A scalar model for MHD turbulence

Physica 17D (1985) 154-182 North-Holland, Amsterdam

A SCALAR

MODEL FOR MHD TURBULENCE

C. GLOAGUEN CNET-92131

Issy-Les-Moulineaux,

France

J. LkORAT Universitb Paris VII and Observatoire de Paris-Meudon,

92195 Meudon Principal Cedex, Frunce

A. POUQUET CNRS and Observatoire de Nice, 06007 Nice, France

R. GRAPPIN Observatoire

de Paris-Met&on,

92195 Meudon, France

Received 15 October 1984 Revised manuscript received 1 May 1985

A model for homogeneous MHD turbulence is proposed. Nonlinear interactions acts between nearest neighbours in a discretized wavenumbers space and conservation properties (total energy and v-b correlation) are verified. The model can be truncated at will. With three modes, a bifurcation analysis is given. In the turbulent case (dissipation and kinetic forcing are present) one obtains time fluctuations at all scales and time-averaged power law spectra, the small scales exhibit intermittency effects. Typical MHD phenomena such as the dynamo effect or the increase of v-b correlation in the decaying cases are also observed.

1.

Introduction

MHD turbulence occurs in conducting when both kinetic (R”) and magnetic Reynolds numbers

fluids (R”)

R” = UL/v, RM = UL/X,

(1.1)

are large compared to unity (U and L are characteristic velocity and length of the flow; v and X are the kinematic viscosity and magnetic diffusivity of the fluid). These numbers are a measure of the relative intensity of the non-linear interactions compared to the viscous and Joule losses (linear terms).

Some of the fundamental questions concerning MHD turbulence are recalled below: -How can a mean magnetic field, or mean magnetic energy be sustained by turbulence generated through a velocity field instability (dynamo effect) (see for example Moffatt [l])? -What are the spectral properties of the turbulence (recall that Kraichnan [2] has proposed a - 3/2 law in the magnetic inertial range instead of the -5/3 Kolmogorov law in the absence of magnetic field)? -How does the correlation of magnetic and velocity fields vary in decaying MHD turbulence when correlation is close to unity (as in the Solar Wind)? Are the spectral properties and the decay law of MHD turbulence modified in the fully

0167-2789/85/$03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

C. Gloaguen ef (11./A scalar model for MHD turbulence

correlated case (Dobrowolny et al. [3], Grappin et al. [4, S])? Definite answers to these questions are still lacking (Leorat et al. [29]). MHD turbulence is quite common in the universe. The reason is that in astrophysical situations the magnetic Reynolds numbers are often greater than 106, whereas in laboratory experiments RM is generally below unity. On the other hand, numerical experimentation using the full MHD equations in three dimensions can reliably reach (as of today) Reynolds numbers up to a hundred (Meneguzzi et al. [6]). Another possibility to study MHD turbulence is to use a closure scheme in which Reynolds numbers of the order of lo6 can be obtained (Pouquet et al. [7]). However, one deals only with averaged quantities and information concerning temporal fluctuations is lacking in such models. The main difficulty in the numerical study of turbulence is the high number of degrees of freedom which are coupled together; for example, this number varies as (Rv)9’4 for a 3D hydrodynamic turbulence following the Kolmogorov law. In the closure approach, a reduction in the number of degrees of freedom is obtained with the help of the hypothesis of homogeneity and isotropy. Accordingly, the mean spectra depend only on two scalar variables (time and wavenumber), and a logarithmic discretization in wavenumber space is possible, because solutions to the closure equations are well behaved. Thus the number of degrees of freedom varies only as log RV. However, closures yield to a set of coupled nonlinear integrodifferential equations which, at the highest resolution, require the use of large computers. There are simpler ways yet to modelize turbulent behaviour. In this paper, we study a discrete model for homogeneous isotropic MHD turbulence which on the one hand, uses the same reduction in the number of degrees of freedom achieved in closures, and on the other hand, allows the study of time fluctuations of the turbulent kinetic and magnetic excitations.

155

This model has been introduced by Grappin, Leorat, Pouquet (1981, unpublished). A first account has been published in Frisch [43]. Most of the material presented here is contained in the unpublished thesis by Gloaguen [14]. The idea stems from a model for hydrodynamic turbulence proposed by Desnyanski and Novikov [8, 91. They consider homogeneous turbulence assuming periodic boundary conditions. In Fourier space, the hydrodynamic equations of incompressible fluids reads &J,(k, t)/&

+ Vk*UJk, t)

where the coupling factor Aabc depends only upon the geometry of the triangle constructed from k and p. Evolution of the u-component (a = 1,2,3) with wave vector k depends upon all pairs of modes whose wave vectors can form a triangle with k. Concerning the efficiency of these triad interactions, it is generally accepted that the average dominant contribution in the integral on the r.h.s. of eq. (1.2) comes from wave vectors p such that IpI - Ikl and jk-pl - Ikl. In other words, transfer of energy occurs predominantly through interactions between similar scales; interactions between different scales are of advection or diffusion type, with a relatively reduced energy transfer. Kolmogorov phenomenology for example depends crucially upon this fundamental hypothesis known also as the “local (in wavenumbers space) energy cascade” hypothesis. The nonlinear time scale (“turn over time”) associated with this local process is (lkl. lu(k)l)-‘. The main idea of Desnyanski and Novikov is to modelize the “local” cascade of energy using a discretization of wavenumber space into shells (labelled by n) with mean radius k, growing in a geometric progression, and limiting the non-linear interaction to “nearest neighbours” shells (modes n f 1). Thus the model equations of hydrodynamics reads (in

C. Gloaguen et al./A scalar modelfor MHD turbulence

156

An important symmetry property of the MHD equations (2.1) is the invariance under a magnetic field reversal (b -+ -6). Rewriting the equations (2.1) with the Elslsser variables

shell space) du,(t)/dt

+ %&,(t)

= k,CA,_+,+,U,+.,. 1.J (1.3)

The A,J are coupling constants between adjacent shells (I, J = - 1,O or + l), independent of wavenumbers, and are determined by the conservation of energy (- x,,uz). The u, field is a collective variable representing the root mean square energy in the nth shell and remains positive (see also Bell and Nelkin [ll]). We extend this hydrodynamic model to the MHD case and examine its main features, particularly its time fluctuations. This paper is organized as follows: the equations of the MHD scalar model and its basic properties in the inviscid limit are given in section 2. Section 3 concentrates on the transition to turbulence when the number of modes is small. The case of fully developed turbulence is presented in section 4 and the conclusion appear in the last section 5.

2.

The equations of the scalar model of MHD

turbulence

(2.2)

the nonlinear terms take a more symmetric form, ~z+/~r-~@z+-~~v2z-=

-(z-•V)Z+-vp,

az-/at-v#--vJ&+=

-(z+T)z--VP,

(2.3) v,=(v+Q/2,

v,=(v-Q/2.

This shows that z + and z- fields do not interact directly between themselves. A trivial consequence is that the maximally correlated flows (b = + u or b = -u) are exact solution of the inviscid equations (the question of the u - b correlation will be examined further in section 4). In the nondissipative three-dimensional case (v = A = 0) the MHD equations have three quadratic invariants: the total energy u2 + b2) d3x,

E=fl(

the total correlation

We look for a model having in common with the original problem the main structural properties such as symmetries and conservation laws. Let us first review these constraints, assuming homogeneous, isotropic, fully developed MHD turbulence. The MHD equations for an incompressible fluid read

au/at-vv2u=

z’=u+b -

-(u*v)u+(b*v)b-vp.

f%/&-Xv26=curl(uX6),

(2.1)

where u and b are the velocity and magnetic fields (measured in AlfvCn velocity unit); the density has been taken equal to one, and p is the total pressure (kinetic and magnetic).

and the total magnetic helicity H=

/

a*curlad3x,

where a is the vector potential (b = curl a). This latter invariant vanishes when turbulence is mirror symmetric; this conservation property relies only upon the magnetic component of eqs. (2.1). We consider istotropic MHD turbulence with periodic boundary conditions, in a cubic box of length L. The velocity and magnetic fields are expanded in Fourier series. The MHD equations,

C. G&pen

et al. / A scalar model for MHD turbulence

which take a form analogous to eqs. (1.2) will be modelized now. . Following the suggestion of Desnyanski and Novikov [8], we replace all the Fourier wavevectors k in the shell with mean radius k,, k,/Jt;s

Ikl < k,JI;

(2.5)

=h,

where h > 1 is a constant, which has been taken equal to two in the numerical applications (cf. sections 3 and 4). We shall assume in this paper that the fields corresponding to the mode n, u,(t), b,(t) or z,*(t) = u, f b,,, have a single component. Note however that this is not essential to the model (see subsection 2.3 below). The model equations are obtained with the following prescriptions, inspired by the properties of the primitive MHD equations (2.3): -nonlinear interactions act only between nearest neighbours in model space; -the nonlinear interactions are quadratic between z+ and z- variables; - the nonlinear time scale associated with mode n is the local turn over time: (k,Jz: I)-‘; - the linear dissipative terms vary as kt. The above constraints lead to the following equations: dz;/dt

+ v,k,‘z;

+ v,,,k,zz, = knzAIJz,=,z;+J, IJ

dz,/dt

+ v,,k;z,

+ v,,,k;z; = k,xBIJz,+rz,=J, IJ (2.6)

where A,J and B, are coupling constants and I, J = + 1, - 1 or 0. In order to preserve the z+/zsymmetry of the primitive equations (2.3), we take B JJ E A,J. Writing now that the total energy, ET=EV+EM

=c(d+b,2)/2,

and the total correlation, C= &b,,, n

(2.8)

are conserved by the nonlinear equivalently that

interactions

(or

(2.4)

by a unique “scalar” mode labelled by n, with a “ wavenumber” k, varying geometrically with n: kn+i/k,

157

(2.7)

E’=x(z’)*/2=ET&C n

(2.9)

are conserved), we see that only two independent coefficients remain. Denoting them by (Y and /3, the equations for the z,’ finally read dz;/dt

+ v,,k;z:

= “(k,z,+_,z,,

+ v,,,k;z, - kn+lz;+1z;)

+B(k nz,-1 + z-” - k,+lz,=lz;+&

(2.10)

The time evolution of the z- field is obtained by exchanging the + and - in eq. (2.10). We will mainly use the corresponding equations for the kinetic and magnetic field variables which read du,/dt

+ Vk;u, = a(k,&

-k,b;-,

- kn+lunun+l

+ k,+,bnbn+d

+P(k nun-1un -kn+14+1 -k,bnbn-,

+ kn+lb;+d,

dbn/dt + hk:bn = akn+l(un+,bn - u,b,,+,) +Pk,(u,bn-, u=(z$+v,)/2,

- un-lbn),

(2.11)

X=(v,-V&2.

A diagrammatic representation of eqs. (2.10) and (2.11) is given in fig. 1. More general equations for the case of complex u and b-fields are derived in appendix A and the limit k,,+/k, + 1 is examined in appendix B. In the nonmagnetic case, when (Y= 0 one recovers the equation first obtained by Obukhov [12]; when j3 = 0, one recovers the equation of Desnyansky and Novikov [8]. With one-dimensional field variables, no magnetic helicity conservation property can be prescribed. However, notice that the non-linear terms in the magnetic equations in (2.11) have a conservation property which is reminiscent magnetic helicity

C. Gloaguen et ol./A scalar model for MHD turbulence

158

in the nondissipative case. This property was not imposed as a constraint for the derivation of the model equations and, in fact, it is not verified in the hydrodynamic case (z: = 2;). Recall that the Liouville property is shared by the primitive hydrodynamic and MHD equations (write (2.1) in Fourier space and take the real and imaginary parts of each fields as coordinates). (ii) In the limit of an infinite number of modes and when v = h = 0, the equations admit stationary solutions of the form

‘@,

‘.

I

t

b,, - u, - k;?

Fig. and and and

1. a) Nonlinear contributions to a) dzi/dt;b) du,/dt; c) db,,/dr. Continuous lines join the fields coupled by a dashed lines join the fields coupled by /3 (see eqs. (2.10) (2.11)).

conservation: when ah + p = 0.

(2.14)

Notice that equipartition of kinetic and magnetic energy, due to the “Alfven effect” (see below), is not implied by these stationary spectra. (iii) The “AlfvCn effect”, i.e. the nonlocal interaction of large scale magnetic fields with the small kinetic and magnetic scales has been explicitly eliminated when the nonlinear interactions have been restricted to nearest neighbours in the model space. The linearisation of the primitive MHD equations (2.3) with a constant large magnetic field B,(B, x=+u, b; v = X = 0) gives the following advection equations for the z * fields

(2.12) Although G is not a quadratic quantity, b,,/k, could be defined as a “magnetic potential”, and an inverse cascade of G analogous to the magnetic helicity inverse cascade (Pouquet et al. [7]) may occur. In this paper, we set ah + p # 0. This choice has the advantage that fundamental conservation properties will not depend upon the arbitrary discretization parameter h. 2.1. Some general properties of the inviscid scalar model equations (i) A Liouville theorem holds for the system (2.11). With N modes, the phase space is 2Ndimensional (first order evolution equations for z’) and a(dz;/dt)

a~,+ = 0

(2.13)

a2 */at

(2.15)

T B*VZ += 0,

which conserve total energy and correlation. According to the above described procedure, one could write scalar equations corresponding to eq. (2.15) dz’/dt

T k,B,z>=

0,

which, however cannot be used to generalize eqs. (2.10) since they violate the conservation properties. This is simply due to the fact that we have taken u and b fields with only one component. Conservation of energy and correlation can easily be recovered by taking two-dimensional vector fields. The two components can be taken as the real and imaginary part of two complex fields

C. Gloaguen et al./ A scalar model for MHD turbulence

with energies defined as

159

energy to b,_, is impossible, whereas transfer to is allowed: thus inverse transfer of magnetic energy towards large wavenumbers cannot occur when (Y= 0. More generally the interpretation of P/(Y based on the hydrodynamic situation cannot be extended to the magnetic one. In conclusion, we notice that the choice of (Y= 0 (or p = 0) imposes restrictions on the direction of the nonlinear energy fluxes when kinetic or magnetic modes vanish outside a finite interval. These restrictions may not apply when there is non vanishing excitation at all scales. Although a study of the inviscid system is interesting per se (see appendix C for more details), we shall now turn to the dissipative case in presence of forcing in order to study the turbulent solutions to the scalar model. bs+l

II (where - means a complex quantity and * means complex conjugate). A nonlocal AlfvCn effect based on (2.15) may then be explicitly introduced in the model equations, dz”/dt

r ik, B,,Z’ = 0.

(2.16)

This formulation permits energy and correlation conservation but doubles the number of independent variables. The study of properties of MHD turbulence in the presence of Alfven waves using the complex scalar model will not be examined further in this paper but the equations are given in appendix A. (iv) The number of modes in eqs. (2.11) may be chosen arbitrarily: any quadratic quantity such as energy or correlation are conserved per pair of adjacent modes (corresponding to the detailed conservation property per triad of the primitive MHD equations). Suppose that kinetic excitation is initially limited to a finite number of modes pertaining to the interval [I, S] and examine the role of the coefficients (Yand @ in the transfer of the excitation outside the interval [I, S]. This is most easily seen with the help of diagrams (fig. l), showing the contributions to the derivatives du,/dt or db,,/dt, when n is taken equal to Z - 1 or S+ 1. In the nonmagnetic case if (Y= 0, it is easily that no transfer of kinetic energy to us+i can occur and only transfer to uI_i is possible. Thus direct transfer towards wavenumbers larger than S is impossible when a = 0; when /3 = 0 inverse transfer towards wavenumbers smaller than Z cannot occur. The constant /3/e may be considered as a measure of the relative ability of inverse transfer in the hydrodynamic scalar model. We now consider the magnetic situation. Suppose first that the magnetic excitation is limited to the interval [I, S] and that kinetic excitation is present everywhere. If (Y= 0, transfer of magnetic

2.2.

Equations in the forced dissipative case

To obtain a statistical equilibrium in presence of dissipation, we choose to feed the system on the first mode (largest scale) with kinetic energy. Suppose that this mode ui interacts according to eq. (2.11) with an external mode u,, taken constant. The evolution equation of the first mode reads du,/dt

+ yk&

+ (c&u;)

= (du,/dt),,

+ (Bk,u,)u,,

(2.17)

where (duJdt)NL represents the nonlinear terms of eqs. (2.11). The second term in the r.h.s. is a constant acceleration acting upon the largest scale available (first mode) and the last term modelizes a constant velocity gradient. One can show following Lorenz [13] that the total energy of the system remains finite when this last source term is zero; otherwise, forcing with a term proportional to ui leads to a divergence of the total energy and has thus been discarded. The three parameters of the model (a, p and the constant acceleration A = ak,ui) can be reduced to one. Suppose for example a # 0 and multiply eqs. (2.11) by (Y.One can define adimensionalized quantities (t’, k’, u’, b’) by choosing new units of

C. Gloaguen et al./ A scalar model for MHD turbulence

160

time and length based on acceleration and box size L: t = t’( L/(&41)“*, k, = k;/L,

au, = u’(lcwAJL)l’*,

(2.18) 3.1.

cub,,= b;(:(laA)L)“*

and eqs. (2.11) become (with the nonlinear terms within brackets not detailed here): du;/dt’

- v’k,$:, = k,{ t/u’, b’b’)

+ P/rxk,, { u’u’, b’b’} + 1, db;/dt’-X’k,zb:,=k,{u’b’}+(/?/ar)k,{’ub’},

= L(LlcrA()‘/*/v and X’-’ = are the kinetic and magnetic Reynolds numbers based on the large scale L and the characteristic velocity uO. The equations (2.19) depend on the two Reynolds numbers as the primitive equations (2.3) and also on one arbitrary nonlinear coupling constant (/3/a). v’-’

L( L(cuA 1)1/2/A

3.

Hydrodynamic states

Consider first the nonmagnetic regime with a! = 0. The fixed points HN(O, 8; v), with components u lr.. . , iiN( ii, Z 0) are solutions of the following system of equations (h = k,+Jk, = 2): /3[0-2u;]

(2.19)

where

a great number of regions, even when N = 3. A detailed discussion, which we shall attempt to summarize in this section, can be found in Gloaguen [ 141.

Transition to turbulence

In this section we consider essentially the case of low Reynolds numbers where only a small number of modes are excited so that analytical studies may be performed. We shall first find time independent solutions. For the hydrodynamic (b, = 0) and the magnetic regimes respectively, we shall denote by HN(qj3; v) and MN(o,p; v, A) the fixed points for which excitation is identically zero for modes larger than N. These fixed points may experience various instabilities leading eventually to other steady states: for example HN + HN+l, or HN --+ MN (corresponding to a dynamo effect). When the Reynolds numbers increase, we have found that pitchfork bifurcations first lead to the excitation of a greater number of modes. Timedependent solutions such as limit cycles and chaotic regimes then appear for larger Reynolds numbers. The parameter space is in fact divided in

-vu,+1=0,

j3 [ UlU2 - 2S323- 2vii, = 0, p[u,u,-25iq]

(3.1)

-4vu,=o,

/3[ iiN_luN - o] - 2+$v

= 0.

Since UN# 0, the last equation gives li,_, = 2N-‘~/j3, the (N - 1)th equation gives ii,_, as a function of ( iiN)2 taken as unknown, and so on. The resulting polynome must have real roots; this leads to critical values of the viscosity above which HN does not exist. The components I&, ii,, i& are given in table I for N = 1,2,3. When N grows, it is more difficult to get explicit expressions; H4(0, 1; v) (resp. H,(O, 1; v)) exists only when v-l > &iJ (resp. v-l > 0X%). Two types of instability of HN may be distinguished: (i) instability of HN due to perturbations restricted to the N modes. A linear stability analysis shows that HN is always stable with N modes when N = 1 or 2. With N = 3, one obtains the following characteristic equation for the growth rate a of the instability: a3 + a*(69 - 2/3/v*)v + 2a(31/3/v2 - 990)~~

+ 364v( p/v2 - 34) = 0.

(3.2)

Instability occurs when there is a root with a positive real part, i.e. when v-l > (34.41/p)‘/‘, using the Routh criterion. This is a Hopf bifurca-

Fixedpoints

aii2/v solOtio0 of

zq-’ - h(16d + X2)/16a%)1’2

iQ - 2a-zJ15”

iit- (L/ZV)( a2 - 3OV4)

i?l- *((A+

ii, - l/( A + V)

& = f (C: - 4v2/a2)‘/2

ii, = v/a 33-0

ii, - (17/4)Lav

X/a

iP,= 0 =

Up= A2/16av

u2 -h/2a

& = *((2h/@)(l-

&=o

z2 = 0

ix - 0

h = (W4a)h

I,= 8, = -4A//3

iJ3-0

f4aW(a2(v+X))-2-vX-~/2)“2

q-0

P, - aiiQ4v

U2= aN( /I/( h + V) + 2X)

- a2/2r4 - 0

xs + x’ + (17/4)xX + 4x2 + x

x -

32A2/j9- 2~~/B))l’~

ii, =o

P, = au:/(2v - /3q

+ 82)

-vh-jt/2zO

Y’ < d/200 L = ( a2 + 225~‘)~’

A- *,

( y’ < 4a2/17, if X = V)

Xv(r+X)2(1+A2/16~2)ca2

8’0

32X2+ 2v2 < /I

a2(“+x)-2

N=(@+4a2)-’

one realpositiveroot x

c = -4v/(4a2

b=4(/3+“2)/(4a2+gz)

(1- -j?(fi+4v2)/(4a+jP)r

l/V > (34//3)“2, j3 ) 0

u, = 4Y/fi

- 68”2)l’2

i& = *(l/8)(28

5, = *(l/&0(/2

B> 0

l/V > (2/8)“2,

Ua -0 ids - 0

ii,=0 - u2y

Mode 3

Mode 2

q-2N((2a2/((X+v)-/?X)

Modei

Table I FixaIpointsofthetNluted&lnodeltith3modes

162

C. Gloaguen et cd/ A scularmodelfor MHD turbulence

tion which gives rise to a limit cycle (see below). (ii) instability of HN due to perturbation of the (N + 1)th mode. Hi is unstable with respect to the second mode when Y-’ > (2/p)‘/*. At vP1 = (34/P) ‘12, H: becomes unstable whereas H; remains stable for all viscosities. More generally, instability of the N + 1 mode at the fixed point HN sets in when v = /3iiN/kn+l, which is precisely the relation used to get the fixed point H,,,+l. For N greater than 3, closed expressions for the components of the fixed points have not been obtained due to the high order of the polynomial equations, and thus stability has not been analytically investigated. Numerical evidence for N up to 10 and /3 = 1 with various initial conditions led to conjecture that there is at least one stable fixed point H;(O, 1; V) when v c V,= m = 0.17047 and N 2 3. For example, with N = 3 and initial conditions close to the unstable Hc(O, 1; v,), for v in the range [0.170 46, 0.1621 the system is attracted to a limit cycle with a radius increasing as (l/v - l/v,) ‘I2 . When the limit cycle becomes sufficiently large in phase space, the system may fall onto the basin of attraction of another fixed point (or cycle). Indeed, we observe that for v = 0.161, the system evolves towards the fixed point H;, but we have not attempted to follow the fate of the limit cycle and its basin of attraction in more details as the Reynolds number grows. We now turn to the case /3 = 0. The fixed points HN(cq 0, v) have already been discussed by Desnyansky and Novikov [8, 91: one can show using Bendixson’s theorem that H,(a,O, v) is stable for all values of the viscosity. When N + oc and v -+ 0, the fixed point approaches the Kolmogorov solution Ii, - k-'13. Table 1 gives the coordinates of H,(a,O; v). I; is easy to see that all (~ii, are positive and unique, by inspection of the fixed points equation. Recall from section 2 that if kinetic excitation is limited initially to a finite range, it can be transferred to higher ,modes only when a Z 0 and that this can occur only through external perturbations. When (Y and p are both non zero, analytical results have been obtained only for N I 2 (table

I); with N = 3, ii, verifies a polynomial equation of degree 8, with coefficients depending on a, p and v and no explicit expression has been found for ii,. For large N, spectral self-similarity properties have been investigated both numerically and analytically by Bell and Nelkin [15], for ~$3< 0 and with either a constant forcing on the wavenumber k,, or without forcing and total initial energy on k,. With forcing, if - 1 < p/&/3 < 0, they obtain a Kolmogorov stationary solution for k, > k, and a different power law for k, <: k,; if p/&t”3 < - 1, two power laws are recovered but stationarity is lost due to an inverse cascade of energy. Without forcing, when - 1 c /3/&13 -C0, a Kolmogorov spectrum is again obtained but when P/cxh'/3 -c - 1, the simple power law scaling relations seems to break down. The case c@ > 0 has not been considered by Bell and Nelkin because they identify the variables u,(t) to r.m.s. kinetic energy and positivity of U, is not preserved by the equation of motion in this case. However, this restriction is not compulsory and in the following, we shall let the u,, variables fluctuate freely around zero, as actual velocity components would. When c$ > 0 temporal fluctuations (with change of sign) do arise. To study numerically these fluctuations we have taken (Y= 1, p = 2, v = 0.1, starting from the origin (ui = . . . =u N = 0) and varying the number of modes. With N = 2, one obtains a fixed point H,(l, 2; 0.1) (Ui = 9.125 x 10-2, U2 = 0.47541). With N = 3, a limit cycle is observed and with N = 4 a fixed point is recovered. To characterize the qualitative difference between N = 3 and N = 4 for a fixed Reynolds number, we observe that the condition for the formation of a dissipation range (i.e. characteristic dissipation time (vki)-’ -=K characteristic nonlinear time (k,u,)-') is violated for N = 3. This is an example of the dependence on truncation of the temporal properties of a dynamical system. In the context of the modelization of a turbulent flow the conditions stated above must be fullfilled (see also section 4 in the turbulent MHD case). To study hydrodynamical

C. Gloaguen et al./A

scalar model for MHD rurbulence

turbulence with many excited scales, we have also tried some runs with N = 9, Y= 10e3 and various values of /3/a. Taking initial conditions of the Kolmogorov type ( u,,( t = 0) - k; ‘13) and p/a = 0.5, the system relaxes uniformly towards a fixed point with a Kolmogorov inertial range and a dissipation range for N 2 7 analogous to the case p = 0 seen above. Increasing P/a the same type of fixed point is obtained after a phase of oscillations (mainly in the u2 component) lasting roughly a few large scale eddy turn-over times (taken as the unit of time). With /?/cu = 2, the system undergoes violent oscillations, which are shown on fig. 2 for the component ui. However, when starting from the origin (ui = . . - =u,=O at t=O), a fixed point is recovered at time = 6.

163

We thus are led to conjecture that in the nonmagnetic case, for all values of /3/a! there exists a fixed point which attracts nearly all trajectories when the viscosity is large enough to permit the formation of a dissipation range. Recall that the nonmagnetic equations do not verify the Liouville theorem. Plausibly, the oscillations described above are transients. Occurrence and disparition of limit cycles or invariant torii with a number of modes greater than 3 is not easy to study analytically (nor numerically), because of the lack of information on the width of basins of attraction. 3.2.

Magnetic states

Existence and stability of magnetic fixed points MN when the number of modes or the Reynolds

I 5

10

15

Fig. 2. Time variation of the amplitude of the fundamental mode u, in the nonmagnetic case (V = 10m3; 9 modes). that these oscillations are transient. Compare with evolution of u,(r) in fig. 4 and fig. 8 for example.

It is conjectured

C. Gloaguen et al./A

164

scalar model for MHD

numbers vary are obtained with similar methods. When (Y= 0, b, may always be chosen equal to zero initially since it would otherwise decay exponentially to zero at a rate Ak:. The equation for b, reads db,/dt

= -/3k,u,b,

- 4Ab,.

(3.3)

b cannot change sign; when fi > 0, b, oscillates within approximately exponentially decaying envelopes (remember that the mean value of ui, which is equal to the rate of energy injection must be positive). Notice that when N = 2 and b, = 0, one recovers the hydrodynamic equations changing in eq. (3.3) b, to u2, h to v and fi to -/3; when N = 3 the same property holds changing b, tou,,Xtovandu,to-u,whenb,=b,=u,=O. Corresponding changes naturally work among the coordinates of the fixed points M, and H,, M3 and H3 which can be found in table I. Two types of instability may be distinguished in the magnetic situation: instability of HN with respect to a magnetic perturbation (H + M) and instability of MN. (i) instability H + M. This transition is related to the dynamo effect. It is relatively easy to study since the characteristic equation of the linearized system (about a purely hydrodynamical state HN) is factorized into kinetic and magnetic parts. The magnetic part in the (Y= 0 case of N modes reads (-hk:-a)(-/3k,ii,-hk,2-a) . ..(-pk.u,_,-Akl:-a)=O,

(3.4)

where a represents the growth rate. The critical magnetic Reynolds number ( - X-‘) for dynamo effect will depend upon the kinetic Reynolds number since HN is v dependent. For fi < 0, HI is magnetically unstable and leads to M, when (vh))’ > -2/p (H, does not exist)? For /? > 0, H; turns unstable when 2v2 + 32X2< p and M3 appears. H,* are always magnetically stable when /3 > 0 since Ui and ii, are positive. Determination of the critical magnetic Reynolds number for N > 3

turbulence

would require knowledge of the explicit value of iiN_ I as a function of v. (ii) N modes instability of MN. When M2(0,p; v, X) exists (/I < 0, (vX)-’ > 2/p), it is always stable. Linear instability of M3 occurs when one of the roots of the characteristic equation u3v + a2(5v2 + 64X2- 2/3) +u[v(129P+

4v2 - 2fl) + 64h( /I - 2v2 - 32X2]

+ 64Xv( p - 32X2- 2v2) = 0,

(3.5)

has a positive real part, which defines a critical curve I’(v, h). In the (v2, A2) plane, this curve lies below the straight line (A,) which limits the existence domain of M,(2v2 + 32X2- p < 0). If 5v2 + 64X2 - 2p 2 0, the coefficients of the characteristic equations are all of the same sign so that, using Routh criterion, the critical curve (r) must lie above a second straight line (A2). Further determination of I’( v, A) has been done only numerically. All the results concerning the case N = 3 (Y= 0, /? > 0 are summarized in a diagram (fig. 3) on which the different stable states appear in the (v2, A2) plane (notice that scales are not respected on this figure). There are ten regions, which we describe below: I: only one stable state, for all X; II: above (A,), two stable states H; and H2+; II’: between (A,) and (r), H2* are stable only to kinetic perturbations, M,’ are stable; II”: below (r), the magnetic state is unstable. Numerical simulations show that the system is attracted to H2+; III: above (A,), three stable states (H3* and H,-); III’: between (A,) and r, H3* are stable only to kinetic perturbations; M,* are stable; III”: below (r), the system is attracted to the stable H3* ; IV: above (A,), one stable state, H; ; IV’: between (A,) and (r), H; is stable to kinetic perturbations; M3* are stable; IV”: below (r), no stable state.

C. Gloaguen et al./A scalar mode/for MHD Mndence

i 1

.Ll

113L

(D

v2/p Fig. 3. Stability diagram (not to scale) of the 3 modes magnetic model when a = 0 and /3 = 1 in the plane (v’, A’). Thin straight lines connect stable fixed points. The thick straight lines correspond either to pitchfork bifurcations or to Hopf bifurcations (dotted lines); the nature of the bifurcations on curve (r) is not known. Notice the increase in the number of excited modes with decreasing Y or X and the absence of fixed points at sufficiently high Reynolds numbers.

In region IV”, a chaotic magnetic regime is obtained, which may be explained qualitatively. Recall that b, and b, both decrease towards 0 when /I > 0. After a few dissipation times of the k,-mode, b, is set equal to zero by the numerical algorithm and system (2.11) (with forcing) and k,+ i/k, = 2 reduces to du,/dt

= - 2~; - vu1 + 1,

du,/dt

= -4~:

du,/dt

= 4u,u, - 16vu,,

db,/dt

= -4u,b,

+ 4b: + 2u,u, - 4vu,,

(3.6)

- 16hb,.

The trajectory (fig. 4) oscillates between the two unstable fixed points Hz*(O, /3; v) in the following manner: in the vicinity of H;(u, and b, small, u2 > 0), the system is unstable to us perturbations, and uj exhibits a sudden growth; as a consequence, du,/dt becomes negative, uz changes sign leading to a rapid decrease of uj and to the system

165

being attracted by H;. Near H; , the b, instability sends the system back towards H2+, and so on. The reduced equations (3.6), which do not verify the Liouville theorem, allow to understand the absence of steady state and the qualitative properties of the attractor, somewhat reminiscent of the 3-modes Lorenz model trajectories. We now consider the magnetic states when (Y# 0 and /3 = 0. Coordinates of the fixed points can be obtained explicitly only for N = 2. When N = 3, 6, decreases exponentially; there is a first type of fixed point M,(cu,O; v, X) with b, z 0, b, = b3 = 0. If b2 # 0, in order to simplify the analysis, we take v = A to find the expression of another magnetic fixed point M{(cu, 0; v, h = v). The coordinates of Mz, M3 and M; are given in table I. Analysis of stability yields the following results: (i) dynamo instability: HN + MN. Hz + M,(a,O; v, A) when Xv(v + h)2 < (Ye, H3 --* M3(a, 0; v, A) when Xv(v + X)*(1 + A2/16v2) < (Y*. For larger values of N and fl= 0, the magnetic part of the characteristic equations factorizes in the following way: (2ak,Z2

- hk; - a)(2ak,ii,

.+hk,:-a)=O.

- Ak; - a)

(3.7)

However this is of little use since the general expressions of iii,, . . , ii, are not known. (ii) instability of M,,,. M2* are always stable for all v and X, provided that M2* exist. Stability of M3( a, 0; v, A) has been proved only when h = v. Numerical evidence shows that M3 is always stable and M3) is always unstable. In order to compare the two limit cases (a = 0, /3 # 0) and (a # 0, p = 0) with 3 modes, the results concerning the existence and stability of the hydrodynamic and magnetic fixed points has been summarized on fig. 5 when the kinetic and magnetic Reynolds numbers, taken equal, are increased. The absence of stable states when the magnetic and kinetic Reynolds numbers are high enough and when a = 0, /3 > 0 (region (IV “) of

C. Gloaguen et al./A scalar model for MHD turbulence

b&t)

u,(t)

Fig. 4. The 3 modes magnetic attractor when a = 0, /3 = 1( b, = b, = 0). The time evolution is given up to t = 100(time unit the large scale turn-over time). The two top figures give uz( u,) and u,( u,). The two unstable fixed points Hz*) are shown (with +). The four other figures give the time evolution of the four fields u,, u2. u, and b,.

C. Gloaguen et d/A

H;-C_-_-_ % --_-

_I+,---

II!0

mr)

(III

(1)

scular model for MHD turbulence

partition of the (v, X)-space similar in complexity to the one of fig. 3 (o = 0, p = 1) obtains; this problem can be tackled only through numerical integration. We take 1y= lo-‘, /? = 1 so that some of the fixed points can be calculated analytically as perturbations of known fixed points with (Y= 0, and X = 0.1 in order to explore a region of type IV” of high magnetic Reynolds number for which a dynamo effect is expected. Starting always with the same set of initial conditions (near H3(0, 1; 0.15), with small magnetic excitation), we have done several runs with decreasing Y. Results are summarized in fig. 6 which we now comment. For v 2 0.8 a fixed point H,(cy, fl; v) is obtained, which is a perturbation of H,(O, p; v). For v = 0.3 or 0.19, another fixed point &(a, 8, v) is obtained, which is a perturbation of H,(O, /3, v). At vr = 0.17581, an hydrodynamic limit cycle appears (recall that for (Y= 0, /3 = 1 a similar cycle occurs for v, = l/fim = 0.1705). At another critical viscosity v2 in the interval (0.13,015), magnetic energy ceases to be dissipated and dynamo effect is obtained. No stable magnetic fixed point could be found although after integration was pursued up to 25,000 turn-over times. Remark that in a similar situation, when passing from region III” to region IV” of fig. 3(a = 0, /3 = l), one would also observe a transition from a fixed stable kinetic fixed point to magnetic chaos. Component z+(t) has a continuous noisy spectrum shown in fig. 7a and 7b. For v = 0.1215, bands may be recognized in the Fourier spectrum, corresponding to harmonics of the fundamental frequency f= 0.26. The trajectory in phase space turns from chaotic to

mr”)

.------

--_---

--_ --_

~_--------

(200)'"

117/L)"L

l/V.l/X

1

(b)

Fig. 5. Comparison of fixed points stability with 3 modes when a/3 = 0 and 1,‘~ = l/X. Stable fixed points are connected by continuous lines and unstable ones by interrupted lines.

fig. 4) appears particularly interesting as far as turbulence is concerned, contrary to the case /3 = 0 which leads to fixed points. Notice that vanishing of b, and b2 is a consequence of (Y= 0, and vanishing of b, is obtained when /3 = 0: such peculiar evolutions are due to the lack of nonlinear terms which could feed these modes (cf. section 2). Turning now to the general case o/3 # 0, we can find the coordinates of the magnetic fixed points when N = 2 (table I). M** are always stable fixed points. The case N = 3 could only be studied numerically and the results are given below. 3.3. The three-mode magnetic attractor We now examine whether a chaotic domain without fixed point also exists for N = 3 when ap # 0. We expect that for every value of p/o, a

blagnatic

Magnetic

Chaos

I1

I

0.001

It 0.09

Limit cycles

II 0.1155

I "4

0.115a

1 1 0.120

v3

llagnetic

Kinetic

Chaos

LiMit.cycle

I

I1

0.124

0.130

Fig. 6. Stability diagram (not to scale) for the 3 mode magnetic caSe a = 0, B = 1 given in fig. 3 (where a constant X corresponds

161

v2

II

II

0.150

0.176

Kinetic Fixed Points Hi

3

"2

I

I

VI

0.8

I 2.5

I system (with A = 0.1, OL= lO_‘, to a horizontal line).

p = 1). Compare

with the simpler

168

C. Gloaguen et al./ A scalar model for MHD turbulence

Y= 0.12145

v=O116

v=o.1155 e

Fig. 7. Comparison of ut spectra when viscosity changes in the 3 mode magnetic system (X = 0.1, a = 10e2, /3 = 1). As viscosity is decreased, one can observe a transition from magnetic chaos (figs. 7a and 7b) to magnetic limit cycles (figs. 7c. 7d, 7e) and to magnetic chaos again (fig. 7f). Notice the presence of bands with harmonics of frequency f when Y= 0.1215 (f = 0.26) and subharmonics of f when Y= 0.12 (f/4) and Y= 0.116 (f/16).

C. Gloaguen et al./A

scalar model for MHD turbulence

periodic when v is below another critical viscosity vj, with V~ close to 0.1214. The spectral lines correspond to harmonics of f/4 (see fig. 7c, for v = 0.12). Decreasing v, a succession of period doubling can be observed, Fig. 7d shows for example the Fourier spectrum of ui(t) for v = 0.116, were one can easily find the frequency f/16. There are not enough period doubling bifurcations to check an eventual Feigenbaum sequence. Notice however that the ratio of successive subharmonic intensities does not follow a power law. When v decreases below v, (where v, lies in the interval [0.1155, 0.1158]), noisy spectra develop again. The limit cycle disappears and the lines corresponding to it are somewhat broadened (fig. 7e); when v = 0.11 and smaller (fig. 7f) no spectral lines can be observed anymore in the spectra. Do the features which have been described above survive when N is increased and /?/a vary? We have verified that the two main magnetic regimes observed with N = 3 (periodic and chaotic) also remain present with N = 4 (X = 0.1). For v = 0.117, a periodic attractor with the same visual appearance obtains; for v = 0.11, a chaotic trajectory occurs again. Amplitudes of uq and b4 are below lo-’ so that the conditions (l/r&i) < (l/k4xq) (with x = u or b) for the formation of a dissipation range are fullfilled. Thus, we are confident that the formation of the observed chaotic magnetic regime at these kinetic and magnetic Reynolds numbers is a stable phenomenon with respect to increasing the number N of degrees of freedom. We shall investigate the domain of higher Reynolds number in the next section.

4. 4.1.

Turbulence in the &alar model

Scope

This section deals with the modelization of fully developed MHD turbulence by the scalar equations (2.11). One may then ask whether the threemode system is adequate to describe as many as

169

possible of the main properties of a turbulent flow, or whether we have to go to higher resolutions. It is already known that several properties of dynamical systems, in particular those concerning the relaminarisation windows, do not survive tnmcation (Mashke and Saramito [16]). The reason is that, as the parameter (Reynolds or Rayleigh number) is increased, more small scales are created by the nonlinear interactions which are not taken into account in the truncated model. What Reynolds number can then be associated with the scalar model, for a given truncation at N modes? A little reminder of the Kolmogorov [17] phenomenology which will be heretofore abbreviated as K41, is in order to answer this question. In the velocity spectrum of a turbulent flow, three main regimes can be identified. The large scales (small wavenumbers) represent the energy range, in which most of the energy is concentrated. Through the nonlinear interactions, energy is transferred to smaller scales, in the inertial range. Finally in the dissipation range, dissipation sets in. In the inertial range, the energy spectrum follows a power-law, such as the one proposed initially by Kolmogorov, i.e. E(k) - ke5j3. It obtains by mere dimensional analysis by stating that E(k) depends only in the inertial range on wavenumber and injection rate of energy. This law is verified within experimental bounds in the laboratory, although slight corrections to the 5/3 law may occur which takes into account the intermittency of the small scales. We can now compute easily at what scale dissipation sets in for a given viscosity by assuming that the nonlinear (t,, = f/V,) and dissipative (to = ,*/v) times are equal for 1= I,. One easily finds that It, - v - 3/4. When the minimum scale I,, in the computation is larger than the dissipation scale 1, (for a given viscosity), dissipation in the numerical simulation of the dynamical system will be insufficient. When the inequality I,, > I, is only slightly violated, a tail will appear in the spectrum, indicative of truncation errors. If the Reynolds number were to be augmented much beyond this truncation-dependent value, the numerical system would probably evolve towards

170

C. Gloaguen et al./A

scalar model for MHD turbulence

a state of equipartition between the modes (recall that the property of detailed conservation holds for the model as well as for the primitive MHD equations). This final equilibrium, however, does not preclude temporal fluctuations of a chaotic type but is not representative of the turbulence we wish to describe. The scaling relation between the number of modes in the calculation and the Reynolds number obtains to within a numerical constant which must be determined empircally. This constant is adjusted by ensuring that the energy spectrum does not level off at the highest wavenumbers. We thus found that, with nine wavenumbers only, a Reynolds number RV = 1000 can be reached. It is the exponential discretisation in the model that permits treatment of such a high Reynolds with so few modes. The same advantage holds for twopoint closures but their integro-differential structure renders their numerical integration trickier. There lies the main difference with an actual turbulent flow, for which the density of wavenumbers in the small scales varies as k2 in dimension three. 4.2. The 9-mode system In the preceding section, we examined the various bifurcations taking place in the three-mode system. We shall now augment the number of modes up to nine, covering a wavenumber range [1,512] which leaves room for a reasonable inertial range. Computational limitations arise, though, in particular since it is desirable to treat accurately the small scales. An explicit temporal fourth order scheme is chosen. Taking the characteristic time of the large-scale (low mode) equal to unity, a computation up to a large scale turn-over time (t = l), for the nine-mode system takes 4 seconds of CPU on a Vax 11-780. Numerical experiments were thus performed with the following set of parameters: (Y= 10e2, p = 1, v =X = 1.2 x 10m3, with a time step of the order of 10 - 3. The overall temporal aspect of the modes is still chaotic. To see this, we plot in fig. 8

the time evolution of u and b modes once stationary average spectra have been obtained. Looking now at the modal variation of energy, we can define an average energy in shell n as fl

E,; = L 1 Zx,2dt,

t 0

(4.1)

with x = u or b and n E [l, N]. The averages converge after a time of the order of 100. The resulting spectra, plotted in fig. 8 for the kinetic energy (solid line) and the magnetic energy (dashed line) follow approximately a - 5/3 Kolmogorov law. New questions then arise, related to intermittency. Note that the derivation of K41 does not make use of the precise form of the nonlinear interactions of the Navier-Stokes equations and can thus be also found for approximations to these equations, such as two-point closures, and the scalar model. The K41 phenomenology also predicts power-law variation for the higher moments of the velocity field. For example, the flatness factor is defined as

F,x=

(x.$0:)‘,

(4.2)

where x = u or b. For the fourth-order moment, K41 predicts a -4/3 exponent whereas experimental data yields -1.22 (Anselmet et al. [18]). The discrepancy between experiment and K41 seems to augment for higher order moments, although these determinations become increasingly difficult, because they weight heavily on the smallest scales of the signal, in which noise becomes progressively predominant. Departure from the K41 law is related to intermittency, that is to the fact that at high Reynolds number, the energy associated with the small scales in a turbulent flow is distributed in an irregular fashion in space, and the volume to which this energy is confined diminishes together with the size of the structures. This intermittency could be due for example to the stretching of vortex lines (Kraichnan [19]). Twopoint closures show, basically by construction, no

C. Gloaguen et a/./A scalar modelfor MHD lurbulence

171

I ,t

\“.

\’

Fig. 8. 9 modes MHD turbulence (a = lo-*, p = 1, Y = A = 1.2 X lo-‘) between time T- 1024 and T- 1049 (100 samples, about 25 turn-over times). The time evolution of kinetic and magnetic modes are quite similar, except in the first two (compare u1 and b,). Notice the intermittency of 7th and 8th modes, which are in the dissipation range, as can be verified from the time averaged spectra of magnetic and kinetic energies (upper right diagram).

C. Glooguen ei d/A

112

sculur

departure from the Kolmogorov law and direct numerical simulations do not give enough accuracy for determining the spectral index of the inertial range. Several models have been constructed often in an ad-hoc fashion, to describe the possible effect of intermittency on the inertial range (Kolmogorov [20]; Kraichnan [21]; Frisch et al. [22]; Frisch [23]). In these models, the probabilistic element is introduced arbitrarily. Other models have displayed an intrinsic intermittency, such as the one proposed by Kerr and Siggia [24] for the Burgers equation. In their work, the velocity modes are taken complex, and the dissipation is artificially constrained to the last wavenumber. It seems that the observed intermittency is due to this peculiar dissipation function, since it disappears when a Laplacian viscous term is used (Lee [25]). However, another difference with our model is worth mentioning, namely the fact that the variables are interpreted as r.m.s. energies, and thus constrained to remain positive, whereas we interpret them as field-components which can fluctuate freely around zero. In the MHD scalar model (2.11), we do observe temporal intermittency as shown in fig. 8 in which the 7th and 8th magnetic modes are represented, after they have reached a statistically steady state. A time-span of three hundred large scale (mode 1) eddy turn-over time is shown, during which the bursts are more visible as higher modes are considered. This effect is particularly visible for the mode 8 for which the signal consists of long quiescent periods interrupted by strong sudden bursts occur-

modelfor MHD turbulence

ring at random times. Such intermittency is not due to truncature effects, since we have checked the existence of a dissipation zone of the spectrum. Table II gives the growth of the flatness factor (eq. (4.2)) with wavenumber at t = 1,6OO(v= 3 X 10e4, X = 3.2 x 10P4). A higher resolution computation on a Crayl may help to study the variation of the flatness factor at higher Reynolds numbers. One may ask whether the intermittency found in the scalar system is “robust”: does it persist when modifications to the functional form of the equations (2.11) are performed? A change that comes to mind which is commonly used in direct numerical simulations would be to extend the resolution by confining dissipation to a narrow range of wavenumbers: one may try a higher than unity power of the Laplacian (vk2 changed to vkZn with (Y> 1). This modified dissipativity may easily be introduced in the scalar model, and it is not clear whether the turbulent properties (chaos and intermittency) will persist. It is possible that such a turbulence is associated with a viscosity-dependent destabilization of the Kolmogorov fixed point (Frisch, private communication). 4.3.

Magnetohydrodynamic properties

We give here a basic summary of three of the main properties of MHD turbulence that are shared by the scalar model. The first feature concerns the dynamo problem. It is known (see for example, Moffatt [l], for review) that above a critical magnetic Reynolds

Table II Variation of the mean kinetic and magnetic components and of the flatness factors with wavenumber (9 modes, Y= 3 X 10e4X = 3.2 X 10e4, OL = 10-s, B = 1, averages taken over 1600 time units). Notice the increase of the flatness factors with n, which characterizes intermittency.

k,

1 (n=l)

2 (n=2)

4 (n=3)

8 (n=4)

16 (n=5)

32 (n=6)

64 (n=7)

128 (n=8)

(u&m

0.635

0.463

0.409

0.305

0.212

0.098

0.023

0.008

(b,)/m

0.996

0.598

0.458

0.327

0.216

0.083

0.010

0.002

2.72 1.7

2.45 4.08

3.25 3.69

3.51 3.56

4.48 4.21

6.09 5.52

8.74 8.27

14.88 14.02

256 (n=9) 0.005 0.002 58.05 58.23

C. Gloaguen et al. /A scalar model for MHD turbulence

number Ry and for a high enough kinetic Reynolds number, a seed magnetic field is enhanced by line stretching due to velocity gradients. Whether this effect will persist for long times depends on several factors, among which the dimensionality of the flow, the presence of helicity (velocity-vorticity correlations) in the small scales, and on the respective values of the kinetic and magnetic Reynolds numbers R" and R". In twopoint closures (Leorat et al. [26]) and in direct numerical simulations of the MHD equations (Meneguzzi et al. [6]), it is found that RF is of the order of a few tens in the nonhelical case. The kinetic helicity HV = (u w), where o = curl V is the vorticity, cannot be included property in our model which takes into account on by nearestneighbour interactions with an exponential discretization. Plotting the magnetic energy in the steady state for various Reynolds numbers gives a rough estimate of the critical magnetic Reynolds number above which dynamo action can take place. This evaluation of RF is in agreement with the aforementioned results but vary with P/a. We also find, in accordance with closures, that the maximum of the magnetic energy spectrum is at a wavenumber somewhat larger than the injection wavenumber (see fig. 8). A related problem concerns the lack of exact equipartition of small scale kinetic and magnetic energy. The spectra displayed in fig. 8 show that there is a slight relative enhancement of magnetic excitation for the high modes. Equipartition is expected in an MHD flow because of the action of AlfvCn waves that propagate along the lines of force of the large scale magnetic field (either external or the mean turbulent field). However, as was mentioned earlier, the specific effect of Alfven waves is not included explicitely in our model (this would double the number of modes at a given Reynolds number, by making the field variables complex, see appendix A). The main expected consequence of such an effect is to change the Kolmogorov spectrum into the one predicated by Kraichnan [2]: a shallower - 3/2 spectrum indicating less efficient nonlinear transfer due to the l

173

incoherence of interactions introduced by the waves. However, hopefully, this modification of the scalar model would not alter the other structural properties (namely, chaos, intermittency, dynamo and the growth of correlation to be mentioned below). Alfven waves are due to a zero-order effect (see for example Pouquet et al. [7]) which include only part of the nonlinear transfer. Indeed, in two-point closures, one observes at high Reynolds number a slight excess of magnetic energy (by roughly a factor two in the inertial range) over its kinetic counterpart. This excess is also seen in direct numerical simulations (Meneguzzi et al. [6]) and, more surprisingly, in the observed spectra in the solar wind (Mathaeus and Goldstein [27]). The fact that this property is also found in the scalar model may indicate that the local interactions (between scales of comparable sizes) play an important role in enforcing the small scale enhancement of the magnetic field. Finally, on the scalar model, one can also exhibit the phenomenon of the temporal growth of the correlation coefficient between the velocity and the magnetic field. In-situ satellite date in the solar wind indicate that the fluctuating velocity and magnetic components, as well as the density are strongly correlated for long periods of times (over one day). The solar wind then appears as consisting of a superposition of Alfven waves. However, Dobrowolny et al. [3] raised the possibility, on a phenomenological basis, that this correlation might be due to nonlinear interactions, and numerous studies tend to support their conjecture (Grappin et al. [4, 51; Pouquet et al. [28]; Leorat et al. [29], Mathaeus et al. [30]). Define the correlation coefficient as (4.3)

i.e. as the ratio of the two inviscid invariants (correlation and total energy). We found that in the scalar model, p augments with time, in unforced computation. This result holds for different initial values of p and the increase in p occurs by

C. Gloaguen et al./

174

A scalar

bursts, associated with changes of sign of the kinetic components. The typical time scale of the growth of the correlation coefficient does not appear to be independent of initial and running conditions. As in the two-point closure calculations of Grappin et al. [4, 51, the spectrum of the correlation does not keep a constant sign, but fluctuates around zero. The negative (say) small scale correlation then acts in fact as a source of correlation of the opposite sign in the large scales. Note that when p approaches its maximum value, the nonlinear terms in the MHD equations become weaker and weaker, and the turbulence evolves at high Reynolds number at the slow dissipative characteristic time.

5.

Summary and conclusion

An essential property of the scalar model is that, contrary to Navier - Stokes (or MHD) truncations of the Lorenz [13] type, it can be easily extended to an arbitrarily high number of modes, i.e. to high Reynolds numberst. The model we propose here for MHD shares with a turbulent flow its scaling properties, hence its name. We have shown that it allows a detailed numerical study at high kinetic and magnetic Reynolds numbers, thanks to an exponential discretisation in wavenumber. This model, contrary to its hydrodynamical counterpart, displays chaos and intermittency in the small scales; as suggested before, these properties may be eventually recovered in the nonmagnetic situation (see further below). Notice that chaos has also been observed in the n-disk dynamo (see Miura et al. [31] and references therein), but existence of scaling properties in such a model have not been investigated. A more detailed study of the chaotic properties of the scalar model is in progress. In particular, a quantitative characterization of the chaotic structure of the underlying attractor can be obtained by tacking to the system a fractal dimension. Several ?Note however that the numerical time step scales in the Kolmogorov case as d/*, i.e. as 2-2Nfi for N octaves.

model

forMHD

turbulence

algorithms have been introduced to compute the dimension of the set on which the representative point of the dynamical system settles in phase space. They are reviewed for example in Farmer, Ott and Yorke [32]. Computations become rapidly quite heavy as soon as the number of degrees of freedom of the system goes beyond the usual three. They have already been performed on the scalar model (Pouquet et al. [33]) for a small set of parameters, going up to 8 modes (16 degree of freedom), using either the point-wise or the correlation algorithm (Grassberger and Procaccia [34]). An extension of this work, computing the dimension of the attractor by the formula of Kaplan and Yorke [35] which makes use of the Lyapounov exponents, is now in progress. This will allow us to give an estimate of the variation of dimension of the dynamic system with Reynolds number, and to see in particular whether or not there is saturation of dimension. Several extensions of the model are envisageable. The simplest one is to augment the resolution. If the three-mode system can be simulated on a microcomputer, the twenty-mode system would need a Cray, or an equivalent machine. Indeed, in the latter case, the ratio of minimum to maximum wavenumber is close to 106. The eddy turn over time, for a Kolmogorov range, scales as t,, k-‘13. For twenty modes, the ratio in t,, for the extrema in wavenumbers would be of the order of 104. This sets a severe limitation on the computation that can be done, treating all time scales explicitely to ensure that no average is performed on the small scales (which are thought to be responsible for the intermittency). A study of the effect of the “connectance” of the dynamical system on the appearance and on the characteristic properties of chaos would also be of interest. The connectance of a system measures, in a normalized way, how a mode is related (through the evolution equation of the dynamical system) to the other modes (the normalization is such that the connectance C is unity when a mode relates to all the others). It is believed that the higher the connectance (i.e. the more the modes interact between

115

C. Gloaguen et al./A scalar model for MHD turbulence

themselves), the more ergodic the system is. The study of Froeschle [36] suggests that there is a critical connectance below which an important part of phase space is occupied by invariant torii (for an integrable Hamiltonian system) and above which it is no more the case, and the system becomes ergodic. The connectance of the scalar model is low since it only connects to nearest neighbours. Note that by shifting from the purely hydrodynamical scalar equations to the MHD ones, we double the connectance of this system, since a given variable is related to three (including itself) in the former case, and to six in the latter (excluding border-line effects which are marginal). It is not clear whether the chaos appears because of the doubling of C, or rather because the positivity constraint on the variables has been released. This can be checked by doing either one of two things. One is to include interactions with the next-to-nearest neighbours (modes n rt 2). The influence of helicity can be studied in that framework, helicity being defined as either the correlation between velocity and vorticity or magnetic field and magnetic potential. An alteration of the magnetic energy spectrum, which may undergo an inverse cascade to the low wavenumbers may then be observed, as suggested by analytical and phenomenological studies of helical MHD (Pouquet et al. ]71). The connectance can also be augmented by rendering the field variables complex. The complex equations of the scalar model with nearest neighbour interactions are given in appendix A. We conjecture that chaos will appear in the complex hydrodynamical model when both coupling constants of (Yand @type are considered. We have noticed in section 2 that with complex variables, we can explicitly introduce the AlfvQ effect, by including the linearized Alfven waves propagating along a uniform magnetic field. The energy spectra may in that case display a k-3/2 instead of a kvsj3 range. It would be of interest to see in what respect this Alfvtn effect would affect the result of slight excess of magnetic energy in the small scales.

Finally we may suggest to use the scalar model as a mean to achieve a subgridscale modelization of turbulence (Chollet J.P. and Lesieur [37]. To be more specific, suppose that a direct numerical simulation is performed by a spectral method with Fourier modes up to wave number K. The space of wave vectors k may be divided into three domains: a) ]k 1 _ 1): this corresponds to the large scales of the flow, which are the most important for the main transport properties. These scales are treated using the Navier-Stokes or MHD equations. b) K/h I Ik( I K: in this transition shell of intermediate scales, the flow is coupled to a scalar model whose fundamental mode is k, = K/a. Let U, and u, denote respectively the Fourier and scalar components. One can write in this shell

aU,Jat = WJ,/dt),,,-- w,@Jd,,

c WJ&

du,/dt = @u,/dt),, + Y(

(5.1) (5.2)

where indices a and b recall the wave vectors domains, SM stands for scalar model and y is a coupling constant. These equations conserve total energy by construction. c) 1k 1 > K: these scales are smaller than the grid and are described by the scalar model only, which acts essentially as an energy drain. When the scalar model is integrated, the U, may be assumed constant (constant acceleration). Contrary to other parametrizations of turbulence, energy is not dissipated in the smallest explicitely treated scales of the spectral simulation, which lie in the inertial range of the scalar model. This parametrization depends on two adjustable constants: the cut-off wavenumber K and the coupling constant y. In the coupling shell, some isotropy must verified for the parametrization proposed here to be meaningful. It is also clear that this parametrization will generate errors which will cascade towards large scales (predicibility problem, cf. Leith and

model for MHD turbulence

176

Kraichnan [38]). In this respect, notice however that although a butterfly will undoubtly affect the atmospheric flow, experience shows that the measure of its influence on, say, the Trade Winds, appears as negligible. Inclusion of the scalar model in a simulation, may also lead to negligible perturbations of the largest scales. Detailed comparisons between different parameterizations with help of numerical experiments would be justified.

and z + fields characterize the rows and columns:

q-F-

3 0

0

0

0

(‘4.1) Acknowledgements

We are thankful to A. Ameodo, U. Frisch, B. Legras and P. Manneville for useful discussions.

Appendix A

The scalar model equations with complex fields The coupling constant A,J of the nonlinear terms (cf. eq. (2.6)) are determined below when the kinetic and magnetic components of the scalar model are complex quantities. The evolution equation for zf(t) involves three modes (n - 1, n and n + l), which appear only in z+z- products. Taking complex conjugate fields into account (z*) there are a priori 6 X 6 complex coupling constants A,, which are such that the + energies are conserved by the nonlinear terms. The nonlinear contributions to the equation for zz (t) reads

k,-‘(dz;,‘dt),, + A,zz,,z,‘-l

The coefficients Ai, may be grouped by blocks as follows: a) Only nearest neighbours interactions are allowed, thus A,,=A,,=A,,=A,,=A,,=A,,=A,,=A,,=O. b) We discard nonlinear interaction involving only a single wavenumber (energy transfer to nearest neighbours is null in the case). We thus have set A,,, A34, A,,, and A, equal to zero. c) Writing the energy conservation condition

= A,,z,,z; xz,+*dz;/dt ”

+ . +. . .

.

The discussion is rendered caster if the A,J coefficients are defined using a table where the z-

+ z: dz,+*/dt = 0,

it is easily seen that the following relations must hold:

C. Gloaguen ei al./A

k,A,, k,A,,

+ k,_,A& = 0 + kn-,A3,c, = 0

k,A,, k,A,,

+ k,_,A$ + k,_,A,

k,A,, k,A,,

+ k,-,A& = 0 + kn-1AS6 = 0

k A,,+k k:A,,

this paper: (dzn/dt+)N, 64.2)

=0 = 0I

P=A,, 64.3)

+ k n-1A66 = 0 I

A,,+A&=O A,,=0

(A.4)

A,,=0

1

A,,+A;3=0 A,,=0 A,,=0

.

(A.5) i

Notice that independence on n of (A.2) and (A.3) imposes that the wavenumbers vary geometrically with n, whereas (A.4) and (A.5) are compatible with any discretization. The complete nonlinear terms in the evolution equation for 2, finally read [dzn+/dt],,

= A,,k,z,,z,+_,

- A;lkn+lz;*z:+l

+Anknz;-1zn+-*1-

Ankn+~z,z:+~

+A,,k,z,=*,z,+_,

- A:,k,+,z,z;+,

+A,,k,z,=*,z,+_*,

- A22kn+1z,*zn++*l

+A,,k,z,z,+_,

- AT,k,+,z,;*,z,=,

+Auknznzn+-*I - A,,k,+,z,,z,=*, +Aaknz;*z,+-1 +A,,k,z,

-Ai%k,+,z,,z,=,

*z;-*~ - A,,k,+,z,;:z,=*,

+A53knz;;*1zn+ - A&knzn;*lzn+ +A,,k,z,,z;

+P(k .z,z,‘-

= cx(k,z,,z,+_, 1 - kn+A+d+d,

where 1~=A,, +&+A21

n_ 1 A*55 =0

177

scalar model for MHD turbulence

- A;,k,z,S*,z,+. 64.6)

A Liouville theorem is satisfied only when A,, and A,, are either purely imaginary or zero. In the simplified case where the coefficient Ajj and the fields are real, the last two lines vanish, and we recover the equations which have been studied in

- k.+lz,z,=& (A.71

+&,

+A,, +A41 +&.

The complex nonmagnetic case is obtained by identifying z: and z; to 2,. In this case, again Liouville theorem is satisfied only when A12, A,,, A,,, A,,, A,, and A,, are either purely imaginary or zero. The choice A,, = -i with other constants equal to zero, yields the non-linear terms of the scalar equation studied by Kerr and Siggia [24] and Lee 1251, (dz,/dtJNL = -i(k,,zz-i + kn-+iQn+i). These authors differ by the choice of the linear dissipation terms. Lee has shown that the time fluctuations observed by Kerr and Siggia disappear when dissipation is modelized with a Laplacian, as in the primitive equations (1.2). The respective contributions of (Yand j3 terms may be pictured in wave vector space as follows. Recall that nonlinear interactions in the primitive equations (written in Fourier space) occur only between wave vectors forming a triangle. In the scalar model the infinite number of interacting triangles has been reduced to four types; two corresponding to (Yterms and two to /i terms. In fig. 9 these triangles are OA,K, OA,K, OB,K, OB,K and OB,K and their symetrics with respect to the media&ix of’ K'K: they have as common side the wavevectors OK or (OK’) such that ]OK] = IOK’] = k,. Consider now eq. (A.7) (or eq. (A.6)): -interacting wavevectors with summits A, or A’, are modelized by az;_iz,‘_i terms; - interacting wavevectors with summits AZ or A; are modelized by ~z;.z~+i terms; -interacting wavevectors with summits B,, B, or Bi, B; are modelized by /3z;z,‘_i terms; -interacting wavevectors with summits B, or B; are modelized by Pz;+ izT+ i) terms.

C. Gloaguen et al./A scalarmodelfor MHD turbulence

178 OA, = kn_l

where

OK = k,

0% =kn+l

k:, =

fi

kJ

and

k; = k,fi.

Since

r(k,,,)=z[h*‘k,] =~(k,,)+~k,,[~z/~k]k,,+6’(8~),

03.2)

eqs. (2.10) become in the limit 6 + 0 i9z+(k)/&+vpk2z+(k)+v,k2z-(k) =

K'

A;

A!,

0

A,

K

In incompressible turbulence, contributions corresponding to the flat triangles OA,K and OA,K vanish. They occur in compressible situations such as in the Burgers equation. It is not surprising that Lee [25] obtained only this type of contribution since he was modelizing this last equation. 1: would be interesting to study the nonmagnetic scalar model in this complex case including coupling constants of the “/I ” type (A 31rA r2, A 41, A,, cf. eq. (A.6)). We have seen in this paper that they are necessary for the scalar model to exhibit turbulent properties with real components. Recall also that complex equations are interesting in the magnetic case because AlfvCn effect is easily included (section 2). Appendix B

When the distribution of scalar modes is continuous, the corresponding equations are obtained by taking the limit h = k,+i/k, + 1 (cf. Bell and Nelkin [15]). Let h = es and suppose 6 + 0. To obtain the correspondence between the discrete field z, and the continuous field z(k), evaluate the energy in the n th mode 22 =

k’z2(k)dk=z2(k,)knS+0(62),

/ k:,

(B.1)

2kz-i?z+/ak

+kz+az-/ak],

A2

Fig. 9. Representation of the a- and p-nonlinear interactions in wave vectors space. The three circles correspond to shells with wave numbers k,_,, k, and k,+,(k,/k,_, = k,+,/k, = 2). The dots are the sum_mits of triangles which can be formed with OK or OK’( 1OK 1 = 1O& 1 = k,) and the other sides of length km_,, k, or k,+,. The flat triangles (summits A,, A,, Ai and A’,) correspond to a-terms whereas the others are of @-type.

-yk3”[(5/2)z+z-+

03.3)

where y= limit of (a+ p)S312 when 6 -+O. (If a + p = 0, a development to second order in S is necessary). Remark that z(k) has not the dimension of a Fourier mode (z 2( k) is a spectral energy density). Conservation of f energies in the non dissipative case is more easily seen on the following resulting equation: a(~+~/2)/& =

-b’(k

+

vpk2z+2 + v,k2z+z-

5’2z+2z-)/ak.

(B.4)

Notice that the direction of the energy flux due to the nonlinear interactions depends upon the sign of the z * fields (we take y = 1 by resealing the fields: z + -+ z */y ). If the initial functions z+(k,O) and z-( k,O) are positive, equations (B.4) can be written in terms of the spectral energy densities: E*(k,t)=z*‘(k,t)/2, b’E+(k, =

t)/&

+ 2vpk2E++

-23/2&(k5/2E+F).

2v,,,k2E-

(B.5)

Eqs. (B.5) is in fact a closure equation, which reduce to the Kovaznay [39] equation in the hydrodynamic case (E+ = E-). The nonlinear transfer terms derive from energy fluxes rr *(k): T’(k,r)=

-

a

Z,*(k,r),

~*(k,t)=23”2k5/2E+fi,

03.6)

and the total energy transfered to wavenumbers larger than K is mT+(k,r)dk=Ir*(K,t)

/K

C. Gloaguen et aL/A

scalar model for MHD turbulence

assuming that B *(K, t) --, 0 when K tends to infinity. Thus a direct cascade of energy (towards large wavenumbers) obtains if z- and z+ remains positive. When the flux 72* are independent of k, one recovers the Kolmogorov power law spectrum E f(k)

= C +e+-5/3,

?r+= 2VC *e,

(B-7)

179

conservation is not possible in such simple equations without violating galilean invariance. On the other hand, equation (B.ll) are written in the space of wavenumbers and conserve both energy and correlation, proportional respectively to

Jdk (ii2+h2+h2)k-5’3 and

where the C * are Kolmogorov constants. Eqs. (B.3) can also be written adimensionnalized field variables J, * z*(k)

using I

= $*(k)&“3k-s’6,

such that #* inertial range,

03.8)

are constant

in the Kolmogorov

= -El/3k5/3[2q-a++/ak

+ ++a+-/ak]. (B-9)

ic = (4’+

kinetic and magnetic fields

G-)/2,

(B.lO)

&=($+-J/-)/2, verify afi/at + vk2ii = -3E1/3k5/3[~afi/ak a8/a+hk2i,=

Appendix C

The inviscid limit of the scalar model equations

a++/at + vpk2J,++ vmk2#-

The~“compensated”

dkfiik-5/3.

- h aijak],

- C3k5/3[ ii ah/ak - i, aiijak]. (B.11)

These equations remind the equations written by Thomas [40, 411 (see also Sulem et al. [42]) in the framework of a one-dimensional model for MHD turbulence:

Quantitative results are difficult to obtain analytically due to the nonlinear terms of the equations: at a given time, a mode n interacts with its nearest neighbours (n - 1 and n + 1). We may tentatively explain some qualitative properties of the model by imposing restrictions upon the nonlinear interactions of the scalar model as follows: we divide time in successive cycles of length that are small compared to any dynamical time; the mode (n) interacts only with (n - 1) during the first half of the cycle. Thus at a given time the whole system is an ensemble of decoupled pairs, each interacting non-linearly; the conservation properties allow to reduce the number of degrees of freedom to only two in this “dichotomous scalar model” (DSM). Consider for example the pair of modes (n, n + 1) such that

z,‘=z*cos(e*-~)~ h/at

ab/dt

-

V a2/aX2

- h a2b/a2X

=

-

[U

h/ax

-

3b ab/aX],

Zfn+l

=Z*,.sin(8*-G),

Cl)

= - [U i?b/ax - b &/ax].

(B.12) Thomas’s equations are written in physical space and conserve energy; it appears that correlation

where 8 is a constant defined below and 2 * are positive. During the first half of the cycle, the f energies of the pair (n, n + l), ((~,9~)+(z,f+r)~)/2, are

C. Gloaguen et ul./A scular modelfor MHD turbulence

180

conserved and the amplitudes 2 * remain constant. The evolution of the pair is thus given by the two equations: dB+/dr = yk,+,Z-

sine-,

= yk,+,Z+

sine+,

de-/dt

(C.2)

d*(f3,+ +8;)/dt2=y2G2sin(8++e-). = y2G2sin(0+-

e-1,

(C.3)

where a2 = k,‘+,Z-Z+. Let us first discuss the nonmagnetic case. There is only one dynamical variable 8 = 8+= r3-(Z = Z-= Z’) which verifies the integrable equation (cf. eq. (C.2)) dB/dt = yk,+,ZsinB.

cc.41

There are two fixed points 8 = 0 (unstable) and 0 = r (stable). During the second half of the cycle, the mode (n) interacts with mode (n - 1) and the mode (n + 1) interacts with mode (n + 2), thus when n and n + 1 interact again the initial phase 0 and amplitude Z differ from their formal final values. However, the point 6 = 7~is always attracting and corresponds to a ratio u,+t/u, = -a//I which is independent of the pair considered. Thus the fixed point such that u,,+r/u,, = -a/P eventually attracts all trajectories in the D.S.M., in the nonmagnetic situation. One can rewrite eqs. (2.11) in order to show explicitly that this fixed point exits also in the scalar model du,/dt

= knz+(cw,_l

-k,+1%+1

u,=o, ak,ii_,

where (Y=ysinJ/, p=ycos#(y>O). Combining the + variables, two decoupled non-linear pendulum equations obtain

d*(8, - e,)/dt*

ple, when aP > 0, (C.6) - ~k,+l~,z+,

= 0,

(where U denotes a fixed point). Numerical experiments show that when a = 0 (resp. p = 0) the first type of fixed point is reached: all the energy fills the lowest n = 1, (resp. highest n = N)) mode. When a//I = lo-* or 1, the second type of fixed point (C.6) has been obtained, with the signs of U,,, t/U,_ 1 varying with initial condition: the phase space is divided in many bassins of attraction. It is clear that the DSM approach is not relevant in the latter case (eq. (C.6)) in which the two d modes adjacent to n conspire to drain it off. Consider now the application of the DSM to the magnetic case. The trajectories in the (e’, e-) plane, corresponding to eq. (C.2) are sketched on fig. 10. Notice that there are no more stable fixed points. When 8-Z 8+, the trajectories are periodic during the first half of the cycles; in the limit of small amplitudes, the pulsation is close to yk,+,dm and different trajectories in the (e’, 8-) plane are followed for each first half of the cycles. In the average, equipartition of (z,‘)~

+ Pu,)

(au, + Ij%+1).

cc.51

Other trivial fixed points can be found. For exam-

Fig. 10. Phase trajectories in the dichotomous scalar model. Such trajectories are followed only during a half cycle. After another intermediate half cycle, a new trajectory begins from an initial point which is not the final one of the former half cycle.

C. Glwguen

5

10

15

et al./A

scalar modelfor

20

Fig. 11. Time evolution of the inviscid magnetic scalar model (4 modes, o = 1, fi = 0). Only the 2, g u, + h, are represented during 20 time units. Notice the scaling of the characteristic evolution time scales with n. Taking time averages, equipartition of energy is verified. Compare with a turbulent case (fig. 8, where dissipation and forcing are present. Recall also that in the inviscid nonmagnetic case, the system is attracted by a fixed point.

and (z;)’ is expected, and also equipartition of (z’)’ and (z,‘,,)’ (cf. (C.l)). A numerical integration of the scalar model gives the following results: each mode keeps on fluctuating with a characteristic time of order (k,u,)-’ or (k,b,)-’ (fig. ll), equipartition of energy between kinetic and magnetic excitations, and between the modes is verified within 5% after t = 1000 (taking the characteristic time of the lowest mode as unit).

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[6] M. Meneguzzi, U. Frisch and A. Pouquet, Phys. Rev. Lett. 47, (1981) 1060. [7] A. Pouqust, U. Frisch and J. Ltorat, J. Fluid Mech. 77 (1976) 321. [8] V.N. Desnyansky and E.A. Novikov, Prikl. Mat. Mech. 38 (1974) 507. 191 V.N. Desnyansky and E.A. Novikov, Izvestya Atm. and Oceanic Phys. 10 (1974) 127. [lo] R.H. Kraichnan, J. Fluid Mech. 47 (1971) 525. [ll] T.L. Bell and M. Nelkin Physics of Fluids 20 (1977) 345. [12] A.M. Obukhov, Izv. Akad. N. SSSR, Fig. Atm. Okeana 7 (1971) 471. [13] E.N. Lorenz, “Deterministic Non-Periodic Flow”, Journal of the Atmospheric Sciences 20 (1963) 130. [14] C. Gloaguen, “Un modele scalaire de la turbulence MHD”, These de 3eme cycle, Universitt Paris VII et Observatoire de Meudon, 1983. [15] T.L. Bell and M. Nelkin, J. Fluid Mech. 88 (1978) 369. i16j E.K. Mashke and B. Saramito, Physica Scripta (SAS) (1982) 410. C.R. Acad. Sci. URSS 30 (1941) 301. [171 A.N. Kolmogorov, E.J. Hopfinger and R.A. Antonia, “High [181 F. Anselmet, order velocity structure functions in turbulent shear flows”, preprint Institute de Mtcanique de Grenoble (1983). J. Fluid Mech. 62 (1974) 305. 1191 R.H. Kraichnan, J. Fluid Mech. 13 (1962) 82. ]201 A.N. Kolmogorov, in: Statistical Mechanics: Nex Concepts, 1211 R.H. Kraichnan, New Problems, Now applications, J.A. Bite, K.F. Breed and J.C. Light, eds. (Univ. of Chicago Press, Chicago, 1972) p. 201. 1221 U. Frisch, P.L. Sulem and M. Nelkin, J. Fluid Mech. 87 (1978) 719. singularities and ]231 U. Frisch, Fully developed turbulence, intermittency, Ecole d’ett des Houches (1981). ]241 R.M. Kerr and E.D. Siggia, J. Stat. Physics 19 (1978) 543. ~251 J. Lee, J. Fluid Mech. 101 (1980) 349. ]261 J. Leorat, A. Pouquet and U. Frisch, J. Fluid Mech. 104 (1981) 417. 1271 W.M. Mattheus and M.L. Goldstein, J. Geophys. Res. 87 (1982) 6011. ]281 A. Pouquet, M. Meneguzzi and U. Frisch, The Growth of Correlation in MHD Turbulence, preprint, Observatoire de Nice (1983). R. Grappin, A. Pouquet and U. Frisch, J. ]291 J. Leorat, Geophys. Astrophys. 2 (1981) 67. M.L. Goldstein and D. Montgomery, ]301 W.H. Mathaeus, Phys. Rev. Lett. 51 (1983) 1484. [311 T. Miura and T. Kai, Phys. Letters 1OlA (1984) 450. ]321 J.D. Farmer, E. Ott and J.A. Yorke, Physica 7D (1983) 153. [331 A. Pouquet, C. Gloaguen, J. LCorat and R. Grappin, in: Chaos and Statistical Methods, Kuramoto, ed. (Springer), Berlin, (1983) p. 205. and I. Procaccia, Phys. Rev. Lett. 50 (1983) [341 P. Grassberger 345. (351 J. Kaplan and J.A. Yorke, Lectures Notes in Math., vol. 730, H.O. Peltgen and H.O. Walther, eds. (Springer, Berlin, 1979) p. 228. 1361 C. FroeschlC, Phys. Rev. Al8 (1978) 277. ]371 J.P. Chollet and M. Lesieur, J. Atm. Sci. 38 (1981) 2747.

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[38] C.E. Leith and R.H. Kraichnan, J. Atm. Sci. 29 (1972) 1041. [39] L.G.S. Kovaznay, J. Aero. Sci. 15 (1948) 745. [40] J.H. Thomas, Phys. Fluids 11 (1968) 1245. [41] J.H. Thomas, Phys. Fluids 13 (1970) 1877. [42] C. Sulem, Foumier, U. Frisch and P.L. Sulem, C.R.Ac. SC.

Paris 288 (1979) 571. [43] U. Frisch, Fully developed turbulence and intermittency in Turbulence and Predictability in Geophysical Fluid Dynamics and Climate dynamics (1985), Sot. Itahana di Fisica, Bologna. [44] E.D. Siggia, Phys. Rev. Al7 (1978) 1166.