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Planetary waves in a polar ocean LeBlond, Paul Henri 1964

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PLANETARY WAVES IN A POLAR OCEAN  by . PAUL HENRI LEBLOND  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE  REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in  t h e Department of Physics  We a c c e p t required  THE  t h i s t h e s i s as conforming  to the  standard  UNIVERSITY OF BRITISH APRIL,  1964.  COLUMBIA  In the  presenting  r e q u i r e m e n t s f o r an  British  for reference  mission for extensive p u r p o s e s may his  be  of  granted  written  Department  of  and  by  the  study,  the  for  Head  Date  J H i ^ s t «-S  M-ti-iOt  Library  o f my  Columbia,  3  ^  r  /frG-^  fulfilment  University  shall  I further  agree for  of  of  make i t f r e e l y that  per-  scholarly  Department  that  f i n a n c i a l gain  5  the  this thesis  permission*  The U n i v e r s i t y o f B r i t i s h Vancouver 8 Canada  in partial  degree at  I t i s understood  this thesis  w i t h o u t my  that  c o p y i n g of  representatives.  cation  advanced  Columbia, I agree  available  this thesis  or  c o p y i n g or  s h a l l not  be  by publi-  allowed  ii  ABSTRACT.  The dynamics of t h e A r c t i c ocean a r e s t u d i e d a polar is  projection  idealized  of the sphere.  as a t w o - l a y e r s y s t e m ,  formulation  i s developed  latitudinal  and  asymmetries  i n the b o u n d a r i e s  the d e n s i t y  structure  are p r e s e n t . levels  The and  density  structure  a general  which allows i n c l u s i o n  l o n g i t u d i n a l depth v a r i a t i o n s of the ocean.  of  as w e l l For  i s n e g l e c t e d when d e p t h  Time d e p e n d e n t  on  as  simplicity,  variations  displacements from  equilibrium  a r e assumed t o be waves o f c o n s t a n t z o n a l wave  number; no  radial  propagation i s considered.  Amplitude  e q u a t i o n s a r e d e r i v e d f o r these d i s p l a c e m e n t s , s u b j e c t to the assumption  that  t o keep o n l y a f i r s t  t h e p o l a r b a s i n i s s m a l l enough  a p p r o x i m a t i o n t o the c u r v a t u r e of  the E a r t h . A semi-qualitative solutions  i s made i n t h e c a s e  t h e Method for  o f S i g n a t u r e s , and  the s o l u t i o n s  constant,  f o r such motions symmetrical  the r e s u l t s Goldsbrough,  a) .  of r a d i a l  o n . p l a n e t a r y waves,  I t i s found  are  depth  found  variations.  explicit  i n the s i m p l e s t case  boundaries)  possible  basin, using  existence c r i t e r i a  of o t h e r i n v e s t i g a t o r s 1914  of the  of a symmetrical  i n the presence  Concentrating thereafter solution  investigation  (depth  a l l o w s comparison  with  ( L o n g u e t - H i g g i n s , 1964 that  the p o l a r  b;  projection  iii  and f i r s t approximation to the c u r v a t u r e give q u i t e results,  so that t h i s  method may be a p p l i e d to p o l a r  i n the same way as the ft -plane The g e n e r a l e f f e c t s discussed  of B a l l  regions  i s used i n m i d - l a t i t u d e s .  of r a d i a l bottom slopes are  and a simple example t r e a t e d more  Some theorems  good  (1963) on the motions  explicitly. of  shallow  rotating fluids  i n p a r a b o l o i d a l basins  are found to hold  f o r such b a s i n s  i n the p o l a r plane approximation to  the  sphere.  G.L. Pickard  ACKNOWLEDGEMENTS.  I wish t o express and  suggestions  this Dr.  my a p p r e c i a t i o n  o f D r . R.W.  xvork w a s c o n d u c t e d . G.L. P i c k a r d ,  Stewart,  D r . R.W.  for  amiable  years.  financial of  I must a l s o  assistance  the National  of British  guidance and c o o p e r a t i o n acknowledge w i t h  of the Institute  Research  Council  supervision  the director,  B u r l i n g , and t h e s t a f f  o f Oceanography, U n i v e r s i t y  three  u n d e r whose  I a l s o wish t o thank  Institute their  f o r the advice  during  ofthe Columbia the last  gratitude the  o f O c e a n o g r a p h y and  o f Canada.  i V  TABLE  OF  CONTENTS Page  TITLE  PAGE  i  ABSTRACT  TABLE  OF  iii. iv  CONTENTS  vi  List  of  Tables  List  of Figures  viii xi  ACKNOWLEDGEMENTS.  I  INTRODUCTION.  II  BATHYMETRY  III  IV  V  AND  T H E POLAR  i  i i  i i i  VII  OCEANOGRAPHY  P L A N E T A R Y WAVES:  FORMULATION  VI  1 OCEAN,  5  BACKGROUND,  10  PROBLEM.  16  OF T H E  PROJECTION.  Transformation  of  Time  solutions,  dependent  Phvsics  SYMMETRIC  SYMMETRIC i  OF T H E A R C T I C  o f Rossbv  OCEAN;  OCEAN;  32  Ll  waves.  GENERAL CONSIDERATIONS.  FLAT  S o l u t i o n i n terms functions.  26*  equations.  BOTTOM  43  SOLUTIONS.  of confluent  hypergeometric 71  ii  Approximate s o l u t i o n  i n terms of  Bessel 33  functions.  iii  VIII IX  sphere.  102  RADIAL DEPTH V A R I A T I O N S .  113  Comparison w i t h r e s u l t s  SYMMETRIC  OCEAN;  on the  A S Y M M E T R I C A L BOTTOM TOPOGRAPHY.  123  X CONCLUSIONS.  130  XI  133  APPENDIX I .  BIBLIOGRAPHY  136  vi  L I S T OF TABLES. Table I  Page Numerical values used  II  of basic  physical  parameters  i n t h i s work.  27  Eigenfrequencies  o f Rossby  waves i n a  ocean w i t h a f l a t  bottom,  as c a l c u l a t e d  the  confluent  hypergeometric  symmetric from  solution  ('homogeneous' mode). III  Eigenfrequencies o c e a n v/ith a f l a t the  Bessel  87  o f Rossby  waves i n a  bottom,  as c a l c u l a t e d  function  solution  symmetric from  ('homogeneous'  mode). IV  94  Eigenfrequencies symmetric  of i n t e r n a l  ocean w i t h a f l a t  from t h e B e s s e l f u n c t i o n V  VI  Computed e i g e n f r e q u e n c i e s  R o s s b y waves i n a bottom  (calculated 95  solution). of Rossby  a  symmetric  a  s p h e r e , a c c o r d i n g t o L o n g u e t - H i g g i n s ' model.  Comparison  polar basin with a f l a t  waves, f o r  of eigenfrequencies  waves i n a s y m m e t r i c bottom,  as o b t a i n e d  bottom  on 105  of planetary  p o l a r ocean v/ith a from d i f f e r e n t  flat  methods.  -  109  V l l  VII  Some e i g e n f r e q u e n c i e s  of p l a n e t a r y waves f o r  a symmetrical ocean with a r a d i a l 2 v a r i a t i o n . H ' = 1 + px /Z .  ,  bottom 117  viii  LIST  OF  FIGURES.  Fig.  Page  1  Bathymetry  2  Arctic  3  The  density  spherical  velocity 4  The  c o o r d i n a t e s and  the c o r r e s p o n d i n g 21  adopted  of the A r c t i c  on  density 23  of the s u r f a c e  of the 31  to a plane.  Phase p a t h s  7  Asymptotic signature  f o r the  ocean.  Orthographic projection  6  3  9  structure.  t w o - l a y e r system  sphere  6*  ocean.  components.  structure 5  of the A r c t i c  i n segments  of v a r i o u s s i g n a t u r e s .  b e h a v i o u r o f phase p a t h s (+,-)  as  £ —• +  Phase p a t h w i t h compound  of 55  GO s i g n a t u r e (-,+)(-,-)  ( + ,-)(+,+ ) 9  The  56  diagnostic  diagram  and  i t s subdivision  into 57  four regions. 10  Phase p a t h s  11  Bottom in  area  f o r area  profile I.  54  and  61  I.  phase p a t h s  for sloping  walls 64  ix  12  Phase p a t h s  f o r area  II.  66  13  Phase p a t h s  f o r area  III.  67  14  Phase p a t h s  f o r area  IV.  69  15  The c o n f l u e n t h y p e r g e o m e t r i c ^F^(a;b;y) of  16  as a f u n c t i o n  o f y f o r a few v a l u e s  76  a and b .  The e x p r e s s i o n (85 r h s ) as a f u n c t i o n o f frequency,  17  function  co' .  The e x p r e s s i o n  e  79  /2  as a f u n c t i o n  of frequency  . 3 0  OJ'. 18  The p a r a m e t e r  ' a ' (86) p l o t t e d  OJ' f o r a few v a l u e s o f s ( s 19  The r e l a t i v e  positions  hypergeometric  20.  The s i g n  of  subdivisions  21  8  <  ( a s g i v e n by  81  84  (94))in t h e four  of the diagnostic  diagram.  of surface amplitude  contours  f o r p l a n e t a r y waves i n a  Sketches  0).  of the confluent  Sketches  wave number,  frequency  f u n c t i o n s at the e i g e n f r e q u e n c i e s ,  ocean w i t h a f l a t  22  versus  89  (100))  ( -q , f r o m  symmetrical  b o t t o m f o r a few v a l u e s o f  s, and i n d e x number, n .  of the zonal  (v) and r a d i a l  components o f t h e v e l o c i t y  field  93 (u)  (from  (101)  X  and  (102)  23  Ball's  24  A  ) for  s =  definition  'distension'  -2,  n =  1.  99  of v e r t i c a l dimensions.  113  ( B a l l , 1963).  122  \  1  I.  INTRODUCTION. C o n s i d e r a b l e progress has been made i n r e c e n t  i n the d e s c r i p t i v e a s p e c t s of A r c t i c  oceanography.  only l o g i c a l t h a t the next step i n A r c t i c  There being however no o b s e r v a t i o n s  large scale,  phenomena i n those r e g i o n s ,  I t seems  oceanographic  r e s e a r c h should be a study of the dynamics of the ocean.  it  Arctic  of l o n g - p e r i o d , is  impossible  to analyse the s i t u a t i o n from an o b s e r v a t i o n a l b a s i s . problem w i l l of  view,  therefore  years  be t a c k l e d from a t h e o r e t i c a l  The point  so as to o b t a i n an i d e a of what to look f o r i n  a f u t u r e o b s e r v a t i o n a l program.  T h i s i s not an uncommon  approach i n oceanographic r e s e a r c h : data c o l l e c t i o n and r e d u c t i o n are e s s e n t i a l , and  it  i s necessary  aims before going t o The  but d i f f i c u l t  t o have some d e f i n i t e  observational  sea.  problem of f i n d i n g the p r o p e r t i e s  characteristic  of  the  free motions of the A r c t i c ocean i s  formulated so as to a l l o w the t o p o l o g i c a l features and  and time consuming,  here  i n c l u s i o n of some of the main  of the A r c t i c b a s i n ; both l a t i t u d i n a l  l o n g i t u d i n a l bottom and boundary v a r i a t i o n s can be  f o r m a l l y i n c l u d e d i n the model.  Most of the work a p p l i e s  to motions w i t h i n a very wide frequency range, but one type of  o s c i l l a t i o n was chosen f o r c l o s e r  s c r u t i n y : the  long  p e r i o d v o r t i c i t y waves c a l l e d p l a n e t a r y or Rossby waves.  2  This choice was motivated not only by a p a r t i c u l a r , interest  in that  type of motions,  but a l s o because they  can be considered as good t e s t m a t e r i a l f o r some of approximation methods used i n the a n a l y s i s . useful, regions,  when s t u d y i n g oceanographic to be able t o c o n s i d e r the  c u r v a t u r e only p a r a m e t r i c a l l y , as i s i n the s o - c a l l e d ft -plane  It would be  phenomena i n the p o l a r influence  of the E a r t h ' s  done i n m i d - l a t i t u d e s  ( V e r o n i s , 1963).  The a n a l y s i s  would be g r e a t l y s i m p l i f i e d and the p h y s i c s more e x t r a c t e d from the mathematics. t r a n s f e r r e d from the s u r f a c e  The problem i s  easily thus  of the sphere to a plane  p o l a r p r o . i e c t i o n of the A r c t i c r e g i o n s ,  the c u r v a t u r e of  the Earth being only kept as a s m a l l c o r r e c t i o n . p l a n e t a r y waves depend i n t i m a t e l y upon that (through i t s vector)  effect  f o r t h e i r e x i s t e n c e and i n t h e i r  Since  curvature  on the l o c a l component of the  comparison between r e s u l t s  the  rotation  properties,  obtained f o r Rossby waves i n  the p o l a r p r o j e c t i o n w i t h s i m i l a r r e s u l t s  derived  entirely  i n the s p h e r i c a l geometry w i l l g i v e an i n d i c a t i o n of  the  a p p l i c a b i l i t y of t h i s a p p r o x i m a t i o n . Before t a c k l i n g the the A r c t i c  ocean,  problem of the dynamics of  i t w i l l be u s e f u l  to review some of  the  i n f o r m a t i o n a v a i l a b l e on A r c t i c oceanography and bathymetry (section II),  and to look i n t o the background of  r e l a t e d to p l a n e t a r y waves ( s e c t i o n I I I ) . f o r m u l a t i o n of the problem i s developed  investigations  The mathematical  i n s e c t i o n IV,  3  where the assumptions  l e a d i n g t o s i m p l i f i c a t i o n of  problem ( l i n e a r i z a t i o n , are examined i n d e t a i l . approximation i s  hydrostatic  pressure)  In s e c t i o n V , the p o l a r plane  i n t r o d u c e d , and amplitude equations  derived f o r solutions The p h y s i c s  i n v i s c i d flow,  the  c o r r e s p o n d i n g to f r e e  are  zonal waves.  of p l a n e t a r y waves are a l s o re-examined  in  more d e t a i l . Some g e n e r a l c o n c l u s i o n s properties  of the s o l u t i o n s  obtained i n s e c t i o n V I .  as to the e x i s t e n c e and  i n a symmetrical ocean are  These apply not only to  planetary  waves but to a l l modes of f r e e  zonal o s c i l l a t i o n s  s i m p l i f i e d model.  of t h i s  The r e s u l t s  last section  used as a g u i d i n g beacon i n the search f o r a n a l y t i c a l solutions investigated  explicit  of i n c r e a s i n g l v complex bathymetry.  In s e c t i o n V I I p l a n e t a r y e i g e n s o l u t i o n s symmetric b a s i n w i t h a f l a t  are found f o r a  bottom, f i r s t  by s o l v i n g  s t r i c t amplitude equation i n terms of c o n f l u e n t functions,  of B e s s e l  functions.  are  f o r p l a n e t a r y waves, which are  in basins  geometric  i n the  the  hyper-  then s o l v i n g approximately i n terms The eigenfrequencies  are more  a c c u r a t e l y computed through the second s o l u t i o n , and. comparing them w i t h t h e i r e q u i v a l e n t s geometry  ( L o n g u e t - H i g g i n s , 1964 b ) ,  it  in spherical appears that  p o l a r plane approximation g i v e s good r e s u l t s enough p o l a r  basins.  the  f o r small  The discussed  general effect  1,  (1963)  The a p p l i c a b i l i t y  example t r e a t e d  o f some t h e o r e m s  of  (on t h e f i n i t e m o t i o n o f s h a l l o w f l u i d i n a  rotating  paraboloid)  complete  i n the polar plane  i s investigated  r e m a r k s a r e made i n s e c t i o n case.  bottom s l o p e s i s  i n s e c t i o n V I I I , and a s i m p l e  more e x p l i c i t l y . Ball  of r a d i a l  and f o u n d  projection.  t o be  Finally,  IX a b o u t t h e g e n e r a l  a  few  asymmetric  5  II.  BATHYMETRY AND The  had  e x p l o r a t i o n of the  been m o t i v a t e d  Northeast  by t h e  passage, took  orientation.  The  continental interest  The  bathymetric  has  areas  oceanographic  broadened  same t i m e  1961;  as  v a r i a b l e s was  Arctic  the  1962).  the  and  Canadian  The  E u r a s i a n and  the as  an  while  geophysics.  ocean has  distribution  become more  of the  idea of  main  the  Greenland  ocean i s not  bounded  72°N i n t h e  ocean  into  Canadian b a s i n s  the  m.  The  i s through  Sea.  two on  deep the  Russian They being  the  of  sill  o n l y deep the  death  noted  deep w a t e r  north  the  connection  relatively  I t must a l s o be  Sea,  the  Eurasian basin  symmetrically,  Beaufort  that of  ridge respectively.  4 km,  1500  of water  (Gordienko,  main d i s c o v e r y was  i t s counterpart;  r i d g e i s about  t o the  f a r as  into  bottom topography  s i d e s of the  deeper than  other bodies  straits  the  i c e have r e v e a l e d many h i t h e r t o  the  have a mean d e p t h o f a r o u n d  with  new  the deep b a s i n s ,  mapped and  f e a t u r e s of the  basins,  bisecting  decades a  have e x t e n d e d f r o m  into  Lomonosov r i d g e w h i c h d i v i d e s t h e  slightly  few  or a  obtained.  Ostenso,  the  which f o r c e n t u r i e s  from geography  Soundings from the unsuspected  ARCTIC OCEAN.  f o r a Northwest  past  explored  p i c t u r e of the  a t the  oceanography  i n the  THE  Arctic,  search  s h e l v e s northward  the  detailed  OCEANOGRAPHY OF  narrow that  reaching  of A l a s k a ,  but  6 only t o &>5°N at the northern the c h a r t  t i p of Greenland.  ( F i g u r e 1) w i l l r e v e a l a l l the  A model pretending  A glance at  important  t o i n c l u d e the  features.  topographic  f e a t u r e s having some i n f l u e n c e on the l a r g e s c a l e "kinematics of the A r c t i c ocean w i l l have t o i n c l u d e a r i d g e  and  asymmetrical boundaries; a simpler  symmetrical model w i l l  however be s t u d i e d  to gain some i n s i g h t i n t o  the p h y s i c s  first,  i n order  of the problem.  The  s i m p l i f i e d model w i l l have  only l a t i t u d i n a l depth v a r i a t i o n s and w i l l be bounded at a g i v e n p a r a l l e l of l a t i t u d e . The not  narrow but deep opening t o the A t l a n t i c w i l l  be i n c l u d e d  be considered  i n t h i s study, and  the A r c t i c basin  closed; t h i s i s j u s t i f i e d  such a narrow opening should  will  on the b a s i s t h a t  have only a s m a l l  kinematic  i n f l u e n c e , although i t cannot be n e g l e c t e d  when f o r c e d  motions are  Atlantic tide  s t u d i e d , and  the e f f e c t of the  included. Oceanographic sampling i n the A r c t i c has  revealed  t h a t , as i n other  oceans, a number of d i f f e r e n t water massed  can be recognized  i n a v e r t i c a l column of water, these  water masses being and  salinity  c h a r a c t e r i z e d mainly by t h e i r temperature  (Coachman, 1962).  We  are here i n t e r e s t e d  mainly i n the d e n s i t y s t r u c t u r e of the A r c t i c waters, which depends mostly on surface melting  or f r e e z i n g of the  s a l i n i t y v a r i a t i o n s produced i c e cover. Figure  observed range of s a l i n i t i e s  i n the upper 300  by  2 a) shows the metres i n the  w i n t e r and the summer.  The deep waters  (below 5 0 m) are  then almost homogeneous and w i l l be considered of uniform density.  The d e n s i t y  structure is  i d e a l i z e d by a two-  l a y e r system (Figure 2 b ) . The top l a y e r i s  shallow  ( 5 0 m)  _o  and has an average d e n s i t y  of 1 . 0 2 5 gm cm  ( cr^ = 2 5 ) ,  c o r r e s p o n d i n g to a temperature of 0 ° C and a s a l i n i t y 32° °; / o  the bottom l a y e r i s much t h i c k e r , extending to  of the  —3 bottom of the ocean,  and i s  s l i g h t l y denser:  1 . 0 2 8 gm cm  ( 0 ° C and 3 5 % ° ; °> = 2 8 ) .  The s t r a t i f i c a t i o n when the intensity of the  ice  is melting;  it  i s most pronounced i n the summer, i s a l s o bound to vary i n  from p l a c e to place because of the n o n - u n i f o r m i t y  ice cover.  These v a r i a t i o n s  not be taken i n t o account, considered uniform and  i n time and space w i l l  and the s t r a t i f i c a t i o n w i l l be  constant.  3  FIGURE I. BATHYMETRY OF THE ARCTIC OCEAN (AFTER  OSTENSO , 1961 )  DEPTHS in  METRES .  9  SALINITY 30  DENSITY,  in % o 32  24  34  100  26  <r  t  28  100 h "  CD  E  200  200 RANGE in  —  o_ UJ Q  SUMMER WINTER  300  300  OBSERVED SALINITY S T R U C T U R E  IDEALIZED DENSITY STRUCTURE  a)  FIGURE 2.  b)  ARCTIC  DENSITY  STRUCTURE  10  III.  PLANETARY WAVES. The  who  term p l a n e t a r y wave was  encountered them i n the study  coined by C.G.  Rossby  of time-ciependent  i n a b a r o t r o p i c atmosphere (Rossby 1939).  He  motions  identified  them as v o r t i c i t y waves a s s o c i a t e d with the s p h e r i c i t y the E a r t h , and  gave f o r them a s e m i - d e s c r i p t i v e  of  definition  as " q u a s i - h o r i z o n t a l . . . . wave motions whose shape, wavel e n g t h and  displacements  are c o n t r o l l e d by the  of the C o r i o l i s parameter w i t h l a t i t u d e . " Rossby f i r s t  i n c l u d e d however, and without for  (Rossby, 19L9)•  explained the dynamics of p l a n e t a r y  waves by s t u d y i n g the e f f e c t on a zonal c u r r e n t .  variation  The  of a v e l o c i t y  perturbation  zonal stream does not have t o  the v o r t i c i t y equation  for a fluid  any net t r a n s p o r t of matter w i l l account very  the mechanisms i n v o l v e d i n p l a n e t a r y waves.  look at the p h y s i c s of these  o s c i l l a t i o n s w i l l be  a f t e r the problem has been mathematically ( s e c t i o n V ) . For the moment, they roughly being  can be  be  well  A closer taken  formulated, considered  analogous to short g r a v i t y waves, t h e i r motion  i n the h o r i z o n t a l plane and  about some l a t i t u d e  e q u i l i b r i u m v o r t i c i t y r a t h e r than i n the v e r t i c a l  of  plane  about a f r e e e q u i l i b r i u m s u r f a c e . P l a n e t a r y waves were f i r s t and  s t u d i e d by  meteorologists,  t h e i r importance i n atmospheric c i r c u l a t i o n has  been  closely  investigated;  t h e y a r e now r e c o g n i z e d  as p l a y i n g  a m a j o r r o l e i n t h e h e a t and momentum b a l a n c e atmosphere, properties Their (500  as e f f i c i e n t between  presence mb.)  account  This  scale  low and h i g h  i s immediately  synoptic  chart,  study  exchangers o f these  apparent  and they  of planetary  1959).  l a t i t u d e s (Rossby, on any h i g h  level  have t o be t a k e n  i n any n u m e r i c a l f o r e c a s t i n g The  degree  large  of the  into  scheme.  waves has n o t r e a c h e d  such a  o f completeness and s o p h i s t i c a t i o n i n oceanography.  i s m o s t l y due t o t h e p r e s e n c e  do  not allow  as  i n t h e a t m o s p h e r e , and a l s o  theoretical  very  large  studies.  oceanographic  of continents,  scale recognizable complicate  waves t o d e v e l o p , considerably  The t i m e s c a l e between  measurements i s a l s o u s u a l l y  many t i m e d e p e n d e n t  which  any  consecutive  so l a r g e  that  phenomena a r e n o t o b s e r v e d .  Some t h e o r e t i c a l work h a s however been done on planetary and  waves.  Stommel  investigated wind  One s h o u l d  (Veronis,  1956; Veronis  the response  s t r e s s e s , and f o u n d  that f o rperiods  since the  i t was d i s c o v e r e d  western  portion  into quasi-geostrophic by Stommel  intensification  d e p e n d s on t h e c u r v a t u r e  of Veronis 1 9 5 6 ) who  and Stommel,  o f an i n f i n i t e  pendulum d a y , a c o n s i d e r a b l e transferred  mention t h e s t u d i e s  ocean t o v a r i a b l e o f more t h a n one  o f t h e energy i s  planetary  waves.  (Stommel,  1948) that  o f wind-induced  of the Earth,  ocean  Also,  currents  a s do t h e R o s s b y  waves, some work has been done on t h e i n f l u e n c e  o f Rossby  12 waves on ocean intensified Although  (Moore, 1963)  a role  atmospheric importance this  a n d on t h e  boundary c u r r e n t s themselves  p l a n e t a r y waves m i g h t  important  and  circulation  possibly  (Warren,  n o t p l a y as  i n oceanic c i r c u l a t i o n , a s  circulation,  they  will  t h e y do i n  seem t o be o f g r e a t  i n many t i m e - d e p e n d e n t  importance  1963).  oceanographic  p r o b a b l y be r e a l i z e d  phenomena,  more  fully  as more r e s e a r c h i s done on t h e s u b j e c t . I t was r e c o g n i z e d some t i m e a f t e r R o s s b y ' s work 1957)  (Stommel, literature  t h e r e were a n t e c e d e n t s  i n the  f o r t h e s e m i - g e o s t r o p h i c p l a n e t a r y waves;  oscillations by t i d a l  that  o f v e r y much t h e same n a t u r e had b e e n  t h e o r i s t s under the a p p e l l a t i o n  o f t h e Second C l a s s . o f an ocean  This c l a s s i f i c a t i o n  of constant depth  of O s c i l l a t i o n s of the O s c i l l a t i o n s  on a r o t a t i n g  sphere  c l a s s e s was made by Hough  after  integration  e q u a t i o n under  of the Laplace t i d a l  conditions yielded by t h e a s y m p t o t i c of r o t a t i o n of the F i r s t  he f o u n d  two c a t e g o r i e s o f s o l u t i o n s , behaviour  of t h e i r  of the E a r t h decreases  treated  into  two  that h i s these characterized  p e r i o d s as t h e f r e q u e n c y  t o z e r o . The  Oscillations  C l a s s have f r e q u e n c i e s w h i c h t e n d t o f i n i t e  v a l u e s as the r o t a t i o n  of the Earth vanishes;  i n those of  t h e Second C l a s s t h e f r e q u e n c i e s t e n d t o z e r o and t h e y become  steady motions  Poincare',  on a n o n - r o t a t i n g . g l o b e  (see a l s o  1910).  This  classification  has by some a u t h o r s  (e.g. Eckart,  I960; C h a p t e r X V I I , page 275) b e e n a t t r i b u t e d t o L a p l a c e , but  Laplace's  three  entirely different,  species  o f t i d e s were  reflecting  o f t h e components o f t h e t i d e  t h e l o n g i t u d i n a l symmetry producing  t o do v * i t h t h e a s y m p t o t i c  nothing rotation Laplace  f o r c e s , and had  limit  at v a n i s h i n g  r a t e s w h i c h Hough i n v e s t i g a t e d . Oscillations  something  o f t h e Second C l a s s  The l a b e l i s therefore  e r r o n e o u s , and i t would be p r e f e r a b l e t o a t t a c h name t o p l a n e t a r y The  waves.  work done i n t h e t h e o r y  a v a i l a b l e t o oceanographers  study  geometry.  This d i f f i c u l t y himself,  ou  a l l o f i t was done i n  i s quite appropriate  that  f o r enclosed  was e v i d e n t  to'the  speaking  of the general  Little  r o t a t i o n ) n'est  1799;  time.  p a s n u l , e t ou l a mer a une l e s f o r c e s de l ' a n a l y s e ; "  Premiere p a r t i e ,  L i v r e IV, C h a p i t r e I ) .  has been made i n t h a t Some s p e c i a l , c a s e s  latitude  (Goldsbrough, (Goldsbrough,  equation,  dans l e c a s g e n e r a l  respect  have b e e n  G o l d s b r o u g h f o r an o c e a n o f c o n s t a n t  meridians  and L a p l a c e  amplitude  f  progress  Laplace's  tidal  " L i n t e g r a t i o n de 1 ' e q u a t i o n  n (Earth's  (Laplace,  oceans o f v a r i a b l e d e p t h .  from t h e beginning,  profondeur v a r i a b l e , surpasse  by  investigations of  o f phenomena on a s p h e r e , b u t makes t h e a n a l y s i s  extremely d i f f i c u l t  said  This  o f t i d e s was, t h e n ,  in their  p l a n e t a r y waves; u n f o r t u n a t e l y , spherical  Hough's  since studied  d e p t h bound by  1933), o r b y two p a r a l l e l s o f 1914 b ) ,  or of rectangular  form  14 (Goldsbrough, 1 9 3 D .  Love (1913) s t u d i e d the case of a  s m a l l c i r c u l a r ocean, and the only case of s l o p i n g bottom was  t r e a t e d by Proudman (1916), the depth v a r y i n g only  with l a t i t u d e . was  I t i s then not s u r p r i s i n g t h a t t i d a l t h e o r y  no more u s e f u l t o the study of p l a n e t a r y xvaves i n a c t u a l  oceans than i t had been i n p r e d i c t i n g the amplitudes of the t i d e s i n these same oceans, and f o r the same reasons. Rossby's a n a l y s i s was assumption  v e r y much s i m p l i f i e d  t h a t the c h a r a c t e r i s t i c e f f e c t s of the  by h i s terrestrial  c u r v a t u r e c o u l d be preserved by making the C o r i o l i s v a r y l i n e a r l y w i t h l a t i t u d e over a s h o r t d i s t a n c e . approximation has been termed  parameter This  the /9 -plane approximation,  and has been used e x t e n s i v e l y , both i n oceanography and i n meteorology,  s i n c e i t s i n t r o d u c t i o n by Rossby (Rossby,  1939).  The t h e o r e t i c a l work done on p l a n e t a r y waves i n oceanography has n e a r l y a l l been performed  on the /3 -plane; the  mathematics are thus q u i t e s i m p l i f i e d and a c l e a r of the p h y s i c a l happenings  idea  over a r e s t r i c t e d band of l a t i t u d e s  can be a r r i v e d at without the use of s p h e r i c a l geometry. C o n c u r r e n t l y with t h i s work, Longuet-Higgins  (1964  b)  has i n t r o d u c e d another v e r y u s e f u l s i m p l i f i c a t i o n which allows even c l e a r e r p h y s i c a l understanding of p l a n e t a r y waves.  Taking advantage  of the q u a s i - h o r i z o n t a l nature of  the waves, he n e g l e c t s the i n f l u e n c e of s u r f a c e displacements i n the v o r t i c i t y balance, r e d u c i n g the problem t o a  two-  d i m e n s i o n a l one, which can be solved i n terms of a stream  f u n c t i o n . T h i s approximation i s v a l i d f o r wave l e n g t h s s m a l l e r than the r a d i u s of the E a r t h , Longuet-Higgins uses i t w i t h the /3 -plane t o examine the behaviour of p l a n e t a r y waves i n b a s i n s o f a v a r i e t y o f shapes,  and  shows t h a t i n the l i m i t o f s m a l l wave l e n g t h s the f r e q u e n c i e s of the o s c i l l a t i o n s  i n the ft -plane are  i d e n t i c a l t o those on the sphere. T h i s approximation i s a l s o used to advantage  on the sphere; i n p a r t i c u l a r ,  waves between two p a r a l l e l s o f l a t i t u d e are c o n s i d e r e d , of which waves i n a p o l a r ocean are j u s t a s p e c i a l case. The problem it  i s not t r e a t e d i n d e t a i l however, and  since  i s e s s e n t i a l l y two-dimensional, no depth v a r i a t i o n s  can be  included. Bottom v a r i a t i o n s are i n c l u d e d i n my work, and  the a n a l y s i s i s performed  in a special projection  about  the pole t o a v o i d the c o m p l e x i t i e s of s p h e r i c a l geometry. Longuet-Higgins' approximation cannot be made i f the bottom i s not f l a t , problem  so t h a t the f o r m u l a t i o n of the  i s i n t e r m e d i a t e i n complexity between the s t r i c t  t i d a l problem of Laplace and Longuet-Higgins' t h a t the motions are p u r e l y  two-dimensional.  assumption  16  IV. FORMULATION OF THE PROBLEM. The  motions r e p r e s e n t a t i v e o f the A r c t i c ocean w i l l  be those depending, e i t h e r f o r t h e i r e x i s t e n c e or i n m o d i f i c a t i o n s of t h e i r p r o p e r t i e s , on the geometry o r the d e n s i t y s t r u c t u r e o f t h e A r c t i c ocean. Short g r a v i t y waves c e r t a i n l y do not f a l l  i n t o t h a t category,  and we  can assume t h a t the wave motions which w i l l r e f l e c t i n t h e i r c h a r a c t e r i s t i c s the p r o p e r t i e s of the A r c t i c  will  be o f p e r i o d s o f a t l e a s t many hours and of s c a l e s o f more than a few k i l o m e t r e s ; t h i s i n c l u d e s p l a n e t a r y waves at the lower frequency  l i m i t . With t h i s fundamental  assumption and a few secondary ones the mathematical f o r m u l a t i o n can be c o n s i d e r a b l y The  simplified.  movements o f an i n c o m p r e s s i b l e  fluid  ina  r o t a t i n g frame o f r e f e r e n c e are d e s c r i b e d by t h e N a v i e r Stokes  equation:  + u . V u + 2Q x u +_lVP -*V u + g = 0  d  2  at and  P  t h e c o n t i n u i t y equation  V-u i n which u(x,t)  ,  P  =  0  ,  (2)  i s the f l u i d v e l o c i t y r e l a t i v e t o t h e  c o o r d i n a t e system £2  (1)  x  r o t a t i n g w i t h an angular  i s the pressure,  p  the f l u i d  velocity  density, the  net  force  v  the  k i n e m a t i c v i s c o s i t y of the  acceleration  due  to  the  due  Earth's  Assuming t h a t scale the  U,  to  a time s c a l e  gravity  division  the r  and  a horizontal  . JL  i  horizontal  1  .  balance  i n most  t h e s e t e r m s and  i n keeping  when we longer its it  term f o r p e r i o d s interested  periods,  influence contains  This  we  by  one  of  viscous  £  = and  V  are,  xu.  the  our  be  vorticity  VxfuxU  When, as w i l l terms are  justified the  other  a day,  but  motions  of  period  of  term  the  the even  much  t e r m , however  momentum b a l a n c e ,  time d e r i v a t i v e  -  we  is  comparable t o  order of  unknowns, t h e  at the  non-linear  Are  that  .  oceanic flows  i n time-dependent  of the  looking  of  horizontal  d j d t  where  L,  (1)  terms of  neglecting  term w i l l  cannot n e g l e c t  i n the  importance  evident  only  time d e r i v a t i v e  are  scale  JL- , _JL_  .  P  here  Coriolis  a velocity  length  successive  Coriolis forces.  The  centrifugal  r e l a t i v e m o t i o n has  between p r e s s u r e and  ones?  g  by  JL  The  the  and  rotation.  r e l a t i v e magnitudes of the  after  and  fluid  small  because motion.  i s made more  equation:  + 2 ^ ) 1 - y C, 2  v  be  assumed below,  of  little  = 0 ,  the  importance,  the only terms l e f t  i n the v o r t i c i t y equation are the time  r a t e of change term and the C o r i o l i s t e r m . If, is  as o r i g i n a l l y assumed,  the s c a l e of motion, L ,  l a r g e enough, the v i s c o u s terms w i l l be of  influence  i n the N a v i e r - S t o k e s equation:  sec  v  nL This is  little  IO"  2  c e r t a i n l y true i f 2  ( 0 . 0 1 cm  4  L  v  <<  1  2  i s the molecular  viscosity  -1  sec  f o r water);  6  2  even f o r an eddy  viscosity  —1  as high as 10  cm sec  , the i n e q u a l i t y holds  f o r L > 10km.  The i n f l u e n c e  of v i s c o u s  t h e r e f o r e be ignored and the f l u i d There a r e no d i s s i p a t i v e  forces  strongly will  t r e a t e d as i n v i s c i d .  processes and, i f the ocean  bounded, no energy can be r a d i a t e d away so that the  is total  energy of the water mass w i l l be c o n s t a n t .  ^  The amplitude of the motions t o be s t u d i e d  will  be assumed s m a l l enough to make the n o n - l i n e a r terms of negligible  influence: U  the Rossby number i s  ~  U sec  IU  «  1  lO"*- L  Since £l v a r i e s v e r y l i t t l e  over the area c o n s i d e r e d ,  r a t i o of v e l o c i t y to h o r i z o n t a l s c a l e relative  the  (a measure of the  importance of the l o c a l v e r t i c a l component of  v o r t i c i t y of the f l u i d most c r i t i c a l position,  s m a l l and  to the p l a n e t a r y v o r t i c i t y )  parameter i n t h i s  and i f s o ,  study;  is  the the  i t may depend on  i t must remain f i n i t e  everywhere  in  19  the domain i n which we use the l i n e a r i z e d equations.  Also,  since the t o t a l energy content of the system i s an a r b i t r a r y parameter and has a d i r e c t influence on the amplitude of the v e l o c i t i e s encountered, the Rossby number i s determined only w i t h i n an a r b i t r a r y m u l t i p l i e r .  L i n e a r i z a t i o n i s thus  appropriate t o low energy systems, and i t s v a l i d i t y w i l l have t o be v i n d i c a t e d a p o s t e r i o r i by showing that the Rossby number remains a n a l y t i c over the whole A r c t i c ocean. A f u r t h e r assumption which can be made when the periods are long enough i s t o neglect v e r t i c a l a c c e l e r a t i o n s of the f l u i d compared t o g r a v i t y .  Using the c r i t e r i a given  by Proudman (1953; Chapter XI, p 223)  f o r the v a l i d i t y of  t h i s s i m p l i f i c a t i o n , one must have, i n a homogeneous column of water, 2 2  g i n which  1  H  dt  1  i s the displacement of the surface from i t s  equilibrium position, (here 4 km.).  g i s g r a v i t y , and H the depth  This w i l l hold f o r periods longer than  10 minutes, so that the c o n d i t i o n i s not very r e s t r i c t i v e . The equivalent c r i t e r i o n f o r the motions of the i n t e r f a c e of a two-layer system i s (Proudman, 1953; Chapter XV, p 336) 2  P  n A2k. / 02  2 x at / &V Lp 2  _ &P  ~  P  P  H  gr  ;  T~  « i  20  i n which f) thickness  i s the displacement o f the i n t e r f a c e , H  2  the  o f t h e lower l a y e r (again n e a r l y 4 km.), the  d e n s i t y o f the lower l a y e r , and A p  the d e n s i t y d i f f e r e n c e  **3 between the two-layers  (3.10  ~3 gm cm  i s more b i n d i n g than the p r e v i o u s are i n g e n e r a l  ).  This condition  one, but i n t e r n a l waves  of longer p e r i o d than s u r f a c e waves of t h e  same dimensions;  the p e r i o d i n t h a t case must be longer  than 12 hours. In order t o take the pressure  as  completely  h y d r o s t a t i c , i t i s a l s o necessary t h a t the v e r t i c a l component o f the C o r i o l i s f o r c e be much s m a l l e r than gravity: U sinQ  <<  1  g  i n which 0 i s as d e f i n e d  i n F i g u r e 3 . T h i s depends on the  t o t a l energy, as does the Rossby number, but the c o n d i t i o n t h a t i t remains a n a l y t i c a t t h e pole s i n c e sinO vanishes  i s less stringent  there.  A f t e r having been subjected  t o a l l the above»  s i m p l i f i c a t i o n s , equations (1) and (2) appear as below; they are w r i t t e n out i n the s p h e r i c a l p o l a r  coordinates  shown i n F i g u r e 3« dv + 2flucos9  -2flwsin9  =  dt du dt  -1 />rsin0  - 2flvcos9  =  -1 pr  d P <^ aP dQ  (3)  21  ap  =  - p  (5)  g  dr  £l > r s i n 9 d\  1  ^uslnQ  rsinQ  a©  +  _£w d  =  0  (6} \  z  The assumptions now about t o be made are based o n t h e geometry  and s i t u a t i o n of t h e A r c t i c ocean. In any  ocean, the maximum depth i s extremely small compared t o t h e radius of the Earth.  In t h e above equations,  be r e p l a c e d by a constant  r w i l l then  r a d i u s , R, and d/dr by d/dz,  where z i s the l o c a l v e r t i c a l , measured upwards from the  F i g u r e 3.  The s p h e r i c a l c o o r d i n a t e s  corresponding  v e l o c i t y components.  and the  22  water s u r f a c e . Furthermore, s i n c e we a r e working c l o s e enough t o the pole f o r tan9 t o be q u i t e s m a l l , and s i n c e the h o r i z o n t a l v e l o c i t i e s a r e i n g e n e r a l  much  l a r g e r than the v e r t i c a l v e l o c i t i e s f o r long waves, we w i l l assume that Ucos9  »  Wsin9  ,  so that only one C o r i o l i s term i s r e t a i n e d i n ( 3 ) . The  pressure  expressions  f o r c e s can be r e p l a c e d  c o n t a i n i n g the g r a d i e n t s  (z = 77^) and i n t e r f a c e ( z = V^)  by e q u i v a l e n t  of the s u r f a c e  displacements from an  e q u i l i b r i u m l e v e l . I f a two-layer s t r u c t u r e as i l l u s t r a t e d i n F i g u r e 4 i s adopted, and the constant sure n e g l e c t e d ,  atmospheric  i n t e g r a t i o n o f the h y d r o s t a t i c  ( 5 ) , g i v e s f o r the pressures  pres-  equation,  i n the top and bottom s t r a t a  respectively P  The  x  =  g  (7)  - z)  momentum equations f o r the two l a y e r s a r e as f o l l o w s  when displacement g r a d i e n t s pressure  gradients;  are substituted f o r  the c o n t i n u i t y equation (6) i s  unchanged and has the same form i n both l a y e r s ( u , ~ 1 and u a r e the v e l o c i t i e s i n t h e upper and lower l a y e r ) .  23  Figure  4.  The t w o - l a y e r system adopted f o r the  density structure 1 + 2flu cos9 1  of t h e A r c t i c ocean. =  (9)  a t  RsinG  d  ^  au - 2 flv cosO  1  =  (10)  a t  av at a  R  2 + 2l2u cosG = 2  -g ~ u  Rsih©  aG  i  '1 R  *L??1  +  ax  2  ^ 2  a  di)  x  u  at  2  - 2 <ft v^cosG =  -g R  (12)  a  r  e  The f o l l o w i n g boundary c o n d i t i o n s d e s c r i p t i o n o f the p h y s i c a l s i t u a t i o n .  2  a  9  complete the  The v e l o c i t y  24 component will  perpendicular  vanish  at that  t o a f i x e d boundary  boundary:  In p a r t i c u l a r , a t t h e  ( u« n  and  R  —  dO  _  1  RsinO  H  constant:  since  a\  z-  -H  0^,  (14)  a\  1  depth of the  i n t e r f a c e , H^,  i t i s d e t e r m i n e d by s u r f a c e  by any mean c u r r e n t s  (13)  v.  a©.  v sin9  The a v e r a g e  )  aH  -1  at a v e r t i c a l w a l l at l a t i t u d e  u  normal n )  bottom,  •1 i j i u , z- -H  =0  (with  (here  zero),  i t will  is  considered  phenomena  n o t be  and  influenced  by d e p t h v a r i a t i o n s . A linear surface  boundary  condition  i s used:  (15)  w., d t and  a s i m i l a r c o n d i t i o n at the i n t e r f a c e , together  continuity  of v e r t i c a l  w  with  velocities:  =  (16 )  w z = -H. + V 1 2  dt  25  No n o n - s l i p c o n d i t i o n s have been imposed, e i t h e r at the boundaries or at the i n t e r f a c e , since there i s no v i s c o s i t y i n the model,  A f u r t h e r general requirement i s  that a l l v a r i a b l e s remain f i n i t e over the e n t i r e ocean, and e s p e c i a l l y at the pole, where s i n g u l a r i t i e s are to  likely  occur. The equations  equation  (6) are now  (9)-(12)  together w i t h the c o n t i n u i t y  integrated over t h e i r r e s p e c t i v e  l a y e r s , subject to the above c o n d i t i o n s .  One  then has f o r  the top l a y e r : 1 + 2&u d t du  1  -g RsinG d X  cosQ =  -g  - 2 f l v cosG =  a t ,  7  1  Rsin9 RsinG and f o r the bottom l a y e r : V  f + 2 Aiu^cosO d t a  _J  =  _ ftv cos9 = 2>  2  1 ^ 2 "2 Rsin9 a X  ^_  $Vi  P  ^  dX  2  p 2  1 ^ 2 2sin9 RsinG a 9 H  dy P  2  , A/>  d Q  ,  &p  +  [" * l * \  U  +  />i  -g Rsin9  R  at  V  = 0  <19)  x  n  9  (13)  lsin9 + 1 _ i J 7 a 9 H <H  +  x  l  ae  R  d  D  d 7 )  (17)  2  P  2  dV  =0 d t  d  (20)  X  ^2 D  z  (21)  Q  (22)  26 These are the equations I s h a l l attempt t o s o l v e , although i n a d i f f e r e n t coordinate system.  Before  performing the transformation, a few words should be said concerning the i n f l u e n c e of an i c e cover on surface waves. Such a s i t u a t i o n has been studied by Ewing and Crary  (1934);  t h e i r r e s u l t s show that the e f f e c t i s n e g l i g i b l e when the wave length i s very large compared w i t h the thickness of the i c e . We are already l i m i t e d by the h y d r o s t a t i c assumption t o periods longer than 10 minutes;  waves of  such periods w i l l have phase speeds s l i g h t l y l e s s than 200 m sec~^" i n a 4 km deep ocean, and correspondingly a scale of over a hundred k i l o m e t e r s .  There i s t h e r e f o r e  no reason to worry about d i s t o r t i o n due t o the i c e cover in the r e s t of t h i s work.  For s h o r t e r periods however,  f l e x u r a l - g r a v i t y waves w i l l be observed to d i f f e r from pure g r a v i t y waves;  observation of such shorter period  motions has been made by Hunkins (1962) from f l o a t i n g research s t a t i o n s i n the A r c t i c . A t a b u l a t i o n of the values of the p h y s i c a l parameters used i n t h i s study w i l l be q u i t e u s e f u l , and I f i n i s h t h i s s e c t i o n w i t h such a l i s t (Table I ) .  27  TABLE I .  Values of the b a s i c  physical  parameters  used i n t h i s work. QUANTITY.  SYMBOL.  NUMERICAL VALU* 1  Angular v e l o c i t y  of the E a r t h .  "  0,7*10  sec  Radius of the E a r t h .  R  6370 km.  Gravitational attraction.  g  -> 10 cm sec"*^  H  4 km,  0  Depth of A r c t i c ocean ( a v e r a g e ) . Thickness  of  surface  layer.  50 m  0  Radius of A r c t i c basin  (average).  Densitv s t r u c t u r e ,  r^  1500 km*  A_P  3xio~^  28  V.  THE  POLAR  PROJECTION.  i)  Transformation  We  h a v e now  o f the  formulated  the  d y n a m i c s o f a p o l a r b a s i n on  (17)-(22)  equations in  section  theorists of such form not  vindicates Laplace's  a g e n e r a l p r o b l e m . The  so d r a s t i c  mena. The  as  analytical  variable  topography;  the is  y9  -plane  of l a t i t u d e of l a t i t u d e s  Coriolis  by  a Mercator  of the  sphere  the  can  t o such  t o the  p o l e , and i n the  should pheno-  the  g e o m e t r y and  the  i n t r o d u c e d the  /5-  the d i f f i c u l t y . s u r f a c e of the  In sphere  projection,  so t h a t  lines,  i f a n a r r o w band  be  and,  reproduced  by  c u r v a t u r e would be  the  making  the  latitude.  a p l a n e , where o n l y  projection,  parallels  i n f l u e n c e of  f u n c t i o n of  some o t h e r  present  tidal  difficulty  stems f r o m  spherical  (1939)  of  physical  seems t h e r e f o r e n a t u r a l t o t r y and  being a Mercator  used  however  simplification  i s c o n s i d e r e d , t h e main  approximations  be  any  per  i n i t s present  interesting  1963),  become s t r a i g h t  p o l a r r e g i o n s on  near the  problem  the  as  seen  o p i n i o n of the  to circumvent  parameter a l i n e a r  It  plane  Rossby  (Veronis,  transposed  curvature  have a l s o  intractability  i n f l u e n c e s of the  approximation  but  to hide  combined  plane  We  sphere,  I I I t h a t t h e work o f f o u r g e n e r a t i o n s  seems a b i t h o p e l e s s , be  g e n e r a l problem of  a rotating  (14).  and  equations.  project  the  first  k e p t . The  /3 -  i t i s not a p p l i c a b l e  projection w i l l  p r o b l e m . I f we  are able  have t o to  29 show t h a t the a n a l y s i s gives r e s u l t s  of the motions on such a p r o j e c t i o n  not too d i f f e r e n t  obtained on the  sphere,  from those which have been  we then have a t o o l which can be  used i n the p o l a r regions with the  same confidence  ft - p l a n e i s used i n m i d - l a t i t u d e s . amount of l i g h t  as  the  Judging from the  shed by the ft - p l a n e analyses on the  m e t e o r o l o g i c a l and oceanographic  situations  in mid-latitudes,  i t would indeed be v e r y v a l u a b l e t o have such a method available  i n the p o l a r  regions.  There are many ways of p r o j e c t i n g the s u r f a c e the sphere  i n the p o l a r r e g i o n s  on to a plane;  here what I t h i n k i s the s i m p l e s t  one.  and of  (Figure 5 ) .  If  t o the  plane p o l a r c o o r d i n a t e s  r  the mapping imposes the f o l l o w i n g r e l a t i o n s h i p with coordinates: dr = R cos9 d9 ; dci  =  r = R sin9  (23)  dX  The working a p p r o x i m a t i o n , which w i l l after a l l differentiations  n e g l e c t i n g terms of order .( r / R )  d(Cos©)/dr  be introduced  have been performed, w i l l w i t h respect  Cos9 then becomes equal t o u n i t y when not and  sphere  cp are used on the plane of p r o j e c t i o n , the geometry •  the s p h e r i c a l  of  I choose  An o r t h o g r a p h i c  p r o j e c t i o n i s made on t o a plane tangent at the pole  of  i s approximately - r / ^ 2 .  consist  to u n i t y .  differentiated,  Only  first  approximations to the c u r v a t u r e are then k e p t , and t h i s  30 will  e v i d e n t l y be v a l i d  only  f o rbasins  extent  about t h e pole.  Arctic  ocean i s i n t h e Beaufort  than  13°, and s i n ^ 9  neglected  The  than  the  Coriolis  /3  parameter  with  o f (23)  c3v,  -plane  dt  (17)-(22)  i n t h e new  1  , cos9 r  cos9  =  a  V  =  d r  £ + 2AZ u^cos9 = -^g at  r  1  d  2  H  2 ^ cos9  d<f>  plane  d 4>  d ?!  -g cos9  (25)  7  J L ±( v  fl  d  Jl  f>2 d<f> P  . a r  X  PZ d  \ at  1  dV 2  Af.  +  (26)  - 7 ) 7  2  - 2 i i v „ c o s 9 = -g c o s Q 2 V  to the polar  geometry:  9  a t  but v a r i e s  dr  v  c  projection  (24)  a o  a u  plane  2X2^0039  - 2flv  d cfi  of  i s that the derivative of  of equations  <3u  1  an o r d e r  basin.  r  1  less  t h e terms  u n i t y by a t l e a s t  dt  r  so t h a t  i s not a constant,  gives  +  1  of the  latitude.  Transfer means  0.095,  main d i f f e r e n c e between t h i s  the conventional  by  less  latitudinal  S e a , a t 72°N: 9 i s t h e n  t h e whole A r c t i c  and  linearly  The s o u t h e r n m o s t c o r n e r  a r e smaller than  magnitude over  of small  (27)  Pz dcf> _  dV  1  a  _ 0  d  p  V2  A  p  .  (23)  2 a r (29)  31  F i g u r e 5.  Orthographic p r o j e c t i o n of the  surface  of the sphere on to a p l a n e . , Note t h a t two a d d i t i o n a l terms i n v o l v i n g c o s © appear, b e s i d e s those a s s o c i a t e d w i t h the C o r i o l i s parameter; they account f o r two g e o m e t r i c a l c h a r a c t e r i s t i c s of the mapping.  Equal areas on the sphere map on t o  progressively  more exiguous areas of the plane as the l a t i t u d e also,  decreases;  whereas on the sphere the angle between two meridians  decrease  s t e a d i l y from the pole to lower l a t i t u d e s ,  remains constant  on the plane of p r o j e c t i o n .  it  32 ii)  Time d e p e n d e n t  solutions.  Now l e t u s l o o k f o r t i m e  dependent  solutions  of  (24)-(29) o f t h e f o r m  TT.  in  which  cu  =  i s the frequency,  w a v e n u m b e r , F^( determined,  -  t  s  <  ^  s the constant  and t h e s u b s c r i p t  respectively.  i ( a J  r,<jb ) a n a m p l i t u d e  1 o r 2 t o denote  written  F.( r,<p) e  azimuthal  function  to be  j i s used w i t h  surface or interface  The d e r i v a t i v e s  the values  displacements,  of the displacements are  as  dV  =  iu)V  (31)  at d 7  a  )  =  - io-??  _ _ i _a_F  ; a  F ar  r  l  =  dr  - i ^ ?  ;/  7  = s _ i _ +  d<f> The  subscripts  these the  solved  F  have been  left  displacements:  1  =  of j .  g  for the velocities  the pairs G  x  V  1  ;  With  x  =  g J  since  the help of c a n be  i n terms o f  (24)-(25) a n d ( 2 7 ) - ( 2 3 ) v  (33)  dcf>  ( 3 0 ) - ( 3 3 ) , t h e momentum e q u a t i o n s  explicitly  n  JUL  o u t o f t h e above  r e l a t i o n s a r e independent  relations  (32)  x  ^  become  (74)  33  (35) ^2 The  functions G and  ^2  J are abbreviations  temporary convenience and  ^  defined  used f o r  by  ( -to cr cos9 + i v 2,0, cos9/* )  L_  =  _  A  7  ( 4^ cos 9 2  ( J  2  j  =  cu )  2  v ui/  2  - i cr 2X2 c o s 9 2  r  J  (36)  •)  (37)  J  ( 4l2 cos 9 2  2  2  )  Complete e l i m i n a t i o n of the v e l o c i t i e s i s achieved by s u b s t i t u t i o n of (34) and  (35)  c o n t i n u i t y equations (26) and  In terms of the  abbreviated n o t a t i o n , G and  (29).  i n t o the  J , the top l a y e r equation  becomes fcos9 / i  {  a G  [~  l  +  ,c G 1  . iG  -±  1+  r  \ v  r -!}  and the bottom l a y e r equation  J  1  \TJ 1  r  \)  = g H l  '  (33)  0  0  3  9  '  i  d  + < r  i  a  iV  xV  iG  dr cosQi  i  a G  P  1  r  r  P  2  /2 2l A/° 2  + V 2 + i 2\  2  J  \  X  G  J  ,?7  P  L H  2  a  2  2  d H I A/°  i 2 J  2 +  ar  H r  a<£J  2  g H  is  constant  will  us look f i r s t  ?  ^g 2  V.  0)  Let  (3?)  H r ad>J /°  R  + J icosQ _a_H G  H2,  2  at the case where the  and the ocean symmetric; the  be r e t a i n e d only f o r t h i s  d i s a p p e a r from (39), i t or F  2  depth  stratification  simple geometry.  though the terms c o n t a i n i n g d e r i v a t i v e s  eliminate  2  Even  of the depth  i s not f o r m a l l y p o s s i b l e  from (33) and (39). I f ,  to  however,  one  c o n s i d e r s t h a t the r a t i o of the amplitudes of the  surface  and  density  i n t e r f a c e displacements  structure that  depends  only upon t h e  ( r e l a t i v e depths and d e n s i t i e s of l a y e r s ) ,  f o r a constant  depth and uniform s t r a t i f i c a t i o n  r a t i o i s not a f u n c t i o n of p o s i t i o n ,  the s u b s c r i p t s  so this are  35 no longer necessary i n (33) and  (39).  Since now  F (r)/F (r) 1  2  i s a c o n s t a n t , the l o g a r i t h m i c d e r i v a t i v e of e i t h e r of these amplitudes  i s the same, and the v a r i a b l e s  cr  and J  and G assume the same value f o r j = 1 and J 3 Equation (39) then reduces t o  P  2.  j=  I  2  y  and  gH  2  E l i m i n a t i n g the terms c o n t a i n i n g G and J between (33) and  ( 4 0 ) , a q u a d r a t i c i n the r a t i o of the  amplitudes  is obtained:  AP _  -  - AP_ Pn  H  2  H  p  2  Provided, as i s the case here, t h a t \/P P  z  that  — 1,  =0  (41)  and  H/H^ — 1, good approximations t o the r o o t s of  (41)  are F  F 1  F The f i r s t  2  =  I  H  H  2  =  1  F  - A?  .  2  root corresponds t o o s c i l l a t i o n s of the water  mass as i f i t were homogeneous; the two displacements are i n phase and n e a r l y equal i n amplitude.  The second  root  can be i d e n t i f i e d w i t h i n t e r n a l o s c i l l a t i o n s at the i n t e r f a c e ; the s u r f a c e amplitude  i s much s m a l l e r and  out  of phase with t h e i n t e r f a c e a m p l i t u d e . Note t h a t i t i s p o s s i b l e t o d e r i v e t h i s without the above m i l d assumption  result  on the constancy of the  36 r a t i o of the amplitudes; one from  (1964),  i f , f o l l o w i n g Rattray  e l i m i n a t e s the displacements r a t h e r than the v e l o c i t i e s (24)-(29),  i t i s p o s s i b l e t o separate f o r m a l l y the  homogeneous and i n t e r n a l o s c i l l a t i o n s by i n t r o d u c i n g new dependent v a r i a b l e s u'  =  u  + 1  =  u 2 2 -, H'  H„ u 2 1 H  L  P  H  H2  u.  J  T h i s i s indeed an i d e a l method when t h e bathymetric  effects  are n e g l e c t e d , but s i n c e u" i m p l i c i t l y c o n t a i n s the depth, I p r e f e r t o r e t a i n the other f o r m u l a t i o n as more convenient i n s t u d y i n g the e f f e c t s of bottom v a r i a t i o n s . S u b s t i t u t i n g f o r the constant r o o t s o f ( 4 1 ) ,  (33)  can be w r i t t e n as cosQ ( i d G + o dr  i n which  1 / V^  G  +  i G \ s J r / r  i s the  +  =  — gH^ (  1  ' 2 ^ 1^ » V  (42)  constant appropriate e i t h e r to the  homogeneous or t o the i n t e r n a l mode; t h e r i g h t hand s i d e p  takes the form two c a s e s .  ^/gH and  gH]_  r e s p e c t i v e l y i n those  Equation ('42) i s an o r d i n a r y d i f f e r e n t i a l  equation i n only one v a r i a b l e , F ( r ), and w i l l be s o l v e d i n s e c t i o n VI-I; i t needs however be put i n a more e x p l i c i t form, and i t i s more convenient t o do so r i g h t  now.  37 in (36),  R e p l a c i n g G and J by t h e i r d e f i n i t i o n intermediate  equation  follows:  ( iojcr c o s 9 + 2X2cos 9 s )(2  £X2 sin9 )  2  + ( 4X2 cos 9 - oJ ) 2  2  2  2  jojo- tan9 R  - ito cos9 der + dr  + 2X2 t a n £ j3 - coo cos9+ 2X2cos9cr is r  - cu  sj '  r  - icuo* cos9  -  R  an  (43)  r  - 2X2cos 9 iscr 2  =  It the  ( 4i2 cos 9 2  i s at t h i s  2  -O) )_OJ_( 2  1. - V / 2  V )  p o i n t that the approximation of  P - p l a n e type i s made; a l l d i f f e r e n t i a t i o n s  have  been performed, and no i n f o r m a t i o n w i l l be l o s t by w r i t i n g cos9  =  sin9  =  1;tan9 =  Hence, the C o r i o l i s parameter, f, respect  r/R.  and i t s  d e r i v a t i v e with  to r , /3 , which, when expanded i n r / R , are f =  2ncos9  = -2il tan9  •R  =  =  2&(l  -  r /R  -2X2 r ( l  R'  2  -  )^  2  r /R 2  2  38  f  become  n 2X2  /3 a  -2>Q r  R2 i n ( r / R ) w i t h r e s p e c t t o 1,  N e g l e c t i n g terms  (43)  simplifies to -ar -  idcr  2  - i a  i a r -  +  dr  •2l2 (4X2 +cu ) a; 2 2  R  (4X2 -  2  after writing  F, t h r o u g h  l d £ \  diA F d r ' sf  r  -  l d F / l d F  +  I  function,  l  (4^2  2  + co )r 2  \  (4^2 „ 2 ) 2 /  r  w  R  4X2 -rqj ) 2  S  2  R ( 4X22-60 2) 2  2  -co )( 2  1 -  V  much s m a l l e r t h a n  been k e p t  values  +  F dr^ F dr  Note t h a t a l t h o u g h  has  T^)  1 - V  cr i n t e r m s o f t h e a m p l i t u d e  2X2 (  2  ( 4X2  considered  (44)  2  (32), becomes  d_/  -  2  ircr  gHi  R  which,  2  r  2  w )(  2  s  s  o f co .  since  7  (45) ? i  )  (r/R)  =  0  has c o n s i s t e n t l y  2  1, (4X2 + to  2  )/(4X2  i t may n o t be n e g l i g i b l e  This w i l l  2  been  -co )V 2  f o r suitable  p r o b a b l y n o t be t h e c a s e f o r  R o s s b y waves, w h i c h I e x p e c t  t o be o f low f r e q u e n c y , b u t  39  may w e l l be so f o r o t h e r types o f motion t o which  this  equation a p p l i e s . We have i n (45) an equation g o v e r n i n g the  amplitudes o f waves i n a two-layer system i n t h e  absence o f any bottom v a r i a t i o n s and asymmetries. When t h e r e are bottom slopes., i n .view o f the t  complexity o f e q u a t i o n ( 3 9 ) , we w i l l riot c o n s i d e r the internal  o s c i l l a t i o n s and l i m i t our study t o t h e motions  of a homogeneous ocean. The s t r a t i f i c a t i o n i s then dropped, and only equation (39) remains; when H^= 0 , (39)  becomes cosG .( i d G + o- G + i G ) •+ dr +  l i J i i rH  d  +  y_J  r a.H  i G cosQ  4>  dr  H  r  =  a) gH  An amplitude equation can be d e r i v e d from (46) by f o l l o w i n g the same procedure used t o o b t a i n (42):  t h e a b b r e v i a t i o n s G and J a r e r e w r i t t e n i n terms o f (36) and ( 3 7 ) , the approximation  their explicit definitions, [T/R) « F  (45) from  1 i s made, and cr and y  are expanded  i n terms o f  . A f u r t h e r assumption i s now t h a t the amplitude f u n c t i o n  F and the depth H a r e p e r i o d i c f u n c t i o n s o f l a t i t u d e , F  CC  e  i (  p  <  £ \  (47)  where p may be a f u n c t i o n o f r . E x p r e s s i o n s o f the form i .F  dF  & 4>  and  d H are t h e r e f o r e r e a l ; the ensuing H d(f>  40 amplitude e q u a t i o n  a F  dF  2  a 2  +  a  r  r  a F 1 r a<£ _rF  +  F  1  is  1 aH  +  (4x2 + c u ) r , 2ix2 / i  +  r  H  ( 4 X 2 _ co 2) R  ar  aF + a<p rH  2  a H -2is deb  - s i -2X2 ( 4 X 2 - r c o ) 2  2  W  - 2iX2  2  a F  co r dcfid  r  H a  CO  s_ - ( 4 X 2 - G J ) 2  g  R  torH  H  r H dep  ar  2  T  applicable to a l l cases,  function F(r,<£ ) s a t i s f y i n g  i s not  complete idealized  A general  is that  the  (48) be continuous and  over the whole extent of the p r o j e c t i o n of  ocean,  restriction be s a t i s f i e d  aH  (48)  s i t u a t i o n r e p r e s e n t e d by equation ( 4 8 ) .  Arctic  2  a H -is  without boundary c o n d i t i o n s a p p r o p r i a t e t o the  analytic  2  0  The d e s c r i p t i o n of the motions  prescription,  d cp  H  + cu ) 2 ( 4 X 2 2 _ 2) co  -2SX2  2  2 2  =  1  - i a_H 2 £ _ -2X2 ir(4X2  ( 4 X 2 - c coo)) R2 R2 2  aF 1 a H  OJ \ F ^  2  i n c l u d i n g the boundary, r = r ^ ^ ^  The  on the v e l o c i t y at a v e r t i c a l w a l l must whenever such a s i t u a t i o n a r i s e s ;  the  also  neglecting  2 r  i/R  2  with r e s p e c t  projection:  to u n i t y ,  (14) becomes,  i n the p o l a r  41  u(r )  =  1  and,  -ioj  F  dr  r  _  amplitude  then  a  iii) Now  r  d<f>  l\  =  equations,  first  be k e p t  Physics that  causes only:  x  2fl r  0  Coriolis  J\ 2  a<£  (50)  solving  cases (45),  Although  more  calculations  waves.  in stricter  local  d  o f Rossby waves.  oscillations  shows t h a t  r  i n mind, s p e c i f i c  the formulation  Consideration (25),  r  situations, (43).  f o r planetary  problem s t a t e d  sustain  -  i n the simplest  has been  form,  study the e f f e c t s of the various or  - 1 +1W  F  o f t h i s work i s c o n c e r n e d w i t h  solutions w i l l  done o n l y  the  i  i n more c o m p l i c a t e d  general are  rest  +. a  r a^> '  u) s  x  The  a r.  -  2«Q  r  F(r ) / is  +  (34)  F , using  (49) becomes  d  the  (49)  i n terms o f t h e a m p l i t u d e f u n c t i o n , (35),  and  111  v(r^)  e s t a b l i s h e d and  i t becomes e a s i e r t o  f o r c e s at  work t o p r o d u c e  i n the basin. o f t h e momentum e q u a t i o n s , accelerations  (24) and  a r e p r o d u c e d b y two  and g r a v i t y f o r c e s .  The e f f e c t o f  42  g r a v i t y i s w e l l known, and w i l l tend t o e l i m i n a t e any d e v i a t i o n s from the free e q u i l i b r i u m surface of the f l u i d ; i f g r a v i t y alone a c t s , f a m i l i a r g r a v i t y waves r e s u l t .  The  a c t i o n of the C o r i o l i s force i s a l s o w e l l known: i n the northern hemisphere i t acts as a f o r c e p u l l i n g a moving body to the r i g h t of i t s d i r e c t i o n of motion. Both fundamental f o r c e s may be of s i m i l a r importance to the motions; when t h i s happens i n a steady s t a t e , geostrophic currents r e s u l t , i n which pressure e x a c t l y balance the C o r i o l i s f o r c e . and e a s i l y v i s u a l i z e d .  gradients  This i s s t i l l  simple  What happens when the C o r i o l i s  parameter i s not constant, but v a r i e s with l a t i t u d e , as i t does on the Earth, and i n general on any curved r o t a t i n g surface ?  Instead of studying the problem on the Earth,  l e t us examine i t on the p o l a r p r o j e c t i o n introduced above. Although  i t i s somewhat u n r e a l i s t i c t o have a plane  c h a r a c t e r i z e d by d i f f e r e n t r o t a t i o n r a t e s at d i f f e r e n t p o s i t i o n s , the construct i s u s e f u l i n that i t reproduces the p r o p e r t i e s of the s p h e r i c a l surface w i t h simpler symbolism.  In order t o b r i n g the v a r i a t i o n of the C o r i o l i s  parameter i n t o the equations and see what i t s i n f l u e n c e i s , l e t us examine the equation f o r the v e r t i c a l component of vorticity,£ . When v i s c o s i t y i s neglected but the nonl i n e a r terms r e t a i n e d , the f o l l o w i n g v o r t i c i t y equation can be derived by c r o s s - d i f f e r e n t i a t i o n of the n o n - l i n e a r i z e d momentum equations  (Stommel, I960; ch. V I I I , p. 103):  43  d( £ + f)  f+f  dt  H + -7 . dt  d( H-f^ )  (51)  To obtain (51), use i s a l s o made of the i n t e g r a t e d c o n t i n u i t y equation i n i t s n o n - l i n e a r i z e d form: (H+77 )  cosQ d u r  d(H + V ) a^  (52)  dt  Equation (51) a p p l i e s on the polar plane; f i s the variable Coriolis  parameter ( 2 f l c o s 0 ) , H the t o t a l  e q u i l i b r i u m depth, 17 the surface displacement  from  e q u i l i b r i u m , and £ the v o r t i c i t y component i n the v e r t i c a l d i r e c t i o n , defined by  cos9  a vr - au a r  (53)  4>  d  The v o r t i c i t y equation (51) can be immediately integrated t o y i e l d f +  t  constant  (54)  H + V  This w i l l be recognized as a form of the p o t e n t i a l v o r t i c i t y conservation theorem, of frequent use i n  44  m e t e o r o l o g y and  oceanography. I t s t a t e s t h a t the p o t e n t i a l  vorticity  of a given  left  side of  hand  (54),  o f w a t e r moves a b o u t a  special  angular  volume o f w a t e r , as  statement  does not  i n the of the  momentum i n t h e  Now,  i n most  of v o r t i c i t y than the  due  exceptions  f l u i d . This  absence  flows,  vorticity  near the  i n boundary  currents  able  are  like  (54)  as  equation First is  see  imposes  consider  constant,  vorticity  i n the  the  that  balance.  local  motions  still  of small  Rossby can  .  be  Equation  f / ( H + 7)  const.  7  potential vorticity  affect  If a perturbation  and  i s introduced  do  the  this  .  depth  the p o t e n t i a l  a p p l i c a t i o n of torques,  h o l d . Suppose we  £  that  affairs.  s i t u a t i o n where t h e  i t does n o t  remain constant,  (54)  and  o f t h e f l u i d when f » £  a  vorticity  we  s t a t e of  approximate form  the  of  cases,  times,  vorticity  and  consider-  b e e n made a b o v e . U/L  system without will  special  assumption  for simplicity  so  smaller  l o c a t i o n ( f );  what c o n s t r a i n t s t h e on  ) i s much  disrupt this  a measure o f t h e  us  Gulf  the  w h i c h has  i s then used Let  by  component  S t r e a m , where  motions s t u d i e d w i l l not  regarded  at t h a t  torques.  local  (£  only  of  the  the  number, U / L X 2 « 1 ,  i s of course  a t a l l p o s i t i o n s and  justified  volume  where f v a n i s h e s ,  assume t h a t  is also  the  equator,  can  This  the  found. B a r r i n g these f» £  that  of a p p l i e d  t o w a t e r movements  occur  shears  change as  by  theorem of c o n s e r v a t i o n  oceanic  planetary  defined  the  the p o t e n t i a l  approximate by  in  changing  form the  45  surface  elevation V  ; t h e r e must t h e n be a n e q u i v a l e n t  change i n t h e d e n o m i n a t o r That  t o keep t h e r a t i o  c a n be a c h i e v e d o n l y by c h a n g i n g  changing  the The  fluid  t h u s moves t o a d i f f e r e n t the r i g h t  v a l u e f o r the p o t e n t i a l  with  and f i n d  i n e r t i a , and i t w i l l  itself  i n t h e same  on t h e o t h e r s i d e . The s i t u a t i o n g r a v i t y waves a b o u t  a free  v o r t i c i t y waves a b o u t  investigation  kind  of  s u r f a c e : we have  imbalance to  instead  a l o n g such g e n e r a l l i n e s  o f p l a n e t a r y waves(1939).  =  that  latitude.  t h e waves a s p u r e l y t w o - d i m e n s i o n a l ,  d_l_  overshoot  i s then analogous  a critical  Rossby proceeded first  latitude,  b a l a n c e t o be r e e s t a b l i s h e d . The f l u i d i s  however g i f t e d latitude  f , that i s  latitude.  where f has j u s t vorticity  constant.  so t h a t  inhis  He c o n s i d e r e d (51)  reduces t o  - £v ,  (55)  dt where ft i s t h e r a t e tant  i n the plane p r o j e c t i o n used  plane).  In t h e  v are v e l o c i t i e s £ z z d_v - dn and to  o f change o f f w i t h l a t i t u d e ,  t o the east  cons-  ( t h e /? -  - p l a n e , and i n R o s s b y ' s n o t a t i o n ,  u and  ( x ) and n o r t h ( y ) and  . Assuming motions  linearizing  by R o s s b y  a  independent  (55), a s i m p l e wave e q u a t i o n  of  latitude,  applicable  t h e s t u d y o f p l a n e t a r y waves i n t h e / ? - p l a n e  results:  46 For a simple v o r t i c i t y wave of the form <f = (56) g i v e s a phase v e l o c i t y  Although many o f the Rossby's a n a l y s i s  c— —  of  sink(x-ct),  •  simplifications  present  in  have not been made i n d e r i v i n g the  amplitude equations  (45) and (43). so that  they are  c o n s i d e r a b l y more complicated than the simple wave equation  (56), t h e i r s o l u t i o n s  which depend on the  v a r i a t i o n o f the C o r i o l i s parameter w i l l s a t i s f y (54) e x p r e s s i n g the c o n s e r v a t i o n of p o t e n t i a l  expression  vorticity.  E x t r a c o m p l i c a t i o n s over Rossby's problem w i l l be due t o ft not being a c o n s t a n t , with l a t i t u d e ,  and the presence  bottom topography. solutions  the amplitude of the waves v a r y i n g  i n the  of boundaries and v a r i a b l e  We can then expect p l a n e t a r y wave  present  problem t o be c o n s i d e r a b l y more  complicated than those found by Rossby. C l o s e r c o n s i d e r a t i o n of  (54) shows that  depth  v a r i a t i o n s can have e x a c t l y the same kind of i n f l u e n c e  as  v a r i a t i o n s of p l a n e t a r y v o r t i c i t y ; motions of a nature s i m i l a r t o p l a n e t a r y waves can then occur i n the of depth g r a d i e n t s a l o n e .  presence  Such waves on c o n t i n e n t a l  have r e c e n t l y been i n v e s t i g a t e d  shelves  by Robinson (1964). When  both f and H are allowed to v a r y , as w i l l be the case here, t h e i r respective  e f f e c t s can r e i n f o r c e or c a n c e l each  There w i l l then be two main f o r c e s fluid  i n the model used:  influence  other.  a c t i n g on the  g r a v i t y and C o r i o l i s f o r c e .  The  of g r a v i t y i s w e l l known; we have seen however,  f o l l o w i n g an a n a l y s i s s i m i l a r t o Rossby's (1939), that i f the C o r i o l i s parameter or the depth of the basin v a r i e s with p o s i t i o n , there can a r i s e o s c i l l a t i o n s governed mainly by the v a r i a t i o n of f and/or H.  These  are the s o - c a l l e d planetary waves t o which we w i l l pay particular attention.  43  VI.  SYMMETRIC OCEAN; GENERAL  CONSIDERATIONS.  Before attempting to f i n d e x p l i c i t the equations  solutions  d e s c r i b i n g the v a r i a t i o n s of the wave  amplitude w i t h p o s i t i o n and f o r d i f f e r e n t v a l u e s wave parameters, equations and  for  of the  i t would be wise t o examine these  together w i t h the r e l e v a n t boundary c o n d i t i o n s  to see under what circumstances they admit  This w i l l  solutions.  prevent l o o k i n g f o r s o l u t i o n s where they cannot  be f o u n d , and a l s o g i v e some q u a l i t a t i v e idea of t h e i r aspect.  The a n a l y s i s  can be done much more simply when  the  problem has complete  for  the e ^ ^ s  symmetry around the pole  (except  p a r t ) , as only o r d i n a r y d i f f e r e n t i a l  equations  remain. In the absence (45)  of a l l l o n g i t u d i n a l v a r i a t i o n s ,  and ( 4 3 ) become so s i m i l a r i n form t h a t they w i l l be  analysed as  one:  4^ + "  d F  f dF  dr  Ldr  2  2  - F f s lr  2  2  2  U  a  -  2  , 2fls  cu  2  r  2  4^ +<u 2  wR 4ft -oj 2  2  1(1+ rdH)l  +  2  2  H dr J  (  , ( 4 f l - co ) . 2 f l s 2  2  4  S  H  5  7  dHJ= drJ  The f i r s t term of ( 4 5 ) has been expanded, b r i n g i n g i n a  )  0  slight simplification;  when t h e r e  i s no bottom s l o p e ,  dH/dr vanishes i n (57); the constant to g H,hp/p  gH must be changed  i f i n t e r n a l o s c i l l a t i o n s are to be  Some non-dimensional v a r i a b l e s and are now  introduced  UJ  so as t o make (57)  abbreviations  more compact:  cu/2X2 ,  =  1  studied.  H'  =  H/H  ; H  x  =  r / ^  q  =  Q  depth at the  pole,  , 2  1 - co' M  =  4X2 R 2  S  ,  2  +  - u' )M  (1  2  In t h i s modified x d F 2  2  dx  2  +  xdF  (58)  gH  S£  =  R2  2  (€ x + 2  notation, H  2  2  (57)  .  is  how  1 + 2L_ dH' )  dx - F ( s  rf/R  + Sx  2  T  dx  + xs_ cu'H'  dH' ) dx  =  0  T h i s l a s t form w i l l be u s e f u l when we looking f o r e x p l i c i t a n a l y t i c a l solutions.  .  (59)  come t o  The  problem  i s completed by a statement of the boundary c o n d i t i o n s : F must be f i n i t e and and  a n a l y t i c everywhere i n the  on i t s boundaries, and  basin  at a v e r t i c a l w a l l at x • 1,  the f o l l o w i n g must hold f o r u to v a n i s h  (by  (33),(34),  50  (35))dF  _ x =1  dx  (60)  F(l)  s CO  Note t h a t g r a v i t a t i o n a l e f f e c t s are r e p r e s e n t e d by M,  and c u r v a t u r e e f f e c t s by e  ;  the r e l a t i v e  magnitudes of e x p r e s s i o n s c o n t a i n i n g these q u a n t i t i e s w i l l decide which of the two,  c u r v a t u r e or g r a v i t y , i s the  most i n f l u e n t i a l f o r a p a r t i c u l a r type of motion. In order t o f i n d f o r what values of  frequency,  wave number or bottom s l o p e s the system ( 5 9 ) - ( 6 0 ) admits s o l u t i o n , i t w i l l be necessary t o r e c a s t the problem i n a d i f f e r e n t form. defined  A new  variable, £  , i s introduced  by  I  L i k e any  =  1/x  .  l i n e a r o r d i n a r y d i f f e r e n t i a l equation of second  order, (59) can be r e w r i t t e n as a system of two order equations and a new  dF  dZ  one  _ f-  U  d £  _  1  =  Z d e f i n e d by the equations  )1z  =  A ( £  ,  co  '  )  (6l)  Z  J  ( st+ 2  2  8  dH')exp( c /2 £ )11  + s£  3  - B(£  of the system  2  H'  - f H'  first  i n the o r i g i n a l dependent v a r i a b l e (F)  exp(- e / 2 £  \C  d£  and  ^' ' H  , co',  s) F  .  F  2  dx  J (62)  51  The boundary  condition  (60)  is similarly  r e f o r m u l a t e d , i n terms of F and Z, as F(l)  =  Z(l)  aS_  exp (- e / 2 )  (63)  sH'  The method used t o study e x i s t e n c e c o n d i t i o n s i s sometimes c a l l e d the "Method of S i g n a t u r e s " , and i s w i d e l y used i n the Hamiltonian f o r m u l a t i o n of c l a s s i c a l mechanics;  I w i l l b r i e f l y review the p r i n c i p l e s (Eckart, I 9 6 0 ;  following Eckart's presentation  Consider a system of two dependent  and  involved,  Chapter one  independent v a r i a b l e s as f o l l o w s : dF  =  A(£  dZ  =  -B( £  , X  )  z  ,  (64)  , X ) F .  (65)  . H T h i s i s e x a c t l y the form i n which the present problem has been transformed i n (61)  and  (62).  The parameter X  stands f o r one or more independent parameters of the problem.  Equations (64)  (65)  and  canonic form, and t h i s may d e f i n i n g a Hamiltonian, H -  are s a i d to be i n  be brought out by f o r m a l l y |(  AZ  2  + B F ) , i n terms of 2  which the above system takes a form i d e n t i c a l t o t h e c a n o n i c a l Hamiltonian equations of mechanics,,  XIV).  52  The  (Z,F) space w i l l be c a l l e d the phase space,  and a phase path i s d e f i n e d as the curve t r a c e d i n phase space by a s o l u t i o n of ( 6 4 ) - ( 6 5 ) as £ v a r i e s ; the instantaneous p o s i t i o n o f the phase path i s c a l l e d a phase point.  Some theorems can be e s t a b l i s h e d about the behaviour  of the phase paths a c c o r d i n g t o the s i g n s of the f u n c t i o n s A and B (see E c k a r t ) ; they w i l l not be proven here, but, except f o r t h e f i r s t It  i s understood  one, they are a l l f a i r l y  i n the f o l l o w i n g theorems t h a t t h e  f u n c t i o n s A and B are f i n i t e THEOREM I . Let  obvious.  and continuous i n £ .  The phase paths a r e continuous curves.  Z^,F^ be any p o i n t i n phase space other than the  o r i g i n , and £  an a r b i t r a r y f i n i t e  value of £ ;  then  t h e r e e x i s t s o n l y one phase path p a s s i n g through the point Z^,F^  for £  °  ^ i * Moreover, no phase path goes through  the o r i g i n f o r f i n i t e values o f £ , but some may approach it  as £ — ±  CO  .  THEOREM I I .  A l l phase paths i n t e r s e c t the  c o o r d i n a t e axes a t r i g h t a n g l e s . THEOREM I I I .  If £  i s a r o o t o f A ( o r B ) , then  the phase path has a h o r i z o n t a l  ( o r v e r t i c a l ) tangent f o r  that value o f £ . These l a s t two theorems are i m p l i c i t (64)  i n equations  and ( 6 5 ) . The theorems t o f o l l o w w i l l be made more (64)-(65)  e v i d e n t by w r i t i n g  i n plane p o l a r c o o r d i n a t e s :  Z  =  R cos0 ,  F  =  R sinQ »  53  de  -  (66)  A cos 9 + B sin 9 2  2  d£ 1 dR  ( A - B ) s i n 9 cos9  =  (67)  R d£  D e f i n i n g the s i g n a t u r e of a segment of phase path as the p a r t i c u l a r combination of signs of A and B along that segment, and e x p r e s s i n g i t s y m b o l i c a l l y as ( s i g n of A, s i g n o f B), i t becomes apparent that much of the behaviour of phase paths can be deduced from knowledge of the s i g n a t u r e s a l o n g the path. Note that  (A), as given by ( 6 1 ) ,  i s always n e g a t i v e . THEOREM IV.  In (+,+)  segments, the a n g u l a r v e l o c i t y  ( d9/d£ ) of the phase point i s p o s i t i v e ; it  i s negative.  i n (-,-)  segments  Furthermore, f o r such s i g n a t u r e s , once  the phase path has entered any quadrant, i t s d i s t a n c e t o the a x i s p e r p e n d i c u l a r to the one i t has j u s t crossed must not  increase. THEOREM V.  away from the o r i g i n third  In (+,-) segments, the phase point moves (and from both axes) i n the f i r s t or  quadrants, and towards i t i n the other two.  situation  i s reversed i n (-, + ) segments.  These l a s t two theorems a r e i m p l i c i t (64)  The  to ( 6 7 ) .  The (+,+)  and (-,-)  i n equations  segments a r e o f t e n  called  o s c i l l a t o r y segments, and the other ones n o n - o s c i l l a t o r y . Behaviour of the phase paths under d i f f e r e n t illustrated  i n Figure 6 .  signatures i s  54  V  J  r  )  v .  r  F i g u r e 6.  Phase paths  signatures; of the  i n segments of  the phase point moves i n the  oscillatory,  velocity  tends to  or the  or the  depending on the s i g n s and the £  .When the  a l o n g l i n e s given by tan 9^ =  l i m (-A/B)  path  angular or  functional segment i s  the phase paths have asymptotic  2  I f the  path may become a c i r c l e  behaviour of A and B at l a r g e oscillatory,  infinity?  i t w i l l s p i r a l i n or out,  may v a n i s h ,  an e l l i p s e ,  direction  arrow.  What happens when £ is  various  non-  directions  55 T h i s i s of course subject t o the s t i p u l a t i o n s of theorem V; F i g u r e 7 shows what happens to segments of s i g n a t u r e the s i t u a t i o n i s r e v e r s e d f o r (-,+)  £ -  -co  (+,-);  segments, or when  .  F  Z  F i g u r e 7.  Asymptotic behaviour of phase paths of  s i g n a t u r e ( + ,-)  as £—• +00  .  I f A a n d / o r B have r o o t s i n £ , the s i g n a t u r e s change a l o n g a path;  segments of d i f f e r e n t s i g n a t u r e s  be j o i n e d t o g e t h e r where A a n d / o r B have r o o t s ; to theorem I I I , respectively. is  this  will will  according  i s at h o r i z o n t a l and v e r t i c a l tangents  An example of a path with compound s i g n a t u r e  given i n F i g u r e 8 . s  \  56  F i g u r e 8. (-,+)  (-,-)  This i s analysis  Phase path with compound s i g n a t u r e : (+,-) (+,+).  a l l the theory needed f o r the  of the amplitude equations  (61)  and ( 6 2 ) ; i t s  a p p l i c a t i o n can be e x p l a i n e d i n a few words. A and B (and t h e r e f o r e i n terms of £ conditions  the s i g n a t u r e s )  £  The f u n c t i o n s  are known e x p l i c i t l y  and the parameters grouped under  X  . Boundary  impose a s t a r t i n g point on the phase path and  the r e s t r i c t i o n t h a t the s o l u t i o n be f i n i t e of  intended  ; solutions  will  for a l l  values  then e x i s t only f o r those values  the parameters r e s u l t i n g i n the r i g h t combination of signatures,  a l l o w i n g the phase path t o remain i n  where the amplitude i s f i n i t e . very s i m p l e ,  it  Although t h i s  method  i s q u i t e powerful, and w i l l g i v e  i n d i c a t i o n s as to where the s o l u t i o n s  regions is  clear  can be found;  it  of  57 even r e v e a l s what the s o l u t i o n s w i l l look l i k e : and  extrema of the amplitude  the phase  zeros  can be read d i r e c t l y  from  paths.  I t w i l l be convenient,  i n order t o systematize the  a n a l y s i s t o d i v i d e the wave number-frequency  ( s , OJ ) space 1  i n t o f o u r domains, a c c o r d i n g t o t h e s i g n o f s and t h e magnitude of  a>';  e x i s t e n c e c r i t e r i a w i l l then be examined  i n t u r n i n the f o u r r e g i o n s of the d i a g n o s t i c diagram the s- a) space i s c a l l e d ) . 1  (as  Even though we are concerned  p r i m a r i l y w i t h p l a n e t a r y waves, which a r e o f low frequency, *  and to  hence presumably found extend  i n areas I and I I I , i t i s easy  the a n a l y s i s t o t h e whole d i a g n o s t i c diagram.  III  +  IV  t S  o  -OJ  II  F i g u r e 9.  S u b d i v i s i o n of t h e d i a g n o s t i c diagram  into four regions. Before we proceed  to i n v e s t i g a t e the existence of  s o l u t i o n s i n t h e s u b d i v i o n s o f the d i a g n o s t i c diagram, we should make a survey o f the p r o p e r t i e s of phase  paths  53 which are i n v a r i a n t over the whole ( s , cu') one  The  first  i s the asymptotic angle t o which n o n - o s c i l l a t i n g segments  tend at l a r g e v a l u e s of £ . by  space.  (6L)  yield  (62)  and  its definition  tan 9 2  f  The f u n c t i o n s A and B, as given  f o r the f i n a l a n g l e , a c c o r d i n g to  in  (63),  =  lim  { —  — A  £o/.H' dx  The f i n a l angle i s thus d e f i n e d f o r a l l areas of the d i a g n o s t i c diagram. depth, H' and s  i f the  non-dimensional  i t s d e r i v a t i v e are f i n i t e and continuous near  the p o l e , (£ — +CO s e r i e s around  Furthermore,  x=0,  ), so t h a t i t can be reduced t o a MacLaurin the f i n a l angle then s i m p l i f i e s t o  tan 9  =  2  F  1/s  (69)  2  T h i s i s e a s i l y shown t o be v a l i d under the above s t i p u l a t i o n s : as x—*0,  H — 1 , and dH'/dx OC f  n . x ~ \ where n  n i s p o s i t i v e and the lowest power of x i n the s e r i e s expansion of H . !  v a r i e s as x  n  The  second term of the denominator  when x — 0 ,  so t h a t  i t vanishes i n the  Another more s p e c i a l r e s u l t  therefore limit.  can be d e r i v e d c o n c e r n i n g  the behaviour of phase segments at l a r g e v a l u e s of £ the magnitude of the f i n a l angle i s d e f i n e d by  (69),  ; only and  we  59  can obtain more d e t a i l e d information as f o l l o w s . Let us expand d9/d£ through (61) and (62); at large values of £ d9 d£  , and f o r a constant depth (H' — _  1), we f i n d that  cos 9(-l + s tan 9 + tan 9 2  2  2  t  In (-70)terms i n £ ~  . n  )  2  £  (70)  2  have been neglected f o r n > 2.  I t should be remarked that since the terms neglected i n d e r i v i n g the expression f o r the asymptotic angle are of —7 7 2 -2 order £ (s "tan'9 - 1) i s of order £ and has the same £ dependence as the l a s t term of (70). I t .is u s e f u l i n the present d i s c u s s i o n t o d i v i d e phase space i n two regions as i l l u s t r a t e d i n Figure 9 a . In region I , tan9< l / | s | ; i n region I I , t a n 9 > l / | s l F  Figure 9 a . Phase oaths at large £ .  /  .  59a  Since £ paths we  i s very large,  discussed w i l l  will  a l s o denote  w o r r y i n g about that I  i t will  d9  I  _  have u s e d Only  situation  exact form  (m  2  (69)  axes,  being s i m i l a r  t h e y do n o t  of the  JL.  )  last  will  on t h e  (-, + ) w i l l  except  be  (71)  term  depart from  of  of the F  axis.  s i g n a t u r e (-, + ) i n the  origin  t o the  line  t o any  eigensolutions.  F = -Z/ |s|  quadrant,  tend t o the  (70).  d i s c u a s e d , the  other side  r e g i o n I of the f i r s t  signature  c o n s t a n t m,  •  the  Z>0  will  correspond  brevity,  by m/^.2, w i t h o u t  a l l phase p a t h s w i t h  being asymptotic  In  +  space  f o u r t h quadrant  (-,+). F o r  becomes  to simplify  the h a l f  phase  i n region I I , negative i n region  , (70)  cos 9  From t h e o r e m V, the  assumed t o be  positive  . In terms of m  s i g n a t u r e of the  (s t a n 9 -1)  the  be  be  the  origin  and  , so  that  phase p a t h s along the  p r o v i d e d t h e y have d9/d£ > 0 a t l a r g e £ . As  will  l a t e r when we  (£=1)  in that  quencies path  not  too near the  i s t o go  quadrant, implies  c o n s i d e r phase paths which  region,d9/d£ < 0  for£ =  inertial  be  we  start  for a l l fre-  frequency. I f the  back t o the a s y m p t o t i c  t h e r e must t h e n  1  line  i n the  a root  o f A,  o f w h i c h t h e r e i s none. No  solutions w i l l  exist  f o r such  phase  first  a h o r i z o n t a l tangent:  phase paths  with  line  F = Z/|S| see  both  this  eigen-  either.  60  Finally, quadrant  phase p a t h s  will  i n region I I of the f i r s t  tend t o the o r i g i n  p r o v i d e d d9/d£  n e g a t i v e a t l a r g e £ , w h i c h i s t h e c a s e when S enough. I f | S | i s t o o s m a l l ,  tive  and l a r g e  will  go i n t o t h e s e c o n d  the phase path w i l l in  quadrant;  approach  line  so c l o s e l y  i t again, but never  AREA I . I n t h i s propagation  then  that  exist paths  t h e y do  crossing that  than the  inertial  ( o,' = l ) ;  phase path w i l l  quadrant, for  region of the diagnostic  i s t o w a r d s t h e .west  since  from  start  F(l)/Z(l)  (s<0)  line.  be t a k e n  point w i l l  be l o c a t e d  (-,+) sign. will  (H'(l) =  the boundary c o n d i t i o n  (63)  i n the fourth  i s chosen,  i n the f o u r t h . Again  solutions  corresponding  from  (63),  that  i s much l e s s t h a n a t t h e be e i t h e r  (-,-) o r  n e g a t i v e and o n l y B c a n change  theorem V s t a t e s t h a t  i n that  and t h e s t a r t i n g '  between t h e l i n e s F = - Z / s and F = 0,  s i n c e A i s always  move away f r o m  or the second  i s < 0 there. I t i s immaterial  1 ) . The s i g n a t u r e w i l l  When B > 0 ,  diagram, less  u n l e s s t h e depth a t t h e boundary pole  path.  and f r e q u e n c y  the a n a l y s i s which quadrant  point w i l l  path  and wander  any g i v e n s and GJ' t h e r e i s b u t one s u c h phase  For  no  line  cu' w h i c h a l l o w t h e p h a s e  the asymptotic  not depart from  t h e phase  Eigensolutions w i l l  only f o r those values of to  i s nega-  i f i t i s too large,  cross the asymptotic  the f o u r t h quadrant.  is  the o r i g i n  t h e phase  path  and b o t h a x e s . T h e r e a r e  area of the d i a g n o s t i c  diagram  t o s u c h v a l u e s o f wave number and f r e q u e n c y  61  F i g u r e 10. such t h a t B > 0  Phase paths f o r area I .  f o r a l l values o f £ .  In order t h a t t h e phase path remain i n r e g i o n s o f phase space where F and Z a r e f i n i t e , B must be n e g a t i v e f o r a wide enough range of v a l u e s of £ oscillatory  t o have an  segment of l e n g t h s u f f i c i e n t t o a l l o w the phase  p o i n t t o leave t h e f o u r t h quadrant. the phase path must be i n t h e f i r s t  For a s o l u t i o n t o e x i s t , o r t h i r d quadrant when  B changes s i g n . B i s n e g a t i v e when  s£ 2  2  +  sj  "J*  €  +  I_  dH»)  H» dx  +  M(l  -  O  J  '  2  )  ^  "^2  <  0  .  (73)  62 This i n e q u a l i t y w i l l be s a t i s f i e d  e i t h e r f o r very low  f r e q u e n c i e s o r , c o n s i d e r i n g the d e f i n i t i o n of 6 , ( 5 8 ) , f o r nearly i n e r t i a l frequencies.  We a n t i c i p a t e t h a t t h e low  frequency s o l u t i o n s w i l l be p l a n e t a r y waves.  I t i s doubtful  t h a t the present l i n e a r i z e d f o r m u l a t i o n can t r e a t the n e a r l y i n e r t i a l s o l u t i o n s adequately, r e g u l a r a t t h a t frequency  i n the present model.  I t appears a t once from i f the i n e q u a l i t y i s s a t i s f i e d w  s,o of  (73) and theorem IV t h a t f o r one v a l u e of frequency,  say, then i t i s g o i n g t o be s a t i s f i e d  s u c c e s s i v e l y lower f r e q u e n c i e s .  i s the f i r s t of  s i n c e the v e l o c i t i e s a r e not  by an i n f i n i t y  L e t us say t h a t  frequency t h a t i s found  i  s, o  t o produce a segment  s i g n a t u r e (-,-) long enough t o a l l o w B t o change s i g n i n •  the t h i r d  quadrant,  i n which theorem V r u l e s t h a t t h e phase  path w i l l move towards t h e o r i g i n as £ — + CD allow s o l u t i o n s t o e x i s t .  '  , and thus  C l e a r l y , i f a lower  frequency,  ^ i s chosen such t h a t B remains negative f o r l a r g e r  values o f £ ( i . e . , in  ^  i t takes longer f o r t h e s  2  £  2  term  (73) t o catch up w i t h t h e negative terms), t h e phase  path w i l l remain of o s c i l l a t i n g range of £  and, i f  reach the f i r s t  i  w  s, 1  quadrant  nature f o r an extended  i s p r o p e r l y chosen, i t w i l l b e f o r e B'changes s i g n .  An i n f i n i t e  s e r i e s o f s o l u t i o n s can then e x i s t f o r a same v a l u e of wavenumber, s,: t h e system i s degenerate i n s. Denoting the s o l u t i o n s by F ( x ) , with f r e q u e n c i e s to (with sjn sjn n = 0,1,2....),  the phase path w i l l wind around i n a  63 c l o c k w i s e d i r e c t i o n r e q u i r i n g an e x t r a IT r a d i a n s to reach the o r i g i n between each s u c c e s s i v e s o l u t i o n ,  »  and the  i  f r e q u e n c i e s w i l l s t e a d i l y decrease: ai , < a) . The ^ s,n+l s,n solution F (x) w i l l have n zeros between the pole and s ,n J  the boundary s i n c e  the phase path c r o s s e s the F a x i s n times;  n i s then an index number d e n o t i n g t h e number of nodes of the  radial  solution.  The i n f l u e n c e  of bottom topography on the  can be examined through t h e i r e f f e c t  on the  solutions  inequality  Let us c o n s i d e r n e a r l y uniform bottom v a r i a t i o n s f o r  (73).  the  moment; the slope  having the same s i g n f o r a l l x and being  nearly Constant.  I f the depth o f the b a s i n decreases from  the pole to the boundary ( d H ' / d x < 0 ) , t h e n , as increases,  the negative  part of B i s made more  | dH'/dx negative,  and i t w i l l take a s m a l l e r frequency to b r i n g the phase path to the same p o i n t f o r the same value of no bottom s l o p e .  Negative  slopes  £  as when there  (with r e s p e c t  v a r i a b l e x) have t h e r e f o r e the e f f e c t  to  is  the  of r e d u c i n g the  frequency of p l a n e t a r y waves; slopes of t h e opposite s i g n w i l l of course have the opposite e f f e c t . If dH'/d: is the  l a r g e enough so t h a t -jj-, d H ' / d x + € < 0 f o r a l l values i n e q u a l i t y ( 7 3 ) w i l l not hold f o r any v a l u e of  i n the f i r s t  r e g i o n of the d i a g n o s t i c d i a g r a m .  then a maximum uniform slope  of £  frequency  There i s  ( d H ' / d x < 0 ) a l l o w i n g westward  propagating p l a n e t a r y waves t o e x i s t i n a symmetric p o l a r basin.  It  i s g i v e n by IdH'/dx  > e H'(l) .  ,  64  I f we do not i n s i s t t h a t t h e bottom slope be uniform i n x, s o l u t i o n s can be found f o r bathymetries  that w i l l  i n c l u d e l o c a l slopes g r e a t e r than the maximum uniform The  only c o n d i t i o n t o be s a t i s f i e d  phase path  i s a f t e r a l l t h a t the  escape from t h e i n i t i a l quadrant and t h a t B has  i t s l a s t r o o t when the phase p o i n t i s i n the t h i r d first  slope.  quadrant.  The f u n c t i o n B can have more than one r o o t  i f the depth v a r i e s s t r o n g l y enough; a simple a situation,  o r the  example of such  i l l u s t r a t e d below, a l s o shows the e f f e c t o f  s l o p i n g w a l l s a t t h e same time. Consider  a b a s i n bounded by w a l l s w i t h s l o p e s  than the maximum average s l o p e a l l o w i n g westward waves;  propagating  the bottom i s f l a t beyond the s l o p i n g w a l l s and  remains so u n t i l the pole i s reached ( F i g . 1 1 ) . will  greater  The s i g n a t u r e  i n i t i a l l y be (-,+) and t h e r a d i u s o f the phase p o i n t w i l l  increase.  When a value of £  corresponding  t o the r a p i d  decrease i n slope i s reached, B w i l l become negative frequency  i f the  i s s m a l l enough, and the phase path can be c a r r i e d  F  Figure 1 1 .  Bottom p r o f i l e and phase paths f o r s l o p i n g  w a l l s i n Area I of the d i a g n o s t i c diagram.  65  t o the t h i r d  or t o the f i r s t quadrant, where B w i l l have i t s  second r o o t .  The  s o l u t i o n s w i l l t h e r e f o r e be s i m i l a r t o  those f o r a f l a t bottom, but w i t h is  lower f r e q u e n c i e s .  There  then a wide c l a s s of bathymetries a l l o w i n g westward  propagating  p l a n e t a r y waves i n a symmetric p o l a r b a s i n ,  the  g e n e r a l r e s t r i c t i o n being t h a t f o r some wave number s and bottom c o n f i g u r a t i o n H'(x) frequency which allows  there e x i s t s a r e a l value  B t o have i t s l a s t r o o t i n a quadrant  of the phase space where the phase path w i l l tend o r i g i n f o r large values II.  AREA  frequency i s now is  again  The  of £  > 1, 2  (1-  l i n e F = -Z/k  r  expressed by  '/R  2  dominates or when  terms ( l / R  might t h i n k ; f o r s = -1,  in  with  (73), but  the  since  T h i s may  dH/dx>o and  co '  e  ), The  large.  more important  frequency necessary  i s however not as l a r g e as  (73)  one  i s already s a t i s f i e d f o r  then be a type o f motion f o r which i t 2  j u s t i f i a b l e to neglect The  very  ( F i g . 12). Once  or bottom s l o p e s , are now  t o make t h e ' i n e q u a l i t y hold  (59).  point  i  than c u r v a t u r e  co' =1.5.  now  s o l u t i o n s f o r B > 0;  2  to' )M  not  starting  the  (making € < 0) t h i s w i l l hold e i t h e r when  G r a v i t y terms i n M,  is  The  taken i n the f o u r t h quadrant, but may  more, t h e r e w i l l not e x i s t any  now  i s s t i l l westward, but  g r e a t e r than i n e r t i a l .  i n e q u a l i t y B< 0 i s s t i l l  t o the  •  propagation  l i k e l y be l o c a t e d below the  is  of  e x  as compared t o 1, i n  lowest frequency which makes B< 0 w i l l  i n c r e a s i n g k («|s|) and  r , ; the  increase  i n f l u e n c e of bottom  66  F  F i g u r e 12*  Phase paths f o r area I I .  s l o p e s i n the I n e q u a l i t y ( 7 3 ) i s as i n area I , but the frequency  i s a l t e r e d i n a d i f f e r e n t manner s i n c e B i s now  made n e g a t i v e by terms i n which  i s large instead of  terras i n which i t i s s m a l l , as i n area I . radially frequency  A depth d e c r e a s i n g  now causes an i n c r e a s e i n s t e a d o f a decrease i n over t h e c o r r e s p o n d i n g f l a t bottom s o l u t i o n .  The  s o l u t i o n s are a l s o s u b j e c t e d t o an i n f i n i t e degeneracy i n wave number; and  they are c o n t r o l l e d mostly by g r a v i t y  cannot be c a l l e d p l a n e t a r y waves, AREA I I I .  frequency  Propagation  i s l e s s than i n e r t i a l .  be taken i n the f i r s t will to  effects/  quadrant,  be found under t h e l i n e  the e x p l i c i t  i s now t o t h e e a s t , w h i l e The s t a r t i n g p o i n t w i l l and, u n l e s s H ' ( l ) <<  1, i t  F = Z/s ( F i g . 1 3 ) . A c c o r d i n g  expression" f o r B ( 6 2 ) , the s i g n a t u r e w i l l  67  F i g u r e 13. (B)  Phase paths f o r area I I I . ( A ) . F l a t bottom.  Large p o s i t i v e depth g r a d i e n t s : dH'/d£ > 0 .  always be (-,+ ) u n l e s s there i s a s u f f i c i e n t l y depth g r a d i e n t , Let  l a r g e negative  £H.\ dx  us f i r s t  look f o r f l a t  bottom s o l u t i o n s .  The  s i g n a t u r e i s (-,+) and, a c c o r d i n g t o theorem V, the phase path moves towards t h e o r i g i n f o r that s i g n a t u r e i n t h e f i r s t quadrant.  There w i l l then be s o l u t i o n s i f t h e phase path  does not escape t o t h e neighbouring quadrants. not  There w i l l  be any s o l u t i o n s i f the phase path moves i n t o t h e second  quadrant and s t a y s t h e r e (theorem V ) ; but once i n quadrant 11$  63 the  phase p a t h  cannot loop back i n t o t h e f i r s t  because t h i s would  imply  none. F o r f r e q u e n c i e s the  angular  velocity  1  for £ =  a root  quadrant,  o f B, o f w h i c h t h e r e i s  not t o o near t h e i n e r t i a l d9/d£  o f t h e phase p a t h ,  of the sign of (s +8 2  )  w  '  /  2  -  ?  1  frequency, ,  is  ; this i s  s ' < 0,97  negative  for  page 59a  , there w i l l then  and  1  o/ <  then cal  w  • According  be no e i g e n s o l u t i o n s f o r s > 0  i n the absence o f bottom s l o p e s . There a r e  no e a s t w a r d p r o p a g a t i n g  p l a n e t a r y waves i n a s y m m e t r i -  p o l a r ocean i n t h e absence o f bottom When l a r g e n e g a t i v e  present,  nature  radial  depth d e r i v a t i v e s are  i s a segment o f phase p a t h  enabling the f i r s t  or t h i r d  a manner c o n s i s t e n t w i t h  The  slopes.  i t i s p o s s i b l e t o choose a frequency  so t h a t t h e r e  in  to the analysis of  same m u l t i p l i c i t y  of o s c i l l a t i n g  q u a d r a n t t o be  the existence  as  frequencies  o f s o l u t i o n s i s found  the s l o p e  of equivalent  ( aj'> 1 ) ,  and p r o p a g a t i o n  ing  i s now f o u n d  above t h e l i n e F = Z / s , than as for  unity.  as i n a r e a s  i s now  such  s o l u t i o n s w i l l now  increase  increases.  AREA I V . The f r e q u e n c y  point  reached  of s o l u t i o n s .  I and I I . The i n f l u e n c e o f b o t t o m s l o p e s that  l o w enough  i s here  l a r g e r than (s>0).  t o the east  (again  i n the f i r s t  provided  H (l)  14.  There  i s then  The  start-  quadrant)  i s n o t much l a r g e r  f  F o r B> 0 f o r a l l £ , t h e r e  shown i n F i g u r e  inertial  may be e i g e n s o l u t i o o s s  o n l y one e i g e n s o l u t i o n  e a c h wave number and no m u l t i p l i c i t y  as i n t h e p r e v i o u s  69  Figure  1 4 . Phase p a t h s  f o r Area IV. /  instances. the t h i r d  When t h e f r e q u e n c y term  o f phase p a t h ,  general  as t h o s e  be f o u n d . the s i g n  solutions  encountered  Depth v a r i a t i o n s  mostly  i n areas  I and I I w i l l  have t h e same i n f l u e n c e  by g r a v i t y  (-,-)  o f t h e same  o f B as t h e y have i n a r e a I I I ; as i n a r e a  waves a r e c o n t r o l l e d ifiable  enough t o make  o f (73) d o m i n a n t , and t h e s i g n a t u r e  o y e r a segment nature  i s large  on II,  and a r e n o t c l a s s -  a s p l a n e t a r y waves. I  have o b t a i n e d i n t h i s  q u a n t i t a t i v e Method  the  section,  by t h e s e m i -  of Signatures, valuable information  70  as t o where i n the d i a g n o s t i c diagram s o l u t i o n s  compatible  with the approximations and the boundary c o n d i t i o n s can be found.  Much i n f o r m a t i o n about t h e q u a l i t a t i v e aspect o f  s o l u t i o n s i s r e v e a l e d by t h e behaviour of phase paths, and e x t e n s i v e degeneracy i n wave number has been d i s c o v e r e d i n the p o s s i b l e s o l u t i o n s . t h a t low frequency  In p a r t i c u l a r , i t has been d i s c o v e r e d  o s c i l l a t i o n s c o n t r o l l e d mostly by curvature  e f f e c t s and presumably i d e n t i f i a b l e w i t h p l a n e t a r y waves can be found only i n area  I of the d i a g n o s t i c diagram i n t h e  absence o f bottom s l o p e s . L i t t l e more i n f o r m a t i o n can be obtained Method of S i g n a t u r e s , and the amplitude equation be s o l v e d e x p l i c i t l y t o f i n d their  frequencies.  from the must now  the form of the s o l u t i o n s and  71  VII.  FLAT BOTTOM SOLUTIONS: i)  Solution  SYMMETRIC BASIN.  i n terms o f c o n f l u e n t  hypergeometric  functions. S t a r t i n g with been  studied  in different 1964  Longuet-Higgins, polar plane  the  b)  simplest  formulations and  of a basin with  boundaries.  In t h e  2  dx  while  xdF( e x  2  +  the  we  2  + 1)  waves, i n w h i c h  simplified  ex  and  can  with be  other  first  b o t t o m and  - F(  reduces  =  s + 8x ) 2  2  (60)  1914  a,  the the  symmetrical  to  0  ,  (74)  our cu'  is unaltered.  a t t e n t i o n on i s so  respect  small  t o 1,  negligible,  future  and  long  i n area  (74)  I I of the 2  f o r which  ex  (74)  period  t h a t we is  i n terms of B e s s e l  were m o t i o n s f o r w h i c h  i t s entirety.  possible  made, I s t u d y  (59)  solved  have s e e n however t h a t  diagram there  in  equation  concentrate  neglect  entirely  be  a flat  boundary c o n d i t i o n  planetary  We  (Goldsbrough,  dx  2  If  safely  already  absence of depth v a r i a t i o n s , the  amplitude  x d F  w h i c h has  t h r o u g h w h i c h a c h e c k on  a p p r o x i m a t i o n can  oscillations  symmetric  case,  considerably functions.  diagnostic  might not  should  be  It w i l l  t h e n be  u s e f u l , i n view  extension  of t h i s  work t o t y p e s  than p l a n e t a r y  x^aves, t o s o l v e  the  can  more  be  kept of  of  exact  motion  72  equation its  (74),  partly to i l l u s t r a t e  possible application  a priori (74)  justification  t h e method u s e d  i n other cases,  of the  assumption  when p l a n e t a r y waves a r e s t u d i e d .  pages,  I will  equation,  then  (74);  of c o n t i n u i t y  find  these  and  partly  solutions  pages may  t o be  t h e r e a d e r may,  as  an  made i n  In the f o l l o w i n g  o f t h e more  be  and  exact  omitted without i f he  solution  wishes so,  proceed 2  t o page 8 8 is  , where t h e a p p r o x i m a t i o n  t r e a t e d , as  a p p r o p r i a t e t o the  Equation form  of the  (74)  F  =  resulting  x  ± S  u(y)  ; y = - ex /2  i n which primes to  F(x)=x" pole  (also  called  .  (75)  is  -u(±s€ -S  ) =  0  indicate d i f f e r e n t i a t i o n with  ( x =  vanishes, 0).  y =  of  or at l e a s t  Furthermore,  (76)  ,  (76)  respect  =  e  isc£  positive  or  ?  s  o  i n order to  such  that  at  the  insure  p o l e , one  t h a t s must be  negative.  must be  remains f i n i t e ,  of the s o l u t i o n s around the  ZTT + cfi ) s  integer,  appropriate solution  u(y)  continuity i(  standard  y. The  e  1  <<  variables 2  ± s - y)  ex  into the  equation  change o f  equation f o r u(y)  with  o f p l a n e t a r y waves.  transformed  by t h e  yu"+u'(l  (74)  study  confluent hypergeometric  Kummer's e q u a t i o n )  The  c a n be  to  zero or  must an  have  7 3  W r i t i n g Rummer's equation i n g e n e r a l form, w i t h parameters a and b, i t i s yu" +  u'(b  -  y)  -  ay  =  0  ( 7 7 )  .  Compariton o f (76) and (77) shows t h a t here, b = and a = ( ± s integer; the  - 8)/2  .  1 ± s,  The parameter b i s t h e r e f o r e an  i n such a case (77) has two independent  solutions:  one i s f i n i t e a t the pole and i s ^ F ^ ; °; y ) »  first  3  and the second one i s a l o g a r i t h m i c s o l u t i o n which y = 0  at  ( S l a t e r , I960; Chapter I ) .  diverges  The only s o l u t i o n of  (74) which w i l l remain f i n i t e a t the pole f o r i n t e g r a l v a l u e s of s i s then  F(x) =  cx  k  F  (kg - 8 ) ; 1+k; -tx /2 2  , (73)  2  i n which c i s an a r b i t r a r y c o n s t a n t and k i s |sl . The f u n c t i o n •^F^(a: b; y) i s the standard hypergeometric n o t a t i o n f o r the  series  1  +  a b  y +  a ( a +1)  y  b(b + l)  25  2  +  a(a  +l)(a  b ( b + i ) ( b  + 2) + 2)  _jr + . . . 3  (79)  3»  The boundary c o n d i t i o n (60) can a l s o bje expressed i n terms o f the s o l u t i o n o f the problem.  U s i n g the  f  d i f f e r e n t i a t i o n formula f o r , F,  ( S l a t e r , I960; Chapter 2)  74  d_  1  F ( a ; b; y) = a ^ ( a  dy and  1; b  1  1; y) ,  (  d  Q  }  b  a r e c u r s i o n formula  a^F  (a+l;b+l;  y)= ( a - b ^ U ;  b+1; y j + b ^ a ;  b; y)  (31) the boundary c o n d i t i o n  ^ ( ( k f ^ ) ;  (60) becomes  * + 2; - « / 2 )  ^  (  k  +  1  )  (  a  /  u  |  , ,  k  +  g  )  (k/2 +1)€ +S/2 (( k e - 8); k+1;- / e  F  1  >•  2  O £  )  /  2 e  (32)  Confluent hypergeometric f u n c t i o n s are t a b u l a t e d only f o r p o s i t i v e values of the argument y, so that when e > 0, i t w i l l be more convenient t o express the s o l u t i o n i n terms of functions of + e x / 2 . 2  Kummer's f i r s t  This i s done by using  theorem:  e ' ^ F ( a ; b; y) =  ^(b-a;  b;-y)  (33)  Instead of (73) we then have f o r s o l u t i o n F(x)= c x e x p ( - € x / 2 ) ^ k  2  k/2+l+ /2e ; k+1; € x / 2 ) . ( 3 4 ) S  2  and  f o r the boundary  condition  */ )  k+2;  2  f k/2 + 1 + / e; k + 1;  F  S  _  2  {  k  +  1  )  (  k  . /o,M s  /2  €  2  (d'5)  Before a t t e m p t i n g t o e x t r a c t boundary c o n d i t i o n s  ( 8 2 ) or ( 8 5 ) , i t w i l l be i n s t r u c t i v e  t o review a few of t h e d e s c r i p t i v e •jF-^. ( S l a t e r , I 9 6 0 ; Chapter X). it  i s not,  a n y t h i n g from the  properties  of t h e s e r i e s  Chapter 6 . Jahnke and Emde, 1945;  The v a r i a b l e y w i l l be assumed p o s i t i v e ; i f the f u n c t i o n  can always be transformed v i a ( 8 3 )  t o make i t so. 1- There w i l l not e x i s t any r o o t s  of  u n l e s s e i t h e r a or b or both are n e g a t i v e . b (=k+l) 2-  F 1  1  (a;  i s always p o s i t i v e . I f b > 0 and a =  -n+"0,  where n = 1 , 2 , 3 . . . and  ]Jf-jJa; b; y) has e x a c t l y  y  at t h e l a r g e s t v a l u e o f y, y ~ *  i s the r o o t  zeros, and i f 0 0  >  ^  a; b; y) >  F ( a; b+1; y)  <  ^  a; b; y) .  b)  1  ]  1  4- I f y^ and y  2  are roots  -.F-, ( a ; b ; y ) r e s p e c t i v e l y ,  of ^ i ( i ? F  and a, and a  as n—- CD .  i f a and b a r e > 0 ,  a) F _( a+1; b; y) 1  p  n  n  3- I t i s evident from (79) that  9  ; y)  In t h i s work,  0 < © < 1 , then n  b  a  ?  0  b-^; y) and  are both n e g a t i v e ,  76  F i g u r e 15, 1  F (a; 1  The c o n f l u e n t  hypergeometric f u n c t i o n  b; y) as a f u n c t i o n of y f o r a few v a l u e s  a and b .  of  77 then, a) I f l a ^ > | a i w h i l e  = b , then y  2  b) I f  a  c) I f y  1  =  a  and 0 < b < b  2  1  i s the m  m  2  t  n  2  , then y <  >  (from  ; y  2  .  r o o t (from the o r i g i n o f y)  of -,F, ( a; b; y) then, when b 0 , as a---GO, y — 5- At y = 0 ,  <  (79))  the Confluent  0.  hypergeometric  s e r i e s i s equal t o u n i t y f o r a l l v a l u e s of a and b. 6- As y — + CO , the s e r i e s tends  e x p o n e n t i a l l y t o +00  i f t h e r e i s an even number of r o o t s , t o -CO o t h e r w i s e . F i g u r e 15. i l l u s t r a t e s some of the above p o i n t s and g i v e s an idea o f some aspects o f t h e c o n f l u e n t  hypergeometric  series. We found with a f l a t  i n s e c t i o n VI t h a t i n a symmetric  bottom low frequency  t o p l a n e t a r y waves could e x i s t  ocean  s o l u t i o n s corresponding  only i n area I of t h e  d i a g n o s t i c diagram ( F i g u r e 9 ) .  C o n c e n t r a t i n g our a t t e n t i o n  on the p l a n e t a r y wave s o l u t i o n s , we see t h a t s i n c e i n area I of the d i a g n o s t i c diagram s < 0,  cu ' < 1,  then e > 0, and  the eigenvalue problem c o n s i s t s i n f i n d i n g the v a l u e o f frequency  cu ' which w i l l s a t i s f y t h e boundary c o n d i t i o n  i n the form  (35).  a characteristic frequency  Such a frequency equation.  equation i s o f t e n c a l l e d  I f we wished t o study the. h i g h  s o l u t i o n s which can be found  of the d i a g n o s t i c diagram  (where  i n areas I I and IV  cu ' > 1,  and consequently  € < 0) the same method as w i l l be used below would be f o l l o w e d , but s t a r t i n g from  equation  (32).  73 Since  i n (85)  the parameters of the  confluent  hypergeometric f u n c t i o n as w e l l as the r i g h t hand s i d e depend on frequency, i t w i l l be wise t o examine the boundary c o n d i t i o n as a f u n c t i o n of frequency.  The  frequency dependent e x p r e s s i o n  on the r i g h t hand s i d e of  (85),  (85  h e r e a f t e r r e f e r r e d t o as  -Figure  16 a g a i n s t  t o area  frequency f o r values  of s c o r r e s p o n d i n g  I of the d i a g n o s t i c diagram ( s < 0) . (85  ( = Is| ) >1, 2 order  rhs), i s plotted in  10  inertial  rhs)  i s always p o s i t i v e and  f o r values frequency  ( co'=l).  This i s a l s o true for  - 1 , except i n a range of v a l u e s  (85  rhs)  i s l a r g e and  negative.  of f r e q u e n c i e s  the r a t i o of the  two  we  frequency  confluent  hypergeometric f u n c t i o n s w i l l be l a r g e and s = -1,  where  From s e c t i o n VI,  p l a n e t a r y waves w i l l be of low  so t h a t f o r k > l ,  for  at l e a s t of  of frequency not too near the  s =  expect t h a t the  For k  that r a t i o w i l l be l a r g e a l s o  positive; ( > 10 ) , but 2  w  i t s s i g n w i l l depend on whether the the r i g h t or t o the The confluent  €  s i n g u l a r i t y i n (85  v a r i a t i o n of the t h i r d v a r i a b l e o f  /2,  17.  i s s m a l l and The of the  of the  hypergeometric f u n c t i o n s on the  of ( 8 5 ) , . Figure  left  eigenfrequency i s t o  /2  e  i s q u i t e simple and  v a r i e s only very s l o w l y near  l e f t hand s i d e of (85)  the hand s i d e  is illustrated  becomes l a r g e only near  f i r s t v a r i a b l e , 'a',  left  rhs).  o) = l , T  co'  =  i n each of the  i s , when 8  in and  0. functions  i s expanded i n  Figure  16.  frequency,  The  expression  co » 1  (6*5 rhs) as a f u n c t i o n  of  80  F i g u r e 17. The e x p r e s s i o n e/2 frequency  as a f u n c t i o n of  a) ' .  terms of i t s d e f i n i t i o n i n (58),  'a' = l+k(  1 -  2  M ( 1 - a;' ) 2  2 ( 1  This expression negative  +  (  g  6  )  +a>' ) 2  i s p l o t t e d a g a i n s t frequency f o r a few  values of s ( F i g u r e 18).  The v a l u e of M ( =  20)  a p p r o p r i a t e t o the homogeneous mode of o s c i l l a t i o n has been u s e d , but the shape of the curves i s v e r y s i m i l a r , although on a d i f f e r e n t  scale,  when i n t e r n a l  oscillations  i  are s t u d i e d .  With M = 0.53x10 , the v e r t i c a l s c a l e  m u l t i p l i e d by about 10^" and the r o o t s moved t o the  is extreme  31  F i g u r e 18.  The parameter ' a '  (86)  frequency  at' f o r a few v a l u e s of s ( s < 0).  plotted  versus  32  left  of t h e frequency s c a l e . From F i g u r e 18 and the d e s c r i p t i v e c o n s i d e r a t i o n s on  the p r o p e r t i e s o f c o n f l u e n t hypergeometric f u n c t i o n s , t h i s much can be concluded about t h e l e f t hand s i d e of ( 8 5 ) . higher than t h e root of ( 8 6 ) , a > 0 ,  For a l l f r e q u e n c i e s  b > 0 , and from property but l e s s than u n i t y .  3b the r a t i o w i l l be p o s i t i v e  For  co between zero and the r o o t , 1  a < 0 , b > 0 and thus, from p r o p e r t y  2, the r a t i o can  assume e i t h e r s i g n and amplitudes from zero t o CO „ o f ( 8 5 rhs) has a l r e a d y r e v e a l e d t h a t the  Consideration  r a t i o of c o n f l u e n t hypergeometric f u n c t i o n s w i l l be a t 2  l e a s t o f order the and  l a t t e r region to'=0) It  (86)  10  i n magnitude, so that i t i s only i n 18  (between the r o o t o f 'a' i n F i g u r e  t h a t s o l u t i o n s o f ( 8 5 ) w i l l be p o s s i b l e . i s t o be n o t i c e d t h a t f o r s = - 1 the root o f  ( F i g u r e 18) occurs a t a lower value  than t h e f i r s t  s i n g u l a r i t y i n ( 8 2 rhs)  of frequency  ( F i g u r e 1 6 ) , so  that i n the frequency range where s o l u t i o n s can e x i s t the r a t i o o f the c o n f l u e n t hypergeometric f u n c t i o n s must be l a r g e and p o s i t i v e . Curves o f t h e f u n c t i o n s ^F^( a; b 1; ^F-^( a; b; y) w i l l  y) and  f o l l o w each other f a i r l y c l o s e l y when  p l o t t e d a g a i n s t y ( F i g u r e 1 9 a ) , and the f u n c t i o n s have t h e same number of r o o t s .  will  The r a t i o of two such  f u n c t i o n s , as i n ( 8 5 ) , can be of order 1 0 o r 2  only near a r o o t of the denominator.  greater  The l e f t hand  of ( 8 5 ) w i l l be l a r g e and p o s i t i v e a t the p o s i t i o n s  3ide  y-j (,i = 1,2..) j u s t to the l e f t of zeros of ^F^( a; b; y) (Figure 19a).  The p r e c i s e value of the p o s i t i o n s y. w i l l 3  of course depend on the value of the other parameters ('a' and 'b') of the confluent hypergeometric f u n c t i o n s , i . e . , w i l l depend on the frequency d e f i n i t i o n of 'a' i n ( 8 6 K v a r i a b l e y (=  €  co ' through the  At low frequencies, the  / 2 ) i s n e a r l y constant:  e  / 2 — r£/^2 .  Although the number of p o s i t i o n s y . i s equal t o the number of roots of i F, , only that p o s i t i o n y.( co ') — r / 2 J 1 2R  will  give an eigenfrequency. I t i s apparent from d e s c r i p t i v e p r o p e r t i e s 1 and 4c) t h a t , as co' decreases and 'a becomes more negative f  (Figure lo*) , the m  tn  root (m f i x e d ) of F (a;b;y) tends t o  the o r i g i n of y. But as each m  root ' f i r s t ' appears  (when 'a' decreases t o -m), i t does so at a higher value of y than f o r the (m-1)  root.  Hence, as co' decreases,  these r o o t s must i n t u r n , i n order m, s u c c e s s i v e l y move to t h e l e f t along the y-axis and pass, through y = r f / 2 . O D  For a s e r i e s of decreasing values of frequency the l e f t hand side of t h e boundary c o n d i t i o n (85) passes through a high p o s i t i v e value equal t o the r i g h t hand s i d e ; t h i s happens at the p o s i t i o n s i n d i c a t e d as eigenvalues i n Figure 19b and 19c. We then recognize the degeneracy predicted by the Method of Signatures, there being an i n f i n i t e number of eigenfrequencies f o r each value of wave number s.  F i g u r e 19.  The  hypergeometric  r e l a t i v e p o s i t i o n s of the c o n f l u e n t f u n c t i o n s a t the e i g e n v a l u e s ,  p o s i t i o n s y . where the r a t i o of the two and  positive,  eigenvalue.  b) The f i r s t  eigenvalue,  a)  The  functions i s large c)  The  second  85 The eigenfrequencies w i l l be c a l c u l a t e d on the assumption that we are c l o s e enough t o the root of the denominator the n k + 1; 1  of the l e f t hand side of ( 8 5 ) t o approximate  eigenfrequency by the n  t n  €  /2).  t h  root of F ( k / 2 +1 + $/2e 1  1  A f i r s t approximation t o the n ^ root of n  F ( a; b; y) i s ( S l a t e r , I960; Chapter 6) 1  3/4J TTH n + b/2 --3/4)'  =  n  (2b  -  2  4a)  This i s v a l i d when (-a + b/2) » 1.  I t w i l l be v e r i f i e d  a p o s t e r i o r i that t h i s approximation i s a good one here. S u b s t i t u t i o n f o r the dummy v a r i a b l e s a,b,y i n terms of the a c t u a l v a r i a b l e s gives f o r the -root =  •/2  "  Ms,n  (83|  2 + 2 S/e  2 (4n+2k -1) . When S and e are ' th expanded i n terms of frequency, by means of (58), a 5 .  where  2 = tr /  ^  s  n  1 6  degree algebraic equation i n c*j' r e s u l t s . I f however the " frequency i s s m a l l enough, cu'2 << 1, a l i'n e a r expression gives the eigenfrequency  s, n  explicitly:  86 A few e i g e n f r e q u e n c i e s ,  as c a l c u l a t e d from t h i s  formula,  2 are l i s t e d  i n Table  II.  I t i s c e r t a i n l y t r u e that  so t h a t we may s a f e l y pass from (88) t o (89). small values  of co , the e x p r e s s i o n 1  cu'  =1,  Also, f o r  (-a + b/2), which has  t o be l a r g e f o r approximation (87) t o b e . v a l i d , i s found to be equal t o  MR / 2 s,n r  ; the adopted value o f  ±  r / ^2 i n the case of a symmetrical 2  so t h a t the r e l e v a n t e x p r e s s i o n smallest value  of  ^  basin i s l / 2 0 (Table I ) ,  i s l a r g e , even f o r the ( = 15.33).  , which i s H-  1,1  s,n  I t can f i n a l l y be v e r i f i e d , by a c t u a l computation of the r a t i o o f c o n f l u e n t hypergeometric s e r i e s on t h e left  hand s i d e of (85), t h a t approximating the e i g e n f r e q u e n c i e s  by the r o o t s o f the denominator g i v e s q u i t e good r e s u l t s , at l e a s t f o r s m a l l n and c a l c u l a t i o n gives  F o r example, a more p r e c i s e  u>' l,  =  0.00317 ( *a = ,s  - 146), which  i.  d i f f e r s only by about 3% from the approximate value  by (89) :  0.00306 ( 'a' = - 152.3).  given  The exact value o f  'a' which s a t i s f i e s the boundary c o n d i t i o n (85) i s between -146  and -147, so t h a t f o r s m a l l n and k, approximating  the e i g e n f r e q u e n c i e s  by the r o o t s of ^F-jJ a; b; y) i s a  b e t t e r approximation than t h a t used t o f i n d those  roots  (87). This approximation however g r a d u a l l y l o o s e s i t s v a l i d i t y as k i n c r e a s e s . Although a l o n g c h a i n o f approximations i s necessary t o f i n d the eigenvalues ( oo' ) when the s, n s o l u t i o n i s expressed i n the form of c o n f l u e n t hyperJ  37  TABLE I I . homogeneous  E i g e n f r e q u e n c i e s of Rossby waves i n a symmetrical ocean with a f l a t  as c a l c u l a t e d from the c o n f l u e n t solution,.  bottom,  hypergeometric  OJ' given by ( 3 9 ) . s,n -2  •1  s =  -3  Period (days) n=l  .00306  (164)  .00319  .00294  2  .00093  (510)  .00132  .00142  3  .00048  (1040)  .00072  .00084  4  .00028  (1785)  .00045  .00055  5  .00018  (2775)  .00031  .00039  geometric f u n c t i o n s , possible for  the  the above example shows t h a t i t  t o o b t a i n approximations t o the case of a f l a t bottom even when  neglected  compared to u n i t y .  is  eigenfrequencies 2 ex  is  hot  T h i s i s not necessary  when  p l a n e t a r y waves are under i n v e s t i g a t i o n ^ : but the method has been developed  i n p r e v i s i o n of f u r t h e r work i n c l u d i n g  types of o s c i l l a t i o n s where i t (74)  into  ( 9 0 ) below.  is  impossible to  simplify  We have seen i n s e c t i o n V I . t h a t  t h e r e may be such time dependent motions i n areas I I and IV of the d i a g n o s t i c diagram. 2 The approximation e x  «  1 w i l l now be made, and  the problem w i l l a c q u i r e much c l a r i t y by so d o i n g .  88  ii)  Approximate s o l u t i o n i n terms of Bessel functions,  Neglecting  ex  i n ( 7 4 ) , the amplitude  2  equation f o r  a symmetrical ocean with a f l a t bottom becomes x d F 2  +  2  dx  x dF  s  - F(  2  + 8x ) = 0 2  ,  (90)  dx  2  which has s o l u t i o n s i n terms of Bessel f u n c t i o n s , ( Z ) , of the general form (Whittaker and Watson, 1 9 2 7 ; Chapter XVII) F(x)  =  Z  ( i V8  ± s  x)  (91)  .  The boundary c o n d i t i o n ( 6 0 ) i s unchanged; at a v e r t i c a l w a l l at x = 1 , Z' (1^8 z ±s  s  x)  (92).  (iVSx)  l a  '<*  i n which the prime i n d i c a t e s d i f f e r e n t i a t i o n with respect to the argument of the Bessel f u n c t i o n .  Equation ( 9 2 )  becomes, using the formula - 2 Z (y) +  Z' (y) = p  y  Z  (y)  P  which i s a p p l i c a b l e t o a l l Bessel f u n c t i o n s , Z  + g  _ (i- /8 1  x)  v  Z. ( i V S  x)  =  _ k j l + s_±) iV8 koj'  .  (93)  39 a ; ' i s not n e a r l y equal to 1 (which i s the  When  c o n d i t i o n f o r the s i m p l i f i c a t i o n be approximated  parameter  8  S__  + M(l -  , as given by  6J  )  !  i s s m a l l enough:  1  cu  1  < 0.1  F i g u r e 20.  for s =  o>' <  -2, and  (94), w i l l be n e g a t i v e i n  0.05  negative;  i n area I I I , s > 0,  positive.  Finally,  provided  and  a/  in  1, and  8  i s always  co' < 1, and  8  i s always  >  cu'>l), 8  will  cu' i s v e r y near u n i t y .  Ill  s  >0,cu'<l  s>0,ai>l  8 >0  8 <O  8 <  The  jyr  0  8 <0  8>0  s<0,a/<l  F i g u r e 20.  f o r s = -1,  i n area IV ( s > 0 ,  be negative u n l e s s  cu'<l),  so on. T h i s i s i l l u s t r a t e d  In area I I , s < 0,  I  can  (94)  area I of the d i a g n o s t i c diagram ( s < 0 , oj  ^  by  s =  The  l e a d i n g t o (90))  s<0,u/>|  s i g n of 8  JJ  (as given by  (94)) i n  the s u b d i v i s i o n s of the d i a g n o s t i c diagram.  90  For negative v a l u e s  of S , the B e s s e l f u n c t i o n J^l^y/l"^  x)  i s r e a l and w i l l be used where the g e n e r a l e x p r e s s i o n f o r a B e s s e l f u n c t i o n appears above.  The magnitude of s,  k,  is  chosen as the o r d e r of the B e s s e l f u n c t i o n : s i n c e s i s an integer,  i n which case Z  and Z  are not l i n e a r l y independent^  i t does not matter which one of these two f u n c t i o n s i s f o r the  chosen  solution. When 8  is positive,  the f u n c t i o n  x)  is  real  and w i l l be chosen f o r the s o l u t i o n of ( 9 0 ) . T h i s f u n c t i o n i s r e l a t e d to the o r d i n a r y B e s s e l f u n c t i o n as l  =  I (y) k  It  i"  k  J (iy)  follows:  .  k  i s not necessary to i n c l u d e B e s s e l f u n c t i o n s of  the second kind to complete the s o l u t i o n (as should be done when s i s an i n t e g e r )  because these d i v e r g e at x = 0, where  we want our s o l u t i o n to be r e g u l a r . For  co' <  1, the parameter  §  i s negative  when s < 0 ( area I of the d i a g n o s t i c diagram ) . now see t h a t s o l u t i o n s  of  only  We s h a l l  ( 9 0 ) i n that case correspond t o  the p l a n e t a r y wave s o l u t i o n s  of (74) which we found above.  Let us look f i r s t a t the completely symmetric case:  s = 0 . We can show t h a t there w i l l be no p l a n e t a r y  waves e x h i b i t i n g such symmetry. &  By (94), when s = 0,  becomes 8  =  (1 - oo' ) 2  M  7  r  2  ,  91 which i s p o s i t i v e f o r f r e q u e n c i e s boundary c o n d i t i o n (93)  l e s s than i n e r t i a l .  The  then takes the form  1^78  )  =  0 .  (95)  I j j y ) b e i n g a p o s i t i v e monotonic i n c r e a s i n g f u n c t i o n of y, (Jahnke and Emde, 1945; 8=0,  only when  Chapter V I I I ) , (95)  i . e . , when  OJ ' = 1.  can be s a t i s f i e d  This s o l u t i o n i s i  however e n t i r e l y i n c o n s i s t e n t with the assumption which l e d t o (90),  ex << 1,  so t h a t t h e case s = 0 cannot be p r o p e r l y (90).  d i s c u s s e d by the approximate equation go back (74)  Z  We must then  and f i n d what the c o n f l u e n t hypergeometric  s o l u t i o n p r e d i c t s when s ( and k) = 0;  from  (89), 0,n  r  i s zero f o r a l l n, so t h a t there a r e no p l a n e t a r y waves with  zero wave number. I t should be noted t h a t i f higher f r e q u e n c i e s a r e  considered s=0.  ( w' > 1)  s o l u t i o n s o f (90)  can be found f o r  The boundary c o n d i t i o n then becomes  (96)  which i s s a t i s f i e d when  «' = The  constant ft , l,n  course  (1 +  i s the n  t h  $  2  l,n  (97)  ) .  root o f J , ( y ) .  1  This i s of  J  o n l y a l i m i t i n g case o f the g r a v i t y c o n t r o l l e d  o s c i l l a t i o n s of areas I I and IV of the d i a g n o s t i c diagram. We l i m i t o u r s e l v e s here t o a d i s c u s s i o n o f p l a n e t a r y waves  92  i n a p o l a r ocean, so t h a t nothing more w i l l be s a i d about these g r a v i t y waves. For non-zero n e g a t i v e v a l u e s o f s, when  8 i s negative  co' i s s m a l l enough ( F i g u r e 2 0 ) , and the s o l u t i o n o f  (90) i s  F(x)  =  (93)  oyTiSl x) .  The boundary c o n d i t i o n i s g i v e n by ( 9 3 ) , w i t h r e p l a c i n g Z^. Even when  cu' i s not v e r y s m a l l , the r i g h t  hand s i d e o f ( 9 3 ) w i l l be c o n s i d e r a b l y l a r g e r than u n i t y , and  one f i n d s t h a t the v a l u e of 8 which s a t i s f i e s the  boundary c o n d i t i o n d i f f e r s from the r o o t o f the denominator of  the l e f t hand s i d e o f ( 9 3 ) only i n the t h i r d  f i g u r e . The e i g e n f r e q u e n c i e s w i l l then be  significant  approximately  g i v e n by •  k  o , 2 M + R^/r  s.n  The  constant  J^;  k,n k takes the v a l u e s 1 ,  s = -1, km)  i s the n  -2, -3,... 2  1500  B  2,  2 1  /3  (99) k,n  r o o t o f the B e s s e l f u n c t i o n 3,...  corresponding t o  . From Table I (R = 6 3 7 0 km, r ^ =  •  2  R /r  = 2 0 . Some o f the e i g e n f r e q u e n c i e s , as  calculated from(99) are l i s t e d  i n Table I I I . They depart  from the e i g e n f r e q u e n c i e s as c a l c u l a t e d from t h e c o n f l u e n t hypergeometric  s o l u t i o n a t l a r g e r v a l u e s o f k. T h i s i s not  due  t o the approximation  large  ex  o f (93) ( t h i s  stems f r o m  (85).  those  retained.  their  n, t h i s  approximation These r e s u l t s  analysis  last  a value  of the curvature will  be n e c e s s a r y  best  seen  comparison internal periods  of the Earth with  geometry.  s i m i l a r when t h e i n t e r n a l ; t h e c o n s t a n t M, w h i c h  i s now however  changed  value  adopted:  M =  means t h a t l o w e r  0.53x10^.  frequencies  t o make t h e i n f l u e n c e o f t h e t e r r e s t r i a l  c u r v a t u r e comparable is  where  i n the spherical  of gravity,  gravity  results  l a t e r be compared  i s entirely  will  i n Table I I I ,  o f 20 t o a new and much h i g h e r  smaller effective  We  i n a p o l a r plane  corresponding t o the s t r a t i f i c a t i o n  will  approximation  I I , as c h a r a c t e r i s t i c  i s investigated  represents the effect  The  of the confluent  by t h e r o o t s o f t h e denominator  o f Table  mode o f o s c i l l a t i o n  from  the eigenvalues  e q u i v a l e n t s as c a l c u l a t e d ,The  hand  i m p r o v e s a s k i n c r e a s e s ), b u t  a n a l y s i s when p e r f o r m e d  only a f i r s t  ) or to  the roots of the l e f t  the eigenfrequencies l i s t e d  r a t h e r than  is  from  cu' d e c r e a s e s  l o o s e s i n e x a c t i t u d e as k i n c r e a s e s .  adopt  of t h i s  solution  For a constant  gradually then  also  approximating  hypergeometric of  1 becoming worse f o r  k ( i t becomes b e t t e r , s i n c e  estimating the eigenvalue side  <<  i n importance  i n the d e f i n i t i o n  to gravity  of 8  forces;  ( 9 4 ) , where a  o f t h e two i n f l u e n c e s c a n be made.  frequencies w i l l  direct  Some  mode f r e q u e n c i e s a r e t a b u l a t e d i n T a b l e corresponding t o these  this  IV; t h e be o f t h e  94 TABLE I I I .  Eigenfrequencies  of RosSby waves i n a  homogeneous symmetrical ocean with a f l a t c a l c u l a t e d from the B e s s e l '  solution.  -2  -3  -8  -5  1  .00324  .00366  .00359  .00320  .00261  2  .00099  .00139  .00155  .00163  .00152  3  .00048  .00073  .00083  .00101  4  .00028  .00045  .00057  .00069  5  .00018  .00031  .00040  .00051  order of a m i l l e n i u m and more.  R e a l i z i n g that the  subjected t o annual v a r i a t i o n s a s s o c i a t e d  atmospheric c o n d i t i o n s ,  it  w i l l not remain constant it  and  that  the  interface  is  seen t h a t the  d u r i n g long p e r i o d  i s not s t r i c t l y c o r r e c t to t r e a t as constant  in time.  s e c u l a r change,  since  with  oscillations, the depth of  I f the c o n d i t i o n s is  i t might be p e r m i s s i b l e t o take the  s t r a t i f i c a t i o n as c o n s t a n t ,  surface  stratification  p u r e l y p e r i o d i c over a y e a r l y p e r i o d , and t h e r e  is  as  given by (99).  s, n  -1  layer is  function  bottom,  the p e r i o d of  are  little average  variations  so s m a l l compared w i t h the p e r i o d of the p l a n e t a r y waves.  Needless to say, measurement  there  is  little  hope of any d i r e c t  of the l o n g - p e r i o d i n t e r n a l waves.  95  TABLE I V . for  Eigenfrequencies  a two-layer  as c a l c u l a t e d w  '  s |n  of i n t e r n a l Rossby waves  symmetrical ocean w i t h a f l a t  from the  Bessel function  g i v e n by ( 9 9 ) . M =  0.53xi0  -1  s =  bottom,  solution.  .  6  -2  -3  Period (years) 1.86*xl0~  n=l  6  (2915)  3.76xl0"  4  l.o7x "  (2930)  3.74x "  9.30x "  9  1.83x "  (2995)  3.64x  9.00x  Before comparing the above  "  geometry,  let  what are the  us see  what the  properties  look l i k e ,  =  c J  k  and  frequencies  From (30) and ( 9 8 ) , and ( 3 4 ) ,  ( f o r the  homogeneous mode)  c o r r e s p o n d i n g v e l o c i t i e s are e x p l i c i t l y  V  with  spherical  of those waves, the  which we j u s t c a l c u l a t e d .  the s u r f a c e displacement  solutions  6  "  eigenfrequencies  those obtained f o r a s i m i l a r b a s i n i n the  of  9.40xl0~  6  ( (k/^-Mj^x ) e  i  U  t  -  and the  given by  S  *  )  (100)  R  u=  - i cgc'[ 2X2r  n  d J  k  I dx  (  )  }  cy'x  I  e *"*-*) 1  (101)  96  v  ill  2  dimensions  of  of  the  is  same i n  length,  system. (101)  and  ellipse  Ju(  )\e  i  ^ -  (  t  S  ^ )  (  >  as  and  and  boundary.  argument in  (102)  of  as  by  the  in  c has  the  the  total  Bessel  functions  (100), and  has  i n a p e r i o d i c manner a r o u n d velocities The  time v a r i e s , at  eccentricity  constant  i s determined  The  varying  amplitude the  and  the  not  explicitly.  Besides  pole  '  above e x p r e s s i o n s ,  been w r i t t e n i n  the  OJ  (102)  a  energy  pole,  S  dx  In the  the  i  djk(  of t h i s  have, n o d e s b e t w e e n  velocity any  ellipse  vector  fixed  traces  location;  i s given  the  an  the  by  )  « kU  _s  dx  1 - lul  the  )  u  OJ'X  ,  Ivl dJ  ( k_  ~  s  OJ  (103)  k  t  dx  so  that  i t will  frequency, the  cu' , a n d  ellipse  (radially), clockwise clockwise,  vary  i s traced and  will as  with  distance  wave number, will  s.  The  also vary  from  bands where the alternate with  one  progresses  from the  velocity  x,  direction in place vector  bands where from the  pole,  pole  to  which  place  rotates  i t rotates towards  counter-  the  97 boundary ( F i g u r e 2 2 c ) .  Sketches  contour's f o r a few values  of amplitude and v e l o c i t y  of s and n, are found i n  figures  21 and 2 2 . k-1  k Being of order x  and x  r e s p e c t i v e l y at  of x , the amplitude and v e l o c i t i e s  values  over the whole A r c t i c ocean and i t s k  remain f i n i t e  boundaries provided  > 0 , which i s the case f o r the p l a n e t a r y waves s t u d i e d  now.  The v o r t i c i t y i n the v e r t i c a l d i r e c t i o n , £  given by ( 5 2 ) , becomes i n terms of f u n c t i o n s  (  small  =  ££_  2^1  At f i r s t  Vi  i f ^ L dx  sight,  of x  '-s!* '\ • "' J  *  2  it  , as  seems t h a t t h i s  (  1<  J  2  (104)  e x p r e s s i o n would be of  k-2 order x at s m a l l x ; i f however we r e p l a c e the independent v a r i a b l e x by the.argument of the B e s s e l f u n c t i o n s , *\/\h\ x ,  2 noting that k =  2  2flr'  Since J  1  I  2 s , we f i n d  k  that  |S, 2  k  V  i-lx  VIOIx  is a Bessel f u n c t i o n ,  e q u a t i o n , whatever i t s  it  k  j  satisfies  (  Bessel's  argument:  2  2  y J £ ( y ) + yJ (y) + ( y - k ) J (y) = 0 2  R  k  .  1  Q  5  )  98  S = - 2 , n»l. F i g u r e 21.  Sketches  ( 7 , from (100))  S = - 3 , n=2. of s u r f a c e amplitude  contours  f o r p l a n e t a r y waves i n a symmetrical  ocean w i t h a f l a t bottom, f o r a few values of wave number, s, and index number, n. The p a t t e r n s r o t a t e c l o c k w i s e with angular v e l o c i t y  ^  / s. Dotted l i n e s are nodal l i n e s .  99  c.) - F i g u r e 22,  a)  (u)  components  for  s =  angular  -2,  n  and  b)  Sketches  of the v e l o c i t y =  1.  velocity  The w  field  patterns / 2,  1  of the  zonal (from  (v) (101)  and and  rotate clockwise  Dotted  lines  are  radial (102))  with  nodal  lines,  —2,1 and  the  velocities  c)  The  direction  an  ellipse  : -  are  l a r g e r where t h e  i n which the f o r clockwise,  arrows are  local velocity +  otherwise.  vector  longer, traces  100 The primes i n (105) and i n Bessel's equation i n d i c a t e d i f f e r e n t i a t i o n with respect t o the argument of the f u n c t i o n . The three terms i n brackets i n (105) then reduce t o - J^i^/lcTl x) , and the v o r t i c i t y becomes  f  =  -jsll 2^  J.ivGi  x).  .  K  (  1  0  6  )  r^  which i s of order x  at s m a l l x. In s p i t e of the neglect  of v i s c o s i t y , there i s no s i n g u l a r i t y i n v o r t i c i t y a t the pole f o r the planetary waves considered, and the Rossby number remains f i n i t e everywhere. To f i r s t order, t h e average energy t r a n s p o r t due the wave motion i n a v e r t i c a l column of water i s <  / z  p v dz  >  ,  (107)  i n which the brackets i n d i c a t e average over a c y c l e .  When  the pressure i s h y d r o s t a t i c , and the v e l o c i t y does not depend on Z, (107) becomes  <  P&VZ  H  >  (108)  101  The time average of the r a d i a l energy t r a n s p o r t v a n i s h e s ; the zonal component becomes p r o p o r t i o n a l to H  |  (1 - cu  dF  2  dx  - s cu'F  2  (109)  ,x  The net energy t r a n s p o r t i n the z o n a l d i r e c t i o n i s i n t e g r a l of  the  (109) from the pole t o the boundary; when the  depth i s c o n s t a n t ,  this  is  p r o p o r t i o n a l to 1  F (1) - s  c u '  J  0  F dx  .  (110)  x  At the boundary, the amplitude i s very s m a l l ; as a matter of f a c t ,  we have approximated the e i g e n f r e q u e n c i e s  those v a l u e s at x  =1.  of frequency which make the amplitude v a n i s h The second term of  (110) w i l l t h e r e f o r e dominate,  and the energy w i l l propagate on the. average and over a l l values opposite t o that  of the r a d i a l coordinate)  group v e l o c i t y  ( over a c y c l e in a d i r e c t i o n  i n which the phase moves. . The energy  t r a n s p o r t so c a l c u l a t e d d i f f e r s  Higgins,  by  from t h a t g i v e n by the  only by a n o n - d i v e r g e n t v e c t o r (Longuet-  1964 a ) ;  method i n c l o s e d  it  i s more convenient t o use the  basins.  present  102  The the  case S and  8  case >  p l a n e t a r y waves j u s t d e s c r i b e d >  0.  0 in (94).  Let us  From F i g u r e  2 0 , t h i s can  s m a l l bands of areas I and  diagram.  The  look b r i e f l y at  IV of the  boundary c o n d i t i o n (93)  1  k  "  •  1  I (Vo k  V§  )  :  to  the  occur i n area  III  diagnostic  becomes, when  .(1+s  k  correspond  )  &  0,  >  (Ill)  .  k co'  Since the f u n c t i o n s  I, (x) and I, , (x) are monotonic k k-1 i n c r e a s i n g f u n c t i o n s of x such t h a t I. ..(x) > I (x) K"* JL  the r i g h t hand s i d e of (111)  must be p o s i t i v e and  than u n i t y f o r the r e l a t i o n t o be  0,  >  k  greater  satisfied.  O s c i l l a t i o n s of t h i s kind w i l l be very K e l v i n waves, i n the sense t h a t they w i l l hug  s i m i l a r to  the  sides  of  the b a s i n , t h e i r amplitude i n c r e a s i n g very r a p i d l y near x = 1, are now  according being  t o the behaviour o f I ^ i v^S  studied  i n more d e t a i l by H.G.  the U n i v e r s i t y of Washington therefore  I  existence.  Comparison w i t h r e s u l t s on a sphere.  Let us now t h e i r equivalents the v a l i d i t y  They  Farmer, at  (Farmer, 1964), and  l i m i t myself to a mention of t h e i r iii)  x).  compare the  on the sphere.  r e s u l t s obtained I f the two  are  above with compatible,  of the a n a l y s i s performed i n the p o l a r  plane  103  approximation t o the sphere w i l l be e s t a b l i s h e d i n the study of a l l p o s s i b l e motions of the contained f l u i d . is  This  so because p l a n e t a r y waves, b e i n g dependent on the  c u r v a t u r e of the E a r t h f o r t h e i r e x i s t e n c e w i l l be more strongly affected  by any departures from the exact c u r v a t u r e  than any other type of motions. for  p l a n e t a r y waves,  I f the approximation works  i t w i l l then work and give r e l i a b l e  r e s u l t s f o r a l l other motions of the A r c t i c The  s i m p l e s t b a s i s of comparison i s the work of  Longuet-Higgins ( 1 9 6 4 b ) .  By assuming that the  displacements have a n e g l i g i b l e balance,  ocean.  surface  i n f l u e n c e on the v o r t i c i t y  L o n g u e t - H i g g i n s has been a b l e t o formulate the  problem of p l a n e t a r y waves i n two dimensions and t o it  i n terms of a stream f u n c t i o n .  T h i s approximation w i l l  give good r e s u l t s provided the wave l e n g t h i s the r a d i u s of the E a r t h .  solve  In p a r t i c u l a r ,  s m a l l e r than  he g i v e s f o r the  stream f u n c t i o n c h a r a c t e r i z i n g the p l a n e t a r y waves i n a p o l a r b a s i n on the sphere  ^ in  which  =  P £ (cos9)  (112)  k  P (cos9) i s the Legendre f u n c t i o n of arguments 2/  k, v , and cosG; k i s defined as above, and so i s 0 , w h i l e v  i s a p o s i t i v e r e a l number (not n e c e s s a r i l y an i n t e g e r )  which allows the boundary c o n d i t i o n t o be  ?  v  (cosS,) = 0 ;  9, =  sin  satisfied:  -1  (113)  104 The  frequency of the p l a n e t a r y wave i s then given by t  (114)  v ( v + 1)  Equation (114)  i s not very d i s s i m i l a r i n form t o  the e v a l u a t i o n of v  (99),  but  i s much more complicated than f i n d i n g  k p.  .  The Legendre f u n c t i o n P ( x ) ,  (but not  when x i s r e a l and k  v  ) i s an i n t e g e r ,  v  is  v  i  P^(x)(-2)  J7(r + k  kl  given by the o k/2  +  -  expression  x^)  ,  (  n  5  )  v  -k + 1)  ?{v  = F (1+k-t-z/,k-z/ ; k + 1; £ - £ x ) 2  where ^F-^ i s the geometric  series,  1  standard n o t a t i o n f o r the u s u a l h y p e r T(y)  is  the gamma f u n c t i o n :  co  r(y)  =  JeS " 7  dt .  1  o  The  expression  hypergeometric  (114) w i l l have r o o t s s e r i e s vanishes;  behaviour of hypergeometric Chapter II) of for  that  there  it  series  is  in  v o n l y when the  q u i t e c l e a r from the  ( E r d e l y i et a l . , 1953;  i s an i n f i n i t y of v a l u e s  of  v ,  ever i n c r e a s i n g magnitude, which makes the s e r i e s constant  k and x.  zero  The same degeneracy then e x i s t s  as  105 TABLE V . for  Computed e i g e n f r e q u e n c i e s  of Rossby waves  a symmetric p o l a r b a s i n with a f l a t  sphere,  bottom on a  a c c o r d i n g to L o n g u e t - H i g g i n s ' model.  n =1.  s = l  was found i n the p o l a r  co' , s,l  =  0.00344  2  0.00378  3  0.00369  5  0.00324  8  0.00268  12  0.00224  16  0.00179  plane.  There i s no a n a l y t i c a l formula a l l o w i n g of  the r o o t s ,  and I have estimated  computing the sum of the f i r s t i n c r e a s i n g v a l u e s of The  v  the f i r s t  of s i g n  c o s i n e of the angle c o r r e s p o n d i n g to  i s 0.975; the  root by  20 terms of the  u n t i l a change  calculation  series  occured.  (r-^/R)  = l/20  i n t e r v a l between s u c c e s s i v e v a l u e s of v  was taken as 0.5 except f o r the l a s t two e s t i m a t e s , i t was 1.0.  for  The value of v  where  corresponding t o the r o o t was m  then estimated  by l i n e a r i n t e r p o l a t i o n .  root was obtained t h i s way,  Only the  first  s i n c e only comparison i n one  d i r e c t i o n was deemed n e c e s s a r y ,  and because of  the  106  considerable time necessary f o r computing.  The c a l c u l a t i o n s  were done on the U n i v e r s i t y of B r i t i s h Columbia's IBM 1620„ and the program w r i t t e n f o r t h i s purpose i s given i n Appendix The r e s u l t s are l i s t e d i n Table V . of t a b l e s I I I and V shows that the  Comparison  eigenfrequencies c a l c u l a t e d on a polar plane model d i f f e r from Longuet-Higgins' two-dimensional approximation on the sphere by small but appreciable v a l u e s .  To see whether t h i s  discrepancy a r i s e s from the polar plane assumption or-from n e g l e c t i n g the i n f l u e n c e of surface displacements, we can apply t h i s l a s t approximation t o the polar plane, and see whether the r e s u l t s are c l o s e r t o the s t r i c t e r polar plane r e s u l t s or t o the r e s u l t s on the sphere.  In- the f i r s t case,  the discrepancy a r i s e s from the use of the polar plane, i n the second, from the neglect of surface displacements. Fpr a two-dimensional problem i n the angles 9 and X, we can w r i t e the v e l o c i t i e s i n terms of a stream function  ; i n s p h e r i c a l p o l a r coordinates, u  d y\f  =  Rsin9 =  -1  df  R  a 9  aX  where u and v are r a d i a l and zonal v e l o c i t i e s r e s p e c t i v e l y . E l i m i n a t i o n of the pressure g r a d i e n t s from the momentum equations ( 9 ) and ( 1 0 ) and of the v e l o c i t i e s through the  I.  107  continuity  equation  (6)  gives  a-vorticity  becomes  terms  the  above  defined  in  dVH a  which  is  the  working to  d±_  ©  ax  the  in  polar  the  of  plane  new  stream  =  which  function  0  ,  (117)  Longuet-Higgins. by  means  of  the  relations  (23),  geometry  a^L  +  a  equation  a</>2  2  _df  2  fi_ i ± _  at l r  1  -  " R sine  becomes  a  ft  t  Transforming (117)  of  equation,  i a_^_ ajrcosoil + 2x2coso a^_ = 0 .  +  d r  2  r  a  a  r  J  r  R  2A <  £ (1.18)  Let  us  look  for ft  since  V  th is  (3  0),  is the  by  replacing  of  (119)  Z  d  2  of  dx  +  =  ft  (r)  the  by  V  (118)  d  This  .  r  l  x  R  x  equation  w  t  form of  of  -  s  as  f  2  2  an  the  £  <  (119)  the  postulated  obtained  x  as  amplitude  ) -  fo(  s  +  2  .  resembles  form  )  are  Defining  gives  x d f o ( l +  2  1  same  f  waves e *  derivatives  into  %  v o r t i c i t y  in  from  (58),  s x  2  much  r  l  )  (3l)-(33)  substitution  equation  ^*R  very  elevation  for  =  ft : Q  0  (120)  2  (74); i n  fact,  2 when two  the  frequency  differ  only  in  is the  low  enough  d e f i n i t i o n  so of  that the  £x  <<1,  constant  the 3  0  108  Putting  now  *  2  1  = Hi cV 2 R  we  see t h a t i n t h i s case t h e r e i s no g r a v i t y i n f l u e n c e  (as represented by the M term i n ( 9 4 ) ) .  T h i s i s of  course  a d i r e c t consequence of n e g l e c t i n g the s u r f a c e e l e v a t i o n s : the problem i s t r e a t e d as two-dimensional, no departures  and  there a r e  from the e q u i l i b r i u m l e v e l on which g r a v i t y  can a c t . When  $^  i s n e g a t i v e , p l a n e t a r y wave s o l u t i o n s  f o r the amplitude  equation of the stream w i l l  analogous to those f o r the displacement + (x) o The  =  J  k  (V^il v  boundary c o n d i t i o n i s now v// (l) o  =  0 =  x)  Comparison of (122)  J (vf8",l k  of k and/or n, the two  )  (122)  g i v e n by the  formula  ( 3) 12  P  k,n  (99) shows t h a t f o r l a r g e values'  formulae w i l l  Some v a l u e s c a l c u l a t e d from (122)  give very s i m i l a r  are l i s t e d  where they a r e compared w i t h some of the d e r i v e d by other methods.  (121)  k  20  with  i n (94):  s l i g h t l y d i f f e r e n t , and i s  = s,n  amplitude  0  so t h a t the e i g e n f r e q u e n c i e s are now  co'  exist  i n Table  results. VI,  eigenfrequencies  109 TABLE V I . Comparison of eigenfrequencies of planetarywaves i n a symmetrical p o l a r ocean w i t h a f l a t bottom, as obtained from d i f f e r e n t methods.  P-P means p o l a r  plane approximation; 2-D i s Longuet-Higgins' twodimensional approximation. s =  T OJ  —  3,1  P-P  2-D  P-P and 2  Goldsbrough  -1  .00324  .00344  .00342  .00494  -2  .00366  .00378  .00380-  .00735  -3  .00359  .00369  .00369  -5  .00320  .00324  .00326  -8  .00261  .00268  .00268  -12  .00222  .00224  .00225  -16  .00179  .00179  .00179  Calculated from equation (99)  (114)  (122)  I t appears from Table VI that the eigenfrequencies OJ'  S  1  t e n d  t o  ^  e  s a r n e  values whatever the mode of  c a l c u l a t i o n , provided k i s large enough.  Furthermore, the  discrepancies between the s o l u t i o n s on the p o l a r plane and the approximate two-dimensional s o l u t i o n s on the sphere cannot be a t t r i b u t e d t o the imperfection of the mapping on the plane, because when the problem i s formulated as two-  110 d i m e n s i o n a l on the plane, almost i d e n t i c a l r e s u l t s are obtained as on the sphere. assumption  Since Longuet-Higgins*  t h a t the s u r f a c e displacements are of n e g l i g i b l e  i n f l u e n c e on the v o r t i c i t y balance i s known t o be v a l i d only f o r wave lengths a p p r e c i a b l y s m a l l e r than the r a d i u s of the E a r t h , the d i f f e r e n c e w i t h the r e s u l t s on the p o l a r plane would seemingly be caused by the inadequacy d i m e n s i o n a l assumption  of the  at s m a l l wave numbers.  The  twodifference  between r e s u l t s on the sphere and those on the p o l a r plane u s i n g Longuet-Higgins approximation i s not d e t e c t a b l e , so that one must conclude that the r e s u l t s of column 1 i n t a b l e VI are more p r e c i s e than those of the f o l l o w i n g two  columns.  When the diameter of the p o l a r ocean i s s m a l l enough t o drop (r,/R)  with r e s p e c t t o 1, the r e s u l t s provided by  the p o l a r plane approximation are as p r e c i s e as those provided b y Longuet-Higgins method on the sphere, and even more p r e c i s e at low wave numbers.  Another advantage i s t h a t the e i g e n -  f r e q u e n c i e s a r e much e a s i e r t o c a l c u l a t e on the p o l a r plane; f i n a l l y , my  f o r m u l a t i o n a l l o w s c o n s i d e r a t i o n of bathymetric  v a r i a t i o n s i n the model. restricted  T h i s a n a l y s i s i s of course  to p o l a r r e g i o n s , and does not have the g e n e r a l  a p p l i c a b i l i t y t o a l l l a t i t u d e s t h a t Longuet-Higgins' method possesses. Another b a s i s of comparison i s the work of Goldsbrough  on the dynamics of t i d e s i n p o l a r b a s i n s  (Goldsbrough, 1914 a ) ; h i s work i s done e n t i r e l y i n  I l l  s p h e r i c a l p o l a r c o o r d i n a t e s , and the two f r e q u e n c i e s which can be compared t o the r e s u l t s of the present work are presented  i n the f o u r t h column of Table V I .  They depart  c o n s i d e r a b l y from the c o r r e s p o n d i n g v a l u e s i n the o t h e r t h r e e columns.  The process o f c a l c u l a t i n g a n y t h i n g but a  f i r s t approximation  t o Goldsbrough's f r e q u e n c i e s i s q u i t e  i n v o l v e d , s i n c e the e i g e n f r e q u e n c i e s a r e t o be e v a l u a t e d from an i n f i n i t e determinant. been attempted, of i n t e r e s t .  A second  approximation has  but does not y i e l d v a l u e s o f  No apparent  co ' near  reason has been found f o r the  d i s c r e p a n c y i n the magnitude ,of t h e e i g e n v a l u e s . Longuet-Higgins'  those  Comparing  simple and c l e a r f o r m u l a t i o n w i t h  Goldsbrough's i n v o l v e d s e r i e s s o l u t i o n s and i n f i n i t e determinants  one i s tempted t o g i v e more f a i t h t o the  r e s u l t s of the former. In view of the good agreement of p o l a r plane with s p h e r i c a l geometry r e s u l t s as d e r i v e d from  results  Longuet-  H i g g i n s , and i n s p i t e of the not so good agreement w i t h Goldsbrough's v a l u e s , f o r which the b a s i s o f comparison i s narrower (two f r e q u e n c i e s ) , I then conclude t h a t the p o l a r plane w i l l be q u i t e u s e f u l i n s t u d y i n g the motions o f f l u i d s in r e s t r i c t e d  p o l a r b a s i n s , and g i v e q u a n t i t a t i v e l y p r e c i s e  results, The  p o l a r plane, d e f i n e d as the p r o j e c t i o n o f  F i g u r e 5 together w i t h the r e t e n t i o n o f only a f i r s t approximation used  t o t h e E a r t h ' s c u r v a t u r e , can t h e r e f o r e be  i n the A r c t i c r e g i o n s i n the same manner as the  112  ft -plane i s used i n m i d - l a t i t u d e s . T h i s s e c t i o n has  d e s c r i b e d the  o s c i l l a t i o n s of the s i m p l e s t a p a r e l l e l of l a t i t u d e and  characteristic  p o l a r b a s i n : bounded  without any  depth v a r i a t i o n s .  T h i s i s f a r from d e s c r i b i n g the a c t u a l A r c t i c and  i n the next s e c t i o n , an added degree of  w i l l be  introduced  I t may  complexity  i n the form of r a d i a l depth v a r i a t i o n s .  waves as d i s c o v e r e d  (1956)  bathymetry,  be asked whether such long p e r i o d  even i n a simple  along  above are of any  planetary  dynamic s i g n i f i c a n c e ,  symmetric b a s i n . V e r o n i s  and  Stommel  have shown ( i n the ft -plane f o r m u l a t i o n  ) that  f o r winds a c t i n g over a p e r i o d of more than h a l f a pendulum day  a s i g n i f i c a n t p o r t i o n of the t o t a l energy  i s t r a n s f e r r e d i n t o long period  semi-geostrophic  p l a n e t a r y waves. T h i s r e s u l t does not depend on p a r t i c u l a r p r o j e c t i o n used for  the p o l a r plane.  , and w i l l  hold  the  j u s t as w e l l  P l a n e t a r y waves can then be  generated by f l u c t u a t i n g winds over such a symmetrical b a s i n as s t u d i e d above.  113  VIII.  SYMMETRICAL OCEAN: RADIAL DEPTH .VARIATIONS. The  next step  i n the s c a l e of i n c r e a s i n g  i s the i n c l u s i o n of r a d i a l depth v a r i a t i o n s : dH'/a<£ = 0 . considered  complexities  dH'/dx?* 0 ;  The water content of t h e p o l a r b a s i n  i s now  v e r t i c a l l y homogeneous; the amplitude i s  determined by equation ( 5 9 ) and the boundary c o n d i t i o n ( 6 0 ) . We have seen i n s e c t i o n VI that  i t i s possible f o r solutions  t o e x i s t i n the presence of a wide v a r i e t y o f bottom configurations;  i t i s not easy however t o s o l v e  explicitly  the amplitude equation when H ( x ) i s s u b s t i t u t e d f  We w i l l t h e r e f o r e  in i t .  have t o be s a t i s f i e d w i t h the s i m p l e s t  bottom topography i n order t o o b t a i n  explicit  solutions.  T h i s w i l l s u f f i c e however t o show the nature of the e f f e c t s of the depth v a r i a t i o n s . The  f o l l o w i n g simple depth dependence  illustrates  very w e l l the i n f l u e n c e of bottom topography on p l a n e t a r y waves.  L e t us assume t h a t the depth v a r i e s v e r y l i t t l e  the e x t e n t of the b a s i n ,  so that  i t can be considered  over  constant  when not d i f f e r e n t i a t e d ; i t s r a d i a l dependence i s of the same form as t h a t  of the C o r i o l i s H»  i n which p/2 «  1.  =  parameter:  (1 + px /2 2  )  When ( 1 2 3 ) i s s u b s t i t u t e d  (123)  i n t o ( 5 9 ) , the  amplitude equation assumes t h e same form as when there a r e no depth v a r i a t i o n s ; only some of the c o n s t a n t s a r e changed:  114  x d_F dx  xdF [ l + U + p J x ] 2  +  2  d  F  (8  [s + 2  +JDS)X ]= 2  can be s o l v e d  i n terms o f c o n f l u e n t  geometric f u n c t i o n s , as ( 7 4 ) was, but t h i s when p l a n e t a r y waves a r e c o n s i d e r e d , 2 possible to neglect the reduced X V F  dx  hyper-  i s not necessary  s i n c e i t i s then  p)x.. with r e s p e c t t o 1 and use  (e +  equation - F [ S + (8+JDS)X ]= 0 ,  + xdF 2  d  2  (125)  2  x  which i s i d e n t i c a l i n form with similar  (124)  *>'  x  T h i s equation  0 .  solutions.  ( 9 0 ) , and has t h e r e f o r e  Examining the p l a n e t a r y wave s o l u t i o n s ,  which, by analogy to ( 9 0 ) , occur where  8 +  p s / cu ' < 0 ,  the amplitude i s then  FU)  =  J (V[8+£s| x)  Expanding t h e constant frequency,  because of the low f r e q u e n c i e s  cu'  ~  order terms i n cu  1  of p l a n e t a r y waves, one has  2 (p + ^ l ) s + 2 cu' R  (126)  8 + p s / cu' i n terms o f  and keeping only t h e f i r s t  S + p_s  .  k  2 M ^ l 2 R  <  0 .  (127)  115 In equation (127) one can see the r o l e played by a bottom c o n f i g u r a t i o n of the form (123): i f p i s p o s i t i v e ,  so t h a t  th.e depth i n c r e a s e s towards the b o u n d a r i e s , then s must be negative and propagation t o the west,  i n o r d e r t o keep (127)  negative.  I f p is negative,  radially,  and l a r g e enough t o make (p + r-^/^2) negative  also,  so t h a t the depth  decreases  then the wave number s w i l l have to be p o s i t i v e  (127) to h o l d ( f o r low f r e q u e n c i e s , part of (127) w i l l dominate). the e a s t .  This is  for  the frequency dependent  Propagation i s t h e r e f o r e  e x a c t l y what the method of  to  signatures  p r e d i c t e d i n s e c t i o n V I : p l a n e t a r y waves p r o p a g a t i n g towards dH  1  the east can e x i s t i f  is  to a negative  observes  p.  One a l s o  l a r g e and n e g a t i v e , i n (127) t h a t  v a r i a t i o n of C o r i o l i s parameter (the r^/^2 term)  corresponding the produces  asymmetries between waves c o r r e s p o n d i n g to equal but  opposite  depth g r a d i e n t s . The boundary c o n d i t i o n  is  (128)  and,  as i n the f l a t bottom c a s e ,  closely left  the eigenfrequency  is  approximated by the r o o t of the denominator of the  hand s i d e .  T h i s g i v e s f o r the  frequencies  116  p +  s(  cu  2  2  s ,n /? k,n 2  i n which  r /R )  ft  i s d e f i n e d as  +  M  r  l  /  R  '  2  i n (99).  (  A few  1  2  9  )  eigenfrequencies  are t a b u l a t e d i n t a b l e V I I . The s o l u t i o n s  c o r r e s p o n d i n g to  ( 8 + ps/  cu ) 1  >  have not been i n v e s t i g a t e d , ,but they w i l l be an e x t e n s i o n the r e s u l t s  for a f l a t  bottom when  8 >0  it  illustrates  of  (111).  Although the bathymetry adopted i n the example i s very s i m p l e ,  0  above  c l e a r l y the  influence  of bottom v a r i a t i o n s on the p r o p e r t i e s of p l a n e t a r y waves. For more complicated t o p o g r a p h i e s ,  i t may not be  possible  to f i n d an e x p l i c i t a n a l y t i c s o l u t i o n of the amplitude equation; H'(x),  it  may however be i n t e g r a t e d n u m e r i c a l l y , given  the frequency being a d j u s t e d u n t i l a value  satisfying  the boundary c o n d i t i o n i s f o u n d . Some g e n e r a l theorems c o n c e r n i n g the motion of shallow r o t a t i n g l i q u i d s on a p a r a b o l o i d have been demonstrated by B a l l  i n a recent a r t i c l e  (Ball,  1963);  a s p e c i a l case has a l s o been t r e a t e d by M i l e s and B a l l  (1963).  Ro these theorems apply t o an A r c t i c b a s i n i n the  p o l a r p r o j e c t i o n used i n t h i s  study?  B a l l ' s r e p r e s e n t a t i o n of the problem i s different  from that used up t o now; the reference  slightly level  117  TABLE V I I . for H  1  Some e i g e n f r e q u e n c i e s  a symmetrical ocean with a r a d i a l bottom  1 + px /2.  =  2  a;'  c a l c u l a t e d from s,  p  p  0.1  =  k =  of  CO  slope.  (129).  1,  s <0  for  of p l a n e t a r y waves  = - 0.1 3  >0  1  s, n 0.00954  0.00313  2  0.0110  0.00366  5  0.00963  0.00321  the v e r t i c a l coordinate i s taken at the maximum depth the b a s i n , the  elevation  of the bottom above that  b e i n g Z, and t h e s u r f a c e displacement and the  location  local  e q u i l i b r i u m depth being grouped under the same v a r i a b l e h . This i s  i l l u s t r a t e d in Figure The  first  displacement independent  23.  theorem proven by B a l l  of the centre  occurs w i t h i n the  of g r a v i t y .  His b a s i c  is  liquid  equations,  with c o o r d i n a t e s x,  y and v e l o c i t i e s u , v to the  north r e s p e c t i v e l y ,  are then  Du + g _d__(h + Z)  the  of g r a v i t y of the l i q u i d  of the motion t h a t  r e l a t i v e t o the c e n t r e  i s that  east and  = fv  (130)  = -fu  (131)  Dt  Dv + Dt  g  ^  _d_(h + Z)  ay  Figure 23.  B a l l ' s d e f i n i t i o n of v e r t i c a l dimensions.  Dh+h(ce_u c3v) Dt dx dj +  =  0.  (132  In the above, f i s the C o r i o l i s parameter, 2£l cos9; the D /Dt are t o t a l time d e r i v a t i v e s , the non-linear terms, being included.  The proof of the theorem involves  m u l t i p l y i n g the equations of motion by the depth and i n t e g r a t i n g t o obtain expressions i n v o l v i n g the coordinates  119  of the  centre  of g r a v i t y ; performing these o p e r a t i o n s  the C o r i o l i s a c c e l e r a t i o n terms,  on  one has (133)  the i n t e g r a l b e i n g over an area e n c l o s i n g a given amount the boundaries moving w i t h the l i q u i d t  of l i q u i d ,  C o r i o l i s parameter i s as f = f  Q  4- f^,  f  Q  not a c o n s t a n t ,  being a constant,  If  the  but can be w r i t t e n but not f^,  the  integral  ( 1 3 3 ) becomes ( u s i n g B a l l ' s n o t a t i o n , w i t h Q =  constant  volume of f l u i d ,  of  X , Y , the c o o r d i n a t e s  of t h e  total  centre  gravity) (134) dt  Ball is  i m p l i c i t l y assumes t h a t e i t h e r the C o r i o l i s parameter  constant,  so that value,  or t h a t the area of the  basin is  s m a l l enough  i t does not depart v e r y much from some average the second term of  ( 1 3 4 ) being t h e r e f o r e  Since the v a r i a t i o n of C o r i o l i s importance i n p l a n e t a r y waves,  parameter i s  of  negligible. primordial  i t would seem t h a t the  theorem of B a l l would not apply t o them.  first  We have seen  however t h a t as f a r as p l a n e t a r y waives are concerned a change i n depth has an e f f e c t (see  s i m i l a r to a change i n f  the p o t e n t i a l v o r t i c i t y e q u a t i o n ,  parameter can t h e r e f o r e  (54)).  The C o r i o l i s  be c o n s i d e r e d c o n s t a n t ,  its  v a r i a b l e p a r t being r e p r e s e n t e d by a depth v a r i a t i o n . For example,  i f the C o r i o l i s  parameter were c o n s t a n t ,  but  a depth v a r i a t i o n of the  H« =  ( 1 +  form ( 1 2 3 ) e x i s t e d , 2 r / R 2  2  ? x ),  the f r e e  eigensolutions  the f l a t  bottom case i n s e c t i o n V I I .  therefore applies  liquid is  r  f = constant,  would be e x a c t l y those found f o r Ball's first  i n cases where f v a r i e s ,  1) the displacement  with p = ] / g 2 ,  of the c e n t r e  so  theorem  that'  of g r a v i t y of the  independent of the motion that occurs w i t h i n the  l i q u i d r e l a t i v e to the c e n t r e 2) the equations r e l a t i v e to i t s  of g r a v i t y , and  governing the motion of the  centre  liquid  of g r a v i t y have e x a c t l y the same  form as the o r i g i n a l equations  of motion.  The other fundamental theorem demonstrated by B a l l concerns the v a r i a t i o n s of the angular momentum; h i s equations  are now put i n p o l a r form:  Du + g _a_(h + Z) = ( f + / r ) v v  Dt  (135)  dr m  Dv Dt  +  £_ J_th + Z) r d4>  =  -(f + v/r)u  (136)  121  When ( 2 4 ) , ( 2 5 ) and ( 2 9 ) ( i n which the i n d i c e s a r e dropped) are  formulated i n the geometry  of F i g u r e 2 3,  the r e s u l t i n g  equations are i d e n t i c a l t o t h e l i n e a r i z e d forms of the above except f o r a cos9 m u l t i p l i e r attached t o the second equation ( 1 3 5 ) .  term o f the f i r s t  The cos9 which appears  i n the c o n t i n u i t y e q u a t i o n can be n e g l e c t e d i n r/R) s i n c e i t i s not d i f f e r e n t i a t e d .  ( t o f i r s t order  How w i l l  this  s l i g h t d i f f e r e n c e a f f e c t the r e s u l t ? The t o t a l energy o f the l i q u i d  and i t s t o t a l  a b s o l u t e angular momentum about t h e p o l a r a x i s a r e c o n s t a n t s , as i n B a l l ' s work.  With t h e moment o f i n e r t i a , I, about  a v e r t i c a l a x i s through the o r i g i n d e f i n e d as i n B a l l : I = the  Jh  r  2  dS  (138)  ,  theorem s t a t i n g t h a t i f I and i t s time d e r i v a t i v e a r e  initially  known, they a r e determined u n i q u e l y a t a l l times  thereafter s t i l l  holds.  The ensuing d i s t e n s i o n t h e o r y i s  v a l i d , and i t i s t h e r e f o r e p o s s i b l e t o s e p a r a t e the motion of  the l i q u i d  i n two p a r t s :  first  a distension,  defined  as an i s o t r o p i c two-dimensional d i l a t a t i o n and r o t a t i o n (Figure 2 4 ) , and second, motions superimposed  on t h e  distension. To quote B a l l ,  "the main e f f e c t s o f t h e d i s t e n s i o n  on these (superimposed) motions .are, f i r s t , or  speeding up of every aspect o f the motion  a slowing down (whether  v o r t i c e s or g r a v i t y waves) a c c o r d i n g as the l i q u i d as a  122  Extreme  Figure 2 4 . whole i s  Dilatation.  A 'distension'  ( B a l l (1963)).  s t r e t c h e d or c o n t r a c t e d , and s e c o n d l y ,  stabilization,....".  T h i s a l s o a p p l i e s t o the  a general motions  s t u d i e d i n a p a r a b o l o i d a l p o l a r b a s i n : i f p l a n e t a r y waves coexist with a d i s t e n s i o n ,  energy w i l l be interchanged  between the two modes of motion at a r a t e depending on the r a t e of d i l a t a t i o n a s s o c i a t e d with the When the f l u i d  contracts  distension.  (water p i l e s up at the  energy w i l l be e x t r a c t e d by the waves from the and the f r e q u e n c i e s w i l l  pole), distension  i n c r e a s e ; when water i s h i g h at  the edges, the f r e q u e n c i e s w i l l  decrease.  123  I t can be shown very simply t h a t the concerning the i n f l u e n c e oscillations  (I963),  so t h a t the  of  studied  w i l l be a p p l i c a b l e only when the  of v a r i a t i o n s of the C o r i o l i s parameter  negligible,  case  of r o t a t i o n , o n the f r e q u e n c i e s  i n a shallow r o t a t i n g p a r a b o l o i d , as  by M i l e s and B a l l influence  special  results  is  t h e r e i n cannot be a p p l i e d  to the study of p l a n e t a r y waves i n the  corresponding  geometry. The and  b a s i n c o n s i d e r e d i s a p a r a b o l o i d of r e v o l u t i o n ,  the depth i s given by  H' =  (1 -x)  .  2  (139)  E l i m i n a t i o n of a l l v a r i a b l e s except s u r f a c e amplitude i n (Lamb, 1 9 3 2 ; p 209)  the s h a l l o w water l i n e a r i z e d equations gives,  i n the n o t a t i o n of ( 5 8 )  x ( l - x )ofF dx 2  2  2  xdF ( 1 - 3 x ) dx 2  +  2  - F [ s ( l - x ) + ( - 2 s MJjJx ] = 0 2  2  2  +  «'  R2  (HO)  1  The series,  solution is  found i n term of a hypergeometric  and the c o n d i t i o n t h a t t h i s  boundaries y i e l d s  s e r i e s converge at  a frequency c o n d i t i o n from which M i l e s  and  B a l l deduce t h a t the f r e q u e n c i e s  for  a z i m u t h a l wave numbers s = 0 and s = 1 are  of  the  of t h e dominant modes independent  the frequency of r o t a t i o n f o r an observer i n a non-  r o t a t i n g frame of r e f e r e n c e ,  and t h a t  the f r e q u e n c i e s  of  124  all  other axisymtnetric modes a r e decreased by r o t a t i o n . To see i f these p r o p e r t i e s are a l s o a p p l i c a b l e  t o a b a s i n i n which the r o t a t i o n v a r i e s with p o s i t i o n , l e t us s u b s t i t u t e the depth, as given by (139) i n t o the amplitude equation (59):  x (l  - x )dfp  2  xdF [ e ( x + l ) ( l - x ) - 2 x ]  2  dx  2  +  2  d  2  2  x  - F|(,S 4-SX € )(1 - x ) +[(1- O/ )M_HL - 2 s ] x i = 2  I  2  w  2  2  »  0  2  R  2  w  »  (141)  J  2 Assuming that t h e f r e q u e n c i e s a r e such t h a t neglected  with r e s p e c t t o 1, and t h a t  ex  can be  < u ' « 1 (141)  becomes x (l 2  - X )£F 2  d x 2  (s+ 2  xdF (1 - 3 x ) 2  +  d  sA)  (142)  x  (1 - x ) + [ j M j f - 2 s ] i 0 2  x  T h i s w i l l apply t o the study periods;  o f p l a n e t a r y motions o f long  (142) would be i d e n t i c a l t o (140) were i t not f o r  *  2  the presence of a term i n sx coefficient  of F i n (142).  € /  i n the f i r s t  F o r t h e low f r e q u e n c i e s  encountered i n p l a n e t a r y waves, t h i s term i s comparable  125  to s  and i s not n e g l i g i b l e .  When the e f f e c t s of t e r r e s t r i a l  c u r v a t u r e ( i . e . , v a r i a t i o n o f t h e C o r i o l i s parameter) 'cannot be n e g l e c t e d , t h e r e i s a s i g n i f i c a n t d i f f e r e n c e between motions i n the p o l a r plane and those i n M i l e s model.  and B a l l ' s  Equation (142) does not, " l i k e (140) have a s o l u t i o n  i n terms o f a known s e r i e s , and the c o n d i t i o n  for finiteness  of the s o l u t i o n s w i l l i n g e n e r a l be d i f f e r e n t . The paraboloidal not  A r c t i c b a s i n does not of course have a s i m p l e ' bottom topography: the depth v a r i a t i o n s a r e  even symmetrical around the pole,  so t h a t  results  a p p l i c a b l e t o a f l a t bottom ocean or to an ocean w i t h symmetrical bottom slopes  are only  of academic i n t e r e s t as  f a r as the a c t u a l A r c t i c ocean i s concerned. contorted  But s i n c e the.  geometry o f t h e A r c t i c does not y i e l d e a s i l y t o  a n a l y s i s , i t i s necessary t o understand the s i t u a t i o n i n simplified  s i t u a t i o n s before even l o o k i n g at the more  complex-cases.  I t might a l s o be p o s s i b l e t o deduce some  qualitative properties  o f the s o l u t i o n s  i n the more complex  s i t u a t i o n s through knowledge of the p h y s i c s topographies  i n the simple  studied.  We have v e r i f i e d solutions predicted  i n t h i s s e c t i o n that t h e e i g e n -  by t h e Method of S i g n a t u r e s ( s e c t i o n VI)  i n the presence of s l o p i n g bottoms indeed e x i s t and t h a t the  influence  the  eigensolutions  section VI.  of the bottom s l o p e s  on the f r e q u e n c i e s  of  i s as expected from the a n a l y s i s o f  I f the e f f e c t o f the depth v a r i a t i o n s i n the  126 potential vorticity balance is' in the same direction as that of the Coriolis parameter (dH'/dx > 0), the frequency „,' is increased over that of the flat bottom solutions, s,n If the influence of the depth is in the opposite direction w  (dH'/dx.< 0), the frequency is decreased until i t becomes • 2 zero for dH'/dx — -]/g2x..... For steeper negative depth r  gradients the direction of propagation is reversed and planetary waves with positive wave number (s > 0) cart exist. The applicability of some recent theorems of Ball (1963) has been investigated; they have been found to apply quite generally, so that Ball's separation of the motions of the fluid in a shallow rotating paraboloid into three parts applies to the case of such basins in the polar plane. These three parts are as follows: 1) The motion of the centre of gravity, which is entirely independent of the motions relative to i t . 2) An isotropic two-dimensional dilatation and rotation, which Ball calls a 'distension' (Figure,24). 3) The motions that remain after the removal of the velocity fields associated with the preceding motions. These theorems apply only to paraboloidal basins. Concerning the effects of radial depth variations we can then make the following conclusions.  As seen in the  potential vorticity equation (54)., depth variations have , very much the same influence as variations in the Coriolis  127  parameter.  Using t h e Method of Signatures  (section VI),  i t can be determined whether e i g e n s o l u t i o n s w i l l e x i s t or not f o r any g i v e n r a d i a l depth v a r i a t i o n H * ( x ) . Namely, s o l u t i o n s w i l l e x i s t when H'(x), through i t s i n f l u e n c e on the s i g n a t u r e s , allows the phase path t o terminate o r i g i n o f the phase diagram.  When H'(x)  i s given  a t the explicitly,  the s i g n a t u r e s can be found f o r a l l v a l u e s of £ ( = l / x ) , and it  so can the p o s i t i o n of the phase path.  In p r i n c i p l e ,  i s then p o s s i b l e t o draw c o n c l u s i o n s on the e i g e n s o l u t i o n s  when t h e r a d i a l depth v a r i a t i o n i s known. The equations simple  actual e x p l i c i t  s o l u t i o n of the amplitude  i s p o s s i b l e only i n very simple  case (H' =  cases;  one such  (1 + px / 2 ) ) has been examined above.  In the s p e c i a l case o f p a r a b o l o i d a l b a s i n s , some g e n e r a l theorems due t o B a l l , and s t a t e d above, a p p l y t o the motions of the contained The considered  f l u i d , provided  i t i s shallow.  case o f symmetric depth v a r i a t i o n s can then be  t o be r e s o l v e d i n p r i n c i p l e , s i n c e even though  i t may not be p o s s i b l e t o s o l v e the amplitude e x p l i c i t l y , the e x i s t e n c e  equation  ( o r non-existence) o f s o l u t i o n s  can be a s c e r t a i n e d , and t h e amplitude equation n u m e r i c a l l y t o f i n d the e i g e n f r e q u e n c i e s .  solved  128  IX.  ASYMMETRICAL TOPOGRAPHY. The problem becomes enormously more complex when  asymmetries a r e allowed, e i t h e r i n the bathymetry o r i n the boundaries.  A glance at F i g u r e 1 shows t h a t the  asymmetries a r e very important and w i l l p l a y a dominant r o l e i n the dynamics The simple  i n a l l probability  of the A r c t i c  ocean.  s o l u t i o n s of s e c t i o n s V I I and V I I I w i l l then not  be d i r e c t l y a p p l i c a b l e t o the a c t u a l A r c t i c ocean, and the more complete equation (48) must be s o l v e d .  U s i n g some o f  the a b b r e v i a t i o n s d e f i n e d i n (58), (48) becomes  dtf> \j dtp  +  H  d<f>  HOJ  a  T  0  r  (143)  129  The boundary c o n d i t i o n i s s t i l l as given by ( 5 0 ) . I t i s however d o u b t f u l whether the assumed form f o r the surface displacements, (30),' which i s appropriate t o the d e s c r i p t i o n of waves i n a c y l i n d r i c a l b a s i n , w i l l be of any u t i l i t y when important departures from c y l i n d r i c a l symmetry are present.  I t might be necessary i n that case t o reformulate  the problem i n a coordinate system more appropriate t o the new symmetry, and i n which i t might be p o s s i b l e t o separate the amplitude equation. Asymmetries i n the boundaries w i l l then not be included, i n which case (50) reduces t o  (i  cu ) d F  r = r1  -  + 1_ JJL d . r 4> x  i s F( ) r i  r  r  = x r  =  0  (144)  I  No s o l u t i o n s of (143) have been found, even i n very s p e c i a l cases; i t i s not even known whether the system (143)-(144) has any s o l u t i o n s at a l l . In. view of t h i s u n c e r t a i n t y and of the n o n - l i n e a r i t y of the p a r t i a l d i f f e r e n t i a l equation (143), i t i s d o u b t f u l whether one should pursue t h i s l i n e of attack any f u r t h e r .  The problem  might be more t r a c t a b l e with a l e s s i n c l u s i v e f o r m u l a t i o n , •but on the other hand, i t might be necessary t o study the non-symmetrical s i t u a t i o n w i t h more q u a l i t a t i v e  arguments.  130  X.  CONCLUSIONS. The problem  o f the dynamics of the A r c t i c  ocean  has been formulated i n a geometry a p p r o p r i a t e t o t h e p o l a r r e g i o n s by t r a n s f o r m i n g the equations of motion and of c o n t i n u i t y from the sphere t o a p o l a r plane.  Although  this  mapping a r i s e s q u i t e n a t u r a l l y i n the study of g e o p h y s i c a l phenomena, i t seems t h a t nobody had taken advantage  of i t  in that respect previously. The main advantage  of t h e mapping i s a c o n s i d e r a b l e  r e d u c t i o n i n mathematical complexity; f o r a r e s t r i c t e d  polar  cap, one can r e t a i n only a f i r s t approximation t o the t e r r e s t r i a l c u r v a t u r e , thus a n a l y s i n g the problem modified b e t a - p l a n e .  in a  A p p l y i n g the transformed equations  to t h e simple case o f a symmetrical ocean w i t h a f l a t bottom, one f i n d s t h a t the p l a n e t a r y wave e i g e n s o l u t i o n s compare r e a s o n a b l y w e l l i n . f r e q u e n c y and appearance  v*ith  t h e i r e q u i v a l e n t s i n a s i m i l a r b a s i n on the sphere, as d e r i v e d by Goldsbrough  and Longuet-Higgins.  The s o l u t i o n s  i n the p o l a r p r o j e c t i o n are furthermore e a s i e r t o r e p r e s e n t and t h e i r f r e q u e n c i e s c a l c u l a t e d with a minimum o f l a b o u r . Having found t h a t the s o l u t i o n s f o r p l a n e t a r y waves were v e r y s i m i l a r i n t h e p o l a r plane and i n the s p h e r i c a l geometry, we can conclude t h a t the a n a l y s i s i n t h e approximate p o l a r plane w i l l y i e l d almost u n d i s t o r t e d r e s u l t s f o r a l l  131  p o s s i b l e modes of motion o f a s m a l l A r c t i c a c t u a l one'is s m a l l enough).  ocean (the  T h i s i s so because the  p l a n e t a r y waves, depending i n t h e i r e x i s t e n c e and p r o p e r t i e s on the curvature  o f the E a r t h , are most l i k e l y t o be  d i s t o r t e d by any d e p a r t u r e s p o l a r plane  from the s t r i c t  p r o j e c t i o n can t h e r e f o r e be used as r e l i a b l y  as the o r d i n a r y beta-plane  used i n m i d - l a t i t u d e s .  The r e s u l t s concerning flat  g e o i d . . The  the symmetrical, ocean w i t h  bottom are not new; t h i s work goes" beyond t h a t of  Goldsbrough and Longuet-Higgins i n f o r m u l a t i n g the problem to The  i n c l u d e v a r i a b l e bathymetry and asymmetrical symmetrical  ocean i s o f course  boundaries.  much s i m p l e r t o d i s c u s s ,  the amplitude then being determined by an o r d i n a r y d i f f e r e n t i a l equation. some g e n e r a l c r i t e r i a  Using t h e Method o f S i g n a t u r e s , can be e s t a b l i s h e d concerning the  e x i s t e n c e and p r o p e r t i e s of the e i g e n s o l u t i o n s of a symmetrical  basin.  For a given bottom c o n f i g u r a t i o n ,  H ' ( r ) , one can determine whether e i g e n s o l u t i o n s w i l l be found o r not;  i n p a r t i c u l a r , , i t i s found t h a t no p l a n e t a r y  waves can propagate towards t h e east when t h e depth o f the symmetric ocean i s c o n s t a n t . The asymmetrical,  r e a l A r c t i c ocean i s o f course  grossly  and i f motions dependent on t h e topography  are i n v e s t i g a t e d , i t can c e r t a i n l y not be approximated by a c y l i n d r i c a l l y symmetrical  basin.  No c o n c l u s i o n s have been  reached i n t h i s more g e n e r a l s i t u a t i o n , mostly because o f  132  the i n c r e a s e d a n a l y t i c a l complexity: the amplitude d i f f e r e n t i a l equation i s now  p a r t i a l and n o n - l i n e a r .  It  might be necessary t o t r e a t the more g e n e r a l case by more q u a l i t a t i v e methods s i n c e the mathematical  difficulties  have not sb f a r been surmounted. Needless t o say, much work remains t o be done b e f o r e the dynamic oceanography i s understood i n d e t a i l .  of the a c t u a l A r c t i c  ocean  I hope then t h a t t h i s work w i l l  serve as a b a s i s as w e l l as a stimulus f o r f u r t h e r developments  and t h a t the p o l a r plane approximation here  i n t r o d u c e d w i l l a l s o prove u s e f u l i n f u r t h e r r e s e a r c h .  133  APPENDIX I .  F0RTRAN in ' v  language program t o f i n d the f i r s t r o o t  of the hypergeometric s e r i e s  J  (1+ 2? 1 The s e r i e s i s  k + v ,k  summed ( f i r s t  l a r g e r v a l u e s of  - v k +1 ;  ;i-  £x )  .  20 terms) f o r l a r g e r and  u n t i l the s i g n of the sum changes;  v  the same procedure i s then repeated f o r another v a l u e of k.  The n o t a t i o n i s not the same i n the program below as'  i n the above s e r i e s : v i s c a l l e d RNU  C  1+ k = C, k - v  = B, and 14- k + v = A;  The program below i s the one used f o r  f i n d i n g the f i r s t root when k = 1,2,3,5 and &; s l i g h t m o d i f i c a t i o n s are i n t r o d u c e d f o r f i n d i n g the r o o t when k = 12 and 16.  $F0RTRAN RNU =101 90  READ 100,C  100  F0RMAT(  F12.0)  D = C - 1. PRINT 110,D  110.  F0RMAT(1OX,5HM = , F4.0) PRINT 120  120  F0RMAT(5X,2HNU,7X,6HSUMSER) X = 0.0125  190  A = C + RNU  134 B = D - RNU DIMENSION Y(20) Y(l) = A*B*X/C SUMSER = 1. + Y ( l )  D0 200 I = 2,20 P = I Y(I) = 200  Y(I-1)*((A P-1.)*(B P-1.)*X)/((P D)*P)  SUMSER = SUMSER + Y(I) PRINT 210,RNU,SUMSER  210  F0RMAT(3X,*F5.1,4X,F1O.5) IF(SUMSER) 230,220,220  220  RNU = RNU + 0.5  G0 T0 190 230  G0 T0 90 END  $DATA •  2. 3. 4. 6. 9. A f t e r c a l c u l a t i n g the f i r s t root f o r a value of  k, RNU i s not brought back t o 10; the l a s t value of RNU used i s the f i r s t one used i n c a l c u l a t i n g the s e r i e s f o r the next value of k.  This i s possible because the value  of the root increases monotonically with k. When the f i r s t root i s c a l c u l a t e d f o r k = 12 and 16, the program i s s t a r t e d w i t h RNU = 60.0 t o save time. A l s o , statement 220  i s changed t o RNU = RNU + 1.0 f o r the same reason;  the  d e f i n i t i o n i s decreased by i n c r e a s i n g t h e width o f the interval,  but l i t t l e percent p r e c i s i o n  i s much l a r g e r f o r those v a l u e s of k.  i s l o s t since  RNU  136  BIBLIOGRAPHY B a l l , F . K . , 1 9 6 3 . Some general theorems concerning the f i n i t e motion of a shallow r o t a t i n g l i a u i d l y i n g on a p a r a b o l o i d . J o u r . F l u i d Mech., 17, (2) : 240-256. Coachman,L.K., 1962. On the water masses of the A r c t i c ocean. Doctoral d i s s e r t a t i o n , Dept. of Oceanography, Univ. of Washington, r e f . no M62-11. E c k a r t , C , I960. Hydrodynamics of oceans and atmospheres. Pergamon Press, New York. 290 pp. E r d e l y i . A . , M.Magnus, F. Oberhettinger and F. T r i c o m i , 1 9 5 3 . Higher transcendental f u n c t i o n s , v o l . I . McGraw-Hill, New York. . Ewing,M. and A.P.Crary, 1 9 3 4 . Propagation of e l a s t i c waves i n i c e , 2 . Physics, 5 : 181-184. Farmer,H.G., 1 9 6 4 . B a r o c l i n i c o s c i l l a t i o n s i n a p o l a r sea. Oral communication to the 1 3 annual meeting of the P a c i f i c Northwest Oceanographers, S e a t t l e , 7 - 8 Feb. 1964. t n  Goldsbrough,G.R., 1914 a ) . The dynamical theory of t i d e s i n a r>olar ocean. Proc. Lon. Math. S o c , Ser.2, 14 : 31-66'  ~  Goldsbrough,G.R., 1914 b ) . The dynamical theory of t i d e s i n a zonal ocean. Proc. Lon. Math. Soc., Ser. 2, 14 : 207-229. Goldsbrough,G.R., 1931. The t i d a l o s c i l l a t i o n s i n rectangular basins. Proc. Roy. S o c , A132 : 689-701. Goldsbrough,G.R., 1933. The t i d e s i n oceans on a r o t a t i n g globe. P r o c Roy. S o c , A140 : 2 4 1 - 2 5 3 . Gordienko,P.A., 1961. The A r c t i c ocean. S c i e n t i f i c American, 204 , ( 5 ) " : 88-102. Hough,S.S., 1898. On the a p p l i c a t i o n of harmonic a n a l y s i s to the dynamical theory of t i d e s . Part I I : On the general "integration of Laplace's dynamical equations. P h i l . Trans. A191 : 139-186. Hunkins,K., 1962. Waves on the A r c t i c ocean. Jour. Geophys. Res.,  67,  (6)  :  2477-2489.  137  Jahnke,E. and F. Emde, 1945. Tables of f u n c t i o n s . Dover, New York. Lamb Jl.,,1932. Hydrodynamics. Dover, New Laplace,P.S., 1799. T r a i t e de mecanique Paris.  York.  738  pp.  c e l e s t e . Crapelet,  Longuet-Higgins,M.S., 1964 a ) . On group v e l o c i t y and energy f l u x i n p l a n e t a r y wave motions. Deep Sea Res., 11, ( 1 ) :  35-43.  '  ~~  Longuet-Higgins,M.S., 1964 b). P l a n e t a r y waves on a r o t a t i n g sphere. Proc. Roy. Soc. A ( i n p r e s s ) . Love,A.E.H., 1913. Notes on the dynamical theory of t i d e s . Proc. Lon. Math. S o c , Ser. 2, 12 : 309-314. Miles,J.W. and F.K. B a l l , 1963. On f r e e s u r f a c e o s c i l l a t i o n s i n a r o t a t i n g p a r a b o l o i d . Jour. F l u i d Mech., 17, (2) :  257-267.  Moore,D.W., 1963. Rossby waves i n ocean c i r c u l a t i o n . Deep  Sea Res., 10,. (6)  : 735-749.  Ostenso,N.A., 1962. Geomagnetism and g r a v i t y i n the A r c t i c , b a s i n . P r o c e e d i n g s o f t h e A r c t i c b a s i n symposium, Oct. 1962, A r c t i c I n s t i t u t e of North America, pp 9-40. Tide water p u b l i s h i n g co. ;  Poincare',H., 1910. Le§ons de mecanique c e l e s t e . G a u t h i e r V i l l a r s , P a r i s . (Tome I I I , T h e o r i e des marees). Proudman,J., 1916. On the dynamical equation of the t i d e s . Part I I I : Oceans on a sphere. Proc. Lon. Math. S o c ,  Ser. 2, 18 : 51-68.  Proudman,J., 1953. Dynamical Oceanography. London; Wiley, New York.  Methuen and  Co.,  Rattray,M. J r . , I 9 6 4 . Time dependent motion i n an ocean; a u n i f i e d t w o - l a y e r beta plane approximation. Hidaka Anniversary Volume. (In p r e s s ) . Robinson,A.R., 1964. C o n t i n e n t a l s h e l f waves and the response of sea l e v e l t o weather systems. Jour. Geophys. Res.,  69_, (2) : 367-368..  Rossby,C.G., 1939. R e l a t i o n s between v a r i a t i o n s i n the i n t e n s i t y of the z o n a l c i r c u l a t i o n o f the atmosphere and the displacements of the semi-permanent c e n t r e s of a c t i o n . Jour. Marine Res., 2 : 38-55.  '  138  Rossby,C.G., 1949. On the d i s p e r s i o n of p l a n e t a r y waves i n a b a r o t r o p i c atmosphere. T e l l u s , 1 : 54-5$. Rossby,C.G., 1959. Currents problems i n meteorology. The atmosphere and the sea i n motion: s c i e n t i f i c c o n t r i b u t i o n s t o the Rossby memorial volume. Bert B o l i n e d i t o r . N.Y. R o c k e f e l l e r I n s t i t u t e Press and Oxford Univ. P r e s s .  S l a t e r , L . J . , I960. The c o n f l u e n t hypergeometric Cambridge Univ. P r e s s . 243  PP«  function.  Stommel, H., 1948. The westward i n t e n s i f i c a t i o n of wind d r i v e n ocean c u r r e n t s . Trans. Amer. Geophys. Union,  29 : 202-206.  Stommel,H., 1957.  A survey of ocean c u r r e n t t h e o r y . DeeD  Sea Res., 4 : 149-184.  Stommel,H., I960. The G u l f Stream. Univ. of C a l i f o r n i a Press, Berkeley and Los Angeles; Cambridge U n i v e r s i t y Press, London. Veronis,G., 1956. P a r t i t i o n of energy between g e o s t r o p h i c and non-geostrophic motions. Deep Sea Res., 3. : 157-177. Veronis,G,, 1963. On the approximations i n v o l v e d i n t r a n s f o r m i n g the equations of motion from a s p h e r i c a l s u r f a c e t o the b e t a - p l a n e . I . B a r o t r o p i c systems. Jour. Marine Res., 21, (2) : 110-125. Veronis,G. and H.Stommel, 1956. The a c t i o n of a v a r i a b l e wind s t r e s s on a s t r a t i f i e d ocean. Jour. Marine Res.,  15 : 43-75.  Warren,B.A., 1963. Tomographic i n f l u e n c e s on the path of the G u l f Stream. T e l l u s , 15, (2) : 167-184. Whittaker,E.T. and G.N. Watson, 1927. A course i n modern a n a l y s i s . Cambridge Univ. P r e s s . 608 pp.  

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