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Sawada, Yasuji

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Title Spatio-temporal Chaos in Fluid SystemsAuthor(s)Sawada, YasujiCitation物性研究 (1992), 58(5): 504-509Issue Date1992-08-20URL http://hdl.handle.net/2433/94926RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversitySpatio-temporal Chaos in Fluid SystemsYasuji SawadaResearch Institute of Electrical Communication,Tohoku University, Sendai 980Fluid systems are known to undergo series of changes ofmacroscopic states when they are driven far from equilibrium. Atypical example is convection system where fluid is confinedamong two parallel boundaries and heated from below. As thetemperature di fference is increased, the system changes from aquiescent state, steady convective structure, oscillatory,When the horizontal size ischaotic, and so on. The bifurcation route to chaos were prettymuch clarified for a system whose horizontal size is comparableto t he ve r ticaId i me n s ion ( 1 ) .extended one-dimensionally or two-dimensionally, the spatialdegree of freedom contributes to increase the effective number ofdynamical variables, and the bifurcation sequence of the systembecomes more complicated.It is, therefore, questioned whether the behavior of aspatially extended system can still be efficiently looked at as adeterministic dynamical system, or should it be more effectivelytreated as a statistical system. The spatio-temporal chaos is aname for the complex behavior of spatially extended systemsincluding fluids taking the former view point. This brief noteoverviews the recent works on spatio-temporal chaos of fluidsystems driven far from equilibrium.- 504-rmtfit!\m~O)~;jj\ c:1i:~ - STATPHYS 19~=JQ] ~t-c - JS. Sato et. all (2) showed that fluids in a one-dimensionallyextended rectangu'lar thermal convection cell whose aspect ratiois 15xl exhibited an inhomogeneity in oscillatory strength anddimension of chaotic attractors when the controll parameter isrelatively low. Whether the localization of activity isthere.intrinsic or caused by a boundary effect remained unclarifiedRecently T.Kato et.al.(3) undertook a similar experimentusing an annular cell to avoid boundaries. They found that thelocalization of oscillation strength is intrinsic and related tothe deformation of the underlying periodic structure.At higher control parame.ter regions, spatio-temporalintermittency, which is a mixed state of laminar and turbulentdomains is, generally obserbed It was predicted, first byK.Kaneko(4) based on numerical simulation of coupled map latticemodel and later by H. Chate et. a1 . (5) based on numericalsimulation of. a phase equation, that turbulence is achieved inone-dimensional system by spatio-temporal intermittency.The experimental evidence was provided first by S.Cilibertoet. a1. ( 6 ) and later by F.Daviaud ( 7 )et . al . . They have shownthat the spatio-temporal intermittency shows an interestingstatistical behavior. It was found that the number of laminarregion of a given laminar length is a power of the length andthat the exponent behaves similar to the second-order phasetransition. The experimental results are partially in accordancewith the numerical simulation of the model systems. S.Ci1ibertoet.al. tried to construct a thermodynamics of spatio-temoralchaos based on these experimental observations(8).- 505-~~~¥IH5Spatio-temporal chaos has also been studied in two dimensionalsystem. As two-dimensional experimental systems ,e lec t ro-hydrodynamic convection phenomena of nematic liquid crystals isoften used because of the time constant faster than the ordinarythermal convection. The phenomena was introduced into p!lysicscommunity by Kai et.al.(9) as the one which shows vaiiety ofinteresting pattern formation. The electro-hydrodynamicconvection of nematic liquid crystal shows a more complicatedbifurcation diagram compared to the thermal convection(lO).However, one may simply say that the system undergoes from aquiescent state to a regular parallel role state to a fluctuatingparallele role state by choosing a proper experimental parameters.Then,after some complicated states one can reach a twodimensional oscillating lattice state.Here " two problems may be relevant. One is how to understanddeterministically the. fluctuation and defects in the parallelroll state. Theoretically, it is shown that the spatially andtemporally slow variable just above a bifurcation point can bedescribed by a Ginzburg-Landau type of equation with complexcoefficients. Furthermore, for a phenomena in which only phasevariation, not phase variation, is relevant, the complex GLequation is reduced to Kuramoto-Shivashinski equation(ll). Thisequation should be useful to describe a phenomena such asfluctuating parallel role of convection structure.P.Coullet et.al. (12) have made a large scale computer.simulation on the GL equation wi th complex coefficients. Theirresults showed that as soon as the phase stability condition isbroken the amplitude variation occurs and from time to time zero- 506-r~~t!ltn~(7)~;tktH!:fI'l - STATPHYS 19~:rttJ~t"( - Jamplitude spots appear and create pairs of defects. It may meanthat phase-turbulence without defects may be not stable. RecentlyH.Chate is carrying out simulation over wider parameterspace(13), so the situation will become clearer in near future. Aclear result by experimental research is also needed.The second point is self-organ'ization of a pacemaker whichemits target pattern of phasewaves in an oscillating grid latticestate(14). This phenomena is interesting in two aspects. One isthat it is similar to the pacemaker of Belouzov-Zhabotinskisystem, yet it is intrinsic in the sense that it is createdwithout any spatial inhomogeneity. The second is that the emittedwave is spatially bounded. It was experimentally verified thatthe convection pattern is spatially expandede in the pacemakerregion. Recently H.Sakaguchi(15) and independently F.Daviaudet.al.(16) derived phase equations based on coupled equations ofoscillatory field and underlying spatially periodic structureproposed by P.Coullet et.al.(17). Detailed experimentalmeasurement by M.Sano et.al.(18) verified that the theorywell explains experimental resul ts. However, the physicalcoupling mechanism between the oscillatory field and theunderlying convection structure is unknown at present stage.Spatio-temporal chaos is after all a deterministic view pointfor complex phenomena. In other words, according to this viewpoint, the complex phenomena is a collection of yet unknownorders. It would be interesting to know how far one can continuethis view point to understand turbulence. On the other hand- 507-biological system is certainly complex but generally con~ideredto be in a complex ordered state. The research to understand theliving state(19) is also considered to be along the same linewith the present subject.REFERENCE1) For example, P.Berge, Y.Pomeau and Ch.Vidal, Order withinChaos(wiley, New York, 1987)2) S.Sato, M.Sano and Y.Sawada, Phys. Rev. A37,1679(1988)3) T.Kato and Y. Sawada, submitted4) K.kaneko, Prog. Theor. Phys. 21, 1033(1985)5) H.Ghate and P.Manneville, Phys. Rev. Letters 58, 112(1987-)6) S.Ciliberto and P.Bigazzi, Phys. Rev. Letters §Q, 286(1988)7) F.Daviaud,M.Bonetti and M.Dubois, Phys. Rev. A42, 3388(1990)8) S.Ciliberto and M.Caponeri, Phys. Rev. Letters 60, 2775(1990)9) S.Kai and K.Hirakawa, Suppl. Prog. Theor. Phys. No.64,212(1978)10) S.Sasa, Prog. Theor. Phys. 83, 824(1990); 84, 1008(1991);S.Nasuno and S.Kai, Europhys. Lett. li, 779(1991); S.Nasuno, Ph.Dthesis(Tohoku Univ. 1989 unpublished)11) Y.Kuramoto, Chemical Oscillations, Waves, and Turbulence(Springer-Verlag, Berin, Heidelberg, New York, Tokyo, 1984)12) P.Coullet, L.Gil and j.Lega, Physica D, n, 91(1989)13) private communication14.) S. Nasuno, M.Sano and Y. Sawada, J. Phys .. Soc. Jpn. 58,1875(1989); S.Nasuno and Y.sawada, Suppl. Prog. Theor.99450(1989); M.Sano, K.Sato, S.Nasuno and H.Kokubo, Phys. Rev. A1992 to appear; M.Sano, K.Sato and B.Janiaud, Pattern Formationin Complex Dissipative System(World Sience, to appear 1992 ed. byS.Kai)15) H.Sakaguchi, Prog. Theor. phys. ~, 241(1992)- 508-16}F.Daviaud, J.Lega, P.Berge, P.Coullet and M.Dubois, Physica D,.§.Q, 287(1992)17} P.Coullet and a.looss, Phys. Rev. Letters, 64, 866(1990)18} M.Sano, K.Sato, S.Nasuno and H.Kokubo, Phys. Rev. A,appear1992 to19) Y.Sawada, Proc. 4-th Asia Pacific Phys. Conf., 69~(WorldScientific 1991); Pattern formation in Complex DissipativeSystems(World $cience, to appear 1992)- 509-

流体における時空カオス

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流体における時空カオス

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