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J . Fluid Mech. (1970), vol. 44, art 4,pp. 705-719

Printed in Great Britain

7 05

Dynamo action associated with random inertialwaves in a rotating conducting fluid

By H. K. MOFFATT

Department of Applied Mathematics and Theoretical Physics,

Silver Street, Cambridge

(Received 26 March 1970)

1 It is shown that a random superposition of inertial waves in a rotating conductingfluid can act as a dynamo, i.e. can systematically transfer energy to a magnetic1 field which has no source other than electric currents within the fluid. Dynamo

action occurs provided the statistical properties of the velocity field lack re-

flexional symmetry, and this occurs when conditions are such that there is a net

energy flux (positive or negative) in the direction of the rotation vector Q.

If the magnetic field grows from an infinitesimal level, then the mode of maxi-

mum growth rate dominates before the back-reaction associated with the Lorentz

force becomes significant. This mode is first determined, and then the back-

reaction associated with it alone is analysed. It is shown that the magnetic energygrows exponentially during the stage when the Lorentz forces are negligible, then

reaches a maximum depending on the values of the parameters

R, = u0l/h, Q = !212/h,

(uo= initial r.m.s. velocity, 1 = length scale characteristic of the velocity field,

h = magnetic diffusivity) and ultimately decays as t-l (equation ( 5 . 1 5 ) ) .This

decay is coupled with a decay of the velocity field due to ohmic dissipation, and

it occurs because there is no external source of energy for the fluid motion.

1. Introduction

In a previous paper (Moffatt 1970, hereafter referred to as I), he effect of

turbulence on a weak magnetic field in an electrically conducting fluid was con-

sidered; and it was shown that when the magnetic Reynolds number R, = uol/h

is small (uo= r.m.s. velocity, 1= lengthscaleof energy containingeddies,h = mag-

netic diffusivity), exponentially growing magnetic modes are possible, provided

the statistical properties of the turbulence lack reflexional symmetry. The grow-

1 ing magnetic field has no source other than electric currents flowing within the’ fluid itself. It has a length scale L large compared with 1 ( L= O ( R G ~ )) and a

time scale t , (doubling time) large compared with the time scale to = l/uo harac-

teristic of the turbulence (tl = O ( R G ~ )o).The exponential growth continues only

for so long as the back-reaction of the Lorentz force on the fluid can be neglected;

this (dynamic) aspect of the problem was not investigated in I.

In the present paper the investigation is extended and specialized to a situa-

tion that may be of particular relevance and importance in geophysical and

45 FLM 44

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706 H . K . Moffatt

astrophysical contexts. We suppose th at the ‘turbulent velocity field u(x, )

consists of a random superposition of inertial waves in a fluid rotating with uni-

form angular velocity 51.It will be supposed that the Rossby number is small, i.e.

( 1 . 1 )

so that inertial interactions between waves of different wave-numbers may be

neglected. Further, viscosity will be neglected, on the grounds that viscous dis-

sipation is likely to be dominated by ohmic dissipation in situations of practical

interest. It will first be shown that such a motion is capable of amplifying a

magnetic field B(x, )on scales L,T arge compares with 1and t , through much the

same mechanism as described in I. The nature of the amplification processes is

largely governed by the values of th e parameter Q = Q12/h= RmR i l . If

R, = Uo/Ql < 1,

& < 1 , (1 .2 ) ’

(sothat,by(l .1) R m < 1 (1.3)

also), then the process is identical with th at of I; but ifc

& + I , (1.4)

(the value of R, being unrestricted) there are some important differences; e.g.

the growth of the magnetic field shows strong directional preferences, even if the

amplitudes of the inertial waves are isotropically distributed.

As the field grows in strength, it begins to react back upon the constituent

inertial waves of the velocity field in a manner that can be explicitly taken intoaccount. The most significant effect is that each inertial wave, through its

coupling with the magnetic field (which may be regarded as uniform and steady

over scales characteristic of the velocity field), loses energy to the ohmic sink.

I n the absence of any body force distribution, the velocity field therefore decays

as the magnetic field grows. Ultimately th e magnetic field must likewise decay

to zero. We are primarily interested in the maximum level at tained by the field

before this ultimate decay sets in. In the absence of any input of energy, the mag-

netic field clearly cannot acquire more energy than is released from the velocity

field; for a random magnetic field, this implies

(pp)-l (B2) < U: -2, (1 .5)

where U, is now the ‘initial’ ( t +-CO) r.m.8. velocity, p and p are respectively

the magnetic permeability and the density of the fluid, and the angular brackets

(...) are used to mean an average over scales large compared with L.Actually,

it will turn out that in none of the circumstances considered can the magnetic

field acquire more than a small fraction of the initial kinetic energy associated

with the wave motion.

As in I, it will be convenient to write

B(x, ) = B,(x, )+b(X ) , (1.6)

where B, = varies only on scales L, T, nd b is the small fluctuation field in -

duced by the motion U across B,; it will emerge in $ 2 th at the condition (1.1)

ensures tha tIbl IBoL (1.7)



Dyn amo action in a rotating conducting jluid 707

v

tIt will further be convenient to use the equivalent Alfvh velocities

, h, = (pd-4 B,, h = (pp ) - ib , (1 .8 )

as a measure of the magnetic field. As in I, the large scale field h, evolves accord-ing to the equation

(1 .9)

and the main objective is to find an expression for U A h in terms of the initial

properties of the velocity field, and of h, and t ; and then t o solve (1 .9) . To find

U A h we need to investigate some of the detailed properties of inertial waves in an

(apparently) uniform steady magnetic field h,, and this is done in the following

section.The possibility of dynamo action due to a mechanism of the type analysed in

this paper was first explored by Steenbeck, Krause & R&dler (1966). The idea

was further developed in a series of papers by the same authors, full references to

which are given in I. In none of these papers however was the back-reaction of

the growing magnetic field taken into account; and an understanding of this

process is the chief objective of the present investigation.

I -h, = V A U A h+hV2h,,at

,

;-

,

2. Inertial waves in the presence of a uniform steady magnetic field

field h, are (Lehnert 1954; Chandrasekhar 1961, $50 )

The linearized equations governing inertial waves modified by a magnetic

a U / a t + 2 n A U = -VX+h,.Vh, (2.1)

(2 .2)

V . U = V.h = 0. (2 .3)

The linearization is valid provided IuI 4 dl and Ihl 4 hol.~ ( x ,) is a reduced

pressure distribution, modified by the centrifugal force, and the irrotational part

(2 .4)f the Lorentz force:

(2 .5)

(2 .6)

(2.7)

(2.8)

ah/at= h, .VU +hV2h,

x = p / p+ 2p) -I B2- S2 A x ) ~ .

Equations (2.1)-(2.3) admit plane wave solutions of the form

(U, ,x)= (Q,h, 2)exp {i(k.x-wt) ] ,

- i d + 2S2 A Q = - kf+i(h,. k)h,- w i = i(h, .k)Q- k26,

where

nk .Q= k . h = 0.

Such waves degenerate to pure inertial waves in the limit h, -+ 0, and to dampedAlfv h waves in the limit S2 -+ 0. Equation (2.7) gives the important relation

between h and Q; and (2 .6) then gives

where

(2.10)

(2.11)

45-2



708 H . K . Moffatt

From (2.10)using k . Q= 0, we have

i c ~ k ~ Q - 2 ( k . Q ) Q0,

and, crossing again with k ,id2Q+(k .8 )k A Q = 0,

so that , eliminating Q,

and, correspondingly,

v = f ( k . Q ) / k ,

Q = i k A Q = f k Q ,

where Q is the vorticity Fourier component corresponding to 0.

(2.12)

(2.13)

(2.14)

(2.15)

T he situation when h, -+0

When h, = 0, CT= - U , so that from (2.14),

U =T 2 ( k . ~ ) / k T 2 ~ ~ ~ s e ,

(2.16)

where 8 is the angle between 8 nd k.This is the well-known dispersion relation

for pure inertial waves (Greenspan 1968). The phase velocity is wk / k 2 ,and the

group velocity is

(2.17)

where QL is the component of 8 erpendicular to k (figure l (a)) . Hence, if

k . 8 > 0, the upper and lower signs correspond to propagation ‘downwards’

and ‘upwards’ relative to the direction of Q.

2

Ic3cg= V ~ U T - Qk2- (a. )k) = T 2 8 J k ,

(4 . (b )

FIGURE.(a) he phase velocity c = 2(k.$2)k/k3, nd the group velocity c , = 2kr\ (Q A k) /pfor inertial waves with energy flux in the positive z-direction. ( b ) The velocity field in a

single inertial wave (with or without magnetic field).

A random superposition of such waves will exhibit a lack of reflexional gym-metry only if there is a net energy flux upwards or downwards, i.e. only if there

Etre more of the upward propagating waves than the downward (or vice-versa).

Such a situation could arise, for example, if the waves were generated in the half

space x > 0 by random mechanical excitation on the plane z = 0; in this case,

only the upward propagating waves would be present. R‘e shall assume that only

such waves are present in what follows.



Dynamo action in a rotating conducting fluid 709

The velocity field in a typical upward propagating wave is indicated in figure

1 b ) . The streamlines are straight, and their direction rotates clockwise in the

direction of the wave-vector k . The particle paths are circles in planes perpen-

dicular to k .A measure of the lack of reflexional symmetry in a single wave is provided by

t h e h eZ i c i t yu . o = u . ( V ~ u ) .rom (2.15),

u.o=+aa.o f & k l Q 1 2 , (2.18)

so that the helicity of upward propagating waves is negative. A random super-

position of upward propagating waves gives a velocity field.

-u(x,t) 92 Q(k)exp{i(k.x-w(k)t))d3k, (2 .19)

swherein we may suppose k.8 0, and where, in the limit h, -+ 0,

w(k) = 2(k .Q) /k= 2L2cos8. (2.20)

The spectrum tensor of this velocity field is

Qi3(k)= lim a:(k) Qj(k) sk.dsk-+O

If the amplitudes of the waves are isotropically distributed, this must take theform

i F ( k )Qij(k)= E (k2& - , kj)fmi j k k k ,

2Tk4( 2 . 2 2 )

where E ( k )= .lrk2Qi,(k)= r k 2 lim IQ(k)I2d3k (2 .23)

is the energy spectrum function, andd'k+O

(2 .24)

is the helicity spectrum function. The assumption that there are only upwardpropagating waves means that

so that

ik A Q = Q(k)= - kQ(k) ,

F ( k )= - kE ( k ) .

The velocity scale U , and length scale 1 characteristic of the wave field may now

be defined by

W =

Iom( k ) k = +U&

-

(2 .27)

and ( 2 . 2 8 )

t When the overbar appears in such an expression, it must be interpreted as an en-

$ More generally, if a mixture of upward and downward propagating waves were oon-semble average, which is identical with the space average for homogeneous turbulence.

sidered, a relation of the form

where l a (k ) l c 1, would hold.~ ( k )2 4 k )~ ( k ) ,



7 1 0 H . K . Moffatt

grounds

and (2 .2 7 )and ( 2 . 2 8 ) then become

If U , and 1 are the only scales characterizing the field, then on dimensional

(2 . 2 9 )(k)= &:Zf(kl),

(2 . 3 0 )

A particular form off ( r ) , atisfying these constraint's, that will be used by way of

illustration in what follows, is

f ( r ) 4 7 - - 1 ) , ( 2 . 3 1 ) t

corresponding to a sea of waves, randomly oriented, bu t all having the same wave-

length 27~1.

Dynamic inJluenceof the magnetic f ield

When h, $: 0, (2 . 1 1 ) and ( 2 . 1 4 ) (lower sign) give

(2 . 3 2 )

The condition U, < C21, or equivalently u, k < C2 for all k giving significant con-

tributions t o the integral ( 2 . 2 7 ) ,and the gross energy constraint (from ( 1 . 5 ) ) ,

lhol < U,,

together imply that Iho.k[< C 2 .

(2 . 3 3 )(2 . 3 4 )

Hence (2 .29) implies that , except possibly when 6' E =&T,he magnetic field causes

a small modification in w :

w x 2C2cos6'+ (h, k)22 Q cos6'+ ihk2'

(2 . 3 5 )

and it is evident that this approximation is va'lid provided

1h, .kl < ( 4 Q 2cos26'+h2k4)4. (2 . 3 6 )

We shall use (2 . 3 5 ) in what follows, and investigate the limits of its validity in

It is evident from (2 . 3 5 ) that when hoe =l=, w becomes complex, implying a

damping of the waves; indeed if w = w , + w i , then the lowest approximation for

9 5 .

-(h,-,.k)2hk2, and wi is

wi 4 ~ 2os2 e+m 4w, w 2C 2COSB ) (2 . 3 7 )

Note that waves for which h,, k = 0 are not damped; such waves show no ten-

dency to bend the magnetic lines of force, and they therefore cannot feel the

effect of ohmic dissipation.

t A more realistic form, satisfying (2.30) and the necessary kinematic constraint near

9 = 0, might be

bu t there seems little point in complicating the analysis in this paper by the use of such a

function.

f(7) C74e-6rl, C = 55/4!,



Dynamo action in a rotating conducting Jluid 711

Evidently, A,, is a real symmetric tensor, depending on the parameters Q and

h!t/h,and on the orientation of the vector harelative to SZ. We shall f i s t consider

(in 3 4) the form of A , when h, is so weak tha t the exponential factor in the inte-

grand is effectively unity; and we shall show th at in these circumstances dynamoaction occurs for all values of the parameter Q. This means that h, grows expo-

nentially so that a t some stage the exponential factor in the integrand becomes

important in restricting the growth of ho.This effect will be examined in detail

in 95.

1j

3. The expression for ur\h

4. Dynamo action during the stage of negligible Lorentz forces

Neglect of the Lorentz force is equivalent to taking the limit h, + 0 in (3.8),

i.e. t o omitting the exponential factor. The $-integration is then trivial and we

have(4 .1)

I

4 j utl[ao(Q) ij+ YO(Q) -ao(Q)) i Qj/Q21,

We are now in a position to calculaten

ur\h = &% IQ A h*d3k.J

Using (2.9) we have

and, using (2.25) an d k .Q ( k ) 0,

Hence hk(h,.k )kE(k) auit

nk21w+ihk212 *

%'Qr \ i ; *=

Under the condition (2.36) for all relevant k,we therefore have

~

(U h)i = h-lAiihoj,

(3.4)

(3.5)

where (3.6)

If t he spectrum function E(k)has the form ( 2 . 29 ) ,withf(7) given by (2.31), henputting

(3.6) becomes

k = k(sin 0 cos4 , sin 8sin 4, cos 8) = hk, (3.7)

1 .ircos28sin0d8= 1-[l--tan-'2Q].

Y O ( Q ) 5 I a 1+ 4Q2cos28 8Q2 2Q(4.3)



712 H . K . Moffatt

These functions are sketched in figure 2. Note tha t

ao(0)= YO(0)= &,

a,(&)> yo(Q).

and that, as Q -+ CO, a,(&) n-/2&, yo(Q)N l / 8Qz .

Also,forallQ > 0,

Q

FIGURE. The functions a,,(&)nd yo(&)defined in (4.2), (4.3).

In the limit Q +-0,Ail becomes isotropic:

Aij = & ~ $ l S i j , (4.7)and this limit corresponds to the situation considered in I. For Q + 0 however,

Ai3 s anisotropic (though still axisymmetric about the direction ofS) ven if the

amplitudes of the inertial waves are isotropically distributed; this arises of course

as a direct result of the anisotropy of the dispersion relation (2.16).

From ( 3 . 5 )and (4.1)we now have

__

U A h = A-’ [alho+ (71- a,) (a.,) Q/Q2], (4.8)

where a1= uiZa,,y, = uily,, and so equation (1.9)becomes

an equation linear in h,, with constant coefficients. This equation admits ‘wave-

type ’ solutions of the form

h, = 9 ( & o e i K * x e m t ) ,K.h , = 0, (4.10)

m=-

K2 h-l{a,y,(Kt +K i )+a;K$}3. (4.11)The upper sign corresponds to an exponentially growing magnetic mode whenever

a1y1(K4+RZ)+a2,Kl > h4R4, (4.12)

and there are certainly wave-vectors K fo rwhich this inequality holds (figure3 ( a ) ) .It may be noticed in passing that the same separation of wave-number

space into a region of amplification and a region of decay occurs in dynamo

where (cf.I, 8 6 )



Dynamo action in a rotating conducting$uid 7 13

models in which the velocity field is a simple periodic function of the space

variables (Roberts 1969; Childress 1969) .

Of particular interest is the mode having maximum growth rate, since it is the

one which will dominate in a magnetic field which has been amplified from aninfinitesimal level. Since y1 < al,he maximum value of m, for given IKI (taking

the upper sign in (4 . 11 ) )occurs for

K = (0, ,K ) , so that h, = (Aol, AO2,0) (4 . 13 )

and then m = -hK2+alK/h. (4 .14)

Clearly m > 0 , indicating dynamo action, if K < al/h2.Substitution of (4 .13)

and (4 . 14 )in (4 . 9 ) gives

and so ( 4 . 1 0 )becomes

Ao2 = iA,,,

h, = h,,(cos K ( z- , ), - in K ( z- , ), 0) em t (4 . 15 )

for some z,. This is a ‘force-free’ magnetic field (Roberts 1967)with straight lines

of force, whose direction rotates (anticlockwise) with increasing z (figure 3 ( b ) ) .

FIGURE. ( a )The surface a,y,(K: +K i )+a!K! = A K4 in K-space separating t..e region o

amplification of magnetic modes proportional to exp ( iK .x) from the region of decay.

( b ) The magnetic mode of maximum growth rate, given by (4.15).

For a magnetic field that has grown from a n infinitesimal level, the mode for

which m has a maximum value will dominate long before the Lorentz forcebecomes significant. From ( 4 . 1 4 ) , he maximum value m, of m occurs at K = K,

K, = a1/2h2, m, = m(KJ = af /4h3. (4 .16)here

For consistency, these values must satisfy

K,Z& 1 and m,51-1& 1; (4 . 17 )

the first of these ensures that the length scale of variation of the growing field,

L = O ( K ; l ) , is large compared with the length scale of the background velocity



714 H . K . Moflatt

field, and the second ensures that the growth rate is slow relative to the time-scale

of the velocity field, so tha t the treatment of Q Q 2 and 3, in which h, is treated as

locally uniform and steady, is legitimate. With a, = uglao(Q),he conditions

(4 .17)becomeR i 29 Q2ao(Q) nd R i 49 Q3(a0(Q))2. (4.18)

When Q = O(1) or less, these conditions are both implied by the assumed in-

equality (1.1). f Q 9 1, however, the first inequalityof (4.18),with a,(&)N n/2Q,

becomesR i 29 Q B 1, (4.19)

a somewhat stronger condition than ( 1 .1 ) . If Q 9 R i 29 1, there may still be

dynamo action, but the 'double-length-scale analysis of this paper would not

then appear to be legitimate.

5. The effect of the Lorentz force

The exponential growth of the magnetic field described in Q 4 cannot continue

indefinitely; no matter how weak the initial field may be, at some stage the

back-reaction of the Lorentz force on the fluid motion must be taken into account.

We shall suppose that there has been sufficient time for the emergence of a

dominant mode of the form (4 . 1 3 ) , ( 4 . 1 5 ) .The field h, defined by (4.15)has the

property that ht is independent of position, and this means that the principal

values of the tensor A ij defined by ( 3 . 6 ) or ( 3 . 8 ) remain independent of x evenwhen the influence of h, is included. The principal axes of A ij (with h, , 2 = 0)

are in the directions SZ, h, and S2 A h,, and Aijnow has the form

where a,p and y will now depend on the parameter S ( t )= hgt/A as well as on Q.

From ( 3 . 5 ) ,we then have

so that it will be sufficient to calculate a .

-A h = utl(a/A) ,, ( 5 . 2 )

Choosing axes so that, locally,

= (090,a), , = (h,, 0, ) , (5.3)

and with k given by (3 .7 ) , we have

I m p . (5 .4)5 sin2B cos2 5

1+4Q2cos26A , , - 111sin3Bcos2q5

ugl 47r 1+4Q2c0s26(Q,S)=- -

The asymptotic forms of this function for Q 3 0 and Q -+ O are obtained inthe appendix. Note th at

Substitution of (5 .2 )in (1.9)gives

ah, u21a-- 0 A ho+hV2h,,at A

( 5 . 5 )



Dynamo action in a rotating conducting $fluid 7 15

and we now take account of the dependence of a on h$.The growing mode, selected

on the linear analysis of $ 4 , satisfies

V A h, = K,h,, V2h, = -K2,h0, (5.7)

and since a s independent of x (though dependent on t ) , his behaviour persists

for all t , if only this single magnetic mode of maximum growth rate is considered.

Hence ( 5 . 6 )becomes

h, -hK: h,, (5 .8 )ho flullaKcat h

or, in terms of the magnetic energy density

M ( t )= +h$, ( 5 . 9 )

(5.10)

where now s = ZtM( t ) / h .

For small t , S < 1, a M a,(&),and M increases exponentially as described in

$ 4 . As t increases, S therefore increases, and so from ( 5 . 4 ) , a(Q,S ) ecreases.

However, a cannot decrease permanently below the level +a,(&), ecause if it

did, equation (5.10)would imply an exponential decrease of M and so a decrease

of S and so an immediate increase of a. Hence for t --f CO, we must have

S+ So(Q)

a(&,So) = &a,(&),

where S,(Q) is determined by(5.11)

(5.12)

and is O(1) for all Q (see appendix).

The maximum value of M ( t )attained before the ultimate decay

M ( t )N +AS,(&)-l (5 . 13 )

sets in, may now be estimated. For values of t such that S(t)< 1,

M ( t )=MO xp {&h-3(a1(Q))2} , a,= flu$Za,, (5 .14)

where MO is the initial energy density in the mode of maximum growth rate

(assumed small). Thefunctions (5.13)and 5.14)have the same order of magnitude

M1 = s S o ( Q ) (5.15)1

(5.16)

and the maximum value atta ined by the magnetic energy density is therefore

of order Ml(Q). or Q < 1, the function (5.16) has the behaviour

Ml RLu;,

while for Q $ 1, it has the behaviour

Ml Q-2 R U: = R$u$.

(5.17)T

(5.18)

t The symbol k is used to cienote an asymptotic dependence with constants of order unity

omitted.



716 H . K. Mojfatt

I n both limits, Ml < U:, so that the magnetic field does not in fact acquire

more than a small fraction of the initially available kinetic energy of the motion,

The rate of dissipation of kinetic energy via the Lorentz force to the ohmic sink

is an order of magnitude greater than the rate of conversion of kinetic energy to

magnetic energy. In this sense, the form of dynamo action considered is grosslyinefficient, but even an inefficient dynamo is of course more significant than no

dynamo at all.

0 -001)- t

FIGURE.A sketch of the development of the magnetic energy density M ( t ) n th e mode ofmaximum initial growth rate. During state I, t < t,, Lorentz forces are negligible; during

stage I1 9 ,, the magnetic energy decays because the velocity field which feeds it decays

through ohmic dissipation.

We may now check on the validity of the condition (2 . 36 ) ,which will be satis-

lhol < (5 . 19 )ied for all k provided

for all t , or equivalently provided

NI4 V 2 = R,'U~. (5.20)

When Q < 1 (so that R, < l ) , this is certainly satisfied by virtue of (5 .19) .

When Q 3 1 , from (5 .18) , it requires that

Rz2 RL = Q2R:, i.e. R i a3 Q2 , (5 .21)

consistent with the requirement (4 . 19 ) . It should be noted that when Q 1,

i t is those inertial waves for which 6'M *n, os0 M 0, which contribute most t o the

principal values of A i j ,so th at the conditions (5 . 19 ) and ( 2 . 3 6 )are virtually the

same for those Fourier components of the velocity field which are of most crucial

importance in the analysis.

6. Discussion

The analysis of the foregoing sections shows that a random superposition of

inertial waves in a rotating fluid is certainly capable of transferring energy to an

initially weak magnetic field, and it describes one mechanism by which this trans-

fer may ultimately be limited by the intervention of Lorentz forces. The model

however, suffers from the defect that it cannot predict the development of a



Dynamo action in a rotating conducting Juid 7 17

steady s ta te in which transfer of energy to the magnetic field is exactly balanced

by ohmic dissipation. This is because no sources of energy are present in the

model, and the presence of dissipation implies that ultimately the sum of kinetic

and magnetic energies decreases to zero. I n this sense, the model of this paperbears the same relation to the observed phenomenon of steady (or a t least quasi-

steady) geophysical and astrophysical dynamos as the theory of decaying homo-

geneous turbulence bears to the observed phenomenon of statistically steady

shear flow turbulence : th e results are suggestive and intrinsically interesting bu t

are otherwise not of great value.

There are two ways in which the model may be modified so that a steady

dynamomayresult, but both modifications lead to major difficulties. The first, and

simplest expedient, would be to introduce a random body force distribution

f(x, ) on the right-hand side of ( 2 . 1 ) ;but then in order to have a turbulent field

of finite energy in the limit h, -+ 0, we have t o include viscous dissipation also.

A further difficulty is that the velocity field and so the properties of the

growing magnetic field will be determined by the statistical properties of the

assumed field f(x, ) ;and unless some information is available concerning this,

the labour involved in carrying ou t the calculation is hardly justified.

The second, and more realistic, way to supply energy to the fluid is to do so

through the fluid boundaries, either by thermal or by mechanical means. (In

the case of the fluid in the earth's core, both mechanisms are probably present.Thermal convection has long been considered an important mechanism in

driving the irregular core motions that are inferred from, for example, secular

variations of the surface magnetic field. Mechanical excitation can arise through

relative motion of the core fluid and irregularities on the inner boundary of the

mantle; and a recent analysis of the correlation between magnetic and gravita-

tional pertiirbations on the surface of the earth (Hide & Malin 1970) suggests

strongly that this also is an important mechanism.) A statistically steady state

is then certainly conceivable, but unfortunately the idealization of spatial

homogeneity must be abandoned, since the wave energy of the backgroundturbulence must necessarily attenuate in the direction of energy propagation.

The mechanism of control of the growth of magnetic energy is (in this paper)

very simple: where the growth is most rapid, the dissipation of the velocity

field (which feeds the growing magnetic field) is likewise most rapid, and sothe growth weakens. In the case of a steady dynamo, with a mechanical source

of energy, as envisaged in the two preceding paragraphs, the control mechanism

would be more subtle. The vital term V A U A h in ( 1 . 9 )arises essentially because

U and h are out of phase; but as h, grows in strength, the phase difference between

U and h decreases (for a non-dissipative Alfvbn wave, it vanishes altogether),

and so V A U A h will decrease until some kind of balance with thedissipative term

AV", of (1.9)is possible. The ultimate level of magnetic energy attained in these

circumstances may well be very much larger than the maximum level H,at ta ined

under the conditions of § 5 of this paper.

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718 H. K. MoJffatt

Appendix

function defined in (5.4), viz.

We have to obtain the asymptotic behaviour for small and large Q of the

1 2” sin3Bcos2$ -SSsin2 3 cos2$

a(Q,S) GS, S, 1+4Q2cos2 3 1+4Q2cos2 8

(i) Q < 1

I n this limit, explicit dependence on Q disappears, and the integral may be most

conveniently simplified by using polar angles 8’, $’ measured from ho; he half-

space I 3 < in-becomes the half-space 0 < $’ < n-, and ( A l )becomes

~ ( Q , s ) ~ ( o , s )4j os213’e-25(30828’sin13’a13’

=- (25)-%erf(2S)fr-e-25].

1 ”0

(A 2)nfr

168

(ii) Q 9 1

For general Q, the $-integration in (A1)may be carried out in terms of the associ-

ated Bessel function I&) (Gradshteyn & Rijzhik 1965, $3.388):

where

For Q 9 1, the dominant contribution comes from the neighbourhood of ,U = 0,

and here

(A5 )

Changing the variable of integration in (A3) from p to x = q/S, we have, for

&-+a,

s~ Q P (slq-‘ 1+4Q2p2’

a: N g(f9/32Q, (A 6)

where

The functions defined by (A2 ) and (A7) are monotonic decreasing functions

of S (as is clearly the general expression (Al) ) . The function S,(Q) defined by

(5.12) is clearly O(1) as Q--f

0 end as Q+CO; and since a(&,S)decreases morerapidly with S when Q is large than when Q is small (the integral (A 1)being then

dominated by contributions from the neighbourhood of I 3 = in-), ,(Q) is also

monotonic decreasing, and O(1)for all Q.

P



Dyn amo action in a rotating conducting $uid 7 19

R E F E R E N C E S

CHANDRASEKHAR,. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford Univer-

sity Press.CHILDRESS,S. 1969 A class of solutions of the magnetohydrodynamic dynamo problem.

Reprinted from The Application of Modern Physics to the Earth and Planetary Interiors

(ed. S.K. Runcorn). New York: Interscience.

GRADSHTEYN,. S. & RIJZHIK, . M. 1965 Tables of Integrals, Series and Products. New

York : Academic. ’

GREENSPAN,. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.

HIDE, R. & MALIN,S. R. C. 1970 Novel correlations between global features of the

Earth’s gravitational and magnetic fields. Nature, 225, 605.

LEHNERT, . 1954 Magnetohydrodynamic waves under the action of the Coriolis force.

Astrophys. J . 119, 647 .MOFFATT,H. K . 1970 Turbulent dynamo action at low magnetic Reynolds number.

J . Fluid Mech. 41, 435.

ROBERTS, . 0. 1969 Periodic dynamos. Ph.D. thesis, Cambridge University.

ROBERTS,. H. 1967 An introduction to Magnetohydrodynamics. London : Longmans.

STEENBECK,., KRAUSE, . & RADLER, . H. 1966 Berechnung der mittleren Lorentz-

Feldstarke v A B fur ein elektrisch leitendes Medium in turbulenter, durch Coriolis-

KrBfte beeinflusster Bewegung. 2.Naturf. 21a , 369.

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