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Kinematic and thermodynamic stability of vortices in superfluids Chapman, David M. F. 1977

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THE KINEMATIC AND  THERMODYNAMIC STABILITY OF VORTICES IN SUPERFLUIDS BY DAVID M.F. CHAPMAN B.Sc,  University  A THESIS SUBMITTED  o f Ottawa, 1975  IN PARTIAL FULFILLMENT OF  THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (DEPARTMENT  OF PHYSICS)  We a c c e p t t h i s t h e s i s as to the required  April,  conforming  standard.  1977  C o p y r i g h t , David M.F. Chapman, 1977  In p r e s e n t i n g t h i s  thesis  an advanced degree at  further  agree  fulfilment  of  the  requirements  the U n i v e r s i t y of B r i t i s h Columbia, I agree  the L i b r a r y s h a l l make it I  in p a r t i a l  freely  available  for  t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f  of  representatives.  this thesis for  It  financial  this  thesis  of  P  h  y  s  l  c  g a i n s h a l l not be allowed without my  s  The U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5  D  a  t  e  A p r i l 26, 1977  or  i s understood that copying or p u b l i c a t i o n  written permission.  Department  that  reference and study.  f o r s c h o l a r l y purposes may be granted by the Head of my Department by h i s  for  (i)  ABSTRACT  The fluid  equations  are d e r i v e d , and  are d i s c u s s e d . Two  of m o t i o n of v o r t i c e s i n an i d e a l  the laws of c o n s e r v a t i o n a s s o c i a t e d w i t h  r e g i o n s of flow a r e c o n s i d e r e d ,  r e g i o n bounded on the o u t s i d e by a c i r c l e . momentum of a v o r t e x f l u i d formalism  i s introduced  v o r t e x systems and bility  of r i g i d l y  superfluid of and  i n these  two  The  an i n f i n i t e  r e g i o n and  r e g i o n s are c a l c u l a t e d . A  a  angular Lagrangian  i n o r d e r t o d i s c u s s the symmetry t r a n s f o r m a t i o n s  rotating polygonal  kinematic  of  sta-  c o n f i g u r a t i o n s of q u a n t i z e d v o r t i c e s  the s t a t e s of thermodynamic e q u i l i b r i u m of a r o t a t i n g  are found  f o r low  angular v e l o c i t i e s ,  the spectrum of c r i t i c a l a n g u l a r v e l o c i t i e s three v o r t i c e s .  the m o t i o n  k i n e t i c energy and  t h e i r a s s o c i a t e d c o n s e r v a t i o n laws. The  i s determined, and  two-dimensional  resulting  i n the  calculation  f o r the c r e a t i o n of one,  two,  (ii) TABLE OF CONTENTS Page 1. INTRODUCTION 1.1  A B r i e f Review o f t h e R e l e v a n t L i t e r a t u r e  1  1.2  The Two-Fluid Model o f L i q u i d Helium  2  1.3  Some M a t h e m a t i c a l P r e l i m i n a r i e s  4  1.4  The O b j e c t o f t h i s Work  6  2. MOTION OF VORTICES IN A TWO-DIMENSIONAL 2.1  FLUID  8  E q u a t i o n s o f M o t i o n and C o n s e r v a t i o n Laws i n an I n f i n i t e F l u i d  2.2  8  The V e l o c i t y P o t e n t i a l o f a V o r t e x F l u i d I n s i d e the U n i t C i r c l e  2.3  11  E q u a t i o n s of M o t i o n  and C o n s e r v a t i o n Laws  I n s i d e the U n i t C i r c l e  12  3. DYNAMICAL QUANTITIES OF A VORTEX FLUID 3.1  14  K i n e t i c Energy  14  3.2 . A n g u l a r Momentum  16  3.3  R e n o r m a l i z i n g t h e Energy  17  3.4  I n t h e L i m i t of an I n f i n i t e F l u i d  18  4. THE LAGRANGIAN MECHANICS OF VORTEX SYSTEMS  20  4.1  The L a g r a n g i a n Formalism  20  4.2  The Symmetry T r a n s f o r m a t i o n s and C o n s e r v a t i o n Laws of V.ortex M o t i o n  22  5. THE STABILITY OF RIGIDLY ROTATING CONFIGURATIONS  25  5.1  The E q u i l i b r i u m  5.2  The L i n e a r i z e d  State of a Rotating Vortex F l u i d Equations of Motion  of the  Perturbations 5.3  The S t a b i l i t y  25  27 of the P o l y g o n a l C o n f i g u r a t i o n s  30  (iii) TABLE OF CONTENTS  (continued) Page  6. THERMODYNAMIC EQUILIBRIUM OF A ROTATING SUPERFLUID  33  6.1  One Vortex i n a Rotating Cylinder  34  6.2  The Spectrum of C r i t i c a l Angular V e l o c i t i e s i n He I I  35  7. CONCLUSION  39  BIBLIOGRAPHY  41  APPENDIX A:  STREAM FUNCTION APPROXIMATION FOR ENERGY CALCULATION  43  APPENDIX B:  SOME RELEVANT TRIGONOMETRIC SUMS  44  APPENDIX C:  THE FREE ENERGY OF REGULAR POLYGONAL CONFIGURATIONS  45  APPENDIX D:  SOME MISPRINTS IN HAVELOCK'S PAPER  46  '  LIST OF TABLES TABLE I  Some Relevant Trigonometric Sums  TABLE II The Values of r c  and fi . f o r the Stable Polygons mm  32 32  LIST OF FIGURES Figure 1: The equilibrium free energies of one, two, and three v o r t i c e s  38  (iv)  ACKNOWLEDGMENTS  I thank F.A. Kaempffer f o r h i s s u g g e s t i o n  of my r e s e a r c h  and  f o r h i s s u p e r v i s i o n d u r i n g my p e r i o d o f study a t UBC. I am a l s o  ful  f o r t h e knowledge and u n d e r s t a n d i n g o f p h y s i c s  i n f o r m a l d i s c u s s i o n s we have had, I am i n d e b t e d valuable on  suggestions  the subject  and from h i s  I have gained  from t h e  t o my c o - s t u d e n t s G. Mertz and D. H a l l y f o r t h e our many d i s c u s s i o n s  motion.  I thank the N a t i o n a l Research C o u n c i l o f Canada f o r t h e i r support  y  grate-  lectures.  and i d e a s they forwarded d u r i n g  of v o r t e x  topic  i n t h e form of a P o s t g r a d u a t e  Scholarship.  financial  1 1. INTRODUCTION  1.1  A B r i e f Review of the R e l e v a n t  Literature  I n t e r e s t i n the motion of v o r t i c e s f l u i d was  initiated  such as H e l m h o l t z ,  i n the l a t e  i n a two-dimensional  ideal  19*"* c e n t u r y by m a t h e m a t i c i a n s and  physicists  1  K i r c h o f f , L o r d K e l v i n , Stokes, and Routh. Much of  e a r l y work i s c o v e r e d  i n the c l a s s i c  and Milne-Thomson(1968). While  this  t e x t s on hydrodynamics by Lamb(1932)  c o n s i d e r a b l e e f f o r t was  invested i n finding  s o l u t i o n s of the e q u a t i o n s of motion of v o r t i c e s i n v a r i o u s c o n f i g u r a t i o n s with various boundaries,  i t became apparent  t h a t c e r t a i n c o n f i g u r a t i o n s of  v o r t i c e s moved as a r i g i d body, p r e s e r v i n g t h e i r r e l a t i v e  displacements,  g e n e r a l l y by r o t a t i n g w i t h a c o n s t a n t a n g u l a r v e l o c i t y . The rigidly  rotating  c o n f i g u r a t i o n s against small deformations  Thomson(1883) s t u d i e d the s t a b i l i t y the v e r t i c e s of r e g u l a r p o l y g o n s . He up  to and  i n c l u d i n g N=6  independent  stability became of  these  interest.  of v o r t i c e s of e q u a l s t r e n g t h p l a c e d at found  t h a t such c o n f i g u r a t i o n s a r e  stable  (N b e i n g the number of v o r t i c e s i n the c o n f i g u r a t i o n )  of the s i z e of the p o l y g o n , whereas a l l c o n f i g u r a t i o n s w i t h  are u n s t a b l e , N=7  b e i n g of i n d e t e r m i n a n t  stability  to f i r s t  rotating  f i g u r a t i o n s has been done by Morton(1934,1935) and H a v e l o c k ( 1 9 3 1 ) . on the motion of v o r t i c e s i n two by L i n ( 1 9 4 3 )  Interest  a l s o reviews  A  with a r b i t r a r y boundaries some of the e a r l y  i n the motion of v o r t i c e s was  by Onsager(1949) and allowed p e r s i s t e n t  who  dimensions  Feynman(1955) t h a t l i q u i d  r e k i n d l e d by  helium  con-  treatise has  literature. the c o n j e c t u r e  i n the s u p e r f l u i d  c u r r e n t s of c i r c u l a t o r y f l o w whose c i r c u l a t i o n would  q u a n t i z e d i n u n i t s of h/m,  N>7  order i n p e r t u r -  b a t i o n t h e o r y . F u r t h e r work on the s t a b i l i t y of v a r i o u s r i g i d l y  been produced  of  m b e i n g the mass of the h e l i u m atom. T h i s  state be  curl-free  2 flow with net c i r c u l a t i o n could ing the f l u i d ,  o n l y be s u p p o r t e d by s i n g u l a r v o r t i c e s  such t h a t t h e f l o w would be i r r o t a t i o n a l everywhere except at  the v o r t e x l i n e i t s e l f .  This conjecture  Vinen(1961) who demonstrated t h a t be  was v e r i f i e d  experimentally  t h e c i r c u l a t i o n i n He I I c o u l d  by  indeed  only  i n u n i t s of h/m. The presence of v o r t e x l i n e s c l e a r e d up many m y s t e r i e s  about H e ' l l i n r o t a t i o n , e s p e c i a l l y t h e " r o t a t i o n paradox", d i s c u s s e d W i l k s ( 1 9 7 0 ) i n c h a p t e r s 7 and 8 of h i s i n t r o d u c t o r y The  thread-  discovery  p r e c i p i t a t e d much r e s e a r c h  m o t i o n of v o r t i c e s i n c y l i n d r i c a l  on  book on l i q u i d  by  helium.  r o t a t i n g s u p e r f l u i d s and on t h e  containers.  The t h e o r e t i c a l work o f Hess  (1967) and the e x p e r i m e n t a l work of Reppy, D e p a t i e , and Lane(1961) and Reppy and  Lane(1965) i s r e l e v a n t  bility  t o t h i s work. Tkachenko(1966) determined t h e s t a -  of i n f i n i t e , doubly p e r i o d i c v o r t e x l a t t i c e s . Putterman and Uhlenbeck  (1969) produced the d e f i n i t i v e paper on t h e thermodynamics of r o t a t i n g s u p e r f l u i d s upon which t h i s work i s based. The d e t e c t i o n  o f t h e p r e s e n c e of d i s -  c r e t e q u a n t i z e d v o r t i c e s i n He I I was a c c o m p l i s h e d by Packard and Sanders(1969) and  t h e s p a t i a l p o s i t i o n s of v o r t i c e s i n r o t a t i n g He I I were photographed by  W i l l i a m s and P a c k a r d ( 1 9 7 4 ) , showing no d i s c e r n a b l e and  Sanders(1969) a l s o demonstrated t h a t  velocity  there  pattern.  The work o f Packard  exists a c r i t i c a l  angular  of r o t a t i o n , below which no v o r t i c e s may be p r e s e n t i n the f l u i d a t  equilibrium.  1.2  The Two-Fluid Model o f L i q u i d Helium I t i s not i n t e n d e d to cover the theory of l i q u i d h e l i u m i n t h i s  work, but some of the r e s u l t s and b a s i c  concepts of the t w o - f l u i d model due  to Landau(1941) w i l l be s t a t e d . The r e a d e r i s a g a i n r e f e r e d f o r a more complete  to Wilks(1970)  introduction.  In the t w o - f l u i d model of He I I i t i s p o s t u l a t e d s u p e r f l u i d may be d e s c r i b e d  that  the a c t u a l  by two n o n - i n t e r a c t i n g , i n t e r p e n e t r a t i n g  f l u i d s , so  t h a t at each p o i n t and  also a  of the  there  "superfluid"  fluid  i s p=p  n  +p  s  equilibrium ratio-P /P and  with density  , and  the  a  viscosity,  the  and  velocity V -n  total  density  i s i=p V +p V . n-n s-s  lambda t r a n s i t i o n temperature  to be  component may  e n t r o p y of the f l u i d  the  not  The  at (above  experience  z e r o , whereas the normal component resides  s u p e r f l u i d flow i s also postulated  t h i s c o n d i t i o n b e i n g s a t i s f i e d by  i n the  t o have no  normal  turbulence,  statement curlV" =0 ( t h a t i s , the s  flow  "irrotational") . The  the f l o w He  , the  a s u p e r f l u i d ) . The..superfluid  does e x p e r i e n c e v i s c o s i t y and  is  n  v e l o c i t y V^-. The  t o t a l momentum d e n s i t y  i t s entropy i s postulated  component a l o n e . The  p  f u n c t i o n of temperature a l o n e , b e i n g u n i t y  d r o p p i n g to z e r o as T->-T  which h e l i u m i s not  and  J  is  g  T=0,  i s a "normal" f l u i d w i t h d e n s i t y  r e s u l t s which are needed i n t h i s work concern the  of b o t h normal and  I I to be  the  fluid  tum  c o n s t a n t . The (a) The  conditions  t o t a l mass, t o t a l energy, and on the v e l o c i t y f i e l d s  normal f l u i d  container,  Putterman and Uhlenbeck use  rotates  as a r i g i d  reference  of  entropy  of  t o t a l a n g u l a r momen-  are:  body w i t h the a n g u l a r  velo-  i.e.  V = -n - (b) The  on  a variational  e q u i l i b r i u m c o n d i t i o n s , maximizing the  while maintaining  c i t y of the  s u p e r f l u i d components f o r a r o t a t i n g v e s s e l  i n thermal e q u i l i b r i u m .  p r i n c i p l e to determine the  conditions  fixr  (1-1)  s u p e r f l u i d v e l o c i t y f i e l d i s s t a t i o n a r y i n the  i n which the normal f l u i d  frame of  i s at r e s t , i . e .  3V  —g , (c) The  -z— = 0 ( i n the r o t a t i n g frame) o t superfluid velocity f i e l d is irrotational, i.e.  (1-2)  VxV = 0 -s (d) With the the  a p p r o x i m a t i o n t h a t .p  (1-3) and  c o n d i t i o n f o r thermodynamic e q u i l i b r i u m of the  are fluid  everywhere c o n s t a n t , i s that  the  "free  energy" of t h e s u p e r f l u i d  component be a t an a b s o l u t e minimum, i . e .  F = E-fi.L = minimum  (1-4)  In which E i s the k i n e t i c energy o f t h e s u p e r f l u i d  and L i s t h e a n g u l a r  momentum o f t h e s u p e r f l u i d . T h i s statement i s e q u i v a l e n t t o the statement t h a t , i n the r o t a t i n g frame, t h e energy o f t h e s u p e r f l u i d i s a t an a b s o l u t e minimum. Conditions  (b) and (c) a r e s a t i s f i e d by s i n g u l a r v o r t i c e s o f t h e  c l a s s i c a l type b e i n g p r e s e n t i n the s u p e r f l u i d  and r o t a t i n g  i n rigid  c o n f i g u r a t i o n s w i t h a n g u l a r v e l o c i t y Q. I f t h e v o r t e x s t r e n g t h s a r e not q u a n t i z e d , but t a k e any v a l u e i n a c o n t i n u o u s range, c o n d i t i o n  (d) would  be s a t i s f i e d by an i n f i n i t e number o f v o r t i c e s of i n f i n i t e s i m a l s t r e n g t h , a p p r o a c h i n g a s t a t e o f s o l i d body r o t a t i o n . The problem of s a t i s f y i n g tion  (d) becomes i n t e r e s t i n g when i t i s r e c o g n i z e d t h a t t h e v o r t e x s t r e n g t h s  a r e i n r e a l i t y q u a n t i z e d , and t h a t  1.3  condi-  t h e r e i s a minimum .non-zero  strength.  Some M a t h e m a t i c a l P r e l i m i n a r i e s The experiments on r o t a t i n g He I I g e n e r a l l y c o n t a i n t h e l i q u i d  i n c y l i n d r i c a l v e s s e l s whose diameter i s much s m a l l e r than t h e i r l e n g t h , so that  the e f f e c t s o f the v a p o u r - l i q u i d  boundary  at the bottom  i n t e r f a c e at t h e top and t h e s o l i d  are i g n o r a b l e . Furthermore, a t h e o r e t i c a l  assumption  i s made t h a t the v o r t e x l i n e s a r e r e c t i l i n e a r and p a r a l l e l i n t h e f l u i d , not s p a g h e t t i - l i k e t a n g l e , so t h a t t h e f l u i d has t r a n s l a t i o n a l symmetry a l o n g the a x i s of r o t a t i o n of the c o n t a i n e r . A c r o s s - s e c t i o n of the f l u i d d i c u l a r to the a x i s may be regarded as t y p i c a l o f the f l u i d the hydrodynamics  perpen-  as a whole, and  of t h e f l u i d may be reduced to a two-dimensional  problem,  i n w h i c h v o r t e x l i n e s a r e s i n g u l a r p o i n t s i n the two-dimensional v e l o c i t y field.  The a l l o w s the give  the  c o n d i t i o n t h a t the  superfluid velocity field  components of  = | i  :  1  v  =  2  | i  3xj In a d d i t i o n ,  the  a p p r o x i m a t i o n P = c o n s t a n t i s tantamount s  which i s s a t i s f i e d  to  = 0  i n two  ,  dimensions by  the  (1-6)  i n t r o d u c t i o n o f a stream  function  writing l  v  Equating  dip  A  = 3x2 ^  (1-6)  v  and  (1-7)  gives  imaginary p a r t s  2  = "  ..  3i!i 9xj  (1-7)  e q u a t i o n s which may  Riemann e q u a t i o n s g o v e r n i n g the  i s the  d-5)  5x2  V.V  and  irrotational  i n t r o d u c t i o n of a v e l o c i t y p o t e n t i a l 4>,the d e r i v a t i v e s of which  V!  ij» and  is  be viewed as  conjugate functions  <f> and  the  Cauchy-  which are  the  real  of a complex v e l o c i t y p o t e n t i a l 0 whose complex d e r i v a t i v e  complex v e l o c i t y f i e l d  d<5>  —  w:  = V!-iv  2  = w(z)  (1-8)  C o n s e q u e n t l y , the d i f f e r e n t i a l e q u a t i o n g o v e r n i n g  the  p o t e n t i a l $ i s Laplace's  equation V$ 2  = 0  w i t h the boundary c o n d i t i o n the f l o w , i . e . ^ c o n s t a n t be  on  (1-9) t h a t any  s o l i d boundary must be  the boundary. A f l u i d  expressed i n such a manner i s c a l l e d an i d e a l The  whose v e l o c i t y f i e l d fluid.  hydrodynamics of a t w o - d i m e n s i o n a l , i d e a l f l u i d may  approached u s i n g functions.  a streamline  This  the  t e c h n i q u e s of  i s the  starting  complex v a r i a b l e t h e o r y and  point  now  be  analytic  of s e c t i o n 2 on v o r t e x m o t i o n .  of may  1.4  The Object o f t h i s Work The  q u e s t i o n s which must be answered a r e "what c o n f i g u r a t i o n s of  v o r t e x l i n e s i n s i d e a r o t a t i n g c y l i n d e r r o t a t e r i g i d l y w i t h the c y l i n d e r and are  stable against  small perturbations,  s t a t e of thermodynamic e q u i l i b r i u m The  first  question  forming r e g u l a r  and which o f t h e s e r e p r e s e n t  at a given  t i o n s w i t h N<7 a r e s t a b l e i f t h e r a d i u s of t h e polygon i s s m a l l e r that  angular v e l o c i t y of r o t a t i o n ? "  i s answered f o r c o n f i g u r a t i o n s  polygons: c o n f i g u r a t i o n s  of e q u a l s t r e n g t h  w i t h N>7 a r e u n s t a b l e , of the c i r c l e  above which t h e c o n f i g u r a t i o n  vortices  and  configura-  formed by the v e r t i c e s  than a C e r t a i n f r a c t i o n o f t h e c y l i n d e r  f r a c t i o n depending on N. T h i s  the  radius,  t r a n s l a t e s i n t o a minimum a n g u l a r v e l o c i t y  i s s t a b l e . The s t a t e of thermodynamic  equili-  brium as a f u n c t i o n of ft i s determined f o r low v a l u e s of ft, and the c a l c u l a t i o n of t h e c r i t i c a l a n g u l a r v e l o c i t i e s f o r t h e c r e a t i o n of one, two, and t h r e e v o r t i c e s i s performed. In s e c t i o n 2 the e q u a t i o n s o f m o t i o n of v o r t i c e s i n an i n f i n i t e fluid  and i n a f l u i d  discussed. culated  contained  within  a c i r c u l a r boundary a r e d e r i v e d and  The k i n e t i c energy and a n g u l a r momentum o f a v o r t e x f l u i d  are c a l -  i n s e c t i o n 3, showing t h e i r r e l a t i o n t o the c o n s t a n t s o f t h e motion  of v o r t e x systems. In s e c t i o n 4 a L a g r a n g i a n f o r m a l i s m i s i n t r o d u c e d , the  c o n s t a n t s of t h e motion t o be d e r i v e d  The  e q u a t i o n s o f motion of v o r t i c e s p e r t u r b e d s l i g h t l y  configurations configurations.  are derived  from symmetries of the L a g r a n g i a n .  i n s e c t i o n 5 and s o l v e d  from r i g i d l y  f o r the r e g u l a r  rotating polygonal  In s e c t i o n 6 the s t a t e of thermodynamic e q u i l i b r i u m of He I I  at low v a l u e s of ft i s determined. S e c t i o n work.  allowing  7 contains  the c o n c l u s i o n  of t h i s  7 Various portions  of the m a t e r i a l  contained  i n t h i s work can be  found i n the l i t e r a t u r e c i t e d . The o r i g i n a l c o n t r i b u t i o n s  of t h e a u t h o r  include  the method of c a l c u l a t i n g the k i n e t i c energy and the a n g u l a r momentum i n sections of  3.1 and 3.2, and t h e L a g r a n g i a n t e c h n i q u e i n s e c t i o n 4.2. The method  c a l c u l a t i o n of t h e spectrum of c r i t i c a l  i s due t o Putterman(1974), but the a c t u a l the  literature.  angular v e l o c i t i e s i n s e c t i o n  6.2  c a l c u l a t i o n s a r e not a v a i l a b l e i n  8  2. MOTION OF VORTICES IN A TWO-DIMENSIONAL IDEAL FLUID  2.1  Equations  o f M o t i o n and C o n s e r v a t i o n Laws i n an I n f i n i t e  The v e l o c i t y and  incompressible)  field  of a two-dimensional  can be expressed  ideal fluid  Fluid (inviscid  as the d e r i v a t i v e of a complex,  scalar  potential function *(z)  = <|>(z)+iiKz).  (2-1)  tj) and IJJ b e i n g r e a l - v a l u e d f u n c t i o n s . The v e l o c i t y w(z) Of i n t e r e s t  = v ( z ) - i v ( z ) =^|^1 dz 1  i s obtained  .  from (2-2)  i s a v e l o c i t y p o t e n t i a l o f t h e form  $(z) = which g i v e s r i s e  2  field  ln(z)  t o the v e l o c i t y w(z)  (2-3)  field  =  (2-4)  which, i n p o l a r c o o r d i n a t e s , i s v = r  T h i s i s the v e l o c i t y  0,  field  v = Q  J  .  (2-5)  o f a v o r t e x s i t u a t e d a t z=0 i n an i n f i n i t e  f l u i d . The f l o w i s i r r o t a t i o n a l ,  s i n c e c u r l v = 0 everywhere except  p o l e a t the o r i g i n . The c i r c u l a t i o n o f the f l u i d  about any c l o s e d  at the  contour  e n c l o s i n g the p o l e i s T  =  o  hence y i s c h a r a c t e r i s t i c vortex.  y.d£  =  2TTY  of t h e v o r t e x and i s c a l l e d  (2-6)  the s t r e n g t h of t h e  9 Of more i n t e r e s t  i s a c o n f i g u r a t i o n o f N v o r t i c e s s i t u a t e d a t time  t at p o s i t i o n s z ^ { t ) (k=l,2,...N) w i t h s t r e n g t h s y. •, whose v e l o c i t y  potential  appears as the' sum '  ( z )  and whose c o r r e s p o n d i n g  =  i  E k  l n  Y  ( z  instantaneous  k "  z k  )  (2-7)  velocity  field is  k  The v o r t i c e s themselves move i n t h e . f l u i d , due.the presence v o r t i c e s . The u s u a l p r e s c r i p t i o n f o r o b t a i n i n g t h e e q u a t i o n s the v o r t i c e s , a v a i l a b l e i n any of t h e c l a s s i c a s s i g n to t h e v o r t e x a t p o s i t i o n of  the v e l o c i t y  of the other of motion of  t e x t s on hydrodynamics, i s t o  the v e l o c i t y o b t a i n e d by a s u p e r p o s i t i o n  c o n t r i b u t i o n s o f a l l the o t h e r v o r t i c e s , e v a l u a t e d a t z=z . n  That i s , • 'n dt d  y' k f z (t)-z, (t) k n k  ( t )  (2-9)  Y  1  where Z' i s the sum over a l l k=l,2,...N e x c l u d i n g k=n. E q u a t i o n sents N ordinary d i f f e r e n t i a l  equations,  a r e the v o r t e x t r a j e c t o r i e s . N i n i t i a l  of f i r s t  (2-9) r e p r e -  o r d e r i n t , whose s o l u t i o n s  c o n d i t i o n s are r e q u i r e d , the i n i t i a l  p o s i t i o n s of t h e v o r t i c e s . U n l i k e the mechanics of p o i n t p a r t i c l e s , t i a l v e l o c i t i e s a r e n o t needed, s i n c e t h e e q u a t i o n s a r e o f f i r s t The tial  i n i t i a l v e l o c i t i e s o f t h e v o r t i c e s a r e u n i q u e l y determined p o s i t i o n s by the e q u a t i o n s The  equations  of motion  order i n t .  by t h e i r  ini-  (2-9) .  o f motion can be put i n t o c a n o n i c a l form through the  i n t r o d u c t i o n of t h e v o r t e x stream V  the i n i -  -  function  ^ l .n k  V  k  ^  W  (  2  -  1  a l o n g w i t h the c a n o n i c a l c o o r d i n a t e s and momenta  V  V  p  k Vk =  ( 2 _ 1 1 )  0  )  10 It i s e a s i l y v e r i f i e d  that equations  (2-9) and t h e i r  complex  conjugates  may be w r i t t e n An . dq. k = dt  1  (2-12a)  o 9Pi  n  . dp, 84< k = o dt  (2-12b)  I  suggesting that  be i d e n t i f i e d  as t h e H a m i l t o n i a n o f t h e v o r t e x  T h i s c l a i m w i l l be s u b s t a n t i a t e d i n s e c t i o n 4, which d e a l s w i t h and  c o n s e r v a t i o n laws of the v o r t e x system from  a Lagrangian  system.  symmetries  viewpoint.  By a s i m p l e a p p l i c a t i o n of t h e c h a i n r u l e o f d i f f e r e n t i a t i o n and the c a n o n i c a l e q u a t i o n s of motion, t h e time r a t e o f change o f any f u n c t i o n A(q^,p^,t)  may be w r i t t e n  1  dt  1 A  'V  +  3A X  9A  i n which  L «k 3  i s the Poisson Bracket cit  (2-13)  9t o  3A  k  3 p  3  ^k  (P.B.) of A and ¥  time dependence a r e conserved  o 9  (2-14)  V  . F u n c t i o n s A which have no e x p l i -  q u a n t i t i e s of t h e motion i f  {A,¥ } = 0  (2-15)  o  The  f o l l o w i n g conserved  q u a n t i t i e s o f v o r t e x motion can be found:  C e n t r e of C i r c u l a t i o n  Z  Moment of C i r c u l a t i o n  0  Vortex  1 = — o y o  Stream F u n c t i o n f  r )y z i k k o k  (2-16a) k  = TY, ~ k k  (2-16b)  = -Y Y'v n k  Y , I n z -z n k  (2-16c)  A n g u l a r Moment of C i r c u l a t i o n w  =  o As an independent  1 v "T7 ZYI ( 2i , k k L  Z  I  Z. - Z . Z  k k  )  k k  check, these can be shown t o be c o n s t a n t by d i r e c t  s u b s t i t u t i o n of t h e e q u a t i o n s of motion.  (2-16d)  11 2.2  The V e l o c i t y P o t e n t i a l of a V o r t e x F l u i d  I n s i d e the U n i t  Circle  The p r e v i o u s s e c t i o n t r e a t e d v o r t i c e s i n an i n f i n i t e realistic  f l u i d . A more  s i t u a t i o n i s to have the v o r t i c e s c o n t a i n e d i n a bounded f l u i d ,  and  of p a r t i c u l a r i n t e r e s t i n t h i s work i s the case of a c i r c u l a r boundary, s i n c e many experiments  on l i q u i d He use  c y l i n d r i c a l v e s s e l s . For an i d e a l f l u i d ,  v e l o c i t y p o t e n t i a l must be the s o l u t i o n of the d i f f e r e n t i a l  the  equation  V $(z) = 0  (2-17)  2  w i t h the c o n d i t i o n a t the boundary t h a t t h e r e be no normal component of the velocity  f i e l d w ( z ) . T h i s i s e q u i v a l e n t to r e q u i r i n g  s t r e a m l i n e of the f l o w , i . e .  ijj(z)  = 0 a l o n g the boundary.  of a c o n s t a n t t o the p o t e n t i a l does not a f f e c t t a n t which p a r a m e t r i z e s  t h a t the boundary be a  the v e l o c i t y  the boundary s t r e a m l i n e may  then be expressed $(z) =$(z)  The  and  addition  the  cons-  zero.  c l o s e d . The  boun-  as  a l o n g the boundary.  (2-18)  problem of f i n d i n g the a p p r o p r i a t e v e l o c i t y p o t e n t i a l f o r a  vortex f l u i d with a r b i t r a r y boundaries but  field,  be chosen t o be  I t has a l s o been assumed t h a t the boundary i s c o n t i n u o u s dary c o n d i t i o n may  S i n c e the  f o r simple b o u n d a r i e s  has been approached by  Lin(1943),  such as the c i r c u l a r boundary the t e c h n i q u e  conformal mapping i s more c o n v e n i e n t . The boundary v a l u e problem  of  (2-17) w i t h  (2-18) can be mapped i n t o a r e g i o n f o r which the s o l u t i o n i s a l r e a d y known o r i s e a s i e r t o s o l v e ; the i n v e r s e mapping g i v e s the c o r r e c t  s o l u t i o n i n the  o r i g i n a l r e g i o n . In t h i s c a s e , the r e g i o n i n s i d e the u n i t  circle  mapped onto  can  be  the upper h a l f p l a n e u s i n g the t r a n s f o r m a t i o n g(z) = i  'l  -  (2-19)  1 + zj  The  same t r a n s f o r m a t i o n maps p o l e s i n the z-plane  i n t o p o l e s i n the  and  the c i r c u l a r boundary i s mapped i n t o the r e a l a x i s . The  g-plane,  s o l u t i o n to  i n the upper h a l f p l a n e i s well-known, c f . Milne-Thomson(1968), and i s  (2-17)  = I lY ln< -g ) - J lY ln(g-I ) k k  *(g)  Except at the p o l e s , it  gives  8  k  to show t h a t , on  '(2-20)  k  the r e a l a x i s  s a t i s f y i n g the c o n d i t i o n  the r e q u i r e d  k  t h i s i s a s o l u t i o n to the p o t e n t i a l e q u a t i o n  i s straightforward  totally real,  k  *(z) = j where a r e a l c o n s t a n t  has  \)  lY ln(l-z k  k  k  been dropped. T h i s  boundary c o n d i t i o n , which can be  transformation  (2-19)  circle:  ln(z-z ) - ±  £Y k  and  (g=g), the s o l u t i o n i s  (2-18). U s i n g the  s o l u t i o n i n s i d e the u n i t  (2-17),  (2-21)  solution also s a t i s f i e s  checked u s i n g  the f a c t  the  t h a t z=l/z on  the  boundary. I t i s i n t e r e s t i n g to note t h a t the c o n f o r m a l mapping method i s not u s e f u l i n d e t e r m i n i n g the p o t e n t i a l o u t s i d e d e r i v i n g the c i r c l e transformation  jacobian not  2.3  theorem of Milne-Thomson (page 157  (2-19) maps the o u t s i d e  h a l f g-plane, but  of the u n i t c i r c l e  at t h a t  vanishes there  and  Conservation  by  tex i n an at z=z  n  i n t o the  lower  ( g = - i ) , so  the  i n v e r s e mapping i s  Laws I n s i d e the U n i t  Once the v e l o c i t y p o t e n t i a l f o r a v o r t e x  obtained  The  point.  E q u a t i o n s of M o t i o n and  mined f o r any  the  that i s , i n  of r e f e r e n c e ) .  the d e r i v a t i v e (dg/dz) v a n i s h e s a t z-*°  of the t r a n s f o r m a t i o n  defined  the u n i t c i r c l e ,  region,  the e q u a t i o n of motion of the v o r t e x  s u b t r a c t i n g from the p o t e n t i a l the  infinite fluid,  to the v o r t e x  dt  f l u i d has  and  assigning  been  c o n t r i b u t i o n of the  is  vor-  the v e l o c i t y of the r e s u l t i n g f i e l d J  dz  deter-  at z , say, n  at z . T h i s i s expressed m a t h e m a t i c a l l y n n  Circle  '(z) - -ry l n ( z - z ) i n n z=z  as (2-22)  13 U s i n g the v e l o c i t y p o t e n t i a l from i n s i d e the u n i t  T h i s may  circle  (2-21), the e q u a t i o n s of motion  are  be i n t e r p r e t e d as each v o r t e x h a v i n g i n d u c e d an image v o r t e x of  p o s i t e s i g n at the r e c i p r o c a l p o i n t . Note t h a t the second term k=n,  c o n t a i n s the  1  the e q u a t i o n s of motion  c a l form by the i n t r o d u c t i o n of the v o r t e x stream T  o e q u a t i o n s of motion  canonical equations  = ^"I^ 'I'Y Y , l n z -z. n'k n k  (2-16) remain  (2-23) and  (2-12a) and  7h  Yi  l  ^f'n'k n k  can be d e t e r m i n e d .  4.)  canoni-  function  (2-12b) . U s i n g r e l a t i o n ,  c o n s t a n t s of the motion  see s e c t i o n  can be put i n t o  n  1-z  z. n k  (2-24)  t h e i r complex c o n j u g a t e s f o l l o w from  ( T h i s i s because the e q u a t i o n s of motion l a t i o n i n space;  +  n k  ed q u a n t i t i e s of the motion in  sum  op-  r e p r e s e n t i n g the i n t e r a c t i o n of the n*"^ v o r t e x w i t h i t s image. As i n s e c t i o n 2.1,  The  of v o r t i c e s  (2-15), the  conserv-  A l l of the q u a n t i t i e s d e f i n e d  e x c l u d i n g the c e n t r e of are no  the  circulation.  l o n g e r i n v a r i a n t under  trans-  14  3. DYNAMICAL QUANTITIES OF A VORTEX FLUID  The will  kinetic be The  energy.and a n g u l a r momentum of a t w o - d i m e n s i o n a l v o r t e x  calculated  f o r the case of the f l u i d  d e n s i t y i s assumed c o n s t a n t  a b l e assumption f o r the s u p e r f l u i d dropping and  rapidly  tum  3.1  stream  be  He,  reason-  this density cores  function, ¥  , and q  and  ener-  t h a t the a n g u l a r momenso t h e s e  dynamical  a r e f u n c t i o n s o n l y of the c o n f i g u r a t i o n o f  vortices  fluid.  Kinetic  Energy  The  kinetic  energy of an i d e a l f l u i d  *3P  (Vijj) dxdy R  HP  V. (ipVijOdxdy R  of c o n s t a n t  density p i s  2  = hp Hm.fC)c\i  (3-D  !c  i n which ij; i s the stream flow, and  f u n c t i o n of the f l o w  C i s the contour  i d e n t i t y V-(AyB) second l i n e , The  circle.  shown t h a t the k i n e t i c  t o the moment of c i r c u l a t i o n , 0 ,  q u a n t i t i e s of the f l u i d i n the  the f l u i d , which i s a  component of l i q u i d  Putterman,1974). I t w i l l  to the v o r t e x  i s related  throughout  i n s i d e the u n i t  t o zero o n l y w i t h i n m i c r o s c o p i c d i s t a n c e s from v o r t e x  boundaries.(see  gy i s r e l a t e d  contained  fluid  contour  e n c l o s i n g R,  ( I m $ ( z ) ) , R i s the r e g i o n of  w i t h outward normal fi. The  = yA.yB + Av 2 B a l o n g w i t h  the f a c t  that  V IJJ=0 2  gives  w h i l e a p p l i c a t i o n of Green's Theorem g i v e s the f i n a l C i s chosen to be  e<<l s u r r o u n d i n g  the u n i t  the p o l e s a t z=z  circle,  and  ; the s t r a i g h t  vector  small c i r c l e s  segments j o i n i n g  the  line. of r a d i u s the  outer  K.  and  inner contours  are ignorable, since t h e i r  c o n t r i b u t i o n t o the  integral  c a n c e l s . Small d i s k s of r a d i u s £ have been e x c l u d e d  from the r e g i o n of f l o w .  i n c a l c u l a t i n g t h e i n t e g r a l , but  on p h y s i c a l grounds s i n c e  r e a l v o r t i c e s have f i n i t e one  would be u n j u s t i f i e d  the f l u i d ,  t h i s i s reasonable  c o r e s c o n t a i n i n g no  this  fact,  i n s h r i n k i n g e to below an i n t e r - a t o m i c d i s t a n c e i n  the d i s t a n c e at which the use  q u e s t i o n a b l e . The  l i q u i d . Even w i t h o u t  approximation  of c l a s s i c a l hydrodynamics becomes  used here  i s a c c e p t a b l e as l o n g as the v o r -  t i c e s do not approach each o t h e r o r t h e boundary a t d i s t a n c e s of t h e o r d e r s i n c e then the energy of d e f o r m a t i o n account.  The  parameter e  c a l angular v e l o c i t y i n l i q u i d He The circle,  becomes important  i n the c a l c u l a t i o n of the  f o r the appearance of a s i n g l e q u a n t i z e d v o r t e x  critiline  j u d i c i o u s c h o i c e of $ ( z ) i n (2-21), i n which ^(z)=0 on the the i n t e g r a l E = -%pl m  which c o n s i s t s of the sum  (3-1)  <n|;(V^.n)dS,  (3-2)  m  of a l l the c o n t o u r  ( i . e . C_ surrounds m  z-z  mk so t h a t ijj(z) may  r  k  vortex  of t h e s e  integrals  to d e f i n e the q u a n t i t i e s  =  Z  k (3-3)  z -z m k  mk  z -1/z, m k  be expanded i n powers of z about z . The m i n Appendix A.  s m a l l , o n l y l n e and  sion, obtaining  i n t e g r a l s around the  z j ) . In o r d e r t o e v a l u a t e one m  m  c a l c u l a t i o n are found  unit  to  about the v o r t e x at z , say, i t i s c o n v e n i e n t  very  into  I I , i n s e c t i o n 6.  reduces  positions  of the v o r t e x c o r e s must be t a k e n  e,  zero  order  d e t a i l s of  this  In a n t i c i p a t i o n of a l l o w i n g e to become terms i n e are r e t a i n e d i n t h i s  expan-  * = -Y ln m  £  -]'Y lnr k  S i m i l a r l y , i n the v i c i n i t y ViJj.ndJl = 1^- d e Combining  (3-4)  j ^ l n r ^ + j ^ l n ^ + 0(e)  +  m k  of z , m  ed6 = - ( y + 0 ( e ) ) d 8 m  (3-5)  these i n ( 3 - 1 ) , and a g a i n r e t a i n i n g o n l y z e r o o r d e r and l n e terms, 2TT *3P  =  ijjd6  ZY  m m  (3-6)  P T T { - J y y Y-, I n r , + J Jy y, l n r ' ^ ^ m k mk f ' m' k mk m k m k 1  1  L  which a c q u i r e s a f a m i l i a r form when complex  E = pu{- I  I ' Y Y l n z -z m  k  m k =  PTT(¥  m  k  set  2  n o t a t i o n i s adopted:  + T Ty y. m k  I n 1-z z, m k  - lY lne} m 2  L  J  2  (3-7) appears as t h e sum o f what has been  identi  as t h e H a m i l t o n i a n o f the v o r t e x system and a term which, f o r a g i v e n of v o r t i c e s  first the  - yy ln£:l m ' m  L  1  - Ey lne)  Thus the energy o f a v o r t e x f l u i d fied  + T jy y . l n r , f'm'k k m k  (y's c o n s t a n t ) , does not depend  on t h e c o n f i g u r a t i o n . The  sum i n (3-7) r e p r e s e n t s t h e energy o f i n t e r a c t i o n between t h e v o r t i c e s  second sum i s t h e energy o f i n t e r a c t i o n between t h e v o r t i c e s and a l l t h e  images, w h i l e t h e t h i r d  sum may be c o n s i d e r e d as t h e s e l f - e n e r g y o f t h e  v o r t i c e s . T h i s term d i v e r g e s l o g a r i t h m i c a l l y as e-»-0, b u t an argument has a l ready been g i v e n t h a t t h i s parameter must remain f i n i t e , a l t h o u g h s m a l l .  3.2  Angular Momentum The a n g u l a r momentum o f a two-dimensional i d e a l f l u i d i s L = p  (rxV)r.drde = p R  U s i n g the v e l o c i t y  Re{izw}r.drd9  (3-8)  . R f i e l d f o r v o r t i c e s i n s i d e the unit  ReUzw} = ly  circle,  (Re{z/(z-z )} + Re{ zz / (1-zz )})  the integrand (3-9)  17 The  first  the p o i n t s  s e t o f terms c o n t a i n s i n g u l a r i t i e s i n t h e r e g i o n of i n t e g r a t i o n a t z = z k  >  s  they must be expanded i n d i f f e r e n t  o  i n t h e two r e g i o n s  convergent  series  |z|'<|z^| and |z|>|z^|:  Re{z/(z-z )} = - I n=! k  )  z < z.  cos n(e-e,)  z > z k  cos n(e-e  K.  CO  = I  k  n=0  The  remaining  (3-10)  1  terms do n o t c o n t a i n p o l e s i n the r e g i o n of i n t e g r a t i o n  |z|<l,  so one s e r i e s s u f f i c e s , namely Re{zz / ( 1 - z z )} =  The  I ( r r ) c o s n(0-6 ) n=l n  (3-11)  |z|<l  o n l y terms i n these s e r i e s t h a t c o n t r i b u t e t o the a n g u l a r momentum  a r e those f o r which n=0, due t o the f a c t 2TT  2TT  cos n(0-6 )d6 = { ~. rt k 0  that n=0 n^O  (3-12)  so the a n g u l a r momentum i n t e g r a l becomes rl  L = p£y k  k  2TT r d r = p u ^ U - r ^ ) ''r k k  = TT(Y -0 ) o o  (3-13)  P  The  a n g u l a r momentum i s the sum of a c o n f i g u r a t i o n - f r e e term .(total  circu-  l a t i o n ) and t h e n e g a t i v e of the moment of c i r c u l a t i o n . Note t h a t the p o l e s o f the v e l o c i t y f i e l d  do n o t p r e s e n t  3.3  t h e Energy  Renormalizing  Expression stream  i n the f i n a l  result.  (3-7) f o r the k i n e t i c energy d i f f e r s from the v o r t e x  f u n c t i o n (2-24) o n l y by what has been i d e n t i f i e d as the s e l f - e n e r g y  contributions stream  any d i v e r g e n c e s  -lysine  from the v o r t i c e s . In cases f o r which the v o r t e x  f u n c t i o n s e r v e s as the h a m i l t o n i a n of the system, the y 's a r e K.  18 regarded  as b e i n g f i x e d , hence the energy may  be  " r e n o r m a l i z e d " by  t r a c t i n g the s e l f - e n e r g i e s , the energy and v o r t e x stream identical.  Since Ey  2  sub-  f u n c t i o n becoming  i s a p o s i t i v e d e f i n i t e q u a n t i t y , i t may  not v a n i s h by  a  m s u i t a b l e c h o i c e of y's.  F o r use i n a p p l i c a t i o n s to l i q u i d h e l i u m p h y s i c s ,  the c o n t r i b u t i o n of the s e l f - e n e r g i e s may  not be i g n o r e d , s i n c e Putterman's  c r i t e r i o n f o r the thermodynamic e q u i l i b r i u m of s u p e r f l u i d s (Putterman Uhlenbeck(1969)) r e q u i r e s the comparison of f r e e e n e r g i e s of v o r t e x c o n f i g u r a t i o n s w i t h u n e q u a l t o t a l c i r c u l a t i o n s . Even f o r q u a n t i z e d w i t h each y, due  s t r e n g t h , the £y  an i n t e g r a l m u l t i p l e of some u n i t  2  line systems  may  change  to a change i n t o t a l c i r c u l a t i o n , or a r e - d i s t r i b u t i o n of v o r t e x  g i v i n g the same t o t a l c i r c u l a t i o n . Thus the r e n o r m a l i z e d only i n d e t e r m i n i n g  strengths  energy i s u s e f u l  the r e l a t i v e s t a b i l i t y of c o n f i g u r a t i o n s h a v i n g  t o t a l c i r c u l a t i o n and  and  the same  d i s t r i b u t i o n of v o r t e x s t r e n g t h s , t h a t i s , f o r those  which d i f f e r o n l y i n t h e i r g e o m e t r i c a l c o n f i g u r a t i o n . In t h i s work ( s e c t i o n only q u a n t i z e d v o r t i c e s of u n i t s t r e n g t h w i l l be c o n s i d e r e d , so t h a t y <*N o Z y N , where N i s the t o t a l number of v o r t i c e s , ra  6)  and  2o:  3.4  In the L i m i t of an I n f i n i t e The  may  energy and  a n g u l a r momentum of an i n f i n i t e v o r t e x  fluid  be o b t a i n e d by r e p e a t i n g the p r e v i o u s c a l c u l a t i o n s , r e p l a c i n g the bound-  ary at  |z|=l by a boundary at  necessary sult  kinetic  Fluid  to  repeat  z^ by  r e v e a l i n g the b e h a v i o u r  z^/a,  a n  Y'Y  l  YI  1  e by  e/a  i n expressions  (3-7)  and  (3-13)  n  l  z  ~, z  I ~ IVlne +  y lna 2  + 0(a  )  (3-14a)  m  = -Ty |z I m m m u  d  re-  a t l a r g e v a l u e s of a to be  m k  L/pu  t a k i n g the l i m i t as a->-°°. I t i s not  the c a l c u l a t i o n , s i n c e the a p p r o p r i a t e e x p r e s s i o n s  from r e p l a c i n g  E/p-rr = -T  |z|=a, and  1  2  + Y a o  2  (3-14b)  19 Unless  the t o t a l c i r c u l a t i o n i s z e r o ,  entum of an i n f i n i t e v o r t e x f l u i d and and  the k i n e t i c  contain i n f i n i t e  energy  and a n g u l a r mom-  terms which behave as l n a  2 ' a , r e s p e c t i v e l y . These d i v e r g i n g terms appear i n t h e form o f t h e energy a n g u l a r momentum of a s i n g l e v o r t e x o f s t r e n g t h y , due t o t h e n e t c i r o  c u l a t i o n of t h e f l o w a t l a r g e d i s t a n c e s . I f t h e c o n d i t i o n Y the i n f i n i t e fields case Y  0 i s imposed,  terms v a n i s h , c o r r e s p o n d i n g t o t h e c a n c e l l a t i o n of t h e v e l o c i t y  of t h e v o r t i c e s a t l a r g e d i s t a n c e s . The p r e s e n c e Q  = o  ^0 i s tantamount t o t h e statement  duced i n t o an i n f i n i t e v o r t e x f l u i d  of these terms i n t h e  t h a t no n e t c i r c u l a t i o n may be i n t r o -  due t o t h e i n f i n i t e  energy  and a n g u l a r  momentum t h a t would be r e q u i r e d t o do s o . Apart tween t h e energy is  from  these i n f i n i t e  terms, t h e o n l y d i f f e r e n c e r e m a i n i n g be-  and the v o r t e x stream  f u n c t i o n (2-10) f o r an i n f i n i t e  t h e c o n t r i b u t i o n from t h e s e l f - e n e r g i e s of t h e v o r t i c e s . A g a i n ,  may be " r e n o r m a l i z e d "  , although  fluid  t h e energy  the procedure  seems u n c e r t a i n i n t h i s  case,  s i n c e t h e q u e s t i o n o f r e n o r m a l i z i n g t h e energy  t o exclude the i n f i n i t e  terms  i s r a i s e d . T h i s q u e s t i o n , and whether i t makes sense q u a n t i t i e s s e r i o u s l y , w i l l be l e f t  open.  t o d i s c u s s such  infinite  20 4. THE LAGRANGIAN MECHANICS OF VORTEX SYSTEMS  4.1  The L a g r a n g i a n  Formalism  I t i s p o s s i b l e t o view the e q u a t i o n s o f motion as a r i s i n g from a p r i n c i p l e o f l e a s t  of point o b j e c t s  a c t i o n , i n cases f o r which  f u n c t i o n may be d e f i n e d f o r the system.  a Lagrangian  Once a L a g r a n g i a n has been d e f i n e d ,  the symmetry p r o p e r t i e s of t h e system and t h e c o n s e r v a t i o n laws o f the motion may be e x t r a c t e d w i t h a minimum o f e f f o r t . F o r a r e v i e w o f t h e r e l a t i o n between Hamilton's of m a t h e m a t i c a l  p r i n c i p l e o f l e a s t a c t i o n and t h e c o n s e r v a t i o n theorems  p h y s i c s , see H i l l ( 1 9 5 1 ) . The b a s i c concepts o f t h e f o r m a l i s m  are i n t r o d u c e d below.  The L a g r a n g i a n f o r a system of p o i n t o b j e c t s i s a f u n c t i o n L ( q , q , t ) of t h e c o o r d i n a t e s q ( t ) and t h e i r time d e r i v a t i v e s q ( t ) , and p o s s i b l y o f t itself.  ( I n the. f o l l o w i n g , a l l the c o o r d i n a t e s have been denoted  generically  by q, and the sums over c o o r d i n a t e l a b e l a r e o m i t t e d , f o r c l a r i t y . ) The a c t i o n i s the f u n c t i o n a l (4-1)  computed between t h e two p o i n t s ( q ^ ( t ^ ) , q ^ ( t ^ ) ) g u r a t i o n space a l o n g a l l curves j o i n i n g  and (q^ (t^), q^ (t^) )  t h e p o i n t s . The t r a j e c t o r y f o r which  the v a l u e o f S i s a minimum, compared w i t h a l l o t h e r t r a j e c t o r i e s , t u a l motion  of c o n f i -  i s the ac 1  o f the system between the p o i n t s . The t e c h n i q u e s of v a r i a t i o n a l  c a l c u l u s , a l o n g w i t h t h i s p r i n c i p l e , show that t h e e q u a t i o n s of motion may be o b t a i n e d from the L a g r a n g i a n by the E u l e r - L a g r a n g e 3L 3q  d 3Li = 0 dt[3qj  equations (4-2)  21  Under a t r a n s f o r m a t i o n t->-t' , q->-q', t h e f u n c t i o n a l form of the L a g r a n g i a n a l t e r to p r e s e r v e  i t s numerical  l e a v e the L a g r a n g i a n  i n v a r i a n c e , however, some t r a n s f o r m a t i o n s  f.orm-invariant, o r a t most add the t o t a l d e r i v a t i v e of  some f u n c t i o n of t h e c o o r d i n a t e s  (which l e a v e s <SS i n v a r i a n t ) . That i s ,  (4-3)  L'(q') = L ( q J ) + ~TT"A (q ' ) Such t r a n s f o r m a t i o n s  must  a r e symmetry t r a n s f o r m a t i o n s ,  t r a n s f o r m a t i o n of a s o l u t i o n of the e q u a t i o n s s o l u t i o n of t h e e q u a t i o n s  s i n c e they r e s u l t i n t h e  of motion i n t o another  of m o t i o n . As shown by H i l l ,  possible  the t e s t f o r a symmet-  ry transformation i s that  (4-4) where 61, 6q, and 6q are i n f i n i t e s i m a l q u a n t i t i e s . The i n t e r p r e t a t i o n of (4-4) is  that the RHS must be e x p r e s s i b l e as the t o t a l d e r i v a t i v e of some  infini-  t e s i m a l f u n c t i o n 6 A ( q , t ) . A g i v e n i n f i n i t e s i m a l symmetry t r a n s f o r m a t i o n of L, w i t h the equations  of motion d e r i v e d from an a c t i o n p r i n c i p l e , g i v e s r i s e to  an a s s o c i a t e d c o n s e r v a t i o n law, w r i t t e n  (4-5) In t h i s way, each symmetry t r a n s f o r m a t i o n of L l e a d s to a c o n s e r v a t i o n law, although  the r e v e r s e i s not n e c e s s a r i l y t r u e . There e x i s t c o n s e r v a t i o n laws  which do not correspond  to symmetries of the L a g r a n g i a n ,  such as the Runge-  Lenz v e c t o r of the K e p l e r problem, which i s d i s c u s s e d by Greenberg(1966).  1  22 4.2  The  Symmetry T r a n s f o r m a t i o n s and C o n s e r v a t i o n Laws of V o r t e x That  result  t h e e q u a t i o n s of motion  (2-8) of v o r t i c e s i n an i n f i n i t e  v v v V •i i  z  -1 J * V k  kV  k  is  m  the E u l e r - L a g r a n g e e q u a t i o n s  (regarding z  l n  lv kl z  ( 4 _ 6 )  k  and  z  as independent  variables)  s t r a i g h t f o r w a r d , i n f a c t , t h i s L a g r a n g i a n has been c o n s t r u c t e d so t h a t  would be so. The Lagrangian(4.-6) d i s p l a y s some u n u s u a l f e a t u r e s : The do not appear  i n q u a d r a t u r e , but i n b i l i n e a r  ensuring that  the e q u a t i o n s of motion  is  fluid  from an a c t i o n p r i n c i p l e u s i n g the L a g r a n g i a n L (  via  Motion  the sum  of two  Lagrangian i t s e l f  o r d e r i n time  i s a c o n s t a n t of the motion.  In Newtonian p o i n t  which i s not c o n s e r v e d ; t h i s L a g r a n g i a n a l l o w s no  mechanics,  and p o t e n t i a l e n e r g i e s ,  such i n t e r p r e t a t i o n ,  o f the system h a v i n g been shown t o be ¥  i n mind, the L a g r a n g i a n may  . The L a g r a n g i a n .  ( p r e v i o u s l y shown), W and ¥ , so the •o o  the L a g r a n g i a n i s the d i f f e r e n c e between the k i n e t i c  t o t a l energy  velocities  combination w i t h the c o o r d i n a t e s ,  are f i r s t  c o n s t a n t s of the motion  this  q  the  a l o n e . With t h e s e comments  be i n v e s t i g a t e d f o r i t s symmetries,  which are  a l r e a d y known from s e c t i o n 2, w i t h the i n t e n t of a s s o c i a t i n g a c o n s e r v a t i o n law w i t h each symmetry t r a n s f o r m a t i o n . The w r i t t e n , from  4  test  f o r a symmetry t r a n s f o r m a t i o n of t h i s L a g r a n g i a n i s  (4-4),  t  -  (  6  k  and  the a s s o c i a t e d oT^o  6 t  +  t  )  +  m  k  m  (  K V (  i  k  m  (6z -6z, c o n s e rmv akt i o n law i s k  n  k  - -^(SA) k zK-z V k zA -z .k= V vzA)  +  (4-7)  f i z  k '  6z -6z, -\  >  6A  =  0  ( A  -  8 )  k These e x p r e s s i o n s w i l l now t r a n s f o r m a t i o n s and  be used  to t e s t proposed  infinitesimal  t o f i n d t h e i r a s s o c i a t e d c o n s e r v a t i o n laws.  symmetry  23 (a) Space T r a n s l a t i o n Symmetry and C o n s e r v a t i o n The  of Centre  of C i r c u l a t i o n  transformation St  = 0,  Sz = a, 6z = 0  (a i n f i n i t e s i m a l )  (4-9)  i s a symmetry t r a n s f o r m a t i o n on L , b u t does not l e a v e L i n v a r i a n t i n form, r e q u i r i n g the i n t r o d u c t i o n of the i n f i n i t e s i m a l f u n c t i o n 6A  The  -k  ( a E  kVk -  5  z  k V k  )  (  4  -  1  0  )  a s s o c i a t e d c o n s e r v a t i o n law f o l l o w s from a l l o w i n g a t o be a r b i t r a r y .  L e t t i n g a be c o m p l e t e l y  r e a l or c o m p l e t e l y  imaginary  r e s u l t s i n the conser-  v a t i o n of the imaginary  or r e a l p a r t s , r e s p e c t i v e l y , o f t h e c e n t r e o f  circulation, Z . o  (b) R o t a t i o n a l Symmetry and t h e C o n s e r v a t i o n The  o f Moment of C i r c u l a t i o n  transformation 6t  = 0, ^ z ^ ^ -  a z  i ' c  ^  z  = k  ^  a z  k ^  a  i i n  t  e  s  i  m  a  l )  (4-11)  l e a v e s L f o r m - i n v a r i a n t , r e s u l t i n g i n t h e c o n s e r v a t i o n of t h e moment o f circulation 0 . o  (c) Time t r a n s l a t i o n Symmetry and C o n s e r v a t i o n The  of V o r t e x  Stream  Function  transformation 6t  = x,  6 z = 0, k  l e a v e s L form i n v a r i a n t , r e s u l t i n g  6z = 0 k  ( T infinitesimal)  i n t h e c o n s e r v a t i o n of v o r t e x  (4-12)  stream  function, Y . o  These t h r e e symmetries of L a r e the ones which, i n Newtonian mechanics, l e a d t o the c o n s e r v a t i o n o f l i n e a r momentum, a n g u l a r momentum, and energy, r e s p e c t i v e l y . I t i s easy t o see t h a t i n a v o r t e x zero, due to the symmetry of the v e l o c i t y  field  fluid,  t h e l i n e a r momentum i s  c o n t r i b u t e d by each v o r t e x .  24 I t has been shown t h a t t h e moment of c i r c u l a t i o n , 0 , i s a measure of t h e o a n g u l a r momentum i n an i n f i n i t e is  t h e energy o f t h e f l u i d  fluid,  and t h a t t h e v o r t e x stream  (minus t h e s e l f  energy o f t h e v o r t i c e s ) . A l s o , i n  Newtonian m e c h a n i c s , i t i s t h e H a m i l t o n i a n which g e n e r a t e s time,  so t h i s j u s t i f i e s system.  (d) The Absence of a Symmetry T r a n s f o r m a t i o n L e a d i n g At t h e time  of w r i t i n g ,  remaining  an e x t r a  t h e c o n s e r v a t i o n of W . There i s one  2 3  t,  z  = e z , 6  k  k  z  = e" £  (g r e a l )  e  k  k  which i s a s c a l e t r a n s f o r m a t i o n , and does not l e a d  (4-13)  t o a c o n s e r v a t i o n law.  i s a c o n s t a n t of t h e motion which a p p a r e n t l y does n o t r e f l e c t  metry o f L, i t may b e l o n g by  Q  symmetry o f t h e e q u a t i o n s o f m o t i o n , namely, t h e t r a n s f o r m a t i o n  : t' = e  q  t o C o n s e r v a t i o n of W  i t has n o t been p o s s i b l e t o f i n d  symmetry t r a n s f o r m a t i o n a s s o c i a t e d w i t h  Since W  translations i n  t h e i d e n t i f i c a t i o n of ¥ w i t h t h e H a m i l t o n i a n of t h e o  J  vortex  function, ¥ , o  a sym-  t o t h a t c l a s s o f c o n s e r v a t i o n laws e x e m p l i f i e d  t h e Runge-Lenz v e c t o r , mentioned e a r l i e r . T h i s q u e s t i o n remains open.  As unit  circle  a f i n a l note,  the Lagrangian  i s o b t a i n e d by adding  f o r t h e v o r t e x system bounded by t h e  t o L t h e terms n e c e s s a r y  t o a l t e r 4* , i . e -  I Iv Y , l n | l - z z. | k ^ m k ' m k m k  (4-14) .  1  The  correct  The  a d d i t i o n o f t h e s e terms reduces  Lagrangian  equations  o f motion a r e o b t a i n e d from  the Euler-Lagrange  t h e symmetry o f t h e system, s i n c e the  i s no l o n g e r i n v a r i a n t under t r a n s l a t i o n s  v e r i f i e d . T h i s agrees with the r e s u l t of c i r c u l a t i o n ,  Z , i s not conserved o  equations.  i n space,  stated i n section i n a fluid  T h i s r e s u l t may be g e n e r a l i z e d t o a l l bounded  as i s e a s i l y  2.3, t h a t the c e n t r e  bounded by t h e u n i t  fluids.  circle,  25 5.  THE  STABILITY OF RIGIDLY ROTATING VORTEX CONFIGURATIONS  In t h i s s e c t i o n , i t w i l l be the v o r t e x f l u i d configurations  i s one  shown t h a t  of r i g i d  must then be  the  s t a t e of k i n e m a t i c e q u i l i b r i u m  r o t a t i o n of the v o r t e x c o n f i g u r a t i o n .  examined f o r s t a b i l i t y  from k i n e m a t i c e q u i l i b r i u m , which i s e f f e c t e d by motion  i n small  q u a n t i t i e s about a r i g i d l y  ed v o r t i c e s l y i n g e q u a l l y c l e . The  the  The  (as a f i r s t  2  Equilibrium The  of degree 2N,  State  of a R o t a t i n g  Vortex  component i s one  irrotational  at ft, and mum.  obtaining  approximation). of  quantiz-  r < l i n s i d e the u n i t  cir-  that  Fluid  I f the  been t r e a t e d by  of r i g i d  r o t a t i o n (Yn= stationary  " f r e e energy" of the  frame r e q u i r e s  that  the  Putterman  e q u i l i b r i u m s t a t e of the £ ^)> x  i n the  w h i l e the  s u p e r f l u i d , F=E-ftL, must be  the c o n d i t i o n of s t a t i o n a r y V  configuration  in a  normal  s u p e r f l u i d must  frame of r e f e r e n c e  c o n d i t i o n of i r r o t a t i o n a l f l o w i s s a t i s f i e d by  v o r t e x l i n e s i n the f l u i d ,  the  r e s u l t s show t h a t , f o r a s u p e r f l u i d c o n t a i n e d  ( c u r l V = 0) and s the  of  of r o t a t i n g s u p e r f l u i d s based on  v e s s e l r o t a t i n g w i t h a n g u l a r v e l o c i t y ft, the  be  of  where N i s the number of v o r t i c e s .  thermodynamic e q u i l i b r i u m  Uhlenbeck(1969). T h e i r  fluid  equations  i s reduced to f i n d i n g the r o o t s  Landau(1941,1947) macroscopic, t w o - f l u i d model has and  perturbations  polygonal configurations  a c i r c l e of r a d i u s  c r i t e r i o n of s t a b i l i t y  polynomials i n r  5.1  spaced on  expanding the  These  symmetry of t h e s e c o n f i g u r a t i o n s makes the normal modes of v i b r a t i o n .  apparent, and two  case of r e g u l a r  small  rotating configuration,  l i n e a r e q u a t i o n s of motion of the p e r t u r b a t i o n s T h i s method i s used i n the  against  of  rotating  at a  the presence i n the  miniof  rotating  of v o r t e x l i n e s r o t a t e r i g i d l y  with  a n g u l a r v e l o c i t y ft. This  can be  shown e a s i l y f o r a two-dimensional v o r t e x f l u i d ,  for  26 which the e q u a t i o n s of motion and known. The V  o  conserved  , the energy,  ¥ , keeping o o  0 o  0  the c o n s t a n t s of the m o t i o n a r e a l r e a d y  q u a n t i t i e s f o r the v o r t e x f l u i d  , and W . The o o  and W  i n s i d e the u n i t c i r c l e  s t a t e of e q u i l i b r i u m of the system must  The  q  the c o n s e r v a t i o n of  s t a t e of s t a b l e e q u i l i b r i u m i s g i v e n by F  = y + ft0 = minimum o o  (5-1)  i n which ft i s , f o r t h e moment, an undetermined m u l t i p l i e r . The t i v e s of F w i t h r e s p e c t to the independent l e a d i n g immediately  to the d i f f e r e n t i a l z=  iftz  k  and  their  v a r i a b l e s z^ and  >, == k J  z  z° ik,  Z  z^ must v a n i s h ,  (5-2)  k  are (5-3)  e  i n which the z£ a r e c o n s t a n t s , and ft may v e l o c i t y of a r i g i d l y  f i r s t deriva-  equations  complex c o n j u g a t e s , whose s o l u t i o n s z  is  to  c o n s t a n t i n t h e v a r i a t i o n a l problem  q  as the s o l u t i o n o b t a i n e d thereby a u t o m a t i c a l l y guarantees W.  minimize  c o n s t a n t as a u x i l i a r y c o n d i t i o n s . I t i s s u f f i c i e n t  r e l a x these c o n d i t i o n s , r e t a i n i n g only 0  now  be i d e n t i f i e d  as the  angular  r o t a t i n g c o n f i g u r a t i o n of v o r t i c e s . The v a l u e of ft  c a l c u l a t e d by s u b s t i t u t i n g  (5-3)  i n t o the e q u a t i o n s of m o t i o n , o b t a i n i n g a  c o n s t r a i n t on the z° i n the b a r g a i n , namely  z°ft = E.'Y; ( Z O - Z . ) " + Z - V ^ U - z ^ , ) " . n k'k n k k'k k n k * 0  This  1  0  0  unit  (5-4)  1  i s a k i n d of e i g e n v a l u e problem, the g e n e r a l s o l u t i o n to which would  enumerate a l l p o s s i b l e c o n f i g u r a t i o n s which r o t a t e r i g i d l y and lar  are  velocities.  I f t h e problem i s r e s t r i c t e d  strength vortices,  to the i n f i n i t e f l u i d  i t can be shown t h a t the r i g i d l y  determinant.  case w i t h  rotating configura  t i o n s of N v o r t i c e s can be put i n t o one-to-one correspondence symmetric m a t r i c e s w i t h zero  t h e i r angu-  w i t h the  N*N  27 R e t u r n i n g a g a i n t o W^,  i t i s evident  W =fi0 , and i s t h e r e f o r e o o  5.2  automatically  for rigidly  looking  stable r i g i d l y  r o t a t i n g systems,  c o n s e r v e d , as expected, r  The L i n e a r i z e d E q u a t i o n s of M o t i o n of t h e The  Perturbations  rotating configurations  a r e now t o be found by  for. o s c i l l a t o r y motions of t h e v o r t i c e s about t h e i r  positions. This  equilibrium  i s e f f e c t e d by expanding t h e e q u a t i o n s of m o t i o n i n terms of  small perturbations the  that  about the r i g i d l y  expansion a t the f i r s t  the p e r t u r b a t i o n s  r o t a t i n g c o n f i g u r a t i o n , and  o r d e r terms, so t h a t  terminating  l i n e a r e q u a t i o n s of m o t i o n of  r e s u l t . The s u b s t i t u t i o n z  k  =  ( z  k V +  t))e±nt  (5_5)  i s made i n (2-23), and the e q u a t i o n s of m o t i o n a r e  k  k  k Due to the s u b s t i t u t i o n ( 5 - 5 ) , frame of r e f e r e n c e  the v o r t i c e s a r e now b e i n g viewed from the  r o t a t i n g a t i l , so the e q u i l i b r i u m p o s i t i o n s z£ a r e  stationary. It  i s n e c e s s a r y t h a t t h e p e r t u r b e d system l e a v e  motion ¥ , 9 , and W o o o (5-6)  unchanged t o f i r s t  t h e c o n s t a n t s of the  o r d e r i n the b ' s , so the s o l u t i o n s t o k  need some c o n s t r a i n t s . The c o n s t a n t s of the motion may be expanded about  a r i g i d l y r o t a t i n g c o n f i g u r a t i o n i n terms o f t h e b ^ s , and t h e f i r s t terms must v a n i s h  f o r the c o n s e r v a t i o n  laws to remain v a l i d . The r e s u l t i n g  equations of c o n s t r a i n t a r e  Vo w  Vk kV kV =  :  °o  (z  :  2  0  ^\ kV kV = (i  z  order  0  28 which combine to g i v e  J  The if  t h e y's  that  the  simpler,  k V k  tions  of r a d i u s  r<l  system. For  already  of some u n i t  (  been p o i n t e d  strength,  c l a s s of  out,  rigidly  rotating  lying equally  these c o n f i g u r a t i o n s  positions z  0  of =  r e  N coupled equations  solutions  configuraa  2 T r  the normal modes  to  (5-6)  for  may  , . . i(2Tr/N)kr D (.L,t; = e l L a  are ( 5  be  uncoupled by  iwt  e  i(2Tr/N)Lk  e  p r o p o s i n g the  , °L  k  e  -iut  -i(2 r/N)Lk J T  e  a  and  c  1  e  l a J t  and  yields  e  the  l t J t  are  two ( P T  2  linearly  initial  conditions.  S i n c e the  independent, s u b s t i t u t i o n of  are  real  L  Li  the  (5-9)  functions into  (5-6)  homogeneous e q u a t i o n s -  p  _  l  s  L  +  l  )  a  L "  (T  N-L+1 "  "  n  U ) C  L  =  ° (L=l,2,..,N-1)  -(T  9 )  (5-9)  i n which oo i s r e a l , L l a b e l s the mode of v i b r a t i o n , and d e t e r m i n e d by  _  solutions  (L=1,2,..,N)  c o n s t a n t s to be  the  investigated.  /N)k  (5-6)  circle  polygon.The  the u n p e r t u r b e d v o r t i c e s f o r m i n g an N-gon  i(  )  however,  spaced on  suggests that  p o s s e s s a s i m i l a r symmetry. The w i l l be  8  a c e r t a i n symmetry i s  ( s i n c e the boundary i s at r = l ) , f o r m i n g an N - s i d e d  polygonal configurations  "  5  r e a d i l y apparent, e s p e c i a l l y  of N v o r t i c e s of e q u a l s t r e n g t h  of v i b r a t i o n w i l l  The  i s not  example, one  h i g h degree of symmetry of  The  (5-6)  of a r b i t r a r y v a l u e , I t has  the  i s that  '  g e n e r a l s o l u t i o n to  are  constraint  k - °  b  i f a l l the v o r t i c e s are  imposed on  single  L + 1  -ft - . ) a  L  +  ( T P  2  -P  _ 1  S _ N  L + 1  )c  (5-10)  =0  L  2  i n which p=r in closed For  and  the  trigonometric  form i n T a b l e I, the  non-trivial solutions  sums T  quadratic  equation  , and ft are  defined  and  given  c a l c u l a t i o n s f o r which appear i n Appendix  to e x i s t f o r a  and Lt  of the  , S  , OJ must be  c Li  the  solution  B.  29  "2  - » W  W  +  fi(T  +  N-L l  W  +  +  The  r o o t s of t h i s  ( P T  2 -P  _ 1 S  L 1  ) 2  +  " N-L+r L l T  +  °  =  T  (  "  5  U  )  equation are o  03 = C ± /AS  (5-12)  where 2pA  = N p - -^±2J2  N  L  9  0  2 B P  = N  „ p T  2  N  L  L.2 -^_L  l-p  n  ^  i  1-P Applying  N  - L(N-L)  ^  l  (1-P )  the c o n s t r a i n t (5-8) t o the s o l u t i o n s (5-9) g i v e s , ioit . - i o j t . p i(2iT/N)Lk _ (a^e + c e ))e =0 L  The  (5-13)  1-P  .H  .  l-p  W  L N-L + NLP - P  (1-p )  2 p C  4N - E — - 2(N-1) + L(N-L)  + NL*-^  (1-pV  L  constraint i s identically  c o n d i t i o n s on a  and c  . ... (5-14)  /c  k  satisfied  f o r L=1,2,...N-1 and p l a c e s no  f o r these modes. However, f o r L=N, i t i s n e c e s s a r y  that a =c =0 f o r ,the c o n s t r a i n t to be s a t i s f i e d . I n o t h e r words, the L=N mode must be removed from c o n s i d e r a t i o n i f t h e p e r t u r b a t i o n s a r e not t o alter  the c o n s t a n t s  o f t h e motion.  Havelock(1931) has used a s i m i l a r approach t o the problem and h i s e x p r e s s i o n s g i v i n g t h e f r e q u e n c i e s o f t h e normal modes a r e t h e same, a p a r t from some m i s p r i n t s i n t h e p u b l i c a t i o n . These m i s p r i n t s a r e l i s t e d i n Appendix D.  %  30 5.3  The S t a b i l i t y of t h e P o l y g o n a l For  the polygonal  Configurations  configurations  t o have s t a b l e o s c i l l a t o r y motion  about e q u i l i b r i u m , i t i s n e c e s s a r y t h a t a> be r e a l f o r a l l a l l o w e d modes, so the p r o d u c t AB must be p o s i t i v e f o r t h e s e modes. To demonstrate i n s t a b i l i t y , it  s u f f i c e s t o f i n d one mode which i s u n s t a b l e .  From t h e q u a d r a t i c  equation  (5-11) g o v e r n i n g u>, A and B may be w r i t t e n i n t h e form :  0 This  - ^  expression  ( T  N-L 1  +  +  W  "  f i± ( P T  ls  i s more c o n v e n i e n t than  "  15)  (5-13) i n d e m o n s t r a t i n g t h e f a c t  A and B a r e symmetric under t h e i n t e r c h a n g e S  (5  -P" W  2  o f L and N-L ( r e c o g n i z i n g  has t h i s symmetry i t s e l f ) . The consequence of t h i s i s t h a t  that  that  the l e a s t  s t a b l e mode or modes are. the " c e n t r a l " ones,that i s , L=N/2 f o r N even and L=(N±l)/2 f o r N odd. The s t a b i l i t y  of the polygonal  configurations  i s then  determined by t h e s t a b i l i t y o f t h e s e modes, which a r e t o be examined  separ-  a t e l y f o r N even and N odd. (a) N even Let  N=2n,  L=n,  N-L=n  (n=l,2,...)  (5-16)  S u b s t i t u t i n g t h e s e i n t o (5-13) g i v e s , f o r t h e l e a s t s t a b l e mode a = 2p(l-p ) A= N  2  (n -4n+2) + 4 n p 2  2  n  + 2(3n -2)p 2  + (n +4n+2)p 2  2 n  +  4n p 2  3 n  4n  (5-17)  „ , N.2 2 n.4 b = 2 p ( l - p ) B = -n (1-p ) S i n c e b<0 always, t h e c r i t e r i o n f o r i n s t a b i l i t y f o r a l l p f o r n>4. S i n c e  i s a>0, which i s s a t i s f i e d  a(p=0)<0 f o r n=l,2,3 and ^ ™ a(p)>0, t h e r e  a r o o t P ( ) of a. F o r p < p ( n ) , t h e c o n f i g u r a t i o n s n  c  c  exists  n=l,2,3 a r e s t a b l e ;  f o r p > p ( n ) , they a r e u n s t a b l e . c  (b) N odd Let  N=2n+1,  L=n,  N-L=n+1  (n=l,2,...)  (5-18)  31 Then, f o r the l e a s t  s t a b l e modes,  a = n(n-3) + n ( 2 n + l ) +  +  n P  (n+l)(2n+l)p  (n+1)(2n+l)p +n(2n+l)p  3 n + 1  +  n + 1  -  3 n + 2  2(3n +3n-l)p 2  <n+l)(n+4)p  2 n + 1  4 n + 2  (5-18) b = -n(n+l) + n ( 2 n + l ) p + A g a i n , b<0 to a>0,  (n+l)(2n+l)p  +  n  3 n + 1  (n+1)(2n+l)p + n(2n+l)p  3 n + 2  -2(3n +3n+l)p  n + 1  2  -  n(n+l)p  always i n the r e g i o n 0<p<l, r e d u c i n g the c r i t e r i o n  which i s s a t i s f i e d  f o r a l l p f o r n>3.  n  c  c  4 n + 2  for i n s t a b i l i t y  A g a i n , a s i g n change i n a o c c u r s  f o r the s p e c i a l cases n=l,2, so t h e r e e x i s t s a r o o t P ( ) c o n f i g u r a t i o n s a r e s t a b l e i f p < p ( n ) , and  2 n + 1  such t h a t  these  are unstable i f p>p (n). c  In summary, the r e g u l a r p o l y g o n a l c o n f i g u r a t i o n s i n s i d e the u n i t c i r c l e a r e u n s t a b l e f o r a l l v a l u e s of r , except  f o r N=l,2,3,4 ,5 , and  6.  For  t h e s e v a l u e s of N t h e r e e x i s t s a c r i t i c a l v a l u e of r , c a l l e d r , such t h a t c i n the r e g i o n x <*c>  the c o n f i g u r a t i o n i s s t a b l e , and  g u r a t i o n i s u n s t a b l e . The v a l u e of r  c  for * * > >  c  the  f o r the d i f f e r e n t polygons  confi-  i s available  from a n u m e r i c a l c a l c u l a t i o n of the r o o t s of the p o l y n o m i a l a i n each The r e s u l t s of such a computation  case.  appear i n T a b l e I I . In a d d i t i o n , r ^ g i v e s a  minimum v a l u e of ft f o r which the p o l y g o n  i s s t a b l e , v i a the r e l a t i o n i n  T a b l e I. These v a l u e s of ft . f o r each N a l s o appear i n T a b l e I I . mm  N-l  1  J l  ( 1  _ i(2 r/N)Lk e  T  _li(2WN)  N  k )  -Js(L-2)(N-L)  2  i(2-rr/N)Lk 2  J l  ( 1  _p i(2,/N)k e  ) 2  N-L -±—T,N 2 ri d - p ^)  N-L - N(L-l)-£ N 1-p  N-l p  1  S  1  TABLE I  + T  1  -  P  T  1  SOME RELEVANT TRIGONOMETRIC SUMS (defined  N  TABLE I I  hp ( N - l ) + N  2  and i n c l o s e d  r  form)  n . mm  c  2  .462  2.79  3  .567  3.43  4  .574  4.70  5  .588  5.86  6  .547  8.37  THE VALUES OF r  c  AND  . FOR THE STABLE POLYGONS mm  33 6. THERMODYNAMIC EQUILIBRIUM OF A ROTATING SUPERFLUID The  s t a t e s of s t a b l e k i n e m a t i c e q u i l i b r i u m o f q u a n t i z e d v o r t i c e s  r e g u l a r polygons were found' i n t h e l a s t  s e c t i o n , however, t h i s i s i n s u f f i c i e n t  to determine t h e s t a t e o f thermodynamic e q u i l i b r i u m fluid. and  of the r o t a t i n g vortex  The c r i t e r i o n f o r thermodynamic e q u i l i b r i u m a c c o r d i n g  Uhlenbeck(1969) i s t h a t  forming  t o Putterman  t h e v o r t e x f l u i d must have an a b s o l u t e  of t h e f r e e energy F=E-ftL. E x p e r i m e n t a l l y ,  vessels  minimum  of H e l l a r e r o t a t e d a t  some c o n s t a n t a n g u l a r v e l o c i t y ft, so t h e minima o f F f o r t h e v a r i o u s  confi-  gurations  must be found f o r g i v e n ft, and t h e s e minima compared i n o r d e r t o  find  c o n f i g u r a t i o n which r e p r e s e n t s  that  As ft i s i n c r e a s e d  from z e r o ,  v o r t i c e s represents  there  the absolute  minimum o f F f o r t h a t ft.  i s a range o f ft f o r which the  t h e s t a t e of thermodynamic e q u i l i b r i u m . Above t h e c r i t i c a l  a n g u l a r v e l o c i t y ftj, t h e f r e e energy o f one v o r t e x at t h e c e n t r e the  absence o f  i s l e s s than  f r e e energy o f z e r o v o r t i c e s , so t h i s becomes t h e s t a t e o f thermodynamic  e q u i l i b r i u m . As ft i s i n c r e a s e d  f u r t h e r , there  appears a spectrum o f c r i t i c a l  a n g u l a r v e l o c i t i e s as i t becomes e n e r g e t i c a l l y f a v o u r a b l e v o r t i c e s i n t o the f l u i d .  to introduce  new  That such i s t h e case i n r e a l i t y has been shown by  the work of Packard and Sanders(1969). The purpose o f t h i s s e c t i o n i s t o show that  such a spectrum can be r e p r o d u c e d from t h e o r y , a t l e a s t f o r s m a l l  numbers  of v o r t i c e s . S i n c e the energy of t h e v o r t e x f l u i d strengths  y^,  i t i s r e a s o n a b l e t o assume t h a t  f l u i d would r e q u i r e strength  involves  products of the vortex  the e q u i l i b r i u m  s t a t e of the  t h a t a l l t h e v o r t i c e s have t h e minimum non-zero quantum  ( i . e . ti/m f o r h'elium) . F o r s m a l l numbers o f v o r t i c e s , a t t e n t i o n  i s r e s t r i c t e d t o t h e r e g u l a r p o l y g o n s , whose s t a b i l i t y has been determined, or r e g u l a r polygons w i t h an e x t r a v o r t e x a t t h e c e n t r e . the  The s t a b i l i t y o f  l a t t e r has not been i n v e s t i g a t e d . The c a l c u l a t i o n s o f t h e minimum o f  34 F as a f u n c t i o n of Q, i s a t a s k i n n u m e r i c a l  a n a l y s i s f o r every c o n f i g u r a t i o n  except  be  i n the case of one v o r t e x , which may  important of one  This  c a l c u l a t i o n r e l a t e s the c r i t i c a l a n g u l a r v e l o c i t y f o r the c r e a t i o n  v o r t e x , &i,  t o the v o r t e x core r a d i u s parameter, e.  The u n i t of l e n g t h has r a d i u s . The u n i t A dimensionless I being  treated a n a l y t i c a l l y .  a l r e a d y been chosen to be a, the  of time w i l l be ma /Y> where y 2  F i s i n t r o d u c e d by c h o o s i n g  the depth of the l i q u i d , and  i s the u n i t quantum s t r e n g t h .  the u n i t  p being  of energy t o be  r a t i o of the polygon  2  £ ,  (in  component of  r a t i o of the v o r t e x core r a d i u s to the c y l i n d e r r a d i u s  the v o r t e x c o r e r a d i u s i s R=ea. The  p 7 r y  t h e d e n s i t y of the l i q u i d  t h i s case the d e n s i t y used i s the d e n s i t y of t h e s u p e r f l u i d H e l l ) . The  cylinder  is  e,,so  r a d i u s to the  cylin-  der r a d i u s w i l l c o n t i n u e t o be r .  6.1  One  Vortex The  i n a Rotating Cylinder  f r e e energy of a s i n g l e v o r t e x i n the c y l i n d e r  i s , from(3-7)  and (3-13), F  = E-ftL = l n ( l - r ) - l n e -  fi(l-r )  2  2  (6-1)  For a g i v e n ft, e q u i l i b r i u m i s g i v e n by  f=° The  ->  f£i-i,  second extremum does not  B  •  «-»  e x i s t f o r ft<l. The  first  extremum i s a minimum  f o r ft>l, whereas the second extremum i s always a maximum. The of a s i n g l e v o r t e x i s at the c e n t r e , f o r ft>l. The  stable position  e q u i l i b r i u m f r e e "energy of  a s i n g l e v o r t e x as a f u n c t i o n of ft becomes  F1 The  = -ft - l n e  f r e e energy of z e r o v o r t i c e s i n the f l u i d  (6-3) i s z e r o , so the c o n d i t i o n t h a t  35 one  vortex  present  at the  centre  i s t h e r m o d y n a m i c a l l y s t a b l e over z e r o  is  T  = -ft - l n e < 0  l  l e a d i n g t o the d e f i n i t i o n of the of one  conventional  In the r e g i o n s i n c e that but  (6-4)  c r i t i c a l angular v e l o c i t y f o r the  creation  vortex &l  (In  vortices  = -lne  units, this  l<ft<fti, one  (6-5) i s 1^ =  vortex  (y/a )ln(a/R). 2  a t the  centre  configuration i s stable against  i t does not  represent  minimize  F absolutely.  the f l u i d  represents  the  i s s a i d t o be  metastable,  s m a l l perturbation's  from  equilibrium,  s t a t e of thermodynamic e q u i l i b r i u m , which must  In t h i s r e g i o n  the  )  of a n g u l a r v e l o c i t y , z e r o v o r t i c e s i n  s t a t e of thermodynamic e q u i l i b r i u m :  the  superfluid  component i s at r e s t . The be  1.6  sec  q u a n t i t y ft^ has  The  fti=1.0  sec  value 1  of t h e v o r t e x  Sanders' work t h a t  e f f e c t s of m e t a s t a b i l i t y and measurement of subsequent  The  2  e t c -was  A n g u l a r V e l o c i t i e s i n He  of R  c a l c u l a t e d from  d e t a i l s appear i n A p p e n d i x C , the  (3-3) result  since  i s evidence i n subject  to  strong  The  reproducible.  II  f r e e energy of N v o r t i c e s i n a r e g u l a r p o l y g o n a l be  £.  same c y l i n d e r r a d i u s .  a good d e t e r m i n a t i o n  c e r t a i n l y not  5  from W i l k s ( 1 9 7 0 ) ,  t h e i r e x p e r i m e n t a l system was  ft ,ft3,  i n a v e s s e l r o t a t i n g at ft may ( 6 - 4 ) . The  g i v i n g R=5*10  the measurement of ftj i s i n some doubt.  Spectrum of C r i t i c a l The  and  considered  Sanders(1969) to  i s R=0.3 X,  s e n s i t i v e to s m a l l v a r i a t i o n s i n ft}. A l s o , t h e r e  Packard and  6.2  core radius  ( 1 6 . 6 i n d i m e n s i o n l e s s u n i t s ) w i t h the  measurement of ft^ cannot be  R i s very  Packard and  ^(25.2 i n d i m e n s i o n l e s s u n i t s ) w i t h a=0.05 cm,  A more r e a l i s t i c giving  been measured by  and  configuration  (3-13) u s i n g  (5-9)  being  F (ft.p) = N l n ( l - p ) -!sN(N-l)lnp -Nft(l-p) -NlnN +Nft N  x  (6-6)  36 F o r N=l,2,3,4,5,6 the polygons  a r e s t a b l e , so t h i s  minimum of F w i t h r e s p e c t to d e f o r m a t i o n s  expression represents a  of the p o l y g o n .  and 0, a r e independent i n t h i s e x p r e s s i o n , but  i t was  The v a r i a b l e s p=r  shown t h a t i n e q u i l i b r i u m  the p o l y g o n must r o t a t e w i t h the same a n g u l a r v e l o c i t y as the v e s s e l , the  2  giving  relation  2  p  p  F o r a g i v e n N and ft, (6-6) the s o l u t i o n t o  i -  N P  i s the f r e e energy of t h e c o n f i g u r a t i o n , p b e i n g  ( 6 - 7 ) , t a k i n g care t h a t ft i s g r e a t e r than ft . mxn  T a b l e I I , s e c t i o n 5. O p e r a t i o n a l l y , (6-7) may of ft by an i t e r a t i v e  technique;  be  given i n  solved for a s p e c i f i e d  the v a l u e of p o b t a i n e d  value  i s substituted into  (6-6) w i t h ft to o b t a i n F. Thus a curve of t h e e q u i l i b r i u m v a l u e of F v s ft,may be computed f o r each c o n f i g u r a t i o n . The energy and  a d d i t i o n of a v o r t e x at the c e n t r e a l t e r s the form of the  a n g u l a r v e l o c i t y . The  free  f r e e energy of an. N-gon w i t h a c e n t r a l  vortex  is F ( f t , p) = N l n ( l - p ) -JsN(N+l)lnp N  N  + w i t h the new  The  (N+l)ft! -ft(l+N(l-p>)  a n g u l a r v e l o c i t y , due  2  c u r v e s of F vs ft may  P  P  -NlnN  1-  (6-8)  to the c e n t r a l v o r t e x  N P  be o b t a i n e d  i n the same way  as above.  Once these curves have been p l o t t e d , the s t a t e s of t h e r m a l brium can be determined  by v i s u a l i n s p e c t i o n , and  the c r i t i c a l  equili-  angular  v e l o c i t i e s o b t a i n e d by  the i n t e r s e c t i o n o f t h e a p p r o p r i a t e c u r v e s . The  e n e r g i e s of one  ( F ^ ) , two  lateral  triangle  vortex (F^)>  and  vortices  (F^)  , t h r e e v o r t i c e s i n an e q u i -  three c o l i n e a r v o r t i c e s  f u n c t i o n of ft i n F i g u r e 1, u s i n g R=0.3 X. The g i v e the t h e o r e t i c a l v a l u e s of  fii,^,  a n <  l  %t°  free  O^)  &  re  plotted  i n t e r s e c t i o n of these be  16.6,  20.1,  and  as a curves  22.1  37 respectively Figure  (1.05  sec  ^,  1.27  sec  and  1 a l s o shows t h a t , i n t h i s r e g i o n  1.40  sec  * i n conventional  of a n g u l a r v e l o c i t y , the  units).  triangular  c o n f i g u r a t i o n of t h r e e v o r t i c e s i s t h e r m o d y n a m i c a l l y s t a b l e o v e r t h r e e  co-  l i n e a r v o r t i c e s , although both c o n f i g u r a t i o n s  to  small perturbations  from  equilibrium.  are  stable with respect  38  Figure  1: The  equilibrium  f r e e e n e r g i e s of one,  v o r t i c e s . F and ft are p l o t t e d The  curves  F^, "F^,  F^,  i n dimensionless  and F^  two, units  and  three  (see t e x t ) .  are the e q u i l i b r i u m  free  of one v o r t e x , two v o r t i c e s , t h r e e c o l i n e a r v o r t i c e s , and vortices ft}, 0,2, two,  and  i n an e q u i l a t e r a l t r i a n g l e , r e s p e c t i v e l y . and  The  energies three  quantities  0 3 are the c r i t i c a l v e l o c i t i e s f o r the c r e a t i o n  three v o r t i c e s ,  respectively.  of  one,  39 7. The  CONCLUSION  r e s u l t s of s e c t i o n 5 c o n c e r n i n g  configurations  i n an  the  stability  i d e a l f l u i d bounded by  f o r the s o l u t i o n of the g e n e r a l  of the r e g u l a r  a c i r c l e are a d e p a r t u r e  problem of the  stability  r o t a t i n g c o n f i g u r a t i o n s . B e f o r e t h i s i s attempted, the sifying  a l l possible r i g i d l y  acceptance of q u a n t i z e d it  strengths  sub-problem of  s i m p l i f i e s t h i s task  i s expected t h a t the r e s u l t i n g c o n f i g u r a t i o n s w i l l  point  of a l l r i g i d l y  r o t a t i n g c o n f i g u r a t i o n s must be  vortex  polygonal  solved.  clasThe  somewhat, s i n c e  e x h i b i t some degree of  symmetry; t h e s e symmetries would i n d i c a t e the normal modes of v i b r a t i o n the s o l u t i o n of the e q u a t i o n s of motion of the in a straightforward One research  question  I t has  n a t u r a l l y from the associated  to W  o  which has  of c o n s e r v a t i o n  symmetries of the  has  law,  but  the o t h e r  the  100  conservation  case t h a t  considered.  The i n He  relevance  I I depends on  p r o p e r t i e s of He  once and  symmetry an  others.  If  the  of He  this  subject.  for a l l .  Su(1973) and  Pointin  and  bear more  fruit  physics.  of t h i s work to the a c t u a l b e h a v i o u r of v o r t e x  I I : that  inde-  prove cumbersome when l a r g e numbers of  Recent work by  the v a l i d i t y  macroscopic p r o p e r t i e s  t h a t no  t h i s i s not  Lundgren(1976) i n v o l v i n g methods of s t a t i s t i c a l mechanics may f o r workers i n l i q u i d h e l i u m  laws a r i s e  y e a r s of l i t e r a t u r e on the  to have t h i s c l e a r e d up  approach used here may  v o r t i c e s are b e i n g  be  this  of a n g u l a r moment of  t h a t i t i s a consequence of the  emerged i n the  satisfying  a u t h o r throughout  system of v o r t i c e s , but  been found. I t may  i s a f a c t , i t . h a s not  The  law  been nagging the  been demonstrated t h a t  pendent c o n s e r v a t i o n  I t would be  would p r o c e e d  manner.  i s the n a t u r e of the  circulation.  perturbations  and  of the  assumptions made c o n c e r n i n g  lines  the  t w o - f l u i d model i s adequate i n d e s c r i b i n g I I , that  the problem i s e s s e n t i a l l y two-  the  dimen-  40 s i o n a l , t h a t the normal f l u i d sible, The  and s u p e r f l u i d may be t r e a t e d as  incompres-  and t h a t the d e t a i l s o f what o c c u r s a t t h e v o r t e x c o r e i s i g n o r a b l e .  ability  t o reproduce  q u a n t i t a t i v e l y t h e spectrum  of c r i t i c a l  v e l o c i t i e s f o r v o r t e x c r e a t i o n a t low a n g u l a r v e l o c i t y  angular  seems t o support the  assumptions made - more a c c u r a t e e x p e r i m e n t a l r e s u l t s a r e needed b e f o r e a s t r o n g e r statement  can be made.  41 BIBLIOGRAPHY R.P. 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L e t . 33_, 280, (1974)  A3 APPENDIX A:  The  STREAM FUNCTION APPROXIMATION  stream f u n c t i o n f o r a v o r t e x f l u i d  FOR ENERGY CALCULATION  i n s i d e the u n i t c i c l e i s  4)(z) = - Z y In | z-z. | + Z , , l n | l - z i ' y  The  following notation z-z  m  i s introduced:  ig = ee m  z, = k  i6 . z -z, = r , e mk m k mk In t h e v i c i n i t y  ' k  i6. k  r, e  . ,ie' z -1/z. = r ,e mk m k mk  of z , t h e two terms i n t h e stream f u n c t i o n may be w r i t t e n m  l n l z - z , I = l n l (z-z )+(z -z, ) I = l n l e e ^ m + r . e ^mk| k m m k ' mk 1  1  1  1  1  = hln(e 2  r  i  ln|l-zi | k  1  + r + 2 e r , c o s ( g -6 . ) ) mk mk m mk 2  lne l n r . + 0(e/r . ) mk mk  (k=m) (k#n)  = ln|z | + ln|z-l/i | k  = ln|z" | + l n | ( z - z ^ + C z ^ l / z " ^ |  k  k  = l n r , + l n l e e ^ m + r',e^mk| k mk 1  1  = lnr, + ^ l n ( e k  2  + r ' + 2 e r \ cos(B -6', ) ) mk mk m mk 2  = lnr, + l n r ' , + 0(e/r \ ) k mk mk If  e<<r , and e<<r', then t h e stream f u n c t i o n i n t h e v i c i n i t y mk mk  written  * "V - 'k V mk k k mk k^k k ° =  n£  nr  +E  Y  lnr  +E  lnr +  (£)  of z may be ' m  44 APPENDIX B:  It  SOME RELEVANT TRIGONOMETRIC SUMS  i s n e c e s s a r y t o make u s e of t h e r e l a t i o n f  i(2WN)mk  e  1=12  and t h e s e r i e s oo  (1-x)"  2  =  oo  £(n+l)x n=0  x(l-x)"  n  1  = I x n=l  n  oo  x(l-x)  = £ nx n=l  11  E v a l u a t i o n of T : N T  f  L  i(2Tr/N)Lk  J l ( l - p ^  =  i(n+l)p e n  ^ )  2  2  P  <  1  L = 1  ' >--' 2  N  1 ( 2 7 T / N ) ( L + n ) k  k=l n=0  Nl<n l)p 6 n=0  * = 1,2,...  n  +  = N p" 2  L  I  '  *p * - N ( L - l ) p N  A=l  (1-P  N  £  £=1  N-L  0  I p  L  N-L  )  1-P  E v a l u a t i o n of S^: N-l y L  =  k=l  i(2Tr/N)Lk l-e ( l -  lim r N + N(N-l)p P+l '  i  2^ JiT  (  N L  N  ^(N-L)(L-2)  7  7  2  n  N  /  N  - N p N.2 (1-P )  ( -L)P~ ~ N  2  )  e  l  =  (  L  N  L  k  )  2  _  limr P^  1  N  + N(L-l) p  H  , L  ( l - p " ) ,  1  I  - 2N(n-L)(L-l)p" } L  (using l ' H o p i t a l ' s r  U  l  6  )  APPENDIX C:  It  THE FREE ENERGY OF REGULAR POLYGONAL  CONFIGURATIONS  i s n e c e s s a r y t o u s e t h e same r e l a t i o n used i n Appendix B and t h e s e r i e s oo  x -  l n ( l - x ) = -I 1 n=l  1  n  1  to f i n d t h e f o l l o w i n g sums N-l ln.]l-e  j  1 ( 2 T r / N  > |.=  m=l °°  N-l  oo 1  ,. lim p+1  n=l  "(N6 n,£N '  V  £N £ P_ _ P_  £  r  tl  1 z  lim  In  P+l  v  I  £  T  r  /  N  )  m  -  n  + c.c.}  1)1  £ = 1 2  1  = InN  1-p  N £  2  £  I  N = -Jj £  (  u  N N I I ln|l-pe m=l k=l  and  1  n  { ,n7 L  (using l ' H o p i t a l ' s r u l e )  i ( 2 7 T / N ) ( m  - ) k  °° n J £ i(2ir/N)(m-k)n e  +  m=l k=l n=l  , n n=l = Nln(l-p  )  " - m k IV kl Z  Z  l n  %N(N-l)lnp and  c. c,  £ = 1,2,.  n,£N '  Using these i t i s s t r a i g h t f o r w a r d  E  |j  n  { I I J e F*l n= 1 m=1  lim P+l  1(2ir/N)m  m=l  P  , lim 2  {Yl4l-Pe  ^  m  Z  +  t o show t h a t  (with z  W ^ - V J -V  -NlnN + N l n ( l - p  that L = 1 (1-z z ) = N ( l - p ) m mm ' v  n  ) - Nine  £  i =  e  (  2  7  T  / ) N  k )  46 APPENDIX D: SOME MISPRINTS IN HAVELOCK'S  The p u b l i s h e d v e r s i o n o f t h e work by Havelock(1931)  PAPER  on the s t a b i l i t y of  v a r i o u s c o n f i g u r a t i o n s o f v o r t i c e s c o n t a i n s some m i s l e a d i n g m i s p r i n t s . The important  ones a r e l i s t e d h e r e ( t h e e q u a t i o n numbers a r e H a v e l o c k ' s ) .  ( i ) In (23), t h e second term i n the l a s t ,  s h o u l d read ...  2 n-1 p -f-p d-p  (ii)  line  )  I n (25), the second term i n Q s h o u l d read a missing  " - 2 ( n - l ) ", and t h e r e i s  term: _  4  i  n n  1-P (iii)  In ( 2 8 ) , the second term s h o u l d from  read  " - 2 ( n - l ) ", but the term m i s s i n g  (25) has been r e p l a c e d . The f i r s t  term s h o u l d r e a d "  (%n)  2  ".  

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