F Daviaud

M Bonetti

M Dubois

Transition to turbulence via spatiotemporal intermittency in one-dimensional Rayleigh-Benard convection

Rayleigh-Benard convection is studied in quasi-one-dimensional geometries. Fixed and periodic boundary conditions are imposed using a rectangular and an annular cell, respectively. The desta- bilization process of the homogeneous convective pattern is studied for increasing Rayleigh number A. The first time-dependent behaviors are given by the appearance of coupled oscillators. At larger A values, the spatial breakdown appears through the propagation of spatial defects, which appear to be solitary waves. This spatiotemporal destabilization is followed at higher % by a spatiotem- poral intermittent regime, which corresponds to a dramatic decrease of the spatial coherence and to a mixing of turbulent patches within laminar domains. This last regime is studied within the frame of phase transitions. The statistical analysis evidences a second-order phase transition at least in the rectangular geometry (fixed boundary conditions), while this transition looks imperfect in the annu- lar geometry (periodic boundary conditions). Nevertheless, the essential qualitative features shown by theoretical and numerical models are observed in both geometries. Comparison with a simple model of directed percolation shows that the imperfect nature of the transition in the annulus could be the consequence of some mechanism of self-generation of the turbulent domains. This mecha- nism is, however, unknown but is probably related to the influence of the boundaries.

I. INTRODUCTION The evolution towards turbulent states of one- dimensional (ID) hydrodynamical systems has recently received much interest.

In fact, these systems are classified between the highly confined systems where deterministic chaotic dynamics might be observed and the extended systems in which developed turbulence gen- erally takes place. Theoretical studies on these systems, which depend mainly on one space variable, have shown that a steady periodic cellular state may destabilize and become turbulent when a control parameter is varied. In this context, numerical simulations of phase equations, such as those derived from the Kuramoto-Sivashinsky equation, ' coupled map lattices, ' and cellular auto- mata show that turbulence can be reached via spatiotem- poral intermittency (STI). This STI regime corresponds to a mixture of organized and turbulent domains which exchange themselves in space and time.

In very recent years, experiments were done on 1D sys- tems in which the basic state is spatially periodic. In this frame, Rayleigh-Benard convection was studied in cells with narrow gaps (one horizontal dimension is much larger than the other) where only one space variable is in- volved. Specific spatial and dynamical properties were observed, as well as STI (Ref. 6) at high values of the Rayleigh number. Statistical analysis of STI regimes, performed under more adequate experimental conditions, ' has given results similar to those of numencal simulations.

Nevertheless, the nature of the transition, which could be reminiscent of a direct percolation pro- cess, left some questions unanswered.

In order to get a deeper understanding of the dynamical regimes involved in this transition to turbulencevia STIin 1D systems, we report detailed experimental re- sults of convection in annular (periodic boundary condi- tions) and in rectangular (fixed boundary conditions) geometries. The two flows exhibit STI above a given Rayleigh number, but whereas the transition is almost perfect in the rectangular case, it turns out to be imperfect in the annulus, probably owing to the existence of local instabilities preceding the sustained STI regime.

In the following, we first describe the experimental set- up and discuss the conditions relevant to the experiments (Sec. II). In Sec. III we present the regimes leading to STI and the existence of solitary waves. The statistical properties of STI in both geometries are discussed in Sec. IV. The use of a spatial criterion to discriminate laminar domains (LD's) from turbulent domains (TD's) allows us to perform a detailed statistical analysis in the case of the annular (Sec. IV A) and rectangular (Sec. IV B) geometries.

In Sec. V we show numerical results of directed percolation and compare them with experimen- tal data. Finally, in Sec. VI, the convective regimes in both geometries and the nature of the transition to STI are discussed.

II. EXPERIMENTAL CONDITIONS

Experiments were performed both in annular and rectangular cells filled with silicon oil of Prandtl number P=7. Both geometries had vertical walls in Plexiglass.

The annular and rectangular cells had sapphire and copper horizontal plates, respectively. In order to get a 3388 1990

The American Physical Society quasi-1D geometry, ' the annular cell had a circumferen- tial aspect ratio I, =2vrr /d =35 and a radial aspect ra- tio I "=Dr/d =0.29 where r, b, r, and d are the mean radius, the radial gap, and the cell depth (d =7 mm for the two cells). In the rectangular cell the longitudinal as- pect ratio was I "=L"/d =25.7 and the transverse as- pect ratio was I =L /d =0.43. In the two sets of ex- periments the temperature difference across the fluid layer was kept constant to within 0.01 K by means of circu- lating thermal regulated ~ater.

The convective structures were visualized by shadow- graphic imaging. A parallel light beam crossed vertically the annular cell through the horizontal sapphire plates, while in the rectangular cell, the beam was shone perpen- dicularly through the horizontal longest side L . In this way, the beam deflection was integrated along the cell depth (circular cell) and along the smallest horizontal di- mension (rectangular cell). One had therefore the top view of the convective pattern in the circular geometry and the front view in the rectangular one (see Fig. 1).

The temporal evolution of the annular roll pattern was recorded using either a circumferentially moving photo- diode positioned at the place of the shadowgraphic image or a video camera recording system. With the former technique, the annular convective structure was azimu- thally scanned every second over 260 spatial points giving near the onset of STI a spatial resolution of -6 points/wavelength.

The video technique was used in both geometries and, as the convective dynamics is quasi-1D, the shadowgraphic image was digitized along a circle of approximately 1200 pixels (annular cell) or an horizontal line of 512 pixels (rectangular cell), giving a spatial resolution of -30 pixels and -16 pixels per wavelength, respectively. The circle and line were ap- propriately selected in order to get a representative skeleton of the convective pattern, without losing too much information on the flow dynamics. The acquisition frequency could be increased up to 5 circles/sec and 20 lines/sec for the annulus and rectangle, respectively.

In both geometries the time lapse between each new line acquisition was set according to which dynamics, i.e. , a short or a long time evolution, was studied. For exam- ple, when we analyzed the spatiotemporal intermittency,

we looked particularly at the long time evolution of the roll pattern. In the rectangular cell, up to 1900 lines were recorded with a time interval generally set to 3 sec; in the annular cell, time series up to 2000 sec were recorded.

The total acquisition time is to be compared to a charac- teristic time, namely, the basic period To of the rolls os- cillators. As To --1 and 2 sec in the rectangular and an- nular cells, respectively, this yields a total acquisition time of about 6000 and 1000 basic oscillations of the rolls.

After each temperature increase, dynamical equilibri- um must be reached. In both experiments, we waited as long as 24 h up to 48 h before each new data acquisition.

As a matter of fact, these delays revealed themselves sufficient to obtain a new equilibrium dynamical regime. The comparison with the phase diffusion time = L, /D ~~cannot easily be done since D ~~is not known for these narrow geometries and high Rayleigh numbers.

Nevertheless, taking D~~= 2.2X10 cm sec ' as calcu- lated (and measured) near the convective onset, " these delays are considered sufficient (rD =2 and 3 days for the rectangular and the annular geometries, respectively).

III. FLOW EVOLUTION

BELOW THE SPATIOTEMPORAL INTERMITTENT REGIME

Spatial and dynamical properties specific to the narrow channels, (i.e. , I, (0.5), have been already reported else- where. ' They are summarized as follows. When R exceeds a critical value %, =f(I ~), perfect stationary patterns are observed with the roll's axis perpendicular to the longest side of the container. An increase of A in- duces generally the appearance of new rolls, making the wavelength A, of the pattern much smaller than the one observed in large containers. ' Wavelengths down to 2L can be obtained. The value of k depends on the Rayleigh number and on the thermal history. Near the onset of STI and in a reproducible manner, 30 Q,O=0. 43k, with A, , =2d) and 40 (10=0.44k. , ) wavelengths Ao are present in the rectangular channel and the annular channel, respectively. In the annular geometry, at lower % values, a stable spatial inhomogeneity of the local wavelength is generally observed and up to now, no clear explanation of this phenomenon has been given.

Above a given e value [@=A/A, (1"~) -I] which de- pends on the transverse aspect ratio I and on the present wavelength, the pattern becomes time dependent with the appearance of thermal oscillators (domain la- beled by D in Fig. 2).

In the rectangular channel, at a=310, hot plumes ap- FIG.

1. Shadowgraphic image of 1D Rayleigh-Benard convection in a rectangular container with I ~=0.3 at @=45. Bright and dark lines are cold and warm currents, respectively. pear periodically in a pattern with a wavelength A, =0. 45k, The system is at first monoperiodic, i.e. , all the plumes have the same frequency everywhere and the convective rolls are similar to a chain of coupled oscilla- tors.

In the annular cell, collective oscillations are observed for e & 200 and are related to a mechanism of Uacillations, as it has been described in Ref. 12. We recall that these vacillations, which correspond to a periodic displacement, in phase opposition, of the ascending and descend- ing streams around their mean position, are specific to the narrow channels with Iy 0 40 When the Rayleigh number is further increased, the collective periodic be- havior may become chaotic with properties similar to those of a pure dynamical system, while the spatial prop- erties remain unchanged. Still in the annulus, a small evolution of the pattern, probably induced by a slow and a nonhomogeneous drift of the phase of the rolls can also be observed, but globally the spatial features of the pat- tern remain unchanged with a wavelength dispersion as already mentioned.

At a larger value of e, new events leading to a spatial symmetry breaking appear. These events were observed in a stationary pattern present in a rectangular channel between horizontal glass plates. ' The behavior is the fo11owing: in an otherwise perfect pattern, a local desta- bilization produces a wavelength larger than its neigh- bors. This defect, which extends over two or three wave- lengths, propagates along the pattern and can be reflected at the lateral walls while keeping its topological aspect (Fig. 3). Therefore it looks like a solitary wave. These waves, which have been further observed within the experimental conditions of the results reported here, are difficult to detect because their velocity is very low [(10 -5 X 10 )Ao sec ' with Ao the basic wavelength of

the pattern] and their dynamic is combined with the be- haviors of the oscillators. In the studied rectangular channel where hot plumes are present, the defect propa- gates only in part of the layer and can be reflected by or- ganized domains that are very robust, at least for a given time [Fig. 4(a)].

In the annular channel, these solitary waves have also FIG. 3. Time evolution of the shadowgraphic minima in a rectangular cell (between horizontal glass plates) with I =21

and I "=0. 35. Notice the collapse of two solitary waves which gives rise to two local large cells.

been observed just below the threshold of STI. In fact, these waves appear in an e domain which corresponds also to a homogeneity of the local wavelengths that are no longer dispersed at the threshold of STI. There could be a relation between the two phenomena though it has not been evidenced. Figure 5(a) shows the existence of such waves in different directions. It is difficult to know whether they propagate all around the cell and cross each other or are reflected by domains of oscillating cells.

In all the experiments, these waves are observed just below the onset of local turbulent events and it seems likely that they have a fundamental role in their appear- ance. Indeed, when two defects propagating in opposite directions collapse, they give rise to large local cells that are origins for turbulent patches (Fig. 3). Their interac- tion with oscillators can also destabilize locally the pat- tern. The convective pattern enters then in the regime of spatiotemporal intermittencies.

IV. SPATIOTEMPORAL INTERMITTENCY

In both geometries, above a given e value, the pattern shows a mixture of organized laminar domains and in- coherent turbulent domains (later defined) that are fiuc- tuating in space and time. The first events of intermitten- cy are observed for e'=450 in the annular cell and e"=350 in the rectangular channel (Fig. 2). Here, super- scripts a and r will refer, respectively, to the annular and rectangular geometries. Annular channel.

In the range e' & e (500, the dynamical regime in the annulus is characterized by tur- bulent patches which are localized both in space and time. These TD's can reach a spatial extension up to ten rolls with a lifetime shorter than 100TO, with To the typi- cal time period of the roll's oscillation. They can propa- tained in the rectangular channel from the shadowgraphic im- ages. The light intensity is plotted using a gray scale with 256 levels. Spatial digitization is made over 512 pixels (horizontal axis). The total observation time (vertical axis) is 1536 sec. Vertical dark lines correspond to stationary warm ascending currents. (a) @=350,(b) @=409,(c) @=561.

gate around the container by contamination but they do not spread. Moreover, they can be separated by very long periods of complete ordered regimes (up to several hours near e'). When e is increased, the spatial and tem- poral extension of these domains increases while the periods of ordered regimes become more and more sel- dom. Around @ =500, a Auctuating regime made of LD's and TD's is always present in all the container and is characteristic of sustained intermittency. The propaga- tion of the TD's through the pattern of LD's has a tree- like structure (Fig. 6) as is observed in the spatiotemporal diagrams of directed percolation (see Sec. V).

Rectangular channel. The transition in the rectangular cell from the organized "laminar" state to the STI can be observed in Fig. 4. The behavior looks qualitatively the same as in the annular cell, at least near the thresh- old, with the coexistence of laminar and turbulent domains, however with a stronger spatial confinement of the turbulent regions. As can be seen in Fig. 4(b), tur- bulent patches are sandwiched between laminar domains and hardly propagate throughout the cell.

The simultaneous coexistence of two kinds of qualitatively different domains implies the search for a criterion 4 but with the annular convective pattern at a=520. The total observation time is 1000 sec (along the horizontal axis, time is going from left to right) and the spa- tial digitization is made over 250 pixels. Notice the contamina- tion process of the turbulent domains throughout the laminar ones. periodicity of the roll structure is preserved, see Fig. 4(b). Furthermore, in LD's the time evolution of the shadow- graphic intensity I(x, t) at a fixed given point is either stationary or oscillating in time leading to a local mono- periodic or quasiperiodic regime with a small noise level. We recall that the oscillations correspond to the vacilla- tions in the annulus and to ascending plumes in the rec- tangular cell. On the contrary, TD's are regions without spatial coherence (see Sec. IV B) and their time evolution is chaotic. This difference between LD's and TD's is well evidenced in Fig. 4(c).

The choice of a spatial criterion based on the local wavelength is therefore justified to discriminate LD's from TD's without ambiguity.

A reduction of the signal can thus be performed, first by finding the extrema of the intensity I(x, t) (a maximum corresponding to a cold stream and a minimum to a hot one). A region in the convective pattern between two consecutive extrema x; and x, +, of I is then called laminar if

A, o( -, ' -b, ) & x;+, -x; & A, o( -, ' +6 ),
6 being a tolerance factor and A, o the mean wavelength, otherwise it is called turbulent. We have checked that the result does not depend on 6 provided -, ' ~6 ~-, '. Such a choice is suggested by the peaked distribution of the wavelengths as shown in Fig. 5(b). The signal can then be reduced to a binary representation of I (laminar or tur- bulent) which allows us to perform statistics on LD's and TD's. %'e have also used a temporal criterion to validate the previous data reductions. Intensity differences be- tween successive acquisitions were computed and compared to a cutoff value 6. A pixel i is called laminar if I(i, t +1) I(i, t) «5, - otherwise it is called turbulent. The results are similar, but with a larger dispersion, to those obtained with the spatial criterion. In the following, we present results which have been computed using the spatial criterion.

In order to characterize the global degree of chaos, we have computed the mean turbulent fraction F, : F, refers to the averaged total length occupied by the turbulent cells divided by the total length of the container. F, is plotted as a function of e for the rectangular [Fig. 7(a)] and the annular [Fig. 7(b)] geometry. The difference be- tween the two geometries is striking even if they both

show a continuous increase of F, with e. In the rectangu- lar case, the transition looks like a quasiperfect phase transition and a fit of the experimental data gives the re- lation F, -(e -eF )~, with eF = 360 and P"=0. 3+0.05. In the annular geometry, the transition to STI appears not well defined.

Notice that in both geometries, no hysteretic phenomenon has been observed when increasing or de- creasing the Rayleigh number.

In order to get more information on the nature of the transition and to understand the difference observed in the two geometries, a detailed statistical analysis has been made. We present the results separately for each case in the following.

A. STI in annular geometry

The distributions of spatial lengths of LD's have been computed. Figure 8 shows the histograms of the number N(L) of LD's with length L for different values of e. One must notice that the length L is computed in term of lam- inar cells, so L can take values between 1 and 80 in the annular geometry. When e is varied, two kinds of distri- butions are evidenced. In the range 460& e & 500, the histogram shows a power-law decay, in a domain of lengths L which is e dependent.

The power-law decay which can be estimated on the largest available range of scales at e=e, '=480+5 has a characteristic exponent p' = l. 7+0. l [Fig. 8(a)]. Accord- ing to a theoretical definition, ' this value e, ' could be defined as the threshold of STI. If we plot the relation Ft (E E )~w ith P=P"=0.3 as computed in the rec- tangular channel, we can fit the experimental points only at the large values of e. The discrepancy observed at the low values of e could be related to an imperfect nature of the transition as will be discussed in Sec. V.

For e& e, ' the histogram can be divided into two parts: a first part in the range of the small scales shows an alge- braic decay with the same exponent p, ' while at large scales an exponential decay is evidenced [Fig. 8 over the whole range of lengths. Figure 8(c} shows this exponential decay for a=530. If m, is the slope of the curve ln[N (L )] versus L, the characteristic length 1, =1/m, is observed to decrease when e increases. A plot of m, as a function of e [Fig. 9(a)] reveals two kinds of behavior: at the lowest values of e for which m, can be computed, m, has a nearly constant value, while at larger values of e, m, increases almost linearly as a function of e. This linear behavior is similar to the one observed in the rectangular channel and in the experiment of Ciliber- to and Bigazzi. However, the global behavior remains complex and if there exists a transitionin the sense of phase transitionthese experimental results evidence its imperfect nature.

The distributions of the lifetime of LD's have also been studied. By analyzing our space-time series with the same spatial criterion to discriminate laminar from tur- bulent cells, we have performed histograms giving the number N(r) of laminar periods lasting r. This temporal approach displays features similar to those obtained in the spatial distributions.

The histogram shows an alge- braic decay for e', =490+10 with a characteristic ex- ponent p', =2+0. 1, and an exponential decay for e& 500.

The threshold of STI is indeed the same (E, '=e', ) as for the spatial statistics even if the exponents are slightly different. In the case of an exponential decay, a charac- teristic time l, =1/m, is defined and m, varies also linearly with e [Fig. We have performed the same spatiotemporal statistical analysis for the rectangular geometry. The results hap- pen to be similar to those of the annular geometry, except for the fact that the transitionvia STIin this linear array of convective rolls is quasiperfect.

The evolution towards sustained STI has been first quantified via frequencywave-number power spectra computed from the temporal evolution of the rolls of Fig.

4

. Near the threshold of STI and for increasing %, a large broadening, in wave number, around the mean wave number is noticed as shown in Fig. 10(a). At larger %, a broadening of the peaks takes place in the frequency domain. We conjecture that this is mainly due to an un- correlated dynamics of the rolls and differs strongly from the time fluctuations of the oscillators. From these 2D Fourier transforms, no propagation of defects, such as the solitary waves observed near the threshold of STI, can be revealed. Such frequencywave-number spectra also show that the transition toward sustained STI is at first initiated by a spatial destabilization of the uniform convective pattern giving a wavelength dispersion, fol- lowed, at higher A, by time fiuctuations of the new spa- tial pattern.

The distributions of spatial and temporal lengths of LD's exhibit qualitatively the same kind of behavior as in the annular cell. The histograms show a power-law de- cay near e=e, "=360+10, while they reveal an exponen- tial decay for e) 380 [Fig. 11(a)]. The characteristic length 1, =1/m, and time 1, = 1/m, have a well-defined dependence on e, as r m, -(e -e, ") ', a, "=0. 5+0.05, Probably due to finite-size effects (the container con- tains only 60 rolls), the power-law distributions are only defined on a small range of scales and therefore it is diffiicult to find a precise value of p, " or p, ". An estimation gives p, "=1. 6+0.2 and p, "=2. 0+0.2. On the other hand, and contrary to the annular case, a unique threshold for STI is defined. In fact, the value of the threshold e, " ob- tained from the distribution displaying a power law and the value eF defined from the law of evolution of I', on e axis are wave-number and frequency axis, respectively. The maximum wave number and frequency are 1/8 cm ' and 1/6 Hz. The logarithmic power spectra are plotted above a given threshold. {a) @=380,(b) a=409.

are the same, e, '=eF--360. This value also corresponds to the first events of turbulence which can be observed in the experiment (e"=350). Accordingly, the transition via STI appears to be almost perfect in the rectangular con- tainer, but with exponents which do not differ within the error bars from these of the annular geometry.

To get more information on the transition process, we have also computed the correlation length g of our sys- tern. g is defined through the spatial correlation function:

C(r) =([r(r, t) -(r &][r(O,t) -(r &] &/(r'),
where I(r, t) and I(o, t) are the light intensity at two points separated by a distance r and ( ) denotes the en- semble average. The envelope of the correlation function C(r) is expected to decay as C ( r ) -exp( r lg ), where g is the correlation length to be compared to the length L, of the system. As stressed by Hohenberg and Shraiman, " when g& L, which is the case for all small systems, the fluctuations may be regular or chaotic in time but are still coherent in space. On the other hand, for ( &L"the dynamical behavior is incoherent in space and at moderate Rayleigh number is typical of a regime of weak turbulence without any energy cascade. This is the case for the convective regime studied in the two geometries.

Correlation functions of the intensity were computed to measure the correlation length g for different values of e. The envelopes of these functions have an exponential decay with the distance r, from which g can be extracted. Fig. 11(b) shows the dependence of g on e and reveals a change of behavior near a= 370, where a sudden decrease of the correlation length is observed. Below the thresh- old of STI, the correlation length is of the order of the length of the system: /=20k, -L"while far beyond e, ", (=A, &L, which corresponds to the observed regime of spatiotemporal chaos.

A characteristic frequency was also computed from the first moment of the averaged frequency power spectra. ' A monotonic increase of the frequency is observed as a function of e; however, no saturation of the frequency value can be seen beyond the sustained STI regime.

V. COMPARISON OF EXPERIMENTS WITH DIRECTED PERCOLATION SIMULATIONS

The results of spatiotemporal analysis performed in the rectangular geometry are typical of a second-order phase transition, but the transition turns out to be imperfect in the case of the annular geometry. The nature and the characteristics of the transition will be discussed in Sec. VI. In this section we compare first the experimental measurements with available numerical results. Some statistical properties obtained with a very simple model constructed from direct percolation are then given.

First of all, the experiments reveal the same qualitative route to turbulence via STI as in the numerical simula- tions of some partial differential equations (PDE s), cou- pled map lattices, and directed percolation (DP) and most of the characteristic properties observed in the critical re- gion around the threshold of STI are similar to those ob- tained numerically. The Kuramoto-Sivashinsky equation with an additional damping term reveals a second-order transition, with an algebraic decay close to the threshold, in the histograms giving the distribution of lengths of LD's, ' although the critical exponents are different from the experimental ones. A one-dimensional array of maps coupled by diffusion exhibits also the same behavior. In this last case, Chate and Manneville have shown that coupled map lattices display critical properties analogous to DP, but that they do not belong to the same universali- ty class. The relationship between DP and the transi- tion to turbulence via STI had been pointed out by Pomeau, who has placed the transition to turbulence for weakly confined systems in the field of critical phenomena.

To go further in this comparison between DP and ex- periments, we have performed calculations with a simple model of cellular automaton. In our problem, the au- tomaton is defined on a 2D lattice: one space dimension made of N sites (standing for the 1D geometry of our ex- periments) plus time. Each site has two possible states: a laminar state (L) and a turbulent state (T) by identification, respectively, with a laminar and a tur- bulent cell in the experiments. The probabilistic rules are defined on a three-site neighborhood, because the experi- ments suggest the inAuence of the two nearest convective rolls on the dynamical regime of the roll they surround.

The state of a cell i at time t +1 thus depends on its state and on the states of its neighbors i -1 and i + 1 at time t.

In DP, the laminar state is absorbing, i.e. , it cannot be- come turbulent if its parents are laminar: p (LLL ~T)=0. However, since the experimental data show that a turbulent region may originate from a laminar one around the threshold of STI, a small probability p is in- troduced so that p(LLL~T)=p .

This probability stands for the spontaneous apparition of turbulent spots in LD's.

The automaton is defined by a set of eight elementary probabilities Ip"~k =0, 7I: p0=p(LLL~T), p, =p(LLT~T), p2 =p (LTL ~T), p3 =p (LTT~T), p4=p(TLL ~T), p~= p(TLT~T), p6 =p ( TTL ~T), p7 =p (TTT~T) .

The complementary probabilities are given by the rela- tion p(XYZ~L)=1p(XYZ~T) and because of left- right symmetries, we have p, =p4 and p3 =p6. %e have chosen the case of directed bond percolation, which seems to be more representative of our experiments than site percolation, because of the diffusion process. More- over, in order to make the calculations easier, we have taken totalistic rules, ' so that the following relations are satisfied:

5'I P2 P4

We are thus dealing with only one control parameter p plus the probability p0=p. For the numerical simula- tions, the number of sites N was 10 and the duration of the run was several 10 . The statistical analyses were per- formed only after some 10 iterations to skip the transient regimes, especially near threshold. These values are not large enough to ensure accurate results of the exponents, which also depend on the size of the lattice. However, they allow a description of the main characteristics of the studied behaviors.

First of all, when @=0, the results show a transition for p =p, =0.44. The fraction of turbulent sites (the time average of the fraction of the system occupied by TD's) varies as F, -(p -p, )' with @=0.28 and, for p =p"the histograms giving the distribution of LD's of length L show a power-law decay with a characteristic exponent p, (p=0)=1.5. More- over, for p ) p" the histograms exhibit an exponential de- cay with a characteristic length l, =1/m, which depends on p. The evolution of I, can be fitted by Q m, -(p -p, ) ', a, =l .

The distributions of the lifetime of LD's evidence the same transition at p =p, and give an exponent p, (p = 0) = 1.6 and a characteristic time l, = 1/m, for p )p, . with v, =0.5. A second-order phase transition is therefore observed when p =0, with well-defined threshold and characteristic exponents.

The results obtained when pWO display several impor- tant differences from the case p =0. First of all, the spa- tiotemporal representations of the automaton exhibit features very similar to the experimental results. It shows the importance of the spontaneous appearance of turbulent sites for the generation of localized TD's near the threshold p"while for p &&p" the role of p is totally hidden by the process of contamination of LD's by TD's. The statistical properties are also changed by the presence of p. If it is small enough (p (10 ), an algebraic decay is still observed in the histograms giving the distri- bution of LD's, but in a range of length scales which shortens when p increases. The characteristic exponent also depends on p. With p=5X10, the histograms show an algebraic decay around p =0.44 on 100 sites with a characteristic exponent p, (p =5X10 ) =1.4. On the other hand, no noticeable change is observed in the distributions of lifetime of LD's, provided p is small enough: p, (p= 0)=p, , (P= 5X1 0 ). When p is in- creased further, no transition in the functional form of the distributions of LD's is observed: all the histograms show an exponential decay. This last result is similar to the conclusions of Bagnoli et al. , ' who have also compared the behavior of the experimental data of Ciliberto and Bigazzi with that of suitable chosen automaton rules.

When p&0, a change of behavior of the turbulent frac- tion is still observable (Fig. 12). The value of I', for p =p, is different from zero and increases with p. In fact, if p & 10,F, evolves slowly with p and reveals no transi- tion. The evolution of m, (or m, ) with p is also affected by an increase of p, as the evolution of F, .

In conclusion, these numerical results show the ex- istence of a well-defined transition in the case of directed percolation, i.e. , when the laminar state is absorbing (p =0). When the laminar state is quasiabsorbing (P&0), the transition disappears in a strict sense or becomes im- perfect. The evolution of the turbulent fraction becomes FIG. 12. Mean turbulent fraction F, as a function of the probability p for different p (see text). Squares, P'=0; crosses, p = 5 X 10;diamonds, p = 10 ', triangles, p =2 X 10 smooth; meanwhile, the algebraic decay, which is characteristic of the critical transition, disappears when p in- creases.

Finally, these numerical results performed on a very simple model of DP give an interesting insight on the ex- perimental data. The rectangular geometry is well represented by pure DP; meanwhile, the annular case can be compared to DP with pAO. Though the physical na- ture of p is not explained, it gives a good representation of the spontaneous apparition of turbulent cells. As a power-law decay is present in the experimental distribu- tions, it turns out that the probability p is small in the ex- periments, as shown by the numerical results. The lami- nar state of the convective rolls is thus quasiabsorbing.

VI. DISCUSSION

The experimental results discussed in this paper have shown that in quasi-1D Rayleigh-Benard convection, the evolution of the convective state to sustained spatial tur- bulence is achieved through the development of spatiotemporal intermittency, as expected from theoretical and numerical studies. The observation and quantitative analysis of STI regimes have been previously performed in convection experiments, but in that case, the 1D character was not well fulfilled, and the criterion to separate LD's from TD's was different from that used in the present study. Nevertheless, many properties remain similar in the two cases, showing the robustness of the STI regimes.

In the present study, two different geometries were used. Periodic boundary conditions were obtained using an annular cell, while fixed boundary conditions prevailed in the rectangular channel. The transverse aspect ratio was slightly different for the two cases, but this difference seemed to influence only the dynamical regimes preced- ing STI. As a matter of fact, the first time-dependent convective behavior was given by the appearance of oscil- lators in the convective patternplumes in the rectangu- lar channel, vacillations in the annulusin agreement with the general observation that vacillations do not exist when I" )0.4. Nevertheless, the qualitative behavior of the sustained STI regime was exactly the same in both geometries, therefore it is independent of the nature of the oscillators in the convective pattern. Their infiuence on the STI threshold through eventual specific mecha- nisms of spatial destabilization is unknown, but it is likely that the observed difference (e"=360, e'=480) is related only to the value of I' itself (in the experiment reported in Ref. 7, the threshold value e'=200 with I = 1).

In the two geometries, the spatiotemporal intermittent regime corresponds to a loss of spatial coherence and ap- pears as an interspersing of turbulent patches within lam- inar domains. Very close to the STI threshold, turbulent patches spontaneously appear from time to time and, after a while, relax and disappear. The intermittent be- havior of a complete ordered pattern with a locally tur- bulent one is observed within a very narrow e interval in the rectangular channel, and within a larger e domain in the annulus, as is clearly evidenced by the variation of the turbulent fraction as a function of e. Moreover, this vari- ation shows that in the former case, we have a quasiper- fect second-order phase transition. In the case of the an- nulus, the nature of the transition from laminar state to STI is more complex and looks imperfect, though the statistics concerning the length of the laminar domains (power law at the "effective" threshold, scaling law for the characteristic length) are similar to the statistics ex- pected for a well-defined transition. A somewhat similar behavior has been reported in Ref. 7, where the convec- tion is also built up in an annular container (with I =1).

The striking difference between the behaviors in the an- nular and rectangular geometries is probably due to the influence of the boundaries which stabilizes the phase of the rolls in the rectangular channel. On the contrary, in the annular geometry, an intrinsic phase roll instability together with a possible large scale flow could induce the observed imperfect transition. In fact, it is only in this geometry that turbulent patches propagate near the threshold.

The experimental results agree globally with those ob- tained by numerical simulation of PDE or coupled maps and also with directed percolation. The quantitative comparison between the values of the exponents numeri- cally and experimentally measured (see Table I) is not straightforward, for recent studies" show that the ex- ponents are not universal. They depend, in particular, on the size of the system, but also on the underlying mecha- nisms which govern the transition. Nevertheless, the ex- ponents P'for the evolution of the turbulent fraction in the rectangular geometryand p,of the algebraic decaydeduced from the experimental observations, may agree with those of DP and coupled maps. To our knowledge, the prediction concerning the u exponent describing the evolution of the inverse of the characteris- tic length as a function of e, has not yet been done. The exponent obtained through our numerical simulation of directed percolation is not the same as from the experi- mental data. Furthermore, no clear correspondence be- tween the two control parameters p and e can be estab- lished. A recent phenomenologica1 study of a 1D chain of rolls, performed in the frame of the Landau- Ginsburg equation, ' could give a transition similar to our experimental observations.

The recent studies of 1D periodic systems have re- vealed the richness of their spatial and dynamical proper- ties. In a certain range of the control parameter, they present an analogy with a chain of coupled oscillators, and so, are linked to dynamical systems. Then, the first symmetry breaking of the space translation takes place by the appearance of propagative solitary waves. These waves, observed in the convective narrow channels, have been also evidenced in directional solidification ' and in the periodic front of the meniscus of a fiuid between two rotating cylinders.

In this last experiment, spatiotemporal intermittency is also currently under investigation.

Therefore it seems that a large amount of the properties we have found is generic to 1D systems. a,

  FIG. 2. Phase diagram of the convective state as a functionof e in the annular cell (a) and in the rectangular channel (b). S stands for a stationary pattern, D for a time and/or spatial os- cillating pattern, STI for spatiotemporal intermittency, and T for a complete disorganized and turbulent pattern.

  FIG. 5. (a) Same as in Fig.4but with the annular channel.The total observation time is 1500 sec and @=420. Notice at the right and left of the frame the crossing of solitary waves. Dark lines indicate the trajectory of some solitary waves.(b) Wavelength distribution at a=420.

  FIG. 7. Mean turbulent fraction F, as a function of e. (a) Rectangular channel, (b) annular cell. See text for the full curves.

  FIG. 8. Histograms X(L) of laminar domains of length L. (a) Power-law distribution at e=e, '=480, {b) @=500, (c) exponential distribution at e= 530.

  9(b)] if we skip the low values of 6.

  FIG. 9. Annular cell. (a) Square of the slope m, computed from the exponential decay of the histograms of the laminar domains, as a function of e. Crosses and rectangles correspond to increasing and decreasing e, respectively. (b) Square of the slope m, computed from the exponential decay of the histo- grams of the laminar domains with lifetime ~as a function of e.

  FIG. 11. Rectangular cell. (a) Slope m, computed from the exponential decay of the histograms of the laminar domains, as a function of e. (b) Spatial correlation length g as a function of e. Notice the strong decrease of g when the convective pattern enters into the spatiotemporal regime at a= 370.

ACKNOWLEDGMENTS

We wish to acknowledge G. Balzer, P. Berge, R. Bi- daux, Y. Pomeau, and P. Manneville for fruitful and stimulating discussions. We would like to thank also P. Hede for the development of the image processing and B.

Ozenda for his technical assistance.

H. Chate and P. Manneville, Phys. Rev. Lett. 58, 112 (1987).

According to a private communication with one of these au- thors, P. Manneville, the knowledge of the exact nature of the transition, studied in this paper, requires more studies.


F Daviaud

A Pocheau

Inhibition of Phase Turbulence Close to Onset of Convection by Permeable Lateral Boundary Condition for the Mean Flow

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

We show that the mechanisms which govern the onset of time dependence in usual Rayleigh-Benard convection at low Prandtl number may be inhibited by a suitable choice of the lateral boundary condition for the sole mean flow. We first build a boundary condition which behaves like a rigid one for the roll flow and like a permeable one for the mean flow. We then observe that phase turbulence is inhibited close to the onset of convection. We understand this effect by solving the mean-flow field from the Cross-Newel1 equations. Our experimental result together with its interpretation demonstrates indirectly the existence of mean flows and enlightens the ways by which mean flows destabilize patterns.

In the past decade, Rayleigh-Benard convection in extended systems has brought about a puzzling paradox: at low Prandtl number (Pr = 0.7), the threshold of time dependence is roughly one order of magnitude lower for distorted patterns ( E = O(O.l)) [l, 21 than for straight ones ( E = 3) [3] ( E = [Ra -Ra,]/Ra,, where R a is the Rayleigh number and Ra, its value at the onset of convection). Thus pattern distortion represents quite a dangerous mode, capable of leading to an overdestabilization. The origin of this mechanism has been traced back to the existence of large scale parallel flows (Le. mean flows) which are produced by the roll flows, whenever the roll structure is distorted [4]. It has already been proved that large-scale flows play a crucial role in the instabilities of straight roll patterns [5] and in the transition to both time dependence and turbulence in a cylindrical container [6-91. In this letter, we aim at showing that the spatio-temporal properties of convective patterns, including their route to turbulence, may be changed by suitable action on the sole large-scale flow. In particular, we have been able to inhibit phase turbulence close to onset of convection in a cylindrical container and thus to obtain stationary distorted patterns at values of E one order of magnitude greater than in usual configurations. This result proves indirectly the existence of mean flows and enlightens the mechanisms by which pattern overdestabilization arises.

Let us first recall the mechanisms of time dependence in usual convection experiments, i.e. in closed containers. For cylindrical geometries and at Pr=0.7, visual observations show that rolls tend to end perpendicularly to the walls so that two centres of pattern curvature arise near the onset of convection (similarly to fig. la)). Roll curvature then where R is defined as the ratio of the radius of the cylindrical cell to its depth. In this system, mean flows and roll distortion are coupled in a nonlocal way: roll distortion produces mean flows but mean flows may in turn produce distortion, as demonstrated by the study of forced mean flow effects on a chain of rolls [lo].

Nonlocality is a consequence of mass conservation (any distorted roll of the convective field participates in producing a mean flow at a given point) and of the scale difference (mean-flow effects accumulate from roll t o roll). This nonlocal coupling is described by the Cross-Newel1 equations [ll]. An exact perturbative solution of these equations has been found in a closed cylindrical container for patterns displaying two opposite bendings [8]. In this solution, a back-mean flow is then generated by a pressure gradient on the line joining the foci to the centre of the pattern (y-axis) (fig. 24). In agreement with experiments, it sustains dangerous compression modes in the phase equation which finally trigger roll pinchings by localized instabilities at E = O(O.1). We emphasize that the main modes leading to time dependence are roll curvature for producing mean flow and the back flow for yielding overdestabilization. We finally remark that the back flow is a direct consequence of the lateral boundary conditions.

In our experiment, we aim at modifying this mechanism of time dependence by acting on the sole large scale flow independently of the underlying roll structure. To achieve this, we alter the lateral boundary condition for the mean flow, but we keep that for the roll flow unchanged. We therefore expect the rolls to still end perpendicularly to the boundary so that curvature is maintained. As described below, our new boundary stops convection but is permeable to mean flows. We therefore expect the pressure gradient and thus the back flow together with the route to time dependence, to be modified. Since mean flows are generated in a nonlocal way by the whole pattern, inhibiting convection outside a given domain should not stop them, provided the boundaries are not rigid walls. In order to satisfy this condition, we slightly lower the cell depth outside a cylindrical domain by inserting a thin sheet at quarter of the cell height (fig. 3). This trick gives rise to negligible thermal perturbations but to strong flow modifications: only conduction can take place in the outer region and the sheet is expected to let a large part of the mean flows pass. Visual observations ensure that the sheet is sufficient to give rise to roll curvature. Finally, as in usual cells, lateral walls protect from external thermal or flow perturbations. We work in argon gas under a pressure of 30 bars (Pr = 0.7). We use an experimental apparatus similar to that described in [2], the extension of the visual field having been increased, however. The walls are cylindrical and the cell aspect ratio is R' = 25. The cell depth is 1.6" so that the critical temperature is 3 5 °C . The thin sheet has an annular geometry. It gives rise to a cylindrical convective region of aspect ratio R = 12.5 (fig. 3).

c o n d u c t i o n L a t e r a L ------) waLL5 a n n u L a r sheet' convection
At onset of convection, rolls are straight. As E is increased, two centres of curvature appear and the curvature grows but saturates at ~~0 . 2 . This is in agreement with observations in a closed container, except that no dangerous compression occurs at the centre of the pattern. Moreover, the pattern remains stationary up to surprisingly high values of E (fig. la)). Such stationary states are reached after transient evolution, whatever the initial condition is. Sustained time dependence occurs only at E = 1.2 and involves repetitive localized processes which are reminiscent of the skewed-varicose and cross-roll instabilities of Busse and Clever [12]. The threshold of time dependence of distorted patterns is thus one order of magnitude greater in an <<open container. than in a closed one. Time dependence and phase turbulence are therefore inhibited over a large range of E.

The system undergoes several restabilization processes towards stationary patterns in its route to turbulence. The first process leads to a wave number selection. It arises at E = 0.56 by a roll pinching at the centre of the pattern (fig. lb)) which leads to the nucleation of a pair of dislocations which eventually disappear at the foci. The whole pattern has then lost one wavelength. It remains stationary up to E = 0.74, where a new roll pinching occurs at the centre of the pattern. Further additional defect nucleations yield a long transient spatio-temporal evolution which relaxes towards a complex stationary pattern. Stationarity persists until E = 1.2, where localized phase oscillations occur by periodic cross-roll-like instabilities or grain boundary motions. Finally, at E = 1.5, observations on a time scale of the order R . T ~, where q, is the horizontal diffusion time, show phase turbulence. Focus singularities near the boundary then display phase travelling waves which either generate new rolls or force rolls to disappear. The latter feature is never observed in a closed container, presumably because mean flows cannot escape the convective domain. Measurements of the roll wave number distribution allow us to compare quantitatively the pattern behaviour in closed and open cells. As shown in fig. 4, the first two dislocation nucleations occur at a local wave number which, as in a closed container, is unstable with respect to the skewed-varicose instability, the shift with the marginal stability being probably due to finite-size effects produced by the localization. However, this local wavenumber crosses the straight roll stability domain at values of E five times greater in open cells than in closed cells. This corroborates the observation of a delay in the occurrence of time dependence. In addition, we note that the decrease of the wavenumber allows the restabilization of patterns. For both these reasons, although the pattern is distorted, its behaviour is similar to that of straight rolls [3].

Moreover, in contrast with the case of closed containers, the wave numbers at the foci and at the centre of the pattern have quite similar values. This means that the compression of the structure has been largely reduced, so that no dangerous large wave number is encountered close to the convective threshold. This indicates that, as expected, the meanflow field has been modified by the change of lateral boundary in such a way that it does not induce destabilizing distortions. In particular, our measurements evidence that the amplitude of the back flow is strongly lowered.

In order to corroborate these experimental results, we have determined [131 the meanflow field by expansion of the Cross-Newel1 equations in a way similar to that performed in a closed container [8]. The phase field p is thus modelled in a polynomial space p(x, y) = = ko(l + A ) y(1-ax2/R2 + by2/R2 + cy4/R4 + dx2 y2/R2) where the relation between (b,e,d,A ) and a has to be found by expansion at first order in a2 and 1/R, assuming R >> 1 and a << 1 and where ko is close to k, = 3.117 at Pr = 0.7. In a closed container, we recall that the mean-flow vertical vorticity reads 0, = 2wxy/R4 (w is positive and roughly inversely proportional to the Prandtl number). Moreover, the orientation being defined such that the y-axis (0 = k d 2 ) joins the foci (fig. la)), the mean-flow field F takes the following form in polar coordinates: where ,5 is only related to a divergence-free pressure gradient.

Since the container is closed, the normal flow Fr must vanish at the boundary r = R, so that ,5 = -1. In an <<open, container, we consider, as a first step, that mean flows cross the boundary of the convective domain quite freely. In order to determine the mean-flow field in both the conductive and convective domains, we first assume that no additional source of large-scale vertical vorticity 0, is generated by the new boundary. The boundary conditions therefore involve the continuity at r = R of the pressure, the vorticity 0, and the radial flux F * r and the vanishing at r = R ' of the radial flux. In the convective domain, 0, is similar to that of a closed container but it must diffuse and vanish in the conductive domain, since there are no longer mean-flow sources there. For large R, we find that the mean-flow field takes the previous form (1) inside the convective domain with ,5 = -1/2(1 + F -~) and p=R'IR, and the following form in the conductive region [13]:

As shown in fig. 2b), the mean-flow streamlines are strongly modified by the change of lateral boundary. In particular, a shear flow appears at the boundary of the convective domain so that additional vorticity is created. However, since it remains localized, it does not modify the former determination away from the boundary of the convective domain. Instead of the strong back flow present in closed containers, the mean flow reverses its direction on the line joining a focus to the centre of the pattern and its amplitude is now weaker. Introducing its value in the phase equation enables us to determine the stationary conditions on the large scale. Compared to a closed container [8], the relation between (b, e, d, A ) and a is unchanged except that b is multiplied by -p, i.e. 1/2(1 + ,c-~). For small p-4(p-4 << l), the maximum wave number is then ko + o(u2/R) and it is reached at both the foci and the centre of the pattern. The latter prediction is close to the experimental results (fig. 4). However, the additional viscous stress produced by the sheet in the conductive domain produces an additional back flow which may be taken into account by introducing a transmission factor. The agreement with the experiment is then even better (fig. 4).

Our solution of the Cross-Newel1 equations thus nicely recovers the experimental results and agrees with their interpretation. In particular, it confirms that the back flow is weakened. Accordingly, though mean flows are still present, the conditions leading to .dynamical frustration. [6] are less fulfilled than in a closed container. This explains why time dependence is delayed.

We have demonstrated that opening the container to flows other than the primary convective flows changes the spatio-temporal properties of an extended cylindrical convective layer and inhibits phase turbulence in a large range of the control parameter E. We have solved the Cross-Newel1 equations for this situation at the dominant order in the amplitude of the pattern distortion. The mean-flow field of this solution enables us to understand the delay observed in the appearance of time dependence. These results prove indirectly the existence of mean flows and c o n f m the mechanisms of phase turbulence based on the mean flow-roll distortion interaction. In particular, they point out the nonlocal link between mean flows and boundary conditions. At a more general level, our study shows, as an example, that spatio-temporal turbulence may be inhibited, provided that the right phenomena are understood. We envision to extend this control of turbulence to other cellular patterns and to systems exhibiting interactions between more than two scales.

Fig. 1 .

 1 Fig. 1. -Pictures of the open container. a) Stationary pattern at e = 0.55. The y-axis is defined such that it joins the foci. b) Occurrence of the first instability at E = 0.56. Notice the skewed-varicose-like distortion.

F

  . DAVIAUD et al.: INHIBITION OF PHASE TURBULENCE CLOSE TO ONSET ETC.

Fig. 2 .

 2 Fig. 2. -Mean-flow field solutions of the Cross-Newel1 equations. a) The boundary of the convective domain (small circle) is nonpermeable to mean flows. Notice the back flows along the y-axis (the orientation is defined in fig. la)). b) The boundary of the convective domain is permeable to mean flows. The mean flow amplitude is then weaker than in the closed configuration. The shear at the boundary of the convective domain generates localized additional vertical vorticity, not represented on the picture.

Fig. 3 .

 3 Fig. 3. -Sketch of the experimental set-up leading to a permeable lateral boundary condition for the mean flows.

Fig. 4 .

 4 Fig. 4. -Stability diagram of infinite straight rolls at Pr=0.7 displaying the marginal (MI, the Eckhaus (E) and the skewed-varicose (SV) stability curves. We have plotted the local wave numbers for closed and open containers. Black squares correspond to the band of wave numbers in a closed container and triangles, crosses and open squares to the wave numbers in an open container, respectively at the boundary on the x-axis, the focus and the centre. The line corresponds to the maximum wave number of the solution of the Cross-Newel1 equations, the sheet viscous stress being taken into account.

We wish to acknowledge P. PELCE, P. BERGE, P. CLAVIN, P. MANNEVILLE, H. CHATE, M. DUBOIS for discussions and M. LABOUISE and B. OZENDA for technical assistance.


M Dubois

R Da Silva

F Daviaud

P Berge

A Petrov

Collective Oscillating Mode in a One-Dimensional Chain of Convective Rolls

Keywords: 47, 25Q -Convection and heat transfer, 47, 20 -Hydrodynamic stability and instability, 05, 70 -Thermodynamics

Experiments of Rayleigh-BBnard convection in intermediate geometries show that very small wavelengths can be stabilized at very high values of the Rayleigh number when the fluid layer is quasi one-dimensional. A new collective oscillating instability is then observed; this instability may be a first step toward spatio-temporal chaos.

Rayleigh-BBnard convection has shown itself to be a good system for the study of the transition to turbulence. In small geometries, observations of the dynamical evolution with the Rayleigh number have led to a precise understanding of chaotic behaviours (for example [l-31). In large geometries, mechanisms leading to turbulence near the onset have been observed, in particular in low Prandtl number fluids; they are related to the spatial properties of the system (for example [4,5]). The remaining domain, that of intermediate geometries, has been little studied to date; these geometries are nevertheless of interest as they provide systems for the study of the approach to turbulence, starting from the now well-understood concepts of deterministic chaos [6] and adding progressively the notion of space.

Recent theoretical work in this direction is exemplified by numerical studies of coupled map lattices ['7l and of partial differential equations derived from the 91. The experimental work parallel to the former corresponds to the study of a chain of oscillators, whereas the latter can be applied to observations in a onedimensional chain of convective rolls; this kind of convection can be achieved experimentally in a convective structure for which one of the horizontal extensions is small and the other great, in comparison with the depth. We report here some striking features, in particular a new collective oscillating state observed in Rayleigh-BBnard convection in these intermediate geometries.

The experimrttal results. -The Rayleigh-BBnard convection is achieved in two different geometries: one is rectangular ((180 x 2 x 6 ) mm), with apeet ratios I', = 36 and I ' , = 0.4 and filled with silicon oil of Prandtl number 217.5. The other cell is annular, with internal diameter 120 mm and external diameter 126 mm thus having a gap of 3 mm. Its depth is 10mm, giving an aspect ratio TR=0.3 along the radius and r,=38.6 along the circumference. This cell is filled with silicon oil of Prandtl number 2 : 22. The walls of both cells are of plexiglass and are between upper and lower copper plates.

The shadowgraphic images of the convective states are observed for both geometries, the beam crossing the cell along the direction parallel to the rolls' axes. The images, observed on a screen (fig. l), give then a picture of the pattern in a vertical plane. These images are recorded with a CCD camera on a video recorder. In the rectangular cell, a part of the image, along a line parallel to the great extension of the cell, is focused onto a photodiode array of 256 pixels. We may then record the intensity of the pattern at any time by digitizing the photodiodes' signals with an 12-bit ADC controlled by a microcomputer. The resulting time series allow us to follow the spatio-temporal evolutions of the pattern. In the case of the annular geometry, the same kind of information is given by a photodiode which sweeps the circumference at a chosen frequency. In both cases, the array or the mobile photodiode, the intensity is such that a maximum corresponds to cold streams and a minimum to hot streams in the vertical plane.

The convective pattern consists, above the threshold, of a perfect chain of parallel rolls with axes perpendicular to the long side of the cell. At a given Rayleigh number the actual wavelength depends on the previous thermal history; in particular, it is known that when the fluid is subjected to a quick increase of the temperature difference, a stable structure can be obtained with a wavelength lower than Ac[lO]. After such an increase in the temperature difference, we have observed that unusually short wavelengths (as short as hJ3, with A,=2d) can be stabilized in the cells. This situation may be explained,by the natural tendency for these geometries to favour wavelengths shorter than In addition, it is seen that the smaller the r, value, the smaller the minimal stabilized wavelength; at least in a certain domain the system appears to exhibit a resonance between the observed short wavelength and the cell width. We may then observe a very nice periodic oscillation of the hot and cold streams as the first time-dependent behaviour above a given value of the Rayleigh number. This kind of oscillating instability corresponds to a fine collective behaviour as can be seen in fig. 2. If we look at the uprising streams near the hot bottom plate, their individual position oscillates around a mean value. In addition, as cold streams and hot streams oscillate with phase opposition [14], the rolls'size seems to change continuously, from smaller to greater value compared to the mean size. Moreover, there is a phase delay between the streams' motion near the copper plates and their motion at the centre of the fluid layer, so the streams appear to undulate (or to vacillate) in the vertical plane.

The amplitude of the oscillation varies, depending on the roll position in the cell. As seen in fig. 2, some rolls do not oscillate (like nodes of vibrations) defining domains of oscillation inside which there is a smooth spatial modulation of the displacement amplitude. (For the The shaded region corresponds to the domain of oscillation. Wavelengths smaller than 0.3& have not been stabilized.

example shown in fig. 2, the maximum displacement is 61 = AB.) In the rectangular cell the positions for which the oscillation amplitude is zero are periodic in space and could define a stationary wave with a new characteristic length A --6A0 if A. is the actual wavelength. The frequency of the oscillations is exactly the same along the entire cell (=0.27Hz at Ra = 1.48 -lo6 which gives a period of around 1.1 -@IDT, where d2/DT is the vertical thermal diffusion time) and this perfect coherence can be surprising in a structure numbering 94 rolls. A second very low frequency can also be present (typically =h/50 with fi the fundamental frequency of the rolls' oscillations). In this case, the second frequency is due to a periodic small displacement of the nodes, i.e. of the position of the nonoscillating streams in the cell.

In the case of the annular geometry (I'v = 0.3 and Pr = 22), the same basic phenomenon is observed (with frequency =0,13Hz at Ra=4.8.106 which gives a period around 0.58.

10-2d2/DT). In addition, the relative spacing between periodic nodes is approximately the same for the two geometries. However, for the annular geometry, the wavelength is generally not homogeneous all along the circumference and, in this case, some parts are oscillating and others not. In fact this inhomogeneity reveals in a striking way that the collective oscillation of the streams is very sensitive to the local wavelength. Observations show that the oscillations are present in a very restricted domain of the local wavelength, near the value AJ2.5. The exact domain of stability seems to depend slightly on the Prandtl number and perhaps also on the transverse aspect ratio.

Discussion. -It is important to note that in the studied geometries, which attempt to simulate a one-dimensional chain of rolls, the first time-dependent behaviour which is observed when the Rayleigh number is increased corresponds to a well-ordered motion of the rolls, in particular in the case of the rectangular geometry. As the motion of the hot and cold streams is exactly the same along the entire length of the rolls' axes (observations from above [15]), there appears a tendency for the structure to adapt its local short wavelength to a greater value (I). So we are probably in the presence of an instability in the plane [Ra, a], which, to our knowledge, has not yet been revealed by any model. It could be of a nature similar to that of the instabilities studied by Bolton et al. [16], in the case of convection with a moderate Prandtl number fluid. But, in this later case, the horizontal extensions of the fluid layer are considered to be quasi-infinite, i.e. far from our experimental conditions (Tv < 1). Our geometry is closer to Hele-Shaw type, for which an oscillatory instability has been predicted, according to a model of porous medium [17], but for small wave numbers

From another point of view, a one-dimensional chain of rolls may be described by equations derived from the Swift-Hohenberg model and for which similar oscillations have been observed numerically [lS]. So this collective oscillating instability of the roll pattern, which is also evidenced in experiments using lower Prandtl number fluids

* * *

The authors thank particularly M, A. RUBIO for his active contribution, and P.

Fig. 1 .

 1 Fig. 1. -Shadowgraphic picture of the rolls in the whole rectangular cell.

Fig. 2 .

 2 Fig. 2. -U) Space-time evolution of the light intensity measured in the middle of the rectangular cell (21 rolls). Each continuous line is separated by 350 ms. Ra = 1.48 -lo6. b) Space-time evolution of the corresponding minima intensity (hot streams).

Fig. 3 .

 3 Fig. 3. -Domain of stability of the collective oscillating mode, as observed in the annular geometry.

  [19], seems to be a first step in the destabilization of quasi-one-dimensional chains of rolls, and a prelude to spatio-temporal intermittencies [ZO-221.only.

MANNEVILLE

  and H. CHATJ~ for constructive discussions. C. SCHOUKROUN and B. OZENDA are also acknowledged for their very efficient assistance.

(l) The boundary layer appears to have a leading role in initiating the oscillation phenomena discussed. The following observation tests this assumption: if the oil has not been thoroughly degassed, at a high-temperature difference very small bubbles (-0.1 mm) may appear at the lower plate. For a while they remain trapped within the boundary layer, wherein they describe a striking periodic motion in the horizontal direction. The motion of the bubbles attests for an oscillating motion in the boundary layer.