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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 i n the Department of P h y s i c s We accept t h i s t h e s i s as conforming t o the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA APRIL, 1964. In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study, I f u r t h e r agree that per-m i s s i o n f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i -c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n permission* Department of J H i ^ s t «-S  The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8 5 Canada Date M - t i - i O t 3 r^ / f r G - ^ i i ABSTRACT. The dynamics of the A r c t i c ocean a r e studied on a p o l a r p r o j e c t i o n of the sphere. The d e n s i t y s t r u c t u r e i s i d e a l i z e d as a two-layer system, and a g e n e r a l f o r m u l a t i o n i s developed which a l l o w s i n c l u s i o n of l a t i t u d i n a l and l o n g i t u d i n a l depth v a r i a t i o n s as w e l l as asymmetries i n the boundaries of the ocean. For s i m p l i c i t y , the d e n s i t y s t r u c t u r e i s n e g l e c t e d when depth v a r i a t i o n s are p r e s e n t . Time dependent displacements from e q u i l i b r i u m l e v e l s are assumed t o be waves of constant z o n a l wave number; no r a d i a l propagation i s c o n s i d e r e d . Amplitude equations are d e r i v e d f o r these displacements, s u b j e c t t o the assumption t h a t the p o l a r b a s i n i s s m a l l enough t o keep o n l y a f i r s t a pproximation to the c u r v a t u r e of the E a r t h . A s e m i - q u a l i t a t i v e i n v e s t i g a t i o n of the p o s s i b l e s o l u t i o n s i s made i n the case of a symmetrical b a s i n , u s i n g the Method of S i g n a t u r e s , and e x i s t e n c e c r i t e r i a are found f o r the s o l u t i o n s i n the presence of r a d i a l depth v a r i a t i o n s . C o n c e n t r a t i n g t h e r e a f t e r o n .planetary waves, e x p l i c i t s o l u t i o n f o r such motions i n the s i m p l e s t case (depth constant, symmetrical boundaries) a l l o w s comparison with the r e s u l t s of other i n v e s t i g a t o r s (Longuet-Higgins, 1964 b; Goldsbrough, 1914 a) . I t i s found t h a t the p o l a r p r o j e c t i o n i i i and f i r s t approximation to the curvature give quite good r e s u l t s , so that t h i s method may be applied to polar regions in the same way as the ft -plane i s used in mid - la t i tudes . The general e f fects of r a d i a l bottom slopes are discussed and a simple example treated more e x p l i c i t l y . Some theorems of B a l l (1963) on the motions of shallow ro ta t ing f l u i d s in parabolo idal basins are found to hold for such basins i n the polar plane approximation to the sphere. G.L. Pickard ACKNOWLEDGEMENTS. I w i s h t o e x p r e s s my a p p r e c i a t i o n f o r t h e a d v i c e and s u g g e s t i o n s o f D r . R.W. S t e w a r t , u n d e r whose s u p e r v i s i o n t h i s xvork was c o n d u c t e d . I a l s o w i s h t o t h a n k t h e d i r e c t o r , D r . G.L. P i c k a r d , D r . R.W. B u r l i n g , and t h e s t a f f o f t h e I n s t i t u t e o f O c e a n o g r a p h y , U n i v e r s i t y o f B r i t i s h C o l u m b i a f o r t h e i r a m i a b l e g u i d a n c e and c o o p e r a t i o n d u r i n g t h e l a s t t h r e e y e a r s . I must a l s o a c k n o w l e d g e w i t h g r a t i t u d e t h e f i n a n c i a l a s s i s t a n c e o f t h e I n s t i t u t e o f O c e a n o g r a p h y and o f t h e N a t i o n a l R e s e a r c h C o u n c i l o f C a n a d a . i V TABLE OF CONTENTS T I T L E PAGE ABSTRACT TABLE OF CONTENTS L i s t o f T a b l e s L i s t o f F i g u r e s ACKNOWLEDGEMENTS. I INTRODUCTION. I I BATHYMETRY AND OCEANOGRAPHY OF THE ARCTIC OCEAN, I I I PLANETARY WAVES: BACKGROUND, IV FORMULATION OF THE PROBLEM. V THE POLAR PROJECTION. Page i i i i . i v v i v i i i x i 1 5 10 16 i T r a n s f o r m a t i o n o f e q u a t i o n s . i i Time dependent s o l u t i o n s , i i i P h v s i c s o f R o s s b v waves. V I SYMMETRIC OCEAN; GENERAL CONSIDERATIONS. 26* 32 L l 43 V I I SYMMETRIC OCEAN; FLAT BOTTOM SOLUTIONS. i S o l u t i o n i n t e r m s o f c o n f l u e n t h y p e r g e o m e t r i c f u n c t i o n s . 71 i i Approximate so lut ion in terms of Bessel funct ions . 33 i i i Comparison with r e s u l t s on the sphere. 102 V I I I SYMMETRIC OCEAN; RADIAL DEPTH VARIATIONS. 113 I X ASYMMETRICAL BOTTOM TOPOGRAPHY. 123 X CONCLUSIONS. 130 X I APPENDIX I . 133 BIBLIOGRAPHY 136 v i LIST OF TABLES. Table Page I Numerical v a l u e s of b a s i c p h y s i c a l parameters used i n t h i s work. 27 I I E i g e n f r e q u e n c i e s of Rossby waves i n a symmetric ocean with a f l a t bottom, as c a l c u l a t e d from the c o n f l u e n t hypergeometric s o l u t i o n ('homogeneous' mode). 87 I I I E i g e n f r e q u e n c i e s of Rossby waves i n a symmetric ocean v/ith a f l a t bottom, as c a l c u l a t e d from the B e s s e l f u n c t i o n s o l u t i o n ('homogeneous' mode). 94 IV E i g e n f r e q u e n c i e s of i n t e r n a l Rossby waves i n a symmetric ocean with a f l a t bottom ( c a l c u l a t e d from the B e s s e l f u n c t i o n s o l u t i o n ) . 95 V Computed e i g e n f r e q u e n c i e s of Rossby waves, f o r a symmetric p o l a r b a s i n with a f l a t bottom on a sphere, a c c o r d i n g t o Longuet-Higgins' model. 105 VI Comparison of 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 i n a symmetric p o l a r ocean v/ith a f l a t bottom, as obtained from d i f f e r e n t methods. - 109 V l l VII Some eigenfrequencies of planetary waves for , a symmetrical ocean with a r a d i a l bottom 2 v a r i a t i o n . H ' = 1 + px /Z . 117 v i i i LIST OF FIGURES. F i g . Page 1 Bathymetry of the A r c t i c ocean. 6* 2 A r c t i c d e n s i t y s t r u c t u r e . 9 3 The s p h e r i c a l 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 v e l o c i t y components. 21 4 The two-layer system adopted f o r 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 ocean. 23 5 Orthographic p r o j e c t i o n of the s u r f a c e of the sphere on to a p l a n e . 31 6 Phase paths i n segments of v a r i o u s s i g n a t u r e s . 54 7 Asymptotic behaviour of phase paths of s i g n a t u r e (+,-) as £ —• + GO 55 3 Phase path w i t h compound s i g n a t u r e (-,+)(-,-) ( + ,-)(+,+ ) 56 9 The d i a g n o s t i c diagram and i t s s u b d i v i s i o n i n t o f o u r r e g i o n s . 57 10 Phase paths f o r area I. 61 11 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 . 64 i x 12 Phase paths f o r area I I . 66 13 Phase paths f o r area I I I . 67 14 Phase paths f o r area IV. 69 15 The c o n f l u e n t hypergeometric f u n c t i o n ^F^(a;b;y) as a f u n c t i o n of y f o r a few valu e s of a and b. 76 16 The e x p r e s s i o n (85 rhs) as a f u n c t i o n of frequency, co' . 79 17 The e x p r e s s i o n e/2 as a f u n c t i o n o f frequency OJ'. . 3 0 18 The parameter 'a' (86) p l o t t e d versus frequency OJ' f o r a few v a l u e s of s (s < 0). 81 19 The 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 hypergeometric 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 , 84 20. The s i g n of 8 (as given by (94))in the f o u r 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. 89 21 Sketches of s u r f a c e amplitude ( -q , from (100)) 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 valu e s of wave number, s, and index number, n. 93 22 Sketches of the z o n a l (v) and r a d i a l (u) components of the v e l o c i t y f i e l d (from (101) X and ( 1 0 2 ) ) f o r s = - 2 , n = 1. 99 23 B a l l ' s d e f i n i t i o n o f v e r t i c a l d i m e n s i o n s . 113 24 A ' d i s t e n s i o n ' ( B a l l , 1 9 6 3 ) . 122 \ 1 I. INTRODUCTION. Considerable progress has been made in recent years in the descr ip t ive aspects of A r c t i c oceanography. It seems only l o g i c a l that the next step i n A r c t i c oceanographic research should be a study of the dynamics of the A r c t i c ocean. There being however no observations of long-per iod , large sca le , phenomena i n those regions, i t i s impossible to analyse the s i t u a t i o n from an observat ional b a s i s . The problem w i l l therefore be tackled from a t h e o r e t i c a l point of view, so as to obtain an idea of what to look for in a future observat ional program. This i s not an uncommon approach i n oceanographic research: data c o l l e c t i o n and reduct ion are e s s e n t i a l , but d i f f i c u l t and time consuming, and i t is necessary to have some de f in i t e observat ional aims before going to sea. The problem of f ind ing the propert ies of the c h a r a c t e r i s t i c free motions of the A r c t i c ocean i s here formulated so as to allow the i n c l u s i o n of some of the main t o p o l o g i c a l features of the A r c t i c basin; both l a t i t u d i n a l and l o n g i t u d i n a l bottom and boundary var ia t ions can be formally included i n the model. Most of the work appl ies to motions wi th in a very wide frequency range, but one type of o s c i l l a t i o n was chosen for c lo ser scrut iny : the long period v o r t i c i t y waves c a l l e d planetary or Rossby waves. 2 This choice was motivated not only by a p a r t i c u l a r , in teres t in that type of motions, but also because they can be considered as good test mater ia l f or some of the approximation methods used i n the a n a l y s i s . It would be u s e f u l , when studying oceanographic phenomena in the polar regions , to be able to consider the influence of the Earth ' s curvature only parametr ica l ly , as i s done in mid- lat i tudes i n the so - ca l l ed ft -plane (Veronis , 1963). The analys i s would be great ly s i m p l i f i e d and the physics more eas i ly extracted from the mathematics. The problem i s thus trans ferred from the surface of the sphere to a plane polar pro.iection of the A r c t i c regions , the curvature of the Earth being only kept as a small c o r r e c t i o n . Since planetary waves depend int imately upon that curvature (through i t s ef fect on the l o c a l component of the r o t a t i o n vector) for t h e i r existence and in t h e i r propert i e s , comparison between re su l t s obtained f o r Rossby waves in the polar projec t ion with s imi lar re su l t s derived e n t i r e l y in the spher i ca l geometry w i l l give an ind ica t ion of the a p p l i c a b i l i t y of t h i s approximation. Before t a c k l i n g the problem of the dynamics of the A r c t i c ocean, i t w i l l be useful to review some of the information ava i lab l e on A r c t i c oceanography and bathymetry (sect ion I I ) , and to look into the background of inves t igat ions re la ted to planetary waves (sect ion I I I ) . The mathematical formulation of the problem i s developed in sect ion IV, 3 where the assumptions leading to s i m p l i f i c a t i o n of the problem ( l i n e a r i z a t i o n , i n v i s c i d flow, hydrostat ic pressure) are examined i n d e t a i l . In sect ion V, the polar plane approximation i s introduced, and amplitude equations are derived f o r so lut ions corresponding to free zonal waves. The physics of planetary waves are a lso re-examined in more d e t a i l . Some general conclusions as to the existence and propert ies of the so lut ions in a symmetrical ocean are obtained in sect ion V I . These apply not only to planetary waves but to a l l modes of free zonal o s c i l l a t i o n s in the s i m p l i f i e d model. The resu l t s of th i s l a s t sect ion are used as a guiding beacon in the search f o r e x p l i c i t a n a l y t i c a l so lut ions for planetary waves, which are invest igated in basins of increas ing lv complex bathymetry. In sect ion VII planetary eigensolutions are found for a symmetric basin with a f l a t bottom, f i r s t by so lv ing the s t r i c t amplitude equation i n terms of confluent hyper-geometric funct ions , then s o l v i n g approximately in terms of Bessel funct ions . The eigenfrequencies are more accurate ly computed through the second so lu t ion , and. comparing them with t h e i r equivalents in spher i ca l geometry (Longuet-Higgins, 1964 b ) , i t appears that the polar plane approximation gives good re su l t s f or small enough polar bas ins . The g e n e r a l e f f e c t of r a d i a l bottom s l o p e s i s d i s c u s s e d i n s e c t i o n V I I I , and a simple example t r e a t e d more e x p l i c i t l y . The a p p l i c a b i l i t y of some theorems of B a l l (1963) (on the f i n i t e motion of shallow f l u i d i n a r o t a t i n g p a r a b o l o i d ) i s i n v e s t i g a t e d and found t o be complete i n the p o l a r plane p r o j e c t i o n . F i n a l l y , a few 1, remarks are made i n s e c t i o n IX about the g e n e r a l asymmetric case. 5 I I . BATHYMETRY AND OCEANOGRAPHY OF THE ARCTIC OCEAN. The e x p l o r a t i o n of the A r c t i c , which f o r c e n t u r i e s had been motivated by the search f o r a Northwest or a Northeast passage, took i n the past few decades a new o r i e n t a t i o n . The areas explored have extended from the c o n t i n e n t a l shelves northward i n t o the deep b a s i n s , while the i n t e r e s t has broadened from geography i n t o geophysics. The bathymetric p i c t u r e of the A r c t i c ocean has become more d e t a i l e d a t the same time as the d i s t r i b u t i o n of the main oceanographic v a r i a b l e s was mapped and an idea of the oceanography o b t a i n e d . Soundings from the i c e have r e v e a l e d many h i t h e r t o unsuspected f e a t u r e s of the bottom topography (Gordienko, 1961; Ostenso, 1962). The main d i s c o v e r y was t h a t of the Lomonosov r i d g e which d i v i d e s the ocean i n t o two deep b a s i n s , the E u r a s i a n and the Canadian b a s i n s on the Russian and the Canadian s i d e s of the r i d g e r e s p e c t i v e l y . They have a mean depth of around 4 km, the E u r a s i a n b a s i n b e i n g s l i g h t l y deeper than i t s c o u n t e r p a r t ; the s i l l death of the b i s e c t i n g r i d g e i s about 1500 m. The o n l y deep conne c t i o n with other bodies of water i s through the r e l a t i v e l y narrow s t r a i t s t o the Greenland Sea. I t must a l s o be noted t h a t the ocean i s not bounded symmetrically, deep water r e a c h i n g as f a r as 72°N i n the Beaufort Sea, n o r t h of Alaska, but 6 only to &>5°N at the northern t i p of Greenland. A glance at the chart (Figure 1) w i l l reveal a l l the important features. A model pretending to include the topographic features having some influence on the large scale "kinematics of the A r c t i c ocean w i l l have to include a ridge and asymmetrical boundaries; a simpler symmetrical model w i l l however be studied f i r s t , in order to gain some insight into the physics 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 variations and w i l l be bounded at a given p a r a l l e l of l a t i t u d e . The narrow but deep opening to the A t l a n t i c w i l l not be included in t h i s study, and the A r c t i c basin w i l l be considered closed; t h i s i s j u s t i f i e d on the basis that such a narrow opening should have only a small kinematic influence, although i t cannot be neglected when forced motions are studied, and the effect of the Atlantic t i d e included. Oceanographic sampling i n the A r c t i c has revealed that, as in other oceans, a number of d i f f e r e n t water massed can be recognized in a v e r t i c a l column of water, these water masses being characterized mainly by t h e i r temperature and s a l i n i t y (Coachman, 1962). We are here interested mainly i n the density structure of the A r c t i c waters, which depends mostly on surface s a l i n i t y variations produced by melting or freezing of the ice cover. Figure 2 a) shows the observed range of s a l i n i t i e s in the upper 300 metres in the winter and the summer. The deep waters (below 50 m) are then almost homogeneous and w i l l be considered of uniform dens i ty . The density s tructure i s idea l i zed by a two-layer system (Figure 2 b ) . The top layer i s shallow ( 5 0 m) _o and has an average density of 1 . 0 2 5 gm cm ( cr^ = 2 5 ) , corresponding to a temperature of 0°C and a s a l i n i t y of 3 2 ° / o ° ; the bottom layer i s much t h i c k e r , extending to the —3 bottom of the ocean, and is s l i g h t l y denser: 1 . 0 2 8 gm cm ( 0°C and 35 % ° ; °> = 2 8 ) . The s t r a t i f i c a t i o n i s most pronounced i n the summer, when the ice i s melt ing; i t i s a l so bound to vary in i n t e n s i t y from place to place because of the non-uniformity of the ice cover. These var ia t ions in time and space w i l l not be taken into account, and the s t r a t i f i c a t i o n w i l l be considered uniform and constant. 3 FIGURE I. BATHYMETRY OF THE ARCTIC OCEAN (AFTER OSTENSO , 1961 ) DEPTHS in METRES . 9 SALINITY in % o 30 32 34 100 200 RANGE in — SUMMER WINTER 300 24 DENSITY, <rt 26 28 100 h" CD E o_ UJ Q 200 300 OBSERVED SALINITY STRUCTURE IDEALIZED DENSITY STRUCTURE a) b) F I G U R E 2. ARCT IC DENSITY S T R U C T U R E 10 I I I . PLANETARY WAVES. The term planetary wave was coined by C.G. Rossby who encountered them in the study of time-ciependent motions in a barotropic atmosphere (Rossby 1939). He i d e n t i f i e d them as v o r t i c i t y waves associated with the sphericity of the Earth, and gave f o r them a semi-descriptive d e f i n i t i o n as "quasi-horizontal.... wave motions whose shape, wave-length and displacements are controlled by the v a r i a t i o n of the C o r i o l i s parameter with l a t i t u d e . " (Rossby, 19L9)• Rossby f i r s t explained the dynamics of planetary waves by studying the e f f e c t of a v e l o c i t y perturbation on a zonal current. The zonal stream does not have to be included however, and the v o r t i c i t y equation f o r a f l u i d without any net transport of matter w i l l account very well for the mechanisms involved in planetary waves. A closer look at the physics of these o s c i l l a t i o n s w i l l be taken a f t e r the problem has been mathematically formulated, (section V). For the moment, they can be considered roughly analogous to short gravity waves, t h e i r motion being in the horizontal plane and about some la t i t u d e of equilibrium v o r t i c i t y rather than in the v e r t i c a l plane about a free equilibrium surface. Planetary waves were f i r s t studied by meteorologists, and t h e i r importance in atmospheric c i r c u l a t i o n has been c l o s e l y i n v e s t i g a t e d ; they are now re c o g n i z e d as p l a y i n g a major r o l e i n the heat and momentum balance of the atmosphere, as e f f i c i e n t l a r g e s c a l e exchangers of these p r o p e r t i e s between low and high l a t i t u d e s (Rossby, 1 9 5 9 ) . T h e i r presence i s immediately apparent on any h i g h l e v e l ( 5 0 0 mb.) sy n o p t i c c h a r t , and they have t o be taken i n t o account i n any numerical f o r e c a s t i n g scheme. The study of p l a n e t a r y waves has not reached such a degree of completeness and s o p h i s t i c a t i o n i n oceanography. T h i s i s mostly due t o the presence of c o n t i n e n t s , which do not a l l o w very l a r g e s c a l e r e c o g n i z a b l e waves to develop, as i n the atmosphere, and a l s o complicate c o n s i d e r a b l y any t h e o r e t i c a l s t u d i e s . The time s c a l e between c o n s e c u t i v e oceanographic measurements i s a l s o u s u a l l y so l a r g e t h a t many time dependent phenomena are not observed. Some t h e o r e t i c a l work has however been done on p l a n e t a r y waves. One should mention the s t u d i e s of V e r o n i s and Stommel (V e r o n i s , 1 9 5 6 ; V e r o n i s and Stommel, 1 9 5 6 ) who i n v e s t i g a t e d the response of an i n f i n i t e ocean t o v a r i a b l e wind s t r e s s e s , and found that f o r pe r i o d s of more than one pendulum day, a c o n s i d e r a b l e p o r t i o n of the energy i s t r a n s f e r r e d i n t o q u a s i - g e o s t r o p h i c p l a n e t a r y waves. A l s o , s i n c e i t was d i s c o v e r e d by Stommel (Stommel, 1 9 4 8 ) t h a t the western i n t e n s i f i c a t i o n of wind-induced ocean c u r r e n t s depends on the c u r v a t u r e of the E a r t h , as do the Rossby waves, some work has been done on the i n f l u e n c e of Rossby 12 waves on ocean c i r c u l a t i o n (Moore, 1963) and on the i n t e n s i f i e d boundary c u r r e n t s themselves (Warren, 1963). Although p l a n e t a r y waves might p o s s i b l y not p l a y as important a r o l e i n oceanic c i r c u l a t i o n , a s they do i n atmospheric c i r c u l a t i o n , they seem t o be of great importance i n many time-dependent oceanographic phenomena, and t h i s importance w i l l probably be r e a l i z e d more f u l l y as more r e s e a r c h i s done on the s u b j e c t . I t was r e c o g n i z e d some time a f t e r Rossby's work (Stommel, 1957) t h a t t h e r e were antecedents i n the l i t e r a t u r e f o r the semi-geostrophic p l a n e t a r y waves; o s c i l l a t i o n s of very much the same nature had been t r e a t e d by t i d a l t h e o r i s t s under the a p p e l l a t i o n of O s c i l l a t i o n s of the Second C l a s s . T h i s c l a s s i f i c a t i o n o f the O s c i l l a t i o n s of an ocean of constant depth on a r o t a t i n g sphere i n t o two c l a s s e s was made by Hough a f t e r he found t h a t h i s i n t e g r a t i o n of the Laplace t i d a l equation under these c o n d i t i o n s y i e l d e d two c a t e g o r i e s of s o l u t i o n s , c h a r a c t e r i z e d by the asymptotic behaviour of t h e i r p e r i o d s as the frequency of r o t a t i o n of the E a r t h decreases t o zero. The O s c i l l a t i o n s of the F i r s t C l a s s have f r e q u e n c i e s which tend t o f i n i t e v a l u e s as the r o t a t i o n o f the E a r t h vanishes; i n those of the Second C l a s s the f r e q u e n c i e s tend t o zero and they become steady motions on a non-rotating.globe (see a l s o Poincare', 1910). T h i s c l a s s i f i c a t i o n has by some authors (e.g. E c k a r t , I960; Chapter XVII, page 275) been a t t r i b u t e d t o Laplace, but L a p l a c e ' s t h r e e s p e c i e s of t i d e s were something e n t i r e l y d i f f e r e n t , r e f l e c t i n g the l o n g i t u d i n a l symmetry of the components of the t i d e producing f o r c e s , and had no t h i n g t o do v * i t h the asymptotic l i m i t at v a n i s h i n g r o t a t i o n r a t e s which Hough i n v e s t i g a t e d . The l a b e l Laplace O s c i l l a t i o n s of the Second C l a s s i s t h e r e f o r e erroneous, and i t would be p r e f e r a b l e t o a t t a c h Hough's name to p l a n e t a r y waves. The work done i n the theory of t i d e s was, then, a v a i l a b l e t o oceanographers i n t h e i r i n v e s t i g a t i o n s of p l a n e t a r y waves; u n f o r t u n a t e l y , a l l of i t was done i n s p h e r i c a l geometry. T h i s i s q u i t e a p p r o p r i a t e t o ' t h e study of phenomena on a sphere, but makes the a n a l y s i s extremely d i f f i c u l t f o r enclosed oceans of v a r i a b l e depth. T h i s d i f f i c u l t y was evident from the beginning, and Laplace h i m s e l f , speaking of the g e n e r a l t i d a l amplitude equation, s a i d t h a t " L f i n t e g r a t i o n de 1'equation dans l e cas g e n e r a l ou n (E a r t h ' s r o t a t i o n ) n'est pas n u l , et ou l a mer a une profondeur v a r i a b l e , surpasse l e s f o r c e s de l ' a n a l y s e ; " (Laplace, 1799; Premiere p a r t i e , L i v r e IV, C h a p i t r e I ) . L i t t l e progress has been made i n that r e s p e c t s i n c e L a p l a c e ' s time. Some s p e c i a l , cases have been s t u d i e d by Goldsbrough f o r an ocean of constant depth bound by meridians (Goldsbrough, 1933), or by two p a r a l l e l s of l a t i t u d e (Goldsbrough, 1914 b ) , or of r e c t a n g u l a r form 14 (Goldsbrough, 1 9 3 D . Love (1913) studied the case of a small c i r c u l a r ocean, and the only case of sloping bottom was treated by Proudman (1916), the depth varying only with l a t i t u d e . It i s then not surprising that t i d a l theory was no more useful to the study of planetary xvaves in actual oceans than i t had been in predicting the amplitudes of the tides in these same oceans, and f o r the same reasons. Rossby's analysis was very much si m p l i f i e d by his assumption that the c h a r a c t e r i s t i c e f fects of the t e r r e s t r i a l curvature could be preserved by making the C o r i o l i s parameter vary l i n e a r l y with l a t i t u d e over a short distance. This approximation has been termed the /9 -plane approximation, and has been used extensively, both i n oceanography and in meteorology, since i t s introduction by Rossby (Rossby, 1939). The t h e o r e t i c a l work done on planetary waves in oceanography has nearly a l l been performed on the /3 -plane; the mathematics are thus quite s i m p l i f i e d and a cl e a r idea of the physical happenings over a r e s t r i c t e d band of latitudes can be arrived at without the use of spherical geometry. Concurrently with t h i s work, Longuet-Higgins (1964 b) has introduced another very useful s i m p l i f i c a t i o n which allows even clearer physical understanding of planetary waves. Taking advantage of the quasi-horizontal nature of the waves, he neglects the influence of surface displacements in the v o r t i c i t y balance, reducing the problem to a two-dimensional one, which can be solved i n terms of a stream function. This approximation i s v a l i d f o r wave lengths smaller than the radius of the Earth, Longuet-Higgins uses i t with the /3 -plane to examine the behaviour of planetary waves in basins of a var i e t y of shapes, and shows that i n the l i m i t of small wave lengths the frequencies of the o s c i l l a t i o n s in the ft -plane are i d e n t i c a l to those on the sphere. This approximation i s also 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 of l a t i t u d e are considered, of which waves i n a polar ocean are just a s p e c i a l case. The problem i s not treated i n d e t a i l however, and since i t i s e s s e n t i a l l y two-dimensional, no depth variations can be included. Bottom variations are included i n my work, and the analysis i s performed i n a s p e c i a l projection about the pole to avoid the complexities of spherical geometry. Longuet-Higgins' approximation cannot be made i f the bottom i s not f l a t , so that the formulation of the problem i s intermediate in complexity between the s t r i c t t i d a l problem of Laplace and Longuet-Higgins' assumption that the motions are purely two-dimensional. 16 IV. FORMULATION OF THE PROBLEM. The motions representative of the Ar c t i c ocean w i l l be those depending, either for t h e i r existence or i n modifications of t h e i r properties, on the geometry or the density structure of the A r c t i c ocean. Short gravity waves c e r t a i n l y do not f a l l into that category, and we can assume that the wave motions which w i l l r e f l e c t in t h e i r c h a r a c t e r i s t i c s the properties of the A r c t i c w i l l be of periods of at least many hours and of scales of more than a few kilometres; t h i s includes planetary waves at the lower frequency l i m i t . With t h i s fundamental assumption and a few secondary ones the mathematical formulation can be considerably s i m p l i f i e d . The movements of an incompressible f l u i d i n a rot a t i n g frame of reference are described by the Navier-Stokes equation: d + u . V u + 2Q xu +_lVP -*V2u + g = 0 (1) a t P and the continuity equation V-u = 0 , (2) i n which u(x,t) i s the f l u i d v e l o c i t y r e l a t i v e to the coordinate system x rota t i n g with an angular v e l o c i t y £2 , P i s the pressure, p the f l u i d d e n s i t y , v the kinematic v i s c o s i t y of the f l u i d and g the net a c c e l e r a t i o n due t o g r a v i t y and the c e n t r i f u g a l f o r c e due t o the E a r t h ' s r o t a t i o n . Assuming t h a t the r e l a t i v e motion has a v e l o c i t y s c a l e U, a time s c a l e r and a h o r i z o n t a l l e n g t h s c a l e L, the r e l a t i v e magnitudes of the s u c c e s s i v e terms of ( 1 ) are, a f t e r d i v i s i o n by J L . J L i 1 . P . JL- , _JL_ . The h o r i z o n t a l balance i n most oceanic f l o w s i s between pressure and C o r i o l i s f o r c e s . Are we j u s t i f i e d here i n keeping only these terms and n e g l e c t i n g the other ones? The time d e r i v a t i v e term w i l l be comparable t o the C o r i o l i s term f o r p e r i o d s of the order of a day, but even when we are i n t e r e s t e d i n time-dependent motions of much longer p e r i o d s , we cannot n e g l e c t t h a t term, however smal l i t s i n f l u e n c e i n the h o r i z o n t a l momentum balance, because i t c o n t a i n s one of our unknowns, the p e r i o d of the motion. T h i s importance of the time d e r i v a t i v e term i s made more evid e n t by l o o k i n g at the v o r t i c i t y equation: d j - V x f u x U + 2 ^ ) 1 - vy2C, = 0 , d t where £ = V x u . When, as w i l l be assumed below, the v i s c o u s and n o n - l i n e a r terms are of l i t t l e importance, the only terms l e f t in the v o r t i c i t y equation are the time rate of change term and the C o r i o l i s term. I f , as o r i g i n a l l y assumed, the sca le of motion, L , i s large enough, the viscous terms w i l l be of l i t t l e influence in the Navier-Stokes equation: v sec << 1 I O " 4 L 2 This i s c e r t a i n l y true i f v i s the molecular v i s c o s i t y 2 -1 ( 0.01 cm sec for water); even for an eddy v i s c o s i t y 6 2 —1 as high as 10 cm sec , the inequa l i ty holds s trongly for L > 10km. The influence of viscous forces w i l l therefore be ignored and the f l u i d treated as i n v i s c i d . There are no d i s s i p a t i v e processes and, i f the ocean i s bounded, no energy can be radiated away so that the t o t a l energy of the water mass w i l l be constant. ^ The amplitude of the motions to be studied w i l l be assumed small enough to make the non- l inear terms of n e g l i g i b l e inf luence: the Rossby number is small and U ~ U sec « 1 I U lO"*- L Since £l var ies very l i t t l e over the area considered, the r a t i o of v e l o c i t y to hor i zonta l scale (a measure of the r e l a t i v e importance of the l o c a l v e r t i c a l component of the v o r t i c i t y of the f l u i d to the planetary v o r t i c i t y ) i s the most c r i t i c a l parameter in t h i s study; i t may depend on p o s i t i o n , and i f so, i t must remain f i n i t e everywhere in n L 2 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 di 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 within an a r b i t r a r y m u l t i p l i e r . Linearization 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 to be vindicated a p o s t e r i o r i by showing that the Rossby number remains analytic over the whole A r c t i c ocean. A further assumption which can be made when the periods are long enough i s to neglect v e r t i c a l accelerations of the f l u i d compared to gravity. Using the c r i t e r i a given by Proudman (1953; Chapter XI, p 223) for 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 dt2 H 1 g 1 i n which i s the displacement of the surface from i t s equilibrium position, g i s gravity, and H the depth (here 4 km.). This w i l l hold f o r periods longer than 10 minutes, so that the condition 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 interface of a two-layer system i s (Proudman, 1953; Chapter XV, p 336) 2 Pn0A2k. / _ &P H ; « i 2 2 x ~ P T~ at / &V2Lp P g r 20 in which f) i s the displacement of the interface, H 2 the thickness of the lower layer (again nearly 4 km.), the density of the lower layer, and A p the density difference **3 ~3 between the two-layers (3.10 gm cm ). This condition i s more binding than the previous one, but i n t e r n a l waves are in general of longer period than surface waves of the same dimensions; the period in that case must be longer than 12 hours. In order to take the pressure as completely hydrostatic, i t i s also necessary that the v e r t i c a l component of the C o r i o l i s force be much smaller than gravity: U sinQ < < 1 g in which 0 i s as defined in Figure 3 . This depends on the t o t a l energy, as does the Rossby number, but the condition that i t remains an a l y t i c at the pole i s less stringent since sinO vanishes there. After having been subjected to 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 written out in the spherical polar coordinates shown in Figure 3« dv + 2flucos9 -2flwsin9 = -1 d P (3) dt />rsin0 <^ du - 2flvcos9 = -1 aP dt pr dQ 2 1 ap = - p g (5) dr £ l > 1 ^uslnQ + _£w = 0 (6} rs in9 d\ rsinQ a © d z The assumptions now about to be made are based o n the geometry and s i t u a t i o n of the A r c t i c ocean. In any ocean, the maximum depth i s extremely small compared to the radius of the Earth. In the above equations, r w i l l then be replaced by a constant radius, 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 \ Figure 3. The spherical coordinates and the corresponding v e l o c i t y components. 22 water surface. Furthermore, since we are working close enough to the pole f o r tan9 to be quite small, and since the horizontal v e l o c i t i e s are in general much larger 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 retained in (3). The pressure forces can be replaced by equivalent expressions containing the gradients of the surface (z = 77^) and interface ( z = V^) displacements from an equilibrium l e v e l . If a two-layer structure as i l l u s t r a t e d in Figure 4 i s adopted, and the constant atmospheric pres-sure neglected, integration of the hydrostatic equation, (5), gives f o r the pressures i n the top and bottom strata respectively P x = g - z) (7) The momentum equations f o r the two layers are as follows when displacement gradients are substituted f o r pressure gradients; the continuity equation (6) i s unchanged and has the same form in both layers ( u , ~ 1 and u are the v e l o c i t i e s in the upper and lower l a y e r ) . 23 Figure 4. The two-layer system adopted f o r the density structure of the A r c t i c ocean. 1 + 2flu 1cos9 = a t RsinG d ^ (9) au 1 - 2 flv cosO = a t R a G (10) av at a u a t 2 + 2l2u cosG = -g 2 ~ u-Rsih© ' 1 i + *L??1 R2 ax ^ 2 a x 2 - 2 <ft v^cosG = -g R r 2 a 9 a e d i ) (12) The following boundary conditions complete the description of the physical s i t u a t i o n . The v e l o c i t y 24 component p e r p e n d i c u l a r t o a f i x e d boundary (with normal n ) w i l l v a n i s h at that boundary: ( u« n = 0 ) In p a r t i c u l a r , a t the bottom, z- -H •1 i j i u , R dO 1 - 1 a H v. _ H RsinO a\ (13) z- -H and at a v e r t i c a l w a l l at l a t i t u d e 0^, u — v a©. (14) s i n 9 1 a\ The average depth of the i n t e r f a c e , H^, i s considered constant: s i n c e i t i s determined by s u r f a c e phenomena and by any mean c u r r e n t s (here z e r o ) , i t w i l l not be i n f l u e n c e d by depth v a r i a t i o n s . A l i n e a r s u r f a c e boundary c o n d i t i o n i s used: w., d t (15) and a s i m i l a r c o n d i t i o n a t the i n t e r f a c e , t o g e t h e r w i t h c o n t i n u i t y of v e r t i c a l v e l o c i t i e s : w = w z = -H. + V 1 2 dt (16 ) 2 5 No non-slip conditions have been imposed, either at the boundaries or at the interface, since there i s no v i s c o s i t y i n the model, A further general requirement i s that a l l variables remain f i n i t e over the entire ocean, and especially at the pole, where s i n g u l a r i t i e s are l i k e l y to occur. The equations ( 9 ) - ( 1 2 ) together with the continuity equation (6) are now integrated over t h e i r respective layers, subject to the above conditions. One then has for the top layer: 1 + 2&u cosQ = -g (17) d t RsinG d X d u 1 - 2 f l v cosG = -g d 7 ) l (13) a t , R ae + l s i n 9 Rsin9 d x RsinG a 9 H x <H and f o r the bottom layer: + 1 _ i J 7 7 1 - V = 0 <19) D V 9 n f + 2 Aiu^cosO = -g d t Rsin9 />i $Vi ^ &p dyz + P2 dX P 2 d X a _ J _ 2 >ftv 2cos9 = ^ _ [" * l * \ , A/> ^ 2 a t R p 2 d Q P2 D Q (20) (21) 1 ^ 2 " 2 + 1 ^ U2 H2sin9 , dV2 =0 (22) Rsin9 a X RsinG a 9 d t 26 These are the equations I s h a l l attempt to solve, 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 influence of an ice 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 effect i s negligible when the wave length i s very large compared with the thickness of the i c e . We are already limited by the hydrostatic assumption to periods longer than 10 minutes; waves of such periods w i l l have phase speeds s l i g h t l y less than 200 m sec~^" in a 4 km deep ocean, and correspondingly a scale of over a hundred kilometers. There i s therefore no reason to worry about d i s t o r t i o n due to the ice cover in the rest of t h i s work. For shorter 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 gravity waves; observation of such shorter period motions has been made by Hunkins (1962) from f l o a t i n g research stations i n the A r c t i c . A tabulation of the values of the physical parameters used i n t h i s study w i l l be quite useful, and I f i n i s h t h i s section with such a l i s t (Table I ) . 27 TABLE I . Values of the basic phys i ca l parameters used i n th i s work. Angular v e l o c i t y QUANTITY. SYMBOL. NUMERICAL VALU*1-of the E a r t h . " 0,7*10 sec Radius of the E a r t h . R 6370 km. G r a v i t a t i o n a l -> 0 a t t r a c t i o n . g 10 cm sec"*^ Depth of A r c t i c ocean (average). H 4 km, Thickness of surface l a y e r . 50 m0 Radius of A r c t i c basin (average). r^ 1500 km* Densitv s t r u c t u r e , A_P 3xio~^ 28 V. THE POLAR PROJECTION. i ) Transformation of the e q u a t i o n s . We have now formulated the g e n e r a l problem of the dynamics of a p o l a r b a s i n on a r o t a t i n g sphere, as per equations (17)-(22) and (14). We have a l s o seen however i n s e c t i o n I I I t h a t the work of f o u r g e n e r a t i o n s of t i d a l t h e o r i s t s v i n d i c a t e s L a p l a c e ' s o p i n i o n of the d i f f i c u l t y of such a g e n e r a l problem. The problem i n i t s present form seems a b i t hopeless, but any s i m p l i f i c a t i o n should not be so d r a s t i c as to hide i n t e r e s t i n g p h y s i c a l pheno-mena. The a n a l y t i c a l i n t r a c t a b i l i t y stems from the combined i n f l u e n c e s of the s p h e r i c a l geometry and the v a r i a b l e topography; Rossby (1939) i n t r o d u c e d the /5-plane approximation to circumvent the d i f f i c u l t y . In the y9 -plane ( V e r o n i s , 1963), the s u r f a c e of the sphere i s transposed by a Mercator p r o j e c t i o n , so t h a t p a r a l l e l s of l a t i t u d e become s t r a i g h t l i n e s , and, i f a narrow band of l a t i t u d e s i s c o n s i d e r e d , the main i n f l u e n c e of the c u r v a t u r e of the sphere can be reproduced by making the C o r i o l i s parameter a l i n e a r f u n c t i o n of l a t i t u d e . I t seems t h e r e f o r e n a t u r a l t o t r y and p r o j e c t the p o l a r r e g i o n s on to such a plane, where only f i r s t approximations t o the c u r v a t u r e would be kept. The /3 -plane b e i n g a Mercator p r o j e c t i o n , i t i s not a p p l i c a b l e near the p o l e , and some other p r o j e c t i o n w i l l have t o be used i n the present problem. I f we are a b l e t o 2 9 show that the analys is of the motions on such a projec t ion gives re su l t s not too d i f f eren t from those which have been obtained on the sphere, we then have a t o o l which can be used i n the polar regions with the same confidence as the ft -plane i s used in mid - la t i tudes . Judging from the amount of l i g h t shed by the ft -plane analyses on the meteorological and oceanographic s i tuat ions in mid- la t i tudes , i t would indeed be very valuable to have such a method ava i lab l e in the polar regions . There are many ways of projec t ing the surface of the sphere in the polar regions on to a plane; I choose here what I think i s the simplest one. An orthographic projec t ion is made on to a plane tangent to the sphere at the pole (Figure 5). I f plane polar coordinates r and cp are used on the plane of pro jec t i on , the geometry • of the mapping imposes the fol lowing r e l a t i o n s h i p with the s p h e r i c a l coordinates: dr = R cos9 d9 ; r = R sin9 (23) dci = dX The working approximation, which w i l l be introduced a f ter a l l d i f f e r e n t i a t i o n s have been performed, w i l l consist of neglect ing terms of order .( r/R) with respect to u n i t y . Cos9 then becomes equal to uni ty when not d i f f e r e n t i a t e d , and d ( C o s © ) / d r i s approximately - r/^2. Only f i r s t approximations to the curvature are then kept, and t h i s 30 w i l l e v i d e n t l y be v a l i d o n l y f o r b a s i n s o f s m a l l l a t i t u d i n a l e x t e n t a b o u t t h e p o l e . The s o u t h e r n m o s t c o r n e r o f t h e A r c t i c o c e a n i s i n t h e B e a u f o r t S e a , a t 72°N: 9 i s t h e n l e s s t h a n 13°, and s i n ^ 9 l e s s t h a n 0 . 0 9 5 , s o t h a t t h e t e r m s n e g l e c t e d a r e s m a l l e r t h a n u n i t y b y a t l e a s t an o r d e r o f m a g n i t u d e o v e r t h e w h o l e A r c t i c b a s i n . The m a i n d i f f e r e n c e b e t w e e n t h i s p l a n e p r o j e c t i o n and t h e c o n v e n t i o n a l /3 - p l a n e i s t h a t t h e d e r i v a t i v e o f t h e C o r i o l i s p a r a m e t e r i s n o t a c o n s t a n t , b u t v a r i e s l i n e a r l y w i t h l a t i t u d e . T r a n s f e r o f e q u a t i o n s ( 1 7 ) - ( 2 2 ) t o t h e p o l a r p l a n e by means o f (23) g i v e s i n t h e new g e o m e t r y : c3v, d t <3u 1 + 2 X 2^0039 1 - 2 f l v cos9 1 d t 1 , cos9 a V r d cfi r d r r d 4> = -g cos9 d 7 ? ! d r = J L ± ( v2 - 77 1) a v o £ + 2AZ u^cos9 = -^ g a t r dV 2 fl dJl + Af. f>2 d<f> Pz dcf> _ a u 9 c - 2 i iv„cos9 = -g cosQ 2 a t PX dV1 Ap dV2 PZ a p 2 a r 1 d V 2 H 2 ^ cos9 . d \ _ 0 . (24) (25) ( 2 6 ) (27) (23) (29) d<f> a r a t 31 Figure 5. Orthographic projec t ion of the surface of the sphere on to a plane. , Note that two a d d i t i o n a l terms invo lv ing cos© appear, besides those associated with the C o r i o l i s parameter; they account f o r two geometrical 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 to progress ive ly more exiguous areas of the plane as the l a t i t u d e decreases; a l so , 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 , i t remains constant on the plane of p r o j e c t i o n . 32 i i ) Time d e p e n d e n t s o l u t i o n s . Now l e t us l o o k f o r t i m e d e p e n d e n t s o l u t i o n s o f (24)-(29) o f t h e f o r m TT. = F.( r,<p) e i ( a J t - s < ^ i n w h i c h cu i s t h e f r e q u e n c y , s t h e c o n s t a n t a z i m u t h a l wave number, F^( r,<jb ) a n a m p l i t u d e f u n c t i o n to be d e t e r m i n e d , and t h e s u b s c r i p t j i s u s e d w i t h t h e v a l u e s 1 o r 2 t o d e n o t e s u r f a c e o r i n t e r f a c e d i s p l a c e m e n t s , r e s p e c t i v e l y . The d e r i v a t i v e s o f t h e d i s p l a c e m e n t s a r e w r i t t e n a s dV = iu)V (31) a t d 7) = - io-?? ; a _ _i _a_F (32) a r F a r drl = - i ^ 7 ? ; / = s + _ i _ JUL (33) d<f> F dcf> The s u b s c r i p t s h a v e been l e f t o u t o f t h e abo v e s i n c e t h e s e r e l a t i o n s a r e i n d e p e n d e n t o f j . W i t h t h e h e l p o f t h e r e l a t i o n s (30)-(33), t h e momentum e q u a t i o n s c a n be s o l v e d e x p l i c i t l y f o r t h e v e l o c i t i e s i n t e r m s o f d i s p l a c e m e n t s : t h e p a i r s (24)-(25) a n d ( 2 7 ) - ( 2 3 ) become n1 = g G x V1 ; v x = g J x ^ (74) 33 (35) ^2 ^2 2 The functions G and J are abbreviations used f o r temporary convenience and defined by ( -to cr cos9 + i v 2,0, cos9/* ) ^ = L_ 7A _ (36) ( 4 ^ 2 c o s 2 9 - c u 2 ) ( - v ui/r - i cr 2X2 cos 29 •) J j = J J (37) ( 4 l2 2cos 29 - 2) Complete elimination of the v e l o c i t i e s i s achieved by substitution of (34) and (35) into the continuity equations (26) and (29). In terms of the abbreviated notation, G and J, the top layer equation becomes fcos9 / i a G l ,c G . iG \ v J \TJ { [ ~ + 1 1+-± r1-!} 1 = \) (33) r r g H l ' and the bottom layer equation 0 0 3 9 ' i d a i + < r i V iGxV P X \ cosQi d r r 1 r J P 2 i a G 2 + V 2 + i G 2 \ /2 J2l A/° ,?72 P 2 L H 2 a R H 2 r ad>J /°2 + J icosQ _a_H G 2 + i J 2 d H I A/° ^ g H2, a r H 2 r a<£J ? 2 0) V. (3?) g H 2 Let us look f i r s t at the case where the depth is constant and the ocean symmetric; the s t r a t i f i c a t i o n w i l l be retained only for t h i s simple geometry. Even though the terms containing der ivat ives of the depth disappear from (39), i t i s not formal ly poss ib le to e l iminate or F 2 from (33) and (39). I f , however, one considers that the r a t i o of the amplitudes of the surface and interface displacements depends only upon the density s tructure ( r e l a t i v e depths and dens i t i e s of l a y e r s ) , so that f or a constant depth and uniform s t r a t i f i c a t i o n t h i s r a t i o i s not a funct ion of p o s i t i o n , the subscripts are 35 no longer necessary in (33) and (39). Since now F 1 ( r ) / F 2 ( r ) i s a constant, the logarithmic derivative of either of these amplitudes i s the same, and the variables cr and y and J and G assume the same value f o r j = 1 and j = 2. J 3 Equation (39) then reduces to P 2 I g H 2 Eliminating the terms containing G and J between (33) and (40), a quadratic in the r a t i o of the amplitudes is obtained: AP - H _ p2 H 2 - AP_ =0 (41) Pn Provided, as i s the case here, that P\/Pz — 1, and that H/H^ — 1, good approximations to the roots of (41) are F F 1 = H I 1 = - A? . F 2 H 2 F 2 The f i r s t root corresponds to 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 in phase and nearly equal in amplitude. The second root can be i d e n t i f i e d with i n t e r n a l o s c i l l a t i o n s at the interface; the surface amplitude i s much smaller and out of phase with the interface amplitude. Note that i t i s possible to derive t h i s r e s u l t without the above mild assumption on the constancy of the 36 r a t i o of the amplitudes; i f , following Rattray ( 1 9 6 4 ) , one eliminates the displacements rather than the v e l o c i t i e s from ( 2 4 ) - ( 2 9 ) , i t i s possible to separate formally the homogeneous and i n t e r n a l o s c i l l a t i o n s by introducing new dependent variables u' = u + u 1 2 = H„ u 2 1 H L H P H' 2 -, H 2 J u. This i s indeed an id e a l method when the bathymetric effects are neglected, but since u" i m p l i c i t l y contains the depth, I prefer to retain the other formulation as more convenient in studying the ef f e c t s of bottom va r i a t i o n s . Substituting f o r the constant roots of ( 4 1 ) , (33) can be written as cosQ ( i d G + o - G + i G \ + s J = — ( 1 ' V 2 ^ 1^ » dr r / r gH^ (42) in which 1/ V^ i s the constant appropriate e i t h e r to the homogeneous or to the in t e r n a l mode; the ri g h t hand side p takes the form ^/gH and gH]_ respectively i n those two cases. Equation ('42) i s an ordinary d i f f e r e n t i a l equation in only one variable, F(r ), and w i l l be solved in section 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 to do so ri g h t now. 37 Replacing G and J by t h e i r d e f i n i t i o n in ( 3 6 ) , an intermediate equation fol lows: ( iojcr cos 29 + 2X2cos29 s )(- £ X 2 2sin9 ) + ( 4X22cos29 - oJ2) - ito cos9 der + jojo- tan9 - cu sj dr R r' + 2X2 t a n £ j3 - coo- cos9+ 2X2cos9cr is - icuo* cos9 R r r (43) - 2X2cos29 iscr = ( 4i2 2 cos 29 -O) 2)_OJ_( 1. - V2/ V ) It i s at t h i s point that the approximation of the P -plane 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 information w i l l be l o s t by w r i t i n g cos9 = 1;-sin9 = tan9 = r / R . Hence, the C o r i o l i s parameter, f, and i t s d e r i v a t i v e with respect to r , /3 , which, when expanded in r / R , are f = 2ncos9 = 2&(l - r 2 / R 2 )^ = -2il tan9 = -2X2 r ( l - r 2 / R 2 •R R' 38 become f n 2X2 /3 a -2>Q r R2 N e g l e c t i n g terms i n (r/R) w i t h r e s p e c t t o 1, (43) s i m p l i f i e s to -ar2- idcr - i a + i a r - s 2 dr R 2 r 2 i r c r (44) • 2 l 2 s ( 4 X 2 2 + c u 2 ) a; R2 ( 4 X 2 2 - w 2)( 1 - V T^ ) I gHi which, a f t e r w r i t i n g cr i n terms of the amplitude f u n c t i o n , F, through (32), becomes d_/ l d £ \ + l d F / l d F + l (4^2 2 + c o 2 ) r \ diA F d r ' F dr ^  F dr r ( 4 ^ 2 „ w 2 ) R2 / - s f - 2 X 2 S ( 4 X 2 2-rqj 2) r 2 R 2( 4X22-60 2) (45) ( 4 X 2 2 - c o 2 ) ( 1 - V 7 ? i ) = 0 Note t h a t a l t h o u g h (r/R) has c o n s i s t e n t l y been 2 2 2 c o n s i d e r e d much s m a l l e r than 1, (4X2 + to ) / ( 4X2 - c o 2 ) V has been kept s i n c e i t may not be n e g l i g i b l e f o r s u i t a b l e v a l u e s of co . T h i s w i l l probably not be the case f o r Rossby waves, which I expect t o be of low frequency, but 39 may well be so f o r other types of motion to which t h i s equation applies. We have i n (45) an equation governing the amplitudes of waves in a two-layer system in the absence of any bottom variations and asymmetries. When there are bottom slopes., i n .view of the t complexity of equation ( 3 9 ) , we w i l l riot consider the in t e r n a l o s c i l l a t i o n s and l i m i t our study to the 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 + iG ) •+ y_J d r r r + l i J i i + iG cosQ a.H = a) rH d4> H dr gH An amplitude equation can be derived from (46) by following the same procedure used to obtain (45) from ( 4 2 ) : the abbreviations G and J are rewritten in terms of t h e i r e x p l i c i t d e f i n i t i o n s , (36) and ( 3 7 ) , the approximation [T/R) « 1 i s made, and cr and y are expanded i n terms of F . A further assumption i s now that the amplitude function F and the depth H are periodic functions of l a t i t u d e , F CC e i ( p < £ \ (47) where p may be a function of r . Expressions of the form i d F and d H are therefore r e a l ; the ensuing . F & 4> H d(f> 40 amplitude equation i s a 2 F a r 2 + d F a r 1 + 1 aH + (4x2 + c u ) r , 2ix2 / i a F 1 1 a H r H ar ( 4 X 2 2 _ co 2) R 2 OJ r\F ^ H d cp a F 1 a F r a<£ _rF a<p + a H - 2 i s - i a_H 2£_ -2X2 ir(4X2 2 + cu 2) rH deb H a CO R 2 ( 4 X 2 2 _ co 2) + F -si -2X2 ( 4 X 2 2 - r c o 2 ) s_ - ( 4 X 2 2 - G J 2 ) -2SX2 a H - i s a H  2 W ( 4 X 2 2 - c o 2 ) R2 g H torH a r r 2 H dep co 2 ) 2 - 2iX2 a F = 0 co r dcfidT (48) The descr ip t ion of the motions i s not complete without boundary condit ions appropriate to the idea l i zed s i t u a t i o n represented by equation (48). A general p r e s c r i p t i o n , appl icable to a l l cases, i s that the funct ion F(r ,<£ ) s a t i s f y i n g (48) be continuous and ana ly t i c over the whole extent of the project ion of the A r c t i c ocean, inc lud ing the boundary, r = r ^ ^ ^ The r e s t r i c t i o n on the v e l o c i t y at a v e r t i c a l wal l must a l so be s a t i s f i e d whenever such a s i t u a t i o n ar i s e s ; neg lect ing 2 r i/R 2 with respect to u n i t y , (14) becomes, i n the polar p r o j e c t i o n : 41 u ( r 1 ) = v ( r ^ ) 111 (49) and, i n terms of the amplitude f u n c t i o n , F , u s i n g (34) and ( 3 5 ) , (49) becomes d F d r -ioj - a r. + . a F r _ r 2«Q r i a^> ' d<f> -1 +1W dJ\ - r r x 2 f l r 2 a<£ + F ( r x ) / i s u) s a r l \ = 0 (50) The r e s t of t h i s work i s concerned w i t h s o l v i n g the amplitude equations, f i r s t i n the simplest cases ( 4 5 ) , then i n more complicated s i t u a t i o n s , ( 4 3 ) . Although more g e n e r a l s o l u t i o n s w i l l be kept i n mind, s p e c i f i c c a l c u l a t i o n s are done only f o r p l a n e t a r y waves. i i i ) Physics of Rossby waves. Now t h a t the f o r m u l a t i o n has been e s t a b l i s h e d and the problem s t a t e d i n s t r i c t e r form, i t becomes e a s i e r t o study the e f f e c t s of the v a r i o u s f o r c e s at work t o produce or s u s t a i n o s c i l l a t i o n s i n the b a s i n . C o n s i d e r a t i o n of the momentum equations, (24) and ( 2 5 ) , shows t h a t l o c a l a c c e l e r a t i o n s are produced by two causes only: C o r i o l i s and g r a v i t y f o r c e s . The e f f e c t of 42 gravity i s well known, and w i l l tend to eliminate any deviations from the free equilibrium surface of the f l u i d ; i f gravity alone acts, f a m i l i a r gravity waves r e s u l t . The action of the C o r i o l i s force i s also well known: in the northern hemisphere i t acts as a force 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 forces may be of s i m i l a r importance to the motions; when t h i s happens i n a steady state, geostrophic currents r e s u l t , i n which pressure gradients exactly balance the C o r i o l i s force. This i s s t i l l simple and e a s i l y v i s u a l i z e d . What happens when the C o r i o l i s parameter i s not constant, but varies with l a t i t u d e , as i t does on the Earth, and in 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 polar projection introduced above. Although i t i s somewhat u n r e a l i s t i c to have a plane characterized by d i f f e r e n t rotation rates at di f f e r e n t positions, the construct i s useful in that i t reproduces the properties of the spherical surface with simpler symbolism. In order to bring the v a r i a t i o n of the C o r i o l i s parameter into the equations and see what i t s influence i s , l e t us examine the equation f or 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 non-lin e a r terms retained, the following 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 non-linearized momentum equations (Stommel, I960; ch. V I I I , p. 103): 43 d( £ + f) dt f + f d( H-f^ ) H + -7 . dt (51) To obtain (51), use i s also made of the integrated continuity equation in i t s non-linearized form: (H+77 ) cosQ dur a^ d(H + V ) dt ( 5 2 ) Equation (51) applies on the polar plane; f i s the variable C o r i o l i s parameter (2flcos0), H the t o t a l equilibrium depth, 17 the surface displacement from equilibrium, 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 - a u a r d4> (53) The v o r t i c i t y equation (51) can be immediately integrated to y i e l d f + t H + V constant (54) This w i l l be recognized as a form of the po t e n t i a l v o r t i c i t y conservation theorem, of frequent use i n 4 4 meteorology and oceanography. I t s t a t e s t h a t the p o t e n t i a l v o r t i c i t y of a given volume of water, as d e f i n e d by the l e f t hand s i d e of ( 5 4 ), does not change as t h a t volume of water moves about i n the f l u i d . T h i s i s of course only a s p e c i a l statement of the theorem of c o n s e r v a t i o n of angular momentum i n the absence of a p p l i e d t o r q u e s . Now, i n most oceanic f l o w s , the l o c a l component of v o r t i c i t y due t o water movements (£ ) i s much s m a l l e r than the p l a n e t a r y v o r t i c i t y at t h a t l o c a t i o n ( f ); exceptions occur near the equator, where f v a n i s h e s , and i n boundary c u r r e n t s l i k e the G u l f Stream, where c o n s i d e r -a b l e shears are found. B a r r i n g these s p e c i a l cases, we can assume t h a t f » £ at a l l p o s i t i o n s and times, and that the motions s t u d i e d w i l l not d i s r u p t t h i s s t a t e of a f f a i r s . T h i s i s a l s o j u s t i f i e d by the assumption of s m a l l Rossby number, U / L X 2 « 1 , which has been made above. U/L can be regarded as a measure of the l o c a l v o r t i c i t y £ . Equation ( 5 4 ) i s then used i n the approximate form f / ( H + 7 7) c o n s t . Let us see what c o n s t r a i n t s the p o t e n t i a l v o r t i c i t y equation imposes on the motions of the f l u i d when f » £ . F i r s t c o n s i d e r f o r s i m p l i c i t y a s i t u a t i o n where the depth i s constant, so t h a t i t does not a f f e c t the p o t e n t i a l v o r t i c i t y b a l a n c e . I f a p e r t u r b a t i o n i s i n t r o d u c e d i n the system without the a p p l i c a t i o n of torques, the p o t e n t i a l v o r t i c i t y w i l l remain constant, and the approximate form of ( 5 4 ) s t i l l h o l d . Suppose we do t h i s by changing the 45 s u r f a c e e l e v a t i o n V ; there must then be an e q u i v a l e n t change i n the denominator t o keep the r a t i o c o n s t a n t . That can be achieved only by changing f , t h a t i s changing the l a t i t u d e . The f l u i d thus moves t o a d i f f e r e n t l a t i t u d e , where f has j u s t the r i g h t value f o r the p o t e n t i a l v o r t i c i t y balance 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 w i t h i n e r t i a , and i t w i l l overshoot t h a t l a t i t u d e and f i n d i t s e l f i n the same kind of imbalance on the other s i d e . The s i t u a t i o n i s then analogous t o g r a v i t y waves about a f r e e s u r f a c e : we have i n s t e a d v o r t i c i t y waves about a c r i t i c a l l a t i t u d e . Rossby proceeded a l o n g such g e n e r a l l i n e s i n h i s f i r s t i n v e s t i g a t i o n of p l a n e t a r y waves(1939). He considered the waves as p u r e l y two-dimensional, so t h a t (51) reduces t o d _ l _ = - £ v , (55) dt where ft i s the r a t e of change of f w i t h l a t i t u d e , a cons-t a n t i n the plane p r o j e c t i o n used by Rossby ( the /? -p l a n e ) . In the -plane, and i n Rossby's n o t a t i o n , u and v are v e l o c i t i e s t o the east (x) and n o r t h (y) and £ z z d_v - dn . Assuming motions independent of l a t i t u d e , and l i n e a r i z i n g (55), a simple wave equation a p p l i c a b l e t o the study of p l a n e t a r y waves i n the /?-plane r e s u l t s : 46 For a simple v o r t i c i t y wave of the form <f = s i n k ( x - c t ) , (56) gives a phase v e l o c i t y of c — — • Although many of the s i m p l i f i c a t i o n s present in Rossby's analys i s have not been made in d e r i v i n g the amplitude equations (45) and (43). so that they are considerably more complicated than the simple wave equation (56), t h e i r solut ions which depend on the v a r i a t i o n of the C o r i o l i s parameter w i l l sa t i s fy expression (54) expressing the conservation of po ten t ia l v o r t i c i t y . Extra complications over Rossby's problem w i l l be due to ft not being a constant, the amplitude of the waves varying with l a t i t u d e , and the presence of boundaries and var iab le bottom topography. We can then expect planetary wave so lut ions in the present problem to be considerably more complicated than those found by Rossby. Closer considerat ion of (54) shows that depth v a r i a t i o n s can have exactly the same kind of influence as v a r i a t i o n s of planetary v o r t i c i t y ; motions of a nature s i m i l a r to planetary waves can then occur in the presence of depth gradients alone. Such waves on cont inenta l shelves have recent ly been invest igated by Robinson (1964). When both f and H are allowed to vary, as w i l l be the case here, t h e i r respect ive e f fects can re in force or cancel each other. There w i l l then be two main forces ac t ing on the f l u i d in the model used: grav i ty and C o r i o l i s force . The influence of grav i ty i s wel l known; we have seen however, following an analysis s i m i l a r to Rossby's (1939), that i f the C o r i o l i s parameter or the depth of the basin varies with position, 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 so-called planetary waves to which we w i l l pay p a r t i c u l a r attention. 43 V I . SYMMETRIC OCEAN; GENERAL CONSIDERATIONS. Before attempting to f ind e x p l i c i t so lut ions for the equations descr ib ing the var ia t ions of the wave amplitude with pos i t i on and f o r d i f f e r e n t values of the wave parameters, i t would be wise to examine these equations together with the relevant boundary condit ions and to see under what circumstances they admit s o l u t i o n s . This w i l l prevent looking f o r so lut ions where they cannot be found, and a l so give some q u a l i t a t i v e idea of t h e i r aspect . The analys i s can be done much more simply when the problem has complete symmetry around the pole (except for the e^ s ^ p a r t ) , as only ordinary d i f f e r e n t i a l equations remain. In the absence 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 , ( 4 5 ) and ( 4 3 ) become so s i m i l a r in form that they w i l l be analysed as one: d 2 F f dF 4 ^ 2 + " 2 + 1 ( 1 + r d H ) l d r 2 Ldr U a 2 - cu 2 r H dr J ( 5 7 ) - F f s 2 , 2 f l s 4 ^ 2 + < u 2 , ( 4 f l 2 - co 2) 4. 2 f l s dHJ= 0 l r 2 w R 2 4 f t 2 - o j 2 S H drJ The f i r s t term of ( 4 5 ) has been expanded, br ing ing in a s l i g h t s i m p l i f i c a t i o n ; when there i s no bottom slope, dH/dr vanishes in (57); the constant gH must be changed to g H,hp/p 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 studied. Some non-dimensional variables and abbreviations are now introduced so as to make (57) more compact: UJ1 = cu/2X2 , H' = H/H q ; H Q = depth at the pole, x = r / ^ , 2 1 - co ' 2 R2 (58) M = 4X2 2 R 2 , - gH S = S £ + (1 - u ' 2 ) M r f / R 2 . In t h i s modified notation, (57) i s how x 2 d 2 F + xdF ( € x 2+ 1 + 2L_ dH' ) dx 2 dx HT dx - F ( s 2 + S x 2 + x s _ dH' ) = 0 . (59) cu'H' dx This l a s t form w i l l be useful when we come to looking f o r e x p l i c i t a n a l y t i c a l solutions. The problem i s completed by a statement of the boundary conditions: F must be f i n i t e and analytic everywhere i n the basin and on i t s boundaries, and at a v e r t i c a l wall at x • 1, the following must hold f o r u to vanish (by (33),(34), 50 (35))-dF dx x =1 _ s CO F ( l ) (60) Note that g r a v i t a t i o n a l effects are represented by M, and curvature effects by e ; the r e l a t i v e magnitudes of expressions containing these quantities w i l l decide which of the two, curvature or gravity, 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 to f i n d f o r what values of frequency, wave number or bottom slopes the system (59)-(60) admits solution, i t w i l l be necessary to recast the problem in a d i f f e r e n t form. A new variable, £ , i s introduced and defined by I = 1/x . Like any l i n e a r ordinary d i f f e r e n t i a l equation of second order, (59) can be rewritten as a system of two f i r s t order equations in the o r i g i n a l dependent variable (F) and a new one Z defined by the equations of the system dF _ f - 1 exp(- e/2£ 2)1z = A ( £ , co ' ) Z ( 6 l ) d £ U H ' J dZ _ - f H' ( s2t2+ 8 + s£ dH')exp( c /2 £ 2)11 F d£ \C3 ^' H' dx J = - B(£ , c o ' , s) F . (62) 51 The boundary condition (60) i s s i m i l a r l y reformulated, in terms of F and Z, as F ( l ) = aS_ exp (- e/ 2 ) (63) Z ( l ) sH' The method used to study existence conditions i s sometimes c a l l e d the "Method of Signatures", and i s widely used in the Hamiltonian formulation 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 involved, following Eckart's presentation (Eckart, I 9 6 0 ; Chapter X I V ) . Consider a system of two dependent and one independent variables as follows: dF = A ( £ , X ) z , (64) dZ = -B( £ , X ) F . (65) . H This i s exactly the form in which the present problem has been transformed i n (61) and ( 6 2 ) . The parameter X stands f o r one or more independent parameters of the problem. Equations (64) and (65) are said to be in canonic form, and t h i s may be brought out by formally defining a Hamiltonian, H - |( AZ 2 + B F 2 ) , in terms of which the above system takes a form i d e n t i c a l to the canonical Hamiltonian equations of mechanics,, 52 The (Z,F) space w i l l be called the phase space, and a phase path i s defined as the curve traced i n phase space by a solution of ( 6 4 ) - ( 6 5 ) as £ varies; the instantaneous position of the phase path i s called a phase point. Some theorems can be established about the behaviour of the phase paths according to the signs of the functions A and B (see Eckart); they w i l l not be proven here, but, except f o r the f i r s t one, they are a l l f a i r l y obvious. It i s understood i n the following theorems that the functions A and B are f i n i t e and continuous in £ . THEOREM I. The phase paths are continuous curves. Let Z^,F^ be any point in phase space other than the or i g i n , and £ an arb i t r a r y f i n i t e value of £ ; then there exists only one phase path passing through the point Z^,F^ f o r £ ° ^ i * Moreover, no phase path goes through the o r i g i n f o r f i n i t e values of £ , but some may approach i t as £ — ± CO . THEOREM I I . A l l phase paths intersect the coordinate axes at r i g h t angles. THEOREM I I I . I f £ i s a root of A (or B), then the phase path has a horizontal (or v e r t i c a l ) tangent f o r that value of £ . These l a s t two theorems are i m p l i c i t in equations ( 6 4 ) and ( 6 5 ) . The theorems to follow w i l l be made more evident by writing ( 6 4 ) - ( 6 5 ) in plane polar coordinates: Z = R cos0 , F = R sinQ » 53 de - A cos 29 + B s i n 2 9 ( 6 6 ) d£ 1 dR = ( A - B )sin9 cos9 ( 6 7 ) R d£ Defining the signature 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 expressing i t symbolically as (sign of A, sign of B), i t becomes apparent that much of the behaviour of phase paths can be deduced from knowledge of the signatures along the path. Note that (A), as given by ( 6 1 ) , i s always negative. THEOREM IV. In (+,+) segments, the angular v e l o c i t y ( d9/d£ ) of the phase point i s positive; i n (-,-) segments i t i s negative. Furthermore, f o r such signatures, once the phase path has entered any quadrant, i t s distance to the axis perpendicular to the one i t has just crossed must not increase. THEOREM V. In (+,-) segments, the phase point moves away from the o r i g i n (and from both axes) in the f i r s t or t h i r d quadrants, and towards i t in the other two. The si t u a t i o n i s reversed i n (-, + ) segments. These l a s t two theorems are i m p l i c i t in equations ( 6 4 ) to ( 6 7 ) . The (+,+) and (-,-) segments are often call e d o s c i l l a t o r y segments, and the other ones non-oscillatory. Behaviour of the phase paths under d i f f e r e n t signatures i s i l l u s t r a t e d in Figure 6 . 54 J r v . V r ) Figure 6. Phase paths i n segments of various signatures; the phase point moves in the d i r e c t i o n of the arrow. What happens when £ tends to i n f i n i t y ? I f the path is o s c i l l a t o r y , i t w i l l s p i r a l in or out, or the angular v e l o c i t y may vanish, or the path may become a c i r c l e or an e l l i p s e , depending on the s igns and the func t iona l behaviour of A and B at large £ .When the segment is non-o s c i l l a t o r y , the phase paths have asymptotic d i rec t i ons along l i n e s given by tan 2 9 ^ = lim (-A/B) 55 This i s of course subject to the s t i p u l a t i o n s of theorem V; Figure 7 shows what happens to segments of s ignature (+,-); the s i t u a t i o n is reversed for (-,+) segments, or when £ - - c o . F Z Figure 7. Asymptotic behaviour of phase paths of s ignature ( + ,-) as £—• +00 . I f A and/or B have roots in £ , the signatures w i l l change along a path; segments of d i f f e r e n t signatures w i l l be joined together where A and/or B have roots; according to theorem I I I , t h i s i s at hor i zonta l and v e r t i c a l tangents r e s p e c t i v e l y . An example of a path with compound signature i s given in Figure 8s. \ 56 Figure 8. Phase path with compound signature: (-,+) (-,-) (+,-) (+,+). This i s a l l the theory needed for the intended analys is of the amplitude equations (61) and ( 6 2 ) ; i t s a p p l i c a t i o n can be explained in a few words. The functions A and B (and therefore the signatures) are known e x p l i c i t l y in terms of £ and the parameters grouped under X . Boundary condit ions 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 that the so lu t ion be f i n i t e for a l l values of £ ; so lut ions w i l l then exist only for those values of the parameters r e s u l t i n g i n the r i g h t combination of s ignatures , a l lowing the phase path to remain in regions where the amplitude i s f i n i t e . Although t h i s method i s very simple, i t i s quite powerful, and w i l l give c l ear ind ica t ions as to where the solut ions can be found; i t 57 even reveals what the solutions w i l l look l i k e : zeros and extrema of the amplitude can be read d i r e c t l y from the phase paths. It w i l l be convenient, in order to systematize the analysis to divide the wave number-frequency (s, OJ1) space into four domains, according to the sign of s and the magnitude of a>'; existence c r i t e r i a w i l l then be examined in turn in the four regions of the diagnostic diagram (as the s- a)1 space i s c a l l e d ) . Even though we are concerned primarily with planetary waves, which are of low frequency, * and hence presumably found i n areas I and I I I , i t i s easy to extend the analysis to the whole diagnostic diagram. + t S o III I V II -OJ Figure 9. Subdivision of the diagnostic diagram into four regions. Before we proceed to investigate the existence of solutions in the subdivions of the diagnostic diagram, we should make a survey of the properties of phase paths 53 which are invariant over the whole (s, cu') space. The f i r s t one i s the asymptotic angle to which n o n - o s c i l l a t i n g segments tend at large values of £ . The functions A and B, as given by (6L) and (62) yield f o r the f i n a l angle, according to i t s d e f i n i t i o n in (63) , t a n 2 9 f = lim { — — A £o/.H' dx The f i n a l angle i s thus defined f o r a l l areas of the diagnostic diagram. Furthermore, i f the non-dimensional depth, H's and i t s derivative are f i n i t e and continuous near the pole, (£ — +CO ), so that i t can be reduced to a MacLaurin series around x=0, the f i n a l angle then s i m p l i f i e s to t a n 2 9 F = 1/s 2 (69) This i s e a s i l y shown to be v a l i d under the above s t i p u l a t i o n s : as x—*0, Hf — 1 , and dH'/dx OC n.x n~\ where n i s p o s i t i v e and the lowest power of x in the series expansion of H !. The second term of the denominator therefore varies as x n when x — 0 , so that i t vanishes in the l i m i t . Another more sp e c i a l r e s u l t can be derived concerning the behaviour of phase segments at large values of £ ; only the magnitude of the f i n a l angle i s defined by (69), and we 59 can obtain more detailed information as follows. Let us expand d9/d£ through (61) and (62); at large values of £ , and fo r a constant depth (H' — 1), we fi n d that d9 _ c o s 2 9 ( - l + s 2t a n 29 + tan 29 ) (70) d£ t . £ 2 In (-70)terms i n £ ~ n have been neglected f o r n > 2. It should be remarked that since the terms neglected i n deriving the expression for 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). It .is useful in the present discussion to divide phase space i n two regions as i l l u s t r a t e d i n Figure 9a. In region I, tan9< l / | s | ; i n region I I , tan9>l/|sl . F Figure 9a. Phase oaths at large £ . / 59a Since £ i s v e r y l a r g e , the s i g n a t u r e of the phase paths d i s c u s s e d w i l l be assumed to be (-,+). For b r e v i t y , we w i l l a l s o denote (s t a n 9 -1) by m/^ .2, without worrying about the exact form of the constant m, except t h a t i t w i l l be p o s i t i v e i n r e g i o n I I , n e g a t i v e i n r e g i o n I . In terms of m , (70) becomes d9 _ c o s 2 9 (m + JL. ) • (71) I have used (69) to s i m p l i f y the l a s t term of (70). Only the h a l f space Z > 0 w i l l be d i s c u a s e d , the s i t u a t i o n being s i m i l a r on the other s i d e of the F a x i s . From theorem V, a l l phase paths w i t h s i g n a t u r e (-, + ) i n the f o u r t h quadrant w i l l depart from the o r i g i n and both axes, being asymptotic t o the l i n e F = -Z/ |s| , so t h a t they do not correspond to any e i g e n s o l u t i o n s . In r e g i o n I of the f i r s t quadrant, phase paths with s i g n a t u r e (-, + ) w i l l tend t o the o r i g i n a long the l i n e F = Z/|S| provided they have d9/d£ > 0 at l a r g e £ . As we w i l l see l a t e r when we c o n s i d e r phase paths which s t a r t ( £ = 1 ) i n t h a t region,d9/d£ < 0 for£ = 1 f o r a l l f r e -quencies not too near the i n e r t i a l frequency. I f the phase path i s t o go back to the asymptotic l i n e i n the f i r s t quadrant, there must then be a h o r i z o n t a l tangent: t h i s i m p l i e s a r o o t of A, of which t h e r e i s none. No e i g e n -s o l u t i o n s w i l l e x i s t f o r such phase paths e i t h e r . 60 F i n a l l y , phase paths i n r e g i o n I I of the f i r s t quadrant w i l l tend t o the o r i g i n provided d9/d£ i s negative at l a r g e £ , which i s the case when S i s nega-t i v e and l a r g e enough. I f | S | i s too s m a l l , the phase path w i l l go i n t o the second quadrant; i f i t i s too l a r g e , the phase path w i l l c r o s s the asymptotic l i n e and wander i n the f o u r t h quadrant. E i g e n s o l u t i o n s w i l l then e x i s t only f o r those values of cu' which a l l o w the phase paths t o approach the asymptotic l i n e so c l o s e l y t h a t they do not depart from i t again, but never c r o s s i n g t h a t l i n e . For any g i v e n s and GJ' t h e r e i s but one such phase path. AREA I . In t h i s r e g i o n of the d i a g n o s t i c diagram, propagation i s towards the .west (s<0) and frequency l e s s than i n e r t i a l ( o,' = l ) ; from the boundary c o n d i t i o n (63) the phase path w i l l s t a r t i n the f o u r t h or the second quadrant, s i n c e F ( l ) / Z ( l ) i s < 0 t h e r e . I t i s immaterial f o r the a n a l y s i s which quadrant i s chosen, and the s t a r t i n g ' p o i n t w i l l be taken i n the f o u r t h . Again from (63), t h a t p o i n t w i l l be l o c a t e d between the l i n e s F = -Z/ s and F = 0, u n l e s s the depth a t the boundary i s much l e s s than at the pole ( H ' ( l ) = 1). The s i g n a t u r e w i l l be e i t h e r (-,-) or (-,+) s i n c e A i s always negative and only B can change s i g n . When B>0, theorem V s t a t e s that the phase path w i l l move away from the o r i g i n and both axes. There are no s o l u t i o n s i n t h a t area of the d i a g n o s t i c diagram c o r r e s p o n d i n g t o such v a l u e s of wave number and frequency 61 Figure 10. Phase paths f o r area I. such that B>0 f o r a l l values of £ . In order that the phase path remain i n regions of phase space where F and Z are f i n i t e , B must be negative f o r a wide enough range of values of £ to have an o s c i l l a t o r y segment of length s u f f i c i e n t to allow the phase point to leave the fourth quadrant. For a solution t o exi s t , the phase path must be i n the f i r s t or t h i r d quadrant when B changes sign. B i s negative when s2£ 2 + s j € + I_ dH») + M(l - O J ' 2 ) ^ < 0 . (73) " J * H» dx "^ 2 62 This inequality w i l l be s a t i s f i e d either for very low frequencies or, considering 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 ant i c i p a t e that the low frequency solutions w i l l be planetary waves. It i s doubtful that the present l i n e a r i z e d formulation can treat the nearly i n e r t i a l solutions adequately, since the v e l o c i t i e s are not regular at that frequency in the present model. It appears at once from (73) and theorem IV that i f the inequality i s s a t i s f i e d f o r one value of frequency, w say, then i t is going to be s a t i s f i e d by an i n f i n i t y s,o i of successively lower frequencies. Let us say that ^ s, o i s the f i r s t frequency that i s found to produce a segment of signature (-,-) long enough t o allow B to change sign in • the t h i r d quadrant, in which theorem V rules that the phase path w i l l move towards the o r i g i n as £ — + CD , and thus allow solutions to e x i s t . Clearly, i f a lower frequency, ^ i s chosen such that B remains negative f o r larger ' 2 2 values of £ ( i . e . , i t takes longer f o r the s £ term in (73) to catch up with the negative terms), the phase path w i l l remain of o s c i l l a t i n g nature f o r an extended i range of £ and, i f w i s properly chosen, i t w i l l s, 1 reach the f i r s t quadrant before B'changes s i g n . An i n f i n i t e series of solutions can then exist f o r a same value of wave-number, s,: the system i s degenerate in s. Denoting the solutions by F (x), with frequencies to (with s j n s j n n = 0 , 1 , 2 . . . . ) , the phase path w i l l wind around in a 63 clockwise d i r e c t i o n r e q u i r i n g an extra IT radians to reach the o r i g i n between each successive so lu t ion , and the » i frequencies w i l l s t ead i ly decrease: ai , < a) . The ^ J s ,n+l s ,n so lu t ion F (x) w i l l have n zeros between the pole and s ,n the boundary since the phase path crosses the F axis n times; n i s then an index number denoting the number of r a d i a l nodes of the s o l u t i o n . The influence of bottom topography on the solut ions can be examined through t h e i r effect on the inequa l i ty (73). Let us consider nearly uniform bottom v a r i a t i o n s for the moment; the slope having the same s ign f o r a l l x and being nearly Constant. I f the depth of the basin decreases from the pole to the boundary (dH'/dx < 0 ) , then, as | dH' /dx increases , the negative part of B is made more negative, and i t w i l l take a smaller frequency to br ing the phase path to the same point f o r the same value of £ as when there i s no bottom s lope . Negative slopes (with respect to the var iab l e x) have therefore the ef fect of reducing the frequency of planetary waves; slopes of the opposite s ign w i l l of course have the opposite e f f e c t . I f dH' /d : i s large enough so that -jj-, dH'/dx + € < 0 f or a l l values of £ , the inequa l i ty (73) w i l l not hold for any value of frequency i n the f i r s t region of the d iagnost ic diagram. There i s then a maximum uniform slope ( dH'/dx< 0) a l lowing westward propagating planetary waves to ex i s t in a symmetric polar b a s i n . It i s given by IdH'/dx > e H '(l) . 64 I f we do not i n s i s t that the bottom slope be uniform in x, solutions can be found f o r bathymetries that w i l l include l o c a l slopes greater than the maximum uniform slope. The only condition to be s a t i s f i e d i s a f t e r a l l that the phase path escape from the i n i t i a l quadrant and that B has i t s l a s t root when the phase point i s i n the t h i r d or the f i r s t quadrant. The function B can have more than one root i f the depth varies strongly enough; a simple example of such a s i t u a t i o n , i l l u s t r a t e d below, also shows the effect of sloping walls at the same time. Consider a basin bounded by walls with slopes greater than the maximum average slope allowing westward propagating waves; the bottom i s f l a t beyond the sloping walls and remains so u n t i l the pole i s reached ( F i g . 1 1 ) . The signature w i l l i n i t i a l l y be (-,+) and the radius of the phase point w i l l increase. When a value of £ corresponding to the rapid decrease in slope i s reached, B w i l l become negative i f the frequency i s small enough, and the phase path can be carried F Figure 1 1 . Bottom p r o f i l e and phase paths f o r sloping walls i n Area I of the diagnostic diagram. 65 to the t h i r d or to the f i r s t quadrant, where B w i l l have i t s second root. The solutions w i l l therefore be s i m i l a r to those f o r a f l a t bottom, but with lower frequencies. There is then a wide class of bathymetries allowing westward propagating planetary waves in a symmetric polar basin, the general r e s t r i c t i o n being that f o r some wave number s and bottom configuration H'(x) there exists a r e a l value of frequency which allows B to have i t s l a s t root in a quadrant of the phase space where the phase path w i l l tend to the o r i g i n f o r large values of £ • AREA I I . The propagation i s s t i l l westward, but the frequency i s now greater than i n e r t i a l . The s t a r t i n g point i s again taken in the fourth quadrant, but may now very l i k e l y be located below the l i n e F = -Z/k ( F i g . 12). Once more, there w i l l not exist any solutions f o r B > 0; the inequality B< 0 i s s t i l l expressed by (73), but since co ' i s now > 1, (making € < 0) t h i s w i l l hold either when 2 2 i (1- to' )M r ' / R 2 dominates or when dH/dx>o and large. Gravity terms in M, or bottom slopes, are now more important than curvature terms ( l / R in e ), The frequency necessary to make the'inequality hold i s however not as large as one might think; f o r s = -1, (73) i s already s a t i s f i e d f o r co' =1.5. This may then be a type of motion f o r which i t 2 i s not j u s t i f i a b l e to neglect e x as compared to 1, in (59). The lowest frequency which makes B< 0 w i l l increase with increasing k («|s|) and r,; the influence of bottom 66 F Figure 12* Phase paths f o r area I I . slopes in the Inequality (73) i s as i n area I, but the frequency i s altered i n a d i f f e r e n t manner since B i s now made negative by terms i n which i s large instead of terras i n which i t i s small, as i n area I. A depth decreasing r a d i a l l y now causes an increase instead of a decrease i n frequency over the corresponding f l a t bottom solu t i o n . The solutions are also subjected to an i n f i n i t e degeneracy i n wave number; they are controlled mostly by gravity e f f e c t s / and cannot be c a l l e d planetary waves, AREA I I I . Propagation i s now to the east, while frequency i s less than i n e r t i a l . The s t a r t i n g point w i l l be taken i n the f i r s t quadrant, and, unless H'(l) << 1, i t w i l l be found under the l i n e F = Z/s (F i g . 1 3 ) . According to the e x p l i c i t expression" f o r B ( 6 2 ) , the signature w i l l 67 Figure 13. Phase paths f o r area I I I . (A). Flat bottom. ( B ) Large positive depth gradients: dH'/d£ > 0 . always be (-,+ ) unless there i s a s u f f i c i e n t l y large negative depth gradient, £H.\ dx Let us f i r s t look f o r f l a t bottom solutions. The signature i s (-,+) and, according to theorem V, the phase path moves towards the o r i g i n for that signature in the f i r s t quadrant. There w i l l then be solutions i f the phase path does not escape to the neighbouring quadrants. There w i l l not be any solutions i f the phase path moves into the second quadrant and stays there (theorem V); but once in quadrant 11$ 63 the phase path cannot loop back i n t o the f i r s t quadrant, because t h i s would imply a root of B, of which t h e r e i s none. For f r e q u e n c i e s not too near the i n e r t i a l frequency, the angular v e l o c i t y of the phase path, d 9/d£ , i s f o r £ = 1 of the s i g n of (s 2 + 8 ) w ' 2 / ? - 1 ; t h i s i s s negative f o r w ' < 0 ,97 • A c c o r d i n g t o the a n a l y s i s of page 59a , there w i l l then be no e i g e n s o l u t i o n s f o r s > 0 and o/ < 1 i n the absence of bottom s l o p e s . There are then no eastward propagating p l a n e t a r y waves i n a symmetri-c a l p o l a r ocean i n the absence of bottom s l o p e s . When l a r g e n e g a t i v e r a d i a l depth d e r i v a t i v e s are present, i t i s p o s s i b l e t o choose a frequency low enough so t h a t there i s a segment of phase path of o s c i l l a t i n g nature e n a b l i n g the f i r s t or t h i r d quadrant t o be reached i n a manner c o n s i s t e n t w i t h the e x i s t e n c e of s o l u t i o n s . The same m u l t i p l i c i t y of s o l u t i o n s i s found as i n areas I and I I . The i n f l u e n c e of bottom s l o p e s i s now such t h a t f r e q u e n c i e s of e q u i v a l e n t s o l u t i o n s w i l l now i n c r e a s e as the s l o p e i n c r e a s e s . AREA IV. The frequency i s here l a r g e r than i n e r t i a l ( aj'> 1 ) , and propagation t o the east ( s > 0 ) . The s t a r t -i n g p o i n t i s now found (again i n the f i r s t quadrant) above the l i n e F = Z / s , provided H f ( l ) i s not much l a r g e r than u n i t y . For B> 0 f o r a l l £ , th e r e may be eigensolutiooss as shown i n F i g u r e 1 4 . There i s then only one e i g e n s o l u t i o n f o r each wave number and no m u l t i p l i c i t y as i n the previous 69 F i g u r e 1 4 . Phase paths f o r Area IV. / i n s t a n c e s . When the frequency i s l a r g e enough t o make the t h i r d term of (73) dominant, and the s i g n a t u r e (-,-) oyer a segment of phase path, s o l u t i o n s of the same ge n e r a l nature as those encountered i n areas I and I I w i l l be found. Depth v a r i a t i o n s have the same i n f l u e n c e on the s i g n of B as they have i n area I I I ; as i n area I I , the waves are c o n t r o l l e d mostly by g r a v i t y and are not c l a s s -i f i a b l e as p l a n e t a r y waves. I have obtained i n t h i s s e c t i o n , by the semi-q u a n t i t a t i v e Method of S i g n a t u r e s , v a l u a b l e i n f o r m a t i o n 7 0 as to where in the diagnostic diagram solutions compatible with the approximations and the boundary conditions can be found. Much information about the q u a l i t a t i v e aspect of solutions i s revealed by the behaviour of phase paths, and extensive degeneracy in wave number has been discovered in the possible s o l u t i o n s . In p a r t i c u l a r , i t has been discovered that low frequency o s c i l l a t i o n s controlled mostly by curvature effects and presumably i d e n t i f i a b l e with planetary waves can be found only in area I of the diagnostic diagram i n the absence of bottom slopes. L i t t l e more information can be obtained from the Method of Signatures, and the amplitude equation must now be solved e x p l i c i t l y to fi n d the form of the solutions and t h e i r frequencies. 71 V I I . FLAT BOTTOM SOLUTIONS: SYMMETRIC BASIN. i ) S o l u t i o n i n terms of c o n f l u e n t hypergeometric f u n c t i o n s . S t a r t i n g w i t h the s i m p l e s t case, which has a l r e a d y been s t u d i e d i n d i f f e r e n t f o r m u l a t i o n s (Goldsbrough, 1914 a, Longuet-Higgins, 1964 b) and through which a check on the p o l a r plane approximation can be made, I study f i r s t the o s c i l l a t i o n s of a b a s i n w i t h a f l a t bottom and symmetrical boundaries. In the absence of depth v a r i a t i o n s , the symmetric amplitude equation (59) reduces to x 2 d 2 F + xdF( e x 2 + 1) - F( s 2 + 8x 2) = 0 , (74) d x 2 dx while the boundary c o n d i t i o n (60) i s u n a l t e r e d . I f we c o n c e n t r a t e our a t t e n t i o n on l o n g p e r i o d p l a n e t a r y waves, i n which cu' i s so s m a l l that we can s a f e l y n e g l e c t e x with r e s p e c t to 1, (74) i s c o n s i d e r a b l y s i m p l i f i e d and can be s o l v e d i n terms of B e s s e l f u n c t i o n s . We have seen however t h a t i n area I I of the d i a g n o s t i c 2 diagram there were motions f o r which e x might not be e n t i r e l y n e g l i g i b l e , and f o r which (74) should be kept i n i t s e n t i r e t y . I t w i l l then be u s e f u l , i n view of p o s s i b l e f u t u r e e x t e n s i o n of t h i s work to types of motion other than p l a n e t a r y x^aves, t o s o l v e the more exact 72 equation (74), p a r t l y t o i l l u s t r a t e the method used and i t s p o s s i b l e a p p l i c a t i o n i n other cases, p a r t l y as an a p r i o r i j u s t i f i c a t i o n of the assumption t o be made i n (74) when p l a n e t a r y waves are s t u d i e d . In the f o l l o w i n g pages, I w i l l then f i n d s o l u t i o n s of the more exact equation, (74); these pages may be omitted without s o l u t i o n of c o n t i n u i t y and the reader may, i f he wishes so, proceed 2 t o page 8 8 , where the approximation to (74) w i t h e x << 1 i s t r e a t e d , as a p p r o p r i a t e t o the study of p l a n e t a r y waves. Equation (74) can be transformed i n t o the standard form of the c o n f l u e n t hypergeometric equation ( a l s o c a l l e d Kummer's equation) by the change of v a r i a b l e s F = x ± S u ( y ) ; y = - ex2/2 . (75) The r e s u l t i n g equation f o r u(y) i s y u " + u ' ( l ± s - y) -u(±s€ -S ) = 0 , (76) i n which primes i n d i c a t e d i f f e r e n t i a t i o n with r e s p e c t t o y. The a p p r o p r i a t e s o l u t i o n of (76) must be such t h a t F ( x ) = x " u(y) v a n i s h e s , or at l e a s t remains f i n i t e , at the pole ( x = y = 0 ). Furthermore, i n order to i n s u r e c o n t i n u i t y of the s o l u t i o n s around the pole, one must have e i ( ZTT + cfi )s = eisc£ ? s o t h a t s must be zero or an i n t e g e r , p o s i t i v e or n e g a t i v e . 7 3 Writing Rummer's equation in general form, with parameters a and b, i t i s yu" + u ' ( b - y ) - a y = 0 . ( 7 7 ) Compariton of (76) and (77) shows that here, b = 1 ± s, and a = ( ± s - 8)/2 . The parameter b i s therefore an integer; in such a case (77) has two independent solutions: the f i r s t one i s f i n i t e at the pole and i s ^ F ^ 3 ; °; y ) » and the second one i s a logarithmic solution which diverges at y = 0 (Slater, I960; Chapter I ) . The only solution of (74) which w i l l remain f i n i t e at the pole f o r i n t e g r a l values of s i s then F(x) = c x k F (kg - 8 ); 1+k; -tx2/2 , (73) 2 i n which c i s an a r b i t r a r y constant and k i s |sl . The function •^F^(a: b; y) i s the standard hypergeometric notation f o r the series 1 + a y + a ( a +1) y 2 + a ( a + l ) ( a + 2) _ j r 3 + . . . (79) b b ( b + l ) 25 b ( b + i ) ( b + 2) 3» The boundary condition (60) can also bje expressed i n terms of the solution of the problem. Using the f d i f f e r e n t i a t i o n formula f o r , F, (Slater, I960; Chapter 2) 74 d_ 1 F 1 ( a ; b; y) = a ^ ( a 1; b 1; y) , ( d Q } dy b and a recurs ion formula a^F ( a + l ; b + l ; y)= ( a - b ^ U ; b+1; y j + b ^ a ; b; y) (31) the boundary condi t ion (60) becomes ^ ( ( k f ^ ) ; * + 2; - « / 2 ) ^ ( k + 1 ) ( a / u | , , k + g ) (k/2 +1)€ +S/2 F ( ( k e - 8 ) ; k+ 1 ; - e / 2 ) 1 >• O £ / 2 e (32) Confluent hypergeometric functions are tabulated only for positive values of the argument y, so that when e > 0, i t w i l l be more convenient to express the solution in terms of functions of + ex 2/2. This i s done by using Kummer's f i r s t theorem: e ' ^ F (a; b; y) = ^ ( b - a ; b;-y) (33) Instead of (73) we then have for solution F(x)= cx kexp(-€x 2/2) ^ k/2+l+ S/2e ; k+1; €x 2/2).(34) and f o r the boundary condition k + 2 ; */ 2 ) _ 2 { k + 1 ) ( k . s/o,M F f k/2 + 1 +S/2e; k + 1; €/2 Before attempting to extract anything from the boundary conditions (82) or ( 8 5 ) , i t w i l l be i n s t r u c t i v e to review a few of the desc r i p t i v e properties of the series •jF-^. (Slater, I 9 6 0 ; Chapter 6 . Jahnke and Emde, 1945; Chapter X). The variable y w i l l be assumed po s i t i v e ; i f i t i s not, the function can always be transformed vi a (83) to make i t so. 1- There w i l l not exist any roots of 1 F 1 ( a ; b ; y) unless either a or b or both are negative. In t h i s work, b ( = k + l ) i s always p o s i t i v e . 2 - If b > 0 and a = - n + " 0 , where n = 1 , 2 , 3 . . . and 0 < © < 1 , then ]Jf-jJa; b; y) has exactly n zeros, and i f y n i s the root at the largest value of y, y n ~ * 0 0 as n—- CD . 3- It i s evident from (79) that i f a and b are > 0 , a) 1F ]_( a+1; b; y) > ^ a; b; y) > 0 b) 1 F 1 ( a; b+1; y) < ^ a; b; y) . 4- If y^ and y 2 are roots of ^ F i ( a i ? b-^ ; y) and -.F-, (a 9;b p;y) respectively, and a, and a ? are both negative, (d'5) 76 Figure 15, The confluent hypergeometric funct ion 1 F 1 ( a ; b; y) as a funct ion of y for a few values of a and b . 77 then, a) I f l a ^ > |a 2i while = b 2, then y < ; b) If a 1 = a 2 and 0 < b 1 < b 2 , then y < y 2 . c) I f y m i s the m t n root (from the o r i g i n of y) of -,F, ( a; b; y) then, when b > 0 , as a---GO, y — 0. 5- At y= 0 , (from (79)) the Confluent hypergeometric series i s equal to unity f o r a l l values of a and b. 6- As y — + CO , the series tends exponentially to +00 i f there i s an even number of roots, to -CO otherwise. Figure 15. i l l u s t r a t e s some of the above points and gives an idea of some aspects of the confluent hypergeometric s e r i e s . We found in section VI that in a symmetric ocean with a f l a t bottom low frequency solutions corresponding to planetary waves could exist only in area I of the diagnostic diagram (Figure 9 ) . Concentrating our attention on the planetary wave solutions, we see that since in area I of the diagnostic diagram s < 0, cu ' < 1, then e > 0, and the eigenvalue problem consists in f i n d i n g the value of frequency cu ' which w i l l s a t i s f y the boundary condition in the form (35). Such a frequency equation i s often c a l l e d a c h a r a c t e r i s t i c equation. I f we wished to study the. high frequency solutions which can be found in areas II and IV of the diagnostic diagram (where cu ' > 1, and consequently € < 0) the same method as w i l l be used below would be followed, but s t a r t i n g from equation (32). 73 Since in (85) the parameters of the confluent hypergeometric function as well as the r i g h t hand side depend on frequency, i t w i l l be wise to examine the boundary condition as a function of frequency. The frequency dependent expression on the r i g h t hand side of ( 8 5 ) , hereafter referred to as (85 rhs), i s plotted i n -Figure 16 against frequency f o r values of s corresponding to area I of the diagnostic diagram ( s < 0) . For k ( = Is| ) >1, (85 rhs) i s always posi t i v e and at least of 2 order 10 for values of frequency not too near the i n e r t i a l frequency ( c o ' = l ) . This i s also true for s = - 1 , except in a range of values of frequencies where (85 rhs) i s large and negative. From section VI, we expect that the planetary waves w i l l be of low frequency so that f o r k > l , the r a t i o of the two confluent hypergeometric functions w i l l be large and p o s i t i v e ; for s = -1, that r a t i o w i l l be large also ( > 10 2) , but w i t s sign w i l l depend on whether the eigenfrequency i s to the right or to the l e f t of the s i n g u l a r i t y in (85 r h s ) . The v a r i a t i o n of the t h i r d variable of the confluent hypergeometric functions on the l e f t hand side of ( 8 5 ) , . € / 2 , i s quite simple and i s i l l u s t r a t e d in Figure 17. e/2 becomes large only near o)T = l , and i s small and varies only very slowly near co' = 0. The f i r s t variable, 'a', in each of the functions of the l e f t hand side of (85) i s , when 8 i s expanded i n Figure 16. The expression (6*5 rhs) as a function of frequency, co 1» 80 Figure 17. The expression e/2 as a funct ion of frequency a) ' . terms of i t s d e f i n i t i o n in (58), ' a ' = l + k ( 1 - + M ( 1 - a;' 2) ( g 6 ) 2 2 ( 1 +a>'2) This expression i s p lot ted against frequency f o r a few negative values of s (Figure 18). The value of M ( = 20) appropriate to the homogeneous mode of o s c i l l a t i o n has been used, but the shape of the curves i s very s i m i l a r , although on a d i f f eren t sca le , when i n t e r n a l o s c i l l a t i o n s i are s tudied . With M = 0.53x10 , the v e r t i c a l scale i s m u l t i p l i e d by about 10^ " and the roots moved to the extreme 31 Figure 18. The parameter ' a ' (86) p lot ted versus frequency at' for a few values of s ( s < 0). 32 l e f t of the frequency scale. From Figure 18 and the descriptive considerations on the properties of confluent hypergeometric functions, t h i s much can be concluded about the l e f t hand side of ( 8 5 ) . For a l l frequencies higher than the root of ( 8 6 ) , a > 0 , b > 0 , and from property 3b the r a t i o w i l l be positive but less than unity. For co1 between zero and the root, a < 0 , b > 0 and thus, from property 2, the r a t i o can assume either sign and amplitudes from zero to CO „ Consideration of (85 rhs) has already revealed that the r a t i o of confluent hypergeometric functions w i l l be at 2 least of order 10 in magnitude, so that i t i s only in the l a t t e r region (between the root of 'a' in Figure 18 and t o ' = 0 ) that solutions of (85) w i l l be possible. It i s to be noticed that f o r s = -1 the root of (86) (Figure 18) occurs at a lower value of frequency than the f i r s t s i n g u l a r i t y in ( 8 2 rhs) (Figure 1 6 ) , so that in the frequency range where solutions can exist the r a t i o of the confluent hypergeometric functions must be large and p o s i t i v e . Curves of the functions ^F^( a; b 1; y) and ^F-^( a; b; y) w i l l follow each other f a i r l y c l o s e l y when plotted against y (Figure 1 9 a ) , and the functions w i l l have the same number of roots. The r a t i o of two such functions, as in ( 8 5 ) , can be of order 1 0 2or greater only near a root of the denominator. The l e f t hand 3 i d e of (85) w i l l be large and po s i t i v e at the positions y-j (,i = 1,2..) just to the l e f t of zeros of ^ F^( a; b; y) (Figure 19a). The precise value of the positions y. w i l l 3 of course depend on the value of the other parameters ('a' and 'b') of the confluent hypergeometric functions, i . e . , w i l l depend on the frequency co ' through the d e f i n i t i o n of 'a' i n (86K At low frequencies, the variable y (= € / 2 ) i s nearly constant: e/2 — r£/^2 . Although the number of positions y . i s equal to the number of roots of i F, , only that position y.( co ') — r / 2 w i l l J 1 2R give an eigenfrequency. I t i s apparent from descriptive properties 1 and 4c) that, as co' decreases and 'a f becomes more negative (Figure lo*) , the m t n root (m fixed) of F (a;b;y) tends to the origin of y. But as each m root ' f i r s t ' appears (when 'a' decreases to -m), i t does so at a higher value of y than f o r the (m-1) root. Hence, as co' decreases, these roots must in turn, in order m, successively move to the l e f t along the y-axis and pass, through y = r f / O D 2 . For a series of decreasing values of frequency the l e f t hand side of the boundary condition (85) passes through a high positive value equal to the right hand side; t h i s happens at the positions indicated 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 or each value of wave number s. Figure 19. The r e l a t i v e positions of the confluent hypergeometric functions at the eigenvalues, a) The positions y. where the r a t i o of the two functions i s large and p o s i t i v e , b) The f i r s t eigenvalue, c) The second eigenvalue. 85 The eigenfrequencies w i l l be calculated on the assumption that we are close enough to the root of the denominator of the l e f t hand side of (85) to approximate the n t n eigenfrequency by the n t h root of 1F 1(k/2 +1 + $/2e k + 1; €/2). A f i r s t approximation to the n^ n root of 1 F 1 ( a; b; y) i s (Slater, I960; Chapter 6) -3/4J 2 (2b - 4a) = TTH n + b/2 - )' n 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. Substitution f o r the dummy variables a,b,y i n terms of the actual variables gives f o r the -root •/2 = " Ms,n 2 + 2 S/e ( 8 3 | 2 2 where ^ = tr / (4n+2k -1) . When S and e are s ' n 1 6 th expanded i n terms of frequency, by means of (58), a 5 . degree algebraic equation i n c*j' r e s u l t s . I f however the 2 ' " frequency i s small enough, cu' << 1, a l i n e a r expression gives the eigenfrequency e x p l i c i t l y : s, n 86 A few eigenfrequencies, as calculated from t h i s formula, 2 are l i s t e d i n Table I I . It i s c e r t a i n l y true that cu' = 1 , so that we may safely pass from (88) to (89). Also, f o r small values of co 1 , the expression (-a + b/2), which has to be large f o r approximation (87) to be.valid, i s found to be equal to M- R / 2 ; the adopted value of s,n r ± r 2 / ^2 in the case of a symmetrical basin i s l/20 (Table I ) , so that the relevant expression i s large, even f o r the smallest value of ^ , which i s H- ( = 15.33). s,n 1,1 It can f i n a l l y be v e r i f i e d , by actual computation of the r a t i o of confluent hypergeometric series on the l e f t hand side of (85), that approximating the eigenfrequencies by the roots of the denominator gives quite good r e s u l t s , at least f o r small n and For example, a more precise c a l c u l a t i o n gives u>' = 0.00317 ( *a , s= - 146), which l , i . d i f f e r s only by about 3% from the approximate value given by (89) : 0.00306 ( 'a' = - 152.3). The exact value of 'a' which s a t i s f i e s the boundary condition (85) i s between -146 and -147, so that f o r small n and k, approximating the eigenfrequencies by the roots of ^F-jJ a; b; y) i s a better approximation than that used to f i n d those roots (87). This approximation however gradually looses i t s v a l i d i t y as k increases. Although a long chain of approximations i s necessary to f i n d the eigenvalues ( oo' ) when the J s, n solution i s expressed i n the form of confluent hyper-37 TABLE I I . Eigenfrequencies of Rossby waves i n a homogeneous symmetrical ocean with a f l a t bottom, as ca lcu la ted from the confluent hypergeometric solution,. OJ' given by ( 3 9 ) . s ,n s = •1 Period (days) - 2 - 3 n = l . 0 0 3 0 6 (164) . 0 0 3 1 9 . 0 0 2 9 4 2 . 0 0 0 9 3 (510) . 0 0 1 3 2 . 0 0 1 4 2 3 . 0 0 0 4 8 (1040) . 0 0 0 7 2 . 0 0 0 8 4 4 . 0 0 0 2 8 (1785) . 0 0 0 4 5 . 0 0 0 5 5 5 . 0 0 0 1 8 (2775) . 0 0 0 3 1 . 0 0 0 3 9 geometric funct ions , the above example shows that i t i s poss ible to obtain approximations to the eigenfrequencies 2 for the case of a f l a t bottom even when e x i s hot neglected compared to u n i t y . This i s not necessary when planetary waves are under investigation^: but the method has been developed i n prev i s ion of further work inc lud ing types of o s c i l l a t i o n s where i t i s impossible to s i m p l i f y (74) into (90) below. We have seen i n sect ion V I . t h a t there may be such time dependent motions i n areas II and IV of the diagnost ic diagram. 2 The approximation e x « 1 w i l l now be made, and the problem w i l l acquire much c l a r i t y by so doing. 8 8 i i ) Approximate solution i n terms of Bessel functions, Neglecting e x 2 i n ( 7 4 ) , the amplitude equation for a symmetrical ocean with a f l a t bottom becomes x 2 d 2 F + x dF - F( s 2 + 8 x 2 ) = 0 , (90) dx 2 dx which has solutions i n terms of Bessel functions, (Z), of the general form (Whittaker and Watson, 1 9 2 7 ; Chapter XVII) F(x) = Z ± s ( i V8 x) . ( 9 1 ) The boundary condition ( 6 0 ) i s unchanged; at a v e r t i c a l wall at x = 1 , Z' (1^8 x) s z ( i V S x ) l a'<* ±s ( 9 2 ) . in which the prime indicates d i f f e r e n t i a t i o n with respect to the argument of the Bessel function. Equation ( 9 2 ) becomes, using the formula Z' p(y) = - 2 Z (y) + Z (y) y P which i s applicable to a l l Bessel functions, Z + g _ 1 ( i - v / 8 x) = _ k j l + s_±) . ( 9 3 ) iV8 koj' Z. ( i V S x) 39 When a ; ' i s not nearly equal to 1 (which i s the condition for the s i m p l i f i c a t i o n leading to (90)) ^ can be approximated by s = S__ + M(l - 6 J ! ) (94) The parameter 8 , as given by (94), w i l l be negative in area I of the diagnostic diagram ( s<0, cu'<l), provided oj1 i s small enough: o>' < 0.05 for s = -1, and cu1 < 0.1 f o r s = -2, and so on. This i s i l l u s t r a t e d in Figure 20. In area II, s < 0, a / > 1, and 8 i s always negative; in area I I I , s > 0, co' < 1, and 8 i s always p o s i t i v e . F i n a l l y , i n area IV (s>0, cu'>l), 8 w i l l be negative unless cu' i s very near unity. I l l s >0,cu ' < l 8 > 0 8 < O I s<0,a/<l 8 > 0 s > 0 , a i > l jyr 8 < 0 8 < 0 s<0,u/>| J J Figure 20. The sign of 8 (as given by (94)) i n the subdivisions of the diagnostic diagram. 9 0 For negative values of S , the Bessel funct ion J^l^y/l"^ x) i s r e a l and w i l l be used where the general expression for a Bessel funct ion appears above. The magnitude of s, k, i s chosen as the order of the Bessel funct ion: since s i s an integer , in 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 functions i s chosen for the s o l u t i o n . When 8 i s p o s i t i v e , the funct ion x) i s r e a l and w i l l be chosen for the so lu t ion of (90). This funct ion is re la ted to the ordinary Bessel funct ion as fo l lows: l I k ( y ) = i " k J k ( i y ) . It i s not necessary to include Bessel funct ions of the second kind to complete the so lut ion (as should be done when s is an integer) because these diverge at x = 0, where we want our so lu t ion to be regu lar . For co' < 1, the parameter § i s negative only when s < 0 ( area I of the d iagnost ic diagram ) . We s h a l l now see that so lut ions of (90) in that case correspond to the planetary wave so lut ions of (74) which we found above. Let us look f i r s t at the completely symmetric case: s = 0. We can show that there w i l l be no planetary waves exh ib i t ing such symmetry. By (94), when s = 0, & becomes 8 = (1 - oo'2) M r 2 , 7 91 which i s p o s i t i v e for frequencies less than i n e r t i a l . The boundary condition (93) then takes the form 1^78 ) = 0 . (95) I j j y ) being a positive monotonic increasing function of y, (Jahnke and Emde, 1945; Chapter VIII), (95) can be s a t i s f i e d only when 8= 0 , i . e . , when OJ ' = 1. This solution i s i Z however e n t i r e l y inconsistent with the assumption ex << 1, which led to (90), so that the case s = 0 cannot be properly discussed by the approximate equation (90). We must then go back (74) and fin d what the confluent hypergeometric solution predicts when s ( and k) = 0; from (89), r 0 , n i s zero f o r a l l n, so that there are no planetary waves with zero wave number. It should be noted that i f higher frequencies are considered ( w' > 1) solutions of (90) can be found f o r s = 0 . The boundary condition then becomes (96) which i s s a t i s f i e d when «' = (1 + $2 ) . (97) l,n The constant ft , i s the n t h root of J,(y). This i s of l,n 1 J course only a l i m i t i n g case of the gravity controlled o s c i l l a t i o n s of areas II and IV of the diagnostic diagram. We l i m i t ourselves here to a discussion of planetary waves 9 2 in a polar ocean, so that nothing more w i l l be said about these gravity waves. For non-zero negative values of s, 8 i s negative when co' i s small enough (Figure 2 0 ) , and the solution of ( 9 0 ) i s F(x) = oyTiSl x) . ( 9 3 ) The boundary condition i s given by ( 9 3 ) , with replacing Z^. Even when cu' i s not very small, the r i g h t hand side of ( 9 3 ) w i l l be considerably larger than unity, and one finds that the value of 8 which s a t i s f i e s the boundary condition d i f f e r s from the root of the denominator of the l e f t hand side of ( 9 3 ) only in the t h i r d s i g n i f i c a n t f i g u r e . The eigenfrequencies w i l l then be approximately given by • k s.n o , 2 2 ( 9 9 ) M + R^/r 1 /3 k,n The constant B i s the n root of the Bessel function k,n J^; k takes the values 1 , 2 , 3 , . . . corresponding to s = - 1 , - 2 , - 3 , . . . . From Table I (R = 6 3 7 0 km, r^= • 2 2 1 5 0 0 km) R / r = 2 0 . Some of the eigenfrequencies, as calculated from ( 9 9 ) are l i s t e d in Table I I I . They depart from the eigenfrequencies as calculated from the confluent hypergeometric solution at larger values of k. This i s not due t o the approximation e x << 1 becoming worse f o r l a r g e k ( i t becomes b e t t e r , s i n c e cu' decreases ) or t o e s t i m a t i n g the eigenvalue from the r o o t s of the l e f t hand s i d e o f (93) ( t h i s a l s o improves as k i n c r e a s e s ), but stems from approximating the eigenvalues of the c o n f l u e n t hypergeometric s o l u t i o n by the r o o t s of the denominator of (85). For a constant n, t h i s l a s t approximation g r a d u a l l y 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 . We w i l l then adopt the e i g e n f r e q u e n c i e s l i s t e d i n Table I I I , r a t h e r than those o f Table I I , as c h a r a c t e r i s t i c r e s u l t s of t h i s a n a l y s i s when performed i n a p o l a r plane where only a f i r s t approximation of the curva t u r e of the E a r t h i s r e t a i n e d . These r e s u l t s w i l l l a t e r be compared w i t h t h e i r e q u i v a l e n t s as c a l c u l a t e d i n the s p h e r i c a l geometry. ,The a n a l y s i s i s e n t i r e l y s i m i l a r when the i n t e r n a l mode of o s c i l l a t i o n i s i n v e s t i g a t e d ; the constant M, which r e p r e s e n t s the e f f e c t of g r a v i t y , i s now however changed from a va l u e of 20 t o a new and much h i g h e r value c o r r e s p o n d i n g t o the s t r a t i f i c a t i o n adopted: M = 0.53x10^. The s m a l l e r e f f e c t i v e g r a v i t y means t h a t lower f r e q u e n c i e s w i l l be necessary t o make the i n f l u e n c e of the t e r r e s t r i a l c u r v a t u r e comparable i n importance t o g r a v i t y f o r c e s ; t h i s i s b e s t seen i n the d e f i n i t i o n o f 8 (94), where a d i r e c t comparison of the two i n f l u e n c e s can be made. Some i n t e r n a l mode f r e q u e n c i e s are t a b u l a t e d i n Table IV; the p e r i o d s c o r r e s p o n d i n g t o these f r e q u e n c i e s w i l l be of the 94 TABLE I I I . Eigenfrequencies of RosSby waves i n a homogeneous symmetrical ocean with a f l a t bottom, as ca lcu lated from the Bessel funct ion s o l u t i o n . ' given by (99). s, n -1 -2 -3 -5 -8 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 millenium and more. R e a l i z i n g that the surface layer i s subjected to annual var ia t ions associated with atmospheric condi t ions , i t i s seen that the s t r a t i f i c a t i o n w i l l not remain constant during long period o s c i l l a t i o n s , and that i t i s not s t r i c t l y correct to treat the depth of the in ter face as constant in t ime. I f the condit ions are purely per iodic over a yearly per iod , and there i s l i t t l e secular change, i t might be permissible to take the average s t r a t i f i c a t i o n as constant, s ince the period of var ia t ions i s so small compared with the period of the planetary waves. Needless to say, there i s l i t t l e hope of any d i r e c t measurement of the long-period i n t e r n a l waves. 95 TABLE IV. Eigenfrequencies of i n t e r n a l Rossby waves for a two-layer symmetrical ocean with a f l a t bottom, as ca lcu la ted from the Bessel funct ion s o l u t i o n . w ' given by ( 9 9 ) . M = 0 . 5 3 x i 0 6 . s | n s = - 1 - 2 - 3 Period (years) n = l 1 . 8 6 * x l 0 ~ 6 ( 2 9 1 5 ) 3 . 7 6 x l 0 " 6 9 . 4 0 x l 0 ~ 6 4 l . o 7 x " ( 2 9 3 0 ) 3 . 7 4 x " 9.30x " 9 1 . 8 3 x " ( 2 9 9 5 ) 3 . 6 4 x " 9 . 0 0 x " Before comparing the above eigenfrequencies with those obtained for a s i m i l a r basin i n the spher ica l geometry, l e t us see what the so lut ions look l i k e , and what are the propert ies of those waves, the frequencies of which we just c a l c u l a t e d . From (30) and (98), and (34), the surface displacement ( for the homogeneous mode) and the corresponding v e l o c i t i e s are e x p l i c i t l y given by V = c J k ( ( k / ^ - M j ^ x ) e i U t - S * ) (100) R u = - i c g c ' [ d J k ( } ) I e 1 * " * - * ) ( 1 0 1 ) 2 X 2 r n I dx cy'x 9 6 v 2 ill djk( i d x S OJ ' J u ( ) \ e i ( ^ t - S ( ^ > ) (102) I n t h e a b o v e e x p r e s s i o n s , t h e c o n s t a n t c h a s t h e d i m e n s i o n s o f a l e n g t h , and i s d e t e r m i n e d by t h e t o t a l e n e r g y o f t h e s y s t e m . The a r g u m e n t o f t h e B e s s e l f u n c t i o n s i s t h e same i n (101) and i n (102) a s i n ( 1 0 0 ) , a n d h a s n o t b e e n w r i t t e n i n e x p l i c i t l y . B e s i d e s v a r y i n g i n a p e r i o d i c manner a r o u n d t h e p o l e , t h e a m p l i t u d e and v e l o c i t i e s have, n o d e s b e t w e e n t h e p o l e and t h e b o u n d a r y . The v e l o c i t y v e c t o r t r a c e s a n e l l i p s e a s t i m e v a r i e s , a t a n y f i x e d l o c a t i o n ; t h e e c c e n t r i c i t y o f t h i s e l l i p s e i s g i v e n b y 1 - lul Ivl d x ) _s u k O J ' X « U ) , (103) d J ( k_ d x ~ s O J t k s o t h a t i t w i l l v a r y w i t h d i s t a n c e f r o m t h e p o l e , x, f r e q u e n c y , cu' , and wave number, s. The d i r e c t i o n i n w h i c h t h e e l l i p s e i s t r a c e d w i l l a l s o v a r y f r o m p l a c e t o p l a c e ( r a d i a l l y ) , and b a n d s w h e r e t h e v e l o c i t y v e c t o r r o t a t e s c l o c k w i s e w i l l a l t e r n a t e w i t h b a n d s where i t r o t a t e s c o u n t e r -c l o c k w i s e , a s one p r o g r e s s e s f r o m t h e p o l e t o w a r d s t h e 97 boundary (Figure 2 2 c ) . Sketches of amplitude and v e l o c i t y contour's f o r a few values of s and n, are found i n f igures 21 and 22. k k-1 Being of order x and x r e spec t ive ly at small values of x , the amplitude and v e l o c i t i e s remain f i n i t e over the whole A r c t i c ocean and i t s boundaries provided k >0, which is the case f o r the planetary waves studied now. The v o r t i c i t y in the v e r t i c a l d i r e c t i o n , £ , as given by ( 5 2 ) , becomes in terms of functions of x ( = ££_ i f ^ L V i '-s! J*( '\ •1<"' 2 ^ 1 dx 2 * 2 J (104) At f i r s t s ight , i t seems that t h i s expression would be of k-2 order x at small x ; i f however we replace the independent var iab le x by the.argument of the Bessel funct ions , *\/\h\ x , 2 2 noting that k = s , we f i n d that 2 1 k k |S,V2 k 2 f l r ' I VIOIx i - l x j ( 1 Q 5 ) Since J i s a Bessel func t ion , i t s a t i s f i e s Bessel 's equation, whatever i t s argument: 2 2 y 2 J £ ( y ) + yJ R (y ) + ( y - k ) J k ( y ) = 0 . 98 S = - 2 , n»l. S = -3 , n=2. Figure 21. Sketches of surface amplitude contours ( 7  , from (100)  f o r planetary waves in a symmetrical ocean with a f l a t bottom, f o r a few values of wave number, s, and index number, n. The patterns rotate clockwise with angular v e l o c i t y ^ / s. Dotted lines are nodal l i n e s . 99 c.) -Figure 22, a) and b) Sketches of the zonal (v) and r a d i a l (u) components of the v e l o c i t y f i e l d (from (101) and (102)) f o r s = -2, n = 1. 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 w1 / 2, Dotted l i n e s are nodal l i n e s , —2,1 and the v e l o c i t i e s are l a r g e r where the arrows are l o n g e r , c) The d i r e c t i o n i n which the l o c a l v e l o c i t y v e c t o r t r a c e s an e l l i p s e : - f o r c l o c k w i s e , + otherwise. 100 The primes i n (105) and in Bessel's equation indicate d i f f e r e n t i a t i o n with respect to the argument of the function. The three terms i n brackets i n (105) then reduce to - J^i^/lcTl x) , and the v o r t i c i t y becomes f = -jsll J.ivGi x ) . . K ( 1 0 6 ) 2^ r ^ which i s of order x at small x. In spite 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 at the pole for the planetary waves considered, and the Rossby number remains f i n i t e everywhere. To f i r s t order, the average energy transport due the wave motion i n a v e r t i c a l column of water i s < / p v dz > , (107) z in which the brackets indicate average over a cycle. When the pressure i s hydrostatic, and the v e l o c i t y does not depend on Z, (107) becomes < P&VZH > (108) 101 The time average of the r a d i a l energy transport vanishes; the zonal component becomes propor t iona l to H (1 - cu | d F 2 - s cu 'F 2 dx , x (109) The net energy transport in the zonal d i r e c t i o n is the i n t e g r a l of (109) from the pole to the boundary; when the depth is constant, th i s i s proport iona l to 1 F (1) - s c u ' J F dx . (110) 0 x At the boundary, the amplitude i s very small; as a matter of f a c t , we have approximated the eigenfrequencies by those values of frequency which make the amplitude vanish at x = 1 . The second term of (110) w i l l therefore dominate, and the energy w i l l propagate on the. average ( over a cycle and over a l l values of the r a d i a l coordinate) in a d i r e c t i o n opposite to that in which the phase moves. . The energy transport so ca lcu la ted d i f f e r s from that given by the group v e l o c i t y only by a non-divergent vector (Longuet-Higgins , 1964 a); i t i s more convenient to use the present method in closed bas ins . 102 The planetary waves just described correspond to the case 8 > 0 in ( 9 4 ) . Let us look b r i e f l y at the case S > 0 . From Figure 20, t h i s can occur in area III and small bands of areas I and IV of the diagnostic diagram. The boundary condition (93) becomes, when & > 0 , 1 k " 1 • k .(1+s ) . ( I l l ) I (Vo ) V§ : k co' k Since the functions I, (x) and I, , (x) are monotonic k k-1 increasing functions of x such that I. ..(x) > I (x) > 0 , K"* JL k the r i g h t hand side of (111) must be positive and greater than unity f o r the r e l a t i o n to be s a t i s f i e d . O s c i l l a t i o n s of t h i s kind w i l l be very s i m i l a r to Kelvin waves, in the sense that they w i l l hug the sides of the basin, t h e i r amplitude increasing very r a p i d l y near x = 1, according to the behaviour of I ^ i v^S x). They are now being studied in more d e t a i l by H.G. Farmer, at the University of Washington (Farmer, 1964), and I therefore l i m i t myself to a mention of t h e i r existence. i i i ) Comparison with r e s u l t s on a sphere. Let us now compare the results obtained above with t h e i r equivalents on the sphere. I f the two are compatible, the v a l i d i t y of the analysis performed in the polar plane 1 0 3 approximation t o the sphere w i l l be establ ished in the study of a l l poss ible motions of the contained f l u i d . This i s so because planetary waves, being dependent on the curvature of the Earth for t h e i r existence w i l l be more s trongly affected by any departures from the exact curvature than any other type of motions. I f the approximation works for planetary waves, 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 ocean. Longuet-Higgins ( 1 9 6 4 b ) . By assuming that the surface displacements have a n e g l i g i b l e influence on the v o r t i c i t y balance, Longuet-Higgins has been able to formulate the problem of planetary waves in two dimensions and to solve i t in terms of a stream f u n c t i o n . This approximation w i l l give good r e s u l t s provided the wave length i s smaller than the radius of the E a r t h . In p a r t i c u l a r , he gives f o r the stream funct ion charac ter i z ing the planetary waves i n a polar basin on the sphere in which P2/(cos9) i s the Legendre funct ion of arguments k, v , and cosG; k i s defined as above, and so i s 0 , while v i s a p o s i t i v e r e a l number (not necessar i ly an integer) which allows the boundary condi t ion to be s a t i s f i e d : The simplest basis of comparison i s the work of ^ = P £ (cos9) ( 1 1 2 ) k ? v (cosS,) = 0 ; - 1 9, = s in ( 1 1 3 ) 104 The frequency of the planetary wave i s then given by t v ( v + 1) (114) Equation (114) i s not very d i s s i m i l a r in form to (99), but the evaluat ion of v is much more complicated than f ind ing k p . . The Legendre funct ion P v ( x ) , when x i s r e a l and k (but not v ) i s an integer , i s given by the expression i v o k/2 P ^ ( x ) ( - 2 ) J7(r + k + - x^) , v  ( n 5 ) k l ?{v -k + 1) = 2F 1(1+k-t-z/,k-z/ ; k + 1; £ - £ x ) where ^F-^ i s the standard notat ion for the usual hyper-geometric ser i e s , T(y) i s the gamma funct ion: co r(y) = JeS7"1 dt . o The expression (114) w i l l have roots in v only when the hypergeometric ser ies vanishes; i t i s quite c l ear from the behaviour of hypergeometric ser ies ( E r d e l y i et a l . , 1953; Chapter II) that there i s an i n f i n i t y of values of v , of ever increas ing magnitude, which makes the ser ies zero for constant k and x. The same degeneracy then exis ts as 105 TABLE V. Computed eigenfrequencies of Rossby waves for a symmetric polar basin with a f l a t bottom on a sphere, according to Longuet-Higgins' model. n =1. s = l co' , = 0.00344 s , l 2 0.00378 3 0.00369 5 0.00324 8 0.00268 12 0.00224 16 0.00179 was found in the polar plane. There i s no a n a l y t i c a l formula al lowing c a l c u l a t i o n of the roots , and I have estimated the f i r s t root by computing the sum of the f i r s t 20 terms of the ser ies for increas ing values of v u n t i l a change of s ign occured. The cosine of the angle corresponding to (r-^/R) = l/20 i s 0.975; the i n t e r v a l between successive values of v was taken as 0.5 except for the l a s t two estimates, where i t was 1.0. The value of v corresponding to the root was m then estimated by l i n e a r i n t e r p o l a t i o n . Only the f i r s t root was obtained t h i s way, since only comparison in one d i r e c t i o n was deemed necessary, and because of the 106 considerable time necessary f o r computing. The calculations were done on the University of B r i t i s h Columbia's IBM 1620„ and the program written f o r t h i s purpose i s given in Appendix I . The results are l i s t e d i n Table V . Comparison of tables I II and V shows that the eigenfrequencies calculated on a polar plane model d i f f e r from Longuet-Higgins' two-dimensional approximation on the sphere by small but appreciable values. To see whether t h i s discrepancy arises from the polar plane assumption or-from neglecting the influence of surface displacements, we can apply t h i s l a s t approximation to the polar plane, and see whether the r e s u l t s are closer to the s t r i c t e r polar plane r e s u l t s or to the r e s u l t s on the sphere. In- the f i r s t case, the discrepancy arises from the use of the polar plane, i n the second, from the neglect of surface displacements. Fpr a two-dimensional problem in the angles 9 and X, we can write the v e l o c i t i e s in terms of a stream function ; i n spherical polar coordinates, u = d y\f R s i n 9 aX = -1 d f R a 9 where u and v are r a d i a l and zonal v e l o c i t i e s respectively. Elimination of the pressure gradients from the momentum equations ( 9 ) and ( 1 0 ) and of the v e l o c i t i e s through the 107 c o n t i n u i t y e q u a t i o n (6) g i v e s a - v o r t i c i t y e q u a t i o n , w h i c h b e c o m e s i n t e r m s o f t h e a b o v e d e f i n e d s t r e a m f u n c t i o n dVH ft - 1 _df d±_ = 0 , (117) a t " R 2 s ine a© ax w h i c h i s t h e w o r k i n g e q u a t i o n o f L o n g u e t - H i g g i n s . T r a n s f o r m i n g t o t h e p o l a r p l a n e b y m e a n s o f t h e r e l a t i o n s ( 2 3 ) , (117) b e c o m e s i n t h e n e w g e o m e t r y a f i _ i ± _ + a^L + i a_^_ ajrcosoil + 2x2coso a^ _ = 0 . at l r 2 a</>2 d r 2 r a r a r J R 2 A < £ (1.18) L e t u s l o o k f o r v o r t i c i t y w a v e s o f t h e f o r m ft = ft ( r ) e 1 * w t - s < £ ) (119) s i n c e t h i s i s o f t h e s a m e f o r m a s t h e p o s t u l a t e d e l e v a t i o n V (3 0 ) , t h e d e r i v a t i v e s o f f a r e o b t a i n e d f r o m (3l)-(33) b y r e p l a c i n g V b y f . D e f i n i n g x a s i n ( 5 8 ) , s u b s t i t u t i o n o f (119) i n t o (118) g i v e s a n a m p l i t u d e e q u a t i o n f o r ftQ: Z d 2 % + x d f o ( l + r l x 2 ) - f o ( s 2 + s x 2 r l ) = 0 d x 2 d x R 2 . ^ * R 2 (120) T h i s e q u a t i o n r e s e m b l e s v e r y m u c h ( 7 4 ) ; i n f a c t , 2 w h e n t h e f r e q u e n c y i s l o w e n o u g h s o t h a t £ x <<1, t h e t w o d i f f e r o n l y i n t h e d e f i n i t i o n o f t h e c o n s t a n t 3 0 108 Putting now * 2 1 = H i c V R2 we see that i n t h i s case there i s no gravity influence (as represented by the M term i n (94)). This i s of course a d i r e c t consequence of neglecting the surface elevations : the problem i s treated as two-dimensional, and there are no departures from the equilibrium l e v e l on which gravity can act. When $^ i s negative, planetary wave solutions fo r the amplitude equation of the stream w i l l exist analogous to those f o r the displacement amplitude i n (94): + (x) = J (V^il x) 0 (121) o k v The boundary condition i s now s l i g h t l y d i f f e r e n t , and i s v//o(l) = 0 = J k (vf8",l ) (122) so that the eigenfrequencies are now given by the formula co' = k ( 1 23) s,n 20 P k,n Comparison of (122) with (99) shows that f o r large values' of k and/or n, the two formulae w i l l give very s i m i l a r results. Some values calculated from (122) are l i s t e d i n Table V I , where they are compared with some of the eigenfrequencies derived by other methods. 109 TABLE VI. Comparison of eigenfrequencies of planetary-waves i n a symmetrical polar ocean with a f l a t bottom, as obtained from d i f f e r e n t methods. P-P means polar plane approximation; 2-D i s Longuet-Higgins' two-dimensional approximation. s = T OJ — 3 , 1 P-P 2-D P-P and 2 -1 .00324 .00344 .00342 -2 .00366 .00378 .00380--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) Goldsbrough .00494 .00735 It appears from Table VI that the eigenfrequencies O J ' S 1 t e n d t o ^ e s a r n e values whatever the mode of ca l c u l a t i o n , provided k i s large enough. Furthermore, the discrepancies between the solutions on the polar plane and the approximate two-dimensional solutions on the sphere cannot be attributed to the imperfection of the mapping on the plane, because when the problem i s formulated as two-110 dimensional on the plane, almost i d e n t i c a l results are obtained as on the sphere. Since Longuet-Higgins* assumption that the surface displacements are of n e g l i g i b l e influence on the v o r t i c i t y balance i s known to be v a l i d only for wave lengths appreciably smaller than the radius of the Earth, the difference with the r e s u l t s on the polar plane would seemingly be caused by the inadequacy of the two-dimensional assumption at small wave numbers. The difference between res u l t s on the sphere and those on the polar plane using Longuet-Higgins approximation i s not detectable, so that one must conclude that the r e s u l t s of column 1 i n table VI are more precise than those of the following two columns. When the diameter of the polar ocean i s small enough to drop (r,/R) with respect to 1, the r e s u l t s provided by the polar plane approximation are as precise as those provided by Longuet-Higgins method on the sphere, and even more precise at low wave numbers. Another advantage i s that the eigen-frequencies are much easier to calculate on the polar plane; f i n a l l y , my formulation allows consideration of bathymetric variations i n the model. This analysis i s of course r e s t r i c t e d to polar regions, and does not have the general a p p l i c a b i l i t y to a l l lati t u d e s that Longuet-Higgins' method possesses. Another basis of comparison i s the work of Goldsbrough on the dynamics of t i d e s in polar basins (Goldsbrough, 1914 a); his work i s done e n t i r e l y i n I l l spherical polar coordinates, and the two frequencies which can be compared to the r e s u l t s of the present work are presented i n the fourth column of Table VI. They depart considerably from the corresponding values i n the other three columns. The process of c a l c u l a t i n g anything but a f i r s t approximation to Goldsbrough's frequencies i s quite involved, since the eigenfrequencies are to be evaluated from an i n f i n i t e determinant. A second approximation has been attempted, but does not y i e l d values of co ' near those of i n t e r e s t . No apparent reason has been found f o r the discrepancy in the magnitude ,of the eigenvalues. Comparing Longuet-Higgins' simple and clear formulation with Goldsbrough's involved series solutions and i n f i n i t e determinants one i s tempted to give more f a i t h to the results of the former. In view of the good agreement of polar plane r e s u l t s with spherical geometry r e s u l t s as derived from Longuet-Higgins, and i n spite of the not so good agreement with Goldsbrough's values, f o r which the basis of comparison i s narrower (two frequencies), I then conclude that the polar plane w i l l be quite useful i n studying the motions of f l u i d s in r e s t r i c t e d polar basins, and give q u a n t i t a t i v e l y precise r e s u l t s , The polar plane, defined as the projection of Figure 5 together with the retention of only a f i r s t approximation to the Earth's curvature, can therefore be used i n the Arctic regions i n the same manner as the 112 ft -plane i s used i n mid-latitudes. This section has described the c h a r a c t e r i s t i c o s c i l l a t i o n s of the simplest polar basin: bounded along a p a r e l l e l of l a t i t u d e and without any depth v a r i a t i o n s . This i s f a r from describing the actual A r c t i c bathymetry, and in the next section, an added degree of complexity w i l l be introduced in the form of r a d i a l depth v a r i a t i o n s . It may be asked whether such long period planetary waves as discovered above are of any dynamic s i g n i f i c a n c e , even in a simple symmetric basin. Veronis and Stommel (1956) have shown (in the ft -plane formulation ) that for winds acting over a period of more than half a pendulum day a s i g n i f i c a n t portion of the t o t a l energy i s transferred into long period semi-geostrophic planetary waves. This result does not depend on the p a r t i c u l a r projection used , and w i l l hold just as well for the polar plane. Planetary waves can then be generated by f l u c t u a t i n g winds over such a symmetrical basin as studied above. 113 VIII. SYMMETRICAL OCEAN: RADIAL DEPTH .VARIATIONS. The next step in the scale of increasing complexities i s the inclusion of r a d i a l depth variations: dH'/dx?* 0 ; dH'/a<£ = 0 . The water content of the polar basin i s now considered v e r t i c a l l y homogeneous; the amplitude i s determined by equation (59) and the boundary condition ( 6 0 ) . We have seen in section VI that i t i s possible f o r solutions to exist i n the presence of a wide variety of bottom configurations; i t i s not easy however to solve e x p l i c i t l y the amplitude equation when H f(x) i s substituted in i t . We w i l l therefore have t o be s a t i s f i e d with the simplest bottom topography i n order to obtain e x p l i c i t solutions. This w i l l s u f f i c e however to show the nature of the effects of the depth v a r i a t i o n s . The following simple depth dependence i l l u s t r a t e s very well the influence of bottom topography on planetary waves. Let us assume that the depth varies very l i t t l e over the extent of the basin, so that i t can be considered 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 that of the C o r i o l i s parameter: H» = (1 + px 2 / 2 ) (123) i n which p/2 « 1. When (123) i s substituted into ( 5 9 ) , the amplitude equation assumes the same form as when there are no depth variations; only some of the constants are changed: 114 x d_F + xdF [ l + U + p J x 2 ] - F [ s 2 + (8 +JDS)X 2]= 0 . (124) dx 2 d x *>' This equation can be solved in terms of confluent hyper-geometric functions, as (74) was, but t h i s i s not necessary when planetary waves are considered, since i t i s then 2 possible to neglect ( e + p)x.. with respect to 1 and use the reduced equation X V F + xdF - F [ S 2 + ( 8 +JDS)X 2]= 0 , (125) dx 2 d x which i s i d e n t i c a l in form with ( 9 0 ) , and has therefore s i m i l a r solutions. Examining the planetary wave solutions, which, by analogy to ( 9 0 ) , occur where 8 + p s / cu ' < 0 , the amplitude i s then F U ) = J k(V[8+£s| x) . (126) Expanding the constant 8 + p s / cu' i n terms of frequency, and keeping only the f i r s t order terms i n cu1 because of the low frequencies of planetary waves, one has 2 2 S + p_s ~ (p + ^ l ) s + M ^ l < 0 . (127) cu' R 2 cu' R 2 115 In equation (127) one can see the ro l e played by a bottom conf igurat ion of the form (123): i f p i s p o s i t i v e , so that th.e depth increases towards the boundaries, then s must be negative and propagation to the west, i n order to keep (127) negative. I f p i s negative, so that the depth decreases r a d i a l l y , and large enough to make (p + r-^/^2) negative a l so , then the wave number s w i l l have to be pos i t i ve for (127) to hold (for low frequencies , the frequency dependent part of (127) w i l l dominate). Propagation i s therefore to the east . This is exactly what the method of signatures predicted i n sect ion VI: planetary waves propagating towards dH 1 the east can ex is t i f i s large and negative, corresponding to a negative p. One a lso observes in (127) that the v a r i a t i o n of C o r i o l i s parameter (the r^/^2 term) produces asymmetries between waves corresponding to equal but opposite depth gradients . The boundary condi t ion i s and, as in the f l a t bottom case, the eigenfrequency i s c l o se ly approximated by the root of the denominator of the l e f t hand s ide . This gives for the frequencies (128) 116 s( p + r 2 / R 2 ) / ? 2 k , n + M r l / R 2 ' ( 1 2 9 ) i n which ft i s defined as i n (99). A few eigenfrequencies are tabulated in table V I I . The so lut ions corresponding to ( 8 + ps / cu 1 ) > 0 have not been inves t igated , ,but they w i l l be an extension of the r e s u l t s for a f l a t bottom when 8 >0 (111). Although the bathymetry adopted in the above example i s very simple, i t i l l u s t r a t e s c l e a r l y the influence of bottom v a r i a t i o n s on the propert ies of planetary waves. For more complicated topographies, i t may not be possible to f i n d an e x p l i c i t ana ly t i c s o l u t i o n of the amplitude equation; i t may however be integrated numerical ly , given H' (x) , the frequency being adjusted u n t i l a value s a t i s f y i n g the boundary condi t ion i s found. Some general theorems concerning the motion of shallow r o t a t i n g l i q u i d s on a paraboloid have been demonstrated by B a l l i n a recent a r t i c l e ( B a l l , 1963); a s p e c i a l case has a l so been treated by Mi les and B a l l (1963). Ro these theorems apply to an A r c t i c basin i n the polar pro jec t ion used in t h i s study? B a l l ' s representat ion of the problem i s s l i g h t l y d i f f e r e n t from that used up to now; the reference l e v e l cu s ,n 117 TABLE VII . Some eigenfrequencies of planetary waves for a symmetrical ocean with a r a d i a l bottom s lope . H 1 = 1 + p x 2 / 2 . a;' ca lculated from (129). s, 1 , p = 0.1 s <0 p = - 0.1 3 > 0 k = 1 2 5 CO s , n 0.00954 0.0110 0.00963 0.00313 0.00366 0.00321 for the v e r t i c a l coordinate i s taken at the maximum depth of the bas in , the e levat ion of the bottom above that loca t ion being Z, and the surface displacement and the l o c a l equi l ibr ium depth being grouped under the same var iab le h. This i s i l l u s t r a t e d in Figure 23. The f i r s t theorem proven by B a l l i s that the displacement of the centre of grav i ty of the l i q u i d i s independent of the motion that occurs wi th in the l i q u i d r e l a t i v e to the centre of g r a v i t y . His basic equations, with coordinates x, y and v e l o c i t i e s u,v to the east and north re spec t ive ly , are then Du + g _d__(h + Z) = f v Dt (130) Dv + g _d_(h + Z) = -fu Dt ^ ay (131) 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. D h +h(ce_u +c3v) = 0 . (132 Dt dx dj 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 de 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 integrating to obtain expressions involving the coordinates 119 of the centre of grav i ty ; performing these operations on the C o r i o l i s acce lera t ion terms, one has the i n t e g r a l being over an area enclos ing a given amount of l i q u i d , the boundaries moving with the l i q u i d t I f the C o r i o l i s parameter i s not a constant, but can be wri t ten as f = f Q 4- f^, f Q being a constant, but not f^, the i n t e g r a l (133) becomes (using B a l l ' s notat ion , with Q = t o t a l constant volume of f l u i d , X , Y , the coordinates of the centre of gravi ty) B a l l i m p l i c i t l y assumes that e i ther the C o r i o l i s parameter i s constant, or that the area of the basin i s small enough so that i t does not depart very much from some average value, the second term of ( 1 3 4 ) being therefore n e g l i g i b l e . Since the v a r i a t i o n of C o r i o l i s parameter i s of pr imordia l importance i n planetary waves, i t would seem that the f i r s t theorem of B a l l would not apply to them. We have seen however that as far as planetary waives are concerned a change in depth has an effect s i m i l a r to a change i n f (see the p o t e n t i a l v o r t i c i t y equation, ( 5 4 ) ) . The C o r i o l i s parameter can therefore be considered constant, i t s var iab le part being represented 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 constant, but (133) (134) dt a depth v a r i a t i o n of the form (123) ex is ted, with p = r]/g2, 2 ? H« = ( 1 + r / 2 R 2 x ) , f = constant, the free eigensolutions would be exact ly those found f o r the f l a t bottom case in sect ion V I I . B a l l ' s f i r s t theorem therefore appl ie s i n cases where f v a r i e s , so t h a t ' 1) the displacement of the centre of grav i ty of the l i q u i d i s independent of the motion that occurs within the l i q u i d r e l a t i v e to the centre of g r a v i t y , and 2) the equations governing the motion of the l i q u i d r e l a t i v e to i t s centre of g r a v i t y have exact ly 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; his equations are now put in polar form: Du + g _a_(h + Z) = ( f + v / r ) v (135) Dt dr m Dv + £_ J_th + Z) = - ( f + v / r ) u (136) Dt r d4> 121 When ( 2 4 ) , (25) and (29) (in which the indices are dropped) are formulated i n the geometry of Figure 2 3, the r e s u l t i n g equations are i d e n t i c a l to the li 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 to the second term of the f i r s t equation ( 1 3 5 ) . The cos9 which appears in the continuity equation can be neglected (to f i r s t order in r/R) since i t i s not d i f f e r e n t i a t e d . How w i l l t h i s s l i g h t difference a f f e c t the result? The t o t a l energy of the l i q u i d and i t s t o t a l absolute angular momentum about the polar axis are constants, as in B a l l ' s work. With the moment of i n e r t i a , I, about a v e r t i c a l axis through the o r i g i n defined as in B a l l : I = J h r 2 dS , (138) the theorem sta t i n g that i f I and i t s time derivative are i n i t i a l l y known, they are determined uniquely at a l l times thereafter s t i l l holds. The ensuing distension theory i s v a l i d , and i t i s therefore possible to separate the motion of the l i q u i d i n two parts: f i r s t a distension, defined as an is o t r o p i c two-dimensional d i l a t a t i o n and rotation (Figure 2 4 ) , and second, motions superimposed on the distension. To quote B a l l , "the main effects of the distension on these (superimposed) motions .are, f i r s t , a slowing down or speeding up of every aspect of the motion (whether vortices or gravity waves) according as the l i q u i d as a 122 E x t r e m e D i l a t a t i o n . Figure 24. A 'd i s t ens ion ' ( B a l l (1963)). whole i s stretched or contracted, and secondly, a general s t a b i l i z a t i o n , . . . . " . This a l so appl ies to the motions studied i n a parabolo ida l polar basin: i f planetary waves coexist with a d i s tens ion , energy w i l l be interchanged between the two modes of motion at a rate depending on the rate of d i l a t a t i o n associated with the d i s t e n s i o n . When the f l u i d contracts (water p i l e s up at the po le ) , energy w i l l be extracted by the waves from the d i s tens ion and the frequencies w i l l increase; when water i s high at the edges, the frequencies w i l l decrease. 123 It can be shown very simply that the spec ia l case concerning the influence of ro ta t ion ,on the frequencies of o s c i l l a t i o n s i n a shallow r o t a t i n g parabolo id , as studied by Miles and B a l l ( I 9 6 3 ) , w i l l be appl i cab le only when the influence of v a r i a t i o n s of the C o r i o l i s parameter i s n e g l i g i b l e , so that the resu l t s there in cannot be appl ied to the study of planetary waves i n the corresponding geometry. The basin considered i s a paraboloid of r e v o l u t i o n , and the depth i s given by H' = ( 1 - x2) . (139) E l i m i n a t i o n of a l l var iab les except surface amplitude i n the shallow water l i n e a r i z e d equations (Lamb, 1932; p 209) gives , i n the notat ion of (58) x 2 ( l - x2)ofF + xdF (1 - 3 x 2 ) dx2 dx 2 - F [ s 2 ( l - x2) + ( - 2 s + MJjJx 2] = 0 (HO) « ' R2 1 The so lu t ion i s found i n term of a hypergeometric s er i e s , and the condi t ion that t h i s series converge at the boundaries y i e lds a frequency condit ion from which Miles and B a l l deduce that the frequencies of the dominant modes for azimuthal wave numbers s = 0 and s = 1 are independent of 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 reference, and that the frequencies of 124 a l l other axisymtnetric modes are decreased by r o t a t i o n . To see i f these properties are also applicable to a basin i n which the rot a t i o n varies with p o s i t i o n , l e t us substitute the depth, as given by (139) into the amplitude equation (59): x 2 ( l - x 2 ) d f p + xdF [ e ( x 2 + l ) ( l - x 2) -2x 2] dx 2 d x - F|(,S24-SX2€ )(1 - x 2) +[(1- O/2)M_HL - 2 s ] x 2 i = 0 (141) I w» R2 w» J 2 Assuming that the frequencies are such that ex can be neglected with respect to 1, and that < u ' « 1 (141) becomes x 2 ( l - X2)£F + xdF (1 - 3x 2) d x 2 d x (142) (s2+ sA) (1 - x 2) + [ jMjf - 2 s ] x i 0 This w i l l apply to the study of planetary motions of long periods; (142) would be i d e n t i c a l to (140) were i t not f o r * 2 the presence of a term i n sx € / i n the f i r s t c o e f f i c i e n t of F in (142). For the low frequencies encountered i n planetary 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 ef f e c t s of t e r r e s t r i a l curvature ( i . e . , v a r i a t i o n of the C o r i o l i s parameter) 'cannot be neglected, there i s a s i g n i f i c a n t difference between motions i n the polar plane and those i n Miles and Ba l l ' s model. Equation (142) does not, "like (140) have a solution in terms of a known series, and the condition f o r fi n i t e n e s s of the solutions w i l l in general be d i f f e r e n t . The Arctic basin does not of course have a simple' paraboloidal bottom topography: the depth variations are not even symmetrical around the pole, so that r e s u l t s applicable to a f l a t bottom ocean or to an ocean with symmetrical bottom slopes are only of academic interest as f a r as the actual A r c t i c ocean i s concerned. But since the. contorted geometry of the Arct i c does not y i e l d e a s i l y to analysis, i t i s necessary to understand the s i t u a t i o n i n si m p l i f i e d situations before even looking at the more complex-cases. It might also be possible to deduce some qu a l i t a t i v e properties of the solutions i n the more complex situations through knowledge of the physics i n the simple topographies studied. We have v e r i f i e d i n t h i s section that the eigen-solutions predicted by the Method of Signatures (section VI) in the presence of sloping bottoms indeed exist and that the influence of the bottom slopes on the frequencies of the eigensolutions i s as expected from the analysis of section VI. If the effect of the depth variations 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, w s,n If the influence of the depth is in the opposite direction (dH'/dx.< 0), the frequency is decreased until it becomes • 2 zero for dH'/dx — -r]/g2x..... For steeper negative depth 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 the Method of Signatures (section VI), i t can be determined whether eigensolutions w i l l exist or not f o r any given r a d i a l depth v a r i a t i o n H*(x). Namely, solutions w i l l exist when H'(x), through i t s influence on the signatures, allows the phase path to terminate at the or i g i n of the phase diagram. When H'(x) i s given e x p l i c i t l y , the signatures can be found f o r a l l values of £ ( = l / x ) , and so can the position of the phase path. In p r i n c i p l e , i t i s then possible to draw conclusions on the eigensolutions when the r a d i a l depth v a r i a t i o n i s known. The actual e x p l i c i t solution of the amplitude equations i s possible only in very simple cases; one such simple case (H' = (1 + px /2)) has been examined above. In the spe c i a l case of paraboloidal basins, some general theorems due to B a l l , and stated above, apply to the motions of the contained f l u i d , provided i t i s shallow. The case of symmetric depth variations can then be considered to be resolved i n p r i n c i p l e , since even though i t may not be possible to solve the amplitude equation e x p l i c i t l y , the existence (or non-existence) of solutions can be ascertained, and the amplitude equation solved numerically to f i n d the eigenfrequencies. 128 IX. ASYMMETRICAL TOPOGRAPHY. The problem becomes enormously more complex when asymmetries are allowed, either i n the bathymetry or i n the boundaries. A glance at Figure 1 shows that the asymmetries are very important and w i l l i n a l l p r o b a b i l i t y play a dominant role i n the dynamics of the Ar c t i c ocean. The simple solutions of sections VII and VIII w i l l then not be d i r e c t l y applicable to the actual A r c t i c ocean, and the more complete equation (48) must be solved. Using some of the abbreviations defined i n (58), (48) becomes dtf> \j dtp H d<f> H O J T a r + 0 (143) 129 The boundary condition i s s t i l l as given by ( 5 0 ) . I t i s however doubtful whether the assumed form for the surface displacements, (30), ' which i s appropriate to the description of waves i n a c y l i n d r i c a l basin, 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 to reformulate the problem i n a coordinate system more appropriate to the new symmetry, and i n which i t might be possible to separate the amplitude equation. Asymmetries i n the boundaries w i l l then not be included, i n which case (50) reduces to - i s F( r i) = 0 (144) ( i cu ) d F r = r + 1_ JJL . r x d4> 1 r = r x r I No solutions 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 solutions at a l l . In. view of t h i s uncertainty and of the non-linearity 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 doubtful whether one should pursue t h i s l i n e of attack any further. The problem might be more tractable with a less i n c l u s i v e formulation, •but on the other hand, i t might be necessary to study the non-symmetrical s i t u a t i o n with more q u a l i t a t i v e arguments. 130 X. CONCLUSIONS. The problem of the dynamics of the A r c t i c ocean has been formulated i n a geometry appropriate to the polar regions by transforming the equations of motion and of continuity from the sphere to a polar plane. Although t h i s mapping arises quite naturally i n the study of geophysical phenomena, i t seems that nobody had taken advantage of i t in that respect previously. The main advantage of the mapping i s a considerable reduction i n mathematical complexity; f o r a r e s t r i c t e d polar cap, one can re t a i n only a f i r s t approximation to the t e r r e s t r i a l curvature, thus analysing the problem i n a modified beta-plane. Applying the transformed equations to the simple case of a symmetrical ocean with a f l a t bottom, one finds that the planetary wave eigensolutions compare reasonably well in.frequency and appearance v*ith t h e i r equivalents i n a s i m i l a r basin on the sphere, as derived by Goldsbrough and Longuet-Higgins. The solutions i n the polar projection are furthermore easier to represent and t h e i r frequencies calculated with a minimum of labour. Having found that the solutions f o r planetary waves were very s i m i l a r in the polar plane and i n the s p h e r i c a l geometry, we can conclude that the analysis i n the approximate polar plane w i l l y i e l d almost undistorted results f o r a l l 131 possible modes of motion of a small A r c t i c ocean (the actual one'is small enough). This i s so because the planetary waves, depending i n t h e i r existence and properties on the curvature of the Earth, are most l i k e l y to be distorted by any departures from the s t r i c t geoid. . The polar plane projection can therefore be used as r e l i a b l y as the ordinary beta-plane used i n mid-latitudes. The r e s u l t s concerning the symmetrical, ocean with f l a t bottom are not new; t h i s work goes" beyond that of Goldsbrough and Longuet-Higgins i n formulating the problem to include variable bathymetry and asymmetrical boundaries. The symmetrical ocean i s of course much simpler to discuss, the amplitude then being determined by an ordinary d i f f e r e n t i a l equation. Using the Method of Signatures, some general c r i t e r i a can be established concerning the existence and properties of the eigensolutions of a symmetrical basin. For a given bottom configuration, H'(r), one can determine whether eigensolutions w i l l be found or not; in particular,, i t i s found that no planetary waves can propagate towards the east when the depth of the symmetric ocean i s constant. The r e a l A r c t i c ocean i s of course grossly asymmetrical, and i f motions dependent on the topography are investigated, 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 conclusions have been reached in t h i s more general s i t u a t i o n , mostly because of 132 the increased 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 non-linear. It might be necessary to treat the more general case by more q u a l i t a t i v e methods since the mathematical d i f f i c u l t i e s have not sb f a r been surmounted. Needless to say, much work remains to be done before the dynamic oceanography of the actual A r c t i c ocean i s understood i n d e t a i l . I hope then that t h i s work w i l l serve as a basis as well as a stimulus for further developments and that the polar plane approximation here introduced w i l l also prove useful in further research. 133 APPENDIX I . F0RTRAN language program to f i n d the f i r s t root i n ' v of the hypergeometric ser ies J (1+ k + v ,k - v ; k + 1 ; i - £ x ) . 2? 1 The ser ies i s summed ( f i r s t 20 terms) for l arger and larger values of v u n t i l the s ign of the sum changes; the same procedure i s then repeated f o r another value of k. The notat ion is not the same i n the program below as' i n the above ser i e s : 1+ k = C, k - v = B, and 14- k + v = A; v i s c a l l e d RNUC The program below i s the one used for 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 modif icat ions are introduced f o r f i n d i n g the root 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) = Y(I-1)*((A P-1.)*(B P-1.)*X)/((P D)*P) 200 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 E N D $DATA 2. • 3. 4. 6. 9. After 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 to 10; the l a s t value of RNU used i s the f i r s t one used i n ca l c u l a t i n g the series 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 calculated f o r k = 12 and 16, the program i s started with RNU = 60.0 to save time. Also, statement 220 i s changed to 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 increasing the width of the i n t e r v a l , but l i t t l e percent precision i s lo s t since RNU is much larger f o r those values of k. 136 BIBLIOGRAPHY Ball,F.K., 1 9 6 3. Some general theorems concerning the f i n i t e motion of a shallow rot a t i n g l i a u i d l y i n g on a paraboloid. Jour.Fluid 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. Eckart,C, I960. Hydrodynamics of oceans and atmospheres. Pergamon Press, New York. 290 pp. Erdelyi.A., M.Magnus, F. Oberhettinger and F. Tricomi, 1 9 5 3 . Higher transcendental functions, 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 in ice,2. Physics, 5 : 181-184. Farmer,H.G., 1 9 6 4 . Baroclinic o s c i l l a t i o n s in a polar sea. Oral communication to the 1 3 t n annual meeting of the P a c i f i c Northwest Oceanographers, Seattle, 7 - 8 Feb. 1964. Goldsbrough,G.R., 1914 a). The dynamical theory of tides in a r>olar ocean. Proc. Lon. Math. S o c , Ser.2, 14 : 31 -66 ' ~ Goldsbrough,G.R., 1914 b). The dynamical theory of tides in 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 tides i n oceans on a rotating globe. P r o c Roy. S o c , A140 : 241 - 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 application of harmonic analysis 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., 6 7 , ( 6 ) : 2 4 7 7 - 2 4 8 9 . 137 Jahnke,E. and F. Emde, 1945. Tables of functions. Dover, New York. Lamb Jl.,,1932. Hydrodynamics. Dover, New York. 738 pp. Laplace,P.S., 1799. Traite de mecanique celeste. Crapelet, Pa r i s . Longuet-Higgins,M.S., 1964 a). On group v e l o c i t y and energy f l u x in planetary wave motions. Deep Sea Res., 11, (1) : 35-43. ' ~~ ' Longuet-Higgins,M.S., 1964 b). Planetary waves on a rot a t i n g sphere. Proc. Roy. Soc. A (in press). 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 free surface o s c i l l a t i o n s in a rotating paraboloid. 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 gravity in the A r c t i c , basin. Proceedings ;of the A r c t i c basin 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 publishing co. Poincare',H., 1910. Le§ons de mecanique celeste. Gauthier-V i l l a r s , P a r i s . (Tome I I I , Theorie 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. Methuen and Co., London; Wiley, New York. Rattray,M. J r . , I 9 6 4 . Time dependent motion i n an ocean; a u n i f i e d two-layer beta plane approximation. Hidaka Anniversary Volume. (In press). Robinson,A.R., 1964. Continental shelf waves and the response of sea l e v e l to weather systems. Jour. Geophys. Res., 69_, (2) : 367-368.. Rossby,C.G., 1939. Relations between variations in the intensity of the zonal c i r c u l a t i o n of the atmosphere and the displacements of the semi-permanent centres of a c t i o n . Jour. Marine Res., 2 : 38-55. 138 Rossby,C.G., 1949. On the dispersion of planetary waves in a barotropic atmosphere. Te l l u s , 1 : 54-5$. Rossby,C.G., 1959. Currents problems in meteorology. The atmosphere and the sea in motion: s c i e n t i f i c contributions to the Rossby memorial volume. Bert Bolin editor. N.Y. Rockefeller I n s t i t u t e Press and Oxford Univ. Press. S l a t e r , L . J . , I960. The confluent hypergeometric function. Cambridge Univ. Press. 243 PP« Stommel, H., 1948. The westward i n t e n s i f i c a t i o n of wind driven ocean currents. Trans. Amer. Geophys. Union, 29 : 202-206. Stommel,H., 1957. A survey of ocean current theory. DeeD Sea Res., 4 : 149-184. Stommel,H., I960. The Gulf Stream. Univ. of C a l i f o r n i a Press, Berkeley and Los Angeles; Cambridge University Press, London. Veronis,G., 1956. P a r t i t i o n of energy between geostrophic and non-geostrophic motions. Deep Sea Res., 3. : 157-177. Veronis,G,, 1963. On the approximations involved i n transforming the equations of motion from a spherical surface to the beta-plane. I. Barotropic systems. Jour. Marine Res., 21, (2) : 110-125. Veronis,G. and H.Stommel, 1956. The action of a variable wind stress on a s t r a t i f i e d ocean. Jour. Marine Res., 15 : 43-75. Warren,B.A., 1963. Tomographic influences on the path of the Gulf Stream. Te l l u s , 15, (2) : 167-184. Whittaker,E.T. and G.N. Watson, 1927. A course i n modern analysis. Cambridge Univ. Press. 608 pp. 

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