J Burguete
email: javier.burguete@fisica.unav.es

N Mukolobwiez

F Daviaud

N Garnier

A Chiffaudel

Buoyant-thermocapillary instabilities in extended liquid layers subjected to a horizontal temperature gradient

We report experiments on buoyant-thermocapillary instabilities in differentially heated liquid layers. The results are obtained for a fluid of Prandtl number 10 in a rectangular geometry with different aspect ratios. Depending on the height of liquid and on the aspect ratios, the two-dimensional basic flow destabilizes into oblique traveling waves or longitudinal stationary rolls, respectively, for small and large fluid heights. Temperature measurements and space-time recordings reveal the waves to correspond to the hydrothermal waves predicted by the linear stability analysis of Smith and Davis ͓J. Fluid Mech. 132, 119 ͑1983͔͒. Moreover, the transition between traveling and stationary modes agrees with the work by Mercier and Normand ͓Phys. Fluids 8, 1433 ͑1996͔͒ even if the exact characteristics of longitudinal rolls differ from theoretical predictions. A discussion about the relevant nondimensional parameters is included. In the stability domain of the waves, two types of sources have been evidenced. For larger heights, the source is a line and generally evolves towards one end of the container leaving a single wave whereas for smaller heights, the source looks like a point and emits a circular wave which becomes almost planar farther from the source in both directions.

I. INTRODUCTION

In the past few decades, there has been a growing interest for buoyant-thermocapillary flows both on fundamental and applied aspects. First, there is the need for understanding and controlling the instabilities of these flows in many applications such as floating zone crystal growth, [START_REF] Preisser | Steady and oscillatory thermocapillary convection in liquid columns with free cylindrical surface[END_REF] electron beam vaporization or laser welding. In particular, experiments are being carried out to explore the possibility of manufacturing crystals in space, where gravity is negligible and only surface tension forces are relevant. As the dynamics observed in these applications is very complex because of the presence of many parameters, it is worth working in a simple configuration with well-controlled experimental conditions, in order to be closer to the theoretical models. Second, theses flows provide very interesting systems of traveling waves which can be described by envelope equations such as the complex Ginzburg-Landau equation. [START_REF] Cross | Pattern formation outside of equilibrium[END_REF] These wave-systems allow a careful study of the transition to spatio-temporal chaos.

Our study concerns a buoyant-thermocapillary flow in a rectangular container of dimensions (L x ,L y ) filled with a Boussinesq fluid layer of depth h, with a free surface and subjected to a horizontal temperature difference ⌬T along x axis. This system is characterized by a set of dimensionless parameters:

͑i͒

Marangoni number M a(␤,d)ϭ␥␤d 2 /; ͑ii͒ Rayleigh number Ra(␤,d)ϭg␣␤d [START_REF] Davis | Thermocapillary instabilities[END_REF] /;

͑iii͒ Prandtl number Pϭ/; ͑iv͒ Dynamic Bond number BoϭRa/M a; ͑v͒ Capillary number Caϭ␥⌬T/; ͑vi͒ Aspect ratios ⌫ x ϭL x /h, ⌫ y ϭL y /h, and ⌫ϭL y /L x , where g is the gravitational acceleration, ␣ the thermal expansion coefficient, the thermal diffusivity, the density of the fluid, the kinematic viscosity, the surface tension and ␥ϭϪ(ץ/ץT). ␤ stands for a temperature gradient and d for a typical length: They will be specified when necessary.

Smith and Davis [START_REF] Smith | Instabilities of dynamic thermocapillary liquid layers. Part 1. Convective instabilities[END_REF][START_REF] Davis | Thermocapillary instabilities[END_REF] ͑SD͒ performed a linear stability analysis of an infinite fluid layer with a free surface, without gravity nor heat exchange to the atmosphere. They found two types of instabilities: Stationary longitudinal rolls and oblique hydrothermal waves depending on the Prandtl number and on the basic flow considered ͑with or without return-flow profile͒. The mechanisms for the hydrothermal wave instability have been described by Smith. [START_REF] Smith | Instability mechanisms in dynamic thermocapillary liquid layers[END_REF] The stability of buoyant-thermocapillary instabilities has been addressed by Laure and Roux [START_REF] Laure | Linear and non-linear analysis of the Hadley circulation[END_REF] for low Prandtl numbers and by Gershuni et al. [START_REF] Gershuni | On the stability of plane-parallel advective flows in long horizontal layers[END_REF] and Parmentier et al. [START_REF] Parmentier | Buoyant-thermocapillary insta-bilities in medium-Prandtl number fluid layers subject to a horizontal temperature gradient[END_REF] for values of the Prandtl number up to 10. The former analysis considers the case of adiabatic surfaces whereas the latter considers conducting boundary conditions. A recent study has been performed by Normand and Mercier [START_REF] Mercier | Buoyant-thermocapillary instabilities of differentially heated liquid layers[END_REF] including gravity and a Biot number introduced to describe the heat transfer at the top free surface. In a conductive state, the Biot number is defined as Biϭ a h/ f h a where a and f are, respectively, the thermal conductivity of air and fluid, and h a is the effective height of the air layer. This study shows the existence of a transition from oscillatory to stationary modes when the Bond number ͑the ratio between the Rayleigh and the Marangoni number͒ is increased. This transition depends on the Biot number.

Priede and Gerbeth [START_REF] Priede | Convective, absolute and global instabilities of thermocapillary-buoyancy convection in extended layers[END_REF] have applied the concepts of convective, absolute, and global instabilities to this problem, to calculate the thresholds of spatial and temporal oscillations of the flow. Finally, a weakly nonlinear stability analysis has been performed by Smith. 11 In particular, he has determined the possible equilibrium wave forms for the instability above threshold and analyzed their destabilization.

The characteristics of two-dimensional ͑2D͒, and more recently three-dimensional ͑3D͒, buoyant-thermocapillary driven flows have been investigated by numerical simulations. Carpenter and Homsy [START_REF] Carpenter | Combined buoyant-thermocapillary flow in a cavity[END_REF] performed simulations for P ϭ1 and large ⌬T varying the Bond number and Ben Hadid and Roux [START_REF] Hadid | Buoyancy and thermocapillary-driven flows in differentially heated cavities for low Prandtl number fluids[END_REF] for low-Prandtl number fluids and various values of the aspect ratio. The latter have shown the existence of a multicellular steady state and a transition to oscillatory convection. Villers and Platten [START_REF] Villers | Coupled buoyancy and Marangoni convection in acetone: experiments and comparison with numerical simulations[END_REF] have carried out both experiments and numerical simulations for Pϭ4 which confirm the existence of the multicellular flow. In a recent study, Mercier and Normand [START_REF] Mercier | Influence of the Prandtl number on the location of recirculation eddies in thermocapillary flows[END_REF] provided a general study of the structure of this multicellular flow with respect to the Prandtl number. They showed the recirculation eddies to originate on the hot ͑resp. cold͒ side of the cell for high ͑resp. low͒ Prandtl number. Mundrane and Zebib [START_REF] Mundrane | Oscillatory buoyant thermocapillary flow[END_REF] have studied in detail the stability boundaries for the onset of oscillatory convection, for different value of P and the aspect ratio. More recently, Xu and Zebib [START_REF] Xu | Oscillatory two-and three-dimensional thermocapillary convection[END_REF] have performed 2D and 3D calculations for fluids with Prandtl number between 1 and 10. They have determined the Hopf bifurcation neutral curves as function of the Reynolds number and the aspect ratio and considered the influence of sidewalls.

The existence of thermocapillary instabilities, and in particular of oscillations, has been demonstrated in numerous experiments, but only few of them have provided a direct evidence of hydrothermal waves ͑see below and Table I͒. In many experiments, the basic flow was observed to destabilize first against a stationary multicellular instability before exhibiting oscillatory behaviors.

Preisser et al. [START_REF] Preisser | Steady and oscillatory thermocapillary convection in liquid columns with free cylindrical surface[END_REF] and Velten et al. [START_REF] Velten | The periodic instability of thermocapillary convection in cylindrical liquid bridges[END_REF] have shown in floating half zones from molten salts ͑Pϭ8 to 9͒ that the primary flow becomes unstable towards an oscillatory traveling instability. Villers and Platten 14 performed careful velocity measurements on a narrow channel ͑L x ϭ30 mm and L y ϭ10 mm͒ in acetone ( Pϭ4.2). They showed the existence of three states: Steady monocellular and multicellular states and oscillatory state. The structure of the oscillating flow is not known but, because ⌫ y is small, one can think that this oscillation does not correspond to hydrothermal waves but to an oscillation of the multicellular pattern. This hypothesis is confirmed by the study of De Saedeleer et al. [START_REF] Saedeleer | The instability of a liquid layer heated from the side when the upper surface is open to air[END_REF] performed in a L x ϭ74 mm, L y ϭ10 mm channel with decane ( Pϭ15). They report that before the appearance of waves, stable corotative rolls span over the whole liquid layer, starting from the hot side. A further increase of the temperature gradient leads to a time periodic pattern. Using a L x ϭ10 cm, L y ϭ1 cm channel filled with fluids with different Prandtl numbers, Garcimartı ´n et al. [START_REF] Garcimartı ´n | Origin of surface waves in surface tension driven convection[END_REF] suggest that this oscillation is due to a hot boundary layer instability. In the same way, the instabilities obtained by Ezersky et al. in a cylindrical geometry [START_REF] Ezersky | Hydrothermal waves in Marangoni convection in a cylindrical container[END_REF] and a rectangular geometry [START_REF] Ezersky | Spatiotemporal structure of hydrothermal waves in Marangoni convection[END_REF] for PӍ60 do not correspond to hydrothermal waves but probably also to an instability of the multicellular flow: The waves propagate from the hot wall towards the cold and do not present the predicted characteristics.

Schwabe et al. [START_REF] Schwabe | Instabilities of shallow thermocapillary liquid layers[END_REF] also observed this multicellular flow before waves showed up, but reported the existence of two different oscillatory instabilities in an annular channel ͑L x ϭ57 mm and L y Ӎ305 mm͒ filled with ethanol ( Pϭ17): A short wavelength instability for small heights (hϽ1.4 mm) with waves traveling in the azimuthal direction and a long wavelength instability for hϾ1.4 mm, with larger surface deformations, and waves traveling in the radial and azimuthal directions. [START_REF] Schneider | Experiments on surface waves in dynamic thermocapillary liquid layers[END_REF] The former seems to be in accordance with the hydrothermal waves predicted by theory, even if some characteristics-e.g., wave number-are not similar, whereas the latter could be related to a surface-wave instability. [START_REF] Smith | Instabilities of dynamic thermocapillary liquid layers. Part 2. Surface-waves instabilities[END_REF] Daviaud and Vince [START_REF] Daviaud | Traveling waves in a fluid layer subjected to a horizontal temperature gradient[END_REF] report the existence of traveling waves in a rectangular channel ͑L x ϭ10 mm and L y ϭ200 mm͒ filled with silicone oil ( Pϭ10) and a transition to stationary rolls when the height of liquid is increased. The waves present all the characteristics of hydrothermal waves except for the angle between the wave vector k and the applied horizontal temperature gradient: The propagation is nearly perpendicular to the applied temperature gradient. We will see in the following that this effect is due to relative shortness of the streamwise aspect ratio.

Riley and Neitzel 27 also observed oblique hydrothermal waves in a rectangular geometry ͑L x ϭ30 mm, L y ϭ50 mm͒ for silicone oil ( Pϭ14) for sufficiently thin layers (h р1.25 mm). But for thicker layers, a steady multicellular flow is evidenced which becomes time-dependent when the temperature gradient is increased. Pelacho and Burguete, [START_REF] Pelacho | Temperature oscillations of hydrothermal waves in thermocapillary-buoyancy convection[END_REF] in a rectangular cell ͑L x ϭ60 mm, L y ϭ50 mm, and Pϭ10͒ observed a multicellular flow and hydrothermal waves. Moreover, they observe a different instability for hу3.5 mm and large ⌬T, which propagates downstream and could be of the same type as the oscillations reported by Villers and Platten, [START_REF] Villers | Coupled buoyancy and Marangoni convection in acetone: experiments and comparison with numerical simulations[END_REF] De Saedeleer et al., [START_REF] Saedeleer | The instability of a liquid layer heated from the side when the upper surface is open to air[END_REF] Garcimartı ´n et al., [START_REF] Garcimartı ´n | Origin of surface waves in surface tension driven convection[END_REF] and Ezersky et al. [START_REF] Ezersky | Hydrothermal waves in Marangoni convection in a cylindrical container[END_REF][START_REF] Ezersky | Spatiotemporal structure of hydrothermal waves in Marangoni convection[END_REF] Kamotani et al. [START_REF] Kamotani | Oscillatory convection due to combined buoyancy and thermocapillarity[END_REF] have performed an experimental study of oscillatory convection in a confined cylindrical container while Gillon and Homsy [START_REF] Gillon | Combined thermocapillary-buoyancy convection in a cavity: An experimental study[END_REF] have studied combined convection in a confined geometry, in a regime where thermocapillary and buoyancy are of equal importance ͑hϭ6.8 mm, L x ϭ10 mm, L y ϭ38 mm, Pϭ9.5͒. They reported a transition from 2D steady convection to 3D steady convection, in agreement with the numerical computations of Mundrane and Zebib. [START_REF] Mundrane | Oscillatory buoyant thermocapillary flow[END_REF] Braunsfurth and Homsy 31 performed velocity measurements for different heights in a confined container with equal aspect ratios ͑L x ϭL y Ӎ1 cm, Pϭ4.4͒. They found the transition from 2D flow to 3D steady longitudinal rolls to depend on the aspect ratio, in agreement with the experimental results of Daviaud and Vince [START_REF] Daviaud | Traveling waves in a fluid layer subjected to a horizontal temperature gradient[END_REF] and the modelization of Mercier and Normand. [START_REF] Mercier | Buoyant-thermocapillary instabilities of differentially heated liquid layers[END_REF] Moreover, they observed an oscillatory flow for higher values of ⌬T.

The nonlinear dynamics of these waves has also been studied: Mukolobwiez et al. [START_REF] Mukolobwiez | Supercritical Eckhaus instability for surface-tension-driven hydrothermal waves[END_REF] have shown the existence of a supercritical Eckhaus instability in an annular container ͑L x ϭ10 mm, L y Ӎ503 mm, and Pϭ10͒ and Burguete et al. [START_REF] Burguete | Hydrothermal wave amplitude holes in a lateral heating convection experiment[END_REF] spatiotemporal defects corresponding to amplitude holes in the rectangular container used in this study.

Finally, let us mention that traveling waves with characteristics similar to hydrothermal waves have been observed in other experimental configurations: hot wire under the surface of a liquid ͑Vince and Dubois [START_REF] Vince | Hot wire below the free surface of a liquid: structural and dynamical properties of a secondary instability[END_REF] ͒ or liquid layer locally heated on its free surface ͑Favre et al. [START_REF] Favre | Instabilities of a liquid layer locally heated on its free surface[END_REF] ͒.

In this article, we consider buoyant-thermocapillary instabilities in a rectangular geometry in which the height and the aspect ratios have been varied over a large range. In particular, the characteristics of oblique traveling waves and stationary longitudinal rolls have been studied in details and compared to available theories. Special attention has been given to the stability domain of the waves where two kinds of sources of waves have been evidenced.

In the following, we first describe in Sec. II the experimental setup and the measurements techniques. Section III is devoted to the experimental results, concerning the basic flow, the oblique traveling waves, the longitudinal stationary rolls, and finally the destabilization of these patterns further from threshold. The discussion of our results in Sec. IV, concerns the relevance of the scaling parameters and a comparison to theoretical and experimental results. A summary and a conclusion are given in Sec. V.

II. EXPERIMENTAL SETUP

The apparatus is an evolution of the one described used by Daviaud and Vince. [START_REF] Daviaud | Traveling waves in a fluid layer subjected to a horizontal temperature gradient[END_REF] It consists of a rectangular container L x long and L y wide as shown in Fig. 1. Most of the experiments have been carried out with L y ϭ250 mm and L x ϭ10, 20, and 30 mm, and in some experiments, L y has been varied down to 30, 50, 75, 90, and 180 mm. For the range of fluid depths examined-1 mmрhр10 mm-these dimensions correspond to streamwise ͑resp. spanwise͒ aspect ratios 1р⌫ x ϭL x /hр30 ͑resp. 3р⌫ y ϭL y /hр250͒. The shorter walls are made of Plexiglas, while the two longer vertical walls are made of copper and are thermally regulated by circulating water at different temperatures T ϩ and T Ϫ . The lower boundary consists of a float glass plate and a Plexiglas plate is inserted a few millimeters above the surface of the fluid to reduce evaporation. With this configuration, the height of liquid remains constant to better than 1% over an experimental run. The same experiment was performed in an annular geometry with a gap between the inner cylinder and the outer cylinder being 10 mm and the mean perimeter 503 mm. [START_REF] Mukolobwiez | Supercritical Eckhaus instability for surface-tension-driven hydrothermal waves[END_REF][START_REF] Mukolobwiez | Etudes de systemes d'ondes propagatives dans de ecoulements thermogravitaires et thermocapillaires[END_REF] In this last experiment, the main characteristics of the wave patterns are similar to those obtained in the rectangular geometry with L x ϭ10 mm.

The working fluid used is a 0.65 cSt Rho ˆne-Poulenc silicone oil whose physical properties are given in Table II. This liquid presents a low susceptibility to surface contamination, a medium Prandtl number Pϭ10 and is transparent to visible light. Another fluid was used ͑2 cSt silicone oil, Pϭ30͒ to check the existence of hydrothermal waves for larger Prandtl numbers.

The fluid depth h is measured with a 0.05 mm precision and a meniscus is present at the boundaries. The horizontal temperature difference ⌬TϭT ϩ ϪT Ϫ , is imposed by the two copper walls and is measured using thermocouples. This parameter is regulated with a stability better than 2ϫ10 Ϫ2 K. Room temperature is kept constant near 20°C and the hot and cold temperatures are adjusted so that the container mean temperature remains close to it.

The flow is characterized by temperature measurements and visualizations associated to image processing. Temperature measurements are performed using a moving Cr-Al ͑type K͒ 0.25 mm thermocouple with a time response small enough to perform temporal temperature series with a 20 Hz maximal sampling rate. The accuracy of these measurements is typically 0.03 K. Shadowgraphy is used to control the perturbation of the flow due to the presence of the probe. We visualize the streamlines by sowing the flow with Merck Iriodin 100 Silver Pearl and by illuminating it with a He-Ne laser light sheet in different vertical planes. The patterns are also visualized by shadowgraphic imaging. A parallel vertical light beam crosses the container from top to bottom and forms a horizontal picture on a screen, due to surface deformations at the oil-air interface and to temperature gradients in the fluid. The surface deformations have been measured, looking at the deflection of a laser beam on the surface of the fluid. These measures give a deformation of less than 15 m in agreement with the value of the capillary number Ca ϭ0.005 K Ϫ1 ϫ⌬T. As the surface deflection is small, the shadowgraphy method gives mainly the temperature gradient field variation. [START_REF] Garnier | Ondes non lineaires a une et deux dimensions dans une mince couche de fluide[END_REF] The spatio-temporal evolution of the structures is recorded using a CCD video camera and the images are digitized to 256 gray levels along a line of 512 pixels perpendicular to the gradient. This method is described in detail elsewhere. [START_REF] Daviaud | Traveling waves in a fluid layer subjected to a horizontal temperature gradient[END_REF] Note that the observed region is limited to 185 mm by the optical setup. Some experiments have been performed in a L y ϭ180 mm reduced cell to avoid this limitation. This will be indicated when needed. To extract the pattern behavior from the spatiotemporal diagrams, space and time Fourier transforms and complex demodulation techniques are used. [START_REF] Burguete | Hydrothermal wave amplitude holes in a lateral heating convection experiment[END_REF] They allow a determination of the amplitude of the oscillations, the wave number, the frequency and thus the phase velocity. The two parameters which control our experimental system are h, the height of fluid in the cell, and ⌬T, the horizontal temperature difference between the two walls. In fact, one must keep in mind that, when varying h, ⌫ x and ⌫ y are also varied. The experimental procedure was the following: For a given L x and height of fluid h, ⌬T is increased from ⌬Tϭ0 by steps and we wait for tϭ30 minϳ100h 2 / at each step to allow the system to stabilize. Above the threshold, ⌬T is then decreased to study the existence of hysteresis. The steps are smaller than 0.5 K.

III. RESULTS

As soon as ⌬T 0, a convective flow is created. The thermal gradient across the cell induces a surface tension gradient on the free surface of the fluid. Because of the thermocapillary effect, this gradient generates a surface flow from the hot side towards the cold side, with a bottom recirculation. Buoyancy forces also drives the flow in the same direction as thermocapillary forces. The basic flow appears thus as a long roll perpendicular to the gradient. In fact, depending on ⌫ x and ⌬T, a unicellular or a multicellular flow with transverse rolls can be observed ͑see below͒. This flow can then destabilize into different patterns, depending on the two parameters ⌬T and h. The results are reported on the stability diagrams of Fig. 2 and show a dependence of the threshold values on L x and h. The region named basic flow ͑BF͒ accounts for both the steady unicellular and multicellular flow. Depending on the height of liquid, two regimes occur. When ⌬Tу⌬T c (h), for small h values, the system exhibits oblique traveling waves ͑TW͒, while for larger h values, stationary longitudinal rolls ͑SR͒ are observed. The value hϭh r which separates TW from SR increases with L x ͑h r ϭ2.7,3.6,4.0 mm for L x ϭ10,20,30 mm, respectively͒ while the threshold ⌬T c does not increase significantly. For each h, the reported ⌬T c value corresponds to the first observed pattern that invades the whole cell. In the following, we first present the basic flow and then the different patterns observed while varying ⌬T and h.

A. Basic flow

The basic flow has been characterized through temperature measurements. Figure 3 presents the reduced horizontal temperature profiles ⌬T ˜(r,t)ϭ͓T(r,t)ϪT Ϫ ͔/⌬T obtained for L x ϭ20 mm and for different depths in two different configurations: hϭ1.2 mm and hϭ3.2 mm. For hϭ1.2 mm, the temperature profile is linear and gives a mean temperature gradient measured in the center of the container ␤ H ϭץT/ץx which is of the order of the imposed temperature gradient ␤ i ϭ⌬T/L x , i.e., ␤ H Ӎ0.8␤ i . Moreover, the profile is similar for all depths. This central temperature gradient ␤ H defines for each depth h an effective temperature difference calculated as ⌬T H ϭ␤ H ϫL x Ͻ⌬T. The difference between effective and imposed temperature difference is due to the presence of thermal boundary layers on both hot and cold sides. This effective gradient ␤ H allows direct comparisons with theoretical analysis by Smith and Davis [START_REF] Smith | Instabilities of dynamic thermocapillary liquid layers. Part 1. Convective instabilities[END_REF] and Mercier and Normand 9 who consider an infinite layer of fluid along x ͑see below͒.

For hϭ4 mm, thermal boundary layers are much more important near the hot and cold walls and the measured temperature gradient is ␤ H Ӎ0.2␤ i . A local inversion of temperature can even be observed near the hot wall in the upper part of the fluid. This is an evidence of a roll in a narrow region near the hot wall. [START_REF] Garcimartı ´n | Origin of surface waves in surface tension driven convection[END_REF][START_REF] Schwabe | Holographic interferometry and flow visualization by tracers applied to buoyant-thermocapillary convection in an open rectangular gap[END_REF] In fact, the ratio between the measured and the imposed temperature gradient ␤ H /␤ i continuously decreases when the height of liquid is increased ͑Fig. 4͒. When hу6 mm, one can consider that there is no more temperature gradient in the middle of the cell; all the gradients take place in the vertical boundary layers.

The vertical temperature profiles obtained in the center of the cell for liquid layers with different height are presented in Fig. 5. For small height (hϭ1.2 mm), the vertical gradient is small and uniform in z. When h increases, this gradient increases and becomes nonuniform along z: A mean vertical temperature gradient ␤ V can be defined. For h above 3.2 mm, an inversion of the vertical gradient is observed in the upper part of the fluid layer. Globally, when h increases, ␤ H decreases while ␤ V increases. For the smallest h ͑resp. highest͒ the gradient is nearly horizontal ͑resp. vertical͒. For high h, due to the return flow, we observe a vertical temperature gradient in the bottom of the cell. This profile is perhaps related to a downwards heat flux accross the glass bottom plate: Higher spatial resolution in the bottom boundary layer should be achieved to answer this question. The thermal boundary condition at the oil-air interface appears to be adiabatic for small h, while an upward-directed heat flux exists for greater h. This heat flux is at the origin of a sublayer below the surface which is unstable against Be ´nard-Marangoni type instability, the surface being colder than deeper in the fluid.

Depending on ⌬T and on the aspect ratio ⌫ x , a multicellular flow can be observed. As it was studied in details in other works, [START_REF] Villers | Coupled buoyancy and Marangoni convection in acetone: experiments and comparison with numerical simulations[END_REF][START_REF] Saedeleer | The instability of a liquid layer heated from the side when the upper surface is open to air[END_REF][START_REF] Pelacho | Temperature oscillations of hydrothermal waves in thermocapillary-buoyancy convection[END_REF][START_REF] Garnier | Ondes non lineaires a une et deux dimensions dans une mince couche de fluide[END_REF] we do not report a detailed study of this instability. Let us mention that visualizations revealed this flow to correspond to corotating rolls. They appear near the hot wall and their number decreases when h increases ͑⌫ x FIG. 6. Basic flow: Co-rotative rolls. ͑a͒ Upper view for hϭ1 mm and ⌬Tϭ6 K visualized with iriodin particles; ͑b͒ horizontal temperature profiles for hϭ2 mm and ⌬Tϭ3 K. Inset: Temperature oscillations around the mean gradient in the gray marked region. TABLE III. Hydrothermal waves: Critical ⌬T and critical Marangoni numbers vs fluid depth h and dynamic Bond number Bo for L x ϭ10, 20, and 30 mm. The Bond number has been computed with the classical definition: Bo(h)ϭ(g␣/␥)h 2 . The Marangoni number has been computed with h and the global thermal gradient ⌬T c /L x for the three values of L x and also with the measured horizontal gradient ␤ H for L x ϭ20 mm. The gradient ␤ H was measured for ⌬Tϭ3 K and 1.2рhр6 mm and the corresponding values have been interpolated. decreases: 8 rolls for hϭ1 mm and L x ϭ20 mm, two rolls for hϭ6 mm͒. These rolls can be visualized on an upper view of the cell ͑Fig. 6͒ and are observed as spatial oscillations on the horizontal temperature profiles. Different mechanisms have been proposed to account for this multicellular flow, 9,13-15 but up to now, no complete answer has been given.

L x ϭ10 mm L x ϭ20 mm L x ϭ30 mm h mm ⌬T c K Bo(h) M a (⌬T c /L x ,h) h mm ⌬T c K ␤ H K•mm Ϫ1 Bo(h) M a (⌬T c /L x ,h) M a (␤ H ,h) h mm ⌬T c K Bo(h) M a (⌬T c /L x ,

B. Traveling waves

Oblique traveling waves are observed for small depth layers (hрh r ) for ⌬Tу⌬T c . ⌬T c depends on h ͑or Bo͒: When increasing h, ⌬T c decreases from small h towards a minimum and then increases up to h r . The positions of the minimum and of h r increase with L x ͑Fig. 2 and Table III͒. As soon as the threshold is crossed, two waves which propagate in opposite direction along the y axis, and towards the hot wall on the x axis appear in the container, separated by a source. The source shape and its evolution with ⌬T above the threshold depend of h.

The waves appear via a supercritical Hopf bifurcation: The frequency is finite at threshold 1,23,26 -28 and the amplitude of the waves behaves as (⌬TϪ⌬T c ) 1/2 ͑Fig. 7͒ [START_REF] Mukolobwiez | Supercritical Eckhaus instability for surface-tension-driven hydrothermal waves[END_REF][START_REF] Garnier | Non-linear transition to a global mode for traveling-wave instability in a finite box[END_REF][START_REF] Garnier | Two dimensional hydrothermal waves in an extended cylindrical vessel[END_REF] ͑see Table IV͒. Using the amplitude critical behavior, a more accurate threshold value can be obtained, smaller than the one given by the observation of spatio-temporal diagrams. In fact, for 1.5 mmՇhϽh r the transition for basic flow to traveling waves invading the cell is relatively sharp, occuring within a few tenth of a degree. However, for smaller depths of fluid, the oscillation that take place first are localized at the end boundaries. In order to study planar waves, we decided to report ͑Fig. 2, Table III͒ the onset in the long cells as the first state where the whole cell oscillates. This state can be as much as one or two degrees above the localized oscillation regime. This overestimation of the onset makes difficult the comparison with theory and experiments in shorter cells [START_REF] Riley | Instability of thermocapillary-buoyancy convection in shallow layers. Part 1. Characterization of steady and oscillatory instabilities[END_REF][START_REF] Pelacho | Temperature oscillations of hydrothermal waves in thermocapillary-buoyancy convection[END_REF] ͑see Discussion͒.

Wave sources

Depending on the fluid depth h, two different types of sources of waves can be distinguished:

͑i͒
For smaller values of h (hϽh c ), the source looks like a point located on the cold wall and emits a circular wave. Due to the confinement by the channel in the x direction, this wave becomes almost planar far from the source ͑Fig. 8͒. ͑ii͒ For higher h (h c ϽhϽh r ), the source looks like a line parallel to the x axis and emits inclined planar waves. At onset, the pattern is symmetric and the source is in the middle of the channel. [START_REF] Garnier | Non-linear transition to a global mode for traveling-wave instability in a finite box[END_REF] For ⌬T slightly above threshold-typically within 0.3 K-we observe the source to stand closer and closer to one end of the channel. Further from threshold, an asymptotic homogeneous state is reached, where just a single wave is present. This state can be seen as a source at one end which emits a wave that dies in a sink at the opposite end ͑Fig. 9͒. The present Paper focuses especially on these homogeneous patterns.

The transition between these two behaviors is not sharp. We observe a crossover domain inbetween and the transition is hysteretic when varying h. To illustrate this phenomenon, let us describe a constant ⌬Tϭ5.5 K experiment, i.e., slightly above onset, with h slowly decreasing in the L x ϭ30 mm-L y ϭ180 mm cell. [START_REF] Garnier | Transition 1D-2D pour des ondes hydrothermales non-line ´aires[END_REF] This experiment is illustrated in Fig. 10 by the position of the source ͑a͒, the period ͑b͒, and the wave number k y ͑c͒ measured far from the source and the transverse phase speed cϭ/k y ͑d͒ which remains nearly constant when h is varied. For hϾh c ϭ1.8 mm a single traveling wave is emitted by a source at the extremity of the channel. For 1.5ϽhϽ1.8 mm the source leaves the extremity and transforms progressively into a circular FIG. 7. Amplitude of the traveling waves vs ⌬T for hϭ1.7 mm, L x ϭ10 mm, and L y ϭ180 mm. Experimental data (ϩ) and fit Aϭ(⌬T Ϫ⌬T c ) 1/2 ͑solid line͒. TABLE IV. Nondimensional characteristics of the hydrothermal waves for some depths just above threshold: Projection of the wave vector along y axis k y , frequency ϭ2/T, and angle . source. At hϭ1.5 mm, we observe a discontinuity of the wave number and frequency. Below this value the source is perfectly circular and continue to move towards the center of the channel. Finally the waves disappear at hϭ0.90 mm. The crossover between the two behaviors can be tracked on spatiotemporal data: For a source of circular waves, the projection of the local wave vector along the y direction changes continuously from a negative value to a positive one when crossing the source; whereas for source of planar waves, we observe a superposition of waves with opposite wave number k y . The superposition occurring in the latter case results in the existence of standing waves ͑along y axis͒ in the core of the source. So we observed the amount of standing waves to smoothly decrease by a factor 10 when h decreases from 1.8 to 1.5 mm. [START_REF] Garnier | Transition 1D-2D pour des ondes hydrothermales non-line ´aires[END_REF] When performing the reverse experiment and increasing h, the source remains located away from the extremities in a small hysteretic region above h c . No systematic study of this transition with parameters h and ⌬T has been performed. We have observed that h c increases with L x : h c ϭ1.1,1.6,1.8 mm for L x ϭ10,20,30 mm, respectively. Other sources and sinks may even appear in the cell when strong increases of ⌬T are made, but they annihilate, leaving a single source. Far above ⌬T c they may even remain stable, like frozen. The asymptotic behaviors described above concern thus a narrow band above ⌬T c . Finally, in the periodic annular channel-L x ϭ10 mm, L y ϭ503 mm-the two types of sources are also observed above and below h c , but only during transients; sources and sinks are both unstable near onset always leaving a single wave. We can thus conclude about the stability of defects that sources are stable at small heights in the rectangular container and sinks are always unstable except at the extremities of the channel.

L x ϭ10 mm L x ϭ20 mm L x ϭ30 mm h ͑mm͒ k y (1/h) (/h 2 ) h ͑mm͒ k y (1/h) (/h 2 ) ͑rad͒ h ͑mm͒ k y (1/h) (/
The two types of sources reported above can be related to the two types of hydrothermal waves observed in a larger aspect ratio experiment. [START_REF] Garnier | Ondes non lineaires a une et deux dimensions dans une mince couche de fluide[END_REF][START_REF] Garnier | Two dimensional hydrothermal waves in an extended cylindrical vessel[END_REF] However, in our very long rectangular channel, the two type of wave can only be differentiated close to the source.

Temperature oscillations measurements

Temperature measurements reveal that the traveling waves correspond to bulk temperature oscillations. As shown in Fig. 11, the corresponding temperature evolution for a defined position x 0 can be described as T x 0 (z,t) ϭA ˜(z)cos( T t)ϩ͗T(x 0 ,z)͘, where T is the frequency of the oscillation, identical to that measured with shadowgraphy , A ˜(z) is the amplitude of the oscillation for each depth and ͗T(x,z)͘ denotes a time-average of the temperature signal in a given spatial position (x,z). This amplitude is less than 0.5 K, and its z-dependence can be seen in Fig. 11͑b͒. The maximum of the amplitude of the oscillations is reached at middle-depth (zӍh/2), and the minimum at the surface (zϭh). This kind of behavior was predicted by Smith [START_REF] Smith | Instability mechanisms in dynamic thermocapillary liquid layers[END_REF] and experimentally observed in another configuration by Pelacho et al. [START_REF] Pelacho | Temperature oscillations of hydrothermal waves in thermocapillary-buoyancy convection[END_REF] This temperature behavior is compatible with the proposed hydrothermal wave instability mechanisms, which is responsible of the up-flow component of the wave number.

Characteristics of the waves at threshold

The waves behave as plane waves near the threshold, i.e., an unique wave number and frequency can describe their evolution. As the wave propagates from the cold to the hot side, the x component k x is negative in our referential, and k y is positive ͑resp. negative͒ for the right ͑resp. left͒ traveling wave. As a consequence, k forms an angle ͑see schematics in Figs. 8 and 9͒ with the x axis, varying between /2Ͻ Ͻ.

When we change the experimental parameters, the wave number k, the angle and the frequency evolve. For the three values L x ϭ10,20,30 mm two regions can be distinguished:

͑i͒
For hՇh c (L x ), the wave number k y and the frequency decrease with h while the angle has a nearly constant value ϳ0.83ϭ2.6 rad ͑Fig. 12͒. Note that if we define the angle with the gradient as ␣ϭ Ϫ, his value is approximatively ␣ϳ/6. ͑ii͒ For h c ՇhϽh r the frequency increases with h, but the angle decreases until a value ϭ0.64ϳ2 rad ͑␣ ϳ/3 for large h, Fig. 12͒. The wave number k y increases slightly for hտh c and saturates when approaching h r .

The different characteristics of the waves for some values of h are summarized in Table IV.

Influence of the aspect ratio

To evaluate the influence of the aspect ratio ⌫ y ϭL y /h or ⌫ϭL y /L x on the instability, various experiments were real- ized for 0.75ϽhϽ2.75 mm and various L y between 30 and 90 mm, leaving L x ϭ30 mm constant, i.e., for horizontal aspect ratio ⌫ from 1 to 3.

For small height, between 0.75 and 1 mm, we observe for ⌫ϳ1, 2, and 3 the waves to nearly propagate along the direction of the thermal gradient from cold side to hot side. This corresponds mainly to the core of point sources as described in Sec. III B 1. A perfectly symmetric point source for L y ϭ90 mm is shown in Fig. 13.

For higher height, between 1.75 and 2.75 mm, we observe for ⌫ϳ1 (L y ϭ30 mm) and ⌫ϳ3 (L y ϭ90 mm), planar waves to propagate obliquely from cold side to hot side along x axis and both side along y axis. Such behavior recalls precisely the observations in longer cells. However, for L y ϭ50 mm, hϭ2.75 mm, and ⌬Tϭ5.8 KϾ⌬T c , we report an inverse dynamics. Traveling waves are observed which propagate from the hot to the cold side, along the gradient direction, and are reminiscent of those observed by Ezersky et al. [START_REF] Ezersky | Hydrothermal waves in Marangoni convection in a cylindrical container[END_REF][START_REF] Ezersky | Spatiotemporal structure of hydrothermal waves in Marangoni convection[END_REF] and Garcimartı ´n et al. [START_REF] Garcimartı ´n | Origin of surface waves in surface tension driven convection[END_REF] Thresholds have been measured for L y ϭ90 mm to be just a few tenth of a degree above those presented above for long channels. A discussion concerning the different thresholds measurement methods is presented in Sec. IV C. For shorter cells, no accurate measurements have been carried out but the threshold appears generally above those for longer channel configuration. This finite size effect as been recently studied by Pelacho et al. [START_REF] Pelacho | Local Marangoni number at the onset of hydrothermal waves[END_REF] Let us note that for small aspect ratio cells the relative importance of menisci with respect to the bulk grows. For low h especially, because ⌬T c is very high, oscillatory instabilities appear in the menisci for ⌬T below the hydrothermal-waves onset in the bulk.

C. Stationary rolls

For hϾh r the basic flow destabilizes in the form of a stationary pattern with a wave vector perpendicular to the horizontal applied gradient. At threshold, the pattern appears only near the hot side and invades the rest of the cell when ⌬T is increased. A typical shadowgraphy of the cell is shown in Fig. 14. These stationary patterns correspond to the destabilization of the primary roll in the y direction as in Be ´nard-Marangoni convection. The wave number associated with this pattern remains nearly constant (kӍ3.2) for a large range of values of hտh r and increases slowly for larger h ͑Fig. 15͒.

Using iriodin particles, we have visualized the flow inside these convective cells: It consists in a superposition of a circulation in the xz plan due to the basic flow and a convective circulation in the yz plan for each cell due to bulk induced convection. The walls of each cell are well defined, and the fluid does not cross them. In a convective cell, because of the primary roll flow, the fluid moves upwards near the hot wall, and downwards near the cold wall.

For intermediate x, the observation of independent cells along the y axis allows to describe the temperature profiles in Fig. 16 as two superimposed layers consisting of counterrotating rolls. Because of the convective flow, the fluid circulates in the yz plan, ascending in some places, and descending in others. These recirculations induce temperature modulations along the z direction inside the upper (2Ͻz Ͻ6 mm) and lower (0ϽzϽ2 mm) layers ͑Fig. 16͒. An isotherm (zϳ2 mm) separates both layers.

The vertical temperature profile below onset ͑Fig. 5͒ exhibits two layers: An upper layer near the surface which is unstable with respect to Be ´nard-Marangoni criterion and a lower layer with stable thermal stratification. Linear stability analysis of uniformly heated layer of depth d predicts a nondimensional wave number kϳ2 in Be ´nard-Marangoni convection. [START_REF] Pearson | On convection cells induced by surface tension[END_REF] In the dϭ2h/3 upper layer, this would lead to kӍ3, close to the kϭ3.2 measured value.

D. Further from threshold

Further from threshold, the different patterns destabilize:

͑i͒
At small depth hՇh c , we observe series of traveling holes [START_REF] Burguete | Hydrothermal wave amplitude holes in a lateral heating convection experiment[END_REF] to be shed continuously in the intermediate regions located between the circular source and the planar-wave regions. Whereas there is almost no defect near onset, their number increases with ⌬T ͑cf. Fig. 2 of Burguete et al. [START_REF] Burguete | Hydrothermal wave amplitude holes in a lateral heating convection experiment[END_REF] ͒. ͑ii͒

For intermediate depth, h c ՇhϽh r , we report the occurrence of a secondary instability in the shape of modulational instability of the single-wave-pattern. This instability is similar to Eckhaus instability but occurs with low but finite wave number. We observe spatially growing wave number-modulations which eventually lead to regular shedding of spatio-temporal dislocations in the cell. This patterns are quasiperiodic in both frequency and space. Downstream the dislocations, a lower-wave number pattern is observed, which is either monoperiodic or chaotic owing to h and L x . These results will be reported elsewhere. ͑iii͒ For large depth hϾh r , the stationary pattern has a secondary instability in the form of an oscillation in phase-opposition ͑optical mode, Fig. 17͒ with a finite frequency at threshold. This behavior has been observed in similar steady pattern for other convection experiments: Rayleigh-Benard convection in narrow gap, [START_REF] Dubois | Collective oscillating mode in a 1D chain of convective rolls[END_REF] and in nonuniform heating, [START_REF] Burguete | Inestabilidades producidas por un calentamiento localizado[END_REF] where a fluid layer is heated from the bottom with a nonuniform temperature profile.

E. Higher Prandtl number fluid

Finally, a restricted set of experiences using a Pϭ30 (ϭ2 cSt) fluid has been realized to evaluate the influence of the Prandtl number. At hϭ2.0 mm no hydrothermal waves were observed up to ⌬Tϭ30 K. For hϾ3 mm, stationary rolls where observed. According to available experimental results in shallow horizontal ͑rectangular or annular͒ layers hydrothermal waves have not been reported above Pϭ17. This annular experiment in ethanol [START_REF] Schwabe | Instabilities of shallow thermocapillary liquid layers[END_REF] has shown hydrothermal waves as well as surface waves depending of h. Never- theless Priede and Gerbeth 10 predicted waves up to P ϭ1000. We believe this prediction to be correct, but it is experimentally very difficult to produce the critical gradient. The higher the Prandtl number, the sharper the temperature variation in the thermal boundary layers at both extremities: The effective horizontal gradient ␤ H keeps small even for high ⌬T.

IV. DISCUSSION

A. Relevant parameters 1. Spatial scales and Bond number

The experimental data show the existence of two transition depths h c and h r . These depths separate three regions where different forces dominate. For small h, surface stresses are dominant and no vertical stratification has been observed ͑Fig. 5͒. Buoyant effects are small. Above h c , buoyant forces are no more negligible, and the temperature profile varies with z ͑Fig. 5͒. Finally, above h r the buoyant forces dominate and the thermal gradient is almost vertical.

The stability diagrams-minima positions, h c , h r ͑Fig. 2͒-and the characteristics of the waves and of the stationary rolls-k, f , ͑Figs. 12 and 15͒-move towards larger depths when L x increases. We tried to reduce the data for different h and L x . The natural idea is to plot the data vs the Bond number BoϭRa/M a. With the classical reduction [START_REF] Mercier | Buoyant-thermocapillary instabilities of differentially heated liquid layers[END_REF] -␤ϭ⌬T/L x and dϭh-the Bond number varies as h 2 and does not reduce the data. Other combinations of h and L x have been tested and h 3 /L x gives the best results: All these curves collapse into single ones ͑Figs. 18 and 19͒. In particular the transition between hydrothermal waves and stationary rolls occurs for h r 3 /L x ϭ2.1Ϯ0.2 mm 2 . As far as the Bond number is believed to be the relevant parameter for this flow, our experimental results impose Bo to be a function of h 3 /L x . A very simple way to obtain this is to build the Marangoni number using exclusively the horizontal scale L x and the Rayleigh number using exclusively the depth h

M a ͩ ⌬T L x ,L x ͪ ϭ ␥⌬TL x , Ra ͩ ⌬T h ,h ͪ ϭ g␣⌬Th 3 .
The Bond number is then

Bo͑h,L x ͒ϭ g␣ ␥ h 3 L x .
For the transitional depth h r , we compute Bo(h r ,L x ) ϭ0. 25 showing buoyant and capillary forces to be of the same order of magnitude.

Thermal gradient

Let us note that even if the experiment control parameter is the horizontal temperature difference ⌬T this study has shown the necessity to measure both horizontal and vertical effective gradients ␤ H and ␤ V to understand the instability mechanisms.

Biot number

In infinite layer theoretical studies, the Biot number is used to control the vertical temperature profile. In our experiment, the temperature profiles have been measured and shown to vary along x. Also, the air layer is typically 1 cm thick and is, therefore, animated by strong convective flows. These two effects make it impossible to calculate an exact value of Bi. It could be possible to measure it from very careful measurement of the temperature field in the air very near the fluid surface.

Using our data, the only possibility is to compare the heat flux across the free surface from vertical temperature profiles ͑Fig. 5͒: The effective Biot number increases with h.

B. Comparison with theoretical results

Hydrothermal waves

In this section, we will compare our measurements with different theoretical results. We will use the classical definitions of Marangoni and Bond numbers, M a Th and Bo Th M a Th ϭM a ͩ ⌬T

L x ,h ͪ ϭ ␥ ⌬Th 2 L x , Bo Th ϭ g␣ ␥ h 2 .
Furthermore, as far as theoretical studies concern only planar waves, we compare them with experimental data collected far from the sources where waves are planar whatever h. Because the thermocapillar effects dominate for small h, we expect the results obtained by Smith and Davis [START_REF] Smith | Instabilities of dynamic thermocapillary liquid layers. Part 1. Convective instabilities[END_REF][START_REF] Davis | Thermocapillary instabilities[END_REF][START_REF] Smith | The nonlinear stability of dynamic thermocapillary liquid layers[END_REF] to describe our experimental results. The nondimensional characteristics predicted by Smith and Davis for the return flow using a fluid with Pϭ10 and Biϭ0 can be summarized as: Wave number ͉k͉ϳ2.5, propagation angle ϳ160°, frequency ϳ0.14 and phase velocity cϳ0.06. It is important to note that the time unit used by Smith and Davis is /␥␤ H . These waves appear for a critical Marangoni number M a c Th ϭ300. Let us consider the case L x ϭ30 mm. ͑The data presented in Table IV for L x ϭ10 or 20 mm give similar results.͒ For hϽh c , a very good agreement with SD is found. For example, for hϭ1.00 mm, and using the Smith and Davis time unit definition, the experimental values are ͉k͉ϭ2.6, ϭ150°, ϭ0.14, and cϭ0.05. Nevertheless, the experimental critical Marangoni number value is M a c Th ϭ668. This value is larger than the predicted value but corresponds to the invasion of the whole cell by the waves. We studied the first occurence of waves in shorter cells allowing a complete visualization and plot the observed threshold in the (M a, Bo) plane in Fig. 20. We report M a Th ϭ434 for the same depth with L y ϭ180 mm. The variation of the critical Marangoni number with Bo is in very good agreement with the predicted limit M a c Th ϭ300 for Boϭ0. On the other side, for hϭ2.74 mmϾh c , the wave appear for M a c Th ϭ2800 which reduces to 840 when based on the estimated effective horizontal temperature gradient ␤ H . The experimental characteristics of the waves above onset are: ͉k͉ϭ1.1, ϭ110°, ϭ0.18, and cϭ0. 16. A discrepancy is found for ͉k͉, and between the experimental values and these values predicted by SD.

Mercier and Normand [START_REF] Mercier | Buoyant-thermocapillary instabilities of differentially heated liquid layers[END_REF] ͑MN͒ have considered the effect of bulk forces. For hϳ1 mm ͑small Bo͒ their results are similar to SD. For Bo Th ϭ1 (hϭ2.83 mm), Pϭ7 and Bi ϭ1 they report M a c ϭ386. Using as time unit h 2 /, they found: kϳ2.5, ϳ100°, ϳ19, cϳ7. Using this time unit, the experimental values for hϭ2.74 mm become: kϭ1.1, ϭ110°, ϭ11, cϭ9. The angle and the phase speed are well recovered while the wave number differs by a factor of 2. In Fig. 12, we show the theoretical data from the Mercier and Normand predictions, Smith and Davis prediction, which does not depend on h, and the experimental data. Qualitatively, a similar behavior is found between the MN values and the experimental ones when h is varied. Nevertheless, a quantitative comparison fails, mainly because of the wave number values, which are always smaller ͑typically one half͒ in the experiment. Two reasons can explain this difference. The first one is the finite size of the experimental cell because it can influence the basic flow even far from the cell walls. This effect is not considered in the numerical analysis. The second one is the discrepancy found between the experimental and numerical temperature z-profiles below the threshold ͑Fig. 5 and MN͒.

Nonlinear behavior

Smith performed a nonlinear stability analysis [START_REF] Smith | The nonlinear stability of dynamic thermocapillary liquid layers[END_REF] of the SD solutions. He predicted the traveling wave ͑resp. mixed wave͒ solution to be stable ͑resp. unstable͒ for Pϭ10. We confirm this prediction for hϾh c , reporting a single inclined traveling-wave. However, for hϽh c , the shape of the source deserves a two-dimensional description [START_REF] Garnier | Transition 1D-2D pour des ondes hydrothermales non-line ´aires[END_REF] which has not been made as far as we know.

Stationary rolls

Stationary patterns for this buoyant-thermocapillary problem have been predicted by Mercier and Normand. [START_REF] Mercier | Buoyant-thermocapillary instabilities of differentially heated liquid layers[END_REF] According to Fig. 1͑b͒ of MN, two unstably stratified layers can develop near the surface and the bottom plate for positive Bi. The above layer destabilizes against Be ´nard-Marangoni instability whereas the bottom layer can be Rayleigh-Be ´nard unstable. This calculation, made for a bottom plate with a fixed linear temperature gradient, predicts the bottom layer to be thicker and to transit first.

In our system, the vertical temperature profile is unstable only in the upper layer ͑Fig. 5͒. This difference can be due to the finite conductivity of the glass bottom plate and thus to possible heat flux across.

C. Comparison with experimental results in large rectangular channels

Experiments in large rectangular containers with similar horizontal gradient have been performed by Riley and Neitzel [START_REF] Riley | Instability of thermocapillary-buoyancy convection in shallow layers. Part 1. Characterization of steady and oscillatory instabilities[END_REF] ͑RN͒ and by Pelacho and Burguete [START_REF] Pelacho | Temperature oscillations of hydrothermal waves in thermocapillary-buoyancy convection[END_REF] ͑PB͒. Both experiments use L y ϭ50 mm, a much smaller value than ours. Onset values have been plotted in Fig. 20 together with our results for L x ϭ30 mm and various L y . The data for L x ϭ10 and 20 mm are not represented ͑cf. Sec. IV A 1͒. All data exhibit the same evolution with Bond number. The only discrepancy concerns the lowest value of Bo ͑small h͒ for our data in Table III. This discrepancy has been noted by RN and is explained by the fact that the waves solution at threshold are strongly inhomogeneous in our long cell. This is reinforced by our choice to report onset only when the whole cell is invaded, in order to compare with the uniform theoretical solution. Complementary studies have considered shorter L y ϭ90 mm and L y ϭ180 mm cells where the first apparition of waves is reported as onset. Onsets are much lower for small h and in perfect agreement with RN and PB data ͑Fig. 20͒. Let us note that the main characteristics of the waves are close to RN and PB data. We also found comparable values for the effective gradient near oscillation threshold for RN data (0.75ϽhϽ1.75). Riley and Neitzel present two oscillating modes: Hydrothermal waves ͑HTW͒ for h Ͻ1.3 mm and oscillarory multicellular cells ͑OMC͒ above. In our longer cell, we clearly observe the hydrothermal waves to superimpose on steady corotating rolls along y axis. [START_REF] Garnier | Ondes non lineaires a une et deux dimensions dans une mince couche de fluide[END_REF] We believe that RN's oscillatory multicellular cells also represent this combination.

V. SUMMARY AND CONCLUSION

Experimental results obtained in a lateral heating experiment have been presented. Varying the experimental parameters h, L x , L y , and ⌬T, various stable states have been characterized. The basic flow that appears as soon as ⌬T is not null, can be destabilized producing hydrothermal waves ͑for hϽh r ͒ or stationary rolls (hϾh r ).

The observed hydrothermal waves correspond to bulk temperature oscillations, and exhibit different behaviors depending on the value of h. For hϽh c , the surface stresses are dominant, and the characteristics of the planar waves are very close to those predicted by Smith and Davis. [START_REF] Smith | Instabilities of dynamic thermocapillary liquid layers. Part 1. Convective instabilities[END_REF] For h Ͼh c , their description is no more applicable because of the vertical temperature gradients that appear across the fluid layer, amplifying buoyancy effects. For these values of h, the predicted results of Mercier and Normand, [START_REF] Mercier | Buoyant-thermocapillary instabilities of differentially heated liquid layers[END_REF] including buoyant forces, are recovered except the wave number value. The quantitative discrepancy could be due to finite size considerations that alter the vertical velocity and temperature profiles. The spatial structure of the wave pattern reveals two types of wave-sources depending on the fluid depth. We believe these sources to trace back the existence of two different types of waves: one-dimensional waves for hϾh c and twodimensional waves for hϽh c . [START_REF] Garnier | Ondes non lineaires a une et deux dimensions dans une mince couche de fluide[END_REF][START_REF] Garnier | Two dimensional hydrothermal waves in an extended cylindrical vessel[END_REF] For large h, when buoyant forces dominate, the basic flow destabilizes in the form of a stationary pattern, which can be related to Be ´nard-Marangoni convection.

This set of experiments brought many quantitative data varying most control parameters of the system. In trying to reduce the set of relevant parameters, we emphasize the role of h 3 /L x as the spatial scale of the problem. This parameter appears in a natural way when the Bond number is expressed using a Marangoni number based on horizontal scale L x and a Rayleigh number based on vertical scale h.

FIG. 1 .

 1 FIG. 1. Schematic drawing of the experimental apparatus. ͑a͒ Rectangular container: Cross section ͑top͒ and top view ͑bottom͒. A liquid layer of depth h is subjected to a horizontal temperature gradient. The origin of coordinates is located on the hot side for x and at the bottom plate for z. ͑b͒ Optical setup used to perform shadowgraphy.

FIG. 2 .

 2 FIG. 2. Stability diagram: temperature difference ⌬T c vs height of liquid h for ͑a͒ L x ϭ10 mm, ͑b͒ L x ϭ20 mm, and ͑c͒ L x ϭ30 mm. BF, TW, and SR, respectively, refer to basic flow, traveling waves, and stationary rolls. Vertical dashed lines correspond to the transitional depth h r .

FIG. 3 .

 3 FIG. 3. Reduced temperature ⌬T ˜ϭ(TϪT Ϫ )/⌬T vs x for different depths z for L x ϭ20 mm and ⌬Tϭ3 K, below the onset of traveling waves. ͑a͒ h ϭ1.2 mm, ͑b͒ hϭ3.2 mm.

FIG. 8 .FIG. 9 .

 89 FIG. 8. Traveling waves, hϽh c . ͑a͒ Shadowgraphic image and ͑b͒ schematic drawing of the pattern in an horizontal plane; ͑c͒ spatiotemporal evolution for L x ϭ30 mm, L y ϭ250 mm, hϭ1.6 mm, and ⌬Tϭ5.6 K. The field of view corresponds to ϳ185 mm.

FIG. 10 .

 10 FIG. 10. Evolution with varying h at ⌬Tϭ5.5 K for L x ϭ30 mm and L y ϭ180 mm. ͑a͒ y position of the source in the cell. ͑b͒ Period T of the wave. ͑c͒ Projection of the wave vector k y away from the source. ͑d͒ Projection of the phase velocity cϭ/k y along y.

FIG. 12 .

 12 FIG. 12. Traveling waves just above threshold: Evolution of ͑a͒ the wave number k y , ͑b͒ the frequency ϭ2/T, ͑c͒ the angle , as a function of h. Experimental data ͑dotted lines͒: ͑᭝͒ L x ϭ10 mm, ͑ᮀ͒ L x ϭ20 mm, (ϩ) L x ϭ30 mm. Theoretical predictions: ͑dashed line͒ Smith and Davis ͑Ref. 3͒; ͑छ and solid line͒ Mercier and Normand ͑Ref. 9͒.

FIG. 15 .

 15 FIG. 15. Stationary rolls: Evolution of the wave number k as a function of the depth of liquid h at threshold (⌬Tϭ⌬T c ).

FIG. 18 .

 18 FIG. 18. ͑a͒ Stability diagram in reduced coordinates: Effective horizontal temperature difference ⌬T H ϭ␤ H L x vs h 3 /L x for different L x . Evolution of ͑b͒ the wave number k y , ͑c͒ the frequency ϭ2/T, and ͑d͒ the angle as a function of h 3 /L x .

FIG. 20 .

 20 FIG. 20. Critical Marangoni M a(⌬T c /L x ,h) number vs dynamic Bond number Bo(h). Open squares stand for L x ϭ30 mm and L y ϭ250 mm data reported in Table III. The closed squares stands for a measurement in the L x ϭ30 mm and L y ϭ180 mm cell and the open circles for the L x ϭ30 mm and L y ϭ90 mm cell. Open triangles stand for Riley and Neitzel's data ͑Ref. 27͒ in L x ϭ30 mm and L y ϭ50 mm cell and closed triangles stand for Pelacho and Burguete's data ͑Ref. 28͒ in L x ϭ60 m and L y ϭ50 mm cell. See text for details.

TABLE I .

 I Different experimental conditions and observations ͑see Introduction for details͒. In each group, entries have been ordered by increasing Prandtl number. The first group contains ''small'' y-aspect-ratio experiments ͑S͒. The second and third group contain ''large'' y-aspect-ratio experiments ͑L͒ that report waves propagating along both x and y. HW stands for hydrothermal waves and BLW for boundary-layer waves. In the annular configurations, L x ϭR o ϪR i and L y ϭ(R o ϩR i ), where R o and R i are, respectively, the outer and inner radius of the cell.

	Experiment	Geometry	L x ͑mm͒	L y ͑mm͒	h ͑mm͒	P	Obs.
	Villers and Platten	rectangular	30	10	1.75-14.25	4.2	S
	Braunsfurth and Homsy rectangular	10	10	1.25-10	4.4	S
	Gillon and Homsy	rectangular	10	38	6.8	9.5	S
	Garcimartı ´n et al.	rectangular	100	10	2-3.5	10, 15, 30 S
	Kamotani et al.	annular	2-15	6 -47	ϳ5 -60	10, 25	S
	De Saedeleer et al.	rectangular	74	10	2.5-4.7	15	S
	Pelacho and Burguete	rectangular	60	50	1.25-3.5	10	L-HW
	Daviaud and Vince	rectangular	10	200	0.6-10	10.3	L-HW
	Mukolobwiez et al.	annular	10	503	1.7	10.3	L-HW
	Riley and Neitzel	rectangular	30	50	0.75-2.5	14	L-HW
	Schwabe et al.	annular	57	305	0.5-3.6	17	L-HW
	Ezersky et al.	rectangular	70	50	1.2-3.1	60	L-BLW
	Ezersky et al.	annular	40	188	2-8	60	L-BLW
	This study	rectangular	10, 20, 30 30-250	0.5-10	10.3	L-HW

TABLE II .

 II Physical properties of Rho ˆne Poulenc silicone oil 0.65 cSt at 20 °C. Surface tension measurements have been performed by J. K. Platten at University of Mons-Hainaut ͑Belgium͒. Other properties are from the technical data sheet.

	v		␣			␥ϭϪץ/ץT
	(m 2 /s)	(kg/m 3 )	(K Ϫ1 )	(m 2 /s)	͑N/m͒	͑N/mK͒	P
	6.5ϫ10 Ϫ7	760	1.34ϫ10 Ϫ3	6.3ϫ10 Ϫ8	1.59ϫ10 Ϫ2	8ϫ10 Ϫ5	10.3

Phys. Fluids, Vol. 13, No. 10, October 2001 Burguete et al.

Phys. Fluids, Vol. 13, No. 10, October 2001 Buoyant-thermocapillary instabilities in liquid layers

ACKNOWLEDGMENTS

The authors thank M. A. Pelacho, A. Garcimartı ´n, H. Mancini, C. Pe ´rez-Garcı ´a, J. F. Mercier, and C. Normand for stimulating discussions and C. Gasquet, M. Labouise, and B. Ozenda for their technical assistance. We also thank J. K. Platten for the measurement of the oil surface tension. J.B. acknowledges a postdoctoral grant from SEUID ͑Spanish Government͒ and financial support through contracts PB98-0208 ͑Spanish Government͒ and PIUNA ͑Universidad de Navarra, Spain͒.


D Bonamy

L Laurent

Ph Claudin

J.-Ph Bouchaud

F Davi- Aud

Electrical conductance of a 2D packing of metallic beads under thermal perturbation

Keywords: . 45.70.-n -Granular systems . . 05.40.-a -Fluctuation phenomena, random processes, noise, and Brownian motion . . 46.55.+d -Tribology and mechanical contacts

Electrical conductivity measurements on a 2D packing of metallic beads have been performed to study internal rearrangements in weakly perturbed granular materials. Small thermal perturbations lead to large non gaussian conductance fluctuations. These fluctuations are found to be intermittent in time, with peaks gathered in bursts. The distributions of the waiting time ∆t between two peaks is found to be a power-law ∆t -(1+αt) inside bursts. The exponent αt is independent of the bead network, the intensity of perturbation and external stress. These bursts are interpreted as the signature of individual bead creep rather than collective vault reorganisations. We propose a simple model giving αt = 1ζ, where ζ is the roughness exponent of the surface of the beads.

Introduction. -Granular materials present interesting and unusual properties [1]. For example, photoelastic visualizations of static confined granular packing under external stress have provided evidence of large inhomogeneities in the distribution of contact forces between grains [2,3], giving rise to strong force chains (vaults or arches) extending on a scale much larger than the size of an individual grain. Recent experiments have revealed the sensitivity of sound transmission in granular media to small perturbations (such as thermal expansion of the grains) [4]. This sensitivity has been interpreted using a simple theoretical model of vault formation, the 'Scalar Arching Model' [5]. This model predicts that large rearrangements of the force chains network can indeed occur, even for small external perturbations. This sensitivity tends to disappear when the media is subjected to a strong external stress [6].

However, both photoelastic visualization and sound transmission techniques require large external stresses and can not give information on a weakly confined granular media where interesting signatures of fragility [7] can be expected. We propose here to study this fragility through conductivity measurements. Experiments on the electrical properties of granular media have been performed in the past : Branly reported [8] the influence of an electromagnetic wave on the electrical resistance of a granular packing. Giraud et al. [9], Marion et al. [10] and Gervois et al. [11] have performed measurements of the electrical conductivity in order to reveal the influence of the contact network on the mechanical properties of the media. More recently, Vandewalle et al. [12] reported power-law distributions of conductivity changes, and argued that this was a signature of large rearrangments of stress paths.

The aim of this work is to investigate whether force chain rearrangements can be observed through the variation of the electrical conductance of a 2D bead packings under small thermal perturbations. Large intermittent fluctuations are indeed observed. They appear in bursts that can be characterized via a statistical analysis of the waiting time separating two fluctuation peaks. The microscopic origin of these electrical fluctuations is then investigated. We argue that most of the interesting statistical features are not related to large force chains rearrangements but to individual microcontacts between two beads that rearrange. The observed non trivial statistics might be related to the self-affine roughness of the surface of the beads.

Experiment. -The experimental set-up is illustrated on fig. 1. It consists mainly of a 2D packing of stainless steel beads of diameter a = 6 mm ± 10 µm confined between two Plexiglas plates (thickness 2 cm). The use of metallic beads of millimetric size allows one to have a good control of geometrical and mechanical properties. We can also change the number of grains in the sample to separate local behaviour from collective behaviour. Beads are carefully arranged to form triangular compact packing. This situation of maximal compacity can easily be reproduced before each experiment. Even if the piling can be viewed as regular, the weak polydispersity and the solid friction between grains is sufficient to entail disorder in the network of contact forces [13]. The size of the packing is W × H where W is its width and H its height in bead size units. Thermal perturbations are induced by a 75 W lamp standing at a distance d = 10 cm. The lamp increases the temperature of the beads and of the Plexiglas plates by typically 3 • C. This leads to an expansion of both the beads and the plates: the thermal expansivity for beads is 2 10 -5 K -1 and 7 10 -5 K -1 for Plexiglas. The bead network can also be vibrated via a buzzer placed against the Plexiglas plate. The packing is connected to a 9 V battery. A resistor R 1 in series insures that the current crossing a contact between any couple of beads is much smaller than 40 mA. Preliminary studies performed on 2 beads indeed show that, as long as this current is below 40 mA, micro-welding is prevented and the contacts behave reversibly like an ohmic resistance. Voltages V 0 , V 1 and V 2 (see fig. 1) are recorded via a 12 bytes AD converter: V 0 is the emf of the battery, V 1 is proportional to the current crossing the packing and V 2 represents the fast fluctuating part (faster than τ = RC < 0.02 s) of V 1 . The following quantities can then be deduced:

• the conductance g p of the packing. For a 33 × 52 vibrated packing, g p ≃ 0.02 Ω -1 .

• the fluctuation of conductance ∆g p defined as the variation of g p between 2 acquisition points. Typically, ∆g p reaches values up to 5.10 -3 Ω -1 to be compared to the detection threshold discussed below, which is typically around 5.10 -6 Ω -1 .

A series of experiments was performed with the bead packing replaced by a 50 Ω resistor in order to determine the level of noise of the acquisition system. This is mainly dominated by digital errors on V 2 equal to 5 µV. A threshold is then defined: ∆g p is considered as relevant when the corresponding ∆V 2 is superior to 50 µV. Sampling rates f from 0.05 Hz up to 10 kHz have been tried: f = 500 Hz is sufficient to separate most successive events (cf fig. 2). For each experiment, data is recorded during a period of 20 minutes. The procedure is the following : (i) the packing is vibrated, (ii) the acquisition is started, (iii) the light is turned on 1 minute after (turned off in experiments with packing prepared with light switched on).

Results. -A packing, initially vibrated, keeps a constant conductance as long as its temperature does not change. As soon as it is thermally perturbed (light turned on or off), g p starts to decrease and to fluctuate. Let us focus firstly on the fluctuations ∆g p for a typical experiment (see fig. 2) on a 33 × 52 packing. Fluctuation peaks appear in an intermittent way. The distribution of ∆g p is to a good approximation symmetrical and can be described (at least for small enough |∆g p |) by the following power-law:

P (|∆g p |) ∼ |∆g p | -(1+αg ) (1)
For this experiment, α g ≃ 1, in agreement with the results reported in [12]. For large |∆g p |, P (|∆g p |) departs from this power-law behaviour. One can also study, for the same experiment, the temporal structure of the apparition of these peaks. The expanded view fig. 2b of ∆g p (t) presents a burst structure, which can be characterised by the distribution of the waiting time ∆t between two successive events (independently of their sign). Two distinct regimes can be observed in fig. 2d:

• The short ∆t part of the probability density P (∆t) (corresponding to events within the same burst) decays as a power law with ∆t:

P (∆t) ∼ ∆t -(1+αt) (2)
The exponent is found to be α t ≃ 0.6.

• for larger ∆t, P (∆t) again departs from this power-law behaviour. The cumulative distribution function F (∆t) is less noisy for large ∆t and shown in the inset of fig. 2d. One clearly sees that F decays exponentially for large ∆t, which is the signature of a Poisson flux. The bursts themselves therefore appear as completely independent events, whereas the events inside bursts are strongly non-Poissonian.

In order to determine the origin of these electrical fluctuations, several experiments have been performed with different packing: (1) regular 33 × 52 packing, (2) disordered packing of 1690 beads, (3) regular 33 × 32 packing, (4) regular 33 × 16 packing, (5) regular 33 × 2 packing. Finally, ( 6) is a regular 33 × 2 packing where all metallic beads have been replaced by insulating glass beads except 3 beads forming a triangle in the middle. For all these experiments, P (|∆g p |) decays as a power-law but the exponent α g depends on the bead network geometry, and lies between 0.8 and 1.6 (see fig. 3a). On the contrary, the power-law decay of P (∆t) does not depend on the packing, and leads to α t ∼ 0.6 (see fig. 3b). For the packing of 3 beads, the power-law behaviour actually extends over the three decades of the time distribution: only few bursts are present. The exponent is found to be α t = 0.6 ± 0.05. When the number of beads increases, the value of ∆t above which P (∆t) starts to deviate from a power-law decreases. It can be interpreted by the increase of independent burst number making the inter-burst waiting time mix more and more with the intra-burst statistics. The intensity of the thermal perturbation has been changed by modifying the distance of the lamp (d = 5 cm, 10 cm and 20 cm) without affecting the value of α t . Moreover, for packing (5), the weight of the upper electrode has been varied (M = 108 g, M = 206 g, M = 304 g and M = 404 g). The number of events decreases, but P (∆t) remains a power-law with the same exponent α t = 0.6.

Discussion. -The origin of these non trivial electric fluctuations should be found in the geometry of electrical paths. A natural idea would be to relate these to force chains rearrangements. Indeed the Scalar Arching Model (SAM) introduced by two of us [5] suggests the existence of large force chains rearrangements in a packing subjected to small thermal perturbations. The SAM predicts a broad distribution of the apparent weight fluctuation ∆W a measured at the bottom of the packing. This broad distribution of ∆W a could in principle be related to the observed broad distribution of ∆g p . Moreover, the model also predicts avalanching contact reorganisations whose electrical signature could indeed be the bursts of fluctuations peaks seen in the thermally perturbed packing, each ∆g p peak corresponding to the creation or the breakdown of a contact somewhere in the packing. However, in this scenario, the intra-burst structure should become less and less pronounced when the number of beads decreases, and should disappear completely for a packing of 3 beads. This is clearly not the case (see fig. 3b).

The fact that the power-law nature of the distributions is independent of the geometry of the packing, and still holds for three beads, suggests that the origin of these electrical fluctuations is local. The direct imaging of the surface of the beads with an Atomic Force Microscope allowed us to measure the average roughness R of the beads. We have found R = 90 nm, to be compared to Hertzian deformation δ in the packing. For contact forces on the order of 10 -2 Newton, δ ≃ 10 nm which is smaller than R. The electrical contact between 2 beads is thus completely dominated by the surface roughness. We propose then the following scenario to explain the observed behaviour of g p and ∆g p in our experiment: when the packing is vibrated, the surface roughness of the two surfaces in contact 'adapt' to one another and the effective contact surface is high. Therefore, g p is maximal. As soon as the packing is heated, beads expand on the order of 0.5 µm. This thermal expansion is hindered by the roughness induced friction between beads. Therefore, most of the time the contact between two beads does not change, until the accumulated stress due to the thermal expansion is sufficient to 'unpin' the rough surfaces and make them slip over each other. This leads to the burst structure of the signal. Inside each burst, individual peaks of ∆g p correspond to the creation or the loss of a micro-contact between the two slipping beads. These micro-contact avalanches globally lead to a smaller effective contact surface, thereby making g p smaller.

Simple models of pinning [14,15] can lead to a power-law distribution of the 'trapping' time between successive events; however, in these models the exponent α t is found to depend on the external driving force. The fact that α t is independent of both the strength of the perturbation and of the external stress suggests a purely geometrical interpretation: if the bead surface can be considered to be self-affine with a Hurst exponent ζ, the distance ∆ℓ separating two successive 'spikes' in a given direction is power-law distributed with an exponent α ℓ = 1-ζ on the interval [a 1 , a 2 ] where a 1 is a microscopic length below which the surface is flat, and a 2 is at most the diameter of an Hertzian contact: a 2 ≃ 1 µm. The typical speed V with which a bead thermally expands is 1 µm per second. If this speed is constant in time, the time between successive micro-contact closing/opening is therefore a power law distribution with an exponent α t = 1 -ζ in the interval [a 1 /V, a 2 /V = 1 s], which is precisely the experimental interval of time scales. We have determined ζ from AFM frames via the moving average method [16]: the moving average of a series z(x) on a length interval L is defined as :

z(x) = 1 L L-1 i=0 z(x + i) (3)
For a self affine signal with an Hurst exponent of ζ, the density ρ of crossing point between two moving averages on length scales L 1 and L 2 > L 1 is found to scale as:

ρ ∼ 1 L 2 [(∆L)(1 -∆L)] ζ-1 (4)
where ∆L = (L 2 -L 1 )/L 2 .

AFM frames are stored as 256 × 256 matrices. L 2 is fixed to 128 and ρ(∆L) has been determined for rows and columns averaging (see fig. 4). In both case, we find ζ = 0.35 ± 0.05, and therefore 1 -ζ ≃ 0.65. This value is consistent with α t = 0.6.

Conclusion. -In summary, a series of electrical measurements has been performed to test the SAM predictions of large force chains rearrangements in a 2D metallic beads packing under small perturbations. As soon as the packing is heated or cooled, its conductance starts to decrease and to fluctuate. These fluctuations are intermittent and gather in bursts as predicted by the SAM. However, the waiting time distribution inside bursts are independent of beads network, perturbation intensity and external applied stress. Consequently, the origin of these fluctuations should be found in local microcontact rearrangements at each beads rather than collective vaults rearrangements, as suggested in [12]. However, some information on these collective rearrangments might lie in the deviation from a pure power-law distribution which is clearly observed on figure 4 when one increases the number of grains. These collective effects might also be needed to account for the variation of the exponent α g with the geometry. Work in this direction is underway. 2), (3), ( 4), (5), and (6) from bottom to top. Data have been shifted for clarity. The straight lines are power-law fits. The exponent αt has been fixed for all P (∆t) to αt = 0.6 minimising errors on the fit for the three bead packing (6). αg is equal, from bottom to top, to 1.0, 1.1, 0.8, 1.6, 0.9, and 0.8 respectively.

Lamp Data acquisition board V 1 V 2 V 0 9V R 1 =196Ω C=10µF R=2kΩ R 2 =20Ω

Fig. 2 -Fig. 3 -

 23 Fig. 1 -experimental set-up

Fig. 4 -

 4 Fig. 4 -(a) AFM picture of the bead surface (b) density ρ as a function of the relative difference ∆L with L2 = 128. o corresponds to row averaging and + correspond to column averaging. Inset : log 10 (ρ) as a function of log 10 (∆L(1 -∆L)). The straight line corresponds to a fit using Eq. 4

We wish to thank F. Mangeant, E. Bonneville and M. Planelles for their participation to the data collection. We also acknowledge discussions with T. Baumberger, M. Bonetti, M. Cates, P. Evesque, E. Kolb, J. Wittmer and G. Zalczcer. We thank C. Gasquet and P. Meininger for technical support. We also acknowledge F. Dubreuil for AFM frames of our beads. Ph. Claudin is currently supported by an Aly Kaufman postdoctoral fellowship.


Javier Burguete

Hugues Chaté

François Daviaud

Nathalie Mukolobwiez

Bekki-Nozaki Amplitude Holes in Hydrothermal Nonlinear Waves

Keywords: numbers: 47.20.Dr, 05.45. -a, 47.27.Te, 47.35. + i

We present and analyze experimental results on the dynamics of hydrothermal waves occurring in a laterally heated fluid layer. We argue that the large-scale modulations of the waves are governed by a one-dimensional complex Ginzburg-Landau equation (CGLE). We determine quantitatively all the coefficients of this amplitude equation using the localized amplitude holes observed in the experiment, which we show to be well described as Bekki-Nozaki hole solutions of the CGLE.

The status and nature of the so-called amplitude equations which can be derived in the vicinity of symmetrybreaking instabilities is now well established [1,[START_REF] Manneville | Dissipative Structures and Weak Turbulence[END_REF]. They are "universal" insofar as they essentially depend on the symmetries of the physical system and of its bifurcated solutions, but also because they often remain valid, at least at a qualitative level, even far away from the instability threshold [3,4]. However, determining accurately the coefficients of the underlying relevant amplitude equation from experimental data remains a difficult task, especially in these far-from-threshold regimes.

The complex Ginzburg-Landau equation (CGLE), which describes the large-scale modulations of the bifurcated solutions near oscillatory instabilities, is perhaps the most-studied amplitude equation [1]. This privileged situation is due to both its relevance to many experimental situations and to the variety of its dynamical behavior, in particular, its spatiotemporal chaos regimes. One of the landmarks of the CGLE is that it possesses localized "defect" solutions. Even in one space dimension, where no topological constraint exists, numerical simulations of the CGLE [5,6] and analytical [7,8] work have revealed the existence and importance of various amplitude hole solutions, which can often be seen as the "building blocks" of the complex spatiotemporal dynamics observed. In particular, the one-parameter family of traveling hole solutions discovered by Bekki and Nozaki [9] has been shown to play an important dynamical role in a large portion of parameter space including in regions where they are linearly unstable [5]. Similar objects have been identified in various experimental contexts of a priori relevance, e.g., Rayleigh-Bénard convection and coupled wakes [10]. However, to our knowledge, there is still no case where a direct comparison with known solutions of the CGLE could be achieved.

In this Letter, we present a quantitative comparison of localized amplitude holes observed in an experiment with hole solutions of the CGLE, the relevant amplitude equation. We use the observed holes to fully determine the coefficients of the underlying CGLE. This provides clear-cut evidence of Bekki-Nozaki holes in an experi-mental context. Our system is a long, straight, and narrow convection cell in which a thin fluid layer with a free surface is subjected to a horizontal temperature gradient. Hydrothermal nonlinear waves appear via a direct Hopf bifurcation, indicating the relevance of the CGLE. The spatiotemporal dynamics of the waves exhibits localized amplitude holes. The basic scales of the equivalent CGLE are determined using the regular part of the wave trains. Data collected in the vicinity of amplitude holes show that they have the structure of Bekki-Nozaki solutions. This also provides estimates of the remaining coefficients of the CGLE, an approach which, we argue, could be efficient in other experimental contexts. Finally, the overall consistency of our results is checked.

The experimental setup is schematically described in Fig. 1. A layer of fluid [silicon oil of viscosity n 0.65 cS (centistoke) and Prandtl number P 10] of height h is confined between two copper blocks maintained at fixed temperatures T 1 and T 2 by thermostated water circulation, and a bottom glass plate. This forms a straight, narrow channel of length L x 25 cm and width L y 2 cm. As soon as the temperature difference DT T 1 2 T 2 is not zero, a basic flow sets in. It consists of a surface flow towards the cold side with a bottom recirculation. Increasing DT , the basic flow becomes unstable to traveling hydrothermal waves [11] via a supercritical Hopf bifurcation [12]. We observe these waves by low-contrast shadowgraphy, which captures the vertical average of the temperature 0031-9007͞99͞82( 16)͞3252(4)$15.00 © 1999 The American Physical Society gradient variations (surface waves exist, but their effect is negligible). In this geometry, the waves propagate away from a "source region" located arbitrarily on the cold wall, and the end boundaries at x 0, L x act as sinks with no apparent reflection (Fig. 1). For h 1.2 mm, corresponding to the experiment reported below, the source region emits curved waves which become planar further away (Fig. 1, bottom) and propagate along the x axis. Figure 2a presents a typical spatiotemporal evolution as obtained from the acquisition, with a fixed-gain camera, of a single 512-pixel line (of negligible width) along the x axis in the center of the cell. Here, the source appears as a rather ill-defined, erratic object. (Closer to the Hopf bifurcation, steady, regular, evolution is observed.) Fourier analysis of diagrams such as Fig. 2a reveals that on each side of the source only waves propagating away from the source are present and that they are approximately monochromatic (the second harmonic is 2 orders of magnitude smaller). More precisely, restricting ourselves to one side of the source (say x $ 90 on Fig. 2a), we can write the recorded physical variable: where k 0 is the dominant wavelength, v 0 is the basic frequency, and A is a one-dimensional complex field describing the (large-scale) modulations of this wave. (On the other side of the source, one has to change the sign of k 0 .) Using complex demodulation techniques, A can be extracted from the experimental data. Figures 2b and2c show the space-time evolution of jAj and k q 6 k 0 , where q ≠ x arg͑A͒. In these pictures, the localized deformations of the waves visible in Fig. 2a clearly appear as propagating amplitude holes across which the phase gradient varies rapidly. At some space-time points, jAj even vanishes and the phase gradient diverges: A spacetime dislocation occurs (Fig. 2a). The amplitude holes can be seen as the objects mediating the evolution to wave patterns more regular than those emitted by the source.

V ͑x, t͒ A͑x, t͒ exp͓i͑k 0 x 2 v 0 t͔͒ 1 c.c. , (1)
Our system clearly calls for a one-dimensional model. The waves arise via a supercritical Hopf bifurcation. Away from the source, they propagate only in one direction. All of this indicates that the evolution of A could be governed by a single CGLE on each side of the source, even though the regime studied here takes place at a finite distance from threshold (for h 1.2 mm, DT c Ӎ 4.3 K, and thus the relative distance to threshold is ´ 0.19 for DT 5.1 K). We thus suppose that A obeys

t 0 ͑≠ t 2 y g ≠ x ͒A ´A 1 j 2 0 ͑1 1 ia͒≠ xx A 2 g͑1 1 ib͒ jAj 2 A , (2)
where y g is the group velocity of the waves, t 0 and j 0 are the basic time and length scales of the wave modulations, and g is a real number. Below, we estimate, from the data of Fig. 2, all of the coefficients of Eq. ( 2) and check the overall consistency of our hypothesis.

The linear part of the variation of the local frequency v with the local wave number k yields our estimate of the group velocity: y g ≠v͞≠k Ӎ 21.16 mm͞s (Fig. 3a). This is consistent with the average value of the velocity of small perturbations, estimated at 21.15 6 0.25 mm͞s, to be compared to the phase velocity y f v 0 ͞k 0 Ӎ 22.8 mm͞s. This confirms that the source is indeed a source, since perturbations do propagate outward.

Figure 3b shows the variation of jAj with k as determined from the portion of Fig. 2 at the left of the source (x # 90 mm). The maximum amplitude is observed for the basic wave number: k 0 Ӎ 21.11 mm 21 . Space-time points away from the localized amplitude holes correspond to the large jAj (say, jAj . 0.5) portion of the curve. Locally, around these points, the solution of ( 2) is expected to be close to one of the phase-winding solutions of wave vector q k 1 k 0 (see, e.g., [1]):

A A q exp͓i͑qx 2 v q t͔͒ with A 2 q ͑´2 j 2 0 q 2 ͒͞g and v q ͓´b 1 ͑a 2 b͒j 2 0 q 2 ͔͞t 0 2 y g q .

(3) The linear variation of jAj 2 with q 2 is confirmed in Fig. 3c, yielding j 0 ͞ p ´Ӎ 2.53 mm and p ´͞g Ӎ 0.00054 (a.u.). Note that we thus have L x ¿ j 0 ͞ p ´ഠ 3͞k 0 : The cell is effectively "infinite" and the variations of A occur on scales significantly larger than the basic length k 21 0 . Time scale t 0 can be estimated from the real part of the spatial linear growth rate of waves near the source, which is equal to ´͞t 0 y g [4]. From Fig. 3d, we find t 0 ͞´Ӎ 8.5 6 0.5 s, about 4 times the basic period 2p͞v 0 Ӎ 2.03 s, confirming that the variations of A are slow compared to the basic oscillations. Note that t 0 is of the same order as the viscous diffusion time h 2 ͞n 2.2 s.

At this stage, all of the basic scales of Eq. ( 2) have been estimated. To determine the remaining two parameters a and b, global quantities deriving from the "wave part" of the data could, in principle, be sufficient. For example, Fig. 3a could be used to extract the expected variation of v q with q. But the data is too noisy to yield any meaningful estimate of a and b. Moreover, as long as the source is not controlled, the "input" waves cannot be varied at will to explore the family of solutions ( 3), contrary to other experimental situations [4]. We now focus instead on the localized amplitude holes already mentioned.

Many localized, propagating objects connecting two phase-winding solutions have been observed numerically in the one-dimensional CGLE [5]. Analytical methods are largely limited, so far, to solutions depending only on the reduced variable j x 2 y h t, where y h is the (constant) velocity of the object [7,8]. Using this ansatz, the CGLE reduces to a third-order ordinary differential equation (ODE) whose fixed points are the phase-winding solutions (3). Localized objects connecting two such solutions of wave vector q L and q R appear, within the ansatz, as homoclinic (q L q R ) or heteroclinic (q L fi q R ) orbits.

The holes observed in Fig. 2 are not stable structures connecting two infinite phase-winding solutions, but they subsist long enough, and can be sufficiently isolated to reveal that their wings are indeed well described by phasewinding solutions. (As a matter of fact, we repeatedly assimilated, above, the large-amplitude regions separating the holes to portions of these solutions.) Figure 4 shows an isolated hole extracted from Fig. 2. One can measure rather accurately the two wave numbers q L 0.13 and q R 20.32 (in the CGLE frame) connected by the central hole, which can thus be tentatively seen as a heteroclinic orbit in the ODE ansatz.

All sufficiently localized structures on Fig. 2 also connect two different wave numbers. This rules out the homoclinic holes recently studied by van Hecke [6], and leave, as possible candidates, the family of hole solutions found by Bekki and Nozaki [9]. The explicit form of these solutions is too lengthy to be given here (see, e.g., [8]). They form a one-parameter family (at fixed a, b) which can be parametrized by, e.g., the velocity y h of the hole. They take the shape of an exponentially localized amplitude hole with a minimum amplitude jAj min accompanied by a rapid phase shift s.

To compare data such as that of Fig. 4 to these solutions, we need to determine the values of a and b and the "optimal" solution of the corresponding family. We proceed as follows: We estimate q L , q R , and jAj min from the data since we found these were the characteristics of the hole for which the most accurate measurement can be made. We find, for all values of the ͑a, b͒ plane where it exists, the Bekki-Nozaki hole solution with the measured value of q L 1 q R . We then select the (codimension 1) subsets of the ͑a, b͒ plane where, moreover, this hole solution possesses the measured value of q L or the estimated value of jAj min (Fig. 5, dashed lines). These two lines intersect, yielding the desired values of a and b. Taking into account the error bars on q L , q R , and jAj min , we find a 21. same token, the hole solution is also uniquely determined (Fig. 4, dashed lines). Its velocity y h , its width, and its phase shift s are all found consistent with the data. We repeated the procedure for other amplitude holes found in Fig. 2. We found almost the same a and b values, to the accuracy of the estimates. This strengthens the confidence in the results, since different objects moving at different velocities, connecting different wave numbers all yield the same parameter values. We also performed a final check, by plotting, for the estimated values of a and b, the variation of jAj min with the local phase gradient at the "bottom" of the hole along the family of solutions (Fig. 3b, solid line). The agreement with the small-jAj values measured on Fig. 2 is very good. This is an additional indication that all of the low-amplitude points are indeed located "inside" Bekki-Nozaki holes.

For the estimated values of a and b, the CGLE is in the parameter region where the phase-winding solutions (3) are linearly stable for jqj small enough, and no sustained spatiotemporal disorder exists in one space dimension [5]. Moreover, the Bekki-Nozaki amplitude holes are linearly unstable [13]. This is not in contradiction with the dynamics observed in the experiment: the Bekki-Nozaki holes, although unstable, exist, and can constitute important building blocks of even chaotic dynamics [5]. The waves emitted by the source can be locally attracted to this family of unstable fixed points before escaping along its unstable manifold (a mechanism also invoked by van Hecke in [6]). The tendency of the waves trains to become more regular away from the source (see Fig. 2) is consistent with disorder being only transient in the CGLE with the estimated parameter values.

In summary, we have presented experimental results on the dynamics of the nonlinear hydrothermal waves traveling in a laterally heated fluid layer. We have shown that, although the regime studied here is rather far from the onset of waves, the large-scale modulations of the basic pattern are governed by a one-dimensional complex Ginzburg-Landau equation and we estimated its full set of coefficients. This was made possible by showing that the localized amplitude holes observed experimentally correspond to the Bekki-Nozaki hole solutions of the CGLE. The overall consistency of our results was checked. Since the operating regime of the CGLE at the estimated parameter values does not exhibit sustained disorder, it would be interesting to analyze experimental data collected at other parameter values in the hope of reaching spatiotemporal chaos regimes of the type exhibited by the CGLE. This is left for future work, together with an attempt to obtain a better control of the system by forcing the behavior of the source. More generally, we believe that using the localized structures or defects of patternforming systems to determine quantitatively their relevant amplitude equations can be a rewarding approach to this difficult experimental problem.

J. B. thanks the Spanish government for support through Project No. PB95-0578A (DGICYT) and a postdoctoral grant (SEUI, Ministerio de Educación y Ciencia).

FIG. 1 .

 1 FIG. 1. Experimental cell and basic wave pattern. Above: Schematic side view. Below: Top view and instantaneous shadowgraphic image of the wave pattern.

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FIG. 2 .

 2 FIG. 2. Spatiotemporal evolution of the wave pattern at h 1.2 mm and DT 5.1 K (yielding a Marangoni number Ma Ӎ 950). Only a central portion of length 18.5 cm is shown during 100 s. (a) Original data; (b) evolution of the modulus jAj (black: jAj 0; white: jAj 1); and (c) phase gradient k (dark: k , 0; bright: k . 0).

3 .

 3 FIG. 3. Analysis of the data of Fig. 2 situated at the left of the source (x # 90 mm). (a) -(c) For each bin around a given k value, all the corresponding space-time points were determined, and averages calculated on each bin. (a) Local frequency v vs local wave number k; the slope of a linear fit gives the group velocity y g . (b) Local amplitude jAj vs k; solid line: see text. (c) jAj 2 vs q 2 [same data as in (b)]. (d) Two instantaneous profiles of jAj ͑x͒ near the source taken at time 0 and 30 s; inset: ln jAj vs x and linear fit (thin line).

FIG. 4 .

 4 FIG. 4. Amplitude hole. Solid lines: Experimental data extracted from a cut of Fig. 2 at t 60 s. Dashed lines: Corresponding Bekki-Nozaki hole solution of the equivalent CGLE (rescaled variable x 0 x p ´͞j 0 ).

FIG. 5 .

 5 FIG. 5. Parameter plane of the CGLE. STI: spatiotemporal intermittency regions (from [5]). B-F: Benjamin-Feir line below which all solutions (3) are unstable. Dashed lines: see text (grey areas: error bars). Circle: estimated ͑a, b͒ values.