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TURBULENCEAND CHAOTIC PHENOMENA IN FLUIDST. Tatsumi (editor)Elsevier SciencePublishers B.V. (North-Holland)8 IUTAM, 1984

SIMPLE TOPOLOGICAL ASPECTS O F TURBULENT V O R T I C I T Y DYNAMICS

223

H . K . M o f f a t t

Depar tmen t o f App l ied Mathemat ics and Th eo re t ic a l Phys icsUn iv er s i t y o f Cambridge

Cambridge, England

The p r o c e s s by w hich t u r b u l e n t v o r t i c i t y i s i n t e n -s i f i e d i s d i s c u ss e d , i n r e l a t i o n t o t h e d im ension-a l i t y of t h e s ub sp ac e i n which v o r t i c i t y i s concen-t r a t e d . F o r a s i n g l e , random ly t w i s t e d v o r t e xf i l a m e n t , no v o r t e x s t r e t c h i n g ca n o c c u r and t h een s t r o p h y rem a in s bounded f o r a l l t i m e . I f t h ev o r t i c i t y i s d i s t r i b u t e d on a r an dom ly co n v o lu t eds u r f a c e , t h e t o t a l h e l i c i t y o f t h e f l ow b e i ng non-z e r o , t h e n t h e e n st r o p h y c a n , and i n g e n e r a l w i l l ,i n c r e a s e w i t h o u t l i m i t due t o l o c a l i n s t a b i l i t i e so f K elv in -H elm ho ltz t y p e , l e a d i n g t o s p i r a l s i n g u l a r -

i t i e s of t h e v o r t e x s h e e t . The r e l a t i o n between t h es t r u c t u r e o f t h e s e s p i r a l s i n g u l a r i t i e s and t h es p e c t r a l e x po ne nt i n t h e i n e r t i a l r an g e of wave-numbers i s demons t ra ted .

I N T R O D U C T I O N

One o f t h e e s s e n t i a l f e a t u r e s of t h e p r oc e ss of e ne rg y t r a n s f e r i nt u r b u l e n t f lo w from l a r g e t o smal l s c a l e s i s t h e a s s o c i a t e d t en de nc yf o r e n s tr o p hy ( i . e . m ean-square v o r t i c i t y ) t o i n c r e a s e . T h i s i n c r e a s ei s u l t i m a t e ly b al an c ed ( i n a s t a t i s t i c a l l y s t e ad y s t a t e ) by v i sc o usd i s s i p a t i o n o f e n s tr op h y; b u t it i s l e g i t i m a t e t o e n q u i r e what h ap pe ns

i n t h e i n v i s c id l i m i t v -+ 0 , and i n p a r t i c u l a r t o i n v e s t i g a t e t h es t r u c t u r e o f t h e v o r t i c i t y f i e l d t h a t deve lops i n t h i s l i m i t , a s t h een s t r o p h y becomes unbounded. Th i s s t r u c t u r e ( i n p h y s i c a l s p ace ) p r e -su mably h a s a b ea r in g on t h e en e r g y sp ect r um t h a t i s e s t a b l i s h e d o ns c a l e s a t which v is co u s e f f e c t s a r e n e g l i g i b l e .

The eq ua t i on f o r mean-square v o r t i c i t y i n homogeneous tu rbu len ce i s

where y = c u r l g.e q u a t i o n i s

The m ost naPve c l o s u r e e n a b l i n g i n t e g r a t i o n o f t h i s

< ? . ( w , . V ) : > = A<;2>3/2 , ( 2 )

where A i s a p o s i t i v e d i m e n s i on l e s s c o n s t a n t ; w i t h v = 0 , (1) t h e ni n t e g r a t e s t o g iv e

- - V~

where < w 2 > o r e p r e s e n t s t h e en st ro ph y a t t = O r and t = A - 1 < ~ 2 > i 4T h i s s o r u t i o n ' bl ow s up' a t t i m e t = to, a behaviour ' tha t i s -proba blyf a i t h f u l t o t h e r e a l dynamics of tu rb ul en ce ( see , f o r e x am p le ,t her e c e n t d i s c u s s io n o f B r a ch e t e t a l . 1983) e ve n a l th o ug h t h e c l o s u r e( 2 ) i s known t o o v e re s t i m a te t h e v o r t i c i t y i n t e n s i f i c a t i o n p ro ce ss .

'

www.moffatt.tc

8/3/2019 H.K. Moffatt- Simple Topological Aspects of Turbulent Vorticity Dynamics

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224 H.K.Moffatt

The same explosion of enstrophy at finite time occurs for a slightlymore sophisticated closure, in which we first decompose the vorticityfield into two ingredients, y = g l + c 2 , the idea being that eachingredient provides the rate-of-strain field that is responsible forthe intensification of the other ingredient. The inviscid enstrophyequations, with a closure analogous to ( Z ) , which encapsulate thisprocess of mutual interaction, are

d $ < g 2 ' > = <,w '>' )These equations have a first integral

<U '>f - <U 2 > $ = - c ,- 1 - 2

and the solution is then given by

(5)

<U1 '> = )exp(-CAt) - 1]-' , (6)

which (like ( 3 ) ) time (provided A > 0).

This behaviour is suggestive in relation to the toroidal vortex sheetmodel which will be discussed below.

RANDOMLY TWISTED VORTEX FILAMENT

Consider first the simplest possible three-dimensional example of a

random vorticity field, i.e. a single randomly twisted vortex fila-ment of strength r . Suppose (for simplicity) that the cross-sectionof the vortex filament is circular, with radius E , which is every-where small compared with the local radius of curvature of the fila-ment. As shown by Da Rios (1906) and Betchov (1965) the motion ofthe vortex filament is determined by the curvature K ( S ) and torsion~ ( s ) , here s is a parameter along the filament. In this approxi-mation, the velocity of the vortex filament is given by

) + cst.) %(s), ( 7 )V ( s ) ,-+ K ( S ) (in(-K ( S ) E

where $ is the unit binormal to the filament.

being the unit tangent vector) the filament behaves like an inexten-sible thread. If it is closed on itself, and its length is L, thenthe associated enstrophy is

Ig'dV = rLL2/V, ( 8 )

where V = ITE'L is the (constant) volume occupied by the vortex fila-ment. Clearly this enstrophy is also constant.

Consider now the further integral invariants associated with thistype of motion. Firstly, the kinetic energy is a conserved quantity.We may estimate this as follows. For points near the vortex filament,the velocity field is approximately that associated with an infiniteline vortex, i.e.

Since f . . V v = 0 ($

for IrK(s) << 1 (9)I l ? ( x ) % E

where r is the distance from the vortex. The corresponding contrib-ution to the kinetic energy of the flow is

(10)~ % pr2L <in(EIK(s)lJ> ,

where the average is over the length of the vortex filament. There*

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Topological Aspects of Turbulent Vorticity Dynamics 225

I

is a second contribution (of order pi"%) to the kinetic energy ofthe flow from points that are more distant from the filament, but itis the contribution (10) which dominates. Since L is constant, it

then follows that < ki I €K ( s ) I-' > is constant also; invariance ofenstrophy implies that, in this sense, energy cannot be transferredto smaller scales, and progressive 'crinkling' of the vortex filamentis impossible.

There is one further invariant associated with the flow, and this isthe helicity

# = /g.$dV, (11)

which, in general, has a topological interpretation associated withthe degree of linkage of vortex lines (Moreau 1961, Moffatt 1969,Arnol'd 1974). This invariant takes a rather interesting form when

the vorticity is concentrated in a single closed filament. Usingthe Biot-Savart law to express in terms of p, we have the alter-native expression

R.gdVdV ,

R3

where f! = 5 - 5 ' .vanishingly small cross-section, this becomes

In going to the limit of a vortex filament of

( 1 3 )2a

i Q i- 2 (1 + - (s)ds) ,

where1 g.d&,,d&

R31 = F / i

(the Gauss integral) The second contribution in (13) involvingthe torsion 'I ( s ) , comes from pairs of points ,x,x,' for which R = O(E).The invariant expression (13) was obtained in a purely topologicalcontext by Calugareanu (1959). (Note that the integral I does con-verge, despite the apparent singularity in the integrand at R = 0 . )

Invariance of .& thus (in a sense) places a constraint on the develop-ment bf torsion in the filament, just as invariance of energy placesa constraint on the development of curvature.

A RANDOMLY CONVOLUTED VORTEX SHEET

The situation is rather different if the vorticity field is confinedto a neighbourhood of a closed surface S. Suppose first that S is atorus (the simplest possibility if the vortex lines are not to beclosed). We use the symbol S also for the area of S; the volume ofits 'E neighbourhood' is then

VE = E S , (15)

and we suppose thatw is non-zero only in V . The strength of thevortex sheet is then-

A U Q E I _ w I I (16)

and this represents the jump in fluid velocity as we cross frominside to outside the torus. Let Vin represent the volume inside thetorus. Then in the inviscid dynamical evolution of the vorticityfield, both Vil! and V, are constant.with the flow is clearly

, The kinetic energy associated

and so AU is also conserved (in order of magnitude). The enstrophyis given by

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226 H. .Moffatt

Now, t h e c o n s t r a in t T = c s t . p la c es no r e s t r i c t i o n on fi which can

i n c r e a s e t hr ou gh i n c r e a s e o f S . An en s t r op hy ex pl os io n ( a -+ m) cani n p r i n c i p l e o cc ur i f a nd o n l y i f S +. -.What can o c c u r u nd er t h e s i n g l e c o n s t r a i n t of e ne rg y c o n s e r v a ti o n i sn o t o f c o u r s e n e c e s s a r i l y t h e same a s w hat d o e s o c c u r un de r t h e f u l lc o ns tr a in ts i m p l i e d by t h e E u l e r e q u a t i o n s . The above argument doeshowever s u g g es t t h a t t h e t o r o i d a l v o r t e x s h e e t i s a b e t t e r c a n d i d a t et h a n t h e c l o s e d v o r t e x l i n e a s t h e p r o t o t y p e s t r u c t u r e w hic h may b ea s s o c i a t e d w i th t h e p r o c es s of e n s tr o p h y e x p l o s i o n i n t u r b u l e n t f lo w.W e p u rs u e t h i s s u g g e st i on i n t h e f o l l o wi ng s e c t i o n s .

KELVIN-HELMHOLZ INSTABILITY AND ENSTROPHY INCREASE

A p la n e v o r te x s h e e t , a s i s well-known, i s a b s o l u t e l y u n s ta b l e t os i n u s o i d a l d i s t u r b a n c e s , t h e g ro wth r a t e o f weak d i s t u r b a n c e s o fwave-number k b e i n g

U Q k A U . (19)

T h is ty p e of d i s p e r s i o n r e l a t i o n s h i p , t o g e t h e r w i t h n o n - l i n e ar i n t e r -a c t i o n m ec ha nis ms ,l ea ds t o t h e a pp ea ra nc e a f t e r a f i n i t e t i m e of weaks i n g u l a r i t i e s ( e .g . d i s c o n t i n u i t i e s o f c u r v a t ur e ) i n t h e s u r f a c es h e e t geomet ry (Moore 1 9 7 9 , Meiron e t a l . 1 9 8 2 ) .

Exper im en ta l ob se r va t i o ns ( e .g . Thorpe 1 9 7 1 ) i n d i c a t e t h a t ani n i t i a l l y s i n u s o i d a l p e r t u r b a t i on of a p l an e v o r t e x s h e e t l e a d s t oa n o v e r tu r n in g p ro c es s ( a s i n d i c a t e d i n f i g u r e 1) a n d h e n c e q u i t er a p id l y t o t h e f orm at ion of s p i r a l s i n g u l a r i t i e s d i s t r i b u t e d a lo ngt h e s h e e t . A v o r te x c o n c en t r at i on t h e n a pp e ar s a t t h e ' e y e ' o feach s p i r a l .

y l

c--t A U

F i g u r e 1 *Development of s p i r a l s i n g u l a r i t i e s on a v o r t e x s h e e t due t oK e l v i n - H e l m h o l t z i n s t a b i l i t y

C l ea r ly t h e a r e a of t h e s h e e t i n c r e a s e s g r e a t l y d u r i n g t h e p ro c es sof s p i r a l f or ma ti on . I n a p u r e l y tw o- di me ns io na l S i t u a t i o n , t h e

3

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Topological Aspe cts of Turbulent Vorticity Dynamics 227

I

ens t rophy i s c o n se r ve d , d e s p i t e t h i s i n c r e a s e o f a r e a . To u nd er -s ta nd t h i s , w e s ho ul d t a k e a cc ou nt o f t h e i n i t i a l s h e e t t h i c k n e s s ,s a y E . A s t i m e p r o gr e ss e s , t h e s h e e t g e t s t h i n n e r o v er t h e s e c t i o n

AB C o f f i g u r e 1, where it i s b e in g m ost s t r e t c h e d . Here t h e r e f o r et h e s h e e t s t r e n g t h d e c r e a se s and t h i s c om pe nsa te s t h e bu il d- up o fv o r te x co n c e nt r a t i on s n e a r t h e s p i r a l s i n g u l a r i t i e s D and E .

S up po se now t h a t , w i t h t h e co o r d i n a t e s y s t em o f f i g u r e 1, t h e r e i s asuperposed veloci ty component w i n t h e z - d i r e c t i o n which i s also d i s -c o n ti n uo u s a c r o s s t h e v o r t e x s h e e t . T h i s v e l o c i t y com ponent i ss im p ly co n vec ted by t h e f lo w in t h e ( x ,y ) p l an e an d so the jump Awi n W a c r o s s t h e s h e e t r em ai ns c o n s t a n t . The a s s o c i a t e d v o r t i c i t ycomponent i n t h e ( x f y ) p l an e p a r a l l e l t o t h e s h e e t t h e n i n c r e a s e s a tp o i n t s w here t h e s h e e t t h i c k n e s s E d ec r e a s e s : and t h e en s t ro p h y th eni n c r e a se s a l s o a s t h e s h e e t a r e a i n c r e a s e s . T h i s b e h av i o ur , a l t h o ug h

c o n c e i v a b l e , i s somewhat a r t i f i c i a l , b ec au se t h e m ost u n s t a b l e d i s -t u r b a n c e o f a p l a n e v o r t e x s h e e t i s a l w a y s o r i e n t e d so t h a t i t s wave-v e c t o r 16 i s p a r a l l e l t o t h e d i s co n t in u i ty o f t h e v e c t o r t a n g e n t i a lv e l o c i t y AU a cr os s t h e s h ee t . For t h i s most u n s t a b l e i s t u r b a n c e ,t h e abo ve G n st ro ph y in c r e a s e can n o t o cc u r .d i f f e r e n t when w e c o ns id e r i n s t a b i l i t i e s t o which a v o r t e x s h ee twrapped on a t o r u s may be su b j e c t .

KELVIN-HELMHOLZ INSTABILITY O F A T O R O I D A L VORTEX SHEET

W e now r e t u r n t o t h e t o r o i d a l c o n f i g u r a t i o n c o n s i d e re d above and w esup pose t h a t t h e v o r te x l i n e s c o ve r t h e se t of t o r u s e s n e s te d i n t h eneighbourhood V, o f S , i . e . e ac h v o r t e x l i n e f o l l o w s a h e l i c a l p a t hround a t o r u s , w i th i r r a t i o n a l p i t c h . I f 8 and Cp a r e a n g u l a r c o-o r d i n a t e s on t h e t o r u s , t h e n w e may a n t i c i p a t e u n s t a b l e d i s t u r b a n c e sp r o p o r t i o n a l t o exp[i(rnO + nCp)lwhere m and n a r e i n t e g e r s - the wavec r e s t s o f su ch d i s t u rb a n c e s a r e h e l i c e s which c l o s e on t h e t o r o i d -a 1 s u r f a c e . Hence t h e w a v e -c r es t s c a n no t now b e o r i e n t e d p a r a l l e l t ot h e vo r t e x l i n e s i n V E ; and by t h e Z n t f t h e p r e v i o u s p a ra g ra p h ,any i n s t a b i l i t y which l e a d s t o e x p l o s i v e grow th o f t h e t o r o i d a l s u r -f a c e a r e a S m ust a t t h e same t i m e l e a d t o e x p l o s i v e gro wt h o f t h et o t a l e ns tr op hy R o f t h e fl ow .

The most u n s t a b l e mode (m,n) w i l l , a s w e h ave s e e n, l ea d t o i n t e n s i f i -c a t i o n o f t h e component o f v o r t i c i t y i n t h e (m,n) d i r e c t i o n on t o r -o i d a l s u r f a c e s . The c o rr e sp o n di n g v e l o c i t y jump i s p r o g r e s s i v e l yo r i e n t e d i n t h e p e r p e n d i c u l a r d i r e c t i o n ( - n f m ) , and a s ec on d ar y modeo f i n s t a b i l i t y w i t h w ave-vec tor i n t h i s d i r e c t i o n t h en a pp ea rs l i k e l y .The v o r t i c i t y f i e l d y may be decomposed i n t o two f i e l d s w = g l + y,w i t h parallel t o (m ,n) and w p a r a l l e l t o ( -n ,m ) . The"prim ary i n -s t a b i l i t y a s s o ci a te d wi th t h e -k i e ld col l e a d s t o i n t e n s i f i c a t i o n o f2 : and t h e s ec on da ry i n s t a b i l i t y a s s o c i a t e d w i t h g 2 l e a ds t o i n t en -s i f i c a t i o n of col.t h i s m utual i n t e n s i f i c a t i o n pr oc e ss .

It i s an e s s e n t i a l f e a t u re o f t h i s p r oc e ss t h a t t h e v o r t ex l i n e s a r en o t c l o s e d , b u t do a c t u a l l y c o v er t o r o i d a l s u r f a c e s . The h e l i c i t y( s e e e q u a t i o n (11)) s s o c i a t e d w i t h s u c h a f l ow i s undoubtedly non-zero f and it might be thou gh t t h a t t h i s v o r t i c i t y d i s t r i b u t i o n i st h e r e f o r e u n t y p i c a l o f t h o s e e xa mp le s o f t u r b u l e n t f lo w ( e . g . g r i d

5.y i s n e v er i d e n t i c a l l y z e r o , and a p l a u s i b l e model o f t h e f l ow maybe provided by a random s up er po s i t i on of f un da m en ta l t o r o i d a l u n i t so f t h e above k i n d , some w i t h p o s i t i v e h e l i c i t y and some w i t h n e g a t i v e

t h e same t i m e s u b j e c t t o c on ve ct io n and d i s t o r t i o n by t h e v e l o c i t y

T h e s i t u a t i o n i s however

Equa t ions ( 4 ) p ro vid e t h e s i m p le s t i d e a l i s a t i o n o f

I t u r b u l e n c e ) f o r which <g.w_>= 0 . However, i n any tu r bu le n t f low,

8 h e l i c i t y , eac h u n i t s u b j e c t t o i t s own i n t r i n s i c i n s t a b i l i t i e s and a t

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228 H.K. Moffatt

f i e l d s a ss o c i a te d w i t h t h e o t h e r u n i t s . T h is i s r e m i n i s c e n t o fa t t e m p t s t o p o r t r a y t u r b u l e n c e a s a random s u p e r p o s i t i o n o f H i l l ' s

s p h e r i c a l v o r t i c e s : b u t t h e non-zero h e l i c i t y o f t h e t o r o i d a l v or-

t i c e s i s h e r e a n e s s e n t i a l c on co m it an t o f t h e p r o c es s o f r a p i di n c r e a s e o f e n s t r o p h y ; t h e u n i t s may mo re ov er b e l i n k e d and t a n g l e dw i t h e a c h o t h e r i n a manner n o t a v a i l a b l e t o H i l l ' s v o r t i c e s .

I t i s o f c o u r s e w e l l known t h a t h e l i c i t y p la y s a c r u c i a l r o l e i nm agn etoh yd rod yn am ic t u r b u l e n ce , i n t h a t , when t h e mean h e l i c i t y i snon-zero, t h e medium i s s u b j e c t t o magne t ic i n s t a b i l i t i e s on a l ls c a l e s l a r g e compared w i t h t h e e n e r gy - c o nt a i ni n g s c a l e s o f t h e t u r -b ul en ce ( f o r f u l l a cc o u n ts o f t h i s i m p o r t an t p ro ble m, see M o f f a t t1978, Krause & R a d l e r 1 9 8 0 ) . I t i s a l s o known t h a t h e l i c i t y c a n ha vea s t r o n g i n f l u e n c e on t h e e f f e c t i v e t u r b u l e n t d i f f u s i v i t y of a con-v e c t ed p a s s i v e s c a l a r f i e l d (K ra ich na n 1 9 7 6 , Drummond, Duane &

Horgan 1983) . The a s s e r t io n made he re i s t h a t h e l i c i t y may be o fp r of o un d i m p or ta n ce a l s o i n t h e n o n - li n e a r dynam ics o f t u r b u l e n c e ;and t h a t ev en when t h e mean h e l i c i t y i s z er o , l o c a l h e l i c i t y f l uc t u -a t i o n s a s s o c i a t e d w i t h t o r o i d a l v o r t i c i t y s t r u c t u r e s ( o r p o s s ib l yw i t h s t r u c t u r e s t h a t a r e t o p o l o g i c a l l y much more com plex) a r e a s s o c i -a t e d i n a n e s s e n t i a l way w i t h t h e f un da me nt al p r o c es s o f i n c r e a s e o fe n s t r o p h y .Lev ich & T s i n o b e r ( 1 9 8 3 ) .

A s i m i l a r c on ce pt is deveLoped i n t h e r e c e n t p a p e r of

SPECTRAL ANALYSIS OF SPIRAL SINGULARITIES

A random s u p e r p o s i t i o n o f p l a n e v o r t e x s h e e t s ( f i g u r e 2 ) p r o v i d e s ap r im i t i v e mode l o f t u rb u l en ce (Townsend 1Y51 ) f o r wh ic h t h e e ne rg yspectrum i s

E ( k ) 0: k- ' f (kRv) ( 2 0 )

where R i s t h e mean t h i c k n e s s of t h e s h e e t s ( c o n t r o l l e d by v i s c o s i t y )and f (x y i s a f u n c t i o n T e p r e s e n t i n g an e x p o n e n t i a l- t y p e v i s c o u s c u t -o f f ( f ( 0 ) = 1 ) . The k- f a c t o r i n ( 2 0 ) i s a s s o c i a t e d w i t h t h e f a c tt h a t any s t r a i g h t l i n e c u t s a f i n i t e number of v o r t e x s h e e t s per u n i tl en gt h; t h e v e l o c i t y f i e l d on t h i s t r a n s v e r s a l th en h f s a f i n i t enumber 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 , a nd t h e k- f a c t o r i n ( 2 0 )

i s a s i m p l e c on se qu en ce o f t h i s f a c t .

F i g u r e 2Random v o r t e x s h e e t s ; t h e v e l o c i t y f i e l d on a t r a n s v e r s a lh 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 on any s e c t i o n A B ,

l e ad in g t o a k-' s p e c t r a l power law .

The s lo w e r f a l l - o f f r ep r e s e n t ed by t h e k '5 /~ -K olm og or ov s p ec tr u ms u gy e st s t h a t , a t l e a s t on s c a l e s o n whiah v is c ou s e f f e c t s a r e neg-l i g i b l e , t h e v o r t i c i t y s t r u c t u r e must i n some s e n s e b e worse t h a nt h a t a s s o c i a t e d w i t h random v o r t e x s h e e t s . T h i s w or se ni ng i s con- L

v e n i e n t l y p r ov id ed by t h e s p i r a l s i n g u l a r i t i e s which d eve lo p a s ar e s u l t o f t h e Kelvin-Helmholz i n s t a b i l i t y , I f t h e s e o c c u r more o rl e ss wherever t h e v o r t e x s h e e t s oc c ur , t he n a s t r a i g h t l i n e t r a n s -v e r s a l w i l l o c c a s i o n a l l y p a ss v er y n e a r a s p i r a l s i n g u l a r i t y . L e t *

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Topological Asp ects of Turbulent Vo rticity Dynam ics 229

us a na ly se t h e e f f e c t t h a t t h i s w i l l have o n t h e r e s u l t i n g e ne rg yspectrum

L e t x be a c o o r d i n a t e al on g t h e t r a n s v e r s a l , and l e t u ( x ) b e av e l o c i t y component p e r p e n d i c u l a r t o t h e t r a n s v e r s a l . Suppose t h i st r a n s v e rs a l p a s s e s n e a r a s p i r a l s i n g u l a r i t y s i t u a t e d on t h e p l an ex = 0 ( f i g u r e 3 ) .

b

F i g u r e 3

T r an sv e rs a l p a ss i n g n e a r a s p i r a l s i n g u l a r i t y ; t h e r e i s t h e na l a r g e number of d i s c o n t i n u i t i e s of t r a n s v e r s e v e l o c i t y n e a rt h e p o i n t 0 .

Then u ( x ) h a s d i s c o n t i n u i t i e s a t a se qu en ce of p o i n t s x o , x , , . . . % ,say , where N i s l a r g e . The F o u r t e r cosine c o e f f i c i e n t s of t h ef u nc t io n u ( x ) i n t h e ra ng e 0 < x 7~ a re g iv en b y

v

0

an = 1 u ( x ) co s nx d x ( 2 1 )

and t h e b eh a vi ou r f o r l a r g e n i s g iven by

I f we now suppose n f i x e d , and con s i de r t h e l i m i t N + m, w e see ' t h a tw e g e t a s i g n i f i c a n t c o n t r i bu t io n t o t h e se r i e s ( 2 2 ) o n ly f r o m th o s eterms f o r which nx = O ( 1 ) o r g r e a t e r . L e t M be t h e number of suchterms .values between -1 and +l, ( 2 2 ) g i v e s i n o r d e r o f m a g ni tu de

Due t o t h e r f a c t o r s i n n x r, w hich i n e f f e c t t a k e s random

( 2 3 )

where Au i s a n a ve ra g e v a l u e o f t h e jump i n / u ( x ) a c r o s s t h e s h e e t .

S up po se , f o r ex am ple, t h a t t h e j um ps o ccu r a t x r=

r-' fo r some s>O.Then M i s given by

n/MS = 0 ( 1 ) , i . e . M % n /S , ( 2 4 )

The cor respond ing s p e c t r a l power law i s

(26)E ( k ) % k-' where A = 2 - ;

The Kolmogorov exponent A = 5 / 3 cor responds t o s = 3 , i . e . t o

v e l o c i t y jumps a t x = z ' ~ , 3 - 3 , 4 - 3 , e t c .

T h i s t y p e o f a n a l y s i s i s i nc om pl et e i n t h a t c l e a r l y t h e d e n s i t y andp r o ba b i l i t y d i s t r i b u t i o n o f s p i r a l s i n g u l a r i t i e s sho uld a l s o p l ay ap a r t i n d e te rm i ni ng t h e r e s u l t i n g e n e rg y sp ec tr um . I t i s n e v e r t h e -l ess i l l u m i n a t i n g , i n t h a t it shows how t h e s t r u c t u r e o f t h e t y p i c a ls p i r a l s i n g u l a r i t y c an h av e a d et er m in in g i n f l u e n c e on t h e s p e c t r a lpower law i n t h e i n e r t i a l ra ng e.known, fro m in dep end en t co n s i d e r a t i o n s , t h e n t h i s g iv e s i n f o r m a t io n

b

C o nv e rs el y, i f t h i s power l a w i s

I abou t t h e s t r u c t u r e and d i s t r i b u t i o n o f s p i r a l s i n g u l a r i t i e s ,

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230 H. .Moffatt

1

assuming that these are indeed the dominant physical feature ofturbulent flows.

FOOTNOTE.

I am indebted to Dr. M. German0 who drew my attention to this early,and little known, paper.

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