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On finite amplitude planetary waves Clarke, Richard Allyn 1970

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ON FINITE AMPLITUDE PLANETARY WAVES by RICHARD ALLYN CLARKE A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Physics I n s t i t u t e of Oceanography We accept t h i s t h e s i s as conforming to the req u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1970 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced d e g r e e at the 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 , I a g r e e t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f Phvsics  The U n i v e r s i t y o f B r i t i s h Co lumbia V a n c o u v e r 8, Canada Abstract F i n i t e amplitude planetary waves are studi e d on a homogeneous f l u i d on both the r o t a t i n g sphere and on a m i d - l a t i t u d e 3-plane. The i n t e g r a t e d equations of motion are rederived both on the r o t a t i n g sphere, i n a s p h e r i c a l p o l a r co-ordinate system whose a x i s i s t i l t e d r e l a t i v e to the r o t a t i o n a x i s , and on a m i d - l a t i t u d e 3-plane. The l i n e a r s o l u t i o n s are re-examined and the e r r o r s a s s ociated w i t h the non-divergent and the 3-plane approximations are each shown to be about 10 to 15% forcwaves of a few thousand kilometers wavelength. Using the i n t e g r a t e d equations of motion both on the sphere i and on the 3-plane, the l i n e a r non-divergent Rossby wave s o l u t i o n s are shown to be exact f i n i t e amplitude s o l u t i o n s . An exact topographic wave s o l u t i o n i s a l s o given f o r the case of an ex p o n e n t i a l depth p r o f i l e . Such behaviour i s not found f o r the divergent waves. Using' a Stokes-type expansion i n terms of an amplitude parameter,the second order s o l u t i o n f o r divergent Rossby waves i s obtained, and i t i s found t h a t , as i n surface g r a v i t y wave theory, the f i r s t order c o r r e c t i o n to the phase v e l o c i t y i s zero. I t i s a l s o shown that the l i n e a r non-divergent Rossby wave s o l u t i o n on a uniformly sheared z o n a l current i s not a f i n i t e amplitude s o l u t i o n , and. the second order s o l u t i o n i s then c a l c u l a t e d . Once again, the phase speed i s c o r r e c t to the f i r s t order. A c l a s s of long waves of permanent form analogous t o the s o l i t a r y i i i and c n o i d a l waves of surface wave theory i s obtained f o r a $-plane channel of e i t h e r constant or e x p o n e n t i a l l y v a r y i n g depth. Such waves are found to e x i s t i n the divergent case i n the absence of any zonal c u r r e n t ; however, i f the divergence i s weak, or i f the non-divergent approximation i s made , then i t i s found, as i t was by Larsen (1965), that these waves w i l l e x i s t only i n the presence of a weakly sheared zonal current. On the e x p o n e n t i a l depth p r o f i l e , such waves e x i s t i n the absence of a sheared zonal c u r r e n t , even i f the non-divergent approximation i s made. I t i s suggested that such i waves may a l s o e x i s t trapped along long ocean ridges or scarps. Table of Contents .. I . I n t r o d u c t i o n 1.1 Aims of t h i s study 1 1.2 P h y s i c a l mechanisms 2 1.3 H i s t o r i c a l background 4 1.4 Oceanic observations of pl a n e t a r y waves 8 1.5 Non-linear e f f e c t s 9 I I . Equations of Motion 11 I I I . Results from L i n e a r Theory 3.1 Rossby waves on the sphere 3.1.1 I n t r o d u c t i o n 18 3.1.2 The non-divergent approximation 19 3.1.3 The divergent s o l u t i o n 21 3.1.4 P r o p e r t i e s of the s o l u t i o n s 25 3.2 The 3-plane s o l u t i o n s ' 27 3.3 topographic waves 31 IV. F i n i t e Amplitude P l a n e t a r y Waves 4.1 I n t r o d u c t i o n 37 4.2 Rossby waves on the sphere 4.2.1 The equations 38 4.2.2 Non-divergent s o l u t i o n s 39 4.2.3 P r o p e r t i e s of the s o l u t i o n s 42 V 4.3 Rossby waves i n a $-plane channel I - Exact s o l u t i o n s 4.3.1 The non-divergent s o l u t i o n 44 4.3.2 P r o p e r t i e s of the s o l u t i o n 46 4.4 F i n i t e amplitude topographic waves 4.4.1 The exp o n e n t i a l p r o f i l e 49 4.4.2, P r o p e r t i e s of the s o l u t i o n 50 4.5 Rossby waves i n a g-plane channel I I - P e r t u r b a t i o n expansions 4.5.1 The p e r t u r b a t i o n equations 53 4.5.2 The f i r s t order s o l u t i o n s 55 4.'5.3 The second order s o l u t i o n s 59 4.5.4 P r o p e r t i e s of the s o l u t i o n s 63 4.6 Rossby waves i n a g-plane channel I I I - Uniformly sheared current 4.6.1 The p e r t u r b a t i o n expansions 66 4.6.2 The; f i r s t order s o l u t i o n s 67 4.6.3 The second order s o l u t i o n 70 4.6.4 P r o p e r t i e s of the s o l u t i o n s 73 4.7' Summary 75 V. Long P l a n e t a r y Waves i n a Zonal Channel 5.1 The s c a l e d equations 77 5.2 The case 6 = 0(1) 5.2.1 D e r i v a t i o n of the long wave equation 81 5.2.2 The transverse eigenfunctions 84 5.2.3 So l u t i o n s to the Korteweg-deVries equation 86 v i 5.2.4 The s o l i t a r y wave 87 5.2.5 The cnoidal waves 89 5.2.6 A s p e c i a l case: one root zero 91 5.2.7 The non-divergent l i m i t 94 5.3 The case of 6 = 0(e) 5.3.1 Introduction 95 5.3.2 The equations 96 5.3.3 The case of uniform shear 99 5.3.4 The s o l i t a r y wave 101 5.3.5 A cnoidal wave 102 5.4 Topographic waves 103 5.5 Summary 106 VI. Concluding Remarks 108 Bibliography 112 Appendixes I - The problem of transformations i n c e r t a i n co-ordinate systems 117 II - Analogous behaviour of i n t e r n a l and planetary waves 119 III - Glossary of symbols 124 List of Tables Table Page I. The percentage difference between divergent and non-divergent phase speeds of Rossby waves on the sphere 24 L i s t of Figures Figure Page 1. Co-ordinate system on the sphere t i l t e d relative to the rotation axis 12 2. Wavelength and phase speed for cnoidal waves 93 Acknowledgements I g r a t e f u l l y acknowledge the advice and suggestions of Dr. P.H. LeBlond, under whose s u p e r v i s i o n t h i s work was done. I a l s o thank the d i r e c t o r , Dr. G.L. P i c k a r d , the f a c u l t y , and f e l l o w graduate students of the I n s t i t u t e of Oceanography f o r many amiable dis c u s s i o n s and much cooperation. I must a l s o acknowledge w i t h g r a t i t u d e the f i n a n c i a l a s s i s t a n c e of the I n s t i t u t e of Oceanography, the U n i v e r s i t y of B r i t i s h Columbia, and the N a t i o n a l Research Council of Canada. I am e s p e c i a l l y g r a t e f u l to Sandra Clarke f o r typi n g t h i s t h e s i s . I , I n t r o d u c t i o n 1.1 Aims of t h i s study In recent years, a body of l i t e r a t u r e (see §1.3) has grown up concerning l i n e a r p l a n e t a r y waves both i n the oceans and the atmosphere and t h e i r importance i n an understanding of the dynamics of both. The purpose of t h i s present i n v e s t i g a t i o n i s to study the p r o p e r t i e s of f i n i t e amplitude planetary waves, p a r t i c u l a r l y w i t h reference to oceanic s c a l e s . To introduce these i n v e s t i g a t i o n s , the p h y s i c a l mechanisms governing planetary waves w i l l be discussed b r i e f l y i n §1.2, the e x i s t i n g l i t e r a t u r e o u t l i n e d _ i n §1.3, and the reported observations of plan e t a r y waves i n the oceans l i s t e d i n §1.4. The f i n i t e amplitude e f f e c t s are i n v e s t i g a t e d by f i r s t r e d e r i v i n g i n Chapter I I the i n t e g r a t e d equations of motion f o r a homogeneous, i n v i s c i d f l u i d on the r o t a t i n g sphere. These equations are f u r t h e r s i m p l i f i e d by assuming that the pressure f i e l d i s h y d r o s t a t i c and that the flow i s b a r o t r o p i c , that i s that the h o r i z o n t a l v e l o c i t y components are independent of depth. The e f f e c t s of the t r a d i t i o n a l non-divergent and 3-plane approximations are then determined from the e x i s t i n g l i n e a r theory i n Chapter I I I i n order to determine t h e i r importance i n the e v a l u a t i o n of the f i n i t e amplitude s o l u t i o n s i n Chapter IV and the long wave ( s o l i t a r y and c n o i d a l wave) s o l u t i o n s i n Chapter V. Some remarks concerning the importance of these s o l u t i o n s are made i n the concluding chapter. 2 The terminology used to describe p l a n e t a r y waves i s f a r from standard. In these i n v e s t i g a t i o n s a p l a n e t a r y wave i s any wave motion i n a r o t a t i n g f l u i d which, i f the r o t a t i o n i s allowed to go to zero, reduces to a steady current. A Rossby wave i s a pl a n e t a r y wave on a , f l u i d of constant depth w i t h non-uniform r o t a t i o n ; a topographic wave i s a planetary wave on a f l u i d of v a r i a b l e depth and e i t h e r uniform or non-uniform r o t a t i o n . 1.2 P h y s i c a l mechanisms A plan e t a r y wave i s , t h e r e f o r e , defined by the p h y s i c a l mechanisms which d r i v e i t . In a shallow, r o t a t i n g , i n v i s c i d , homogeneous f l u i d , the equations of motion can be i n t e g r a t e d over the depth of the f l u i d , then c r o s s - d i f f e r e n t i a t e d to obt a i n the r e s u l t that the p o t e n t i a l r e l a t i v e v o r t i c i t y , 20, twice the r o t a t i o n v e c t o r , H the depth of the f l u i d , ri the sur f a c e e l e v a t i o n , and k a u n i t vector along the l o c a l v e r t i c a l [Greenspan, (1968), p.236]. In the case of non-uniform r o t a t i o n and constant depth, a water column moving w i t h some steady v e l o c i t y along a l i n e , of constant r o t a t i o n may be d i s p l a c e d i n i t i a l l y by an e x t e r n a l f o r c e i n t o a region of h i g h e r r o t a t i o n . I n order to conserve p o t e n t i a l v o r t i c i t y , e i t h e r the water column's r e l a t i v e v o r t i c i t y must decrease or the depth of the water must in c r e a s e through a r i s e i n surface e l e v a t i o n . I t i s found that the f i r s t e f f e c t predominates f o r short wavelength waves, the second f o r long waves. Both e f f e c t s cause the path l i n e to t u r n back conserved where £ i s the towards regions of lower r o t a t i o n and, hence, a r e s t o r i n g force i s provided and a wave motion i s set up about the undisturbed p o s i t i o n of the path l i n e . These waves are the Rossby waves. I f , on the other hand, the r o t a t i o n i s uniform, but the depth i s v a r i a b l e , a water column moving s t e a d i l y along an i s o b a t h may be d i s p l a c e d i n t o a region of decreased depth. This can be seen to have e x a c t l y the same e f f e c t as had moving the column i n t o a region of increased r o t a t i o n . The f l u i d responds by decreasing i t s r e l a t i v e v o r t i c i t y or i n c r e a s i n g i t s surface e l e v a t i o n so as to keep i t s p o t e n t i a l v o r t i c i t y constant. These e f f e c t s again a c c e l e r a t e the f l u i d column towards i t s undisturbed i s o b a t h and so provide a r e s t o r i n g f o r c e f o r the topographic wave. In both these cases, the p l a n e t a r y waves can e x i s t only i n a r o t a t i n g f l u i d . In the absence of r o t a t i o n , the water columns have no p o t e n t i a l v o r t i c i t y i n a uniform undisturbed flow; hence, i f they are d e f l e c t e d , the v o r t i c i t y remains zero and conservation of p o t e n t i a l v o r t i c i t y provides no r e s t o r i n g f o r c e to r e t u r n them to t h e i r o r i g i n a l p o s i t i o n s . In the l i m i t of s m a l l r o t a t i o n , t h e r e f o r e , p l a n e t a r y waves reduce to steady c u r r e n t s . Veronis (19.67a,b) has discussed the analogous behaviour of slow steady flows i n r o t a t i n g and s t r a t i f i e d f l u i d s . . This analogy i s extended i n Appendix I I , t o show that p l a n e t a r y and i n t e r n a l waves e x h i b i t analogous behaviour. This analogy w i l l be p a r t i c u l a r l y u s e f u l i n p r e d i c t i n g the behaviour of topographic waves on bathymetries s i m i l a r t o density p r o f i l e s f o r which i n t e r n a l wave s o l u t i o n s have already been found. 4 1.3 H i s t o r i c a l background The study of planetary waves i n geophysical f l u i d s was i n i t i a t e d by (C.G. Rossby (19 39) i n h i s study of time-dependent motions i n the atmosphere. Using l i n e a r i z e d equations of motion on the (3-plane, he was able to show that a homogeneous f l u i d could support long b a r o t r o p i c waves whose wavelengths and phase speeds were of the same magnitude as disturbances observed i n upper atmosphere m e t e o r o l o g i c a l c h a r t s . This theory was given a f i r m mathematical b a s i s by Haurwitz i (1940a,b), who s o l v e d the l i n e a r i z e d equations both on.the sphere and on the g-plane and showed that the various approximations introduced by Rossby had only s m a l l e f f e c t s on the magnitude of the r e s u l t i n g s o l u t i o n s . In these s t u d i e s , Haurwitz a l s o pointed out that h i s s o l u t i o n on the sphere had, i n f a c t , been obtained p r e v i o u s l y by Margules (1892). P l a n e t a r y wave theory was a p p l i e d to oceanic problems by Arons and Stommel (1956) i n an i n v e s t i g a t i o n of the f r e e periods of m e r i d i o n a l and z o n a l oceans on the $-plane. For Rossby waves they showed that although the phase v e l o c i t y i s always to the west, the group v e l o c i t y may be i n any d i r e c t i o n , and t h e r e f o r e , s t a t i o n a r y wave s o l u t i o n s may be constructed between m e r i d i o n a l boundaries. The amplitudes of these s o l u t i o n s , however, increase without l i m i t northward and southward. Veronis and Stommel (1956) i n v e s t i g a t e d the response of an unbounded two-layer $-plane ocean to moving wind systems. They found s o l u t i o n s f o r both b a r o t r o p i c and b a r o c l i n i c i n t e r n a l f r e e Rossby waves and showed that the frequency of b a r o c l i n i c Rossby waves went through a minimum value f o r wavelengths of the order of s e v e r a l hundred k i l o m e t e r s . This i n v e s t i g a t i o n suggested that f o r m i d - l a t i t u d e s most of 5 the energy from f l u c t u a t i n g winds of periods of one to seven weeks enters the ocean i n the form of b a r o t r o p i c Rossby waves. For longer p e r i o d s j i n c r e a s i n g energy appears i n b a r o c l i n i c motions u n t i l f o r very long periods (at l e a s t 100 y e a r s ) , the response i s pur e l y b a r o c l i n i c . L i g h t h i l l (1969), i n v e s t i g a t i n g the response of the Indian Ocean, to the onset of the monsoon, found that c l o s e to the equator the b a r o c l i n i c response was much qu i c k e r (of the order of one month). Other s t u d i e s of time-dependent motion i n a two-layered, m i d - l a t i t u d e , @-plane ocean were reported by Fofonoff (1962) and Rattray (1964). T h e i r s t u d i e s c l e a r l y show that the frequencies f o r the i n t e r n a l modes are very much l e s s than those of the b a r o t r o p i c modes. Fofonoff '(1962, p.387) f i n d s that f o r a d i f f e r e n c e i n d e n s i t y between the l a y e r s of 2 x 10 g/cm , the minimum periods f o r i n t e r n a l and b a r o t r o p i c waves are about 7 months and 3.6 days r e s p e c t i v e l y . The periods of the b a r o c l i n i c modes are so long, that i t seems l i k e l y that f r i c t i o n a l e f f e c t s must be important. Longuet-Higgins i n a s e r i e s of papers (1964a,b; 1965a; 1966) has e x t e n s i v e l y t r e a t e d the l i n e a r problem of b a r o t r o p i c Rossby waves i n a I homogeneous f l u i d both on the surface of the sphere and on the 3 _plane. In these papers he obtains s o l u t i o n s f o r both; non-divergent and divergent f r e e waves i n an unbounded ocean, and f o r t h e i r r e f l e c t i o n along s o l i d boundaries; using the r e f l e c t i o n p r o p e r t i e s , he found i t p o s s i b l e to sum l i n e a r s o l u t i o n s to f i n d the e i g e n s o l u t i o n s f o r v a r i o u s l y shaped ocean b a s i n s . The e f f e c t of bathymetry on pl a n e t a r y wave s o l u t i o n s was i n v e s t i g a t e d by Veronis (1966). He showed that over most of the ocean, 6 the topographic e f f e c t s were more important than the 3 - e f f e c t , and he a l s o l i n k e d the theory of topographic waves to that of Rossby waves. Topographic wave s o l u t i o n s on d i f f e r e n t bathymetries appear i n the l i t e r a t u r e under s e v e r a l d i f f e r e n t names. Reid (1956) found edge wave s o l u t i o n s , which he c a l l e d edge waves of the second c l a s s ; these are topographic waves on a s l o p i n g s h e l f . His i n v e s t i g a t i o n s were continued by Robinson (1964), Hamon (1966), and Mysak (1967) under the name c o n t i n e n t a l s h e l f waves. Topographic waves along d i s c o n t i n u i t i e s i n depth have been c a l l e d double K e l v i n waves or sea-scarp waves and have been i n v e s t i g a t e d by Longuet-Higgins (1968a,b), Rhines (1969a), and Mysak (1969). Rhines (1969a) a l s o s t u d i e d the r e f l e c t i o n of Rossby waves by submarine ridges and found by c a l c u l a t i o n that the M i d - A t l a n t i c Ridge i s s u f f i c i e n t l y broad to r e f l e c t a l l but the lowest mode Rossby wave i n the North A t l a n t i c . In a d d i t i o n to Veronis and Stommel (1956), other i n v e s t i g a t o r s have s t u d i e d the response of the ocean to f l u c t u a t i n g or moving pressure or wind systems, notably Longuet-Higgins (1965b), Pedlosky (1967), and L i g h t h i l l (1969). T h e i r s t u d i e s , a l l f o r constant depth oceans, confirm the important r o l e Rossby waves must play i n the time-dependent response of the ocean. Hamon (1966) and Mysak (1969) have discussed the generation of c o n t i n e n t a l s h e l f waves and double K e l v i n waves r e s p e c t i v e l y by moving or time-dependent weather systems. Pla n e t a r y l e e waves, generated by steady eastward f l o w i n g currents passing over bottom topography have a l s o been i n v e s t i g a t e d . Warren (1963) demonstrated the r o l e topography plays i n the generation of the 7 Gulf Stream meanders. By i n t e g r a t i n g the v o r t i c i t y equation n u m e r i c a l l y over a bottom topography s i m i l a r to that north of Cape Hatt e r a s , he o' obtained f o r a v a r i e t y of i n i t i a l flows, a v a r i e t y of meander patterns w i t h s i m i l a r shapes, amplitudes and wavelengths to those a c t u a l l y observed. These i n v e s t i g a t i o n s have been continued by N i i l e r and Robinson (1967), and Robinson and N i i l e r (1967). P o r t e r and Rattray (1964) obtained s o l u t i o n s f o r f i n i t e amplitude Rossby l e e wave patterns on steady uniform eastward flows passing over bottom d i s c o n t i n u i t i e s a l i g n e d north to south. A g e n e r a l i z a t i o n of t h e i r model by Clarke and Fofonoff (1969) allowed the c o n s i d e r a t i o n of bottom topography a l i g n e d i n any d i r e c t i o n . This model gave a f i n i t e - a m p l i t u d e l e e Rossby wave s o l u t i o n which increased i n amplitude downstream i f an eastward flow crossed a southeast to northwest step. Such growth of amplitude i s a consequence of the unboundedness of the model 8-plane ocean. Mclntyre (1968), using a Laplace transform technique, i n v e s t i g a t e d the l i n e a r problem of e i t h e r eastward or westward uniform flow over a s i n g l e s m a l l s t e p . For an unbounded ocean he showed that the assumption of no upstream i n f l u e n c e was c o r r e c t f o r eastward flows but i n c o r r e c t f o r flows to the west. I f the ocean i s bounded, as f o r example, the case of a zonal channel, Mclntyre shows th a t the assumption of no upstream i n f l u e n c e can never be made. This r e s u l t i s analagous to i that obtained by Benjamin (1970) i n h i s i n v e s t i g a t i o n of upstream i n f l u e n c e f o r a body moving along the r o t a t i o n axis of a f l u i d contained i n a tube. In t h i s study, Benjamin a l s o found that upstream 8 i n f l u e n c e s were always present, although t h i s cannot be p r e d i c t e d on the b a s i s of energy c o n s i d e r a t i o n alone. 1.4 Oceanic observations of plan e t a r y waves The planetary waves of §1.2 take the form of disturbances ( i n time or space) of current speed or d i r e c t i o n . In the upper troposphere, such waves are e a s i l y observed as wave-like disturbances on charts of i s o b a r i c s u r f a c e s . In f a c t , i t was to e x p l a i n these features that Rossby (1939) f i r s t s t u d i e d these waves that bear h i s name. In the ocean, p l a n e t a r y waves should appear as p e r i o d i c f l u c t u a t i o n s i n long time s e r i e s measurements of v e l o c i t y at s i n g l e p o i n t s or as long wavelength meanders of w e l l - d e f i n e d currents i f observations are completed i n a time much s h o r t e r than the periods of these waves. Few long time s e r i e s records of v e l o c i t y are a v a i l a b l e and t y p i c a l techniques i n s y n o p t i c oceanographic sampling of l a r g e areas obscure the nature of the phenomena; hence, oceanic observations which may be i n t e r p r e t e d as planetary waves are rare. Longuet-Higgins (1965a,p.62) suggests that c e r t a i n deep v e l o c i t y measurements north of Bermuda by Swallow (1961) could be evidence of the presence of i n t e r n a l Rossby waves. He f u r t h e r argues from the magnitude of the v e l o c i t i e s observed ( 38 cm/s) that i f these were Rossby waves, t h e i r amplitudes would be such that the waves would be s u b j e c t to considerable n o n - l i n e a r i t i e s . Wunsch (1967) found some evidence of Rossby waves i n h i s a n a l y s i s of t i d a l records at i s l a n d s t a t i o n s and suggested that these 9 were generated by the f o r n i g h t l y and monthly t i d a l p o t e n t i a l s . Hamon (1966) observed topographic waves i n the form of c o n t i n e n t a l s h e l f waves i n h i s a n a l y s i s of t i d a l records. F i n a l l y Thompson (1969) has found evidence of topographic Rossby waves from the long term current records taken at Woods Hole Oceanographic I n s t i t u t i o n ' s S i t e D. In charts of tr a n s p o r t streamlines f o r regions such as the A n t a r c t i c Ocean [Sverdrup e t a l . , (1963), p.606] and the western boundary regions [Warren, (1963)] wave-like patterns appear which may be pla n e t a r y lee wave p a t t e r n s . The amplitude of the excursions of the streamlines i n these s t a t i o n a r y waves appears to be of s u f f i c i e n t magnitude to expect n o n - l i n e a r e f f e c t s to be important. Even though Rossby waves have not been unequivocally observed i n the deep ocean, t h e o r e t i c a l evidence mentioned i n the previous s e c t i o n suggests that they should be generated i n the oceans by moving or f l u c t u a t i n g atmospheric systems. The l a c k of d e f i n i t e observations of oceanic planetary. waves may be a s c r i b e d to the great e f f o r t and expense required to make the necessary measurements, r a t h e r than to the f a c t the pla n e t a r y waves do not e x i s t i n oceans. An account of some of these o b s e r v a t i o n a l d i f f i c u l t i e s i s given by Thompson (1969). 1.5 Non-linear e f f e c t s I t appears from a few of these observations that the magnitudes of the pla n e t a r y waves i n the ocean may be such that the l i n e a r i z e d theory may not be a p p l i c a b l e and that n o n - l i n e a r i t y must be considered. F i n i t e amplitude s o l u t i o n s already e x i s t i n the form of the l e e wave 10 s o l u t i o n s of P o r t e r and Rattray (1964), and Clarke and Fofonoff (1969). These s o l u t i o n s have been shown t o give reasonable agreement w i t h meander patterns observed i n the A n t a r c t i c Circumpolar Current. This present i n v e s t i g a t i o n w i l l look at f i n i t e amplitude f r e e waves of the same form. I t has a l s o been recognized f o r some time that the i n t e r a c t i o n of p l a n e t a r y waves w i t h ocean currents i s important. K e l l e r and Veronis (1969) i n v e s t i g a t e d the e f f e c t of random currents on planetary waves; however, t h e i r study includes only the advection of the wave by the c u r r e n t s . In t h i s i n v e s t i g a t i o n , the i n t e r a c t i o n of planetary waves w i t h sheared zo n a l currents w i l l be s t u d i e d . F i n a l l y , s t u d i e s of s o l i t a r y and c n o i d a l waves by Lax (1968) have shown that any s o l u t i o n of the time-dependent Korteweg-deVries equation, U T + U U J J + "xxx = 0 (1.1) tends a s y m p t o t i c a l l y to a sum of s o l i t a r y waves; hence, s o l i t a r y waves, where they e x i s t , are an important l i m i t i n g case to f i n i t e amplitude wave motions. I n the f i n a l chapter, s o l i t a r y and c n o i d a l p l a n e t a r y wave s o l u t i o n s w i l l be described. In analogy to the surface g r a v i t y waves, s o l i t a r y p l a n e t a r y waves, i f they e x i s t , could be an important wave form i n the ocean. I I . The Equations of Motion In the f o l l o w i n g i n v e s t i g a t i o n s , wave s o l u t i o n s are sought f o r a homogeneous, i n v i s c i d f l u i d on the s u r f a c e of a r o t a t i n g sphere and on a 3-plane. Furthermore, the waves w i l l be long w i t h respect to the depth of the f l u i d ; t h e r e f o r e the motion w i l l be considered to be two-dimensional (independent of z ) . These waves are the planetary waves and are of two c l a s s e s ; the f i r s t , the Rossby waves, and the second, the topographic waves. In t h i s chapter the equations of motion w i l l be developed i n a general form both on the sphere and on the 3-plane and these w i l l form a b a s i s f o r the i n v e s t i g a t i o n s to f o l l o w . On the sphere, the equations are derived i n a co-ordinate system (see Figure 1) which r o t a t e s about the axis of r o t a t i o n of the sphere w i t h an angular v e l o c i t y , a, r e l a t i v e to the s u r f a c e of the sphere. For a wave of permanent form and phase speed a, t h i s 'frame of reference i s one i n which the motion i s steady. The use of s p h e r i c a l co-ordinates presents some d i f f i c u l t i e s because s p e c i a l assumptions not r e q u i r e d by the physics of the flow, must be made at the poles of the co-ordinates i n order that the mathematical s o l u t i o n s remain well-behaved. I t i s then d i f f i c u l t i n the f i n a l s o l u t i o n s to,separate the s i n g u l a r i t i e s near the poles that are due to the mathematics from those due to the p h y s i c s . This problem i s discussed i n g r e a t e r d e t a i l i n Appendix I . 12 Figure 1. Co-ordinate system on the sphere t i l t e d relative to the rotation axis. 13 In the development of these equations, the axis of the co-ordinate system i s t i l t e d r e l a t i v e to the axis of r o t a t i o n by an angle y i n order that any unique behavior at the poles of r o t a t i o n may be separated from the behavior at the axis of the co-ordinates. This angle y may take any value. Assuming that VQ, and v^ are not functions of r, the equations of motion are smG 3'<j) (2.1) r + g 1 3p_ (D 3r 3t smO 3(J) (2.2) r r _jL 3p_ rp 39 4 + M ) > + v 9 v f r c o t 9 r r (2.3) (siny cos 9 cos<j) - sin9 cosy )v. 1 3p_ rpsin9 3<j> 14 The equation of c o n t i n u i t y i s 1 8 / 2 \ _i_ 1 -2 ^ - ( r z v r ) + — . r 9r r r sxnt f ^ s i n e v e ) + ^ 2d? J = 0 (2.4) and the boundary conditions at the lower and upper boundaries r e s p e c t i v e l y , are v T 3 r 2 + Z 9 i l 2 + Yet 1 ^ 2 (2.5) 3t r 2 86 r 2 sxnB 3cj) at r = r 2 = R - H(0,<J),t) v = 111 + VQ 3 r x  r 3t r 36 r x s l n 0 3<j> p = p 0 = constant (2.6) at r = Tj = R + n(8,<(),t) , where the various symbols are defined i n the Glossary of Symbols contained i n Appendix I I I . In equation (2.5) the f l u i d depth H i s w r i t t e n as a f u n c t i o n of time s i n c e the frame of reference rotates r e l a t i v e to the sphere and, t h e r e f o r e , any depth v a r i a t i o n along the d i r e c t i o n of r o t a t i o n appears i n t h i s frame as a time-dependent depth. These equations must be f u r t h e r s i m p l i f i e d before they are i n a form i n which they may be s o l v e d . I f ft - a - 1Q~^s - 1, VQ - v^ -1 m/s, R - 10 6m, and r i - r 2 - 103m, then i n (2.1) the a c c e l e r a t i o n terms are about 10 _ 9m/s 2, the c e n t r i f u g a l terms 10 _ 6m/s 2, and the C o r i o l i s terms 10 - l fm/s 2 compared to g = 9.8 m/s2. Therefore to a high degree of approximation the pressure f i e l d i s h y d r o s t a t i c and equation (2.1) may be i n t e g r a t e d over r from the free surface r i 15 downwards to give at r p(r,e,<j>,t) = po + g p ( R + n - r ) . (2.7) I f a l l the terms of the c o n t i n u i t y equation (2.4) are to be of the same magnitude, then v r/vg - H/R - 1 0 - 3 . Using t h i s value of v r , i t i s seen that i n (2.2) and (2.3) the c e n t r i f u g a l and the C o r i o l i s terms i n which v r appears may be neglected r e l a t i v e to the other c e n t r i f u g a l and C o r i o l i s terms. The c o n t i n u i t y equation (2.4) may be i n t e g r a t e d over the depth of the f l u i d and the boundary conditions (2.5) and (2.6) a p p l i e d . Making the approximation that r i - Tz - R » H » n, the i n t e g r a t e d c o n t i n u i t y equation i s (2.8) R sinG |^(n + H) + |g[(n + H)sin9 v Q ] + |^[(n + H) v^] = 0 A f u l l e r d e s c r i p t i o n of these approximations i s given by P h i l l i p s (1966). S u b s t i t u t i n g f o r the pressure from (2.7) and making the approximation that r - R, equations (2.2) and (2.3) may be r e w r i t t e n as 3t R 96 R s i n e 9<f> ^ - 2(a + Q) ( s i n y s i n e coscf> + cosy cosG ) v 0 = - -j | 16 If* + + i r ^ l s * + v » V 9 S o t e <2-10> d t R d o R s m 6 dcp R + 2(a + fi)(siny s i n 8 cosd> + COSY c o s 6 ) v A = - „ A -Irr 9 R sm9 3$ Equations (2.8) to (2.10), known as the i n t e g r a t e d equations of motion, form the b a s i s f o r the f o l l o w i n g i n v e s t i g a t i o n s of p l a n e t a r y wave motions on a r o t a t i n g sphere. P l a n e t a r y waves of importance i n t h e o r e t i c a l s t u d i e s of the generation of time-dependent motions i n the oceans have wavelengths considerably s h o r t e r than the width of the ocean b a s i n s . Since most ocean basins have dimensions l e s s than the earth's r a d i u s , such waves have wavelengths considerably s h o r t e r than the earth's r a d i u s . For such waves, i t was shown by Rossby (1939) that the s u r f a c e of the sphere could be mapped onto a tangent plane, the e f f e c t of r o t a t i o n being r e t a i n e d i n a C o r i o l i s parameter l i n e a r i n y, the north-south co-ordinate. Such a transformation allows the use of C a r t e s i a n co-ordinates and, t h e r e f o r e , g r e a t l y s i m p l i f i e s the a n a l y s i s . The e f f e c t s of making the 3-plane transformation have been examined i n some d e t a i l by Veronis (1963). Equations (2.8), (2.9) and (2.10) may be transformed to t h e i r corresponding 3-plane equations by f i r s t s e t t i n g a and y to zero, 1 3 3 1 3 3 then a l l o w i n g R -* °° i n such a way that — - , r= r~x TTT v , R 30 dy R sm6 8<p 3x VQ •+ - v, v^ ->- u, and 2ficos0 f. The equations on the 3-plane are then, 17 3_ 3t (n + H) + 3-[u(n + H) ] + OX 3_ ay [v(n + H)] = o (2.11) 3u 3t + u 3u 3x , 3u j . + v i r - - f v (2.12) 3v 3 f + 3v 3y + fu g 3n 3y (2.13) where f 2ft[sin(y 0/R) + (y/R) cos(y 0/R)] (2.14) = fo + 3y and yo/R i s the l a t i t u d e at which the 3-plane i s tangent to the sphere. Equations (2.8) to-(2.10) and (2.11) to (2.13) describe the depth averaged flow of a shallow, i n v i s c i d and homogeneous f l u i d over a rough bottom both on the r o t a t i n g sphere and on the 3-plane r e s p e c t i v e l y . Using these as a b a s i s , i n the f o l l o w i n g t h e s i s , f i n i t e amplitude pl a n e t a r y waves w i l l be i n v e s t i g a t e d i n a v a r i e t y of cases. I I I . Results from L i n e a r Theory 3.1 Rossby waves on the sphere 3.1.1 I n t r o d u c t i o n The l i n e a r theory of planetary waves has been w e l l developed by Haurwitz (1940), Longuet-Higgins (1964b, 1965a, 1966, 1968a,b), Veronis (1966), and Rhines (1969a,b), as w e l l as others. In t h i s chapter the r e s u l t s of a l l these authors are summarized and the e f f e c t s of the various approximations commonly used i s discussed. In p a r t i c u l a r , the l i n e a r theory w i l l show the magnitude and importance of the e r r o r s introduced by the 3-plane and non-divergent approximations. A knowledge of the e f f e c t s of such approximations i s necessary i f the no n - l i n e a r s o l u t i o n s to be obtained l a t e r are to'be i n t e r p r e t e d . In the'.'following s e c t i o n the s o l u t i o n s f o r Rossby waves i n an ocean of constant depth completely covering the surface of a r o t a t i n g sphere w i l l be given, f o l l o w i n g Longuet-Higgins (1964b, 1965a). The s o l u t i o n s are obtained f i r s t making the non-divergent approximation, then dropping t h i s approximation f o r the divergent case. ;The b a s i c equations of motion are given by (2.8), (2.9), and (2.10). For constant depth, a and y zero, and the v e l o c i t i e s and s u r f a c e e l e v a t i o n s s m a l l , these equations may be l i n e a r i z e d to give = 0 (3.1) 9n 3t + H R sinG fe~(v esine) + If* 19 |2B - ZQcosev^ = - f f l (3.2) + 2fl cosB v f l = - _ g. „ |% (3.3) dt . o R s i n 6 dtp For wave s o l u t i o n s r o t a t i n g about the axis of r o t a t i o n of the sphere the dependent v a r i a b l e s may have t h e i r <() and t dependence expressed by exp i(scf) - a t ) . S u b s t i t u t i n g t h i s i n t o equations (3.1) to (3.3), the p a r t i a l d i f f e r e n t i a l equations are reduced to a s e t of ordinary d i f f e r e n t i a l equations. By d e f i n i n g the f o l l o w i n g v a r i a b l e s ( 3 ' 4 ) D = (1 - y 2 ) - ^ , y = cosB , X = a/2fi , 6 = ^ 2 Longuet-Higgjkns (1965a) reduced these ordinary d i f f e r e n t i a l equations to the s i n g l e equation [V 2 - s' - s , ^ ( g ( ; ^ + 6U 2 - y 2 ) ] ( v e sin0) = 0 (3.5) where s 2 2 • (3.6) 1 - y 3.1.2 The non-divergent approximation Equation (3.5) s t i l l i n c l u d e s i n i t the e f f e c t s of divergence, and, t h e r e f o r e , i t may be s i m p l i f i e d by making the non-divergent approximation. This approximation assumes that the f i r s t term of (3.1) i s much s m a l l e r than the other two and, hence, may be neglected. 20 In terms of the non-dimensional parameters defined by (3.4), t h i s assumption i m p l i e s that 6 << s'. I f t h i s i s t r u e , then (3.5) reduces to [V 2 - s ' ] ( v . sine) = 0 , (3.7) which f o r s 2Qs/a = . - n(n + 1) (3.8) has ,as s o l u t i o n s the s p e r i c a l harmonics of degree n and order s, where s _< n. These non-divergent s o l u t i o n s were f i r s t obtained by Haurwitz (1940). Longuet-Higgins (1964b) g e n e r a l i z e d these r e s u l t s by showing that the axi s of the s p h e r i c a l harmonics could be r o t a t e d through an a r b i t r a r y angle away from the axis of r o t a t i o n of the sphere. P r o v i d i n g these s p h e r i c a l harmonics r o t a t e d about the axi s of r o t a t i o n w i t h an angular v e l o c i t y of - 2£>/n(n+l) , the non-divergent l i n e a r equations are s t i l l s a t i s f i e d . That i s to say, i n the co-ordinate system described by Figure 1 (p. 12), the l i n e a r non-divergent s o l u t i o n c o n s i s t s of spherical'harmonics of degree n and order s where a = - 2fl/n(n+l), and y i s a r b i t r a r y . Therefore, w h i l e the angular phase speed of these waves'.must be about the axis of r o t a t i o n of the sphere, the t i l t i n g of the co-ordinate axis shows that the poles of the co-ordinates (at which the f l u i d v e l o c i t y due to the waves i s zero) do.v not n e c e s s a r i l y c o i n c i d e w i t h the poles of r o t a t i o n . For values of H, R, and 9, corresponding to those of r e a l oceans on the e a r t h , Longuet-Higgins (1965a) gives values of 5 ranging from 15 to 150. For an ocean of 4 km depth, 6 = 22; t h e r e f o r e , i t appears t h a t , i n order that the non-divergent approximation be v a l i d on the sphere, 21 s' must be very l a r g e . From (3.8) s' = 0 ( n 2 ) ; t h e r e f o r e , the non-divergent approximation i s v a l i d only f o r l a r g e n. Ocean basins have h o r i z o n t a l dimensions much l e s s than the earth's circumference. I n many p h y s i c a l oceanic problems such as a i r - s e a energy exchanges w i t h atmospheric disturbances, the wavelengths of i n t e r e s t must be much s m a l l e r than the width of the ocean; hence, s, the number of wavelengths around the equator, must be l a r g e / Since n >. s, then n i s indeed l a r g e ; t h e r e f o r e , the non-divergent Rossby wave s o l u t i o n s may be u s e f u l i n the examination of oceanic phenomena. On the other hand, f o r ,the i n v e s t i g a t i o n s of s t a t i o n a r y Rossby waves i n l a r g e enclosed basins such as the P a c i f i c Ocean, waves whose wavelengths are the same magnitude as the width of the. b a s i n w i l l be important. In t h i s case the non-divergent approximation i s not l i k e l y to be a p p l i c a b l e , and a more reasonable assumption would be that 6/s' < 0 ( 1 ) . 3.1.3 The divergent s o l u t i o n I f 6 = O(s') and a l l terms of 0(1) or l e s s are n e g l e c t e d , then equation (3.5) i s approximated by the s p h e r o i d a l wave equation ( v 0 sin6) = 0 . (3.9) This equation was f i r s t obtained f o r Rossby waves by Longuet-Higgins (1965); i t s s o l u t i o n s are given by the s p h e r o i d a l wave fu n c t i o n s S S (/?,y) where 22 s = - A ( A ) . sn (3.10) The f u n c t i o n A i s given by the s o l u t i o n of a transcendental sn ° J equation i n v o l v i n g continued f r a c t i o n s . Values of A g n ( c ) are tabulated i n S t r a t t o n , Morse, Chu, L i t t l e and Corbato (1956) f o r values of s, n, and c, a l l ranging form 0 to 8. These ranges cover most of the expected v a r i a t i o n of 6; however, the tables do not extend to large enough values of n and s. In the non-divergent l i m i t as 6 -*• 0, A (/£) may be expressed i n terms of a power s e r i e s i n 6 [ S t r a t t o n et a l . , (1956)] given by Therefore, i n the l i m i t of s m a l l 6, (3.10) reduces to (3.8) given by the non-divergent a n a l y s i s . The shape of the waves w i l l be changed from.that given by the non-divergent s o l u t i o n s i f the divergence terms are i n c l u d e d ; however, si n c e these waves are u n l i k e l y to be observed i n d e t a i l , such d i f f e r e n c e s are not of much i n t e r e s t . Of more i n t e r e s t are the d i f f e r e n c e s i n the d i s p e r s i o n r e l a t i o n s between the two cases. From the d e f i n i t i o n of s' given by (3.4) i t i s seen that the angular phase speed of the wave about the a x i s of r o t a t i o n i s given by 2 f i / s ' . Hence, f o r the non-divergent case, the phase speed i s independent of s, the l o n g i t u d i n a l wave number, w h i l e f o r the divergent case the phase speed i s a f u n c t i o n of both n and s. This d i f f e r e n c e i n d i s p e r s i o n r e l a t i o n s has an important e f f e c t on the combination of the wave s o l u t i o n s . I n the non-divergent case, waves of the same n(n + 1) + - 1 " (2s - l ) ( 2 s + 1) (2n - 1) (2n + 3) + 0(62) (3.11) 23 degree n but d i f f e r e n t orders s may be summed to form new l i n e a r wave s o l u t i o n s . Since the phase speeds are a l l the same, these s o l u t i o n s w i l l not disperse as the wave t r a v e l s around the globe. Of course, n o n - l i n e a r i n t e r a c t i o n s between the s o l u t i o n s can be expected to disperse the wave e v e n t u a l l y . On the other hand, f o r the divergent case, the phase speed i s d i f f e r e n t f o r each d i f f e r e n t value of n or s; hence, no such super-p o s i t i o n of s o l u t i o n s i s p o s s i b l e . Any two s o l u t i o n s of the same degree but d i f f e r e n t order w i l l s l o w l y disperse as the wave moves around the sphere, independently of n o n - l i n e a r e f f e c t s . The magnitude of t h i s d i s p e r s i o n can be estimated using the tabulated values of A g n [ S t r a t t o n e t a l . , (1956)]. Using these t a b l e s , Table I was drawn up to give the d i f f e r e n c e between the non-divergent and divergent phase speed as a percentage of the divergent phase speed f o r 6 = 64. This value of 6 i s the h i g h e s t f o r which A (/if) i s tabulated and represents a value that i s l a r g e r than those c a l c u l a t e d f o r most of the world's oceans. Hence, the d i f f e r e n c e s shown i n the t a b l e are l a r g e r than what might be expected f o r an ocean of average depth 4 km. In Table I i t i s seen that the percentage d i f f e r e n c e s i n phase speed between the two cases decrease w i t h i n c r e a s i n g s, and a f t e r an i n i t i a l i n c r e a s e a l s o decrease w i t h i n c r e a s i n g n, except f o r s = 0, which shows no i n i t i a l i n c r e a s e . The minimum percentage d i f f e r e n c e s f o r each n occur along the diagonal given by n = s, and these minimum values a l s o decrease w i t h i n c r e a s i n g n. The maximum value of s f o r which a tabulated value of A g^ was given i s s = 8. This represents a wave 24 TABLE I THE.PERCENTAGE DIFFERENCE BETWEEN DIVERGENT AND NON-DIVERGENT PHASE SPEEDS OF ROSSBY WAVES ON THE SPHERE n(n+l) 100 [ A ^ - n(n + l ) ] / [ n(n + 1)] s = 0 1 2 3 4 5 6 7 8 2 870. 275. 6: 495. 290. 91.7 12 298. 212. 127. 40.8 20 190. 152. 111. 67.0 22.0 30 125. 109. 89.0 65.3 40.0 13.0 42 85.2 80.0 69.8 56.7 41.9 25.7 8.3 56 61.8 59.8 54.6 47.3 38.6 28.6 17.5 5.7 72 47.2 44.4 43.5 39.2 33.8 27.4 20.3 12.5 4.2 whose wavelength at the equator i s approximately 5 x 10 km. or about a t h i r d the width of the P a c i f i c Ocean. Even for this large value of <5, the erro r i n phase speed caused by the non-divergent approximation i s only of the order of 20% i f ( n - s > £ 2 , n = 8 . From Table I an estimate can also be made of the magnitude of the difference i n phase speed between two divergent Rossby waves of the same degree n but d i f f e r e n t orders. The percentage difference i n angular phase speed, ( a - cL ) 100 /a , i s approximately equal to sn sn Sn • • 1 1 (A - A ) 100/[n(n + 1 ) ] , the difference between any two columns of Sn sn Table I. For n = 8, and for a difference i n s of 1, this percentage 25 difference ranges from 3% to 8% as s increases from 0 to 8. This percentage difference also appears to decrease as n increases, and so for the range of large n and s, which i s of the most i n t e r e s t , the e f f e c t i s expected to be n e g l i g i b l y small. However, the fa c t that such a difference i n dispersive behaviour does e x i s t between divergent and non-divergent solutions indicates that t h e i r non-linear behaviour, which w i l l be studied i n l a t e r chapters, may also be d i f f e r e n t . 3.1.4 Properties of the solutions Following Longuet-Higgins (1965a), the spheroidal wave equation (3.9) can be put i n t o the standard L i o u v i l l e form - [ ( s 2 - j ) csc 2G + 6 cos 29 + (s' - j )] V = 0 (3.12) through the transformation v Q = ( sin6 )"2 v(9). (3.13) Setting ( s 2 - j ) csc 26 + 6 cos 26 + (s' - j ) = - v 2 (3.14) i n equation (3.12), i t can be seen that the character of the s o l u t i o n of (3.12) changes from s i n u s o i d a l to exponential as c\)2 goes from p o s i t i v e to negative values. For large s, the f i r s t two terms of the left-hand side of (3.14) are both p o s i t i v e and monotonically increasing increases. Therefore, \) 2 i s p o s i t i v e only i f s' i s both large and negative, and then only i f 9 l i e s between TT — 9 Q and 6 Q where as 26 ( s 2 - j ) c s c 2 0 b + 6 coszQ0 + (s' - j ) = 0 (3.15) The e f f e c t of the non-divergent approximation i s to change the range of 8 over which the s o l u t i o n i s s i n u s o i d a l as w e l l as to change the shape of the s o l u t i o n . I t was. a l s o shown by Longuet-Higgins (1964b) that f o r the non-divergent case, the poles of the s p h e r i c a l harmonics which make up the s o l u t i o n do not have to c o i n c i d e w i t h the sphere's poles of r o t a t i o n . Hence, the e q u a t o r i a l b e l t i n which the waves are s i n u s o i d a l i s a b e l t surrounding the equator of a co-ordinate system, whose a x i s , as i n Chapter I I , may be t i l t e d at an a r b i t r a r y angle y from the ax i s of r o t a t i o n p r o v i d i n g i t rotates about that axis w i t h angular v e l o c i t y - 2J2/n(n + 1 ) . Since the equations are l i n e a r , the sum of s o l u t i o n s i s a l s o a s o l u t i o n . Therefore, i t i s p o s s i b l e to sum many s o l u t i o n s of the same n but d i f f e r e n t s and d i f f e r e n t o r i e n t a t i o n s to give a r e s u l t a n t s o l u t i o n that i s p e r i o d i c i n 0 everywhere. This sum i s not p o s s i b l e i n the divergent case as the waves of d i f f e r e n t orders each move w i t h - a d i f f e r e n t phase speed. In conclusion the l i n e a r s o l u t i o n s show that on the sphere, the e r r o r s introduced by the non-divergent approximation decrease w i t h i n c r e a s i n g wave: .number. Fbrrwavei numbers around 8, the e r r o r introduced i n the phase speed i s about 10 to 20%. The non-divergent approximation eliminates* the v a r i a t i o n of phase speed w i t h the l o n g i t u d i n a l wave number found f o r the divergent s o l u t i o n s ; however, t h i s d i s p e r s i o n i s found to be s m a l l f o r n = 8 and appears a l s o to decrease w i t h both i n c r e a s i n g n and decreasing S • For the wavelengths of i n t e r e s t i n the world oceans s and n are both greater than 8 and the e r r o r introduced by 27 the non-divergent approximation i n . t h e d i s p e r s i o n r e l a t i o n i s , t h e r e f o r e , l e s s than 20%. There i s some i n d i c a t i o n that the non-divergent approximation may have a la r g e e f f e c t when i t comes time to i n v e s t i g a t e the n o n - l i n e a r i t i e s of the s o l u t i o n s i n l a t e r chapters. 3.2 The 3-plane s o l u t i o n s I t has already been s t a t e d that the s o l u t i o n s of most i n t e r e s t i n the study of oceanic problems are those which have wavelengths s m a l l e r than the dimensions of the world oceans. In these cases i t has been shown that the e r r o r s introduced by the non-divergent approximation are not s e r i o u s . For the same range of wavelengths, that i s , those s m a l l e r than the earth's r a d i u s , i t seems l i k e l y that the 3-plane approximation may a l s o be used to s i m p l i f y the s o l u t i o n s s t i l l f u r t h e r . The 3-plane equations are obtained i n Chapter I I by mapping a r e s t r i c t e d area on the surface of the sphere onto a tangent plane, and are given by (2.11), (2.12), and (2.13). I f these equations are l i n e a r i z e d and the depth h e l d constant, they reduce to M + 9t 3v 3y 0 (3.16) 9H 3t f v + g^n 63x 0 (3.17) 0 (3.18) Following Longuet-Higgins (1965a) these equations may be f u r t h e r reduced to the s i n g l e equation 28 §H (f^v 2 + 3 3_ 3x i l 3t 3 + f 2 3 _ 3t 3v 3t ^ = 0 (3.19) I f i t i s assumed, as i t was on the the sphere, that a « 2tt, where a i s the radian frequencey, and a l s o that f i n (3.19) may be t r e a t e d as a constant, then (3.19) has a simple s i n u s o i d a l s o l u t i o n given by v = v D exp i (kx + ly - a t ) (3.20) where the d i s p e r s i o n r e l a t i o n i s k 3 k z + 5/ + f*/gH (3.21) I f the non-divergent approximation i s made by n e g l e c t i n g the f i r s t term of (3.16), then (3.16) to (3.18) reduce to + 3 3_ 3x 0 (3.22) whose s o l u t i o n s i s a l s o (3.20). However, f o r the non-divergent case, the d i s p e r s i o n r e l a t i o n i s given by a k 3 k z + V (3.23) Since the 3 _plane approximation i s v a l i d only over distances which are short r e l a t i v e to the earth's r a d u i s , the 3-plane s o l u t i o n s should show reasonable agreement w i t h the s o l u t i o n s on the sphere only f o r the short wavelength cases. On the sphere i t was f o r these short wavelength cases that the non-divergent approximation was v a l i d . Comparing (3.21) to (3.23), i t i s seen that t h i s i s a l s o the case on the 3-plane. For l a r g e k and £ (short wavelength), the per cent e r r o r i n the zonal phase speed, a/k, introduced by the non-divergent 29 approximation i s approximately 100f 2/gH(k 2 + I 2 ) . For a wavelength of about 1000 km, t h i s e r r o r i s about 10% and w i l l decrease w i t h decreasing wavelength. Near the equator, the phase speed of the Rossby wave s o l u t i o n on the sphere i s given by - 2fiR/A g n and the l o n g i t u d i n a l wavelength by 2T T R / S. I f n = s, the wave c r e s t s are a l i g n e d along the meridians of lon g i t u d e and the corresponding wave on the 3-plane i s given by (3.20) where Z = 0 and k = s/R. Since the assumption t h a t f 2 i s constant i s not v a l i d near the equator, and s i n c e f o r l a r g e n and s and f o r n = s, the non-divergent approximation i s v a l i d , the non-divergent s o l u t i o n s on the sphere and on the 3-plane are compared. Comparing t h e i r phase speeds, i t i s found t h a t , f o r n = s, the percentage d i f f e r e n c e i s approximately 100/s. For s 1 _> 10, that i s , f o r wavelengths l e s s than 1000 km, the e r r o r i n phase speed introduced by the g-plane approximation i s about 10%, and t h i s i s the same order as the e r r o r s introduced by the non-divergent approximations. For longer wavelengths, Longuet-Higgins (1966) shows very good agreement between 3-plane s o l u t i o n s and s p h e r i c a l s o l u t i o n s f o r a h e m i s p h e r i c a l ocean b a s i n centered around the equator. In d i s c u s s i n g the form of the s o l u t i o n s on the sphere i t was pointed out that these s o l u t i o n s are s i n u s o i d a l i n 6 only f o r a range of c o - l a t i t u d e s on each s i d e of the equator of the co-ordinate system. At f i r s t s i g h t t h i s behaviour does not seem to be reproduced by the 3-plane s o l u t i o n s , which seem to remain p e r i o d i c i n both x and y " regardless of the l a t i t u d e . This i s not e n t i r e l y true f o r the divergent s o l u t i o n s , s i n c e the assumption t h a t f 2 may be t r e a t e d as a constant i s v a l i d over d i f f e r e n t ranges of y f o r d i f f e r e n t l a t i t u d e s . 30 For a 3-plane taken around the equator, f = By, and a s o l u t i o n of (3.19) f o r a « f, i s given i n terms of P a r a b o l i c C y l i n d e r functions by v = v 0 exp i ( k x - at) [A U ( A , / ^ y) + B V Q / ^ y ) ] (3.24) whe re A = ^ ( k 2 + f) . (3.25) Since the parameter A i s r e l a t e d to the wavelength of the wave i n the north-south d i r e c t i o n , (3.25) i s the d i s p e r s i o n r e l a t i o n f o r the wave. Even though i n (3.24) the s o l u t i o n v a r i e s i n a n o n - s i n u s o i d a l fashion w i t h l a t i t u d e , i n co n t r a s t to the s o l u t i o n s on the sphere, i t s t i l l remains p e r i o d i c i n y over any range of y. T h i s d i f f e r e n c e i n behaviour i s due to the f a c t that the 3-plane, w h i l e r e s t r i c t e d i n the range over which i t i s v a l i d , i s a c t u a l l y t r e a t e d mathematically as being unbounded. In d i s c u s s i n g the non-divergent s o l u t i o n s on the sphere, i t was noted that the b e l t of c o - l a t i t u d e s f o r which the s o l u t i o n s were p e r i o d i c i n two dimensions could be t i l t e d at any angle to the axis of r o t a t i o n of the sphere. Therefore, anywhere on the surface of the sphere, i t i s p o s s i b l e f o r non-divergent waves that are doubly-p e r i o d i c to e x i s t . What these s o l u t i o n s do r e q u i r e i s that these doubly-p e r i o d i c waves are of f i n i t e l a t e r a l extent. I t i s t h i s f i n i t e l a t e r a l extent that i s missing from the 3-plane s o l u t i o n s . However, i f the width of the e q u a t o r i a l b e l t on the sphere i s l a r g e , then i t may exceed the range over which the 3-plane approximations i s v a l i d ; hence, w i t h i n 31 t h e i r range of a p p l i c a b i l i t y the 3-plane s o l u t i o n s are a good approximation to the s o l u t i o n s on the sphere. Summarizing b r i e f l y , the l i n e a r s o l u t i o n s i n d i c a t e that f o r short wavelengths ( l e s s than 1000 km.) both the 3-plane and non-divergent approximations may be made, the e r r o r s from each not exceeding 10%. 3.3 Topographic waves In each of the preceding s e c t i o n s the f l u i d depth has been h e l d constant and the r e s u l t i n g s o l u t i o n s have been r e f e r r e d to as Rossby waves. In §1.2, i t was shown that v a r i a t i o n s i n depth w i l l support a c l a s s of p l a n e t a r y waves known as topographic waves i n the same way a non-uniform r o t a t i o n f i e l d supports Rossby waves. Veronis (1966) showed that f o r t y p i c a l oceanic v a l u e s , t h i s topographic e f f e c t i s much more important than the 3-effect. The. b a s i c equations are obtained by l i n e a r i z i n g (2.11), (2.12), and (2.13) to give 3t + |-(uH) + |-(vH) = 0 dx dy (3.26) 3u at - f v + 8 3^ o (3.27) 3v 3t + f u + g 3x 0 (3.28) As w i t h Rossby waves, the non-divergent approximation i s made by n e g l e c t i n g the f i r s t term i n ( 3 0 2 6 ) . Following Veronis (1966), 32 the non-divergent approximation i s made, H = h ( y ) , and v « exp i ( k x - a t ) ; then under these c o n d i t i o n s , equations (3.26), (3.27), and (3.28) may be reduced to £(hv) y In equation (3.29) the depth h plays a dual r o l e . In the second term, the v a r i a t i o n of h plays the same r o l e f o r topographic waves as does the v a r i a t i o n of f f o r 3-plane Rossby waves. However, u n l i k e f i n the Rossby wave case, h a l s o appears i n the f i r s t term. This occurs because, independent of the v o r t i c i t y e f f e c t s , the v e l o c i t y must increase or decrease w i t h i n c r e a s i n g or decreasing depth i n order that mass conservation be s a t i s f i e d . For Rossby waves on a m i d - l a t i t u d e 3-plane, f i s always a l i n e a r f u n c t i o n of l a t i t u d e ; however, f o r topographic waves a whole range of d i f f e r e n t depth p r o f i l e s may be chosen, a l l of which model a c t u a l oceanic bathymetries. A simple p r o f i l e , s t u d i e d by Veronis (1966), i s the e x p o n e n t i a l p r o f i l e , h = h Q exp(-Ay). For t h i s p r o f i l e Veronis gives as a s o l u t i o n to (3.29) v = v 0 exp(^Ay) exp i ( k x + ly - at) (3.30) where c = i1 = ~ k* + e + -kA" ( 3 - 3 1 ) and where f has been h e l d constant. The f i r s t f a c t o r i n t h i s s o l u t i o n f o r v i s a growth f a c t o r r e q u i r e d by the presence of h i n the f i r s t term of (3.29) v (3.29) 33 I t was pointed out by Rhines (1969a) that f o r r e a l oceanic s l o p e s , A « k, and, hence, the v a r i a t i o n of h may be neglected i n the f i r s t term of (3.29). Such an approximation has i t s analogue i n the theory of i n t e r n a l g r a v i t y waves. There i t i s t r a d i t i o n a l to make the Boussinesq approximation i n which one negl e c t s the v a r i a t i o n of p where i t appears as an i n e r t i a l mass but r e t a i n s i t s v a r i a t i o n where i t appears m u l t i p l i e d by g and, hence, as p a r t of the body forces on the f l u i d . Veronis (1966) a l s o t r e a t s the case, i n which both f and h are allowed to vary, and o u t l i n e s the d i f f i c u l t i e s which one may encounter, i f the assumption that f may be tr e a t e d as a constant except under d i f f e r e n t i a t i o n i s made without due care. He shows that i f any terms are neglected, care must be taken to n e g l e c t a l l other terms of the same magnitude l e s t terms are r e t a i n e d that may i n d i c a t e that the s o l u t i o n s i s growing i n time. Considering the case of h = h Q exp(- Ax), equations (3.26) to (3.28) may be solved by making the non-divergent approximation, then d e f i n i n g a'transport stream f u n c t i o n , ip, by uh = - |4 , - vh = | i . (3.32) • dy dx In terms of t h i s stream f u n c t i o n Veronis (1966) obtained as a s o l u t i o n to these equations Ax, . r / B , , y f A (3.33) i> = 4>o exp(- — ) exp i[ay + / dy) + (k - )x - at] where 34 ° " 4 ( k 2 + £ 2 + A 2/4) ' ( 3 - 3 4 ) L o c a l l y equation (3.33) may be approximated by \J> = x | J 0 exp(- -| x) exp i(£y + kx - at) (3.35) where I = I + f 0A/2a , k = k - 3/2a (3.36) and t h e r e f o r e the d i s p e r s i o n r e l a t i o n may be w r i t t e n 0 - I 2 + I2 + A 2/4 ( 3 - 3 7 ) which i s s i m i l a r i n form to equation (3.31). The form of equation (3.37) allows the e f f e c t s of bottom topography to be compared w i t h those of non-uniform r o t a t i o n s i n c e both appear. The most l e v e l areas of the 'ocean f l o o r , an abyssal p l a i n , have slopes of about 10 _ l f; f o r c o n t i n e n t a l s l o p e s , r i s e s and shelves and f o r the mid-ocean ridges the slopes are t y p i c a l l y an order or two grea t e r i n magnitude. Even f o r a slope of 10~k and h ^ 10 3 m, A - 10~ 7 m '•}> and, hence, f o r m i d - l a t i t u d e s A f c - 1 0 " 1 1 m"1 s - 1 , the same magnitude as 3 ^ 10 _ 1 1 m"1 s . Therefore, i t i s seen that over much of the ocean ba s i n s , the topographic e f f e c t w i l l be equal to or dominate over the 3~effect. Furthermore, (3.37) shows that f o r given a and i f Afo > 3» the wavelength of the topographic waves i s s h o r t e r than that of the Rossby waves, and a l s o that t h i s wavelength decreases w i t h i n c r e a s i n g bottom slope. Therefore, i t appears that f o r the same range of frequencies the non-divergent and 3-plane approximations may be made 35 w i t h g r e a t e r confidence f o r the topographic waves than f o r the Rossby waves. I f equation (3.29) i s w r i t t e n i n terms of the t r a n s p o r t V = vh, i t takes the form J y kh / f a \h v 0 (3.38) In t h i s equation an analogy, discussed i n . g r e a t e r d e t a i l i n Appendix I I , may be c l e a r l y seen between the behaviour of p l a n e t a r y waves and i n t e r n a l g r a v i t y waves. For i n t e r n a l g r a v i t y waves on a density d i s t r i b u t i o n that v a r i e s only w i t h depth, the v e r t i c a l v e l o c i t y i s governed by the equation [Krauss, (1965)]',. Hz - k 2 .p z O VP; + 1 w (3.39) where w(x,y, z) W(z) exp i ( k x - a t ) (3.40) I f , i n the i n t e r n a l wave case, the depth i s constant and a l s o i f the upper s u r f a c e i s assumed r i g i d then the boundary c o n d i t i o n requires that w be zero at both boundaries. In a f l u i d of i n f i n i t e depth t h i s c o n d i t i o n i s replaced by the c o n d i t i o n that the v e r t i c a l v e l o c i t i e s tend to zero as z -* ± °°. I t i s e a s i l y seen that i f f i s h e l d constant i n (3.38), then the two equations are i d e n t i c a l i n form. Hence, a l l the s o l u t i o n s f o r i n t e r n a l waves on various density p r o f i l e s w i l l have analogous s o l u t i o n s f o r topographic waves on depth p r o f i l e s of the same form. 36 In p a r t i c u l a r , Rhines (1969a) and Longuet-Higgins (1968a,b) have found p l a n e t a r y wave s o l u t i o n s which c o n s i s t of waves trapped along depth p r o f i l e s such as ocean ridges and sea scarps. These s o l u t i o n s are analogous to the i n t e r n a l wave s o l u t i o n s given by Groen (1948) and Krauss (1965) f o r waves trapped on a pycnocline i n a f l u i d of i n f i n i t e depth. So l u t i o n s to (3.38) have been given f o r a v a r i e t y of p r o f i l e s by various authors. Rhines (1969b) has given s o l u t i o n s f o r waves trapped around i s l a n d s and sea mounts; and Mysak (1967) has obtained s h e l f wave s o l u t i o n s ; planetary waves trapped on a s l o p i n g c o n t i n e n t a l s h e l f . For other bathymetries, s t i l l more types of topographic waves can undoubtedly be found. IV. F i n i t e amplitude p l a n e t a r y waves 4.1 I n t r o d u c t i o n F i n i t e amplitude e f f e c t s f o r p l a n e t a r y waves, as f o r any other wave governed by n o n - l i n e a r equations of motion, can be i n v e s t i g a t e d by two fundamentally d i f f e r e n t techniques. In the f i r s t , the e n t i r e n o n - l i n e a r s e t of equations i s manipulated, making only those approximations necessary to f i n d an "exact" s o l u t i o n . Having obtained such s o l u t i o n s , i t i s then p o s s i b l e , a p o s t e r i o r i to determine whether such s o l u t i o n s have any p h y s i c a l s i g n i f i c a n c e . Such a method has been used w i t h great success by Y i h (1960) i n h i s s t u d i e s of s t r a t i f i e d flows over o b s t a c l e s . With the second technique, the i n v e s t i g a t o r must begin w i t h a p h y s i c a l concept of the phenomena of i n t e r e s t so that the terms of the n o n - l i n e a r equations may be p r o p e r l y s c a l e d and s o l v e d using a p e r t u r b a t i o n expansion i n some s m a l l parameter. Both of these techniques w i l l be used i n t h i s chapter to determine the f i n i t e amplitude e f f e c t s on the l i n e a r s o l u t i o n s o u t l i n e d i n Chapter I I I . The f i r s t technique i s used i n sections- 4.2 to 4.4 and exact non-divergent s o l u t i o n s are obtained f o r a constant depth ocean on the r o t a t i n g sphere and f o r a 3-plane channel both f o r uniform depth and f o r an exponential depth p r o f i l e . These Rossby wave s o l u t i o n s on the sphere and the 3-plane are shown to be i d e n t i c a l to the l i n e a r non-divergent Rossby wave s o l u t i o n s obtained by Longuet-Higgins (1964b, 1965a). 38 I f the non-divergent approximation i s not made, the equations cannot be reduced to a form that can be s o l v e d e x a c t l y . In §4.5, the f i n i t e amplitude divergent Rossby wave s o l u t i o n s on the 6-plane are found to the second order i n wave amplitude using a p e r t u r b a t i o n expansion. In §4.6, the important problem of the i n t e r a c t i o n of Rossby waves w i t h shear currents i s i n v e s t i g a t e d . In p a r t i c u l a r , i t i s shown that the l i n e a r non-divergent s o l u t i o n i s no longer a s o l u t i o n to the n o n - l i n e a r equation of motion i n the presence of a weakly sheared zonal current. 4.2 Rossby waves on the sphere 4.2.1 The equations The r e l e v a n t equations of motion f o r i n v i s c i d flow on the sphere have been developed i n Chapter I I and are given by (2.8) to (2.10). For free waves of permanent form r o t a t i n g about the axis of r o t a t i o n of the sphere w i t h angular phase speed a, the motion i s steady i n . t h e frame of reference described i n Chapter I I , that i s , a frame r o t a t i n g w i t h .angular v e l o c i t y , (a + Q), around the r o t a t i o n a x i s of the sphere. In order that the motion be steady i t i s a l s o necessary that the depth be a f u n c t i o n of 8'only, where 8' i s the c o - l a t i t u d e r e l a t i v e to the axis of r o t a t i o n . Under these conditions (2.8) i s w r i t t e n 0 . (4.1) Equation (4.1) allows the d e f i n i t i o n of a stream f u n c t i o n , \Jj(8,(f>), 3_ 88 (ri + H) s i n 8 v f + 8_ 9cj> (n + H) 39 such that 3^  (n + H) sin0 3<j> ' 1 _3j; n + H 36 (4.2) In terms of t h i s stream f u n c t i o n (2.9) and (2.10) are given by 1 3^ 3_ (ri + H)sin6 3<f> 36 1 3J) (ri + H)sin6 3(J> 1 M i _ (n + H ) s i n z 6 36 3(f) 1 JMJJ n + H 3cj) >t e (d±y + H)2\3e/ cot (n  ) + 2(fl + a)R n + H (siny sin6 coscj) + cosy cos6) 3ijj 3n s 36 (4.3) arid 1 _3JJ 3_ (n + H)sin6 3<j> 36 1 _3JJ n + H 36 1 _3JJ 3_ (il + H)sin6 36 3(f) 1 3^ Tl + H 38 + cot 6 3ip di> 2(fl + q)R 3\JJ / - — , „ v 2 — : — 5 - -^rx TTK- - — — r r — r - A " ( s i n y sin6 cos<J> + cosy cos6) TTT (n + H); sm6 3cj) 36 (ri + H)sxn8 v ' 3<p _g_ 3n sin6 3<J) (4.4) 4.2.2 Non-divergent s o l u t i o n s Since i t seems impossible to manipulate (4.3) and (4.4) i n order to get a s i n g l e equation i n e i t h e r ip or n, we s h a l l make the non-divergent approximation. This approximation, discussed p r e v i o u s l y i n Chapter I I I , i s made here by n e g l e c t i n g r| except where i t i s m u l t i p l i e d by g. Then n may be e l i m i n a t e d by c r o s s - d i f f e r e n t i a t i o n between (4.3) and (4.4) to give 40 J H s i n 9 1 _3j; H sinG 3c|> ) + H s i n 6 39 I H 39 1 3_ / s i n 9 3iJ/ (4.5) 2 (ft + g)R H ( s i n y s i n 6 coscj) + cosy cos6) , ijj 0 where J(a,b) i s the Jacobian ^ Q ' ^ . This can be i n t e g r a t e d once to give - 2(ft + a)R s i n 2 6 ( s i n y s i n 6 coscf) + cosy cos6) = F(^)H s i n 2 6 where F(^) appears as an a r b i t r a r y i n t e g r a t i o n f u n c t i o n . P h y s i c a l l y , . e q u a t i o n (4.6) i s an expression of the conservation streamlines c o i n c i d e w i t h p a t h l i n e s , the p o t e n t i a l v o r t i c i t y f i e l d i s a f u n c t i o n of 'the stream f u n c t i o n only. The i n t e g r a t i o n f u n c t i o n F(i|0 i s , t h e r e f o r e , the d i s t r i b u t i o n of p o t e n t i a l v o r t i c i t y . In order to s o l v e (4.6), the f u n c t i o n F(IJJ) must f i r s t be determined. Since F(ip) i s both the v o r t i c i t y d i s t r i b u t i o n due to the wave plus that due to a b a s i c flow, i t s form f o r any p a r t i c u l a r case i s not immediately apparent. For want of any i n f o r m a t i o n of the. shape of F(ijj) , i t may be assumed that i t i s at most a l i n e a r f u n c t i o n of Tp and a l l p o s s i b l e s o l u t i o n s r e s u l t i n g from such an assumption, determined. From these s o l u t i o n s the b a s i c flows f o r which t h i s l i n e a r f u n c t i o n i s the v o r t i c i t y d i s t r i b u t i o n can then be (4.6) of p o t e n t i a l v o r t i c i t y of a f l u i d column. Since, i n steady flow, the found. 41 Since one can always add an a r b i t r a r y constant to a stream f u n c t i o n there are only two p o s s i b l e cases f o r which F(T|J) i s l i n e a r i n \p, these being Case I F(i|0 = d G (4.7) Case I I F(i|i) = - . (4.8) The s o l u t i o n s of i n t e r e s t are f r e e waves i n an ocean of constant depth which completely covers the surface of the sphere. For H constant and F(\p) given by (4.7), equation (4.6) then becomes sine ||) + (4.9) s i n 9 30\ 96/ s i n ^ 6 = d 0H 2 + 2(fi + a)RH(siny s i n 6 cosc}> + cosy cos6) . This equation has no s o l u t i o n s p e r i o d i c i n cj> which are a l s o f i n i t e over the e n t i r e sphere; t h e r e f o r e , Case I gives no wave s o l u t i o n s . An equation of the same form as (4.9) but w i t h a = 0 may have some importance i n the study of steady flows i n channels on the sphere. Turning to Case I I , f o r H constant and F(^) given by (4.8), equation (4.6) becomes s i n 6 9 6 \ s i n " 96/ ' s i n * 6 9cj> 2(fi + a)RH (si n y s i n 6 coscj) + cosy cos8) . The s o l u t i o n of (4.10) , f i n i t e over the e n t i r e sphere, i s given by 42 n = ^ A{J C O S Pn(cos9) (4.11) m=0 . 2(» + g)RH . + — ; — ; — r \ n s m y s i n e costp + C O S Y cos6) n(n + 1) - 2 1 where d i H 2 = n(n + 1) . ' (4.12) 4.2.3 P r o p e r t i e s of the s o l u t i o n s R e c a l l i n g from Figure 1 (p.12) that cosS' = cosy cos9 + s i n y s i n 9 coscp , (4.13) where 6' i s the c o - l a t i t u d e r e l a t i v e to the r o t a t i o n a x i s , i t can be seen that the stream f u n c t i o n as given by (4.11) c o n s i s t s of the sum of surface harmonics of degree n (and of any o r i e n t a t i o n ) plus a steady flow which i s zonal r e l a t i v e to the r o t a t i o n a x i s . I f a l l the wave amplitudes A™ are s e t to zero, equation (4.11) i s reduced to the stream f u n c t i o n f o r the undisturbed flow. This steady zonal flow i s given i n the r o t a t i n g frame by v', = , , . / — t t s i n 9 ' . (4.14) <p n(n + 1) - 2 R e l a t i v e to the surface of the s o l i d sphere, r a t h e r than to the r o t a t i n g frame, the zonal v e l o c i t y i s 2^ + n(n + l ) q j. . A | , . 1 C X v., = — — T~ R sxn9' . (4.15) tp n(n + 1) - I I f t h i s b a s i c zonal current r e l a t i v e to the sphere i s s e t to zero, then from (4.15) 43 20 a fr, -1. 11 * (4.16) n(n + l ; The phase speed of a l i n e a r non-divergent wave on the sphere i s given by equation (3.8) of the previous chapter as o 20 s n(n + 1) (4.17) t h e r e f o r e , f o r the case of zero b a s i c flow, the l i n e a r and n o n - l i n e a r d i s p e r s i o n r e l a t i o n s are i d e n t i c a l . Furthermore, the form of the n o n - l i n e a r s o l u t i o n , being the sum of surface harmonics, i s i d e n t i c a l to that of the l i n e a r s o l u t i o n . Hence the l i n e a r non-divergent on the sphere i s , i n f a c t , an exact s o l u t i o n . This r e s u l t was p r e v i o u s l y obtained by Neamtan (1946) and by B a r r e t t (1958); however, i t does not appear t o be well-known i n the l i t e r a t u r e of oceanic planetary waves. For t h i s reason the a n a l y s i s has been repeated here and l a t e r f o r Rossby waves on the 3-plane. An extension of Neamtan's a n a l y s i s i n §4.4 allows the examination of n o n - l i n e a r topographic wave s o l u t i o n s . Haurwitz (1940b) showed that a zonal wind of the form V CR s i n 9 ' could be added to the l i n e a r equations without changing the form of the s o l u t i o n s . I f the zonal current given by (4.15) i s s e t equal to V DR s i n 6 ' , then 1 - - a = nln^lj • ( 4 ' 1 8 ) n(n + 1) Since n » 1, then (4.18) may be approximated by i For the case of an undisturbed zonal current of the form V 0R sinO', the l i n e a r s o l u t i o n i s then the f u l l s o l u t i o n . In order to get any 44 h i g h e r order i n t e r a c t i o n s i t w i l l be necessary to change the undisturbed v o r t i c i t y f i e l d by the a d d i t i o n of a sheared b a s i c z o n a l current. This would be e q u i v a l e n t to changing the form of F(IJJ) i n equation (4.6). On the other hand, i f the non-divergent approximation i s not made i t appears impossible to reduce (4.3) and (4.4) to a simple equation i n only one of \p and n. Hence, i t i s very u n l i k e l y that the l i n e a r divergent s o l u t i o n s would be exact s o l u t i o n s of the n o n - l i n e a r equation. This can be tes t e d by d i r e c t s u b s t i t u t i o n , but f o r ease of c a l c u l a t i o n t h i s t e s t w i l l be made only on the 3-plane. 4.3 Rossby waves i n a 3-plane channel, I . Exact s o l u t i o n s 4.3.1 The non-divergent s o l u t i o n As discussed e a r l i e r , the 3-plane approximation i n v o l v e s mapping the surface of the sphere onto a tangent plane, and t h e r e f o r e , i s a v a l i d approximation only f o r h o r i z o n t a l s c a l e s much l e s s than the rad"i us of the earth. In order to e s t a b l i s h a h o r i z o n t a l s c a l e , the problem i s t r e a t e d i n a zonal .channel of width L. i • , I f there e x i s t p l a n e t a r y waves of permanent form which have a phase v e l o c i t y , c, i n the x d i r e c t i o n , then i n a frame moving at t h i s phase v e l o c i t y r e l a t i v e to the e a r t h , the motion i s steady.', For such waves to e x i s t , H = H(y) only, s i n c e i f the depth v a r i e s w i t h x, time-dependent terms must enter i n t o the equations as the frame moves from one depth to another. Transforming to such a frame through the transformation s = x - ct (4.20) 4 5 equations ( 2 . 1 1 ) to ( 2 . 1 3 ) become 3_ 3s (u - c) (n + H ) + ^ | v ( n + H ) 3n / s 3u , 3u f (u - c) — + v — - f v + 6 3 s f v 3v , 3v f 3n ( u " c ) 3 l + v 37 + f u + s a? w i t h boundary conditions given by at y 0, L ( 4 . 2 1 ) ( 4 . 2 2 ) ( 4 . 2 3 ) ( 4 . 2 4 ) A tr a n s p o r t stream f u n c t i o n s a t i s f y i n g ( 4 . 1 2 ) may be defined by (u - c) 1 3jjj T) + H 3y v 1 _3j; n + H 3s ( 4 . 2 5 ) S u b s t i t u t i n g ( 4 . 2 5 ) i n t o ( 4 . 2 2 ) and ( 4 . 2 3 ) and c r o s s - d i f f e r e n t i a t i n g between the r e s u l t i n g two equations gives n + H 3s 1 j h j J n + H 3s + n + H 3y 1 3^ n + H 3y ri + H ( 4 . 2 6 ) which can be immediately i n t e g r a t e d once to give 1 3_ n + H 3s 1 _3JJ n + H 3s + n + H 3_ 3y 1 3^ Tl + H 3y n + H ( 4 . 2 7 ) where again F(iJ;) i s an a r b i t r a r y i n t e g r a t i o n f u n c t i o n s p e c i f y i n g the d i s t r i b u t i o n of p o t e n t i a l v o r t i c i t y i n the f l u i d . I n the same way as on the sphere, F(4>) i s chosen to be a l i n e a r f u n c t i o n of ty. Since ljJ i s defined only up to an a r b i t r a r y constant, there are only two cases f o r which F ( T | J ) i s l i n e a r , these corresponding to ( 4 . 7 ) and ( 4 . 8 ) given p r e v i o u s l y f o r the s o l u t i o n s on the sphere. 46 I f the non-divergent approximation i s made by n e g l e c t i n g n r e l a t i v e to H, and i f H i s h e l d constant, then equation (4.27) i s f o r Case I 0 + $ - fH = d.H2 , (4.28) and f o r Case I I 0 + 0 + d l H 2 ^ - fH = 0 , (4.29) w i t h the. boundary c o n d i t i o n that M = 0 at y = 0, L . (4.30) os As on the sphere, there are no s o l u t i o n s f o r case I s a t i s f y i n g the boundary conditions which are al s o p e r i o d i c i n s. On the other hand, case I I has a wave s o l u t i o n given by M = ^ An s i n cos k^s + -^ pg- + B cos (i/djHy + b j ) , (4.31) m=l 1 where k 2 + ^ - d l H 2 = K 2 (4.32) and M 2 < < (M + l ) 2 . (4.33) — TT — 4.3.2 P r o p e r t i e s of the s o l u t i o n As on the sphere, t h i s s o l u t i o n of the n o n - l i n e a r equations i s i d e n t i c a l to to l i n e a r s o l u t i o n as given i n chapter I I I . These r e s u l t s were p r e v i o u s l y obtained by Neamtan (1946) and a p p l i e d to atmospheric processes. The s o l u t i o n s are here r e d e r i v e d i n order that they may be compared to divergent Rossby wave s o l u t i o n s and topographic wave s o l u t i o n s which s h a l l be obtained i n l a t e r s e c t i o n s . 47 2 2 Replacing diH by K , the total wave number, and allowing the wave amplitudes to go to zero, the zonal velocity is given by u-,c = f2 - ^ s i n C K y + bO . (4.34) From equation (4.31) i t appears that a non-divergent Rossby wave can exist in the presence of a sheared basic zonal current providing that current is of the form (4.34). However, in order that the wave be periodic in s, K2 > 2 for some m < M ; therefore, J-J this basic zonal current, i f i t is to be sheared, must have at least as many zeros across the channel as does the wave solution i t s e l f . In the real ocean or atmosphere such a complex basic zonal flow is unlikely to exist; hence, here B w i l l be set to zero. If B were non-zero, the total wave number, K, in this solution would be determined by the wave number of the basic flow; however, for a uniform basic flow, ( B = 0), K is unspecified and waves of any total wave number may exist. Equation (4.31) shows that waves of the same K, though of different m and k, can be summed together with no interactions; however, non-linear interactions may occur between two waves of different K. A case of interest in Chapter V is the case of a weakly sheared zonal basic current. This may be modelled here by looking at the solution for small K . For KL « 1, (4.34) gives a weakly linearly sheared basic current, 3 B K u - c - —2 — [Ky cos bi + sin bi ] ; (4.35) 48 however, (4.32) shows that f o r m j> 1, 2 2 2 2 k^ K - —JT jj < 0 . (4.36) Therefore, i t appears that a weakly sheared current w i l l not support a wave of the form, s i n i^yj c o s ^ m s > a s a wave of permanent form. The question of whether any f i n i t e amplitude wave of permanent form can e x i s t i n t h i s case w i l l be discussed i n §4.6 and Chapter V. Such simple s o l u t i o n s of the n o n - l i n e a r equations of motion are p o s s i b l e only i n the case of non-divergent motions. The presence of divergence b r i n g s i n t o play a whole new s e t of n o n - l i n e a r i n t e r a c t i o n terms, and i t i s no longer p o s s i b l e to f i n d simple s o l u t i o n s to the f u l l equations of motion. While i t i s p o s s i b l e to wriite an equation such as (4.27) which fo r m a l l y appears to be l i n e a r i n IJJ, i t i s not, p o s s i b l e to separate out r| without i n t r o d u c i n g new n o n - l i n e a r i t i e s i n t o the equations. In the l i n e a r s o l u t i o n s , i t was shown that f o r s u i t a b l y s h o r t wavelengths there were n e g l i g i b l e d i f f e r e n c e s between the divergent and non-divergent s o l u t i o n s . However, i n c o n s i d e r i n g the f u l l equations, we see that t h e i r n o n - l i n e a r behaviour i s much d i f f e r e n t . While the non-divergent l i n e a r s o l u t i o n s were shown to be exact s o l u t i o n s , no such behaviour i s i n d i c a t e d f o r the divergent s o l u t i o n s . This d i f f e r e n c e i n behaviour was also suggested by the l i n e a r s o l u t i o n s on the sphere. There, a l l non-divergent s o l u t i o n s of the same degree moved w i t h the same phase speed, suggesting that super-p o s i t i o n of s o l u t i o n s to form a wave of permanent form was p o s s i b l e . On the other hand, the phase speed of the divergent s o l u t i o n s v a r i e d 49 w i t h both degree and order; t h e r e f o r e , any s u p e r p o s i t i o n of s o l u t i o n s would disperse i n time unless n o n - l i n e a r i n t e r a c t i o n s worked to e x a c t l y cancel t h i s d i s p e r s i o n . I f such a wave of permanent form does e x i s t , i t w i l l be a s o l i t a r y or a c n o i d a l wave; such waves w i l l be i n v e s t i g a t e d i n Chapter V. 4.4 F i n i t e amplitude topographic waves 4.4.1 The e x p o n e n t i a l p r o f i l e In Chapter I I I , on the l i n e a r p l a n e t a r y waves, i t was shown that gradients of depth may act i n the same way as gradients of f to support p l a n e t a r y wave motions. In the n o n - l i n e a r case, equation (4.27), f o r H = H(y), w i l l a l s o give wave s o l u t i o n s , even i n the case of uniform f. As i n the theory of i n t e r n a l waves where one f i n d s that the mean density p r o f i l e determines many of the p r o p e r t i e s of the wave s o l u t i o n s , i n the study of topographic waves the choice of depth p r o f i l e has s i m i l a r consequences. Many d i f f e r e n t depth p r o f i l e s may be chosen, but here the problem w i l l be s o l v e d only f o r the e x p o n e n t i a l p r o f i l e , H = Ho exp(- Ay) . (4.37) Once again, making the non-divergent approximation and s e t t i n g K 2 F(ip) = - \rzty, equation (4.27) becomes H 0 % a + e" A y [ e % y ] y + K 2e" 2 A y * - fH.e" A y (4.38) which has as a s o l u t i o n 50 . = C2" { B i Jv(XO + B 2 Y V ( X ? ) } s i n ks (4.39) , f 0 H 0 f^SJnX(g; - t) • -v r a. r. \r + —j^ J ^ dt + Di s i n XC + D 2 cos X? 2 A 3 J - l n 2 t cosX(t - O dt 3H 0 "3 where X = exp(- Ay) , (4.40) X = | , (4.41) V 2 = p + \ , (4.42) and Di and D 2 are a r b i t r a r y constants. 4.4.2 P r o p e r t i e s of the s o l u t i o n The i n t e g r a l s which make up the solu t i o n " ( 4 . 3 9 ) cannot be evaluated a n a l y t i c a l l y ; however, s i n c e f o r a l l f i n i t e y, £ > 0, then the integrands are f i n i t e , and t h e r e f o r e , the i n t e g r a l s themselves are f i n i t e . Once again, i t i s necessary to apply some bounds to the ocean w i t h i n which the 3-plane approximation remains v a l i d . For a zonal channel, equation (4.39) must s a t i s f y the boundary c o n d i t i o n s given by (4.30). These are s a t i s f i e d i f J V ( X ) Y v ( X e " A L ) - Y V(X) J v ( X e " A L ) = 0 . (4.43) I f V i s r e a l , and a and b are p o s i t i v e , Gray and Matthews 51 (1922, p. 82) show that J v ( a x ) Y v(bx) ':- J v ( b x ) Y v ( a x ) i s a s i n g l e -valued, even f u n c t i o n of x whose zeros are a l l r e a l and simple. In (4.42), v may be chosen t o be the p o s i t i v e r o o t , and s i n c e K i s an a r b i t r a r y constant, i t may be chosen of the same s i g n as A so that X , as given by (4.41), i s always p o s i t i v e . Hence, (4.43) has a. sequence of r e a l r o o t s , { X R } . V Abramowitz and Stegun (1965, p.374) give an asymptotic formula f o r determining the r ^ zero of the cross-products i f r i s l a r g e . In terms of the v a r i a b l e s used here, t h i s i s e - A L x 5 , ~ a i + £ + i ^ I L 2 + d - 4pq + 2 P 3  v a i a i a? where ai-4 V 2 - 1 rir r A L ' • V . A L e - 1 8e (4v2 - l ) ( 4v 2 - 25 ) (e 3 A L - 1) 6(4e A L ) 3 ( e A L - 1) (4v2 - 1)(16vJ - 456v2 + 1073)(e 5 A L - 1) . A L 5 A L ~ 5(4e ) (e - 1) A L N , . L — ' ( 4 ' 4 5 ) d = Since (4.40) gives a r e l a t i o n between V and k, the r e f o r e from (4.44) a d i s p e r s i o n r e l a t i o n g i v i n g k i n terms of K , A , L , and r may be obtained. However, because (4.44) i s a transcendental r e l a t i o n v a l i d only f o r l a r g e r , the a c t u a l d i s p e r s i o n r e l a t i o n cannot be obtained a n a l y t i c a l l y . The roots represented by (4.44) have been shown by K l i n e (1948) to reduce i n the l i m i t as e A ^ 0 to the r ^ zerovof J ^ ( x ) . However, i n t h i s l i m i t the non-divergent approximation, which requires that 52 X] « "H, would not be v a l i d . In any p r a c t i c a l problem the f i r s t few roots of (4.43) would have to be c a l c u l a t e d n u m e r i c a l l y f o r the a c t u a l values of K , A, L. The case of uniform r o t a t i o n may be i n v e s t i g a t e d by s e t t i n g 3 = 0 i n (4.39). This e l i m i n a t e s the l a s t i n t e g r a l , and t h e r e f o r e , s i m p l i f i e s the s o l u t i o n somewhat. A l s o i f f i s constant, the s o l u t i o n w i l l h o l d f o r a channel of any o r i e n t a t i o n . The s o l u t i o n given by (4.39) expressed i n terms of the zon a l v e l o c i t y i s A u - c = - H (v + £ ) rrh B i J V ( X ? ) + B 2 Y v(A?)} (4.46) Bi J^+i^O + B 2 YV+T_(A<;)} s i n ks X f 0 f cos A.(t - t) J „ K , „ K •f J dt - Dl - Q cos A? + D 2 s i n A? + ^| 2 l n 2 C + || 2 / l n 2 t s i n X(t - ?) dt . Equation (4.46) may be averaged over a wavelength i n s; however the r e s u l t i n g z o n a l flow s t i l l remains a very complicated f u n c t i o n of y, much more complicated, i n f a c t , than one would expect to e x i s t as a r e a l ocean flow. While (4.39) i s an exact s o l u t i o n to the non-divergent equations, i t i s too complicated to i n t e r p r e t or be u s e f u l as an approximation to r e a l oceanic flow. In summary, i t has been found that a wave of permanent form w i l l e x i s t as an exact s o l u t i o n of the non-divergent equations of motion f o r the case of a channel w i t h bottom p r o f i l e H = H ce A y on the 3-plane, and furthermore, that such a s o l u t i o n w i l l e x i s t even i f the r o t a t i o n i s 53 uniform. However, the b a s i c zonal flow required i n order that t h i s wave e x i s t i s so complicated that i t i s u n l i k e l y that the s o l u t i o n represents a wave l i k e l y to be observed i n e i t h e r the ocean or the atmosphere. Since, i n topographic waves, the wave p r o p e r t i e s depend to a large extent on the p r o p e r t i e s of the topography, i t i s p o s s i b l e that f o r a d i f f e r e n t topography, a simple wave of permanent form may e x i s t without r e q u i r i n g such a complex b a s i c zonal c u r r e n t ; however, such an i n v e r s e problem would be very d i f f i c u l t to s o l v e . The s o l u t i o n described above was obtained by r e q u i r i n g F(iJ;) i n equation (4.27) to be a l i n e a r f u n c t i o n of ty. Since F(i[0 i s the d i s t r i b u t i o n of p o t e n t i a l v o r t i c i t y , and s i n c e f o r the e x p o n e n t i a l p r o f i l e the p o t e n t i a l v o r t i c i t y due to the r o t a t i o n of the f l u i d , f, i s d i s t r i b u t e d e x p o n e n t i a l l y w i t h y, i n order that the b a s i c zonal flow be simple (that i s , at most, l i n e a r i n y) i t would seem l i k e l y that F(tjj) should be some ex p o n e n t i a l f u n c t i o n of . In t h i s case though, (4.27) i s a n o n - l i n e a r equation and d i r e c t s o l u t i o n would be very d i f f i c u l t , p a r t i c u l a r l y s i n c e , f o r n o n - l i n e a r F(I|J) , (4.27) i s 1 no longer separable. 4.5 Rossby waves i n a 3-plane channel, I I . P e r t u r b a t i o n expansions 4.5.1 The p e r t u r b a t i o n equations In t h i s s e c t i o n , the f i n i t e amplitude e f f e c t s on divergent Rossby waves i n a 3-plane channel w i l l be i n v e s t i g a t e d using a Stokes-type p e r t u r b a t i o n expansion. The b a s i c equations ^ governing waves of permanent form i n a 3 -plane channel are given by (4.21), 54 (4.22), (4.23), and the boundary conditions by (4.24) For divergent Rossby waves i n a channel of constant depth, the v a r i a b l e s w i l l be non-dimensionalized through the t r a n s f o r m a t i o n , (u, v, c) = 6L 2(u', v', c') , (4.47) (s, y) = L ( s ' , y') , f = gLf' = 3 L ( f J + y') , (n, H) = H(6n' s i ) where 6 = 3 2L 2/gH, a non-dimensional divergence parameter. S u b s t i t u t i n g these non-dimensional v a r i a b l e s i n t o (4.21) to (4.24), the non-dimensional equations of motion become (on dropping the primes) (u - c) u s + v Uy - f v + n s = 0 (4.48) (u - c) v s + v v y + f u + n y = 0 (4.49) [(u - c ) ( l + 5 n ) ] s + [ v ( l + S n ) ] y = 0 (4.50) v = 0 at y = 0, 1 . (4.51) The various v a r i a b l e s may be expanded i n powers of e, an amplitude parameter, as f o l l o w s u = u 0 ( y ) + £ U ! ( s , y ) + e 2 u 2 ( s , y ) + ... v = e v i ( s , y ) + e 2 v 2 ( s , y ) + ... (4.52) n = rio(y) + e r i i ( s , y ) + e 2ri2(s,y) + ... + ecj + e 2 c 2 55 On s u b s t i t u t i o n of these expansions i n t o the non-dimensional equations and on se p a r a t i o n of terms i n powers of e, the equations of t h A motxon are to zero order, f u 0 + n G y = 0 , (4.53) to f i r s t order ( u 0 - c 0 ) u i s + v i U o y - f v i + n I s = 0 (4.54) (u, - c 0 ) v i s + f u i + H i y = 0 (4.55) (1 + 6 r i o ) u l s + [(1 + 6 n 0 ) v i ] y + 6(u 0 - C o ) n l s = 0 (4.56) v : = 0 at y = 0, 1 , (4.57) to second order (4.58) ( U 0 - C 0 ) u 2 s + V 2 U Q y - f v 2 + n 2 s = - ( U j - C ^ U j g - V j U j y ( u 0 - c Q ) v 2 g + f u 2 + n 2 y = - ( u x - c x ) v l s - v i v i y (4.59) (1 + 6no)u 2 s + [(1 + fin»)v2]y + 6(u 0 - c 0 ) n 2 s (4.60) = - 6 [ ( u x - c 1 ) n 1 ] g - S E v j n J y v 2 = 0 at y = 0, 1 . (4.61) 4.5.2 The f i r s t order s o l u t i o n s In t h i s case of 6 = 0 ( 1 ) , the equations w i l l be s i m p l i f i e d by s e t t i n g the b a s i c c u r r e n t , u D , to zero. Under these circumstances, the f i r s t order equations can be reduced to a s i n g l e equation i n v i by 56 f i r s t e l i m i n a t i n g ri 1 between (4.54) and (4.55), and between (4.54) and (4.56) to give f u i s + c° uisy - c° viss + ( f v i > y = 0 <4-62> and (1 - 6 c 2 ) u l s + v l y - ficofv! = 0 (4.63) r e s p e c t i v e l y , then e l i m i n a t i n g u x between these two equations to leave (1 - 6 c 2 ) v l s s + v l y y - - ^ ( 1 + S c 0 f 2 ) V l = 0 . (4.64) Equation (4.64) i s a non-dimensional form of the l i n e a r equation f o r divergent Rossby waves (3.19) as obtained by Longuet-Higgins (1965a). A s o l u t i o n to (3.19),.given by (3.20) and (3.21), i s obtained by making an approximation equivalent i n (4.64) to n e g l e c t i n g <5c2 w i t h respect to 1, and by t r e a t i n g f 2 as a constant. Making these approximations the s o l u t i o n to (4.64) i s given by Vj = s i n miry cos ks , (4.65) and Co = - [m2TT2 + k 2 + fif2]"1 . (4.66) For a m i d - l a t i t u d e channel such that L - 10 m, H - 10 m, 6 - 10~ 1 1m" 1s and f„ - 10 ^ s " 1 , then (4.47)' gives 6 = 10 " 2, and (4.66) gives c Q - 1 0 _ 1 . Hence, a p o s t e r i o r i , i t i s seen that the e r r o r that these approximations introduce i n t o (4.64) i s approximately 1% f o r t r e a t i n g f as constant, 10~ % f o r n e g l e c t i n g 6 c Q i n the f i r s t term. Equation (4.64) can be solved e x a c t l y , i t s s o l u t i o n being given by v. Y(y) cos ks (4.67) where yy - + <5f2 + (1 - 6 c 2 ) k 2 Co 0 . (4.68) The transformation HE f ( y ) (4.69) transforms (4.68) i n t o 1 , , (1 - 6 c 2 ) k 2 ? 2 2 c 0 A 2~7E 4 0 (4.70) which has as s o l u t i o n s [Abramowitz and Stegun, (1965)] the P a r a b o l i c C y l i n d e r functions U(X,£)» V(A.,C) where 1 _ . (1 - 5 c 2 ) k 2 2^~7T 275" (4.71) The boundary c o n d i t i o n (4.57) i s s a t i s f i e d i f U(A,Ci) VU,C 2) " U(X,? 2) V ( X , d ) (4.72) where ^46 f 0 , and £ 2 = ^46 . (f 0 + 1) , (4.73) i n which case the s o l u t i o n i s given by 58 Y = V t t . C j ) U (A,?) - UCX.d) V (A,0 (A.74) A search of the l i t e r a t u r e was c a r r i e d out, but no tables of zeros of these cross-products nor any in f o r m a t i o n on t h e i r p r o p e r t i e s were found. Tables of values of U ( A , ? ) and V (A,C) are given i n Abramowitz and Stegun (1965) f o r -5 < X < 5, 0<.<;<.5. From these tables i t i s seen that V (A,£) i s monotonic i n c r e a s i n g f o r A > -1.5, and U ( A , £ ) i s monotonic decreasing f o r A > -0.5; t h e r e f o r e , f o r any Ci>?2 s u c n that £ 2 > Cl » t ^ i e cross-product w i l l be p o s i t i v e f o r A i> -0.5 . Thus, a necessary c o n d i t i o n f o r (4.72) to be s a t i s f i e d i s that A < -0.5 . I f , as seems l i k e l y , 6c2, « 1, then the second term of (4.71) i s p o s i t i v e and so, i n order that A < -0.5 , c 0 < 0 . Therefore, i n common w i t h the approximate l i n e a r s o l u t i o n s f o r divergent Rossby waves, the phase v e l o c i t y of these s o l u t i o n s i s always toward the west. Both U ( A , £ ) and V(A,c;) are o s c i l l a t o r y i n A i f A < 0 and 1^ 1 < 2/]A| ; t h e r e f o r e , there w i l l e x i s t an i n f i n i t e sequence, {AM}, of eigenvalues f o r which (4.72) i s s a t i s f i e d . For |A| » £ 2 Abramowitz and Stegun (1965, p.690) give the expansions, (4.75) 59 S u b s t i t u t i n g these expansions i n t o (4.72), i t i s found that the eigenvalues, {X^} , are given by A - m 2 - f r 2 ^ *2 . i n A TTT" ~ ~T fo + (4.76) m 2/6 2where | X j » ? 2 > ? 2 f o r m » 1. S u b s t i t u t i n g from (4.76) f o r X, equation (4.71) becomes c 0 = - [ m 2 7T 2 + (1 - <5c2)k2 + 6 f 2 r 1 (4.77) which, i f 6c 2 « 1, i s the phase speed given by (4.66). Since | X m| » C 2 i m p l i e s that m i s l a r g e and, f u r t h e r , that m2TT2 » 6 f 2 , i t i s seen that f o r l a r g e m and k (short wavelengths), (4.65) and (4.66) are good approximations to the f i r s t order s o l u t i o n s . For t h i s case the non-divergent s o l u t i o n s may als o be v a l i d . 4.5.3 The second order s o l u t i o n s In the same manner as were the f i r s t order equations, the second order equations (4.58) to (4.61) are reduced to a s i n g l e equation i n v 2 . F i r s t n 2 i s e l i m i n a t e d between (4.58) and (4.59), and between (4.58) and (4.60) to give f u 2 g + c G u 2 s y - c 0 v 2 s s + ( f v 2 ) y (4.78) - : [ ( U l - cOv j + v l V l ] + [(u x - c^u, + v l U l ] and (1 - 6 c 2 ) u 2 s + v 2 y -• Sc 0fv 2 (4.79) - 5 { [ ( u 1 - c ^ r i j l g + [V i T i i l + c 0 [ ( u j - c ^ u ^ + V j U ^ 60 r e s p e c t i v e l y , then u 2 i s e l i m i n a t e d between these two to give c 0 ( l - 6 c 0 ) v 2 s s + c 0 v 2 y y . - (1 + 6 c G f 2 ) v 2 (4.80) (1 - 6co){[u,v 1 + V , V i ] - [u.u, + v,u, ] } 1 Ls y s s y y - c 0 6 { ( u 1 n 1 ) s y + K V y y + c 0 ( U l u l s + v l U l y ) y } + C l { ( l - 6 c 2 0 ) ( u l s y - v l s s ) + 6 f ( n l s . + c 0 u l s ) + <Sc 0(n l s y + c 0 u l s y ) } . This equation'may be s i m p l i f i e d by making the approximation that 6c 2 << 1, and then s u b s t i t u t i n g f o r V j , U ! , and n i f r o m (4.67), (4.63), and (4.54). A f t e r some manipulation (4.80) may be w r i t t e n as c 0 ( v 2 g s + v 2 y y ) - (1 + 5 c 0 f 2 ) v 2 (4.81) = 6 f [ ( 3 + c 0 k 2 ) Y 2 - fY Y - c 0 Y 2 ] S ± n k s C O S k s 'y ° y J k — 1 Y cos ks The form of equation (4.81) suggests the s o l u t i o n f o r v 2 i s v 2 ( s , y ) = Z x ( y ) cos ks + Z 2(y) s i n 2ks (4.82) where Z x(y) and Z 2 ( y ) s a t i s f y the equations 61 2 y y - •'+ 6 f 2 .+ 4k 2) Z2 8 f 2kc Q (4.84) (3 + c 0 k 2 ) Y 2 f Y Y y - C o Y 2 as well as the boundary conditions Zi(0) = Z 2 ( 0 ) . = Zi(l) = Z 2(l) = 0 (4.85) If (4.83) is multiplied through by Y, then integrated over y from 0 to 1, i t is found that the left-hand side is identically zero and the right-hand side reduces to 0 dy 0 (4.86) The integrand of (4.86) i s always p o s i t i v e ; t h e r e f o r e , i n order that the equation b e 1 ' s a t i s f i e d , c i = 0. There i s , t h e r e f o r e , no f i r s t order c o r r e c t i o n to the phase speed of the waves and Z\(y) i s zero. The s o l u t i o n to (4.84) i s f o r m a l l y w r i t t e n as /TT 2 U(K,S) f V(K,t) S(t) dt a + /I V(K ,C) J u(K,t) s(t) dt (4.87) where t, is defined by (4.69) r - ^ _ + 4 C o k 2 K " 2 C o/cT S(C) 4kc075o": (3 + c c k2 ) Y 2 CYY^ - 2c 0/5"Y 2 (4.88) (4.89) and a and b are chosen to satisfy the boundary conditions. Since Parabolic Cylinder functions are not easy to manipulate, equation (4.87), 62 w h i l e f o r m a l l y representing the second order c o r r e c t i o n to the s o l u t i o n , w i l l have to be s i m p l i f i e d i n order that the s o l u t i o n be i n t e r p r e t e d . I f 6 i s s m a l l or i f m and k are l a r g e , i t has been shown th a t (4.65) i s a good approximation to the f i r s t order s o l u t i o n . Therefore, (4.84) may be sol v e d by making the same approximation, that i s , that 5 f 2 may be tr e a t e d as a constant i n the l e f t - h a n d s i d e of (4.84). On s u b s t i t u t i n g from (4.65) f o r Y,and from (4.66) f o r c O J (4.84) becomes J 2 y y - (3kz - m2TT2) Z (4.90) 6f 4c„k ( 6 c 0 f2 - 2) cos 2iMry - mirf s i n 2mTry - (2m 2fr 2c 0 + 6 c c f z - 2) The s o l u t i o n t o t h i s i s z 2 ( y ) Sf 4c Qk A, cos 2mTTy + 12c0k(kz+mzTTz) + A, 5f(2m 2-rr 2c 0 + 5 c 0 f 2 - 2) 4c 0k(3k z-m zTr z) s i n 2m7ry (4.91) Sf, 4c Qk r2m 27T 2c 0 + 6 c 0 f 2 - 2 A. 3k z - m TT cos X xy where + 1+f o ^ , —— - COS A-x t o s i n X^y s i n X, m27T2 - 3k 2 4 n 2TT 2 1 3(mzTTz + k z ) Scof2 - 2 - 8m2 TT 2 3 ( 1 ^ + k z ) (4.92) (4.93) 63 A 2 3 ( 1 ^ + k z ) 2 6 c c f2 c L 6k 2 - 10m2TT2 5 + w r p y (4.94) I f X < 0, then the s o l u t i o n i s given by z 2 ( y ) where xi 5f . 0 , Sum 4cTk A l C O S 2 m T ^ + 1 2 c 0 k ( m V + k a ) fz + A, 6f(2m 27T 2c 0 + 6c„f 2 -4c 0k(3k z-m zTT z) 2) s i n 2mTry (4.95) 6 f 0 4c ek 2m 2TT 2c 0 + 6c 0 f 2 - 2 3k z - m zTT z cosh X 2y 1+fo - cosh X; 3k 2 - m 2TT 2 s i n h X 2y si n h X 2 (4.96) 4.5.4 P r o p e r t i e s of the s o l u t i o n s The f i r s t important property of these s o l u t i o n s to the second order i s that there i s no f i r s t order c o r r e c t i o n to the phase v e l o c i t y . Hence, f o r l a r g e m and k, the phase v e l o c i t y i s given by - ( m 2 T T 2 + k z + 6 f t ) - 1 + 0 ( e 2 ) (4.97) This r e s u l t i s s i m i l a r to that found i n the Stokes-expansion of sur f a c e g r a v i t y waves on a f l u i d of i n f i n i t e depth [Lamb, (1945), p.417], and i n the second order expansions of i n t e r n a l g r a v i t y waves on a l i n e a r d ensity p r o f i l e [Thorpe (1968), p.589]; i n each of these cases f i r s t order c o r r e c t i o n f o r the phase speed i s zero. This r e s u l t shows that f o r divergent Rossby waves, the d i s p e r s i o n r e l a t i o n obtained from the l i n e a r equations of motion i s much more accurate than p r e v i o u s l y suspected, having e r r o r s of 0 ( e 2 ) r a t h e r than 64 of 0(e). Although the phase v e l o c i t y i s not changed by the second order s o l u t i o n , the wave p r o f i l e i s . I n terms of the cross channel v e l o c i t y , the wave s o l u t i o n , given by v = e s i n mlTy cos ks (4.98) ^ „2/mTr6[f2 + A,] . „ 6fA, 1 + £ ll2c 0k.(k*-Hn * V ) S i n 2 m ^ y " 4cTk C O S 2 m 7 f y . 6 f(2m 2TT 2c 0 +_ 6 c e f p - .2) 4c 0k(3k z-m zTT z) S f , 4 c ck 2m 2TT 2c 0 + 6 c 0 f o - 2 3k^ - m V A i n c o s x i y + 1+fo , — — - COS A1 f o S l n X i y \ ) s i n 2ks s i n Ai d i f f e r s from the l i n e a r s o l u t i o n , which i s 0(e), w i t h terms of 0(e 2). Any programme attempting to measure Rossby waves i n the ocean would probably i n v o l v e measurements of v e l o c i t y at f i x e d p o i n t s over a pe r i o d of time. On such a re c o r d , a wave p r o f i l e such as (4.98) would appear as v ( t ) : = eDi cos (-kc 0t) + e 2D 2 s i n (-2kc 0t) + 0(e 3) (4.99) where Di and D 2 at any f i x e d p o i n t are constants of order u n i t y , provided y 4 n/m. Therefore, the current record w i l l appear as a s i n u s o i d a l wave of angular frequency k c D which i s steepened at e i t h e r the l e a d i n g or t r a i l i n g edge. On the nodal s u r f a c e , y = n/m, Dj = 0 and the current record appears as a s i n u s o i d a l wave of amplitude 0(e 2) and angular frequency 2 k c 0 . 65 m2TT2 - 3k 2 Equations (4.91) and (4.95) are s o l u t i o n s of (4.90) only i f 4 n 2TT 2. In the s p e c i a l case f o r which \ x = mr, . the question a r i s e s as to whether (4.90) w i l l have s o l u t i o n s which s a t i s f y both boundary conditions;. Supposing that such s o l u t i o n s e x i s t , equation (4.90) may be m u l t i p l i e d through by cos nTTy, then i n t e g r a t e d over y from 0 to 1, w i t h the r e s u l t that the l e f t - h a n d s i d e i s i d e n t i c a l l y zero. I f , on the right-hand s i d e f i s h e l d constant, then the i n t e g r a t e d equation gives 6 f 2 m 2 2c 0k (-)» - 1 4m2 - n 2 0 (4.100) I f n i s odd, then t o t h i s order of approximation, no second order s o l u t i o n can e x i s t which s a t i s f i e s both boundary c o n d i t i o n s . I f n i s even, then (4.100) i s s a t i s f i e d and the s o l u t i o n t o (4.90), i f f i s h e l d constant, i s !My> - ATmI\AZJ^ s i n 2 m ^ - 4 c f t A i c o s 2 ^ < 4 - 1 0 1 ) , Sf(2m 2TT 2c 0 + 6 c 0 f 2 - 2)  + 4c 0k(3k : i - m*-nz) 6 f 0 4c Qk 2m 2TT 2c 0 + 6 c 0 f o - 2 3k z - m TT - A, cos 2pTry where 2„2 4p zT m TT - 3k (p i s an i n t e g e r ) (4.102) O r i g i n a l l y , t h i s problem was solved f o r a 3-plane channel only i n order that the width of the channel provide a h o r i z o n t a l s c a l e , L, w i t h i n which the 3-plane approximation remains v a l i d . A s o l u t i o n l i k e 66 (4.98) i s p e r i o d i c i n y, and t h e r e f o r e , the boundary c o n d i t i o n at y = 0,1 can be r e i n t e r p r e t e d as a p e r i o d i c i t y c o n d i t i o n and the s o l u t i o n considered to be a two dimensional wave p e r i o d i c i n both x and y i n an unbounded ocean. Such an i n t e r p r e t a t i o n i s v a l i d only i f A? > 0, as the s o l u t i o n (4.95) f o r X 2 < 0 i s no longer p e r i o d i c i n y, and, i n f a c t , increases e x p o n e n t i a l l y w i t h y outside of the dimensions of the channel. Returning once more to (4.81), i n the non-divergent l i m i t as 6 -»• 0, the right-hand s i d e goes to zero; hence, there i s no second order c o r r e c t i o n . This i s c o n s i s t e n t w i t h the r e s u l t s of §4.3 which show t h a t , f o r the constant zonal c u r r e n t case, the l i n e a r non-divergent Rossby wave s o l u t i o n i s an exact s o l u t i o n . The f a c t that there e x i s t s a second order c o r r e c t i o n to the l i n e a r divergent Rossby wave s o l u t i o n demonstrates t h a t , u n l i k e the non-divergent case, the l i n e a r divergent s o l u t i o n s are not exact s o l u t i o n s of the equations of motion. I n t h i s way the non-divergent Rossby waves are fundamentally d i f f e r e n t from the divergent s o l u t i o n s . 4.6 Rossby waves i n a g-plane channel, I I I . Uniformly sheared current 4.6.1 The p e r t u r b a t i o n expansions I t has been shown p r e v i o u s l y f o r m i d - l a t i t u d e channels of width 10 3 km, and depth l i km, that 6 ~ 10~ 2. I n view of the complexity of the p e r t u r b a t i o n equations f o r a sheared b a s i c current and 5 = 0 ( 1 ) , perhaps a new expansion i n which 6 = 0(e) would be appropriate. S e t t i n g 6 = ye , (4.103) 67 where ]i = 0 ( 1 ) , and using the expansions f o r u, v, n, and c. i n terms of e , given by (4.52), equations (4.48) to (4.51) may be separated i n th powers of £,to give to the zero order (4.53), to the f i r s t order (4.54), (4.55), (4.57) plus u, + v, = 0 (4.104) 1 S ly and t o the second.order (4.58), (4.59), (4.61), plus U 2 S + v 2 y = _ V i [ ( u ° " c ° ) n l g + ( v f l o ) ] • ( 4 . 1 0 5 ) 4.6.2 The f i r s t order s o l u t i o n s The r e d u c t i o n of the f i r s t order equations to a s i n g l e equation i n v i i s e a s i l y accomplished. F i r s t Hi i s e l i m i n a t e d between (4.54) and (4.55) to give (4.106) [ ( u 0 - c 0 ) u l g ] y - f u l g + [ ( u . - f ) v x ] y - (u, - c 0 ) v l s s = 0 then u J i s e l i m i n a t e d between t h i s and (4.104) to leave (u„ - c 0 ) [ v l s s + v l y y ] + [1. - UoyyK = 0 . (4.107) i I f the b a s i c current i s uniformly sheared, that i s i f u 0 = W0 + ay f: (4.108) where a and W0 are both constants, then the s o l u t i o n f o r v i s Vj = $(y) s i n ks (4.109) where 68 yy U 0 ~ C 0 k2| $ (4.110) S e t t i n g (W0 - Co + ay) (4.111) and Cexp(-C/2) HO (4.112) equation (4.110) may be transformed i n t o K + (2 - 0\ " (1 " 2fkaT = 0 ' (4.113) the confluent hypergeometric equation, the s o l u t i o n s of which, i n the n o t a t i o n of S l a t e r (1960), are given by the confluent hypergeometric 1 1 f u n c t i o n s , i F i ( l - £|fca| >2>£) and U ( l - ^ | , 2 , 5 ) . The boundary c o n d i t i o n (4.57), i n terms of i s given by * ( S o ) (4.114) and i s s a t i s f i e d i f 1 F 1 ( A 3 , 2 , ? 0 ) U(A 3,2,d) - i F i ( A 3 , 2 , C ! ) U(A3,2,Co) = 0 (4.115) where Co = 2 (W0 - Co) , = 2 (Wo .- c 0 + a) A, = 1 -(4.116) 1 2 ka For -n < A 3 < -n+l, 1F 1(A 3,2,<;) and U(A 3,2i'?> each have n p o s i t i v e r e a l zeros [ S l a t e r , (1960), pp.102-106]; hence, k and c 0 may be chosen such that (4.115) i s s a t i s f i e d . The zeros of these func t i o n s are not t a b u l a t e d , and the c a l c u l a t i o n of the a c t u a l 69 d i s p e r s i o n r e l a t i o n i s not of s u f f i c i e n t importance to warrent t h e i r c a l c u l a t i o n here. I f the shear i s weak, that i s , i f a « 1, then (4.110) may be solved using p e r t u r b a t i o n expansions of $ and c 0 i n terms of powers of a. These are given by $ = $ 0 + a$! + a 2 $ 2 + C Q C Q O ™t" 3 . C Q J ~h 3 . C Q 2 "I™ (4.117) Oh s u b s t i t u t i o n of these expansions and s e p a r a t i o n i n powers of a, equation (4.110) i s to the zero order i n a °yy W0 - C 0 0 (4.118) to the f i r s t order lyy Wo -.c, (W0 " C o o ) (4.119) e t c . These equations may be e a s i l y solved subject to the boundary conditions *o(0) * o ( D = * i ( 0 ) = $!(!) = 0 (4.120) to give 2\ 2 . , ay(k z + l / )  1 + " 4 F 'sin Ay + a(y - y 2 ) ( k 2 + £ 2 ) 2 4£ cos Hy + 0 ( a 2 ) , (4.121) and 70 Wo - k 2 I Z2 + f + 0 ( a 2 ) (4.122) where £ = mir ' . (4.123) 4.6.3 The second order s o l u t i o n In the same way, r| 2 may be e l i m i n a t e d from (4.58) and (4.59) to leave ( u 0 - c 0 ) v 2 s s + f u 2 s - [ ( u 0 - C o ) u 2 g ] y - [ ( u Q y - f ) v 2 ] y " " K V L S + v i v i y ] s + f u i u i s + v ! u i y ] y < 4 ' 1 2 4 ) + c i ( v i s s " u i s y > • E l i m i n a t i n g u 2 from (4.124) using (4.105) and also s u b s t i t u t i n g f o r u 1 , v,, and r\1 i n terms of gives ( u 0 - c 0 ' ) ( v 2 s s + v 2 y y ) + v 2 (4.125) c i $ . . - s m ks u 0 - c c + u { [ ( u G - c 0) ( 2 a 2 - af +. c 0) + c 0 f ( a - f ) ] $ + [ r i o ( f - a) - 2a(u G - c 0 ) 2 ' + 2 f u 0 ( u 0 - 2 c G ) ] $ y - ( u 0 - c Q ) [ r i o + ( u 0 - c o ) 2 ] 0 r } s i n ks yy + 2^ ( Myyy '•" Vyy ) s ± n 2 k s 71 The s o l u t i o n f o r v 2 i s therefore of the form v 2 = Zj ( y ) s i n ks + Z 2(y) s i n 2ks • (4.126) where Z1 and Z 2 are functions which s a t i s f y ( u 0 - c G ) Z l y y + [1 - k 2 ( u Q - c 0 ) ] Z 1 ^ 1 2 7 ^ C l $ U 0 - C 0 + y { [ ( u Q - c 0 ) ( 2 a 2 - af + c Q) + c e f ( a - f ) ] $ + [ T l o ( f - a) - 2a(u a - c 0 ) 2 + 2 f u 0 ( u o - 2c0)]$y - ( u 0 - c 0 ) [ r i o + ( u o - c 0 ) 2 ] $ y } and (4.128) ( u o - c 0 ) Z 2 y y + [1 - 4 k 2 ( u 0 - c 0 ) ] Z 2 = ^ ( ^ y y y * y * y y ; A necessary and s u f f i c i e n t c o n d i t i o n t h a t (4.127) have s o l u t i o n s that s a t i s f y the boundary c o n d i t i o n s , Zi ( 0 ) = Z i ( l ) = 0, (4.129) i s obtained by m u l t i p l y i n g (4.127) through by $/(u 0 - co)> then i n t e g r a t i n g over y from 0 to 1. From the boundary conditions i n the l e f t - h a n d s i d e i s i d e n t i c a l l y zero, and the right-hand s i d e i s zero i f 72 l l C 1 J (u, - c 0 ) 2 d y = y J { " [ n ° + ( U o " C°) 2]^yy (4.130) 0 0 + [ri o ( f - a) - 2a(u 0 - c Q ) z + 2 f u 0 ( u 0 - 2 c 0 ) ] U 0 - C 0 + [ ( u 0 - Co) ( 2 a 2 - af + c D) + c c f ( a - f) ] } dy u 0 - c 0 The proof that t h i s c o n d i t i o n i s a s u f f i c i e n t c o n d i t i o n f o r which (4.127) w i l l have s o l u t i o n s s a t i s f y i n g (4.129) i s given by Courant and H i l b e r t (1953, p.359). Equation (4.130) may be i n t e g r a t e d to give ' ^ ~2T2+ fo + f 0 + k ~ (4.131) ( k z + ( k z + £ z ) z ' 3 2V r Wo - (2f% + f 0 + + 8W0 - -rrr2 i \ fk2 + a2 3 0 211 \ 4 I2 + j (3f 0 + 1) + 11 + 0(a) Obtaining an a c t u a l s o l u t i o n t o (4.127) by s u b s t i t u t i n g f o r c i would be a tedious task, g i v i n g i n r e t u r n , only the term which i s of the same zonal wave number as the b a s i c wave. On the other hand, (4.128) may be e a s i l y s o l v e d , to give Z 2 = a ( k 2 J \ i 2 2 ) {3(k 2 + £ 2) + \\. cos 2ly (4.132) - 4£2[cos X l Y + 1 T C ° S X l s i n \lY]} + 0(a 2) s i n A i where again A 2 = I2 - 3k 2 4 n 2 7 T 2 [see (4.92)] . 73 t I f X 2 = (2p + 1 ) 2 T T 2 , there i s no s o l u t i o n to (4.128) which w i l l a l s o s a t i s f y the boundary c o n d i t i o n s . On the other hand, i f X 2 = 4 p 2 T T 2 , then 2 , 2 " a ( 1 2 k X ^ 2 ) ? [ 3 ( k 2 + % 2 ) + X ' ° O S 2 1 7 ' H * ° 0 S 2 p 7 i y ] + 0 ( a 2 ) . (4.133) 4.6.4 P r o p e r t i e s of the s o l u t i o n s Equation (4.132) shows that Z 2 ( y ) i s non-zero only i f the b a s i c current i s uniformly sheared; t h i s i s true despite the f a c t that 6, the divergence parameter, i s non-zero. Since Z 2(y) i s the c o e f f i c i e n t of the " s i n 2ks" term, only i f i t i s non-zero w i l l there be any d e v i a t i o n of the wave p r o f i l e along the axi s of the channel from the l i n e a r s o l u t i o n , at l e a s t at 0 ( e 2 ) . In the previous s e c t i o n , a second order term of wave number 2k was obtained when there was no sheared current present; however, i f 6 = 0(e) i n (4.91) i t i s seen that these terms are then 0 ( e 3 ) . Therefore, i f 6 = 0(e) and i f the b a s i c current i s zero or uniform, one must look at t h i r d order terms i n order to f i n d n o n - l i n e a r i t i e s i n the wave p r o f i l e s . For the case of W0 = 0, a = 0, and s h o r t wavelengths, f 2 » (f 0,l,£~ 2,k" 2) ; hence, the phase v e l o c i t y given by (4.131) and (4.122) may be approximated by c = " F'TT2 l - 6f k 2 + e (4.134) Equation (4.134) i s e x a c t l y the f i r s t two terms of the b i n o m i a l 74 expansion f o r (4.97) where 6 = ey « 1. Hence, f o r 6 = 0 ( e ) , the s o l u t i o n s obtained i n §4.5 reduce t o the s o l u t i o n s obtained here. I f y = 0, the s o l u t i o n s reduce to the non-divergent s o l u t i o n f o r a uniformly sheared zonal current. For y = 0, (4.130) gives C j = 0; hence, there i s no c o n t r i b u t i o n t o the s o l u t i o n s from equation (4.127). The non-divergent s o l u t i o n to the second order i s then given by ( J 1 + — e s i n SLy (4.135) _ ay(y - l ) ( k ' + I2)2 £ c q s ,y + Q ^ s ± n k g + £ 2 a ( kl t 2 i j / 2 ) 2 { 3 ( k 2 + Jl 2 ) + A2 cos 2ly - 4£2[cos X.y IzkAi L , 1 - cos Xi . , •, i . 0 1 ^ . . x S 1 I 1 AiyJJ- sxn 2ks s m Ai and the phase speed i s given by (4.122) to the second order i n e. Since, f o r a constant zonal c u r r e n t , the l i n e a r or f i r s t order non-divergent s o l u t i o n has been already shown to be an exact s o l u t i o n to the non-divergent equations o f motion, i t i s not s u r p r i s i n g that f o r a = 0, (4.135) reduces to the l i n e a r s o l u t i o n . For non-zero a, the second order term introduces a n o n - l i n e a r i t y to the wave p r o f i l e along the a x i s . Depending on the signs of the c o e f f i c i e n t s of the f i r s t and second order terms, t h i s n o n - l i n e a r i t y appears as a steepening of the l e a d i n g (or t r a i l i n g ) edge of the wave. The f a c t that even a weak uniform shear should have such a marked ef f e c t , on the non-divergent wave i s due i n p a r t to the change that such a shear makes i n the v o r t i c i t y f i e l d i n which the wave fi n d s i t s e l f , and a l s o i n part to the p h y s i c a l d i s t o r t i o n such a shear current causes by moving some parts of the wave r e l a t i v e to other p a r t s . Thinking of a t y p i c a l ocean s i t u a t i o n w i t h random currents and random shears, i t seems l i k e l y that any observed Rossby wave f i e l d w i l l be very much a l t e r e d from that t h e o r e t i c a l l y p r e d i c t e d i n such a simple model as a channel w i t h a uniformly sheared c u r r e n t . This model i s v a l u a b l e , however, i n suggesting the importance of the i n t e r a c t i o n s w i t h currents. The phase speed, as given by the l i n e a r theory, i s c o r r e c t to 0(£ ) f o r non-divergent waves i n a uniformly sheared current. Therefore, although the presence of r e a l ocean currents w i l l g r e a t l y d i s t o r t the wave f i e l d s , the t h e o r e t i c a l d i s p e r s i o n r e l a t i o n s given by the l i n e a r theory w i l l give accurate r e s u l t s . This e f f e c t has an analogue i n s u r f a c e g r a v i t y waves where i t i s found that d i s p e r s i o n r e l a t i o n s f o r l i n e a r surface wave theory give accurate r e s u l t s when a p p l i e d to a c t u a l l y observed wave f i e l d s . 4.7 Summary I t has been shown t h a t , i n the presence of a uniform or zero zonal c u r r e n t , the l i n e a r non-divergent Rossby wave s o l u t i o n s are exact s o l u t i o n s of the non-divergent equations of motion both on the sphere and on the 3-plane. Furthermore, l i n e a r non-divergent s o l u t i o n s of the same t o t a l wave number, and hence, of the same phase speed, may be summed together to form new l i n e a r s o l u t i o n s ; these new s o l u t i o n s are a l s o exact s o l u t i o n s of the non-divergent equations of motion. 76 This behaviour i s markedly changed i n the presence of a sheared zonal current. Even i n the s i m p l e s t case of a weak uniform shear, the non-divergent s o l u t i o n s e x h i b i t n o n - l i n e a r i t i e s i n the wave p r o f i l e at 0 ( e 2 ) , although the l i n e a r d i s p e r s i o n r e l a t i o n i s unaffected at 0 ( e ) . K e l l e r and Veronis (1969) have p r e v i o u s l y shown that random currents may s c a t t e r Rossby waves or caus;e them to grow. Here, however, i t i s shown that the presence of current shear can cause energy of a s i n g l e non-divergent Rossby wave to be fed i n t o higher wave numbers. Since the r e a l ocean s i t u a t i o n c o n s i s t s of many currents i n d i f f e r e n t d i r e c t i o n s , t h i s i n t e r a c t i o n between Rossby waves and currents should be very important i n understanding oceanic dynamics. The l i n e a r divergent s o l u t i o n s are shown not to be exact s o l u t i o n s of the g-plane'equations. In the absence of a b a s i c zonal c u r r e n t , the divergent s o l u t i o n s e x h i b i t n o n - l i n e a r i t i e s at 0 ( e 2 ) . Once again the l i n e a r d i s p e r s i o n r e l a t i o n i s c o r r e c t to 0 ( e ) . An exact s o l u t i o n f o r non-divergent topographic waves oh the 3-plane was found; however, t h i s s o l u t i o n requires a very complex b a s i c current p a t t e r n i n order to e x i s t . I t i s f e l t that such a complex s o l u t i o n i s not of much a p p l i c a b i l i t y to r e a l ocean s i t u a t i o n s . V. Long P l a n e t a r y Waves i n a Zonal Channel 5.1 The s c a l e d equations A c l a s s of long n o n - l i n e a r waves, the s o l i t a r y and c n o i d a l waves, has long been known and i n v e s t i g a t e d f o r the case of surface g r a v i t y waves [Korteweg and deVries, (1895); Keulegan and P a t t e r s o n , (1940); Benjamin and L i g h t h i l l , (1957)] and more r e c e n t l y f o r . t h e case of i n t e r n a l g r a v i t y waves [Keulegan (1953); Benjamin (1966); Benney (1966)]. These are waves of permanent form whose wavelengths along the channel are long r e l a t i v e to the width of the channel. Since, as shown i n Appendix I I , there i s a r e s t r i c t e d analogy i n the behaviour of pl a n e t a r y and i n t e r n a l waves, the question a r i s e s whether an analogous c l a s s of waves e x i s t s f o r planetary motions. Using the non-divergent approximation Larsen (1965) showed that s o l i t a r y and c n o i d a l waves could e x i s t i n a zonal channel, p r o v i d i n g there was a l s o present a b a s i c z o n a l current w i t h a weak uniform shear. The f a c t that Larsen found that non-divergent s o l i t a r y and c n o i d a l waves could not e x i s t i f the b a s i c current was uniform i s not s u r p r i s i n g i n the l i g h t of the r e s u l t s obtained i n the previous chapter. Since the l i n e a r s o l u t i o n on a uniform current i s an exact s o l u t i o n t o the non-divergent equations of motion i n t h i s case, they already form a c l a s s of s o l u t i o n s of permanent form. I n the previous chapter i t was a l s o shown that the n o n - l i n e a r behaviour of divergent waves i s much more complex than that of the \ 78 non-divergent waves. In p a r t i c u l a r , the l i n e a r s o l u t i o n s f o r divergent waves are not exact s o l u t i o n s nor were any exact s o l u t i o n s found. Furthermore, i t was shown i n Chapter IV that the non-divergent approximation was v a l i d only f o r short wavelengths. For these reasons Larsen's theory w i l l be extended and s o l i t a r y and c n o i d a l wave s o l u t i o n s sought i n the divergent case. Again the f l u i d is. assumed to be i n v i s c i d and homogeneous, the motion b a r o t r o p i c and h y d r o s t a t i c , and the s o l u t i o n a wave of permanent form moving i n the x - d i r e c t i o n along the axis of the zonal channel. The wavelength of the disturbance w i l l be assumed to be sh o r t enough that the 3 -plane approximation remains v a l i d w h i l e , at the same time, being long w i t h respect to the w i d t h , L, of the channel. The f u l l unsealed equations f o r t h i s case have been discussed, and are given i n Chapter IV by ( 4 . 2 1 ) to ( 4 . 2 4 ) . Non-dimensional v a r i a b l e s are defined by (s , y) = L ( s ' , y') , (u. v, c) = 3L 2(u',v',c') ( 5 . 1 ) f = 3 L f =3L(f 0 + y ' ) , ( n , z) = H (6n',z') LkQ2 where L, H are the width and the depth of the channel, and 6 = ^ , the divergence parameter. On s u b s t i t u t i o n from ( 5 . 1 ) , ( 4 . 2 1 ) to ( 4 . 2 4 ) become (on dropping the primes) (u - c ) u g + vuy - f v = - n s ( 5 . 2 ) (u - c ) v g + w y + f u = - n y ( 5 . 3 ) ( L + 6 n ) ( u s + v ) + 6 [(u - c ) n s + v n y ] = 0 ( 5 . 4 ) v = 0 at y = ± 1 . ( 5 . 4 ) 79. Since the wave s o l u t i o n s are long w i t h respect to the width of the channel, t h e r e f o r e , f o l l o w i n g Larsen (1965), the s co-ordinate i s s t r e t c h e d r e l a t i v e to the y co-ordinate through the transformation £ = e±s ! (5.6) where e i s the amplitude-ordering parameter of the wave and e « 1. The dependent v a r i a b l e s and parameters are expressed i n terms of the f o l l o w i n g p e r t u r b a t i o n expansions: u = u 0 ( y ) + eul(E,,y) + £ 2u 2(£,y) + ••• (5.7) 3 £ v = e^ V l(£,y) + e^v 2(5,y) + ... c = c 0 + E C j + £ 2c 2 + ... n = n 0(y) + en^S.y) + e 2 n 2 ( ? , y ) + ... , where the form of the expansion f o r v . i s chosen i n order t h a t , i f the flow i s non-divergent, that i s , i f <5 = 0, the two remaining terms of the c o n t i n u i t y equation (5.4) are of the same order of magnitude. A f t e r s u b s t i t u t i o n f o r s, u, v, c, and r\ from (5.6) and (5.7), equations (5.2) to (5.5) are ordered i n powers of £ to give to zero order: f u 0 - n 0 y , (5.8) to the f i r s t order: ( U 0 - C 0 ) U X £ . + V i U o y - f v x ' + (5.9) i 80 f U l + m y = o (5.10) ( l + 6 n 0 ) u 1 ? + [(1 + 6 . n 0 ) v 1 ] y + 5 ( u 0 - O n ^ = 0 (5.11) V l = 0 at y = ± 1 , (5.12) and to second order: (5.13) ( u 0 - c 0)u 2£ + ( u 0 y - f ) v 2 + n 2 ^ = - ( u x - c ^ u ^ - v x u l y f u 2 + n 2 v = - ( u 0 - c e ) v 1 (5.14) y (1 + 6rio)u 2 ? + [ ( l + 6 n 0 ) v 2 ] y + 6(u 0 - c 0 ) n 2 ? (5.15) = - &{[(u1 - c ^ n j ^ + [ v ^ J y } v 2 = 0 at y = ± 1 . (5.16) These equations are i n t h e i r most general form, and t h e i r s o l u t i o n without f u r t h e r approximations would be q u i t e complicated. Larsen (1965), s e t t i n g 6 = 0 obtained s o l i t a r y and c n o i d a l wave s o l u t i o n s f o r the non-divergent case i n which the b a s i c current i s weakly sheared. As might be p r e d i c t e d from the r e s u l t s of Chapter IV, i f the b a s i c current i s not sheared, Larsen's a n a l y s i s gives the l i n e a r Rossby wave s o l u t i o n . i Chapter IV i n d i c a t e s that r e t a i n i n g a non-zero 6 i n these equations should give s i g n i f i c a n t l y d i f f e r e n t r e s u l t s from Larsen's non-divergent a n a l y s i s . In p a r t i c u l a r , i t should be p o s s i b l e to o b t a i n s o l i t a r y and c n o i d a l wave s o l u t i o n s even f o r the case of the zonal current zero. I f gL - 10 "5 s " 1 , L - 106m, gH - 10k m/s, then 6 - 10 " 2 ; however, i t s 81 magnitude i s very s e n s i t i v e to v a r i a t i o n s i n the magnitude of L. Here, as i n the previous chapter, two cases are considered: 6 = 0(1) and 6 = 0 ( e ) . In the f i r s t case, i n order to make the c a l c u l a t i o n s more manageable, the zonal current i s s e t to zero. This case w i l l show t h a t , i f the divergent terms are r e t a i n e d , then s o l i t a r y and c n o i d a l waves can e x i s t independently of a zon a l sheared current. In the second case, a b a s i c current w i t h a uniform but weak shear w i l l be r e t a i n e d . The s o l u t i o n s that are obtained w i l l be compared to the non-divergent s o l u t i o n s of Larsen (1965). Furthermore, f o r the second case, the e f f e c t s of bottom topography w i l l be i n c l u d e d i n the equations. 5.2 The case 6 - 0 ( 1 ) 5.2.1 D e r i v a t i o n of the long wave equation S e t t i n g u 0 and r| 0 equal to zero, n l i s e l i m i n a t e d between f i r s t (5.9) and (5.10), and then between (5.9) and (5.11) to give (co|y--+ f)u, + ( f V l ) y = 0 (5.17) (1 - 5cS)u! + v l y - 6 c 0 f V l = 0 , (5.18) and then u1 i s e l i m i n a t e d to give C o v l y y - (1 + 6 c 0 f 2 ) v ! = 0 . (5.19) Equation (5.19) together w i t h the boundary c o n d i t i o n (5.12) determines the v a r i a t i o n of v i across the channel but leaves i t s v a r i a t i o n along the channel completely u n s p e c i f i e d . Therefore, i t i s ' p o s s i b l e to d e f i n e a f u n c t i o n g(£) such that 82 v i = g^CO $ ( y ) (5.20) where (5.19) and (5.12) r e q u i r e that $ y y - ( l / c 0 + 6 f 2 ) $ = 0 (5.21) $(1) = $(-1) = 0 . (5.22) In terms of g ( 0 and $(y) , U i and r i i are then S c 0 f $ - $ ui V = ' I -~6ci"Y • ( 5 ' 2 4 ) In order to determine g(£), the second order equations must be i n v e s t i g a t e d . In the same manner as were the f i r s t order equations, these equations are reduced to a s i n g l e equation i n v 2 . F i r s t T ] 2 i s e l i m i n a t e d from (5.13) to (5.15) to give (C°"d7 + f ) U 2 £ + ( f V 2 ) y = " °lUl£y + ( U 1 U 1 ? + v l u l y > y + C° Vl££ (5.25) and (1 - 6 c 2 ) u 2 ^ + v 2 y - 6c'ofv 2 = 5 c i ( T l ! + c 0 u ^ ) (5.26) - 5 c 0 ( u l U l + v x u l y ) .-fitCUiTii) + ( V i l i f y ] then u 2 i s e l i m i n a t e d between these to leave 83 v 2 y y - ( l / e 0 + 6 f 2 ) v 2 (5.27) - (1 - 6 c 2 ) v 1 1. - 8 Co , . (u,u, + v,u, ) Co 1 'F 1 V y + f H C l - 5 c 2 ) u + ( f + c 0 | - ) 5 ( n l c - + cou )] 9 y / u v " ^ 6 8 which, i n terms of $ and g, may be r e w r i t t e n as -yy - (1/co + 6 f2 ) v 2 - (1 - 6 c 2 ) $ g (5.28) clH + fi{f[3 + 6 c D + 2 ', 6 c 0 f2 ( l + 5 c 2 ) 1 - 6 c 0 S2 , c 0 f ( 3 , - 5 c 2 ) ^2 5 + i - 6 c 2 $ y c 0 ( 4 + 6 c Q f 2 ) + f2 ( l + 36 c 2) 1 - Sci y c 0 ( i - Set) I f equation (5.28) i s m u l t i p l i e d through by $ 5 then i n t e g r a t e d over y from y = -1 to y = 1, i t s l e f t - h a n d s i d e i s i d e n t i c a l l y zero, and i t s right-hand s i d e then gives an equation i n g(£), &iHKK + E2°IG? + E3§8£ o , (5.29) where c 0 ( l - 6 c 2 ) f $ 2 dy, - l l (l/c„) f $ 2 dy - l l 1 T .,r„ . 56c 2 1 - Set •2 / 6 f [ 3 + 1 ( 5 + F2\ 1 - 35 c^ - + 6 c 0 f j ! _ 6 c 2 - l (5.30) (5.31) $ 3dy.(5.32) 84 Equation (5.29) i s the Korteweg-deVries equation, well-known i n the treatment of s o l i t a r y and c n o i d a l surface wave.[Benjamin and L i g h t h i l l , ( 1 9 5 4 ) ] . I f i t i s p o s s i b l e to s o l v e the eigenvalue equation ( 5 . 2 1 ) , the c o e f f i c i e n t s may be determined from (5.30) to ( 5 . 3 2 ) , and hence, s o l u t i o n s f o r (5.29) may be obtained. 5.2.2 The transverse e i g e n f u n c t i o n s Through the transformation, 5 = /2 ^ " f ( y ) , (5.33) equation ( 5 . 2 1 ) becomes Weber's equation, * K - [ 2 ^ + f 2 ] * = ° » ( 5 ' 3 4 ) whose s o l u t i o n s are the P a r o b o l i c C y l i n d e r f u n c t i o n s U ( K , ^ ) , and V(K,£) [Abramowitz and Stegun, ( 1 9 6 5 ) ] where 1 2c 0/6" * (5.35) The boundary c o n d i t i o n (5.22) i s s a t i s f i e d i f U(K,Ci) V ( K , C 2 ) " V ( K , ? 1 ) U ( K , ? 2 ) = 0 (5.36) where C 1 > 2 = VI ( f Q ± 1) . (5.37) This i s the same c o n d i t i o n as was found to be required i n Chapter IV f o r the divergent plane Rossby wave s o l u t i o n i n a 3-plane channel. (See p.58.) Although the zeros of the cross-products of 85 P a r a b o l i c C y l i n d e r f u n c t i o n s a r e n o t t a b u l a t e d , i t was e a r l i e r shown t h a t f o r any v a l u e s o f Ci and C2 (5.36) i s s a t i s f i e d o n l y i f -I X < -0.5. I n terms o f c 0 , t h i s c o n d i t i o n r e q u i r e s t h a t c Q < -6 < 0. T h e r e f o r e , t h e wave's/phase moves i n the n e g a t i v e x - d i r e c t i o n . T r a d i t i o n a l l y , more manageable s o l u t i o n s t o (5.21) a r e o b t a i n e d by h o l d i n g f 2 c o n s t a n t e x c e p t where i t i s d i f f e r e n t i a t e d . A p p l y i n g t h i s a p p r o x i m a t i o n , the s o l u t i o n t o t h e e i g e n v a l u e e q u a t i o n may be w r i t t e n $ = A s i n —^ (y + 1) (5.38) where (5.39) U s i n g t h i s s i m p l e s o l u t i o n , t he c o e f f i c i e n t s f o r e q u a t i o n (5.29) may be e v a l u a t e d from t h e i n t e g r a l s g i v e n by (5.30) t o (5.32). S i n c e f was h e l d c o n s t a n t when t h e e i g e n v a l u e e q u a t i o n was s o l v e d , t o be c o n s i s t e n t , f 2 must be h e l d c o n s t a n t d u r i n g t h e s e i n t e g r a t i o n s . Under t h e s e c o n d i t i o n s t h e c o e f f i c i e n t s a r e g i v e n by e, = c 0 ( l - <5c2) A 2 A 2 / c c f A 3 E mTT A 3 f 0 E mTT (5.40) m even m odd where 86 86 3(1 - Sci) Q , 56c 2 _ (5 + 3 5 c 0 f 2 ) ( l - 36c 2)  J 3 6(1 - Sci) (5.41) 5.2.3 S o l u t i o n s to the Korteweg-deVries equation The Korteweg-deVries equation (5.29) has been the s u b j e c t of considerable recent research, notably s t u d i e s by Miura et a l . (1968), Miura (1968), and Lax (1968). In t h i s study, the s o l u t i o n s of (5.29) w i l l be given f o l l o w i n g the work of Keulegan and P a t t e r s o n , (1940). Equation (5.29) may f i r s t be i n t e g r a t e d twice to give f i g 2 + |3 g3 + £|£l g2 + + _ 5 = Q ( 5 J T 2 ) where e^ and e 5 are a r b i t r a r y i n t e g r a t i o n constants. For the non-divergent case, Larsen (1965) was able to show that e\ depended on the energy of the b a s i c flow and the momentum f l u x . He obtained these r e s u l t s using a quasi-Lagrangian co-ordinate system, a l s o described i n Clarke and Fofonoff, (1969). This co-ordinate system uses the stream f u n c t i o n and time as independent v a r i a b l e s i n p l a c e of the u s u a l space co-ordinates. In the divergent case, however, the two-dimensional stream f u n c t i o n can not be d e f i n e d , and t h e r e f o r e , such an approach i s no longer f e a s i b l e . Hence, e\ and e 5 w i l l be t r e a t e d here as unknown constants w i t h the r e s u l t s of Larsen's non-divergent a n a l y s i s being used to suggest the p h y s i c a l processes from which they a r i s e . Equation ( 5 . 4 2 ) , m u l t i p l i e d through by 2 / e i , becomes H + it'*' + c . ( l - C l6c 2) S 2 + e ^ + e s " 0 .<5'"> 87 where eit = 2 e \ / e i , es = 2e~s/ei. I f i s to be r e a l when g i s zero, then es must be negative. The s o l u t i o n s of (5.43) are to be p e r i o d i c i n g; hence, g^ must be zero f o r at l e a s t two d i f f e r e n t and r e a l values of g. When g^ i s zero, (5.43) i s a cubic i n g. R e c a l l i n g that eg < 0, then the three roots of t h i s cubic f a l l i n one of the f o l l o w i n g three cases: (a) a l l p o s i t i v e and r e a l roots (b) one p o s i t i v e and two negative r e a l roots (c) one p o s i t i v e r e a l and two complex conjugate r o o t s . Case (c) i s not a p p l i c a b l e here s i n c e f o r a p e r i o d i c s o l u t i o n , g must be.zero f o r at l e a s t two d i f f e r e n t r e a l values of g. Both case (a) and case (b) should lead to long planetary wave s o l u t i o n s . 5.2.4 The s o l i t a r y yave The s i m p l e s t s o l u t i o n to (5.43) corresponds to the s o l i t a r y wave. Lax (1968) showed that, 1 i n the l i m i t of long time, any s o l u t i o n of the time-dependent Korteweg-deVries equation [see p.10, (1.1)] tends a s y m p t o t i c a l l y to a sum of s o l i t a r y waves. In general, f o r non-l i n e a r equations, new s o l u t i o n s cannot be created by summing together other s o l u t i o n s ; hence, t h i s s p e c i a l f e a t u r e of the Korteweg-deVries equation was both unexpected and s u r p r i s i n g . The s o l i t a r y wave s o l u t i o n a r i s e s when e^ = es = 0 and i s given by g(Q = s e c h 2 / - ] ^ £ (5.44) and 88 c = - c 2 ( l - 6 c 2 ) - r j f (5.45) where the amplitude of g i s a r b i t r a r i l y set to unity. In order that the s o l u t i o n be r e a l , e 3 / e i must be p o s i t i v e . For the same scales that were used to estimate 6, (that i s , 3L - 10 " 5 s _ 1, f 0 - 10 _ t f s _ 1 , L - 10 6 m, gh * 10h m 2/s 2), c Q - -10'1 m/s; therefore, the sign of e 3 / e i i s the same as that of A ( - ) m / c 0 . Since c Q i s negative, the wave amplitude, A, i s negative i f m i s even, p o s i t i v e i f m i s odd. On s e t t i n g AmTT (5.46) 2(1 - S O the wave p r o f i l e i n terms of the zonal v e l o c i t y i s (5.47) / \m m ' n '/ i i \ 2 6 c 0 f . mTT, , 1 N 1 , 2 / 63 _ u = ( - ) m £[cos -e(y +1) - — — — s i n ~^-(y- + 1)] s e c h V r ^ f - £ z mTT z v I z e j and the phase speed i s c = - + 6 f 2 f - e ( l - 6 c 2 ) c 2 f^3 . (5.48) At the southern boundary, f o r m even, the zonal v e l o c i t y i s i n the opposite d i r e c t i o n to the phase v e l o c i t y ; f o r m odd the zonal v e l o c i t y i s i n the same d i r e c t i o n as the phase v e l o c i t y . At the northern boundary the v e l o c i t y i s westward i n both cases. The phase v e l o c i t y i s increased i n i t s w e s t e r l y d i r e c t i o n by an amount p r o p o r t i o n a l to the wave amplitude. 89 5.2.5 The c n o i d a l waves Again, f o l l o w i n g Keulegan and Patterson (1940), a more general s o l u t i o n to (5.43) may be obtained by s o l v i n g f o r the roots of the cubic, + c 0 ( l - C l 5 c 0 ) S 2 + e 4g + e 5 = 0 . (5.49) I f these roots are g i , g 2 , g 3 , where g i _> g 2 ^ g 3 , then, s i n c e at l e a s t one of the roots must be p o s i t i v e , g i > 0. The remaining two roots may be e i t h e r ' b o t h p o s i t i v e or both negative. This i s u n l i k e the case of s u r f a c e g r a v i t y waves; there, i f the roots are a l l p o s i t i v e , then the s u r f a c e e l e v a t i o n i s always p o s i t i v e . However, the surface e l e v a t i o n i s u s u a l l y defined as the d e v i a t i o n from the average s u r f a c e l e v e l ; hence, such s o l u t i o n s are p h y s i c a l l y u n r e a l i s t i c . In the case of p l a n e t a r y waves, the roots a l l p o s i t i v e requires only that at a given l a t i t u d e the zonal v e l o c i t y i s always i n the same d i r e c t i o n . Therefore, f o r planetary waves no d i s t i n c t i o n need be made between cases (a) and (b) of §5.2.3 . A general s o l u t i o n to (5.43) i s given by g ( Q = g. + B. cn2(P.£/n) (5.50) where i may be any one of 1, 2, or 3, and P l = e3B ±/(12e 1n) (5.51) C l = - c 2 ( l - 6 C o ) ( g . + 4[2n - 1]P 2) (5.52) eif = f ^ g ? + ^ [ ( 1 - n)B. + g . ( 2 n - l ) ] } ' . (5.53) 90 In order that the solution be real, P. must be real. This w i l l 1 also mean that the wavelength of the disturbance, A = (2/P ±) K(n) , (5.54) must also be real, where the wavelength is defined by the requirement that g(£ + X) = g(£). In §5.2.4 i t was shown that the sign of e3/ei is that of (-) mA/c 0; hence, for P^ to be real, AB_^  must be negative for m even, positive for m odd. Setting mTTAB-f 2(1 -! Sc 0) then the zonal v e l o c i t y due to the waves i s , .m+1 u = (-) £ mir, v 2 6 c 0 f . mTT, cos -^-(y+l) ^ — s i n -^(y+l) (5.55) (5.56) f i + cn 2(P ±5/n) L B i and 2 2 m even 12c0m TT n Pi = "/ (5.57) Ef 0 12c 0m TT n m odd An interesting property of this solution arises from the fact that both A and B^ are unspecified parameters. Equation (5.55) specifies one of them as a function of the other, and (5.53) connects them with a third parameter ei+, also unspecified. Hence, i n (5.56), B^, where i t appears, may be any non-zero number. At f i r s t sight this appears to have the effect of permitting a steady zonal current of the form cos y^(y+l) - 2 ^ ° ^ sin y^(y+l) and of any amplitude to be added to the solution without altering the form of the wave. This is 91 not true s i n c e the root i s determined i n pa r t by the constant et,. which i n t u r n i s s p e c i f i e d when B_^  i s s p e c i f i e d . , Larsen (1965) gives a s e r i e s of s o l u t i o n s , s i m i l a r i n form to (5.50), but w i t h B^ s p e c i f i e d as p a r t i c u l a r combinations of the roots of the cubic. Such a procedure has the formal advantage of p e r m i t t i n g P ,. c i , and eit to be determined i n terms of the roots g^ and n only. However, si n c e the roots of the cubic w i l l , i n general, be q u i t e complicated expressions, having the s o l u t i o n parameterized i n terms of them seems of l i t t l e advantage. In order to study these s o l u t i o n s i n gre a t e r d e t a i l i t i s necessary to f i n d a s p e c i a l case i n which the roots of the cubic, or at l e a s t one of them, i s of a simple a n a l y t i c a l form. One such case occurs- i f es i s zero, i n which case one of the roots i s zero and the two remaining roots are given by the roots of a qua d r a t i c . Not only does t h i s case give simple expressions f o r the s o l u t i o n , but i t a l s o contains the s o l i t a r y wave s o l u t i o n s as a s p e c i a l case. 5.2.6 A s p e c i a l case: one root zero I f one of the roots of (5.49) i s zero, then one s o l u t i o n of (5.43) i s g(5) ' = B x cn2(Pi£/n) (5.58) which i s simply (5.50) w i t h g^ = 0. Normalizing the amplitude through the c o n d i t i o n (5.55) where B^ = B i i the zonal v e l o c i t y i s given by 92 = e [cos f ^ ( y - f l ) - 2§£ai S i n M(y+1) ] cn^P^/n) (5.59) where 12c0m*Trzn Ef , 12c 0m TT n m even m odd , (5.60) 4 c 2 ( l - 6c 2)(2n - 1 ) P 2 , (5.61) and the wavelength of the disturbance i s given by (5.54) i n which P^ i s replaced by P i . Tables of e l l i p t i c f u n c t i o n s such as cn(x/n), are given only f o r 0 < n < 1; t h e r e f o r e , f o r n > 1, the wave p r o f i l e i s given by t \m+l m 7 T/ 26c 0f . mTT. u = K-) e[cos — ( y + 1 ) j ^ j - s m y - ( y + l ) ] (5.62) n n 1 / n and the wavelength by For n = 1, s o l u t i o n (5.58) becomes (5.63) g(5) Bj s e c h 2 ( P 1 C ) (5.64) which i s the s o l i t a r y wave s o l u t i o n given p r e v i o u s l y i n §5.2.4 . In Figure 2, 2A~x [P x (1) ] - 1 and c i ( n ) / c i ( l ) are p l o t t e d against n. The wavelength, A, increases s t e a d i l y without l i m i t from zero as n goes from zero to one, then decreases again to an asymptotic l i m i t as n increases beyond one. The s o l i t a r y wave, at n = 1, i s , t h e r e f o r e , the l i m i t of long wavelengths. Since C i ( l ) < 0, the phase speed 94 c o r r e c t i o n , c i ( n ) , i s negative f o r n > -z, p o s i t i v e f o r n < \. Since the b a s i c phase speed c Q i s negative, c i < 0 represents an increase i n the magnitude of the phase speed. 5.2.7 The non-divergent l i m i t S o l u t i o n s s i m i l a r to (5.59) and (5.47) were given by Larsen (1965) f o r the non-divergent case. These non-divergent s o l u t i o n s may be. obtained by s e t t i n g 6 to zero i n equations (5.59) to (5.61). As 6-^-0, E ->• 0 and hence, e 3 -> 0. Since Pj must be non-zero i n order that the wavelength remain f i n i t e , t h e r e f o r e , from (5.51), n 0. Thus, i n t h i s l i m i t of 6 = 0, n = 0, (5.59) becomes u = ( - ) m f l e[cos y k y + D ] c o s 2 ( P ^ ) (5.65) or on r e w r i t i n g i n terms of s u = ( - ) m K L cos f^Cy+l) [1 + cos ( 2 P l V ^ s ) ] . (5.65a) I f k = 2P x/e , ; (5.66) then ec = CoV (5.67) and - —r~2 + —tj—1+ . (5. DO; m IT m l 95 These r e s u l t s are I d e n t i c a l to those obtained by Larsen (1965) when he allowed the b a s i c zonal current i n h i s non-divergent a n a l y s i s to go t o zero. In Chapter IV i t was shown that u = cos Tp-(y+l) cos ks (5.69) i s an exact s o l u t i o n of the non-divergent equations where (5.70) 2 2 ^ .+ k 2 I f k « 7 ^ - , equation (5.68) i s the f i r s t two terms of the b i n o m i a l expansion of (5.70). Therefore, applying the non-divergent approximation, the c n o i d a l wave s o l u t i o n s w i t h no b a s i c zonal current reduce to an expansion of the exact s o l u t i o n s given i n Chapter IV. 5.3 The case of 5 =• 0{e) 5.3.1 I n t r o d u c t i o n The r e s u l t s so f a r have shown that i f divergence i s r e t a i n e d , c n o i d a l and s o l i t a r y waves may e x i s t i n a long channel without the i presence ,of a sheared b a s i c zonal current. This i s i n c o n t r a s t to the non-divergent r e s u l t s of Larsen (1965), who showed that f o r s o l i t a r y and c n o i d a l waves to e x i s t , there must a l s o be present a z o n a l current w i t h at l e a s t a weak shear. The equations f o r the divergent case are too complicated to i n c l u d e a zonal current and s t i l l get simple enough expressions to i n t e r p r e t ; however, r e c a l l i n g the s c a l i n g done p r e v i o u s l y , a reasonable s i m p l i f y i n g assumption i s that 6 = 0 ( e ) . Under t h i s assumption, the equations are s u f f i c i e n t l y simple that the e f f e c t s of a sheared b a s i c 96 current may be s t u d i e d . Furthermore, i n the previous chapters, i t was shown that planetary waves can e x i s t i n a f l u i d of uniform r o t a t i o n w i t h bathymetry; here, the equations are g e n e r a l i z e d to i n c l u d e v a r i a b l e bottom topography. 5.3.2 The equations The b a s i c non-dimensionalized equations are (5.2), (5.3), and (5.5), plus a new c o n t i n u i t y equation [(u - c)(h + 6 n ) ] s + [ v ( h + 6 n ) ] y = 0 (5.71) where h = h(y) i s the non-dimensionalized depth, and H i n (5.1) i s redefined to be the average depth of the channel. D e f i n i n g y = 5/e = 0(1) (5.72) the transformation (5.6) and the p e r t u r b a t i o n expansions (5.7) are once more a p p l i e d to equations (5.2), (5.3), (5.5) and (5.71) th j , to gxve to zero order i n £: f u 0 + rioy = 0 , (5.73) to the f i r s t order: ( u 0 - c 0 ) u 1 ^ + V j U o - y - f v x + n = 0 (5.74) f u i + H l y = 0 (5.75) hu x + (hvj) = 0 (5.76) w1 = 0 at y = ± 1 , (5.77) 97 and to the second order: ( u 0 - c 0 ) u 2 + ( u D y - f ) v 2 + T l 2 = - ( U l - C l ) u 1 ? - V l u l y (5.78) f u 2 + = - (Uo " C Q)V 1 (5.79) hu 2 + ( h v 2 ) y = - U[(u 0 - C o ) ! ! ^ + TioU., + (riov^y] 0 at y = ± 1 (5.80) (5.81) The a n a l y s i s follows that of §5.2. I f the tr a n s p o r t s . a r e defined by Vj = Vjh and H1 = Ujh, then the f i r s t order equations are r e d u c i b l e to ( u 0 - c Q ) v l y y - ( u 0 - c G)^-y vr.. + (1 - u 0 y. y) +-(u 0 y - f ) ^ Y (5.82) Vj = 0 I f the s o l u t i o n i s of the form V 1 = g£(S) *(y) , (5.83) then (5.84) where (5.85) and U, - g(Q * y(y) ( f - u . y ) * + ( u 0 - c 0 ) * y g ( Q t (5.86) (5.87) 98 S i m i l a r l y , u 2 and n 2 may be e l i m i n a t e d from the second order equations, and u l 9 v-y, and r\i s u b s t i t u t e d f o r , i n terms of $ and g,to give ( u 0 - >c0) f(v 2h) 1 + h y ( l - u 0 y y ) + ( u Q y - f) ]jy (5.88) u 0 - c h - " + c* W L 8 - y ( u 0 y - f ) + ( U o - C o ) (Up-Cp) • Tlo^y + U 0 y ( u 0 - C 0 ) + f c 0 + I f (5.88) i s m u l t i p l i e d through by ^>/(u 0-c 0), then i n t e g r a t e d over y from -1 to 1, the l e f t - h a n d s i d e i s i d e n t i c a l l y zero and the right-hand s i d e gives 0 (5.89) where ' f * - i (5.90) - l h \ u 0 - c 0 - V ( U o - C o ) ,2 -1 dy 'u - f ^ "h^ + u°y( u°~ G°) + f c ° (5.91) LUo-Co rio^y + 2 u o y ( u 0 - c 0 ) + 2 f c Q - f u : 2K 7 ^ } dy 99 - l Uo ~ C 0 y L h \ h y-i dy (5.92) Equation (5.89) i s again'the Korteweg-deVries equation. The major d i f f e r e n c e between these equations and the corresponding ones obtained i n §5.2 i s that b 2 , the c o e f f i c i e n t of g^, contains, i n a d d i t i o n to the term i n c i , a second term which a r i s e s from the divergence terms. S o l u t i o n s to (5.89) are f i r s t obtained by s e t t i n g h = 1; these are then compared w i t h those obtained by Larsen w i t h no divergence. The case of topographic waves w i l l be t r e a t e d i n §5.4 . 5.3.3 The case of uniform s h e a r ' For a uniformly sheared zonal v e l o c i t y given by W0 + ay (5.93); where a << 1, equation (5.84) can be solved e x a c t l y i n terms of confluent hypergeometrie functions [see equation (4.110)]; however, an approximate s o l u t i o n gives more workable r e s u l t s . Using p e r t u r b a t i o n expansions f o r both c Q and $ i n the terms of a, equation (5.84) gives as a s o l u t i o n (5.94) = A { s i n £(y+l) + a-| ( y - y 2 ) c o s £(y+l) + -| y s i n £(y+l) + 0(c where * " 2 (5.95) and the phase speed i s 100 2 2 / co = Wo - j2 + § - - ^ f ( l - | 2) + 0 ( a 3 ) . (5.96) From (5.90) to (5.92), the c o e f f i c i e n t s b,, b 2 , and b 3 are b x = A 2 ( l - + 0 ( a 2 ) (5.97) b 2 = A 2 [ £ V 1 - + u£ 2(W G - I 2 ) ( | + f 2 ) + 0(a)] (5.98) b 3 = 2 a | J A 3 [ ; L _ c Q s u + a | ^ 6 + c o s 2£J ] + 0 ( a 2 ) . (5.99) Both b i and b 3 and the expansions f o r * and c 0 are i d e n t i c a l to the c o e f f i c i e n t s that were obtained by Larsen (1965) f o r the non-divergent equations of motion. The e f f e c t of the weak divergence i s f e l t e n t i r e l y i n the b 2 c o e f f i c i e n t which, as pointed.out e a r l i e r , i contains terms which do not co n t a i n c i as a f a c t o r . In the s t r o n g l y divergent case, these terms appear i n thesexpression f o r c 0 s i n c e th they are of zero order. For y = 0, the non-divergent case i s recovered. In the case of a = 0, that i s , f o r uniform zonal c u r r e n t , the b 3 c o e f f i c i e n t i s zero and equation (5.89) i s l i n e a r w i t h general s o l u t i o n g(£) = A + B cos P 2£ (5.100) where P 2 =, i \ - y | ° + ( i - i i 2 w 0 ) ( i - f 0 2) (5.101) Transforming £ back to s , (5.100) becomes .101 g(s) = A + B cos /eP 2s = A + B cos ks (5.102) where k = /eP 2 and (5.103) For the non-divergent case (6 = 0 ) , t h i s reduces to the s o l u t i o n obtained by Larsen, and the phase speed i s the f i r s t few terms of the bi n o m i a l expansion of the phase speed f o r the l i n e a r Rossby wave s o l u t i o n , where both k 2attd6f 2 are sm a l l w i t h respect to I2 and 6/3, r e s p e c t i v e l y . The c o e f f i c i e n t of the n o n - l i n e a r term of (5.89) i s non-zero only i f a i s non-zero; hence, the s o l i t a r y and c n o i d a l wave s o l u t i o n can e x i s t only i f a sheared zonal current i s present. This r e s u l t i s i d e n t i c a l to that o b t a i n e d * i n the non-divergent case by Larsen. S o l u t i o n s of (5.89) take the same form as in"'§5.2 . Again, a general form of s o l u t i o n may be w r i t t e n but t h i s i s not of much i n t e r e s t a n a l y t i c a l l y because i t i n v o l v e s the roots of a cubic equation as w e l l as the i n t r o d u c t i o n of two a r b i t r a r y i n t e g r a t i o n constants. Therefore, the d i s c u s s i o n w i l l be r e s t r i c t e d to two s o l u t i o n s , the s o l i t a r y wave s o l u t i o n and the simple c n o i d a l wave s o l u t i o n f o r which one root of the cubic i s zero. 5.3.4 The s o l i t a r y wave The s o l i t a r y wave s o l u t i o n i n terms of the zonal v e l o c i t y i s u = w0 - e[sgnCa)]m cos &(y+l) sechzCP2?) + °Ca) . C5.104) Wc + + Wc + ( i - Ji 2w 0)(i + f l ) 102 where a l l 1 7af£6 36 m odd m even • (5.105) and the phase speed i s c = Wo - V 4eP2 + I* W0 2 ( l - £2w 0)(f 2 + i ) (5.106) + 0(a) . The only d i f f e r e n c e between t h i s s o l i t a r y wave s o l u t i o n and that f o r tfte non-divergent case i s that the phase speed i s inc r e a s e d or decreased by the term c o n t a i n i n g 6 as a f a c t o r . I f WD i s s e t to zero and i f 1/3 i s neglected r e l a t i v e to f 2 , then (5.106) reduces to an expansion of (5.47) f o r <5 s m a l l , where (5.47) gives the phase speed f o r s o l i t a r y waves i n the s t r o n g l y divergent case. 5.3.5 A c n o i d a l wave The second s o l u t i o n of i n t e r e s t i s chosen so th a t i t contains the s o l i t a r y wave as a s p e c i a l case. This s o l u t i o n i s given by u ,m W0 - e[sgn(a)] cos U y + D cn z(P 2£/n) + 0(a) (5.107) where r 9n 7a 2A 6 36n m odd m even (5.108) and 103 c = W0 - ji - | t ( 2 n - l ) P 2 + |, (5.109) 2 + (1 - £ 2W 0)(f 2 + & + 0(a) Again, except f o r the a d d i t i o n a l terms i n the expression f o r the phase v e l o c i t y , the s o l u t i o n i s the same as that obtained from the non-divergent equations. The v a r i a t i o n of wavelength and phase speed w i t h n i s that given i n Figure 2, except that a term independent of n must be added to each value of c\. Because of the clos e resemblance of these s o l u t i o n s to Larsen's non-divergent s o l u t i o n s , a m o d i f i c a t i o n of quasi-Lagrangian a n a l y s i s might o f f e r some exp l a n a t i o n of the p o s s i b l e p h y s i c a l processes which determine these waves. Such a m o d i f i c a t i o n , however, l i e s beyond the scope of t h i s work. In h i s a n a l y s i s , Larsen obtains an equation of the form of (4.42) i n which a l l the c o e f f i c i e n t s are defined i n terms of the energy and momentum of the b a s i c flow. His a n a l y s i s showed that the generation of the s o l i t a r y wave requires no a d d i t i o n a l energy over that of the b a s i c flow. Waves of moduli n > 1, i are a s s o c i a t e d w i t h a l o s s of energy from the b a s i c flow; those of moduli n < 1, w i t h a gain of energy. This argument may a l s o h o l d f o r the weakly divergent waves. 5.4 Topographic waves As i n the previous chapters, topographic waves w i l l be i n v e s t i g a t e d only f o r the exponential p r o f i l e given by h = exp (-Ay) (5.110) where A « 1. I f i t i s assumed that the topographic e f f e c t i s much 104 more important than the (3-eff e c t , , f can be t r e a t e d as a constant. Once again the b a s i c current i s given by (5.93), that i s a steady zonal current w i t h a weak uniform shear;,and (5.84) i s solved using a p e r t u r b a t i o n expansion f o r $ and c G i n powers of a. The s o l u t i o n s a t i s f y i n g both boundary c o n d i t i o n s , (5.85), i s given by = A a ( l / + A'/4)*y 4FIf exp (--JAy) s i n £(y+l) (5.111) + ^ 4 ^ / M ) (1 - y 2)^xp(- 2Ay),cos-£(y +l); + 0 ( a 2 ) and c o = W0 - z , f A . / 4 + ^ A . / 4 + 0 ( a 2 ) (5.112) where again = Y~ • (5.113) Using t h i s s o l u t i o n , the c o e f f i c i e n t s of the Korteweg-deVries equation (5.89) may be c a l c u l a t e d from the i n t e g r a l s given by (5.90) to (5.92). The c o e f f i c i e n t s are then given by b i = A 2 + 0(a) (5.114) b 2 = c . A 2 ^ ^ (5.115) + ^ A 2{i6a 2' + fI 2/4) [ 2 7 A \ + 4 0 £ 2 a 2 " 1 6 j l " ] ' . ^ £2fW0(16£'t - 5A^) 1 . , . + 8A(£2 + A 2/4) 2 J S i n h a 2 irfW 0(4£ z - 3AZ) i « ^ , yA z + cosh A + 0(a) 105 24& 3(4& 2 + A 2 ) A f(36£z + A z) 3 - cosh A s i n h A m even m odd + 0(a) (5.116) Having these c o e f f i c i e n t s , the s o l u t i o n s to (5.89) f o l l o w i n the same way as they d i d i n the previous two cases. For a = 0, a l l of the c o e f f i c i e n t s remain non-zero; hence, s o l i t a r y and c n o i d a l wave;:solutions w i l l e x i s t even f o r a uniform b a s i c zonal flow. In f a c t , s o l i t a r y and c n o i d a l waves w i l l e x i s t f o r the e x p o n e n t i a l depth p r o f i l e even f o r the non-divergent case, y = 0, and no b a s i c flow, W0 = a = 0. I t should again be emphasized that w i t h topographic waves, the p r o p e r t i e s of the waves are s t r o n g l y dependent on the c h a r a c t e r of the topography. Hence, any property of the waves discussed here i s l i k e l y to be a property only of waves over an e x p o n e n t i a l depth p r o f i l e . For W0 = 0 and a = 0 , the simple c n o i d a l wave s o l u t i o n corresponding to (5.108) i s given by u\ = ( - ) m e c n 2 ( P 2 5 / n ) [ c o s £(y+l) - | r - s i n Uy+1) ]exp(*Ay) (5.117) where m even m odd (5.118) c 4AfPJ(l-2n) (lz + A z/4) •2 - y il 2Af?[27A' t + 40A 2£ 2 - 161,"] 16 (lz + A z/4) 5 (5.119) 106 The s o l i t a r y wave i s contained as the s p e c i a l case of n = 1, i n the above equations. S o l u t i o n s of the same form i n £ have been p r e v i o u s l y discussed i n §5.2.5, §5.2.6, §5.3.4, and §5.3.5 . I t should be noted that the e f f e c t ' of the weak divergence i s f e l t only i n the phase speed,and t h a t - i f y i s zero, a c n o i d a l wave s o l u t i o n s t i l l e x i s t s . Equations (5.83) to (5.87), and (5.89) to (5.92) h o l d f o r an a r b i t r a r y depth p r o f i l e . For any depth p r o f i l e f o r which s o l u t i o n s of the transverse e i g e n f u n c t i o n equation (5.84) e x i s t , s o l i t a r y and c n o i d a l wave s o l u t i o n s should a l s o e x i s t . Benjamin (1967) found that i n t e r n a l s o l i t a r y and c n o i d a l waves of a new form'could e x i s t on density p r o f i l e s i n flui'ds of i n f i n i t e depth, provided that the density v a r i e s only i n a l a y e r whose thickness i s much sm a l l e r thanJthe depth of the f l u i d . Topographic waves, analogous to these s o l u t i o n s , i n c l u d e s h e l f waves and double K e l v i n waves. A p o s s i b l e extension to t h i s present study would be to i n v e s t i g a t e the existence of s u c h ' s o l i t a r y and c n o i d a l topographic waves on an i s o l a t e d topographic f e a t u r e i n an unbounded ocean. 5 L 5 Summary In summary, i t has been shown that a c l a s s of long waves analogous to the s o l i t a r y and c n o i d a l waves of surface wave theory e x i s t i n a channel on the 3 -plane or i n a channel w i t h cross-channel bathymetry f o r a uniformly r o t a t i n g f l u i d . In the Rossby wave case, i t was shown that i f the non-divergent approximation i s made,.or that i f the magnitude of the divergence terms i s of the same order as that of the i n e r t i a l terms, then s o l i t a r y and c n o i d a l waves w i l l e x i s t only 107 i n the presence of a steady sheared current along the channel. Such a current i s not necessary f o r c n o i d a l and s o l i t a r y waves to e x i s t i n the divergent case or f o r topographic waves on an e x p o n e n t i a l p r o f i l e . In a l l cases, the wave p r o f i l e s along the a x i s of the channel are given by the s o l u t i o n s to the Korteweg-deVries equation. In f r e e surface flows, i t i s found that s o l i t a r y and c n o i d a l waves are a p r e f e r r e d form of disturbance i n that they show a remarkable p e r s i s t e n c e of form. Although s o l i t a r y wave disturbances as discussed here have .not been described i n observation of e i t h e r the ocean or the atmosphere, by analogy to surface waves, i t i s f e l t that these s o l u t i o n s may a l s o represent a p r e f e r r e d form of disturbance. VI. Concluding Remarks In t h i s work some of the f i n i t e amplitude behaviour of plan e t a r y waves has been explored i n order to have some understanding of p o s s i b l e n o n - l i n e a r time-dependent motions i n the ocean. With t h i s i n mind, the l i n e a r p l a n e t a r y wave s o l u t i o n s were c l o s e l y examined both on the sphere and the g-plane i n order to determine the magnitude of er r o r s a s s o c i a t e d w i t h the non-divergent and 3-plane approximations. For wavelengths of the order of a few thousand kilometers and l e s s , the e r r o r i n the phase speed associated w i t h the non-divergent approximation both on the sphere and on the 3-plane is--'.about 15%, decreasing w i t h decreasing wavelength. For the same range of wavelengths, the e r r o r a s s o c i a t e d w i t h the, 3-plane i s a l s o about 10%. The l i n e a r , non-divergent s o l u t i o n s on the sphere e x h i b i t the i n t e r e s t i n g property that t h e i r phase speed depends only on the degree of the s p h e r i c a l harmonic and i s independent of the order. This means that any l i n e a r s u p e r p o s i t i o n of waves of the same degree w i l l stay together because they a l l move w i t h the same angular phase speed. Since each of the s p h e r i c a l harmonics making up t h i s sum of s o l u t i o n s may have a d i f f e r e n t a x i s , t h i s s o l u t i o n may become very complex, y et s t i l l be non - d i s p e r s i v e , at l e a s t to the l i m i t s of l i n e a r theory. This property i s not e x h i b i t e d by the divergent waves on the sphere. Here, the phase speed depends on both the degree and order of 109 s p h e r o i d a l harmonics, and t h e r e f o r e , such a l i n e a r s u p e r p o s i t i o n of s o l u t i o n s w i l l d isperse i n time. On the 3-plane, the non-divergent s o l u t i o n s are d o u b l y - p e r i o d i c s i n u s o i d a l waves whose phase speed depends only on the t o t a l wave number. In c o n t r a s t to t h i s , the divergent s o l u t i o n s of the 3-plane, are s i n u s o i d a l i n the d i r e c t i o n of the waves but t h e i r v a r i a t i o n i n y, normal to t h i s d i r e c t i o n , i s i n the form of P a r a b o l i c C y l i n d e r f u n c t i o n s . For short wavelengths i n y, these s o l u t i o n s may be approx-imated by a d o u b l y - p e r i o d i c s i n u s o i d a l wave, whose phase speed i s a f u n c t i o n of the t o t a l wave number only. The e r r o r s a s s ociated w i t h both the 3-plane and non-divergent approximations are s m a l l e r f o r the s h o r t e r wavelength cases than f o r the longer. I t i s shown that f o r bottom slopes commonly found i n the oceans, topographic waves w i l l predominate over Rossby waves, and f u r t h e r , f o r the same range of frequencies the wavelengths a s s o c i a t e d w i t h topographic waves w i l l be much s h o r t e r than those f o r Rossby waves. For t h i s reason the 3-plane arid non-divergent approximation may be used w i t h greater accuracy w i t h topographic waves than w i t h Rossby waves. An exception to t h i s i s the c o n t i n e n t a l s h e l f waves, where, because the depth of the f l u i d goes to zero, the non-divergent approximation may not be used. The f u l l non-divergent'equations of motion on the sphere and on the 3-plane give the l i n e a r non-divergent Rossby wave s o l u t i o n s as exact s o l u t i o n s . Furthermore, s i n c e these exact s o l u t i o n s c o n s i s t of an a r b i t r a r y sum of the l i n e a r s o l u t i o n s of the same phase speed there i s no n o n - l i n e a r i n t e r a c t i o n between l i n e a r non-divergent 110 s o l u t i o n s of the same phase speed. Such behaviour i s not found f o r the divergent wave s o l u t i o n s nor f o r the l i n e a r topographic wave s o l u t i o n s nor f o r non-divergent Rossby wave s o l u t i o n s i n the presence of a uniformly sheared c u r r e n t . In a l l of these cases, the wave p r o f i l e s e x h i b i t n o n - l i n e a r i t i e s to 0 ( e 2 ) where e i s an amplitude parameter; however, s i m i l a r to surf a c e g r a v i t y waves, there i s no f i r s t order c o r r e c t i o n to the phase speed. Such fundamental d i f f e r e n c e s i n n o n - l i n e a r behaviour of divergent and non-divergent waves was not expected. These r e s u l t s suggest that i f one i s stud y i n g n o n - l i n e a r e f f e c t s or i n t e r a c t i o n s between Rossby waves, the non-divergent approximation should be made only w i t h a great d e a l of caution. These r e s u l t s a l s o suggest t h a t i n the mid-ocean, where b a r o t r o p i c currents are perhaps very weak, the depth n e a r l y constant and deep, the Rossby waves once generated w i l l i n t e r a c t w i t h each other only very weakly as the motion w i l l be nearly non-divergent. As these waves move toward the western boundary region they w i l l experience sheared cu r r e n t s , bathymetry and increased divergence due to the decreasing depttt. A l l these e f f e c t s work to make the n o n - l i n e a r i n t e r a c t i o n terms more important. Therefore, these r e s u l t s suggest that the western boundary region i s a region of i n t e n s i f i e d n o n - l i n e a r e f f e c t s f o r Rossby waves. Such an e f f e c t at the western boundary of the Indian Ocean i s a l s o suggested by L i g h t h i l l (1969). F i n a l l y , i t i s shown that a c l a s s o f . l o n g divergent Rossby waves e x i s t s , analogous to the s o l i t a r y and c n o i d a l waves of surface wave theory. Larsen's (1965) c o n c l u s i o n , that such waves could e x i s t i n the I l l non-divergent case only i n the presence of a sheared b a s i c c u r r e n t , i s confirmed and explained i n l i g h t of the exact non-divergent s o l u t i o n s found here. Since these waves are to be long, r e l a t i v e to t h e i r l a t e r a l dimension, i t seems reasonable to expect that the divergence terms should be r e t a i n e d . I t i s al s o shown t h a t s o l i t a r y and c n o i d a l waves can e x i s t on an exp o n e n t i a l depth p r o f i l e , even i n the non-divergent case. 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New York: MacMillan, 306 pp. Appendix I The problem of transformations i n c e r t a i n co-ordinate systems In the mathematical s o l u t i o n of p h y s i c a l problems c e r t a i n systems of co-ordinates introduce i n the s o l u t i o n features which may be d i f f i c u l t to separate from the p h y s i c a l p r o p e r t i e s of the s o l u t i o n i t s e l f . In p a r t i c u l a r , s p e c i a l c o n d i t i o n s are imposed on the s o l u t i o n at the poles or a x i s of s p h e r i c a l p o l a r , c y l i n d r i c a l , or c i r c u l a r p o l a r co-ordinate systems. To i l l u s t r a t e t h i s problem,. consider a f l u i d confined between two c o n c e n t r i c and r o t a t i n g spheres. A n a t u r a l choice of co-ordinates are the s p h e r i c a l p o l a r co-ordinates whose axi s i s the axis of r o t a t i o n ; however, the choice of the axis of the co-ordinates i s , i n f a c t , completely a r b i t r a r y . Having chosen the a x i s of the co-ordinates, both the zonal,v^, and the m e r i d i o n a l , Vg, v e l o c i t y components must be zero at the poles i n order that both be s i n g l e valued and continuously d i f f e r e n t i a b l e i n a neighbourhood of the p o l e s . The p o i n t made here i s that the v e l o c i t y does not have to go to zero by any p h y s i c a l grounds but r a t h e r i s r e q u i r e d mathematically i n order that the co-ordinate system work. P h y s i c a l l y one can c e r t a i n l y allow a v e l o c i t y at the pole although mathematically t h i s could only be described i f one allows v, and v„ to be both m u l t i v a l u e d and discontinuous at the pole. 118 The problems i n v o l v e d become very r e a l and very c l e a r i f one t r i e s to transform from one co-ordinate system to a second, both on the sphere. Suppose i n the f i r s t system the s o l u t i o n i s a steady zonal flow described by v^ = aR s i n 6 , V g = 0. We transform to a new co-ordinate system (0' jCJ)') whose axis l i e s i n the cf> = 0 plane of the f i r s t system and i s i n c l i n e d at an angle y to the o r i g i n a l a x i s . (See Figure 1, p.12). In t h i s co-ordinate system, the v e l o c i t y components at the poles are given by V ' Q , = - aR s i n y sin<f>' (A.l) v 1 ^ , = aR[ cosy s i n 9 ' - s i n y cosO'coso)' ] (A. 2) Although p h y s i c a l l y the v e l o c i t y f i e l d i s continuous and continuously d i f f e r e n t i a b l e , the mathematical d e s c r i p t i o n of i t i s not. Both V ' ^ T and V ' Q , take on a l l values between ± aR s i n y at each p o l e (0' = 0,TT); t h e r e f o r e , one cannot r e a l l y speak of a value of e i t h e r v e l o c i t y component at the poles.. The v e l o c i t y f i e l d i s known to be continuous and s i n g l e - v a l u e d everywhere. Therefore, the s i n g u l a r behaviour at the poles must a r i s e from the behaviour of the co-ordinate system alone. In t h i s way the choice of the co-ordinate systems may have an e f f e c t on the s o l u t i o n which must not be i n t e r p r e t e d as a p h y s i c a l e f f e c t . Appendix I I Analogous behaviour of i n t e r n a l and planetary waves In h i s t r e a t i s e on non-homogeneous f l u i d s , Y i h (1965, ch.VI) gives a general d i s c u s s i o n of the s i m i l a r i t y between the flow of s t r a t i f i e d f l u i d s and f l u i d flow i n an a c c e l e r a t i n g or r o t a t i n g frame. This analogy has been discussed i n greater d e t a i l w i t h references to slow steady flows by Veronis (1967a,b). Here the analogy w i l l be extended to wave motion and hence a p a r a l l e l w i l l be developed between i n t e r n a l waves and non-divergent p l a n e t a r y waves. This p a r a l l e l has been found to be a u s e f u l t o o l i n suggesting the existence or the form of planetary wave s o l u t i o n s f o r bathymetries of the same shape as density p r o f i l e s f o r which i n t e r n a l wave s o l u t i o n s are known. For an incompressible s t r a t i f i e d f l u i d of constant depth, the equation governing i n f i n i t e s m a l amplitude i n t e r n a l g r a v i t y waves i n two dimensions i s given by Lamb (1945, p.378), by t t .0 (B.l) where z i s i n c r e a s i n g upwards u w - t X (B.2) and 120 R = _ 1 Mo . ( B. 3) p 0dz For waves propagating h o r i z o n t a l l y , ^ = W(z) exp i ( k x - cot) , (B.4) hence ( B . l ) becomes W z z " r W z + k 2 ( ^ - 1) W = 0 (B.5) subject to the boundary c o n d i t i o n that W = 0 at a boundary or at z = ± °°. (B.6) Sol u t i o n s are a l s o p o s s i b l e f o r discontinuous density p r o f i l e s . At a d i s c o n t i n u i t y of e i t h e r p or F, both the v e r t i c a l v e l o c i t y and the pressure are continuous across the i n t e r f a c e ; hence, P l W l 7 = ^ 2 ( P 2 " PX> Wt (B.7) > 2 W 2 Z ~ ^ i " i z " 03 where the s u b s c r i p t s r e f e r to values on e i t h e r s i d e of the d i s c o n t i n u i t y . 'In Chapter I I I , the l i n e a r equations f o r pla n e t a r y waves are developed. In p a r t i c u l a r , equation (3]38) governs the y dependence of the zonal t r a n s p o r t of a p l a n e t a r y wave propagating east-west p a r a l l e l to the depth contours. This equation i s given by v - -y v yy h' y k 2 0 (B.8) y w i t h the boundary conditions that V = 0 at a boundary or at y = ± °°. (B.9) 121 At a d i s c o n t i n u i t y of h or f or t h e i r d e r i v a t i v e s , t r a n s p o r t and pressure must be continuous; hence, V x = v 2 V i _ V 2 = fk (hi - h 2) V (B.10) h i y h 2 y a) h i h 2 where again the s u b s c r i p t s r e f e r to values on e i t h e r s i d e of the dis con t i n u i t y . The s i m i l a r i t y between (B.8) and (B.5) plus t h e i r boundary conditions i s at once apparent. I f the Boussinesq approximation i s made i n each of (B.5) and (B.8) (neglect of the f i r s t d e r i v a t i v e term) i t i s apparent that Y and - n^~^ play e x a c t l y e q u i v a l e n t r o l e s y i n i n t e r n a l and i n plan e t a r y waves, r e s p e c t i v e l y . I t then f o l l o w s that s o l u t i o n s of (B.8) w i l l have the same y dependence, W(y), as s o l u t i o n s of (B.5) i f - h f ^ j has the same f u n c t i o n a l dependence as T. \ /y The s i m p l e s t case f o r which t h i s analogy holds i s between i n t e r n a l waves in, a f l u i d w i t h a weakly e x p o n e n t i a l density p r o f i l e contained between r i g i d h o r i z o n t a l planes and Rossby waves i n a 3-plane channel. In both cases (the Boussinesq's approximation being made f o r the i n t e r n a l wave case) the wave form i s given by s i n ^rrz ( s i n ~y f o r Rossby waves) where 1/ n 2TT 2' _ / 3 i7 " V afc" ( B . l l ) kM-^fe - 1) r e s p e c t i v e l y . 122 A f u r t h e r example of t h i s p a r a l l e l i s the analogy between i n t e r n a l waves on the boundary between two unbounded f l u i d s [Lamb, (1945), p.370] and the non-divergent l i m i t of the double K e l v i n wave along a depth d i s c o n t i n u i t y [Rhines, (1969a)]. In both cases the wave amplitudes decay e x p o n e n t i a l l y away from the d i s c o n t i n u i t y . The extension of t h i s analogy to i n c l u d e the f i n i t e amplitude cases i s f a r more tenuous. For i n t e r n a l waves, the motion i t s e l f changes: the density p r o f i l e w h i l e f o r non-divergent p l a n e t a r y waves the (f/h) p r o f i l e s are independent of the waves. There are, however, s t i l l s i m i l a r i t i e s between the n o n - l i n e a r equations. For example, the equation f o r planetary waves of permanent form (4.26) given by n + H 3s 3iJ> H + H 3s + 1 3' H + H 3y 1 3£ r| + H 3y ri + H 0 (B.12) i s s i m i l a r i n form to an equation obtained by Magaard (1965) f o r p r o g r e s s i v e i n t e r n a l waves of permanent form J(V 2^ - g z ^ , = 0 (B.13) where w = ~ » u - c = J p ^ Z ' S = x - ct . (B.14) So l u t i o n s to (B.13) have been given by Magaard (1965) and Y i h (1965, ch. I l l ) and these s o l u t i o n s suggested the procedure l e a d i n g to the p o s s i b l e s o l u t i o n s to (B.12) which were obtained i n §4.2 . 123 Further extensions of the analogy to i n c l u d e i n t e r a c t i o n of waves w i t h currents or long wave s o l u t i o n s are tenuous i n the extreme. The analogy serves to suggest, from research already conducted f o r i n t e r n a l waves, d i r e c t i o n s i n which i n v e s t i g a t i o n s of pla n e t a r y waves might proceed. Appendix I I I Glossary of symbols a angular phase speed on the sphere df 3 = on the 3-plane Y angle between the axis of r o t a t i o n and the axis of the co-ordinate system on the sphere 6 i n §3.1 = 4fi 2R 2/gH, the divergence parameter 6 = 3 2L 2/gH, the divergence parameter e amplitude parameter £ i n §4.4 = exp(-Ay) i n §4.5, §5.2 = kAE ( f c + y) i n §4.6 = 2 (W0 - c 0 + ay) TI surface e l e v a t i o n 8,<f) c o - l a t i t u d e and longitude i n the t i l t e d co-ordinate system 9' ,0)' c o - l a t i t u d e and lon g i t u d e r e l a t i v e to the axis of r o t a t i o n K i n §4.5 = (1 + 4 c o k 2 ) / ( 2 c 0 A ) i n §5.2 = l/(2c0/& ) t o t a l wave number on the 3-plane X i n §3.1 = a/20,, the non-dimensionalized frequency i n §3.2 = f | ( k 2 + ^ - ) 23 O i n §4.4 = K / A c / r 1 - , (1 - 5_Co)k 2 i n § 4 ' 5 = 2 c 7 ^ + 2 7 T i n §5 . 2 , §5.3, §5.4 wavelength 125 A l , A2 y V I K p a 00 r A = ± (m2TT2 - 3k z) i n §4.5, §4.6 i n § 3.1 = - cos i n §4.6, §5.3, §5.4 = 5/e i n §1.2 i n ch. V i n §4.6 i n ch. V i n §4.6 = A 2 + 1 4 r e l a t i v e v o r t i c i t y v e c t o r = /e s density radian frequency of pla n e t a r y waves stream f u n c t i o n radian frequency of i n t e r n a l waves = _ 1 d p 0 P o d z = _ 1 dtti • h dy Vj = $(y) s i n ks v i = $(y) g^(?) * ' = Cexp(- 5/2) f ( 5 ) the r o t a t i o n v e c t o r of the ear t h a. l b. x c d. e. I f g i n §4.6, §5.3 i n §5.3 i n §4.2, §4.3 i n §5.2 du 0 dy i n t e g r a t i o n and phase constants c o e f f i c i e n t s of the Korteweg-deVries equation phase speed on 3-plane constants c o e f f i c i e n t s of the Korteweg-deVries equation = f 0 + 3y> the C o r i o l i s parameter g r a v i t a t i o n a l a c c e l e r a t i o n 126 g(g) i n ch. V h k, I m, n P r s i n §3.1 1 i n ch.IV, ch.V s , : i n §3.1 t u, v, w V V V r v e " v i " v r '"• x, y, z A. , B. l l H L R V 0, W0 W(z) Y(y) i n §4.5 Z (y) i n §4.5, §4.6' v i = $(y) g (O depth of f l u i d wave numbers i n x and y d i r e c t i o n s i n t e g e r s pressure radius zonal wave number = x - c t , transformed x co-ordinate = 20s/o time v e l o c i t y components on g-plane v e l o c i t y components on the sphere v e l o c i t y components r e l a t i v e to the r o t a t i o n a x i s on the sphere co-ordinates on the 3-plane amplitude constants d i s t r i b u t i o n of p o t e n t i a l v o r t i c i t y depth width of the 3-plane channel radius of the earth b a s i c uniform zonal flow z-dependence of the v e r t i c a l v e l o c i t y of i n t e r n a l waves f i r s t order y-dependence of north-south v e l o c i t y second order y-dependence of north-south v e l o c i t y 

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