H.K. Moffatt- Dynamo generation of magnetic fields in fluid conductors

8/3/2019 H.K. Moffatt- Dynamo generation of magnetic fields in fluid conductors

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Fluid Dymm ts Transact ons, ol. 8

Dynamo generation o f m ag ne tic f i e l d s i n f l u i d c on du cto rs

H.K. Moffat t (Cambridge)1. I n t r o d u c t i o n

When e l e c t r i c c u r r e n t s flow i n a n e l e c t r i c a l l y c o n du ctin g f l u i d ,t h e a ss o c f a t e d m a g n et ic f i e l d @[z,t) evo lves accord ing t o t h ei n d u c t i o n eq u a t i o n

h

where $s t h e m agn etic d i f f u s i v i t y o fi s t h e f l u i d v e l o c i t y . T h is e qu a t io n h o ld s

the con ducto r , and g ( E , e )i n t h e f l u i d re gi o n; i f t h i s

r eg i o n i s bounded by a non-conducting re gi on , then i n t h i s non-conductor8 is a p o t e n t ia l f i e l d s a t i s f y i n g v.6 = 0 , a 2 0 .- -

The kinemat ic dynamo problem c on si s t s i n determ ining under whatco n d i t i o n s t h e magn etic ene rg y a s so c i a t ed w i th t h e f i e l d can b e p r ev en t edfrom decaying t o - ze r o t h ro u g h t h e a c t i o n o f ohmict h e f i e l d d e ca ys i n a t im e o f o r d e rsca l e o f t h e co n d u ct in g reg i o n . I t i s now w ell known t h a t i f g has s u i t -a b l e p r o p e r t i e s and > i s n o t t Q o l a r g e , t h en t h i s d ecay can b e p rev en tedand po ss ib ly rev er se d. Growth of th e magnetic energy may be de sc rib ed as"dynamo a c t i o n " . Th is proc ess i s o f c r u c i a l im p or ta nc e i n e x p l a i n in g t h eor ig in and maintenance o f t h e E a r t h ' s m a g ne tic f i e l d (a nd a l s o p o t e n t i a l l yof t h o se o t h e r p l an e t s , M er cu r y , M a r s and J u p i t e r , now known t o haves i g n i f i c a n t l a r g e -s c a l e m ag ne tic f i e l d s ) , a nd a l s o i n e x p la i n i n g t h e

d i f f u si o n . I f g E 0 ,L*/L where L i s t h e t y p i c a l

observed evo lu t ion o f t he l a r ge sc a l e mag net i c f i e l d o f t he sun (whichappear s t o o s c i l l a t e on ro ug hly t h e same t ime-sca l e as t h a t o f the 22-yearsunspo t cyc le - P ar k e r 1 9 7 0 ) .

The l i t e r a t u r e of th e dynamo problem i 3 v z s t , mid ~t J .% s o t p o s s i b l et o review i t a l l a t a l l a de qu ate ly i n a one hour l e o r k l i a . I p?Wpose to

www.moffatt.tc


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- 100 -concen t r a t e on c e r t a i n deve lopment s t h a t have t aken p l ace ove r t he l a s tt e n y e a r s o r so stemming from idea s f i r s t pu t forward by S teenbeck ,Kreise & Radle r (1966) .I s h a l l endeavour t o e l u c i d a t e ) w i t h e a r l i e r a pp r oa ch e s t o t h e problemdeveloped by Parker ( 195 -5 ) and Brag i ns k i i (1964) b u t i t i s o n l y q u i t er e c e n t l y t h a t t h e r e l a t i o n s h i p s b e tw e e n. th e se t h e o r i e s hav e b ee n f u l l yapprec i a t ed .

T hese i d e a s i n f a c t b e a r a c l o s e r e l a t i o n (w hic h

Study of equa t i on (1) f o r p r e s c r ib e d g(E,t) i s of course a pure l ykinem at ic approach , and f o r a f u l l t r e a tm e n t m ust n a t u r a l l y b e s u pp le -mented by an appr op r i a t e equa t i on o f mo tion de s c r i b i ng t he ev o l u t i o n o fg i t s e l f . I n t h e c o n t e x t s d e s cr i be d above t h e f l u i d c an f o r th e m ost

p a r t b e t r e a t e d as incompress ib le and as s a t i s f y i n g t h e N av ie r-S to ke se q u a ti o n i n a r o t a t i n g f l u i d - i . e . w i t h C o r i o l i s f o r c e s i n c lu d e d . TheLorentz force j , , 6 * (where j i s t h e c u r r e n t ) m ust a l s o b e i n c lu d e d .It i s how ever e n t i r e l y l e g i t i m a t e t o d e f e r c o n s i d e r a t i o n o f th i s equa t i ona t t he ou t s e t ; conc l us i on s bas ed on equa t i on (1) a lon e can be of ve ry wideg e n e r a l i t y , a nd p ro v id e u s e f u l g u i d e l i n e s as t o nece s s a ry ( o r a t l e a s th i g h ly d e s i r a b l e ) p r o p e r t i e s o f any v e l o c i t y f i e l d g h , k ) t h a t may b eexpected t o promote dynamo action.

d -

2. H e l i c i t y

One of t h e s e ' h i g h l y d e s i r a b l e ' p r o p e r t i e s t u r n s o ut t o b e a ' l a c ko f r e f l e x i o n a l sym metry' i n t h e v e l o c i t y f i e l d , i . e . i n some se n s e i tm ust e x h i b i t e i t h e r a r ight -handedness or a l e f t-handednes s i n i t s g l o b a lpr op er t i e s . The s im ple s t measure o f t he l a c k o f r e f l e x i on a l symmetry o fa l o c a l i s e d m o tio n o f f i n i t e en erg y i s i t s h e l i c i t y


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- 101 -where 2 V,,% is the vortaci ty and the integral is throughout the fluidvolume V. This quantity is a pseudo-scolar whose sign changes undertransformation from a right-handed to a left-handed frame of reference.Consequently a non-zero value for I certainly implies a lack of reflexiondsymmetry . I is a quantity that is invariant in an inviscid fluid movingunder conservative body forces (Moffatt 1969); this invariance is associa-ted with the interpretation of I as a topological invariant - the 'degreeof knottedness' of the vortex lines o f the motion - a quantity that isconserved essentially by virtue of the fact that under these conditionsvortex lines are frozen in the fluid.

Analogously. when = 0 , equation (1) implies conservation ofthe magnetic helicity

( 3 )= )a result first recognized by ElSaaser (1956). In this perfectlyconducting limit, lines of force (lplinesl) axe frozen in the fluidand the same interpretation of fF1 as a topological invariant applies.It is a feature of most elementary dynamo systems (e.g. the homopolardisc dynamo described by Roberts 1967) that a lack of reflexional symmetryis apparent in the configuration considered; a non-zero value forthough not inevitable, is a frequent consequence.

rH,3 . The mean electromotive force generated by turbulence

In brder'to see how lack of reflexional symmetry can be of relevanceto the problem, 'it. is necessary to develop the notation and approach ofSteenbeck, Krause & Radler (1966), as done in Moffatt (197~,1974). Weconceive of motions on two length scq1e.s L and 1 wjth L>>.( i.e.


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_ - 102 -where i s t h e l a r g e - s c a l e f i e l d a nd %' t h e s m a l l- s ca l e f i e l d(w hich may be tho ug ht of e i t h e r a s t u r b u l e n c e or as a random wave f i e l dw i th s o m e 'd e f in i te d i s p e r s i o n r e l a t i o n ) . S im il ar ly we w ri te

s, = E* x_, t ) + ,bI*_,t). (5)We use angu la r b ra ck e t s t o denote ave rages over t h e s c a l e c , so t h a t 1

( 6 ) ?The average o f equ a t ion (1) i s then

i s a mean e lec t rom ot iv e fo rc e gene ra ted by the sm al l -s ca le mo t ion, andt h e e q u a t i o n f o r i s e v i d e n t l y

\

If we assume t h a tt h a t t h i s e q u a ti o n e s t a b l i s h e s a l i n e a r r e l a t i o n b etw een and E. ,and s o between E =


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- 103 -A n import a nt and i l l um ina t in g ( though ide a l i s e d ) sp e c i a l c a se is

however t h a t i n which 2 0 and t he s t a t i s t i c a l p ro p e rt ie s o f 2'(e .g. ( e" ) ) are stationary and homogeneous - i . e . do not vary with

o r w i th 5 . I n t h i s c as e , %J , F C j k , - - - must be constants ,and i t may be noted that s i n c emay be de r ived on the a ssumpt ion th a t i s uniform, on theassumption that ?B.,/JX, i s un i f o r m, e t c . , i f th i s should prove

dg , a - - - do not depend on 5 *ijconvenient .

A f u r t h e r i d e a l i s a t i o n c o n s i s t s i n t h e a s su mp tio n o f 'n o p r e f e r re dd i re c t i o n ' i n t h e s t a t i s t i c a l p r op e rt ie s o f F'.p r o p e r t i e s a r e i n v a r i a n t u nder r o t a t i o n s o f the frame of r e fe re nc e , then*@ij,ijL, - - -

I f t h ese s t a t i s t i c a l

s h a r e t h i s p r o p e r t y , so t h a t n e c e s s a r il y ,

where Of i s pseudo-sca la r , ' 6 is a s c a l a q & , gh , are pseudo-s c a l a r s , a n d so on. A t t h i s s t a g e , t h e c r u c i a l i m po rt an ce o f the conceptof r ef le x io n al symmetry ( o r l a c k o f i t ) becomes evident.turbulence i s re f lex iona l ly symmetr ic , then a l l s t a t i s t i c a l p ro p er ti es o fth e tu rbu len ce must be in va ri an t under change from a r ight-handed to alef t-handed frame o f r ef er en c e. I n p a r t i c u l a r , d , which i s ( i m p l i c i t l y )a s t a t i s t i c a l p rope r ty of t he tu r bu le nce ( deG endent a l so onthis case van ish . (S im i la r ly a, and 8% must vanish, butt h e r e i s no need f o r t o vanis h (and i t w i l l i n f a c t i nv a ri ab l y b enon-z e ro )).

If t h e

) must i n

fl

Turbulence that e x h i b i t s no p r e f e rr e d d i r e c t i o n b u t t h a t l a c ksr e f l e x i o n a l symmetry mBy be descr ibed as pse udo- iso t r op ic . D e f e r ring f o rth e moment th e imp orta nt qu es ti on of how such a c ond i t ion may b e r e a l i s e dl e t u s f i r s t look a t some of th e consequencies. Equ ations (10 ) and (11)toge the r g ive


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- 104 -

The effect of subsequent terms in this series have not been investigatedand are probably unimportant. Substitution of (12) in (7) (with for themoment 2 = O ) gives

The p -term evidently represent a turbulent diffusivity effect, and itmay confidently be asserted thatgeneral. proof of this appears to be yet available). The Ot -term hasa profoundly different character; the associated electromotive force in(12) is parallel to the local field &Steenbeck, Krause 80 Radlor (1966) - and the resulting term in (13)dominates over the diffusion term when the scale L ofsufficiently large.- of the consequencies of th is phenomenon arebrougth toge ther in the a r t i c l e by KraUsg &bdler

@ is always positive (although no

- this is the Of -effect of6 s*

1971The simplest physical interpretation of this effect is that

originally due to Parker (1955), and illustrated in figure 1.localised motion havin& non-zero helicity, say positive (a 'cyclonic eventtin Parker's terminology).line, ar.d twists the loop so that its normal has a component in thedirection of the original undistorted field. Such a loop of field can be

Consider a

This motion generates a loop in a mean field

1

thought of as associated with a current in the direction of the normal.A random superposition of such events with positive mean helicity may thenbe expected to generate a mean current anti-parallel to the mean field.Diffusion tends to eliminate the field distortion so that the effect will 'be small when A is large; in this limit, d will certainly be negativewhen the mean helicity is positive.

*

The situation is not so clea; whenis small; in this limit loops may be twisted any number of times,


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- 105 -and a twist of one loop through ,&m+f will tend to be cancelled by atwist of another loop through & ~ a r 3 % where H and are integersIf however the lifetime of tho events is very short (as assumed by Parker)then only the limited twist picture of figure 1 will be relevant and againone would expect K to have the opposite s i g n from that of the meanhelici ty .

Let us look more closely at these two limits. If X is in somesense large, then the diffusion termall other terms in the equation linear in b, . As mentioned above, in

Av'h in (9) will dominate overdetermining E< , we may assume that is uniform, and theappropriate limiting form of (9) becomes simply

AV=! 2 - g*.v 9'.In terms of Fourier transforms defined by

this becomes

imd the mean electromotive farce is

Using (16), and translating into spectral terminology, this becornea

where

where f i t l h ) is the usual Eulerian spectrum tensor of the turbulence


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- 106 -(Batchelor 1953). For pseudo-isotropic turbulence,

where E h ) is the energy spectrum function and F(4) is the helicityspectrum function (a pseudo-scalar) satisfying 1

4Substitution of (20) in (19) leads to a(.. t c. . wherebJ Y

Note the dependence of o( on the helicity spectrum and the (expected)appearance of the minus sign in (22). Note also that as expected d 3 0as ,

> >

Consider:now the opposite limit 1j 0 . If we ignore entirelythe effects of diffusion (and it must be admitted that this is adangerous procedure) the classical Cauchy solution if (1) is relevant.Let E(?,k) C- a_ + s ( $ , k ) be the position at time t of the fluidparticle that started at position at time k = 0 . Then, inLagrangian notation, \

B i ( E , t ) = * 8 j ( e , ~ ) x i / ~ a# (23)and so

(24)= < e c j k ? ( e , t ~ B ~ ( G , o )*k/aac > I

where (5,t') (= U; (s, ) ) is the Lagrangian velocity. I n order toderive an expression for o'ijis uniform and that h t 0 d =o . Under this latter assumption itin this limit, we may again assume that


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- 107 -naturally takes time for will be time-dependent (in the previous limit this build-up was effective-ly instantaneous). From (24) we then have

to build up from zero. and so KG

tc 0c.p &)where ijk at> = ~ j k [ ~ ~ ( ~ , ~ )%(%,t)/3ae>c(t (25)

In this form the close relation between this expression and thecorresponding expression for the turbulent diffusion tensor for a scalarfield (Taylor 1921)

is noteworthy; (this expression can be obtained by a method closelyrelated to that described above for H i d ) . In the pseudo-isotropicsituation, again Q = .C& and from (25),

Note again the appearance of a type of helicity correlation, but thistime in terms of the Lagrangian variables; and it may be remarked thatone of the most intractable problems of turbulence is that of expressingLagrangian correlations in terms of the more traditional Eulerianstatistical quantities.

The expression (27) has a recognizably similar structure to theexpression obtained by Parker (1971) in a reexamination andreformulation of his earlier theory. Parkers expression is meaningfulhowever only within the framework of his representation of the turbulenceby random short-lived cyclonic events and there is no guarantee thatturbulence in general admits such a representation.

An expressionfor p comparable with the expression (27 ) for Mhas been obtained by Moffatt (1974). However there are difficulties


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- 108 -whichhav e no t ye t been f u l l y r e so lved asso c ia ted wi th th e convergenceo f i n t e g r a l s su ch as. (27) as t - 4 ; ( t h e d i f f i c u l t i e s a r e more a c ut ef o r t h e i n t e g r a l s a p p ea r in g i n t h e e x p r es s io n f o r /3 ). If t h e i n t e g r a li n (27) does converge as t*& , t he n t h e l i m i a n g v a l ue i s undoubtedlythe a pp r op r i a t e va lue f o r P( i n t h e l i m i t A 4 0 . If t h e i n t e g r a l *does no t converge , then molecu la r d i f fu s io n e f f ec ts must a t some stagein t e rv en e , and the appro pr ia te va lue of o( the depends on A i n a *n o n - t r i v i a l w a y i n t h e l i m i t 1- 0 . It i s d i f f i c u l t to se e how t h i sques t ion can be re so lved by th eo re t i ca l means, bu t numerica l exper imenta-t i o n c ou ld p e rh ap s b e u sed t o s e t t l e t h e m a t t er .

We can now comment on th e r e l a t i o n of th e above approach to t h a t ofBraginski i (1964) .f o r v e l o c i t y and m ag netic f i e l d i n a s h e r i c a 1 c o nd u ct or , b u t wi th t h eangular brack ets rep res ent ing an average over t he aZimUthal a ngle 9 ,so t ha t a nd ,6. me axisymmetric f i e l d s and (LL' and ha r e pe r tu rb a t io n (non-axisymmetric) f i e l d s . Brag nisk i i assumed th a tth es e dep ar tu re s from axisymmetry were weak, and he co nc en tra tedfur thermore on t he weak di f f us io n l i m i t A 4 0 . I n s o l vi n g t h einduc t ion equa t ion he deve loped a pe r tu rb at io n scheme i n powers of' A 'i n which t he magne tic f i e l d a t l e a d i n g o r d er was p u r e l y t o r o i d a l ( i . e . i nthe -d ire c t io n) . On th e b a s i s of the ge ner a l a rguments p resente dabove i t i s t o be e xpec te d th a t amem e le c tr ou io t ive f o r c e l i n e a r l yr e l a t e d in th e mean ,uiagnetic f i e l d w i l l b e g e n er a te d . I n f a c t B r a g i n s k i ifound a mean toroida l e lec tromotive force 6

Bragfnsk i i used a de co mpo sit ion ',f t h e form ( k ) , ( 5 )

given by +=a% whered = f and r i s a ( p se udo- sc a l a r ) qua d r a t i c f unc t iona l of t h e

v e lo c i ty p e r t u rb a t i o n f i e l d 5' .an 6 - e f f e c t , b u t w i th t h e v ery s i g n i f i c a n t d i f f e r e n c e t h a t h i se x p re s si o n f o r & v a ni sh e s i n t h e l i m i t 0 ( un l ike the e xp r e s s io r(27) above vhich does not show any dependence on As i m i l a r expre ss ion given by Parker (1971) ) .

I n o t h er words, Br a g insk i i a l so f ound

, and unl ike the


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- 109 -Br a g insk i i ' s t he o r y has been g re a t ly e lu c id a te d by Soward (1972)

who has shown how B ra g in s k ii 's r e s u l t s may b e o b t a i n e d i n t er m s o f aGagrangian forumla t ion . I n Soward 's fo rm ula t ion , th e reason f o r th eabove dependence of H on 2 i s t r a c e d d i r e c t l y t o t h e tr ansfo rma tiono f th e d i f f u s io n term i n the in duc t ion e qua t ion under t he change t oLa gra ngian va r i a b le s . The va r i a b l e s t h a t Br a gn i sk i i i n t r oduc e d andde sc r ibe d as " e f f e c t i v e v a r i a b l e s " a l s o ap pe ar' i n a much more natural wayfrom Soward's approach .

The c r u c i a l r o l e that d i f f u s i o n may p l ay may b e i l l u s t r a t e d i n v er ysimple te rms as f o ll o w s. C o n si de r t h e e f f e c t o f t h e v e l o c i t y f i e l d

( 2 8 )5 = U, ( 0 ~ 6 k a c - r k ) , epr ( L x - P L ) )on the magnet ic f i e l d ( & ,O ,O ) . The v o r t i c i t y a s s o c i a t e d w i th ( 2 8 ) i s

This i s i n f a c t a motion of 'maximal h e l i c i t y ' (Kraicbnan 1973). I f )ri s equa l to z e r o , t he c o r re spond ing ma gnetic pe r t u r b a t io nby >)?/at '= %>%/)n and i s

h

SO t h a t 0 , and the re i s no -e f fec t . I f howeverthen the phase of i s s h i f te d s l i g h t l y r e l a t i v e t o t h a t ofe,,$ no long er van ishes .

V A < = Mfe ,The value of d does indeeda p p e a r s e s s e n t i a l i f a wave

It may be# S -

va n i sh as

i s g ive n

v e r i f i e d that i n t h i s c a se

2 4 0 . Some d i s s ipa t ionmotion of th e fo rm (25) (o r equ a l ly a random

s u p e r p o s i t i o n of such waves) i s t o p r o v i d e an - e f f e c t . B r a g i n s k i i ' sv e l o c i t y f i e l d s a r e more a k in to t h i s t ype o f mo tion tha n to the s o r t oftu rbu len ce i n which any pair o f p a r t i c l e s d r i f t f u r t h e r and f u r t h e r a p a r t

\


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- 110 -with increasing.time.

4 . Solution of the dynamo equations

If we now focus-attehtion n the equations for the mean field ,we may forget about the background turbulence except in so far as itprovides an o( -effect and an enhanced diffusivity through the -term.In the simplest case described by equation (13) it is easy to see howexponentiuy growing modes can appear. If o( is say positive(corresponding most probably to negative mean helicity) then magnetic modes

P

having a 'force-free' structure satisfying

VZB, = - K'6.A E0 = U& ~ - .mteGidently evolve like C where

(*, = d K - [ x + p ) K "and

i-e

we have dynamo action providedK -= ( i + p ) - ' # ,

provided the scale of the field gm is sufficiently large.example the field given at time b o by

go = So ( 0, CO, K z , k*. K x )cevolves in this way.

Such a field applies only to-a conducting fluid of infinite extent,and serves merely to indicate how easily the &-effect can overcome theeffects of ohmic and turbulent diffusion in promoting dynamo action.The situation of greater relevance in the planetary and solar contexts isthat in which the fluid is confined to a sphere, o( is antisymmetricabout the equatorial plane (for reasons that stem from simple dynamicalconsiderations) and the mean velocity may well be non-zero. The


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- 311 -The e f f e c t iv e ind uc t ion e qua t ion i s then

whore we have dropped t h e s u f f i x on and where h e = A +pi s assumed uniform. T h i s equa t ion , or minor va r i a n t s o f i t , emerges a sde sc ribe d i n $3o f S t e nbeck, Krause and R atiler , and so lu t ion s ha ve bee n e x te ns i ve lyexplored by mmputational means for pr e sc r ibe d c ho ic e s of t h e f h n c t i o n s' #(Eland g(E] - ( s e e pa r t i c u l a r ly Rober ts 1972) .s uc h i n v e s t i g a t i o n s 2 and a r e assumed ax isy mm etr ic. Among R'oberts'amost i n t e r e s t in g c onc lus ions a r e t he f o l low ing :( i )

above from the theories o f Pa r ke r , o f Br a g insk i i , a nd

For t h e m o s t p a r t i n

I f t h e v e l oc i t y f i e l d i s pur e ly t o r id a l and i f the dynamo operatesaccording to the scheme f i r s t proposed by Parker (1955) i n which to ro id a lf i e l d i s genera ted f r o m p o l o i d a l f i e l d by d i f f g e n t i a l r o t a t i o n , w h i l ep o l o i d a l f i e l d i s r e gene r a te d from to r o id a l f i e l d by the o( - e f f e c t , t h e nth e most e a s i l y ex ci te d dynamo mode i s no't s tea dy bu t Y s o s c i l l a t o ry i ntime; when th e s ig n of a and t h e r a d i a l g r a d i e n t o f d i f- ti a l r o t a t i o na r e s u i t a b l y r e l a t e d , th is mode has a d i p o l e s t r u c t u r e and t h e p e r i o d i cevolu t ion co ns is t s o f am pl i f l ing waves propaga t ing f r o m the po les towardsth e equa tor . Such o sc i l la to ry modes can be used t o exp l a in fe a t ur es o fth e Sun's 22-year s unspo t cy cl e i f i t i s assumed t h a t sunspots appear bye r up t ion o f a to ro id a l f i e l d below the s ur f ace when i t reaches a c e r t a i nc .r i t i ca1 le v e l ( se e f o r example Steenbeck and Krause (1969)) .( i i )c i r c u l a t i o n o f s u f f i c i e n t s t r e n g t h and s t r u c t u r e , t h en t h e most e a s i l y

I f i n a d di t i on t o t he d i f f a g l t i a l r o t a t i o n , t h e r e i s a mer id iona l

excited mode i s s t e a dy ; t h e r e l a t i v e s t e a d i n e s s of t h e E a r t h ' s d i p o l e f i e l ds u gg e st s t h a t m er id io na l c i r c u l a t i o n may b e i m po rt an t i n t h i s c o nt e xt .I f th e mer idional c i rc u l a t io n becomes too s t r on g, then t he dynamo f a i l s ,bresumably because the poloida l f lux i s then excluded f rom th e r e e o n o frege nera t ion by the f lux- e x p a s i o n mechanism s tu di ed by Weiss (1966).


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- 112 -It should be mentioned however t h a t thes e con c lus ion s a r e qu i t e

s e n s i t i v e to the p a r t ic u la r a ssumptions made rega rd ing the assumed formsof d and 0 . As Deinzer (1974) has shown, the quest ion of whetherth e pre fe r red mode of e xc i ta t i on i s o s c i l l a t o r y o r s t e a d y c a n dependc r i t i c a l l y on t h e d e g r ee o f e p k i a l s e p a r a t i o n o f t h e r e g i o n s o f & - a c t i v i t yand d i f f g e n t i a l r o t a t i o n a c t i v i t y , as w e l l as on t h e i n f l u e n c e of meridion-a1 c i r c u l a f i o n . It i s of course dynamical cons ide r a t io ns th a t de te rmine

.I

and g(z_] , and t he u l t im a te que s t io n o f w he ther a dynamo w i l lbe o s c i l l a t o r y o r s t e ad y i n c h a r a c t e r c a nn o t t h e r e f o r e b e w ho lly s e p a ra t e dfrom thes e dynamical co ns ide ra t io ns .

3 . Dynamica l equ i l ib ra t ionWhen a magnet ic f i e l d grows exponent ia l ly us i n t h e s im ple

s i t u a t i o n d es cr ib ed i n t h e f i r s t p ar ag ra ph of $4of c o ur se p e r s i s t i n d e f i n i t e l y , b u t o n ly for so l o n g a s t h e v e l o c i t yf i e l d re mains unq f f e c t e d by Lor e n tz f o r c e s . These f o r c e s , be ing qu a d r a t i ci n the magnet ic f i e l d , must ev en tu a l ly in te r ven e .an d modify the mot ioni n such a w a y QS t o p rov ide some k ind o f e qu i l i b r ium , e i t h e r s t e a dy o ros c i l l a t o r y . The Lore n tz f o r c e L A @may have two effects:(i)t o d e cr ea s e t h e - e f f e c t ; t h i s e f f e c t h as been s t u d ie d i n d e t a i l ,(Moffa t t 1970b , 1972 , Soward 1975) i n th e s i t u a t i o n where th e sm al l - sca lev e l o c it y f i e l d c o n s i s t s of a random sup e r po s i t i o n o f i n e r t i a l waves i n ar o t a t i n g c o n d u c t i n g f l u i d . It may be noted t h a t each co ns ti tu en t wave hasa c i r c u l a r l y po l ar is e d s t r u c t u r e l i k e t h a t o f t h e v e lo c it y f i e l d (28),and t h a t suc h waves a r e t he r e f o r e p a r t i c u l a r l y conduc ive t o dynamo action.(ii)s im p le s t r u c t u r e s u ch as t h a t g i v e n by (35) may be f o r c e - f r e e

, t h i s s i t u a t i o n can no t

It may have a damping e f f e c t on the sma l l - s c a l e l o t i o n s thus t e nd ing

It may also i n f lue nc e the mean ve l oc i t y f i e l d ,U(%). A fleld o f(p&8,.),,ga-8) i n which ca s e t h i s e f f e c t d o es n o t a r i s e . But a f i e l d i n a


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- 113 -conductor o f f i n i t e e x te n t and w i th no e x t e r na l sou r c e s c a nno t b eeverywhere fo rce - f ree , and i n t h i s case the e f f e c t must be p res en t . Eveni f we s t a r t w i th a s i t u a t i o n i n whicha s s o c i a t e d w i t h a growing magnet ic f i e l d w i l l d r i v e a growing mean velocityg(%,k )which u l t im a te ly in f lu enc es th e g rowth o f t h e f i e l d t hro ug h i t s

l)=o , th e mean Lorentz f w c e-appearance i n th e mean indu c t io n equa t ion ( 7 ) . The formalism necess aryt o handle such an e f f e c t has been developed by Malkus & Pr oc to r ( 1975) ,and convinc ing computa tiona l ev idence f o r th e approach to eq u i l i b r a t io nhas been givenbby Pr oc tor (1975).

Onaextremely i n te re s t i n g asp ec t o f t h i s s tudy i s th e way i n w hicha c o n s t r a i n t due t o J.B. Taylor (1963) s i n co r po r at ed i n t h e a n a l y s i s .Th i s c o ns t r a in t a r i s e s unde r c i rc ums ta nce s when Cor io l i s and Lorentzf o r c e s dominate over i n e r t i a and v i sc ous f o r c e s so t h a t t h e c o n t r o ll i n gdynamic e qu ati on f o r t h e mean f low i s

Here P i s t he f l u i d d e n s i ty , t h e g lo b al r o t a t i o n r a t e , ispressure , and k and th e mean cu rr en t and mean f i e l d res pe ct i ve -l y . When t h i s e qua t ion ho lds i n f lu id c on taine d i n a ny c a v i ty a xisymmetr icab ou t t h e d i r e c t i o n o f , as shown by T a y l o r , t h e f i e l d must s a t i s f yfor each S t h e c o n s t r a i n t

whereof a c y l i n d e r o f r a d i u s 5

q,7 ) a r e c y l i n d r i c a l p o l a r c o - o r d i n a t e s , and C k S ) i s t h e s u r f a c ei n t e r s e c t i n g t h e c a v i t y b oundary, &d

o h hu4 f f e f h p .at. Now i n g e n e r a l a f i e l d o f t i e f o r m _8(5)e t h a t r e s u l t s from s o l u t i o n

of an equa t ion of the form (36) with ap pro pr ia t e boundary c ond i t ion s w i l l-o t clatisfy t h e c o n s t r a i n t (38).80 modify th e mean v e lo ci ty 9 (and hence th e mean f i e l d 6 ) t h a t t h eI n th i s cas e th e mean Loren tz , fo rce must

r


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- 114 -m o k

c o n s t r a i n t i s l n e a r l y s a t i s f i e d .of the form (37) (magnetogeostrophic balance), or p o s s i b l y t o t o r s i o n a los c i l l a t io ns abou t suc h .a s t a te . Some aspe c t s o f t h i s problem wereanticipate d by Ch ild re ss (1969) and also by Braginskii (1967,1976).Pr oc to r ' s (1975) work i s l i mi t ed t o t h e s i t u a t i o n whereen t i re ly by the mean Lorentz fo rc e and where t he i n i t i a l growth of t hef i e l d i s due en t i re ly t o an & - e f f e c t w i t h r( antisymmetric about theequ ato r ia l p lane . Within t h i s l i m i t a t i o n he has found convincing evidencean approach to eq ui l ib ra t io n under the e ff e ct s of the growing mean Lorentzfo rc e as desc r ibed above. Fu rthe r work along these l i n e s i s s t i l l requ i redto determine whether such phenomena as sudden revgaa l s&i the Earth 'smagnetic f i e l d may a l s o be embraced by th e model.

This w i l l l e a d e i t h e r t o a f o r c e ba la n ce

i s d r i v e n

F i n a l l y , I s h ou l d r e f e r t o p a ra l l e l a tt ac&.o n dynamic a s p ec t s o f t h edynamo problem from th e point ' of view of conv ective i n s t a b i l i t y - heory.When fl u i d between two h o r iz o n ta l pl'ates i s sub jec ted t o a t empera tu regradie nt by hea t ing the lower p la te , thermal convect ion s e t s 'in when th eRa yle igh number i s s u f f i c i e n t l y l a r g e . The n a t u r e of t he i n s t a b i l i t y andt h e s t r u c t u r e of t h e co n v ect iv e c e l l s a r e o f co u r se s en s i t i v e b o th t oC o r i o l i s f o r c e s i f the fZu id i s ro t a t i n g ab ou t an a x i s n ormal t o t h e p l an esand to Lorentz forces i f , s ay , a h o r i zo n t a l mag ne tic f i e l d i s presen t .D i f f e ren t a s p ec t s of this s t a b i l i t y p roblem have been g iven rece n t c lo sestudy by B ra gi ns ki i (19 70) , Busse (19*, El tay eb (197$, and Ro ber ts andStewartson (1974,1975).r o t a t i o n 8l a r g e f o r convec t ion to occur ) i s that t h e h o r i z o n t a l s c a l e i s s m a l lcompared w i t h th e v e r t i c a l s c a l e ( t h e r a t i o be in g o(n- ). Soward(19 74 ) h as ex p l o i t ed t h i s f a c t i n o rd e r t o d ev el o p, by the methods ofthe mean fi e l d electrodynamics of +ac t i o n a s s o c i a t ed w i t h such motions, and he provides evidence f o r th eex ist en ce of such dynamo ac t i on in.w hich th e magnetic energy de ns i ty ,

I I*%] ,nru)A fe at ur e of the convec t ion pa t t er n when the

i s a t r o n g (and yet the Rayleigh number i s s t i l l s u f f i c i e n t l y

.; a U e t a i l e d t re a tm e n t of th e dynamo


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fluctuates about an average value. Busse (1973) similarly demonstratedthe existenca of a convectively driven dynamo; in this case there was nooverall rotation, and the motion consisted of convection in roll-typecells with horigontal axes together with an imposed shear flow parallelto these axes.

These studies (and others too numerous to mention) provideconvincing evidence that the magnetic field of the Earth (and by inferenceof those other planets that have magnetic fields) may indeed be explainedin term of internal dynamo action due to motion in a liquid corehaving both large-scale and small-scale ingredients. The ultimate sourceof energy for these motions is stilla matter for intense debate - whetherconvective as in the models mentioned above, or gravitational due to bulktransfer of matter in the radial direction (Braginskii 1964b),or


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- 316 -

Figure 1 . E f f e c t o f a l o c a l i s e d motion w it h h e l i c i t yon a m agnetic l i n e o f f o r c e .


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- 117 -REFERENCESBatchelor G.K. (1953) The theory of homoReneoue turbulenceCambridge University PressBraginskii S.I. (1 964a) Soviet Physics JETP 20, 726Braginskii S.I. (1964b)Braginskii S.I. (1967)Braginskii S.I. (1970)Busse F.H. (1970)Busse F.H. (1973)Busse F.H. (1975)Childress S. (1969)

Elsasser U.M. (1956)Eltayeb I.A. (1972)

Geomag. Aeron. 2 , 698Geomag. Aeron. 2, 851Geomag. Aeron. 10,J. Fluid Mech. 44, 441J. Fluid Mech. 52, 529J. Fluid Mech. 71, 93The application of modern physics to theEarth and planetary interiors (Ed. S.K.Runcorn, wiley-Interscience) p629Rev. Mod. Phys. 28, 135Proc. Roy,. Soc. U, 29

Eltayeb I.A. (1975) J. Fluid Mech. 2, 61Deinzer U., V. Kusserow H. -U.Astron. Astrophys. 96, 69& Stix M. (1974)Kraichnan R.H. (1973) J. Fluid Mech. z, 745Krause F. & Radler K-H (lf7/) Ergebrisse de Plasmaphysik und dergaselektronik 2, 1 - 153 Akademic-Verlag,BerlinMalkus U.V.R. (1963) J. Geophys. Res. 68 , 2871Malkus U.V.R. &Proctor M.R.E. J. Fluid Mech.

Moffatt H.K. (1969) J. Fluid Mech. 25, 177Moffatt H.K. (1970a) J. Fluid Mech. kl, 435Moffatt H.K. (1970b) J. Fluid Mech. 44, 705Moffatt H.K. (1972) J. Fluid Mech. 2, 85Moffatt H.K. (1974) J. Fluid Mech. 3,Parker E.N. (1955) Aetroph. J. 122, 293Parker E.N. (1970a) Ann. Rev. Astron. Astrophye. E, 1


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- 118 .P a r k e r E.N. (197Ob)P r o c t o r M.R.E. (1975)R ob er ts P.H. (1967 )

R ob er ts P.H. (19 72)Roberts P.H. 6S t e w a r t s o n K. (1974)Roberts P.H. &S t e w a r t s o n K. (1 975)Soward A.M. (1972)Soward A.M. (1974)Soward A.M. (1975)Steenbeck M . , Krause F. &Radler K.H. (1966)T a y l o r G. I. (1 921 )

Ast rophys . J . 163, 65Ph.D. T he s i s , Cambridge U ni ve rs i tyAn i n t r o d u c t i o n t o a ag n et oh y dr od y na m ic LLaymanP h i l . T r a n e . U ,63P h i l . Trans . U ,87J . Fluid Mech. 68, 447Ph i l . Trans . 431nil. Trans . U , 11J . Fluid Mech. 2, 45Z. N a t u r f o r s c h U,69Proc . Lond. Math. Soc. 2 0 , 196

T a y l o r J . B . (19 63) Proc. Roy. SOC .m , 74Weiss N.O. (19 66 ) Proc. Roy. SOC .U , 10

H.K. Moffatt- Dynamo generation of magnetic fields in fluid conductors

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Transcript of H.K. Moffatt- Dynamo generation of magnetic fields in fluid conductors