A formulation of the theory of isotropic hydromagnetic turbulence in an incompressible fluid

ANNALS

OF

PHYSICS:

32, 292-321

(1965)

A Formulation of the Theory of Isotropic Turbulence in an Incompressible LAWRENCE The

University

L.

of Chicago,

Hydromagnetic Fluid”?

LEE‘~ Chicago,

Illinois

The theory of hydromagnetic isotropic turbulence in incompressible fluids is formulated using methods similar to those of quantum field theory. The present formulation is a generalization of Wyld’s formulation of ordinary isotropic turbulence in incompressible fluids. Solutions for the velocity and magnetic field in the form of perturbation series are set up. Terms in the series are then represented by a one-to-one correspondence by diagrams similar to Feynman diagrams. A study of these diagrams reveals that they may be rearranged t.o give integral equations governing the second order correlation functions for the velocity and the magnetic field. These integral equations contain an infinite number of terms. To obtain manageable results, the equations are truncated at finite orders, yielding approximate equations. An approximation at t,he lowest order gives Chandrasekhar’s equations; while a “second approximation” gives a more complicated set of equaf.ions. I. INTRODUCTION

Most of the exist’ing formulations of the theory of turbulence involve plausibility arguments or approximations at an early stage; they are therefore not capable of systemat,ic refinement t,o give more accurate resulk. One except,ion is the formulat’ion by Wyld (I), in which the approximation is introduced at the end of the calculations; and this formulation can, in principle, give exact results. It is the object of this paper to extend Wyld’s formulation to t)he more general case of hydromagnetic t,urbulence. The fluid motion and magnetic field are governed by the standard pair of hydromagnetic equations. Exact solutions of these equations of course describe the motions in all details. However, for a system in the state which we describe as turbulence, such an exact solution is neit.her practical, because of its extreme complexity, nor necessary, because fluctuations wit.h elemenk of randomness are * This research has been supported by the Number Nonr 2121 (24) with the University of t This paper is based on a thesis submitted by requirement,s for a Doctorate at the University of $ Present address: Department of Aeronautics of Minnesot,a, Minneapolis, Minnesota 55455.

tract

292

U. S. Office of Naval Research ConChicago. the author in partial fulfilment of t,he Chicago. and Engineering Mechanics, University

HYDROMAGNETIC

293

TURBULENCE

its essential features. A statistical treatment that a basic feature on which the theory between the components of the velocity at instants of time; such a two point velocity described by the tensor

is clearly suggest’ed. And it appears may concentrate is the correlation two neighboring points and at two correlation (in space-time) will be

Uij(X, t; x’, t’> = (Ui(X, t)zQ(x’, t’>>. Similarly in describing hydromagnetic the correlations

turbulence,

(1)

we might consider, in addition,

Rij(X,

t; X', t')

= (?Ji(X, t)hj(X',

t')),

(2)

Hij(X,

t; X', t')

= (h;(X,

t')),

(3)

and t)hj(X',

between the components of the velocity and the magnetic field or between the components of the magnet,ic field themselves. To simplify the circumstance, we shall formulate the theory under the simplest physical situation, in which bhe system is stationary, homogeneous,and isotropic. Under these conditions, the correlation functions (1 ), (2), and (3) are expressible in t)erms of three scalars U, R and H in the manner (2) [7,(x, t; x’, t’) = curl U(j x - x’ I, ( t - t’ I)eijl(xl - x1’),

Rij(x, t; x', t’) = R(l

-

x

X’

1, 1t -

t’

-

Ij6+(xl

x[),

(4) (5)

and Hjj(X,

t; X', t')

= CUrlH(I

X -

X' 1, 1 t -

tl /)EijE(Xl

-

XL);

(6)

and the problem then reduces to determining the defining scalars U, R, and H. Instead of dealing with t’he velocity and the magnetic field directly, we may consider their Fourier transforms and consider the corresponding correlation functions in the Fourier space; these are related to the correlation functions of Eqs. (4), (5), and (6) by an integral transformation. Therefore, whether we solve the problem in coordinate space-time or in Fourier space, the results should be the same; and the choice is a mat’ter of convenience. It will appear that by working in the Fourier space,we can apply the perturbation techniques developed in quantum field theory. II.

THE

HYDROMAGNETIC

EQUATIONS

The equations of motion for an incompressible conducting fluid in a magnetic field are ‘2 + &

(vivj 3

- hihi)

=

VV’vi

-

g

2

+fi

(7)

294

LEE

and

at+a:. ahi

3

(hivj

- vihj)

= XV2hi,

(8)

where vi (i = 1,2,3) denote the components of the velocity, hi the components of the magnetic field divided by (4rp/~)l’~, fi the components of the external force per unit mass, p the density, p the pressure, and ~.r,v, and u are the coefficients of magnetic permeability, kinematic viscosity, and electrical conductivity respectively and X = $&pa

and

m = p/p + $51 h 12.

(9) Throughout this paper, summation over repeated indices is to be understood. In addition to the equations of motion, we have the divergence condition on the magnetic field

the condition fluid)

on the velocity

(derived

and we will assume that the “forcing

Let us introduce

the Fourier

h&t)

from the assumed incompressibility

term”

of the

fi is also divergence free :

transformations:

= ___ &jl,.

z vi(k,

c+?(~@),

(13)

E, hi(k,

m)ei(k.x-‘“t),

(14)

5 fi(k,

c#‘~‘“-“~),

(15)

and

YJb,t) = ___ (&

z

a(k, w)ei(k.r--ot).

(16)

We have used the normalization appropriate for a box of volume V and a time interval T. At the end of the calculations, V and T will be allowed t.o become

HYDROMAGNETIC

295

TURBULENCE

infinite (as required by homogeneity and stationarity) becomes an integration wit,h the correspondence:

when

the summation

(17) By introducing

the four dimensional

vectors

k = ( k1 , k2 , k8 , - U)

and

x = (x1 , x2 , x3 , t),

(18)

and the four-volume Q = VT, we can rewrite

Eqs. (13)-(17)

(19)

in the simpler forms (ZO)-(23)

and (24) Substituting Eqs. (20)-(23) into Eqs. (7) and (8), multiplying them by properties of the funce-ik’z, integrating over all x and using the orthogonality tions eik’=, we obtain the Fourier transforms of the original equations, namely, -iwvi(k)

+ &,,+ssk

[ki’vi(k’)vi(k”)

- kj’hi(k’)hj(‘“)l (25) = -vk’vi(k)

- &m(k)

+f;(k),

and -&hi(k)

+ -& ,,+T=, [kj’hi(k’)Uj(k’)

- kj’vi(k’)hj(k’)l (26) = -Xk’hi(k).

The divergence equations kivi(k)

(lo)-(

= 0,

12) now require k&(k)

= 0,

and

kifi(k)

= 0.

Multiplied by kike2, and using 6he divergence conditions expressed (27)-( 29), Eq. (25) becomes an explicit equation for o(k) :

a(k) = -A2 In Eqs. (25), (26), and the term in kj” The total pressure making use of Eq.

,,z=, and (30) vanishes a(lc) can (30) ; we

[kikj’vi(k’)vj(k”)

- kikj’ha(lc’)hj(k”)].

(W-(29) by Eqs.

(30)

we may replace ki’ by ki , since kj = kj’ + kj” in view of the orthogonality relations (27)-(29). now be eliminated from Eqs. (25) and (26) by find, aft’er some rearrangements

296

LEE

(-iw

+ &v,(k)

= fi(k)

+ & i

_ kik.k, +-

kjl;,+FCk [vi(k’)v&“)

- hi(k’ (31)

k,gek

[v,#)vj(k”)

- hd~‘M~“)l

,

and (-iti

+ Xk2)hi(k)

= -&

ki ,,+%=g [hi(k’)vj(k”)

These equations can be written

more symmetrically

p&)

- vi(k’)hj(k”)].

(32)

in terms of the tensors

= aij - ‘$,

P:&(k)

= kiPi,(k)

+ k,Pii(k),

(31)

P;,(k)

= kJ’i,(k)

-

(35)

and k,Pij(k).

The tensor Pij(k) is, apart from a scalar multiple, the most general isotropic tensor of the second rank orthogonal t’o k. Similarly Ptj,(k) and P;,(k) are the most general isotropic tensors of the third rank orthogonal to k, and symmetric, or antisymmetric, in the indices j and m. It is easy to show, by straightforward substitution, that Eqs. (31) and (32) can be rewritten as (-iw

+ Vk’)vi(k)

= .fi(k)

- ip;;;2k),,& [vj(k’)v,(k”)- hjUh,d~n)l, (36) and

(-iw + Xk2)hi(k) = -t!?d$

,,g=, Ilvj(k’ha(kn) - hj(~‘hm(~“)l

if’;,(k) = -___2CW III.

(37) c

2vj(kl)h,(kn).

k’+kc=k

PERTURBATION

EXPANSIONS

The method of solution we wish to develop is most, conveniently described in terms of the following equations which are generalizations of Eqs. (36) and (37). We consider vi(k)

= Sil(k)fdk) + KSil(k)P&,(k)

7 [zQ(k’)v,(k

- k’) + hj(k’)h,(k

- k’)],

(38)

HYDROMAGNETIC

297

TURBULENCE

and hi(k)

=

Til(k)gt(k)

+

KTit(k)PGm(k)C

2Vj(k’ibn(k

-

k’),

(39)

where v

Sit(k)

h = ih,

=v;

= S(k)Pil(k);

S(k) = (-iu

T;z(~)

T(k)

+ vk’)-I;

(40)

= T(k)f’il(k),

(41)

= (-iw

(42)

+ Xk’)-‘,

and (43)

and finally gl(k), like j,(k),

will be assumedto satisfy the condition gz(k)kz = 0.

(44)

Since Sil(k) and Til(k) always occur multiplied by jl(k), P$,(k), or gl(k), which are all orthogonal to kl, the berm kiklk-’ in Pil(k) never gives any cont,ribution; therefore, wit#hout loss of generality, we may write Sil(k) = S(k)S7Z9 and It, is (37)

Til(k)

= T(k)Gil .

(45)

clear that Eqs. (38) and (39) become identical with Eqs. (36) and if we let now

K=l

(46)

and gz(k) = 0.

(47)

Equations (46) and (47) will eventually be allowed to be satisfied, but for the present, we shall leave K and gl(/l-) unspecified and expand the solutions of Eqs. (38) and (39) in power seriesin K: U;(k)

=

V;(h-)o

+

Ui(k)lK

+

Vi(/?)zK’

+

V,(k.)sK3

+

‘.’

+

.-.

(48)

and hi(k)

= hi(k),,

+ hi(k)oc

+ hi(k)$

+ hi(k)aK3

.

(49)

Substituting these expansions into Eqs. (38) and (39) and equating the terms with different powers of K, we obtain

298 Vi(k)”

LEE

=

Sil(k)fl(lL),

Gd,

(50)

- m,

and hi(k)0

= Til(k)gl(k), (51)

- ~701, By substituting in the equations for a given order the equations of the earlier orders, we can express the equations in terms of L&(k), Til(lc), &,,(k), fl(lc), and gl(lc) only. The equations governing the higher order terms become increasingly complicated. To simplify the writing, we introduce diagamrepresentations. We shall represent each term in Eqs. (50) and (51) by a diagram and each factor of the term by an element of the diagram. We introduce the solid line element to represent the velocity propagator Sil(lc), and the dotted line element to represent the magnetic propagator Til( Ic). Using ++ as a symbol for representation, and using the Sans Serif capitals to denote tensors whenever no confusion is likely, we write solid line

__ t) S (velocity

propagator),

(52)

and dotted line

.a.... @ T (magnetic propagator).

(53)

In all the equations of (50) and (51), except the two zero-order ones, there are products involving three propagators and a factor P?i,(k). In diagram form, these products are represented by three lines meeting at a vertex. There are two kinds of vertices: a kind, which we shall represent with a simple dot, where three solid lines meet; and a kind, which we shall represent with a small circle, where one solid line meets two dotted lines: simple dot

l

c-) vertex of the first kind,

(54)

and small circle 0 ++ vertex of the second kind.

(55)

The diagrams corresponding to Eqs. (50) and (51) can be written as they appear in Figs, 1 and 2. But if we agree upon the convention that there should always be

HYDROMAGNETIC

v

(hjo

=

t.‘,‘,(k),

=

--f

f

_I 9 t-

-4’

f

FIG.

1. Diagrams

h, (IJo

corresponding

=

299

TURBULENCE

. ..--

to the

..

.. 9

terms

of the perturbation

to the terms

of the perturbation

series

(48)

y _: 3

FIG.

2. Diagrams

corresponding

series

(49)

three lines meeting at every vertex, we may omit all diagram-parts representing a vi(k)0 or an hi(k)0 i.e., all the f and ...... g combinations; the presence of these factors is implied by vertices with less than three lines attached. The diagrams of Figs. 3 and 4 are obtained by omitting all vi(lc)o’s and hi(k)o’s from the diagrams of Figs. 1 and 2. Since the diagrams in this new representation are so much simpler, we give terms up to the fourth order. The diagrams of Figs. 3 and 4 all have the shape of a tree lying on its side. In every one of these diagrams, there is always one and only one straight (solid or dotted) line segment which is connected to a vertex at only one end. We call this part,icular line segment the “trunk” of the diagram. In Figs. 3 and 4, the trunks are all on the extreme left. Every vertex has three sides, although not all of them are shown explicitly in the diagrams of Figs. 3 and 4 in accordance with the convention adopted. On one of the sides of every vertex is attached a chain of straight line segments connecting this vertex to the trunk of the diagram. This particular side will be referred to as the side nearer the trunk. In Figs. 3 and 4, t,he side nearer the trunk of every vertex is on the left. A diagram may be distorted or rotated, but so Iong as the topology remains unchanged, it represents the same analytical term and the preceding definitions are not altered. To make the diagram representation complete, we give the procedure for recovering analytical expressions from diagrams. First we observe that in order to save writing, the wave numbers and subscripts have been omitted from the diagrams. They should first be filled in as labels, using the following rules:

300

LEE b;(h),=

-

u,cq = U,'h)*=

+

-

2-

+ z-

=

a

z

+

+ 8~. +

8--o--

o

+ ~--L--D-

t4<

+4<

+ 16-.-w

4

0..----

+ 4---.-a/'

to the

8 __..
+

8 __.__ 3 .__. o

+

8 .._..

*

$

8

+

8

o--c---c----o

$

8 . . . ..-.....

+

8 .

+

*.....<

+z

+

8.__.. -ys;I

+8

-

. *---cu

terms

-

+ a..... FIG.

4. Simplified

diagrams

“W -*+

"'0

0

perturbation series(48)

of the

+ 8 D p + 8 . . . . -.--o

.

.-

+16

-

-

-

<

+4

. .. . 0

<

+

.. .. 0

. .J

+

16 . .._ o

-

+

Id....-w--o

+

16 . . . . -

o

+ 4 -----4 + 4..--.oy

-“O--G

+ Be.... s, .P

corresponding

. o

o . . 0

. . 0.

IJ

.< 6

. 0

0

$ 4----u/

o 0

+ 4 ...-<.

a ._..

‘--

-. o

+ 4 ._.. 0. .. 0

+

+ 4----c/

o---o

QY

+16-o:

-

+ 4 * . . .+ 4 . . . ..-

“0

o

+a---

+a---&

+8--L

+ 4 -----

-0. ----

+8<._

+ 2.-- -

LJh&

O.--O

+ 16--o

0

corresponding

-

-.a

+16--.-.-o.

6

+“-c&y

2

=

o + 16-+----n

-

i-8-0

-

0 --0.-. 0

+4+

+4
hL(c)> =

L;(h&

L

+ 2-c -0

diagrams

=

:

-

2<

+ 8---o..' 3. Simplified

-

+ E---co.--b--o

+ a -... + 04

$ 8+---m

+ 8~

FIG.

8

-

+a*-.t

0

t 4-------o + 4---.o--o

4P +4--L--a

y(b),

+4--o

..

<

+ 8 ----.
a---- / ..... ---.o

i-16

to the terms

. <

,--. o

of the perturbation

series

(49)

HYDROMAGNETIC

301

TURBULENCE

Rule 1. Wave number is conserved at every vertex, the wave number on the side nearer the trunk is equal to the sum of the wave numbers on the ot’her sides Rule 2. The wave number of the diagram is the same as t.he wave number at the trunk. Rule 3. One index of each element is cont,racted with an index of an adjacent element. The complete analytical expression represented by a diagram equation is the sum of the products of all tensors (and vectors) represented (and implied) by the different diagrams (or terms) with the coefficient given in front of each term. Each term is a summation over all wave numbers not fixed by the requirements of conservation at the vertices. We have not defined the vertices to represent P$m( k) in the most direct manner, but it will become obvious that this method of representation has its advantages when we deal with generalized quant,ities. The vertex of the first kind (simple dot) always represent a P&,(k). The vertex of the second kind (small circle) represents a P&,(k) if a solid line is attached to the side nearer the t,runk, and i% represents a Plj,(lc) if a dotted line is attached to this side. Due to antisymmetry, the order of the indices j and m in Plj,( k) must be carefully attended: m should always be cont,racted wit’h a magnetic propagator or a magnetic field vector. Because of its dual representation, the circle vertex will be called Pyjm(k) (the average of P+ and P-). To avoid confusion, we shall confine P, with the proper superscript, to stand for Phm(k), P&(k), or Pii, OILY, the tensors Pii( PTj,(k), and P;,(k) will always be written in full. Using the foregoing procedure, the analytical term can be obtained from its representative diagram by inspection. We give the following example of how it is done: simple diagram :

(56)

labelled diagram :

(57)

analytical

term:

F

& g

Ti~(k)PjL)(k)T,,(k’)

x P;p(k’)v,(kN)h,(k’

- k”)S,,(k

x P:t,(k The expansions

for the solutions

- k’)v,(k”)v,(k

for vi(k)

and hi(k)

- k’)

(58) - k’ - k”).

were in terms of the

302

LEE

parameter K. By a diagram of the nth order, we have meant one of the diagrams included in the nth term of the expansion with coefficient un. We can consistently redefine the diagram of the nth order as a diagram with n vertices, in which case we do not need the parameter K for distinguishing the order. We shall let K be unity in all future calculations. IV.

CORRELATION

We want to use the perturbation functions

FUNCTIONS

series (48)-(51)

to calculate

the correlation

Uij(k,

k’)

= (Ui(k)Uj(k’)

),

(59)

Hdk,

k’)

= (hi(k)hj(k’)

>,

(60)

Rij(k,

k’)

= +Ji(k)hj(k’)

>,

(61)

and

where the angular brackets denote ensemble averages. The ensembles we average will be homogeneous, isotropic, and stationary as specified in the introduction. These conditions require the correlation functions to be of the forms U;j(ky k’)

= U(k’,

~“)P~j(k)6~,-kr

,

(62)

Hij(k,

= H(k’,

m2)Pij(k)6k,-k,

,

(63)

k’)

and Rij(ky

k’)

= R(k2, u2)tijmlCm6k,-k*,

(64)

where Pij(k) is the isotropic tensor already defined in Eq. (33). We now introduce one more statistical assumption by assuming that the system is maximally random. The principle of maximal randomness was first suggest,ed by Kraichnan, (S), it states that the ensembles should be as random as possible consistent with the hydromagnetic equations; therefore all correlations must arise as a result of the nonlinearity of the equations and not because of boundary condit’ions. It is difficult to apply this condition directly t,o the velocity and the magnetic field components v;(k) and hi(k). However, it can be applied to our perturbation expansions by requiring the zeroth order components Vi( k-j0 and h;(k), t,o be completely random. Therefore, in the maximally random case, all averages over an odd number of vi(lc)o’s or hi(k)o’s will vanish; in particular Rii(ky k’)o = (vi(k)&j(k’)o)

= 0.

(65)

On the other hand, all averages with even numbers of vi(k)o’s expressible as a sum of products of the elementary correlation I Uij(k, k’>o = (v,(k)oVj(k )o) = u(k2, W2)J’ij(k)6k,-k,

,

and hi(k)o’s functions

are

(66)

HYDROMAGNETIC

303

TURBULENCE

and

Hij(k, k’>o = (hi(k)c!Mk’)lJ = H(k*, using a method sometimes known (u~(k)~Uj(k’)~~~(k”)~~,(klll)~)

G)&(k)&,--k>

as “breaking

= Uii(k,

+ Uil(klkfl))oUjm(k’,

of the bars.”

k’)oUl,(k”, k”)o

(67)

, For example,

k”)~ + Uim(kp k”)oUjl(k’,

k”)o .

(68)

Now we can show that Rii(k, k’) vanishes in the maximally random case. From Figs. 3 and 4, by counting the missing elements at the vertices, we find that every term in ui(k) contains an even number of hi(k)o’s and every term in Ai contains an odd number of hi(k)o’s; hence every term in Rii(k, k’) vanishes because it contains an odd number of h&k)o’s. This proof of the vanishing of R in the maximally random caseis a nontrivial achievement of the present formulation. The existence of this correlation function has been a subject of some controversy in the literature. There is an argument, apparently due to Cowling (J), that since replacing h by -h in the hydromagnetic equations (7) and (8) does not change the equations, positive and negative values of h should be equally probable for any given v; therefore, there should be no correlation between v and h. This argument does not appear to have been universally accepted. The recovery of this result from the present discussionin the special caseof maximal randowlness is a confirmation of Cowling’s argument.’ Perturbation seriesfor the correlation functions U and H may be written in a manner similar t’o Eqs. (48)-(51) : I Uij(k, k’) = (vi(k)ouj(k )o) + (~i(k>2~j(k’>o) + (ui(k),vj(k’),)

+ (ui(k)ouj(k’)2)

+ (ui(k)-toj(k’)0) + (Ui(k)aUj(k’)l) + (ui(k)zoj(k’),) + (u;(k)lvj(k’)a)

+ (ui(k)Nj(k’>,)

(69)

+ . *. p

and H+j(k, k’) = (hi(k)&(k’)a) + (hi(k)hj(k’)o)

+ @i(k)

+ (ki(k)ohj(k’)z)

+ (hi(k)&j(k’)o)

+ (hi(k)hj(k’)l)

+ (h(k)zhj(k’)z)

+ @i(k)h

+ (h;(k)&i(k’),)

+ . .* -

(70)

1 Cowling pointed out that the hydromagnetic equations, by themselves, do not require one to suppose that the magnetic field be either polar or axial. Equations (5) and (64) for g would be valid if we assumed h to be axial; on the other hand, if h were assumed to be polar, then the equations for R would be quite analogous to the corresponding equations for IJ and H. It can be readily verified that the proof of the vanishing of R in the maximally random ease is valid on both assumptions.

304

FIG. 5. Diagram equation 9, .‘. , 111, 113 are respectively plane.)

LEE

for

the two-velocity the reflections

correlation function U. (Diagrams 5, 7, of 4, 6, 8, . . , 110, 112 (about a transverse

Introducing the diagram representations tions UO and Ho as wavy

line -f--f

for the elementary

correlation

func-

UO,

(71)

curly line sm e, Ho ,

(72)

and

we can write a diagram equation (Fig. 5) for the perturbation series (69) by taking two tree diagrams of Fig. 3, placing them so that the trunks face away from each other, and connecting the vertices in all possible ways with wavy or curly lines according to the following rule: A wavy line connects two vertices

HYDROMAGNETIC

TURBULENCE

where a factor vi(k)0 is implied, and a curly line connects two vertices where a factor hi(k)0 is implied. The numerical fact’ors are obtained by multiplying the product of the factors in Fig. 3 by the number of different ways the wavy or the curly lines may be inserted to produce the same diag-ram. Similarly, using the diagrams of Fig. 4, we can write the diagram equation for the series (70) ; t’his equation is shown in Fig. 6. This method can be used to obtain diagram equations for higher correlation functions such as the triple velocity-correlation fun&ion (vi(k)~~(k’)v,(X;~) ); which is obtained from Fig. 3 by joining three tree diagrams; but we shall have no need for considering such correlation functions. In Figs. 5 and 6, we have omitted a large class of diagrams exemplified by

306

LEE

and

,9

A diagram belongs to this class if it can be separated into two parts by severing one wavy or one curly line, and when separated, both parts are different from every diagram of Figs. 5 and 6. Using the conservation of wave numbers at the vertices, it is not difficult to show that these diagrams represent Fourier components of zero wave number. Transforming to a coordinate system in which the average velocity of the fluid is zero, and requiring the average magnetic field to vanish because of isotropy, we find that the sum of all Fourier component’s of zero wave number becomes zero. The procedure for recovering analytical expressions from the diagrams of Figs. 3 and 4 applies to the diagrams of Figs. 5 and 6. Since we now have the wavy and the curly lines, we should add the following rule : Rule 4. Because of the delta functions which always accompany the correlation functions (see Eqs. (62) and (63) ), the curly or the wavy line should be labelled by one wave number with opposite signs at the two ends. Thus f(k)W-----c-

An example of its application V. ANALYSIS

i

will be given in the following OF THE

(74)

j

PERTURBATION

section. SERIES

The perturbation seriesrepresented by Figs. 5 and 6 are not useful as they stand because they are not expansions in powers of a small parameter. However, it is possible to regroup the terms to obtain an expansion of a different kind. The method used is similar to Wyld’s method in ordinary turbulence. Both methods are motivated by, and analogous to, the process of renormalization in quantum field theory. The diagrams are composedof the different elements -

t-) S velocity propagator,

.I.....s. t, l

T

magnetic propagator,

t-$ P+ vertex of the first kind,

o +-+ P” vertex of the second kind, and

t+ U. zeroth order (elementary) velocity correlation function,

(75)

HYDROMAGNETIC

307

TURBULENCE

m t--f HO zeroth order (elementary) tion.

magnetic-field

correlation

func-

We want to introduce their generalizations. First let us consider certain combinations of these elements. Some of these combinations may replace or be replaced by each other or by an element in any diagram of Figs. 5 and 6, and the resulting diagram is also included in the same equation. Combinations of elements that satisfy the above interchangeability property are said to connect like one another. A part of a diagram which connects like an element is a generalized diagram-part. These statements may be clarified by the following example: In t’he fourth order diagram

which represent 8 q

the term

g S,l(k)Plt,,(k)~‘~,(k’)U,,(k x P:,Jk”

- k’)Udk”) - k’)P:,,(k’

- k - k”)S,t(k”

x X,&k

- k -

- k’)

kN)P&(

-k)&,(

(77) -k),

t,he part, (78)

4 D

of the diagram connects like a vertex of the first kind. Replacing elementary vertex Pluz/( -k), we obtain the second order diagram

it with

the

which is just diagram 2 of Fig. 5. On the other hand, by denoting momentarily2 this diagram-part (78) by r+, and representing it with a solid circle 0, we find 2 Momentarily is this particular

because triangle.

r+ and the circle

will

represent

many

diagram-parts,

one of which

308

LEE

that the fourth order diagram becomes a second order diagram with ized vertex” :

a “general-

l

and 0

4

=Z

(81)

b Analytically, (105)

=

these equations

are

q 2Xil(k)Pt,,(k)Unu(~‘)U,Y(k:

-

?C’> x r;u,( --E, --Iz’)&,(

-Ii),

(82)

and r;&k)

-4’)

= ~4P:,,(k”

- S’>PL,(k’ x S,,(k”

-

- k - k”)U,,(kN)

k’)S,,(lc’

- k - k”)P;,,(--k).

(83)

Similarly many higher order diagrams (terms) can be reduced to lower order ones using similar generalized diagram-parts. And we expect the equations written in terms of the generalized parts to be much more compact than those shown in Figs. 5 and 6. Having clarified our motive, we now define the generalized diagram-parts: 1. The generalized vertex of the first kind I? is the sum of all diagram-parts which connect like Pf, and which cannot be subdivided into two separate parts by severing one line (solid, dotted, wavy or curly). 2. The generalized vertex of the second kind I” is the sum of all diagram-parts which connect like PO,and which cannot be subdivided into two separated parts by severing one line. 3. The generalized velocity propagator S’ is the sum of all diagram-parts which connect like the elementary velocity propagator S. 4. The generalized magnetic propagator T’ is the sum of all diagram-parts which connect like the elementary magnetic propagator T. 5. The generalized (or complete) two-velocity correlation function U is the sum of all the terms in Fig. 5. 6. The generalized (or complete) two-magnetic-field correlation function H is the sum of all the t’erms in Fig. 6. These definitions should be supplemented by the following representations:

HYDROMAGNETIC

solid circle

.

f--) r+,

double circle

0

t)r”,

thick solid line thick dotted line

l

l

l

309

TURBULENCE

++ s’, l

l

(84)

++ T’,

l

t.hick wavy line

-

c--) u,

thick curly line

-

-

H.

By finding and summing over all diagram parts which satisfy the foregoing definitions, we obtain equations for the generalized diagram parts. The equations, to the fourth order, for rf, r”, S’, T’, U, and H are shown in Figs. 7,8,9, 10,5, and 6 respectively. The diagrams of Figs. 7 and 8’are different from the diagram of (78) by an anticlockwise rotation of 90”. In the diagram for r”, the vertex on the top always connects with an external velocity propagator or correlation function while the other two vertices connect wit,h external magnetic propagators or COTrelation functions. In terms of the generalized diagram parts, we can rewrite the perturbation seriesof Figs. 5 and 6 as equations containing a much smaller number of terms. First let us deal with a special class of diagrams, which following Wyld, we shall call the diagrams of classA. In Fig. 5, these are the diagrams which can be separated into two parts by severing one wavy line (diagrams numbers 1, 4-79, 119 and 120 are included). Using Eqs. (50) and (71) we can write the special wavy

+ FIG,

7. Diagram

@ =

equation

o+

4

+

+ FIG.

8. Diagram

. . . . . .

A

for

+

.4, 4 ,; ‘\ + duub

the generalized

4

A

vertex

I+

vertex

P

+

4:: + &I!!%

. . . . . .

equation

for

the

generalized

310

LEE

FIG. 9. Diagram 27 are respectively

equation for the generalized propagator S’ (Diagrams the reflections of 6,8, 10, . . , 24, 26 about a transverse

FIG. 10. Diagram 25, 27 are respectively

equation for the generalized propagator T’ (Diagrams the reflections of 6,8, 10, . . . , 24, 26 about a transverse

7,9, 11, . . . , 25 plane).

7, 9, 11, --a plane).

line as y”

=

(Ui(lC)OVj(

=

(Sil(lC)fi(lC)fm(--k)S,j(--k))

= -

-k)ll)

FL,(k)

=

Sil(k)Fbn(k)Si9n(--k)

uw

-)

where am,

= (ft(k)fm(

-k)

> = W2,

~“U’zmW.

(86)

,

HYDROMAGNETIC

311

TURBULENCE

In this form, the wavy line contains two velocity propagators which are just what the rest of the diagram (if there is any) needed to become two generalized velocity propagators. Thus all diagrams of class A in Fig. 5 are represented by -F-

(37)

Diagrams of class A in Fig. 6 are those that can be separated into two parts by severing one curly line; they can similarly be shown to be represented by

where G = Gdk)

= (gl(k)gm(---k)

) = G(k2, u2)Pdk).

(89)

From the diagrams of Figs. 5 and 6 which do not belong to class A, we discard all those that can be reduced to a diagram of lower order by replacing parts of the diagram by generalized diagram parts. We are then left with only irreducible diagrams. The series of all irreducible diagrams with all the elements replaced by the corresponding generalized diagram-parts is equivalent to the complete

-t FIG.

11. Diagrams

FIG.

12. Diagrams

. . . . . . corresponding

+

to an integral

equation

for

U

to an integral

equation

for

H

, . . . . . corresponding

312

LEE

t FIG.

13. Diagrams

corresponding

t FIG.

14. Diagrams

. ...*. to an integral

equation

for

r+

equation

for

P

. . . . . .

corresponding

to an integral

series represented in Figs. 5 and 6. The equivalence can be verified by straightforward substitution. The reduced equation for the velocity and magnetic-field correlation functions are shown in Figs. 11 and 12 respectively. The diagram equations for t,he generalized vertices can also be written in terms of generalized diagram-parts. However, it is obvious that the generalized vertex must not be used in the first order terms. The new equations are shown in Figs. 13 and 14; they are very similar to Figs. 7 and 8 because the vertex diagrams up to the third order are all irreducible. Starting from the fifth order, Figs. 7 and 8 would include many reducible diagrams, and would contain many more terms than Figs. 13 and 14. All the fourth order S”s appear to be reducible to the second order and However, upon substitution for the generalized diagram parts, we find that we do not recover the series represented by Fig. 9. Some diagrams (diagrams numbers 4-7, and 24-31) are counted twice.3 The origin of this miscounting is the specially a Similar miscounting occurs in Wyld’s theory of ordinary turbulence. Wyld attempted to avoid the difficulty by converting the propagator into a vertex part through differentiation and using a process leading to equations corresponding to the so-called Ward’s Identities. Wyld showed how such a conversion may be achieved by considering a scalar equation (equation 7 of Wyld’s paper) with a constant coupling coefficient. However, it can be shown that this process cannot be applied to the full three-dimensional equation. We are indebted to Dr. Wyld for his confirmation (private communicatjons) of this observation.

HYDROMAGNETIC

313

TURBULENCE

symmetric shapes of these second order diagrams; and the discrepancy can be corrected by writing the second order diagrams as and

(91)

which are no longer symmetric. Not all the fourth order diagrams are reducible to these forms; and those that cannot be so reduced are the irreducible S”s of the fourth order. The reduced equation for S’ is shown in Fig. 15. Similarly, we find the reduced equation for T'; it is shown in Fig. 16. We find that with these precautions, no miscounting occurs in 6he equations of Figs. 15 and 16 even when they are expanded to include all terms up to the sixth order. The procedure for recovering analytical expressionsfrom the diagrams of Figs. 11-16 are similar to those for the earlier figures. First, we have to define the trunk and its relations with the rest of the diagram for the generalized propagator or for the generalized vertex. We still use the word “trunk” although t’hesediagrams no longer resemble trees. Most of the following definitions are arbitrary choices; but we need to adopt a particular convention in order to distinguish between t,he two kinds of vertices represented by r” and also becauseof the asymmetric way the conservation of wave numbers has to be satisfied at a vertex. We define the first

FJG.

15. Diagrams

. . . . . . ..-...

---.--.---

-2

to an integral

equation

for

.. . ..

+ 16 ----+

. . .. .

+ lb

. ...-

+ 16 -...-+

.....

16. Diagrams

S’

+ 4.-.- .r%..-.._. 0 . . . . .. + 4 . . . .‘4f2&....

•t 16 ----+

+ FIG.

corresponding

-----+

. . . . . . corresponding

to an integral

equation

for T’

314

LEE

elementary propagator on the left as the trunk in the diagram for the generalized propagator. In the vertex diagrams, there is no element that can be suitably described as the trunk. We should, instead, agree upon one side of the diagram as being the side nearer the trunk. In the diagram for rf in Figs. 7 and 13, we choose the vertex at the top of the diagram as being on the side nearer bhe trunk. When the generalized vertex r” corresponds to a P-, we call it a Y’(-), and say that the bottom left vertex of the diagram is on the side nearer the trunk. When I?’ corresponds to a P+, we call it Y’(+) and say that the vertex at the top of the diagram is on the side nearer the trunk. The above definitions agree with the criterion for determining whether a P” refers to a P+ or a P-; the same criterion applies also

to r”. The equations represented by Figs. 11-16 involve the tensors U, H, S’, T’, r+, and r”. We shall now investigate the properties of these tensors. 1. The correlation functions have alreadybeen defined in Eqs. (59) and( 60) ; their properties are apparent from these equations, namely Uij(k)

= U(k’, u2)Pij(k),

(92)

Hii

= H(k2, u2)Pij(k).

(93)

and

2. The propagators are isotropic tensors of the second rank. Because of the conservation of wave numbers at the vertices, the wave numbers of the elements at the two ends are equal and there are summations over all the intermediate wave numbers; therefore the propagators are functions of one wave number only. They are also orthogonal to k because of the factors &(lc) or Tin (see Eqs. (41)) which they always contain. The most general form for the propagator is therefore S:j(k)

= S’(k)Pij(k),

(94)

!f:j(k)

= T’(lc)Pii(k).

(95)

or

The tensors for the correlation functions and propagators all have the same simple form; furthermore, becauseeach of these tensors consistsof only one scalar function multiplied by a known tensor, the equations for U, H, S’, and T’ must reduce to scalar equations for the functions U(k2, w”), H(k2, w2), S’(k), and T’(k) respectively. 3. The generalized vertices represent isotropic tensors of the third rank. Because of the way they connect, one of the three externally given wave numbers is fixed by the conservation of wave numbers; therefore the r’s are functions of two wave numbers only. Writing the vertices as I’&,,,( k, k’) and I$jm(ky k’), where the

HYDROMAGNETIC

315

TURBULENCE

index i and the wave number k are associated with the side nearer the trunk, we find that I’+ and r” are orthogonal to ki because of the tensor Pf or P- which always occur as a factor. (a) The tensor ~‘>,(k, k’) is, furthermore, symmetric in the indices j and m. As may be readily verified, the most general tensor satisfying these properties is I$,(k,

k’) = A, &k,

+ 6i, kj - & &,

(k&n + k&m) (k;k,

1

+ li,k,‘)] (96)

[A&im + Adkj’h’

+ Askjkm

+ A~(k&m’ + 4’km)l, where A1 , A2 , Aa , A., , As, and As are arbitrary scalar functions of k and k’. The tensor equation for I$&(/c, k’) must reduce to six scalar equations for these functions. (b) The tensor I’&,(k, k’) represents r~~~‘( k, k’) if the external line segment attached to its side nearer the trunk is a velocity propagator, and it represents a rij;)( k, k’) if this line segment is a magnetic propagator. This distinction can be made only when r” is inserted as part of a diagram. The diagram equation for r” by itself must therefore represent two equations: one for Y°C+’ and one for . rot+, has the sametensor form as r+. I$j;‘(k, k’) is antisymmetric in the r O(k) indices j and m; and the most general tensor satisfying these properties of I$i;‘(k, k’) is rfj;)(k,

k’) = B,(6ijk,

- Simkj) + Bz(&ik,’ +

Bski - B2 Lzk2

- &mkj’) ki’]

(kjk,’

- kmk;).

(g7)

The diagram equation for r” therefore represents nine scalar equations, six of them for the functions in Y”(+’ and three of them for the functions in rot-). In a conducting fluid in a state of isotropic turbulence, any small electromagnetic perturbation will create a spontaneous magnetic field, which will not depend on the particular manner in which the perturbation is applied. But we want to stressthe fact that there has to be a perturbation. In our problem, the perturbation is represented by the term gl(k) introduced in Eq. (39). It is clear that gl(k) is needed in the perturbation expansions. Ho&ever, in a self-sustaining state represented by the integral equation of Fig. 12, the effect of the perturbation may be eliminated; and we may simply let gl( k) vanish (cf. Eq. (47)) and make our whole discussionapplicable to the case of hydromagnetic turbulence.

316

LEE

The method which we have described enables us to obt)ain equations governing the velocity and the magnetic-field correlation functions. The equations t’o the fourth order have been derived and the procedure for recovering analytical expressions from diagrams has also been explained. The met,hods are sufficieruly general to give equations of any desired order. Since no approximation has yet been introduced, t,he equation including all terms should give exact results, although t’he convergence of the series expansions has not been investigated. The equations do not represent expansions in terms of small parameters in any obvious sense; inst,ead, they appear to represent expansions in terms of the complexity of the interactions. We do not attempt to define rigorously such a difficult concept here; but it roughly corresponds to the fact that diagrams for the propagators and vertices of increasing orders represent interactions with increasing numbers of intermediate states and various terms in the equations for t.he correlation functions represent relations between various moments similar to the normal relations one would obtain from systems in which the velocity and magnetic field obey gaussian distribut#ions. For example, let us consider t’he equation

-c-F--

(98)

for the velocity correlation fun&on in ordinary turbulence, where we have used elementary propagators and vertices for simplicity. The term

2-c2represents a normal relation between the four-velocity relation functions, and the term

(99) and the t,wo-velocit*y

cor-

represents a normal relation between the six-velocity and the two-velocity correlation functions, and so on. This example indicates that, higher order terms represent relations of a more complicated nature. And we may expect, in view of the hypothesis of maximal randomness which we have assumed, that the more complex t’erms will tend to cancel out, leaving the simplest terms as the dominant ones; the equations may then be truncated at low orders and hopefully, will give good results. This optimism can be justified only after a complete proof of convergence has been carried out and terms of different orders have been compared. We do not attempt such an ambitious undertaking here, but content ourselves by showing that the lowest order truncation gives results equivalent to Chandrasekhar’s equations and also obtaining a set of equations in a higher approximation.

HYDROMAGNETIC

VI.

317

TURBULENCE

APPROXIMATIONS

It does not appear that the exact equations for the correlation functions and other auxiliary quantities which we have derived in the preceding sections can be solved by analytical methods. To obtain practical results, we may truncate the series after a finite number of terms, and try to solve the result’ing approximate equations. In this section, the two lowest order approximations which are naturally suggest’ed will be considered. Truncation of the series equations of diagrams (11-16) at t’he lowest, nontrivial order gives OHM,

=

q&+2.+$)-

-F-+2

r66F6

=

4

(101)

-----

-__-

<-

=

-

--.-es..

=

________

(102)

(103) (104)

l

=

.

(105)

Q

=

0

(106)

The last four equations are trivial and may be substituted directly into Eqs. (101) and (102) ; the equations written with all the essential labels filled in are +
=

s

FLp(h)

-j+

+ 2

j

-+2*

(107) +(h-h’)-

and m

=

4 . ..I

0 0

0 I...

(108)

These equations, writt’en analytically, are Uij(k)

= &(k)F1,(k)Si,(

--k)

+ 2 g Xil(li)Pt,,(lz)li,,(k’)U,,(k

- k’v%,Fwj*(--k)

+ 2 c siI(k>Pt;,,(k-)H*,(lct)H,,(k k’

-

w2&~--Ic&,(--k),

(109)

318

LEE

and = 4 F

Hij(k)

T;l(k)P~~n(rc)U,,(Z;‘)H,,(k

-

k’)Piy,(-k)Tjp(-82).

(110)

Now we let the summations become integrations according to Eq. (24) ; we also use the explicit forms for the various tensors given in Eqs. (41), (42)) (43)) (86)) (92), and (93) ; and finally contract the t,ensor equations to obtain the scalar equations. After these reductions we find

-- 1 s

2U(k2, a’> = 2S(k)F(k2,

02)S( -ii)

2(2n)4

1

--

2(27r)4

J

S(lc)S(-k)U(k”,

w’“) U(k”‘,

?)ul(k,

k’) cZ%’

S(k)S(-lc)H(k’2,

ci2)H(kv2,

c?)u~(k,

k’) c&i’,

(111)

and 2H(k2,

a”> = &

1 T(k) T( - k) U(k’2, ci2)H(k”‘,

uN2)uz(k, k’) c?%’

(112)

where k” = k - jq, al(k, k’)

(113)

= P;l(k)P~~,,(k)P,,(k’)p,,(kn)P~pr(-k)Pip(-k) =

_ 2 (k.k’)(k.k”)(k’.k”)

-2k2

(114)

+ 4 (k.k’)2(k.k”)2

k’2k”Z

’

k2k’2k”2

and udk, k’)

= Pil(k)P~~,(k)P,,(k’)P,,(k”)P;;,,( =

-2k2

-k)f’i,(

-k) (115)

+ 2 (k.k’)(k.k”)(k%‘) k’2kN2

The functions al and a2 describe the manner in which the exchange of energy takes place between the velocities and the magnetic-fields of different wave numbers. They may be called the transfer functims. Apart from the term with the external force, Eqs. (111) and (112) are the Fourier transforms of Chandrasekhar’s equations (4), Chandrasekhar deals with the longitudinal correlation functions f(r, t) and g(r, t) which are functions of relative position and time. They are related to the spectral functions we have used by OD

f(r, t) = $$

o k” dk s

jr&r) kr

/

“) . -“) due-“%(k2,

o”),

(116)

HYDROMAGNETIC

319

TURBULENCE

and dr,

t> = &

* jl(kr) o k2 dk kr s

O” --o?doe-““tH(k2, s

a2),

(117)

where jl(kr)

=

sin (kr) k2r2

-

cos (kr) kr

____

(118)

is the spherical Bessel Function of the first order. It can be shown by actually performing the required transformations of Eqs. (111) and (112) that f(r, t) and g(r, t) satisfy the equations

;(.$ - ?D:)f=f;Dsf+g$D5g-2

(119)

and (&A”D:)g=fDag+gDbf+$$

(120)

where 5 the longitudinal correlation function of the external force is the transform of F(k2, w’) by a transformation similar to Eqs. (116) and (117), and (121) is the five dimensional Laplacian operator. Chandrasekhar leaves out the external force term, but otherwise, his equations are precisely our equations (119) and (120). In the next approximation, we retain some nontrivial terms in the equations for the propagators while leaving the diagram equations for the correlation functions and vertices as in the last approximation. (This “second approximation” is similar to Kraichnan’s approximation (3).) The equations for the correlation functions are therefore Eqs. (111) and (112) with S and T replaced by S’ and T’ respectively. The equations for the propagators are Hh-b’)-= (h)

i

I

(122)

i

and ..m.w...

Analytically,

=

-----

these equations

-_

are

+

4

---

(123)

320

LEE

Passing from sums to integrals tions for the defining scalars: 2X’(k)

= 2,9(k)

- h4

and contracting,

/- S(k)U(k”‘,

we obtain t,he:following

w”*)S’(k’)S’(k)b~(k,

k’)

equa-

c141c’ (126)

-~ ,2:,4

S(k)H(k”2,

s

w”’ )T’(k’)S’(k)bz(k,

k’) cl%‘,

and T(W(k”2,

un*)T’(IC’)T’(k)b~(k,

k’) $k’ (127)

-(2i,4

s

T(k)H(k”‘,o”‘)S’(k)T’(k)b,(k,

k’) d41c’,

where k” = k h(k,

k’)

2 k’.(k

- k’)k.(k (k - k’)2

(129)

- k’) ’

= P,,(k)Pt,,(k)P,,(k”)P,,(k’)P,,(k’)P,~(k) = -2k.k’

ba(k, k’)

(128)

= Pil(k)Plt,,(k)P,,(k”)P,,(k’)P~~,(k’)P,i(k) = 2 (k.k’13 k2k’2-

bz(k, k’)

k’,

+

2k.k’ [k.(k k2(k - k’)2

- k’)l’,

(130)

= Pil(k)P~~,(k)P,,(k”)P,,(k’)P,,(k’)P,i(k) = 2k k, _ 2 k’. (k - k’)k. (k - k’) (k - k’)2 ’

(131)

and bdk, k’)

= P~,(k)P,,(k)P,,(kN)P,,(k’)P~~‘,,(k’)P,i(k) = --2k-k’

2k.k’ + k’2(k ___ _ k,)2 [k’ . (k - k’)12.

(132)

HYDROMAGNETTC

321

TURBULENCE

In contrast to the lower approximation in which there are only two scalar equations for the two correlation functions and two transfer functions al(k, k’) and az( k, k’) , in the “second approximation” we have four scalar equations for the correlation functions and propagators, and six transfer functions ol , a2 , bl , bz , ba , and bq . In higher approximations (of which one may consider many variations), we have in general nineteen scalar equations: two for the correlation functions, two for the propagators, six for the vertex of the first kind and nine for t’he vertex of the second kind, and many more transfer functions. Before one considers these higher approximations, it may be more useful to derive the full implications of the approximations we have already obtained. VII.

CONCLUSIONS

The purpose of this paper has been to formulate a set of equations describing stationary, homogeneous, isotropic turbulence in an incompressible conducting fluid, taking magnetic interactions into account. The method used is a generalization of one used by Wyld in his formulation of the theory of ordinary turbulence, though the reduction of the propagator diagrams has been carried out differently. The result is a set of six diagram-equations for the spectral twovelocity and two-magnetic field correlation functions and other related quantities. Terms in these equations all have some intuitive significance, but we have not attempted to describe them rigorously. The difficult problem of solving these equations still remains. To solve these equations in their complete, infinite forms is probably out of the question. For the present, we have been contented with approximations obtained by truncating the equations after finite numbers of terms. Plausibility arguments have been given to support the truncation procedures, these include showing that a truncation at the lowest nontrivial order gives Chandrasekhar’s equations which were obtained by a different method. We have also worked out the next approximation; this higher approximation will provide a basis for comparing terms of different orders. ACHNOWLEDGMENT The author wishes to express his sincere appreciation to Professor his guidance throughout the years when the author was a graduate gesting the thesis topic, and for many valuable discussions. RECEIVED:

August

20, 1964 REFERENCES

1. H. W. WYLD, Ann. 2. S. CHANDRASEKHAR, $. R. H. KRAICHNAN, 4. S. CHANDRASEKHAR,

Phys. (N. Y.) 14,143 (1961). Proc. Roy. Sot. AaO4, 435 (1950). Phys. Rev. 109, 1407 (1958). Proc. Roy. Sot. A233,322 (1955).

S. Chandrasekhar student, and for

for sug-