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Transcript of H.K. Moffatt- Induction in turbulent conductors

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D Y N A M O T H E O R YU

31 IInduction in turbulent conductorsH.K. MOFFATTtSchool of Mathematics, University of B ristol(Received February 11 , 1981; in revised form April 30, 1981)

1 . IntroductionIn this brief review of what is now a well-established theory, I shall attemptto set the scene for some of the subsequent chapters.

Kinematic dynamo theory is governed by Faradays law,a B / a t = - V X E , V * B = 0 ,

Amperes law,j = p , i l V X B ,

and Ohms law,E = - u x B + j / o , (1.3)

where, in standard notation, E and B represent electric and magneticfields,j is electric current density, U is velocity field, and U is the electricalconductivity of the fluid. Elimination of j and E from (1.1)-(1.3) gives thewell-known induction equation

dB/at = V X (U x B) + q V 2 B , (1.4)where q = (pea)-' is the magnetic diffusivity of the fluid (assumeduniform). In the context of planetary interiors, it is certainly reasonable tosuppose that we are dealing with an incompressible fluid, i.e.

Present address: Department of Applied Mathematics and Theoretical Physics, SilverStreet , CambridgeCB39EW, U.K.

www.moffatt.tc

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4 STELLAR AND PLANETARY MAGN ETISMIn the solar context (and in stellar contexts), the incompressibilitycondition (1.5) is reasonable only for scales of motion small compared witha scale height (dlnp/dz)- where p ( z ) is the density distribution as afunction of height z .The first term on the right of (1.4), V x (U x B), represents a tendencyto convect and distort magnetic lines of force (B-lines). In a turbulent flow,any two fluid particles close together tend to move apart; B-lines thereforetend to increase in length and the field strength tends to increase inproportion. The diffusion term in (1 .4) , qV2B, describes diffusion of fieldrelative to fluid, and generally tends to eliminate high field gradients. Therelative importance of these two terms is measured by the magneticReynolds number R,,

where uo and lo are typical scales of velocity and length; if R , + 1 ,diffusion is weak, and the convection effect dominates.

F I G U R E 1 The AlfvCn dynamo mechanism whereby field intensity is doubled by thestretch-twist-fold cycle; the kink in the final double loop can be eliminated only by themolecular diffusion process.

The dynamo process is essentially a process of systematic amplificationof a magnetic field, without ultimate change in its structure. This processhas to be three-dimensional since two-dimensional and axisymmetricdynamo processes are excluded by Cowlings theorem and its variousgeneralisations. The simplest three-dimensional process leading to fieldamplification was conceived by AlfvCn (1950), and is illustrated in Figure 1.We start with a circular flux tube which is then distorted in three stages- tretch, twist and fold- o give a doubled tube as illustrated; a littlediffusion (q # 0 in (1 .4)) will then (presumably) be sufficient to iron outthe kink and to give a field with exactly the original structure, but doublethe intensity. Repetition of this cycle then implies unlimited field

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DYNAMO THEORY 5amplification - n practice the limit is ultimately set by dynamic (orenergetic) considerations.

It is important to recognise the role of diffusion in the above process. Itis of course the erosive effect of diffusion which implies field decay in theabsence of motion, and which makes us seek to explain field maintenancein terms of a dynamo process in the first place. The same non-zerodiffusion turns out to be of crucial importance also in making a dynamo (orregenerative) process a possibility. Without the effect of diffusion in theabove cycle, to iron out the small-scale kink, the field would become moreand more complex in structure as the cycle is repeated; the ultimatestructure would not be the same as the initial structure, and we would nothave a genuine dynamo process.

Modern dynamo theory has been dominated by three overlappingapproaches. The first is that of Parker (1955, 1970, 1979), which is basedon the idea of representing turbulence as a field of random cyclonicevents. The second is that of Braginsky (1964a, b) (subsequentlyelaborated and clarified by Soward, 1972); this approach was developedprimarily in the geophysical context, and is based on the idea that thevelocity field in the core of the Earth is probably nearly axisymmetric aboutthe axis of rotation - o that a theory based on weak departures fromaxisymmetry is indicated. The third is the theory of mean-fieldelectrodynamics initiated by Steenbeck, Krause and Radler (1966; see alsoKrause and Radler, 1980) which provides (in my view) the simplest and yetmost general formulation of the problem.

FIGURE 2vorticity are correlated.Parkers (1955) random cyclonic events, within each of which velocity and

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6 STELLAR AND PLANETARY MAGNETISM2. Som e comm ents on the kinematical approaches ofPa rker(l9 55 )and B raginsky(l964a, b)Parkers (1955) theory rests, as mentioned above, on the concept of therandom cyclonic event which is a motion localised in both x and t , inwhich velocity U and vorticity w are strongly correlated (Figure 2). Thehelicity of such an event is

H ( t ) = J U d V . (2.1)H is a pseudo-scalar (it changes sign under a change from right-handed toleft-handed frame of reference). If H > 0 in a right-handed frame, then thestreamlines have a right-handed helical structure. If H < 0 , they have aleft-handed structure. If the duration of an event is t,, then

I IH(t)dtI - Hob 3 (2.2)where Ho is (say) the maximum value of H( t ) .

Parker conceived of a random distribution of such events, wellseparated in space and in time, so that the net effect of each event on anambient magnetic field could be treated in isolation. A long period of stasis(U= 0) was imagined between events, during which diffusion eliminatedunwanted field perturbations (like the kink in the AlfvCn cycle). Thiselimination takes place on the stasis time-scale

where lo is the length-scale of a typical event.The major achievement of the analysis lay in the conception and

construction of a process by which poloidal field could be generated fromtoroidal field in a spherical geometry. Parker found that the net effect of arandom superposition of cyclonic events was to provide a toroidalelectromotive forceZ Qproportional to the toroidal fieldB , :

(a forerunner of the a-effect of Steenbeck, Krause and Radler, 1966). gqdrives a toroidal current JQwhich acts as the source of a poloidal field Bp.The toroidal field BT = B,CQ can be regenerated from Bp by the very well-known mechanism of differential rotation. Hence we have the possibilityof a closed dynamo cycle:

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DYNAMO THEORY 7cyclonicevents-

T BPUifferentialrotation

or what is now described as an aw-dynamo.The coefficient r in (2.4) is clearly of central importance. It has thedimensions of a velocity, and Parkers usual estimate, backed up by formalcalculations (Parker, 1970,1979; see also Moffatt, 1978,97.10) is

where U, is a typical velocity in a cyclonic event (and one would thenexpect U, - uo= in maximally helical turbulence). The estimate(2.6) may well be correct, (when R, % l), although in this limit there aredifficulties that are to some extent concealed in Parkers analysis. Onedifficulty is this: that, the smaller q is, the larger is the time ts,accordingto (2.3), to eliminate small-scale field- r to restore the status quo asregards ambient field structure. The regenerative process is then effectivein any region only during a fractionof the total time available; as q + 0 this fraction obviously goes to zero,and one would expect the generation coefficient to have this property also;i.e. one would expect that

in contrast to the estimate (2.6), which does not depend on q.Braginskys (1964a, b) theory, by contrast, does lead to an expressionfor r which is proportional to q for smallq. n this approach, it is supposedthat the magnetic Reynolds number R,,, based on the (dominant)axisymmetric toroidal velocity field is large, and perturbation fields U andb are expanded in powers of Ril. When the field U is helical in character(i.e. = 0, the average now being over the azimuthal angle c p )it turns out that

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DYNAMO THEORY 9aBolat = E V x (U Bo) + E V x & + qe2V2Bo, (3.4)

where&= .Here we have included the multipliers E, E, because V here represents

differentiation with respect to X and Ea/ax = a/ax. The potentiallydominant importance of the term E V x & when E is sufficently small isimmediately apparent.

Subtraction of (3.4) from (1.4) gives the fluctuation equationablat - V X (U X b) = V X (UX Bo) + V X G + q V 2 b , (3.5)

where

If we suppose that b(x, 0) = 0, equation (3.5) establishes a h e a rrelationship between b and Bo, and so between % and Boa f Bo were strictlyuniform (instead of slowly varying) this relationship could only take theform

Cei = ij (X) BOj (3.7)where aij(X) is determined (in principle) by the statistical (i.e. x-averaged)properties of the velocity field, and the value of q,which intervenes in thesolution of (3.5) . Allowing for the weak spatial variation of Bo j , 3.7) mustbe regarded as the leading term of a series

s i ( x ) = aij(x) B,~(X)+ . E P ~ , ~ ( X )Bojlaxk + . . . , (3.8)any time derivatives in this series being replaced by space derivatives bymeans of (3.4) . Rapid convergence of the series (3.8) is to be expectedwhen E Q 1.

Two important observations may be made at this point. First, theargument leading to (3.8) is very general and does not require anyassumption of the form I b I Q I Bo 1 . The perturbation field may be aslarge as, or even much larger than, the mean field, without invalidating theargument.Secondly, the argument leading to (3.8) is still valid even in thedynamic regime when the Lorentz force j X B is significant (or evendominant) in the dynamical equation for U. However the coefficients ai j ,P i j k , . . . in (3.8) will then depend (via their dependence on the meanproperties of U) on the local value of B o. A mean electromotive force of the

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10 STELLAR AND PLANETARY MAGNETISMform (3.8) do es in fact app ea r (in som ewha t disguised form ) in dynamicaldynam o theories, such as those of Sow ard (1974) an d Busse (1975) whe respace-averaging in on e form o r an oth er is involved.

4. Symmetric and antisymmetricparts of a i j ( X )W e m ay always decompose aij(X ) into its symm etric an d antisymmetricparts:

a i j (x ) = a\$'@) - &ijkYk(X) (4.1)wh ere ai,(s)(X) is a pseudo-tensor which is non -zero only if th e turbu lencelacks reflexional symm etry. O n t he o ther han d, y is a pu re (polar) vector,which is in general non-zero even fo r reflectionally sym metric turbu lence .A t any point X, we may choose axes so that a?) is diagonal:

(4.2)

an d the corresponding contribution to 5% is

Com parison with (2.7) a nd (2.9) shows that th ere is a m eeting point withth e P ark er a nd Braginsky theo ries , if, say, the 3-direction is identified withth e azimuth (9-) irection, and d3) r. M ore generally, the relationship(4.3) can provide self-exciting fields (in o th er word s, growing solu tions ofth e mean field eq uatio n (3.4) ).Similarly, th e antisymmetric pa rt of (4.1) provides a contribution t o &

= y x B o , (4.4)and a corresponding contribution V X (y X Bo) on th e right-hand side of(3.4). This mean s th at th e effective mean velocity acting o n Bo is no t U butis instead

There is here an important qualitative effect, in that y(X) i sno t ingeneral solenoidal even if the background velocity U = U + U' is strictly

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DYNAMO THEORY 11solenoidal. The mean magnetic field is nevertheless convected anddistorted by this non-solenoidal field, as if it were a real part of the fluidmotion.

This effect was first discovered by Zeldovich (1956) in his study of theeffect of two-dimensional turbulence on an initially uniform magnetic fieldin the plane of the motion. In this situation the vector potential of B hasonly one component, i.e. B = V X (0, 0, A ( x , y ) ) , andA satisfies the heatconduction equation

aAiat + U V A = q v 2 A . (4.6)If we imagine that the fluid is contained in a large cylinder, x2 + y2 < u2,then we have to solve (4.6) subject to a boundary condition specifying thevariation of A on the cylinder. For the corresponding thermal problem,with a non-uniform temperature prescribed on x 2 + y2 = u2 , one wouldexpect the turbulence to establish a uniform temperature in a coreregion, all temperature variation being confined to a thermal boundarylayer. Zeldovich argued by analogy that, when R, * 1,A will tend to aconstant in the core region, and equivalently B must tend to zero. This isthe f lux expulsion effect, later studied in detail analytically by Parker(1963) and numerically by Weiss (1966). In the context of mean-fieldelectrodynamics, this flux expulsion can be interpreted as due to a radialvelocity field y (X)which convects the magnetic field outwards from theturbulent region (Figure 3) . Radler (1968) has described this asdiamagnetic behaviour.

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12 S T EL L AR AN D P L ANE T AR Y MAGNE T IS MIn some circumstances, however, a paramagnetic effect may be

envisaged. As shown by Drobyshevski and Yuferev (1974), topologicalasymmetry in a convecting layer will generally lead to transport (orpumping) of magnetic flux in the direction of connected fluid flow. Inthe solar context, this direction in the convection zone is verticallydownwards, since, on average, rising hot blobs of fluid are disconnected,whereas the falling fluid subsides in a connected network. We maytherefore expect y (X) to be radially inward, and magnetic flux will then bepumped downwards, a paramagnetic behaviour. In Chapter 1.4Nightingale (1982) discusses the important influence that this may have oncertain dynamo models.5. Exp licit calculation of q ( X )There are two limiting situations in which a i j ( X ) may be explicitlycalculated.

The weak turbulence limitIf U is (in some sense) weak, then b will also be weak, and the terminvolving G in (3.5) may then be neglected. There appear to be twosituations in which this important first-order smoothing approximationmay be made:

( a ) I U l lq Q 1 , and (b) I U /w l 4 1 , (5.1)where o s a typical frequency of the U field. The condition ( a ) may be areasonable one in planetary contexts, but it is certainly not satisfied in thesolar context; the condition (b) s not satisfied by turbulence, as normallyobserved, for which I U /ol= 0(1), ut it may be applicable when Urepresents a sea of weakly interacting waves - .g. inertial waves orinternal gravity waves- overned by a dispersion relation o = o k).

In either case, if G is neglected in (3.5) we have to solve a linearinhomogeneous equation for b. Since U varies on a scale large comparedwith that of U, t is self-consistent to treat U as locally uniform, and to referto a frame of reference moving with this velocity.? This has the effect ofeliminating the term V X (U X b) also. We may then solve for b usingsimple Fourier transform techniques, and then construct % =

The effect of the local gradient of U on the small-scale fields can be incorporated (seeKrause and Rad ler, 1980,Chapter 8).

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DYNAMO THEORY 13The coefficient of Bo in this expression provides the tensor aij. It is foundin this approximation that ai jis symmetric (i.e. y = 0) and

k 2 F ( k ,o)= = -l / 3 r l J J o 2 + r 1 2 k 4 d k d o ,where F ( k , CO) is related to the mean helicity by

= SSF(k ,o) k d o , (5.3)that is to say F ( k , o ) is the helicity spectrum function. A very similarexpression to (5.2) was found by Braginsky (1964a, b) for the coefficient rin (2.9); this significant point of contact in the two theories is almostmiraculous in view of the (apparently) totally different initial assumptions!

The order of magnitude of the a calculated above under the assumption( a ) is

and, as noted above, y is zero at this order of approximation. Iterationdoes however give a y-effect involving triple correlations, and in order ofmagnitude

y - R:q/l when R, Q 1 . (5 .5 )The coefficient Pijk(X) in (3.8) may also be calculated by this method,

and it is found that, whenR, Q 1,

This particular combination of the elements of Pijk acts as an eddydiffusivity on Bo, but clearly in this limit (R, Q 1) the effect is swampedby the dominant molecular diffusion process. Radler (1969) hasemphasised the possible importance of other combinations of elements ofpi jk which may also be associated with regenerative (i.e. dynamo) action.My own view (Moffatt, 1978, 97.9) is that such effects are unlikely toprovide more than a small perturbation of the a-effect; this is becausethese effects, like the a-effect, rely for their existence on a lack ofreflexional symmetry in the background turbulence, and when this occursthe p-terms of the expansion (3.8) are generally smaller by a factor E thanthe a-terms- ut see Moffatt and Proctor (1982).

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D Y N A M O T H E O R Y 15than O(k- l ) . The approach works well for the simpler problem of aconvected scalar field, and this provides some encouragement that theresults for the magnetic field problem, although hard to justify rigorously,may nevertheless have som e validity.

ReferencesAlfvCn, H ., Origin of solar magnetic fields, Tellus 2, 74 (1950).Braginsky, S . I . , Self-excitation of a m agnetic field during the motion of a highly conductingfluid, JETP 47,1084-1098 (1964a), Engl. transl. Sov. Phys. J ETP 20,726-735 (1965).Braginsky, S.I., Theory of the hydromagnetic dynamo, JE T P 47,2178-2193 (1964b), Engl.

transl. Sov. Phys. J ETP 20,1462-1471 (1965).Busse, F . H . , A model of the geodynamo, Geophys. J . R oy . astr. Soc. 42 ,4 37 45 9 (1975).Childress, S . , Alpha-effect in fluxropes and sheets, Ph ys . Earth Planet. Inter. 20, 172-180.Childress, S . , Stationary induction by intermittent velocity fields, in these Proceedings,Childress, S., The macrodynamics of spherical dynamos, in these Proceedings, 245-257Drobyshevski, E.M. and Yuferev, V.S., Topological pumping of magnetic flux by three-Forster, D ., Nelson, D. R . and Ste phen , M.J., Large distance and long-time properties of aKrause, F. and Radler, K.-H., Mean-field magnetohydrodynamics and dynamo theory,M offa tt, H .K ., Th e me an electromotive force generated by turbulence in the limit of perfectMoffatt , H.K., Magnetic field generation in electrically conducting fluids, Cambridge:Moffatt, H.K., Some developments in the theory of turbulence, J . Fluid Mech. 1 0 6 , 2 7 4 7Moffatt, H .K. an d Proctor, M .R .E ., The role of the helicity spectrum function in turbulentNightingale, S. J. , Topological pum ping effects on the a2-mechanism, in these Proceedings,Park er, E.N ., Hydromagnetic dynamo models, Astrophys. J . 122,293-314 (1955).Pa rke r, E.N ., Kinematical hydromagnetic theory an d its applications to the low solarParker, E.N., The generation of magnetic fields in astrophysical bodies, I. The dynamoParker , E.N . , Cosmical magneticfields, Clare ndo n Press, Oxford (1979).Radler , K.-H., Zur Elektrodynamik turbulent Beusugler leitender Medien 11.Turbulenzbedingte Leithahigkeits- nd permeabilitatsanderungen, Z . Nuturforsch.23a, 1851-1860 (1968).Radler, K.-H., Zur Electrodynamik in turbulanten, Coriolis - raften untenvorfenenlietenden Medien , Mber. D tsk. Ak ud . Wiss. Berlin 11,194-201 (1969).Soward, A.M., A kinematic theory of large magnetic Reynolds number dynamos. Phil.Tram. Roy . Soc . A272, 431462.Sowa rd, A .M ., A convection-driven dynamo I. Th e weak field case, Phil. Trans. Ro y. Soc.

81-90 (1982a).(1982b).dimensional convection, J . Fluid Mech. 6 5 , 3 3 4 4 ( 1974).randomly stirred fluid, Phys. Rev. A16,732-749 (1977).Akademie-Verlag, Berlin and Pergamon Press, Oxford (1980).conductivity,J. Fh id Mech. 65,l-10 (1974).University Press (1978).(1981)dynamo theory, Geoph ys. Astrophys. Fluid D ynam . 21,265-283 (1982).49-63 (1982).photosphere, Astrophys. J . 138,552-575 (1963).equations, Astrophys. J . 162,665-67 3 (1970).

A275,611-651(1979).Soward. A .M .. Convection-driven dvnamos. in these Proceedings, 237-244 (1982).Steenbeck, M., Krause, F. and Radler, K.-H., BerechnuG der mittleren Lorentz-

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16 S T E LL AR AN D P L ANE T AR Y M AGNE T IS M-eldstarke v x B ur ein elektrisch leitendes Medium in turb ulen ter, d urch Coriolis -Krafte beeinflufiter Bew egun g, Z . Naturforsch. Zla, 369-376 (1966).Taylor, G.I., Diffusion by continuous movements, Proc. Lond . Math. Soc. AZO, 196-211(1921).Weiss, N.O., The expulsion of magnetic flux by eddies, Proc. Roy. Soc. A293, 310-328(1966).Zeldovich, Ya.B., The magnetic field in the two-dimensional motion of a conductingturbulent fluid, JETP 31, 154-155 (1956), Engl. transl. Sov. Phys. J E T P 4, 4 6 0 4 6 2(1957).R e p r i n t e d f r o m :S t e l l a r & P l a n e t a r y M a g n e t i s m( e d . A.M. Soward)Gordon & Breach, London, 1983