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Laboratory rivers adjust their shape to sediment
transport
A. Abramian, O Devauchelle, E. Lajeunesse
To cite this version:
A. Abramian, O Devauchelle, E. Lajeunesse. Laboratory rivers adjust their shape to sediment
transport. Physical Review E , American Physical Society (APS), 2020, 102 (5), �10.1103/Phys-
RevE.102.053101�. �hal-02987014�
Laboratory rivers adjust their shape to sediment transport
A. Abramian,
1, 2, ∗
O. Devauchelle,
1
and E. Lajeunesse
1
1
Universit´e de Paris, Institut de physique du globe de Paris, CNRS, F-75005 Paris, France
2
Institut Jean Le Rond d’Alembert, Sorbonne Universit´e, CNRS, F-75005 Paris, France
(Dated: November 3, 2020)
An alluvial river builds its own bed with the sediment it transports; its shape thus depends not
only on its water discharge, but also on the sediment supply. Here, we investigate the influence of
the latter in laboratory experiments. We find that, as their natural counterpart, laboratory rivers
widen to accommodate an increase of sediment supply. By tracking individual particles as they
travel downstream, we show that, at equilibrium, the river shapes its channel so that the intensity
of sediment transport follows a Boltzmann distribution. This mechanism selects a well-defined width
over which the river transports sediment, while the sediment remains virtually idle on its banks.
For lack of a comprehensive theory, we represent this behaviour with a single-parameter, empirical
model which accords with our observations.
I. INTRODUCTION
The bed of an alluvial river is made of mobile sediment,
such as sand or gravel [1]. Its shape results from the
action of water on this granular bed: the flow entrains
superficial grains and deposit them further downstream,
thus deforming the channel that confines it. With time,
this coupling selects the size and shape of the river.
The width of a river typically scales with its water dis-
charge. Specifically, it follows the empirical law of Lacey:
it is proportional to the square root of the discharge [2].
At leading order, this relationship indicates that the river
bed is near the threshold of motion [3]. Indeed, if we as-
sume that each grain of the bed surface is steady, but
about to move, the river’s cross section should form a co-
sine of prescribed dimensions [4]. Laboratory analogues
of rivers conform to this theory, for both turbulent and
laminar flows [5, 6].
Most rivers, however, carry some sediment, and their
bed is therefore above the threshold of motion [7, 8].
These so-called “active” rivers are generally wider, shal-
lower, and steeper than predicted by the threshold the-
ory, and destabilise into braids beyond a critical sedi-
ment load [9, 10]. Although water discharge is the prime
control on their size, sediment discharge also affects the
shape of an alluvial river [11, 12].
Yet, the role of sediment transport in alluvial rivers
remains obscure. Its investigation has proven challeng-
ing in the laboratory, because most experiments generate
braids, as a result of the unhindered growth of bedforms
[13, 14]. Some experimenters prevent this instability by
adding cohesive sediment [14], or by growing riparian
vegetation [15]. How necessary these ingredients are,
however, remains a matter of debate [16]. Reitz et al.
[17] and Delorme et al. [18] produced active rivers with
moderate sediment supply, but focused on the alluvial
fan they deposit, rather than on their internal dynamics.
Ikeda et al. [19] also maintained an active single channel
∗
anaisabramian@gmail.com
in the laboratory by splitting it into halves with a vertical
wall, but they did not measure the sediment discharge.
To investigate sediment transport per se, it is easier to
confine the flow in a canal or a pipe [20–24]. This con-
figuration, usually referred to as a “flume” experiment,
typically provides a relation between the intensity of the
sediment flux and the shear the fluid exerts on the bed,
τ. It is customary to express this relationship in terms
y
x
z
190 cm
90 cm
Laser
sheet
Camera
Sediment
Q
s
Fluid
Q
w
(a)
1.0 cm
x
y
z
Flow
(b)
FIG. 1. Experimental setup. (a) Laboratory river, with a
top-view camera and inclined laser sheet. Q
w
and Q
s
denote
the flow and the sediment discharges, respectively. (b) Close
view on the river bed. White dashed lines materialize banks.
A few grain trajectories are plotted in pink.
2
of the Shields parameter, defined as the ratio of the fluid
force to the weight of a grain [25]:
θ =
τ
(ρ
s
− ρ
f
)gd
s
. (1)
When the fluid force overcomes the weight of the grain,
the Shields parameter exceeds its threshold value θ
t
, and
bedload transport starts. It then increases linearly with
the excess stress (at least for moderate transport):
q
s
= q
0
(θ − θ
t
) , (2)
where q
0
and θ
t
depend on the fluid and sediment prop-
erties.
At moderate shear stress, the particles move by rolling,
sliding, and bouncing, while gravity maintains them close
to the bed surface. The layer of entrained grains, or
“bedload layer”, is only a few grain-diameters thick [26].
When the shear stress becomes more intense, grains can
be suspended in the bulk of the flow [24], where they
can diffuse across the stream [27]. Here, we focus on
bedload transport. In laboratory flumes that transport
sediment as bedload, the travelling grains collide with
the rough sediment bed underneath [28, 29]. Particle
tracking shows that these collisions turn their trajecto-
ries into random walks across the stream. Collectively,
bedload particles thus diffuse from areas where their pop-
ulation is dense towards less crowded ones [30] —much
like suspended particles.
For the bed to reach equilibrium, another flux must
oppose this diffusive flux. Gravity, which pulls the mov-
ing grains towards the channel’s center, plays this role in
laboratory flumes [31] and most likely in natural rivers
[32]. The balance between gravity and diffusion then sets
the bed’s shape and, surprisingly, its downstream slope.
This statistical equilibrium takes the form of a Boltz-
mann distribution, according to which the sediment flux
q
s
decreases exponentially with the bed elevation h [31]:
q
s
hqi
g
= exp
−
h − hhi
a
λ
B
, (3)
where h·i
a
and h·i
g
are the arithmetic and geometric
means respectively, and λ
B
is the length that measures
the relative importance of diffusion and gravity. To our
knowledge, the latter was measured only once experi-
mentally, for plastic grains entrained by a viscous fluid
flowing in a flume (λ
B
= 0.12 ± 0.02 d
s
, where d
s
is the
grain size) [31].
The same mechanism likely occurs in laboratory rivers
which, unlike confined flumes, can adjust their width. If
so, it should account for the entire shape, too. To test
this hypothesis, we generate single-thread rivers of which
we vary the sediment discharge (section II). We then mea-
sure the bed elevation and the cross-stream profile of the
sediment flux by tracking individual grains (sections III
and IV). Finally, we observe that this distribution se-
lects the width of a river, which increases with sediment
discharge (section VI).
0 5 10 15
Time [h]
0.0
0.5
1.0
1.5
Sediment discharge [g/min]
FIG. 2. Evolution of the sediment discharge in a laboratory
river. Blue dots: sediment discharge measured with particle
tracking (10 min average). Dashed line: exponential relax-
ation with 45-min time constant (fitted to data). Sediment
supply is 0.6 ±0.1 g/min.
II. EXPERIMENTAL SETUP
To generate our laboratory rivers, we use an inclined
plane (90 × 190 cm), covered with a 5 cm-thick layer of
plastic grains [31, Supplemental material] (Fig. 1). All
grains have the same density and size (density ρ
s
=
1490 g/L; median diameter d
s
= 0.82 ± 0.19 mm), but
they come in a variety of colors. We will use the latters
to track the travelling grains and measure the sediment
flux (Sec. IV).
At the beginning of an experiment, we level the sedi-
ment bed with a rake, and carve a straight channel into
it, from the inlet to the outlet. The initial slope of the
sediment bed is about 10
−3
, but we cannot accurately
fix this value. We then inject a mixture of glycerol and
water (density ρ
f
= 1160 ± 5 g/L, viscosity ν = 10 cP).
A tank placed above the experimental setup delivers a
constant discharge Q
w
in the range 0.1–3 L/min. We
measure the density of the fluid every hour, and infer
its viscosity from this measurement. During a run, we
regularly add water to the mixture to compensate for its
evaporation. The Reynolds number of the river remains
close to 10; the flow is therefore laminar.
We also feed the river with sediment using an indus-
trial feeder (Gericke GLD 87), the screw of which pushes
grains into the funnel that guides them towards the in-
let. The rotation speed of the screw controls the sediment
discharge in the range 0.2–20 g/min. Grains then settle
down and concentrate near the bed, as they begin their
travel downstream.
During the first hour of a run, the flow spreads over the
entire bed, and forms an almost uniform sheet of fluid.
Over the next few hours, though, the flow carves a chan-
nel, usually along the one we have incised initially. Dur-
ing this transient, the river continuously entrains more
grains than it deposits, and thus erodes its bed. As a re-
3
sult, the sediment discharge in the channel is larger than
the one we impose at the inlet (Fig. 2). Gradually, the
sediment flux returns to steady state, until it eventually
matches the input Q
s
.
At equilibrium, the flow forms a straight, single-thread
channel, a few centimetres wide (Fig. 1b). This equilib-
rium is dynamical: grains are constantly dislodged from
the bed, while new ones get deposited by the flow. On
average, the sediment discharge is constant, and the river
bed does not change much. Moving grains, however, in-
dicate that the bed remains above the threshold of en-
trainment, in contrast with rivers that are not fed with
sediment [5].
At the beginning of each experiment, we set the slope
of the frame. However, the layer of sediment is thick
enough for the river to later adjust its own slope. The
slope is thus chosen by the system, and not a control
parameter. If the frame’s slope is too steep, the river
incises a gorge into the sediment bed, until it reaches
its equilibrium slope. Conversely, if the frame’s slope is
too shallow, the river deposits its load near the inlet,
where the sediment accumulates into an alluvial fan [18].
To hasten the transient, we adjust the frame’s slope to
match equilibrium. This procedure largely relies on the
experimenter’s intuition.
Beyond a sediment supply of about 1.5 g/min, the river
destabilises into a braid of intertwined, active channels.
This instability, which tightly bounds the sediment sup-
ply in our experiments, might explain why active single-
thread rivers are so sparse in the literature [6, 34]. Its
origin remains debated [35, 36], and this question would
require a dedicated investigation.
Overall, the equilibrium shape of our single-thread
rivers depends on two inputs, the fluid and the sediment
discharges. To investigate the influence of the latter on
the river’s shape, we perform a series of experimental
runs with the same fluid discharge (about 1 L/min), but
for different values of the sediment supply (Tab. I).
Run Sediment Fluid Tracking Number of
label supply input duration trajectories
[g min
−1
] [L min
−1
] [min] longer than 4 d
s
1 0 1.00 - -
2 0.78 0.99 72 15 332
3 0.42 1.00 144 10 538
4 0.22 0.87 177 8 177
5 1.04 0.98 108 29 696
TABLE I. Experimental parameters.
III. CROSS SECTION
After the river has reached steady state, we measure its
cross section with an inclined laser sheet projected onto
its bed (Fig. 1). We first locate the laser line, whose
position is shifted by the fluid. Then, we stop the fluid
and sediment inputs, and let the fluid drain out of the
channel. The travelling grains settle down within a few
seconds, and the bed’s surface appears to freeze. After
the fluid has drained out, we detect the location of the
laser line on the bare surface of the bed. The combination
of the two laser lines, with and without fluid, provide us
with the bed elevation and the flow depth D within an
accuracy δD = 0.5 mm, which is slightly less than a grain
diameter (see addendum for details).
Figure 3 shows the cross section of three laboratory
rivers. For a vanishing sediment supply, the river’s cross
section looks rounded (Fig. 3a). This feature accords
with the observations of Seizilles et al. [5], who inter-
preted them in terms of the threshold theory. According
to this theory, a river that transport no sediment main-
tains its bed at the threshold of motion. Its cross section
should then be [5]:
D(y) = D
0
cos
yS
0
L
, (4)
where D
0
is the maximum depth of the river, S
0
its down-
stream slope. L is a characteristic length depending on
the properties of the sediment and the fluid. It reads:
L =
θ
t
µ
t
(ρ
s
− ρ
f
)d
s
ρ
f
, (5)
where µ
t
is the friction coefficient of the grains, and θ
t
is their threshold Shields number. The threshold theory
prescribes not only the river’s shape, but also its down-
stream slope, which depends on the fluid discharge Q
w
:
S
0
=
4gµ
3
t
L
4
9νQ
w
1/3
. (6)
Likewise, the depth of the threshold river depends on its
fluid discharge through
D
0
=
µ
t
L
S
0
. (7)
In the absence of sediment supply, our experimental
river accords with this threshold theory, without fitting
any parameter (Fig. 3a, dashed line with L = 0.06 d
s
and S
0
= 0.0046).
More intriguingly, when the sediment supply increases,
the river widens and shallows (Fig. 3c and 3e). A sedi-
ment supply of 1 g/min, for instance, doubles the aspect
ratio of the river. The flat cross section of these active
laboratory rivers resembles that of a natural river—more
so than the rounded shape of Fig. 3a [7]. As they adjust
to the sediment supply, our laboratory rivers thus adopt
a more realistic shape.
4
5 mm
Qs = 0 g/min
(a)
2 0 2
y [cm]
0
10
20
30
q
s
[grains/cm/s]
(b)
Qs = 0.2 g/min
(c)
2 0 2
y [cm]
(d)
Qs = 1.0 g/min
(e)
W
T
2 0 2
y [cm]
(f)
q
s
FIG. 3. Bed elevation and sediment-flux profile for runs 1, 4 and 5 (sediment supply increases from left to right). (a) River bed
in the absence of sediment supply. Dashed line: cosine shape predicted by the threshold theory for L = 0.06 d
s
and S = 0.0046
(equation (4)). (c) & (e): River beds with sediment supply. Dashed line: theory of section VI. (b), (d) & (f): Experimental
sediment-flux profiles. Dashed line: empirical model of section VI. Brown arrows in (c) and purple arrows in (d) illustrate
gravity and diffusion sediment fluxes, respectively. Data are available as supplemental material [33].
When setting up a new experimental run, we need to
manually adjust the inclination of the frame to match
the expected slope of the river. As the sediment input
increases, we need to steepen the frame more and more.
This observation suggests that the equilibrium slope of
our rivers increases with sediment discharge, in accor-
dance with previous measurements [18, 37]. Unfortu-
nately, the river slope, of the order of 0.005 in our ex-
periments, induces a change of bed elevation of the order
of 5 mm per meter of river, a value far below the detection
range of our experimental setup.
In the next section, instead, we focus on the mechanism
by which a river adjusts its cross section.
IV. SEDIMENT-FLUX PROFILE
Based on previous observations in confined canals [31],
we suspect that the balance between bedload diffusion
and gravity sets the shape of our rivers. To test this hy-
pothesis, we first need to measure the local sediment flux
q
s
. To do so, we track individual grains entrained by the
flow using the method described in Abramian et al. [31].
With the top-view camera, we first record a 1 hour-long
movie of the grains travelling over the bed surface (50 fps,
see movie as Supplemental material [33]). We then track
independently the motion of the blue, red and orange
grains, based on their color (Fig. 4a), and reconstruct
their trajectories (Tab. I, Fig. 1b).
For each run, and for each grain color, we then cal-
culate the cross-stream profile of the sediment flux. To
account for the proportion of each color in the sediment,
we normalize each profile so that its integral matches the
sediment input Q
s
.
The three independent measurements are consistent
(Fig. 4b), with a variability of less than about 15% (Fig.
4b). The uncertainty about their average depends on
0
1000
Number of pixels
redblueorange
(a)
180 90 0 90 180
Hue
1.5 1.0 0.5 0.0 0.5 1.0 1.5
Cross-stream coordinate,
y
[cm]
0
10
20
Sediment flux
q
s
[grains/s/cm]
(b)
Grain color
red
blue
orange
FIG. 4. (a) Histogram of pixel hue in a single frame for run 3.
(b) Sediment-flux profile for different grain colors (red, blue,
orange). Dashed black line: average.
the number of trajectories, but remains below 5% [31,
supplemental material].
Repeating this procedure with different experimental
runs, we find that the shape of the sediment-flux profile
varies with Q
s
(Fig. 3d and 3f). Its maximum, always
near the center of the channel, increases with the to-
tal sediment discharge, meaning that bedload transport
intensifies. For a high sediment discharge, it is almost
uniform in the center of the channel, where the bed is
virtually flat (Fig. 3f). Bedload transport then quickly
vanishes near the banks.
