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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On finite amplitude planetary waves Clarke, Richard Allyn 1970
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Title | On finite amplitude planetary waves |
Creator |
Clarke, Richard Allyn |
Publisher | University of British Columbia |
Date Issued | 1970 |
Description | Finite amplitude planetary waves are studied on a homogeneous fluid on both the rotating sphere and on a mid-latitude β-plane. The integrated equations of motion are rederived both on the rotating sphere, in a spherical polar co-ordinate system whose axis is tilted relative to the rotation axis, and on a mid-latitude β-plane. The linear solutions are re-examined and the errors associated with the non-divergent and the β-plane approximations are each shown to be about 10 to 15% for waves of a few thousand kilometers wavelength. Using the integrated equations of motion both on the sphere and on the β-plane, the linear non-divergent Rossby wave solutions are shown to be exact finite amplitude solutions. An exact topographic wave solution is also given for the case of an exponential depth profile. Such behaviour is not found for the divergent waves. Using a Stokes-type expansion in terms of an amplitude parameter, the second order solution for divergent Rossby waves is obtained, and it is found that, as in surface gravity wave theory, the first order correction to the phase velocity is zero. It is also shown that the linear non-divergent Rossby wave solution on a uniformly sheared zonal current is not a finite amplitude solution, and the second order solution is then calculated. Once again, the phase speed is correct to the first order. A class of long waves of permanent form analogous to the solitary and cnoidal waves of surface wave theory is obtained for a β-plane channel of either constant or exponentially varying depth. Such waves are found to exist in the divergent case in the absence of any zonal current; however, if the divergence is weak, or if the non-divergent approximation is made , then it is found, as it was by Larsen (1965), that these waves will exist only in the presence of a weakly sheared zonal current. On the exponential depth profile, such waves exist in the absence of a sheared zonal current, even if the non-divergent approximation is made. It is suggested that such waves may also exist trapped along long ocean ridges or scarps. |
Subject |
Wave mechanics Rotating masses of fluids |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-06-08 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0084775 |
URI | http://hdl.handle.net/2429/35301 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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