ESurfEarth Surface DynamicsESurfEarth Surf. Dynam.2196-632XCopernicus PublicationsGöttingen, Germany10.5194/esurf-6-389-2018Advection and dispersion of bed load tracersAdvection and dispersion of bed load tracersLajeunesseErichttps://orcid.org/0000-0002-0950-6054DevauchelleOlivierJamesFrançoisInstitut de Physique du Globe de Paris – Sorbonne Paris Cité, Équipe de Dynamique des Fuides Géologiques, 1 rue Jussieu, 75238 Paris CEDEX 05, FranceInstitut Denis Poisson, Université d'Orléans, Universitéde Tours, CNRS, Route de Chartres, BP 6759, 45067 Orléans CEDEX 2, FranceE. Lajeunesse (lajeunes@ipgp.fr)15May20186238939914September20179November20175March201816April2018This 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/6/389/2018/esurf-6-389-2018.htmlThe full text article is available as a PDF file from https://esurf.copernicus.org/articles/6/389/2018/esurf-6-389-2018.pdf
We use the erosion–deposition model introduced by to
numerically simulate the evolution of a plume of bed load tracers entrained by
a steady flow. In this model, the propagation of the plume results from the
stochastic exchange of particles between the bed and the bed load layer. We
find a transition between two asymptotic regimes. The tracers, initially at rest, are gradually
set into motion by the flow. During this
entrainment regime, the plume is strongly skewed in the direction of
propagation and continuously accelerates while spreading nonlinearly. With
time, the skewness of the plume eventually reaches a maximum value before
decreasing. This marks the transition to an advection–diffusion regime in
which the plume becomes increasingly symmetrical, spreads linearly, and
advances at constant velocity. We analytically derive the expressions of the
position, the variance, and the skewness of the plume and investigate their
asymptotic regimes. Our model assumes steady state. In the field, however,
bed load transport is intermittent. We show that the asymptotic regimes become
insensitive to this intermittency when expressed in terms of the distance
traveled by the plume. If this finding applies to the field, it might provide
an estimate for the average bed load transport rate.
Introduction
Alluvial rivers transport the sediment that makes up their bed. From a
mechanical standpoint, the flow of water applies a shear stress on the
sediment particles and entrains some of them downstream. When the shear
stress is weak, the particles remain close to the bed surface as they travel
. They roll, slide, and bounce over the rough bed until
they settle down . This
process is called bed load transport.
Bed load transport is inherently random . A turbulent
burst or a collision with an entrained grain sometime dislodges a resting
particle. The likeliness of this event depends on the specific arrangement of
the surrounding particles. On average, however, the probability of
entrainment is a function of macroscopic quantities such as shear stress and
grain size . Once dislodged, the velocity of a
particle fluctuates significantly around its average
.
Finally, the particle's return to rest is yet another random event. Overall,
a bed load particle spends only a small fraction of its time in motion.
Altogether, the combination of these stochastic processes results in a
downstream flux of particles. Fluvial geomorphologists measure this flux by
collecting moving particles in traps or Helley–Smith samplers
. The instantaneous sediment
discharge fluctuates due to the inherent randomness of bed load transport.
However, averaging measurements over time yields a consistent sediment flux
.
An alternative approach to sediment flux measurements is to follow the fate
of tracer particles. In November 1960, deposited
18 kg of radioactive sand in the North Loup River, a sand-bed stream located
in Nebraska (USA). Using a scintillator detector, they observed that the
plume of radioactive sand gradually spread as it was entrained downstream.
Tracking cobbles in gravel-bed rivers reveals a similar behavior: tracers
disperse as they travel downstream .
The dispersion of the tracers, expressed as the variance of their location,
results from the randomness of bed load transport.
identify three regimes with distinct timescales. A particle entrained by the
flow repeatedly collides with the bed . At short
timescales, i.e., between two collisions, particles move with the flow, and the variance
increases as the square of time .
This regime is analogous to the ballistic regime of Brownian motion
.
As the particle continues its course, collisions deviate its trajectory. In
this intermediate regime, the variance increases nonlinearly with time
. attribute this behavior to
anomalous super-diffusion, but contest their interpretation.
With time, tracers settle back on the bed, where they can remain trapped for
a long time. How the distribution of resting times influences the long-term
dispersion of tracers remains unknown. The data collected by
are consistent with the existence of a diffusive
regime in which the variance increases linearly .
Other investigators, however, report either sub-diffusion or super-diffusion
. These anomalous diffusion regimes
are sometimes modeled with fractional advection–dispersion equations
.
The variability of the stream discharge further complicates the
interpretation of field data. Bed load transport occurs when the shear stress
exceeds a threshold set by the grain size. Most rivers fulfill this condition
only a small fraction of the time, making sediment transport highly
intermittent . The rate at
which tracers spread thus depends not only on the inherent randomness of
bed load transport, but also on the probability distribution of the river
discharge .
Laboratory experiments under well-controlled conditions isolate these two
effects. For instance, tracked a plume of dyed
particles in an experimental channel. Although the flow was constant in this
experiment, the tracers still dispersed as they traveled downstream. In this
case, dispersion resulted from the inherent randomness of bed load transport
only. We can decompose this randomness into two components. First, the
velocity fluctuations disperse the particles . Secondly, the random exchange
of particles between the bed load layer, where particles travel, and the
sediment bed, where particles are at rest, further disperses the particles
. This effective diffusion
also occurs in chromatography experiments in which a bonded phase exchanges the
analyte with the flow .
In a recent paper, used the erosion–deposition
model introduced by to derive the equations governing the
evolution of a plume of tracers. Neglecting velocity fluctuations, they found
that the second dispersion process, namely the exchange of particles between
the bed load layer and the sediment bed, efficiently disperses the tracers.
They also observed the transition between an initial transient and classical
advection–diffusion. In the present paper, we further this investigation. Our
objective is to formally derive the contribution of the advection exchange of
particles to the dispersion of a plume of tracers. To do so, we briefly
rederive the equations governing the evolution of a plume of tracers (Sect. 2).
We numerically simulate the propagation of a plume of tracers and discuss
the nature of the two asymptotic regimes evidenced in
(Sect. 3). We analyze the long-time
advection–diffusion behavior of the plume and provide an analytical
expression for the diffusion coefficient and the plume velocity (Sect. 4). We
analytically derive the mean, the variance, and the skewness of the tracer
distribution and describe their asymptotic behavior in each regime (Sect. 5).
Finally, we discuss the applicability of these results to the field (Sect. 6).
Entrainment of tracers
In most rivers, sediment is broadly distributed in size. This likely
influences the dispersion of bed load tracers
. For the sake
of simplicity, however, we restrict our analysis to a bed of uniform
particles of size ds. The bed is sheared by a flow, which applies a shear
stress strong enough to entrain some particles. The latter remain confined in
a thin bed load layer.
For moderate values of the shear stress, the concentration of moving
sediments is small, and we can neglect the interactions between particles.
The erosion–deposition model introduced by provides an
accurate description of this dilute regime in which bed load transport is
controlled by the exchange of particles between the sediment bed and the
bed load layer. This exchange sets the surface concentration of moving
particles, nm, through mass balance:
∂nm∂t+V∂nm∂x=E-D,
where we introduce the average particle velocity V. E is the erosion
rate, defined as the number of bed particles set in motion per unit of time
and area. Similarly, the deposition rate D is defined as the number of
bed load particles settling on the bed per unit of time and area
.
To investigate the dispersion of bed load particles, we consider some of
them to be marked (Fig. ). We refer to these marked
particles as “tracers” and assume that their physical properties are the
same as those of unmarked particles. With these assumptions, the mass balance
for the tracers in the bed load layer reads
nm∂ϕ∂t+nmV∂ϕ∂x=Eψ-Dϕ,
where we introduce the proportion of tracers in the moving layer, ϕ.
Similarly, ψ is the proportion of tracers on the bed surface.
When subjected to varying flow and sediment discharges, the bed of a stream
accumulates or releases sediments . Some particles may then be
temporary buried within the bed, inducing streamwise dispersion
. Here, we neglect this
mechanism and restrict our analysis to steady and uniform sediment transport.
Accordingly, we assume that erosion and deposition affects the bed over a
depth of about one grain diameter only. This hypothesis holds if the
departure from the entrainment threshold is small enough. With these
assumptions, the mass balance for the tracers on the bed surface reads
ns∂ψ∂t=Dϕ-Eψ,
where ns is the surface concentration of particles at rest on the bed
surface. Each of them occupies an area of about ds2. The surface
concentration of particles at rest is therefore ns∼ 1/ds2.
For steady and uniform transport, the surface concentration of moving
particles, n, is constant. In addition, erosion and deposition balance each other:
E=D.
Laboratory experiments suggest that the deposition rate is proportional to
the concentration of moving particles:
D=nmτf
where we introduce the average flight duration, τf=ℓf/V, and the
average flight length, ℓf.
The flight length is the distance traveled by a mobile particle between its
erosion and eventual deposition. Similarly, the flight duration is the time a
particle spends in the bed load layer. In practice, measuring these quantities
often proves difficult, since they depend on how one defines the mobile and
the static layer .
Combining Eq. (),
(), (), and ()
provides the set of equations that describe the propagation of the plume:
∂ϕ∂t+V∂ϕ∂x=1τf(ψ-ϕ),∂ψ∂t=-ατf(ψ-ϕ),
where we define α=nm/ns∼nmds2, the
ratio of the concentration of moving particles to the concentration of static
particles. This ratio is smaller than 1. It is proportional to the
intensity qs of bed load transport:
α∼ds2Vqs.
Granular bed sheared by a steady and uniform flow. The bed is a
mixture of marked (red) and unmarked (white) grains.
Complemented with initial and boundary conditions, Eqs. ()
and () describe the evolution of the plume. In
dimensionless form, they read
∂ϕ∂t^+∂ϕ∂x^=ψ-ϕ,∂ψ∂t^=-α(ψ-ϕ),
where t^=t/τf and x^=x/ℓf are dimensionless
variables. For ease of notation, we drop the hat symbol in what follows.
A single parameter controls Eqs. () and (): the
ratio of surface densities α, which characterizes the average distance
between grains in the bed load layer. Since the erosion–deposition model
assumes independent particles, we can only expect it to be valid when moving
particles are sufficiently far away from each other, which is when α is
small or, equivalently, when the Shields parameter is near the threshold.
In the next section, we numerically solve Eqs. () and ().
Propagation of a plume of tracers
Laboratory measurements of bed load often use top-view images
. Unless individual particles
can be tracked, the tracers at rest are usually indistinguishable from those
entrained by the flow. Separating the proportion of tracers in the moving
layer, ϕ, from that on the bed surface, ψ, is practically
impossible. Instead, top-view pictures show the total concentration of tracers:
c=nmϕ+nsψnm+ns=αα+1ϕ+1α+1ψ.
Tracking sediment in rivers poses a similar problem. In general, one records
the position of the tracers when the river stage is below the threshold of
grain entrainment . At the
time of measurement, all tracers are therefore at rest. As a result, the
proportion of mobile tracers vanishes (ϕ= 0), and the total concentration
of tracers reads c=ψ/(α+ 1).
In summary, the proportions of mobile and static tracers, ϕ and ψ,
naturally derive from mass balance (Eq. ) and
Eq. (). However, their measurement proves
difficult during active transport. On the other hand, experimental and field
investigations provide the total concentration of tracers, c. This quantity is
conservative, as the total amount of tracers, M=∫c dx, is
preserved. In the following, we therefore focus on the concentration of tracers, c.
To study the evolution of the tracer concentration, we solve
Eqs. () and () numerically using a finite-volume scheme.
We then compute the tracer concentration using Eq. () (Fig. ).
Evolution of the tracer concentration (α= 0.1) obtained
by numerically solving Eqs. () and ().
(a) Early entrainment regime. (b) Relaxation towards the
diffusive regime. Tracers are initially at rest, forming a symmetric plume of
length L= 0.5 and mass M= 1. The concentration profile
asymptotically tends towards a Gaussian distribution (dotted red
line).
The early evolution of the plume depends on initial conditions. In most field
experiments, tracers are deposited at the surface of the river bed when the
flow stage is low and sediment is motionless .
During floods, the river discharge increases and the shear stress eventually
exceeds the entrainment threshold, setting in motion some of the grains. The
entrainment of particles strongly depends on the arrangement of the bed:
grains highly exposed to the flow move first
. Several authors
find that the tracers they disposed on the bed are more mobile during the
first flood than during later ones . During the
later floods, tracers gradually get trapped in the bed, and their average
mobility decreases. On the other hand, find no
special mobility during the first flood. In the absence of a clear scenario,
we choose the simplest possible initial conditions and assume that
initially all tracers belong to the static layer: ϕ(x, t= 0) = 0.
With these initial conditions, the evolution of the plume follows two
distinct regimes. At early times, the flow gradually dislodges tracers from
the bed and entrains them in the bed load layer. During this entrainment
regime, only a small proportion of the tracers move. Consequently, the plume
develops a thin tail in the downstream direction
(Fig. a). The corresponding distribution of travel
distances is strongly skewed towards the direction of propagation, a feature
commonly observed in field experiments .
With time, the plume moves downstream and spreads both upstream and
downstream. As a result, the concentration rapidly decreases to small levels.
The plume becomes gradually symmetrical and tends asymptotically towards a
Gaussian distribution (Fig. b). This regime is
reminiscent of classical diffusion.
To better illustrate this evolution, we introduce the mean position of the
plume of tracers:
〈x〉=1M∫-∞∞cxdx.
We also characterize its size with the variance,
σ2=1M∫-∞∞c(x-〈x〉)2dx,
and its symmetry with the skewness,
γ=1M∫-∞∞cx-〈x〉σ3dx.
The evolution of these three moments is consistent with the existence of two
asymptotic regimes (Fig. ). At short timescales, the plume
grows a thin tail downstream. This deformation causes the plume's skewness to
increase as t4. During this regime, the average location of the plume
increases as t2 and its variance grows as t3. Although the variance
increases nonlinearly with time, the exponent, 3, is too large for
super-diffusion .
After a characteristic time of the order of τ≈τf, the
skewness of the plume reaches a maximum (Fig. c). This
corresponds to a drastic change in dynamics: the skewness starts decreasing
as the plume becomes gradually more symmetrical. At long timescales, the plume of
tracers advances at constant velocity and diffuses linearly with time
(Fig. a and b). This regime, regardless of the value of
α, corresponds to classical advection–diffusion.
Next, we establish the equivalence between diffusion and the long-time
behavior of the tracers.
(a) Position, (b) variance, and
(c) skewness of a plume of tracers as a function of time for
α= 0.1 and α= 0.001. We compute the evolution of
these three quantities using Eqs. (), (),
and (). The results agree exactly with numerical simulations.
The asymptotic regimes of the skewness are represented with grey lines. Their
intersection provides an estimate of the duration of the entrainment regime
(see Eq. ).
Advection–diffusion at long timescales
The diffusion at work in Eqs. () and () results
from the continuous exchange of particles between the bed load layer, where
particles travel at the constant velocity V, and the sediment bed, where
particles are at rest. The velocity difference between the two layers
gradually smears out the plume and spreads it in the flow direction. This
process occurs in a variety of physical systems in which layers moving at
different velocities exchange a passive tracer. A typical example is Taylor
dispersion, whereby a passive tracer diffuses across a Poiseuille flow in a
circular pipe . The combination of shear rate and
transverse molecular diffusion generates an effective diffusion in the flow
direction. Other examples of effective diffusion include solute transport in
porous media and chromatography .
To formally establish the equivalence between diffusion and the long-time
behavior of the plume, we follow a reasoning similar to the one developed for
chromatography . Equations () and ()
are equivalent to
∂c∂t+αα+1∂c∂x=α(α+1)2∂δ∂x,∂δ∂t+1α+1∂δ∂x+(α+1)δ=∂c∂x,
where we introduce δ=ψ-ϕ, the difference between the
proportion of tracers on the sediment bed and that in the bed load layer.
Eventually, these proportions equilibrate each other. At long timescales, we
therefore expect the solution to Eqs. () and ()
to relax towards steady state, for which δ is of order ϵ≪ 1.
Accordingly, we rewrite these two equations as
∂c∂t+αα+1∂c∂x=ϵα(α+1)2∂δ∂x,∂δ∂t+1α+1∂δ∂x+(α+1)δ=1ϵ∂c∂x.
Introducing T=ϵt and X=ϵx and developing c
and δ with respect to ϵ yields
∂c0∂T+αα+1∂c0∂X=0(α+1)δ0=∂c0∂X
at zeroth order and
∂c1∂T+αα+1∂c1∂X=α(α+1)2∂δ0∂X
at first order.
By multiplying Eq. () by ϵ and summing the result
with Eq. (), we finally get
∂c∂t+αα+1∂c∂x=α(α+1)3∂2c∂x2.
At long timescales, the transport of the tracers follows the advection–diffusion
equation (Eq. ). We identify the advection velocity,
U, which reads
U=αα+1ℓfτf∼αℓfτf.
Likewise, the diffusion coefficient reads
Cd=α(α+1)3ℓf2τf∼αℓf2τf.
This asymptotic equivalence explains the advection–diffusion regime
(Figs. and ).
We interpret this formal derivation as follows. In the reference frame of the
plume, a tracer at rest on the bed moves backward, while a tracer entrained
in the bed load layer moves forward. At long timescales, the proportions of tracers
in each layer equilibrate. Consequently, the probability that a tracer will be
entrained and move forward equals that of deposition. In the reference frame
of the plume, the exchange of particles between the bed and the bed load layer
is thus a Brownian motion driving the linear diffusion of the plume.
In the next section, we investigate the evolution of the location, the size,
and the symmetry of the plume as it propagates downstream.
Location, size, and symmetry of the plume
Concentration, defined as the number of tracers per unit of area, depends on
the area over which it is measured. Its value is meaningful when the
measurement area is much larger than the distance between particles and much
smaller than the plume. During the entrainment regime, the plume develops a
thin tail containing only a small proportion of tracers. Measuring the
concentration profile during this regime is thus challenging. To our
knowledge, only were able to measure consistent
concentration profiles using radioactive sand. In practice, most field
campaigns involve a limited number of tracers (900 at most)
.
It is thus more practical to consider integral
quantities, such as the mean position of the plume 〈x〉, its
variance σ2, and its skewness γ.
Multiplying Eq. () by x and integrating over space provides
the evolution equation for the mean position:
∂〈x〉∂t=αα+1-α(α+1)2〈δ〉,
where
〈δ〉=1M∫δdx
is the average difference between the proportion of tracers on the sediment
bed and in the bed load layer. To solve Eq. (), we need an
equation for 〈δ〉. The latter is obtained by integrating
Eq. () over space:
∂〈δ〉∂t=-(α+1)〈δ〉.
Equations () and () describe the downstream
motion 〈x〉 of the plume. To solve them, we need to specify
initial conditions. As discussed in Sect. , we
consider all tracers to initially belong to the static layer,
i.e., ϕ(x, t= 0) = 0. This condition and the conservation of mass,
〈c〉= 1, provide initial conditions for 〈δ〉:
〈δ〉(t= 0) =α+ 1. With this condition,
Eqs. () and () integrate into
〈x〉-〈x〉0=αα+1t+α(α+1)2e-(α+1)t-1,
where 〈x〉0 is the initial position of the plume.
We now focus on the variance of the plume. Multiplying Eq. () by x2
and integrating over space yields the evolution equation for the second
moment of the tracer distribution:
∂〈x2〉∂t=2α(α+1)〈x〉-2α(α+1)2〈xδ〉,
where
〈xδ〉=1M∫xδdx
is the first moment of δ. To solve Eq. (), we need an
equation for this intermediate quantity. We obtain it by multiplying
Eq. () by x and integrating over space:
∂〈xδ〉∂t=-1-(α+1)〈xδ〉+〈δ〉α+1.
At time t= 0, 〈xδ〉(t= 0) = (α+ 1)〈x〉0.
Equations () and () with this initial
condition provide the expression of the second moment of the tracer distribution:
〈x2〉=〈x2〉0+2α(α+1)3t+2-αα+1e-(α+1)t+α2(α+1)2t2+2α(1-α)(α+1)3t+2α(α-2)(α+1)4,
where 〈x2〉0 is the initial value of the second moment of
the tracer distribution. We then deduce the variance of the plume from
σ2=〈x2〉-〈x〉2.
We follow a similar procedure to derive the skewness of the plume.
Multiplying Eq. () by x3 and integrating over space yields the
evolution equation for the third moment of the tracer distribution:
∂〈x3〉∂t=3α(α+1)〈x2〉-3α(α+1)2〈x2δ〉,
where
〈x2δ〉=1M∫x2δdx
is the second moment of δ. Multiplying Eq. () by x2 and
integrating over space provides the evolution equation for this intermediate quantity:
∂〈x2δ〉∂t=-(α+1)〈x2δ〉+2α+1〈xδ〉-2〈x〉.
At time t= 0, 〈x2δ〉= (α+ 1)〈x2〉0
and 〈x3〉= 0. With these initial conditions,
Eqs. () and () provide the expression of 〈x3〉:
〈x3〉=3αα+1σ02+2α2-8α+2(α+1)4t+3α(α+1)2σ02+2α2-12α+6(α+1)4e-(α+1)t-1+3α(α+1)4t-4α-1α+1te-(α+1)t+α3(α+1)3t-3(α-2)α(α+1)t2
from which we deduce the skewness of the plume as
γ=〈x3〉-3〈x〉σ2-〈x〉3σ3.
Equations (), (), (),
(), and () represent the evolution of the mean, the
variance, and the skewness of the tracer distribution. They describe the
migration, spreading, and symmetry of the plume. They do not require any assumption other
than the ones of the model itself and agree exactly with numerical
simulations (Fig. ).
As discussed in Sect. , numerical simulations reveal a
transient during which the tracers, initially at rest, are gradually set into
motion by the flow (Fig. ). During this entrainment
regime, the plume continuously accelerates, spreads nonlinearly, and becomes
increasingly asymmetrical. To characterize this regime, we expand
Eqs. (), (), (), (),
and () to leading order in time:
〈x〉-〈x〉0∼α2t2,σ2-σ02∼α3t3,γ∼α4σ03t4.
These three equations are consistent with our numerical simulations (Fig. ).
Anomalous diffusion arises from heavy-tailed distributions of either the
step length or the waiting time . The
erosion–deposition model contains no such ingredient. Here the fast increase
in the variance results from the exchange of particles between the sediment
bed and the bed load layer at the beginning of the experiment. Over a time
shorter than the flight duration τf, the tracers entrained by the flow
do not settle back on the bed. They form a thin tail, which leaves the main
body of the plume and moves downstream at the average particle velocity V
(Fig. a). The plume therefore consists of a main
body of virtually constant concentration, followed by a thin tail of
length ∝Vt. Accordingly, we can split the integral that defines its mean
position, Eq. (), into two terms. The first one, obtained
by integrating cx over the main body of the plume, yields the initial
position of the plume 〈x〉0. The second one, obtained by
integrating cx over a tail of length Vt, scales as t2. Summing these
contributions yields Eq. (). Similar reasonings yield
Eqs. () and () for the variance and the skewness.
With time, the plume enters the diffusive regime. Its velocity and its
spreading rate relax towards constants while its skewness decreases
(Fig. ). We derive the corresponding asymptotic
behavior by expanding Eqs. (), (),
(), (), and () in the limit of time being
large:
〈x〉-〈x〉0∼αα+1t∼αtσ2-σ02∼2α(α+1)3t∼2αtγ∼32α1t.
The asymptotic regimes (Eqs. and ) are
consistent with the expressions derived in Sect.
.
The transition between the entrainment and the diffusive regime occurs when
the skewness reaches its maximum value. Equating the skewness estimated from
Eqs. () and () provides the approximate
duration of the entrainment regime, τ. We find
τe=(72)1/9σ02α1/3τf,
which compares well with our numerical simulations (Fig. ).
The duration of the entrainment regime increases with the initial size of the
plume and decreases with the intensity of sediment transport.
The asymptotic regimes (Eqs. , ,
, , , and )
assume that sediment transport is in steady state. In the next section, we
discuss the intermittency of bed load transport in natural streams.
Intermittency of bed load transport
Our description of the plume of tracers is based on the assumption that
sediment transport is in steady state. This hypothesis is often satisfied in
laboratory flumes . In a river, it may be met
for up to a few days . At longer timescales,
however, most rivers alternate between low-flow stages during which sediment
is immobile and floods during which bed particles are entrained downstream
. Bed load transport is thus intermittent.
The intermittency of bed load transport influences the propagation of tracers
in several ways. First of all, sediment transport during a flood modifies the
structure of the bed . As a result,
the proportion of tracers in the bed load layer and in the bed, ϕ and
ψ, likely change from one flood to the next. In a effort to address this
question, P. Allemand and collaborators recently implemented the survey of a
river located on Basse-Terre Island (Guadeloupe archipelago). Their
preliminary observations reveal that the cobbles deposited at the end of a
flood are the first entrained at the beginning of the next (P. Allemand,
personal communication, 30 June 2017). Based on this observation, we
speculate that a tracer belonging to the bed load layer at the end of a flood
will still be part of the bed load layer at the beginning of the next one.
Similarly, a tracer locked in the bed at the end of a flood will belong to
the static layer at the beginning of the next one. In other words, we assume
that tracers freeze between two floods.
(a) Variance and (b) skewness of a plume of
tracers as a function of traveled distance (α= 0.1). These three
quantities are calculated from Eqs. (), (),
and (). Inset: concentration profiles (blue) illustrating the
shape of the plume during the entrainment regime (left), at the transition between the entrainment and the diffusive
regime (center), and in the diffusive regime (right).
If this assumption holds, the simplest way to account for bed load
intermittency is to assume that the river alternates between two
representative stages: (1) a low-flow stage during which tracers are immobile
and
(2) a flood stage characterized by a representative sediment flux qs∼αV/ds2 during which tracers propagate downstream
. Following this model, we may
extrapolate our results to the field, provided we rescale time with respect
to an intermittency factor I=Te/T, where T is the total duration
of elapsed time, and Te is the time during which sediments are
effectively in motion .
In practice, evaluating the intermittency factor requires continuous
monitoring of the river discharge and a correct estimate of the entrainment
threshold. , for instance, monitored the location
of tracer cobbles deposited in the Bouinenc stream (France) during 2 years.
Over this period, the motion of the tracers resulted from 55 floods for a
total duration of 42 days. Sediments were thus in motion less than I= 12 %
of the time.
Here, we suggest another way to circumvent the intermittency of sediment
transport. Plotting the plume variance, (σ2-σ02), and its
skewness, γ, as a function of traveled distance, 〈x〉-〈x〉0, eliminates time from the equations
(Fig. ). In this plot, the position of the plume
acts as a proxy for the effective duration of sediment transport, Te.
The resulting curves are thus filtered from transport intermittency (Fig. ).
The entrainment regime corresponds to small traveled distances. In this
regime, both the size of the plume and its asymmetry increase with traveled
distance (Fig. ). Equations (),
(), and () describe the early evolution
of the plume. Eliminating time by combining them, we find the behavior of the
plume for short traveled distances:
σ2-σ02=8ℓf9α〈x〉-〈x〉03/2,γ=ℓfασ03〈x〉-〈x〉02.
As discussed in Sect. , these scalings
result from the gradual entrainment of the tracers that are initially trapped
in the bed.
After the plume has traveled over a distance roughly equal to the flight
length, its skewness reaches a maximum value and starts decreasing. This
change in dynamics indicates the transition towards the diffusive regime.
Equations (), (), and ()
provide the long-term behavior of the plume:
σ2-σ02∼2ℓf〈x〉-〈x〉0,γ=32ℓf〈x〉-〈x〉0.
The linear increase in the variance with the distance traveled by the plume
is the signature of standard diffusion (see Sect. ).
Equating the skewness estimated from Eqs. ()
and () provides the position 〈x〉max at
which the skewness reaches its maximum:
〈x〉max-〈x〉0∼3α22/5σ06ℓf1/5.
The entrainment regime lasts until the plume has traveled over a distance
comparable to its initial size, which is until 〈x〉-〈x〉0∼σ0.
When expressed in terms of the distance traveled by the plume, the asymptotic
regimes are insensitive to the intermittency of bed load transport. They are
thus a robust test of our model and can help us interpret field data. Let us
assume that a dataset records the evolution of a plume of tracers released in
a river over a distance long enough to explore both the entrainment and the
diffusive regime. During the diffusive regime, the skewness decreases with
the traveled distance. A fit of the data with
Eq. () yields the flight length, ℓf. Knowing the
latter, we could use Eq. () to estimate the
intensity of sediment transport, α, from the evolution of the skewness
during the entrainment regime.
According to Sect. , the skewness reaches a
maximum after a time τe (Eq. ). Taking into account
the intermittency of bed load transport in natural streams, we expect that
this maximum is reached when
t=(72)1/9σ02α1/3τfI,
where I is the intermittency factor. Identifying this maximum in a field
experiment thus yields the ratio τf/I. Combining the latter with our
estimates of the flight length, ℓf, and the intensity of sediment
transport, α, should provide us with the average sediment transport
rate in the river:
qs‾=Iαds2lfτf.
Conclusion
We used the erosion–deposition model introduced by to
describe the evolution of a plume of bed load tracers entrained by a steady
flow. In this model, the propagation of the plume results from the stochastic
exchange of particles between the bed and the bed load layer. This mechanism
is reminiscent of the propagation of tracers in a porous medium
. The evolution of the plume depends on two
control parameters: its initial size, σ0, and the intensity of
sediment transport, α.
Our model captures in a single theoretical framework the transition between
two asymptotic regimes: (1) an early entrainment regime during which the
plume spreads nonlinearly and (2) a late-time relaxation towards classical
advection–diffusion. The latter regime is consistent with previous
observations .
When expressed in terms of the distance traveled by the plume, the asymptotic
regimes are insensitive to the intermittency of bed load transport in natural
streams. According to this model, it should be possible to estimate the
particle flight length and the average bed load transport rate from the
evolution of the variance and the skewness of a plume of tracers in a river.
No data sets were used in this article.
The authors declare that they have no conflict of interest.
Acknowledgements
It is our pleasure to thank Pascal Allemand, David John Furbish, Colin Phillips,
Douglas Jerolmack, and François Métivier
for many helpful and enjoyable discussions.
This work was supported by the French national program
EC2CO-Biohefect/Ecodyn//Dril/MicrobiEn, “Dispersion de contaminants
solides dans le lit d'une rivire”.
Edited by: Patricia Wiberg
Reviewed by: two anonymous referees
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