Emmanuel Audusse

Sébastien Boyaval

Nicole Goutal

Magali Jodeau

Philippe Ung
email: ung@math.cnrs.fr

NUMERICAL SIMULATION OF THE DYNAMICS OF SEDIMENTARY RIVER BEDS WITH A STOCHASTIC EXNER EQUATION

At the scale of a river reach, the dynamics of the river bed is typically modelled by Exner equation (conservation of the solid mass) with an empirical solid flux of transported sediments, which is a simple deterministic algebraic formula function of i) the sediment physical characteristics (size and mass) and of ii) the averaged hydrodynamical description of the ambient water flow. This model has proved useful, in particular through numerical simulations, for hydraulic engineering purposes (like estimating the mass of sediments that is drained through an open dam). Though, the model is also coarse. And its applicability at various space and time scales remains a question of considerable interest for sedimentologists. In particular, physical experiments from the grain scale to the laboratory scale reveal important fluctuations of the solid flux in given hydrodynamical conditions.

This work is a preliminary study of the coupling of a stochastic Exner equation with a hydrodynamical model for large scales. (Stochastic models with a probabilistic solid flux are currently being investigated, but most often only from the viewpoint of theoretical physics at the grain scale.) We introduce a new stochastic Exner model and discuss it using numerical simulations in an appropriate test case.

Introduction

Sediments are particles that can be transported by rivers over large distances. Their transport, deposition or erosion, controls the form of rivers. Depending on the water discharge, the geology, the slope of the area and the supply of sediment, rivers exhibit braid, meander or straight patterns. The study of sediment dynamics is particularly useful to prevent the filling of hydroelectric reservoirs, but also to preserve water intakes in rivers. Furthermore, sediments are a major component defining the habitat of fishes.

Two different kinds of transport processes are usually distinguished for non-cohesive sediments [START_REF] Graf | Hydraulique fluviale Tome 2 Ecoulement non permanent et phénomènes de transport[END_REF] : (i) fine particles (typically smaller than 2 mm) are mainly transported by suspension, contacts with the bed are then rare and particles can be found on the whole vertical of water depth ; (ii) larger particles are transported by bedload near the bed, by rolling or small jumps. This present work is focused on bedload transport.

A usual model to describe bedload is to write (i) a sediment mass conservation equation, the well-known Exner equation [START_REF] Exner | Über die wechselwirkung zwischen wasser und geschiebe in flüssen[END_REF], and (ii) an empirical formula for the flux of sediments, experimentally validated in bedload conditions (moving sediments remain close to the bed).

Most numerical codes modelling the transport of sediments for engineering studies (that is at scales involving a river reach and many days) make use of these equations (for example Telemac-Sisyphe [START_REF] Hervouet | Hydrodynamics of free surface flows[END_REF], or many other codes [START_REF] Papanicolaou | Sediment Transport Modeling Review-Current and Future Developments[END_REF]...). They have correctly predicted useful information such as an approximation of the total mass of sediments transported during a flood. Among other examples, [START_REF] Kadi Abderrezzak | One-dimensional numerical modeling of sediment transport and bed deformation in open channels[END_REF] shows that with a one dimensional sediment transport model, they achieve reasonably good results in reproducing the bed changes due to a large flood event on the Ha!Ha! River (Canada). In [START_REF] Villaret | Morphodynamic modeling using the Telemac finite-element system[END_REF], the authors compare calculations of solid discharge of the Danube River with measurements from a bed-load sampler and show good agreement with the observations. Nevertheless this model remains definitely coarse.

Empirical formulae for the flux of sediments are simply written as algebraic functions of the locally-averaged hydrodynamical conditions above the river bed, since what happens under the river is obviously inaccessible in most practical situations. There are many different established formulae for different flow conditions and sediment sizes, which can still be only determined experimentally at a high cost. Furthermore, the model has proved unable so far to account for wave-like transient phenomena which have been actually observed in rivers at any scale: ripples, dunes. . .

In any case, because of both its advantages and limitations, Exner model is still attracting the attention of many modellers interested by the dynamics of a river bed. In fact, the very notion of solid mass flux (or discharge) in bedload conditions is still discussed, for it strongly depends on the time and space scales at which transport is modelled. Its stochastic features in particular are discussed. On the one hand, a probabilistic framework at the grain scale has indeed been used for a long time, but also more recently, to propose semiempirical deterministic bedload formulae at large scales [START_REF] Einstein | The bed-load function for sediment transportation in open channel flows[END_REF][START_REF] Zhong | A bed-load function based on kinetic theory[END_REF] after direct averaging of the stochastic solid discharge. On the other hand, the solid discharge also exhibits stochastic features at larger scales in laboratory experimental measures [START_REF] Ganti | Subordinated brownian motion model for sediment transport[END_REF][START_REF] Recking | Bed-Load Transport Flume Experiments on Steep Slopes[END_REF], see for instance Figure 1, although the probability distributions reconstructed from the measurements may depend on the sampling [START_REF] Ballio | On the definition of solid discharge in hydro-environment research and applications[END_REF][START_REF] Ganti | Subordinated brownian motion model for sediment transport[END_REF]. Moreover, the idea has also recently emerged that some persistent forms observed at the surface of river beds may result from the nonlinear interactions of small-scale features that should be modelled through the stochastic fluctuations, see e.g. [START_REF] Ancey | Stochastic modeling in sediment dynamics: Exner equation for planar bed incipient bed load transport conditions[END_REF][START_REF] Foufoula-Georgiou | Introduction to special section on stochastic transport and emergent scaling on earth's surface: Rethinking geomorphic transport -stochastic theories, broad scales of motion and nonlocality[END_REF][START_REF] Jerolmack | A unified model for subaqueous bed form dynamics[END_REF][START_REF] Niño | Discrete particle model for analyzing bedform development[END_REF]. Finally, it appears natural to ask whether the solid discharge should still be considered stochastic in the models used for engineering studies, and then on which ground.

In this work, we suggest one possible stochastic Exner model where the solid mass flux is directly defined as a random variable with a probability distribution in keeping with the experimental measurements available in the literature [START_REF] Ganti | Subordinated brownian motion model for sediment transport[END_REF][START_REF] Recking | Bed-Load Transport Flume Experiments on Steep Slopes[END_REF]. Then, we numerically investigate this new model in a standard benchmark test case for sediment transport, and we discuss some mathematical problems that naturally arise during the construction of such probabilistic models. Our goal is to evaluate the potential of new probabilistic models as improvements of the existing codes using Exner model at the scale of a river reach. We do not discuss the physics at the grain scale that could naturally define a stochastic solid discharge, but we demonstrate the need for an adequate mathematical framework before new probabilistic models can be coupled to hydrodynamic models at large Recking. Experiment was performed as described in [START_REF] Recking | Bed-Load Transport Flume Experiments on Steep Slopes[END_REF] .

scales. Much more experimental data (i.e. two-point correlations in space at least) seem necessary in order to correctly achieve that latter goal.

Exner model

The standard Exner model

Let us denote by x ∈ R the abscissa along a one-dimensional (1D) rectilinear infinite river. Exner equation is a local mass balance of sediments under a river bed that is modelled as a graph x → B(t, x) for all times t ≥ 0

φ s ∂ t B = -∂ x Q s (1)
on assuming that the volume fraction φ s of the solid phase remains constant under the river bed. The model is closed after precising the 1D flux of sediments Q s (in [m] 2 .[s] -1 ) flowing through a section close to the river bed (bedload transport). Closure is a very complicated problem in general. River beds are polydisperse (fluidsaturated) granular suspensions with a very complicated rheology. However, one can create flow conditions in laboratory flume experiments such that Q s can be measured.

In experimental settings where sediments are heavy and only move by saltation, rolling and sliding motions close to the river bed [START_REF] Fernandez Luque | Erosion and transport of bed-load sediment[END_REF][START_REF] Parker | Surface-based bedload transport relation for gravel rivers[END_REF], one retrieves a non-dimensional solid mass flux Q s after comparing the data with

Q s = g(ρ s -ρ)d 3 ρ Q s (τ ; τ c ) τ |τ |
where g is the gravity constant, ρ s , ρ are respectively the mass densities of the solid and fluid phases, d is a typical grain diameter for sediments. For instance, Meyer-Peter and Müller [START_REF] Meyer-Peter | Formulas for bedload transport[END_REF] (MPM) have proposed

Q s = 8 (|τ | -τ c ) 3 2 + (2)
for gravel, that is a function of the dimensionless shear stress term also called the Shields parameter [START_REF] Shields | Anwendung der Aehnlichkeitsmechanilr und der Turbulenzforschung auf die Geschiebe bewegung[END_REF]

τ = τ /(g(ρ s -ρ)d),
where τ is the shear stress exerted by the flow on particles at the surface of the river bed (in [kg][m] -1 [s] -2 ≡ P a), and τ c is a critical value for the initiation of motion (tabulated for various grain types). Shields [START_REF] Shields | Anwendung der Aehnlichkeitsmechanilr und der Turbulenzforschung auf die Geschiebe bewegung[END_REF] indeed noted that sediment transport occurs when the bottom shear stress just exceeds the threshold of incipient motion τ c .

In uniform flow conditions, the shear stress τ can be directly computed as a local function of the river bed materials characteristics and of the hydrodynamic conditions above the river bed, namely the water depth H and a "(bed-)shear" velocity U . In particular, following the Darcy-Weisbach law that says that the specific energy loss in uniform flow conditions is proportional to some mean velocity |U | 2 , we use the Manning formula

τ = ρgH Q|Q| H 2 K 2 s R 4/3 h , (3)
where, in the particular case of a rectangular channel with width l, the hydraulic radius R h reads

R h = lH l + 2H
.

For numerical simulations, the dynamics of the whole river (sediments plus water) is finally computed from a model coupling [START_REF] Alvarez-Samaniego | Existence of travelling-wave solutions and local well-posedness of the Fowler equation[END_REF] to the hydrodynamics of the free-surface water flow over the given bed B. Typically, for application to large space scales where Exner equation is complemented with MPM relation [START_REF] Ancey | Entrainment and motion of coarse particles in a shallow stream down a steep slope[END_REF], it is often assumed that a coarse hydrodynamical model like the Saint-Venant nonlinear shallow-water equations

   ∂ t H + ∂ x Q = 0 (4a) ∂ t Q + ∂ x Q 2 /H + gH 2 /2 = -gH∂ x B - τ ρ (4b)
modelling a hydrostatic free-surface shallow flow of water with depth H(t, x) and discharge Q(t, x) above a river bed with small-amplitude variations in the longitudinal direction. Note that the friction term in (4) was defined in (3) and the mean velocity U is defined as the ratio Q/H so the system (1)-(4) with relations (2)-(3) is closed.

A new stochastic Exner model

Let us now propose a new stochastic Exner model. We shall directly introduce some "noise" in the model used at large scale. That is, some variables of the standard Exner model are now made random with a prescribed probability distribution in keeping with experimental measures in stationary conditions. Notice that this requires in particular that the solid discharge at the grain-scale (experimentally measurable) has not only a well-defined probability distribution on each finite space-time intervals of a given scale, but also that the joint probability distribution of various non-overlapping intervals is also well-defined. We are aware that such a scale is not yet fully identified, see e.g. [START_REF] Ballio | On the definition of solid discharge in hydro-environment research and applications[END_REF], but current investigations show promising results [START_REF] Ganti | Subordinated brownian motion model for sediment transport[END_REF], although joint probabilities for instance are not yet accessible. Anyway, assuming a separation of scales is natural when building a mathematical model (here, between microscopic processes at the grain scale and macroscopic transport at Exner scale), and we rely, as in most models, on the assumption of "locally uniform hydrodynamic flow conditions".

To start with, we use one common definition of the solid mass flux Q s , see e.g. [START_REF] Francis | Experiments on the motion of solitary grains along the bed of a water-stream[END_REF]:

Q s = dñV
where d is the particle diameter, ñ is the number of saltating particles per space unit and V their longitudinal velocity. Then, with Bagnold scaling [8]

V = g(ρ s -ρ)d ρ a( √ τ -τ c )
and the local equilibrium condition ñ ∝ (τ -τ c ), one retrieves the usual Meyer-Peter and Müller formula (2) once the empirical "activity" number a is fitted to 8 from the empirical average of experimental measurements with gravel in bedload conditions. Now, experimentally measured time-sequences have also brought out that the activity number a is in fact exponentially-distributed pointwise-in-space, see e.g. [START_REF] Lajeunesse | Bed load transport in turbulent flow at the grain scale: Experiments and modeling[END_REF]. Assuming an ergodic underlying process, we thus suggest to use

Q s = A (|τ | -τ c ) 3 2 + ( 5
)
with a random field A exponentially-distributed pointwise in space and time with mean 8

P(A(t, x) ≤ Ā) = Ā 0 8e -8A dA = 1 -e -8 Ā .
Note that this has also the meaning of an exponentially distributed travel distance λ for individual particles after rewriting nV = Eλ with a (deterministic) pick-up rate E. Furthermore, since we would like our new model to be an enriched version of the standard Exner model and to yield back the standard Meyer-Peter and Müller formula for gravel in the "mean-field" approximation A(t, x) ≈ E P (A), it was natural indeed to choose E P (A) = 8. Though, the model is not fully defined yet. In particular, it is not clear how to define the dependence of the stochastic field A(t, x) on t and x.

In this project, we next explore numerically (i.e. at a discrete level) the implications of the "simple" choice where all the random variables A(t, x) at different t and x are chosen independent one-another, when Exner model is coupled to a hydrodynamical model for large scales (as it is usual in practice), and we explain why, in that frame, such a simple choice turns out to be a problem from the physical viewpoint.

Note that several recent works have already embraced the prospect of building a stochastic model of Exnertype. For instance, in [START_REF] Jerolmack | A unified model for subaqueous bed form dynamics[END_REF], a white noise in time and space has been added to a slightly modified closure of (1) given historically by Exner himself [START_REF] Exner | Über die wechselwirkung zwischen wasser und geschiebe in flüssen[END_REF], when Q s is an algebraic function of B and also of the slope ∂ x B (then (1) becomes a closed equation). The model potentially describes complex bed evolutions, but the postulated solid discharge happens not to be verified in most flow conditions.

Using a microscopic model at the grain scale, and focusing on small variations around a planar bed at rest, [START_REF] Ancey | Entrainment and motion of coarse particles in a shallow stream down a steep slope[END_REF] computes the distribution of -∂ x Q s in uniform flow conditions (i.e. insensitive to a Lagrangian vs. Eulerian description) as a balance between deposition and erosion processes (see also [START_REF] Parker | Probabilistic Exner sediment continuity equation for mixtures with no active layer[END_REF] for details about this reformulation). The deposition and erosion processes are chosen as simple algebraic functions of the total number Ñ of particles in a control volume, with time variations proportional to those of the local bed elevation. In other words, Exner equation is reinterpreted as the master equation of a birth-death process followed by Ñ , using the deposition and erosion processes as birth and death rates. In a formal large-Ñ limit [START_REF] Ancey | Stochastic modeling in sediment dynamics: Exner equation for planar bed incipient bed load transport conditions[END_REF], the pointwise probability distribution of the solid flux is Gaussian and not in keeping with experimental observations [START_REF] Lajeunesse | Bed load transport in turbulent flow at the grain scale: Experiments and modeling[END_REF]. Note also that in this model, the coupling with the surrounding (uniform) hydrodynamics is through the deposition and erosion processes, which require a given particles velocity. The extension to non-uniform flow conditions (and thereby the coupling with hydrodynamics at large scales) is not straightforward.

In the series of papers [START_REF] Furbish | A probabilistic description of the bed load sediment flux: 4. fickian diffusion at low transport rates[END_REF][START_REF] Furbish | A probabilistic description of the bed load sediment flux: 3. the particle velocity distribution and the diffusive flux[END_REF][START_REF] Furbish | A probabilistic description of the bed load sediment flux: 1. theory[END_REF][START_REF] Furbish | A probabilistic derivation of the exponential-like distribution of bed load particle velocities[END_REF][START_REF] Roseberry | A probabilistic description of the bed load sediment flux: 2. particle activity and motions[END_REF], a kinetic approach is used in order to derive Exner equation, from the evolution equation of the probability density of particles possessing some "activity" γ at position x and time t, the time-derivative of that density supposedly being directly proportional to the time-derivative of the bed elevation. This yields a reinterpretation of Exner equation as a nonlinear advection-diffusion equation for the bed elevation, and consequently includes the possibility of a diffusive behaviour that is sometimes observed in experiments (depending on the flows and on the sampling-time of measures). However, this kinetic approach remains rather formal, and does not allow for a precise computation of the advective and diffusive coefficients from an explicit stochastic process satisfied by the particles activity. The coefficients are evaluated empirically. In fact, the same exponential distribution observed in [START_REF] Lajeunesse | Bed load transport in turbulent flow at the grain scale: Experiments and modeling[END_REF] for the longitudinal particles velocity is invoked.

Note that the ability of the approaches above at describing a full erosion/deposition scenario better than the standard Exner model, especially when coupled with a hydrodynamic model that lets the (locally uniform) flow conditions evolve, has not been tested yet. This is exactly what we have in mind here, with our model. Of course, this would be difficult with the model of [START_REF] Jerolmack | A unified model for subaqueous bed form dynamics[END_REF], especially because, whatever the solid discharge is used in Exner equation, adding a source term violates the principle of solid mass conservation. And a computation of the local solid discharge with [START_REF] Ancey | Entrainment and motion of coarse particles in a shallow stream down a steep slope[END_REF][START_REF] Ancey | Stochastic modeling in sediment dynamics: Exner equation for planar bed incipient bed load transport conditions[END_REF] first requires to extend the model to non-uniform flow conditions. Last, with a small diffusive term, the kinetic approach of [START_REF] Furbish | A probabilistic description of the bed load sediment flux: 4. fickian diffusion at low transport rates[END_REF][START_REF] Furbish | A probabilistic description of the bed load sediment flux: 3. the particle velocity distribution and the diffusive flux[END_REF][START_REF] Furbish | A probabilistic description of the bed load sediment flux: 1. theory[END_REF][START_REF] Furbish | A probabilistic derivation of the exponential-like distribution of bed load particle velocities[END_REF][START_REF] Roseberry | A probabilistic description of the bed load sediment flux: 2. particle activity and motions[END_REF] cannot improve on standard Exner model: it is not a new equation and it still requires empirical fitting. Contrary to the latter interpretation of Exner equation starting with a kinetic framework and ending with a fully deterministic closure at the hydrodynamic scale, our model lets the coupled hydrodynamic depend stochastically (through the coupling) on the random particles activity. This is potentially richer, and may allow some "variability in the bedform geometry [to result] from some randomness in sediment flux" [START_REF] Ancey | Stochastic modeling in sediment dynamics: Exner equation for planar bed incipient bed load transport conditions[END_REF]. Of course, this is also potentially dangerous, and the first relevant question to ask is: does such a fully coupled model contain at least the uniform flow conditions used for the matching with experimental measures ? That question still needs to be precised and shall next be investigated numerically. The notion of uniform flow conditions in particular may have to be understood only "in the mean", while the mean behaviour is in fact quite complicated because of the nonlinearities.

A numerical discussion of the stochastic model

= {x ∈ [0, 1)} of R/Z        ∂ t H + ∂ x Q = 0, (6a)
∂ t Q + ∂ x Q 2 H + gH 2 2 + gH∂ x B = - τ ρ , (6b)
∂ t B + ∂ x Q s = 0, (6c)
where g represents the gravitational acceleration, H(t, x) is the water height, Q(t, x) the discharge and B(t, x) the topography. The solid transport flux Q s (t, x) is defined by the stochastic Meyer-Peter-Müller formula [START_REF] Audusse | Numerical simulations of the periodic inviscid Burgers equation with stochastic forcing[END_REF] and the friction term is given by relation [START_REF] Ancey | Stochastic modeling in sediment dynamics: Exner equation for planar bed incipient bed load transport conditions[END_REF].

We now describe how we numerically solve the Saint-Venant-Exner system [START_REF] Ballio | On the definition of solid discharge in hydroenvironment research and applications[END_REF]. Let us first mention some of the recent works [START_REF] Audusse | Sediment transport modelling : Relaxation schemes for Saint-Venant -Exner and three layer models[END_REF][START_REF] Castro Díaz | Sediment transport models in shallow water equations and numerical approach by high order finite volume methods[END_REF][START_REF] Garegnani | On the range of validity of the exner-based models for mobile-bed river flow simulations[END_REF] devoted to the derivation of finite volume methods to handle system (6) at once. Here we do not adopt this strategy and apply a splitting method on a staggered grid in a finite volume -finite difference spirit. Indeed it is easier to preserve a discrete steady state on a constant slope in this way and it also allows us to clearly separate the influence of the stochastic term on the bottom topography on one hand and on the fluid quantities on the other hand.

Let us first introduce some notations. We define two meshes of the domain. We first define a number of semicells equal to 2N x denoted C i,i+1/2 and define the cells

C i = C i-1/2,i ∪C i,i+1/2 and C i+1/2 = C i,i+1/2 ∪C i+1/2,i+1
for the two meshes, i = 0, ..., N x -1. For simplicity we consider uniform meshes and denote the space step ∆x. We define the time step ∆t n later on. The fluid quantities will be defined on cells C i and the bottom topography on cells C i+1/2 .

Then starting from initial quantities defined as the mean values of the initial datas on meshes

C i or C i+1/2 H 0 i = Ci h 0 (x)dx, U 0 i = Ci u 0 (x)dx, B 0 i+1/2 = C i+1/2 b 0 (x)dx
we compute at each time step the numerical solution in two steps (i) first, we compute approximate values for the fluid quantities W = (H, Q) T by classical finite volume formulas on cells C i

W n+1 i = W n i - ∆t n ∆x F n i+1/2 -F n i-1/2 + ∆t n ∆x S( W n i , B n i-1/2 , B n i+1/2 ), (7)
where F i+1/2 is the numerical flux defined by the Rusanov formula in this present case,

F i+1/2 = F ( Wi , Wi+1 ) = F ( Wi ) + F ( Wi+1 ) 2 -c Wi+1 -Wi 2 , (8)
with

F ( W ) = (Q, Q 2 H + gH 2 2 ) T , c = max |U i | + √ gH i , |U i+1 | + gH i+1 and S( W n i , W n i+1
) is a discrete source term written as

S( W n i , B n i-1/2 , B n i+1/2 ) = 0 gH n i (B n i+1/2 -B n i-1/2 ) , (9)
(ii) second, we compute approximate values for the bottom topography by finite difference formulas

B n+1 i+1/2 = B n i+1/2 - ∆t n ∆x Q s (H n i+1 , U n i+1 ) -Q s (H n i , U n i ) , (10)
Since we work on the torus T 1 , we have to deal with periodic limit conditions. For the fluid mesh, we add two ghost cells C -1 and C Nx where we define the periodic limit conditions as follows

H -1 = H Nx-1 , Q -1 = Q Nx-1 , H Nx = H 0 , Q Nx = Q 0 .
For the bottom topography, we similarly define two ghost cells

B -1/2 = B Nx-1/2 , B Nx+1/2 = B +1/2
. Note however that B Nx+1/2 is never actually used in the numerical scheme above because of the discretization choices (explicit fluxes and centered source terms).

We then have to define the time step. It has to satisfy the CFL condition for the Saint-Venant system

∆t n ≤ ∆x max i=1,••• ,Nx |U n i | + gH n i ,
Let us recall that our goal is to perform Monte-Carlo simulations and then to run many realizations. Moreover we also would like to consider different space steps ∆x ∈ {0.5, 0.25, 0.125, 0.0625, 0.03125}. To make the comparison between the results easier, we then choose a quite restrictive time step that will satisfy the CFL condition in any case. The uniform choice ∆t = 10 -2 s is made for all computations in the present study.

Monte-Carlo simulations and numerical results

Let us first mention that all our simulations for the semi-empirical stochastic model are close to a deterministic case that is well understood: stationary uniform flow. Moreover we concentrate on torrential regime where the experiment of Recking [START_REF] Recking | Bed-Load Transport Flume Experiments on Steep Slopes[END_REF] exhibits important fluctuations in solid discharge Q s . Then we first describe the uniform deterministic solution, we second describe the stochastic perturbation of the solid flux and third we comment the results of the Monte-Carlo simulations.

Stationary uniform flow solution to the deterministic model

A particular stationary uniform flow solution to the deterministic model is computed as follows:

• we impose the discharge Q and the Froude number F ,

F = |Q| H √ gH ⇐⇒ H = Q 2 g F 2 1/3 , (11)
which determines the water height H.

• we impose K s , then, the equilibrium between the slope and the friction term reads

gH∂ x B = - τ ρ ⇐⇒ ∂ x B = - Q|Q| H 2 K 2 s R 4/3 h , (12)
which determines the slope

-∂ x B = g H F 2 R 4/3 h K 2 s
as a function of H and F .

In the present paper, we have used the values presented in the table 1.

Parameters Values Domain [0, 75] m l 1 m g 9.8 m.s -2 Q 1 m.s -1 F 1.5 (-) H ≈ 0.36 m U ≈ 2.80 m.s -1 R h ≈ 0.21 m -∂ x B ≈ 7 % K s 30 m 1/3 .s -1
Table 1. Initial conditions in torrential regime.

Description of the stochastic solid flux

In the previous subsection we described the uniform flow. We now have to describe the characteristic of the solid flux Q s in the Exner equation. We use the stochastic Meyer-Peter-Müller formula [START_REF] Audusse | Numerical simulations of the periodic inviscid Burgers equation with stochastic forcing[END_REF] and then the definition of the discrete solid flux is

(Q s ) n i = d g(ρ s -ρ)d ρ A n i (|(τ ) n i | -τ c ) 3 2 sg((τ ) n i )
with the parameters listed in the Table 2.

Parameters Values

ρ water 10 3 kg.m -3 ρ sediment 2.65 × 10 3 kg.m -3 R ρ sediment -ρ water ρ water = 1.65 (-) τ * c 0.047 (-) d 10 -3 m K s 30 m 1/3 .s -1 A 8 (-) τ * ≈ 8.93 (-)
Table 2. Physical parameters of the sediment flux in torrential regime.

The coefficients A n i are, for i = 0 • • • N x -1 and n = 0 • • • N t , independent identically distributed random variables with exponential distribution, as suggested by the experimental data. Moreover, the exponential probability law is chosen with uniform parameter λ = 1/8 = 0.125, so all coefficients A n i have a mean of 1/λ = 8, their usual deterministic value. Note that the variance then reads 1/λ 2 = 64.

Monte Carlo simulations

For the Monte-Carlo simulations, we use 1000 realizations of the stochastic Saint-Venant-Exner system, which is enough to numerically approximate the first moment of the stochastic quantities by the Central Limit theorem considering the estimated magnitude of the variances (see below). This assertion is made more precise by computing the confidence interval using the empirical variance. The length of this interval turns to be smaller than 10 -3 for all the quantities of interest on the considered meshes, see Table 3 (we are aware that the number of realizations is not sufficient to obtain a very precise evaluation of the second moment of the distributions, but it allows us to correctly approximate the order of magnitude of this quantity with the computed empirical variance). Note also that we choose a final time T = N t ∆t large enough such that the empirical variance of all the quantities of interest seems close to long-time stationary values as indicated in Figure 2. Table 3. Characterization of the confidence interval In all the results, we are not interested in the bottom topogaphy itself but in the deviation of the bottom topography, denoted Bapp on the figures, from its initial value, denoted Bex on the figures (that is also the stationary one for the deterministic case).

We present in Figure 3 the empirical mean of the topography as a function of the abcissa x at the fixed time T for two different meshes (N x = 150 and N x = 2400), and in Figure 4 the empirical variance of the same quantity for a series of meshes where the number of points varies between the two limits mentioned above. The empirical variance allows us to evaluate the confidence interval and then to check that the oscillations in the empirical mean are not significant. It follows that we can consider the empirical mean is constant in space. Since the empirical mean appears to be centered around zero we can conclude that the mean value of the disturbed bottom remains equal to the original topography. This is satisfying: when considering our stochastic Saint-Venant-Exner system, we had in mind to require that the expectation of the physical variables coincide with their deterministic values in equilibrium conditions (such as the stationary uniform flow conditions here).

Moreover, we also observe that the long-time limit of the empirical variance of the stochastic bottom topography seems to converge to a uniform value when we refine the mesh, which also indicates the possibility of a physically-interesting long-time behaviour: the long-time solutions of our stochastic Saint-Venant-Exner system may actually model the uncertainty in the location of the river bed at the grain-scale, i.e. ripples. However, let us now turn to the fluid quantities. First, we note that the expectation of the water height and of the bottom topography remain solutions to conservative equations. So the uniform stationary values that E(H) and E(B) can be expected to reach at large times in the present periodic setting are necessarily the same as their initial deterministic values. But as concerns the discharge, it holds (recall (6b))

∂ t E(Q) + ∂ x E Q 2 H + gH 2 2 = -E gH∂ x B + τ ρ
and its total "mass" over the domain is thus not conserved a priori (even in the mean) insofar as the source

S E gH∂ x B + τ ρ ( 13
)
is not zero at all times. (One can see two effects here: i) each trajectory of the stochastic systems departs from equilibrium and has a priori a non-zero spatial average of the source, so nor in the mean for the expectation (13), or ii) nonlinearities imply, for instance, E(H∂ x B) = E(H)E(∂ x B) even though E(H) and E(∂ x B) have uniformly the initial values of H and ∂ x B corresponding to a pointwise-in-space equilibrium with zero source.) Consequently, functions of the velocity or of the discharge, like the energy Q 2 /H + gH(H + B) for instance, are a priori not conserved either along time by the dynamics of our model, for the same reasons.

In Figures 5 and6, we show for the velocity the same curves as for the bottom topography in Figures 3 and4. Here also, when computing the confidence interval from the empirical variance, it appears that the empirical mean is constant in space. Nevertheless this empirical mean is now sensitive to the mesh size: we present in Figure 7 the empirical mean as a function of time for the 150 elements mesh and in Figure 8 the value of the empirical mean at final time for the whole serie of meshes. It clearly appears that the empirical mean decreases when the mesh size ∆x decreases, and does not clearly seem to converge, at least for the meshes that we consider here. Now, recalling that, when considering the stochastic Saint-Venant-Exner system, one would like the expectation of the physical variables to coincide with their deterministic values in equilibrium conditions (such as the stationary uniform flow conditions investigated here), this is of course not satisfactory.

In the rest of the paper, we try to improve our new model. To this aim, we come back in the next section to a simpler problem where we consider the classical Saint-Venant system with a fixed (i.e. time-independent) topography. The bottom topography however remains stochastic in the sense that we consider (time-independent) random perturbations of the constant-slope deterministic uniform reference flow. Before turning to the Saint-Venant system, let us present in Figure 9 the probability density functions for the bottom topography and the velocity on the finest mesh. The PDFs are represented at a given point x (previous figures show that the density function, or at least its two first moments, is not sensitive to space) and for some times t n . We are aware of the lack of convergence for these PDFs due to the relatively small number of realizations we performed but it seems to us interesting to mention two facts: first, both PDFs appear to be centered around their mean value even if the original noise introduced in the solid flux Q s does not follow a centered density function. Second, the PDF for the velocity is another way to exhibit that its empirical mean decreases when time increases, see Figure 9, but it also appears that the PDF just shifts to the left without large deformation (that is also confirmed by the fact the empirical variance remains constant after a short time, see Figure 6).

Stochastic Saint-Venant system

Currently, we have exhibited a phenomenon of energy diffusion, characterized by the decrease of the velocity, when adding noise to the sediment flux of the Saint-Venant-Exner model. To better understand this phenomenon, we propose in the following to treat another stochastic Saint-Venant system inspired by works done for Burgers equation [START_REF] Audusse | Numerical simulations of the periodic inviscid Burgers equation with stochastic forcing[END_REF]. Starting from (4), we propose to consider a stochastic bottom topography B defined by

       B n i+1/2 = B 0 i+1/2 + Bi+1/2 , Bi+1/2 = α √ ∆x N/2 k=1 1 k a k cos 2kπ i + 1/2 N + b k sin 2kπ i + 1/2 N ,
where B 0 i+1/2 corresponds to the non-perturbed initial bottom topography, a k and b k are random variables following a normal law N (0, 1), and α is an imposed amplitude. The process consists on disturbing the bottom topography B at the initial state (t = 0) for each realization. It clearly appears that each coefficient of the added perturbation follows a normal law N (0, α 2 ∆x). With the whole expression of noise, we consider a particular kind of spatial correlation which would formally tend to a white noise when the mesh size tends to zero.

We consider again a torrential flow over a constant sloped bottom topography and the initial parameters are given in Table 1. One recalls that we impose periodic limit conditions at the boundaries of the domain. The different simulations are done with a parameter of amplitude α equal to 10 -1 .

The framework of the test case is the same as the Saint-Venant-Exner case; we work with a 150 points mesh and also simulate 1000 realizations. Given the fact that the bottom topography does not evolve in time, we exhibit the range of values of the injected noise for two realizations in Figure 10. The values oscillate in an interval of length of the order of 10 -1 which is in accordance with the imposed value of the amplitude of the noise. 12. The final values are not the same as in the previous case due to the difference between the range of values taken by the injected noise in the present case and in the Saint-Venant-Exner case. In spite of this observation, the behavior of these curves is similar to the previous ones observed for the Saint-Venant-Exner system. The important fact is that the empirical mean of the velocity decreases as a function of time to a new uniform value, see Figure 11. This result is in accordance with the result obtained for the stochastic Saint-Venant-Exner model. For information purposes only, the PDF of the water height and the velocity are presented on Figure 13 on which we can also note the shift previously mentioned for the PDFs of the velocity. (Note that in this new configuration, the PDFs of the fluid quantities are skewed even if the PDF of the bottom topography is symmetric.)

Now, since the decrease of the discharge (and thereby the energy loss) is due to the imbalance between the slope and the friction terms, we suggest to choose, in the stochastic Saint-Venant models, a Strickler coefficient K s different from the one in the deterministic case. But we propose to proceed in the same spirit as in the deterministic case: the Strickler coefficient K s should in fact be calibrated so as to enforce the equilibrium [START_REF] Einstein | The bed-load function for sediment transportation in open channel flows[END_REF].

In the following, we compute the new coefficient at each time step as a deterministic function of moments of H and Q. To establish an expression, at the discrete level, we require an equality between the expectation of the sum in space of Q n i and

Q n+1 i E Nx-1 i=0 Q n+1 i = E Nx-1 i=0 Q n i
thus imposing that no energy is dissipated by the model. In that way, it is possible to express the Strickler coefficient as a time dependent deterministic coefficient (K s ) n , which can be used explicitly for every realization ω and each time step t n in the momentum conservation equation of [START_REF] Andreotti | Selection of dune shapes and velocities. part 2: A two-dimensional modelling[END_REF]. Precisely, we obtain the formula

(K s ) n =        - E Nx-1 i=0 Q ω,n i |Q ω,n i | H ω,n i (R ω,n h,i ) 4/3 (-∂ x B 0 ) × E Nx-1 i=0 H ω,n i + 1 ∆x E Nx-1 i=0 H ω,n i ( Bω i+1/2 -Bω i-1/2 )        1/2 (14)
where we recall that the expectation of the stochastic bottom slope -∂ x B 0 coincides with the slope (constant and uniform) of the deterministic equilibrium. The new stochastic system ( 7) is now nonlinear in the sense of McKean (the coefficients are functions of the stochastic process), and much more difficult to handle numerically (all realizations in a Monte-Carlo simulation need communicating one-another). Furthermore, it is not clear whether requiring equilibrium through the mean-field approximation above will actually be enough for the long-time expectations in the stochastic Saint-Venant-Exner system to coincide with their deterministic counterparts at equilibrium. We postpone these questions to future works and limit here to the simplified stochastic Saint-Venant system.

For this system, and starting from the deterministic equilibrium where K s = 30, one actually maintains an equilibrium "in the empirical mean" with the formula [START_REF] Exner | Über die wechselwirkung zwischen wasser und geschiebe in flüssen[END_REF] after initially perturbing the bottom topography. We present in Figure 15 the evolution in time of the empirical mean of U for Monte-Carlo simulations. It remains close to the equilibrium. We also exhibit in Figure 14 that K s converges fast in time to a new constant value. This value is higher than the deterministic equilibrium value of K s equal to 30 (around 31.4). This higher value is consistent with the fact that the introduction of a friction coefficient in the Saint-Venant system is often presented as a way to take into account (among other phenomena) a roughness of the topography explicitly introduced in the model, with a view to maintaining uniform flow conditions. On the contrary, when some additional roughness is explicitly included through (stochastic) perturbations of the bottom topography, the parameterized friction effect can decrease, and K s increase.

The fact that the new Strickler coefficient can adapt fast to perturbations of the bottom topography in order to maintain an equilibrium is also very promising for the construction of a consistent stochastic Saint-Venant-Exner model that maintains in the mean the same stationary uniform flows as in the deterministic case. In particular, we have also computed the case where the Strickler coefficient is fixed (time-independent) but with value 31.43, from the long-time limit of the previous time-dependent case. Then, the new equilibrium state close to the deterministic equilibrium is still maintained in the mean for large times, as shown in Figure 16. In addition, as expected, the discrete source term is distributed around the value zero, see Figure 17. (Discretely, the friction term obtained with the new Strickler coefficient compensates well the part of the source term related to the slope of the bottom topography in the mean.) Last, let us again present the variance and the PDFs for the water height and the velocity when replacing the constant value of K s previously used (in every test case, K s = 30 m 1/3 .s -1 ) by a time dependent one as expressed earlier [START_REF] Exner | Über die wechselwirkung zwischen wasser und geschiebe in flüssen[END_REF]. These results are respectively shown in Figures 18 and19. Compared to the results on Figure 12, those on Figure 18 show that the variance for the water depth and the velocity has overall the same behaviour than the former ones. Moreover, the modified Strickler coefficient has corrected perforce the shift previously observed both for the Saint-Venant-Exner model and for the Saint-Venant system: not only E(H) and E(Q) assume uniform values equal to the initial equilibrium conditions for large times, but also the velocity U = Q/H, as shown on Figure 15 (in spite of the nonlinearity). (The empirical mean of the velocity at given x yields an approximation of the expectation in an interval with magnitude of order between 10 -3 and 10 -2 at any time.) Moreover, in the PDFs for the velocity shown at given x and various times in Figure 19, it appears that the most probable values is the same at any time.

Saint-Venant

Saint-Venant-Exner with constant K s with modified Since we are now able to maintain in the mean the deterministic flow, it is also possible to study the variance of the fluid quantities as a function of the amplitude of the injected noise. As the amplitude of the noise grows, the variance of the velocity U increases. It appears that the square root of the variance of U has a linear

K s stochastic Q s E(Bapp -Bex) 0 0 0 V ar(Bapp -Bex) 4

Conclusion

We have proposed a stochastic version of Saint-Venant-Exner model in order to account for the fluctuations of the solid flux observed experimentally in stationary river beds. We have observed numerically that, in the mean, a naive introduction of noise in the standard Saint-Venant-Exner model generates additional dissipative effects compared with the deterministic case, which is a bad news if one wants the expectation of the stochastic model to coincide with the deterministic one, at least in equilibrium conditions (i.e. stationary uniform flows). Then, the focus was placed on the question of the convergence of this dissipation in fonction of the size of the mesh. We have observed that the more refined the mesh is, the more dissipation there is. The next question is thus how to better model the noise introduced in Saint-Venant-Exner model so as to compensate for the dissipation introduced. In particular, the scaling of the injected noise with respect to time and space was very naive. One alternative model could be based on better correlating the space-time increments of some random variables in the system. This is a difficult mathematical task, all the more when it is a priori without physical data. As an alternative, we have thus proposed a numerical method to correct, at discrete level, the undesired added dissipation.

We have suggested to define the Strickler coefficient in a stochastic model so as to maintain exactly in the mean equilibrium conditions that are known a priori. In the case of our finite-volume numerical scheme, the new Strickler coefficient can be computed explicitly, as a function of the whole stochastic process. Though, this modification makes the proposed stochastic Saint-Venant-Exner model quite nonlinear, and more difficult to simulate. In this work, we have thus only checked yet, numerically, that once the bottom topography is perturbed initially (but not evolving in time yet, i.e. a simplified stochastic model), the new Strickler coefficient actually makes sense, and compensates fast enough the mean diffusion due to the stochastic perturbations of the bottom topography.

It remains to see i) whether it is still possible to maintain the deterministic equilibrium in the mean when the bottom topography is perturbed in time in a stochastic Saint-Venant-Exner model, ii) whether the new model is robust as concerns convergence to a continuous time-space model, and iii) the kind of physical information that can actually be supported by the stochastic variable A (characteristic size of ripples. . . ) compared with other ones, e.g. τ .

( a )

 a Time variation of solid discharge. (b) Distribution of solid discharge from figure 1a.

Figure 1 .

 1 Figure 1. Flume experiment of bedload transport on high slopes, data kindly provided by A. Recking. Experiment was performed as described in [41] .

3. 1 .

 1 Finite-Volume discretization of Saint-Venant-Exner modelRecall Saint-Venant-Exner model with periodic boundary conditions on the unit cell S

  Empirical varianceSize of the confidence interval B ∈ [2 × 10 -5 , 2 × 10 -4 ] ∈ [1.4 × 10 -4 , 4.5 × 10 -4 ] U ∈ [5 × 10 -4 , 4 × 10 -3 ] ∈ [7.1 × 10 -4 , 2 × 10 -3 ]

Figure 2 .

 2 Figure 2. Variance of the bottom topography (left) and of the velocity (right) as a function of time and space for the finest mesh

Figure 3 .

 3 Figure 3. Empirical mean of the topography deviation for two meshes

Figure 4 .

 4 Figure 4. Empirical variance of the topography deviation for different meshes

Figure 5 .Figure 6 .Figure 7 .

 567 Figure 5. Empirical mean of the velocity for two meshes

Figure 8 .

 8 Figure 8. Empirical mean of the velocity for different meshes

Figure 9 .

 9 Figure 9. Superimposed PDFs for the topography deviation (left) and the velocity (right) computed on the finest mesh at times [0.01; 0.1] and [1; 2].

Figure 10 .

 10 Figure 10. Added noise as a function of space with α = 10 -1 and N x = 150 for two different realizations

Figure 11 .

 11 Figure 11. Empirical mean of the velocity as a function of time

Figure 12 .

 12 Figure 12. Variance of the water height (left) and the velocity (right) as a function of space and time for N x = 150

Figure 13 .

 13 Figure 13. Superimposed PDFs for the water height at times [0.02; 0.1] and [1; 10] (left), and the velocity at times [0.01; 0.05] and [3; 10] (right) with N x = 150.

Figure 14 .Figure 15 .

 1415 Figure 14. Values of K s as a function of time

Figure 16 .

 16 Figure 16. Empirical mean of the velocity as a function of time with K s = 31.43

Figure 17 .

 17 Figure 17. PDF of the discrete source term with K s = 31.43

Figure 18 .Figure 19 .

 1819 Figure 18. Variance of the water height (left) and the velocity (right) as a function of space and time for N x = 150

Figure 20 .

 20 Figure 20. Log-log graph of the square root of the variance of U as a function of the amplitude of the noise α

Table 4 .

 4 Comparison of the results obtained for the three test cases with N x = 150: Saint-Venant + perturbed B and constant K s , Saint-Venant + perturbed B and modified K s at each time step, and stochastic Saint-Venant-Exner.

Résumé. A l'échelle d'un bras de rivière, la dynamique du lit de la rivière est généralement modélisée par l'équation d'Exner (conservation de la masse solide) avec un flux solide empirique pour le transport de sédiments, flux défini par une formule algébrique et déterministe dépendant i) des caractéristiques physiques des sédiments (taille et masse) et ii) de la description hydraulique moyenne de l'écoulement local. Ce modèle s'est avéré utile, notamment à travers des simulations numériques, pour des applications en ingénierie hydraulique (par exemple, estimation de la masse de sédiments mobilisée lors d'une vidange de barrage). Néanmoins, le modèle est également grossier. Et son utilisation pour des échelles de temps et d'espace variées reste une question d'un intérêt considérable pour les sédimentologues. En particulier, les expériences physiques allant de l'échelle du grain à celle du laboratoire révèlent d'importantes fluctuations du flux solide sous des conditions hydrodynamiques données.Ce travail est une étude préliminaire du couplage d'une équation d'Exner stochastique avec un modèle hydrodynamique pour les grandes échelles. (Des modèles stochastiques prenant en compte un flux solide probabiliste sont actuellement étudiés, mais le plus souvent seulement du point de vue de la physique théorique à l'échelle du grain.) On introduit un nouveau modèle d'Exner stochastique possible puis on le discute à l'aide de simulations numériques à travers un cas test approprié.

Acknowledgements

The authors thank A. Recking for the experimental data. The authors greatly thank the CIRM for its warm welcome during the CEMRACS. EA and PU thank EDF R&D and Laboratoire d'Hydraulique Saint-Venant for financially supporting their participation to CEMRACS 2013.


A Govindaraj

E Flahaut

Ch Laurent

A Peigney

A Rousset

C N R Rao

An investigation of carbon nanotubes obtained from the decomposition of methane over reduced Mg 1-x M x Al 2 O 4 spinel catalysts

Carbon nanotubes produced by the treatment of Mg 12x M x Al 2 O 4 (M Fe, Co, or Ni; x 0.1, 0.2, 0.3, or 0.4) spinels with an H 2 -CH 4 mixture at 1070 ± C have been investigated systematically. The grains of the oxide-metal composite particles are uniformly covered by a weblike network of carbon nanotube bundles, several tens of micrometers long, made up of single-wall nanotubes with a diameter close to 4 nm. Only the smallest metal particles (, 5 nm) are involved in the formation of the nanotubes. A macroscopic characterization method involving surface area measurements and chemical analysis has been developed in order to compare the different nanotube specimens. An increase in the transition metal content of the catalyst yields more carbon nanotubes (up to a metal content of 10.0 wt% or x 0.3), but causes a decrease in carbon quality. The best compromise is to use 6.7 wt% of metal (x 0.2) in the catalyst. Co gives superior results with respect to both the quantity and quality of the nanotubes. In the case of Fe, the quality is notably hampered by the formation of Fe 3 C particles.

I. INTRODUCTION

Carbon nanotubes 1 are attractive materials for use in composites since they exhibit excellent mechanical properties [2][3][4][START_REF] Sinnott | Science and Technology of Fullerene Materials[END_REF][6][7][8] and interesting electrical characteristics [9][10][11][12][13][14][15][16] that are related to their unidimensional nature. Carbon nanotubes are commonly prepared by arc-discharge between carbon electrodes in an inert gas atmosphere. 1,17,18 Transition metals are used as catalysts during the arcdischarge to favor the formation of single-shell nanotubes and also to increase their quantity and length. [18][19][20][21][22][23][24] Nevertheless, products so obtained generally are mixtures of nanotubes and several other carbon forms, including a considerable proportion of amorphous carbon and carbon nanoparticles. While purification becomes necessary, it also decreases the nanotube yield to about 2%. 25,26 In contrast, laser vaporization of transition metal-graphite rods produces "ropes" of single-wall carbon nanotubes with a yield of more than 70%. 27,28 Catalytic decomposition of hydrocarbons [29][30][31][START_REF] Baker | Novel Forms of Carbon II[END_REF][33][34][35] and metallocenes 36 as well as the disproportionation of CO [37][38][39] on small metal particles (Fe, Co, Ni, Cu, Mo, a) Address all correspondence to this author. and Pt) produce carbon filaments among which are some lijima-type nanotubes. Several mechanisms proposed for the formation of tubular carbon species by these methods [39][40][41][42][43] point out that the metal particles are active for nanotube nucleation and growth only if they are sufficiently small (< 20 nm). The minimal internal tube diameter that can be obtained by such means corresponds to that of the catalytic particle. Thus, a very small size of the catalyst particles becomes essential for obtaining single-wall nanotubes. Dai et al. 39 obtained isolated single-wall tubes with diameters ranging between 1 and 5 nm by the disproportionation of CO on Mo particles a few nanometers in size. In order to maximize the nanotube yield with respect to the other forms of carbon, such as carbon nanoparticles and pyrolitic deposits, several authors have investigated the influence of temperature and of the nature of both the catalyst and the conditions of treatment. 29,33,37,42 In particular, Ivanov et al. 33 treated a zeolite-supported Co catalyst in a N 2 -C 2 H 2 atmosphere and obtained carbon tubes of 4 nm diameter (60 mm in length). These authors point out that the longest tubes are also the thickest. Hernadi et al. 43 showed that Co-zeolite catalysts give better results when prepared by impregnation rather than by ion-exchange; these catalysts were superior to Co͞SiO 2 catalysts prepared by impregnation.

The metal particles (Cr, Fe, Co, Ni, and their alloys) obtained by the selective hydrogen reduction of oxide solid solutions are generally smaller than 10 nm in diameter and were located both inside and on the surface of the grains of the matrix oxides such as Al 2 O 3 , Cr 2 O 3 , MgO, and MgAl 2 O 4 . [44][45][46][47][48][49][50][51] When H 2 -CH 4 gas mixture was used instead of pure H 2 for the reduction of a -Al 1.9 Fe 0.1 O 3 , pristine Fe nanoparticles were formed in situ upon reduction. Such metal particles are found to be adequate for the catalytic formation of carbon nanotubes. 52 The resulting carbon nanotube-Fe-Al 2 O 3 composite powder contains a huge amount of singlewall and multiwall nanotubes with diameters in the 1.5-15 nm range. The nanotubes were arranged in bundles smaller than 100 nm in diameter and were more than 100 mm in length, the total bundle length in a gram of the powder being approximately 100,000 km. Studies of alumina-based materials 53,54 have shown that an increase in the reduction temperature (from 900 to 1000 ± C) increases the yield of nanotubes, but decreases the quality. A higher quantity of carbon nanotubes was obtained when a -Al [START_REF] Rao | Chemical Approaches to the Synthesis of Inorganic Materials[END_REF] followed by H 2 reduction gives rise to metal-spinel nanocomposite powders, 49,50 the combustion method being generally well suited to prepare fine particulate materials. We have therefore considered it most worthwhile to investigate the influence of the nature as well as the content of the transition metal (Fe, Co, and Ni) in spinel solid solutions on the yield and the quality of the carbon nanotubes formed by the decomposition of methane in a hydrogen atmosphere. Furthermore, we have employed surface area measurements to quantify the yield of the nanotubes as well as their quality.

II. EXPERIMENTAL

Appropriate amounts of the desired metal nitrates (Mg, Al, Fe, Co, and Ni) were mixed in stoichiometric proportions with urea and dissolved in a minimum amount of water in a Pyrex dish. The transition-metal nitrates were substituted for magnesium nitrate with the aim of preparing Mg 12x M x Al 2 O 4 (M Fe, Co, or Ni; x 0.1, 0.2, 0.3, or 0.4) solid solutions. The stoichiometric composition of the redox mixtures was calculated using the total oxidizing and reducing valency of the metal nitrates (oxidizer) and urea (fuel), so that the equivalence ratio was equal to unity. [START_REF] Rao | Chemical Approaches to the Synthesis of Inorganic Materials[END_REF][START_REF] Kingsley | [END_REF][57][58] The dish containing the solution was placed in a furnace preheated at 600 ± C. The solution immediately started to boil and underwent dehydration. The decomposition of the metal nitrates was accompanied by a large release of gases (oxides of nitrogen and ammonia). The resulting paste frothed and formed a foam which swelled and then blazed. A white flame occurred with the production of a material which swelled to the capacity of the Pyrex dish. The total combustion process was over in less than 5 min. One combustion batch gave about 6 g of the oxide powder. The combustion products were attritionmilled (2000 rpm, 30 min) in an aqueous solution of dispersant using alumina balls and a nylon rotor in a nylon vessel. The product obtained was passed through a sieve using ethanol to wash the alumina balls and the vessel. Excess ethanol was removed by evaporation at 60 ± C in an oven for 24 h. The oxide powders were calcinated in air at 500 ± C for 30 min in order to remove the contamination caused by erosion of nylon during milling.

The calcinated spinels were treated with a H 2 -CH 4 gas mixture (18 mol% CH 4 ) for 6 min (flow rate 250 sccm) at 1070 ± C, to obtain carbon nanotube-metalspinel powders. The flow gas was dried on P 2 O 5 and its composition was controlled using mass-flow controllers. The nanocomposite powders containing the various carbon species along with the catalyst powder (metal particles 1 oxide matrix) were examined by scanning and transmission electron microscopy (SEM and TEM), surface area, and other techniques. The nanocomposite powders (containing the various carbon species) obtained after H 2 -CH 4 treatment were heated in air at 900 ± C for 2 h in order to eliminate all or part of the carbon, as required for the specific surface area study.

Powders for SEM examination were sonicated in ethanol, deposited onto an aluminum sample holder, and coated with Ag to prevent charge accumulation. TEM specimens were sonicated in ethanol, and a drop of the dispersion was deposited onto a holey Cu grid. Phase detection and identification were performed using x-ray diffraction (XRD) with Co K a radiation (l 0.17902 nm). The specific surface areas of the starting spinel oxide powders ͑S ss ͒, of the reduced nanocomposite powders obtained after treatment with H 2 -CH 4 ͑S r ͒ and of the powders heated to 900 ± C in air ͑S o ͒ were measured by the Brunauer, Emmett, and Teller (BET) analysis method using N 2 adsorption at liquid N 2 temperature. The carbon content in the reduced composite powders, C n was determined by flash combustion. For the sake of brevity, the calcinated spinel oxides and the corresponding nanocomposites (obtained after treatment with H 2 -CH 4 ) will hereafter be denoted by OMx and Mx (M Fe, Co, or Ni and x is the compositional coefficient in the starting solid solution), respectively. When discussing the powders containing carbon nanotubes, it is found useful to specify the amount of metallic phase (wt%) in the composite. Oxide specimens with x 0.1, 0.2, 0.3, and 0.4 correspond to composite powders containing 3.3, 6.7, 10.0, and 13.3 wt% of the metallic phase, assuming a total reduction of the transition metal ions (Table I).

III. RESULTS AND DISCUSSION

A. Oxide spinels

X-ray diffraction

Analysis of the XRD patterns (Figs. 123) of the spinel oxide powders (OMx) reveals the presence of a small amount of a -Al 2 O 3 , in all the specimens. Comparison with the XRD patterns recorded prior to attrition-milling of the powders (not shown) suggests that this is a result of erosion of the alumina balls during milling. Besides a -Al 2 O 3 only the spinel phase is detected in the OFex specimens (Fig. 1), whereas small amounts of MgO and NiO are present in addition to the spinel in the OCox (Fig. 2) and ONix (Fig. 3) specimens, respectively. These results are in agreement with those of Quénard et al. 50 on similar compounds, showing that the combustion products are lacunar spinels with an excess of trivalent cations of general formula D 123a T 212a V a O 4 (D: divalent cations, T: trivalent cations, V: vacancies). In the case of Fe-containing oxides, it has been shown that the Fe 21 ions are partly oxidized to Fe 31 ions during the combustion and that the products are monophasic spinels. 50 In the case of Co-and Ni-containing oxides, a fraction of the Mg 21 and Ni 21 ions, respectively, does not enter the spinel lattice and is present as MgO and NiO. 49

Specific surface area

Specific surface areas of the attrition-milled oxide powders ͑S ss ͒ were found to be as follows. For the

B. Carbon nanotubes

X-ray diffraction

Analysis of the XRD patterns (Figs. 456) of the reduced catalyst specimens (Mx) reveals the presence of the metallic phase besides the spinel matrix. The a -Fe (110) reflection (d 0.203 nm) is not clearly detected because of its superimposition with the (400) reflection of the spinel phase (d 0.202 nm) (Fig. 4). The intensity of this peak relative to the spinel (311) peak, however, suggests the presence of a -Fe. Interestingly, Fe 3 C (cementite) is detected in addition to a -Fe and the spinel, the intensity of the corresponding peaks increasing with the increase in Fe content. A wide peak which could correspond to the distance between Quénard et al. 49,50,60 have reported that the size distribution of the Co and Ni particles formed upon H 2 reduction at 1000 ± C is unimodal (approximately 15 nm for products corresponding to Co0.2 and Ni0.2) whereas that of the Fe particles is multimodal, with a second distribution of much larger particles (approximately 200 nm) dispersed on the surface of the matrix grains. Such particles could be too large for the formation of carbon nanotubes and tend to give rise to Fe 3 C. They could also be covered by graphene layers.

Carbon content

Independent of the transition metal (Fe, Co, and Ni), the carbon content ͑C n ͒ increases with the increase in metal content in the starting solid solution (Fig. 7 and Table I). C n is in the 1.8-11.8 wt% range for the Fex composites, the value for Fe0.1 (1.8 wt%) being markedly lower than for the others. C n is in the 2.6-7.1 wt% range for the Cox composites and in the 1.2-5.2 wt% range for the Nix specimens. Clearly, the nature of the transition metal strongly affects the conversion of CH 4 into carbon species during the reduction step. Fe provides the highest yield of carbon and Ni the lowest, Co giving intermediate values. These results are in qualitative agreement with those reported by Jablonski et al. 61 for the deposition of various carbonaceous gases on Fe, Co, and Ni foils.

Electron microscopy

SEM observations of the reduced composite powders (Fig. 8) show that the grains of the oxide matrix, between 0.1 and 2 mm in diameter, are uniformly covered by a weblike network of carbon filaments [Fig. 8(a)], several tens of micrometers long, showing that the reduced powders retain the shape of the reduction vessel. Some nanoparticles, most of which correspond to the metal or the metal carbide covered by a few graphene layers, are observed on the matrix grains. Some of the particles may be onion-like carbon nanostructures. 53 Depending on the nature and quantity of the catalyst, some differences are revealed in the high-magnification SEM images. In the case of the Fe0.2 sample [Fig. 8(b)], most of the carbon filaments are actually bundles comprising smaller ones. These filaments do not exceed 50 nm in diameter, some being smaller than 10 nm. In the case of the Fe0.4 sample, ribbons and other carbon forms [Fig. 8(c)] appear besides the filaments. It was difficult to identify the carbon filaments by SEM in the Co0.2 samples [Fig. 8(d)], probably because the diameters are much smaller and fewer bundles are formed. Most of the filaments are held tight between matrix grains and join each other at nonzero angles. In contrast, bundles clearly appear in the Co0.4 sample. They sometimes form buckles, one of which looking like a ring is shown in Fig. 8(e), but ribbons and short filaments of large diameter were not observed. In the case of the Ni0.2 sample, short, large-diameter filaments appear on some of the matrix grains, in addition to the long, smalldiameter filaments [Fig. 8(f)].

Some of the composite powders were examined by TEM. The influence of the nature and content of the metal catalyst on the carbon species formed cannot be assessed solely from TEM studies. The TEM images in Fig. 9 represent typical examples of the different species present in the composite powders. In the Fe0.2 sample [Fig. 9(a)], we see that the bundles are indeed made up of carbon nanotubes, most of which appear to be singlewalled, with diameters close to 4 nm. Smaller nanotubes (2.5 nm) are also observed. The nanotubes are flexible and some are sharply twisted and bent. Most of the nanotubes appeared to be unstable under the electron beam. Amorphous carbon can be seen decorating the surface of the nanotubes in some places. Fe and/or Fe 3 C particles between 5 and 20 nm in diameter (appearing as dark spots in the image), covered by graphene layers, also decorate the external surfaces of the nanotubes. Clearly, particles in this size range are not connected with the inner part of the nanotubes. With smaller catalyst particles, however, the diameters of the tubes would be small and we therefore do not observe the tips of the nanotubes. A hollow carbon fiber (inner diameter approximately 5 nm) exhibiting the fishbone structure described by Baker and Rodriguez [START_REF] Baker | Novel Forms of Carbon II[END_REF] was also observed in the Fe0.2 sample [Fig. 9(b)]. It would, however, appear that the mechanism responsible for the formation of the carbon nanotubes in the present study is different from that proposed by Baker and Rodriguez. [START_REF] Baker | Novel Forms of Carbon II[END_REF] In Fig. 9(b), we also see thin nanotubes and a hollow carbon particle.

A two-layer nanotube (external diameter equal to 2 nm) is seen bridging two metal-oxide grains in the image of the Co0.1 sample in Fig. 9(c). An image of the Co0.2 specimen [Fig. 9(d)] shows nanotube bundles and carbon cages, one of which contains a Co particle 10 nm in diameter, as well as a closed multiwall tube. There is no catalyst particle at the tip of this tube and the number of concentric layers varies along its length. A coating of amorphous carbon is present in the area between a 9-layer and a 7-layer nanotube section. These results suggest that the extension and thickening of the nanotubes may occur partly by island growth of graphene basal planes on the existing tube surfaces acting as templates. 19,62 The nanotube may also have been damaged by ultrasonic treatment used for TEM specimen preparation. In contrast, observation of the image of the Ni0.2 powder reveals a closed nanotube about 2.5 nm in diameter with a catalyst particle at the tip [Fig. 9(e)]. Thus the size of the catalyst particle could be evaluated to be approximately 2 nm by comparison with the inner diameter of the tube. A 5 wall nanotube with a relatively large inner diameter (approximately 6 nm) observed in the image of Ni0.2 is shown in Fig. 9

(f).

A large proportion of the carbon nanotubes found in the present study appear to be similar to those described by Iijima. This is related to the small size and the nature of the size distribution of the metal particles obtained on reduction of the oxide solid solutions. This observation is consistent with the results of Dai et al. 39 who report that large Mo particles, fully covered by graphite, were inactive for nanotube formation by CO disproportionation. These authors propose that the formation of single-wall nanotubes depends crucially on the very small size of the Mo particles. In the present study, we find both singlewall and multiwall nanotubes because of the presence of a distribution in the size of the catalyst particles.

Specific surface area measurements

The difference DS S r 2 S o between the specific surface area of the nanocomposite powder ͑S r ͒ and that of the same powder after oxidation in air at 900 ± C ͑S o ͒ essentially represents the quantity of nanotube bundles in the composite powder. 52,53 In Fig. 10 andTable I, we have presented the DS values of various samples. DS increases with the increase in transition metal content up to 10 wt% (M0.3) and saturates for a higher metal content (M0.4). It is noteworthy that DS values are much higher for Co samples (9.2-13.6 m 2 ͞g) than for Fe (3.0-8.7 m 2 ͞g) and Ni (2.3-6.4 m 2 ͞g) samples. The markedly low DS for Fe0.1 could be due to the low specific surface area of the corresponding solid solution (Table I).

The values of DS͞C n are useful to obtain a better understanding of the nature of the nanotubes. The increase in specific surface area per gram of carbon, DS͞C n , can be taken to represent the quality of the nanotubes, a higher value denoting a smaller average tube diameter and/or more carbon in tubular form. 52,53 The DS͞C n values of the samples studied by us are reported in Fig. 11 andin Table I. In the case of the Fex and Cox specimens, DS͞C n decreases with the increase in metal content whereas for Nix powders a maximum is observed for Ni0.2. Interestingly, DS͞C n values are much higher for Co (192-354 m 2 ͞g) than for Fe (74 -167 m 2 ͞g) and Ni (123-215 m 2 ͞g) samples. The increase in specific surface area upon the catalytic formation of carbon nanofibers reported in the literature 63,64 are in qualitative agreement with the present results. Hernadi et al. 43 have reported values of 312 and 653 m 2 ͞g for carbon nanotubes treated with KMnO 4 ͞H 2 SO 4 followed by HF. The larger surface areas are because they take into account the inner surfaces, due to the opening of the tubes by the acid. Analysis of the above results shows that increasing the metal content in the catalyst up to 10 wt% yields more carbon nanotubes owing to the presence of a greater number of catalytically active metal particles on the surfaces of the oxide grains. A further increase in metal content was not effective because it gives rise to larger metal particles covered by graphene layers, which would be inactive for nanotube formation. The carbon quality decreases with the increase in metal content, partly because multiwall tubes are obtained in greater proportion rather than single-wall tubes with higher metal content due to a higher average size of the catalyst particles. These results show that a compromise has to be made between quantity and quality of the nanotubes. A good compromise could be 6.7 wt% of the transition metal (i.e., M0.2).

Among the different metals, Co appears to be the best catalyst with respect to both the quantity and quality of the nanotubes. Fe yields more carbon nanotubes than Ni, but the quality is hampered by the formation of Fe 3 C particles. Thus, the present results show that the metallic particles, and not the carbide particles, are the active species for the formation of carbon nanotubes, in contrast to the suggestion of Ivanov et al. 33 In addition, the high proportion of large Fe particles (approximately 200 nm) formed on reduction of the spinel 49,50,60 would be inactive for nanotube formation whether they are simply covered by graphene layers or form the carbide. Since no significant difference was observed regarding the size distribution of the metal particles in Co-and Ni-MgAl 2 O 4 compositions prepared by reduction in pure H 2 , 49 the difference in the yields of carbon nanotubes in the present study reflects an intrinsic effect due to differences in the chemical nature of these metals.

IV. CONCLUSIONS

Carbon nanotubes are obtained in mixture with particles of the metal and the oxide spinel, by the treatment of Mg 12x M x Al 2 O 4 (M Fe, Co, or Ni; x 0.1, 0.2, 0.3, or 0.4) catalysts with H 2 -CH 4 mixtures at 1070 ± C. In the case of Fe, formation of Fe 3 C particles is observed in addition to the metallic particles. Electron microscopy observations reveal that the grains of the metal-oxide composites are uniformly covered by a weblike network of carbon nanotube bundles, several tens of micrometers long. Most of the nanotubes are single-walled with a diameter close to 4 nm. The nanotubes thus have a high aspect ratio and appear to be flexible. Amorphous carbon is present at the surface of some of the nanotubes. TEM observations suggest that the mechanism for nanotube formation by the process employed in the present study is distinctly different from that proposed for the synthesis of hollow carbon fibers. However, the yarmulke mechanism involving the formation of a graphitic cap and template growth 39 is likely to be applicable. Only the smallest metal particles (, 5 nm) seem to be connected with the formation of nanotubes. Macroscopic characterization based on chemical analysis and specific surface area measurements helps to compare the quality of different specimens. Such study shows that an increase in the transition metal content yields more carbon nanotubes up to a metal content of 10 wt% (x 0.3), but decreases the quality. A compromise composition of the catalyst could involve 6.7 wt% of metal (x 0.2). Co gives superior results with respect to both the quantity and quality parameters. In the case of Fe, the quality of the obtained carbon is notably hampered by the formation of Fe 3 C particles. The observed differences between the Co and Ni specimens point to the important role of the metal. Directions for future work include the study of the formation of carbon nanotubes in composite powders containing nanoparticles of Fe͞Co, Fe͞Ni, and Co͞Ni alloys.

FIG. 1 .

 1 FIG. 1. XRD patterns of the Fe-containing spinels (OFex): (a) x 0.1; (b) x 0.2; (c) x 0.3; (d) x 0.4. Indexed peaks are those of the spinel phase. ͑᭢͒ a -Al 2 O 3 contamination from the attrition balls.

FIG. 2 .

 2 FIG. 2. XRD patterns of the Co-containing spinels (OCox): (a) x 0.1; (b) x 0.2; (c) x 0.3; (d) x 0.4. Indexed peaks are those of the spinel phase. ͑᭢͒ a -Al 2 O 3 contamination from the attrition balls; ͑͒ MgO.

FIG. 4 .

 4 FIG. 4. XRD patterns of the Fe-containing nanocomposite powders after treatment with H 2 -CH 4 at 1070 ± C (Fex): (a) x 0.1; (b) x 0.2; (c) x 0.3; (d) x 0.4. ( ≤ ) Fe 3 C; ͑᭢͒ a -Al 2 O 3 contamination from the attrition balls. Cg corresponds to d 002 in multiwall nanotubes and/or in graphite; other peaks: spinel matrix. The a -Fe (110) peak is masked by the (400) spinel peak at 2Q equal to about 52 ± .

FIG. 7 .

 7 FIG. 7. The carbon content ͑C n ͒ versus the transition metal content in the composite powders.

FIG. 8 .

 8 FIG. 8. SEM images of the nanocomposite powders subjected to H 2 -CH 4 treatment at 1070 ± C: (a) Fe0.2; (b) Fe0.2; (c) Fe0.4; (d) Co0.2; (e) Co0.4; (f ) Ni0.2.

FIG. 9 .

 9 FIG. 9. TEM images of the nanocomposite powders subjected to H 2 -CH 4 treatment at 1070 ± C: (a) Fe0.2; (b) Fe0.2; (c) Co0.1; (d) Co0.2; (e) Ni0.2; (f ) Ni0.2.

65

 65

FIG. 10 .

 10 FIG. 10. DS S n 2 S ss versus the transition metal content for the various composite powders.

  1.8 Fe 0.2 O 3 was used as the starting material, but Al 2 O 3 is not a suitable catalyst matrix for use with Co or Ni. On the other hand, solid solutions between the MgAl 2 O 4 spinel and FeAl 2 O 4 , CoAl 2 O 4 , or NiAl 2 O 4 prepared by the combustion method

TABLE I .

 I Some characteristics of the carbon nanotubes-metal-spinel nanocomposite powders. a

	Specimen	Metal content (wt%)	S r ͞m 2 g 21	S o ͞m 2 g 21	C n (wt%)	DS͞m 2 g 21	DS͞C n ͞m 2 g 21
	Fe0.1	3.3	7.1	4.1	1.8	3.0	167
	Fe0.2	6.7	18.2	10.5	5.8	7.7	133
	Fe0.3	10.0	20.6	11.9	9.2	8.7	95
	Fe0.4	13.3	20.1	11.4	11.8	8.7	74
	Co0.1	3.3	19.2	10.0	2.6	9.2	354
	Co0.2	6.7	23.3	10.5	3.8	12.8	337
	Co0.3	10.0	27.5	13.8	5.1	13.7	269
	Co0.4	13.3	29.2	15.6	7.1	13.6	192
	Ni0.1	3.3	11.7	9.4	1.2	2.3	192
	Ni0.2	6.7	13.5	9.2	2.0	4.3	215
	Ni0.3	10.0	16.0	9.4	3.3	6.6	200
	Ni0.4	13.3	16.0	9.6	5.2	6.4	123

a C n : carbon content; S r , S o : specific surface areas of the composite powder after H 2 -CH 4 treatment containing nanotubes and the oxidized powder, respectively; DS S r 2 S o : surface area of carbon for one gram of composite powder, representing the quantity of nanotubes; DS͞C n : specific surface area of carbon, representing the quality of nanotubes.

ACKNOWLEDGMENTS

The authors would like to thank Dr. O. Quénard for his help in the preparation of the oxide solid solutions and Mr. L. Datas for his assistance in the TEM observations. The financial support of the Indo-French Centre for the Promotion of Advanced Research (New Delhi) is gratefully acknowledged.