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-COMMISSARIAT A L1ENERGIE ATOMIQUE CENTRE D'ETUDES NUCLEAIRES DE 5ACLAY CEA-CONF - -9806 Service de Documentation F91191 GIF SUR YVETTE CEDEX LI TRANSITION TO TURBULENCE VIA SPATIOTEMPORAL INTERMITTENCY CHATE H.- MANNEVILLE P. CEA Centre d'Etudes Nucléaires de Saclay, 91 - Gif-sur-Yvette (FR)1 Institut de Recherche Fondamentale Communication présentée à : Proceedings of YKIS'88 Kyoto (JP) 22-26 Aug 1988 O A TRANSITION TO TURBULENCE VIA SPATIOTEMPORAL INTERMITTENCE H. Chaté and P. Manneville Institut de Recherche Fondamentale DPh-G/PSRM, CEN-Saclay F-91191 Gif-sur-Yvette, France The study of low-dimensional dissipative dynamical systems has provided a reasonable understanding of the transition to temporal chaos in strongly confined systems for which the spatial structure can be considered as frozen. The situation is still less advanced for weakly confined systems where r.Laos has both a spatial and a temporal meaning. In order to approach the specificities of the latter, we have chosen to study first a partial differential equation (PDE) displaying a convective-type nonlinear term, steady cellular solutions as in convection, and a transition to spatiotemporal chaos, namely the damped Kuramoto-Sivashinsky (KS) equation: dt4 + v4> + dx*4> + dx*<f> + 2<l>dx<l> = 0 (1) The pure KS equation /l/ is recovered when n = I/KS = 0. A single control parameter then remains: the length L of the interval at which boundary conditions are imposed to the function. In this limit, for L large enough, the solutions are essentially turbulent (phase turbulence). Above T) = 1/4, the trivial solution 4> = 0 is stable. As a matter of fact, around this value equation (1) is better written in the form of a modified Swift-Hohenberg (SH) equation, a well known model of convection /2/. Slightly below 17 = rjsn = 1/4, stable steady cellular solutions exist with nonlinearly selected wavelengths of the order of 2TT\/2 /3/. L being given, the transition to turbulence can be studied by decreasing IJ from I?SH down to rjKs /4/. For L small, vis. ~ 10, the transition is rather reminiscent of what occurs in low dimensional systems. In the intermediate region L ~ 100, chaos is already spatiotemporal in essence but still sizeable end effects come and play a nontrivial role, thus complicating the picture. On the other hand, at larger L's, say for L > 800, the transition to turbulence occurs systematically via spatiotemporal intermittency. Spatiotemporal intermittency (STI) is a fluctuating mixture made of coherent laminar patches embedded in a "sea" of incoherent turbulent domains. Quantitatively, it is character- ized by a statistically stable proportion of laminar and turbulent regions function of the control parameter. Below a well-defined value »JSTI> i<e- on the "phase-turbulence side", this regime remains "for ever" while above 173x1, i.e. on the "convection side", it persists only for a finite time and the system eventually reaches a steady asymptotic state /5/. During the transient, laminar domains reach a "macroscopic" size, of the order of the total width of the system and the domains left spatiotemporally intermittent are seen to recede regularly. Being quite specific of large aspect ratio systems, this scenario can by understood as a contamination process reminiscent of directed percolation, as suggested initially by Pomeau /6/. However, a detailed understanding of the transition has not yet be obtained. First hints for a global theory derive from the recognition that it happens in a parameter range for which an inverse Hopf bifurcation takes place, which manifests itself as a finite amplitude periodic oscillation. Thus, two states are in competition, a steady cellular state and a periodic state and, in sufficiently long systems when confinement effects become weak, coherent nearly steady Proceedings of YKIS'88, Kyoto, August 22-26 SRM/88-67 structures can persist for a long time within a dominantly periodic but mostly incoherent region, well separated from it by domain walls. Moreover, since the bifurcation is not continuous, the coherence length of both states, the width of walls between them, and the relaxation, time, all remain finite. This suggests to consider the whole system as a series of juxtaposed sub-units, interacting with their neighbors at discrete times, i.e. as a chain of coupled maps: where X" is the value at time n of some effective variable X attached to site f, / a map governing the evolution at a given site supposedly isolated, and Wj a coupling weight defined on some neighborhood of each site. As a matter of fact, introducing so-called coupled map lattices (CML), i.e. considering arrays of maps coupled by terms to which a spatial meaning is given, is conceptually the simplest way to increase the number of degrees of freedom in a spatiotemporal context /7/. The transition to turbulence via spatiotemporal intermittency displayed by the PDE has been shown to possess statistical properties reminiscent of critical properties found at a second order phase transition, well in line with an interpretation in terms of directed percolation /8/. However, reliable numerical simulation turning out to be very expensive, we have been obliged to rely on the study of well-chosen CML's /9/ in spite of the conjectural features involved in the reduction. The "minimal requirements" for an adequate modeling of spatiotemporal intermittency are as follows: - first, the local map /(X) must lead to a splitting of the local phase space into two different components, one regular and the other chaotic, the chaotic one being only transient; in our first studies, this was obtained by means of a piecewise-linear map of the form: /PO = rX when O < X < 1/2 O when 1/2 < X < 1 (3) /(X) = fc(X- X*) + X* when X>X >1 1 with r > 2, |*| < 1, and X" = (r + 2)/4. second, a spatial coupling chosen to be merely diffusive for simplicity; when this coupling is restricted to nearest-neighbors in one dimension, the complete system can be written as: /W+1)). (4) A straightforward calculation shows that the uniform laminar state remains stable with respect to infinitesimal delocalized perturbations, as expected for a subcritical instability. On the other hand, localized, finite amplitude perturbations can grow at least initially. In this perspective, spatiotemporal intermittency can be understood as a turbulent state resulting from the conversion of a local transient temporal chaos into a global sustained spatiotemporal chaos. Obviously, more complicated possibilities exist (especially that occurring in the PDE since the bifurcated state is probably not well accounted for by the turbulent part of the local maps) however, if some universality were to be found in the transition to turbulence via spatiotemporal intermittency, relevant features should already be displayed by the considered model. Subsequent work has been devoted to a detailed quantitative analysis of the statistical properties of the CML close to the threshold and to the comparison with corresponding results for probabilistic cellular automata (PCA) /10/. Before proceeding with a presentation of our findings, let us recall that cellular automata are discrete-time, discrete-space systems with a finite number of accessible states per site, most often only two, one active, the other inactive or absorbing. The evolution in time is then defined in terms of transition rules depending on the configuration of the neighborhood of each site, these rules being either deterministic or probabilistic. Directed percolation (DP) is but one concrete realization of a PCA which can model for example the infiltration of a fluid through a porous medium: sites at the ends of pores become wet or dry according to whether they are connected to open or closed pores and to whether some parent sites upstream are wet. The dry state is absorbing since any site connected only to dry sites cannot become wet. Pursuing the analogy in the case of spatiotemporal intermittency we are led, for the local states, to the correspondence: turbulent «->• active and laminar «-•• absorbing. The probability p of having an open pore being the natural control parameter for directed percolation, there exists a critical value pc above which sites at infinity are wet with finite probability while remaining dry below. Moreover, close to pc the fraction of wet sites at infinity grows as (p — Pc)^°F where /?DP is the corresponding critical exponent. Accordingly, one can define a critical coupling ee below which spatiotemporal intermittency is only transient. Our first results were obtained with k = 1, i.e. a laminar state made of a continuum of fixed STI points. In that case, the fraction of turbulent sites was observed to vary as (e - ee)^ . Our best estimates for exponent /?STI are given for r = 3 as a function of space dimension d in the table below /9,11/ together with the corresponding values for directed percolation /8/. d ASTI A5P 1 0.25 0.28 2 0.50 0.60 As can be anticipated from the very existence of exponent /?STI> model (3) with fc = 1 and a diffusive coupling (4) displays a continuous transition via spatiotemporal intermittency. How- ever, the consideration of values given in the table suggest that the process does not belong to the universality class of directed percolation, as initially conjectured by Pomeau /6/. This is confirmed by the study of another critical exponent governing at threshold the distribution A/ of the lengths I of absorbing domains: M(I) — l~ç. For directed percolation one gets fop — 1-75 /8/ while for spatiotemporal intermittency (with r = 3) we have obtained fsTl — 1-99 /9/. The existence of a universality class specific to spatiotemporal intermittency is even questioned by the variation of f with r since when r = 2.1 we get fcTi — 1-78. As a matter of fact, the variation of f may not be a complete surprise since, when r = 2.1, the chance of escaping from a turbulent state toward a laminar state at a given site is smaller than for r = 3.