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UBC Theses and Dissertations

Resonant interactions between continental shelf waves Hsieh, William Wei 1981

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RESONANT INTERACTIONS BETWEEN CONTINENTAL SHELF WAVES WILLIAM B . S c , The U n i v e r s i t y M . S c , the U n i v e r s i t y by WEI J HSIEH of B r i t i s h C o lumbia, 1976 of B r i t i s h C o lumbia, 1978 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY THE FACULTY OF (Department of P h y s i c s and i n GRADUATE STUDIES Department of Oceanography) We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1981 © W i l l i a m Wei H s i e h , 1981 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head of my department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of Physics / Oceanography The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date 24 A p r i l . 1981 i i ABSTRACT P a r t I of t h i s t h e s i s d e v e l o p s a t h e o r y of n o n l i n e a r r e s o n a n t i n t e r a c t i o n s between c o n t i n e n t a l s h e l f waves. From the i n v i s c i d , u n f o r c e d long-wave e q u a t i o n s f o r a r o t a t i n g , homogeneous f l u i d , i t i s shown t h a t r e s o n a n t i n t e r a c t i o n s between t h r e e c o n t i n e n t a l s h e l f waves can o c c u r . E v o l u t i o n e q u a t i o n s g o v e r n i n g t h e a m p l i t u d e and the energy of i n d i v i d u a l waves i n a r e s o n a n t t r i a d a r e d e r i v e d . The n o n l i n e a r i t y i n the g o v e r n i n g e q u a t i o n s a l l o w s energy t o be t r a n s f e r r e d between the waves, but w i t h the t o t a l energy c o n s e r v e d . In p a r t i c u l a r , i n t e r a c t i o n s on an e x p o n e n t i a l s h e l f a r e s t u d i e d . P a r t I I of t h i s t h e s i s compares the t h e o r y from P a r t I w i t h o b s e r v a t i o n s and d a t a from the Oregon s h e l f . R o t a r y s p e c t r a l a n a l y s i s and c r o s s - s h e l f modal f i t t i n g a r e t h e two t e c h n i q u e s used f o r s h e l f wave d e t e c t i o n . Many f e a t u r e s c h a r a c t e r i s t i c of s h e l f waves and of t h e r e s o n a n t t r i a d i n t e r a c t i o n t h e o r y are found i n the c u r r e n t and sea l e v e l d a t a . A l s o , f o r the f i r s t t i m e , s h e l f waves have been unambiguously i d e n t i f i e d i n both the a l o n g s h o r e and c r o s s - s h e l f d i m e n s i o n s . The d a t a i n d i c a t e t h a t the wind g e n e r a t e s l o n g c o n t i n e n t a l s h e l f waves a t low f r e q u e n c i e s . N o n l i n e a r r e s o n a n t i n t e r a c t i o n s then t r a n s f e r energy from the l o w - f r e q u e n c y l o n g waves t o h i g h e r f r e q u e n c y s h e l f waves w i t h much s h o r t e r w a v e l e n g t h s . The good agreement between t h e o r y and o b s e r v a t i o n s u g g e s t s t h a t n o n l i n e a r energy t r a n s f e r may p l a y a s i g n i f i c a n t r o l e i n s h e l f wave dynamics. i i i TABLE OF CONTENTS page ABSTRACT i i LIST OF TABLES i v LIST OF FIGURES • v ACKNOWLEDGEMENT v i i CHAPTER 1. I n t r o d u c t i o n 1 PART I THEORY 2. G o v e r n i n g E q u a t i o n s 6 3. Theory of Resonant I n t e r a c t i o n s between S h e l f Waves 11 4. Energy T r a n s f e r and Energy C o n s e r v a t i o n 16 5. A n a l y s i s of the A m p l i t u d e E q u a t i o n s 20 6. Resonant I n t e r a c t i o n s on an E x p o n e n t i a l S h e l f 24 PART 2 OBSERVATION 7. D i s p e r s i o n Curves and C u r r e n t E l l i p s e s 28 8. R o t a r y S p e c t r a l A n a l y s i s and C r o s s - S h e l f Modal F i t t i n g 43 9. Oregon S h e l f : Summer, 1968 50 10. Oregon S h e l f : Summer, 1972 60 11. Oregon S h e l f : Summer, 1973 75 12. Oregon S h e l f : W i n t e r and S p r i n g , 1975 101 13. D i s c u s s i o n 113 14. Summary and C o n c l u s i o n 118 BIBLIOGRAPHY 122 APPENDIX A: The Group V e l o c i t y 125 APPENDIX B: N u m e r i c a l Technique f o r D i s p e r s i o n Curves .... 126 APPENDIX C: Data P r o c e s s i n g 128 IV LIST OF TABLES T a b l e page 9.1 The c o u p l i n g c o e f f i c i e n t s K j ' s f o r resonant t r i a d s on t h e Oregon s h e l f 58 10.1 Comparison of low f r e q u e n c y energy a t DB-7 and NH-10 70 11.1 P o s s i b l e s h e l f waves c o n t r i b u t i n g a t s e l e c t e d f r e q u e n c i e s 96 11.2 C r o s s - s h e l f modal f i t t i n g a t s e l e c t e d f r e q u e n c i e s .. 97 12.1 Comparison of energy f a l l o f f a t low fr e q u e n c y between CUE-2 and WISP 107 12.2 R a t i o s of c l o c k w i s e t o a n t i c l o c k w i s e energy i n CUE-2 and WISP 107 V LIST OF FIGURES F i g u r e page 7.1 The c u r r e n t e l l i p s e 33 7.2 Sea l e v e l d i s p l a c e m e n t r\e and l l / k f o r 3 s h e l f waves 36 7.3 V e l o c i t y components u,, and v 0 f o r 3 s h e l f waves .... 37 7.4 A n t i c l o c k w i s e and c l o c k w i s e v e l o c i t y components, A and C, f o r 3 s h e l f waves 38 7.5 " P o l a r i z a t i o n " P and " o r i e n t a t i o n " R f o r 3 s h e l f waves 40 7.6 S h e l f wave c u r r e n t e l l i p s e s a c r o s s the c o n t i n e n t a l s h e l f 41 9.1 The e x p o n e n t i a l f i t t o the Oregon s h e l f p r o f i l e .... 51 9.2 D i s p e r s i o n c u r v e s f o r the e x p o n e n t i a l p r o f i l e 52 9.3 O b s e r v a t i o n s made by C u t c h i n and Smith (1973) on the Oregon s h e l f 53 9.4 The resonant t r i a d c o n s i s t e n t w i t h the o b s e r v a t i o n s of C u t c h i n and Smith (1973) 55 10.1 C u r r e n t meter a r r a y i n the CUE-1 experiment 61 10.2 L e n g t h of d a t a r e c o r d s i n CUE-1 62 10.3 Inner c r o s s - s p e c t r u m between the 40 m c u r r e n t a t DB-7 and the 20 m c u r r e n t a t NH-10 64 10.4 Inner c r o s s - s p e c t r u m between t h e 40 m c u r r e n t a t DB-7 and the Newport wind 68 10.5 Inner c r o s s - s p e c t r u m between the 20 m c u r r e n t a t NH-10 and the Newport wind 69 10.6 A u t o s p e c t r a of t h e 20 m c u r r e n t a t NH-10 and the Newport wind 72 10.7 Time s e r i e s and a u t o s p e c t r a of v a t NH-10(60m) and UWIN(66m) 73 11.1 C u r r e n t meter a r r a y i n the CUE-2 experiment .76 11.2 Length of c u r r e n t d a t a r e c o r d s i n CUE-2 77 11.3 C r o s s - s h e l f topography i n the CUE-2 experiment 78 v i 11.4 B a r o t r o p i c s h e l f wave d i s p e r s i o n diagram f o r CUE-2 . 79 11.5 A u t o s p e c t r a of the 40 m c u r r e n t s a t C a r n a t i o n and F o r s y t h i a 80 11.6 Autospectrum of Newport a d j u s t e d sea l e v e l d u r i n g CUE-2 82 11.7 Autospectrum of Newport wind d u r i n g CUE-2 83 11.8 S t a b i l i t y and phase f o r the 40 m c u r r e n t a t A s t e r .. 84 11.9 Inne r c r o s s - s p e c t r u m between the 40 m c u r r e n t s a t A s t e r and C a r n a t i o n 85 11.10 Outer c r o s s - s p e c t r u m between the 40 m c u r r e n t s a t A s t e r and C a r n a t i o n 86 11.11 Inne r c r o s s - s p e c t r u m between the 40 m c u r r e n t a t A s t e r and the a d j u s t e d sea l e v e l a t Newport 88 11.12 Inne r c r o s s - s p e c t r u m between the 40 m c u r r e n t s a t C a r n a t i o n and a t P o i n s e t t i a 90 11.13 Outer c r o s s - s p e c t r u m between the 40 m c u r r e n t s a t C a r n a t i o n and a t P o i n s e t t i a 93 11.14 Inne r c r o s s - s p e c t r u m between the 40 m c u r r e n t a t P o i n s e t t i a and the wind a t Newport 94 12.1 C u r r e n t meter a r r a y i n the WISP experiment 102 12.2 C u r r e n t meter a r r a y and l e n g t h of da t a r e c o r d s i n WISP 103 12.3 C r o s s - s h e l f topography i n the WISP experiment 104 12.4 B a r o t r o p i c s h e l f wave d i s p e r s i o n diagram f o r WISP ..105 12.5 A u t o s p e c t r a of the 25 m c u r r e n t s at Su n f l o w e r and W i s t e r i a 106 12.6 Inne r c r o s s - s p e c t r u m between the 25 m c u r r e n t a t P i k a k e and the Newport a d j u s t e d sea l e v e l 109 12.7 Inne r c r o s s - s p e c t r u m between the 25 m c u r r e n t a t P i k a k e and the Newport wind 110 12.8 A u t o s p e c t r a of the 25 m c u r r e n t a t P i k a k e and the Newport wind 112 v i i ACKNOWLEDGEMENT I would l i k e t o t a k e t h i s o p p o r t u n i t y t o thank a l l my t e a c h e r s i n the Departments of Oceanography, P h y s i c s , and M a t h e m a t i c s — f o r the growth i n knowledge and wisdom e n j o y e d d u r i n g my n i n e y e a r s of study a t the U n i v e r s i t y of B r i t i s h C o l u m b i a . In p a r t i c u l a r , I would l i k e t o e x p r e s s my deepest g r a t i t u d e t o my r e s e a r c h s u p e r v i s o r , P r o f e s s o r L. A. Mysak, f o r h i s e n t h u s i a s t i c and i n s p i r i n g g u i d a n c e , h i s k i n d n e s s and g e n e r o s i t y , which r e n d e r e d my s t u d i e s under him t r u l y f r u i t f u l and memorable. The Oregon s h e l f d a t a were s u p p l i e d g e n e r o u s l y by Dr. A. Huyer of the Oregon S t a t e U n i v e r s i t y a t C o r v a l l i s , and Dr. D. H a l p e r n of the P a c i f i c M a r ine Environment L a b o r a t o r y a t S e a t t l e . V a l u a b l e a d v i c e from Dr. G. S. Pond and Dr. W. J . Emery eased the t a s k of da t a a n a l y s i s . T i d a l removal was performed w i t h the h e l p of Mr. M. G. G. Foreman, Dr. P. B. Crean, and Mr. D. K. Lee. Dr. R. W. B u r l i n g and Dr. P. H. L e B l o n d p r o v i d e d much c o n s t r u c t i v e c r i t i c i s m on a p r e l i m i n a r y d r a f t of t h i s t h e s i s . The a s s i s t a n c e of t e c h n i c a l s t a f f members and f e l l o w g raduate s t u d e n t s , e s p e c i a l l y Mr. D. L a P l a n t e , Mr. R. L a k o w s k i , and Mr. Y. G r a t t o n , i s a l s o g r a t e f u l l y acknowledged. I d e e p l y a p p r e c i a t e the c o n t i n u a l f i n a n c i a l s u p p o r t p r o v i d e d i n the form of s c h o l a r s h i p s by the N a t u r a l S c i e n c e s and E n g i n e e r i n g R e s e a r c h C o u n c i l of Canada, and the K i l l a m F o u n d a t i o n , and the unending moral support p r o v i d e d by my p a r e n t s and s i s t e r s . 1 CHAPTER 1. INTRODUCTION Very l o n g waves w i t h p e r i o d s of s e v e r a l days have o f t e n been o b s e r v e d p r o p a g a t i n g a l o n g c o n t i n e n t a l s h e l v e s , w i t h t h e i r e n e r g i e s t r a p p e d near the c o a s t . The t h e o r y of c o n t i n e n t a l s h e l f waves shows t h a t the c o n t i n e n t a l s h e l f s e r v e s as a wave gui d e f o r low f r e q u e n c y p r o p a g a t i o n , and i n the absence of s t r o n g mean f l o w s , t h e s e waves o n l y propagate w i t h the c o a s t t o t h e i r r i g h t i n the N o r t h e r n Hemisphere, ( i . e . a l o n g the west c o a s t of N o r t h A m e r i c a , s h e l f waves o n l y t r a v e l n o r t h w a r d ) . Mysak (1980) g i v e s a broad review of the r e c e n t t h e o r e t i c a l and o b s e r v a t i o n a l work i n s h e l f wave dynamics. Much of the r e s e a r c h on the t h e o r y of s h e l f waves has been done u s i n g the long-wave e q u a t i o n s i n which the n o n l i n e a r terms have been o m i t t e d . However, d u r i n g the p a s t f i f t e e n y e a r s or so, many i n v e s t i g a t o r s s t u d y i n g o t h e r t y p e s of o c e a n i c wave motion have found t h a t the n o n l i n e a r terms i n the g o v e r n i n g e q u a t i o n s can l e a d t o wave-wave i n t e r a c t i o n s where energy i s t r a n s f e r r e d between d i f f e r e n t wave components. D i s c u s s i o n s of t h i s wave-wave i n t e r a c t i o n mechanism i n the ocean a r e p r e s e n t e d i n Hasselmann (1968), P h i l l i p s (1977), and L e B l o n d and Mysak (1978, Sec. 3 8 ) . In g e n e r a l , the energy exchanges due t o the n o n l i n e a r terms a r e of o r d e r I, -where 6 i s a ( u s u a l l y s m a l l ) parameter a r i s i n g from n o n d i m e n s i o n a l i z i n g the g o v e r n i n g e q u a t i o n s (e.g. f o r s h e l f waves, S i s the Rossby number and i s t y p i c a l l y of o r d e r 10" z ). The n o n l i n e a r energy exchanges a r e , t h e r e f o r e , u s u a l l y s m a l l . However, when c e r t a i n "resonance c o n d i t i o n s " a r e s a t i s f i e d by the waves, the energy exchanges can Chap. 1 INTRODUCTION 2 be a p p r e c i a b l e . The l o w e s t o r d e r r e s o n a n t i n t e r a c t i o n i n v o l v e s t h r e e waves (a t r i a d ) . I t i s w e l l known t h a t r e s o n a n t t r i a d i n t e r a c t i o n s can o c c u r between Rossby waves, i n t e r n a l waves and edge waves, but not between s u r f a c e g r a v i t y waves ( f o r which r e s o n a n t i n t e r a c t i o n s can o c c u r o n l y a t h i g h e r o r d e r , i . e . between f o u r or more waves). In t h i s t h e s i s , i t i s shown t h a t r e s o n a n t i n t e r a c t i o n s between t h r e e u n f o r c e d b a r o t r o p i c s h e l f waves a r e . t h e o r e t i c a l l y p o s s i b l e , and t h a t t h e r e i s o b s e r v a t i o n a l e v i d e n c e of such s h e l f wave i n t e r a c t i o n s on the c o n t i n e n t a l s h e l f o f f Oregon. One of the resonance c o n d i t i o n s i s t h a t the a n g u l a r f r e q u e n c i e s <J, , Q% and CJ3 of the t h r e e s h e l f waves s a t i s f y + C0X - co3 (1.1) C u t c h i n and Smith (1973) o b s e r v e d t h r e e s i g n i f i c a n t peaks at around 0.22, 0.44 and 0.65 cpd ( c y c l e s per day) i n v a r i o u s c r o s s s p e c t r a computed from Oregon s h e l f d a t a c o l l e c t e d d u r i n g the summer of 1968. (See F i g . 9.3). The resonance c o n d i t i o n (1.1) i s f a i r l y w e l l s a t i s f i e d by t h e s e t h r e e o b s e r v e d f r e q u e n c i e s . Buchwald and Adams (1968) noted t h a t f o r s h e l f wave d i s p e r s i o n c u r v e s which have z e r o s l o p e ( i . e . z e r o group v e l o c i t y ) a t c e r t a i n f r e q u e n c i e s , the wave energy cannot p r o p a g a t e away a t t h e s e f r e q u e n c i e s , and t h e r e f o r e , peaks i n t h e f r e q u e n c y spectrum might be e x p e c t e d t h e r e . C u t c h i n and S m i t h (1973) a t t e m p t e d t o a p p l y t h e Buchwald-Adams z e r o group v e l o c i t y resonance mechanism t o t h e i r o b s e r v e d peaks, but c o n c l u d e d : "The r e s t r i c t i o n of s h e l f wave phenomena t o Chap. 1 INTRODUCTION 3 o n l y c e r t a i n f r e q u e n c y bands remains as" one of the most i n t e r e s t i n g and p u z z l i n g o b s e r v a t i o n s made d u r i n g t h i s e x p e r i m e n t . The a u t o s p e c t r a of sea l e v e l s , c u r r e n t s and a t m o s p h e r i c p r e s s u r e a r e r e l a t i v e l y smooth y e t the coherence s p e c t r a a l l e x h i b i t s t r o n g peaks. The ob s e r v e d phase d i f f e r e n c e s between sea l e v e l v a r i a t i o n s a l o n g the c o a s t do not support a s i m p l e e x p l a n a t i o n based on the resonance mechanism suggested by Buchwald and Adams (1968)." The r e s t r i c t i o n of s h e l f wave phenomenon t o c e r t a i n narrow f r e q u e n c y bands s t r o n g l y s u g g e s t s a resonance mechanism a t work on the Oregon s h e l f . Can the t r i a d resonance mechanism succeed where the z e r o group v e l o c i t y mechanism has f a i l e d ? As w i l l be d i s c u s s e d f u l l y i n Chapter 9, i t appears t h a t the r e s o n a n t t r i a d t h e o r y can s a t i s f a c t o r i l y e x p l a i n the o b s e r v a t i o n s of C u t c h i n and Smith (1973). The dominant peak a t 0.22 cpd, presumably g e n e r a t e d by some l a r g e - s c a l e a t m o s p h e r i c system w i t h t h i s f r e q u e n c y , had t o t r a n s f e r some of i t s energy t o the two " p r e f e r r e d " f r e q u e n c i e s a t around 0.40 and 0.65 cpd due t o the n o n l i n e a r terms i n the g o v e r n i n g e q u a t i o n s . At o t h e r f r e q u e n c i e s , the resonance c o n d i t i o n s were not s a t i s f i e d and the t r a n s f e r of energy from the main peak t o these f r e q u e n c i e s was s m a l l . The n o n l i n e a r energy t r a n s f e r e v e n t u a l l y produced a d i s t i n c t i v e l y t r i p l e - p e a k e d (coherence) spectrum, as was o b s e r v e d . One i n t e r e s t i n g consequence of the re s o n a n t t r i a d i n t e r a c t i o n i s the t r a n s f e r of energy from the l o w - f r e q u e n c y w i n d - g e n e r a t e d l o n g s h e l f wave t o s h e l f waves of h i g h e r f r e q u e n c y and much s h o r t e r w a v e l e n g t h . F u r t h e r a n a l y s i s of Oregon s h e l f d a t a i n t h i s t h e s i s r e v e a l s e v i d e n c e f o r the Chap. 1 INTRODUCTION 4 p r e s e n c e of t h e s e s h o r t s h e l f waves. These o b s e r v a t i o n s a r e rema r k a b l e i n t h a t e xcept f o r the case of s u r f a c e g r a v i t y waves ( e . g . see Hasselmann e_t a l . , 1973), the d i r e c t o b s e r v a t i o n of n o n l i n e a r i n t e r a c t i o n s i_n the ocean i s r e l a t i v e l y r a r e . Most o b s e r v a t i o n s of g e o f l u i d d y n a m i c a l wave-wave i n t e r a c t i o n s have been made o n l y i n the l a b o r a t o r y . T h i s t h e s i s complements a number of e a r l i e r s t u d i e s d e a l i n g w i t h n o n l i n e a r e f f e c t s i n s h e l f waves. Sm i t h (1972) and Grimshaw (1977a) have s t u d i e d the m o d i f i c a t i o n of an i n d i v i d u a l s h e l f wave by the n o n l i n e a r terms. Grimshaw (1977b) examined s i d e - b a n d i n s t a b i l i t y f o r s h e l f waves, w h i l e B a r t o n (1977) l o o k e d a t the g e n e r a t i o n of s h e l f waves by the wind v i a resonant i n t e r a c t i o n s . T h i s t h e s i s i s d i v i d e d i n t o two main p a r t s — P a r t I , Theory, and P a r t I I , O b s e r v a t i o n . P a r t I i s s t r u c t u r e d as f o l l o w s : From the n o n l i n e a r g o v e r n i n g e q u a t i o n s i n t r o d u c e d i n Chapter 2, the t h e o r y of resonant t r i a d i n t e r a c t i o n s f o r s h e l f waves i s d e v e l o p e d i n Chapter 3, where, i n p a r t i c u l a r , the e q u a t i o n s g o v e r n i n g the a m p l i t u d e s of the i n d i v i d u a l waves i n a r e s o n a n t t r i a d a r e d e r i v e d . Energy t r a n s f e r and energy c o n s e r v a t i o n a r e examined i n C h a p t e r 4, f o l l o w e d by an e l e m e n t a r y a n a l y s i s of the wave a m p l i t u d e e q u a t i o n s i n Chapter 5. The t h e o r y i s then s p e c i a l i z e d i n Chapter 6 t o the f a m i l i a r e x p o n e n t i a l c o n t i n e n t a l s h e l f p r o f i l e ( f i r s t i n t r o d u c e d by Buchwald and Adams, 1968). P a r t I I i s s t r u c t u r e d as f o l l o w s : In Chapter 7, d i s p e r s i o n c u r v e s and c u r r e n t e l l i p s e s a s s o c i a t e d w i t h s h e l f waves are Chap. 1 INTRODUCTION 5 i n t r o d u c e d . Two t e c h n i q u e s f o r d e t e c t i n g s h e l f w a v e s — r o t a r y s p e c t r a l a n a l y s i s and c r o s s - s h e l f modal f i t t i n g - - a r e p r e s e n t e d i n C h a pter 8. The t r i p l e - p e a k e d spectrum found on the Oregon s h e l f d u r i n g the summer of 1968 by C u t c h i n and Smith (1973) i s i n t e r p r e t e d by the re s o n a n t t r i a d i n t e r a c t i o n mechanism i n Chapter 9, where, i n p a r t i c u l a r , t h e time s c a l e f o r energy t r a n s f e r i s e s t i m a t e d t o be of the o r d e r of 12 days. Chapter 10 examines the CUE-1 experiment on the Oregon s h e l f d u r i n g summer, 1972, and the subsequent paper by Huyer e t a_l. (1975). C h a p t e r s 11 and 12 p r e s e n t t h e o b s e r v a t i o n s on the Oregon s h e l f d u r i n g the CUE-2 (summer, 1973) experiment and the WISP ( w i n t e r and s p r i n g , 1975) e x p e r i m e n t , r e s p e c t i v e l y . C h a pter 13 d i s c u s s e s the problems and l i m i t a t i o n s a s s o c i a t e d w i t h the t h e o r y and w i t h the i n t e r p r e t a t i o n of the d a t a , as w e l l as c r i t i c a l l y e v a l u a t i n g the e v i d e n c e f o r re s o n a n t t r i a d i n t e r a c t i o n s i n l i g h t of o t h e r competing t h e o r i e s . C hapter 14 b r i n g s the t h e s i s t o an end w i t h a summary and a c o n c l u s i o n . PART I . THEORY 6 CHAPTER 2. GOVERNING EQUATIONS The n o n l i n e a r long-wave e q u a t i o n s f o r a r o t a t i n g , homogeneous f l u i d a r e : 1 1^ 4 u u x + vuy - f i / = -$% - -jfV* + ^ r j (2.1) v t + u v x + uv-y + fu. = - S l y — ^ rVy + jw r* ( 2 - 2 ) ( k « 0 x + (ky\ = - ( l u ) x - (2.3) where t i s the t i m e , x the o f f s h o r e c o o r d i n a t e and y the a l o n g s h o r e c o o r d i n a t e ; (u,v) a r e t h e h o r i z o n t a l v e l o c i t y components i n the (x,y) d i r e c t i o n , \ the sea s u r f a c e d i s p l a c e m e n t from e q u i l i b r i u m , p^ the a t m o s p h e r i c p r e s s u r e a t the sea s u r f a c e , ( f|, t^) t h e s u r f a c e w i n d s t r e s s components (mod e l l e d as a body f o r c e ) and h the depth of the ocean; f and g are the C o r i o l i s parameter and the g r a v i t a t i o n a l c o n s t a n t , r e s p e c t i v e l y . We s h a l l assume t h a t the d e n s i t y £ and f a r e c o n s t a n t s , and t h a t the ocean depth h i s a f u n c t i o n of x o n l y . Next, we n o n - d i m e n s i o n a l i z e the v a r i a b l e s {x,y ,h, t ,u,v ,i{, t,, ?x] w i t h r e s p e c t t o the s c a l e s {L,L,H e ,f ', V, V, * [ 0 , f 0 , To}. Assuming t h a t the terms f v and g ^ i n Eq. (2.1) a r e of comparable magnitudes, we l e t 1 Eqs. (2.1) and (2.2) a r i s e from the c o n s e r v a t i o n of momentum i n the x and y d i r e c t i o n s , w h i l e Eq. (2.3) comes from the c o n s e r v a t i o n of mass (or c o n t i n u i t y ) . See e.g. L e B l o n d and Mysak (1978, pp.127-128). Chap. 2 GOVERNING EQUATIONS 7 1 = * V L / 3 (2.4) In a d d i t i o n , we i n t r o d u c e two n o n - d i m e n s i o n a l p a r a m e t e r s : £ = V / L f , /• = ef/gHo (2.5) where £ i s the Rossby number, and JUZ, the d i v e r g e n c e parameter. E x p r e s s i n g the v o r t i c i t y e q u a t i o n ( i . e . 9^  (2.1')— ^ ( 2 . 2 ) ) and Eq. (2.3) i n n o n - d i m e n s i o n a l form y i e l d s : ( * j - v < ) t + e L(uux + v w l ) i j ~ ( M Vx + v v j ) x ] ~ ( v j + U x ) = F (2.6) ( U ) x + M y = - / { * l t + e C M x + O l v - ^ l } (2.7) where F, the w i n d - s t r e s s f o r c i n g term, i s g i v e n by F = ^ U r ^ - ^ + T ? ^ ( 2- 8> w i t h L x an a p p r o p r i a t e l e n g t h s c a l e f o r the w i n d - s t r e s s . We choose L t o be the w i d t h of the c o n t i n e n t a l s h e l f / s l o p e r e g i o n (L ^  100 km) and H„ the s h e l f depth s c a l e (H„ ^ 200 m). For s h e l f waves, V can be tak e n t o be 10"' m s"', and a t -4- —i m i d - l a t i t u d e s f i s t y p i c a l l y 10 r a d s . W i t h th e s e c h o i c e s , our n o n - d i m e n s i o n a l parameters have the f o l l o w i n g v a l u e s : C = 0.01 and yu. = 0.05. T a k i n g the w i n d - s t r e s s s c a l e t0 t o be 10 N m"z, we have %>/(fH 0 fV) = 0.05. Adams and Buchwald (1969) and G i l l and Schumann (1974) have s t u d i e d the g e n e r a t i o n of s h e l f waves by the win d - s t r e s s . . Here, however, we s h a l l n e g l e c t the f o r c i n g term F and examine o n l y the u n f o r c e d n o n l i n e a r e q u a t i o n s . Chap. 2 GOVERNING EQUATIONS 8 F u r t h e r m o r e , we make the n o n d i v e r g e n c e a p p r o x i m a t i o n by n e g l e c t i n g the terms on the r i g h t s i d e of Eq. ( 2 . 7 ) . 1 T h i s a l l o w s us t o i n t r o d u c e a t r a n s p o r t s t r e a m f u n c t i o n such t h a t U = % , U = -% (2 .9) S u b s t i t u t i n g (2 .9) i n t o the u n f o r c e d v e r s i o n of ( 2.6), and w r i t i n g h x as h' , we o b t a i n - + + 4 - + %%„ } ( 2 ' 1 0 ) L o o k i n g f o r waves t r a p p e d near the c o a s t , we impose the boundary c o n d i t i o n s ^-»0 as x-> °o , and hu = 0 a t x = 0. From Eq. ( 2 . 9 ) , the second boundary c o n d i t i o n becomes ^= 0 a t x = 0. Next, we assume ^ can be expanded as f = f ( D ) + e y ( , ) + e a f w + • • • ( 2 . i i ) where the vp^'s a r e independent of £. . We a l s o assume >p = T 7 (x ,y, t , Y,T) where Y and T are the "slow" v a r i a b l e s d e f i n e d by Y = e y , T = t-t (2.12) Thus, i n Eq. ( 2 . 1 0 ) , ^ - |_ + £ |_ «»cV £ -> A + 1 To d i s c a r d terms of 0 ( / t a ) but r e t a i n terms of 0(e) seems u n r e a s o n a b l e at t h i s s t a g e as £. i s s m a l l e r than / t z . However, i n next c h a p t e r , we s h a l l f i n d t h a t when c e r t a i n resonance c o n d i t i o n s a r e s a t i s f i e d , the n o n l i n e a r i n t e r a c t i o n a c t u a l l y o c c u r s a t 0(1) r a t h e r than 0 ( e ) . Chap. 2 GOVERNING EQUATIONS 9 S u b s t i t u t i n g (2.11) i n t o ( 2 . 1 0 ) , we o b t a i n the 0(1) and 0(£) equat i o n s : 0(0 : • + N£? ) = 0 (2.J3) w i t h boundary c o n d i t i o n s r •* 0 as x-»°°, = 0 a t x = 0; and + w<? -<) +[*"- 3^)lj^v;rt + a ^ ' C - fcW - o r • ^  + *K*r - « ( 2 . i 4 , w i t h boundary c o n d i t i o n s >p(IV 0 as x - * o o , and ^ + ^ = 0 a t x = 0. For a s h e l f wave t r a v e l l i n g p a r a l l e l t o a c o a s t , we l e t have the form A(Y,T )ck*)e i a*- w i ) + c c . ( 2 . 1 5 ) where c.c. denotes the complex c o n j u g a t e of the p r e c e d i n g term. S u b s t i t u t i n g Eq. (2.15) i n t o Eq. ( 2 . 1 3 ) , and l e t t i n g c = w/k, we o b t a i n the f o l l o w i n g d i f f e r e n t i a l e q u a t i o n f o r 4>: -(-J-(j>')/+ -£<|, - Jr(J^ )<j> = 0 , 0 <%<"> (2.16) w h i c h , f o r a g i v e n v a l u e of k, t u r n s out t o be o£ the S t u r m - L i o u v i l l e form - [f>Mcj>']'+ <j(x)<f> - Xr(x)d> = 0 (2.17) as g i v e n i n Boyce and D i P r i m a (1969, p.493). Hence, the 0 ( 1 ) e q u a t i o n reduces t o a S t u r m - L i o u v i l l e problem w i t h c~' as the e i g e n v a l u e and w i t h boundary c o n d i t i o n s $ 0 as x * o o , and 0 Chap. 2 GOVERNING EQUATIONS 10 a t x = 0. Eq. (2.16) w i t h an e x p o n e n t i a l s h e l f p r o f i l e has been s o l v e d by Buchwald and Adams (1968) f o r the e i g e n v a l u e s l/c(,° and e i g e n f u n c t i o n s <t>(,° c o r r e s p o n d i n g t o d i f f e r e n t s h e l f wave modes (n = 1,2,...). By v a r y i n g k, the d i s p e r s i o n c u r v e s iJn\k) = kc l f o r the d i f f e r e n t modes were a l s o o b t a i n e d . 11 CHAPTER 3. THEORY OF RESONANT INTERACTIONS BETWEEN SHELF WAVES In t h i s c h a p t e r , we w i l l show t h a t the presence of the n o n - l i n e a r terms i n Eq. (2.14) [ i . e . , the 0 ( e ) e q u a t i o n ] a l l o w s r e s o n a n t i n t e r a c t i o n s t o occur between t h r e e s h e l f waves. We s t a r t w i t h t h r e e d i s t i n c t s h e l f waves of f r e q u e n c i e s cJj and wavenumbers kj ( j = 1,2,3), i . e . V = I , A ; ( Y,T ) ( p.Cx)e t t R^ J + C.C. (3.1) 6 = > J . 5 Each of the <^'s s a t i s f i e s an e q u a t i o n c o r r e s p o n d i n g t o (2.16) : - ( - k 4 0 ' + ~ t - ( f * H = 0 » 0 £ X < o o (3.2) where c^ = ^ j / k j i s the phase speed of the j t h wave. The boundary c o n d i t i o n s a r e 4>. = 0 a t X = O , <fc " > 0 a s X * oo (3.3) We note t h a t the t h r e e waves d e s c r i b e d i n Eqs. (3. 1 ) - ( 3 . 3 ) do not n e c e s s a r i l y c o r r e s p o n d t o t h r e e d i s t i n c t s h e l f waves modes. Hence the s u p e r s c r i p t n p e r t a i n i n g t o the mode number has been dropped. As w i l l be seen l a t e r (Chapter 9 ) , two of the t h r e e waves i n a res o n a n t t r i a d f o r the Oregon s h e l f a r e f i r s t - m o d e s h e l f waves a t two d i f f e r e n t wavenumbers (k, and k 3 ) . S u b s t i t u t i n g (3.1) i n t o ( 2 . 1 4 ) , and u t i l i z i n g ( 3 . 2 ) , we o b t a i n Chap. 3 THEORY OF RESONANT INTERACTIONS 12 In o r d e r t o match the f r e q u e n c i e s and. wavenumbers on the r i g h t s i d e of Eq. ( 3 . 4 ) , ^ ( , ) must a l s o be of the form t ^ Z A l W t f W ^ - ^ t - ,3.5, where the s e t of v a l u e s f o r (k ^ , ^ * ) i s g i v e n by { ( ^ ^ « ) } = {(k5,*5)\j = l , * > 3 } U{Cle,,^t)+ (fem,"*.)|i = 1 . ^3 , n> = i,x , 3 } ( 3 . 6 ) F u r t h e r m o r e , we e s t a b l i s h the f o l l o w i n g c o n v e n t i o n f o r i n d e x i n g : f o r a = 1, 2 , 3 , ( k„ , <0«) s ( k- , ^ ) . We now examine the s i t u a t i o n f o r <y = j = 1,2,3. S u b s t i t u t i n g Eq. (3.5) i n t o the l e f t s i d e of ( 3 . 4 ) , we f i n d t h a t f o r each j , 4^must s a t i s f y an inhomoqeneous e q u a t i o n where f^ denotes the f o r c i n g a t f r e q u e n c y "3- and wavenumber kj from the r i g h t s i d e of ( 3 . 4 ) . [ I n c o n t r a s t , we r e c a l l t h a t ^ s a t i s f i e s the c o r r e s p o n d i n g homogeneous e q u a t i o n (3.2) w h i c h , t o g e t h e r w i t h boundary c o n d i t i o n s ( 3 . 3 ) , c o n s t i t u t e a s t a n d a r d S t u r m - L i o u v i l l e problem.] In nonresonant cases, o n l y the l i n e a r terms on the r i g h t s i d e of (3.4) can c o n t r i b u t e t o the f j ' s ( j = 1,2,3): ih = [^t 3A a T+ (K^2KIS^)4>./\.YVO^kt<) ( 3 ' 8 ) Chap..3 THEORY OF RESONANT INTERACTIONS 13 However, i f any of the f o l l o w i n g resonance c o n d i t i o n s a r e s a t i s f i e d , v i z . , ± ± (^A) + ( " 3 ^ 3 ) =(0,0) ( 3 . 9 ) some of the n o n l i n e a r terms on the r i g h t s i d e of (3.4) a r e a l s o of f r e q u e n c i e s ^ and wavenumbers kj (j= 1,2,3), and hence can c o n t r i b u t e t o the f o r c i n g terms f^ ' s. W i t h no l o s s of g e n e r a l i t y , we w r i t e the resonance c o n d i t i o n s as 6J, + w 2 + " 3 = 0 K,+ kx+ k3 = 0 (3.10) where { } and {k^-} can have e i t h e r p o s i t i v e or n e g a t i v e v a l u e s . The Fredholm a l t e r n a t e theorem from S t u r m - L i o u v i l l e t h e o r y ( s e e , e.g., Boyce and D i P r i m a , 1969, p.506) s t a t e s t h a t the inhomogeneous problem has a s o l u t i o n cp/'^  o n l y i f the f o r c i n g term f j i s o r t h o g o n a l t o the homogeneous s o l u t i o n <kj • That i s , f £. <j>. Joe = O (3.11) When the resonance c o n d i t i o n s (3.10) a re s a t i s f i e d , the Fredholm c o n d i t i o n (3.11) y i e l d s = - ; ci(K«« + "Wtf'C (3-12) where ( j , l , m ) a r e any of the c y c l i c p e r m u t a t i o n s of ( 1 , 2 , 3 ) , and the terms on the r i g h t - h a n d s i d e of (3.12) a r i s e from the n o n l i n e a r terms i n ( 3 . 4 ) , w i t h Chap. 3 THEORY OF RESONANT INTERACTIONS 14 + 3-JL <f»"<|> - i<b'<J>' 4 4>!<b" - <p;> } (3.13) With r e p e a t e d i n t e g r a t i o n by p a r t s and the h e l p of the boundary c o n d i t i o n s ( 3 . 3 ) , can be r e - e x p r e s s e d i n an o t h e r form Without l o s s of g e n e r a l i t y , we can n o r m a l i z e {4^} so t h a t (3.15) Eq. (3.12) then g i v e s us the a m p l i t u d e e q u a t i o n f o r the j t h wave: I t A o + % h A i = - : M X < 3 - 1 6 > where Kj = Ci ( K i l n + <3-17' and c ^ i s the group v e l o c i t y of the j t h wave (see Appendix A f o r j u s t i f i c a t i o n ) , w i t h % = Ci ( , + i r i C 5 k ^ ( 3 - 1 8 ) oo Y- = J i ^ (3.19) The a m p l i t u d e e q u a t i o n (3.16) shows t h a t , a t resonance, the n o n l i n e a r terms p r o v i d e a c o u p l i n g mechanism between t h e a m p l i t u d e s of the t h r e e waves. I f the resonance c o n d i t i o n s a r e Chap. 3 THEORY OF RESONANT INTERACTIONS 15 not s a t i s f i e d , the n o n l i n e a r terms do not c o n t r i b u t e t o the f o r c i n g term f^ ( j = 1,2,3), and the Fredholm c o n d i t i o n (3.11) s i m p l y y i e l d s fr A i + % h A i = ° ( 3 - 2 0 ) where each wave a m p l i t u d e (at l o w e s t o r d e r ) i s independent of the o t h e r s , and propa g a t e a l o n g a t i t s own group v e l o c i t y . Thus, the n o n l i n e a r terms, d e s p i t e t h e i r p resence o n l y i n the 0(£) e q u a t i o n , can n e v e r t h e l e s s d i r e c t l y a f f e c t the 0 ( 1 ) a m p l i t u d e e q u a t i o n s when the resonance c o n d i t i o n s a re f u l f i l l e d . How energy i s exchanged between the t h r e e waves i n a resonant t r i a d i s our next t o p i c . CHAPTER 4. ENERGY TRANSFER AND ENERGY CONSERVATION 16 In t h i s c h a p t e r , we w i l l show t h a t the t o t a l energy of the t r i a d i s c o n s e r v e d even though energy i s t r a n s f e r r e d between the waves. The n ondivergence or " r i g i d l i d " a p p r o x i m a t i o n employed i n Chapter 2 a l l o w s us t o c o n s i d e r o n l y the k i n e t i c e n e r g i e s of the waves, as t h e i r p o t e n t i a l e n e r g i e s a r e v e r y s m a l l by comparison. To l o w e s t o r d e r , the n o n - d i m e n s i o n a l i z e d k i n e t i c energy per u n i t volume i s g i v e n by KE/volume = -I ( u z + (4.1) where the d e n s i t y f = 1, and we have n e g l e c t e d the . s m a l l c o n t r i b u t i o n from the v e r t i c a l v e l o c i t y component. Next we - i n t e g r a t e Eq. (4.1) w i t h the v e r t i c a l c o o r d i n a t e z r u n n i n g from -h t o 0, the a l o n g s h o r e c o o r d i n a t e y from 0 t o 1, and the o f f s h o r e c o o r d i n a t e x from 0 t o ». Assuming u and v t o be depth independent, and i n v o k i n g Eqs. ( 2 . 9 ) , the r e s u l t i n g k i n e t i c energy per u n i t l e n g t h a l o n g s h o r e can be e x p r e s s e d ( t o lowe s t o r d e r ) as f o l l o w s : K E / l e n g t h = ± J dx $ ^ ^[(tf f + (t^f] (4.2) o 0 to) We f i r s t c o n s i d e r the case of a s i n g l e wave, where Y i s g i v e n by Eq. (2.15), and then s u b s t i t u t e d i n t o Eq. ( 4 . 2 ) . D u r i n g the i n t e g r a t i o n of y from 0 t o 1, A = A(Y,T) i s e s s e n t i a l l y c o n s t a n t . With A w r i t t e n as | A | e t e , we f i n a l l y o b t a i n K E / l e n g t h = \t\\%{\[ j Ax \ - a t + e+-fcs* 2 ^ - u * +e)] | s = o Chap. 4 . ENERGY 17 + kC 5 t ^ JLkjJ" w i + e -±smz(ky - ^ t + e)]| ) (4.3) 0 ' S i n c e we a r e i n t e r e s t e d i n the v a r i a t i o n of energy w i t h r e s p e c t t o t h e slow time T, the " s i n e " terms r e p r e s e n t i n g " h i g h " f r e q u e n c y o s c i l l a t i o n s can be e i t h e r averaged out t o z e r o , or s i m p l y i g n o r e d f o r our pur p o s e s . Thus, we have on the average oo K E / l e n g t h = | A|* j <k% \ kY ) (4.4) o I n t e g r a t i n g the f i r s t term by p a r t s , then i n v o k i n g (2.16) and the n o r m a l i z a t i o n c o n d i t i o n ( 3 . 1 5 ) , we f i n a l l y o b t a i n K E / l e n g t h = - -L | a T (4.5) where we remind the r e a d e r t h a t under our c o o r d i n a t e system the phase speed c i s always n e g a t i v e f o r s h e l f waves i n the N o r t h e r n Hemisphere. For the t r i a d c a s e , we s u b s t i t u t e Eq.. (3.1) i n t o ( 4 . 2 ) , and p e r f o r m a s i m i l a r c a l c u l a t i o n . W i t h the o s c i l l a t o r y terms i g n o r e d , we o b t a i n f o r a t r i a d K E / l e n g t h = Z ~ "if. I Ail ( 4-6) 0=' 0 which i s s i m p l y the sum of the k i n e t i c e n e r g i e s of the i n d i v i d u a l waves. I f we now m u l t i p l y the a m p l i t u d e e q u a t i o n (3.16) by — / C j , we have _ _ L , A *aA3 + r / . x u * ^ - A * A * A * Chap. 4 ENERGY 18 Adding (4.7) t o i t s own complex c o n j u g a t e , we a r r i v e a t the energy e q u a t i o n f o r the j t h wave: ff (-V^ )+<1jfr(-*S|AS|,)= >^ M*A](-!) (4.8) where Re{...} denotes the r e a l p a r t of the terms i n s i d e the b r a c e s . We see t h a t i f the n o n l i n e a r c o u p l i n g term on the r i g h t - h a n d s i d e i s not p r e s e n t , the energy of the j t h wave would s i m p l y p r o p a g a t e a l o n g a t i t s group v e l o c i t y c, . To c o n s i d e r the t o t a l energy of the t r i a d , we need t o sum the t h r e e s e p a r a t e energy e q u a t i o n s , i . e . , J, (IT + ^ H j l A j O - 2KeOA,AA)i(~^) (4.9) The c o n s e r v a t i o n of the t o t a l energy i s then e x p r e s s e d by the f o l l o w i n g theorem. THEOREM: itlf^lfX-^W- ° (4.10. P r o o f : Eq. (4.10) f o l l o w s from Eq. (4.9) i f (4.11) From Eq. ( 3 . 1 7 ) , t h i s amounts t o p r o v i n g Z K — K l x 3 + KI31L + K 2 3 | +• « 2 I 3 + K 3 i x + K 3 2, = o ( 4 . i 2 ) W i t h r e p e a t e d i n t e g r a t i o n by p a r t s , and the h e l p of Eqs. (3.3) and ( 3 . 1 0 ) , i n (3.14) can be r e e x p r e s s e d as « = K ( I ) 4 k ( j ) + K ( 3 ) + K W { 4 ' 1 3 ) Chap. 4 ENERGY 19 where C» ~ [ dX {^^P'X - - k k n t t f ^ ) (4.14) oo We note t h a t U 0 ) ' 1,(3) 1.(3) . (4-) ft.) • W V ' 0 , fc^+K^-O, ^ + ^ = 0 (4.18) «"« £ Kf;m = (k,+1,+ w p x 1 = o (4.i9) by the resonance c o n d i t i o n s ( 3 . 1 0 ) . T h e r e f o r e , S^, which can be w r i t t e n as and our theorem f o r the c o n s e r v a t i o n of the t o t a l energy i n a reso n a n t t r i a d i s p r o v e d . 20 CHAPTER 5. ANALYSIS OF THE AMPLITUDE EQUATIONS In r e c e n t y e a r s , the i n v e r s e s c a t t e r i n g method has been used t o s o l v e c o u p l e d p a r t i a l d i f f e r e n t i a l e q u a t i o n s s i m i l i a r i n form t o our a m p l i t u d e e q u a t i o n s ( 3 . 1 6 ) . (Sometimes the e q u a t i o n s a r e f i r s t c o n v e r t e d t o a s t a n d a r d form by s c a l i n g the t h r e e A- 's so t h a t the c o u p l i n g c o e f f i c i e n t s iK^- ' s have the v a l u e s ± 1). A r e c e n t r e v i e w i s g i v e n by Kaup e t a_l. (1979). Here, we s h a l l not attempt t o s o l v e our a m p l i t u d e e q u a t i o n s i n t h e i r f u l l g l o r y . I n s t e a d , we s h a l l o n l y s t u d y t e m p o r a l b e h a v i o u r by ( e v e n t u a l l y ) n e g l e c t i n g the s p a t i a l d e r i v a t i v e 3/3Y i n our e q u a t i o n s . I f we w r i t e where a^ and 0j a r e r e a l f u n c t i o n s , and s u b s t i t u t e t h i s form i n t o our a m p l i t u d e e q u a t i o n s ( 3 . 1 6 ) , the r e a l and the im a g i n a r y p a r t s of the r e s u l t i n g e q u a t i o n s g i v e A J ( V , T ) E ai(Y,T)«ocP[^ j(Y,T)] (5.1) + (5.2) and (5.3) r e s p e c t i v e l y , where 0 = 8, + + @3 . The energy e q u a t i o n (4.8) can a l s o be e x p r e s s e d as (5.4) F o c u s i n g o n l y on t h e t e m p o r a l b e h a v i o u r , we s e t d/dY t o Chap. 5 THE AMPLITUDE EQUATIONS 21 z e r o . Eqs. (5.2) and (5.3) reduce t o ±£i - - K, a „ a w sin e (5.5) . a. = - K. «,0 m «>« * (5.6) 3 dT 3 There a r e many s p e c i a l c a s e s one can c o n s i d e r . For i n s t a n c e , when 6 = 0 or Tr, the s o l u t i o n s a r e a-= c o n s t a n t , and 8. v a r i e s l i n e a r l y w i t h T. That t h i s s i t u a t i o n c o r r e s p o n d s t o z e r o energy exchange can a l s o be seen from the energy e q u a t i o n ( 5 . 4 ) , where the energy t r a n s f e r term on the r i g h t s i d e v a n i s h e s when 0=0 or TT. The case of g r e a t e s t i n t e r e s t o c c u r s when 0 = ^  or H , For th e s e v a l u e s of 0 , the energy exchange i s maximized, as can be seen from ( 5 . 4 ) . A l s o , the phases 9j a r e c o n s t a n t . For © = \ , (5.5) becomes S i n c e the phase speeds f o r f r e e s h e l f waves a r e always n e g a t i v e ( i n the N o r t h e r n Hemisphere under our c o o r d i n a t e s y s t e m ) , Eq. (4.11) i m p l i e s t h a t the t h r e e K- c o e f f i c i e n t s cannot a l l have the same s i g n . Without l o s s of g e n e r a l i t y , we assume K, and Kz t o have the same s i g n , and K 3 t o have the o p p o s i t e s i g n . The s o l u t i o n s of Eqs. (5.7) are the J a c o b i e l l i p t i c f u n c t i o n s . W i t h no l o s s of g e n e r a l i t y , we can choose T = 0 t o be the i n s t a n t when a, > 0, a 2> 0 and a 3= 0. W i t h t h i s c h o i c e , t h e i n i t i a l c o n d i t i o n s a r e a , ( 0 ) s a ) O > 0, and a 1 ( 0 ) = a ! t o > 0 and a 3 ( 0 ) = 0, and the s o l u t i o n s a r e Chap. 5 THE AMPLITUDE EQUATIONS 22 ( T ) = a,0 dn ( O - T / M ) ( 5 < 8 ) «a ( T ) = a z o cn ( O - T / M ) ( 5 > 9 ) a 3 ( r ) = aaof-Ka/Kz^sn^TlM) ( 5 > 1 0 ) where" ( T = a 1 0 ( - K 2 K 3 ) K ; t M = KXo/(K2<) ( 5 . 1 D and dn, c n , sn are the J a c o b i e l l i p t i c f u n c t i o n s i n the n o t a t i o n of Abramowitz and Stegun (1965, Chap. 16 ). U s u a l l y , the e l l i p t i c f u n c t i o n s a re r e s t r i c t e d t o OS M ^ l . For M = 1, the e l l i p t i c f u n c t i o n s reduce t o h y p e r b o l i c f u n c t i o n s , w h i l e f o r M > 1, the f o l l o w i n g t r a n s f o r m a t i o n s can be used: C M O T | M ) = cn (" M * ( T T | M " ' ) (5.12) Cn(o-TJM) = dn ( M ^ c r T | Nf') (5.13) S n ( c r T | M ) = sn ( N ^ O - T | H ~ ' ) ( 5 . 1 4 ) See Abramowitz and Stegun (1965, Chaps. 16 and 17) f o r p r o p e r t i e s of thes e e l l i p t i c f u n c t i o n s . W ithout l o s s of g e n e r a l i t y , assume O s M S l . Then the p e r i o d of energy t r a n s f e r i s g i v e n by the p e r i o d T^ of the e l l i p t i c f u n c t i o n dn (0"T | M) . From Abramowitz and Stegun (1965, Chaps. 16 and 1 7 ) , Td = *K(M)/V= z K ( M ) a ~ o (-KiK3Jy* (5.15) where K(M) i s a complete e l l i p t i c i n t e g r a l of the f i r s t k i n d . Chap. 5 THE AMPLITUDE EQUATIONS 23 The s o l u t i o n s ( 5 .8)-(5.10) c o r r e s p o n d t o the f o l l o w i n g s i t u a t i o n : D u r i n g the time i n t e r v a l 0 < T < ^T^, the t h i r d wave e x t r a c t s energy from the f i r s t two waves, w h i l e d u r i n g JTJL< T < T^ , the t h i r d wave t r a n s f e r s energy back t o the f i r s t two waves u n t i l the i n i t i a l c o n d i t i o n s a t T = 0 are a g a i n reached a t T = . As M -*• 1, K ( M ) - » ° o , i m p l y i n g an i n f i n i t e l y l o n g p e r i o d f o r energy t r a n s f e r . However, except f o r the v a l u e s of M v e r y c l o s e t o u n i t y , K ( M ) i s not a r a p i d l y v a r y i n g f u n c t i o n . For example, as M i n c r e a s e s from 0 t o 0.99, K ( M ) i n c r e a s e s m o n o t o n i c a l l y from 1.57 t o 3.70. U s i n g t h i s f a c t and Eq. ( 5 . 1 5 ) , we s h a l l , i n Chapter 9, o b t a i n a crude e s t i m a t e of the time s c a l e of energy t r a n s f e r f o r resonant s h e l f waves on the Oregon s h e l f . 24 CHAPTER 6- RESONANT INTERACTIONS ON AN EXPONENTIAL SHELF So f a r , the depth h has not been s p e c i f i e d e x p l i c i t l y i n our t h e o r y . In t h i s s e c t i o n , we w i l l use the Buchwald and Adams (1968) e x p o n e n t i a l s h e l f p r o f i l e f o r h, i . e . f H , e * b * o < x < l K = | (6.1) l Hi i <: x < oo where b, H, and H 2 are c o n s t a n t p a r a m e t e r s , w i t h H = H, e Wi t h t h i s c h o i c e of h, Eq. (3.2) becomes 1 2 b ^ ' i _ n 0 — X ~ (6.2) *V ~ ty = 0 |< X < oo The s o l u t i o n of (6.2) i s g i v e n i n Buchwald and Adams. The r e s u l t i n g ' d i s p e r s i o n r e l a t i o n f o r the n t h o f f s h o r e mode i s g i v e n by o f = -ab^ /as/^ +kJ + [?] ( » = ^ , - 0 ( 6 . 3 ) where i s the nth r o o t of the t r a n s c e n d e n t a l e q u a t i o n t « l f = - J ? 7 ( b + l l y l ) , ^ > o (6.4) I t i s customary t o o r d e r the l ^ ' s as f o l l o w s : J Under the n o r m a l i z a t i o n c o n d i t i o n (3.15) , ^  c o r r e s p o n d i n g t o the n t h mode i s g i v e n by Chap. 6 ON THE EXPONENTIAL SHELF 25 M I 3 0 where N f = [ n^/Kasf- . J H a ^ n * (6.6) For c o n v e n i e n c e , we now drop the modal s u p e r s c r i p t n. S u b s t i t u t i n g Eq. (6.5) i n t o (3.14), and p e r f o r m i n g the l e n g t h y but s t r a i g h t f o r w a r d a l g e b r a , one can show t h a t the c o e f f i c i e n t s f o r the e x p o n e n t i a l s h e l f a r e + s ^ - s u ^ ^ n ^ - z ^ i - f ^ + i ^ f t ; - ^ ) ] } (6.7) where I , I 2 and I 3 a r e the f o l l o w i n g i n t e g r a l s : 1x = e b f sin^x s'vnSjL* s W % „ x e k x d % (6.8) o I = € b J S^i$.* C*s5,X S m l X e~k*c{% (6.9) I = e b f C&sl-X SinS** S^S„X e b*p(X (6.10) 3 ^ 3 These i n t e g r a l s can be e v a l u a t e d a n a l y t i c a l l y , y i e l d i n g I v = • t [ ( b A l + b l B 1-t 1c 1 ) - ( b A a+tA--fc aC a) - 0 > A 3 + Ma - b 3 C 3 ) + b + B ^ . - b^ C )^] (6.11) X2 = 4-[(b,A,- bB, + t»C , ) - a a A , - t r 3 2 + b 0 4 ( l,3A3 - b B 3 - f bC3> Chap. 6 ON THE EXPONENTIAL SHELF 26 - (t>^ AH- - b B 4 + \>c*)~] (6.12) I 3 = { [-Ct, A, - b B, + bC,) + ( b 2 A a - b B 2 + b C 2 ) + ( b 3 A 3 - b B 3 + b C 3 ) ~ ( b 4 A 4 - b B 4 + b C 4 ) ] where and LAc .B^C ,- ] = -J - ^ t s l n b i , c o S b ^ e b ] j 1=1,2,3,^ ( 6 . 1 5 ) B e f o r e moving onto P a r t I I , O b s e r v a t i o n , l e t us summarize what has been a c c o m p l i s h e d i n P a r t I . S t a r t i n g from the i n v i s c i d , u n f o r c e d long-wave e q u a t i o n s f o r a r o t a t i n g , homogeneous f l u i d , i t has been shown t h a t r e s o n a n t i n t e r a c t i o n s between t h r e e c o n t i n e n t a l s h e l f waves can o c c u r . The e q u a t i o n s g o v e r n i n g the a m p l i t u d e and the energy of i n d i v i d u a l waves i n a resonant t r i a d have been d e r i v e d . The a n a l y s i s of the energy e q u a t i o n i n Chapter 4 r e v e a l s t h a t energy i s t r a n s f e r r e d between the waves by the n o n l i n e a r terms, but w i t h the t o t a l energy c o n s e r v e d . In Chapter 5 , upon n e g l e c t i n g the a l o n g s h o r e d e r i v a t i v e 9/dY i n the a m p l i t u d e e q u a t i o n s , the wave a m p l i t u d e s a r e g i v e n by e l l i p t i c f u n c t i o n s i n the case of maximum energy exchange. The t h e o r y i s then a p p l i e d t o the f a m i l i a r e x p o n e n t i a l s h e l f p r o f i l e , where the c o u p l i n g c o e f f i c i e n t s a r e ( 6 . 1 3 ) 5, " \ + 5 m ( 6 . 1 4 ) Chap. 6 ON THE EXPONENTIAL SHELF 27 found a n a l y t i c a l l y . The r e s u l t s from C h a p t e r s 5 and 6 w i l l be used i n Chapter 9 t o e s t i m a t e the time s c a l e of n o n l i n e a r energy t r a n s f e r on the Oregon s h e l f . 28 PART 2. OBSERVATION CHAPTER 7. DISPERSION CURVES AND CURRENT ELLIPSES P a r t I I of t h i s t h e s i s compares the t h e o r e t i c a l r e s u l t s from P a r t I w i t h o b s e r v a t i o n s . But b e f o r e a comparison can be made, one must f i r s t t h i n k about ways t o d e t e c t s h e l f waves. In t h i s c h a p t e r , we d i s c u s s two f e a t u r e s of s h e l f waves s u i t a b l e f o r t h e i r d e t e c t i o n and i d e n t i f i c a t i o n , namely, t h e i r d i s p e r s i o n c u r v e s and t h e i r c u r r e n t e l l i p s e s . The next c h a p t e r c o n t i n u e s w i t h two d e t e c t i o n t e c h n i q u e s - - r o t a r y s p e c t r a l a n a l y s i s and c r o s s - s h e l f modal f i t t i n g , f o l l o w e d by f o u r c h a p t e r s of o b s e r v a t i o n s on the Oregon s h e l f . To examine i n d i v i d u a l s h e l f waves, we n e g l e c t the n o n l i n e a r and the a t m o s p h e r i c terms i n Eqs. (2.1) - ( 2 . 3 ) , y i e l d i n g ~ V = - 1 x (7.1) V t + K = -H 3 (7.2) (h«0 x + (hv\ = (7.3) where the e q u a t i o n s have been n o n d i m e n s i o n a l i z e d as i n Chapter 2, and the d i v e r g e n c e parameter /<- i s g i v e n i n ( 2 . 5 ) . One r e c a l l s t h a t t h e r e a r e two s m a l l parameters i n the problem-- £, the Rossby number c h a r a c t e r i z i n g the s i z e of the n o n l i n e a r terms, and /A Z . As p o i n t e d out i n Chapter 2, a t m i d - l a t i t u d e s , t y p i c a l l y £ « 0.01 and / t 1 - 0.05. In P a r t I of t h i s t h e s i s , we were i n t e r e s t e d i n n o n l i n e a r i n t e r a c t i o n s , so we r e t a i n e d the Chap. 7 DISPERSION CURVES 29 terms of 0 ( E ) , but f o r m a t h e m a t i c a l s i m p l i c i t y , we d i s c a r d e d the terms of 0(/f), (which a l l o w e d us t o i n t r o d u c e the stream f u n c t i o n Y ). T h i s i s j u s t i f i e d because we have shown t h a t a t resonance t h e n o n l i n e a r i n t e r a c t i o n s a c t u a l l y p r o c e e d a t 0 ( 1 ) . In t h i s p a r t of the t h e s i s , we a r e i n t e r e s t e d i n d e t e c t i n g i n d i v i d u a l s h e l f waves. For m a t h e m a t i c a l s i m p l i c i t y , we n e g l e c t the n o n l i n e a r terms of 0(£). The s m a l l 0 ( / i 2 ) term w i l l be r e t a i n e d though i t o n l y m o d i f i e s the s o l u t i o n s v e r y s l i g h t l y . F o l l o w i n g C u t c h i n and Smith (1973), we l e t (u.v, 1) = Re { [u 0 ( * ) , v 0 ( * ) , v *>] e ^ i T " " 0 } ( 7 . 4 ) P l a c i n g t h i s form i n t o (7.1) and ( 7 . 2 ) , we o b t a i n U 0 = i (-u>*('. + k%)(^-0'' ( 7 . 5 ) V a = (ukf. - lJ)( (7.6) W i t h ( 7 . 5 ) and (7.6) s u b s t i t u t e d back i n t o ( 7 . 3 ) , we e v e n t u a l l y a r r i v e a t an e q u a t i o n f o r the sea s u r f a c e d i s p l a c e m e n t ^ . (hl.')'+ ["ft* It = 0 (7.7) An i n d i s p e n s a b l e a i d f o r s h e l f wave i d e n t i f i c a t i o n i s the d i s p e r s i o n diagram f o r s h e l f wave modes a l l o w e d on a p a r t i c u l a r s h e l f topography. There a r e two ways t o o b t a i n the d i s p e r s i o n c u r v e s . The f i r s t i s t o f i t the Buchwald-Adams e x p o n e n t i a l p r o f i l e (6.1) t o the r e a l s h e l f p r o f i l e , and then s o l v e Eqs. Chap. 7 DISPERSION CURVES 3 0 ( 6 . 3 ) and ( 6 . 4 ) s i m u l t a n e o u s l y t o o b t a i n the d i s p e r s i o n c u r v e s . The pr e s e n c e of a v e r t i c a l w a l l a t the c o a s t , and the f a c t t h a t not a l l s h e l v e s f i t w e l l t o an e x p o n e n t i a l p r o f i l e , a r e two d i s a d v a n t a g e s of t h i s method. A second and more a c c u r a t e way t o o b t a i n the d i s p e r s i o n c u r v e s i s by n u m e r i c a l i n t e g r a t i o n . I n t r o d u c i n g %o = hfo', Eq. ( 7 . 7 ) can be r e w r i t t e n as two c o u p l e d f i r s t o r d e r l i n e a r d i f f e r e n t i a l e q u a t i o n s : 1 0 = i ^ o ( 7 . 8 ) < = ClA+ fek'+Al-^)]^ ( 7 . 9 ) The boundary c o n d i t i o n hu = 0 a t x = 0 , t o g e t h e r w i t h Eq. ( 7 . 5 ) , r e q u i r e s <*>\l + k f 0 r 0 a t x = 0 ( 7 . 1 0 ) In t he deep ocean r e g i o n , we assume the depth h t o become c o n s t a n t , and ( 7 . 9 ) reduces t o Demanding r ( 0 - » 0 as x •> oo , we have, i n the deep ocean r e g i o n , f o = C0^{-lk\^(l~^)k- (fx} ( 7 . 1 2 ) where C a i s a c o n s t a n t . Appendix B d e s c r i b e s the n u m e r i c a l t e c h n i q u e (based on t h a t 1 The 0(yu 2) term has a c t u a l l y been o m i t t e d i n t h e s e e q u a t i o n s . Chap. 7 DISPERSION CURVES 31 of C a l d w e l l and L o n g u e t - H i g g i n s , 1971) used t o o b t a i n the d i s p e r s i o n diagram, and compares the d i s p e r s i o n diagrams o b t a i n e d by the two methods. D i s p e r s i o n c u r v e s f o r the Oregon s h e l f a r e shown i n l a t e r c h a p t e r s . We next d i s c u s s c u r r e n t e l l i p s e s a s s o c i a t e d w i t h waves. The v e l o c i t y components of a t r a v e l l i n g wave of a n g u l a r f r e q u e n c y observed a t a f i x e d o f f s h o r e s t a t i o n can be d e s c r i b e d by u(-t) = a, cos (sit + S,} = B, c o s i l t + B i S ^ - f i t (7.13) v(-t) = a a s U ( s i t +• s2) = B3 cos£i± + Bf ski-Q-t (7.14) where Si = \o\ . L e t us i n t r o d u c e the complex v e l o c i t y w = u + i v . (7.13) + i ( 7 . 1 4 ) then y i e l d s W - B, Cos Sit + iB^sinfi-t + Bz Sim. Sit + LB^ccsSVt + "k Ci (*3- - < (Bx + B^] £sinQ± + 1 [i(B3-B4) + t(B2+ B3)] ^ 5 ftt = 4 [ (B 1 tB H .V: (B 3 -Bj ]e i n t + i [ ( B l - B l f ) + i(Bs 6 3^] e i J r t :n--t . e x i t = a A e + a c e = v A + i*rc (7.15) Thus the v e l o c i t y v e c t o r w can be decomposed i n t o two v e c t o r s wA and w c, b o t h r o t a t i n g w i t h a n g u l a r f r e q u e n c y Si , but w i t h one Chap. 7 CURRENT ELLIPSES 32 goin g a n t i c l o c k w i s e and the o t h e r , c l o c k w i s e . F o l l o w i n g Mooers (1973), we i n t r o d u c e f o u r r e a l q u a n t i t i e s , A, <f , C, and 9 , d e f i n e d as f o l l o w s : A - K I , <?=a+2(at)j C = lOc-l , ~Q = n(ac) (7.16) Eq. (7.15) can then be e x p r e s s e d as v = A e + C e (7.17) Hence A and C a r e r e s p e c t i v e l y the l e n g t h of the a n t i c l o c k w i s e and c l o c k w i s e r o t a t i n g v e c t o r s . At t = 0, the a n t i c l o c k w i s e v e c t o r makes an a n g l e <f w i t h the x - a x i s , w h i l e the c l o c k w i s e v e c t o r makes an a n g l e -0. F i g . 7.1 shows the c u r r e n t e l l i p s e t r a c e d out by the t i p of the v e l o c i t y v e c t o r . The l e n g t h of the major a x i s i s A+C, and t h a t of the minor a x i s i s |A-C|. The o v e r a l l r o t a t i o n of the v e l o c i t y v e c t o r i s a n t i c l o c k w i s e i f A > C, and c l o c k w i s e i f C > A. When C = A, the minor a x i s has l e n g t h z e r o , and the e l l i p s e d e g e n e r a t e s i n t o a l i n e of o s c i l l a t i o n s . The a n g l e oi d e f i n e d by * = ^{«urg^+«*g(vre)} = i{(sit+9)-(n.t+&)} = £ L § (7.18) i s an i n v a r i a n t w i t h r e s p e c t t o t i m e , which t u r n s out t o be the d i r e c t i o n of the major a x i s (See Mooers, 1973, P.1131). Next we use Eqs. (7.5) and ( 7 . 6 ) , the g o v e r n i n g e q u a t i o n s f o r the s h e l f wave v e l o c i t y components, t o d e r i v e the s h e l f wave c u r r e n t e l l i p s e s . Eqs. (7.4) - (7.6) a l l o w us t o w r i t e u = Re{ipe t^"" t )} = (-f>s£*vkj)cosHt + f>c*sky s$n oJ stnD± (7.19) 33 F i g . 7.1 The current e l l i p s e . Chap. 7 CURRENT ELLIPSES 34 v = R e j / f r e 1 ^ "^ ! = ^ c a r s ^ c d s H t + ^si^k<^ sfnoost^SL± ( 7 . 2 0 ) where f> = (-^ 7„' + k%)(^-\)] ( 7 . 2 1 ) I = f^K - 7o ( 7 . 2 2 ) = |w | , and s g n r t s . E q u a t i n g ( 7 . 1 9 ) , ( 7 . 2 0 ) , w i t h ( 7 . 1 3 ) and ( 7 . 1 4 ) , we have ^ = - b sir*, ky B z = f>cosfcy B 3 = ^ co-s 4^ b\= ft sJ*.fy s^nw ( 7 . 2 3 ) Combining ( 7 . 2 3 ) w i t h ( 7 . 1 5 ) and ( 7 . 1 6 ) , we o b t a i n flA r i ( - f J + ^ S^nw)Csinfey + Cs^n wc*sk^ ) ( 7 . 2 4 ) <*c = -kCl» + *J Syn")f-5i»v|lJJ -f- t s y i ^ C f f S ^ ) ( 7 . 2 5 ) A = - k l - M fr^l , C = i | f + ^ s r ^ | ( 7 . 2 6 ) The phase d i f f e r e n c e <p - 9 can be c a l c u l a t e d e a s i l y by n o t i n g t h a t ^ A * -c" = - S f n (-f) + ^ S ^ n W ) ' S ^ ( f > + | S ^ i " ) ( 7 . 2 7 ) S i n c e the r e s u l t i n g number has no i m a g i n a r y component, the phase d i f f e r e n c e cf - Q can o n l y be 0° or 180° . In f a c t Chap. 7 CURRENT ELLIPSES 35 O0 (7.28) S u b s t i t u t i n g i n Eqs (7.21) and (7.22) we f i n d t h a t - I (7.29) (7.30) S i n c e |u| < 1, both and a r e n e g a t i v e . T h e r e f o r e , the r i g h t s i d e of (7.29) and t h a t of (7.30) have the As p o i n t e d out i n Eq. ( 7 . 1 8 ) , o( , the d i r e c t i o n of the major a x i s , i s s i m p l y (<p-e)/2; so K can o n l y t a k e on the v a l u e 90° or o 0 . Hence the major a x i s of a s h e l f wave c u r r e n t e l l i p s e i s o r i e n t e d e i t h e r p a r a l l e l or p e r p e n d i c u l a r t o the c o a s t . The sea l e v e l d i s p l a c e m e n t ^ ( x ) and ^{x)/k f o r t h r e e s h e l f waves (of modes 1, 2, and 3) on the s h e l f of F i g . 11.3 a r e shown i n F i g . 7.2 . The c o r r e s p o n d i n g v e l o c i t y components u c(x) and v 0(x) a r e s k e t c h e d i n F i g . 7.3, w h i l e the a n t i c l o c k w i s e and c l o c k w i s e components A and C a r e p r e s e n t e d i n F i g . 7.4. We now i n t r o d u c e two u s e f u l q u a n t i t i e s — P,' the " p o l a r i z a t i o n " , and R, the " o r i e n t a t i o n " of the major a x i s — d e f i n e d as f o l l o w s : same s i g n i f | r | c ' | > | k •/„ | , a nd h a v e o p p o s i t e s i g n s I 1o I < I k 1o I • W e t h u s c o n c l u d e i f (7.31) 36 Fig. 7.2 Sea level displacement f\t and 1o/k for 3 shelf waves. The waves are normalized with lo/^- 1 at the coast. A l l three waves are of the same frequency (0.13 cpd). The presence of a hump on the shelf profile (Fig. 11.3) produces humps on the waves at around 20-25 km offshore. 37 g. 7.3 Velocity components u,and v„ for 3 shelf waves, is normalized to 1 at the coast. 38 0.5 I * T T I" mode 1 0.5 mode 2 0.5 i i 1 1 1 1 r 0 40 80 120 160 OFFSHORE DISTANCE (KM) mode 3 Fig. 7.4 Anticlockwise and clockwise velocity components, A and C, for the 3 shelf waves. With v„ normalized to 1 at the coast, A and C are both 0.5 at the coast. Chap. 7 CURRENT ELLIPSES 39 P = ( A - C ) / ( A + C ) (7.32) * = (I1.'|-IM.I)/Ol . ' l + |M.O (7.33) P i n d i c a t e s the r e l a t i v e s t r e n g t h of the a n t i c l o c k w i s e and the c l o c k w i s e components. P > 0 when A > C ( o v e r a l l motion a n t i c l o c k w i s e ) , and P < 0 when C > A ( o v e r a l l motion c l o c k w i s e ) . The extremum v a l u e s , P = 1, o c c u r s when C = 0, and P = - 1 , when A = 0. P = 0 when A = C. A n a l o g o u s l y , from ( 7 . 3 1 ) , we note t h a t R > 0 means f - © = 180° and o< = 90° , w h i l e R < 0 i m p l i e s <f-B = 0° and K = 0 ° . Fig. 7.5 shows P and R as f u n c t i o n s of the o f f s h o r e d i s t a n c e . Two n o t a b l e f e a t u r e s a r e : ( i ) Near the c o a s t , a l l s h e l f waves have P < 0, hence C > A. (However, r i g h t a t the c o a s t , P = 0 and A = C ) . T h i s means t h a t , near the c o a s t , s h e l f waves have l a r g e r c l o c k w i s e - r o t a t i n g v e l o c i t y v e c t o r s . ( i i ) R < 0 o c c u r s o n l y i n s m a l l l o c a l i z e d r e g i o n s , o t h e r w i s e R i s g e n e r a l l y p o s i t i v e , w i t h <?- 6 = 180°, and <x = 90°. T h i s i m p l i e s t h a t f o r most of the s h e l f , the major a x i s of a s h e l f wave c u r r e n t e l l i p s e i s o r i e n t e d p a r a l l e l t o the c o a s t . F i g . 7.6 s c h e m a t i c a l l y i l l u s t r a t e s the s h e l f wave c u r r e n t e l l i p s e s i n the o f f s h o r e x d i r e c t i o n . The t i p of the c u r r e n t v e c t o r s o r b i t around the e l l i p s e s i n a l t e r n a t i n g zones of p r e d o m i n a n t l y c l o c k w i s e and a n t i c l o c k w i s e m o t i o n . For any two p o i n t s s e p a r a t e d o n l y i n the a l o n g s h o r e y d i r e c t i o n , - t h e i r c u r r e n t e l l i p s e s a r e i d e n t i c a l i n shape; o n l y a phase l a g e x i s t s between the r o t a t i n g c u r r e n t v e c t o r s . 40 F i g . 7.5 " P o l a r i z a t i o n " P and " o r i e n t a t i o n " R f o r 3 s h e l f waves. 41 Fig. 7.6 Shelf wave current ellipses across the continental shelf. This schematic diagram illustrates the current ellipses associated with a mode 1 shelf wave. For higher mode waves, the basic pattern is simply repeated offshore. Note that the zone where a= 0P is actually quite narrow compared to the zone where (X= 90°. Chap. 7 CURRENT ELLIPSES 42 Many s h e l f wave p r o p e r t i e s d i s c u s s e d i n t h i s c h a p t e r indeed m a n i f e s t themselves i n the r o t a r y s p e c t r a of l a t e r c h a p t e r s . F u r t h e r m o r e , the c u r r e n t e l l i p s e s a s s o c i a t e d w i t h v a r i o u s s h e l f wave modes (computed by the n u m e r i c a l t e c h n i q u e i n Appendix B) can be f i t t e d by l e a s t squares t o the c u r r e n t e l l i p s e s observed at c r o s s - s h e l f s t a t i o n s , thus a l l o w i n g one t o det e r m i n e which modes a r e p r e s e n t . 43 CHAPTER 8. ROTARY SPECTRAL ANALYSIS AND CROSS-SHELF MODAL FITTING For the purpose of d e t e c t i n g s h e l f waves, r o t a r y s p e c t r a l a n a l y s i s i s f a r s u p e r i o r t o c o n v e n t i o n a l s p e c t r a l a n a l y s i s based on the C a r t e s i a n c o o r d i n a t e system. There a r e two main a d v a n t a g e s : ( i ) The d e v i a t i o n of the c o a s t and the depth c o n t o u r s from s t r a i g h t l i n e s make i t p r a c t i c a l l y i m p o s s i b l e t o choose a c o n s i s t e n t C a r t e s i a n c o o r d i n a t e system. For i n s t a n c e , i f the y - a x i s i s t o p o i n t p a r a l l e l t o the c o a s t , then f o r two s t a t i o n s s e p a r a t e d a l o n g s h o r e , t h e i r y-axes would be o r i e n t e d d i f f e r e n t l y due t o t h i s d e v i a t i o n . As p o i n t e d out i n Mooers (1973), the a u t o s p e c t r a and c o h e r e n c i e s c a l c u l a t e d u s i n g r o t a r y s p e c t r a l a n a l y s i s a r e i n v a r i a n t t o c o o r d i n a t e r o t a t i o n s . ( i i ) S h e l f waves d i s p l a y a l t e r n a t i n g zones of p r e d o m i n a n t l y c l o c k w i s e and a n t i c l o c k w i s e motion a c r o s s the s h e l f . Mooers (1973) s u g g e s t s t h a t the modal s t r u c t u r e of s h e l f waves may be r e s o l v e d from the r o t a r y s p e c t r a of c r o s s - s h e l f s t a t i o n s . The f o l l o w i n g f o r m u l a s f o r r o t a r y s p e c t r a a r e taken d i r e c t l y from Mooers (1973) (but w i t h s e v e r a l e r r o r s c o r r e c t e d ) . I n t r o d u c e the F o u r i e r t r a n s f o r m e d complex v e l o c i t y W(cr) d e f i n e d by ^ { W(cr)e l < r t dcr (8.1) where Chap. 8 ROTARY SPECTRAL ANALYSIS 44 f Ae L < p o->o WC<r)H r -» ' (8.2) and A, C, <p and 6 a r e f u n c t i o n s of the a n g u l a r f r e q u e n c y a. The au t o s p e c t r u m i s f <A*> <r > o S O ) ^ < where <...> denotes e i t h e r an ensemble average over s e v e r a l r e c o r d s , or a band average over n e i g h b o u r i n g f r e q u e n c i e s . G i v e n two time s e r i e s w, and w2, the i n n e r - c o h e r e n c y squared i s f <A,A>a>s(«f l-<R T)> A-H<A,A»sU(«P,-^ > < r i o < A ? X A ' a > < C * X C * > (8.4) and the c o r r e s p o n d i n g i n n e r - p h a s e i s ra . r 3C<A ,A a^4 |- 'P a )> + i<A l A 1 $^(«f| .«P j>] , <r > 0 4 « s (8.5) The i n n e r - c o h e r e n c y and phase measure the coherency and phase d i f f e r e n c e between the a n t i c l o c k w i s e components of the two time s e r i e s , and between t h e c l o c k w i s e components. The o u t e r - c o h e r e n c y squared i s <Ca A , c*s (<P, - e 2) >*+ < C x A, 5^ (<P, - Bjf , <S±° (8.6) Chap. 8 ROTARY SPECTRAL ANALYSIS 4 5 and t h e ou t e r - p h a s e i s a^C<C,Aa<»sC«P1-e«)>-'«.<C,A2 3»*CtP1-oI)>] t cr>o o^[<A ,Cx^C <P.-©a)>^KA 1C as^ f-eO>3 , <r<o (8 .7) The o u t e r - c o h e r e n c y and phase measure the coherency and phase d i f f e r e n c e between the c l o c k w i s e component of one time s e r i e s w i t h the a n t i c l o c k w i s e component of the o t h e r . The coherency and phase between the c l o c k w i s e and a n t i c l o c k w i s e components of each i n d i v i d u a l time s e r i e s w^  ( j = l , 2 ) , can a l s o be d e f i n e d as = a^C<CjA3<*<^-ep>+Kc5AjstKCcpj-aj)>)^ o ( 8 . 9 ) G o n e l l a (1972) has named / t j the " s t a b i l i t y " of the c u r r e n t e l l i p s e , s i n c e c u r r e n t e l l i p s e s a r e w e l l d e f i n e d and s t a b l e i n shape o n l y when t h e r e i s h i g h coherency between the c l o c k w i s e and a n t i c l o c k w i s e r o t a r y v e c t o r s . <o(j> i n d i c a t e s the d i r e c t i o n of the major a x i s f o r the current e l l i p s e . R o t a r y s p e c t r a l a n a l y s i s can • be a p p l i e d t o s c a l a r q u a n t i t i e s ( e . g . the a d j u s t e d sea l e v e l ) as w e l l . In t h a t c a s e , w i s r e a l , A = C, and <P = 0 , and the p o s i t i v e and n e g a t i v e f r e q u e n c y p a r t s of the a u t o s p e c t r u m a r e m i r r o r images of each o t h e r . C r o s s - s p e c t r a between a v e c t o r and a s c a l a r q u a n t i t y ( e . g . c u r r e n t v e r s u s sea l e v e l ) can s t i l l be computed, but the ou t e r - s p e c t r u m i s e s s e n t i a l l y a d u p l i c a t e of the i n n e r - s p e c t r u m . Chap. 8 ROTARY SPECTRAL ANALYSIS 46 Wh i l e the i n n e r - s p e c t r u m has g a i n e d p o p u l a r i t y i n r e c e n t y e a r s , the f u l l p o t e n t i a l of . the o u t e r - s p e c t r u m and the q u a n t i t i e s and <^> have not been e x p l o i t e d . For i n s t a n c e , i f two c u r r e n t meters a r e moored i n the c r o s s - s h e l f d i r e c t i o n ( i . e . p e r p e n d i c u l a r t o the s h o r e l i n e ) such t h a t the f i r s t meter ( c l o s e r t o shore) has C > A , and the second i n a r e g i o n where A > C , (see F i g . 7.9), then t h e h i g h e s t coherency s h o u l d be found i n t h e p o s i t i v e f r e q u e n c y p a r t of the o u t e r - c o h e r e n c y squared spectrum (see Eq. (8.6) ). A l s o , i n the l a s t c h a p t e r , we found t h a t the major a x i s of the c u r r e n t e l l i p s e a s s o c i a t e d w i t h a s h e l f wave i s u s u a l l y a l i g n e d i n the y ( a l o n g s h o r e ) d i r e c t i o n . As seen i n l a t e r c h a p t e r s , < o tj>, the d i r e c t i o n of the major a x i s , i s inde e d o b s e r v e d t o be around 90°. Thus, combining one's knowledge of s h e l f wave d i s p e r s i o n c u r v e s and c u r r e n t e l l i p s e s w i t h the r o t a r y s p e c t r a l t e c h n i q u e a l l o w s one to d e t e c t and d i s t i g u i s h unambiguously s h e l f wave motions from o t h e r t y p e s of waves and t u r b u l e n c e . For n e a r l y twenty y e a r s s i n c e t h e i r f i r s t d e t e c t i o n by Hamon (1962), s h e l f waves have been i d e n t i f i e d by the f o l l o w i n g " s t a n d a r d " p r o c e d u r e : ( i ) f i n d i n g a peak i n the c r o s s - s p e c t r u m between two a l o n g s h o r e s t a t i o n s , ( i i ) d e d u c i n g the c o r r e s p o n d i n g wavelength from the a l o n g s h o r e phase l a g , and ( i i i ) t r y i n g t o see i f the o b s e r v e d (k,<J ) v a l u e s l i e c l o s e t o a p a r t i c u l a r c u r v e on the s h e l f wave d i s p e r s i o n diagram. L i t t l e s t udy has been made of the c r o s s - s h e l f s t r u c t u r e . Chap. 8 CROSS-SHELF MODAL FITTING 47 In r e c e n t y e a r s , c u r r e n t meters have been moored a c r o s s the c o n t i n e n t a l s h e l f i n a d i r e c t i o n p e r p e n d i c u l a r t o the c o a s t l i n e , and t h i s s u g g e s t s a new d e t e c t i o n t e c h n i q u e : The o b s e r v e d c u r r e n t e l l i p s e s a t the c r o s s - s h e l f s t a t i o n s can be f i t t e d by l e a s t s quares t o the t h e o r e t i c a l l y deduced c u r r e n t e l l i p s e s a s s o c i a t e d w i t h v a r i o u s s h e l f wave modes. T h i s c r o s s - s h e l f  modal f i t t i n g may t e l l us which modes are p r e s e n t on the s h e l f . Suppose we have a l i n e of N s t a t i o n s e x t e n d i n g o f f s h o r e . Upon a p p l y i n g d i s c r e t e F o u r i e r t r a n s f o r m t o the c u r r e n t d a t a a t the n t h s t a t i o n (n = 1,...,N), we o b t a i n A n , <PK , C n and 6\ as d e f i n e d i n ( 8 . 2 ) . These f o u r q u a n t i t i e s a r e f u n c t i o n s of f r e q u e n c y , but l e t us f o c u s on one p a r t i c u l a r f r e q u e n c y |<r| . L e t us suppose t h a t from the s h e l f wave d i s p e r s i o n diagram, we f i n d t h a t M s h e l f waves can e x i s t a t t h i s f r e q u e n c y . T h e i r t h e o r e t i c a l v a l u e s a r e d e r i v e d by the n u m e r i c a l i n t e g r a t i o n scheme d i s c u s s e d i n Chapter 7. S i n c e a l l the s t a t i o n s have the same y c o o r d i n a t e , y i n Eq. (7.4) can be s e t t o 0. However, an a r b i t r a r y phase 6 m can be a s s o c i a t e d w i t h the mth s h e l f wave (m = 1,...,M). Thus, analogous t o Eqs. (7.24) and ( 7 . 2 5 ) , the r o t a r y v e c t o r s f o r the mth mode s h e l f wave a r e : = ?mS«« "X s^5« + £ s j n « » s S m ) ( 8.10) CQc)m = i ( frn + fa S ^ n C ^ - S ^ - t IsfinU COS S m ) (8.11) The q u a n t i t i e s A n m , <? n m , C n m , -9nm denote the a m p l i t u d e and phase of the two r o t a r y v e c t o r s f o r the mth wave e v a l u a t e d a t the n t h s t a t i o n , ( w i t h the t i l d e d i s t i n g u i s h i n g t h e s e Chap. 8 CROSS-SHELF MODAL FITTING 48 t h e o r e t i c a l q u a n t i t i e s from t he obs e r v e d o n e s ) . We next i n t r o d u c e the f i t t i n g f u n c t i o n F: + [C„co Se r t - £ am c „c*5 © o mCsm)]+ [c^en-1flmC^sue^Csj"]2J ( 8 > 1 2 ) Here, a t o t a l of M s h e l f waves a r e f i t t e d , and a s s o c i a t e d w i t h each wave a r e two p a r a m e t e r s , the wave a m p l i t u d e a m and the phase . ( I t can be seen from (8.10) and (8.11) t h a t ^ w and 0 n m depend on Sm , w h i l e A„ m and C n n i do n o t ) . The f u n c t i o n F i s composed of f o u r p a r t s , c o r r e s p o n d i n g t o s e p a r a t e f i t s t o the r e a l and i m a g i n a r y components of the a n t i c l o c k w i s e and c l o c k w i s e r o t a r y v e c t o r s . F i s m i n i m i z e d w i t h r e s p e c t t o the parameters a m and (m = 1,...,M). The c o n t r i b u t i o n s from the s t a t i o n s can be weighed d i f f e r e n t l y by a d j u s t i n g the w e i g h t s w a. In a s i t u a t i o n where one has r e l a t i v e l y few s t a t i o n s and s e v e r a l a l l o w a b l e waves, the minimum of F may occ u r w i t h the •waves h a v i n g u n r e a s o n a b l y l a r g e a m p l i t u d e s a m ' s which c a n c e l out a t the s t a t i o n s t o g i v e a good f i t . When t h i s happens, i t i s b e t t e r t o m i n i m i z e i n s t e a d the f u n c t i o n £ where 2 = F + £ & C & 2>B(X*W+ O - (8.13) w i t h E^, the (weighed) t o t a l "energy" o b s e r v e d a t t h e s t a t i o n s , g i v e n by Ew = J . ^ ( A . + O (8.14) The new term on the r i g h t s i d e of Eq. (8.13) i s a " p e n a l t y " or Chap. 8 CROSS-SHELF MODAL FITTING 49 "damping" f u n c t i o n . I t f o r c e s the a m p l i t u d e s a^'s t o be such t h a t the t o t a l energy from the f i t b a l a n c e s the ob s e r v e d energy. The d e v i a t i o n D i s d e f i n e d as D = r/e (8.15) where r i s the r e s i d u a l , +[cBcoS©,- | ( f l « c , m c o s w T + Cc;s^©n- Z a m £ n M s ^ m ( o f } (8. ie) and E, the t o t a l "energy" o b s e r v e d a t the s t a t i o n s , H = £ ( / £ + ( £ ) (8.17) CHAPTER 9. OREGON SHELF: SUMMER, 1968 50 We now t u r n t o the Oregon s h e l f t o see i f o b s e r v a t i o n a l e v i d e n c e f o r res o n a n t s h e l f wave t r i a d s can be found. I n t h i s c h a p t e r , the t r i a d t h e o r y i s a p p l i e d t o the Oregon s h e l f and compared w i t h o b s e r v a t i o n s made d u r i n g the summer of 1968 by C u t c h i n and Smith (1973). ( T h i s paper w i l l h e n c e f o r t h be r e f e r r e d t o as C&S.) F i r s t , we f i t the Buchwald-Adams e x p o n e n t i a l s h e l f p r o f i l e , i . e . , Eq. ( 6 . 1 ) , t o the r e a l d e p th p r o f i l e g i v e n i n C&S. W i t h the l e v e l - o f f depth i n deep water chosen t o be 2.84 km, the o p t i m a l l e a s t - s q u a r e s f i t ( F i g . 9.1 ) y i e l d s the v a l u e 1.65 f o r the ( d i m e n s i o n l e s s ) parameter b. The v a r i a b l e x i s n o n - d i m e n s i o n a l i z e d w i t h r e s p e c t t o the s h e l f / s l o p e w i d t h L (L = 112 km from our f i t ) , and the depth h, w i t h r e s p e c t t o H a, the d e p t h s c a l e of the s h e l f r e g i o n (H 0= 200 m). By s o l v i n g Eqs. (6.3) and (6.4) n u m e r i c a l l y , we o b t a i n the d i s p e r s i o n c u r v e s f o r the e x p o n e n t i a l p r o f i l e . In F i g . 9.2, the c u r v e s c o r r e s p o n d i n g t o the f i r s t f o u r modes a r e shown. (Note t h a t we have a c t u a l l y p l o t t e d -O v e r s u s k, s i n c e s h e l f waves have n e g a t i v e phase speeds i n our c o o r d i n a t e s y s t e m ) . Our d i s p e r s i o n c u r v e s f o r t h e i d e a l i z e d topography resemble f a i r l y c l o s e l y t h e o r i g i n a l c u r v e s i n C&S which have been d e r i v e d u s i n g r e a l t o pography. As mentioned e a r l i e r , C&S found t h r e e peaks a t around 0.22, 0.40, and 0.65 cpd i n t h e i r s p e c t r a l a n a l y s i s of d a t a c o l l e c t e d 51 1 F i g . 9.1 T h e e x p o n e n t i a I f i t t o t h e O r e g o n s h e l f p r o f i l e . T h e d a s h e d c u r v e i s t h e r e a l t o p o g r a p h y , w h i l e t h e s o l i d c u r v e i s t h e e x p o n e n t i a l f i t . F i g . 9.2 D i s p e r s i o n c u r v e s f o r t h e e x p o n e n t i a l p r o f i l e , w i t h b = 1.65. Only t h e f o u r lowest modes a r e shown. 53 125'W 124'W —I—'—' J \ \\ ' M. ' (b) (c) F i g . 9.3 Observations made by Cutchin and Smith on the Oregon sh e l f . (a) The experimental setup. The s i t e of the current meter mooring i s indicated by the square, and that of the tidegauges, by the t r i a n g l e s . The bathymetric p r o f i l e along the l i n e extending from Depoe Bay i s shown as the dashed curve i n F i g . 9.1. (b) Phase and coherency squared between v from the current meter mooring and Newport adjusted sea l e v e l for the period from 5 June to 11 Sept., 1968. Phase i s p o s i t i v e for v leading sea l e v e l . Bandwidth and 95% confidence i n t e r v a l are as shown. (c) Phase and squared coherency between Newport and A s t o r i a adjusted sea l e v e l . (Redrawn from Cutchin and Smith, 1973). Chap. 9 OREGON SHELF: 1968 54 on the Oregon s h e l f d u r i n g the summer of 1968- (see F i g . 9 .3). 1 They a l s o c o n c l u d e d from phase i n f o r m a t i o n t h a t the dominant peak a t 0.22 cpd i s c o n s i s t e n t w i t h a f i r s t mode s h e l f wave w i t h w a v e l e n g t h X.t: 1620 km. Can our resonant i n t e r a c t i o n t h e o r y e x p l a i n the presence of t h e s e o b s e r v e d peaks? To f a c i l i t a t e c o m p a r i s o n , we r e w r i t e the resonance c o n d i t i o n s as u), + u)x = <Oj , k, + k 2 = k 3 (9.1) (which can be c o n v e r t e d back t o the o r i g i n a l forms (3.10) by s i m p l y c h a n g i n g the s i g n s of and k3 ). Eqs. (9.1) can a l s o be e x p r e s s e d as a v e c t o r r e l a t i o n : (klf-fc).) + (fe a, -Ux) = ( k 3 , - " 3 " ) (9.2) I f we now choose -eJ, t o c o r r e s p o n d t o the 0.22 cpd f r e q u e n c y , then the d i s p e r s i o n c u r v e s i n F i g . 9.3 and the v e c t o r r e l a t i o n (9.2) a l l o w us t o deduce.the o t h e r members of the t r i a d , namely ( k a , -oii.) and ( k 3 , - ^ 3 ) , by a s i m p l e g e o m e t r i c c o n s t r u c t i o n ( F i g . 9.4). The v e c t o r a d d i t i o n i n (9.2) i m p l i e s t h a t (k ( , — ^ ) , ( k 2 , -W 2), ( k ^ , -^ 3) a r e the t h r e e v e r t i c e s of a p a r a l l e l o g r a m l y i n g i n t h e (k, -co) p l a n e , w i t h t h e t h r e e v e r t i c e s t o u c h i n g the d i s p e r s i o n c u r v e s , and the f o u r t h v e r t e x f i x e d a t the o r i g i n . S i n c e C&S c o n c l u d e d t h a t the peak at 0.22 cpd i s c o n s i s t e n t w i t h 1 Though the peaks at 0.40 and 0.65 a r e r e l a t i v e l y s m a l l , t h e i r p e r s i s t e n t p r e s ence at v a r i o u s coherency s p e c t r a l e d C u t c h i n and Smith (1973, p.79) t o c o n c l u d e t h a t they a r e s t a t i s t i c a l l y s i g n i f i c a n t . . 9.4 T h e r e s o n a n t t r i a d c o n s i s t e n t w i t h t h e o b s e r v a t i o n s o f C u t c h i n a n d t h . T h i s i s t h e r e s o n a n t t r i a d i n v o l v i n g w a v e s o f t h e l o w e s t p o s s i b l e m o d e s . Chap. 9 OREGON SHELF: 1968 56 a f i r s t mode s h e l f wave, t h i s e s s e n t i a l l y f i x e s our (k, , - u,). L o o k i n g f o r i n t e r a c t i o n s i n v o l v i n g waves of the l o w e s t p o s s i b l e modes, we f i n d t h a t ( k 2 , -W*) b e l o n g t o t h e second mode w h i l e ( k 3 , - M 3 ) b e l o n g t o the f i r s t mode. With t h e s e c h o i c e s f o r the modes, the r e s u l t i n g p a r a l l e l o g r a m of F i g . 9.4 i s unique. The r e s o n a n t t r i a d o b t a i n e d from F i g . 9.4 i s : ( k , , - O,) = (0.382, 0.155), ( k a , = (5.362, 0.281), and ( k 3 , -« 3) = (5.745, 0.436) where t h e s e p a i r s a r e n o n d i m e n s i o n a l i z e d w i t h r e s p e c t t o the s c a l e s (L~', f ) . The c o r r e s p o n d i n g wavelengths and f r e q u e n c i e s f o r the t r i a d a r e : ( X,, V, ) = (1840 km, 0.22 c p d ) , ( "X2, Vx) = (131 km, 0.40 c p d ) , ( A 3 , V 3 ) = (122 km, 0.62 c p d ) . A l t e r n a t i v e l y , we can c o n s t r u c t our p a r a l l e l o g r a m from the o r i g i n a l d i s p e r s i o n c u r v e s of C&S ( d e r i v e d u s i n g r e a l t o p o g r a p h y ) , r e s u l t i n g i n the f o l l o w i n g v a l u e s f o r the t r i a d : (1710 km, 0.22 c p d ) , (138 km, 0.40 c p d ) , and (127 km, 0.62 c p d ) . With e i t h e r c o n s t r u c t i o n , we f i n d e x c e l l e n t agreement w i t h the observe d v a l u e s i n C&S: X,= 1620 km, = 0.22 cpd, Vx= 0.40 cpd, and V3= 0 . 65 cpd. There i s i n s u f f i c i e n t i n f o r m a t i o n i n C&S t o g i v e a r e l i a b l e e s t i m a t e f o r Kz and X 3. The phases a s s o c i a t e d w i t h the second and t h i r d peaks i n F i g . 9.3c a r e a p p r o x i m a t e l y 22° and 43°. However, the t r u e phase can be the ob s e r v e d phase + n-360° (where n i s an i n t e g e r ) . W i t h the two s t a t i o n s 180 km a p a r t , t a k i n g n = 0 y i e l d s A 2= 3000 km, and * 3= 1500 km. But i f n = +1, t h e n Aj= 170 km, and A3= 160 km, which agree v e r y r o u g h l y w i t h t h e t h e o r e t i c a l v a l u e s (131 km and 122 km). C&S at t e m p t e d t o e x p l a i n the presence of the t h r e e peaks by Chap. 9 OREGON SHELF: 1968 57 the z e r o group v e l o c i t y resonance mechanism proposed i n Buchwald and Adams (1968). For d i s p e r s i o n c u r v e s which have z e r o s l o p e a t c e r t a i n f r e q u e n c i e s , the r e s u l t i n g z e r o group v e l o c i t y i m p l i e s t h a t energy cannot, propagate away, and t h e r e f o r e , a c c o r d i n g t o Buchwald and Adams, peaks i n the spectrum might be e x p e c t e d a t t h e s e f r e q u e n c i e s . From the o r i g i n a l d i s p e r s i o n c u r v e s of C&S, the f r e q u e n c i e s of z e r o group v e l o c i t y occur a t 0.29, 0.40, and 0.71 cpd f o r modes I I I , I I and I r e s p e c t i v e l y . . The agreement between t h e s e f r e q u e n c i e s and the o b s e r v e d f r e q u e n c i e s (0.22, 0.40, 0.65 cpd) appears weaker than t h a t between our r e s o n a n t t r i a d f r e q u e n c i e s (0.22, 0.40, 0.62) and the o b s e r v a t i o n s . F u r t h e r m o r e , w i t h the z e r o group v e l o c i t y mechanism, one has t o i d e n t i f y the dominant peak a t 0.22 cpd w i t h a t h i r d mode wave of wavelength < 300 km (see F i g . 9.2). But, C&S d e t e r m i n e d the wavelength a s s o c i a t e d w i t h the dominant peak t o be 1620 km, which i s c o n s i s t e n t o n l y w i t h a f i r s t mode wave. Hence, our r e s o n s a n t t r i a d mechanism seems more c a p a b l e of e x p l a i n i n g the t r i p l e - p e a k e d spectrum i n C&S. No wind d a t a were c o l l e c t e d d u r i n g the C&S 1968 e x p e r i m e n t . However, i n the summer of 1969, both wind and c u r r e n t were m o n i t o r e d on the Oregon s h e l f . Huyer and P a t t u l l o (1972) found t h a t below the d i u r n a l f r e q u e n c y , the coherency between wind and c u r r e n t was h i g h o n l y f o r f r e q u e n c i e s l.ess than 0.4 cpd. Thus we e n v i s a g e the f o l l o w i n g s c e n a r i o on the Oregon s h e l f d u r i n g the summer of 1968. The wind was f e e d i n g energy i n t o the dominant 0.22 cpd peak. The n o n l i n e a r i t y of the medium a u t o m a t i c a l l y began t r a n s f e r r i n g energy from t h i s peak t o the Chap. 9 OREGON SHELF: 1968 58 two o t h e r t r i a d members a t 0.4 ancl 0.65 cpd. (For o t h e r f r e q u e n c i e s , the resonance c o n d i t i o n s were not s a t i s f i e d , and the n o n l i n e a r energy t r a n s f e r was s m a l l ) . The n o n l i n e a r i n t e r a c t i o n t r a n s f e r r e d energy not o n l y t o h i g h e r f r e q u e n c i e s (where t h e c o u p l i n g between wind and c u r r e n t was weak), but a l s o t o much s h o r t e r w a v e l e n g t h s . E v e n t u a l l y , a d i s t i n c t i v e l y t r i p l e - p e a k e d spectrum emerged. Next, we d e r i v e the n u m e r i c a l v a l u e s of the c o u p l i n g c o e f f i c i e n t s Kj f o r the res o n a n t t r i a d i n t e r a c t i o n i n C&S. R e t u r n i n g t o (3.10) as our resonance c o n d i t i o n s , the s i g n s of k 3 and 6J3 must be r e v e r s e d . S u b s t i t u t i n g i n the t r i a d v a l u e s : 0.382, 5.362, -5.745 f o r k,, kz , and k 3, and -0.155, -0.281, 0.436 f o r a, , u>%, and oJ3, Eqs'. ( 6 . 3 ) , ( 6 . 6 ) , ( 6 . 7 ) , (6.11) -(6 . 1 5 ) , and (3.17) g i v e the v a l u e s i n Ta b l e 9.1. Ta b l e 9.1 The c o u p l i n g c o e f f i c i e n t s ( i n n o n d i m e n s i o n a l u n i t s ) f o r the reso n a n t t r i a d on the Oregon s h e l f c o n s i s t e n t w i t h the o b s e r v a t i o n i n C u t c h i n and Smith (1973). j 1 m K5JZm K i / c 3 . 1 2 3 60.6 -39.0 -8.75 21.6 2 3 1 1.48 37.7 -2.05 39.2 3 1 2 -52.7 -8.02 4.61 -60.7 Chap. 9 OREGON SHELF: 1968 59 In C h a pter 5, we found t h a t w i t h d/9Y n e g l e c t e d , and f o r the case of maximum energy exchange, the s o l u t i o n s t o the a m p l i t u d e e q u a t i o n s were g i v e n by e l l i p t i c f u n c t i o n s . The p e r i o d of energy t r a n s f e r , T^, was g i v e n by Eq. (5.15) . We expect a l 0 t o be < 1. ( T h i s i s because the f i r s t wave dominates, hence, i t s v e l o c i t y s h o u l d be comparable i n magnitude t o the measured v e l o c i t y s c a l e V, and thus upon n o n d i m e n s i o n a l i z a t i o n , a l 0 ~ l ) . As a 4 a i s s i g n i f i c a n t l y l e s s than a * , we expect M t o be < 1 from Eq. (5.11) . The e l l i p t i c i n t e g r a l K(M) i n c r e a s e s s l o w l y from 1.57 t o 3.70 as M i n c r e a s e s from 0 t o 0.99. T a k i n g a t y p i c a l v a l u e of 3 f o r K(M), 1 f o r a l 0 , and u s i n g v a l u e s of Kx and K 3 from T a b l e 9.1, we have T ^ 2.0. Remembering T = t t where £ i s the Rossby number, ( £ = 0.01), and t i s n o n d i m e n s i o n a l i z e d w i t h r e s p e c t t o 1/f, (f = 10 sec ) , Tjf i s c o n v e r t e d back t o d i m e n s i o n a l u n i t s upon d i v i s i o n by £f, y i e l d i n g T^ ~ 2.0*10 f e sec or 23 days. Hence, f o r the t r i a d i n C&S, the time s c a l e f o r energy t r a n s f e r ( ~ i: T«i ) i s of o r d e r 12 days. CHAPTER.10. OREGON SHELF: SUMMER, 1972 60 D u r i n g the summer of 1972, the CUE-1 experiment was c a r r i e d out on the Oregon s h e l f . The c u r r e n t meter a r r a y and the l e n g t h of d a t a r e c o r d s a r e shown i n F i g . 10.1 and F i g . 10.2, taken from the d a t a r e p o r t by P i l l s b u r y e t a l . (1974a). Huyer et a l . (1975) ( h e n c e f o r t h r e f e r r e d t o as HHSSP) computed r o t a r y s p e c t r a w i t h d a t a from CUE-1. In F i g . 9 of HHSSP, the c l o c k w i s e p o r t i o n ( i . e . he n e g a t i v e f r e q u e n c y p a r t ) of the coherency squared spectrum between the Depoe Bay sea l e v e l and the 60 m c u r r e n t a t the mooring NH-10 shows f o u r d i s t i n c t (and s t a t i s t i c a l l y s i g n i f i c a n t ) s i g n a l s a t 0.15, 0.27, 0.42, and 0.55 cpd. (These f o u r s i g n a l s a r e a l s o p r e s e n t i n the a n t i c l o c k w i s e p a r t of the s p e c t r u m ) . I f one a g a i n p e r forms the " p a r a l l e l o g r a m " c o n s t r u c t i o n on the d i s p e r s i o n c u r v e s i n C&S, one f i n d s t h a t upon a s s o c i a t i n g the dominant 0.15 cpd peak w i t h a f i r s t mode s h e l f wave, the r e m a i n i n g members of the t r i a d i n v o l v i n g the low e s t p o s s i b l e modes t u r n out t o be: ( X*, = (103 km, 0.40 c p d ) , (second mode), and ( A3, V 3) = (99 km, 0.55 c p d ) , ( f i r s t mode). Thus, i t i s p o s s i b l e t h a t of the f o u r s i g n a l s o b s e r v e d , (0.15, 0.27, 0.42, and 0.55 c p d ) , the l a s t two a r i s e from the resonant t r a n s f e r of energy from the dominant peak a t 0.15 cpd. The c r o s s - s p e c t r u m between the c u r r e n t a t NH-10 and the wind a t Newport ( F i g . 9 of HHSSP) shows h i g h c o r r e l a t i o n a t the 0.15 and 0.27 cpd ( c l o c k w i s e ) f r e q u e n c i e s , s u g g e s t i n g t h a t these s i g n a l s may be wind g e n e r a t e d . So f a r , we have o n l y examined resonances i n v o l v i n g the l o w e s t p o s s i b l e s h e l f wave modes. Resonances i n v o l v i n g h i g h e r 45* 10' 4 5 ' N H44* 3 0 ' 44* 10' 10.1 The current meter array in the CUE-1 experiment. NH-10 NM-IS 80 MM-t0 I 20 m 140 I 70 ! 120 I 20 n I 40 I 60 I 60 OB -13 20 m 40 TO 120 POL -I— I —1— I — • l I i 1 i i ' I 20 tn 40 60 SO -i—i—j—i t _1_ APR MAY JUN S E P V77X TEMPERATURE AND CURRENT DATA E S 3 TEMPERATURE DATA ONLY C = l NO DATA OBTAINED OCT F i g . 10.2 Length of data records i n CUE-1. Chap. 10 OREGON SHELF: 1972 63 modes s h o u l d not be o v e r l o o k e d . In p a r t i c u l a r , the f i r s t t h r e e s i g n a l s i n HHSSP (0.15, 0.27 and 0.42 c p d ) , which a l s o happen t o s a t i s f y the resonance c o n d i t i o n (1.1) , may be a r e s o n a n t t r i a d i n v o l v i n g h i g h e r modes. I f one p e r f o r m s a s i m i l a r " p a r a l l e l o g r a m " c o n s t r u c t i o n on F i g . 9.4, h o l d i n g the dominant 0.15 cpd f i r s t mode wave f i x e d but l e t t i n g the o t h e r two t r i a d members t o be of second mode and t h i r d mode, then ( w i t h k~ 6) t h e i r f r e q u e n c i e s t u r n out t o be 0.26 and 0.41 cpd. The above view was e x p r e s s e d i n H s i e h and Mysak (1980). L a t e r , some CUE-1 d a t a were o b t a i n e d from the Oregon S t a t e U n i v e r s i t y . Appendix C g i v e s a b r i e f d e s c r i p t i o n of the d a t a p r o c e s s i n g i n v o l v e d i n t h i s t h e s i s . F i g . 10.3 shows the i n n e r c r o s s - s p e c t r u m between the 40 m c u r r e n t a t s t a t i o n DB-7 and the 20 m c u r r e n t a t NH-10. 1 S i n c e the s e p a r a t i o n between the s t a t i o n s was o n l y 22 km, the phase d i f f e r e n c e s h o u l d be v e r y s m a l l f o r a l l l o n g s h e l f waves. However, the s h o r t s h e l f waves (~100-150 km wavelength) produced by the r e s o n a n t t r i a d i n t e r a c t i o n s h o u l d d i s p l a y s i z a b l e phase l a g s . One n u i s a n c e i s the v e r y c o m p l i c a t e d topography around the CUE-1 a r r a y ( F i g . 10.1). The c o n t i n e n t a l s h e l f narrows r a p i d l y , and S t o n e w a l l Bank l i e s j u s t s o u t h of the a r r a y . Wang (1980) 1 Throughout t h i s t h e s i s , when c r o s s s p e c t r a between two a l o n g s h o r e s t a t i o n s a r e computed, th e f i r s t t i me s e r i e s w, always comes from the n o r t h e r n s t a t i o n . The CUE-1 d a t a a n a l y z e d i n t h i s t h e s i s runs from J u l y 6 t o August 31, whereas th e d a t a used by HHSSP runs from J u l y 21 t o September 16. o I i I n i — r ~ i — i — i — i — i i i i i i n~i i i i 2 - 0 8 - 0 6 - 0 . 4 - 0 . 2 0 . 0 0 . 2 D . 4 0 . 6 0 . 8 FREQUENCY (CPD) Fig. 10.3 Inner cross spectrum between the 40 m current at DB-7 and the 20 m current at NH-10. The dashed line indicates the 95% significance level (or noise level) according to Groves and Hannan (1968). The uncertainty in the phase becomes large when the coherency is low. Chap. 10 OREGON SHELF: 1972 65 i n v e s t i g a t e d n u m e r i c a l l y the b e h a v i o u r of s h e l f waves p r o p a g a t i n g over bumps and n a r r o w i n g s h e l v e s , and found r a t h e r complex phase p r o p a g a t i o n due t o r e f l e c t i o n and d i f f r a c t i o n . However, downstream away from the t o p o g r a p h i c d i s t u r b a n c e , the c o n s t a n t phase l i n e s a g a i n a l i g n e d themselves p e r p e n d i c u l a r t o the depth c o n t o u r s . T h i s t o p o g r a p h i c s t e e r i n g or r e f r a c t i o n e f f e c t i s w e l l i l l u s t r a t e d i n the s c a t t e r diagram of Kundu and A l l e n (1976, p.186). S i n c e the depth c o n t o u r s a t NH-10 a r e s h i f t e d ~20° w i t h r e s p e c t t o t h a t a t DB-7, t h e second time s e r i e s wx would be m u l t i p l i e d by an e x t r a phase f a c t o r e~ l 6 T , assuming the c o n s t a n t phase l i n e s a l i g n t hemselves p e r p e n d i c u l a r t o the depth c o n t o u r s . From Eqs. (8.1) and (8.2) , i t can be seen t h a t <£-»'P 1-6 T and © 1.-*© i+S T. From ( 8 . 5 ) , t h e i n n e r - p h a s e s h i f t s by the amount 6 T f o r both p o s i t i v e and n e g a t i v e v a l u e s of 0". Thus, the o b s e r v e d i n n e r - p h a s e would be p o s i t i v e l y s h i f t e d by ~20° due t o the c h a n g i n g topography. I f t h e r e i s an a l o n g s h o r e phase l a g due t o a p r o p a g a t i n g s h e l f wave ( w i t h the c o n v e n t i o n 8 W p o s i t i v e when the wave i s t r a v e l l i n g n o r t h w a r d , i . e . n o r t h e r n s t a t i o n l a g s s o u t h e r n s t a t i o n ) , the o b s e r v e d i n n e r - p h a s e X w i l l be: where i s the i n n e r - p h a s e f o r n e g a t i v e <r ( c l o c k w i s e motion) T. (10.1) 1 In the absence of S T , a northward p r o p a g a t i n g s h e l f wave would m a n i f e s t i t s e l f by showing a p o s i t i v e v a l u e f o r X and a n e g a t i v e v a l u e of the same magnitude f o r X+ . Chap. 10 OREGON SHELF: 1972 66 and X+ f o r p o s i t i v e <T ( a n t i c l o c k w i s e ) . S w and S T can be o b t a i n e d e a s i l y from Sw = (X. - TLj/z. ST = (X- + X+Vz (10.2) F i g . 10.3 shows t h a t a t f r e q u e n c i e s £ 0.3 cpd, the in n e r - p h a s e i s g e n e r a l l y p o s i t i v e , m a n i f e s t i n g the t o p o g r a p h i c phase s h i f t e f f e c t . However, S T appears t o be somewhat fr e q u e n c y dependent. At 0.13 cpd, (10.2) g i v e s 5 y = 14° and S w = -5°, w h i l e a t 0.27 cpd, S T = 54* and Sw = -14* . N e g a t i v e v a l u e s of S w would i n d i c a t e southward p r o p a g a t i n g waves, o p p o s i t e t o the d i r e c t i o n of genuine s h e l f waves. However, the 95% c o n f i d e n c e i n t e r v a l f o r the phase as d e t e r m i n e d from J e n k i n s and Watts (1968, p.381) i s i 2 4 . Hence the s m a l l negat i v e S w v a l u e s a r e not s t a t i s t i c a l l y d i f f e r e n t from 0° or a s m a l l p o s i t i v e v a l u e . For s h o r t s h e l f waves, we l o o k f o r l a r g e phase s h i f t s . 1 At -0.67 cpd, the i n n e r - p h a s e X_ i s 95*± 25° . U n f o r t u n a t e l y S w cannot be c a l c u l a t e d from (10.2) s i n c e X+ a t +0.67 cpd i s h i g h l y u n c e r t a i n due t o the low co h e r e n c y . I n s t e a d , assuming a 20° t o p o g r a p h i c phase s h i f t , we use S w = $ T t o o b t a i n S w = 75°± 25° . W i t h 8^ l y i n g w i t h i n the i n t e r v a l (50°, 100°), the c o r r e s p o n d i n g w a v e l e n g t h i s w i t h i n (160 km, 80 km), which a g r e e s w e l l w i t h t h e . p r e d i c t e d s h o r t waves a r i s i n g from r e s o n a n t t r i a d i n t e r a c t i o n s . 1 When the coherency v a l u e i s low, the u n c e r t a i n t y i n - the phase becomes l a r g e . T h e r e f o r e , l a r g e phase s h i f t s a s s o c i a t e d w i t h low coherency a r e s p u r i o u s . Chap. 10 OREGON SHELF: 1972 67 The 0.67 cpd s i g n a l c o u l d be produced by n o n l i n e a r energy t r a n s f e r from the 0.27 cpd s i g n a l . On the d i s p e r s i o n diagram of C&S, a f i r s t mode 0.27 cpd s h e l f wave can r e s o n a n t l y t r a n s f e r energy t o a 0.66 cpd f i r s t mode wave ( w i t h a wavelength of about 150 km) and a 0.39 cpd second mode wave. F i g . 10.4 and F i g . 10.5 show r e s p e c t i v e l y the i n n e r c r o s s - s p e c t r u m between the 40 m c u r r e n t a t DB-7 and Newport wind, and t h a t between the 20 m c u r r e n t a t NH-10 and Newport wind. The p u z z l i n g phenomenon i s t h a t a t f r e q u e n c i e s below 0.25 cpd, the coherency between the DB-7 c u r r e n t and the Newport wind i s much h i g h e r than t h a t between NH-10 and Newport wind, d e s p i t e DB-7 b e i n g f a r t h e r away from Newport. A p o s s i b l e e x p l a n a t i o n i n v o l v e s s e v e r a l s t e p s : ( i ) We note t h a t NH-10 i s v e r y c l o s e t o the S t o n e w a l l Bank ( F i g . 10.1). The wind g e n e r a t e d l o n g wave when e n c o u n t e r i n g the Bank would, a c c o r d i n g t o Wang (1980), s c a t t e r much of i t s . energy t o s h o r t e r waves of the same  f r e q u e n c y . As t h e s e s c a t t e r e d waves a r e not c o h e r e n t w i t h the wind, they would d e c r e a s e the o b s e r v e d coherency between c u r r e n t and wind. ( i i ) Some of the s c a t t e r e d waves may have v e r y s h o r t w avelengths and a r e damped r a p i d l y as they t r a v e l away from the B a n k — hence the i n c r e a s e i n cohe r e n c y a t DB-7. ( i i i ) I n a d d i t i o n , as the l o n g wave p r o p a g a t e s away from the Bank, i t g r a d u a l l y p i c k s up energy from the wind, and the coherency 6 8 O 00 10.4 Inner cross-spectrum between the 40 m current at DB-7 and the Newport wind. 69 o CO o CD ] UJ -J CO - j o _J 9 5 / i r i i i r r - 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 0 . 0 0 2 0 4 FREOUENCr (CPD) 0 . 6 0 . 8 F i g . 10.5 Inner cross-spectrum between the 20 m current at NH-10 and the Newport wind. Chap. 10 OREGON SHELF: 1972 70 r i s e s . 1 T a b l e 10.1 compares the low f r e q u e n c y (0.13 cpd) energy a t DB-7 and NH-10. I t does appear t h a t the low f r e q u e n c y wave Tab l e 10.1 Comparison of low f r e q u e n c y (0.13 cpd) a n t i c l o c k w i s e and c l o c k w i s e energy d e n s i t i e s ( c m 4 s " 2 c p d " 1 ) a t DB-7(40m) and NH-10(20m). S t a t i o n a C DB-7 278 385 1.38 NH-10 207 257 1.24 p i c k s up energy on i t s way from NH-10 t o DB-7, c o n s i s t e n t w i t h ( i i ) . A l s o , i t d i s p l a y s more c l o c k w i s e energy than a n t i c l o c k w i s e energy, as e x p e c t e d f o r a genuine s h e l f wave. The -0.67 cpd s i g n a l a l s o shows up between -0.6 t o -0.67 cpd i n F i g s . 10.4 and 10.5. For t h i s peak, the i n n e r - p h a s e between DB-7 and the wind i s 118°, w h i l e between NH-10 and t h e wind, i t i s 41°. T h i s i n d i c a t e s a l a r g e phase l a g 1 S i n c e t h e r e are o n l y 22 km between NH-10 and DB-7, i t i s not c l e a r i f the l o n g wave can p i c k up a s i g n i f i c a n t amount of wind energy i n such s h o r t d i s t a n c e . G i l l and Schumann (1974, Eq. ( 7 . 8 ) ) showed t h a t t h e l e n g t h s c a l e f o r a l o n g s h o r e energy i n c r e a s e i s comparable t o the wavelength of the w i n d - g e n e r a t e d wave. However, i f the wind system i s t r a v e l l i n g a t the phase speed of t h e s h e l f wave, resonance o c c u r s , ( G i l l and Schumann, Eq. ( 7 . 9 ) ) , and the i n c r e a s e i n wave energy can be s u b s t a n t i a l even o v e r r e l a t i v e l y s h o r t d i s t a n c e . Chap. 10 OREGON SHELF: 1972 71 between the c u r r e n t s a t DB-7 and NH-10, i n agreement w i t h the phase from F i g . 10.3. Though we have i n t e r p r e t e d t h i s peak as due t o a s h o r t wave, and a t t r i b u t e d i t s o r i g i n t o the re s o n a n t t r i a d i n t e r a c t i o n , t h e r e remains t h e p o s s i b i l i t y t h a t t h i s s h o r t wave may be wind g e n e r a t e d s i n c e the cohe r e n c y w i t h the wind i s not low i n F i g s . 10.4 and 10.5. U n f o r t u n a t e l y , t h e r e i s i n s u f f i c i e n t wind d a t a t o a l l o w us t o dete r m i n e i f the wind a t t h i s f r e q u e n c y a l s o has s h o r t a l o n g s h o r e l e n g t h s c a l e . N e v e r t h e l e s s , from F i g . 10.6 which shows the a u t o s p e c t r u m f o r the 20 m c u r r e n t a t NH-10 and t h e Newport wind, we note t h a t a t around 0.65 cpd, the c u r r e n t c o n t a i n s n e a r l y t w i c e as much c l o c k w i s e energy as a n t i c l o c k w i s e energy, whereas the wind d i s p l a y s the o p p o s i t e f e a t u r e - - n e a r l y t w i c e as much a n t i c l o c k w i s e energy as c l o c k w i s e . T h i s mismatch seems t o argue a g a i n s t the 0.67 cpd s i g n a l b e i n g wind g e n e r a t e d . F i g . 10.7 (from Kundu and A l l e n , 1976) shows t h e v e l o c i t y component v a t NH-10(60m) and a t the s t a t i o n UWIN(66m) on the Washington s h e l f , 200 km t o the n o r t h . The time s e r i e s f o r UWIN i s s i m i l a r t o t h a t f o r NH-10, e x c e p t i t has much l e s s " h i g h " f r e q u e n c y o s c i l l a t i o n s . The a u t o s p e c t r a show the two s t a t i o n s t o be s i m i l a r f o r f r e q u e n c i e s below 0.4 cpd; above 0.4 cpd, UWIN has much l e s s energy. As t h e two s t a t i o n s were s e p a r a t e d by o n l y 200 km, i t i s hard t o c o n c e i v e how any (northward p r o p a g a t i n g ) l o n g s h e l f wave can l o s e energy so r a p i d l y . As we s h a l l see below, t h i s phenomenon can be e x p l a i n e d by the dominance of s h o r t waves i n the h i g h e r f r e q u e n c y regime. 72 Fig. 10.6 Autospectra of the 20 m current at NH-10 and the Newport wind (dashed li n e ) . Throughout this thesis, the autospectra for currents are plotted in units of cm* sec" 2cpd - 1 , and in units of m3- sec" 1 cpd"' for the wind. The 95% confidence interval and the bandwidth are as shown. 73 Fig. 10.7 Time series and autospectra of v at NH-10(60m) and UWIN(66m) (from Kundu and Allen, 1976). Chap. 10 OREGON SHELF: 1972 74 The most r e a s o n a b l e p i c t u r e one can assemble i s the f o l l o w i n g : D u r i n g the summer, t h e r e a r e two fr e q u e n c y regimes on the Oregon s h e l f . Below 0.4 cpd, the wind i n f l u e n c e i s s t r o n g , and l o n g s h e l f waves a r e b e i n g g e n e r a t e d . In the f r e q u e n c y regime above 0.4 cpd, the s h o r t waves produced by re s o n a n t n o n l i n e a r energy t r a n s f e r from the l o n g waves a r e a c t u a l l y o b s e r v e d . 1 The l o w - f r e q u e n c y l o n g waves have t h e i r group v e l o c i t i e s i n the same d i r e c t i o n as t h e i r phase v e l o c i t i e s , ( i . e . t h e i r energy p r o p a g a t e s n o r t h w a r d onto the Washington s h e l f ) , whereas the h i g h e r f r e q u e n c y s h o r t waves have t h e i r group v e l o c i t i e s i n the o p p o s i t e d i r e c t i o n — t h e i r energy p r o p a g a t e s southward. On the Washington s h e l f , t he wind i n f l u e n c e i s much l e s s s i g n i f i c a n t (HHSSP, p.3502). In the absence of energy i n p u t from the wind t o the l o n g waves, the n o n l i n e a r energy t r a n s f e r t o the s h o r t waves may a l s o s t a l l , ( i . e . t h e r e i s l i t t l e l o c a l g e n e r a t i o n of e i t h e r l o n g or s h o r t s h e l f waves). T h e r e f o r e , on the Washington s h e l f , the low f r e q u e n c y regime i s dominated by the l o n g waves coming from the Oregon s h e l f , - but the h i g h e r f r e q u e n c y regime i s r e l a t i v e l y b a r r e n . T h i s e x p l a i n s the o b s e r v a t i o n s a t UWIN and NH-10. 1 The t r i a d i n t e r a c t i o n s can a l s o g e n e r a t e s h o r t waves w i t h f r e q u e n c i e s £ 0.4 cpd; however, i t i s d i f f i c u l t t o d e t e c t them as they would be masked by the wi n d - g e n e r a t e d l o n g waves. 75 CHAPTER 11. OREGON SHELF: SUMMER, 1973 The CUE-2 experiment was c a r r i e d out on the Oregon s h e l f d u r i n g the summer of 1973. The c u r r e n t meter a r r a y and the l e n g t h of d a t a r e c o r d s a r e shown i n F i g . 11.1 and F i g . 11.2, taken from the data r e p o r t by P i l l s b u r y e t a l . (1974b). The c r o s s - s h e l f moorings were p l a c e d some d i s t a n c e n o r t h of the CUE-1 a r r a y , i n o r d e r t o enj o y a s i m p l e r topography. A n a l y s e s of b o t h CUE-1 and CUE-2 da t a a r e p r e s e n t e d i n Kundu, A l l e n and Smith (1975), and Kundu and A l l e n (1976 and 1978). The c r o s s - s h e l f topography a l o n g the l i n e of moorings i s shown i n F i g . 11.3, and the b a r o t r o p i c s h e l f wave d i s p e r s i o n diagram o b t a i n e d by n u m e r i c a l i n t e g r a t i o n i s shown i n F i g . 11.4. The d i s p e r s i o n c u r v e s f o r modes 3 and 4 n e a r l y " k i s s e d " a t k ^ 8 . T h i s phenomenon i s p r o b a b l y caused by the presence of humps on the s h e l f p r o f i l e ( F i g . 11.3), as one would e x p e c t s m a l l s c a l e t o p o g r a p h i c p e r t u r b a t i o n s t o a f f e c t t he s h o r t waves. The a u t o s p e c t r a f o r the 40 m c u r r e n t s a t C a r n a t i o n and a t F o r s y t h i a a r e shown i n F i g . 11.5. The water depth i s 100 m a t C a r n a t i o n and 500 m a t F o r s y t h i a . Two s a l i e n t f e a t u r e s from the f i g u r e a r e : ( i ) Energy f a l l s o f f away from the c o a s t — c o n s i s t e n t w i t h the pr e s e n c e of c o a s t a l t r a p p e d waves. ( i i ) The energy f a l l s o f f much f a s t e r a t low f r e q u e n c i e s . For i n s t a n c e , at around 0.12 cpd, the r a t i o of t h e energy d e n s i t y a t C a r n a t i o n t o t h a t a t F o r s y t h i a i s g r e a t e r than 10, w h i l e a t 0.4 cpd, the r a t i o i s around 3. An i n t e r p r e t a t i o n of t h i s i n t e r e s t i n g phenomenon w i l l be g i v e n l a t e r . Not so e a s i l y o b s e r v e d from the l o g - p l o t i s the f a c t t h a t C a r n a t i o n has more c l o c k w i s e energy 76 11.1 Current meter array i n the CUE-2 experiment. J U N -i—I 1—I—r J U L i i—i—i—r A U G - i — i — i — i — r ASTER S E P T—l 1 I T SfC 20m 40 77 * ASTER 8.5m 40 CARNATION m sfc 3 20m 3 40 • 60 3 80 3 95 DAFFODIL EDELWEISS 1 sfc I sfc I 20m I 40 I 80 1120 1180 1195 Ml current meters have U, V, and water temperature. Surface meters have air and water temperatures, and wind U and V. air and water temperature only • water temperature only • ^ 3 no wind U and V, only direction and air and water temperature • no wind U and V, only speed and air and water temperature • fV \ \J FORSYTH IA sfc 3 40 m 3 80 • 120 3 180 • 300 FORGET-ME-NOT 3 20m • 40 3 60 • 80 GLADIOLI S sfc IRIS 20m 3 40 3 60 JASMINE sfc POINSETT IA 3 20m 3 40 3 60 3 80 • • • • 1 •—i—'—<-J U N J U L AUG 1973 S E P F i g . 11.2 Length of.current data records i n CUE-2. 78 O CO r\H I I I | 1 1 1 1 1 1 1 1 1 1 0 20 40 60 80 100 120 140 OFFSHORE DISTANCE (KM) \ Fig. 11.3 Cross shelf topography in the CUE-2 experiment. The crosses indicate data from contour charts. The curve is obtained by f i t t i n g spline functions to the data. (Seamounts and canyons are ignored in the f i t ) . 79 F i g . 11.4 Barotropic shelf wave dispersion diagram for CUE-2. This diagram i s obtained by numerical i n t e g r a t i o n over the shelf shown i n F i g . 11.3. The wavenumber k i s nondimensionalized with respect to L , where L i s chosen to be 120 km. 80 F i g . 11.5 Autospectra of the 40 m currents at Carnation and Forsythia (dashed l i n e ) . Chap. 11 OREGON SHELF: 1973 81 than a n t i c l o c k w i s e energy at p r a c t i c a l l y a l l f r e q u e n c i e s . When the c l o c k w i s e and a n t i c l o c k w i s e e n e r g i e s a r e summed s e p a r a t e l y over f r e q u e n c i e s from 0.05 t o 0.8 cpd, the r a t i o of the t o t a l c l o c k w i s e energy t o a n t i c l o c k w i s e energy i s 1.35. T h i s i s a g a i n c o n s i s t e n t w i t h the presence of s h e l f waves. F i g . 11.6 and F i g . 11.7 a r e r e s p e c t i v e l y the a u t o s p e c t r a of the a d j u s t e d sea l e v e l and the wind v e l o c i t y a t Newport. The wind spectrum r e v e a l s s e v e r a l broad energy "humps". The s t a b i l i t y and i t s phase 2«*j >, from Eqs. (8.8) and ( 8 . 9 ) , a r e p l o t t e d f o r the 40 m c u r r e n t a t A s t e r i n F i g . 11.8. For h i g h v a l u e s of c u r r e n t s t a b i l i t y , the phase 2<»j> i s around 180° . T h i s means < < y^> i s about 90° , c o n s i s t e n t w i t h our f i n d i n g i n Chapter 7 t h a t f o r most of the s h e l f , the major axes of s h e l f waves c u r r e n t e l l i p s e s a r e o r i e n t e d p a r a l l e l t o the c o a s t . The i n n e r and o u t e r c r o s s - s p e c t r u m between the 40 m c u r r e n t s a t A s t e r and C a r n a t i o n a r e shown r e s p e c t i v e l y i n F i g . 11.9 and F i g . 11.10. For r e a s o n a b l y h i g h v a l u e s of c o h e r e n c y , th e i n n e r - p h a s e i s p r a c t i c a l l y z e r o f o r a l l f r e q u e n c i e s , w h i l e the o u t e r - p h a s e i s around 180° . The i n t e r p r e t a t i o n f o r the phases i s as f o l l o w s : N e g l e c t i n g f r i c t i o n a l and t o p o g r a p h i c e f f e c t s , p r o p a g a t i n g s h e l f waves s h o u l d m a n i f e s t <P, = "f\ and ©, =  e z f o r the phases o b s e r v e d a t 2 s t a t i o n s s e p a r a t e d o n l y i n the c r o s s - s h e l f d i r e c t i o n and c l o s e enough t o shore (so t h a t no s h e l f wave nodes o c c u r between them). However, as the major axes of t h e c u r r e n t e l l i p s e s a r e o r i e n t e d a l o n g s h o r e , ( <Xj = <?£>" , and f j - 9- = 2 o(- = 180°), the r e s u l t i s 82 Fig. 11.6 Autospectrum of the Newport adjusted sea level during CUE-2. Since the sea level is a scalar, the two sides of the rotary spectrum are identical. 83 Fig. 11.7 Autospectrum of the Newport wind during CUE-2. 84 o CO I o r-0 0 0 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 FREQUENCY (CPD) Fig. 11.8 Stability and phase for the 40 m current at Aster. 85 O CO _. CO I O U J o FREQUENCT (CPD) Fig. 11.9 Inner cross-spectrum between the 40 m currents at Aster and Carnation. 86 o to CO I Fig. 11.10 Outer cross-spectrum between the 40 m currents at Aster and Carnation. Chap. 11 OREGON SHELF: 1973 ' 87 - « . = f x " K - fl- ©a = ©. = <*°" ( i i . i ) From Eqs. (8.5) and ( 8 . 7 ) , i t i s e a s i l y seen t h a t the i n n e r - p h a s e s h o u l d be 0 w h i l e the o u t e r - p h a s e s h o u l d be 180 , as i s indeed o b s e r v e d . 1 These o b s e r v a t i o n s show t h a t the g e n e r a l c r o s s - s h e l f c u r r e n t b e h a v i o u r on the Oregon s h e l f i s c o n s i s t e n t w i t h the p r e s e n c e of c o n t i n e n t a l s h e l f waves. Next, we examine c r o s s - s p e c t r a between s t a t i o n s s e p a r a t e d i n the a l o n g s h o r e d i r e c t i o n . F i g . 11.11 shows the i n n e r c r o s s - s p e c t r u m between the 40 m c u r r e n t s a t A s t e r and the a d j u s t e d sea l e v e l a t Newport. From the n e g a t i v e f r e q u e n c y ( c l o c k w i s e ) p a r t of the spectrum one n o t e s 5 d i s t i n c t peaks a t 0.14, 0.32, 0.41, 0.57, and 0.66 cpd. The i n n e r - p h a s e tends t o f l u c t u a t e m i l d l y around 90 . I t t u r n s out t h a t f o r a l o n g s h e l f 0 wave, t h e r e i s a 90 l a g between the sea l e v e l and the r o t a r y c u r r e n t components a t the c o a s t . Near the c o a s t , u ^ 0, and c a r e f u l e x a m i n a t i o n of a A and a c i n (7.24) and ( 7 . 2 5 ) , and *\ i n (7.4) shows a phase d i f f e r e n c e of 90°. T h e r e f o r e , the o b s e r v e d s m a l l d e v i a t i o n from 90° s u g g e s t s a s m a l l a l o n g s h o r e phase l a g 6 W between the two s t a t i o n s , and hence the p r e s e n c e of l o n g waves. However, one s h o u l d be c a u t i o n e d t h a t t h i s 90° phase l a g i s a t h e o r e t i c a l r e s u l t and h o l d s o n l y f o r c u r r e n t s s u f f i c i e n t l y c l o s e t o the shore such t h a t |u| < |v|. For l o n g s h e l f waves, x B r i n k and A l l e n (1978) showed t h a t bottom f r i c t i o n can i n t r o d u c e a c r o s s - s h e l f phase l a g . T h i s e f f e c t was obs e r v e d o f f P e r u by B r i n k , A l l e n and Smith (1978). 8 8 O CO _ , cn L U - I c n cn _| o CO i — r I Q L J o — cr V O 0 0 C O a >—CO C_3 • _ U J — U J o — I E O r g O • o i So i——i 957 1 1 1 1 1 1 1 I I I I I - 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 O . f l 0 . 2 0 . 4 FREQUENCY (CPD) 0 . 6 0 . 8 .11 Inner cross-spectrum between the 40 m current at Aster and the adjusted sea. level at Newport. Chap. 11 OREGON SHELF: 1973 89 the assumption |u| < |v| i s g e n e r a l l y good, but f o r s h o r t waves |u| -~ |v j o n l y a l i t t l e d i s t a n c e o f f s h o r e , and the 90° phase l a g no l o n g e r h o l d s . F i g . 11.12 shows t h e i n n e r c r o s s - s p e c t r u m between the 40 m c u r r e n t s a t C a r n a t i o n and a t P o i n s e t t i a , two s t a t i o n s s e p a r a t e d 60 km a l o n g s h o r e . As the depth c o n t o u r a t P o i n s e t t i a i s t i l t e d by about 35° from t h a t a t C a r n a t i o n (see F i g . 1 1 . 1 ) , one e x p e c t s a t o p o g r a p h i c phase s h i f t of S T = +35* i n the i n n e r - p h a s e (and i n the o u t e r - p h a s e ) . Indeed, a t low f r e q u e n c i e s , the in n e r - p h a s e i s s l i g h t l y above 35* f o r n e g a t i v e f r e q u e n c i e s and s l i g h t l y below 35° f o r p o s i t i v e f r e q u e n c i e s , c o n s i s t e n t w i t h n o r t h w a r d p r o p a g a t i n g l o n g waves w i t h c o r r e s p o n d i n g s m a l l S w . For t h e 0.13 cpd peak, S T = 26*110°, ( q u i t e c l o s e t o the 35* ex p e c t e d from the o r i e n t a t i o n of the c o n t o u r s ) , and Sw = 20°. bw = 20 t r a n s l a t e s i n t o a n o r t h w a r d p r o p a g a t i n g wave w i t h a wavel e n g t h of 1080 km, and a wavenumber of 0.70 on the d i s p e r s i o n diagram F i g . 11.4. One sees t h a t a f r e q u e n c y of 0.13 cpd and a wavenumber of 0.70 f i t v e r y w e l l t o a second mode s h e l f wave. T h i s a g r e e s w i t h Kundu and A l l e n (1976), where from the l a g time of maximum c o r r e l a t i o n ( w i t h o u t s p e c t r a l d e c o m p o s i t i o n ) between c u r r e n t s a t a l o n g s h o r e s t a t i o n s , a no r t h w a r d p r o p a g a t i n g speed c o n s i s t e n t w i t h second mode n o n d i s p e r s i v e l o n g s h e l f waves was found. A g a i n , t o f i n d s h o r t s h e l f waves, we l o o k f o r l a r g e phase l a g s . The peak a t -0.55 cpd has an i n n e r - p h a s e of 187°, w i t h a 95% c o n f i d e n c e i n t e r v a l of ± 32° . (Though t h i s s i g n a l i s below the 95% s i g n i f i c a n c e l e v e l i n F i g . 11.12, i t i s not n e c e s s a r i l y 90 o r-CxJ o CD « — 1 LU CO Q_ o o cn i — Q V 11/ 9 5 ; i i i - 0 8 - 0 6 - 0 4 - 0 . 2 0 . 0 0 - 2 . 0 . 4 FREQUENCY (CPD) 0 . 6 0 . 8 F i g . 11.12 Inner cross-spectrum between the 40 m currents at Carnation and at Poins e t t i a . Chap. 11 OREGON SHELF: 1973 91 so i n o t h e r f i g u r e s , e.g. F i g . 11.11). S u b t r a c t i n g o f f 26° f o r 6 T, the phase then l i e s w i t h i n the i n t e r v a l (129°, 193°), and the c o r r e s p o n d i n g w a v e l e n g t h , w i t h i n the i n t e r v a l (167 km, 112 km). 1 As the s h e l f topography i s r a t h e r d i f f e r e n t at the two s t a t i o n s , t h e r e i s some doubt as t o which d i s p e r s i o n diagram t o use f o r t r i a d " p a r a l l e l o g r a m " c o n s t r u c t i o n . The d i s p e r s i o n diagram i n C&S i s f o r topography near P o i n s e t t i a w h i l e F i g . 11.1 i s f o r topography near C a r n a t i o n . As s h e l f waves propagate from south t o n o r t h , the d i s p e r s i o n diagram c o r r e s p o n d i n g t o the so u t h e r n s t a t i o n seems more a p p r o p r i a t e . Upon c o n s t r u c t i n g t r i a d " p a r a l l e l o g r a m s " on the C&S d i s p e r s i o n diagram, we o b t a i n the f o l l o w i n g p o s s i b l e t r i a d s : ( i ) The dominant 0.14 cpd second mode wave g e n e r a t i n g r e s o n a n t l y a f i r s t mode 0.53 cpd wave w i t h 96 km wa v e l e n g t h , and a second mode 0.40 cpd s h o r t wave. ( i i ) The o b s e r v e d 0.30 cpd (second mode) wave g e n e r a t i n g r e s o n a n t l y a f i r s t mode 0.63 cpd wave w i t h 136 km wav e l e n g t h , and a second mode 0.34 cpd wave. The 0.53 cpd wave g e n e r a l l y a g r e e s w i t h the ob s e r v e d c l o c k w i s e s i g n a l a t -0.55 cpd. In a d d i t i o n , we note t h a t b o t h F i g . 11.11 and F i g . 11.12 show a peak a t -0.65 cpd.. T h i s may be the 0.63 cpd wave from t r i a d ( i i ) . 1 Both C a r n a t i o n and P o i n s e t t i a a r e c l o s e t o the shore (*15 km o f f s h o r e ) , and a t 0.55 cpd, o n l y the f i r s t mode can e x i s t ; hence, t h e r e i s no danger of a node f a l l i n g i n between the two s t a t i o n s and i n t r o d u c i n g a 180° phase d i f f e r e n c e . Chap. 11 OREGON SHELF: 1973 92 F i g . 11.13 shows "the c o r r e s p o n d i n g o u t e r c r o s s - s p e c t r u m between the 40 m c u r r e n t s a t C a r n a t i o n and P o i n s e t t i a . At low f r e q u e n c i e s , the outer-phase l i e s c l o s e t o 180°+ 5 T , i n d i c a t i n g s m a l l a l o n g s h o r e phase s h i f t ( S w ) , and hence l o n g waves. However, l a r g e d e v i a t i o n i s obser v e d a t h i g h e r f r e q u e n c i e s . The out e r - p h a s e a t -0.55 cpd i s 31°+ 30° , a g a i n i m p l y i n g an a l o n g s h o r e phase l a g of about 180* , ( a p p r o x i m a t e l y the same as t h a t e s t i m a t e d from the i n n e r - p h a s e ) , and y i e l d i n g a w a v e l e n g t h of ~120 km. F i g . 11.14 shows the i n n e r c r o s s - s p e c t r u m between the 40 m c u r r e n t a t P o i n s e t t i a and the wind a t Newport. We f i n d s i g n i f i c a n t c oherency ( e s p e c i a l l y f o r c l o c k w i s e f r e q u e n c i e s ) between wind and c u r r e n t a t f r e q u e n c i e s below 0.4 cpd, i n agreement w i t h Huyer and P a t t u l l o (1972). The h i g h coherency bands i n F i g . 11.14 g e n e r a l l y c o r r e s p o n d s t o the "humps" i n the wind a u t o s p e c t r u m ( F i g . 11.7). The phase shows the wind t o l e a d t h e c u r r e n t s l i g h t l y ; however, the l e a d i n c r e a s e s t o over 90* a t ~ -0.5 cpd. We next p r o c e e d t o p e r f o r m c r o s s - s h e l f modal f i t t i n g . Four s t a t i o n s , A s t e r (20 m), C a r n a t i o n (40 m), E d e l w e i s s (80 m), and F o r s y t h i a (40 m) a r e u s e d . 1 1 As the water depth i s o n l y 50 m a t A s t e r i a , the 20 m c u r r e n t r e c o r d i s chosen; w h i l e a t E d e l w e i s s , the 40 m c u r r e n t r e c o r d i s d e f e c t i v e , so the 80 m r e c o r d i s used i n s t e a d . 93 O Q Q L U o ' - I cr FREQUENCY (CPD) Fig. 11.13 Outer cross-spectrum between the 40 m currents at Carnation and at Poinsettia. 94 o CO _ Fig. 11.14 Inner cross-spectrum between the 40 m current at Poinsettia and the wind at Newport. Chap. 11 OREGON SHELF: 197 3 95 Upon F o u r i e r t r a n s f o r m i n g the c u r r e n t d a t a , the q u a n t i t i e s A, C, f and 9 from Eq. (8.2) a r e o b t a i n e d as f u n c t i o n s of f r e q u e n c y . 1 For h i g h e r s t a t i s t i c a l s i g n i f i c a n c e , t h e s e q u a n t i t i e s a r e band-averaged over n e i g h b o u r i n g f r e q u e n c i e s u s i n g the f o l l o w i n g f o r m u l a s : C n. = <C 2 > Y* , 6„ = (<Ccose> - l < C s ^ & > ) (11.2) where <...> denotes b a n d - a v e r a g i n g , and the s u b s c r i p t n has been added t o the new q u a n t i t i e s , (n l a b e l s the q u a n t i t y from the n t h s t a t i o n ) . Data f o r A s t e r , C a r n a t i o n and F o r s y t h i a a r e averaged over 5 f r e q u e n c y bands, w h i l e f o r E d e l w e i s s , (due t o i t s s h o r t e r r e c o r d ) , o n l y over 3 bands. The f o l l o w i n g f r e q u e n c i e s a r e s e l e c t e d f o r modal f i t t i n g : 0.13, 0.30, 0.43 and 0.53 cpd. Ta b l e 11.1 l i s t s t he p o s s i b l e s h e l f waves a t these s e l e c t e d f r e q u e n c i e s . The e n t r i e s i n Tab l e 11.1 a r e o b t a i n e d from the i n t e r s e c t i o n s on the d i s p e r s i o n diagram ( F i g . 11.4) between h o r i z o n t a l l i n e s drawn a t the s e l e c t e d f r e q u e n c i e s and t h e d i s p e r s i o n c u r v e s . Note t h a t p o s s i b l e c o n t r i b u t i o n s from waves w i t h k > 10 (\< 75 km) have been n e g l e c t e d . A l s o note t h a t the second mode a c t u a l l y 1 The time s e r i e s must a l l have the same s t a r t i n g t i m e . O t h e r w i s e , the phase of one time s e r i e s w i l l be s h i f t e d by |<r|t w i t h r e s p e c t t o a n o t h e r , where t i s the d i f f e r e n c e i n the s t a r t i n g t i m e s . As the r e c o r d f o r E d e l w e i s s s t a r t e d much l a t e r t han the o t h e r s , the phases f o r E d e l w e i s s had t o be c o r r e c t e d by M t . Chap. 11 OREGON SHELF: 197 3 96 T a b l e 11.1 P o s s i b l e s h e l f waves c o n t r i b u t i n g a t s e l e c t e d f r e q u e n c i e s . Frequency S h e l f wave (cpd) mode wavenumber wavelength (km) 0.13 1 0.26 2900 2 0.82 920 3 1.6 470 4 3.5 210 0.30 1 0.67 1100 2 2.2 340 3 9.7 78 0.43 1 1.1 690 2 4.2 180 2 7.2 100 0.53 1 1.5 502 c o n t r i b u t e d two waves of d i f f e r e n t wavelengths a t 0.43 cpd. A c o m p l i c a t i o n a r i s e s a t t h e 0.53 cpd f r e q u e n c y . In the C&S d i s p e r s i o n diagram, the f i r s t mode c o n t r i b u t e s both a l o n g wave and a s h o r t wave a t 0.53 cpd. However, i n the d i s p e r s i o n d iagram F i g . 11.4 f o r topography a l o n g the c r o s s - s h e l f a r r a y , the f i r s t mode c u r v e a t h i g h wavenumbers f a i l s t o d r o p as much as i n the C&S d i s p e r s i o n diagram ( f o r topography f u r t h e r s o u t h ) . T h i s means the 0.53 cpd t r i a d member, c o n s t r u c t e d from the C&S diagram, can no l o n g e r p r o p a g a t e as a f r e e wave by t h e time i t a r r i v e s a t the CUE-2 c r o s s - s h e l f a r r a y . I t s energy would be s c a t t e r e d i n t o o t h e r f r e e waves of the same f r e q u e n c y . The f i t t i n g was performed by m i n i m i z i n g F i n Eq. (8.12) f o r Chap. 11 OREGON SHELF: 1973 97 each of the s e l e c t e d f r e q u e n c i e s . S i n c e the o f f s h o r e s t a t i o n s have much s m a l l e r wave a m p l i t u d e s than the n e a r s h o r e s t a t i o n s , the w e i g h t s wn have t o be i n c r e a s e d f o r the o f f s h o r e s t a t i o n s . A l s o , the d a t a from E d e l w e i s s were o n l y used f o r the f i t t i n g a t 0.13 cpd; f o r the o t h e r f r e q u e n c i e s , o n l y 3 s t a t i o n s were employed. R e c a l l t h a t each s t a t i o n s u p p l i e s 4 numbers ( A n , C*, tfK, ©a ) , and each s h e l f wave i n t r o d u c e s 2 parameters (a^, and S m ) t o be d e t e r m i n e d from the f i t t i n g . For i n s t a n c e , f o r the 0.43 cpd f r e q u e n c y , we have 3 s t a t i o n s s u p p l y i n g 12 numbers and 3 s h e l f waves w i t h 6 parameters t o be f i t t e d . The r e s u l t s of the f i t t i n g a r e shown i n T a b l e 11.2. T a b l e 11.2 C r o s s - s h e l f modal f i t t i n g a t s e l e c t e d f r e q u e n c i e s . F r e q . ( c p d ) mode >.(km) E m / Z E m D e v i a t i o n 0.13 1 2900 0.17 0.05 2 920 0.70 3 470 0.00 4 210 0.14 0.30 1 1100 0.01 0.09 2 340 0.91 3 78 0.08 0.43 1 690 0.05 0.36 2 180 0.72 2 100 0.23 0.53 1 502 0.33 Chap. 11 OREGON SHELF: 1973 98 The column E m / Z , E m i n d i c a t e s the r e l a t i v e energy c o n t r i b u t i o n of the m-th s h e l f wave i n the f i t . E m , the energy of the m-th wave, ( m = 1,...,M), i s d e f i n e d as and X E m f t h e t o t a l energy of the f i t , i s the sum of the E^'s. For the f r e q u e n c i e s 0.13 and 0.43 c p d , the f i t s were o b t a i n e d by m i n i m i z i n g the f u n c t i o n of ( 8 . 1 3 ) , i n s t e a d of F. (When F was used, s e v e r e " o v e r f i t t i n g " r e s u l t e d a t t h e s e f r e q u e n c i e s , i . e . Z. E m becomes much l a r g e r than E, the t o t a l o b s e r v e d energy at the s t a t i o n s ) . The d e v i a t i o n D, d e f i n e d i n Eqs. (8.15) - (8.17) as the r a t i o of t h e r e s i d u a l a f t e r t h e f i t t o t h e t o t a l o b s e r v e d energy, i s s m a l l when the f i t i s good. From the d e v i a t i o n v a l u e s i n T a b l e 11.2, we see t h a t the f i t i s e x c e l l e n t a t 0.13 c p d , good a t 0.30 cpd, f a i r a t 0.43 and 0.53 c p d . The f i t s a t 0.13, 0.30 and 0.43 cpd r e v e a l t h a t a t low f r e q u e n c i e s the c r o s s - s h e l f s t r u c t u r e i s dominated by the second mode s h e l f wave. 1 T h i s i s i n agreement w i t h t h e a l o n g s h o r e phase l a g measurements mentioned e a r l i e r , which a l s o i n d i c a t e the pre s e n c e of a second mode wave. I t i s a p p r o p r i a t e t o emphasize here t h a t t h i s i s the f i r s t t i m e a s t r o n g s i g n a l ( a t 0.13 cpd) has been unambiguously i d e n t i f i e d as a s h e l f wave from 1 The d e v i a t i o n D i n c r e a s e s w i t h the f r e q u e n c y , i n d i c a t i n g a de c r e a s e i n the s i g n a l t o n o i s e r a t i o as the f r e q u e n c y i n c r e a s e s . (11.3) Chap. 11 OREGON SHELF: 1973 99 a l o n g s h o r e and c r o s s - s h e l f o b s e r v a t i o n s . To check the r e l i a b i l i t y of the f i t a t 0.13 cpd, 6 t r i a l s u s i n g random d a t a were performed. The a m p l i t u d e s A and C were u n i f o r m l y d i s t r i b u t e d between 0 and 1, and the phases, between 0 and 360 . The 6 t r i a l s y i e l d a mean "value of 0.55 f o r D, w i t h a s t a n d a r d e r r o r of ±0.11. F i n a l l y , l e t us r e t u r n t o F i g . 11.5, where we have noted the i n t e r e s t i n g phenomenon t h a t the energy f a l l o f f from the shore i s much more r a p i d a t low f r e q u e n c i e s . T h i s can be e x p l a i n e d as f o l l o w s : At h i g h e r f r e q u e n c i e s , o n l y the f i r s t mode can be e x c i t e d , whereas a t lower f r e q u e n c i e s , h i g h e r modes can a l s o be p r e s e n t . I n g e n e r a l , t h e h i g h e r t h e mode number, the f a s t e r the f a l l o f f . At low f r e q u e n c i e s , we have found t h a t the second mode dominates, hence the f a s t e r f a l l o f f r a t e , as obs e r v e d . T h i s c h a p t e r has demonstrated t h e use of r o t a r y s p e c t r a l a n a l y s i s and c r o s s - s h e l f modal f i t t i n g t o d e t e c t s h e l f waves. Emphasis has been p l a c e d on e x p l o r i n g the c r o s s - s h e l f d i m e n s i o n r a t h e r than the a l o n g s h o r e d i m e n s i o n . The a n a l y s i s i n d i c a t e s the g e n e r a l p i c t u r e i n CUE-2 t o be s i m i l i a r t o t h a t i n CUE-1 a low f r e q u e n c y regime 0.4 cpd) dominated by win d - g e n e r a t e d waves, and a h i g h e r f r e q u e n c y regime where s h o r t s h e l f waves c o n s i s t e n t w i t h the resonant t r i a d t h e o r y a r e found. However, u n l i k e CUE-1, the low fr e q u e n c y waves i n CUE-2 a r e of second mode ( i n s t e a d of f i r s t mode). The c r o s s - s h e l f modal f i t t i n g c o n v i n c i n g l y shows the win d - g e n e r a t e d wave motion a t 0.13 cpd t o Chap. 11 OREGON SHELF: 197 3 100 be a f r e e , second mode c o n t i n e n t a l s h e l f wave. CHAPTER 12. OREGON SHELF: WINTER AND SPRING, 1975 101 The WISP experiment was c a r r i e d out on the Oregon s h e l f d u r i n g the w i n t e r and s p r i n g of 1975. F i g . 12.1 and F i g . 12.2, r e p r o d u c e d from the d a t a r e p o r t by G i l b e r t et a l . (1976), show the c u r r e n t meter a r r a y and the l e n g t h of d a t a r e c o r d s . The s i t e of the a r r a y l a y r o u g h l y h a l f w a y between th o s e i n CUE-1 and CUE-2. The c r o s s - s h e l f topography a l o n g the l i n e of moorings i s shown i n F i g . 12.3, and the n u m e r i c a l l y i n t e g r a t e d s h e l f wave d i s p e r s i o n diagram i s shown i n F i g . 12.4. The d i s p e r s i o n diagram does seem t o f i t somewhere i n between the C&S's d i s p e r s i o n diagram f o r topography near CUE-1 and F i g . 11.4 f o r topography near CUE-2. F i g . 12.5 shows the a u t o s p e c t r a f o r the 25 m c u r r e n t s a t S u n f l o w e r and a t W i s t e r i a . Here, the d e c r e a s e i n energy f o r the deep water s t a t i o n i s no more r a p i d a t low f r e q u e n c i e s than a t h i g h f r e q u e n c i e s - - i n s h a r p c o n t r a s t t o the CUE-2 s i t u a t i o n d e monstrated i n F i g . 11.5. The phenomenon can be examined more q u a n t i t a t i v e l y . C a r n a t i o n i n CUE-2 and S u n f l o w e r i n WISP are b o t h i n 100 m deep water, w h i l e E d e l w e i s s (CUE-2) and W i s t e r i a (WISP) a r e i n 200 m deep wate r . The r a t i o s of the a n t i c l o c k w i s e , c l o c k w i s e and t o t a l e n e r g i e s between the s h a l l o w water s t a t i o n and the deep water s t a t i o n a t low f r e q u e n c y (0.13 cpd) a r e shown i n T a b l e 12.1. The t h e o r e t i c a l f a l l o f f r a t i o s f o r the f i r s t and second mode s h e l f waves a r e a l s o l i s t e d i n the l a s t two columns. Note t h e v e r y good agreement between the f a l l o f f r a t e s o b s e r v e d d u r i n g CUE-2 and t h o s e p r e d i c t e d by a second mode wave. D u r i n g WISP, the much slowe r f a l l o f f r a t e s a t 102 125° 3 0 ' I25" 124 ° 3 0 ' 124* F i g . 12.1 Current meter array i n the WISP experiment. 103 LONGITUDE 10' 12 5" W 50' 40' 30' 20' 10' 124° W 1 1 1 1 1 1 1 1 1 1 1 1 1 1 » 1 I / i • I-1 1 < y p . 1 1 W i 1 1 1 1 /• • ' . 1 1 45°00 'N i i 80 60 40 20 KILOMETERS 100 200 300 400 500 Wisteria Sunflower Pikoke Hydro Sections 28 25m I— 50 I— 100 I— 150 I— 200 I— 25ml-50 l-75 I-90 I-25mt-50 t-28-29 4-5 • s,e,T,c,p-• S,G,T,P • s,e,T,c,p • s,e,T,c — • s,e,T,c — JAM FEB S,6,T,P,C s,e,T,p,c 26 3-4 M 19 1-2 H 17-IS H MAR 1975 APR 16 —I 18-20 H MAY F i g . 12.2 Current meter array and length of data records i n WISP. 104 Fig. 12.3 Cross-shelf topography in the WISP experiment. 105 CO o 0 . 0 2 . 0 4 . 0 6 . 0 6 . 0 10.0 WflVENUMBER K F i g . 12.4 Barotropic shelf wave dispersion diagram f o r WISP. 106 F i g . 12.5 Autospectra of the 25 m currents at Sunflower and at Wisteria (dashed l i n e d ) . Chap. 12 OREGON SHELF: 1975 107 Tab l e 12.1 Comparison of energy f a l l o f f between s h a l l o w water s t a t i o n and deep water s t a t i o n a t low f r e q u e n c y (0.13 cpd) between CUE-2 and WISP. 1 C a r n . / E d e l . S u n f . / W i s t . mode 1 mode 2 A a 10. 2.6 1.6 8.0 C 2 5.2 3.6 1.5 4.7 A* + C 1 6.7 3.1 1.5 5.7 low f r e q u e n c i e s suggest t h a t the f i r s t mode i s p r o b a b l y dominant over the second mode. A s i m i l a r c o n c l u s i o n i s reached upon exa m i n i n g the r a t i o s of c l o c k w i s e t o a n t i c l o c k w i s e energy at thes e s t a t i o n s , as demonstrated i n T a b l e 12.2. Tab l e 12.2 R a t i o s of c l o c k w i s e t o a n t i c l o c k w i s e energy (C^/A 1) a t the s t a t i o n s i n CUE-2 ( C a r n a t i o n and E d e l w e i s s ) and i n WISP (S u n f l o w e r and W i s t e r i a ) . C V A " water depth C U E - 2 WISP mode 1 mode 2 100 m 1.30 1.11 1.06 1.23 200 m 2.42 0.88 1.15 2.12 The 25 m currents are used at Sunflower and Wisteria, 40 m at Carnation and 80 m at Edelweiss. (The rapid f a l l o f f between Carnation and Edelweiss at low frequency is not due to the currents being taken from different depths, as Fig. 11.5 showing the 40 m currents at Carnation and Forsythia can verify). Chap. 12 OREGON SHELF: 1975 108 F i g . 12.6 shows the i n n e r c r o s s - s p e c t r u m between the 25 m c u r r e n t a t P i k a k e and the a d j u s t e d sea l e v e l a t Newport. The c oherency i s g e n e r a l l y h i g h , w i t h broad peaks at 0.12, 0.30, and 0.50 cpd. The phase i s c e n t e r e d around 90° . At v e r y low f r e q u e n c i e s {< 0.1 c p d ) , the i n n e r - p h a s e %. < ; Eq. (10.2) then i m p l i e s a n e g a t i v e Sw and southward p r o p a g a t i o n , ( o p p o s i t e t o the d i r e c t i o n of f r e e s h e l f waves), w i t h a phase speed of around 115 km/day. At f r e q u e n c i e s above 0.1 cpd, the phase d i s p l a y s a g e n e r a l l i n e a r t i l t t h a t i n d i c a t e s n o n d i s p e r s i v e 2 n o r t h w a r d p r o p a g a t i o n a t a speed of around 230 km/day. For k < 2, the f i r s t mode has a n o n d i s p e r s i v e phase speed of 250-300 km/day, w h i l e the second mode has a speed of ^120 km/day. The s i t u a t i o n i s not u n l i k e t h a t found by Mooers and Smith (1968) on the Oregon s h e l f d u r i n g the summer of 1933, where a 0.1 cpd wave moved southward w i t h the a t m o s p h e r i c p r e s s u r e system, w h i l e a 0.35 cpd wave p r o p a g a t e d northward as a f r e e c o n t i n e n t a l s h e l f wave. G i l l and Schumann (1974) have dem o n s t r a t e d t h e o r e t i c a l l y t h a t the wind may g e n e r a t e e i t h e r f r e e s h e l f waves or f o r c e d waves t h a t move w i t h the wind. The i n n e r c r o s s spectrum between the 25 m c u r r e n t a t P i k a k e and Newport wind i s shown i n F i g . 12.7. The c l o c k w i s e p a r t of the spectrum has much h i g h e r coherency v a l u e s , and f o r which the wind l e a d s the c u r r e n t , w i t h the phase d i f f e r e n c e i n c r e a s i n g 2 A l o n g s h o r e phase l a g »c X cx: k. O b s e r v i n g a l i n e a r r e l a t i o n between phase and f r e q u e n c y i m p l i e s k << u , or c = ° / k i s a c o n s t a n t . Hence n o n d i s p e r s i v e waves. 109 ~] 1 1 1 1 1 1 1 I I I I - 0 . 6 - 0 . 6 - 0 . 4 - 0 . 2 0 . 0 0 . 2 0 . 4 FREQUENCY (CPD) 0 . 6 0 . 8 F i g . 12.6 Inner cross-spectrum between the 25 m current at Pikake and the Newport adjusted sea l e v e l . 110 F i g . 12.7 Inner cross-spectrum between the 25 m current at Pikake and the Newport wind. Chap. 12 OREGON SHELF: 1975 111 6 O from about 0 a t low f r e q u e n c i e s t o 90 a t h i g h f r e q u e n c i e s . Both the c u r r e n t and the wind have g e n e r a l l y more c l o c k w i s e energy, as shown i n F i g . 12.8. In g e n e r a l , the a u t o s p e c t r a show t h a t the w i n t e r and s p r i n g winds and c u r r e n t s have r o u g h l y 2-10 t i m e s as much energy (over a l l f r e q u e n c i e s ) as t h e i r c o u n t e r p a r t s have i n summer. In summary, the h i g h coherency between wind and c u r r e n t s u g g e s t s a l l the main peaks o b s e r v e d i n WISP were wind g e n e r a t e d . At f r e q u e n c i e s above 0.1 cpd, the a l o n g s h o r e phase l a g and the c r o s s - s h e l f f a l l o f f r a t e r e v e a l the p r e s ence of n o n d i s p e r s i v e f i r s t mode s h e l f waves. At f r e q u e n c i e s below 0.1 cpd, t h e r e appeared t o be a f o r c e d wave p r o p a g a t i n g southward. The s h o r t s h e l f waves from t r i a d i n t e r a c t i o n s , i f p r e s e n t , seems t o be masked by the v e r y s t r o n g wind i n f l u e n c e d u r i n g the w i n t e r and s p r i n g s e a s o n s . 112 Fig. 12.8 Autospectra of the 25 m current at Pikake and the Newport wind.(dashed line ) . 2 — 2 . * ~ i 2 2 " " ' (Autospectra in units of cm sec" cpd for current and m sec" cpd for wind). 113 CHAPTER 13. DISCUSSION In t h i s c h a p t e r , we d i s c u s s the problems and l i m i t a t i o n s a s s o c i a t e d w i t h the t h e o r y i n P a r t I and w i t h the i n t e r p e t a t i o n of the d a t a i n P a r t I I . The l i m i t a t i o n s of the t h e o r y a r e : no d e n s i t y s t r a t i f i c a t i o n , no bottom f r i c t i o n or i r r e g u l a r i t i e s , no a l o n g s h o r e v a r i a t i o n s i n the s h e l f w i d t h , and no a t m o s p h e r i c f o r e i n g . W h i l e C&S r e p o r t e d b a r o t r o p i c ( i . e . depth independent) motion i n t h e i r measurements, HHSSP found s i g n i f i c a n t b a r o c l i n i c ( i . e . depth dependent) components i n some of t h e i r s i g n a l s , i n d i c a t i n g d e n s i t y s t r a t i f i c a t i o n may be i m p o r t a n t . F u r t h e r m o r e , the phase speed of 356 km/day f o r the 0.22 cpd s i g n a l i n C&S i s i n e x c e l l e n t agreement w i t h the t h e o r e t i c a l phase speed of a l o w e s t mode b a r o t r o p i c s h e l f wave, w h i l e i n c o n t r a s t , the phase speeds i n HHSSP seem s u b s t a n t i a l l y h i g h e r than the t h e o r e t i c a l phase speeds f o r b a r o t r o p i c s h e l f waves. S t r a t i f i c a t i o n i s known t o modify, the d i s p e r s i o n c u r v e s r e s u l t i n g i n h i g h e r phase speeds. (See Mysak, 1980, S e c . 3 ) . Bottom f r i c t i o n and i r r e g u l a r i t i e s i n bottom topography (see A l l e n , 1980, Sec.4) have a l s o been n e g l e c t e d i n the t h e o r y , but c o u l d be i m p o r t a n t f a c t o r s . Due t o t h e i r s h o r t l e n g t h s c a l e s , the s h o r t waves g e n e r a t e d by the r e s o n a n t t r i a d i n t e r a c t i o n may be p a r t i c u l a r l y s e n s i t i v e t o t o p o g r a p h i c i r r e g u l a r i t i e s and f r i c t i o n . The a c t u a l b a l a n c e between f r i c t i o n a l d i s s i p a t i o n and n o n l i n e a r energy t r a n s f e r r e q u i r e s f u r t h e r i n v e s t i g a t i o n . We have f o c u s e d p r i m a r i l y on r e s o n a n t t r i a d s i n v o l v i n g Chap. 13 DISCUSSION 114 waves of the l o w e s t p o s s i b l e modes. Resonant i n t e r a c t i o n s i n v o l v i n g h i g h e r modes a r e a l s o p o s s i b l e . Bottom f r i c t i o n and i r r e g u l a r i t i e s may d e c i d e which of t h e s e t r i a d s would be e x c i t e d more r e a d i l y than o t h e r s . F u r t h e r m o r e , the two secondary members of the t r i a d e x t r a c t i n g energy from the dominant member may i n t u r n i n t e r a c t r e s o n a n t l y w i t h o t h e r waves, forming a d d i t i o n a l t r i a d s , and t h i s p r o c e s s may be r e p e a t e d . The a l o n g s h o r e v a r i a t i o n i n the s h e l f w i d t h cannot be o v e r l o o k e d . The changes i n the w i d t h and shape of the c o n t i n e n t a l s h e l f a l t e r the d i s p e r s i o n c u r v e s , e s p e c i a l l y i n the s h o r t wave ( l a r g e k) regime. Thus, the s h o r t waves would be s c a t t e r e d r e a d i l y . A t m o s p h e r i c f o r c i n g has a l s o been o m i t t e d i n the t h e o r y . However, i t must be emphasized t h a t when a p p l y i n g the t r i a d t h e o r y t o the Oregon s h e l f , i t i s i m p l i c i t l y assumed t h a t the dominant member of the t r i a d i s g e n e r a t e d by (and r e c e i v i n g energy from) the a t m o s p h e r i c system. Without t h i s i n p u t of energy, a l l the waves w i l l e v e n t u a l l y be damped out by f r i c t i o n . The works of Adams and Buchwald (1969) and G i l l and Schumann (1974) have d e a l t w i t h the g e n e r a t i o n of s h e l f waves by the wind. T h i s t r i a d t h e o r y i s f o r i n t e r a c t i o n s between t h r e e s h e l f waves of d i s c r e t e f r e q u e n c i e s and wavenumbers. An a l t e r n a t i v e a pproach i s t o s t u d y s h e l f wave i n t e r a c t i o n s t h a t o ccur c o n t i n u o u s l y over a broad band i n cj-k space ( e . g . , see Hasselmann, 1968). The continuum approach i n g e n e r a l y i e l d s s m a l l e r growth r a t e s than the d i s c r e t e a p proach. S i n c e d i s t i n c t Chap. 13 DISCUSSION 115 peaks a r e a c t u a l l y p r e s e n t i n the o b s e r v e d s p e c t r a , the d i s c r e t e approach employed i n t h i s t h e s i s i s p r o b a b l y adequate. The i r r e g u l a r topography and the s t r o n g wind i n f l u e n c e on the Oregon s h e l f c o n s i d e r a b l y c o m p l i c a t e d the t a s k of i n t e r p r e t i n g the o b s e r v a t i o n s i n P a r t I I of t h i s t h e s i s . The i r r e g u l a r topography i n t r o d u c e s not o n l y r e f r a c t i o n e f f e c t s (such as t o p o g r a p h i c phase s h i f t ) t o the o r i g i n a l wave, but a l s o g e n e r a t e s new s c a t t e r e d waves. (See Mysak, 1980, S e c . 6 ) . However, the s c a t t e r e d waves a r e a l l of the same f r e q u e n c y as the o r i g i n a l wave. S c a t t e r i n g of a l o n g s h e l f wave by topography o n l y t r a n s f e r s energy t o s h o r t e r w a v e l e n g t h s , but not t o o t h e r f r e q u e n c i e s . In s h a r p c o n t r a s t , the resonant t r i a d i n t e r a c t i o n t r a n s f e r s energy t o waves of d i f f e r e n t w avelengths and d i f f e r e n t f r e q u e n c i e s . As t h e wind a u t o s p e c t r a r e v e a l much more energy a t low f r e q u e n c i e s , and the c u r r e n t s a r e g e n e r a l l y c o h e r e n t w i t h the wind a t f r e q u e n c i e s £ 0.4 cpd, i t appears c e r t a i n t h a t the wind t r a n s f e r s energy t o the c u r r e n t s a t low f r e q u e n c i e s . The wi n d - g e n e r a t e d d i s t u r b a n c e s may pro p a g a t e e i t h e r as f r e e s h e l f waves or as f o r c e d waves ( G i l l and Schumann, 1974). Some r e s e a r c h e r s b e l i e v e the d i s t u r b a n c e s propagate as f o r c e d waves r a t h e r than f r e e s h e l f waves as a consequence of the s t r o n g wind i n f l u e n c e . However, the i n v e s t i g a t i o n i n t o the c r o s s - s h e l f c u r r e n t s t r u c t u r e and energy f a l l o f f i n t h i s t h e s i s shows t h a t except f o r some i s o l a t e d f r e q u e n c y r e g i o n s , 1 the c o n t i n e n t a l s h e l f wave model f i t s v e r y w e l l t o the o b s e r v e d wave motion a t Chap. 13 DISCUSSION 116 low f r e q u e n c y , i n the p r e s e n c e of s t r o n g wind. The s h o r t s h e l f waves p r e d i c t e d from the resonant t r i a d t h e o r y a r e indeed o b s e r v e d i n both CUE-1 and CUE-2. U n f o r t u n a t e l y , the coherency w i t h the wind a t t h e s e f r e q u e n c i e s a r e not n e g l i g i b l e , and t h e r e i s i n s u f f i c i e n t d a t a t o r u l e out the p o s s i b i l i t y t h a t t h e s e s h o r t waves might a l s o be wind g e n e r a t e d . One can, however, e x c l u d e the p o s s i b i l i t y of t h e s e s h o r t waves b e i n g produced from the l o w - f r e q u e n c y l o n g waves by t o p o g r a p h i c s c a t t e r i n g p r o c e s s e s , because they a r e found i n a h i g h e r f r e q u e n c y regime, w h i l e s c a t t e r e d waves must have the same f r e q u e n c y as the o r i g i n a l wave. One more c r i t i c i s m c o n c e r n s the f a c t t h a t the t r i a d t h e o r y p r e d i c t s two s h o r t s h e l f waves, whereas o n l y the h i g h e r f r e q u e n c y s h o r t wave has been o b s e r v e d . The problem w i t h d e t e c t i n g the second s h o r t wave i s t h a t i t has a f r e q u e n c y <0.4 cpd, and would be masked by the wind g e n e r a t e d waves. Resonant t r i a d s i n v o l v i n g h i g h e r modes would a l s o produce s h o r t waves a t lower f r e q u e n c i e s , making them d i f f i c u l t t o d e t e c t . F i n a l l y , l e t us r e t u r n t o the z e r o group v e l o c i t y resonance mechanism proposed by Buchwald and Adams (1968) where energy i s supposed t o accumulate a t the z e r o s l o p e r e g i o n s of the d i s p e r s i o n c u r v e s . As the c o n t i n e n t a l s h e l f p r o f i l e and the c o r r e s p o n d i n g d i s p e r s i o n diagram do not change r a p i d l y w i t h t i m e , t h i s mechanism p r e d i c t s a v e r y s t a t i c s i t u a t i o n on the 1 The v e r y low f r e q u e n c y motion (< 0.1 cpd) i n WISP seemed t o p r o p a g a t e southward. Chap. 13 DISCUSSION 117 s h e l f . But the o b s e r v a t i o n s show a time v a r y i n g b e h a v i o u r on the s h e l f . For i n s t a n c e , C&S found s i g n a l s a t 0.22, 0.40 and 0.65 cpd i n 1968, w h i l e a t about the same l o c a t i o n , HHSSP found s i g n a l s a t 0.15, 0.27, 0.42 and 0.55 cpd i n 1972. In the summer of 1972, the low fr e q u e n c y motion p r o p a g a t e s a t the f i r s t mode n o n d i s p e r s i v e phase speed (Kundu and A l l e n , 1976), w h i l e i n the f o l l o w i n g summer, i t p r o p a g a t e s a t the second mode phase speed. The time v a r y i n g b e h a v i o u r on the Oregon s h e l f i s i n c o m p a t i b l e w i t h the v e r y s t a t i c p i c t u r e p o r t r a i t e d by the z e r o group v e l o c i t y resonance mechanism. CHAPTER 14. SUMMARY AND CONCLUSION 118 In P a r t I , s t a r t i n g from the i n v i s c i d , u n f o r c e d long-wave e q u a t i o n s f o r a r o t a t i n g , homogeneous f l u i d , i t i s shown t h a t r e s o n a n t i n t e r a c t i o n s between t h r e e c o n t i n e n t a l s h e l f waves can o c c u r . The e q u a t i o n s g o v e r n i n g the a m p l i t u d e and the energy of i n d i v i d u a l waves i n a r e s o n a n t t r i a d a r e d e r i v e d . The energy e q u a t i o n r e v e a l s t h a t energy i s t r a n s f e r r e d between the waves by the n o n l i n e a r terms, but w i t h the t o t a l energy c o n s e r v e d . Upon n e g l e c t i n g the a l o n g s h o r e d e r i v a t i v e d/3Y i n the a m p l i t u d e e q u a t i o n s , the wave a m p l i t u d e s a r e g i v e n by e l l i p t i c f u n c t i o n s i n t h e case of maximum energy exchange. The t h e o r y i s then a p p l i e d t o the f a m i l i a r e x p o n e n t i a l s h e l f p r o f i l e , where the c o u p l i n g c o e f f i c i e n t s a r e o b t a i n e d a n a l y t i c a l l y . P a r t I I b e g i n s w i t h an e x a m i n a t i o n of the d i s p e r s i o n diagram and the c u r r e n t e l l i p s e s a s s o c i a t e d w i t h s h e l f waves, f o l l o w e d by two d e t e c t i o n t e c h n i q u e s — r o t a r y s p e c t r a l a n a l y s i s and c r o s s - s h e l f modal f i t t i n g . The resonant t r i a d t h e o r y i s then used t o i n t e r p r e t the o b s e r v a t i o n s made on the Oregon s h e l f by C u t c h i n and Smith (1973), where a dominant s i g n a l a t 0.22 cpd, and two secondary ones a t 0.40 and 0.65 cpd were found. For i n t e r a c t i o n s i n v o l v i n g waves of the l o w e s t p o s s i b l e modes, a 0.22 cpd f i r s t mode wave can, i n t h e o r y , i n t e r a c t r e s o n a n t l y w i t h (and t r a n s f e r energy t o ) two s h o r t s h e l f waves of f r e q u e n c i e s 0.40 and 0.62 c p d — i n e x c e l l e n t agreement w i t h the o b s e r v e d f r e q u e n c i e s . Oregon s h e l f d a t a from CUE-1 (summer, 1972), CUE-2 (summer, 1973) and WISP ( w i n t e r and s p r i n g , 1975) a r e a l s o a n a l y z e d . For Chap. 14 SUMMARY AND CONCLUSION 119 the summer months, the wind i n f l u e n c e on the c u r r e n t s i s s t r o n g a t f r e q u e n c i e s £ 0.4 cpd, w h i l e i n the w i n t e r and s p r i n g s e a s o n s , the wind i n f l u e n c e dominates t h r o u g h o u t . In a d d i t i o n t o a l o n g s h o r e phase l a g s , an e x a m i n a t i o n of the c r o s s - s h e l f c u r r e n t s t r u c t u r e and the energy f a l l o f f r a t e s r e v e a l t h a t e x c e p t f o r c e r t a i n i s o l a t e d f r e q u e n c y r e g i o n s , the wi n d - g e n e r a t e d s i g n a l s indeed p r o p a g a t e as f r e e c o n t i n e n t a l s h e l f waves. Other i n t e r e s t i n g f e a t u r e s such as the o r i e n t a t i o n of the major axes i n s h e l f wave c u r r e n t e l l i p s e s and the t o p o g r a p h i c i n d u c e d phase s h i f t s have a l s o been o b s e r v e d . The s h o r t waves (100-150 km wavelength) p r e d i c t e d by the reso n a n t t r i a d t h e o r y have been found i n the h i g h e r f r e q u e n c y regime i n b o t h the CUE-1 and CUE-2 d a t a . At lower f r e q u e n c y and i n w i n t e r , the s h o r t waves a r e not o b s e r v e d , as they would be masked by the s t r o n g e r w i n d - g e n e r a t e d waves. In c o n c l u s i o n , the resonant t r i a d mechanism appears t o be im p o r t a n t on the Oregon s h e l f d u r i n g summer. Energy i s t r a n s f e r r e d from the low fr e q u e n c y regime (< 0.4 c p d ) , where t h e r e i s much i n p u t of energy from the wind, t o the h i g h e r f r e q u e n c y regime where t h e r e i s l i t t l e wind i n p u t . The low fre q u e n c y regime i s dominated by l o n g waves, w h i l e the h i g h e r f r e q u e n c y regime, by s h o r t waves. U n l i k e the l o n g s h e l f waves, the s h o r t waves have group v e l o c i t i e s o p p o s i t e l y d i r e c t e d t o t h e i r phase v e l o c i t i e s . Hence, the d i r e c t i o n of energy p r o p a g a t i o n i n the two frequ e n c y regimes may a l s o be d i f f e r e n t -n o r thward a t low f r e q u e n c i e s and southward a t h i g h e r f r e q u e n c i e s . Chap. 14 SUMMARY AND CONCLUSION 120 The n o n l i n e a r r e s o n a n t t r i a d i n t e r a c t i o n mechanism i s not u nique as an energy t r a n s f e r mechanism on t h e c o n t i n e n t a l s h e l f . But i t does have c e r t a i n p r o p e r t i e s t h a t s t a n d out among competing t h e o r i e s : ( i ) I t i s c a p a b l e of t r a n s f e r r i n g energy t o d i f f e r e n t w a v elengths and d i f f e r e n t f r e q u e n c i e s , w h i l e t o p o g r a p h i c s c a t t e r i n g t h e o r i e s are i n c a p a b l e of t r a n s f e r r i n g energy t o d i f f e r e n t f r e q u e n c i e s . ( i i ) I t can a c c e p t c h a n g i n g s c e n a r i o s on the c o n t i n e n t a l s h e l f from one y e a r t o a n o ther (or one season t o a n o t h e r ) , w h i l e a v e r y s t a t i c mechanism, l i k e the z e r o group v e l o c i t y resonance, c a n n o t . The s h o r t s h e l f waves, due t o t h e i r s h o r t e r l e n g t h s c a l e s , may i n t e r a c t much more r e a d i l y w i t h s m a l l s c a l e f e a t u r e s and e v e n t s on the c o n t i n e n t a l s h e l f than the l o n g waves. The l e n g t h s c a l e of the 100-150 km s h o r t waves i s 1/k = *-/rn ~ 20 km. In a d d i t i o n t o t o p o g r a p h i c i r r e g u l a r i t i e s , the s h o r t waves may i n t e r a c t w i t h c o a s t a l u p w e l l i n g , (though t h i s would r e q u i r e the i n c o r p o r a t i o n of d e n s i t y s t r a t i f i c a t i o n i n t o the t h e o r y ) . Thus, the s h o r t s h e l f waves may have an a c t i v e r o l e i n c o n t i n e n t a l s h e l f dynamics. F i n a l l y , t h i s t h e s i s d e m o n s t r a t e s t h a t the c r o s s - s h e l f d i m e n s i o n of c o n t i n e n t a l s h e l f waves c o n t a i n s a m a z i n g l y r i c h s t r u c t u r e s — a l t e r n a t i n g zones of dominant c l o c k w i s e and a n t i c l o c k w i s e m o t i o n , modal dependent energy f a l l o f f r a t e s , v a r y i n g shapes and o r i e n t a t i o n s of t h e c u r r e n t e l l i p s e s — a l l of which can be p r o f i t a b l y e x p l o i t e d by the o b s e r v a t i o n a l i s t t o Chap. 14 SUMMARY AND CONCLUSION 121 complement h i s a l o n g s h o r e coherency and phase measurements. Perhaps the e x c e l l e n t agreement found between the c r o s s - s h e l f modal f i t t i n g and t h e a l o n g s h o r e phase measurement f o r t h e dominant 0.13 cpd s i g n a l i n CUE-2 w i l l succeed i n c o n v e r t i n g the l a s t s k e p t i c s on the e x i s t e n c e of c o n t i n e n t a l s h e l f waves. 122 BIBLIOGRAPHY Abramowitz, M., and I . A. Stegun, Ed., 1965: Handbook of  M a t h e m a t i c a l F u n c t i o n s . Dover, 1046 pp. Adams, J . K., and V. T. Buchwald, 1969: The g e n e r a t i o n of c o n t i n e n t a l s h e l f waves. F l u i d Mech., 35, 815-826. A l l e n , J . S., 1980: Models of w i n d - d r i v e n c u r r e n t s on the c o n t i n e n t a l s h e l f . Annual Review of F l u i d Mechanics, V o l . 1 2 , Annual Reviews, I n c . , 389-433. B a r t o n , N. G., 1977: Resonant i n t e r a c t i o n s of s h e l f waves w i t h w i n d - g e n e r a t e d e f f e c t s . Geophys. A s t r o p h y s . F l u i d  Dyn., 9, 101-114. Bendat, J . S., and A. G. P i e r s o l , 1971. Random Data: A n a l y s i s  and Measurement P r e c e d u r e s . W i l e y , 407 pp. Boyce, W. E., and R. C. D i P r i m a , 1969: E l e m e n t a r y D i f f e r e n t i a l  E q u a t i o n s and Boundary V a l u e Problems, 2nd ed. W i l e y , 533 pp. B r i n k , K. H., and J . S. A l l e n , 1978: On the e f f e c t of bottom f r i c t i o n on b a r o t r o p i c motion over the c o n t i n e n t a l s h e l f . Phys. Oceanogr., 8, 919-922. B r i n k , K. H., J . S. A l l e n , and R. L. S m i t h , 1978: A study of l o w - f r e q u e n c y f l u c t u a t i o n s near the Peru c o a s t . J .  Phys. Oceanogr., 8,1025-1041. Buchwald, V. T., and J . K. Adams, 1968: The p r o p a g a t i o n of c o n t i n e n t a l s h e l f waves. P r o c . R. Soc. Lond., A305, 235-250. C a l d w e l l , D. R., and M. S. L o n g u e t - H i g g i n s , 1971: The e x p e r i m e n t a l g e n e r a t i o n of double K e l v i n waves. P r o c .  Roy. Soc. London, A326, 39-52. C u t c h i n , D. L., and R. L. S m i t h , 1973: C o n t i n e n t a l s h e l f waves: Low f r e q u e n c y v a r i a t i o n s i n sea l e v e l and c u r r e n t s over the Oregon c o n t i n e n t a l s h e l f . Phys. Oceanogr. , 3, 73-82. Foreman, M. G. G., 1977: Manual f o r T i d a l H e i g h t s A n a l y s i s and P r e d i c t i o n . P a c i f i c M a r i n e S c i e n c e Report 77-10. I n s t i t u t e of Ocean S c i e n c e s , P a t r i c i a Bay, V i c t o r i a , B. C. 97 pp. " U n p u b l i s h e d m a n u s c r i p t " . G i l b e r t , W. E., A. Huyer, E. D. B a r t o n , and R. L. S m i t h , 1976: P h y s i c a l Oceanographyic O b s e r v a t i o n s o f f the Oregon  C o a s t , 1975: WISP and UP-75. Ref. 76-4, Oregon S t a t e U n i v e r s i t y S c h o o l of Oceanography, C o r v a l l i s . 189pp. BIBLIOGRAPHY 12 3 G i l l , A. E., and E. H. Schumann, 1974: The g e n e r a t i o n of l o n g s h e l f waves by the wind. Phys. Oceanoqr., 4, 83-90. G o n e l l a , J . , 1972: A rotary-component method f o r a n a l y z i n g m e t e o r o l o g i c a l and o c e a n o g r a p h i c v e c t o r time s e r i e s . Deep-Sea Res., 19, 833-846. Grimshaw, R., 1977a: N o n l i n e a r a s p e c t s of l o n g s h e l f waves. Geophys. A s t r o p h y s . F l u i d Dyn., 8, 3-16. Grimshaw, R., 1977b: The s t a b i l i t y of c o n t i n e n t a l s h e l f waves. I . S i d e band i n s t a b i l i t y and l o n g wave resonance. J . A u s t r a l . Math. Soc. , S e r . B, 2_0, 13-30. Groves, G. W. And E. J . Hannan, 1968: Time s e r i e s r e g r e s s i o n of sea l e v e l on weather. Rev. Geophys., 6, 129-174. Hamon, B. V., 1962: The spectrums of mean sea l e v e l a t Sydney, C o f f ' s Harbour, and L o r d Howe I s l a n d . J_;_ Geophys.  Res. , 67, 5147-5155. ( C o r r e c t i o n , J_j_ Geophys. Res. , 68, 4635, 1963). Hasselmann, K., 1968: Weak i n t e r a c t i o n t h e o r y of ocean waves. B a s i c Developments i n F l u i d Dynamics. V o l . 2 , M. H o l t , Ed., Academic P r e s s , 117-182. Hasselmann, K., T. P. B a r n e t t , E. Bouws, H. C a r l s o n , D. E. C a r t w r i g h t , K. Enke, J . A. Ewing, H. Gienapp, D. E. Hasselmann, P. Kruseman, A. Meerburg, P. M u l l e r , D. J . O l b e r s , K. R i c h t e r , W. S e l l and H. Walden, 1973: Measurement's of wind-wave growth and s w e l l decay d u r i n g the j o i n t N o r t h Sea wave p r o j e c t (JONSWAP). D t s c h . Hydrogr. 2, A 8 ( S u p p l . ) H s i e h , W. W., and L. A. Mysak, 1980: Resonant i n t e r a c t i o n s between s h e l f waves, w i t h a p p l i c a t i o n s t o the Oregon s h e l f . J_j_ Phys. Oceanoqr. , 10, 1729-1741. Huyer, A. B., B. M. H i c k e y , J . D. Sm i t h , R. L. Smith and R. D. P i l l s b u r y , 1975: A l o n g s h o r e coherence a t low fr e q u e n c y i n c u r r e n t s o b s e r v e d over the c o n t i n e n t a l s h e l f o f f Oregon and Washington. J_j_ Geophys. Res., 80, 3495-3505. Huyer, A., and J . G. P a t u l l o , 1972: A comparison between wind and c u r r e n t o b s e r v a t i o n s over the c o n t i n e n t a l s h e l f o f f Oregon, summer 1969. Geophys. Res., 77, 3215-3200. J e n k i n s , G. M. And D. G. Watts, 1968: S p e c t r a l A n a l y s i s and i t s A p p l i c a t i o n s . Holden-Day, 525 pp. Kaup, D. J . , A. Reiman, and A. B e r s , 1979: Space-time e v o l u t i o n BIBLIOGRAPHY 124 of n o n l i n e a r three-wave i n t e r a c t i o n s . I . I n t e r a c t i o n s i n a homogeneous medium. Rev. Mod. Phys., 51, 275-309. Kundu, P. K.,. and J . S. A l l e n , 1976: Some t h r e e - d i m e n s i o n a l c h a r a c t e r i s t i c s of l o w - f r e q u e n c y c u r r e n t f l u c t u a t i o n s near the Oregon c o a s t . JL Phys. Oceanogr., 6, 181-199. Kundu, P. K., and J . S. A l l e n , 1978.: on the momentum, v e l o c i t y and mass b a l a n c e on the Oregon s h e l f . J ^ Phys.  Oceanogr., 8, 13-27. Kundu, P. K., J . S. A l l e n , and R. L. Sm i t h , 1975: Modal d e c o m p o s i t i o n of the v e l o c i t y f i e l d near the Oregon c o a s t . Phys. Oceanogr., 5, 683-704. L e B l o n d , P. H., and L. A. Mysak, 1978: Waves i n the Ocean. E l s e v i e r , 602 pp. Mooers, C. N. K., 1973: A t e c h n i q u e f o r the c r o s s spectrum a n a l y s i s of p a i r s of co m p l e x - v a l u e d time s e r i e s , w i t h emphasis on p r o p e r t i e s of p o l a r i z e d components and r o t a t i o n a l i n v a r i a n t s . Deep-sea Res., 20, 1129-1141. Mooers, C. N. K., and R. L. S m i t h , 1968: C o n t i n e n t a l s h e l f waves o f f Oregon. Geophys. Res., 73, 549-557. Mysak, L. A., 1980: Recent advances i n s h e l f wave dynamics. Rev.  Geophys. Space Phys., 18, 211-214. P h i l l i p s , O. M., 1977: The Dynamics of the Upper Ocean. 2nd ed. Cambridge U n i v e r s i t y P r e s s , 336 pp. P i l l s b u r y , R. D., J . S. B o t t e r o , R. E. S t i l l , and W. E. G i l b e r t , 1974a: A A C o m p i l a t i o n of O b s e r v a t i o n s from Moored  C u r r e n t meters: V o l . V I ; Oregon C o n t i n e n t a l S h e l f ,  A p r i l - O c t o b e r 1972. Ref. 74-2, Oregon S t a t e U n i v e r s i t y , S c h o o l of Oceanography, C o r v a l l i s . 230 pp. P i l l s b u r y , R. D., J . S. B o t t e r o , R.. E. S t i l l , and W. E. G i l b e r t , 1974b: A A C o m p i l a t i o n of Observat i o n s from Moored  C u r r e n t meters: V o l . V I I ; Oregon C o n t i n e n t a l S h e l f ,  J u l y - A u g u s t 1973. Ref. 74-7, Oregon S t a t e U n i v e r s i t y , S c h o o l of Oceanography, C o r v a l l i s . 87 pp. Smi t h , R., 1972: N o n l i n e a r K e l v i n and c o n t i n e n t a l s h e l f waves. J_j_ F l u i d Mech. , 52, 379-391. Wang, D.-P., 1980: D i f f r a c t i o n of c o n t i n e n t a l s h e l f waves by i r r e g u l a r a l o n g s h o r e geometry. J ^ Phys. Oceanoqr., 10, 1187-1199. 125 APPENDIX A: THE GROUP VELOCITY In Chapter 3, we c l a i m e d t h a t the c<^ which shows up i n the a m p l i t u d e e q u a t i o n (3.16) and d e f i n e d by Eqs. (3.18) and (3.19) i s i n d e e d the group v e l o c i t y of the j t h wave, i . e . l ^ i = ( A . l ) The p r o o f i s as f o l l o w s : F i r s t m u l t i p l y Eq. (3.2) by . r e p l a c e c^ by ^ j / k ^ , and then d i f f e r e n t i a t e the e n t i r e e q u a t i o n w i t h r e s p e c t t o . From t h i s , s u b t r a c t o f f Eq. (3.2) m u l t i p l i e d by 2 9<t>;j /3 kj , y i e l d i n g : Next, we i n t e g r a t e t h i s e q u a t i o n w i t h r e s p e c t t o x from 0 t o » . I n t e g r a t i n g by p a r t s and a p p l y i n g the boundary c o n d i t i o n s ( 3 . 3 ) , the f i r s t two terms c a n c e l , l e a v i n g oo Under the n o r m a l i z a t i o n c o n d i t i o n (3.15) , t h i s reduces t o ( A . l ) w i t h eg d e f i n e d by Eqs. (3.18) and (3.19). APPENDIX B: NUMERICAL TECHNIQUE FOR DISPERSION CURVES 126 T h i s appendix d e s c r i b e s the t e c h n i q u e used t o o b t a i n d i s p e r s i o n diagrams by the second method mentioned i n Chapter 7 i . e . by d i r e c t n u m e r i c a l i n t e g r a t i o n . A comparison i s a l s o made between the d i s p e r s i o n diagrams o b t a i n e d by t h i s method and by the e x p o n e n t i a l p r o f i l e method. B e f o r e one can i n t e g r a t e n u m e r i c a l l y , the d i s c r e t e depth d a t a o b t a i n e d from c o n t o u r c h a r t s must be r e p r e s e n t e d as a f u n c t i o n h ( x ) w i t h c o n t i n u o u s f i r s t d e r i v a t i v e . T h i s i s a c h i e v e d by f i t t i n g c u b i c s p l i n e f u n c t i o n s ( p i e c e w i s e c u b i c p o l y n o m i a l s ) t o the d i s c r e t e d a t a . To o b t a i n the d i s p e r s i o n c u r v e s , one uses a s h o o t i n g method as f o l l o w s : F i r s t f i x k, then guess a t a v a l u e f o r id . S t a r t n u m e r i c a l l y i n t e g r a t i n g the f i r s t o r d e r system (7.8) - (7.9) from the deep ocean r e g i o n where the s o l u t i o n i s g i v e n by ( 7 . 1 2 ) , and pr o c e e d towards the s h o r e . As Eq. (7.8) p r e v e n t s us from i n t e g r a t i n g . r i g h t t o the shore where h = 0, s t o p when h = 2 metres. How w e l l the boundary c o n d i t i o n (7.10) i s s a t i s f i e d i s measured by the f u n c t i o n F ( ^ ) = -ufo + Mo' U s i n g a r o o t - f i n d i n g a l g o r i t h m , one t r i e s v a r i o u s v a l u e s f o r <^  ( w h i l e h o l d i n g k f i x e d ) and r e p e a t e d l y p e r f o r m the above i n t e g r a t i o n , u n t i l one a r r i v e s a t an <o v a l u e which makes F(co) = 0. T h i s p a i r of (k, -£0) v a l u e s r e p r e s e n t s a p o i n t l y i n g on one of the d i s p e r s i o n c u r v e s f o r the s h e l f wave modes (see e.g. F i g . 11 . 4 ) . W i t h 1„(x) and l . f x ) known from the i n t e g r a t i o n , u 0(x) and "^ c(x) a r e o b t a i n e d v i a Eqs. (7.5) and ( 7 . 6 ) . T h i s second method f o r f i n d i n g the d i s p e r s i o n c u r v e s i s 127 c o m p u t a t i o n a l l y more e x p e n s i v e than the f i r s t ( t h e e x p o n e n t i a l p r o f i l e method). When c l o s e t o the shore* the d e pth h becomes s m a l l , and the i n t e g r a t i o n must be c a r r i e d out i n s m a l l s t e p s . I t i s found t h a t t h e r e i s no s i g n i f i c a n t d i f f e r e n c e between the d i s p e r s i o n c u r v e s o b t a i n e d w i t h i n t e g r a t i o n t e r m i n a t e d a t h = 2 m and those o b t a i n e d w i t h s m a l l e r t e r m i n a l d e p t h s . However, i f i n t e g r a t i o n i s t e r m i n a t e d a t g r e a t e r d e p t h , (which i s e q u i v a l e n t t o h a v i n g a l a r g e r v e r t i c a l w a l l a t the c o a s t ) , then the r e s u l t i n g d i s p e r s i o n c u r v e s appear t o drop s l i g h t l y below the c o r r e c t ones. A comparison between the d i s p e r s i o n c u r v e s o b t a i n e d by the f i r s t method (where the e x p o n e n t i a l p r o f i l e has a v e r t i c a l w a l l of o r d e r 50-100 m a t the c o a s t ) and t h o s e o b t a i n e d by the second method shows t h a t i n g e n e r a l , the c u r v e s o b t a i n e d by method 1 l i e somewhat below tho s e from method 2. Only f o r the f i r s t mode, and a t s m a l l t o i n t e r m e d i a t e wavenumbers, does the d i s p e r s i o n c u r v e from method 1 l i e above t h a t from method 2. 128 APPENDIX C: DATA PROCESSING The c u r r e n t and wind h o u r l y d a t a r e c o r d s had t h e i r mean and l i n e a r t r e n d removed by the average s l o p e method d e s c r i b e d i n Bendat and P i e r s o l (1971, p.289). The o r i g i n a l Newport a d j u s t e d sea l e v e l d a t a , when F o u r i e r t r a n s f o r m e d , showed s u b s t a n t i a l l e a k a g e of power from the d i u r n a l t i d e s t o f r e q u e n c i e s below 0.8 cpd. Thus, i t was n e c e s s a r y t o p e r f o r m a t i d a l harmonic a n a l y s i s on the sea l e v e l d a t a . The t i d a l c o n s t i t u e n t s were d e t e r m i n e d by a l e a s t squares f i t u s i n g the program i n Foreman (1977), and then s u b t r a c t e d from the sea l e v e l d a t a . These p r o c e s s e d d a t a were then used f o r r o t a r y s p e c t r a l a n a l y s i s and c r o s s - s h e l f modal f i t t i n g . The c u r r e n t r e c o r d f o r the s t a t i o n P o i n s e t t i a i n CUE-2 has a one-week gap. When p e r f o r m i n g s p e c t r a l a n a l y s i s i n v o l v i n g P o i n s e t t i a , the two segments were F o u r i e r t r a n s f o r m e d s e p a r a t e l y , then assembled t o g e t h e r i n the f r e q u e n c y space, and band averaged t o produce the d e s i r e d spectrum. 

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