Table Of ContentLecture Notes in Physics
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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
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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
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© Springer-Verlag Berlin Heidelberg 1991
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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 +-..
