H.K. Moffatt and M.R.E. Proctor- Topological constraints associated with fast dynamo action

8/3/2019 H.K. Moffatt and M.R.E. Proctor- Topological constraints associated with fast dynamo action

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J . FZud Mech. (1985),d.54. p p . 493-507Printed in Great Br itai n

493

Topological constraints associated with fastdynamo actionBy H. K. MOFFATT A N D M. R. E. PROCTOR

Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge(Received 23 May 1984)

The conjecture of Vainshtein & Zeldovich (1972) concerning the existence of a fastdynamo (i.e. one whose growth rate is independent of magnetic diffusivity 7 in thelimit 7 -to) s discussed with particular reference to ( i ) the stretch-twist-fold cyclewhich can double the strength of a magnetic flux tube, and ( i i ) the space-periodicBeltrami flow of maximal helicity, which has been shown to be capable of space-periodic dynamo action with the same period as the velocity field, by Arnold &Korkina (1983) and by Galloway & Frisch (1984). The topological constraintassociated with conservation of magnetic helicity is shown to preclude fast dynamoaction unless the scale of the magnetic field is almost everywhere of order 7; as 7 + O ;in this case, the field structure is severely singular in the limit. A steady incompressiblevelocity field, quadratic in the space variables, is shown to mimic the action of thestretch-twist-fold cycle, and is proposed as a plausible candidate for fast dynamoaction.

1. IntroductionThe term dynamo action in magnetohydrodynamics is generally used to describethe systematic and sustained generation of magnetic energy as a result of the

stretching action of the velocity field u(x, ) on the magnetic field B ( x , ) . This actionis described by the induction equation

( 1 . 1 )B = ~( U A B ) + ~ V ~ B (v.B=o),atwhere 7 is the magnetic diffusivity of the fluid (see e.g. Moffatt 1978).

I n a purely kinematic approach to the dynamo problem, the velocity field U isregarded as k n o w n , and in particular the back-reaction of the magnetic field on U (viathe Lorentz force distribution) is assumed negligible. This k n o w n velocity field 7naysatisfy certain dynamic constraints (e.g. hose imposed by the Euler equations or theNavier-Stokes equations, with or without Coriolis forces, buoyancy forces, etc.) butit is convenient to adopt a general approach in which U is reed from any such dynamicconstraints, and we simply investigate the general behaviour of solutions of ( 1 . 1 ) fora wide class of velocity fieldsU,which are supposed to satisfy only the mild kinematicconstraint of incompressibility v.u = 0. (1.2)

In the particular impor tant situation in which U is steady, i.e. U = u ( x ) , 1 . 1 )admits(1.3)(1.4)

solutions of the form B ( x , ) = Re {&x) ept},p 8 = v A (U 8 ) 7 V28.here

www.moffatt.tc


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494 H . K. Moffatt and M. R. E. ProctorHere p is the (possibly complex) growth rate associated with the field structure B(x)(which satisfies V . 8 = 0). In conjunction with appropriate boundary conditions on&x), (1.4) constitutes an eigenvalue problem; and if any of the eigenvaluesp l ,p,, p,, . have positive real part, then the corresponding field structuresB,(x),8,(x),B,(x), ..,grow exponentially in time. The associated dynamo is oscilla-tory or non-oscillatory in type according as the imaginary part of p is non-zero orzero respectively.The distinction between fast and slow dynamos has been introduced byVainshtein & Zeldovich (1972),and the distinction provides the basis for much ofthe discussion in the recently published monograph of Zeldovich, Ruzmaikin &Sokoloff (1983).Suppose that the velocity field u ( x ) s characterized by a lengthscale1, and a velocity scale U,, so that the timescale characteristic of the motion (theturnover time ) is

The magnetic Reynolds number associated with the flow ist o = ZO/UO. (15 )

R m = uo V T , (1.6)and we are particularly concerned in astrophysical contexts with the limitingbehaviour when R, +m. A dynamo with growth ra te p = p , + pit is said to be slowifand it is said to be fast if

p,t,+O as R m + m . ( 1 . 7 )

p,t,+const > 0 as R m + CC (1.8)(Zeldovich & Ruzmaikin 1980). For a slow dynamo. the mechanism of fieldgeneration is diffusive in character (or at least involves magnetic diffusion in anessential way). All dynamos with laminar velocity fields u ( x )for which detailed andrigorous calculations have been carried out are of the slow type ; typically, for a slowdynamo, p,t ,=O(R;;P) asR,+m, wh er eO


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Topological constraints on fa st dy na mo action 495

FIGVRE. The s t re tch- twist - fo ld cyc le ; i s th is a fast d yn am o?way, and it can do this only if the field B varies on a scale O($) so that 7 V2B= 0(1)as 7+0. We shall show in $3 hat , if the relative helicity of a fast-dynamo magneticfield is O ( l ) , hen this scale-refinement effect must occur throughout a fraction of theavailable volume that remains O ( 1 ) as 7+0 (and no t, for example, only in theneighbourhood of a set of singular surfaces).

These results do not prove the existence of a fast dynamo - hey merely describewhat it must look like if it does exist. It is desirable to have a much more detailedpicture, and for this purpose there are two candidates for fast-dynamo action whichdeserve detailed study.( a ) The stretch, twist and fold dy na mo . This is the prototype fast dynamo proposedby Vainshtein & Zeldovich (1972).An initially circular flux tube of small cross-sectionis subjected to a stretch , twist and fold sequence as indicated in figure 1. like thedoubling of an elastic band. T o get back exactly to the initial configuration, with adoubling of the field st rength, a li ttle diffusion is evidently needed to eliminate thecrossing of field lines in the neighbourhood of the point P; but if it is accepted thatthis can be achieved, then the doubling time should be of order lo/uo= to , thetimescale for the stretch-twist-fold cycle, independent of 7 . JT7e shall study thisprocess closely in $4, and show that the effect of diffusion is crucial in determiningthe field structure th at may develop under many i terations of the cycle. We shall alsoconstruct an Eulerian velocity field which incorporates the stretch, twist and foldingredients, and which is proposed as a candidate for a localized fast dynamo.

( b ) Th e space-periodic Bel trami dyn am o. A second veliocity field that has at trac tedrecent attention in the fast-dynamo context (Arnold 81 Korkina 1983: Galloway &Frisch 1984) is the space-periodic field

U = ( U 3 inkz+ C, cosky. C, s i n k + C , coskz, C, sink y+ C, coskx), (1.13)which satisfies the Beltrami condition

o = V A U = k U , (1.14)and which is therefore a field of maximal mean helicity ( u * o ) = k ( u 2 ) (angularbrackets indicating an average over a cube of side 2 x l k ) .The field (1.13) s of intrinsicinterest because of the chaotic character of the streamlines when C, C, C, + 0 (Arnold1965; HBnon 1966), a property th at may be conducive to fast-dynamo action.Mean-field and first-order-smoothing techniques (Roberts 1970; Childress 1970: seealso Moffatt 1978, chap. 7 ) may be used to show tha t the velocity field (1.13) willact as a dynamo when R, is small , the field B then growing on a scale L large comparedwith 1, - -l . As R, increases, the scale L decreases, and ultimately the techniquesof mean-field electrodynamics are inapplicable. The approach of Arnold & Korkina(1983) and of Galloway & Frisch (1984) is to restrict attention to fields B ( x , ) thatare space-periodic with the same period 2 n l k as U (and with zero mean over the basiccube), and to compute the field evolution. Results obtained for R, up to 200


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496 H . K . Moffatt and M . R. E. Proctor(Galloway & Frisch 1984) are suggestive of fast-dynamo action, although a moredetailed analysis of field structure than is yet available will be required to confirmthis behaviour. We return to this problem in $5, where some general aspects ofspace-periodic dynamos are discussed.

2. Topological constraints on a non-diffusive fast dynamothe frozen field equationConsider first the perfectly conducting situation in which r ] = 0 and B ( x , ) satisfies

- V A ( U A B ) .B_ -atSuppose that B is localized in the sense that

e"~BI-+O a s r = I x I - t c ~for some k > 0, and let A ( x , ) be a vector potential for B . Then it is well known(Woltjer 1958) that

X = A - B dV = const. ( 2 . 3 )sThis invariant, the helicity of the field B, is essentially topological in character(Moffatt 1969)'and is in fact a generalization of the Hopf invariant, described as theasymptotic Hopf inva riant by Arnold (1974).A magnetic field with non-zero helicity is one for which there is a net linkage oflines of force. The fact that lines of force are frozen in the fluid implies that this ne tlinkage cannot change, and this is reflected mathematically in the conservation of2. t is therefore obvious th at a field that has non-zero helicity cannot be amplifiedby dynamo action, since this would imply a corresponding exponential increasein X .This argument does not exclude the possibility that a field for which X = 0 (i.e.for which the net linkage is zero) may be amplified by dynamo action, with at mosttime-periodic change of structure, when r ] = 0. This possibility may however beeliminated by the following argument.We are concerned with the existence of solutions of (1.10)withp $: 0 for given u ( x ) .We may include the possibility of compressible flow by introducing a density fieldp ( x ) , nd a (steady)mass conservation equation

V . ( p u )= 0. (2.4)We shall consider two cases:which u'n = 0 ;surface we again denote by S.I n either case, let the volume interior-to S be denoted by V .

case A . u ( x ) s localized in the sense that there exists a finite closed surface S oncase B. U and B are space-periodic with the same basic cuboid of periodicity, whose

Now if A is a vector potential for B , we may 'uncurl' (1.10) o obtainp A = U A B - v q ? (2.5)

p a l = U A B, v A A , = 2, (2.6)fcr sope scalar field @. If p $: 0, we may introduce the change of gaugeA , = A +p-' V@, so that


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Topological constraints o n fast dyna mo action 497from which it immediately follows that

A ; ( V AA , ) = 0. ( 2 . 7 )If A,(x) were a real vector field, then this would be recognized as the condition forthe existence of a family of surfaces g(x)= const., everywhere orthogonal to A , , i.e.for the existence of scalar functions f ( x ) ,g(x) such that1, V g , B= Q f A V g . ( 2 . 8 )In fact, the result (2.8)still holds t when 2, s complex, but now f ( x )and g(x) arecomplex scalar fields. Substituting (2.8)back in (2.6)gives

pf vg = U A (vfA v g ) = vf U ' v g - V g U ' v f , (2.9)and crossing this with V f givesc' ( p f + u . V f ) B = 0. (2.10)

A

Hence at every point of space, either B = 0 orp f + u * V f = 0. (2.11)Suppose first that B is non-zero throughout V , so that (2.11)holds thoughout V .We easily deduce that

c ,.In case A the surface integral vanishes because u'n = 0 on S ; n case B it vanishesby periodicity. So in either case it follows that p , = ? j (p+ p*)= 0 , and so we do nothave a fast dynamo.Now suppose that there exists a surface S, inside S on which B = 0, but such thatB =l= in the volume V, interior to S,. Then B ( x , ) = 0 (all t ) on S, and B ( x , ) + 0in V,. By AlfvBn's theorem, i t follows that u-n 0 on S , (since otherwise theseconditions could not persist). We may then apply the above argument to the volumeV,, and again we conclude that p , = 0.

We may conclude therefore that, in all cases considered, a non-diffusive fastdynamo is impossible.Sote that , if p = ipi + 0, then from (2.9)) t all points where + 0, we must have

} V f A V g + O .f + u . V f = 0,U - v g = 0 (2.13)This can happen only if the streamlines of the flow are the intersections of the surfacesg, = const,gi = const, i.e. only for a very special (non-generic)class of velocity fields.

3. Topological constraints on a diffusive fast dynamofact, from ( 1 . 1 ) and the 'uncurled ' equationIn the presence of weak molecular diffusivity, helicity is no longer conserved. In

aA- U A B - V V - T V A (V A A )att This is a consequence of Frobenius' theorem in C3 (see e.g. Boothby 1975, p. 159).The resultmay be proved by elementary methods (following Ince 1925, p. 52) by regardinga, as an analyticfunction of 3 complex variables (z, y , z ) and ultimately restricting to the real axes.


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498we may readily obtain the equation

H . K. Moffatt and M. . . Proctor

5 [A * B V = -2 7 B * VA (V A A ) dV,dt swhere the integrals are throughout all space. Hence, even if 7 is very small, thehelicity can change significantly when the field gradient becomes large.We may obtain an upper bound on the growth rate p , of a dynamo as follows. Firstwe average ( 3 . 2 )over a time 2n/pi; denoting this average by a n overbar. ( 3 . 2 ) hengives

p , s m d V = -7 sB - V A (V A A) dV. ( 3 . 3 )By the Schwarz inequality,

SB*VA(VAA) V< ( S B i d V ) ' ( S ( V A ( V A A ) ) ' d ~ ) ' . ( 3 . 4 )Defining the lengthscale 1 characteristic of the field B by

j ( V A ( V A A ) ) z d V =Zi4sZ d Vand the relative helicity xRsatisfying I XR < 1 by

j m d VxR = (SP d V S F dV)"

we easily obtain from ( 3 . 3 ) - ( 3 . 6 )

or equivalently

( 3 . 5 )

( 3 . 7 )( 3 . 8 )

Now in general a velocity field that has non-zero helicity will tend to generate amagnetic field with non-zero helicity, so that in general there is no reason to expectthat ZRhould be small. If IXR = O(1) then ( 3 . 8 ) implies that ZB/l, is at mostO(&) ; from the definition ( 3 . 5 ) t then seems likely th at the scale of B must be O(R2 )or less over an O(1) fraction of the flow domain.

The diffusive fast dynamo (if it exists !) therefore generates a magnetic field whosegradient is typically O(&m), and which evidently becomes non-differentiable over asubstantial par t of the flow domain in the limit E,+ 03 ( q + O ) . The Lorentz forcedistribution in such a dynamo is

F ( x , ) = j A B =,u;l(V AB) B - B2/,u, I,, (3.9)and this also will vary on the scale I,. This force will ultimately generate a n additionalvelocity field u , ( x , ) on this same lengthscale. a process which must ultimately beresponsible for equilibration of the growing field.

Such a dynamo is totally different from the slow dynamos that emerge from, forexample, the two-scale analysis of mean-field electrodynamics. In these dynamos, thegrowing mean magnetic field has a scale L large compared with I, , and there is alsoa fluctuating ingredient on the scale 1, driven directly by the velocity field U. TheLorentz force acts either to suppress the turbulence (Moffatt 1972) or to drive alarge-scale mean velocity (Malkus & Proctor 1975) . In the fast dynamo considered


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Topological constraints on fas t dyna mo ac tion 499

0 Cross-sectionI0 ross-section

FIQURE. The stretch U,@) leads to flattening of the flux-tube cross-section to elliptic fo rm. Thisis followed by the twist u 2 ( x ) nd the plane strain u 3 ( x ) o bring the points A and B towards eachother.here, the scale of the magnetic field is small compared with the scale of the velocityfield u ( x ) ,and the Lorentz force acts to generate small-scale velocity fluctuationsul(x, t ) . This is indeed a novel situation in the dynamo context.I n $4we examine in detail the st retch , twist and fold dynamo described in $ 1, witha view to understanding just how it is that large field gradients can develop.

4. Stretch, twist and fold dynamoLet us consider again the distortion process depicted in figure 1, but now takingaccount of the finite cross-section of the flux tube. Suppose that the centreline of theflux tube is initially the circle x2+y2= a: in the plane z = 0, and tha t its cross-sectionis initially a circle of radius c 4 ,. We shall suppose first th at 7 = 0, i.e. th at diffusionis totally negligible.

The initial process of stretching (figure 2 ) may be achieved by the uniformincompressible straining field

U,@) = (czx,ay,- 2 4 , (4.1)with a > 0. Cnder the action of this field, the radius of the flux tube increases

( 4 4exponentially : a ( t )= a , eZt,being doubled after a time t , = a-l ln2. At the same time the cross-section of thetube is flattened by the s tra in into an ellipse

(4.3)with semiaxes in the ratio 8 : 1 . Sote that the volume of the flux tube V x 2n2c2a,remains constant.

Consider now the twist stage. A twist about the x-axis is well-represented by the(4.4)velocity field

where w(x) is antisymmetric about x = 0. the simplest possibility (uniform twist)being w ( x )= - x, where f is constant (f > 0 for a right-handed twist), so that

- ~ ~ ) '162' = c',

UZ(X) = (0, - 4 x ) z , W ) y ) ,

U&) = (O,fXZ, --fXY). (4.5)


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500 H . K . Moffatt and M . R . E. Proctor

4 econnect

FIGURE. Twisting and reconnection of paper tape, or flux tube of elliptic cross-section.

This is a twist of the kind tha t is applied to an elastic band in the doubling process,but i t fails to mimic this process in tha t the points (0 , f 2 a 0 , ) ( A and B in figure 2 )remain fixed under the velocity field ( 4 . 5 ) and do not approach each other assuggested in figure 1. This approach in an experiment with elastic band or papertape is a natural consequence of the resistance of these materials to stretching, afeature that does not arise for the magnetic flux tube if the Lorentz force is negligible.(It is interesting to note here that a strong magnetic field subjected to twist wouldpresumably respond in a nearly inextensible manner, so th at , in a dynamic regimein which Lorentz forces are important, the elastic-band analogy may be morerelevant .)

I n order to achieve the approach of the points A and B , we have to supplementthe twist field ( 4 . 5 )by a further strain field which compresses along the y-axis, butleaves the scale of the loop along the x-axis undisturbed. The two-dimensional strain

( 4 . 6 )fieldwith P > 0 will do for this purpose. If the fields U, and U, act simultaneously for atime t , = n / 4 a o , hen the distance between A and B will be reduced to

u3(4 = (0, -PY, P 4

(4.7)= 2a0 e-& = 2a e-~3/4aof.0Kow, however, we are twisting not simply a closed curve, bu t a flux tube with initiallyelliptic cross-section. The twisting of a paper tape provides a better analogy. A s asimple experiment will demonstrate, a right-handed twist applied to a paper tapeinduces a left-handed twist of the tape about its own centreline (figure 3 ) . tIf the tapeis broken and reconnected a t the points A and B (simulating the diffusion process),then the two loops thus created have the form of Mobius strips, each one having anet left-handed twist of n.

Similarly the flux tube will develop what may be described as intrinsic twist asa result of the action of the velocity fieldu z ( x )+u , (x ) .When reconnection takes place,this intrinsic twist manifests itself as helicity of the magnetic field. A poloidalmagnetic field has been generated, superposed on the original toroidal field round each

t Certain features of this process have been recently considered by Berger & Field (1984).


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Topological constraints on f a s t dynamo action 501

FIGURE.Helical field B in a wis t ed t u b e of fo rce ; t h e p o loid al p a r t of th e f ie ld B , isassocia ted wi th a to ro idal cur re n tj , which ha s zero in tegral over th e tu be cross-sectionIY , y

/ I II

/- oops in(y,2)-plane- -e x-/d IFIQURE. Fold ing of two loops in to near -co incidence, fol lowed b y t rans la t ion Can d ro t a t i o n w t o r e tu rn t o co n fig u ra t io n of figure 2.

loop. Thus the process is not unlike the process by which poloidal loops of field aregenerated by cyclonic eddies (Parker 1955), he process which underlies the a-effectof Steenbeck, Krause & Radler (1966). n the present context , the poloidal field isassociated with a toroidal current flowing round each flux tube ; the net flux of currentalong the tube is however zero, since the poloidal field is confined to the neighbourhoodof the flux tube (figure 4).To complete the stretch-twist-fold cycle, we require a velocity field U , (X ) thatrepresents the action of folding the two loops of figure 5 into near-coincidence, A fieldthat will achieve this is %(X) = ( - yx, Y+9"2 , O ) ,with y > 0, g > 0. The gx2 ingredient deforms the loops out of the (x, )-plane, andthe remaining (plane-strain) part of (4.8)compresses both loops towards each otheron the (y,z)-plane. A small value of g (ga , < y ) will suffice t o achieve the necessaryeffect, in a time t , satisfying yt4We now have a double loop in the ( y ,2)-plane.To complete the process and to re turnto the initial configuration, we require a translation (of order a,) and a rotationabout the y-axis; the velocity field

1 .

U , ( X ) = (0,w ,0) A X- (0,U ,0) (4.9)with I; x 2aoo/7t will achieve this effect in a time t, = 2 w / n .

The velocity fields u , ( x ) ,u 2 ( x ) ,..,u , ( x )should thus, in succession, and applied forsuitable time intervals, achieve an approximate doubling of the initial magnetic field,but a t the cost of generating a net twist in both of the new flux tubes. If these velocityfields act simultaneously, so th at we have the steady velocity field

u ( x )= u , ( x ) + u 2 ( x ) + .. + U 6 ( X )= (a ' x+w z , p 'y+gx2+fxz- U , y'z-wx-fxy), (4.10)

where a/ a- y , p' = a+ y - p , y / = -2a+p, (4.11)


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502 H . K. Moffatt and M . R. E . Proctorthen we may expect that the initial loop will be continuously deformed in a similarway, and indeed that the doubling process should be repeated again and again if thevelocity field (4.10) s sustained. The particle trajectories associated with (4.10)aregiven by the dynamical system

dx- ax +wz ,dtdY-t = py+gx2+fxz- c, (4.12)dzdt- - yz -ox- xy,

where a+p +y = 0, and where approaching of the two loops in the twist and foldprocess is achieved if a < 0 and /3 < 0. This volume-preserving ( V - U= 0) systemclearly deserves close study, for various values of a, , w , g , f , U ; t seems highlylikely tha t, in general, the trajectories are chaotic.

The vorticity associated with the velocity field (4.10) sw ( x )= ( -2 fx , 2w+fy, 2gx+fz), (4.13)

and the helicity, integrated over any sphere I x I < R, is

The motion therefore has a net right-handed or left-handed sense according as a < 0or > 0. It is the helicity (4.14) which, in conjunction with weak diffusion, isresponsible for generating helicity (of opposite sign!) in the magnetic field.

The motion (4.10) s of course unbounded a t infinity, and there is no guarantee thatthe trajectories of fluid particles will remain within a sphere r < R, no matter howlarge R may be. It is easy, however, to modify the velocity field (4.10)so that nearlyall of the trajectories do all return to the neighbourhood of the origin. To do this,let A(x)be the vector potential of ~ ( x ) ,cubic function of the coordinates: in fact

A ( x )= [gx*z- L%+;fx(z2+y2), yz x -~ (x2+ 2*) , Pxy]. (4.15)Sow define the modified vector potential

A (x ) = A ( x ) prIR (4.16)and the modified (solenoidal) velocity field

UM(X) = v A AM(X ) . (4.17)Then clearly uMcoincides with U for r 6 R, and yet is exponentially small for r 2 ,so that nearly all of the streamlines are forced to re turn to the interior of the spherer = R. Any magnetic field that is initially confined to the sphere r 5 R will thenprobably remain so confined for all time (under the frozen field assumption).Let us now consider what happens when the stretch-twist-fold cycle is repeated.The stretch is now applied to two adjacent flux tubeseach of 8 : 1 elliptical cross-sectiontwisted in the form of a Mobius strip. The initial stretch in the (x, )-plane againflattens the cross-section : where the long axis of the ellipse is initially parallel t o theplane, the ellipse is further stretched till it s axes are in the ratio 64 : 1 , and where thelong axis is initially perpendicular to the plane, the cross-section returns to theoriginal circular form (bu t with the original radius). The twist about the x-axis


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Topological constraints on fas t dyn am o ac t ionI I \ U = (ax,ay, - 2 a z ) 503

IFIGURE. Section of flux tu be w hen diffusion l imits comp ression.

again induces additional intrinsic twist in the flux tube - he twisting and reconnectionof a Mobius strip leads to a Mobius strip (twist 7c) and a strip with twist 27c; in thecase of flux tubes, if symmetry is maintained between the two daughter tubes, theneach will have a twist of !7c. The folding stage again leads t o the superposition of thesetwo tubes, which will moreover be linked with the neighbouring cousin tubes.It is clear that , even after only two stretch, twist and fold cycles, we have generateda field of considerable complexity. To be sure, the toroidal flux has increased fourfold;but poloidal field varying on a scale &c has been generated - and after n cycles thescale of variation would bf: c / Z ~ ~ .his is in effect an exponential decrease of scale1 , c e-tto, where to is the timescale of the stretch-twist-fold cycle. Clearly,molecular diffusion, neglected in the discussion so far , must intervene to eliminatethese field variations as well as to achieve the reconnection of flux tubes. It seemsclear that we are dealing with a fast dynamo of the diffusive rather than thenon-diffusive type (see$ 3 ) .Let us now consider the effects of diffusion in more detail.For the sake of argument, suppose that the initial stretch (4.1) s maintained fora long time until the smaller dimension of the cross-section of the tube is reducedto O ( ~ / c t ) a , t which diffusion becomes important. This lengthscale does not thenreduce further (figure6) .However, the larger dimension of the cross-section continuesto increase like eZt,and, since the to tal toroidal flux in the tube is constant duringthe stretch process, the field intensity must decrease like eTZt. n fact , if we move withthe flux tube, the relevant local solution of ( 1 . 1 ) with the velocity field (4.1) is

B(z, ) = B, e-ot e-az*T. (4.18)= ( 0 ,B(z, ) ,0 ) ,withThis type of behaviour in which the decrease of scale in one direction is limited bymolecular diffusion was discussed in the context of scalar-field diffusion by Batchelor(1959); its importance in the dynamo context has recently been emphasized byZeldovich et a l . (1984).When we are on this small lengthscale, it is evident t hat, in the stretch-twist-foldcycle, the toroidal field is not doubled - t is halved ! The toroidal fiuz is neverthelessdoubled, because there are now two adjacent flux tubes each with double the originalcross-section. Repetition of the cycle leads to continued increase of the net cross-section, and the structure of the field that finally emerges from many repetitions willundoubtedly be very different from the initial simple circular flux tube of smallcross-section.


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504 H . K. Moffatt and M . R. E. Proctor5. The space-periodic Beltrami dynamo

A Beltrami flow is one for which V A U = k, U,where k, is constant. Such a flowhas maximal helicity, and for this reason is of particular interest in the dynamocontext. The helical wave

(5.1)l ( x , t )= (0,U , sin(k,x-w,t), U, cos(k,x-w,t))satisfies the Beltrami condition, and has helicity

X1= ( u ; V A u , ) = k, U:.Similarly, defining

u 2 ( x , t ) ( U , cos(k,y-w,t),O, U , sin(k,y-ww,t),u , ( x , t )= ( U , sin(k,z-w,t), U , cos(k,z-w,t),O),

( 5 . 3 )(5.4)

the velocity field ( 5 . 5 )satisfies V A U = k, U and has helicity

X = ( u ' V A U ) = ko(U:+U,2+U:). (5 .6)The flow (5 .5 ) s an exact solution of the Euler equation in a rotating fluid; in a framerotating with angular velocity a this equation may be written

(5.7)and it is easily verified that this is satisfied by ( 5 . 5 )provided

= -a(@,, w2 ,U , ) , ( 5 . 8 )and provided p is suitably chosen. When w1 = w2 = w , = 0, we have the steady flow(1.14)discussed briefly in 1 .It is well known that flows of this type are capable of dynamo action onlengthscales L large compared with k;l (Childress 1970;Roberts 1970). Indeed, thereis an a-effect associated with the motion ( 5 . 5 ) ,which, on first-order-smoothing theory,is given by the tensor

where

(5.9)

(5.10)(Moffatt 1978, $ 7 . 7 ) . When the frequencies w , are non-zero, d n ) + O as q + O , aproperty that persists a t all higher orders of perturbation theory (Dillon 1974).If, for simplicity, we consider the isotropic situation

(5.11)1 -w1 = w2 = 0, w ,then aij= dij,here (5.12)


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Topological constraints o n fas t dy na mo action 505Similarly, there is an augmentation of the molecular diffusivity given by

( 5 . 1 3 )where E = :U2 is the mean kinetic-energy density of the motion (3 2 k , E ) . Theimportant point here is that -a l = O ( T )21 = O ( k , ) ,P (5.14)and, although this is a result of first-order-smoothing theory, it may be expected tohold at higher orders of perturbation theory also.The mean-field equation, describing evolution of a field B, on a scale large

aB0 (5.15)compared with k; l , is

This has non-oscillatory dynamo solutions of force-free structure (V A Bo= K B , )whose growth rate p is given byp = aK-(v+P)K2. (5.16)

- V A B,+ (7+p)V 2 B , .atThe maximum growth rate occurs for

and, if we adopt the expressions (5.12)and ( 5 . 1 3 ) , hen( 5 . 1 7 )

(5.18)so that K , 6 k , (as required for self-consistency of the approximation (5.15))

w 2 + y 2 k ;9 C 2 k i . ( 5 . 1 9 )rovidedAs Cincreases (for fixedw and v ) , the preferred scale of growth of the fieldB, decreasestowards the scale k;l of the velocity field, and the methods of mean-field theorybecome progressively less reliable.The alternative approach (Arnold& Korkina 1983 :Galloway & Frisch 1984) is thento restrict a ttention to a field B ( x , ) with the same periodicity as the field U and withzero mean over a basic cube of side 27c/k,. The results ( 2 . 5 )and (2.6)apply equallyif V is taken to be this cube - so. if 7 = 0, the magnetic helicity in the cube is constant.The arguments of 992 and 3 then imply that, if we have a fast dynamo ( p , = O ( 1 ) )with non-zero magnetic helicity, then the scale of variation of the magnetic field mustbe of the order R$ k;l nearly everywhere. In spectral terms, the spectrum of B maypzak at wavenumbers of order E , . but the spectrum of V AB must have strongcontributions a t wavenumbers of order fi, k ,. The situation is consistent with thestatement of Galloway & Frisch ( 1 9 8 4 ) t that spectra of the growing or decayingmodes show that the value of the wavenumber k a t which the energy peaks issurprisingly insensitive to R, hough the length of the tail appears to increase roughlyas the square root of R. In the range of wavenumbers k , < k < & E , , the magneticspectrum function M ( k )presumably has a power-law dependence of the form

M ( k ) V k - Q . 0 < q < 3 . ( 5 . 2 0 )t Galloway & Frisch restrict attentio n to t he steady case (q w2 = wg= 0) and present resultsfor R(= R- I < 200.


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506 H . K. 3loffatt and M . R. E. ProctorIf the process of field distortion is qualitatively similar to that of the stretch-twist-fold dynamo of $4 hen it would appear that the fluctuation of V A B hat isgenerated a t any scale k- l is related to the twist effective a t that scale, and for themotion ( 5 . 5 ) onsidered, this twist is independent of k . This suggests that ( k 2 M ( k ) )should be independent of k , i.e. tha t q = 3 in (5.20).The integral of k 2 M ( k ) s thenlogarithmically divergent as 71+0, corresponding to the non-analytical character ofthe magnetic field in this limit. The results available a t present (Galloway. privatecommunication) suggest t ha t M ( k ) n fact falls off more slowly than K 3 n the rangeE , 4 4krn, ; the reasons for this are not a s yet clear.

6. ConclusionsThe fast dynamo is a dynamo whose growth rate p , is independent of 11 in the limit

? + O S I ts existence, as a phenomenon distinct from the more familiar slow dynamo,has not yet been rigorously established for a steady flow. This paper has been devotedto a discussion of its structural properties, if it does exist. We have shown firstly thata non-diffusive ast dynamo does not exist, and secondly that , for a diffusive fastdynamo, the scale of variation of the magnetic field of a growing helical mode musttypically be O(R2)and diffusion must play the key role in resolving the conflictbetween magnetic-helicity invariance and exponential field growth. The stretch-twist-fold dynamo has been examined in some detail, and the mechanism by whichfine structure appears in the magnetic field has been revealed. A similar fine structuremust appear in the space-periodic dynamo of Arnold & Korkina (1983) f, as suggestedby the computations of Galloway & Frisch (1984), this is indeed a fast dynamo inthe limit R, +CO,

R E F E R E N C E SARKOLD,. I . 1965 Sur la topologie des Bcoulements stationnaires des fluides parfaits. C. R. Acad.Sci. Paris 26 1, 17-20.ARNOLD, . I. 1974 The asymptotic Hopf invariant and its applications (in Russian). In Proc.Sum me r School in Differential Equa tions, Erevan 1 9 7 4 . Armenian SSR Acad. Sci.ARNOLD,. I. & KORKINA.. I. 1983 The growth of a magnetic field in a three-dimensional steadyincompressible flow (in Russian). Vest. M o s k . Un . T a . S e r. 1, Ma t. Mec. 3 , 43-46.BATCHELOR,. K. 1959 Small-scale variations of convected quantities like temperature inturbulent fluid. Part 1 . General discussion and the case of small conductivity. J . Fluid Xech .BERGER, . A. & FIELD,. B. 1984 The topological properties of magnetic helicity. J . Fluid Aech.BOOTHBY, . M . 1975 An Introduction to Differentiable Manifolds and Riemannian Geometry.Academic.CHILDRESS,. 1970 New solutions of the kinematic dynamo problem. J . X a t h . P h ys . 11,3063-3076.DILLON,R. F. 1974 Gravity and magnetic field correlation and related geomagnetic topics,GALLOWAY,. & FRISCH,. 1984 A numerical investigation of magnetic field generation in a flowHENON, . 966 Sur la topologie des lignes de courant dans un cas particulier. C. R. Acad. Sci .INCE,. L. 1925 Ordinary Differential Equations. Dover.MALKCS, . V. R. & PROCTOR,. . E . 1975 The macrodynamics of cc-effect dynamos in rotating

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chap. 6. Ph.D. thesis. Cambridge University.with chaotic streamlines. Geophys. Astrophys. Fluid D yn . 29. 13-18.Paris 262, 312-314.

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H.K. Moffatt and M.R.E. Proctor- Topological constraints associated with fast dynamo action

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Transcript of H.K. Moffatt and M.R.E. Proctor- Topological constraints associated with fast dynamo action