H.K. Moffatt- Geophysical and Astrophysical Turbulence

8/3/2019 H.K. Moffatt- Geophysical and Astrophysical Turbulence

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Reprint fromAdvances in TurbulenceEditors: G. Comte-Bellot and J. Mathieu0 Springer-Verlag Berlin Heidelb erg 1987Printed in Germany. N ot for Sale.Reprint only allowed with permission from Springer-Verlag

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Geophysical and Astrophysical TurbulenceH .K . Moffa t tDepartment of Applied Mathematics and Theoretical Physics,University of Cambridge, Silver Street, Cambridge CB3 9EW, UK

1 . I n t r o d u c t i o nT u rb u le nc e i n l a r g e - s c a l e g e o p h y si c a l and a s t r o p h y s i c a l c o n t e x t s i sd om in at ed o n t h e o ne han d by C o r i o l i s f o r c e s a s s o c i a t e d w i t h g l o b a lr o t a t i o n , and on t h e o t h e r by L o re n tz f o r c e s a s s o c i a t e d w i t h t h ec u r r e n t s t h a t f lo w i n t h e e l e c t r i c a l l y co n d uc t i n g medium of a s t r o -p h y s i c a l b o d i e s and t h e m ag ne ti c f i e l d s t o w hich t h e s e c u r r e n t s g i v er i s e .

C o r i o l i s f o r c e s i n t h e up pe r atm osp he re o f t h e E a r t h c o n s t r a i n t h et u r b u l e n c e t o b e a p p r o x i m a t e ly tw o -d im e ns io na l i n h o r i z o n t a l p l a n e s .D i r e c t n um e ri c al s i m u l a t i o n ( e .g . M c W i l l i a m s 1 9 8 4 ) has p rovede f f e c t i v e i n r e v e a l in g t h e c h a r a c t e r i s t i c s t r u c t u r e s t h a t emergeu nd er e v o l u t i o n t h a t i s c o ns t r a i ne d t o r e ma in two- d im e nsiona l . T her ea re two a p p a r e n t l y c o n f l i c t i n g t h e o r i e s o f tw o- di me ns io na l t u r b u l e n c e ,t h e ' v o r t e x p a t c h ' t h e o r y of S affm an ( 1 9 7 1 ) w h i c h y i e l d s a k-4 e ne r gys pe c t r um , a nd t h e e ns t r op hy c a s c a de t he o r y o f Kr ai c hna n ( 1 9 6 7 ) andB a t c h e l o r ( 1 9 6 9 ) w h i c h y i e l d s a k - 3 spectrum. W e s h a l l ar g ue t h a tt h e s e t h e o r i e s a r e n o t i r r e c o n c i l a b l e , and t h a t a k s pe c t r um w i l lg r a d u a l l y e v o l v e t o w a rd s a k-3 s p ec t ru m t h r o u gh i n t e r a c t i o n o f v o r t e xp a t c h e s l e a d i n g t o t i g h t l y wound-up s p i r a l s t r u c t u r e s .

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I n s e c t i o n 3 , w e b r i e f l y r e v ie w t h e t h e o ry d e v el op ed i n M o ff a t t( 1 9 8 5 , 1 9 8 6 a , b ) wh ic h r e g a r d s f u l l y t hr e e -d i m e n si o n a l t u r b u l e n c e asa n a g g l om e r a ti o n o f 'r an do m v o r t e x s h e e t s an d c o h e r e n t h e l i c a l s t r u ct u r e s ' . The s t a r t i n g p o i n t i s t h e E u l e r e q u a t i o n , r e ga r de d as ad y n am i ca l s y st em e v o l v i n g i n an i n f i n i t e - d i m e n s i o n a l f u n c t i o n s p a c e.The f i x e d p o i n t s of t h i s s y st em a r e t h e s t e a d y s o l u t i o n s o f t h e E ul e re q u a t i o n s ( o r ' E u l e r f l o w s ' ) , and by c o n s i d e r a t i o n o f t h e pr ob le m o fm a g ne t ic r e l a x a t i o n t ow a rd s a n al og ou s m a g n e t o s t a t i c e q u i l i b r i a , it mayb e shown t h a t t h e r e e x i s t E u l e r f lo w s o f a r b i t r a r i l y complex to po lo g y.These f lows a re c h a r a c t e r i s e d by t o r o i d a l b l o b s w i t h i n which t h es t r e a m l i n e s a re e r g o d i c a nd t h e f l ow h a s m ax im al h e l i c i t y , s e p a r a t e dby f a m i l i e s o f s t r e a m - v o r t e x s u r f a c e s on w hi ch v o r t e x s h e e t s may b el o c a t e d . W e t e n t a t i v e l y i d e n t i f y t h e t o r o i d a l b lo bs a s t h e ' co he re n t

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s t r u c t u r e s ' o f t h e t u r b u l e n t f lo w t h a t e v o lv e s from t h e E u l e r fl owt h ro u g h K e lv in -H e lm ho lt z i n s t a b i l i t y of t h e v o r t e x s h e e t s . Thesec o he r en t s t r u c t u r e s a re p e r s i s t e n t s im pl y b ec au se t h e v o r t i c i t y r e ma in sn e a r l y p a r a l l e l t o t h e v e l o c i t y s o t h a t n o n l in e a r en er gy t r a n s f e r t os m a l l e r s c a l e s r e m a in s w eak. V i s c ou s d i s s i p a t i o n i s c o n c e n t r a t e d i nt h e v o r t e x s h e e t s ; and t h e i n e r t i a l r a ng e , w i t h i t s a s s o c i a t e dKolmogorov spectrum i s t o b e f ound i n t h e s p i r a l s t r u c t u r e s a r i s i n gf ro m t h e Ke lv in -H e lm h ol tz i n s t a b i l i t y ( c f Lu ndg re n 1 9 82 , M o f f a t t 1 9 8 4 ) .

I n s e c t i o n 4 , w e c o n s i d e r b r i e f l y t h e p r o c e s s by which c u r r e n t -s h e e t d i s c o n t i n u i t i e s may a p pe a r d u r i n g t h e m ag n et ic r e l a x a t i o n p ro bl emm e nt io ne d a b ov e . T h i s c a n o c c u r e v en i n t w o- di m en s io n al s i t u a t i o n s ,i n w hich t h e t o p o l o g i c a l c o n s t r a i n t s a r e more e a s i l y a pp re he nd ed . Thef o rm a t io n o f c u r r e n t s h e e t s h as b ee n n o t e d i n a number o f d i r e c tn u m e r i c a l s i m u l a t i o n s of 2 - di m e ns i on a l MHD t u r b u l e n c e (e.cr. F r i s c h e ta(I): 983) and i s known t o b e a s s o c i a t e d w i t h t h e p r e s e n c e o f X -type ( o rh y p e r b o l i c ) n e u t r a l p o i n t s o f t h e m a gn e ti c f i e l d , which of c o u r s e o c c u rrandomly when th e f i e l d i s random.

I n s e c t i o n 5 , w e c o n s i d e r w ha t s t i l l a pp ea rs t o be t h e c e n t r a lproblem of dynamo th eo ry , a t l e a s t i n a s t r o p h y s i c a l c o n t e x t s , n amelyd e t e r m i n a t i o n o f t h e p a r a m e te r s a ( g e n e r a t i o n c o e f f i c i e n t ) and Bt u r b u l e n t d i f f u s i v i t y ) i n me an -f ie ld e le c tr od y na m ic s ( s e e , f o r ex am p l e ,M o f f a t t 1 9 7 8 , K r au s e & R ad le r 19 80) f o r t u r b u le n c e w i t h h e l i c i t y i nt h e l i m i t of l a rg e magne t i c Reyno lds number Rm . I f , a s f r eq u e n t l ym ai nta in ed i n t h e a s t r o p h y s i c a l l i t e r a t u r e ,

a - u 0 , B -.uoR as R m + m ,where U i s t h e r m s t u r b u l e n t v e l o c i t y , a nd R i s a c h a r a c t e r i s t i ct u r b u l e n t l e n g t h - s c a l e , t h e n t h e mean f i e l d dynamo e q u a t i o n s im plye x p o n e n t i a l g ro wt h o f l a r g e - s c a l e m a gn e ti c f i e l d w i t h a g ro wt h r a t e p

0

i n d ep en d en t o f Rm i n t h e l i m i t R m + O0 .dynam o, i n t h e s en s e o f V a i n s h t e i n & Z e l ' d o v i c h ( 1 9 7 2 ) . I t has been

The dynamo i s t h e n a ' f a s t '

b y M o f f a t t & P r o c t o r (19 85) t h a t , f o r t h e c a s e o f a s t e a d yf i e l d , a f a s t dynamo h a s t h e p r o p e r t y t h a t t h e f i e l d becomes

n o n - d i f f e r e n t i a b l e n e a r l y ev er yw h er e as R + - , and i t s e e m s p r o b ab l et h a t t h i s p r o pe r t y w i l l c a r r y o ve r t o un st ea dy (a nd t u r b u l e n t ) v e l o c i t yf i e l d s a l s o . The s o l a r dynamo ( f o r w hic h Rm - 1 0 ) m ust i n e v i t a b l y beof t h i s t y pe . It i s t h e r e f o r e of v i t a l im po rt an ce t o c l a r i f y t h eb e h a v i o ur o f t u r b u l e n t dynamos i n t h e l a r g e Rm l i m i t .l i s a t i o n - g r o u p p r o c e d u r e d e s c r i b e d by M o f f a t t ( 19 83 , s e c t i o n 11) s t i l la p pe a rs t o be t h e b e s t ap proa ch t o a do pt , a l t ho ug h d i f f i c u l t t o j u s t i f yw i t h a ny d e g r e e o f r i g o u r ; t h i s p r oc e d ur e i s r ev ie w ed i n s e c t i o n 5 .

m4

The renorma-

F i n a l l y , i n S e c t i o n 6 , w e c o n s i d er c e r t a i n f e a t u r e s of what may be229


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.d e sc r i b e d as ' c h r o m o s p h e r i c t u r b u l e n c e ' , i . e . t ur bu le nc e i n t h e s o l a ra tm osph er e o u t s i d e t h e p h o t o sp h e r e . Here, t h e d e n s i t y i s low, andthe dynamics i s c o n t r o l l e d by t h e m ag ne ti c f i e l d , whose f i e l d l i n e sa r e ' r o o t e d ' on t h e s o l a r su r f a c e . The f o o t p o i n t s move r an do ml y w i t ht h e s u b - ph o t os p h e ri c t u r b u l e n t m o t io n , and t h e f i e l d i n t h e chromo-s p h e r e i s t he n i n s t a n t a n eo u s l y d et er mi ne d. P a r t i c u l a r i n t e r e s ta t t a c h e s t o t h e q u es t i o n o f f or ma ti on o f d i s c o n t i n u i t i e s ( c u r r e n ts h e e t s ) i n t h i s r e gi on ; s uc h s h e e t s a re t h e s i t e of s t rong Ohmich e a t i n g , an d may p r o v id e a n e x p l a n a t i o n f o r s o l a r f l a r e s a nd s i m i l a ri n t e n s e b u r s t s o f a c t i v i t y . One m echanism by whi ch d i s c o n t i n u i t i e scan appear i s p r e s en t e d i n s e c t i o n 6 .

2. S p i r a l S t r u c t u r e s i n Two- dim en sio na l T u r b ul e n ceI t i s well-known ( see , f o r example, Aref 1983) t h a t when 4 o r morep o i n t v o r t i c e s move un de r t h e i r mu tu al i n f l u e n c e , t h e c o r r es p o n di n gdynamical system i s i n g e n e r a l no n - in t e gr a b le , and ea ch p o i n t v o r t e xmoves on a c h a o t i c t r a j e c t o r y . I t may t h e r e f o r e b e r e a d i l y a p p re -c i a t e d t h a t f l u i d p a r t i c l e s i n a tw o- di me ns io na l t u r b u l e n t f i e l d- = ( U x , y , t ) , v ( x , y , t ) , 0 ) w i l l f o l l o w an e q u a l l y c h a o t i c p a t h , a ndt h a t any s m a l l ' p a t c h ' o f f l u i d marked by d ye w i l l r a p i d l y s p r e a du nde r t h e random d i s t o r t i n g e f f e c t .

There a re two a p p a r e n t l y c o n f l i c t i n g t h e o r i e s o f tw o- dim en si on alt ur bu le nc e; t h e f i r s t i s t h e ' v o r t e x p a t c h ' t h e o r y o f S aff ma n ( 1 9 7 1 )which conce ives o f tu rb u lenc e as t h e m u tu al i n t e r a c t i o n o f a l a r g enumber o f p a t c h e s i n ea c h o f w hich t h e v o r t i c i t y t a k e s a c o n s t a n tv a l u e , a s i t u a t i o n t h a t p e r s i s t s s i n ce t h e v o r t i c i t y e q ua t i on red uce st o D U / D t = 0. The v o r t i c i t y on any s t r a i g h t - l i n e t r a n s v e r s a l th e nh a s a f i n i t e number o f d i s c o n t i n u i t i e s p e r u n i t l e n g t h , and so t h ev o r t i c i t y s pe ct ru m k E ( k ) f a l l s o f f a s k-'.cou rse prov ide an exp on ent ia l cu t -o ff f o r wave-numbers kv = 2

2 V i s c o u s e f f e c t s w i l l of

c h a r a c t e r i s i n g t h e r e a l w id th o f t h e ' d i s c o n t i n u i t i e s ' , b u t w et h a t v i s c o s i ty i s s o weak t h a t t h i s e f f e c t may b e i g n o r e d f o r a l lt i m e s o f i n t e r e s t .

The second theory i s t h e ' e n s t r o p h y c a sc a d e ' t h e o r y o f Kr ai ch na n( 1 9 6 7 ) and Ba tc he lo r (1969) . Th i s s t a r t s fro m t h e e n s t r o ph y ( o rmean- sq uar e v o r t ic i y equation,- 2 2d t = - g < ( v w ) >

where q i s t h e r a t e o f su p p l y of e n s t r o p h y a t low wave-numbers ofo r d e r ko ( an d e q u a l l y t h e r a t e o f d i s s i p a t i o n o f e n s t r o p h y when as t a t i s t i c a l l y s t e ad y s t a t e i s a t t a i n e d ) . I t i s t h e n ar g u ed , i np a r a l l e l w i t h t h e Kolmogorov t h e o r y o f 3 -d im en si on al t u r b u l e q c e , t h a t230


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e n s tr o p h y c a s c ad e s t h r ou g h t h e s p ec tr u m a t a r a t e t ow a rd s t h e v i s c o u ss i n k a t h i g h k . On d i m e n si o n a l g r ou n d s, t h e e n e r gy s p ec t ru m i n t h ei n e r t i a l r an ge k


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Fig . 2 . Form o f en e r g y s p ec t r u m r e s u l t i n g f ro m random d i s t r i b u t i o n o fs p i r a l v o r t e x s t r u c t u r e s ; a s p i r a l r an g e w i t h s p e c t r a l e xp on en t -11/3a p p e a r s ( A . D . G i l b e r t 1 9 86 ) .

b e r e p e a t e d , a s i n d i c a t e d i n F ig 3 : i f a n o th e r s t r o n g v o r t e x p a tc hs h o u l d w and er i n t o t h e n ei gh bo ur ho od o f an a l r e a d y f orm ed s p i r a ls t r u c t u r e , t h en it w i l l i n d u ce a s e co nd a ry s p i r a l d e f or m a ti o n, t h u si n c r e a s in g f u r t h e r t h e s p a t i a l c om plex ity o f t h e v o r t i c i t y f i e l d andl e a d i n g t o a d e c re a s e i n t h e r e l e v a n t i nd e x X i n t h e s p i r a l r ang e .R epeated s p i r a l d i s t o r t i o n o f t h i s k i nd w i l l l e a d t o a v o r t i c i t y f i e l ds t r u c t u r e t h a t i s s e l f - s i m i l a r on a l l s c a l e s , and it i s t h i s p r op e r tyt h a t c h a r a c t e r i s e s a k-3-spectrum.s p ec t r u m w i l l emerge f rom r ep e a te d s p i r a l d i s t o r t i o n s o f t h e k in di n d i c a t e d a b o v e .t h e o r i e s may t h u s b e r eco g n i s ed .

W e t h u s c o n j e c tu r e t h a t a k-3-

A p o s s i b l e r e c o n c i l i a t i o n be tw ee n t h e k-4 and k-3

F i g . 3. I n t e r a c t i o n o f a s e co n d s t r o n g v o r t e x w i t h a p r e - e x i s t i n gs p i r a l s t r u c t u r e :n i n g s o f a f r a c t a l s t r u c t u r e may b e di s ce r n ed .

a s ec on da ry s p i r a l i s formed i n wh ich th e beg in -

0W e n o te t h a t s u b s t a n t i a l e vi de nc e f o r t h e ' s t a c k i n g ' o f v o r t i c i t yd i s c o n t i n u i t i e s i s p r o v i d e d by t h e n u m e r i c a l s i m u l a t i o n s o f B r ac h e t e ta1 ( 1 9 86 ) , p r e se n t ed i n t h e p o s t e r s e s s i o n ' a t t h i s m e e ti ng .f o rm a ti on o f s p i r a l s i s one mechanism by which suc h s ta c k in g may beu n d e r s t o o d .

The

3. F i xe d P o i n t s o f t h e E u l e r E q u a t i o nsT u r n i n g now t o t h r ee - d i m en s i o n a l t u r b u l en ce , it i s n a t u r a l t o c on si de rt h e g e n e r a l s t r u c t u r e o f s o l u t i o n s of t h e E u l e r eq u at io ns ( v = 01,r e c o g n i z in g t h a t v o r t e x s h e e t s may b e p r e s e n t i n s uc h s o l u t i o n s , a nd232


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t h a t v i s c o s i ty w i l l e v e n t u a l ly r e s o l v e t h e s t r u c t u r e o f s uc h s h e e t s( c f . Townsend 1951 ) . The Eul e r equ a t i on s may be w r i t t e n i n t h e form

2where = c u r l and h = + 4 , and w e r eg ar d t h i s a s an i n f i n i t e -d imens iona l dynamica l sys tem, ana logou s t o a f i n i t e d i m e ns io n al s y st emof the fo rm

- -L!d t - E (U)where 5 = ( u l ( t ) u2 ( t ) . . u , ( t ) ) .n a t u r a l t o l o c a t e f i r s t t h e f i x ed p o i n t s of t h e s ys te m (where ( g ) =0 ) , and t h e n t o c o n si d e r t h e s t a b i l i t y o f t h e s e f i x e d p o i n t s . Even i ft he y a r e a l l u n s t a b l e , and t h e r e f o r e i n a s e n s e u n p h y si c al , t h e i r l o ca -t i o n p r o v i d e s v a l u a b l e i n f o r m a t i o n a b o u t t h e e v o l. u ti o n o f t h e s y s t e m i nt s pha s e - s pac e , a s r e f e r e nc e t o s uc h well- known e xa mple s a s t h e L or enzsystem w i l l make c l e a r .

From such a v i e w p o i n t , it i s

The f i x e d p o i n t s zE ( 5 ) o f t h e E u l er e q u a t i o n s , d e s c r i b e d as E u l e r( 3 . 2 )

0f l o w s , a r e s o l u t i o n s of t h e s t e a d y e q u a t i o n s

U x E Z V h .An a p pr o ac h , s u g ge s te d f i r s t by A rn ol d ( 1 9 7 4 ) , has been dev e loped byMoffa t t (1985, 1986 a , b ) , a nd i s r em a rk a bl y po w er fu l i n r e v e a l i n g a ni n c r e d i b l e r i c h n e s s i n t h e s t r u c t u r e of E u l e r f lo w s. W e f i r s t n o t eth e well-known e x a c t a na logy w i th t h e p r ob l em o f m a gn e t o s t a t i c s o f ap e r f e c t l y c on du ct in g f l u i d

-x B = V p ( 3 .3 )w i t h = c u r l E, V.g = 0. Here, ; e p r e s e n t s ma gn et ic f i e l d , and -J,e l e c t r i c c u r r e n t . The a na lo gy i s b etw een t h e v a r i a b l e s

( 3 . 4 )B , g t.--, 2, h -p.-L o ca ti o n o f s o l u t i o n s o f ( 3 .3 ) l e n d s i t s e l f t o a method of 'ma gn et i cr e l a x a t i o n ' t h a t i s n o t a v a i l a b l e ( wi th ou t i n t r o d u ci n g v er y a r t i f i c i a lp h y s i c s ) f o r t h e problem ( 3 . 2 ) . N e v e rt h el e ss , t o e v er y s o l u t i o n o f

e ( 3 . 3 ) f oun d i n t h i s way, t h e r e c o r r e sp o n d s an E u l e r f l ow , v i a t h ee x a c t a n al o gy ( 3 . 4 ) . The a n al o gy d o es n o t e x t e n d t o q u e s t i o n s o fs t a b i l i t y of t h e s t e a d y s t a t e s t hus de t e r m ine d .

T he m a gne t i c r e l a xa t i on a r gum e n t p r oc e e ds as f o l l o w s . W e supposet h a t g o ( x ) i s an a r b i t r a r y i n i t i a l f i e l d ( a t t = 0 ) n a domain D(w hich may be i n f i n i t e ) a nd t h a t t h e f l u i d i s p e r f e c t l y c o n d u c t i n g(so t h a t E - li n es a re f r o ze n i n t h e f l u i d ) b u t v is c o us ( s o t h a t e ne rg yi s d i s s i p a t e d when t h e f l u i d m o ve s) . S uch a s su m p t io n s a re p h y s i c a l l ya r t i f i c i a l , b u t no mat te r : t h e y a re m er el y an a r t i f i c e , a means to-w ar ds a n e nd . I n g e n e r a l t h e L o r e n t z f o r c e -J x i s r o t a t i o n a l a tt i m e t = 0 , so t h e f l u i d w i l l move. The t o t a l e ne r gy ( m a gne t i c p l u s

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9/18

1

Only i n t h i s c i r cu m s ta n c e c an t h e s t r e a m l i n e s be e r g o d i c i n D1 ( o r i nsome subdomain of D1).ofm ust b e s a t i s f i e d . T h i s i s i m p or t an t , b ec au se t h e a r b i t r a r y r e f e r e n c ef i e l d g(x) w i l l i n g e n e r a l ha ve e r g o d i c r e g i o n s w hich w i l l map i n t oc o r r e s pond ing r e g ions D1 , D2, - - -, n w hich t h e a s s o c i a t e d E u l e rf low i s l i k e w i s e e r g o d i c . The p i c t u r e i s i n d i c a t e d s ch e m at i ca l ly i nF ig . 4 : b l o b s o f maxim al h e l i c i t y (E = a U ) i n which t h e s t r e a m l i n e sa r e e r g o d i c a r e s e p a r a t e d by f a m i l i e s o f s t re a m -v o r te x s u r f a c e s , o nwh ic h, t h r ough c on f lue nc e o f s t r e a m - s u r f a c e s , vo r t e x s h e e t s may bel o c a t e d .

C o n v er s el y , i f w e know t h a t t h e s t r e a m l i n e sa re e r g o d i c i n a subdomain D1, t h e n t h e B e l t r a m i p r o p e r t y ( 3 . 9 )

n -

F ig . 4 . G e ne ri c s t r u c t u r e o f E u l e r fl o w : e r g o d i c b l o b s a re s e p a r a t e dby f a m i l i e s o f s t r e a r n - vo r t e x s u r f a c e s on wh ic h vo r t e x s he e t s may bel o c a t e d .

W e have obs e r ve d above t h a t t he a na logy betwe en E u le r f l ows a nda a g n e t o s t a t i c e q u i l i b r i a does n o t e x te n d t o qu e s t i on s o f s t a b i l i t y .

The m a g n e t o s ta t i c e q u i l i b r i a , a r r i v e d a t t h r o u g h a r e l a x a t i o n p r o c e s s ,may b e e x p ec t ed t o be s t a b l e w i t h i n t h e p e r f e c t c o n d u c t i v i t y frame-work , t h e m a gne t i c e ne rgy be ing m in im al i n t h e s ubs pa c e of t o p o l o g i -c a l l y a c c e s s i b le f i e l d s . The ana logous Eu le r f low sEmay however beu n st ab le i n t h e d i f f e r e n t s ub sp ac e of v e l o c i t y f i e l d s whose v o r t i c i t yf i e l d s W_ are t o p o l o g i c a l l y a c c e s s i b l e from O_ This dichotomy hasb ee n e x p l o r e d i n d e t a i l ( M o f f a t t 1 98 6a ) w he re it i s shown, by way ofexample, t h a t th e ABC-flow s t u d i e d by Dombr6 e t a 1 (1986) i s u n s t a b l et o l a r g e - s c a l e h e l i c a l p e r t u r b a t i o n s , w h er ea s t h e a n a lo g o us m agneto-s t a t i c e q u i l i b r i u m i s s t a b l e .

E .

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When v o r t e x s h e e t s a r e p r e s e n t i n t h e E u l e r f l ow , t h e s e a r e o fc o u r se s u b j e c t t o t h e f un da me nt al K elv in -H elm ho ltz i n s t a b i l i t y l e a d in gt o i n t r i n s i c and in e sc a pa b le w ind-up i n t o t h e f a m i l i a r d ou bl e s p i r a ls t r u c t u r e s . When t h e w a v e- c re s ts o f t h e i n s t a b i l i t y a re i n c l in e d t ot h e v o r t i c i t y i n t h e s h e e t a s i s i n e v i t a b l e f o r a g e n e ra l t o r o i d a lt op ol og y ( M of fa tt 1 9 8 4 ) , t h e i n s t a b i l i t y s t r e t c h e s t h e v o r t i c i t y p er -p e nd i cu l ar t o t h e c r e s t s , so t h a t e n st r op h y i n c r e a s e s ( u n l i k e t h etw o- dim en sio na l s i t u a t i o n ) ; t h i s o f c o u r se i s an e s s e n t i a l f e a t u r eo f 3 - d im e ns iona l t u r bu l e nc e .

The e l em en ta l p i c t u r e o f random v or te x s h ee t s (Townsend 1951) g iv esa k- ene rgy spec t rum, modi f ie d by an ex po ne nt ia l cu t - o f f on th einner (Kolmogorov) sca l e . W e c o n j e c t u r e t h a t t h e d o ub le s p i r a l wind-up a s s o c i a t e d w i t h t h e K elvin-H elm holtz i n s t a b i l i t y , p o s s i b l y o c c u r r i n gon a s u c c e s s i o n of d e c r e a s i n g l e n g t h - s c a l e s , w i l l c on ve rt t h i s s p e c tt o t h e s lo we r k-I3 f a l l - o f f r e q u i r e d by t h e ca sc ad e ( s i m i l a r i t y )th e o r y of Kolmogorov ( see , f o r e xa mp le , B a tc h el o r 1 95 3) i n p a r a l l e lw i th t h e c o r r e s pond ing 2 - d im e ns ional p r ob le m de s c r ibe d i n 42 above.H e r e , t h e p i c t u r e i s c l os e t o t h a t o f Lundgren (1982) who has suc -c ee de d i n e x t r a c t i n g a k-53-spectrum from a s t r a i n e d s p i r a l v or te xmode l o f tu rbu lence .

119)

A s r e g a r d s t h e e r g o d i c bl o b s o f m aximal h e l i c i t y , w e may tenta-t i v e l y a s s o c i a t e t h e s e w i t h t h e c o h e re n t s t r u c t u r e s t h a t h ave beeni d e n t i f i e d i n a v a r i e t y o f t u r b u l e n t f l ow s. T o b e s u r e , t h e c on -d i t i o n w_ = a n U_ i s no l on ge r e x a c t ly s a t i s f i e d i n t h e b lo b D n whenp e r t u r b e d by t h e i n f l u e n c e o f K elvin-Helmholz i n s t a b i l i t i e s i na d j a c e n t l a y e r s ; b u t i f , i n some s e n se , t h e r e a l t u r b u l e n t f l o wf o l l o w s a t r a j e c t o r y ( i n f u n c t i o n s p ac e ) aro und a n u n s t a b l e f i x e dp o i n t ( a s s u g g e s t e d by t h e b e h a v io u r o f s i m p l e r f i n i t e - d i m e n s i o n a ls y s te m s ) t h e n e r g o d i c b l o b s i n w hic h 2 i s n e a r l y p a r a l l e l t o are t ob e e x pe c te d ; w i t h i n t h e s e b l o b s , n o n l i n e a r e n er g y t r a n s f e r r e pr e -s e n t e d by t h e t e r m 5 x 2 o f t h e E u l e r e q u a t i o n ( 3 . 1 ) i s then weak,and so t h e y w i l l b e r e l a t i v e l y p e r s i s t e n t , and t h e r e f o r e i d e n t i f i a b l e .Independent ev idence has been prov id ed by Ts inobe r & Levich (1983)t h a t c o he re nt s t r u c t u r e s i n r e a l t u r b u l e n t f l o w s do in d e e d g e n e r a l l ye x h i b i t n on -z er o h e l i c i t y i n t h e a p p r o p r i a t e co-moving f ram e o fr e fe rence ; and(Sht i1man e t a1 1985) t h a t c o h e r e n t s t r u c t u r e s o f non-z e r o h e l i c i t y ( p o s i t i v e and n e g a t i v e ) d o emerge i n d i r e c t n um er ic als i m u l a t i o n s o f c e r t a i n s p a ce - p e ri o d i c f l o w s , of which t h e Tay lor -Green v o r te x may be re ga rd ed as t h e p r o t o t y p e (see a l s o L e v i ch 1985) .

a

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,

4 . App eara nce o f D i sc o n t i n u i t i e s i n t h e Ma gn et ic R e l a x a t i o n P r ob le mThe a p p ea r a nc e o f d i s c o n t i n u i t i e s i s a v e r y i m p o r t a n t a s p e c t of t h emagne ti c r e l ax a t io n p roblem, fo reshadowed by th e d i sc us s i on o f Arno ld( 1 9 7 4 ) who s im pl y p o i n t e d o u t t h a t t h e p r o bl em wo uld i n g e n e r a l h a veno s o l u t i o n ' w i t h i n t h e c l a s s o f smooth v e c t o r f i e l d s ' . D is co n-t i n u i t i e s form e ve n i n t wo -d im en si on al s i t u a t i o n s , by a mechanismt h a t w e s h a l l now i n d i c a t e .

F ig .Oine5. R el ax at io n o f f i e l d w i th one e l l i p t i c n e u t r a l p o i n t : eachof fo r ce becomes a c i r c l e .C on sid er f i r s t t h e v e r y s i mp le t o po l og y i n d i c a t e d i n F i g. 5 , i n

which t h e i n i t i a l f i e l d _B,(x)w i t h i n a c y l i n d e r o f c i r c u l a r c r o s s-s e c t i o n h as a s i n g l e e l l i p t i c ( o r ' 0 - ty p e ' ) n e u t r a l p o i n t i n t h ei n t e r i o r o f t h e dom ain. Ma gn et ic r e l a x a t i o n p r o c e e d s t h r o u g hc o n t r a c t i o n o f E- l i ne s , b u t h e r e s u b j e c t t o t h e e s s e n t i a l l y t op ol o-g i c a l c o n s t r a i n t t h a t t h e a r ea w i t h i n e ac h co nv ec te d f i e l d - l i n er em ai ns c o n s t a n t . E q u i l i b r i u m i s a t t a i n e d when t h e l e n g t h o f e a c h- -line i s m inim ised s u b j e c t t o t h i s c o n s t r a i n t , i . e . when each E-l inei s c i r c u l a r ; moreover t h e s e c i r c l e s must b e c o n c e n t r i c w i t h t h ec y l in d e r i n o r d e r t o e q u a l i s e t h e m ag ne ti c p r e s s u r e a lo n g E - li n e s.There i s t h u s a u n iq u e e q u i l i b r i u m to wa rd s wh ich t h e f i e l d r e l a x e s ,and t h i s h a s no d i s c o n t i n u i t i e s .

Consider now th e more complex topol ogy of f i g u r e 6 i n which t h ei n i t i a l f i e l d _B,(x) a s two e l l i p t i c n e u t r a l p o i n t s a nd o ne h y p e r b o l i c

o r 'X -t yp e' ) n e u t r a l p o i n t i n t h e i n t e r i o r o f t h e domain. A ga in ,.'a ch f i e l d l i n e would l i k e t o become c i r c u l a r i n o r d e r t o min im is el en g th s u b j e c t t o c o n s t a n t e n cl o se d a r e a , b u t t h i s i s no longer com-p a t i b l e w i t h t h e t o p o lo g y . The sy s te m m us t s e e k a n a l t e r n a t i v e r o u t et o e q u i l i b ri u m . The d i s p o s i t i o n o f L o r en t z f o r c e s i n t h e n eig hb ou r-hood o f t h e X-type n e u t r a l po in t i s such a s t o e j e c t f l u i d f rom t h ea c u t e a n g l e s , an d s o t o d e c re a s e t h e s e a n g l e s . T h i s p r o ce s s c o n-t i nu e s u n t i l ( a s t * . a f i e l d d i s c o n t i n u i t y ( o r c u r r e n t s h e e t ) mustform. Th i s e f f e c t has been conf i rmed num er ica l ly by Ba le r (1986 , i np r e p a r a t i o n ) who f i n d s a n a sy m p t o t i c c o n f i g u r a t i o n w i t h c u sp ed s t r u c -t u r e as i n d i c a t ed i n F i g. 6 .

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F ig . 6 . R e la x at i on of f i e l d w i th two e l l i p t i c n e u t r a l p o i n t s and oneh y p e r bo l i c p o i n t : a c u r r e n t s h e e t d i s c o n t i n u i t y forms w it h a cuspeds t r u c t u r e a t i t s e nd - po in t s ( K . B a j e r 1986) .

F or a more g e n e r a l t op o lo g y , a nd i n p a r t i c u l a r f o r a random mag-n e t i c f i e l d , su ch be ha vi ou r i s t o be e xpe c t e d i n t he neighbour hood o fe v e r y X-type n e u t r a l p o i n t . T h i s i nde e d & t h e b eh av io ur t h a t h a sbeen r e v e a l e d i n t h e d i r e c t nu m er ic al s i m u l a t i o n s o f F r i s c h e t a1( 1983 ) . T he se s im u l a t i on s were c a r r i e d o u t w i t h no n- ze ro m a gn e ti cd i f f u s i v i t y rl ( i n f a c t w i th 1? = v ) , b u t t h e t e n d e n c y t o form near-d i s c o n t i n u i t i e s w a s n e v e r t h e l e s s a p pa r en t .

i n t h e s o l a r a tm os ph er e, as p e r s u a s i v e l y a r gu e d by P a r k e r (1979 ,chap. 1.4) who c o n s i d e rs t h e p o s s i b l e s t r u c t u r e o f f o r c e - f r e e f i e l d sand of m a g n e t o s t a t i c e q u i l i b r i a o f p r e s c r i b e d n o n - t r i v i a l t op ol og y,w i t h t h e c on cl us io n t h a t i n g e n e r a l t h e s e must e x h i b i t d i s c o n t i n u i t i e s .T h i s i s e n t i r e l y c o n s i s t e n t wi t h t h e p o i n t o f view develo ped i n t h ep r e s e n t p a p e r . W e r e t u r n t o t h e s o l a r atmo sphere c o nt e xt i n s e c t i o n6 below.

0Form ation o f c u r r e n t s h e e t d i s c o n t i n u i t i e s i s a n im por t a n t p r oc e s s

5. The F a s t Turbulent DynamoA s i n d i ca t e d i n t h e i n t ro d u c t io n , a t u r b u l e n t dynamo a c t i n g i n t h ec onve c t i on zone o f a ny l a t e - t y pe s t a r ( e . g . t h e S u n ) , an d i n w hich t h emagnetic Reynolds number i s o f o r d e r 1 0 4 o r g r e a t e r , m us t p re suma blyb e of ' f a s t ' t yp e i n t h e s en s e o f V a in sh te in & Zel 'dov ich (1972) . Thegrowth r a t e of such a dynamo ( ne g l e c t i n g ba c k - r e a c t i on e f f e c t s ) i si nde pe nde n t of Rm i n t h e l i m i t Rm -+ 4 3 , i . e . it e x h i b i t s a 'magne t icReynolds number s i m i l a r i t y ' ana logous to t h e Reynolds number s i m i l a r i t yt h a t i s f a m i l i a r i n c o nv e nt i on a l t u rb u le n ce . I t has been shown byM o f f a t t & P r o c to r ( 19 85 ) t h a t i f t h e m ag ne ti c d i f f u s i v i t y q i s z e r o( i . e . Rm = a), then dynamo act ion i s i m p o s s i b l e w i t h a s t e a d y v e l o c i t yf i e l d ~ ( x ) ,ven i f t h i s i s a random f un c t i on o f po s i t i on . I t seemsl i k e l y t h a t t h i s i s t r u e a l s o f o r u ns te ad y ( t u r b u l e n t ) f i e l d s g ( x , t ) ,a l t h o u g h o p i n i o n s a re d i v i d e d on t h i s mat te r .

I t seems i n t u i t i v e l y c l e a r t h a t , as Rm -t a, t h e m a gn et ic f i e l df l u c t u a t i o n s w i l l h av e p r o g r e s s i v e l y f i n e r s c a l e , and w i l l i n e v i t a b l ybe l a r g e r ( by some p o s i t i v e power o f Rm) t h a n t h e mean f i e l d .238

0

This


13/18

has been used by P idd in g ton ( 1 9 7 2 ) a s a n arg um ent a g a i n s t t h e a p p l i c a -b i l i t y o f t he me thods o f m e a n - f i e ld e l e c t r odyna m ic s de ve lope d bySteenbeck, Krause & Radle r ( 1 9 6 6 ) ( f o r a comprehens ive account oft h i s a nd s u b se q u en t p a p e r s , see Krause & R s d l e r 1 3 8 0 ) ; t h e c r i t i c i s mi s however n o t sound:t h e f l u c t u a t i o n be smal l compared w it h t h e mean. Th is a r e q u i r e -ment o f t h e f r e q u e n t l y u s ed ' f i r s t - o r d e r s m o ot h in g ' a p pr o x i m a t io n(see Mo ffa t t 1978, Chap.7) which i s c e r t a i n l y n o t v a l i d when R >> 1.What i s needed when Rm >> 1 i s a n a l t e r n a t i v e a p p r o a a h , s t i l l w i t h i nth e ge ne ra l f ramework of t h e mean-f ie ld method. Thi s means esse n-t i a l l y t h a t w e need a good t h e o r y f o r t h e d e t e r m i n a t i o n of t h e' g e n e r a t i o n c o e f f i c i e n t a ( i n t h e ' a - e f f e c t ' ) and t h e tu r b u l e n tm a g n e t i c d i f f u s i v i t y 6 , which r e m a ins va l i d no m at t e r how large Rm

t h e mean-f ield method doe s not r e qu i r e t h a t

m

ay be.One poss ible approach, which seems t o i n co r p or a te t h e r i g h t p hy si cs ,

i n v o l v es t h e r e n o r m a l i s a t i o n gr ou p t e c h n i q u e, a s d e s c r i b e d by M o f f a t t( 1 9 8 3 ) . Here w e s im p ly re v i e w th e e l e m e n t s of t h i s p r o c e d u re . Thee n e rg y s p ec t ru m E ( k ) a nd t h e h e l i c i t y s p e c t r u m H ( k ) ( on w hi ch t h ea - e f f e c t d e p e n d s ) a re f i r s t d i s c r e t i s e d , and f i r s t o r d e r smoothing i sa p p l i e d t o t h e sma l l e s t s c a l e s ( o r l a r g e s t wave- number s) , whoseas so ci a t ed magn et ic Reynolds number Rm(k) is s m a l l .' m ea n -f i el d ' e q u a t i o n i n v o l v i n g a n a - e f f e c t a nd a ( w e a k ) t u r b u l e n td i f f u s i v i t y w hi ch s u p pl e m en t s t h e ( weak) m o l e c u la r d i f f u s i v i t y . T h i sp r oc e s s i s t h en r e p e a t e d : a n o th e r ' b i t e ' i s t a k e n of t h e e n e r g y a n dh e l i c i t y s p e c t r a , t h e s e b ei n g p r o g r e s s i v e l y d ev ou re d fro m t h e 'k = 00'end. The mean-f ield eq ua ti on i s m o d i f i ed o n l y t h ro u g h c ha ng e i n t h ev a l u e s o f a and ( i . e . t hr ou g h r e n o r m a l i s a t i o n ) . A f t e r n s i m i l a rs t e p s , w e o b t a i n v a l u e s a n f f3, i n terms of an- l ,H( kn ) .j u s t i f y w i th any r i g o u r , t h a t c u m ul at iv e e r r o r s ar'e s m a l l , a n d t h a t

T h i s y i e l d s a

f3n-l E (k,) andNow, w e make t h e e s s e n t i a l a ss u m pt i on , w hi ch i s h a rd t o

w e may r e t u r n t o continuum ( r a t h e r t h a n d i s c r e t e ) s p e c t r a . T h i s.yields d i f f e r e n t i a l e q u a t io n s f o r a ( k ) an d B ( k) t h e e f f e c t i v e a and

f3 a s s o c i a t e d w it h a l l s ca l e s of mot ion sma l l e r than k- ' .These equa t ions are

da - 1 2aE(k) + (1? + B)H(k)a k - 3 k2( q+8I 2 - a2 ( 5 . 1 )

a n d t h e s e are t o b e i n t e g r a t e d i n from k = 00, w i th ' i n i t i a l ' condi -t i o n s a ( & ) = 0 , @ ( a ) = 0 .

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The i n t e r e s t i n g t h i n g now i s t h a t a l i t t l e m o le c ul ar d i f f u s i v i t y qi s needed t o , a s it were, g e t t h e sys t em s t a r t e d : i f rl = 0 , t h e f i r s ts t e p o f t h e s mo oth in g p r o c e ss c a nn o t b e j u s t i f i e d . A t t h e same t i m e ,i f rl > 0 (no m a t te r how s m a l l ) , t h en L a ( k ) 1 i n c r e a s e s as k d e c r e a s e st o a maximum v a l u e loc(k) in de pe nd e nt o f q. I n f a c t , q +B may bemaxe l i m i n a t e d f r o m ( 5 . 1 ) g i v i n g2 ( 5 - 2 )a 2 = 2 a 2 ( 2 a E - h H )3 (kh - o r2 )k (kh + a 2 )

00

w h e r e h ( k ) = $ 5 k-2 H(k) dk . (5 . 3)kWhen k i s l a r g e ( a nd ( aJ m a l l ) , (5 .2) becomesS up po se , f o r ex am pl e, t h a t H (k )- k-m f o r l a r g e k , s o t h a t

and( k ) - $Then a2 - k-2 'm + as k + W . ( 5 . 6 )

( m + ] ) - ' k'(m+l)a2 - -2 k - l ( m + 1 ) a 2 . ( 5 . 5 )dk

Thus, as k d e c r ea s e s f ro m 0 0, a* w i l l ' l a t c h o n ' t o t h i s s o l u t io n . Thet y p i c a l b eh a vi o ur o f a 2 i s i n d ic a t e d i n f i g u r e 7 .

I km kF i g . 7 . Form o f t h e f u n c t i o n a 2 ( k ) a s d et er mi ne d by i n t e g r a t i o n o fe q u a t i o n ( 5 . 2 ) .

I n a r e a l s y s t e m s u c h as t h e S un, it i s o f c o ur se u n r e a l i s t i c t op ur su e t h e i n t e g r a t i o n a l l t he way t o k = 0 . R at h e r , o n e s h o u l ds t o p around t h e wave-number km where a2 i s maximal and use eq ua t i on s( 5 . 1 ) t o d et er mi ne a ( k m ) , B(km)m o d e l l i n g ) .n u m e r i c a l s i m u l a t i o n , i n t h e m anner o f Gilman & M i l l e r ( 1 9 8 1 ) .

( t h i s i s j u s t a form o f sub - g r idS c a l e s l a r g e r t h a n km ls hould i d e a l l y be t r e a t e d by f u l l

6 . C h ro m os ph er ic T u r bu l en ce, an d t h e F o r ma t io n o f D i s c o n t i n u i t i e sO u t s id e t h e v i s i b l e s u r f a c e o f t h e Sun ( t h e p h ot o sp h er e ) t h e d e n s i t yof t h e s o l a r atmosphere f a l l s o f f v e r y r a p i d l y , and t h e dyna mics i st o t a l l y do mi na te d by t h e m ag ne ti c f i e l d em a na ti ng f ro m t h e s u r f a c e .240


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T h is f i e l d pre su mab ly a d j u s t s i t s e l f more o r l e s s i n s t a n ta n e o u s l y t ob e f o r c e - f r e e (1 = 01, s i n c e o th er wi se i n f i n i t e a c c e l e r a t i o n sw ould o c c u r . A c t u a l l y , t h e p r o c e s s by w hic h t h i s a d j u s tm e n t o c c u r si n r e sp o n se t o a c h an gin g p h o to s p h e r i c f i e l d i s n o t a t a l l c l e a r ; l e tus assume however , f o l low ing Low ( 1 9 7 7 ) , t h a t it d o e s o c c u r a nd t h a tt h e f i e l d i n t h e c hro mosp he re i s e s s e n t i a l l y f o r ce - f re e a t a l l t i m e s .The f i e l d l i n e s emerg e fr om t h e p h o t o s p h e r e , w he re t h e y a re r o o te d t oth e sub -p ho tos pher i c p l asma wh ich may be assumed p e r fe c t l y cond uc t ingon the t ime-scale c h a r a c t e r i s t i c o f s o l a r m ag ne ti c a c t i v i t y .f i e l d h a s a complex random s t r u c t u r e , wh ich i s c o n s t a n t l y c h a n g i n g ast h e f o o t p o i n t s o f t h e f i e l d - l i n e s move w i t h t h e pla sm a.

A do pti ng , f o r s i m p l i c i t y , a t wo -d im e ns io na l m odel o f t h i s f i e l d ,

T h i s

w e may ex pe c t a random d i s t r i b u t i o n o f 0 - t y p e and X-type n e u t r a lo i n t s be low t h e p h o to s p he r e . The f i e l d t o po l og y i n t h e n e i gh b ou r-4 od o f any X-type n e u t r a l p o i n t i s a s i n d i c a t ed i n f i g u r e 8 . C l e a r l yone f i e l d l i n e ABC a bo ve t h e n e u t r a l p o i n t m us t t ou c h t h e p h o to s p h e r e

( a t B ) .and below t h i s ' c r i t i c a l ' f i e l d l i n e . A s t h e f o o t p o i n t s move du e t os ub -p ho to sp he ri c t u r b u le n c e , t h e f i e l d i ns t a n t a ne o u s l y a d j u s t s i t s e l ft o r em ain f o r c e - f r e e . T h i s means t h a t t h e f i e l d i n t h e c hr om os ph er ei s i n s t a n t a n e o u s l y d e te r m in e d by t h e m oti on o f t h e f o o t p o i n t s , i . e .by t h e mot ion of A ' and C ' f o r t h e f i e l d l i n e A ' C ? , and by A1 and B1f o r t h e f i e l d l i n e AIB1.i n d ep en d en t , w e must t h e r e f o r e e x p ec t i n g e n e r a l t h a t a t a n g e n t i a ld i s c o n t i n u i t y ( o r c u r r e n t s h e e t ) w i l l f or m on AB and l i k ew i s e on B C.D e t a il e d c a l c u l a t i o n c o nf ir m s t h i s c o n cl u si o n.

C o n s i d e r now two f i e l d l i n e s A ' C ' and A1B1B2C2 j u s t above

S i n c e t h e m o t io n s of B1 and C ' a r e q u i t e

F i g . 8. F i e l d s t r u c t u r e above a h y p e rb o li c n e u t r a l p o i n t j u s t belowt h e p h o t o s p h e r e : w hen t h e f o o t p o i n t s move, a c u r r e n t s h e e t a p pe a r s ont h e c r i t i c a l l i n e o f f o r c e ABC.

The i m p or t a nc e o f X -type n e u t r a l p o i n t s i n t h e c h r o m os p h er i c con-t e x t was f i r s t p o in t e d o u t by Sw eet ( 1 9 6 9 ) and Sy rova t sky ( 1 9 6 6 ) -see a l s o P r i e s t ( 1 9 8 2 , chap .10 ) . Thebelow th e pho tosph ere may e q ua l l y havec hr om os ph er ic a c t i v i t y h a s n o t how everC u rr en t s h e e t d i s c o n t i n u i t i e s , and t h e

f a c t t h a t s uc h n e u t r a l p o i n t sa n i m p o r t a n t i n f l u e n c e onb ee n p r e v i o u s l y r e c o gn i z e d .means by which th e y mgy a pp ea r,

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ha ve an im por t a n t be a r i ng on t h e phenomenon o f s o l a r f l a r e s ands i m i l a r b u r s t s o f m ag ne ti c a c t i v i t y on t h e Sun.8. SummarvT h i s r e vi e w o f g e o p h y s i c a l a nd a s t r o p h y s i c a l t u r b u l e n c e h a s n e ce s -s a r i l y been v e ry s e l e c t i v e , and w e have focu ssed main ly on asp ec t s o fbo th two-d ime nsiona l a nd t h r e e - d im e ns iona l t u r bu l e nc e t h a t may f a i r l yb e d e s c r i b e d a s t o p o l o g i c a l i n c h a r a c t e r . When a m a g n e t i c f i e l d i sp r e s e n t ( a s i t a l m o s t i n e v i t a b l y i s i n a h i g h c o n d u c t i v i t y pl as m a)t h e f a c t t h a t - l i n e s are f r oz e n i n t h e f l u i d on t im e - sc a le s s m a l lcom pared w i t h t h e ( l o n g ) d i f f u s i o n t i m e makes t o po lo g i c a l ( a s oppos edt o p u r e l y a n a l y t i c ) ar gu me nt s e ve n more a p p e a l in g . T h i s p o i n t ofview has been urged by Parker ( 1 9 7 9 ) and o t h e r s ; and t h e f a c t t h a th e l i c i t y , a l r e a d y known t o be o f v i t a l i m po r ta nc e i n t h e dynamoc o n t e x t , h a s a t o p o l o g ic a l i n t e r p r e t a t i o n , l en ds f u r t h e r f o rc e t ot h i s s t a n d p o i n t . The method d e s c r i b e d i n 9 3 , i n v o l v i n g t o p o l o g i c a li d e a s i n a n e s s e n t i a l w ay, i s r emarkably p ower fu l i n p rov id i ng newi n s i g h t s b ot h f o r t h e pro blem of t u r b u l e n c e , and f o r c e r t a i n c l a s s i c a lproblems of l ami nar f l ow th eo ry (M of fa t t 198633). I t seems t h a t w emay be on t h e th re sh o l d of m aj or a dv an ce s i n b a s i c u n d e r s t a n d i n g ,which w i l l ha ve a p r o f ound i n f l ue nc e on f u r t h e r a n a l y t i c a l and nume-r i c a l work.

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