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Vortex motion in thin films Hally, David 1980

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VORTEX  MOTION  IN  THIN  FILMS  by DAVID B.Sc.  The U n i v e r s i t y  A THESIS THE  HALLY  SUBMITTED  IN  REQUIREMENTS  of Toronto,  PARTIAL FOR THE  1976  F U L F I L M E N T OF DEGREE OF  DOCTOR OF P H I L O S O P H Y i n THE  FACULTY THE  DEPARTMENT OF  We a c c e p t to  OF GRADUATE  this  the  thesis  required  THE U N I V E R S I T Y  0  David  PHYSICS as  conforming  standard  OF B R I T I S H  December,  STUDIES  COLUMBIA  1979  Hally,  1979  In  presenting  this  an a d v a n c e d  degree  the  shall  I  Library  f u r t h e r agree  for  scholarly  by h i s of  at make  that  it  written  for  may  financial  of  University  P of  'ri  IC  British  2075 W e s b r o o k P l a c e V a n c o u v e r , Canada V6T 1WS  6  is  Apr.  23.  ?  19  SO  of  British for  for extensive  be g r a n t e d  It  fulfilment of  available  by  gain  ^ Columbia  shall  the  that  not  requirements  Columbia,  I  agree  r e f e r e n c e and copying  t h e Head o f  understood  permission.  Department  Date  freely  permission  purposes  thesis  in p a r t i a l  the U n i v e r s i t y  representatives.  this  The  thesis  of  or  that  study.  this  thesis  my D e p a r t m e n t  copying  for  or  publication  be a l l o w e d w i t h o u t  my  ABSTRACT The been  generalized  depth  on  curved  to  lowest  of  fluid  radii  order is  of  vortex tion  classical to  theory  include  surfaces. in  of  A number  small  is of  criteria  on  surfaces  revolution  the both  result  of  von  staggered  effects radial  of  elliptical theory  finite  symmetry  introduce  to  and  small core  in  Karman,  are  core  size  core  treated  atmospheric  rings  0  In  the in  detail. and  are  examined  with  the  principal  a  generalized  of  the  generate  of  are  depth  conserva-  described.  rings  of  vortices  contradistinction streets)  may  be  motion.  are The  superfluid  in The  from  shown case  of  of  the  vortices  are  Applications  to  stable.  Departures  distributions  vortex  varying  which  (vortex  examined.  has  in  to  configurations are  of  motion  systems  stability  found.  cyclones  di s c u s s e d .  used  vortex  vorticity in  of  motion  fluids  Existence  and  double  symmetric  thin  comparison  the  vortex  expansion  surface.  for  wobbles is  in  simple  particular,  in  equations  proved  In  of  vortices The  the  streamfunction  laws.  rectilinear  a perturbation  considered  curvature  of  to an  CONTENTS Page ABSTRACT  i i  LIST  OF F I G U R E S  LIST  OF T A B L E S  v vi  ACKNOWLEDGEMENTS  vii  I.  INTRODUCTION  1  II.  COORDINATES  6  1. 2. 3. 4. III.  IDEAL 1. 2. 3.  IV.  3. 4. 5. 6. V.  15  FLUIDS  2. 3.  IN T H I N  FILMS  The V o r t e x V e l o c i t y F i e l d The V e l o c i t y o f a V o r t e x i n a o f U n i f o r m Depth The V e l o c i t y o f a V o r t e x i n a o f V a r y i n g Depth The V o r t e x S t r e a m f u n c t i o n Conformal Transformations Constants of the Motion  SIMPLE 1.  15 17 19  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 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 The T h i n F i l m A p p r o x i m a t i o n  THE MOTION OF V O R T I C E S 1. 2.  6 7 12 14  Harmonic Coordinates Surfaces of R e v o l u t i o n Thin Film Coordinates V e c t o r Components: Notation  . . . .  Fluid Fluid  23 23 30 36 43 48 50 54  VORTEX SYSTEMS  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 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 b) P l a n e S u r f a c e , N o n - U n i f o r m D e p t h 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 The M o t i o n o f a V o r t e x P a i r : Yi -Y2 =  i i i  54 54 55 58 61  i v Contents 4. 5.  VI.  VIII.  p  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 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 of R e v o l u t i o n a) S t a g g e r e d V o r t e x S t r e e t s b) S y m m e t r i c V o r t e x S t r e e t s c ) E x a m p l e : The C y l i n d e r d) E x a m p l e : The S p h e r e  VORTICES 1. 2. 3. 4. 5.  VII.  (Cont'd)  WITH  g  66 72 73 81 85 90  CORES  96  The P o s i t i o n a n d V e l o c i t y o f a V o r t e x C i r c u l a r Cores 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 E l l i p t i c a l Cores P e r t u r b a t i o n s of Plane S o l u t i o n s  96 98 102 104 113  APPLICATIONS  FINITE  a  TO  1.  Atmospheric  2.  Superfluid  CONCLUSION  .  REAL  Cyclones Vortices  ., .  BIBLIOGRAPHY APPENDICES A. S y m p l e c t i c B. E v a l u a t i o n  FLUIDS  118 118 122 124 126  Systems o f Sums  128 128 129  e  LIST  OF F I G U R E S Page  I.  The S u r f a c e  of Revolution  II.  Vortex  in a Fluid  III.  Streamlines a  Core  Fluid  Near  of Varying  t h e Core  of Varying  IV.  Fluid with  Depression  V.  Fluid with  Surface  p = f(z)  8  Depth  of a Vortex  . . .  37  in  Depth  40  near  the Origin  Curvature  . . . .  near the  Origin VI.  The P a t h  VII.  Staggered  VIII.  Surfaces  56  59 of a Vortex Vortex  Street:  f o r which  Definite  Pair  65 T h e Modes  t h e Mode  M=N . . .  78  M=N h a s  Stability  79  IX.  Staggered  Vortex  Street:  T h e Modes  M=%N  . .  80  X.  Symmetric  Vortex  Streets:  The Modes  M=N  .  .  84  XI.  Symmetric  Vortex  Streets:  T h e Modes  M=%N .  .  86  XII.  y ( x ) and y"(x)  XIII.  A Vortex  with  XIV.  The P a t h  of a Vortex  XV.  Velocity Vortex  and V o r t i c i t y o f a  89  +  an E l l i p t i c a l with  Core an E l l i p t i c a l  110 Core  111  Quasi-Steady 116  v  LIST  OF  TABLES Page  1.  Regions of S t a b i l i t y of S t r e e t s on a C y l i n d e r  Staggered  Vortex  2.  Regions of S t a b i l i t y of S t r e e t s on a S p h e r e  Symmetric  Vortex  Regions of S t a b i l i t y of S t r e e t s on a S p h e r e  Staggered  3.  90 91  Vortex 94  vi  ACKNOWLEDGEMENT I  would  like  to  thank  this  work  and  for  initiating  Prof. his  F.A.  advice  Kaempffer and  for  criticism  throughout. The  financial  assistance  British  Columbia  (Macmillan  Natural  Sciences  and  Canada  (Postgraduate  of  Graduate  both  University  Fellowship)  Engineering  Research  Fellowship)  is  vi i  the  and  Council  gratefully  of  the  of  acknowledged.  1 I.  INTRODUCTION In  beginning that  in  point are  1858 of  Helmholtz  the  a fluid  is  the  in  of  which  gradient  advected.  a paper  vorticity  in  the  of  went  N isolated  points,  equations  describing  the  to  order  system  the  positions  describing the  original associated  of  superposition: of  the  with  Each  field,  is  called  vortex  is  not  over  some  velocity vortex  refer  of  core.  the  theorems tion fluid to  of  to  region  t h o s e " made  by  as  the  position  and  (1869) the  C is:  later  T  c  the  of =  Although  of  vorticity  the  principle  points  the with  the  its  is  The  velocity of  is  change  velocity  position field) of  similar of  a  of  and a  position  Helmholtz'  , where  circulation  of  distributed  c i r c u l a t i o n ; the  under  the of  vorticity  but  jv_ • ds_  that,  equations  point  reformulated  idea  showed  Helmholtz,  rate  reduced  with  core.  the  zero  be  field  point,  the  with  can  velocity  practice,  a single  is  is  differential  obeys  v o r t i c i t y , along  confused  Kelvin  each  lines  fluid  vorticity.  which  total  be  a contour  velocity.  field  In  the  differential of  each  vortex  vorticity  fluid  the  showed  at  if  partial  the  associated  known  introduced  the  points  the  a vortex.  Kelvin  and  around  to to  Lord  fields  point  its  that  marked  He  production  non-linear,  is,  fluids.  show  and  of  the  velocity  confined  small (not  a  are  that  velocity  vorticity.  to  ordinary of  which  p o t e n t i a l , the  then  motion  of  equations  is  sum  on  two-dimensional,  at  a 2Nth  momentum  a scalar  Helmholtz  incompressible, except  study  published  any  circula-  v_  is  the  assumptions advected  2 contour  is  constant  enclose  the  same  set  strength,  y  around  contour  any  , of  Routh function the  (a  vortex  Kirch offAs  in  esimal  with  vortex  addition, angular  moment  Section  IV.6. Since  little much in  work  has  been  is  his  work  Hell  sparked day.  the  flows  of  time  formal on  interest  applications bluff  was  are  to  It  the  Lin of  in are  to  the to  bodies  of  vortex  of  (e.g.,  there  real  which  the  centre  moment  of of  In  as  in  ( 1949)  atmospheric  infinit-  p.530).  quantized  subject  laws  of  the  detail  has  motion.  to  ( 1943).  conservation  discussed  (1943)  by  systems  conservation  Onsager  vortices  Lin  under  vortex  application of  and  q u a n t i t y known  is  governing  generalized  (1931)  (1967),  conserved  stream-  equations much  translations  theory  other.  conservation  (Batchelor  circulation  conservation  For  to  of  no  Hamiltonian  leads  prediction  support  past  the  both The  the  vortex  of  Masotti  there  rotations  another  written  renewed  street).  system,  publication  the  would  Other to  (1921),  circulation.  p a r t i c u l a r , the  fluid  and  although  to  and  the  system  transformations.  of  on  vortex  the  streamfunction there  that  C  = r^, .  for  under  C and  proportional  translations  and  if  then  introduced  invariances  under  circulation,  is  first  c i r c u l a t i o n , under  the  vortices  Lagally  coordinate  Moreover,  enclosing  Hamiltonian  invariance of  of  Hamiltonian motion)  associated  time.  a vortex  (1881)  ( 1876),  any  in  in  been However,  fluid that  systems;  super-  circulation persists  weather  Karman  to  this  systems  vortex  3  Almost that and  the  vortices  parallel  flow  in  the  briefly  pal  the  is  of  thesis  depth  to  an  Lamb  (1916),  in  his  is  gradient  perturbation  scheme  III).  Vortex  are  examined  of  shown  to  which  casts  relation energy  be  the  is  shown  the and  invariances  under  generated.  In  of  circulation  to  in  of  that  with  is  more  text,  motion  comparison It  long  two-dimensional  the  the  depth  the  the  princi-  purpose  detail  non-uniform  coordinates II),  the  (one  which the  in  and  of  to  but  in  in  useful  in  momentum  the  depth  to  the  IV  flow.  motion  and  of  field  which  are  into  laws  particular,  the  is  be  equations  of  vortices  are  streamfunction  and  the  form.  of  to  angular  invariance  A  kinetic  transformations  r e l a t e d to  the  additional  corresponding  conservation  a  small  order  symplectic  streamfunction  to  and  assumed  the  vortex  production  fluid  lowest  Systems  the  subjected  the  under  infinitesimal  to  the  derived  conservation  shown  which  a generalized  vortex  simple  governing  Section  of  prove  equations  are  solutions  by  which  velocity  equations  between  infinitely  assumption  surface.  of  incompressible  governed  in  assumed  small.  in of  the  systems  a scalar)  (Section  assumption  the  such  is  component  motion  of  small  has  well-known  determining under  fluids  vertical  of  and  fluid  of  for  surface  to  ideal  is  reduced  (Section  of  that  date  is  defining  sections  to  problem  examine  variation  vortices  the  a method  further  on  rectilinear:  uniform  motion the  that  work  curvature  After later  are  a curved  depth  generalize  the  plane.  on  radii  this  so  outlines  vortices fluid  all  are moment  under  4 scale  transformations.  shown  that  conformal  the  The  vortex  Section  variation  motion  of  V,  on  of  the  some  a single  simple vortex  surface  Also,  s t a b i 1 i t i e s o f some  of  vortices  results Kelvin  on  are  vortices  vorticity  the that  cores  The  solution.  tion  an  the  is  of  must  have  in  are  of  in  it  is  under  are  depth  examined.  are  derived  Karman  be  of  analyzed.  configurations The  by  In  to  symmetrically  stable  Lord  (1912).  contradistinction  rings  past  determined.  first  von  and  vortices  rotating  cores  of  are  from  each  of  while  finite  the placed  similar  circular  vortex,  travels  systematic  examined of  the  a vortex that  length  a precise  of  in  the  drift  if  can  the  that,  order  The  of  the  is  introduce for the  core  simplest  be e x a m i n e d  plane  a vortex  time  form.  cores  but  of  It  of  times mean  closely case  of  detail.  core in  distributions  examined.  a c i r c u l a r core.  shown  appreciable  simply  curvature  systems pairs  those  effects  vortex  the  having  It  the  motion  is  of  and  may  vortex  evolution  scheme  of  departures  that  vortex  revolution  double  VI,  separation,  uniform,  unstable.  the  small  into  elliptical  for  are  surface  rigidly  that,  streets)  within  approximates  ation  Karman,  of  and  (1883)  found  Section  enough  vortex  is  rings  that  wobbles long  it von  In  argued  Thomson  (vortex  staggered  of  is  transforms  depressions  generalizations  particular, of  and  surfaces  (1878),  results  bumps  depth  coordinates.  effects  localized the  fluid  streamfunction  transformations  In depth  When t h e  then  is its  as to  its  in  a  lowest  remain  vorticity  perturborder  circular distribu-  5 The attempt in  to  which  just  as  every the  the a  linear  ent  model  Coriolis  e.g.,  of  only  order the  other  in  in  The  are  was  earth of  is  or  a cone.  cyclones effects  since  important of  atmospheric  is  of  only  surface  by  Almost  variation  the  of  of  system  curvature  via  curvathe  not  variations if  not  derived  is  an  cyclones  surface  approximation  (1963a))  largely  treated exactly,  latitudinal  B-plane  effects  for  a plane  which  the  motivated  model  terrestrial  Veronis  parameter  that  of  approximation  parameter.  (see,  enough  a first  1 1 - VI  perturbation  included  Coriolis  Sections  curvature  analytic  are  of  provide  6-plane  ture  work  are  consistthe  is  large  also  important. Our forces tion  have  if  one  cyclone) in curvature  approach been  alone.  compromise.  model  including  are  discussed  Another A rotating fluid of  revolution  in  depth  of  the  theory is  of  of Its  liquid  Sections  discussed.  inherent  Coriolis  motivating  is  et.al.  helium  is  this  is  assump-  a terrestria1 of  surface  cyclone  in  must  a simple  affect  vortex  curvature  also  the  VII.2  work  known  effects  How  applicable  be  does and  extent to  the  following.  support  to  motion  the  is  to  known  (1963)).  affect  Section  11 - VI  a drastic  effects  surface  Coriolis  VII.l.  surface  In  of  a large  and  liquid Hell  (Osborne  vortices? of  The  problem  vortices.  the  of  problems  direction;  (clearly  a good model  highlight  Section  bucket  other  A good model  both  in  to  the  entirely  expects  attempt  some  from  neglected  really an  is  to  a  superparaboloid  the  variation  distribution which  superfluid  the vortices  6 II.  COORDINATES Throughout  cular  tractable.  discussed  II.l  M be a t w o - d i m e n s i o n a l ,  nates  such  (x,y)  g^-U.y) h(x,y)  called  harmonic  and to  with  surface  will  always  over  with  2  dropped  i f  of these  coordinates  by  In  has  Riemannian  to choose  coordi-  the form:  that  D .  a torus)  is  M  .  2  part  In  no c h a n c e  it  completely  to multiply More  connected  complex  are defined  by: ( 1 1 . 1 . 2)  ty is:  + r dty ) z  of  considered.  y = r si n  2  be  to a r e s t r i c t i o n  or sphere.  (r,cf>)  are  For s i m p l i c i t y  amounts  n o t be  only  that  parametrized  coordinates  in. terms  (D i s  equivalent  plane  (x,y)  we s h a l l  flows).  This  will  ;  = h(x,y)  of  non-negative  (1909)).  general D  topologically  is  possible  Eisenhart  (x,y)  harmonic  there  oriented,  d i f f e r e n t i ab l e  the f l u i d  = h* (r,cj>)(dr  h*(r,<J>)  the mathe-  (II.1.1)  sub-domain  which  element  is  the m e t r i c  of t h e complex  (e.g.,  ds  parti-  2  x = r c o s (J> line  t o make  a  .h (x,y)  i j  be a s s u m e d  Polar  The  It  a real-valued  some  surfaces  sub-domains surfaces  .  coordinates.  unambiguously those  that  = 6  is  g..  (see, f o r example,  concerned  necessary  that  Coordinates  metric  function  is  be f o u n d  length.  manifold with  where  i t will  Here' the p r o p e r t i e s  at  Harmonic Let  the  work  choice! of c o o r d i n a t e s  matics are  this  (II.1.3)  2  future  the a s t e r i s k  of confusion  will  between  be  h* and h .  II.2  Surfaces The  in  the  r  <J)  element  = g (u)du  2  2  is  = r(u)  an  and ds  so  line  Revolution of  a surface  of  revolution  can  be  put  form: ds  where  of  angular  require  g|(u)dct  2  +  2  (r,d>)  2  =  gj(u)du  2  +  period  be  harmonic  2TT .  Let  polars.  Then:  2  h (r)((r'(u)) du 2  of  r d<j) )  =  2  (II.2.1)  2  coordinate  that  = h (r)(dr  2  .+  2  2  +  r (u)d<J) ) 2  2  (II.2.2)  g (u)dd, 2  2  that: r'(u) r ( u)  =  9u(u) g (u)  (II.2.3)  (J)  when ce : U  r(u)  =  h(r(u))  It  will  prove  p  i ( r )  exp  9u(s) 9cpt s J  =  h ' ( r) r  =  (II.2.4)  -^y  convenient  \  ds  -rhr"+  to  (II  .2.5)  (II  .2.6)  define  , . 1  Then : r (u) g ( u j r ' ( u) 2  p(r(u))  =  (J)  • gj,(u) = 3  guT u T  _d_ rg<b( du [ r ( u)  +  1  (II.2.7)  Al s o :  rp  1  -  (r)  r ( u)  r ' (u)  gj>(u)i [g (u)  du  u  g<f>(u) _d_ f-gj>(u) gu(u) du [ g ( u ) J  (II.2.8)  u  The  Gaussian  curvature i k  *  of a surface 3 1k n r  in  2  9x'  L  +  r  h  9 x  is  defined  m _ i m kn r  r  n  by  m i k mn_  (11. 2 .9 )  r  with .km 9 2  ln In  harmonic  8  3g mn  9jm 9x  Sx"  n  3  8xi a surface  of  h r 2  The  h = h(r)  only  interested  3  consider Figure  line  3X  ^  p,  apply  determine  in  r,  f o r any m a n i f o l d  applications  manifolds  which  h(r)  of a function  revolution  these  z  and  d>  one i s  p(r) about  an  by:  in  surfaces  axis. in  IR  and  3  p = f(z)  to parametrize is:  generally  for  coordinates  defined  coordinates  such  c a n be i m b e d d e d  and  a n d be c y l i n d r i c a l  One c a n u s e  element  (II.2.12)  2  in those  of  and  rh (r)  For p h y s i c a l  cj)  2  h = h(r)  definitions  .  (II.2.11)  V £vrch  P'(r)  r. dh_ h dr  the surface  I).  (II.2.10)  n  1  by t h e r e v o l u t i o n Let  The  d  We t h e r e f o r e  obtained  n  revolution  above  that  "IR .  i  coordinates: -1  For  9  8xm  1  the  (see  surface  9  Fig.I:  The  Surface  of  Revolution  p =  f(z)  10  ds whence,  = (1 + ( f ( z ) ) ) d z  2  2  using  + f (z)d<j)  2  2  (II.2.13)  2  (II.2.4-6)  r = exp  rz (1 +  (f'(s)) r ds fjil 2  2  (II.2.14)  Zo  r h ( r) = f ( z ) p(r)  =  Notice that then  2  (II.2.16)  (f'U))*)*  -1 £ p ( z ) _< 1 .  I f the slope  of  f(z) is  tanO  .  Similarly,  ds  f ( z )  (1 +  p(z) = sine  z = b(p) .  (II.2.15)  one can d e f i n e a s u r f a c e  The l i n e e l e m e n t  is:  = (1 + ( b ' ( p ) ) ) d p  2  2  of r e v o l u t i o n by:  + p d<j> 2  (II .2.17)  :  and (1 + ( b ( s ) ) s 1  r = exp  2  )" ds 2  (II.2.18)  Po  rh(r)  = p  p( r )  I f the slope The 1ater.  (II.2.19)  (1 + of  (II .2.20)  2  b(p)  function  is  tane  then  -jp(&wh(r))  p ( r ) = cose  will  also  be o f  . importance  N o t i ce t h a t , : jL(*«h(r))  since:  1 (b'(p)) )^  p(r) < 1 .  =  1) < o  (II.2.21)  11 Special a)  Cases  Plane For  the plane  b(p)  = const,  whence:  r  p  = 1  ;  = 0  -  ;  h(r)  rp (r) 1  ;  (II.2.22)  K = 0 b)  Cylinder For r  the c y l i n d e r  = exp(z/R) p(r)  c)  ;  =0  = R = const,  h(r).=  R/r  ; rp'(r)  whence:  = Rexp(-z/R)  = 0  ;  ;  K = 0  (II.2.23)  Sphere For  0  f(z)  is  the sphere  g (0)  the c o l a t i t u d e .  r  = tan%9  = R  Q  o  ;  9.(9) (p  ;  h(r)  = ( i + yU)  Mr )  K  =  =  R^T  JT+YTJ  .where  Then:  2cos %0  =  =  c  o  s  8  ;  r  p  '  (  r  ;  2  2  P(r)  = RsinB  )  =  4r (1 + r ) 2  =  -  s  i  n  2  e  (II.2.24)  12 II.3  Coordinates  for  a Thin  Consider,  now  a fluid  having  similar  Mi  M  and  fluid  is  2  is  topological  much  z.  requiring surfaces  on  z so  each  that of  x,y  that  two  their  radii  on  such  g  the  surfaces, The  of  is  z z  line  the  Mi  distance  M  3  g  z z  depth  the  nearly  are  2  be  x,y  surfaces  z=0,z=l.  (x,y)  element  are  are  of  The  are  harmonic. constant  x,y  defined  orthogonal  in  of  z z 9z  the  to  +  thin  3  g dy yy  +  2  J  the  independent  s  g zz  dz  z coordinate of  II.3.1  2  can  be  chosen  z.  n.  fluid,  the  '•  1  1  <<  by  k*(x,y)  is  3.2  then:  1 /g  k*(x,y) 0 The  thinness  comparison y  2  between  curvature:  3g  The  and  form:  3  is  M  that  z=constant  constant  2g dxdy xy  fluid  and  them t o  The has  +  2  3  Mi  Mi  surfaces  of  z.  then  = g dx xx  2  the  lines  constant  Since that  of  the  coordinates ds  so  than  s i m p l i c i t y we c h o o s e  coordinates  these  by  characteristics.  coordinates  a coordinate  For  bounded  "thin". Choose  Choose  smaller  Film  1  dz  zz  /g 3  of  the  fluid  with  the  distance  appreciably.  ?  II.3.3  1  ZZ  requires over  that which  the a  depth  g ,g „ , g xx' xy yy a  a  is  small vary J  in  13 zz g 3  grad  3  3  'XX  'zz  grad  g xx  <<  g  <<  'xy The  z component  Therefore  g  v  9  v  and  g xy  n  r  a  d  k*  It varies  3h 3x" will  <<  (II.3.4)  3z  independent are  of z.  harmonic: (II.3.5)  h (x,y) 2  components 3 8x  of grad  y h(x,y)  +  «  2  be a s s u m e d  only  over  1  I Hi*, h 3y  approximately  9 8y  the f l u i d  k * 3h_ F " dy  are then  is  (II.3.6) thin  become: ( r i . 3.7)  1 that  the depth  distances  much  of the  larger  fluid  than  the  i.e.,  h  3x  <<  Equations whe r e b y :  ~  y y  that  1  also  appreciably  depth:  9  x ~ h(x,y)  the requirements  W  k*(x,y)  vanishes  horizontal  hor  is:  are approximately  '  )  1  (II.3.3)  Mi t h e ( x , y ) c o o r d i n a t e s  x  <<  yy  on  J  g  yy  zz  ( ' y  2  g  Xjf  nearly  a  The  and  ~  grad  1  9z  , g . . ,g__,  v  since  xx  xy  /g—«  XX  Moreover,  5  o f grad  g r a d.  'zz  1  II.3.2.8  <<  suggest  1  (II.3.8) a perturbation  scheme  14  where the  g  x x  (x,y,z)  = h^x.y)  g  x y  (x,y,z)  -  g  y y  (x,y,z)  = h (x,y) + A g  g  z z  (x,y,z)  = Xk (x,y)  + l9[  a small  parameter  which  A is  vertical The  k*(x,y)  1  X g ^ U . y . z )  2  Z)  z  scales  of the f l u i d ,  = Xk(x,y)  1 y y  2  )  +  2  and h o r i z o n t a l depth  + A g ^  to  ( x ,y , z )  +  ...  (II.3.8a)  ...  (II.3.8b)  > (x ,y , z )  +  (x ,y , z ) +  measures of the lowest  ...  (II.3.8c)  ...  the r a t i o  (II.3.8d)  of  system. order,  is (II.3.9)  15 II.4  Vector In  in three  will  always  be d e n o t e d  (e.g.,  letters  subscripts  by  lower  case  components  then  will  be  the l i n e  ds  = hfdx  v  x  -  h l  case  components  letters  by  F o r t h e most  with  upper  V^) , a n d p h y s i c a l  v^).  element + hfdy  2  case  components part  physical  = ^  , v  is: + h dz  2  2  (II.4.1)  2  3  components  V*  Contravariant  used.  If  the vector  (e.g.,  c a n be  components,  by u p p e r  1  (e.g.,  a vector  components.  V ) , covariant  letters  system  contravariant  and p h y s i c a l  superscripts with  coordinate  ways:  components,  components  Notation  any o r t h o g o n a l  represented covariant  Components:  are r e l a t e d  y  -  h V* 2  - ^  by: , v  -  z  h V 3  z  =  ^ (II.4.2)  Covariant and  the symbol  differential  derivatives  V  operator  8a  for  any v e c t o r V  2  =  will  a^ .  JlL- +  will  be d e n o t e d  be r e s e r v e d  defined  so  f o r the  by  semi-colons  two-dimensional  that:  8a  Also: (TT  44)  16 III.  IDEAL  III-l  Field An  stress The  FLUIDS  Equations  ideal  f o r an I d e a l  fluid  is  c a n be d e r i v e d  equation  defined  Fluid  t o be one f o r w h i c h t h e  as t h e g r a d i e n t  describing  momentum  of a s c a l a r  conservation  is  function then  n  (in  c o v a r i ant form) :  9  V  i  . „k +  dt where  V\  velocity  V  i;k = "  V  and  V  fields  n  ; i  (III.1.1)  are the c o v a r i a n t  1  respectively  and c o n t r a v a r i a n t  and s e m i - c o l o n s  denote  covariant  d e r i v a t i ves . The  f f  where  + (PV ).,1  is  p In  replace is  equation  o f mass  density  (III.1.2)  of the  of vortices  (III.1.1)  is:  = 0  t h e mass  studies  conservation  i t is  fluid. most  convenient  by t h e v o r t i c i t y e q u a t i o n .  to  The v o r t i c i t y  defined by:  = £  w' 1  i n k  v . g" ' i  k  (in.1.3)  2  n  ink where e  1  by  2  3  is  e  = 1 , and  taking  with  e  i  n  the antisymetric g = d e t g. . .  the c o v a r i a n t k  and using  tensor  density  An e q u a t i o n  derivative (III.1.2)  of  for  having W  (III.1.1)  to e l i m i n a t e  1  is  obtained  contracting V  1  . : »i  17  3t  (III.1.2),  {' P  Note  a r e now r e g a r d e d  as t h e  of motion.  (III.1.4)  may be w r i t t e n  i n terms  of  derivatives:  k 3 3x since  (III.1.4)  ;k  and ( I I I . 1 . 4 )  equations that  i  p  IP J  (III.1.3)  fundamental  ordinary  J  W  + V'  the terms  F  ,w\ ^ P  '  W "  k  P  in the connections  3V  1  3x  k  ( I I I . 1 .5)  r .. c a n c e l Jk 1  18  III.2  The K e l v i n In  earlier  1869, Lord  work.  definition  Circulation Kelvin  H i s most  reformulated  important  of the c i r c u l a t i o n  Circulation Let  much  contribution  and h i s p r o o f  of  Helmholtz  was t h e  of the  Kelvin  Theorem. C be a c l o s e d  C(t)  = {x  (s,t)  contour  x (s,t)  , k = 1,2,3  of  s  and  such  all  t  t  in  the  fluid:  : 0 < s < 1}  where  .  Theorem  a r e smooth  that:  The c i r c u l a t i o n  (III.2.1)  x (0,t)  = x (l,t)  k  around  real-valued , k -  k  C  is  functions  defined  1,2,3  by: (III.2.2)  V (x (s,t),t)^-(s,t)ds 1  o  If  C is  k  advected  9  x  3t  then  i t c a n be p a r a m e t r i z e d  such  that:  (s,t) = V (x (x,t) ,t) k  (III.2.3)  i  There fore : dr.. n ^ C _d_ q dt dt J  . . k v (x (s,t))^-(s,t)ds 1  k  ^ ( x \ t )  V  +  (x ,t)| (s t) 1  k  ;  n  f  )  ,2..k  '1 ,o  -V (x ,t)V,. (x ,t) k ;n n  i  i  r i  ds  +  n. (x\t) k  3x •(s,t) 3s r  + +  V  (x ,t)V (x ,t) i  k  ;  n  n  i  V (x ,t)fL(x ,t) i  k  i  for  ds  3S  •(s,t)  19  = nU^i.tht) -  Thus, This  - nix  the Kelvin  ( o , t ) ,t)  (V V )(x (0,t),t) k  i  k  the c i r c u l a t i o n around is  1  +  (v v )(x (i,t),t) k  = 0  any a d v e c t e d c o n t o u r  Circulation  Theorem.  i  k  (III.2.4)  is  conserved.  20 III.3  The T h i n In  introduced powers  this  parameter  approximations  The f l u i d i s " t h i n "  c)  is  o  equations  in  A of Sec.II.3.  so t h a t  a r e as  follows:  the approximation  of equations  = p  +  Q  i s nearly p  ( 1 )  constant:  (x,y,z,t)  + ...  (III.3.1)  constant,  We d e f i n e : v  =V h x  x  X  where  V  , v  =v h , v y  y  V ,V ,V J  To l o w e s t  so  to this  order,  physical  velocity It  velocity  ( I I I . 3.2)  2  i s assumed  little  v  order x  of the contravariant  i n X, t h e m e t r i c  , v , v  z  is  diagonal  a r e t h e components  ofthe  field.  i s small  varies  ='v k*  a r e t h e components  field.  that,  z  z  velocity  and  of the f i e l d  t o be i m p o s e d  of the f l u i d  p(x,y,z,t) p  of approximations are  may be u s e d .  The d e n s i t y  where  a number  the expansion  of the small  (11.3.8) b)  Approximation  section  allowing  The a)  Film  that  the v e r t i c a l  i n comparison with  v (x,y,z,t) x  = v  with  component  ofthe  the horizontal  velocity  height. 0 ) x  (x,y,t)  + X v ^ > ( x ,y , t ) 1  + ... (III.3.3a)  v (x,y,z,t) y  = v  0 ) y  (x,y,t)  + Av  ! ) y  (x,y,t)  + ... (III.3.3b)  v (x,y,z,t) z  = Xv  ! ) z  (x,y,t)  + A v 2  2 ) z  (x ,y , z , t ) + . . . (III.3.3c)  Similarly,  vorticity  components  are defined:  21  w  = W h  ;  X  x  (III.1.3)  w = W h y  and ( I I I . 3 . 3 ) v  (  1  then  X  w  v 1 •i i-+:-Xw[  f -  9  (  1  w = W k* z Z  (III.3.4)  imply:  )  w  1  ;  y  ^ ( x - y . z . t )  +  (III.3^5a)  + •••  )  (x,y z t)  1 )  5  3(hv[ w  0 )  )  l  +  l  3(hv<°')  3x  ...  (II1.3.5b)  + ^l^U.y.z.t) +  3x  (III.3.5c) Written  e x p l i c i t l y in  terms  of  w ,w ,w x y z  becomes w _3_ X 3 t P.h I i w  f X + ir- — w h 3x Ph I J  3 3x  fw ] y 3 t Ph  ' V  x h  v  x h  w _ x _3_ ph 3x  JL 3t  w. Pk  which  W  ~  f  + ph 4- 3y- i ~ VI 3 y 3x ph  J  _L h 3x Ph  JL  order  X  h  I  w  »  + _?. _L pk 3z r  w  ~  V  X  h  X  h  J  (III.3.6a)  >  w. 3y  w z _L h 3z Ph  J L  w. 3y  in  (III.3.6b)  pk ' 3 z  + JL JL h 3y ph  w ph  Ph  \  V  v  ..+ _z _L k 3z  X  (III.1.5) '  w v w + JL JL y • + -L _ L y k 3y Ph h 3y Ph v  ph  to the lowest  ^ 3 h ay  +  w  ^ _3_ ph 3x  w  N  .  v  ,  ph A becomes:  3z  (III.3.6c)  22  _3_ w. 3t  To  lowest  w. h  order  3(hkv  3x  h  the equation  0 ) x  )  9(bkv  3x  The w ^  equation of this  completely  v  A  u  The  equations  this  order  0  order.  for w  of  x  the production  In  form:  in  that  i t is  approximated  necessary  assumed  that  to a r b i t r a r y  x  so  analyticity  of f l u i d  through  at the boundaries;  v °> = v that  y  ° > = 0 a t z the only  conditions  is  will  *  w z  be d i s r e g a r d e d  to lowest  remain  to  solution  the t r i v i a l  conditions  fwhich  is,  require ((  )  with v ^ =  may be  function,  are that  " "  l , b u t ^ 3z  compatible  for  i n the term  .There  conditions  solution  be o f t h e  b y an a n a l y t i c  0  =  at the  any f u n c t i o n  may be a b s o r b e d  no s l i p  A l l physical  order.  valid  function  any b o u n d a r y .  = 0 a n d z  be o m i t t e d .  (Since  accuracy  The assumed b o u n d a r y  slip  a n c y  w  a l l boundaries  analytic.  order  flux  v  A.  x #  f o r some  from  no  x  v x ', v y a n d w_ o f o r d e r  ( 1 1 1 . 3 . 3 a , b ,c)  departures A).  >  v  equations  in  may t h e r e f o r e  y  to  i s , changes  be a s s u m e d t o be t a k e n  order  is  according  y  changes  f ( x , y ) + 0 ( A ) = const,  simplicity  X  approximation.  will  boundaries,  (  and w : t h a t  x  H e n c e f o r t h , the s u p e r s c r i p t s quantities  o f w ° > and  of approximation, of w  and w  is  (III.3.8)  However,  order  induce  (III.1.2)  conservation  = 0  independent  w x ,* w y o f o r d e r  (III.3.7)  )  expressing  ( 1 1 1 . 3 . 5a , b) , t o t h i s are  Q ) y  = 0  3z  o f mass  3y  are also  y  w  ^  3z no s l i p 0.  of  there  however, that = 0 boundary  Notice  that  is  23 since  v z  ^  »  0  =  a n c  '  surfaces  z = const.,  with  flux  zero  The reduced  to  boundaries independent  t  through  problem of  n  upper  e  the  lowest  these  of  z  than .  w  x  lower  order  boundaries  solution  is  are consistent  surfaces.  finding  a two-dimensional (other  and  the  fluid  problem and  w  y  motion  since )  are,  all to  has  now  fields lowest  been  and order,  24 IV.  VORTICES In  in  are  which  f i l m of  equation  of  the  3(hkv )  9x  3y  general  mixed  real  partial  =  defined  except  at  = 0  z  (x',y')  r  that  1  1  if  ,  2TTY  :  second  flow. to  be  The the  ;  by  the  k(x,y)?  (III.3.8)  that  is,  9y  2TTY<S  is:  1.1)  y  (IV.1.2)  j  having  continuous  ip(x,y) field  incompressible the  is  of  a  the vortex  velocity  region  of  (IV.1.1)  field  flow,  D ,  and:  x  ^  6(x-x')6(y-y')  (IV.1.3)  a contour  (x-x ' ) 6 (y-y  1  3(hv )  3(hv )  9x  8y  y  '  satisfying  9x  is  u  velocity  with  then:  =  function  order.  throughout  9(hv )  C  9 x  ^(x,y)  3(hv^  2  (x ,y )eG  the  of  everywhere  w_ = h r-T  Notice  hk  derivatives  at  w  described  of  (IV,  function  for  having  a system  0  9y  valued  is  depth  of  is  solution:  streamfunction (x',y')  fluid  thesis  y  hk  some  the  motion  conservation  = _L i i ( x , y ) for  of  of  Field  mass  3(hkv )  has  problem  ideal  and  Velocity  x  main  equations  h(x,y)  Vortex  The  the  the  a thin  function The  FILMS  section  what  N vortices  IV.1  THIN  this  addressed:  metric  IN  ) dxdy  x  dxdy  interior  G  such  that  Thus, Y is  by  the  Kelvin  c a l l e d the  (I V . 1 . 3 )  one  C i r c u l a t i o n Theorem  vortex  strength.  x , y ;x  unit of  strength  1  ,y ' )  at  is  (x  the  region  1  of  1  flow  is  for is  1  a vortex  of  a Green's  function  operator.  {(x,y)eD>  where  the  boundary  is:  boundary y =  M U 3-D i=0  3.DH3.D =  conditions  for  is  3-D  often =  ^  Theorem  ^  1  a  i  given,  D  assumed, o  that  ¥ ( x ,y ; x ^ , y . )  const,  although  =  implies  .  (IV.1.6)  on  are:  3^ D, i =0 , . . . , M ;  i = l , . . . ,M  i l»---»M  r  <j> , i / k  K  (unspecified) r  r  s t r e a m f u n c t i on  differential  1  1  It  into  (IV.1.5)  1  a self-adjoint elliptic  3D =  The  (IV.1.2)  = -2TTS ( x - x ) 6 ( y - y ' )  x,y;x ,y )  D  constant.  Substituting  ,y ).  The of  a  has :  £ v v ( x , y ; x ' , y ' )] where  y is  the  •  (IV.  this  Notice  boundary  is  not  that  necessary,  the  conditions  Kelvin are  1.7)  that  Circulation  constant  in  t i me. The (IV.1.5)  existence  and  the  boundary  ^(x , y ; x ' , y ) 1  is  well  known  demonstrate  and  =  (Courant  the  ^ and  existence  uniqueness  of  ¥  satisfying  conditions: = const, Hilbert and  on  3.D,i=0 , . . . , M  (1962)).  uniqueness  up  We u s e to  an  (IV.1.8) this  to  additive  m  constant  of  ¥  Set  ¥  M -tuples, of  the  space  satisfying = 0 and  0  ( Y ^,. . . , f ^ )  boundary of  of  map  f:  the A-*B  having  1  prove  constant  Y  under  the  to  arises  necessary  to  to  possible  the  B be  and  is  Y  unique  an  show  a unique  and  3D  to  the  r  does  that  (r_.  g_  equal  D  Clearly up  to  f  an  n  is  (the  to is  it  linear.  is  only  freedom  fix  Yo  ,...,I\  n  d  the  additive  (IV.1.7)  not  There  M  )  of  ).  the  Thus (0,...,0)  =  U  pre-image.  Suppose (IV.1.5)  vector  on  d •U  has  values  unique  = l  inverse  one  the Y F..  .  conditions  when  the  of  (IV.1.7).  finding  setting  this  has  A  corresponding  condition  for  f  space  Let  )  (IV.1.7).  vector  conditions  boundary that  conditions  M  , and  0  exists  constant  only  3 D  9^D  prove  Q  g  boundary  boundary  around  that  necessary  is  (IV.1.8).  , d e f i n e d by  3.j D, i - 1 , . . . , M, Y = 0 on circulation  the  corresponding  condition  values  y ( x ,y ;x ' ,y )  it  consider  M  a natural  additive  boundary  M- tirpl:e.s, ( r ^ , . . . , r i  possible  To  the  ¥ ( x ,y ;x  (IV.1.7)  k  9  1  ,y )  and  1  with  n  Y * (x ,y ;x  1  ,y ) 1  satisfy  r\ = 0,i = l,...,M. 3.U dxdy D n  lyjy-y*)  Then:  » v ( d x d y (IV.1.9)  where left  denotes a d e r i v a t i v e normal t o the b o u n d a r y . dn. s i d e v a n i s h e s : , s i n c e from ( I V . 1 . 2 ) and ( I V . 1 . 7 ) :  3  (Y-Y*)  h.D 1  k  3  n  (  ¥  _ y * )  d  s  =  c  o  n  s  t .  xT„ a  = 0,  n  i  i=1  The  M  U  (I.V.I.10)  27 a n d when  i  the  of  right  = 0  0  (IV.1.9)  on  vanishes  9 D 0  by  Iy(Y-Y*) • V(Y-Y*)dxdy whence  V ( Y-Y*) Thus,  image  and  (IV.1.5)  = 0  and,  r  r  therefore and  there  (IV.1.7)  term  on  Hence:  (• I V. 1 . 1 1 ) on  (0,...,0)  =  is  first  (IV.1.15).  V = Y*  exists  which  The  = 0  since  ( 3D»---» 3[))  .  0  has  a Y( x ,y ;x  unique  9 D  1  up t o  , Y = Y*  a unique  ,y ' )  .  pre-  satisfying  an  additive  constant. Y ( x , y ; x ' ,y ) a l s o  has  1  V(x,y;x' ,y') and  can  the  reciprocity  property:  =• Y ( x ' , y ' ; x , y )  be w r i t t e n i n  the  (IV-.1.12)  form:  V ( x ,y ;x ' ,y ) = - A ( x ,y ; x ' ,y ' )lnr + B ( x ,y  x',y )  1  1  ( I V. 1 . 1 3 ) with  r  =  D if  k is  [(x-x ) 1  analytic  circulation 2TT  +  2  around  f  =  27T  (y-y ) ] 1  (see, the  I 91  k  9r  2  for  and w i t h  A and  B analytic  example,  Sommerfeld  contour  r = e  is:  2 ^ A ( x ' , y ' ; x ' ,,y' ) k~(7 Ty T  as  small rd6  2  r  _ T  in  (1949)).  The  e + 0  whe nee A ( x ' ,y  1  ; x \ y ' ) = k(x' ,y' )  Substituting as  r  (IV.1.13)  into  (IV.1.14) (IV.1.5)  one  finds  that  0  _9_ 9r  , +  1 9A k 97 -  „ 0  (IV.1.15)  2Q when ce : VA(x' ,y' ;x' , y ' )  The with I  total  positions  = %vk(x',y' )  streamfunction  (* ,y ) n  (IV.1.16)  f o r a system  and r e s p e c t i v e  n  of N  strengths  vortices  y ,n = l,...,N n  s: =  <r(x,y)  n  n  where  ty*{x,y)  flows  (e.g.,  N z Y Y ( x , y ; x ,y ) + n n n  =  is  a uniform  and H i l b e r t  (1962))  The s p e c i a l (e.g.,  Let  be a c l o s e d  vortex  lies  tacitly  of D to i n f i n i t y  surface C  ty*(x,y)  must  case  a sphere)  on  presents  and w i l l  of vortices  i s , however,  contour  C .  assumed t h a t  on s u c h  Denote  no r e a l  n o t be  what  is  by  C  e  x  t  the e x t e r i o r o:h(v  o f what  i s , of course,  dx + v d y ) =  c  is  such  C. . l nt  unbounded interest. that  no  and i t s  t h e i n t e r i o r and  arbitrary).  Then:  K  dxdy J  considered  a surface, by  bounded.  problems  of p a r t i c u l a r  its interior  (the choice  D is  on a c l o s e d  J  exterior  satisfy:  (IV.1.18)  f a r i t has been  extension  further.  stream).  imposed  o  So  (Courant  (IV.1.17)  t h e s t r e a m f u n c t i on due t o o t h e r  V,"  The  t*{x,y)  1  Cint  =  s Y n z „ne C~ i n t ( I V . 1 . 19)  29 Integrating -o h ( v C  around  dx + v dy)  J  C in =  Z n  y  the  other  d i r e c t i o n one  y  c  finds:  (IV.1.20)  ext  whence E y n=l  = 0  Thus, The  velocity  must at  be  (IV.1.21)  n  there  cannot  and  1  a single  vortex  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  i n t e r p r e t e d as  (x'y )  be  one  of  that  due  negative  on  such  a  surface.  ¥ satisfying  (IV.1.5)  to  a vortex  of  unit  strength  unit  strength  at  infinity.  (Note  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 a n y o t h e r point:  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 . )  position vortex  of  at  put  streamfunctions  infinity  If (we  N such  the  satisfying  the  disappears.  depth  of  the  fluid  is  constant,  simplicity)  and  x ,y ; x ' , y  of  the  Laplacian.  ¥ can  then b e - w r i t t e n :  Y(x,y;xiy')  = -Inr  Sommerfeld  1  )  k is  for  e.g.,  super-  (IV.1.21)  k= l  function  (see,  Upon t h e  constant  becomes  the  + B(x,y;x',y')  (1949)),  whence,  Green's  (IV.1.22) upon  comparison  with  (IV.1.13) A(x,y;x' ,y') Moreover, are  no  boundaries,  = 1  (IV.1.23)  since:  V £nr  =  2  2TT6  (x-x ' ) 6 (y-y ' ) , i f  there  then:  B ( x , y ; x ' ,y') = 0 ; ^ ( x , y ; x ' , y ' )  = -h&n [ ( x - x ) + ( y - y ' ) ] 2  2  (IV.1.24) When  the  flow  induced  constant  fluid by  there  depth  is  constant  B to  be  due  is  no  such  to  one  the  simple  can  therefore  boundaries.  decomposition  If of  regard k is the  the  not  flow.  30 IV.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 The m o t i o n  of the v o r t i c e s  i s governed  by e q u a t i o n  (III.3.7): _3_ 3t which  h  implies  Zl  3x  that  remains  n  W  of  that  (IV.2.1)  i s , each  zl  of a vortex  system:  i . e . , the s t r e a m f u n c t i o n  sati sfies:  but t h e  x  N = -2i-2  V^J  F  and  n  c a r r i e d along  y  Y 5(x-x )fi(y-y ) n  n  (IV.2.2)  n  are time-dependent,  n  moving  t h e v e l o c i t y o f each  expand the v e l o c i t y f i e l d near i t s s i n g u l a r i t y . (uniform It  depth  will  fluid)  prove  is treated  convenient  variable  z = x + iy  variable  z  of Section  are then  z  and  symbols  (there  should  II.3).  z-= x - i y .  in arguments.  x  -  Equation  1 v  y  2i h ( z ,z)  3z  The c a s e  first.  be no c o n f u s i o n  The i n d e p e n d e n t  with the  variables  For s i m p l i c i t y the present  (IV.1.2)  ^  v o r t e x we  t o i n t r o d u c e t h e complex  are r e t a i n e d f o r a l l f u n c t i o n s  v  as i f  by t h e f l o w .  In o r d e r t o d e t e r m i n e  k=l  fluid  c o n c e n t r a t e d and t h e f l o w r e t a i n s t h e  characteristics always  = 0  as i t moves a r o u n d i n 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  fluid.  (x^,y )  rW.  is advected:  -rH  e l e m e n t -mai n t a i ns i t s v a l u e the  z 3 h dy  +  is  (z,I)  despite  their  change  then: (I V . 2 . 3 )  31  Which,  upon  substitution  (IV.1.17)  of  N  r  (IV.1.22)  and  -i.y .  3B(z,z;z.  z-z,.  T  k  + If  |z  - z  is  n  ,z. )  3z  2 i ^ ( z , z )  \  (IV.2.4)  small : 8h n - Yn 8z  •y. TV  -  becomes  y  z-z  +  +  n  i|>*(z,z)  97  2  Z Y k^n  , k  +  8z  Y B(z,z;z ,z ) n  i'(z,z;z  K  -1  dh  (z-z ) V n'  _ 1  z k  )  }  k  n  n  +  ° d - n z  z  z= z (IV.2.5)  where  h  = h(z  n  The  n  ,z  ) n'  velocity  field:  concentric  about  z  a  to  vortex.  velocity The  flows the  velocity arise By  velocity  radially  vortex  the  .  n  either.  field  Prefering  field  from  z It  should  n  not  projecting  the  flow  onto  introduced  The  velocity  giving field:  i v  seem  contain  does  -  z  no  hence  might  v_  are  v  and  because  terms  i v. y  x  =  y  h  (  v  cannot  ,z  n  V«_v = 0  plane to  v  it  cannot  3h  a  that  such  as  )  {z-z  velocity a  radial  =  They -v/V£nh  source  terms.  to  vortex  this.  but  "fictitious"  the  - 1  impart  1 ^ )  impart  terms  is  )(z-z ) n' n'  i"Y ^ )  paradoxical  satisfy  rise  v  direction  radial  the  hjz  V  x  "  i  y  v  = hf f  ^  3h"  1  n  ^  32 +  2  Z Y Y(z,z;z k?n  +  uniform  and t h e r e f o r e  carries  4>*{z,z) +;  £  ,z ) • z=z z= z„  k  K  is  i  Y. B(z,z;z ,z ) n  induce vortex  other  terms  any motion is  vanish  the vortex  as  in the vortex  z  at  +  2  at  z  z  therefore:  ° that  s n  .  n  with  3  they  cannot  The v e l o c i t y  of the  J  +  ifi*(z,z)  Z k^n  alonq  n  1  9h  +  n  r  it. All  n  Yk^Cz.zjz. K  k  ,z. ) . k  Y B(z,z;z ,z ) n  n  n  (IV.2.6) r z=z, z = z  But u = h x x n n  ,  u = h y y n^n  ( I V. 2 . 7 )  Therefore : i l _ L nr, - £nh(z,z) h 9z n 2  +  or, r e v e r t i n g  + ip*(z,z)  r  Z k^n  Y B(z,z;z ,z ) n  n  z.) • z= z •n  y.V(z,z;z K  +  k  (IV.2.8)  k  z = z  n  r  to (x,y) coordinates:  3n. h (x ,y ) 2  n  n  -1  3y  h .(x ,y ) 2  n  = n y=yn x  x  n  ^ n 9x x =xn y yn =  (IV.2.9)  33 with Y fi  =  n  ^-Jinh(x,y)  ;  +  These  Z k^n are  +  T ^ * ( X ,y)  y B ( x ,y ; x  +  n  ,y  n  )  p  Y . V ( x ,y ;x. ,y. ) K  K  the  (IV.2.10)  K  equations  of  motion  for  the  vortex  system. If  there  are  no  boundaries  then  from  (IV.2.8)  and  (IV.1.24)  if  = X h  n  Example:  on  the  method  n  order  vortex  to  (From to  This  which  in  and  is  its  , and  8 z  r  "  u  1  (IV.2.9-10)  determine  also  be with  vortex  strength  of of  in  south  The  the  harmonic pole). and  z  harmonic  coordinates:  n  -  '  •  ii p  -Yh'(r') 2h(r')  at -y  Since  of  i  "  by  an  give  of  a  the vortex  alternative  (x',y') at  the  there  ? , )  -y(p(r')-l) 2r'  the  corresponds are  no  ij;* = 0  depth, vortex  on  infinity.  coordinates  - hTT^7rT Tr' '-  y  velocity  y  uniform  velocity  indeed  first.  strength  boundaries .  the  do  determined  agreement  the  no  k =  polar  we  can  infinity  flows,  u  a  , i.e.,  % " '" in  (IV.2.11)  n  that  counterpart  (II.2.4),  external  or,  check  sphere.  0 = ^  B = 0  h  velocity,  Consider sphere  n  z  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 a n d U n i f o r m D e p t h In  correct  *  , , k/n V  is  ,  therefore:  < - ' ' IV  2  12  (IV.2.13)  34 Using  (II.2.4) :  u r  This  = %  = 0  c a n be d e r i v e d  ;  u„ p = £ Rt ^  a n  (IV.2.14)  ^  alternatively,  for this  case  only,  is  zero,  as  f o 11ows: Since consider Y  the v e l o c i t y  t o be t h a t  vorticity upon The  t h e sum o f a l l v o r t e x  such  that  superposing velocity  there  is  o f each  owing  the vortex  ordinary  has  vortex  pole  1 Rsi n 0  w  •3( v  evaluated  and t h e  to  one g e t s  ay . zero  completely  Then vorticity  symmetric  of the sphere;  motion. then  strength  hence,  The v e l o c i t y  satisfies  field  (using  sin-9)  3v„-,  3 i<J>  89  •ay (IV.2.15)  solution  si n 0 If  is  of  coordinates):  0  which  fields  vortex  at the south  polar  and equal  t o t h e symmetry  no s ^ l f - i n d u c e d  vortex  incompressible  constant  the v e l o c i t y  field  i t s core  of a single  i t is  i s everywhere  about  of  field  strengths  v^  is  t o be b o u n d e d  by r e q u i r i n g  • 2 iry  Li m  f  2TT  cos 6  at  B const.  0=0  , 6=1  (IV.2.16)  a  is  that:  VxRsin0d<j> = L i m - 2 i r a Y R ( 1 - c o s e ) 2  0+TT  •4fTaYR  0+7T  :  (IV.2.17)  35  Therefore : 1 a = 2R  (IV.2.18)  2  and  (IV.2.19)  v  The given  by  velocity  (IV.2.14)  of the vortex  since  the  vortex  at the south  (Of  course,  the v o r t e x  that is  at  (IV.2.19) the  is  pole.)  only  i t  is  pole.  at  carried  at  is  1  along  in  The two methods  at the south correct  (e ,^')  pole  is  the flow are  also  the i n s t a n t  therefore  in  moving  this  of  agreement. so  vortex  36 IV.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 of Varying Depth The  is  more  owing  velocity  complicated  t o t h e more  V(x , y ; x ' , y )  .  1  meet  the upper  that  the boundaries  curved must fluid of  no l o n g e r  core  will  reason  curve  uniform  is  the  vortex  in order  (See F i g u r e  so II);  the v e l o c i t y  fast.-  T h u s , one  of a vortex  i n some way on t h e  in a structure  core.  velocity  as  in Section  field  II.2,  by e x a m i n i n g  in the neighbourhood  of  z^.  ( I V . 1 . 2 ) , ( I V . 1 . 1 5 ) , a n d (I V . 1 . 1 7 ) :  2i y  h ( z , z ) k ( z , z )  9z  -y  2i ( h k ) ( z  n  , z  A ( z  +  Y  "  n  n  l  'n  z  '  z  A ( z  ;  n  , z  n  ; z  n  , z  n  )  )  , z  n  : z  T h k ) ( z  n  , z  , z  n  )  9 v( h k ) v ( z  '  =  , z  n 9z  )  9 A ( z , z ; z ,z ) n n 9z \L Ln z=z  f z-z.  n  ^  z- z  n  3A(z  Y,  •  ,z  ;z 9z  n  to  an i n f i n i t e s i mal l y , s m a l l  infinitely  depend  in  perpendicularly  surfaces.  propagate  depth  depth  that  slightly  surfaces  ( 1358) t h a t  must  of  for this  b u t must  that  of .varying  of the s i n g u l a r i t y  are streaming  depth  We p r o c e e d physical  in a fluid  bounding  at the outset  of varying  its  straight  and lower  in a f l u i d  nature  by H e l m h o l t z  vortex  expect  that  The p h y s i c a l  is  was s h o w n  than  complex  core  It  of a vortex  ,z ) n  z=z z=z  n  x n  &  n  l " n z  z  )  the. From  Fig.  II  Vortex  Core  in  a F l u i d of  Varying  Depth  38 +  Y  Z  )  n  +  3(hk),, 3z W  " n' n (hk)(z ,z ) A ( Z  n  i *(z,z)  K  +  From  one  K  n  and  = h(z  n  3 Y B ( z ,z ;z ,z ) 3F n n n'  k  n  n  0(|z-z Un|z-z |)  (IV.1.14)  h  .  E Y | ^ ( z , z ;z. ,z. ) z=z k^n z=z  +  r  ,  (  v  (IV.1.16)  ,z  n  (IV.3.1)  n  n'  )  ; k  and  k(z  E  n  writing  ,z ) n n  v  (IV.3.2)  gets :  iv  k ' n n  -y  2i h k n n  y  I T ^ rnr'  +  ,Y  v  'n  n.  Y  3z  n  n  n  ,z  n  )  +  h  iii* v (  r  n  'z- z }  n 3z  n  n  z- n J z  k 3h _n_ n h 3z  n 3z  "2  B(z,z;z  ,  9k  Y  _JT_  ,  i  E Y vp(z,z;z  +  n  3k  Y  3k  !  n  z ,z )  .zjl-  n  k^n  z=z  n  Z-rZfj  +  As no  before,  velocity  to  the  0(|z-zj£n|z-zj)  the  terms  vortex.  of  the  term  in  carry  vortex  with  them.  source  of  section.  the It  n  difficulties is  remaining  &n|z-z |  divergent  The  as  are  term  outlined z->z  z- z  an d  z- z  The  exception the  in  (IV.3.3)  terms  at but  with  uniform  in  can  z - z„  and  the has  beginning a  the therefore is  in\z-z^\  impart  the  of  the  definite  n direction implies  (along  that  the  the  curves  vortex  of  moves  constant with  k  infinite  ).  It  therefore  velocity  along  39 k = const. small  but  surface and  If,  however,  finite  of  the  the  -iy r—n  3k r—  V n  carried  along  v.. x  is  assumed  c i r c u l a r core  core  eq u a l s :  it  in  iv.. ' y  velocity  Ine  radius due  to  .  The  n  Z Y.Y(z,z;z k^n  +  n  the  volume  advected  of  each (h  h  where  n n n  X  k  £  is  and S e c t i o n  = %  the  III.3.  increases  k  the  streamlines  of  not  concentric  (See  the  vortex  Section in  VI  a more  that  this  when  (X  the  appears of  0  small  As  calculation  +  this the  and,  core  the  uniform  is  therefore  h  ^  3z  n  (IV.3.4)  k  =  since  must  be  z  n  n  the  fluid  is  constant.  incompress-  Hence: (IV.3.5)  perturbation  parameter  0(k|V£nh|)) of  the  move  rigorous  manner.  t e r m may  be  a  in  direction  finite  Suffice  incorporated  is  n  shrinking,  term w i l l of  Section  approximately In  the  of  it by  a  be  the  centre  -Vk  .  cores  say,  However,  circular,  deferred  sized to  of  II.3  constant.  decreases.  III).  velocity effects  , and core  flow, while  Figure to  is  e X] n n k n  radius  the  at  a  ty*{z,z)  ,z.) Jz=z  k  z  ible,  +  n  k  is  has  flow:  n  w  term  boundary  Y B(z,z;z ,z )  Now,  , then this  ^-r-(-2~ -^-(Ane^-l) h k \ 2 3z  +  vortex  n  uniform  n  the  n  3 2  the  of  that  for  are of  The  until are the  "renormalizing"  treated present, :  Fi g.111  Streamlines in  a  Fluid  near of  the  Varying  Core  of  Depth  a  Vortex  41 i.e.,  by  replacing  a  n  =  a  n  by  •3  e  (IV.3.6)  n  whe r e 'cn * n n Y  E  is  c n  the k i n e t i c  velocity and  k  of  (IV.3.7) p  energy  the vortex  r e n o r m a l i z i ng  a  3  u  x  "  1 u  y  "  within  is,  nth  substituting  core.  (IV.3.5)  Thus, into  the  (IV.3.4)  :  n  2i . I • 'n h k [ 4 n  n 9z  n  f In  a* -2 h k n n  n  K  2  Y B(z,z;z ,z )  9z +  the  n  +  n  3z  ^*(z,z)  S Y * ( z , z ; z , ,z, ) k?n l An z=z  (IV.3.8)  k  K  k  k  J  =  n  whence,  using  (IV.2.7)  and s i m p l i f y i n g  the equations  of  motion  a re:  X  n  "  , 1 h k 2  n  9ft* n 9y  ;  y  -1 h k  n  d  90*  n 9x  2  n  (IV.3.9)  with: Y = -^  ft*  h (x,y)k(x,y) 2  .k(x,y)JU  + ijj*(x,y)  +  that  constant)  if  (IV.3.10) one p u t s  n  reduces k=l  y  n  B (x ,y ;x ,y ) ' n n' v  , J  s Y ^(x,y;x ,y ) k^n n  n  Notice  +  .  n  to  J  (IV.3.10)  n  n  (IV.2.10)  (within  an  additive  The that  the  circular that  velocity  core  is  cores  for  will  in  cease  the  approximations,  the  vortex  c i r c u l a r and  (IV.3.9-10)  estimates  of  periods  are  remains  general to  given  be  be  been  which  Section  derived  circular.  distorted  valid.  over in  has  Order  by of  However, advection  magnitude  (IV.3.9-10) VI.3.  assuming  are  good  so  43 IV.4  The  Vortex  It vortex  is  Streamfunction  now s h o w n t h a t  one c a n d e r i v e  a  generalized  streamfunction. From  (IV.1.14)  A(x,y;x' ,y')  and ( I V . 1 . 1 6 ) :  = k(x\y')  + &=f±  3  9k(x' ,y')  + iXzlll 2  k  (  +  3y  ^ '  Q (  y  2 ) v  ,  )  '  k ( x , y ) k ( x ' ,y') + 0 ( r ) ,  =  h  r  2  h  =  [(x-x ) 1  +  2  (y-y ) ] 1  ( I V. 4 . 1)  2  whence : 3A ( x , y ' ;x ,y) 3x  M (x,y;x' ,y' ) 3x |y=y and  a similar  both  obey  equation  in  the r e c i p r o c i t y  3B_ (x ,y ;x ,y ' ) 3x  [y=y -r—.  = c  n  be w r i t t e n  1 3ft Y h k 3y ' n n n n  . 'n  2  J  since  3B ( x ' ,y ' ;x ,y) 3y  x=x |y=y  can t h e r e f o r e  Thus,  Y  1  and  Inr  property:  1  (IV.3.9)  (IV.4.2)  1  =  in  the  (IV.4.3) x=x |y=y  form:  -1 3ft Y h k 3x n n n n  (IV.4.4)  2  1  whe r e ft  =  \l y +  2  x  k  *  k (x o  n  W k ^ V V V V ,y  )  *j ^ n ^ n '  +  ' h ( x ,y )k(x ,y ) ., n n n n  y  n  >n>n x  y  2  &n  17  J  +  2Y ^ * ( X ,y ) n n n' (IV.4.5)  }  44 is given may  by  be  C.C.  put  in  x = o where  x^  2-form,  is  of G  Denote cores  the  symplectic  As  is  vortex  shown  in  streamfunction  Appendix  A,  (IV.4.6)  is  the  vortex  exterior  The  {(x,y)  boundary  core  :  [(x-x )  by  3G  may  Gn +  2  n  a  symplectic  derivative.  streamfunction  fluid.  is  a  be  of  the  ( y - y j  The  related  2  ] *  region  nth  to  the  vortex  < e > of  is  cores  is:  fluid  outside  the  is:  assumed t h a t are  the  kinetic  e  energy  's  are  sufficiently  of  the  fluid  small  that  the  in  D*  is:  (v +v )h kdxdy 2 D* y 2  L  (IV.4.8)  disjoint.  The  2  2  x  p(  ?  k  I  rati 2 + 3x  since:  kinetic  (IV.4.7)  H  N D \ U 9G n n= l It  (IV.4.4-5)  form  Vft  V  =  its  (1943).  the  -1  the  n  Lin  of  a 2 N - d i mens i o n a 1 v e c t o r ,  and The  e n e r g3yj  a generalization  k  3x  'dip'  2"  M 3y  dxdy  Ik  9yJ  dxdy  ( I V. 4 . 9)  45 Applying  Green's  where  is  to  (IV.4.9):  (I V . 4 . 1 0 )  2  L  Theorem  3D  1  the d i r e c t i o n a l  derivative  normal  3D*  to  Si nce : N * and  k-^k^k^'  =  since  (using  <?  y  ;  k' k  x  y  = ds  3D  one  -  =  E.  n ds 3D  1  k  3n  d  s  (IV.4.12)  3n  N  n=l  first ty*  term  n  y  r  k  ^  f  3^  ds  k  the energy by  E^*  3 D * may be  i  n  3n  (IV.4.13)  3D*  be d e n o t e d  M  3D*  k  -ds  k=l  boundary  3D* =  y.  n= l  represents  and w i l l The  y  N  8 ¥  • p "  ^ d s  k  + p X  so  n  D'  ^  3D^  by  , y ))  n  has:  •*  The  x  -  r  ( I V . 4 . 11)  ^*( 'y)  +  ~ ^( x , y ; x  n  V l)rr  1  )  r  N  U 3.D u U 3G. lj =l U=l 1 J  due t o t h e f l o w  induced  .  decomposed.  i  (IV.4.14)  that  3D  Since const.,  ty*  *  is  k  3n  ds  =  i=0  constant  the f i r s t  N  z  on  k  3Di  3D  n  ds 3n  z  j-1  3D.n k is  3^„  ,4.  3G-  1 !Inds  and  t e r m on t h e r i g h t  +•  3n  constant.  ii _Ji k  3n  a  d s  s  (IV.4.15) " YDn r  Since  = !  e  46is  very  small,  ¥" *  Therefore  K  ty* ^ n ds  3G,-:  x , y ) on f] n n n  K  *  k  3n  k  3ri •ds =  3G„ . n  n  2TT^*(x ,y ) j  (IV.4.16)  j  S i mi 1 a r l y :  3D  *  M 1  ,  Y  3.  n  N  Y  {  3Y  n  =1 (IV.4.17)  and t h e f i r s t %  term i s c o n s t a n t  - Y(Xj,y..;x ,y ) n  on  n  .n ^ k I T -37T -3Gj  36^  as b e f o r e .  If  n^j , then  and:  y  If  n=j  n 3G  and n  but  *  d s  k^j  (IV.4.18)  ^ V ^ V V  one can p r o c e e d as i n ( I V . 4 . 1 2 )  k , whence  (IV.4.18)  holds  unless  n=k=j  to  .  transpose  S i n c e on  : k £ n r + B ( x , y ; x , y ) n n n n n J  Y  ,  J  (IV.4.19)  3Y  3G, '  2 7 T  1 3 17 3 f - n (  f  k  A  n  r  +  B  ( >y; x  x n  >y ) n  ende r-.e,  = -2rr(k £ne n  Therefore,  using  n  - B( x ,y ;x ,y ) ) n  R  n  (I V. 4.1 3,15 ,17 ,18 ,20) one f i n d s : N  f  E* = E, * + TTp £ { 2 y ^ * ( x 1 -  (IV.4,20)  n  k £n n  £ n  ,y ) + y B ( x 2  ,y ;x ,y )  N  | + T T P ^  k  ^ Y Y ^ ( x , y ; x , y ) + const. (IV.4.21) n  n  n  k  k  47 The  kinetic  energy  of a l l the f l u i d  is  therefore:  N E =  +  2TTP-U E  k.= l N  n=lL  which,  upon  2  N  V(x ,y ;x ,y ) n  n  k  k  r  YnB ( x n 2  +  2 Y k^n  x  ,y 17  n  Y ^"(x ,y ) n  n  n  +  n  substitution  ;x  of  ,y  17  n  7  )  +  y -^-ln 2  'n 2  a  n  const.  (IV.4.22)  (IV.3.7) and (IV.3.6)  for E  c n  be come s : E = E ^ * + 2-n-pft + Thus, of  Q is  the flow which  is  const.  proportional  to the energy  due t o t h e v o r t i c e s .  of that  part  4,8 IV.5  Conformal  Transformations  Since, function  of  Laplacian,  when  the  a single it  is  fluid  vortex  natural  depth  is  to  the  ask  a conformal  transformation  since  these  the  first  considered  by  Lin  tion  (1943)  the  who  vortex  surfaces under  N  denote  We now of  the  that  constant  function  vortex  and  under  the  in  the changes ,  question  more  conformal  transforms  of  z+z  This  later,  stream-  motion  coordinates,  invariant.  (1881)  was  generality,  transforma-  as:  dz  In  dz  n  (IV.5.1) n  transformed  show  of  the  the  that  depth  quantities  (IV.5.1)  remains  provided  the  K = 2iry  (Lin's valid  surface  is  on  and  curved  invariant  transformation.  Let  z = f(z)  coordinates. harmonic  showed  - h Z Y n=l  tildes  W = -27rfi) .  Routh  streamfunction  Si = where  by  Laplacian  constant,  Green's  how  under  leave  is  be  Notice  that  coordinates  ds  2  a conformal (x,y)  transformation  such  that  x +  iy  of  complex  = z  are  since:  = h (z,z)(dx 2  2  - h (z,z~)dzdf ~ TTzlT'TiT" 2  + dy )  =  2  h (z,7)dzdz 2  _ h (z T)(dx + f'(z)f'(z) 2  2  dy ) 2  , (IV.5.2) ?  )  Th us : h(z,z)  The  dz dz  =  streamfunction  n  for  $(2,f ; 2 \ Y ) r  =  (z,z)  a vortex  (IV.5.3)  in  ¥ ( z , z ; z ' ,z ) 1  the  transformed  system  is:  (IV.5.4)  49  and  ifi*(2,z) since  (IV.5.5)  r(2,z)  =  the Laplacian  is  invariant.  + B(z,l;z' ,f')  -ln\z-T'\  From  (IV.1.22) +  = -ln\z-z'\  B(z,z;z ,z') 1  (IV.5.6) o r: B ( z , z ; z ' ,z"') = B ( z , z ; z ' , 7 ' ) + "An z - z z-z  (IV.5.7)  The r e f o r e : B ( z ' , z ' ;z' , ! ) 1  B ( z , z ; z ' , z ' ) + In  = Lim z+z  B(z' ,z' ;z' ,z') In the  the transformed  transformed  °  =  k  y  =  Q  (IV.5.8)  the motion  ^ 1 J/nV<VVV'V +  dz dz  In  is  derived  from  steamfunction:  2  ^ n h  2  ( z  ,z  N  verifying  system  +  z-z' z-z  -  (IV.5.1)  h  Z  k=l  )  +  2Y  Y*(Z  +  * j [ y J B ( 2 , f ; z~ , t„ )  ,Z z  i  n  n  n  )  n' n);  dz In  dz  (IV.5.8)  .50 IV.7  Constants As  laws  J.M.  the  and  vortex  the an  Motion  symplectic  system,  streamfunction  ( 1969)).  depth  In  streamfunction  vortex  k  must  and  also  induce  all  positions. N  +  -^X  is  a constant  In  particular,  of ft  Consider  the is  the  = x.. + n  metric  invariant.  be  function  are  transformation  be  )  the  e.g.,  invariant  then  the  Symmetries  of  laws.'  a real  valued  function  |_G,fl]  (IV.  of  # -n y  V -  a  9G 9x n  n ^ n ^ n  G  ,y  n  (see,  which  Then:  r *r  Z  conservation  boundaries  conservation  G'( x i ,y i , . . . , x  d_G dt  are  invariant  particular, if  function  f l u i d therefore  there  infinitesimal transformations  infinitesimal coordinate  Let the  the  with  vortex  Souriau  under  the  any  associated  leave  h  in  of  9ft 3y n  9G  J  motion  itself  9 y  if  9G 9y n  h k ' n n n  J  n and  =  nonly  if  [G,ftJ  =  7.1)  0..  conserved.  infinitesimal  Y  9ft 9 x  '  nn  transformation:  = y n J  e  Y h k 2  'n n n  9G 9x n (IV.7.2)  for  some  small  transformation.  »ni  e  .  G is  c a l l e d the  generator  Then:  ..,5 »n ) n  n  fi(xi,yi  x y ) n >  n  of  the  51 N Z „ ,Yh k n=1 n n  =  for  some  -[G,ft]  9 x  n  n  3 y  3G  3ft 3 y  3  n  + ou ) 2  V  + 0(e .)  (IV.7.3)  2  e  small  3G  dtt  1  1  .  Examples: a)  If  hEh(x)  x = const, The  , kEk(x)  then  generator  ft  of  which  is  plane  and  is  of  b)  hsh(r)  r  If  £ ^n  = x n  are  x  n  e  -  J  is: (IV.7.4)  2  n  conserved.  constant  , ksk(r)  then  =  curves  . n = y 'n n  n  When t h e  this  yields  surface  the  of  flow  conservation  circulation.  = const,  ^n  boundaries  under:  transformation  h (x)k(x )dx  n  therefore  centre  only  (  Z y  k  the  invariant  this  N  G =  is  , and  +  £ y  ft  is  n  '  , and  all  boundaries  invariant  n  n  =  y  n  "  £ X  under  n  The r e f o r e  /  are  curves  is of  a  52  y  which  can  be  "  '  ( I V  - - ' 7  5  rewritten  2Y h'(r )k<r )r ) n  n  n  (IV.7.6)  ^ = 0  ;  n  Thus N 2 Z y n=l  G =  is  conserved.  yields  the  2  For  )k(r  n  /  )r n n  v  dr  flow  in  the  conservation  of  moment  Vortex another  h (r n  systems  conserved  in  the  quantity  n  plane  known  with  of  plane as  (I V . 7 . 7 ) \ /  -  k = const,  this  circulation. with the  k = const, angular  exhibit  moment  of  ci rcu1ati on: N  ^Y (x y -x y ) n  One  v  :  can  n  n  to  Suppose  and  Physically  h  h(ax,ay) this  means  transformations preserved  under must  ft(axi in  order  that  appropriately original  =  curved k  surfaces  are  a h(x,y)  , k(-ax.ay)  that  fluid  is  .  the  y  the  n  motion  system.  follows: functions  of  = a k(x,y)  .  v  invariant  then  order  under  scale  boundaries  are  the  stream-  vortex  also  as:  ,ay!,... ,ax ,ay )  scaled  If  transformation,  transform  the  as  homogeneous  ( x ,y)->-(ax , a y ) the  (IV.7.8)  = const.  n  generalize  i.e.,  function  n  = a fi(x, ,y u  n  in  time)  the is  x  ,...,x ,y )  transformed  similar  Differentiating  by  n  n  system  +  b(a)  (with  to  the  motion  a  and  then  in  the  setting  53 N  9ft  E  n= l Using  'n 9x  (IV.4.4)  + y  9ft  lift +  nay„  and t h e f a c t  const  that  ft  is  itself  conserved,  one . h a s :  f^n n n[Vn- n n^ n  n  The therefore  k  x  y  conservation  of  =  c  o  n  s  angular  r e l a t e d to the invariance  transformations.  That  vortices  pointed  has  been  this  is  (IV.7.10)  t  moment  of  circulation  of the f l u i d  the case  o u t by Chapman  for  under  rectilinear  (1978).  is  scale  54 V.  SIMPLE  VORTEX  The examined  behaviour  in  order  differences varying V.l  in  depth  to  the  conserved  some  gain  of  its  the  path  Y -  2  insight  into on  vortices  systems the  a  now  qualitative  a curved  on  is  surface  with  plane.  Vortex: Field motion  is  2ft  vortex  vortices  a Single Velocity  first  simple  some  rectilinear  The M o t i o n o f No E x t e r n a l  is  of  motion  and  Consider ft  SYSTEMS  of  given  a single  vortex.  Since  by:  'h (x,y)k(x,y) _ k(x,yUn 2  - = B(x,y;x,y) '+  =  const.  (V.l.l) The  vortex  are  two  a)  path  may  special  Curved  cases  Surface,  If  the  one  depth  B=0  supposes  and  the  explicitly,  of  +  is  vortex  may  cross  itself.  There  Depth  uniform  that  moves  one  can  put  k=l  and:  = const.  2  velocity  never  interest.  £nh (x,y)  further  its  but  Uniform  B(x,y,x,y) If  close  there  along  are  (V.l.2) no  a curve  boundaries h=const.  then More  is:  v = -yzxVh(x,y).  (V.  1.3)  h (x,y) 2  The in  motion the  ledge  is  in  plane, of  determine  which  h(x,y) v  marked  .  in  contrast  remains the  to  the  stationary.  vicinity  of  the  motion  of  Notice core  is  a  that  vortex a  know-  sufficient  to  55 b)  Plane  Surface, If  the  Non-Uniform  surface  B(x,y;x,y)  Now,  however,  necessarily solving  +  even  in  are  It the  fluid  the  origin  is  The  vortex  Let  the  is  x(x , y )  Y  ^  One For  can  of  only  put  _  h=l  so  (V.l.4)  boundaries, B  k  in  very  few  B  does  general  uniform Figure ,  Q  depth  r  > e  assumed  to  be  +  the  to  but  treat  for  the  a small  k = k ( r)  ,  at  =  (x,y)  the  x  flow  case  in  which  depression  r  <  near  (-a,0)  (V.1.5) with  a >>  e  be:  (x,y)  (V.1.6)  s t r e a m f u n c t i on  if  k= k  everywhere.  Q  Then  sati sfies :  e  the  is  (V.l.7)  region  s u f f i c i e n t l y small in  which  -k'(r)  Y  nmrff form:  by  IV).  ^(x ,y) = Y ( x , y )  If  since:  not  analytic  VY,  across  that:  constant  obtain  however,  for  is  may  fk(x,y))  absence  streamfunction  0  l  one  non-constant  0  where  i  possible,  (see' = k  planar  known.  has  k  is  the  vanish.  (IV.1.5).  solutions  k  Depth  Y (x,y) 0  k ~ ~  -yk £n[(x +a) 0  V*F  varies.  2  -k. k'(r) 0  k  °(r)a  + y ]^. 2  is  0  nearly  constant  Thus:  Y  c  o  s  *  x therefore  ^ has  A  - ^ the  vortex  core  Fig.IV:  Fluid  with  Depression at  the  Origin  57 X = Since  Vx  ff(r)coscj>  d  = 0  2  some c o n s t a n t X  .  s  at  ( x  .  0  r  > e  depending  Thus,  , y )  0  for  i f  , f(r) ^  only  as  d~  at  (x-x )  +  induced  d , where  d is  The d i r e c t i o n  to  the l i n e  the vortex  joining  the depth  the vortex  is  there  along  induced  Notice  It  is  fluids those  largely of  vortex  and the  depression  is  curves  that,  of constant  depth depth.  is of  varying  is  property  a r e more  is  perpendicular  at the  zxVk(x,y) k.  vortex  position  velocity:  (V.1.10)  This  the previous  of flow  of the  varies  depression.  of the vortex  unlike  variation  of the v e l o c i t y  of constant  due t o t h i s  varying  the distance  a component  the region  (V.1.9)  by t h e d e p t h  of the f l u i d  by t h e b e n d i n g  k throughout  and b  2  and t h e  + 1 directec  e  0  the depression.  of  At the  (x,y)  (y-y )  to  When  >>  by  of the vortex  for large  3  is  r  then:  0  velocity  on k ( r ) .  the vortex  B ( x , y ; x , y ) *  The  for  velocity  core. case,  necessary that  component  to determine  vortex  difficult  a knowledge  systems  to analyze  in than  of v_ .  5 8  V.2  The M o t i o n  of a Single  Consider constant is  radially is  the motion  thickness  curved  near  the o r i g i n ;  by  (so  that  The  polar  one  chooses  harmonic p  V).  is  in a f l u i d of  planar  at i n f i n i t y but  i t is  One may t h e n  smooth  1 1 . 2 )  r  is  given  assumed  suppose where  at the o r i g i n  coordinate  so  0  is  (see Section  b ( p )  the surface  that  Stream  vortex  for simplicity  (Figure  z =  in a Uniform  of a single  on a s u r f a c e  symmetric  defined  Vortex  by  and  the  surface  b ' ( 0 ) = 0  as  b+0  p+<*>  and i f  ( I I . 2 . 1 8 )  that:  /l+]b' (s)  -  Po  ds  =  1  ( V . 2 . 1 )  Po  then  as  p - > - ° ° , r - > p , k - > - l . The  as  p ^ oo  .  external If  field  is  one c h o o s e s  such  that  i t is  its direction  a uniform  t o be t h e  stream  x-direction  then :  ty*(r,<$>) = Uy = Ursine}) The  path  of the vortex  JUh(r)  where  y = y  0  +  U  is  Clearly, stream  -  ( y  ^  the path y = y  do n o t d e p e n d  all.  general,  the o r i g i n  0  of the vortex is  path  depend  the time upon  far  upstream.  of the vortex  f a r down-  at i n f i n i t y , the paths  on t h e c u r v a t u r e  though.,  will  the  y + psincj)  infinity  pass  therefore:  = 0  y o )  t o o , and s i n c e  In  is  ( V . 2 . 2 )  near  taken  the o r i g i n  f o r the vortex  the d e t a i l s  at at to  o f the c u r v a t u r e .  59  F i g . V:  Fluid  with  Surface  Curvature  Near  the  Origin  60 The  equations  • _  yyh ' ( r) h  3  U  -  >_0  z  there  x = 0  ,  y  motion  of  U  :  2h (r)r  x  Assuming  of  is  =  r  v  ( r )  the =  vortex  are  YXh'(r)  (V.2.4)  2h (r)r  y  3  a stationary  point  at (V.2.5)  0  whe r e : h'(r ) h( r ) 0  -2U y  =  0  (Note  that,  point  is  from  stable  (II.2.21), if  has  Q,  h'(r)/h(r)  < 0  a maximum o r  ).  minimum  The  singular  there:  i.e.,  i f: 3 ft(o,r ) 2  0  9 ft(o , r )  T (p(ro)-l)  2  2  0  9 ft(o,r)  d Q(o,r)  2  2  dydx  0  2rT  dxdy  dx-  Y Crop-'(ro)-/(tJ(.irb)-l)3 g  27T  dr  (V.2.7) is  positive  or  negative  definite.  Since  p(r)  < 1  this  occurs  when: r o P (<"o) 1  Since  the  (V.2.8) b(p)  is  -  Gaussian  predicts  0  +  1 < 0  curvature  of  s t a b i l i t y only  decreasing  sufficiently  p( r )  close  this to  the  occurs  (V.2.8) the  surface  if  K(r )  only  origin.  0  if  is:  > 0 .  the  K = ~Pr/ ) rh ( r ) r  ,  If  stationary  point  is  61  V.3  The  Motion, of Consider  but  opposite  than of  the  in  are  (using  of  of  by  -y  2  vortices  a distance depth.  with  much The  equal  smaller  equations  notation) :  ix i b i  +  h?  2  y\'=  of  constant  complex  h?(zi-z )  Pair:  a pair  separated  a fluid  ii  Zi  Vortex  motion  strengths  |V£nh|  motion  a Close  z  3z  iy  hf  hi(zj-z )  2  2  3h : 3z 2  ( V.3.1) Therefore,  defining  Z = 1  Z = h{z\  - " „ 1 4d h ( Z + d )  .  n  +  z ) 2  , d = %(zi - z ) 2  1 h ( Z + d)  1  h (Z-d)  2  z  3  1 h 3(Z-d7  IX 4d  1  '  1  h (Z+d)  "  2  3h(Z-d) az  2  1 hMZ-d) (The  argument  Expanding  and  Z =  Note  Z  of  dropping  2dhIz) 2  has terms  TT a  been of  3h(Z+d) 3z  dropped  order  - i Y,  h3(Z)  3h(Z) 3z  for  3h(Z+d) 3 z. (V.3.2a)  3h(Z+d) 3z  (V.3.2b) simplicity)  ^  d d  (V.3.3)  that: d d  so  h  1  • Y h 3 ( Z + d7  h (Z-d)  :  that:  2  2dh (Z) 2  hTzT  3h(Z)  3z  -2Z  3h(Z)  hTzT 3z  ( V. 3 . 4 )  62 +  d _  d\; =  J  d  Z  " [HTZT  3h(Z)  , . I  3z  hTD"  3h(Z)l  3z  -2-—£nh(Z)  (V.3.5)  = E = const.  (V.3.6)  whence: h (Z)dd 2  This  is  the  conservation  of  energy  to  this  level  of  approxima-  t i on. From  v  The  pair  2  (V.3.3)  and  = h (Z)ZZ  = ^  2  moves One  by  with  can  = const.  constant  eliminate  differentiating  '  (V.3.6) :  d  Suppose  from  the  equations  of  motion  (V.3.3):  2  L  speed.  d  - i>7 _ -Y • ~ 4 E •" 2Eh T Z T  V  (V.3.7)  3  hEh(r)  3h(Z) 3z  Z _ Z  put  Z =  and  7 2 3£nh(_Zj_ ^ 3z ' 2  ,  .  Z  Re  1 $  .  Then  (V.3.8)  becomes: R* R$  R$ +  2R$  y /4E  is  |  whence :  -  R $ )^£nh(R) 2  (V.3.9a)  2  = -2RR^£nh(R)  2  another  • •  2  = h (R)(R  2  There  = - (R  2  +  2  constant  (V.3.9b)  R $ ) 2  of  (V.3.10)  2  the  motion  since:  •  + X  =  -  2  R ^  h  (  R  )  (V.3.11)  63 R $ h ( R) E J = 2  Using  (V.3.12) D2  to eliminate J  +  R h"rr7  the surface  and  (V.3.12)  is  (V.3.12) in (V.3.10)  I  Y  _  2  2  K  If  const,  2  gives  2  ( V. 3 . 1 3 )  4Eh (rT z  Z = b(p)  (see Section  III.2)  then  may be r e w r i t t e n :  p<S> = J  (V.3.14)  2  i i _ J  motion  2  (V.3.15)  a+r (p)D  P "  4E  (V.3.14)  (V.3.11)  b ,  2  and (V.3.15)  are i d e n t i c a l to the equations  of a p a r t i c l e moving  under  the influence  of  of the c e n t r a l  potential :  For with of  each  the vortex d£ d<J)  a  pair  P  is  given  the  is  fixed  l  also  and i s  c  a  is  never  n  exactly  one c i r c u l a r  be l e s s - t h a n  a.  orbit  The .path  by:  (V.3.17)  TT+XbTpTpT^ that  the distance  p = a .  the envelope  of closest  The c i r c u l a r o r b i t o f a l l the p o s s i b l e  approach  is  b^-const,  as  qualitatively similar  p -> °° to that  the path shown  orbits  of the vortex  in Figure  to  therefore for  E and J . If  is  (V.3.16)  2  2  2  one f i n d s  unstable  +  •  from which origin  (P)]  (l [b'(p)] )  9*  fp y _ 4J E 2  =  V  2  J and E there  2 J /E~ = ~ ~ =  p  J  Xl 4E  V(p)  VI.  pair The  64  impact parameter l -  £ V  is:  Y  -  a  ( V . 3 . 1 8 )  65  Fig.VI:  The  Path  of  a Vortex  Pair  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 on a S u r f a c e o f Revolution Historically  vortices deal  which  of  vortices  results  concluded seven  or  in  more  that  vortices  at  (Havelock  or  insufficient of  this  notably,  polygonal  the by  vpirtices  Thomson,  (1933)  is  was  who  stable  while  erred  in  Thomson's  in  1977  the  method  of  a  that  configurations  streets  with  (1977))  polygons.  stability  only  equal  Mertz  stable.  rigid  vortex  vertices  (1883)  showed t h a t  It  great  regular  are  the  a  examined  however,  determine  other  the  of  J . J . Thomson  such  to  first  of  of  von  a circular  and  infinite  has  also  Karman  (1912),  container vortex  lattices  (1966)).  configurations for  positions  received  vertices  Chapman  Experimental  obtained  has  at  configuration of  configurations  (1878)  vortices.  stability  Vortices  Kelvin  fewer  Morton  of  bodies  unstable.  (1931),  (Tkachenko  rigid  Lord  six are  studied,  stability  completed  heptagon  The been  as  placed  case.  fact  regular showed  were  that  heptagonal was  move  attention.  strength His  the  of  of  confirmation  within vortices the  a circular in  Hell,  vortices  are  by  of  the  stability  boundary  has  a technique  actually  of  some  recently  been  in  the  which  photographed  (Yarmchuk,  et.al.(1979)). We c o n s i d e r , sidered  originally  ring  vortices  of  Let initially  at:  N  on  now,  by  an  Lord  extension  Kelvin  a surface  vortices  of  of  equal  and  of  the  question  Thomson:  revolution strength  is  a  consingle  stable? y  be  placed  6.7 2 TT i n  on  a surface  fluid  of  whose  revolution  depth  is  (V.4.1)  described  constant  (k=l)  by  h '=  h(r)  and  has  no  and  in  a  boundary  N  (B = 0)  .  1 y j= 0 t h e s u r f a c e c a n n o t be n=1 of motion a r e , from ( I V . 2 . 1 1 ) :  Since  closed.  n  The  equations  •n "  The of  iYZ  '  - T n  h (r )  lk*nV n  of  initial  2  n  symmetry the  E  1  the  z  h (r  ) n'  1  n  v  (V.4.2)  2r„h(rJ n n v  configuration  suggests  a  solution  form:  z  =  n  "2TTJ n  r(t)exp  Substituting  into  r( t)  =  r  0  OJi t  +  N  (V.4.2)  one  (V.4.3)  finds 1-1  _Y_  ;wi  r  2  h M r  )  0  k  ^  r h'(r V 0  l-exp(2irik/N)  0  2h(r ) 0  ( V. 4. 4) The  sum  is  k  so  evaluated  ^  B  giving  (V.4.5)  2  that  where,  %[N-P(r )]  for  convenience,  r h (r )/y 2  2  0  ring  revolution  stability  (V.4.6)  0  The of  Appendix  l - e x p ( 2 7 T i k/N)  CO  be:  in  of  unit  of  time  has  been  taken  to  the  axis  . of  with the  the  vortices angular  rotates  velocity  configuration  rigidly u>  consider  1  .  To  small  about  examine  the  deviations  from  68 the  motion  : (t)  =  n  Substituting e  [r exp(27Tin/N)  + e (  0  into  (V.4.2) and  tjje '" .* 1  n  expanding  (V.4.7)  1  to  first  order  in  the  ' s (e  k  -e. ) n k' J (l-exp(2TT(k-n)/N))2  + P( r  +  Q(r  ,u )e exp(4Trin/N)  0  1  )e  -a)  o  n  i '  n  iexp(-4xrin/N)  n  (V.4.8)  where:  P(r,w) = % r p ' ( r ) + ( p ( r) - 1 ) (u-Jg) Q(r,o))  The  solutions  e  = %rp'(r)  to  of  the  form  1  b exp[2Tri (l-M)n/N-i  (V.4.9)  Q(r .Wi))a  +  0  S  + +  A t]  M  and  (V.4.11)  M  equating  coefficients  of  e  1  ^  s e p a r a t e l y . y i e l ds :  M  ( l  M  into  M U  are  (V.4.10)  M  Substituting e  (V.4.8)  p(r)oj  = a e x p [ ( 2 T r i (l+M)n/N)+i A t ]  n  +  and  +  (V.4.9)  P  M  +  M  ( o> i)) M r  w  a  (S _ 2  +  M  (" M A  +  P(r ,a) ))b 1  M  = 0  +  Q ( ^ ^  ) ) b  M  = 0  0  1  (V.4.12a)  (V.4:12b)  with : c L  N-l y ^  = "  l-exp(2TriLk/N) (l-exp(2~Trik/N))  2  -  Js(N-L) ( 2 - L )  , L = l , . . . ,N ,( V . 4 . 1 3 )  (see  Appendix  (V. 4.12)  if  B).  and  There  only  if:  are  non-trivial  solutions  of  69 " Q (r ,  X  )  2  M  0  W l  + (S  + P(r ,o) ))(S _  1+ M  0  1  1  + P(r ,  M  0  U l  ))  = 0  (V.4.14) The  M t h mode  Q  2  is  stable  K>^)  "  i f  ( l S  is  +  p  + M  real;  K ' S > H  s  i.e.,i f :  i - M  ' P < r  +  A ) )  0  >;  0  (V.4.15) Using  (V.4.6),  (V;4.9),  (V.4.10)  and ( V . 4 . 1 3 ) ,  (V.4.15)  becomes: ^  g  (  o  )  r  °  + p(r )(N-p(r )) 0  0  - ^i!iL  > 0  , M=1,...,N-1 (V.4.16)  In  a linear s t a b i l i t y analysis  symplectic  system  solutions: system  along  first  its orbit,  is  have  a stable  mode;  of  Here  been  stability  P  .  neglected  displacements  corresponding  the second  is  modes  in (V.4.16).  ft  forbidden  are the They  of the  to small  o f f the hyper-surface  these  of a  =  const.  by t h e  M=N modes  cannot  a f f e c t the  of the c o n f i g u r a t i o n .  (V.4.16) M=J5(N±1)  ft  solution  two z e r o - f r e q u e n c y  to small  the other  o f the system  conservation which  expects  one c o r r e s p o n d i n g  displacements The  one a l w a y s  o f any p e r i o d i c  h a s a m i n i m u m when  N odd.  Hence,  '  + p(r )[N-p(r )]  g  (  o  )  r  °  M=%N , N e v e n  f o r s t a b i l i t y of the ring  0  0  - x  >  0  N  or of  e  v  e  vortices:  n  (V.4.17) >: -1 Notice  that,  i n c o n t r a d i s t i n c t i o n to the system  the  s t a b i l i t y i s enhanced  at  r  is  0  .  By m a k i n g  possible  N odd  by n e g a t i v e  the curvature  to accomodate  at  in Section V.2,  curvature  of the surface  r  and negative  0  large  a r b i t r a r i l y large  numbers  of  i t  vortices  in  a stable  ring.  Examples. a)  Plane For  For  the plane:  p(r) =l , p ' ( r ) = 0  (Section  II.2).  stabi1i ty: -N  2  + 8N - 8 > 0  N even  > -1 whence  there  criterion  is  is  inconclusive  perturbation case  stability  theory  on w h i c h  N odd i f  N < 7 .  If  a n d one must  to determine  Thomson  (V.4.18) N=7  the s t a b i l i t y  use h i g h e r  the s t a b i l i t y .  order This  is the  erred.  b) ~ C y l i n d e r A cylinder the  cylinder  there  is  extends  p(r)=0  around  -(l+aHn((x-x' + a£n(x  a perfectly  circulation around  r=r'  valid  around as  ( V . 4 . 2 ) we h a v e c r i t e r i on i s  r of  However, 0  since  a n d as  r  °° ,  ¥( x , y ;x , y ) 1  r=r'  as  r'  )  (y-y') )  1  °° a n d  r=r'  as is  arbitrarily  r  2  +  2  %  + y)  2  2  streamfunction  r'+O  then :  as  .  words :  Y U . y ^ ' ,y' ) =  is  both  in the choice  on t h e c i r c u l a t i o n In o t h e r  , rp'(r)=0  to i n f i n i t y  an a r b i t r a r i n e s s  depending r ' -»• 0 .  has:  1  °°  -2Tra . chosen  (V.4.19)  h  f o r a unit is  vortex.  2 TT ( 1 + a )  In w r i t i n g a = 0 .  while the  The that  equations  The s t a b i l i t y  71 N whence  < 0  2  all  rings  The is a  of  the  velocity  vortices:  ,  N even of  velocity  form: field  v  r  vortices field = 0  cannot  (V.4.20)  ,  is  N  2  < -1  are  by  the  for  odd  non-zero  v^ = c o n s t .  affect valid  N  unstable.  induced  ,  ,  The  stability  all  a  .  a  in  addition of  a  ring  (V.4.19) of of  such  72 V.5  The S t a b i l i t y Surfaces of If  flow to  is  the  also the  in  addition  invariant axis  of  rotate  of Vortex Streets Revolution to  under  rotational  then  about  (1912) Let  p =  f(z)  Then,  these  the  r  .  z  of  II.2)  the  vortices  rotation.  In  can  view of  of  von  streets.  revolution f  of  perpendicular  configurations  vortex  with  of  surface  be  described  an., e v e n  by:  function.  D *  e x p f  Q  (V.5.1)  0  = 0  r(-z) Moreover,  to  a plane  rings of  the  (II.2.14)  Z  Choosing  axis  called  surface  (Section  from  are  double  the  qualitative similarities  Karman  symmetry  r e f l e c t i o n in  rotation  rigidly  on  and  since  f(s)  =  f(-s):  = yr^j  (V.5.2)  ( I I. 2 . 1 5 )  implies:  h(i) r h  (V.5.3)  (r)  is  = —f-  the  invariant  upon  r=l  %  (z=0)  (V.5.3)  condition  r e f l e c t i o n in  We e x a m i n e strength  y  There  two  are  st r e e t s .  which  at  the  r (z ) 0  0  distinct  implies  the  plane  s t a b i l i t y of and cases:  N  of  that  the  surface  containing  a ring  of  strength  staggered  and  N -y  the  is  curve  vortices at  symmetric  of  -^-(-z ). 0  vortex  73 a)  Staggered The  initially  Vortex  Streets  vortices  of  a staggered  vortex  street  are  situated  at:  r  = r  n  , <j>  0  = 2-iTn/N  , n=l,...,N  , strength  y (V.5.4a)  r  =  m  » *  (2m+l)TTi/N  =  n  , m=l , . . . ,N , s t r e n g t h  -y  (V.5.4b) In the  the  absence  equations  of of  boundaries motion  n  h 2 (  r  n  )  in  a  fluid  N  1X_ 'z_-z  of  uniform  depth  become:  N z  and  I I  +  ,k=l n " k Lk^n z  z  £  i +  m=l "m""n  n 2r h ( r ) n n Y  Z  n  h  ,  (  r  }  v  (V.5.5a) 1 h (r  "m  The of  2  symmetry the  N  of  hi  )  the  ,  =i .k^m  N  z'm- z , k.  k  initial  1  n=l  m  i Yz h ( r ) ' m n' 2r h ( r ) n n (V.5.5b) v  _LX_ zm „ - z .n  configuration  suggests  solutions  form: r  =  n  r(t)  =  <j>m  Substituting  , *  2 ir i n / N  =  n  (2m+l)Tri/N  into  (V.5.5)  =  ; o)  and N-l  r(t)  r  Q  2  -  ^£  +  N E m  =  1  +  o* t 2  , r  m  = ^  ,  + oo t  (V.5.6)  2  making  use  of  (V.5.3)  gives  1 i-exp(2Trik/N) 1 l-r- exp((2m+l)TTi/N) " 2  r h'(r ) 2h(r ) 0  0  0  (V.5.7) The  first  second  sum  sum was is  encountered  (Appendix  B):  in  the  last  section  (V.4.5).  The  74 N Z m=l  1 l - x e x p ( (2m+l)TTi/N)  ( V.5.8) 1+x  1  Thus 03  The  unit  L(r?  of To :  time  =  +  has  again  order  the  taken  in  the  the  e 's  v  k  t  n  of  6 's  and  P(r ,oj )e 0  2  motion  and  (V.5.10b) expanding  v  1  +. Q ( r  0  to  :  -e, ) k  n  set:  (V.5.10a)  1 a )  m  :  kfn +  2  configuration  m  ( l - e x p ( 2 i r i (k-riJ/N)y  1  2  i u)„ t 6 (t)]e 2  +  )'/y  r h (r  , a )  n  equations  (e  be:  i w„ t e (t).]e 2  +  [r7exp( ( 2 m + l ) T r i / N ) into  to  s t a b i l i t y . o f the  0  Substituting  ( V. 5 . 9)  n  been  [r exp(27Tin/N)  "m  first  P(r )  N  determine  n  +  r; ,  I (1 - ro-?exp(('.2(nwi)+l.)'Mi / N ) )  u )e exp(4Trin/N)  J  2  nr  n  n  i e x p ( - 4fri n/N) ( V. 5 . 11 a)  N  m  \l  x  m- k (l-exp(2Tri(k-n)/N))' ( 6  6  (6 m-e n)' l-r§TxpTj2Tn-niH)Tri/N))2-  )  v  +  E  (V.5.lib)  where  P  and P(r)  so  Q  are  = -p(i)  d e f i n e d by ;  rp'(r)  (V.4.9,10).  Using  = ip'(i)  (V.5.3): (V.5.12)  that: P(p-u)  = P(r)  + 2u + p ( r )  ;  Q(r,u))  =  Q(£.-o>) (V.5.13)  75 The  solutions  e  to  (V.5.11)  are  of  = a e . x p [ 2 T r i ( l + M)n/N  +  M  n  +  the  form:  i X t] M  b exp[2Tri(l-M)n/N  -  M  iX^i] ( V . 5 . 14a)  = c e x p [ - ' ( 2 m + l ) ( l + M)Tri/N  6 n  +  M  i A t] M  + • d e x p [ (2m+l.)(l-M)iri/N  -  M  i X t] M  (V.5.14b) Substituting of  e  l  A  r  ,  and  t  into  " '^ 1  e  M  (V.5.11)  and e q u a t i n g  separately  t  yields  coefficients  (using  (V.5.9)  and  (V.5.13)) :  U  + Q(r  M  *M  A  l +  T  T  (" M  +  ,u ))a +  0  A  2  l-M  (  r  r  2 )  o  )  a  M  r*J  M  Q( ." ))V  +  M o V ( r  Ab +  M  (  +  0  " M A  n  (  r ) d 2  J l +M T  ( r  QK'^^V  +  V  A  r  2  u  ( X  M  +  ^  r  o '  W  2  )  o  M  = 0  M'  ) £  =  «  M  M  d  =  (V.5.15b)  0  Ad =0  )  (V.5.15a)  (V.5.15c)  (V.5.15d)  0  where A = S  1 +  M  + P(r ,u) ) 0  2  + T (r )  = Q(r ,u> )  2  N  0  —IT (r  2  M  1  +  N  0  T i  (x) = - x " T u ; x i 2  L  -  N  _  L  +  2  fx"M U ; -  ,N-L Nx - ((L-l)x M  L  M  -  7 ( 1  N  0  ^M(N-M)  (V.5.16) )  2  exp((2k + l ) U i / N ) _ e p((2k + l)ui/N)) X  2  X  (N-L+l))  |\J  (1 + x'V  ^  r  -  , 5  L  N 1 ,.. .,  (V.5.17)  76 (see  Appendix  trivial  B).  S  solutions  of  is  L  defined  (V.5.15)  in  (IV.4.13).  only  if  a = ±r  There  are  d , , an d; b .=  2  non±r  2  c  Then : U  + Q(r ,u> )  M  0  Aa +  (-X  M  whence  for ±  r  2  o(Ti  M  M  ±  2  r T  ; M  AF, = 0  ( r )) b  2  2  1 +  M  (V.5.18)  M  M  = 0  (V.5.19)  solutions:  (r )  T _ (r ))A  -  2  +  +  2  1  0  (Q(r0,u>2)  -  r T _ (r ))a  + Q(r ,a3 )  M  non-trivial  M  A  ±  2  ±  1  2  M  M  A  +  2  r T _ (r ))(Q(r ,o3 ) ± 2  2  1  M  0  2  r T 2  (r -)) 2  1  +  | y |  = 0  (V.5.20) The  Mth  modes  o"  r  ( T  l +M  ± which  can  ( r  ^oT  be  (2C  are  stable  o  "  }  T  is  l - ( o ) ) r  2  M  (r ))(Q(r ,a)) 2  1+  M  0  simplified ±  if  D)(4q  2  real,  > 4(A ±  2  i.e.,  -  r ' T ^ f r  if:  (Q(r ,u>) 0  ) )  2  (V.5.21)  to:  -  2C  ±  D)  2  = %x(l  > 0  (V.5.22)  whe r e : C  E (Q  -  D -= r ( T  A)/N  2  (r )  V  2  1  +  M  +  _ xcosh((l-x)y)  q  If T  (2C  l M +  ( r  o)  = Q(r ,co )/N  2  0  x  = M/N  ±  D)(4q =  T  l-M  2  ( r  (r  M  -  -  2  x)  -  %sech (%y)  (V.5.23)  2  ))/N  2  (l-x)cosh(xy)  M/  C O A\  (V.5.25)  2  2  ,  y  =  Nfcnr ,  -  2C  ±  D)  o)  •  2  = 0  the  solution  is  stable  unless:  (V.5.26)  77  r  -+ r  0  1 Q  on l y  y  The  stability  and  M -> N - M  >  0  and  The  Nth  mode  (Fig.VII;  L  lower signs).  frequency  the  modes.  It  is  is  invariant  under  t h e r e f o r e s u f f i c i e n t to  consider  1 .  U denotes The  to  the  perturbations  upper  t w o L modes  U modes  . L^lLA  Ut  (V.5.22)  corresponds  The  .. ,  Q(r  .  h <x£  types the  criterion  are  .  signs  are  the  stable  if:  r + r 0  in  four  V.5.22,  the  expected zero  Bi^.mrUp L  of  +p(ri)  '  < 0  0  (V.5.27) If  > 0  p(r ) 0  stable.  If  unstable.  are in  p(r ) 0  Other  If they  N is  tend to shown  the  and  in  0  < 0 and cases  even  be  the  0  must  then  IX.  be e x a m i n e d  the to  modes go  They  criterion  Villa)  < 0 (Figure  K(r )  first  Figure  stability  > 0 (Figure  K(r )  which  VI11b)  M = ^N  all is  mode  then  it  are  important  The  four  similar.  the  this  is is  separately.  unstable.  are  then  same  possibilities  This  for  as  is  upper  reflected and  lower  s i gns : (h If  q  is  -  This the if  = 2arccosh is of  surface of  case  (as  von of  the  > 0  sech ^y) 2  N-*-«>)  then  (V.5.28)  stability  flow as  sphere,  is  around:  /2  (V.5.29)  analogous  Karman  k + k  when  region  infinite  of  2  occurs  a small  stability  those the  to  criterion  the  -  2  small  restricted y  h sech %y)(4q  is  to  the  staggered  von  Karman  vortex  a c y l i n d e r , our  N+°°)  .  (V.5.28)  condition  streets  (in  for fact,  results  must  However,  as  be s e e n  is  incompatible with  often  will  approach for  78  2rr  Fig.VII:  Staggered Vortex S t r e e t : The Modes M=N. Dots i n i t i a l positions, crosses perturbed positions  denote  79  a)  Staggered vortex streets streets unstable.  stable.  Symmetric  vortex  p(r )<0, K(r )<0 o  b.)  Staggered vortex streets vortex streets stable.  Fig.VIII:  o  unstable.  S u r f a c e f o r which the Definite Stability  Mode  Symmetric  M=N  has  80  u  St-  u  0  Fig.XI:  —  Staggered Vortex S t r e e t : The Modes M=JgN. dots denote i n i t i a l p o s i t i o n s , the crosses perturbed positions.  The  81 (V.5.27)  leading  to  overall  O t h e r mo.des a r e be  examined  b)  Symmetric The  initially  considerably  more  for  surfaces  separately  Vortex  instability.  specific  complicated of  and  must  flow.  Streets  vortices  of  a symmetric  vortex  street  are  located  at:  r  =  n  r  ;  Q  <|>  = 2irin/N  ;  n=l,...,N  , strength  y  (V.5.30) cf>  m  m  ;  27Tim/N  =  -m=l,...,N  , s t r e n ggtthh  -y -y  (V.5.31) Trying  a s o l u t i o n , to z  =  n  (V.5.5)  r(t)exp(2TTin/N  of +  the  form  u t)  ;  3  z  m  =  r"  1  ( t ) e x p ( 2 T H m/N  +  u>.  (V.5.32) one  finds: r(t)  =  r  0  ;  N(r w  N  r" ) N  +  (V.5.33)  3  A o-r  )  r  where  use  has  k =  (see mined  been  of  B).  The  1  and:  (V.5.34)  _ N':  stability  x  of  the  configuration  is  deter-  p u t t i ng: z  =  n  r exp(2Trin/N 0  - i  r exp(2uin/N 0  m substituting e 's  (V.4.5)  (l-xexp(27rik/N))  1  Appendix by  made  0  and  into  6 's e  = n  and  (V.5.5), trying  +  E„(t))e  + 6 ; ( t ) ) ei m  to  solutions +  (V.5.35a)  u,3 t  of  i X t] M  ,t  l w  expanding  a e x p [ 2 T r i ( l + M)n/N n  i  to  (V.5.35b) first  the  order  in  the  form:  + • b ^ x p [ 2 i r i (1 -M) n/N> i  \t]  (V.5.36a)  82 6  = c e x p [ 2 T r i (1 + M)m/N + i X t ] +: d ^ e x p [ 2 i r i ( 1 - M ) m / N - i X t ] f l  m  M  M  (V.5.36n) The  solutions (X  satisfy:  +  M  Q(r ,o) ))a  Ba :+(-A M  R  l +M  R  l-M  (  (  r  o  )  M  0  (  +  M  a  ^  M  (  + Q(r ,aj3))b -:+ ^R  M  o ^ M  r  + BF --+ ^  3  0  +  ~ M A  %  B  :  (  0  X  +  M  Q  r  +  (r )5  M  + B d  M  o '  W  = 0  3  )  )  d  M  =  M  M  (V.5.37a)  = 0  2  1  3  (  ^  2  Q(r .u ))c --  +  +  r  r  (V.5.37b) (V.5.37c)  = 0  (V.5.37d)  0  where  S  =  B  T _ M -  R  M ( ^  2  )  P  +  K >  3 )  W  " R R  L  =  (*) U  J  x~ R fx"M N-L+.2 ' R  Nx " [(N-M+l) N  =  U  ( 1 _  Equations so  that  the  (V.5.21):  (V.5.37)  stability  N X  3  (V.5.38)  2  exp(27TJLk/N) (l-exp(2irik/n))2  (M-l)x ]  , M= 1, . . . ,N  N  (V.5.39)  ) 2  are  exactly  criterion  is  the  analogous equation  to  (V.5.15)  analogous  to  i.e.,  r MR ( r S ) ±  can  be  R^U )) 2  +  1 + M  0  which  +  M  , w ) - JsM(N-M)  0  ^W^KJ  = y ~ klx  2  X  Q(r  =  r R 2  3  > 4(B  (r ))(Q(^ ,a) ) 2  1  +  M  0  simplified  (2E ± F ) ( 4 q  2  -  3  2  ±  -  (Q(r ,a> )  r ^ ^ r  0  2  3  ) ) )  (V.5.40)  to:  2E ± F)  > 0  ,  M=1,...,N  (V.5.41)  with: E  = Jgx(l-x)  + %csch (%y) 2  (V.5.42)  83 (l-x)cosh(xy) + xcosh((l-x)y) 2sinh (%y)  F  Q(r  ,o) )/N  0  one  may  The  y  four  signs  the  modes  are  signs  N£nr  that  modes  lower  upper  =  suppose  denotes L  (V.5.44)  2  3  M/N  x Again  (V.5.43)  2  the  y >_ 0  with  in  (V.5.45)  2  M=N  V.5.41,  , h±x<_\  .  are  in  shown  U upper  expected zero  Figure  signs).  As  X  .  (L  before,  f r e q u e n c y modes.  The  give:  > 0 (V.5.46) If i s  p(r )  < 0  0  true,  and  K(r )  p(r )  > 0  0  Other  (see  Figure  VI11b)  then  (V.5.46)  si nee: +  If  < 0  0  cases  p ( r ) > N - l > 0  and  must  K(r )  > 0  0  be  then  Q(r  ,w )  0  < 0  3  determined separately  for  .. '  each  r  surface  of  interest. We now criterion  may  Let  show be  that  simplified  g(x,y)  -  for  (sinh(xy)  y  >_ 0  and  +  the  stability  +1  -  (l-x)cosh(xy)  xcosh(l-x)y -  sinh(xy)  s i nh ( ( 1 - x J y ) )  % j< x  that  further. 2  = x ( l - x ) [sinhy  Now:  so  = 2x(l-x)sinh %y  ; Then:  2 E ± F ^ 0  <_ 1  since  -  s i n h ( (1-x) )y]  = y(.sinh(xy)  sinhx  is  -  s i nh ( ( l - x ) y ) )  increasing.  84  U  0 —  U  0 —  0 —  0 — Fig.X.  Symmetric Vortex S t r e e t : The Modes M=N. Dots denote i n i t i a l p o s i t i o n s , crosses perturbed p o s i t i o n s .  85 Therefore: whence  hsinhhy  < sinh(xy)  |£ > 0 . dy  + sinh((l-x)y)  <  sinhy  Therefore:  —  g(x,y)  > g(x,0)  = l-(l-x)-x  = 0  The r e f o r e : 2E The  ± F > 2E -  stability 4q  The M = HN  y  z  > 0  y  i f y > 0  c r i t e r i o n may t h e r e f o r e  -  3  l\^ l  F =  2E - F > 0  critical  (N e v e n ;  ( V . 5 . 4 7)  be s i m p l i f i e d t o :  , M=1,...,N-1  modes  (V.5.48)  are in general  see Figure  XI)  f o r which  t h e L modes  with  the s t a b i l i t y  c r i t e r i on i s : > (1 + c o s h ^ y )  Q  q  For  sufficiently  c)  large  N  The  N^  0 0  , q •+ 0  so t h a t  3  a symmetric  vortex  street  4  g  -^  )  j  for  will  always  The  Cylinder  stability  of vortex  streets  on t h e c y l i n d e r  is  examined. For  (V.5.49) streets  the c y l i n d e r  then are  immediately  4C  2  For modes.  p(r) = 0 implies  , p'('r) that  so  that  q  a l l symmetric  2  = q  3  = 0  vortex  unstable.  Staggered  tion  As  5  unstable.  Example:  now  > yg- .  y  lv,3  2  q  3  (  16tanh Uy)  2  stability  become  =  16 s i n h % y  1  3  1  2  -  D  2  vortex  are stable  i f :  < 0  M = 1  T h e L modes  o f moment  streets  (V.5.50) a l l four have  modes  already  of c i r c u l a t i o n  now y i e l d  been  and  zero  explained.  angular  frequency The..conserva-  86  U  U  Fig. XI:  Symmetric Vortex S t r e e t : denote i n i t i a l p o s i t i o n s , positions.  The the  Modes M % N . The d o t s c r o s s e s pe r t u r b e d ;  87 moment  of c i r c u l a t i o n Z  y  n  n  £  n  r  n  S-Y 4> n  =  c  conservation  These  modes  and  must  N is  i.e.,  From  (from  - \  by t h e M = N . ,  be n e g l e c t e d  U modes.  a n d do n o t a f f e c t t h e  is  M=%N(x = Jg)  .  D is  then  zero,  (V.5.22)).  sech (J- y) 2  2  i  2  unstable  0  (V.5.53)  unless:  /2  (V.5.54)  and ( V . 5 . 2 6 ) ,  the separation  R is  =  the separation  of the rings  of  =  c  within  each  ring  of the c y l i n d e r .  only  is:  Therefore:  arccosh/2"  with  c  (V.5.56)  2TTR/N  the s t a b i l i t y  The  / ^ (V.5.55)  of the v o r t i c e s  the radius  agreement  determined  N  are v i o l a t e d  Rarccosh/2 N  =  d -T—  that  (V.5.52)  is:  d  in  (V.5.51.)  o n e may p u t  = 2arccosh  ,  where  -  laws  even  |  (II.2.23)  v o r t i ces  while  _  t h e mode y  t  configuration.  for stability • 2  s  therefore  of the  If  n  = const.  These  stability  o  f o r the c y l i n d e r a r e :  ,,. (V.5.57) c  von K a r m a n . of this  by h i g h e r  stabilities  When  mode order  is  M=%,  T^  + M  (x)  = T^_ (x) m  indeterminate.  perturbation  of the other  modes  It  c a n be  theory.  f o r N even  and f o r  o d d a r e now d e t e r m i n e d . -^[(D-2C)cosh isy] 2  =  x  (  \ ~  x ]  [s i nh ( (1 - x ) y )  -2cosh(%y)sinh(Jgy)]  so  -  sinh(xy) < 0  88 since  sinhx  is  increasing  (D-2C)cosh %y  = x  2  i f  x>0  > 0  2  .  when  y = o  as  y + °°  -*- -°° ( x ^ 1) Thus,  i f XT*1,  Denote  these  D-2C h a s e x a c t l y zeros:  one z e r o  x(1-x) 2  2  =  3y  +  (l-e ) 8 2  ( l - e  where  2  )  These  y"(x)  when  2  inh(Jgy)  > 0  y -»• °°  one z e r o  f o r each  x  (x^l).  y (x). +  y = arccosh/2  , C=0 a n d D_>0  2arccosh/7T< y ( x )  and  +  y"(x)  sinh(xy)  y=0  as  oo  are denoted  <  2cosh(Jsy)s  -  Moreover:  D + 2C h a s e x a c t l y  When  sinh((l-x)y)  2sinh(%ye)cosh(%y)+2cosh(Jay)sinh(i2y)  D + 2C = - ( 1 - x )  Therefore  x.  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 )  e = 2x-l .  ->  f o r each  y"(x).  3(D+2C)cosh (%y)  _  But:  <  y  ;  therefore  D + 2C _> 0  when:  < y (x)  %<x<l  +  (V.5.58)  Define y" The  vortex y"  = max y " ( x ) x street  is  < y  +  Figure  < y XII  and y~ c o r r e s p o n d  ,  stable  y  +  = min y ( x ) x  (V.5.59)  +  i f : (v.5.60)  is  a graph  o f y ~ ( x ) and y ( x ) .  to the values  +  of x closest  t o h.  Clearly  y  +  For N odd  90 this  is:  Thus, around given  y in  if  Table  N =  1:  3  x = h  a small  first  . region  of  few N t h e s e  stability  regions  are  Regions of S t a b i l i t y of Staggered V o r t e x S t r e e t s on a C y l i n d e r < 2,079 4  A =  1.6 2 78 < y  <  1.9 344  A = 0 . 0 30 7  ,  1 . 6 6 34 < y  < 1.8806  A = 0.. 01-55  ,  1.6842  <  A =  N = 9 is  is  the  7  the  actual  < y  < y  1.8525  width  of  these  stability  is  only  regions  0.0,879  0.0094  measured  in  radi i :  +  A  y  If  N is  one  even  higher  somewhat  of  more  (II.2.24) ,  Q(r ,o) ) 0  perturbation  if  theory  y=arccosh/2, to  check  Sphere  analysis  From  order  possible  ^N. The  The is  -y 2N  needs  M'=  Example:  sphere  For  ,  cylinder  d)  .  there  1.5 5 2.2  A  mode  odd  N even:  ,  N =  the  For  1.  N = 5  but  N is  = arccosh/2  Table  A=0  + k).  x = %(1  3  =  -h  the  stability  of  c o m p l i c a t e d than (V.5.33)  -H  and  tanh (y/2N) 2  vortex  streets  on  cylinder.  the  on  the  (V.5.45):  + -JgNtanh(y/-2N) c o t h ( J g y )  (V.5.61) Since:  t a n h ( a x ) c o t h x : <^ 1  if  a < 1 :  91 Moreover: E =-Jsx(l-x)  + %cosech (%y)  > Jgx(l-x)  2  ;  F>0  (V.5.63)  Therefore: 4q If  N is  even 4q  If  N is  for  2E  -  3  2E  there  4q  -  3  -  there  odd  Therefore The  -  3  2E  -  F < is  a mode  -  x=%  ,  2  a mode  F < -(N -8N+3)/N 2  symmetric  vortex  stability  criterion  (V.5.48)  (V.5.64)  whence 2  < 0  , if  ,  whence  < 0  , if  x=%(l+^)  all  N = 2 , . . . ,6  x(l-x)  F < -(N -8N+4)/N is  -  + ^  2  streets has  with  been  N>7  N>7 N>7  are  examined  unstable  numerically  .  Table  2:  Regions of S t a b i l i t y of S t r e e t s on a S p h e r e  N = 2  ,  stable  if  y  > 4.2451  ,  9 <  38.1727°  N =  3  ,  stable  if  y  > 5.3020  ,  9 <  44.9072°  N = 4  ,  stable  if  y  > 7.5957  ,  9 <  42.3078°  N = 5  ,  stable  if  y  > -10 ..4306 ,  0 <  38.8225°  N = 6  ,  stable  if'  y  > 1 7.7602 ,  9 <  25.6478°  N _> 7  ,  unstable  From : ( 1 1 . 2 . 2 4 ) ,  Q(r ,oo ) 0  2  = -k  (V.5.9)  and  -%tanh (y/2N) 2  Symmetric  Vortex  (V.5.26):  + %N t a n h ( y / 2 N ) t a n h (%y) (V.5.65)  The r e f o r e :  92 §  -  sech (y/2N)  ^  2  [  t  a  n  h  (  y  )  _l  t  + ^tanh(y/2N)sech (Jgy) f o r each N there  Denote  it  by  y Let  y(N)  > y(N)  a = y/2N  .  is The  (see  .  exactly vortex  n  n  (  > 0  2  Thus,  a  y  /  2  N  )  ]  for y > 0  one y  such  street  is  that  (V.5.66) Q = 0  unstable  if:  (V.5.27)).  Then  N = y/2a  and:  1 + tanh a - —tanhatanh(%y) 2  a  tanha 4a  M_ N  o  ( l +tanh a)a  (V.5.67)  a  2  tanha  w  ytanh^y  >  >  < 1 : i.e., if  ^ , 1  y  y(N)  > 1.543  Let  f(a) = a -  x u  •  £  Therefore  < 1.543  .  f(0)  = 0  a(l+tanh a) tanha  .  (l+a)tanha  a  ytanh(%y) Let  y*(N)  .  =  +  .  a  t  a  n  , h  <  a  < 0  if  1  l  0  1 +  2  a  •  T  T  ,  h  u  a > 0  and: .  s  1  f  :  , then Q > 0 .  largest  y*(N)tanh(%y*(N))  Then:  a < (l+a)tanha  > 1 + 2a = 1 + be t h e  if  (V.5.68)  Therefore:  2  . ,  n  Q < 0  Therefore:  f ' ( a ) =• ( 1 - t a n h a ) ( 1 - ( 1 + a ) (1 + t a n h a ) ) But  n  value = 1 +  of y  such  that (V.5.69)  Then : 1.543  < y(N)  Differentiating  < y*(N)  (V.5.69)  by  (V.5.70) N:  93 tanh(Jgy*(N))  Since  y*(N)  y*(N)  <  One  can  always  check  < y(N)  N > 4  ,  that  y*(4)  A staggered vortex  street  on  and  > 1.6  is  M=%N Let  N _> 4  now s h o w n  or  =  0.648  that  M=^(N±1)  f(x,y)  < 1.6  .  Thus  and:  for  unstable i f  modes  numerically  < 1.6  It  -y*(N) N  ^  Q  for all  1.543  _  > t a n h (JgXl. 5 43)  2  **|M  9y*(N) 9N  :  + % y * ( N ) s e e n (*gy*(N) )  < 1.6  ±  2  > 1.543  tanh(Jgy*(N)) whence  + %y*(N)sech (%y*(N) ) -  y  if  are  N > 4  y  the  (V.5.71)  sphere  is  therefore  .  < 1.6  and  N >^ 6;  then  the  unstable.  = cosh(y(l-x))  +  (1-f)cosh(yx)  .  Then:  A )  3f  1  = ^ycoshUy) [ l - x y t a n h ( x y ) ] + y ( s i n h ( x y ) -  With f(x,y)  y  < 1.6  sinh((l-x)y))> (V.5.72)  > f(h,y)  = 0  holds  i f  0 for  h < x  D = %xf(x,y)csch (h>y)  x  < 3/4  > 0  2  i f  i f  xy < 3/4 .  < 1.1997 .  (V.5.72)  Thus:  Therefore:  h<x  < 3/4  , y < 1.6 (V.5.73)  and: f£  =  % c s c h  2  ( J a x ) ^ | ^ -  >  0  i f  h < x  < 3/4  ,  y  <  1.6  (V.5.74) Moreover, q  from  2  (V.5.65)  1 - 7m  and  (V.5.66): ( V . 5 . 75)  94 The  modes  M=J§N  or  M=%(N + 1)  have:  x <  whence C = Jsx(l-x)  -  hsech (hy)  < h - ^ -  z  -  %sech (%y) 2  (V.5.76) and,  using 4q  ( V . 5 . 73 ,75 , 7 6 ) : "  2  2C + D > ^  +  ;sech (J5X1.6) 2  3  4N2  +  0.0297  (V.5.77) if  y  < 1.6 4q  Moreover, y =  1.6  for  y  .  The r e f o r e :  -2C + D > 0  2  one c a n c h e c k and  Using fore on  has  unstable  the sphere  been  ,  t h e modes  (V.5.22),  The  , N > 6  < 1.6  then:  (V.5.78)  i f  2C + D < 0  .  • Since  —  y < 1 . 6 M=%N  (V.5.78)  i f with  y  < 1.6  or  ,  J s < x < 0 . 5 8 M=%(N + 1)  and ( V . 5 . 7 9 ) .  N ^> 6  Hence are  determined numerically  3:  have  these  modes  a l l staggered  ( V . 5 . 79) x < 0.58. are t h e r e vortex  streets  unstable.  s t a b i l i t y of staggered  Table  < 0  dX  3 , x < j :  < 1.6  N _> 6  y  numerically that  x = 0.58  2 C + D < 0 For  i f  vortex  (Table  streets  for  N <_ 5  3).  Regions of S t a b i l i t y o f Staggered V o r t e x S t r e e t s on a S p h e r e  N = 2  stable  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 . 5 5 2 2 < y < 1.6 306 , 7 4 . 6 1 7 1 ° < 8 <  N > 3  unstable  75.3404°  95  These cylinder vortex  (or  results von  streets  streets.  differ  Karman  exhibit  q u a l i t a t i v e l y from  vortex greater  streets) stability  where than  those  the  on  the  staggered  symmetric  vortex  96 VI.  VORTICES  WITH  FINITE  CORES  Until  now,  it  been  i n f i n i t e s i m a 1ly finite ular, core the  small  is  is  shown  not  motion  The  that  the  the  the  Y =  the  are  then  all  section  vortices  the  of  flow  a wobble  is is  In  Velocity extends  of  the  of  a  Vortex  over  a  finite  vortex  region  G  into  one  can  w (x,y,t)xh (x,y)dxdy  (VI.1.1a)  w_(x,y,t)yh (x,y)dxdy  (VI.1.1b)  w_(x,y,t)h (x,y)dxdy  (VI.  2  2TTY  the  by:  2  2TTY  and  introduced  2  1  of  partic-  curved  z  2TTY  have  effects  considered.  surface  symmetric  core  1  this  vorticity if  and  position  X =  In  that  vortex.  Position  When define  of  radially of  assumed  cores.  distributions it  VI.1.  has  1.2)  Since: _1_ h  w.  9(hv )'  9(hv )  x  y  3(hv )  3(hv )  y  2 Try  (VI.1.3)  3y  2  x  dxdy  3x  = o j  h(v 9  G  x  dx  +  v dy) y  =  r db a r  ( VI.1.4) so  that  taining depend  2TTY  the on  is core.  still By  the  circulation  Kelvin's  Circulation  time.  The  velocity  of  the  around  core  is:  a contour  Theorem  y  condoes  not  97  U  X  =  3w (x,y,t) z  X -  xh (x,y)dxdy  1  2  2 Try J  2TTY  3t  w .(x,y,t)xh(x,y) [(v -v '9G x  G x  )dy  -  (v -v g  G y  )dxJ  (VI.1.5a) 9w U  y  = Y =  1  2iryJ  2 Try  9G  fx,y,t) -yh 3t  (x ,y) dxdy  2  w (x,y,t)yh(-x,y)[(v >  x  v  Q x  )dy  - (v  -v  Q  )J.dx  (VI.1.5b) where on  v_g  9G  is  the  the  v e l o c i t y of  boundary  terms  the  core  vanish/  boundary.  Using  Since  (III.3.7)  w = 0  and  (III.3.8): x IT =  fW  -1 2TTY  1  • V  V  -1  z xhkdxdy  k r w.  2TTY  1 2nyj  V • (vxhk)dxdy  w v hdxdy  (VI.1.6a)  w  (VI.1.6b)  z  x  Similarly:  2TTY  In  terms  of  the  G  v hdxdy z y  s t r e a m f u n c t i on 1  2  h k  -  i r h *ik  ^  ITY  2TTY  2  G  ty  d e f i n e d by  (IV.1.2)  |^dxdy  ( VI.1.7a)  ffdxdy  (VI.1.7b)  98  VI. 2  Circular A  derived  Cores  "circular"  vortex  is  defined  by a s t r e a m f u n c t i on o f t h e  iMx,y)  = -yA(x,y;X  0  t o be one w h i c h  is  form:  ,Y )f(r)  + y B ( x ,y ; X  0  ,Y„) +  0  ty*{*,y) (VI.2.1)  where  r =  [(x-X)  f ( r)  f  has  continuous).  one  and  w = 0  r  (VI.2.1),  and  1 5  >  (VI.2.2) (so A  that  and  the v e l o c i t y  B  are  those  field  is  defined  by  satisfies:  ty*  =o  for  r  (VI.2.3)  > e  and t h e c o r e  (VI.1.2)  and ( V I . 1 . 3 )  boundary  is:  to evaluate  r = e  (VI.1.1)  finds: X = X  where of  ,  The f u n c t i o n s  k  Using  2  derivatives  „ . f1V * *  Thus,  (y-Y) ]  = Inr  continuous  (IV.1.13)  +  2  is  v  + 0(v)  0  a small  the core  radius  ;  Y = Y  parameter  +  0  of  0(v)  order  and t h e d i s t a n c e  (VI.2.4) e|V£nh|  over  which  , the  ratio  h and k  vary  appreciably. We w i s h vanish  as  e -> 0  f(r) There f o r e : dxdy  to  ^  determine with  y  O(Ane)  w ^ 0  ^ 0(e ) 2  ,  h e  one  2  :  finds.  U  and U  x  up t o t e r m s  y  which  constant.  f'(r) v U ^  ^ 0(f) ^  ,  T^-  ,  O(-^r)  , so  f"(r) and that  % 0(A") •  since terms  of  99 order e°  e~  in  must  VIJJ  be r e t a i n e d  and t e r m s  of  k  in  V*  order  V  0  2  ,  e  f(r)  in  +  terms  h  .  of  order  Us i ng  (IV. 4.1)  0(£ne)  (VI.2.5)  k (x,y) 2  9k, 2  dy  9k  dy  T [ r >  k  r  K  o  T  +  ^  0  2  Tx"  2  9y  (VI.2.6)  1  1  h k '2 2  h^k  3  '  3/ 0  2  J-  2x 3_h_o 9x  uh  -  3x_ 2k  0  9j<o_ 9x  0  _2y_ 3h_o_ • J3J/_ 9k_p_' h 3y " 2 k 9y 0  0  + 0(e )  (VI.2.7)  2  For zero  convenience denote  (X,Y)  has  evaluation  been  at  made  the o r i g i n  the o r i g i n .  and  subscripts  Substituting  into  (VI.1.7a) :  U'  f 2TT  1  1 k 2  h  2TTY  o  J  " 0  0  9k l 0  3y  'k with  x = rcoscj)  U'  , y  1 k FT"  ( Z  2  K  +  0  0 "0  + yk  2  0  2x_ 9 h ~h 9x  0  0  0  yk r  J  x  j  3/  = rsincf>  9y J  ' 2  .  9k 2y 3x ~ h  9h 9y  zXfrriALo2 dy "  IX r  0  0  d frdf(r)] dr dr  l!io. + Z iko.1 9x  3x " 2k  0  T  +  '  {  YIP^ "9y Y  r  )  +  '  9i  9y  d  rdf dr  h  0  9h iL 9y  +  n  9y  n  -1 2k  0  9k, dr 3y JJ  r  2  (VI.2.8)  J  2  ( ) dr f  rd<|>dr +• 0 ( e £ n e )  Therefore:  _d f r d f ( r ) l dr dr  0  9y  +  9y  0(e£n e) 2  100  koh*  Y9Bo  3^-"  3y  3y  I  _1_ 3 h 3y  0  1 2ko  +  [ho  Xjkj  2  3B  ko " o L 3y K  0  3 ^  +  2  Y3k, 3y  df dr  'o  J  +  2  0  (e£  y3k 3y  0  +  °  0  y3k 2 3y  0  3y  4  k  e )  2  2  dr  •  2  y  Y_k_o_ 3 £ n ( h k ) _  +  y  3  (ef'(e))  ef(e)f'(e) 2  fdf(r) l dr dr  ay J lh  8k o 3y J  •  In e  (VI.2.9)  0(e£n e) 2  D e f i ne :  re  fdf] dr  r 0  The  kinetic-energy V£  c  dr  of  ( V I . 2 . .10)  the  2  o 'o  'o 'o  of  the  Section core  £  v h krdcj)dr  2  2  fdrt  k„Y  IV.3,  is  the  _ r£  2  2  in  in  C2TT  r e r TT  As  fluid  a  r dr  d<J)dr  = h k £ 2  0  2  0  constant.  J„ '0  [  Y  2 1 T  '0.  =  core .  2  H  +  3x  k  7TpY k 2  is  is  0  3  the  if/  2"  d<j>dr  by,  (VI.2.  0  constant  Therefore,  '3  since  velocity  the of  11)  volume  the  core  i s: ,x U'  1 k h 0  2  3 3y  Y  B(x,y;X,.Y)  r  +  \k(x,y)ln  similarly:  ^*(x,y)  h (x,y)k(x,y)l 2  a, and  +  +  Jx=X  y^  0(££n £) 2  (VI.2.12a)  101 y  _  - i  k„h  U  YB(x,y;X,Y)  3x  2  +  ty*(x,y)  fh (x,y)k(x,y) 2  +  in  agreement  been  Y  fk(x y)£n s  included  in  propagate  as  a  with  vortex  provided  its  core  the  will  be  core  within the in  it.  effects  The  the  the  e is  < k not  than  of  effects  finite  a core  of  and  due  (VI.  of  other  size  2.12b)  vortices  remains  to  of  will  have  therefore  infinitesimal  the  the  size  circular.  advection  therefore,  distortion  the  of  A v ^  induced it  is  the  to  of  have  core.  (D  =  by V  in  vortex  In the  some  This  induced  pj-|_V£nh| ^  is  the  approximation  the  surface  necessary larger motion  then the  v_  that than  general, vorticity  idea  of  discussed  is  to  e >>  the  V  the  :  depth by  be m o r e  the  The II.3  Section  curvature k  surface .  the  approximation  (D, (D . o.  the  if  induced must  of  Thus,  v  by  by  v(°^e|V£nh|  ^ j f ^ °^ I VJlnh | .  vortex  suggested  the  order:  neglected  negligible,  V  with  of  necessary,  considerably but  a core  circular  velocity  negligible, be  the  order:  vortex  must  of  is  velocities of  is  beX  (the  distorted  It  2  VI.3.  Section  curvature  is  0(eiln e)  (VI.2.12)).  in  ty*  A vortex  a*  (IV.3.9)  with  +  of  motion be  nearly  (III.3.4a):  radius  fluid.  surface  of  non-  core  the  are  If  curvature  horizontal i.e.,  102 VI. 3  The  core  of  Validity  of  the  Circular  Suppose  that  the  a vortex  is  nearly  Approximation  vorticity  distribution  circular:  that  within  is:  w. where  v  (VI.3.1)  is  the  v = v  h = h  W  The  z  the  0  o  (r)  )  +  + vh  0  parameter  ( l )  vv  ( 1 )  of  velocity  (VI.2.12).  Within  the  r,(j))  (VI.3.3) (VI.3.4)  fe-*0(v)  of  the  core  (VI.3.2)  (r,<M  = ^  )  U =  (  small  vortex  is:  1 [ w vhdxdy 2 Try 'G " z  hn  ,  2 Try  v  ( 0  )w(°)dxdy  v  +  +v v  ^w^M )  (v_  2TTYJ  1  first  core. even  term  The small  relatively  in  other  z  is  and  advected  periodic  terms  deviations large  However, w (r)  (VI.3.5)  from  changes  w with  v( )w  1 z  )h  +  +  w  0(v ) 2  v( )w 1  o +  (o) (l)  v  ,h )dxdy  ( 0 ) z  0  (VI.3.5)  0(v ) 2  obtained  a circular  core  order  can  of  a  circular  the  w  the  z  core  ' on  — r ^ — r where  w ir ; n  so  that  produce vortex.  distribution  perturbation around  for  magnitude  velocity  a circular  of  ( l )  0  comparable  the  h  h )dxdy  0 )  of  carried  period  z  ( o W d )  that  localized is  w  v  is  in  consider  a small  orbit  are  (  o  +  + The  ( 1 )  f  of  vorticity  . o .S i,n c, e^ a  nearly r  is o  the  103 radius  at  orbits  the  w^^ ment is  which core,  also of  localized initially.  direction  through  vortex  zero  is  the  sweeps  the  almost  w ^ z  due  (i.e.,  an  to it  the  angle  this is  of  of  As  velocity  2TT .  velocity  in V  order:  The  w ^ z  induced net  the  displace-  time  2 (1)  —>—r  V  —T-rn—^ w '  v  by  2  w^  )  £  0 J  u  z  so  that  the  time  averaged  the  departure  from  The  motion  the  the  path  of  of  a c i r c u l a r core vortex  a circular  Similarly, that  the  motion  position motion  of  of  of  of  order:  vv/  The at  most  vortex.  of  net  X ~  the  vortex  small  is  (to for is,  vortex.  The  the only  of  due  order  to  v^v} ^  .  0  wobble  about  v).  more to  velocity  a periodic  order  frequencies  complicated  lowest of  order,  order  amplitude  of  w^^ a  super-  y/e z  upon  the  wobbles  the is  2  =  v  which  e  systematic  order:  For  vortex  with  of  therefore  expects  a circular e  is  one  wobbles (o)  component  vU v.  this  quite  addition  where c  is  is  U  is  to  negligible. the  the  c neglible.  vortex  velocity  velocity of  a  is  circular  104 VI.4  Elliptical The  explicitly  wobble if  distribution This  is  (see,  Cores  the  its  in  the  core  within  motion  is  it.  of  a  elliptical The  depth  vortex with  of  the  g e n e r a 1 i z a t i on o f Ki, r c h h o f f ' s  e.g.,  Lamb  (1916)  p.226)  to  can  be  a uniform fluid  is  vorticity constant.  elliptical  non-planar  demonstrated  vortex  surfaces.  Suppose : w  z  = w =  Q  = const.  (x,y) £ G  0  (x,y)£  G  (VI  .4.1)  with: G = {(x,y)  If  there  are  no  ;  (f)  2  (£] <  boundaries  V iMx,y) 2  or  -w h (x,y)  =  2  0  + =  0  i j  2  +  external  flows:  = -w h [l  + ax  0  0(v ),(x,y) 2  ,  (x,y)  i  (  €  0  +  V  I  gy]  G  G  (VI  .4.3)  whe r e :  It may  is  convenient  use  to  introduce  complex  conformal  transformations.  f^ffU.z)  = -%w h 0  =  0  2 0  coordinates  so  that  one  Then:  [ l + %(a-iB)z + Js(a+i3)z]  , (z,z)  £  G  (z,I)  £  G  (VI.4.5)  105 whence: i>(z,z)  = -%w h  [zz+J (a-i3)z z+%(a+i3)zz ]  2  2  0  +  4>. ( z )  * e  =  (  z  2  s  )  +  V  +  (z,z) € G  *.(z)  z  (z,i) $  >  G  (VI.4.6) the  subscripts  the  interior  e  and  i  denoting  and e x t e r i o r  Consider  the  of the  terms dz d?  Moreover: Therefore if  r,  ell—d-  =  n  (a-b)  UTTbT  t 0  1  field  P *»S +  for  core.  a+b  the core  £2  the mapping  the v e l o c i t y  4>  of '  potentials  mapping:  c = In  complex  if  boundary U l > 1  is: since  is  conformal  outside  is  to  at  z  n=l  vanish  |?|=1 "| d| < 1 .  the core.  Thus,  infinity: (VI.4.7)  Pn?"  Moreove r: I q z  n  n  =  I  n=l  The that  coefficients the v e l o c i t y  q c  (VI.4.8)  n  n  n= l  p  n  and  field  q  n  are  determined  be c o n t i n u o u s  by  requiring  on t h e b o u n d a r y .  One  f i n ds : n > 4  Pn  (VI.4.9)  wo ho d ^[a(2+d)-i3(2-d)]  (VI.4.10)  wo ho d 8  (VI.4.11)  L  4  106 ,w h c (l 2  q i  =  WphocM  Po  d )  2  0  4  2  +  ( 1  +  2  =  w h c (l-d )d 8  p  3  =  w h c^(l-d )d 48  2  w  is  z  +  i g ( 1  _  i\/ " V  a ( 1  +  d  )  2  +  i  B  (  d ) 2  -|  (VI.4.14)  1  _  d  )  2  j  uniformly future  of the core  is  ( V I . 4. 16)  over  the core  times  therefore  since  it  it  is  determined  boundary.  con t r a v a r i a n t  l  2  d i s t r i b u t e d at a l l  y  "  )  distributed  by t h e shape o f t h e  x  d  2  The s t r u c t u r e  V  (VI.4.12)  d ) : ]  (VI.4.15)  L  uniformly  v  _  2  0  Since  The  i e ( 1  0  2  solely  _  d )  2  p  advected.  +  (VI.4.13)  0  2  remains  1  z  w h c^(l-d )^  =  ,  l-d )  2  P l  [ a (  velocity  f i e l d on  the b o u n d a r y  is:  lij 3?|  2 1  h (z,z) 2  i w o c (1 - d ) l-(a-iB)z 2 2  \  +  (a+iB)z  ^ (a(l+d) +i6(l-d) )] 2  T  i wo ab  Ca~+~bT  -  •aa+i b B + e "  1  +  2  T  9  +|  2i e  | ? [ = 1  0(v ) 2  aa(d-3)-iBb(d+3) (VI.4.17)  where  e is The  tional  d e f i n e d by: first  velocity:  c = se  two t e r m s  1 9  are  constant  implying  a  transla-  107 u  x  Consider, Only can  the  . ,,y l  U  =  now,  _ i wo ab 2(a + b ) C the  term:  component  change  V  its  =  x  (e  ve 1 o c i t y  1  is  to  the  )  surface  of  iVw  -de  is:  i Woabe  ( V I . 4 . 19)  -1  field  = -icoi  of  this  of  a uniform  rotation  with  = -icoc(e  velocity  _ 1  9  +de  field  n  9  )  ,  |?|=1  perpendicular  i e  sin -de-  2w ab (a+b)  to  29 i 0  the  velocity  third given  The V  (VI.4.19)  to  the  if: (VI.4.22)  2  term by  causes  the  ellipse  to  rotate  with  angular  (VI.4.22).  pe r p e n d i cu 1 a r • c o m p o n e n t w ab 4(a+b)  x  (VI.4.20)  (VI.4.21)  |  0  Thus,  angular  i s : 2codc  =  core  is:  identical  w  the  a+b  1 B  |e which  = - —  y  component  -de  (a+b)le  co  J  i 9  sin28  boundary V  x  2woabd  component  core  (VI.4.18)  g]  V -iV  -de-  9  velocity  v£ The  l b  This  e  The  -  perpendicular  shape.  Re  a a  0  of  the  aa(d-3)(sin6-dsin39)  fourth +  term  is:  6b(d+3)(cos 8-dcos30) (VI.4.23)  The  terms  in  sin9  and  cos0  imply  an  additional  contribution  to  108 the  trans 1ationa1  velocity  of  the  vortex  B(d+3)-ia(d-3) Since  the  field,  perpendicular  U -iU x  The  j  =  x  (  a  +  )  D  velocity  from  taken  (VI.4.24)  y  a distortion  TTW ab  a constant  U bcos9+U asin9  a+b  contribution  causes  of  is:  y  V"  component  (VI.4.23)  of  for  the  terms  the  ellipse.  the  in  ellipse  sin  and  However, to  1  39  rotate  cos  in  the  once  39 time  the  displacer  0  ment D ^  of e2a  which  ^  changes  .  through by  The  small  The  net  The  l  U  as  the  terms seen  words,  displacement D n„ e ^ t+ . ~ v 2 £  times  of  distorting of  order  w  term w i l l  h  order: in  appears  phase  the  Thus  the  a frame  of  the  of  .  of  from  gradient  other  is  of  a  boundary  the  core  to  +  iB  thereremains  0  henceforth  translation  of  the  be  core  disregarded, is,  from  (VI.4.23) :  _  -  circulation  2Try  net  above  time,  in  velocity  x  U  (or,  for  the  rest,  cancels:  elliptical  lj _iijy  to  this  at  The  nearly  and  is  2TT  2TT) .  (VI.4.18)  due  During  ellipse  very  nearly  boundary  ve  the  rotate  fore  the  =  w ab 4 ( a + b) 9  around  Wo  h  2  3(b-2a)+ia(a-2b) the  dxdy  core  (VI.4.25)  is 2  T r a b w  0  h  0  +  0(v2)  (VI.4.26)  Therefore :  Y  :  Wp  ab 2  (VI . 4 . 2 7 )  109 Using the  core  is  X  for  that  y  TlT+bT  when  circular  one  chooses  and  9ho_ 3y  o is  the v e l o c i t y  3  i (a-2b)  3h_o_' 3x  h3 n  i n agreement  XIII)  X  derived  axes  the x-axis  U -iU  with  using  of  (VI.4.28)  the result  t o be a l o n g  Vh  an a n g l e  0 with  of the vortex  -iy 2h  1 + 3(a-b)  9h 3x  3  the e l l i p s e  0  rotates  with  angular  whose  instead, the major  the  x-axis  is:  2i  Ta+BT  1  If,  so t h a t  the v e l o c i t y  y  coordinates  of the e l l i p s e .  now m a k e s  0  Since  h3  has been  of the e l l i p s e  (Figure  "(b-2a)  a=b t h i s  are the p r i n c i p l e  axis  a  cores.  (VI.4.28) axes  of  then:  U -iU  Notice  the definitions  (VI.4.29)  velocity  co g i v e n  by  (VI.4.22):  (^ Uu - i U ) ( t ) = -2Xh _3 Ilia 8x x  Over  many  y  periods  considered  nearly Y 2uh  v Y  3y  A  changes  constant. 9h 3x  .  0 w  t  +  very  little  0  i  n  2  a  (VI.4.30)  so t h a t  Integrating  s  3ho ( a - b ) ^ ^ ^  2icot  T a + bT  1  . 3(a-b) • , 2(a + b )  " 4^hJ 3 7 T i W  This amplitude  3 0  h  1 + 3(a-b)  then  gives:  .1 j  (VI.4.31a)  t  .  (VI.3.31b)  C O S 2 c j t  i s the equation  of a trochoid  (Figure  of the o s c i l l a t i o n  i n the motion  is:  .  ^  4a)h 3 n  jjutejq.,. 3(.;-b') 3x (a + b) 16 h,  i t may be  j  ^  3x  ,  0  (  v  e  )  X I V ) . The  110  Fig.XIII:  A Vortex  with  an  Elliptical  Core  The The but  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 Core. e c c e n t r i c i t i e s of the e l l i p s e s are correct, t h e i r s i z e s a r e much r e d u c e d .  112 It  is  interesting  (VI.4.23)  the  core  constant  at  with  a much  obey  the  terms  slower  causing  note  distortion  rate  than  other  c i r c u l a r approximation time.  that  vorticity within  lengths  of  core  found which  is  to  In  the has  next  it  cores for  one  puts  disappear. is  therefore  and w i l l  another vorticity  d=0  in  A circular distorted  therefore  correspondingly  section  a continuous  if  such  greater circular  distribution.  113 VI.5  Perturbations Several  uniform By  depth  using  exact  flows  the  to  uniform  depth  flows  general  vortex  anid l e t flow.  is  LI c a n  the be  is  are  example,  of  such  known  Batchelor  a solution for  obtain  for  perturbation  (1967)  as  a  to  but  this  expansions  p.534).  first  non-planar  solutions  planar  more  in  the  the the  s t r e a m f u n c t i on known  of  the  s t r e a m f u n c t i on  non-planar  of  the  flow  planar  that: 0  by  can  the  for  ty  so  few  that  h  of  their.  terms  and  k  Taylor  :  (x - U t , t ) of  r > e  first  expand  =  velocity  + v^  translation  1  ;  (x,t)  of  the  +...  (VI.5.1)  zeroth  order  solution.  expanded:  also  equation _3_  at  ( 1 )  + v U 2  ( 2 )  +. ..  (VI  .5.2)  expanded:  h = h The  be  2  One  U = vll h  can  -v ^ ) + o  -ty{x,t) is  for  as  be  approximated  expansions.  v_U  one  assumed =  solutions  .  0 ;  0  be  v  Solutions  streamfunction  ty(x,t)  w^ ) may  (see,  problem  if/ (x_,t) It  vortex  the  parameter Let  Planar  streamfunction  approximation  small  of  (  1  )  + ...  governing  1  h2  Substituting  + vh  0  V  2  ij,  (VI.5.1)  h  ty  2  = h  0  is  (from  l i JL  [3y  and  3x  »  "  +  T  x^JL  expanding  y  y^fl-  dy  one  —2  h  +  •^ y9y * '  (III.3.7)  l i JL 3x  +  3>T" 3x  X_  V v  has  2  +  <  and  $  to  V I  - - > 5  3  (IV.1.2)): 0  (VI.5.4)  lowest  order:  :  114  vV 2  9 ^ ° > _3_ 9y 9x  hi  which  is  uni form  the requirement  a..  9X  0)  (VI.5.5)  ,(0) ijr be a s o l u t i o n  that  for planar,  ;  depth f1ow. To  order  v :  _i  _9_ 9t  9^ 9y  h 2  ,  I 9 9x  ( 0 )  u X ( l ) V _9_  _  9x  3y  9^ 9x (  Q  a'  )  f ^  ,••(1).  9y  (  1  ) +  _9_ uy(D 9y  ax  2!jiiL»»«»-  0  t  no  (VI.5.6) This boundary  is  a linear  In only  general  subject  to the  initially  as  r •>  ;  = 0  the solution  is  (VI.5.7)  , t=0  c o m p l i c a t e d a n d c a n be  numerically.  We w i s h  long  t o be s o l v e d  conditions: -> 0  found  equation  to look  for solutions  c i r c u l a r and r e t a i n  periods  of time.  Thus,  their we l o o k  to (VI.5.4)  circular  form  for solutions  which are for relatively of  (VI.5.6)  having:  ^ ° ) '= * < ° > ( r )  a n d ty  (  (VI.5.6)  {1)  = 0 .  becomes :  r  (0) rdty dr dr  (0) 1r d Ar r d ty dr  (0) dr (0) d ty dr  9h 9x 9h  0  (  •dr[r  (0) rdty dr dr  s l n'  d I cTr r  (0) A rd^ dr dr  c o s ty = 0  n y ( l ) • ' j d . f l _1 - U  x(l)  (VI.5.8)  115 There  is  a solution '_d_ r a > dr dr [dr dr  U  with:  a  Putting  f(r)  y ( 1 )  (  if: W j dr J dr  0  )  0  ) + a  ,(0) —  ( (  0  (VI.5.9)  -1  \f^f)  = £  (0) d frdifr dr dr  _d_n dr r  (VI.5.10) and  u.-=  r  , (VI.5.9)-  2  becomes  ( V I . 5 . 11) The r  boundary  -* oo - f/  of  s  conditions 0  r  as  require  r -> 0  .  that:  There  f -> c o n s t .  is  as  a one-parameter  family  solutions: x  4u _ 4r a +u a + r 2  _  2  2  (VI.5.12)  2  The r e f o r e : (0)  d±  _  =  (0)  4ar (a +r )  dr  2  2  i f ;  ;  (  0  )  2a£n(a + r ) 2  =  2  (VI.5.13) The  strength  of the  vortex  ,(0)  _  Y = Lim r v  is:  •4a  (VI.5.14)  r->oo  so  that  —  (0)  =  a +  r  2  ,y(D  2  -Y  2h  0  liLo. dx  ,  i i x ( l )  _x_  =  2h  u  0  l b . dy  (VI.5.15) The  vorticity  w The  (0)  radius  initial  of  is, _  to  2 a TT2 +  lowest  order:  :  (VI.5.16)  Y  r ) 2  the core  vorticity  is  of order  distribution  a  has. t h e  .  A vortex form  whose  (VI.5.16)  will  116  1.0  .5  1.0  0  2.0  3.0  r a Fig.XV:  The  Velocity  and  Vorticity  of  a Quasi-Steady  Vortex  therefore vortex  remain  might In  be  circular called  general  it  is  not  depth  of  fluid  of  not  good a p p r o x i m a t i o n  they  a  satisfy  long  periods  of  time.  Such  quasi-steady.  analysis be  the  for  possible  to  is 4>  different equations:  to  perform  varying  since  since  outside  V i|J 2  0 ;  =  0  ,  a  similar need  the V*  core TJV;^]  118 VII.  APPLICATIONS  TO  VII. 1 Atmospheric As for an  this  atmospheric  the  curvature  mid-latitudes the a  Coriolis  dominant  also  phere  the  the  of  in  taking  earth. is  the  the  provide  full  the  of  the  observation  neglected e n t i r e l y ,  Coriolis  may  motion  be  this  is  assumption relaxed  for  the  evidence  section,  force  of  that  we  In  is  for  effects  geostrophic  have  motivation  model  roughly  unrealistic  of  the  of  a simple  account  However,  the  density  equations  to  atmosphere.  including that  introduction, part  attempt  wind  constant  The  an  the  f o r c e , which  shown  has  in  cyclone  role  possibility is  was  of  FLUIDS  Cyclones  explained  thesis  REAL  that  plays the  examined. that  at  It  the  atmos-  may  be  somewhat.  atmosphere  wri tten : sV  |= where the  v_  + and  V  hydrostatic  centrifugal gravity the  v-vv  curl  and of  VP  = - = ^ - - 2 w x v - £ are  pressure  force  are  G J is  obtains  the  three-dimensional  and e f f e c t s  ignored.  the  (VII.1.1)  -|| + V - ( p v ) one  both  angular and  using  £  of  is  the  vectors.  viscosity  the  velocity  = 0  vorticity  (VII.1.1)  of  P  and  acceleration the  equation  of  earth.  is  the due  to  Taking  continuity:  (VII.1.2) equation:  119 We s u p p o s e order, that  as  tangential  the density  earth.  While  a much  III  that  to the surface is  a function  a very  (VI I . 1 . 2 )  lowest  is  + W p  at  the v e l o c i t y  then  i s , to  of the e a r t h .  only  than  that  satisfied  this  above is  the density i f :  V/v = 0  suppose the  certainly  is  uniform.  .  However:  = 0  order.  lowest  We a l s o  of the height  r e s t r i c t i v e assumption  better approximation  Equation  to  in Section  (VII.1.4)  One c a n t h e n  show  ^)'l )  0  (see, f o r example,  Veronis  (1963b))that:  f(_  9 t + v- V which  i n terms  becomes  + 1*.  at  with  h  equation  by  w  ax  The so  w  Thus,  i t  is  force  does.  isolated  vortex  a region  2u is  acts  no l o n g e r  is  (VII.1.7)  the density  in the equations  parameter  vorticity as  II.3  (VII .1.6)  and ( I I I . 3 . 7 )  allowing  a n d on t h e s p h e r e  that  = 0  o  are (III.3.6)  no c h a n g e s  Coriolis  of Section  continuity:  ay  Coriolis  meter  ay  h  ax  z  coordinates  w +2co z z  JL  a(hkv )  produces the  ll  +  A.  a(hkv )  + co .  film  (VII.1.5)  order.  of  equations z  p  p  x  These  2  of the thin  to f i r s t JL  +  in  z  is  as  vortex  of i s o l a t e d  as  where  a constant  possible  to  t o have cores.  w  vary  of motion,  known  2cocos0  with  replaced  z  with  but  including  the C o r i o l i s 9  is  source flows  para-  the c o l a t i t u d e of v o r t i c i t y  i n which the  The w h o l e  v o r t i c i t y then  height  idea  breaks  of a  down.  The  120 only k  satisfactory co  and  effects  are both  of surface Thus  this  thesis  Coriolis  the  that  to neglecting  all  variation.  the v o r t i c e s  in a simple  on a c u r v e d  vorticity  that  throughout  discussed  in  way t o a c c o u n t  rotating  the flow  for  surface  and n o t  will just  core.  been  modelled  way,  one m i g h t  for  the C o r i o l i s  Y  large  force  numbers  vorticity  vortices.  vortex  IV.  by r e q u i r i n g  v(x  r>—  y  n  n  )  a  by a l a r g e  It  is  that  In  the vortex  have  similar  number  possible  to  more  k  simply  k(x ,y )  +  n  n  (VII.1.8)  W  (  ^ n l ^ V V  +  n o  •  as:  • no Y  =  c  o  n  s  t  '  '  *nl  =  c  o  n  s  t  -  (VII.1.9) Unfortunately, Ey  constraint  such n  = 0  a system  does  .  Thus,  there  will  pole)  whose  fixed  at  infinity  (the south  wanes  as  the flow  progresses.  very  large  negligible model (which  the e f f e c t  of this  and the system  f o r the flow. would  involve  If  solving  still  the  appear  vortex  the equations  waxes  of  vortices  will  become  provide  investigation  required  t o be a  strength  t h e number  single  should  A proper  not respect  a  vortex and is  satisfactory  of this of motion  system by  of  account  strengths  „  n " FTvV^ Y  extended  of point  in Section  2  n "  with  to:  may be r e s t a t e d  Y  flows  an a t m o s p h e r i c  discussed  according -  planar  model  vortices  ,*,  many  by  the  which  amounts  and depth  conclude  A vortex  t o be t h e a p p r o x i m a t i o n  which  be e x t e n d e d  effects.  However,  vary  constant  one must  have  seems  curvature  cannot  necessarily at  resolution  121  computer initial  f o r l a r g e numbers o f v o r t i c e s , l o n g times and conditions)  i s beyond  the scope of t h i s t h e s i s .  varied  122 IV.2  Superfluid In  Vortices  recent  almost  exclusively  Hell.  While  tions  which  completely that  years in  work  relation  the complete describe  of  of  component  Vortices  vortex  having  are observed.  The c o r e s  ized,  n o t by r e g i o n s  of  singular arity. of  one o r two  describe  £  equations  m = mass  is  zero  constant  not  incomin  of are  is  is  units helium  characterin  truly  at the  and i s  which  singul-  the  order  Angstroms.  expects,  n n  nearly  The v o r t i c i t y  of the core  the motion  h  velocity  quantized  is  equa-  accepted  b u t by a r e g i o n  of the f l u i d  liquid  is  of the v o r t i c e s  rapidly.  of  been  are s t i l l it  very  constant,  vorticity,  but the d e n s i t y  On  The  decreases  The r a d i u s  Hell  circulation  h/m = 1 0 " crrr/se.c (h = P l a n c k ' s  density  liquid  cores,  has  differential  of the f l u i d  atom)  the  to the b e h a v i o u r  (see Putterman(1974))  irr.otationa 1 and, o u t s i d e pressible.  vortices  set of p a r t i a l  the flow  understood  the s u p e r f l u i d  on i s o l a t e d  =  6  n  =  therefore, of  c  o  vortices  n  s  t  of motions o- c< dk 2i n n h k l2 az  but that  (IV. 3.4)  should  now:  (VII.2.1)  f o r a vortex  in  •h. n  n  +  equation  '  1  2  + >*(z,z>  that  +1  n  +  system  ..n h n  E Y^(Z,Z;Z. k^n k  K  n  are  then:  a Y B (z ,z ;z ,z ) n n n az  +  az  ,z, )  (VII.2.2)  K  z=z„ There the  is  no v o r t e x  fluid  outside  streamfunction the core  is  and the k i n e t i c  not conserved.  energy  However,  in  of  123 both  the  vortex  case  of  uniform  streamfunctions  flow  exist.  and  These  of  flow  are,  a  ,y  2  h(x  2  ,y  ) +  2y  ^ * ( X ,y  J  1  r  '  J  )1  k(x ,y )(£n6 -l) n  n  n  (VII.2.3) v  N E iy B(x =i l f  ,y  2  n  n  2y ^x ,y )}  +  ; x . ,y. )  n  , k= l  , J  ^  plane  respectively:  ' n n n' n " n n J N ft = % E E Y y.^x ,y ; x . ,y. ) + n = l k/n n n k k' -  on  N r E y if. *F(x ,y ; x , ,y. ) + % E i y B ( x k/n n'^n k'-'k' n=l^  N ft = h E n=l + Y  depth  n  n  n  ,  n  n  ; x , ,y. ) k k'  h=l (VII.2.4)  tion  of  The  conservation  the  k i n e t i c energy  These problems date.  in  equations  more  of  it  difficult  is  methods  the  seem  free to  of  corresponds the  now  be  used  geometries  with  surface solve  necessary.  of for  to  the  fluid external to  than  k = k  0  is  + ar  a rotating ¥ ( x ,y : x ,y 1  2  been  that  1  )  of  since  fluid. and  conserva-  to  discuss  have  particular interest  a cylindrical container of  ft  can  general  A problem  shape  of  the  core.  vortex used  to  vortices this  is  in  the  Unfortunately, numerical  124 VIII.  CONCLUSION In  this  vortex  motion  fluids  of  has  small  To a similar fluid  thesis been  but  the  the  generalized  varying  author's  nature  is  by  "orthomorphic  two  The  fields  in  shown in  to  purely  it  fluids be  a small  horizontal He  has  is  has  vortex  motion.  in  only  also  describing If  the  the  and  to  a plane  or  a sphere,  explicitly  in  terms  As  did  shown  the  more  that  (1943) there  is  The  of the  be  Lamb  may  only  be example  The  the  vortex  approximations  ratio  of  equations  there  is  of of  the  of  velocity  perturbation  the  of  choice  used  expansion  vertical are  to  examined.  a function surface  are  h(x,y)  be  curvature  fluid  is  uniform,  flow  is  topologica11y  similar  of  motion  written  equations  there  on  may  be  are  h(x,y).  of  case  motion  elliptic  for  might  motion  order  effects  the  if  plane."  a suitable  determine  that  general  of  fluid  a systematic  surface  of  equations  functions Lin  the  by  depth.  depth  boundaries  Green's  to  lowest  shown  no  the  that,  varying  scales;  the  that,  fields  the  work  sphere.  p a r a m e t e r A r e p r e s e n t ! * ng  for  depth,  a  shown  consistent  suited  In  on  possible  of  onto  published  two-dtrriension.aT.flow.  vortices  author  coordinates,  projection  j u s t i f i c a t i o n that  discussed  suggested  velocity  in  surfaces.  only  vortex  rectilinear  vortices  the  depth,  by  by  curved  who  obtained  is  on  of  include  (1916)  constant  approximated  to  theory  Lamb  of  without  depth  knowledge,  is  assumes  conventional  with  can  boundaries  be w r i t t e n  partial  terms  differential  rectilinear vortices,  a vortex  in  and  streamfunction  varying of  the  operators.  the  author  for  these  has equations  125  of  motion,  fluid, under  that  and  that,  examined depth  extend  cores  cause  discussed.  It  cores  is  core much  that  it  energy  of  the  transforms  surface  of  these  simply  the  has  been  curvature  and  differences  stability  qualitative  of  rigidly  differences  also  stability.  was  of  shown  depends  is  systems  qualitative  effects  introduces  the  vortex  marked  of  the  wobble  kinetic  uniform,  Studies  VI  asymmetries  the  showing  showed t h a t  Section  If  the  is  simple  systems.  symmetric  vortex.  of  of  considerations  were  radially  depth  V,  can  systems  In  a  Section  vortex  to  the  behaviour  in  rotating  r e l a t e d to  transformations.  variation  between  is  if  conformal The  that  it  non-infinitesimal that  only  small  on  than  the  velocity  their  wobbles  approximately  smaller  the  into  of  strength the  circular core  vortex  motion  the  radius  but of  amplitude,  and  may  be  neglected. The  theory  developed  in  Sections  etical  interest  as  a generalization  vortex  motion.  It  is  applied seems and is  to  atmospheric  impossible  of  surface  quite  that  model  of  to  reconcile  these  that  important  in  such  an  role  in  vortices  a cyclone.  an  in  in  atmosphere  Of  As the  simple  would  course,  cyclogenesis, be  of  vortices but  provide if  ask  in  effects  the then  greatly  is  of  classical  to  shown  a similar,  would  the  however,  motion.  curvature  possible  phere  natural,  of  1 1 1 - VI  how  Section the of  theortheory  it  the  first force  incidence  suppressed.  it force  type.  non-rotating,  Coriolis  be  VII.l,  Coriolis  this  a simple  may  of  It  atmosorder plays of  cyclones  126  Vortices theory that  of  the  longer  Section core  cases  IV  of  flow with  of  calculating  one  remain  uniform  does  makes  the  depth  depth  can  In  for  the  and  of  planar  the  most  there  is  still  V  described  and  general,  flow,  For  functions  be  a modification  constant.  exist.  varying  field.  if  helium  streamfunction  of  streamfunction  velocity  superfluid  radii  a vortex  special  in  B  but flow  to  for a  which  the  require  there  interesting the  by  is  no  the vortex problems  practical describe  problem the  127 BIBLIOGRAPHY B a t c h e l o r , G . K . ( 1 9 6 7 ) An I n t r o d u c t i o n t o F l u i d (Cambridge U n i v e r s i t y P r e s s , Cambridge, Chapman,  D.M.F. (1977) C o l u m b i a)  Chapman,  D.M.F.  Courant,  ( 1978)  M.Sc. J.  Thesis  Math.  (University  Phys.  R. , a n d H i l b e r t , D. ( 1 9 6 2 ) P h y s i c s (John W i l e y & Sons, Sec.IV, p.290.  Eisenhart, of Havelock,  19,  (1931)  Phil.Mag.  of  British  p.1988.  Methods o f M a t h e m a t i c a l New Y o r k ) V o l . 1 1 , C h . I V ,  L . P . ( 1 9 0 9 ) A T r e a t i s e on t h e C u r v e s and S u r f a c e s ( G i n n a n d T.H.  Dynamics England)  1_1,  D i f f e r e n t i a l Geometry Co., Boston), p.93.  p.617.  H e l m h o l t z , H. ( 1 8 5 8 ) C r e l l e ' s J o u r n a l , t r a n s l a t e d : S e r . 4 , No.226, Supp.33, p.485 (1867) Kelvin,  Lord  ( 1878),  Nature  Kelvin,  Lord  ( 1869),  T r a n s . Roy . S o c . , E d i n .  Karman,  T.von  (1912)  18.,  p.13  Phys.Zeits.,  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 Mechanik ( L e i p z i g ) p.255 Koege,  p.  Lagally, Lamb, Lin,  H.  (1918) M.  Acta  (1921)  Math.  41,  Math.Seits.,  Phil.Mag.  2_5  13,  p.49  uber  mathematishe  Physik,  p.306 10.,  p.231  (1916) Hydrodynamics (Cambridge U n i v e r s i t y Cambridge, England) 4th ed. , A r t . 8 0 , Ch.IV,  CC.  ( 1 9 4 3 ) On t h e M o t i o n ( U n i v e r s i t y of Toronto  o f V o r t i c e s i n Two Press, Toronto)  Press, p.101. Dimensions  M a s o t t i , A.  (1931)  Atti.Pontif.Accad.Sci.Nuovi.Lincei  Mertz,  ( 1978)  Phys.Fluids.  G.J.  Morton,  W.B.  Onsager,  L.  Osborne,  D.V.  ( 1935) ( 1949)  2_1,  P r o c . R o y . I r i s'h Nuovo  et.al.  Cim.  (1963)  Supp.  84,  p.209.  p.1092 Acad. 6_,  Can.J.Phys.  A 4_2,  p.21  p.249 4J_, p . 8 2 0  P u t t e r m a n , S . J . (1974) S u p e r f l u i d Hydrodynamics P u b l i s h i n g Co. , .Amsterdam, 1974) p . l 9 f f  (North Holland and p.267ff.  128  Routh,  E.J.,  Proc.L.M.S.,  1_2  p.83  Sommerfeld, A.(1949) P a r t i a l D i f f e r e n t i a l Equations ( 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 . Souriau  J . M . ( 1969) S t r u c t u r e des P a r i s , 1969)  Thomson,  J . J . (1883) (Adams P r i z e  Tkachenko,  V.  ( 1966)  T r e a t i s e on Essay 1882, Sov.Phys.  the Motion of Vortex Rings M a c M i l l a n , London) p.95.  JETP  G. ,  ( 1963a)  J.Mar.Res.  Zl_  Veronis,  G. , ( 1 9 6 3 b )  J.Mar.Res.  21,  E.J.,  et.al.  (1979)  Physics  S y s t e m e s Dynami ques ( D u n o d ,  Veronis,  Yarmchuk,  in  t  2_3,  p.1049  p.110 p.199  Phys.Rev.Lett. 43,  p.214  129 APPENDIX  A:  MATHEMATICAL  Let with  where and  k  N  g  .  One may  = e(-det  e is  the  is  extension M  M be a t w o - d i m e n s i o n a l  metric o*  FORMALISM  define  a scalar o*  , namely,  the  Riemannian  a two-form  manifold  by:  a*  g)\  (Al)  antisymmetric  of  smooth  tensor  function  to  on  a two-form  unique  M . on  two-form  There  the  N  with is  a  e  =  1 2  1  natural  2N-dimensiona1  a  (dx'x...xdx ,dy'x...xdy ) N  density  manifold  satisfying: N E yo*(dx ,dy ) n=l  =  n  (A2)  n  n  where and  the a  y  is  ,n=l,...,N  are  constants.  Note  d i f f e r e n t i ab 1 e e v e r y w h e r e ,  a symplectic non-trivial  structure  on  three-forms  M  .  N  cannot  a  (Note exist  a  ker  therefore  that  on  that  Va*  a=0  induces  = 0  since  two-dimensional  mani f o l d . ) Suppose A natural  flow  that is  ft  is  induced,  some the  scalar  function  equations  of  on  motion  M of  .  N  which  a re : HY  -1  ^| = a  A  Vfi  = grad n  (A3) N  where  denotes  x  derivative.  Notice  4 f = Vft so and  t h a t ft i s (A3)  matics  of  position  M  and  V  is  the  exterior  that: = a(fl,n) = 0  conserved.  becomes  on  In  (IV.4.4).  symplectic  systems  harmonic For see  coordinates  further Souriau  reference (1969).  o* = h ( x , y ) k ( x , y ) e 2  on  the  mathe-  130 APPENDIX  B:  E V A L U A T I O N OF  All Section  the  V  may  V  ~  sums  be e v a l u a t e d  N R, ( z )  special  SUMS  e x p ( 2TTJ  necessary easily  =  calculations  of  Ln/N)  (l-zexp(27rin/N7T "  1  the  once:  '  2  n  for  Z  c  o  m  P  l  e  >  x  L =  1  (Bl) is  known.  Suppose  R. ( z )  The  infinite  reordering  R, ( z )  The  =  of  00 E k z  the 1/1  K  -  is  absolutely  n=l  second  sum  vanishes  unless  E N( r N - L + l ) z r 1  r  N  _  L+k-1  =  L  N-fa  N((N-L+l)z ' +(L-l)z N  L  (l-z ) N  right  regions hence,  sides  of by  the  of  both  complex  analytic  To  = n =  convergent  (Bl)  and  plane  (B3)  exp(  z  ~  = rN  , r an  If  I I-  L  )  (B4) are  excluding (B4)  is  z=xe  -rri/n  (2n + l ) T r i L / N )  N  +  M  (l x ) N  analytic  the  l (l-exp((2n + l)Tri/N))  +  Nth  valid  , -  (L-l)x  e  2  2 N  ^  = "  L  -  N  v n  1  l,  l - e x p ( 2-TTi L k / N ) (l-exp(2Trik/N)r  =  for  '  - )  . . ^ m  / <N n  R  (  / \ " L  Z }  R  V  all of  all  ( x e  one: z.  7  7  1  ^  M  ' (B5)  2  L  in  boots  +  c  integer.  ,_N-L + 1  (B4)  put  L  N(-(N-L l)x -  S  allowing  2  T (x)  N T, ( x )  2 N  continuation,  evaluate  (B2)  e x p ( 2 T r i (L + k - l ) n / N )  =  The  Then:  N  k-1  =  < 1 .  sums:  E  i  |z|  ,k-l £ kz e x p ( 2 77 i ( k - 1 ) / N) k=l  series  L  R. ( z )  that  N E exp(2-rriLn/N) n=l  =  L  the  first  /  n (  Z  }  w )  131  L  i  N ( l + ( N - l ) z  m  z-1  N  - ( N - L + l ) z  ( l - z  i n  N ( N ( N - l ) z  m  N  "  zll  1  N  -  - ( L - l ) z  L  - ( N - L + l ) ( N - L ) z N  -  1  ( 1 - Z  N  z-1  2 ( 1 - Z  +l) (N-L) -  L  )  N  N  N  ~  L  "  1  - ( L - l ) ( 2 N - L ) z  2  N  "  L  "  1  )  )  (N(N-l)-(N-L + l)(N-L)z~  KN-L  -  2  )  - 2 N Z  Mm  N  L  -(L-l);(2N-L)z  N  -  L  )  )  (N-L)(L-1)(2N-L) 2N  Sg(N-L) ( 2 - L )  L ' H o p i t a l ' s  [ij r^^wm'  ( B 6 )  rule  has  s  been  used  > • '»' - > N 1  twice.  (B7)  

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