H.K. Moffatt- Aspects of Dynamo Theory

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    CHAPTER 7ASPECTS OF DYNAMO THEORY

    H.K. MoffattDepartnwnt of Applied Mathematics and Theoretical Pliysics

    University of CambridgeEnyland CB3 9EW

    The elements of dynamo theory are discussed, w i t h pazticular at ten-tio n t o the par ticu lar problem tha t a ri se when, as i n the solar con-te xt , tlic laagrietic d if fu s iv it y i s very small. The growth of the d ip ol emvimnt of a localised current system i s essen t ia l ly d i f fus ive i n charac-t e r ; i n the l i m i t of vanishing dif fus ivi ty, the sp at ia l s tructu re of a n ydynamo m u s t become increasingly complex; t h i s i s the 'fast dynamo'l i m i t .

    When convective eddies a re p e rs i s te n t , th e phenomena of f l u x expul-sion and topological pumping play a n important part i n the dynanm pzo-ces s. These ef fe ct s appear i n the ' m an -f ie ld ' theozy of the turbulerltdynamo vi a an 'e ff ec ti ve ve lo ci ty ' of transp ort of the mew magneticf i e ld m t i v e t o the f l u i d .

    These effe ct s are a l l discussed i n the context of the so la r dynamo,regarded as a dynamo of a w - t y p e , with magnetic buoyancy providing anequilibration mechanism.

    \7 . 1 'rm HOMOPOLRR DISC DYNAMOSome p ec u li ar it ie s of dyriaurw theory a le very well i l lu st ra te d b y theprototype example of s e l f -e x c i t in g dynania ac ti on , v iz . the homopvlaxdisc dynamo sketched i n figure 7.1. The conducting disc rotates abouti t s axis under the action of a n applied torque G . A wire, twistedabout the axis i n the manner shown, makes sl id in g con tac t w i t h the disca t A , and with the axis a t 8, and carries a current / ( t ) . The magneticf ie ld associated with t h i s current h a s a f l u x @ = M I across the disc,where M i s the mutual inductance between the wire and the rim of the

    172

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    CHAPTER I: ASPECTS OF DYNAMO THEORY 173

    Figure 7 . 1

    disc. The r o t a t i o n of th e d i s c i n t h e presence of t h i s flux prov ides an nr ad ia l e l ec t rom ot ive fo rce %@ - I which d r i ve s t he cu r r en t 1 . On2 nt h i s s i m p l i s t i c d e s c r i p ti o n , the equa t ion fo r / i s

    suppose that n i s main tained cons tan t by su i t ab le ad jus tment o fP tt h e d r i v i n g torque. lien (7.1) has exponen t i a l so lu t i on I ( t ) = I ( 0 ) ewhere

    where R i s the t o t a l r e s i s t a n c e of the c i r c u i t a n d L i t s s e l f -inductance.\

    and we have ex po ne nt ia l growth of / ( t ) a n d so of the magnetic f i e l d t ow h i c h i t g i v e s r ise ( i . e . w e have dynamo a c t i o n ) provided M n > 2 r R ,i . e . provided the d i s c ro t a t e s r ap id ly enough .

    Appealing though t h i s des cr ip t ion i s i n i t s s i m p l i c i t y , i t cannotbe c o r r e c t ( a l t h o u g h i t w i l l be found i n many t e x t s and review a r t i -c les l ) . For c o n s i d e r t h e l i m i t i n g s i t u a t i o n of a ~ r E g c _ t l y onductingd i s c a d Kir_e, in which case R = 0 . Then, o n the one hand, ( 7 . 2 ) g i v e sp = M f l / 2 n l so t h a t w e s t i l l have dynamo action. B u t on the o therhand, the rim of the d i s c i s a closed circui t moving wi th a perfect con-ductor , and Al fvens theorem ( t h e most basic theorem i n magnetohydro-dynamics) t e l l s us t ha t t he flux @ through t h i s c i r c u i t m u s t be con-s t a n t . There is an obvious con t rad ic t ion . What has gone wrong?

    U

    -1 Jl-p = l l z r n - R I

    The answer i s t h a t we have neglected the c u r r e n t s t h a t Elow azimu-t h a l l y i n the d i s c - i . e . the v e r y c u r r e n t s t h a t are associated w i t h t h e

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    174 H . K MOFrATTdiffusion of flux across the rim of the disc. These currents becomeparticularly important in the lunit R - 0 , and they completely invali-date the above description. The paradox can be resolved by supposingthat the azimuthal current J ( t ) is constrained to flow round the rim ofthe disc (by a suitable distribbtion of radial insulating strips). Thenthe flues through the I and J circuits are given by

    4, = L I t M J (7.3)

    and the equations governing the current flow are

    d132- = - R ' Jd twhere R ' , 1' refer to the J-circuit. This system still admits exponen-tial solutions, ( I , J ) a e , and tlir criterivri for dyriaito activn isstill M n > 2 n R . Now however, p - 0 as R' + 0 , and so the descriptionis consistent with Alfven's theorem. Lletails may be found in Moffatt( 1979) where the nonlinear dynamical system (including the equation forn ( t ) for constant torque G is considered. As shown by Knobloch (1981),a rescaling of the variables for this problem yields the Lorenz systemwith the now familiar chaotic characteristics. It is noteworthy thatthis simplest prototype dynamo system already contains the seeds ofchaos (provided the formulation is self-consistent).

    P t

    It is important to note that, while dynamo action requires that theresistance of the circuit R be low, i.e. that the conductivity U ofdisc and wire be high, we lose the dynamo if we go to the limit U - C Q ,because then the field cannot diffuse into the region in which inductionis operative. cffc.l-e-tjt & n m rgquir_es a c~nd&ctiy&y Lh& Ls large.but nqt t-w arge.

    \

    7 2 "E S'I'RE'l~H-'lWC5'P-POLD D Y K M

    The magnetic field B ( E _ , ~ )volves in 'a conducting fluid of diffusivity 7)moving with velocity u(_x, t )according to the induction equation

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    CHAPTER 7 : ASPECTS OF DYNAMO THEORY

    Figure 7 . 2

    175

    ( 7 . 5 )

    I n the perfectly conducting l i m i t ( 7 - 0 ) , the magnetic lin es of forc e( 'E- l ines ' ) a re f rozen i n the fluid, and i f the motion i s incompressi-b le ( V . g = 0 ) , then stret ch ing of E-lines implies proportio nate i n t e n -si fi ca ti on . The simplest 'h eu ris tic ' dynamo i s based on t h i s ef fec t : amagnetic t h of force can be doubled i n in tens i ty by the stretch-twist-fold cycle indicated i n f igure 7 . 2 (Vainshtein 6 Zel'dovich 1 9 8 2 ) .

    Clearly, as recognized by Vainshtein & Zel'dovich, a l i t t l e d i f fus ion i sneeded to 'ge t back to square on e' , but n eve rthe less the doubling t in efor the process does not apparently depend on diffusivity; i n this sensethe dynamo i s a ' f a s t ' dynanm.

    Here again, however, there i s a danger of over-simplification.When account i s taken of the tube structure, and the way tha t t h i sevolves under repeated ap pl ic at io n of the cycle of fig ure 7 . 2 (see Mof-f a t t & Proctor 1 9 8 4 ) , a h i g h l y complex field structure emerges, and theindicat ions are that the f ie ld E(& t ) develops increasingly fine-scalestru ctu re as the cycle continues, righ t down to the dif fus ive sc aleO(7)"). I n the l i m i t ?) - 0 , the field becomes non-differentiableevezywhere. So here a lso, although the doubling process of fig ure 7 . 2i s non-diffusive i n character, the fast dynamo, i f i t exists, depends i na subt le way on th e ac tion of di ffus io n even i n the l i m i t T ) - 0 .7 . 3 BEHAVIOUR OF THE DIPOLE WMENT IN A CONPINED SYSTEMThis v i t a l inf luence of d i f fus iv i ty i n s.-t&&9 dynamo action i s evi-dent also from the classical results of Bondi 6 Gold (1950) concerningthe Cipole moment ~ ( t ) ssociated with electric currents confined to a

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    176 n. K.MOPFATTsphere of radius R of conducting fluid.field, then two equivalent expressions for k ( t ) are

    If 5 is the resulting magnetic

    From the second of those expressions, it is easy to obtain an upperbound on l c r l , viz

    where 4 is the total flux of S entering the sphere, i.e. the integralof B . over that part of S on which B . p > 0. If 'I) = 0 , then4 = C s t . (Alfven's theorem again) and so exponential increase of & iscertainly impossible; no matter what the velocity field g(&. ) may be,the inequality (7.7) controls the situation.

    Diffusivity however may release this control. Using (7.6), andsome elementaiy manipulation, we have

    When 7) = 0, the first term redistributes the flux on r = R , butrespects the inequality (7.7). When T ) Z 0, provided the velocity fieldis such as to maintain a predominantly positive value of[ - & . 2 X ( V l3) over the surface r = R , diffusion will provide a sus-tained (and potentially unbounded) increase oi lg l . Here therefore thep K b a n mechanism for dynamo action is diffusion, and the growth rate pmay be expected to depend on 7), with p -0 a3 7) 0. This is a 'slow'dynamo in the terminology of Vainshtein h Zel'dovich (1982). In factall known dynamos that have been rigorously established are of the'slow' variety. Frequently p = O ( T ) ' ) with 0 < q < 1, ay T ) - 0.7 . 4 TKE PROS AND CONS OF DYNAMO ACPIONAs mentioned in 87.1, dynamo action can occur only if the fluid conduc-tivity is 'sufficiently large', i.e. only if = (koa)-' is suffi-ciently small. How small is sufficient? A partial answer is providsdby two classical results obtained by manipulation of the equation formagnetic energy associated with electric currents in a sphere r 4 f l :

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    CHAPTER 7 : ASPECTS 01: DYNAMO THEORYn e c e s s a r y c o n d i t i o n s for dynamo act ion are

    ( Backus 1958

    (Ch i ld res s 1969

    177

    (7.9)

    ( 7 . 1 0 )

    where U, is t h e m a x i m u m v a l u e of 1x1 i n r R , and e i s th e m a x i m u mo f th e largest p r i n c i p l e rate o f s t r a i n i n r < R . F r eq u en t l ye,R = O(U,,,), so t h a t (7.9) and ( 7 . 1 0 ) are comparable, though not t h esame. I t may happen however t h a t e m R a U,,, ( i f t h e v e l o c i t y g r a d i en t sare everywhere h igh as i n a t u r b u l e n t f low), and then (7.10) i s a muchs t r o ng e r r e s u l t s .

    m

    I t must be emphasised t h a t (7.9) and (7.10) are n eces s a r y f o rdynamo ac t i on , bu t by no means su f f i c i e n t , A simple s u f f i c i e n t c o n d i - 't i o n c an be f or mu la te d o n l y f o r t u r b u l e n t flow ( s e e 57.6 below).

    T h e r e s u l t s (7.9) nd (7.10), which have been s t rengthened by Proc-t o r (3977), are th e 'pros ' of dynamo action. The ' c o n s ' are provided byt h e various ant i -dynamo theorems, m ai nl y v a r i a n t s a nd g e n e r a l i s a t i o n s ofCowling 's (1934) theorem which s ta tes t h a t " s t e a d y axisymmetric dynamoa c t i o n i s impossible". A s y s t e m a t i c t r e a t m e n t of t h i s class of theoremsis p f ov i ded b y t h e r e cen t w ork of Hide 6i Palmer (1982).

    7.5 FLUX EXPULSION AND TOPOLOGICAL P U M P I N GA f u r t h e r e f f e c t which m i t i g a t e s a g a i n s t e f f i c i e n t dynamo a c t i o n when ?)i s small i s t h e e f f e c t of t h e e x pu l si o n of mag n e t i c f l u x from an y r eg i o no f closed s t r e a m l i n e s . J u s t as f o r the homopolar d i s c dynamo, i f mag-n e t i c f l u x c a nn o t p e n e t r a t e s uc h a r e gi o n, t h en a ny i nd u c t iv e e f f e c t i nt h a t r e g i o n w i l l be-qgte impotent .

    F l ux e x p u l si o n o cc u r s be ca us e t h e v e l o c i t y f i e l d w in ds u p th e mag-n e t i c f i e l d , g en er a l ly i n t o a t i g h t double s p i r a l , i n t h e r eg i o n o fclosed s t r e a m l i n e s . D i f f u si o n t h e n ac t s t o e l i m i n a t e t h e f i e l d f r o mt h i s r e g i o n . The p r o c e s s is w e l l i l l u s t r a t e d b y th e made1 problems ke tc h ed i n f i g u r e 7.3: (see M o f f a t t 6i K a n k a r 1983). Here t h e i n i t i a lf i e J d ( 0 , b o c o s h x , 0 ) i s s h ea r ed b y t h e ve loc i ty f i e l d-- = ( a y , , 0 ) . The problem i s e a s i l y solved i n t e r ms o f the vectorp o t e n t i a l ( 0 , 0 , A ) o f S which sa t is f ies t h e c o n v e c t i o n - d i f f u s i o n

    0

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    178e q u a t i o n

    H . K.MOFFATT

    aA-- t U. V A = vV2A ,atw i t h i n i t i a l condi t ion

    - 10 0 0A ( x , y , O ) = - k b s i n k x .

    The s o l u t i o n here i sA ( x , y , t ) = - k o 1 B0 I m [a(r)ek(t)+-],

    where

    It i

    an d

    a ( t ) = expt h e t -term i n t h e l a t t e r

    ( 7 . 1 1 )

    ( 7 . 1 2 )

    ( 7 . 1 3 )

    ( 7 . 1 4 )

    2- T k 0 ( t t $a3r3) 1 . (7.15)expression w h ic h encapsulates t h e f lux-

    e x p u l s i o n e f f e c t . The time-scale o f t h i s f i e l c l - e l i m i n a t i o n process i se v i d e n t l y

    2rn 0w h e r e R = a / k ( > > 1 ) i s t h e m a g n e t i c R e y n o ld s number a s s o c i a t e dw i t h t h e shear. This estimate i s c o n s i s t e n t w i t h t h a t i n f e r r e d i n t h epioneering s t u d y o f Weiss ( 1966 ) .

    \-#F i g u r e 7 . 3

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    CHAPTER 1:ASPECTS Or DYNAMO THEORY 179I f t h e shear i s localised ( f i g u r e 7 . 4 ) t h e n f l u x e x pu l si o n acts o n l y i nt h e r eg i o n o f shear, and reconnec t ion of l i n e s o f f o r ce i s i n e v i t a b l e ,as i n d i ca t e d i n th e f i g u r e .

    This however i s n o t the whole s t o r y . R h i n e s 6, Young ( 1 9 8 3 ) haver e c e n t l y s t u d i e d ( 7 . 1 1 ) i n t h e c o n t e x t o f scalar d i f f u s i o n , and h av eobserved t h a t a r e s i d ua l f i e l d may s u rv i ve i n a r eg i o n o f closed stream-l i n e s o ve r t h e o r d in a ry d i f f u s i v e time-scale td = a - l R m . It i s e a s yt o see how t h i s may occ ur i n th e mag n e t i c co n t ex t co n s i d e r ed here . I ft h e !-lines co in ci de wi t h t h e g - l i n e s i n t h e r eg i o n of closed g - l i n e s ,t h e n there is no winding-up e f f e c t ( f i g u r e 7 . 5 ) .

    F i g u r e 7 . 4 F i g u r e 7 .5A f i e l d of t h i s k in d w i l l d i f f u s e ,po t h a t i t w i l l n o t r ema i n ex ac t l ya l i g n e d w i t h g ; b u t as shown b y v i n e s 6. Young, t h e s t ro ng shear inge f f e c t of th e _u_-field j s always such as t o mai n t a i n a & f i e l d t h a t i s( t o l e a d i n g orde r ) a l i g n e d w i t h U, and t h i s f i e l d does i n d eed s u r v i v ei n t h e r eg io n o f closed E - l i n e s o n t h e l o ng time-scale t d ,

    J t i s an o pen q u es t i o n wh e th e r f l u x ex p u l s i o n o cc u r s , or n ot , i nmore complex t h ree - di mens i on a l s i t u a t i o n s . One s i t u a t i o n of p a r t i c u l a rc u r r e n t i n t e r e s t is t h a t i n which t h e !--lines are e r g o d i c ( s p ace -f i l l i n g ) i n some r e g i o n V of R . Can a magnet ic f i e l d survive i n d e f i n -i t e l y i n such a region (when 71 # 0 ) or i s it expelled b y a quasi-two-dimensional mechanism on th e F?3-timescale? N o genera l answer t o t h i sq u e s t i o n i s as y e t known.

    3

    m

    An i n t e r e s t i n g t h ree - di mens i on a l v a r i an t of t h e f l u x e x p ul s io n pro-cess i s t h e t op ol og ic al pumping mechanism, id e n t i f i e d by Drobyshevski& Yuferev ( 1 9 7 4 ) . I n t h e t o p o l o g i c a l l y asymmetric motion associatedwi th hexagonal c e l l s i n a B en ar d l ay e r , h o r i zo n t a l f l - l i n e s can be con-vected downwards, but cannot be convected upwards, s i n c e th e r e g i o n s of

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    180 t l . K.MOFPATTupward moving f l u i d are disconnected. There i s t he re fo re a net pumpingef fec t downwards which becomes more effective as R increases froms m a l l values. m

    Recent computations #f or lar ge r R m ( 100-200) by Galloway andProctor (1984) and by Arter (1984) have shown tha t h ere a l so t h e e f f e c t sare much more subtle than or ig ina l ly realised. Not only i s f l u xapparently pumped downwards, but by some mysterious mechanism that i snot a l together clear , reverse4 f lux i s generated n e a r t he top o f t h elayer. (Could t h i s phenomenon have some bearing on the as yet unex-plained appearance of reve rse f i e l d i n th e Reversed-Field Pinch (Bodin &Newton 1980) ? ? ) .

    7.6 MEAN-FIELD ELECTRODYNAMICSThere can be no disp ute t h a t t h e major advances i n dynamo theory overt h e past 20 years have been associated with t h e development of mean-field electrodynamics, i n .a turb ulen t con tex t, whose o ri g in s may bet raced to the work of Parker ( 1955), Braginski i ( 1964) and Steenbeck,Krause 6 Radler (1966). This theory i s f u ll y described by Moffatt(1978) and by Krause & Radler (1980), and i t w i l l be s u f f i c i e n t here t odi sc us s c e r t a i n key po in ts of th e the ory , and t o comment on some weakp o i n t s which c a l l f o r f u r t h e r i nves t iga t ion .

    The theory i s based on a decomposition of th e t o t a l ve loc i ty f ie ldU and t o t a l magnetic f i e l d 6 i n to mean and f lu ct ua ti ng parts- to t - to t

    The mean of t h e induction equation i s then(7.16)

    (7.17)

    where E = ( g X b > i s the electromotive force associated with the tur-b u l e n c e . Consideration of th e equation for th e f l u c t u a t i n g f i e l d _4establishes (on qui t e general grounds) a l i ne ar re la t io ns hi p between Eand E ; and provided there i s a sca le separa t ion ( t h e sca le of t he f luc-tuating fields being small compared with the scale of the mean f i e l d s )t h i s l i nea r re l a t ionsh ip takes the form

    (7.18)

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    CHAPTER 7: ASPECTS OF DYNAMO THEORY 181where a and Bilk are p s e u do - t en s o r s , d e t e r m in e d ( i n p r i n c i p l e ) b y t h es t a t i s t i c s o f t h e t u r b u l e n c e , a nd t h e parameter g , When the scale ofjs s u f f i c i e n t l y l a r g e , t h e series (7.18) may be e x p e c t e d t o c o n v e r g er a p i d l y ! a nd i n practice o n l y th e f i r s t t w o t e r m s are r e t a i n e d . I t i showever q u i t e common i n dynam o models t o f i n d t h a t t h e &term i n (7.18)is comparable i n importance w i t h the a - t e r m , a n d o n e may detec t heret h e seeds of a c e r t a i n i n c o n s i s t e n c y : i f th e f i r s t t w o t e r m s are compar-ab le , t h e n what a b o u t t h e t h i r d t e r m , t o s a y n o t h i n g of the nth term?

    I I

    T h e f i r s t problem i n mean-field e l e c t r o d y n a m i c s ( a n a l o g o u s t o t r a n -sport problems i n s t a t i s t i c a l p h y s i c s ) i s t h e n t o o b t a i n expl i c i te x p r e s s i o n s f o r a Pl ik i n t e r m s of g a n d of s t a t i s t i c a l properties of--. The a s t r o p h y s i c a l l y i n t e r e s t i n g s i t u a t i o n i s t h a t i n which g - 0(or, more s t r i c t l y , i n which th e t u r b u l e n t m a g n e t i c R e y n ol ds n um ber is1arge)r u n f o r t u n a t e l y t h i s i s th e l i m i t i n which t heo re t i ca l a n a l y s i s is .p e c u l i a r l y d i f f i c u l t 1 I f t y p i c a l m a g n i t u d e s of a /, Biik are, d e n o t e d b ya a n d 4 , a n d i f these are i n d e p e n d e n t of g i n t h e l i m i t g -4 0 , t h e n o nd i m e n s i o n a l g r o u n d s o n e w o u l d , e x p e c t t h a t

    a = O ( u o ) , 4 = O ( U O l l O ) , (7.19)where uo = lh n d 1, i s a characteristic scale of t h e t u r b u l e n c e ;a n d i n d e e d t h e estimates (7.19) are commonly used ( w i t h s u i t a b l e n u m eri-c a l c o e f f i c i e n t s ) i n th e a s t r o p h y s i c a l l i t e r a t u r e . B u t w e have a l r e a d yn o t e d t h e subtleties of t h e l i m i t g -* 0 i n t h e l a m i n a r c o n t e x t ; a ndthe re i s n o r e a s o n t o s u p p o s e t h a t t h e b e h a v i o u r w i l l be a n y l ess subtlei n t h e t u r b u l e n t c o n t e x t . I f a s t r o p h y s i c a l d ynamo models h a v e t o d e p e n do n l y o n t h e d i m e n s i o n a l j u s t i f i c a t i o n of (7.19). thzs- i s a s h a k y f o u n d a -t i o n f o r a n e no rm ou s s u p e r s t r u c t u r e 1

    There i s however some e v i d e n c e from n u m e r i c a l s i m u l a t i o n experi-m e n t s t h a t (7.19) may, desp i te th e a p p a r e n t n a i v e t y , be e s s e n t i a l l ycorrect . F o r m a l l y exact e x p r e s s i o n s for all a n d B I i k were o b t a i n e d b yI a n g r a n g i a n a n a l y s i s b y Moffatt (1975) a n d these were u s e d i n a n u m e r i -c a l s i m u l a t i o n b y K r a i ch n an (1976) who showed t h a t , except p o s s i b l y i nth e a r t i f i c i a l case of ' f r o z e n ' t u r b u l e n c e , a a n d 4 s e t t l e down t ov a l u e s of o r d e r uo a n d u o L o r e s p e c t i v e l y . C u r r e n t w o r k o f D r u m m o n d ,Duane 6, H o r g an (1984), which i n c o r p o r a t e s weak d i f f u s i o n v i a a B r o w n i an' j i g g l e ' s u p e r p o s e d o n t h e t u r b u l e n c e , f i n d s r e s u l t s so far c o n s i s t e n t

    w i t h K r a i c h n a n ' s s t u d y , and t h i s i s a t least r e a s s u r i n g . The c a l c u l a -t i o n s are however a t th e l i m i t o f available c o m p u t e r p o w e r, a n d o n e m u s tq u e s t i o n whether t r u e asymptotic ( t -. ") c o n d i t i o n s are a t t a i n e d i n

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    these c o m p u t a t i o n s .The case of i so t rop ic t u r b u l e n c e ( s t a t i s t i c a l l y i n v a r i a n t u nd er

    r o t a t i o n s of t h e f ra m e o f r e f e r e n c e ) d e s e r v e s p a r t i c u l a r com ment. I nt h i s case, aii a n d P l j k are i so t rop ic , i . e .

    ( 7 . 2 0 )

    where, now, a i s a p s e u d o - s c a l a r a n d 4 i s a scalar . This d i f f e r e n c e ish i g h l y s i g n i f i c a n t : a c a n be n on -z ero o n l y i n t u r b u l e n c e t h a t l a c k sr e f l e x i o n a l symmetry; 4 , o n th e other h a n d , i s g e n e r a l l y n o n - z e r o ,w h e t h e r t h e t u r b u l e n c e lacks r e f l e x i o n a l sy m me try o r n o t .

    i l k = a i l k! = a 6 i j ,i i

    The simplest m e a s u r e of t h e l a c k of r e f l e x i o n a l symmetry i n a f i e l dof t u r b u l e n c e i s t h e mean h e l i c i t y

    H = < -U_.u r l g , . ( 7 . 2 1 )A t l o w t u r b u l e n t m a g n e t i c R e yn o ld s num ber, t h e r e i s a d i r e c t r e l a t i o n -s h i p b e t w e e n a a n d H:a! i s a w e ig h t e d i n t e g r a l o f t n e s p e c t ru m of H( M o f f a t t 1978, 5 7 . 8 ) .

    I t i s known t h a t , when 11 = 0 a n d a # 0 , e q u a t i o n ( 7 . 1 7 ) admitsdynam o s o l u t i o n s p ro v i d e d Ia ( / ( I) t 4) xceeds a c r i t i c a l v a l u e d e p e n -d e n t o n l y o n t h e shape of t h e f l u i d d o m a i n , where R i s a typical scaleof t h i s d o m a i n . H e n c e , a s u f f i c i e n t c o n d i t i o n fo r dynam o a c t i o n i n s u c ha d o m a i n i s t h a t Ja I be n o n - z e r o a n d R be s u f f i c i e n t l y l a r g e ; thef o r m e r c o n d i t i o n i s g e n e r a l l y sa t i s f i e d i f t h e t u r b u l e n c e i n t h e d o m a i nJ a c k s r e f l e x i o n a l s y m m e t r y . This i s t h e s u f f i c i e n t c o n d i t i o n referredt o i n 57 .4 above.

    \aii a n d B l i k. 7 SOME PROPERTIES O F THE PSEUDO-TENSORSJ f t h e t u r b u l e n c e i s n o t i so t rop ic ( a n d it s e l d o m i s l ) t h e n t h e r e arec e r t a i n other ef fec t s c o n c e a le d i n a a n d 4 i n a d d i t i o n t o t h e simplea-effect a n d t h e e d d y d i f f u s i v i t y ( p - ) e f fec t t h a t are p r e s e n t i n iso-t ropic c o n d i t i o n s . F i r s t l y , a i . n e e d n o t be symmetric; i f w e decomposei t i n t o symmetric a n d antisylmne4ric parts , i . e .

    i i i lk

    ( 7a22 )

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    CHAPTER 7:ASPECTS OF D Y N A M O THEORYthen it is evident t h a t 2 i s a polar vector which need not varefl ex io nal ly symmetric turbulence. The symmetric part a!.') doever vanish unless the turbulence lacks reflexional symmtry./

    I n the 'f ir st -o rd e r smoothing approximation' i n which te rd r a t i c i n f luctuating quantit ies are neglected i n the fluctuatict i o n , it t u r n s out tha t a..& symmetric, i . e . 2 = 0 . A t torder, however, ' second-order smothing , z can be expresseweighted int eg ral of t r i p l e spectra ( i . e . Fourier transforms ofveloci ty cor re la t ions) , and i s i n general non-zero. A more i ns i tua t ion i s perhaps that i n which the turbulence i s inhome=t h i s case a contribution t o _r_ obtained a t the f irs t-or der snl eve l , i n th e di re ct io n of decreasing turbulence inte nsit y:

    11

    where again L i s th e sca le of t he turbulence, and k i s a dimenconstant of order unity; the fac tor 8 - l i s a product of the fir:smoothing approximation. Note t h a t for inhomogeneous turbulenvector 2 given by (7.23) w i l l be a function of position, 2p e n s ub st it ut ed i n the mean-field equation, v ia (7.22) and (7gives a contribution

    0

    II as- = V X ( 2 X @ ) t . . .at

    i . e . _r_ acts l i k e a n ef fe ct iv e ve loc ity , transpo rting the mear e l a t i v e t o t h e f l u i d . I t i s important however t o note t ha tgene ral non-solenoidal, i . e . V . 2 # 0 , so t h a t the qualitativeof i s quite different from t h a t of t h e actual f l u i d mean velorwhich i s assumed t o s a t i s f y V . U = 0 . I n fact, the x-effectfied here i s none other than th e flux-expulsion e ff ec t (incorFtopological pumping also), reappearing w i t h i n the mean-field fram

    Turning now t o Bilk, a fir st -o rd er smoothing ana lysi s g icontributions (Moffatt h Proctor 1 9 8 2 ) . The f i r s t is a win te g ra l of t he symmetric pa rt of t he spectrum tens or of the t u r band admits interpretation as an anisotropic eddy d i f f u s i v i t )second part is a weighted in te gr al over the he li ci ty spectrum f

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    184 n. K. M O I T A ~

    scenario for dynamo action i s indicated i n figure (7.6). The rotationof the Sun has an important double influence on th e convective cells i n

    This f u l l expression i s given here j u s t to indica te the measure of t e n -so ri al complexity th at ar is es even a t the lowest order of approximation.I n the special case of axisymmetric turbulence, it can be shown that theexpression (7.25) contains the Radler effect (Radler 1969):

    aailk axk(2)--1 = R ( g J ) i + . . . (7.26)

    where i. i s the mean current, _e _ i s a u n i t vector along the axis of sym-metry, and R i s the Radler coeff icient ( a pseudo-scalar) . ' As shown byMoffatt 6 Proctor (1983), i f the turbulence i s statist ically symmetrica b o u t a plane perpendicular t o the axis of symmetry, then ( a t f i r s t -order smoothing level), a . , = 0 b u t R # 0; i n t h i s situation the Radleref fec t may be important fo r f ie ld generation.1

    Pigure 7.6

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    CHAPTER I : ASPECTS 01: YNAMO THEORY 185th e convection zonet f i r s t , C o r io l i s forces cause a d e f l e c t i o n of r i s i n gblobs of f l u i d ; t h i s causes t h e genera t ion of a Reynolds stress d i s t r i -but ion , which i n t ur n is bel ieved t o be respons ib le fo r th e d i f f e r e n t i a lr o t a t i o n w ( r , e ) of t h e Sun. Secondly, as blobs r ise, they expand andthnrefore t end t o rotate more s lowly (conservi ng t h e i r i n t r i n s i c a n g ul armomentum); t h i s establishes a co r re l a t i on be tween ve r t i ca l ve loc i ty anclver t ica l v o r t i c i t y , i . e . a h e l i c i t y d i s t r i b u t i o n, whjch i n t ur n leads t o

    8 an a-effect. Thus, the two i ng red i en t s of an aw-dynamo, th e a-effectand d i f f e r e n t i a l r o ta t i o n , are bo th a consequence of C o r i o l i s fo rces ;from a dynamical point of view, w e are not free t o s p e c i f y a ( r , e ) andw ( r , e ) independently - they should both be der ived in a se l f - cons i s t en tmanner from the governing dynamical equations. This desirable aim hasnot as ye t been a t t a ined .

    \

    I

    L e t us however look a t th e t w o processes in a l i t t l e more d e t a i l .The equat ion of moti.on, whatever else it may contain, contains aCoriolis force, 1

    ( 7 . 2 7 )-a" = - 2 Q X K + . . .atwhere, i n l oc a l Car t e s i an ' coord ina t es ( sou th , east , a n d v e r t i c a l l y u p )a t c o l a t i t u d e 8 ,

    - - . = I ( - n s i n e , o , n c o s e ) . ( 7 . 2 8 )With y- = ( u , v , w ) , and w i t h U and v i n i t i a l l y zero , w e f i n d a n i n i t i a ltendency ( from ( 7.27 ) )

    3v = - 2 w s i n 8 . Rt t O ( t ) , ( 7 . 2 9 )2 4U = - 2 w c o s e s i n e ( p t ) t O ( t ) , (7.30)

    so t h a t t h e Reynolds stress is3 2 2 5( u v ) = 4 ( R t ) < w > c o s e s i n e + O ( t ) . ( 7 . 3 1 )

    This sugges t s t h a t a rea sona ble approximation i n a s t a t i s t i c a l l y st ead ys t a t e should be

    3 2 2 = 4(nt ) < w P c o s e s i n e , ( 7 . 3 2 )C

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    186 H . K. lOPFATT5where tc i s a coherence t i m e f o r t h e r i s ing b lobs ( t c = 3 x 10 s,nt = 0 . 2 for supergranular sca l e s ) . This genera tes d i f f e ren t ja l ro ta -

    t i o n w whose &dependence i s given byC

    aw = < u v >' T Ss ( 7 . 3 3 )2where v 7 is an eddy viscosity (

    and sub-granular scal es . In tegra t in g ( 7 . 3 3 ) gives80 km 1 s ) associated w i t h granular

    23vT

    4 c w > 3 3 4w ( r , e ) = --- ( R t c ) ( s i n 8 - y;) , ( 7 . 3 4 )where t h e constant of integ rat i on i s chosen so t h a t < w ) = 0 , i . e .W(f ,8 ) r ep resen ts th e f luctuat ion about the mean. The expression( 7 . 3 4 ) ind ica tes equa to r ia l acce le ra t ion , as observed i n t h e Sun, andindeed t h e d i f fe rence in w between equator and poles,

    n 4 2 -7 - 12( r , - - ) -. w ( r , O ) = < w > (nt )3 6 . 6 X 10 S , ( 7 . 3 5 )C3 v ~

    which compares very favourably w i t h th e observed value( 7 . 9 x 1 0 - ~ - l ) .

    Consider now th e mechanism of generation of a n a- ,effect (Steen-beck, Krause & R a d l e r 1966 ) . As a blob rises i n to a region of decreas-ing dens i ty , th e vertical component of ( w t 2 f l ) / p t ends t o be con-served ( where g i s the vo r t ic i t y) . Hence fo r small t ,

    ( 7 . 3 6 )

    where p ( 2 ) is t h e b a s i c den s it y s t r a t i f i c a t i o n , and so t h e h e l i c i t y is02 ( 7 . 3 7 )

    where H is the dens i ty scale-height . The associated a-e ffe ct (on thesimplest theory) i s

    PQ ' = = - ( n t C ) < w> c o s e / vP

    1 1 2 23 c 3 c P '= - - - H t = - . n t c o s e c w > / H ( 7 . 3 8 )

    quations ( 7 . 3 4 ) and ( 7 . 3 8 ) provide a pair of dynamically consistentexpressions for a and w, which could usefully be employed i n numericalinvestigation of dynamo modes.

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    CHAPTER 7 : ASPECTS O F DYNAMO THEORY 18 7

    7.9 MAGNETIC BUOYANCY AS AN EQUILIBRATION MECHANISM

    I t is well-known t h a t when Rm i s l a r g e as i n t h e Sun, t h a t a - e ff e c t i nc on junc t ion w i t h d i f f e r e n t i a l r o t a ti o n w i l l y i e l d s o l u t i o n s of (7.17) na spherical geometry having an osci l la tory dynamo character , i . e .

    (7.39)

    where p, > 0 , p i # 0 . The f i e l d then grows in in tens i ty f rom one cyc leo f i t s pe r io d ic behav iou r t o the ne x t , a nd u l t im a te ly it must react backupon th e dynamical system through some equilibration mechanism. Thereare three possibi l i t ies here: ( i ) s t rong f i e l d w i l l t e nd t o s u p p re s st h e tu r bu len t convect ion, and thus t o decrease t h e a-effect, a n e f f e c ts tud ie d by Moffat t (1972)~ i i ) ikewise, a s t ro n g f i e l d w i l l react upont h e mean v e l o c i t y f i e l d , and i n p a r t i c u l a r w i l l t e nd t o damp t h e d i f -f e r e n t i a l r o ta t io n ; t h i s mechanism w a s f i r s t s t u d i e d by Malkus and Proc-t o r (1975), and it has re ce nt ly been i de n t i f i ed by Gi.lman (1984) n hismonumental numerical investiqation of t h e solar dynamo, as a mechanismo f crucia l importance, The t h i r d mechanism, not incl uded i n t h e Gilmanmodel, i s probably equal ly important : t h i s i s magnetic buoyancy (Parker1955). When a s t r o n g to ro ida l magnetic f ie ld El7 is genera ted deep i n

    _ . i gu re 7.7 (from Nigh t inga le 1985

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    188 H. K.MOFPATTct h e so la r convection z o n e , it is s u b j e c t t o a s e lf - i n d u ce d i n s t a b i l i t ywhich causes f l u x tubes t o r i se and b u r s t t h r o u g h t h e photosphere. I fdownward topological pumping is present, then t h i s magnetic b u o y a n c y ,i n s t a b i l i t y is what m u s t l i m i t t h e accumulation of t o r o i d a l f l u x neart h e bottom o f the convect ion zone . Magne t ic buoyancy can be incor-p o r a t e d i n a n aw-dynamo v ia t h e y - e f f e c t d e s c r i b ed i n 5 7 . 7 above, andw i t h a ver t i ca l e f f e c t i v e v e l o c i t y p r o p o r t i o n a l t o - 2 ( B T )( N i g h t i n g a l e 1 9 8 5 ) . The b o un d ar y c o n d i t i o n a d o p te d o n the p h o t o s p h e r i as u r f a c e r = H m u s t be such as t o allow t h e t o r o i d a l f i e l d t o escapewhen it g e t s there - e . g . a b o un d ar y c o n d i t i o n oE t h e form

    d 2

    = o o n r = R ( 7 . 4 0 )aBTB y t R - a r -i s one p o s s i b i l i t y . F i g u re ( 7 . 7 ) shows contours of BT ( r , 8 , f ) a t a f i x e dv a l u e of r i n the ( e , t ) plane ( b u t t e r f l y d ia gr am s) , fo r a par t icularchoice of a, w and 7 . The i n i t i a l e x p o n e n t i a l growth is clear , as isthe e q u i l i h r a t i o n a t c ons ta n t a m p l i t u d e i n d uc e d b y t h e magnetic buoyancyterm i n th e equat ions. N i g h t i n g a l e ' s c h o i c e of a and w w a s b a s e d o nth e previous p u r e l y k i n e m a t i c s t u d y o f Roberts ( 1 9 7 2 ) , a n d is not dynam-i c a l l y c o n s i s t e n t i n t h e sense of 5 7 . 8 above - n e v e r t h e l e s s it d o e ss uc ce ed i n e s t a b l i s h i n g t h a t magnetic buoyancy c a n e q u i l j b ra t e , and itp o i n t s t h e way fo r f u t u r e s t u d i e s t h a t s h o u l d a i m i n a d d i t i o n a t dynami-cal c o n s i s t e n c y .

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    Bodin , H.A.13. and Newton , A . A . : 1980 Nuclear F u s i o n 20 , 1 2 5 5 .Bondi , H. a n d G o ld , T . : 1 9 5 0 Mon. Not. EsyI A s t r , Soc. 110, 607-611.B r a g i n s k i i , S . i . : 1 96 4 S ov . P h vs . JEW 20, 726-735.C h i l d r e s s , S . : 1969 -&ectureg gg Dvnamo Theorv I n s t . H e n r i goincare,P a r i s .C o w l i n g , T.G.: 1934 Mon. Not, A s t r . SOC, 94, 39-48.Drummond, I . T . , Duane, S . and Horgan, R . R. : 1984 &leech, 138,Gal loway , D . J . a n d P r o c t o r , m . R . E . : 1983 GeoDh. A s t r . F l u i d 34,

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    CHAPTER I: ASPECTS OF DYNAMO THEORY 189Kraichnan, R.h.: 1976 Hech, 77, 753-768.K r a u s e , F. and Radler, K . - H . : 1980 Mean-f ie ld mgm tohvd ro dv D m iB axlg y m h e o n . Pergamon.Malkus, W.V.R. and Proctor , M.R.E.: 1975 J. Fluid Mech, 67, 417-444.Moffa t t , H . K . : 1972 Fluid 53, 385-399.Moffa t t , H . K . : 1974L Fluid 65, 1-10,Mo f f a t t , H.K.: 1978 mane t i q f a l A aenera t i o j in e l e c t r i c a l l y a n d u c t inqfl u i dg . Cambridge Un iver s i ty Press.Moffa t t , H . K . : 1979 GeoPhv8 , A S t K . F l u i a 14, 147-166.Moffa t t , H . K . and K a m k a r , H.: 19183 I n SgeLlar and P--aa Maanetism( e a . A . D . Soward), Gordon & Breach, 91-98.Moffa t t , H.K. and Proctor , M . R . E . : 1983 Geophhvs. A s t r , PyLLr 21,

    Moffa t t , H . K . and Proctor, M . R . E . : 3984 pluid Mech, 154, 493-507.Nigh t inga le , S.: 1985 Maanetis flux pumpinq a d p a n t i q Fuovancy am a n -f ie ld dynamos Ph.D, Thesis, Cambridge Universi ty , i n p rep a ra t i o n .Parker , E . N . : 1955 Astrophvs, L 122, 293-314.mines, P.E. and Youngs, w . R . : 1983 L Fluid Mech, 133, 133-145.R o b e r t s , P . H . : 1972 p h i l . T r a n s . ROY. SOC, A 272, 663-698.Steenbeck, M . , Krause, F. and Rarller, K . - H . : 1966 & p a tu r fo r s ch , 21a,Vajnsh te in , S. and Zel 'dovich, Y a . B . : 1978 Sov. Phvs. Use, 15, 159-172.Weiss, N . O . : 1966 Proc, ROY. SOC, A293, 310-328.

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