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: metivier@ipgp.jussieu.fr (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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