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

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ON THE STABILITY AND PROPAGATION OF BAROTROPIC MODONS IN SLOWLY VARYING MEDIA by GORDON EDWIN SWATERS B.Math.(Honours), University Of Waterloo, 1980 M.Sc, University Of B r i t i s h Columbia, 1983 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY THE FACULTY OF GRADUATE STUDIES Department Of Mathematics Insti t u t e Of Applied Mathematics Department Of Oceanography We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA i n July 1985 V © Gordon Edwin Swaters, 1985 In presenting t h i s thesis in p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Mathematics The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: July 1, 1985 i i A b s t r a c t Two a s p e c t s of the t h e o r y of b a r o t r o p i c modons a r e examined i n t h i s t h e s i s . F i r s t , s u f f i c i e n t n e u t r a l s t a b i l i t y c o n d i t i o n s a r e d e r i v e d i n the form of an i n t e g r a l c o n s t r a i n t f o r westward and e a s t w a r d - t r a v e l l i n g modons. I t i s shown t h a t e a s t ward-t r a v e l l i n g and w e s t w a r d - t r a v e l l i n g modons are n e u t r a l l y s t a b l e t o p e r t u r b a t i o n s i n which t h e energy i s c o n t a i n e d m a i n l y i n s p e c t r a l components w i t h wavenumber magnitudes (|T?|) s a t i s f y i n g |TJ|<K and |TJ|>K, r e s p e c t i v e l y , where K i s the modon wavenumber. These r e s u l t s imply t h a t when K/|T?|>1 the s l o p e of the n e u t r a l s t a b i l i t y c u r v e proposed by M c W i l l i a m s e t a l . ( l 9 8 l ) f o r e a s t w a r d - t r a v e l l i n g modons must b e g i n t o i n c r e a s e as K/|T?| i n c r e a s e s . The n e u t r a l s t a b i l i t y c o n d i t i o n i s computed w i t h mesoscale wavenumber eddy energy s p e c t r a r e p r e s e n t a t i v e of the atmosphere and ocean. E a s t w a r d - t r a v e l l i n g a t m o s p h e r i c modons a r e n e u t r a l l y s t a b l e t o the ob s e r v e d s e a s o n a l l y - and a n n u a l l y -averaged a t m o s p h e r i c e d d i e s . The n e u t r a l s t a b i l i t y of westward-t r a v e l l i n g a t m o s p h e r i c modons and o c e a n i c modons cannot be i n f e r r e d on t h e b a s i s of the ob s e r v e d wavenumber eddy energy s p e c t r a f o r t h e atmosphere and ocean. Second, a l e a d i n g o r d e r p e r t u r b a t i o n t h e o r y i s deve l o p e d t o d e s c r i b e the p r o p a g a t i o n of b a r o t r o p i c modons i n a s l o w l y v a r y i n g medium. Two problems a r e posed and s o l v e d . A p e r t u r b a t i o n s o l u t i o n i s o b t a i n e d d e s c r i b i n g the p r o p a g a t i o n of an e a s t w a r d - t r a v e l l i n g modon modulated by a weak bottom Ekman boundary l a y e r . The r e s u l t s p r e d i c t t h a t the modon r a d i u s and t r a n s l a t i o n speed decay e x p o n e n t i a l l y and t h a t the modon wavenumber i n c r e a s e s e x p o n e n t i a l l y , r e s u l t i n g i n an e x p o n e n t i a l a m p l i t u d e decay i n the s t r e a m f u n c t i o n and v o r t i c i t y . These r e s u l t s agree w i t h the n u m e r i c a l s o l u t i o n of M c W i l l i a m s e t a l . ( l 9 8 l ) . A l e a d i n g o r d e r p e r t u r b a t i o n t h e o r y i s a l s o d e v e l o p e d d e s c r i b i n g modon p r o p a g a t i o n over s l o w l y v a r y i n g topography. N o n l i n e a r h y p e r b o l i c e q u a t i o n s a r e d e r i v e d t o d e s c r i b e the e v o l u t i o n of the s l o w l y v a r y i n g modon r a d i u s , t r a n s l a t i o n speed and wavenumber f o r a r b i t r a r y f i n i t e - a m p l i t u d e topography. To l e a d i n g o r d e r , t h e modon i s u n a f f e c t e d by m e r i d i o n a l g r a d i e n t s i n topography. A n a l y t i c a l p e r t u r b a t i o n s o l u t i o n s f o r the modon r a d i u s , t r a n s l a t i o n speed and wavenumber are o b t a i n e d f o r s m a l l - a m p l i t u d e topography. The p e r t u r b a t i o n s t a k e the form of westward and e a s t w a r d ^ t r a v e l l i n g t r a n s i e n t s and a s t a t i o n a r y component p r o p o r t i o n a l t o the topography. The g e n e r a l s o l u t i o n i s a p p l i e d t o r i d g e - l i k e and e s c a r p m e n t - l i k e t o p o g r a p h i c c o n f i g u r a t i o n s . Table of Contents Abstract i i L i s t of Figures v Acknowledgement v i Chapter I INTRODUCTION 1 Chapter II STABILITY OF BAROTROPIC MODONS 11 2.1 Integral Constraints For Perturbations Of Barotropic Modons ....14 2.2 Discussion And Application To The Atmosphere And Ocean 18 Chapter III MODON PROPAGATION IN A SLOWLY VARYING MEDIUM 29 3.1 Perturbation Solution For Modon Propagation Over A Bottom Ekman Boundary Layer 32 3.1.1 Formulation And Solution Of The Dissipation Problem 33 3.1.2 Discussion Of The Dissipation Solution 42 3.2 Modon Propagation Over Slowly Varying Topography ..55 3.2.1 Perturbation Solution For Modon Propagation Over Slowly Varying Topography 55 3.2.2 Discussion Of The Small-Amplitude Topographic Solution 70 3.2.3 Gaussian-ridge Topography 78 3.2.4 Tanh-escarpment Topography 97 Chapter IV CONCLUSIONS 116 BIBLIOGRAPHY 120 APPENDIX A - CALCULATION OF SOLVABILITY INTEGRALS IN TOPOGRAPHIC PROBLEM 126 V L i s t of Figures 1. Three dimensional contour plot of the surface displacement associated with an eastward-travelling modon 10 2. S t a b i l i t y regime diagram for an eastward-travelling modon 26 3. Two dimensional atmospheric kinetic energy spectrum. .27 4. Scalar wavenumber spectra of barotropic k i n e t i c energy for the ocean 28 5. Sequence of contour plots of the modon pathlines for the Ekman di s s i p a t i o n problem. 45 6. Sequence of contour plots of the modon v o r t i c i t y for the Ekman di s s i p a t i o n problem 50 7. Sequence of space-like s l i c e s in space-time showing the evolution of the modon tra n s l a t i o n speed for modon propagation over a slowly varying gaussian ridge 82 8. Sequence of space-like s l i c e s in space-time showing the evolution of the modon radius for modon propagation over a slowly varying gaussian ridge 87 9. Sequence of space-like s l i c e s in space-time showing the evolution of the modon wavenumber for modon propagation over a slowly varying gaussian ridge 92 10. Sequence of space-like s l i c e s in space-time showing the evolution of the modon tr a n s l a t i o n speed for modon propagation over a slowly varying hyperbolic-tangent escarpment 101 11. Sequence of space-like s l i c e s in space-time showing the evolution of the modon radius for modon propagation over a slowly varying hyperbolic-tangent escarpment 106 12. Sequence of space-like s l i c e s in space-time showing the evolution of the modon wavenumber for modon propagation over a slowly varying hyperbolic-tangent escarpment. 111 Acknowledgement It i s a pleasure to thank Dr. Lawrence A. Mysak for many discussions on applied mathematics and physical oceanography, and for his enthusiastic encouragement of and interest in t h i s research. The author also thanks Dr. Paul LeBlond, Dr. Kevin Hamilton, Dr. Uri Ascher and Dr. Brian Seymour for many helpful discussions pertaining to his research. This research was supported by Natural Sciences and Engineering Research Council of Canada and U.S. Off i c e of Naval Research grants awarded to Dr. Lawrence A. Mysak, and by a U.B.C. Department of Mathematics Teaching Assistantship. Of special note I wish the oceanography beer garden regulars and in pa r t i c u l a r Don Dunbar, Karen - Perry, Keith Thomson and Tom Kessler long l i f e and prosperity. If i t was not for you people the whole thing would not nearly have been as much fun. v i i I have r e s o l v e d t o q u i t o n l y a b s t r a c t geometry, t h a t i s t o say, the c o n s i d e r a t i o n of q u e s t i o n s t h a t s e r v e o n l y t o e x e r c i s e the mind, and t h i s , i n o r d e r t o study a n o t h e r k i n d of geometry, which has f o r i t s o b j e c t the e x p l a n a t i o n of the phenomena of n a t u r e . Rene D e s c a r t e s 1 I . INTRODUCTION Rings and e d d i e s p l a y an i m p o r t a n t r o l e i n the o v e r a l l dynamics of the ocean. F l i e r l ( l 9 7 7 ) has a r g u e d , f o r example, t h a t i n the n o r t h w e s t e r n A t l a n t i c Ocean the obser v e d d i s t r i b u t i o n of f l u c t u a t i o n k i n e t i c energy and a t h e o r e t i c a l e s t i m a t e of t h i s d i s t r i b u t i o n based on q u a s i g e o s t r o p h i c dynamics (QGD) of e d d i e s and t h e i r d i s p e r s i v e wave f i e l d a r e comparable. L a i and Ri c h a r d s o n ( 1 9 7 7 ) e s t i m a t e from h i s t o r i c a l d a t a t h a t the l i f e s p a n of a G u l f Stream eddy i s t y p i c a l l y between 2 t o 3 y e a r s , the average t r a n s l a t i o n speeds a r e on the o r d e r of 5 km d a y " 1 , the r a d i i a r e about 100 km and t h e v e r t i c a l e x t e n t i s up t o one k i l o m e t e r . The number of r i n g s and e d d i e s o b s e r v e d a t any g i v e n time was a p p r o x i m a t e l y 15 w i t h maximum c o u n t s about t w i c e t h i s v a l u e . These o b s e r v a t i o n s s u g g e s t t h a t eddy and r i n g dynamics must be i m p o r t a n t f o r t h e r m a l , v o r t i c i t y and n u t r i e n t m i x i n g i n the ocean on mesoscales ( i . e . , l e n g t h s c a l e s on the o r d e r of 100 km). E d d i e s and r i n g s can be g e n e r a t e d from a v a r i e t y of f l o w -geometry c o n f i g u r a t i o n s . O b s e r v a t i o n s of e d d i e s have been t h e o r e t i c a l l y d e s c r i b e d by c o a s t a l c u r r e n t i n s t a b i l i t y (Ikeda e t a l . , 1984), topographic-mean f l o w i n t e r a c t i o n s (Swaters and Mysak, 1985), p l a n e t a r y wave r e f l e c t i o n o f f c o a s t a l geometry ( W i l l m o t t and Mysak, 1980) and the meandering of western boundary c u r r e n t s ( L a i and- R i c h a r d s o n , 1977 and Csanady, 1979). In a d d i t i o n t o t h e above mechanisms, p l a n e t a r y e d d i e s can a l s o be o b t a i n e d when t h e e f f e c t s of phase d i s p e r s i o n 2 b a l a n c e a m p l i t u d e d i s p e r s i o n i n p l a n e t a r y waves ( C l a r k e , 1971) t o produce the s o l i t a r y p l a n e t a r y waves. M a l a n o t t e - R i z z o l i ( 1 9 8 2 ) c l a s s i f i e d models of a t m o s p h e r i c and o c e a n i c s o l i t a r y p l a n e t a r y waves i n t o one of two c a t e g o r i e s a c c o r d i n g t o the n o n l i n e a r i t y of the p o t e n t i a l v o r t i c i t y e q u a t i o n (PVE). S t e a d i l y t r a n s l a t i n g s o l u t i o n s of the PVE reduce t o d e s c r i b i n g the p o t e n t i a l v o r t i c i t y as a f u n c t i o n (P) of the p a t h l i n e s . Rossby s o l i t o n s and s o l i t a r y waves are d e s c r i b e d by assuming P t o be a n a l y t i c . Examples of t h i s approach i n c l u d e the Maxworthy and Redekopp(1976) d e r i v a t i o n of Korteweg-De V r i e s (KdV) and m o d i f i e d - K d V e q u a t i o n s from QGD i n z o n a l c h a n n e l models w i t h a sheared z o n a l f l o w . S i m i l a r d e r i v a t i o n s by M a l a n o t t e - R i z z o l i and H e n d e r s h o t t ( 1 9 8 0 ) and M a l a n o t t e - R i z z o l i ( 1 9 8 4 ) w i t h c r o s s - c h a n n e l t o p o g r a p h i c v a r i a t i o n a l s o r e s u l t e d i n a KdV e q u a t i o n and hence a s o l i t a r y wave s o l u t i o n . Also,• F l i e r K 1 9 7 9 ) has o b t a i n e d r a d i a l l y symmetric q u a s i g e o s t r o p h i c s o l i t a r y wave s o l u t i o n s on an i n f i n i t e /3-plane assuming P t o be a n a l y t i c . The second c a t e g o r y i s t o s p e c i f y P as a n o n a n a l y t i c f u n c t i o n . The s o l u t i o n s t h a t t h i s p r o c e d u r e y i e l d s , h e r e a f t e r c a l l e d modons, have the p r o p e r t y t h a t t h e s t r e a m f u n c t i o n i s o n l y d i f f e r e n t i a b l e t o some f i n i t e o r d e r ( u s u a l l y two). S t e r n ( l 9 7 5 ) o b t a i n e d t h e p r o t o t y p e modon s o l u t i o n (and named them as such) by assuming P t o be a l i n e a r f u n c t i o n i n a bounded c i r c u l a r domain on the /3-plane w i t h the streamf u n c t i o n v a n i s h i n g i d e n t i c a l l y o u t s i d e t h i s r e g i o n . These s o l u t i o n s were c o n t i n u o u s but not d i f f e r e n t i a b l e a t the boundary and had a 3 d i p o l e v o r t e x s t r u c t u r e . L a r i c h e v and R e z n i k ( l 9 7 6 ) o b t a i n e d what i s now c a l l e d the b a r o t r o p i c modon by d e t e r m i n i n g a form f o r P i n the e x t e r i o r r e g i o n when the s t r e a m f u n c t i o n v a n i s h e s a t i n f i n i t y (see F i g u r e 1 f o r a t h r e e - d i m e n s i o n a l p l o t of a modon). These s o l u t i o n s have smooth v o r t i c i t y everywhere e x c e p t a t the boundary where the v o r t i c i t y has a f i n i t e s t e p d i s c o n t i n u i t y i n i t s r a d i a l d e r i v a t i v e . These s o l u t i o n s have been s u b s e q u e n t l y g e n e r a l i z e d t o t w o - l a y e r f l u i d s ( F l i e r l e t a l . , 1980) and s p h e r i c a l geometry ( T r i b b i a , 1984 and V e r k l e y , 1984). Kloeden(1985a,1985b) c l a i m s t o have e s t a b l i s h e d a theorem t h a t s t a t e s t h a t the modon i s the unique l o c a l i z e d s e p a r a b l e s o l i t a r y wave s o l u t i o n of the PVE assuming t h a t P i s p i e c e w i s e l i n e a r . S t u d i e s of modon dynamics have p r i m a r i l y f o c u s s e d on n u m e r i c a l i n t e g r a t i o n s of t h e PVE. For example, M c W i l l i a m s e t a l . ( l 9 8 l ) n u m e r i c a l l y c a l c u l a t e d the e f f e c t of Ekman d i s s i p a t i o n on modon p r o p a g a t i o n and c o n c l u d e d t h a t the decay was a p p r o x i m a t e l y e x p o n e n t i a l and shape p r e s e r v i n g . The d e t a i l s of the decay i n the a m p l i t u d e and t r a n s l a t i o n speed c o u l d not be e x p l i c i t l y d e t e r m i n e d e x c e p t t h a t they appeared t o occ u r on the modon " d i s p e r s i o n c u r v e " . Other n u m e r i c a l e x p e r i m e n t s w i t h random v o r t i c i t y p e r t u r b a t i o n s i n d i c a t e d t h a t the o n set of i n s t a b i l i t y was d e t e r m i n e d by the l e n g t h s c a l e of the p e r t u r b a t i o n s . M c W i l l i a m s and Zabusky(1982) n u m e r i c a l l y s i m u l a t e d c o l l i s i o n s between b a r o t r o p i c modons. Depending on the parameter v a l u e s , a wide range of i n t e r a c t i o n p o s s i b i l i t i e s were 4 o b s e r v e d , from s o l i t o n - l i k e , f u s i o n - l i k e and f i s s i o n - l i k e i n t e r a c t i o n s t o the complete a n n i h i l a t i o n of the v o r t i c e s . The m u l t i t u d e of i n t e r a c t i o n p o s s i b i l i t i e s r e s u l t e d i n some debate as t o whether or not the modon i s i n f a c t a t w o - d i m e n s i o n a l s o l i t o n ( F l i e r l e t a l . , 1980 and M c W i l l i a m s , 1980). No t h e o r e t i c a l framework has y e t been d e v e l o p e d from which t o u n d e r s t a n d t h e s e i n t e r a c t i o n s . Mied and Lindemann(1982) and M c W i l l i a m s ( 1 9 8 3 ) have c o n s i d e r e d the problem of modon g e n e s i s . I t had been suggested ( F l i e r l , 1976) t h a t a pure b a r o c l i n i c eddy would n a t u r a l l y tend t o d e v e l o p i n t o a b a r o t r o p i c modon. Mied and Lindemann(1982) s u b s e q u e n t l y n u m e r i c a l l y c a l c u l a t e d the e v o l u t i o n of a b a r o c l i n i c eddy w i t h a t i l t e d v e r t i c a l a x i s and c o u n t e r - r o t a t i n g upper and lower l a y e r s . T h e i r r e s u l t s i n d i c a t e d t h a t when the v e r t i c a l a x i s l i e s i n the n o r t h - s o u t h p l a n e modon g e n e s i s ensues u n l e s s the h o r i z o n t a l s e p a r a t i o n between the c e n t e r s of the v o r t i c e s i s t o o l a r g e . I f the v e r t i c a l a x i s l a y i n o t h e r d i r e c t i o n s the two v o r t i c e s tended t o s e p a r a t e w i t h no c o u p l i n g t a k i n g p l a c e . McWi11iams(1983) was a l s o a b l e t o c r e a t e modon-l i k e d i p o l e s by the c o l l i s i o n of two q u a s i g e o s t r o p h i c v o r t i c e s . L a b o r a t o r y s t u d i e s of modon g e n e s i s ( F l i e r l e t a l . , 1983) i n d i c a t e d t h a t d i p o l e v o r t e x f o r m a t i o n l i k e l y e v o l v e s from v e r y g e n e r a l i n i t i a l c o n d i t i o n s . T h i s c o n c l u s i o n was f o r m u l a t e d as a theorem ( F l i e r l e t a l . , 1983) which s t a t e s t h a t any s l o w l y v a r y i n g i s o l a t e d d i s t u r b a n c e on a /3-plane must have z e r o net a n g u l a r momentum. The modon i s one of the s i m p l e s t n o n t r i v i a l f l o w c o n f i g u r a t i o n s w i t h t h i s p r o p e r t y . 5 In summary, p r e v i o u s r e s e a r c h on modon dynamics has i n d i c a t e d the f o l l o w i n g r e s u l t s . Modon s t a b i l i t y i s dependent on the s t r u c t u r e of the p e r t u r b a t i o n f i e l d . Modon g e n e s i s i s c o n j e c t u r e d t o e v o l v e from r a t h e r g e n e r a l i n i t i a l c o n d i t i o n s and i s one of t h e s i m p l e s t r e a l i z a t i o n s of i s o l a t e d s t e a d i l y t r a n s l a t i n g f l u i d motions on a /3-plane. Modons i n a d i s s i p a t i v e environment appear t o p r e s e r v e t h e i r shape, a t l e a s t i n i t i a l l y . Modon-modon i n t e r a c t i o n s appear t o have many p o s s i b i l i t i e s , some of' which a r e s o l i t o n - l i k e o t h e r s n o t . However, s e v e r a l q u e s t i o n s i n modon dynamics remain t o be answered b e f o r e we can u n d e r s t a n d the r o l e of modons t o a t m o s p h e r i c and o c e a n i c dynamics. No t h e o r e t i c a l or a n a l y t i c a l framework has y e t been d e v e l o p e d which can be used t o d e s c r i b e the above n u m e r i c a l r e s u l t s . Such a framework would seem t o be r e q u i r e d i f modon dynamics i s t o be u n d e r s t o o d i n terms of known g e o p h y s i c a l f l u i d dynamic mechanisms. In p a r t i c u l a r , n o t h i n g i s known about b a s i c problems such as how modons might i n t e r a c t w i t h t h e i r e n vironment. For example, the i n t e r a c t i o n of modons w i t h c u r r e n t s and Rossby waves have y e t t o be s t u d i e d . I n a d d i t i o n , q u e s t i o n s r e l a t i n g t o the s t a b i l i t y of modons a r e unanswered. F or example, t h e s p e c i f i c a t i o n of s t a b i l i t y c o n d i t i o n s and the s o l u t i o n of the l i n e a r i z e d s t a b i l i t y problem a r e unknown. T h i s t h e s i s examines two a s p e c t s of b a r o t r o p i c modon dynamics. The f i r s t a s p e c t i s examined i n Chapter I I , i n which s u f f i c i e n t n e u t r a l s t a b i l i t y c o n d i t i o n s a re d e r i v e d f o r normal-mode s m a l l - a m p l i t u d e p e r t u r b a t i o n s of westward and e a s t w a r d -6 t r a v e l l i n g modons in the form of an integral constraint. These conditions are derived from the s p a t i a l l y - i n t e g r a t e d perturbation energy and enstrophy equations (Charney and F l i e r l ( l 9 8 l ) and M a l a n o t t e - R i z z o l i ( 1 9 8 2 ) ) for steadily translating quasigeostrophic f l u i d motions. It i s shown that eastward-travelling modons are neutrally stable to normal-mode perturbations that are solely composed of spectral components with wavelength magnitudes larger than 2 K / K where K i s the modon wavenumber (see Chapter 2 ) . Westward-travelling modons are neutrally stable to normal-mode perturbations s o l e l y composed of spectral components with wavelengths smaller than 2TI/K. These results imply that when K / | T J | > 1 , where |T?| i s the magnitude of the perturbation wavenumber vector TJ= (77,, TJ 2 ) , the slope o f the neutral s t a b i l i t y curve proposed by McWilliams et a l . ( l 9 8 l ) should begin to increase as | rj \ decreases. A similar trend in the neutral s t a b i l i t y curve has been numerically determined for topographically-forced planetary s o l i t a r y waves (Malanotte-Rizzoli, 1 9 8 2 ) . The neutral s t a b i l i t y i n t e g r a l i s tested using observed eddy energy spectra for the atmosphere and ocean. For the atmospheric c a l c u l a t i o n , seasonally- and annually-averaged mid-latitude 3 0 0 , 5 0 0 and 7 0 0 mb two-dimensional wavenumber eddy energy spectra were inferred from Tomatsu(1979), Saltzman and F l e i s h e r ( 1 9 6 2 ) and Eliasen and Machenhauer (1965). It i s argued on the basis of the calculations contained in t h i s thesis that eastward-travelling atmospheric modons are neutrally stable to the observed seasonally- and annually-averaged fluctuations in 7 the atmosphere. W e s t w a r d - t r a v e l l i n g a t m o s p h e r i c modons may not s a t i s f y the n e u t r a l s t a b i l i t y c o n d i t i o n and t h u s t h e i r s t a b i l i t y or i n s t a b i l i t y cannot be i n f e r r e d . The o n l y p u b l i s h e d b a r o t r o p i c wavenumber eddy energy spectrum a v a i l a b l e f o r the oceans i s c o n t a i n e d i n F U ( 1 9 8 3 ) . Simple s c a l i n g arguments a r e p r e s e n t e d t o suggest t h a t o n l y e a s t w a r d - t r a v e l l i n g b a r o t r o p i c modons w i l l have r e a l i s t i c t r a n s l a t i o n and p a r t i c l e speeds i n the ocean. The s t a b i l i t y of e a s t w a r d - t r a v e l l i n g o c e a n i c modons cannot be i n f e r r e d from the F U ( 1 9 8 3 ) spectrum. Arguments a r e p r e s e n t e d s u g g e s t i n g t h a t t h i s c o n c l u s i o n s h o u l d o n l y be c o n s i d e r e d a v e r y p r e l i m i n a r y e s t i m a t e of o c e a n i c modon s t a b i l i t y due t o the l i m i t a t i o n s of the F U ( 1 9 8 3 ) spectrum. The second a s p e c t of modon dynamics examined i n t h i s t h e s i s i s c o n t a i n e d i n Chapter I I I , i n which a p e r t u r b a t i o n t h e o r y i s proposed t o d e s c r i b e modon p r o p a g a t i o n i n a s l o w l y v a r y i n g medium. The p e r t u r b a t i o n methods de v e l o p e d a r e t w o - d i m e n s i o n a l g e n e r a l i z a t i o n s of o n e - d i m e n s i o n a l s l o w l y v a r y i n g s o l i t a r y wave c a l c u l a t i o n s ( e . g . , L u k e ( l 9 6 6 ) , Grimshaw(1970, 1971, 1977, 1979a,b, 1981), Zakharov and Rubenchik(1974) and Kodama and A b l o w i t z ( 1 9 8 0 , 1 9 8 1 ) , among o t h e r s ) . T h i s c a l c u l a t i o n r e p r e s e n t s the f i r s t a p p l i c a t i o n of t h e s e methods t o a f u l l y two-d i m e n s i o n a l s o l i t a r y wave. Two problems a r e posed and s o l v e d . In S e c t i o n 3.1 a p e r t u r b a t i o n t h e o r y d e s c r i b i n g the p r o p a g a t i o n of an e a s t w a r d -t r a v e l l i n g modon over a weak Ekman bottom boundary l a y e r i s d e v e l o p e d . T h i s c a l c u l a t i o n was done i n o r d e r t o compare the 8 r e s u l t of the l e a d i n g order s o l u t i o n with the numerical s o l u t i o n of McWilliams et a l . d 9 8 l ) . The numerical s o l u t i o n suggested that the modon r a d i u s and t r a n s l a t i o n speed decay e x p o n e n t i a l l y and the modon wavenumber i n c r e a s e s e x p o n e n t i a l l y . Throughout the decay the modon parameters (to l e a d i n g order) s a t i s f i e d the modon d i s p e r s i o n r e l a t i o n s h i p . In the f i n a l stages of the decay the modon d i s s o l v e d i n t o a f i e l d of w e s t w a r d - t r a v e l l i n g p l a n a r Rossby waves. The l e a d i n g order p e r t u r b a t i o n s o l u t i o n o b t a i n e d here agrees with these r e s u l t s but i s unable t o d e s c r i b e the f i n a l d egeneration i n t o the Rossby waves. T h i s c a l c u l a t i o n has been summarized i n Swaters(1985). In S e c t i o n 3.2 a l e a d i n g order p e r t u r b a t i o n theory i s developed to d e s c r i b e modon propagation over slowly v a r y i n g topography. As i n previous work on the e f f e c t s of v a r i a b l e topography on b a r o t r o p i c p l a n e t a r y waves (e.g., V e r o n i s ( 1 9 6 6 ) , Rhines( 1969a,b), C l a r k e d 9 7 l ) and LeBlond and Mysak(l978; Sec. 20), among others) the theory i s developed i n the context of the r i g i d - l i d shallow water equations on the | 3 -p l a n e . N o n l i n e a r h y p e r b o l i c equations are d e r i v e d f o r the slow e v o l u t i o n of the l e a d i n g order modon r a d i u s , t r a n s l a t i o n speed and wavenumber. These equations are v a l i d f o r a r b i t r a r y slowly v a r y i n g f i n i t e - a m p l i t u d e topography. In general they must be sol v e d n u m e r i c a l l y . I t i s shown t h a t to l e a d i n g order the e v o l u t i o n of the modon i s independent of the m e r i d i o n a l (North-South) topographic s t r u c t u r e . T h i s r e s u l t i s i n t e r p r e t e d i n 9 terms of s i m p l e v o r t i c i t y arguments. A n a l y t i c a l p e r t u r b a t i o n s o l u t i o n s a r e o b t a i n e d f o r s m a l l -a m p l i t u d e topography ( r e l a t i v e t o t h e f l u i d d e p t h , w h i c h i s the case i n many a t m o s p h e r i c and o c e a n i c a p p l i c a t i o n s ) . The p e r t u r b a t i o n s t a k e the form of e a s t w a r d and w e s t w a r d - t r a v e l l i n g h y p e r b o l i c t r a n s i e n t s and a s t a t i o n a r y component p r o p o r t i o n a l t o the topography. The g e n e r a l p r o p e r t i e s of the s o l u t i o n a re d e s c r i b e d i n S u b s e c t i o n 3.2.2. S u b s e c t i o n s 3.2.3 and 3.2.4 d e s c r i b e t h e s l o w l y v a r y i n g modon f o r t h e s p e c i f i c examples of a t o p o g r a p h i c r i d g e and escarpment, r e s p e c t i v e l y . The work c o n t a i n e d i n t h i s t h e s i s i s summarized i n Chapter IV. In the Appendix d e t a i l s of the many i n t e g r a l c a l c u l a t i o n s r e q u i r e d i n the t o p o g r a p h i c s o l u t i o n a r e g i v e n . F i g u r e 1. Three d i m e n s i o n a l c o n t o u r p l o t of the s u r f a c e d i s p l a c e m e n t a s s o c i a t e d w i t h an e a s t w a r d - t r a v e l l i n g modon. For atmosphe r i c s c a l e s , the h o r i z o n t a l d i s t a n c e between the maximum and minimum d e f l e c t i o n i s about 800 km w i t h a d e f l e c t i o n of the g e o p o t e n t i a l i s on the order of 100 m. For o c e a n i c s c a l e s , the h o r i z o n t a l d i s t a n c e between the the maximum and minimum d e f l e c t i o n i s about 80 km w i t h a d e f l e c t i o n on the o r d e r of 10 cm. The c o o r d i n a t e system i s r o t a t i n g w i t h n o n d i m e n s i o n a l a n g u l a r v e l o c i t y 1+e5y where e i s the Rossby number c ( f a ) " 1 and 8 i s the p l a n e t a r y v o r t i c i t y f a c t o r 0a2c~1 w i t h a, c, f and 0 the modon r a d i u s , modon t r a n s l a t i o n speed, l o c a l C o r i o l i s parameter and northward g r a d i e n t i n the C o r i o l i s p a r a m e t e r , r e s p e c t i v e l y . 11 I I . STABILITY OF BAROTROPIC MODONS M c W i l l i a m s ( 1 9 8 0 ) m o d e l l e d an ob s e r v e d a t m o s p h e r i c b l o c k i n g event w i t h a b a r o t r o p i c modon. Among s e v e r a l q u e s t i o n s posed was whether or not the modon s o l u t i o n i s s t a b l e f o r t y p i c a l l y o b s e r v e d a t m o s p h e r i c f l u c t u a t i o n s . S u b s e q u e n t l y , M c W i l l i a m s e t a l . ( 1 9 8 1 ) n u m e r i c a l l y examined the s t a b i l i t y of e a s t w a r d - t r a v e l l i n g modons when p e r t u r b e d by a random v o r t i c i t y f i e l d . T h e i r r e s u l t s i n d i c a t e t h a t f o r a g i v e n v o r t i c i t y p e r t u r b a t i o n a m p l i t u d e ( e ) , i n c r e a s i n g the p e r t u r b a t i o n w a velength l e a d s t o i n s t a b i l i t y ; a l s o , f o r a g i v e n p e r t u r b a t i o n w a v e l e n g t h , i n c r e a s i n g e l e a d s t o i n s t a b i l i t y . T h i s c h a p t e r d e s c r i b e s s u f f i c i e n t n e u t r a l s t a b i l i t y c o n d i t i o n s f o r westward and e a s t w a r d - t r a v e l l i n g modons. M o d i f i c a t i o n s t o the regime ' diagram proposed by M c W i l l i a m s e t a l . ( l 9 8 l ) a r e d i s c u s s e d i n l i g h t of t h e s e r e s u l t s . The s t a b i l i t y c o n d i t i o n i s c a l c u l a t e d w i t h t y p i c a l a t m o s p h e r i c and o c e a n i c energy s p e c t r a . I t i s shown t h a t a t m o s p h e r i c e a s t w a r d - t r a v e l l i n g b a r o t r o p i c modons a r e n e u t r a l l y s t a b l e t o t h e observed s e a s o n a l l y - and a n n u a l l y - a v e r a g e d 3 0 0 , 500 and 7 0 0 mb t r a n s i e n t e d d i e s . A s i m i l a r s t a b i l i t y c a l c u l a t i o n i s unable t o det e r m i n e the s t a b i l i t y of w e s t w a r d - t r a v e l l i n g a t m o s p h e r i c modons. A n e u t r a l s t a b i l i t y c a l c u l a t i o n i s a l s o done f o r an o c e a n i c b a r o t r o p i c k i n e t i c energy spectrum. On the b a s i s of t h i s c a l c u l a t i o n the s t a b i l i t y or i n s t a b i l i t y of o c e a n i c modons cannot be i n f e r r e d . Charney and F l i e r M 1 981 ) d e r i v e d a s t a b i l i t y theorem f o r 12 normal-mode i n f i n i t e s i m a l p e r t u r b a t i o n s of stea d y q u a s i g e o s t r o p h i c f l o w based on the c o n s e r v a t i o n of energy and e n s t r o p h y ( v o r t i c i t y s q u a r e d ) . T h i s method g i v e s s t a b i l i t y c o n d i t i o n s s i m i l a r t o t h e Blumen(l968) f i n i t e - a m p l i t u d e r e s u l t based on e s t a b l i s h i n g s u f f i c i e n t c o n d i t i o n s f o r which the mean f l o w i s a s t a b l e extremum of a s u i t a b l y c o n s t r u c t e d ( A r n o l ' d , 1965) e n e r g y - e n s t r o p h y f u n c t i o n a l . S u f f i c i e n t n e u t r a l s t a b i l i t y c o n d i t i o n s f o r b a r o t r o p i c modons a r e d e r i v e d by s i m i l a r methods. E a s t w a r d - t r a v e l l i n g modons a r e n e u t r a l l y s t a b l e t o p e r t u r b a t i o n s s o l e l y composed of s p e c t r a l components w i t h wavenumbers 17 = ( T J 1 , T J 2 ) s a t i s f y i n g |??|<K where K i s the modon wavenumber ( i . e . , t h e parameter d e s c r i b i n g the f u n c t i o n a l dependence of the p a t h l i n e s on the p o t e n t i a l v o r t i c i t y .in the modon i n t e r i o r ) . W e s t w a r d - t r a v e l l i n g modons a r e n e u t r a l l y s t a b l e t o p e r t u r b a t i o n s s o l e l y composed w i t h s p e c t r a l components w i t h wavenumbers s a t i s f y i n g |T?|>K. For e a s t w a r d - t r a v e l l i n g modons, the n e u t r a l s t a b i l i t y c o n d i t i o n i m p l i e s t h a t when e i s f i n i t e (though p o s s i b l y o n l y ' s m a l l ' ) and the dominant p e r t u r b a t i o n s p e c t r a l component c o n s i s t s of s c a l e s K/|TJ|>1 the modon i s n e u t r a l l y s t a b l e . Thus when K/|TJ|>1 the s l o p e of t h e n e u t r a l s t a b i l i t y c u r v e proposed by M c W i l l i a m s e t a l . ( l 9 8 l ) f o r e a s t w a r d - t r a v e l l i n g modons s h o u l d b e g i n t o i n c r e a s e as K/|T?| i n c r e a s e s . T h i s p r o p e r t y i n the n e u t r a l s t a b i l i t y c u r v e has been n u m e r i c a l l y d e t e r m i n e d f o r t o p o g r a p h i c a l l y - f o r c e d s o l i t a r y p l a n e t a r y e d d i e s ( M a l a n o t t e -R i z z o l i , 1982). 13 In r e a l i t y , a modon would be s u b j e c t e d t o p e r t u r b a t i o n s d e s c r i b e d by a spectrum of wavenumbers. The d e r i v e d s t a b i l i t y c o n d i t i o n s u g g e s t s ( f o r a modon r a d i u s on the o r d e r of the e x t e r n a l d e f o r m a t i o n r a d i u s ) t h a t an a t m o s p h e r i c westward-t r a v e l l i n g modon i s n e u t r a l l y s t a b l e when the eddy energy of the s u r r o u n d i n g f l u i d i s c o n t a i n e d m a i n l y i n g l o b a l wavenumbers 1 w i t h magnitudes g r e a t e r than a p p r o x i m a t e l y 16. However, e a s t w a r d - t r a v e l l i n g modons a r e n e u t r a l l y s t a b l e i n f l o w regimes i n which the energy i s c o n t a i n e d m a i n l y i n wavenumbers w i t h magnitudes l e s s than a p p r o x i m a t e l y 16. C o n s e q u e n t l y e a s t w a r d -t r a v e l l i n g modons ar e n e u t r a l l y s t a b l e f o r the o b s e r v e d mid-l a t i t u d e eddy e n e r g e t i c s because t y p i c a l o b s e r v a t i o n s show |TJ| - 8 (see F i g u r e 3) ( E l i a s e n and Machenhauer, 1965; Saltzman and F l e i s h e r , 1962 and Tomatsu, 1979). The o n l y p u b l i s h e d b a r o t r o p i c wavenumber eddy energy spectrum a v a i l a b l e f o r t h e oceans i s c o n t a i n e d i n F u ( l 9 8 3 ) . S i m p le s c a l i n g arguments a r e p r e s e n t e d t o suggest t h a t o n l y e a s t w a r d - t r a v e l l i n g b a r o t r o p i c modons w i l l have r e a l i s t i c t r a n s l a t i o n and p a r t i c l e speeds i n the ocean. The n e u t r a l s t a b i l i t y c o n d i t i o n s u g g e s t s t h a t o c e a n i c e a s t w a r d - t r a v e l l i n g modons w i l l be s t a b l e i f the eddy energy of the s u r r o u n d i n g f l u i d i s c o n t a i n e d m a i n l y i n wavelengths g r e a t e r than about 160 km. The n e u t r a l s t a b i l i t y of e a s t w a r d - t r a v e l l i n g o c e a n i c modons cannot be i n f e r r e d from the F U ( 1 9 8 3 ) spectrum. I t i s 1 The g l o b a l wavenumber i s d e f i n e d so t h a t a wavenumber magnitude of one c o r r e s p o n d s t o a wavelength e q u a l l i n g the c i r c u m f e r e n c e around a l a t i t u d e c i r c l e . 14 noted t h a t t h i s c a l c u l a t i o n s h o u l d o n l y be c o n s i d e r e d a very p r e l i m i n a r y e s t i m a t e of o c e a n i c modon s t a b i l i t y due t o the l i m i t a t i o n s of the F U ( 1 9 8 3 ) spectrum. 2.1 I n t e g r a l C o n s t r a i n t s F o r P e r t u r b a t i o n s Of B a r o t r o p i c Modons Co n s i d e r the no n d i m e n s i o n a l b a r o t r o p i c p o t e n t i a l v o r t i c i t y e q u a t i o n ( P e d l o s k y , 1979) (A - F)i// + J U + y, Ai// - Fxfj + 6y) = 0 (2.1) t where \}/ i s t h e g e o s t r o p h i c p r e s s u r e , J(«,*) i s the J a c o b i a n d e t e r m i n a n t 3 ( • ,* ) / b(£ ,y) and A=3 + 9 where £ i s the U yy t r a n s l a t e d c o o r d i n a t e £ = x - t w i t h x, y and t the u s u a l e a s t , n o r t h and time c o o r d i n a t e s . The c o e f f i c i e n t s 6 = /3a 2/c and F = f 2 a 2 / ( g H ) a r e the p l a n e t a r y v o r t i c i t y f a c t o r and r o t a t i o n a l Froude number r e s p e c t i v e l y , w i t h f , 0, g, H, a and c the l o c a l C o r i o l i s p a r a meter, n o r t h w a r d g r a d i e n t of the C o r i o l i s p arameter, g r a v i t a t i o n a l a c c e l e r a t i o n , f l u i d d e p t h , modon r a d i u s and modon t r a n s l a t i o n speed r e s p e c t i v e l y . The l e n g t h , time and speed s c a l i n g s a r e a, a/c and c, r e s p e c t i v e l y . We note t h a t c may be p o s i t i v e or n e g a t i v e . Steady s o l u t i o n s of (2.1) have the v o r t i c i t y A\p - F\[/ + 6y ex p r e s s e d as a f u n c t i o n of the p a t h l i n e s \J/ + y, v i z . P(i// + y ) . For the b a r o t r o p i c modon P i s d e f i n e d by ( F l i e r l e t a l . , 1980) P(z ) = 8z f o r r>1 (2.2a) 15 P ( z ) = -U2 + F ) z f o r r<1, (2.2b) r e s u l t i n g i n </  = - K , [ ( 6 + F) 1 / 2 r ] s i n ( c 9 ) / K j (6 + F ) 1 / 2 ] r>1 (2.3a) xP = (5 + F ) J , ( K r ) s i n ( 6 ) / ( K 2 J 1 ( K ) ) -U2 + F + 5)K-2rsin(t9) r<1 (2.3b) where r 2 = £ 2 + y 2 , t a n ( 0 ) = y/£ and where J , and K t are the o r d i n a r y and m o d i f i e d B e s s e l f u n c t i o n s of o r d e r one r e s p e c t i v e l y (see F i g u r e 1 ) . The parameter K ( h e n c e f o r t h c a l l e d the modon wavenumber) i s det e r m i n e d by r e q u i r i n g c o n t i n u i t y of V>// on r=l and i s the f i r s t nonzero s o l u t i o n of the modon ' d i s p e r s i o n ' r e l a t i o n -(6 + F ) 1 / 2 J 2 U ) K 1 [ ( 6 + F ) 1 / 2 ] = K J , ( I C ) K 2 [ ( 6 + F ) l / 2 ] . (2.3c) I t t u r n s out (see F l i e r l e t a l . , 1980) t h a t K i s a s l o w l y v a r y i n g f u n c t i o n of 6 + F ( i n p a r t i c u l a r 6 + F = 0 ( 1 ) - > K = < 4 ) . To o b t a i n the s t a b i l i t y c o n d i t i o n c o n s i d e r \p = * + exp(ot ) i / / ' (x,y) w i t h l ^ ' l ^ l * ! where * i s the modon s o l u t i o n ( 2 . 3 ) . S u b s t i t u t i n g i n t o ( 2 . 1 ) , l i n e a r i z i n g and e x p l o i t i n g (2.2) r e s u l t s i n t h e e i g e n v a l u e problem a(A - F)\p + U 0 * V [ ( _ - F - P)i//] = 0 (2.4) 16 where t h e prime has been dropped, P = d P ( z ) / d z and U 0 = (-• -y 1, • ). E q u a t i o n (2.4) i s d e f i n e d on the d i s c o n n e c t e d open i n t e r v a l s 0<r<1 and 1<r<" w i t h P o b t a i n e d from ( 2 . 2 ) . On r=1 \p and V\// a r e assumed t o be c o n t i n u o u s . I t f o l l o w s from (2.4) t h a t the s p a t i a l l y i n t e g r a t e d energy and en s t r o p h y e q u a t i o n s a r e , r e s p e c t i v e l y 7T 0 0 (a + a*) J f \V4>\2 + F|(//|2 r d r d f l = -TT 0 It » - J / (U 0-Vi//*)A^ + ( U o - V ^ ) ^ * r d r d f l , (2.5) It 0° (a + a*) J S - F^| 2/P rdrdc9 = -it 0 It oo J / ( U Q - V I / / * ) ^ + (U 0 -V<//)A^* r d r d f l , (2.6) -7T 0 ~ where i s the complex c o n j u g a t e of ^ . E q u a t i o n s (2.5) and (2.6) have been d e r i v e d by o b t a i n i n g energy and e n s t r o p h y e q u a t i o n s i n the e x t e r i o r ( r > l ) and i n t e r i o r (r<1) and ad d i n g the r e s u l t s t o g e t h e r . In t h i s d e r i v a t i o n c e r t a i n i n t e g r a t i o n by p a r t s a r e r e q u i r e d which r e s u l t i n boundary i n t e g r a l s on r=1. These terms a r e e i t h e r i d e n t i c a l l y z e r o s i n c e U o*n=0 on r=1 where n i s the u n i t normal on r=1 or sum t o z e r o when the e x t e r i o r and i n t e r i o r i n t e g r a l s 17 a r e added t o g e t h e r (see Charney and F l i e r l ( l 9 8 l ) and M a l a n o t t e -R i z z o l i ( 1982)). The a d d i t i o n of (2.5) and (2.6) g i v e s (a + a * ) J J {\V\f/\2 +F|<//|2 + |Av// - 1 2/P} r d r d f l = 0. (2.7) - it 0 I f the i n t e g r a l i n (2.7) i s nonzero then the Re(a)=0 and the modon i s n e u t r a l l y s t a b l e ( D r a z i n and R e i d , 1981). Thus a s u f f i c i e n t c o n d i t i o n f o r the n e u t r a l s t a b i l i t y of b a r o t r o p i c modons i s t h a t the i n t e g r a l i n (2.7) be nonzero. I t i s noted here t h a t f o r i n s t a b i l i t y or f o r a s y m p t o t i c s t a b i l i t y the i n t e g r a l must be i d e n t i c a l l y z e r o ( s i n c e i n e i t h e r case Re(a) i s n o n z e r o ) . The c o n d i t i o n t h a t the i n t e g r a l i n (2.7) i s z e r o can be r e a r r a n g e d t o g i v e it °= J / (|V^/|2 + F|<//|2 " \W ~ F i ^ l V U 2 + F)} rdrdt? = - I , / 6 (2.8) -it 0 It °° S S { I V i ^ I 2 + F|^| 2 + |A<J/ - Fi / / j 2/6) } r d r d S = I 2 / 6 (2.9) -it 0 w i t h it 00 I , - (6 + F + K 2 ) U 2 + F ) " 1 / / - F^| 2 r d r d f l £ 0 -it 1 * 1 1 2 = (5 + F + K 2 ) ( K 2 + F ) " 1 / / |Vtf - F i / / | 2 rdrdt9 £ 0, -it 0 18 s i n c e 6 + F>0 f o r s o l u t i o n s of the form ( 2 . 3 ) . E q u a t i o n s (2.8) and (2.9) can be r e w r i t t e n J |tfT|*U2 - | T J | 2 ) ( | T , | 2 + F) drj = - 4 T T 2 ( K 2 + F ) l , / 6 (2.10) / I ^ I M |r?| 2 + F ) ( 5 + F + | r j | 2 ) dr? = 4 T T 2 I 2 (2.11) due t o P a r s e v a l ' s e q u a l i t y , where \pT i s the F o u r i e r t r a n s f o r m of and r] the wave number v e c t o r ( T ? , , T J 2 ) . E q u a t i o n (2.10) forms the b a s i s f o r the re m a i n i n g a n a l y s i s . 2.2 D i s c u s s i o n And A p p l i c a t i o n To The Atmosphere And Ocean The i n t e g r a l (2.10) p r o v i d e s s u f f i c i e n t c o n d i t i o n s on the wavenumber spectrum of the p e r t u r b a t i o n f i e l d i f the modon i s t o be n e u t r a l l y s t a b l e . C o n s i d e r the case c < 0 ( i . e . , a westward-t r a v e l l i n g modon). I t f o l l o w s from (2.10) t h a t i f the p e r t u r b a t i o n i s s o l e l y composed of wavenumbers s a t i s f y i n g |TJ| > K the LHS and RHS of (2.10) a r e of d i f f e r e n t s i g n ( s i n c e 5<0). T h e r e f o r e the i n t e g r a l i n (2.7) cannot be z e r o and the w e s t w a r d - t r a v e l l i n g modon must be n e u t r a l l y s t a b l e . E q u a t i o n (2.11) p l a c e s no c o n s t r a i n t on the p e r t u r b a t i o n wavenumbers s i n c e b o t h the LHS and RHS a r e n o n n e g a t i v e . T h i s s t a b i l i t y c o n d i t i o n s u g g e s t s t h a t w e s t w a r d - t r a v e l l i n g modons a r e n e u t r a l l y s t a b l e when t h e p e r t u r b a t i o n f i e l d i s dominanted by wavenumbers w i t h magnitudes l a r g e r than the modon wavenumber. E a s t w a r d - t r a v e l l i n g modons (c > 0 hence 6 > 0) a r e n e u t r a l l y s t a b l e i f the p e r t u r b a t i o n i s s o l e l y composed of 19 wavenumbers s a t i s f y i n g < K. As b e f o r e , (2.11) p l a c e s no r e s t r i c t i o n on the p e r t u r b a t i o n wavenumbers. T h i s s t a b i l i t y c o n d i t i o n s u g g e s t s t h a t an e a s t w a r d - t r a v e l l i n g modon w i l l be n e u t r a l l y s t a b l e when the s u r r o u n d i n g f l u i d i s dominated by wavenumbers s m a l l e r than the modon wavenumber. The s u f f i c i e n t s t a b i l i t y c o n d i t i o n s j u s t d e s c r i b e d a r e i n f a c t v a l i d f o r f i n i t e - a m p l i t u d e p e r t u r b a t i o n s (Charney and F l i e r l , 1981, B e n z i et a l . , 1982 and P u r i n i and S a l u s t i , 1984) a l t h o u g h the a n a l y s i s may be o n l y v a l i d f o r s m a l l - a m p l i t u d e p e r t u r b a t i o n s depending on the d i s t r i b u t i o n of extremums t o the Blumen(l968) energy-enstrophy f u n c t i o n a l ( D r a z i n and R e i d , 1981). However t h i s a n a l y s i s i s not a r i g o r o u s d e m o n s t r a t i o n of f u l l n o n l i n e a r s t a b i l i t y . E b i n and Marsden( 1970), Marsden and Abraham( 1970) and Holm et a l . d 983) have p o i n t e d out t h a t the A r n o l ' d ( 1 9 6 5 ) argument i s not a p r o o f of n o n l i n e a r s t a b i l i t y due t o i n c o n s i s t e n c i e s between the t o p o l o g y of the H i l b e r t space of which i|/ i s a member and the t o p o l o g y of the second v a r i a t i o n of the e n e r g y - e n s t r o p h y f u n c t i o n a l . A r i g o r o u s p r o o f of a n o n l i n e a r s t a b i l i t y theorem f o r p l a n e c u r v i l i n e a r f l o w s can be g i v e n u s i n g c o n v e x i t y arguments (see A r n o l ' d , 1969 and Holm et a l . , 1983). M c W i l l i a m s e t a l . ( l 9 8 l ) p r o p osed a regime diagram (see F i g u r e 2; adapted from F i g u r e 10 i n M c W i l l i a m s e t a l . , 1981) f o r e a s t w a r d - t r a v e l l i n g modons. T h e i r n u m e r i c a l l y d etermined n e u t r a l s t a b i l i t y c u r v e i s shown as a s o l i d l i n e . The s t a b i l i t y c o n d i t i o n g i v e n above ( f o r c>0) s u g g e s t s t h a t as K / | T J | i n c r e a s e s 20 f o r K/|T?|>1 ( i . e . , a d i s t u r b a n c e dominated by wavenumbers s m a l l e r than the modon wavenumber) a r e g i o n of s t a b i l i t y s h o u l d e x i s t ( a t l e a s t f o r s m a l l - a m p l i t u d e p e r t u r b a t i o n s ) . Thus when K/|TJ|>1 the s l o p e of the n e u t r a l s t a b i l i t y c u r v e s h o u l d b e g i n t o i n c r e a s e as | TJ | d e c r e a s e s , a l l o w i n g a r e g i o n of s t a b i l i t y a d j a c e n t t o t h e wavenumber a x i s . T h i s t r e n d i n the n e u t r a l s t a b i l i t y c u r v e i s q u a l i t a t i v e l y shown by the dashed l i n e i n F i g u r e 2 (where i t i s assumed t h a t f o r the two c u r v e s s h o u l d be c l o s e t o each o t h e r ) . S i m i l a r n e u t r a l s t a b i l i t y c u r v e b e h a v i o u r has been d e t e r m i n e d f o r t o p o g r a p h i c a l l y - f o r c e d p l a n e t a r y s o l i t a r y e d d i e s ( M a l a n o t t e - R i z z o l i , 1982). In the LHS of (2.10), the o n l y e x p l i c i t r e f e r e n c e t o the modon i s g i v e n by K and F, both of which a r e s p a t i a l l y c o n s t a n t f o r a g i v e n t r a n s l a t i o n speed (c) and modon r a d i u s ( a ) . For at m o s p h e r i c s c a l e s (see M c W i l l i a m s , 1980) t h e modon r a d i u s i s on the o r d e r of t h e d e f o r m a t i o n r a d i u s and |c| 0(10) m/s. Thus F «* 1 , 6 0(1 ) 2 and c o n s e q u e n t l y K « 4 ( c f . (2.3c) see a l s o F l i e r l e t a l . ( 1 9 8 0 ) ) . In m i d - l a t i t u d e s t h e s e parameter v a l u e s i m p l y t h a t w e s t w a r d - t r a v e l l i n g a t m o s p h e r i c modons w i l l be n e u t r a l l y s t a b l e i f the energy i n the t r a n s i e n t s i n the s u r r o u n d i n g f l u i d a r e c o n t a i n e d m a i n l y i n s c a l e s w i t h ( g l o b a l ) wavenumber magnitudes g r e a t e r than a p p r o x i m a t e l y 16 (16 * »cRcos(</>)/a where R and <P a r e the E a r t h r a d i u s and l a t i t u d e , r e s p e c t i v e l y ) . E a stward-2 The p l a n e t a r y v o r t i c i t y f a c t o r must be 0(1) f o r phase d i s p e r s i o n t o b a l a n c e a m p l i t u d e s t e e p i n g i n (2.1) (see Charney and F l i e r l , 1981). 21 t r a v e l l i n g modons, on the o t h e r - h a n d , w i l l be n e u t r a l l y s t a b l e i n f l o w regimes i n which t h e energy i s c o n t a i n e d m a i n l y i n s c a l e s w i t h wavenumbers l e s s than a p p r o x i m a t e l y 16. However, the dominant c o n t r i b u t i o n t o the energy spectrum of e d d i e s i n the m i d - l a t i t u d e 300 mb, 500 mb and 700 mb atmosphere (which are r e p r e s e n t a t i v e of the b a r o t r o p i c f l o w ) r e s i d e s i n z o n a l wavenumbers w i t h magnitudes l e s s than about 4-5 and m e r i d i o n a l wavenumbers (see F i g u r e 3) 6-8 g i v i n g a ( g l o b a l ) wavenumber eddy energy peak a t a p p r o x i m a t e l y 8 ( c a l c u l a t e d from E l i a s e n and Machenhauer, 1965; Saltzman and F l e i s h e r , 1962 and Tomatsu, 1979). Consequently the LHS of (2.10) w i l l be p o s i t i v e when e v a l u a t e d f o r the obse r v e d 300 mb, 500 mb and.700 mb eddy energy s p e c t r a . As an example, (2.10) can be e v a l u a t e d f o r the Tomatsu(1979) s e a s o n a l l y - and a n n u a l l y - a v e r a g e d 500 mb eddy k i n e t i c energy spectrum. To c a l c u l a t e (2.10) we approximate | \pT | 2 by 2E(7j, , r}2) 1? | " 2 w i t h E the n o n d i m e n s i o n a l eddy k i n e t i c energy spectrum. A m i d - l a t i t u d e n o r t h - s o u t h ( c a r t e s i a n ) wavenumber was i n f e r r e d w i t h the a p p r o x i m a t i o n r j 2 - a{-D2P^0)/p£U)}1/2/(RcosU)) e v a l u a t e d a t 45°N, where P^(0) i s the a s s o c i a t e d Legendre f u n c t i o n , D2P®(<p) = d 2 P 5 U ) / d * 2 and m = tj, Rc os (*) /a the n o n d i m e n s i o n a l z o n a l wavenumber. The c o n t r i b u t i o n of the P « ( 0 ) harmonic t o the Tomatsu(1979) mth z o n a l wavenumber s p e c t r a was i n f e r r e d from the E l i a s e n and Machenhauer(1965) spectrum so t h a t 22 the p e r c e n t a g e c o n t r i b u t i o n s remained the same (see F i g u r e 3) i n bot h . These c a l c u l a t i o n s imply a LHS of (2.10) of +35.57, +42.17 and +28.43 f o r the a n n u a l , w i n t e r and summer 3 eddy energy s p e c t r a , r e s p e c t i v e l y ; w i t h e x p e c t e d s t a n d a r d d e v i a t i o n s of a p p r o x i m a t e l y 14.88, 13.47 and 11.51, r e s p e c t i v e l y based on per c e n t a g e s t a n d a r d d e v i a t i o n s c o n s i s t e n t w i t h Saltzman and F l e i s h e r ( 1 9 6 2 ) . (These c a l c u l a t i o n s a re n o n d i m e n s i o n a l . ) C a l c u l a t i o n s based on the 300 mb and 700 mb eddy energy s p e c t r a a r e q u a l i t a t i v e l y s i m i l a r . T h e r e f o r e the LHS and t h e RHS of (2.10) a r e of d i f f e r e n t s i g n ( t o two s t a n d a r d d e v i a t i o n s ) f o r c>0 and the c o n c l u s i o n i s made t h a t e a s t w a r d - t r a v e l l i n g modons are n e u t r a l l y s t a b l e f o r t y p i c a l ( i . e . , observed p e r t u r b a t i o n ) m i d - l a t i t u d e e n e r g e t i c s . A s t a b i l i t y or- i n s t a b i l i t y i n f e r e n c e cannot be made f o r w e s t w a r d - t r a v e l l i n g modons s i n c e b o t h s i d e s of (2.10) a r e of the same s i g n and thus i t i s p o s s i b l e t h a t the n e u t r a l s t a b i l i t y c o n d i t i o n i s not s a t i s f i e d . The a v a i l a b l e d a t a d e s c r i b i n g o c e a n i c mesoscale wavenumber v a r i a b i l i t y p a l e s i n comparison t o the at m o s p h e r i c r e c o r d . To the a u t h o r ' s knowledge, F U ( 1 9 8 3 ) c o n t a i n s the o n l y p u b l i s h e d ( b a r o t r o p i c ) g e o s t r o p h i c k i n e t i c energy wavenumber spectrum f o r the oceans. Fu's c a l c u l a t i o n i s based on SEASAT a l t i m e t e r measurements of the sea s u r f a c e v a r i a b i l i t y , f o r wavelengths between 100-1000 km. A c a l c u l a t i o n t e s t i n g (2.10) u s i n g Fu's dat a s h o u l d o n l y be c o n s i d e r e d a p r e l i m i n a r y e s t i m a t e of modon The seasons a r e d e f i n e d so t h a t w i n t e r i s September through t o Fe b u r a r y and summer i s March t h r o u g h t o August. 2 3 s t a b i l i t y i n the ocean. Note t h a t s o l u t i o n s of the form (2.3) r e q u i r e 6 + F > 0 (or e l s e t h e m o d i f i e d B e s s e l f u n c t i o n s have i m a g i n a r y argument). The t r a n s l a t i o n speed f o r w e s t w a r d - t r a v e l l i n g modons t h e r e f o r e s a t i s f i e s the c o n s t r a i n t c < -/3a2F-1. Assuming 0*1 .6-1 0- 1 1 n r ' s ' 1 , a=O05m, f ^ l O ^ s " 1 a n d H ~ 5 - 1 u 3 m i t f o l l o w s t h a t F*10~ 2 and c o n s e q u e n t l y t h a t c < -10 ms"'. T h i s l a r g e ( i n a b s o l u t e v a l u e ) t r a n s l a t i o n speed (and hence p a r t i c l e speeds) s u g g e s t s t h a t b a r o t r o p i c w e s t w a r d - t r a v e l l i n g modons a r e not l i k e l y t o be observ e d i n the mid-ocean. Under the a p p r o x i m a t i o n s c>0 and F—>0 (2.10) reduces t o / | * T | 2 | T ? | 2 U 2 - | r j | 2 ) dTj < 0. (2.12) As i n t h e a t m o s p h e r i c c a l c u l a t i o n , l ^ 1 , 2 i s approx i m a t e d by 2E (7} , , 7i2) | TJ | " 2 where E ( T J 1 , T J 2 ) i s t h e two d i m e n s i o n a l k i n e t i c energy spectrum. S i n c e h o r i z o n t a l mesoscale v a r i a b i l i t y i n the oceans i s n e a r l y i s o t r o p i c ( B e r n s t e i n and Whit e , 1974; Mode Group, 1978 and R i c h a r d s o n , 1983) i t f o l l o w s t h a t (Fu, 1983) E(i?,,i? 2> = E 0 ( | T j | ) ( 2 7 r | T ? | ) - 1 where E 0 ( | r ? | ) i s the s c a l a r wavenumber spectrum f o r the g e o s t r o p h i c k i n e t i c energy (see F i g u r e 4, adapted from F i g u r e 8 i n Fu, 1983). Thus the i n e q u a l i t y (2.12) t a k e s the form OB 2 / E 0 ( | T J | ) [ K 2 - M 2 ] d | 7 ? | < 0 . (2.13) 24 C l e a r l y i f (2.13) i s t o be c o n t r a d i c t e d ( i . e . , a d e m o n s t r a t i o n of n e u t r a l s t a b i l i t y ) , the dominant c o n t r i b u t i o n t o the i n t e g r a l must come from |T?| < K. For a=l00 km, C=*10 cms" 1 and /3<*1 .6 • 10" 1 1 i t f o l l o w s t h a t K=3.9616, c o r r e s p o n d i n g t o a wav e l e n g t h of about 160 km (=2007T/K km). Thus o c e a n i c b a r o t r o p i c modons w i l l be n e u t r a l l y s t a b l e t o p e r t u r b a t i o n s f o r which the energy i s c o n t a i n e d m a i n l y i n wavelengths g r e a t e r than about 160 km. The i n t e g r a l i n (2.13) was c a l c u l a t e d u s i n g the F U(1983) spectrum (see F i g u r e 4) w i t h E 0 s e t i d e n t i c a l l y z e r o f o r wavelengths o u t s i d e the 100-1000 km band. For the h i g h energy spectrum (see F i g u r e 4 ) , c o r r e s p o n d i n g t o d a t a o b t a i n e d near major c u r r e n t systems (see Fu, 1983), the LHS of (2.13) was computed t o be -56.06. For the low energy spectrum (see F i g u r e 4 ) , c o r r e s p o n d i n g t o d a t a o b t a i n e d i n r e g i o n s remote from major c u r r e n t systems (see Fu, 1983), the LHS of (2.13) was computed t o be -44.54. As a measure of the e r r o r , s t a n d a r d d e v i a t i o n s of .6 and .5 were computed f o r the above e s t i m a t e s , r e s p e c t i v e l y , based on the 95% c o n f i d e n c e i n t e r v a l s f o r the h i g h and low energy s p e c t r a i n F U ( 1 9 8 3 ) . (These r e s u l t s a r e n o n d i m e n s i o n a l . ) Based on the s e c a l c u l a t i o n s , t h e n e u t r a l s t a b i l i t y of b a r o t r o p i c modons i n the oceans cannot be i n f e r r e d . There a r e , however, l i m i t a t i o n s t o the F U ( 1 9 8 3 ) spectrum which may r e s t r i c t i t s a p p l i c a b i l i t y h e r e . For example, the d a t a used t o compute the s p e c t r a r e p r e s e n t s energy o n l y a t p e r i o d s l e s s than 24 days (Fu, 1983), whereas W u n s c h ( l 9 8 l ) has shown t h a t the dominant energy c o n t a i n i n g e d d i e s have p e r i o d s 25 between 50 and 150 days. A l s o , F U ( 1 9 8 3 ) argued t h a t because of the s h o r t d u r a t i o n of the d a t a , the i n t e g r a t e d k i n e t i c energy o b s e r v e d i s about 5 times l e s s than t h a t r e p o r t e d by W y r t k i e t a l . ( l 9 7 6 ) . I t i s t h e r e f o r e c o n c e i v a b l e t h a t a wavenumber spectrum computed w i t h l o n g e r r e c o r d s would be q u i t e d i f f e r e n t from the F U(1983) spectrum t o the degree t h a t the o p p o s i t e c o n c l u s i o n g i v e n above may be reached. C l e a r l y , f u r t h e r work on the mesoscale wavenumber spectrum of the ocean i s r e q u i r e d b e f o r e the c o n s t r a i n t (2.13) can be r e a l i s t i c a l l y t e s t e d . E q u a t i o n (2.1) can a l s o be i n t e r p r e t e d as a n o n l i n e a r r e d u c e d - g r a v i t y or 1-1/2 l a y e r model (White and Saur, 1981 and Mysak, 1983) where \[/ i s the i n t e r f a c i a l d i s p l a c e m e n t and F i s the i n t e r n a l Froude number. C o n s e q u e n t l y , F * 0(1) and w e s t w a r d - t r a v e l l i n g modons w i l l have r e a l i s t i c t r a n s l a t i o n speeds (c=*-10" 1ms" 1 ). In t h i s c o n t e x t , (2.10) c o u l d be c a l c u l a t e d u s i n g a two d i m e n s i o n a l b a r o c l i n i c f l u c t u a t i o n energy wavenumber spectrum o b t a i n e d from g e o s t r o p h i c v e l o c i t y f i e l d s t h a t a r e r e l a t i v e t o a l e v e l of no motion ( e . g . , from i s o t h e r m d e f l e c t i o n d a t a c o l l e c t e d from a s p a t i a l a r r a y of XBTs). A c a l c u l a t i o n of a low frequency energy wavenumber spectrum a l o n g t h e s e l i n e s i s c u r r e n t l y i n p r o g r e s s (K. A. Thomson, 1985; p e r s o n a l c o m m u n i c a t i o n ) . 26 10 0.1 INSTABILITY STABILITY * = | i . l 0 2 T r / | ? | F i g u r e 2. S t a b i l i t y regime diagram f o r an e a s t w a r d - t r a v e l l i n g modon. The v e r t i c a l c o o r d i n a t e e, i s the a r e a - a v e r a g e d p e r t u r b a t i o n v o r t i c i t y a m p l i t u d e . The h o r i z o n t a l c o o r d i n a t e i s s c a l e d so t h a t | TJ | = TT c o r r e s p o n d s t o a wavelength e q u a l l i n g one modon d i a m e t e r (see S e c t i o n 2.2). The s o l i d l i n e i s the n u m e r i c a l l y d e t e r m i n e d n e u t r a l s t a b i l i t y c u r v e of M c W i l l i a m s et a l . d 9 8 l ) . The dashed l i n e q u a l i t a t i v e l y i l l u s t r a t e s t he expected i n c r e a s i n g s l o p e i n the n e u t r a l s t a b i l i t y c u r v e ( c f . (2.10)) f o r is/ \ 771 » K / K 0.8 under the assump t i o n t h a t when | T ? | ^ K the two c u r v e s a r e s i m i l a r . Figure 3. The two-dimensional atmospheric kinetic energy spectrum used to compute the LHS of (2.10) based on annually-averaged s t a t i s t i c s . The zonal wavenumber m, i s scaled so that m=1 corresponds to a wavelength equalling the circumference of a latitude c i r c l e . The (cartesian) meridional wave number n, i s scaled so that n=1 corresponds to wavelength equalling the longitudinal circumference ( i . e . geodesic circumference through the poles). The north-south wavenumber T?2 is related to n v i a Tj2=an/R (see section 2.2). The energy density amplitude has been normalized so that SS E(m,n) dndm = 1. The dominant contribution to the spectrum comes from m=1, 2, 3, 4, 5 and 6 and n=6, 7, 8 and 9, and accounts for 70% of the energy. 2 8 F i g u r e 4. S c a l a r wavenumber spectrum of b a r o t r o p i c k i n e t i c energy f o r the ocean used t o compute the LHS of ( 2 . 1 3 ) . The c u r v e l a b e l l e d H c o r r e s p o n d s t o d a t a c o l l e c t e d from h i g h energy r e g i o n s (eg. near major c u r r e n t s y s t e m s ) . The c u r v e l a b e l l e d L c o r r e s p o n d s t o d a t a c o l l e c t e d from low energy r e g i o n s (eg. away from major c u r r e n t s y s t e m s ) . The two l a r g e v e r t i c a l marks on the wavenumber a x i s c o r r e s p o n d t o 1000 km and 100 km from l e f t t o r i g h t , r e s p e c t i v e l y . The p o i n t | T J | = K i s i n d i c a t e d w i t h a dot on the wavenumber a x i s . 2 9 I I I . MODON PROPAGATION IN A SLOWLY VARYING MEDIUM Whitham(1965) showed t h a t the slow m o d u l a t i o n of n o n l i n e a r waves i n a d i s p e r s i v e medium c o u l d be d e s c r i b e d by the slow v a r i a t i o n of the wave parameters (such as f r e q u e n c y , wavenumber and a m p l i t u d e ) w i t h i n an averaged (over one wave p e r i o d ) L a g r a n g i a n f o r m u l a t i o n of the g o v e r n i n g e q u a t i o n s . Subsequent r e s e a r c h developments i n s l o w l y v a r y i n g n o n l i n e a r waves have g e n e r a l l y tended t o u t i l i z e p e r t u r b a t i o n methods ( d e s c r i b e d below) or when p o s s i b l e an i n v e r s e s c a t t e r i n g t r a n s f o r m a t i o n ( e . g . , Kaup and N e w e l l , 1978; and Karpman and Maslov I979a,b). In t h i s c h a p t e r a l e a d i n g o r d e r p e r t u r b a t i o n t h e o r y i s d e v e l o p e d d e s c r i b i n g modon p r o p a g a t i o n i n a s l o w l y v a r y i n g medium. The s o l u t i o n s p r e s e n t e d here r e p r e s e n t the f i r s t a p p l i c a t i o n of t h e s e methods t o a f u l l y t w o - d i m e n s i o n a l s o l i t a r y wave. Two problems a r e posed and s o l v e d i n t h i s c h a p t e r . In S e c t i o n 3.1 a p e r t u r b a t i o n method i s d e v e l o p e d t o s t u d y the e f f e c t of an Ekman bottom boundary l a y e r on an e a s t w a r d -t r a v e l l i n g modon. T h i s problem was chosen i n o r d e r t h a t the a n a l y t i c a l r e s u l t s o b t a i n e d c o u l d be compared w i t h a n u m e r i c a l s o l u t i o n f o r the same problem ( M c W i l l i a m s e t a l . , 1981). ( T h i s p r o v i d e s a t e s t f o r the p e r t u r b a t i o n method.) In S e c t i o n 3.2 the p e r t u r b a t i o n method i s used t o d e s c r i b e the i n t e r a c t i o n between a b a r o t r o p i c modon and s l o w l y v a r y i n g topography. Grimshaw(1970,1971) dev e l o p e d a p e r t u r b a t i o n method t o d e s c r i b e the slow e v o l u t i o n of the B o u s s i n e s q s o l i t a r y wave as i t t r a v e l s over s l o w l y v a r y i n g topography. Johnson(1973) and 30 Grimshaw(1977,1978,1979a,b,1981) d e v e l o p e d s i m i l a r p e r t u r b a t i o n t h e o r i e s t o d e s c r i b e o t h e r s l o w l y v a r y i n g s o l i t a r y waves i n a v a r i e t y of p h y s i c a l problems. Warn and B r a s n e t t ( 1 9 8 3 ) have a p p l i e d t h e s e methods t o model a t m o s p h e r i c b l o c k i n g as the slow m o d u l a t i o n of a t m o s p h e r i c s o l i t o n s o v e r v a r i a b l e topography. In the u s u a l f a s h i o n , the p e r t u r b a t i o n s o l u t i o n i s o b t a i n e d by i n t r o d u c i n g slow s p a t i a l and t e m p o r a l v a r i a b l e s ( f o l l o w i n g Whitham, 1965 and Luke, 1966) r e f l e c t i n g the s c a l i n g of the s l o w l y v a r y i n g medium r e l a t i v e t o a t y p i c a l w a v e l e n g t h . The dependent v a r i a b l e s and wave phase a r e expanded i n a p e r t u r b a t i o n s e r i e s i n a s m a l l parameter c h a r a c t e r i z i n g the d i f f e r e n t s c a l e s a s s o c i a t e d w i t h t h e wave and v a r i a b l e medium, w i t h the 0(1) s o l u t i o n taken t o be t h e s o l i t a r y wave. In the Johnson and Grimshaw a n a l y s i s , e v o l u t i o n e q u a t i o n s f o r the wave parameters a r e d e t e r m i n e d by demanding t h a t the c o e f f i c i e n t s of the i n h o m o g e n e i t i e s l e a d i n g t o h i g h e r o r d e r s e c u l a r i t i e s v a n i s h . S i n c e t h e s e c o e f f i c i e n t s c o n t a i n d e r i v a t i v e s of the wave parameters w i t h r e s p e c t t o t h e slow v a r i a b l e s , d i f f e r e n t i a l e q u a t i o n s a r e o b t a i n e d d e s c r i b i n g the slow e v o l u t i o n of the s o l i t a r y wave. L u k e d 9 6 6 ) , A b l o w i t z ( 1 971), Zakharov and Rubenchik( 1 974), Ko and K u e l h ( l 9 7 8 ) , A b l o w i t z and Kodama(1979), A b l o w i t z and S e g u r ( l 9 8 l ) and Kodama and A b l o w i t z ( 1 9 8 0 , 1 9 8 1 ) have d e v e l o p e d an a l t e r n a t e p e r t u r b a t i o n method f o r d e s c r i b i n g v a r i o u s one-d i m e n s i o n a l s l o w l y v a r y i n g s o l i t a r y wave problems. The i n i t i a l f o r m u l a t i o n of the two p e r t u r b a t i o n problems i s i d e n t i c a l ; however, the d i f f e r e n t i a l e q u a t i o n s d e s c r i b i n g t h e slow 31 m o d u l a t i o n of the l e a d i n g o r d e r wave parameters a r e o b t a i n e d by e x p l o i t i n g a s o l v a b i l i t y c o n d i t i o n on the f i r s t o r d e r p e r t u r b a t i o n e q u a t i o n . The s o l v a b i l i t y c o n d i t i o n i s t h a t the inhomogeneity i n the p e r t u r b a t i o n e q u a t i o n be o r t h o g o n a l t o the homogeneous s o l u t i o n of the a d j o i n t problem a s s o c i a t e d w i t h the f i r s t o r d e r p e r t u r b a t i o n o p e r a t o r ( i . e . , t he Fredholm a l t e r n a t i v e theorem). When th e f i r s t o r d e r p e r t u r b a t i o n o p e r a t o r i s s e l f - a d j o i n t , as i n the n o n l i n e a r K l e i n - G o r d o n ( A b l o w i t z , 1971), n o n l i n e a r S c h r o e d i n g e r and n o n l i n e a r sine-Gordon (Kodama and A b l o w i t z , 1980, 1981) e q u a t i o n s , t h e two methods a r e f o r m a l l y the same. However, f o r the ( r e g u l a r and m o d i f i e d ) Korteweg-de V r i e s (Kodama and A b l o w i t z , 1981), Kadomstev and P e t v i a s h v i l i ( A b l o w i t z and Segur, 1981) and p o t e n t i a l v o r t i c i t y e q u a t i o n s the p e r t u r b a t i o n o p e r a t o r i s not s e l f - a d j o i n t . In the Johnson and Grimshaw a n a l y s i s the s o l u t i o n of the f i r s t o r d e r p e r t u r b a t i o n e q u a t i o n s must be o b t a i n e d i n o r d e r t o d e t e r m i n e the s e c u l a r terms. I n the problem c o n s i d e r e d here t h i s d i d not prove t o be t r a c t a b l e . However, the homogeneous s o l u t i o n of the a d j o i n t problem a s s o c i a t e d w i t h the f i r s t o r d e r p e r t u r b a t i o n e q u a t i o n s was e a s i l y seen t o be the z e r o t h o r d e r s o l u t i o n . Thus the t h e o r y d e v e l o p e d here f o l l o w s the l a t t e r a n a l y s i s . 32 3.1 P e r t u r b a t i o n S o l u t i o n For Modon P r o p a g a t i o n Over A Bottom  Ekman Boundary L a y e r M c W i l l i a m s e t a l . ( l 9 8 l ) n u m e r i c a l l y c a l c u l a t e d the e f f e c t of ( l i n e a r , Newtonian and b i h a r m o n i c ) v o r t i c i t y d i s s i p a t i o n on an e a s t w a r d - t r a v e l l i n g modon and c o n c l u d e d t h a t the decay was a p p r o x i m a t e l y e x p o n e n t i a l and shape p r e s e r v i n g . The modon param e t e r s ( i . e . , t he r a d i u s , t r a n s l a t i o n speed and wavenumber) e v o l v e d ( t o a f i r s t a p p r o x i m a t i o n ) i n such a manner as t o p r e s e r v e the modon d i s p e r s i o n r e l a t i o n s h i p . I n the f i n a l s t a g e s of the d i s s i p a t i o n the modon degenerated i n t o a f i e l d of w e s t w a r d - t r a v e l l i n g p l a n a r Rossby waves. These o b s e r v a t i o n s s u ggest t h a t the d i s s i p a t i o n of a modon due t o bottom f r i c t i o n can be t h e o r e t i c a l l y viewed i n the c o n t e x t of the slow e v o l u t i o n of s o l i t a r y waves. T h i s S e c t i o n d e s c r i b e s a t h e o r y f o r a n a l y t i c a l l y o b t a i n i n g t h e l e a d i n g o r d e r s o l u t i o n of an e a s t w a r d - t r a v e l l i n g modon i n the presence of a bottom boundary l a y e r . The s o l u t i o n we o b t a i n a g r e e s w i t h the n u m e r i c a l c a l c u l a t i o n of M c W i l l i a m s e t a l . ( l 9 8 l ) f o r the d i s s i p a t i o n of a modon a l t h o u g h i t i s unable t o d e s c r i b e t h e t r a n s i t i o n t o w e s t w a r d - t r a v e l l i n g Rossby waves i n the f i n a l s t a g e s of the decay. For t y p i c a l o c e a n i c and a t m o s p h e r i c modon s c a l e s ( d e c r i b e d i n S u b s e c t i o n 3.1.1) the e f f e c t of bottom f r i c t i o n i s an order of magnitude s m a l l e r than the i n e r t i a l and d i s p e r s i v e terms i n th e p o t e n t i a l v o r t i c i t y e q u a t i o n . Thus the d i s s i p a t i o n of a b a r o t r o p i c modon when the e f f e c t s of a bottom Ekman l a y e r a re 33 i n c l u d e d i n the v o r t i c i t y e q u a t i o n can be d e s c r i b e d by a s l o w l y v a r y i n g s o l i t a r y wave c a l c u l a t i o n . The p arameters which d e s c r i b e the modon s o l i t a r y wave are a l l o w e d t o be f u n c t i o n s of a slow time and the s o l v a b i l i t y c o n d i t i o n l e a d s t o i n i t i a l - v a l u e problems f o r the l e a d i n g o r d e r t r a n s l a t i o n speed, r a d i u s and wavenumber. The method d e v e l o p e d i n t h i s S e c t i o n i s used t o d e s c r i b e modon p r o p a g a t i o n over s l o w l y v a r y i n g topography i n S e c t i o n 3.2. 3.1.1 F o r m u l a t i o n And S o l u t i o n Of The D i s s i p a t i o n Problem The n o n d i m e n s i o n a l b a r o t r o p i c p o t e n t i a l v o r t i c i t y e q u a t i o n i n which the i n t e r i o r of the f l u i d i s a s y m p t o t i c a l l y matched t o a bottom Ekman boundary l a y e r i s ( P e d l o s k y , 1979) A<// + J(t/>, A^ + 6 2y) = -eW (3.1.1) t where \p i s t h e g e o s t r o p h i c p r e s s u r e f i e l d , J(*,«) i s the J a c o b i a n d e t e r m i n a n t 9 ( * , • ) / 9 ( x , y ) w i t h x, y and t the u s u a l ( p o s i t i v e ) e a s t w a r d , ( p o s i t i v e ) northward and time c o o r d i n a t e s and where A i s the h o r i z o n t a l L a p l a c i a n . The parameters 8 2 = / 3 a 0 2 / c 0 and e = E 1 / 2 / ( 2 r 0 ) are the p l a n e t a r y v o r t i c i t y f a c t o r and damping c o e f f i c i e n t r e s p e c t i v e l y , w i t h E the v e r t i c a l Ekman number 2 p f ~ 1 H ~ 1 where *>, f and H a r e the v e r t i c a l eddy v i s c o s i t y , C o r i o l i s parameter and f l u i d d e p th r e s p e c t i v e l y , and where r 0 i s the Rossby number c 0 f ~ 1 ( a 0 ) ~ 1 w i t h a 0 and c 0 the undamped modon r a d i u s and t r a n s l a t i o n speed r e s p e c t i v e l y . The 34 space, time and v e l o c i t y s c a l i n g s have been chosen as a 0 , a 0 / c 0 and c 0 r e s p e c t i v e l y . F o r t y p i c a l o c e a n i c ( a t m o s p h e r i c ) modon parameter v a l u e s of 0 , a 0 , c 0 , v, H and f of 1.6•10~ 1 1 m~ 1s~ 1, 100 (1000) km, 10" 1 (10) ms" 1, 10' 2 (10) m 2s' 1, 4 (10) km and l O " 0 s " 1 r e s p e c t i v e l y , i t f o l l o w s t h a t e 10" 1. Thus f o r modon s c a l e s t h e RHS of (3.1.1) can be vie w e d as a s m a l l p e r t u r b a t i o n . The v a l u e s of a 0 and c 0 were chosen t o g i v e an o r d e r u n i t y p l a n e t a r y v o r t i c i t y f a c t o r w h i l e s a t i s f y i n g q u a s i g e o s t r o p h y . E q u a t i o n (3.1.1) does not i n c l u d e t h e f r e e s u r f a c e e f f e c t , s i n c e f o r o c e a n i c a p p l i c a t i o n s the l e n g t h s c a l e a 0 i s much s m a l l e r than t h e e x t e r n a l d e f o r m a t i o n r a d i u s ( g H / f ) 1 / 2 * 2000 km. For a l e a d i n g o r d e r s o l u t i o n t h e f a s t v a r i a b l e s (Ko and K u e h l d 978), Grimshaw(1 979,1 981 ) and Kodama and A b l o w i t z (1 981)) a r e g i v e n by T i = x - e- 1 ; c ( f ) d t ' o y = y and a slow time i s g i v e n by T = et. Thus £ =-c(T) -and £ =1. S u b s t i t u t i o n of the s e v a r i a b l e s i n t o t x (3.1.1) g i v e s 35 J(\l> + c y , + 8 2y) = - e A t f - eAtf (3.1.2) T where the J a c o b i a n i s taken w i t h r e s p e c t t o £ and y. As i n Grimshaw(1979a,b,1981) and Kodama and A b l o w i t z ( 1 9 8 1 ) a p e r t u r b a t i o n s o l u t i o n t o (3.1.2) i s c o n s t r u c t e d i n the form * - * ( 0 > U,y?T) + e<//(1)(£,y;T) + ... . The 0(1) problem i s J(<//( 0 1 + c y , A ^ ( 0 ' + 6 2y) = 0, the s o l u t i o n of which i s ta k e n t o be the modon ( F l i e r l e t a l . , 1980) ^ ( 0> = - c a K , ( 5 c - 1 / 2 r ) s i n ( e ) / K , ( 6 a c " 1 / 2 ) A<// ( 0 ) = - 5 2 a K 1 ( 6 c - l / 2 r ) s i n ( e ) / K 1 ( 8 a c - 1 / 2 ) A i / / ( 0 ) = ( 6 2 / c ) i / / < 0 ) r>a (3.1.3) ^ ( 0 ) = 62K-2aJi (Kr)sin(©)/J, (/ca) - (62 + K 2 C ) K' 2rsin (6) Ai// ( 0> = - 6 2 a J , Ur)sin(0)/J, (/ca) A ^ ( 0 ) = - K 2 I / / ( 0 ) - ( 6 2 + c/c2)rsin(0) r<a (3.1.4) 36 where J , and K, a r e the o r d i n a r y and m o d i f i e d B e s s e l f u n c t i o n s of o r d e r one, w i t h the p o l a r c o o r d i n a t e s r and 8 d e f i n e d by r 2 = U - £ 0 ( T ) ] 2 + y 2 and tan(6*) = y / [ * - * 0 ( T ) ] . The term i j 0 ( T ) i s an 0(e) phase s h i f t ( i n comparison t o the l e a d i n g o r d e r phase £) and i s determined by f i r s t o r d e r p e r t u r b a t i o n energy c o n s i d e r a t i o n s (Ko and K u e l h , 1978, Grimshaw, I979a,b and Kodama and A b l o w i t z , 1981). For the 0(1) a n a l y s i s p r e s e n t e d here i t remains undetermined and i s e v e n t u a l l y chosen as a c o n s t a n t ( the x - c o o r d i n a t e of the wave c e n t e r a t T=0). I t i s f o r m a l l y i n c l u d e d as a s l o w l y v a r y i n g q u a n t i t y a t t h i s stage because i t appears i n the 0(e) e q u a t i o n s . The modon wavenumber K i n (3.1.4) i s the f i r s t nonzero s o l u t i o n of the d i s p e r s i o n r e l a t i o n ( o b t a i n e d by r e q u i r i n g c o n t i n u i t y of Vi|>( 0 > on r=a) -6J 2 U a)K, ( 8 a c - 1 / 2 ) = c 1 ' 2 KJ , Ua ) K 2 ( Sac" 1 ' 2 ). (3.1.5) The modon r a d i u s , t r a n s l a t i o n speed and wavenumber a, c and K r e s p e c t i v e l y , a r e a l l o w e d t o be f u n c t i o n s of the slow time ( f o l l o w i n g the g e n e r a l t h e o r y of Grimshaw, 1979, Kodama and A b l o w i t z , 1981 and A b l o w i t z and Segur, 1981) w i t h the i n i t i a l c o n d i t i o n s a(0)=1, c(0)=1 and K{0)=KO where K 0 s o l v e s the modon d i s p e r s i o n r e l a t i o n f o r a=c=1 (*c0 = 3.9226, based on 6=1). The 0(e) problem a s s o c i a t e d w i t h (3.1.2) i s J ( f 0 > + cy, _W/< 1 ' ) + 37 J ( \ / / < 1 ) , _ t y ( 0 ) + 6 2y) = - _ _ i / / ( 0 > - A ^ ( 0 ) , T which f o r r>a can be r e w r i t t e n as (see (3.1.3)) J ( f °» + c y , Ai£< 1 > - 5 2 c - 1 i r / ( 1 ) ) = -&yp{ 0 ' - A i / / ( 0 ) . (3.1.6) T The homogeneous a d j o i n t e q u a t i o n a s s o c i a t e d w i t h (3.1.6) i s (A - 6 2 c - 1 ) J ( < / / < 0 ) + c y , u) = 0 f o r w hich u = i/> ( 0 )(r>a) i s a s o l u t i o n . The s o l v a b i l i t y c o n d i t i o n on ^< 0' f o r r>a i s t h e r e f o r e (see the Luke and A b l o w i t z c i t a t i o n s g i v e n above) IT oo J J ^ ' ° ' ( A ^ ( 0 ) + A<//(0)) rdrdt? = 0. (3.1.7) -TT a T The 0(e) problem f o r r<a can be w r i t t e n as (see (3.1.4)) J ( i / / ( 0 ) + c y , A i J / ( 1 ) + K 2(// ( 1>) = -Ai/> ( 0 > - A i / / ( 0 ) , (3.1.8) T w i t h the r e l a t e d homogeneous a d j o i n t e q u a t i o n (A + K 2 ) J ( I / / ( 0 ) + c y , u) = 0 f o r which u = \// < 0 )(r<a) i s a s o l u t i o n . The s o l v a b i l i t y 38 c o n d i t i o n on \pl0) f o r r<a i s t h e r e f o r e 7r a J J ,/,< 0 > (A<//< °> + A^ ( 0>) rdrdfl = 0. (3.1.9) -TT 0 T I t i s noted here that A<J/(0) = { a - 1 a - [DK, ( 6 a c - " 2)/K, ( f i a c " 1 / 2 ) ]8ac" 1 / 2 [ a " 1 a -T T T ( 2 c ) ' 1 c ]}A<//(0> + c 1 / 2 ( r c ' 1 / 2 ) A,//(0> -T T r cos(0)$o A ^ ( 0 ) + r ' 1 s i n ( e ) £ 0 A<//(0), r>a T r T 6 A i / / ( 0 ) = { a- 1a - [ D J , ( K a ) / J , (/ca) ]iea[a- 1 a + T T T K" 1/c ]}A^< 0 > + K- 1 (/cr) A(//( 0 ' -T T r cos(0)£ o A<//(0> + r- 1sin(0)£ o A ^ ( 0 ) , r<a T r T 6 where D K ^ S a c " 1 ' 2 ) and DJ^fca) are the d e r i v a t i v e s of K ^ S a c - 1 ' 2 ) and J,(/ca) with r e s p e c t to t h e i r arguments 8 a c " 1 / 2 and /ca, r e s p e c t i v e l y . The f a s t v a r i a b l e r i s r e t a i n e d i n the terms ( r c " 1 / 2 ) and (/cr) because of the slowly v a r y i n g a(T) i n T T the i n t e g r a t i o n l i m i t s i n (3.1.7) and (3.1.9). Note t h a t the l a s t two terms i n both equations express the slow e v o l u t i o n of the phase s h i f t term. I t t u r n s out that these terms i n t e g r a t e to zero i n the s o l v a b i l i t y c o n d i t i o n s due to the p e r i o d i c i t y i n 6. 39 A f t e r some algebra i t can be shown that (3.1.7) and (3.1.9) imply r e s p e c t i v e l y A a " 1a - [A - 1] ( 2 c ) " 1 c = -1 (3.1.10) T T B a " 1 a + [B - 1] K ' 1 K = -1 (3.1.11) T T where A = 7 K 2 ( 7 ) / K i ( 7 ) " 1 " K 2(7)/D, 1 D! = -7" 1(7K 2<7> " 2K 0(7)K,(7) ~ 7^(7)} 1 o B = 1 - { k J 0 ( k ) D 2 / j 3 ( k ) -1 2 ( k J 0 ( k ) / J , ( k ) + 2)(1 + k 2 7 " 2 ) J 2 ( k ) / ( k J i ( k ) ) + 1 + 2 k 2 7 - 2 } / ( D 2 / J 2 ( 7 ) - 2(1 + k 2 7 - 2 ) J 2 ( k ) / ( k J 1 ( k ) ) } i D 2 = k - 1 { k J 2 ( k ) - 2 J 2 ( k ) J , ( k ) + k J 2 ( k ) } 2 1 where 7 = ( 6 2 a 2 / c ) 1 / 2 k = «a. Equations (3.1.10) and (3.1.11) are two eqautions i n three unknowns. A t h i r d equation i s obtained by d i f f e r e n t i a t i n g (3.1.5) with r e s p e c t to T y i e l d i n g 40 K- 1K = Nta-'a - (2c)-'c ] - (2c)" 1c (3.1.12) T T T T where N = -{ 7R + k2R/7}/{4 + 7 / R + k 2 R A } and R = K 2 ( T J / K , (7). Eliminating K" 1K between (3.1.11) and (3.1.12) gives T M a" 1a - [ M - 1] (2c)- 1c = -1 (3.1.13) T T where M = BN+B-N. Provided MT*A the unique solutions (3.1.10), (3.1.11) and (3.1.13) are eas i l y seen to be a = -a c = -2c K = K i . e . , T T T a = exp(-et) c = exp(-2et) K = K 0exp(et). (3.1.14) The solutions (3.1.14) are the p r i n c i p a l result of our calculation in this Section. Concomitant with the i n t u i t i v e expectation that the RHS of (3.1.3) must result in a exponential-like decay in the modon, (3.1.14) implies that the v o r t i c i t y and streamfunction amplitudes decay as exp(-T) and exp(-3T) respectively. The exponential decay that the solutions (3.1.14) predict can be seen as the result of the energy and enstrophy equations associated with (3.1.1). It follows from (3.1.1) that It 00 Tt 00 9 S S Vtf-Vtf rdrd0 = -2e / / Vtf-Vtf rdrdfl t "TT 0 -it 0 41 3 J / |Ai//| 2 r d r d S = -2e J J |_v//|2 r d r d S . t -TT 0 -rr 0 Thus the s p a t i a l l y i n t e g r a t e d energy and v o r t i c i t y e q u a t i o n s p r e d i c t e x p o n e n t i a l decay as i m p l i e d by (3.1.14). The s o l u t i o n s (3.1.14) appear non-unique when M=A, s i n c e when t h i s o c c u r s t h e r e a r e o n l y two independent e q u a t i o n s f o r t h r e e unknowns ( i . e . , (3.1.10) and (3.1.13) a r e i d e n t i c a l ) . However, n u m e r i c a l c a l c u l a t i o n s showed t h a t 7=5.776 and k=4.4835 were the o n l y v a l u e s which c o u l d s a t i s f y the d i s p e r s i o n r e l a t i o n (3.1.5) and M=A(=.5073). The f o l l o w i n g argument shows t h a t the d i s c r e t e n e s s of these v a l u e s and the c o n t i n u i t y of a, c and K imply the uniqueness of (3.1.14) i r r e s p e c t i v e of A and M. Suppose A=M a t T = T and t h e r e e x i s t s o l u t i o n s a, c and K such t h a t 7 ( T = T ) * 0 ( r e c a l l 7 2 = 6 2 a 2 / c ; the p r o o f works e q u a l l y T w e l l e x p l o i t i n g k=«a). I t f o l l o w s t h a t t h e r e e x i s t s a>0 f o r which 7 ( T ) * 7 ( T ) f o r the i n t e r v a l T<T<r+a. Hence M*A i n t h i s i n t e r v a l and thus (3.1.14) a r e the s o l u t i o n s i n t h i s i n t e r v a l . Suppose the same hypotheses but t h a t 7 ( T = T ) = 0 . But then e i t h e r T 7 =0 i n some nonzero i n t e r v a l (r,T+a) or n o t . I f n o t , the T p r e v i o u s r e s u l t a p p l i e s on t h i s i n t e r v a l . I f t r u e , from the d e f i n i t i o n of 7 and from (3.1.10) and (3.1.11) i t f o l l o w s a _ 1 a = ( 2 c ) _ 1 c = - K _ 1 K = -1 a r e the s o l u t i o n s on t h i s T T T i n t e r v a l . However r i s a r b i t r a r y so the proo f i s c o m p l e t e . 42 3 . 1 . 2 D i s c u s s i o n Of The D i s s i p a t i o n S o l u t i o n The s o l u t i o n s f o r a, c and K s a t i s f y (*ca) = 0 , ( a c " 1 / 2 ) = 0 and T T ( K C 1 / 2 ) = 0 . T h e r e f o r e ( 3 . 1 . 5 ) reduces t o T - 6 J 2 ( K 0 ) K , ( 5 ) = K 0 J I ( K 0 ) K 2 ( 5 ) f o r a l l T i m p l y i n g t h a t t h e d i s p e r s i o n r e l a t i o n s h i p i s i n v a r i a n t d u r i n g the decay. Thus t h e modon remains d y n a m i c a l l y e q u i v a l e n t t o i t s i n i t i a l s t a t e , a t l e a s t i n i t i a l l y and t o 0 ( 1 ) , as any WKB-like t h e o r y must p r e d i c t . The e x p o n e n t i a l decay of the s t r e a m f u n c t i o n and v o r t i c i t y and t h e i n v a r i a n c e of the d i s p e r s i o n r e l a t i o n w hich we have o b t a i n e d i s i n agreement w i t h the n u m e r i c a l s o l u t i o n of ( 3 . 1 . 1 ) ( M c W i l l i a m s e t a l . , 1 9 8 1 ) f o r a modon i n i t i a l s t a t e . The c o m p a t i b i l i t y c o n d i t i o n s ( 3 . 1 . 7 ) and ( 3 . 1 . 9 ) a r e i n f a c t s u f f i c i e n t t o e l i m i n a t e the s e c u l a r i t y i n \pL 1 ' (note t h a t _VJ/ ( 0 ) i s a homogeneous s o l u t i o n t o ( 3 . 1 . 6 ) and ( 3 . 1 . 8 ) ) s i n c e A\// ( 0 ) + A i / / ( 0 ) i s i n d e n t i c a l l y z e r o as a consequence of ( 3 . 1 . 1 4 ) T ( i n t r o d u c e t h e change of v a r i a b l e r —> a ( T ) r i n ( 3 . 1 . 3 ) and ( 3 . 1 . 4 ) ) . F i g u r e 5 i s a sequence of c o n t o u r p l o t s of the p a t h l i n e s ^,(0) + c ( T ) y as T i n c r e a s e s . The o b s e r v e r i s i n the r e f e r e n c e frame of the modon so t h a t as d i s s i p a t i o n o c c u r s the s u r r o u n d i n g f l u i d appears t o slow down, as i n d i c a t e d by the i n c r e a s i n g s e p a r a t i o n of the c o n t o u r s . F i g u r e 6 i s a sequence of c o n t o u r p l o t s showing the decay i n the v o r t i c i t y f i e l d as T i n c r e a s e s . 43 The o b s e r v e r i s f i x e d w i t h r e s p e c t t o the f l u i d a t i n f i n i t y . The modon moves t o the r i g h t w i t h speed c ( T ) . An upper bound on the d i s t a n c e over which the d i s s i p a t i n g modon t r a v e l s as a modon can be o b t a i n e d from the c h a r a c t e r i s t i c e q u a t i o n dx/dt = c ( T ) which i n t e g r a t e s t o x ( t ) = £ 0 + (1 - e x p ( - 2 e t ) ) / ( 2 e ) so t h a t the modon t r a v e l s a maximum d i s t a n c e ( 2 e ) _ 1 (about 5 modon r a d i i ) b e f o r e b r e a k i n g up i n t o a f i e l d of Rossby waves. M c w i l l i a m s et a l . ( l 9 8 l ) e s t i m a t e t h a t f o r t < 15 the modon decays as a modon (based on s i m i l a r parameter v a l u e s f o r 6 2 and r) and when t * 15 a modon Rossby wave t r a n s i t i o n o c c u r s . Based on the s c a l i n g i n t h i s S e c t i o n , the t r a n s i t i o n t a k e s p l a c e when T * 1.5 (note t h a t F i g u r e 5 o n l y goes up t o T=1.39). At t h i s s t age the a m p l i t u d e s of ^ ( 0 ) and Ai/> ( 0 ) are v e r y s m a l l (see F i g u r e s 5 and 6) and thus the above s o l u t i o n q u a l i t a t i v e l y d e s c r i b e s the p r i n c i p a l decay mechanism. For o c e a n i c s c a l e s of a 0 and c 0 of 100 km and 0.1 m s ~ 1 r e s p e c t i v e l y the above p e r t u r b a t i o n s o l u t i o n w i l l be a s y m p t o t i c a l l y v a l i d f o r a time s c a l e of 100 days, whereas f o r a t m o s p h e r i c s c a l e s of a 0 and c 0 of 1000 km and 10 m s " 1 , r e s p e c t i v e l y (see M c W i l l i a m s , 1980) the p e r t u r b a t i o n s o l u t i o n w i l l be a s y m p t o t i c a l l y v a l i d on a time s c a l e of 10 days 45 RRDIUS =1.0000 SPEED = 1 .0000 TIME = 0.0000 KflPPR = 3.9226 STREAM FUNCTION FIELD + CY -5.0 -3.0 T -1.0 1.0 X A X I S 3.0 5.0 F i g u r e 5a. Sequence of co n t o u r p l o t s of the p a t h l i n e s ^ , ( 0 ) + C ( T ) V f o r the Ekman d i s s i p a t i o n problem. The o b s e r v e r i s f i x e d w i t h r e s p e c t t o a c o o r d i n a t e system a t t a c h e d t o the modon. The c o n t o u r i n t e r v a l s are ± 0 . 2 . The z e r o c o n t o u r i s marked w i t h a 0 . The v a l u e s of a, c and K a t each slow time T a r e l i s t e d i n the upper l e f t hand c o r n e r . 46 RADIUS =0.7046 TIME = SPEED = 0.4965 KfiPPfi = 0.3500 5.5664 STREAM FUNCTION FIELD + CY o i n . ' o C O X cr >-c=> o o i n . -5.0 -3.0 1 1 -1.0 1.0 X A X I S 3.0 5.0 F i g u r e 5b. Modon p a t h l i n e s a t T=0.35 under Ekman d i s s i p a t i o n , 47 RRD1US =0.4990 TIME = 0.6950 SPEED = 0.2490 KRPPfl = 7.8597 STREAM FUNCTION FIELD + CY o O 00 -m X c r o ro _ CD uo. -5.0 -3.0 -1.0 1.0 3.0 5.0 x n x i s F i g u r e 5c. Modon p a t h l i n e s a t T=0.695 under Ekman d i s s i p a t i o n . 48 RADIUS =0 .3534 TINE = 1.0400 SPEED = 0.1249 KfiPPfl = 11.0979 STREAM FUNCTION FIELD + CY o LO " o ro" UO X CE > - ° i o ro. LO . 1 1 1 1 : | 5.0 -3.0 -1.0 1.0 3.0 5.0 X A X I S F i g u r e 5d. Modon p a t h l i n e s a t T=1.04 under Ekman d i s s i p a t i o n . 49 RRDIUS =0.2490 TIME = 1.3900 SPEED = 0.0620 KRPPR = 15.7487 S T R E A M F U N C T I O N F I E L D + C Y o O ro • c n I 1 X d o ro. o in. -5.0 -3.0 -1.0 1.0 3.0 5.0 X R X 1 S F i g u r e 5e. Modon p a t h l i n e s at T=1.39 under Ekman d i s s i p a t i o n . 50 RADIUS =1.0000 TIHE = 0.0000 SPEED = 1.0000 KflPPfl = 3.9226 V O R T I C I T Y F I E L D o i n ' o r o ' C O t 1 X c r ^ ° i o r o . o t n . -5.0 -3.0 •1.0 1.0 X A X I S 3.0 5.0 Figure 6a. Sequence of contour plots of the v o r t i c i t y A i / / < 0 ) for the Ekman d i s s i p a t i o n problem. The observer i s fixed with respect to the f l u i d at i n f i n i t y . The contour intervals are ±2.0. The zero contour is marked with a 0. The values of a, c and K at each slow time T are l i s t e d in the upper l e f t hand corner. 51 RADIUS =0.7046 TIME = 0.3500 SPEED = 0.4965 KRPPR = 5.5664 VORTICITY FIELD o i n ' o n o ' cn x >- ° o i n . t 1 1 1 1 1 I -2.483 - 0 . 4 8 3 1.517 3.517 5.517 7.517 X n x i s F i g u r e 6b. Modon v o r t i c i t y a t T=0.35 under Ekman d i s s i p a t i o n . 52 RADIUS =0.4990 TIME = 0.6950 SPEED = 0.2490 KflPPfl = 7.8597 V O R T I C I T Y F I E L D o m l o i n . - 1 . 2 4 5 ~1 0.755 - I 1 1 2.755 4.755 6.755 X n x i s 8 . 7 5 5 gure 6c. Modon v o r t i c i t y a t T=0.695 under Ekman d i s s i p a t 53 RRDIUS SPEED = =0.3534 0.1249 T I H E = K A P P f l = 1.0400 11 .0979 V O R T I C I T Y F I E L D o ID ' O CD X c r o r o . o i n . I I I 1 -0.525 1.375 3.375 5.375 7.375 X R X I S 9.375 F i g u r e 6d. Modon v o r t i c i t y a t T=1.04 under Ekman d i s s i p a t i o n . 54 RADIUS =0.2490 TIME = 1.3900 SPEED = 0.0620 KAPPA = 15.7487 V O R T I C I T Y F I E L D o in" to X c r o ro_| ! j j j , 0.31 1.69 3.69 5.69 7.69 9.69 X R X I S F i g u r e 6e. Modon v o r t i c i t y a t T=1.39 under Ekman d i s s i p a t i o n . 55 3.2 Modon P r o p a g a t i o n Over S l o w l y V a r y i n g Topography In t h i s S e c t i o n a l e a d i n g o r d e r p e r t u r b a t i o n t h e o r y f o r modon p r o p a g a t i o n over s l o w l y v a r y i n g topography i s d e v e l o p e d . T h i s s o l u t i o n , which i s independent of any f u n c t i o n a l form of the topography, i s a p p l i e d t o two s p e c i f i c t o p o g r a p h i c c o n f i g u r a t i o n s . In S u b s e c t i o n 3.2.1 the g e n e r a l t h e o r y i s de v e l o p e d which i s v a l i d f o r f i n i t e - a m p l i t u d e s l o w l y v a r y i n g topography ( i . e . , t o p o g r a p h i c a m p l i t u d e s on the o r d e r of the depth of the f l u i d ) . A l s o i n S u b s e c t i o n 3.2.1, a n a l y t i c a l p e r t u r b a t i o n s o l u t i o n s f o r the modon r a d i u s , t r a n s l a t i o n speed and wavenumber a r e o b t a i n e d f o r s m a l l - a m p l i t u d e topography. These a n a l y t i c a l s o l u t i o n s a r e d e s c r i b e d i n S u b s e c t i o n 3.2.2. S u b s e c t i o n s 3.2.3 and 3.2.4 a p p l y the s m a l l - a m p l i t u d e s o l u t i o n s t o a modon t r a v e l l i n g over a m e r i d i o n a l r i d g e ( m o d e l l e d as a g a u s s i a n i n the x - c o o r d i n a t e d i r e c t i o n ) and an escarpment (modelled as a h y p e r b o l i c tangent i n the x - c o o r d i n a t e d i r e c t i o n ) , r e s p e c t i v e l y . 3.2.1 P e r t u r b a t i o n S o l u t i o n For Modon P r o p a g a t i o n Over S l o w l y  V a r y i n g Topography As i n Veronis(1966), Rhines(1969a,b), Clarke(1971), LeB l o n d and M y s a k d 9 7 8 ; Sec. 2 0 ) , M a l a n o t t e - R i z z o l i and Hendershott(1980), Matsuura and Yamagata(1982) and Yamagata(1982) t h e e f f e c t s of v a r i a b l e topography on p l a n e t a r y waves are m o d e l l e d w i t h the s h a l l o w water e q u a t i o n s on the 0-plane. The n o n d i m e n s i o n a l r i g i d - l i d s h a l l o w water p o t e n t i a l v o r t i c i t y e q u a t i o n can be 56 w r i t t e n as (LeBlond and Mysak, 1978; Sec. 20) V - f H - ' V i / / ) + J[<K H-'V-(H- 1V^) + f H " 1 ] = 0. (3.2.1) t A l l symbols are d e f i n e d as i n S e c t i o n 3.1 with the f o l l o w i n g e x c e p t i o n s ; t// i s the t r a n s p o r t streamf u n c t i o n -Hu = tf/ Hv = \p , y x where u and v are the ( p o s i t i v e ) eastward and ( p o s i t i v e ) northward v e l o c i t y components r e s p e c t i v e l y , and where H(ex,ey) = 1 - uh(ex,ey) i s the s l o w l y v a r y i n g topography with e = a 0 / ( l e n g t h s c a l e of topography) « 1 and u i s the topographic amplitude parameter. 4 In t h i s S ubsection i t i s assumed that 0<e<<jx^1. The l e a d i n g order s o l u t i o n w i l l be v a l i d f o r f i n i t e - a m p l i t u d e topography ( i . e . , y=*0(l)), although the smallness of u w i l l be e v e n t u a l l y demanded ( i . e . , 0<e«ji«1) i n order to o b t a i n a n a l y t i c a l s o l u t i o n s . The C o r i o l i s parameter i s f = ( r 0 ) " 1 + 6 2y. The e f f e c t s of p l a n e t a r y v o r t i c i t y dominate topographic s t e e r i n g i f ( 6 2 r 0 ) " 10(VH/H) =* e ( 8 2 r 0 ) ' 1 < 1 (LeBlond and Mysak, 1978; Sec. 20). T y p i c a l slopes of the ocean f l o o r away from mid-ocean r i d g e s , l a r g e seamounts and escarpment breaks give e =* 10" 3, * The topographic parameter i s the maximum abs o l u t e value of the h e i g h t of the topography d i v i d e d by the mean depth. The theory a p p l i e s to topographic d e p r e s s i o n s (e.g., trenchs) as w e l l as to topographic p r o t r u s i o n s (e.g., r i d g e s ) . 5 7 thus e ( 8 2 r 0 ) ~ 1 * 10" 1 ( f o r s c a l i n g s d e s c r i b e d i n S e c t i o n 3.1). We note that l a r g e r bottom slopes can be c o n s i d e r e d as r 0 i s allowed to in c r e a s e (e.g., atmospheric a p p l i c a t i o n s or i n e q u a t o r i a l r e g i o n s ) . As i n L u k e d 9 6 6 ) , Gr imshaw ( 1 970, 1 971 , 1 979a, b, 1 981), Kodama and Ablowitz(1980) and the work i n S e c t i o n 3.1 a s o l u t i o n to (3.2.1) i s found i n the form \P = H(X,Y){ A(X,Y,T) + »//< °> (£,y;X,Y,T) + ei/'1 1 ' U,y;X,Y,T) + ... }, (3.2.2a) with the r e l a t e d p e r t u r b a t i o n v e l o c i t y f i e l d u = u< 0 ' U,y;X,Y,T) + eu< 1 > U ,y;X, Y,T) + ... , (3.2.2b) v = v l 0 > U,y;X,Y,T) + ev< 1 > ( £ ,y;X, Y,T) + ... , (3.2.2c) where £ = -c(X,Y,T) and £ = 1 and where the slow v a r i a b l e s t x X, Y and T are d e f i n e d by X=ex, Y= ey and T=et. S u b s t i t u t i o n of these v a r i a b l e s i n t o (3.2.1) y i e l d s -cH ' 1 A i / / + 8 2H - V + H " 2 J U , Ai//) = e{ -H" 1Ai// + 2cH - 1 i / / + i i T u x -2cH~ 1 \jj - 8 2H " V - H ' 2 J (\l>, At//) - H - 2 J (i//, Ai//) -£yY X X Y 58 2 H - 2 J U , ^ ) - 2H"2J(i//, <// ) - cH" 2H \p - cH" 2H ^ + £X yY X U Y £y 2H- 3(Ai / / )J (,//, H) + 2H" 3(Ai//)j U , H) + f H " 2 J U , H) + X Y X f H - 2 J U , H) + H"3H J U , i// ) + H-3H J(i//, »//)} + O U 2 ) , (3.2.3) Y X I Y y where J(-,*) = 3(•,*)/3(£,y), J (•,*) = 3(•,*)/3(X,y) and X J (•,*) = 3 ( • , * ) / d ( t , Y ) . Y The f o r m u l a t i o n of the phase v a r i a b l e £ does not i n c l u d e a x - d i r e c t i o n wavenumber, say k. I t can shown that i f one d e f i n e s £ such t h a t £ =-ck and £ =k the l e a d i n g order r e s u l t s presented t x here are not changed s i n c e the x - d i r e c t i o n wavenumber f a c t o r s out of the governing equations and only c remains. T h e r e f o r e , at l e a s t to t h i s order such a parameter i s f r e e and one can set i t equal to u n i t y . Moreover, we argue that the r o l e played by a slowly v a r y i n g wavenumber i n the d e f i n i t i o n of the phase v a r i a b l e i s played by the slowly v a r y i n g modon r a d i u s and wavenumber. A l s o , i t would be i n c o r r e c t to attempt to i n c o r p o r a t e the y v a r i a b l e i n t o the phase v a r i a b l e £ s i n c e i t would no longer be p o s s i b l e to d e f i n e the p o l a r c o o r d i n a t e s r e q u i r e d to write the modon s o l u t i o n . The 0(1) problem i s J ( f 0 ) + cy, A\pi0) + 6 2y) = 0, the s o l u t i o n of which i s taken to be the modon (3.1.3), (3.1.4) 59 and (3.1.5). The comments reg a r d i n g the phase s h i f t term | 0 ( X , Y , T ) made i n S e c t i o n 3.1 apply here and thus e v e n t u a l l y | 0 w i l l be chosen to be the x-coordinate of the wave center at T=0, al l o w i n g the topography to be centered at X=0. The wave parameters a, c and K are slowly v a r y i n g f u n c t i o n s of X , Y and T that s a t i s f y the d i s p e r s i o n r e l a t i o n s h i p (3.1.5). For the i n i t i a l v a lue problem, a ( X , Y , 0 ) = l , C ( X , Y , 0 ) = 1 and K ( X , Y , 0 ) = K O / where (as i n Se c t i o n 3.1) K0 i s obt a i n e d from the d i s p e r s i o n r e l a t i o n (3.1.5) when T=0 (K 0=3.9226 when 6=1). The 0(e) problem can be w r i t t e n as J ( i / / < 0 ) + cy, AiJ< ( 1 )) + J ( v / / ( 1 ) , A i / / ( 0 ) + 6 2y) = -6 2A -X 6 2H" 1AH - H" 1J (HA, A ^ l 0 ) ) - H" 1 J (HA, Ai/> ( 0 )) - A^< 0 » + X X Y T 2c^ < 0> - 8 2 ^ ( 0 > - J U ( 0 ) , A\//<0)) - 2J ( t / / ( 0 ) , i//(0)) + | | X X X | X 2 c ^ ( 0 ) - J U < 0 > , A<//(0>) - 2 J ( ^ ( 0 > , i//(0)) + lyY Y yY . c(H-'H ) ^ ( 0 ) - 8 2(H- 1H ) V ( 0 ) - (H-'H ) J ( * < 0 > , <//< 0>) -x n X X | (Ai/> ( 0 ) + f ) ( H " 1 H ) ^ ( 0 ) - (H-'H ) 0 ' ° > A ^ ( O ) + X y X y C(H" 1H ) ^ < 0 > - (H-'H ) J ( i / / ( 0 > , ^ ( 0 ) ) + Y |y Y y (Ai//<0> + f ) ( H " 1 H )^ ( 0> + (H- 1H )^<°>A^ ( 0'. (3.2.4) Y I Y | As i n Lu k e d 9 6 6 ) , Grimshaw( 1 970, 1 971 , 1 979a,b, 1 981) and Kodama and Ablowitz(1981), ^ ( 1 } i s assumed to have the pro p e r t y 1>->0 as r->°=. Consequently, i n the l i m i t as r—>» (3.2.4) reduces to 60 A + (H' 1H )A = 0 , ( 3 . 2 . 5 ) X X ( r e c a l l t//( 0 >->0 as r->=>). The gene r a l s o l u t i o n of ( 3 . 2 . 5 ) i s A = g(Y,T)H" 1 f o r some f u n c t i o n g(Y,T). Thus the f i r s t term i n the p e r t u r b a t i o n expansion ( 3 . 2 . 2 a ) ( i . e . , AH) i s simply g(Y,T) which w i l l be l e f t undetermined i n the l e a d i n g order a n a l y s i s developed here. Note that s i n c e AH = g(Y,T), J (HA, A i / / ( O > ) = 0 and X J (HA, ,Ai// ( 0 > )=-g A i / / ( 0 ) . I t turns out t h i s l a s t remaining term Y Y i i n t e g r a t e s to zero i n the imposed s o l v a b i l i t y c o n d i t i o n s due to the p e r i o d i c i t y i n 6 (see the Appendix). A l s o , note, that g(Y,T) w i l l not c o n t r i b u t e to the 0 ( 1 ) v e l o c i t y f i e l d s i n c e u t 0 ) = _^,( 0 ) v ( 0 > = V { 0 ' . y * Th e r e f o r e , with no l o s s of g e n e r a l i t y , g(Y,T) can be set equal to zero i n t h i s l e a d i n g order a n a l y s i s . I t i s f o r m a l l y r e t a i n e d at t h i s stage because of i t s appearance i n the 0(e) problems. For r>a, ( 3 . 2 . 4 ) can be r e w r i t t e n as (see ( 3 . 1 . 3 ) ) J ( i / / ( 0 > , A I / - ( 1 ) - 6 2 c " V ( 1 > ) = ( 6 2 / c ) H - 1 g i//(0> - A i / / ' 0 ' + Y I T 2 c i / / ( 0 ) - 6 2 »// ( 0 > + ( 8 2 / c ) ^ / ( 0 ) ^ ( 0 ) - 2 J ( l / / ( 0 ) , i// ( 0 ) ) + ££X X X y £X 61 C ( H " 1 H ) ^ ( 0 > - 6 2 ( H " 1 H )\J/(0> - ( H " 1 H ) J ( ^ ( 0 ) , ^ ( 0 ) ) -x a x x i [ ( r 0 ) " 1 + 8 2y](H-'H ) ^ ( 0 ) - ( 2 8 2 / c ) ( H - 1 H ),/,< 0 > ° > + X y X y 2 c < / / ( 0 ) - ( 8 2 / c ) i / / ( 0 ^ ' ° > - 2 J ( i / / ( 0 ) , ^ ( 0 ) ) + ijY Y i yY C ( H " 1 H )i// ( 0> - ( H - ' H ) J U ( 0 > , <//(0)) + Y iy Y y [ ( r o ) " 1 + 5 2 y ] ( H - 1 H ) ^ ( 0 > + 2 ( 8 2 / c ) ( H - 1 H ) ,/,< ° > ,/,< 0 > . ( 3 . 2 . 6 ) Y i Y i The homogeneous a d j o i n t equation a s s o c i a t e d with ( 3 . 2 . 6 ) i s (A - 6 2 c " 1 )J(i//< °> + cy, u) = 0 f o r which u = i// ( 0 )(r>a) i s a s o l u t i o n . The s o l v a b i l i t y c o n d i t i o n on \ J / ( 0 ) f o r r>a i s t h e r e f o r e (see the Luke and Ablowitz c i t a t i o n s ) rr CD / J i//< 0 ) {RHS ( 3 . 2 . 6 ) } rdrdfl = 0. ( 3 . 2 . 7 ) -it a For r<a, ( 3 . 2 . 4 ) can be r e w r i t t e n as (see ( 3 . 1 . 4 ) ) j ( ^ < ° > , Ai//( 1 > + K 2 ^ ( 1 ) ) = - K 2 H " 1 g t// ( 0 ) - A\l/{0) + Y i T 2 c i / / ( 0 > + K2c\p(0) - U 2 ) ^ (0>^<°> - 2 J ( i / / ( 0 ) , \^ t 0 ) ) + i i X X X y £X c ( H - 1 H ) V ( 0 ) + K 2 C ( H " 1 H )^/<°> - ( H " 1 H )j(<//<0), i// (0 )) -x H x x i [ ( r 0 ) - 1 - K 2 c y ] ( H " 1 H ) ^ , 0 > + 2 K 2 ( H - 1 H ) ^ < ° > ^ < ° > + X y X y 62 2cypi0) + U 2 ) ^<0>tf<°> + [ r s i n ( 0 ) ( 6 2 + K 2 C ) ] <// ( 0 ) -SyY Y i Y £ 2 J ( ^ ( 0 ) , ^ ( 0 > ) + c ( H " ' H ) ^ ( 0 > - ( H " 1 H ) J ( V t 0 ) , * l 0 > ) + yY Y £y Y y [ ( r 0 ) - 1 - K 2 c y ] ( H - 1 H ) ^ ( 0 ) - 2 K 2 ( H - 1 H )^<°)^<0'. (3.2.8) Y % Y £ The homogeneous a d j o i n t equation a s s o c i a t e d with (3.2.8) i s (A + K 2 ) J U < 0 > + cy, u) = 0 f o r which u = i// ( 0 ,(r<a) i s a s o l u t i o n . The s o l v a b i l i t y c o n d i t i o n on ^ < 0 ) f o r r<a i s t h e r e f o r e TT a / / ^' 0>{RHS(3.2.8)} rdrdfl = 0. (3.2.9) The c a l c u l a t i o n of the c o m p a t i b i l i t y c o n d i t i o n s (3.2.7) and (3.2.9) i s t e d i o u s but e n t i r e l y s t r a i g h t f o r w a r d . These computations are d e s c r i b e d i n the Appendix. Four o b s e r v a t i o n s are made here about these c a l c u l a t i o n s . I t has been noted that the f i r s t term i n the RHS of (3.2.6) and (3.2.8) ( c o n t a i n i n g g(Y,T)) i n t e g r a t e s to zero due to the p e r i o d i c i t y i n 6 (see the Appendix). A l s o , a l l terms that c o n t a i n d e r i v a t i v e s of the phase s h i f t term £ 0(X,Y,T) with respect t o X,Y and T i n t e g r a t e t o zero f o r the same reason (see the Appendix). In a d d i t i o n , terms which c o n t a i n the Rossby number as a c o e f f i c i e n t i n (3.2.6) and (3.2.8) a l s o i n t e g r a t e t o zero due to the p e r i o d i c i t y i n 6 (see the Appendix). Thus the 63 e q u a t i o n s we d e r i v e to d e s c r i b e the slow v a r i a t i o n of the modon parameters are independent of the Rossby number, $ 0(X,Y,T) and g(Y,T). Of p h y s i c a l i n t e r e s t i s the o b s e r v a t i o n t h a t each i n d i v i d u a l term i n the RHS's of ( 3 . 2 . 6 ) and ( 3 . 2 . 7 ) that c o n t a i n s a d e r i v a t i v e with respect to the slow v a r i a b l e Y i n t e g r a t e s to i d e n t i c a l l y zero due to the p e r i o d i c i t y i n 6 (see the Appendix). Therefore the m e r i d i o n a l s t r u c t u r e of the slowly v a r y i n g topography appears i n parametric form i n the l e a d i n g order s o l u t i o n and the l e a d i n g order e v o l u t i o n of the modon parameters i s s o l e l y determined by the east-west topographic s t r u c t u r e . However, only the l e a d i n g order s o l u t i o n has a parametric dependence On the m e r i d i o n a l topographic s t r u c t u r e . I t i s easy to check that higher order d e r i v a t i v e s with respect t o Y do not s a t i s f y t h i s o r t h o g o n a l i t y p r o p e r t y . (Such terms are 0 ( e 2 ) and have been ignored i n ( 3 . 2 . 3 ) . ) M e r i d i o n a l g r a d i e n t s i n the topography t h e r e f o r e give r i s e t o 0 ( e 2 ) modulations i n the modon parameters. A p h y s i c a l e x p l a n a t i o n f o r t h i s l e a d i n g order parameteric dependence on the m e r i d i o n a l topographic s t r u c t u r e can be given based on the f o l l o w i n g v o r t i c i t y arguments. Equation ( 3 . 2 . 1 ) s t a t e s t h at the p o t e n t i a l v o r t i c i t y (v -u +f)/H i s a conserved — x y — q u a n t i t y . Consider a p a r t i c l e of f l u i d d i s p l a c e d p a r a l l e l to the x - a x i s . I f H i s non-zero then the r e l a t i v e v o r t i c i t y must x change i n response to changes i n H (f being constant when y. i s 6 4 c o n s t a n t ) . Thus the l e a d i n g order r e l a t i v e v o r t i c i t y adjustment t o z onal displacements occurs i n response to z o n a l topographic v a r i a t i o n s . However, the l e a d i n g order v o r t i c i t y adjustment i s not due t o topographic v a r i a t i o n s i n m e r i d i o n a l displacements. The s c a l i n g of (3.2.1) (see a l s o (3.2.3)) has assumed that changes i n p l a n e t a r y v o r t i c i t y dominate topographic v a r i a t i o n s . Thus the l e a d i n g order v o r t i c i t y adjustment to m e r i d i o n a l displacements occurs i n response to changes i n the p l a n e t a r y v o r t i c i t y ( i . e . , the y - c o o r d i n a t e ) and topographic v a r i a t i o n s are second order in comparison. A f t e r c o n s i d e r a b l e a l g e b r a (see the Appendix), (3.2.7) and (3.2.9) r e s u l t i n the f o l l o w i n g two d i f f e r e n t i a l equations f o r a(X,Y,T) and c(X,Y,T) c A 3 a - 1 a = -cE,H"'H X (3.2.10) X (B, - 1 ) ( 2 c ) - 1 c ] + c B 2 ( 2 c ) " 1 c + X X cB 3a"'a = -cE 2H" 1H X (3.2.1 1 ) X r e s p e c t i v e l y , where A,, A 2, A 3, E 1 f B 1 f B 2, B 3 and E 2 are 6 5 nondimensional f u n c t i o n s of 7 and k ( r e c a l l 7 2 = 5 2 a 2 / c and k=«a) which are d e r i v e d i n the Appendix. The system (3.2.10) and (3.2.11) can be r e w r i t t e n as a " 1 a + c ( A n + l / 2 ) a ' 1 a + c A 1 2 ( 2 c ) - 1 c = - C F T H ^ H (3.2.12) T X X X (2c)" 1 c + c A 2 , a - 1 a + T X c ( A 2 2 + l / 2 ) ( 2 c ) " , c = - c F 2 H - 1 H , (3.2.13) X X where A,, - [(1 " B,)A 3 + (A, - 1 ) B 3 ] / ( A , - B , ) A 1 2 = [(1 - B , ) A 2 + (A, ~ 1)B 2]/(A, - B , ) A 2 1 = ( - B , A 3 + A , B 3 ) / ( A , - B,) A 2 2 = ( - B , A 2 + A,B 2)/(A, - B,) F , = [(1 - B,)E, + (A, - 1 ) E 2 ] / ( A 1 - B , ) F 2 = (-B,E, + h,E2)/(k, - B , ) . The e v o l u t i o n of K ( X , Y , T ) i s determined by s o l v i n g (3.1.5) at each space-time c o o r d i n a t e given a ( X , Y , T ) and c ( X , Y , T ) , the s o l u t i o n s of (3.2.12) and (3.2.13). 66 Whitham(1965) argued that h y p e r b o l i c d i f f e r e n t i a l equations ought to govern the e v o l u t i o n of the slowly v a r y i n g wave so that the parameter modulations c o u l d propagate i n space-time. The eigenvalues of the matrix [M ] with e n t r i e s M 1 1=A,,+l/2, i j M, 2=A , 2 , M 2 i = A 2 i and M 2 2=A 2 2+1/2 (denoted with the usual convention) were n u m e r i c a l l y determined to be r e a l and d i s t i n c t f o r a l l value s of 7 and k ( c o n s i s t e n t with the d i s p e r s i o n r e l a t i o n (3.1.5)) except f o r when A ^ B , ( i n which case the matrix [M ] i s not d e f i n e d ) . A ^ B , only when 7=5.776 and i j k=4.4835 (A, and B, are i n f a c t i d e n t i c a l to the f u n c t i o n s A and M d e f i n e d i n S e c t i o n 3.1, r e s p e c t i v e l y ) . S i m i l a r arguments to those presented i n S e c t i o n 3.1 can show that i f there e x i s t s a s o l u t i o n to the i n i t i a l - v a l u e problem a s s o c i a t e d with (3.2.10) and (3.2.11) then the c o n t i n u i t y of a, c and K and the d i s c r e t e n e s s of the 7 and K f o r which A,=B, imply that the s o l u t i o n s s a t i s f y (3.2.12) and (3.2.13) (except at the set of d i s c r e t e space-time c o o r d i n a t e s f o r which the s i n g u l a r i t y o c c u r s ) . The equations (3.2.12) and (3.2.13) t h e r e f o r e form a no n l i n e a r h y p e r b o l i c system which i s v a l i d f o r f i n i t e - a m p l i t u d e topography. In p r a c t i c e , the problem must be so l v e d n u m e r i c a l l y . A n a l y t i c a l s o l u t i o n s can be obtained f o r a, c and K by demanding the smallness of the topographic amplitude parameter u. When 0<e«jz<<1 (which i s the case i n many oceanic and atmospheric a p p l i c a t i o n s ) , s o l u t i o n s to (3.2.12), (3.2.13) and (3.1.5) can be obtained i n the form 67 a = 1 + ual 1 > (X,Y,T) + 0 ( / i 2 ) c = 1 + MC< 1>(X,Y,Y) + 0 ( M 2 ) K = K0[ 1 + U K 1 1 ' (X,Y,T) + 0(/z 2) ]. Note that a < 1 > , c ( 1 ) and K { 1 ' are ^ - p e r t u r b a t i o n s and are not to be confused with the r o l e played by ^ ( 1 ' , u ( 1 } and v ( 1 ' i n the e - p e r t u r b a t i o n expansion (3.2.2). Since H(X,Y) = 1~Mh(X,Y) i t f o l l o w s H"1H = -uh + 0(u2). X X The 0(u) terms i n (3.2.12) and (3.2.13) are a ( 1 > + (A 0 1, + l / 2 ) a ( 1 ' + A 0 1 2 c ( 1 ) / 2 = F 0 1 h (3.2.14) T X X X c l 1 , / 2 + A 0 2 i a ( 1 ) + ( A 0 2 2 + 1 / 2 ) C ( 1 ) / 2 = F 0 2 h , (3.2.15) T X X X where A 0 , , , A 0 , 2 , A 0 2 1 , A 0 2 2 , F 0 1 and F 0 2 are the v a l u e s of A,,, A , 2 , A 2 , , A 2 2 , F, and F 2 r e s p e c t i v e l y , evaluated f o r 7=8 and k = K 0 « Expanding the d i s p e r s i o n r e l a t i o n s h i p (3.1.5) i n a T a y l o r s e r i e s about u=0 g i v e s at 0(u) K< 1 » = N 0[a< 1 ' - c ( 1>/2] - c< 1'/2, (3.2.16) 68 where N 0 = -{5R + U 0) 2R/6}/{4 + 8/R + U 0) 2R/6} with R=K 2(6)/K,(6). From (3.2.14) and (3.2.15) i t f o l l o w s that a'1> + p,a<1> + p 2 a ( 1 > = v,h (3.2.17) TT XT XX XX C C D + P l c ( 1 > + p 2 c ( 1 ) = 2u2h , (3.2.18) TT XT XX XX where P i = A01 1 + A 0 2 2 + 1 P2 = ( A 0 2 2 + 1 / 2 ) ( A 0 I I + 1/2) - A 0 i 2 A 0 2 1 V\ = ( A 0 2 2 + 1 / 2 ) F 0 1 - A 0 1 2 F 0 2 v2 = ( A 0 1 1 + 1 / 2 ) F 0 2 " A 0 2 1 F 0 1 . The s o l u t i o n s to (3.2.17) and (3.2.18) sub j e c t t o the i n i t i a l c o n d i t i o n s a ( 1 ) ( X , Y , 0 ) = 0, a ( 1 ) ( X r Y , 0 ) = F 0 1 h (3.2.19a) T X c ( 1 ) ( X , Y , 0 ) = 0, c ( 1 ) ( X , Y , 0 ) = 2 F 0 2 h (3.2.19b) T X are g i v e n by 69 a ( 1 ) ( X , Y , T ) = X t h U , ? ) + X 2h(X-a 1T,Y) + X 3 h ( X - a 2 T , Y ) , (3.2.20) c ( 1 ) ( X , Y , T ) = X„h(X,Y) + X 5h(X-a,T rY) + X 6 h ( X - 0 2 T , Y ) , (3.2.21) where the c o e f f i c i e n t s are d e f i n e d as X, = v, (a,a 2) • 1 X 2 = ( F o t o , - » 1 ) [ o 1 ( a 2 - ffi-)]"1 X 3 = (v, - F 0 i o 2 ) [ o 2 ( a 2 ~ a , ) ] ' 1 Xft = 2v2(o)a2)'1 X 5 = 2 ( F 0 2 a 1 - v 2 ) [ o , ( o 2 ~ a , ) ] - 1 X 6 = 2{v2 - F 0 2 o 2 ) [ a 2 ( a 2 - a , ) ] " 1 , and where the c h a r a c t e r i s t i c speeds a, and o2 are given by a, - {p, - [ ( p , ) 2 - 4 p 2 ] 1 ' 2 } / 2 c 2 = {p, + [ ( P i ) 2 - 4 p 2 ] 1 / 2 } / 2 . The i n i t i a l c o n d i t i o n s on a ( 1 } and c ( 1 ) (see (3.2.19)) are T T obtained by e v a l u a t i n g (3.2.14) and (3.2.15) at T=0, r e s p e c t i v e l y . The s o l u t i o n f o r K ( 1>(X,Y,T) i s determined by 70 (3.2.16), (3.2.20) and (3.2.21) y i e l d i n g K ( 1 ) ( X , Y , T ) = X 7 h ( X , Y ) + X 8 h ( X - a , T , Y ) + X 9 h ( X - a 2 T , Y ) , (3.2.22) with X 7, X B and X 9 given by X 7 = N 0(X, - X,/2) - X„/2 X 8 = N 0 ( X 2 - X 5/2) - X 5/2 X 9 = N 0 ( X 3 - X 6/2) - X 6/2. 3.2.2 D i s c u s s i o n Of The Small-Amplitude Topographic S o l u t i o n The slowly v a r y i n g modon w i l l be d e s c r i b e d (to l e a d i n g order) by (3.1.3) and (3.1.4) with a, c and K given by a = 1 + M a ( 1 ' ( X , Y , T ) (3.2.23) c = 1 + ucl1>(X,Y,T) (3.2.24) K = K0[1+ UK1 1 ' ( X,Y,T) ] , (3.2.25) where a ( 1 ' , c ( 1 ) and K ( 1 ' are giv e n by (3.2.20), (3.2.21) and (3.2.22) r e s p e c t i v e l y , e v a l u a t e d at the modon ce n t e r X at time T. The p o s i t i o n of the modon c e n t e r as a f u n c t i o n of time i s obtained from the c h a r a c t e r i s t i c e quation 71 dX/dT = c(X,0,T), (3.2.26) w r i t t e n i n slow v a r i a b l e s . (Note t h a t the modon center to l e a d order occurs on Y=0; see (3.1.3) and (3.1.4).) In g e n e r a l , (3.2.26) i s not separable so X(T) i s d e f i n e d i m p l i c i t l y and must be obtained n u m e r i c a l l y . F o r m a l l y , the s o l u t i o n of (3.2.26) i s w r i t t e n T X(T) = e £ 0 + J c[X(T),0,T]dT (3.2.27) o where X ( 0 ) = e £ o (see Subs e c t i o n 3.1.1). However i f a t t e n t i o n i s r e s t r i c t e d to T<<1 ( i . e . , t < < e ~ 1 ) then C(X,0,T) « c ( e £ o , 0 , 0 ) + c ( e £ o , 0 , 0 ) T + 0 ( T 2 ) . T Consequently, (3.2.27) and (3.2.15) i m p l i e s that the modon center w i l l be approximately given by X = e £ 0 + T + MF 0 2h ( e £ o , 0 ) T 2 + 0 ( T 3 ) . (3.2.28) X Using (3.2.28) the space v a r i a b l e X can be e l i m i n a t e d from the s o l u t i o n s (3.2.23), (3.2.24) and (3.2.25) t o o b t a i n a n a l y t i c asymptotic ( i . e . , T<<1) s o l u t i o n s i n the s i n g l e slow time v a r i a b l e T. 72 As a t y p i c a l c a l c u l a t i o n of the s o l u t i o n parameters l e t 6=1.0, hence: /c0 = 3.9226 (from 3.1.5), N o=-0.9636, p,=-7.39, p 2=-39.44, J>,=2.00, * 2=-9.20, a,=-l0.98, a 2=3.59, F O i=4.04 and F 0 2 = -0.63 from which i t f o l l o w s X,=-0.05, X 2=0.29, X 3=-0.24, X„=0.46, X 5=-0.20, X 6=-0.26, X 7=0.04, X8=-0.28 and X 9=0.24. Other values of 6 give q u a l i t a t i v e l y s i m i l a r r e s u l t s (except near the s i n g u l a r value 6=5.776). The i n i t i a l - v a l u e s o l u t i o n t h e r e f o r e takes the form of eastward and w e s t w a r d - t r a v e l l i n g h y p e r b o l i c waves and a s t a t i o n a r y component p r o p o r t i o n a l t o the topography. Note that i n consequence of the i n i t i a l c o n d i t i o n s (3.2.19), the parameters s a t i s f y X, + X 2 + X 3 = 0 (3.2.29a) X« + X 5 + X 6 = 0 (3.2.29b) X 7 + X 8 + X 9 = 0. (3.2.29c) These r e l a t i o n s are c l e a r l y r e q u i r e d i f h = constant i s to r e s u l t i n only the t r i v i a l s o l u t i o n ( i . e . , a ( 1 ) , c ( 1 > and K ( 1 ' are a l l z e r o ) . Consider the s o l u t i o n f o r a ( 1 ) ( X , Y , T ) given by (3.2.20). Since X 3<0, the r a d i u s - p e r t u r b a t i o n a s s o c i a t e d with the e a s t w a r d - t r a v e l l i n g wave ( i . e . , h(X-o 2T,Y)) a c t s to decrease the modon r a d i u s . S i n c e X 2>0, the r a d i u s - p e r t u r b a t i o n a s s o c i a t e d with the w e s t w a r d - t r a v e l l i n g wave ( i . e . , hfX-c^TjY)) a c t s to i n c r e a s e the r a d i u s . Since X,<0, the r a d i u s -p e r t u r b a t i o n a s s o c i a t e d with the s t a t i o n a r y component ( i . e . , h(X,Y)) a c t s to decrease the modon r a d i u s . These r e s u l t s have 73 i m p l i c i t l y assumed a p o s i t i v e topography ( i . e . , h ( X , Y ) £ 0 ) . I f one c o n s i d e r s a topographic d e p r e s s i o n ( i . e . , h ( X , Y ) £ 0 ) , the r e s u l t s are simply reversed. S i m i l a r reasoning a p p l i e s to the t r a n s l a t i o n speed c ( X , Y , T ) s o l u t i o n (3.2.21). Since X5<0 and X 6<0, the speed-perturbations a s s o c i a t e d with the w e s t w a r d - t r a v e l l i n g wave and eastward-t r a v e l l i n g wave act to decrease the t r a n s l a t i o n speed. Since X„>0, the spe e d - p e r t u r b a t i o n a s s o c i a t e d with the s t a t i o n a r y component of the s o l u t i o n a c t s to i n c r e a s e the t r a n s l a t i o n speed. As b e f o r e , i f one c o n s i d e r s a topographic depression these r e s u l t s are reversed. The q u a l i t a t i v e behaviour of the wavenumber-perturbations a s s o c i a t e d w i t h the i n d i v i d u a l s o l u t i o n components i n (3.2.22) i s opposite t o that of the r a d i u s - p e r t u r b a t i o n s . Since X7>0 and X9>0, the wavenumber-perturbations a s s o c i a t e d with the s t a t i o n a r y and e a s t w a r d - t r a v e l l i n g wave components act to increase the modon wavenumber. Since X8<0 the wavenumber-p e r t u r b a t i o n a s s o c i a t e d with the w e s t w a r d - t r a v e l l i n g wave a c t s to i n c rease the modon wavenumber. As b e f o r e , i f one c o n s i d e r s a topographic d e p r e s s i o n these r e s u l t s are reversed. When the topography s a t i s f i e s h ( X , Y ) — > A *( Y) and A ~ ( Y ) f o r X —>+» and X —>-» r e s p e c t i v e l y , i t i s meaningful to speak of a ' l o c a l ' ( i . e . , i n a neighbourhood of X=0) s t e a d y - s t a t e s o l u t i o n ( i . e . , T->») f o r a ( X , Y , T ) , c ( X , Y , T ) and K ( X , Y , T ) . Note that i f one d e f i n e s h ( X , Y ) = A ~ ( Y ) + h ' ( X , Y ) , the terms- c o n t a i n i n g A " ( Y ) i n (3.2.20), (3.2.21) and (3.2.22) sum to zero due to (3.2.29). Thus with no l o s s of g e n e r a l i t y we set A~=0 and assume h ( X , Y ) i s 74 r e l a t i v e to the upstream topography (which may depend on Y but i s a s y m p t o t i c a l l y constant i n X ) . Under these c o n d i t i o n s (3.2.20), (3.2.21) and (3.2.22) w i l l imply l o c a l s t e a d y - s t a t e s o l u t i o n s of the form a * 1 + M { X 2 [ A * - h ( X , Y ) ] - X 3 h ( X , Y ) } (3.2.30) c * 1 + u{\s[h* - h ( X , Y ) ] - X 6 h ( X , Y ) } (3.2.31) + K 0 M { X 8 [ A + - h ( X , Y ) ] - X 9 h ( X , Y ) } (3.2.32) ( i n a neighbourhood of X=0 f o r s u f f i c i e n t l y l a r g e t i m e ) . Two cases are of i n t e r e s t . Consider the s t r u c t u r e of the approximate l o c a l s t e a d y - s t a t e s o l u t i o n s f o r i s o l a t e d (or compact) topography ( i . e . , A*=0). Then (3.2.30), (3.2.31) and (3.2.32) f u r t h e r reduce to a =* 1 + M X , M X , Y ) (3.2.33) c ~ 1 + M X , h ( X , Y ) (3.2.34) K — KQ + M K 0 X 7 h ( X , Y ) (3.2.35) where (3.2.29) has been used. Since X,<0, (3.2.33) i m p l i e s that a^1. T h e r e f o r e the r a d i u s of the modon decreases as i t t r a v e l s over i s o l a t e d p o s i t i v e topography ( i g n o r i n g the t r a n s i e n t waves s i n c e T>>1 i s assumed). I f the modon were t r a v e l l i n g over an 75 i s o l a t e d d e p r e s s i o n , say a t r e n c h , i t s r a d i u s would i n c r e a s e ( i . e . , a£l) s i n c e h(X,Y)£0. From (3.2.34), c>1 over p o s i t i v e i s o l a t e d topography and c<1 over i s o l a t e d depressions s i n c e X„>0. Modon p a r t i c l e speeds are on the same order as the t r a n s l a t i o n speed so the f l u i d a c c e l e r a t e s over i s o l a t e d p o s i t i v e topography and d e a c c e l e r a t e s over i s o l a t e d d e p r e s s i o n s , i n l i n e with i n t u i t i v e c o n t i n u i t y arguments. The modon wavenumber a d j u s t s t o i s o l a t e d topography i n j u s t the reverse manner as the modon r a d i u s (as might be expected s i n c e a wavenumber i s d i m e n s i o n a l l y the in v e r s e of a l e n g t h ) . From (3.2.35), the modon wavenumber w i l l s a t i s f y K>K0 over p o s i t i v e i s o l a t e d topography and s a t i s f i e s K<K0 over i s o l a t e d d e p r e ssions s i n c e X 7>0. In summary, the f o l l o w i n g q u a l i t a t i v e s t r u c t u r e develops as the modon t r a v e l s over i s o l a t e d slowly v a r y i n g topography ( i g n o r i n g the t r a n s i e n t waves). I f the topography i s p o s i t i v e , the modon t r a n s l a t i o n speed i n c r e a s e s and hence the p a r t i c l e speeds i n c r e a s e , the r a d i u s decreases and the wavenumber i n c r e a s e s . The modon t h e r e f o r e c o n t r a c t s and a c c e l e r a t e s over p o s i t i v e topography. Over i s o l a t e d d e p r e s s i o n s the reverse happens. The modon d i l a t e s and d e a c c e l e r a t e s . Subsection 3.2.3 i l l u s t r a t e s the e v o l u t i o n of the modon over p o s i t i v e i s o l a t e d topography by c o n s i d e r i n g a m e r i d i o n a l r i d g e modelled with a gaussian i n the x-coordinate d i r e c t i o n as an example. When A +(Y) i s not zero the approximate s o l u t i o n s (3.2.30), (3.2.31) and (3.2.32) depend on the f a r f i e l d ( i . e . , X » 1 ) 76 topographic s t r u c t u r e . The terms with A* represent the t r a n s m i s s i o n of t h i s f a r f i e l d i n f o r m a t i o n i n t o the X=0 neighbourhood by way of the w e s t w a r d - t r a v e l l i n g h y p e r b o l i c wave. In g e n e r a l , of course, the a n a l y s i s w i l l be d i f f e r e n t f o r each assumed form f o r A * ( Y ) . To i l l u s t r a t e the d i f f e r e n c e s and s i m i l a r i t i e s between i s o l a t e d and n o n i s o l a t e d topography c o n s i d e r A* as a nonzero constant. A l l o w i n g a m e r i d i o n a l i dependence does not complicate the e s s e n t i a l ideas as the s o l u t i o n s depend only p a r a m e t r i c a l l y on Y. Consider the case 0£h£A*<» (henceforth c a l l e d i n c r e a s i n g topography). I t f o l l o w s from (3.2.30) that a>1 f o r a l l X s i n c e X2>0 and X 3<0. As X i n c r e a s e s , h—>A* so a—>1-/xX 3A*£1 . As X decreases, h->0 so a->1+M X 2 A*£1. However, s i n c e |X2|>|X3| i t f o l l o w s that there w i l l ' b e a r e l a t i v e decrease i n the modon ra d i u s from X<<0 to X » 0 f o r s u f f i c i e n t l y l a r g e T. From (3.2.31), c>1 f o r X>0 s i n c e X 6<0, and c<1 f o r X<0 s i n c e X 5<0. Because |X6|>|X5| the i n c r e a s e i n the t r a n s l a t i o n speed f o r X>0 i s l a r g e r than the decrease in the t r a n s l a t i o n speed fo r X<0. The p a r t i c l e speeds w i l l a l s o have t h i s p r o p e r t y . The behaviour of the modon wavenumber i s again reversed to that of the modon r a d i u s . There i s a g l o b a l decrease in the wavenumber s i n c e a l l terms i n (3.2.32) are n e g a t i v e . However, s i m i l a r reasoning to that given f o r the modon r a d i u s shows there i s a r e l a t i v e i n c r e a s e i n the modon wavenumber from X<0 to X>0 s i n c e |X8|<|X9|. In Subsection 3.2.4 the s o l u t i o n s f o r a topographic escarpment modelled as a h y p e r b o l i c tangent i n the 77 x - c o o r d i n a t e d i r e c t i o n are presented as an example of a topographic c o n f i g u r a t i o n d i s p l a y i n g the above p r o p e r t i e s . The behaviour of the s o l u t i o n s f o r a d e c r e a s i n g topographic c o n f i g u r a t i o n (-°><A + £h£0, i . e . , a topographic depression) g i v e s the o p p o s i t e r e s u l t s to those s t a t e d above. The modon r a d i u s decreases f o r a l l X with a r e l a t i v e i n c r e a s e from X<0 to X>0. The t r a n s l a t i o n speed w i l l s a t i s f y c>1 f o r X<0 and c<1 f o r X>0 with the decrease i n c f o r X<0 l a r g e r than the i n c r e a s e i n c f o r X>0. The p a r t i c l e speeds w i l l a l s o have t h i s p r o p e r t y . The modon wavenumber w i l l i n c r e a s e f o r a l l X with a r e l a t i v e decrease from X<0 to X>0. The l e a d i n g order s o l u t i o n 0 } ( £ ,y ;X,Y,T) i s not expected to be u n i f o r m l y v a l i d as £—>±°°. I t i s e a s i l y seen that 0 ' i s a homogeneous s o l u t i o n to the 0(e) problem (3.2.4)'and that c e r t a i n terms in the RHS's of (3.2.4) and (3.2.6) are p r o p o r t i o n a l to A\//(0). Thus resonance w i l l i n general occur and the expansion (3.2.2) i s not expected to be v a l i d f o r a l l £. In one dimensional s o l i t a r y waves t h i s n o n u n i f o r m i t y manifests i t s e l f as a ' s h e l f ahead and behind of the s o l i t a r y wave (e.g., Ko and Kuelh; 1978 and Grimshaw; 1979a,b and Kodama and A b l o w i t z ; 1980). The s h e l f appears as a consequence of <^1'—>B* and B " as £—>+» and -» r e s p e c t i v e l y , where B * and B " are c o n s t a n t s . In g e n e r a l , i t i s assumed that ahead of the s o l i t a r y wave the f l u i d (to l e a d i n g order) i s u n d i s t u r b e d , thus B*=0 (e.g., Grimshaw, 1979a,b). The s h e l f r e g i o n behind the wave i s removed by the i n t r o d u c t i o n of an outer expansion v a l i d f o r £ =* 0 ( e " n ) (the power n i s determined i n the problem) (e.g., 78 Grimshaw; 1979a,b and Kodama and Ablowtiz, 1980). However, s i n c e i t i s the term 1' that determines p a r t of the 0(e) v e l o c i t y f i e l d , t h i s constant does not c o n t r i b u t e to ( u ( 1 ) , v ( 1 ) ) , consequently a nonuniformity of t h i s type does not seem to be r e l e v a n t here. 3.2.3 G a u s s i a n - r i d g e Topography In t h i s Subsection the s o l u t i o n f o r modon propagation over a slowly v a r y i n g r i d g e i s d e s c r i b e d . The topography i s given by H(X,Y) = 1 - <xexp(-X 2), (3.2.36) hence h(X,Y) = e x p ( - X 2 ) . Since h->0 as |X|->°° the q u a l i t a t i v e behaviour of the s o l u t i o n s f o r a, c and K w i l l f o l l o w the a n a l y s i s given i n Subsection 3.2.3 f o r the case A* = 0. F i g u r e s 7, 8 and 9 i l l u s t r a t e the s p a t i a l s t r u c t u r e of the modon t r a n s l a t i o n speed c=1+uc ( 1 ), modon r a d i u s a=1+Ma ( 1 ) and modon wavenumber K = K 0 + M « O K ( ' 1 r e s p e c t i v e l y , with ^=0.2 f o r the sequence of slow times T=0.0, 0.2, 0.6, 1.4 and 4.0. These f i g u r e s r epresent s p a c e - l i k e s l i c e s ( i . e . , T h e l d c o n s t a n t ) of the modon parameters i n space-time. The p a r t i c u l a r v a l u e of a, c and K t h a t a p a r t i c u l a r modon experiences at a g i v e n time i s , of course, determined by the c h a r a c t e r i s t i c i n space-time the modon i s propagating on. T h i s c h a r a c t e r i s t i c i s the unique s o l u t i o n of (3.2.26), which i s approximately (3.2.28) ( f o r T « 1 ) , with the given h(X) i n (3.2.36). F i g u r e 7a shows the i n i t i a l c o n d i t i o n on the t r a n s l a t i o n 79 speed ( i . e . , C(X,0,0)=1). F i g u r e 7b shows the s p a t i a l s t r u c t u r e of c at T=0.2. The w e s t w a r d - t r a v e l l i n g and e a s t w a r d - t r a v e l l i n g wave t r a n s i e n t s are j u s t becoming d i s t i n g u i s h a b l e from the s t a t i o n a r y component i n (3.2.21). In F i g u r e 7c the se p a r a t i o n i s more apparent. The w e s t w a r d - t r a v e l l i n g wave i s f u r t h e r removed from X=0 than i s the e a s t w a r d - t r a v e l l i n g wave si n c e |o 1|>|a 2|. The amplitude of the e a s t w a r d - t r a v e l l i n g wave i s l a r g e r than the w e s t w a r d - t r a v e l l i n g wave si n c e |\6|>|X5|. By the time T=1.4 (Figure 7d) the w e s t w a r d - t r a v e l l i n g wave has l e f t the d i s p l a y e d space domain ( i . e . , -10 ^ X^10). The more slowly moving e a s t w a r d - t r a v e l l i n g wave has completely separated from the s t a t i o n a r y component. F i g u r e 7e shows the s t a t i o n a r y component of the s o l u t i o n (3.2.21). T h i s i s the l o c a l steady-s t a t e s o l u t i o n f o r c ( X,Y,T) d e s c r i b e d l a s t S e c t i o n f o r i s o l a t e d topography. F i g u r e s 8a, 8b, 8c, 8d and 8e show the s p a c e - l i k e s t r u c t u r e i n the modon r a d i u s . Since the t r a n s l a t i o n speeds f o r the w e s t w a r d - t r a v e l l i n g and e a s t w a r d - t r a v e l l i n g t r a n s i e n t waves i n ,(3.2.20) are the same as i n the s o l u t i o n s f o r c ( 1 > (see (3.2.21)), the s e p a r a t i o n behaviour i n the t r a n s i e n t s i s the same as that given above. The e a s t w a r d - t r a v e l l i n g wave (see F i g u r e s 8c and 8d) a c t s to decrease the modon wavenumber ( i f the modon had an i n i t i a l p o s i t i v e X p o s i t i o n ) s i n c e X 3<0. The w e s t w a r d - t r a v e l l i n g t r a n s i e n t (see F i g u r e s 8b and 8c) a c t s to in c r e a s e the modon r a d i u s s i n c e X 2>0. The s t a t i o n a r y component of the modon r a d i u s s o l u t i o n (see F i g u r e 8e) a c t s to reduce the modon r a d i u s over the topography s i n c e X^O. 80 F i g u r e s 9a, 9b, 9c, 9d and 9e show the s p a c e - l i k e s t r u c t u r e of the modon wavenumber. The i n i t i a l c o n d i t i o n i s shown i n Fi g u r e 11a. The s e p a r a t i o n behaviour i s the same as i n the above comments s i n c e the t r a n s l a t i o n speeds are the same. The w e s t w a r d - t r a v e l l i n g t r a n s i e n t (see F i g u r e s 9b and 9c) a c t s to decrease the wavenumber si n c e X 8<0. The e a s t w a r d - t r a v e l l i n g t r a n s i e n t (see F i g u r e s 9c and 9d) a c t s to i n c r e a s e the modon wavenumber s i n c e X 9>0. The s t a t i o n a r y component (see F i g u r e 9e) ac t s to i n c r e a s e the modon wavenumber over the topography. In summary, co n s i d e r the f o l l o w i n g q u a l i t a t i v e behaviour of the modon i f i t s i n i t i a l p o s i t i o n i s , say, e£ o=-l0. The modon w i l l i n i t i a l l y propagate eastward u n a f f e c t e d by the topography (see F i g u r e s 7b, 8b and 9b). When the c h a r a c t e r i s t i c a s s o c i a t e d with the w e s t w a r d - t r a v e l l i n g t r a n s i e n t i n t e r s e c t s the modon c h a r a c t e r i s t i c , the modon t r a n s l a t i o n speed i s reduced, i t s ra d i u s i s in c r e a s e d and i t s wavenumber decreases. The modon continu e s to propagate eastward s i n c e c>0 (see F i g u r e 7) throughout t h i s i n i t i a l i n t e r a c t i o n . A f t e r s u f f i c i e n t time the w e s t w a r d - t r a v e l l i n g wave propagates completely through the modon c h a r a c t e r i s t i c and subsequently does not c o n t r i b u t e to the s o l u t i o n . Some time l a t e r the modon c h a r a c t e r i s t i c i n t e r s e c t s the space-time region c o n t a i n i n g the c h a r a c t e r i s t i c s a s s o c i a t e d with the s t a t i o n a r y component of the s o l u t i o n . The modon t r a n s l a t i o n speed i n c r e a s e s , the r a d i u s decreases and the wavenumber in c r e a s e s (see F i g u r e s 7e, 8e and 9e). Subsequent to t h i s i n t e r a c t i o n there i s no f u r t h e r i n t e r a c t i o n with the topography 81 or the t r a n s i e n t waves. The e a s t w a r d - t r a v e l l i n g t r a n s i e n t propagates with speed a 2=3.59 which i s gr e a t e r than the modon t r a n s l a t i o n speed (see F i g u r e 7) and thus the two c h a r a c t e r i s t i c s ( i . e . , of the modon and the e a s t w a r d - t r a v e l l i n g wave) never i n t e r s e c t f o r T>0 i f £ 0 < n « T h i s a n a l y s i s has assumed that h(X)>0. I f the 'ri d g e ' was a d e p r e s s i o n ( i . e . , h(X)<0) the above r e s u l t s would be rev e r s e d . If So-0 (say e £ 0 = 1 u ) , the q u a l i t a t i v e a n a l y s i s i s s i m i l a r to the above except that the e f f e c t s of the e a s t w a r d - t r a v e l l i n g t r a n s i e n t must taken i n t o account. C l e a r l y i n t h i s s c e n a r i o the w e s t w a r d - t r a v e l l i n g wave does not a f f e c t the modon si n c e f o r s p a t i a l c o o r d i n a t e s g r e a t e r than the c u r r e n t l o c a t i o n of the modon center the topography i s i d e n t i c a l l y z e r o . Hence c h a r a c t e r i s t i c s o r g i n a t i n g from s p a t i a l c o o r d i n a t e s p o s i t i v e of the modon center at any given time have zero amplitude and t h e r e f o r e do not a f f e c t the modon. 82 in IT) CM Q U J O Q_ cn ID LT) CD ' -10.0 -6.0 -2.0 X 2.0 6.0 10.0 F i g u r e 7a. Sequence of s p a c e - l i k e s l i c e s showing the space-time e v o l u t i o n of the modon t r a n s l a t i o n speed f o r modon propagation over a slowly v a r y i n g gaussian r i d g e centered at X=0. T h i s p l o t shows the i n i t i a l c o n d i t i o n ( i . e . at T=0.0) on the t r a n s l a t i o n speed ( i . e . C ( X , Y , 0 ) = 1 ) . 83 in in CM in CD in <=> I I I I 1 1 -10.0 -6.0 -2.0 2.0 6.0 10.0 X F i g u r e 7b. S p a c e - l i k e s t r u c t u r e of the modon t r a n s l a t i o n speed i n d u c e d by a g a u s s i a n r i d g e a t T=0.2. 84 F i g u r e 7c. S p a c e - l i k e s t r u c t u r e of the modon t r a n s l a t i o n speed induced by a gaussian r i d g e at T=0.6. 85 F i g u r e 7d. S p a c e - l i k e s t r u c t u r e of the modon t r a n s l a t i o n speed induced by a gaussian ridge at T=1.4. 86 i n LT) CD Q_ CO in o in oH 1 1 — 1 i i - 1 0 . 0 - 6 . 0 - 2 . 0 2 .0 6 .0 10.0 X F i g u r e 7e. S p a c e - l i k e s t r u c t u r e of the modon t r a n s l a t i o n speed induced by a gaussian r i d g e at T=4.0. 87 in in CNJ CE in in o •10.0 -6.0 -2.0 X 2.0 6.0 10.0 F i g u r e 8a. Sequence of s p a c e - l i k e s l i c e s showing the space-time e v o l u t i o n of the modon radiu s f o r modon propagation over a slowly v a r y i n g gaussian r i d g e c e n t e r e d at X=0. T h i s p l o t shows the i n i t i a l c o n d i t i o n ( i . e . at T=0.0) on the r a d i u s ( i . e . a(X,Y,0)=1). 88 F i g u r e 8b. S p a c e - l i k e s t r u c t u r e of the modon r a d i u s induced by a g a u s s i a n r i d g e a t T=0.2. 89 F i g u r e 8c. S p a c e - l i k e s t r u c t u r e of the modon r a d i u s induced by a gaussian r i d g e at T=0.6. 90 i n in CO =>< CD' CX in r-in •10.0 -6.0 -2.0 X 2.0 6.0 10.0 F i g u r e 8d. S p a c e - l i k e s t r u c t u r e of the modon r a d i u s induced by a gaussian r i d g e at T=1.4. 91 i n tn CNJ CO a— cr in r -o in CD 10.0 -6.0 -2.0 X 2.0 6.0 10.0 F i g u r e 8e. S p a c e - l i k e s t r u c t u r e of the modon r a d i u s induced by a gaussian r i d g e at T=4.0. 92 IT) tn CM CC 0_<=> CC ^1 in r-i n cn' -10.0 -6.0 -2.0 X 2.0 6.0 10.0 F i g u r e 9a. Sequence of s p a c e - l i k e s l i c e s showing t h e space-time e v o l u t i o n of t h e modon wavenumber f o r modon p r o p a g a t i o n over a s l o w l y v a r y i n g g a u s s i a n r i d g e c e n t e r e d a t X=0. T h i s p l o t shows the i n i t i a l c o n d i t i o n ( i . e . a t T=0.0) on the wavenumber ( i . e . K ( X , Y , 0 ) = K O ) • 93 F i g u r e 9b. S p a c e - l i k e s t r u c t u r e of the modon wavenumber induced by a g a u s s i a n r i d g e a t T=0.2. 94 F i g u r e 9c. S p a c e - l i k e s t r u c t u r e of the modon wavenumber induced by a gaussian r i d g e at T=0.6. 95 F i g u r e 9d. S p a c e - l i k e s t r u c t u r e of the modon wavenumber induced by a gaussian r i d g e at T=1.4. 96 in in CNJ CE CE in r-in •10.0 -6.0 -2.0 X 2.0 6.0 10.0 F i g u r e 9e. S p a c e - l i k e s t r u c t u r e of the modon wavenumber induced by a gaussian r i d g e at T=4.0. 97 3.2.4 Tanh-escarpment Topography In t h i s S u b s e c t i o n the s o l u t i o n f o r modon propagation over a slowly v a r y i n g escarpment i s d e s c r i b e d . The topography i s given by H(X,Y) = 1 - u{] + tanh(X)}/2, (3.2.37) hence h(X,Y) = {1 + tanh(X)}/2. Since h->0 and h->1 as X->-» and X—>°° r e s p e c t i v e l y , the q u a l i t a t i v e behaviour of the s o l u t i o n s w i l l f o l l o w the a n a l y s i s given i n Subsection 3.2.3 f o r the case 0£h^A*<°°. F i g u r e s 10, 11 and 12 i l l u s t r a t e the s p a t i a l s t r u c t u r e of the modon t r a n s l a t i o n speed C=1+MC ( 1 ), modon radiu s a=1+Ma ( 1 ) and modon'wavenumber K = K 0 + ( I K 0 K ( 1 ' r e s p e c t i v e l y , with u=0.2 f o r the sequence of slow times T=0.0, 0.2, 0.6, 1.4 and 4.0. These f i g u r e s represent s p a c e - l i k e s l i c e s ( i . e . , T he l d constant) i n space-time of the s o l u t i o n s f o r the modon t r a n s l a t i o n speed, modon r a d i u s and modon wavenumnber. As s t a t e d i n Subsection 3.2.3 the p a r t i c u l a r values of a, c and K that the modon experiences a t a given time are determined by the i n t e r s e c t i o n of the c h a r a c t e r i s t i c on which the modon i s propagating and the c h a r a c t e r i s t i c s d e f i n i n g the i n d i v i d u a l s o l u t i o n components i n (3.2.20), (3.2.21) and (3.2.22). The modon c h a r a c t e r i s t i c i s d e f i n e d by (3.2.26) or (3.2.27) or a s y m p t o t i c a l l y ( i . e . , T<<1) by (3.2.28) with h(X) given i n (3.2.37). 98 The q u a l i t a t i v e a n a l y s i s of the i n t e r a c t i o n between the escarpment topography i n t h i s Subsection f o l l o w s that of the d e s c r i p t i o n given l a s t S u bsection. Thus we only b r i e f l y comment on the s i m i l a r aspects and h i g h l i g h t the d i f f e r e n c e s . F i g u r e 10 shows s p a c e - l i k e s l i c e s i n space-time of the slo w l y v a r y i n g modon t r a n s l a t i o n speed. As i n Subsection (3.2.3) the westward-t r a v e l l i n g t r a n s i e n t a c t s to reduce the modon t r a n s l a t i o n speed (see F i g u r e 10a) f o r X<0 and the e a s t w a r d - t r a v e l l i n g t r a n s i e n t a c t s to in c r e a s e the t r a n s l a t i o n speed f o r X>0. S i m i l a r l y , the modon ra d i u s (wavenumber) i s i n c r e a s e d (decreased) by the we s t w a r d - t r a v e l l i n g t r a n s i e n t and decreased (increased) by the ea s t w a r d = t r a v e l l i n g t r a n s i e n t . However the f a c t that h(X)—>1 as X—>+°° i s of c r u c i a l importance here. The escarpment f o r c e s . a permanent deformation i n the modon once i n i t i a l i n t e r a c t i o n begins. The westward-t r a v e l l i n g and e a s t w a r d - t r a v e l l i n g t r a n s i e n t s have the r o l e of t r a n s m i t t i n g i n f o r m a t i o n of the eastward and westward topographic c o n f i g u r a t i o n s . If a modon has i n i t i a l p o s i t i o n e£ o<0, a f t e r s u f f i c i e n t time the modon c h a r a c t e r i s t i c w i l l i n t e r s e c t the c h a r a c t e r i s t i c s c o n t a i n i n g the X>0 topographic i n f o r m a t i o n . However s i n c e the upstream topography does not e v e n t u a l l y approach zero a l l subsequent w e s t w a r d - t r a v e l l i n g c h a r a c t e r i s t i c s w i l l c a r r y a nonzero s i g n a l . Therefore the s p a t i a l s t r u c t u r e of the modon parameters i s c o n t i n u o u s l y deformed. T h i s s p a t i a l deformation i s shown i n Fi g u r e s 10, 11 and 12. The w e s t w a r d - t r a v e l l i n g t r a n s i e n t r e s u l t s i n a continued 99 r e d u c t i o n i n the modon t r a n s l a t i o n speed f o r X<0 and a continued i n c r e a s e i n the modon t r a n s l a t i o n speed f o r X>0 (see F i g u r e s 10b, 10c and 1Od f o r the t r a n s i e n t behaviour and F i g u r e 1Od f o r the l o c a l s t e a d y - s t a t e s t r u c t u r e ) . The modon r a d i u s i s i n c r e a s e d f o r the e n t i r e domain upstream of the l e a d i n g edge of the w e s t w a r d - t r a v e l l i n g t r a n s i e n t . There i s r e l a t i v e decrease i n the r a d i u s across X=0 sin c e the c o e f f i c i e n t of the s t a t i o n a r y component of the s o l u t i o n i s negative (see Subsection 3.2.3). F i g u r e s 11b, 11c and 11d i l l u s t r a t e the t r a n s i e n t behaviour and F i g u r e 11e the l o c a l s t e a d y - s t a t e s t r u c t u r e . The modon wavenumber i s decreased f o r the e n t i r e domain upstream of the l e a d i n g edge of the w e s t w a r d - t r a v e l l i n g t r a n s i e n t . There i s a r e l a t i v e i n c r e a s e i n the modon wavenumber across X=0 s i n c e the amplitude of the s t a t i o n a r y component i s p o s i t i v e (see Subsection 3.2.3). F i g u r e s 12b, 12c and 12d i l l u s t r a t e the t r a n s i e n t s t r u c t u r e and F i g u r e 12e the l o c a l s t e a d y - s t a t e s o l u t i o n . As an example of the q u a l i t a t i v e behaviour of the slow e v o l u t i o n of the modon as i t goes up and over a escarpment of the form (3.2.39) suppose i n i t i a l l y t h a t the modon center i s at e £ o = _ i n « T n e modon i n i t i a l l y propagates eastward u n a f f e c t e d by the upstream topography. E v e n t u a l l y a f t e r s u f f i c i e n t time the modon c h a r a c t e r i s t i c i n t e r s e c t s the c h a r a c t e r i s t i c s c a r r y i n g the nonzero upstream topographic i n f o r m a t i o n . The modon t r a n s l a t i o n speed decreases, i t s r a d i u s i n c r e a s e s and i t s wavenumber decreases. Since the upstream topography remains nonzero as X 100 i n c r e a s e s and the w e s t w a r d - t r a v e l l i n g t r a n s i e n t c a r r i e s t h i s i n f o r m a t i o n , the modon c h a r a c t e r i s t i c continues to i n t e r s e c t n o n t r i v i a l w e s t w a r d - t r a v e l l i n g t r a n s i e n t c h a r a c t e r i s t i c s . Thus the t r a n s l a t i o n speed remains reduced, the r a d i u s remains i n c r e a s e d and the wavenumber remains decreased. A f t e r s u f f i c i e n t f u r t h e r time the modon propagates up over the escarpment and the t r a n s l a t i o n speed i n c r e a s e s , the ra d i u s decreases and the wavenumber i n c r e a s e s s i n c e the modon c h a r a c t e r i s t i c i s i n t e r s e c t i n g the c h a r a c t e r i s t i c s a s s o c i a t e d with the s t a t i o n a r y p a r t of the p e r t u r b a t i o n s o l u t i o n s . T h i s q u a l i t a t i v e a n a l y s i s assumed that the escarpment i n c r e a s e d .from X<0 to X>0. I f the reverse had been assumed, the above behaviour would be r e v e r s e d . I f the i n i t i a l p o s i t i o n of the modon i s p o s i t i v e (e.g., e £ o > 0 ) , the modon w i l l i n i t i a l l y be u n a f f e c t e d by the topography. I f e £ 0 i s la r g e (e.g., 10), the c h a r a c t e r i s t i c s a s s o c i a t e d with the w e s t w a r d - t r a v e l l i n g t r a n s i e n t and the s t a t i o n a r y component c a r r y constant i n f o r m a t i o n ( i . e . , h(X)==1 f o r X>10). However as the modon c h a r a c t e r i s t i c begins to i n t e r s e c t c h a r a c t e r i s t i c s a s s o c i a t e d with the topographic s t r u c t u r e f o r X<0, the modon t r a n s l a t i o n speed w i l l i n c r e a s e , the modon r a d i u s w i l l i n c r e a s e and the modon wavenumber w i l l decrease. 101 in in Q LxJo Q_ CO in in o •10.0 -6.0 -2.0 X 2.0 6.0 10.0 F i g u r e 10a. Sequence of s p a c e - l i k e s l i c e s showing the space-time e v o l u t i o n of the modon t r a n s l a t i o n speed f o r modon propagation over a slowly v a r y i n g h y p e r b o l i c - t a n g e n t escarpment c e n t e r e d at X=0. T h i s p l o t shows the i n i t i a l c o n d i t i o n ( i . e . at T=0.0) on the t r a n s l a t i o n speed ( i . e . C ( X , Y , 0 ) = 1 ) . 102 LD ID in o in o •10.0 -6.0 -2.0 X 2.0 6.0 10.0 F i g u r e 10b. S p a c e - l i k e s t r u c t u r e of the modon t r a n s l a t i o n speed induced by a h y p e r b o l i c - t a n g e n t escarpment at T=0.2. 103 F i g u r e 10c. S p a c e - l i k e s t r u c t u r e of the modon t r a n s l a t i o n speed induced by a h y p e r b o l i c - t a n g e n t escarpment at T=0.6. 104 LT> LT) CXI ID o LT) CD -10.0 -6.0 -2.0 X 2.0 6.0 10.0 F i g u r e I0d. S p a c e - l i k e s t r u c t u r e of the modon t r a n s l a t i o n speed induced by a h y p e r b o l i c - t a n g e n t escarpment at T=1.4. 105 i n in CM O LLJCD LU_; Q_ CO in CD in o -10.0 -6.0 -2.0 X 2.0 6.0 10.0 F i g u r e I0e. S p a c e - l i k e s t r u c t u r e of the modon t r a n s l a t i o n speed induced by a h y p e r b o l i c - t a n g e n t escarpment at T=4.0. 106 in in CO CE in tn o' •10.0 -6.0 -2.0 2.0 6.0 10.0 Y F i g u r e 11a. Sequence of s p a c e - l i k e s l i c e s showing the space-time e v o l u t i o n of the modon r a d i u s f o r modon p r o p a g a t i o n over slowly v a r y i n g h y p e r b o l i c - t a n g e n t escarpment c e n t e r e d at X=0. T h i s p l o t shows the i n i t i a l c o n d i t i o n ( i . e . a t T=0.0) on the r a d i u s ( i . e . a(X,Y,0)=1). 107 in in o m o 10.0 -6.0 -2.0 X 2.0 6.0 10.0 F i g u r e 11b. S p a c e - l i k e s t r u c t u r e of the modon r a d i u s induced by a h y p e r b o l i c - t a n g e n t escarpment at T=0.2. 108 in in CM in oH I I : I I I -10.0 -6.0 -2.0 2.0 6.0 10.0 X F i g u r e 11c. S p a c e - l i k e s t r u c t u r e of the modon r a d i u s induced by a h y p e r b o l i c - t a n g e n t escarpment at T=0.6. 109 in in CM CO cr cn in tn o •10.0 -6.0 -2.0 X 2.0 6.0 10.0 F i g u r e 11d. S p a c e - l i k e s t r u c t u r e of the modon r a d i u s induced by a h y p e r b o l i c - t a n g e n t escarpment a t T=1.4. 110 in in CO ZD o CE en in CD in o •10.0 -6.0 -2.0 X 2.0 6.0 10.0 F i g u r e 11e. S p a c e - l i k e s t r u c t u r e of the modon r a d i u s induced by a h y p e r b o l i c - t a n g e n t escarpment at T=4.0. 111 in in CM cr cr m in ro' •10.0 -6.0 -2.0 X 2.0 6.0 10.0 F i g u r e 12. Sequence of s p a c e - l i k e s l i c e s showing the space-time e v o l u t i o n of the modon wavenumber f o r modon propagation over a slowly v a r y i n g h y p e r b o l i c - t a n g e n t escarpment c e n t e r e d at X=0. T h i s p l o t shows the i n i t i a l c o n d i t i o n ( i . e . at T=0.0) on the wavenumber ( i . e . K(X, Y, 0) = /c0 ). 112 F i g u r e 12b. S p a c e - l i k e s t r u c t u r e of the modon wavenumber induced by a h y p e r b o l i c - t a n g e n t escarpment at T=0.2. 113 F i g u r e 12c. S p a c e - l i k e s t r u c t u r e of the modon wavenumber induced by a h y p e r b o l i c - t a n g e n t escarpment at T=0.6. F i g u r e 12d. S p a c e - l i k e s t r u c t u r e of the modon wavenumber induced by a h y p e r b o l i c - t a n g e n t escarpment at T=1.4. 115 LT) IT) CNJ C E C E in m 10.0 -6.0 -2.0 X 2.0 6.0 10.0 F i g u r e 12e. S p a c e - l i k e s t r u c t u r e of the modon wavenumber induced by a h y p e r b o l i c - t a n g e n t escarpment at T=4.0. 116 IV. CONCLUSIONS T h i s t h e s i s has examined two aspects of the theory of b a r o t r o p i c modons. The f i r s t aspect was examined i n Chapter II i n which a s u f f i c i e n t n e u t r a l s t a b i l i t y c o n d i t i o n ( i n the form of an i n t e g r a l c o n s t r a i n t ) f o r b a r o t r o p i c modons was obtained. E a s t w a r d - t r a v e l l i n g modons are n e u t r a l l y s t a b l e to p e r t u r b a t i o n s s o l e l y composed with wavenumber magnitudes (|*?|) s a t i s f y i n g |TJ|<K where K i s the modon wavenumber. Wes t w a r d - t r a v e l l i n g modons are n e u t r a l l y s t a b l e to p e r t u r b a t i o n s composed s o l e l y of s p e c t r a l components s a t i s f y i n g |T ? | > K (or e l s e the modon i s s t a b l e ) . The e a s t w a r d - t r a v e l l i n g modon n e u t r a l s t a b i l i t y c o n d i t i o n i m p l i e s that when K/|T?|>1 the slope of the n e u t r a l s t a b i l i t y curve proposed by McWilliams et a l . ( l 9 8 l ) should begin to increase as 1171 decreases. A s i m i l a r trend i n the n e u t r a l s t a b i l i t y curve has been n u m e r i c a l l y determined f o r t o p o g r a p h i c a l l y - f o r c e d p l a n e t a r y eddies (Malanotte-R i z z o l i , 1982). As an atmospheric a p p l i c a t i o n , the s t a b i l i t y c o n d i t i o n was c a l c u l a t e d based on the 300, 500 and 700 mb eddy k i n e t i c energy spectrum d e s c r i b e d by Tomatsu(1979), Saltzman and F l e i s h e r ( 1 9 6 2 ) and E l i a s e n and Machenhauer(1965). E a s t w a r d - t r a v e l l i n g modons s a t i s f y the s t a b i l i t y c o n d i t i o n and thus we conclude that for t y p i c a l m i d - l a t i t u d e 700 mb to 300 mb atmosphere e n e r g e t i c s e a s t w a r d - t r a v e l l i n g modons are n e u t r a l l y s t a b l e . A s i m i l a r c a l c u l a t i o n f o r w e s t w a r d - t r a v e l l i n g modons f a i l s to s a t i s f y the 117 s t a b i l i t y c o n d i t i o n and thus the s t a b i l i t y or i n s t a b i l i t y f o r w e s t w a r d - t r a v e l l i n g modons cannot be determined. Simple s c a l i n g arguments suggested that only eastward-t r a v e l l i n g b a r o t r o p i c modons would have r e a l i s t i c t r a n s l a t i o n and p a r t i c l e speeds i n the ocean. The s t a b i l i t y i n t e g r a l was t e s t e d with the FU(1983) mesoscale wavenumber eddy energy spectrum. The s t a b i l i t y of oceanic modons c o u l d not be i n f e r r e d . However due to the short p e r i o d over which the data f o r the FU(1983) spectrum was c o l l e c t e d (24 days) arguments were presented to suggest that the Fu(l983) spectrum may not be r e p r e s e n t a t i v e of e i t h e r the t o t a l eddy energy or i t s wavenumber d i s t r i b u t i o n . Therefore the oceanic c a l c u l a t i o n u s i n g the FU(1983) spectrum should only be considered a very p r e l i m i n a r y estimate of modon s t a b i l i t y i n the oceans. The second aspect of t h i s t h e s i s i s c o n t a i n e d i n Chapter I I I . A l e a d i n g order p e r t u r b a t i o n theory was developed to d e s c r i b e the propagation of a b a r o t r o p i c modon i n a slo w l y v a r y i n g medium. The p e r t u r b a t i o n method developed here r e p r e s e n t s an extension to a two-dimensional s o l i t a r y wave of v a r i o u s c a l c u l a t i o n s made of slowly v a r y i n g one dimensional s o l i t a r y waves (e.g., Luke(l966), Grimshaw(1970, 1971, 1977, 1978, 1979a,b, 1981), Zakharov and Rubenchik(1974) and Kodama and Ablowitz(1980,1981), among o t h e r s ) . Two problem were posed and s o l v e d . In S e c t i o n 3.1 a p e r t u r b a t i o n s o l u t i o n f o r the propagation of an eastward t r a v e l l i n g modon with a bottom boundary l a y e r was o b t a i n e d . The ge o s t r o p h i c pressure has been expanded i n the damping 118 c o e f f i c i e n t e = E 1 / 2 / ( 2 r 0 ) * 10" 1 with E the v e r t i c a l Ekman number and r 0 the Rossby number. The modon r a d i u s ( a ) , t r a n s l a t i o n speed (c) and wavenumber ( K ) are allowed to be f u n c t i o n s of the slow time T = et. The 0(e) equations r e q u i r e a necessary c o m p a t i b i l i t y c o n d i t i o n on the 0(1) s o l u t i o n s (taken to be an e a s t w a r d - t r a v e l l i n g modon) r e s u l t i n g i n n o n l i n e a r i n i t i a l - v a l u e problems f o r the modon parameters. The s o l u t i o n s a = exp(-T), c = exp(-2T) and K = K 0 e x p ( T ) leave the modon d i s p e r s i o n r e l a t i o n s h i p i n v a r i a n t d u r i n g the decay. The amplitude of the modon streamfunction and v o r t i c i t y decays l i k e exp(-3T) and exp(-T), r e s p e c t i v e l y . The maximum d i s t a n c e over which the modon t r a v e l s before complete d i s s i p a t i o n i s about 5 modon ( i n i t i a l ) r a d i i . Based on a comparison with a numerical s o l u t i o n (McWilliams et a l . , 1981) f o r the f r i c t i o n a l d i s s i p a t i o n of an e a s t w a r d - t r a v e l l i n g modon the asymptotic s o l u t i o n obtained here d e s c r i b e s the decay over a 100 day time s c a l e f o r oceanic parameters and a 10 day time s c a l e f o r atmospheric parameters. In S e c t i o n 3.2 a l e a d i n g order p e r t u r b a t i o n theory i s developed to d e s c r i b e modon propagation over slowly v a r y i n g topography. As i n pre v i o u s work on the e f f e c t s of v a r i a b l e topography on b a r o t r o p i c p l a n e t a r y waves (e.g., Veronis(1966), Rhines(1969a,b), C l a r k e d 9 7 1 ) and LeBlond and Mysak(l978), among others) the theory i s developed i n the context of the r i g i d - l i d shallow water equations on the /3-plane. Nonlinear h y p e r b o l i c equations are d e r i v e d f o r the slow 119 e v o l u t i o n of the l e a d i n g order modon r a d i u s , t r a n s l a t i o n speed and wavenumber. These equations are v a l i d f o r a r b i t r a r y slowly v a r y i n g f i n i t e - a m p l i t u d e topography. In general they must be sol v e d n u m e r i c a l l y . I t i s shown th a t to l e a d i n g order the e v o l u t i o n of the modon i s independent of the m e r i d i o n a l topographic s t r u c t u r e . T h i s r e s u l t i s i n t e r p r e t e d using simple v o r t i c i t y arguments. A n a l y t i c a l p e r t u r b a t i o n s o l u t i o n s f o r the modon r a d i u s , t r a n s l a t i o n speed and wavenumber are obtained f o r s m a l l -amplitude topography (which i s the case i n many atmospheric and oceanic a p p l i c a t i o n s ) . The s o l u t i o n s take the form of eastward and w e s t w a r d - t r a v e l l i n g h y p e r b o l i c t r a n s i e n t s and a s t a t i o n a r y component p r o p o r t i o n a l t o the topography. The general p r o p e r t i e s of the s o l u t i o n are d e s c r i b e d i n Subsection 3.2.2. Subsections 3.2.3 and 3.2.4 d e s c r i b e the slowly v a r y i n g modon fo r the s p e c i f i c examples of a topographic r i d g e and escarpment, r e s p e c t i v e l y . 120 BIBLIOGRAPHY 1. Ab l o w i t z , M. J . , 1971: A p p l i c a t i o n of slowly v a r y i n g n o n l i n e a r d i s p e r s i v e wave t h e o r i e s . Stud. i n Appl.  Math.,, 54, 329-344. 2. A b l o w i t z , M. J . and Kodama, Y., 1979: Transverse i n s t a b i l i t y one dimensional t r a n s p a r e n t o p t i c a l p u l s e s i n resonant media. Phys. L e t t . , A70, 83-86. 3. A b l o w i t z , M. J . , and H. 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Geophys. Res., 81, 2641-2646. 75. Yamagata, T., 1982: On n o n l i n e a r p l a n e t a r y waves: A c l a s s of s o l u t i o n s missed by the t r a d i t i o n a l q u a s i g e o s t r o p h i c approximation. Oceanogr. Soc. Japan, 38, 263-244. 76. Zakharov, V. E., and A. M. Rubenchik, 1974: I n s t a b i l i t y of waveguides and s o l i t o n s i n n o n l i n e a r media. Sov. Phys.  JETP, 38, 494-501. 126 APPENDIX A - CALCULATION OF SOLVABILITY INTEGRALS IN TOPOGRAPHIC PROBLEM Thi s appendix d e s c r i b e s the c a l c u l a t i o n s of the c o m p a t i b i l i t y c o n d i t i o n s (3.2.7) and (3.2.9). The e x t e r i o r computations ( i . e . , (3.2.7)) are presented f i r s t and the i n t e r i o r computations ( i . e . , (3.2.9)) f o l l o w . E x t e r i o r C a l c u l a t i o n s The c o m p a t i b i l i t y c o n d i t i o n (3.2.7) i n v o l v e s 18 i n t e g r a l s . The terms are d e s c r i b e d i n the order of t h e i r appearance i n (3.2.6) and are denoted I\ through t o l i s * R e c a l l that f o r r>a i//<°> = - c a K , ( 6 c - 1 ' 2 r ) s i n ( 0 ) / K 1 ( 6 a c " 1 / 2 ) A^<°> = - 6 2 a K 1 ( 6 c " 1 / 2 r ) s i n ( e ) / K l ( 6 a c - 1 / 2 ) i . e . , A ^ ( 0 ) = (62/c)i//< 0 > . I t w i l l be convenient to l e t R(r) = - c a K 1 ( 6 c - 1 / 2 r ) / K 1 ( 6 a c - 1 / 2 ) , so that i / / ( 0 ) = R ( r ) s i n ( t 9 ) , and d e f i n e the operat o r s D = d/dr and D 0F[(»)] = dF[(•)]/£(•) ( i . e . , d i f f e r e n t a t i o n with respect to r and arguments, r e s p e c t i v e l y ) . R e c a l l r 2 = ( £ _ £ 0 ) 2 + Y2 a n & tan(0)=y / U - £ o ) , hence (3 , 3 , 3 )r = - c o s ( 0 ) ( 3 , 3 , 3 )£ 0 T X Y T X Y (3 , 3 , 3 )6 = f'sinU)(3 , 3 , 3 )£ 0 . T X Y T X Y And f i n a l l y i t i s h e l p f u l to r e c a l l t h a t by d e f i n i t i o n D 2R + DR/r - ( r - 2 + 6 2/c)R = 0. The i n t e g r a l c a l c u l a t i o n s f o r r>a are as f o l l o w s ; I i : 7T » I, = ( 6 2 / c ) H " 1 g J f f 0 l f 0 1 rdrdfl Y -IT a i 127 Note that i / y ( 0 ) = (DR - R / r ) s i n ( 0 ) c o s ( 0 ) . Therefore the t r i g o n o m e t r i c part of the i n t e g r a l c o n t a i n s c o s ( 0 ) s i n 2 ( 0 ) which i n t e g r a t e s to zero due to the p e r i o d i c i t y in 0. Thus I, = 0. I 2 : it » 12 = ~ / J 0 >Ai//< 0 > rdrd0 -it a T Note that A<//(0> = { a ^ a - [ D 0 K 1 ( 8 a c - 1 / 2 ) / K 1 ( 6 a c - , / 2 ) ] 8 a c - , / 2 [ a - 1 a -T T T ( 2 c ) - 1 c ]}A<//(0> + c 1 ' 2 ( r c - 1 ' 2 ) A^ < 0> -T T r (6 2/c)cos(0)sin(0)£ o DR + ( 6 2 / c ) r - 1 sin(0)cos(0)£ 0 R. T T The t r i g o n o m e t r i c component of the i n t e g r a l a s s o c i a t e d with the l a s t two terms ( i . e . , the terms c o n t a i n i n g £ 0 ) i s s i n 2 ( 0 ) c o s ( 0 ) T and thus i n t e g r a t e s to zero . The remaining c a l c u l a t i o n g i v e s I 2 = -7r6 2ca"Q[A 1a-'a - (A, - 1 ) ( 2 C ) " 1 C ] T T where A i = 7 K 2 ( 7 ) / K I ( T ) ~ (2Q)" 1- 1 (A1a) Q = -[7K 2(7> " 2K 0(7)K,(7) ~ 7 ^ ( 7 ) ) / ( 2 7 ^ ( 7 ) ] , ( A l b ) 1 o 1 r e c a l l i n g 7 2 = 8 2 a 2 / c . I 3 : We r e q u i r e it » 1 3 = 2c / J 1 0 0 > rdrdS -it a ££X r// ( 0 ) = s i n ( 0 ) c o s 2 ( 0) (D 2 2DR/r - 2R/r 2) + ££X X s i n 3 ( 0 ) ( D R / r - R/r 2) + X 128 r - 1 ^ ( c o s 3 ( 0 ) s i n ( 0 ) - 2 s i n 3 ( 0 ) c o s ( 0 ) ( D 2 R - 2DR/r - 2R/r 2) -X £ 0 c o s 3 ( 0 ) s i n ( 0 ) D ( D 2 - 2DR/r - 2R/r 2) + X S r - ^ o s i n 3 (6)cos(6) (DR/r - R / r 2 ) -X X £o c o s ( 0 ) s i n 3 ( e ) D ( D R / r - R / r 2 ) . X The t r i g o n o m e t r i c component of the i n t e g r a l s c o n t a i n i n g £ 0 have X c o s 3 ( 0 ) s i n 2 ( 8 ) or s i n 4 ( 0 ) c o s ( 0 ) both of which i n t e g r a t e to zero. The remaining i n t e g r a t i o n g i v e s I 3 = ( 1 / 2 ) f f 6 2 c 2 a , t Q [ A 1 a - 1 a - (A, - l ) ( 2 c ) " 1 c ]. X X I , : It => I« = ~6 2 / J ^ < 0 > V l 0 ) rdrd0 -it a X The c a l c u l a t i o n of I« i s e s s e n t i a l l y the same as I 2 (modulo the 6 2/c f a c t o r a s s o c i a t e d with At/> < 0 ) compared t o ^ ( 0 } and the X-de'rivative r a t h e r than the T - d e r i v a t i v e ) . The r e s u l t i s I, = - 6 2 i r c 2 a ' , Q [ A 1 a - 1 a - (A, - 1 ) ( 2 c ) - 1 c + c ' 1 c ]. X X X I 5 : It « I 5 = (6Vc) / J ^ ( 0 ) ^ , ( 0 ) ^ ( 0 ) rarde X -it a y We note that ^ ( 0 ) ^ ( 0 ) ^ , ( 0 ) = O / 3 ) [ s i n 0 D + r - 1 c o s ( 0 ) 3 / 9 0 ) R 3 s i n 3 ( e ) y which giv e s I 5 = -(6 27rc 2aV2) (2c)" \c . X 129 I 6 : I, = -2 J / ,/,<0'j(<//(0), ^ ( 0 ) ) rdrdfl -TT a £X We r e q u i r e ( r - V 0 ) ) = (cosMe) - s i n 2 ( 0 ) ( D R / r - R / r 2 ) -ie X X 2r" 1sin 2(0)cos(0)£ o (DR/r - R/r 2) -X cos(0)(cos 2 ( 6 ) - s i n 2 ( 0 ) £ o D(DR/r - R / r 2 ) X and ( i / / ( 0 ) ) = cos ( 0) s i n ( &) (D 2R - DR/r + R / r 2 ) -it X X r " 1 s i n ( 0 ) ( c o s 2 ( 6 ) - sin 2(0))£ o (D 2R - DR/r + R / r 2 ) -X cos 2(0)sin(0)£ 0 D(D 2R - DR/r + R / r 2 ) . X Note that j(^<°>, i//<0)) = V ( 0 ) ( r - V < 0 > ) ~ r" V 0 ' U l 0 > ) . £X r £0 X 0 £r X The t r i g o n o m e t r i c component of the terms c o n t a i n i n g £ 0 i s X s i n * ( 0 ) c o s ( 0 ) or s i n 2 ( 0 ) c o s 3 ( 0 ) , both of which i n t e g r a t e to zero. The remaining i n t e g r a t i o n g i v e s I 6 = - 7 r a 2 c 3 7 K 2 ( 7 ) ( 2 K 1 ( 7 ) ) - 1 [ ( 2 c ) - 1 c + B ( a - ' a - ( 2 c ) - 1 c )] X X X where B = 7 [ K 2 ( 7 ) / K i ( 7 ) - K , ( 7 ) / K 2 ( 7 ) " 3 / 7 ) ] . ( A 2 ) I 7 : 130 It » I 7 = c(H- 1H ) J J ^'°>^' 0> rdrdfl X -it a i£ We r e q u i r e \//(0> = s i n ( 0 ) c o s 2 (6) (D 2R - 2DR/r + 2R/r 2) + s i n 3 ( 6 ) ( D R / r - R/r 2) which r e s u l t s i n I 7 = ( 1 /4 ) 6 27ra',c 2QH" 'H . X I 8 : It » I 8 = -6 2(H" 1H ) / J ^<°)^'°> rdrdfl X - IT a S t r a i g h t f o r w a r d c a l c u l a t i o n r e s u l t s i n I, = - 5 2 a * c 2 7 r Q H - 1H . X I 9 : it 00 I 9 = -(H" 1H ) / / , / / ( 0 ) J U ( 0 > , ^'°') rdrdfl X —it a £ We r e q u i r e i / / ( 0 ) = ( c o s 2 ( 0 ) - s i n 2 ( 0 ) ) ( D R - R/r) ie I J > ( 0 ) = c o s ( 0 ) s i n ( 0 ) (D 2R - DR/r + R / r 2 ) , ir which g i v e s I 9 = - i r a 2 c 3 7 K 2 ( 7 ) ( 4 K 1 ( 7 ) ) - 1 H - 1 H . X I 1 0 : 131 it * I,o - -(H-'H ) / / ^ ( 0 ) [ ( r 0 ) - 1 + 6 2y]<// ( 0 ) rdrd0 X - 7 r a y We r e q u i r e </<(0) = s i n 2 ( 0 ) D R + c o s 2 ( 0 ) R / r . y The t r i g o n o m e t r i c components of the i n t e g r a l with the Rossby number as a c o e f f i c i e n t are s i n 3 ( 0 ) or c o s 2 ( 0 ) s i n ( 0 ) both of which i n t e g r a t e to zero. The remaining i n t e g r a t i o n g i v e s I 1 0 = (3/8)6 2ira"c 2H-'H + (1/2) 6 2ira t tc 2QH- 1H . X X I n ' : It 00 I , , = - ( 2 6 2 / c ) ( H - 1 H ) / J ^(°>^< 0'^<°) rdrd0 X - i t a y The e v a l u a t i o n of I,, i s s i m i l a r to I 5 the r e s u l t i s I,,= - ( l / 2 ) 5 2 7 r a 4c 2H- 1H . X I 1 2 : It oo I , 2 = 2c / S 0 >i//( 0 > rdrd0 - i t a £yY We r e q u i r e i / / ( 0 > = c o s ( 0 ) s i n 2 ( 0 ) (D 2R - 2DR/r + 2R/r 2) + SyY Y c o s 3 ( 0 ) ( D R / r - R / r 2 ) + Y r - 1 ( c o s 2 ( 0 ) s i n 2 ( 0 ) - s i n a(0))£ o (D 2R - DR/r + 2R/r 2) -Y c o s 2 ( 0 ) s i n 2 ( 0 ) £ 0 D(D 2R - 2DR/r + 2R/r 2) -Y 3 r " 1 c o s 2 ( 0 ) s i n 2 ( 0 ) £ o (DR/r - R/r 2) - cos f l(0)£ o D(DR/r - R / r 2 ) . Y Y 132 The t r i g o n o m e t r i c part of these terms i s i n t e g r a t e d a g a i n s t s i n ( 0 ) ( b e l o n g i n g to ^ ( 0 ) ) . I t i s easy to see that each term i n t e g r a t e s t o zero due to the p e r i o d i c i t y i n 0, consequently I12-O. l i s : It OD I 1 3 = - ( 6 2 / c ) ; J ,/,<<> >,/,<<> ),/,<<>> rdrdS Y -TT a i We r e q u i r e VJ> ( 0 ) = c o s ( 0 ) s i n ( 0 ) (DR - R / r ) . When t h i s term i s i n t e g r a t e d a g a i n s t the s i n 2 ( 0 ) t r i g o n o m e t r i c component of (i//(0))2 the r e s u l t i n g i n t e g r a l vanishes, consequently I 1 3 = 0 . 11«: We r e q u i r e I , , = -2 J J V 0 ) W 0 ' , i//(0)) rdrd0 - it a yY (r-V<0)) = 2sin(0)cos(0) (DR/r - R/r2) + y0 Y Y 2 r - 1 ( c o s 2 ( 0 ) s i n ( 0 ) - si n 3 ( 0 ) ) £ o (DR/r - R / r 2 ) -Y 2sin 2(0)cos(0)£ o D(DR/r - R / r 2 ) . Y U<°>) = s i n 2 ( 0 ) ( D 2 R ) + cos 2(0.)(D 2R - R / r 2 ) -yr Y Y Y sin 2(0)cos(0)£ o (D 3R - 2D 2R/r) -Y 2r- 1sin 2(0)cos(0)£ o (D 2R - R / r 2 ) - c o s 3 ( 0 ) £ o D(D 2R - R / r 2 ) Y Y 133 The f i r s t term i s m u l t i p l i e d by (DR)Rsin 2(0) which i n t e g r a t e s to zero due to the t r i g o n o m e t r i c i n t e g r a t i o n . The second term i s m u l t i p l i e d by R 2 s i n ( 0 ) c o s ( 0 ) which i n t e g r a t e s to zero due to the t r i g o n o m e t r i c i n t e g r a t i o n . Therefore I , , = 0. l i s : I 1 5 = c(H- 1H ) J J V ' 0 ' ' / ' ' 0 ' rdrdS Y -TT a £y We r e q u i r e <iV 0 ) = c o s ( 0 ) s i n 2 ( 0 ) (D 2R - 2DR/r + 2R/r 2) + c o s 3 ( 0 ) ( D R / r - R / r 2 ) which when i n t e g r a t e d a g a i n s t Rsin(0) i s zero due to the t r i g o n o m e t r i c i n t e g r a t i o n . T h e r e f o r e I, 5=0. l i s : 7T » I , 6 = -(H" 1H ) J J °'J(^< °', ^<°») rdrd0 Y -7r a y We r e q u i r e t/>(0) = s i n 2 ( 0 ) D 2 R + c o s 2 ( 0 ) [ D R / r - R / r 2 ) , yr 0co> = 2 r " ' s i n ^ J c o s U ) (DR/r - R / r 2 ) . The f i r s t term i s i n t e g r a t e d a g a i n s t R 2 s i n ( 0 ) c o s ( 0 ) and the second term i s i n t e g r a t e d a g a i n s t (DR)Rsin 2(0) which i n t e g r a t e to zero due to the t r i g o n o m e t r i c i n t e g r a t i o n . T h e r e f o r e I, 6=0. 1 , 7 : 7T » I , 7 = (H- 1H ) / / ^ ( 0 ) [ ( r 0 ) - 1 + 6 2 y ] ^ ( 0 ' rdrd0 Y -ir a £ The i n t e g r a n d i s ( ( r 0 ) _ 1 + 8 2 r s i n ( 0 ) ) R s i n 2 ( 0 ) c o s ( 0 ) ( D R - R/r) 1 3 4 which i n t e g r a t e s to zero due to the t r i g o n o m e t r i c i n t e g r a t i o n . T h e r e f o r e I, 7=0. I1 8 ' it °° I 1 8 = 2 ( 6 2 / c ) ( H - 1 H ) / J V ( 0 , ^ ( 0 > ^ ( 0 ) r d r d f i . Y -it a I T h i s i n t e g r a l i s t r i g o n o m e t r i c a l l y i n d e n t i c a l t o I 1 3 . T h e r e f o r e I 18 = 0. The c o m p a t i b i l i t y c o n d i t i o n (3.2.7) i s t h e r e f o r e it °= 1 8 0 = J f V 0 ' ( R H S [ 3 . 2 . 6 ] ) rd r d e = 1 1 , -it a n= 1 n which a f t e r a l i t t l e a l g e b r a r e s u l t s i n ( 3 . 2 . 1 0 ) where A , i s g i v e n by ( A 1 ) and A 2 = 2 + Q/2 - K 2 ( 7 ) Q [ 2 T K 1 ( 7 ) ] * 1 ( B - 1 ) ( A 3 ) A 3 = K 2 ( 7 ) Q [ 2 T K I ( 7 ) ] " ' B ( A 4 ) E , = 1 / 4 + Q / 8 + K 2 ( 7 ) Q t 2 7 K 1 ( 7 ) ] - 1 ( A 5 ) with B i s given by ( A 2 ) . I n t e r i o r C a l c u l a t i o n s The i n t e r i o r c o m p a t i b i l i t y c o n d i t i o n ( 3 . 2 . 9 ) c o n t a i n s 1 9 i n t e g r a l s denoted I, through I , 9 . R e c a l l that f o r r<a ^ ( 0> = 8 2 K " 2 a J 1 ( K r ) s i n ( e ) / J , U a ) - ( 6 2 + K 2 C ) / C" 2 r s i n ( 0 ) A i / / ( 0 ) = - 6 2 a J ! (/cr)sin(0)/J, U a ) i . e . , A<// ( 0 ) = - K 2^<o) _ ( 6 2 + C K 2 ) r s i n ( 0 ) . As i n the pr e v i o u s c a l c u l a t i o n i t w i l l be convenient to l e t R(r) = 5 2 K - 2 a J , U r ) / J , U a ) - ( 6 2 + « 2c)»c- 2r so that i p { 0 ) = R ( r ) s i n ( 0 ) , and d e f i n e the operators D = d/dr and D 0F[(«)1 = dF[(•)]/£(•) ( i . e . , d i f f e r e n t a t i o n with r e s p e c t to r and arguments, r e s p e c t i v e l y ) . R e c a l l r 2 = (£-£o) 2 + Y2 and tan(e)=y/(£-£ 0)> hence 135 (3 , 3 , 3 )r = -c o s ( 0 ) ( 3 , 3 , 3 )£ 0 T X Y T X Y ( 3 , 3 , 3 ) 0 = r " 1 s i n ( 0 ) ( 3 , 3 , 3 ) £ 0 . T X Y T X Y And f i n a l l y i t i s h e l p f u l to r e c a l l t h a t by d e f i n i t i o n D 2R + DR/r - ( r - 2 - K 2 ) R = 0. The i n t e g r a l s are given as - f o l l o w s . I i : it a I, = - K 2 H - 1 g J J ^<°>^/<°) rdrdfl Y -it 0 £ T h i s i n t e g r a l i s t r i g o n o m e t r i c a l l y i d e n t i c a l t o I, i n the e x t e r i o r c a l c u l a t i o n s . T h e r e f o r e I ^ O . I 2 : it a I 2 = ~ / / ^ 0 >A<//< 0 ' rdrdfl -it 0 T Note that A^<°> = { a ' 1 a - [ D 0 J , U a ) / J , U a ) ] « a [ a " 1 a + T T T K ~ 1 K ]}A<//< 0 > + K- 1 (KT) Al//< 0 ) " T T r cos(0)£ o Ai/> ( 0 ) + r " 1 s i n ( 0 ) £ o Ai// ( 0>. T r T 8 The t r i g o n o m e t r i c component of the terms with £ 0 i s T s i n 2 ( 8 ) c o s ( 6 ) which i n t e g r a t e s to zero due to the p e r i o d i c i t y i n 8. The remaining c a l c u l a t i o n g i v e s 1 2 = Ga" 1 a + L[a* ! a + K " 1 K ] T T T — where G = C 2 [ D 2 - (1 + k 2 / 7 2 ) J 2 ( k ) [ k J 1 ( k ) ] - 1 (A6) L = C 2 [ - k D 2 / J 2 ( k ) + ( k J 0 ( k ) / J , ( k ) + 2 ) 0 + k 2 7 " 2 ) -1 136 J 2 ( k ) [ k J 1 ( k ) ] - 1 - 1/2 - ( k / 7 ) 2 ] (A7) C 2 = 6*a f l7r»c- 2 (A8) D 2 = ( 1 / 2 ) [ J 2 ( k ) - 2 J 2 ( k ) j 1 ( k ) k - 1 + J 2 ( k ) ] / J 2 ( k ) (A9) 2 1 1 r e c a l l i n g k=«a, I 3 : We r e q u i r e ir a I 3 = 2c J J ^ t 0 > ^ ( 0 > rdrdfl -* 0 ^ ( 0 ) = s i n ( 0 ) c o s 2 ( 6 ) ( D 2 - 2DR/r - 2R/r 2) + ax x s i n 3 ( 6 ) ( D R / r - R/r 2) + X r " 1 £ 0 ( c o s 3 ( e ) s i n ( f l ) - 2 s i n 3 ( 0 ) c o s ( 0 ) ( D 2 R - 2DR/r - 2R/r 2) X £ 0 c o s 3 ( 0 ) s i n ( 0 ) D ( D 2 - 2DR/r - 2R/r 2) + X 3 r " 1 £ 0 s i n 3 ( 0 ) c o s ( 0 ) ( D R / r - R / r 2 ) -X X £ 0 c o s ( 0 ) s i n 3 ( 0 ) D ( D R / r - R / r 2 ) . X The t r i g o n o m e t r i c component of the i n t e g r a l s c o n t a i n i n g £ 0 have X c o s 3 ( 0 ) s i n 2 ( 0 ) or s i n * ( 0 ) c o s ( 6 ) both of which i n t e g r a t e to zero. The remaining i n t e g r a t i o n g i v e s I 3 = -( c / 2 ) G a " 1 a - ( c / 2 ) L ( a " 1 a + K " 1 * ) . X X X TT a I4 = K2C J J tl0)*pl0) rdrd0 -TT 0 X 137 Note that IJ> ( 0 ) = { a' 1a - [ D 0 J i ( K a ) / J , («<a) ]»ca[a-'a + X T X K " 1 K ] } l / / ( 0 > + K " 1 ( KT ) ^ ( 0 ) " X X r cos(0)£ o ^ < 0 > + r- 1sin(0)£ o i/> < 0 ). X r X 6 The t r i g o n o m e t r i c component of the terms with £ 0 i s X s i n 2 ( 0 ) c o s ( 0 ) which i n t e g r a t e s to zero due to the p e r i o d i c i t y 6. The remaining c a l c u l a t i o n g i v e s I , = cGa~ 1a + c L [ a _ 1 a + K _ 1 K ] - 2 C G K _ 1 K + X X X X c C 2 [ - J 2 ( k ) / ( k J , ( k ) ) + (1 + ( k / 7 ) 2 ) / 4 ] . U 2 a " 1 6 - ' { U 2 + c « 2 ) a K - 2 } . X it a Is = - < K 2 ) S S ^ ( 0 0 } * ( 0 } rdrd'0 X -TT 0 y We note t h a t ^ ( 0 ) ^ ( 0 ) ^ , ( 0 ) = ( i / 3 ) [ s i n 0 D + r" ! c o s ( 0 ) 3 / 3 0 ) R 3 s i n 3 (6) y which g i v e s I 5 = c C 2 ( k / 7 ) « K - 1 K /2. X I 6 : it a I 6 = -2 / / i/>< ° ' JU ( 0 > r i / / ( 0 ) ) rdrd0 -it 0 £X We r e q u i r e 138 ( r - V 0 ) ) = ( c o s 2 ( 0 ) - s i n 2 ( 0 ) ( D R / r - R / r 2 ) -ie X X 2r~ 1sin 2(0)cos(0)£ o (DR/r - R/r 2) -X c o s ( 0 ) ( c o s 2 ( 0 ) - s i n 2 ( 0 ) £ o D(DR/r - R / r 2 ) X and (<//<0)) = cos ( 0) s i n (0) (D 2R - DR/r + R / r 2 ) -£r X X r " 1 s i n ( 0 ) ( c o s 2 ( 0 ) - sin 2(0))£ o (D 2R - DR/r + R / r 2 ) -X cos 2(0)sin(0)£ o D(D 2R - DR/r + R / r 2 ) . X Note that j(,//<0>, V ( 0 ) ) = 0 ' (r- 1i//< 0 ' ) - r" V ( 0 ' U ( 0 ' ) • £X r £0 X 0 $r X The t r i g o n o m e t r i c component of the terms c o n t a i n i n g £ 0 i s X s i n " ( 0 ) c o s ( 0 ) or s i n 2 ( 0 ) c o s 3 ( 0 ) , both of which i n t e g r a t e to zero. The remaining i n t e g r a t i o n g i v e s I 6 = c J 2 ( k ) C 2 k ( 2 J 1 ( k ) 7 2 ) - 1 [ - K - 1 / c + B 2 ( a " 1 a + K ' 1 K ) ] X X X where B 2 = k [ J , ( k ) / J 2 ( k ) + J 2 ( k ) / J , ( k ) - 3 / k ] . I 7 : IT a I 7 = c(H- 1H ) J J ^<°»^(°» rdrd0 X -TT 0 a We r e q u i r e i//(0> = s i n ( 0 ) c o s 2 ( 0 ) (D 2R - 2DR/r + 2R/r 2) + a 139 s i n 3 ( 6 ) ( D R / r - R/r 2) which r e s u l t s i n I 7 = -cC 2(G/4)H"'H . X it a I 8 = K 2 C ( H - 1 H ) J S ^ l 0 ) ^ l 0 ) rdrdfl X "TT 0 S t r a i g h t f o r w a r d c a l c u l a t i o n r e s u l t s i n I B = c C 2 [ D 2 - 2(1 + ( k / 7 ) 2 ) J 2 ( k ) / ( k J , ( k ) ) + (1 + ( k / 7 ) 2 ) 2 / 4 ] H - 1 H . . X I 9 : it a I 9 = -(H" 1H ) ; / <//< 0 ' J(i//< 0 ' , \pl0)) rdrdfl X - 7 T 0 i We r e q u i r e i / / ( 0 ) = ( c o s 2 ( 0 ) - s i n 2 ( 0 ) ) ( D R - R/r) ie <//t0) = c o s ( 0 ) s i n ( 0 ) (D 2R - DR/r + R / r 2 ) , which g i v e s I, = c C 2 k ( 4 7 2 ) - 1 ( J 2 ( k ) / J 1 ( k ) ) H " 1 H . X it a I 1 0 = (H-'H ) J f f " ' [ ( r o ) " ' - K 2 c y ] ^ ( 0 ) rdrd0 X -it 0 y 140 We r e q u i r e vV ( 0 ) = s i n 2 ( 0 ) D R + c o s 2 ( 0 ) R / r . y The t r i g o n o m e t r i c components of the i n t e g r a l with the Rossby number as a c o e f f i c i e n t are s i n 3 ( 0 ) or c o s 2 ( 6 ) s i n ( 8 ) both of which i n t e g r a t e to zero. The remaining i n t e g r a t i o n g i v e s I , 0 = c C 2 [ ( 3 / 8 ) ( k / 7 ) * - D 2/2 + (1 + ( k / 7 ) 2 ) J 2 ( k ) / ( k J 1 ( k ) ) - ( 1 / 8 ) 0 + ( k / 7 ) 2 ) 2 ] H ' ' H . X I n : it a I,, = 2 K 2 ( H " 1 H ) / J 0 >iiV( 0 '(//< 0 ' rdrdfl X -TT 0 y The e v a l u a t i o n of I 1 t i s s i m i l a r to I 5 , the r e s u l t i s I 1 i = - 0 / 2 ) c C 2 ( k / 7 ) 1 , H - 1H . X I , 2 : it a I 1 2 = 2c J J i/>( 0 0 > rdrdfl -it 0 £yY T h i s i n t e g r a l i s t r i g o n o m e t r i c a l l y i n d e n t i c a l to I,2 i n the e x t e r i o r c a l c u l a t i o n s . T h e r e f o r e I 1 2=0. I i 3 : it a i i 3 = u 2) ; / ^ « o ) ^ ( o > ^ ( 0 ) r d r d e Y -it 0 i T h i s i n t e g r a l i s t r i g o n o m e t r i c a l l y i d e n t i c a l to the I 1 3 i n t e g r a l i n the e x t e r i o r c a l c u l a t i o n . T h e r e f o r e I 1 3=0. 11»: 141 it a I i a = J / I*1 0> [ r s i n ( f i ) ( 8 2 + K 2 C ) ] <//(0> rdrd0 -it 0 Y £ We r e q u i r e [ r s i n ( 0 ) ( 5 2 + K 2 C ) ] = - c o s ( 0 ) s i n ( 0 ) | 0 ( 5 2 + * 2 c ) + Y Y cos(0)sin(0)£ o ( S 2 + K 2 C ) + Y s i n ( 0 ) [ r ( 5 2 + K 2 C ) ] , Y and i / / ( 0 ) = (DR - R / r ) s i n ( 0 ) c o s ( 0 ) . The t r i g o n o m e t r i c component t of the i n t e g r a l i s t h e r e f o r e s i n 3 ( 0 ) c o s 2 ( 0 ) or s i n 3 ( 0 ) c o s ( 0 ) , both of which i n t e g r a t e to zero due to the p e r i o d i c i t y i n 0. The i n t e g r a l s I 1 5 through I 1 9 are t r i g o n o n m e t r i c a l l y i n d e n t i c a l to the e x t e r i o r i n t e g r a l s I H through I , 8 , r e s p e c t i v e l y . Therefore a l l the remaining i n t e g r a l s are zero. l i s : it a I 1 5 = -2 S S * ( 0 > J U < 0 ) , ^ ( 0 ) ) rdrd0 -it 0 yY I i it a I , 6 = c ( H " 1 H ) J J f ° ' f 0 1 rdrd0 Y -Tr 0 £y 1 7 it a I 1 7 = - ( H " 1 H ) ; J i//( °>J( ,// ( 0 ) , i//<°>) rdrd0 Y -it 0 y 1 8 I , 8 = ( H - 1 H ) J J < K 0 ) [ ( r 0 ) - 1 - i c 2 c y ] ^ ( 0 > rdrd0 Y -it 0 | 142 I 1 9 J it a I 1 9 = -2/c 2(H" 1H ) J J ,/,< 0 >,/,< ° >,/>< 0 > rdrdS. Y -Tr 0 £ The c o m p a t i b i l i t y c o n d i t i o n (3.2.9) i s t h e r e f o r e it a 19 0 = / J 0 ' (RHS[3.2.8]) rdrdfl = 1 1 . (A10) - it 0 n=1 n In order to o b t a i n (3.2.11) the terms K " 1 K and K _ 1 K i n (A10) T X are e l i m i n a t e d i n favour of a " 1 a , (2c)''c , a _ 1 a and ( 2 c ) " 1 c T T X X by d i f f e r e n t i a t i n g the d i s p e r s i o n r e l a t i o n (3.1.5) to obt a i n K " 1 K = Nfa-'a - (2c)-^c ] - (2c)-*c T T T T K " 1 K = N[a" 1a - (2c)-'lc ] - ( 2 c ) " 1 c X X X X where N = -{7R + k 2R/7}/{4 + 7/R + k 2R/7} and R=K 2(7)/K,(7). A f t e r a l i t t l e a l e g b r a , (3.2.11) i s obtained from (A10) where B,, B 2, B 3 and E 2 are given by the f o l l o w i n g h i e r a r c h y of def i n i t i o n s ; M = L/G + 1, E = [ - J 2 ( k ) / ( k J , ( k ) ) + (1 + ( k / 7 ) 2 ) / 4 ] / G , B, = NM + M - N, (A11 ) |B2 = 2(1 + N) + E(N + 2 + 2 ( k / 7 ) 2 ) + (k/7)•(2G)" 1(N - 1) + k J 2 ( k ) [ 2 7 2 J 1 ( k ) G ] " 1 ( l ~ B,N - B, + N), (A12) B 3 = -2N + E(1 - N + (k/7) 2) + k*N(27 i ,G)- 1 + k ( 2 7 2 J , ( k ) G ) " 1 J 2 ( k ) [ B , N + B, - N], (A13) E 2 = 1/4 + E/2 - k ,(87*G)" 1 + k j 2 ( k ) [ 4 7 2 J , ( k ) G ] " 1 . (A14) 

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