Volume 26A, number l0 P H Y S I C S L E T T E R S 8 A p r i l 1968
N O T E O N T H E V O R T E X D E N S I T Y I N R O T A T I N G H E L I U M I I D. S T A U F F E R
Max-Planck-Institut JZdr Physik und Astrophysik, Munich, Germany and
F. P O B E L L , W. S C H O E P E
Physik Department der Technischen Hochschule, Munich, Germany
Received 8 March 1968
Computer calculations show that there exist no large stable deviations from the homogeneous vortex den- sity in an annulus or cylinder; the solutions of Masson and Tien are unstable.
M a s s o n and T i e n [1] h a v e r e c e n t l y c a l c u l a t e d in a c o n t i n u u m a p p r o x i m a t i o n the v o r t e x d e n s i t y in a r o t a t i n g annulus f i l l e d with He II. In addition to the e n e r g e t i c a l l y m o s t f a v o u r a b l e h o m o g e n e o u s d i s t r i b u t i o n they found s o l u t i o n s with a s p a t i a l l y o s c i l l a t i n g v o r t e x density. F o r s m a l l o s c i l l a t i o n a m p l i t u d e s the d i s t a n c e b e t w e e n two m a x i m a of v o r t e x d e n s i t y i s only of the o r d e r of the v o r t e x m e a n s p a c i n g s = (K/2~)½, (K = h / m , ~2 = a n g u l a r v e l o c i t y ) . T h e r e f o r e t h e s e " s m a l l - a m p l i t u d e "
s o l u t i o n s a r e not d i f f e r e n t f r o m the h o m o g e n e o u s solution. The o s c i l l a t i o n s with l a r g e a m p l i t u d e s and g r e a t e r d i s t a n c e s b e t w e e n two p e a k s in the v o r t e x d e n s i t y r e m a i n to be i n v e s t i g a t e d . We w i l l g e n e r a l i z e the s o l u t i o n s of t h i s type g i v e n in ref.1 and d i s c u s s t h e i r s t a b i l i t y .
In such e q u i l i b r i u m c o n f i g u r a t i o n s with widely s e p a r a t e d n a r r o w p e a k s e a c h p e a k r e p r e s e n t s a r i n g c o m p o s e d of v o r t i c e s with s p a c i n g << s; the d i s t a n c e b e t w e e n the d i f f e r e n t c o n c e n t r i c r i n g s i s >> s. It w a s shown by e x a c t c o m p u t e r c a l c u l a - t i o n s without a v e r i g i n g [2] that v o r t i c e s a l w a y s t e n d to f o r m a s y s t e m of c o n c e n t r i c r i n g s . In the e n e r g e t i c a l l y m o s t l y f a v o u r e d c o n f i g u r a t i o n s the s p a c i n g of both the d i f f e r e n t r i n g s and the v o r t i c e s within e a c h r i n g i s ~ s.
In o r d e r to find o t h e r e q u i l i b r i u m c o n f i g u r a - t i o n s we a s s u m e a s y s t e m of r i n g s , e a c h with N i v o r t i c e s and a r a d i u s Ri. We t r e a t the d i f f e r e n t r i n g s s e p a r a t e l y , but a p p r o x i m a t e the i n t e r a c t i o n b e t w e e n t h e m by s m e a r i n g out the d i s c r e t e v o r t e x s t r u c t u r e within e a c h ring. T h i s m e a n s an " e x a c t "
c a l c u l a t i o n in the r a d i a l d i r e c t i o n and an a v e r a g e c a l c u l a t i o n along the c i r c u m f e r e n c e s of the r i n g s . T h u s e a c h r i n g p r o d u c e s a z e r o v e l o c i t y i n s i d e , and the v e l o c i t y Ni K/2~r outside. The s e l f - i n -
duced v e l o c i t y ( N i - 1) K / 4 n R i at its r a d i u s i s n e a r l y the a v e r a g e of the i n n e r and o u t e r v e l o c i - t i e s . At e q u i l i b r i u m e a c h r i n g is at r e s t in the r o t a t i n g f r a m e ; t h i s condition y i e l d s the a p p r o x i - m a t e r e l a t i o n b e t w e e n Ni and Ri:
~ R i = (F1 + ~ ( N i - 1 ) K ) / 2 n R i 1 (1) H e r e F 1 is the t o t a l c i r c u l a t i o n within the i - t h r i n g , F 1 = F o + K AJk= 1 Nk, i-1 w h e r e F o is the c i r - c u l a t i o n around the i n n e r c y l i n d e r in an annulus o r a r o u n d the c e n t e r v o r t e x in a c y l i n d e r . F o r e x a m p l e , in a c y l i n d e r with 37 v o r t i c e s two of the s t a b l e c o n f i g u r a t i o n s a r e given by Ni =
= 6 , 12, 18 and N i = 7 , 12, 17 [2, fig. 1]. In the m o s t e x t r e m e c a s e the c e n t e r v o r t e x might be s u r r o u n d e d by a s i n g l e r i n g with 36 v o r t i c e s : Ni = O, 36, 0. S i m i l a r l y , f o r l a r g e r v o r t e x n u m - b e r s N i may be v a r i e d r a t h e r a r b i t r a r i l y as long as a l l R i l i e within the bucket. T h e r e f o r e eq. (1) with n e a r l y a r b i t r a r y c h a n g e s of R i s e e m s to be an a p p r o p r i a t e g e n e r a l i z a t i o n of the solution of ref. 1 ( n e a r l y p e r i o d i c v a r i a t i o n s of the v o r t e x density).
H o w e v e r , not a l l of t h e s e e q u i l i b r i u m c o n f i - g u r a t i o n s a r e s t a b l e a g a i n s t s m a l l (not n e c e s s a - r i l y a x i a l l y s y m m e t r i c ) p e r t u r b a t i o n s . A r i n g of m o r e than 6 v o r t i c e s without c e n t e r v o r t e x is u n - s t a b l e [3]; with c e n t e r v o r t e x the boundary is 9.
A s y s t e m of w e l l s e p a r a t e d r i n g s with c l o s e l y s p a c e d v o r t i c e s is s i m i l a r to a s y s t e m of c l a s s i - c a l v o r t e x s h e e t s which a r e u n s t a b l e [4,5]. If we a s s u m e that the v o r t e x r i n g s a r e w e l l s e p a r a t e d we c a n u s e d eq. (14) of ref. 3 to show that each r a d i u s R i of the h o m o g e n e o u s solution can be e n - l a r g e d in s t a b l e s y s t e m s only by ARt < 0.1 S.
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Volume 26A, n u m b e r 10 P H Y S I C S L E T T E R S 8 April 1968 T h i s s h o w s , t h a t t h e l a r g e d e v i a t i o n s
( A R i >> s) s u g g e s t e d b y r e f . 1 a r e u n s t a b l e . P e r - h a p s t h i s l i m i t f o r A R / i s q u a n t i t a t i v e l y i n c o r r e c t b e c a u s e n o w t h e r i n g s a r e n o g w e l l s e p a r a t e d . H e n c e , we p e r f o r m e d e x a c t c o m p u t e r c a l c u l a t i o n s u s i n g t h e i t e r a t i o n p r o g r a m of r e f . 2. N e g l e c t i n g i m a g e f o r c e s t h e f o l l o w i n g c a s e s w e r e s o l v e d : t h e a n n u l u s (2 r i n g s ) w i t h 30, 42 a n d 54 v o r t i c e s ( F o / K = 7, 19 a n d 37, r e s p e c t i v e l y ) a n d t h e c y - l i n d e r w i t h 19, 37 a n d 61 v o r t i c e s (2, 3 a n d 4 r i n g s , r e s p e c t i v e l y ) . T h e c a l c u l a t i o n s s h o w t h a t t h e R i f r o m eq. (1) a r e a c c u r a t e w i t h i n 0 . 0 2 s i f N i >i 5. S h i f t i n g s o m e v o r t i c e s a m o n g t h e d i f f e r - e n t r i n g s we f o u n d t h e c o n f i g u r a t i o n s t o b e c o m e u n s t a b l e if t h e g r e a t e s t d e v i a t i o n I / ~ R / I / s e x c e e d s a v a l u e b e t w e e n 0 . 0 9 a n d 0.08. T h i s n u m b e r a g r e e s w i t h t h e a b o v e e s t i m a t e a n d s h o w s n o t e n - d e n c y t o i n c r e a s e w i t h i n c r e a s i n g v o r t e x n u m b e r . T h e r e f o r e t h e r a d i i of a l l v o r t e x r i n g s c a n b e c h a n g e d a t m o s t b y a s m a l l f r a c t i o n of t h e v o r - t e x m e a n s p a c i n g f r o m i t s e n e r g e t i c a l l y m o s t
f a v o u r a b l e v a l u e ; t h i s m e a n s t h a t t h e v o r t e x d e n s i t y i s a l w a y s h o m o g e n o u s .
W e t h a n k D r . M a s s o n a n d P r o f . T i e n f o r c o r - r e s p o n d e n c e . O n e of u s (D. S.) i s i n d e b t e d t o t h e M a x - P l a n c k - I n s t i t u t f o r a g r a n t .
R e f e r e n c e s
1. B. M a s s o n and C. L. Tien, Phys. Rev. 165 (1968) 300.
2. D. Stauffer and A. L. F e t t e r , Phys. R e v . , to be pub- lished.
3. T.H. Havelock, Phil. Mag. (VII) 11 (1931) 617.
4. A. Michalke and A. T i m m e , J. Fluid Mech. 29 (1967) 647.
5. J. R. Weske and T. M. Rankin, Phys. Fluids 6 (1963) t397, fig. 5.
N E W F O R M F O R T H E T H E R M O D Y N A M I C P O T E N T I A L O F H e I I N E A R T ; t D. J . A M I T
D e p a r t m e n t o f T h e o r e t i c a l P h y s i c s , H e b r e w U n i v e r s i t y , J e r u s a l e m , I s r a e l Received 4 M a r c h 1968
A new t h e r m o d y n a m i c potential function is p r e s e n t e d for t e m p e r a t u r e s j u s t below T~t which r e p r o d u c e s the {T~t - T)~ b e h a v i o r of Ps and the c r i t i c a l velocity Vc.
A c c u r a t e m e a s u r e m e n t s of t h e s u p e r f l u i d d e n - s i t y P s [1] a s o n e a p p r o a c h e s T~t f r o m b e l o w r ~ - v e a l a b e h a v i o r of t h e f o r m P s = 1 . 4 4 p ( T ~ - T)~.
T h i s i m p l i e s , i m m e d i a t e l y , t h a t t h e t h e o r y of G i n z b u r g a n d P i t a e v s k i i [2] w h i c h p r e d i c t s a l i n e a r b e h a v i o r of P s a s a f u n c t i o n of T k - T h a s t o b e a m e n d e d . T h i s h a s b e e n r e c e n t l y t a k e n u p b y M a m a l a d z e [3] w h o k e e p s t h e L a n d a u [4,5] e x - p a n s i o n of t h e t h e r m o d y n a m i c p o t e n t i a l b u t c h a n g e s t h e t e m p e r a t u r e d e p e n d e n c e of t h e c o - e f f i c i e n t s i n t h e e x p a n s i o n . T h e e x p a n s i o n i s c a r r i e d o u t i n p o w e r s of t h e o r d e r p a r a m e t e r w h i c h i s c o n n e c t e d t o t h e s u p e r f l u i d n u m b e r d e n - s i t y P s v i a P s = I$12 *- If t h e s y s t e m i s a t r e s t w e
* ~ is connected to J o s e p h s o n ' s ~ v i a ~b = r n ~ . It would be p r o p o r t i o n a l to J o s e p h s o n ' s l~ if ?7 t u r n e d out to be z e r o .
466
c a n t a k e ~ t o b e r e a l a n d p o s i t i v e . W e t h e n w r i t e M a m l a d z e ' s [3] r e s u l t i n t h e f o l l o w i n g f o r m :
• ( P , T) = ~ I ( P , T) + A ~ 2 + B@ 4 , (1) w i t h A = - ~ (T;t - T ) ~ , B = ~ (T;t - T ) ~ , a , f l > 0 ,
~ I i s t h e t h e r m o d y n a m i c p o t e n t i a l of HeI. T h e e q u i l i b r i u m v a l u e of ~ o r P s i s d e t e r m i n e d b y m i n i m i z i n g • w i t h r e s p e c t t o ~. a > 0 f o l l o w s f r o m t h e r e q u i r e m e n t t h a t ~ = 0 w i l l n o t b e m i n i - m u m at T < Tk e v e n t h o u g h i t s o l v e s ~ / a ~ = 0.
In t h e o l d t h e o r y [ 2 , 4 , 5 ] A w a s l i n e a r i n T~t - T a n d B a c o n s t a n t . T h e s i g n of B w a s d i c t a t e d b y t h e r e q u i r e m e n t t h a t ~ = 0 w i l l b e a s t a b l e m i n i - m u m at T = TX. T h i s l a s t r e q u i r e m e n t c a n n o t b e s a t i s f i e d w i t h eq. (1) [3] s i n c e b o t h c o e f f i c i e n t s v a n i s h a t T = T~t. I n t h i s c a s e P s a t e q u i l i b r i u m i s ( a / 2 ~ ) (T~t - T ) ~ a n d to m a k e p s > 0 w e m u s t
