Computation of the dimension of a model of fully developed turbulence

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Computation of the dimension of a model of fully developed turbulence

R. Grappin, J. Léorat, Annick Pouquet

To cite this version:

R. Grappin, J. Léorat, Annick Pouquet. Computation of the dimension of a model of fully developed turbulence. Journal de Physique, 1986, 47 (7), pp.1127-1136. �10.1051/jphys:019860047070112700�.

�jpa-00210300�

1127

Computation ^{of the} dimension of a model of fully developed ^turbulence

R. Grappin (1), J. Léorat (1) ^{and A.} Pouquet (2)

(1) ^CNRS Observatoire de Meudon, DAF, ^F-92190 Meudon, ^France (2) ^CNRS Observatoire de Nice, BP252, F-06007 Nice Cedex, ^France

(Reçu le 27 décembre 1985, accepté ^{le 24}

1986)

Résumé. 2014 On calcule la dimension de l’attracteur d’un modèle scalaire de turbulence MHD développée ^{selon la}

méthode des exposants ^de Lyapounov ^d’une part, ^{et d’autre} part ^à l’aide d’un algorithme ^de comptage. ^{On obtient} numériquement que la dimension de Lyapounov ^{est à peu} près égale

nombre total de modes présents ^{dans la}

zone

inertielle, ^et que l’exposant maximal, c’est-à-dire le taux de divergence ^des trajectoires proches, ^{est donné}

par l’inverse du temps caractéristique ^le plus ^court présent ^{dans la zone} inertielle. La dimension de corrélation,

d’autre part, ^est égale ^{à environ un} tiers du nombre de modes présents ^{dans la zone} inertielle. Les deux estimations de la dimension

coincident qu’à faible nombre de Reynolds, c’est-à-dire à basse dimension (égale ^{à environ} 3).

Extrapolés ^{à la} turbulence réelle à 3 dimensions (Navier-Stockes), ^nos résultats numériques ^sont compatibles

avec

les deux lois d’échelle : 1) l’exposant ^de Lyapounov maximal varie

Re1/2, 2) ^la dimension de Lya-

pounov ^et l’entropie ^de Kolmogorov ^{varient comme Re9/4} (pas de saturation de la dimension de l’attracteur

turbulent).

Abstract

We compute numerically ^{the attractor} dimension of

model of fully developed ^MHD turbulence both

using Lyapounov exponents, ^{and via}

counting algorithm. ^The Lyapounov ^{dimension is found to be} essentially given by ^{the total number} of modes which lie in the inertial range; the maximal exponent, ^{i.e. the} divergence ^rate

of nearby trajectories, ^is given by the inverse of the shortest time scale available in the inertial range. The correlation

dimension,

^{the other} hand, ^{is found to be} given by roughly

third of the modes in the inertial range. Both evaluations of dimension coincide only ^at very low Reynolds, ^{i.e. at low dimension} (equal ^{to about} 3). ^{If extra-} polated ^to real 3-dimensional Navier-Stokes turbulence,

^results

consistent with the two scaling ^{laws :} 1) the maximal Lyapounov exponent ^scales

Re1/2, 2) ^the Lyapounov dimension and Kolmogorov entropy scale

Re9/4 (no saturation of the dimension of the turbulent attractor).

J. Physique ⁴⁷ (1986) ^{1127-1136 JUILLET} 1986,

Classification Physics ^Abstracts

03.20 - 47.25

1. Introduction

The description ^of homogeneous turbulent flows at

high Reynolds ^number requires ^a great ^{number of} variables, ^which may exceed the capacity ^{of even the} biggest computers : ^{there is no} hope ^{in the near}

future to simulate the smallest scales of most geophy-

sical or astrophysical flows. Turbulence modelling

is a way ^to circumvent this difficulty, by reducing

the number of degrees of freedom taken into account in the calculations. The minimum number of inde-

pendent variables which is _necessary to represent ^a turbulent flow depends ^{on the} Reynolds number,

but also on the method used to model turbulence.

For instance, direct numerical integration ^{of the} primitive equations ^{at a} given Reynolds ^number requires ^more variables than the integration ^of spectral equations ^obtained by ^an isotropic ^closure model, ^with geometric discretisation of wavenumbers.

Can we justify any eventual reduction of degrees

of freedom in homogeneous turbulence modelling ?

One _way would be to show that the dimension of the turbulent attractor is smaller than the total dimension of phase space. Laminar flows, which alone are

accessible to detailed analysis, ^{have a} small number of effective degrees ^of freedom; ^new excited modes appear during the transition to turbulence, ^the precise ^scenario depending ^{on the flow} (B6nard, Couette, etc...). Review of work in that line is given

in Abraham et al. [1].

Since Lorenz, several models have allowed to better understand the first bifurcations from laminar to turbulent (o ^chaotic ») states, but their extension to

fully developed turbulence seems difficult (see ^for

example Francheschini and Tebaldi [2]). ^{On another}

side, using ^a rigorous analytical study of the Navier- Stokes equations, Constantin et al. [3a, 3b] ^have recently ^{found an} upper bound for the number of

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019860047070112700

pertinent ^{modes. We} propose here ^to compute ^nume-

rically the dimension of a model of fully developed

turbulence.

The model we shall study ^{is a} generalization ^to

MHD of a hydrodynamical model introduced by Desniansky and Novikov [4, 5] : ^{in its} simplest ^for- mulation, ^{it involves N}

^{2 S real variables} Xn (S kinetic modes and S magnetic modes). Spatial

scales are discretized by ^{octaves and} appear in the

equations ^{via the} wavenumbers k,, - ^{2n -1. The name}

of the model stems from the fact that all physical quantities ^are scaled, using the association of the n-th mode with wavenumber kn. ^The equations ^read formally (see Gloaguen et ^al. [6] ^for details) :

In equation (1) ^{the constant} coefficients Aij ^couple only neighbouring ^scales (i, j

^{n, n ±} 1) ^{and are}

chosen so as to conserve the two inviscid invariants of MHD equations, total energy and velocity-magnetic

fied correlation. This constraint reduces the number of free coupling parameters ^{to two} coupling ^constants

a and 3 (actually only ^{the ratio} 0153lP matters). ^The

coefficient ^Q stands for a dissipation coefficient,

either the viscosity ^v (for ^kinetic modes), ^{or the} magnetic diffusiviiy 11 (for magnetic modes). ^The

third term is a forcing (constant acceleration) ^which only ^{acts on} the first (large scale) kinetic mode.

The complete equations ^are given ^{in the} Appendix.

Let us summarize briefly ^the properties ^{of the}

model (see Gloaguen et ^al. [6] ^for details). ^{If one} neglects ^the dissipation ^and forcing ^{terms, it is a}

conservative system, ^{i.e. it} conserves an N-volume in phase space. Gibbs statistical equilibrium obtains,

with equipartition ^of energy between all modes.

If on the other hand we put ^{non zero} dissipation ^and forcing, ^{then if N is} high enough, and with suitable values of coupling ^{constants a} and fl (we ^{shall come}

back to this point ^{in the next} Section), ^two regions

appear in the wavenumber _space. The large ^scales (o ^{inertial »} zone) show chaotic but persistent ^fluc- tuations ; long-time averages of _energy distribution

reproduce very nearly ^the Kolmogorov power-law spectrum. Small scales (dissipative range) ^{are on the}

other hand characterized by excitation appearing

very intermittently, ^{and an} energy spectrum decaying

much more quickly (quasi-exponentially) ^{with wave-}

number. The boundary between both zones goes towards small scales when the viscosity ^{v and} magnetic diffusivity ^decrease.

Note that the coexistence of spatial scaling ^laws

and temporal ^{chaos is an} original property ^{of this} model; ⁱⁿ particular, ^the hydrodynamical ^version

of this model (Desnyansky and Novikov [4, 5]) ^has

the Kolmogorov spectrum ^{as a} stable fixed point,

and shows thus no chaotic behaviour.

Since the scalar model verifies several characteristic basic properties ^of fully developed turbulence, ^{it is}

a good ^{candidate to} investigate how its dimension scales with the Reynolds number. There are several methods to measure the dimension of an attractor

numerically. ^The simplest ^is probably the «natural»

(or «pointwise») ^{measure which} amounts to count the number of points ^{of the} trajectory contained in a

disk of radius R around a given ^centre (average ^is eventually made later on the position ^{of the centre :}

see Grassberger ^{and Procaccia} [7]). If the number of

points grows ^as RD, ^{then D is the dimension of the}

attractor. A detailed account of results obtained via this method (of ^{which a} preliminary presentations ^has

been previously published ⁱⁿ Pouquet et ^al. [8])

will be given in section 4.

Another method, which will be presented ^now, brings ^more information than a mere evaluation of dimension. It consists in computing ^the Lyapounov exponents (Benettin et ^al. [10]). (A ^{first account of}

some results obtained on the scalar model is given

in Grappin et ^al. [9].) ^{These exponents} generalize

to nonlinear dynamics ^the concept ^of eigenvalues ^{of a}

linear system ^{with constant} coefficients. Let the initial state X ° (which ^{is a vector with N} coordinates)

in equation (1) ^be replaced by ^{X ° +} 6X ’; linearizing

about X’, ^{we have}

where A (X’) ^{is an N x N} matrix. The trace of matrix A is a (negative) ^{constant :}

which gives ^{the rate} of contraction of a parallelepiped

with N orthogonal ^sides 6X’,..., 03B4Xtn. Lyapounov exponents Ai ^are time-averages ^{of the} eigenvalues ^of

matrix A, ^{defined as :}

where the 6 Yj ^are defined from the 6X! by ^{an ortho-} gonalization procedure (Benettin ^{et al.} [10]), 11 Y 11 being ^{the norm of Y. An ordered} spectrum ^of expo- nents is obtained in this _{way :} ;’1 > Å2 > ... > AN,

the first exponent being positive (if ^the system ^is

chaotic) ^and measuring ^the average divergence ^of

two nearby trajectories, ^{and at} least the last exponent being negative, ^{in order to} satisfy equation (3). ^{If we}

start from a segment ^{and build} parallelepipeds ^with

a growing number of dimensions _n, we see from

equation (3) that there must be a critical number Nr

such that the parallelepiped ^with Nc dimensions is still expanding, ^i.e. Jl( N c) = ;’1 ^{+ ... +} ÅN c >

0,

while the one with N c ^{+ 1} dimensions shrinks with

time, ^i.e. Ii(Nc ⁺ 1) = A 1 ^{+ ... +} ÅNc+ 1 ^{0. The} Lya-

pounov dimension of the attractor is defined by

Kaplan and Yorke [11] ^{as a linear} interpolation

1129

between these two successive integers :

Figure ¹ illustrates this definition; ^it shows, ^{for two}

different Reynolds numbers, ^{the curve} of cumulated exponents (M(n) ^{as a} function of n) ^obtained by ^inte- grating system (1) : ^the Lyapounov dimension is

given by the abcissa of the intersection of the curve

p(n) with the horizontal axis.

Note that relation (5) implies that the conservative

dynamical system which obtains when forcing ^and dissipation ^{are absent} (v = 11

0) has dimension

DL

N. At first sight, ^{this is a} paradoxical result,

since system (1) possesses ^two invariants in that case, hence it should have N - 2 degrees of freedom left,

not N. However, ^the paradox disappears ^{if we recall}

that the Lyapounov exponents explore the dimensions of the linearized system (2), ^{which has no reason to}

have _any invariant. Thus, ⁱⁿ the conservative _case, different Lyapounov exponents could result from

calculating ^N nearby ^nonlinear trajectories ^from

system (1), ^without any linearization. Remark however that this problem disappears ^{in the} dissipative ^case,

since there are no invariants _any more.

2. Identification of attractors at large Reynolds.

In order that solutions of system (1) exhibit both chaos and power-laws ^for spectra averaged ^{with time,}

several conditions must be met. First, ^as in simula-

tions, ^{we want} the truncation kS 1 ^to be smaller than the dissipation scale. This is obtained by imposing

a viscosity ^{v and} magnetic diffusivity 11 large enough,

so that at the smallest scale, ^the dissipation ^{terms are}

greater than the nonlinear terms : and

where Xs ^{is the} (kinetic ^or magnetic) excitation at the smallest available scale. Otherwise, excitation will try ^to flow into scales which are not present ^{in the} truncated N-system, energy will accumulate at the smallest scale and the system will resemble a conser-

vative system. Assuming Kolmogorov scaling Xn N kn 1/3, the above condition (6) ^{leads to}

Second, substantial magnetic excitation must be present, ^{since the} hydrodynamic version of system (1) (Desnyansky ^and Novikov, [4, 5]) ^{is not turbulent.}

Last, kinetic and magnetic excitation should not be

completely correlated, (i.e. ^{we should not have} Un =bn

for all n or un = - bn ^{for all} n), otherwise nonlinear terms in equation (1) will vanish : this property ^is shared with the primitive ^MHD equations. ^{These last}

two conditions are governed by ^{the choice of two} parameters ^{a and} f3 ^which weight ^two groups of nonlinear terms in the evolution equations (1) (see

Fig. ^1.

^Cumulated Lyapounov exponents. The ordinate

n

is the

of the first n Lyapounov exponents p,(n) = L ^03BBi,

which measures the growth ^rate

parallelepiped ⁱⁿ phase

space,

^a function of its dimension

The Lyapounov

dimension is given by the intersection of the

with the horizontal axis. Two

shown, ^with parameters

a ⁼

10- 2, fl

l, viscosities v

^{l OT 4} and v =6.25 x 10- 6,

and respectively ^N

18 and 24 modes. Integration ^times

are

up to T

1 024 and T

512, respectively. ^{Note the} large plateau, which grows with Reynolds number, ^of quasi-

null Lyapounov exponents ^{in both}

Appendix). ^The properties of the model have been

investigated in detail at small N(N

^{4 to} 6) by (Gloaguen et ^al. [6]); these authors have also looked at one special value of the ratio of coupling parameters

«(Xlf3

0.01) ^{at N}

^18. Analysis of the bifurcation

properties of the model when (XI f3 varies and N is

large ^is difficult, ^{so that numerical} integration ^is

needed.

A predictor-corrector numerical scheme of order four was used to integrate ^{the main} trajectory (Eq. (1)),

while the linear system ^{of N} equations (2) ^{was inte-} grated by calculating ^a fourth-order Taylor expansion

of the propagator exp(A(Xt) H), where H is the time step ^{used to} integrate equation (1). (Since the matrix A is tridiagonal, ^the computation of the first _powers of matrix A is performed rapidly). Time-step ^{is chosen}

to be small enough, ^{so that} equality (3) is satisfied with

a relative precision ^{of 10- 5. The method was tested}

on the Lorenz system. Calculations _up to T =1024 with N

18 modes (ks/k1

256), ^v

10- 3, 11 = 1.2 ^x 10- 3, ^time step ^H

^{2 x} 10- 3, ^{took a little}

less than 10 min of CPU time on a Cray ¹ computer.

For most of the _runs, initial conditions were as

follows. Magnetic excitation was small at all scales

(bn

10-4), and kinetic excitation was substantial

only ^at large ^scale (exponential spectrum, u,,

exp( - kn)).

Rapid exploration ^of system (1) with these values of parameters, varying the value of the ratio of cou-

pling ^constants a/P, ^was undertaken. Integration ^was stopped ^{at time T}

^{64, or in} some cases at time T

128. Such an integration ^time proved ^{to be} large enough ^{to reveal} unambiguously ^{the nature of}

the attractor. We found only ^{three different} types depending ^on the value of ratio a/# (see ^Table I). ^For oc/# smaller than about 0.3, ^a high dimension for the attractor is the rule, ^{with a} large ^{inertial zone,} large fluctuations, ^and equipartition ^{of kinetic and} magnetic

energy. For a/fl ^{between 0.5 and} 1, ^{the attractor is a} non-magnetic, ^{stable fixed} point (non-turbulent, ^cons-

tant Kolmogorov ^kinetic spectrum). ^For all larger

than 2, the correlation coefficient _p

2 E v. ^bn/

E(V2 ^{+ bn) is seen to grow rapidly at} almost maximal

values, with dimension decreasing accordingly ^with

time : the _energy spectra ^are decreasing ^{much more} rapidly ^{than a} power-law, and less and less small scales are excited with time. We interpret these results

as indicating ^{that the attractor is a fixed} point ^with

maximal correlation (p

^{1 or} p

1), ^{and thus}

its Lyapounov dimension is zero. Note that, ^since

nonlinear terms vanish when I p I

1, time scales become infinite as [ p approaches its maximal value : hence the slow convergence in the above data.

It thus seems (see ^Table I) ^{that the} picture ^{of bifur-}

cations (in a/fl space) ^{is much} simpler ^at large ^{N than}

at small N, showing ^{one stable fixed} point ^at large all (magnetic ^at sufficiently large a/fl, non-magnetic ^at

moderate a/fl) and, ^at sufficiently ^{low ratio} alP, ^one high dimensional turbulent attractor. No attempt ^has been made to have a complete ^{view of} the bifurcation

landscape which has been skeched above (in particular,

we have kept fixed initial conditions).

We shall now concentrate on the magnetic ^turbu-

lent attractor properties, ^and study in detail the case

{3

^{1 and a}

0.01, ^{a case} already considered in

(Gloaguen et ^al. [6]). ^{Note that the a and} fl coupling parameters represent respectively ^two groups of

interacting wavenumber triads (k, p, q) (with k

p ⁺ q) : ^{the a} parameter corresponds ^to (almost)

«flat triangles (i.e. quasi-parallel wavenumbers), whereas fl represents triangles such that k ;= _{p = q.}

Thus the choice of a/fl

^0.01 is consistent with

incompressible turbulence, for which strictly ^flat triangles ^{do not} contribute to nonlinear terms.

3. Spectral properties ^and Lyapounov exponents ^{of the}

turbulent attractor.

To obtain reasonably converging ^{values of} Lyapounov

exponents, ^we integrated system (1) ^at long ^times (up ^{to T}

1024),.with ^a

10-2, #

¹ (see Fig. 2).

Table I.

Short time exploration of correlation coefficients p and Lyapounov ^dimension DL ^{with various values} of ^the coupling parameters ^ratio alP. The correlation coefficients p is 2 Ev,, b,,l E(V2 ⁺ b 2). ^Its initial value is 0.001 in all runs. The’Lyapounov ^dimension DL ^is given by equation (5). ^{a and} fl ^{are two} coupling parameters ⁱⁿ system (1) (see Appendix). ^Number of modes ^{is N}

18, viscosity ^{is v}

10-3, magnetic diff usivity ^is ~

1.2 ^x 10- 3. The

correspond ^to non-calculated data. Note the three regions. First, almost maximal correlation obtains for alp>

^2,

and dimension decreasing with ^time (with probable ⁰ limit, ^see text). Second, ^the stable fixed point for a/fl between 0.5

and 1 (there ^is vanishing magnetic energy in this case). Third, large ^values of dimension, ^which grows slowly ^with

time, for ^{small values} of a/fl (see text). ^Case a/03B2

^{0.01 is studied} extensively in section 3.

1131

Fig. ^2.

Convergence ^of Lyapounov dimension with time.

Note the growth of dimension with time, corresponding

to more and

modes becoming turbulent. Turn-over time (characteristic ^{time of} largest scale) is of order 1.

The large time scale in system (1) (which would be the

«turn-over time » of the larger ^{eddies in a classical}

turbulence description) ^is given by 1 /(k 1 u 1 )

I /u 1,

which was close to unity ^{in all runs. We thus} integrated

about 1 000 turn-over times, ^{which seems} necessary to completely explore ^the attractor. Note however that the time-step ^{is not} determined by ^the large ^scale

characteristic time, ^but by the small scale time, ^which depends ^{on the} viscosity, and may be much smaller.

Four values of viscosities, ranging ^{from 6.25 x 10- 6}

up ^to 10- 2, have been considered. In all runs, magnetic diffusivity ^and viscosity ^were taken with comparable

values (see ^Table II). The total number of modes

was 18 in all but one case, where it was 24.

Figure ¹ gives ^{the cumulated} Lyapounov exponents distribution u(n) ^{at time T}

^{1 024, for the two} higher Reynolds ^{numbers. Note the} large plateau, ^{due to a}

majority ^of exponents being very close _{to zero,} which leads to a high Lyapounov ^dimension (Eq. (5)).

Figure ² gives ^the Lyapounov ^{dimension obtained}

as a function of time, for different values of dissipation

coefficient. Note the slow rise of dimension with time, showing ^{that a} growing ^{number of} degrees ^{of freedom}

come into play : ^it may be a consequence of intermit- tency in the smallest scales.

Time-averaged spectra ^show large inertial ranges with Kolmogorov-like power-laws (Fig. 3), ^{both for}

kinetic and magnetic energy. A strong ^non-linear dynamo ^{is at work : one observes} equipartition

between both field at all but the largest scales, ^where magnetic energy remains almost absent. Note that initial magnetic energy is rather small ( 10- 4 ^{at all} modes) ^{and that we have} only ^kinetic forcing, (see Eq. (1)).

Inspection ^of figure ³ shows that decreasing ^the viscosity lengthens, ^as expected, ^{the inertial} range.

Let us recall the standard Kolmogorov phenome- nological prediction. ^The largest « inertial » wave-

number k* is determined by approximate ^balance

Fig. ^3.

Time-average ^{of kinetic and} magnetic spectra.

Abcissa : n

log2(kn) ; ordinate : kinetic energy un /2 ^and magnetic energy b;/2 (right). Straight ^{line :} Kolmogorov

law k; 2/3. ^Note equipartition between kinetic and magnetic spectra, ^at all but the largest magnetic scales, ^{and the} slight departure ^from Kolmogorov ^{law. All}

^{with 18} modes, except with the smallest viscosity.

Table II.

Maximal Lyapounov exponents ^and Lyapounov ^{dimension at} long times, for ^{case a}

10- 2, 03B2

^1.

The number of ^{modes in} the inertial _{range is} found by ^visual inspection of figure ^{3 : one counts} the modes which

follow ^a power-law( forinstance, ^{in the run with} viscosity ^6.25 x 10-6, ^{all modes are} _{kept, except} the two large ^scale

magnetic modes).

between linear dissipation ^terms and nonlinear terms in equation (1) (cf. [6]). ^{This leads to :}

and, ^{as a} corollary, ^{we see} that the shortest time scale in the inertial _range scales as T * -(I lvk *2) - ^{v 1/2.}

A corresponding scaling ^is observed for both the maximum Lyapounov exponent and the Kolomo- gorov entropy ⁸ (sum ^over positive Lyapounov exponents) : 03BB1- ^{E v-ll2} (see Fig. 4) suggesting

that the divergence ^of nearby ^{states is} governed by ^the

time scale of the smallest scale available in inertial range. Note however that the divergence ^{rate at} large scale, (which is related to the predictability problem

and is outside the _scope of this paper), ^{needs not be}

related in _any simple way ^to this time scale.

The fact that smaller scales do not contribute to the

divergence ^of trajectories may be viewed as due to the dominance of viscous damping ^{at such} scales, ^for

most of the time : these scales ^are only intermittently excited, ^when nearby (larger) scales have their maxi-

mum amplitude, ^which gives ^{rise to} transient nonlinear bursts. Figure ^{5 shows the} spectrum of the flatness factor Fn = XI >/ X; )2 for kinetic and magnetic

modes. We make two remarks. First, the flatness

« spectrum >> ^{is flat} (about 10, ^{a value} substantially higher than the Gaussian value 3) ^{in a} large range of wavenumbers. Second, this flat _range corresponds reasonably ^{well to} the inertial _range, i.e. to the _range where the corresponding energy spectrum ^{follows a} power-law. ^At all« non-inertial » scales, ^{which are the}

Fig. ^4.

^Maximum Lyapounov exponent and Kolmo- gorov entropy

viscosity. Straight line is the curve

1 /. ^{(see text).}

Fig. ^5.

^{Kinetic and} magnetic flatness factor. Abcissa :

n ⁼

log2(kn); ordinate : Fn = Xn ) j( XI >2, ^where Xn

is the kinetic (left)

magnetic (right) fluctuation. Note that the flat part ^{of the} spectrum corresponds ^to the inertial ranges of figure ^3.

dissipation scales and the large magnetic scales, ^the intermittency ^is significantly higher.

Developed turbulence _may thus be viewed as being

made of pertinent ^modes (those in the inertial range),

and non-pertinent, intermittent modes. More precisely,

we mean that the detailed number of non-inertial modes represented ^{in the} system ^{does not} change ^the

dimension : only ^the number of «inertial modes matters. To show this, ^we plot ^the Lyapounov ^dimen-

sion DL ^{as a} function of viscosity; ^results agree well with the curve :

(See Fig. 6). ^This may be interpreted ^as follows. The number of kinetic (or magnetic) inertial modes is

approximately given (see Eq. (7)) by :

(recall kn ’" 2n-l); the ^{total number} N* of inertial modes is (forgetting ^{the one or two} large scale inter- mittent magnetic modes) ^twice that number :

Hence equation (8) ^{indicates that}

which is what we wanted to show (see ^Table II)

4. Dimension obtained by counting algorithms.

A direct evaluation of dimension is provided by simply

sampling ^{a set of N} points ^{of the} trajectory ^{and deter-}

mining, ^{as has been} proposed by Grassberger ^and

1133

Fig. ^6.

Lyapounov and correlation dimensions

viscosity. Abcissa : n

log2(kn). Straight ^{line : -} 3/2 log2(viscosity) (see text). ^The good parallelism between this line and the Lyapounov ^dimension DL ^{shows that} DL grows

as

the number of modes following

Kolmogorov power- law (see text). ^{Note that} Lyapounov and correlation dimen- sions coincide only ^{at low dimension} (about 3).

Procaccia [7], the correlation function :

where H(x) ^{is 0 when x 0 and 1 when x} >

0.

Assuming C(r) - rD, ^{D is} the dimension (henceforth

called correlation dimension Dc) ^{of the attractor} (see also Mandelbrot [12]). ^The point-wide ^dimension

is obtained in the same way, but keeping ^{one of the} points (say ^X

Xo) ^fixed (with normalization factor

equal ^to I/N). ^The point-wise dimension looks at how the number of points ^{in balls of} increasing ^{radii r}

centred on X o augments ^{with r. It} naturally requires ^N

times less operations ^than (11) ⁱⁿ principle (see below)

but is a local defmition and its value may vary from

point ^to point.

Numerically, ^the algorithm requires ^the storage of the relative distance in phase ^space between the N

points; ^this array is of order N 2 and brings ^{an im-}

mediate limitation to the algorithm. ^In pratice, ^one

stores the coordinates of the points ^{on the attractor,}

with a sampling ^time which is of the order (but ^smaller than) ^the large-scale eddy ^{turn-over time.}

The code was checked ^on the Lorenz system : ^the dynamical dimension calculated with 104 points ^was

found to agree with the Lyapounov dimension, ^{for the} standard values of parameters. We also checked that,

for parameter values in the scalar model, ^{such that}

either a fixed point ^{or a limit} cycle obtains, ^{the corre-}

lation dimension found numerically ^{is as} expected (0 ^and 1).

An important feature of these counting algorithms

is that it is _{very easy} to obtain, ^{on a} log-log plot, ^a straight line, ^{even when} convergence is not established

Figure ^7a shows the correlation functions (11) ^obtained

on the 6-mode system ^with cx = 0.01, # = 1, v = tj = 0.1769, varying ^the integration ^time (T

1000, 2 000, 20 000) with fixed sampling time interval (At

0.2).

This low-dimensional system has for these values of parameters ^a stable fixed point (see Gloaguen [13]).

However, ^{with a} given value of the time-step, ^the

numerical solution shows persistent oscillations (whose amplitude diminish when the time-step ^is reduced)

around the fixed point, ^{hence the non-zero} slopes ^of

the correlation functions observed ^on the figure ^at

small scales.

The resulting ^dimension appears ^to converge from below with the integration ^time (or ^{number of sam-} pling points). Figure ^{7b shows the case} of the 8-modes system ^{with v = q}

0.0102 and respectively ^T

3 000, 4 000, 5 000. The resulting dimensions are

summarized in table III. These data show that a small

sample strongly underestimates the actual dimension of the attractor but ^one needs not double the number of points, ^{since a} small increase suffices to indicate the rate of convergence of the algorithm.

We now come to the evaluation of dimension at

high Reynolds numbers, ^{with a}

0.01, P

^{1 as in}

the calculations of the preceeding section. As the number of modes is increased, ^the Reynolds ^number

is increased, by decreasing viscosity ^and magnetic diffusivity according ^to inequality (6). ^{Table IV and} figure 6 show the variation of both Lyapounov ^and

correlation dimensions with viscosity, ^{with for corre-}

lation dimension a number of modes between 6 and 18, and a viscosity ^{between 2.5 x 10-2 and 2.5 x 10-4.}

Table III.

Dimension calculated by counting algo-

rithms : variation with number of sampling points for

low-dimensional systems. All dimensions have been obtained via the correlation function (11), except ^the

fourth ^result of ^run A, ^{which is a} point-wise ^dimension (see text) ^with averaging ^{on 2fl} points ^and calculating

on 105 sampling points. ^{Run A :} 6-modes system;

run B : 8-modes system (see text). ^Note that reasonable

convergence obtains at about 30 000 points.

Fig. ^7.

Correlation functions versus number of sampling points. ^{Abcissa :} log, (distance); ordinate : log. (correlation function). The correlation function is defined in equation (11); it is obtained by counting

^{subset of} points

^the computed trajectory. ^Its slope gives ^the

correlation dimension » (see text).

a) ^6-mode system. ^{A best fit on} small scales give ^the following dimensions : number of points : ^{5 000 10 000 100 000}

correlation dimension : 1.68 2.22 2.56

b) ^8-mode system. ^{A best fit} gives :

number of points : ^{15 000 20 000 25 000}

correlation dimension : 2.86 3.93 4.06

Note in all cases the straight ^{lines in} log-log plot,

^{at the} lowest resolutions.

Table IV.

Dimension calculated by counting algorithm : ^{variation with} Reynolds number. All dimensions have been obtained via the correlation function (11). Viscosity ^and diffusivity ^are equal. Reynolds ^{is about} I /viscosity.

The correlation dimension varies from 3.3 to 8.3, but it must be stressed that, ^as the number of modes is increased, ^the time needed to achieve reasonable convergence ^on the computer ^becomes large (of ^the

order of one hour on a Cray 1 A for the last case).

5. Discussion

We have studied in this _paper a model of fully ^deve- loped ^MHD turbulence which shares _many structural

properties with real fluid equations. ^{This model}

exhibits «intrinsic stochasticity ^», i.e. does not rely

on external forcing ^{to reach a turbulent} regime. ^Note

that the constant acceleration ^on large ^{kinetic scale}

in system (1) ^was imposed only ^{in order to obtain}

statistical stationarity.

Computation ^of Lyapounov dimension has also

’ been done recently for another model of developed turbulence, governed by ^the Kuramoto-Sivashinsky equation (see Manneville [14]). ^However, the results

so obtained cannot be easily extrapolated ^{to fluid} turbulence, since there is no other common properties

between this model and real fluid than the Galilean invariance and the possibility ^{to control} the number of

«turbulent degrees of freedom by fixing ^{an external} parameter.

On the other hand, ^direct study of the Navier- Stokes or MHD equations ^are time-consuming, ^and

it seems to be impractical ^{to calculate more than the}

first Lyapounov exponents ^{on the} primitive equations.

Legras et ^al. (1985, private communication) ^have

recently calculated the first two Lyapounov exponents

1135

of 2-dimensional Navier-Stokes flow. Their results

seem to be compatible ^{with the} relation that is shown in figure ^4, namely Ål being comparable ^to the inverse of the shortest time-scale in the inertial _range.

Let us summarize our results. A rapid exploration, varying ^the coupling parameters ^{indicated that for a} large ^{number of} degrees of freedom (N

18, ^corres- ponding ^{to k} max/k ^min

256), system (1) ^{has three}

attractors : one hydrodynamical (non-magnetic) ^fixed point, ^{a maximal} velocity-magnetic correlation attrac- tor which is probably ^{a fixed} point, ^{and a turbulent}

attractor.

A detailed study ^{of this turbulent} attractor, varying

the viscosity ^and using ^{both the} Lyapounov ^and

correlation algorithms, ^{has shown} that its dimension is growing ^with Reynolds number with apparently ^no saturation, ^thus confirming the indication given ⁱⁿ Pouquet et ^al. [8]. However, ^the resulting ^dimension

estimations differ greatly ^at large Reynolds, depen- ding ^{on which} technique ^is employed ^{For the} Lya-

pounov dimension, ^{all «} inertial modes » are relevant, while for the correlation dimension, only ^{a fraction} (around ^one third) is relevant. Actually, extrapolating

the results plotted ⁱⁿ figure 6 shows that both esti- mations probably ^coincide only ^at very low Reynolds,

for an attractor’s dimension equal ^to about 3. A

comparison between different methods of measuring

dimensions of dynamical systems ^{has been} given by

Termonia and Alexandrowicz [ 15] : these authors also find that the Lyapounov dimension exceeds the correlation dimension for dimensions larger ^than

about 3 (they ^find Dc - ^{7.5, while} DL ^{10 in the}

case of a delay differential Eq.). ^This discrepancy may be explained by ^{arguments such as those} developed

in Atten et al. [16], ^{that the} counting algorithms

underestimate the dimension more and more when the dimension _grows, due to the simple geometric ^effect that, ^{in a} phase space with large number of dimen- sions, almost all couples ^of points accumulate at or near to the maximum distance available (a ^border effect).

The results obtained by computing Lyapounov exponents may be summarized as follows (see Figs. ⁴

and 6) :

(i) ^first Lyapounov exponent ^and Kolmogorov entropy ^{scale as} 10,

(ii) ^dimension DL ^{is close to the} number N * of modes in the inertial _range,

(iii) ^dimension DL ^{scales as -} 3 j2 log2 ( v).

It has been shown in section 3 that points (ii) ^and (iii) were.consistent with standard Kolmogorov pheno- menology, ^which predicts ^{for the} logarithmic ^discre-

tization of modes used in the scalar model that N* = - 3/2 log2(v). (Note ^that Kolmogorov phe- nomenology ^{«works » in the framework of our}

model (which ^{is a} model of MHD turbulence), ^while

we would perhaps expect ^{the MHD} phenomenology

of Kraichnan [17] ^to work here. This is due to the lack,

in our model, of interactions between widely separated

modes, ^which suppresses the possibility ^of any Alfv6n effect (see Gloaguen et ^al. [6D).

These results are in line with those of Ruelle [18],

who found by studying the Navier-Stokes equation, using analytical ^and phenomenological arguments, that intermittent flows should have an infinite number of null characteristic (Lyapounov) exponents per unit volume (see ^the large plateau ⁱⁿ Fig. 1). Equivalently,

we may say that all degrees of freedom contribute

to the dimension, except the intermittent modes

(which ^are essentially ^{in the} dissipation range ^at small scales).

Constantin et al. [2] ^have proposed, ^{as an} upper

bound, that the dimension of the Navier-Stokes attractor would scale as Re . It is _easy to see that relation (ii) ^{leads to the same} result, ^when extrapolated

to 3-dimensional Navier-Stokes turbulence. Indeed,

the number N* of inertial modes ^{is then} (k*)3, ^or

v- 9/4 (see Eq. (7)).

We are thus tempted ^to generalize ^our results and to

conjecture ^that for turbulence with Kolmogorov spectra :

(1) ^{the first} Lyapounov, ^and Kolmogorov entropy, scale as Re1/2

(2) ^the Lyapounov dimension scales as Re9/4.

That the dimension of the scalar model is so large (indeed, ^the largest possible) ^{raises some} questions.

First, the model presents temporal intermittency,

both at large ^and small scales (see Fig. 5), and this does not produce any noticeable reduction of the dimension

(see ^{the arguments} of Kraichnan [19] ^{in the context}

of spatial intermittency, ^{in favour of such a} reduction).

Second, ^a dissipative ^turbulent system ^{and a conser-} vative system ^{with the same} number of excited modes may thus have almost the same dimension, although

their statistical properties ^are qualitatively ^different.

This limits the usefulness of the concept of dimension.

Finally, if the dimension of real Navier-Stokes tur- bulence is indeed maximal, ^as conjectured above, ^all

modes are pertinent ^to describe the turbulent attractor

(except ^{the most} « intermittent » modes which lie in the dissipative range plus ^the largest scales in the MHD case), and there would thus be no hope ^for any substantial reduction of the number of degrees ^of

freedom at large Reynolds.

We are of the opinion ^{that the} importance ^of

« pertinent >> modes, ^{as measured} by ^the Lyapounov dimension, ^{must not be} exaggerated Even if the turbulent attractor asks for an infinite number of

degrees ^{of freedom, its} spatio-temporal properties

may be correctly described with a reduced number of

degrees of freedom with _proper (yet ^{to be} found) averaging. ^Solitons provide ^an example ^{of such a}

reduction in the context of integrable systems. ^{One of} the remaining ^most interesting questions ^{on the scalar}

model is the possibility (under study) ^{to model struc-}

tures like random bursts of excitation propagating

towards both ends of the spectrum, which could build

power spectra ^{in the} long ^term.

Acknowledgments.

We want to thank J. Curry, ^U. Frisch, ^C. Froeschl6,

C. Gloaguen ^{and D.} Russel for useful discussions at

early stages of this work. We are also grateful ^to

P. Manneville for pointing ^{to us} the work of Ruelle

[18] and Atten et al. [16], and for his constructive

criticism of an early version of this _paper. Part of the

computational facilities were provided by the National Center for Atmospheric ^Research (Boulder), ^and part

by the Scientific Commitee of the Centre de Calcul Vectoriel _pour la Recherche (CCVR), ^{which are both}

thanked

Appendix.

EQUATIONS ^{OF THE SCALAR MODEL} (SEE GLOAGUEN et al. [6]).

^We give ^{here the} details of equation (1), ^which

describe the one-dimensional model (o scalar model ») of the MHD equations ^{studied in this} paper. The equa- tions for the kinetic modes read :

The equations ^{for the} magnetic modes read :

The coefficients v and q ^are respectively ^the viscosity ^and magnetic diffusivity. ^{Note that a} and P ^{are free} para- meters ; actually, only ^{the ratio} al P matters, since, given (XI/3, choosing ^a amounts to choose a time unit.

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