UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Vortex motion in thin films Hally, David 1980

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1980_A1 H34.pdf [ 5MB ]
Metadata
JSON: 831-1.0085760.json
JSON-LD: 831-1.0085760-ld.json
RDF/XML (Pretty): 831-1.0085760-rdf.xml
RDF/JSON: 831-1.0085760-rdf.json
Turtle: 831-1.0085760-turtle.txt
N-Triples: 831-1.0085760-rdf-ntriples.txt
Original Record: 831-1.0085760-source.json
Full Text
831-1.0085760-fulltext.txt
Citation
831-1.0085760.ris

Full Text

VORTEX MOTION IN THIN FILMS by DAVID HALLY B . S c . The U n i v e r s i t y o f T o r o n t o , 1976 A THES IS SUBMITTED IN PART IAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES THE DEPARTMENT OF PHYSICS We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERS ITY OF BR IT I SH COLUMBIA D e c e m b e r , 1979 0 D a v i d H a l l y , 1979 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Co lumb ia , I a g ree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s tudy . I f u r t h e r agree 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 c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d that c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i thout my w r i t t e n p e r m i s s i o n . Department o f P 'ri I C ^ The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1WS Date Apr. 23. ? 19 S O 6 ABSTRACT The c l a s s i c a l t h e o r y o f r e c t i l i n e a r v o r t e x m o t i o n has been g e n e r a l i z e d t o i n c l u d e v o r t i c e s i n t h i n f l u i d s o f v a r y i n g d e p t h on c u r v e d s u r f a c e s . The e q u a t i o n s o f m o t i o n a r e e x a m i n e d t o l o w e s t o r d e r i n a p e r t u r b a t i o n e x p a n s i o n i n w h i c h t h e d e p t h o f f l u i d i s c o n s i d e r e d s m a l l i n c o m p a r i s o n w i t h t h e p r i n c i p a l r a d i i o f c u r v a t u r e o f t h e s u r f a c e . E x i s t e n c e o f a g e n e r a l i z e d v o r t e x s t r e a m f u n c t i o n i s p r o v e d and u s e d t o g e n e r a t e c o n s e r v a -t i o n l a w s . A number o f s i m p l e v o r t e x s y s t e m s a r e d e s c r i b e d . In p a r t i c u l a r , c r i t e r i a f o r t h e s t a b i l i t y o f r i n g s o f v o r t i c e s on s u r f a c e s o f r e v o l u t i o n a r e f o u n d . In c o n t r a d i s t i n c t i o n t o t h e r e s u l t o f von K a r m a n , d o u b l e r i n g s ( v o r t e x s t r e e t s ) i n b o t h s t a g g e r e d and s y m m e t r i c c o n f i g u r a t i o n s may be s t a b l e . The e f f e c t s o f f i n i t e c o r e s i z e a r e e x a m i n e d . D e p a r t u r e s f r o m r a d i a l s y m m e t r y i n c o r e v o r t i c i t y d i s t r i b u t i o n s a r e shown t o i n t r o d u c e s m a l l w o b b l e s i n t h e v o r t e x m o t i o n . The c a s e o f an e l l i p t i c a l c o r e i s t r e a t e d i n d e t a i l . A p p l i c a t i o n s o f t h e t h e o r y t o a t m o s p h e r i c c y c l o n e s and s u p e r f l u i d v o r t i c e s a r e d i s c u s s e d . 0 CONTENTS Page ABSTRACT i i L I ST OF FIGURES v L I ST OF TABLES v i ACKNOWLEDGEMENTS v i i I. INTRODUCTION 1 I I . COORDINATES 6 1. H a r m o n i c C o o r d i n a t e s 6 2. S u r f a c e s o f R e v o l u t i o n 7 3. T h i n F i l m C o o r d i n a t e s 12 4. V e c t o r C o m p o n e n t s : N o t a t i o n 14 I I I . IDEAL FLUIDS 15 1. F i e l d E q u a t i o n s f o r an I d e a l F l u i d 15 2. The K e l v i n C i r c u l a t i o n T h e o r e m 17 3. The T h i n F i l m A p p r o x i m a t i o n 19 IV . THE MOTION OF VORTICES IN THIN FILMS . . . . 23 1. The V o r t e x V e l o c i t y F i e l d 23 2. The V e l o c i t y o f a V o r t e x i n a F l u i d o f U n i f o r m Depth 30 3. The V e l o c i t y o f a V o r t e x i n a F l u i d o f V a r y i n g Depth 36 4. The V o r t e x S t r e a m f u n c t i o n 43 5. C o n f o r m a l T r a n s f o r m a t i o n s 48 6. C o n s t a n t s o f t h e M o t i o n 50 V. S IMPLE VORTEX SYSTEMS 54 1. The M o t i o n o f a S i n g l e V o r t e x : No E x t e r n a l V e l o c i t y F i e l d 54 a) U n i f o r m D e p t h , C u r v e d S u r f a c e s 54 b) P l a n e S u r f a c e , N o n - U n i f o r m Depth 55 2. The M o t i o n o f a S i n g l e V o r t e x i n a Uni f o r m S t r e a m 58 3. The M o t i o n o f a V o r t e x P a i r : Yi = - Y 2 61 i i i i v C o n t e n t s ( C o n t ' d ) p a g e 4 . The S t a b i l i t y o f a S i n g l e R i n g o f V o r t i c e s on a S u r f a c e o f R e v o l u t i o n 66 5. The S t a b i l i t y o f V o r t e x S t r e e t s on S u r f a c e s o f R e v o l u t i o n 72 a) S t a g g e r e d V o r t e x S t r e e t s 73 b) S y m m e t r i c V o r t e x S t r e e t s 81 c) E x a m p l e : The C y l i n d e r 85 d) E x a m p l e : The S p h e r e 90 V I . VORTICES WITH F IN ITE CORES 96 1. The P o s i t i o n and V e l o c i t y o f a V o r t e x 96 2. C i r c u l a r C o r e s 98 3. The V a l i d i t y o f t h e C i r c u l a r A p p r o x i m a t i o n 102 4. E l l i p t i c a l C o r e s 104 5. P e r t u r b a t i o n s o f P l a n e S o l u t i o n s 113 V I I . APPL ICAT IONS TO REAL FLUIDS 118 1. A t m o s p h e r i c C y c l o n e s 118 2. S u p e r f l u i d V o r t i c e s 122 V I I I . CONCLUSION . ., . 124 B IBL IOGRAPHY 126 APPENDICES 128 A. S y m p l e c t i c S y s t e m s 128 B. E v a l u a t i o n o f Sums 129 L IST OF FIGURES Page I. The S u r f a c e o f R e v o l u t i o n p = f ( z ) 8 I I . V o r t e x Co re i n a F l u i d o f V a r y i n g Depth . . . 37 I I I . S t r e a m l i n e s N e a r t h e Co re o f a V o r t e x i n a F l u i d o f V a r y i n g Depth 40 IV. F l u i d w i t h D e p r e s s i o n n e a r t h e O r i g i n . . . . 56 V. F l u i d w i t h S u r f a c e C u r v a t u r e n e a r t h e O r i g i n 59 V I . The P a t h o f a V o r t e x P a i r 65 V I I . S t a g g e r e d V o r t e x S t r e e t : The Modes M=N . . . 78 V I I I . S u r f a c e s f o r w h i c h t h e Mode M=N has D e f i n i t e S t a b i l i t y 79 IX . S t a g g e r e d V o r t e x S t r e e t : The Modes M=%N . . 80 X. S y m m e t r i c V o r t e x S t r e e t s : The Modes M=N . . 84 X I . S y m m e t r i c V o r t e x S t r e e t s : The Modes M=%N . . 86 X I I . y + ( x ) and y " ( x ) 89 X I I I . A V o r t e x w i t h an E l l i p t i c a l Co re 110 X IV . The P a t h o f a V o r t e x w i t h an E l l i p t i c a l Co re 111 XV. V e l o c i t y and V o r t i c i t y o f a Q u a s i - S t e a d y V o r t e x 116 v L I ST OF TABLES Page 1. R e g i o n s o f S t a b i l i t y o f S t a g g e r e d V o r t e x S t r e e t s on a C y l i n d e r 90 2. R e g i o n s o f S t a b i l i t y o f S y m m e t r i c V o r t e x S t r e e t s on a S p h e r e 91 3. R e g i o n s o f S t a b i l i t y o f S t a g g e r e d V o r t e x S t r e e t s on a S p h e r e 94 v i ACKNOWLEDGEMENT I w o u l d l i k e t o t h a n k P r o f . F .A. K a e m p f f e r f o r i n i t i a t i n g t h i s work and f o r h i s a d v i c e and c r i t i c i s m t h r o u g h o u t . The f i n a n c i a l a s s i s t a n c e o f b o t h t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a ( M a c m i l l a n G r a d u a t e F e l l o w s h i p ) and t h e N a t u r a l S c i e n c e s and E n g i n e e r i n g R e s e a r c h C o u n c i l o f Canada ( P o s t g r a d u a t e F e l l o w s h i p ) i s g r a t e f u l l y a c k n o w l e d g e d . v i i 1 I. INTRODUCTION In 1858 H e l m h o l t z p u b l i s h e d a p a p e r w h i c h m a r k e d t h e b e g i n n i n g o f t h e s t u d y o f v o r t i c i t y i n f l u i d s . He showed t h a t i n a f l u i d i n w h i c h t h e momentum p r o d u c t i o n a t e a c h p o i n t i s t h e g r a d i e n t o f a s c a l a r p o t e n t i a l , t h e v o r t e x l i n e s a r e a d v e c t e d . H e l m h o l t z wen t on t o show t h a t i f t h e f l u i d i s i n c o m p r e s s i b l e , t w o - d i m e n s i o n a l , and i t s v o r t i c i t y i s z e r o e x c e p t a t N i s o l a t e d p o i n t s , t h e n t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n s d e s c r i b i n g t h e m o t i o n o f t h e f l u i d can be r e d u c e d t o a 2Nth o r d e r s y s t e m o f o r d i n a r y d i f f e r e n t i a l e q u a t i o n s d e s c r i b i n g t h e p o s i t i o n s o f t h e p o i n t s o f v o r t i c i t y . A l t h o u g h t h e o r i g i n a l e q u a t i o n s a r e n o n - l i n e a r , e a c h p o i n t o f v o r t i c i t y i s a s s o c i a t e d w i t h a v e l o c i t y f i e l d w h i c h o b e y s t h e p r i n c i p l e o f s u p e r p o s i t i o n : t h a t i s , t h e t o t a l v e l o c i t y f i e l d i s t h e sum o f t h e v e l o c i t y f i e l d s a s s o c i a t e d w i t h t h e p o i n t s o f v o r t i c i t y . Each p o i n t o f v o r t i c i t y , a l o n g w i t h i t s v e l o c i t y f i e l d , i s c a l l e d a v o r t e x . In p r a c t i c e , t h e v o r t i c i t y o f a v o r t e x i s n o t c o n f i n e d t o a s i n g l e p o i n t , b u t i s d i s t r i b u t e d o v e r some s m a l l r e g i o n known as t h e c o r e . The p o s i t i o n and v e l o c i t y ( n o t t o be c o n f u s e d w i t h t h e v e l o c i t y f i e l d ) o f a v o r t e x r e f e r t o t h e p o s i t i o n and r a t e o f change o f p o s i t i o n o f t h e c o r e . L o r d K e l v i n ( 1 869 ) l a t e r r e f o r m u l a t e d H e l m h o l t z ' t h e o r e m s and i n t r o d u c e d t h e i d e a o f c i r c u l a t i o n ; t h e c i r c u l a -t i o n a r o u n d a c o n t o u r C i s : Tc = jv_ • ds_ , whe re v_ i s t h e f l u i d v e l o c i t y . K e l v i n showed t h a t , u n d e r s i m i l a r a s s u m p t i o n s t o t h o s e " made by H e l m h o l t z , t h e c i r c u l a t i o n o f any a d v e c t e d 2 c o n t o u r i s c o n s t a n t i n t i m e . M o r e o v e r , i f C and C b o t h e n c l o s e t h e same s e t o f v o r t i c e s t h e n = r ^ , . The s t r e n g t h , y , o f a v o r t e x i s p r o p o r t i o n a l t o t h e c i r c u l a t i o n a r o u n d any c o n t o u r e n c l o s i n g t h a t v o r t e x and no o t h e r . Rou th ( 1 8 8 1 ) f i r s t i n t r o d u c e d t h e v o r t e x s t r e a m -f u n c t i o n (a H a m i l t o n i a n f o r t h e s y s t e m o f e q u a t i o n s g o v e r n i n g t h e v o r t e x m o t i o n ) a l t h o u g h h i s work was much g e n e r a l i z e d by K i r c h o f f - ( 1 8 7 6 ) , L a g a l l y ( 1 9 2 1 ) , M a s o t t i ( 1 931 ) and L i n ( 1 9 4 3 ) . As i n any H a m i l t o n i a n s y s t e m , t h e r e a r e c o n s e r v a t i o n l a w s a s s o c i a t e d w i t h i n v a r i a n c e s o f t h e H a m i l t o n i a n u n d e r i n f i n i t -e s i m a l c o o r d i n a t e t r a n s f o r m a t i o n s . F o r v o r t e x s y s t e m s i n v a r i a n c e u n d e r t r a n s l a t i o n s l e a d s t o c o n s e r v a t i o n o f c e n t r e o f c i r c u l a t i o n , u n d e r r o t a t i o n s t o c o n s e r v a t i o n o f moment o f c i r c u l a t i o n , and u n d e r t i m e t r a n s l a t i o n s t o c o n s e r v a t i o n o f t h e v o r t e x s t r e a m f u n c t i o n ( B a t c h e l o r ( 1 9 6 7 ) , p . 5 3 0 ) . In a d d i t i o n , t h e r e i s a n o t h e r c o n s e r v e d q uan t i t y known as t h e a n g u l a r moment o f c i r c u l a t i o n . I t i s d i s c u s s e d i n d e t a i l i n S e c t i o n I V . 6 . S i n c e t h e p u b l i c a t i o n o f L i n ( 1 9 4 3 ) t h e r e has been l i t t l e work on t h e f o r m a l t h e o r y o f v o r t e x m o t i o n . H o w e v e r , much has been w r i t t e n on t h e a p p l i c a t i o n t o r e a l f l u i d s y s t e m s ; i n p a r t i c u l a r , t h e p r e d i c t i o n o f O n s a g e r ( 1949) t h a t s u p e r -f l u i d H e l l w o u l d s u p p o r t v o r t i c e s o f q u a n t i z e d c i r c u l a t i o n s p a r k e d r e n e w e d i n t e r e s t i n t h e s u b j e c t w h i c h p e r s i s t s t o t h i s d a y . O t h e r a p p l i c a t i o n s a r e t o a t m o s p h e r i c w e a t h e r s y s t e m s and t o f l o w s p a s t b l u f f b o d i e s ( e . g . , t h e Karman v o r t e x s t r e e t ) . 3 A l m o s t a l l t h e work on v o r t i c e s t o d a t e has a s sumed t h a t t h e v o r t i c e s a r e r e c t i l i n e a r : t h a t i s i n f i n i t e l y l o n g and p a r a l l e l so t h a t t h e p r o b l e m i s r e d u c e d t o t w o - d i m e n s i o n a l f l o w i n t h e p l a n e . Lamb ( 1 9 1 6 ) , i n h i s w e l l - k n o w n t e x t , b r i e f l y o u t l i n e s a me thod f o r d e t e r m i n i n g t h e m o t i o n o f v o r t i c e s on a c u r v e d s u r f a c e u n d e r t h e a s s u m p t i o n t h a t t h e f l u i d d e p t h i s u n i f o r m and s m a l l i n c o m p a r i s o n w i t h t h e p r i n c i -p a l r a d i i o f c u r v a t u r e o f t h e s u r f a c e . I t i s t h e p u r p o s e o f t h i s t h e s i s t o e x a m i n e s u ch s y s t e m s i n more d e t a i l and t o g e n e r a l i z e f u r t h e r t o f l u i d s o f n o n - u n i f o r m d e p t h b u t i n w h i c h t h e d e p t h v a r i a t i o n i s s m a l l . A f t e r d e f i n i n g c o o r d i n a t e s w h i c h p r o v e u s e f u l i n l a t e r s e c t i o n s ( S e c t i o n I I ) , t h e e q u a t i o n s g o v e r n i n g t h e m o t i o n o f an i d e a l f l u i d (one i n w h i c h t h e momentum p r o d u c t i o n i s t h e g r a d i e n t o f a s c a l a r ) a r e d e r i v e d and s u b j e c t e d t o a p e r t u r b a t i o n scheme i n w h i c h t h e d e p t h o f t h e f l u i d and t h e v e r t i c a l c omponen t o f t h e v e l o c i t y f i e l d a r e a s s umed s m a l l ( S e c t i o n I I I ) . V o r t e x s o l u t i o n s t o t h e l o w e s t o r d e r e q u a t i o n s o f m o t i o n a r e e x a m i n e d i n S e c t i o n IV u n d e r t h e a d d i t i o n a l a s s u m p t i o n o f i n c o m p r e s s i b l e f l o w . S y s t e m s o f v o r t i c e s a r e shown t o be g o v e r n e d by a g e n e r a l i z e d v o r t e x s t r e a m f u n c t i o n w h i c h c a s t s t h e e q u a t i o n s o f m o t i o n i n t o s y m p l e c t i c f o r m . A r e l a t i o n b e t w e e n t h e v o r t e x s t r e a m f u n c t i o n and t h e k i n e t i c e n e r g y i s shown and c o n s e r v a t i o n l aw s c o r r e s p o n d i n g t o i n v a r i a n c e s u n d e r s i m p l e i n f i n i t e s i m a l t r a n s f o r m a t i o n s a r e g e n e r a t e d . In p a r t i c u l a r , t h e c o n s e r v a t i o n o f a n g u l a r moment o f c i r c u l a t i o n i s shown t o be r e l a t e d t o i n v a r i a n c e u n d e r 4 s c a l e t r a n s f o r m a t i o n s . When t h e f l u i d d e p t h i s u n i f o r m , i t i s shown t h a t t h e v o r t e x s t r e a m f u n c t i o n t r a n s f o r m s s i m p l y u n d e r c o n f o r m a l t r a n s f o r m a t i o n s o f c o o r d i n a t e s . In S e c t i o n V, t h e e f f e c t s o f s u r f a c e c u r v a t u r e and d e p t h v a r i a t i o n on some s i m p l e v o r t e x s y s t e m s a r e e x a m i n e d . The m o t i o n o f a s i n g l e v o r t e x and o f p a i r s o f v o r t i c e s p a s t l o c a l i z e d s u r f a c e bumps and d e p r e s s i o n s i n d e p t h a r e a n a l y z e d . A l s o , t h e s t a b i 1 i t i e s o f some 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 o f v o r t i c e s on s u r f a c e s o f r e v o l u t i o n a r e d e t e r m i n e d . The r e s u l t s a r e g e n e r a l i z a t i o n s o f t h o s e f i r s t d e r i v e d by L o r d K e l v i n ( 1 8 7 8 ) , Thomson ( 1 883 ) and von Karman ( 1 9 1 2 ) . In p a r t i c u l a r , i t i s f o u n d t h a t , i n c o n t r a d i s t i n c t i o n t o t h e r e s u l t s o f von K a r m a n , d o u b l e r i n g s o f s y m m e t r i c a l l y p l a c e d v o r t i c e s ( v o r t e x s t r e e t s ) may be s t a b l e w h i l e s i m i l a r s t a g g e r e d r i n g s a r e u n s t a b l e . In S e c t i o n V I , t h e e f f e c t s o f f i n i t e d i s t r i b u t i o n s o f v o r t i c i t y w i t h i n t h e v o r t e x c o r e s a r e e x a m i n e d . I t i s a r g u e d t h a t s m a l l d e p a r t u r e s f r o m c i r c u l a r c o r e s i n t r o d u c e w o b b l e s i n t o t h e m o t i o n o f e a c h v o r t e x , b u t t h a t , f o r t i m e s l o n g enough t h a t t h e v o r t e x t r a v e l s t h e o r d e r o f t h e mean v o r t e x s e p a r a t i o n , t h e s y s t e m a t i c d r i f t o f t h e c o r e c l o s e l y a p p r o x i m a t e s t h a t o f a c i r c u l a r c o r e . The s i m p l e s t c a s e o f e l l i p t i c a l c o r e s i s e x a m i n e d i n d e t a i l . The e v o l u t i o n o f t h e c o r e can be e x a m i n e d i n a p e r t u r b -a t i o n scheme h a v i n g a v o r t e x i n t h e p l a n e as i t s l o w e s t o r d e r s o l u t i o n . I t i s shown t h a t i f a v o r t e x i s t o r e m a i n c i r c u l a r f o r an a p p r e c i a b l e l e n g t h o f t i m e t h e n i t s v o r t i c i t y d i s t r i b u -t i o n must have a p r e c i s e f o r m . 5 The work o f S e c t i o n s 1 1 - VI was m o t i v a t e d l a r g e l y by an a t t e m p t t o p r o v i d e a f i r s t o r d e r mode l f o r a t m o s p h e r i c c y c l o n e s i n w h i c h t h e c u r v a t u r e o f t h e e a r t h i s t r e a t e d e x a c t l y , n o t j u s t as a l i n e a r p e r t u r b a t i o n o f a p l a n e o r a c o n e . A l m o s t e v e r y a n a l y t i c mode l o f t e r r e s t r i a l c y c l o n e s i s d e r i v e d v i a t h e 6 - p l a n e a p p r o x i m a t i o n i n w h i c h e f f e c t s o f s u r f a c e c u r v a -t u r e a r e i n c l u d e d o n l y i n t h e l a t i t u d i n a l v a r i a t i o n o f t h e C o r i o l i s p a r a m e t e r . The B - p l a n e a p p r o x i m a t i o n i s n o t c o n s i s t -e n t ( s e e , e . g . , V e r o n i s ( 1 9 6 3 a ) ) s i n c e v a r i a t i o n s o f t h e C o r i o l i s p a r a m e t e r a r e i m p o r t a n t o n l y i f t h e s y s t e m i s l a r g e enough t h a t o t h e r e f f e c t s o f s u r f a c e c u r v a t u r e a r e a l s o i m p o r t a n t . Our a p p r o a c h i s f r o m t h e o t h e r d i r e c t i o n ; C o r i o l i s f o r c e s have been n e g l e c t e d e n t i r e l y ( c l e a r l y a d r a s t i c a s s u m p -t i o n i f one r e a l l y e x p e c t s a good mode l o f a t e r r e s t r i a 1 c y c l o n e ) i n an a t t e m p t t o h i g h l i g h t t h e e f f e c t s o f s u r f a c e c u r v a t u r e a l o n e . A good mode l o f a l a r g e c y c l o n e must a f f e c t some c o m p r o m i s e . The p r o b l e m s i n h e r e n t i n a s i m p l e v o r t e x mode l i n c l u d i n g b o t h C o r i o l i s and s u r f a c e c u r v a t u r e e f f e c t s a r e d i s c u s s e d i n S e c t i o n V I I . l . A n o t h e r p r o b l e m m o t i v a t i n g t h i s work i s t h e f o l l o w i n g . A r o t a t i n g b u c k e t o f l i q u i d H e l l i s known t o s u p p o r t s u p e r -f l u i d v o r t i c e s . I t s s u r f a c e i s a l s o known t o be a p a r a b o l o i d o f r e v o l u t i o n ( O s b o r n e e t . a l . ( 1 9 6 3 ) ) . How does t h e v a r i a t i o n i n d e p t h o f l i q u i d h e l i u m a f f e c t t h e m o t i o n and d i s t r i b u t i o n o f t h e v o r t i c e s ? In S e c t i o n V I I . 2 t h e e x t e n t t o w h i c h t h e t h e o r y o f S e c t i o n s 11 - VI i s a p p l i c a b l e t o s u p e r f l u i d v o r t i c e s i s d i s c u s s e d . 6 I I . COORDINATES T h r o u g h o u t t h i s work i t w i l l be f o u n d t h a t a p a r t i -c u l a r c h o i c e ! o f c o o r d i n a t e s i s n e c e s s a r y t o make t h e m a t h e -m a t i c s t r a c t a b l e . H e r e ' t h e p r o p e r t i e s o f t h e s e c o o r d i n a t e s a r e d i s c u s s e d a t l e n g t h . I I . l H a r m o n i c C o o r d i n a t e s L e t M be a t w o - d i m e n s i o n a l , o r i e n t e d , R i e m a n n i a n m a n i f o l d w i t h m e t r i c g . . . I t i s p o s s i b l e t o c h o o s e c o o r d i -n a t e s ( x , y ) s u c h t h a t t h e m e t r i c has t h e f o r m : g^-U .y) = 6 i j . h 2 ( x , y ) ( I I . 1 . 1 ) whe re h ( x , y ) i s a r e a l - v a l u e d d i f f e r e n t i ab l e n o n - n e g a t i v e f u n c t i o n ( s e e , f o r e x a m p l e , E i s e n h a r t ( 1 9 0 9 ) ) . ( x , y ) a r e c a l l e d h a r m o n i c c o o r d i n a t e s . In g e n e r a l we s h a l l o n l y be c o n c e r n e d w i t h some s u b - d o m a i n D o f M (D i s t h a t p a r t o f t h e s u r f a c e o v e r w h i c h t h e f l u i d f l o w s ) . F o r s i m p l i c i t y i t w i l l a l w a y s be a s s umed t h a t D i s p a r a m e t r i z e d c o m p l e t e l y and u n a m b i g u o u s l y by ( x , y ) . T h i s amount s t o a r e s t r i c t i o n t o t h o s e s u r f a c e s t o p o l o g i c a l l y e q u i v a l e n t t o m u l t i p l y c o n n e c t e d s u b - d o m a i n s o f t h e c o m p l e x p l a n e o r s p h e r e . More c o m p l e x s u r f a c e s ( e . g . , a t o r u s ) w i l l n o t be c o n s i d e r e d . P o l a r h a r m o n i c c o o r d i n a t e s a r e d e f i n e d b y : x = r c o s (J> ; y = r s i n ty ( 1 1 . 1 . 2) The l i n e e l e m e n t i n . t e r m s (r,cf>) i s : d s 2 = h * 2 ( r , c j > ) ( d r 2 + rzdty2) ( I I . 1 . 3 ) w i t h h*(r,<J>) = h ( x , y ) . In f u t u r e t h e a s t e r i s k w i l l be d r o p p e d i f t h e r e i s no c h a n c e o f c o n f u s i o n b e t w e e n h * and h . I I . 2 S u r f a c e s o f R e v o l u t i o n The l i n e e l e m e n t o f a s u r f a c e o f r e v o l u t i o n can be p u t i n t h e f o r m : d s 2 = g 2 ( u ) d u 2 .+ g | ( u ) d c t 2 ( I I . 2 . 1 ) whe re <J) i s an a n g u l a r c o o r d i n a t e o f p e r i o d 2TT . L e t r = r ( u ) and r e q u i r e t h a t (r,d>) be h a r m o n i c p o l a r s . T h e n : d s 2 = h 2 ( r ) ( d r 2 + r 2d<j) 2) = h 2 ( r ) ( ( r ' ( u ) ) 2 d u 2 + r 2 (u )d<J ) 2 ) = g j ( u ) d u 2 + g 2 ( u ) d d , 2 - ( I I . 2 . 2 ) so t h a t : r ' ( u ) = 9 u ( u ) r ( u) g ( J ) (u) ( I I . 2 . 3 ) when ce : r ( u ) = exp U 9 u ( s ) 9 c p t s J ds ( I I . 2 . 4 ) h ( r ( u ) ) = -^y ( I I . 2 . 5 ) I t w i l l p r o v e c o n v e n i e n t t o d e f i n e i \ h ' ( r ) r , . p ( r ) = - r h r " + 1 ( I I . 2 . 6 ) Then : p ( r ( u ) ) = _d_ rg<b( g ( J ) (u j r ' ( u) du [ r ( u) r 2 ( u ) + 1 • gj,(u) = g T u T 3 u ( I I . 2 . 7 ) Al s o : rp 1 ( r ) - r ( u) gj>(u)i r ' (u ) du [ g u ( u ) g<f>(u) _d_ f-gj>(u) g u ( u ) du [ g u ( u ) J ( I I . 2 . 8 ) The G a u s s i a n c u r v a t u r e o f a s u r f a c e i s d e f i n e d by * 2 i k i n L 9x' 3 r 1 k + r h r m _ r n r m 9 x n im kn i k mn_ ( 1 1 . 2 . 9 ) w i t h l n 9 2 .km 8 9 j m 9 x n 3g mn 3 9 i n Sx"1 8xm ( I I . 2 . 1 0 ) In h a r m o n i c c o o r d i n a t e s : 8 x i n 3X 1 -1 V2£vrch ( I I . 2 . 1 1 ) F o r a s u r f a c e o f r e v o l u t i o n h = h ( r ) and h 2 r d ^ r. dh_ h d r P ' ( r ) r h 2 ( r ) ( I I . 2 . 1 2 ) The above d e f i n i t i o n s a p p l y f o r any m a n i f o l d s u c h t h a t h = h ( r ) . F o r p h y s i c a l a p p l i c a t i o n s one i s g e n e r a l l y o n l y i n t e r e s t e d i n t h o s e m a n i f o l d s w h i c h can be i m b e d d e d i n "IR3. We t h e r e f o r e d e t e r m i n e r , h ( r ) and p ( r ) f o r s u r f a c e s o b t a i n e d by t h e r e v o l u t i o n o f a f u n c t i o n a b o u t an a x i s . L e t p , cj) and be c y l i n d r i c a l c o o r d i n a t e s i n IR 3 and c o n s i d e r t h e s u r f a c e o f r e v o l u t i o n d e f i n e d b y : p = f ( z ) ( s e e F i g u r e I ) . One c a n use z and d> t o p a r a m e t r i z e t h e s u r f a c e The l i n e e l e m e n t i n t h e s e c o o r d i n a t e s i s : 9 F i g . I : The S u r f a c e o f R e v o l u t i o n p = f ( z ) 10 d s 2 = (1 + ( f ( z ) ) 2 ) d z 2 + f 2 (z)d<j) 2 ( I I . 2 . 1 3 ) whence, u s i n g ( I I . 2 . 4 - 6 ) rz r = exp r h ( r ) = p ( r ) = (1 + ( f ' ( s ) ) 2 r 2 d s f j i l Z o f ( z ) f ( z ) (1 + ( f ' U ) ) * ) * ( I I . 2 . 1 4 ) ( I I . 2 . 1 5 ) ( I I . 2 . 1 6 ) N o t i c e t h a t -1 £ p ( z ) _< 1 . I f the s l o p e o f f ( z ) i s tanO then p ( z ) = s i n e . S i m i l a r l y , one can d e f i n e a s u r f a c e of r e v o l u t i o n by: z = b(p) . The l i n e e lement i s : d s 2 = (1 + ( b ' ( p ) ) 2 ) d p 2 + p2d<j>: and r = exp Po (1 + ( b 1 ( s ) ) 2 )"2ds s r h ( r ) = p p( r ) 1 (1 + ( b ' ( p ) ) 2 ) ^ ( I I .2.17) ( I I . 2 . 1 8 ) ( I I . 2 . 1 9 ) ( I I .2.20) I f the s l o p e of b(p) i s tane then p ( r ) = cose . The f u n c t i o n -jp(&wh(r)) w i l l a l s o be of impo r tance 1 a t e r . No t i ce that,: j L ( * « h ( r ) ) = 1) < o ( I I . 2 . 2 1 ) s i n c e : p ( r ) < 1 . 11 S p e c i a l Ca se s a) P l a n e F o r t h e p l a n e b ( p ) = c o n s t , w h e n c e : r - p ; h ( r ) = 1 ; r p 1 ( r ) = 0 ; ( I I . 2 . 2 2 ) K = 0 b) C y l i n d e r F o r t h e c y l i n d e r f ( z ) = R = c o n s t , w h e n c e : r = e x p ( z / R ) ; h ( r ) . = R/r = R e x p ( - z / R ) ; p ( r ) = 0 ; r p ' ( r ) = 0 ; K = 0 ( I I . 2 . 2 3 ) c ) S p h e r e F o r t h e s p h e r e g Q ( 0 ) = R ; 9 . ( 9 ) = R s i n B .where o ( p 0 i s t h e c o l a t i t u d e . T h e n : r = t a n % 9 ; h ( r ) = ( i + yU) = 2 c o s 2 % 0 ; M r 2 ) 4 r P ( r ) = JT+YTJ = c o s 8 ; r p ' ( r ) = (1 + r 2 ) = - s i n 2 e K = R^T ( I I . 2 . 2 4 ) 12 I I . 3 C o o r d i n a t e s f o r a T h i n F i l m C o n s i d e r , now a f l u i d bounded by two s u r f a c e s , Mi and M 2 h a v i n g s i m i l a r t o p o l o g i c a l c h a r a c t e r i s t i c s . The d i s t a n c e b e t w e e n Mi and M 2 i s much s m a l l e r t h a n t h e i r r a d i i o f c u r v a t u r e : t h e f l u i d i s " t h i n " . Choose c o o r d i n a t e s x , y on Mi s u ch t h a t x , y a r e h a r m o n i c . Choose a c o o r d i n a t e z so t h a t Mi and M 2 a r e s u r f a c e s o f c o n s t a n t z . F o r s i m p l i c i t y we c h o o s e them t o be z = 0 , z = l . The x , y c o o r d i n a t e s on e a c h o f t h e s u r f a c e s z = c o n s t a n t a r e d e f i n e d by r e q u i r i n g t h a t t h e l i n e s o f c o n s t a n t ( x , y ) a r e o r t h o g o n a l t o t h e s u r f a c e s o f c o n s t a n t z . The l i n e e l e m e n t i n 1 '• t h e s e c o o r d i n a t e s t h e n has t h e f o r m : d s 2 = g d x 2 + 2g dxdy + g d y 2 + g d z 2 I I . 3 . 1 3 x x 3 x y 3 y y J s z z S i n c e t h e f l u i d i s t h i n t h e z c o o r d i n a t e can be c h o s e n so t h a t g z z i s n e a r l y i n d e p e n d e n t o f z . 3g 3 z z g z z 9z << 1 n . 3.2 The d e p t h o f t h e f l u i d , k * ( x , y ) i s t h e n : 1 k * ( x , y ) /g 1 dz ? /g 1 I I . 3 . 3 0 zz 3ZZ The t h i n n e s s o f t h e f l u i d r e q u i r e s t h a t t h e d e p t h i s s m a l l i n c o m p a r i s o n w i t h t h e d i s t a n c e o v e r w h i c h g ,g „ ,g v a r y y a x x ' a x y a y y J a p p r e c i a b l y . 13 3 z z g r a d g g 3 3 x x 'XX << 1 ' z z 5 y y g r a d g y y << 1 ' z z ' x y g r a d g xy << 1 ( I I . 3 . 3 ) The z c o m p o n e n t o f g r a d i s : g r a d. /g—« 9z k * ( x , y ) 3z ( I I . 3 . 4 ) T h e r e f o r e g v v , g v . . ,g__, a r e a p p r o x i m a t e l y i n d e p e n d e n t o f z . X X Xjf z z M o r e o v e r , s i n c e on Mi t h e ( x , y ) c o o r d i n a t e s a r e h a r m o n i c : 9 x x ~ n 2 ( x ' y ) ' 9 y y ~ h 2 ( x , y ) and g n e a r l y v a n i s h e s a x y J ( I I . 3 . 5 ) The h o r i z o n t a l c o m p o n e n t s o f g r a d a r e t h e n a p p r o x i m a t e l y ( I I . 3 . 6 ) x 3 y 9 g r a d h o r ~ h ( x , y ) 8x + h ( x , y ) 8y and t h e r e q u i r e m e n t s t h a t t h e f l u i d i s t h i n become : k* 3h W 3x" << 1 k* 3h_ F 2 " dy « 1 ( r i . 3 .7 ) I t w i l l a l s o be a s sumed t h a t t h e d e p t h o f t h e f l u i d v a r i e s a p p r e c i a b l y o n l y o v e r d i s t a n c e s much l a r g e r t h a n t h e d e p t h : i . e . , h 3x << 1 I H i* , h 3y << 1 ( I I . 3 . 8 ) E q u a t i o n s I I . 3 . 2 . 8 s u g g e s t a p e r t u r b a t i o n scheme whe r e b y : 14 g x x ( x , y , z ) = h ^ x . y ) + A g ^ 1 ) (x ,y , z ) + . . . ( I I . 3 . 8 a ) g x y ( x , y , z ) - X g ^ U . y . z ) + . . . ( I I . 3 . 8 b ) g y y ( x , y , z ) = h 2 ( x , y ) + A g y y 1 > (x ,y , z ) + . . . ( I I . 3 . 8 c ) g z z ( x , y , z ) = X k 2 ( x , y ) + l29z[Z) (x ,y , z ) + . . . ( I I . 3 . 8 d ) whe re A i s a s m a l l p a r a m e t e r w h i c h m e a s u r e s t h e r a t i o o f t h e v e r t i c a l and h o r i z o n t a l s c a l e s o f t h e s y s t e m . The d e p t h o f t h e f l u i d , t o l o w e s t o r d e r , i s k * ( x , y ) = X k ( x , y ) ( I I . 3 . 9 ) 15 I I . 4 V e c t o r C o m p o n e n t s : N o t a t i o n In any o r t h o g o n a l c o o r d i n a t e s y s t e m a v e c t o r c an be r e p r e s e n t e d i n t h r e e w a y s : c o n t r a v a r i a n t c o m p o n e n t s , c o v a r i a n t c o m p o n e n t s , and p h y s i c a l c o m p o n e n t s . C o n t r a v a r i a n t c o m p o n e n t s w i l l a l w a y s be d e n o t e d by u p p e r c a s e l e t t e r s w i t h s u p e r s c r i p t s ( e . g . , V 1 ) , c o v a r i a n t c o m p o n e n t s by u p p e r c a s e l e t t e r s w i t h s u b s c r i p t s ( e . g . , V^) , and p h y s i c a l c o m p o n e n t s by l o w e r c a s e l e t t e r s ( e . g . , v ^ ) . F o r t h e most p a r t p h y s i c a l c o m p o n e n t s w i l l be u s e d . I f t h e l i n e e l e m e n t i s : ds = h f d x 2 + h f d y 2 + h 2 3 d z 2 ( I I . 4 . 1 ) t h e n t h e v e c t o r c o m p o n e n t s a r e r e l a t e d b y : v x - h l V * = ^ , v y - h 2 V * - ^ , v z - h 3 V z = ^ ( I I . 4 . 2 ) C o v a r i a n t d e r i v a t i v e s w i l l be d e n o t e d by s e m i - c o l o n s and t h e s y m b o l V w i l l be r e s e r v e d f o r t h e t w o - d i m e n s i o n a l d i f f e r e n t i a l o p e r a t o r d e f i n e d so t h a t : 8a 8a f o r any v e c t o r a^  . A l s o : V 2 = JlL- + ( T T 4 4 ) 16 I I I . IDEAL FLUIDS I I I - l F i e l d E q u a t i o n s f o r an I d e a l F l u i d An i d e a l f l u i d i s d e f i n e d t o be one f o r w h i c h t h e s t r e s s can be d e r i v e d as t h e g r a d i e n t o f a s c a l a r f u n c t i o n n The e q u a t i o n d e s c r i b i n g momentum c o n s e r v a t i o n i s t h e n ( i n c o v a r i a n t f o r m ) : 9 V i . „k dt + V V i ; k = " n ; i ( I I I . 1 . 1 ) whe re V\ and V 1 a r e t h e c o v a r i a n t and c o n t r a v a r i a n t v e l o c i t y f i e l d s r e s p e c t i v e l y and s e m i - c o l o n s d e n o t e c o v a r i a n t d e r i v a t i ves . The e q u a t i o n o f mass c o n s e r v a t i o n i s : f f + ( P V 1 ) . , - = 0 ( I I I . 1 . 2 ) whe re p i s t h e mass d e n s i t y o f t h e f l u i d . In s t u d i e s o f v o r t i c e s i t i s most c o n v e n i e n t t o r e p l a c e ( I I I . 1 . 1 ) by t h e v o r t i c i t y e q u a t i o n . The v o r t i c i t y i s d e f i n e d b y : w1' = £ i n k v k . n g " i ' 2 ( i n . 1 . 3 ) i n k w h e r e e i s t h e a n t i s y m e t r i c t e n s o r d e n s i t y h a v i n g e 1 2 3 = 1 , and g = d e t g. . . An e q u a t i o n f o r W1 i s o b t a i n e d by t a k i n g t h e c o v a r i a n t d e r i v a t i v e o f ( I I I . 1 . 1 ) c o n t r a c t i n g w i t h e i n k a nd u s i n g ( I I I . 1 . 2 ) to e l i m i n a t e V 1 . : »i 17 3t {' P J + V' I P J W i p ;k ( I I I . 1 . 4 ) ( I I I . 1 . 2 ) , ( I I I . 1 . 3 ) and ( I I I . 1 . 4 ) a r e now r e g a r d e d as t h e f u n d a m e n t a l e q u a t i o n s o f m o t i o n . N o t e t h a t ( I I I . 1 . 4 ) may be w r i t t e n i n t e r m s o f o r d i n a r y d e r i v a t i v e s : k 3 , w \ W k 3V 1 3x F ^ P ' " P 3 x k ( I I I . 1 .5) s i n c e t h e t e r m s i n t h e c o n n e c t i o n s r1.. c a n c e l J k 18 I I I . 2 The K e l v i n C i r c u l a t i o n T h e o r e m In 1 8 6 9 , L o r d K e l v i n r e f o r m u l a t e d much o f H e l m h o l t z e a r l i e r w o r k . H i s most i m p o r t a n t c o n t r i b u t i o n was t h e d e f i n i t i o n o f t h e c i r c u l a t i o n and h i s p r o o f o f t h e K e l v i n C i r c u l a t i o n T h e o r e m . L e t C be a c l o s e d c o n t o u r i n t h e f l u i d : C ( t ) = {x ( s , t ) : 0 < s < 1} ( I I I . 2 . 1 ) whe re x ( s , t ) , k = 1 ,2 ,3 a r e smooth r e a l - v a l u e d f u n c t i o n s o f s and t s u c h t h a t : x k ( 0 , t ) = x k ( l , t ) , k - 1 ,2 ,3 f o r a l l t . The c i r c u l a t i o n a r o u n d C i s d e f i n e d b y : o V k ( x 1 ( s , t ) , t ) ^ - ( s , t ) d s ( I I I . 2 . 2 ) I f C i s a d v e c t e d t h e n i t can be p a r a m e t r i z e d s u c h t h a t : 3 t T h e r e f o r e : 9 x ( s , t ) = V k ( x i ( x , t ) , t ) ( I I I . 2 . 3 ) dr.. J n . . k q v k ( x 1 ( s , t ) ) ^ - ( s , t ) d s ^ C _d_ d t d t ^ ( x \ t ) + V k ; n ( x 1 , t ) | f ( s ) t ) 3S • ( s , t ) ,2..k ds '1 , o - V n ( x i , t ) V , . r i ( x i , t ) + n . k ( x \ t ) k ; n + V k ; n ( x i , t ) V n ( x i , t ) 3x r 3 s • ( s , t ) + V k ( x i , t ) f L ( x i , t ) ds 19 = n U ^ i . t h t ) - n i x 1 (o , t ) ,t) + ( v k v k ) ( x i ( i , t ) , t ) - ( V k V k ) ( x i ( 0 , t ) , t ) = 0 ( I I I . 2 . 4 ) T h u s , t h e c i r c u l a t i o n a r o u n d any a d v e c t e d c o n t o u r i s c o n s e r v e d . T h i s i s t h e K e l v i n C i r c u l a t i o n T h e o r e m . 20 I I I . 3 The T h i n F i l m A p p r o x i m a t i o n In t h i s s e c t i o n a number o f a p p r o x i m a t i o n s a r e i n t r o d u c e d a l l o w i n g t h e e x p a n s i o n o f t h e f i e l d e q u a t i o n s i n power s o f t h e s m a l l p a r a m e t e r A o f S e c . I I . 3 . The a p p r o x i m a t i o n s t o be i m p o s e d a r e as f o l l o w s : a) The f l u i d i s " t h i n " so t h a t t h e a p p r o x i m a t i o n o f e q u a t i o n s ( 1 1 . 3 . 8 ) may be u s e d . b) The d e n s i t y o f t h e f l u i d i s n e a r l y c o n s t a n t : p ( x , y , z , t ) = p Q + p ( 1 ) ( x , y , z , t ) + . . . ( I I I . 3 . 1 ) whe re p i s c o n s t a n t , o c ) We d e f i n e : v x = V x h , v y =v y h , v z ='v2k* ( I I I . 3 .2 ) X V z whe re V ,VJ ,V a r e t h e c o m p o n e n t s o f t h e c o n t r a v a r i a n t v e l o c i t y f i e l d . To l o w e s t o r d e r i n X, t he m e t r i c i s d i a g o n a l so t h a t , t o t h i s o r d e r , v x , v , v z a r e t h e c o m p o n e n t s o f t h e p h y s i c a l v e l o c i t y f i e l d . I t i s a s sumed t h a t t h e v e r t i c a l c o m p o n e n t o f t h e v e l o c i t y i s s m a l l i n c o m p a r i s o n w i t h t h e h o r i z o n t a l v e l o c i t y and v a r i e s l i t t l e w i t h h e i g h t . v x ( x , y , z , t ) = v x 0 ) ( x , y , t ) + X v ^ 1 > ( x ,y , t ) + . . . ( I I I . 3 . 3 a ) v y ( x , y , z , t ) = v y 0 ) ( x , y , t ) + A v y ! ) ( x , y , t ) + . . . ( I I I . 3 . 3 b ) v z ( x , y , z , t ) = X v z ! ) ( x , y , t ) + A 2 v z 2 ) (x ,y , z , t ) + . . . ( I I I . 3 . 3 c ) S i m i l a r l y , v o r t i c i t y c omponen t s a r e d e f i n e d : 21 w = W Xh ; w = W y h ; w = W Z k * x y z ( I I I . 3 . 4 ) ( I I I . 1 . 3 ) and ( I I I . 3 . 3 ) t h e n i m p l y : w w v ( 1 ) X 1 9 f - + ^ ( x - y . z . t ) + ••• v ( 1 ) 1 • i 5 i - + : - X w [ 1 ) ( x , y l z l t ) + . . . ( I I I . 3 ^ 5 a ) ( I I 1 . 3 . 5 b ) w 3 ( h v [ 0 ) ) 3 (hv<° ' ) 3x 3x + ^ l ^ U . y . z . t ) + ( I I I . 3 . 5 c ) W r i t t e n e x p l i c i t l y i n t e r m s o f w ,w ,w , ( I I I . 1 . 5 ) x y z ' becomes _3_ 3t w v . f N w w v ~ w X + ir- — X + ^ 3 X ..+ _z _L X P.h h 3x Ph h ay Ph k 3z h I i I J I J w ' V W ~ f \ V w » V 3 x + 4- - i - X + _?. _L X 3x h ph 3y h pk 3z h ( I I I . 3 . 6 a ) 3t fw ] v ~ VI v r > w v w y Ph x 3 h 3x y ph J + JL JL h 3y y Ph • + -L _ L k 3y y Ph w _x _3_ ph 3x w ph 3y w. pk ' 3 z ( I I I . 3 . 6 b ) JL 3t w. Pk JL _ L h 3x Ph + JL JL h 3y ph J L _ L h 3z w z Ph ^ _3_ ph 3x w ph 3y w. ph 3z ( I I I . 3 . 6 c ) w h i c h t o t h e l o w e s t o r d e r i n A b e c o m e s : 2 2 _3_ 3t w. h 3x w. h 3z w = 0 ( I I I . 3 . 7 ) To l o w e s t o r d e r t h e e q u a t i o n o f mass c o n s e r v a t i o n ( I I I . 1 . 2 ) i s 3 ( h k v x 0 ) ) 3x 9 ( b k v y Q ) ) 3y = 0 ( I I I . 3 . 8 ) The e q u a t i o n e x p r e s s i n g t h e p r o d u c t i o n o f w ( ° > and X w y ^ a r e a l s o o f t h i s o r d e r . H o w e v e r , a c c o r d i n g t o e q u a t i o n s ( 1 1 1 . 3. 5a , b) , t o t h i s o r d e r o f a p p r o x i m a t i o n , v x > v y a n c * w z a r e c o m p l e t e l y i n d e p e n d e n t o f w x and w y : t h a t i s , c h a n g e s i n w v , w u o f o r d e r A 0 i n d u c e c h a n g e s i n v x # , v w and w_ o f o r d e r A. x* y x ' y The e q u a t i o n s f o r w x and w y may t h e r e f o r e be d i s r e g a r d e d t o t h i s o r d e r o f a p p r o x i m a t i o n . H e n c e f o r t h , t h e s u p e r s c r i p t s w i l l be o m i t t e d . A l l p h y s i c a l q u a n t i t i e s w i l l be a s s umed t o be t a k e n t o l o w e s t o r d e r . In o r d e r t h a t ( 1 1 1 . 3 . 3 a , b , c ) r e m a i n v a l i d a t t h e b o u n d a r i e s , i t i s n e c e s s a r y t h a t a l l b o u n d a r i e s be o f t h e f o r m : f ( x , y ) + 0 ( A ) = c o n s t , f o r some f u n c t i o n f w h i c h f o r s i m p l i c i t y i s a s sumed a n a l y t i c . ( S i n c e any f u n c t i o n may be a p p r o x i m a t e d t o a r b i t r a r y a c c u r a c y by an a n a l y t i c f u n c t i o n , d e p a r t u r e s f r o m a n a l y t i c i t y may be a b s o r b e d i n t h e t e r m o f o r d e r A ) . The a s s umed b o u n d a r y c o n d i t i o n s a r e t h a t t h e r e i s no f l u x o f f l u i d t h r o u g h any b o u n d a r y . . T h e r e i s , h o w e v e r , s l i p a t t h e b o u n d a r i e s ; no s l i p c o n d i t i o n s r e q u i r e t h a t " " (( = 0 v x ° > = v y ° > = 0 a t z = 0 a n d z = l , b u t ^ 0 ) ^ 3z 3z so t h a t t h e o n l y s o l u t i o n c o m p a t i b l e w i t h no s l i p b o u n d a r y c o n d i t i o n s i s t h e t r i v i a l s o l u t i o n v ^ = 0 . N o t i c e t h a t 23 s i n c e v z ^ = 0 » a n c ' t n e u p p e r and l o w e r b o u n d a r i e s a r e s u r f a c e s z = c o n s t . , t h e l o w e s t o r d e r s o l u t i o n i s c o n s i s t e n t w i t h z e r o f l u x t h r o u g h t h e s e s u r f a c e s . The p r o b l e m o f f i n d i n g t h e f l u i d m o t i o n has now been r e d u c e d t o a t w o - d i m e n s i o n a l p r o b l e m s i n c e a l l f i e l d s and b o u n d a r i e s ( o t h e r t h a n w x and w y ) a r e , t o l o w e s t o r d e r , i n d e p e n d e n t o f z . 24 IV. VORTICES IN THIN FILMS In t h i s s e c t i o n t h e ma in p r o b l e m o f t h e t h e s i s i s a d d r e s s e d : wha t a r e t h e e q u a t i o n s o f m o t i o n o f a s y s t e m o f N v o r t i c e s i n a t h i n f i l m o f i d e a l f l u i d d e s c r i b e d by t h e m e t r i c f u n c t i o n h ( x , y ) and t h e d e p t h f u n c t i o n k ( x , y ) ? IV .1 The V o r t e x V e l o c i t y F i e l d The e q u a t i o n o f mass c o n s e r v a t i o n ( I I I . 3 . 8 ) i s : 3 ( h k v x ) 9x 3 ( h k v y ) 3y = 0 ( I V , 1.1) w h i c h has g e n e r a l s o l u t i o n : = _L i i hk 9y ( x , y ) hk 9 x u ' y j ( I V . 1 . 2 ) f o r some r e a l v a l u e d f u n c t i o n ^ ( x , y ) h a v i n g c o n t i n u o u s m i x e d p a r t i a l d e r i v a t i v e s o f s e c o n d o r d e r . i p ( x , y ) i s t h e s t r e a m f u n c t i o n f o r t h e f l o w . The v e l o c i t y f i e l d o f a v o r t e x a t ( x ' , y ' ) i s d e f i n e d t o be t h e i n c o m p r e s s i b l e v e l o c i t y f i e l d h a v i n g w z = 0 e v e r y w h e r e t h r o u g h o u t t h e r e g i o n o f f l o w , D , e x c e p t a t ( x ' , y ' ) : t h a t i s , s a t i s f y i n g ( I V . 1 . 1 ) a n d : w_ = r-T h 2 r 3 ( h v ^ 9 ( h v x ) 9x 9y ^ 6 ( x - x ' ) 6 ( y - y ' ) ( I V . 1 . 3 ) N o t i c e t h a t i f C i s a c o n t o u r w i t h i n t e r i o r G s u c h t h a t ( x 1 , y 1 ) e G , t h e n : 2 T T Y = ; 2TTY<S ( x - x ' ) 6 ( y - y 1 ) dxdy 3 ( h v y ) 3 ( h v x ) 9x 8y dxdy T h u s , by t h e K e l v i n C i r c u l a t i o n T h e o r e m y i s a c o n s t a n t . Y i s c a l l e d t h e v o r t e x s t r e n g t h . S u b s t i t u t i n g ( I V . 1 . 2 ) i n t o ( I V . 1 . 3 ) one has : £ v v ( x , y ; x ' , y ' ) ] = -2TTS ( x - x 1 ) 6 ( y - y ' ) ( I V . 1 . 5 ) w h e r e x ,y ;x 1 ,y ' ) i s t h e s t r e a m f u n c t i on f o r a v o r t e x o f u n i t s t r e n g t h a t (x 1 , y 1 ) . x , y ; x 1 , y 1 ) i s a G r e e n ' s f u n c t i o n o f a s e l f - a d j o i n t e l l i p t i c d i f f e r e n t i a l o p e r a t o r . The r e g i o n o f f l o w i s { ( x , y ) e D > w h e r e t h e b o u n d a r y o f D i s : M 3D = U 3-D 3.DH3.D = <j> , i / k . ( I V . 1 . 6 ) i =0 1 1 K The b o u n d a r y c o n d i t i o n s f o r ¥ ( x ,y ;x^ , y . ) a r e : y = ( u n s p e c i f i e d ) c o n s t , on 3^ D, i =0 , . . . , M ; r a i D g i v e n , i = l , . . . ,M ( I V . 1.7) I t i s o f t e n a s s u m e d , a l t h o u g h t h i s i s n o t n e c e s s a r y , t h a t r 3 - D = ^ ^ o r i = l » - - - » M • N o t i c e t h a t t h e K e l v i n C i r c u l a t i o n Theo rem i m p l i e s t h a t t h e b o u n d a r y c o n d i t i o n s a r e c o n s t a n t i n t i me. The e x i s t e n c e and u n i q u e n e s s o f ¥ s a t i s f y i n g ( I V . 1 . 5 ) and t h e b o u n d a r y c o n d i t i o n s : ^ ( x , y ; x ' , y 1 ) = ^ = c o n s t , on 3 .D , i =0 , . . . , M ( I V . 1 . 8 ) i s w e l l known ( C o u r a n t and H i l b e r t ( 1 9 6 2 ) ) . We use t h i s t o d e m o n s t r a t e t h e e x i s t e n c e and u n i q u e n e s s up t o an a d d i t i v e m c o n s t a n t o f ¥ s a t i s f y i n g t h e b o u n d a r y c o n d i t i o n s ( I V . 1 . 7 ) . S e t ¥ 0 = 0 and c o n s i d e r t h e v e c t o r s p a c e A o f M - t u p l e s , ( Y ^, . . . , f ^ ) c o r r e s p o n d i n g t o t h e p o s s i b l e v a l u e s o f t h e b o u n d a r y c o n d i t i o n ( I V . 1 . 8 ) . L e t B be t h e v e c t o r s p a c e o f M - t i r p l : e . s , ( r ^ M , . . . , r g Q ) c o r r e s p o n d i n g t o t h e i M p o s s i b l e v a l u e s o f t h e b o u n d a r y c o n d i t i o n ( I V . 1 . 7 ) . T h e r e i s a n a t u r a l map f : A-*B , d e f i n e d by f i n d i n g t h e u n i q u e y ( x ,y ;x ' ,y 1 ) h a v i n g b o u n d a r y c o n d i t i o n s Y = lF.. on 3.j D, i - 1 , . . . , M, Y = 0 on 3 0 D , and s e t t i n g r g _ D e q u a l t o t h e c i r c u l a t i o n a r o u n d 9^D f o r t h i s Y . C l e a r l y f i s l i n e a r . To p r o v e t h a t Y e x i s t s and i s u n i q u e up t o an a d d i t i v e c o n s t a n t u n d e r t h e b o u n d a r y c o n d i t i o n s ( I V . 1 . 7 ) i t i s o n l y n e c e s s a r y t o p r o v e t h a t f has an i n v e r s e ( t h e f r e e d o m o f t h e a d d i t i v e c o n s t a n t a r i s e s when one does n o t f i x Yo ). Thus i t i s o n l y n e c e s s a r y t o show t h a t (r_. n , . . . , I \ n ) = ( 0 , . . . , 0 ) d • U d M U has a u n i q u e p r e - i m a g e . S u p p o s e ¥( x ,y ;x 1 ,y 1 ) and Y * (x ,y ;x 1 ,y 1 ) s a t i s f y ( I V . 1 . 5 ) and ( I V . 1 . 7 ) w i t h r \ n = 0 , i = l , . . . , M . T h e n : 3.U 3D k 9 n D dxdy l y j y - y * ) » v ( d x d y (IV.1.9) where d e n o t e s a d e r i v a t i v e n o r m a l t o t he b o u n d a r y . The dn. l e f t s i d e v a n i s h e s : , s i n c e f r o m ( I V . 1 . 2 ) and ( I V . 1 . 7 ) : ( Y - Y * ) 3 ( ¥ _ y * ) d s = c o n s t . xT„ n = 0 , i = 1 M h.D k 3 n a i U  1 ( I . V . I . 1 0 ) 27 and when i = 0 0 on 9 0 D . The f i r s t t e r m on t h e r i g h t o f ( I V . 1 . 9 ) v a n i s h e s by ( I V . 1 . 1 5 ) . H e n c e : I y ( Y - Y * ) • V ( Y - Y * ) d x d y = 0 (• I V. 1.11) whence V ( Y - Y * ) = 0 a n d , s i n c e V = Y * on 9 0 D , Y = Y * . T h u s , ( r 3 D » - - - » r 3 [ ) ) = ( 0 , . . . , 0 ) has a u n i q u e p r e -image a n d t h e r e f o r e t h e r e e x i s t s a Y( x ,y ;x 1 ,y ' ) s a t i s f y i n g ( I V . 1 . 5 ) and ( I V . 1 . 7 ) w h i c h i s u n i q u e up t o an a d d i t i v e c o n s t a n t . Y ( x , y ; x ' ,y 1 ) a l s o has t h e r e c i p r o c i t y p r o p e r t y : V ( x , y ; x ' , y ' ) =• Y ( x ' , y ' ; x , y ) ( I V - . 1 . 12 ) and can be w r i t t e n i n t h e f o r m : V (x ,y ;x ' ,y 1 ) = - A ( x ,y ; x ' ,y ' )lnr + B (x ,y x ' , y 1 ) ( I V. 1.13) w i t h r = [ ( x - x 1 ) 2 + ( y - y 1 ) 2 ] 2 and w i t h A and B a n a l y t i c i n D i f k i s a n a l y t i c ( s e e , f o r e x a m p l e , S o m m e r f e l d ( 1 9 4 9 ) ) . The c i r c u l a t i o n a r o u n d t h e s m a l l c o n t o u r r = e i s : 2TT = f 2 7 T I 91 k 9 r r d6 2 ^ A ( x ' , y ' ; x ' ,,y' ) k ~ ( 7 r T y _ T T as e + 0 whe nee A ( x ' ,y 1 ; x \ y ' ) = k ( x ' , y ' ) ( I V . 1 . 1 4 ) S u b s t i t u t i n g ( I V . 1 . 1 3 ) i n t o ( I V . 1 . 5 ) one f i n d s t h a t as r 0 _9_ 9 r , 1 9A „ + k 97 - 0 ( I V . 1 . 1 5 ) when ce : 2Q V A ( x ' , y ' ; x ' , y ' ) = % v k ( x ' , y ' ) ( I V . 1 . 1 6 ) The t o t a l s t r e a m f u n c t i o n f o r a s y s t e m o f N v o r t i c e s w i t h p o s i t i o n s ( * n , y n ) and r e s p e c t i v e s t r e n g t h s y n , n = l , . . . , N I s : N <r(x,y) = z Y n Y ( x , y ; x ,y ) + t*{x,y) n = 1 n n n ( I V . 1 . 1 7 ) whe re ty*{x,y) i s t h e s t r e a m f u n c t i on due t o o t h e r i m p o s e d f l o w s ( e . g . , a u n i f o r m s t r e a m ) . ty*(x,y) must s a t i s f y : V," o ( I V . 1 . 1 8 ) So f a r i t has been t a c i t l y a s s umed t h a t D i s b o u n d e d . The e x t e n s i o n o f D t o i n f i n i t y p r e s e n t s no r e a l p r o b l e m s ( C o u r a n t and H i l b e r t ( 1 9 6 2 ) ) and w i l l n o t be c o n s i d e r e d f u r t h e r . The s p e c i a l c a s e o f v o r t i c e s on a c l o s e d u n b o u n d e d s u r f a c e ( e . g . , a s p h e r e ) i s , h o w e v e r , o f p a r t i c u l a r i n t e r e s t . L e t C be a c l o s e d c o n t o u r on s u c h a s u r f a c e , s u c h t h a t no v o r t e x l i e s on C . Deno te i t s i n t e r i o r by C. . and i t s J l n t e x t e r i o r by C e x t ( t h e c h o i c e o f w h a t i s t h e i n t e r i o r and what i s t h e e x t e r i o r i s , o f c o u r s e , a r b i t r a r y ) . T h e n : o : h ( v dx + v dy) = c K J C i n t dxdy = s z„eC Y n n ~ i n t ( I V . 1 . 19) 29 I n t e g r a t i n g a r o u n d C i n t h e o t h e r d i r e c t i o n one f i n d s : -o h ( v dx + v dy) = Z c y ( I V . 1 . 2 0 ) JC y n e x t whence E y = 0 n = l n ( I V . 1 . 2 1 ) T h u s , t h e r e c a n n o t be a s i n g l e v o r t e x on s u ch a s u r f a c e . The v e l o c i t y f i e l d due t o t h e s t r e a m f u n c t i o n ¥ s a t i s f y i n g ( I V . 1 . 5 ) must be i n t e r p r e t e d as t h a t due t o a v o r t e x o f u n i t s t r e n g t h a t ( x ' y 1 ) and one o f n e g a t i v e u n i t s t r e n g t h a t i n f i n i t y . ( N o t e t h a t on a c l o s e d s u r f a c e t h e p o i n t i n f i n i t y i s j u s t l i k e any o t h e r p o i n t : e . g . , on t h e s p h e r e i t i s t h e s o u t h p o l e . ) Upon t h e s u p e r -p o s i t i o n o f N s u c h s t r e a m f u n c t i o n s s a t i s f y i n g ( I V . 1 . 2 1 ) t h e v o r t e x a t i n f i n i t y d i s a p p e a r s . I f t h e d e p t h o f t h e f l u i d i s c o n s t a n t , k i s c o n s t a n t (we p u t k = l f o r s i m p l i c i t y ) and x ,y ; x ' , y 1 ) becomes t h e G r e e n ' s f u n c t i o n o f t h e L a p l a c i a n . ¥ can t h e n b e - w r i t t e n : Y ( x , y ; x i y ' ) = -Inr + B ( x , y ; x ' , y ' ) ( I V . 1 . 2 2 ) ( s e e , e . g . , S o m m e r f e l d ( 1 9 4 9 ) ) , w h e n c e , upon c o m p a r i s o n w i t h ( I V . 1 . 1 3 ) A ( x , y ; x ' , y ' ) = 1 ( I V . 1 . 2 3 ) M o r e o v e r , s i n c e : V 2 £ n r = 2TT6 ( x - x ' ) 6 ( y - y ' ) , i f t h e r e a r e no b o u n d a r i e s , t h e n : B ( x , y ; x ' , y ' ) = 0 ; ^ ( x , y ; x ' , y ' ) = -h&n [ (x-x) 2 + ( y - y ' ) 2 ] ( I V . 1 . 2 4 ) When t h e f l u i d d e p t h i s c o n s t a n t one can t h e r e f o r e r e g a r d t h e f l o w i n d u c e d by B to be due t o t h e b o u n d a r i e s . I f k i s n o t c o n s t a n t t h e r e i s no s u c h s i m p l e d e c o m p o s i t i o n o f t h e f l o w . 30 IV.2 The V e l o c i t y o f a Vo r tex i n a F l u i d of Un i f o rm Depth The mot ion o f the v o r t i c e s i s governed by e q u a t i o n ( I I I . 3 . 7 ) : _3_ 3t h 3x Zl + z 3 rW. h dy = 0 ( I V . 2 . 1 ) which i m p l i e s t h a t e lement -mai n t a i ns i t s va lue of -rH i s a d v e c t e d : t h a t i s , each f l u i d W z l as i t moves around i n the f l u i d . Thus , the v o r t i c i t y c o n c e n t r a t e d at the p o i n t s ( x ^ , y n ) remains c o n c e n t r a t e d and the f l ow r e t a i n s the c h a r a c t e r i s t i c s o f a v o r t e x s y s tem: i . e . , the s t r e a m f u n c t i o n a lways s a t i s f i e s : N F V^J = - 2 i - 2 Y n 5 ( x - x n ) f i ( y - y n ) ( I V . 2 . 2 ) but the x n and y n a re t i m e - d e p e n d e n t , moving as i f c a r r i e d a l o n g by the f l o w . In o r d e r t o dete rmine the v e l o c i t y of each v o r t e x we expand the v e l o c i t y f i e l d nea r i t s s i n g u l a r i t y . The case k=l ( u n i f o r m depth f l u i d ) i s t r e a t e d f i r s t . I t w i l l p rove c o n v e n i e n t t o i n t r o d u c e the complex v a r i a b l e z = x + i y ( t h e r e s h o u l d be no c o n f u s i o n w i t h the v a r i a b l e z o f S e c t i o n I I . 3 ) . The i ndependen t v a r i a b l e s are then z and z-= x - i y . For s i m p l i c i t y the p r e s e n t symbols are r e t a i n e d f o r a l l f u n c t i o n s d e s p i t e t h e i r change i n a rguments . E q u a t i o n ( I V . 1 . 2 ) i s t h e n : v x - 1 v y 2i h(z ,z) 3z ^ ( z , I ) (I V.2.3) 3 1 W h i c h , upon s u b s t i t u t i o n o f ( I V . 1 . 1 7 ) and ( I V . 1 . 2 2 ) becomes N r - i . y . 3 B ( z , z ; z . , z . ) z - z , . T k 3z I f |z - z n i s s m a l l : •y. - T V y z - z - Y n 8z + 2 i ^ ( z , z ) \ ( I V . 2 . 4 ) 8 h _ 1 ( z - z ) n V n ' dh - 1 n 8z + 297 i|>*(z,z) + Y n B ( z , z ; z n , z n ) + Z Y k , i ' ( z , z ; z z ) k^n K k k } + ° d z - z n z= z ( I V . 2 . 5 ) whe re h = h ( z , z ) n n n ' The v e l o c i t y f i e l d : i v. h j z , z ) ( z - z ) v n n ' v n ' i s x y c o n c e n t r i c a b o u t z n . P r e f e r i n g no d i r e c t i o n i t c a n n o t i m p a r t a v e l o c i t y t o t h e v o r t e x . i"Y 3 h - 1 {z-z ) The v e l o c i t y f i e l d v z - i v y = h ( v ^ ) 1 ^ ) f l o w s r a d i a l l y f r o m z n and h e n c e c a n n o t i m p a r t a v e l o c i t y t o t h e v o r t e x e i t h e r . I t m i g h t seem p a r a d o x i c a l t h a t a v o r t e x v e l o c i t y f i e l d s h o u l d c o n t a i n r a d i a l t e r m s s u c h as t h i s . They a r i s e b e c a u s e v_ does n o t s a t i s f y V«_v = 0 b u t = - v / V £ n h By p r o j e c t i n g t h e f l o w o n t o t h e p l a n e " f i c t i t i o u s " s o u r c e t e r m s a r e i n t r o d u c e d g i v i n g r i s e t o t h e r a d i a l t e r m s . The v e l o c i t y f i e l d : 3h" 1 V x " i v y = h f f ^ n ^ + 2 i £ 32 4>*{z,z) +; Y . n B ( z , z ; z n , z n ) + Z Y k Y ( z , z ; z ,z ) k?n K • z = z r z= z„ i s u n i f o r m and t h e r e f o r e c a r r i e s t h e v o r t e x a t z a l o n q w i t h n 3 i t . A l l o t h e r t e r m s v a n i s h as z + 2 n s ° t h a t t h e y c a n n o t i n d u c e any m o t i o n i n t h e v o r t e x a t z . The v e l o c i t y o f t h e n J v o r t e x i s t h e r e f o r e : 9h 1 i f i * ( z , z ) + Y n B ( z , z ; z n , z n ) + Z Y k ^ C z . z j z . ,z. ) k^n K k k . z = z r z=z, ( I V . 2 . 6 ) Bu t u = h x , u = h y x n n y n^n ( I V. 2 . 7 ) T h e r e f o r e : i l _ L h 2 9z n nr, - r £ n h ( z , z ) + i p * ( z , z ) + Y n B ( z , z ; z n , z n ) + Z y.V(z,z;z z . ) k^n K k k • z = z •n ( I V . 2 . 8 ) z = z r o r , r e v e r t i n g t o ( x , y ) c o o r d i n a t e s : 3n. h 2 ( x n , y n ) 3y x = x n y=yn -1 ^ n h 2 . ( x n , y n ) 9x x = x n y = y n ( I V . 2 . 9 ) w i t h 33 Y fin = ; ^ - J i n h ( x , y ) + T ^ * ( X , y ) + y n B ( x ,y ; x n , y p ) + Z Y . V ( x ,y ;x. ,y. ) ( I V . 2 . 1 0 ) k^n K K K The se a r e t h e e q u a t i o n s o f m o t i o n f o r t h e v o r t e x s y s t e m . I f t h e r e a r e no b o u n d a r i e s t h e n f r o m ( I V . 2 . 8 ) and ( I V . 1 . 2 4 ) if = X * n h , , n k/n Vz n h n 8 z ( I V . 2 . 1 1 ) E x a m p l e : The V e l o c i t y o f t h e V o r t e x on a S p h e r e w i t h No B o u n d a r i e s and U n i f o r m Depth In o r d e r t o c h e c k t h a t ( I V . 2 . 9 - 1 0 ) do i n d e e d g i v e t h e c o r r e c t v o r t e x v e l o c i t y , we d e t e r m i n e t h e v e l o c i t y o f a v o r t e x on t h e s p h e r e . T h i s can a l s o be d e t e r m i n e d by an a l t e r n a t i v e me thod w h i c h i s i n a g r e e m e n t w i t h t h e f i r s t . C o n s i d e r a v o r t e x o f s t r e n g t h y a t ( x ' , y ' ) on t h e s p h e r e and i t s c o u n t e r p a r t o f s t r e n g t h - y a t i n f i n i t y . ( F rom ( I I . 2 . 4 ) , i n f i n i t y i n h a r m o n i c c o o r d i n a t e s c o r r e s p o n d s t o 0 = ^ , i . e . , t h e s o u t h p o l e ) . S i n c e t h e r e a r e no e x t e r n a l f l o w s , no b o u n d a r i e s and u n i f o r m d e p t h , ij ;* = 0 , B = 0 , and k = 1 . The v e l o c i t y o f t h e v o r t e x i s t h e r e f o r e : % " ' " y - h T T ^ 7 r T i T r ' z ' - ? , ) < I V- 2 ' 1 2 ' o r , i n p o l a r h a r m o n i c c o o r d i n a t e s : - n • ii - - Y h ' ( r ' ) - - y ( p ( r ' ) - l ) ( I V . 2 . 1 3 ) u r " u ' p 2 h ( r ' ) " 2 r ' 34 U s i n g ( I I . 2 . 4 ) : ur = % = 0 ; u„ = £ t p R ^ a n ^ ( I V .2. 1 4 ) T h i s can be d e r i v e d a l t e r n a t i v e l y , f o r t h i s c a s e o n l y , as f o 11ows : S i n c e t h e sum o f a l l v o r t e x s t r e n g t h s i s z e r o , c o n s i d e r t h e v e l o c i t y f i e l d o f a s i n g l e v o r t e x o f s t r e n g t h Y t o be t h a t s u c h t h a t i t i s i n c o m p r e s s i b l e and t h e v o r t i c i t y i s e v e r y w h e r e c o n s t a n t and e q u a l t o ay . Then upon s u p e r p o s i n g t h e v e l o c i t y f i e l d s one g e t s z e r o v o r t i c i t y The v e l o c i t y f i e l d o f e a c h v o r t e x i s c o m p l e t e l y s y m m e t r i c a b o u t i t s c o r e o w i n g t o t h e s y m m e t r y o f t h e s p h e r e ; h e n c e , t h e r e i s no s ^ l f - i n d u c e d v o r t e x m o t i o n . The v e l o c i t y f i e l d o f t h e v o r t e x a t t h e s o u t h p o l e t h e n s a t i s f i e s ( u s i n g o r d i n a r y p o l a r c o o r d i n a t e s ) : 0 w h i c h has s o l u t i o n w 1 R s i n 0 •3( v s i n -9 ) 8 9 3v„-, i3<J> •ay ( I V . 2 . 1 5 ) s i n 0 cos 6 B c o n s t . ( I V.2 . 1 6 ) I f v^ i s t o be b o u n d e d a t 0=0 , 6=1 e v a l u a t e d by r e q u i r i n g t h a t : f 2TT a i s • 2 iry L i m 0+TT VxRsin0d<j> = L i m-2iraYR 2 ( 1 - c o s e ) 0+7T •4fTaYR : ( I V . 2 . 1 7 ) 35 T h e r e f o r e : 1 ( I V . 2 . 1 8 ) a = 2 R 2 and v ( I V . 2 . 1 9 ) The v e l o c i t y o f t h e v o r t e x a t ( e 1 , ^ ' ) i s t h e r e f o r e g i v e n by ( I V . 2 . 1 4 ) s i n c e i t i s c a r r i e d a l o n g i n t h e f l o w o f t h e v o r t e x a t t h e s o u t h p o l e . The two method s a r e i n a g r e e m e n t . (O f c o u r s e , t h e v o r t e x a t t h e s o u t h p o l e i s a l s o m o v i n g so t h a t ( I V . 2 . 1 9 ) i s o n l y c o r r e c t a t t h e i n s t a n t t h i s v o r t e x i s a t t h e p o l e . ) 36 I V . 3 The V e l o c i t y o f a V o r t e x i n a F l u i d o f V a r y i n g Depth The v e l o c i t y o f a v o r t e x i n a f l u i d o f . v a r y i n g d e p t h i s more c o m p l i c a t e d t h a n t h a t i n a f l u i d o f u n i f o r m d e p t h o w i n g t o t h e more c o m p l e x n a t u r e o f t h e s i n g u l a r i t y i n V(x , y ; x ' , y 1 ) . The p h y s i c a l r e a s o n f o r t h i s i s t h a t t h e v o r t e x c o r e i s no l o n g e r s t r a i g h t b u t must c u r v e s l i g h t l y i n o r d e r t o meet t h e u p p e r and l o w e r b o u n d i n g s u r f a c e s p e r p e n d i c u l a r l y so t h a t t h e b o u n d a r i e s a r e s t r e a m i n g s u r f a c e s . ( See F i g u r e I I ) ; I t was shown by H e l m h o l t z ( 1358) t h a t an i n f i n i t e s i mal l y , s m a l l c u r v e d v o r t e x c o r e w i l l p r o p a g a t e i n f i n i t e l y f a s t . - T h u s , one must e x p e c t a t t h e o u t s e t t h a t t h e v e l o c i t y o f a v o r t e x i n a f l u i d o f v a r y i n g d e p t h must depend i n some way on t h e s t r u c t u r e o f i t s c o r e . We p r o c e e d as i n S e c t i o n I I . 2 , by e x a m i n i n g t h e . p h y s i c a l v e l o c i t y f i e l d i n t h e n e i g h b o u r h o o d o f z^. From ( I V . 1 . 2 ) , ( I V . 1 . 1 5 ) , and ( I V . 1 . 1 7 ) : 2 i y h ( z , z ) k ( z , z ) 9z l z ' z ; 2 i - y A ( z , z ; z , z ) ' n n n n n ( h k ) ( z n , z n ) A ( z n , z : z , z ) 9 ( h k ) ( z , z ) " n n n v ' v n n + Y T h k ) ( z , z ) 9z 9 A ( z , z ; z , z ) n n 9z \L=Ln z = z n f z - z . ^ z - z Y, 3A ( z , z ; z , z ) • n n 9z z = z n x & n l z " z n z = z n F i g . I I V o r t e x Co re i n a F l u i d o f V a r y i n g Depth + Y " A ( Z n ' Z n ) 3 ( h k ) , , - , . 3 ( h k ) ( z n , z n ) 3z ( W 3F 38 Y B ( z , z ; z , z ) n n n' + i r * ( z , z ) + E Y | ^ ( z , z ; z . , z . ) k^n K K k + 0 ( | z - z n U n | z - z n | ) z = z n z = z n ( I V . 3 . 1 ) From ( I V . 1 . 1 4 ) and ( I V . 1 . 1 6 ) and w r i t i n g h = h ( z , z ) ; k E k ( z , z ) n v n n ' n v n n ( I V . 3 . 2 ) one g e t s : i v 2 i y h k n n - y k ' n n , I T ^ r r + Y n v n ' 3k ! n , n n h 3z n 'z- z } n z- z n J Y 3k ' n n . i Y 9k k 3h , _JT_ n _n_ n "2 3z h n 3z 3z Y B ( z , z ; z , z ) + i ii* ( z , z ) n n n r v + E Y n v p ( z , z ; z . z j l -k^n z = z n Z-rZfj + 0 ( | z - z j £ n | z - z j ) ( I V . 3 . 3 ) As b e f o r e , t h e t e r m s i n z - z an d z - z z - z„ can i m p a r t no v e l o c i t y t o t h e v o r t e x . The r e m a i n i n g t e r m s w i t h t h e e x c e p t i o n o f t h e t e r m i n & n | z - z n | a r e u n i f o r m and t h e r e f o r e c a r r y t h e v o r t e x w i t h t h e m . The t e r m i n in\z-z^\ i s t h e s o u r c e o f t h e d i f f i c u l t i e s o u t l i n e d a t t h e b e g i n n i n g o f t h e s e c t i o n . I t i s d i v e r g e n t as z->z b u t has a d e f i n i t e n d i r e c t i o n ( a l o n g t h e c u r v e s o f c o n s t a n t k ). I t t h e r e f o r e i m p l i e s t h a t t h e v o r t e x moves w i t h i n f i n i t e v e l o c i t y a l o n g 39 k = c o n s t . I f , h o w e v e r , i t i s a s sumed t h a t t h e v o r t e x has a s m a l l b u t f i n i t e c i r c u l a r c o r e o f r a d i u s , t h e n a t t h e s u r f a c e o f t h e c o r e t h e v e l o c i t y due t o t h i s t e r m i s u n i f o r m and eq u a l s : - i y 3 k n r — n r — Ine . The b o u n d a r y i s t h e r e f o r e V n 3 2 n c a r r i e d a l o n g i n t h e u n i f o r m f l o w : v.. - i v . . - ^ - r - ( - 2 ~ - ^ - ( A n e ^ - l ) + ^ x ' y h n k n \ 2 3z h n 3z Y n B ( z , z ; z n , z n ) + ty*{z,z) + Z Y . Y ( z , z ; z , z . ) k^n k k k J z = z n ( I V . 3 . 4 ) z = z n w Now, i s a d v e c t e d a n d , s i n c e t h e f l u i d i s i n c o m p r e s s -i b l e , t h e vo l ume o f e a c h c o r e must be c o n s t a n t . H e n c e : ( I V . 3 . 5 ) h n k n £ n = % + 0 (h e X] n n k n whe re X i s t h e s m a l l p e r t u r b a t i o n p a r a m e t e r o f S e c t i o n I I . 3 and S e c t i o n I I I . 3 . (X 0 ( k | V £ n h | ) ) , and a n i s a c o n s t a n t . As k i n c r e a s e s t h e r a d i u s o f t h e c o r e d e c r e a s e s . H o w e v e r , t h e s t r e a m l i n e s o f t h e f l o w , w h i l e a p p r o x i m a t e l y c i r c u l a r , a r e n o t c o n c e n t r i c ( See F i g u r e I I I ) . In s h r i n k i n g , t h e c e n t r e o f t h e v o r t e x a p p e a r s t o move i n t h e d i r e c t i o n o f -Vk . The c a l c u l a t i o n o f t h i s v e l o c i t y t e r m w i l l be d e f e r r e d u n t i l S e c t i o n VI when t h e e f f e c t s o f f i n i t e s i z e d c o r e s a r e t r e a t e d i n a more r i g o r o u s m a n n e r . S u f f i c e i t t o s a y , f o r t h e p r e s e n t , t h a t t h i s t e r m may be i n c o r p o r a t e d by " r e n o r m a l i z i n g " : F i g . 1 1 1 S t r e a m l i n e s n e a r t h e C o r e o f a V o r t e x i n a F l u i d o f V a r y i n g Depth 41 i . e . , by r e p l a c i n g by a n = a n e •3 n ( I V . 3 . 6 ) whe re ' c n * Y n k n p ( I V . 3 . 7 ) E c n i s t h e k i n e t i c e n e r g y w i t h i n t h e n t h c o r e . T h u s , t h e v e l o c i t y o f t h e v o r t e x i s , s u b s t i t u t i n g ( I V . 3 . 5 ) i n t o ( I V . 3 . 4 ) and re n o r m a l i z i ng a : 3 n 2 i . I • ' n n u x " 1 u y " h n k n [ 4 9z f In a* h 2 k -2 n n K 3z 9z Y n B ( z , z ; z n , z n ) + ^ * ( z , z ) + S Y k * ( z , z ; z , , z , ) k ?n K k k J l = A n z = z n ( I V . 3 . 8 ) w h e n c e , u s i n g ( I V . 2 . 7 ) and s i m p l i f y i n g t h e e q u a t i o n s o f m o t i o n a r e : , 9ft* 1 n X n " h 2 k n 9y ; y 90* -1 d n n h 2 k n 9x ( I V . 3 . 9 ) w i t h : Y ft* = - ^ . k ( x , y ) J U h2 ( x , y ) k ( x , y ) + y B (x ,y ;x ,y ) n v , J ' n J n ' + i j j * ( x , y ) + s Y n ^ ( x , y ; x n , y n ) k^n n n n ( I V . 3 . 1 0 ) N o t i c e t h a t ( I V . 3 . 1 0 ) r e d u c e s t o ( I V . 2 . 1 0 ) ( w i t h i n an a d d i t i v e c o n s t a n t ) i f one p u t s k = l . The v e l o c i t y o f t h e v o r t e x has been d e r i v e d a s s u m i n g t h a t t h e c o r e i s c i r c u l a r and r e m a i n s c i r c u l a r . H o w e v e r , c i r c u l a r c o r e s w i l l i n g e n e r a l be d i s t o r t e d by a d v e c t i o n so t h a t ( I V . 3 . 9 - 1 0 ) c e a s e t o be v a l i d . O r d e r o f m a g n i t u d e e s t i m a t e s f o r t h e p e r i o d s o v e r w h i c h ( I V . 3 . 9 - 1 0 ) a r e good a p p r o x i m a t i o n s , a r e g i v e n i n S e c t i o n V I . 3 . 43 I V . 4 The V o r t e x S t r e a m f u n c t i o n I t i s now shown t h a t one can d e r i v e a g e n e r a l i z e d v o r t e x s t r e a m f u n c t i o n . From ( I V . 1 . 1 4 ) and ( I V . 1 . 1 6 ) : A ( x , y ; x ' , y ' ) = k ( x \ y ' ) + &=f± 3 k ( ^ ' y , ) + iXzlll 9 k ( x ' , y ' ) + Q ( 2 ) 2 3y v ' = k h ( x , y ) k h ( x ' , y ' ) + 0 ( r 2 ) , r = [ ( x - x 1 ) 2 + ( y - y 1 ) 2 ] ( I V. 4 . 1) whence : M 3x ( x , y ; x ' , y ' ) 3A 3x ( x1 , y ' ;x ,y) |y=y [y=y 1 ( I V . 4 . 2 ) and a s i m i l a r e q u a t i o n i n -r— . T h u s , s i n c e Y and Inr b o t h obey t h e r e c i p r o c i t y p r o p e r t y : 3B_ 3x (x ,y ;x 1 ,y ' ) x = x |y=y 3B 3y ( x ' ,y ' ;x ,y) x = x |y=y ( I V . 4 . 3 ) ( I V . 3 . 9 ) can t h e r e f o r e be w r i t t e n i n t h e f o r m : = 1 3ft cn Y h 2 k 3y ' n n n J n . = -1 3ft 'n Y h 2 k 3x 1 n n n n ( I V . 4 . 4 ) whe re ft = \ l x k * n W k ^ V V V V + * j ^ n ^ n ' y n >xn > yn } y 2 k (x ,y ) + o &n ' h 2 ( x ,y )k(x ,y ) ., n 17 n n J n + 2Y ^ * (X ,y ) n n n ' ( I V . 4 . 5 ) 44 i s a g e n e r a l i z a t i o n o f t h e v o r t e x s t r e a m f u n c t i o n g i v e n by C . C . L i n ( 1 9 4 3 ) . As i s shown i n A p p e n d i x A , ( I V . 4 . 4 - 5 ) may be p u t i n t h e s y m p l e c t i c f o r m -1 x = o Vft ( I V . 4 . 6 ) whe re x^  i s a 2 N - d i mens i ona 1 v e c t o r , a i s a s y m p l e c t i c 2 - f o r m , and V i s t h e e x t e r i o r d e r i v a t i v e . The v o r t e x s t r e a m f u n c t i o n may be r e l a t e d t o t h e k i n e t i c e n e r g y o f t h e f l u i d . The c o r e G o f t h e n t h v o r t e x i s : 3 j n G n = { ( x , y ) : [ ( x - x n ) 2 + ( y - y j 2 ] * < e H > ( I V . 4 . 7 ) Deno te i t s b o u n d a r y by 3G c o r e s i s : N D \ U 9G n = l n The r e g i o n o f f l u i d o u t s i d e t h e ( I V . 4 . 8 ) I t i s a s s umed t h a t t h e e ' s a r e s u f f i c i e n t l y s m a l l t h a t t h e c o r e s a r e d i s j o i n t . The k i n e t i c e n e r g y o f t h e f l u i d i n D* i s : L 2 ( v2 + v 2 ) h 2 k d x d y D* x y p ( I ? k rati 2 + 'dip' 2" M dxdy 3x k 3x 3y Ik 9yJ dxdy ( I V. 4 . 9) s i n c e : 45 A p p l y i n g G r e e n ' s Theo rem t o ( I V . 4 . 9 ) : L 2 3D 1 ( I V . 4 . 1 0 ) whe re i s t h e d i r e c t i o n a l d e r i v a t i v e n o r m a l t o 3D* S i nce : N * = k - ^ k ^ k ^ ' y ; x k ' y k ) + ^ * ( x ' y ) ( I V . 4 . 11) and s i n c e ( u s i n g <?n ~ ^( x , y ; x n , y n )) 3D 1 V l)rrds = - r - n ds D' 3D 1 k 3n d s ( I V . 4 . 1 2 ) one h a s : • * = E. ^ ^ d s • p " 3D^ k 3n y. n = l 8 ¥ n 3D* k 3n ds + p X N N y n y k f ^ 3 ^ k n=l k=l 3D* -ds ( I V . 4 . 1 3 ) The f i r s t t e r m r e p r e s e n t s t h e e n e r g y due t o t h e f l o w i n d u c e d by ty* and w i l l be d e n o t e d by E^* . The b o u n d a r y 3D* may be d e c o m p o s e d . r M i r N i 3D* = U 3.D u U 3 G . U = l 1 J l j = l ( I V . 4 . 1 4 ) so t h a t 3D * k 3n ds = z i =0 3D i k 3n N ds +• z j - 1 S i n c e ty* i s c o n s t a n t on 3 D n and 3D.n 1 !In k 3n 3G-ds , 4 . 3^„ i i _ J i d s k 3n a s ( I V . 4 . 1 5 ) " r Y D n = ! c o n s t . , t h e f i r s t t e r m on t h e r i g h t i s c o n s t a n t . S i n c e e i s very s m a l l , ¥ " * x n , y n ) on 3G„ . K K f] n n n T h e r e f o r e 3G,-: ty* ^ n k 3n ds * 2 T T ^ * ( x j , y j ) 46-( I V . 4 . 16 ) S i mi 1 a r l y : 3D * k 3ri •ds = M , Y n 3 . N { Y n 3Y 1 = 1 ( I V .4 .17 ) and the f i r s t term i s c o n s t a n t as b e f o r e . I f n^j , then % - Y ( X j , y . . ; x n , y n ) on 36^ and: -3Gj y.n ^ k I T - 3 7 T d s * ^ V ^ V V ( I V . 4 . 18 ) I f n=j but k^j one can p roceed as i n ( I V . 4 . 12 ) to t r a n s p o s e n and k , whence ( I V . 4 . 18 ) ho ld s un l e s s n=k=j . S i n c e on 3G n : 3G, k £ n r + B ( x , y ; x , y ) , n n J n n J n Y 3Y ( I V . 4 . 19 ) ' 2 7 T 1 3 f 17 3 f ( - k n A n r + B ( x > y ; x n > y n ) e n d e r-.e, = - 2 r r ( k n £ n e n - B ( x n , y R ; x n , y n ) ) ( I V . 4 ,20 ) T h e r e f o r e , u s i n g (I V. 4.1 3,15 ,17 ,18 ,20) one f i n d s : N f E* = E, * + TTp £ { 2 y ^ * ( x ,y ) + y 2 B ( x ,y ;x ,y ) 1 N - k n £ n £ n | + T T P ^ k ^ Y n Y ^ ( x n , y n ; x k , y k ) + c o n s t . ( I V . 4 . 21 ) 47 The k i n e t i c e n e r g y o f a l l t h e f l u i d i s t h e r e f o r e : N E = + 2TTP-U E 2 Y N V ( x n , y n ; x k , y k ) k.= l k^n N r n = l L Y 2 B ( x ,y ;x ,y ) + y2-^-ln n x n 17 n n 17 n 7 ' n 2 a n + 2 Y n ^ " ( x n , y n ) + c o n s t . ( I V . 4 . 2 2 ) w h i c h , upon s u b s t i t u t i o n o f ( IV.3.7) and ( IV.3.6) f o r E c n be come s : E = E^* + 2-n-pft + c o n s t . T h u s , Q i s p r o p o r t i o n a l t o t h e e n e r g y o f t h a t p a r t o f t h e f l o w w h i c h i s due t o t h e v o r t i c e s . 4,8 I V . 5 C o n f o r m a l T r a n s f o r m a t i o n s S i n c e , when t h e f l u i d d e p t h i s c o n s t a n t , t h e s t r e a m -f u n c t i o n o f a s i n g l e v o r t e x i s t h e G r e e n ' s f u n c t i o n o f t h e L a p l a c i a n , i t i s n a t u r a l t o a sk how t h e v o r t e x m o t i o n c h a n g e s u n d e r a c o n f o r m a l t r a n s f o r m a t i o n o f c o o r d i n a t e s , z+z , s i n c e t h e s e l e a v e t h e L a p l a c i a n i n v a r i a n t . T h i s q u e s t i o n was f i r s t c o n s i d e r e d by Rou th ( 1 8 8 1 ) and l a t e r , i n more g e n e r a l i t y , by L i n ( 1 9 4 3 ) who s howed t h a t u n d e r t h e c o n f o r m a l t r a n s f o r m a -t i o n t h e v o r t e x s t r e a m f u n c t i o n t r a n s f o r m s a s : N Si = - h Z Y In n = l n dz dz n ( I V . 5 . 1 ) whe re t i l d e s d e n o t e t r a n s f o r m e d q u a n t i t i e s ( L i n ' s K = 2iry and W = -27rfi) . We now show t h a t ( I V . 5 . 1 ) r e m a i n s v a l i d on c u r v e d s u r f a c e s o f c o n s t a n t d e p t h p r o v i d e d t h e s u r f a c e i s i n v a r i a n t u n d e r t h e t r a n s f o r m a t i o n . L e t z = f ( z ) be a c o n f o r m a l t r a n s f o r m a t i o n o f c o m p l e x c o o r d i n a t e s . N o t i c e t h a t ( x , y ) s u c h t h a t x + i y = z a r e h a r m o n i c c o o r d i n a t e s s i n c e : d s 2 = h 2 ( z , z ) ( d x 2 + d y 2 ) = h 2 ( z , 7 ) d z d z - h 2 ( z , z ~ ) d z d f _ h 2 ( z T ) ( d x 2 + d y 2 ) , ? ) ~ T T z l T ' T i T " f ' ( z ) f ' ( z ) ( I V . 5 . 2 ) Th us : dz h ( z , z ) = n ( z , z ) ( I V . 5 . 3 ) dz The s t r e a m f u n c t i o n f o r a v o r t e x i n t h e t r a n s f o r m e d s y s t e m i s : $ ( 2 , f ; 2 \ Y r ) = ¥ ( z , z ; z ' , z 1 ) ( I V . 5 . 4 ) 49 and ifi*(2,z) = r ( 2 , z ) ( I V . 5 . 5 ) s i n c e t h e L a p l a c i a n i s i n v a r i a n t . From ( I V . 1 . 2 2 ) -ln\z-T'\ + B ( z , l ; z ' , f ' ) = -ln\z-z'\ + B ( z , z ; z 1 , z ' ) ( I V . 5 . 6 ) o r : B ( z , z ; z ' ,z"') = B(z,z;z' , 7 ' ) + "An z - z z - z ( I V . 5 . 7 ) The re f o r e : B ( z ' , z ' ;z' , ! 1 ) = L i m z+z B ( z , z ; z ' , z ' ) + In z - z ' z - z B ( z ' , z ' ; z ' , z ' ) + In dz dz ( I V . 5 . 8 ) In t h e t r a n s f o r m e d s y s t e m t h e m o t i o n i s d e r i v e d f r o m t h e t r a n s f o r m e d s t e a m f u n c t i o n : ° = ^ 1 k J / n V < V V V ' V + * j i [ y J B ( 2 n , fn ; z~n , t„ ) + y 2 ^ n h 2 ( z , z ) + 2Y Y * ( Z ,Z ) n ' z n ) ; N = Q - h Z In k=l dz dz ( I V . 5 . 8 ) v e r i f y i n g ( I V . 5 . 1 ) .50 I V . 7 C o n s t a n t s o f t h e M o t i o n As i n any s y m p l e c t i c s y s t e m , t h e r e a r e c o n s e r v a t i o n l a w s a s s o c i a t e d w i t h i n f i n i t e s i m a l t r a n s f o r m a t i o n s w h i c h l e a v e t h e v o r t e x s t r e a m f u n c t i o n i n v a r i a n t ( s e e , e . g . , J . M . S o u r i a u ( 1 9 6 9 ) ) . In p a r t i c u l a r , i f t h e m e t r i c f u n c t i o n h and t h e d e p t h f u n c t i o n k and a l l b o u n d a r i e s a r e i n v a r i a n t u n d e r an i n f i n i t e s i m a l c o o r d i n a t e t r a n s f o r m a t i o n t h e n t h e v o r t e x s t r e a m f u n c t i o n mu s t a l s o be i n v a r i a n t . S y m m e t r i e s o f t h e f l u i d t h e r e f o r e i n d u c e c o n s e r v a t i o n l a w s . ' L e t G'( x i ,y i , . . . , x n ,y ) be a r e a l v a l u e d f u n c t i o n o f t h e v o r t e x p o s i t i o n s . T h e n : d_G d t N r * r Z - ^ X + # - y aV n-n ^ n ^ n 9G 9ft 9x 3y n J n 9G 9ft 9 y n 9 x n -= |_G,fl] ( I V . 7 .1 ) G i s a c o n s t a n t o f t h e m o t i o n i f and o n l y i f [G,ftJ = 0 . . In p a r t i c u l a r , ft i s i t s e l f c o n s e r v e d . C o n s i d e r t h e i n f i n i t e s i m a l t r a n s f o r m a t i o n : = x.. + 9G n Y h k 9y ' ' n n n J n n = y n J n e 9G Y h 2 k 9x ' n n n n ( I V . 7 . 2 ) f o r some s m a l l e . G i s c a l l e d t h e g e n e r a t o r o f t h e t r a n s f o r m a t i o n . T h e n : »ni ..,5 n»n n) - fi(xi,yi x n > y n ) 51 N Z 1 „ ,Yh k n = 1 1 n n dtt 3G 3ft 3G 9 x n 3 y n 3 y n 3 V + ou2) = -[G,ft] + 0(e2.) ( IV .7 . 3 ) f o r some s m a l l e . E x a m p l e s : a) I f h E h ( x ) , k E k ( x ) , and t h e o n l y b o u n d a r i e s a r e c u r v e s x = c o n s t , t h e n ft i s i n v a r i a n t u n d e r : £ = x . n = y - e ^n n 'n J n The g e n e r a t o r o f t h i s t r a n s f o r m a t i o n i s : N ( G = Z y n h 2 ( x ) k ( x n ) d x n ( IV .7 . 4 ) w h i c h i s t h e r e f o r e c o n s e r v e d . When t h e s u r f a c e o f f l o w i s a p l a n e and k i s c o n s t a n t t h i s y i e l d s t h e c o n s e r v a t i o n o f c e n t r e o f c i r c u l a t i o n . b) I f h s h ( r ) , k s k ( r ) , and a l l b o u n d a r i e s a r e c u r v e s r = c o n s t , t h e n ft i s i n v a r i a n t u n d e r ^n = x n + £ y n ' n n = y n " £ X n The re f o r e / 5 2 y " ' ( I V - 7 - 5 ' w h i c h can be r e w r i t t e n 2 Y n h ' ( r n ) k < r n ) r n ) ; ^ = 0 (IV.7.6) Thus N G = 2 Z y n= l n h 2 ( r ) k ( r ) r d r ( I V . 7 . 7 ) n / v n n n - \ / i s c o n s e r v e d . F o r f l o w i n t h e p l a n e w i t h k = c o n s t , t h i s y i e l d s t h e c o n s e r v a t i o n o f moment o f c i r c u l a t i o n . V o r t e x s y s t e m s i n t h e p l a n e w i t h k = c o n s t , e x h i b i t a n o t h e r c o n s e r v e d q u a n t i t y known as t h e a n g u l a r moment o f c i r c u 1 a t i o n : N ^ Y n ( x n y n - x n y n ) = c o n s t . ( I V . 7 . 8 ) One can g e n e r a l i z e t o c u r v e d s u r f a c e s as f o l l o w s : S u p p o s e h and k a r e homogeneous f u n c t i o n s o f o r d e r v : i . e . , h ( a x , a y ) = a y h ( x , y ) , k ( - a x . a y ) = a v k ( x , y ) . P h y s i c a l l y t h i s means t h a t t h e f l u i d i s i n v a r i a n t u n d e r s c a l e t r a n s f o r m a t i o n s (x ,y)->-(ax , a y ) . I f t h e b o u n d a r i e s a r e a l s o p r e s e r v e d u n d e r t h e t r a n s f o r m a t i o n , t h e n t h e v o r t e x s t r e a m -f u n c t i o n must t r a n s f o r m a s : ft(axi , a y ! , . . . , a x n , a y n ) = a u f i ( x , , y x , . . . , x n , y n ) + b ( a ) i n o r d e r t h a t t h e m o t i o n i n t h e t r a n s f o r m e d s y s t e m ( w i t h a p p r o p r i a t e l y s c a l e d t i m e ) i s s i m i l a r t o t h e m o t i o n i n t h e o r i g i n a l s y s t e m . D i f f e r e n t i a t i n g by a and t h e n s e t t i n g 53 N E n = l 9ft 'n 9x + y 9ft nay„ lift + c o n s t U s i n g ( I V . 4 . 4 ) and t h e f a c t t h a t ft i s i t s e l f c o n s e r v e d , one . has : n f ^ n n n k n [ V n - x n y n ^ = c o n s t ( I V . 7 . 1 0 ) The c o n s e r v a t i o n o f a n g u l a r moment o f c i r c u l a t i o n i s t h e r e f o r e r e l a t e d t o t h e i n v a r i a n c e o f t h e f l u i d u n d e r s c a l e t r a n s f o r m a t i o n s . T h a t t h i s i s t h e c a s e f o r r e c t i l i n e a r v o r t i c e s has been p o i n t e d o u t by Chapman ( 1 9 7 8 ) . 54 V. S IMPLE VORTEX SYSTEMS The b e h a v i o u r o f some s i m p l e v o r t e x s y s t e m s i s now e x a m i n e d i n o r d e r t o g a i n some i n s i g h t i n t o t h e q u a l i t a t i v e d i f f e r e n c e s i n t h e m o t i o n o f v o r t i c e s on a c u r v e d s u r f a c e w i t h v a r y i n g d e p t h and r e c t i l i n e a r v o r t i c e s on a p l a n e . V . l The M o t i o n o f a S i n g l e V o r t e x : No E x t e r n a l V e l o c i t y F i e l d C o n s i d e r f i r s t t h e m o t i o n o f a s i n g l e v o r t e x . S i n c e ft i s c o n s e r v e d i t s p a t h i s g i v e n b y : 2ft - 2- = B(x,y;x,y) '+ k(x,yUn ' h 2 ( x , y ) k ( x , y ) = c o n s t . Y _ ( V . l . l ) The v o r t e x p a t h may c l o s e b u t may n e v e r c r o s s i t s e l f . T h e r e a r e two s p e c i a l c a s e s o f i n t e r e s t . a) C u r v e d S u r f a c e , U n i f o r m Depth I f t h e d e p t h i s u n i f o r m one c an p u t k = l a n d : B ( x , y , x , y ) + £ n h 2 ( x , y ) = c o n s t . ( V . l . 2 ) I f one s u p p o s e s f u r t h e r t h a t t h e r e a r e no b o u n d a r i e s t h e n B=0 and t h e v o r t e x moves a l o n g a c u r v e h = c o n s t . More e x p l i c i t l y , i t s v e l o c i t y i s : v = - y z x V h ( x , y ) . ( V . 1.3) h 2 ( x , y ) The m o t i o n i s i n m a r k e d c o n t r a s t t o t h e m o t i o n o f a v o r t e x i n t h e p l a n e , w h i c h r e m a i n s s t a t i o n a r y . N o t i c e t h a t a know-l e d g e o f h ( x , y ) i n t h e v i c i n i t y o f t h e c o r e i s s u f f i c i e n t t o d e t e r m i n e v . 55 b) P l a n e S u r f a c e , N o n - U n i f o r m Depth I f t h e s u r f a c e i s p l a n a r one may p u t h= l so t h a t : f k ( x , y ) ) _ B ( x , y ; x , y ) + k i ^ l c o n s t a n t ( V . l . 4 ) Now, h o w e v e r , e ven i n t h e a b s e n c e o f b o u n d a r i e s , B does n o t n e c e s s a r i l y v a n i s h . One can o n l y o b t a i n B i n g e n e r a l by s o l v i n g ( I V . 1 . 5 ) . Fo r n o n - c o n s t a n t k v e r y few a n a l y t i c s o l u t i o n s a r e known . I t i s p o s s i b l e , h o w e v e r , t o t r e a t t h e c a s e i n w h i c h t h e f l u i d has u n i f o r m d e p t h b u t f o r a s m a l l d e p r e s s i o n n e a r t h e o r i g i n ( s e e ' F i g u r e I V ) . k = k Q , r > e k = k ( r ) , r < ( V . 1 . 5 ) The v o r t e x i s a s sumed t o be a t ( x , y ) = ( - a , 0 ) w i t h a >> e L e t t h e s t r e a m f u n c t i o n f o r t h e f l o w b e : ^ ( x ,y ) = Y 0 ( x , y ) + x ( x , y ) ( V . 1 . 6 ) whe re Y 0 i s t h e s t r e a m f u n c t i on i f k = k Q e v e r y w h e r e . Then x(x ,y ) s a t i s f i e s : VY, ( V . l . 7 ) I f e i s s u f f i c i e n t l y s m a l l V*F0 i s n e a r l y c o n s t a n t a c r o s s t h e r e g i o n i n w h i c h k v a r i e s . T h u s : s i n c e : Y 0 ( x , y ) f o r m : - k ' ( r ) Y ~ ~ - k . 0 k ' ( r ) Y nmrff - k ° ( r ) a c o s * ^ A - ^ - y k 0 £ n [ ( x + a ) 2 + y 2 ] ^ . x t h e r e f o r e has t h e vortex core F i g . I V : F l u i d w i t h D e p r e s s i o n a t t h e O r i g i n 57 X = f f ( r ) c o s c j > d S i n c e V2x = 0 f o r r > e , f ( r ) ^ f o r r >> e and b some c o n s t a n t d e p e n d i n g o n l y on k ( r ) . A t t h e v o r t e x X s . T h u s , i f t h e v o r t e x i s a t ( x , y ) and t h e d e p r e s s i o n a t ( x 0 , y 0 ) . t h e n : B ( x , y ; x , y ) * ( x - x 0 ) + ( y - y 0 ) 2 by ( V . 1 . 9 ) The v e l o c i t y o f t h e v o r t e x i n d u c e d by t h e d e p t h v a r i a t i o n v a r i e s as d ~ 3 f o r l a r g e d , whe re d i s t h e d i s t a n c e o f t h e v o r t e x t o t h e d e p r e s s i o n . The d i r e c t i o n o f t h e v e l o c i t y i s p e r p e n d i c u l a r t o t h e l i n e j o i n i n g t h e v o r t e x and t h e d e p r e s s i o n . When t he d e p t h o f t h e f l u i d i s v a r y i n g a t t h e p o s i t i o n o f t h e v o r t e x t h e r e i s a c o m p o n e n t o f v e l o c i t y : d i r e c t e c a l o n g c u r v e s o f c o n s t a n t k. T h i s v e l o c i t y componen t i s i n d u c e d by t h e b e n d i n g o f t h e v o r t e x c o r e . k t h r o u g h o u t t h e r e g i o n o f f l o w i s n e c e s s a r y t o d e t e r m i n e v_ . I t i s l a r g e l y due t o t h i s p r o p e r t y t h a t v o r t e x s y s t e m s i n f l u i d s o f v a r y i n g d e p t h a r e more d i f f i c u l t t o a n a l y z e t h a n t h o s e o f c o n s t a n t d e p t h . + 1 z x V k ( x , y ) ( V . 1 . 1 0 ) N o t i c e t h a t , u n l i k e t h e p r e v i o u s c a s e , a k n o w l e d g e o f 5 8 V . 2 The M o t i o n o f a S i n g l e V o r t e x i n a U n i f o r m S t r e a m C o n s i d e r t h e m o t i o n o f a s i n g l e v o r t e x i n a f l u i d o f c o n s t a n t t h i c k n e s s on a s u r f a c e t h a t i s p l a n a r a t i n f i n i t y b u t i s c u r v e d n e a r t h e o r i g i n ; f o r s i m p l i c i t y i t i s a s sumed r a d i a l l y s y m m e t r i c ( F i g u r e V ) . One may t h e n s u p p o s e t h e s u r f a c e i s d e f i n e d by z = b ( p ) ( s e e S e c t i o n 1 1 . 2 ) w h e r e b ' ( 0 ) = 0 ( s o t h a t t h e s u r f a c e i s smooth a t t h e o r i g i n a nd b + 0 as p+<*> The p o l a r h a r m o n i c c o o r d i n a t e r i s g i v e n by ( I I . 2 . 1 8 ) and i f one c h o o s e s p 0 so t h a t : Po / l + ] b ' ( s ) - ds = 1 ( V . 2 . 1 ) Po t h e n as p - > - ° ° , r - > p , k - > - l . The e x t e r n a l f i e l d i s s u c h t h a t i t i s a u n i f o r m s t r e a m as p ^ oo . I f one c h o o s e s i t s d i r e c t i o n to be t h e x - d i r e c t i o n t h e n : ty*(r,<$>) = Uy = Ursine}) ( V . 2 . 2 ) The p a t h o f t h e v o r t e x i s t h e r e f o r e : J U h ( r ) + U - ( y ^ y o ) = 0 where y = y 0 i s t h e p a t h o f t h e v o r t e x f a r u p s t r e a m . C l e a r l y , y = y 0 i s t h e p a t h o f t h e v o r t e x f a r down-s t r e a m t o o , and s i n c e y + psincj) a t i n f i n i t y , t h e p a t h s a t i n f i n i t y do n o t depend on t h e c u r v a t u r e n e a r t h e o r i g i n a t a l l . In g e n e r a l , t h o u g h . , t h e t i m e t a k e n f o r t h e v o r t e x t o pa s s t h e o r i g i n w i l l d epend upon t h e d e t a i l s o f t h e c u r v a t u r e . 59 F i g . V: F l u i d w i t h S u r f a c e C u r v a t u r e N e a r t h e O r i g i n 60 The e q u a t i o n s o f m o t i o n o f t h e v o r t e x a r e • _ y y h : ' ( r ) U  x 2 h 3 ( r ) r h z ( r ) U v = Y X h ' ( r )  y 2 h 3 ( r ) r A s s u m i n g - >_0 t h e r e i s a s t a t i o n a r y p o i n t a t ( V . 2 . 4 ) x = 0 , y = r 0 whe re : h ' ( r 0 ) = - 2U h( r 0 ) y ( V . 2 . 5 ) ( N o t e t h a t , f r o m ( I I . 2 . 2 1 ) , h ' ( r ) / h ( r ) < 0 ). The s i n g u l a r p o i n t i s s t a b l e i f Q, has a maximum o r min imum t h e r e : i . e . , i f : 3 2 f t ( o , r 0 ) 9 2 f t ( o , r 0 ) dx- dxdy d2Q(o,r) 9 2 f t ( o , r ) dydx dr T 2 ( p ( r o ) - l ) 2rT 0 YgCrop-'(ro)-/(tJ(.irb)-l)3 2 7 T ( V . 2 . 7 ) i s p o s i t i v e o r n e g a t i v e d e f i n i t e . S i n c e p ( r ) < 1 t h i s o c c u r s when : r oP 1 (<"o) - p( r 0 ) + 1 < 0 ( V . 2 . 8 ) S i n c e t h e G a u s s i a n c u r v a t u r e o f t h e s u r f a c e i s : K = ~ P r / r ) , rh ( r ) ( V . 2 . 8 ) p r e d i c t s s t a b i l i t y o n l y i f K ( r 0 ) > 0 . I f b ( p ) i s d e c r e a s i n g t h i s o c c u r s o n l y i f t h e s t a t i o n a r y p o i n t i s s u f f i c i e n t l y c l o s e t o t h e o r i g i n . 61 V . 3 T h e M o t i o n , o f a C l o s e V o r t e x P a i r : y\'= - y 2 C o n s i d e r t h e m o t i o n o f a p a i r o f v o r t i c e s w i t h e q u a l b u t o p p o s i t e s t r e n g t h s s e p a r a t e d by a d i s t a n c e much s m a l l e r t h a n |V£nh| i n a f l u i d o f c o n s t a n t d e p t h . The e q u a t i o n s o f m o t i o n a r e ( u s i n g c o m p l e x n o t a t i o n ) : Z i i i + ix i b i h ? ( z i - z 2 ) h? 3z z 2 h i ( z j - z 2 ) i y 3h 2 : hf 3z ( V . 3 . 1 ) T h e r e f o r e , d e f i n i n g Z = h{z\ + z 2 ) , d = % ( z i - z 2 ) : Z = n -1 4d " „ 1 . 1 h 2 ( Z + d ) h z ( Z - d ) 1 3h(Z+d) h 3 ( Z + d) 3 z. 1 3 h ( Z - d ) h 3 ( Z-d7 az ( V . 3 . 2 a ) I X 4d 1 1 h 2 ( Z + d ) " h 2 ( Z - d ) • Y ' 1 3h(Z+d) h 3 ( Z + d7 3z 1 3h(Z+d) h M Z - d ) 3z (The a r g u m e n t Z o f h has been d r o p p e d f o r s i m p l i c i t y ) ( V . 3 . 2 b ) E x p a n d i n g and d r o p p i n g t e r m s o f o r d e r ^ Z = 2dh2Iz) TT - - i Y, 3h (Z ) d a h 3 ( Z ) 3z d ( V . 3 . 3 ) Note t h a t : d d 2 d h2 ( Z ) 2 3h ( Z ) hTzT 3z - 2Z 3h ( Z ) hTzT 3z ( V. 3 . 4 ) so t h a t : 62 + d _ J Z 3h (Z ) , . I 3 h ( Z ) l d\; d " [HTZT 3z hTD" 3z = - 2 - — £ n h ( Z ) ( V . 3 . 5 ) w h e n c e : h 2 ( Z ) d d = E = c o n s t . ( V . 3 . 6 ) T h i s i s t h e c o n s e r v a t i o n o f e n e r g y t o t h i s l e v e l o f a p p r o x i m a -t i o n . From ( V . 3 . 3 ) and ( V . 3 . 6 ) : v 2 = h 2 ( Z ) Z Z = ^ = c o n s t . ( V . 3 . 7 ) The p a i r moves w i t h c o n s t a n t s p e e d . One c a n e l i m i n a t e d f r o m t h e e q u a t i o n s o f m o t i o n by d i f f e r e n t i a t i n g ( V . 3 . 3 ) : ' V - i>7 _ - Y 2 • 3 h ( Z ) Z _ 2 72 3£nh(_Zj_ , . L ~ 4 E d •" 2Eh 3 T Z T 3z Z ^ Z 3 z ' S uppo se h E h ( r ) and p u t Z = R e 1 $ . Then ( V . 3 . 8 ) b e c o m e s : R* - R $ 2 = - ( R 2 - R 2 $ 2 ) ^ £ n h ( R ) ( V . 3 . 9 a ) R $ + 2R$ = - 2 R R ^ £ n h ( R ) ( V . 3 . 9 b ) y 2 / 4 E = h 2 ( R ) ( R 2 + R 2 $ 2 ) ( V . 3 . 1 0 ) T h e r e i s a n o t h e r c o n s t a n t o f t h e m o t i o n s i n c e : • • • | + X = - 2 R ^ h ( R ) ( V . 3 . 1 1 ) whence : R 2 $ h 2 ( R) E J = c o n s t , U s i n g ( V . 3 . 1 2 ) t o e l i m i n a t e I i n ( V . 3 . 1 0 ) g i v e s D 2 + J 2 _ Y 2 K R 2 h " r r 7 4 E h z ( r T 63 ( V . 3 . 1 2 ) ( V. 3. 13) I f t h e s u r f a c e i s Z = b ( p ) ( s e e S e c t i o n I I I . 2 ) t h e n ( V . 3 . 1 1 ) and ( V . 3 . 1 2 ) may be r e w r i t t e n : p2<S> = J i i _ 4E J 2 P 2 " a + r b , (p ) D ( V . 3 . 1 4 ) ( V . 3 . 1 5 ) ( V . 3 . 1 4 ) and ( V . 3 . 1 5 ) a r e i d e n t i c a l t o t h e e q u a t i o n s o f m o t i o n o f a p a r t i c l e m o v i n g u n d e r t h e i n f l u e n c e o f t h e c e n t r a l p o t e n t i a l : V ( p ) X l 4E J 2 9* V (P ) ] ( l + [ b ' ( p ) ] 2 ) ( V . 3 . 1 6 ) w i t h p F o r e a c h J and E t h e r e i s e x a c t l y one c i r c u l a r o r b i t 2 J /E~ = ~ ~ = a • P c a n n e v e r be l e s s - t h a n a . The .path o f t h e v o r t e x p a i r i s g i v e n b y : d£ = f p 2 y 2 _ l d<J) 4 J 2 E ( V . 3 . 1 7 ) T T + X b T p T p T ^ f r o m w h i c h one f i n d s t h a t t h e d i s t a n c e o f c l o s e s t a p p r o a c h t o t h e o r i g i n i s a l s o p = a . The c i r c u l a r o r b i t i s t h e r e f o r e u n s t a b l e a n d i s t h e e n v e l o p e o f a l l t h e p o s s i b l e o r b i t s f o r f i x e d E and J . I f b ^ - c o n s t , as p -> °° t h e p a t h o f t h e v o r t e x p a i r i s q u a l i t a t i v e l y s i m i l a r t o t h a t shown i n F i g u r e V I . The 6 4 i m p a c t p a r a m e t e r i s : l - £ - - a V Y ( V . 3 . 1 8 ) 65 F i g . V I : The P a t h o f a V o r t e x P a i r 66 V.4 The S t a b i l i t y o f a S i n g l e R i n g o f V o r t i c e s on a S u r f a c e o f R e v o l u t i o n H i s t o r i c a l l y t h e s t a b i l i t y o f c o n f i g u r a t i o n s o f v o r t i c e s w h i c h move as r i g i d b o d i e s has r e c e i v e d a g r e a t d e a l o f a t t e n t i o n . L o r d K e l v i n ( 1 8 7 8 ) f i r s t e x a m i n e d e q u a l s t r e n g t h v o r t i c e s p l a c e d a t t h e v e r t i c e s o f r e g u l a r p o l y g o n s . H i s r e s u l t s were c o m p l e t e d by J . J . Thomson ( 1 8 8 3 ) who c o n c l u d e d t h a t s i x o r f e w e r s u c h v p i r t i c e s a r e s t a b l e w h i l e s e v e n o r more a r e u n s t a b l e . T h o m s o n , h o w e v e r , e r r e d i n t h e h e p t a g o n a l c a s e . M o r t o n ( 1 9 3 3 ) s howed t h a t T h o m s o n ' s m e t h o d was i n f a c t i n s u f f i c i e n t t o d e t e r m i n e t h e s t a b i l i t y o f a r e g u l a r h e p t a g o n o f v o r t i c e s . I t was o n l y i n 1977 t h a t M e r t z showed t h a t t h i s c o n f i g u r a t i o n i s s t a b l e . The s t a b i l i t y o f o t h e r r i g i d c o n f i g u r a t i o n s has a l s o been s t u d i e d , n o t a b l y , t h e v o r t e x s t r e e t s o f von Karman ( 1 9 1 2 ) , v o r t i c e s a t p o l y g o n a l v e r t i c e s w i t h a c i r c u l a r c o n t a i n e r ( H a v e l o c k ( 1 9 3 1 ) , Chapman ( 1 9 7 7 ) ) and i n f i n i t e v o r t e x l a t t i c e s ( T k a c h e n k o ( 1 9 6 6 ) ) . E x p e r i m e n t a l c o n f i r m a t i o n o f t h e s t a b i l i t y o f some c o n f i g u r a t i o n s w i t h i n a c i r c u l a r b o u n d a r y has r e c e n t l y been o b t a i n e d f o r v o r t i c e s i n H e l l , by a t e c h n i q u e i n w h i c h t h e p o s i t i o n s o f t h e v o r t i c e s a r e a c t u a l l y p h o t o g r a p h e d ( Y a r m c h u k , e t . a l . ( 1 9 7 9 ) ) . We c o n s i d e r , now, an e x t e n s i o n o f t h e q u e s t i o n c o n -s i d e r e d o r i g i n a l l y by L o r d K e l v i n and T h o m s o n : i s a s i n g l e r i n g o f v o r t i c e s on a s u r f a c e o f r e v o l u t i o n s t a b l e ? L e t N v o r t i c e s o f e q u a l s t r e n g t h y be p l a c e d i n i t i a l l y a t : 2 TT i n 6.7 ( V . 4 . 1 ) on a s u r f a c e o f r e v o l u t i o n d e s c r i b e d by h '= h ( r ) and i n a f l u i d whose d e p t h i s c o n s t a n t ( k = l ) and has no b o u n d a r y N (B = 0) . S i n c e 1 y j= 0 t h e s u r f a c e c a n n o t be c l o s e d . n = 1 n The e q u a t i o n s o f m o t i o n a r e , f r o m ( I V . 2 . 1 1 ) : •n " h 2 ( r n ) E 1 - n T i Y Z h 1 ( r ) ' n v n ' l k *nV z n 2 r „ h ( r J n v n ( V . 4 . 2 ) The s ymmet r y o f t h e i n i t i a l c o n f i g u r a t i o n s u g g e s t s a s o l u t i o n o f t h e f o r m : z n = r ( t ) e x p " 2 T T J n N + OJi t S u b s t i t u t i n g i n t o ( V . 4 . 2 ) one f i n d s 1-1 r ( t ) = r 0 ;wi _Y_ r 2 h M r 0 ) ( V . 4 . 3 ) r 0 h ' ( r 0 V k ^ l - e x p ( 2 i r i k / N ) 2 h ( r 0 ) ( V. 4. 4) The sum i s e v a l u a t e d i n A p p e n d i x B g i v i n g k ^ l - e x p ( 2 7 T i k/N) 2 so t h a t CO % [ N - P ( r 0 ) ] ( V . 4 . 5 ) ( V . 4 . 6 ) w h e r e , f o r c o n v e n i e n c e , t h e u n i t o f t i m e has been t a k e n t o b e : r 2 h 2 ( r 0 ) / y . The r i n g o f v o r t i c e s r o t a t e s r i g i d l y a b o u t t h e a x i s o f r e v o l u t i o n w i t h a n g u l a r v e l o c i t y u>1 . To e x a m i n e t h e s t a b i l i t y o f t h e c o n f i g u r a t i o n c o n s i d e r s m a l l d e v i a t i o n s f r o m 68 the m o t i o n : n ( t ) = [ r 0 e x p ( 2 7 T i n / N ) + e n ( t j j e 1 ' " 1 . * ( V . 4 . 7 ) S u b s t i t u t i n g i n t o (V.4.2) and e x p a n d i n g t o f i r s t o r d e r i n t h e e ' s ( e -e. ) n k ' k J n ( l - e x p ( 2 T T ( k - n ) / N ) ) 2 + Q ( r 0 , u 1 ) e n e x p ( 4 T r i n / N ) + P( r -a) )e o i ' n i e x p ( - 4 x r i n / N ) (V.4.8) w h e r e : P(r,w) = % r p ' ( r ) + ( p ( r) - 1 ) (u-Jg) Q(r,o) ) = % r p ' ( r ) + p ( r)oj The s o l u t i o n s t o ( V . 4 . 8 ) a r e o f t h e f o r m e n = a M e x p [ ( 2 T r i ( l + M ) n / N ) + i A M t ] + b M e x p [ 2 T r i ( l - M ) n / N - i A M t ] ( V . 4 . 9 ) ( V . 4 . 1 0 ) ( V . 4 . 1 1 ) S u b s t i t u t i n g i n t o ( V . 4 . 9 ) and e q u a t i n g c o e f f i c i e n t s o f e 1 ^ 1 M and e s e p a r a t e l y . y i e l ds : U M + Q ( r 0 . W i ) ) a M + ( S 2 _ M + P ( r 0 , a ) 1 ) ) b M = 0 ( V . 4 . 1 2 a ) ( S l + M + P ( r o > w i ) ) a M + ( " A M + Q ( ^ ^ 1 ) ) b M = 0 ( V . 4 : 1 2 b ) w i t h : N - l c = y l - e x p ( 2 T r i L k / N ) L " ^ ( l - e x p ( 2 ~ T r i k / N ) ) 2 - Js (N-L) ( 2 - L ) , L = l , . . . ,N ,( V . 4 . 1 3 ) ( s e e A p p e n d i x B ) . T h e r e a r e n o n - t r i v i a l s o l u t i o n s o f (V . 4 .12) i f and o n l y i f : 69 XM " Q 2 ( r 0 , W l ) + ( S 1 + M + P ( r 0 , o ) 1 ) ) ( S 1 _ M + P ( r 0 , U l ) ) = 0 ( V . 4 . 1 4 ) The Mth mode i s s t a b l e i f i s r e a l ; i . e . , i f : Q 2 K > ^ ) " ( S l + M + p K ' S > H s i - M + ' P < r 0 A ) ) > ; 0 ( V . 4 . 1 5 ) U s i n g ( V . 4 . 6 ) , ( V ; 4 . 9 ) , ( V . 4 . 1 0 ) and ( V . 4 . 1 3 ) , ( V . 4 . 1 5 ) b e c o m e s : ^ ( g o ) r ° + p ( r 0 ) ( N - p ( r 0 ) ) - ^ i!iL > 0 , M=1 , . . . ,N - 1 ( V . 4 . 1 6 ) In a l i n e a r s t a b i l i t y a n a l y s i s o f any p e r i o d i c s o l u t i o n o f a s y m p l e c t i c s y s t e m one a l w a y s e x p e c t s two z e r o - f r e q u e n c y s o l u t i o n s : one c o r r e s p o n d i n g t o s m a l l d i s p l a c e m e n t s o f t h e s y s t e m a l o n g i t s o r b i t , t h e o t h e r c o r r e s p o n d i n g t o s m a l l d i s p l a c e m e n t s o f t h e s y s t e m o f f t h e h y p e r - s u r f a c e ft = c o n s t . The f i r s t i s a s t a b l e mode; t h e s e c o n d i s f o r b i d d e n by t h e c o n s e r v a t i o n o f ft . H e r e t h e s e modes a r e t h e M=N modes w h i c h have been n e g l e c t e d i n ( V . 4 . 1 6 ) . They c a n n o t a f f e c t t h e s t a b i l i t y o f t h e c o n f i g u r a t i o n . ( V . 4 . 1 6 ) has a min imum when M=%N , N e v e n o r M=J5(N±1) N o d d . H e n c e , f o r s t a b i l i t y o f t h e r i n g o f v o r t i c e s : P ' ( g o ) r ° + p ( r 0 ) [ N - p ( r 0 ) ] - x > 0 N e v e n ( V . 4 . 1 7 ) >: -1 N odd N o t i c e t h a t , i n c o n t r a d i s t i n c t i o n t o t h e s y s t e m i n S e c t i o n V . 2 , t h e s t a b i l i t y i s e n h a n c e d by n e g a t i v e c u r v a t u r e o f t h e s u r f a c e a t r 0 . By m a k i n g t h e c u r v a t u r e a t r 0 l a r g e and n e g a t i v e i t i s p o s s i b l e t o a c c o m o d a t e a r b i t r a r i l y l a r g e number s o f v o r t i c e s i n a s t a b l e r i n g . E x a m p l e s . a) P l a n e F o r t h e p l a n e : p ( r ) = l , p ' ( r ) = 0 ( S e c t i o n I I . 2 ) . F o r s t a b i 1 i t y : - N 2 + 8N - 8 > 0 N e v e n > -1 N odd ( V . 4 . 1 8 ) whence t h e r e i s s t a b i l i t y i f N < 7 . I f N=7 t h e s t a b i l i t y c r i t e r i o n i s i n c o n c l u s i v e and one must use h i g h e r o r d e r p e r t u r b a t i o n t h e o r y t o d e t e r m i n e t h e s t a b i l i t y . T h i s i s t h e c a s e on w h i c h Thomson e r r e d . b) ~ C y l i n d e r A c y l i n d e r h a s : p ( r ) = 0 , r p ' ( r ) = 0 . H o w e v e r , s i n c e t h e c y l i n d e r e x t e n d s t o i n f i n i t y b o t h as r 0 and as r °° , t h e r e i s an a r b i t r a r i n e s s i n t h e c h o i c e o f ¥( x ,y ;x 1 ,y 1 ) d e p e n d i n g on t h e c i r c u l a t i o n a r o u n d r = r ' as r ' °° and r ' -»• 0 . In o t h e r wo rd s : Y U . y ^ ' , y ' ) = -(l+aHn((x-x' ) 2 + ( y - y ' ) 2 ) % + a £ n ( x 2 + y2)h ( V . 4 . 1 9 ) i s a p e r f e c t l y v a l i d s t r e a m f u n c t i o n f o r a u n i t v o r t e x . The c i r c u l a t i o n a r o u n d r = r ' as r 1 °° i s 2 TT ( 1+a ) w h i l e t h a t a r o u n d r = r ' as r ' + O i s -2Tra . In w r i t i n g t h e e q u a t i o n s ( V . 4.2) we have a r b i t r a r i l y c h o s e n a = 0 . The s t a b i l i t y c r i t e r i on i s t h e n : 71 N 2 < 0 , N e v e n , N 2 < -1 , N odd whence a l l r i n g s o f v o r t i c e s a r e u n s t a b l e . The v e l o c i t y f i e l d i n d u c e d by n o n - z e r o a i n ( V . 4 . 1 9 ) i s o f t h e f o r m : v r = 0 , v ^ = c o n s t . The a d d i t i o n o f s u c h a v e l o c i t y f i e l d c a n n o t a f f e c t t h e s t a b i l i t y o f a r i n g o f v o r t i c e s : ( V . 4 . 2 0 ) i s v a l i d f o r a l l a . 72 V.5 The S t a b i l i t y o f V o r t e x S t r e e t s on S u r f a c e s o f R e v o l u t i o n I f i n a d d i t i o n t o r o t a t i o n a l s y m m e t r y t h e s u r f a c e o f f l o w i s i n v a r i a n t u n d e r r e f l e c t i o n i n a p l a n e p e r p e n d i c u l a r t o t h e a x i s o f r o t a t i o n t h e n d o u b l e r i n g s o f v o r t i c e s can a l s o r o t a t e r i g i d l y a b o u t t h e a x i s o f r o t a t i o n . In v i e w o f t h e q u a l i t a t i v e s i m i l a r i t i e s t o t h e c o n f i g u r a t i o n s o f von Karman ( 1 912 ) t h e s e a r e c a l l e d v o r t e x s t r e e t s . L e t t h e s u r f a c e o f r e v o l u t i o n be d e s c r i b e d b y : p = f ( z ) ( S e c t i o n I I . 2 ) w i t h f an., e ven f u n c t i o n . T h e n , f r o m ( I I . 2 . 1 4 ) r . e x p f D * (V.5.1) Z 0 C h o o s i n g z Q = 0 and s i n c e f ( s ) = f ( - s ) : r ( - z ) = yr^j ( V . 5 . 2 ) M o r e o v e r , ( I I. 2 . 1 5 ) i m p l i e s : h(i) r h ( r ) = —f- ( V . 5 . 3 ) ( V . 5 . 3 ) i s t h e c o n d i t i o n w h i c h i m p l i e s t h a t t h e s u r f a c e i s i n v a r i a n t upon r e f l e c t i o n i n t h e p l a n e c o n t a i n i n g t h e c u r v e r = l ( z=0) % We e x a m i n e t h e s t a b i l i t y o f a r i n g o f N v o r t i c e s o f s t r e n g t h y a t r 0 ( z 0 ) and N o f s t r e n g t h - y a t - ^ - ( - z 0 ) . T h e r e a r e two d i s t i n c t c a s e s : s t a g g e r e d and s y m m e t r i c v o r t e x s t r e e t s . 73 a) S t a g g e r e d V o r t e x S t r e e t s The v o r t i c e s o f a s t a g g e r e d v o r t e x s t r e e t a r e s i t u a t e d i n i t i a l l y a t : r n = r 0 , <j> = 2-iTn/N , n = l , . . . , N , s t r e n g t h y ( V . 5 . 4 a ) r m = » * n = ( 2 m +l)TTi/N , m=l , . . . ,N , s t r e n g t h -y ( V . 5 . 4 b ) In t h e a b s e n c e o f b o u n d a r i e s and i n a f l u i d o f u n i f o r m d e p t h t h e e q u a t i o n s o f m o t i o n b e c o m e : z n h 2 ( r ) n 1 N I I + £ ,k=l z n " z k m=l "m""n Lk^n N , N N 1 X _ + i Y Z n h , ( r n } ' z _ - z 2 r h ( r ) n v n ( V . 5 . 5 a ) "m h 2 ( r ) hi _LX_ i Y z h 1 ( r ) ' m v n ' k = i z m - z , . z „ - z . .k^m 'm k n = l m n 2 r h ( r ) n n ( V . 5 . 5 b ) The s ymmet r y o f t h e i n i t i a l c o n f i g u r a t i o n s u g g e s t s s o l u t i o n s o f t h e f o r m : r n = r ( t ) , * n = 2 ir i n / N + o * 2 t , r m = ^ , <j>m = ( 2 m + l ) T r i / N + oo 2t ( V . 5 . 6 ) S u b s t i t u t i n g i n t o ( V . 5 . 5 ) and m a k i n g use o f ( V . 5 . 3 ) g i v e s N - l £ 1 r ( t ) = r Q ; o ) 2 - ^ i - e x p ( 2 T r i k / N ) N + E 1 r 0 h ' ( r 0 ) m = 1 l - r - 2 e x p ( ( 2 m + l ) T T i / N ) " 2 h ( r 0 ) ( V . 5 . 7 ) The f i r s t sum was e n c o u n t e r e d i n t h e l a s t s e c t i o n ( V . 4 . 5 ) . The s e c o n d sum i s ( A p p e n d i x B ) : N Z m=l 1 l - x e x p ( (2m+l)TTi/N) 1+x1 74 ( V . 5 . 8 ) Thus 03 L ( r ? + r ; N , + P ( r n ) ( V. 5 . 9) The u n i t o f t i m e has a g a i n been t a k e n t o b e : r 2 h 2 ( r ) ' /y To d e t e r m i n e t h e s t a b i l i t y . o f t h e c o n f i g u r a t i o n s e t : i w„ t : n = [ r 0 e x p ( 2 7 T i n / N ) + e n ( t ) . ] e , a ) 2 i u)„ t "m [r7exp( (2m+l)Tri/N) + 6 m ( t ) ] e 1 a ) 2 ( V . 5 . 1 0 a ) ( V . 5 . 1 0 b ) S u b s t i t u t i n g i n t o t h e e q u a t i o n s o f m o t i o n and e x p a n d i n g t o f i r s t o r d e r i n t he e ' s and 6 ' s : (e -e, ) v n k 1 k t 1 ( l - e x p ( 2 i r i (k-riJ/N)y kfn v n nr m: I (1 - ro-?exp(('.2(nwi)+l.)'Mi /N) ) m + P ( r 0 , o j 2 ) e n +. Q ( r 0 J u 2 ) e n e x p ( 4 T r i n / N ) N ( 6 m - 6 k ) \lx ( l - e x p ( 2 T r i ( k - n ) / N ) ) ' + E i e xp ( - 4fri n/N) ( V. 5 . 11 a) (6 -e ) v m n ' l-r§TxpTj2Tn-niH)Tri/N))2-( V . 5 . l i b ) w h e r e P and Q a r e d e f i n e d by ( V . 4 . 9 , 1 0 ) . U s i n g ( V . 5 . 3 ) : P ( r ) = - p ( i ) ; r p ' ( r ) = i p ' ( i ) ( V . 5 . 1 2 ) so t h a t : P ( p - u ) = P ( r ) + 2u + p ( r ) ; Q(r ,u)) = Q(£. -o>) ( V . 5 . 1 3 ) 75 The s o l u t i o n s t o ( V . 5 . 1 1 ) a r e o f t h e f o r m : e n = a M e . x p [ 2 T r i ( l + M)n/N + i X M t ] + b M e x p [ 2 T r i ( l - M ) n / N - iX^i] ( V . 5 . 14a) 6n = c M e x p [ - ' ( 2 m + l ) ( l + M)Tr i/N + i A M t ] + • d M e x p [ ( 2 m + l . ) ( l - M ) i r i / N - i X M t ] ( V . 5 . 1 4 b ) S u b s t i t u t i n g i n t o ( V . 5 . 1 1 ) and e q u a t i n g c o e f f i c i e n t s o f e l A r , t and e " 1 ' ^ M t s e p a r a t e l y y i e l d s ( u s i n g ( V . 5 . 9 ) and ( V . 5 . 1 3 ) ) : U M + Q ( r 0 , u 2 ) ) a M + Ab M + r*J u n ( r 2 ) d M = 0 ( V . 5 . 1 5 a ) A * M + ( " A M + Q( r0."2))V r J T l + M ( r o ) £ M ' = 0 ( V . 5 . 1 5 b ) T l + M ( r o 2 ) V ( " A M + Q K ' ^ ^ V A d M = 0 « ( V . 5 . 1 5 c ) T l - M ( r o ) a M + A V ( X M + ^ r o ' W 2 ) ) d M = 0 ( V . 5 . 1 5 d ) whe re A = S 1 + M + P ( r 0 , u ) 2 ) + T N ( r 2 ) = Q(r 0 , u > 2 ) - ^M(N-M) —IT1 M ( V . 5 . 1 6 ) ( r N + r N ) 2 0 0 T ( x ) = - x " 2 T f x " M - 7 e x p ( ( 2 k + l ) U i / N ) i L u ; x i N _ L + 2 U ; - ^ ( 1 _ X e X p ( ( 2 k + l ) u i / N ) ) 2 ,N-L - N x M - L ( ( L - l ) x M - ( N - L + l ) ) , |\J 5 L 1 , . . . , N (1 + x'V ( V . 5 . 1 7 ) 76 ( s e e A p p e n d i x B ) . S L i s d e f i n e d i n ( I V . 4 . 1 3 ) . T h e r e a r e non-t r i v i a l s o l u t i o n s o f ( V . 5 . 1 5 ) o n l y i f a = ± r 2 d , , an d; b .= ± r 2 c Then : U M + Q ( r 0 , u > 2 ) ± r 2 T 1 _ M ( r 2 ) ) a ; M + AF,M= 0 ( V . 5 . 1 8 ) Aa M + ( - X M + Q ( r 0,a3 2) ± r 2 T 1 + M ( r 2 ) ) b M = 0 ( V . 5 . 1 9 ) whence f o r n o n - t r i v i a l s o l u t i o n s : A M ± r o ( T i + M ( r 2 ) - T 1 _ M ( r 2 ) ) A M + A 2 - ( Q ( r 0 , u > 2 ) ± r 2 T 1 _ M ( r 2 ) ) ( Q ( r 0 , o 3 2 ) ± r 2 T 1 + | y | ( r 2 - ) ) = 0 ( V . 5 . 2 0 ) The Mth modes a r e s t a b l e i f i s r e a l , i . e . , i f : r o " ( T l + M ( r o } " T l - M ( r o ) ) 2 > 4 ( A 2 - (Q( r 0 ,u> ) ± ^ o T 1 + M ( r 2 ) ) ( Q ( r 0 , a ) ) ± r ' T ^ f r 2 ) ) ( V . 5 . 2 1 ) w h i c h can be s i m p l i f i e d t o : (2C ± D ) ( 4 q 2 - 2C ± D) > 0 ( V . 5 . 2 2 ) whe r e : C E (Q - A ) / N 2 = % x ( l - x ) - % s e c h 2 ( % y ) ( V . 5 . 2 3 ) D -= r 2 ( T 1 + M ( r 2 ) + V M ( r 2 ) ) / N 2 _ x c o s h ( ( l - x ) y ) - ( l - x ) c o s h ( x y ) M/ C O A\ q 2 = Q ( r 0 , c o 2 ) / N 2 ( V . 5 . 2 5 ) x = M/N , y = Nfcnr 2, I f (2C ± D ) ( 4 q 2 - 2C ± D) = 0 t h e s o l u t i o n i s s t a b l e u n l e s s : T l + M ( r o ) = T l - M ( r o ) • ( V . 5 . 2 6 ) 77 The s t a b i l i t y c r i t e r i o n ( V . 5 . 2 2 ) i s i n v a r i a n t u n d e r r 0 -+ r Q 1 and M -> N -M . I t i s t h e r e f o r e s u f f i c i e n t t o c o n s i d e r on l y y > 0 and h < x £ 1 . The Nth mode c o r r e s p o n d s t o p e r t u r b a t i o n s o f f o u r t y p e s ( F i g . V I I ; t h e U d e n o t e s t h e u p p e r s i g n s i n V . 5 . 2 2 , t h e L t h e l o w e r s i g n s ) . The two L modes a r e t h e e x p e c t e d z e r o f r e q u e n c y modes . The U modes a r e s t a b l e i f : Q(r..Ut, . L^lLA . B i ^ . m r U p + p(ri)' L r 0 + r 0 < 0 ( V . 5 . 2 7 ) I f p ( r 0 ) > 0 a nd K ( r 0 ) > 0 ( F i g u r e V i l l a ) t h e n t h i s mode i s s t a b l e . I f p ( r 0 ) < 0 and K ( r 0 ) < 0 ( F i g u r e V I 1 1b ) t h e n i t i s u n s t a b l e . O t h e r c a s e s must be e x a m i n e d s e p a r a t e l y . I f N i s e v e n t h e n t h e modes M = ^N a r e i m p o r t a n t as t h e y t e n d t o be t h e f i r s t t o go u n s t a b l e . The f o u r p o s s i b i l i t i e s a r e shown i n F i g u r e I X . They a r e a l l s i m i l a r . T h i s i s r e f l e c t e d i n t h e s t a b i l i t y c r i t e r i o n w h i c h i s t h e same f o r u p p e r and l o w e r s i gns : (h - h s e c h 2 % y ) ( 4 q 2 - k + k s e c h 2 ^ y ) > 0 ( V . 5 . 2 8 ) I f q i s s m a l l ( a s o c c u r s when N-*-«>) t h e n s t a b i l i t y i s r e s t r i c t e d t o a s m a l l r e g i o n a r o u n d : y = 2 a r c c o s h /2 ( V . 5 . 2 9 ) T h i s c r i t e r i o n i s a n a l o g o u s t o t h e von Karman c o n d i t i o n f o r t h e s t a b i l i t y o f i n f i n i t e s t a g g e r e d v o r t e x s t r e e t s ( i n f a c t , i f t h e s u r f a c e o f f l o w i s a c y l i n d e r , o u r r e s u l t s must a p p r o a c h t h o s e o f von Karman as N+°° ) . H o w e v e r , as w i l l be s e e n f o r t h e c a s e o f t h e s p h e r e , ( V . 5 . 2 8 ) i s o f t e n i n c o m p a t i b l e w i t h 78 2rr F i g . V I I : S t a g g e r e d V o r t e x S t r e e t : The Modes M=N. Dots i n i t i a l p o s i t i o n s , c r o s s e s p e r t u r b e d p o s i t i o n s d e n o t e 79 a) S t a g g e r e d v o r t e x s t r e e t s s t a b l e . S y m m e t r i c v o r t e x s t r e e t s u n s t a b l e . p(r o)<0, K(ro)<0 b.) S t a g g e r e d v o r t e x s t r e e t s u n s t a b l e . S y m m e t r i c v o r t e x s t r e e t s s t a b l e . F i g . V I I I : S u r f a c e f o r w h i c h t h e Mode M=N has D e f i n i t e S t a b i l i t y 80 u St-u 0 — F i g . X I : S t a g g e r e d V o r t e x S t r e e t : The Modes M=JgN. The d o t s d e n o t e i n i t i a l p o s i t i o n s , t h e c r o s s e s p e r t u r b e d p o s i t i o n s . 81 ( V . 5 . 2 7 ) l e a d i n g t o o v e r a l l i n s t a b i l i t y . O t h e r mo.des a r e c o n s i d e r a b l y more c o m p l i c a t e d and must be e x a m i n e d s e p a r a t e l y f o r s p e c i f i c s u r f a c e s o f f l o w . b) S y m m e t r i c V o r t e x S t r e e t s The v o r t i c e s o f a s y m m e t r i c v o r t e x s t r e e t a r e l o c a t e d i n i t i a l l y a t : r n = r Q ; <|> = 2 i r i n / N ; n = l , . . . , N , s t r e n g t h y ( V . 5 . 3 0 ) g t h - y ( V . 5 . 3 1 ) m c f > m = 27Tim/ N ; - m = l , . . . , N , s t r e n g t -yT r y i n g a s o l u t i o n , t o ( V . 5 . 5 ) o f t h e f o r m z n = r ( t ) e x p ( 2 T T i n / N + u 3 t ) ; z m = r " 1 ( t ) e xp ( 2 T H m/N + u>. one f i n d s : r ( t ) = r 0 ; w 3 N ( r N + r " N ) Aro-r0 ) ( V . 5 . 3 2 ) ( V . 5 . 3 3 ) whe re use has been made o f ( V . 4 . 5 ) a n d : ( V . 5 . 3 4 ) k = 1 ( l - x e x p ( 2 7 r i k / N ) ) 1 _ x N ' : ( s e e A p p e n d i x B ) . The s t a b i l i t y o f t h e c o n f i g u r a t i o n i s d e t e r -m i n e d by p u t t i n g : i to , t z n = r 0 e x p ( 2 T r i n / N + E „ ( t ) ) e - i i u, t m r 0 e x p ( 2 u i n / N + 6 ; m ( t ) ) e l w 3 ( V . 5 . 3 5 a ) ( V . 5 . 3 5 b ) s u b s t i t u t i n g i n t o ( V . 5 . 5 ) , e x p a n d i n g t o f i r s t o r d e r i n t h e e ' s and 6 ' s and t r y i n g s o l u t i o n s o f t h e f o r m : en = a n e x p [ 2 T r i ( l + M)n/N + i X M t ] + • b ^ x p [2 i r i (1 -M) n/N> i \ t ] ( V . 5 . 3 6 a ) 82 6 m = c f l e x p [ 2 T r i (1 + M)m/N + i X M t ] +: d^exp [2 i r i (1 -M )m/N - i X M t ] ( V . 5 . 3 6 n ) The s o l u t i o n s s a t i s f y : ( X M + Q ( r 0 , o ) 3 ) ) a + BFM--+ ^ ^ ( r 2 ^ = 0 ( V . 5 . 3 7 a ) Ba M : + ( - A M + Q ( r 0 , a j 3 ) ) b M - : + r ^ R 1 + M ( r 2 ) 5 M = 0 ( V . 5 . 3 7 b ) R l + M ( r o ^ M + ( ~ A M + Q ( r 0 . u 3 ) ) c M - - + B d M = 0 ( V . 5 . 3 7 c ) R l - M ( r o ) a M + B % : + ( X M + Q ( r o ' W 3 ) ) d M = 0 ( V . 5 . 3 7 d ) whe re B = S T _ M - R M ( ^ 2 ) + P K > W 3 ) = Q ( r 0 , w 3 ) - JsM(N-M) " ^W^KJ2 ( V . 5 . 3 8 ) R (*) = x ~ 2 R f x " M = y exp (27TJLk/N) R L U J X R N - L + . 2 U ' ~ klx ( l - e x p ( 2 i r i k / n ) ) 2 = N x N " M [ ( N - M + l ) + ( M - l ) x N ] , M= 1, . . . ,N ( 1 _ X N ) 2 ( V . 5 . 3 9 ) E q u a t i o n s ( V . 5 . 3 7 ) a r e e x a c t l y a n a l o g o u s t o ( V . 5 . 1 5 ) so t h a t t h e s t a b i l i t y c r i t e r i o n i s t h e e q u a t i o n a n a l o g o u s t o ( V . 5 . 2 1 ) : i . e . , r0MR1 + M ( rS) + R ^ U 2 ) ) 2 > 4 ( B 2 - (Q(r 0 ,a> 3 ) ± r 2 R 1 + M ( r 2 ) ) ( Q ( ^ 0 , a ) 3 ) ± r ^ ^ r 2 ) ) ) ( V . 5 . 4 0 ) w h i c h can be s i m p l i f i e d t o : (2E ± F ) ( 4 q 3 - 2E ± F) > 0 , M=1 , . . . ,N ( V . 5 . 4 1 ) w i t h : E = J g x ( l - x ) + % c s c h 2 ( % y ) ( V . 5 . 4 2 ) 83 F ( l - x ) c o s h ( x y ) + x c o s h ( ( l - x ) y ) 2 s i n h 2 ( % y ) ( V . 5 . 4 3 ) Q ( r 0 ,o ) 3 ) / N 2 ( V . 5 . 4 4 ) x M/N y = N £ n r 2 ( V . 5 . 4 5 ) A g a i n one may s u p p o s e t h a t y >_ 0 , h±x<_\ . The f o u r modes w i t h M=N a r e shown i n F i g u r e X . (L d e n o t e s l o w e r s i g n s i n V . 5 . 4 1 , U u p p e r s i g n s ) . As b e f o r e , t h e L modes a r e t h e e x p e c t e d z e r o f r e q u e n c y modes . The u p p e r s i g n s g i v e : ( V . 5 . 4 6 ) I f p ( r 0 ) < 0 and K ( r 0 ) < 0 ( s e e F i g u r e V I11b ) t h e n ( V . 5 . 4 6 ) i s t r u e , s i n e e : I f p ( r 0 ) > 0 and K ( r 0 ) > 0 t h e n Q ( r 0 , w 3 ) < 0 .. ' r O t h e r c a s e s must be d e t e r m i n e d s e p a r a t e l y f o r e a c h s u r f a c e o f i n t e r e s t . We now show t h a t 2 E ± F ^ 0 so t h a t t h e s t a b i l i t y c r i t e r i o n may be s i m p l i f i e d f u r t h e r . L e t g ( x , y ) = 2 x ( l - x ) s i n h 2 % y +1 - ( l - x ) c o s h ( x y ) ; - x c o s h ( l - x ) y T h e n : = x ( l - x ) [ s i n h y - s i n h ( x y ) - s i n h ( ( 1 - x ) ) y ] Now: ( s i n h ( x y ) + s i nh ( ( 1 - x J y ) ) = y ( . s i n h ( x y ) - s i nh ( ( l - x ) y ) ) f o r y >_ 0 and % j< x <_ 1 s i n c e s i n h x i s i n c r e a s i n g . > 0 + p ( r ) > N - l > 0 84 U 0 — U 0 — 0 — 0 — F i g . X . S y m m e t r i c V o r t e x S t r e e t : The Modes M=N. Dot s d e n o t e i n i t i a l p o s i t i o n s , c r o s s e s p e r t u r b e d p o s i t i o n s . 85 T h e r e f o r e : hsinhhy < s i n h ( x y ) + s i n h ( ( l - x ) y ) < s i n h y whence | £ > 0 . T h e r e f o r e : dy — g ( x , y ) > g ( x , 0 ) = l - ( l - x ) - x = 0 The r e f o r e : 2E ± F > 2E - F = zl\^yly > 0 i f y > 0 ( V . 5 . 4 7) The s t a b i l i t y c r i t e r i o n may t h e r e f o r e be s i m p l i f i e d t o : 4 q 3 - 2E - F > 0 , M = 1 , . . . , N - 1 ( V . 5 . 4 8 ) The c r i t i c a l modes a r e i n g e n e r a l t h e L modes w i t h M = HN (N e v e n ; s ee F i g u r e X I ) f o r w h i c h t h e s t a b i l i t y c r i t e r i on i s : Q > (1 + c o s h ^ y ) 2 = 1 ( y 5 4 g ) q 3 1 16 s i n h 2 % y 16tanh 2Uy) l v , 3-^ j F o r s t a b i l i t y q 3 > yg- . As N ^ 0 0 , q 3 •+ 0 so t h a t f o r s u f f i c i e n t l y l a r g e N a s y m m e t r i c v o r t e x s t r e e t w i l l a l w a y s become u n s t a b l e . c ) E x a m p l e : The C y l i n d e r The s t a b i l i t y o f v o r t e x s t r e e t s on t h e c y l i n d e r i s now e x a m i n e d . F o r t h e c y l i n d e r p ( r ) = 0 , p ' ( ' r ) so t h a t q 2 = q 3 = 0 ( V . 5 . 4 9 ) t h e n i m m e d i a t e l y i m p l i e s t h a t a l l s y m m e t r i c v o r t e x s t r e e t s a r e u n s t a b l e . S t a g g e r e d v o r t e x s t r e e t s a r e s t a b l e i f : 4 C 2 - D 2 < 0 ( V . 5 . 5 0 ) F o r M = 1 a l l f o u r modes now y i e l d z e r o f r e q u e n c y modes . The L modes have a l r e a d y been e x p l a i n e d . T h e . . c o n s e r v a -t i o n o f moment o f c i r c u l a t i o n and a n g u l a r 86 U U F i g . X I : S y m m e t r i c V o r t e x S t r e e t : The Modes M ; d e n o t e i n i t i a l p o s i t i o n s , t h e c r o s s e s p o s i t i o n s . % N . The d o t s pe r t u r b e d 87 moment o f c i r c u l a t i o n f o r t h e c y l i n d e r a r e : Z y n £ n r n = c o n s t - (V .5 .51. ) n S-Y 4> = c o n s t . ( V . 5 . 5 2 ) n T h e s e c o n s e r v a t i o n l aws a r e v i o l a t e d by t h e M=N., U mode s . T h e s e modes must t h e r e f o r e be n e g l e c t e d and do n o t a f f e c t t h e s t a b i l i t y o f t h e c o n f i g u r a t i o n . I f N i s e v e n one may p u t M=%N(x = Jg) . D i s t h e n z e r o , and f o r s t a b i l i t y ( f r o m ( V . 5 . 2 2 ) ) . | - \ s e c h 2 ( J - 2 y ) • 2 _ 2 i 0 ( V . 5 . 5 3 ) i . e . , t h e mode i s u n s t a b l e u n l e s s : y = 2 a r c c o s h /2 ( V . 5 . 5 4 ) From ( I I . 2 . 2 3 ) and ( V . 5 . 2 6 ) , t h e s e p a r a t i o n o f t h e r i n g s o f v o r t i ces i s : , R a r c c o s h / 2 / c c ^ = N ( V . 5 . 5 5 ) w h i l e t h e s e p a r a t i o n o f t h e v o r t i c e s w i t h i n e a c h r i n g i s : d = 2T TR/N ( V . 5 . 5 6 ) whe re R i s t h e r a d i u s o f t h e c y l i n d e r . T h e r e f o r e : d a r c c o s h / 2 " ,,. c -T— = ( V . 5 . 5 7 ) i n a g r e e m e n t w i t h von Ka rman . When M=%, T ^ + M ( x ) = T ^ _ m ( x ) so t h a t t h e s t a b i l i t y o f t h i s mode i s i n d e t e r m i n a t e . I t can be d e t e r m i n e d o n l y by h i g h e r o r d e r p e r t u r b a t i o n t h e o r y . The s t a b i l i t i e s o f t h e o t h e r modes f o r N e v e n and f o r N odd a r e now d e t e r m i n e d . - ^ [ ( D - 2 C ) c o s h 2 i s y ] = x ( \ ~ x ] [s i nh ( (1 - x ) y ) - s i n h ( x y ) - 2 c o s h ( % y ) s i n h ( J g y ) ] < 0 88 s i n c e s i n h x i s i n c r e a s i n g i f x > 0 . B u t : ( D - 2 C ) c o s h 2 % y = x 2 > 0 when y = o -*- -°° ( x ^ 1) as y + °° T h u s , i f X T * 1 , D-2C has e x a c t l y one z e r o f o r e a c h x . Deno te t h e s e z e r o s : y " ( x ) . s i n h ( ( l - x ) y ) - s i n h ( x y ) 3 ( D + 2 C ) c o s h 2 ( % y ) = x ( 1 - x ) 3y 2 + 2co sh ( J s y ) s i n h ( J g y ) _ (l-e2) 8 ( l - e 2 ) s i n h { h y ( l - e ) ) - s i nh{hy(1 + e ) ) + 2 c o s h ( % y ) s i n h ( % y ) 2 s i n h ( % y e ) c o s h ( % y ) + 2 c o s h ( J a y ) s i n h ( i 2 y ) > 0 whe re e = 2 x - l . M o r e o v e r : D + 2C = - ( 1 - x ) 2 when y=0 -> oo as y -»• °° T h e r e f o r e D + 2C has e x a c t l y one z e r o f o r e a c h x ( x ^ l ) . T h e s e a r e d e n o t e d y + ( x ) . When y = a r c c o s h / 2 , C=0 and D_>0 ; t h e r e f o r e y " ( x ) < 2 a r c c o s h / 7 T < y + ( x ) and D + 2C _> 0 w h e n : y " ( x ) < y < y + ( x ) %<x<l ( V . 5 . 5 8 ) D e f i n e y " = max y " ( x ) , y + = min y + ( x ) ( V . 5 . 5 9 ) x x The v o r t e x s t r e e t i s s t a b l e i f : y " < y < y + ( v . 5 . 6 0 ) F i g u r e X I I i s a g r a p h o f y ~ ( x ) and y + ( x ) . C l e a r l y y + and y~ c o r r e s p o n d t o t h e v a l u e s o f x c l o s e s t t o h. F o r N o d d 90 t h i s i s : x = %(1 + k). F o r N e v e n : x = h . T h u s , i f N i s odd t h e r e i s a s m a l l r e g i o n o f s t a b i l i t y a r o u n d y = a r c c o s h / 2 . F o r t h e f i r s t few N t h e s e r e g i o n s a r e g i v e n i n T a b l e 1. T a b l e 1: R e g i o n s o f S t a b i l i t y o f S t a g g e r e d V o r t e x S t r e e t s on a C y l i n d e r N = 3 , 1.5 5 2.2 < y < 2 , 079 4 A = 0.0,879 N = 5 , 1.6 2 78 < y < 1.9 344 A = 0 .0 30 7 N = 7 , 1.66 34 < y < 1 .8806 A = 0.. 01-55 N = 9 , 1 .6842 < y < 1 .8525 A = 0 . 0 0 9 4 A i s t h e a c t u a l w i d t h o f t h e s e r e g i o n s m e a s u r e d i n c y l i n d e r r a d i i : A + y - y 2N I f N i s e ven s t a b i l i t y i s o n l y p o s s i b l e i f y = a r c c o s h / 2 , A=0 b u t one needs h i g h e r o r d e r p e r t u r b a t i o n t h e o r y t o c h e c k t h e mode M'= ^ N . d) E x a m p l e : The S p h e r e The a n a l y s i s o f t h e s t a b i l i t y o f v o r t e x s t r e e t s on t h e s p h e r e i s somewhat more c o m p l i c a t e d t h a n on t h e c y l i n d e r . From ( I I . 2 . 2 4 ) , ( V . 5 . 3 3 ) and ( V . 5 . 4 5 ) : Q ( r 0 , o ) 3 ) = -h -H t a n h 2 ( y / 2 N ) + -JgNtanh(y/-2N) c o t h ( J g y ) ( V . 5 . 6 1 ) S i n c e : t a n h ( a x ) c o t h x : <^  1 i f a < 1 : 91 M o r e o v e r : E = - J s x ( l - x ) + % c o s e c h 2 ( % y ) > J g x ( l - x ) ; F>0 ( V . 5 . 6 3 ) T h e r e f o r e : 4 q 3 - 2E - F < + ^ - x ( l - x ) ( V . 5 . 6 4 ) I f N i s e v e n t h e r e i s a mode x=% , whence 4 q 3 - 2E - F < - ( N 2 - 8 N + 4 ) / N 2 < 0 , i f N>7 I f N i s odd t h e r e i s a mode x=%( l+^ ) , whence 4 q 3 - 2E - F < - ( N 2 - 8 N + 3 ) / N 2 < 0 , i f N>7 T h e r e f o r e a l l s y m m e t r i c v o r t e x s t r e e t s w i t h N>7 a r e u n s t a b l e The s t a b i l i t y c r i t e r i o n ( V . 5 . 4 8 ) has been e x a m i n e d n u m e r i c a l l y f o r N = 2 , . . . ,6 . T a b l e 2: R e g i o n s o f S t a b i l i t y o f S y m m e t r i c V o r t e x S t r e e t s on a S p h e r e N = 2 , s t a b l e i f y > 4 . 2 4 5 1 , 9 < 3 8 . 1 7 2 7 ° N = 3 , s t a b l e i f y > 5 .3020 , 9 < 4 4 . 9 0 7 2 ° N = 4 , s t a b l e i f y > 7 . 5 9 5 7 , 9 < 4 2 . 3 0 7 8 ° N = 5 , s t a b l e i f y > -10 ..4306 , 0 < 3 8 . 8 2 2 5 ° N = 6 , s t a b l e i f ' y > 1 7 . 7602 , 9 < 2 5 . 6 4 7 8 ° N _> 7 , u n s t a b l e From : ( 1 1 . 2 . 2 4 ) , ( V . 5 . 9 ) and ( V . 5 . 2 6 ) : Q ( r 0,oo 2) = -k - % t a n h 2 ( y / 2 N ) + %N t a n h ( y/2N) t a n h (%y) ( V . 5 . 6 5 ) The re f o r e : 92 § - s e c h 2 ( y / 2 N ) [ t a n h ( ^ y ) _ l t a n n ( y / 2 N ) ] + ^ t a n h ( y / 2 N ) s e c h 2 ( J g y ) > 0 f o r y > 0 ( V . 5 . 6 6 ) T h u s , f o r e a c h N t h e r e i s e x a c t l y one y s u c h t h a t Q = 0 Deno te i t by y ( N ) . The v o r t e x s t r e e t i s u n s t a b l e i f : y > y ( N ) ( s e e ( V . 5 . 2 7 ) ) . L e t a = y/2N . Then N = y / 2 a a n d : 1 + t a n h 2 a - — t a n h a t a n h ( % y ) a t a n h a 4a ( V . 5 . 6 7 ) M _ ( l + t a n h 2 a ) a a ^ , x u £ n n . , N o w t a n h a > > 1 • T h e r e f o r e Q < 0 i f y t a n h ^ y < 1 : i . e . , i f y < 1 .543 . T h e r e f o r e : y ( N ) > 1 .543 ( V . 5 . 6 8 ) L e t f(a) = a - ( l + a ) t a n h a . T h e n : f'(a) =• ( 1 - t a n h a ) (1 - (1+a) (1 + t a n h a ) ) < 0 i f a > 0 Bu t f ( 0 ) = 0 . T h e r e f o r e : a < ( l + a ) t a n h a a n d : a ( l + t a n h 2 a ) a . , 1 l 0 T, . t a n h a = + a t a n h a < 1 + 2 a • T h u s 1 f : y t a n h ( % y ) > 1 + 2a = 1 + , t h e n Q > 0 . L e t y * ( N ) be t h e l a r g e s t v a l u e o f y s u c h t h a t y * ( N ) t a n h ( % y * ( N ) ) = 1 + ( V . 5 . 6 9 ) Then : 1 .543 < y ( N ) < y * ( N ) ( V . 5 . 7 0 ) D i f f e r e n t i a t i n g ( V . 5 . 6 9 ) by N: 93 t a n h(Jgy * ( N ) ) + % y * ( N ) s e c h 2 ( % y * ( N ) ) - ± 9 y * ( N ) _ - y * ( N ) 9N N S i n c e y * ( N ) > 1.543 : t a n h ( J g y * ( N ) ) + % y * ( N ) s e e n 2 (*gy*(N) ) > t a n h ( J gX l . 5 43) = 0 . 6 4 8 whence **|M < Q ^ One can c h e c k n u m e r i c a l l y t h a t y * ( 4 ) < 1.6 . Thus y * ( N ) < 1.6 f o r a l l N > 4 , a n d : 1 .543 < y ( N ) < 1.6 f o r N > 4 ( V . 5 . 7 1 ) A s t a g g e r e d v o r t e x s t r e e t on t h e s p h e r e i s t h e r e f o r e a l w a y s u n s t a b l e i f N _> 4 and y > 1.6 . I t i s now shown t h a t i f y < 1.6 and N >^  6; t h e n t h e modes M=%N o r M = ^ ( N ± 1 ) a r e u n s t a b l e . L e t f ( x , y ) = c o s h ( y ( l - x ) ) + ( 1 - f ) c o s h ( y x ) . T h e n : A ) 3 f 1 = ^ycoshUy ) [ l - x y t a n h ( x y ) ] + y ( s i n h ( x y ) - s i n h ( ( l - x ) y ) ) > 0 i f xy < 1 . 1997 ( V . 5 . 7 2 ) W i t h y < 1.6 ( V . 5 . 7 2 ) h o l d s f o r x < 3/4 . T h u s : f ( x , y ) > f(h,y) = 0 i f h < x < 3/4 . T h e r e f o r e : D = % x f ( x , y ) c s c h 2 ( h > y ) > 0 i f h<x < 3/4 , y < 1.6 ( V . 5 . 7 3 ) a n d : f £ = % c s c h 2 ( J a x ) ^ | ^ - > 0 i f h < x < 3/4 , y < 1.6 ( V . 5 . 7 4 ) M o r e o v e r , f r o m ( V . 5 . 6 5 ) and ( V . 5 . 6 6 ) : q 2 1 - 7m ( V . 5 . 75) 94 The modes M=J§N o r M=%(N + 1) h a v e : x < whence C = J s x ( l - x ) - hsechz(hy) < h - ^ - - % s e c h 2 ( % y ) ( V . 5 . 7 6 ) a n d , u s i n g ( V . 5 . 73 ,75 , 76) : 4 q 2 " 2C + D > ^ + ; s e c h 2 ( J 5 X 1 . 6 ) 3 + 0 . 0 2 9 7 4 N 2 ( V . 5 . 7 7 ) i f y < 1.6 . The re f o r e : 4 q 2 - 2C + D > 0 i f y < 1.6 , N > 6 ( V . 5 . 7 8 ) M o r e o v e r , one can c h e c k n u m e r i c a l l y t h a t i f • y = 1.6 and x = 0 . 5 8 t h e n : 2C + D < 0 . S i n c e — < 0 dX 3 f o r y < 1.6 , x < j : 2 C + D < 0 , y < 1 . 6 , J s < x < 0 . 5 8 ( V . 5 . 79) F o r N _> 6 t h e modes M=%N o r M=%(N + 1) have x < 0 . 5 8 . U s i n g ( V . 5 . 2 2 ) , ( V . 5 . 7 8 ) and ( V . 5 . 7 9 ) t h e s e modes a r e t h e r e -f o r e u n s t a b l e i f y < 1.6 . Hence a l l s t a g g e r e d v o r t e x s t r e e t s on t h e s p h e r e w i t h N >^ 6 a r e u n s t a b l e . The s t a b i l i t y o f s t a g g e r e d v o r t e x s t r e e t s f o r N <_ 5 has been d e t e r m i n e d n u m e r i c a l l y ( T a b l e 3 ) . T a b l e 3: R e g i o n s o f S t a b i l i t y o f S t a g g e r e d V o r t e x S t r e e t s on a S p h e r e N = 2 s t a b l e i f 0 < y < 1. 7 6 2 7 , 65 .•5.302°<e < 9'0° N = 3 s t a b l e i f 1.5522<y < 1.6 306 , 7 4 . 6 1 7 1 ° < 8 < 7 5 . 3 4 0 4 ° N > 3 u n s t a b l e 95 The se r e s u l t s d i f f e r q u a l i t a t i v e l y f r o m t h o s e on t h e c y l i n d e r ( o r von Karman v o r t e x s t r e e t s ) whe re t h e s t a g g e r e d v o r t e x s t r e e t s e x h i b i t g r e a t e r s t a b i l i t y t h a n s y m m e t r i c v o r t e x s t r e e t s . 96 V I . VORTICES WITH F IN ITE CORES U n t i l now, i t has been a s s umed t h a t a l l v o r t i c e s have i n f i n i t e s i m a 1 l y s m a l l c o r e s . In t h i s s e c t i o n t h e e f f e c t s o f f i n i t e d i s t r i b u t i o n s o f v o r t i c i t y a r e c o n s i d e r e d . In p a r t i c -u l a r , i t i s shown t h a t i f t h e s u r f a c e o f f l o w i s c u r v e d and t h e c o r e i s n o t r a d i a l l y s y m m e t r i c t h e n a w o b b l e i s i n t r o d u c e d i n t o t h e m o t i o n o f t h e v o r t e x . V I . 1 . The P o s i t i o n and V e l o c i t y o f a V o r t e x When t h e c o r e e x t e n d s o v e r a f i n i t e r e g i o n G one can d e f i n e t h e p o s i t i o n o f t h e v o r t e x b y : 1 X = Y = 2 T T Y 2 T T Y 1 2 T T Y w z ( x , y , t ) x h 2 ( x , y ) d x d y w _ ( x , y , t ) y h 2 ( x , y ) d x d y w _ ( x , y , t ) h 2 ( x , y ) d x d y ( V I . 1 . 1 a ) ( V I . 1 . 1 b ) ( V I . 1.2) S i n c e : w. _1_ h 2 2 Try 9 ( h v y ) 3 ( h v y ) 3x 9 ( h v x ) ' 3y 3 ( h v x ) ( V I . 1 . 3 ) dxdy = o h ( v dx + v dy) = r a r j 9 G x y db ( V I . 1 . 4 ) so t h a t 2 T T Y i s s t i l l t h e c i r c u l a t i o n a r o u n d a c o n t o u r c o n -t a i n i n g t h e c o r e . By K e l v i n ' s C i r c u l a t i o n Theo rem y does n o t d e p e n d on t i m e . The v e l o c i t y o f t h e c o r e i s : 97 U X = X - 1 2 Try J 3 w z ( x , y , t ) 3 t x h 2 ( x , y ) d x d y 2 T T Y w '9G . ( x , y , t ) x h ( x , y ) [ ( v x - v G x ) d y - ( v g - v G y ) d x J ( V I . 1 . 5 a ) U y = Y = 1 2 i r y J 9w f x , y , t ) 3 t - y h 2 (x , y ) dxdy 2 Try 9G w > ( x , y , t ) y h ( - x , y ) [ ( v x - v Q x ) d y - (v - v Q )J.dx ( V I . 1 . 5 b ) whe re v_g i s t h e v e l o c i t y o f t h e c o r e b o u n d a r y . S i n c e w on 9G t h e b o u n d a r y t e r m s v a n i s h / U s i n g ( I I I . 3 . 7 ) and ( I I I . 3 . 8 ) : = 0 x -1 IT = 2 T T Y 1 f W -1 V • V z k x h k d x d y r w. 2 T T Y 1 V • ( v x h k ) d x d y 2nyj w z v x h d x d y ( V I . 1 . 6 a ) S i m i l a r l y : 2 T T Y w v hdxdy G z y ( V I . 1 . 6 b ) In t e r m s o f t h e s t r e a m f u n c t i on ty d e f i n e d by ( I V . 1 . 2 ) 1 2 I T Y G i r i h 2 k -2 T T Y h*k ^ | ^ d x d y f f d x d y ( V I . 1 . 7 a ) ( V I . 1 . 7 b ) 98 V I . 2 C i r c u l a r C o r e s A " c i r c u l a r " v o r t e x i s d e f i n e d t o be one w h i c h i s d e r i v e d by a s t r e a m f u n c t i on o f t h e f o r m : iMx,y) = - y A ( x , y ; X 0 , Y 0 ) f ( r ) + y B ( x ,y ; X 0 , Y„ ) + ty*{*,y) ( V I . 2 . 1 ) w h e r e r = [ ( x - X ) 2 + ( y - Y ) 2 ] 1 5 and f ( r ) = Inr , r > ( V I . 2 . 2 ) f has c o n t i n u o u s d e r i v a t i v e s ( s o t h a t t h e v e l o c i t y f i e l d i s c o n t i n u o u s ) . The f u n c t i o n s A and B a r e t h o s e d e f i n e d by ( I V . 1 . 1 3 ) and ty* s a t i s f i e s : „ . f 1 k V * * = o ( V I . 2 . 3 ) T h u s , w = 0 f o r r > e and t h e c o r e b o u n d a r y i s : r = e U s i n g ( V I . 2 . 1 ) , ( V I . 1 . 2 ) and ( V I . 1 . 3 ) t o e v a l u a t e ( V I . 1 . 1 ) one f i n d s : X = X 0 + 0 ( v ) ; Y = Y 0 + 0 ( v ) ( V I . 2 . 4 ) whe re v i s a s m a l l p a r a m e t e r o f o r d e r e | V £ n h | , t h e r a t i o o f t h e c o r e r a d i u s and t h e d i s t a n c e o v e r w h i c h h and k v a r y a p p r e c i a b l y . We w i s h t o d e t e r m i n e U x and U y up t o t e r m s w h i c h v a n i s h as e -> 0 w i t h y c o n s t a n t . f ( r ) ^ O ( A n e ) , f ' ( r ) ^ 0(f) , f " ( r ) % 0(A") • T h e r e f o r e : w ^ 0 h 2 e : v ^ T ^ - , and s i n c e dxdy ^ 0 ( e 2 ) one f i n d s . U ^ O(-^r) , so t h a t t e r m s o f 99 o r d e r e~ must be r e t a i n e d i n V* , t e r m s o f o r d e r e° i n VIJJ and t e r m s o f o r d e r e i n h . Us i ng ( I V . 4 . 1 ) k 0 V 2 k 2 ( x , y ) f ( r ) + 0 ( £ n e ) ( V I . 2 . 5 ) dy 9k, 2 T [ r > dy r k + ^ K o T 2 9 k 0 T x " 2 9y ( V I . 2 . 6 ) 1 1 h 2 k 3'2 h^k 0 3 /2 ' 2x 3_h_o J- - u 3x_ 9j<o_ _2y_ 3h_o_ • J3J/_ 9k_p_' h 0 9x 2 k 0 9x h 0 3y " 2 k 0 9y + 0 ( e 2 ) ( V I . 2 . 7 ) Fo r c o n v e n i e n c e ( X , Y ) has been made t h e o r i g i n and s u b s c r i p t s z e r o d e n o t e e v a l u a t i o n a t t h e o r i g i n . S u b s t i t u t i n g i n t o ( V I . 1 . 7 a ) : U' f 2TT 1 1 2 T T Y h k 3/2 " 0 0 o J 0 j 2x_ 9 h 0 3x 9 k 0 2y 9 h 0 ~ h 0 9x " 2 k 0 3x ~ h 0 9y z X f r r i A L o - I X d f ( r ) 2 T { r ) dy " r d r 9 k 0 l y k 0 d f r d f ( r ) ] 3 y J r d r d r 'k0 + x l!io. + Z iko.1 + Y I P ^ + 9i 2 9x ' 2 9y J ' Y " 9 y ' 9y w i t h x = rcoscj) , y = rsincf> . T h e r e f o r e : ( Z J f r d f ( r ) l rd<|>dr +• 0 ( e £ n 2 e ) ( V I . 2 . 8 ) U' 1 k FT2" K 0 " 0 _d d r d r 2 n n 9y 9y 9y + yk r d f d r h 0 9h iL + -1 9k, 9y 2 k 0 3y JJ d r + 0 ( e £ n 2 e ) 100 koh* Y9Bo 3 ^ - " • ( e f ' ( e ) ) 2 • I 3y 3y J 2 y k ° _1_ 3 h 0 + 1 8k o [ho 3y 2ko 3y J X j k j ay J l f d f ( r ) 2 l d r d r e f ( e ) f ' ( e ) y 3 k 0 2 3y + 0 ( e £ 2 e ) k h 2  K o " o 3B 0 + 3 ^ + Y_k_o_ 3 £ n ( h 0 2 k 0 ) _ y3k L 3y 3 y 4 3y 2 3y In e Def i ne : Y3k, 3y re f d f ] r 0 d r 'o d f d r d r + 0 ( e £ n 2 e ) d r ( V I . 2 . 9 ) ( V I . 2 . .10) The k i n e t i c - e n e r g y o f t h e f l u i d i n t h e c o r e i s c 2 V£ C2TT _ r£ Y 2 1 T . v 2 h 2 k rdc j )d r o 'o '0 '0 £ [ H 2J„ . k 2 + ' 3 if/ 2" 3x b y , d<j>dr r e r 2 TT 'o 'o k „ Y r fdrt d r d<J)dr = 7 T p Y2 k 0 3 0 ( V I . 2 . 11) As i n S e c t i o n I V .3, a 0 = h 2 k 0 £ 2 i s c o n s t a n t s i n c e t h e vo l ume o f t h e c o r e i s c o n s t a n t . T h e r e f o r e , t h e v e l o c i t y o f t h e c o r e i s : ,x 1 3 U' k 0 h 2 3y + \k(x,y)ln Y B ( x , y ; X , . Y ) + ^ * ( x , y ) r h 2 ( x , y ) k ( x , y ) l a, Jx=X y^  + 0 ( £ £ n 2 £ ) ( V I . 2 . 1 2 a ) and s i m i l a r l y : 1 0 1 y _ - i U k „ h 2 3x Y B ( x , y ; X , Y ) + ty*(x,y) + Y f k ( x s y ) £ n f h 2 ( x , y ) k ( x , y ) a * + 0 ( e i l n 2 e ) beX ( V I . 2 . 1 2 b ) i n a g r e e m e n t w i t h ( I V . 3 . 9 ) ( t h e e f f e c t s o f o t h e r v o r t i c e s have been i n c l u d e d i n ty* i n ( V I . 2 . 1 2 ) ) . A v o r t e x w i t h a c o r e o f f i n i t e s i z e w i l l t h e r e f o r e p r o p a g a t e as a v o r t e x w i t h a c o r e o f i n f i n i t e s i m a l s i z e p r o v i d e d i t s c o r e i s c i r c u l a r and r e m a i n s c i r c u l a r . In g e n e r a l , t h e c o r e w i l l be d i s t o r t e d due t o t h e a d v e c t i o n o f t h e v o r t i c i t y w i t h i n i t . I t i s n e c e s s a r y , t h e r e f o r e , t o have some i d e a o f t h e e f f e c t s o f t h e d i s t o r t i o n o f t h e c o r e . T h i s i s d i s c u s s e d i n S e c t i o n V I . 3 . The v e l o c i t y o f t h e v o r t e x i n d u c e d by t h e s u r f a c e c u r v a t u r e i s o f t h e o r d e r : pj-|_V£nh| ^ v ( ° ^ e | V £ n h | . The v e l o c i t i e s n e g l e c t e d i n t h e a p p r o x i m a t i o n o f S e c t i o n I I . 3 a r e o f t h e o r d e r : A v ^ ^ j f v ^ °^ I VJlnh | . T h u s , i f t h e m o t i o n o f t h e v o r t e x i n d u c e d by t h e s u r f a c e c u r v a t u r e i s t o be n o n -n e g l i g i b l e , i t i s n e c e s s a r y t h a t e >> k : t h e c o r e r a d i u s must be c o n s i d e r a b l y l a r g e r t h a n t h e d e p t h o f t h e f l u i d . I f e < k b u t t h e v o r t e x m o t i o n i n d u c e d by t h e s u r f a c e c u r v a t u r e i s n o t n e g l i g i b l e , t h e n v_ must be more n e a r l y h o r i z o n t a l t h a n s u g g e s t e d by t h e a p p r o x i m a t i o n ( I I I . 3 . 4 a ) : i . e . , V ( D = V ( D , V ( D . o . 102 V I . 3 The V a l i d i t y o f t h e C i r c u l a r A p p r o x i m a t i o n S uppo se t h a t t h e v o r t i c i t y d i s t r i b u t i o n w i t h i n t h e c o r e o f a v o r t e x i s n e a r l y c i r c u l a r : t h a t i s : w. ( V I . 3 . 1 ) whe re v i s t h e s m a l l p a r a m e t e r o f ( V I . 2 . 1 2 ) . W i t h i n t h e c o r e v = v ( o ) ( r ) + v v ( 1 ) r , ( j ) ) h = h 0 + v h ( l ) ( r , < M W z 0 ) = ^ fe-*0(v) The v e l o c i t y o f t h e v o r t e x i s : ( V I . 3 . 2 ) ( V I . 3 . 3 ) ( V I . 3 . 4 ) U = 1 [ 2 Try hn 2 Try v 2 T T Y J w v h d x d y 'G z " , v ( 0 ) w ( ° ) d x d y + v f ( v ( o W d ) h ( l ) + v ( o ) w ( l ) + v ( 1 ) w z 0 ) h 0 ) d x d y + 0 ( v 2 ) (v_ ^ w ^ M 1 ) + v ( o ) w z 1 ) h o + v ( 1 ) w z ( 0 ) , h 0 ) d x d y ( V I . 3 . 5 ) + 0 ( v 2 ) The f i r s t t e r m i n ( V I . 3 . 5 ) i s t h a t o b t a i n e d f o r a c i r c u l a r c o r e . The o t h e r t e r m s a r e o f c o m p a r a b l e m a g n i t u d e so t h a t e ven s m a l l d e v i a t i o n s f r o m a c i r c u l a r c o r e can p r o d u c e r e l a t i v e l y l a r g e c h a n g e s i n t h e v e l o c i t y o f t h e v o r t e x . H o w e v e r , c o n s i d e r a c i r c u l a r d i s t r i b u t i o n o f v o r t i c i t y w z ( r ) and a s m a l l l o c a l i z e d p e r t u r b a t i o n w z ' . o . , , ^ i s a d v e c t e d w S i n c e i s c a r r i e d a r o u n d t h e c o r e on a n e a r l y p e r i o d i c o r b i t w i t h p e r i o d o f o r d e r — r ^ — r whe re r i s t h e w irn; o 103 r a d i u s a t w h i c h w ^ i s l o c a l i z e d i n i t i a l l y . As w ^ z z o r b i t s t h e c o r e , t h e d i r e c t i o n o f t h e v e l o c i t y i n d u c e d by w ^ ^ a l s o sweeps t h r o u g h an a n g l e 2TT . The n e t d i s p l a c e -ment o f t h e v o r t e x due t o t h i s v e l o c i t y i n t h e t i m e —>—r V 2 V ( 1 ) w ^ 0 J i s a l m o s t z e r o ( i . e . , i t i s o f o r d e r : — T - r n — ^ v 2 £ ) w z u ' so t h a t t h e t i m e a v e r a g e d c o m p o n e n t o f t h e v e l o c i t y due t o t h e d e p a r t u r e f r o m a c i r c u l a r c o r e i s o n l y o f o r d e r v^v}0^ . The m o t i o n o f t h e v o r t e x i s t h e r e f o r e a p e r i o d i c w o b b l e a b o u t t h e p a t h o f a c i r c u l a r v o r t e x ( t o o r d e r v ) . S i m i l a r l y , one e x p e c t s f o r more c o m p l i c a t e d w ^ ^ t h a t t h e m o t i o n o f t h e v o r t e x i s , t o l o w e s t o r d e r , a s u p e r -p o s i t i o n o f w o b b l e s w i t h f r e q u e n c i e s o f o r d e r y/e z upon t h e m o t i o n o f a c i r c u l a r v o r t e x . The a m p l i t u d e o f t h e w o b b l e s i s ( o ) e 2 o f o r d e r : v v/ X ~ = v e w h i c h i s q u i t e n e g l i g i b l e . The n e t s y s t e m a t i c a d d i t i o n t o t h e v o r t e x v e l o c i t y i s a t mos t o f o r d e r : vU whe re U i s t h e v e l o c i t y o f a c i r c u l a r c c v o r t e x . F o r s m a l l v. t h i s i s n e g l i b l e . 104 V I . 4 E l l i p t i c a l C o r e s The w o b b l e i n t h e m o t i o n o f a v o r t e x can be d e m o n s t r a t e d e x p l i c i t l y i f i t s c o r e i s e l l i p t i c a l w i t h a u n i f o r m v o r t i c i t y d i s t r i b u t i o n w i t h i n i t . The d e p t h o f t h e f l u i d i s c o n s t a n t . T h i s i s t h e gene r a 1 i z a t i on o f Ki, r c h h o f f ' s e l l i p t i c a l v o r t e x ( s e e , e . g . , Lamb ( 1 916 ) p . 2 26 ) t o n o n - p l a n a r s u r f a c e s . S uppo se : w z = w Q = c o n s t . ( x , y ) £ G = 0 ( x , y ) £ G (VI . 4 . 1 ) w i t h : G = { ( x , y ) ; (f) 2 + (£]2< i j ( V I I f t h e r e a r e no b o u n d a r i e s o r e x t e r n a l f l o w s : V 2 iMx,y) = - w 0 h 2 ( x , y ) = - w 0 h 0 [ l + ax + gy] + 0 ( v 2 ) , ( x , y ) € G = 0 , ( x , y ) i G (VI . 4 . 3 ) whe r e : I t i s c o n v e n i e n t t o i n t r o d u c e c o m p l e x c o o r d i n a t e s so t h a t one may use c o n f o r m a l t r a n s f o r m a t i o n s . T h e n : f^ffU . z ) = - % w 0 h 0 2 [ l + % ( a - i B ) z + J s ( a + i 3 ) z ] , ( z , z ) £ G = 0 ( z , I ) £ G ( V I . 4 . 5 ) 105 w h e n c e : i>(z,z) = - % w 0 h 2 [ z z + J s ( a - i 3 ) z 2 z + % ( a + i 3 ) z z 2 ] ( z , z ) € G + 4>. ( z ) + * . ( z ) = * e ( z ) + V z > ( z , i ) $ G ( V I . 4 . 6 ) t h e s u b s c r i p t s e and i d e n o t i n g c o m p l e x p o t e n t i a l s f o r t h e i n t e r i o r and e x t e r i o r o f t h e c o r e . C o n s i d e r t h e m a p p i n g : c = a + b ( a - b ) UTTbT In t e r m s o f r, t he c o r e b o u n d a r y i s : |?|=1 M o r e o v e r : dz d? e l l — d - 1 ' £ 2 t 0 i f U l > 1 s i n c e "| d| < 1 . T h e r e f o r e t h e m a p p i n g i s c o n f o r m a l o u t s i d e t h e c o r e . T h u s , i f t h e v e l o c i t y f i e l d i s t o v a n i s h a t i n f i n i t y : 4> = P n*»S + z Pn? " n = l ( V I . 4 . 7 ) Mo reove r : I q n z n = I q n c n n = l n = l ( V I . 4 . 8 ) The c o e f f i c i e n t s p n and q n a r e d e t e r m i n e d by r e q u i r i n g t h a t t h e v e l o c i t y f i e l d be c o n t i n u o u s on t h e b o u n d a r y . One f i n ds : Pn w o h o d n > 4 L4 ^ [ a ( 2 + d ) - i 3 ( 2 - d ) ] w o h o d 8 ( V I . 4 . 9 ) ( V I . 4 . 1 0 ) ( V I . 4 . 1 1 ) q i = , w 0 h 2 c 2 ( l + d 2 ) [ a ( , 1 + d ) _ i e ( 1 _ d ) : ] Po W p h o c M l - dz ) 4 P l = w 0 h 2 c ^ ( l - d 2 ) ^ ( 1 + d ) 2 + i g ( 1 _ d ) 2 - | p 2 = w 0 h 2 c 2 ( l - d 2 ) d 8 106 ( V I . 4 . 12 ) ( V I . 4 . 13 ) (V I . 4 . 14 ) ( V I . 4 . 15 ) w 0 h 2 c ^ ( l - d 2 ) d L - a ( 1 + d ) 2 + i B ( 1 _ d ) 2 j ( V I . 4. 16) p 3 = 48 S i n c e w z i s d i s t r i b u t e d u n i f o r m l y o ve r the co re i t remains u n i f o r m l y d i s t r i b u t e d at a l l f u t u r e t imes s i n c e i t i s a d v e c t e d . The s t r u c t u r e o f the co re i s t h e r e f o r e de te rm ined s o l e l y by the shape o f the boundary. The con t r a v a r i ant v e l o c i t y f i e l d on the boundary i s : v x i \ / y - 2 1 l i j V " l V " h 2 ( z , z ) 3?| i w o c (1 - d 2 ) 2 l - ( a - i B ) z - ( a+ i B ) z \ + T ^ T ( a ( l + d ) 2 + i 6 ( l - d ) 2 ) ] | ? [ = 1 + 0 ( v 2 ) i w o a b Ca~+~bT •aa+i bB+e"1 9 + | 2i e a a ( d - 3 ) - i B b ( d + 3 ) (V I . 4 . 17 ) where e i s d e f i n e d by: c = s e 1 9 The f i r s t two terms are c o n s t a n t i m p l y i n g a t r a n s l a -t i o n a l v e l o c i t y : u x . ,,y _ i wo ab l U = 2 ( a + b ) C a a - l b g ] 107 ( V I . 4 . 1 8 ) C o n s i d e r , now, t h e t e r m : V x - i V y = - — O n l y t h e c o m p o n e n t p e r p e n d i c u l a r t o t h e s u r f a c e o f t h e c o r e can c h a n g e i t s s h a p e . T h i s c o m p o n e n t i s : V x = Re ( e1 9 - d e - i 9 ) e - d e i W o a b e a + b 2woabd s i n 2 8 ( a + b ) l e 1 B - d e -1 ( V I . 4 . 19) The v e l o c i t y f i e l d o f a u n i f o r m r o t a t i o n w i t h a n g u l a r ve 1 o c i t y co i s : v£ - i V w = - icoi = - i c o c ( e _ 1 9 + d e n 9 ) , |?|=1 (VI. 4 . 2 0 ) The c o m p o n e n t o f t h i s v e l o c i t y f i e l d p e r p e n d i c u l a r t o t h e c o r e b o u n d a r y i s : V J 2codc s i n 2 9 | e i e - d e - i 0 | ( V I . 4 . 2 1 ) w h i c h i s i d e n t i c a l t o (VI. 4 . 1 9 ) i f : 2 w 0 a b w = ( a + b ) 2 ( V I . 4 . 2 2 ) T h u s , t h e t h i r d t e r m c a u s e s t h e e l l i p s e t o r o t a t e w i t h a n g u l a r v e l o c i t y g i v e n by ( V I . 4 . 2 2 ) . The pe r p e n d i cu 1 a r •component o f t h e f o u r t h t e r m i s : V x w 0 ab 4 ( a+b ) a a ( d - 3 ) ( s i n 6 - d s i n 3 9 ) + 6 b ( d+ 3 ) ( c o s 8 - d c o s 3 0 ) (VI. 4 . 2 3 ) The t e r m s i n s i n 9 and c o s 0 i m p l y an a d d i t i o n a l c o n t r i b u t i o n t o 108 t h e t r a n s 1 a t i o n a 1 v e l o c i t y o f t h e v o r t e x B ( d + 3 ) - i a ( d - 3 ) ( V I . 4 . 2 3 ) S i n c e t h e p e r p e n d i c u l a r c o m p o n e n t o f a c o n s t a n t v e l o c i t y f i e l d , U x - i U y i s : V" a + b Ux b c o s 9 + U y a s i n 9 ( V I . 4 . 2 4 ) The c o n t r i b u t i o n f r o m t h e t e r m s i n s i n 39 and co s 39 c a u s e s a d i s t o r t i o n o f t h e e l l i p s e . H o w e v e r , i n t h e t i m e j = ( a + D ) t a k e n f o r t h e e l l i p s e t o r o t a t e once t h e d i s p l a c e -T T W 0 a b 1 r ment o f t h e b o u n d a r y due t o t h e above t e r m s i s o f t h e o r d e r : D ^ e 2 a ^ ve . D u r i n g t h i s t i m e , as s e e n f r o m a f r a m e i n w h i c h t h e e l l i p s e i s a t r e s t , t h e g r a d i e n t o f h a p p e a r s t o r o t a t e t h r o u g h 2TT ( o r , i n o t h e r w o r d s , t h e pha se o f a + i B c h a n g e s by 2TT) . The n e t d i s p l a c e m e n t o f t h e b o u n d a r y t h e r e -f o r e v e r y n e a r l y c a n c e l s : D „ ^ + . ~ v 2 £ . Thus t h e c o r e r e m a i n s n e t n e a r l y e l l i p t i c a l f o r t i m e s o f o r d e r w 0 The s m a l l d i s t o r t i n g t e r m w i l l h e n c e f o r t h be d i s r e g a r d e d , The n e t v e l o c i t y o f t r a n s l a t i o n o f t h e c o r e i s , f r o m ( V I . 4 . 1 8 ) and ( V I . 4 . 2 3 ) : l j x _ i i j y - w 9 a b  U l U _ 4 ( a + b) 3 ( b - 2 a ) + i a ( a - 2 b ) ( V I . 4 . 2 5 ) The c i r c u l a t i o n a r o u n d t h e c o r e i s 2 T r y = W o h 2 dxdy 2 T r a b w 0 h 0 + 0 ( v 2 ) T h e r e f o r e : Y : W p ab 2 ( V I . 4 . 2 6 ) (V I . 4 . 2 7 ) 109 U s i n g t h e d e f i n i t i o n s o f a and 3 t h e v e l o c i t y o f t h e c o r e i s t h e n : U X - i U y TlT+bT " ( b - 2a ) 9ho_ i ( a - 2 b ) 3h_o_' 3y hn3 3x h3o ( V I . 4 . 2 8 ) N o t i c e t h a t when a=b t h i s i s i n a g r e e m e n t w i t h t h e r e s u l t f o r c i r c u l a r c o r e s . ( V I . 4 . 2 8 ) has been d e r i v e d u s i n g c o o r d i n a t e s whose a xe s a r e t h e p r i n c i p l e a x e s o f t h e e l l i p s e . I f , i n s t e a d , one c h o o s e s t h e x - a x i s t o be a l o n g Vh so t h a t t h e m a j o r a x i s o f t h e e l l i p s e now makes an a n g l e 0 w i t h t h e x - a x i s ( F i g u r e XI I I) t h e v e l o c i t y o f t h e v o r t e x i s : U X - i U y - i y 9 h 0 2h 0 3 3x 1 + 3 ( a - b ) 2 i 1 T a + B T ( V I . 4 . 2 9 ) S i n c e t h e e l l i p s e r o t a t e s w i t h a n g u l a r v e l o c i t y co g i v e n by ( V I . 4 . 2 2 ) : ( U x - i U y ) ( t ) = - X _ Ilia ^ u 2h3 8 x 1 + 3 ( a - b ) 2 icot 1 T a + bT ( V I . 4 . 3 0 ) O v e r many p e r i o d s h c h a n g e s v e r y l i t t l e so t h a t i t may be c o n s i d e r e d n e a r l y c o n s t a n t . I n t e g r a t i n g t h e n g i v e s : Y 9 h 0 2uh 03 3x . . 3 ( a - b ) • , .1 w t + 2 ( a + b ) s i n 2 a j t ( V I . 4 . 3 1 a ) v - 3y 3ho ( a - b ) ^ ^ ^ 0 . Y " 4^hJ 37 T i W C O S 2 c j t ( V I . 3 . 3 1 b ) T h i s i s t h e e q u a t i o n o f a t r o c h o i d ( F i g u r e X I V ) . The a m p l i t u d e o f t h e o s c i l l a t i o n i n t h e m o t i o n i s : A . ^ jju t e j q . , . 3 ( . ; - b ' ) j ^ , 0 ( v e ) 4a)hn3 3x (a + b) 16 h, 3x 110 F i g . X I I I : A V o r t e x w i t h an E l l i p t i c a l Co re The P a t h s o f a V o r t e x w i t h an E l l i p t i c a l C o r e . The e c c e n t r i c i t i e s o f t h e e l l i p s e s a r e c o r r e c t , b u t t h e i r s i z e s a r e much r e d u c e d . 112 I t i s i n t e r e s t i n g t o n o t e t h a t i f one p u t s d=0 i n ( V I . 4 . 2 3 ) t h e t e r m s c a u s i n g d i s t o r t i o n d i s a p p e a r . A c i r c u l a r c o r e w i t h c o n s t a n t v o r t i c i t y w i t h i n i t i s t h e r e f o r e d i s t o r t e d a t a much s l o w e r r a t e t h a n o t h e r c o r e s and w i l l t h e r e f o r e obey t h e c i r c u l a r a p p r o x i m a t i o n f o r c o r r e s p o n d i n g l y g r e a t e r l e n g t h s o f t i m e . In t h e n e x t s e c t i o n a n o t h e r s u c h c i r c u l a r c o r e i s f o u n d w h i c h has a c o n t i n u o u s v o r t i c i t y d i s t r i b u t i o n . 113 V I . 5 P e r t u r b a t i o n s o f P l a n a r S o l u t i o n s S e v e r a l e x a c t v o r t e x s o l u t i o n s a r e known f o r p l a n a r u n i f o r m d e p t h f l o w s ( s e e , f o r e x a m p l e , B a t c h e l o r ( 1 967 ) p . 5 3 4 ) . By u s i n g t h e s t r e a m f u n c t i o n o f s u c h a s o l u t i o n as a f i r s t a p p r o x i m a t i o n t o t h e s t r e a m f u n c t i o n f o r n o n - p l a n a r b u t u n i f o r m d e p t h f l o w s one can o b t a i n s o l u t i o n s t o t h i s more g e n e r a l v o r t e x p r o b l e m as p e r t u r b a t i o n e x p a n s i o n s i n t h e s m a l l p a r a m e t e r v . L e t ty(x,t) be t h e s t r e a m f u n c t i on o f t h e n o n - p l a n a r f l o w anid l e t i f / 0 ; ( x _ , t ) be t h e known s t r e a m f u n c t i on o f t h e p l a n a r f l o w . I t i s a s sumed t h a t : w ^ 0 ) = - v 2 ^ 0 ) + o f o r r > e so t h a t h and k may be a p p r o x i m a t e d by t h e f i r s t few t e r m s o f t h e i r . T a y l o r e x p a n s i o n s . One can e x p a n d ty : -ty{x,t) = (x - U t , t ) + v ^ 1 ; ( x , t ) +... ( V I . 5 . 1 ) v_U i s t h e v e l o c i t y o f t r a n s l a t i o n o f t h e z e r o t h o r d e r s o l u t i o n . LI can be e x p a n d e d : U = v l l ( 1 ) + v 2 U ( 2 ) + . . . (V I . 5 . 2 ) h i s a l s o e x p a n d e d : » T X_3>T" y ^ y + * ' < V I - 5 - 3 > h = h 0 + v h( 1 ) + . . . = h 0 + x ^ J L + y ^ f l - + 3x • 9y The e q u a t i o n g o v e r n i n g ty i s ( f r o m ( I I I . 3 . 7 ) and ( I V . 1 . 2 ) ) : : 0 ( V I . 5 . 4 ) _3_ at 1 V 2 ij, h 2 h 2 l i JL l i JL [3y 3x " 3x dy — V 2 $ h 2 v S u b s t i t u t i n g ( V I . 5 . 1 ) and e x p a n d i n g one has t o l o w e s t o r d e r : h i 9 ^ ° > _3_ 9y 9x a.. 9X v2V0) 114 ( V I . 5 . 5 ) ,(0) w h i c h i s t h e r e q u i r e m e n t t h a t ijr ; be a s o l u t i o n f o r p l a n a r , un i f o r m d e p t h f 1 o w . To o r d e r v : _9_ 9 t _i _ u X ( l ) V I 3y _9_ 9x f ^ ( 1 ) + uy(D ax _9_ 9y h 2 , 9 ^ ( 0 ) 9 9 ^ ( Q ) a' 9y 9x 9x 9y , • • ( 1 ) . 2!jiiL»»t«»-n o 0 ( V I . 5 . 6 ) T h i s i s a l i n e a r e q u a t i o n t o be s o l v e d s u b j e c t t o t h e b o u n d a r y c o n d i t i o n s : -> 0 as r •> ; = 0 , t=0 ( V I . 5 . 7 ) In g e n e r a l t h e s o l u t i o n i s c o m p l i c a t e d and can be f o u n d o n l y n u m e r i c a l l y . We w i s h t o l o o k f o r s o l u t i o n s t o ( V I . 5 . 4 ) w h i c h a r e i n i t i a l l y c i r c u l a r and r e t a i n t h e i r c i r c u l a r f o r m f o r r e l a t i v e l y l o n g p e r i o d s o f t i m e . T h u s , we l o o k f o r s o l u t i o n s o f ( V I . 5 . 6 ) h a v i n g : ^ ( ° ) '= *<°>(r ) and ty{1) = 0 . ( V I . 5 . 6 ) becomes : r d r 1 A r d r rdty (0 ) r d ty d r ( 0 ) ( 0 ) d r d r ( 0 ) d ty 9 h 0 9x 9h ( n y ( l ) • ' j d . f l _1 • d r [ r d r rdty (0 ) d r - U x ( l ) d cTr I A r d r rd^ d r ( 0 ) s l n' d r cos ty ( V I . 5 . 8 ) = 0 115 T h e r e i s a s o l u t i o n i f : w i t h : a '_d_ r a > ( 0 ) W 0 ) + _d_n d f r d i f r ( ( d r d r j d r a d r r d r d r [ d r d r J d r U y ( 1 ) \ f ^ f ) ,(0) ( 0 ) 0 ( V I . 5 . 9 ) -1 ( V I . 5 . 1 0 ) P u t t i n g f ( r ) = £ — and u.-= r 2 , ( V I . 5 . 9 ) - becomes ( V I . 5 . 11) The b o u n d a r y c o n d i t i o n s r e q u i r e t h a t : f -> c o n s t . as r -* oo -s f/r 0 as r -> 0 . T h e r e i s a o n e - p a r a m e t e r f a m i l y o f s o l u t i o n s : x _ 4u _ 4 r 2 a 2 + u a 2 + r 2 ( V I . 5 . 1 2 ) The re f o r e : (0 ) = _ d± (0 ) d r 4 a r ( a 2 + r 2 ) ; The s t r e n g t h o f t h e v o r t e x i s : ,(0) _ Y = L i m r v r->oo •4a so t h a t (0) = — a 2 + r 2 ,y(D i f ; ( 0 ) = 2 a £ n ( a 2 + r 2 ) The v o r t i c i t y i s , t o l o w e s t o r d e r : w (0 ) _ 2 Y a : TT2 + r 2 ) ( V I . 5 . 1 3 ) ( V I . 5 . 1 4 ) -Y l iLo. , i i x ( l ) = _x_ l b . 2 h 0 dx u 2 h 0 dy ( V I . 5 . 1 5 ) ( V I . 5 . 1 6 ) The r a d i u s o f t h e c o r e i s o f o r d e r a . A v o r t e x whose i n i t i a l v o r t i c i t y d i s t r i b u t i o n has. t h e f o r m ( V I . 5 . 1 6 ) w i l l 116 1.0 .5 0 1.0 2.0 3.0 r a F i g . X V : The V e l o c i t y and V o r t i c i t y o f a Q u a s i - S t e a d y V o r t e x t h e r e f o r e r e m a i n c i r c u l a r f o r l o n g p e r i o d s o f t i m e . Such v o r t e x m i g h t be c a l l e d q u a s i - s t e a d y . In g e n e r a l i t i s n o t p o s s i b l e t o p e r f o r m a s i m i l a r a n a l y s i s o f t h e d e p t h o f f l u i d i s v a r y i n g s i n c e need n o t be a good a p p r o x i m a t i o n t o 4> s i n c e o u t s i d e t h e c o r e t h e y s a t i s f y d i f f e r e n t e q u a t i o n s : V 2 i | J 0 ; = 0 , V * T JV ; ^ ] 118 V I I . APPL ICAT IONS TO REAL FLUIDS V I I . 1 A t m o s p h e r i c C y c l o n e s As e x p l a i n e d i n t h e i n t r o d u c t i o n , p a r t o f t h e m o t i v a t i o n f o r t h i s t h e s i s was an a t t e m p t t o p r o v i d e a s i m p l e mode l f o r an a t m o s p h e r i c c y c l o n e t a k i n g f u l l a c c o u n t o f t h e e f f e c t s o f t h e c u r v a t u r e o f t h e e a r t h . H o w e v e r , t h e o b s e r v a t i o n t h a t a t m i d - l a t i t u d e s t h e w i n d i s r o u g h l y g e o s t r o p h i c i s e v i d e n c e t h a t t h e C o r i o l i s f o r c e , w h i c h we have n e g l e c t e d e n t i r e l y , p l a y s a d o m i n a n t r o l e i n t h e a t m o s p h e r e . In t h i s s e c t i o n , t h e p o s s i b i l i t y o f i n c l u d i n g t h e C o r i o l i s f o r c e i s e x a m i n e d . I t i s a l s o shown t h a t t h e u n r e a l i s t i c a s s u m p t i o n t h a t t h e a t m o s -p h e r e has c o n s t a n t d e n s i t y may be r e l a x e d s omewha t . The e q u a t i o n s o f m o t i o n f o r t h e a t m o s p h e r e may be w r i t t e n : s V VP |= + v - v v = - = ^ - - 2 w x v - £ ( V I I . 1 . 1 ) whe re v_ and V a r e b o t h t h r e e - d i m e n s i o n a l v e c t o r s . P i s t h e h y d r o s t a t i c p r e s s u r e and e f f e c t s o f v i s c o s i t y and t h e c e n t r i f u g a l f o r c e a r e i g n o r e d . £ i s t h e a c c e l e r a t i o n due t o g r a v i t y and G J i s t h e a n g u l a r v e l o c i t y o f t h e e a r t h . T a k i n g t h e c u r l o f ( V I I . 1 . 1 ) and u s i n g t h e e q u a t i o n o f c o n t i n u i t y : -|| + V - ( p v ) = 0 ( V I I . 1 . 2 ) one o b t a i n s t h e v o r t i c i t y e q u a t i o n : 119 We s u p p o s e as i n S e c t i o n I I I t h a t t h e v e l o c i t y i s , t o l o w e s t o r d e r , t a n g e n t i a l t o t h e s u r f a c e o f t h e e a r t h . We a l s o s u p p o s e t h a t t h e d e n s i t y i s a f u n c t i o n o n l y o f t h e h e i g h t above t h e e a r t h . W h i l e a v e r y r e s t r i c t i v e a s s u m p t i o n t h i s i s c e r t a i n l y a much b e t t e r a p p r o x i m a t i o n t h a n t h a t t h e d e n s i t y i s u n i f o r m . E q u a t i o n (V I I . 1 . 2 ) i s t h e n s a t i s f i e d i f : V/v = 0 . H o w e v e r : at + W p = 0 ( V I I . 1 . 4 ) t o l o w e s t o r d e r . One can t h e n show ( s e e , f o r e x a m p l e , V e r o n i s ( 1 9 6 3 b ) ) t h a t : 9 t + v- V f ( _ + 2^)'lp) p 0 ( V I I . 1 . 5 ) w h i c h i n t e r m s o f t h e t h i n f i l m c o o r d i n a t e s o f S e c t i o n I I . 3 becomes t o f i r s t o r d e r . JL + 1*. A. + ll JL at h ax h ay w i t h e q u a t i o n o f c o n t i n u i t y : a ( h k v w ) w +2co z z = 0 ( V I I . 1 . 6 ) a ( h k v x ) ax ay o ( V I I . 1 . 7 ) T h e s e e q u a t i o n s a r e ( I I I . 3 . 6 ) and ( I I I . 3 . 7 ) w i t h w z r e p l a c e d by w z + coz . T h u s , a l l o w i n g t h e d e n s i t y t o v a r y w i t h h e i g h t p r o d u c e s no c h a n g e s i n t h e e q u a t i o n s o f m o t i o n , b u t i n c l u d i n g t h e C o r i o l i s f o r c e d o e s . 2 u z i s known as t h e C o r i o l i s p a r a -m e t e r and on t h e s p h e r e i s 2cocos0 w h e r e 9 i s t h e c o l a t i t u d e The C o r i o l i s p a r a m e t e r a c t s as a c o n s t a n t s o u r c e o f v o r t i c i t y so t h a t i t i s no l o n g e r p o s s i b l e t o have f l o w s i n w h i c h t h e v o r t i c i t y i s i s o l a t e d i n v o r t e x c o r e s . The w h o l e i d e a o f a v o r t e x as a r e g i o n o f i s o l a t e d v o r t i c i t y t h e n b r e a k s down. The 120 o n l y s a t i s f a c t o r y r e s o l u t i o n seems t o be t h e a p p r o x i m a t i o n t h a t k and co a r e b o t h c o n s t a n t w h i c h amounts t o n e g l e c t i n g a l l e f f e c t s o f s u r f a c e c u r v a t u r e and d e p t h v a r i a t i o n . Thus one mus t c o n c l u d e t h a t t h e v o r t i c e s d i s c u s s e d i n t h i s t h e s i s c a n n o t be e x t e n d e d i n a s i m p l e way t o a c c o u n t f o r C o r i o l i s e f f e c t s . A v o r t e x on a c u r v e d r o t a t i n g s u r f a c e w i l l n e c e s s a r i l y have v o r t i c i t y t h r o u g h o u t t h e f l o w and n o t j u s t a t t h e c o r e . H o w e v e r , many p l a n a r f l o w s w i t h e x t e n d e d v o r t i c i t y have been m o d e l l e d by l a r g e numbers o f p o i n t v o r t i c e s . In a s i m i l a r way , one m i g h t mode l an a t m o s p h e r i c v o r t e x by a l a r g e number o f t h e v o r t i c e s d i s c u s s e d i n S e c t i o n IV . I t i s p o s s i b l e t o a c c o u n t f o r t h e C o r i o l i s f o r c e by r e q u i r i n g t h a t t h e v o r t e x s t r e n g t h s v a r y a c c o r d i n g t o : v ( x y ) ,*, - r>— n n „ Y n " 2 F T v V ^ • k ( W ( V I I . 1 . 8 ) w h i c h may be r e s t a t e d more s i m p l y a s : + ^ n l ^ V V  Y n " Y n o + k ( x n , y n ) • Y n o = c o n s t ' ' * n l = c o n s t -( V I I . 1 . 9 ) U n f o r t u n a t e l y , s u c h a s y s t e m does n o t r e s p e c t t h e r e q u i r e d c o n s t r a i n t E y n = 0 . T h u s , t h e r e w i l l a p p e a r t o be a v o r t e x f i x e d a t i n f i n i t y ( t h e s o u t h p o l e ) whose s t r e n g t h waxes and wanes as t h e f l o w p r o g r e s s e s . I f t h e number o f v o r t i c e s i s v e r y l a r g e t h e e f f e c t o f t h i s s i n g l e v o r t e x w i l l become n e g l i g i b l e and t h e s y s t e m s h o u l d s t i l l p r o v i d e a s a t i s f a c t o r y mode l f o r t h e f l o w . A p r o p e r i n v e s t i g a t i o n o f t h i s s y s t e m ( w h i c h w o u l d i n v o l v e s o l v i n g t h e e q u a t i o n s o f m o t i o n by 121 c o m p u t e r f o r l a r g e n u m b e r s o f v o r t i c e s , l o n g t i m e s a n d v a r i e d i n i t i a l c o n d i t i o n s ) i s b e y o n d t h e s c o p e o f t h i s t h e s i s . 122 I V . 2 S u p e r f l u i d V o r t i c e s In r e c e n t y e a r s work on i s o l a t e d v o r t i c e s has been a l m o s t e x c l u s i v e l y i n r e l a t i o n to t h e b e h a v i o u r o f l i q u i d H e l l . W h i l e t h e c o m p l e t e s e t o f p a r t i a l d i f f e r e n t i a l e q u a -t i o n s w h i c h d e s c r i b e t h e f l o w o f l i q u i d H e l l a r e s t i l l n o t c o m p l e t e l y u n d e r s t o o d ( s e e P u t t e r m a n ( 1 9 7 4 ) ) i t i s a c c e p t e d t h a t t h e s u p e r f l u i d c o m p o n e n t o f t h e f l u i d v e l o c i t y i s i r r . o t a t i o n a 1 a n d , o u t s i d e v o r t e x c o r e s , v e r y n e a r l y i n c o m -p r e s s i b l e . V o r t i c e s h a v i n g c i r c u l a t i o n q u a n t i z e d i n u n i t s o f h/m = 10 " crrr/se.c (h = P l a n c k ' s c o n s t a n t , m = mass o f h e l i u m atom) a r e o b s e r v e d . The c o r e s o f t h e v o r t i c e s a r e c h a r a c t e r -i z e d , n o t by r e g i o n s o f v o r t i c i t y , b u t by a r e g i o n i n w h i c h t h e d e n s i t y d e c r e a s e s r a p i d l y . The v o r t i c i t y i s t r u l y s i n g u l a r b u t t h e d e n s i t y o f t h e f l u i d i s z e r o a t t h e s i n g u l -a r i t y . The r a d i u s o f t h e c o r e i s c o n s t a n t and i s t h e o r d e r o f one o r two A n g s t r o m s . On e x p e c t s , t h e r e f o r e , t h a t e q u a t i o n ( I V . 3 .4 ) s h o u l d d e s c r i b e t h e m o t i o n o f v o r t i c e s b u t t h a t now: h n £ n = 6 n = c o n s t ' ( V I I . 2 . 1 ) The e q u a t i o n s o f m o t i o n s f o r a v o r t e x s y s t e m a r e t h e n : •h. o- c< dk 2 i 1 n n h 2 k n l 2 az in n n + 1 + ..n n + a h n az az Y B ( z , z ; z , z ) n n n + > * ( z , z > + E Y ^ ( Z , Z ; Z . , z , ) k^n k K K z = z „ ( V I I . 2 . 2 ) T h e r e i s no v o r t e x s t r e a m f u n c t i o n and t h e k i n e t i c e n e r g y o f t h e f l u i d o u t s i d e t h e c o r e i s n o t c o n s e r v e d . H o w e v e r , i n 123 b o t h t h e c a s e o f u n i f o r m d e p t h f l o w and o f f l o w on a p l a n e v o r t e x s t r e a m f u n c t i o n s e x i s t . T h e s e a r e , r e s p e c t i v e l y : N N r ft = h E E y if. *F(x ,y ;x, ,y. ) + % E i y 2 B ( x ,y ;x. ,y. ) n= l k/n n ' ^ n k ' - ' k ' n = l ^ n + Y 2 h ( x ,y ) + 2y ^ * (X ,y )1 , k = l ( V I I . 2 . 3 ) ' n n J n ' 1 n r " n J n ' J v N N f ft = % E E Y y . ^ x ,y ;x. ,y. ) + E i y 2 B ( x ,y ;x, ,y. ) n = l k/n n , J n k k ' n = i l n n n k k ' - ^ k ( x n , y n ) ( £ n 6 n - l ) + 2 y n ^ x n , y n ) } , h= l ( V I I . 2 . 4 ) The c o n s e r v a t i o n o f ft c o r r e s p o n d s t o t h e c o n s e r v a -t i o n o f t h e k i n e t i c e n e r g y o f t h e f l u i d e x t e r n a l t o t h e c o r e . T h e s e e q u a t i o n s can now be u s e d t o d i s c u s s v o r t e x p r o b l e m s i n more g e n e r a l g e o m e t r i e s t h a n have been u sed t o d a t e . A p r o b l e m o f p a r t i c u l a r i n t e r e s t i s t h a t o f v o r t i c e s i n a c y l i n d r i c a l c o n t a i n e r w i t h k = k 0 + a r 2 s i n c e t h i s i s t h e s hape o f t h e f r e e s u r f a c e o f a r o t a t i n g f l u i d . U n f o r t u n a t e l y , i t i s d i f f i c u l t t o s o l v e f o r ¥ (x ,y : x 1 ,y 1 ) and n u m e r i c a l me thod s seem n e c e s s a r y . 124 V I I I . CONCLUSION In t h i s t h e s i s t h e c o n v e n t i o n a l t h e o r y o f r e c t i l i n e a r v o r t e x m o t i o n has been g e n e r a l i z e d t o i n c l u d e v o r t i c e s i n f l u i d s o f s m a l l b u t v a r y i n g d e p t h on c u r v e d s u r f a c e s . To t h e a u t h o r ' s k n o w l e d g e , t h e o n l y p u b l i s h e d work o f a s i m i l a r n a t u r e i s by Lamb ( 1 916 ) who s u g g e s t e d t h a t , i f t h e f l u i d i s o f c o n s t a n t d e p t h , v o r t e x v e l o c i t y f i e l d s m i g h t be o b t a i n e d by " o r t h o m o r p h i c p r o j e c t i o n o n t o t h e p l a n e . " Lamb a s sumes w i t h o u t j u s t i f i c a t i o n t h a t t h e f l u i d m o t i o n may be a p p r o x i m a t e d by p u r e l y t w o - d t r r i e n s i o n . a T . f l o w . The o n l y e x a m p l e d i s c u s s e d i s two v o r t i c e s on a s p h e r e . The a u t h o r has shown t h a t , by a s u i t a b l e c h o i c e o f c o o r d i n a t e s , i t i s p o s s i b l e t o d e t e r m i n e t h e v o r t e x v e l o c i t y f i e l d s i n f l u i d s o f v a r y i n g d e p t h . The a p p r o x i m a t i o n s u s e d a r e shown to be c o n s i s t e n t i n a s y s t e m a t i c p e r t u r b a t i o n e x p a n s i o n i n a s m a l l p a r a m e t e r A r e p r e s e n t ! * ng t h e r a t i o o f v e r t i c a l t o h o r i z o n t a l s c a l e s ; o n l y l o w e s t o r d e r e q u a t i o n s a r e e x a m i n e d . He has a l s o shown t h a t t h e r e i s a f u n c t i o n h ( x , y ) be s u i t e d f o r d e s c r i b i n g t h e e f f e c t s o f t h e s u r f a c e c u r v a t u r e on v o r t e x m o t i o n . I f t h e d e p t h o f f l u i d i s u n i f o r m , t h e r e a r e no b o u n d a r i e s and t h e s u r f a c e o f f l o w i s t o p o l o g i c a 1 1 y s i m i l a r t o a p l a n e o r a s p h e r e , t h e e q u a t i o n s o f m o t i o n may be w r i t t e n e x p l i c i t l y i n t e r m s o f h ( x , y ) . In t h e more g e n e r a l c a s e w i t h b o u n d a r i e s and v a r y i n g d e p t h , t h e e q u a t i o n s o f m o t i o n can be w r i t t e n i n t e r m s o f t h e G r e e n ' s f u n c t i o n s o f e l l i p t i c p a r t i a l d i f f e r e n t i a l o p e r a t o r s . As d i d L i n ( 1 943 ) f o r r e c t i l i n e a r v o r t i c e s , t h e a u t h o r has shown t h a t t h e r e i s a v o r t e x s t r e a m f u n c t i o n f o r t h e s e e q u a t i o n s 125 o f m o t i o n , t h a t i t i s r e l a t e d t o t h e k i n e t i c e n e r g y o f t h e f l u i d , and t h a t , i f t h e d e p t h i s u n i f o r m , i t t r a n s f o r m s s i m p l y u n d e r c o n f o r m a l t r a n s f o r m a t i o n s . The b e h a v i o u r o f s i m p l e v o r t e x s y s t e m s has been e x a m i n e d i n S e c t i o n V, s h o w i n g t h a t s u r f a c e c u r v a t u r e and d e p t h v a r i a t i o n can c a u s e m a r k e d q u a l i t a t i v e d i f f e r e n c e s b e t w e e n v o r t e x s y s t e m s . S t u d i e s o f t h e s t a b i l i t y o f r i g i d l y r o t a t i n g s y s t e m s showed t h a t t h e s e q u a l i t a t i v e d i f f e r e n c e s a l s o e x t e n d t o c o n s i d e r a t i o n s o f s t a b i l i t y . In S e c t i o n VI t h e e f f e c t s o f n o n - i n f i n i t e s i m a l v o r t e x c o r e s we re d i s c u s s e d . I t was shown t h a t t h e v e l o c i t y o f r a d i a l l y s y m m e t r i c c o r e s depend s o n l y on t h e i r s t r e n g t h b u t t h a t a s y m m e t r i e s i n t r o d u c e s s m a l l w o b b l e s i n t o t h e m o t i o n o f a v o r t e x . I f t h e c o r e i s a p p r o x i m a t e l y c i r c u l a r t h e a m p l i t u d e , o f t h e w o b b l e i s much s m a l l e r t h a n t h e c o r e r a d i u s and may be n e g l e c t e d . The t h e o r y d e v e l o p e d i n S e c t i o n s 1 1 1 - VI i s o f t h e o r -e t i c a l i n t e r e s t as a g e n e r a l i z a t i o n o f t h e c l a s s i c a l t h e o r y o f v o r t e x m o t i o n . I t i s n a t u r a l , h o w e v e r , t o a sk how i t may be a p p l i e d t o a t m o s p h e r i c m o t i o n . As shown i n S e c t i o n V I I . l , i t seems i m p o s s i b l e t o r e c o n c i l e t h e e f f e c t s o f t h e C o r i o l i s f o r c e and o f s u r f a c e c u r v a t u r e i n s i m p l e v o r t i c e s o f t h i s t y p e . I t i s q u i t e p o s s i b l e t h a t i n a s i m i l a r , b u t n o n - r o t a t i n g , a t m o s -p h e r e t h a t t h e s e v o r t i c e s w o u l d p r o v i d e a s i m p l e f i r s t o r d e r mode l o f a c y c l o n e . Of c o u r s e , i f t h e C o r i o l i s f o r c e p l a y s an i m p o r t a n t r o l e i n c y c l o g e n e s i s , t h e n t h e i n c i d e n c e o f c y c l o n e s i n s u c h an a t m o s p h e r e w o u l d be g r e a t l y s u p p r e s s e d . 1 2 6 V o r t i c e s i n s u p e r f l u i d h e l i u m can be d e s c r i b e d by t h e t h e o r y o f S e c t i o n IV i f one makes a m o d i f i c a t i o n t o r e q u i r e t h a t t h e c o r e r a d i i r e m a i n c o n s t a n t . In g e n e r a l , t h e r e i s no l o n g e r a v o r t e x s t r e a m f u n c t i o n f o r t h e f l o w , b u t f o r t h e s p e c i a l c a s e s o f u n i f o r m d e p t h and o f p l a n a r f l o w a v o r t e x s t r e a m f u n c t i o n does e x i s t . F o r t h e most i n t e r e s t i n g p r o b l e m s o f f l o w w i t h v a r y i n g d e p t h t h e r e i s s t i l l t h e p r a c t i c a l p r o b l e m o f c a l c u l a t i n g t h e f u n c t i o n s V and B w h i c h d e s c r i b e t h e v e l o c i t y f i e l d . 127 B IBL IOGRAPHY B a t c h e l o r , G.K. ( 1 967 ) An I n t r o d u c t i o n t o F l u i d Dynam ic s ( C a m b r i d g e U n i v e r s i t y P r e s s , C a m b r i d g e , E n g l a n d ) Chapman, D.M.F. ( 1 977 ) M . S c . T h e s i s ( U n i v e r s i t y o f B r i t i s h C o l u m b i a) Chapman , D.M.F. ( 1978) J . M a t h . P h y s . 1 9 , p . 1 9 8 8 . C o u r a n t , R. , and H i l b e r t , D. ( 1962) Me thod s o f M a t h e m a t i c a l P h y s i c s ( J o h n W i l e y & S o n s , New Y o r k ) V o l . 1 1 , C h . I V , S e c . I V , p . 2 9 0 . E i s e n h a r t , L . P . ( 1 909 ) A T r e a t i s e on t h e D i f f e r e n t i a l G e o m e t r y o f C u r v e s and S u r f a c e s ( G i n n a n d C o . , B o s t o n ) , p . 9 3 . H a v e l o c k , T . H . ( 1 9 3 1 ) P h i l . M a g . 1_1, p . 6 1 7 . H e l m h o l t z , H. ( 1 858 ) C r e l l e ' s J o u r n a l , t r a n s l a t e d : P h i l . M a g . S e r . 4 , N o . 2 2 6 , S u p p . 3 3 , p . 485 ( 1 8 6 7 ) K e l v i n , L o r d ( 1 8 7 8 ) , N a t u r e 18., p . 1 3 K e l v i n , L o r d ( 1 8 6 9 ) , T r a n s . Roy . S o c . , E d i n . 2_5 K a r m a n , T . v o n ( 1 912 ) P h y s . Z e i t s . , 1 3 , p .49 K i r c h h o f f , G. ( 1 8 7 6 ) , V o r l e s u n g e n u b e r m a t h e m a t i s h e P h y s i k , M e c h a n i k ( L e i p z i g ) p .255 K o e g e , p. ( 1 918 ) A c t a M a t h . 4 1 , p .306 L a g a l l y , M. ( 1 9 2 1 ) M a t h . S e i t s . , 10., p .231 Lamb, H. ( 1 916 ) H y d r o d y n a m i c s ( C a m b r i d g e U n i v e r s i t y P r e s s , C a m b r i d g e , E n g l a n d ) 4 t h e d . , A r t . 8 0 , C h . I V , p . 1 0 1 . L i n , C C . ( 1943) On t h e M o t i o n o f V o r t i c e s i n Two D i m e n s i o n s ( U n i v e r s i t y o f T o r o n t o P r e s s , T o r o n t o ) M a s o t t i , A. ( 1 9 3 1 ) A t t i . P o n t i f . A c c a d . S c i . N u o v i . L i n c e i 8 4 , p . 2 0 9 . M e r t z , G . J . ( 1978) P h y s . F l u i d s . 2_1, p .1092 M o r t o n , W.B. ( 1935) P r o c . Roy. I r i s'h A c a d . A 4_2, p.21 O n s a g e r , L. ( 1949) Nuovo C i m . S u p p . 6_, p .249 O s b o r n e , D.V. e t . a l . ( 1 9 6 3 ) C a n . J . P h y s . 4J_, p . 820 P u t t e r m a n , S . J . ( 1 974 ) S u p e r f l u i d H y d r o d y n a m i c s ( N o r t h H o l l a n d P u b l i s h i n g Co . , . A m s t e r d a m , 1974) p . l 9 f f and p . 2 6 7 f f . 1 2 8 R o u t h , E . J . , P r o c . L . M . S . , 1_2 p .83 S o m m e r f e l d , A . ( 1 9 4 9 ) P a r t i a l D i f f e r e n t i a l E q u a t i o n s i n P h y s i c s ( A c a d e m i c P r e s s I n c . , New Y o r k (N . Y .)) p . 5 0 . S o u r i a u J . M . ( 1969) S t r u c t u r e des Sy s temes Dynami ques ( Dunod , P a r i s , 1969) T h o m s o n , J . J . ( 1 883 ) T r e a t i s e on t h e M o t i o n o f V o r t e x R i n g s (Adams P r i z e E s s a y 1 8 8 2 , M a c M i l l a n , L o n d o n ) p . 9 5 . T k a c h e n k o , V. ( 1966) S o v . P h y s . J ETP 2_3, p . 1049 V e r o n i s , G. , ( 1963a ) J . M a r . R e s . Zl_t p . 110 V e r o n i s , G. , ( 1963b ) J . M a r . R e s . 2 1 , p .199 Y a r m c h u k , E . J . , e t . a l . ( 1 9 7 9 ) P h y s . R e v . L e t t . 4 3 , p . 214 129 APPENDIX A: MATHEMATICAL FORMALISM L e t M be a t w o - d i m e n s i o n a l smooth R i e m a n n i a n m a n i f o l d w i t h m e t r i c g . One may d e f i n e a t w o - f o r m a* b y : o* = e ( -det g)\ ( A l ) w h e r e e i s t h e a n t i s y m m e t r i c t e n s o r d e n s i t y w i t h e 1 2 = 1 and k i s a s c a l a r f u n c t i o n on M . T h e r e i s a n a t u r a l e x t e n s i o n o f o* t o a t w o - f o r m on t h e 2 N - d i m e n s i o n a 1 m a n i f o l d M N , n a m e l y , t h e u n i q u e t w o - f o r m a s a t i s f y i n g : N ( d x ' x . . . x d x N , d y ' x . . . x d y N ) = E yo*(dxn,dyn) (A2 ) n = l n where t h e y , n = l , . . . , N a r e c o n s t a n t s . No te t h a t k e r a=0 and a i s d i f f e r e n t i ab 1 e e v e r y w h e r e , a t h e r e f o r e i n d u c e s a s y m p l e c t i c s t r u c t u r e on M N . ( N o t e t h a t V a * = 0 s i n c e n o n - t r i v i a l t h r e e - f o r m s c a n n o t e x i s t on a t w o - d i m e n s i o n a l mani f o l d . ) S u p p o s e t h a t ft i s some s c a l a r f u n c t i o n on M N . A n a t u r a l f l o w i s i n d u c e d , t h e e q u a t i o n s o f m o t i o n o f w h i c h a r e : H Y -1 ^ | = a A V f i = g r a d n (A3 ) N whe re x d e n o t e s p o s i t i o n on M and V i s t h e e x t e r i o r d e r i v a t i v e . N o t i c e t h a t : 4f = Vft = a(fl,n) = 0 so t h a t ft i s c o n s e r v e d . In h a r m o n i c c o o r d i n a t e s o* = h 2 ( x , y ) k ( x , y ) e and (A3) becomes ( I V . 4 . 4 ) . F o r f u r t h e r r e f e r e n c e on t h e m a t h e -m a t i c s o f s y m p l e c t i c s y s t e m s see S o u r i a u ( 1 9 6 9 ) . 130 APPENDIX B: EVALUATION OF SUMS A l l t h e s p e c i a l sums n e c e s s a r y f o r t h e c a l c u l a t i o n s o f S e c t i o n V may be e v a l u a t e d e a s i l y o n c e : N R, ( z ) exp ( 2TTJ Ln/N) V ~ n = 1 ( l - zexp(27r in/ N 7 T 2 " ' Z c o m P l e x > L = 1 i s known. S u p p o s e f i r s t t h a t | z| < 1 . T h e n : N , k - l R. ( z ) = E e xp (2 - r r i Ln/N ) £ k z ex p ( 2 77 i ( k - 1 ) / N) L n= l k = l ( B l ) (B2 ) The i n f i n i t e s e r i e s i s a b s o l u t e l y c o n v e r g e n t a l l o w i n g t h e r e o r d e r i n g o f t h e sums: N 00 1/1 R, ( z ) = E k z K - i E e x p ( 2 T r i (L + k - l ) n / N ) L k-1 n= l (B3 ) The s e c o n d sum v a n i s h e s u n l e s s L+k-1 = rN , r an i n t e g e r . R. ( z ) = E N( r N - L + l ) z r N _ L = N-f-r = 1 a z , _N - L + 1 I I- If N ( ( N - L + l ) z N ' L + ( L - l ) z 2 N ~ L ) ( l - z N ) 2 (B4 ) The r i g h t s i d e s o f b o t h ( B l ) and (B4 ) a r e a n a l y t i c i n a l l r e g i o n s o f t h e c o m p l e x p l a n e e x c l u d i n g t h e N th b o o t s o f o n e : h e n c e , by a n a l y t i c c o n t i n u a t i o n , (B4 ) i s v a l i d f o r a l l z . - r r i / n To e v a l u a t e T L ( x ) p u t z=xe T, ( x ) = N e x p ( (2n + l ) T r i L / N ) n = l ( l - e x p ( ( 2 n + l ) T r i / N ) ) , 2 - e ^ ' V ( x e7 7 1 ^ N ( - ( N - L + l ) x N - M + ( L - l ) x 2 N - M ) ( l + x N ) 2 c = " N v 1 l - e x p ( 2-TTi Lk/N) . . / n / \ n / w S L - nl, ( l - e x p ( 2 T r i k / N ) r = L ^ m < R N ( Z } " R L ( Z } ) ' ( B5 ) 131 L i m N ( l + ( N - l ) z N - ( N - L + l ) z N - L - ( L - l ) z 2 N - L ) z - 1 ( l - z N ) i n m N ( N ( N - l ) z N " 1 - ( N - L + l ) ( N - L ) z N ~ L " 1 - ( L - l ) ( 2 N - L ) z 2 N " L " 1 ) zll - 2 N Z N - 1 ( 1 - Z N ) M m ( N ( N - l ) - ( N - L + l ) ( N - L ) z ~ L - ( L - l ) ; ( 2 N - L ) z N - L ) z - 1 2 ( 1 - Z N ) K N - L + l ) ( N - L ) - ( N - L ) ( L - 1 ) ( 2 N - L ) 2 N Sg (N-L ) ( 2 - L ) ( B 6 ) L ' H o p i t a l ' s r u l e has been u sed t w i c e . [ij r^^wm' s> • '»'N-1> (B7) 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0085760/manifest

Comment

Related Items