Volume 31A, number 3 P H Y S I C S L E T T E R S 9 February 1970

placed, doing work aga ins t the ex te rna l p r e s - s u r e Poo which may be r e t r i e v e d when the c o r e is shor tened giving an ef fec t ive potent ia l energy:

V = Poo Vc = ¼P K 2 R . (11)

This vor t ex has total energy E = K+ U given by eqs. (1) and (11),

E = ½pK2R[ ln ( 8 R / a ) - {] , (12)

sure . We propose that these equat ions should be applied to quant ized vo r t ex r ings as d i scussed in the expe r imen t of Rayf ie ld and Reif [3]. On r e - analyzing the i r data, we find the appropr ia te value of the co re radius is

a = 1.28 + 0 .13A ( T = 2.28OK) (14)

The analys is given he re r e m o v e s a t roub lesome incons i s tency in r e s e a r c h on superf luidi ty .

so that

v = (OE/gp) a = (K/4trR)[ln ( 8 R / a ) - ½] , (13)

which coinc ides with eq. (2). E x p r e s s i o n s (12) and (13) a r e seen to be r e a -

sonably independent of the spec i f ic model chosen, and to obey Hami l ton ' s equaUon at constant p r e s -

R e f e r e n c e s 1. w. M. Hicks, Phil. Trans. Roy. Soe. London 175

(1884) 161, pp. 183 and 190. 2. H. Lamb, Hydrodynamics, Sixth ed. (Dover Publi-

cations, New York, 1965) p. 241. 3. G.W. Rayfield and F. Reif, Phys. Rev. 136 (1964)

Al194.

A G E N E R A L K I N E T I C E Q U A T I O N F O R W E A K L Y T U R B U L E N T H O M O G E N E O U S S Y S T E M S

W. P. M. M A L F L I E T Association Euratom-FOM, FOM-Instituut voor Plasma-Fysica, Rijnhuizen, Jutphaas,

The Netherlands

Received 23 December 1969

A kinetic equation has been found which describes the non-linear interaction between waves, including the cases of unstable or damped waves.

It is wel l known that a k inet ic equation for a weakly turbulent sy s t em could be de r ived by s e v e r a l methods [1-3] in the case of s table and undamped waves . However , these methods a r e only va l id when the e igenf requenc ies resu l t ing f rom the l i n e a r i z e d theory p rove to be rea l ; d i ss ipa t ion or ins tab i l i ty e f fec t s could only be in t roduced a p o s t e r i o r i . On the o ther hand we recen t ly showed that a dynamical equat ion for the ene rgy spec t rum could be der ived , in the case of the B u r g e r s equation (homogeneous hydrodynamic turbulence) , by applying the Bogol iubov-expans ion method. In that case , the sys t em was phys ica l ly d e t e r m i n e d only by i ts v i s c o s i t y (diss ipat ion!) .

The used p r o c e d u r e could be extended to the genera l case a s soc i a t ed with unstable or damped waves . The co r re spond ing ana lys i s wil l be expla ined here for the following one -d imens iona l model equat ion [3]

+ o O

8C(k, t) = _ i f V ( k ; k ' , k - k ' ) C ( k ' , t ) C ( k - k ' , t ) e x p { - i ( o o ( k ' ) + w ( k - k ' ) - o ~ ( k ) ) t } d k ' , (1) 8t _~o

where C(k, t ) is the ampl i tude of an individual k wave of the fo rm e x p { i k x - iw(k)t}; quite genera l ly we he re a s s u m e a d i spe r s ion re la t ion

138

Volume 31A, number 3 P H Y S I C S L E T T E R S 9 February 1970

~(h) = ~ ( k ) - iv(k) (v> 0) (2)

and, in o r d e r to avo id c o m p l e x i t y , a l s o the r e l a t i o n s WR(-k) = - 0aR(k) a n d v (~) = V (-k). Eq. (1) d e - s c r i b e s the e v o l u t i o n of a p a r t i c u l a r a m p l i t u d e due to n o n l i n e a r i n t e r a c t i o n s wi th a l l o t h e r m o d e s . An a p p l i c a t i o n of the r a n d o m - p h a s e a p p r o x i m a t i o n which i s e .g . , r e q u i r e d in the c a s e of h o m o g e n e o u s t u r b u l e n c e , i n v o l v e s p r o p e r t i e s fo r the w a v e c o r r e l a t i o n s such as :

( c(k 1 , t ) c(k 2, t)) = a(k 1 , t) ~ (k 1 +k 2 ) , (3a)

( C(kl, t) C(k 2, t) C(k3, t)) = H(kl, h2, t ) 5 ( k 1 + k2+ k3) , . . . (3b)

With the a id of eq. (1) and the r e l a t i o n s (3) a h i e r a r c h y of d y n a m i c a l m o m e n t e q u a t i o n s can be s e t up. T h i s h i e r a r c h y can be c l o s e d by m e a n s of the de f in i t ion of w e a k t u r b u l e n c e : G ~ O ( 0 , H a t l e a s t of o r d e r ¢ 2, and so on. S ince , in v i e w of eq. (2), we a l s o a d m i t a p o s s i b l e d i s s i p a t i o n we i n t r o d u c e the s p e c t r u m func t ion E(k, t) = G(k, t) exp {-2~(k) t} .

The s y s t e m of e q u a t i o n s i s now s u i t a b l e f o r an a p p l i c a t i o n of the B o g o l t u b o v - e x p a n s i o n m e t h o d [5] a s s u m i n g the e x i s t e n c e of a f o r m a l k ine t i c equa t i on

aE = 2 ~ , °A i (k; ~ ) (4) at i=O

and a t t i m e t = 0 the " f u n c t i o n a l a n a s a t z " H = H(kl, k2; G(t=O)), a c c o r d i n g to wh ich H i s c o m p l e t e l y f ixed by the e x p l i c i t f o r m of G at t = 0.

App ly ing a l l th i s i n f o r m a t i o n to the m o m e n t equa t ions and a s s u m i n g the c o n v e n t i o n a l cond i t i ons fo r V we a r r i v e a t the fo l l owing d y n a m i c a l equa t ion fo r the t w o - w a v e c o r r e l a t i o n func t ion (that i s eq. (4) up to s e c o n d o r d e r ) :

0 0

E(k,t)_-2T(k)E(k)+4 f dle' ]V(k;k',k-k')f2t E(k')E(k-k')r(k',k-k';k) + ~t _~ t r ~ , ~ )

- r 2 ~ ~ k ; k) r2(k, k-k'; k) + n2(k', k-k'; k)~ (5)

w h e r e ~(kl,k2;k 3) = O~R(k 3) - ¢OR(k 1) - o~R(k 2) and F(kl,k2;k 3) = v(k3) - v ( k l ) -Y(k2).

H e r e the l i m i t F ~ + 0 l e a d s to the w e l l - k n o w n n o n l i n e a r m o d e - m o d e coupl ing k i n e t i c equa t ion , wh i l e f o r fl -~ 0 we can , in p r i n c i p l e , r e f i n d ou r B u r g e r s d y n a m i c a l equa t i on [4]. T h i s t h e o r y can a l s o be u s e d f o r c a s e s w h e r e , i n s t e a d of eq. (2), a~ = w R + iv ( l i n e a r i n s t ab i l i t y ) , wi th the r e s t r i c t i o n tha t E(k) r e m a i n s of O(e) in v i e w of the u s e d t echn ique .

The a u t h o r i s g r a t e f u l to P r o f e s s o r H. B r e m m e r and Dr . M. P. H. Ween ink f o r u se fu l d i s c u s s i o n s .

T h i s w o r k was p e r f o r m e d a s p a r t of the r e s e a r c h p r o g r a m m e of the a s s o c i a t i o n a g r e e m e n t of E u r a t o m and the "S t i ch t i ng v o o r F u n d a m e n t e e l O n d e r z o e k d e r M a t t e r i e n (FOM) wi th f i nanc i a l s u p p o r t f r o m the " N e d e r l a n d s e O r g a n i s a t i e v o o r Z u i v e r - W e t e n s c h a p p e l i j k O n d e r z o e k w (ZWO) and E u r a t o m .

References 1. B. Coppi, M.N. Rosenbluth and R. N. Sudan, Ann. Phys. (1969) 207. 2. R. C. Davidson, J . Plasma Phys. (1967) 341. 3. R. Z. Sagdeev and A. A Galeev, in : Nonlinear plasma theory, eds. T. M. O'Neil and S. L. Book (W. A. Benjamin,

Inc., New York, 1969). 4. W. P. M. Malfliet, Physica 45 (1969) 257. 5. N.N. Bogoliubov, in : Studies of s tat is t ical mechanics, Vol. I, eds. J. De Boer and G. E. Uhlenbeck (North-Holland,

Amsterdam, 1962). 139