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Input and output mass flux correlations in an experimental braided
stream. Implications on the dynamics of bed load transport
F. Metivier*, P. Meunier
Laboratoire de Dynamique des Systemes Geologiques, Groupe Geomorphologie et eaux Continentales, Institut de Physique de Globe de Paris,
Universite de Paris VII, 75251 Paris, France
Received 18 May 2001; revised 1 October 2002; accepted 4 October 2002
Abstract
Through the development of a model experiment it is shown that there exists a correlation between input and output sediment
fluxes in a micro scale braided stream that remains valid regardless of the stability of the braided river. This correlation has
some important consequences on the mechanics of bed load transport by braided rivers. It enables the definition of both a
dimensionless stream power and a dimensionless transport efficiency. These dimensionless variables in turn permit the
definition of a braided river stability criterion with regard to bed load transport. The existence of such a correlation also suggests
that the average critical shear stress or slope of motion may depend on the flux of mass input to the system. Using these findings
together with a one-dimensional Exner equation for the conservation of mass, a kinematic wave equation for the average
evolution of the riverbed is eventually derived and its significance analyzed.
q 2002 Elsevier Science B.V. All rights reserved.
Keywords: Braided river; Bed load transport; Micro scale experiment
1. Introduction
1.1. Scientific background
Braided rivers remain among the least well-known
fluvial systems (Bridge, 1993; Bristow and Best,
1993). As an example, the definition itself of what is
unambiguously considered to be a braided river has
long been and still is a subject of debate (Bridge,
1993; Richardson, 1997). For the purpose of our study
we will only consider braided rivers as streams that
are characterized by an alluvial plain over which an
unstable (time and space varying) but persistent
network of interlaced channels flows. Why a single
stream destabilizes to produce such extraordinary
patterns has been one of the essences of research on
braided streams since the work of Leopold and
Wolman (1957). Achievement of such a knowledge
has been encouraged both by scientific, engineering
and economic perspectives (Bristow and Best, 1993).
From the scientific point of view braiding is directly
related to mass transport along the bed and fluid-
sediment interaction (Engelund, 1970; Parker, 1976;
Ashmore, 1988; Ashworth et al., 1994). Thus research
on braiding mechanisms has long been driven in
conjunction with research on bed load transport by
Journal of Hydrology 271 (2003) 22–38
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* Corresponding author. Tel.: þ33-1-44-27-28-19; fax: þ33-1-
44-27-81-48.
E-mail address: [email protected] (F. Metivier).
rivers. Still neither of these two problems has received
a definitive solution as no theoretical formalism
resting on natural and experimental evidence has
encountered success in both generally predicting bed
load transport and destabilization of a river channel.
Bed load equations are numerous. But as shown by
Brownlie (1981a,b), although they are important to
our understanding of the gross pattern of sediment
transport none of the equations derived since the work
of Einstein has been able to describe in a truly
satisfactory way the movement of sediment along a
river bed. Thus the lack of such an equation for
sediment transport prevents the scientific community
from establishing reliable equations for the evolution
of stream morphology.
Quantitative experimental approaches to braiding
and bed load transport have developed rapidly in the
eighties at the University of Colorado (e.g. see
Schumm and Khan, 1972; Schumm et al., 1987;
Germanoski and Schumm, 1993), at the University of
Alberta (e.g. Ashmore, 1982,1988) and at the
University of Lincoln (New Zealand) (e.g. see
Warburton, 1996; Young and Warburton, 1996;
Warburton et al., 1996, and references therein).
These experiments proved the possibility to correctly
model gravel bed braided rivers. Among major
findings was the discovery that braiding could occur
at constant water discharge (Ashmore, 1982,1988)
and that confluence scours were major dynamic
structures of a braided river (Ashmore, 1982,1988).
Researches also developed on the form of sedimen-
tary structures associated to braid plain formation
(Ashmore and Parker, 1983; Ashworth, 1996). More
recently Sapozhnikov and Foufoula-Georgiou (1997)
showed that dynamic and spatial scaling seemed to be
a characteristic feature of braided river mechanics
suggesting that scale invariant processes related to
transport and deposition were taking place in multi-
channel rivers.
1.2. Bed load transport by small scale braided rivers
and stream power
Only few experiments exist that focus on sediment
transport by braided streams although these rivers are
known to transport a significant portion of their load
as bed material. Three important experimental
studies explicitly focused on bed load transport by
experimental braided streams, the studies of Ashmore
(1988), Young and Davies (1991), Warburton and
Davies (1994). Young and Davies followed by
Warburton and Davies looked at the relationship
existing between bed load transport and different
morphological parameters like the braiding intensity.
They also suggest that for a given slope bed load
transport was related to discharge through a power
law relationship. The study of Ashmore suggested that
sediment transport could be related to the stream
power of the braided network, there again through a
power law relationship of the type
Qs / ðV2VcÞa ð1Þ
where V ¼ rf gQS (Fig. 1) is the total stream power
(in kg.m/s3 or Watt/m; rf : density of water, g:
acceleration of gravity, Q: discharge, S: bed slope),
Vc is a threshold stream power for bed material
entrainment and a is an empirical exponent (see Table
1 for a summary of the symbols used hereafter). The
experiments were conducted with a sediment input
flux that was held equal to the output flux through
recirculation. Correlations were found to be signifi-
cant although the power law exponent a changes from
one author to the other and even in the same study for
steady and unsteady flow conditions. The fact that the
value of the exponent changes among authors
constitutes an important drawback in the achievement
of a general relationship for bed load transport using
the stream power. We therefore feel it is important to
go back to the original derivation of the stream power
relationship because it can be shown that the value a
should be equal to 1 in order for the stream-power law
to be coherent with physically based shear stress
approaches to bed load transport.
The definition of the stream power relationship for
bed load transport was first introduced by Bagnold
(1973,1977,1980) and developed by many researchers
after him (e.g. see Raudkivi, 1990; Yalin, 1992).
According to the concepts developed by Bagnold, the
sediment flux per unit width of the river bed (Qs=Wc in
kg/m/s), can be expressed as follows
Qs
Wc
¼bub
gs
ðt0 2 tcÞ; ð2Þ
ub is the bottom velocity in the boundary layer where
initiation and transport of the bed material takes place,
t0 and tc are the bottom shear stress and the critical
F. Metivier, P. Meunier / Journal of Hydrology 271 (2003) 22–38 23
shear stress for movement initiation respectively,
gs ¼ ðrs 2 rf Þg=rf ; b is a non dimensional pre-factor
that depends mainly on grain size, and Wc is the width
of the channel.
The main problem with Eq. (2) is that it relates
mass movement to quantities which are seldom
measured both in micro scale braided streams and in
natural conditions. It is therefore useful to rearrange it
in order to derive a simpler, though approximate, form
related to macroscopic variables that can be easily
measured.
Eq. (2) can be rearranged using relationships that
are first order approximations in natural systems
t0 , rf ghS ð3Þ
Q , Wchu ð4Þ
u , cffiffiffiffiffighS
p¼ cu: ð5Þ
Eq. (3) represents the expression of the shear stress at
the boundary between flow and sediment as the result
of the flow weight, Eq. (4) stands for the mass
conservation of the flow in a rectangular channel of
section Wch; and Eq. (5) is the classic Chezy formula
that relates the average flow velocity (u ), and the
boundary shear flow (u , ub), to the flow height h,
slope S and to the Chezy friction factor c which is an
equivalent of the Darcy-Weissbach friction factor f
(c ¼ 8=f ) (Raudkivi, 1990).
Using these relationships into Eq. (2), leads to
Qs ,bfrf
8ðrs 2 rf Þgrf gQS 2 rf gQ
tc
rf gh
!ð6Þ
The ratio tc=rf gh is dimensionless and has the sense of
a critical slope for initiation of movement. We can
thus define
Sc ¼tc
rf ghð7Þ
and simplify Eq. (6) into
Qs , jðrf gQS 2 rf gQScÞ ð8Þ
Fig. 1. Experimental data showing the linear correlation between bed load transport rate and total stream power data V ¼ rf gQS: Modified from
Ashmore (1988), Young and Davies, (1991) and Warburton and Davies (1994).
F. Metivier, P. Meunier / Journal of Hydrology 271 (2003) 22–3824
where
j ¼bfrf
8ðrs 2 rf Þgð9Þ
Defining the total stream power and the critical stream
power according to V ¼ rf gQS and Vc ¼ rf gQSc
respectively, we end up with
Qs ¼ jðV2VcÞ ð10Þ
This relationship is equivalent to Eq. (1) with a ¼ 1:
At this point it is very important to note that Eqs.
(2) and (10) are equivalent to (Raudkivi, 1990, see for
example)
Qs
Wc
¼ C½ðt0 2 tcÞ3=2� ¼ C½ðup 2 upcÞ
3� ð11Þ
where C is a linear function, up ¼ffiffiffiffiffiffit0=rf
pis the shear
velocity of the flow, t0 is the shear stress on the bed, tc
and upc are critical shear stress and shear velocity for
the initiation of movement respectively. An example
of such a relationship is given by the well-known
Meyer-Peter and Mueller equation for bed load
transport
Qs
Wcrs
ffiffiffiffiffigsd
pd
¼ 8t0 2 0:047
ðrs 2 rf Þgd
!3=2
ð12Þ
Many other experimental bed load transport equations
exist but, as shown by Ref. Yalin (1992), most of these
equations present the 3/2 power law exponent. This
dependence has been derived by many researchers
using either dimensional considerations, balance of
the forces exerted on grains, analysis of the work done
Table 1
Variables and physical parameters used in this study. Prefactors and empirical constants are not included (nm: not measured)
Symbol Definition Dimension Range in experiment
rf Density of water [M][L]23 1000 kg/m3
rs Density of sediments [M][L]23 2500 kg/m3
g Acceleration of gravity [L][T]22 9.81 m/s2
gs Reduced gravity [L][T]22 14.71 m/s2
n Kinematic viscosity of water [L]2[T]21 1026 m2/s
g Surface tension of water [M][T]22 or [F][L]21 0.07 N/m
Ql Fluid discharge [L]3[T]21 0.0155–0.0416 l/s
rfQl Fluid discharge [M][T]21 15.5–41.6 g/s
Qe Input flux of sediments [M][T]21 0.03–0.7 g/s
Qs Output flux of sediments [M][T]21 0.14–0.88 g/s
S Slope – 0.033–0.0925
D Grain size [L] 500 mm
H Flow depth [L] 0.1–1 cm
Wc Channel width [L] 1–10 cm
Wt Total flow width [L] 3.5–27.8 cm
L Braid plain length [L] 1 m
u Flow velocity [L][T]21 0.23–0.56 m/s
up Shear velocity [L][T]21 nm
ub , up Bottom velocity [L][T]21 nm
t0 Bottom shear stress [M][L]21[T]22 nm
Se Critical Slope – nm
vs Fall velocity of the sediment [L][T]21 0.13 m/s
V Stream power [M][L][T]23 or Watt/m 5–37.7 mW/m
Vc Critical Stream power [M][L][T]23 or Watt/m nm
V p Dimensionless stream power – 2–60
QSp Dimensionless transport efficiency – 0.2–30
kzl Width averaged bed elevation [L] nm
qs local 2D transport rate [M][L]21 [T]21 nm
c Chezy friction factor – nm
f Darcy Weissbach friction factor – nm
F. Metivier, P. Meunier / Journal of Hydrology 271 (2003) 22–38 25
or energy considerations (for a discussion see Parker
and Klingeman, 1982; Raudkivi, 1990). Thus stream
power transport relationships of the form depicted by
Eq. (1) are coherent with local transport equations
only if the exponent a is equal to one. Let us
eventually note that, ironically, Bagnold, after having
derived this linear relationship (Eqs. (12), (13) and
(16) in Bagnold (1973)), suggested on an empirical
basis that a non linear transport equation could be
used to get a better agreement with a set of natural
data (Bagnold, 1980). This probably is the reason why
many researchers have been looking for variable
values of the exponent a:
From the data of the experimental studies Ash-
more, 1988, Young and Davies (1991) and Warburton
and Davies (1994) we can see that the general form of
Eq. (10) can be used to fit the experimental data with
good regression coefficients. Only the values of j and
Vc change because of the differences in geometric
scales used for the different experiments (e.g. grain
size, length and width). The fact that in all of these
experiments the bed load transport can be explained
by the linear stream power relationship brings further
support to the general validity of this concept.
1.3. Possible influence of the input flux on transport
dynamics
One feature that seems very important in natural
rivers and that is not taken into account by a simple
stream power law such as the one defined above, is the
possibility for a stream to carry different loads for the
same stream power as bed material transport has long
been described as an intermittent and wave-like
process (e.g. Weir, 1983; Pickup et al., 1983; Ashmore,
1988; Davies, 1987; Hoey and Sutherland, 1991;
Warburton and Davies, 1994; Lane et al., 1996).
Although measurements uncertainties are partly
responsible for the observed variations of nearly an
order of magnitude in bed material load for a given
stream power (e.g. Hubbell, 1987), these variations
may have a physical explanations. Two main par-
ameters may at first sight influence the relationship
between bed load and stream power, the grain diameter
and its distribution and the input mass flux the river
carries when it enters its floodplain and starts to braid.
In the following we concentrate on the influence
the input mass flux may bear on river transport. To our
knowledge, this possible influence of a continuous
input flux of mass to a floodplain has never been
studied experimentally. The experimental studies
cited above either used the same input flux for all
the experiments (Warburton and Davies, 1994) or
either adjusted the input flux in order that there be no
net aggradation or degradation at the flume entrance
thereby constantly changing the boundary conditions
for the mass flux (Young and Davies, 1991) or either
recirculated the sediment at the outlet to the inlet
(Ashmore, 1988). These studies were seeking to
achieve long-term equilibrium to understand the
general features of bed load transport by braided
streams. As sediment supply to natural floodplains
changes continuously it is not clear what conse-
quences these variations may bear on the dynamics of
bed load transport.
To avoid interference with grain sizes and
distribution we shall use an approximately uniform
distribution of sediment. Although restrictive this
simplification of the problem is also important
because it enables the testing of bed load transport
formulas developed for a characteristic particle size
(the usual D50 or D90 diameters). The experiment
reported hereafter is not a Froude scaled model of a
specific prototype but a generic model of braided
streams (Ashmore, 1988), although it shall be shown
through a dimensional analysis of the bed load
transport problem, that our results are realistic and
should, to some extent, be representative of natural
processes.
2. Experimental apparatus and observations
2.1. Setup
Fig. 2 shows the experimental apparatus used to
reproduce micro scale braided streams. The rivers
were reproduced using a mobile bed in altuglass
(plastic) of 1m length and 0.5 m width. The sediment
used are 500 mm (distribution between 400 and
600 mm), glass spheres of density rs ¼ 2:5: As
explained above, we used sediment with an approxi-
mate single grain size for two reasons. First we
wanted to look at the relationship between fluxes of
water and mass without having to take the sediment
distribution into account. Using a single grain size can
F. Metivier, P. Meunier / Journal of Hydrology 271 (2003) 22–3826
best do this. Second it has long been usual to define
the influence of the sediment in the dynamics of
transport through the variable D50 that is the median
grain diameter. If this approximation is to be
reasonable a uniform grain size should lead to results
comparable in many extents to what is known of
natural streams. The mobile bed rested on an
adjustable elevating tablet that enabled control of
the bed slope. Slopes were measured using a digital
level device. The inflow took place through a 2.5 cm
wide canal. Water pressure was controlled by a
pressure gauge and discharge controlled using a flow
meter. During each experiment the discharge was
maintained constant within less than 5%. Average
flow velocity was measured using dye injections. An
endless screw driven by a variable speed electric
power drive controlled inflow of grains. This system,
designed for granular material and powders, permits
constant input of grains during several hours at the
inlet of the river system. The bed material load was
measured at the outlet through the use of an
overflowing sedimentation tank. The tank was main-
tained at constant water level and grains were allowed
to settle. It was positioned on a high precision
weighting device (2 gr precision) connected to a PC
that collected the weight of the tank every 10 s. This
gave us access to a direct and precise measure of the
cumulated mass curve at the outlet of our braided
system. As outflow was allowed to take place all along
the box width (0.50 m), the river was braided at the
outlet. Therefore collection of the mass represents the
integral over the width of the floodplain of the
true simultaneous transport of particles by several
channels. A digital video camera was used to film the
experiments. Typical experiment duration was
between 2 and 5 h. The initial topography of the
floodplain was either flat or an initial canal was carved
in order to see the destabilization of the banks.
2.2. Evolution of the experiment during a run
Fifty three experiments where conducted at the St
Maur laboratory under varying Discharge
(Ql , 0:0155 2 0:0416 l=s½56 2 150 l=h�), slope
(S , 0:033 2 0:0925), and sediment feed rates
(Qe , 0:03 2 0:7gr=s). After flow of water Ql and
sediment Qe started it took a few tens of second until a
channel was carved by the overland flow of water in
the case where no initial channels was present. In both
case the channel quickly widened and the output flux
of mass began to stabilize. Instabilities rapidly began
to develop inside the main channel and mid-channel
bars began to form. Diversion occurred once the bar
height was of the same order as flow depth or either
through avulsion in a meander bends. Confluence
scours formed and a stable braided system developed
(Fig. 3). Depending on the flow conditions up to ten
channels could be seen to form the braided system.
The braided river remained active during all the
experiment duration.
Fig. 4 the evolution of both the cumulated mass
and the mass flux at the outlet of the experimental
braided stream. Experiment initiation and establish-
ment of a stable braided pattern in the box
corresponds to the time in which strong fall of the
sediment flux is observed.
Fig. 2. Experimental setup, not to scale.
F. Metivier, P. Meunier / Journal of Hydrology 271 (2003) 22–38 27
The flux curve shows that on average sediment flux
carried by the stream to the outlet tends towards a
‘steady state’. At that time initial conditions are totally
forgotten. Oscillations around the average equilibrium
flux Qs (see the closeup in Fig. 4b) are significant of a
wave like transport of sediments inside the exper-
imental braid plain. This ‘steady state’ is not a true
equilibrium in most experiments because the braided
stream either aggrades or degrades (as will be shown
later, the ratio of the output to the input flux is not
equal to one). Thus changes in the mass transport are
expected on the long term. But, until a significant
change of the average slope propagates from one end
to the other, the river can be approximated with a very
good precision (regression coefficients on order of
0.99-1), to achieve a steady state. Thus although a
river is not in equilibrium conditions regarding net
erosion or aggradation, the average flux of mass
carried throughout the braid plain may remain
constant for a significant period of time.
2.3. Dimensional analysis of the bed load transport
problem and similarity of the experiments
To understand the mechanics of sediment transport
by an alluvial river it is of importance to define
Fig. 3. Views of a braided experiment run taken at different time steps, Ql ¼ 13:3g=s; Qe ¼ 0:18g=s; S ¼ 0:0875:
F. Metivier, P. Meunier / Journal of Hydrology 271 (2003) 22–3828
the basic parameters involved in the process and
perform a dimensional analysis in order to be able to
see (i) what characteristic scales can be defined in the
problem, (ii) the functional dependence that is to be
expected from the different forces driving the
dynamics of the system, and (iii) if the small scale
experiment can reasonably account for natural
processes.
Fig. 4. Typical cumulated sediment volume (a) and sediment flux (b) measured at the outlet of the experimental braided stream. Sediment input
is noted in dashed line for comparison.
F. Metivier, P. Meunier / Journal of Hydrology 271 (2003) 22–38 29
Looking at the problem of bed material load, we
seek characterization of geometric and cinematic
scales together with physical parameters of the fluid
and materials. Physical properties include densities of
both the fluid and sediment rf and rs and dynamic
viscosity of the fluid m: The gravity field is of major
importance and scales according the acceleration of
gravity g. Near the bed there are two cinematic scales
one related to the fluid and one to the sediment. They
are respectively the shear velocity u of the fluid which
clearly stands out as the scaling parameter and the fall
velocity of the sediment vs: Two geometric scales at
least are involved, the grain diameter D and the bed
width Wc: The thickness of the boundary layer is often
assumed to scale with the grain size. The flow depth
is, to the first order, related to the shear velocity, the
bed slope and the acceleration of gravity according to
up ,ffiffiffiffiffighS
p: It is therefore not an independent
parameter and does not have to be integrated in the
dimensional analysis. The same can be said for the
shear stress at the bottom t0 as by definition u ¼ffiffiffiffiffiffit0=rf
p: Eventually the slope of the bed is the last
fundamental parameter of the problem.
Assuming these parameters are necessary and
sufficient the problem of sediment transport can to
the first order be written as
Qs ¼ Cðrs; rf ;m; g; u; vs;D;Wc; SÞ ð13Þ
Eq. (13) is composed of 9 independent variables (in
parentheses). 8 of these variables are dimensional and
3 dimensions are involved (length L, mass M and time
T ). The dependant variable is the sediment flux QS:
The problem can be reduced to a problem involving 6
independent dimensionless parameters and one
dependant dimensionless variable (Barenblatt,
1996). This leads to
Qs
rs
ffiffiffiffigD
pD2
¼ C Sh;Rep;Ro; St;rs
rf
; S
!ð14Þ
where
Sh ¼rsu
2p
ðrs 2 rf ÞgDð15Þ
is the Shields number,
Rep ¼rf upD
mð16Þ
is the grain Reynolds number,
Ro ¼vs
kup
ð17Þ
is the Rouse number defining the characteristic
transport mode of sediment by the stream (k ¼ 0:4
is the universal Von Karmn constant), and eventually
St ¼ frsD
2
18m
2up
Wc
ð18Þ
is the Stokes number of the flow (Eaton and Fessler,
1994; Crowe et al., 1997). The stokes number
quantifies the response of particles to the drag exerted
by the flow. For low stokes number the particles
essentially follow streamlines whereas for larger
stokes number the particle’s inertia makes their path
differ significantly from the streamlines a property
that is essential in understanding preferential concen-
tration of particles in areas of the flow. The coefficient
f is dependant and the particle Reynolds number
(Eaton and Fessler, 1994). Here we take an expression
of the form f ¼ 1=ð1 þ 0:15Re0:687p Þ (Eaton and
Fessler, 1994; Crowe et al., 1997).
Recall that the choice of the non-dimensional
parameters we use is arbitrary provided they form an
independent family, that is none of these parameters
can be written as a product of the others. In our case
our choice is guided by the fact that we are interested
in sediment transport, sediment-flow interactions and
its link with macroscopic morphology of the
streambed. It is therefore of crucial importance to
concentrate on dimensionless ratios that are repre-
sentative of such interactions. Hence our decision to
focus on dimensionless parameters where both
variables related to grain movement and variables
related to the flow appear.
Looking at Table 2 we can see that our experiment
falls nicely in the range of natural gravel bed rivers for
the Shields, the Rouse and the Stokes number
although the range covered by the last two numbers
is narrower than what can be found in natural streams.
Departure from natural streams is observed for the
grain Reynolds number. As has been shown before
(Ashmore, 1988; Ashworth et al., 1994) values of Repdown to 15 are reasonable approximations for mass
transport in the boundary layer although slight viscous
effects may occur. This should not affect the form of
the solutions and analyses derived because, as has
F. Metivier, P. Meunier / Journal of Hydrology 271 (2003) 22–3830
been shown by Francis (1973), the form of bed
material load transport seems not to be affected by
change in flow dynamics from turbulent to inter-
mittent or even laminar flow. This property explains
in part why very realistic behavior may be observed at
very small scales even in Non-fully turbulent systems.
When comparing with the torrent de St Pierre (French
Alps), which was surveyed twice a day during five
days at the beginning of snowmelt in June 1999,
departure from experimental and natural values occur
also for the Rouse number. This should not be very
problematic as a Rouse number greater then 1
indicates that most of the sediment making the bed
is transported as bed load, whereas the Rouse number
becomes crucial only when suspension is predominant
(see Raudkivi, 1990).
2.4. Micro-scale models and surface tension
In the previous analysis, surface tension was not
taken into account because in natural streams the
amplitudes of surface tension forces are orders of
magnitudes less than either buoyancy or inertia.
Surface tension forces take place at the interface
between two fluids that is at the free surface of the
flow. As Bed material is transported on the bottom of
the channel where only one fluid is present surface
tension can not have a direct influence on bed load.
Despite this surface tension may influence flow
dynamics and therefore exert an indirect control on
sediment transport. Two dimensionless numbers
quantify the relative influence of surface tension to
either gravity or inertia: the Bond and Weber numbers.
The Bond number quantifies the deformation of a fluid
drop resting on a solid plane due to Buoyancy (Guyon
et al., 2001). It is defined as
Bo ¼Drgr2
d
gð19Þ
where Dr is the density contrast between the two
fluids in contact (here water and air), rd is the radius of
the contact line between the drop and the solid plane,
and g is the surface tension of water. In the case of a
gravity driven flow in a channel of width Wc; an
analogue to the Bond number can be defined where
rg , Wc=2
Bo ,DrgW2
c
4gð20Þ
This number quantifies the respective influence of
buoyancy and surface tension on the shape of the
section. The Weber number quantifies the ratio of
inertia to surface tension in a moving fluid (Peakall
and Warburton, 1996, e.g.). It is defined as
We ¼rf u
2h
gð21Þ
where u and h are respectively the average fluid
velocity and the average depth of flow. In natural
streams both the Bond and Weber number are very
high indicating that surface tension forces are
negligible compared to both gravity and Inertia. In
micro scale models of alluvial rivers both flow depth,
width and velocity are reduced. Surface tension forces
may therefore influence the flow dynamics. Studies of
flow dynamics in analogue models of braided rivers
suggest that a critical Weber number exist below
Table 2
Comparison between dimensionless products in natural gravel bed streams and experiment. Sources for natural rivers come from (Bagnold,
1973; Ashmore, 1988; Ashworth et al., 1994; Brownlie, 1981a) and measurements made on a proglacial braided stream in the French Alps the
torrent de St Pierre
Dimensionless product Order of magnitude in typical braided streams Value in the torrent de St Pierre Value in experiment
Qs=rs
ffiffiffiffiffiffigD50
pD2
50 0.01–100 – 1–5
Sh 0.01–0.8 0.01–0.1 0.013–0.66
Rep 103–104 4315–14436 15–35
Ro 5–80 23–77 6–15
St 0.01–1 0.06–0.17 0.05–0.1
S 10–2–1024 0.025 0.033–0.0925
Bo 105–107 8 £ 105 3.5–350
We 103–105 3450 1.3–35.7
F. Metivier, P. Meunier / Journal of Hydrology 271 (2003) 22–38 31
which surface tension may have a non negligible
effect on the flow pattern (Peakall and Warburton,
1996). It has been hypothesized that in models where
We is less than about 10–100, flow dynamics is
influenced by surface tension although this influence
remains to be quantified. No such definition exists for
Bo. However, until quantification of the influence of
surface tension in hydraulic models is available, it is
important, as pointed out by Peakall and Warburton
(1996), to assess this possible influence by reporting
the order of magnitude of both Bo and We numbers.
Table 2 shows the range of values of both Bo and We
in our experiment and in natural rivers. These values
are several orders of magnitude smaller in exper-
iments than in natural streams. Despite this difference
buoyancy forces are on average always an order of
magnitude higher than surface tension forces. This
explains why wide and very shallow channels may be
reproduced in small-scale experiments. Table 2 also
shows that the Weber number lies between 1.3 and
35.7, a common range in small scale braided rivers
experiments (see Table 2 in Peakall and Warburton,
1996). The lowest value corresponds to the smallest
channels where both the width and the velocity are
small. In these channels surface tension forces may be
of the same order of magnitude than inertial forces
and buoyancy forces. They can therefore probably not
be neglected in flow computations. But as no
significant transport seems to occur in these very
small channel the small value of We should not be
critical. In the larger channels, where most of the
transport takes place, the Weber number indicates that
inertia is an order of magnitude higher than surface
tension forces. In conclusion our analysis suggests
that surface tension can to the first order be neglected
in the experiments described here although it may
locally influence the dynamics of the micro scale
braided streams.
3. Influence of the input flux of sediment on bed
material transport
3.1. Direct comparison of bed load transport and
stream power
For each experiment (see description above) we
obtained a cumulative mass flux at the outlet and from
this cumulated mass we derived the flux QsðtÞ by time
derivation. Fig. 4 shows the two curves. As can be
seen Qs tends toward a constant value which indicates
that the flux of mass becomes stationary. We then
fitted the flux curve to get the value for this
‘equilibrium’ regime.
Knowing the value of the ‘equilibrium’ mass flux
leaving our braided stream we then tried to evaluate
the adequacy of Eq. (6) directly in terms of Qs; rf Ql
and S. The results are shown on Fig. 5 (rf Ql is the
water flux expressed in mass per unit time)
As can be seen from Fig. 5, the correlation is as
usual relatively sound, that is the predicted correlation
plots inside the cloud of points on the graph
(R2 ¼ 0:82), although more than half of the points
do not intersect the correlation line within the
experimental uncertainties. One evident problem in
this graphical representation is that it does not take
into account the possible influence of the input flux of
mass on the behavior of the river and especially on the
capacity to erode its bed given an existing bed load.
3.2. Dimensional collapse
From the analysis derived in Section 1.2 sediment
transport is supposed to scale according to the stream
power of the stream (Eq. (10)). Looking at the
independent macroscopic variables of our experiment
we seek a relationship of the form
Qs ¼ Fðrf Ql; S;QeÞ ð22Þ
Applaying the Buckingham theorem (Barenblatt,
1996) and putting Eq. (22) in dimensionless form
leads to
Qs
Qe
¼ Frf Ql
Qe
; S
� �ð23Þ
If the stream power is a significant variable in our
problem as suggested by Eq. (10) our experimental
results should, to the first order, follow a dimension-
less relationship of the form
Qs
Qe
¼ Frf QlS
Qe
� �ð24Þ
Eq. (24) defines the relationship between a dimen-
sionless stream power Vp ¼ rf QlS=Qe and a dimen-
sionless sediment flux Qps ¼ Qs=Qe that represents the
transport efficiency of the braided stream.
F. Metivier, P. Meunier / Journal of Hydrology 271 (2003) 22–3832
Eq. (6) can be put in dimensionless form by the
input flux of mass according to
Qs
Qe
, jgrf QlS
Qe
2rf QlSc
Qe
� �ð25Þ
which is a linear form of Eq. (24). If sediment
transport dynamics essentially follows the concept of
stream power entrainment F must then be a linear
function.
Fig. 6 shows the result. The data fit is linear with a
regression coefficient of 0.98 which indicates a very
good adjustment of our data to the behavior predicted
by Eq. (6). Furthermore most of the points intersect
the correlation line within the experimental
uncertainties.
In the case of our experiment the coefficients are
jg ¼bfr2
8aðrs 2 rf Þ, 0:47 ð26Þ
and
jgQlSc ¼ 0:58Qe ð27Þ
or
Sc ¼ 1:2Qe
gQl
ð28Þ
Eventually Eq. (24) may be rewritten as
Qps ¼ 0:47Vp
2 0:58 ð29Þ
Eqs. (29) and (28) imply that sediment transport by
our micro-scale braided stream not only depends on
stream power as predicted by Eq. (10) but also on the
input flux of mass to the floodplain where braiding
develops. This implies that the average critical slope
for motion of grains in the system depends on the ratio
of the input sediment flux to the fluid flux. Said in
other terms it shows that the concentration of bed load
exerts an influence on the capacity of the river to
entrain further grains from the bed. At this point it has
to be noted that the dimensionless coefficients derived
above may depend on the distance between the two
ends of the river system. That is sediment dispersion
may occur (Pickup et al., 1983; Madej and Ozaki,
1996; Hoey, 1996) that could change the value of
these coefficients (especially the second one).
Fig. 5. Comparison between output mass flux and total stream power (both in g/s) for the micro scale braided river experiments.
F. Metivier, P. Meunier / Journal of Hydrology 271 (2003) 22–38 33
3.3. Wavelike transport of sediments
Eq. (29) represents the correlation existing
between the average flux at the outlet and the both
the stream power and the sediment discharge at the
inlet of the micro-scale floodplain. It also represents
an integral of the fluxes along the width of the
floodplain (see the experimental setup). Dividing both
sides of Eq. (29) by the average width Wt of the active
channels, that is the time and space averaged total
width of flow, and multiplying by Qe we get an
equation for the average flux per unit width of the
experimental braiding plain
Qs
Wt
¼ c1
rf QlS
Wt
2 c2
Qe
Wt
ð30Þ
where c1 ¼ 0:47 and c2 ¼ 0:58 are the coefficients of
Eq. (29). Eq. (30) is equivalent to the description of a
one-dimensional evolution of sediment transport by
our experimental braided stream. it can be written as
1
Wt
ðQs 2 QeÞ ¼1
Wt
ðrf QlS 2 ½1 þ c2�QeÞ ð31Þ
The slope S represents the average slope of the
braided river. Locally the slope can vary according to
the succession of scours and bars. S can therefore be
written
S ¼ 21
L
ðL
0
›kzl›x
dx ð32Þ
where k; zl is the river-width averaged elevation at a
given distance from the inlet (Fig. 7a).
The flux difference on the left-hand side of Eq. (31)
can also be written as an integral namely
1
Wt
ðQs 2 QeÞ ¼ðL
0
›qs
›xdx ð33Þ
There again ›qs=›x represents the width averaged
spatial variation of the sediment flux at a distance x
from the inlet. The notation q is intended to remind
that this is a 2-dimensional flux of sediment. Using a
one dimensional Exner equation for the conservation
of mass (Fig. 7b) one can write
›qs
›x¼ 2rsð1 2 pÞ
›kzl›t
ð34Þ
Fig. 6. Dimensional collapse of the experiment showing the relationship between the dimensionless transport efficiency and the dimensionless
stream power.
F. Metivier, P. Meunier / Journal of Hydrology 271 (2003) 22–3834
where p is the porosity of the sediment. Using Eqs.
(34) and (32) together with Eq. (31) leads toðL
0
›kzl›t
2c1rf Ql
rsð1 2 pÞWtL
›kzl›x
2ð1 þ c2ÞQe
rsð1 2 pÞWtL
�
dx ¼ 0
ð35Þ
which is an integral form of a wave equation. Eq. (35)
is in many extent similar to one-dimensional wave
equation derived by Ref. (Weir, 1983) from the
shallow water equations and applied with some
success to the East Fork river (a single channel gravel
bed stream, Wyoming). Our experimental results
therefore add to the observations made by others
(e.g. Weir, 1983; Pickup et al., 1983; Madej and
Ozaki, 1996; Meade, 1983; Hubbell, 1987; Gomez
et al., 1989; Hoey, 1996; Lane et al., 1996) on the
wavelike nature of bed load transport by gravel bed
streams. First Eq. (35) shows that a one dimensional
wave equation, which is similar to equations devel-
oped for single channel rivers, may apply to braided
streams. Second it also shows that two characteristic
wave velocities can be defined: one linked to the fluid
flow
vfl ¼c1rf Ql
rsð1 2 pÞWtL0; ð36Þ
and one linked to the input sediment discharge
vsed ¼ð1 þ c2ÞQe
rsð1 2 pÞWtL: ð37Þ
The dependence of vsed on the input discharge is clearly
new as it cannot be derived from the Saint Venant
equations which have been of common use in studies of
sediment waves. It implies that if the sediment input is
changed at the entrance of our stream the information
will be carried by the braided river to the other end of
the system through and aggradation or degradation
wave of characteristic velocity vsed; whereas change in
the discharge would induce a traveling wave of
velocity vfl: Eventually when equilibrium transport is
achieved according to Eq. (29) the evolution of the
average elevation of the experimental braided stream
defined by Eq. (35) becomes
ðL
0
›kzl›t
¼ 0 ð38Þ
Eq. (38) expresses the fact that it is possible for a
stream to both braid and maintain on average a
dynamic equilibrium regarding transport provided
that the bed oscillations averaged over the width of
the flow are periodical in L.
Eventually Eq. (35) implies that further research
on bed load transport has to focus on determining
Fig. 7. Schematic representation of a width averaged (a) profile of bed elevation and (b) conservation of mass.
F. Metivier, P. Meunier / Journal of Hydrology 271 (2003) 22–38 35
the exact nature of wavelike motion of grains along
the bottom. The derivation of a wavelike equation for
the evolution of the streambed averaged over the
width of the floodplain has been made assuming that
the dimensionless numerical coefficients obtained
experimentally are constant. Because of possible
dispersion of the sediment waves (Pickup et al.,
1983; Hoey, 1996; Madej and Ozaki, 1996) the values
of these coefficients c1 and c2 may depend on the
position at which the flux is measured. These possible
variations would lead to a more complicated equation
but would not change its general form. At this point
more experiments are needed in order (i) to constrain
the scaling of the numerical coefficients we obtained
using our experimental setup, (ii) to ascertain the
functional form of Eq. (35), and (iii) to derive and test
solutions of this wave equation. This implies changing
the length of the box and the grain size of the
sediments used.
4. Conclusion
Despite all the restrictions mentioned above, our
set of experiments shows that braiding is essentially
independent of grain size distribution as it is possible
to reproduce braided flow patterns using a uniform
sediment. In other words, the fact that a braided
stream can develop on micro beads of uniform
diameter suggests that the mechanisms by which a
river braids are probably not controlled by the grain
size distribution or dispersion but by a single
characteristic size that remains to be defined. This
adds to the findings of Ashmore (1982,1988)) who
showed that braiding was also essentially independent
of discharge fluctuations. Second we show the
existence of a correlation between both sediment
fluxes at the boundaries and flow parameters as
defined by Eq. (29). This correlation enables the
definition of a dimensionless stream power
Vp ¼rf QlS
Qe
ð39Þ
that defines the effective stream power of the
experimental braided stream. This dimensionless
stream power is a new and direct quantification of
the balance between eroding power and transport
capacity developed by geomorphologists in the past
(see (Bull, 1991; Burbank and Anderson, 2001) for a
review). It is very useful because it defines the
aggradation/degradation state of the river between
two measurement points. In the case of our exper-
iments if Vp . 3:36 the river is on average eroding its
bed, if Vp , 3:36 the river is aggrading. Eventually
our experimental braided stream is in true equilibrium
regarding transfer of mass when V ¼ 3:36: In that
case the river on averages neither aggrades nor
degrades its bed.
We have further shown how this correlation is
coherent with a bed load transport by braided streams
that essentially follows a wave like mechanism of the
form depicted by Eq. (35). Two wave velocities are
defined that depend on the fluid or the sediment
discharge. This should be taken into account in our
attempt to understand bed load transport by natural
streams and especially in the case of braided rivers,
which are the typical streams reproduced in this
experiment. This correlation between fluxes at a
distance may depend on the distance at which it is
measured. Further research has to focus on the
distance that separates the points where the mass
flux is measured and its possible influence on the
correlation we have established.
Eventually our results imply that prediction of
mass fluxes with flow variables only cannot lead to
precise results unless we may properly define the
fundamental characteristics of what happens to look
like a typical wave transfer mechanism : length,
velocity and amplitude of sediment waves as func-
tions of the parameters of the flow and sediment input
to a stream.
Acknowledgements
This study was supported by french research
programs PNSE and PNRH, by the Institut Francais
des Petroles (IFP) and by IPGP. The Parc National des
Ecrins is acknowledged for authorising us to work on
the torrent de St Pierre. Joel Faure was of invaluable
assistance to us on the field. A. Howard and L. Pham
helped in collecting hydraulic and topographic data.
We benefited from the technical assistance of
G. Bienfait, C. Carbonne, K. Mahiouz, Y. Gamblin
and A. Viera. T.B. Hoey, J. Warburton and an
F. Metivier, P. Meunier / Journal of Hydrology 271 (2003) 22–3836
anonymous reviewer made constructive and useful
comments. This is IPGP contribution N 1850.
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