A general kinetic equation for weakly turbulent homogeneous systems

Volume 31A, number 3

P HY SI C S L E T T E R S

pla c e d, doing w o r k a g a i n s t the e x t e r n a l p r e s s u r e Poo which m ay be r e t r i e v e d when the c o r e is s h o r t e n e d giving an e f f e c t i v e p o te n t ia l e n e r g y :

V = Poo Vc = ¼P K 2 R .

(11)

T h i s v o r t e x has total e n e r g y E = K+ U given by eqs. (1) and (11), E = ½ p K2 R [ ln ( 8 R / a ) - {] ,

(12)

9 February 1970

s u r e . We p r o p o s e that t h e s e equations should be applied to q u an t i zed v o r t e x r i n g s as d i s c u s s e d in the e x p e r i m e n t of R a y f i e l d and Reif [3]. On r e a n a l y z i n g t h e i r data, we find the a p p r o p r i a t e v al u e of the c o r e r a d i u s is a = 1.28 + 0 . 1 3 A

(14)

( T = 2.28OK)

Th e a n a l y s i s g i v en h e r e r e m o v e s a t r o u b l e s o m e i n c o n s i s t e n c y in r e s e a r c h on superfluidity.

so that

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

(13)

w hic h c o i n c i d e s with eq. (2). E x p r e s s i o n s (12) and (13) a r e s e e n to be r e a sonably independent of the s p e c i f i c m o d e l chosen, and to obey H a m i l t o n ' s equaUon at c o n s t a n t p r e s -

A GENERAL

KINETIC EQUATION HOMOGENEOUS

References 1. w. M. Hicks, Phil. Trans. Roy. Soe. London 175 (1884) 161, pp. 183 and 190. 2. H. Lamb, Hydrodynamics, Sixth ed. (Dover Publications, New York, 1965) p. 241. 3. G.W. Rayfield and F. Reif, Phys. Rev. 136 (1964) Al194.

FOR WEAKLY SYSTEMS

TURBULENT

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 w e l l known that a k i n e t i c equation f o r a weakly t u r b u l e n t s y s t e m could be d e r i v e d by s e v e r a l m e t h o d s [1-3] in the c a s e of s t a b l e and undamped w a v e s . H o w e v e r , t h e s e m e t h o d s a r e only v a l i d when the e i g e n f r e q u e n c i e s r e s u l t i n g f r o m the l i n e a r i z e d t h e o r y p r o v e to be r e a l ; d i s s i p a t i o n o r i n s t a b i l i t y e f f e c t s could only be i n t r o d u c e d a p o s t e r i o r i . On the o t h e r hand we r e c e n t l y showed that a d y n a m i c a l equation f o r the e n e r g y s p e c t r u m could be d e r i v e d , in the c a s e of the B u r g e r s equation (homogeneous h y d r o d y n a m i c t u r b u l e n c e ) , by applying the B o g o l i u b o v - e x p a n s i o n method. In that c a s e , the s y s t e m was p h y s i c a l l y d e t e r m i n e d only by its v i s c o s i t y ( d i s s i p a t i o n ! ) . The u s e d p r o c e d u r e could be e x t e n d e d to the g e n e r a l c a s e a s s o c i a t e d with u n s t a b l e o r d a m p e d w a v e s . The c o r r e s p o n d i n g a n a l y s i s will be e x p l a i n e d h e r e f o r the following o n e - d i m e n s i o n a l m o d e l equation [3] +oO

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

+ w(k-k')-o~(k))t}dk'

,

(1)

w h e r e C(k, t ) is the a m p l i t u d e of an individual k wave of the f o r m e x p { i k x - iw(k)t}; quite g e n e r a l l y we here assume a dispersion relation

138

Volume 31A, number 3

~(h) = ~ ( k )

PHYSICS

- iv(k)

LETTERS

9 February 1970

(2)

( v > 0)

and, in o r d e r to a v o i d 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 w i t h 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 w h i c h 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 f o r the w a v e c o r r e l a t i o n s s u c h a s :

(3a)

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

( C(kl, t) C(k2, t) C(k3, t)) = H(kl, h2, t ) 5 ( k 1 + k2+

(3b)

k3) , . . .

W i t h the a i d 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 c a n b e s e t up. T h i s h i e r a r c h y c a n be c l o s e d by m e a n s of the d e f i n i t i o n of w e a k t u r b u l e n c e : G ~ O ( 0 , H at l e a s t of o r d e r ¢ 2, and s o on. S i n c e , in v i e w of eq. (2), w e 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 f u n c t i o n E(k, t) = G(k, t) e x p {-2~(k) t } . T h e s y s t e m of e q u a t i o n s is 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 i n e t i c e q u a t i o n aE

at

= 2 ~ , °A i (k; ~ )

(4)

i=O

and at 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 w h i c h H i s c o m p l e t e l y f i x e d by the e x p l i c i t f o r m of G at t = 0. A p p l y i n g a l l t h i s i n f o r m a t i o n to the m o m e n t e q u a t i o n s and a s s u m i n g the c o n v e n t i o n a l c o n d i t i o n s f o r V w e a r r i v e at the f o l l o w i n g d y n a m i c a l e q u a t i o n f o r the t w o - w a v e c o r r e l a t i o n f u n c t i o n (that i s eq. (4) up to s e c o n d o r d e r ) : 00

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

_~

- r 2 ~ where

~(kl,k2;k 3)

~ k

E(k')E(k-k')r(k',k-k';k) t

r

~

,

~

+

) (5)

; k) r2(k, k-k'; k) + n2(k', k-k'; k)~

= 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 c o u p l i n g k i n e t i c e q u a t i o n , w h i l e f o r fl -~ 0 we c a n , in p r i n c i p l e , r e f i n d o u r B u r g e r s d y n a m i c a l e q u a t i o n [4]. T h i s t h e o r y c a n 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 a b i l i t y ) , w i t h the r e s t r i c t i o n t h a 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 e c h n i q u e . T h e 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 D r . M. P. H. W e e n i n k f o r u s e f u l d i s c u s s i o n s .

T h i s w o r k w a s 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 c h t i n g 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) w i t h f i n a n c 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 statistical mechanics, Vol. I, eds. J. De Boer and G. E. Uhlenbeck (North-Holland, Amsterdam, 1962). 139