Large scale spatial structures in two-dimensional turbulent flows

Nuclear PhysicsB (Prec. Suppl.) 2 (1987) 453-484 North-Holland, Amsterdam

453

LARGE SCALE SPATIAL STRUCTURES IN TWO-DIMENSIONAL TURBULENT FLOWS Bast1 Nicolaenko (Nichols)* Mathematical Sciences Institute, CorneU University, Ithaca, NY 14853. Permanent address: Center for Nonlinear Studies and T h e o r y Division, Los Alamos National Laboratory, Los Alamos, NM 87545 The organized motion of coherent structures in two-dlmenslonal turbulent flows is vested with a quasi-determlnlstlc behavior. The spontaneous formation of large-scale structures could be accounted for as a manifestation of long-wave instability of the corresponding small-scale flow. For a class of such flows, dynamics of coherent structures are modeled by the KolmogorovSpiegel-Sivashinsky family of partial differential equations (PDE's). We present mathematical and computational investigations of the latter. W e outline how locally negative viscosity mechanisms generate coherent structures with a finite, small number of degrees of freedom. Both bursting streaks and spatially localized, temporally intermittent structures are evidenced. Although an infinity of small scales a priori dissipate energy, these are slaved by a finite number of inertialmodes. Only finite dimensional dynamics and chaos sabslst. I. I N T R O D U C T I O N

AND SUMMARY

OF RESULTS

A most intriguing problem in the theory of hydrodynamic turbulence is the formation of large-scale structures in a flow performing random turbulent motion at small scales. Generally, coherent structures (C.S.) and small scale turbulence may b e viewed as the main features of a double structure:

Although the C.S. are not especially energetic nor

long lived, they are important in transport of heat, mass and momentum I°2. They are responsible for the i n t e r m i t t e n t n a t u r e o f turbulence such as observed in the outer edges of turbulent shear flows, specifically wakes and jets 3. Indeed Townsend realized that

the

C.S. (large eddies in turbulent

quasi-deterministic behavior.

Alternatively, the

shear flow) ought to have a

earliest observations of organized

motion were made in the turbulent boundary l a y e r along the wall where the motion is most complex 4. Alternating arrays of high and low speed regions called °'streaks" have been observed to interact with the outer portions of the flow. The streaks lift up, oscillate suddenly and break up; this sequence o f events is called ~bursttn~ '4. These observations illuminate an important link b e t w e e n a quasi-deterministic, repeatable unsteady motion and the production and maintenance of mean turbulent t r a n s p o r t l . This raises a challenge:

dynamical chaos and dissipative dynamical systems have

been phenomenally successful in predicting interrnittencies and 'q:)urstlng streaks" in convective systems where the number of degrees o f freedom is small. Furthermore, for weakly confined systems with many degrees of f r e e d o m excited, the w e a k turbulence 0920-5632/87/$03.500 Elsevier Science Publishers B.V. (North-Holland PhysicsPublishingDivision)

454

B. Nicolaenko / Large scale spatial structures

becomes truly s p a t l o - t e m p o r a l with competition b e t w e e n temporal chaos and coherent spatial structures S. The m a t h e m a t i c a l models are partial differential equations (PDE's) vested a priori with an infinite number of degrees of freedom 5. We can now phrase the outstanding problem in bridging the gap b e t w e e n dynamical chaos and turbulence with many degrees of freedom: can we predict the quasi-deterministic (chaotic??) organized motion of C.S. in turbulence through some deterministic field PDE coupled with a stochastic modeling of v e l o c i t y fluctuations at smaller scales?

Such a coupling could

possibly involve an external stochastic force, and/or stochastic coefficients. Here we shall investigate b o t h computationally and m a t h e m a t i c a l l y a family of such PDE's which are prime candidates for modeling c o h e r e n t structures in specific contexts of two-dimensional turbulence:

namely, 1) the Kolmogorov Flow (a turbulent system of

s m a l l - s c a l e eddies supported b y external energy sources) and 2) the l a r g e - s c a l e structure of compressible, non-Boussinesqian convection ( l a r g e - s c a l e turbulent solar convection). The theory of homogeneous turbulence s o m e t i m e s uses an approach which assumes t h a t the flow is supported b y an external space-homogeneous stochastic f o r c e 7. At least in the two-dimew'tonal case, the energy is known t o be able to pass along the spectrum 8

f r o m s m a l l - s c a l e motions to l a r g e - s c a l e ones . Coherent structures m a y arise whose c h a r a c t e r i s t i c size is higher t h a n t h a t of the external force field inhomogeneity.

Randomness of the external f o r c e field is not

essential: it is replaceable b y an appropriately chosen periodic v e c t o r field. In a series of seminal articles G. I. Sivashinsky, L. Shtilman and V, Yakhot 9-13 pinned the genesis of C.S. as a manifestation of long wave instability of a system of s m a l l - s c a l e eddies maintained by an external periodic force field.

The simplest hydrodynamic system of

this t y p e is the so-called Kolmogorov flow (Kolmogorovl4; Arnold and Meshalkinl5; Obukhovl6):

this

is the

two-dimensional

flow of a viscous liquid

unidirectional force field periodic in one of the coordinates.

induced by a

The g e n e r a t e d system of

s m a l l - s c a l e eddies turns out to be unstable to l o n g - w a v e instabilities. Sivashinsky et al. d e m o n s t r a t e t h a t the e f f e c t i v e viscosity of t h e

corresponding l a r g e - s c a l e flow is

negative (provided the flow is sufficiently anisotropic) and generalize this t o small-scale three-dimensional periodic flows 13. In completely unrelated work, E. A. Spiegel and coworkers M. C. Depassier, J. P. Poyet 17-19, reached strikingly similar conclusions for the large scale structure of compressible convection in a plane-parallel l a y e r of p e r f e c t gas (polytroptc static state).

Mixed f l u x - t y p e boundary

considered. effects.

conditions

and non-Boussinesqian

e f f e c t s were

Finite amplitude instability again g e n e r a t e s C.S. through n e g a t i v e viscosity

Turbulent convection with very large horizontal extent is e x p e c t e d to be

prevalent a t significant amplitudes, and significant in the interpretation of l a r g e - s c a l e solar convection. In this article, we propose a general (by no m e a n s universal) class of PDE's as candidates to model the dynamics of C.S. in a broad c o n t e x t of 2D-turbulent flows:

455

B. Nicolaenko / Large scale spatial structures

a-~-+ a¥ 52 ~ + B~/+ > [ v ~ [2

+ div {[a - 6 1V~I 2] V~} +c A(¥2)+~ A(IV~I2)=0

.

(i)

where ~(x, t) is a large-scale field variable, x is a rescaled direction in which the effective viscosity of the large scale flow is negative and a, B, ¥, 6, ¢, ~ positive physics/parameters. These are generalizations of field PDE's previously derived by the above-mentloned authors; we shall designated (I) as the Kolmogorov-Splegel-Sivashinsky family of PDE's (K.S.S.). The K.S.S. family is characterized by locally negative viscosity effects.

We

conjecture that the competition between locally negative viscosity (pumpins of energy into large scales) and convective mechanisms (coupling with small scales) together with dissipation generate spatio-temporal intermittencies. We also conjecture that physics compel such PDE's to be mathematically equivalent to finite (albeit large) dimensional dynamical systems. We prove these conjectures for a simpler one-dimensional version of

(1): a_+ at

+

a~+ ax4

s ~ . y laxa_+l 2

+If ++I'+]

+~-£

a - 8 ax

=o

(2)

which still models effective negative viscosity in a certain direction x of a large-scale flow. For K.S.S. equation (2) w e demonstrate that: -1) it is strictly equivalent to a finite dimensional dynamical system on an "ine.'rtial manifold"20; the latter is a finite dimensional, Lipschitz manifold invariant under the dynamics of the PDE and attracting exponentially all trajectories. -2) it possesses a universal (strange) a t t r a c t o r whose fractal

and Hausclorff

dimensions are rigorously proportional to the number of linearly unstable long-wave modes. (This is a strong non-linear mathematical result.) -3) high precision computer simulations evidence "bursting streaks" for many parameter ranges; we show these "streaks" to be generated by multiple Shilnikov homoclinic bifurcations 21. Shiinikov homoclinic loops are classical in low dimensional dynamical systems. Here we encounter them even for large turbulent flow parameters (generalized Reynolds numbers). -4) computer simulations evidence spatially localized, temporally intermittent structures. Relatively quiescent large scale spatial subdomains appear and y e t coexist at a given time with complementary subdomains o f homogeneous turbulence.

Gradual

outflow of these localized structures ends in sudden oscillations and breakup.

We

456

B. Nicolaenko / Large scale spatial " structures

conjecture

that

such

spatio-temporally

intermittent

structures

are

the

coherent

structures of the l a r g e - s c a l e turbulent flow, at l e a s t for m o d e r a t e l y large Reynolds numbers. We outline a possible dynamical chaos m e c h a n i s m for such structures: chaotic orbits wandering n e a r m e t a s t a b l e t h r e e - f r e q u e n c i e s Tori (actually travelling waves with two frequencies of modulations). M a t h e m a t i c a l results 1) and 2) were obtained jointly with C. Foias and R. T e m a m and will be detailed in a s e p a r a t e publication 22. The K.S.S. equations are r a t h e r generic: our computational simulations predict the same m e c h a n i s m s for '~oursting streaks" than a v e r y low-dimensional dynamical model for boundary layer turbulence n e a r a wall, derived

by

N.

Aubry,

J.

Lumley,

P.

Holmes

and

E.

Stone 23.

The

excessive

low-dimensionality of their Galerkin s y s t e m p r e v e n t e d these authors f r o m observing the l a r g e - s c a l e i n t e r m i t t e n t s t r u c t u r e s of result 4). The K.S.S. equations a r e also robust when confronted with e x p e r i m e n t a l measurements:

experiments by S. Ciliberto 24,25 on

small-aspect

30),

ratio,

high-Prandtl

number

(=

Rayleigh-Benard

convection,

corroborate the scenario predicted by the K.S.S. models with an uncanny qualitative fit! These experiments were carried up to 300 t i m e s the critical Rayleigh number (Ra critical = 1200). The reader is encouraged to survey the article by S. Ciliberto in these proceedings. 2. BACKGROUND

AND

MATHEMATICAL

PROPERTIES

OF

THE

K OLMOG O ROV-SPIEGEL-SIVASHINSKY MODELS The m o s t general K.S.S. model (1) was derived by Spiegel et al.17-19 through s y s t e m a t i c multiple-scales analysis of a compressible solar convective l a y e r zone, where x is a l a r g e - s c a l e horizontal variable (perpendicular across the rolls) and ~(x, t) represents the l a r g e - s c a l e horizontal fluctuations

of the ter~perature field, a f t e r

subtacting the m e a n field (which depends only upon t h e v e r t i c a l z-variable):

a+ a4 ¥ ~--+ .:~--+ B , ax

++I+-+(++ +I /a+\ 2

a2

+,,

a2

a~ 2

t_- o

(3a)

+

Vt(x + L, t) - +(x, t), periodic B.C.; the

same

interpretation

carries

over

(3b) to

Ciliberto's

experiments; 24-25

only the

fluctuating t e m p e r a t u r e field has b e e n averaged along the y-axis of the t'mite rolls.

B. Nicolaenko / Large scale spatial structures

457

In Eq. (3), the term ~x~xx generates higher-order viscous damping of small-scales; 8 is a classical linear damping,

c(~2)xx , ¢ << 1, is a hydrodynamic blow-up term

(vortex singularities); if the model reduces to: aT a+ + a-t + a ~ + i~+ a2

+ c ~

+-(v/2) = o

,

(4)

then classical mathematical blow-up theory shows that ~(x, t) in (4) would exhibit a singularity in finite time. Fortunately, in the context of (3), locally negative viscosity mechanisms are dominant, and overpower the term ¢(~2)~o~. The last term ~ [ (~x)2_]xx corresponds to non-Boussinesqian compressibility effects. The two key terms in (3) &re fL~st:

~(,x) z

(5)

which accounts for convection of energy between large and small scales (notice that it is really a Burgers' nonlinearity ~0 ~Ox,if we rewrite (3) for the gradient q)(x, t) = ~x(X, t)). The second key t e r m is the locally negative viscosity:

9

indeed, if in some s p a c e - t i m e subdomain ¥~ is small, the local e f f e c t i v e viscosity is negative (large wave band instability) and energy is locally fed from small scale to unstable

large

scales; however

as ~

increases locally, a non-linear

saturation

mechanism sets in which r e v e r t s to a regime of locally positive viscosity and dissipation of energy to small-scales.

Mechanisms (5) and (6) act as a cyclic oscillator for energy

transfer between small and large scales. Of course, the linearized equation (3) around ~ = 0 with periodic boundary conditions on 0 < x < L, is vested with the wave band of linearly unstable modes

km~ <_k <_kmax

,

(7a)

where ~min = [2L--~(s/C) t/2] + 1 kmax = [ 2~ °tl/2(1 - 2-~a2)]

,

(~b) (7c)

458

B. Nicolaenl¢o / Large scale spatial structures

where [r] designates the integer component of a real number r, and where we take the approximation: 81a 2 << i

(7d)

Essentially, the number Nu of linearly unstable modes is:

N u-- C1

L 2--v

a ,

(8)

where C 1 is a dimensionless constant depending upon B/a 2. The simpler equation (2) has been derived by G. I. Sivashinsky 12 for t h e Kolmogorov flow (the two-dimensional viscous flow included b y a unidirectional external force field periodic in one of the coordinates. Figure 1 shows streamlines and velocity profile of the corresponding laminar flow which obeys a periodic sine-law along the y-axis.

N.F.

Bondarenko, M. Z. Gak and F. V. Dolzhansky 26 have reproduced the Kolmogorov flow under laboratory conditions in a thin layer of electrolyte: the periodic force field is realized as a Lorentz force. Their experiments show that, beyond the critical value of the Reynolds Number R c, the unidirectional flow b e c o m e s unstable and a regular system of stationary eddies appear (Fig. 2). At larger R, the steady system o f eddies starts oscillating periodically (Fig. 3); for R >> 3 R c, turbulence sets in.

X

•

_

=

FIGURE 1 Velocity profile and streamlines of the Kolmogorov flow, f r o m 26.

B. Nicolaenko / Large scale spatial structures

459

b

FIGURE 2 Secondary-flow pattern in the range just above criticalR, from 26.

b

FIGURE 3 Secondary flow pattern for R > 1.25 R c from 26. For a unidirectional l a r g e - s c a l e flow parallel to a certain direction (where the e f f e c t i v e viscosity of the s e c o n d a r y flow is negative), G. I. Sivashinsky derives the following equation:

a__+_

.a+. +

a t + a x4 a + b-~ {[2

+ t~," a_~ 2 -

6

(ax)

] a_~ } = 0 ax

.

(9)

where ~ is the ( r e n o r m a l i z e d ) l a r g e - s c a l e s t r e a m function, a f t e r s u b t r a c t i o n of the m e a n periodic field component; x is a rescaled p r e f e r e n t i a l direction of n e g a t i v e viscosity (not to be confused with the original x in Fig. I).

•

¢

B. Nicolaenko / Large scale spattal structures

460

Siv~'s

equation is a special case of:

¢t + ~xxxx + {[4 - 6~x 2] Cx}x +B ~+Y(@x)2=0

,

~(x + L. t) - ¢(x. t)

(10)

In a joint work with C. Foias and R. T e m a m 2 2 we demonstrate that solutions of the K.S.S. Eq. (2) and (10) remain uniformly bounded f o r all times t ~ 0 and are a t t r a c t e d in finite times by a bounded "absorbing" ball in the functional space H2(O, L): this is the Hilbert space of functions square integrable on [O, L], with square integrable first and second derivatives. This ensures the existence of a universal, maximal a t t r a c t o r A 6 for the trajectories of (10). We then demonstrate t h a t the Hausdorff dimension d H and the f r a c t a l dimension d F of the (strange) a t t a c t o r A are bounded:

THEOREM I: d~A) <_ d~A) <_ c2

41/2

.

(11)

equivalently d~A) <_d~A) <_ C3 N u

,

(12)

where C 2 and C 3 are dimensionless constants depending on a, ~, y, 8 and N u is the number of linearly unstable modes as in Eq. (8). This result is striking:

i t asserts that the number of degrees of freedom of the

asymptotic nok~linear dynamics of (10) on the a t t r a c t o r A is the same as the number of

N u of linearlyunstable large-scale modes' This is in fact a deep Rlobal!ynon1~ear result whose rigorous mathematical proof fully follows the "relaxation" properties of the locally negative viscosity mechanism (6). The nordinear mechanism of pumping energy back and forth between large and small scale modes ultimately results in an attractor A with the same dimension as the unstable wave-band'. What we effectively demonstrate mathematically is that locally negative viscosity does generate org~dZed motion of coherent structureswith a small number of degrees of freedom d F. This still does not imply that the K.S.S. P D E (10) is strictly equivalent to a finite dimensional dynamical system of Ordinary Differential Equations: remember that A characterizes only asymptotic dynamics as t -* + oo. Deeper properties of the dynamics are required to establish the above.

The approach (introduced in27) consists in

constructing a finite dimensional Lipschltz manifold ~ called inertialin the phase space of the P D E such that:

B. Nicolaenko / Large scale spatial structures

(i)

461

Y~ is invariant and has c o m p a c t support; t h a t is ff (S(t, ")t>0 is the nonlinear

semigroup associated with t h e initial value p r o b l e m for the equations, t h e n S(t, Y.) is contained in 7. for all t >_ 0. (ti)

All solutions converge exponentially to }~. In particular the universal a t t r a c t o r ,

A is included in 7. and the dissipative s y s t e m r e d u c e s on Z to a finite s y s t e m (called an inertial

ODE).

(iii) Asymptotic c o m p l e t e n e s s holds:

for e v e r y initial value for t h e full K.S.S.

equation, t h e r e exists some initial point on the inertial manifold Y. such t h a t the relative distance b e t w e e n the full infinite dime~.~ional t r a j e c t o r y and the t r a j e c t o r y on decreases exlxmentially to zero. pO The l a s t point does fully establish the equivalence b e t w e e n the PDE and the inertial ODE on ~.

Concretely, given a chaotic t r a j e c t o r y f o r the e x a c t PDE, t h e r e is a finite

dimensional chaotic t r a j e c t o r y for the inertial ODE, such t h a t the t w o t r a j e c t o r i e s converge exponentially: In Ref. 2P, we do prove: THEOREM 7-. The K.S.S. PDE possesses an inertial manifold of dimension din, with: din >__

C4(2~ ~'-a)5

'

(13)

where C 4 is a dimensionless constant depending upon a, p, y, 5. It m u s t b e emphasized t h a t the m a t h e m a t i c a l e s t i m a t e (13) for din is f a r f r o m optimal, as opposed to e s t i m a t e s (11-12) f o r the a t t r a c t o r ' s dimensions. Nevertheless w e h a v e established t h e key: Proposition.

the w e a k turbulence of c o h e r e n t structures modeled b y the K.S.S.

equation is strictly equivalent t o finite dimensional chaos on a finite dimensional inertial dynamical system. We are currently extending these m a t h e m a t i c a l results to the m o s t general K.S.S. PDE's (I) and (3).

Whereas the K.S.S. models a r e invarlant under O ( 2 ) - s y m m e t r y

(translations and reflections in x), we are also investigating the b i f u r c a t i o n s of solutions f r o m the s y m m e t r y breaking point of view. 3. OVERVIEW

OF

COMPUTATIONAL

RESULTS

AND

BIFURCATIONS

OF

THE

ATTRACTOR S p a t i o - t e m p o r a l turbulence still involves complex mechanisms within t h e bifurcations of the a t t r a c t o r and the inertial manifold. The s y s t e m does r e m e m b e r its PDE c h a r a c t e r in generating coherent s p a t i o - t e m p o r a l structures. To unravel these m e c h a n i s m s we m u s t obtain a c l e a r picture of those f e w nonlinear s t a t e s (spatial structures) which f o r m a reduced nonlinear coordinate basis for the manifold. A nonlinear r e p r e s e n t a t i o n of the inertial manifold must be constructed, based on reduced coordinates patches, with the goal o f establishing low dimensional inertial normal f o r m s for the inertial parameter. turbulence.

O.D.E.'s, valid f o r some range of t h e bifurcation

These inertial n o r m a l f o r m s control t h e global v e c t o r field bifurcations into

B. Nicolaenko / Large scale spatial structures

462

We have systematically searched for classical dynamical systems bifurcations and for multiple basins of attractions for the K.S.S. model for some selected values of B, y and 8. The interactions of multiple basins through their fractalized boundaries have been evidenced.

Intermittencies in turbulent time series are one of the key mechanisms in

bridging the gap between PDE's and dynamical systems. unstable manifolds of key hyperbolic points or Tori.

They enable us to t r a c k the

These intermittencies are random

time windows where dynamics remain highly o s c u l a t o r y y e t are confined in a relatively small neighborhood of some metastable point or Torus.

Such critical s t a t e s are the

natural candidates for local nonlinear coordinates o f the inertial manifold. At the point a word o f caution is necessary.

We did isolate a typically low

dimensional behavior in the vicinity of a few fixed points and traveling waves. However, this does not contradict the f a c t that the fractal dimension of the chaotic a t t r a c t o r is growing with L. Indeed, the system can be low dimensional in the vicinity of the stable or metastable states, but these states might be sparsely distributed in a manifold of larger dimensions. Moreover, the number of the relevant a t t r a c t o r s or m e t a s t a b l e states seems to grow with the bifurcation parameter. We have

investigated

Sivashinsky's original

Eq.

(9) with

the

linear

coefficient set to B = 0.1 and the negative viscosity set to 8 = 0,0025.

damping We have

normalized the K.S.S. equation to an interval length 2~T; and appropriately rescaled other parameters, the original equation with bifurcation p a r a m e t e r L:

~t + d~mo(:x + {(2 - 0.0025 ~x2)dpx } + (4)x) 2 + 0.I

~} = O, 0 <__x K__L

,

(14)

now becorfles:

*t

+ 4

+

+

) 2) 0
¢(x + 2~, t)

= ¢(x,

t)

,

(1.5)

where =

8(L/2~) 2

= 4B(L/21T)

4

8

4a

10-2 (16)

and x and t appropriately rescaled. The limiting unstable wave number kmi n and I ( become:

max

B. Nicolaenko / Large scale spatial structures

kmaX

Ca/4)1/2 1

463

2_J ct

-- 0.875 (a/4) 1/2 kmin _- (~/~) 1/2 = (E/4) 1/2 (40) -1/2 and the number of linearlyunstable wave numbers N u is given by: N u ~- 0.717(a/4)112

(17)

In view of (17),we shall consider ~ as a convenient bifurcation parameter. Although the wave number k = 0 is always stable, it is convenient to s u b t r a c t f r o m (15) the dynamics of the m e a n value of the solution to Eq. (15):

re(t) = ~v j.2~

(18)

(x, t) dx

o

which satisfies the drift equation:

2w

(19) To normalize this drift to zero, we numerically solved the equation for

(20)

~(x, t) = ~(x, t) - re(t)

t h a t is the K.S.S. equation w i t h zero means:

(Pt + 4 (Px}c~ + ~(~XX + ~ X ) 2)

- 10_2( 3x)x ÷

(~x)2 d x = O

- ~

(21)

o

We h a v e scanned the domain 40 <_ ~ <__160, i.e.. 2.26 <__Nu <__4.53. No chaos has been observed for

~

<__ 43.

In the

calcttlatious, w e used discrete

Fourier t r a n s f o r m

pseudospectral approximations to the spatial d e r i v a t i v e s on grids ranging f r o m 64 to 512 mesh points in singie precision (14 digits) on a C r a y XMP computer. all-purpose code developed b y J. M. Hyman a t

Los Alamos 28.

We used an

The solution was

i n t e g r a t e d in t i m e using a v a r i a b l e order, v a r i a b l e t i m e step backward differentiation m e t h o d in t i m e t h a t r e t a i n e d an absolute error t o l e r a n c e between 10 -6 and 10 -10 per unit time. The runs p r e s e n t e d h e r e took b e t w e e n 104 and 106 t i m e steps. The implicit equation was solved on each t i m e step with a quasi Newton I t e r a t i v e algorithm. Since these

equations

were not

introduced in the calculation.

solved exactly,

a s y m m e t r y breaking p e r t u r b a t i o n was

B. Nicolaenko / Large scale spatial structures

464

We systematically t r a c k e d the d o m a ~

of stability of each a t t r a c t o r w i t h r e s p e c t to

the bifurcation p a r a m e t e r b y varying ~ and reinitializing ~(x, 0) to the final solution f r o m the previous run with a different ~.

Many problems were r e c a l c u l a t e d several

times with different grid resolutions and t i m e t r u n c a t i o n error criteria to ensure t h a t the numerical s o l u t i o ~ wee converged within an a c c e p t a b l e accuracy. A r e m a r k a b l e f e a t u r e of the K.S.S. equation, is the alternating sequence of intervals in ~ containing laminar behavior (some fixed point is u l t i n ~ t e l y a t t r a c t i n g ) with intervals of persistent oscillatory and/or chaotic behavior. interval.

Let Ij = [aj, a j + l ] be the jth

For j even, I. is c h a r a c t e r i z e d by the u l t i m a t e decay to a globally a t t r a c t i n g J

fixed point uL(x ), q = j/2 + 1, j >_ 2. These fixed points have most o f their energy concentrated

in the

qth mode.

The higher harmonics

appear with

exponentially

decreasing energy and the fixed poin~ has a lacunary Fourier expanmon ~q (x)

= a l q cos qx + 2 + e

~ a2qCos 2qx

n-i a3q cos 3qx + ... + e anq cos nqx + ....

(22)

where q = j/2 + 1 and the solution has been n o r m a l i z e d by trav-qlatlon. Numerically we lave found t h a t a l q is 0(1) and e = 10 -1. We call these sinks associated with lj, j even, cellular states. When the Fourier expression (22) o f a cellular s t a t e is dominated by cos qx we call it a q-modal cellular state. They are invariant unclear 0(2) s y m m e t r y . The relaminarization intervals I~, j even, are consistent with Ciliberto's experiments at m o d e r a t e Ka~leigh nurabers 24'2~. '~ Moreover, as j and a increase, the u l t i m a t e decay follows (extremely) long periods of transient chaos.

Transient chaos is observed in the

K.S.S. equations beginning in the interval 14, provided enough modes are excited in the initial data.

Moreover, as ~ increases, the m e a n l i f e t i m e of transient chaos increases

exponentially in L: this growth makes transient chaotic intervals indistinguishable in

practice f r o m chaotic intervals in the strongly chaotic regimes (say, w h e n the f r a c t a l dimension of the universal a t t r a c t o r , A, for the flow is large, dimF(A) >__10). When j is odd, t h e intervals I~ have persistent oscillatory and/or chaotic behavior. For m o d e r a t e values of ~ (say, up to 17), the quaslperiodic and/or chaotic behavior r e f l e c t s a competition b e t w e e n the previous (j + 1)/2 cellular state, dominated by the cos((j + l)x/2) mode, and the (j + 3)/2 cellular s t a t e , dominated by cos ((j + 3)x/2). This competition creates a complex interplay b e t w e e n t e m p o r a l chaos and spatial coherence. In some sense, the (low-dimensional) temporal chaos corresponds to a d j u s t m e n t f r o m one [tow dimensional) space p a t t e r n to the next one.

Unfortunately, this simpie picture is

not borne by our computations a t m o d e r a t e l y chaotic regimes where a zoo of strange hyperbolic fixed points a p p e a r in intermittencles.

Their strangeness resides in t h a t they

are not cellular in the sense of (22) and possess a broad energy spectrum band covering all the unstable modes up to k m a x. This is reflected into increasing spatial complexity of the chaotic solution trajectories (see Section 5).

B. Nicolaenko / Large scale spatial structures

465

4. LOW DIMENSIONAL CHAOS FOR THE KOLMOGOROV-SPIEGEL-SWASHINSKY EQUATION In this section, we describe the behavior of t h e solutions to the K.S.S. equation (15) and equivalently Eq. (21), f o r p a r a m e t e r values in t h e intervals I2, I3, 14 w h e r e 23 < a < 73, or 1.72 < Nu < 3.06. Our preliminary catalogue f o r these i n t e r m e d i a t e values of ¢ is presented in Fig. 4. The sequence of "laminar" intervals and intervals with complex oscillatory and/or chaotic b e h a v i o r is 12

= 23 < ¢ < 4315:

1 3 = 43.5 <

a < 58:

14 = 58 < ~ < 73:

a 2-cellular s t a t e globally a t t r a c t i n g complex oscillatory b e h a v i o r

a 3-cellular s t a t e globally a t t r a c t i n g Traveling wave train A

Torus T1x t

(

TorusT 2

Torus T 2 • • Perturbed homochnmc orbit

Homoclinic orbit

HOPF x

! TorusT2'

i

I

Bimodal fixed point j (global attractor)! i

43.5 ,,

(global attractor) I

I

48

!

I

I3

I

I

53

I

(local attractor) I I I

~

I

58 i

I4

FIGURE 4 Multipe a t t r a c t o r s f o r small values of ~ already generate chaos. In the discussion below, the "energy" is the integral of ($y)2 and the " e n e r g y in mode k" is the modulus of the k - t h Fourier component of $, k >__1. Within these intervals, w e see canonical v e c t o r field befurcatinns a l r e a d y leading to quasi-periodic motions and chaos.

Especially i n t e r e s t i n g are f o r e r m m e r s of "bursting

streaks" and i n t e r m i t t e n t c o h e r e n t structures observed f o r larger ~ (cf. section 5): i) a persistent homoclintc loop a t t r a c t s orbits for 43.5 < ~ <_ 48; it) a r o t a t i n g w a v e appears a t ~ = 53, undergoes transition to a traveling beating wave (single frequency of modulation) and generates chaos f o r 56 < ~ <_ 58; this peculiar s p a t i o - t e m p o r a l chaos is dominated b y bits and pieces of the traveling b e a t i n g w a v e alternatively propagating to the l e f t or the right. The bimodal cellular s t a t e ~2(~) b i f u r c a t e s a t ~ = 43.5.

The b i f u r c a t i o n yields a

persistent homocltnic loop, c o n s t r u c t e d f r o m two h e t e r o c l i n i c connections b e t w e e n ~2(x) and ~2(x + ~). The existence o f such heteroclinic connections results f r o m the action of

466

B. Nicolaenko / Large scale spatial struc tures

the diedral group D 4 on some isotropic subgroups o f the group O(2), in a fashion similar to the s y m m e t r y breaking studies of the Kuramoto-Sivashinsky equation in. 29 This is explored in Figs. 5-7, ~" = 44 where the initial value ~0o is 72(44 ) = 2.18 cos 2x + 0.52 cos 4x .... In Fig. 5 (energy vs. time), we se t h a t t h e orbit spends a long t r a n s i t t i m e in a small neighborhood of the b/modal hyperbolic points ~2(x) and ~2(x + ~) b e f o r e bursting around the homoclinic loop. The motion around t h e loop is triggered by a v e r y sensitive exchange of energy b e t w e e n the odd and even modes. In Fig. 6, the e n e r g y in mode one bursts quickly f r o m 10 -6 to 2.5: this b e t r a y s unstable directions along odd modes for the hyperbolic points ~2(x) and ~2(x + ~). A corresponding dip is observed in the energy of mode two (Fig. 7): the unstable manifolds of 72 involve only odd Fourier components. At ~ -- 48, the homocltnic loop looses its basin o f attraction. As shown in Fig. 8, the dynamics first take place n e a r a perturbed homoclinic loop, with less and less t i m e spent in the neighborhood of the hyperbolic point; at t ~ 1.5, the trajectories l o c k onto a Torus T 2.

Close examination of Fig. 9 (energy in m o d e one) reveals i n t e r m i t t e n c i e s in the

regime 1.5 < t < 3, where t h e orbit briefly wanders b a c k to ~2" This b e t r a y s complex interwinding b e t w e e n the unstable manifold of ~2 and the basin of a t t r a c t i o n boundary for T 2. 35

tlJ

I

15

I

I

I

I

t

t

I 3.0

0

TINL:

FIGURE 5 The energy bursts along the homoclintc loop (~. = 44).

B. Nicolaenko / Large scale spatial structures

I

467

iLL Jl I

I

I

I

TIME

I

3.0

FIGURE 6 The energy in the first m o d e oscillates o v e r a range of i 0 - 6 ( ~ = 44).

o z >-,

o-I

I

I

I

I TIME

I

I

I

I 3.0

FIGURE 7 The energy in the s e c o n d mode is nearly c o n s t a n t b e t w e e n bursts (~ = 44).

B. Nicolaenko / Large scale spatial structures

468

lo

I

t,

I

I

I

I ,,I

I 3.0

TIME

FIGURE

8

The orbit first bursts with a decreasing frequency and then locks onto the T 2 Torus (~ = 48).

,

I

I

c) z

uJ

0 0

3.0

TIME

FIGURE 9 With

dynamics

essentially

near the torus i n t e r m i t t e n t l y near the b t m o d a l state (~ = 48)

T 2,

the

~bl.t

stUl

wanders

B. Nicolaenko / Large scale spatial structures

469

At ~ = 53, the orbit suddenly locks into a t r a v e l i n g (rotating) w a v e . possesses

a characteristic

structure

with

two humps (Fig.

The l a t t e r

i0), is endowed with

near-eq111partion of energy on the unstable w a v e number band, t r a v e l s a t a constant speed (frequency) on the spatial torus 0 <__ x <__ 21r. At ~ = 54, it undergoes Hopf bifurcation in its moving f r a m e of r e f e r e n c e and b e c o m e s a traveling beating wave (modulated rotating wave) w i t h a single f r e q u e n c y of rotation.

Technlcally, this is

another T2-torus, albeit a s p a t i o - t e m p o r a l one! At ~ = 56, the traveling b e a t i n g wave undergoes transition to s p a t l o - t e m p o r a l chaos. The evolution of energy vs. t i m e in Fig. 11 is deceivingly simple: the amplitude of the modulation increases gradualiy, t h e n suddenly relaxes. These cyclic r e l a x a t i o n s are not exactly periodic. The full p i c t u r e emerges in Fig. 12, where we plot t h e contour levels of ~0(x, t), as t increases f r o m 0 to 1.20. We o b s e r v e t r a j e c t o r i e s which alternatively follow closeiy a modulated rotatirt8 wave traveling to the l e f t or to the right.

Careful

examination reveals t h a t r e v e r s a l of propagation coincides with the t i m e s of relaxation in Fig. 11.

This is clearly s p a t i o - t e m p o r a l chaos in the neighborhood of a pair of

metastable

(hyperbolic) m o d u l a t e d

rotating w a v e s

(spatlo-temporal

Torus?).

Such

c o u n t e r - p r o p a g a t l n g chaotic w a v e s have b e e n o b s e r v e d by S. Cillberto 24'25 through spatial correlation m e a s u r e m e n t s .

~a

-4

I

I

I

I n

I

I

I

FIGURE 10 The rotating w a v e possesses a typical t w o - h u m p e d profile (5 -- 53).

I

2n

470

B. Nicolaenko / Large scale spatial structures

45

c~ t~ t~

20 2.0 T IME

FIGURE 11 A deceiving pseudo-Van-tier Pol oscillator (~ -- 56). q' { x , I )

C{)NTOIIII,~;

1.28

0.96

O. 6 4 tL1

0.32

0 0

2~ X

Really, spatio-tmporal traveling waves (~ = 56).

chaos

F I G U R E 12 generated by

counter-propagating

beating

B. Nicolaenko / Large scale spatial structures

5. B U R S T I N G S T R E A K S A N D I N T E R M I T T E N T C O H E R E N T

471

STRUCTURES

We now s y s t e m a t i c a l l y i n v e s t i g a t e the bifurcations of the K.S.S. Eq. (21) f o r 73 < < 160, where the number of linearly unstable m o d e s N u varies b e t w e e n 3.06 and 4.50. From T h e o r e m I, the f r a c t a l dimension d F of the a t t r a c t o r A is proportional to these numbers. Within these intervals, we evidence v e c t o r field bifurcations which generate '~oursting streaks" and i n t e r m i t t e n t coherent s t r u c t u r e s (C.S.). The m e c h a n i s m which we pin down are truly generic:

we observe t h e m in s y s t e m s with much l a r g e r numbers of

cells and larger f r a c t a l dimensions (up to 50). The classical homocltnic loop bifurcation fro a saddle-point observed around ~ = 43.5 does n o t reoccur.

Rather i t is t r a n s m u t e d

into a Shilnikov homoclinic loop 21, where the t r a j e c t o r y bursts into a spiral oscillatory s t r e a k away f r o m and b a c k into some "saddle-focus" unstable cellular fixed point. Between the bursts, the c o h e r e n t cellular s t r u c t u r e s occupy the full spatial domain. Yet c o m p l e t e l y different and novel s p a t i o - t e m p o r a l i n t e r m i t t e n c i e s d o m i n a t e at higher regimes:

then, coherent s t r u c t u r e s occur as usual randomly in time; b u t at a given

m o m e n t , t h e y do only v a r t i a l l y occupy some subset of the spatial interval, whereas homogeneous turbulence intermittent

dominates

elsewhere.

C.S. are found in m e t a s t a b l e ,

Harbingers of such s p a t i o - t e m p o r a l

strange

two-humped f i x e d points (the

precursors of which were a l r e a d y shadowed f r o m ~ ffi 53 to ~ = 58). Our preliminary catalogue for these larger values of a, 73 < ~ < 160, is presented in Figure 13. The sequence of l a m i n a r intervals and i n t e r v a l s with chaos r e a d s as: 15=73<

~ <94:

I 6 = 94 < ~ < 128: 17=128<

chaos a 4 - c e l l u l a r s t a t e is globally a t t r a c t i n g

~ < 149: chaos

18 = 149 < ~ < ?:

a 5 - c e l l u l a r s t a t e is globally a t t r a c t i n g .

The trtmodal cellular s t a t e ~3 (~-~)is a global a t t r a c t o r in 14 until it b i f u r c a t e s at ~ = 72.8, where complex conjugate eigenvalues cross t h e imaginary axis. By c a r e f u l scanning of the r a n g e 71 < ~

< 73, we pin down a subcritical Hopf bifurcation:

periodic orbit b i f u r c a t e s a w a y f r o m ~a for ~

<_ 72.8.

a metastable

For ~[ g r e a t e r , the dynamics

consist of a Shilnikov homoclinic orbit 2 r a s y m p t o t i c to the saddle-focus point ~3" This is explored in Figs. 14-16, w h e r e ~ = 73, ~3 = 2.74 cos 3x + 0.43 cos 6x + .... and the initial d a t a was ~3 + 0.3 sin x. Fig. i4 evidences t h e dichotomy: the orbit spirals around ~3 with a high frequency, bursts away on the trustable manifold of ~3' p u f f s into a spiked i n t e r m t t t e n c y (or a m o r e complex chaotic window) and is reir0ected into ~3" The energy in the f i r s t mode (Fig. 15) is low during the small oscillations around t h e saddle-focus and nearly quadruples during the chaotic bursts around the loop. The e n e r g y in the third mode (Fig.16) is the mirror i m a g e of Fig. 14: it is, on the average, m u c h lower during the chaotic spikes.

Careful analysis of the d a t a show t h a t the t r a j e c t o r y is not

B. Nicolaenko / Large scale spatial structures

472

" ~¢ . o

L0

I

•-~ O

~<

(0 o

¢.O

o '-r¢..9

~8 I,.,,,

0) oo

> 0

~.

09

_0

o

7_

- -

T-

~

m

_~

~D

~

¢D

re D O O >

O3

>

O <

I O "O c00 >

I -4-

g

g'~

>.

CO-

ttl

~~--° I0

~_

O

I

~=

I

o_ -r i

-c7~ I .~=~

~o

I ~'-'-

o~.

~c,)~ I'H ~ B I :,E o ~ i ~'r''

Eo "v__ l.-,x ,+-

O~ 0) CO CO

ta

B. Nicolaenko / Large scale spatial structures

473

7 0 ~

c~ u~

f

'

II

i

1.2 TINE

FIGURE 14 The Shiintkov homocUrdc loop reinjects the orbit into the v i c i n i t y o f the trimodal s t a t e (~= 73).

o z

I

I

I

I

I

I

I

I

1.2

TIME

FIGURE 15 The energy in the first m o d e is l o w during t h e small oscillations near the trimodal saddle-focus (~= 73).

B. Nicolaenko / Large scale spatial structures

474

07 1.2

0

T I/ql!

FIGURE 16 In the bursting streaks, the orbit wanders b e t w e e n the trimodal cellular state and some other metastable s t a t e (~= 73). reinjected into exactly the same 73, but rather into successive ~ shifts thereof. There is a complex interplay b e t w e e n the symmetry group O(2), the diedral group D(3) of 2~ 3 rotations and the diedral group D(6) of ~/3 rotations. These s y m m e t r y mechanisms account for a persistent Shiinikov loop until ~

= 76. Finally, two f u r t h e r caveats to

"simplify" the picture: i) the saddle-foci are actually drifting with a large period (slow rotation on 0 < x <_ 2~) and the Shtinikov picture is stricto sensu c o r r e c t in a moving reference frame; and ii) the chaotic excursion f o r 0.45 < t < 0.75 reveals an orbit wandering chaotically between 73 (this is especially clear In Fig. 16) and some other metastable state; the l a t t e r turns out to be an ominous two-humped strange state. There is evidence of a tangle between the unstable manifold of 73 and t h e stable one of the strange state. At ~ = 76, the strange two-humped fixed point q~*(~) suddenly b e c o m e s a globally a t t r a c t i n g sink. It is not related to any cellular s t a t e (Fig. 17), its Fourier expansion is rather fiat, with energy present in all first six modes. It is a stronger nordinear version of a similar two-humped modulated rotating wave observed from ~ = 53 to ~ = 58. Possibly, ~0"(~) has undergone reverse bifurcations back to stability. generates

crisis of the

Shilntkov chaos (Fig.

18).

It certainly

A global system of nonlinear

B. Nicolaenko / Large scale spatial structures

475

-4 0

II

Z II

X

F I G U R E 17 The strange fixed point ~* b e c o m e s a global attractor at ~ = 76, w i t h a characteristically strongly nonlinear structure.

7{~ l -

-

t_l _

~,~t 0

I

I

I

I

I

I

T I r,11~

F I G U R E 18 Crisis of ShilnLkov chaos (~ -- 76).

I

I

O.7

476

B, Nicolaenko / Large scale spatial structures

coordinates for the Shilnikov mechanism (71 < ~

< 76) could be c o n s t r u c t e d with the

strange s t a t e ~* and various discrete images of t h e cellular s t a t e ~3 under the diedral group D(6): this will be explored in another work. The strange fixed point ~0" persists as a global sink until

~

-- 80, where it undergoes

Hopf bifurcation (with unstable

eigenstates localized in Fourier modes five and seven).

A1 ~

~ 84, it undergoes a

secondary Hopf bifurcation (two frequencies of modulation) and close to t h a t value of a, undergoes a further transition to a slowly r o t a t i n g wave (technicaUy, the rotation frequency is zero at the critical value of ~). This is further explored in Fig. 19; ~ = 85. Technically, we have a travelling beating wave (two frequencies of modulation, one of rotation).

In the conventional terminology of dynamical systems, this is a T 3 Torus, a

supposedly non-generic route to chaos: Yet the contour levels of Fig. 19 reveal a most peculiar s p a t i o - t e m p o r a l idiosyncrasy of this wave:

the rapid temporal oscillations are

strictly localized in space, on the tip of the higher hump of the strange s t a t e . The lower hump is barely fluctuating, w i t h oniy the slow drift (rotation) left; this is especially clear when tracking the contour level line zero in Fig. 19. Alternatively, the t w o - f r e q u e n c i e s modulations of the higher h u m p are probably already weakly chaotic.

Such a wave has

been nicknamed "the beUy dancer." We have an example of rotating w a v e carrying weak chaos strongly localized in space.

Experimentally, such spatially localized oscillations,

alternating with spatial subdomain of near p e r f e c t

quiescence, h a v e been clearly

1.0

0.635

O. 270 ()

r

~ T

FIGURE 19 The temporal oscillations are sharply localized spatially on the slowly rotating wave (~ = 85).

477

B. Nicolaenko / Large scale spatial structures

observed by both Berg630 and Ciliberto 24'25. The K.S.S. model is the f i r s t one knowledge

to

mathematically

and

computationally

observations accurately (as opposed to

reproduce these

to our

experimental

crude mimicking by conventional coupled

lattice-maps, 31 which somewhat miss oscillatory negative viscosity mechanisms). The further breakdown of this rotating wave c a r r i e r of strongly localized oscillations yields turbulent coherent s t r u c t u r e s (C.S.), which dominate the dynamics in 87 < ~ < 94. This is explored in detail in a p r o t r a c t e d time-series, at ~ = 88, 0 < t < 2 (random initial conditions filtered out), Figs. 20-25. In Fig. 20 the energy (plotted only until t = 1.2, for sake of clear resolution) clearly betrays two temporal i n t e r m i t t e n c i e s within chaos, first for 0 <__to < 0.19, secondly for 0.32 < t o ~ 0.57. During these intermittencies the orbit is wandering and oscillating around some metastable state.

The surprisingly

strong spatio-temporal c h a r a c t e r of these i n t e r m i t t e n c i e s is revealed in the contour plots of ~ ( x , t ) , a s t runs f r o m t = 0 t o t = 2, in Figs. 21-25. In these figures, ~(x, t) must 2v be identified to 9(x, t) from Eq. 21, where J'o 9(x, t) dx -- 0. These figures clearly show bits and pieces of travelling waves with alternating arrays of low-speed and high speed regions. Quiescent subdomains coexist at a given time with complementary spatial

zones of homogeneous turbulence.

Sudden high-oscillatory

7.

1.2

T I ME

FIGURE 20 The strong spatical coherence of the i n t e r m i t t e n c i e s is not evident f r o m the t i m e series (R = 88).

B. Nicolaenko / Large scale spatial structures

478

subdomains, in the vicinity of the steepest part of the C.S., cause break-up of the large scale-structure.

In Fig. 21 (0 < t < 0.512), this is clearly evidenced at t = 0.19, near x =

0; also note the nucleation p h e n o m e n a building up near the contour level line ~0 = O, 0.1 < t < 0.2. In Fig. 22 (0.126 < t < 0.638), the C.S. reappears at t = 0.32, with at first a much

smaller velocity of rotation; a small patch of "contamination" of the laminar

subdomain by the coherent one is observed at t -- 0.38, 1.5 < x < 3. This confirms that the C.S. is not just a simple "clean" piece of the modulated rotating w a v e Dancer") observed at ~ = 84.

("Belly

Rather, as confirmed in Fig. 23 (0.832 < t < 1.34), the

orbits wander chaoticaUy near the C.S. generated by the n o w metastable modulated rotating waves.

Computer

movies seem to indicate homogeneous

chaos based on a

metastable T 2 - torus (centered on a trimodal state) at 1.2 < t < 1.3. Fig. 24 (1.34 < t < 1.85) confirms that periods of homogeneous turbulence can be rather protracted.

Fig. 25

(1.61 < t < 2.0) confirms that the C.S. can very well restart without any appreciable drift or rotation (1.7 < t < 1.8). Nucleation of defects again appears at t = 1.8 and triggers homogeneous

chaos.

Both Bergd 30 and Ciliberto 24'25 have observed these localized

spatio-temporal coherent structures at large Rayleigh numbers (up to 600 R a critical).

T IX,I

)

(ONI~HIIU;

(}.512

O. 2 5 6

.IIJ X

F I G U R E 21 C o h e r e n t S t r u c t u r e s , 0 < t < 0.512 (~ = 88).

B. Nicolaenko / Large scale spatial structures

'P ( x , t )

479

CON'I'OI)I~S

t). l) 3 8

0. 382 tm F-.

O. 120 211

II ×

FIGURE 22 The C.S. reappears, 0.126 < t < 0.638 (~ = 88). '~ ( x , t )

CON+I'OURS

.34

L. 09

[-~

O. 832 0

11

2H

X

FIGURE 23 The C.S. is g e n e r a t e d b y t h e r o t a t i n g m o d u l a t e d w a v e s , n o w m e t a s t a b l e (0.832 < t < 1.34, R = 88).

B. Nicolaenko / Large scale spatial structures

480

T(:'. , t )

{:I)N'I'IHIR~

FIGURE 24 Periods o f h o m o g e n e o u s t u r b u l e n c e can be p r o t r a c t e d (1.34 < t < 1.85, ~ = 88). I' I x , t }

(:()g'l'(llll.~';

Z. 0

I .80

1.61 2tl

,(

FIGURE 25 N u c l e a t i o n o f d e f e c t s dooms t h e C.S. (I.61 < t < 2.0, ~ = 88).

B. Nicolaenko / Large scale spatial structures

481

A laminar regime 16, 94 < t < 127.5 reappears with a quadrlmodal cellular state ~4 (5) as the global sink. The genericity of the Shilnikov mechanism as a trigger of strong chaos is demonstrated in Fig. 26-27, ~ = 128. Here, the initial condition was chosen as ~4 (x) + 0.1 sin x, where 74 (x) = 2.7 cos 4x + 0.40 cos 8x + ....

Note that the chaotic

bursts are more protracted in Fig. 26 (energy), with clear oscillatory build-up of the streaks. Fig. 27 (energy in m o d e

nature of 7 4 (R~, ~ >__ 127.5.

4) clearly demonstrate the Shlinikov saddle-focus

The i m p a c t of t h e s y m m m e t r y isotropy subgroups D 4

(dietral subgroup of the square) and D 8 (dietral subgroup of the octagon) can be evidenced. 6. CONCLUSIONS AND FUTURE DIRECTIONS Both the Shilnikov m e c h a n i s m s for bursting s t r e a k s and m o r e complex i n t e r m i t t e n t coherent structures have b e e n observed in c o m p u t e r simulations up to • -- 800, Nu -- 10 and dim F A > 10.

The qualitative a g r e e m e n t w i t h experiments d e m o n s t r a t e s the

robustness and the v e r s a t i l i t y of the K.S.S. models f o r large scale structures. Yet m o r e tantalizing is the scenario which emerges for t h e genesis of such C.S.: strong local relaxation phenomena of oscillatory nonlinear v i s c o s i t y physics are e f f e c t i v e l y translatod into turbulent dynamics which randomly explore a limttod set of strongly spatially

120 - -

~a

40 0.23

I

I

I

I

T I ME

FIGURE 26 The energy history is typical of Shtlnikov bursts (~ = 128).

I

I

1.0

B. Nicolaenko / Large scale spatial structures

482

0 0,23

] .0 "t I ~II;

FIGURE 27 The energy in m o d e four betrays a quadri-cellular saddle-focus (~ = 128).

coherent structures.

The l a t t e r are islands of m e t a s t a b l e hyperbolic structures (fixed

points, T 2 and T 3 Tori?) on a chaotic sea imbedded in a much larger dimensional inertial manifold.

Permeating

fluctuations.

the

whole structure

are

small-scale

turbulent

(stochastic)

Are C.S. quasi-deterministically ruled by a much lower number of local

nonlinear coordinates than the Kolmogorov estimates for the number of degrees of freedom for turbulence would indicate? Such is the challer~e. We are currently actively investigating more turbulent (less ,%vtmpy") systems where Nu > 30 and dim F (A) > 100, where percolation and nucleation phenomena actively compete for generating intermittencies. 32'33 investigation

of lattice

systems

Another very promising avenue is the

obtained by discrettzing the

K.S.S.

models and

coarse-graining the discrete systems thus obtained.

In this spirit we encourage the 34 reader to refer to the contribution by Minh-Duong-van in these proceedings. ACKNOWLEDGEMENT We wish to thank P. Berg6, H. Chat6, S. Ciliberto, M. Dubois, C. Foias, M. Golubitsky, J. Gluckenhelmer, P. Holmes, M. Kruskal, P. Mazmeville, Y. Porneau, E. Spiegel, Mirth Duong-van, and S. Zaleski for interesting and fruitful discussions.

Our

computations would not have been possible without the all-purpose, friendly code developed by James M. Hyman at Los Alamos. The author was supported by the U.S. Department of Energy under c o n t r a c t W-7405-ENG-36, by NATO c o n t r a c t 85/0509 and by the Mathematical Sciences Institute at Come11 University.

B. Nicolaenko / Large scale spatial structures

483

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Chaos

in

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the

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