h.k. moffattt and m.r.e. proctor- the role of the helicity spectrum function in turbulent dynamo...

8/3/2019 H.K. Moffattt and M.R.E. Proctor- The Role of the Helicity Spectrum Function in Turbulent Dynamo Theory

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266 H. K. MOFFATT AND M. R. E. PROCTORbe present also, and that this will then swamp any influence of the fi A Jingredient of the mean electromotive force.It nevertheless appears possible that under particular additionalsymmetry conditions, the a-effect may vanish while the coefficient Rappearing in the C2 A J-effect remains non-zero, for the simple reason thata and R are both expressible as weighted integrals of the helicity spectrumfunction of the turbulence (and possibly higher order spectral functions)but with differentweight functions. The main purpose of the present paperis to explore this possibility, and to clarify the circumstances (if any)under which the C2 A J-effect may be of dominant importance.In order to focus attention on the problem, certain idealisations are inorder. To begin with, we shall adopt a strictly kinematic point of view, thestatistical properties of the velocity field u(x, t ) being assumed known.(Dynamical effects associated with the action of Coriolis forces in arotating system will be considered later in Section 7.) We shall supposethat these statistical properties are homogeneous and stationary in time,and that ( U ) =0, the angular brackets representing an ensemble average.We suppose that the dominant length-scale of u(x, t ) is l,, and we considthe evolution of a mean magnetic field B(x, t) on a much larger length-scale L(>>l0).If the to ta l field is B + b where (b)=O, then the meanelectromotive force generated by the turbu lence is

e.& = ( U A b), (1.1)

and the mean field evolves according to the equation

It is also well-known that d is linearly related to B by an equation of theform m&i(x,t)=JdT dzKij(T,z)Bj(x- T , t - z ) , (1.3)0where Kij(& ) is a tensor (or, more precisely, a pseudo-tensor) determined(in principle) by the statistical properties of the turbulence and themagnetic diffusivity of the fluid q.see (1.2) above an d (1.4) below-that its time varia tion is slow) allows usto expand B(x-T,t-z) about t =O , z =O , an d t o integrate (1.3) term byterm. This results in a series of the form

The assumption that the scale L of B is large (with the consequence--.)


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HELICITY SPECTRA AND DYNAMO THEORY 267where

etc. It is generally believed th at the series (1.4) converges rapidly providedl,/L is sufficiently small.In the special case of isotropic turbulence, the coefficientsai j , i j k , a{;),. . . must also be isotropic, i.e.

Correspondingly, Eq. (1.2) then becomesaB/a t=aV A B + ( q + p ) V 2 B + a c ) a ( V A B)/a t+ ... (1.8)0

The term aB in (1.7) represents the a-effect of Steenbeck, Krause andRadler (1966), an d the p-term provides an eddy-diffusivity effect. The a( ) -term was discussed by Radler (1968) and interpreted as a capacitativeeffect.If the scale of B is so large that

in (1.7), then the time derivatives in (1.7) may be replaced iteratively byspace derivatives using (1.8); this procedure gives

8 aB- p- a())V A B + (1.10)where the O ( L - 2 ) terms include a contribution d ) ( q+p)VZB. It wouldappea r from (1.10) th at when a and a() are non-zero, we have a modifiedturbulent diffusivity

p(W = p- aa()*The corresponding modification of the tensor b i j k in (1.4) is given by

(1.11)


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268 H. . MOFFATT AND M. R. E. PROCTORWe shall find that determination of 0113 is coupled very naturally withdetermination of P i j k , so that the modifications (1.11) an d (1.12) are easilyevaluated. However, as pointed out by a referee, the procedure by which(1.11) an d (1.12) are obta ined is somewhat suspect if (as is frequently thecase in dynamo models) the terms a B and PV A B of (1.7) are of the sameorder of magnitude. We shall not therefore pursue the implications of(1.11) and (1.12) in this paper.

2. KINEMATICS OF REFLEXIONALLY NON-SYMMETRICTURBULENCEWe adopt the usual Fourier representation for u(x, t ) , viz.

u(x,t)=SG(k,w) ei4dA, (2.1)where cp= k x-ot and dA = dk d o , the integral being a four-fold integralwith inverse 0

B(k,o) ( 2 7 ~ - ~u(x, t) e - @ x dt. (2.2)Reality of U implies tha t

B( - k , - ~ ) = i i * ( k , ~ ) , (2.3)where * denotes the complex conjugate. We restrict attention toincompressible velocity fields for which V U = 0, so that

k * a(k, CO)=0 (all k,0). (2.4)The spectrum tensor Qij(k,w ) of the turbulence is given by the well-knownrelation

(GT(k, w)iij(k,U)) = CDij(k, w)6(k - k ) d ( w - ~ ) , (2.5)and it satisfies the conditions [corresponding to (2.3) an d (2.4)]

Qij(k,CO)= Q j i ( - , -0) CD;(k, CO), (2.6)kiCDij(k,w)=0, kjQij(k,w)= 0 (all k,0). (2.7)

and


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HELICITY SPECTRA AND DYNAMO THEORY 269Let us now decompose Q ij into its symmetric and antisymmetric partswith respect to i-j [superscripts (s) and (a)],

where, by virtue of (2.6), @!) is real, and @$) is pure imaginary. Writing

where P ( k , o ) is a real pseudo-vector, the conditions (2.7) imply tha tk A P= 0, i.e.P=tk H (k, w)/k2 (2.10)

for some real pseudo-scalar function H(k,w). Hence

e n d , eciprocally,H(k,o) - krneijmQij(k,). (2.12)

This is the helicity spectrum function of the turbulence, with the property( u . V A u ) = i H ( k , o ) d A . (2.13)

It is noteworthy that (without any assumption of isotropy oraxisymmetry) the single scalar function H(k,w) is sufficient to determinethe asymmetric part of the spectrum tensor.The energy spectrum function E(k,w) is given byE(k, CO)=$@ii(k,O) ( 2 ), (2.14)

and has the property+(U) = J E(k,U )dA. (2.15)

The function H satisfies an important inequality which follows fromCramers (1940) theoremQij(k,w)XiXj*>=0 (all complex X). (2.16)

Taking X = p + i q , where p and q are real unit vectors such that (k ,p ,q)


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270 H. K. MOFFATT AND M . R. E. PROCTORforms an orthog onal triad, we have

QijXiXT =2E(k, U )f -'H(k, U), (2.17)and it follows from (2.16) that

(H(k,U ) [5 2kE(k,w). (2.18)This inequality is well-known for the case of isotropic turbulence, but doesnot seem to have been recognized previously for the general homogeneousturbulence considered here. From (2.6), (2.12) and (2.14), it is evident thatE(k,w ) and H(k,w ) satisfy the symmetry conditions

E(- , -0) E(k,CO), H (- , -0) H(k,CO). (2.19)It follows that, for any function A(k,w) with the property

A (- , -CO) = -A(k,w),we have

J A(k, w)H(k,U )dA = 0, (2.21)(and similarly with H replaced by E ). Repeated use will be made of thistype of result in the following sections.

3. CALCULATION O F ail UNDER THE FIRST-ORDERS M O O T H ING APPROXI MAT10NIn calculating cxi j in (1.4), it is clearly legitimate to assume that B is strictlyuniform, and therefore steady, since d [in (1.2)] is then also uniform. Thefollowing calculation is well-known, but is reproduced briefly here as anessential preliminary to the subsequent calculation of P i j k .

The p erturbation field b satisfies (Moffatt 1978, Section 7.5)a b p t = v A (U A B ) + V A b'+qV2b,

where & = U A b- (u A b ) . The first-order smoothing approximation,which we adopt here, consists in neglecting the term V A 8' n (3.1), whichthen becomes


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HELICITY SPECTRA AND DYNAMO THEORY 27 1with Fourier transform

(- io+qk2)6= (B .k)B. (3.3)The mean electromotive force d s then given by

(3.4)where cp =k * x -ot, and we use the shorthand notation U = B ( k , 0).Hence, using (3.3) and (2.5), and carrying out the trivial integration overA, we obtain i = a i j B j where

d Ak i k j H ( k ,o)I J s -io+qk2 -s k 2 (- w +qk2)k . i&. Q, ( k , o )a , ,= J l P k Pkor

where we have used (2.21) to discard the imaginary

(3.5)

part-of course theresult must be real from the definition of ai j . The result ( 3 4 , which hasbeen obtained previously in very similar form by Soward and Roberts(1 76), holds for general homogeneous turbulence, with no assumptionsconcerning directional symmetry. Note that a i j= a j i under first-ordersmoothing. An antisymmetric part of aij (and an associated field pumpingeffect) appears only at higher orders, involving cubic velocity spectra(Krause and Radler, 1980, Section 7.2).

4. CALCULATION O F jijkN D a$)This calculation proceeds in a similar manner, but with the assumptionthat B has a uniform gradient, and in consequence a uniform time-rate-of-change, i.e.

where B,=B(O,O), and B j , l and B j are constants. Fro m (1.4) this gives


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272an d substitu tion of (4.1) an d (4.2) in (1.2) gives

H. . MOFFATT AND M. R. E. PROCTOR

Since the time and space gradients of B are related in this way,determination of b i j k is necessarily coupled with a simultaneousdetermination of ae ) .Again neglecting the term V A 8, 3.1) now becomes

with Fou rier transform

whereG (k ,w)=( - o + q k 2 ) - l , (4.6)

Substitution in (3.4) gives

e(4-4)dA dA. (4.7)Consider first the term involving the operator dldk,. We have

-1 k m @ j k 6 ( k - k ) 6 ( o - w ) - a { G e i ( @ - @ ) )AdAak,

= - s km@ jk[ (aG /ak , )+x lG ] dA. (4.8)The term involving ajaw in (4.7) may be manipulated in a similar way.


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HELICITY SPECTRA AND DYNAMO THEORY 273Gathering the various terms together, we find

The first integral here is just a imBm( x , t ) ,where aim is as determined inSection 3 and B m ( x , t ) s given by (4.3) .The second integral gives P im l anda{;) directly in the form

P im l= &ijk ! ( dG /dk l ) km@jkA - ijm.I G D j I A , (4.10)and

a{;)= -&ijkj(aG/&o)km@jkA. (4 .11)These expressions may be further simplified. The decomposition (2.8) of

O i j into its symmetric and antisymmetric parts provides a similar.decomposition of P i m l ,viz.P i m l = PE,+Pt% (4.12)

The first ingredient here is given by/ ? E , = E ~ ~ ~ D , ,ith D p ,= Jy k2 ( m 2 y2k4) - @,b ;) dA . ( 4 .13 )

D,, s a symmetric positive-definite tensor (and it is actually the symmetricpart of the diffusion tensor that acts on a convected scalar field withmolecular diffusivity q, under first-order smoothing). The correspondingcontribution to d s

since V * B = O , so that we have here also a pure eddy-diffusivitycontribution in the mean-field equation (1.2).The second ingredient of (4 .12) simplifies, using (2.1 ), to the form


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274 H. K. MOFFATT AND M. R. E. PROCTORwhich, like aij, is a weighted integral of the helicity spectrum function.This term has the property

Ei j , / 3 $ = 0 , (4 .17)and therefore makes no contribution to eddy diffusivity (see Section 5below). It does however contain the Radler A2 A J-effect, although ofcourse dependence on A2 (the rotation rate of the frame of reference)appears only when we take account of the dynamical influence of Coriolisforces on the function H(k,o)-see Section 7 below.Finally, the expression (4 .11)for a!;) may be simplified to the form

(U -q2k4)k ik ,H dA ,s ( ~ + q k ~ ) ~ k a!;)= - (4 .18)again a weighted integral of the helicity spectrum function.

5. THE CASE OF ISOTROPIC TURBULENCEIn the case of isotropic turbulence (i.e. turbulence whose statisticalproperties are invariant under rotations, but not necessarily underreflexions, of the frame of reference), we have

0

E = E (k ,U ) , H =H ( k ,U ) , (5 .1)where k = k / . From ( 3 4 , we then have

with d A= 4rc.k dk do. imilarly, from (4.13),D i j=Dd,, (5.4)

hereD = @ i i = ( 2 q / 3 ) J k 2 ( U 2 + q 2 k 4 ) - E ( k , w ) d A . ( 5. 5)

It follows from (4 .13) hatpjj1= B ( S ) E i j k


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HELICITY SPECTRA AND DYNAMO THEORY 275where

p( )=& . .j k p ! ? ) / 6 = s D i i = D .jk (5 .7)The integral (4.16) for ,@$ vanishes, by symmetry, in the isotropic

situation, but (4.18)becomes

6. THE CASE OF A XISYMMETR IC TU R B U LEN C ESuppose now that the turbulence is statistically axisymmetric about anaxis defined by the unit vector e . Let

and the symm etry conditions (2 .19)become

From (3.5), aij is now given bycrij=adij+a1(dij-3eiej),

where, as before,

and

Similarly, from (4.13),D i j=Ddij+D,(di j- 3eie j ) (6 .8)


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276 H. K. MOFFATT AND M . R. E. PROCTORwhere

Dije ie j A ,k21 2 + q 2 k 4E ( k , p , u ) d A , D - 2 D , = q21 2+ q 2 k 4D =+qand

The expression (4.16) for @$ takes the form

where

0nd the corresponding contribution to 8 s&)=Re A (V A B ) - ( R 1 + 2 R ) V ( e . B ) . (6.13)

The second term, being the gradient of a scalar, makes no contribution tothe mean-field equation. The first term does make a contribution,however, an d in fact it is the contribution that Radler (1969) described asthe a J-effect. (We shall refer to R as the Radler coefficient.) Th ederivation given here however should make it clear that this effect (likethe a-effect) will arise in general when (and only when) the helicityspectrum H(k,0) s non-zero. Both effects depend indirectly on therotation vector !2 of the frame of reference only insofar as Coriolis forcesare responsible for the generation of a non-zero H ( k ,0).The expressions (6.6) an d (6.12a) for a an d R differ only in the weightingfactors in the integral. It is clear that if H satisfies the condition

[note the distinction between this and the automatic condition (6.4b)lthen, in general,a = O a n d R # O . (6.15)

Physically, the condition (6.14), which may be described as the conditionof up-down symmetry means that positive helicity associated with wavesfor which wp>O is compensated by equal negative helicity associated with


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HELICITY SPECTRA AND DYNAMO THEORY 277waves for which wp


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278 H. K. MOFFATT AND M. R. E. PROCTORab ou t the direction of SL, i.e.

where p= co sO = k. e/ k as before, e being the unit vector in the directionof SL. Then the axisymmetric forms obtained in Section 6 are relevant; inparticular from (6.6),cc=(4@22/3)Sk'p0(~0~r2k4)-11Al-2F(k,p,~)dA, (7.6)

and from (6.12)(7.7)=(2q2R/5) S k 8 2 CO2(w' + q 2 k4) -21AI-2F(k ,p,0)A,

where R =52.e (a pseudo-scalar). I t is now clear tha t ifF(k ,- ,4 = J'(k ,4,

i.e. if the forcing is symmetric with respect t o the directions f e , then cc=Obut, in general, R # 0.The above results are consistent with the discussion of Krause andRadler (1980, Section 7.5) who however restrict attention to a situation inwhich the anisotropy is entirely due to the influence of Coriolis forces,assumed weak. In the above treatment [as in Moffatt (1972)l there is nosuch limitation, and indeed the linearised model on which (7.1) is basedbecomes more justifiable when IRI is large. The limiting form of theintegrals (7.6) and (7.7) as IRl+co (keeping all other parameters fixed) is ofparticular interest. Since

the dominant contributions to both integrals come from theneighbourhood of p = &a reflection of the strong anisotropy associatedwith the Taylor-Proudm an theorem. Pu tting ( = 2kRp, and d< = 2kRdp,the limits p= _+ 1 become (for n+co) (= f O; integration over ( in both

which may be evaluated by con tou r integration to giveZ(k,0) z/2vk2. (7.11)


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HELICITY SPECTRA AND DYNAMO THEORY 279Using this result, the asym ptotic forms of (7.6) an d (7.7) are

andk 5 c 0 2 F ( k ,0,co)R - S j{ d k dco,10vR2 (w2+q 2 4)

(7.12)

(7.13)

as IR1-m. Both c( and R decrease like R - as I R / + m . Note tha t ainvolves a F / a p evaluated at p = 0; of course this vanishes under thecondition of up-dow n symm etry (7.8). By contras t, R involves only thefunction F itself evaluated at p = 0.Having established the formal circumstances in which the Radler effectmay be expected to occur in the absence of an a-effect, the question stillremains as to whether such circumstances can ever arise in a natural@physical system. Turbulen t therm al convection in a stellar convection zoneis frequently modelled by Boussinesq convection in a plane layer- z o < z < z o rotating with angular velocity R about the vertical. The factthat heat is transported upwards by the turbulence does not in itself breakthe up-down symmetry, because as far as the velocity field is concerned, arising ho t blob an d a falling cold blob are quite equivalent+.g. theyentrain fluid and conserve angular momentum in much the same manner.In stability studies, it is in fact found that the helicity ( u . ( V A U)) (theaverage being over horizontal planes) is an antisymmetric function of z,and that when a magnetic field is present, an antisymmetric a(z) appearsalso [see, e.g. Soward (1974), Moffatt (1978, Section 12.3)]. The coefficientR will also be z-dependent in these circumstances, but there is no reasonto believe that it will vanish on z = 0. If the turbulence is approximatelyhomogeneous in a core region (the inhomogeneity being concentrated inthermal boundary layers on z = i o) then it is at least conceivable thatthe R-effect may dominate over the a-effect throughout the core region.This situation is still of course somewhat idealised. In a stellarconvection zone, the spherical geometry in itself is enough to break theup-down symmetry; moreover compressibility plays an important part inthe process by which a rising blob acquires helicity, and the fact that theambient pressure increases with depth is clearly responsible for a furtherstrong departure from up-down symmetry. The assertion of Moffatt (1978,p. 156) that it seems almost inevitable th at whenever the R adler-effect isoperative, it will be dominated by the a-effect is hard to refute in thestellar context.

0


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280 H. K. MOFFATT AND M. . E. PROCTOR8. FREE MODES ASSOCIATED WITH THE RADLER EFFECT IN

CONJUNCTION WITH MEAN SHEARDespite the above reservations, it seems worthwhile to investigate whetherthe Radler effect (w ithout any a-effect) can, in conjunction with meanshear (or differential rotation), give rise to dynamo excitation of a meanfield. Numerical results for spherical models of this type are reported byKrause and Radler (1980, Section 16.5) and it app ears th at both steadyand oscillatory dynamos are possible. Here we investigate local solutionsof the mean field equation

with 8i = &kdBj/dXk, following the free-mode approach of Parker (1955).We suppose that B = B(x,z , t ) and that

U= (0,G z ,0), (8.2)where G = constant. Then

[V A (U A B)J = G i j B j , 0(8.3)where G i j = G 6i26 j3 .Equation (8.1) admits solutions of the form

B = B, exp ( p t+ K .x), K 9 B, = 0, (8.4)( p + q K Z ) B i + A i j B j = O , (8.5)

Ai j= EipkflkjmKpKm Gij . (8.6)

where

with

The possible values of p are then determined by

For simplicity, consider the particular case when K = K , ,O) (whichcorresponds to horizontal field variation in the presence of a verticallysheared horizontal velocity; the result below that growing modes are non-oscillatory is unaffected by this simplification). Then (8.7) becomes


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HELICITY SPECTRA AND DYNAMO THEORY 28 1where p = p + q K 2 ; the root p =0 is spurious (since by (8.4b), B,, = 0); theother two roots are given byN ow from (8.6) we have

an d , from (4.12), (4.13) an d (4.16),

whereP1= 4 q 2 J u k , 2 k 3 ( u 2+q 2 k 4 ) - H d A . (8.12)

@Also

(8.13)No te th at, by the Schwartz inequality,

p? S P 2 2 1 P 3 3 1 . (8.14)Substitution of (8.10) and (8.11) in (8.9) gives

where we have written j2 or j221nd P 3 fo r p 3 3 1 . If Gf12>0 ,both roots(8.15) have negative real p arts an d represent decaying solutions. If GPz < Ohowever, on e of the ro ots (8.15) is real an d positive providedK 2 < -GB2 / [ ( q+Dl l ) 2+ ( f 12P3 -P? )1 , (8.16)

i.e. provided the wavelength of the field is sufficiently large. Note that, in0 the case of turbulence that is axisymmetric about the direction of the unitvector e, from (6.10) an d (6.11).P 2 = P 3 = (8.17)


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282 H. K. M O F F A T T AND M. R. E. P R O C T O Rwhere R is the Radier coefficient. The result (8.16) therefore confirms thatnon-oscillatory dynamo action can occur as a result of an interactionbetween mean shear and the Radler effect.The possibility of oscillatory dynamo action does not emerge from thissimplified local analysis. The existence of oscillatory modes in a sphericalgeometry (Radler, 1976; Krause and Radler, 1980, Section 16.5) musttherefore be associated with the global geometry, or with spatial variationof the shear rate G and the Radler coefficient R.

9. CONCLUSIONSThe main aim of this paper has been the elucidation of circumstances inwhich the a-effect due to a turbulent velocity field vanishes, but the Radlera J-effect does not. The two effects are closely related, since they bothdepend on the helicity spectrum function H(k,w) alone, even forfluctuating fields with no isotropy o r symmetry . In spite of this, t h aintegrals (3.5) and (4.16) from which the coefficients aij and j~~~recalculated are invariant under different symmetry operations, with theresult that there are circumstances for which a i j vanishes and p$i is non-zero. Two examples are the propagation of helical inertial waves in auniformly rotating fluid if there is no net flux of momentum in thedirection of the rotation axis 0, and the well-mixed turbulent core thatoccurs far from boundaries in convection in a rotating layer. Although theprecise symmetries that are needed to make aij vanish will not be exactlysatisfied in any real system, one would expect that if they areapproximately true, then the effect of ciij will be no more than comparablewith the Radler effect (of nominally smaller order in the expansion). Thusthe apparenrly contradictory stances taken by M offatt (1978) an d K rau seand Radler (1980) on the importance of the Radler effect can bereconciled. Some properties of growing solutions of the mean fieldequations due to the interaction of shear and Radler effect are alsodiscussed.An interesting secondary point to emerge from the developmentconcerns the necessity to calculate the coefficient of dB/& in the ansatz(1.4). Here we have assumed, consistent with the spatial form of B, thatI B I K ~ .More generally, one would expect that a more accurate descriptioncan be obtained by using the form Bccexp (ik .x +pt). Although the resultswould be unaffected to the order we have taken them, it is possible toproduce a renormalized theory, taking effects at all orders into account.The results of this promising new approach will be presented elsewhere.


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HELICITY SPECTRA AND DYNAMO THEORY 283ReferencesCramer, H., O n the theory of stationary random processes, Ann. Math . 41, 215-230 (1940).Kraichnan, R. H., Diffusion of passive-scalar and magnetic fields by helical turbulence, J .Krause, F. and Radler, K.-H., Me an Field Magnetohydrodynamics and Dynamo T heo ry,Moffatt, H. K., An approach to a dynamic theory of dynamo action in a rotatingMoffatt, H. K., The mean electromotive force generated by turbulence in the limit of perfectMoffattt, H. K., Magnetic Field Generation in Electrically Conducting Fluids, CambridgeParker, E. N., Hydromagnetic dynamo models, Astrophys. J . 122, 293-314 (1955).Radler, K.-H., O n the electrodynamics of conducting fluids in turbulent motion, 11.Turbulent conductivity and turbulent permeability, Z . Naturforsch. 23a, 1851-1860.

(1968). (English Trans lation: Rob erts an d Stix 1971, pp. 267-289.)Radler, K.-H., On the electrodynamics of turbulent fluids under the influence of Coriolisforces, Mona ts. Dt. A kad. Wiss. Berlin 11, 19 42 01 (1969a). English translation: Roberts &Stix 1971, pp . 291-299.)Radler, K.-H., A new turbulent dynamo, I, Monats. Dt. Akad. Wiss. Berlin 11, 272-279(1969b). (English transla tion: Roberts an d Stix, 1971, pp. 301-308.)Radler, K.-H., Mean field magnetohydrodynamics as a basis for solar dynamo theory, in:Basic Mechanisms of Solar Activity (Ed. V. Bumba and J. Kleczek) D. Reidel Pub. Co.,Dordrech t, Holland/Boston , U.S.A., pp. 323-344 (1976).Roberts, P. H. and Stix, M., The turbulent dynamo: a translation of a series of papers by F.Krause, K.-H. Radler an d M. S teenbeck, Tech. No te 60, NCA R, Boulder, Colorado(1971).Soward, A. M., A convection driven dynamo I, The weak field case, Philos. Trans. R. Soc.

Soward, A. M. and Roberts, P. M., Recent developments in dynamo theory, Magn.Gidrodin. 1, 3-51 (1976).Steenbeck, M., Krause, F. and Radler, K.-H., A calculation of the mean electromotive forcein an electrically conducting fluid in turbulent motion, under the influence of Coriolisforces, Z . Naturforsch. 21a, 369-376 (1966) (English transla tion: Roberts & Stix 1971,pp. 2947 .)

Fluid Mech. 77, 753-768 (1976).Pergamo n Press (1980).conducting fluid, J . Fluid Mech. 53, 385-399 (1972).conductivity, J . Fluid Mech. 65, 1-10 (1974).University Press (1978).

A275, 611-651 (1974).

h.k. moffattt and m.r.e. proctor- the role of the helicity spectrum function in turbulent dynamo...

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