nonlinear dynamics of waves and modulated waves in 1d thermocapillary flows. ii. convective/absolute...

Physica D 174 (2003) 30–55

Nonlinear dynamics of waves and modulated waves in 1Dthermocapillary flows. II. Convective/absolute transitions

Nicolas Garnier∗, Arnaud Chiffaudel, François DaviaudGroupe Instabilités et Turbulence, Service de Physique de l’Etat Condensé, Direction des Sciences de la Matière,

CEA Saclay, CNRS URA 2464, Bˆat. 772, Orme des Merisiers, 91191 Gif-sur-Yvette, France

Received 30 October 2001; received in revised form 25 May 2002; accepted 5 June 2002

Abstract

We present experimental results on hydrothermal waves in long and narrow 1D channels. In a bounded channel, we describethe primary and secondary instabilities leading to waves and modulated waves in terms of convective/absolute transitions.Because of on the combined effect of finite group velocity and of the presence of boundaries, the wave patterns are non-uniformin space. We also investigate non-uniform wave patterns observed in an annular channel in the presence of sources and sinksof hydrothermal waves. We connect our observations with the complex Ginzburg–Landau model equation in the very sameway as in the first part of the paper [Nonlinear dynamics of waves and modulated waves in 1D thermocapillary flows. I.General presentation and periodic solutions, Physica D (2003), this issue].© 2002 Elsevier Science B.V. All rights reserved.

Keywords:Hydrothermal waves; Ginzburg–Landau equation; Eckhaus instability; Convective/absolute transition; Modulated waves

1. Introduction

Nonlinear traveling waves have exhibited a fas-cinating variety of behaviors and patterns. Wavessystems have been studied in binary-fluid convection(subcritical traveling waves bifurcation)[2,3], oscil-latory instability in low Prandtl number convection[4], oscillatory rotating convection[5] and cylinderwake[6]. One of the main source of richness in wavepatterns is the existence of two different regimes: theconvective and the absolute one[7,8]. This distinc-tion arises when the group velocity of the waves innon-zero. The convective/absolute transition betweenthose two regimes leads to critical phenomena of im-

∗ Present address: Laboratoire de Physique, Ecole NormaleSuperieure de Lyon, CNRS, UMR 5672, 46 Allee d’Italie, 69364Lyon Cedex 07, France.E-mail address:[email protected] (N. Garnier).

portant relevance in wave systems. We present herethe first complete description of convective/absolutetransitions for traveling waves in finite a box for bothprimary and secondary instability onset.

Most 1D physical systems produce right- andleft-propagating nonlinear waves[9]. Nonlinear com-petition and reflections at the boundaries then leadto a central-source pattern[10–13] due to counter-propagating exponentially growing waves as the firstglobal mode at onset. But the global mode may alsobe produced at the convective/absolute transition aswe will illustrate. So far, this phenomenon was de-scribed for single waves[7,14,15], i.e., with brokenleft–right symmetry.

Hydrothermal waves[16–19] provide very inter-esting and generic systems of traveling waves whichcan be modeled by envelope equations such as thecomplex Ginzburg–Landau (CGL) equation[20,21].

0167-2789/03/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved.PII: S0167-2789(02)00681-4

N. Garnier et al. / Physica D 174 (2003) 30–55 31

The present paper is the second part of a paper de-voted to the connection between hydrothermal wavesand those amplitude equations. We will further referto the companion paper[1] as I.

In I, we present the experimental hydrothermalwaves systems, and the periodic solutions obtainedin an annular cell. We will often refer to this paper,but we recall here inSection 2the main character-istics of our systems and some important results.The present paper is mainly devoted to the case of arectangular cell, i.e., of a long narrow channel withnon-periodical boundary conditions. We show thathydrothermal waves realize as in I an ideal super-critical nonlinear wave system in accordance withtheoretical predictions in such a bounded geometry:convective and absolute transitions have to be takeninto account to describe the effect of the group ve-locity over the onsets of both primary and secondaryinstabilities. We also present results in an annularcell with periodic boundary conditions in the specialcase where the periodicity is broken by the patternitself.

In Section 3, we study first the critical behavior atthe primary onset for a finite cell with low-reflectionboundaries. When the wave pattern appears, it isqualitatively very different from the uniform hy-drothermal waves (UHWs) pattern observed withinperiodic boundary conditions and described in I,through the governing equations are the same. Thisresults from the broken Galilean invariance due to theexistence of boundaries. We will show that the firstglobal mode, instead of being constructed by suc-cessive reflections in the convective regime, resultsfrom the onset of absolute instability. The onset ofthis mode—corresponding to the convective/absolutetransition—is shifted above the value correspondingto convective instability. This is illustrated in Fig. 4of I and we will detail qualitatively and quantitativelyhow the transition occurs. Some experimental criticalexponents are discussed in the framework of existingtheoretical descriptions and a quantitative comparisonwith CGL model is proposed.

Section 4 is devoted to modulated waves. Forhigher values of the control parameter, the wavepattern undergoes a secondary modulational instabil-

ity; we present the convective/absolute transition forthis instability in the bounded channel. This instabil-ity is of the same nature as the Eckhaus instabilityoccurring on UHW in the annular channel and de-scribed in the companion paper I: it leads also to awavenumber selection in the rectangular channel. Wewill introduce fronts of spatio-temporal defects andlink them to the convective and absolute nature of theinstability.

In Section 5, we show that the convective/absolutetransitions have reminiscent effects in the annular ge-ometry, when sources and sinks exist that break theGalilean invariance as physical boundaries do. Thegroup velocity term cannot be cancelled out of theequations anymore where such objects are present.This leads to qualitative and quantitative behaviorspredicted theoretically[9] and observed in our exper-imental system; we then confirm the pertinence of theconvective/absolute instability transitions.

2. The rectangular geometry and the wavemodel

The experimental system has been described inSection 2.1 and Fig. 1 of the companion paper I. Westudy the traveling waves instability of a thermocap-illary flow obtained when applying a horizontal tem-perature gradient over a thin liquid layer with a freesurface. Let us emphasize that we use annular andrectangular channels of both the same width 10 mm,and that the fluid height ish = 1.7 mm for all experi-ments reported in the present paper. The curvature isnegligible[22], and we have shown by stability anal-ysis [23] that its effects on critical values are of order3×10−2, so both wave pattern reports can be directlyconnected. Please note that the channel length willbe notedLb for the bounded (rectangular) channeland Lp for the periodic (annular) channel. Withoutsubscript,L will concern the current channel, andL∗

the non-dimensional lengthL/ξ0 whereξ0 = 5.1 mmis the coherence length of the pattern (see Section3.3 in I). We haveLb = 180 mm andLp = 503 mm;the aspect ratios in both cells ensure that patterns areone-dimensional.

32 N. Garnier et al. / Physica D 174 (2003) 30–55

The control parameter is the horizontal temperaturedifference�T between the two long sides of the con-tainer. The fluid is observed using shadowgraphy andtaking care of being in the linear regime (see I), i.e.,we record the shadowgraphic image on a screen lo-cated at a distance much larger than the focal distanceof the convection pattern, even for larger values ofthe temperature constraint[22]. The first bifurcationof the basic thermocapillary flow towards hydrother-mal waves occurs in the annulus forh = 1.7 mmat �Tc = (3.1 ± 0.1)K as described in Section 2.2of I. The bifurcated pattern for hydrothermal wavesis then a uniform-amplitude traveling wave of crit-ical wavenumberk0 = (0.684 ± 0.003)mm−1 andcritical frequencyf0 = 0.237 Hz. In the rectangulargeometry, boundaries act as sources or sinks and infact both right and left-traveling waves are present atthreshold. So hydrothermal waves must be modeledby two slowly varying amplitudesA andB obeyingtwo CGL equations:

τ0(AT + sAX) = ε(1 + ic0)A+ ξ20 (1 + ic1)AXX

−g(1 + ic2)|A|2A−g(λ+ iµ)|B|2A,τ0(BT − sBX) = ε(1 + ic0)B + ξ2

0 (1 + ic1)BXX

−g(1 + ic2)|B|2B−g(λ+ iµ)|A|2B,(1)

ε = (�T − �Tc)/�Tc is the reduced distance fromthreshold. Boundary conditions forA andB should beincluded with this description. Perturbations are veri-fied to travel at the group velocityvg in the rectangularbox as well as in the annular one. Please note that inthe CGLequation (1), s denotes the value of the groupvelocityvg at the convective onset. The coefficientsτ0,c0, c1, c2, g, λ andµ are all real and commented in I.

In order to quantitatively connect results in annularand rectangular geometry, we propose inFig. 1a rep-resentation of stable states in the annulus together withall states obtained in the rectangle. In this figure, as inall the text in this paper,ε is defined for both exper-iments using the onset in the annulus. Wavenumbersare expressed in such a way that they do not dependon the size of the system, so we use dimensional unit(mm−1). In the annulus, we observe UHWs, i.e., waveswith constant amplitude, wavenumber and frequency

and modulated waves, i.e., waves with modulated am-plitude, wavenumber and frequency. The existence ofUHW and modulated waves suggest that the wave sys-tem may be described using phase equations. Pleasenote that in the annulus, no particular selection of thewavenumber exists within the stability zone of UHW,except the restriction of that the wavenumber must bean integer when expressed in(2π/L) units. In the rect-angle, no such restriction exists and the wavenumbercan take any value.

Let us describe the states observed in the rectan-gular channel. First, we note that betweenε = 0and ε = εa = 0.18, we never observe waves. Thisfact will be detailed inSection 3presenting the pri-mary onset. When increasing the control parameterfrom the threshold valueεa, the wavenumberk in-creases while exploring the whole band of allowedwavenumbers.

Aboveε � 0.45, a smooth selection process occursand the wavenumberk is of order of 21(2π/Lb) �0.73 mm−1: this correspond to the vertical branch onthe right of the stability diagram. Forε > εm,c ∼0.758 ± 0.01, the waves are modulated and tworegions are observed in the cell, corresponding totwo different mean wavenumbers. The first regioncorresponds to the previous right branch and theother to the left branch of the diagram, withk ∼17(2π/Lb) � 0.59 mm−1. This selection processis rather sharp. Aboveεm,a, any state in the cell isrepresented by a point on each of the two branches.Betweenεm,c and εm,a, the left branch is only pop-ulated by transients. The splitting of the(k, ε) curvein two branches is due to the Eckhaus instability andwill be carefully detailed inSection 4.

Please note that far from onset, wavenumbers areeither greater or smaller than the critical one and in factthe two branches are away fromk = kc = (0.684±0.003)mm−1 (Section 3.6 of I).

Let us also note that in the rectangle, due to thenon-periodic boundary conditions, the amplitude isforced to vanish or have very low values, thus jeopar-dizing any phase description. Our wave system is suchthat the boundaries atx = 0, L are poorly reflective[17]: we have indications that the reflection coefficientis within the range 10−3 to 10−1.


Fig. 1. Experimental stability diagram for hydrothermal wave ath = 1.7 mm. Data are presented for both annular (�) and rectangular (�)geometry. In the annulus, small circles stand for homogeneous waves. Modulated waves are depicted with an additional cross (+). In therectangle, no waves are observed belowεa = 0.18, and forε > εm,c = 0.76 points are grouped on two different vertical branches. Thosebranches correspond to two different wave-trains at the same time within the cell. An additional+ in the squares denotes that those wavesare modulated.εm,c andεm,a stands for the onset of convective and absolute modulational instability (Section 4). The solid parabola is themeasured marginal stability curve.

Please note that for sufficiently highε (≥ 1), weobserve the same asymmetrical selection of wavenum-bersk < kc, as in the annulus. This global selectionconfirm the relevance of higher order terms perturbingthe amplitude equation (Section 3.3 in I). However,let us emphasize that when considering the primaryonset, i.e., small value ofε and small amplitudesAandB of the waves, we may neglect the higher or-der terms. In contrast, when presenting the Eckhausinstability leading to the selection of lower wavenum-bers, one should include those higher order terms toperform a correct analysis of the secondary instability.

The main result fromFig. 1is the perfect overlap ofthe zones of existence. Stable waves—Stokes solutionsof the CGLequation (1)—are observed in the rectan-gle and in the annulus in the same region of the(k, ε)plane. Moreover, modulated waves, and so the Eck-haus instability, occur in the same regions. Though thespatial extents of the cells are different, they are bothextended and the global wavenumber selection pro-cess looks the same: asε is increased, the wavenum-ber is expected to be lower. The overlap also confirmthat coefficients in amplitude equations are likely tobe identical in both geometries (Section 3.3 in I).


3. Onset of primary wave instability in therectangle

We describe here how the wave pattern appearsand evolves near the primary onset. As explained, forsmall values ofε as those considered here, higher or-der terms can be neglected in the amplitude equationsand we believe that usual CGL equations for travel-ing waves are sufficient for the description of the pri-mary onset. In the following we choose, for clarity,to present the major wave asA (right-traveling) andthe minor wave asB (left-traveling), but the reversesituation has been observed with equal probability.

3.1. Description of the patterns

Typical amplitude profiles in the rectangle forAandB are shown inFigs. 2 and 3for various tem-perature differences. For�T < �Ta = 3.65 K (ε <εa = 0.177), no wave is observed and soA = B =0. Just above onset, for�T = 3.66 K, we observe asymmetric wave pattern (Fig. 2). As�T is increasedabove�Ta, i.e.,ε is increased aboveεa, one wave be-comes dominant and invades a larger region of the cell

Fig. 2. Amplitude profiles of the traveling waves at differentε. Top figure presents right (A) and left (B) amplitude profiles forε = 0.18(�T = 3.66 K) and bottom figure forε = 0.21 (�T = 3.75 K).

(Fig. 3). The waves compete up toε � 0.25, abovewhich the smallest wave becomes negligible with re-spect to the dominant wave (Fig. 4). The wave enve-lope of the dominant wave may be seen as composed ofthree domains: (i) just after the wallX = 0 which mayalso be called source, where both amplitudeA andBare nearly zero, there is a front where the amplitude isexponentially growing; this is illustrated inFig. 3; (ii)after the front, a plateau is present for higher valuesof �T ; (iii) finally, just before the end wallX = L,the amplitude profile has a bump and maximum am-plitude is reached here, in what is called a wall-mode.

In the front region, we can compute for each realiza-tion the spatial growth rate of the amplitude envelope|A(X)| from the source, as shown inFig. 3. The front

can be described by an exponential envelope eξ−1F X,

whereξ−1F is the growth rate. The characteristic length

ξF is linked to the front critical behavior[17] and willbe described in the next paragraph. Please note thatthe spatial growth rate in the source region is well de-fined both close to the onset, and far from the onset.

The plateau is vanishing in the vicinity of the thresh-old and so it cannot be used to quantify the criticalbehavior. Quantitative description of the bifurcation


Fig. 3. Amplitude profiles of the dominant traveling wave for growingε up to 0.53 (�T = 4.75 K). On both figures, the data is the same;left: linear scale; right: logarithmic scale. All dominant waves are represented as right-traveling waves. Following the wave, we encounterthree domains: an exponential growth or front, a saturated plateau (for largerε), and a sharp wall-mode. The linear fits on the lin-log plotgive the value of the exponential spatial growth rateξ−1

F . Similarly, spatial growthξ−1WM and decay rateξ−1

down can be fitted on both sidesof the wall-mode on the right side of the plot.

are given in terms of the wall-mode amplitude and thefront spatial growth rate.

Please note that the amplitude is almost vanishingat the boundariesX = 0 andX = L. This is an

Fig. 4. Amplitude ratioB/A for both averaged amplitude〈A〉 over the cell (�) and maximum amplitudeAmax (+), see text for details.Solid line is an exponential fit. Asε is increased higher above onset, the minor wave disappear.

indication for a very low reflection coefficient in oursystem[24].

When ε � 0.3, the states are not always sta-tionary and we observe quasi-periodic realizations


Fig. 5. Blinking state: spatio-temporal diagram of the local amplitude showing the periodical oscillations of the source separating theright- and left-traveling waves forε = 0.36 (�T = 4.21 K). Left: left-traveling waves; right: right-traveling waves. The wave amplitude isproportional to the gray level of the images but for the minor (left) wave, amplitude has been multiplied by a factor of 2. Black correspondsto zero amplitude. A low-frequency beating is observed with a period around 320 s.

corresponding to a beating of minor and majorwaves. Those states were called “blinking states”[10,12,13,25]. Such a realization is presented inFig. 5.We believe that the occurrence of such quasi-periodicpatterns is due either to reflections at the boundariesbetween both counter-propagating waves and/or to thenonlinear competition between counter-propagatingwaves. This scheme is detailed inSection 3.3.

3.2. Quantitative results

Fig. 6presents a typical wavenumber measurement.We obtain its value everywhere the amplitude is largeenough to allow measurements: we discard regions inwhich the amplitude is less than 0.5 a.u. We then notethan except in the wall-mode region where amplitudevariations are large, the local wavenumber is almostconstant in the cell. The mean wavenumber is com-

puted by averaging values far from the boundaries. Itis close tok ∼ 21(2π/Lb) � 0.73 mm−1. Measuredvalues of frequency and wavenumber are presented inFig. 7. We note that they match the annulus critical val-ues at onset, as seen in the stability diagram (Fig. 1).For higher values of the control parameter, frequencyand wavenumber are multivalued; this results from theexistence of a new branch of solutions as presentedin Fig. 1. This new branch appears when the primarywavenumber becomes unstable with respect to modu-lational instability (Section 4).

Computing the velocity at which perturbations areadvected allows us to compute the group velocity. Thecorresponding data is represented inFig. 8, togetherwith the phase velocity of the waves. Here again,for clarity, we distinguish between waves of differ-ent wavenumbers, i.e., we distinguish between the twobranches in the stability diagram. Of importance is the


Fig. 6. Spatial profile of the local wavenumberk in the rectangle together with the amplitude profile, for�T = 3.75 K. Though noisy, thewavenumber is well defined far from the source region.k is fairly constant except in the wall-mode region where amplitude variations arelarge.

fact that the group velocity is always finite and large.Note that the value ofvg at εa is the same as the oneat onset in the annulus, i.e.,vg(εa) = s. The phasevelocity is about twice the group velocity.

Let us now present the critical behavior of the frontusing the spatial growth rateξ−1

F defined in the previ-ous paragraph. InFig. 9 are presented such data andpossible fits. Following the simplest intuition, we canpropose a linear fit for the data; this leads to an es-timation for the threshold at 3.65 K. A closer obser-vation of the experimental points at 3.66 K suggeststhis may not be the best description very close to on-set: we have conducted several experiment, atε = εa,and all led to a small but finite value ofξ , larger thanthe error bar. We also tried a square root law, perti-nent close to the onset but jeopardized for values ofε a bit larger. Following the authors of Refs.[15,26],we then searched for a linear relation betweenξ andln (ε− εa) and found a better agreement for all exper-imental points within a broader range, and most of all

close to the threshold. The physical meaning of thislogarithmic critical behavior will be discussed below.

Let us look at the amplitude inFig. 3and search todefine another order parameter of the transition. Theamplitude being non-uniform, two quantities are easilyextracted: the averaged and the maximum value ofthe amplitude profile, both represented inFig. 10. Theaverage amplitude〈|A(X)|〉[0,L] evolves linearly withrespect toε (Fig. 10b), but shows a small finite stepat εa. On the other hand, the maximumAmax whichoccurs near the downstream boundary, at the top ofthe wall-mode, behaves like(ε−εa)1/2 (Fig. 10a). Webelieve it to be the order parameter of this supercriticalbifurcation. Please note that this quantity is studiedhere very carefully around its onsetεa, while a globalview is given in I over a wide range ofε in order tocompare to the annular geometry which bifurcate atε = 0. Finally, as noted in the previous paragraph, thesaturation amplitude within the plateau region cannotbe measured close to the transition. This does not allow


Fig. 7. Frequency and wavenumber in the rectangle versus�T . Full horizontal lines show the critical values from annular cell data. Onthe left are shown zooms on the region [3.5 K, 4.5 K] corresponding to the primary instability and represented on the right figures bya segment. Different symbols are used for the two branches of solutions for higher values of�T ; those branches correspond to meanwavenumber 21(2π/Lb) = 0.73 mm−1 and 17(2π/Lb) = 0.59 mm−1, see alsoFig. 1.

Fig. 8. Phase velocityvφ and group velocityvg in the rectangle versus�T . � stands for waves of mean wavenumber close to 21(2π/L)(the right branch inFig. 1) and � for waves of mean wavenumber close to 17(2π/L) (the left branch inFig. 1). The group velocity isalways finite and lower than the phase velocity. The symbol atε = 0 stands fors, the group velocity at onset in periodical geometry andthe dashed line recalls the linear fit obtained in periodical geometry (Fig. 12 in I).


Fig. 9. Critical behavior of the spatial growth rate characterizing the front region. Left graph shows experimental data (� for stationarystates and+ for blinking states) together with a linear fit, a square root fit (solid lines) and a logarithmic fit (dashed line). Fits areperformed on stationary data (�) only. The graph on the right present the logarithmic fitξF ∝ ln (ε − εa), see text for details.

one to conclude anything about the threshold value; alinear fit of the squared amplitude versusε even leadsto an incorrect value of the onset (Fig. 10c).

Finally, the ratio of the two amplitudesA andB,plotted inFig. 4, behaves like exp(−α(ε−εa)/εa)withα = 3.0. This is observed both for the averaged am-plitudes(〈|A|〉, 〈|B|〉) and the maximum (wall-mode)

Fig. 10. Critical behaviors in the rectangle for: (a) the maximum amplitudeAmax, i.e., the modulus of the wall-mode; (b) the meanamplitude〈|A(X)|〉[0,L] ; (c) the amplitude of the plateau. The vertical line represent the onset valueεa = 0.18 (�T = 3.65 K). Only theamplitude of the wall-mode has a critical behavior in accordance with the observed onset value.

amplitudes(Amax, Bmax). The decrease of the minorwave is effective as soon asε > εa; this is differentfrom the behavior observed in[10]. It denotes a strongcompetition between the two waves. Due to the powerlaw increase of the maximum amplitude, this revealsthe exponential disappearance of the minor wave when�T is increased.


3.3. Discussion

3.3.1. Onset shiftThe onset shift between the annular cell and rect-

angular cell experiments is well interpreted using theconvective/absolute transition. This shift along withthe critical behavior of the front have been described in[17,27]. On the hydrodynamic point of view, a linearstability analysis of the thermocapillary problem in afinite box[28] has revealed a particular spatial growthrate—an envelope of the main unstable mode—at on-set, together with a shift of this onset from the valuecomputed in infinite geometry.

The shift of the onset is very large: 0.18 inε or0.55 K in �T . Usual finite size effect are known tobe of order(π/L∗)2, i.e., 0.01 inε for theL∗ = 35rectangular channel. This law has been verified witha good accuracy for hydrothermal waves in a variablerectangular cell[29]. Another difference between thetwo cells is the curvature. This can be quantified by alinear stability analysis of the thermocapillary problem[22,23]: in the annulus, the curvature increases theonset by 0.03 inε. None of these effects explains the0.18 shift.

So what in the rectangle makes the first global mode,or self-excited wave, observed aboveεa = 0.18? Be-low εa, no waves are observed and we unsuccessfullytried to trigger wave-trains with mechanical perturba-tions (but not with thermal perturbations which shouldbe more efficient as tested within a hot-wire experi-ment[30]).

3.3.2. Global eigenmode of the CGL modelIn the periodic channel, once the convective onset

is crossed, a traveling wave self-organizes after suc-cessive rounds in the cell: the basic uniform travel-ing wave is a global mode and both convective andabsolute onset collapse[7]. This corresponds mathe-matically to the Galilean invariance of the problem inthe annulus: one can eliminate the group velocitys bystudying the problem in a frame moving at velocitys.In the non-periodic channel however,s is finite and theproblem has to be solved in the laboratory frame. Thefirst global mode is then observed when the growthrate is large enough not to have the waves envelope

advected away by the group velocity. This correspondto the transition between convective and absolute in-stabilities of the primary wave pattern. This transitionoccurs in the CGL equation whenε reaches the criti-cal valueεabs given by

εabs= 1

1 + c21

(sτ0

2ξ0

)2

. (2)

Belowεabs, waves are convectively unstable in infinitegeometries[31,32]and no global mode exists in finitenon-periodical geometry[14]. For ε = εa, the globalmode in a bounded system of sizeL is of the form

A(x) = Agmsin

(πX

L

)e(1−ic1)ξ

−1absX,

ξabs= (1 + c21)

2ξ20

sτ0.

In the vicinity of this threshold, the global mode am-plitude is predicted to behave asAgm ∝ (ε − εa)1/2.

We proposed[17] that our system exhibit the transi-tion from convective to absolute instability at the onsetof the waves in the rectangular cell. This conjecturecan be summarized as

εa = εabs. (3)

The spatial structure of the predicted global eigen-mode is an exponential growth with a spatial growthrate ξ−1

abs independent ofε. The main feature of theobserved pattern, however, is the fast varyingξF ofthe front region (Fig. 9). Very close toεa, we ob-serve that the wall-mode is well visible (Fig. 3). Thiswall-mode shows a spatial growth, measured betweenthe plateau and the maximum, which is independentof ε (Fig. 11a). This measure is difficult to performbecause the wall-mode growth is almost hidden by thefront mode, but it appears clearly that the wall-modeis a good candidate for the global eigenmode. Quanti-tative measurements[17] confirms this hypothesis al-lowing to measure the value of characteristic time co-efficient τ0.

After the maximum of the wall-mode, one canfinally measure a spatial damping rateξ−1

down =(39± 4)L−1 = (1.36± 0.14)mm−1 for the envelopeof the wave in the wave sink in the vicinity ofX = L


Fig. 11. (a) Spatial growth rateξWM of the wall-mode measured upstream to the wall-mode as the spatial growth rate of the bump. (b)Spatial damping rateξdown of the envelope of the wave measured downstream of the bump, close toX = L. ξWM and ξdown are almostconstant with respect to the control parameters.

(Fig. 11b). Within the error bars, this quantity is in-dependent ofε, and is comparable to the wavelengthof the hydrothermal wave. This result is similar to theobservation of the damping in the core of wave sinksby Pastur et al.[33]. One may suspect non-adiabaticeffects to be responsible for this property. CGLmay not be the right model to describe the sinkcores.

Note also that for higher values ofε, the theory[14]predicts a wavenumber selection by the front, exactlyas we observe:k � 21(2π/L) � 0.73 mm−1 insteadof kc = 0.68 mm−1.

3.3.3. Effect of wave reflectionsThis scenario is not the only way to explain a shift

in the threshold. Cross et al.[12,13,34] proposed adifferent mechanism involving the two opposite trav-eling waves and their mutual reflections at the bound-ariesx = 0 andx = L. It is found that waves can beobserved only forε > εr, where

εr = −sτ0L−1 ln (r)+ O(L−2), (4)

where 0≤ r ≤ 1 is a real number, interpreted as a re-flection coefficient of one wave into the opposite oneat the boundary. Martel and Vega[35,36] exploredthis theory much further, explaining how a globalmode is constructed for two waves in a boundedgeometry. Their analysis also used the reflection co-efficient r between the two waves at the boundariesand the nonlinear coupling between the two waves.

They recovered the experimental succession of statesdescribed by Croquette and Williams[10], i.e., theexistence of a full range inε where the two-wave pat-tern is symmetrical. Furthermore, they described thesecondary instability; but they discarded the effect ofthe group velocity, i.e., the convective/absolute transi-tion, through it is of great importance in finite boxes[14].

The conjecture(3) is equivalent toεabs< εr for thehydrothermal waves system. Owing to poor reflectionsin the rectangular cell with Plexiglas ends, we believethat this is pertinent and thatεr is large.

Finally, another experimental fact confirms our con-jecture. The experimental sequence of competition be-tween right- and left-waves withε (Figs. 2, 3 and 5)is the following: symmetric state→ asymmetric state→ quasi-periodic blinking state→ “filling” state, i.e.,a single wave. This sequence fits the description incase of reflection-controlled mechanism[12,13,35,36]except that the initial symmetrical pattern exists onlyfor ε = εa instead of existing over a finite range as isthe case in the reflection-controlled mechanism, andobserved experimentally in that case[10,13].

Blinking states are observed in our experiment forε � 0.3, we may suggest that this value could be anupper estimate forεr. It can also be interpreted as thefact that nonlinear interactions become important forsuch higher values of the control parameter.

At the present time, a complete model taking intoaccount both the convective/absolute transition and the


role of the reflection coefficient and the nonlinear cou-pling remains to be done in the case of a two-wavesystem.

3.3.4. The nature of the convective/absolute transitionThe last question we wish to address concerns

the nature of the convective/absolute transition. Weobserved that the front size behaves logarithmicallyaroundεa (Fig. 9). This behavior has been detectedexperimentally by Gondret et al.[15] in nonlinearsurface waves and is the signature, in the sense ofChomaz and Couairon[26], of a nonlinear convec-tive/absolute transition. A linear transition[14,26]would show the front size to behave as(ε − εa)−1/2.This has a practical consequence on the experimentalobservation: since all derivatives of the spatial growthrate are infinite atεa, the front appears much fasterwith ε than the eigenmode. This explains why theeigenmode is almost hidden by the front and maybe detected only as a faint wall-mode in the closevicinity of εa.

For a better observation of the eigenmode one cansuggest to increase the lengthL of the channel. Theresults of Pastur et al.[33] for a long hot-wire exper-iment could have lead to such observation. However,since εr decreases withL (Eq. (4)), these authorsreport another result: belowεa (referred to byεso),they observe fluctuating sources due to the amplifi-cation of experimental noise by the convective insta-bility. Near εa, a crossover leads to non-fluctuatingsources, comparable to the front we observe: thesources are located in the middle of the channel andnot at one end, but their size also critically dependof ε. However, since waves and sources are alreadypresent in the convective region belowεa, the tran-sition itself is also hidden and its nature cannot bedetermined.

By increasing the length of our cell and/or by reduc-ing the length of Pastur et al. cell[33] we are confidentthat one could make the connection between the twoexperimental observations: to observe the eigenmodeas clearly as we do below a certain critical channellength, and then analyze how the convective wave be-comes visible belowεa whenL gets long enough forεr to become smaller thanεa.

4. Onset of secondary modulational instability inthe rectangle

Experiments on nonlinear TWs have been fre-quently carried in annular cells[2,4–6,16] for thesimplicity of the underlying wave pattern. In suchperiodic geometries, despite an eventual shift of theonset[37,38], the Eckhaus instability is always abso-lute. Nevertheless, this instability, as the primary one,may be convective when the group velocity cannotbe canceled out of the equations[39,40]. Moreover,the main specificity of our wave system is to be-come Eckhaus unstable for increasing values of thecontrol parameter, i.e., as a first step on the route tospatio-temporal chaos[1,16].

As shown inFig. 1, this secondary instability isnon-symmetrical ink − kc; we proposed in Section3.3 of I that higher order terms should be included inthe amplitude equation. Such terms are of importancewhen considering secondary instability becauseε isthen no longer small.

Evaporation is limiting the duration of experi-ments and refills strongly perturb the patterns. So,in order to get long data series close to onsets, wegenerally used a protocol in which refills are madejust before acquisition starts and long thermal stabi-lization time are avoided: the temperature gradientis first established in the vessel, the cell is refilledand the fluid is agitated to break the thermal gradi-ents. Within a few seconds the hydrothermal wavesreappear without history, i.e., without any specialvalues of the wavenumber, or any special positionof dislocations within the cell. However, such aprotocol makes it difficult to test the presence ofhysteresis at the transitions with control parameterε.

4.1. Wave system

From the previous section we know that forε >0.45 (�T > 4.5 K), a single wave is present in thecell with constant amplitude, wavenumber and fre-quency (Fig. 3). This pattern constitutes now our basicstate and the present section focus on the secondaryinstability of this single wave-train for higher values


Fig. 12. Spatio-temporal diagrams of the local and instantaneouswavenumberk(x, t) of the wave: temporally stabilized regimesfor: (a) ε = 0.79, �T = 5.54 K; (b) ε = 0.82, �T = 5.65 K.The waves propagate from left to right. The mean wavenumbercan be estimated visually by the mean gray level and is labeled(units 2π/L) in the upstream (ku � 21(2π/L)) and downstream(kd � 17(2π/L)) regions. A uniform wavenumber (a) correspondsto a homogeneous state and illustrates both stable and convectiveregimes below�Tm,a. It is the asymptotic regime obtained afterthe transient shown inFig. 13 (left). The modulated state (b) isthe global mode of the Eckhaus instability. Each black to whitetransition of the wavenumber value atxF/L = 0.32 is due to aphase jump in the core of a defect. The defect front is stable alongtime. (c) By Hilbert demodulation of phase gradient image (b) weget the spatial profile of the amplitudeAmod of the modulation,presented here in logarithmic units.

of ε. We choose it to be a right-traveling wave andwrite its amplitude asA.

In the absence of modulations, this wave-train isuniform. This is illustrated inFig. 12a: the local andinstantaneous wavenumber is homogeneous in the celland constant in time. When modulations are present,as inFig. 12b, we perform a second demodulation to

compute the amplitude of this modulation, as well asits wavenumber and frequency. This second Hilberttransform is computed over a spatio-temporal im-age of the carrier phase derivative (wavenumber orfrequency).

As the group velocity is finite, all perturbations, in-cluding the modulational ones, are advected. We willshow the relevance of a new object, namely a front: adislocations front or a modulations front. In periodicconditions (Section 4 of I), this modulational instabil-ity occurs at the lowest possible wavenumberKmod =2π/Lp [16,37]: it is strictly an Eckhaus[41] instabil-ity. In the rectangular channel, we will also refer toEckhaus instability, although the wavenumber of themodulational instability modes are somewhat bigger:typicallyKmod ∼ 4(2π/Lb); it is a finite wavenumberinstability.

In the following, we describe the observed regimesstarting from the absolute one obtained for highervalues of the control parameter; we then present theconvective and seemingly stable regimes for smallervalues ofε.

4.2. Absolute instability and corresponding states

Figs. 12 and 13present the three states whichsupport our discussion. Forε > εm,a = 0.79 or�T > �Tm,a = (5.56 ± 0.03)K, the observed pat-tern (Fig. 12b) can be described as a wave composedof two wave-trains of mean wavenumbersku andkd.The wavenumber, frequency and amplitude of bothwave-trains are modulated in space and time. ThewavenumberKmod of the modulation is of order of|ku − kd|. Waves are emitted by one side of the cellwith wavenumberku ∼ 21(2π/L) � 0.73 mm−1

and propagate along the cell at the phase velocity.The phase modulation of this wave-train, traveling atthe group velocity, is spatially growing. InFig. 12c,we clearly see the exponential growth of the localwavenumber modulation amplitudeAmod alongx. Ata fixed, finite distancexF from the source-boundary,the wavenumber modulation is so large that it al-lows the wavenumber to change fromku to kd bytime-periodic phase slips. Forx > xF, the meanwavenumber iskd ∼ 17(2π/L) � 0.59 mm−1. In this


Fig. 13. Spatio-temporal diagrams of the spatial phase gradient (local and instantaneous wavenumber) for transients leading to a homogeneousstate like the one ofFig. 12a. Initial conditions follow the protocol described in the text, they have been prepared att = 0. A dislocationfront is slowly advected out of the cell. The modulations grow alongx but vanish alongt : this is the signature of a convective instabilityregime. Left: right-traveling wave forε = 0.79 (�T = 5.44 K), the front position moves monotonically and leads to the final stagepresented inFig. 12a. Right: left-traveling waveε = 0.78 (�T = 5.52 K), the front position oscillates before being evacuated at a roughlyconstant speed.

second region, the modulation is damped (Fig. 12c):we conclude thatku (resp. kd) waves are unstable(resp. stable) with respect to modulations.

We calldislocation frontthe set of spatio-temporalloci where spatio-temporal dislocations occur. Forε > εm,a, the positionxF of this object is station-ary; Fig. 14 shows the relation between the controlparameter and the front position which remains lo-cated in the first half of the cell whateverε. Steadydislocation fronts have already been observed fortraveling waves in a Taylor–Dean experiment[42]. Ingeneral, hysteresis as not been investigated. From themodulation amplitude profilesAmod(x) (Fig. 12c),we also extract the spatial growth rate of the modula-tions: this quantity will be discussed below together

with convective and stable regimes (Section 4.4andFig. 17).

We pretend those stationary states to be the globalmodes for the modulational Eckhaus instability. Allperturbations leave the structure of those modes un-affected; the front position is always the same at agiven value ofε. The structure of these global modesis very original: nothing seems to saturate the mod-ulations except the break-up of the underlying wavepattern, i.e., the abrupt change of the mean wavenum-ber downstream the dislocations. Similar patterns havebeen observed numerically in semi-infinite[43] andclosed cells[14]. Like Couairon and Chomaz[43] weobserve the nonlinear global threshold and the abso-lute instability threshold to be identical.


Fig. 14. Spatial positionxF of the dislocation front for absolutely unstable states versus�T . Stable and convectively unstable states withoutpermanent dislocation front are represented as realizations atxF = L (arbitrary choice).

4.3. Convective instability states

For �T < �Tm,a, dislocation fronts are not ob-served on asymptotic states. The asymptotic regime(Fig. 12a) is a homogeneous wave, of uniform unmod-ulated wavenumberku ∼ 21(2π/L). However, tran-sients obtained after control parameter changes showpropagating dislocation fronts (Fig. 13). These frontsare slowly advected out of the channel: those statesare convectively unstable states with respect to themodulational Eckhaus instability. Note that the frontposition can increase monotonically or with an oscil-lation at a given low frequency. It seems that this os-cillation may result from the bouncing of the front onthe boundary. Although, there is no possible reflec-tion of the modulation on the boundary because thereis not support for a backward traveling modulationon the right-traveling carrier. In both cases, we mea-sured the velocity of the front at the end of the processwhere the velocity is almost constant. Those movingobjects are observed in the small gap betweenεm,c =0.76 (�Tm,c = 5.45 K) andεm,a = 0.79 (�Tm,a =5.56 K).

4.3.1. Noisy source statesDue to the special hydrodynamics of the thermo-

capillary problem, there is another way of observing

transients and measuring spatial growth rates. The sil-icon oil used in the experiments is very volatile andduring long experimental runs, the fluid depth in thecell decreases. Of course, we keep working with fluiddepth almost constant aroundh = 1.7 mm, but dur-ing some runsh can decrease down to 1.65 mm. Forthis small depth—small but still within the error barswe allowed—the boundary emitting waves, acting asa source, becomes larger in space and noisy. This maybe interpreted as follows: the system has at the be-ginning of the run a rather well-fixed wavenumber—around 21(2π/L)—and wants to keep locked on thisvalue for a givenε, even if the fluid depthh decreases;this is only possible if the effective size of the systemis reduced by shifting the source from the boundaryof the cell to the interior of the cell. This gives an ap-parently large source because the opposite travelingwave is very damped and almost invisible. The posi-tion of the displaced source is not well defined andthis may explain the fluctuations observed. Moreover,the source may emit modulations, so we call it a noisysource.

Although hydrodynamical considerations may bet-ter explain the exact nature of this source, we onlylook at it in the present study as a source of modula-tions over the upstream wavenumberku, the stabilityof which we are considering. The aim is to study the


Fig. 15. Modulations emitted by a noisy source forε = 0.70 (�T = 5.27 K). The noisy source appears at timet � 1000 s atx/L � 0.1.Left: spatio-temporal diagram of the local frequency obtained after using Hilbert demodulation (400 s< t < 2800 s). Right: spatio-temporaldiagram obtained after a second demodulation applied on the left-side diagram, showing the amplitude of the modulation (0 s< t < 6400 s).Before time t � 1000 s, the right-traveling hydrothermal wave is unmodulated: uniform gray level for the frequency (left) and black orzero amplitude for the modulation (right). Aftert � 1000 s, the wave is modulated. One can see on the amplitude of the modulation (rightdiagram) that, after a transient (t � 4000 s), the perturbations emitted by the source are spatially and temporally damped: this is illustratedby the darker zone in the upper right part of the image. It is the signature of the hydrothermal wave being stable with respect to modulations.

response of our system under a continuous forcing andsuch a source provides a convenient forcing.Fig. 15shows the response of the system to the apparition ofa noise source. InFig. 15, a uniform right-travelingwave is displayed fort � 1000 s. Att � 1000 s,hdecreases under a critical value and a source appearsnearx/L � 0.1. Then, modulations are continuouslyemitted from the source. InFig. 15 (left) these mod-ulations appear as waves on the local frequency data.The amplitude of this modulation is viewed inFig. 15(right) obtained after demodulation of the left diagram:black represents unmodulated waves (zero amplitude)and white represents the maximum modulation levelreached. After a complex transient betweent � 1000

and 4000 s, we observe the modulation amplitude todecay along both spatial and temporal axis. In thiscase, the basic unmodulated waves are thus stable withrespect to the modulations induced by the presence ofthe noisy source.

In Fig. 16several profiles of the modulation ampli-tude are presented for variousε. Such profiles are ex-tracted at the end of diagrams similar toFig. 15(right),i.e., in the asymptotic state where the modulation am-plitude decays along time. Again, the amplitude of thefrequency modulation is obtained by a second Hilberttransform performed on the local frequency data. Weobserved that forε > εm,c perturbations are spatiallyamplified whereas they are damped forε < εm,c.


Fig. 16. Superposition of several asymptotic profiles of modulation amplitude in the case where a noisy source is present. The source is onthe left (x = 0) and the waves are right-propagating. Those profiles are extracted at the end of the time series, but still during the transientwhich may take a very long time to completely relax to zero (seeFig. 15, right). The signal portion just downstream of the source core(x/L � 0.2) reveals the stability of the modulation. From bottom to top,ε = 0.690, 0.732, 0.752 and 0.765. For the two lower values ofε, the modulations relax to zero as they propagate: the unmodulated wave is stable. For the two higher values ofε, the modulations arespatially amplified: the unmodulated wave is convectively unstable. The critical on set value for the transition isεm,c = 0.758± 0.010 or�Tm,c = (5.45± 0.03)K.

4.4. Stable/convectively unstable transition

Forε < εm,c, asymptotic states are uniform and dis-location fronts do not exist in the absence of forcing.In the presence of forcing by a noisy source, the per-turbations are spatially damped as they are advectedaway from the source (Fig. 15). Close toεm,c verylong transients are often observed. These transientpatterns are also slightly modulated; this is illustratedin Fig. 15. Most often, the modulations do not reachthe critical amplitude producing dislocations. Asymp-totically, the modulation amplitude is decreasing(negative spatial growth rate) along the downstreamdirection for ε < εm,c, and increasing forε > εm,c.Anyway, sinceε < εm,a, we observe the modulationamplitude to decay along time. On the modulationamplitude image inFig. 15 (right), a dark corner onthe upper right part on the image signals the spatialand temporal damping of the modulation after a longrelaxation. The complete relaxation may last muchlonger than the experimental running time, and thoseresults have to be considered with care. Aboveεm,c,asymptotic states in the presence of forcing are notuniform but constitute global modes under an externalforcing.

On a quantitative point of view, we measured spa-tial and temporal growth rates of the modulation.We will present these data for the unstable upstreamwave-train. Thetemporal growth rate for modula-tions in the laboratory frame is negative belowεm,aand positive above. It is also close to zero aroundthe convective transition where very long transientsare reported. The spatial growth rate of the upstreamku wave-train for all three regimes is presented inFig. 17a. It is positive for both unstable states butthe slope is seemingly different in the convective andabsolute regimes. It is negative belowεm,c.

4.5. Perturbed states

In order to test the above description, we perturbedthe uniform states by either plunging a thin needlein the convective layer or by dropping a cold or hotdroplet of fluid. Two examples are reproduced inFig. 18. The frequency contents of those perturbationsdiffer from the above reported transients or forcedstates: the modulation wave-trains contains only afew wavelengths and appears to be advected down-stream at roughly the group velocity. All observedperturbations show positive spatial growth rate and


Fig. 17. (a) Evolution of the spatial growth rate of the modulation with the control parameter for spontaneously modulated wave patterns,transient (�) or steady (�). Linear fits of the three regimes—stable, convective and absolute—are presented. They intersect atεm,c andεm,a. These data concerns the modulations of the upstream region of the cell whose mean wavenumber isku ∼ 21(2π/L). Correspondingdata for the downstream region are negative while�T ≤ 8 K. (b) Idem for perturbation initiated wave-packets in the stable (+) andconvectively unstable (�) regimes. The solid lines recall the fits of (a) to allow quantitative comparisons: the same selection of the growthrate is observed in both cases for the convective regime.

negative temporal growth rate in the laboratory frame.The spatial growth rates are presented inFig. 17b. Inthe convective regime, the growth rate appears to beselected at the same value than in spontaneous tran-sients or forced states. In the stable regime, however,the data are very dispersed but remain positive.

Fig. 18. Spatio-temporal diagrams of the local frequency showing the response to an external mechanical perturbation. Left:ε = 0.735(�T = 5.38 K); a perturbation is induced at timet = 130 s. Right:ε = 0.764 (�T = 5.47 K); a perturbation is induced at timet = 0 s.Modulations are spatially growing, but are advected away very quickly. So they disappear after a few hundred seconds.

4.6. Discussion

First, let us point out that the upstream and down-stream wavenumbers are incommensurate, and so arethe associated frequencies. InFig. 7, two branches arevisible on the frequency data, corresponding to the


Fig. 19. Ratio of the frequency in the upstream region by the frequency in the downstream region versusε. This ratio is almost constant,and its value(1.130± 6)× 10−3 is far from any simple fraction.

two different wavenumbersku andkd. The mean ratiobetween the frequency ofku andkd is (1.130± 6)×10−3, i.e., far from any simple rational fraction. Thisis shown inFig. 19. We then conclude that those twofrequencies are not resonant and that no locking occursin our system.

Second, let us point out that the modulation ampli-tudeAmod never saturates. All observedAmod profilesappear locally exponential alongx. No nonlinear sat-uration effect is thus observed. The dislocation onset,for a givenAmod ∼ |ku −kd| is the only limit to expo-nential growth. This is a strong argument for the Eck-haus instability to behave subcritically in this closedcell. Remember it is supercritical in the annular cell[16] (see I for discussion).

Third, the modulation wave system is a perfect sin-gle wave system: reflection of the modulations at theboundaries are irrelevant for there is no possibility forreflected information to travel back to the source with-out being supported by a counter-propagating wave.

The observational facts related above are coherentwith the interpretation in terms of convective and ab-solute instability. The striking point is the positive spa-tial growth rates for perturbations in the seeminglystable regime belowεm,c. As for spontaneously mod-ulated patterns, we would expect those modulationwave-packets to decrease in space exactly as the stablekd wave-trains do in the absolute regime (Fig. 12c).

We may explain this in the following way: supposethe convective instability is subcritical as suggestedpreviously, then, aboveεm,c, the transient evolves on

an unstable branch (Fig. 13), close to the absolutebranch (Fig. 12b). However, belowεm,c, a secondunstable branch co-exists, which can be reachedonly by perturbing the flow: this description can besupported by the schematic representation shown inFig. 20 inspired by zero group velocity instabilities.These branches present very different patterns. Theupper branch exhibits extended modulations over thewhole cell, with slow evolution and, for high enoughamplitudes—the generally observed case aboveεm,c—dislocation fronts. The lower branch exhibits fasttraveling narrow modulation wave-trains and cannotbe reached spontaneously by varyingε.

This hypothesis can explain the very different as-pect of spontaneous and induced transients in thestable regime belowεm,c. It is also known that theshape of induced nonlinear patterns below subcritical

Fig. 20. Schematic representation of the observed regimes, basedon the usual representation of a subcritical bifurcation with zerogroup velocity. The control parameterε is in abscissa, while theordinate is only qualitative. Solid heavy lines represents the steadystates, bifurcated or not, above or belowεm,a = 0.794 K. The thindashed lines may account for two different transient modes (seetext).


Fig. 21. Front velocity around the convective/absolute transition. The circles (�) show the velocity of dislocation fronts in transientconvective regimes belowεm,a. The (negative) velocities of transient modulation fronts invading the cell from downstream, aboveεm,a,are shown by squares (�). For comparison, the group velocity is 0.90 mm s−1.

instabilities depends on the forcing amplitude[44],so the dispersion inFig. 17b may be due to both theeffect of amplitude and the presence of two branches.

Another observation of the convective branch is in-triguing. We report inFig. 21the asymptotic velocityof the dislocation fronts betweenεm,c and εm,a, i.e.,the tangent to the space–time trajectory when the frontquits the cell as inFig. 13. The observation is sur-prising: the closer the absolute instability onset, thefaster the front moves! And then jumps below zeroabove�Tm,a. A contrario, aroundεm,c, the front ve-locity is zero, leading to infinitely long transients, i.e.,temporal marginality. This quantifies the experimen-tal complexity of carrying the experiment around thispoint.

What is the meaning of the velocity jump atεm,a?Is the convective/absolute transition also subcritical?Probably it is: while our protocol did not allow toexplore all branches by varyingε up and down fromone state to another, a test have been made to transitdirectly from an absolute state to a stable state justbelow εm,c: the absolute modulation profile remainsfixed in the cell. This can be either due to hysteresis,due to the vanishing front velocity, etc., which makesthe system marginal in this region. This point wouldneed to be addressed with an improved experimentaldevice.

Let us note that for higherε values, the front po-sition xF(t) exhibits chaotic behaviors and can thusbe viewed as the order parameter for the modula-tional instability up to the transition to spatio-temporalchaos. An example of disordered state is presented inFig. 22which suggest that the spatio-temporal behav-ior can be described using this front as a dynamicalsystem.

Last but not least, modulations in the rectangle canbe connected to modulations in the annulus describedin Section 4 of I. First, in periodic boundary condi-tions, the Eckhaus secondary instability is supercriti-cal for wavenumbers close to the critical one, whereasit is rather subcritical far from the critical wavenumberkc. This is confirmed by observations in non-periodicalboundary conditions: the selected wavenumbers arefar from kc in the rectangle and the modulational in-stability is subcritical. This opens the following ques-tion: can the observed modulations in the rectanglebe described as modulated amplitude waves (MAWs)[45–47]? Such modulations have already been seen byKolodner et al.[2,3,5]. As pointed out before, mod-ulated waves in the rectangle are not saturated, nei-ther should be corresponding MAWs; moreover, thespectral richness of the modulations is weak in therectangle: modulations are almost monochromatic. So,speaking of MAWs, we are facing solutions of the


Fig. 22. Transient forε = 1.74 with ku = 19(2π/L) = 0.65 mm−1 and kd = 16(2π/L) = 0.56 mm−1. A period doubling occurs for themodulation front: one modulation over two explodes in a dislocation forming a first dislocation front in space–time, then one modulationover four explodes forming another dislocation front, then one over eight forms another front, and so on until all those dislocations frontsmerge together leaving a state in whichku is absolutely unstable.

CGL equation that connect two non-saturated modu-lated waves.

5. Sources and sinks in the annulus

Coming back to the annulus will allow us to showthat the convective/absolute transition advocated forin the rectangle is also relevant in periodic boundaryconditions, when the Galilean invariance is brokendue to the presence of both right- and left-travelingwaves. The interpretation of bifurcations in the rect-angle in terms of convective/absolute transitions isthen reinforced.

5.1. Obtaining source/sink pairs

Let us describe how patterns form whenε is rapidlyincreased from a negative value to a supercritical valueεf ; describing those transients allow us to distinguishbetween different behaviors. In all such experiments,the waves first appear in small patches at several placesin the cell. The envelopes of those patches propagate atgroup velocity while their spatial extension increase.After a short transient, waves have invaded all the cellbut sources and sinks are presents which are reminis-cent of the boundaries of the initial patches. The num-ber of initial patches, and so the number of sources andsinks depends on the time derivative of the ramp inε


leading toεf : the faster the control parameter is variedacross the threshold of waves, the more sources/sinkspairs are present in the cell; a typical realization showstwo or three pairs. Quasi-static variations ofε showsat least one pair.

This first “invasion” time is followed by a secondtransient regime where sources and sinks interact.Fig. 23 presents a typical competition leading to theannihilation of two source/sink pairs. For this exper-iment, we start from a supercritical valueε = 0.14(�T = 3.52 K) and a single right-traveling wave andthen reduce the control parameter by switching off theelectrical heating of the inner block. So�T decreasesand the right-traveling wave disappear as the onset iscrossed from above. As soon as the wave disappear,we switch the heating on again. Waves reappears in

Fig. 23. Spatio-temporal diagrams of the local amplitude showing the initial competition between right- and left-propagating waves in theannulus. Left: amplitude of left-traveling waves; right: amplitude of right-traveling waves. The gray scale is proportional to the amplitude(increasing from black to white).�T is increased slowly from a (slightly) subcritical value to 3.60 K (εf = 0.16). Two couples ofsource/sink appear quickly at timet = 2400 s; then they interact and only a single wave remains after timet = 2600 s. Note that initially(beforet = 1800 s), only the right-traveling wave is present, and aftert = 2700 s, only the left-traveling wave is present; on that remainingleft wave, a transient modulation is present.

four different patches forming two sources and twosinks. Then, after a 5 min transient, the left waveremains alone.

A nice observation is done by looking around thevalue εa. If the operating valueεf is close to 0, allpairs annihilate within the transient and the asymp-totic pattern is always a single homogeneous travel-ing wave. For 0< εf < εa = 0.18, no source/sinkpattern has been seen asymptotically. In contrast,when εf > εa, i.e., the ending value of the im-posed control parameter is highly supercritical, it ispossible—but not mandatory—to have a source/sinkpattern frozen on long times. We have observed thatthe higher theεf is, the easier the source/sink statesare frozen[16]. For εf > εa, we have observedsome realizations in which sources and sinks are


present after times much longer than the diffusiontime.

So in the annular experiment, sources and sinkshave been observed during transients for all pos-sible values ofε. But a striking result is that thecouples of sources and sinks are always unstablebelow ε < εa = 0.18; in that case, all source/sinkpair collapse and only a homogeneous single wavesubsists. In contrast, sources and sinks have beenobserved in a stable way forε > εa, i.e., at leastone such pair lasts as long as the experiment isperformed.

5.2. Convective/absolute onset in the annulus

Below εa, source/sink couples are unstable in thesense that they collapse together, but sources can alsobe qualified as unstable because they emit modula-tions. Those modulations are also present close to theonset of waves.

It has been theoretically and numerically[9,48] es-tablished that stables sources only exists in a 1D pe-riodical geometry when the control parameterε isaboveεabs, i.e., when the instability is absolute. Thismay be understood remembering that in the convec-tive regime, the wave system acts as a noise amplifier.The two different regimes for the sources are to be dis-cerned: belowεabs, sources are noisy and their widthis large; aboveεabs, sources select a wavenumber andhave a smaller spatial extend.

Measurements of the source width have been per-formed recently in a very long rectangular geometry[33],1 leading to a nice confirmation of this transition.Here, we use the qualitative observation of stablesources aboveεa = 0.18 and the disappearance of allsources belowεa to confirm thatεabs = 0.18 in theannulus. Using expression ofεabs in Eq. (2), and as-suming that the parameters involved are the same inboth rectangle and annular geometry, we are confidentthat the overall description using convective/absolutetransition for the onset shift in the rectangle isrelevant.

1 In Ref. [33], the geometry is non-periodical, but very extended.This allows to conjecture thatεr < εabs and so that waves existfor ε < εabs.

5.3. Discussion

As previously said, modulations are emitted by thesources, even ifε is close to 0. Moreover, when a sin-gle traveling wave is finally produced by the collapseof the last sink within the last source, modulations arealso emitted. So the single unmodulated uniform trav-eling wave pattern appears always after a transient inwhich initial modulations are present but damped. Thistransient is studied in I (Section 4.1.1 and Fig. 7). Thedetailed study of modulations emitted by the sourcesremains to be done, but one can expect that sources inthe annulus behave like the one in the rectangle. Forexample, if their position is not well fixed as it is thecase of noisy sources in the rectangle (Section 4.3.1),it may explain the same emission of modulations.

As studied in Refs.[9,49] the effect of the group ve-locity is also a key point for a better understanding oftraveling wave systems in both non-periodic and pe-riodic geometries. Other parameters such as the com-plex coupling coefficient (λ + iµ in Eq. (1)) shouldalso be tracked for its influence on the observed pat-terns in both geometries, thus suggesting that the CGLequations cannot be only described using simply twoparameters (c1 andc2 in Eq. (1)).

6. Conclusion

Owing to their apparition via a supercritical in-stability with finite frequency, finite wavenumberand finite group velocity, hydrothermal waves werepreviously shown to be very well modelized by anamplitude equation of the CGL type (paper I). In thepresent paper, we used a one-dimensional hydrother-mal wave system as an experimental expression ofthe one-dimensional CGL equation or of a system ofcoupled one-dimensional CGL equations, to explorethe effect of the boundary conditions.

We have presented the global mode appearing ina rectangular box at the absolute instability thresh-old for hydrothermal traveling waves. Qualitativeand quantitative comparisons have been performedto distinguish from the case of a reflection-controlledglobal mode. The relevance of the convective/absolute


distinction was demonstrated by accurate comparisonof threshold values and critical behaviors of the orderparameters. Those measurements have revealed thatthe transition is fully nonlinear in the sense of Chomazand Couairon[26] and is well connected to the pre-dictions of Tobias et al.[14] for a single wave pattern.

For higher control parameter values in the rectan-gular bounded cell, we observe a quasi one-directionaltraveling-pattern which undergoes an Eckhaus sec-ondary instability leading to traveling modulations.These modulated patterns behave as nonlinear frontswhose dynamics reveals convective and absoluteregimes as well[18,27]. Thus we have observed bothstable/convective and convective/absolute transitionsfor the modulational instability. The stable/convectivetransition is subcritical and is characterized by zerospatial growth rate for the modulation, together withzero advection velocity of the modulated pattern,which can be viewed as spatial and temporal marginal-ity. The convective/absolute transition is characterizedby the dynamics of dislocation fronts. According tofront velocity data, we suggest this transition to besubcritical as well which qualitatively differs from thecase of the annulus. This question deserves a theoret-ical support which remains yet, as far as we know,unexplored.

Finite group velocity in the presence of boundaries,leading to the transition from convective to absolute,are linked with important qualitative and quantitativechanges of the global structure of the wave patterns inrectangular geometry. We showed that this is also truein the annulus where sources emitting waves are verysensitive to the convective or absolute nature of the pri-mary waves. The description of one-dimensional hy-drothermal waves using Ginzburg–Landau equationsappears to be very complete and satisfactory. One mayexpect even more connections with future theoreticalwork on these equations.

Acknowledgements

We wish to thank Lutz Brusch, Jean-Marc Fles-selles, Joceline Lega, Carlos Martel, Wim van Saar-loos, Luc Pastur, Alessandro Torcini and Laurette

Tuckerman for interesting discussions. Special thanksto Vincent Croquette for providing us his powerfulsoftware XVin. Thanks to Alexis Casner and FrédéricJoly who contributed to the data acquisition, and toCécile Gasquet for her efficient and friendly technicalassistance.

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nonlinear dynamics of waves and modulated waves in 1d thermocapillary flows. ii. convective/absolute...

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