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On the stability and propagation of barotropic modons in slowly varying media Swaters, Gordon Edwin 1985

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ON THE STABILITY AND PROPAGATION OF BAROTROPIC MODONS IN SLOWLY VARYING MEDIA by GORDON EDWIN SWATERS B.Math.(Honours), U n i v e r s i t y Of Waterloo, 1980 M . S c , U n i v e r s i t y Of B r i t i s h Columbia, 1983  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  i n  THE FACULTY OF GRADUATE STUDIES Department Of Mathematics I n s t i t u t e Of A p p l i e d Mathematics Department Of Oceanography  We accept t h i s t h e s i s as conforming to the r e q u i r e d  standard  THE UNIVERSITY OF BRITISH COLUMBIA July  ©  1985  Gordon Edwin Swaters, 1985  V  In  presenting  this  thesis  in  partial  fulfilment  of  the  requirements f o r an advanced degree at the U n i v e r s i t y  of  British  Columbia,  I  it  freely  available  for  permission  agree  for  purposes may or  her  that  the  Library  shall  reference  and  study.  I  extensive  p u b l i c a t i o n of t h i s t h e s i s allowed without my  Department of  written  Date:  July  1,  It for  is  1985  agree  Department or  understood  financial  permission.  Mathematics  The U n i v e r s i t y of B r i t i s h 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5  further  that  c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y  be granted by the Head of my  representatives.  make  Columbia  gain  that  by  copying  shall  not  his or be  i i  Abstract  Two a s p e c t s in  this  are  derived  and  thesis.  First,  o f b a r o t r o p i c modons a r e e x a m i n e d  sufficient  neutral stability  i n t h e form o f an i n t e g r a l c o n s t r a i n t  eastward-travelling  travelling to  of t h e theory  modons.  and w e s t w a r d - t r a v e l l i n g  perturbations  in  which  the  It  is  modons a r e  results  stability  imply  curve  increases.  modons  wavenumber  neutrally  must  travelling inferred spectra  varying  increase  condition  is  the  basis  and  as  K/|T?|  computed  representative atmospheric  oceanic  for  with of the modons  and a n n u a l l y of westward-  modons  cannot  be  o f t h e o b s e r v e d wavenumber e d d y e n e r g y  f o r t h e atmosphere and ocean.  the  propagation  medium.  perturbation an  to  The n e u t r a l s t a b i l i t y  modons  Second, a l e a d i n g order describe  of the neutral  stable t o the observed seasonally-  atmospheric on  begin  Eastward-travelling  averaged atmospheric eddies.  satisfying  i s t h e modon wavenumber.  eddy e n e r g y s p e c t r a  atmosphere and ocean.  stable  McWilliams et a l . ( l 9 8 l )  The n e u t r a l s t a b i l i t y  mesoscale  are  by  eastward-  mainly i n  (|T?|)  t h a t when K/|T?|>1 t h e s l o p e  proposed  eastward-travelling  K  westward  neutrally  energy i s contained  and | T J | > K , r e s p e c t i v e l y , where  These  for  shown t h a t  s p e c t r a l c o m p o n e n t s w i t h wavenumber m a g n i t u d e s |TJ|<K  conditions  of  Two  barotropic  problems  s o l u t i o n i s obtained  eastward-travelling  boundary l a y e r . translation  perturbation  modon  are  theory  i s developed t o  modons posed  in  and  decay  slowly  solved.  describing the propagation  A of  m o d u l a t e d by a weak b o t t o m Ekman  The r e s u l t s p r e d i c t t h a t t h e modon  speed  a  exponentially  and  that  radius the  and modon  wavenumber i n c r e a s e s amplitude  decay  results  agree  exponentially,  in  the  also  streamfunction  with  the  McWilliams et a l . ( l 9 8 l ) .  resulting  exponential  and v o r t i c i t y .  numerical  A leading order  describe  Nonlinear  the  translation  perturbation  evolution  of  the  equations slowly  are  varying  s p e e d a n d wavenumber f o r a r b i t r a r y  topography. meridional  hyperbolic  To  leading  gradients  in  order,  the  topography.  modon  obtained  f o r small-amplitude  is  theory i s  topography.  general  solution  topographic  proportional  to  The  the  derived  modon  to  radius,  unaffected  by  perturbation  perturbations t r a n s i e n t s and  topography.  i s a p p l i e d t o r i d g e - l i k e and  configurations.  varying  s p e e d a n d wavenumber  t a k e t h e form of westward and e a s t w a r d ^ t r a v e l l i n g a s t a t i o n a r y component  of  finite-amplitude  Analytical  s o l u t i o n s f o r t h e modon r a d i u s , t r a n s l a t i o n  These  solution  d e v e l o p e d d e s c r i b i n g modon p r o p a g a t i o n o v e r s l o w l y  topography.  are  i n an  The  escarpment-like  Table of Contents  Abstract L i s t of F i g u r e s Acknowledgement Chapter I INTRODUCTION  i i v vi  1  Chapter II STABILITY OF BAROTROPIC MODONS 11 2.1 I n t e g r a l C o n s t r a i n t s For P e r t u r b a t i o n s Of B a r o t r o p i c Modons ....14 2.2 D i s c u s s i o n And A p p l i c a t i o n To The Atmosphere And Ocean 18 Chapter I I I MODON PROPAGATION IN A SLOWLY VARYING MEDIUM 29 3.1 P e r t u r b a t i o n S o l u t i o n For Modon Propagation Over A Bottom Ekman Boundary Layer 32 3.1.1 Formulation And S o l u t i o n Of The D i s s i p a t i o n Problem 33 3.1.2 D i s c u s s i o n Of The D i s s i p a t i o n S o l u t i o n 42 3.2 Modon Propagation Over Slowly V a r y i n g Topography ..55 3.2.1 P e r t u r b a t i o n S o l u t i o n For Modon Propagation Over Slowly V a r y i n g Topography 55 3.2.2 D i s c u s s i o n Of The Small-Amplitude Topographic Solution 70 3.2.3 G a u s s i a n - r i d g e Topography 78 3.2.4 Tanh-escarpment Topography 97 Chapter IV CONCLUSIONS  116  BIBLIOGRAPHY  120  APPENDIX A - CALCULATION OF SOLVABILITY INTEGRALS IN TOPOGRAPHIC PROBLEM  126  V  List  of F i g u r e s  1. Three dimensional contour p l o t of the s u r f a c e displacement a s s o c i a t e d with an e a s t w a r d - t r a v e l l i n g modon  10  2. S t a b i l i t y modon  26  regime diagram f o r an e a s t w a r d - t r a v e l l i n g  3. Two dimensional  atmospheric k i n e t i c energy spectrum. .27  4. S c a l a r wavenumber s p e c t r a of b a r o t r o p i c k i n e t i c f o r the ocean  energy 28  5. Sequence of contour p l o t s of the modon p a t h l i n e s f o r the Ekman d i s s i p a t i o n problem. 45 6. Sequence of contour p l o t s of the modon v o r t i c i t y Ekman d i s s i p a t i o n problem  f o r the 50  7. Sequence of s p a c e - l i k e s l i c e s i n space-time showing the e v o l u t i o n of the modon t r a n s l a t i o n speed f o r modon propagation over a slowly v a r y i n g gaussian r i d g e 82 8. Sequence of s p a c e - l i k e s l i c e s i n space-time showing the e v o l u t i o n of the modon r a d i u s f o r modon propagation over a slowly v a r y i n g gaussian r i d g e 87 9. Sequence of s p a c e - l i k e s l i c e s i n space-time showing the e v o l u t i o n of the modon wavenumber f o r modon propagation over a slowly v a r y i n g gaussian r i d g e 92 10. Sequence of s p a c e - l i k e s l i c e s i n space-time showing the e v o l u t i o n of the modon t r a n s l a t i o n speed f o r modon propagation over a slowly v a r y i n g h y p e r b o l i c - t a n g e n t escarpment 101 11. Sequence of s p a c e - l i k e s l i c e s i n space-time showing the e v o l u t i o n of the modon r a d i u s f o r modon propagation over a slowly v a r y i n g h y p e r b o l i c - t a n g e n t escarpment 106 12.  Sequence of s p a c e - l i k e s l i c e s i n space-time showing the e v o l u t i o n of the modon wavenumber f o r modon propagation over a slowly v a r y i n g h y p e r b o l i c - t a n g e n t escarpment. 111  Acknowledgement  It  isa  pleasure  discussions and  on  to  thank  applied  Dr. Lawrence  A.  Mysak  f o r many  mathematics and p h y s i c a l oceanography,  f o r h i s e n t h u s i a s t i c encouragement of and i n t e r e s t  research.  The  author  also  in  thanks Dr. Paul LeBlond, Dr. Kevin  Hamilton, Dr. U r i Ascher and Dr. Brian Seymour f o r many discussions  pertaining  supported by N a t u r a l and  to  Sciences  this  h i s research. and E n g i n e e r i n g  This  helpful  research  Research  was  Council  of  Canada  U.S. O f f i c e of Naval Research grants awarded to  Dr.  Lawrence A. Mysak, and by a U.B.C. Department of Mathematics  Teaching A s s i s t a n t s h i p . Of s p e c i a l note I wish the beer  garden r e g u l a r s and i n p a r t i c u l a r Don Dunbar, Karen  K e i t h Thomson and Tom K e s s l e r was  oceanography  not  f o r you  been as much f u n .  long l i f e and  prosperity.  -  Perry, If i t  people the whole t h i n g would not n e a r l y have  vi i  I have r e s o l v e d t o q u i t the  consideration  mind, and t h i s , which  has  for  o n l y a b s t r a c t geometry, t h a t  of q u e s t i o n s  i n order  to  i t s object  that serve  study  another  the explanation  only kind  i s to  say,  to exercise the of  geometry,  o f t h e phenomena o f  nature. Rene D e s c a r t e s  1  I.  INTRODUCTION  R i n g s a n d e d d i e s p l a y an i m p o r t a n t  role  of t h e ocean.  F l i e r l ( l 9 7 7 ) has argued,  northwestern  Atlantic  fluctuation  kinetic  distribution and  their  based  of  a  Gulf  average radii  are  Stream  about  distribution estimate  field  are  on  of  eddy i s t y p i c a l l y  comparable.  of this  between 2 t o 3 y e a r s , t h e  speeds a r e on t h e o r d e r of and  L a i and  data that the l i f e s p a n  the v e r t i c a l  1  5 km d a y " ,  a t any g i v e n  15 w i t h maximum c o u n t s a b o u t  observations suggest  mesoscales  (i.e.,  twice  t h a t eddy a n d r i n g  f o r thermal, v o r t i c i t y  the  e x t e n t i s up t o one  The number o f r i n g s a n d e d d i e s o b s e r v e d  These  ocean  and a t h e o r e t i c a l  wave  100 km  must be i m p o r t a n t the  observed  on q u a s i g e o s t r o p h i c d y n a m i c s (QGD) o f e d d i e s  t i m e was a p p r o x i m a t e l y value.  the  dynamics  f o r example, t h a t i n the  e s t i m a t e from h i s t o r i c a l  translation  kilometer.  energy  dispersive  Richardson(1977)  Ocean  i n the o v e r a l l  this  dynamics  and n u t r i e n t m i x i n g  in  l e n g t h s c a l e s on t h e o r d e r o f  100 km). E d d i e s a n d r i n g s c a n be g e n e r a t e d geometry  configurations.  theoretically (Ikeda e t a l . ,  described 1984),  from a v a r i e t y  Observations by  coastal  of  eddies  (Willmott  and  western  boundary  currents  Csanady, 1979).  M y s a k , 1980)  flow  and  the  been  instability interactions  ( S w a t e r s a n d M y s a k , 1 9 8 5 ) , p l a n e t a r y wave r e f l e c t i o n geometry  flow-  have  current  topographic-mean  of  o f f coastal  meandering  ( L a i and- R i c h a r d s o n ,  1977  of and  I n a d d i t i o n t o t h e above mechanisms,  planetary  e d d i e s c a n a l s o be o b t a i n e d when t h e e f f e c t s o f p h a s e  dispersion  2  balance  amplitude  to produce the  d i s p e r s i o n i n p l a n e t a r y waves ( C l a r k e ,  solitary  planetary  Malanotte-Rizzoli(1982) and  oceanic  according equation  waves.  classified  models  s o l i t a r y p l a n e t a r y waves i n t o one  to  the  nonlinearity  (PVE).  Steadily  of  the  translating  of  atmospheric  of two  categories  potential  vorticity  solutions  of  reduce t o d e s c r i b i n g the p o t e n t i a l v o r t i c i t y  as a  of  solitary  the  pathlines.  Rossby  assuming  solitons  described  by  approach  i n c l u d e t h e M a x w o r t h y and  Korteweg-De V r i e s zonal  channel  derivations  P  (KdV)  and  models  by  in  a  KdV  solitary  a s s u m i n g P t o be  analytic.  second category  function.  The  to  t o some f i n i t e  obtained  prototype  on  identically continuous  the  /3-plane  outside but  and  specify  not  this  a  in  nonanalytic  These at  hereafter i s only  Stern(l975)  named them as bounded  streamf unction  region.  /3-plane  streamfunction  a  wave  symmetric  infinite  as  and  variation  radially  ( u s u a l l y two).  function  differentiable  in  Similar  hence a s o l i t a r y  P  that the  the  QGD  topographic  obtained  order  with  this  Hendershott(1980)  modon s o l u t i o n (and  by a s s u m i n g P t o be a l i n e a r domain  from  s o l u t i o n s that t h i s procedure y i e l d s ,  differentiable  of  flow.  wave s o l u t i o n s on an  is  (P)  waves a r e  Examples  zonal  and  has  c a l l e d modons, have t h e p r o p e r t y  the  sheared  equation  Also,• F l i e r K 1 9 7 9 )  PVE  function  modified-KdV equations a  the  Redekopp(1976) d e r i v a t i o n of  with cross-channel  quasigeostrophic  The  and  analytic.  Malanotte-Rizzoli  resulted  solution.  be  with  Malanotte-Rizzoli(1984) also  to  1971)  such)  circular vanishing  solutions  t h e b o u n d a r y and  were had  a  3  dipole vortex Larichev barotropic region Figure  and R e z n i k ( l 9 7 6 )  modon  when 1  have  boundary  by  the  for a  solutions  its  structure.  determining  three-dimensional  plot  smooth  to  vorticity  geometry  of  has a f i n i t e  fluids  claims  (see  modon).  These  except  (Flierl  been  numerical  modon  e t a l . , 1980) a n d  and  Verkley,  integrations  have  of  the  McWilliams et a l . ( l 9 8 l ) numerically Ekman  dissipation  on  d e c a y was a p p r o x i m a t e l y details  of  the  decay  separable  c o u l d n o t be e x p l i c i t l y occur  on  the  experiments with the the  onset  modon  exponential  primarily PVE.  calculated  modon p r o p a g a t i o n  in  1984).  t o have e s t a b l i s h e d a theorem  dynamics  linear.  focussed  For the  that  solitary  on  example, effect  of  and c o n c l u d e d t h a t t h e  a n d shape  preserving.  The  t h e a m p l i t u d e and t r a n s l a t i o n speed  determined except that they appeared "dispersion  random v o r t i c i t y  of i n s t a b i l i t y  the  subsequently  wave s o l u t i o n o f t h e PVE a s s u m i n g t h a t P i s p i e c e w i s e of  at  step d i s c o n t i n u i t y i n  s t a t e s t h a t t h e modon i s t h e u n i q u e l o c a l i z e d  Studies  the  infinity  a  everywhere  ( T r i b b i a , 1984  Kloeden(1985a,1985b)  at  These s o l u t i o n s have  two-layer  called  a form f o r P i n t h e e x t e r i o r vanishes  radial derivative.  spherical  what i s now  streamfunction  where t h e v o r t i c i t y  generalized  obtained  curve".  Other  perturbations  to  numerical  indicated  that  was d e t e r m i n e d by t h e l e n g t h s c a l e o f  perturbations. McWilliams  collisions  and  between  parameter v a l u e s ,  Zabusky(1982) barotropic  numerically  modons.  Depending  simulated on  the  a wide range of i n t e r a c t i o n p o s s i b i l i t i e s  were  4  observed,  from  interactions multitude as  to  soliton-like,  fusion-like  whether  or  (Flierl  theoretical  fission-like  t o the complete a n n i h i l a t i o n of the v o r t i c e s .  of i n t e r a c t i o n p o s s i b i l i t i e s  soliton  and  not  some  debate  t h e modon i s i n f a c t a t w o - d i m e n s i o n a l  e t a l . , 1980  framework  resulted in  The  has  and  yet  McWilliams,  been  developed  1980).  No  from which t o  understand these i n t e r a c t i o n s . Mied  and  considered (Flierl, to  McWilliams(1983) I t h a d been  baroclinic and  vertical  into  a b a r o t r o p i c modon.  numerically  lower l a y e r s .  vertical  Their  i n the north-south  large.  If  place.  evolution  Laboratory indicated  p l a n e modon g e n e s i s  between the  the  vertical  initial  s t u d i e s o f modon  conditions.  theorem  (Flierl  varying  isolated  angular  momentum.  genesis  formation  disturbance  a  which on  ensues of  likely  w i t h no  coupling  t o c r e a t e modonvortices.  e t a l . , 1983)  evolves  from  was f o r m u l a t e d  states  the  a x i s l a y i n other  (Flierl  This conclusion  e t a l . , 1983)  that  any  very as a  slowly  a /3-plane must h a v e z e r o n e t  The modon i s one o f t h e  flow c o n f i g u r a t i o n s with t h i s  centers  o f two q u a s i g e o s t r o p h i c  that dipole vortex  of  a x i s and c o u n t e r - r o t a t i n g  M c W i 1 1 i a m s ( 1 9 8 3 ) was a l s o a b l e  l i k e d i p o l e s by t h e c o l l i s i o n  tend  r e s u l t s i n d i c a t e d t h a t when t h e  the h o r i z o n t a l separation i s too  the  d i r e c t i o n s t h e two v o r t i c e s t e n d e d t o s e p a r a t e  general  suggested  Mied and Lindemann(1982)  calculated  eddy w i t h a t i l t e d  axis lies  vortices  taking  have  1976) t h a t a p u r e b a r o c l i n i c eddy w o u l d n a t u r a l l y  subsequently  unless  and  t h e p r o b l e m o f modon g e n e s i s .  develop  upper  Lindemann(1982)  property.  simplest  nontrivial  5  In  summary,  indicated on  the  previous  the following structure  research  results.  on  modon  dynamics  Modon s t a b i l i t y  of t h e p e r t u r b a t i o n  field.  is  one  of  translating  the  simplest  realizations  f l u i d m o t i o n s on a /3-plane.  of  dependent  Modon g e n e s i s i s  c o n j e c t u r e d t o e v o l v e from r a t h e r g e n e r a l i n i t i a l is  c o n d i t i o n s and  isolated  Modons i n a  steadily  dissipative  environment appear t o p r e s e r v e t h e i r shape, a t l e a s t Modon-modon i n t e r a c t i o n s  has  initially.  a p p e a r t o have many p o s s i b i l i t i e s ,  some  of' w h i c h a r e s o l i t o n - l i k e o t h e r s n o t . However, answered  several  before  atmospheric  we  q u e s t i o n s i n modon d y n a m i c s r e m a i n t o be can  understand  and oceanic dynamics.  the  role  of  No t h e o r e t i c a l  above n u m e r i c a l r e s u l t s .  required  Such a framework  to  describe  w o u l d seem t o be  i f modon d y n a m i c s i s t o be u n d e r s t o o d i n t e r m s o f known  geophysical  f l u i d dynamic mechanisms.  known a b o u t b a s i c p r o b l e m s s u c h a s with  to  or a n a l y t i c a l  f r a m e w o r k h a s y e t been d e v e l o p e d w h i c h c a n be u s e d the  modons  t h e i r environment.  how  interact  For example, the i n t e r a c t i o n  o f modons  have  addition,  to  unanswered.  For  conditions  relating  example,  and t h e s o l u t i o n  the  modons  nothing i s  might  w i t h c u r r e n t s a n d R o s s b y waves questions  In p a r t i c u l a r ,  yet the  to  be  studied.  stability  specification  In  o f modons a r e of  stability  of the l i n e a r i z e d  stability  two  barotropic  problem  a r e unknown. This dynamics. sufficient mode  thesis  examines  The f i r s t  a s p e c t i s examined  neutral stability  small-amplitude  aspects  of  i n Chapter I I , i n which  conditions are derived  perturbations  of  modon  f o r normal-  westward and e a s t w a r d -  6  t r a v e l l i n g modons i n the form of an i n t e g r a l c o n s t r a i n t . conditions  are  perturbation  energy  Flierl(l98l)  derived and  from  the  enstrophy  spatially-integrated  equations  (Charney  Malanotte-Rizzoli(1982))  and  These  for  f l u i d motions.  eastward-travelling  are n e u t r a l l y s t a b l e t o normal-mode  perturbations  that are s o l e l y composed  of  is  steadily  t r a n s l a t i n g quasigeostrophic modons  It  and  shown  spectral  that  components  with wavelength magnitudes l a r g e r than 2 K / K where K i s the modon wavenumber neutrally  (see  Chapter 2 ) .  Westward-travelling  s t a b l e to normal-mode p e r t u r b a t i o n s  magnitude slope  of  of  the p e r t u r b a t i o n the  neutral  McWilliams et a l . ( l 9 8 l ) decreases.  stability should  A s i m i l a r trend  been n u m e r i c a l l y  wavenumber  stability  eddy energy s p e c t r a  for  the  the  TJ= (77,, TJ ) , the 2  proposed increase  by  as  curve  determined f o r t o p o g r a p h i c a l l y - f o r c e d  neutral  atmospheric  to  |T?| i s  i n the n e u t r a l s t a b i l i t y  s o l i t a r y waves ( M a l a n o t t e - R i z z o l i , The  vector  curve  begin  2TI/K.  than where  K/|TJ|>1,  are  s o l e l y composed of  s p e c t r a l components with wavelengths smaller These r e s u l t s imply that when  modons  | rj \ has  planetary  1982).  integral atmosphere  i s tested using and  ocean.  observed For  the  c a l c u l a t i o n , s e a s o n a l l y - and annually-averaged mid-  l a t i t u d e 3 0 0 , 5 0 0 and  7 0 0 mb  energy  i n f e r r e d from T o m a t s u ( 1 9 7 9 ) , Saltzman and  spectra  Fleisher(1962) on  the  were  two-dimensional  wavenumber  and E l i a s e n and M a c h e n h a u e r ( 1 9 6 5 ) .  b a s i s of the c a l c u l a t i o n s c o n t a i n e d  It i s  observed  argued  i n t h i s t h e s i s that  e a s t w a r d - t r a v e l l i n g atmospheric modons are n e u t r a l l y the  eddy  s e a s o n a l l y - and a n n u a l l y - a v e r a g e d  stable  to  fluctuations in  7  the atmosphere. satisfy or  Westward-travelling atmospheric  the n e u t r a l s t a b i l i t y  instability The  c a n n o t be  only  Simple  scaling  translation  and  barotropic  f o r the oceans  arguments  eastward-travelling  are  is  contained  presented  barotropic  particle  modons  Arguments are p r e s e n t e d  oceanic  FU(1983) The  only  realistic stability  of  inferred  from  the  suggesting  that  this  o n l y be c o n s i d e r e d a v e r y p r e l i m i n a r y e s t i m a t e  modon  stability  second aspect  proposed  due  to  the  limitations  of  to The  calculations 1979a,b,  describe  (e.g.,  the  Zakharov  perturbation  of  theory  are  two-dimensional  1971,  Rubenchik(1974)  these  and  This c a l c u l a t i o n methods  is  a slowly varying  Grimshaw(1970,  among o t h e r s ) .  solitary  problems  and  in  thesis  slowly varying solitary  Luke(l966),  application  dimensional  propagation  of o n e - d i m e n s i o n a l  Ablowitz(1980,1981),  Two  modon  p e r t u r b a t i o n methods d e v e l o p e d  1981),  first  of modon d y n a m i c s e x a m i n e d i n t h i s  i n Chapter I I I , i n which a p e r t u r b a t i o n  generalizations  the  that  spectrum.  is contained  medium.  modons c a n n o t be  The  energy  FU(1983).  in  have  speeds i n the ocean.  FU(1983) spectrum.  of  stability  eddy  t o suggest  will  oceanic  should  their  wavenumber  eastward-travelling  conclusion  thus  not  inferred.  published  spectrum a v a i l a b l e  c o n d i t i o n and  modons may  to  a  wave 1977,  Kodama  and  represents  fully  two-  wave. are  posed  and  solved.  theory d e s c r i b i n g the p r o p a g a t i o n  travelling  modon  over  a  weak  developed.  T h i s c a l c u l a t i o n was  In  S e c t i o n 3.1  of  an  eastward-  Ekman b o t t o m b o u n d a r y l a y e r done i n o r d e r  to  a  compare  is the  8  result of  of the l e a d i n g order  McWilliams The  solution  translation  speed  increases  exponentially.  waves. agrees  into a The  with  summarized  the  these into  to  of  modon  wavenumber  the  decay  the  satisfied  the  modon  dispersion  stages  of  the  decay  but i s unable  t h e R o s s b y waves.  to  modon  t h e modon  planar  perturbation solution  Rossby  obtained  describe  the  This calculation  here final  h a s been  Swaters(1985). a  describe  leading modon  order  on  perturbation  propagation  As i n p r e v i o u s work  topography  t h a t t h e modon r a d i u s and  westward-travelling  order  results  S e c t i o n 3.2  topography.  solution  Throughout  final  field  leading  in  developed  the numerical  decay e x p o n e n t i a l l y a n d t h e  In  degeneration  In  suggested  (to leading order)  relationship. dissolved  with  et a l . d 9 8 l ) .  numerical  parameters  solution  on  the  over  R h i n e s ( 1969a,b),  Clarked97l)  Mysak(l978;  Sec. 20),  among  the c o n t e x t  of t h e r i g i d - l i d  and  of  variable  Veronis(1966),  LeBlond  others) the theory shallow  is  slowly varying  effects  b a r o t r o p i c p l a n e t a r y waves ( e . g . ,  theory  and  i s developed i n  water e q u a t i o n s  the | 3 -  on  plane. Nonlinear evolution and  hyperbolic  equations  are  d e r i v e d f o r t h e slow  o f t h e l e a d i n g o r d e r modon r a d i u s ,  wavenumber.  These e q u a t i o n s  are v a l i d  translation  speed  for arbitrary  slowly  varying  f i n i t e - a m p l i t u d e topography.  In g e n e r a l  solved  numerically.  that  evolution South)  It  is  shown  o f t h e modon i s i n d e p e n d e n t  topographic  structure.  This  they  must  be  t o l e a d i n g order the  of the meridional result  (North-  i s interpreted in  9  terms of simple  vorticity  arguments.  Analytical perturbation amplitude case  topography  in  many  hyperbolic the  take  describe  in  oceanic  small-  depth, which i s the  applications).  The  t h e form of e a s t w a r d and w e s t w a r d - t r a v e l l i n g  The g e n e r a l Subsection  properties  3.2.2.  of  the  Subsections  solution  are  3.2.3 a n d 3.2.4  t h e s l o w l y v a r y i n g modon f o r t h e s p e c i f i c  topographic The  and  for  t r a n s i e n t s a n d a s t a t i o n a r y component p r o p o r t i o n a l t o  topography.  described  (relative to the f l u i d  atmospheric  perturbations  solutions are obtained  examples of a  r i d g e and escarpment, r e s p e c t i v e l y . work  Chapter IV. calculations  contained In  the  required  in  Appendix  this  thesis  details  i n the topographic  of  i s summarized the  many  in  integral  solution are given.  F i g u r e 1. T h r e e d i m e n s i o n a l c o n t o u r p l o t o f t h e s u r f a c e d i s p l a c e m e n t a s s o c i a t e d w i t h an e a s t w a r d - t r a v e l l i n g modon. F o r a t m o s p h e r i c s c a l e s , t h e h o r i z o n t a l d i s t a n c e b e t w e e n t h e maximum and minimum d e f l e c t i o n i s a b o u t 800 km w i t h a d e f l e c t i o n o f t h e g e o p o t e n t i a l i s on t h e o r d e r o f 100 m. For oceanic s c a l e s , the h o r i z o n t a l d i s t a n c e between t h e t h e maximum a n d minimum d e f l e c t i o n i s a b o u t 80 km w i t h a d e f l e c t i o n on t h e o r d e r o f 10 cm. The c o o r d i n a t e s y s t e m i s r o t a t i n g w i t h n o n d i m e n s i o n a l a n g u l a r v e l o c i t y 1+e5y where e i s t h e R o s s b y number c ( f a ) " a n d 8 i s t h e p l a n e t a r y v o r t i c i t y f a c t o r 0a2c~1 w i t h a , c , f a n d 0 t h e modon r a d i u s , modon t r a n s l a t i o n s p e e d , l o c a l C o r i o l i s parameter and n o r t h w a r d g r a d i e n t i n t h e C o r i o l i s p a r a m e t e r , respectively. 1  11  II.  McWilliams(1980)  S T A B I L I T Y OF BAROTROPIC MODONS  m o d e l l e d an o b s e r v e d a t m o s p h e r i c b l o c k i n g  w i t h a b a r o t r o p i c modon. whether  or  not  observed  the  Among  modon  solution  atmospheric  McWilliams et al.(1981)  several  questions  is  field.  Their  perturbation  results amplitude  wavelength  leads  wavelength,  increasing  This  indicate  westward  to  McWilliams et a l . ( l 9 8 l ) The and  stability oceanic It  energy  given  the  sufficient  perturbation perturbation  neutral  stability  regime  ' diagram in light  i s calculated with  of these  typical  300,  stability  to  500  the  calculation  is  of w e s t w a r d - t r a v e l l i n g  stability  c a l c u l a t i o n i s a l s o done  stability  energy  spectrum.  or i n s t a b i l i t y  observed  7 0 0 mb  and  stability  kinetic  by  results.  atmospheric  spectra.  modons a r e n e u t r a l l y s t a b l e  similar  modons.  proposed  i s shown t h a t a t m o s p h e r i c e a s t w a r d - t r a v e l l i n g  annually-averaged  vorticity  eastward-travelling  are discussed  condition  vorticity  instability.  and  the  for a  a l s o , f o r a given  to  describes  for  Modifications  e leads  the s t a b i l i t y of  by a random  increasing  to instability;  chapter  conditions  (e),  that  was  Subsequently,  examined  e a s t w a r d - t r a v e l l i n g modons when p e r t u r b e d  posed  stable for typically  fluctuations. numerically  event  seasonally-  transient  unable  to  determine  oceanic  and  eddies.  a t m o s p h e r i c modons. f o r an  barotropic  A the  A neutral barotropic  On t h e b a s i s o f t h i s c a l c u l a t i o n t h e  of oceanic  modons c a n n o t be i n f e r r e d .  C h a r n e y a n d F l i e r M 1 981 ) d e r i v e d  a  stability  theorem  for  12  normal-mode  infinitesimal  quasigeostrophic  flow  perturbations  b a s e d on t h e c o n s e r v a t i o n  enstrophy  (vorticity  conditions  s i m i l a r t o the Blumen(l968)  based  establishing sufficient  flow  on is  a  (Arnol'd,  similar  squared).  stable  extremum  conditions  for  perturbations wavenumbers  17  wavenumber  modon  gives  a  and  stability result  f o r w h i c h t h e mean  suitably  constructed  Sufficient neutral  modons  =  modons  are  (TJ ,TJ )  s a t i s f y i n g |??|<K where  1  (i.e.,  2  the  wavenumbers  derived  by  stable  to  and  the  consists  of s c a l e s  dominant  the slope  to  increase  stability  as  curve  topographically-forced Rizzoli,  1982).  modons,  the  perturbation t h e modon  of the neutral  by M c W i l l i a m s e t a l . ( l 9 8 l )  the  the  modon  functional  vorticity are  with  .in the neutrally  s p e c t r a l components  |T?|>K.  when e i s f i n i t e  K/|TJ|>1  is  K  modons  s o l e l y composed w i t h  that  'small')  neutral  on t h e p o t e n t i a l  satisfying  components  describing  Westward-travelling  implies  K/|TJ|>1  spectral  parameter  eastward-travelling  condition  of  neutrally  composed  stable to perturbations  For  are  solely  interior).  begin  energy  finite-amplitude  barotropic  dependence o f t h e p a t h l i n e s  when  method  conditions  of  of  steady  methods.  Eastward-travelling  with  This  1965) e n e r g y - e n s t r o p h y f u n c t i o n a l .  stability  of  neutral  (though p o s s i b l y spectral  stability  curve  increases.  has  been  solitary  This  numerically planetary  Thus  proposed  modons  property  should i n the  determined  eddies  only  component  i s neutrally stable.  for eastward-travelling  K/|T?|  stability  for  (Malanotte-  13  In r e a l i t y , described  by  a modon w o u l d  be  subjected  a s p e c t r u m o f wavenumbers.  condition  suggests  ( f o r a modon  external  deformation  radius  radius)  that  to  The on  an  derived  the  with  i s contained  magnitudes  greater  eastward-travelling in  which  the  mainly than  modons a r e  energy  is  latitude  eddy  |TJ| - 8 and  modons  (see  Figure  only  spectrum  3)  and  available  Simple s c a l i n g arguments are eastward-travelling t r a n s l a t i o n and stability  condition  modons w i l l be fluid 160  km.  modons  1  is  contained The  neutral  cannot  be  f o r the  with  eastward-  observed  observations  wavenumber  to  modons in that  i f the  eddy  mainly  in  inferred  regimes  M a c h e n h a u e r , 1965;  presented  speeds  stability  However,  in flow  midshow  Saltzman  eddy  oceans i s c o n t a i n e d  suggests  stable  16.  1  1979).  barotropic  particle  the  wavenumbers  i n wavenumbers  typical  barotropic the  the  westward-  Consequently  stable  because  Tomatsu,  for  global  mainly  ( E l i a s e n and  published  in  16.  neutrally  energetics  F l e i s h e r , 1962 The  are  of  eddy e n e r g y o f  neutrally stable  magnitudes l e s s than approximately travelling  order  approximately  contained  stability  atmospheric  t r a v e l l i n g modon i s n e u t r a l l y s t a b l e when t h e surrounding f l u i d  perturbations  the  suggest will  that  only  realistic  The  neutral  eastward-travelling of  the  wavelengths greater of  Fu(l983).  have  ocean.  oceanic energy  in  energy  surrounding than  eastward-travelling  from the F U ( 1 9 8 3 ) spectrum.  about  oceanic It is  The g l o b a l wavenumber i s d e f i n e d so t h a t a wavenumber magnitude of one corresponds to a wavelength e q u a l l i n g the c i r c u m f e r e n c e around a l a t i t u d e c i r c l e .  14  noted that t h i s c a l c u l a t i o n preliminary  estimate  of  should  only  oceanic  be  modon  considered stability  a  very  due t o t h e  l i m i t a t i o n s of the FU(1983) spectrum. 2.1 I n t e g r a l C o n s t r a i n t s F o r P e r t u r b a t i o n s Of B a r o t r o p i c Modons  Consider  the  equation  nondimensional  (Pedlosky,  (A  -  barotropic  potential  vorticity  1979)  F)i//  +  J U  +  y,  Ai// -  Fxfj +  6y)  =  0  (2.1)  t  where \}/ i s t h e g e o s t r o p h i c 3 ( • , * ) / b ( £ ,y)  determinant  pressure, and  J(«,*)  A=3  + 9  U translated coordinate north  and 2  time  coordinates.  The  Coriolis  where  Jacobian  £  i s the  yy 6 = /3a /c and  f a c t o r and  rotational  r e s p e c t i v e l y , w i t h f , 0, g , H, a a n d c t h e l o c a l  parameter,  northward  gradient  of  the  Coriolis  p a r a m e t e r , g r a v i t a t i o n a l a c c e l e r a t i o n , f l u i d d e p t h , modon and  east, 2  coefficients  2  number  the  £ = x - t w i t h x, y a n d t t h e u s u a l  F = f a / ( g H ) a r e the p l a n e t a r y v o r t i c i t y Froude  is  modon t r a n s l a t i o n  speed r e s p e c t i v e l y .  radius  The l e n g t h , t i m e a n d  s p e e d s c a l i n g s a r e a , a/c a n d c , r e s p e c t i v e l y .  We n o t e  that  c  may be p o s i t i v e o r n e g a t i v e . Steady expressed For  solutions  o f ( 2 . 1 ) have t h e v o r t i c i t y  A\p - F\[/ + 6y  a s a f u n c t i o n o f t h e p a t h l i n e s \J/ + y, v i z .  t h e b a r o t r o p i c modon P i s d e f i n e d by ( F l i e r l  P(z)  = 8z  f o r r>1  P(i// + y ) .  e t a l . , 1980)  (2.2a)  15  P(z)  -U  =  2  for  + F)z  (2.2b)  r<1,  resulting in  <// = - K , [ ( 6 + F)  xP =  U where  2  r  ordinary  = £  r ] s i n ( c 9 ) / K j (6 +  2  K  2  2  + F + 5) - rsin(t9)  ]  (K) )  1  (2.3a)  r>1  -  (2.3b)  r<1  K  2  + y ,  t a n ( 0 ) = y/£ Bessel  and  where J , a n d K  f u n c t i o n s of order K (henceforth  1 ) . The p a r a m e t e r  wavenumber)  1 / 2  F)  (5 + F ) J , ( r ) s i n ( 6 ) / ( K J  and m o d i f i e d  (see F i g u r e  and  2  1 / 2  t  are the  one r e s p e c t i v e l y  called  the  modon  i s d e t e r m i n e d by r e q u i r i n g c o n t i n u i t y o f V>// on r = l  i s the f i r s t  nonzero  solution  of  the  modon  'dispersion'  relation  -(6  It  +  F )  turns  varying 6 + F = To  1  /  2  J  2  U ) K  out  1  [ ( 6  + F)  (see F l i e r l  function  1 / 2  ]  =  KJ,(IC)K [(6 2  e t a l . , 1980)  of  + F)  K  that  6 + F  l  (in  /  2  ].  (2.3c)  i s a slowly particular  0(1)->K=<4). obtain  the  \p = * + e x p ( o t ) i / / ' ( x , y )  with  stability l ^ ' l ^ l * ! into  where  solution  (2.3).  exploiting  (2.2) r e s u l t s i n t h e eigenvalue  a(A  Substituting  condition *  (2.1),  i s the linearizing  modon and  problem  - F)\p + U * V [ ( _ - F - P)i//] = 0 0  consider  (2.4)  16  where  t h e p r i m e h a s been d r o p p e d , P = d P ( z ) / d z a n d U  1, • ) .  Equation  intervals and  (2.4) i s d e f i n e d  on  the disconnected  0<r<1 a n d 1<r<" w i t h P o b t a i n e d f r o m  V\// a r e assumed t o be c o n t i n u o u s .  the s p a t i a l l y  i n t e g r a t e d energy  and  = (-• y  0  On r=1 \p  (2.2).  I t f o l l o w s from enstrophy  open  (2.4) t h a t  equations a r e ,  respectively  0 0  7T  2  (a + a*) J f \V4>\ + F|(//|  2  rdrdfl =  0  -TT »  It  - J / (U -Vi//*)A^ + ( U o - V ^ ) ^ *  rdrdfl,  0  It  0° 2  (a + a*) J S -it  It  where  0  (UQ-VI//*)^ ~  +  (U -V<//)A^* 0  i s t h e complex c o n j u g a t e Equations  energy interior  and  (2.5) and  enstrophy  (r<1) and  derivation certain result  i n boundary  identically on  - F ^ | / P rdrdc9 =  0  oo  J / -7T  r=1  (2.5)  (2.6)  o f ^.  ( 2 . 6 ) h a v e been d e r i v e d by o b t a i n i n g  equations adding  rdrdfl,  i n the e x t e r i o r  the  integration integrals  results  by  parts  (r>l)  together. are  In  required  and  this which  on r = 1 . T h e s e t e r m s a r e e i t h e r  z e r o s i n c e U *n=0 on r=1 where n i s t h e u n i t o  o r sum t o z e r o when t h e e x t e r i o r a n d i n t e r i o r  normal  integrals  17  are  added t o g e t h e r  ( s e e Charney and F l i e r l ( l 9 8 l )  and  Malanotte-  R i z z o l i ( 1982)). The  a d d i t i o n of (2.5) and (2.6) g i v e s  a*)  (a +  J - it  If  {\V\f/\  J  the integral  modon  2  +F|<//|  2  modons  condition  i s that  here that  2  i n (2.7) i s nonzero then  i s neutrally  sufficient  1 /P} r d r d f l = 0.  + |Av// -  (2.7)  0  the  Re(a)=0  stable  (Drazin  and  f o r the  neutral  stability  the integral  for instability  1981). of  i n ( 2 . 7 ) be n o n z e r o .  or  i n t e g r a l must be i d e n t i c a l l y  Reid,  f o r asymptotic  zero  and t h e Thus a  barotropic I t i s noted  stability  the  ( s i n c e i n e i t h e r c a s e Re(a) i s  nonzero). The  condition  that  the  integral  i n (2.7) i s zero  c a n be  rearranged to give  it  °= 2  ~ Fi^lVU  2  J / (|V^/| + F|<//| " \W -it  2  + F ) } rdrdt? = - I , / 6 ( 2 . 8 )  0  It  °°  S S { I Vi^ I  2  -it  + F|^|  + |A<J/ - Fi// j / 6 ) } r d r d S = I / 6  2  2  2  (2.9)  0  with it  K  I , - (6 + F +  2  ) U  + F)"  2  1  00  / / -it  - F^|  2  rdrdfl £ 0  1  * 1 1  2  = (5 +  F  +  K  2  ) ( K  2  +  F ) "  1  -it  /  / 0  |Vtf -  Fi//|  2  rdrdt9 £ 0,  18  since and  6 + F>0 f o r s o l u t i o n s o f t h e f o r m ( 2 . 3 ) .  Equations (2.8)  ( 2 . 9 ) c a n be r e w r i t t e n  T  2  J |tf |*U - | T J |  2  )(|T,|  2  + F ) drj =  -4TT (K 2  / I ^ I M |r?| + F ) ( 5 + F + | r j | ) dr? = 2  2  + F)l,/6  2  4TT I  2  T  the  basis  ( T ? , , T J ) . Equation 2  And A p p l i c a t i o n To The A t m o s p h e r e  The  (2.10)  integral  provides  sufficient  spectrum of t h e p e r t u r b a t i o n  conditions  field  travelling  I t follows  modon).  perturbation  i s solely  composed  from of  (2.10)  wavenumbers  modon must be  (2.11)  constraint  no  s i n c e b o t h t h e LHS a n d RHS  stable  i f the satisfying  sign  (since  t h e i n t e g r a l i n ( 2 . 7 ) c a n n o t be z e r o a n d t h e  westward-travelling places  a westward-  that  |TJ| > K t h e LHS a n d RHS o f ( 2 . 1 0 ) a r e o f d i f f e r e n t Therefore  on t h e  i f t h e modon i s t o  Consider thecase c < 0 ( i . e . ,  condition  (2.10) forms  A n d Ocean  be n e u t r a l l y s t a b l e .  5<0).  transform of  f o r the remaining a n a l y s i s .  2.2 D i s c u s s i o n  wavenumber  (2.11)  2  due t o P a r s e v a l ' s e q u a l i t y , where \p i s t h e F o u r i e r and r] t h e wave number v e c t o r  (2.10)  on  neutrally  theperturbation  are nonnegative.  suggests that westward-travelling  when t h e p e r t u r b a t i o n  stable.  field  Equation wavenumbers  This  stability  modons a r e n e u t r a l l y  i s d o m i n a n t e d by wavenumbers  w i t h m a g n i t u d e s l a r g e r t h a n t h e modon wavenumber. Eastward-travelling neutrally  stable  modons  (c > 0  i f the perturbation  hence  6 > 0)  are  i s s o l e l y composed o f  19  < K.  wavenumbers s a t i s f y i n g restriction condition  on  the  perturbation  s u g g e s t s t h a t an  neutrally  stable  As  when  before,  (2.11)  wavenumbers.  eastward-travelling  the  surrounding  places  This  stability  modon  fluid  no  will  be  i s dominated  by  wavenumbers s m a l l e r t h a n t h e modon wavenumber. The fact  sufficient  valid  Flierl,  1981,  although  for  finite-amplitude  Benzi  the  s t a b i l i t y conditions just described  e t a l . , 1982  analysis  may  perturbations  and  be  Purini  only v a l i d  p e r t u r b a t i o n s d e p e n d i n g on t h e d i s t r i b u t i o n Blumen(l968) Reid,  energy-enstrophy  in  (Charney  and  and  Salusti,  for  small-amplitude  1984)  of e x t r e m u m s t o  functional  the  (Drazin  and  1981). However t h i s a n a l y s i s i s n o t  full  nonlinear  Abraham( 1970)  stability.  and  Ebin  w h i c h |i/ i s a member and  given  stability  using  and  of  Marsden( 1970), Marsden out  of n o n l i n e a r  topology  theorem f o r p l a n e arguments  A  and  that  the  stability  of t h e H i l b e r t  of t h e  functional.  convexity  demonstration  have p o i n t e d  the t o p o l o g y  energy-enstrophy  nonlinear  rigorous  a proof  i n c o n s i s t e n c i e s between t h e  the  a  Holm e t a l . d 983)  A r n o l ' d ( 1 9 6 5 ) argument i s not to  are  due  s p a c e of  second v a r i a t i o n  rigorous  curvilinear (see  proof flows  Arnol'd,  of  of can  1969  a be and  Holm e t a l . , 1 9 8 3 ) . McWilliams et a l . ( l 9 8 l ) Figure  2; a d a p t e d f r o m F i g u r e  eastward-travelling neutral  proposed  modons.  s t a b i l i t y curve  10  a  regime  i n M c W i l l i a m s e t a l . , 1981) Their  numerically  i s shown a s a s o l i d  c o n d i t i o n g i v e n a b o v e ( f o r c>0)  diagram  line.  s u g g e s t s t h a t as  (see for  determined  The  stability  K/|TJ|  increases  20  for  (i.e.,  K/|T?|>1  smaller exist  a  disturbance  increase  as  | TJ |  decreases,  allowing  a d j a c e n t t o t h e wavenumber a x i s . stability  curve  This  i s qualitatively  F i g u r e 2 ( w h e r e i t i s assumed t h a t s h o u l d be c l o s e t o e a c h o t h e r ) . behaviour  has  been  planetary s o l i t a r y  for  region  trend  in  of s t a b i l i t y the  neutral  shown by t h e d a s h e d l i n e i n for  the  two  curves  for  curve  topographically-forced 1982). to the  i s g i v e n by K and F, b o t h o f w h i c h a r e s p a t i a l l y  constant  speed  ( c ) and  of the deformation  order 6  0(1 )  2  and  modon  radius  For  1980) t h e modon r a d i u s i s on  r a d i u s and | c |  consequently  (a).  0 ( 1 0 ) m/s.  K « 4 ( c f . (2.3c)  Thus  see a l s o  et al.(1980)).  In  mid-latitudes  westward-travelling t h e energy  contained  these  i n the transients  mainly  Earth  radius  parameter  atmospheric  values  modons w i l l in  the  imply  that  be n e u t r a l l y  stable  surrounding  i n s c a l e s w i t h ( g l o b a l ) wavenumber  g r e a t e r than approximately  2  when  reference  the  the  a  t h e LHS o f ( 2 . 1 0 ) , t h e o n l y e x p l i c i t  s c a l e s (see M c W i l l i a m s ,  if  Thus  curve should begin t o  (Malanotte-Rizzoli,  atmospheric  Flierl  should  Similar neutral stability  determined  eddies  a given t r a n s l a t i o n  F «* 1 ,  wavenumbers  f o r small-amplitude perturbations).  the s l o p e of the n e u t r a l s t a b i l i t y  K/|TJ|>1  modon  by  t h a n t h e modon wavenumber) a r e g i o n o f s t a b i l i t y  (at least  In  dominated  and  The p l a n e t a r y v o r t i c i t y to balance amplitude Flierl, 1981).  fluid  are  magnitudes  16 (16 * »cRcos(</>)/a where R a n d <P a r e  latitude,  respectively).  Eastward-  f a c t o r must be 0 ( 1 ) f o r p h a s e d i s p e r s i o n s t e e p i n g i n (2.1) (see Charney and  21  travelling in  modons,  flow regimes  scales the  with  on  the other-hand, w i l l  i n which  the  wavenumbers  dominant  representative  contained  of  peak  Machenhauer,  at  barotropic  flow)  l e s s than about  8  Saltzman  and  f o r t h e o b s e r v e d 300 mb,  meridional eddy  and  be p o s i t i v e  500 mb and.700 mb  eddy  spectra.  As  an  example,  (2.10)  Tomatsu(1979)  seasonally-  kinetic  spectrum.  T  zonal  F l e i s h e r , 1962  when  | \p |  in  ( c a l c u l a t e d from E l i a s e n and  C o n s e q u e n t l y t h e LHS o f (2.10) w i l l  energy  in  (which are  and  Tomatsu, 1979). evaluated  eddies  resides 4-5  in  However,  3) 6-8 g i v i n g a ( g l o b a l ) wavenumber  approximately  1965;  of  500 mb a n d 700 mb a t m o s p h e r e  the  (see Figure  mainly  l e s s t h a n a p p r o x i m a t e l y 16.  wavenumbers w i t h m a g n i t u d e s  energy  is  c o n t r i b u t i o n t o the energy spectrum  t h e m i d - l a t i t u d e 300 mb,  wavenumbers  energy  be n e u t r a l l y s t a b l e  2  energy  energy  2  2  spectrum.  wavenumber  A  and To  by 2E(7j, , r} ) 11? | "  can  be  annually-averaged  calculate  with  evaluated  (2.10)  we  for 500 mb  north-south  eddy  approximate  E t h e n o n d i m e n s i o n a l eddy  mid-latitude  the  kinetic  (cartesian)  was i n f e r r e d w i t h t h e a p p r o x i m a t i o n  2  1/2  r j - a{-D P^0)/p£U)} /(RcosU)) 2  evaluated  at  45°N, 2  function,  D P®(<p)  where  P^(0)  = d2P5U)/d*  n o n d i m e n s i o n a l z o n a l wavenumber. harmonic inferred  to  is 2  the  and  associated  Legendre  m = tj, Rc o s (*) / a  The c o n t r i b u t i o n o f t h e  t h e T o m a t s u ( 1 9 7 9 ) mth z o n a l wavenumber  the  P«(0)  spectra  was  from t h e E l i a s e n and Machenhauer(1965) spectrum so t h a t  22  t h e p e r c e n t a g e c o n t r i b u t i o n s r e m a i n e d t h e same ( s e e F i g u r e both.  These c a l c u l a t i o n s imply  a  LHS  of  +42.17 a n d +28.43 f o r t h e a n n u a l , w i n t e r spectra,  respectively;  approximately  14.88,  percentage standard Fleisher(1962).  with  expected  13.47  and  deviations (These  (2.10)  of  a n d summer standard  11.51,  3  +35.57,  eddy  Therefore sign made  typical A  r e s p e c t i v e l y b a s e d on  consistent  calculations  the  LHS  with are  Saltzman  deviations)  nondimensional.)  be  made  for  modons s i n c e b o t h s i d e s o f ( 2 . 1 0 ) a r e o f t h e that  a v a i l a b l e data d e s c r i b i n g pales  author's  (barotropic) oceans.  the  neutral  stability  knowledge, geostrophic Fu's  100-1000 km.  data should  only  o c e a n i c m e s o s c a l e wavenumber  i n comparison t o the atmospheric FU(1983) kinetic  calculation  measurements of t h e sea between  cannot  energetics.  i s not s a t i s f i e d .  variability  the  mid-latitude  inference  same s i g n a n d t h u s i t i s p o s s i b l e  the  is  modons a r e n e u t r a l l y s t a b l e f o r  or- i n s t a b i l i t y  westward-travelling  The  spectra  f o r c>0 a n d t h e c o n c l u s i o n  ( i . e . , observed p e r t u r b a t i o n )  condition  and  a n d t h e RHS o f (2.10) a r e o f d i f f e r e n t  eastward-travelling  stability  of  similar.  ( t o two s t a n d a r d that  energy  deviations  C a l c u l a t i o n s b a s e d on t h e 300 mb a n d 700 mb eddy e n e r g y are q u a l i t a t i v e l y  3) i n  surface A  contains  record.  the only  published  e n e r g y wavenumber s p e c t r u m is  based  To  for  on SEASAT a l t i m e t e r  variability,  for  wavelengths  c a l c u l a t i o n t e s t i n g (2.10) u s i n g Fu's  be c o n s i d e r e d  a preliminary  estimate  of  modon  The seasons a r e d e f i n e d so t h a t w i n t e r i s September t h r o u g h t o F e b u r a r y a n d summer i s M a r c h t h r o u g h t o A u g u s t .  23  stability  i n t h e ocean.  Note t h a t else The  s o l u t i o n s of t h e form  the modified  Bessel  (2.3)  require 6 + F > 0 (or  f u n c t i o n s have i m a g i n a r y  t r a n s l a t i o n speed f o r w e s t w a r d - t r a v e l l i n g  satisfies 0*1 .6-1 0that  the 11  1  2  value)  suggests  that  likely  to  -1  5  1  andH~5-1u  3  t r a n s l a t i o n speed barotropic  be  ( a n d hence  in  the  i t follows This  large (in  particle  westward-travelling  observed  Assuming  m  a n d c o n s e q u e n t l y t h a t c < -10 ms"'.  absolute  therefore  c < -/3a F .  n r ' s ' , a=O0 m, f ^ l O ^ s "  F*10~  modons 2  constraint  argument).  speeds)  modons  mid-ocean.  a r e not  Under  the  a p p r o x i m a t i o n s c>0 a n d F—>0 ( 2 . 1 0 ) r e d u c e s t o  /  |  2  | T ? |  2  U  | TJ  2  | " where  spectrum.  E ( T J  Since  oceans i s n e a r l y  1  , T J  |rj| )  dTj < 0.  (Bernstein  kinetic  i s a p p r o x i m a t e d by  dimensional  and White,  = E (|Tj|)(27r|T |)0  i s the scalar  0  i n F u , 1983).  i s t h e two  2  1983) i t f o l l o w s t h a t  2  geostrophic  )  isotropic  E(i?,,i? >  E (|r?|)  2  (2.12)  kinetic  h o r i z o n t a l mesoscale v a r i a b i l i t y  G r o u p , 1978 a n d R i c h a r d s o n ,  where  2  -  2  i n t h e atmospheric c a l c u l a t i o n , l ^ ,  ( 7 } , , 7i2)  energy  T  1  As 2E  | *  energy  (see  i n the  1974; Mode  ( F u , 1983)  1  ?  wavenumber Figure  spectrum  f o r the  4, a d a p t e d f r o m F i g u r e 8  Thus t h e i n e q u a l i t y (2.12) t a k e s  t h e form  OB  2  /  E  0  ( | T J | ) [ K  2  -  M2]  d|7?|  <  0.  (2.13)  24  Clearly of  i f  neutral  must  i s t o be c o n t r a d i c t e d  stability),  come  /3<*1 .6 • 1 0 "  (2.13)  |T?| < K.  from  11  t h e dominant  i t follows  wavelength  of  about  barotropic  modons w i l l  which the energy about  For  (i.e.,  a demonstration  contribution  t o the integral  a=l00  K=3.9616,  that 160 km  (=2007T/K  be n e u t r a l l y  C=*10 c m s "  km,  corresponding km).  Thus  1  and to  a  oceanic  stable to perturbations for  i s contained mainly i n wavelengths greater  than  160 km. The  integral  spectrum  (see  i n ( 2 . 1 3 ) was c a l c u l a t e d  F i g u r e 4)  with  E  set  0  using  FU(1983)  the  identically  zero  w a v e l e n g t h s o u t s i d e t h e 100-1000 km b a n d .  For the  spectrum  t o data o b t a i n e d near  major  (see Figure 4 ) ,  current  computed  to  systems be  corresponding  (see F u , 1983),  -56.06.  For  the  the low  LHS  high  for  of  energy  energy  (2.13)  spectrum (see  F i g u r e 4 ) , c o r r e s p o n d i n g t o d a t a o b t a i n e d i n r e g i o n s remote major  current  computed  to  systems be  (see Fu, 1983),  -44.54.  As  the  LHS  and  low  energy  Based  FU(1983).  results  are  inferred.  used  to  compute  the  l i m i t a t i o n s t o the FU(1983)  i t sapplicability the  p e r i o d s l e s s t h a n 24 d a y s that  calculations,  f o r the high  o f b a r o t r o p i c modons i n t h e o c e a n s c a n n o t be a r e , however,  these  (These  estimates,  neutral  w h i c h may r e s t r i c t  shown  in  on  was  the  There  data  spectra  (2.13)  above  b a s e d on t h e 9 5 % c o n f i d e n c e i n t e r v a l s  nondimensional.) stability  of  from  a measure of t h e e r r o r , s t a n d a r d  d e v i a t i o n s o f .6 a n d .5 were c o m p u t e d f o r t h e respectively,  was  dominant  spectra  here.  For  spectrum  example,  the  represents energy only at  ( F u , 1983), whereas  Wunsch(l98l)  energy c o n t a i n i n g e d d i e s have  has  periods  25  b e t w e e n 50 and the  short  observed Wyrtki  duration is  A l s o , FU(1983) argued t h a t because  150 d a y s .  about  of t h e d a t a , t h e i n t e g r a t e d 5  times  et a l . ( l 9 7 6 ) .  It  less is  than  different  from  opposite further is  the  conclusion work  FU(1983) given  that  therefore  wavenumber s p e c t r u m c o m p u t e d w i t h l o n g e r  may  energy  reported  conceivable  t o the degree be  reached.  (2.13)  can  be  a  quite  that  the  Clearly,  on t h e m e s o s c a l e wavenumber s p e c t r u m o f t h e  required before the c o n s t r a i n t  by  that  r e c o r d s w o u l d be  spectrum  above  kinetic  of  ocean  realistically  tested. Equation  (2.1)  reduced-gravity or M y s a k , 1983) the  1-1/2  where  internal  Froude  1  (c=*-10" ms"  1  also  be  interpreted  l a y e r model  ( W h i t e and  number.  Saur,  modons ).  In  Consequently, will  this  have  nonlinear 1981  context,  (2.10)  spectrum  obtained  could  energy  from g e o s t r o p h i c v e l o c i t y  fields  o f no m o t i o n  deflection  data  collected  from  calculation  o f a low f r e q u e n c y e n e r g y wavenumber s p e c t r u m  personal  is  currently  communication).  in  be  fluctuation  to a level  lines  and  translation  that are r e l a t i v e  these  and  F * 0(1)  realistic  c a l c u l a t e d u s i n g a two d i m e n s i o n a l b a r o c l i n i c wavenumber  as a  \[/ i s t h e i n t e r f a c i a l d i s p l a c e m e n t a n d F i s  westward-travelling speeds  can  a  (e.g.,  from  isotherm  s p a t i a l a r r a y of X B T s ) .  progress  (K. A. Thomson,  A  along 1985;  26  10  INSTABILITY  STABILITY  *=|i.l  0.1  0  2 Tr/| | ?  F i g u r e 2. S t a b i l i t y r e g i m e d i a g r a m f o r an e a s t w a r d - t r a v e l l i n g modon. The v e r t i c a l c o o r d i n a t e e, i s t h e a r e a - a v e r a g e d p e r t u r b a t i o n v o r t i c i t y a m p l i t u d e . The h o r i z o n t a l c o o r d i n a t e i s s c a l e d s o t h a t | TJ | = TT c o r r e s p o n d s t o a w a v e l e n g t h e q u a l l i n g one modon d i a m e t e r ( s e e S e c t i o n 2 . 2 ) . The s o l i d l i n e i s t h e n u m e r i c a l l y determined n e u t r a l s t a b i l i t y curve of McWilliams et a l . d 9 8 l ) . The d a s h e d l i n e qualitatively i l l u s t r a t e s the expected i n c r e a s i n g slope i n the n e u t r a l s t a b i l i t y c u r v e ( c f . ( 2 . 1 0 ) ) f o r is/ \ 771 » K / K 0.8 u n d e r t h e a s s u m p t i o n t h a t when | T ? | ^ K t h e two c u r v e s a r e s i m i l a r .  Figure 3. The two-dimensional atmospheric k i n e t i c energy spectrum used to compute the LHS of (2.10) based on a n n u a l l y averaged s t a t i s t i c s . The zonal wavenumber m, i s s c a l e d so that m=1 corresponds to a wavelength e q u a l l i n g the c i r c u m f e r e n c e of a latitude c i r c l e . The ( c a r t e s i a n ) m e r i d i o n a l wave number n, i s s c a l e d so that n=1 corresponds t o wavelength e q u a l l i n g the l o n g i t u d i n a l circumference ( i . e . geodesic c i r c u m f e r e n c e through the p o l e s ) . The north-south wavenumber T? i s r e l a t e d t o n v i a Tj =an/R (see s e c t i o n 2.2). The energy d e n s i t y amplitude has been normalized so that SS E(m,n) dndm = 1. The dominant c o n t r i b u t i o n to the spectrum comes from m=1, 2, 3, 4, 5 and 6 and n=6, 7, 8 and 9, and accounts f o r 70% of the energy. 2  2  28  F i g u r e 4. S c a l a r wavenumber s p e c t r u m o f b a r o t r o p i c k i n e t i c e n e r g y f o r t h e o c e a n u s e d t o c o m p u t e t h e LHS o f ( 2 . 1 3 ) . The curve l a b e l l e d H c o r r e s p o n d s t o d a t a c o l l e c t e d from h i g h energy r e g i o n s ( e g . n e a r m a j o r c u r r e n t s y s t e m s ) . The c u r v e l a b e l l e d L c o r r e s p o n d s t o d a t a c o l l e c t e d f r o m l o w e n e r g y r e g i o n s ( e g . away from major c u r r e n t s y s t e m s ) . The two l a r g e v e r t i c a l m a r k s on t h e wavenumber a x i s c o r r e s p o n d t o 1000 km a n d 100 km f r o m l e f t to r i g h t , r e s p e c t i v e l y . The p o i n t | T J | = K i s i n d i c a t e d w i t h a d o t on t h e wavenumber a x i s .  29  III.  MODON PROPAGATION IN A SLOWLY VARYING MEDIUM  W h i t h a m ( 1 9 6 5 ) showed t h a t t h e  slow m o d u l a t i o n of n o n l i n e a r  i n a d i s p e r s i v e medium c o u l d be of  the  wave  parameters  (such  a m p l i t u d e ) w i t h i n an a v e r a g e d formulation  of  the  described as  (over  governing  by  the  slow  frequency, one  developments i n s l o w l y v a r y i n g n o n l i n e a r  variation  wavenumber  wave p e r i o d )  equations.  waves  Lagrangian  Subsequent  waves  and  have  research generally  t e n d e d t o u t i l i z e p e r t u r b a t i o n methods ( d e s c r i b e d b e l o w ) o r when possible Newell, a  an  inverse  1978;  leading  and order  modon p r o p a g a t i o n  scattering transformation  Karpman and  Maslov  perturbation in a slowly  theory varying  presented  here represent  to a f u l l y  two-dimensional s o l i t a r y  Two  problems are  Section effect  3.1  a  of  an  travelling  solution provides  Ekman  This  a test  medium.  travels  The  c o u l d be  this  solutions  chapter.  on  an  compared w i t h  the  a  that  the  numerical  et a l . , 1981).  interaction  (This 3.2  the  between  topography. a  perturbation  method  to  wave  as  Johnson(1973)  and  slow e v o l u t i o n of the B o u s s i n e s q s o l i t a r y over  the  eastward-  chosen i n order  (McWilliams  developed  In  developed to study layer  was  slowly varying  Grimshaw(1970,1971)  it  describing  f o r t h e p e r t u r b a t i o n method.) I n S e c t i o n  a b a r o t r o p i c modon and  the  in  is  p e r t u r b a t i o n method i s u s e d t o d e s c r i b e  describe  chapter  wave.  boundary  same p r o b l e m  i s developed  solved  problem  In t h i s  and  a p p l i c a t i o n of t h e s e methods  method  bottom  results obtained f o r the  first  p o s e d and  perturbation  modon.  analytical  the  I979a,b).  ( e . g . , Kaup  slowly v a r y i n g topography.  30  Grimshaw(1977,1978,1979a,b,1981) theories  to  describe  v a r i e t y of p h y s i c a l applied  other  developed s i m i l a r  slowly varying  problems.  Warn  and  solitary  fashion, the p e r t u r b a t i o n  by i n t r o d u c i n g  slow s p a t i a l and  temporal  Whitham,  and  reflecting  1965  slowly varying dependent  L u k e , 1966)  medium r e l a t i v e  variables  perturbation different  and  series in  scales  a  to  a  wave small  associated  s o l u t i o n i s obtained variables the  typical phase  with  the  wave  parameters  c o e f f i c i e n t s of the secularities  wavelength. expanded  Since  differential  characterizing  L u k e d 9 6 6 ) , A b l o w i t z ( 1 971),  Segur(l98l) alternate dimensional formulation  a  the  wave.  evolution  leading  these  equations  Ablowitz  to  higher  coefficients  with  equations  slow e v o l u t i o n of the s o l i t a r y  Kuelh(l978),  in  a r e d e t e r m i n e d by d e m a n d i n g t h a t t h e  d e r i v a t i v e s o f t h e wave p a r a m e t e r s variables,  The  t h e wave a n d v a r i a b l e medium,  inhomogeneities  vanish.  (following  s c a l i n g of the  are  parameter  I n t h e J o h n s o n and Grimshaw a n a l y s i s ,  however,  have  topography.  w i t h t h e 0 ( 1 ) s o l u t i o n t a k e n t o be t h e s o l i t a r y  and  in a  t h e s e methods t o m o d e l a t m o s p h e r i c b l o c k i n g a s t h e s l o w  In the u s u a l  Ko  waves  Brasnett(1983)  modulation of atmospheric s o l i t o n s over v a r i a b l e  for  perturbation  respect  are obtained  to  order contain  the  slow  d e s c r i b i n g the  wave. Zakharov and  and  Rubenchik( 1 974),  Kodama(1979), A b l o w i t z  and  a n d Kodama a n d A b l o w i t z ( 1 9 8 0 , 1 9 8 1 ) h a v e d e v e l o p e d an perturbation  method  slowly varying of the  the  two  for  describing  various  s o l i t a r y wave p r o b l e m s .  perturbation  differential  problems  equations  is  describing  The  oneinitial  identical; the  slow  31  modulation of the l e a d i n g order exploiting  a  perturbation  solvability  wave p a r a m e t e r s a r e o b t a i n e d condition  perturbation the  equation  adjoint  perturbation When in  the the  operator  first  Schroedinger  the Vries  perturbation  Klein-Gordon  and Segur, operator  the  nonlinear  analysis.  first  order  operator  i s s e l f - a d j o i n t , as 1971),  sine-Gordon t h e two  nonlinear  (Kodama  methods  are  and  formally  and  Petviashvili  1981) a n d p o t e n t i a l v o r t i c i t y  equations the  Johnson  the  1981),  Kadomstev  i s not s e l f - a d j o i n t . and Grimshaw a n a l y s i s t h e s o l u t i o n of t h e  perturbation secular  e q u a t i o n s must be o b t a i n e d terms.  However,  e q u a t i o n s was e a s i l y s e e n t o be Thus  the  theory  i n order  I n the problem considered  of the a d j o i n t problem a s s o c i a t e d  perturbation solution.  the  (Ablowitz,  t h i s d i d n o t p r o v e t o be t r a c t a b l e . solution  with  However, f o r t h e ( r e g u l a r and m o d i f i e d ) Korteweg-de  order  determine  the  the Fredholm a l t e r n a t i v e theorem).  1980, 1981) e q u a t i o n s ,  perturbation  first  order  (Kodama a n d A b l o w i t z ,  In  order  t o t h e homogeneous s o l u t i o n  associated  (i.e.,  and  same.  (Ablowitz  first  i s that the inhomogeneity i n  be o r t h o g o n a l  problem  nonlinear  Ablowitz,  the  equation.  The s o l v a b i l i t y c o n d i t i o n  of  on  by  the  with  here  homogeneous  the f i r s t  the  to  zeroth  order order  developed here f o l l o w s the l a t t e r  32  3.1 P e r t u r b a t i o n Ekman B o u n d a r y  S o l u t i o n F o r Modon P r o p a g a t i o n  Layer  McWilliams et a l . ( l 9 8 l ) (linear,  Newtonian  numerically  parameters  and  exponential  (i.e.,  calculated  concluded  and  shape  preserve  t h e modon d i s p e r s i o n r e l a t i o n s h i p . dissipation  westward-travelling  approximation)  the  modon  planar  suggest that the d i s s i p a t i o n can of  be t h e o r e t i c a l l y v i e w e d solitary This  in  The  was modon  a manner a s t o  into  a  These  o f a modon due t o  stages  field  of  observations  bottom  friction  of t h e slow  evolution  foranalytically  obtaining  i n the context  waves. Section describes solution  a theory  leading order  the  presence of a bottom boundary l a y e r .  agrees with the numerical  the  on an  decay  In the f i n a l  waves.  the  for  such  degenerated  Rossby  of  s p e e d a n d wavenumber)  (to a  the  the  preserving.  the radius, translation first  effect  dissipation  that  evolved  of  the  and b i h a r m o n i c ) v o r t i c i t y  e a s t w a r d - t r a v e l l i n g modon approximately  Over A B o t t o m  the d i s s i p a t i o n  o f an  eastward-travelling  calculation  The s o l u t i o n  we  in  obtain  of McWilliams e t a l . ( l 9 8 l )  o f a modon a l t h o u g h  transition to westward-travelling  modon  i t i s unable t o describe R o s s b y waves i n t h e f i n a l  s t a g e s of t h e decay. For in  typical  Subsection  oceanic  3.1.1)  of magnitude s m a l l e r the  potential  a n d a t m o s p h e r i c modon s c a l e s  the e f f e c t  of bottom f r i c t i o n  than t h e i n e r t i a l  vorticity  equation.  b a r o t r o p i c modon when t h e e f f e c t s  and d i s p e r s i v e  (decribed i s an  terms  Thus t h e d i s s i p a t i o n  of a bottom  Ekman  order  layer  in of a are  33  included  i n the v o r t i c i t y  varying  solitary  describe  wave  the  problems f o r the wavenumber.  The  described  The  method  translation  developed over  by a  t o be  initial-value  speed,  varying  which  f u n c t i o n s of  radius  in this Section  slowly  slowly  parameters  s o l v a b i l i t y c o n d i t i o n leads to  d e s c r i b e modon p r o p a g a t i o n Section  be  wave a r e a l l o w e d  leading order  The  can  calculation.  t h e modon s o l i t a r y  a s l o w t i m e and  3.1.1  equation  and  i s used  topography  to in  3.2.  Formulation  And  nondimensional  which the  interior  S o l u t i o n Of  barotropic of the  The  D i s s i p a t i o n Problem  potential vorticity  fluid  i s a s y m p t o t i c a l l y matched  b o t t o m Ekman b o u n d a r y l a y e r i s ( P e d l o s k y ,  + J(t/>, A ^ + 6 y )  to  in a  1979)  = -eW  2  A<//  equation  (3.1.1)  t  where  \p i s  Jacobian  the  geostrophic  determinant  ( p o s i t i v e ) eastward, and 8  2  where  A  2  = /3a /c 0  f a c t o r and Ekman  0  is  and  9(*,•)/9(x,y)  the e = E  1  field,  horizontal 1 / 2  /(2r )  2pf~ H~  1  the  where  *>, f and  where  c f~ (a )~  0  is  t h e R o s s b y number  undamped modon r a d i u s and  parameters  planetary  vorticity  translation  vertical  the v e r t i c a l  eddy  depth r e s p e c t i v e l y ,  1  0  usual  The  H are  fluid  the  coordinates  r e s p e c t i v e l y , w i t h E the  v i s c o s i t y , C o r i o l i s p a r a m e t e r and r  is  t the  time  Laplacian.  are  0  J(*,«)  w i t h x, y and  ( p o s i t i v e ) n o r t h w a r d and  damping c o e f f i c i e n t  number  pressure  0  speed  1  with a  0  and  c  respectively.  0  and the The  34  space, and  c  t i m e and v e l o c i t y s c a l i n g s h a v e been c h o s e n a s a , a / c 0  respectively.  0  parameter 100 lO"0  1  oceanic  (atmospheric)  o f 0 , a , c , v, H a n d f o f 1.6•10~  values  0  ( 1 0 0 0 ) km, 1 0 " s"  For typical  1  1  10'  2  2  1  (10) m s' ,  respectively, i tfollows that  values  planetary Equation for  of  a  0  vorticity  and  c  factor  0  applications  the  than t h e e x t e r n a l deformation radius For  a leading order  K u e h l d 978), are  perturbation.  fast  unity  quasigeostrophy.  scale a  (gH/f)  solution the  and  1 0 " . T h u s f o r modon  satisfying  length  1  1  e  (3.1.1) does n o t i n c l u d e t h e f r e e s u r f a c e  oceanic  1  m~ s~ ,  were c h o s e n t o g i v e a n o r d e r while  0  modon  4 ( 1 0 ) km  s c a l e s t h e RHS o f ( 3 . 1 . 1 ) c a n be v i e w e d a s a s m a l l The  11  0  (10) ms" ,  0  1 / 2  0  effect,  i s much  since smaller  * 2000 km.  variables  (Ko a n d  G r i m s h a w ( 1 979,1 981 ) a n d Kodama a n d A b l o w i t z (1 9 8 1 ) )  g i v e n by  i  = x - e-  1  T ;  c  ( f )dt'  o  y = y  and  a s l o w t i m e i s g i v e n by  T =  Thus  £ =-c(T) -and t  (3.1.1)  gives  £ =1. x  et.  S u b s t i t u t i o n of these v a r i a b l e s i n t o  35  + 8 y ) = -eAtf -  J(\l> + c y ,  2  eAtf  (3.1.2) T  where t h e J a c o b i a n As  i s t a k e n w i t h r e s p e c t t o £ a n d y.  i n G r i m s h a w ( 1 9 7 9 a , b , 1 9 8 1 ) a n d Kodama a n d  a perturbation  solution  * - *  (0 >  Ablowitz(1981)  t o (3.1.2) i s c o n s t r u c t e d i n t h e form  (1)  U,y?T) + e<// (£,y;T) + ... .  The 0 ( 1 ) p r o b l e m i s  + c y , A ^ ( ' + 6 y ) = 0,  (0 1  0  J(<//  the  solution  (Flierl  of  which  is  taken  to  be  the  modon  e t a l . , 1980)  ( 0  ^ >  A<// ( 0 )  =  2  =  =  -5 aK (6c-  a J i  = -K I// 2  ( 0 )  r)sin(e)/K,(6ac"  r)sin(e)/K (8ac-  = (6 /c)i//<0) 2  = -6 aJ, 2  - (6  2  1 / 2  1 / 2  1  (Kr)sin(©)/J,  Ai// ( 0 >  A^(0)  l / 2  1  62-2 K  1 / 2  -caK,(5c-  Ai//(0)  ^ ( 0 )  2  +  )  )  r>a  (/ca)  2  - (6  (3.1.3)  +  K  2  C )  K'  2  rsin (6)  Ur)sin(0)/J, (/ca)  2  c/c)rsin(0)  r<a  (3.1.4)  36  where  J,  of o r d e r r  a n d K, a r e t h e o r d i n a r y a n d m o d i f i e d  one, w i t h t h e p o l a r c o o r d i n a t e s  = U-£ (T)]  2  2  0  The leading  + y  and tan(6*)  phase energy  Grimshaw, I979a,b analysis  here as  a constant  of  the dispersion relation 0  by and  first  For the 0(1)  undetermined  (thex-coordinate as  a  order  K u e l h , 1978,  1981).  K i n (3.1.4) i s t h e f i r s t  and  is  o f t h e wave  slowly  varying  nonzero s o l u t i o n  ( o b t a i n e d by r e q u i r i n g c o n t i n u i t y o f  > on r = a )  -6J Ua)K, ( 8 a c -  1 / 2  2  The  by  because i t appears i n t h e 0 ( e ) equations.  modon wavenumber  (  (Ko  i t remains  The  Vi|>  i s determined  I t i s formally included  q u a n t i t y a t t h i s stage  defined  ( i n comparison t o the  a n d Kodama a n d A b l o w i t z ,  chosen  a t T=0).  and  considerations  presented  eventually  £)  8  0  0  order  and  functions  = y/[*-* (T)].  t e r m i j ( T ) i s an 0 ( e ) p h a s e s h i f t  perturbation  center  2  r  Bessel  modon  radius,  ) = c ' KJ , Ua ) K ( S a c " 1  2  translation  r e s p e c t i v e l y , are allowed  to  (following  theory  Ablowitz,  the  general  1981 a n d A b l o w i t z  conditions  The  functions of  (3.1.5)  O  the  slow  with 0  the  initial  s o l v e s t h e modon  (*c = 3.9226, b a s e d on 6=1). 0  1  time  G r i m s h a w , 1979, Kodama a n d  where K  K{0)=K  0(e) problem a s s o c i a t e d with  0  of  a n d S e g u r , 1981)  f o r a=c=1  J(f  2  ' ).  s p e e d a n d wavenumber a , c a n d K  be  a ( 0 ) = 1 , c(0)=1 and  dispersion relation  1  2  (3.1.2) i s  > + c y , _W/< ' ) +  37  J(\//  <  1  _ty  ,  )  ( 0 )  2  + 6 y) = - _ _ i / /  ( 0 >  -  A^  ( 0 )  ,  T  w h i c h f o r r>a c a n be r e w r i t t e n a s ( s e e ( 3 . 1 . 3 ) )  J ( f  1  °»  2  1  ) = -&yp{  ( 1 )  + c y , Ai£< > - 5 c - i / r  0  ( 0 )  ' - Ai//  .  (3.1.6)  T  The homogeneous a d j o i n t  (A  for  2  -  6 c-  )J(<//<  1  (0)  which  u = i/> (r>a)  condition Ablowitz  equation associated  0  on  ^< '  is  for  IT  a  r>a  c i t a t i o n s given  0 )  +  cy,  with  u)  =  solution.  is  (3.1.6) i s  0  The  therefore  solvability  (see  t h e Luke and  above)  oo  J J ^'°'(A^ -TT a  ( 0 )  (0)  + A<// ) rdrdt? = 0. T  (3.1.7)  The 0 ( e ) p r o b l e m f o r r<a c a n be w r i t t e n a s ( s e e ( 3 . 1 . 4 ) )  J(i//  ( 0 )  + cy, AiJ/  + K (// >) = -Ai/>  ( 1 )  2  (1  ( 0 >  - Ai//  ( 0 )  ,  (3.1.8)  T  with  t h e r e l a t e d homogeneous a d j o i n t  (A + K ) J ( I / / 2  for  which  <0)  u = \// (r<a)  ( 0 )  is  equation  + c y , u)  a  = 0  solution.  The  solvability  38  condition  on  \p  l0)  f o r r<a  J J It  A<J/ T  ,/,<  0  here  1  =  { a- a  -  rdrdfl  ( 0  > (A<//< °>  + A^ >) T  0  i s noted  (0)  therefore  a  7r -TT  is  1  that  2  [DK, ( 6 a c - " ) / K , ( f i a c "  (0  ]}A<// >  + c  1  /  2  1  (rc'  /  2  cos(0)$o  Ai//  ( 0 )  A^ T  r  {  a- a  =  cos(0)£  terms  1  2  ' )  /ca,  -  1 / 2  1  [a" a  A,// > T r  /c ]}A^< T (0  o  and  1 / 2  0  K-  > +  1  (/cr)  r>a  1  ]iea[a- a  and  A^  o  and  DJ^fca)  The  (/cr)  fast  ' -  ( 0 )  are  variable the  r<a  the  to t h e i r  b e c a u s e of  ,  6  derivatives  arguments  8ac"  r is retained  slowly  varying  of  in  1 / 2  the  a(T)  in  Note t h a t  the  evolution  of  T  the  integration  last  two  the  phase  limits  in  (3.1.7) and  (3.1.9).  terms i n b o t h e q u a t i o n s e x p r e s s the shift  i n the  +  r  T  J,(/ca) w i t h r e s p e c t  )  ( 0  A(//  T + r- sin(0)£  respectively.  (rc"  (0)  1  A<// > T r  2  -  A<// , T 6  [ D J , ( K a ) / J , (/ca)  T  T  1  T  6.  1  1  DK^Sac" ' )  to zero  ) ]8ac"  T K"  where  + r' sin(e)£0  ( 0 )  T  and  1 / 2  (0  )  T  - 1  (3.1.9)  T (2c)' c  K^Sac  = 0.  term.  It turns  solvability  out  conditions  that due  slow  these terms to  the  integrate  periodicity  in  39  After imply  some a l g e b r a  i t c a n be shown t h a t  (3.1.7) and  (3.1.9)  respectively  1  A a" a  1  - [A -  1] ( 2 c ) " c  T  = -1  (3.1.10)  T  1  B a" a  + [B -  1  1] K ' K  T  = -1  (3.1.11)  T  where  2  A = 7K (7)/Ki(7)  " 1 " K (7)/D,  2  1  D!  1  2  = -7" (7K <7>  1  " 2K (7)K,(7) ~ 0  3  B = 1 - {kJ (k)D /j (k) 0  2  7^(7)} o  -  1 2(kJ (k)/J,(k) 0  2  2  2  (D /J ( ) i 2  D  - 2(1 + k  7  1  2  2  + 2)(1 + k 7 " ) J ( k ) / ( k J i ( k ) ) + 1 +  2  2k  2  2  7  - }/  2  7  - )J (k)/(kJ (k))} 2  1  2  = k- {kJ (k) - 2J (k)J,(k) + 2  2  2  kJ (k)}  1  where 2  2  7=(6 a /c)  Equations  (3.1.10) and  unknowns. (3.1.5) w i t h  A  third  1 / 2  (3.1.11) equation  respect to  T  k = «a.  are is  yielding  two  obtained  eqautions by  in  three  differentiating  40  1  K- K  = Nta-'a T  where  T  ] - (2c)" c T  2  N = -{ R + k R/7}/{4 + 7  Eliminating  1  - (2c)-'c  7/R  +  (3.1.12) T  k  2  R A }  and  R = K  (TJ/K,  2  1  K " K between (3.1.11) and (3.1.12) g i v e s T  1  1  M a " a - [ M - 1] ( 2 c ) - c = -1 T T  where  M  (7).  = BN+B-N.  Provided  (3.1.13)  the unique s o l u t i o n s  MT*A  (3.1.10),  (3.1.11) and (3.1.13) a r e e a s i l y seen t o be  a  = -a  a = exp(-et)  The  in  expectation  that  this  exponential-like  =  Section.  the  RHS  decay  of  K  i.e.,  T K = K exp(et).  c = exp(-2et)  s o l u t i o n s (3.1.14) are  calculation  K  c = -2c T  T  (3.1.14)  0  the  principal  result  of our  Concomitant with the i n t u i t i v e (3.1.3)  must  result  in  a  i n the modon, (3.1.14) i m p l i e s that the  v o r t i c i t y and streamfunction  amplitudes  decay  as  exp(-T)  and  exp(-3T) r e s p e c t i v e l y . The  exponential  decay that the s o l u t i o n s (3.1.14) p r e d i c t  can be seen as the r e s u l t of the energy and enstrophy a s s o c i a t e d with (3.1.1).  It  9 t  00  I t f o l l o w s from (3.1.1) that  Tt  00  S S Vtf-Vtf rdrd0 = -2e / / Vtf-Vtf rdrdfl "TT  0  -it  0  equations  41  J /  3 t  Thus  the  predict  -TT  |Ai//|  J J |_v//| -rr  0  spatially  integrated  this  occurs  there  three  unknowns ( i . e . ,  a r e only  (3.1.10)  (3.1.13)  H o w e v e r , n u m e r i c a l c a l c u l a t i o n s showed t h a t  a n d M=A(=.5073).  discreteness  when  M=A,  since  two independent e q u a t i o n s f o r  and  values which c o u l d  equations  by ( 3 . 1 . 1 4 ) .  (3.1.14) appear non-unique  when  (3.1.5)  rdrdS.  energy and v o r t i c i t y  decay a s i m p l i e d  solutions  were t h e o n l y  2  0  exponential  The  r d r d S = -2e  2  satisfy  are  identical).  7=5.776 a n d k=4.4835  thedispersion r e l a t i o n  T h e f o l l o w i n g a r g u m e n t shows t h a t t h e  of these v a l u e s and t h e c o n t i n u i t y of  and K  a, c  i m p l y t h e u n i q u e n e s s o f ( 3 . 1 . 1 4 ) i r r e s p e c t i v e o f A a n d M. Suppose such that  A=M (T=T)*0  7  T=T  at  (recall  e x i s t s o l u t i o n s a, c and K  and there 2  2  2  7 =6 a /c; the proof  works  equally  T well  exploiting  which  7 ( T ) * 7 ( T )  interval  k=«a).  I t  follows  f o r t h e i n t e r v a l T<T<r+a.  and thus  there  =0 i n some n o n z e r o i n t e r v a l  or  M*A  i n this  7 (T=T)=0. T  (r,T+a)  e x i s t s a>0 f o r  Hence  (3.1.14) a r e t h e s o l u t i o n s  S u p p o s e t h e same h y p o t h e s e s b u t t h a t 7  that  in  interval.  But then  not.  this  either  I f not,the  T previous definition _ 1  a a  result of 7  = (2c) T  interval.  _ 1  applies and = - K  c T  from _  1  K  on  this  interval.  (3.1.10) = -1  are  and the  I f true,  (3.1.11)  i t  solutions  from t h e follows on  T  However r i s a r b i t r a r y s o t h e p r o o f  i s complete.  this  42  3.1.2  Of The D i s s i p a t i o n S o l u t i o n  s o l u t i o n s f o r a, c and K s a t i s f y  The ( K C  Discussion  1  /  2  = 0.  )  (3.1.5)  Therefore  (*ca) = 0 , T  (ac"  1 / 2  = 0 T  )  and  reduces t o  T -6J (K )K,(5)  for  a l l T implying  during  the decay.  to i t s i n i t i a l WKB-like  streamfunction  the  _VJ/ A\//  must  predict.  vorticity  ) K  2  ( 5 )  The and  and  + Ai// T  (introduce  of  i s i n agreement  any  the with  e t a l . , 1981) f o r  (McWilliams  conditions  (3.1.7)  and ( 3 . 1 . 9 ) L1  t o e l i m i n a t e t h e s e c u l a r i t y i n \p '  i s indentically  the  as  decay of the  invariance  homogeneous s o l u t i o n t o ( 3 . 1 . 6 )  ( 0 )  0(1),  to  exponential  the  equivalent  state.  compatibility  is a  (0)  0  Thus t h e modon r e m a i n s d y n a m i c a l l y  s o l u t i o n of ( 3 . 1 . 1 )  sufficient  (0)  J I ( K  that the dispersion r e l a t i o n s h i p i s invariant  and  a modon i n i t i a l  fact  0  r e l a t i o n w h i c h we h a v e o b t a i n e d  numerical  The  K  state, at least i n i t i a l l y  theory  dispersion  =  0  2  change  of  (note  and ( 3 . 1 . 8 ) )  zero as a consequence o f variable  are i n  r —> a ( T ) r  that since  (3.1.14)  i n ( 3 . 1 . 3 ) and  (3.1.4)).  Figure ^,(0)  +  c  (T)y  5 i s a sequence of c o n t o u r p l o t s as  T increases.  The o b s e r v e r  of  the  pathlines  i s i n the reference  f r a m e o f t h e modon s o t h a t a s d i s s i p a t i o n o c c u r s  the  surrounding  f l u i d appears t o slow  the  increasing  separation  of  down,  the contours.  as  indicated  Figure  by  6 i s a sequence of c o n t o u r  p l o t s showing t h e decay i n t h e v o r t i c i t y  f i e l d as T  increases.  43  The The  observer  is  fixed with  respect  modon moves t o t h e r i g h t w i t h An  t o the f l u i d  speed  u p p e r bound on t h e d i s t a n c e  infinity.  c(T).  over which  modon t r a v e l s a s a modon c a n be o b t a i n e d  at  the d i s s i p a t i n g  from t h e c h a r a c t e r i s t i c  equation  dx/dt = c ( T )  which i n t e g r a t e s t o  x(t)  so  that  the  modon  modon r a d i i ) b e f o r e Mcwilliams decays  = £  + (1 -  0  exp(-2et))/(2e)  t r a v e l s a maximum d i s t a n c e breaking  up i n t o a f i e l d  et a l . ( l 9 8 l ) estimate  that  (2e)  f o r t < 15 t h e  a s a modon ( b a s e d on s i m i l a r p a r a m e t e r v a l u e s  the s c a l i n g i n t h i s Section,  T * 1.5  (note  that  Figure ( 0 )  stage the amplitudes of ^ Figures  5  and  describes a  0  and  of  and  thus  and the  Ai/>  (0)  c  of  0  of  100 km  and  solution will  are  occurs.  very  and  Based when  At t h i s  small  (see  above s o l u t i o n q u a l i t a t i v e l y  0.1 m s ~  1  For oceanic  scales  respectively  be a s y m p t o t i c a l l y v a l i d  for a  1  be a s y m p t o t i c a l l y  valid  of  t h e above  1000 km a n d 10 m s " , r e s p e c t i v e l y ( s e e M c W i l l i a m s , solution will  2  takes place  100 d a y s , w h e r e a s f o r a t m o s p h e r i c s c a l e s o f a  perturbation  modon  for 6  g o e s up t o T = 1 . 3 9 ) .  t h e p r i n c i p a l decay mechanism.  perturbation scale  6)  the t r a n s i t i o n  5 only  (about 5  o f Rossby waves.  r ) a n d when t * 15 a modon R o s s b y wave t r a n s i t i o n on  _ 1  on  time and c  0  0  1980) t h e a  time  s c a l e of  10 d a y s  45  RRDIUS =1.0000  TIME = 0.0000  SPEED = 1 .0000  KflPPR =  3.9226  STREAM FUNCTION FIELD + CY  T  -5.0  -3.0  -1.0 X  1.0  3.0  5.0  AXIS  F i g u r e 5a. Sequence o f contour p l o t s o f t h e p a t h l i n e s ^,(0) ( T ) V f o r t h e Ekman d i s s i p a t i o n p r o b l e m . The o b s e r v e r i s f i x e d w i t h r e s p e c t t o a c o o r d i n a t e s y s t e m a t t a c h e d t o t h e modon. The c o n t o u r i n t e r v a l s a r e ± 0 . 2 . The z e r o c o n t o u r i s m a r k e d w i t h a 0 . The v a l u e s o f a , c a n d K a t e a c h s l o w t i m e T a r e l i s t e d i n the upper l e f t hand c o r n e r . +  C  46  RADIUS  =0.7046  SPEED = 0.4965  TIME = KfiPPfi =  0.3500 5.5664  STREAM FUNCTION FIELD + CY o in.'  o  CO X  cr >-c=>  o  o i n .  1 -5.0  -3.0  -1.0 X  1 1.0  3.0  5.0  AXIS  F i g u r e 5b. Modon p a t h l i n e s a t T=0.35 u n d e r Ekman  dissipation,  47  RRD1US = 0 . 4 9 9 0  TIME =  SPEED = 0.2490  KRPPfl =  0.6950 7.8597  STREAM FUNCTION FIELD + CY o  O  00  -  m  X cr  o ro _  CD  uo.  -5.0  -3.0  -1.0  1.0  3.0  5.0  x nxis  F i g u r e 5 c . Modon p a t h l i n e s a t T=0.695 u n d e r Ekman  dissipation.  48  RADIUS =0 .3534 SPEED = 0.1249  TINE = 1.0400 KfiPPfl = 11.0979  STREAM FUNCTION FIELD + CY o LO "  o  ro"  UO X CE  >-° i  o ro.  LO .  1  5.0  -3.0  1  -1.0 X  1  1  1.0  3.0  : |  5.0  AXIS  F i g u r e 5 d . Modon p a t h l i n e s a t T=1.04 u n d e r Ekman  dissipation.  49  RRDIUS  =0.2490  TIME =  SPEED = 0.0620  STREAM  KRPPR  1.3900 =  FUNCTION  15.7487  FIELD  +  CY  o  O  ro •  cn I  1  X  d  o ro.  o in. -5.0  Figure  5e.  -3.0  -1.0 X  1.0  3.0  5.0  RX1S  Modon p a t h l i n e s a t T=1.39 under Ekman  dissipation.  50  RADIUS =1.0000  TIHE =  SPEED = 1.0000  KflPPfl =  VORTICITY  0.0000 3.9226  FIELD  o i n '  o ro'  CO t  1  X  cr  ^° i  o ro .  o t n .  -5.0  -3.0  •1.0 X  1.0  3.0  5.0  AXIS  < 0 )  F i g u r e 6a. Sequence of contour p l o t s of the v o r t i c i t y A i / / for the Ekman d i s s i p a t i o n problem. The observer i s f i x e d w i t h respect to the f l u i d at i n f i n i t y . The contour i n t e r v a l s are ±2.0. The zero contour i s marked with a 0. The values of a, c and K at each slow time T are l i s t e d i n the upper l e f t hand corner.  51  RADIUS =0.7046 SPEED = 0.4965  TIME = 0.3500 KRPPR = 5.5664  VORTICITY FIELD o  in'  o  no'  cn x >- °  o  in.  t  1 -2.483  1  -0.483  1  1  1.517  3.517  1  5.517  I  7.517  X nxis  F i g u r e 6b.  Modon v o r t i c i t y a t T=0.35 u n d e r Ekman  dissipation.  52  RADIUS =0.4990 SPEED = 0.2490  VORTICITY  TIME = 0.6950 KflPPfl = 7.8597  F I E L D  o  m  l  o  in.  -1.245  ~1 0.755  -I 2.755  1 4.755  1 6.755  8.755  X nxis  g u r e 6 c . Modon v o r t i c i t y a t T=0.695 u n d e r Ekman  dissipat  53  RRDIUS =0.3534 SPEED = 0.1249 VORTICITY  TIHE =  1.0400 11 .0979  KAPPfl =  F I E L D  o  ID '  O  CD X  cr  o  ro .  o  in.  I  -0.525  1.375  I  I  3.375 X  F i g u r e 6d.  Modon v o r t i c i t y  5.375  1  7.375  9.375  RXIS  a t T=1.04 u n d e r Ekman  dissipation.  54  RADIUS =0.2490 SPEED = 0.0620  VORTICITY  TIME = 1.3900 KAPPA = 15.7487  F I E L D  o in"  to X  cr  o ro_|  !  0.31  1.69  j  j  3.69 X  F i g u r e 6e.  5.69  j  7.69  , 9.69  RXIS  Modon v o r t i c i t y a t T=1.39 u n d e r Ekman  dissipation.  55  3.2 Modon P r o p a g a t i o n O v e r S l o w l y V a r y i n g  In  this  Section  propagation solution,  over  a l e a d i n g o r d e r p e r t u r b a t i o n t h e o r y f o r modon s l o w l y v a r y i n g topography  which  topography,  Topography  is  is  independent  applied  i s developed.  o f any f u n c t i o n a l  to  two  specific  This  form of t h e topographic  configurations. In is valid  Subsection  amplitudes  i n Subsection  t h e modon for  the general theory  f o r finite-amplitude slowly varying  topographic Also  3.2.1  Subsection  ridge  These a n a l y t i c a l  3.2.2.  Subsections  obtained  solutions are  3.2.3  and  3.2.4  s o l u t i o n s t o a modon t r a v e l l i n g o v e r a  (modelled  d i r e c t i o n ) a n d an e s c a r p m e n t  as  a gaussian  (modelled as a  in the x-coordinate d i r e c t i o n ) ,  3.2.1  (i.e.,  of the f l u i d ) .  s p e e d a n d wavenumber a r e  topography.  apply the small-amplitude meridional  which  3.2.1, a n a l y t i c a l p e r t u r b a t i o n s o l u t i o n s f o r  small-amplitude in  topography  on t h e o r d e r o f t h e d e p t h  radius, translation  described  i s developed  i n the x-coordinate hyperbolic  tangent  respectively.  P e r t u r b a t i o n S o l u t i o n F o r Modon P r o p a g a t i o n O v e r  Slowly  V a r y i n g Topography  As  i n V e r o n i s ( 1 9 6 6 ) , Rhines(1969a,b), Clarke(1971), L e B l o n d and  M y s a k d 9 7 8 ; S e c . 2 0 ) , M a l a n o t t e - R i z z o l i and Matsuura  and  Yamagata(1982)  v a r i a b l e topography shallow rigid-lid  water  and Yamagata(1982) t h e e f f e c t s o f  on p l a n e t a r y waves  equations  shallow water  on  Hendershott(1980),  the  potential  are  0-plane. vorticity  modelled The  with  the  nondimensional  equation  can  be  56  written  as  (LeBlond  V-fH-'Vi//  and  Mysak,  1978;  Sec.  1  ) + J[<K  20)  1  H-'V-(H- V^) +  fH" ]  = 0.  (3.2.1)  t  All  symbols a r e d e f i n e d as t// i s t h e  exceptions;  in  S e c t i o n 3.1  =  tf/  northward  v  are  the  velocity  H(ex,ey) =  components  scale  0  topographic  of  amplitude  smallness order  of u w i l l  topography  be  planetary 2  Coriolis  1  parameter  is f =  dominate  2  1  ridges,  Typical large  s l o p e s of  seamounts  and  where  u  with is  Subsection  solution  will  the  it be  (i.e.,  is  valid  y=*0(l)), a l t h o u g h  (r )"  1  0  2  + 6 y.  topographic  (LeBlond  0  20).  and  the  0<e«ji«1)  in  The  of  solutions.  ( 6 r ) " 0(VH/H) =* e ( 8 r ) ' < 1 Sec.  (i.e.,  (positive)  topography  1  this  e v e n t u a l l y demanded  vorticity  0  In  leading order  to obtain a n a l y t i c a l The  4  and  respectively,  topography) «  The  finite-amplitude  eastward  slowly varying  parameter.  assumed t h a t 0<e<<jx^1. for  (positive)  1 - uh(ex,ey) i s the  e = a /(length  following  = \p , x  Hv  y  and  the  t r a n s p o r t streamf u n c t i o n  -Hu  where u  with  the  and  ocean  and  floor  escarpment  steering Mysak,  away from  breaks  effects  give  if 1978;  mid-ocean e =*  3  10" ,  * The t o p o g r a p h i c p a r a m e t e r i s t h e maximum a b s o l u t e v a l u e of the height o f t h e t o p o g r a p h y d i v i d e d by t h e mean d e p t h . The t h e o r y a p p l i e s t o t o p o g r a p h i c d e p r e s s i o n s ( e . g . , t r e n c h s ) as w e l l as t o topographic p r o t r u s i o n s (e.g., r i d g e s ) .  57  2  thus We  e(8 r )~ note  allowed  * 10"  that to  equatorial As and  1  0  1  (for scalings  larger  increase  described  (e.g.,  atmospheric  applications  or  0  is in  regions).  i n L u k e d 9 6 6 ) , Gr imshaw ( 1 970, 1 971 , 1 979a, b, 1 9 8 1 ) , Kodama  (3.2.1) i s f o u n d  Section  1  ei/'  1  A(X,Y,T)  velocity  0  > U,y;X,Y,T) + ev<  £ = -c(X,Y,T) t  X, Y and T a r e d e f i n e d  and  1  2  i  + ... ,  + ... ,  a n d where t h e s l o w  Y= ey  +  and  1  £yY  (3.2.2c) variables  into  H " J U , Ai//) = 2  T=et.  (3.2.1) y i e l d s e{ -H" 1 Ai//  i  -2cH~ \jj  (3.2.2b)  by  of these v a r i a b l e s + 8 H-V  (3.2.2a)  > ( £ ,y;X, Y,T)  £ = 1 x  X=ex,  -cH'1Ai//  to  field  1  Substitution  solution  + »//< °> ( £ , y ; X , Y , T ) +  u = u< ' U,y;X,Y,T) + eu< > U ,y;X, Y,T)  l0  a  ' U,y;X,Y,T) + ... },  the r e l a t e d perturbation  v = v  3.1  i n t h e form  \P = H(X,Y){  where  3.1).  b o t t o m s l o p e s c a n be c o n s i d e r e d a s r  A b l o w i t z ( 1 9 8 0 ) and t h e work i n  with  in Section  + 2cH-1i//  T  - 8 H"V  - H ' J (\l>, At//) - H - J (i//, Ai//) 2  2  X  X  2  Y  +  ux  58  2  2H- JU, ^  2  2H" J(i//, <// ) - c H " H  ) £X  yY  -  \p  2  2  cH" H  U  X  ^ + Y £y  2  2H- (Ai//)J (,//, H) + 2 H " ( A i / / ) j U , H) + f H " J U , H) + 3  3  X  Y  2  f H - J U , H) + H" H  2  J U , i// ) + H- H  3  J(i//, »//)} + O U ) , (3.2.3)  3  Y  I  X  Y  J(-,*) = 3(•,*)/3(£,y),  where  X  y  J (•,*)  = 3(•,*)/3(X,y)  and  X J  (•,*)  =  3(•,*)/d(t,Y).  Y The  formulation  x-direction £  of t h e phase v a r i a b l e £ does not i n c l u d e  wavenumber,  such t h a t  £ =-ck and £ =k t x  h e r e a r e n o t changed out  of  it  equal  to unity. varying  incorporate  required The  order  wavenumber by  Also,  i t  the  slowly  would  results  c remains.  longer to write  be  possible  t h e modon  defines  presented  wavenumber  factors  Therefore,  i s f r e e and one c a n that  the r o l e  definition varying  be  i f one  to define  the  modon  incorrect  to  the p o l a r  set  played  of  t h e y v a r i a b l e i n t o t h e phase v a r i a b l e  by a phase  radius attempt £  since  and to i t  coordinates  solution.  0(1) problem i s  J ( f  the  and o n l y  in  the  order  x-direction  M o r e o v e r , we a r g u e  i s played  wavenumber.  no  the  such a parameter  to this  would  the leading  equations  least  variable  since  I t c a n shown t h a t  the governing  at  slowly  s a y k.  a  0  )  s o l u t i o n of which  + c y , A\p  i s taken  i0)  + 6 y) 2  =  0,  t o be t h e modon  (3.1.3),  (3.1.4)  59  and  (3.1.5).  | (X,Y,T)  made  0  will  T h e comments  be c h o s e n  allowing  the  regarding  i n S e c t i o n 3.1 a p p l y  here  t o be t h e x - c o o r d i n a t e topography  t o be  t h e phase  shift  and thus  term  eventually |  0  o f t h e wave c e n t e r a t T=0, a t X=0.  centered  The wave  p a r a m e t e r s a, c and K a r e s l o w l y v a r y i n g f u n c t i o n s o f X , Y and T that  satisfy  relationship  v a l u e problem, a ( X , Y , 0 ) = l ,  initial where  the dispersion  K  ( a s i n S e c t i o n 3.1)  (3.1.5).  C(X,Y,0)=1  K(X,Y,0)=K /  and  i s obtained  0  from  For the O  the d i s p e r s i o n  (3.1.5) when T=0 ( K = 3 . 9 2 2 6 when 6=1).  relation  0  The  0 ( e ) p r o b l e m c a n be w r i t t e n a s  J(i//  < 0 )  + c y , AiJ<  (1)  )  + J(v//  ( 1 )  , Ai//  ( 0 )  2  2  + 6 y) = -6 A X  2  1  1  6 H" AH  - H " J (HA, A ^ X  )  (  )  < 0  2  - 8 ^  ( 0 >  2c^  lyY (  ) ^  0  )  (Ai/>  < 0 >  ,  2  1  1  + f)(H" H  <  0  ) ^ Y |y 1  + f)(H" H  >  >->0  i n Luked966),  )^  reduces t o  (0)  ( 0  (0)  i// ) +  >,  0  )  - (H-'H ) J ( *  < 0  >,  0  <//< > ) -  X  ) ^ y  ( 0  I  (  0  )  |  - (H-'H ) 0 ' ° > A ^ X y ( 0 >  ,  (  ^ y  1  0  )  )  ( O )  +  +  (0  > + (H- H )^<°>A^ '. Y |  (3.2.4)  Grimshaw( 1 970, 1 971 , 1 979a,b, 1 981)  and A b l o w i t z ( 1 9 8 1 ) , a s r->°=.  (  - (H-'H ) J ( i / / Y  Y  1  (0  X  (0)  1  Kodama  (0)  2 J ( t / / , i// ) +  A<// >) - 2 J ( ^  - 8 (H- H ) V  C(H" H  <0  <0)  A\// ) -  yY .  X  (Ai// >  ,  0  ) - A^< » + T  |X  n  x  0  (0)  X  - J U Y  ( 0 )  U  - J  X  ||X  As  1  - H" J (HA, A i / > Y  X  2c^ >  c(H-'H  l 0 )  ^  (  1  Consequently,  }  i s assumed  and  t o have t h e p r o p e r t y  i n the l i m i t  as  r—>»  (3.2.4)  60  A  + ( H ' H )A = 0 , X X  t// >->0 a s r->=>).  (recall  some  s o l u t i o n of ( 3 . 2 . 5 ) i s  The g e n e r a l  (0  A  for  = g(Y,T)H"  1  function  g(Y,T).  Thus  perturbation  expansion  (3.2.2a)  (i.e.,  which  be  will  developed  (3.2.5)  1  left  undetermined  the  first  AH)  term  is  i n the  simply  i n the leading  g(Y,T)  order  analysis  here.  Note  that  since  J  AH = g ( Y , T ) ,  (HA, A i / /  ( O >  )=0  and  X J  (HA, ,Ai// > )=-g A i / / Y Y i 0 (  integrates the  t o zero  I t turns  i n t h e imposed  out t h i s  t  0 )  =  _^,(  0  )  v  (  Also,  > = V  0  y  Therefore, zero  in this  at  this  stage  J(i//  leading  ( 0 >  ,  AI/-  -  ( 1 )  order  analysis.  c a n be r e w r i t t e n  6 c"V 2  (  since  '.  1  >  )  ( 0 )  -  6 »// > X 2  (0  + (8 /c) 2  i// >  1  X  ^ y  ( 0 )  retained  as (see ( 3 . 1 . 3 ) )  2  ( 0 )  equal  i n the 0(e) problems.  = (6 /c)H- g  ^/  set  I t i s formally  -  (0  Y I 2ci// ££X  g(Y,T)  *  because o f i t s appearance  r>a, ( 3 . 2 . 4 )  term  due t o  note, t h a t  w i t h no l o s s o f g e n e r a l i t y , g ( Y , T ) c a n be  to  For  { 0  remaining  conditions  to the 0 ( 1 ) v e l o c i t y f i e l d  not contribute  u  last  solvability  i n 6 (see the Appendix).  periodicity  will  .  ( 0 )  -  2J( // l  ( 0 )  ,  Ai//' ' T 0  i// ) £X (0)  +  +  61  C(H" H  )^ >  1  x [(r )"  -  ( 0  6 (H" H 2  2  + 8 y](H-'H  0  2c<//  -  (0)  0  -  )  )i// >  1  2  (H-'H ) Y  iy  + 5 y](H- H  )^ >  1  2J(i//  JU  ( 0 >  >  0  ° > +  ,  y ^ ) yY (  0  (0)  y  2  ) ,/,< ° > ,/,<  1  Y  equation  +  )  , <// ) +  i  The homogeneous a d j o i n t  ( 0 )  + 2(8 /c)(H- H  ( 0  Y  ),/,<  1  i  -  (0  Y  -  ( 0  ) -  i  (28 /c)(H- H 2  ( 0 )  , ^  X  Y  C(H" H  1  1  i// ^'°>  2  ( 0 )  )J(^  (H" H  x (  )^ X y  (8 /c)  ijY  [(ro)"  -  (0  x  a  1  )\J/ >  1  associated  0  >.  (3.2.6)  i  (3.2.6) is  with  (A - 6 c " )J(i//< °> + c y , u) = 0 2  for  u = i//  which  condition  on  \J/  (0)  1  (r>a)  is a  f o r r>a  ( 0 )  is  solution.  therefore  The  solvability  (see t h e  Luke  and  Ablowitz c i t a t i o n s )  rr  CD  J i//< -it a 0  /  For  )  {RHS(3.2.6)}  r<a, ( 3 . 2 . 4 )  j(^<°>,  2ci//  Ai//  ( 1  c a n be r e w r i t t e n  > +  + K c\p  ( 0 >  2  iiX 1  x [(r )0  -  (0)  2  ^  (  1  ( 0 )  +K  = - K  2  2  1  C(H"  1  H  -  A\l/  +  {0)  T  -  £X ( H "  X  1  H  0)  (0)  )j(<//< , i// ) -  x )^  1  t0)  y  )^/<°>  K cy](H" H 2  (0)  ( 0 )  x -  t// Y i  g  (0  X 2  H "  as (see ( 3 . 1 . 4 ) )  U ) ^ >^<°> - 2 J ( i / / , \ ^ ) +  H 1  )  )  X  )V  c(H- H  K  (3.2.7)  r d r d f l = 0.  , 0  y  >  +  2 K  2  ( H -  i 1  H  )^<°>^<°>  X  y  +  62  2cyp  2  0  + U )  i0)  ^< >tf<°>  SyY ( 0 )  2J(^  ( 0 >  ^  yY [(r )-  )^  Y  - K cy](H- H  1  2  0  <//  2  1  The homogeneous a d j o i n t  -  (0)  Y £  + c(H"'H  )  + K C)]  i  Y  ,  2  + [rsin(0)(6  ) ^ Y %  (  -  ( 0 >  1  £y 0  t 0 )  )J(V , *  (H" H  Y  - 2K (H- H  )  2  l 0 >  ) +  y 0  )^<°)^< '.  1  Y  (3.2.8)  £  equation associated  with  (3.2.8) i s  (A + K ) J U < > + c y , u) = 0 2  for  u = i//  which  condition  <  on ^  0  )  (0,  (r<a)  0  is  a  solution.  The  solvability  f o r r<a i s t h e r e f o r e  a / / ^' >{RHS(3.2.8)} r d r d f l = 0. TT  0  The c a l c u l a t i o n (3.2.9)  is  of  the  tedious  (3.2.9)  compatibility  conditions  but  straightforward.  computations are d e s c r i b e d  entirely  has been n o t e d  (3.2.8)  (containing  periodicity contain  in  6  derivatives  respect  that  g(Y,T))  of  due  the  the  In a d d i t i o n ,  number a s a c o e f f i c i e n t  These  calculations.  t e r m i n t h e RHS  integrates  to  Appendix).  phase  t o X,Y and T i n t e g r a t e  the A p p e n d i x ) .  zero  the f i r s t  (see  and  i n the Appendix.  F o u r o b s e r v a t i o n s a r e made h e r e about t h e s e It  (3.2.7)  shift  t o zero  terms  o f (3.2.6) and  zero  Also, term  due  to  the  a l l terms  that  £ (X,Y,T) 0  f o r t h e same r e a s o n ( s e e  which  contain  i n (3.2.6) a n d (3.2.8) a l s o  to the p e r i o d i c i t y  with  the  Rossby  integrate to  i n 6 (see the Appendix).  Thus t h e  63  equations  we d e r i v e t o d e s c r i b e t h e slow v a r i a t i o n  parameters  a r e independent  o f t h e Rossby number,  o f t h e modon $ (X,Y,T) and 0  g(Y,T). Of  physical  individual  interest  term  i n t h e RHS's  contains a derivative integrates  varying order  with  to identically  the Appendix).  Therefore  topography solution  parameters  i s the  i ssolely  (3.2.6)  of  respect  to  and  t h e slow  that  each  (3.2.7)  that  variable  z e r o due t o t h e p e r i o d i c i t y  Y  i n 6 (see  the m e r i d i o n a l s t r u c t u r e of t h e slowly  appears  and  observation  i n parametric  the leading determined  form  order  by  i n the leading  e v o l u t i o n o f t h e modon  the east-west  topographic  structure. However,  only  the l e a d i n g order  solution  d e p e n d e n c e On t h e m e r i d i o n a l t o p o g r a p h i c to  check  satisfy have  that higher  this  been  topography  order  derivatives  orthogonality property. ignored  in  (3.2.3).)  therefore give r i s e  has a p a r a m e t r i c  structure. with  It i s  easy  r e s p e c t t o Y do n o t terms a r e 0 ( e ) and 2  (Such  Meridional  gradients i n the  t o 0 ( e ) modulations 2  i n t h e modon  parameters. A physical dependence based  explanation f o r this  on t h e m e r i d i o n a l t o p o g r a p h i c  on t h e f o l l o w i n g v o r t i c i t y  states  leading  that  the p o t e n t i a l  the  Consider  x-axis.  change  a particle  Equation  (3.2.1)  (v - u + f ) / H i s a c o n s e r v e d — x  quantity.  parameteric  s t r u c t u r e c a n be g i v e n  arguments.  vorticity  order  of f l u i d  y— displaced  parallel  to  I f H i s n o n - z e r o t h e n t h e r e l a t i v e v o r t i c i t y must x i n r e s p o n s e t o c h a n g e s i n H ( f b e i n g c o n s t a n t when y. i s  64  constant). to  Thus t h e  zonal  l e a d i n g order  displacements  occurs  relative  i n response  vorticity  adjustment  to zonal  topographic  variations. However, t h e to  topographic  scaling in the  of  vorticity  leading  order  displacements  are  variations  second  order  After  (3.2.3))  a ( X , Y , T ) and  in  response  displacements.  The  assumed  changes  variations. to  t o changes and  that  Thus  meridional  i n the p l a n e t a r y  topographic  variations  comparison.  i n the  (see the Appendix),  f o l l o w i n g two  differential  (3.2.7)  equations  and for  c(X,Y,T)  1  cA a- a X  (B, -  (3.2.10)  = -cE,H"'H  3  X  1  1  1)(2c)- c  ] + cB (2c)" c  cB a"'a  X  -cE H" H 2  X  where  A,,  (3.2.1 1 )  1  =  3  +  2  X  respectively,  due  adjustment  considerable algebra  (3.2.9) r e s u l t  has  i s not  dominate t o p o g r a p h i c  the y - c o o r d i n a t e ) in  adjustment  meridional  vorticity  occurs  (i.e.,  vorticity  in  (3.2.1) (see a l s o  planetary  vorticity  leading order  X  A , 2  A , 3  E  1 f  B  1 f  B , 2  B  3  and  E  2  are  65  nondimensional which  are  derived  (3.2.11) can  1  a" a  functions  be  + c(A  in  the  rewritten  k  7 = 5 a / c and  (recall  Appendix.  l/2)a' a  + cA  The  system  (2c)- c X  1  + cA ,a- a  and  (3.2.12) X  +  2  X  T  = -cF H- H  ,  +  k=«a)  (3.2.10)  X  1  (2c)" c  2 2  2  = -CFTH^H  1  1 2  T  c(A  2  2  as  1  +  n  7 and  of  l/2)(2c)" c  ,  1  2  X  (3.2.13)  X  where  A,,  A  -  =  1 2  F,  =  evolution  " B,)A  of  (A,  -  1)B ]/(A,  A  2 1  =  (-B,A  3  + A,B )/(A,  -  B,)  A  2 2  =  (-B,A  2  + A,B )/(A,  -  B,)  2  of  =  +  2  - B,)E,  (A,  ~  (A,  2  (3.2.12) and  2  2  (-B,E, + h,E )/(k,  K(X,Y,T)  1)B ]/(A,  3  +  -  1)E ]/(A 2  -  (3.2.13).  1  B,)  -  B,)  -  B,)  B,).  i s determined given  -  3  - B,)A  each space-time c o o r d i n a t e solutions  +  3  [(1  [(1  F  The  [(1  a(X,Y,T)  by  solving and  (3.1.5) a t  c(X,Y,T),  the  66  Whitham(1965) a r g u e d ought the  t o govern  the evolution  parameter  eigenvalues  that  modulations  of  the  hyperbolic d i f f e r e n t i a l  equations  o f t h e s l o w l y v a r y i n g wave s o t h a t c o u l d propagate  matrix  [M  ]  with  i n space-time.  entries  The  M =A,,+l/2, 11  i j M, =A,2,  M i=A i 2  2  convention) for  matrix  of  (3.1.5)) [M ] ij  k=4.4835  M =A +1/2 2 2  (denoted  2 2  were n u m e r i c a l l y d e t e r m i n e d  a l l values  relation  and  2  7  except  i s not  and  k  for  when  (A, a n d B, a r e i n f a c t  t o be r e a l  (consistent  defined).  presented  solution and  discreteness solutions discrete  the  space-time  identical  The  equations  In  the  Similar  arguments  of  a,  i fthere exists a  c  A,=B,  a n d (3.2.13)  to  with  (3.2.10) K  and imply  and t h e  that  the  (except a t t h e s e t of  f o r which  the  singularity  (3.2.12) a n d (3.2.13) t h e r e f o r e form a  h y p e r b o l i c system  topography.  case  t o t h e f u n c t i o n s A and  associated  f o r which  coordinates  the dispersion  o n l y when 7=5.776 and  problem  (3.2.12)  distinct  A^B,  continuity  o f the 7 and K  satisfy  occurs). nonlinear  then  and  usual  ( i n which  i n S e c t i o n 3.1 c a n show t h a t  to the i n i t i a l - v a l u e  (3.2.11)  with  the  A^B,  M d e f i n e d i n S e c t i o n 3.1, r e s p e c t i v e l y ) . those  with  which  practice,  i svalid  the  for  problem  finite-amplitude must  be  solved  numerically. Analytical demanding u.  the  solutions  c a n be o b t a i n e d  smallness  of the topographic  When 0<e«jz<<1  atmospheric  (which  applications),  (3.1.5) c a n be o b t a i n e d  is  the  case  solutions  i n t h e form  for  in  a,  c  and  amplitude many  t o (3.2.12),  K  by  parameter  oceanic  and  (3.2.13) a n d  67  a = 1 + ua  1  l  c = 1 + MC< > ( X , Y , Y ) 1  K =  Note t h a t be  a  <  1  >  ,  confused  K[  (  c  1  )  with  e-perturbation  1 + UK  0  K  and  {  1  S i n c e H(X,Y)  1  2  a r e ^ - p e r t u r b a t i o n s and a r e n o t t o  played  = -uh X  a  >  +  l 1 ,  /2  (A ,  + l/2)a  01  + A  0  2  0  2  A ,, 2  0  A  2 2  2  ,  ( 1  '  + A  i a  (  1  )  +  (A  0  2  0 1 2  c  ( 1 )  + 1/2)C X  2  A  A  0  F, and F  2  0  2  1  ,  2  2  ,  F 1 0  1  }  and v  /2  Expanding the d i s p e r s i o n  series  a b o u t u=0  gives at  K< » = N [ a < ' 1  1  0  = F  0 1  ( 1  h  X  ( 1 )  '  i n the  /2  (3.2.14) X  = F  0 2  h  ,  (3.2.15)  X  and F  0  respectively,  k=K « 0  (  (3.2.13) a r e  X  where A , , , A , ,  ', u  0(u ).  X  T  A, ,  ( 1  2  +  T  c  by ^  X  t e r m s i n (3.2.12) and  ( 1  2  = 1~Mh(X,Y) i t f o l l o w s  1  0(u)  + 0(M )  (3.2.2).  H" H  The  0(/i )  ' ( X , Y , T ) + 0(/z ) ] .  '  the r o l e  expansion  1  2  > (X,Y,T) +  2  are the v a l u e s evaluated  relationship  of A,,,  for 7=8  (3.1.5) i n a  and Taylor  0(u)  - c  ( 1  1  > / 2 ] - c< ' / 2 ,  (3.2.16)  68  where  N  = -{5R + U ) R / 6 } / { 4 + 8/R + U ) R / 6 } 2  0  2  0  0  with  R=K (6)/K,(6). 2  From  (3.2.14) a n d (3.2.15) i t f o l l o w s  1  1  a' > TT  C  ( 1  + p,a< > + p a > XT XX  = v,h XX  2  C D TT  +  P l  c  that  (3.2.17)  > + p c = 2u h , XT XX XX  ( 1  ( 1 )  (3.2.18)  2  2  where  Pi  P2  ( A  =  V\  v  2  0  2  =  =  1  0  2  ( A  ( A  0  0  2  1  1  1/2)F  +  0  22  + 1/2) -  1/2)F  +  2  A  +  1/2)(A II  + 0  A1  =  0  -  0 1  2  "  A  +  A  A  0  1  2  0  2  1  F  1  0  i  2  F  0  2  0  1  .  A  The s o l u t i o n s t o (3.2.17) and (3.2.18) s u b j e c t  0  2  1  t o the i n i t i a l  conditions  a  ( 1 )  (X,Y,0)  = 0,  a  ( 1 )  (X Y,0) r  = F  0  T c  ( 1 )  (X,Y,0)  = 0,  c  ( 1 )  T  are  g i v e n by  (X,Y,0) = 2F  0 2  h X  (3.2.19a)  h X  (3.2.19b)  1  69  a  ( 1 )  c  ( 1 )  (X,Y,T)  = XthU,?)  + X h(X-a T,Y)  + X h(X-a T,Y),  (3.2.20)  (X,Y,T)  = X„h(X,Y) + X h ( X - a , T Y )  + X h(X-0 T,Y),  (3.2.21)  2  1  5  where t h e c o e f f i c i e n t s  3  r  are d e f i n e d  2  6  2  as  X, = v, ( a , a ) •  1  2  X  2  =  (F  X  3  = (v,  o t  o,  X  and  = 2(F  5  = 2{v  6  0 2  c  initial  2  a  ~  2  2  2  - v )[o,(o 2  - F  0 2  o )[a (a 2  2  1  ffi-)]"  2  2v (o)a )'  1  - {p, -  ~  2  by  respectively.  evaluating The  a,)]  [(p,)  on  a  (  2  2  -  a,)]" ,  o  1  (3.2.14)  solution  for  by  2  2  - 4p ]  }  are given  2  4p ] ' }/2  1 / 2  2  1  - 1  1  -  2  1  1  and  T obtained  a,)]'  s p e e d s a, and  = {p, + [ ( P i )  conditions  2  =  where t h e c h a r a c t e r i s t i c  a,  The  2  1  2  0  ft  X  1  - F io )[o (a  X  -  - » )[o (a  c  }/2.  (  1  )  (see (3.2.19)) a r e  T and ( 1  (3.2.15)  K >(X,Y,T)  at  T=0,  i s determined  by  70  (3.2.16),  K  (  with  1  )  (3.2.20) and (3.2.21)  (X,Y,T)  X ,  X  7  B  = X h(X,Y)  9  X  slowly  9  2  (3.2.22)  9  0  = N (X  8  0  = N (X 0  - X /2) -  2  X /2  5  5  - X /2) - X /2.  3  6  6  Of The S m a l l - A m p l i t u d e  varying  + X h(X-a T,Y),  = N ( X , - X,/2) - X„/2  7  X  3.2.2 D i s c u s s i o n  8  g i v e n by  X  The  + X h(X-a,T,Y)  7  and X  yielding  modon w i l l  Topographic  be d e s c r i b e d  Solution  (to leading  o r d e r ) by  (3.1.3) and (3.1.4) w i t h a, c a n d K g i v e n by  a = 1 + a  (1  M  '(X,Y,T)  (3.2.23)  c = 1 + uc >(X,Y,T) l  (3.2.24)  1  K  where a  (  (3.2.22) T.  1  ', c  (  1  )  0  and K  respectively,  The p o s i t i o n  obtained  = K [1+  (  1  UK1  1  ' (X,Y,T) ] ,  ' are given evaluated  by  (3.2.21)  a t t h e modon c e n t e r  o f t h e modon c e n t e r  from the c h a r a c t e r i s t i c  (3.2.20),  (3.2.25)  as a function  equation  of  and  X a t time time  is  71  dX/dT = c ( X , 0 , T ) ,  written order  i n slow occurs  (3.2.26)  variables. on  Y=0;  (Note  see  (3.2.26)  that  t h e modon c e n t e r t o  (3.1.3)  and (3.1.4).)  In g e n e r a l ,  i s not separable so X(T) i s d e f i n e d i m p l i c i t l y  be o b t a i n e d n u m e r i c a l l y .  Formally, the s o l u t i o n  lead  a n d must  o f (3.2.26)  is  written  X(T)  X(0)=e£  where  However  = e £  T J c[X(T),0,T]dT o  +  0  (3.2.27)  (see S u b s e c t i o n 3.1.1).  o  i f attention  i s restricted  t o T<<1  t<<e~1)  (i.e.,  then  C(X,0,T)  Consequently, center  will  « c(e£ ,0,0) o  (3.2.27)  and  0  + T +  MF  0 2  o  (3.2.15)  be a p p r o x i m a t e l y  X = e £  + c (e£ ,0,0)T T  +  implies  2  0(T ).  that  the  modon  g i v e n by  (e£ ,0)T  h  o  2  3  + 0(T ).  (3.2.28)  X  Using  (3.2.28) t h e s p a c e  solutions  (3.2.23),  asymptotic  (i.e.,  variable  T.  variable  (3.2.24)  T<<1)  X c a n be e l i m i n a t e d  and  solutions  from  (3.2.25) t o o b t a i n in  the  single  the  analytic  slow  time  72  As  a  typical  6=1.0, h e n c e : 39.44,  calculation  /c = 3.9226  (from  0  o  2  i t follows  5  6  6  singular  qualitatively  value  6=5.776).  are  in  only  values  near  the  solution  therefore hyperbolic  t o the topography.  conditions  (3.2.19), the  X, + X  2  + X  3  = 0  (3.2.29a)  X«  5  + X  6  = 0  (3.2.29b)  = 0.  (3.2.29c)  7  + X + X  8  + X  clearly  the t r i v i a l  9  required solution  i f  h = constant  (i.e.,  a  (  1  )  (  , c  1  >  i s to and K  (  1  a l l zero). Consider  Since  the s o l u t i o n f o r a  X <0,  the  3  eastward-travelling the  =-  satisfy  relations are  result  proportional  2  (except  westward-travelling  i n consequence of t h e i n i t i a l  X  These  results  The i n i t i a l - v a l u e and  Other  9  0  X„=0.46,  3  a n d X =0.24.  8  waves and a s t a t i o n a r y component  parameters  O  2  X =-0.28 similar  t h e form o f e a s t w a r d  Note t h a t  a = 3 . 5 9 , F i = 4 . 0 4 and F 2  7  give  2  X,=-0.05, X =0.29, X =-0.24,  X =-0.20, X =-0.26, X =0.04,  takes  3 . 1 . 5 ) , N =-0.9636, p,=-7.39, p = -  J>,=2.00, * =-9.20, a , = - l 0 . 9 8 ,  0.63 from w h i c h  of  of the s o l u t i o n parameters l e t  modon  associated acts  to  h(X,Y))  wave  acts  2  Since  X >0, 2  associated  the with  to decrease  by  (3.2.20).  associated  ( i . e . , h(X-o T,Y))  the westward-travelling  increase  perturbation  ( X , Y , T ) given  radius-perturbation  radius. with  ( 1 )  acts  with  to  the  decrease  the  radius-perturbation  wave  (i.e.,  radius.  Since  the  stationary  t h e modon r a d i u s .  X,<0,  hfX-c^TjY)) the  radius-  component  (i.e.,  These  r e s u l t s have  '  73  implicitly one  assumed a p o s i t i v e  considers  results  a  topographic  a r e simply  Similar solution  topography  depression  reasoning  (3.2.21).  Since  X <0 and X <0, t h e 5  westward-travelling  travelling  wave  act  to decrease  X„>0, t h e  speed-perturbation  component  of  The  stationary increase  topographic When  with  a  behaviour  speed.  the  Since  stationary  depression  of the wavenumber-perturbations  solution  components  eastward-travelling modon  wavenumber.  a s s o c i a t e d with t h e modon depression  the  topography  wave Since  X <0  X >0 and  As b e f o r e ,  results  satisfies  with act  to  wave  acts  are reversed. h ( X , Y ) — > A *(Y)  and A ~ ( Y ) f o r  to  speak  o f X=0) s t e a d y - s t a t e  of  a  solution  Note t h a t  if  t h e terms- c o n t a i n i n g A " ( Y ) i n  (3.2.20),  (3.2.21) and (3.2.22) sum t o z e r o  Thus w i t h  no l o s s  o f g e n e r a l i t y we  the  i f one c o n s i d e r s a  c ( X , Y , T ) and K ( X , Y , T ) .  defines h(X,Y)=A~(Y)+h'(X,Y),  7  t h e wavenumber-  8  i n a neighbourhood  T->») f o r a ( X , Y , T ) ,  Since  components  'local'  (i.e.,  (3.2.22)  the w e s t w a r d - t r a v e l l i n g  wavenumber. these  in  associated  r e s p e c t i v e l y , i t i s meaningful  one  eastward-  topographic  X — > + » and X — > - »  (i.e.,  and  increase the t r a n s l a t i o n  wavenumber-perturbations  the  increase  to  considers  the i n d i v i d u a l  and  perturbation to  wave  t o t h a t of the r a d i u s - p e r t u r b a t i o n s .  the  9  acts  c(X,Y,T)  are reversed.  with  opposite  X >0,  i f one  speed  the t r a n s l a t i o n  associated  solution  qualitative  associated is  the  results  the  speed-perturbations  6  the  these  h(X,Y)£0),  (i.e.,  a p p l i e s to the t r a n s l a t i o n  with  As b e f o r e ,  If  reversed.  associated  speed.  h(X,Y)£0).  (i.e.,  s e t A~=0  due  to  (3.2.29).  and assume h ( X , Y )  is  74  relative is  to  t h e upstream  asymptotically  (3.2.20), solutions  topography  constant  (3.2.21)  and  in  X ) .  c  *  u{\ [h*  1 +  +  cases  approximate  K M{X [A 0  over since  +  8  (3.2.30)  -  (3.2.31)  - h(X,Y)]  are of i n t e r e s t . steady-state  topography  X h(X,Y)} 6  -  X h(X,Y)}  large  time).  Consider the structure solutions  A*=0).  (i.e.,  (3.2.32)  9  Then  for  of the  isolated  (3.2.30),  (or  (3.2.31) a n d  reduce t o  K  a^1.  local steady-state  - X h(X,Y)}  o f X=0 f o r s u f f i c i e n t l y  local  (3.2.32) f u r t h e r  where  conditions  3  - h(X,Y)]  s  a neighbourhood  compact)  imply  these  o f t h e form  2  Two  Under  (3.2.22) w i l l  a * 1 + M { X [ A * - h(X,Y)]  (in  ( w h i c h may depend on Y b u t  a  =* 1 + M X , M X , Y )  (3.2.33)  c  ~  1 + MX,h(X,Y)  (3.2.34)  — KQ + M K X h ( X , Y ) 0  (3.2.29) h a s been u s e d .  (3.2.35)  7  S i n c e X,<0, (3.2.33) i m p l i e s  that  T h e r e f o r e t h e r a d i u s o f t h e modon d e c r e a s e s a s i t t r a v e l s isolated T>>1  positive is  topography  assumed).  (ignoring  the transient  I f t h e modon were t r a v e l l i n g  waves o v e r an  75  isolated (i.e.,  depression,  a£l) s i n c e  From  say a t r e n c h ,  c<1  over  are  on t h e same o r d e r  accelerates over  over  isolated  would  increase  h(X,Y)£0.  (3.2.34), isolated  i t s radius  c>1  over  depressions  positive isolated  since  X„>0.  Modon p a r t i c l e  as t h e t r a n s l a t i o n  isolated  topography and  speed  so  speeds  the  fluid  p o s i t i v e topography and d e a c c e l e r a t e s  depressions,  in line  with  intuitive  continuity  arguments. The the  modon wavenumber a d j u s t s  reverse  manner  as  to isolated  t h e modon r a d i u s  since  a wavenumber i s d i m e n s i o n a l l y  From  (3.2.35),  positive  isolated  depressions In the  modon  speeds  wavenumber  will  topography and s a t i s f i e s  positive  The  speed  topography  length). K>K0  over  over  isolated  s t r u c t u r e develops as  slowly  varying  I f the topography  topography is  positive,  i n c r e a s e s and hence t h e p a r t i c l e decreases  and  the  wavenumber  modon t h e r e f o r e c o n t r a c t s a n d a c c e l e r a t e s Over  The modon d i l a t e s  illustrates  When  isolated  the radius  topography.  happens.  gaussian  over  translation  increases.  a  7  the t r a n s i e n t waves).  increase,  be e x p e c t e d  satisfy  K<K0  i n just  X >0.  travels  modon  (as might  the inverse of  summary, t h e f o l l o w i n g q u a l i t a t i v e  (ignoring the  since  t h e modon  topography  the evolution  isolated  +  A ( Y ) i s not zero  (3.2.31) a n d (3.2.32) depend  the  reverse  and d e a c c e l e r a t e s .  S u b s e c t i o n 3.2.3  o f t h e modon o v e r  positive isolated  by c o n s i d e r i n g a m e r i d i o n a l  i n thex-coordinate  depressions  over  direction  ridge  with  a  a s an e x a m p l e .  t h e approximate on  modelled  solutions  the f a r f i e l d  (3.2.30),  (i.e.,  X » 1 )  76  topographic  structure.  transmission  of  neighbourhood  by  In  this way  general,  e a c h assumed  far  of  the  A*  as  a  with  A*  into  X=0  hyperbolic  wave.  be  To  differences  illustrate and  constant.  the  the  the  the a n a l y s i s w i l l  isolated  nonzero  represent  information  westward-travelling  of c o u r s e ,  between  terms  field  form f o r A * ( Y ) .  similarities consider  The  different  for and  nonisolated  topography  Allowing  meridional  a  i  dependence  does  not  complicate  s o l u t i o n s depend o n l y Consider  X >0 and  follows radius  As  3  decreases,  0£h£A*<»  It follows  X <0.  2  h->0  that  X  (3.2.31),  since  X <0.  Because  speed  for  X>0  speed  for  X<0.  5  is  relative  for  X>0  |X |>|X | the 6  5  l a r g e r than The  so  large  since  speeds  the i n the  will  2  3  the  modon  T.  6  in  X  |X |>|X | i t  in  X <0, and  increase  since As  3  the d e c r e a s e  particle  the  increasing  a—>1-/xX A*£1 .  decrease  for s u f f i c i e n t l y  c>1  as  for a l l X  However, s i n c e  2  will'be a  called  (3.2.30) t h a t a>1  a->1+MX A*£1.  from X<<0 t o X » 0  From  from  ideas  Y.  (henceforth  i n c r e a s e s , h—>A*  so  there  essential  p a r a m e t r i c a l l y on  the case  topography).  the  c<1  for  X<0  translation translation  also  have  this  property. The that  of  behaviour  t h e modon r a d i u s .  wavenumber similar is  since  reasoning  a relative  since  of t h e  8  topographic  9  There  a l l terms i n to that  increase  |X |<|X |.  modon wavenumber  In  given  is a  i s again  global  (3.2.32) a r e  decrease negative.  f o r t h e modon r a d i u s  i n t h e modon wavenumber Subsection  3.2.4  reversed  the  e s c a r p m e n t m o d e l l e d as a h y p e r b o l i c  in  the  However,  shows  f r o m X<0  to  solutions tangent  to  there X>0 for a  in  the  77  x-coordinate topographic The  direction  behaviour  of t h e +  results  to those  speed  with  the decrease  i n c f o r X<0  modon  particle  decrease The  f r o m X<0  be  a  homogeneous  manifests  ^ < 1 — ' > B * and are constants.  1980). B"  for  i s removed by n  £ =* 0 ( e " )  the  have  topographic gives  modon r a d i u s  X<0 and  to c<1  X>0. for  X>0  increase in c for  this  all  property.  X  with  a  ( £ ,y ;X,Y,T) i s n o t  It i s easily 0(e)  of  expected solitary  'shelf  The  relative  shelf  £—>+» and  and  fluid  1979a,b).  the  Grimshaw;  (3.2.6)  are  as  that  shelf  i n t r o d u c t i o n of an  ( t h e power n i s d e t e r m i n e d  nonuniformity the  solitary  1979a,b and a  and  f o r a l l £.  of  -» r e s p e c t i v e l y ,  The  is that  this  behind  (to leading order)  '  in general occur  waves  assumed  0  (3.2.4)'and  t o be v a l i d  appears  expected  that  and  will  ahead and  1978  seen  problem  (3.2.4)  Thus r e s o n a n c e  The  as  0 }  In g e n e r a l , i t i s  wave t h e  The  f o r X<0  than  for  the  RHS's  Kuelh;  B*=0 ( e . g . , Grimshaw, wave  to  dimensional  and  also  £—>±°°.  the  i t s e l f as a  Ablowitz;  solitary  as  (3.2.2) i s not  ( e . g . , Ko  c>1  larger  solution  (0)  one  a  X>0.  t o A\// .  the expansion In  in  of  depression)  i n c r e a s e from  satisfy  increase  solution  terms  proportional  and  to  uniformly valid  certain  wave  will  l e a d i n g order  to  will  speeds w i l l  wavenumber  example  t h e above p r o p e r t i e s .  s t a t e d above.  for a l l X with a r e l a t i v e  translation  The  an  i . e . , a topographic  The  X>0.  as  solutions for a decreasing  (-°><A £ h £ 0 ,  opposite  decreases  presented  configuration displaying  configuration the  are  Kodama  consequence  of  where B * and  B"  ahead  of  the  i s undisturbed,  thus  region  behind  outer expansion i n the problem)  the valid  (e.g.,  78  Grimshaw; since  1979a,b  it  is  velocity (u  (  1  )  ,v  (  In  the  field, 1  )  ),  seem t o be  3.2.3  and  slowly  1  term this  relevant  Subsection  the  determines does  for  analysis  of  the  given  Figures  sequence of  times  and  of  course, determined  solution  of  with  Figure  to  does  3.2.3  c=1+uc (  not  1  ( 1 )  propagating (3.2.26),  i s given  spatial  0.6,  This  which  g i v e n h(X)  7a  shows  the  in  is  the  A* = 0.  with  of  ^=0.2  and  the  ( 1 )  and  f o r the  4.0.  These  T held constant)  particular  value  modon e x p e r i e n c e s a t a g i v e n  on.  by  follow  structure  1.4  (i.e., The  a  qualitative  , modon r a d i u s a = 1 + M a  the c h a r a c t e r i s t i c  the  the K will  and  respectively,  in space-time.  a particular  c  the  over  (3.2.36)  f o r the case  T=0.0, 0.2,  by  propagation  |X|->°°  a,  space-like slices  c  is  type  topography  as  for  M«OK '  +  0  t h e modon p a r a m e t e r s K that  S i n c e h->0  speed  K=K  represent  The  9 illustrate  modon wavenumber  figures  0(e)  contribute  2  solutions  8 and  translation  slow  the  1 - <xexp(-X ),  in Subsection  7,  not  modon  varying ridge i s described.  behaviour  However,  p a r t of  a n o n u n i f o r m i t y of t h i s  solution  2  T«1),  that  1980).  Topography  hence h(X,Y) = e x p ( - X ) .  modon  Ablowtiz,  here.  H(X,Y) =  modon  '  and  constant  consequently  Gaussian-ridge  this  Kodama  in  a,  time i s ,  space-time  characteristic approximately  of  of  i s the (3.2.28)  the unique (for  (3.2.36).  initial  condition  on  the  translation  79  speed of  C(X,0,0)=1).  (i.e.,  c a t T=0.2.  wave  is  more  removed  The  than  t i m e T=1.4  the  displayed  the  is  the  amplitude  (Figure space  stationary  eastward-travelling  In F i g u r e  7c  separation  wave  i s further wave  the e a s t w a r d - t r a v e l l i n g wave  since  since wave i s  |\ |>|X |. 6  7d) t h e w e s t w a r d - t r a v e l l i n g  By  5  wave has l e f t  (i.e.,  -10^X^10).  The more  slowly  wave has  completely  separated  from  domain  component.  of the s o l u t i o n  from t h e  the  eastward-travelling  of  structure  distinguishable  Figure  (3.2.21).  7e  shows  This  solution for c(X,Y,T) described  state  and  westward-travelling  eastward-travelling  component  becoming  the westward-travelling  the  moving  The  X=0 t h a n  2  larger  just  i n (3.2.21).  apparent.  |o |>|a |. 1  are  component  from  7b shows t h e s p a t i a l  The w e s t w a r d - t r a v e l l i n g  transients  stationary  Figure  the  i s the  last  stationary  local  Section  steady-  for isolated  topography. Figures in  8 a , 8b, 8 c , 8d and 8e show t h e s p a c e - l i k e  t h e modon  radius.  westward-travelling ,(3.2.20) a r e (3.2.21)), same a s t h a t Figures modon  and  the  the  Since  same  given  an  as  above.  initial  in  The  radius  t h e modon  modon  radius  radius over  for  c  (  1  >  (see  i n the t r a n s i e n t s i sthe  position) since  X >0. 2  s o l u t i o n (see F i g u r e  the topography  t r a n s i e n t waves i n  solutions  (see F i g u r e s  since  f o r the  wave  (see  t h e modon wavenumber ( i f t h e X  positive  increase  speeds  eastward-travelling  t o decrease  transient  of  the  behaviour  westward-travelling t h e modon  translation  eastward-travelling  separation  8c a n d 8d) a c t s had  the  structure  since  8b a n d  8c)  The s t a t i o n a r y 8e) a c t s X^O.  X <0. 3  acts  The to  component  t o reduce  the  80  Figures of  the  9a, 9b, 9 c , 9d and 9e show t h e s p a c e - l i k e  modon  Figure  11a.  above  comments  wavenumber.  The s e p a r a t i o n since  westward-travelling the  transient  (see Figures  acts  In  modon  will  initially  the  is  continues throughout  X <0.  9b and  i s shown i n as  i n the  9c)  The  acts  to  The e a s t w a r d - t r a v e l l i n g  8  to  increase  t h e modon wavenumber o v e r  the  modon  (see F i g u r e  9e)  the topography.  the f o l l o w i n g q u a l i t a t i v e behaviour of p o s i t i o n i s , say, e£ =-l0.  The modon  o  propagate eastward u n a f f e c t e d  the  topography  7b, 8b and 9 b ) . When t h e c h a r a c t e r i s t i c  associated  westward-travelling  characteristic, radius  same  The s t a t i o n a r y component  9  i f its initial  (see F i g u r e s with  since  summary, c o n s i d e r  the  the  (see F i g u r e s  9c and 9d) a c t s  X >0.  to increase  is  condition  t h e t r a n s l a t i o n s p e e d s a r e t h e same.  wavenumber  since  initial  behaviour  transient  decrease  wavenumber  The  structure  t h e modon  increased to  this  translation  and  propagate initial  transient  by  intersects  speed  is  reduced,  i t s wavenumber d e c r e a s e s .  eastward  since  interaction.  c>0  After  wave p r o p a g a t e s c o m p l e t e l y  characteristic  subsequently  does  not  its  The modon  (see  Figure  sufficient  westward-travelling and  t h e modon  7)  time t h e  t h r o u g h t h e modon  contribute  to  the  solution. Some space-time the  time region  later  containing  s t a t i o n a r y component  speed  increases,  increases interaction  the  (see Figures there  the  modon c h a r a c t e r i s t i c  the c h a r a c t e r i s t i c s a s s o c i a t e d  of the s o l u t i o n . radius 7e,  i n t e r s e c t s the  decreases 8e  i s no f u r t h e r  and  with  The modon t r a n s l a t i o n and  9e).  the  wavenumber  Subsequent  i n t e r a c t i o n with  the  to this  topography  81  or  the  transient  propagates with translation  speed  wave) n e v e r  (i.e.,  (i.e.,  except must  taken  i n t o account.  modon  center  characteristics  therefore  < n 0  effects  wave  does  the  center  of  the  current  topography  is  do n o t a f f e c t  t h e modon.  was  reversed.  i s s i m i l a r to the  eastward-travelling s c e n a r i o the  t h e modon s i n c e f o r location  identically  from s p a t i a l given  two  I f the 'ridge'  Clearly in this  the  any  the  r e s u l t s w o u l d be  than  at  modon  eastward-travelling  h(X)>0.  affect  orginating  the  «  not  greater  than  transient  thus  the q u a l i t a t i v e a n a l y s i s the  coordinates  and  o f t h e modon a n d t h e  that  spatial  modon  7)  h ( X ) < 0 ) t h e above  westward-travelling  the  Figure  a n a l y s i s h a s assumed t h a t  0  transient  i s greater  2  (see  So-0 (say e £ = 1 u ) ,  above  eastward-travelling  i n t e r s e c t f o r T>0 i f £  a depression If  The  a =3.59 w h i c h  speed  characteristics  This  waves.  coordinates  t i m e have z e r o  zero.  of  the  Hence  positive  of  a m p l i t u d e and  82  in  IT) CM  Q UJO Q_  cn ID  LT)  CD '  -10.0  -6.0  -2.0  2.0  6.0  10.0  X  F i g u r e 7a. S e q u e n c e o f s p a c e - l i k e s l i c e s s h o w i n g t h e s p a c e - t i m e e v o l u t i o n o f t h e modon t r a n s l a t i o n s p e e d f o r modon p r o p a g a t i o n o v e r a s l o w l y v a r y i n g g a u s s i a n r i d g e c e n t e r e d a t X=0. T h i s p l o t shows t h e i n i t i a l c o n d i t i o n ( i . e . a t T=0.0) on t h e t r a n s l a t i o n speed ( i . e . C ( X , Y , 0 ) = 1 ) .  83  in  in CM  in CD  in <=> I  -10.0  I  -6.0  I  -2.0  I  1  2.0  6.0  1  10.0  X  F i g u r e 7b. S p a c e - l i k e s t r u c t u r e o f t h e modon t r a n s l a t i o n i n d u c e d by a g a u s s i a n r i d g e a t T=0.2.  speed  84  F i g u r e 7 c . S p a c e - l i k e s t r u c t u r e o f t h e modon t r a n s l a t i o n s p e e d i n d u c e d by a g a u s s i a n r i d g e a t T=0.6.  85  F i g u r e 7d. S p a c e - l i k e s t r u c t u r e o f t h e modon t r a n s l a t i o n s p e e d i n d u c e d by a g a u s s i a n r i d g e a t T=1.4.  86  in  LT)  CD Q_ CO  in o  in  oH -10.0  1  -6.0  1 —  -2.0  1  i  2.0  6.0  i  10.0  X  F i g u r e 7e. S p a c e - l i k e s t r u c t u r e o f t h e modon t r a n s l a t i o n s p e e d i n d u c e d by a g a u s s i a n r i d g e a t T=4.0.  87  in  in CNJ  CE  in  in o  •10.0  -6.0  -2.0  2.0  6.0  10.0  X  F i g u r e 8a. Sequence of s p a c e - l i k e s l i c e s showing t h e space-time e v o l u t i o n o f t h e modon r a d i u s f o r modon p r o p a g a t i o n o v e r a s l o w l y v a r y i n g g a u s s i a n r i d g e c e n t e r e d a t X=0. T h i s p l o t shows t h e i n i t i a l c o n d i t i o n ( i . e . a t T=0.0) on t h e r a d i u s ( i . e . a(X,Y,0)=1).  88  F i g u r e 8b. S p a c e - l i k e s t r u c t u r e o f t h e modon r a d i u s a g a u s s i a n r i d g e a t T=0.2.  induced  by  89  F i g u r e 8 c . S p a c e - l i k e s t r u c t u r e o f t h e modon a g a u s s i a n r i d g e a t T=0.6.  radius  induced  by  90  in  in  CO  =><  CD' CX  in r-  in •10.0  -6.0  -2.0  2.0  6.0  10.0  X  F i g u r e 8 d . S p a c e - l i k e s t r u c t u r e o f t h e modon a g a u s s i a n r i d g e a t T=1.4.  radius  induced  by  91  in  tn CNJ  CO  a— cr in ro  in CD  10.0  -6.0  2.0  -2.0  6.0  10.0  X  F i g u r e 8e. S p a c e - l i k e s t r u c t u r e o f t h e modon a g a u s s i a n r i d g e a t T=4.0.  radius  induced  by  92  IT)  tn CM  CC  0_<=> CC ^1  in r-  in  cn' -10.0  -6.0  -2.0  2.0  6.0  10.0  X  F i g u r e 9a. Sequence of s p a c e - l i k e s l i c e s showing t h e space-time e v o l u t i o n o f t h e modon wavenumber f o r modon p r o p a g a t i o n o v e r a s l o w l y v a r y i n g g a u s s i a n r i d g e c e n t e r e d a t X=0. T h i s p l o t shows t h e i n i t i a l c o n d i t i o n ( i . e . a t T=0.0) on t h e wavenumber ( i . e . K(X,Y,0)=K ) • O  93  F i g u r e 9b. S p a c e - l i k e s t r u c t u r e o f t h e modon wavenumber by a g a u s s i a n r i d g e a t T=0.2.  induced  94  F i g u r e 9c. S p a c e - l i k e s t r u c t u r e by a g a u s s i a n r i d g e a t T=0.6.  o f t h e modon  wavenumber  induced  95  F i g u r e 9d. S p a c e - l i k e s t r u c t u r e by a g a u s s i a n r i d g e a t T=1.4.  o f t h e modon wavenumber  induced  96  in  in CNJ  CE CE  in r-  in •10.0  -6.0  -2.0  2.0  6.0  10.0  X  F i g u r e 9e. S p a c e - l i k e s t r u c t u r e by a g a u s s i a n r i d g e a t T=4.0.  o f t h e modon  wavenumber  induced  97  3.2.4 T a n h - e s c a r p m e n t  In  this  slowly  Topography  Subsection the s o l u t i o n varying  escarpment  for  modon  i s described.  propagation  The t o p o g r a p h y  over  a  i s given  by  H(X,Y) = 1 - u{]  + tanh(X)}/2,  (3.2.37)  hence h ( X , Y ) = {1 + t a n h ( X ) } / 2 .  S i n c e h->0 a n d  and  qualitative  X—>°°  solutions the  case  will  u=0.2  follow  0£h^A*<°°.  structure  a=1+Ma  respectively,  the  the a n a l y s i s  for  the  behaviour  F i g u r e s 10, 11 and 12 i l l u s t r a t e  a n d modon'wavenumber K = K sequence  o f slow  as  X->-» of  the  g i v e n i n S u b s e c t i o n 3.2.3 f o r  o f t h e modon t r a n s l a t i o n  (1)  h->1  speed + 0  (IK K  C=1+MC (  1  0  '  ( 1 )  the s p a t i a l  , modon  radius  respectively,  with  t i m e s T=0.0, 0.2, 0.6, 1.4 and  4.0. These  figures  represent space-like  constant)  in  translation  speed,  stated  S u b s e c t i o n 3.2.3 t h e p a r t i c u l a r  that  in  modon  of  the  radius  and  propagating solution  of  the  characteristic  and  the  characteristics  components  i n (3.2.20),  characteristic  modon  asymptotically (3.2.37).  (i.e.,  on  defined  T<<1)  by  by  for  held  the  modon As  v a l u e s of a, c and K are determined which  and  (3.2.26)  (3.2.28)  T  wavenumnber.  defining  (3.2.21)  is  (i.e.,  solutions  t h e modon e x p e r i e n c e s a t a g i v e n t i m e  intersection  modon  space-time  slices  with  the  by t h e  modon  is  the i n d i v i d u a l (3.2.22).  The  o r (3.2.27) o r h(X)  given  in  98  The  qualitative  escarpment topography description on  given  the s i m i l a r  shows  Subsection  slices  in  speed.  follows  T h u s we o n l y  and h i g h l i g h t  10a)  radius  As i n S u b s e c t i o n  (wavenumber)  is  speed  f o r X>0.  increased  transient.  However  the  importance here. modon  travelling  topographic  that  h(X)—>1  The e s c a r p m e n t once  initial  as  interaction  sufficient  However approach  characteristics  time  speed  by  the  (increased)  by  the  i s of c r u c i a l  begins.  deformation The w e s t w a r d -  zero  structure  of  eastward  modon the  has  modon  the  a  the  role  westward position  characteristic X>0  of  will  topographic  u p s t r e a m t o p o g r a p h y does n o t  nonzero  modon  and  initial  the  a l l subsequent  carry  the  (decreased)  X—>+°°  containing  since  will  the  If a  the c h a r a c t e r i s t i c s  eventually  10  varying  Similarly,  f o r c e s . a permanent  of  configurations.  information.  spatial  fact  information  after  intersect  comment  Figure  and e a s t w a r d - t r a v e l l i n g t r a n s i e n t s have t h e  transmitting  o  briefly  the  f o r X<0 and t h e e a s t w a r d - t r a v e l l i n g t r a n s i e n t  eastward=travelling  e£ <0,  of  (3.2.3) t h e westward-  t r a n s i e n t and d e c r e a s e d  the  that  of the s l o w l y  westward-travelling  in  between t h e  the d i f f e r e n c e s .  space-time  to increase the t r a n s l a t i o n  modon  the interaction  t r a n s i e n t a c t s t o r e d u c e t h e modon t r a n s l a t i o n  Figure  acts  of  Subsection.  aspects  modon t r a n s l a t i o n  (see  in this  last  space-like  travelling  analysis  westward-travelling  signal.  parameters  Therefore the  is  continuously  deformed. This The  spatial  deformation  westward-travelling  i s shown i n F i g u r e s  transient  results  10, 11 and 12.  in  a  continued  99  reduction increase 10b, the  i n t h e modon t r a n s l a t i o n i n t h e modon t r a n s l a t i o n  10c and 1Od f o r t h e t r a n s i e n t b e h a v i o u r a n d F i g u r e  1Od f o r  modon of  radius the  There  since  coefficient  the  of  of  transient.  of  decrease the  positive  (see  leading  the  i n the radius across component  3.2.3).  Figures  Subsection  for  of  the  increase of t h e  3.2.3). structure  the  Figures and  the  example  e  of  the  qualitative  upstream  initially  modon i n i t i a l l y topography.  modon c h a r a c t e r i s t i c  decreases,  decreases.  12c  12e  behaviour  Since  that  a  11c  11e t h e  domain  is  a n d 12d  the  local  o f t h e slow  escarpment  t h e modon c e n t e r  propagates eastward u n a f f e c t e d Eventually  after  sufficient  i n t e r s e c t s the c h a r a c t e r i s t i c s  nonzero upstream topographic speed  11b,  component  12b,  Figure  (3.2.39) s u p p o s e n  the  entire  stationary  the  T  of  i n t h e modon wavenumber  o f t h e modon a s i t goes up a n d o v e r  «  X=0  westward-travelling  evolution  _ i n  domain  solution.  an  form  entire  westward-travelling  stationary  edge  i s a relative  the t r a n s i e n t  steady-state  the  structure.  the  There  illustrate  e£o=  X>0  t h e t r a n s i e n t b e h a v i o u r and F i g u r e  X=0 s i n c e t h e a m p l i t u d e  As  for  modon wavenumber i s d e c r e a s e d  upstream  across  edge  (see Subsection  illustrate  steady-state  increased  i s relative  i s negative  11d  is  leading  transient.  The  for  steady-state structure).  upstream  solution  speed  continued Figures  The  local  f o r X<0 and a  (see  local  and  speed  information.  i t s radius  increases  the upstream topography  i s at by  time the  c a r r y i n g the  The modon and  of  its  translation wavenumber  remains nonzero as X  100  increases  and the  information, nontrivial the  the  modon  increased  and  sufficient  speed  the  further  escarpment  and  decreases  and  characteristic  wavenumber  remains  This  the  modon  translation  i s intersecting  qualitative  the  analysis  unaffected  by  characteristics  associated  transient  and  the  information  (i.e.,  characteristic with speed  o  begins  the  over the  since  0  radius  the  modon  associated  solutions. the  escarpment  had been assumed, t h e  i s large  the  position  of  initially  be  (e.g.,  10),  the  westward-travelling  component  X>10).  the  carry  However  as  constant the  modon  intersect characteristics associated  structure  increase,  modon wavenumber w i l l  to  up  t h e modon w i l l  with  for  After  I f the i n i t i a l  If e£  stationary  h(X)==1  the topographic will  e£ >0),  the topography.  decreased.  that  I f the reverse  i s p o s i t i v e (e.g.,  remains  characteristics  assumed  Thus  radius  increases  above b e h a v i o u r would be r e v e r s e d . modon  to intersect  increases,  of the p e r t u r b a t i o n  i n c r e a s e d .from X<0 t o X>0.  the  the  propagates  speed  wavenumber  the s t a t i o n a r y part  this  transient characteristics. reduced,  the  carries  continues  remains  time  the  transient  characteristic  westward-travelling  translation  with  westward-travelling  f o r X<0,  modon  decrease.  the  modon  radius w i l l  translation  increase  and t h e  101  in  in  Q LxJo Q_ CO  in  in o  •10.0  -6.0  -2.0  2.0  6.0  10.0  X  F i g u r e 10a. Sequence o f s p a c e - l i k e s l i c e s s h o w i n g t h e s p a c e t i m e e v o l u t i o n o f t h e modon t r a n s l a t i o n s p e e d f o r modon p r o p a g a t i o n over a s l o w l y v a r y i n g h y p e r b o l i c - t a n g e n t escarpment c e n t e r e d a t X = 0 . T h i s p l o t shows t h e i n i t i a l c o n d i t i o n ( i . e . a t T=0.0) on t h e t r a n s l a t i o n s p e e d ( i . e . C(X,Y,0)=1).  102  LD  ID  in o  in o  •10.0  -6.0  2.0  -2.0  6.0  10.0  X  F i g u r e 10b. S p a c e - l i k e s t r u c t u r e o f t h e modon t r a n s l a t i o n i n d u c e d by a h y p e r b o l i c - t a n g e n t e s c a r p m e n t a t T=0.2.  speed  103  F i g u r e 10c. S p a c e - l i k e s t r u c t u r e o f t h e modon t r a n s l a t i o n i n d u c e d by a h y p e r b o l i c - t a n g e n t e s c a r p m e n t a t T=0.6.  speed  104  LT>  LT) CXI  ID  o  LT)  CD  -10.0  -6.0  2.0  -2.0  6.0  10.0  X  F i g u r e I 0 d . S p a c e - l i k e s t r u c t u r e o f t h e modon t r a n s l a t i o n i n d u c e d by a h y p e r b o l i c - t a n g e n t e s c a r p m e n t a t T=1.4.  speed  105  in  in CM  O LLJCD  LU_; Q_ CO  in CD  in o -10.0  -6.0  2.0  -2.0  6.0  10.0  X  F i g u r e I 0 e . S p a c e - l i k e s t r u c t u r e o f t h e modon t r a n s l a t i o n i n d u c e d by a h y p e r b o l i c - t a n g e n t e s c a r p m e n t a t T=4.0.  speed  106  in  in  CO  CE  in  tn o'  •10.0  -6.0  -2.0  2.0  6.0  10.0  Y  F i g u r e 11a. Sequence o f s p a c e - l i k e s l i c e s s h o w i n g t h e s p a c e t i m e e v o l u t i o n o f t h e modon r a d i u s f o r modon p r o p a g a t i o n over slowly varying hyperbolic-tangent e s c a r p m e n t c e n t e r e d a t X=0. T h i s p l o t shows t h e i n i t i a l c o n d i t i o n ( i . e . a t T=0.0) on t h e r a d i u s ( i . e . a(X,Y,0)=1).  107  in  in o  m o  10.0  -6.0  2.0  -2.0  6.0  10.0  X  F i g u r e 11b. S p a c e - l i k e s t r u c t u r e o f t h e modon a h y p e r b o l i c - t a n g e n t e s c a r p m e n t a t T=0.2.  radius  induced  by  108  in  in  CM  in  oH -10.0  I -6.0  I -2.0  :  I 2.0  I 6.0  I 10.0  X  F i g u r e 11c. S p a c e - l i k e s t r u c t u r e o f t h e modon a h y p e r b o l i c - t a n g e n t e s c a r p m e n t a t T=0.6.  radius  induced  by  109  in  in CM  CO  cr cn  in  tn o  •10.0  -6.0  -2.0  2.0  6.0  10.0  X  F i g u r e 11d. S p a c e - l i k e s t r u c t u r e o f t h e modon r a d i u s i n d u c e d by a h y p e r b o l i c - t a n g e n t e s c a r p m e n t a t T=1.4.  110  in  in  CO  ZD  o  CE  en  in CD  in o  •10.0  -6.0  2.0  -2.0  6.0  10.0  X  F i g u r e 11e. S p a c e - l i k e s t r u c t u r e o f t h e modon a h y p e r b o l i c - t a n g e n t e s c a r p m e n t a t T=4.0.  radius  induced  by  111  in  in CM  cr cr m  in ro' •10.0  -6.0  -2.0  2.0  6.0  10.0  X  F i g u r e 12. Sequence o f s p a c e - l i k e s l i c e s s h o w i n g t h e s p a c e - t i m e e v o l u t i o n o f t h e modon wavenumber f o r modon p r o p a g a t i o n o v e r a slowly varying hyperbolic-tangent e s c a r p m e n t c e n t e r e d a t X=0. T h i s p l o t shows t h e i n i t i a l c o n d i t i o n ( i . e . a t T=0.0) on t h e wavenumber ( i . e . K ( X , Y, 0) = /c ) . 0  112  F i g u r e 12b. S p a c e - l i k e s t r u c t u r e o f t h e modon wavenumber i n d u c e d by a h y p e r b o l i c - t a n g e n t e s c a r p m e n t a t T=0.2.  113  F i g u r e 12c. S p a c e - l i k e s t r u c t u r e o f t h e modon wavenumber i n d u c e d by a h y p e r b o l i c - t a n g e n t e s c a r p m e n t a t T=0.6.  F i g u r e 12d. S p a c e - l i k e s t r u c t u r e o f t h e modon wavenumber i n d u c e d by a h y p e r b o l i c - t a n g e n t e s c a r p m e n t a t T=1.4.  115  LT)  IT) CNJ  CE CE  in  m  10.0  -6.0  -2.0  2.0  6.0  10.0  X  F i g u r e 12e. S p a c e - l i k e s t r u c t u r e o f t h e modon wavenumber i n d u c e d by a h y p e r b o l i c - t a n g e n t e s c a r p m e n t a t T=4.0.  116  IV.  This  CONCLUSIONS  t h e s i s h a s examined two a s p e c t s o f t h e t h e o r y  modons.  The f i r s t  sufficient integral  aspect  neutral  stability  constraint)  for  Eastward-travelling solely |TJ|<K  composed where K  modons  was examined  barotropic  wavenumber  the  modon  are neutrally stable  spectral  condition  II i n which  ( i n the  modons  magnitudes  wavenumber.  |T?|>K  a  o f an  obtained. perturbations  (|*?|) s a t i s f y i n g Westward-travelling  to perturbations  components s a t i s f y i n g  form  was  modons a r e n e u t r a l l y s t a b l e t o  with  is  i n Chapter  of b a r o t r o p i c  (or  composed  else  s o l e l y of  the  modon  is  stable). The  eastward-travelling  implies  that  when  curve  proposed  stability  condition  the  neutral  stability  McWilliams et a l . ( l 9 8 l )  should  K/|T?|>1 by  stability  curve  has  topographically-forced Rizzoli, As  A  of  similar  been  trend  numerically  planetary  in  begin  the  neutral  determined  eddies  to  for  (Malanotte-  1982). an a t m o s p h e r i c  calculated  Eliasen  condition  by T o m a t s u ( 1 9 7 9 ) , S a l t z m a n and  and M a c h e n h a u e r ( 1 9 6 5 ) .  satisfy  the s t a b i l i t y  typical  mid-latitude  eastward-travelling calculation  application, the s t a b i l i t y  b a s e d on t h e 300, 500 and 700 mb eddy k i n e t i c  spectrum d e s c r i b e d and  the slope  a s 1171 d e c r e a s e s .  increase  modon n e u t r a l  condition 700 mb  modons  to  are  300 mb  neutrally  f o r westward-travelling  conclude atmosphere stable.  modons f a i l s  energy  Fleisher(1962)  Eastward-travelling  and t h u s we  was  modons  that  for  energetics A  similar  t o s a t i s f y the  117  stability  c o n d i t i o n and thus  westward-travelling Simple travelling and  arguments  speeds  with  of  presented  to  suggest  representative  FU(1983)  the  the  III.  aspect  medium.  The  r e p r e s e n t s an e x t e n s i o n  oceanic  not the  spectrum  energy  be data  were  may n o t be  o r i t s wavenumber  calculation  waves ( e . g . ,  Luke(l966),  1981),  Ablowitz(1980,1981), were  modon w i t h pressure  of  slowly  Zakharov  using  the  contained  method  in  in  developed a  slowly  developed  solitary  v a r y i n g one  Grimshaw(1970,  here  wave  of  dimensional 1971,  1977,  a n d R u b e n c h i k ( 1 9 7 4 ) a n d Kodama  among o t h e r s ) . posed  solution  is  t o a two-dimensional  solitary  geostrophic  which  was  energy  (24 d a y s ) a r g u m e n t s  thesis  perturbation  made  travelling  could  o f a b a r o t r o p i c modon  calculations  perturbation  eddy  p e r t u r b a t i o n t h e o r y was  various  Two p r o b l e m  eddy  integral  wavenumber  Fu(l983)  this  A l e a d i n g order  1979a,b,  translation  The s t a b i l i t y  modons  eastward-  i n the oceans.  of  to describe the propagation varying  for  s h o u l d o n l y be c o n s i d e r e d a v e r y p r e l i m i n a r y  o f modon s t a b i l i t y second  only  realistic  oceanic  the t o t a l  Therefore  spectrum  The  and  that  have  was c o l l e c t e d  that  of e i t h e r  distribution.  1978,  instability  However due t o t h e s h o r t p e r i o d o v e r  the F U ( 1 9 8 3 ) spectrum  Chapter  suggested  mesoscale  stability  or  determined.  i n the ocean.  FU(1983)  the The  inferred.  estimate  be  b a r o t r o p i c modons w o u l d  spectrum.  for  modons c a n n o t  scaling  particle  tested  the s t a b i l i t y  for  and the  solved. propagation  a bottom boundary has  been  layer  expanded  In  S e c t i o n 3.1  of  an  eastward  was o b t a i n e d . in  the  a  The  damping  118  coefficient  e = E  number a n d r The  1 / 2  0  t h e Rossby  0  modon  (K)  are  0(e)  equations  the  0(1)  radius  allowed  resulting  /(2r )  * 10"  1  with  E  the  vertical  number.  (a), translation  speed  ( c ) and  t o be f u n c t i o n s o f t h e slow t i m e  require a necessary  solutions  Ekman  wavenumber  T = e t . The  compatibility  condition  on  ( t a k e n t o be an e a s t w a r d - t r a v e l l i n g modon)  in nonlinear  initial-value  problems  for  the  modon  parameters. The leave  solutions  t h e modon d i s p e r s i o n  decay.  The a m p l i t u d e  decays  like  distance  i s about  comparison  with  for  5  solution  100  scale  scale  developed  to  topography. topography  modon  solution o f an  a  leading modon  A s i n p r e v i o u s work  the theory  shallow  water  maximum complete  Based  on  a  e t a l . , 1981) modon  d e s c r i b e s the decay over a  p a r a m e t e r s a n d a 10 day t i m e  order  perturbation  propagation on  the  C l a r k e d 9 7 1 ) and L e B l o n d  others)  Nonlinear  (McWilliams  i s developed  equations  over effects  b a r o t r o p i c p l a n e t a r y waves  Rhines(1969a,b),  The  before  radii.  the  and v o r t i c i t y  eastward-travelling  here  oceanic  during  parameters.  describe  on  travels  (initial)  obtained  for  f o r atmospheric In S e c t i o n 3.2  invariant  respectively.  modon  dissipation  the asymptotic time  the  a numerical  the f r i c t i o n a l  day  relationship  exp(-T),  which  dissipation  0  o f t h e modon s t r e a m f u n c t i o n  exp(-3T) and over  c = exp(-2T) and K = K e x p ( T )  a = exp(-T),  (e.g.,  theory  slowly varying of  variable  Veronis(1966),  and M y s a k ( l 9 7 8 ) ,  i n the context  is  of the  among  rigid-lid  on t h e /3-plane.  hyperbolic  equations  are  d e r i v e d f o r t h e slow  119  evolution and  of the l e a d i n g order  wavenumber.  modon r a d i u s ,  These e q u a t i o n s  are v a l i d  translation  speed  for arbitrary  slowly  varying  f i n i t e - a m p l i t u d e topography.  In g e n e r a l  solved  numerically.  that  evolution  of  topographic vorticity  the  amplitude  shown  is  independent  This  perturbation  speed  topography  and  result  be  t o l e a d i n g order the of  the  meridional  i s i n t e r p r e t e d using  westward-travelling  properties  proportional  solutions  wavenumber  (which  applications).  component  for are  i s the case  hyperbolic to  3.2.3  the s p e c i f i c  respectively.  and  the  simple  small-  in  a  stationary The  Subsection  the slowly  examples o f a t o p o g r a p h i c  and  t h e form o f e a s t w a r d  topography.  3.2.4 d e s c r i b e  for  radius,  i n many a t m o s p h e r i c  t r a n s i e n t s and  the  modon  obtained  The s o l u t i o n s t a k e  of the s o l u t i o n are described  Subsections for  is  must  arguments.  translation  and  modon  structure.  Analytical  oceanic  It  they  general 3.2.2.  v a r y i n g modon  r i d g e and e s c a r p m e n t ,  120  BIBLIOGRAPHY  1.  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R u b e n c h i k , 1974: w a v e g u i d e s and s o l i t o n s i n n o n l i n e a r m e d i a . J E T P , 38, 494-501.  Atmospherically J^ Phys.  1976: Eddy e n e r g y i n 2641-2646.  I n s t a b i l i t y of Sov. Phys.  126  APPENDIX  A - CALCULATION OF SOLVABILITY PROBLEM  INTEGRALS  IN TOPOGRAPHIC  This appendix d e s c r i b e s the c a l c u l a t i o n s of the c o m p a t i b i l i t y conditions (3.2.7) a n d ( 3 . 2 . 9 ) . The e x t e r i o r c o m p u t a t i o n s ( i . e . , (3.2.7)) a r e p r e s e n t e d f i r s t and the i n t e r i o r computations ( i . e . , (3.2.9)) f o l l o w . Exterior  Calculations  The c o m p a t i b i l i t y c o n d i t i o n (3.2.7) i n v o l v e s 18 i n t e g r a l s . The t e r m s a r e d e s c r i b e d i n the o r d e r of t h e i r appearance i n (3.2.6) a n d a r e d e n o t e d I \ t h r o u g h t o l i s * Recall  that  f o r r>a  i//<°> = - c a K , A^<°>  =  2  be c o n v e n i e n t R(r)  1 / 2  1  A^  will  2  1 / 2  )  1 / 2  )  1  -6 aK (6c"  i.e.,  It  1  (6c- ' r)sin(0)/K (6ac"  ( 0 )  r)sin(e)/K (6acl  2  = (6 /c)i//<  0  .  >  to l e t  =  -caK (6c-  1 / 2  1  r)/K (6ac-  1 / 2  1  ),  so t h a t i / / = R ( r ) s i n ( t 9 ) , and d e f i n e t h e o p e r a t o r s D = d / d r a n d D F [ ( » ) ] = dF[(•)]/£(•) ( i . e . , d i f f e r e n t a t i o n with respect t o r and a r g u m e n t s , r e s p e c t i v e l y ) . Recall r = (£ £ ) Y & t a n ( 0 ) = y / U - £ o ) , hence ( 0 )  0  2  _  2  +  2  0  (3 T  , 3 , 3 )r = -cos(0)(3 , 3 , 3 )£ X Y T X Y  0  (3 , 3 , 3 )6 = f'sinU)(3 , 3 , 3 ) £ . 0  T And  finally  X  i t i s helpful 2  D R  The  Y  T to r e c a l l  X  that  2  Y  by d e f i n i t i o n  2  + DR/r - ( r - + 6 / c ) R = 0.  integral calculations  f o r r>a a r e a s  follows;  Ii:  7T  I,  2  1  = (6 /c)H" g  J Y  -IT  »  f f a  0  l  f  0  i  1  rdrdfl  a n  127  Note  that  i/y  ( 0 )  = (DR - R / r ) s i n ( 0 ) c o s ( 0 ) .  Therefore  the 2  trigonometric part i n t e g r a t e s to zero  of t h e i n t e g r a l c o n t a i n s c o s ( 0 ) s i n ( 0 ) w h i c h due t o t h e p e r i o d i c i t y i n 0. Thus I , = 0.  I : 2  it  12  » 0  >Ai//< > r d r d 0 T  a  -it  Note  0  = ~ / J  that (0  1 / 2  , / 2  , / 2  1  A<// > = { a ^ a - [D K (8ac)/K (6ac)]8ac[a- a T T T ( 2 c ) - c ]}A<// > + c ' ( r c - ' ) A^ > T T r (6 /c)cos(0)sin(0)£ DR + ( 6 / c ) r - s i n ( 0 ) c o s ( 0 ) £ R. T T 0  1  1  1  (0  1  2  1  2  2  2  < 0  1  o  0  The t r i g o n o m e t r i c component o f t h e i n t e g r a l a s s o c i a t e d w i t h t h e l a s t two t e r m s ( i . e . , t h e t e r m s c o n t a i n i n g £ ) is sin (0)cos(0) T and t h u s i n t e g r a t e s t o z e r o . The r e m a i n i n g c a l c u l a t i o n g i v e s 2  0  I  2  = -7r6 ca"Q[A a-'a  2  -  1  (A,  1)(2C)" C  -  ]  1  T  T  where = 7K (7)/KI(T)  Ai  2  Q = -[7K (7>  1  ~  2  (2Q)" -  7  2  =  (Alb)  0  o 2  (A1a)  " 2K (7)K,(7) ~ 7 ^ ( 7 ) ) / ( 2 7 ^ ( 7 ) ] ,  1  recalling  1  1  2  8 a /c.  I : 3  it  1  We  3  »  = 2c / J -it a  1  0  0  > rdrdS ££X  require (0)  2  r// = s i n ( 0 ) c o s ( 0) ( D ££X 3  sin (0)(DR/r  2  2  2DR/r - 2 R / r ) X 2  - R/r )  + X  +  128  r -  1  3  ^  3  2  (cos (0)sin(0) - 2sin (0)cos(0)(D R  2  - 2DR/r - 2 R / r ) -  X £  3  cos (0)sin(0)D(D  0  2  2  - 2DR/r - 2 R / r ) +  X Sr-^o  sin  3  (6)cos(6)  (DR/r  2  - R/r )  X  -  X 3  2  £o c o s ( 0 ) s i n ( e ) D ( D R / r  - R/r ).  X The  trigonometric  component o f t h e i n t e g r a l s c o n t a i n i n g  £  have  0  X 3  2  4  c o s ( 0 ) s i n ( 8 ) or s i n ( 0 ) c o s ( 0 ) both of which The r e m a i n i n g i n t e g r a t i o n g i v e s I  2  3  2  , t  1  integrate to zero.  1  = (1/2)ff6 c a Q[A a- a  - (A, - l ) ( 2 c ) " c ] .  1  X  X  I,:  It  I«  = ~6  2  =>  / J ^ -it a  < 0  >V  l 0 )  rdrd0  X  The c a l c u l a t i o n o f I« i s e s s e n t i a l l y t h e same a s I (modulo t h e 6 /c f a c t o r a s s o c i a t e d w i t h At/> compared t o ^ a n d t h e Xd e ' r i v a t i v e r a t h e r than t h e T - d e r i v a t i v e ) . The r e s u l t i s 2  2  <0)  2  2  ,  1  (0 }  1  I, = - 6 i r c a ' Q [ A a - a  - (A, - 1 ) ( 2 c ) - c  1  X  1  + c' c  X  ].  X  I : 5  I  5  It  =  X We n o t e  «  / J  (6Vc) -it  ^(0)^,(0)^(0)  a  r  that  ^(0)^(0)^,(0)  O/3)[sin0D +  =  1  gives  I  2  5  2  3  3  r- cos(0)3/90)R sin (e)  y  which  arde  y  = - ( 6 7 r c a V 2 ) (2c)" \c . X  129  I : 6  0  I,  We  (0)  ( 0 )  = -2 J / ,/,< 'j(<// , ^ ) -TT a £X  rdrdfl  require  (r-V  0  )  ie  2  )  2  = (cosMe) - sin (0)(DR/r  - R/r ) -  X  X 1  2  2r" sin (0)cos(0)£  2  (DR/r - R / r ) -  o  X cos(0)(cos (6) 2  2  - sin (0)£  2  D(DR/r - R / r )  o  X  and (i//  ( 0 )  it  2  )  = c o s ( 0) s i n ( &) ( D R  2  - DR/r  + R/r ) X  X  r" sin(0)(cos (6) 1  2  -  2  sin (0))£  2  (D R  o  - DR/r  -  2  + R/r ) -  X 2  cos (0)sin(0)£  2  D ( D R - DR/r  0  +  2  R/r ).  X  Note  that <0)  i// ) = V  j(^<°>,  £X  The  trigonometric  ( 0 )  (r-V  r  < 0 >  £0  ~ r" V  ) X  0  'U  0  component of t h e t e r m s c o n t a i n i n g 2  2  6  = -7ra c  3  1  K (7)(2K ( ))- [(2c)- c + B(a-'a X X 2  1  X  is X integrate to  3  1  7  >) .  £r  s i n * ( 0 ) c o s ( 0 ) or s i n ( 0 ) c o s ( 0 ) , b o t h of which zero. The r e m a i n i n g i n t e g r a t i o n g i v e s I  l 0  7  £  0  1  - (2c)- c  )] X  where B  I : 7  =  7[K (7)/Ki(7) 2  -  K,(7)/K (7) 2  "  3 /  7  ) ] .  (A2)  130  »  It  I  1  = c(H- H  7  0  )  J J ^'°>^' > -it a i£  X We  rdrdfl  require \//  (0>  2  2  2  = s i n ( 0 ) c o s (6) (D R  - 2DR/r + 2 R / r ) +  3  sin (6)(DR/r which  results  2  -  R/r )  in I  ( 1 /4 ) 6 7ra' c QH" 'H X 2  =  7  ,  2  .  I : 8  »  It  I  2  1  = -6 (H" H  8  ) X  Straightforward  calculation I,  / J ^<°)^'°> -IT a  results  in  -5 a*c 7rQH- H 2  2  =  rdrdfl  1  .  X  I : 9  I  9  it  )  1  = -(H" H X  We  00  ( 0 )  ( 0  / / ,// JU >,  —it  ^'°') £  a  rdrdfl  require i// ie  ( 0 )  IJ> ir  which  ( 0 )  =  2  2  (cos (0)  - sin (0))(DR -  2  = c o s ( 0 ) s i n ( 0 ) (D R  R/r)  2  - DR/r  + R/r ),  gives I  2  9  3  1  1  = -ira c 7K (7)(4K ( ))- H- H 2  1  .  7  X  I 10:  131  *  it  I,o  - -(H-'H  )  / / ^ -7r a  X We  (  0  )  [(r )-  1  2  + 6 y]<// y  0  (0)  rdrd0  require (0)  2  </< y  = sin (0)DR  2  +  cos (0)R/r.  The t r i g o n o m e t r i c components o f t h e i n t e g r a l w i t h t h e R o s s b y number a s a c o e f f i c i e n t a r e s i n ( 0 ) o r c o s ( 0 ) s i n ( 0 ) b o t h of which i n t e g r a t e t o z e r o . The r e m a i n i n g i n t e g r a t i o n g i v e s 3  I  2  =  1 0  2  2  (3/8)6 ira"c H-'H  2  +  t t  2  1  (1/2) 6 i r a c Q H - H  X  .  X  In': It 00 2  I,,  1  0  = -(26 /c)(H- H )  / J ^(°>^< '^<°)  X The e v a l u a t i o n  -it  of I , , i s s i m i l a r I,,=  a  y  to I 2  rdrd0  the r e s u l t  5  is  -(l/2)5 7ra c H- H . 4  2  1  X I  12: It  I,  We  = 2c  2  oo  / S -it a  0  >i//  (  > rdrd0 £yY 0  require i// = c o s ( 0 ) s i n ( 0 ) (D R SyY 2  ( 0 >  2  3  2  - 2DR/r + 2 R / r )  2  cos (0)(DR/r  + Y  - R/r )  + Y  1  2  2  r- (cos (0)sin (0)  a  - sin (0))£  2  (D R  o  2  - DR/r  + 2R/r )  -  Y 2  2  cos (0)sin (0)£  2  2  D ( D R - 2DR/r + 2 R / r )  0  -  Y 1  2  2  3r" cos (0)sin (0)£  (DR/r  o  Y  2  - R/r )  f l  - cos (0)£  D(DR/r  o  Y  -  2  R/r ).  132  The t r i g o n o m e t r i c p a r t o f t h e s e t e r m s i s i n t e g r a t e d a g a i n s t sin(0) (belonging to ^ ) . I t i s easy t o see t h a t each term i n t e g r a t e s t o z e r o due t o t h e p e r i o d i c i t y i n 0, c o n s e q u e n t l y I12-O. (  0  )  l i s :  It  I  We  2  = -(6 /c)  1 3  OD  ; J ,/,<<> >,/,<<> ),/,<<>> r d r d S Y -TT a i  require VJ>  (0)  = c o s ( 0 ) s i n ( 0 ) (DR - R / r ) .  2  When t h i s t e r m i s i n t e g r a t e d a g a i n s t t h e s i n ( 0 ) t r i g o n o m e t r i c component o f (i//(0))2 t h e r e s u l t i n g i n t e g r a l v a n i s h e s , consequently I =0. 1 3  11«:  I,,  We  = -2 J J - it a  V  W  0 )  ' , i// ) r d r d 0  0  (0)  yY  require <0)  ( -V ) = 2sin(0)cos(0) (DR/r y0 1  R/r ) + 2  -  r  Y  Y  2  3  2r- (cos (0)sin(0)  - sin (0))£  2  (DR/r - R / r ) -  o  Y 2  2  2sin (0)cos(0)£  U<°>) yr  2  2  = sin (0)(D R) Y  o  D(DR/r - R / r ) . Y  2  2  2  + cos (0.)(D R - R / r ) Y Y  2  sin (0)cos(0)£  3  2  (D R - 2D R/r) -  o  Y 1  2  2r- sin (0)cos(0)£  2  2  3  (D R - R / r ) - c o s ( 0 ) £  o  Y  2  2  D(D R - R / r )  o  Y  133  2  The f i r s t t e r m i s m u l t i p l i e d by ( D R ) R s i n ( 0 ) w h i c h i n t e g r a t e s t o z e r o due t o t h e t r i g o n o m e t r i c i n t e g r a t i o n . The s e c o n d t e r m i s m u l t i p l i e d by R s i n ( 0 ) c o s ( 0 ) w h i c h i n t e g r a t e s t o z e r o due t o t h e t r i g o n o m e t r i c i n t e g r a t i o n . T h e r e f o r e I , , = 0. 2  lis:  I  1  = c(H- H  1 5  J J V' ''/'' ' -TT a £y  )  0  Y We  rdrdS  0  require <iV  2  2  2  = c o s ( 0 ) s i n ( 0 ) (D R  0)  3  cos (0)(DR/r  - 2DR/r + 2 R / r ) +  2  -  R/r )  w h i c h when i n t e g r a t e d a g a i n s t R s i n ( 0 ) trigonometric i n t e g r a t i o n . Therefore  i s zero I, =0.  due  to  the  5  lis: »  7T  I,  We  1  = -(H" H  6  ) Y  J J -7r a  2  2  °'J(^< °',  ^<°») y  rdrd0  require t/> yr  (0)  2  = sin (0)D R  0co>  =  + cos (0)[DR/r  2 r " ' s i n ^ J c o s U ) (DR/r  2  - R/r ),  2  - R/r ).  2  The f i r s t term i s i n t e g r a t e d a g a i n s t R s i n ( 0 ) c o s ( 0 ) and t h e second term i s i n t e g r a t e d a g a i n s t ( D R ) R s i n ( 0 ) which i n t e g r a t e t o z e r o due t o t h e t r i g o n o m e t r i c i n t e g r a t i o n . T h e r e f o r e I , = 0 . 2  6  1, : 7  7T  I,  7  1  =  (H- H  ) Y  The  integrand  »  / / ^ -ir a  (  0  )  [(r )0  1  2  ( 0  + 6 y]^ ' £  rdrd0  is  ((r ) 0  _ 1  2  2  + 8 rsin(0))Rsin (0)cos(0)(DR  -  R/r)  134  w h i c h i n t e g r a t e s t o z e r o due t o t h e t r i g o n o m e t r i c T h e r e f o r e I, =0.  integration.  7  I1 8 '  I  2  1  1  = 2(6 /c)(H- H  8  ) Y  This  integral  it °° / J V -it a  (  0  ,  ^  (  0  >  ^  (  0  )  rdrdfi.  I  i s trigonometrically indentical  to I  1  3  .  Therefore  I 18 = 0.  The c o m p a t i b i l i t y c o n d i t i o n  0  it °= J f V '(RHS[3.2.6]) rdrde -it a  which a f t e r a l i t t l e g i v e n by ( A 1 ) a n d =  2  algebra  +  2  A  E,  with B i s given  Interior  18  0  =  A  (3.2.7) i s t h e r e f o r e  =  Q/2 =  3  K  2  K  results  2  (  7  (3.2.10)  in  ) Q [ 2 T K  (  1  = 1 1 , n=1 n  ) ] *  7  ( B - 1 )  1/4 + Q/8 + K (7)Qt2 K (7)]7  2  (A3)  1  (A5)  1  by ( A 2 ) .  Calculations  = 8 K" aJ (Kr)sin(e)/J,Ua) 2  2  = - 2^<o)  (0)  K  _  (  6  = 5 K- aJ, Ur)/J, Ua) 2  + K C)/C" r s i n ( 0 ) 2  2  2  i n the previous c a l c u l a t i o n R(r)  2  19  = - 6 a J ! (/cr)sin(0)/J, U a )  ( 0 )  A<//  i.e.,  contains f o r r<a  ( 3 . 2 . 9 )  that  - (6  1  Ai//  As  is  (A4)  9  ( 0  A ,  ( 7 ) Q [ 2 T K I ( 7 ) ] " ' B  The i n t e r i o r c o m p a t i b i l i t y c o n d i t i o n i n t e g r a l s denoted I, through I , . R e c a l l ^ >  1  where  2  2  2 +  C K  )rsin(0).  i t will - (6  2  be c o n v e n i e n t 2  to l e t  2  + « c)»c- r  so t h a t i p = R ( r ) s i n ( 0 ) , and d e f i n e t h e o p e r a t o r s D = d / d r and D F[(«)1 = dF[(•)]/£(•) ( i . e . , d i f f e r e n t a t i o n w i t h r e s p e c t t o r and a r g u m e n t s , r e s p e c t i v e l y ) . Recall r = (£-£o) Y and t a n ( e ) = y / ( £ - £ ) > hence { 0 )  0  2  0  2  +  2  135  (3 T  , 3 , 3 )r = -cos(0)(3 , 3 , 3 )£ X Y T X Y 1  ( 3 , 3 , 3 ) 0 = T X Y And  finally  2  + DR/r  i n t e g r a l s are given  as  -  that  (r- - K 2  -  , 3 , 3 )£ . X Y 0  T  i t i s helpful to r e c a l l DR  The  r" sin(0)(3  0  2  by  ) R  definition  =  0.  follows.  Ii:  it a J J ^<°>^/<°)  = -K H- g 2  I,  1  Y -it  0  rdrdfl  £  This integral i s trigonometrically identical exterior c a l c u l a t i o n s . Therefore I^O.  t o I, i n the  I : 2  I  it a = ~ / / ^  2  -it  Note  0  0  >A<//< ' r d r d f l T  0  1  that  1  A^<°> = { a ' a - [D J,Ua)/J,Ua)]«a[a" a + T T T K ~ K ]}A<//< + K- (KT) Al//< " T T r cos(0)£ Ai/> +r" sin(0)£ Ai// >. T r T 8 0  1  0 >  1  ( 0 )  0  1  (0  o  The  trigonometric  o  component  of the terms w i t h  s i n ( 8 ) c o s ( 6 ) which i n t e g r a t e s to zero 8. The r e m a i n i n g c a l c u l a t i o n g i v e s 2  1  1  due  1  + L[a* a  T  is T to the p e r i o d i c i t y  + K" K  !  = Ga" a  2  )  T  £  0  in  ] T—  where G = C [D 2  -  2  2  L = C [-kD /J (k) + 2  2  1  2  (1 + k /  2 7  )J (k)[kJ (k)]2  (kJ (k)/J,(k) 0  1  (A6)  1  + 2)0  +  k  2  2  7  " )-  136  J (k)[kJ (k)]2  C  D  2  = (1/2)[J (k) 2  2  recalling  1  1  6*a f l 7r»c-  =  2  2  - 1/2 - ( k / 7 ) ] 2  (A8)  - 2J (k)j (k)k2  (A7)  1  2  2  + J (k)]/J (k) 1 1  1  (A9)  k=«a,  I : 3  I  ir a = 2c J J ^ -*  3  t  0  >  ^  (  0  >  rdrdfl  0  We  require ^  = sin(0)cos (6)(D  ( 0 )  2  ax  2  1  sin (6)(DR/r  3  (cos (e)sin(fl)  0  +  x  3  r" £  2  - 2DR/r - 2 R / r )  2  - R/r ) + X  3  2  - 2sin (0)cos(0)(D R  2  - 2DR/r - 2 R / r )  X 3  £0  cos (0)sin(0)D(D  2  2  - 2DR/r - 2 R / r ) +  X 1  3r" £  3  2  sin (0)cos(0)(DR/r  0  - R/r ) X  X 3  £0  cos(0)sin (0)D(DR/r  2  - R/r ).  X  The  trigonometric  component  of the i n t e g r a l s containing  c o s ( 0 ) s i n ( 0 ) or s i n * ( 0 ) c o s ( 6 ) both of which The r e m a i n i n g i n t e g r a t i o n g i v e s 3  2  I  1  3  1  = -(c/2)Ga" a  - (c/2)L(a" a X  a J J t *p TT  I4  2  = KC  l0)  -TT  0  l0)  X  have X integrate to zero.  1  + K" * ). X X  rdrd0  £  0  137  Note  that IJ> X  (0)  1  =  { a' a  -  ]}l// >  1  +  ( 0  X cos(0)£  ^  o  X  trigonometric  K"  1  ( KT )  ^  < 0 >  1  r  X  of  the  terms w i t h  2  1  + cL[a  _ 1  a  X  2  1  2  _  K  is  0  X to the  due  1  £  ]  2CGK  -  periodicity  K  _ 1  X +  2  U a" 6-'{U  K  + X  cC [-J (k)/(kJ,(k)) 2  <0)  i/> . 6  o  component  = cGa~ a  "  ( 0 )  X r + r- sin(0)£  s i n ( 0 ) c o s ( 0 ) which i n t e g r a t e s to zero 6. The r e m a i n i n g c a l c u l a t i o n g i v e s I,  +  0  X K" K  The  [ D J i ( K a ) / J , («<a) ]»ca[a-'a  T  + X  (1  +  2  (k/ ) )/4]. 7  2  2  + c« )aK- }  . X  Is  = -<K ) 2  X  We  note  it a S S ^ -TT  ( 0  0  }  *  (  0  0 }  rdrd'0  y  that  ^(0)^(0)^,(0)  !  3  ( i / 3 ) [ i n 0 D + r" cos(0) 3/30)R sin  =  s  y  which  gives I  = cC (k/7)«K- K 1  5  /2.  2  X  I : 6  I  6  it a / / i/>< ° ' J U  = -2 -it  We  require  0  (  0 >  ( 0 )  r  i// ) £X  rdrd0  3  (6)  138  0  ( r - V  )  2  )  ie  2  = (cos (0)  2  - sin (0)(DR/r  - R/r )  X  X  1  2  2r~ sin (0)cos(0)£  2  (DR/r  o  - R/r )  -  X 2  2  cos(0)(cos (0)  - sin (0)£  2  D(DR/r -  o  R/r )  X  and <0)  2  (<// ) = c o s ( 0) s i n (0) (D R £r X 1  2  r" sin(0)(cos (0)  2  - sin (0))£  - DR/r  2  (D R  o  2  + R/r ) X  -  2  - DR/r  + R/r )  -  X 2  cos (0)sin(0)£  2  D ( D R - DR/r  o  +  2  R/r ).  X  Note  that  j(,//< 0 >,  V(0))  =  £X  The  trigonometric  ' (r- 1 i//< ' )  0  0  r  £0  - r" V (  X  0  0  component o f t h e t e r m s c o n t a i n i n g 2  6  = cJ (k)C k(2J 2  0  2  (k)7 )- [-K- /c 2  1  1  X  is X integrate to  3  1  ' ) •  $r  s i n " ( 0 ) c o s ( 0 ) or s i n ( 0 ) c o s ( 0 ) , both of which zero. The r e m a i n i n g i n t e g r a t i o n g i v e s I  ' U(  £  + K ' K )] X X  1  + B (a" a  1  2  X  where B  2  = k[J,(k)/J (k) + J (k)/J,(k)-3/k]. 2  2  I : 7  a J J ^<°»^(°»  IT  I  1  7  = c(H- H  ) X  We  0  -TT  a  rdrd0  require (0  i// >  a  2  2  0  2  = s i n ( 0 ) c o s ( 0 ) ( D R - 2DR/r + 2 R / r )  +  139  sin (6)(DR/r 3  which  2  -  R/r )  results in I  = -cC (G/4)H"'H  7  .  2  X  it  I  K  =  8  2  C ( H -  H  1  "TT  X  Straightforward I  B  calculation  = cC [D 2  a  J S ^  )  l 0 )  ^  l 0 )  rdrdfl  0  results in 2  - 2(1 + ( k / 7 ) ) J ( k ) / ( k J , ( k ) ) +  2  2  (1  +  2  2  1  (k/ ) ) /4]H- H . . X 7  I : 9  it  I  1  9  = -(H" H X  We  a ; / <//< ' J(i//< 0  )  -7T  0  ' ,  \p ) l0)  rdrdfl  i  0  require i//  ( 0 )  2  2  = (cos (0)  - sin (0))(DR  - R/r)  ie t0)  <//  which  2  = c o s ( 0 ) s i n ( 0 ) (D R  2  - DR/r  + R/r ),  gives I,  2  1  1  = cC k(47 )- (J (k)/J (k))H" H 2  2  .  1  X  it  I  1  0  = (H-'H  ) X  J -it  a f f"'[(ro)"' - K cy]^ y 2  0  (  0  )  rdrd0  140  We  require vV  (0)  2  2  = sin (0)DR  + cos (0)R/r.  y  The t r i g o n o m e t r i c components o f t h e i n t e g r a l w i t h t h e R o s s b y number a s a c o e f f i c i e n t a r e s i n ( 0 ) o r c o s ( 6 ) s i n ( 8 ) b o t h o f which integrate t o zero. The r e m a i n i n g i n t e g r a t i o n gives 3  I, (1  = cC [(3/8)(k/ )*  0  2  2  - D /2 +  7  2  + (k/ ) )J (k)/(kJ (k)) 7  2  -  1  2  (1/8)0  + (k/ ) ) ]H''H 2  7  .  2  X  In: it  I,,  = 2K (H" H 2  a / J  )  1  X  The  evaluation  of I  -TT  >iiV  ( 0  '(//< ' r d r d f l 0  0  y  to I ,  i ssimilar  1 t  0  5  I 1i = -0/2)cC (k/ ) 2  7  1 ,  the r e s u l t i s 1  H- H . X  I, : 2  I  it a = 2c J J i/>  0 (  1  2  -it  0  > rdrdfl £yY  0  This integral i strigonometrically exterior calculations. Therefore  I  t o I,  indentical =0.  2  i n the  1 2  Ii : 3  it  i i  3  =  ;  2  u ) Y  -it  a / 0  ^«o)^(o>^(0)  r  d  r  d  e  i  This integral i strigonometrically i d e n t i c a l to the I in the exterior calculation. Therefore I =0. 1 3  11»:  1 3  integral  141  Iia  We  it a  J / I* > [ r s i n ( f i ) ( 8 10  =  -it  2  0  + K C ) ] <// Y £  (0>  2  rdrd0  require [rsin(0)(5  + K C ) ] = -cos(0)sin(0)| (5 Y Y  2  2  2+  0  cos(0)sin(0)£  2  * c) +  (S + K C) + 2  o  2  Y sin(0)[r(5  + K C)]  2  ,  2  Y  and  i//  ( 0 )  = (DR - R / r ) s i n ( 0 ) c o s ( 0 ) .  The t r i g o n o m e t r i c  component  t 3  2  3  of t h e i n t e g r a l i s t h e r e f o r e s i n ( 0 ) c o s ( 0 ) o r s i n ( 0 ) c o s ( 0 ) , b o t h o f w h i c h i n t e g r a t e t o z e r o due t o t h e p e r i o d i c i t y i n 0. The i n t e g r a l s I through I are trigononmetrically indentical to theexterior integrals I through I , , respectively. Therefore a l l the remaining i n t e g r a l s are zero. 1  5  1  9  H  8  l i s :  I  it a = -2 S S  1 5  -it  *  0  ( 0 >  JU  < 0 )  , ^  ( 0 )  )  rdrd0  yY  I i  I,  = c(H" H  6  it a J J f  )  1  Y  -Tr  ° ' f £y  0  0  rdrd0  1  1 7  I  1 7  = -(H" H  )  1  Y  it a  J i// °>J(,//  ;  -it  (  (  0  )  0  , i//<°>) y  rdrd0  1 8  I,  = (H- H ) J J <K Y -it 0 1  8  0 )  [(r )0  1  -  ic cy]^ > 2  ( 0  |  rdrd0  142  I  1 9  J  I  2  1  = -2/c (H" H  1 9  it a J J ,/,< >,/,< ° >,/>< > r d r d S . 0  )  -Tr  Y  £  The c o m p a t i b i l i t y c o n d i t i o n it a / J - it  0 =  0  0  (3.2.9) i s t h e r e f o r e  19 ' (RHS[3.2.8]) r d r d f l = 1 1 . n=1 n  0  0  (A10)  (3.2.11) t h e t e r m s K " K and K K i n (A10) T X are e l i m i n a t e d i n favour of a " a , ( 2 c ) ' ' c , a a and ( 2 c ) " c T T X X by d i f f e r e n t i a t i n g t h e d i s p e r s i o n r e l a t i o n (3.1.5) t o o b t a i n In o r d e r  to obtain  1  _  1  K" K  = Nfa-'a  1  T K" K 1  _ 1  - (2c)-^c  1  (2c)-*c  T  1  l  - (2c)-' c  T ] -  1  (2c)" c  X  2  where N = -{7R  ] -  T  = N[a" a X X  1  X  2  + k R/7}/{4 + 7/R  + k R/7} and R = K ( 7 ) / K , ( 7 ) . 2  A f t e r a l i t t l e a l e g b r a , (3.2.11) i s o b t a i n e d from (A10) where B,, B , B and E a r e g i v e n by t h e f o l l o w i n g h i e r a r c h y o f def i n i t i o n s ; 2  3  2  M = L/G  E = [-J (k)/(kJ,(k))  + (1 +  2  B, = NM |B  = 2(1 + N)  2  2  1  3  ~ B,N  1  = -2N  + E(1 - N +  2  2  1  2  2  = 1/4  + E/2  ,  - k (87*G)"  - B, + N ) ,  (k/7) ) + k * N ( 2 7 G ) -  k(27 J,(k)G)" J (k)[B,N  E  (A11 ) + ( k / 7 ) • ( 2 G ) " ( N - 1) +  7  2  7  1  + E(N + 2 + 2 ( k / ) )  2  2  (k/ ) )/4]/G,  + M - N,  kJ (k)[27 J (k)G]" (l  B  + 1,  1  i,  (A12)  1  +  + B, - N ] ,  + kj (k)[4 2  2 7  (A13)  1  J,(k)G]" .  (A14)  

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