F Daviaud

J Hegseth

P Berge

Subcritical Transition to Turbulence in Plane Couette Flow

Keywords: numbers: 47, 20, k, 47, 15, x

The transition to turbulence in plane Couette flow was studied experimentally.

The subcritical aspect of this transition is revealed by the stable coexistence of laminar and turbulent domains. By perturbing the flow, a critical Reynolds number has been determined, above which an artificially triggered turbulent spot can persist. The study of the spatiotemporal evolution of these spots shows, among other things, the existence of waves traveling away from the turbulent regions.

for which the systems experience an abrupt and often lo- calized transition from a regular state to a highly disor- dered one [START_REF] See | Physical Fluid Dynamics[END_REF]. Following an analogy with bifurcation theory, the former can be called supercritical and the latter subcritical [START_REF] Pomeau | [END_REF]. Many examples of the two types of transition to turbulence can be found in the literature. Circular Couette flow with the outer cylinder at rest or with the two cylinders corotating [3] is a good example of a system showing a supercritical transition, while coun- terrotating circular Couette flow [4] or plane Poiseuille flow [5] are known to exhibit turbulent spots coexisting with the otherwise laminar flow. This metastability of two statesone regular (laminar) and one irregular (turbulent)is typical of a subcritical bifurcation. How- ever, very few experiments involving plane Couette flow have been performed, though it appears to be one of the simplest flows likely to display a subcritical transition to turbulence.

Plane Couette flow (PCF) corresponds to a linear ve- locity profile which results from the relative motion of two infinite parallel walls at a given distance apart. It is an exact solution of the governing nonlinear equations and it is believed to remain linearly stable at all R [START_REF] See | Hydrodynamic Stability[END_REF] (R =Uh/ v, where U is the speed of either wall, h =d/2 is half the gap, and v is the kinematic viscosity). This theoretical result suggests that instability in PCF has a nonlinear origin and can be generated by finite-amplitude perturbations.

Orszag and Kells have shown numerically

[7] that only three-dimensional disturbances can drive a transition to turbulence at Reynolds numbers larger than R=1000 (with our definition of R) and Nagata [8] has discovered the existence of 3D finite-amplitude solutions of PCF for R & 125, which provides a lower bound for in- stability and turbulence. Recently, Lerner and Knobloch [9] and Dubrulle and Zahn [10] have studied the influence of a small defect on the onset of instability for inviscid and viscous PCF, respectively. The development of turbulent spots in PCF was also studied by means of direct numerical simulations by Lundbladh and 3ohans- son [11], and a critical Reynolds number R=375 was found, above which localized turbulent regions can persist. Facing these theoretical and numerical works, only a few experimental studies of transition in PCF have been carried out, probably due to the experimental difficulties encountered when running and making measurements in a PCF apparatus. Reichardt [12] has observed tur- bulence for R & 750 and Leutheusser and Chu [13] for R & 280. More recently, Tillmark and Alfredsson have reported the existence of a transitional Reynolds number R=360, which is the lowest R for which a transition to turbulence occurs if a strong enough disturbance is ap- plied [14]. In this Letter, we report experimental results

concerning the transition to turbulence in a PCF. By per- turbing the flow, we have both determined a critical Reynolds number above which turbulence persists and characterized the transition. We have also studied the spatiotemporal evolution of the turbulent spots, which ex- hibits the existence of waves traveling away from the tur- bulent regions.

Several different geometries have been tried in attempts to make PCF [12][13][14][15][16]. Our system utilizes a geometry which results in no mean flow, i.e. , the walls move in opposite directions with the same velocity [START_REF] Hegseth | Ordered and Turbulent Pattern in Taylor-Couette Flow[END_REF].

In this geometry the two vertically oriented parallel walls move in a horizontal direction. We define these walls by using an endless transparent plastic film belt (363.0 cm long, 25.4 cm wide, and 0.16 mm thick), two pairs of small rotating Plexiglas cylinders, and two large rotating

Plexiglas cylinders as shown schematically in Fig. 1. The endless belt is driven, through friction, by one of the two large cylinders (12.00 cm in diameter) which are placed at opposite ends of the system. The cylinder which drives the belt is mechanically coupled to an ac motor which drives the system. The two "infinite" parallel planes are defined by guiding the endless belt through two pairs of smaller cylinders 7.0 mm apart, all of which are parallel to the large cylinders as shown in Fig. 1 ~The belt is also guided by two glass plates, one of which is black and the other transparent, defining a test section of 1 m. Using h =d/2 as a length scale, the transverse aspect ratio I, (dimensionless width of the channel) is I, = 70 and the longitudinal aspect ratio is (dimensionless length of the 1992 The American Physical Society 2511 channel) I, = 285. We have chosen to use a transparent belt so that optical methods in data acquisition, such as laser Doppler velocimetry (LDV) and flow visualization with image processing, may be used. The entire assembly is placed in a glass tank which is filled with water, our working fluid, which has a viscosity of v=0.01 stoke at 293 K.

We visualize the laminar and turbulent regions by seeding the flow with a dilute solution of Merck Iriodin 100 Silver Pearl which consists of thin and flat reflective mica platelets (0.1 pm thick and 10 ium across) which align, on average, with the stream planes of the flow. They also respond very quickly to any local change in the flow field, giving an almost instantaneous change in the light reflectance field whenever there is a change in the velocity field. This makes Iriodin an excellent indicator of turbulence on all scales [START_REF] Schwarz | [END_REF]. When the flow is seeded with Iriodin, turbulence is indicated by a relatively rapid fluctuation in the reflected light field whereas in the lami- nar flow the light reflectance is steady.

The light reflectance field may then be recorded using a video camera and subsequently digitized and analyzed by image processing. We have also visualized the velocity field in two dimensions by seeding the flow with a white ceramic powder (Pyroceram 7575, 100 Mesh) and illuminating the flow with an argon laser light sheet in a vertical or horizontal plane. The light scattered by the particles is reflected by a mirror into a camera for recording. In this manner, we have checked that the velocity profile is linear for R up to 300 by following the particles in the stream- wise cross section [START_REF] Hegseth | Ordered and Turbulent Pattern in Taylor-Couette Flow[END_REF].

Artificially triggered disturbances are made by a tur- bulent jet which is injected into the laminar flow from a hole machined in the dark glass plate. This injection is controlled by an electrovalve that is activated when the 2-mm hole in the plastic belt is aligned with the hole of the plate. The amplitude of the disturbance can also be controlled by adjusting the pressure of the jet.

The system has first been studied without being artificially disturbed, in order to study the intrinsic noise of the experiment. For R (400 the turbulence is local- ized at the entrances of the test section and does not in- vade the laminar flow during observation times of more than 30 min. For R) 400, turbulence invades the sys- tem under the form of two fronts coming from each side of the test section and contaminating the laminar flow. The size of the part that is contaminated by turbulence increases with R while the time of invasion decreases with R. For R=450 and after 5 min, all of the test section has been invaded [19].

Turbulent spots were initiated at given values of R, as described above, and their evolution was recorded in the (x,z) plane (x and z refer to the streamwise and span- wise directions, respectively). As expected, they are not advected with a mean velocity, but remain in the test section of the experiment. Moreover, as they are created in the center of the belt, they appear as quasisymmetrical ellipsoids, their shape being stretched in the streamwise direction (cf. Fig. 2).

We have studied the evolution of the spots with R and determined the minimum R, for the subcritical transition to turbulence.

For R & R, = 370+ 10, the turbulent spots generated in the laminar flow relax in short times [cf. Fig. 2(a)], while for R ~R" they are self-sustaining for arbitrarily long times [cf. Fig. 2(b)]. The mean relax- ation time r of the spots was measured by repeating the procedure a large number of times for each value of R. The evolution of r with R exhibits a divergence of r near R, (cf. Fig. 3). Moreover, the fluctuations of the relaxa- tion times around the mean value increase when approaching R,. The transition thus appears to have some characteristics of critical phenomena [START_REF] See | Phase Transitions and Critical Phenomena[END_REF] and work is in progress to characterize it quantitatively [19].

The influence of the amplitude of the perturbation on the transition has been studied by adjusting the velocity of the turbulent jet that creates the perturbation. This amplitude does not seem to significantly affect the thresh- old R"but it does, however, influence the aspect of the transition: The mean relaxation time increases with the amplitude of the perturbation, especially near R" [19].

When R & R"the turbulent spots first spread very quickly (spreading rate -0.3h sec ') and saturate to a macroscopic size which depends on the value of R. Then, they have a slow spatial and temporal evolution, but occupy the same mean space. Their position and shape FIG. 2. Evolution of a turbulent spot after a perturbation; the time interval between photographs is 5 sec. (a), (b), (c) R 350; (d), (e),(f) R =380. evolve more slowly than the time for fluctuations of light reflectance within the turbulent regions. Successive im- ages have been digitized in the (x,z) plane and a rms cal- culation shows that the maximum light fluctuations correspond to the turbulent spot. We have also observed vortical structures in the spanwise (y, z) plane [19].

Two types of regions (laminar and turbulent) can easi- ly be distinguished from an experimental viewpoint. Turbulence is seen to invade the laminar flow by a process of contamination of the regions adjacent to the spot. The edges of the turbulent patches are usually either curved or straight and inclined to the horizontal. Throughout the turbulent region horizontal streaks can be seen, and these streaks propagate outward from the spot in the vertical (spanwise) direction. The spatiotemporal evolu- tion of a spot and the existence of traveling waves are displayed in the diagram of Fig. 4. This diagram is ob- tained by digitizing the image along a vertical line (z) sit- uated in the middle of the test section, and plotting the evolution of the intensity along this line as a function of time. The wavelength of these waves is -2h in the streamwise direction and their phase velocity is -2h sec ' near R,. They could play an important role in the propagation of turbulence, particularly in the spanwise direction.

As R is further increased, the spots can move, grow, split into two spots, or decay and a regime reminiscent of spatiotemporal intermittency is observed [START_REF] Daviaud | [END_REF]. This re- gime will be characterized elsewhere, but the mean fraction of total space which is turbulent can be defined and it appears to increase with R (at R=450, nearly all the test section is turbulent). Near R =400, the turbulence 200-I le . may take a form that is similar to the spots that are ob- served in plane Poiseuille flow [5] or in the Blasius boundary layer [22]. They can also form straight in- clined patches across the (x,z) plane, a shape similar to that of the spiral turbulence observed in circular Couette flow [4].

A PCF apparatus has been constructed in which we have observed the subcritical transition to turbulence at R =R"byperturbing the flow. Below R"the laminar state is stable to finite-amplitude perturbations, and above R"the perturbations do not relax and a regime of intermittent turbulence is observed. Although many aspects are still to be investigated, the transition appears to be a critical phenomenon.

After submitting this paper, we became aware of similar results obtained by Tillmark

and Alfredsson [23]. By perturbing a PCF by different means (air bubble situated at the bottom), they have found a value of R, very close to our measurements.

They have also noticed the creation of waves at the wing tips of the spots, which appear to be similar to those displayed in Fig. 4. Our experimental results are also in very good agreement with the numerical results obtained by Lundbladh and Johansson [11],especially for the criti- cal Reynolds number R, . However, the existence of waves was not predicted by their simulations nor was the intermittent regime which is observed above the thresh- old. Work is currently under progress in this direction.

FIG. 1 .

 1 FIG. 1. Schematic drawing of the experimental apparatus in a horizontal plane. (a) Large cylinders, (b) tank, (c) water, (d) clear plastic belt, (e) small cylinders, and (f) glass plates.

FIG. 3 .

 3 FIG.3. Evolution of the relaxation time r of the artificially triggered turbulent spots as a function of the Reynolds number R.

We wish to thank B. Bonetti, G. Balzer, P, Manneville, and Y. Pomeau for stimulating discussions and M. La- bouise, P, Hede, and B. Ozenda for their technical assis- tance.


F Daviaud

A Burnol

Ronsin

Travelling and Standing Waves in a Spatially Forced 2D Convection Experiment

Keywords: . 47.20 -Hydrodynamic stability and instability. . 47.25 -Turbulent flows, convection, and heat transfer

Rayleigh-BBnard convection is studied in a rectangular geometry with a spatial forcing induced in one direction by electric wires. When using fluids of relatively large Prandtl numbers, this forcing allows the existence of a perfect one-dimensional pattern until the onset of bimodal convection. The transition to bidimensional convection is studied for increasing Rayleigh number and reveals the existence of different spatio-temporal regimes depending on the value of the forcing. At the onset of the transition, a stationary pattern is observed for weak forcing, while travelling waves are evidenced for strong forcing. Both behaviours give place to collective oscillations at higher Rayleigh number.

In the past few years, many studies both experimental and theoretical have been devoted to the problem of the evolution toward turbulent states of convective systems with large aspect ratios [l-41. However, these systems exhibit defects and display imperfect patterns when the Rayleigh number is increased above the onset of convection. The resulting spatiotemporal regimes are generally very complex and difficult to analyse, except in some particular cases (see, e.g., [5,6]). On the other hand, the study of one-dimensional periodic systems has recently provided a new approach to the problem of transition to turbulence in weakly confined systems. In these systems, one dimension is larger than the others so that the physical quantities only depend on one space parameter plus time. Related experiments involving Rayleigh-B6nard slot convection [7] and theoretical studies of partial differential equations [8] and coupled map chains [9,10] have revealed a transition to spatio-temporal chaos through spatio-temporal intermittency. In order to increase the number of space parameters while keeping a simple spatial structure, interest has also concentrated on the dynamical regimes of two-dimensional (2D) periodic patterns which destabilize when a control parameter is varied. Experiments involving parametrically forced surface waves [ll], convection in binary mixtures [12] and in nematic liquid crystals [131 have revealed a richness of pattern formation and dynamical behaviours.

We present here an experimental study of Rayleigh-Benard convection in a rectangular geometry with a spatial forcing which allows the existence of a perfect 2D cellular pattern.

We fist describe the experimental set-up and the data acquisition techniques. The different spatio-temporal regimes which are observed when the Rayleigh number Ra is increased above the threshold of bimodal convection are then described. We finally analyse and discuss these results.

The experiment was performed in a rectangular cell filled with silicon oil of Prandtl number 30 at 293 K. The vertical side walls were made of plexiglass (d = 5.00 mm high), while the horizontal plates were made of glass to allow visualization of the convective structures. The rectangular cell had a longitudinal aspect ratio T,=L,ld=28 and a transverse aspect ratio I', = L,/d = 14. The temperature difference across the fluid layer was kept constant to within 0.01 K by means of circulating thermal regulated water. After each temperature increase, dynamical equilibrium was awaited for as long as 24 hours up to 48 hours before each new set of measurement. This time is to be compared with the horizontal diffusion time (zH = LE/K= 20 h, with K the thermal diffusivity).

The spatial forcing was achieved by electric wires regularly disposed parallel to the small side of the cell (y-axis) and near the horizontal hot plate (cf. fig. 1). The distance between them was fixed to e = 2 cm = 2A,, with A, the critical wavelength, at the onset of convection (Ra = Ra,). This gap was chosen because it corresponds to the natural wavelength of the system at the threshold of bimodal convection (Ra = RaII), when locally measured in the absence of forcing. Experiments realized with a gap e = A, have indeed revealed a mismatch between the forcing and the preferred pattern leading to spatial defects [14]. The tension U, applied to the wires, and thus the forcing, could be varied.

The convective structures were visualized by shadowgraphic imaging. A parallel light beam crossed the cell vertically through the horizontal glass plates. The beam deflection was thus integrated along the cell depth and a top view of the convective pattern was obtained (cf. fig. 3). The spatio-temporal evolution of the convective pattern was recorded using a video camera. The image was digitized in a matrix of 512 x 512 pixels with a grey scale of 256 levels (see ref. [7] for details). The data acquisition was generally performed along a line of 512 pixels parallel to the y-axis and the acquisition frequency was varied according to the natural frequency of the system. The transition from one-dimensional (1D) convection above Ra, to two-dimensional (2D) convection above RaII is studied as a function of the two control parameters U. and E = (Ra -RaJRa,. The results are reported on the stability diagram of fig. 2. Bimodal convection appears under the form of second rolls, which start to grow at Ra = RaII and are oriented at right angles with the basic roll pattern. This phenomenon is caused by the instability of the thermal boundary layers and can only be observed in fluids with a relatively large Prandtl number (see ref. [l] for details). First of all, in the absence of spatial forcing (Uo = 0), the 1D convective structure displays defects and instabilities such as the cross-roll and the skewed-varicose instability before the threshold of 2D convection. The transition to bimodal convection occurs for E = 14 (the exact value depends on the local wavelength) but the pattern is inhomogeneous and unstable. These observations are in agreement with the results of Busse [l] and Krishnamurti [2] for the corresponding Prandtl number. As E is increased further, the two-dimensional pattern destabilizes and spokes are observed (cf. ref. [l]). In fact, even when U. = 0, the electric wires slightly force the pattern along the x-axis and the destabilization first occurs along the y-axis (cf. fig. 3a)). On the contrary, the presence of forcing (Uo # 0 V) allows the existence of a stable and stationary one-dimensional pattern (domain labelled by 1DS in fig. 2) until the onset of bimodal convection E = qI. The wavelength A, of this first set of rolls is fixed by the electric wires and does not change with E , at least until E = 35 for U. = 5 V (A, = 2A,). A similar result can be obtained, but only near the onset of convection, by using electric wires at the lateral boundaries [XI. For E > E ~~, a second set of rolls appears and a quasi-perfect 2D pattern is observed (cf. fig. 3b)) but its temporal evolution strongly depends on the value of the forcing. As is shown on the stability diagram of fig. 2, a stationary pattern is observed for a weak forcing (Uo < 6 V, domain labelled by 2DS), while travelling waves are evidenced for a strong forcing (Uo > 6 V, domain labelled by 2DTW). However, in both cases, the pattern is stable over periods of several horizontal diffusion times zH and no defects are observed. The threshold of bimodal convection sII slightly increases with the forcing Uo, whereas the wavelength A, of the second set of rolls, which are parallel to the large side of the cell (xaxis), decreases when U. or E increases (A, = 0.78AC for U. = 4 V and E = 20). The transition from 1DS to 2DS or 2DTW is well defined and no hysteresis has been observed. When E is increased further, both behaviours give place to collective oscillations of the second set of rolls (domain labelled by 2DCO). However, the transition from 2DS or 2DTW to 2DC0 is imperfect and hysteretic, and mixed states of two different dynamical regimes can be observed. In the following, we describe the different spatio-temporal regimes that are observed, depending on the forcing.

In the case of a weak forcing and above cII, the 2D pattern is first stationary for cII < E < €111 and then displays a collective oscillating mode for E > qII. The phase of the rolls of the second set oscillates along the direction of the first set of rolls (y-axis) around a mean position with a maximum amplitude A,/3. Hot streams and cold streams oscillate in phase opposition and the frequencyfis the same in all the container. This frequency slightly increases with E as EO.^ (f = 0.03 Hz, i.e. T = l/f= 0.08d2/~ for E = 26 and U. = 5 V). The values of the oscillation frequencies f are in agreement with the results obtained by Busse and Clever [l] and Krishnamurti [2], who have also observed an increase offwith E , however with a different exponent (f-P). This regime is represented on the spatio-temporal diagram of fig. 4a). The data acquisition has been performed over 512 pixels along a vertical line parallel to the y-axis, between two electric wires and near a cold stream of the first set of rolls. Thus, only the cold streams (bright lines) of the second set of oscillating rolls can be seen in fig. 4a), but the behaviour of the hot streams is similar. A phase delay can be noticed between the middle of the container and the edges. The behaviour is the same between each couple of electric wires and so there is no x-dependence in the amplitude of the oscillations.

Only a phase delay has been observed along the x-axis. This regime is similar to the optical mode which had been observed in a one-dimensional chain of convective rolls [16]. However, the threshold qII is not so well defined here because of hysteresis and inhomogeneous oscillating regimes can be observed. In fact, the amplitude of oscillation depends on E and on the local wavelength A, . The appearance of a new pair of rolls in the y-direction, e.g. when E is increased, can locally stop the oscillations [17]. Such a behaviour can be seen in fig. 2 for Uo=4V: when E is varied from ~= 2 7 to ~= 2 8 , the 2DC0 regime (represented by a triangle) corresponding to A, = 0.78Ac is replaced by a stationary regime (represented by a cross) corresponding to A, = 0.70Ac. On the contrary, a 2DC0 regime obtained for U. = 5 V by increasing E up to 30 can remain stable down to E = 27 when E is decreased. In the case of a strong forcing and for E > eII, bimodal convection appears as a second set of rolls which propagate along the first set, parallel to the electric wires. More precisely, this regime consists of two waves coming from the large sides of the container (sources) and going to the centre (sink) as is shown in fig. 4b). The sink is situated near the centre when the two waves the same amplitude but can move near one edge (e.g., the right edge) when the amplitude of one wave (the left wave) is greater than the other. This regime is homogeneous in all the container and the phase velocity w of these waves slightly increases with E but does not decrease to zero near the threshold. The corresponding frequency is similar to that observed for the oscillating regime at the same E value, except for a small increase due to Uo. The threshold qI is well-defined and no hysteretic phenomenon has been observed. On the contrary, when E is increasing, the transition between 2DTW and 2DC0 is complex and mixed states showing simultaneously both propagation and oscillation can be present (cf. fig. 4c)). The same phenomenon occurs when U, is varied and E kept constant along this transition line.

In both cases, only collective oscillations were observed above ~= 3 5 . The dynamical behaviour then becomes more complex for E > 40. The convective structure displays intermittent oscillations, whose amplitude is a function of space and time, before showing spokes structures and finally entering spatio-temporal chaos.

This study shows that Rayleigh-Bknard convection in a rectangular geometry with a spatial forcing allows the existence of perfect and stable 1D and 2D periodic patterns. The similarity between the structure of bimodal convection and a 2D crystal lattice is striking and leads to study its stability ws. the two control parameters. New spatio-temporal regimes have been evidenced. The oscillating mode, which corresponds to a phase oscillation, can be compared with the regime observed by Busse and Whitehead [l], though their experimental procedure is different and only allows the existence of transients. In fact, the patterns studied in ref.

[ 11 are created by controlled initial conditions (illumination of a grid over in an otherwise normal convection container) and are not stable over long time periods. The oscillating bimodal cells thus cannot be generated in a strictly homogeneous pattern. Busse and Clever have performed a stability analysis of the fully 3D nonlinear problem of bimodal convection [l]. Their numerical results explain the collective oscillation of the bimodal cells in terms of an oscillatory instability and the destabilization of the bimodal pattern to spokes in terms of a knot instability. In the case of a weak forcing, our experimental results seem to agree, at least qualitatively, with this approach. However, the frequencies obtained in their calculations (for a Prandtl number 7) are inferior to our experimental values.

The results concerning the propagation of convective rolls are to our knowledge unknown in Rayleigh-Bhard convection with pure fluids at high Prandtl numbers. They remind us of the results obtained in the study of binary fluid mixtures [18] and oscillatory convection in low Prandtl number fluids [19], though the physical mechanism is different. In fact, this regime can be interpreted within the frame of amplitude equations and work is currently under progress in this direction [17]. By means of complex demodulation techniques (cf. ref.

[19]), we have measured the amplitude profiles of the waves observed in the different spatio-temporal regimes. Figure 5 displays the profiles of the left-going and right-going waves corresponding to the travelling waves shown in fig. 4b). The amplitude of the leftgoing wave is large in the left part of the y-axis (source on the left) and decreases down to the sink after which it is dominated by the right-going wave. A phenomenological model based on two coupled Swift-Hohenberg equations is also being developed and the first results look promising [20]. However, the exact role of a local forcing on convection is still unknown and more theoretical work is necessary to explain, for instance, the propagations of rolls.

In conclusion, 2D forced convection exhibits a rich variety of spatial and dynamical behaviours. It shows both stationary and propagative phenomena at the onset of bimodal convection, depending on the values of the control parameters. The possibility to have a perfect 2D periodic oscillating pattern seems also very interesting to study the transition to spatio-temporal chaos.

* * *

Fig. 1 .Fig. 2 .

 12 Fig. 1.

Fig. 3 .

 3 Fig. 3. -Shadowgraphic pictures of the convective structure; a) U0 = 0 V and E = 25, b) U0 = 5 V and E = 25.

Fig. 4 .

 4 Fig. 4. -Space-times evolution of the convective pattern; spatial digitization is made over 512 pixels along a vertical line parallel to an electric wire and the light intensity is plotted using a grey scale with 256 levels. "he total observation time is 512 seconds; a) 2DCO regime, b) ZDTW regime, c ) mixed state.

Fig. 5 .

 5 Fig. 5. -Wave amplitude profiles A corresponding to the travelling-wave state of fig. 4b).

We wish to acknowledge G. BALZER, P. BERGE, V. CROQUETTE, M. DUBOIS, and C. PEREZ-GARCIA for fruitful discussions, and P. HEDE and B. OZENDA for their technical assistance.


F Daviaud

J Legab

P Berg&

P Coulletb

M Dubois

Spatio-temporal intermittency in a 1D convective pattern: theoretical model and experiments

We describe the occurrence of spatio-temporal intermittency in a one-dimensional convective system that first shows time-dependent patterns.

We recall experimental results and propose a model based on the normal form description of a secondary Hopf bifurcation of a stationary periodic structure. Numerical simulations of this model show spatio-temporal intermittent behaviors, which we characterize briefly and compare to those given by the experiment.

Introduction

Rayleigh-BCnard experiments in both channel and annulus of small transverse aspect ratio [l-3] have shown transitions from laminar states toward spatio-temporal intermittency (ST11 [4]. The latter is described as a regime resulting from the competition between two metastable states, namely one fully turbulent and one completely laminar, and shows both spatially and temporally localized turbulent patches that appear in laminar domains. It has been observed in Kuramoto-Sivashinsky type equations [4] as well as in coupled maps [5], and widely analyzed [START_REF] Chat& | For a review[END_REF]. So far, there is no clear understanding of the link between this kind of behavior and experimental results, but still they seem analogous. On the other hand, it has been early recognized [71 that normal forms or amplitude equations [S--101 give good descriptions of behaviors above first bifurcation thresholds, and, in some particular cases, they can be computed from "microscopic" equations [g-12]. The aim of this paper is to shed some light on a possible link between ST1 and experiments by giving a normal form description of the destabilization of a stationary periodic pattern, and by showing it may give rise to spatio-temporal intermittent behaviors. In the first section, we recall the main experimental results about ST1 in a convective pattern. Section 3 is devoted to the derivation of the model equations and their numerical simulations. Finally, we conclude with a short discussion of our results.

Summary of experimental results

In the following, we summarize experimental results on Rayleigh-BCnard convection in quasi onedimensional (1D) systems. Detailed studies in both rectangular and annular cells have been reported elsewhere [2,3], and we shall focus on the dynamical regimes observed in an annulus (periodic boundary 0167-2789/92/$05.00 0 1992 -El sevier Science Publishers B.V. All rights reserved conditions). When increasing the Rayleigh number, the periodic stationary convective pattern first exhibits oscillatory regimes and propagation of spatial defects. Then, a transition to turbulence via spatio-temporal intermittency (ST11 occurs, which corresponds to a mixing of turbulent patches within laminar domains.

Experimental setup

We have studied Rayleigh-Btnard convection in an annular cell with a narrow gap. The latter was filled with silicon oil of Prandtl number 7. The vertical and horizontal walls were respectively made of Plexiglass and sapphire. The transverse aspect ratio was r, =,0.29, which is small enough to get a 1D geometry, and the circumferential aspect ratio was r, = 35. The temperature difference across the fluid layer was controlled at a precision of 0.01 K, by means of thermally regulated circulating water. After each temperature increase, dynamical equilibrium was awaited before making new data acquisition.

The convective structures were visualized by shadowgraphic imaging and their spatio-temporal evolution was recorded using a video camera. The image was digitized along a circle of 1024 pixels, giving a resolution of 2.5 pixels per wavelength (at the onset of STI, 40 wavelengths are present in the cell). The acquisition frequency was chosen between 1 image every 10 seconds and 10 images per second, in order to obtain information on the long-time evolution of the pattern, or on the exact dynamics of a turbulent event. Time series of several thousands of T,, were needed to perform statistics, where T, = 1 s is the basic period of the oscillating rolls.

Regimes observed

When Ra exceeds a critical value that depends on r,, perfect stationary patterns of rolls are observed; the rolls axis is then perpendicular to the longer side of the container (domain labeled by S in fig. 11. The number of convective rolls increases with the Rayleigh number, making the wavelength of the pattern much smaller than that observed in large containers. Moreover, a stable spatial inhomogeneity of the local wavelength is generally observed. This phenomenon, which is characterized by the coexistence of small and large cells, has also been observed in rectangular geometry, and a precise study is currently in progress [13]. However, near the onset of STI, the pattern is homogeneous, i.e. the distribution of wavelengths shows a sharp peak at A, = 0.44h,.

Above a given value of the control parameter E = Ra/Ra, -1, the pattern becomes time dependent through the appearance of vacillations (domain D in fig. 1). The threshold value for this phenomenon depends on r, and on the present wavelength A,,. This regime, which has been described in detail in ref. [14], consists of oscillations of the hot and cold streams about their mean position (cf. fig. 2a1, cold and hot streams being out of phase. Near threshold (E = 200), the oscillations are monoperiodic, with a frequency f = 0.5 Hz. When E is increased further, a bi-periodic regime is observed. The oscillations strongly depend on the spatial structure of the pattern. More precisely, some parts of the cell oscillate and others do not, depending on the local wavelength. When the pattern is homogeneous, a quasi-perfect regime of oscillations can be observed, with the same frequency all over the cell. However, nodes of oscillations are present. In the monoperiodic regime, they correspond to points at rest, and are intermittency in a ID conoective partern associated with a standing wave for the amplitude of oscillations [14]. The bi-periodic regime corresponds to a periodic displacement of these nodes.

At a larger value of E, new events leading to a spatial symmetry breaking occur. They are characterized by the propagation of defects (or solitary waves, cf. refs. [2,31). Then, above aa = 450, the interaction of these waves with the oscillators destabilizes the spatial structure, and turbulent patches are observed: the convective pattern enters the regime of STI. The pattern shows a mixture of organized laminar domains and incoherent turbulent domains (defined later), that are fluctuating in space and time.

At the onset of ST1 (cf. domain labeled by pre-ST1 in fig. 0, turbulent domains of small size seem to appear spontaneously. In the annular geometry, they are localized in space and time, and can propagate through the container with a constant velocity (0.8 h,,/T,, where A, is the mean wavelength). They may also locally break the spatial pattern of rolls. In the latter situation (see the spatio-temporal diagram of fig. 2b), a turbulent zone can spread by contaminating the next cells, or relax to a laminar state. Above a certain threshold (Ed = 480), turbulent domains never relax. We then have a regime of sustained intermittency (cf. domain labeled by ST1 in fig. 1). A typical spatio-temporal diagram is shown in fig. 2c.

Statistical measurements

As is well evidenced in fig. 2, turbulent patches have no spatial coherence, and a chaotic time evolution. On the contrary, laminar domains show a regular behavior in space and time; the periodicity of the pattern is preserved, and the shadowgraphic intensity at a given point is either stationary or periodic in time. A spatial criterion was taken to decide if locally the behavior is laminar or turbulent. If the local wavelength (i.e. the size of the cells) is such that A, -AA < A < A, + AA, where A, is the basic wavelength of the pattern and AA a tolerance factor, the behavior is said to be laminar; otherwise it is turbulent. Provided +A, < AA < $A,, AA has no major incidence on the statistics calculated from this binary reduction. Calculations performed (see ref. [3]) with a criterion based on the local temporal evolution gave similar results. In this case, the criterion was as follows: a cell i is said to be laminar if the shadowgraphic intensity Z(i, t) satisfies IZ(i, t + 1) -Z(i, t>l I 6, where 6 is a cutoff value properly chosen, i.e. such that no major change occurs in the results when 6 is varied.

A global characterization of the regime is given by the evolution of the turbulent fraction F,: F, is calculated as the averaged total length occupied by the turbulent cells, divided by the length of the container. The results show an increase of F, with E, but the transition looks imperfect (cf. fig. 3). This imperfection may be due to the annular geometry (for more detail and comparison with a rectangular cell, see ref. [3]), or come from the way measurements are made. Note that, in rectangular geometry, the dependence of F, on E gives a well-defined transition: Ft -(E -E,)~ with p = 0.3 [3] (the full curve in fig. 3 corresponds to a fit of the experimental points with @ = 0.3). Nevertheless, a power law decay is observed in the annulus at the threshold value E = Ed, for the distributions of the number N(L) of laminar domains of length L, and of the number N(r) of laminar domains lasting T. Above threshold, these distributions become exponential, defining characteristic length and time which diverge like (E -520)-l'* (cf. fig. 4).

The experiments reveal the same qualitative route to turbulence via ST1 as in the numerical simulations of some partial differential equations like the Kuramoto-Sivashinsky equation [hl, coupled map lattices [5] and directed percolation [15], and most of the characteristic properties observed in the critical region around the threshold of ST1 are similar [2, 31. Nevertheless, there is no clear understanding of the link between experimental and numerical results. Here, a starting point is that the experiment described above shows that ST1 occurs after the appearance of temporal regimes. Besides, the perfect vacillation pattern results from a Hopf bifurcation of the periodic 1D structure. Therefore, an approach to this problem consists in the study of the secondary Hopf bifurcation of a 1D pattern.

In the next section, we show that spatio-temporal intermittent behaviors may be related to a phase instability of the order parameters associated with this secondary bifurcation.

Description through model equations

We consider a one-dimensional periodic pattern that undergoes a secondary Hopf bifurcation, thus leading to an oscillatory regime. We first characterize this bifurcation in order to get vacillations above threshold, and then describe the resulting dynamics by means of amplitude equations.

Our intent is to show that, depending on external parameters, the latter equations model localized destruction of the vacillation pattern.

More precisely, we shall show that a phase instability of the vacillation solution provides a mechanism for spatio-temporal intermittent-like behaviors in the system.

Description of vacillations

Vacillations result from the growth of an oscillatory mode that adds to the basic periodic structure, i.e. to stationary rolls. This mode is such that convective extrema display periodic oscillations for which maxima and minima are rr out-of-phase. This corresponds to oscillations of the second harmonic of a mode of same period as the basic pattern, but of opposite parity. Indeed, a mode of same period and same parity as the basic pattern would lead to beatings in the amplitude of convection, but not to periodic displacements of extrema, whereas a mode with same period and opposite parity would result in in-phase displacements of minima and extrema.

Model equations

We now analyze the Hopf bifurcation of a parity-invariant periodic pattern, for which the growing mode above threshold has a parity opposite to that of the initial pattern. In other words, we consider a 1D periodic pattern, described by U = U&X + cp), where U,(x) is an even function, and cp stands for the phase shift due to an arbitrary choice of the origin of space coordinates. This degree of freedom is reminiscent of the space translation invariance of the system before the first bifurcation, namely the one giving rise to the periodic pattern U,. We then assume that this pattern undergoes a Hopf bifurcation characterized by a temporal frequency o. Near this secondary bifurcation threshold, the quantity U can be written in the form [16, 171

U(x,t,cp,A) =Uo(x+'p)
+exp(i,t)A(x,t)5(x+cp) +c.c.+ . . . .

where C(x + cp) is the most unstable mode at threshold and is periodic in X, A is the complex amplitude associated with the Hopf bifurcation, c.c denotes complex conjugate, and the dots stand for higher order corrective terms. Since we want vacillations to occur above threshold, we assume that l has the same period as the basic pattern (which is in particular the case for a second harmonic), and an opposite parity. Thus, U depends on two quantities, namely cp and A, which are both assumed to vary slowly in space and time. The fact that cp, which can be thought of as the "phase" of U,, is now allowed to depend on space or time is crucial for the following, and simply comes from the fact that U, is marginal with respect to homogeneous "phase" perturbations, and therefore, its large scale "phase" variations should be taken into account in this description.

The dynamics of U is then ruled by that of 9 and A, which obey two coupled equations, whose form can be deduced by means of symmetry arguments. Namely, since U, is of given parity, i.e. since the system before the second bifurcation exhibits a reflection symmetry, then if U(x, t, cp, A) is a solution, UC-x, t, -cp, -A) should also be an admissible solution. The negative sign in front of A in the latter expression is due to the odd parity of 5. Therefore, these equations should be invariant under the transformation x-+ -x, (p-+ -cp and A+ -A.

Another symmetry comes from the invariance of the system under time translations before the second bifurcation. Then, if U(x, t, cp, A) is a solution, U(x, t + T, cp, A) should also be a solution, and therefore the evolution equations must be left invariant by the following transformation: Here, p measures the distance from the Hopf bifurcation threshold, (Y and p express dispersion and nonlinear renormalization of the temporal frequency, y, 6, 5 and n are coupling coefficients, and K is the diffusion coefficient of the phase cp. Some prefactors have been set to one after scaling the space coordinate and A. The other coefficients are of order 1 a priori. The first part of eq. (la) is the classical Ginzburg-Landau equation associated with a Hopf bifurcation [START_REF] Kuramoto | Chemical Oscillations, Waves and turbulence[END_REF]. The coupling with the phase of the basic pattern U, simply means that the growth rate of A (cf. the term proportional to -y> and its temporal frequency (term in S) are modified if the basic pattern is compressed or streched (i.e. if its phase gradient @/ax is negative or positive). The coupling terms in eq. (lb) mean that the phase is sensitive to amplitude (term in 5) and phase (term in 7) variations of A. Let us mention that we would have obtained the same equations if 5 had had the same parity as the basic pattern, or if its period had been twice as big as that of the initial structure [16].

Above equations possess a "laminar state" (LS), given by A=A,=aexp(-iP@+i&), (p='po, whose stability is investigated below. Here, $. and (p. are arbitrary constants. Notice that this solution corresponds to vacillations of the periodic pattern, according to the description of U we gave above. For LS, eq. (la) is uncoupled to eq. (lb), i.e. A obeys the complex Ginzburg-Landau equation. But, if 1 + c+? < 0, this latter equation is known to produce phase turbulence [18, 191, and eventually leads to the destruction of the solution A,, through the appearance of zeroes of A. Hence, here is a possible way for spatio-temporal intermittent behaviors to occur in the system, and therefore, our guideline for the followmg will be to choose the parameters such that the laminar solution phase perturbations, and see whether the coupling with the phase cp of the instability or dampens it.

is unstable with respect to basic pattern enhances this

Linear stability of the laminar state -conditions for phase instability

Since LS is marginal with respect to homogeneous phase perturbations of 50 and of the phase of A denoted I/J, and since amplitude perturbations of A are damped (see appendix), the dynamics of the system about LS can be reduced, at least when LS is stable, to two coupled phase equations [START_REF] Kuramoto | Chemical Oscillations, Waves and turbulence[END_REF] for cp and I/J, Their derivation, together with the linear stability analysis of LS, is carried out in the appendix. These equations reflect the symmetries of eqs. (1). Namely, they are invariant under the transformation Moreover, without coupling, i.e. with y = 0 and 6 = 0, eq. ( 2a) is the well-known phase equation [START_REF] Kuramoto | Chemical Oscillations, Waves and turbulence[END_REF] or Kuramoto-Sivashinsky [20, 211 equation associated with the complex Ginzburg-Landau equation. The two phase eigenvalues, expanded as a power series of any fourier mode k, read (see appendix)

h=h(k)= +/i,k-;[l+@+K+y(f+e)]k2+...,
where A: = 277&y -6). Therefore, if c = np(py -6) < 0, A, is pure imaginary and LS is stable as soon as D = 1 + CY~ + K + y(& -5) is positive. Hence, the coupling between the complex amplitude A and the phase cp of the basic pattern may delay the occurrence of phase instability for A, since 1 + (YP may be negative whereas D is positive. But, when D becomes negative, LS is phase unstable, and may then display STI. Therefore, our ST1 control parameter will be D. When c is positive, LS is always unstable, whatever D may be, and here again, ST1 may occur.

Numerics

Before describing the dynamics of eqs. (1) when LS is phase unstable, let us first consider some numerical aspects of the problem. Since only spatial derivatives of cp are relevant, it is more convenient for the numerics to deal with r = +/ax instead of cp, and therefore, we consider the following equations, which are deduced from eqs. ( 2 As an initial condition, we take the laminar solution LS, slightly randomly perturbed. The numerical simulations have been performed on a CRAY-YMP, using a pseudo-spectral code with periodic boundary conditions. The number of collocation points was 1000, and the box length was 300 units of length. Now, we need to overcome two problems: how to cope with the nucleations or annihilations of cells in the basic pattern, and how to distinguish laminar cells from turbulent ones.

Since we only take into account the phase of the basic pattern, these equations cannot describe the existence of nucleations of new pairs of cells, or their annihilation. Nevertheless, when the phase gradients of cp become high enough, two cells should annihilate, or a new pair should be created. The occurrence of one of these two events depends on the sign of the phase gradient, i.e. if the pattern is compressed or streched. In order to implement these nucleation events in our model, we introduce a cutoff value r, above which r is set to zero. This stands for the fact that when phase gradients (r) of cp are too high, annihilations or nucleations occur in order to relax the phase, and therefore r gets smaller. The after-cutoff discontinuity in r will be smoothed by the numerics, and this ad hoc perturbation is assumed to describe the one which is introduced by the nucleation or annihilation of cells in the basic pattern. So, for each time step, and at each point, if T(x) >TO then T(x) =O.

We have chosen r, = 1 and checked that such perturbations do not bring aliasing into the numerics.

The second stage is to choose a criterion to separate laminar cells from turbulent ones. Like in the experiment, we define "turbulent" as "which departs from laminar", and since LS is characterized by IA I = 6 and (aA/at > + iPpA = 0, our criterion will be as follows:

if laA/at + @pAI I4 < s then the cell is laminar, if laA/at + @pAI IAI r s then the cell is turbulent.

(da)

In other words, a cell is said to be "turbulent" if A is small or if the local behavior is no longer periodic at the right frequency. In the simulations, we have chosen s = 0.65, and checked the results are not sensitive to the very value of this threshold. Note that, given the fact that amplitude equations cannot describe the periodic pattern at the scale of its period, this criterion is close to the one used in the experiment. Indeed, since A is the order parameter associated with the Hopf bifurcation, a small value of A corresponds to a spatial destruction of vacillations (therefore a cell should be turbulent if l/ IA I is large), whereas a change in its temporal frequency means temporal irregularity in the oscillations of the pattern (whence the use of IaA/at + i/?pAl as a second criterion). The conjunction of the two in ( 4) allows us to define a turbulent patch as showing irregularity either in time or in space, which is the definition used in the experiment.

Occurrence of spatio-temporal intermittent behaviors

We have seen that there are two ways of destabilizing the laminar solution LS, either by choosing D < 0 or by taking c > 0. We first consider these two situations separately.

When c < 0 and D < 0, the instability of LS is analogous to the Benjamin-Feir [lo] instability of the complex Ginzburg-Landau (CGL) equation, except that D is now given by D = 1 + CY@ + K + -y(/3~~ -51, whereas it is 1 + a@ for CGL alone. This instability gives rise to the appearance of dips in the amplitude of A, that travel with a constant velocity, at least close to threshold. Fig. 5 shows a spatio-temporal diagram of the system when D = -0.3. Black spots correspond to "turbulent" cells according to criterion (4b), and white regions are laminar domains, that satisfy (4a). Black or "turbulent" lines correspond to above-mentioned traveling dips, which are reminiscent of the traveling hole solutions to CGL described by Bekki and Nozaki [22]. The structure of one of these objects is detailed in fig. 6. A rapid variation in the phase of A occurs at the center of the depression and, depending on the asymptotic wavevectors, the dip propagates to the right or to the left. These two kinds of propagations correspond to lines slanted to the right or to the left on the spatio-temporal diagram of fig. 5. The traveling hole solution described in [22] is characterized by the fact that its velocity is related to the phase shift between the asymptotic waves. Because of the phase instability, the dips we observe always evolve in time and we cannot make accurate measurements of their asymptotic wave numbers. However, it is likely that, close to the phase instability threshold, these wave numbers are related to the wave vector k,,, whose growth rate is maximum. This could explain why the velocity of the dips is almost constant near threshold. Indeed, the wavenumbers on both sides of the dips should be close to kk,, and, as a consequence, their velocities would be nearly equal in absolute value. At this stage, we cannot give any quantitative check of this possible explanation. When IDI is increased further, the density of dips gets higher and their velocity is not as well defined as before. Fig. 7 shows a spatio-temporal diagram analogous to that of fig. 5, but for D = -2.25. For this picture, the mean value of r is about 0.12. Thus, the binary picture defined by ( 4) is less meaningful than before, since the amplitude of A and its temporal frequency are modified by the phase gradients (r) of the basic pattern. Fig. 8 shows a spatio-temporal diagram for the amplitude of A for the same value of D. Finally, let us mention that the dynamics described here is similar to that which is observed for CGL alone when its homogeneous solution is Benjamin-Feir unstable [231.

When c > 0 and D > 0, the instability is stronger and results in persistent localized destruction of the pattern, that shows less propagation and few branching. The number of "nucleations" (i.e. the number of times r gets higher than the cutoff value To) is bigger than before. In these regimes, the binary picture defined by ( 4) is not very informative, and we only show spatio-temporal diagrams for the amplitude of A. Fig. 9 is such a diagram for D = 0.3 and c = 0.1, where we see that the main feature is the existence of localized amplitude variations. When c is increased further, the number of "nucleations" gets higher again, and the mean value of r is no longer close to zero.

Finally, fig. 10 shows the behavior of the amplitude of A for D = -0.15 and c = 0.35 (i.e. both conditions for instability are satisfied). The pattern is now strongly destroyed and few branching phenomena occur. Here again, the number of "nucleations" is high and the man value of r (?; = 0.23) is not close to zero. Because of the increasing number of "nucleations" in the basic pattern, we cannot push further the analysis of this regime since the description with amplitude equations would no longer be valid.

Measurements

We focus on the regime where D < 0 and c < 0, which is characterized by the propagation of dips in the amplitude of A. The simulations have been performed with p < 0 variable and p = 1, (Y = 1, y = 0.5, 6 = 0.3, ,$ = 1, 77 = 1 and K = 1. Then c is always negative, and D is given by D = 1.5(1 + p). For each value of p, we compute the distribution of lengths of turbulent and laminar cells as well as their mean values, together with the turbulent fraction F,, measured in the ST1 regime, and defined as

F, = number of turbulent cells number of cells > f'
where ( . >, means average over time. Since LS is stable when D > 0, Ft is zero when p > -1. The turbulent fraction increases as lpl gets larger, and its behavior is given in fig. 11. The near-threshold behavior resembles ST1 in the sense that Ft scales like a power of the control parameter, though the exponent here is close to one. Besides, the picture we get is close to that which is got experimentally. However, there is no imperfect transition here, since the ST1 threshold is quite definite. The mean length of turbulent regions increases with IDI a -(l + p). More interesting is the mean length of laminar regions, which decreases as IDI a -(l + /3) increases, i.e. as ST1 goes stronger. Since the real parts of the phase eigenvalues (see appendix) are given by

Re(h) = h,k2 + h,k4,

where A, > 0 and A, < 0, the larger excited mode is and therefore, laminar regions of length I such that 21r/l> k,,, should remain laminar, since they are now stable. Close to threshold, A4 is finite and k,,, scales like 6 = d-.

For the particular . Hence, we have the following picture: laminar regions larger than I,,, are phase unstable, which give rise to patches of "turbulent" cells. Their appearance results in shortened laminar regions, which are now stable since they are too short for phase instability to develop. If laminar regions become too large, new turbulent patches appear, and the system goes, in space and in time, from laminar to "turbulent states", in an intermittent way.

Finally, we have computed the distributions of laminar and turbulent lengths. A typical distribution behavior is given in fig. 13. We have always observed an exponential decay, even for values of D close to the threshold value D = 0. However, we cannot go too close to D = 0 since the time then required for ST1 to develop gets too long. Nevertheless, fig. 14 shows the behavior as a function of D of the inverse of the square of the characteristic length associated with the exponential decay of these distributions. This characteristic length gets very large for D small, which leaves open the possibility of an algebraic decay at threshold. But we cannot give a definite answer to this question since we do not have points close enough to threshold (this would require boxes of increasing length and much longer computations), and also because the measure of the characteristic length from the distributions is not highly accurate.

Conclusion

We have shown that the two coupled equations describing a secondary Hopf bifurcation of a 1D periodic pattern can display spatio-temporal intermittent behaviors. The basic idea is that a phase instability of the order parameters associated with the secondary bifurcation can create localized patches of "turbulent" domains. The latter appear randomly in space, and their lifetime is also arbitrary. This coexistence of laminar and turbulent patterns is characteristic of spatio-temporal intermittency. However, this picture does not correspond to a competition between two "metastable" states, one turbulent and one laminar, since the laminar state is definitively unstable when D < 0 or c > 0, at least in a box whose length is large enough. Here, the term spatio-temporal intermittency denotes an alternance in space and time of laminar and "turbulent" behaviors, and is not as specific as it was originally [4, 61. The mechanism we propose qualitatively reproduces the experimental observations. For instance, for D > 0 and c < 0, the system features turbulent cells that propagate through the system at a constant velocity near threshold. This is reminiscent of the propagation of turbulent patches observed experimentally in the pre-ST1 regime (cf. section 2.2). Fully developed ST1 seems to be qualitatively well described by the regime corresponding to c > 0 and D < 0. Finally, the evolution of the turbulent fraction and of the characteristic length are similar (cf. figs. 3 and 11, figs. 4, 13, and 14). Thus, a phase instability associated with a secondary Hopf bifurcation of the convective pattern could be a possible explanation of ST1 in the Rayleigh-BCnard experiment.

Nevertheless, many questions are still open. For instance, we have assumed a Hopf bifurcation towards idealized vacillations. But the existence of regimes without oscillations, or the presence of a standing wave for the amplitude of oscillations have not been considered. This latter situation might though be described by assuming that the Hopf bifurcation occurs for a wavevector k which is non-commensurable with that of the basic pattern. This would then lead to three coupled equations (see ref. [16]), that are likely to give behaviors analogous to those of the two equations we have considered. More importantly, we have not proved that a phase instability does occur in the experiment, even though this work concludes it is likely. For these reasons, the next stage of this study is to start from the hydrodynamic equations (which turn out to be tractable since we have a 1D problem), and numerically follow the convective solution till its secondary bifurcation occurs. This should then allow a complete characterization of the secondary bifurcation for this experiment, and determine the signs of c and D for the envelope equations. Experimental and numerical work in this direction is in progress. where Ai = 2~77(py -6). The first eigenvalue, which is associated with amplitude perturbations of A, is negative for small k, and therefore amplitude perturbations are damped. The two other eigenvalues correspond to phase perturbations of cp or $. They vanish for homogeneous perturbations (k = 0) and may become positive if c = p~(py -6) > 0 or D = 1-t a/3 + K + y(/3q -5) < 0. When c < 0 and D < 0, it is often useful to push further the expansion of A + to see how it decays for larger k. Since the computation is somehow cumbersome, we will only show that A, is pure imaginary (when c < 0) and A, is real. The right hand side is real, and since A, is pure imaginary, A, is also pure imaginary; moreover, A, and A, have same sign.

At order 5, the equation for A, reads 1

Here again, the RHS is real and therefore, A, is real. Thus, when c < 0, the two phase eigenvalues are given by A+= fA,k+A2k2+A,k"+A4k4+ . . . Close to IST threshold, A, = 0, and Since A, and A, have same sign, A, is negative, as soon as (A,/A,Xv6 -qPy -~5) is not too large. Then, the real part of A + behaves like Re(A,) = lA,lk2-IA,lk4+@(kh).

Since amplitude perturbations are damped (cf. A, < O), the dynamics about LS can be described [START_REF] Kuramoto | Chemical Oscillations, Waves and turbulence[END_REF] by that of the two phases cp and $, when LS is stable. In the following, we derive these phase equations up to order two in space derivatives. We look for a solution to eqs. (1) in the form A=(fi+ EU, + E2U2 + . ..)exp [-i&+i$] where E is a small parameter, and we assume that cp, IJ and the ui's vary slowly in space and time. Therefore, the space and time derivatives are transformed as follows:

at at azy aT2

We plug this into eqs. ( 1) and identify at each order in E. Order 0 is satisfied since we look for solutions about LS, which is a solution to eqs. (1).

At order 1, we get

  Fig. 1. Schematic phase diagram of the convective state as a function of 8. S stands for a stationary pattern, D for dynamical regimes, ST1 for spatio-temporal intermittency and T for a complete disorganized and turbulent pattern.

Fig. 2 .

 2 Fig. 2. Space-time evolution of the convective pattern obtained in the annular geometry. Spatial digitization is made over 1024 pixels. The light intensity is averaged over the nearest neighbors and plotted every other pixel, using a 256-level grey scale. (a) Vacillations; E = 400, (b) Pre-STI; E = 470, (c) STI; E = 540.

Fig. 4 .

 4 Fig. 4. (a) Histogram N(L) of laminar domains of length L, showing an exponential distribution at E = 530; (b) Square of the slope m, (computed from the exponential decay of the histograms of laminar domains) as a function of E.

  A -+A exp(i$), where I,!J is an arbitrary real constant. Thus, the generic form of these two equations is [16, 171 aA -=pA+(l+io)$-(l+i/3)]AI*A-(y+i~?)gA, at (la> (lb)

294F.

  Daviaud et al. /Spatio-temporal intermittency in a ID conuectice pattern They read (2b)

  ): aA -=~A+(l+iru)$-(l+i/?)IA~2A-(y+i8)E4, at (3b)

F.

  Fig. 5. Numerical simulation of eqs. 3, with p = 1, a = 1, p = -1.2, y = 0.5, 6 = 0.3, .( = 1, 77 = 1 and K = 1. The evolution of the pattern is represented here during 1300 units of time. The entire box is shown, which corresponds <to 300 units of length in the x direction. Laminar regions are white, turbulent domains are black, according to (4).

Fig. 6 .

 6 Fig. 6. Amplitude (a) and phase (b) of one of the traveling dips in fig. 5. The value of IA 1 at the center of the dip and the amplitude of the "phase jump" change as the dips propagates.

Fig. 7 .

 7 Fig. 7. Same as fig. 5, but with p = -2.5.

Fig. 8 .

 8 Fig. 8. Spatio-temporal diagram for the amplitude of A. Minima arc in dark and maxima correspond to lighter regions. Only one half of the box is shown, and during a period of 1000 units of time. The parameters are the same as in fig. 7.

Fig. 9 .

 9 Fig. 9. Same as fig. 8, but with p = 1, (Y = 1, p = -0.8, y = 0.5, 6 = -0.5, 5 = 1, 77 = 1 and K = 1.

Fig. 11 .

 11 Fig. 11. Behavior of the turbulent fraction F, as a function of the order parameter -Cl + p) a ID I. Measurements have been made from samples consisting of 3000 consecutive states (distant from each other by one unit of time) of the 1D system, and for various values of /I?. Other parameters were kept constant (see text).

FFig. 12 .

 12 Fig. 12. Behavior of the mean length of laminar domains, as a function of [ -(1 + p)]-'/2.

.

  _,'...,....,....,....

Fig. 14 .

 14 Fig. 14. Inverse of the square of the characteristic length of 0 0.5 1 1.5 2 2.5 the distribution of laminar lengths, as a function of -(l + p) -(l+P)

  -+[l+C@+K+y(/3~--)] ifh"=o. Thus, the three eigenvalues associated with the linearized problem about LS read AR=-21L-[(1-(YP)-Y(P?7-~)lk2+~(k4), A,= kA,k-;[l+ap+K+y(Pq-[)]k2+8(k3),

At order 4 ,

 4 we get, assuming A, = 0 -2~[A~+2A,A3+A2(1 -iia+~) +~(l -icy)] -A,[2A,A~+Al(l -ia+K)] -(l+i(~+A~)[A~+2i~~(y-i6)] +~(1-$)[-2i(~K-2ic-uA~] + a( it -7) [2cu( y -iS)fi -h2[ A, + 277( 6 -Pr)] + 77(6 -h')(l + K) -K-Y?7P+ff(Y77Y--LYPK+5~~+ty.

-

  2~[2AzA3+2A,A4+Aj(1-icu+~)] -A,[A~+2AlA3+AZ(1--ia+~)+~(1-i~)] -rs] -3A', -4A, -2~h, -2K -(1 + a').

F

$

  exp(-i&t+i$)])&exp($~t-ig)].Since the phases cp and $ as well as the ui's are assumed constant at fast space and time scales, ij?) (36 u: + ~PL+) exp( -iP@ + i(cr)

Acknowledgements

We are grateful to H. Chat&, S. Fauve, P. Manneville, T. Passot and Y. Pomeau for fruitful discussions. We acknowledge the EEC twinning project Spatio-temporal chaos in extended systems (No. SCl-0035-C (CD)) for financial support. Two of us (P.C. and J.L.) wish to acknowledge the Arizona Center for Mathematical Sciences (ACMS) for support. ACMS is sponsored by AFOSR contract FQ8671-9000589

F. Daviaud et al. / Spatio-temporal intermittency in a ID convectiue pattern

(AFOSR-90-0021) with the University Research Initiative Program at the University of Arizona. Finally, we thank the Pittsburgh Supercomputing Center where the numerical simulations have been performed, under grant number DMS900005P.

Appendix

In this appendix, we perform the linear stability analysis of the laminar state (LS) given by A=A,=fiexp(-iPpLt+i$,), cp='pO, which is a solution to eqs. (1):

Writing we get for the linearized system about LS: aa at= -p(l+ip)(u+Z)+(l+iu)$-(v+ia)&+!j$. and the eigenvalues A associated with the Fourier mode k are solutions to Finally, writing

we have