Two-point, two-time analysis of the decay of turbulence in the presence of a magnetic field

ht. J. Engng

Sci., 1975,Vol.

13,pp. 393405.

Pergamon Press. Printed in Great Britain

TWO-POINT, TWO-TIME ANALYSIS OF THE DECAY OF TURBULENCE IN THE PRESENCE OF A MAGNETIC FIELD P. KUMAR, Department

of Mathematics,

S. R. PATEL

Indian

Institute

and J. S. RAWAT

of Technology, India

Hauz

Khas,

New Delhi-l 10029,

Abstract-A two-point, two-time approach is attempted for the final period of decay of turbulence in an external, homogeneous magnetic field. Two-point, two-time correlation and spectral equations are obtained by considering the equations of fluid and electrodynamics for two points in a turbulent fluid at two different times. Solutions, corresponding to any given initial distribution, are obtained by assuming that the turbulence is weak enough for second-order truncation approximations to be applicable. The analysis shows that pronounced axisymmetric properties are developed in this case, turbulence elements with small extensions in the direction of the field and with large time separations being damped relatively rapidly under normal physical conditions. In its final period, the decay is governed by both periodic and non-periodic motions. 1. INTRODUCTION STUDY of turbulence in electrically conducting fluids has received considerable interest during the last twenty five years [l-8]. Bulk of these studies have been made in view of their applications in some astrophysical and geophysical problems such as sun spot theory, motion of the intersteller gas, origin of earth magnetism etc. In most applications, the electrical conductivity is high enough for electro-dynamic forces to play an essential role and for turbulence to be governed by the laws of magnetohydrodynamics. This paper deals with the formulation of the law of decay of homogeneous turbulence as modified by the introduction of an external magnetic field. A two-point, two-time analysis of the turbulent fields is carried out on the lines analogous to that of Deissler [9]. As mentioned in reference [9], the multi-point, multi-time correlations in ordinary homogeneous turbulence have been treated by many authors. However, the problem is slightly complicated in magnetohydrodynamics, inasmuch as one has to consider both the turbulent fluctuations in the magnetic field as well as those in the velocity field and their interaction. The present study gives an insight into some of the dynamical aspects of magneto-fluid dynamic turbulence. It is assumed that the fluid is incompressible. The regions considered are sufficiently small so that the assumption of a homogeneous external field is justified. Also, the usual Maxwell equations and Constitutive relations are assumed to be valid, apart from the modifications required by the hydrodynamic motion. Solutions for the final period are obtained by neglecting the triple correlations in the two-point, two-time equations and it is shown that the motion becomes axisymmetric with respect to the direction of the field. THE

2. THE GOVERNING Under follows:

the above

assumptions

Du Dt

EQUATIONS

the fundamental

1 --vp P

+ &+j 393

equations

j x b,

of this problem

are as

(1)

394

P. KUMAR. S. R. PATEL and J. S. RAWAT

v.u = 0,

(2)

j = PLO’Vx b,

(3)

V.b = 0, Vx E =

(4)

--ablat,

(5)

j=v(E+uxb-e;‘nL’jxb),

(6)

where D signifies a substantial derivative, u the instantaneous velocity, t the time, p the density, p the instantaneous pressure, j the instantaneous electrical current, b the instantaneous magnetic field, E the instantaneous electrical field, u the kinematic viscosity, p. the permeability of free space, u the electrical conductivity, e, the charge on an electron, and n, the number density of electrons. The velocity field u is supposed to be nonrelativistic and Giorgi units are used throughout. The last term in equation (6) is the Hall current. Taking the curl of equation (6), and using equations (2)~(5), we get (7) where l is the usual alternating tensor. Einstein’s double suffix summation convention is adopted. Similarly, equation (l), with the help of equation (3), takes the shape [Jk

axkaxk

The instantaneous

pop

1

~---

2pop

dxk

a(hbd axi

(8)

*

field can be written as bi = bi + Bfy

(9)

where b, is the fluctuating component and B, is the steady mean component of the magnetic field. Now, making use of relation (9) and subtracting from equation (8) that obtained by averaging it, we get &.4i = --&

at

(UIuk

-

- +-@ &

U,Uk)

-

;

s

I

+

V&

,;,

+-Xax

ak

(bib, - bib, + biBk + b,B,)

(10)

(bkbk - bkbk+ ZbrB,),

where the overbars indicate average values. A similar treatment of equation (7) supplies -

!!$$(.bk-u,bk+U&-b,u,+biu~-~iu~)+--!--~

- (ecnepo(+)-16Jk-

k

(b,br

- b,b,+

b&

+ b&h

a* dx,ax, (11)

Two-point, two-time analysis of the decay of turbulence in the presence of a magnetic field

395

The last two equations were derived by Deissler[Q In what follows, we will employ the following notations:

(12) where h characterizes the magnetic field and is essentially the velocity of Alfven’s wave. Thus, the equations for the velocity and magnetic field at a point P in the fluid can be written in the forms

-;;(hkhk

-hkhk+2hkHk)

(13)

and $=$(U,hk

-_Uihk‘t

UiHk-hi&

+&k-Hi&)+

As k

k

k

-

axa;x

(ecne)-‘(~Lop)“*XEijk-

I I

(hrhk - hrh, + hrH, + h&),

(14)

where A = l/(crpo), the magnetic diffusivity. The corresponding equations for the point P’, separated from P by the distance vector r and the time increment Vt, are au; 8=--&u;*; + -$

;

C3’l.l;

-u;up$~+v-

(hjh:

-

dX:%X;

h(h; + h;H,: + h;H;) - ; &

I

(h;h: - h:h: + 2h:H;)

(15)

and ah; d at’=ax’(u;h;-i$+u;H;-h;u;+h;ul-H;u;)+A+$ ;

k

-

,,“h:

(e,n,)-‘(~~p)“‘AEj,k-

3. TWO-POINT,

TWO-TIME

m

;

(h:h: - h;h;+ h(HL + h:H;).

CORRELATION

(16)

EQUATIONS

We now multiply equation (13) by u \ and equation (15) by Ui,add and take average values, to get

auiu;= at

a -UiUk&-

--

p

axk

+

&

(m k

~ a 2uiu;

1 apuf - -++----

+mH,

axi

&k&k

+

hkUjf&)

-f

&

--

(hkhkU;

+ 2hkUiHk)

(17)

P. KUMAR,

396

S. R. PATEL

and J. S. RAWAT

and

auiu; -=__

a ~ ax;

at’

1 ap’ui

uiu;u; - p -+vax(

azuiu;

ax;ax:

+&uih;h:+u,h:Il* +uih:H;)-;&(Ulh:h:+2u,hlHI), (18) : I

where the fact that quantities at xi and t are independent of x( and t’ was used. By transformations aldxi = -alar,, a/ax; = alar,, introducing the (a/at),, = (a/at),, - a/avt, and a/at’ = a/aVt, the following equations are obtained from equations (17) and (18): c?UiUj

-

at

a = ar, (zmd

+ uiu;ul) + ;

_ _ (uih;h; - hihkUD

a-+H,nr(U,hl_hiun+~~,pu:+~(h,h,u:+2h,u:H,)I k

(19) and auiu; -=$avt

a

k

-

a-la-p-

(Ukhih;-UiU~U!J)+Hk~

k

~

Uih(-5ar-[PfU'+3(Uih;h;+

2Uih;Hk)]+

J

ahuj

VP

ar,ar, ’ (20)

for homogeneous turbulence and a uniform mean magnetic field. Indeed, the last two equations may be interpreted as the space-time equivalents of the generalized Von Karman-Howarth equation (in the presence of an external magnetic field). Further, it can easily be shown that the pressure and magnetic pressure terms occuring in equations (19) and (20) vanish if the triple correlations are neglected. (See equations (23) and (24), reference [8] and equation (14), reference [IO]). Thus, by neglecting triple correlations, these equations reduce to a2.ku;

a--

2v-

-=Hkay(uih;-hiu~+

at

!i

a 3~;

(21)

arkark

and

auiu;

-= avt

respectively. Proceeding exactly along equations may be constructed

El,;=+

k

ah4; lJar t k

the same lines, the following from equations (13)-(16):

cm additional

correlation

Two-point, two-time analysis of the decay of turbulence in the presence of a magnetic field

397

-

-

-

ara;r

(e,n.)-‘(l_Lop)“*h4mk-

m I

(uih:EL + uih:fi),

(23)

(uih:Hk + uih:R), am -= at

- (e,n,)-‘(l*~p)“*h~imk &

m I

(h,uj%

(24)

+ hku(R), (25)

-

(26)

- (e,n,)-‘(p,,p)“*h

a* -

-

_

[~i,,,kar ar (hrhlf& + hth23)

+

m I

6mk

&

m I

(hih;Z& + hih:R )I,

(27)

-

,PB,

(e,n,)-‘(CLoP)“‘A~ji,t-

m I

-

(hih:Hk + hih:H,).

(28)

It may be noticed that the equations (21), (23), (25) and (27) could also be obtained directly from the set of equations (19)-(22) of reference [8] if the triple correlation terms are neglected in the latter set and use is made of equation (12). 4. SPECTRAL

EQATIONS

In order to convert equations (21)-(i8) to spectral form, we define the following three-dimensional Fourier transforms: uiu((r,Vt,t) hihj(r,Vt,t)

m

=

h,ui(r,Vt,t)

J_

dk e””

(29)

u,hl(r,Vt,t) where k is a wave number vector and dk = dktdkzdkx. By introducing these relations, the Fourier transforms obtained as

of equations

(21)-(28) are

P. KUMAR S. R. PATEL and J. S. RAWAT

398

-

a&

avt $

dp?j -= avt f$

= ikkH&

= ikkHk(&

ikJ$&

(31)

- vk2$ij,

- pii) - (A + v)k’P;

+ (ecne)-‘(kop)1’2A

- XkZPyj + (ecne)-‘(~op)“‘hkmklEjmk

= ikkHk(Pij - +ij) - (A + v)k’P$

X kmk,Ejmk(HkPZ -t HIP~)~

X (&P’Ir + H,P’jk),

+ (e,n,)-‘(~~p)“‘hk,kl

X eimk(HkP:j + H&G),

(32)

(33)

(34)

(35)

api, = ikkHk(P:j - PC) - 2Ak2Pij + at

X

a& -= dvt

(ecn,)-‘(plop)“2A

[k,k,ei,nk (Hkp,j + H@nj) + krnklejrnk(H&l

jktH,$;i

- Ak*/&j + (e,n,)~‘(CLop)“‘Ak,kl~j,k

+

H&II.

X (H&l

(36)

+ RPik).

(37)

In principle, it should be possible to find an analytic solution of the system of equations (30)-(37) but, obviously, it will be very difficult to handle it. We will come to this point in the next section. However, if the Hall current terms are neglected (for large FL or a), a considerable simplification results. These currents are negligible for a dense plasma under the influence of a moderate magnetic field. Thus, omitting the terms corresponding to Hall currents in equations (30) to (37), one gets %

= iF(P:$ - p;i) - 2b&j,

ap,, .

-=

lF(P(,

%=

iF(/3ij -$ij)-(a

at

- P:l)-2apij,

(38)

(39)

+ b)P(i,

(40)

~=iF(g,i-p,j)-(n-h)PG,

(41)

d+ij

(42)

-

avt

= iF/3$ - b&,

a/3::= iF&, - afi?;,

(43)

fF$

(44)

avt

= iFPii

-

b/jii,

Two-point, two-time analysis of the decay of turbulence in the presence of a magnetic field apiJ

-

avt

399 (45)

= iFp$ - @ii,

where a = hk*; b = vk’; F = kkHk.

5. SOLUTIONS

The set of equations

b(k,Vt,t) Pii(kVt,t)

FOR THE FINAL

(46)

PERIOD

(38)-(41) can be solved to give a solution in the form

exp(m,t) = exp(md)

i%(kVt,t) /3$(k,Vt,t)

exp(md)

4:;‘(k,Vt)

c.$;f(k,Vt)

+:;‘(k,Vt) X

-(i/()(1

- s)$Ii”(k,Vt)

(i/5)(1 - s)6i,!‘(k,Vt)

-(i/5)(1

+ s)4C’(k,Vt)

(i/5)(1 + s)4ii(k,Vt)

&“(k,Vt)

(2i/lJ4f’(k,Vt)+

4Ii*‘(k,Vt)

wherem1,2=-(a+b)~[(a-b)2-4FZl”2,~~=-(~+b)

(47) (48)

and 5 = 2F/(a

-b)

= 2kkHk/[kZ(h - v)] and s = (1 - [z)“z.

(49)

The remaining four equations possess the following solution ,

(50)

P’~i(k,Vt,t)=$(l-s)Aij(k,t)exp(~)+~(l+s)B,j(k,t)exp(~),

(51)

B$(k, Vt,t) = Cij(k, t) exp

(52)

p,,(k,Vt,t)=~(l-s)Cii(k,

t)exp(y)+i(l+s)&(k.t)exP(y).

(53)

In order to obtain a solution which contains both t and Vt explicitly, we observe that +:,?(k,Vt) etc. must be finite for Vt = 0. If we compare the solutions (47) and the solutions (50) to (53), we get four identities. Setting Vt = 0 in each of these identities, the resulting equations can be solved for Aij(k,t), &(k,t), C,(k,t) and &(k,t). Substituting the values of Aij(k,t) etc. so obtained in equations (50)-(53), we get the following general solution for the final period:

400

P. KUMAR.

S. R. PATEL

and J. S. RAWAT

4ii (k,Vt,t ) Pii(kVt,t)

(s;

/3:;(k,Vt,t)

-j(l

/3::(k,Vt,t)

i:r:; -S>

$(I

j(l-S)

I

I

1

1

-$(l+S)

+S)

j(l+s)

4)j’(k)exp

1

-$(1-s)

$(1-S)

[m,(r

+?/)I

4if’(k)

exp [ rn,(t

+?)I

4:;‘(k)

exp [ nr7t + y]

j(l+s)

X

(54) ’

c$i?(k) exp [ m,t + y]

where

we have written [~~j”(k,Vt)lor=o = 4::‘(k) (and similar

notations

c$;j’(k) = -(1/2s)[(l

-

- s)+:;‘(k)

in the sequel),

@‘+?(Wl

(5% (56)

and 4;p’(k) = (1/2s)[(l

+ s)4!:‘(k)

- ~54~;‘(Wl.

(57)

In equation (54) +if’(k), +f’(k), 4Ijs’(k) and 4::‘(k) are constants which depend on initial conditions. Here we remark that even if the Hall current terms are retained, it is reasonable to expect an analytic solution of equations (30)-(37) by combining the present technique with that of Reference [ 111. The parameter 5 defined by equation (49) enters into the foregoing solutions in a manner which seems to be natural for this type of analysis. It serves the role of an useful similarity parameter, especially, in discussing the behaviour of the solutions in the limiting case, as we shall see in what follows. A non-dimensional entity in itself, it can also be associated with the form H,L,/(h - v) where H, is a characteristic wave velocity and L, a length characteristic of the scale of variation of the variables. Furthermore, one may connect it with parameters used in references [3], [12] and [13]. For small values of [ the form (54) reduces to

=

1

1

1

1

Cl4

4/C2

1

1

- i{/2

&Cl2

-2i/c

2il5

-2ilc

6512

- ill2

Z/5

Two-point,

two-time

analysis

of the decay of turbulence

Im”(k)exp

[m(f

in the presence

+F)]

of a magnetic

we have used

401

1

X

where

field

(5fv

the abbreviations

4:?(k) = -(<*/4)4:;‘(k) + (i&./2)+;;‘(k)

(59)

I$ I;‘(k) = 4 j;‘(k) - (i5/2)4 :4’(k).

(60)

and

The defining structure of 5 obviously suggests that it can assume small values in a number of physical situations. Let us, for example, analyse the case on laboratory scale where the largest turbulence elements hardly exceed 10m2 meter. Thus, considering mercury at room temperature as typical of the fluids to which we may want to apply the theory under terrestrial conditions, we have A = 9.43 m*/sec.,

v = 1.2 X lo-’ m’lsec.

and H = 1 m/set. When the mercury column is acted upon by a magnetic field strength of lo3 gauss (see references [ 131 and [ 141). In accordance with the above data the maximum value of &’comes out to be of the order of 10-3( 4 1) and, hence, we infer that the solutions (58) are valid for the whole spectral range. Similar will be the case for liquid sodium and other liquid metals. In particular, solutions (58) should hold good for small scale components of turbulence (large k) for small as well as moderate ratios H/(A - v). All quantities have to be finite at 5 = 0, and therefore, solutions (58) suggest that, when 5

tends to zero, 4:;‘(k) = [&lo(k) + OK),

(61)

P?(k) = (4/5*)~:f’(k) = [Piilo(k) + O(b),

(62)

(PI)“‘(k)

= W/5)+:?(k)

= [/3L lo(k) + O(C)

(63)

and (p::)‘“’ = (2i/&b:?(k)

= [P:lok) + O(l),

(64)

where [&lo(k), [&lo(k), [P$],,(k) and [P$Mk) are independent

of 5 and O({“) are terms at least of order n. The remaining coefficients of the exponential terms are either O(l) or O(<‘), and consequently, tend to zero as 5 -+O. Hence for < = 0, we get

h(k,Vt,t)

= [&lo(k) exp [-2uk’(t

L%(kVt,t) = [Pij]o(k) exp [-2hk’(l PL(k,Vt,t)

+;)I, +:)I,

= (/3i,Mk) exp [-(A + v)k’t

- vk*Vt]

(66) (67)

402

P. KUMAR,

S. R. PATEL

and J. S. RAWAT

and PC(k,Vt,t)

= [P$lo(k) exp [-(h

+ v)k*t - Ak’Vt).

(68)

These solutions represent the final period of decay of homogeneous turbulence. For Vt = 0, equations (65)-(68) reduce to the corresponding expressions in the final period, which involve only one time (e.g. [7], equations (52)-(54)). In general, the situation t = 0 corresponds to the case of vanishing velocity of the so-called magneto-hydrodynamic wave i.e. the field-free case. Equations (65) and (66), therefore, lead us to the conclusion that if the external sources are taken away the turbulent fields of the magnetic and the kinetic energies decay independently of each other. A simple fact emerges out from equation (65). Let us assume that the magnetic field is so weak that it does not affect the turbulent velocity field which we consider as isotropic. Then, if the magnetic field is switched off at any stage t = to, we can start with the initial approximation that, for weak turbulence, = (J~/12~*)(kZ6ij - kikj), (69) TC=o t=t,> where Jo is a constant that depends on initial conditions and Sii is the Kronecker delta (e.g. [15], equation (43)). Applying the foregoing initial condition in equation (65), the energy spectrum function is obtained as

[hj(k,Vt,t)l

E = 2rk24ij

= (Jok4/3r)

exp

[

-2vk’

(

t - t +s ”

2)l'

(70)

which is the same as obtained by Deissler ([91, equation (17)) for ordinary homogeneous turbulence. Hence, we conclude that if a turbulent motion is unaffected by a weak magnetic field and, then, if the external sources are taken away, the subsequent turbulent motion continues to remain ordinary in its final period. This, of course, is to be anticipated on physical grounds. Corresponding to a finite time increment, the decay times 7k for the space-time spectral tensors are given by l/71.2 = k2(h + v) -+ k2(h - u)(l - J’)“‘; l/73 = k2(A + v). The following three possibilities arise: (i) J’< 1, giving three non-periodic solutions, (ii) 5’ > 1, giving two periodic solutions, both ~3 = l/lk*(h (iii) 5’ = 1, all spectral

tensors

6. PHYSICAL

decay

all with different decay times; with the same real damping T? where

+ v)];

(72)

with the same time constant

INTERPRETATION

(71)

OF THE

DECAY

T, =

73.

LAW

Following Lehnert [7], we consider a plane state of motion in a liquid between two infinitely conducting planes at a distance L. If a homogeneous magnetic field b,, is introduced in the z-direction, i.e. perpendicular to the planes, then, for small amplitudes, we have

a a _=_=

ax ay

0; u = (O,u,O); h = (O&,0).

(73)

Two-point, two-time analysis of the decay of turbulence in the presence of a magnetic field

In view of equation (73), equations

(13) and (14) are transformed

403

into (74)

and (75) where we have neglected the Hall currents. Similar equations are obtained corresponding to the point P’. Now, by using the fact that the magnetic lines and the liquid are attached to the infinitely conducting walls, it can be shown that u admits a solution of the form . , k = TklL),

(76)

provided m2+ 4k2HZ+ 2(h + v)k’m Similarly, corresponding

+ 4Avk4 = 0.

to the point P’, we get k = Irk/L).

equation (77) possesses

(77)

(78)

the roots m ,.t = -(A + v)k’T

(A - v)k*(l - c2)“‘,

(79)

which are the two first values given by equation (48). Interpretation of this is given in reference [7]. Constructing correlations from equations (76) and (78), we obtain time factors of the form T(t).

T(t’) = exp [1/2(m1.2t + m1.2t’)l

(80)

for a given k. These factors are consistent with those given by equations (48) and (54). Again, if we set t’ = t + Vt = t i.e. Vt = 0 in equation (80), this reduces to the equation (67) of reference [7]. We now formulate an asymptotic law for $ii (k,Vt,t). Equation (54) shows that, in the final period, this tensor can be written as the sum of four tensors. However, during the largest part of the final period of decay, it will be represented by the only term corresponding to which the argument of the exponential term is maximum (for a given Vt, of course). It can be easily shown that, for sufficiently small values of 5 and for large ratios A/V, m,(t +iVt) is much greater than the remaining arguments. Now, equation (61) shows that the corresponding factor, 4$“(k), contains a term of zero order in 5 which tends to zero in this limiting case. Consequently, during the largest part of the final period of decay, &(k,Vt,t) will be represented by the asymptotic law &(k,Vt,t)

= 4\;‘(k) exp (-2[vk’+

k:H2/k2(A

-

v)](t +F)},

(81)

404

P. KUMAR,

S. R. PATEL

and J. S. RAWAT

where the x1-axis is chosen in the direction of the field and equation (49) is used. The conditions of this limiting situation are met by the already cited example of mercury where < = 10~’ and h/v = 10’. They are also satisfied in certain astrophysical circumstances where, for instance, in the hydrogen-convective zone in the solar atmosphere and for eddy sizes, say, of the order of IO’ meters (for bigger eddies the asymtotic law may cease to hold), < = IO ’ and h/u = IO4 (see references [7] and 1161). Similar analysis may be carried out corresponding to the case A < V. Hence, in view of equation (81). the conclusions of reference [7] could be generalised to state that during the turbulent decay of a liquid with A % u, all periodic turbulence elements as well as the aperiodic ones, with small extensions in the direction of the field and with large time separation are damped out relatively rapidly. Here too, the asymtotic state of decay is two-dimensional with respect to the direction of the field. Further, for finite time separations, the only vortices left in the final state are corresponding to k, = 0, for which the damping is entirely by viscosity. REFERENCES [l] G. [2] S. 131 S. 141 T. [SI I,. [hl T. 171 B. 181 R. 191 R. [IO] R. [II] Y. [ 121 B. [I31 S. [ 141 S. [IS] R. [ I61 S.

K. BATCHELOR, Proc. R. Sot. A201, 406 (1950). CHANDRASEKHAR. Proc. R. Sot. A204. 435 (1951). LUNDQUIST, Arkic f. fysik 5, 338 (1952). TATSUMI, Rrr. Mod. Phys. 32, 807 (1960). S. G. KOVASZNAY, Rrr. Mod. Phys. 32. 815 (1960). G. COWLING, Magtretohydrodynamics. Interxience (1957). LEHNERT, Q. J. appl. Math. 12, 321 (1955). G. DEISSLER. Phvs. Fluids 6, 1250 (1963). G. DEISSLER. NASA TR R-96, 2 (1961). G. DEISSLER. Phys. Fluids 3. 176 (1960). PAUL, Tensor, New Series 18. 212 (1967). LEHNERT, Arkiu f. fysik 5, 69 (1951). CHANDRASEKHAR, Proc. R. Sot. A216, 293 (19.53). ESKINAZI, Vector Mechanics of Fluids und Magnetofluids, Academic G. DEISSLER, Phys. Fluids 4, 1187 (1961). CHANDRASEKHAR. Phi/. Msg. 43. 501 (1952). (Received

IXJanuury

Press

p. 468 (1967).

1974)

Resume-Un abordage deux points deux temps est tente pour la periode finale de decadence de turbulence dam un champ magnttique homogene externe. Une correlation deux points deux temps et des equations spectrales sont obtenues en considerant les equations de fluide et d’electrodynamique pour deux points dans un fluide turbulent a deux moments differents. Des solutions sont obtenues correspondant a toute distribution initiale donnte, en supposant que la turbulence est suffisamment faible pour que des approximations de tronquement de second ordre soient applicables. L’analyse montre que des proprietes prononcees d’axisymmetrie sont developpees dans ce cas, les elements de turbulence avec petites extensions dans le sens du champ et avec grandes separations de temps Ctant amorties relativement rapidement sous des conditions physiques normales. Dans sa ptriode finale, la decadence est gouvernte par des mouvements aussi bien periodiques que nonperiodiques.

Sommario-Un accostamento a due punti ed a due tempi, per la fase finale dello smorzamento della turbolenza in un campo magnetic0 esterno omogeneo. La correlazione a due punti ed a due tempi e le equazioni spettrali si ottengono considerando le equazioni del fluid0 e dell’elettrodinamica per due punti. in un fluid0 turbolento, a due tempi diversi. Si ottengono soluzioni, corrispondenti a qualsiasi prestabilita distribuzione iniziale, presupponendo the la turbolenza sia sufficientemente indebolita per applicare troncamenti approssimati di second’ordine. L’analisi dimostra the, in quest0 case, si sviluppano pronunciate proprieta assisimmetriche poiche vengono smorzati con relativa rapidita gli elementi di turbolenza con modeste propagazioni in direzione del campo e con grandi distanziamenti di tempo, in normali condizioni fisiche. Nella fase finale lo smorzamento e controllato da movimenti periodici e non periodici.

Two-point, two-time analysis of the decay of turbulence in the presence of a magnetic field

405

Zusammenfassung-Eine zwei-Punkte, zwei-Zeiten Annaherung wird fur die Endperiode des Abklingens einer Turbulenz in einem %uSeren, homogenen magnetischen Feld versucht. Zwei-Punkte, zwei-Zeiten Korrelation und Spektralgleichungen werden durch Berticksichtigung der Gleichungen von Fltissigkeit- und Elektrodynamik fiir zwei Punkte in einer turbulenten Fhissigkeit zu zwei verschiedenen Zeiten erhalten. Es werden Losungen, entsprechend jeder gegebenen anflnglichen Verteilung durch Annahme erhalten, dal3 die Turbulenz schwach genug ist, ftir Abrundungsannlherungen zweiten Grades anwendbar zu sein. Die Analyse zeigt, da8 bestimmte achsensymmetrische Eigsnschaften in diesem Falle entwickelt werden, daR Turbulenzelemente mit kleinen Erweiterungen in Richtung des Feldes und mit langen zeitlichen Trennungen verhlltnismPRig schnell unter normalen physikalischen Verhaltnissen gedlmpft werden. In der Schlugperiode wird das Abklingen durch periodische wie durch nicht-periodische Bewegungen geregelt.