J. Stat. Mech. - UCSBSwift–Hohenberg model and from experiment for domain chaos near the onset of rotating Rayleigh–B´enard convection. Using both global Fourier analysis and

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ournal of Statistical Mechanics:An IOP and SISSA journalJ Theory and Experiment

Local wave director analysis of domainchaos in Rayleigh–Benard convection

Nathan Becker and Guenter Ahlers

Department of Physics and iQCD, University of California, Santa Barbara,CA 93106, USAE-mail: [email protected] and [email protected]

Received 4 August 2006Accepted 10 November 2006Published 4 December 2006

Online at stacks.iop.org/JSTAT/2006/P12002doi:10.1088/1742-5468/2006/12/P12002

Abstract. We present new results both from numerical simulations using theSwift–Hohenberg model and from experiment for domain chaos near the onsetof rotating Rayleigh–Benard convection. Using both global Fourier analysisand local wave director (LWD) analysis, the domain chaos patterns werecharacterized. Several image analysis techniques are discussed and were appliedto the director field obtained from LWD analysis. They yielded statisticalparameters, averaged over long time series, that characterize several aspects ofthe chaotic patterns. The effect of the finite image size on the determination ofthe domain switching frequency obtained with Fourier analysis, as well as withLWD analysis, was investigated.

Using the LWD method, we studied ξθ, a correlation length obtained from thetime-averaged autocorrelation of the local angle field θ. The defects and domainsin the pattern were examined locally and their average area fraction, averagearea, and average major axis length were obtained. A periodic component wasobserved in the time series of the instantaneous correlation length ξθ and of thedefect area, but was found not to be present in the time series of the domainarea. We determined a parameter D, defined as the average dot product of thewave director and the major domain axis, that characterizes the orientation ofthe domains relative to the roll patterns within them. The results for D wereindependent of the control parameter ε and of the image size. Simulations andexperiment gave remarkably similar results for D.

Keywords: hydrodynamic instabilities, patterns

c©2006 IOP Publishing Ltd and SISSA 1742-5468/06/P12002+39$30.00

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Local wave director analysis of domain chaos in Rayleigh–Benard convection

Contents

1. Introduction 2

2. The Egolf–Melnikov–Bodenschatz (EMB) and Cross–Meiron–Tu (CMT) algo-rithms 6

3. Accuracy tests for the Egolf–Melnikov–Bodenschatz (EMB) and Cross–Meiron–Tu (CMT) algorithms 7

4. Configuration of the simulation and experiment 10

5. Frequencies determined from the time–angle correlation functions of the Swift–Hohenberg (SH) simulations 12

6. Details of the image analysis techniques 146.1. Angular correlation function . . . . . . . . . . . . . . . . . . . . . . . . . . 146.2. Defects and domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

7. Local wave director (LWD) results from the Swift–Hohenberg (SH) simulations 197.1. Angular correlation function . . . . . . . . . . . . . . . . . . . . . . . . . . 197.2. Defects and domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

7.2.1. Areas of defects and domains. . . . . . . . . . . . . . . . . . . . . . 217.2.2. Numbers of defects and domains. . . . . . . . . . . . . . . . . . . . 227.2.3. Area fraction of defects and domains. . . . . . . . . . . . . . . . . . 237.2.4. Lengths of defect major axes. . . . . . . . . . . . . . . . . . . . . . 257.2.5. Relative orientation of domains. . . . . . . . . . . . . . . . . . . . . 25

8. Local wave director (LWD) results from experiment 258.1. Angular correlation function . . . . . . . . . . . . . . . . . . . . . . . . . . 25

8.1.1. The time-averaged quantities. . . . . . . . . . . . . . . . . . . . . . 258.1.2. The time dependent correlation length. . . . . . . . . . . . . . . . . 27

8.2. Defects and domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

9. Summary 36

Acknowledgments 37

References 37

1. Introduction

Rayleigh–Benard convection (RBC) is one of the most popular systems for the studyof pattern formation [1]. It is extremely attractive because it is accessible from botha theoretical and an experimental perspective, and because it provides a wealth ofinteresting pattern formation phenomena [2]. The equations of motion (the Navier–Stokesequations) are known, potentially allowing a detailed understanding of the observationsand measurements [3, 4]. In the present work, we focus on a state of RBC known asdomain chaos (DC), which occurs in RBC with rotation about a vertical axis [5]–[15].

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Figure 1. A snapshot of size 60d×60d of domain chaos with ε = 0.125, Ω = 16.25,and Γ = 61.5.

This state is particularly interesting because it exhibits spatio-temporal chaos at theonset of convection where there are quantitative specific predictions from theory that canbe tested experimentally.

The RBC system consists of a layer of fluid, with thickness d, confined between twoplates. The bottom plate is heated and the top plate is cooled, resulting in a temperaturedifference ΔT . When ΔT reaches a critical value ΔTc, the heat transport between theplates makes a transition from being purely conductive, aside from fluctuations [16]–[18],to being both conductive and convective.

When the sample is rotated about a vertical axis at a rate Ω greater than a criticalΩc, the Coriolis force induces an instability of the convection rolls, known as the Kuppers–Lortz instability [5, 6], that leads to domain chaos even at the onset of convection. Thena set of rolls is unstable to a new set that is oriented at a particular angle relative to thefirst. However, once this new set overcomes the original one, it is also unstable to anotheroriented again at an angle relative to it. This leads to persistent and chaotic switchingdynamics of the convection roll pattern. During their brief lifetime, the rolls form moreor less coherent patches known as domains. A snapshot of DC obtained from experimentis shown in figure 1. The dynamics are generally much more complex than the simple rollswitching picture explained above, because they are dominated by motion of the domainwalls and of travelling defects peppered liberally throughout the pattern.

Since domain chaos appears as a supercritical bifurcation from the conductionstate [7, 11], it offers a unique opportunity for theoretical study because weakly non-linear theories are expected to be applicable. These theories, in the form of Ginzburg–Landau (GL) or Swift–Hohenberg (SH) equations, by virtue of their general structure andfrom their numerical solutions, make a prediction for instance about the dependence of acorrelation length ξ and a characteristic frequency f on ε ≡ ΔT/ΔTc−1. The correlationlength is the inverse half-width at half-height of the structure factor (SF, the spatial powerspectrum of the pattern). The prediction is that ξ should vary as ε−ν with ν = 1/2 as εvanishes [19]–[27], and that it is the only diverging length scale in the problem. Similarly,f should vary as εμ with μ = 1 and its inverse should be the only long timescale in

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the problem. Thus it was particularly disappointing that experimental measurements fordomain chaos disagreed with this expectation [10]. Determinations of a correlation lengthξ based on the variance of the SF found ξ ∝ ε−νeff with νeff � 1/4 rather than 1/2.1

Similarly the frequency f was found to vary with ε as f ∝ εμeff with μeff � 1/2. Recentlythis discrepancy between experiment and theory was resolved [28, 29]. In the case of ξ, itwas found that conventional Fourier analysis, as was typically used to obtain ξ [10, 24],was unable to overcome the surprisingly difficult obstacle of the finite size of the patternimages, which hampered the accurate extraction of ξ. The maximum entropy method(MEM) [30, 31] for the power spectrum estimation provided the necessary advantageover the standard Fourier-transform (FT) method that made it possible to defeat thisobstacle [28] and to accurately determine ξ from the necessarily finite pattern images. Inthe case of f , the centrifugal force turned out to play an important role in decreasingμ [29]; the theory typically neglects this force.

Although these basic disagreements are resolved, several interesting features of DCremain to be explored. In particular, both ξ and f only address the global properties ofthe patterns. However, the patterns typically contain a rich local structure. In the presentwork we address some of these local details. Our primary technique was to extract thelocal wave director (LWD) field from the pattern images. From this field we determineda multitude of additional quantities.

We used the LWD field to extract correlation functions and statistical informationabout domains and defects from the experimental pattern images. From the correlationsof the wave director angle θ we extracted a time series of the length scale, ξθ, which turnedout to possess a periodic time component that was related to the switching frequency ofthe domains. Additionally we found that the time series of the area occupied by defectsad contained this same periodic component, but the time series of the area occupiedby domains aD did not2. We made several comparisons between the results from SHsimulations and the experiment as a function of the width Γ∗ (measured in units of thesample spacing d) of a pattern image. Generally there was good agreement with theory atthe largest Γ∗ accessible in each case. However, the actual values of Γ∗ where agreementwith predictions for the infinite system was found were different for experiment and SHsimulation, implying that the finite image size may play a more important role in one orthe other. Notably however, we found that the domain ‘orientation’ D, defined as theaverage dot product of the wave director and the domain major axis, agreed extremelywell between simulation and experiment at all Γ∗ and ε that were investigated.

The technique of LWD analysis has gradually become ubiquitous in the field of patternformation over the past 25 years or so. Although somewhat crude by today’s standards, itwas applied to Taylor-vortex flow in order to analyse wavenumber readjustment [32] as farback as 1983. A semi-automated brute force method was used to study striped patternsin RBC a couple years later [33, 34]. A technique based on cross correlations, which was

1 The overline in ξ denotes the fact that it was obtained using numerical moments of the SF, rather than using acurve fit to a function implied by the SH equations [28].2 To clarify the notation we emphasize that for quantities with a tilde, ξθ, aD, and ad, the tilde denotes the factthat a time series of values was obtained by processing each successive image. For example ad ≡ ad(t), whereeach value of ad was determined by averaging over all defect areas in the corresponding image at that time step.The quantities without a tilde, such as ξθ, aD, and ad were obtained by an ensemble average of all images in agiven ε step.

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the precursor for one of the methods [23] used in the present work, was applied to Turing-type chemical spatial patterns a decade and a half ago [35]. In the same year, analysis ofthe stripe axis skeleton yielded statistics of defects and identification of domain walls inpatterns obtained via polarization microscopy from ferrimagnetic garnet films [36]. Theresulting overlay of orientation field and pattern image made an eye-catching picture onthe cover of Science. Several years later, fits of line segments to local contours were usedto determine the curvature as well as other features of spirals in the spiral defect chaosstate [37] of RBC [38, 39]. More recently, other brute force methods of fitting segmentswere applied to study spiral defect chaos [40, 41]. A demodulation technique to directlyextract the phase of the pattern, and then determine the LWD field from that, was appliedto travelling waves observed in RBC consisting of a binary fluid mixture [42, 43], and halfa decade later that system was studied using an algorithm developed by Egolf et al [46, 47](the EMB algorithm) which is discussed below [44]. Local techniques were also used toobserve wavenumber modulation of spirals in the Belousov–Zhabotinsky reaction [45].

The modern approach to LWD analysis utilizes several different algorithms. Ideally,all techniques would give the same result, but, as we will discuss below, some are betterthan others for certain tasks. Thus they serve to complement each other. The EMBalgorithm mentioned above uses a mathematical trick with derivatives that is reminiscentof separation of variables in the solution of partial differential equations. Another method,due to Cross, Meiron and Tu [23] (the CMT algorithm), uses a Fourier filtering schemeto locally demodulate the pattern and could be thought of as a simplified Gabor-wavelettransform [48]. It was based on an earlier technique mentioned above [35] as well as amore formal wavelet treatment [49]. In [23] it was applied to DC patterns obtained froma simulation of the SH model. Recently, it was used to study wavenumber selection bytarget patterns in RBC [50] and also to quantify centrifugal force effects in DC observedin RBC [29]. We primarily used the CMT algorithm in the present work. However belowwe provide some comparisons between the EMB and CMT algorithm results3.

There are some additional algorithms which may be regarded as modern techniques,but we did not investigate them for the present work. One, using a much moresophisticated wavelet transform than in the CMT algorithm, was proposed in [51, 52],and it was applied to some striped RBC patterns. Another technique does not providethe wave director field, but instead provides an invariant measure of local disorder [53].

The EMB method was first applied to spiral defect chaos in RBC [46], and hasbeen further used to study that state recently [54, 55]. It has also been applied todetermine the curvature of RBC coarsening patterns [56, 57]. Wavenumber selectionin an RBC sample with a radial ramp was studied experimentally [58] using the EMBalgorithm, and the ramped system was investigated theoretically [59] using a brute forcemethod to obtain the wavenumber. It was also used to study RBC in an inclinedlayer [60, 61]. The EMB algorithm has been applied to many systems other than RBC.It was used to characterize the pitch of a chiral sample in electroconvection [62] and alsoapplied to the study of domain coarsening in electroconvection [63, 64]. Wavenumberinstabilities were characterized in striped patterns observed in a vertically oscillatedgranular system [65] using the EMB algorithm, as well as spiral relaxation [66] and other

3 We note that in the case of the EMB algorithm, we followed the formulae given in [47] rather than in [46], exceptfor a minor addition of a factor of k as discussed in [81].

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coarsening events [67, 68]. It was also applied to wavenumber selection and defects inBenard–Marangoni convection [69, 70]. Striped patterns observed in wrinkles formed on anelastomer surface during thermal expansion were studied using the EMB algorithm [71]–[74]. It was also used to study coarsening in relaxation of striped patterns formed byannealing of diblock copolymers [75]–[77], and to study lattice spacing and orientation inpolymer crystals [78]. Sink and source defects along with wavenumbers and amplitudeswere studied in the printer’s instability [79, 80].

2. The Egolf–Melnikov–Bodenschatz (EMB) and Cross–Meiron–Tu (CMT)algorithms

Here we provide a brief overview of the inner workings of the EMB and CMT algorithms.For more detail see [23, 46, 47, 81]. The essence of the EMB algorithm is that, at each loca-tion, a spatial derivative contains the product of the pattern field u(x) and of a componentof k(x), such that for instance |kx(x)|2 = −[∂2

xu(x)]/u(x) under the approximation thatu is sinusoidal. By taking various spatial derivatives and dividing the results by the orig-inal pattern it is possible to reconstruct the components of k. An improvement put forthin [47] sometimes divides the third derivative by the first derivative when the function u issmaller that its first derivative. This helps avoid problems with machine precision wherek would otherwise accidentally diverge at some pixels. We note that we always use thisimprovement when applying the EMB algorithm, although we made a slight change toequations (5)–(12) of [47] by including a factor of k in the four tests in order to make theunits consistent [81]. Because the algorithm relies on a numerical division, there are stillsome accidental divergences. These are replaced by the average values from neighbouringpixels, and then a Gaussian blur is applied to smooth everything out.

The CMT algorithm proceeds by looking at the local contributions to the Fouriertransform of the pattern. The image is locally demodulated to determine the angle ofthe mode with the strongest contribution. This is done as follows. The image is Fouriertransformed, and an azimuthal Gaussian filter is applied along the θ axis so as to filterout all modes except for those with an orientation near a specific angle θ0. This step isrepeated for nθ different values of θ0 spaced evenly from 0 to π. The result is a sequence ofnθ Fourier transforms, representing demodulated straight rolls with a director orientationclose to θ0, that each possess half the total power of all the rolls in the pattern directed atthe corresponding angle. The filter automatically demodulates these rolls because it filtersout the complex conjugate of the peak, only keeping the signal over the range 0 < θ < π.Applying an inverse FT results in a complex field. The real part is an image consistingof straight rolls, with an orientation close to θ0, in all regions where the original patternexhibited such rolls, and consisting of blurred out patches of small amplitude elsewhere.The imaginary part looks similar, but is phase shifted by 90◦ so that the magnitude of thecomplex field is an amplitude of the rolls that have an angle close to θ0. The amplitudefields are held in memory so that the maximum amplitude as a function of θ0 can bedetermined at every pixel. Interpolation is used to determine θmax with good accuracyfor a relatively small nθ. In the section 3 we give examples that suggest that nθ � 32 issufficient to yield an excellent representation of the probability distribution function ofthe wave director angle field θ(x).

There are various advantages to each of the two algorithms. Although the EMB algo-rithm can be much faster than the CMT algorithm when both are implemented using the

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Figure 2. Local θ field of the pattern shown in figure 1 using the colour palettefor θ as indicated. (a) Determined with the CMT algorithm. (b) Determinedwith the EMB algorithm and a Gaussian blur.

fast Fourier transform [82], it is not quite as accurate, as discussed below. Given enoughcomputation time, the CMT algorithm provides the best representation of θ(x), and alsoprovides the local pattern amplitude field A, which the EMB algorithm does not provide.However, the CMT algorithm does not give the magnitude k(x) of the wave director. Wedid not use k in the present work, but in other cases where we did want it we combinedthe CMT algorithm with a brute force technique to form a hybrid algorithm [81].

As an example of LWD analysis, we consider the domain chaos pattern shown infigure 1. Applying the EMB and CMT algorithms to determine θ(x) yielded the imagesof figure 2. A visual inspection shows that the CMT algorithm produces θ fields that aremuch smoother than the EMB results. Further, the CMT result was obtained withoutblurring, while the EMB result was smoothed by blurring over the range of a wavelength.In fact, we never considered unblurred EMB results because they are simply too full ofspeckly noise.

3. Accuracy tests for the Egolf–Melnikov–Bodenschatz (EMB) andCross–Meiron–Tu (CMT) algorithms

In this section we consider two benchmarks of the accuracy provided by the LWDalgorithms. First we examine concentric rolls with k = 3.117, as shown in figure 3.

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Figure 3. A computer-generated test pattern of concentric rolls with the pixelspacing chosen such that k = 3.117.

This pattern was generated using the equation

u(x, y) = cos(k√

x2 + y2)

(1)

where x and y are the coordinates and k is the wavenumber.We applied the CMT and EMB algorithms to the circular region of the image to

determine the θ field, and examined the distribution function P (θ) as a test of the accuracyof the algorithms. For infinitely extended ideal concentric rolls all angles are equallyrepresented, so we would expect P∞(θ) = 1/π. However, the test image is not idealbecause it is finite with limited statistics. So, we also determined the true distributionPt(θ) for the finite number of pixels in the test image. It is given by the green curvein figure 4. Figure 4 also shows P (θ) for both algorithms. There are 512 bins for P (θ)over the range 0 < θ < π. Both the CMT and the EMB algorithm did a decent job ofreproducing the true distribution. The CMT algorithm is very slightly more accurate inthe sense that the deviations from the true distribution are smaller. This can be seenquantitatively by computing the standard deviation

std =

√1

σP

∑[Pt(θ) − P (θ)]2 (2)

of P (θ) from Pt(θ) (σP is the number of θ bins in the distribution). For a perfectly accuratealgorithm, the standard deviation would approach zero.

Figure 5 shows the result of applying equation (2). This serves as both a test of whichalgorithm is more accurate, and also indicates what value of nθ is optimal for the CMTalgorithm. Recall that the CMT algorithm uses the method of locally demodulatingthe pattern for a sequence of nθ orientations, equally spaced from 0 to π, in order todetermine the strength of each orientation in the image. We achieved convergence afternθ � 32 orientations were used, and the standard deviation was just slightly smaller forthe CMT algorithm than for the EMB algorithm. This indicates that for concentric rollswe should barely prefer the CMT algorithm over the EMB algorithm if computation time

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Figure 4. Distribution of angles for the test pattern shown in figure 3 using 512bins over the range 0 < θ < π. Red: P (θ) determined by the EMB algorithm.Black: P (θ) determined by the CMT algorithm with nθ = 32. Green: P (θ) givenby equation (1) for the actual data points in the image. Blue: P (θ) = 1/π.

Figure 5. Standard deviation of angular distributions in figure 4 from the truevalue, as given by equation (2). Circles: results from the CMT algorithm. Solidline: result from the EMB algorithm.

is not an issue. It also shows that choosing 8 ≤ nθ ≤ 12, as dictated by the speed ofcomputers at the time of the work reported in [23], does not achieve the best possibleresult with the CMT algorithm.

In a different test, we applied the LWD algorithm to domains of ideal straight rolls.Figure 6 shows a synthetic image of nine domains of straight rolls. Of course, this imagedoes not closely resemble domain chaos, but it does challenge the algorithms to determinethe orientation of many differently oriented roll patches. Both the EMB and the CMTalgorithms did an excellent job of determining the orientations. Ideally we would find aP (θ) that contained nine sharp peaks, each of probability 1/9 when P (θ) dθ was integratedover the size of one bin. Figure 7 shows that the CMT algorithm came slightly closer toreaching this idealization. The green line in the figure is the height that the peaks would

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Figure 6. Test image of nine patches of straight rolls.

Figure 7. P (θ) determined from the image in figure 6. Black: result from theCMT algorithm. Red: result from the EMB algorithm. Green: height that thepeaks would reach in the ideal case.

reach if all the probability was deposited in the correct bin, indicating that no mistakeswere made, and the CMT result is closer to this line than the EMB result is. The peaksfrom the EMB algorithm are slightly wider than the results from the CMT algorithm,further convincing us that for the purpose of computing the θ field, we generally preferthe CMT algorithm.

4. Configuration of the simulation and experiment

We studied the impact of the finite size of pattern images on the LWD results and thetime–angle correlation results by using simulations of the Swift–Hohenberg model fordomain chaos [23] as test images. We utilized the algorithm given in [23] and periodic

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Figure 8. A solution of equation (3) for ε = 0.12. This is a 512× 512 (Γ∗ = 150)cutout from the centre of a 1024 × 1024 image with Γ∗

max = 300.

boundary conditions to solve the equation

∂tψ = εψ − (∇2 + 1)2ψ − g1ψ3 + g2z · ∇ × [(∇ψ)2∇ψ] + g3∇ · [(∇ψ)2∇ψ] (3)

for ψ(x), a field that can be used to represent for instance the temperature of theconvection sample at the mid-plane. Figure 8 shows a sample of the result for ψ at oneinstant of time. The simulation was done for 1024×1024 pixels, corresponding to an aspectratio (width over sample ‘thickness’) Γ∗

max = 300. This choice of pixel spacing roughlycorresponded to that used in the experiment, thus it facilitated convenient comparisonbetween the simulation and the experiment. Central squares of various sizes Γ∗ were usedin the studies to be discussed below. The time step for numerical integration was 0.1.The initial condition for ψ was a grid of straight-roll patches like that shown in figure 6,but using a 12 × 12 grid of patches with random orientation and also with white noisesuperposed. At each ε, 10 000 warm-up time steps were performed followed by either 256snapshots of ψ recorded at an interval of 1250 time steps in the case of all the analysisexcept when the characteristic frequency f was determined, and 1024 snapshots of ψrecorded at an interval of 50 time steps in the case where f was determined. The differentchoice of the interval between images was necessary because, for resolving frequencies,we needed excellent time resolution, but for achieving good statistics of spatial quantitiessuch as the ξθ, ad, or aD we needed a large time series of relatively uncorrelated images.The pixel spacing was chosen to reflect a non-unity sample thickness, unlike the choicein [23], in order for the wavenumber and ξθ to be nearer the values in the experiment.

The control parameter ε in the simulation is related to the experimental controlparameter ε by ε = (4/k2

cξ20)ε � 2.60ε [1], where kc is the critical wavenumber and ξ0 is

the curvature of the neutral curve4. The numerical value 2.60 corresponds to Ω = 17.5.5

The parameters g1, g2, and g3 can be chosen to model a specific Ω and Kuppers–Lortzangle θKL. In the present work we used g1 = 1, g2 = −2.4534, and g3 = 0.522 which are

4 See equation (159) of chapter III of [3] for the neutral curve of rotating RBC.5 See [28] for a graph of ξ2

0 and 4/k2c ξ2

0 versus Ω.

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the same values used in figure 1(b) of [23] as well as in [28] and which correspond toθKL = 51◦, and roughly to Ω � 17.5 for the Prandtl number σ of the present experiment.Note that equation (3) assumes infinite σ but permits a variable Ωc, so this roughvalue of Ω comes from comparing Ω/Ωc with the σ dependence of Ωc of the physicalsystem.

We used two different sample configurations, in an apparatus described elsewhere [83],to obtain the experimental results of the present work. Both samples were cylindricaland differed from each other mainly in their aspect ratio Γ ≡ r/d (r is the radius ofthe sample)6. We changed Γ by changing the sample thicknesses d, keeping r roughlyconstant. The fluid was sulfur hexa-fluoride (SF6), a compressed gas under our operatingconditions, and the pressure was chosen so that the fluid properties permitted theBoussinesq approximation to apply well for both of the samples.

The larger sample had Γ = 61.5, a thickness of 720 μm, a pressure of 20.00 bars,a mean temperature of 38.00 ◦C, and σ = 0.87. The smaller sample had Γ = 36,was 1230 μm thick, operating at 12.34 bars, with a mean temperature of 38.00 ◦C, andσ = 0.82.

5. Frequencies determined from the time–angle correlation functions of theSwift–Hohenberg (SH) simulations

Although it is based entirely on FT rather than on LWD analysis, for completenesswe present here the results from the SH simulation of the effect of the finite imagesize on the frequency of the KL dynamics. This frequency was determined from thetime–angle correlation of the FTs of an image sequence. This technique was usedpreviously [10, 12, 29, 84], but in the case of the experiment it yielded data that disagreedwith the prediction from the GL model for the ε dependence of the frequency. Asmentioned above, it was shown recently that the discrepancy was due to the presence ofthe centrifugal force in the experiment, which is neglected in the GL and SH models [29].However, it was also shown recently [28] that the finite size of the pattern image affects theaccuracy of the SF, and thus may skew the results for quantities determined from it. Inorder to gain insight into the consequences of using the time–angle correlation technique onfinite images, we applied it to images obtained from SH simulations, i.e. from equation (3).

The time–angle correlation was obtained as follows. The FT was computed and thecorresponding SF was radially averaged (i.e. over k) at each time step. These SFs werearranged consecutively in time in a two-dimensional (wave director angle)–time image,where the intensity of each image pixel corresponded to the height of the k-averaged SFcurve at the given θ bin. We then found the autocorrelation of this 2D image, usingperiodic boundaries along the θ axis and a square window along the time axis. Thisresulting autocorrelation function was represented by an image where the horizontal axiscorresponded to displacements in θ and the vertical axis corresponded to displacementsin time. An example is shown in figure 9.

The time–angle autocorrelation contained a series of peaks which corresponded to adisplacement of time and angle for a typical domain switching event, provided that the

6 For clarity, we emphasize that the difference between Γ and Γ∗ is that Γ refers to the physical size the samplein the experimental apparatus or in a Boussinesq simulation with realistic boundary conditions, while Γ∗ refersto the size of a pattern image or a central cutout from a pattern image.

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tΔ

Δθ

Figure 9. Time–angle autocorrelation image for ε = 0.1 and Γ∗ = 58.6. Blackand white correspond to large and small correlations respectively.

Figure 10. Domain switching frequency determined from the radially averagedSF with 128 bins per 180◦ and for various image sizes Γ∗. Triangles: Γ∗ = 300.Diamonds: Γ∗ = 225. Squares: Γ∗ = 150. Circles: Γ∗ = 75. Crosses: Γ∗ = 58.6.Stars: Γ∗ = 42.2. Pluses: Γ∗ = 25.8. The line has a slope μ = 1.

size of the cutout image was sufficiently small that multiple domains were not includedin the SF. We determined the frequency f from the average spacing between the peaksalong the time axis. Some examples of the results are shown in figure 10.

By fitting the power law f ∝ εμ to data like those shown in figure 10, we determinedthe effect of the finite image size on μ. The result is shown in figure 11. We did observean effect of the image size on this frequency scaling, but the effect appeared as the imagesbecame too large rather then too small, opposite to the effect on ξ as explored in [28]. Thisis indicated by the fact that for small and moderate Γ∗ the value for μ approaches unity,

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Figure 11. Effect of the image size on μ. Triangles, squares, and circles representthe results from the radial average of the SF using 32, 64, and 128 bins per 180◦

respectively.

as is predicted by the GL model. This is a fortunate situation because it means that themeasurement of the frequency from experimental data is not limited by the finite imagesize available in a realistic experiment, as was the case for the correlation length ξ [28].Although the choice of boundary conditions used when computing the autocorrelationintegral affected the results for the exponent μ quantitatively the main conclusion, thatμ approaches one at small enough Γ∗, was not affected. For Γ∗ � 100, the binning of theSF becomes important. This causes the non-systematic variation in μ versus Γ∗ at largeΓ∗. Recall that in order to determine a radially averaged SF from the 2D power spectrumof a pattern image, we must arrange the SF intensity into bins corresponding to a givenband of angles. Fortunately the number of bins is not that important when the imagesize of the pattern is sufficiently small. Thus the binning effect is not really an issue, themain conclusion of this analysis being that it is important to choose a sufficiently smallimage cutout size when determining f .

6. Details of the image analysis techniques

Using the CMT algorithm of LWD analysis (see section 2 above) we decomposed eachimage, either from SH simulation or from experiment, into two fields. One contained theorientation θ of the wave director k and the other held the amplitude A of the dominantsinusoidal component of the pattern. The CMT algorithm can be combined with othertechniques to obtain the modulus k of k [81]. However, we did not need k for the presentwork because we focused on results obtained using θ in order to determine autocorrelationfunctions related to the typical domain size, and on results obtained using A in order tolocate defects and domains and to quantify their statistics.

6.1. Angular correlation function

The autocorrelation of the orientation field θ was determined from the discrete analogue of

C(δx) =

∫cos{2[θ(x) − θ(x + δx)]} dx (4)

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which was utilized previously for domain chaos [23, 29]. The azimuthal average of C(δx)yielded C(δr) as shown below in figures 16 and 28. The half-width at half-maximum ofC(δr) yielded a correlation length ξθ as explained in more detail below.

We considered two normalizations of equation (4). Both use a constant at the originchosen such that C(0, 0) = 1. However, at increasing δx, the biased correlation function isnormalized by the same constant factor, while the unbiased correlation function is dividedby a factor M(δx), the number of data points used in the integral to find the correlation atthe spatial separation δx. The latter normalization corrects for the diminished quantityof data available at increasingly separated locations. Both approaches have merits, inparticular the biased form must be used for computing power spectra7. As discussed insection 7, we found that the biased normalization yielded more consistent ε dependenceof the correlation length ξθ. However, the unbiased normalization yielded more consistentvalues of ξθ at moderate to large ε. Thus we prefer the biased normalization for extracting εdependence, but might sometimes prefer the unbiased normalization for moderate ε undercircumstances where the main concern was not the ε dependence8.

6.2. Defects and domains

In the study of defects, we focused on detecting patches of diminished amplitude. Thiswas explored previously in simulations of the SH model of domain chaos [23]. We useda thresholding criterion similar to that of [23] to label sections of the amplitude field aseither consisting of defects or not. We went further than in [23] by also locating domainwalls along strong amplitude regions on the basis of the magnitude of the gradient ofthe orientation field, |∇ exp(2iθ)|. Combining the amplitude information, the gradientinformation, and the known domain orientation we also detected patches of independentdomains. From our database of defects and domains we determined quantities such asthe defect and domain area fraction, the number of defects and domains, their sizes, andin the case of domains the orientation of the wave director relative to the long axis of thedomain. Most importantly, we were able to separate the domain and defect length scalesin order to determine their relative contribution to the dynamics.

As an example of the processing involved in studying amplitude defects, we considerthe pattern shown in figure 1. Its local amplitude A is shown in figure 12(a). We applieda threshold to A such that all pixels below 75% of the mean value of A were marked asbeing amplitude defects [23]. The resulting defect field is shown in figure 12(b).

To go further, we made the approximation that each blob (black area surroundedcompletely by white pixels) in figure 12(b) corresponded to a single defect. This was notnecessarily true in the sense that many regions of diminished amplitude at times clumpedtogether so there was not a clear definition of where one defect ended and another began.As a first approximation, we simply treated each connected blob as a single defect. Wemade a database of all the blob locations, areas, and axis lengths.

A quantity that has been studied previously in striped patterns [38, 39, 56] is the

curvature κθ = ∇ · k, where k = k/|k|. The curvature is useful for locating domain wallsbecause it identifies regions of the θ field that have large spatial variation. Figure 13(a)

7 For a discussion of biased and unbiased normalizations, see [85].8 For an example where we preferred the unbiased normalization see [29].

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Figure 12. (a) Local amplitude of the pattern shown in figure 1. Whitecorresponds to large amplitude and black corresponds to small amplitude.(b) Amplitude defects determined by thresholding the image on the left. Blackcorresponds to defects and white corresponds to non-defects.

Figure 13. Curvature and magnitude of the gradient of θ for the pattern shownin figure 1. (a) Curvature. (b) Magnitude of the gradient. Black corresponds tolarger values.

shows the curvature field determined from the CMT results for the θ field of the patternin figure 1. However, we found that the magnitude of the gradient, |∇ exp(2iθ)|, wasmore effective in determining sharp lines along domain wall boundaries. This quantity isshown in figure 13(b). It shows lines that are similar to κθ in figure 13(a), but the linesare heavier, more clearly outlining the domain structure.

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Figure 14. (a) Edge detection applied to the θ field of the pattern shown infigure 1. (b) Overlay of the θ field and the detected edges.

In order to determine well-defined domain boundaries from the θ field, we applied athreshold to the gradient magnitude in figure 13(b) so that we only kept lines that weredarker than two standard deviations above the mean value of the gradient magnitude.This resulted in removing many of the softer lines that did not correspond to true domainwalls. A speckle of pixels remained which was removed by deleting any pixels that hadall white neighbours. The result was an edge detection of the domain walls as shown infigure 14.

We also applied a blob-database technique to identify unique domains in the pattern,but with the added complication of not only taking into account the connectedness ofwhite pixels in figure 12(b), but also utilizing the local θ information in figure 2(a) andthe θ edge detection in figure 14(a). We laid the edge data on top of the defect datafrom figure 12(b), so that differently oriented domains would be separated by black pixelsin our defect field even if the domain walls consisted of large amplitude regions. Fromthis combined field, we made a database of the domain locations, areas, and axis lengths.Figure 15(a) shows an overlay of the edges and domains with the pattern and θ field as areference for where the domains are. Figure 15(b) shows the processed domain database,where each domain is coloured corresponding to its average θ value, and the perimetersof the domains are highlighted in black.

We determined the defect area fraction

ρd =Ndad

Ndad + NDaD(5)

where ad is the ensemble-averaged area occupied by defects, aD is the ensemble-averagedarea occupied by domains, Nd is the total number of these defects at a given ε, and ND

is the total number of these domains at a given ε. In computing these quantities wesummed only over those blobs that did not overlap the edge of the image, since those

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Figure 15. (a) Overlay of the pattern, the θ field, the amplitude defects, andthe θ edge detection. (b) The detected domains, where solid lines denote domainboundaries, white regions denote defects, the domains are coloured to representtheir mean orientation θd, and a single white pixel in the centre of each domaindenotes the centre pixel of that domain.

Figure 16. Examples of C(δr) for Γ∗ = 75. Solid and dotted lines: unbiasedC(δr). Dashed and dashed–dotted lines: biased C(δr). Solid and dashed lines:ε = 0.05. Dotted and dashed–dotted lines: ε = 0.12.

overlapping the edge were likely to be larger than we could see. We also considered thesimilar quantity

ρD =NDaD

Ndad + NDaD

(6)

for the domain area fraction. As mentioned above, we determined the lengths of the majorand minor axes of the blobs. For this we made the assumption that a blob is roughlyelliptical in shape (a similar approach was followed in [84]). We also investigated the

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relationship between the orientation of the major axis of a domain and the wave directororientation within that domain. This was accomplished by considering the dot-product-like quantity

D = 〈cos 2[θM − θd]〉 (7)

where θM is the angle of the major axis of the domain, θd is the average wave directororientation at the pixels in the domain, and the brackets indicate an ensemble averageover all domains in all images at a particular ε step. We note that D does not depend onan explicit assumption that the domains are elliptical in shape. It only depends on theLWD field orientation and an estimate of the orientation of the long axis of the domain,thus any shapes with a clear major axis are accounted for.

7. Local wave director (LWD) results from the Swift–Hohenberg (SH) simulations

In this section we derive a number of parameters from the SH simulations. Although thesimulations were limited to the relatively narrow range 0.02 � ε � 0.05 by the availablecomputational facilities, this range is roughly in the middle of the range 0.01 � ε � 0.1explored by experiment and permits the determination of effective exponents from powerlaw fits. These exponents permit a meaningful examination of trends with Γ∗ and permita comparison with experimental results from the somewhat wider ε range and withBoussinesq simulations over similar ranges. To what extent any of these results representthe asymptotic behaviour ε → 0 is of course open for discussion.


The correlation length ξθ from the θ component of the LWD field need not be the same asthe two-point correlation length ξ determined from the SF and provided in [28]. However,since the theoretical model for domain chaos [22] only contains one length scale, it isreasonable that ξθ should have the same ε dependence as ξ.

The value of ξθ was extracted from the width at half the maximum of C(δr). Examplesof C(δr) are shown in figure 16. One sees that C(δr) is well fitted by an exponentialfunction from small up to moderately large δr. Fitting an exponential function or justfinding the δr value where C(δr) dropped to 1/2 gave nearly identical results for ξθ,provided that the result from the half-maximum was divided by the required factor ofln 2. Finding the 1/2 level is more convenient, and was used previously in domain chaosin a very similar situation [23], so we used that technique for determining ξθ.

Figure 17 shows the ξ and ξθ data together on one plot for Γ∗ = 75. The ξθ datado not land on top of the ξ data, although this may be related to our choice of definingξθ as the width at half the maximum instead of some other fraction of the maximum.Choosing a different cut-off would not affect the ε dependence, but just shift all the ξθ

values by a constant factor. There are two determinations of ξθ because we consideredboth a biased and an unbiased C(δr). Although not shown in the figure, at large Γ∗ thebiased and unbiased forms approach each other. As a function of Γ∗, the biased ξθ valueshave a more consistent ε dependence, while the unbiased ξθ values do not exhibit as largeof a shift in magnitude as Γ∗ increases. Since we are limited to fairly small Γ∗ in theexperiment, we want to determine which normalization gives the more accurate result.

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Figure 17. Determination of ξ given in figure 7 of [28] from the SF with theMEM (open circles), and also determinations of ξθ from many C(δr) like thoseshown in figure 16, all with Γ∗ = 75 (solid symbols). Diamonds: unbiased C(δr).Squares: biased C(δr). Solid lines: fits of the power law ξ ∝ ε−ν to the data overthe range shown.

Figure 18. Results for ν and νθ from fits like those shown in figure 17. Opencircles: from the MEM results in [28]. Diamonds: from the unbiased C(δr).Squares: from the biased C(δr). Dashed line: the ν = 1/2 prediction from modelequations.

The exponent νθ was determined from fits of ξθ ∝ ε−νθ to the data as shown infigure 17. The Γ∗ dependence of νθ is shown in figure 18. The biased C(δr) gave νθ

values that are within our resolution independent of Γ∗. At very large Γ∗ these valuesmet with results obtained from the unbiased C(δr). This is a good indication that weshould prefer the biased C(δr) when extracting the ε dependence of ξθ at small Γ∗. It isnotable that νθ approaches roughly 0.4 at large Γ∗, somewhat shy of the prediction fromthe GL model of ν = 0.5 as well as the result of ν � 0.5 at large Γ∗ given by the SFtechnique. One possible explanation for the discrepancy is that the SF is affected by both

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Figure 19. Average defect area ad given by the defect analysis of the SHsimulation data. Triangles: Γ∗ = 300. Diamonds: Γ∗ = 225. Squares: Γ∗ = 150.Pluses: Γ∗ = 112.5. Circles: Γ∗ = 75. Crosses: Γ∗ = 58.6. Solid lines: fits of apower law ad ∝ ε−νd to the data.

the magnitude and orientation of the wave director, while C(δr) was determined explicitlyfrom the orientation field alone.


The defect and domain detection was applied to the simulations. This yielded the averagedefect area ad, the average domain area aD, the number of defects Nd, the number ofdomains ND, the defect area fraction ρd from equation (5), and D from equation (7).

7.2.1. Areas of defects and domains. In view of the different scaling of the correlation lengthin the x and y directions that is predicted by the Newell–Whitehead–Segel equation [86, 87]and incorporated in the GL model of Tu and Cross [22], one expects that aD ∼ ε−3/4.We are not aware of a similar prediction for ad. Power law fits to the defect and thedomain areas yielded effective exponents νd for defects and νD for domains. The resultswere severely affected by the finite sizes of the images.

As shown in figure 19, the average defect area ad converges to a Γ∗ independent valuefor large enough Γ∗. However, this does not happen until Γ∗ � 100, which is beyond therange accessible with the present experiment. This behaviour is in contrast to that for theaverage domain area aD which still increases with increasing Γ∗ even for the largest imagesizes as indicated by the successively shifted aD versus ε curves shown in figure 20(a). Infact, it does not seem that aD is approaching Γ∗ independence, as indicated by the nearlylinear shape of aD versus Γ∗ shown in figure 20(b).

The results for νd and νD are summarized in figure 21. They suggest that νD stillincreases even for the largest Γ∗ with no clear indication that it will converge on 3/4as predicted. For the defects, the data suggest that νd will converge to a value close to0.5. This result is consistent with the value reported for simulations of the Boussinesqequations [84], although that work used a considerably smaller Γ∗.

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Figure 20. Average domain area aD given by the domain analysis of the SHsimulation data. (a) ε dependence at constant Γ∗. Triangles: Γ∗ = 300. Stars:Γ∗ = 262.5. Diamonds: Γ∗ = 225. Circles: Γ∗ = 187.5. Squares: Γ∗ = 150.Pluses: Γ∗ = 112.5. Solid lines: fits of a power law aD ∝ ε−νD to the data.(b) Γ∗ dependence at constant ε. Triangles: ε = 0.046. Circles: ε = 0.019. Lines:connect points as a guide to the eye.

Figure 21. Effective exponents for blob areas shown in figures 19 and 20. Solidcircles: νd. Open circles: νD.

7.2.2. Numbers of defects and domains. Figures 22 and 23 show the number of defects perunit area Nd/Γ∗2 and ND/Γ∗2 respectively. For large Γ∗ these results converge toward aconstant value at each ε, suggesting that Nd and ND are extensive quantities. This isshown explicitly in figures 22(b) and 23(b), where curves at constant ε are plotted againstΓ∗. One sees that both Nd/Γ∗2 and ND/Γ∗2 depend on Γ∗ for small Γ∗, but that theytend toward a constant value as Γ∗ increases beyond about 250. Particularly with regardto the behaviour at small Γ∗ we note that simply dividing Nd or ND by Γ∗2 is not quitecorrect because the value of the effective system size was more complicated because theblobs overlapping the boundaries were neglected (see section 6).

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Figure 22. Defect number, given by the defect analysis of the SH simulationdata, roughly collapsed by plotting Nd/Γ∗2. (a) ε dependence at constant Γ∗.Triangles: Γ∗ = 300. Diamonds: Γ∗ = 225. Squares: Γ∗ = 150. Pluses:Γ∗ = 112.5. Circles: Γ∗ = 75. Crosses: Γ∗ = 58.6. (b) Γ∗ dependence atconstant ε. Triangles: ε = 0.046. Squares: ε = 0.031. Circles: ε = 0.019. Lines:connect points as a guide to the eye.

Figure 23. Domain number, given by the domain analysis of the SH simulationdata, roughly collapsed by plotting ND/Γ∗2. (a) ε dependence at constant Γ∗.Triangles: Γ∗ = 300. Diamonds: Γ∗ = 225. Squares: Γ∗ = 150. Pluses:Γ∗ = 112.5. Circles: Γ∗ = 75. Crosses: Γ∗ = 58.6. (b) Γ∗ dependence atconstant ε. Triangles: ε = 0.046. Squares: ε = 0.031. Circles: ε = 0.019. Lines:connect points as a guide to the eye.

7.2.3. Area fraction of defects and domains. The area fractions ρd and ρD are shown infigure 24. As ε increased, the patterns became more dominated by domains. That isnot to say that defects became less plentiful with increasing ε. Instead, defects shrank insize more quickly than they gained in number such that ρd decreased with ε because Nd

increased more slowly than ad decreased. In contrast for domains, ND increased while aD

decreased such that ρD increased with ε.The Γ∗ dependence of ρd and ρD is shown in figure 25. This may help explain why

we were unable to find Γ∗ independent values for aD or νD. Not only did the patterns

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Figure 24. Defect area fraction ρd and domain area fraction ρD, given by thedefect and domain analysis of the SH simulation data, plotted again ε. Solidsymbols: defects. Open symbols: domains. Triangles: Γ∗ = 300. Squares:Γ∗ = 150. Circles: Γ∗ = 75. Lines: connect points as a guide to the eye.

Figure 25. Defect area fraction ρd and domain area fraction ρD, given by thedefect and domain analysis of the SH simulation data, plotted again Γ∗. Solidsymbols: defects. Open symbols: domains. Triangles: ε = 0.046. Circles:ε = 0.019. Black lines: fits of ρd ∝ Γ∗νg . Red lines: determined from thefits shown in black using the relationship ρd + ρD � 1 implied by equations (5)and (6).

become hugely populated with domains, but also ρd seemed to drop off with a power lawdependence on Γ∗. The black solid lines shown in figure 25 are fits of ρd ∝ Γ∗νg , with0.6 < νg < 0.65 depending on ε. By neglecting the contribution of the domains whichwere removed because they overlapped the boundary of the image, equations (5) and (6)imply that ρd + ρD = 1. The red lines that pass through the ρD data were determinedusing this relationship directly from the fits to ρd. Thus it seems that the cut-off domainsare negligible. Further, if the power law fall-off of ρd persists as Γ∗ increases, then we mayexpect aD to never reach Γ∗ independence.

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Figure 26. Average defect major axis length Sd given by the defect analysis ofthe SH simulation data. Triangles: Γ∗ = 300. Diamonds: Γ∗ = 225. Squares:Γ∗ = 150. Pluses: Γ∗ = 112.5. Circles: Γ∗ = 75. Crosses: Γ∗ = 58.6. Solid lines:fits of a power law Sd ∝ ε−νSd to the data.

7.2.4. Lengths of defect major axes. We also located the major and minor axes of the defectsand domains using the techniques described in section 6. Unfortunately the lengths ofthese axes were severely affected by the finite image size. Figure 26 shows the length Sd ofthe defect major axis averaged over all such axes at a given ε step. A fit of the power lawSd ∝ ε−νSd to the data yielded νSd � 0.25 for Γ∗ = 300. Although not discussed in detailin the present work, we also obtained νSd � 0.25 in the case of the Γ = 61.5 experimentalsample. This is not too far from the value of 0.36 given by [84] for a Boussinesq simulation.

7.2.5. Relative orientation of domains. Finally, we considered the dot-product-like quantityD in equation (7) as a measure for the domain orientation relative to the orientation of therolls within it. We found that the wave director of the rolls tends not to be parallel to themajor axis of the domains. The value of D was consistently about 0.73 regardless of ε orΓ∗. This corresponds to an angle of about 22◦. Figure 27 shows that D is independent ofthe parameters. There was also excellent agreement for D between these results obtainedfrom the SH simulation and the results obtained from the experimental data as shown infigure 43 below.

8. Local wave director (LWD) results from experiment


8.1.1. The time-averaged quantities. In order to analyse the LWD results from theexperimental patterns, we followed the same procedure that was used on the patternsfrom the SH simulation, as discussed in section 7. Briefly, we extracted a length scale ξθ

from the width of the azimuthally averaged and time-averaged autocorrelation functionC(δr) = 〈C(δr, t)〉t of the local θ field. Some examples of C(δr) are shown in figure 28.Unlike the result from the SH simulation shown in figure 16, the C(δr) data fromexperimental patterns did not seem to closely follow an exponential form at small δr.

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Figure 27. D given by the domain analysis of the SH simulation data. Triangles:Γ∗ = 300. Diamonds: Γ∗ = 225. Squares: Γ∗ = 150. Pluses: Γ∗ = 112.5. Circles:Γ∗ = 75. Crosses: Γ∗ = 58.6.

Figure 28. Time-averaged C(δr) for Ω = 17.7 and Γ = 61.5. Solid and dottedlines: determined by processing a 60d×60d cutout from the centre of the sample.Dashed and dashed–dotted lines: determined by processing a 44d × 44d cutoutfrom the centre of the sample. Solid and dashed lines: ε = 0.05. Dotted anddashed–dotted lines: ε = 0.009.

Nevertheless, we still extracted a length scale ξθ by determining the width at half themaximum.

From figure 29 one sees that the image size had a stronger effect on ξθ in theexperiment than in the simulation. For the Γ∗ = 60 image size, a fit of ξθ ∝ ε−νθ to thedata yielded νθ = 0.41, but the same data with an image cutout of size Γ∗ = 44 yieldedνθ = 0.25. This is a significantly larger effect on νθ than would have been expected onthe basis of the SH simulation, as indicated by comparing the Γ∗ = 60 results with theΓ∗ = 44 results shown in figure 18. As discussed in [29], the domain size is smaller in theexperiment than in the SH simulation due to the omission of the centrifugal force in theSH model. Thus, it is not surprising that the Γ∗ dependence differs between experiment

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Figure 29. Values of ξθ for Ω = 17.7 and Γ = 61.5 extracted from C(δr) at variousε like those shown in figure 28. Circles: determined by processing a 60d × 60dcutout from the centre of the sample. Squares: determined by processing a44d × 44d cutout from the centre of the sample. Solid lines: fits of ξθ ∝ ε−νθ tothe data. Circles yielded νθ = 0.41 and squares yielded νθ = 0.25.

and simulation because more domains are contained in a given region in the experimentalpatterns than in the simulation patterns.

The physical sample size Γ also affected the results. Although not shown, ξθ

determined from the Γ = 36 sample contained wild fluctuations and provided unreasonablevalues for νθ. As a result, we will not report detailed results for that sample here. Apossible cause of this problem is the relatively small image size; the Γ = 36 sample islimited to an image size of Γ∗ = 44.

8.1.2. The time dependent correlation length. The time series of widths ξθ(t) determinedfrom C(δr, t) of consecutive individual images not only fluctuated over time, but alsoexhibited a periodic component. An example is shown in figure 30. The fluctuationsof ξθ extended over a rather large range; sometimes they were several times larger thanindicated by the short time segment shown in figure 30. Although there was quite a bitof fluctuation in ξθ, there was still a periodic component as indicated by the dashed lineswhich are drawn at an interval corresponding to a characteristic period determined fromthe Fourier power spectrum shown in figure 32.

The large fluctuations in ξθ are illustrated by the pattern examples shown in figure 31.We observed both very disordered and remarkably organized patterns over a broad range ofε, as indicated by the fact that the range of ξθ overlapped considerably for these differentε. As indicated by comparing figure 31(b) with figure 31(c), especially at small ε thesystem exhibited a pattern containing much more disorder than more organized patternsat large ε. Sometimes the pattern would organize by chance into a very large domain asshown in figure 31(a). We emphasize that this picture is a snapshot from the middle of avery long time series, in which there were some patterns observed earlier and later thatresembled the very disordered pattern shown in figure 31(b). Thus, it is not a transientor steady-state pattern.

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Figure 30. A short section, with an arbitrary origin, of a much longer time seriesof ξθ determined from consecutive shadowgraph patterns from experiment forΩ = 17.7, ε = 0.05, and Γ = 61.5. Dashed lines are separated along time t at aperiod 1/f , corresponding to the domain switching period determined from thepeak in the spectrum shown in red in figure 32.

The Fourier power spectrum for the time series of ξθ was determined. There wasclearly a dominant peak in each of the spectra as shown in figure 32. These peakscorrespond to a periodic variation in the average length scale of the domains. Presumablythis variation is caused by the KL switching events, thus the characteristic frequencycorresponds to the domain switching frequency as was previously measured via time–anglecorrelations [10, 12]. In order to verify this, we compared the ε dependence of f that we

extracted using ξθ, to the quantity ωa reported by [12] to be proportional to the switchingfrequency. This comparison is shown in figure 33 for parameters in the experiment fromthe present work and in [12] that are nearly identical. Although the f values from thepresent work do not land directly on top of the ωa data from [12], they both have thesame ε dependence, giving an exponent of μ � 0.5 for the power law f ∝ εμ, significantlydifferent from the prediction of μ = 1 from the domain chaos model. (As noted earlier,this discrepancy is due to the presence of the centrifugal force in the experiment but notin the model [29].) The solid line shown in figure 33 is not a fit, but a power law drawnthrough the data by eye using the exponent μ � 0.52 reported by [12]. We expected ωa todiffer from f by at least a factor of the switching angle because ωa was obtained from theslope of a line on a time–angle correlation plot, thus it has units of frequency multipliedby angle. We note that coincidentally, the switching angle in the experiment was shownto be the same as the KL angle, for infinite σ, of about 58◦ [12], even though the switchingprocess is not likely to be dominated by a purely KL instability event. Dividing ωa by2π/3 collapses it onto our present frequency measurements determined from the spectra

of ξθ. We are not sure why this particular factor appeared, but we note that it is roughlythe supplement to the switching angle.

As a test of the Γ dependence of the switching frequency, we compared f determinedfrom both the Γ = 36 and 61.5 samples at Ω = 17.7 as shown in figure 34. Although theε dependence of f changed only a little with different Γ, the actual values of f failed toland on top of each other. This result qualitatively agrees with that reported in [12] as

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Figure 31. Examples of a 60d × 60d cutout from the pattern image fromexperiment for Ω = 17.7 and Γ = 61.5. Upper: ε = 0.017. Lower: ε = 0.10.Left: large ξθ. Right: small ξθ.

indicated by comparing figures 13 and 14 of that work, which show different values of ωa,for similar Ω and ε, but with Γ = 40 and 23. We emphasize that this difference is notan artifact of the analysis in the sense that it is not caused by the choice of image cutoutsize Γ∗. Although not shown in the figure, we in fact found that analysing the Γ = 61.5system using an image size cutout of 60d × 60d yielded data that landed directly on topof the circles in figure 34, i.e. it had no effect on the value of f . In the present work andin [12], the frequency shifted up with decreasing Γ. It is likely that this shift with Γ iscaused by the effect of the centrifugal force, which has been shown to not only affect thevalue of the frequency, but also affect the scaling exponent μ [29].

Qualitatively we found similar behaviour for ξθ in the SH simulation as in theexperiment. As indicated by figure 35, the value of ξθ fluctuated in time and containeda periodic component. Just like in the experiment, the pattern in the simulation could

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Figure 32. Spectrum of ξθ time series from experiment for Ω = 17.7. Black:ε = 0.017. Red: ε = 0.05. Blue: ε = 0.10.

Figure 33. Switching frequency determined with different techniques. Circles:present work with Ω = 16.25 and Γ = 36, determined from many spectra likethose shown in figure 32 of the time series of ξθ. Plus signs: ωa from the plussigns in figure 13 of [12] at Ω = 16.5 and Γ = 40. Crosses: same as plus signs butdivided by 2π/3. Solid line: f = 0.15ε0.52.

exhibit both a relatively ordered and a relatively disordered state at the same ε. Somesnapshots of these fluctuations are shown in figure 36. We also observed a peak in thepower spectrum of the ξθ time series in the SH simulation data, as shown in figure 37.Unfortunately, this peak was not detectable over a large enough range of ε to determinethe dependence of f on ε for the simulation.


We studied the defects and domains in the experimental patterns using an approach similarto the one used on the SH simulation data as discussed in section 7.2. Unfortunately,the simulation indicated that the finite image size has a huge effect on the defect anddomain results, especially at the small image sizes available to the experiment. We report

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Figure 34. Switching frequency determined from the spectra of the time seriesof ξθ with Ω = 17.7, but with different Γ and both determined from a cutout ofsize 44d×44d from the centre of the sample. Circles: Γ = 61.5. Squares: Γ = 36.Solid line: f = 0.17ε0.65. Dashed line: f = 0.26ε0.59.

Figure 35. A short section, with an arbitrary origin, of a time series of ξθ fromthe SH simulation for ε = 0.1. Dashed lines are separated along time t at a period1/f , corresponding to the domain switching period determined from the peak inthe spectrum shown in figure 37.

our results for many quantities here, and even note various agreements between the SHsimulation and the experiment. However, we emphasize that without either significantlylarger image sizes in the experiment or great improvement in the detection algorithms,these results cannot be considered definitive, although in many cases they are quiteintriguing.

The ensemble-averaged area of defects ad and domains aD is shown in figure 38. Theresult for domains is strongly affected by the small image size available to the experimentas indicated by the fact that aD is nearly independent of ε. This result is contrary to theprediction of aD ∝ ε−νD with νD = 3/4. A similar effect appeared for comparable imagesizes in the simulation where aD for small Γ∗ was also roughly independent of ε.

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Figure 36. Examples of a Γ∗ = 150 cutout from the SH simulation data withε = 0.1. Upper: large ξθ. Lower: small ξθ.

Figure 37. Spectrum of ξθ time series from the SH simulation from a longer timeseries of the data shown in figure 35.

The defect area ad in the experimental patterns yielded an ε dependence that roughlyagreed with the simulation. As indicated by figure 21, the exponent νd from fitting apower law ad ∝ ε−νd to the defect area seems to be approaching νd = 0.5 in the case ofthe simulation. The experiment yielded a value of νd = 0.41, which is about twice as largeas the simulation found for a similar Γ∗, but is near to the large Γ∗ value of νd → 0.5.It is also quite close to the value νd = 0.51 reported by [84] in the case of Boussinesqsimulations.

Figure 39 shows the defect area fraction ρd and domain area fraction ρD. As εincreased, the domains dominated the pattern just like in the SH simulation. The actualvalues of ρd and ρD roughly agreed with the values obtained from the simulation withΓ∗ = 300, although in the experiment Γ∗ = 60. The decreasing size of the defects ad,rather than the presence of fewer defects, was what caused ρd to decrease as ε increased.

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Figure 38. Defect and domain areas averaged over all blobs at a given ε step forΩ = 17.7 and Γ = 61.5 from a 60d × 60d cutout from the centre of the sample.Solid diamonds: average defect area ad. Open diamonds: average domain areaaD. Solid lines: fits of power laws where ad ∝ ε−νd with νd = 0.41 and aD ∝ ε−νD

with νD = 0.09.

Figure 39. Defect and domain area fraction for Ω = 17.7 and Γ = 61.5 from a60d × 60d cutout from the centre of the sample. Solid circles: ρd. Open circles:ρD.

In fact, the average number of defects per image Nd grew as ε increased as indicatedby figure 40. This behaviour is qualitatively the same as that found in the SH simulation.Likewise, as in the simulation the average number of domains per image ND grew with ε.For large enough Γ and Γ∗ one would expect Nd and ND to be extensive quantities. Inthe case of the simulation Nd and ND for different Γ∗ roughly collapsed when divided bythe appropriate value of Γ∗, as shown in figures 22 and 23.

As a comparison with the ξθ time series shown in figure 30, the time dependence ofthe defect area ad and domain area aD time series was determined by averaging over allthe defects and domains for each consecutive image. Figure 41 shows an example of asegment of such a time series. We note that both ad and aD fluctuate over time, however

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Figure 40. Average number of defects and domains per image for Ω = 17.7 andΓ = 61.5 from a 60d × 60d cutout from the centre of the sample. Solid circles:Nd. Open circles: ND.

Figure 41. Short sections, all with the same arbitrary origin, of time series ofξθ, ad, and aD for Ω = 17.7, ε = 0.05, and Γ = 61.5. Dashed lines are separatedalong t by a time 1/f , corresponding to the domain switching period determinedfrom the peak in the spectrum shown in red in figure 32. Red: ξθ. Black: ad.Blue: aD.

ad seems to have a periodic component like ξθ does, while aD seems to fluctuate with noparticular period.

The Fourier spectrum of the ad and aD time series, shown in figure 42, confirms thatthis is the case. The defect area contains a time periodic component that is in agreementwith the domain switching frequency, while the domain area seems to possess no dominantfrequency component. This is indicative of the mechanism for domain switching. Duringa switching event, defects form in the interior of the domain and grow in area. The defectsare diminished after the switch, so the average defect area decreases. This defect growthcycle explains why ad contains a periodic component commensurate with the switchingfrequency. The domain area aD does not seem to contain this periodic component. This

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Figure 42. Spectra of longer time series like the ones shown in figure 41 forΩ = 17.7 and ε = 0.05. Red: from ξθ. Black: from ad. Blue: from aD.

Figure 43. ε dependence of D for the Γ = 61.5 sample and several Ω valuesdetermined from a 60d × 60d cutout from the centre of the sample. Plus signs:Ω = 15. Circles: Ω = 16.25. Squares: Ω = 17.7. Triangles: Ω = 19.5. Diamonds:Ω = 21.7.

is either because the area is truly more stable during the course of a switching event orthe value of Γ∗ is too small to detect the periodicity. Regardless of the exact explanation,it is clear that the defects play the dominant role in a domain switching event.

The values for the dot-product-like quantity D, which measures the angle betweenthe major axis of the domain and the wave director, agreed remarkably well between theSH simulation and the experiment. As discussed in section 7.2, we found that D wasindependent of both ε and Γ∗. Independence from Γ∗ is welcome, especially consideringthe image size difficulty with accurately determining other quantities from defect anddomain detection. In the experiment we found values 0.7 � D � 0.8 which correspondto an angle of about 20◦ between the major axis and wave director of the domain. Wealso found that D decreased slightly with increasing ε. As an illustration of the domainshape, figure 44 shows a picture of a domain detected by the software, with the majorand minor axes and the wave director drawn on top.

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Figure 44. An example of a domain detected by the software from a 26d × 26dcutout from a pattern image with Ω = 17.7, ε = 0.05, and Γ = 61.5. Blue line:major axis. Red line: minor axis. Black line with white arrowheads: averagewave director for all pixels in the domain. Black outline: perimeter of domain.Colour: θ field using the palette shown in figure 2.

9. Summary

Our main topic has been the application of the local wave director (LWD) technique to thedomain chaos (DC) state of Rayleigh–Benard convection. Data from both experiment andsimulation were used to study this state. The LWD technique provided the orientation andamplitude of the wave director for the striped patterns of DC. Two main algorithms forcomputing the LWD were considered, and the Cross–Meiron–Tu algorithm was found tobe slightly better than the Egolf–Melnikov–Bodenschatz algorithm for our purposes. Fromthese fields, numerous secondary quantities were determined such as the autocorrelationlength ξθ of the orientation field, a characteristic frequency f corresponding to a periodiccomponent in ξθ, the area, number and area fraction of defects and domains, and aquantity D that represents the orientation of domains relative to the wave directors ofthe rolls within them.

The primary purpose of the simulation was to help facilitate a detailed understandingof the effect of a finite image on the results. Because the experiment was limited torelatively small image sizes, this was important for the interpretation of the data. Inaddition, it was interesting to compare the physical DC system with the Swift–Hohenbergmodel for DC.

The simulation indicated that the image size needed to be very large before resultsfrom the LWD analysis converged on a Γ∗ independent result. In the case of the scalingexponent νd for the defect area ad, the simulation converged on νd � 0.5 for large Γ∗.This differs only modestly from the result νd = 0.41 obtained from the experiment. Thedomain area aD from simulation was not Γ∗ independent at even the largest available Γ∗.This prevented a careful test of the scaling exponent νD, which is predicted to be 3/4 bythe Ginzburg–Landau model. The simulation also indicated that Nd and ND are extensivefor Γ∗ � 250. Finally, the quantity D, which measures the orientation of the domains

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relative to the local wave director, was consistent between the experiment and simulationat all Γ∗ and ε. The value for D indicated that the major axis of the domain made anangle of about 20◦ with the wave director.

For completeness, results for f determined from the structure factors (SF) of thesimulation were also provided. This was for the purpose of understanding the effect ofthe finite image size on obtaining f . It was previously found that the image size had acrucial effect on the determination of ξ from the SF [28]. Unlike ξ, where extremely largeimage sizes or use of the maximum entropy method was required to overcome the finiteimage size, f was best obtained with the Fourier-transform method at small image sizes.This result further justifies the use of small image size to obtain f elsewhere [29].

Finally, a periodic component was observed in the time series of ξθ in the experiment.The frequency corresponded to that obtained earlier with the SF [12]. The same periodictime dependence was present in the simulation, although the signal was not strong enoughto measure f at all ε. Further, the defect area ad in the experiment exhibited periodicityat the same frequency as ξθ, but the domain area aD did not. This is indicative of thedominant role that defects play in a domain switching event.

Acknowledgments

We are grateful for numerous stimulating conversations with a number of people, includingespecially E Bodenschatz, M C Cross, P C Hohenberg, A Newell, M Paul, W Pesch, andJ D Scheel. This work was supported by the US National Science Foundation throughGrant DMR02-43336.

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Documents

J. Stat. Mech. - UCSBSwift–Hohenberg model and from experiment for domain chaos near the onset of rotating Rayleigh–B´enard convection. Using both global Fourier analysis and