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APS/123-QED Hexaroll chaos in inclined layer convection Will Brunner Max Planck Institute for Dynamics and Self Organization Göttingen, Germany Jonathan McCoy Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York Werner Pesch University of Bayreuth, Bayreuth, Germany Eberhard Bodenschatz 6 ∗ Max Planck Institute for Dynamics and Self Organization 0 0 Göttingen, Germany and 2 Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York (Dated: February 6, 2008) n a Wereportexperimentalobservationsofhexarollchaosininclinedlayerconvection. Twoseparate J populations of defect complexes characterize the defect turbulent state, one the classical penta- 5 hepta defect of hexagonal planform patterns, and the other a short-lived complex capable of self- annihilation. Bymeasuringthedefectstatistics weshow thatshort-liveddefect complexesgiverise ] tolineardefectdestructionrates. Thisobservationexplainspreviousresultsinotherdefect-turbulent S pattern forming systems. P . n PACSnumbers: 47.20.Bp,47.20.Ky,47.27.Te,47.54.+r i l n [ Systems that are driven out of equilibrium often show Here we report experimental results from non- similar patterns although the underlying processes can Boussinesq inclined layer convection (ILC) where a in- 1 be quite different [1]. One challenge is to find measures clined layer is heated at one side and and cooled at the v that can quantitatively assess the similarity of different other. It is well known [15] that a shear flow is set up 6 1 patterns. A convenient starting point for a classification with a flow that is up at the warm side and down at the 0 are topological defects, as has long been known in the cold. Fornon-Boussinesqconditionsinahorizontallayer 1 field of fingerprint recognition [2]. Much of the previous a hexagonal planform would be observed [1]. However, 0 work [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] is concerned hereduetotheshearflowonerollorientationispreferred 6 withtheparticle-likeappearanceofdefectsandwiththeir leading to a pattern that is similar to the hexarolls de- 0 / chaoticspatiotemporaldynamics,i.e., defectturbulence. scribedforcylindricalrotatingconvection[16,17]. Inour n A key feature of defect turbulence is the reaction dy- case,weobserveadefectturbulentstateofspatiotempo- i l namics between different types of defects. Gil et al. [3] ral chaos (STC) (Fig. 1) that we term Hexaroll Chaos. n : showedthatthetopologicalpropertycarriedbydefectsis We show that hexaroll chaos has two separate popula- v conserved, causing them to behave as charged particles tions of defect complexes, one the classical penta-hepta i X in reactions. This leads to a generalized Poisson dis- defect of hexagonal planform patterns, and the other a r tribution for defect populations, and reaction rates that short-livedcomplexcapableofselfannihilation. Bymea- a obeythemassactionlaw,wheredefectsarecreatedwith suringthedefectstatisticsweshowthatshort-liveddefect rates independent of N, the number of defects present, complexes give rise to linear defect destruction rates ex- and are annihilated at a rate proportional to N2. This plainingtheearlierobservationsinotherpatternforming result was confirmed [5] in a thermal convection experi- systems. Hexaroll chaos can be described by amplitude ment,wheretheyalsoaccountedfordefectsenteringand equations of the form leaving through the boundaries. Recently, however, two resultshavebeenbroughtforwardthatappeartocontra- 3 dict the notion that mass action is followed. The first of ∂tA1 =σ1A1−η1A∗2A∗3−A1Xa1j|Aj|2, (1a) these is a Swift-Hohenberg type model of rotating ther- j=1 mal convection [9], and the second is a calculation of 3 thecomplexGinzburg-Landauequationwithnoiseadded ∂tA2 =σ2A2−η2A∗1A∗3−A2Xa2j|Aj|2, (1b) [14]. Both produce defect statistics where defect annihi- j=1 lation rate has a part proportional to the number of de- 3 fectspresentinadditiontotheexpectedN2 dependence. ∂tA3 =σ3A3−η3A∗1A∗2−A3Xa3j|Aj|2. (1c) To date an explanation of this phenomenon is lacking. j=1 2 Here the convection amplitude is a linear superposition 3 of the three modes: V(x) = A exp[iq x]+c.c., Pj=1 j j where the vectors q define a hexagonal lattice, q1 = (√23q,−q/2),q2 = (0,q),q3 = (−√23q,−q/2), and the vectorxrepresentsposition. Theconvectioncellistilted upwardalongthe x=(1,0)direction,breakingthe sym- metryoftheA2mode,buttheothertworemainsymmet- rical,sothefollowingcoefficientrelationscanbeinferred: η1 =η3∗,a12 =a∗32,a11 =a∗33,a13 =a∗31 (2) where a star denotes complex conjugate. The linear growth rates are given by: σ1 =ǫ−∆ǫ+iω,σ2 =ǫ,σ3 =σ1∗ (3) whereǫ≡ ∆T ∆Tc isthereducedcontrolparameter,∆T ∆−Tc isthetemperaturedifferencedrivingconvection,and∆T c is the value of ∆T at onset of convection. ω > 0 repre- sents the downhill drift of modes one and three. The implication of these equations is that convection rolls of mode two appear at ǫ = 0 and the other two modes appear at ǫ = ∆ǫ. We observe ∆ǫ = 1.0×10 3 ±10 4 − − which agrees well with the value of 9.7×10 4 obtained − from numerical solution of the Navier-Stokes equations. Dislocationsprimarilyariseinmodesoneandthreeinre- gions of low amplitude. These can be pairedinto defects of two distinct types. One type is the classical penta- hepta defect (PHD), where a dislocation of one topolog- FIG.1: (a)Asampleimagefromǫ=0.054. Uphilldirection ical sign in one mode is bound to one of opposite sign istoright. (b)Oneobliquemodeonly,producedbyapplying in the other mode. Additionally, we observe a new type anangularFourierfilterto(a). Contrast-enhancedforclarity. of defect complex where short-lived pairs of dislocations of opposite sign are found in the same mode. We term these dislocation pairs same-mode complexes (SMCs). We collected data by quasistatically increasing ǫ in We use a gas convection apparatus [18] with a work- steps of 10−3 to ǫ = 0.054, where we acquired 106 ing fluid of CO2 at (4.240×106±103) Pa regulated to shadowgraph images of the region of interest at 5 ±300Paandthemeantemperatureof(31±0.05)◦Creg- frames/second,foranobservationtimeof357100τv. (See ulated to ±5 × 10 4 C. We tilt the plane of the fluid Ref. 18 for details of shadowgraphic technique.) An ex- − ◦ layer to θ = 5 from the horizontal. The fluid layer ample frame appears in Fig. 1. Note the variation in ◦ is d = 308±2µm high, and the vertical diffusion time obliquerollamplitudethatgivestheappearanceofmixed τ =0.56±0.01s. Prandtlnumberisσ ≡κ/ν ≈1,where rolls and hexagons, and that oblique-mode dislocations v κ is thermal diffusivity and ν is kinematic viscosity. We are restricted to those areas of low amplitude. find onset to rolls at ∆T =20.68±0.01 C and onset to We processed all images by first separating them into c ◦ hexarolls drifting downward with ω =0.8τ s/rad above threeindividualmodesbyapplyinganangularFourierfil- v that at ∆ǫ. We choose a region of interest 142d×95d, ter. We identified dislocations in each mode using previ- with the long edge parallel to the direction of tilt, lim- ously describedtechniques [8]. Dislocations in mode two iting spatial inhomogeneity of ǫ to < 10% during the (with stripes parallelto tilt) are rare,and were recorded experiment. The large value of ∆T makes convection but not processed further. After all dislocations were c strongly non-Boussinesq, with non-Boussinesq parame- identifiedforallframes,weassignedeachtoeitheraPHD ters (see, e.g., Ref. 10) of γ0 = 0.1653, γ1 = −0.2885, or to an SMC. We did this by iterating through all dis- γ2 = 0.2053, γ3 = 0.0085, and γ4 = −0.2491, where locations,and searchingwithin a thresholdradius for an γ0 and γ1 are the first- and second-order coefficients for opposite charge dislocation in either the same mode or the dependence of density on temperature, and γ2–γ4 theothermode. Ifsuchamatchwasfound,bothdisloca- arethefirst-ordercoefficientsforthetemperaturedepen- tionsweremarkedasbeingbound,andthesearchmoved dence of viscosity, heat conductivity, and specific heat, on to the unbound dislocations. After all dislocations respectively. werevisited,weincreasedthethresholdandrepeatedthe 3 FIG. 2: Histogram of separation distance of dislocations in defect complexes FIG. 3: PHD Creation, destruction, leaving and entering rates, (C,D,L,E, respectively) along with fits. Number (N) is the geometric mean of the populations of the two types of search over all unbound dislocations. This process was PHDs,whichareapproximatelyequal. Ratemeansnumberof continueduntilamaximumthresholdof∼12disreached eventsforeachN,dividedbythenumberofframescontaining exactlyN.Errorbarsrepresentstandarddeviation,calculated (Fig. 2). Typically,∼90%ofdislocationsareaccounted by subdividing the data set, and are used for weighting fits. for in bound pairs, leaving the other 10% unbound at Please note that the mean rate is overestimated when the the edges. PHDs are roughly four times more prevalent standard deviation is on the order of the mean because all thanSMCsinall106images. Thenextstepofprocessing valuesare positive. wastotrackthecenterpointsofPHDsandSMCsbythe dislocation-tracking method of Huepe et al. [12]. This allowed us to tally defect creation and destruction rates The reactions are: andratesofenteringandleavingtheframe(Figs.3&4). We calculated the rate for each N by dividing the num- 1+1 +3+3 ↔1+3 +1 3+, (5a) − − − − berofeventsbythe totalnumberofframescontainingN 1+1 ↔0, (5b) defects, where N is the geometric mean of the two types − of either PHDs or SMCs. We split the data set into 100 3+3− ↔0. (5c) equal parts to calculate standard deviation of rates. Where (5a) is the realignment of two SMCs into two Let us consider the types of reactions that SMCs and PHDs, and (5b) and (5c) are the self-annihilation of PHDs can undergo. This will illuminate later descrip- SMCs. Here we have two sets of rates, those for PHDs tions of defect creation and destruction rates. The classical case of a symmetrical hexagonal pattern (see,e.g., and those for SMCs. PHDs are createdby (5a) proceed- Ref.19)hassixtypesofPHDs: 1+2 ,1 2+,1+3 ,1 3+, ing to the right. The rate for this reaction is indepen- − − − − 2+3 and 2 3+ where, e.g., 1+ is a positive dislocation dent of the number of PHDs because there are no PHDs − − as reactants, and populations of SMCs and PHDs are in mode 1. These follow reactions of the forms uncorrelated. PHDs are destroyed by the same reaction movingintheoppositedirection,witharateproportional 1+2 +1 3+ ↔2 3+, (4a) − − − tothenumberofPHDssquared. SMCsarecreatedbyall 1+2−+1−2+ ↔0, (4b) three reactions going to the left, which gives a constant rate law. Finally, SMCs are destroyedby these reactions whichyieldninepossiblereactionsoncepermutationsare going to the right, giving a linear rate from (5b) and considered. Note that all destruction reactions (those (5c) with a squared term from (5a). In practice, we find whichproceedtotheright)involvetworeactants,sofrom that this squared term is negligible, so it is set to zero massactionweexpectaratethatdependsonthenumber when constructing fits. Otherwise, these rate relations of PHDs squared. Creationreactions follow a linear rate and those for entering and leaving given by Daniels and law, with a nonzero intercept due to (4b). Now, if we Bodenschatz [8] fit the data in Figs. 3 & 4 with good consider the asymmetric case of hexaroll chaos, we must results. disallow dislocations in mode two. There are now two Using detailed balance, we recursively define a modi- types of PHDs possible: 1+3 and 1 3+. In addition, fied Poisson distribution of the probability of finding N − − there are two types of SMCs possible: 1+1 and 3+3 . defects in the frame at a giventime in terms of creation, − − 4 existingframeworkofreactionsandratelawsifweintro- duce a new type of short-lived defect complexes (SMCs) that usually self-annihilate, but that may stabilize by binding into long-lived PHDs. These complexes appear to arise because of broken symmetry which changes the allowed ways of creating and destroying PHDs. We sug- gestthatthismayalsobethecaseinothersystems,such as the rotating hexagons of Young and Riecke [9] where theiransatzofmixedlinearandquadraticratelawsmay be separable into two sets of laws, one for SMCs and one for PHDs. Similarly, Wang’s STC state with noise added[14]maycontainapopulationofshort-lived,noise- induceddefects thatprimarilyannihilate withthe defect they were created with, thus following a linear rate law. More detailed study of these systems is called for to de- termine if there is a generality that connects symmetry FIG.4: SMCcreation,destruction,leavingandenteringrates breaking with altered rate laws. (C,D,L,E, respectively). Number (N) is the geometric mean of the populations of thetwo typesof SMCs ∗ URL: http://milou.msc.cornell.edu/; Electronic ad- dress: [email protected] [1] M. C. Cross and P. C. Hohenberg, Reviews of Modern Physics 65, 851 (1993). [2] F. Galton, Finger Prints (Macmillan and Co., 1892). [3] L. Gil, J. Lega, and J. L. Meunier, Phys. Rev. A 41, 1138 (1990). [4] D. A. Egolf, Physical ReviewLetters 81, 4120 (1998). [5] G. D. Granzow and H. Riecke, Physical Review Letters 87, 174502 (2001). [6] P.Colinet, A.A.Nepomnyashchy,andJ.C.Legros, Eu- rophysics Letters 57, 480 (2002). [7] P.Tóth, N.E´ber,T. M.Bock, A´.Buka,andL.Kramer, Europhysics Letters 57, 824 (2002). [8] K.E.DanielsandE.Bodenschatz,PhysicalReviewLet- ters 88, 034501 (2002). FIG.5: Probabilitydistributionfunctions(PDFs)foraverage [9] Y. Young and H. Riecke, Physical Review Letters 90, numbers of each type of SMCs and PHDs. Boxes are mea- 134502 (2003). sured,andcurvesarecalculatedfromfitsinFigs. 3&4using [10] Y. Young, H. Riecke, and W. Pesch, New Journal of detailed balance. Physics 5, 135 (2003). [11] K. E. Daniels, C. Beck, and E. Bodenschatz, Physica D Nonlinear Phenomena 193, 208 (2004). destruction, leaving, and entering rates: [12] C.Huepe,H.Riecke,K.E.Daniels,andE.Bodenschatz, Chaos 14, 864 (2004). N [13] T.Walter,W.Pesch,andE.Bodenschatz,Chaos14,933 C(j−1)E(j−1) P(N)=P(0)Y (6) (2004). j=1 D(j)L(j) [14] H. Wang, Physical Review Letters 93, 154101 (2004). [15] K.E.Daniels,B.B.Plapp,andE.Bodenschatz,Physical where P(0) is supplied as a normalization condition. Review Letters 84, 5320 (2000). Note that for the case of SMCs where C and E are con- [16] M.Auer,F.H.Busse,andR.M.Clever,JournalofFluid Mechanics 301, 371 (1995). stants,andDandLarelinearfunctionsofN,(6)reduces [17] O. Brausch, Ph.D. thesis, Universität Bayreuth (2005). to the Poission distribution. For the PHD case, (6) re- [18] J. R. de Bruyn, E. Bodenschatz, S. W. Morris, S. P. duces to the squared Poisson distribution of Gil et al. Trainoff,Y.Hu,D.S.Cannell, andG.Ahlers,Reviewof [3] with the finite-size extension of Daniels and Boden- Scientific Instruments67, 2043 (1996). schatz [8] (Fig. 5). We conclude that defect statistics [19] L.S.Tsimring,PhysicalReviewLetters74,4201(1995). of hexaroll chaos may be easily understood within the