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For further information please contact Springer-Verlag, Physics Editorial Department ,V Tiergarten- strasse ,71 W-6900 Heidelberg, FRG Jean-Daniel Foumier Pierre-Louis Sulem (Eds.) Large Scale serutcurtS in Nonlinear Physics Proceedings of a Workshop Held in Villefranche-sur-Mer, France 13-18 January 1991 galreV-regnirpS Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest Editors Jean-Daniel Fournier Pierre-Louis Sulem CNRS, URA 1362 Observatoire de la C6te d'Azur B. E 139, F-06003 Nice C6dex, France ISBN 3-540-54899-8 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-54899-8 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1991 Printed in Germany Typesetting: Camera ready by author Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 58/3140-543210 - Printed on acid-free paper Preface This volume contains the proceedings of a workshop initiated by a suggestion of Her- bert M. Fried to bring together scientists whose research focuses on large-scale structures in various fields of non-linear physics. The meeting, organized by Jean-Daniel Fournier, Herbert M. Fried and Pierre-Louis Sulem, was held at the "Citadelle" in Villefranche sur Mer (France), 31 - 81 January 1991 and was attended by 54 participants. Coherent states, convective and turbulent patterns, inverse cascades, interfaces and cooperative phenomena in fluids and plasmas were discussed, together with the imple- mentation of concepts of statistical mechanics to particle physics and nuclear matter. Special attention was devoted to phenomena such as mixing, fast dynamo and pre- dictability, which display macroscopic features, even though generated by small-scale dynamical processes. In this context, homoclinic structure, the KAM theorem, Lya- punov stability and singularities were addressed. A lecture was delivered on a new perturbative technique for non-linear classical and quantum fields. Finally, new results concerning the analysis of hierarchically organized objects were presented. In collaboration with Springer-Verlag, a special effort was made in order that these proceedings be attractive to a large audience. Authors were asked to put their contri- bution in perspective. Moreover, about one third of the articles are extended versions of conference talks, intended to provide an introductory background to the reader. To all the authors and to the Managing Editor of Lecture Notes in Physics, we wish to express our gratitude. We also thank all the lecturers and participants for the quality of their presentations and the interest of their comments. Finally, we are grateful to Tran Thanh Van for valuable advice. The meeting benefitted from partial support from the CNRS (astrophysics and mechanical engineering departments), the ACCES program of the MinistSre de la Recherche et de la Technologic and the City of Nice through the LSpine Committee. J.-D. Fournier and P.-L. Sulem Nice, July 1991 Co~e~s The phase-diffusion equation and its regularization for natural convective patterns 1 Y. Passot and A.C. Newell Asymptotic time behavior of nonlinear classical field equations 12 Y. Pomeau Nonlinear waves and the KAM theorem: nonlinear degeneracies 73 W. Craig and C.E. Wayne Homoclinic structures in open flows 05 .71 Rom-Kedar Large-scale structures in kinetic plasma turbulence 3"7 D.F. Eseande Supersonic homogeneous turbulence 501 D.H. Porter, A. Pouquet and P.R. Woodward Large-scale instabilities and inverse cascades in ordinary and MttD flows at low Reynolds numbers 621 B. Galanti and P.L. Sulem From quantum fields to fractal structures: intermittency in particle physics 741 R. Pesehanski Hot QCD and the quark-gluon plasma .161 M. Le Bellac New approach to the solution of nonlinear problems 091 C.M. Bender Numerical study of the transition to chaotic convection inside spherical shells 112 L. Valdettaro and M. Rieutord Direct numerical simulations of natural and controlled transition of three- dimensional mixing layers ,222 P. Comte, M. Leaieur and E. Lamballai8 Organized vortices as maximum entropy structures 232 J. Sommeria Spectral degeneracy and hydrodynamic stability 242 L Goldhirseh Nonlinear stability of plane Couette flow 252 B. Dubrulle Continued-function solutions and eikonal approximations 262 H.M. Fried Quantum chaos and Sabine's law of reverberation in ergodic rooms 762 .O Legrand and .)1 Sornette Self-organized criticality, earthquakes and plate tectonics 572 .)1 Sornette, A. Sornette and .C Vanneste IIIV Psi-series and their summability for nonintegrable dynamical systems 278 J.-D. Fournier and M. Tabor Bound solitons in the nonlinear SchrSdinger/Ginzburg-Landau equation 288 B.A. Malomed The colour of the force in the renormalized Navier-Stokes equation: A free parameter? 295 D. Carati Structure of homogeneous turbulence observed in a direct numerical simulation 303 M. Meneguzzi and A. Vincent New results on the fine scale structure of fully developed turbulence 312 M. Vergassola and .U Frisch On fast dynamo action in steady chaotic flows 313 A.D. Gilbert Magnetic structures in fast dynamo 318 A. A. Ruzmaikin Rigorous wavelet analysis for multifractal sets 320 J.M. Ghez and S. Vaienti Eigenfunction analysis of turbulent mixing phenomena 329 RM. Everson, L. Sirovich, M. Winter and T.J. Barber Wavelets and the analysis of astronomical objects 340 A. Bijaoui List of participants 349 THE PHASE-DIFFUSION EQUATION AND ITS REGULARIZATION FOR NATURAL CONVECTIVE PATTERNS T. Passot and A.C. Newell Department of Mathematics, University of Arizona, Tucson AZ, 85721 Abstract We first present the phase diffusion and mean drift equation which describe con- vective patterns in large aspect ratio containers and for arbitrary Rayleigh and Prandtl numbers. Some applications are presented such as the prediction of the selected wave- number or the instability of loci. We propose in a second step a regularized form of the phase diffusion equation able to reproduce the formation and dynamics of defects. I. Introduction Many interesting behaviors of pattern-forming systems described by non-linear par- tial differential equations can be captured by standard perturbative techniques when the stress parameter is small and/or there exists a spectral gap between the microscopic and the macroscopic scales. We are mainly interested here in the latter case, and more particularly in the case of Rayleigh-Bhnard convection where only rolls are formed. A physical system close to a bifurcation can be described by a low-dimensional dynamics if the number of modes whose eigenvalue cross the imaginary axis at the bi- furcation threshold (the order parameters) is small. Although these order parameters, which determine the macroscopic order of the system, still interact with the damped modes, the "slaving principle", (rigorously demonstrated for ODE's), states indeed that these decaying modes can be eliminated and explicitly expressed by the order parame- ters. In the class of systems leading to pattern formation, this may be compromised due to the spatial extension of the system (an infinite number of modes are squeezed in the immediate vicinity of the critical point). When one direction can be privileged (either because the system is not rotationaly invariant as it is the case for liquid crystals, or because one chooses to look at a particular direction only), it is possible to derive ampli- tude equations (such as the Ginzburg-Landau or the Newell-Whitehead-Segel equations [1]) which describe the behavior of the order parameters on a slow time scale. These equations contain slow spatial derivatives to account for the mode coupling in the band- width around each normal mode and thus are called envelope equations. They can be reduced to a generic form depending only on the type of bifurcation considered and the symmetries of the problem. Another approach can be adopted to treat the fully two-dimensional case [2], where equations for the order parameters are derived which contain the fully spatio-temporal dynamics on a fast varying scale (such as in the Swift- Hohenberg equation 3). It will be observed in the conclusion that the approach we propose in this paper is close to this one but it applies even far from the bifurcation point. Usually an order parameter is a complex field ,4~ corresponding to an amplitude A and a phase ~. It often happens that, due to some additional gauge invarianees of the amplitude equations 4~( +-- .ZIexp(i~)), corresponding to physical invariances of the system (translational invariance), the dynamics of A, close to a particular solution .d0, can be reduced to the evolution of the phase variable b~ only. This reduction is possible if the amplitude A remains slaved to phase gradients on the so-called phase branch, graph of the phase eigenvalue ~(q) in terms of the wavenumber q of the phase perturbation. Homogeneous phase perturbations of a given solution 04~ are marginal so that ~(0) = 0. The phase branch determines the linear terms of the phase equations; the nonlinear terms are determined through the elimination of the damped modes. It is however now well-known 4 that the phase description may break down when phase instabilities are present. For example the phase equation corresponding to the complex Ginzburg-Landau equation is the Kuramoto-Sivashinski equation which is known to develop a form of weak turbulence whereas the long term development of the phase instability in the complex Ginzburg-Landau equation always leads to the appearance of defects in the system, due to the coupling of phase and amplitude modes. When the stress parameter is well above its critical value leading to the onset of the spatial pattern under investigation, it is still possible (and in fact it is only possible) to derive a large-scale description of the dynamics in terms of phase variations. The undisturbed pattern is indeed translationally invariant, and thus the phase mode is still a marginal mode. The amplitude of the nonlinear waves building the pattern is however always slaved to phase gradients, except possibly near defects. An exact derivation of the phase equation for rolls with no privileged direction has been achieved recently in the ease of Rayleigh-B~nard convection by Cross and Newell 5 and Newell, Passot and Souli 6-7. The technique was already known in other fields: for modulated nondissipative nonlinear train waves as developed by Whitham 8 or for modulated traveling waves in reaction-diffusion equations as exposed by Howard and Kopell 9 (see also Kuramoto 10). In all these examples, use is made of the inverse aspect ratio e as an expansion parameter, and not of the stress parameter as previously. This relies on the observation that almost everywhere in the convective pattern, a local wavevector k can be defined which changes slowly throughout the container. Therefore, ignoring the dependence in the vertical direction, the field variables can be represented by locally periodic functions f(O; R(1) ) where f is r~2 periodic in 0 and R(1) represents the collection of stress parameters of the system. The field variable f varies over distances of the order of the roll size whereas the amplitude A (the norm of f e.g.) or the wavevector (the gradient of 0) vary over distances of the size of the box. This fact allows for the definition of a large-scale phase O(X, ,TI T) = e0 where X = ex, Y = ey and T = e2t are the scales of the long wavelength disturbances. The phase 0 refers here to the total phase of the pattern whereas ~q defined above corresponds to the deviation about the phase of the basic pattern 00 = kox. Note also at this point that in contrast to the case of the Newell- Whitehead-Segel equations, direction Y is here scaled in the same way as direction X since we want to preserve rotational invariance. To leading order, and when no mean flow effects are present, the phase obeys a universal quasi-linear diffusion equation T(k)OT + V" kB(k) = 0 (1) where > 0 and B(k) are calculable functions of the waven ber k = N. To this order of approximation, the amplitude A of the rolls is slaved to the wavenumber k through the relation d 2 = #2(k) (2) Equation (1) generalizes the fixed orientation equation of Pomeau and Manneville 11. At finite Prandtl numbers a large scale horizontal mean flow, generated by the curvature of the rolls, advects the phase contours. Its presence is due to the existence of another marginal mode, namely a large-scale varying pressure field. The next section is devoted to a derivation of these coupled phase-mean drift equations and Section 3 is concerned with some predictions made on the behavior of convective patterns using the previous equations. In particular it is shown that all previous theories are contained in the present formalism and that moreover some new instabilities can be analyzed, such as the one of circular target patterns. The difficulty with equation (1) is that it is ill-posed for some values of the wavenum- ber, outside the nonlinear stability region. Section 4 is devoted to the definition of a regularizing scheme for that equation, illustrated on a microscopic pattern forming model, the Swift-Hohenberg equation. Section 5 exposes some numerical experiments on the regularized equation and Section 6 is the conclusion. II. The phase diffusion-mean drift equations The starting point of the calculation is the determination of stable fully nonlinear straight parallel roll solutions of the following Oberbeck-Boussinesq equations: (3~) a(&u+u. Vu) =-Vp+T~+ V2u (3b) O,T + uVT = Rw + V2T V.u=0 (3c) where the parameter R represents the Rayleigh number and rc is the inverse Prandtl number. The temperature is T, the pressure p and the velocity u has components (u, v, w). Rigid-rigid boundary conditions are considered which state that u = T = 0 at the top and bottom boundaries: z = 4-1. The existence of such a r~2 periodic solution v = f(0, z, k) (where v = (u, v, w, T,p) and 0 = kx) is assured by the previous work of Busse and Colleagues 12-19; f is calculated, using a Galerkin technique as in 12. Now we must look for modulated solutions and solve the linear equations obtained after inserting in (3a-3c) the following expansions in powers of the inverse aspect ratio zqO ~---- zO 0x ~ oOk + ¢Vx QG t 4--- e0T08 q- e20T v = vo + evl + e2v2 +-..