H.K. Moffatt- The mean electromotive force generated by turbulence in the limit of perfect...

8/3/2019 H.K. Moffatt- The mean electromotive force generated by turbulence in the limit of perfect conductivity

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J . Fluid Mech. (1974), vol. 65 , part 1 , pp. 1-10Printed in Great Britain

The mean electromotive force generated by turbulencein the limit of perfect conductivity

By H. K. MOFFATTDepartment of Applied Mathematics and Theoretical Physics,University of Cambridge

(Received 28 September 1971 and in revised form 16 November 1973)When homogeneous turbulence acts upon a non-uniform magnetic field B,(x, ) ,a mean electromotive force 8(x, ) s in general established owing to the corre-lation between the velocit5 field and the fluctuating magnetic field that isgenerated. The relation between 8 and B, is linear, and i t is generally believedthat , when the scale of inhomogeneity of B, is sufficiently large, it can be ex-pressed as a series involving successively higher spatial derivatives of B, :

cfi = ailB , +barnB,,/ax, + .where the tensors ail,barn,.. are determined (in principle) by the statisticalproperties of the turbulence and the magnetic diffusivity h of the fluid. Thesetensors are of crucial importance in turbulent dynamo theory. In this paper thequestion of their asymptotic form in the limit h -+ 0 is considered. By putt ingh = 0 and assuming that the field is non-random at an initial instant t = 0, hedeveloping form of the tensors ail(t) nd Pil,(t) is determined. The expressionsinvolve time integrals of Lagrangian correlation functions associated with thevelocity field, which are comparable in structure with the eddy diffusion tensorin the analogous turbulent diffusion problem (Taylor 1921) . Some doubts areexpressed concerning the convergence of the time integrals as t -+ CO, and it isconcluded that a satisfactory treatment of the problem will require the inclusionof weak diffusion effects (as recognized by Parker 1 9 5 5 ) .

1. IntroductionThe object of this paper is to point out certain fundamental difficulties in

the theory developed by Parker ( 1 95 5 , 1 9 70 , 1 9 7 1 ~ - f ) to explain the generationof magnetic fields in astrophysical bodies. The theory is in effect a high con-ductivity (or small magnetic diffusivity) counterpart of the theory of Krause(1967) and Radler ( 1 9 6 8 a , b ) . t Certain aspects of the low conductivity limithave been considered by Moffatt ( 1 9 7 0 ~ ) .

Let u(x, ) be a solenoidal turbulent velocity field, statistically homogeneousin x and stationary in t , and having zero mean. Let B(x, ) be a magnetic fieldt These papers, and others on the same topic by F. Krause, K. H. Riidler and M. Steen-beck, have been translated into English by P. H. Roberts & M . Stix, and are available asTechnical Report TN/lA-60 (June 1971) of the National Center for Atmospheric Research,Boulder, Colorado.I F L M 65

www.moffatt.tc


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2 H . K . Moflatthaving no sources other than electric currents within the fluid. The evolution of Bis governed by the solenoidal condition V .B = 0 and the induction equation

aB/at = V A (U A B)+hV2B, (1 .1)where h is the magnetic diffusivity of the fluid. Of particular interest is theevolution of the ensemble average field

B,(x,t)= (B(x,t)). (1.2)

(1.3)where b = B - B,. A primary aim of turbulent dynamo theory is to obtain anexpression for the mean electromotive force 8'= (U A b) in terms of B,, so that( 1 .3 ) can be integrated.

This satisfies the averaged equationaB,/at = V A (U A b) +AV2B,,

The equation satisfied by b is2{b} E b/at- V A (U A b-(u A b)) -hV2b = V A (U A B,). (1 .4 )

We shall suppose that b(x,0) = 0so that (1.4)establishes a linear relationship between b and B,, and so between(U A b) and B,. On the assumption th at the scale L of inhomogeneity of B , islarge ( L9 ,, where lc is a typical correlation length) this linear relationshipmay be developed as a series (see, for example, Roberts 1971),

L$(x,~) = (U Ab), = au(t)B,l+p,im(t)aB,,/axm+.., (1 .6 )where ail@), i zm( t ) , . are pseudo-tensors determined solely by the statisticalproperties of the turbulence and the parameter A ; their dependence on t arisesthrough the condition (1 .5 ) , which clearly implies a,(O) = 0, pil,(0) = 0.

When $he magnetic Reynolds number R,, defined byR, = u, lc /h , U, = (U"*,

is small, it is known that ail(t), i im(t ) , quickly settle down to asymptoticsteady values; in particular (Moffatt 1970aT) in the case of turbulence that isrotationally symmetric,

(1.8)ail= asil, a N - - / k - 2 F ( k ) d k ,3Awhere F ( b ) s the helicity spectrum function, with the property

( u . V A U ) = F ( k ) d k .s omMoreover, in this limit, the term involving pi,, makes a negligible contributionto ( 1 .3 ) compared with the strong diffusion term AV2B,.

by 4.t The factor Q n equation (3 .11) of Moffatt (1970a ) s incorrect, and should be replaced


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The electromotive force generated by turbulence 3When R, is large, and in particular in the limit R, -+ CO, the situation is far

less clear. First, it is by no means certain that, in the limit h 3 0, ail(t)andpil,(t) do settle down to asymptotic steady values as t + CO. If they do, then ondimensional grounds

a - uo, p - U O L (1.10)where a and p are typical components of ail,pi l , - ay= 1 i i , p = 1%3k. p..j k ; ( l . l l a , )

the estimates (1.10) arise in models developed by Parker (1955, 1970, 1971b).If they do not, then the ultimate values of ail,p i t , must depend on h throughrelationships of the form

a - uof(Rm), P - uo4g(R,) (1.12a, b )and i t would then be desirable to obtain the asymptotic form of the functionsf(R,) and g(R,) for R, + 0. In the case of rotationally symmetric turbulence,,8 is just the eddy diffusivity for magnetic field and a relation of the form (1.12b )is certainly not out of the question; a qualitative theory of Moffatt (1961) infact gave

P - UOU R!L (1.13)in the limiting situation R, -+ CO, RIR, -+m, where R = uolc/v nd v is thekinematic viscosity; this theory was based on estimating the spectrum ofmagnetic fluctuations and calculating the consequent increase in ohmic dissipa-tion of magnetic energy.tThe effectiveness of turbulent dynamos when R,, > 1 is highly dependent onthe appropriate value of the eddy diffusivity, and i t is important t o be able todistinguish between such alternatives as (1.10b ) and (1.13).The question of theappropriate asymptotic value of a is likewise of crucial importance in suchtheories as that of Steenbeck & Krause (1 9 6 6 ) for the generation of the solarand terrestrial magnetic fields. The uncertainty in the value ofa an be illustratedin physical terms through reference to figure 1, which illustrates the processdiscussed by Parker (1970). Parker describes as a cyclonic event a velocityfield u(x, )which is localized in space and of limited duration and whose linearand angular momenta are related in such a way as to give a definite sense oftwisting; i t is evident th at the total helicity

I = U . ( V A U ) d 3 Xs (1.14)for such a motion must be non-zero (positive in the situation illustrated infigure 1); i t can be argued on dynamical grounds th at such events are possible,and perhaps probable, in a thermally convecting rotating fluid. The event tendsto distort aline of force in the manner indicated in figure 1 ( a ) , nd a loop appearshaving a projection on the plane perpendicular to the local (undisturbed) field B .

t Rlidler (1968b ) has made the point tha t different definitions of eddy diffusivity maywell lead to different values.1-2


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4 H . K . Moflatt

o0IB

& U I(a ) (4

FIGURE. Possible distortionof a line of force by a cyclonic event.From Ampikes law, this loop of field can be conceived as being due to a current jhaving a component parallel to B; i.e.

j . B = a B 2 , (1.15)the a here being identifiable with the a introduced above. If the situation isstrongly diffusive or if the event is sufficiently short-lived, then the perturbationfield remains weak compared with the unperturbed field, and when I > 0, a isevidently negative consistent with the result (1.8).In the weak diffusion limit however, and for more persistent events, the loopmay be twisted any number of times (figure 1b ) and the appropriate value of amay then apparently be positive or negative. For a random superposition ofcyclonic events (with I > 0 in each), the asymptotic value of a may dependin a very sensitive way on the precise statistics of the velocity field. If the eventsare sufficiently short-lived and if different events are uncorrelated, then thelimited twist picture of figure l (a ) will presumably apply, so that a will benegative; on these specific assumptions, Parker (1 970) obtained the followingexpression for ail:

ail = eijk X,(a)aXj/aa, d3a , (1.16)where X(a) is the total displacement during an event of the particle initiallya t a, the angular brackets indicate an average over events, and v1 represents themean rate of occurrence of events per unit volume. Discussion of (1.16) s deferreduntil the comparable formula ( 3 . 3 )below has been derived.

A possible approach to the weak diffusion limit has been suggested by Roberts(1971); this is simply to restrict attention to the situation uOte Zc when it maybe reasonable to neglect the awkward term

(S )

G = v A (U b-(U h b)) (1.17)in (1.4), and to evaluate b (in the limit A + 0 ) rom the equation

ab/at = V A (U A B,,). (1.18)


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The electromotive force generated b y turbulence 5Neglect of the term G (the first-order smoothing approximation) has alsobeen advocated by Lerche (1971a, b ) in an attempt to find solutions of (1.3)without exploiting an expansion of the form (1.6).Unfortunately there appearto be few circumstances when the neglect of G when h -+0 is justified, unlesspossibly attention is restricted to a sea of random waves of small amplitude(e.g. inertial waves in a rotating fluid as considered by Moffatt 1970b), ratherthan to turbulence as normally understood. Moreover if G is neglected, then theexpression obtained for a in fact vanishes in the limit h -+0. The reason isas follows (cf. the discussion of 3 4 of Moffatt 1970a):since (1.6)is valid for anyfield distribution, ail may be calculated on the simple assumption that B, isuniform, in which case 6, = ailB,, is also uniform and so [from ( 1 .1 ) ] B, isconstant in time. Equation (1.18)then becomes

a q a t = B,. vu (1.19)and, defining the Fourier transform of $(x, ) by

n n$(k,o) = JJ @(x,t)exp{i(k.x-ot)}d3kdw, (1.20)this becomes ~6 = - k.B,) 6 . (1.21)The fact tha t 6 and ii are exactly in phase then implies that

(u A b) = //(ii A 6 * ) d 3 k dw= 0. (1.22)A little dissipation (i.e. h + 0 ) ,or a little nonlinearity (i.e. G + 0 ) s essential toprovide an a-effect. In the case of random inertial waves, this result is evidentin the formulae (4.1)-(4.5)of Moffatt (1970b).

2. Lagrangian treatment of the induction equationco-ordinates. Let

In the limit h -+0 , he solution of (1 . 1 ) is best expressed in terms of Lagrangianx = a + X ( a , t ) (2.1)

X(a,O)= 0. (2.2)Let v(a , ) = aX/at = u(x(a, ) , ) (2.3)

X, ,Ja, t )= aXi/aaj (2.4)

be the equation of the path of the fluid particle satisfying

be the Lagrangian velocity, and let

be the Lagrangian deformation tensor, which gives a measure of the totaldeformation of an infinitesimal element of fluid initially a t the point a during thetime interval [0 , ] .Also let

(2.5),,Ja, ) = aX,,Ja, ) /a t = av,/aajbe the Lagrangian rate of deformation tensor.


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6 H . K . MoflattCertain properties of the random function v(a, ) in stationary homogeneous

turbulence have been obtained by Lumley (1962) . In particular, it is known thatthe single-particle statistics of v(a,t)are stationary in time, but that the n-particle statistics are in general non-stationary for n < 2 . For example, thecorrelation coefficient

r(a1- a2,t ) = Wa,, t ) v(a2, t)>/(v"a,4) (2 .6 )certainly tends to zero as t --f CO, since any two particles ultimately drift infinitelyfar apart with probability 1. However, r(al-a2, 0) is arbitrarily near to 1when la,-a,l is sufficiently small (e.g. smaller than the Kolmogorov innerscale). Hence &/a t $. 0 in general, and so the two-particle statistics of v(a,t)are non-stationary .

This would seem to imply also that , although v(a, ) is stationary for fixed a,v iJa , t ) s in general non-stationary for fixed a since it involves the joint statisticsof v a t two points a and a+Sa in the limit Sa --f 0. This conclusion is consistent

avi auiax,ith the statementaaj aX,aa,

and the fact that aui/ax, is presumably a stationary random function o f t evenif evaluated on the random path (2.1) while ax,/aa, is certainly a non-stationaryrandom function of t.

The solution of ( 1 . 1 )with A = 0 (cf. Cauchy's solution of the vorticity equation)is

(2 .7)= - -

B i ( x , ) = (Xi,Ja, )+ Sij)Bj(a, ). (2.8)Now d ? = ( U A b ) = ( U A B ) , (2 . 9 )&i(x,) = cijr,(vj(a, ) (Xk,l(a, )+ &]cl) Bl(a,O)) , (2 .10)

where, on th e right-hand side, a(x,t) s to be regarded as a random function(for given x,t) arying from one realization of the flow to another. From theinitial condition (1 .5) , (2.10)becomes

(2 . 11 )

and so, from (2.8),

&i(x,) = eijk(vj(a, ) (X,,l(a, )+ Ski) Bda, 0)).3. Evaluation of ail(t)

In order to evaluate aiZ(t),s mentioned in 3 1, we may simply consider thespecial case in whichin which case B, remains also constant in time. Then (2 . 11 )becomesgi= cril(t)B,,where

B,(x,0) = constant, ( 3 . 1 )

= %jk(vj(a,)X,,,(a, ) ) ( 3 4since (v) = 0. It is instructive t o express this as a time integral,

( 3 . 3 )


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The electromotive force generated by turbulence 7and to compare with Taylors (1921) expression for the eddy diffusivity tensorfor a passive scalar field

r t

Owing to the stationarity of v(a , ) for fixed a, the Lagrangian correlationR,(t - ) (v,(a, )%(a, )) (3.5)

is a function only of the time difference t - , and provided only that Ri2(tl)sintegrable in [0,CO],

r mD, N J RLl(tl)dt,as t -+CO.0

The Lagrangian correlation time t , may be defined byr m

and the asymptotic result (3.6) s accurate for t > ,.It is tempting to apply the same formalism to (3.3))but we are now faced with

the difficulty that vk,Ja,7)s not stationary in r ($2))so th at the Lagrangian

may depend on t and 7 independently (and not just on the difference t - - 7 ) . Itis reasonable t o suppose that

dj,(t,7 )+ 0 a t t- --f CO (3.9)but this is not sufficient to guarantee the convergence of the integral (3 .3 ) ast -+ CO. For example, if

dj,(t, 7) C .kl coswt e-lc(t-T), (3 . 10 )(which is not implausible bearing in mind the discussion in 3 1 of the physicalprocess represented by figure 1 ) ) then

ail(t)= &$jl ,c jklk- coswt, (3.11)which does not settle down to a constant value as t -+ CO.Unfortunately, the result (3.3) s of interest only if the integral does convergeas t + CO; for if not, then in a situation in which B,(x, t ) is changing with time,the e.m.f. 8(x, ) would depend on values of B,(x, ) for arbitrarily large valueso f t- , and the assumptions underlying an instantaneous expansion of the form(1.6)would be invalid. Of course it is possible to conceive of kinematically possiblevelocity fields (as in effect done by Parker 1955)which get round the difficultyof the possible divergence of (3.3); for example, suppose that v ( a , t ) s firstswitched on for a time of order t,, then switched off for a long time of ordert, = 0(1;4/h)o allow the small-scale field to be eliminated by diffusion, and thatthe whole process is then repeated, the statistical properties of v ( a , t )beingperiodic with the period of the cycle. Then there is an effective cut-off whent = O(t,) to the integral (3.3). The need to invoke molecular diffusion was


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8 H . K . Moffattemphasized by Parker (1955) and by Parker & Krook (1956), but the moreformal treatment of Parker (1970) in fact makes no appeal to effects ofmolecular diffusivity and t.he expression (1.16) obtained by Parker (1970) inconsequence shows no explicit dependence on A.

The expression (3.3) may be compared with Parkers expression (1.16),withwhich i t clearly has a certain formal resemblance. The advantage of (3.3) s thatthis expression is determinate fo r a n y turbulent velocity jkZd and is (in principle)measurable by following the continuous deformation of small dye spots ina turbulent flow and averaging over spots.? The expression (1 .16) is defined onlyin terms of a rather artificial model of turbulence (random cyclonic events) andit is not measurable, even in principle; the concept of a mean rate of occurrenceof events is particularly elusive for a fully developed turbulent flow in whichevents merge in a distinctly nonlinear manner in both space and time. Moreover,the expression (1.15) conceals the possibility that aiz(t) s calculated on a A = 0basis may in fact diverge as t --f 00, and th at finite h effects may have to be con-sidered in this eventuality to determine the correct asymptotic behaviour forlarge t.

The structure of (3.3) s of particular interest when the turbulence is rotationallysymmetric, in which case a,(t) = a(t) il,where

a(t)= Qaii(t)= -- (v(a, ).0, v(a,7)) 7.x (3.12)Note the appearance of the Lagrangian helicity correlation under the integralsign. If (U.V A U) > 0 say, then i t is certain th at a(t)< 0 for t < tc; but thereseems no obvious reason why a(t)as defined by (3.12) should remain negativeas t increases. The expression (3.12)may be compared with the expression (1.8)obtained in the strong diffusion limit.

4. Evaluation of Pil,(t)From (1.3)and (1 . 6 ) ,

a B O i / a t = % jk %cl B O l , j $-%jk PkLm BOl,jm f + BO i , k k ,aBoi,p/t = e i j k a k l % , j p + i j k P kl m B o ~ , j m p+ . + V 2 B o ~ , p k k .

(4.1)( 4 4nd 80

Hence if the field gradient tensor BOi,ps uniform initially then, according to(4.2), in strictly homogeneous turbulence BOi,p emains constant in time. Itthen follows from (4.1)tha t a t any point x

B o i ( X , t , = B 0 d x , O ) s i j k BOZ,j/:akZ(7) d7. (4.3)In order to evaluate Pilm(t),we may suppose that Bol,ms uniform (and so con-stant), and that Bda ,0) = B o d ~)+ (x-a)mBOt,m* (4.4)

A numerical experiment to test the large time behaviour of components of Sikl(t ,7)or of the function a(t)defined by (3.12)would be of great interest.


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and Djm(t)s given by (3.4) above. Using (4.3), (4 .5) may be expressed in therequired form

4 ( x , ) = @il ( t )B,@, t )+PiZ,(t) &,m, (4.7)t0where PiZ,(t) = Ptzrn(t)-.,&/ %Z(d d t- (4.8)

This expression for Pizm(t)now reveals a double difficulty. First, the doubleintegral of the triple Lagrangian correlation in (4.6)s even less likely t o convergethan the integral (3.3)defining ai,(t).Second, if a k z ( t )ends to a non-zero constanttensor as t -+ GO (asenvisaged by Parker 1970) then the integral in (4.8)cer t a i dydiverges, so that the ultimate level of PiZman only be determined by consideringthe effects of weak diffusion.

It is hard to escape the conclusion that when h -+ 0 the eddy diffusivity /3defined by (1.11b ) n general increases without limit, and that a simple asymptoticrelation of the form ( 1 . o b ) sunlikely to be correct. This conclusionis a t variancewith th at of Parker (1971b ) who argued th at the eddy diffusivity for a magneticfield in the limit h -+ 0 should be identical with that for a scalar field (such astemperature) viz. QDii(co).Parker's argument, however, relied implicitly on thesplitting of the correlation

(vj(a, t )X,(a, t )X,,z(a, 0 ) + (vj(a, )X,(% t ) )


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10 H . K . MoflattPARKER, . N. 1970 A s t r o p h y s . J . 162, 665.PARKER, . N. 1971a Astrophys. J . 163, 255.PARKER,. N. 1971b A s t r o p h y s . J . 163, 279.PARKER,. N. 1971c A s t r o p h y s . J . 164, 491.PARKER,. N. 1971d A s t r o p h y s . J . 165, 129.PARKER,. N. 1971e A s t rophys . J . 166, 295.PARKER,. N. 1971 Astrophys. J . 168, 239.PARKER,. N. & KROOK,M . 1956 As t rophy s . J . 120, 214.RADLER, .-H. 1968a 2 . Natur forsch. A 23, 1841.RADLER,K.-H. 1968b 2 . Natur forsch. A 23, 1851.ROBERTS,P. H. 1971STEENBECK, . & KRAUSE,F. 1966 2. Natur forsch. 21a, 1285.TAYLOR,G. I. 1921 Proc. L o n d . Math. Soc. 20, 196.

Lectures in Applied Mathemat ics (ed. W. H. Reid). Providence:Am. Math. Soc.

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