopping and
sliding, is the dominant mode of transport in rivers for particles larger
than 10 mm ().
Bed-load flux equations lose their predictive power as fluid stress decreases
toward the threshold of motion (), where sediment
transport becomes increasingly intermittent and exhibits fluctuations across
a wide range of length scales and timescales
(). Gravel-bed rivers organize
their bankfull geometry such that they are always near the threshold
(). There are two potential
causes of intermittency in near-threshold bed load: (i) variability in the
driving stress due to turbulent eddies near the bed () and (ii) variability in the resistive force of the
bed due to structural arrangements of the grains (). While
the role of turbulence has received the most attention, granular
contributions to bed-load dynamics are increasingly being recognized
(). One
of the defining features of granular systems is a continuous transition from
flowing to static regimes, known as the jamming transition. On approach to
jamming, particle motion becomes progressively slower and more heterogeneous;
the variance in fluctuations of particle displacements grows rapidly
(). Experiments show that the
onset of bed-load transport has the hallmarks of a jamming transition
().
Near-threshold transport rates exhibit strong correlations and intermittency,
while fluxes at rates far above the threshold are uncorrelated and smooth
().
Given the many-body nature of the problem, one sensible approach is to
examine bed-load transport in a probabilistic framework after
. Einstein defined a bed-load flux function of the form qx=ELx‾. In this formulation of bed-load flux qx, the
entrainment rate function E assumes a fixed timescale for the exchange of
an inactive particle with an active one; Lx‾ is the mean hop
length. Importantly, this formulation assumes that both entrainment and
deposition of particles are a time-independent Poisson process and that
particles do not interact. The discussion above, however, indicates that
bed-load transport has characteristics that deviate from the
time-independent, non-correlated process assumed by Einstein. Indeed,
experimental and field observations have revealed extreme fluctuations in
particle activity/flux above the mean (i.e., extreme variance)
(), collective grain motion () and anomalous diffusion of
particles ().
Starting with , a series of models for bed-load
transport have been proposed that posit that the probability of entrainment
depends on the number of moving particles. These models propose modifications
of Einstein's entrainment function that take this correlated behavior into
account through the introduction of a collective entrainment rate, μ,
that leads to a characteristic correlation length scale, lc
(). As the mean transport rate is lowered, the relative
contribution of the model-derived μ parameter must increase in order to
reproduce the observed growth in variance of bed-load activity (i.e., the
number density of moving grains). Collective entrainment is thus hypothesized
to be the primary driver of observed intermittent and correlated bursts in
bed-load transport near the threshold. The dynamical origins of this apparent
collective behavior, however, are unknown because it has not been directly
observed in terms of grain motion. have taken a
more generalized approach to modeling stochastic bed-load transport by
viewing all probabilistic formulations of bed-load flux that incorporate
diffusivity as an approximation of a master equation that exactly conserves
both probability and mass. One key to making this approximation effective is
a deep understanding of the underlying assumptions used to construct the
effective diffusivity of the particles. For example, recently
showed that the apparently anomalous behavior in the
diffusivity of bed-load particles at short times is actually a by-product of
the nonlinear growth in the variance of particle hop lengths as particle
travel times are shortened. Furbish and colleagues' statistical mechanics
framework is the most general model for bed-load transport; given knowledge
of the microscopic and probabilistic motions of particles, one may derive
continuum-like expressions for the macroscopic behavior. Collective particle
motion could be incorporated into this framework, but this requires an
intimate understanding of the associated scales and correlations.
While the probabilistic approach has proven valuable for describing the
nature of transport near the threshold, it is vital to link this description to
the physical origins of the stochastic behavior. If collective entrainment is
the primary cause of bed-load flux intermittency, then what leads to it? One
possible mechanism for collective motion is entrainment due to collisional
impulses. Collisions are widely recognized as drivers of bed-load transport
in aeolian systems where separate thresholds for entrainment without
collision (the fluid threshold) and with collisions (the impact threshold) have been defined (). In aeolian
systems these collisions are accompanied by dramatic “splash” events where
numerous particles are ejected at once (i.e., collectively). Recently, it has
been proposed that entrainment in subaqueous systems has a significant
collisional component as well, especially in the case of large Stokes numbers
(). The Stokes number is the ratio of a particle's
inertial forces to the viscous forces of the fluid and, for binary collision
between same-sized spheres, is given by :
St=(1/9)RDusν. Here, R is the submerged
specific density, us is particle velocity, ν is fluid viscosity and
D is particle diameter. For St>102, viscous damping of
collisions is negligible () and thus collisions from
saltation are expected to impart significant momentum to both the bed and
neighboring particles for D≥10-2 m. Thus, it is likely that in
coarse gravel streams, colliding particles cause a subdued splash similar
to aeolian systems. If the analogy with aeolian systems holds, then this
splash entrainment will involve many particles becoming entrained at once.
This hypothetical, collision-induced collective entrainment could be strong
enough to be a primary driver of burstiness in bed-load flux near the threshold.
There are other physical systems examined previously that organize themselves
near a threshold and display intermittent mass flux; the behavior of
avalanching sand and rice piles comes to mind
(). These systems have
been extensively studied and display intermittent transport at the limit
where they are slowly driven past a threshold (in this case a critical
angle). In the intermittent regime, the size and duration of avalanches is
indeterminate (). As the sand pile is driven
harder this intermittent regime gives way to continuous flow down the heap
with an approximately constant flux. showed how
this transition into continuous flow can be viewed as a merger of the
intermittent, avalanching events. Might bed load fit into a class of more
generic avalanching systems that transition from intermittent to continuous
transport as they are taken from slowly driven to continuously driven?
In this paper we use the slowly to continuously driven limits as end-members
to explore how the nature of bed-load transport in an idealized experiment
changes as the frequency of mean transport is varied. This is achieved by
using a system that allows for precise control of the sediment feed rate
while all other parameters (slope, fluid discharge, etc.) are held constant,
while particle motion is tracked using sequential images. The imposed feed
rate is analogous to a driving frequency. We replicate the previously
observed growth in the intermittency of transport as the imposed sediment
feed-rate/driving frequency is slowed. Our major contribution is the direct
observation of collective entrainment, which reveals that collisions release
spatially grouped clusters from the bed that are analogous to avalanches. We
note here that the term avalanche is used in this paper in the generic manner
of dynamical systems, i.e., bursts of transport that exhibit some kind of
spatially extended correlation. We relate the scales of collective
entrainment to the kinetic energy deposited in the bed by colliding
saltators. This lends credence to the hypothesis that saltator–bed collisions
play a large role in entrainment (both collective or otherwise). In our
experiments, the growth in intermittency in bed-load transport appears to
arise primarily from the nonlinear growth in the waiting times between
transport events as the driving rate is slowed.
Mean flow conditions observed during the experiments. All values are
means taken around the range of flow conditions observed over all
experiments. h is the mean flow depth, u‾f is the mean flow
velocity, Sh is the Shields number, Fr is the Froude
number, Re is the Reynolds number and St is the Stokes
number of the large diameter grains. St for the small diameter
grains is also much larger than the viscously damped limit.
The experiments are conducted using a narrow, quasi 2-dimensional (2-D) flume
in which all the grains in the subsurface and surface can be monitored. The
flume channel is 2.3 m long and 20 mm wide. For all experiments, two
different sized spherical glass beads, D1=12 mm and D2=16 mm in
diameter (D), are fed into the channel in an even mixture. The two
different sizes are chosen to ensure a randomly packed bed. The “quasi 2-D”
nature of the experiment arises from the fact that the small glass beads have
significant overlap with one another along the axis orthogonal to the viewing
window. All experiments are conducted in a flume slope of 6 % and with a
fixed discharge of 37.9 L min-1, while the feed rate at which the
particle mixture is introduced to the channel is varied. The feed rate is the
control parameter used in the experiments, and throughout the rest of the
paper will be referred to as the driving frequency. The driving frequencies
used for the experiment were: 40, 60, 80, 160 and 200 marbles per minute.
Throughout the paper the abbreviation MPM will be used for marbles per
minute.
At the flow rate used, all flows in the channel are turbulent, with Reynolds
numbers greater than 104. Flow depths were found to be uniform with the
exception of 10–15 cm near the inlet and outlet of the flume. The flow is
supercritical with Froude numbers greater than 1, though any bedforms that
would be present at these flow conditions are suppressed due to the
narrowness of the channel. Experiments are in the high Stokes number regime
where collisions are expected to be important, in order to mimic the
conditions of gravel-bed rivers. Although collision velocities vary (they are
quantified below), they scale roughly with settling velocity, using terminal
fall velocity as a scale parameter, St>102 for all experiments.
Details about the flow parameters observed during the experiments are given
in Table . Only mean flow parameters are listed as the flow
parameters are kept approximately constant across experiments. This was
verified during the experimental runs where the range of flow conditions that
occurred during a single experiment was similar to the variability in
conditions seen across experiments. This flume is thought to represent the
simplest possible physical model of collision-driven bed-load transport. A
diagram of the experimental setup can be seen in Fig. . The
experiment is very similar to that used by
. This is intentional so that their results
can guide our study and our findings can be compared to their data.
Schematic of the experimental setup. The system is 2.3 m long and
20 mm wide. This quasi-two dimensional channel is fed at a constant water
discharge for all experiments. The slope is kept constant at 6 %. The
sediment feed is uncoupled from the fluid discharge and is introduced from
above using a custom-designed feeder built at the Penn Sediment Dynamics Laboratory (PennSeD) laboratory. A viewing
window of approximately 35 cm wide by 30 cm high is selected two-thirds of
the way down the flume. The window is backlit and the resulting images can be
seen in the figure inset.
A camera is situated approximately 100 cm downstream of the flume inlet. The
viewing window of the camera is 35 cm for all experiments. This section of
the flume is backlit using a white LED panel array that outputs at 300 lumen.
This produces a sharp silhouette of all the grains in the viewing window that
can then be used to acquire approximate particle centers. Images are acquired
at a rate of 120 fps and streamed continuously to a computer. This
acquisition rate is necessary to adequately capture the trajectories of
individual particles as they move through the viewing window.
Data acquisition and analysis
Once images are acquired, approximate particle centers are located using a
hybrid form of the algorithms outlined in and
. Using this method it is possible to obtain
particle centers that are accurate to better than 1 mm. However, the method
is highly sensitive to the degree of occlusion that the particles in the bed
experience, and thus centers can sometimes be less accurate. Once particle
centers are obtained, particles are linked together from image to image to
obtain particle trajectories using the method outlined by
. An example trajectory that is the final output of
this process can be seen in Fig. .
With a particle trajectory affixed to each particle that enters and leaves
the viewing window, it becomes possible to analyze the dynamics of mobile
particles over a wide variety of timescales. Emigration events sampled in the
viewing window are also easy to obtain from these trajectories. Emigration
series are obtained by choosing a fixed along-stream distance, x, to sample
along the viewing window for all experiments in question. If a particle
center crosses this position in the downstream direction, it is counted as a
positive emigration event. If it crosses this position in the upstream
direction, it is counted as a negative emigration event. This definition is
identical to that used in . To study the active particles
within the viewing window, it was necessary to set a threshold for particle
mobility. To do this, particle trajectories were analyzed over 1/10 of a
second. If the particle was displaced 2.4 mm within this time window, then the
particle was deemed to be mobile.
Particle transport is sampled at a fine enough timescale that emigration is
binary; there are either 0 or 1 particle(s) passing the sampled line at any
instant in time, making the time series of transport highly intermittent. One
approach to estimating the intermittency of the series of emigration events
obtained during an experiment (see Fig. ) is to look at how long
one needs to sample to arrive at a threshold standard deviation. In the case
of a uniform, low-intermittency time series, this sampling timescale will be
very short, whereas in the case of a highly intermittent series, a long
sampling time will be needed. This timescale is referred to throughout the
rest of the paper as tconv. It is computed directly from the
obtained emigration series for all of the driving frequencies studied, by
incrementally increasing the time, τ, used to sample from the emigration
series. For a given τ, 500 samples from the emigration series are
randomly resampled from the emigration series in question using a bootstrap
technique. The standard deviation of these samples is then computed and
normalized by the mean emigration rate for the samples taken from the series.
As τ grows, the standard deviation approaches the value chosen as the
threshold standard deviation, tconv. When the threshold
standard deviation is reached, this value is interpreted as tconv.
This approach is identical to that used in .
An example saltator trajectory obtained during the experimental run
with a sediment feed rate of 40 MPM; hydraulic conditions listed in
Table . Trajectories are created for all particles present in
the sampling area of the experiment.
Waiting times are sampled from the emigration series as well. They are
interpreted to be the time periods in between active emigration events over
position x. A waiting time period is started after an emigration event over
position x occurs, and it ends when the next emigration event happens.
Activity within the whole viewing window sampled in the experiments is
characterized in the paper through two different event-based metrics. One
type of event is referred to specifically as a “collective entrainment
event”. This is defined as a group of one or more mobile particles (mobility
was determined using the criteria above) moving within one large-grain
diameter of each other. This analysis is a simplified version of that used to
identify mobile clusters in . An example of a
collective entrainment event is given in Fig. . In this
example, the four large grains that are in color would be considered a
collective entrainment event. Collective entrainment events were identified
directly from analysis of the mobile particle trajectories sampled in the
viewing window for a given experiment. For a given time step, all N mobile
grain trajectories were identified. For i=1 to i=N, the distance of
the ith mobile grain to all the rest of the mobile grains was computed. A
clustering algorithm was then employed to identify clusters of grains that
were within a threshold distance of one another. This algorithm is capable of
identifying an arbitrary number of mobile clusters occurring at the same time
within the viewing window. A single mobile cluster of grains is defined as a
collective entrainment event. This cluster analysis was performed for the
entirety of the time steps available for a given experiment. This allowed us
to gather statistics of all of the collective entrainment events that
occurred in the viewing window for a given experiment.
Schematic side view of the experiment showing an example of a
collective motion event, sampled at four different times during the event.
Black circles are outlines of non-moving particles; colors correspond to
particles that are displaced actively during the event and are color coded
according to their position at the time step (t1, t2, etc.) as shown in
the figure. At t1 only two particles are moving. The large particle
collides with three particles on the bed at t2, and these three particles
are displaced at t3 and t4. The four large particles would be
classified as moving together collectively.
Example time series of emigration events for Qi=40 MPM. A
position x along the bed viewing window (as seen in Fig. ) is
monitored during the experimental run. When a particle passes position x, it
is considered an emigration event. This is a simple measure of particle
activity that can be converted into a time-averaged flux. Time series of
emigration sampled at a fixed position x along the bed were determined for
all experimental runs.
Determination of convergence time for experiments at all driving
rates. (a) Standard deviation of an ensemble of samples over a given
Δt. As Δt grows, the standard deviation decreases and
approaches the threshold standard deviation. This value of Δt is
interpreted to be the convergence time tconv. The standard
deviation is normalized by the mean emigration rate Q. The legend indicates the feed rate in marbles per minute (MPM). (b) The time Δtconv necessary for flux measurements to converge to a threshold
standard deviation of 10 %, as a function of the driving frequency in
number of marbles per minute. The dotted markers are the actual observed
convergence times, while the dashed red line displays the trend that one
would expect the convergence time to take if it were simply a function of the
feed frequency (Δtconv=110/finput; see text
for details).
Complementary cumulative probability plots of waiting times between
emigration events; (a) data for all experiments and
(b) the same data normalized by the driving frequency of each
respective experiment. Expectation from an exponential distribution is shown
for comparison with dashed line. Legend as in Fig. .
The other type of event is a “transport event”; it is more general and
contains collective entrainment events within it as a subset. It is defined
as a time period where there is at least one mobile grain within the viewing
window. As long as this is the case, an event is said to be taking place.
Once there are no mobile grains within the viewing window, then the transport
event has stopped. A portion of a transport event is pictured in
Fig. . Here, all the grains that are colored are considered to
be part of the current transport event that is taking place. It is possible
to see that there are time instances in Fig. where collective
motion is not occurring but particle activity is still ongoing. These time
instances with no collectively moving particles would be counted as part of a
transport event but not as part of a collective motion event.
To analyze the effects of impacts during events, saltating grains were
separated from the rest of the mobile population for a given event. The x and
y components of the saltator trajectories were then numerically
differentiated twice to obtain acceleration components which were then
converted to magnitudes of grain accelerations through time. A change-point
detection algorithm was then employed to identify the spikes in the
acceleration series representative of impact events.
Results
We computed Δtconv for all driving rates used in the
present study and found that it declines monotonically with increasing feed
rate (Fig. a). A simple expectation for the decrease in the
averaging time is that it should be proportional to the inverse of the
driving frequency; that is, Δtconv=a/Qi, where a is
a scaling parameter that depends on the percent standard deviation threshold
chosen to determine tconv. This relation can be thought of as
marking the growth in time required to count a fixed number of particles
emigrating past a line if driving frequency were the only thing that
mattered. This should be the case at high transport rates where we expect
smoother transport; accordingly, we choose a=110 such that the relation
matches the observed data for the highest feed-rate experiment. We see that
the inverse relation describes the convergence time reasonably well for the
three highest feed-rate experiments (Fig. ). For the two
slowest feed rates in the study, however, the actual increase in averaging
time with a reduction in Qi is more rapid than this expectation.
The waiting times between all observed emigration events for a given
experiment were sampled and used to compute empirical complementary
cumulative distributions (Fig. a). We compare these
distributions to an exponential distribution with a value λ=1,
which is on the order of the mean waiting times seen in the experiments. The
exponential distribution is expected for a Poisson process and was chosen
for comparison because of the extensive body of literature showing its
fitness for modeling uncorrelated random processes
(). If the waiting times were uncorrelated and truly
random, they should follow an exponential distribution; however, the measured
waiting times decay much more slowly (Fig. a). As expected, the
waiting times between emigration events seem to be a function of the driving
frequency. When the former are non-dimensionalized by the latter, the
variance among the experiments is significantly reduced
(Fig. b). Moderate dispersion remains among the different
experiments, however, indicating that driving rate is not the only factor
controlling the waiting times. We compute the average waiting time for each
experiment; the naive expectation is that this waiting time is precisely the
inverse of the driving frequency. The data follow this expectation for the
three highest driving frequencies; however, the mean waiting times for the
two slowest-driven experiments are significantly larger than expected
(Fig. ).
Relation between the driving frequency finput and mean
waiting time between observed emigration events W for each experimental
run. Note the full waiting time distributions for all experiments are shown
in Fig. . The dashed red line shows the expected relation that the
mean waiting time is the inverse of the driving frequency, W=1/finput.
The above results demonstrate that driving frequency has a strong effect on
the timing of emigration events and the timescale required for averaging. To
determine if this frequency also effects how particles are transported, it is
necessary to look at the particle kinematics during times when particle
activity is present. We examine here the complementary cumulative
distributions of particle speed for all experiments (Fig. ).
The speed of a given particle is computed as up2+wp2, where
up and wp are the respective horizontal and vertical components of the
particle velocity. The tails of the speed distributions do not vary strongly
as a function of driving frequency. This is sensible as the slow speeds (<0.1 mm s-1) are associated with essentially immobile particles, while
the fast speeds (>100 mm s-1) are almost exclusively associated
with saltators. As the fluid discharge is kept constant across experiments,
we do not expect to see large differences in the speed of saltating grains.
We do see an effect of driving frequency, however, for the intermediate
speeds (Fig. ). As the driving frequency declines, the
transition between mobile (fast) and immobile (slow) particles appears to
grow more abrupt; this is manifest as a growing kink in the curves. In other
words, the distribution of particle speeds is more continuous at high driving
frequencies and becomes more bimodal at low driving frequencies as motion
separates more distinctly into slow and fast particles.
Complementary cumulative distributions of active particle speeds for
all experiments. Legend as in Fig. .
Thus far, we have examined the motion of individual particles. Here, we
consider collective entrainment events – in particular, the size
distribution of particles that have been determined to be moving together.
These mobile clusters are analogous to avalanches in other granular systems,
although here they are restricted to a thin surficial layer of grains.
Interestingly, the distribution of mobile clusters does not vary
significantly with driving frequency (Fig. ). All experiments
show a roughly exponential distribution of cluster sizes, with a mean size
that varies only slightly with driving frequency.
Complementary cumulative distributions of mobile cluster sizes, for
each experiment at a different driving frequency. A cluster is defined as a
group of mobile particles moving together in space. The dashed black line is
an exponential trend, plotted for the sake of comparison. Note logarithmic
y axis.
Particle mobility as a function of kinetic energy (KE) deposited in the bed for an event. (a) Cumulative displacement for all
mobile particles during an event increases linearly with KE deposited.
(b) Probability distribution of the amount of deposited KE necessary
to entrain a given number of mobile particles. The mean of the distribution
is displayed as a red cross, and the medians are shown as green squares.
We observe qualitatively that almost all entrainment is associated with
impacts. However, the exact nature of this relationship is extremely
difficult to untangle for individual entrainment events. Entrainment can
happen immediately after an event or after an unpredictable time delay. In
addition, because the disordered bed absorbs and transmits momentum in a
complex way, a particle can be entrained as a result of an impact that
happened a significant distance (≫D) upstream. To avoid these issues,
while still gaining insight into the effects of impacts on entrainment, an
attempt to look at all impacts for a given period of particle activity in the
observational window of an experiment was performed. An event was defined as
a period where at least one particle was always mobile. Once all particles in
the observation window become immobilized, the event is deemed to be over (see
above). For a given event we computed (i) the amount of kinetic energy (KE)
deposited in the observed section of the bed, (ii) the cumulative
displacement of all mobilized particles and (iii) the number of particles
mobilized. Here, the cumulative displacement of all mobilized particles
includes the displacement of impacting saltators, which are considered to be
part of the mobile population. Deposited KE was determined by identifying the
points in time when an entrained saltator collided with the bed. The saltator
velocities immediately before and after the collision were used to obtain the
difference in KE of the saltating particle that occurs as a result of the
collision. This difference in KE was interpreted as occurring because of the
inelastic collision of the saltator with the bed and can be interpreted as
being the kinetic energy transferred (or deposited) by the saltator into the
bed. The total deposited KE for each event is the sum of the deposited KE
associated with each impact observed during that event. Deposited KE is a
positive value in all observed impacts; in other words, all observed impacts
resulted in the impacting particle losing KE into the bed. The KE deposited,
the cumulative displacement and the number of particles mobilized were compiled
for all events and for all driving rates, in order to determine the extent to
which particle mobility may be understood from collision energetics. The data
reveal a remarkably clear, linear relation between the total KE deposited and
the cumulative displacement of mobilized particles (Fig. a). We
also see that the number of mobilized particles systematically increases with
KE deposited, though there is significant variability (Fig. b).
Discussion and conclusion
For all driving frequencies, both the magnitude of collective entrainment
events (Fig. ) and the speed of saltating (fast) particles
(Fig. ) are similar. This indicates that collision dynamics do
not vary significantly across the range of sediment feed rates probed here.
Roughly, the intermittency of transport is controlled by the growth in the
mean waiting time as the driving frequency is slowed (see
Fig. ). Changing the driving rate appears to primarily affect
how quickly events happen and not the fundamental nature of entrainment. In
the slowly driven limit, (collective) entrainment events are infrequent and
may be considered discrete bursts in transport. As the system is driven
harder, events occur more frequently and begin to overlap with one another.
At the fastest driving rates, events become indistinguishable from one
another and continuous transport emerges. This picture aligns with behavior
seen in avalanching systems that display an intermittent to continuous
transition (). A sand pile
model by showed that overlapping avalanches may
interact, introducing correlations in the flux output of the system. The
observed changing distribution of particle speeds with driving rate in our
experiments may be an indication of this kind of complex behavior.
While much of the difference in entrainment rate and intermittency can be
related simply to the driving rate, some of it cannot. In particular, at low
driving rates we see waiting and averaging times that are significantly
longer than expected, suggesting that the first-order kinematic avalanching
model described above is incomplete. One timescale that has not been
considered is the relaxation time of avalanches, which for our experiments
would be the deposition timescale of mobile clusters following an entrainment
event. This timescale may not be independent of driving rate, and it becomes
impossible to measure when avalanches overlap in time. Another complicating
factor at low driving rates is the influence of creep, which has been
demonstrated to drive bursty bed-load transport in the viscous flow regime
(). Movies of our experiments reveal the presence
of slow creep also, but quantifying its significance is beyond the scope of
the present paper.
Indeterminate, complex behavior (such as the possible scenario outlined
above) is often an inherent feature of many-bodied, driven and strongly
dissipative systems (). For our system of a turbulent
fluid driving marbles that collide with a bed, it is not possible to predict
the response of a collision. Some collisions result in a strong rebound of
the saltator and no (observable) response of the bed; others drive an
immediate splash as several grains are entrained; and yet others lead to a
delayed response, in which a large number of grains become destabilized and
slowly accelerate to become entrained. Knowledge of the kinetic energy of an
impact is not sufficient to understand entrainment, due to the complicated
nature of energy dissipation. Knowledge of energy dissipation, however,
allows for significant predictive power. The strong relations between energy
deposited and the size and cumulative displacement of entrained particles provide some mechanistic basis for understanding collective entrainment and
burstiness in collision-driven bed load.
The similar size distributions of collective entrainment events across
varying driving rates show that, while collective entrainment is present,
its associated length scale does not vary as a function of intermittency.
While more analysis remains to be done, it is likely that collision-induced
momentum transfer into the bed is what sets the scale of collective
entrainment. Since fluid discharge did not vary in our experiments, the
velocity of saltating grains (and hence impact energy) remained roughly
constant for all driving rates. The approximately constant exponential trend
(Fig. ) aligns with the expectation that momentum transfer due
to saltator–bed impacts should be a primary driver of entrainment in this
system. were correct in positing a length scale
for collective entrainment; we see definite evidence for a length scale of
particle motion that is larger than that of a single particle. While this
length scale does not vary in these experiments, it is challenging to
extrapolate to other systems. At smaller Stokes numbers, collisions are
damped and turbulence becomes an important driver of collective entrainment
(). The shapes and size distributions of
natural particles and the roughness of the river bed (e.g., bed forms) will
also likely influence collective entrainment in ways that are difficult to
anticipate. Nevertheless, collision-driven entrainment should be the norm for
gravel-bed rivers (), and collective entrainment has
already been observed in the field ().
Incorporating this length scale into the general probabilistic framework
proposed by Furbish (e.g., ) will be important in
the effort to build statistical–mechanical models of bed-load transport that
start with correct assumptions of the underlying dynamics that govern
bed-load particle trajectories. Understanding the scales of bursty bed-load
transport will also inform the requisite times for bed-load sampling in the
field and laboratory ().
The principle data product derived from images of the
experiments is particle trajectories: horizontal and vertical coordinates of
particle centers through time. All trajectories used for this paper, as well
as the data products generated to produce the plots in this paper, are freely
available at 10.6084/m9.figshare.7356569.v1 ().
DBL conducted, analyzed and interpreted the experiments
and wrote the paper. DJ supervised the research and assisted in interpreting
the results and writing the paper.
The authors declare that they have no conflict of
interest.
Acknowledgements
We thank Carlos Ortiz and Morgane Houssais for discussion and help with data
analysis and David Furbish and Christoph Ancey for very helpful reviews that
clarified and improved the text. This research was partially supported by the
US National Science Foundation (NSF) grant EAR-1224943 and the US Army
Research Office, Division of Earth Materials and Processes grant
64455EV. Edited by: Jens
Turowski Reviewed by: David J. Furbish and Christophe Ancey
References
Ancey, C. and Heyman, J.: A microstructural approach to bed load transport:
mean behaviour and fluctuations of particle transport rates, J. Fluid Mech.,
744, 129–168, 2014.
Ancey, C., Davison, A., Böhm, T., Jodeau, M., and Frey, P.: Entrainment
and
motion of coarse particles in a shallow water stream down a steep slope,
J. Fluid Mech., 595, 83–114, 2008.
Bagnold, R. A.: The physics of wind blown sand and desert dunes, Methuen,
London, 1941.
Charru, F., Mouilleron, H., and Eiff, O.: Erosion and deposition of particles
on a bed sheared by a viscous flow, J. Fluid Mech., 519, 55–80,
2004.Crocker, J., Crocker, J., and Grier, D.: Methods of Digital Video Microscopy
for Colloidal Studies, J. Colloid Interf. Sci., 179,
298–310, 10.1006/jcis.1996.0217, 1996.
Dade, W. B. and Friend, P. F.: Grain-size, sediment-transport regime, and
channel slope in alluvial rivers, J. Geol., 106, 661–676,
1998.
Dinehart, R. L.: Correlative velocity fluctuations over a gravel river bed,
Water Resour. Res., 35, 569–582, 1999.Diplas, P., Dancey, C. L., Celik, A. O., Valyrakis, M., Greer, K., and Akar,
T.: The role of impulse on the initiation of particle movement under
turbulent flow conditions, Science, 322, 717–20,
10.1126/science.1158954, 2008.
Drake, T. G., Shreve, R. L., Dietrich, W. E., Whiting, P. J., and Leopold,
L. B.: Bedload transport of fine gravel observed by motion-picture
photography, J. Fluid Mech., 192, 193–217, 1988.
Einstein, H. A.: The Bed-Load Function for Sediment Transportation in Open
Channel Flows, Soil Conservation Service, US Department of Agriculture Washington, DC, 1–31, 1950.
Fathel, S., Furbish, D., and Schmeeckle, M.: Parsing anomalous versus normal
diffusive behavior of bedload sediment particles, Earth Surf. Proc. Land.,
41, 1797–1803, 2016.
Frette, V., Christensen, K., Malthe-Sørenssen, A., Feder, J., Jøssang,
T., and Meakin, P.: Avalanche
dynamics in a pile of rice, Nature, 379, 49 pp., 1996.Frey, P. and Church, M.: Bedload: a granular phenomenon, Earth Surf. Proc.
Land., 36, 58–69, 10.1002/esp.2103, 2011.Furbish, D. J., Haff, P. K., Roseberry, J. C., and Schmeeckle, M. W.: A
probabilistic description of the bed load sediment flux: 1. Theory, J. Geophys. Res.-Earth,
117, F03031, 10.1029/2012JF002352,
2012.
Furbish, D. J., Fathel, S. L., Schmeeckle, M. W., Jerolmack, D. J., and
Schumer, R.: The elements and richness of particle diffusion during sediment
transport at small timescales, Earth Surf. Proc. Land., 42,
214–237, 2017.Ganti, V., Meerschaert, M. M., Foufoula-Georgiou, E., Viparelli, E., and
Parker, G.: Normal and anomalous diffusion of gravel tracer particles in
rivers, J. Geophys. Res.-Earth, 115, F00A12, 10.1029/2008JF001222, 2010.
Gomez, B. and Phillips, J. D.: Deterministic uncertainty in bed load
transport,
J. Hydraul. Eng., 125, 305–308, 1999.
Heyman, J., Mettra, F., Ma, H., and Ancey, C.: Statistics of bedload
transport
over steep slopes: Separation of time scales and collective motion,
Geophys. Res. Lett., 40, 128–133, 2013.
Heyman, J., Ma, H., Mettra, F., and Ancey, C.: Spatial correlations in bed
load
transport: Evidence, importance, and modeling, J. Geophys.
Res.-Earth, 119, 1751–1767, 2014.Houssais, M., Ortiz, C. P., Durian, D. J., and Jerolmack, D. J.: Onset of
sediment transport is a continuous transition driven by fluid shear and
granular creep, Nat. Commun., 6, 6527, 10.1038/ncomms7527, 2015.Houssais, M., Ortiz, C. P., Durian, D. J., and Jerolmack, D. J.: Rheology of
sediment transported by a laminar flow, Phys. Rev. E, 94, 062609, 10.1103/PhysRevE.94.062609,
2016.
Hwa, T. and Kardar, M.: Avalanches, hydrodynamics, and discharge events in
models of sandpiles, Phys. Rev. A, 45, 7002–7023, 1992.
Jerolmack, D. J. and Brzinski, T. A.: Equivalence of abrupt grain-size
transitions in alluvial rivers and eolian sand seas: A hypothesis, Geology,
38, 719–722, 2010.
Keys, A. S., Abate, A. R., Glotzer, S. C., and Durian, D. J.: Measurement of
growing dynamical length scales and prediction of the jamming transition in a
granular material, Nat. Phys., 3, 260–264, 2007.Khan, H. A. and Maruf, G. M.: Counting clustered cells using distance
mapping, 2013 International Conference on Informatics, Electronics and
Vision, ICIEV 2013, 10.1109/ICIEV.2013.6572677, 2013.
Lawler, G. F. and Limic, V.: Random walk: a modern introduction, Vol. 123,
Cambridge University Press, Cambridge, UK, 2010.Lee, D. B. and Jerolmack, D.: MarbleData,
10.6084/m9.figshare.7356569.v1, 2018.
Lemieux, P.-A. and Durian, D.: From avalanches to fluid flow: A continuous
picture of grain dynamics down a heap, Phys. Rev. Lett., 85, 4273–4276,
2000.
Liu, A. J. and Nagel, S. R.: The jamming transition and the marginally jammed
solid, Annu. Rev. Conden. Ma. P., 1, 347–369, 2010.
Ma, H., Heyman, J., Fu, X., Mettra, F., Ancey, C., and Parker, G.: Bed load
transport over a broad range of timescales: Determination of three regimes of
fluctuations, J. Geophys. Res.-Earth, 119,
2653–2673, 2014.
Martin, R. L. and Kok, J. F.: Field measurements demonstrate distinct
initiation and cessation thresholds governing aeolian sediment transport
flux, arXiv preprint arXiv:1610.10059, 2016.
Martin, R. L., Purohit, P. K., and Jerolmack, D. J.: Sedimentary bed
evolution
as a mean-reverting random walk: Implications for tracer statistics,
Geophys. Res. Lett., 41, 6152–6159, 2014.
Maurin, R., Chauchat, J., and Frey, P.: Dense granular flow rheology in
turbulent bedload transport, J. Fluid Mech., 804, 490–512, 2016.
Nelson, J. M., Shreve, R. L., McLean, S. R., and Drake, T. G.: Role of
near-bed
turbulence structure in bed load transport and bed form mechanics, Water
Resour. Res., 31, 2071–2086, 1995.Nikora, V., Habersack, H., Huber, T., and McEwan, I.: On bed particle
diffusion
in gravel bed flows under weak bed load transport, Water Resour. Res.,
38, 17-1–17-9, 10.1029/2001WR000513, 2002.Pähtz, T. and Durán, O.: Fluid forces or impacts: What governs the
entrainment of soil particles in sediment transport mediated by a Newtonian
fluid?, Phys. Rev. Fluids, 2, 074303, 10.1103/PhysRevFluids.2.074303, 2017.
Papanicolaou, A., Diplas, P., Dancey, C., and Balakrishnan, M.: Surface
roughness effects in near-bed turbulence: Implications to sediment
entrainment, J. Eng. Mech., 127, 211–218, 2001.
Parker, G., Wilcock, P. R., Paola, C., Dietrich, W. E., and Pitlick, J.:
Physical basis for quasi-universal relations describing bankfull hydraulic
geometry of single-thread gravel bed rivers, Journal of Geophysical Research:
Earth Surface (2003–2012), 112, 2007.Parthasarathy, R.: Rapid, accurate particle tracking by calculation of
radial
symmetry centers, Nat. Methods, 9, 724–726, 10.1038/nmeth.2071, 2012.
Phillips, C. B. and Jerolmack, D. J.: Self-organization of river channels as
a
critical filter on climate signals, Science, 352, 694–697, 2016.Phillips, C. B., Martin, R. L., and Jerolmack, D. J.: Impulse framework for
unsteady flows reveals superdiffusive bed load transport, Geophys.
Res. Lett., 40, 1328–1333, 10.1002/grl.50323, 2013.
Prancevic, J. P. and Lamb, M. P.: Unraveling bed slope from relative
roughness
in initial sediment motion, J. Geophys. Res.-Earth,
120, 474–489, 2015.Rajchenbach, J.: Flow in powders: From discrete avalanches to continuous
regime, Phys. Rev. Lett., 65, 2221, 10.1103/PhysRevLett.65.2221, 1990.Recking, A.: A comparison between flume and field bed load transport data
and
consequences for surface-based bed load transport prediction, Water
Resour. Res., 46, W03518, 10.1029/2009WR008007, 2010.Regev, I., Lookman, T., and Reichhardt, C.: Onset of irreversibility and
chaos
in amorphous solids under periodic shear, Phys. Rev. E, 88, 062401, 10.1103/PhysRevE.88.062401,
2013.Schmeeckle, M. W. and Nelson, J. M.: Direct numerical simulation of bedload
transport using a local, dynamic boundary condition, Sedimentology, 50,
279–301, 2003.
Schmeeckle, M. W., Nelson, J. M., Pitlick, J., Bennett, J. P., and
Elastohydrodynamic, A.: Interparticle collision of natural sediment grains
in water, Water Resour. Res., 37, 2377–2391, 2001.Singh, A., Fienberg, K., Jerolmack, D. J., Marr, J., and Foufoula-Georgiou,
E.:
Experimental evidence for statistical scaling and intermittency in sediment
transport rates, J. Geophys. Res., 114, F01025,
10.1029/2007JF000963, 2009.
Sumer, B. M., Chua, L. H., Cheng, N.-S., and Fredsøe, J.: Influence of
turbulence on bed load sediment transport, J. Hydraul. Eng.,
129, 585–596, 2003.Tucker, G. E. and Bradley, D. N.: Trouble with diffusion: Reassessing
hillslope
erosion laws with a particle-based model, J. Geophys. Res.-Earth, 115,
F00A10,
10.1029/2009JF001264,
2010.Yager, E., Kirchner, J., and Dietrich, W.: Calculating bed load transport in
steep boulder bed channels, Water Resour. Res., 43, W07418, 10.1029/2006WR005432, 2007.