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UBC Theses and Dissertations

Topographically induced baroclinic eddies along a coastline Swaters, Gordon E. 1983

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TOPOGRAPHICALLY INDUCED BAROCLINIC EDDIES ALONG A COASTLINE by GORDON EDWIN SWATERS B . M a t h . ( H o n o u r s ) , U n i v e r s i t y Of W a t e r l o o , 1980 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES D e p a r t m e n t Of M a t h e m a t i c s I n s t i t u t e Of A p p l i e d M a t h e m a t i c s And S t a t i s t i c s D e p a r t m e n t Of O c e a n o g r a p h y We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF B R I T I S H COLUMBIA A u g u s t 1983 © G o r d o n E d w i n S w a t e r s , 1983 I n 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 o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e 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 a nd s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f M a t h e m a t i c s The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 D a t e : A u g u s t 1, 1983 i i A b s t r a c t A m a t h e m a t i c a l model i s f o r m u l a t e d which d e s c r i b e s the i n t e r a c t i o n between a b a r o c l i n i c c u r r e n t and o r d e r Rossby number topography a l o n g a c o a s t l i n e . The l e a d term s o l u t i o n , i n an a s y m p t o t i c e x p a n s i o n i n t h e Rossby number, i s o b t a i n e d f o r t h e p r e s s u r e , d e n s i t y , v e l o c i t y and mass t r a n s p o r t f i e l d s . The l e a d term s o l u t i o n i s found u s i n g a normal mode a n a l y s i s and a Green's f u n c t i o n t e c h n i q u e . The s o l u t i o n i s a p p l i e d t o t h e p o s s i b l e t o p o g r a p h i c g e n e r a t i o n of the S i t k a eddy i n the n o r t h e a s t P a c i f i c Ocean. The n u m e r i c a l c a l c u l a t i o n s of the model and t h e o b s e r v e d l o c a t i o n , d i m e n s i o n s , v e l o c i t i e s and t r a n s p o r t s of the S i t k a eddy a r e i n v e r y good agreement. T a b l e o f C o n t e n t s A b s t r a c t i i L i s t o f F i g u r e s i v A c k n o w l e d g e m e n t s v i i i C h a p t e r I INTRODUCTION 1 C h a p t e r I I FORMULATION OF THE MATHEMATICAL MODEL 5 2.1 C u r r e n t s And G e o m e t r y Of The N o r t h E a s t P a c i f i c O cean 5 2.2 The B a s i c E q u a t i o n s Of M o t i o n And Mass 9 2.3 Q u a s i - G e o s t r o p h i c P o t e n t i a l V o r t i c i t y E q u a t i o n ....17 C h a p t e r I I I SOLUTION OF THE F I E L D EQUATIONS 31 3.1 H o r i z o n t a l A m p l i t u d e F u n c t i o n s 34 3.2 F a r F i e l d V e r t i c a l C u r r e n t S t r u c t u r e 40 3.3 V e r t i c a l Mode E i g e n f u n c t i o n s And E i g e n v a l u e s 44 3.4 F o r m u l a e F o r The P r e s s u r e , V e l o c i t y , D e n s i t y And Mass T r a n s p o r t F i e l d s 49 C h a p t e r I V PARAMETER S E N S I T I V I T Y ANALYSIS 59 4.1 The S o l u t i o n F o r The O b s e r v e d P a r a m e t e r s 63 4.2 H o r i z o n t a l C u r r e n t S h e a r And R o s s b y Number 83 4.3 T o p o g r a p h i c P a r a m e t e r s 93 4.4 B r u n t - V a i s a l a F r e q u e n c y 97 4.5 H o r i z o n t a l C u r r e n t S h e a r And S u r f a c e C u r r e n t 102 4.6 H o r i z o n t a l C u r r e n t S h e a r And B o t t o m C u r r e n t 107 C h a p t e r V APPLICATION TO THE SITKA EDDY 177 C h a p t e r V I CONCLUSIONS 182 BIBLIOGRAPHY 187 i v L i s t o f F i g u r e s 1. B a t h y m e t r y o f t h e n o r t h e a s t P a c i f i c Ocean 26 2. H o r i z o n t a l a nd v e r t i c a l c u r r e n t s t r u c t u r e i n t h e n o r t h e a s t P a c i f i c Ocean 27 3. B r u n t - V a i s a l a f r e q u e n c y 28 4. Mean s t a t e d e n s i t y p r o f i l e 29 5. C o n t o u r p l o t o f m o d e l l e d t o p o g r a p h y 30 6. F a r f i e l d v e r t i c a l c u r r e n t s t r u c t u r e 55 7. G r a p h o f G 0 ( z ) v s . z 56 8. G r a p h o f G, ( z ) v s . z 57 9. G r a p h o f G 2 ( z ) v s . z 58 10. L o c a t i o n o f c o m p u t e d v e r t i c a l p r o f i l e s r e l a t i v e t o t h e t o p o g r a p h y 111 11. H o r i z o n t a l c o n t o u r p l o t o f t h e s t r e a m f u n c t i o n on z = 1 112 12. H o r i z o n t a l c o n t o u r p l o t o f t h e s t r e a m f u n c t i o n on z = .9 113 13. H o r i z o n t a l c o n t o u r p l o t o f t h e s t r e a m f u n c t i o n on z = .8 114 14. H o r i z o n t a l c o n t o u r p l o t o f t h e s t r e a m f u n c t i o n on z = .7 115 15. H o r i z o n t a l c o n t o u r p l o t o f t h e s t r e a m f u n c t i o n on z = .6 116 16. V e r t i c a l c o n t o u r p l o t o f p ( 0 ) on y = .2 117 17. V e r t i c a l c o n t o u r p l o t o f p ( 0 ' on y = . 4 118 18. V e r t i c a l c o n t o u r p l o t o f p ( 0 ) on y = .6 119 19. V e r t i c a l c o n t o u r p l o t o f p ( 0 > on y = .8 120 20. V e r t i c a l c o n t o u r p l o t o f p ( 0 ) on y = 1 121 2 1 . S t i c k p l o t o f t h e h o r i z o n t a l v e l o c i t y f i e l d on z = 1 122 V 2 2 . S t i c k p l o t o f t h e h o r i z o n t a l v e l o c i t y f i e l d on z = .9 123 23. S t i c k p l o t o f t h e h o r i z o n t a l v e l o c i t y f i e l d on z = .8 124 24. S t i c k p l o t o f t h e h o r i z o n t a l v e l o c i t y f i e l d on z = .7 125 2 5 . S t i c k p l o t o f t h e h o r i z o n t a l v e l o c i t y f i e l d on z = .6 126 26. S t i c k p l o t o f t h e mass t r a n s p o r t f i e l d 127 27. H o r i z o n t a l c o n t o u r p l o t o f p ( 0 ) on z = 1 128 28. H o r i z o n t a l c o n t o u r p l o t o f p ( 0 ) on z = . 9 129 29. H o r i z o n t a l c o n t o u r p l o t o f p ( 0 ) on z = .8 130 30. V e r t i c a l c o n t o u r p l o t o f p ( 0 ) on y = .2 131 31 . V e r t i c a l c o n t o u r p l o t o f p ( 0 ) on y = . 4 132 32. V e r t i c a l c o n t o u r p l o t o f p ( 0 ) on y = .6 133 3 3 . V e r t i c a l c o n t o u r p l o t o f p ( 0 ) on y = . 8 134 34. V e r t i c a l c o n t o u r p l o t o f p ( 0 ) on y = 1 135 35. V e r t i c a l p r o f i l e o f p < 0 > a t ( x , y ) = (-2,.1) 136 36. V e r t i c a l p r o f i l e o f p ( 0 ' a t ( x , y ) = (-2,.1) 137 37. V e r t i c a l p r o f i l e o f u ( 0 ) a t ( x , y ) = (-2,.1) 138 38. V e r t i c a l p r o f i l e o f v ( 0 ) a t ( x , y ) = (-2,.1) 139 39. V e r t i c a l p r o f i l e o f w ( 0 ) a t ( x , y ) = (-2,.1) 140 40. V e r t i c a l p r o f i l e o f p ( 0 ) a t ( x , y ) = (-.4,.1) 141 4 1 . V e r t i c a l p r o f i l e o f p ( 0 ) a t ( x , y ) = (-.4,-1) 142 42. V e r t i c a l p r o f i l e o f u < 0 ) a t ( x , y ) = (-.4,.1) 143 4 3 . V e r t i c a l p r o f i l e o f v ( 0 ) a t ( x , y ) = (-.4,.1) 144 44. V e r t i c a l p r o f i l e o f w ( 0 ) a t ( x , y ) = (-.4,-1) 145 45. V e r t i c a l p r o f i l e o f p ( 0 ) a t ( x , y ) = ( 0 , . 7 5 ) 146 46. V e r t i c a l p r o f i l e o f p l 0 ) a t ( x , y ) = ( 0 , . 7 5 ) 147 v i 47. V e r t i c a l p r o f i l e o f u ( 0 > a t ( x , y ) = ( 0 , . 7 5 ) 148 48. V e r t i c a l p r o f i l e o f v ( 0 ) a t ( x , y ) = ( 0 , . 7 5 ) 149 49. V e r t i c a l p r o f i l e o f w' 0 ) a t ( x , y ) = ( 0 , . 7 5 ) 150 50. V e r t i c a l p r o f i l e o f p l 0 > a t ( x , y ) = (.75,.75) 151 51 . V e r t i c a l p r o f i l e o f p ( 0 ) a t ( x , y ) = (.75,.75) 152 52. V e r t i c a l p r o f i l e o f u ( 0 ) a t ( x , y ) = (.75,.75) 153 53. V e r t i c a l p r o f i l e o f v ( 0 > a t ( x , y ) = (.75,.75) 154 54. V e r t i c a l p r o f i l e o f w ( 0 » a t ( x , y ) = (.75,.75) 155 55. H o r i z o n t a l c o n t o u r p l o t o f t h e s t r e a m f u n c t i o n on z = 1 w i t h ( a , e ) = ( 1 0 , . 1 ) 156 56. H o r i z o n t a l c o n t o u r p l o t o f t h e s t r e a m f u n c t i o n on z - 1 w i t h ( a , e) = ( 1 0, .01 ) 1 57 57. H o r i z o n t a l c o n t o u r p l o t o f t h e s t r e a m f u n c t i o n on z = 1 w i t h ( a , e) = (1 , . 1 ) 1 58 58. H o r i z o n t a l c o n t o u r p l o t o f t h e s t r e a m f u n c t i o n on z = 1 w i t h ( a , e) = ( 1 , .01 ) 1 59 59. V e r t i c a l c o n t o u r p l o t o f p ( 0 > on y = .6 w i t h ( a , e ) = (1 ,.1) ..160 60. V e r t i c a l c o n t o u r p l o t o f p ( t M on y = .6 w i t h (a,e) = ( 1 , . 0 1 ) 161 61 . H o r i z o n t a l c o n t o u r p l o t o f t h e s t r e a m f u n c t i o n on z = 1 w i t h no seamount 162 62. H o r i z o n t a l c o n t o u r p l o t o f t h e s t r e a m f u n c t i o n on z = 1 w i t h no s l o p e p r o t r u s i o n 163 6 3 . V e r t i c a l c o n t o u r p l o t o f p ( 0 > on y = .8 w i t h ( N 0 , 7 * ) = (.01s- 1,300"'m" 1) 164 64. V e r t i c a l c o n t o u r p l o t o f p ( 0 ) on y = .2 w i t h ( N 0 , 7 * ) = ( . 0 2 s - 1 , 2 2 5 - 1 m " 1 ) 165 65. V e r t i c a l c o n t o u r p l o t o f p ( 0 ) on y = .8 w i t h ( N 0 , 7 * ) = ( . 0 2 s " 1 , 2 2 5 " 1 m - 1 ) 166 66. V e r t i c a l c o n t o u r p l o t o f p< 0> on y = .8 w i t h ( N 0 , 7 * ) = ( . 0 2 s - 1 , 3 0 0 " 1 m - 1 ) 167 67. H o r i z o n t a l c o n t o u r p l o t o f t h e s t r e a m f u n c t i o n on z = 1 w i t h Z( 1 ) = .05 168 v i i 68. V e r t i c a l c o n t o u r p l o t o f p ( 0 > on y = .2 w i t h Z ( 1 ) = .05 169 69. V e r t i c a l c o n t o u r p l o t o f p ( 0 ) on y = .8 w i t h Z ( 1 ) = .05 170 70. H o r i z o n t a l c o n t o u r p l o t o f t h e s t r e a m f u n c t i o n on z = 1 w i t h Z( 1 ) = 1 171 7 1 . H o r i z o n t a l c o n t o u r p l o t o f t h e s t r e a m f u n c t i o n on z = 1 w i t h Z( 1 ) = .01 172 72. V e r t i c a l c o n t o u r p l o t o f p ( 0 ' on y = .2 w i t h Z ( 1 ) = .01 173 73. V e r t i c a l c o n t o u r p l o t o f p ( 0 > on y = .8 w i t h Z ( 1 ) = .01 174 74. H o r i z o n t a l c o n t o u r p l o t o f t h e s t r e a m f u n c t i o n on z = 1 w i t h Z ( 0 ) = .001 175 7 5 . V e r t i c a l c o n t o u r p l o t o f p < 0 ' on y = .2 w i t h Z ( 0 ) = .001 176 v i i i A c k n o w l e d g e m e n t I t i s a p l e a s u r e t o t h a n k my a d v i s o r P r o f e s s o r L a w r e n c e A. Mysak f o r s u g g e s t i n g t h i s r e s e a r c h t o p i c and f o r h i s g u i d a n c e d u r i n g t h e p r e p a r a t i o n o f t h i s t h e s i s . P r o f e s s o r P a u l H. L e B l o n d d e s e r v e s s p e c i a l t h a n k s f o r t h e many d i s c u s s i o n s we had a b o u t p h y s i c a l o c e a n o g r a p h y a nd t h i s t h e s i s d u r i n g t h e 1982-83 a c a d e m i c y e a r when P r o f e s s o r Mysak was on s a b b a t i c a l l e a v e . I w o u l d l i k e t o e x p r e s s my g r a t i t u d e t o P r o f e s s o r Mysak f o r t h e g e n e r o u s f i n a n c i a l s u p p o r t p r o v i d e d me f r o m a g r a n t a w a r d e d h i m by t h e U n i t e d S t a t e s D e p a r t m e n t o f N a v a l R e s e a r c h . I w o u l d a l s o l i k e t o t h a n k t h e D e p a r t m e n t o f M a t h e m a t i c s f o r t h e T e a c h i n g A s s i s t a n t s h i p s a w a r d e d me f r o m 1981 t h r o u g h t o 1983, a n d t h e D e p a r t m e n t o f O c e a n o g r a p h y f o r u s e o f t h e i r c o m p u t i n g f a c i l i t i e s . 1 I . INTRODUCTION I n t h e u p p e r n o r t h e a s t o f t h e P a c i f i c O c e a n , a few h u n d r e d k i l o m e t r e s o f f s h o r e f r o m S i t k a , A l a s k a , t h e r e e x i s t s a m e s o s c a l e b a r o c l i n i c a n t i c y c l o n i c eddy ( T a b a t a ; 1 9 8 2 ) . T h i s e d d y , w i t h r a d i u s a b o u t 150 km a n d e x t e n d i n g a b o u t 1000 m e t r e s i n t o t h e w a t e r c o l u m n , seems t o o c c u r a n n u a l l y d u r i n g t h e N o r t h e r n H e m i s p h e r e summer. The eddy seems t o d i s s i p a t e d u r i n g t h e w i n t e r m o n t h s . T a b a t a ( l 9 8 2 ) h a s d e f i n i t e l y d o c u m e n t e d i t s e x i s t e n c e d u r i n g 1958, 1960 a n d 1961, a n d h a s p r o v i d e d c o m p e l l i n g e v i d e n c e f o r i t s e x i s t e n c e i n o t h e r y e a r s . T h i s e d d y , r e f e r r e d t o a s t h e S i t k a e d d y , r e o c c u r s i n t h e same g e o g r a p h i c l o c a t i o n d u r i n g t h e y e a r s i t h a s been o b s e r v e d . A n a l y s i s o f t h e w i n d s t r e s s d a t a f o r t h i s r e g i o n showed t h a t t h e s p e c t r a l e n e r g y o f t h e a n n u a l f r e q u e n c y was an o r d e r o f m a g n i t u d e l a r g e r t h a n a l l o t h e r f r e q u e n c i e s bands ( B a k u n ; 1 9 7 8 ) . T h i s f a c t , c o u p l e d t o t h e o b s e r v a t i o n t h a t t h e c u r l o f t h e w i n d s t r e s s s h o u l d g e n e r a t e a n t i c y c l o n i c m o t i o n s i n t h e r e g i o n , n a t u r a l l y l e a d t o t h e a t t e m p t t o c o r r e l a t e t h e S i t k a eddy t o a t m o s p h e r i c f o r c i n g . H o wever, i t was n o t p o s s i b l e t o c o r r e l a t e t h e o c c u r a n c e o f t h e S i t k a eddy w i t h t h e a n n u a l c h a n g e i n t h e c u r l o f t h e w i n d s t r e s s ( T a b a t a ; 1 9 8 2 ) . W i l l m o t t a n d M y s a k ( l 9 8 0 ) showed t h a t a s i x y e a r p e r i o d i c a t m o s p h e r i c f o r c i n g o f t h e n o r t h e a s t P a c i f i c O cean w i l l g e n e r a t e e d d i e s s i t u a t e d , more o r l e s s , a l o n g t h e A l a s k a n p a n h a n d l e . T h e s e e d d i e s were t h e r e s u l t o f r e f l e c t i o n s o f b a r o c l i n i c R o s s b y w a v e s , c r e a t e d by t h e a t m o s p h e r i c f o r c i n g o f 2 the ocean, o f f the A l a s k a n - B r i t i s h Columbian c o a s t l i n e . The W i l l m o t t and My s a k ( l 9 8 0 ) t r e a t m e n t was unable t o r e s o l v e two key f e a t u r e s of the S i t k a eddy. F i r s t , the S i t k a eddy o c c u r s a l o n g t h a t p a r t of the A l a s k a n c o a s t l i n e which i s a d j a c e n t t o B r i t i s h Columbia and not a l o n g the panhandle. Second, t h e i r t r e a t m e n t produced many e d d i e s whereas the a v a i l a b l e d a t a suggested t h e r e were few. T h e r e f o r e even su p p o s i n g t h a t the S i t k a eddy i s a t m o s p h e r i c a l l y i n d u c e d some o t h e r f o r c i n g mechanism i s o p e r a t i n g t o s e l e c t out the ob s e r v e d eddy. One o b v i o u s c a n d i d a t e i s an i n t e r a c t i o n p r o c e s s between the r e g i o n a l topography and the l o c a l mean f l o w . T h i s t h e s i s examines t h e p o s s i b i l i t y t h a t t h e l o c a l mean summertime c u r r e n t f i e l d can i n t e r a c t w i t h the prominent r e g i o n a l topography t o produce mesoscale a n t i c y c l o n i c b a r o c l i n i c e d d i e s . In o r d e r t o i n v e s t i g a t e t h i s c o n j e c t u r e a m a t h e m a t i c a l model i s de v e l o p e d f o r the t o p o g r a p h i c f o r c i n g of a b a r o c l i n i c c u r r e n t a l o n g a c o a s t l i n e . The model i s f o r m u l a t e d i n Chapter I I . P r i o r t o d i s c u s s i n g the d e t a i l s of the m a t h e m a t i c a l d e r i v a t i o n , a b r i e f d e s c r i p t i o n of the oce a n o g r a p h i c and b a t h y m e t r i c d a t a f o r the n o r t h e a s t P a c i f i c Ocean i s g i v e n . The s t r u c t u r e and o r i g i n s of the r e g i o n ' s c u r r e n t s i s d e s c r i b e d as a r e the p r i n c i p l e t o p o g r a p h i c f e a t u r e s . A q u a l i t a t i v e d e s c r i p t i o n of the S i t k a eddy i s a l s o g i v e n . Based on t h i s e x a m i n a t i o n of the a v i a l i b l e d a t a the i n v i s c i d , s t r a t i f i e d , s t e a d y , i n c o m p r e s s i b l e , B o u s s i n e s q and f -p l a n e e q u a t i o n s of motion a r e s c a l e d v i a g e o s t r o p h y t o a s c e r t a i n 3 t h e q u a l i t a t i v e n a t u r e o f t h e f l u i d d y n a m i c s . The r e l e v a n t p a r a m e t e r i s t h e R o s s b y number. The s m a l l n e s s o f t h e R o s s b y number i s e x p l o i t e d by c o n s t r u c t i n g t h e l e a d o r d e r s o l u t i o n s f o r t h e p r e s s u r e , d e n s i t y , v e l o c i t y a n d mass t r a n s p o r t f i e l d s i n an a s y m p t o t i c e x p a n s i o n i n t h e R o s s b y number. The o r d e r one p r e s s u r e f i e l d , w h i c h a c t s as a s t r e a m f u n c t i o n , must c o n s e r v e p o t e n t i a l v o r t i c i t y . C h a p t e r I I c o n c l u d e s w i t h t h e f o r m u l a t i o n o f t h e a p p r o p i a t e b o u n d a r y c o n d i t i o n s w h i c h t h e s o l u t i o n o f t h e p o t e n t i a l v o r t i c i t y e q u a t i o n must s a t i s f y . C h a p t e r I I I c o n t a i n s t h e a n a l y t i c a l s o l u t i o n o f t h e p r o b l e m . A s o l u t i o n i s s o u g h t i n w h i c h t h e o r d e r one p r e s s u r e f i e l d i s g i v e n a s t h e sum of t h e u p s t r e a m s t r e a m f u n c t i o n a nd a p r e s s u r e f i e l d r e p r e s e n t i n g t h e t o p o g r a p h i c mean f l o w i n t e r a c t i o n . T h i s i n t e r a c t i o n p r e s s u r e f i e l d i s o b t a i n e d by u s i n g a n o r m a l mode a n a l y s i s d e s c r i b e d by Chao e_t a l . ; ( 1 9 8 0 ) . The n o r m a l modes a r e f o u n d t o be g i v e n by B e s s e l f u n c t i o n s o f o r d e r o n e , w i t h t h e s o l u t i o n f o u n d v i a a method d e s c r i b e d i n B r y a n a n d R i p a ( l 9 7 8 ) . The h o r i z o n t a l a m p l i t u d e f u n c t i o n s a r e f o u n d u s i n g a G r e e n ' s f u n c t i o n t e c h i n q u e . C h a p t e r I I I c o n c l u d e s w i t h t h e e x p l i c i t f o r m u l a e g i v e n f o r t h e o r d e r one p r e s s u r e , v e l o c i t y , d e n s i t y and mass t r a n s p o r t f i e l d s . C h a p t e r I V h a s two f u n c t i o n s . The s o l u t i o n i s d e s c r i b e d when e v a l u a t e d f o r t h e s e t of p a r a m e t e r s t h a t were o b t a i n e d as e s t i m a t e s f r o m t h e d a t a f o r t h e n o r t h e a s t P a c i f i c O c e a n . The r e s u l t s o f v a r y i n g t h e p a r a m e t e r s i s a l s o d e s c r i b e d . I n a d d i t i o n t o d e s c r i b i n g t h e n u m e r i c a l c h a n g e s r e s u l t i n g f r o m p a r a m e t e r v a r i a t i o n s , p h y s i c a l e x p l a n a t i o n s a r e a l s o g i v e n , 4 b a s e d o n v o r t i c i t y a r g u m e n t s . C h a p t e r V d i s c u s s e s t h e a p p l i c a t i o n o f t h e m o d e l t o g e n e r a t i o n o f t h e S i t k a e d d y . C h a p t e r V I s u m m a r i z e s t h e w o r k c o n t a i n e d i n t h i s t h e s i s . 5 I I . FORMULATION OF THE MATHEMATICAL MODEL 2.1 C u r r e n t s And G e o m e t r y Of The N o r t h E a s t P a c i f i c Ocean The m o t i v a t i o n f o r c r e a t i n g a m a t h e m a t i c a l m o d e l f o r t o p o g r a p h i c a l l y i n d u c e d e d d i e s a l o n g a c o a s t l i n e i s t h e p o s s i b l e a p p l i c a t i o n o f t h e model i n u n d e r s t a n d i n g t h e d y n a m i c s o f t h e S i t k a e d d y . I t i s t h e r e f o r e e s s e n t i a l t o h a v e a t l e a s t a q u a l i t a t i v e a p p r e c i a t i o n o f t h e o c e a n o g r a p h y a n d b a t h y m e t r y o f t h i s r e g i o n . I n t h i s s e c t i o n a b r i e f s u r v e y o f t h e p h y s i c a l o c e a n o g r a p h y and g e o m e t r y o f t h e n o r t h e a s t P a c i f i c Ocean i s p r e s e n t e d . S p e c i f i c a l l y , t h a t r e g i o n w h i c h i s b o u n d e d by t h e l i n e s o f l o n g i t u d e 130°W a n d 145°W, and t h e l i n e s o f l a t i t u d e 53°N a n d 59°N. I n t h e s u b s e q u e n t d i s c u s s i o n t h i s i s t h e g e o g r a p h i c a l a r e a r e f e r r e d t o a s ' t h e r e g i o n ' . T h i s summary i s l a r g e l y drawn f r o m t h e work o f T a b a t a ( l 9 8 2 ) a n d B e n n e t t ( 1 9 5 9 ) . F i g u r e 1 i s a b a t h y m e t r i c map o f t h e n o r t h e a s t P a c i f i c O c e a n . I n t h e r e g i o n o f i n t e r e s t , t h e o c e a n f l o o r c a n be d e s c r i b e d a s a s l i g h t l y s l o p i n g a b y s s a l p l a i n w i t h s e v e r a l i r r e g u l a r l y s p a c e d s e a m o u n t s . Of p a r t i c u l a r n o t e i s t h e c o l l e c t i o n o f s e a m o u n t s i n t h e i m m e d i a t e v i c i n i t y o f t h e P r a t t s e a m o u n t , l o c a t e d a t 142°W 56°N. T h e s e s e a m o u n t s h a v e h e i g h t s on t h e o r d e r o f 2500 m e t r e s i n a b o u t 3500+ m e t r e s o f w a t e r . T o w a r d t h e s o u t h e a s t t h e b o t t o m p r o f i l e becomes h i g h l y i r r e g u l a r w i t h many t o p o g r a p h i c p r o t r u s i o n s . T h e i r s i z e s h owever a r e somewhat s m a l l e r t h a n t h e c o l l e c t i o n n e a r t h e P r a t t s e a m o u n t . The A l a s k a n - B r i t i s h C o l u m b i a n c o a s t l i n e , i n t h e a b o v e r e g i o n , i s more o r l e s s s t r a i g h t , i n c l i n e d a b o u t 45° t o t h e we s t o f a l i n e o f c o n s t a n t l o n g i t u d e . N o r t h w a r d o f t h e r e g i o n t h e 6 A l a s k a n c o a s t l i n e t u r n s 90° t o t h e w e s t , g i v i n g t h e i m p r e s s i o n o f f o r m i n g a b o u n d a r y o f a q u a r t e r p l a n e r e g i o n . S o u t h e a s t w a r d o f t h e r e g i o n , t h e g r o s s f e a t u r e s o f t h e B r i t i s h C o l u m b i a n c o a s t l i n e r e m a i n q u a l i t a t i v e l y s t r a i g h t . The c o n t i n e n t a l s l o p e i n t h i s a r e a h a s a n i n t e r e s t i n g c h a r a c t e r i z a t i o n . I n t h e n o r t h t h e s h e l f b r e a k o c c u r s w i t h i n 50 km a n d p a r a l l e l t o t h e c o a s t . However n e a r t h e l o c a t i o n o f t h e S i t k a eddy t h e s l o p e r e g i o n b r o a d e n s g i v i n g t h e i m p r e s s i o n o f a h o r i z o n t a l p r o t r u s i o n o f t h e s h e l f o u t i n t o t h e d e e p e r o c e a n . S o u t h e a s t o f t h e c o n t i n e n t a l s l o p e bump, t h e s h e l f b r e a k r e s u m e s i t s n o r t h e r n p a t t e r n . T h i s 'bump' a l o n g t h e s h e l f b r e a k i s r o u g h l y s y m m e t r i c a b o u t t h e n o r m a l t o t h e c o a s t l i n e t a k e n a t l a t i t u d e 56°N a n d l o n g i t u d e 135°W. I n F i g u r e 1 t h e 1600 f a t h o m (3000 m e t r e ) c o n t o u r i s m a r k e d , p r o v i d i n g a b e n c h mark f o r t h e e x t e n t o f t h e c o n t i n e n t a l s l o p e bump, o u t t o a b o u t 140°W 55°N. Thus i n t h e r e g i o n i n w h i c h t h e S i t k a e ddy o c c u r s t h e t o p o g r a p h y f o r m s t h e f o l l o w i n g i d e a l i z e d p i c t u r e . F i r s t t h e c o a s t l i n e i s more o r l e s s s t r a i g h t f a l l i n g o f f q u i c k l y t o an a b y s s a l p l a i n o f a b o u t 3500 m e t r e s d e p t h . N e a r S i t k a , A l a s k a t h e r e i s a l i m i t e d p r o t r u s i o n o f t h e c o n t i n e n t a l s l o p e w h i c h f a l l s o f f somewhat more s l o w l y t o t h e a b y s s a l p l a i n b e l o w t h a n t h e s u r r o u n d i n g c o n t i n e n t a l s h e l f . I f t h e 3000 m e t r e c o n t o u r i s t a k e n a s t h e e x t e n t o f t h e h o r i z o n t a l p r o t r u s i o n t h e n t h e maximum e x t e n t o f t h i s t o p o g r a p h i c f e a t u r e i s n e a r l y o u t t o t h e P r a t t s e a m o n t . The s e c o n d p r o n o u n c e d t o p o g r a p h i c f e a t u r e i s t h e c o l l e c t i o n o f s e a m o u n t s i n t h e i m m e d i a t e a r e a o f t h e P r a t t seamount w i t h t h e s u r r o u n d i n g t e r r a i n i n c o m p a r i s o n a p p e a r i n g 7 r e l a t i v e l y f l a t . The c i r c u l a t i o n a l o n g t h e B r i t i s h C o l u m b i a n and A l a s k a n c o a s t n e a r t h e s o u t h o f t h e a b o v e r e g i o n c o n s i s t s o f a b r o a d weak p o l e w a r d c u r r e n t w i t h a s p e e d on t h e o r d e r o f 10 cm s " 1 . T h i s c u r r e n t h a s i t s o r i g i n s i n t h e e a s t w a r d f l o w i n g S u b - A r c t i c P o l a r c u r r e n t o f t h e P a c i f i c Ocean s i t u a t e d on a b o u t t h e 50°N l i n e o f l a t i t u d e . Upon r e a c h i n g t h e c o n t i n e n t a l s h e l f o f N o r t h A m e r i c a t h i s c u r r e n t b i f u r c a t e s i n t o a p o l e w a r d a n d e q u a t o r w a r d c o m p o n e n t . The p o l e w a r d f l o w i n g c u r r e n t b e i n g t h e c o a s t a l c u r r e n t m e n t i o n e d a b o v e . T h i s n o r t h w a r d f l o w c o n t i n u e s u n t i l i t i s d i r e c t e d s o u t h w e s t w a r d by t h e A l a s k a n s h e l f w here i t f o r m s i t s e l f i n t o a n a r r o w c o a s t a l j e t known a s t h e A l a s k a n S t r e a m . T homson(1972) has shown t h a t t h i s s t r e a m i n g i s d y n a m i c a l l y s i m i l i a r t o t h e i n t e n s i f i c a t i o n o f w e s t e r n b o u n d a r y c u r r e n t s . The v e r t i c a l s t r u c t u r e o f t h e n o r t h w a r d f l o w i n g c o a s t a l c u r r e n t v a r i e s a s t h e d i s t a n c e f r o m t h e c o a s t l i n e i n c r e a s e s . B e n n e t t ( 1 9 5 9 ) c l a s s i f i e d t h e v e l o c i t y p r o f i l e s i n t o f o u r c a t e g o r i e s . F i g u r e 2, t a k e n f r o m B e n n e t t ' s p a p e r , shows t h a t t h e n e a r c o a s t a l c u r r e n t ( g r o u p 4) i s s t r o n g l y a t t e n u a t e d by d e p t h , w i t h a maximum s u r f a c e s p e e d o f a b o u t 10 cm s " 1 . G r o u p 2 c u r r e n t s , s i t u a t e d t o t h e i m m e d i a t e w e s t o f t h e g r o u p 4 c u r r e n t s a l s o m o n t o n i c a l l y d e c a y w i t h d e p t h a l t h o u g h t h e a t t e n u a t i o n i s n o t a s s e v e r e a s w i t h t h e g r o u p 4 c u r r e n t s . T y p i c a l l y , g r o u p 2 c u r r e n t s h a v e s p e e d s on t h e o r d e r o f 5 cm s " 1 . G r o u p 3 c u r r e n t s , l o c a t e d more o r l e s s w e s t w a r d o f t h e g r o u p 2 c u r r e n t s have s m a l l e r s p e e d s t h a n e i t h e r g r o u p 4 o r 8 g r o u p 3 c u r r e n t s . T y p i c a l l y t h e y h a v e m a g n i t u d e s on t h e o r d e r o f 3 cm s " 1 . One i n t e r e s t i n g f e a t u r e o f t h e c o n t o u r b e t w e e n g r o u p 2 and g r o u p 3 c u r r e n t s i s t h e s u g g e s t i o n o f a t o n g u e o f g r o u p 2 c u r r e n t s w h i c h p r o t r u d e s n o r t h w a r d i n t o a n o m i n a l l y g r o u p 3 r e g i m e . T h i s p r o t r u s i o n l i e s p r e c i s e l y o v e r t h e P r a t t s e a m o u n t , a s i g n i f i c a n t o r o g r a p h i c f e a t u r e o f t h e r e g i o n a l b a t h y m e t r y . F i g u r e 2 a l s o g i v e s t h e q u a l i t a t i v e i m p r e s s i o n t h a t most o f t h e t r a n s p o r t o c c u r s i n t h e u p p e r l a y e r s o f t h e o c e a n . C a l c u l a t i o n s by T a b a t a ( l 9 8 2 ) c o n f i r m t h i s by r e p o r t i n g t h a t a b o u t 75 p e r c e n t o f t h e t r a n s p o r t o c c u r s i n t h e f i r s t 500 m e t r e s o f o c e a n . T a b a t a ' s ( 1 9 8 2 ) a n a l y s i s o f g e o p o t e n t i a l a n o m a l y d a t a f o r t h i s r e g i o n f o r t h e y e a r s 1 954 t o 19.67 h a s shown t h e e x i s t e n c e o f a m e s o s c a l e b a r o c l i n i c a n t i c y c l o n i c e d d y , s i t u a t e d a t a b o u t 57°N 138°W, r e f e r r e d t o a s t h e S i t k a e d d y . T h i s l o c a t i o n w o u l d p l a c e t h e S i t k a eddy s l i g h t l y t o t h e e a s t a n d n o r t h o f t h e P r a t t s e a m o u n t , a nd s l i g h t l y t o t h e w e s t a n d n o r t h o f t h e c o n t i n e n t a l s h e l f bump. T h i s l o c a t i o n i s shown i n F i g u r e 1. The S i t k a eddy i s o b s e r v e d t o h a v e a t y p i c a l r a d i u s o f be t w e e n 200 and 300 km. O b s e v e d s u r f a c e s p e e d s r a n g e f r o m 15 cm s " 1 t o 37 cm s " 1 a t 50 km f r o m t h e c e n t r e o f t h e e d d y . E s t i m a t e s o f t h e s u r f a c e s p e e d , b a s e d on d r i f t i n g - b u o y t r a j e c t o r i e s , go a s h i g h a s 110 cm s " 1 70 km f r o m t h e eddy c e n t r e . The d e p t h t o w h i c h t h e eddy o c c u r s i s t y p i c a l l y a b o u t 1000 m e t r e s , a l t h o u g h e v i d e n c e s u p p o r t s e s t i m a t e s up t o 2000 m e t r e s . 9 The S i t k a eddy seems t o be most d e t e c t a b l e d u r i n g t h e N o r t h e r n H e m i s p h e r e summer a n d l e s s s o i n t h e w i n t e r . T a b a t a ( l 9 8 2 ) h a s i d e n t i f i e d i t d u r i n g t h e l a t e s p r i n g t h r o u g h l a t e summer o f t h e y e a r s 1958, 1960 a n d 1961. I n o t h e r y e a r s , d u r i n g t h e same s e a s o n , T a b a t a ' s a n a l y s i s h a s shown t h a t t h e r e i s some a m b i g u i t y i n a t t e m p t i n g t o r e s o l v e t h e S i t k a e d d y . H o w e v e r , w i t h i n some o f t h e s e y e a r s , T a b a t a ' s maps o f t h e g e o p o t e n t i a l a n o m a l y show a m a r k e d c l o c k w i s e f l o w a t t h e l o c a t i o n o f t h e S i t k a e d d y . E v e n s t i l l , s i g n i f i c a n t ' v a r i a b i l i t y h a s been o b s e r v e d i n t h e r e g i o n ' s c u r r e n t s . I n t e r e s t i n g l y , B e n n e t t ( 1 9 5 9 ) h a s a f i g u r e o f t h e t r a n s p o r t f o r 1955 w h i c h c l e a r l y i n d i c a t e s a s t r o n g c y c l o n i c eddy i n t h e r e g i o n where t h e S i t k a eddy n o r m a l l y o c c u r s . On b a l a n c e , t h e e v i d e n c e i n d i c a t e s t h a t t h e S i t k a eddy i s a more o r l e s s a n n u a l e v e n t o c c u r i n g i n t h e l a t e s p r i n g t h r o u g h t o l a t e summer. The r e l a t i v e l y s t a b l e g e o g r a p h i c a l l o c a t i o n o f t h e S i t k a eddy n a t u r a l l y s u g g e s t s t h a t t o p o g r a p h y may be an i m p o r t a n t f o r c i n g m e c h a n i s m i n i t s g e n e r a t i o n a nd m a i n t a i n a n c e . W i t h t h i s p o s s i b i l i t y i n m i n d t h i s t h e s i s e x a m i n e s t h e f o l l o w i n g c o n j e c t u r e : t h a t t h e mean f l o w t y p i c a l l y o b s e r v e d i n t h i s r e g i o n d u r i n g t h e s p r i n g a n d summer c a n i n t e r a c t w i t h t h e p r o m i n e n t r e g i o n a l t o p o g r a p h y t o p r o d u c e m e s o s c a l e a n t i c y c l o n i c e d d i e s . 2.2 The B a s i c E q u a t i o n s Of M o t i o n And Mass T h i s s e c t i o n i s c o n c e r n e d w i t h f o r m u l a t i n g t h e a p p r o p r i a t e m a t h e m a t i c a l m o d e l f o r t o p o g r a p h i c a l l y i n d u c e d b a r o c l i n i c e d d i e s . The b a s i c e q u a t i o n s f o r momentum a n d mass 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 m a c r o s c a l e s i n o r d e r t o a s c e r t a i n t h e 10 q u a l i t a t i v e n a t u r e o f t h e f l u i d d y n a m i c s . The s u b s e q u e n t a n a l y s i s o c c u r s on a f - p l a n e . I n o t h e r w o r d s a r i g h t - h a n d e d u n i f o r m l y r o t a t i n g c a r t e s i a n c o o r d i n a t e s y s t e m . I f t h e o r d e r e d t r i p l e ( x * , y * , z * ) i s a p o i n t i n t h i s s p a c e , t h e n p o s i t i v e x* p o i n t s n o r t h w a r d , p o s i t i v e y* p o i n t s w e s t w a r d a n d p o s i t i v e z* p o i n t s u p w a r d . The l o c a t i o n o f t h e o r i g i n o f t h i s c o o r d i n a t e s y t e m i s m o t i v a t e d by t h e g e o m e t r y o f t h e p r o b l e m d i s c u s s e d i n t h e l a s t s e c t i o n . The p l a n e y* = 0 i s t a k e n t o be t h e c o a s t l i n e . The p l a n e z* = 0 i s t a k e n a s t h e a b y s s a l p l a i n d e s c r i b e d l a s t s e c t i o n . The o r i g i n i s t a k e n a s t h e p o i n t on t h e i n t e r s e c t i o n o f t h e a b o v e two p l a n e s c o r r e s p o n d i n g t o l a t i t u d e 55°N. The a n g u l a r v e l o c i t y o f t h e f -p l a n e i s g i v e n by ( 0 , 0 , f / 2 ) where f = 2 | £21 s i n ( 0) i s t h e C o r i o l o s p a r a m e t e r , t h e m a g n i t u d e o f t h e a n g u l a r v e l o c i t y o f t h e E a r t h ' s r o t a t i o n and 6 t h e l a t i t u d e . L e t ( u * , v * , w * ) , p * , p* be t h e v e l o c i t y , p r e s s u r e a n d d e n s i t y f i e l d s r e s p e c t i v e l y . The s t e a d y , i n v i s c i d , i n c o m p r e s s i b l e , s t r a t i f i e d , B o u s s i n e s q a n d f - p l a n e d i m e n s i o n a l i z e d e q u a t i o n s o f m o t i o n c a n be w r i t t e n ( L e B l o n d a n d Mys a k ; 1978) a s : u*9*u * + v * 9 * u * + w*9*u* - f v * = - p 0 - 1 9 * p * 1 2 3 1 u * g * v * + y * 9 * v * + w*9*v* + f u * = - p 0 " 1 9 * p * 1 2 3 2 p 0 ( u * 9 * w * + v*9*w* + w*3*v*) + gp* = - 3 * p * 1 2 3 3 9*u* + 9*v* + 9*w* = 0 1 2 3 11 u*9*p* + v * 9 * p * + w*9*p* = 0. 1 2 3 The a b o v e e q u a t i o n s h a v e been w r i t t e n w i t h t h e f o l l o w i n g c o n v e n t i o n s . The s u p e r s c r i p t '*' i m p l i e s t h a t t h e v a r i a b l e i m m e d i a t e l y p r e c e e d i n g i t i s d i m e n s i o n a l . The q u a n t i t i e s 9*, 9* 1 2 a n d 9* a r e t h e f i r s t p a r t i a l d e r i v a t i v e s w i t h r e s p e c t t o t h e 3 d i m e n s i o n a l i z e d v a r i a b l e s x *, y* and z* r e s p e c t i v e l y . The B o u s s i n e s q a p p r o x i m a t i o n h a s been i m p l e m e n t e d t h r o u g h d e f i n i n g Po a s a c o n s t a n t r e f e r e n c e d e n s i t y . However p* i s d e f i n e d a s a d i m e n s i o n a l i z e d v a r i a b l e d e n s i t y . The f i r s t t h r e e e q u a t i o n s a r e t h e momentum e q u a t i o n s i n t h e x*, y* and z* d i r e c t i o n s r e s p e c t i v e l y . The f o u r t h a n d f i f t h e q u a t i o n s e x p r e s s t h e f a c t t h e f l o w i s i n c o m p r e s s i b l e . S i n c e t h e f l o w i s i n h o m o g e n e o u s b u t i n c o m p r e s s i b l e t h e u s u a l r e q u i r e m e n t t h a t t h e v e l o c i t y f i e l d i s s o l e n o i d a l ( t h e f o u r t h e q u a t i o n ) i s s u p p l e m e n t e d by t h e f i f t h e q u a t i o n w h i c h e x p r e s s e s t h e f a c t t h a t t h e d e n s i t y o f a f l u i d p a r t i c l e r e m a i n s c o n s t a n t f o l l o w i n g i t s m o t i o n . I n o r d e r t o a s s e s s t h e i m p o r t a n c e o f e a c h t e r m i n t h e a b o v e e q u a t i o n s i t i s a p p r o p r i a t e t o n o n d i m e n s i o n a l i z e e a c h v a r i a b l e u t i l i z i n g q u a l i t a t i v e g e o p h y s i c a l f l u i d d y n a m i c b a l a n c e s . F o r e x a m p l e , t h e o c e a n t o l o w e s t o r d e r i s a t r e s t a n d i n h y d r o s t a t i c b a l a n c e . T h e r e f o r e t h e mean p r e s s u r e a n d d e n s i t y f i e l d s a r e i n h y d r o s t a t i c e q u i l i b r i u m , w h i c h s u g g e s t s an a p p r o p r i a t e s c a l i n g f o r them. L a r g e s c a l e m o t i o n s , s u c h a s b a r o c l i n i c e d d i e s , a r e 1 2 p r i m a r i l y i n g e o s t r o p h i c b a l a n c e . I t f o l l o w s t h a t the dynamic p r e s s u r e f i e l d a s s o c i a t e d w i t h f l u i d motion s h o u l d be n o n d i m e n s i o n a l i z e d u s i n g g e o s t r o p h i c s c a l i n g . A s s o c i a t e d w i t h the dynamic p r e s s u r e f i e l d i s a dynamic d e n s i t y f i e l d which i s s c a l e d so t h a t t h e s e two f i e l d s a r e i n h y d r o s t a t i c b a l a n c e . The h o r i z o n t a l c o o r d i n a t e s x* and y* a r e s c a l e d by L, a c h a r a c t e r i s t i c l e n g t h o b t a i n e d from g e o m e t r i c a l c o n s i d e r a t i o n s , d i s c u s s e d l a t e r i n t h i s s e c t i o n . The v e r t i c a l c o o r d i n a t e z* i s s c a l e d by the mean ocean d e p t h , say H. The h o r i z o n t a l v e l o c i t y f i e l d u* and v* i s s c a l e d w i t h a c h a r a c t e r i s t i c speed U which i s o b t a i n e d from upstream f l o w c o n d i t i o n s . The s c a l i n g f o r the v e r t i c a l v e l o c i t y i s deduced from the f a c t t h a t the v e l o c i t y f i e l d i s n o n d i v e r g e n t . Suppose f o r the moment t h a t a l l t h r e e terms i n V*•(u*,v*,w*)=0 a r e the same o r d e r of magnitude. (Here V* i s the d i m e n s i o n a l i z e d L a p l a c i a n . ) I t f o l l o w s t h a t w* s h o u l d be s c a l e d w i t h UHLT 1. However i t t u r n s out t h a t t h i s i s an over e s t i m a t e of the o r d e r of magnitude of the v e r t i c a l v e l o c i t y . S i n c e the h o r i z o n t a l v e l o c i t y f i e l d i s . s c a l e d v i a g e o s t r o p h y the r e s u l t i n g f l o w i s e s s e n t i a l l y h o r i z o n t a l l y n o n d i v e r g e n t . T h i s i m p l i e s t o a f i r s t a p p r o x i m a t i o n t h a t t h e r e i s no v e r t i c a l shear i n the v e r t i c a l v e l o c i t y . So t h a t i f ever w*=0 then throughout the water column w*=0. For q u a s i - g e o s t r o p h i c dynamics a b e t t e r e s t i m a t e of w* i s U 2 H ( f L 2 ) - 1 . The n o n d i m e n s i o n a l ( u n a s t e r i s k e d ) v a r i a b l e s a r e d e f i n e d as f o l l o w s : 13 ( x * , y * ) = L ( x , y ) , z* = Hz ( u * , v * ) = U ( u , v ) , w* = cUHL-'w P* = p 0 [ p ( z ) + e F p ( x , y , z ) ] p* = p 0 g H p ( z ) + p 0 f U L p ( x , y , z ) ,' w i t h e d e f i n e d a s t h e R o s s b y number U ( f L ) - 1 and F t h e s q u a r e d r a t i o o f t h e l e n g t h s c a l e t o t h e e x t e r n a l R o s s b y d e f o r m a t i o n r a d i u s , i e . L 2 f 2 / ( g H ) . The t e r m s p ( z ) a n d p ( z ) a r e t h e mean d e n s i t y a n d p r e s s u r e f i e l d s r e s p e c t i v e l y , w h i c h d e p e n d o n l y t h e v e r t i c a l c o o r d i n a t e . T h e s e two f i e l d s a r e i n h y d r o s t a t i c b a l a n c e , a r e f l o w i n d e p e n d e n t a n d must be o b t a i n e d f r o m o b s e r v a t i o n . When t h e ab o v e s c a l e d v a r i a b l e s a r e s u b s t i t u t e d i n t o t h e e q u a t i o n s o f m o t i o n , t h e y r e s u l t i n : e ( u 9 , u + v 9 2 u + ew9 3u) - v + 9,p = 0 e ( u 9 , v + v 9 2 v + e w 9 3 v ) + u + 3 2 p = 0 e 2 6 2 ( u 9 1 w + v 9 2 w + ew9 3w) + p + 9 3 p = 0 9,u + 9 2 v + e9 3w = 0 u9,p + v 9 2 p + ew3 3p = S ( z ) w , w i t h t h e q u a n t i t i e s 9 1 f 9 2 a n d 9 3 d e f i n e d a s t h e f i r s t p a r t i a l s w i t h r e s p e c t t o x, y and z r e s p e c t i v e l y . The p a r a m e t e r 6 i s t h e a s p e c t r a t i o d e f i n e d a s H L _ 1 . The a s p e c t r a t i o i s t h o u g h t o f a s e s t i m a t i n g t h e r a t i o o f t h e l e n g t h s c a l e a s s o c i a t e d w i t h v e r t i c a l m o t i o n s t o t h e l e n g t h s c a l e a s s o c i a t e d w i t h h o r i z o n t a l m o t i o n s . I f t h e a s p e c t r a t i o i s 14 s m a l l c o m p a r e d t o u n i t y t h e n t h e f l o w i s p r i m a r i l y h o r i z o n t a l . The R o s s b y number e m e a s u r e s t h e d e g r e e t o w h i c h t h e n o n l i n e a r i t y i n t h e e q u a t i o n s o f m o t i o n f o r c e a d e p a r t u r e f r o m s t r i c t g e o s t r o p h y . The h y d r o s t a t i c a p p r o x i m a t i o n r e q u i r e s t h a t e8 << 0 ( 1 ) . L a t e r i n t h i s s e c t i o n i t i s shown t h a t 6 ^ 0 ( e ) w i t h e << 0 ( 1 ) . The p a r a m e t e r F h a s t h e p h y s i c a l i n t e r p e t a t i o n o f m e a s u r i n g t h e c o n t r i b u t i o n t o t h e t o t a l p o t e n t i a l v o r t i c i t y o f t h e v o r t i c i t y a s s o c i a t e d w i t h b a r o t r o p i c i s o b a r i c d e f l e c t i o n s c o m p a r e d t o t h e r e l a t i v e v o r t i c i t y 3 T V - 3 2 u . Thus i f F << 0 ( 1 ) , t h e n t h e s e a s u r f a c e c a n be a p p r o x i m a t e d a s a r i g i d l i d . However i f F 0 ( 1 ) t h e n t h e v o r t i c i t y a s s o c i a t e d w i t h s e a s u r f a c e d e f o r m a t i o n i s n o t a n e g l i g i b l e a s p e c t o f t h e p o t e n t i a l v o r t i c i t y . L a t e r i n t h i s s e c t i o n i t shown t h a t F 0 ( e ) . The q u a n t i t y S ( z ) i s d e f i n e d S ( z ) = - F - 1 D 3 p w i t h D 3 t h e o r d i n a r y d e r i v a t i v e w i t h r e s p e c t t o z. W i l l m o t t a n d M y s a k ( l 9 8 0 ) c a l c u l a t e d a l e a s t s q u a r e s f i t o f a B r u n t - V a i s a l a f r e q u e n c y t y p i c a l l y o b s e r v e d i n t h e n o r t h e a s t P a c i f i c O c e a n , w i t h an e x p o n e n t i a l o f t h e f o r m , [ N * ( z * ) ] 2 = N 0 2 e x p [ 7 * ( z * - H ) ] . T h i s e x p r e s s i o n h a s been w r i t t e n i n d i m e n s i o n a l f o r m , w i t h z*=0 c o r r e s p o n d i n g t o t h e o c e a n f l o o r i n t h e a b s e n c e o f t o p o g r a p h y 15 and z*=H t o the ocean s u r f a c e . The l e a s t s quares procedure r e s u l t e d i n N 0 = .011045 s _ 1 and 7 * = ( 2 5 4 . 5 1 ) ' 1 nr 1. S i n c e the d e f i n i t i o n of t h e B o u s s i n e s q B r u n t - V a i s a l a f r e q u e n c y i s [ N * ( z * ) ] 2 = - g ( H p 0 ) - 1 D 3 t p 0 p ( z ) ] i t f o l l o w s t h a t D 3 p ( z ) = - N 0 2 H g - 1 e x p [ 7 ( z - 1 ) ] where 7 = 7*H, and c o n s e q u e n t l y t h a t S ( z ) = [ ( N 0 H ) / ( f L ) ] 2 e x p [ 7 ( z - 1 ) ] . I t i s c l e a r t h a t S ( z ) can be i n t e r p e t e d as a nond i r n e n s i o n a l B r u n t - V a i s a l a f r e q u e n c y . The parameter ( N 0 H ) / ( f L ) i s a s t r a t i f i c a t i o n or Burger number. I t measures the r a t i o of the i n t e r n a l Rossby r a d i u s t o the g e o m e t r i c l e n g t h s c a l e L. In the s i t u a t i o n where S << 0(1) the motion i s b a r o t r o p i c i m p l y i n g t h a t the e f f e c t s of s t r a t i f i c a t i o n a r e n e g l i g i b l e . On the o t h e r hand when S >> 0 ( 1 ) b a r o c l i n i c i t y i s a dominant f e a t u r e of the f l u i d dynamics. I n s e c t i o n one of t h i s c h a p t e r a g e n e r a l d i s c u s s i o n of the b a t h y m e t r i c and oc e a n o g r a p h i c d a t a i n t h i s r e g i o n was p r e s e n t e d . Based on t h i s d i s c u s s i o n i t i s now p o s s i b l e t o make a r e a s o n a b l e 16 q u a l i t a t i v e e s t i m a t e o f t h e p a r a m e t e r s e, 6, F and s 0 = I ( N 0 H ) / ( f L ) ] 2 . The d e p t h o f t h e a b s s y a l p l a i n i s on t h e o r d e r o f 3500 m e t r e s . T h i s i s c h o s e n t o be H. The d i s t a n c e b e t w e e n t h e c o a s t l i n e a n d t h e P r a t t seamount i s on t h e o r d e r o f 400 km. T h i s c h o s e n t o be L. The m a g n i t u d e o f E a r t h ' s r o t a t i o n v e c t o r i s 7.292«10- 5 s " 1 , s o t h a t a t 55°N t h e C o r i o l o s p a r a m e t e r i s 1.2-10"* s _ 1 . T a b a t a ( l 9 8 2 ) r e p o r t e d c u r r e n t s a s h i g h a s 1 ms"', s o we t a k e U = 1 ms" 1. The v a l u e f o r N 0 i s t a k e n t o be t h a t c a l c u l a t e d by W i l l m o t t a n d M y s a k d 9 8 0 ) . W i t h t h i s s c a l i n g t h e n o n d i m e n s i o n a l p a r a m e t e r s a r e : F=0.07, e=0.02, 6=0.01 a n d s o = 0 . 6 5 . C o n s e q u e n t l y i t i s e x p e c t e d t h a t t h e d y n a m i c s o f t h e r e g i o n i s g o v e r n e d by s 0 - 0 ( 1 ) , 6 = 0 ( e ) , F * 0 ( e ) a n d t h a t e << 0 ( 1 ) . T h i s p a r a m e t e r r e g i m e i m p l i e s t h a t t h e e f f e c t s o f v o r t e x t u b e s t r e t c h i n g i s an i m p o r t a n t s o u r c e o f v o r t i c i t y i n t h e w a t e r c o l u m n . H owever, t h e b a r o t r o p i c r e s p o n s e o f t h e o c e a n , w h i c h i s m a n i f e s t e d i n a s e a s u r f a c e s l o p e i s n e g l i g i b l e . T h e r e f o r e t h e o c e a n s u r f a c e c a n be a p p r o x i m a t e d a s a r i g i d l i d . The a s p e c t r a t i o a p p e a r s t o be a t most t h e same o r d e r o f m a g n i t u d e a s t h e R o s s b y number. C o n s e q u e n t l y a h y d r o s t a t i c b a l a n c e b e t w e e n t h e p r e s s u r e a n d d e n s i t y f i e l d s h o l d s t o a r e m a r k a b l y good a p p r o x i m a t i o n . F i g u r e 3 i s a g r a p h o f t h e n o n d i m e n s i o n a l B r u n t - V a i s a l a f r e q u e n c y S ( z ) v s . z. The l a r g e s t g r a d i e n t i n S ( z ) o c c u r s i n t h e t o p 2 5 % o f o c e a n , i m p l y i n g t h a t i n t h e u p p e r r e g i o n s o f t h e w a t e r c o l u m n t h e r e s t s t a t e d e n s i t y v a r i e s r a p i d l y . I n t h e 1 7 i n t e r i o r o f t h e w a t e r c o l u m n ( z = 0 t o z .5) S ( z ) * 0, so t h a t t h e d e n s i t y i s a p p r o x i m a t e l y u n i f o r m . F i g u r e 4 i s a g r a p h o f t h e n o n d i m e n s i o n a l d e n s i t y f i e l d p v s . z . The s c a l e d e n s i t y p 0 i s c h o s e n t o be an a v e r a g e s u r f a c e d e n s i t y , r e s u l t i n g i n p ( 0 ) = 1. T y p i c a l v a l u e s f o r t h e s u r f a c e d e n s i t y i n t h e n o r t h e a s t P a c i f i c Ocean ( T a b a t a ; 1982) p l a c e Po - 1025 kg n r 3 . The d e n s i t y p r o f i l e i n F i g u r e 4 i n c r e a s e s a b o u t . 3 % i n t h e t o p 2 5% o f o c e a n , 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 c o n t a i n e d i n T a b a t a d 982) and B e n n e t t (1 959) . "Below z ~ .75 (900 m e t r e s ) t h e d e n s i t y i s u n i f o r m w i t h a v a l u e o f a b o u t 1.0032 t i m e s i t s s u r f a c e v a l u e . 2.3 Q u a s i - G e o s t r o p h i c P o t e n t i a l V o r t i c i t y E q u a t i o n The r e l a t i v e e s t i m a t e s o f t h e p a r a m e t e r s F, 6, s 0 and e s u g g e s t s t h a t t h e s i g n i f i c a n t p a r a m e t e r i s t h e R o s s b y number. The s m a l l n e s s o f e i s e x p l o i t e d by c o n s t r u c t i n g t h e l e a d t e r m s o f a n a i v e a s y m p t o t i c e x p a n s i o n f o r u, v , w, p and p i n t h e R o s s b y number. C o n s i d e r a s o l u t i o n o f t h e f o r m : u = u ( 0 ' + e u ( 1 ' + c 2 u l 2 ' ... v = v ( 0 ' + e v ( 1 > + e 2v< 2 ' ... w = w l 0 ) + e w ( 1 ) + e 2 w ( 2 > ... p = p< o) + e p ( 1' + e 2 p c 2 > ... p = p< 0 ' + ep' 1> + e 2 p ( 2 » ... . 18 S u b s t i t u t i o n o f t h e a b o v e e x p a n s i o n s i n t o t h e s c a l e d e q u a t i o n s o f m o t i o n i m p l i e s t h a t t h e 0 ( 1 ) f i e l d e q u a t i o n s a r e : 9 , p ( 0 > - v < 0 > = 0 2.1 u l 0 » + 9 2 p ( 0 > = 0 2.2 p< 0 » + 93p< o) = o 2.3 3 , u ( 0 ' + 9 2 v < 0 > = 0 2.4 u ' 0 ^ ^ ' 0 ' + v ( 0 , 9 2 p < ° > = S ( z ) w ( 0 » . 2.5 T h e u ( 0 ) a n d v ( 0 ) a r e i n g e o s t r o p h i c b a l a n c e w i t h t h e p ( 0 > p r e s s u r e f i e l d . H o w e v e r s i n c e t h e 0 ( 1 ) p r e s s u r e f i e l d t r i v i a l l y s a t i s f i e s ( 9 , , 9 2 ) • ( u ( 0 ' , v ( 0 ' ) = 0, t h e a b o v e s y s t e m o f e q u a t i o n s i s u n d e r d e t e r m i n e d , r e f e r r e d t o a s g e o s t r o p h i c d e g e n e r a c y . T h e f i e l d e q u a t i o n s a r e c l o s e d b y c o n s t r u c t i n g t h e q u a s i - g e o s t r o p h i c p o t e n t i a l v o r t i c i t y e q u a t i o n . T o b e g i n w i t h , t h e r e l e v a n t 0 ( e ) e q u a t i o n s a r e : u< 0 > 9 ^ ' 0 > + v ( 0 ) 9 2 u ( 0 ) - v< 1 >" = - 9 , p ( 1 > 2.6 u ( 0 > a i V ( 0 ) + v < o ) a 2 V ( 0 > + u ( i > = - 9 2 p ( D 2.7 9, u ( 1 > + 9 2 v ( 1> + 9 3w< 0' = 0. 2.8 19 T a k i n g 3 , ( 2 . 7 ) - 3 2 ( 2 . 6 ) a nd u s i n g 2.8 a n d 2.4 i t f o l l o w s t h a t , u < o ) V 2 v ( 0 ) _ V ( 0 ) V 2 U ( 0 > = a 3 w < ° > , 2.9 where V 2 i s t h e h o r i z o n t a l L a p l a c i a n g i v e n by 3 t 1 + 3 2 2 . R e w r i t i n g 2.9 i n t e r m s o f t h e p r e s s u r e f i e l d y i e l d s J [ p ( 0 > , V 2 p ( 0 » + 3 3 ( S " ' 3 3 p ( ° ' ) ] = 0 , where J [ A , B ] i s t h e J a c o b i a n o f A a n d B i e . 3,A3 2B - 3 2 A 3 , B . The t e r m s on t h e r i g h t i n t h e J a c o b i a n f o r m a l i s m f o r m t h e p o t e n t i a l v o r t i c i t y . The f i r s t o f t h e s e i s t h e v o r t i c i t y a s s o c i a t e d w i t h t h e a n g u l a r momentum o f t h e f l u i d a n d t h e s e c o n d i s t h e v o r t i c i t y a s s o c i a t e d w i t h t h e s t r e t c h i n g o f v o r t e x t u b e s . S i n c e t h e J a c o b i a n b e t w e e n t h e s t r e a m f u n c t i o n a n d t h e p o t e n t i a l v o r t i c i t y i s z e r o , t h e p o t e n t i a l v o r t i c i t y c a n be w r i t t e n i n t e r m s o f t h e p r e s s u r e f i e l d . I n o t h e r w ords t h e p o t e n t i a l v o r t i c i t y i s c o n s e r v e d a l o n g s t r e a m l i n e s . T h i s i m p l i e s t h a t t h e e s s e n t i a l p h y s i c a l f e a t u r e i n c o r p o r a t e d i n t o t h e m a t h e m a t i c a l m o d e l i s t h a t t h e c o n s e r v a t i o n o f a n g u l a r momentum i s p r i m a r i l y a b a l a n c e b e t w e e n t h e r e l a t i v e v o r t i c i t y ( i e . t h e z-component o f t h e c u r l o f t h e v e l o c i t y f i e l d ) a n d t h e a n g u l a r momentum i n d u c e d by b a r o c l i n i c v o r t e x t u b e s t r e t c h i n g . I n v i e w o f t h e a b o v e r e m a r k s t h e q u a s i - g e o s t r o p h i c p o t e n t i a l v o r t i c i t y e q u a t i o n c a n be w r i t t e n a s : V 2 p ( 0 ' + 3 3 [ S " ' S a p ' 0 ) ] = f u n c [ p ( 0 > ] , 2 0 where 'func' means f u n c t i o n . G e n e r a l l y s p e a k i n g the p r e c i s e f u n c t i o n a l form of f u n c [ p < 0 ) ] i s d e t e r m i n e d by the upstream flow c o n d i t i o n s . T h i s t h e s i s c o n s i d e r s a f u n c t i o n of the form - / c p ( 0 > , w i t h K a r e a l number. I t w i l l become o b v i o u s t h a t even a " l i n e a r f u n c t i o n as t h i s w i l l g i v e r i s e t o a n o n t r i v i a l upstream v e r t i c a l c u r r e n t s h e a r . The r e a l advantage of the l i n e a r i t y i s , of c o u r s e , t h a t i t p e r m i t s a n a l y t i c a l s o l u t i o n s . T h e r e f o r e the c o n s e r v a t i o n of p o t e n t i a l v o r t i c i t y reduces t o f i n d i n g the stream f u n c t i o n which s a t i s f i e s [ V 2 + d 3 ( S - } b 3 ) + K ] p t 0 > = 0 2 . 1 0 s u b j e c t t o a p p r o p r i a t e boundary c o n d i t i o n s . I d e a l l y the domain i n which the above e q u a t i o n s h o u l d be s o l v e d i s the s e m i - i n f i n i t e domain which i s bounded below by the bottom, above by the s u r f a c e and t o one s i d e by the c o a s t l i n e . However assuming t h a t the topography has compact su p p o r t and t h a t any t o p o g r a p h i c mean f l o w i n t e r a c t i o n must decay w i t h i n c r e a s i n g d i s t a n c e from the source topography i t f o l l o w s t h a t a s u i t a b l y l a r g e c h a n n e l p a r a l l e l t o the c o a s t l i n e can mimick the i d e a l domain. M a t h e m a t i c a l l y t h i s assumption a l l o w s the c r o s s stream c u r r e n t s t r u c t u r e t o be s o l v e d i n terms of c r o s s stream o r t h o g o n a l b a s i s f u n c t i o n s . Suppose t h a t the bottom of the ocean i s g i v e n by z = h ( x , y ) . When t h e r e i s no topography h ( x , y ) = 0 . The p o t e n t i a l v o r t i c i t y e q u a t i o n 2 . 1 0 i s s o l v e d i n the domain g i v e n by: 21 { ( x , y , z ) : -»<x<+», 0<y<2, h ( x , y ) < z < 1 }. The s e a w a r d c h a n n e l w a l l h a s been c h o s e n a s y=2. N u m e r i c a l c a l c u l a t i o n s , p r e s e n t e d i n c h a p t e r I V , c o n f i r m t h i s c h o i c e as s u i t a b l e . The b o u n d a r y c o n d i t i o n s w h i c h t h e p r e s s u r e f i e l d must s a t i s f y a r e now c o n s i d e r e d . L e t t h e u p s t r e a m o r f a r f i e l d c u r r e n t be g i v e n by u 0.= e x p ( - a y ) Z ( z ) v 0 = 0, where Z ( z ) r e p r e s e n t s t h e v e r t i c a l s t r u c t u r e o f t h e u p s t r e a m f l o w f i e l d . S i n c e t h e f a r f i e l d c u r r e n t i s assumed 0 ( 1 ) t h e n a s |x| -> », u t 0 ) —> u 0 and v ( 0 > —> 0. I n t e r m s o f t h e p r e s s u r e f i e l d t h e 0 ( 1 ) f a r f i e l d b o u n d a r y c o n d i t i o n i s : p ( 0 > _> a - i ( e x p [ - a y ] - 1)Z (Z) a s |x| -> » . The u p s t r e a m f l o w c o n d i t i o n c o r r e s p o n d s t o a h o r i z o n t a l l y a nd v e r t i c a l l y s h e a r e d c u r r e n t . The h o r i z o n t a l s h e a r i s e x p o n e n t i a l w i t h an e - f o l d i n g l e n g t h o f a " 1 . I n d i m e n s i o n a l n o t a t i o n t h i s w o u l d be a " 1 L m e t r e s . The v e r t i c a l s h e a r must be c h o s e n s o t h a t t h e p o t e n t i a l v o r t i c i t y a s s o c i a t e d w i t h t h i s f l o w i s c o n s i s t e n t w i t h l i n e a r i z a t i o n done t o t h e q u a s i - g e o s t r o p h i c p o t e n t i a l v o r t i c i t y e q u a t i o n . T a k i n g - 9 2 ( 2 . 1 0 ) i t f o l l o w s t h a t : 22 D 3 ( S " 1 D 3 Z ) + ( a 2 + K ) Z = 0 . A l o n g the b o u n d a r i e s of the c h a n n e l the normal component of the v e l o c i t y ' f i e l d must v a n i s h . On y=0 and y=2 t h i s means t h a t v t 0 ) must be i d e n t i c a l l y z e r o . On z=1, the sea s u r f a c e , w ( 0 ) = 0. In terms of the p r e s s u r e f i e l d : 3,p<°> = 0 on y=0 and y=2 J [ p < 0 1 , S - 1 9 3 p < 0 ' ] = 0 on z=1. 2.11 The boundary c o n d i t i o n s a t z = 1 and z = 0 a r e c a s t i n t o the form s u g g e s t e d by H o g g ( l 9 8 0 ) . The s u r f a c e boundary c o n d i t i o n i m p l i e s t h a t 3 3 p ( 0 > i s c o n s e r v e d a l o n g s t r e a m l i n e s on z=1. For th o s e s t r e a m l i n e s which o r i g i n a t e upstream 2.11 i n t e g r a t e s t o Z3 3p< 0 ' - p< 0 >D 3Z = 0 on z=1. The upstream v e r t i c a l c u r r e n t s t r u c t u r e f u n c t i o n t r i v i a l l y s a t i s f i e s t h i s boundary c o n d i t i o n . C o n s e q u e n t l y the boundary v a l u e of Z a t z=1 i s a f r e e parameter, chosen from o b s e r v a t i o n . A l o n g the bottom of the c h a n n e l the normal component of the v e l o c i t y f i e l d must v a n i s h . In d i m e n s i o n a l v a r i a b l e s t h i s can be w r i t t e n : w* = ( u * , v * ) ' V * h * ( x * , y * ) on z* = h * ( x * , y * ) . 23 D e f i n i n g h * ( x * , y * ) = e H h ( x , y ) , r e s u l t s i n t h e n o n d i m e n s i o n a l b o t t o m b o u n d a r y c o n d i t i o n : w = (u,v)»Vh(x,y) on z = e h ( x , y ) . L e t t h e maximum h e i g h t o f h * ( x * , y * ) be h 0 . F o r h ( x , y ) 0(1) i t i s r e q u i r e d t h a t e " 1 ( h 0 / H ) 0(1). T h i s i m p l i e s t h a t t h e h e i g h t o f t h e t o p o g r a p h y i s a t most 0 ( c ) w i t h r e s p e c t t o t h e mean d e p t h o f t h e o c e a n . T h i s p r o v i d e s f o r m a l j u s t i f i c a t i o n f o r e x p a n d i n g t h e b o t t o m b o u n d a r y c o n d i t i o n i n a T a y l o r s e r i e s a b o u t z=0 a s f o l l o w s : (w<°> + ...) + eh3 3(w<°> + ...) + ... = [ ( u ( 0 > + ... , v ( 0 » + ...) + e h 3 3 ( u ( 0 > + ... , v ( 0 ' + ...) + . . . ] - V h on z=0. The 0(1) b o t t o m b o u n d a r y c o n d i t i o n i s t h e r e f o r e g i v e n b y : w ( o > = ( u < o > f V < o ) ) . v h o n z = 0 f w h i c h i n t e r m s o f t h e p r e s s u r e f i e l d i s w r i t t e n , J [ p ( 0 > , 3 3 p ( 0 > + S ( 0 ) h ( x , y ) ] = 0 on z = 0. T h i s c a n be i n t e g r a t e d u p s t r e a m t o y i e l d Z 3 3 p ( 0 ) - p ( 0 ) D 3 Z = - Z ( z ) S ( z ) h ( x , y ) on z=0. 24 S i n c e h ( x , y ) h a s c o m p a c t s u p p o r t , a s |x| -> °°; h ( x , y ) = 0 i m p l y i n g Z ( z ) t r i v i a l l y s a t i s f i e s t h i s b o u n d a r y c o n d i t i o n . T h e r e f o r e t h e b o u n d a r y v a l u e o f Z ( z ) f o r z=0 i s a f r e e p a r a m e t e r o b t a i n e d f r o m o b s e r v a t i o n . The i d e a l i z e d t o p o g r a p h y h ( x , y ) h a s t h e f o r m , h ( x , y ) = h , c o s ( 7 r x ) c o s ( 7 r y ) f o r |x|<.5 0<y<.5 2.12 = h 2 c o s [ 4 7 r(x-. 6) ] c o s [ 4 7 r ( y - . 7 5 ) ] f o r |x-.6|<.125 a n d |y-.75|<.125 2.13 = 0 f o r a l l o t h e r x and y, 2.14 where h, and h 2 a r e t h e maximum h e i g h t s o f t h e c o n t i n e n t a l s l o p e p r o t r u s i o n a n d t h e P r a t t seamount r e s p e c t i v e l y , s c a l e d by t h e q u a n t i t y eH. F i g u r e 5 i s a c o n t o u r map o f t h e i d e a l i z e d t o p o g r a p h y , w i t h c o n t o u r i n g i n t e r v a l s o f 5 u n i t s (5eH m e t r e s ) . The s u p p o r t o f h ( x , y ) was o b t a i n e d by e x a m i n g F i g u r e 1 and e s t i m a t i n g t h e r e l e v a n t c o o r d i n a t e l e n g t h s . The f o r m o f h ( x , y ) g i v e n by 2.12 c o r r e s p o n d s t o t h e c o n t i n e n t a l s l o p e p r o t r u s i o n . The f o r m o f h ( x , y ) g i v e n by 2.13 m o d e l s t h e t h e c o l l e c t i o n o f s e a m o u n t s i n t h e i m m e d i a t e v i n c i n t y o f t h e P r a t t seamount a s a s m o o t h o r o g r a p h i c f e a t u r e w i t h t h e n o n d i m e n s i o n a l h e i g h t o f t h e P r a t t s e a m o u n t . The f o r m f o r h ( x , y ) g i v e n by 2.14 c o r r e s p o n d s t o t h e a b y s s a l p l a i n . F i n a l l y , t h e p r o b l e m f o r t h e 0 ( 1 ) f l o w f i e l d r e d u c e s t o 25 f i n d i n g t h e s o l u t i o n o f t h e f o l l o w i n g i n h o m o g e n e o u s H e l m o l t z b o u n d a r y v a l u e p r o b l e m : [ V 2 + a 3 ( s - 1 3 3 ) + K ] p ( 0 > = 0 2.15 s u b j e c t t o t h e b o u n d a r y c o n d i t i o n s d,p{0) = 0 on y=0 and y=2 2.16 z a 3 p ( 0 > - p ( 0 ) D 3 Z = 0 on z=1 2.17 Z 3 3 p ( 0 ) - p ( 0 ) D 3 Z = -ZSh on z=0 2.18 p ( 0 ' -> a - 1 ( e x p [ - a y ] - 1 ) Z ( Z ) a s |x| -> » 2.19 w i t h Z ( z ) t h e s o l u t i o n o f D 3 ( S " 1 D 3 Z ) + U + a 2 ) Z = 0 2.20 s u b j e c t t o t h e b o u n d a r y c o n d i t i o n s Z ( 0 ) = b and Z ( 1 ) = a. 2.21 27 A p p r o x i m a t e R e g i o n a l O c c u r r e n c e of V e l o c i t y P r o f i l e T y p e s 160°W 145°W 130°W G e o s t r o p h i c C u r r e n t (cm/sec) Re la t i ve to 2 0 0 0 d e c i b a r s 0 5 10 15 20 25-10-5 0 -5 0 5 -10-5 0 G r o u p I G roup II G r o u p III G roup IV F i g u r e 2 - H o r i z o n t a l and v e r t i c a l c u r r e n t s t r u c t u r e i n the n o r t h e a s t P a c i f i c Ocean F i g u r e 3 - B r u n t - V a i s a l a frequency CI F i g u r e 4 - Mean s t a t e d e n s i t y p r o f i l e 30 31 I I I . SOLUTION OF THE F I E L D EQUATIONS The l i n e a r i t y o f t h e b o u n d a r y v a l u e p r o b l e m 2.15 t h r o u g h 2.21 i s e x p l o i t e d by c o n s t r u c t i n g a s o l u t i o n i n t h e f o r m p(o> = a - 1 [ e x p ( - a y ) - 1 ] Z ( z ) + p ( x , y , z ) . S u b s t i t u t i n g t h i s e x p r e s s i o n i n t o 2.15 t h r o u g h 2.19 i m p l i e s t h a t p ( x , y , z ) must s o l v e : [ V 2 + 3 3 ( S " '83) + /c]p = 0 3.1 s u b j e c t t o t h e b o u n d a r y c o n d i t i o n s , 1 3,p = 0 on y=0 a n d y=2 3.2 Z 3 3 p - pD 3Z = 0 on z=1 3.3 Z 3 3 p - pD 3Z = -ZSh on z=0 3.4 p - > 0 a s | x | -> °°, 3.5 w i t h Z ( z ) t h e s o l u t i o n o f 2.20 a n d 2.21. The b o u n d a r y c o n d i t i o n s 3.2 a n d 3.5 i m p l y t h a t p=0 on y=0 an d y=2. T h i s i s b e c a u s e t h e b o u n d a r y p l a n e s y=0 a n d y=2 a r e p a r a l l e l t o t h e x - a x i s a n d t h e b o u n d a r y c o n d i t i o n 3.2 i m p l i e s t h a t t h e g r a d i e n t o f p i n t h e x d i r e c t i o n v a n i s h e s on t h e s e 32 p l a n e s . S i n c e p -> 0 a s |x| -> » t h e n p=0 i n d e n t i c a l l y on y=0 and y=2. The p h y s i c a l i m p l i c a t i o n i s t h a t no n e t down c h a n n e l t r a n s p o r t c a n be c r e a t e d f r o m t o p o g r a p h i c mean f l o w i n t e r a c t i o n . The f u n c t i o n p i s t h e 0 ( 1 ) e f f e c t o f t h e t o p o g r a p h y on t h e f a r f i e l d s t r e a m f u n c t i o n . No r e s t r i c t i o n i s made on t h e o r d e r o f m a g n i t u d e o f p ( x , y , z ) . I n f a c t p * o ( a " 1 { e x p [ - a y ] - 1}Z) i f f l o w r e v e r s a l i s t o o c c u r . The f u n c t i o n p ( x , y , z ) i s s o l v e d f o r v i a t h e f o l l o w i n g n o r m a l mode d e c o m p o s i t i o n (Chao e t a l . ; 1980) w i t h G ( z ) t h e ' n ' t h o r t h o n o r m a l e i g e n f u n c t i o n s o l u t i o n o f : OS p = I P ( x , y ) G ( z ) n=0 n n 3.6 n D 3 ( S " 1 D 3 G ) + X G = 0 3.7 n n n s u b j e c t t o t h e b o u n d a r y c o n d i t i o n s , ZD 3G - G D 3Z = 0 n n on z=0 a n d z=1 3.8 J G ( z ) G ( z ) d z = 6 3.9 nm where 6 i s t h e K r o n e c k e r d e l t a f u n c t i o n b e t w e e n n and m. nm 33 The g o v e r n i n g e q u a t i o n s f o r t h e c r o s s s t r e a m f u n c t i o n s P ( x , y ) a r e o b t a i n e d by m u l t i p l y i n g 3.7 by p ( x , y , z ) and n i n t e g r a t i n g f r o m 0 t o 1 w i t h r e s p e c t t o z i e . , J D 3 [ S - 1 D 3 G ]p dz + X } G p dz = 0. 0 n n 0 n I n t e g r a t i n g t h e f i r s t i n t e g r a n d by p a r t s t w i c e a n d u t i l i z i n g 3.1, 3.3 a n d 3.4 i t f o l l o w s t h a t : ( V 2 + K - X ) j G p dz = -G ( 0 ) h ( x , y ) . n 0 n n S i n c e t h e f u n c t i o n p i s a l i n e a r c o m b i n a t i o n o f t h e e i g e n f u n c t i o n s t h e a b o v e e q u a t i o n i m p l i e s t h a t f o r e a c h n ( V 2 + K - X )P = -G ( 0 ) h ( x , y ) 3.10 n n n s u b j e c t t o t h e b o u n d a r y c o n d i t i o n s P = 0 on y=0 a n d y=2 3.11 n P -> 0 a s |x| -> oo. n 3.12 34 The n o r m a l mode s o l u t i o n 3.6 h a s t h e p r o p e r t y t h a t 3 3 ( p / Z ) i s d i s c o n t i n u o u s a t z=0. The b o u n d a r y c o n d i t i o n 3.4 c a n be r e w r i t t e n i n t h e f o r m 3 3 ( p / Z ) = - S ( z ) h ( x , y ) / Z ( z ) on z=0 and OB Z P ( 3 . 8 ) c a n be w r i t t e n i n t h e f o r m 3 3 ( p / Z ) = 0 on z=0. n=0 n A s s u m i n g t h a t Z ( z ) assumes i t s b o u n d a r y c o n d i t i o n a t z=0 s m o o t h l y i t f o l l o w s t h a t 3 3 p ( z = 0 * ) - 3 3 p ( z = 0 ) = - S ( 0 ) h ( x , y ) . C o n s e q u e n t l y u ( 0 ) , v < 0 ) a n d p ( 0 > a r e c o n t i n u o u s a t z=0 w h i l e p < 0 > a n d w ( 0 ) , w h i c h d e p e n d on 3 3 p a r e n o t . The o r d e r o f m a g n i t u d e o f t h e d i s c o n t i n u i t y i n 3 3 p i s a b o u t 3 0 ' e ~ 1 f l r e s u l t i n g i n a n u m e r i c a l l y i n s i g n i f i c a n t c o r r e c t i o n t o t h e p ( 0 ) a n d w ( 0 ) f i e l d s a t z = 0. I n t h e r e m a i n d e r o f t h i s c h a p t e r t h e s o l u t i o n s f o r P ( x , y ) , n G ( z ) a n d Z ( z ) a r e d e r i v e d . I n s e c t i o n one t h e s o l u t i o n o f n 3.10, 3.11 and 3.12 i s o b t a i n e d . I n s e c t i o n two t h e p r o b l e m 2.20 a n d 2.21 i s s o l v e d . I n s e c t i o n t h r e e t h e p r o b l e m 3.7, 3.8 an d 3.9 i s s o l v e d . I n s e c t i o n f o u r t h e f o r m u l a e f o r t h e 0 ( 1 ) p r e s s u r e , v e l o c i t y , mass t r a n s p o r t a n d d e n s i t y f i e l d s a r e c o m p u t e d . 3.1 H o r i z o n t a l A m p l i t u d e F u n c t i o n s I n t h i s s e c t i o n t h e ' n ' t h c r o s s s t r e a m , o r t h e h o r i z o n t a l a m p l i t u d e f u n c t i o n P i s o b t a i n e d . T h e s e f u n c t i o n s a r e t h e n s o l u t i o n o f t h e two d i m e n s i o n a l i n h o m o g e n e o u s H e l m h o l t z p r o b l e m 3.10, 3.11 a n d 3.12. The i n h o m o g e n e i t y o r i n o t h e r w o r d s t h e f o r c i n g t e r m , c o r r e s p o n d s t o t h e e x c i t e m e n t o f t h e ' n ' t h e i g e n m o d e by t h e t o p o g r a p h y . I n t h e a b s e n c e o f t o p o g r a p h y i t i s 35 expected that the interact ion pressure f i e l d p (x ,y , z ) vanishes. Consider the following reformulation of 3.12. For every c > 0 there ex i s t s 6 < » such that for every |x | > 8 the maximum over ye[0,2] of |P (x ,y ) | s a t i s f i e s |P ( x , y ) | < e. It i s shown n n la ter in th i s section that X 0 = a 2 + K, hence X - K > 0 for a l l n n. In order to avoid d i f f i c u l t i e s with the s e m i - i n f i n i t e domain consider the problem of so lv ing 3.10, with h(x,y) = 0, in the domain given by {(x,y): |xj < 6 and 0 < y < 2], subject to 3.11 and the reformulated 3.12 for a given e. Applying the maximum p r i n c i p l e to th i s problem i t i s c lear that the maximum of |P | n must occur on |x | = 5. However th i s can be made a r b i t r a r i l y smal l , by e —> 0 and 6 —> °°. L i n e a r i t y and uniqueness therefore imply that the re lated homogeneous problem has only the t r i v i a l s o l u t i o n . This fact in turn implies that there exis ts a unique Green's function for the hor izonta l amplitude equations. The so lut ion of 3.10, 3.11 and 3.12 is constructed from the Green's funct ion , defined as g ( x , y | x 0 , y 0 ) . The Green's function s a t i s f i e s : (V 2 + K - X )g = -G ( 0 ) 6 ( x - x o ) 6 ( y - y o ) , 3.13 n n with 5(x-x 0 ) and 6(y - yo) the Dirac de l ta functions centered at x = x 0 and y = y 0 re spec t ive ly . Let F(g) be the Fourier transform with respect to x of 36 g(x,y|xo,y 0) r i e . OS F ( g ) = j" g ( x , y | x 0 , y 0 ) e x p ( i k x ) d x , " OB where i 2 = - 1 and k i s t h e t r a n s f o r m v a r i a b l e . T a k i n g t h e F o u r i e r t r a n s f o r m o f 3.13 a n d 3.11 w i t h r e s p e c t t o x y i e l d s , ( 3 2 2 + K - X - k 2 ) [ F ( g ) ] = -G ( 0 ) 6 ( y - y o ) e x p ( i x o k ) 3.14 n n s u b j e c t t o t h e b o u n d a r y c o n d i t i o n s : F ( g ) = 0 on y=0 a n d y=2. 3.15 The F o u r i e r t r a n s f o r m o f t h e G r e e n ' s f u n c t i o n i s o b t a i n e d i n t h e f o r m : oo F ( g ) = £ A ( x 0 , y 0 , k ) s i n ( m 7 r y / 2 ) . 3.16 m= 1 nm M o d e l l i n g t h e d o m ain a s a c h a n n e l f i n d s i t s f u l l e x p r e s s i o n i n 3.16. S i n c e 3.14 i s d e f i n e d on a b o u n d e d y i n t e r v a l , i t i s p o s s i b l e t o e x p r e s s t h e s o l u t i o n o f 3.14 a n d 3.15 a s a l i n e a r c o m b i n a t i o n o f a c o m p l e t e s e t o f b a s i s f u n c t i o n s . The o r t h o g o n a l b a s i s f u n c t i o n s sin(m7ry/2) a r e s u c h a s e t w h i c h t r i v i a l l y s a t i s f y t h e b o u n d a r y c o n d i t i o n s . 37 S u b s t i t u t i o n of 3.16 i n t o 3.14 i m p l i e s t h a t : OO I {U - X - k2 - ( 7 r m / 2 ) 2 ) A sin(m7ry/2)} = m=1 n nm - G (0)exp(ix 0k)6(y-y 0). n The o r t h o g o n a l i t y of the c r o s s stream b a s i s f u n c t i o n s i m p l i e s t h a t the c o e f f i c i e n t f u n c t i o n s A ( x 0 , y 0 , k ) i n 3.16 are g iven by nm A = 2G ( 0 ) s i n ( m 7 r y o / 2 ) e x p ( i x o k ) [ (m7r /2 ) 2 + k 2 + X - K ] " 1 . nm n n The G r e e n ' s f u n c t i o n i s r e l a t e d to i t s F o u r i e r t r a n s f o r m v i a g ( x » y | x o # y o ) = (2TT ) - 1 / F ( g ) e x p ( - i x k ) d k . - oo S u b s t i t u t i n g 3.16 i n t o t h i s e x p r e s s i o n y i e l d s , g = I {TT" 1 s i n (m7ry/2) s i n ( m 7 r y 0 / 2 )G ( 0 ) ' m=1 n OS / [ ( m 7 r / 2 ) 2 + k 2 + X - /cJ-'expf-Hx-XoUldk}. » n 38 T h e r e a r e no p o l e s i n t h e a b o v e i n t e g r a l s i n c e X 0 = a 2 + K. To s e e t h i s c o n s i d e r t h e f o l l o w i n g a r g u m e n t . D e f i n e a f u n c t i o n * ( z ) s u c h t h a t G ( z ) = * ( z ) Z ( z ) . S u b s t i t u t i n g t h i s f o r m o f G i n t o 3.7, 3.8 and 3.9, u t i l i z i n g 2.20, r e s u l t s i n D 3[Z 2(D 3<I> )/S] + Z 2 [ X - ( a 2 + i c ) ] * = 0, n n n D3<t> = 0 on z = 0 and z = 1 , n / Z24> $ dz = 6 . 0 n m nm I t f o l l o w s t h a t f o r e a c h n X = a 2 + K + } Z 2 S " 1 ( D 3 * ) 2 d z . The minimum e i g e n v a l u e X 0 i s o b t a i n e d by m i n i m i z i n g t h e abov e i n t e g r a l o v e r t h e s e t o f * 's f o r w h i c h D 3 $ = 0 on z = 0 n n and z = 1. S i n c e t h e i n t e g r a l i s p o s i t i v e d e f i n i t e ( S ( z ) > 0) t h e n i t s minimum i s c e r t a i n l y a c h i e v e d i f i t v a n i s h e s . I t v a n i s h e s f o r c o n s t a n t $ . The o n l y c o n s t a n t <i> t h a t c a n s a t i s f y n n t h e o r t h o n o r m a l i z a t i o n c o n d i t i o n i s [ f Z 2 d z ] " 1 / 2 . I t f o l l o w s 39 t h e r e f o r e t h a t G 0 ( z ) = Z ( z ) [ J Z 2 d z ] - ' / 2 a n d X 0 = a 2 + K . o E v a l u a t i n g t h e a b o v e G r e e n ' s f u n c t i o n i n t e g r a l i n t h e a b s e n c e o f a n y p o l e s t h e r e f o r e y i e l d s : g = I {G ( 0 ) [ (m7r/2) 2 + X - K ] - 1 s i n (m7ry/2 ) s i n (m7ry 0/2 ) • m= 1 n n e x p [ - | x - x 0 | ( ( m 7 r / 2 ) 2 + X - K ) 1 / 2 ] } . 3 .17 n T h e ' n ' t h h o r i z o n t a l a m p l i t u d e f u n c t i o n i s r e l a t e d t o i t s G r e e n ' s f u n c t i o n b y , P <x,y ) = / J* g ( x , y | x 0 , y o ) h ( x o , y o ) d y 0 d x o . 3 .18 n -oo o S u b s t i t u t i n g i n t o t h i s e q u a t i o n h ( x , y ) g i v e n b y 2 . 1 2 , 2 . 13 a n d 2.14 y i e l d s , .s s P ( x , y ) = h , J* / g ( x , y | x 0 ,y 0 ) c o s ( 7 r x 0 ) c o s (7 r y 0 ) d x 0 d y 0 -n © -s" h 2 / / g ( x , y | x 0 , y 0 ) c o s [ 4 7 r ( x 0 - . 6 ) ] c o s ( 4 7 r y 0 ) d x 0 d y 0 . 3 . 1 9 ..ur .Mir E q u a t i o n 3 . 1 9 i s t h e h o r i z o n t a l a m p l i t u d e f u n c t i o n a s s o c i a t e d w i t h ' n ' t h v e r t i c a l mode e i g e n f u n c t i o n G ( z ) . T h e s e f u n c t i o n s n d e s c r i b e t h e c r o s s c h a n n e l s t r u c t u r e o f t h e t o p o g r a p h i c a l l y 40 i n d u c e d p r e s s u r e f i e l d . 3.2 F a r F i e l d V e r t i c a l C u r r e n t S t r u c t u r e I n t h i s s e c t i o n t h e s o l u t i o n f o r t h e v e r t i c a l s t r u c t u r e o f t h e f a r f i e l d o r u p s t r e a m c u r r e n t i s o b t a i n e d . The m e t h o d o f s o l u t i o n f o r 2.20 and 2.21 i s s i m i l i a r t o t h a t u s e d by B r y a n and R i p a ( 1 9 7 8 ) i n t h e i r a n a l y s i s o f v e r t i c a l t e m p e r a t u r e s t r u c t u r e i n t h e n o r t h e a s t P a c i f i c O c e a n . L e t V ( z ) be t h a t f u n c t i o n d e f i n e d a s D 3 V ( z ) = Z ( z ) . S u b s t i t u t i n g t h i s d e f i n i t i o n i n t o 2.20 f o r Z ( z ) a l l o w s t h a t e q u a t i o n t o be i n t e g r a t e d o n c e t o y i e l d , D 3 3 V + (K + a 2 ) S ( z ) V ( z ) = c S ( z ) , 3.20 w i t h c a c o n s t a n t o f i n t e g r a t i o n . The s o l u t i o n o f 3.20 i s , o f c o u r s e , t h e sum o f t h e homogeneous s o l u t i o n a n d a p a r t i c u l a r s o l u t i o n . The p a r t i c u l a r s o l u t i o n c a n be w r i t t e n a s C / ( K + a 2 ) . S i n c e Z ( z ) = D 3 V ( z ) t h e r e i s no c o n t r i b u t i o n f r o m t h e p a r t i c u l a r s o l u t i o n t o t h e f u n c t i o n Z ( z ) . W i t h no l o s s o f g e n e r a l i t y t h e c o n s t a n t i n 3.20 c a n t h e r e f o r e be s e t i d e n t i c a l l y e q u a l t o z e r o . I t t u r n s o u t t h a t t h e c r i t i c a l p a r a m e t e r i n t h e q u a l i t a t i v e d e s c r i p t i o n o f Z ( z ) i s t h e s i g n o f (K + a 2 ) . C o n s i d e r e d now i s t h e c a s e when (K + a 2 ) > 0. A new i n d e p e n d e n t v a r i a b l e t i s d e f i n e d so t h a t t = 2 T - ' [ U + a 2 ) S ( z ) ] " 2 V ( t ) = V [ z ( t ) ] . 41 T h i s t r a n s f o r m a t i o n r e s u l t s i n 3.20 b e i n g r e c a s t i n t o t h e f o r m : w h e r e V a n d V ' a r e t h e f i r s t a n d s e c o n d d e r i v a t i v e s o f V ( t ) w i t h r e s p e c t t o t , r e s p e c t i v e l y . E q u a t i o n 3.21 i s B e s s e l ' s e q u a t i o n o f o r d e r z e r o . T h e g e n e r a l s o l u t i o n o f 3.21 i s t h e r e f o r e g i v e n b y w h e r e J 0 a n d Y 0 a r e B e s s e l f u n c t i o n s o f t h e f i r s t a n d s e c o n d k i n d o f o r d e r z e r o r e s p e c t i v e l y , a n d w h e r e c , a n d c 2 a r e c o n s t a n t s o f i n t e g r a t i o n . T a k i n g t h e d e r i v a t i v e o f 3.22 r e s p e c t t o z i m p l i e s t h a t Z ( z ) c a n b e w r i t t e n : Z ( z ) = e x p ( 7 z / 2 ) { C , J , [ X e x p ( 7 z / 2 ) ] .+ C 2 Y , [ X e x p ( 7 z / 2 ) ] } , 3.23 w h e r e J , a n d Y, a r e B e s s e l f u n c t i o n s o f t h e f i r s t a n d s e c o n d k i n d o f o r d e r o n e r e s p e c t i v e l y , a n d C, a n d C 2 a r e c o n s t a n t s d e t e r m i n e d b y b o u n d a r y c o n d i t i o n s , a n d f i n a l l y w h e r e t 2 V ' + t V + t 2 V = 0, 3.21 V ( t ) = c , J 0 ( t ) + c 2 Y 0 ( t ) , 3.22 X = ( 2/ 7)[s 0U + a 2 ) e x p ( - 7 ) ] W 2 . 3.24 A p p l i c a t i o n o f t h e b o u n d a r y c o n d i t i o n s 2.21 i m p l i e s : C, = C 0 { b Y , [ X e x p ( 7 / 2 ) ] - a Y , [ X ] e x p ( - 7 / 2 ) } 3.25 42 C 2 = C 0 { a J , [ X ] e x p ( - 7 / 2 ) - b J , [ X e x p ( 7 / 2 ) ] } 3.26 w h e r e C 0 = ( J j X j Y j X e x p ^ ) ] - J 1 [ X e x p ( 7 / 2 ) ] Y , [ X ] } - 1 . 3.27 I n t h e c a s e w hen K + a 2 < 0 t h e n t h e s o l u t i o n i s w r i t t e n i n t e r m s I< a n d K 1 f t h e m o d i f i e d B e s s e l f u n c t i o n s o f t h e f i r s t a n d s e c o n d k i n d o f o r d e r o n e r e s p e c t i v e l y . S p e c i f i c a l l y , t h e s o l u t i o n s a r e g i v e n b y 3 . 2 3 , 3 . 2 5 , 3.26 a n d 3.27 w i t h t h e c h a n g e s I n t h e c a s e when K + a 2 = 0 t h e b a r o c l i n i c s o l u t i o n t o 2.20 a n d 2.21 i s e a s i l y c o m p u t e d t o b e : J - > I i a n d Y1 —> K X = ( 2 / 7 ) [ S 0 | K + a 2 | e x p ( - 7 ) ] V 2 . Z ( z ) = C , e x p ( 7 z ) + C 2 3.28 C, = ( b - a ) C 0 3 . 2 9 C 2 = [ a - b e x p ( 7 ) ] C 0 3.30 C 0 = [1 - e x p ( 7 ) ] " 1 . 3.31 43 The a b o v e s o l u t i o n h a s a s a s p e c i a l c a s e t h e s i t u a t i o n where t h e u p s t r e a m c u r r e n t i s b a r o t r o p i c . I n s u c h a s i t u a t i o n , Z ( z ) = c w i t h a = b = c w i l l i m p l y t h a t C, = 0 a n d C 2 = c . I n f a c t , a s s u m i n g t h e u p s t r e a m c u r r e n t t o be b a r o t r o p i c i m p l i e s t h a t K = - a 2 . T h i s c a n be s e e n by e x a m i n i n g 2.20. I f Z ( z ) = c t h e n D 3Z = 0. The o n l y way 2.20 c a n be s a t i s f i e d i s i f c = 0 ( w h i c h i s u n i n t e r e s t i n g ) o r i f K + a2 = 0. N u m e r i c a l c a l c u l a t i o n s show t h a t t h e q u a l i t a t i v e s hape o f Z ( z ) c h a n g e s l i t t l e w i t h e i t h e r o f t h e f i r s t two c a s e s . T y p i c a l p a r a m e t e r v a l u e s f o r t h e n o r t h e a s t P a c i f i c Ocean s u g g e s t t h a t K + a2 > 0. C o n s e q u e n t l y f u r t h e r a n a l y s i s i s r e s t r i c t e d t o t h a t c a s e . F i g u r e 6 i s a g r a p h o f Z ( z ) v s . z f o r t h e c a s e K + a2 > 0, w i t h a = .1 and b = . 0 1 . I n t h e r e g i o n where S ( z ) 0 i t i s e x p e c t e d t h a t Z ( z ) = c o n s t a n t b e c a u s e , r o u g h l y s p e a k i n g , i n t h i s r e g i m e V ( z ) , t h e s o l u t i o n o f 3.20, must be n e a r l y l i n e a r i n z i m p l y i n g t h a t Z ( z ) i s c o n s t a n t . The p h y s i c a l i n t e r p e t a t i o n f o r Z ( z ) b e i n g a p p r o x i m a t e l y c o n s t a n t i n t h i s r e g i o n i s t h a t i n t h i s r e g i m e t h e f l u i d i s v i r t u a l l y homogeneous i m p l y i n g t h a t t h e h o r i z o n t a l v e l o c i t y f i e l d i s n e a r l y b a r o t r o p i c . R a p i d c h a n g e s i n Z ( z ) a r e t h e r e f o r e c o n s t r a i n e d t o a r e g i o n o f n o n z e r o B r u n t -V a i s a l a f r e q u e n c y . F i g u r e 6 i l l u s t r a t e s t h i s w i t h t h e s h a p e s t g r a d i e n t i n Z ( z ) o c c u r i n g i n t h e u p p e r o c e a n . 44 3.3 V e r t i c a l Mode E i g e n f u n c t i o n s And E i g e n v a l u e s The o r t h o n o r m a l e i g e n f u n c t i o n s a r e t h e s o l u t i o n s o f 3.7, 3.8 and 3.9. T h e s e e q u a t i o n s a r e s o l v e d u s i n g t h e same t e c h n i q u e u s e d t o o b t a i n t h e u p s t r e a m v e r t i c a l c u r r e n t s t r u c t u r e f u n c t i o n s . F i r s t t h e f u n c t i o n V ( z ) i s d e f i n e d so t h a t n D 3V ( z ) = G ( z ) . S u b s t i t u t i n g t h i s d e f i n i t i o n i n t o 3.7 i m p l i e s n n t h a t s o l v i n g 3.7 r e d u c e s t o s o l v i n g , D 3 3 V ( z ) + X S ( z ) V ( z ) = 0. 3.32 n n n The t r a n s f o r m a t i o n t = ( 2 / 7 ) [ X S ( z ) ] 1 ' 2 n V ( t ) = V [ z ( t ) ] f n n a l l o w s 3.32 t o be r e w r i t t e n t 2 ( V ) " + t ( V )' + t 2 V = 0 , n n n w h e r e , a s b e f o r e , (V ) ' ' a n d (V )' a r e t h e f i r s t a n d s e c o n d n n d e r i v a t i v e s o f V ( t ) w i t h r e s p e c t t o t , r e s p e c t i v e l y . T h i s i s , n o f c o u r s e , B e s s e l ' s e q u a t i o n o f o r d e r z e r o . T h e r e f o r e t h e g e n e r a l s o l u t i o n f o r V ( t ) i s g i v e n b y , n 45 v ( t ) = c , j 0 ( t ) + c 2 y 0 ( t ) , n which i n t u r n i m p l i e s t h a t G (z) i s g i v e n by: n G (z) = e x p ( 7 z / 2 ) { A J , [ c e x p ( 7 z / 2 ) ] + B Y,[c e x p ( 7 z / 2 ) ] } , 3.33 n n n n n c = ( 2 / 7 ) [ X S o e x p C ' - y ) ] 1 / 2 . 3.34 n n Here J 0 , J i and Y 0,Y, are B e s s e l f u n c t i o n s of the f i r s t and second k i n d of o r d e r z e r o and one r e s p e c t i v e l y . The requirement t h a t the G 's be o r t h o n o r m a l ( i e . 3.9) n i m p l i e s t h a t 1 = (A ) 2 } e x p ( 7 z ) { J 0 [ c e x p ( 7 z / 2 ) ] } 2 d z + n 0 n (B ) 2 J e x p ( 7 z ) { Y 0 [ c e x p ( 7 z / 2 ) ] } 2 d z . 3.35 n 0 n These i n t e g r a l s a re e a s i l y e v a l u a t e d by i n t r o d u c i n g the change of v a r i a b l e x = c e x p ( 7 z / 2 ) . L e t a and b be the numbers n n n d e f i n e d as Cn «Xpl»/2) a = 2 [ ( c ) 2 7 ] ' 1 / x { J 1 ( x ) } 2 d x n n c„ 46 b = 2[(c ) 2 7 ] - 1 / x{Y,(x')}2dx n n r which can be integrated exactly to give, a = {(c ) 2 7 } - , [ { x J 0 ( x ) } 2 - 2 J 0 ( x ) J 1 ( x ) + {xJ,(x)} 2] 3.36 n n r C n€Xp(Jf/2) b = {(c ) 2 7 } " 1 [ { x Y 0 ( x ) } 2 - 2Y 0(x)Y 1(x) + {xj,(x)} 2] . 3.37 n n -With the aid of 3.36 and 3.37 the orthonormal condition 3.35 can be succinctly rewritten as (A ) 2a + (B ) 2b = 1. 3.38 n n n n The boundary conditions 3.8 implies that at z=0 A {(c 7 / 2 ) J 0 ( c ) - [Z'(0)/Z(0)]J,(c )} + n n n n B {(c 7 / 2 ) Y 0 ( c ) - [Z'(0)/Z(0)]Y,(c )} = 0, 3.39 n n n n and that at z=1 A {(c 7 / 2 ) e x p ( 7 / 2 ) J 0 [ c exp( 7/2)] -n n n [Z'(l)/Z(1)]J,[c exp( 7/2)]} + n 47 B {(c 7/2)exp(7/2)Y 0 [c exp(>/2)] -n n n [ Z ' ( 1 ) / Z ( l ) ] Y , [ c exp( 7 /2)]} = 0. 3.40 n Viewing 3.39 and 3.40 as a system of two equations in the unknowns A and B , the condit ion that there exis t n o n t r i v i a l n n solut ions for A and B is that the determinant of the n n c o e f f i c i e n t s must vanish i e . {(c 7 /2 )exp( 7 / 2 )Y 0 [ c exp( 7 /2 ) ] - [Z ' (1 ) /Z(1 ) ]Y , [ c exp( 7 / 2 ) ] } . n n n {(c 7 / 2 ) J 0 ( c ) - [ Z , ( 0 ) / Z ( 0 ) ] J , ( c )} = n n n {(c 7 / 2 ) e x p ( 7 / 2 ) J 0 [ c exp( 7 /2 ) ] - [ Z ' ( 1 ) / Z ( 1 ) ] J , [ c exp( 7 /2) ]} -n n n {(c 7 / 2 ) Y 0 ( c ) - [Z ' (0 ) /Z(0 ) ]Y , ( c )}. 3.41 n n n Equation 3.41 forms a transcendental equation for the parameter c . The allowed c 's form a d i scre te denumerable set of rea l n n numbers, with the eigenvalue X re la ted to c by inver t ing 3.34 n n to obtain A = [ 7 c / 2 ] 2 ( s 0 ) - ' e x p ( y ) . 3.42 n n The constants A and B are obtained from the normalization n n 48 c o n d i t i o n 3.38 a n d e i t h e r 3.39 o r 3.40. T h i s c a l c u l a t i o n i m p l i e s t h a t : A = { ( c 7 / 2 ) Y 0 ( c ) - [ Z * ( 0 ) / Z ( 0 ) ] Y , ( c ) } / C , 3.43 n n n n n B = - { ( c 7 / 2 ) J 0 ( c ) - [ Z ' ( 0 ) / Z ( 0 ) ] J , ( c ) } / C , 3.44 n n n n n w i t h C = {b [ ( c 7 / 2 ) J 0 ( c ) - [ Z * ( 0 ) / Z ( 0 ) ] J , ( c ) ] 2 + n n n n n a [ ( c 7 / 2 ) Y 0 ( c ) - [ Z ' ( 0 ) / Z ( 6 ) ] Y , ( c ) ] 2 } 1 ' 2 . 3.45 n n n n To s u m m a r i z e , t h e v e r t i c a l e i g e n f u n c t i o n s a r e g i v e n by 3.33 where A and B a r e g i v e n by 3.43 and 3.44 r e s p e c t i v e l y , and n n where t h e e i g e n v a l u e s X a r e t h e s o l u t i o n s o f 3.41 and 3.42. n F i g u r e s 7, 8 a n d 9 a r e g r a p h s o f G 0 ( z ) , G , ( z ) a n d G 2 ( z ) v s . z, r e s p e c t i v e l y f o r a = .1 a n d b = .01. The q u a l i t a t i v e comments made i n t h e l a s t s e c t i o n a b o u t t h e s t r u c t u r e o f Z ( z ) a r e a l s o v a l i d f o r t h e e i g e n f u n c t i o n s . I n t h e r e g i o n where S ( z ) 0 i t i s e x p e c t e d t h a t G ( z ) s h o u l d v a r y n s l o w l y . C o n s e q u e n t l y t h e z e r o s o f t h e e i g e n f u n c t i o n s c o n c o m i t a n t w i t h o s c i l l a t o r y b e h a v i o u r s h o u l d o c c u r i n t h e u p p e r o c e a n . T h i s c a n be s e e n i n G 1 ( z ) w i t h a z e r o a t z = .9 and i n G 2 ( z ) w i t h i t s two z e r o s a t a p p r o x i m a t e l y .95 a n d .7. G r a p h s c o m p u t e d o f t h e h i g h e r modes a l s o h a v e t h i s f e a t u r e . 49 3.4 F o r m u l a e F o r The P r e s s u r e , V e l o c i t y , D e n s i t y And Mass  T r a n s p o r t F i e l d s W i t h t h e s o l u t i o n f o r t h e i n t e r a c t i o n s t r e a m f u n c t i o n o b t a i n e d i n t h e p r e v i o u s s e c t i o n s , i t i s now p o s s i b l e t o d e s c r i b e t h e p h y s i c a l v a r i a b l e s : p r e s s u r e , v e l o c i t y , d e n s i t y and mass t r a n s p o r t . R e c a l l t h a t t h e 0 ( 1 ) p r e s s u r e f i e l d i s g i v e n by; > ( 0 ) = a ' 1 [ e x p ( - a y ) - 1 ] Z ( Z ) + Z P ( x , y ) G ( z ) , 3.46 n=0 n n where t h e f u n c t i o n s Z ( z ) , G ( z ) and P (x , y ) have been o b t a i n e d n n i n t h e t h r e e p r e v i o u s s e c t i o n s . The r e m a i n i n g c a l c u l a t i o n r e q u i r e d f o r t h e s t r e a m f u n c t i o n i s t h e e v a l u a t i o n of t h e i n t e g r a l 3.19, wh i c h y i e l d s : P = h,G (0)sinUy)R ( x ) [ 2 7 r ( i r 2 + X - K)(2TT 2 + X - K ) ] " 1 + n n n2 n n Z h,G (0 ) s i n ( m 7 r y / 2 ) R (x)[m/2 - s i n ( m 7 r / 4 ) ] m=1 n nm U(m2/4 - D ( (m7r/2) 2 + X - K ) ( ( m 7 r / 2 ) 2 + TT2 + X - K ) ] ' 1 n n OD Z 4 h 2 G ( 0 ) s i n ( m 7 r y / 2 ) S (x ) [sin(7mn/1 6) + s i n (5?rm/l 6) ] m=1 n nm 50 [ 7 r ( m 2/4 - i ) ( ( m 7 r / 2 ) 2 + X - K)(16TT 2 + ( m i r / 2 ) 2 + X - K ) ] ' \ n n where t h e f u n c t i o n s R a n d S a r e g i v e n by nm nm R ( x ) = 27rexp[-|x | ( ( m 7 r/2) 2 + X - K ) 1 ' 2 ] . nm n c o s h [ ( ( m f f / 2 ) 2 + X - K ) 1 / 2 / 2 ] i f |x| > .5 n R ( x ) = 2 ( ( m 7 r/2) 2 + X - K ) 1 / 2 COS(TTX) + nm n 27rexp[-( (m7r/2) 2 + X - K ) 1 / 2 / 2 ] « n c o s h [ x ( ( m j r / 2 ) 2 + X - K ) 1 / 2 ] i f | x | < .5 n S ( x ) = 87rexp[-|x-.6| ((m7r/2) 2 + X - K ) 1 ' 2 -nm n c o s h [ ( ( m 7 r/2) 2 + X - K ) 1 ' 2 / 8 ] i f | x - . 6 | > 1/8 n S ( x ) = 2 ( ( m 7 r / 4 ) 2 + X - K) " 2 c o s [ 4TT{ X-. 6) ] + nm n 87rexp[-( (m7r/2) 2 + X - K) 1 ' 2 / 8 ] -c o s h [ ( x - . 6 ) ( ( m 7 r / 2 ) 2 + X - K ) " 2 ] i f | x - . 6 | < 1/8. n The 0 ( 1 ) a l o n g s h o r e v e l o c i t y u < 0 > i s g i v e n by 2.2 a n d 3 . 4 6 , i m p l y i n g ; 51 u ( 0 > = Z ( z ) e x p ( - a y ) - Z 9 2 P G , 3.47 n = 0 n n where 9 2 P i s g i v e n b y : n 9 2 P = h,G ( 0 ) c o s ( i r y ) R [ 2 ( T T 2 + X - K ) ( 2 I T 2 + X - K ) ] ' 1 + n n n2 n n OD Z h,mG ( 0 ) c o s ( n \ 7 r y / 2 ) R [m/2 - sin (m7r/4)]» m= 1 n nm Tr\tZ [ 2 ( ( m i / 2 ) 2 + X - * ) ( m 2 / 4 - 1 ) ( ( m 7 r/2) 2 + TT2 + X - K ) ] " 1 -n n CD Z 2h 2mG ( 0 ) c o s ( m 7 r y / 2 ) S [ s i n ( 77rm/l 6 ) + s i n (5mjr/i 6) ] • m= 1 n nm [ ( m 2 / 4 - 16 ) ( ( m 7 r/2) 2 + X - »c)((mir/2)2 + 16TT 2 + X - K ) ] " 1 . n n The 0 ( 1 ) c r o s s c h a n n e l v e l o c i t y v ( 0 ) i s g i v e n by 2.1 a n d 3.46, i m p l y i n g ; v ( 0 > = Z 3,P G , 3.48 n = 0 n n where 9,? i s g i v e n b y : n 3,P = h,G ( 0 ) s i n ( 7 r y ) D 1 R [ 2 * r ( j r 2 + X - K ) ( 2 T T 2 + X - / c ) ] " 1 + n n n2 n n 52 L h,G ( 0 ) s i n ( m 7 r y / 2 ) D 1 R [m/2 - s i n ( m i r / 4 ) ] » m=1 n nm m*2 [ 7 r(m 2/4 - 1 ) ( ( m 7 r / 2 ) 2 + X - K ) ( ( m 7 r / 2 ) 2 + i r 2 + X - K ) ] ' 1 n n oo I 4 h 2 G ( 0 ) s i n ( m 7 r y / 2 ) D 1 S [ s i n ( 7irm/1 6 ) + s i n ( 57rm/l 6 ) ] m=1 n nm m*2 [;r(m 2/4 - l ) ( ( m i r / 2 ) 2 + X - K)( 1 6 T T 2 + (m7r/2) 2 + X - * ) ] n n where t h e d e r i v a t i v e s o f R a n d S w i t h r e s p e c t t o x a r e nm nm by; D,R ( x ) = - 2 i r ( (nnr/2) 2 + X - K ) 1 ' 2 -nm n s g n ( x ) e x p [ - | x | ( (m7r/2) 2 + X - K ) 1 ' 2 ] . n c o s h [ ( ( n n r / 2 ) 2 + X - K)'/2/2] i f |x| > .5 n D,R ( x ) = -2TT( ( m 7 r/2) 2 + X - K) 1 > 2 ( s i n (TTX) -nm n e x p [ - ( (m7r/2) 2 + X - K ) 1 ' 2 / 2 ] -n s i n h [ x ( (m7r/2) 2 + X - K ) 1 ' 2 ] ) i f | x n D,S ( x ) = -8JT( ( m i r / 2 ) 2 + X - / c ) 1 / 2 -nm n s g n ( x - . 6 ) e x p [ - | x - . 6 | ( ( m 7 r / 2 ) 2 + X - K ) 1 ' 2 . n 53 cosh[ ( ( m 7 r / 2 ) 2 + X - K)''2/B] i f |x-.6| > 1/8 n D,S (x) = -8TT( (m»r/4) 2 + X - K ) 1 / 2 ( s i n [ 4TT (x-. 6 ) ] -nm n e x p [ - ( (m7r/2) 2 + X - / c ) 1 ' 2 ^ ] -n s i n h [ (x-.6) ( (m7r/2) 2 + X - K ) 1 ' 2 ] ) i f |x-.6| < 1/8, n w i t h sgn(x) b e i n g the s i g n of the v a r i a b l e x. The 0(1) d e n s i t y f i e l d p ( 0 ) i s o b t a i n e d from 2.3 and 3.36, i m p l y i n g t h a t ; OS p ( o > = a-'M - e x p ( - a y ) ] D 3 Z - L P D 3G 3.49 n = 0 n n where D 3 Z ( z ) and D 3G (z) are g i v e n by n D 3G (z) = (c 7 / 2 ) e x p ( 7 Z ) ( A J 0 [ c e x p ( 7 z / 2 ) ] + B Y 0 [ c e x p ( 7 z / 2 ) ] ) , n n n n n n D 3 Z ( z ) = ( X 7 / 2 ) e x p ( 7 z ) ( C 1 J 0 [ X e x p ( T z / 2 ) ] + C 2 Y 0 [ X e x p ( 7 z / 2 ) ] ) , w i t h the c o n s t a n t s A , B , C, and C 2 d e f i n e d by 3.43, 3.44, 3.25 n n and 3.26 r e s p e c t i v e l y . The 0(1) v e r t i c a l v e l o c i t y w ( 0 ) g i v e n by 2.5 and 3.46, can be r e w r i t t e n i n the form; 54 w ( 0> = [ S ( z ) ] - 1 J [ p ( 0 ) , p ( 0 > ] 3.50 w i t h p < 0 ! g i v e n by 3.49. The v e c t o r v a l u e d n o n d i m e n s i o n a l 0 ( 1 ) mass t r a n s p o r t i s d e f i n e d a s M = ( m 1 , m 2 ) , w i t h t h e c o m p o n e n t s m, and m 2 g i v e n by; m, = } u l 0 >dz 3.51 o m 2 = } v ( 0 ' d z . 3.52 o T h e s e i n t e g r a l s c a n be e v a l u a t e d t o y i e l d m, = ( X 7 / 2 ) " 1 e x p ( - a y ) [ C , J 0 ( x ) + C 2 Y 0 ( x ) ] Z 3 2 P (c 7 / 2 ) - 1 [ A J 0 ( x ) + B Y 0 ( x ) ] " , n = 0 n n n n cne*pltf/2-) 0 0 C r\ m 2 = Z 3,P (c 7 / 2 ) - 1 [ A J 0 ( x ) + B Y 0 ( x ) ] n=0 n n n n CnexpU/2) w i t h x t h e dummy i n t e g r a t i o n v a r i a b l e . F i g u r e 6 - Far f i e l d v e r t i c a l c u r r e n t s t r u c t u r e a •tf: CD CM CO I 1 LD a i a I 0 .B75 — 1 0 .75 ~ l 0 .625 1 0.5 Z RXJS "~1 0 .375 ! .0 0 .25 0 .125 F i g u r e 7 - Graph of G 0 ( z ) v s . z CD .1 .0 — 1 1 — 0.675 0.75 ~1 0 .625 - 1 0.5 AXIS T T 0.375 0.25 ~ l 1 0.125 0. F i g u r e 8 - Graph of G,(z) v s . z o F i g u r e 9 - Graph of G 2 ( z ) v s . z 59 I V . PARAMETER S E N S I T I V I T Y ANALYSIS -The i n t e n t o f t h i s c h a p t e r i s two f o l d . F i r s t , a d e s c r i p t i o n o f t h e s o l u t i o n s o b t a i n e d l a s t c h a p t e r i s g i v e n b a s e d on p a r a m e t e r v a l u e s o b t a i n e d f r o m e x a m i n i n g t h e a v a l i a b l e d a t a f o r t h e n o r t h e a s t P a c i f i c O c e a n . S e c o n d , t h i s c h a p t e r d i s c u s s e s t h e e f f e c t o f v a r i a t i o n s i n t h e p a r a m e t e r s on t h e s t r u c t u r e o f t h e p r e v i o u s l y o b t a i n e d s o l u t i o n s . I n p a r t i c u l a r f i v e t a b l e s o f p a r a m e t e r c a l c u l a t i o n s r e p r e s e n t i n g 97 n u m e r i c a l s i m u l a t i o n s o f t h e e n t i r e s o l u t i o n were p r e f o r m e d . T h e s e c a l c u l a t i o n s w e r e p r e f o r m e d on t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a ' s Amdol 470 u s i n g d o u b l e p r e c i s i o n w a t f i v . N u m e r i c a l v a l u e s f o r t h e n o n d i m e n s i o n a l o b s e r v a b l e s w e re c o m p u t e d a t g r i d l o c a t i o n s d e n o t e d ( i , j , k ) l o c a t e d i n t h e d o m a i n a t (-2 + i 6 x , j 6 y , k 6 z ) w i t h ( 6 x , 6 y , 5 z ) = (.2,.1,.1) and i , j , k t a k i n g on t h e i n t e g e r r a n g e s 0 ^ i £ 20, 0 £ j ^ 20 a n d 0 ^ k < 10. The number o f c r o s s c h a n n e l modes u s e d t o compute 3.16 was 20 and t h e number o f v e r t i c a l e i g e n m o d e s u s e d t o compute 3.6 was 6. A d d i n g a d d i t i o n a l modes h a d a b s o l u t e l y no e f f e c t on t h e c o m p u t e d o b s e r v a b l e s t o d o u b l e p r e c i s i o n a c c u r a c y . I n a d d i t i o n t o n u m e r i c a l v a l u e s t h e 0 ( 1 ) s t r e a m f u n c t i o n a n d d e n s i t y f i e l d s were c o n t o u r e d and s t i c k d i a g r a m s o f t h e 0 ( 1 ) v e l o c i t y a n d mass t r a n s p o r t f i e l d s were c o m p u t e d t o p r o v i d e a g r a p h i c a l r e p r e s e n t a t i o n ' o f t h e r e s u l t i n g f l o w f i e l d . The f i v e p a r a m e t e r p a i r g r o u p s e x a m i n e d , i n o r d e r o f t h e i r d i s c u s s i o n , a r e : t h e f a r f i e l d h o r i z o n t a l c u r r e n t s h e a r and t h e R o s s b y number ( a , e ) , t h e t o p o g r a p h i c p a r a m e t e r s ( h 1 , h 2 ) , t h e 60 B r u n t - V a i s a l a frequency parameters (N 0,7*), the f a r f i e l d h o r i z o n t a l c u r r e n t shear and the f a r f i e l d s u r f a c e c u r r e n t (a,a) and f i n a l l y the f a r f i e l d h o r i z o n t a l c u r r e n t shear and the f a r f i e l d bottom c u r r e n t ( a , b ) . In each parameter p a i r a n a l y s i s the other parameters were he l d f i x e d . The standard values assumed were based on the topographic and oceanographic o b s e r v a t i o n s c o n t a i n e d i n Bennett(1959) , Tabata(1982), Emery e t a L (1983) and Will m o t t and Mysak(l980) of the north east P a c i f i c Ocean, d i s c u s s e d i n s e c t i o n 2.1. The s c a l i n g parameters H, L and U were h e l d f i x e d at 3500 m, 400 km and 1 m s " 1 r e s p e c t i v e l y , r e s u l t i n g i n a Rossby number e of 2.1-10 - 2. The standard set of valu e s f o r the parameters N 0, ( 7 * ) ~ 1 , a, b, a, K, h, and h 2 were .011045 s ~ 1 , 254.51 m, 0.1, 0.01, 5.0, 0.0, 10.9 and 34.1 r e s p e c t i v e l y . T h i s set of parameters y i e l d numerical values of the f i r s t s i x eige n v a l u e s of: X 0 = 25, X, = 382.86, X 2 = 2219.01, X 3 = 5584.99, X, = 10394.55 and X 5 = 16642.86. In the standard set of parameters the upstream v o r t i c i t y constant K has been set i d e n t i c a l l y equal to z e r o . A b r i e f comment i s necessary to o u t l i n e the reasons f o r t h i s c h o i c e . The parameter K cannot be set equal to zero i f the upstream c u r r e n t i s b a r o t r o p i c . In S e c t i o n 3.2 i t was shown that K = - a 2 i f Z ( z ) i s given by a c o n s t a n t . However, examining F i g u r e 2, i t i s obvious that the c u r r e n t p r o f i l e s i n the north east P a c i f i c are b a r o c l i n i c . The parameter K e f f e c t s a b a r o c l i n i c upstream c u r r e n t 61 t h r ough the q u a n t i t y a2 + K i n 2.20. N u m e r i c a l v a l u e s f o r a based on the o b s e r v a t i o n s d e s c r i b e d i n T a b a t a ( l 8 9 2 ) and Benn e t t ( 1 9 5 9 ) suggest t h a t a 5, so t h a t the e - f o l d i n g d i s t a n c e i n the h o r i z o n t a l l y s heared f a r f i e l d c u r r e n t i s about 80 km. N u m e r i c a l c a l c u l a t i o n s of the f a r f i e l d c u r r e n t show t h a t the v e r t i c a l c u r r e n t s t r u c t u r e v a r i e s l i t t l e f o r d i f f e r e n t v a l u e s of K p r o v i d e d |K| << a 2 . The p r i n c i p l e mechanism by which K e f f e c t s the t o p o g r a p h i c a l l y i n d uced f l o w f i e l d i s i n the denominator of the Green's f u n c t i o n g ( x , y | x 0 , y 0 ) g i v e n by 3.17. I t was shown i n Chapter I I I t h a t X 0 = a2 + K and t h a t G 0 ( z ) = Z ( z ) [ ) Z 2 ( z ) d z ] - 1 / 2 . o For a g i v e n ( x , x 0 , y , y 0 ) the magnitude of the Green's f u n c t i o n | g ( x , y | x 0 , y 0 ) I w i l l be maximized when the denominator i n 3.17 i s m i n i m i z e d , o c c u r i n g f o r n = 0. T h i s i s c o n s i s t e n t w i t h the p h y s i c a l i n t u i t i o n t h a t the p r i n c i p l e response of the upstream f l o w t o the t o p o g r a p h i c f o r c i n g w i l l the g r a v e s t (n=0) mode. When n = 0 the denominator of 3.17 w i l l s a t i s f y (m7r/2) 2 + X 0 - K £ a 2 . C o n s e q u e n t l y the g r a v e s t mode response of the ocean t o the topography i s v i r t u a l l y u n a f f e c t e d by K. R i g o r o u s l y s p e a k i n g the parameter K does e f f e c t the n = 0 eigenmode by way of i t s e f f e c t on the f a r f i e l d c u r r e n t and c o n s e q u e n t l y on G 0 ( z ) . The e f f e c t of K on the h i g h e r modes i s e q u a l l y i n s i g n i f i c a n t s i n c e the e i g e n v a l u e s form an i n c r e a s i n g sequence and X Q / X , << 1. T y p i c a l l y t h i s r a t i o i s on the o r d e r of .07 62 i m p l y i n g t h a t X, =* 10X0. T h e r e f o r e the parameter K has o n l y minor e f f e c t on the h i g h e r modes which themselves make a p r o g r e s s i v e l y s m a l l e r c o n t r i b u t i o n t o t h e f l o w f i e l d . The l a c k of any s i g n i f i c a n t e f f e c t on the f l o w f i e l d f o r nonzero 0(1) K p r o v i d e d | a 2 + K\ » 0 s u g g e s t s t h a t t h e r e i s no l o s s of a p p l i c a b i l i t y of the m a t h e m a t i c a l model t o the n o r t h e a s t P a c i f i c Ocean i f K = 0. The h i g h l y n o n - t r i v i a l way t h a t the p arameters appear i n the s o l u t i o n s d e r i v e d l a s t c h a p t e r makes an a n a l y t i c a l i n v e s t i g a t i o n of t h e e f f e c t of t h e i r v a r i a t i o n e x t r e m e l y d i f f i c u l t . Indeed such an e x a m i n a t i o n would not s e r v e t o i l l u m i n a t e the e s s e n t i a l dynamics i n the problem. Of more u t i l i t y i s t o argue the e f f e c t of v a r i a t i o n s i n the parameters on the s o l u t i o n based on the p h y s i c s and geometry of the problem. However w h i l e the f o l l o w i n g d i s c u s s i o n i s q u a l i t a t i v e r a t h e r than q u a n t i t a t i v e , i t w i l l be e x t e n s i v e c o n s i s t i n g of examining s e v e r a l v e r t i c a l and h o r i z o n t a l s e c t i o n s i n the p r e s s u r e , d e n s i t y and v e l o c i t y f i e l d s . W i t h t h i s p h i l o s o p h y i n mind the e x a c t n u m e r i c a l v a l u e s of a l l the v a r i a b l e s i s not so i m p o r t a n t as the o r d e r s of magnitudes and q u a l i t a t i v e f e a t u r e s of t h e c a l c u l a t e d f l o w f i e l d . T h i s c h a p t e r b e g i n s w i t h a d e s c r i p t i o n of the f l o w f i e l d f o r the s t a n d a r d s e t of p a r a m e t e r s . T h i s d i s c u s s i o n w i l l c o n c e n t r a t e on g i v i n g a q u a l i t a t i v e but complete d e s c r i p t i o n of the r e s u l t i n g f l o w f i e l d . The s o l u t i o n d e s c r i b e d i n t h i s s e c t i o n w i l l be the s t a n d a r d t o which the s o l u t i o n s f o r the v a r i e d parameters w i l l be compared. In the s e c t i o n s concerned 63 w i t h t h e s t r u c t u r e o f t h e s o l u t i o n f o r t h e v a r i e d p a r a m e t e r s , t h e d i s c u s s i o n i s two f o l d . F i r s t a q u a l i t a t i v e a n a l y s i s i s g i v e n d e s c r i b i n g t h e e f f e c t o f v a r i a t i o n s b a s e d on v o r t i c i t y a r g u m e n t s , and s e c o n d one o r two e x a m p l e s a r e d e s c r i b e d i l l u s t r a t i n g t h e s e a r g u m e n t s . 4.1 The S o l u t i o n F o r The O b s e r v e d P a r a m e t e r s H o r i z o n t a l s e c t i o n s o f s o l u t i o n f o r t h e o b s e r v e d p a r a m e t e r s were t a k e n a t d e p t h i n t e r v a l s o f .1 (350 m e t r e s ) . V e r t i c a l p r o f i l e s were c o m p u t e d a t t h e t e n l o c a t i o n s g i v e n by ( x , y ) c o o r d i n a t e s ( - 2 , . 1 ) , ( - . 7 5 , - 1 ) , ( - . 4 , . 1 ) , ( 0 , . 1 ) , ( 0 , . 7 5 ) , ( . 4 , . 1 ) , ( . 6 , 1 . 2 ) , ( . 7 5 . . 1 ) , (.75,.75) and ( 2 , . 1 ) . T h e s e t e n l o c a t i o n s a r e p l o t t e d i n F i g u r e 10 s h o w i n g t h e i r l o c a t i o n r e l a t i v e t o t h e t o p o g r a p h y . I n a d d i t i o n v e r t i c a l s e c t i o n s p a r a l l e l t o t h e c o a s t l i n e were c o n t o u r e d of p < 0 ) a n d p ( 0 ' f o r y r a n g i n g f r o m 0 t o 2 i n i n c r e m e n t s o f .2. The d i s c u s s i o n p r o c e e d s a s f o l l o w s . F i r s t , a d e s r i p t i o n i s g i v e n o f t h e h o r i z o n t a l s t r u c t u r e o f t h e f l o w f i e l d f o r z = 1, .9, .8, .7 and .6. The o r d e r o f t h e d i s c u s s i o n o f t h e o b s e r v a b l e s i s : p ( 0 > , ( u ( 0 ) , v < 0 ) ) , (m,,m 2), a n d p ( 0 ) . F o l l o w i n g t h e d e s c r i p t i o n o f t h e h o r i z o n t a l v a r i a t i o n t h e f o c u s o f t h i s s e c t i o n i s s h i f t e d t o d e s c r i b i n g t h e v e r t i c a l v a r i a t i o n o f t h e f l o w f i e l d a t v a r i o u s f i x e d ( x , y ) c o o r d i n a t e s . F i g u r e s 11, 12, 13, 14 a n d 15 a r e c o n t o u r p l o t s o f t h e s t r e a m f u n c t i o n p ( 0 ' a t z = 1, .9, .8, .7 and .6 r e s p e c t i v e l y . The c o n t o u r s were p l o t t e d a t i n t e r v a l s o f + o r - . 0 1 , r e l a t i v e t o t h e z e r c s t r e a m l i n e . The z e r o s t r e a m l i n e , i n F i g u r e 11, b i f u r c a t e s a t t h e 64 u p s t r e a m s t a g n a t i o n p o i n t s i t u a t e d a p p r o x i m a t e l y a t ( - . 6 , 0 ) . One b r a n c h r e m a i n s on t h e c o a s t l i n e w h i l e t h e o t h e r t u r n s i n t o t h e i n t e r i o r o f t h e d o m a i n r e j o i n i n g t h e c o a s t l i n e a t t h e d o w n s t r e a m s t a g n a t i o n p o i n t a t a p p r o x i m a t e l y ( - . 6 5 , 0 ) . The s t a g n a t i o n p o i n t s a r e n o t s y m m e t r i c a b o u t x = 0 b e c a u s e h ( x , y ) i s n o t s y m m e t r i c a b o u t x = 0. The z e r o s t r e a m l i n e f o r m s t h e o u t e r b o u n d a r y o f a r e g i o n o f c l o s e d s t r e a m l i n e s . The i n t e r i o r r e g i o n i n F i g u r e 11 f o r m s t h e s u r f a c e e x p r e s s i o n o f a b a r o c l i n i c e d d y , w i t h an a p p r o x i m a t e d i a m a t e r o f 1 (400 km). The s t r e a m f u n c t i o n i n t h e i n t e r i o r o f t h i s r e g i o n i s p o s i t i v e , s u g g e s t i n g t h a t t h e i n w a r d r a d i a l g r a d i e n t of t h e s t r e a m f u n c t i o n i s p o s i t i v e i m p l y i n g c l o c k w i s e o r a n t i c y c l o n i c c i r c u l a t i o n . The s t r u c t u r e o f t h e s t r e a m l i n e s i n t h e i n t e r i o r o f t h e eddy i s somewhat c o m p l i c a t e d . Over t h e seamount and s h e l f p r o t r u s i o n a r e two w e l l d e v e l o p e d a n t i c y c l o n i c e d d i e s . I n t h e r e g i o n b e t w e e n t h e s e two e d d i e s , i n t h e n e i g h b o u r h o o d o f ( . 4 , . 4 ) , i t f o l l o w s t h a t a s t r o n g c u r r e n t s h e a r e x i s t s . A r e a s o f s i g n i f i c a n t c u r r e n t s h e a r a r e g e n e r a l l y a s s o c i a t e d w i t h r e g i o n s o f h i g h b i o l o g i c a l a c t i v i t y . F i g u r e 11 a l s o p i c t u r e s s t r e a m l i n e s r a d i a l l y i n w a r d o f t h e z e r o s t r e a m l i n e w h i c h e n c i r c l e t h e two l o c a l a n t i c y c l o n i c e d d i e s . Thus i n a d d i t i o n t o t h e l o c a l e d d i e s t h e r e i s a l a r g e r s c a l e a n t i c y c l o n i c c i r c u l a t i o n . T h i s f l o w c o r r e s p o n d s t o t h e t o p o g r a p h i c s t e e r i n g o f t h e u p s t r e a m c o a s t a l c u r r e n t by t h e s h e l f p r o t r u s i o n o u t t o a r e g i o n where i t c a n be t o p o g r a p h i c a l l y s t e e r e d by t h e s e amount. The r e t u r n c o a s t a l f l o w i s 65 c o n c e n t r a t e d i n a n a r r o w b a n d o f w i d t h a p p r o x i m a t e l y .15 (60 km) n e a r x = 0. I n t h e i m m e d i a t e r e g i o n o f t h i s r e t u r n f l o w t h e c u r r e n t s h ave t h e l o c a l a p p e a r a n c e a s ban d s o f a l t e r n a t i n g c u r r e n t s . The maximum v a l u e o f p ( 0 ' o c c u r s on z = 1 o v e r t h e seamount w i t h an a p p r o x i m a t e v a l u e o f . 1 . T h i s maximum o c c u r s o v e r t h e seamount r a t h e r t h a n o v e r t h e s h e l f p r o t r u s i o n b e c a u s e p ( 0 ) i s p r o p o r t i o n a l t o h, and h 2 a n d h 2 > h,. T h e r e i s , h o w e v e r , a l o c a l maximum o v e r t h e c o n t i n e n t a l s h e l f bump a t a b o u t ( 0 , . 2 5 ) w i t h an a p p r o x i m a t e v a l u e b e t w e e n .07 and .08. I n a d d i t i o n t h e r e must e x i s t a s a d d l e p o i n t i n t h e s t r e a m f u n c t i o n f i e l d i n t h e i m m e d i a t e v i n c i n t y o f ( . 4 , . 4 ) , a s s o c i a t e d w i t h t h e f l u i d t r a j e c t o r i e s i n t h a t a r e a . F i g u r e s 12, 13, 14 a n d 15 a r e h o r i z o n t a l s e c t i o n s o f t h e s t r e a m f u n c t i o n f i e l d a t z = .9, .8, .7 a n d .6. T h i s s e q u e n c e o f f i g u r e s shows t h a t t h e eddy d e c a y s r a p i d l y w i t h d e p t h . The d e c a y i s most r a p i d b e t w e e n z = 1 a n d z = .9 w i t h t h e p r e s s u r e f i e l d o v e r t h e seamount d e c a y i n g f r o m i t s maximum o f .1 on z = 1 t o a b o u t .035 on z = .9. The v e r t i c a l d e c a y c o n t i n u e s u n t i l a b o u t z = .6 where t h e d e c a y becomes n e g l i g i b l e a n d t h e r e m a i n i n g eddy s t r u c t u r e c o n t i n u e s more o r l e s s u n i f o r m l y down t o t h e b o t t o m a t z = 0. The v e r t i c a l s t r u c t u r e o f t h e d e c a y w i l l be c o m p l e t e l y d e s c r i b e d l a t e r i n t h i s s e c t i o n . The h o r i z o n t a l g r a d i e n t s i n t h e p r e s s u r e f i e l d on z = 1 a r e t h e l a r g e s t i n t h e r e t u r n c o a s t a l f l o w . C o n s e q u e n t l y , t h e p a r t i c l e v e l o c i t i e s w i l l be l a r g e s t i n t h a t r e g i o n . The r e l a t i v e l y l a r g e s p e e d s i n t h i s a r e a a r e t h e r e s u l t o f t h e 66 c r o w d i n g of the c o a s t a l s t r e a m l i n e s t o g e t h e r by the e x i s t e n c e of the a n t i c y c l o n i c eddy over the s h e l f p r o t r u s i o n and the c o a s t a l boundary. The h o r i z o n t a l g r a d i e n t s i n the p r e s s u r e f i e l d a r e l a r g e r i n the l o c a l eddy over the seamount than i n the l o c a l eddy over the s h e l f p r o t r u s i o n . The reason i s t h a t i n t h i s r e g i o n the v e l o c i t y f i e l d i s r e l a t e d t o the p r e s s u r e f i e l d t h r o ugh the g r a d i e n t s of p ( 0 ) , and t h e s e i n t u r n w i l l be p r o p o r t i o n a l t o V h ( x , y ) . F i g u r e 5 which i s a c o n t o u r map of the f u n c t i o n h ( x , y ) c l e a r l y shows t h a t Vh(x,y) i s l a r g e s t over the seamont. The weakening of the g r a d i e n t s i n the p < 0 > f i e l d w i t h depth i s i l l u s t r a t e d i n F i g u r e s 11 t h r o u g h 15 by the f a c t t h a t as the depth i n c r e a s e s the number of c o n t o u r s d e c r e a s e . C o n s e q u e n t l y , the a n g u l a r speed of the eddy d e c r e a s e s w i t h d e p t h . F i g u r e s 16, 17, 18, 19 and 20 a r e v e r t i c a l s e c t i o n s of p ( 0> taken a t y = .2, .4, .6, .8 and 1 p a r a l l e l t o the c o a s t l i n e . On each s e c t i o n the p r e s s u r e f i e l d was c o n t o u r e d a t i n t e r v a l s of + or - .01 r e l a t i v e t o the 0 p r e s s u r e c o n t o u r ^ For t h i s s e r i e s of graphs the f u n c t i o n p ( 0 ) i s b e s t d e s c r i b e d as the 0(1) p r e s s u r e than as the stream f u n c t i o n . The l a t t e r t e r m i n o l o g y s u g g e s t s t h a t the f l u i d t r a j e c t o r i e s l i e a l o n g l i n e s of c o n s t a n t p r e s s u r e . In the v e r t i c a l s e c t i o n s d e s c r i b e d here t h i s i n t e r p e t a t i o n i s i n c o r r e c t , whereas f o r the h o r i z o n t a l s e c t i o n s p r e v i o u s l y d e s c r i b e d i t i s c o r r e c t . Comparing F i g u r e 16 w i t h F i g u r e 11, the v e r t i c a l s e c t i o n a l o n g y = .2 i s seen t o pass through the seaward r e g i o n of the r e t u r n c o a s t a l f l o w and t h r o u g h the i n t e r i o r of the l o c a l eddy 67 p r o d u c e d by t h e s h e l f p r o t r u s i o n . The h o r i z o n t a l b o u n d a r i e s o f t h e l a r g e s c a l e a n t i c y c l o n i c c i r c u l a t i o n i s f o r m e d by t h e 0 p r e s s u r e c o n t o u r . A l o n g y = .2 t h i s o c c u r s a t x * -.6 and x * .7. F i g u r e 16 t h r o u g h 20 show t h i s c o n t o u r t o be v e r t i c a l l y s t r a i g h t , w h i c h i s a c o n s e q u e n c e o f t h e f a c t t h a t t o l e a d o r d e r t h e r e i s no v e r t i c a l v e l o c i t y . I n t h e i n t e r i o r o f t h e r e g i o n b o u n d e d by t h e 0 p r e s s u r e c o n t o u r t h e p r e s s u r e i s e v e r y w h e r e p o s t i v e . C a l c u l a t i o n s show t h a t f o r e v e r y z t h e i n w a r d r a d i a l g r a d i e n t o f p ( 0 ) i n t h e i n t e r i o r i s p o s i t i v e , i m p l y i n g t h a t t h e e n t i r e i n t e r i o r r e g i o n r o t a t e s a n t i c y c l o n i c a l l y . I n l i n e w i t h t h e o b s e r v a t i o n made f o r t h e h o r i z o n t a l s e c t i o n s , F i g u r e s 16 t h r o u g h 20 show t h a t t h e l a r g e s c a l e a n t i c y l o n i c c i r c u l a t i o n i s . e s s e n t i a l l y s u r f a c e t r a p p e d . I n F i g u r e 16, t h e f l o w f i e l d i s o n l y s l i g h t l y e f f e c t e d by t h e seamount s i n c e t h e p r e s s u r e c o n t o u r s a r e r o u g h l y s y m m e t r i c a b o u t x = 0. The v e r t i c a l s e c t i o n t a k e n on y = .4 p a s s e s j u s t s e a w a r d o f t h e l o c a l e ddy f o r m e d o v e r t h e s h e l f p r o t r u s i o n a n d c u t s t h r o u g h t h e c o a s t w a r d e x t r e m e o f t h e l o c a l eddy p r o d u c e d by t h e sea m o u n t . The i n c r e a s i n g i n f l u e n c e o f t h e seamount i s s e e n i n F i g u r e 17 w i t h t h e p r e s s u r e c o n t o u r s b e c o m i n g somewhat a s y m m e t r i c a b o u t x = 0. I n t h i s i n t e r m e d i a t e r e g i o n t h e v e r t i c a l e x t e n t o f t h e e d d y , a s m a n i f e s t e d i n p ( 0 ) , h a s s h a l l o w e d . A l o n g y = .2 t h e .01 c o n t o u r r e a c h e s a maximum d e p t h o f a b o u t .7 (1000 m e t r e s ) , w h e r a s a l o n g y = .4 t h i s i s r e d u c e d t o a b o u t .2 (800 m e t r e s ) . E x a m i n i n g F i g u r e 5 shows t h a t a l o n g 68 y = .4 t h e s l o p e h e i g h t i s a b o u t 1 (70 m e t r e s ) r e s u l t i n g i n s l i g h t t o p o g r a p h i c f o r c i n g a t t h i s p o i n t . I n t h e F i g u r e s 18 a n d 19 t h e i n f l u e n c e o f t h e seamount becomes t h e d e n o m i n a t e f e a t u r e i n t h e f o r c i n g o f t h e f l o w . I n F i g u r e 18, t a k e n a l o n g y = .6 t h e v e r t i c a l e x t e n t h a s i n c r e a s e d b a c k down t o a d e p t h o f Of .25 (900 m e t r e s ) . The h o r i z o n t a l b o u n d a r i e s o f t h e eddy h a s s h i f t e d t o w a r d p o s i t i v e x w i t h t h e c i r c u l a t i o n c o n t a i n e d b e t w e e n x =* -.4 and x =* 1 . The v e r t i c a l e x t e n t o f t h e eddy i s s u b s t a n t i a l l y i n c r e a s e d a l o n g y = .8, shown i n F i g u r e 19. F o r t h e f i r s t t i m e t h e .01 p r e s s u r e c o n t o u r i s n o t c l o s e d a n d i n f a c t r e a c h e s t h e b o t t o m . C o n s e q u e n t l y t h e h o r i z o n t a l g r a d i e n t s f o r p ( 0 ) a r e much s t r o n g e r a n d e x t e n d d e e p e r down o v e r t h e seamount t h a n e l s e w h e r e . As a r g u e d e a r l i e r t h i s i s a c o n s e q u e n c e o f t h e i n c r e a s e d h e i g h t a n d l a r g e r g r a d i e n t s a s s o c i a t e d w i t h t h e seamount c o m p a r e d t o t h e s l o p e p r o t r u s i o n . The f i n a l v e r t i c a l s e c t i o n shown o f p ( 0 ' , F i g u r e 20, a l o n g y = 1 shows t h e r a p i d d e c a y o f t h e p r e s s u r e f i e l d as t h e d i s t a n c e t o t h e s u p p o r t o f h ( x , y ) i n c r e a s e s . F i g u r e s 2 1 , 22, 23, 24 a n d 25 a r e s t i c k p l o t s o f t h e h o r i z o n t a l v e l o c i t y f i e l d on z = 1, .9, .8, .7 a n d .6 r e s p e c t i v e l y . E a c h s t i c k r e p r e s e n t s a h o r i z o n t a l v e l o c i t y v e c t o r . The v e l o c i t y was c a l c u l a t e d a t ( x , y ) c o o r d i n a t e s (-2 + i S x , j 6 y ) w i t h i , j , 6x and 6y g i v e n e a r l i e r i n t h i s c h a p t e r . The t a i l o f e a c h s t i c k o r v e l o c i t y v e c t o r s i t s on one o f t h e s e p o i n t s . The s c a l i n g o f t h e v e c t o r s i s done s o t h a t a s t i c k l e n g t h o f one i n c h e q u a l s f i f t y c e n t i m e t r e s p e r s e c o n d . By c o m p a r i n g t h e s t i c k p l o t s t o t h e a s s o c i a t e d s t r e a m f u n c t i o n p l o t 69 t h e d i r e c t i o n o f f l o w becomes c l e a r . I n t h e f a r f i e l d u p s t r e a m r e g i o n , s a y a l o n g t h e l i n e x = -2, t h e c u r r e n t i s u n a f f e c t e d by t h e t o p o g r a p h y a nd i s g i v e n by u ( o > _ e x p ( - a y ) Z ( z ) v ( 0 > = 0 a s d e s c r i b e d i n C h a p t e r I I . On t h e s u r f a c e (z=1) t h e c u r r e n t a l o n g x = - 2 , shown i n F i g u r e 21 i s a maximum a t t h e c o a s t w i t h a m a g n i t u d e o f .1 (10 cm s " 1 ) . The c u r r e n t e x p o n e n t i a l l y d e c a y s away f r o m t h e c o a s t w i t h an e-f o l d i n g l e n g t h o f a " 1 (80 km). F i g u r e 21 shows t h a t t h e p a r t i c l e s p e e d s i n c r e a s e a s t h e u p s t r e a m c u r r e n t i n t e r a c t s w i t h t h e t o p o g r a p h y . I t i s p o s s i b l e t o see t h e two i n t e r i o r a n t i c y c l o n i c e d d i e s i n F i g u r e 21 by c o m p a r i n g t h a t f i g u r e w i t h F i g u r e 11 . N u m e r i c a l c a l c u l a t i o n s show t h a t t y p i c a l s p e e d s a t t h e e x t e r i o r o f t h e eddy i n F i g u r e 21 ( c o r r e s p o n d i n g t o t h e o f f s h o r e 0 s t r e a m l i n e i n F i g u r e 11) a r e a b o u t .2 (20 cm s ~ 1 ) , r e p r e s e n t i n g a b o u t a 700% i n c r e a s e i n p a r t i c l e s p e e d s . I m m e d i a t e l y o v e r t h e seamount t h e t y p i c a l s u r f a c e s p e e d s a r e on t h e o r d e r o f .4 (40 cm s " 1 ) . T y p i c a l s p e e d s c a l c u l a t e d o v e r t h e c o n t i n e n t a l s h e l f bump a r e on t h e o r d e r o f .2 (20 cm s " 1 ) , c o n s i s t e n t w i t h t h o s e s p e e d s c a l c u l a t e d f o r t h e l a r g e r e n c i r c l i n g a n t i c y c l o n i c c i r c u l a t i o n . The n a r r o w c o a s t a l b a nd o f r e t u r n f l o w h a s t h e l a r g e s t s p e e d s w i t h t h e c a l c u l a t i o n s s h o w i n g t h a t a t on t h e s u r f a c e a t ( x , y ) c o o r d i n a t e ( 0 , 0 ) t h e s p e e d s a r e i n e x c e s s o f .50 (50 cm s ~ 1 ) d e c a y i n g t o a b o u t .3 (30 cm s " 1 ) a t ( 0 , . 1 ) . F i g u r e 21 a l s o c l e a r l y shows t h a t t h e t o p o g r a p h i c a l l y i n d u c e d f l o w d e c a y s r a p i d l y a s t h e h o r i z o n t a l d i s t a n c e f r o m t h e 70 s u p p o r t o f h ( x , y ) i n c r e a s e s . I f t h e c e n t e r o f t h e eddy i s c h o s e n a s ( x , y ) c o o r d i n a t e (.25,.5) t h e n a t t h e c e n t e r t h e s u r f a c e s p e e d s a r e t y p i c a l l y on t h e o r d e r o f .2 (20 cm s ~ 1 ) . F u r t h e r o u t f r o m t h e c e n t e r s a y a t (.2,.6) t h e s p e e d i s a b o u t .15 (15 cm s " 1 ) . The d i s t a n c e b e t w e e n t h e s e two p o i n t s i s a b o u t 40 km. A t ( . 2 , 1 ) , a b o u t 250 km f r o m t h e c e n t e r , t h e s u r f a c e s p e e d i s on t h e o r d e r o f .05 (5 cm s " 1 ) . However a s F i g u r e 21 i l l u s t r a t e s t h e eddy p r o d u c e d f r o m t h e c o m b i n e d i n t e r a c t i o n o f t h e c o a s t a l t o p o g r a p h y and t h e seamount d o e s n o t p r o d u c e a s y m m e t r i c e d d y . C o n s e q u e n t l y f r o m t h e v i e w p o i n t o f an o b s e r v e r m o v i n g N o r t h w e s t w a r d , t o w a r d t h e s eamount, f r o m t h e c e n t e r (.2,.6) t h e p a r t i c l e s p e e d s w o u l d seem t o i n c r e a s e a s t h e seamount i n d u c e d f l o w i s e n c o u n t e r e d . F i g u r e s 22, 23, 24 a n d 25 a r e s t i c k p l o t s o f t h e h o r i z o n t a l v e l o c i t y f i e l d a t z = .9, .8, .7 a n d .6 r e s p e c t i v e l y . T h e s e f i g u r e s show t h e v e r t i c a l d e c a y o f t h e h o r i z o n t a l v e l o c i t y a s t h e d e p t h i n c r e a s e s . The most s i g n i f i c a n t d e c a y o c c u r s b e t w e e n t h e s u r f a c e z = 1 and z = .9. I n g e n e r a l t h e p a r t i c l e s p e e d s a t z = .9 a r e h a l f t h o s e a t t h e s u r f a c e . F o r e x a m p l e i n t h e n e a r c o a s t a l r e t u r n f l o w a s s o c i a t e d w i t h t h e l a r g e a n t i c y c l o n i c c i r c u l a t i o n t h e s p e e d s a t z = .9 a r e on t h e o r d e r o f .2 (20 cm s " 1 ) c o m p a r e d t o t h e 50 cm s ~ 1 c o m p u t e d f o r t h e s u r f a c e . O v e r t h e seamount c u r r e n t s p e e d s a r e r e d u c e d f r o m 40 cm s " 1 on z = 1 t o a b o u t 21 cm s " 1 on z = .9. The s p e e d s o v e r t h e c o n t i n e n t a l s h e l f p r o t r u s i o n a r e r e d u c e d f r o m a s u r f a c e maximum o f a b o u t 25 cm s " 1 t o a b o u t 12 cm s " 1 on z = .9. The h o r i z o n t a l d e c a y o f t h e v e l o c i t y f i e l d i s i n d e p e n d e n t 71 o f d e p t h a s a c o n s e q u e n c e o f t h e v e r t i c a l mode s o l u t i o n t e c h n i q u e . The v e r t i c a l d e c a y o f t h e v e l o c i t y f i e l d c o n t i n u e s u n t i l a b o u t z = .6 where t h e h o r i z o n t a l v e l o c i t y f i e l d becomes v i r t u a l l y d e p t h i n d e p e n d e n t . The d e p t h a s s o c i a t e d w i t h z = .6 i s a b o u t 1400 m e t r e s . B e l o w t h i s d e p t h t h e f l o w a p p e a r s d e p t h i n d e p e n d e n t a n d t h e r e f o r e b a r o t r o p i c . B a s e d on t h e c o m p u t a t i o n s o f t h i s s e c t i o n , u s i n g a l e v e l o f no m o t i o n on t h e o r d e r o f 1500 t o 2500 m e t r e s , t h e v e r t i c a l e x t e n t o f t h e eddy w o u l d be on t h e o r d e r o f 1000+ m e t r e s . T y p i c a l s p e e d s c o m p u t e d f o r t h e d e e p i n t e r i o r o f t h e w a t e r c o l u m n (z<.5) o v e r t h e seamount a r e a b o u t .01 (1 cm s ~ 1 ) . Over t h e c o n t i n e n t a l s h e l f bump, t y p i c a l s p e e d s a r e on t h e o r d e r o f .02 (2 cm s " 1 ) . The c o a s t a l r e t u r n f l o w h a s s p e e d s on t h e o r d e r o f .1 (10 cm s " 1 ) . C u r r e n t s i n t h e s e a w a r d e x t e r i o r r e g i o n o f t h e e d d y , a s s o c i a t e d w i t h t h e 0 s t r e a m l i n e , have m a g n i t u d e s on t h e o r d e r o f .005 (.5 cm s " 1 ) . Thus w h i l e t h e eddy d e c a y s r a p i d l y i n t h e v e r t i c a l d i r e c t i o n t h e m o d e l c a l c u l a t i o n s p r e d i c t c o n t i n u e d a n t i c y c l o n i c f l o w e v e n i n t h e d e e p i n t e r i o r o f t h e w a t e r c o l u m n . F i g u r e 26 i s a s t i c k p l o t o f t h e mass t r a n s p o r t v e c t o r ( m 1 f m 2 ) g i v e n by 3.51 and 3.52. The s c a l i n g o f t h e s t i c k l e n g t h s i s s u c h t h a t 1 i n c h o f s t i c k l e n g t h c o r r e s p o n d s t o a b o u t 7 • 1 0 6 kg m"1 s~ 1 , b a s e d on an a v e r a g e d e n s i t y o f 1025 kg m~3. To c o n v e r t t o v o l u m e t r a n s p o r t u n i t s ( i e . S v e r d r u p s ; 1 Sv = 1 0 6 m 3 s " 1 ) m, and m 2 a r e i n t e g r a t e d o v e r a h o r i z o n t a l d i s t a n c e o f .25 (100 km). Thus a c o n s t a n t n o n d i m e n s i o n a l 72 t r a n s p o r t o f one c o r r e s p o n d s t o a v o l u m e t r a n s p o r t o f 350 Sv w h i c h i n t u r n w o u l d be r e p r e s e n t e d by a s t i c k l e n g t h o f 2 i n c h e s . I n t h e f a r f i e l d t h e t r a n s p o r t i s p a r a l l e l t o t h e c o a s t , d e c a y i n g e x p o n e n t i a l l y away from' t h e c o a s t w i t h t h e n o n d i m e n s i o n a l e - f o l d i n g l e n g t h o f o"1. On y=0 m, = .02 (7 Sv) a n d m 2 = 0. The t r a n s p o r t i n c r e a s e s a s t h e f l o w i n t e r a c t s w i t h t h e t o p o g r a p h y . Over t h e seamount t h e m a g n i t u d e o f t h e t r a n s p o r t i s a b o u t .06 (20 S v ) . The n e a r c o a s t a l r e t u r n f l o w h a s a t r a n s p o r t on t h e o r d e r o f .06 (20 S v ) . The c o n t i n e n t a l s h e l f bump i n d u c e s a t y p i c a l t r a n s p o r t o f .04 (14 S v ) . The o u t e r e x t e r i o r o f t h e eddy h a s a t y p i c a l t r a n s p o r t m a g n i t u d e o f .01 (4 S v ) . The h o r i z o n t a l d e c a y of t h e t r a n s p o r t f i e l d i s q u a l i t a t i v e l y t h e same a s t h e t h e h o r i z o n t a l d e c a y o f t h e v e l o c i t y f i e l d . C o n s e q u e n t l y t h e t o p o g r a p h i c a l l y i n d u c e d mass t r a n s p o r t q u i c k l y d e c a y s away t o z e r o a s t h e d i s t a n c e i n c r e a s e s f r o m t h e s u p p o r t o f h ( x , y ) . F i g u r e s 27, 28 a n d 29 a r e c o n t o u r e d p l o t s o f h o r i z o n t a l s e c t i o n s o f t h e 0 ( 1 ) d e n s i t y f i e l d p ( 0 ) f o r z = 1, .9, .8 r e s p e c t i v e l y . The c o n t o u r i n g i n c r e m e n t i s + o r - .1 r e l a t i v e t o t h e 0 d e n s i t y c o n t o u r . F i g u r e 27, i n w h i c h p ( 0 > i s c o n t o u r e d f o r z = 1, c l e a r l y shows t h a t t h e d e n s i t y c o n t o u r s c l o s e l y f o l l o w t h e s t r e a m f u n c t i o n c o n t o u r s o f F i g u r e 11. I n t h e f a r f i e l d , s a y a l o n g x = -2 o r +2, t h e 0 ( 1 ) d e n s i t y i n c r e a s e s a s t h e d i s t a n c e f r o m t h e c o a s t i n c r e a s e s . A l o n g t h e c o a s t l i n e p < 0 ) = 0, o w i n g t o t h e 73 f a c t t h a t p ( 0 ) = 0 a l o n g y = 0 and t h a t p ( 0 ) = - 9 3 p < 0 ) . I t f o l l o w s f r o m t h e t h e r m a l w i n d r e l a t i o n 9 3 u ( 0 ) = 9 2 P < 0 ) t h a t t h e 0 ( 1 ) f a r f i e l d d e n s i t y s h o u l d i n c r e a s e a s y i n c r e a s e s s i n c e t h e m a g n i t u d e o f t h e 0 ( 1 ) a l o n g s h o r e v e l o c i t y u ( 0 > d e c r e a s e s w i t h i n c r e a s i n g d e p t h . The t o p o g r a p h y b e g i n s t o e f f e c t t h e d e n s i t y f i e l d a r o u n d x = - 1 . T h e r e i s a b i f u r c a t i o n o f t h e 0 d e n s i t y c o n t o u r a t a b o u t x = - . 6 , s i m i l i a r t o t h e b i f u r c a t i o n i n t h e 0 p r e s s u r e c o n t o u r , r e j o i n i n g t h e c o a s t l i n e d o w n s t r e a m a t a b o u t x = .65. T h i s 0 d e n s i t y c o n t o u r i s a n o t h e r i n d i c a t o r o f t h e h o r i z o n t a l e x t e n t o f t h e eddy c i r c u l a t i o n on z = 1. The 0 ( 1 ) d e n s i t y r a d i a l l y i n w a r d o f t h i s c o n t o u r i s n e g a t i v e , i n d i c a t i v e o f t h e downward d e f l e c t i o n o f t h e i s o p y c n a l s a s s o c i a t e d w i t h an a n t i c y c l o n i c c i r c u l a t i o n t h a t d e c a y s w i t h i n c r e a s i n g d e p t h . I n t h e i n t e r i o r o f t h e r e g i o n b o u n d e d by t h e 0 d e n s i t y c o n t o u r , t h e d e n s i t y c o n t o u r s c l e a r l y mark o u t t h e two s m a l l e r a n t i c y c l o n i c e d d i e s o v e r t h e seamount and c o n t i n e n t a l s h e l f p r o t r u s i o n . The minimum i n t h e 0 ( 1 ) d e n s i t y f i e l d o c c u r s o v e r t h e seamount w i t h a v a l u e o f a b o u t -1.2 . R e c a l l t h a t t h e f l o w r e l a t e d d e n s i t y f i e l d was s c a l e d by e F p 0 =* 2 kg n r 3 , h e n c e t h e minimum i n t h e s u r f a c e 0 ( 1 ) d e n s i t y f i e l d c o r r e s p o n d s t o a c h a n g e i n t h e t o t a l d e n s i t y o f a b o u t -2.5 kg m~3. T h e r e i s a n o t h e r l o c a l minimum i n t h e 0 ( 1 ) s u r f a c e d e n s i t y f i e l d l o c a t e d o v e r t h e c o n t i n e n t a l s l o p e p r o t r u s i o n a t a b o u t ( x , y ) c o o r d i n a t e ( 0 , . 2 5 ) , w i t h v a l u e a p p r o x i m a t e l y -.9 (-1.8 kg m " 3 ) . C o r r e s p o n d i n g t o t h e s a d d l e p o i n t i n t h e s t r e a m f u n c t i o n f i e l d i n t h e n e i g h b o u r h o o d o f (.4,.4) t h e r e i s a l o c a l maximum i n t h e 74 0 ( 1 ) s u r f a c e d e n s i t y f i e l d o f a b o u t -.4 (-.8 kg n r 3 ) . C o n c o m i t a n t w i t h t h e i n c r e a s i n g u n i f o r m i t y o f t h e s t r e a m f u n c t i o n a s t h e d e p t h i n c r e a s e s , F i g u r e s 28 a n d 29 p i c t u r e t h e h o r i z o n t a l v a r i a t i o n i n p ( 0 ) f l a t t i n g o u t . F i g u r e 29 w h i c h maps t h e c o n t o u r s p ( 0 ) on z = .8 (700 m d e p t h ) shows t h a t t h e d e n s i t y g r a d i e n t s d e c a y r a p i d l y w i t h d e p t h . I n f a c t a t t h i s d e p t h t h e d e n s i t y v a r i a t i o n s a r e b ounded by + o r - .1 (.2 kg n r 3 ) , s i n c e o n l y t h e 0 c o n t o u r i s mapped. The v e r t i c a l d e f l e c t i o n o f t h e i s o p c y n a l s i s shown i n F i g u r e s 30, 3 1 , 32, 33 and 34. T h e s e f i g u r e s a r e v e r t i c a l s e c t i o n s o f t h e 0 ( 1 ) d e n s i t y f i e l d t a k e n p a r a l l e l t o t h e c o a s t l i n e a t y = .2, .4, .6, .8 a n d 1, r e s p e c t i v e l y . By c o m p a r i n g F i g u r e 30 t o F i g u r e 27, t h e v e r t i c a l s l i c e a l o n g y = .2 i s s e e n t o p a s s t h r o u g h b o t h t h e r e t u r n c o a s t a l f l o w and t h e f l o w a s s o c i a t e d w i t h t h e a n t i c y c l o n i c e ddy o v e r t h e c o n t i n e n t a l s h e l f bump. Thus t h e d e f l e c t i o n s shown i n F i g u r e 30 a r e l a r g e l y due t o t h e s l o p e t o p o g r a p h y and c o a s t a l r e t u r n f l o w r a t h e r t h a n t h e s e amount. The e x t e r n a l b o u n d a r y o f t h e e ddy c i r c u l a t i o n i s m a r k e d by t h e 0 d e n s i t y c o n t o u r . I t e x t e n d s down t o t h e b o t t o m b e c a u s e t h e a n t i c y c l o n i c c i r c u l a t i o n d o e s so a s w e l l . However i n t h e d e e p i n t e r i o r o f t h e w a t e r c o l u m n (z<.6) t h e d e f l e c t i o n o f t h e i s o p y c n a l s , r e l a t e d t o t h e v e r t i c a l s h e a r i n t h e , p r e s s u r e f i e l d becomes q u i t e s m a l l (<.2 kg m" 3) due t o t h e u n i f o r m i t y w i t h r e s p e c t t o d e p t h o f t h e s t r e a m f u n c t i o n p ( 0 > . F i g u r e s 3 1 , 32 and 33 show t h e p r o g r e s s i v e i n f l u e n c e o f t h e seamount on t h e v e r t i c a l d e f l e c t i o n o f t h e i s o p c y n a l s . A l o n g 75 y = .8, shown i n F i g u r e 33, t h e r e i s o n l y a m i n i m a l c o n t r i b u t i o n of the s l o p e p r o t r u s i o n t o the f l o w as m a n i f e s t e d i n the 0(1) d e n s i t y f i e l d . F i g u r e 34, taken on y = 1, shows the g r a d u a l decay of the d e f l e c t i o n s of the i s p c y n a l s as the d i s t a n c e from the s u p p o r t of h ( x , y ) i n c r e a s e s . Computations f o r y > 1.2 c o n t a i n v i r t u a l l y no i s o p y c n a l d e f l e c t i o n . The s e r i e s of F i g u r e s 30 th r o u g h t o 34 su p p o r t the c o n c l u s i o n t h a t the a n t i c y c l o n i c c i r c u l a t i o n produced by the the topography i s e s s e n t i a l l y s u r f a c e t r a p p e d . The maximum downward p e n e t r a t i o n of the eddy t a k e s p l a c e over the seamount, c o n s i s t e n t w i t h the extrema o b s e r v e d i n the 0(1) p r e s s u r e , d e n s i t y , v e l o c i t y and mass t r a n s p o r t f i e l d s over t h i s r e g i o n . Based on F i g u r e 33 the s h a r p e s t v e r t i c a l g r a d i e n t s i n the d e n s i t y f i e l d t a k e p l a c e i n a p p r o x i m a t e l y the t o p 30% of ocean. Roughly s p e a k i n g the v e r t i c a l e x t e n t of the eddy can be a s s o c i a t e d w i t h s h a r p e s t v e r t i c a l g r a d i e n t s , r e s u l t i n g i n the de p t h t o which the a n t i c y c l o n i c c i r c u l a t i o n o c c u r s on the o r d e r of 1000 metres. D e s c r i b e d now i s the change i n the v e r t i c a l p r o f i l e of the upstream f l o w f i e l d as i t i n t e r a c t s w i t h the s u p p o r t of h ( x , y ) . The p r e v i o u s l y g i v e n has shown t h a t the f l o w f i e l d a l o n g x = -2 i s u n a f f e c t e d by the topography. F i g u r e s 35, 36, 37, 38 and 39 a r e the v e r t i c a l p r o f i l e s a t (x,y) = (-2,.1) of p ( 0 ) , p ( 0 ) , u ( 0 > , v ( 0 ) and w ( 0 ) r e s p e c t i v e l y . The l a c k of any i n t e r a c t i o n between the topography and the upstream f l o w a t t h i s l o c a t i o n i m p l i e s t h a t v ( 0 > = w ( 0' = 0 throughout the water column as shown i n F i g u r e s 38 and 39. The 76 u p s t r e a m down c h a n n e l v e l o c i t y u ( 0 > , shown i n F i g u r e 37, d e c a y s w i t h i n c r e a s i n g d e p t h . A t ( x , y , z ) = ( - 2 , . 1 , 1 ) , u ( 0 ) = e x p ( -. 2 ) / l 0 = .06 (6 cm s " 1 ) . The u p s t r e a m c u r r e n t i s e s s e n t i a l l y s u r f a c e t r a p p e d w i t h v i r t u a l l y no c u r r e n t (< .1 cm s " 1 ) f o r z < .85 ( d e e p e r t h a n a b o u t 500 m e t r e s ) . The p r e s s u r e f i e l d p r o f i l e shown i n F i g u r e 35 i s a minimum a t t h e s u r f a c e and i n c r e a s e s w i t h i n c r e a s i n g d e p t h . A t t h e s u r f a c e p ( 0 ) =* - . 0 1 , i n c r e a s i n g t o a p p r o x i m a t e l y -.001 a t t h e b o t t o m . T h i s i s a c o n s e q u e n c e o f t h e f a c t t h a t t h e c u r r e n t d e c a y s w i t h i n c r e a s i n g d e p t h i m p l y i n g t h a t t h e c r o s s c h a n n e l g r a d i e n t i n p ( 0 ) d e c a y s w i t h d e p t h , w h i c h i n t u r n i m p l i e s t h a t p ( 0 1 i n c r e a s e s w i t h i n c r e a s i n g d e p t h . The 0 ( 1 ) d e n s i t y p < 0 > , shown i n F i g u r e 36, s a t i s f i e s 3 3 p < 0 > > 0. T h i s i s a c o n s e q u e n c e o f t h e t h e r m a l w i n d r e l a t i o n 3 3 u < 0 ) = 3 2 p ( 0 ) . S i n c e t h e v e r t i c a l g r a d i e n t i n t h e a l o n g c h a n n e l v e l o c i t y component i s p o s i t v e i t f o l l o w s t h a t t h e c r o s s c h a n n e l g r a d i e n t i n p < 0 ) i s a l s o p o s i t i v e . The v e r t i c a l g r a d i e n t i n u ( 0 ) d e c r e a s e s w i t h i n c r e a s i n g d e p t h s o t h a t p ( 0 ) i s more p o s i t i v e a t t h e s u r f a c e t h a n d e e p e r i n t h e w a t e r c o l u m n a n d h e n c e t h e shape o f p < 0 > i n F i g u r e 36. T h i s f a c t d o e s n o t i m p l y t h a t t h e w a t e r i s u n s t a b l y s t r a t i f i e d . From C h a p t e r I I t h e n o n d i m e n s i o n a l i n s i t u d e n s i t y f i e l d w i l l be g i v e n by p + e F p < 0 > + 0 ( e 3 ) . N u m e r i c a l c a l c u l a t i o n s c o n f i r m t h a t 3 3 ( p + e F p ( 0 ) ) < 0 s o t h a t t h e w a t e r c o l u m n i s s t a b l y s t r a t i f i e d . F i g u r e s 40, 4 1 , 42, 43 and 44 a r e t h e v e r t i c a l p r o f i l e s o f 77 p ( 0 ) , p ( 0 ) , u ( 0 ) , . v < 0 ) a n d w ( 0 ) c o m p u t e d a t ( x , y ) = ( - . 4 , . 1 ) . T h i s p o i n t i s a b o u t .4 u n i t s (64 km) away f r o m t h e c e n t r e (.2,.6) o f t h e e n t i r e a n t i c y c l o n i c c i r c u l a t i o n . E x a m i n i n g F i g u r e 10 t h e c o o r d i n a t e s (-.4,.1) a r e s e e n t o be l o c a t e d s l i g h t l y d o w n s t r e a m f r o m t h e u p s t r e a m edge o f t h e c o a s t a l p r o t r u s i o n . The h e i g h t o f t h e s l o p e p r o t r u s i o n a t t h i s l o c a t i o n i s a b o u t 2.5 (175 m e t r e s ) . F i g u r e 11 shows t h a t t h i s v e r t i c a l p r o f i l e i s s i t u a t e d i n t h e b o u n d a r y r e g i o n b e t w e e n t h e c o a s t a l r e t u r n f l o w a n d t h e l o c a l a n t i c y c l o n i c eddy p r o d u c e d by t h e s l o p e p r o t r u s i o n . F i g u r e 40 of t h e 0 ( 1 ) p r e s s u r e f i e l d i s an t y p i c a l e x a m p l e o f p ( 0 ) i n t h e i n t e r i o r o f t h e r e g i o n b o u n d e d by t h e 0 s t r e a m l i n e . A t t h e s u r f a c e t h e 0 ( 1 ) p r e s s u r e i s p o s i t i v e w h i c h d e c a y s a s t h e d e p t h i n c r e a s e s . The s u r f a c e p r e s s u r e i s a p p r o x i m a t e l y .02 a t t h i s l o c a t i o n d e c a y i n g t o a b o u t .004 a t z = .8. The 0 ( 1 ) d e n s i t y i s n e g a t i v e t h r o u g h o u t t h e w a t e r c o l u m n w i t h i t s minimum a t z = 1 o f a b o u t -.27 i n c r e a s i n g a s t h e d e p t h i n c r e a s e s . As a r g u e d e a r l i e r t h e i n t e r i o r d e n s i t y f i e l d must be n e g a t i v e a s c o n s e q u e n c e o f t h e t h e r m a l w i n d r e l a t i o n w h i c h h a s t h a t t h e d e n s i t y must i n c r e a s e a s t h e d i s t a n c e t o t h e c e n t r e o f t h e eddy i n c r e a s e s b e c a u s e t h e v e r t i c a l g r a d i e n t i n t h e v e l o c i t y f i e l d i s p o s i t i v e . F i g u r e s 42 and 43 a r e t h e v e r t i c a l p r o f i l e s o f t h e h o r i z o n t a l v e l o c i t y c o m p o n e n t s u ( 0 ) a n d v ( 0 ) r e s p e c t i v e l y . C o n s i s t e n t w i t h F i g u r e 11 t h e s e f i g u r e s show u ( 0 ) < 0 and v ( 0 ) > 0 a t t h i s l o c a t i o n . A t t h e s u r f a c e t h e s p e e d 78 [ ( u ( 0 ) ) 2 + ( v ( 0 ) ) 2 ] 1 / 2 i s a b o u t .2 (20 cm s ' 1 ) d e c a y i n g t o a b o u t .04 (4 cm s " 1 ) a t z = .8 (700 m e t r e s d e e p ) . The v e r t i c a l p r o f i l e o f t h e v e r t i c a l v e l o c i t y w ( 0 ) a t (-.4,.1) i s shown i n F i g u r e 44. The v e r t i c a l v e l o c i t y i s n e g a t i v e e v e r y w h e r e i n t h e w a t e r c o l u m n b e c a u s e t h e h o r i z o n t a l c o o r d i n a t e s a r e s i t u a t e d i n a r e g i o n where t h e f l o w i s down t h e c o a s t a l s l o p e p r o t r u s i o n . The v e l o c i t y i s z e r o a t t h e s u r f a c e a s a c o n s e q u e n c e o f t h e r i g i d l i d a p p r o x i m a t i o n . The minimum i n w ( 0' o c c u r s a t a p p r o x i m a t e l y .75 (900 m e t r e s d e e p ) . The e x t r e m a i n w ( 0 ) i s e x p e c t e d t o o c c u r i n t h e u p p e r r e g i o n o f t h e w a t e r c o l u m n s i n c e i n t h i s r e g i o n t h e B r u n t -V a i s a l a f r e q u e n c y h a s i t s l a r g e s t g r a d i e n t s . I n t h e d e e p i n t e r i o r o f t h e w a t e r c o l u m n S ( z ) « 0 m e a n i n g t h a t t h e w a t e r c o l u m n i s a p p r o x i m a t e l y homogeneous. I n t h i s r e g i o n t h e v e r t i c a l v e l o c i t y i s t h e r e f o r e q u i t e s m a l l . F i g u r e 44 s u g g e s t s a v a l u e o f a b o u t .005 ( I O " " cm s _ 1 ) a t z = .25 (2600 m e t r e s d e e p ) . F u t h e r m o r e i n t h e d e e p i n t e r i o r where S ( z ) * 0, w ( 0 ) i s a p p r o x i m a t e l y l i n e a r , c o n s i s t e n t w i t h t h e r e m a r k s made i n C h a p t e r I I I a b o u t Z ( z ) a n d G ( z ) i n t h i s r e g i o n . n F i g u r e 44 c o n f i r m s t h e q u a l i t a t i v e a s s e r t i o n t h a t t h e d i c o n t i n u i t y i n 3 3 p ( 0 ) a t z = 0 i s n o t s i g n i f i c a n t n u m e r i c a l l y . The v e r t i c a l v e l o c i t y i s g i v e n by w ( 0 ) = J [ p ( 0 » , p < 0 ' / S ( z ) ] . I n C h a p t e r I I I i t was shown t h a t t h e v e r t i c a l mode d e c o m p o s i t i o n o f t h e i n t e r a c t i o n p r e s s u r e f i e l d i n d u c e s a d i s c o n t i n u i t y i n 3 3 p ( 0 ) o f o r d e r 3 0 « e ~ 1 a = 2 • 1 0 " 5 a t z = 0 f o r ( x , y ) c o o r d i n a t e s i n t h e s u p p o r t o f h ( x , y ) . S i n c e p ( 0 ) = - 3 3 p ( 0 ) , i t f o l l o w s t h a t w ( 0 ) w i l l be d i s c o n t i n u o u s a t z = 0 w i t h a d i s c o n t i n u i t y 79 0 ( 1 0 - 5 ) . However s i n c e w ( 0 ) * 0 ( 1 0 " 2 ) , then the d i s c o n t i n u i t y i s t h r e e o r d e r s of magnitude s m a l l e r than the t y p i c a l v a l u e s w t 0 ) assumes i n the water column. T h e r e f o r e a c o n t i n u o u s l y d i f f e r e n t i a b l e stream f u n c t i o n would p r o b a b l y o n l y m a r g i n a l l y d i f f e r from the s o l u t i o n o b t a i n e d h e r e , and then o n l y so near the bottom. S i n c e one o b j e c t i v e of t h i s t h e s i s i s t o c h a r a c t e r i z e the m i d d l e t o upper ocean p r e d i c t i o n s of t h i s model, i n a d e s i r e t o u n d e r s t a n d a p o r t i o n of the dynamics of the S i t k a eddy, t h i s m a t h e m a t i c a l p r o p e r t y i s o n l y a f o r m a l d i f f i c u l t y and i s not an e s s e n t i a l weakness i n the m a t h e m a t i c a l model. F i g u r e s 45, 46, 47, 48 and 49 a r e the v e r t i c a l p r o f i l e s of p ( 0 ) , p ( 0 ) , u ( 0 ) , v l 0» and w t 0> taken a t (x,y) = ( 0 , . 1 ) . These p r o f i l e s a r e r e p r e s e n t a t i v e of the o u t e r r e g i o n f l o w of the l a r g e r a n t i c y c l o n i c c i r c u l a t i o n t h a t e n c l o s e s the two s m a l l e r e d d i e s . Examining F i g u r e 11 i t i s seen t h a t i n t h i s r e g i o n the h o r i z o n t a l g r a d i e n t s i n the stream f u n c t i o n are s m a l l e r than i n the narrow c o a s t a l r e t u r n f l o w . I t f o l l o w s t h a t s i n c e the v e r t i c a l shear i n the o b s e r v a b l e s i s independent of h o r i z o n t a l p o s i t i o n the v e r t i c a l p r o f i l e s w i l l be f l a t t e r a t t h i s l o c a t i o n . F i g u r e s 45 t h rough 49 have t h i s p r o p e r t y . The r e l a t i v e l y weak h o r i z o n t a l g r a d i e n t s i n p ( 0 ' a t t h i s l o c a t i o n r e s u l t i n a s u r f a c e speed of about .1 (10 cm s " 1 ) . The speed decays q u i c k l y t o about .02 (2 cm s" 1) a t z = .8 (700 metres deep). There i s a n e g l i g i b l e v e r t i c a l v e l o c i t y i n the water column at t h i s p o i n t because t h i s h o r i z o n t a l l o c a t i o n i s not i n the support of h ( x , y ) . T h i s f a c t a l s o i m p l i e s t h a t 80 w ( 0 ) a n d p ( 0 ) a r e c o n t i n u o u s a t z = 0. The f i n a l s e t o f v e r t i c a l p r o f i l e s e x a m i n e d a r e t h o s e f o r ( x , y ) = ( . 7 5 , . 7 5 ) . F i g u r e s 50, 5 1 , 52, 53 and 54 a r e t h e g r a p h s o f t h e v e r t i c a l p r o f i l e s o f p ( 0 > , p ( 0 > , u ( 0 > , v ( 0 ) and w ( 0 » r e s p e c t i v e l y c o m p u t e d a t ( . 7 5 , . 7 5 ) . T h i s l o c a t i o n i s on t h e u p s t r e a m edge o f t h e s e a m o u n t , a s c a n be s e e n i n F i g u r e 10. The h e i g h t o f t h e seamount a t t h i s l o c a t i o n i s a b o u t 5 (350 m e t r e s ) . F i g u r e 50, o f p ( 0 ' shows o v e r t h e seamount a r e g i o n o f i n t e n s e h i g h p r e s s u r e i s f o r m e d , on a c c o u n t o f t h e f a c t t h a t h 2 > h,. The f a c t t h a t t h e v e r t i c a l e x t e n t o f t h e eddy i s d e e p e r h e r e t h a n o v e r t h e s l o p e p r o t r u s i o n c a n be s e e n by c o m p a r i n g F i g u r e 50 t o F i g u r e 40. A s s o c i a t e d w i t h t h e h i g h e r p r e s s u r e i s a c o r r e s p o n d i n g i n c r e a s e s i n t h e v e r t i c a l d e f l e c t i o n o f t h e i s o p y c n a l s . T h i s i s d e d u c e d by c o m p a r i n g F i g u r e 51 w i t h F i g u r e 41 a n d F i g u r e 46. The i n c r e a s e i n t h e m a g n i t u d e o f p ( 0 > means t h a t t h e d e n s i t y o b s e r v e d a t a p a r t i c u l a r i n F i g u r e s 41 and 46 w i l l o c c u r a t a d e e p e r l o c a t i o n i n F i g u r e 5 1 . F i g u r e 52 o f u ( 0 ) shows t h a t t h e r e i s v i r t u a l l y no down c h a n n e l f l o w a t t h i s l o c a t i o n . C o m p a r i n g F i g u r e 11 t o F i g u r e 10 i t i s s e e n t h a t a t t h i s l o c a t i o n t h e s t r e a m l i n e s a p p e a r t o be p a r a l l e l t o t h e t o p o g r a p h i c c o n t o u r s , w h i c h a r e o r i e n t e d i n t h e c r o s s c h a n n e l d i r e c t i o n . A t t h e s u r f a c e u ( 0 ' * .02 (2 cm s " 1 ) a n d v ( 0 > -.4 (-40 cm s " 1 ) . The v e r t i c a l e x t e n t o f t h e eddy i n t h i s l o c a t i o n c a n be o b s e r v e d i n t h e v e r t i c a l p r o f i l e o f v ( 0 > w h i c h i n c r e a s e s f r o m i t s s u r f a c e minimum o f -40 cm s " 1 t o 5 cm s " 1 a t z = .25 (900 m e t r e s d e e p ) . F i g u r e 54 o f w < 0 > shows t h a t t h e v e r t i c a l v e l o c i t y i s s m a l l 81 a t t h i s l o c a t i o n . T h r o u g h o u t t h e w a t e r c o l u m n w ( 0 ) = 0 ( 2 • 1 0" 3) ( 4 * 1 0 " 5 cm s " 1 ) . The e x t r e m a i n w ( 0 ) o c c u r s a t a b o u t 350 m e t r e s w i t h m a g n i t u d e a b o u t 10"* cm s " 1 . The v e r t i c a l v e l o c i t y i s n e g a t i v e s i n c e a t t h i s l o c a t i o n t h e f l o w i s p a s s e d t h e maximum h e i g h t and i s now f l o w i n g down t h e b a c k o f t h e s e a m o u n t . The f o u r s e t s o f v e r t i c a l p r o f i l e s j u s t e x a m i n e d e f f e c t i v e l y c h a r a c t e r i z e t h e f l o w f i e l d . The f l o w f i e l d a t (.4,.1) i s s i m i l i a r t o t h a t a t ( - . 4 , . 1 ) . I n t h i s r e g i o n h o w e v e r t h e c u r r e n t s a r e n o t a s s t r o n g a s t h o s e c o m p u t e d a t ( - . 4 , . 1 ) , w i t h a s u r f a c e c u r r e n t on t h e o r d e r o f 18 cm s " 1 . The f l o w f i e l d a t (.75,1.2) i s s i m i l i a r t h a t t h a t o f ( 0 , . 7 5 ) , h o w e v e r h e r e a g a i n t h e f l o w i s w e a k e r w i t h a s u r f a c e s p e e d a b o u t 2 cm s " 1 . I n summary t h e f o l l o w i n g p h y s i c a l d e s c r i p t i o n e merges o f t h e i n t e r a c t i o n b e t w e e n t h e t o p o g r a p h y and t h e u p s t r e a m c u r r e n t . The b a r o c l i n i c c o a s t a l c u r r e n t b e g i n s t o i n t e r a c t w i t h t h e t o p o g r a p h y i n t h e v i n i c i t y o f x = - . 7 5 . The t o p o g r a p h y f o r c e s t h e c u r r e n t t o u p w e l l . The u p w e l l i n g i s c o n s t r a i n e d by t h e r i g i d l i d s o t h a t t h e v o r t e x t u b e s a r e t h e n c o m p r e s s e d . The c o m p r e s s i o n o f t h e v o r t e x t u b e s c o u p l e d w i t h t h e c o n s e r v a t i o n o f p o t e n t a l v o r t i c i t y i m p l i e s t h a t t h e a n g u l a r momentum o f t h e v o r t e x t u b e must d e c r e a s e . T h i s i s a c c o m p l i s h e d w i t h an i n c r e a s e d a n t i c y c l o n i c c i r c u l a t i o n , s i n c e f o r c l o c k w i s e c i r c u l a t i o n d , v { 0 ) -9 2 u t 0 > < 0. Hence t h e f l u i d t r a j e c t o r i e s a r e f o r c e d o u t i n t o t h e c h a n n e l . F o r some o f t h e s t r e a m l i n e s t h e s t r o n g g r a d i e n t s i n t h e c o a s t a l p r o t r u s i o n t o p o g r a p h i c a l l y t r a p them so a l o c a l 82 a n t i c y c l o n i c eddy i s s e t up. For s t r e a m l i n e s o u t s i d e of some c r i t i c a l g r a d i e n t of h ( x , y ) the s l o p e p r o t r u s i o n cannot t r a p them and they c o n t i n u e out i n t o the ch a n n e l where some of them ar e t r a p p e d by the seamount and o t h e r s o u t s i d e some c r i t i c a l r a d i u s c o n t i n u e down stream. The seamount a g a i n i n d u c e s an u p w e l l i n g of the water column w i t h the conc o m i t a n t i n c r e a s e d a n t i c y c l o n i c r o t a t i o n of the v o r t e x t u b e s . Thus the f l o w i s d i r e c t e d back toward the c o a s t l i n e where i t i s e i t h e r t r a p p e d by the s l o p e p r o t r u s i o n or i s f o r c e d by the c o a s t l i n e i n the downstream d i r e c t i o n . The r e t u r n f l o w t r a p p e d by the s l o p e p r o t r u s i o n i s a g a i n u p w e l l e d , d e c r e a s i n g i t s a n g u l a r momentum where upon i t t u r n s c l o c k w i s e and p roceeds on the above c i r c u i t a g a i n . The s u r f a c e i n t e n s i f i c a t i o n of the f l o w has a s i m i l i a r s i m p l e e x p l a n a t i o n i n the c o n s e r v a t i o n of p o t e n t i a l v o r t i c i t y . In the absence of any s t r a t i f i c a t i o n the topography would induce a v e r t i c a l l y u n i f o r m d e c r e a s e i n the a n g u l a r momentum of the v o r t e x t u b e s . T h i s i s the s i t u a t i o n i n the lower r e g i o n of the water column. In the deep i n t e r i o r of the water column the r e l a t i v e homogeneity of the mean d e n s i t y f i e l d i m p l i e s t h a t the f l u i d cannot s u s t a i n s i g n i f i c a n t v e r t i c a l g r a d i e n t s . Thus i n t h i s r e g i o n of the water column t h e r e i s , p r a c t i c a l l y s p e a k i n g , no b a r o c l i n i c v o r t e x tube c o m p r e s s i o n . The absence of any s i g n i f i c a n t s t r a t i f i c a t i o n i n the lower water column has the m a t h e m a t i c a l i m p l i c a t i o n t h a t the topography f o r c e s the v o r t i c i t y term 3 3 ( S ~ 1 3 3 p ( 0 > ) t o be p o s i t i v e but s m a l l . The 83 c o n s e r v a t i o n of p o t e n t i a l v o r t i c i t y i m p l i e s t h a t i n the lower water column r h e r e w i l l be a c o r r e s p o n d i n g d e c r e a s e i n the a n g u l a r v e l o c i t y of the v o r t e x t u b e s . Near the s u r f a c e , where the mean d e n s i t y g r a d i e n t i s the l a r g e s t , t h e u p w e l l i n g of the water column r e s u l t s i n a l a r g e r b a r o c l i n i c c ompression of the v o r t e x tubes than i n the i n t e r i o r of the water column. C o n s e q u e n t l y the a n g u l a r momentum of the v o r t e x t u b e s i s s t r o n g l y reduced i n t h i s r e g i o n r e s u l t i n g i n an i n c r e a s e d c l o c k w i s e c i r c u l a t i o n of the f l u i d p a r t i c l e s , r e l a t i v e t o the i n t e r i o r . 4.2 H o r i z o n t a l C u r r e n t Shear And Rossby Number The Rossby number [e = U ( f L ) " 1 ] and the parameter g o v e r n i n g the f a r f i e l d c u r r e n t shear a a r e two of the more i m p o r t a n t parameters c o n t r o l l i n g the q u a l i t a t i v e s t r u c t u r e of t h e f l o w f i e l d . The s o l u t i o n s o b t a i n e d i n Chapter I I I were computed f o r a t a k i n g on the range of v a l u e s 10, 5, 2, 1 and .1, c o r r e s p o n d i n g t o e - f o l d i n g l e n g t h s - of 40, 80, 200, 400 and 4000 km, r e s p e c t i v e l y . The Rossby number assumed the v a l u e s .01, .05, .1 and .5. For each v a l u e of the Rossby number the e f f e c t of the f i v e d i f f e r e n t e - f o l d i n g l e n g t h s was examined. Thus, i n t o t a l 20 n u m e r i c a l e x p e r i m e n t s were preformed t e s t i n g the e f f e c t of v a r i a t i o n s i n a and e. The Rossby number has two main e f f e c t s on the f l o w f i e l d . O b v i o u s l y , w i t h o u t 0(e) << 1 the e n t i r e a s y m p t o t i c a n a l y s i s has no v a l i d i t y . The p r o p e r t y which i s most germain t o our d i s c u s s i o n , however, i s the r e l a t i o n s h i p between e and the t o p o g r a p h i c parameters h, and h 2 . The n o n d i m e n s i o n a l h e i g h t s of 84 the s l o p e p r o t r u s i o n h t and the seamount h 2 have been o b t a i n e d as h 0 ( H e ) ~ 1 w i t h the term h 0 d e f i n e d as the maximum h e i g h t of the s l o p e p r o t r u s i o n f o r h, and as the maximum h e i g h t of the P r a t t seamount f o r h 2 . I n c r e a s i n g the Rossby number w i l l t h e r e f o r e have the e f f e c t of r e d u c i n g the magnitude of h, and h 2 . The r e d u c t i o n of the s e two parameters w i l l r e s u l t i n r e d u c i n g the o r d e r of magnitude of the t o p o g r a p h i c f o r c i n g term i n 3.10 and c o n s e q u e n t l y w i l l a f f e c t the h o r i z o n t a l a m p l i t u d e f u n c t i o n s P ( x , y ) . n The a f f e c t of i n c r e a s i n g the Rossby number on the f l o w f i e l d i s n o t , however, u n i f o r m . Huppert(1975) has p r o v i d e d a n e c e s s a r y c o n d i t i o n f o r the f o r m a t i o n of a s t r a t i f i e d T a y l o r column over an o r d e r Rossby number topography. S i m p l y s t a t e d , Huppert's c o n d i t i o n i s t h a t the parameter h 0 ( H e ) ~ 1 must be g r e a t e r than some c r i t i c a l 0(1) v a l u e based on the t o p o g r a p h i c geometry,.flow and domain geometry. The c r i t i c a l v a l u e i s o b t a i n e d by ex a m i n i n g the c o n d i t i o n s under which a s t a g n a t i o n p o i n t can occur i n the f l o w , i e . under what c o n d i t i o n s can v p ( 0 ) = (0,0) somwhere i n the domain. T h i s a n a l y s i s was a t t e m p t e d f o r t h e p r e s e n t problem, however the topography, domain and upstream f l o w c o n s p i r e d t o make t h i s an u n p r o d u c t i v e e x e r c i s e . Huppert's a n a l y s i s assumed an a x i s y m m e t r i c topography i n a h o r i z o n t a l l y unbounded f l u i d w i t h a b a r o t r o p i c upstream c u r r e n t . N e v e r t h e l e s s , one q u a l i t a t i v e f a c t remains c l e a r . The c o n t i n e n t a l s l o p e p r o t r u s i o n and the seamount would have c r i t i c a l v a l u e s f o r h, and h 2 , r e s p e c t i v e l y f o r which l o c a l 8 5 e d d i e s would form. S i n c e t h e r e i s no reason t o assume t h a t the c r i t i c a l v a l u e s would be the same f o r each i n d i v i d u a l t o p o g r a p h i c f e a t u r e , i t f o l l o w s t h a t i n c r e a s i n g the Rossby number w i l l r e s u l t i n a nonuniform change i n the f l o w as one or the o t h e r of h, and h 2 n e a r s i t s c r i t i c a l v a l u e . C o n s e q u e n t l y , t h e r e c o u l d be p h y s i c a l l y r e a l i s t i c Rossby numbers f o r which an eddy forms over one of the t o p o g r a p h i c f e a t u r e s but not the o t h e r . In f a c t , depending on the v a l u e of the o t h e r p a r a m e t e r s , i n c r e a s i n g the Rossby number t o about 0.1 p r e v e n t s an eddy from f o r m i n g over the s l o p e p r o t r u s i o n , but a l l o w s an eddy t o form over the seamount. In the s i t u a t i o n where e d e c r e a s e s the t o p o g r a p h i c parameters i n c r e a s e so t h a t the i n t e r a c t i o n between the f l o w and the support of h(x,y) s h o u l d i n t e n s i f y . The e f f e c t of the Rossby number on the i n t e r a c t i o n between the upstream c u r r e n t and the support of h(x,y) has a s i m p l e p h y s i c a l i n t e r p e t a t i o n . For a f i x e d l e n g t h s c a l e and l a t i t u d e i n c r e a s i n g e i s e q u i v a l e n t t o i n c r e a s i n g the speed s c a l e of the f l o w . I n t u i t i v e l y , i f the upstream f l o w i s sped up i t i s p o s s i b l e t o t h i n k of the upstream c u r r e n t has h a v i n g a reduced sense of the bottom topography. In o t h e r words, by the time a f l u i d p a r c e l can respond t o v a r i a t i o n s i n the bottom i t s speed has f o r c e d i t t o shoot p a s t the support of h ( x , y ) . The Rossby number can a l s o be i n c r e a s e d f o r f i x e d speed s c a l e and l a t i t u d e by d e c r e a s i n g the l e n g t h s c a l e . ( R e c a l l t h a t the l e n g t h s c a l e i s chosen as the d i s t a n c e from the c o a s t l i n e to the P r a t t seamount.) D e c r e a s i n g the l e n g t h s c a l e i n t u i t i v e l y 86 l e a d s t o t h e e x p e c t a t i o n t h a t t h e u p s t r e a m c u r r e n t e x p e r i e n c e s t o p o g r a p h i c c h a n g e s o v e r t o o s h o r t an i n t e r v a l t o i n d u c e c l o s e d s t r e a m l i n e c i r c u l a t i o n . The p a r a m e t e r a d e t e r m i n e s t h e u p s t r e a m h o r i z o n t a l c u r r e n t s h e a r . I f a i n c r e a s e s t h a n t h e c u r r e n t a s sumes t h e s h a p e o f a n a r r o w c o a s t a l b a r o c l i n i c j e t . D e c r e a s i n g a c r e a t e s a b r o a d e r c o a s t a l f l o w . The d i m e n s i o n a l e q u i v a l e n t o f a i s g i v e n by a L " 1 m"1. C h o o s i n g L = 400 km i m p l i e s t h a t a = 5 c o r r e s p o n d s t o a c u r r e n t w i t h a h o r i z o n t a l e - f o l d i n g d i s t a n c e o f L a " 1 = 80 km. The m a i n m a t h e m a t i c a l m e c h a n i s m by w h i c h a e f f e c t s t h e f l o w i s i t s e x i s t e n c e i n t h e d e n o m i n a t o r o f t h e G r e e n ' s f u n c t i o n g ( x , y | x 0 , y 0 ) i n 3.17. S i n c e X 0 = a 2 + K t h e d e n o m i n a t o r o f 3.17 s a t i s f i e s ( m r / 2 ) 2 + X - K > {mn/2)2 + a2. T h e r e f o r e n r e d u c i n g a s h o u l d r e s u l t i n i n t e n s i f y i n g t h e i n t e r a c t i o n b e t w e e n t h e u p s t r e a m f l o w a n d t h e t o p o g r a p h y . I n c r e a s i n g a w i l l d e c r e a s e t h e d e n o m i n a t o r a n d c o n s e q u e n t l y r e d u c e t h e o r d e r o f m a g n i t u d e o f t h e i n t e r a c t i o n s t r e a m f u n c t i o n p ( x , y , z V . The i n c r e a s e d i n t e r a c t i o n , i e . g r e a t e r a n t i c y c l o n i c m o t i o n i n t h e w a t e r c o l u m n , f o r d e c r e a s e d a h a s t h e f o l l o w i n g p h y s i c a l i n t e r p e t a t i o n . The p h y s i c s b e h i n d t h e r e s p o n s e o f t h e e ddy f l o w f i e l d t o v a r i a t i o n s i n a i s o b t a i n e d by e x a m i n i n g t h e p o t e n t i a l v o r t i c i t y . The r e l a t i v e v o r t i c i t y o f t h e u p s t r e a m c u r r e n t i s g i v e n by - 9 2 u 0 = a e x p ( - a y ) Z ( z ) , w h i c h i s p o s i t i v e f o r t h e a and Z ( z ) c o n s i d e r e d i n t h i s t h e s i s . The p o r t i o n o f t h e f a r f i e l d c u r r e n t w h i c h i s w i t h i n an e - f o l d i n g d i s t a n c e ( y < a ~ 1 ) o f t h e 87 c o a s t l i n e (y=0) e x p e r i e n c e s d e c r e a s i n g r e l a t i v e v o r t i c i t y f o r d e c r e a s i n g a. The f a r f i e l d c u r r e n t o u t s i d e t h i s r e g i o n ( i e . y > a " 1 ) e x p e r i e n c e s i n c r e a s e d r e l a t i v e v o r t i c i t y f o r d e c r e a s i n g a. V a r i a t i o n s i n a, h o w e v e r have no e f f e c t on t h e mean s t a t e d e n s i t y p r o f i l e . Hence t h e c h a n g e i n v o r t i c i t y a s s o c i a t e d w i t h t h e c o m p r e s s i o n o f t h e mean s t a t e i s o p c y n a l s , a s t h e f l o w e n c o u n t e r s t h e t o p o g r a p h y , c a n be v i e w e d a s a f i x e d p o s i t i v e number. F o r t h o s e u p s t r e a m s t r e a m l i n e s f o r w h i c h y < a " 1 , c o n s e r v a t i o n o f p o t e n t i a l v o r t i c i t y t h e r e f o r e i m p l i e s t h a t t h e r e l a t i v e v o r t i c i t y o f t h e eddy f i e l d d e c r e a s e s f o r d e c r e a s i n g a. I n t h e r e g i o n y > a " 1 t h e r e l a t i v e v o r t i c i t y o f t h e eddy f i e l d w i l l i n c r e a s e a s a d e c r e a s e s . I n o t h e r w o r d s , d e c r e a s i n g a f o r c e s t h e a l o n g c o a s t l i n e c u r r e n t t o i n c r e a s e i t s a n t i c y c l o n i c m o t i o n b u t f o r c e s t h e more h o r i z o n t a l l y i n t e r i o r f l o w t o d e c r e a s e i t s a n t i c y c l o n i c m o t i o n d u r i n g i n t e r a c t i o n w i t h h ( x , y ) . The d e c r e a s e of t h e r e l a t i v e v o r t i c i t y o f t h e n e a r c o a s t l i n e c u r r e n t i s a c h i e v e d by t h e t u r i n g o f t h e s t r e a m l i n e s f u r t h e r o u t i n t o t h e h o r i z o n t a l i n t e r i o r o f t h e d o m a i n . T h i s t e n d e n c y i s m i t i g a t e d a g a i n s t by t h e r e d u c e d t e n d e n c y o f t h e more i n t e r i o r s t r e a m l i n e s t o c h a n g e t h e i r r e l a t i v e v o r t i c i t y . The s u b s e q u e n t c r o w d i n g t o g e t h e r o f s t r e a m l i n e s i n c r e a s e s t h e h o r i z o n t a l g r a d i e n t s i n t h e p r e s s u r e f i e l d r e s u l t i n g i n t h e i n c r e a s e d a n t i c y c l o n i c r o t a t i o n o f t h e eddy i n t e r i o r . F i g u r e s 55, 56, 57 a n d 58 a r e c o n t o u r p l o t s o f p ( 0 ' on z = 1 f o r t h e ( a , e ) p a i r s ( 1 0 , . 1 ) , ( 1 0 , . 0 1 ) , ( 1 , . 1 ) and ( 1 , . 0 1 ) 88 r e s p e c t i v e l y . F i g u r e s 55 a n d 56 i l l u s t r a t e t h e e f f e c t o f i n c r e a s i n g a f r o m i t s s t a n d a r d v a l u e o f 5 t o 10, f o r an i n c r e a s e d e o f .1 and a d e c r e a s e d e o f .01 f r o m i t s s t a n d a r d v a l u e o f .02. F i g u r e s 57 a n d 58 d e p i c t t h e e f f e c t o f d e c r e a s i n g a f r o m 4 t o 1 f o r t h e same s e t o f R o s s b y n u m bers. I n t h i s s e r i e s o f c a l c u l a t i o n s t h e o r d e r o f m a g n i t u d e o f p ( 0 ) v a r i e d s u b s t a n t i a l l y . T h e r e f o r e t h e c o n t o u r i n g i n t e r v a l s h a ve been a l t e r e d f o r e a c h p a r t i c u l a r c a s e . I n F i g u r e 55 t h e c o n t o u r i n g i n t e r v a l s a r e + o r - .001 r e l a t i v e t o t h e 0 c o n t o u r . I n F i g u r e s 56 a n d 58 t h e i n t e r v a l s a r e + o r - .01 r e l a t i v e t o t h e 0 c o n t o u r . I n F i g u r e 57 t h e c o n t o u r i n t e r v a l s a r e + o r - .1 r e l a t i v e t o t h e 0 p r e s s u r e c o n t o u r . F i g u r e 55, t h e c o n t o u r p l o t o f p ( 0 ' f o r ( a , e ) = ( 1 0 , . 1 ) on z = 1, shows t h e e f f e c t s o f i n c r e a s i n g t h e s h e a r i n t h e u p s t r e a m c u r r e n t a n d t h e R o s s b y number. B a s e d on t h e p h y s i c s o f t h e p r o b l e m i t i s e x p e c t e d t h a t f o r t h i s p a r a m e t e r r e g i m e t h e s i z e a n d s t r e n g t h o f t h e eddy p r o d u c e d s h o u l d be s m a l l e r t h a n t h e t h a t c a l c u l a t e d f o r t h e s t a n d a r d p a r a m e t e r s . C o m p a r i n g F i g u r e 55 t o F i g u r e 11 t h i s i s c l e a r l y s e e n t o be t h e c a s e . S e v e r a l c h a n g e s t o t h e f l o w f i e l d a r e a p p a r e n t . I n c r e a s i n g t h e R o s s b y number t o 0.1 h a s d e c r e a s e d t h e t o p o g r a p h i c p a r a m e t e r s h, a n d h 2 t o 2.2 a n d 7.1, r e s p e c t i v e l y . F o r t h i s s e t o f ( a , e ) t h e p a r a m e t e r h, i s l e s s t h a n i t s ( H u p p e r t ) c r i t i c a l v a l u e s i n c e no eddy has f o r m e d o v e r t h e s l o p e p r o t r u s i o n . The f a i l u r e t o g e n e r a t e c l o s e d s t r e a m l i n e s o v e r t h e s l o p e p r o t r u s i o n r e s u l t s i n t h e eddy ( c e n t e r e d o v e r t h e s eamount) h a v i n g a s m a l l r e g i o n a l e x t e n t , w i t h an r a d i u s o f o r d e r .25 (100 km). 89 The m a g n i t u d e o f t h e i n t e r a c t i o n p r e s s u r e f i e l d p ( x , y , z ) h a s been s h a r p l y r e d u c e d f o r ( a , e ) = ( 1 0 , . 1 ) . I n f a c t t h e i n t e r a c t i o n i s weakened t o t h e d e g r e e t h a t p ( 0 ) < 0 e v e r y w h e r e i n t h e d o m a i n i n F i g u r e 55. T y p i c a l v a l u e s f o r t h e m a g n i t u d e o f p<°> o v e r t h e seamount a r e -.003 i n F i g u r e 55, w h e r e a s i n F i g u r e 11 p ( 0 ' was t y p i c a l l y +.1 i n t h e same r e g i o n . C o r r e s p o n d i n g t o t h e s h a r p d e c l i n e i n t h e h o r i z o n t a l g r a d i e n t s i n p < 0 ) f o r ( a , e ) = ( 1 0 , . 1 ) , t h e m a g n i t u d e s o f t h e t o p o g r a p h i c a l l y i n d u c e d v e l o c i t i e s h a v e been r e d u c e d . O v e r t h e s l o p e p r o t r u s i o n t h e s p e e d s a r e on t h e o r d e r o f .05 (5 cm s " 1 ) c o m p a r e d t o 20 cm s " 1 i n F i g u r e 11. The seamount i n d u c e s s p e e d s on t h e o r d e r o f .02 (2 cm s " 1 ) i n F i g u r e 55 c o m p a r e d t o 40 cm s " 1 f o r ( a , e ) = ( 1 0 , . 1 ) . The t r a n s p o r t s h a v e s i m i l i a r l y d e c r e a s e d f o r t h i s c h o i c e o f a a n d e. D e f i n i n g t h e t r a n s p o r t m a g n i t u d e a s jM[ = [ ( m , ) 2 + ( m 2 ) 2 ] 1 ' 2 , t y p i c a l t r a n s p o r t s i n F i g u r e 55 a r e a b o u t .006 (2 Sv) a t ( x , y ) = ( 0 , . 1 ) c o m p a r e d t o 20 Sv f o r ( a , e ) = ( 5 , . 0 2 ) . O v e r t h e seamount t h e t r a n s p o r t m a g n i t u d e |M| « .007 (2.5 Sv) c o m p a r e d t o 20 Sv f o r t h e s t a n d a r d p a r a m e t e r s . I n F i g u r e 56 ( a , e ) = ( 1 0 , . 0 1 ) , r e d u c i n g t h e R o s s b y number f o r f i x e d c u r r e n t s h e a r h a s i n t e n s i f i e d t h e i n t e r a c t i o n b e t w e e n t h e f l o w a n d t h e t o p o g r a p h y . The t o p o g r a p h i c p a r a m e t e r s h, and h 2 have been i n c r e a s e d t o 22.8 a n d 71.4, r e s p e c t i v e l y . The p a r a m e t e r h 2 i s c l o s e t o b e c o m i n g 0 ( e ~ 1 ) i m p l y i n g t h a t t h e seamount f o r e = .01 i s n o t 0 ( e H ) . E x a m i n i n g F i g u r e 56 t h e l o c a l a n t i c y c l o n i c eddy o v e r t h e 90 s l o p e p r o t r u s i o n h a s r e a p p e a r e d . A s s o c i a t e d w i t h t h e s l o p e eddy i s t h e r e a p p e a r a n c e o f t h e u p s t r e a m and d o w n s t r e a m s t a g n a t i o n p o i n t s . A l t h o u g h t h e c o n t o u r s i n F i g u r e 56 do n o t i n d i c a t e i t , t h e r e i s an a n t i c y c l o n i c f l o w w h i c h e n c o m p a s s e s b o t h l o c a l e d d i e s , l o c a t e d j u s t r a d i a l l y i n w a r d o f t h e 0 s t r e a m l i n e . The m a g n i t u d e o f t h e i n t e r a c t i o n b e t w e e n t h e t o p o g r a p g h y and t h e u p s t r e a m f l o w h a s i n t e n s i f i e d . O v e r t h e seamount p ( 0 > =* .05 and o v e r t h e s l o p e p r o t r u s i o n p< 0 > =* .04. C o n c o m i t a n t w i t h t h e i n c r e a s e d i n t e r a c t i o n a r e t h e l a r g e r h o r i z o n t a l g r a d i e n t s i n p ( 0 > a s m a n i f e s t e d i n t h e c o m p u t e d v e l o c i t y f i e l d . C u r r e n t s p e e d s o v e r t h e seamount a r e on t h e o r d e r o f .3 (30 cm s ~ 1 ) . The s u r f a c e s p e e d s o v e r t h e s l o p e p r o t r u s i o n have i n c r e a s e d t o a b o u t .25 (25 cm s " 1 ) . T y p i c a l t r a n s p o r t s o v e r t h e seamount a r e on t h e o r d e r o f .07 (25 Sv) a n d o v e r t h e s l o p e p r o t r u s i o n .01 (3.5 S v ) . The v e r t i c a l s t r u c t u r e o f t h e e d d i e s i n F i g u r e s 55 and 56 i s c o n s i s t e n t w i t h t h e v e r t i c a l s t r u c t u r e o f t h e s t a n d a r d p a r a m e t e r s o l u t i o n d e s c r i b e d l a s t s e c t i o n . By t h i s i t i s meant t h a t t h e q u a l i t a t i v e p r o p e r t i e s o f t h e d e c a y o f t h e e d d i e s w i t h i n c r e a s i n g d e p t h r e m a i n s more o r l e s s i n v a r i a n t f o r t h e ( a , e ) p a r a m e t e r p a i r s o f F i g u r e s 56 a n d 57. T h a t b e i n g t h e c a s e i t s u f f i c e s t o s a y t h a t t h e e d d i e s a r e s u r f a c e t r a p p e d , w i t h t h e h o r i z o n t a l v e l o c i t y f i e l d d e c a y i n g t o a b o u t h a l f i t s s u r f a c e v a l u e a t z = .9 (350 m e t r e s d e p t h ) . C o m p a r i n g F i g u r e 55 t o 56 a c o u p l e o f p o i n t s s h o u l d be made. F o r a = 10, i n c r e a s i n g t h e R o s s b y number f r o m .01 t o .1 91 d e c r e a s e s h, t o b e l o w i t s ( H u p p e r t ) c r i t i c a l v a l u e a n d t h u s an eddy f a i l s t o f o r m o v e r t h e s l o p e p r o t r u s i o n . S e c o n d f o r a r e l a t i v e l y n a r r o w c o a s t a l c u r r e n t (a=lO i m p l i e s an e - f o l d i n g l e n g t h o f 40 km) t h e q u a l i t a t i v e a p p e a r a n c e o f t h e i n d u c e d f l o w o v e r t h e s u p p o r t o f h ( x , y ) l o o k s more l i k e two s e p a r a t e e d d i e s , t h a n t h e l a r g e s c a l e f l o w i n t e r a c t i o n s e e n i n F i g u r e 11. I t i s n o t s u r p r i s i n g t o c o n c l u d e , t h e r e f o r e , t h a t a s t h e c u r r e n t becomes b r o a d e r t h e d i s t u r b a n c e s p r o d u c e d by t h e s l o p e p r o t r u s i o n and t h e seamount p r o g r e s s i v e l y i n t e r a c t w i t h e a c h o t h e r a n d p r o d u c e a more c o h e r e n t l a r g e s c a l e eddy c i r c u l a t i o n . I n F i g u r e s 57 and 58 t h e p a r a m e t e r a = 1 a n d e t a k e s on t h e v a l u e s .1 and .01 r e s p e c t i v e l y . H a v i n g a = 1 i m p l i e s t h a t t h e u p s t r e a m c u r r e n t h a s a h o r i z o n t a l e - f o l d i n g l e n g t h o f 400 km. T h e s e two f i g u r e s c l e a r l y i l l u s t r a t e t h a t d e c r e a s i n g a i n c r e a s e s t h e s i z e a n d s t r e n g t h o f t h e e d d y . C a u t i o n must be a p p l i e d t h o u g h when a r g u i n g any a p p l i c a t i o n o f t h e s e two f i g u r e s t o t h e n o r t h e a s t P a c i f i c O c e a n . N u m e r i c a l c a l c u l a t i o n s s u g g e s t t h a t i n F i g u r e 57 t h e no n o r m a l f l o w b o u n d a r y c o n d i t i o n a l o n g y = 2 i s b e g i n i n g t o a f f e c t t h e s t r u c t u r e o f t h e f l o w f i e l d o v e r t h e s u p p o r t o f h ( x , y ) . C a l c u l a t i o n s show t h a t i n F i g u r e 58 t h e i n t e r i o r f l o w h a s been d e f i n i t e l y a l t e r e d by t h e s e a w a r d c h a n n e l w a l l . I n b o t h F i g u r e s 57 a n d 58 t h e r e i s no l o c a l eddy p r o d u c t i o n o v e r t h e s l o p e p r o t r u s i o n . N u m e r i c a l e x p e r i m e n t s were u n a b l e t o d i s t i n g u s h b e t w e e n w h e t h e r t h e i n c r e a s e d i n t e r a c t i o n o f t h e seamount on t h e f l o w s i m p l y d o m i n a t e d any l o c a l e f f e c t o f t h e s l o p e p r o t r u s i o n , o r w h e t h e r t h e f o r t h i s c h o i c e o f a t h e 92 ( H u p p e r t ) c r i t i c a l v a l u e o f h, was so l a r g e t h a t no r e a s o n a b l y s m a l l v a l u e s o f t h e R o s s b y number c o u l d g e n e r a t e a l o c a l eddy o v e r t h e s l o p e p r o t r u s i o n . The maximum v a l u e o f p ( 0 ' o c c u r e d o v e r t h e seamount i n b o t h c a l c u l a t i o n s . I n F i g u r e 57, p < 0 ) * .1 and i n F i g u r e 58 p t 0 ) = 1 o v e r t h e seamount. T y p i c a l s p e e d s f o r ( a , e ) = ( 1 , . 1 ) were on t h e o r d e r .2 (20 cm s " 1 ) o v e r t h e s e a m o u n t . I n t h e r e t u r n c o a s t a l f l o w t h e s p e e d s i n c r e a s e d t o a b o u t .25 (25 cm s " 1 ) . F o r t h e p a r a m e t e r p a i r ( a , e ) = ( 1 , . 0 1 ) t y p i c a l s p e e d s were a b o u t 2 (2 m s " 1 ) o v e r t h e seamount and a b o u t 4 ( 4 m s " 1 ) i n t h e r e t u r n c o a s t a l f l o w . The c o m p u t e d t r a n s p o r t s v a r i e d s u b s t a n t i a l l y a s w e l l . F o r ( a , e ) = ( 1 , . 1 ) t h e t r a n s p o r t m a g n i t u d e was a b o u t .1 (35 Sv) o v e r t h e seamount a n d a b o u t .3 (100 Sv) i n t h e r e t u r n c o a s t a l f l o w . W i t h ( a , e ) = ( 1 , . 0 1 ) t h e t r a n s p o r t m a g n i t u d e i n c r e a s e d t o a b o u t 1.2 (400 Sv) o v e r t h e seamount a n d t o a b o u t 2 (700 Sv) i n t h e r e t u r n c o a s t a l f l o w . D e c r e a s i n g t h e s h e a r i n t h e u p s t r e a m c u r r e n t i n c r e a s e s t h e h o r i z o n t a l e x t e n t o f t h e e d d y . W h i l e F i g u r e s 55 a n d 56 a l m o s t g i v e s t h e i m p r e s s i o n o f two s e p a r a t e f l o w d i s t o r t i o n s o v e r t h e seamount and t h e s l o p e p r o t r u s i o n . F i g u r e s 57 a n d 58 s u g g e s t a s i n g l e l a r g e s c a l e e d d y . I n F i g u r e 57 t h e eddy h a s a t y p i c a l r a d i u s o f a b o u t 300 km a n d i n F i g u r e 58 a r a d i u s o f 400 km. I t was a r g u e d e a r l i e r t h e c a l c u l a t i o n s f o r F i g u r e s 57 and 58 s u g g e s t t h a t f o r t h i s a t h e s e a w a r d b o u n d a r y i s s i g n i f i c a n t l y a f f e c t i n g t h e s t r u c t u r e o f t h e i n t e r i o r f l o w f i e l d . T h i s means t h a t a s t h e s i z e o f t h e eddy i n c r e a s e s a n d t h e a f f e c t s o f t h e 93 b o u n d a r y become i m p o r t a n t t o t h e i n t e r i o r , t h e eddy must e l o n g a t e i t s e l f and b e g i n t o p e n e t r a t e d e e p e r i n t h e w a t e r c o l u m n . T h i s e l o n g a t i o n b e g i n s t o o c c u r i n F i g u r e 57 and i s q u i t e a p p a r e n t i n F i g u r e 58. A s s o c i a t e d w i t h t h e b r o a d e n i n g o f t h e eddy f o r d e c r e a s i n g a i s a r e l a t e d d e e p i n g i n t h e v e r t i c a l e x t e n t o f t h e e d d y . F i g u r e s 59 and 60 a r e a l o n g c o a s t l i n e v e r t i c a l s e c t i o n s o f p < 0 > a t y = .6 f o r ( a , e ) = ( 1 , . 1 ) a n d . ( 1 , . 0 1 ) , r e s p e c t i v e l y . C o m p a r i n g t h e s e two f i g u r e s w i t h F i g u r e 18, t h e c o r r e s p o n d i n g f i g u r e f o r t h e s t a n d a r d s e t o f p a r a m e t e r s , a c o u p l e o f o b s e r v a t i o n s c a n be made. The d e e p e n i n g o f t h e eddy i s a p p a r e n t f r o m t h e f a c t t h a t i n F i g u r e 18 t h e .01 p ( 0 1 d o e s n o t r e a c h t h e b o t t o m , w h e r e a s i n F i g u r e 59 i t d o e s . I n F i g u r e 60 t h e c o n t o u r i n g i n t e r v a l s a r e + o r - . 1 , w i t h t h e .1 c o n t o u r r e a c h i n g t h e b o t t o m , i m p l y i n g t h a t t h e .01 d o e s a s w e l l . Thus t h e h o r i z o n t a l g r a d i e n t s o f t h e p ( 0 ' f i e l d r e m a i n h i g h e r d e e p e r , i n d i c a t i n g t h a t t h e v e r t i c a l s t r u c t u r e p r o t r u d e s f u r t h e r i n t o t h e w a t e r c o l u m n . F i n a l l y , F i g u r e s 59 and 60 show t h e i n c r e a s e d h o r i z o n t a l e x t e n t o f t h e e ddy c o r r e s p o n d i n g t o t h e 0 p r e s s u r e c o n t o u r d i s p l a c e d o u t w a r d f r o m x = 0. 4.3 T o p o g r a p h i c P a r a m e t e r s V a r y i n g t h e t o p o g r a p h i c p a r a m e t e r s h, a n d h 2 d i r e c t l y i s t h e s i m p l e s t way t o i n v e s t i g a t e t h e c o n t r i b u t i o n t o t h e t o t a l e d dy f i e l d o f e a c h i n d i v i d u a l t o p o g r a p h i c f e a t u r e . From C h a p t e r I I a n d l a s t s e c t i o n , t h e t o p o g r a p h i c p a r a m e t e r s h , , h 2 a r e g i v e n by t h e maximum h e i g h t o f t h e s l o p e p r o t r u s i o n and t h e P r a t t 94 seamount h e i g h t r e s p e c t i v e l y , d i v i d e d by eH. The t o p o g r a p h i c p a r a m e t e r s a r e r e q u i r e d t o be 0 ( 1 ) i f t h e T a y l o r e x p a n s i o n o f t h e b o u n d a r y c o n d i t i o n w = ( u , v ) • V h ( x , y ) on z= e h ( x , y ) a b o u t z = 0 i s t o be f o r m a l l y j u s t i f i e d . The maximum h e i g h t s o f t h e P r a t t seamount a n d t h e s l o p e p r o t r u s i o n a r e e s t i m a t e d t o be 2500 m e t r e s a n d 800 m e t r e s r e s p e c t i v e l y f r o m F i g u r e 1. W i t h e ^ .02 and H 3500 t h i s means t h a t h , a n d h 2 a r e a p p r o x i m a t e l y 11 and 34, r e s p e c t i v e l y . T h i s c a l c u l a t i o n i m p l i e s t h a t h, a n d h, a r e s i t u a t e d i n t h e i n t e r m e d i a t e r e g i o n b e t ween 0 ( 1 ) and 0 ( e " 1 ) . Thus t h e T a y l o r e x p a n s i o n i s a p p r o p i a t e , i f o n l y m a r g i n a l l y j u s t i f i e d . N u m e r i c a l s i m u l a t i o n s o f t h e s o l u t i o n s were c o m p u t e d f o r h, t a k i n g on t h e r a n g e o f v a l u e s 0, 5, 10 and 15 a n d h 2 t a k i n g on t h e r a n g e o f v a l u e s 0, 10, 20, 30, 35. T h i s means 20 d i f f e r e n t n u m e r i c a l e x p e r i m e n t s were p r e f o r m e d v a r y i n g t h e t o p o g r a p h i c p a r a m e t e r s . F i g u r e s 61 and 62 a r e c o n t o u r p l o t o f p ( 0 ' f o r ( h , , h 2 ) = ( 1 0 . 9 , 0 ) a nd ( 0 , 3 4 . 1 ) , r e s p e c t i v e l y . The c o n t o u r i n g i n t e r v a l was + o r - .01 r e l a t i v e t o t h e 0 s t r e a m l i n e . T h e s e two s i m u l a t i o n s c o r r e s p o n d t o c o m p u t i n g t h e f l o w f i e l d f o r t h e s t a n d a r d s e t o f p a r a m e t e r s i n t h e a b s e n c e o f t h e seamount ( F i g u r e 61) o r i n t h e a b s e n c e o f t h e s l o p e p r o t r u s i o n ( F i g u r e 6 2 ) . C o m p a r i n g F i g u r e s 61 a n d 62 w i t h F i g u r e 11 t h e r e l a t i v e c o n t r i b u t i o n o f e a c h t o p o g r a p h i c f e a t u r e t o t h e t o t a l t o p o g r a p h i c mean f l o w i n t e r a c t i o n c a n be s e e n . The i n t e r i o r s t r u c t u r e o f t h e l o c a l eddy ( c o n t o u r s o f v a l u e g r e a t e r t h a n 95 +.03) o v e r t h e s l o p e p r o t r u s i o n i s l a r e g l y u n a f f e c t e d by t h e e x i s t e n c e o f t h e seamount. The e x t e r i o r r e g i o n , g i v e n by t h o s e c o n t o u r s o f v a l u e l e s s t h a n a b o u t +.03 a r e s i g n i f i c a n t l y a f f e c t e d by t h e seamount s i n c e t h e f l u i d p a r c e l s a s s o c i a t e d w i t h w i t h t h e s e s t r e a m l i n e s a r e t o p o g r a p h i c a l l y s t e e r e d o u t t o t h e seamount. The e f f e c t o f t h e s l o p e p r o t r u s i o n on t h e l o c a l eddy f o r m e d o v e r t h e seamount i s more i n t e n s e . F i g u r e s 11 and 62 s u g g e s t t h a t t h e s i z e a n d i n t e n s i t y o f t h e l o c a l seamount eddy a r e d e c r e a s e d by t h e s l o p e p r o t r u s i o n . T h i s makes s e n s e s i n c e most o f t h e n e a r c o a s t a l f l o w i s d i r e c t e d i n t o an eddy by t h e s l o p e p r o t r u s i o n . S i n c e t h e f a r u p s t r e a m f l o w d e c r e a s e s a s t h e d i s t a n c e t o t h e c o a s t l i n e i n c r e a s e s i t f o l l o w s t h a t t h e s l o p e p r o t r u s i o n i n t e r a c t s w i t h t h e component o f t h e f l o w w h i c h a c c o u n t s f o r t h e most s i g n i f i c a n t i n t e r a c t i o n . The m a g n i t u d e o f p ( 0 1 o v e r t h e seamount i n F i g u r e 62 i s a b o u t .11 c o m p a r e d t o a b o u t .1 i n F i g u r e 11. Thus t h e p r e s s u r e f i e l d o v e r t h e seamount i s a b o u t 10% t o 20% h i g h e r i n F i g u r e 62 t h a n i t i s i n F i g u r e 11. The c h a n g e i n s u r f a c e s p e e d s o v e r t h e seamount i s s i m i l i a r . I n F i g u r e 11 t h e s p e e d s a r e a r e a b o u t 30 cm s " 1 a t ( x , y ) = (.6,.6) a n d i n F i g u r e 62 t y p i c a l s p e e d s a r e a b o u t 34 cm s " 1 . A l l t h e n u m e r i c a l e x p e r i m e n t s p e r f o r m e d s u g g e s t t h a t i f a l o c a l eddy i s t o o c c u r o v e r t h e s l o p e p r o t r u s i o n t h e s t a g n a t i o n p o i n t s a s s o c i a t e d w i t h eddy must o c c u r on t h e x - a x i s , i m p l y i n g t h a t s t a g n a t i o n s t r e a m l i n e i s t h e 0 p r e s s u r e c o n t o u r . T h i s i s a c o n s e q u e n c e o f t h e c o n s e r v a t i o n o f p o t e n t i a l v o r t i c i t y . 96 S u p p o s e t h a t a l o c a l eddy i s p r o d u c e d o v e r t h e s l o p e p r o t r u s i o n b u t t h a t i t s s t a g n a t i o n p o i n t s a r e i n t h e i n t e r i o r o f t h e d o m a i n . F o r a l l t h o s e s t r e a m l i n e s c o a s t w a r d o f t h e s t a g n a t i o n s t r e a m l i n e t h e f o l l o w i n g two f a c t s a r e most r e l e v a n t . F i r s t , t h e y a r e b o u n d e d s e a w a r d by t h e s t a g n a t i o n s t r e a m l i n e , and s e c o n d h ( x , y ) m o n o t o n i c a l l y d e c r e a s e s w i t h i n c r e a s i n g d i s t a n c e t o t h e c o a s t l i n e . The s e c o n d f a c t i m p l i e s t h a t c o m p r e s s i o n o f t h e i s o p c y n a l s i s i n c r e a s e d c o a s t w a r d o f t h e s t a g n a t i o n s t r e a m l i n e , where upon t h e f i r s t f a c t w i l l i m p l y t h a t t h e f l u i d p a r c e l s c a n n o t s u f f i c i e n t l y d e c r e a s e t h e i r r e l a t i v e v o r t i c i t y i n o r d e r t o c o n s e r v e p o t e n t i a l v o r t i c i t y . T h i s c o n t r a d i c t i o n o f t h e b a s i c p h y s i c s o f t h e m o d e l t h e r e f o r e i m p l i e s t h a t i f a l o c a l eddy e x i s t s o v e r t h e s l o p e p r o t r u s i o n t h e s t a g n a t i o n p o i n t s must o c c u r on t h e 0 p r e s s u r e c o n t o u r l y i n g on t h e c o a s t l i n e . T h i s i s n o t n e c e s s a r i l y t h e c a s e f o r t h e l o c a l eddy f o r m e d o v e r t h e seamount s i n c e t h e s u p p o r t o f h ( x , y ) d o e s n o t e x t e n d t o t h e c o a s t l i n e f o r h, =0. Thus i n t h e a b s e n c e o f t h e s l o p e p r o t r u s i o n t h e s t a g n a t i o n p o i n t s f o r t h e eddy o v e r t h e seamount c a n o c c u r i n t h e i n t e r i o r o f t h e d o m a i n , a s i t d o e s i n F i g u r e 62. N u m e r i c a l e x p e r i m e n t s v a r y i n g t h e v a l u e s o f h, w i t h h 2 = 0 and h 2 w i t h h, = 0 s u g g e s t t h a t t h e ( H u p p e r t ) c r i t i c a l v a l u e o f h, i s a b o u t 4 a n d f o r h 2 i s a b o u t 9. 97 4.4 B r u n t - V a i s a l a Frequency The parameters N 0 2 and ( 7 * ) " 1 c o r r e s p o n d t o the maximum v a l u e ( a t z = 1 ) and the v e r t i c a l a t t e n u a t i o n of the d i m e n s i o n a l i z e d B r u n t - V a i s a l a f r e q u e n c y [ N * ( z * ) ] 2 g i v e n i n s e c t i o n 2 . 1 . T h e i r p h y s i c a l importance r e s i d e s i n c o n t r o l l i n g the c o n t r i b u t i o n t o the p o t e n t i a l v o r t i c i t y of the a n g u l a r momentum a s s o c i a t e d w i t h b a r o c l i n i c v o r t e x tube s t r e t c h i n g . R e c a l l t h a t the n o n d i m e n s i o n a l B r u n t - V a i s a l a f r e q u e n c y was g i v e n by S ( z ) = s 0 e x p [ 7 ( z - 1 ) ] w i t h the Burger number s 0 = [ N 0 H / ( f L ) ] 2 and 7 = 7*H. H o l d i n g 7 f i x e d and d e c r e a s i n g N 0 w i l l d e c r e a s e the Burger number s 0 , the maximum a m p l i t u d e of S ( z ) . P h y s i c a l l y , d e c r e a s i n g N 0 i s e q u i v a l e n t t o r e d u c i n g the v e r t i c a l shear i n the mean s t a t e d e n s i t y f i e l d , t h e r e b y assuming a more homogeneous ocean. C o n s i d e r the consequences of r e d u c i n g N 0. The i n c r e a s e d homogeneity of the ocean r e s u l t s i n r e d u c i n g the degree of compression t h a t can occur t o the i s o p c y n a l s . Thus, as the upstream f l o w e n c o u n t e r s the su p p o r t of h ( x , y ) , b a r o c l i n i c v o r t e x tube compression i s reduced. The c o n s e r v a t i o n of p o t e n t i a l v o r t i c i t y w i l l i m p l y t h a t the magnitude of the a s s o c i a t e d change i n the r e l a t i v e v o r t i c i t y w i l l be e q u i v a l e n t l y reduced. However, s i n c e i s o p c y n a l c o m p r e s s i o n o c c u r e p r i m a r i l y near the s u r f a c e the p r i n c i p l e e f f e c t of d e c r e a s i n g N 0 w i l l be t o d e c r e a s e the degree of s u r f a c e i n t e n s i f i c a t i o n i n the eddy f l o w f i e l d . I n c r e a s i n g N 0 s h o u l d have the o p p o s i t e e f f e c t . The s u r f a c e i n t e n s i f i c a t i o n w i l l i n c r e a s e as the v e r t i c a l g r a d i e n t s i n the 98 mean s t a t e d e n s i t y f i e l d s u r f a c e s h a r p e n . I n c r e a s i n g N 0 a l s o a s t h e p o s s i b i l i t y o f d e e p e n i n g t h e d e p t h t o w h i c h t h e eddy p e n e t r a t e s , s i n c e c h a n g e s i n N 0 u n i f o r m i l y e f f e c t S ( z ) t h r o u g h o u t t h e w a t e r c o l u m n . I f N 0 i s s u f f i c i e n t l y i n c r e a s e d t h e n a p p r e c i a b l e b a r o c l i n i c v o r t e x t u b e c o m p r e s s i o n c a n o c c u r i n i n t e r m e d i a t e r e g i o n s o f t h e w a t e r c o l u m n . T h i s w i l l r e s u l t i n a d e c r e a s e d r e l a t i v e v o r t i c i t y i n t h i s r e g i o n a n d c o n s e q u e n t l y i n an a p p a r e n t d e e p e n i n g o f t h e eddy m o t i o n . I n t h e e x t r e m e c a s e i n w h i c h N 0 = 0 t h e i n t e r a c t i o n s t r e a m f u n c t i o n p ( x , y , z ) i s i n d e p e n d e n t o f z and a T a y l o r c o l u m n w i l l f o r m o v e r t h e r e g i o n where h ( x , y ) * 0. The c i r c u l a t i o n a r o u n d t h e T a y l o r c o l u m n w i l l be a n t i c y c l o n i c a n d w i l l e x t e n d u n i f o r m l y t o t h e s u r f a c e . However i f N 0 * 0 t h e n a s t r a t i f i e d T a y l o r c o l u m n w i l l a p p e a r o v e r t h e r e g i o n h ( x , y ) * 0, w i t h a more c o m p l i c a t e d v e r t i c a l s t r u c t u r e t h a n i n t h e homogeneous c a s e . The v e r t i c a l s h e a r i n S ( z ) i s g o v e r n e d by 7 = 7*H. I n c r e a s i n g 7 * w i l l r e s u l t i n d e c r e a s i n g t h e e - f o l d i n g d i s t a n c e r e q u i r e d f o r S ( z ) t o d e c a y t o s 0 e " ' . Thus i n c r e a s i n g 7 * w i l l i m p l y t h a t t h e most s i g n i f i c a n t v e r t i c a l c h a n g e s i n t h e mean s t a t e d e n s i t y f i e l d o c c u r c l o s e r t o t h e s u r f a c e . N e a r t h e s u r f a c e , t h e r e f o r e , t h e c o m p r e s s i o n o f t h e i s o p c y n a l s i n c r e a s e s a s 7 * i n c r e a s e s . T h i s r e s u l t s i n an i n c r e a s e d s u r f a c e i n t e n s i f i c a t i o n o f t h e eddy t r a p p e d c l o s e r t o t h e s u r f a c e . D e c r e a s i n g 7 * s h o u l d have t h e o p p o s i t e e f f e c t . The s u r f a c e i n t e n s i f i c a t i o n s h o u l d d e c r e a s e somewhat a s t h e g r a d i e n t s i n t h e mean s t a t e d e n s i t y f i e l d a r e s m o o t h e d o u t . The c o m p i l a t i o n by Emery e t a l . ( 1 9 8 3 ) o f t y p i c a l l y 99 o b s e r v e d N o r t h P a c i f i c Ocean B r u n t - V a i s a l a f r e q u e n c i e s s u g g e s t s t h a t t h e p a r a m e t e r N 0 l i e s b e t w e e n 1 0 " 1 s " 1 and 2•10" 1 s " 1 . The d a t a p r e s e n t e d i n Emery e_t a l . ( 1983) a l s o s u g g e s t s t h a t t h e a t t e n u a t i o n o f t h e B r u n t - V a i s a l a f r e q u e n c y w i t h d e p t h i s more o r l e s s g e o g r a p h i c a l l y i n v a r i a n t i n t h e N o r t h e a s t P a c i f i c . I t r e m a i n s , h o w e v e r , d e s i r a b l e t o i n v e s t i g a t e t h e e f f e c t s o f v a r i a b l e v e r t i c a l s h e a r o f t h e B r u n t - V a i s a l a f r e q u e n c y on t h e f l o w f i e l d . The l e a s t s q u a r e s f i t o f t h e p a r a m e t e r ( 7 * ) " 1 by W i l l m o t t a n d M y s a k ( l 9 8 0 ) gave a v a l u e o f 254.51 m, s o t h i s p a r a m e t e r was v a r i e d b e t w e e n 225 and 300 m e t r e s . V a r y i n g ( 7 * ) " 1 o u t s i d e t h i s r a n g e was n u m e r i c a l l y i m p o s s i b l e s i n c e a t c e r t a i n s t a g e s i n t h e c a l c u l a t i o n s some o f t h e v a r i a b l e s e x c e e d e d t h e a b i l i t y o f t h e c o m p u t e r t o h a n d l e , i e . t h e y were t o o l a r g e . The p a r a m e t e r N 0 was v a r i e d b e t w e e n 0.01 s " 1 a n d 0.02 s " 1 . S p e c i f i c a l l y , t h e s o l u t i o n s o b t a i n e d i n C h a p t e r I I were c o m p u t e d f o r N 0 t a k i n g on t h e v a l u e s 0.01 s " 1 , 0.015 s " 1 a n d 0.02 s " 1 and ( 7 * ) - 1 t a k i n g on 225 m, 250 m, 275 m a n d 300 m, g i v i n g 12 p a r a m e t e r p a i r c a l c u l a t i o n s . F i g u r e 63 i s a c o n t o u r p l o t o f a v e r t i c a l s e c t i o n o f p ( 0 ) c o m p u t e d a l o n g y = .8 f o r ( 7*,N 0) = (3 0 0 " 1 n r 1 , . 01 s " 1 ) . T h i s v a l u e o f ( 7 * ) " 1 c o r r e s p o n d s t o a c h a n g e o f a b o u t +12% f r o m i t s s t a n d a r d v a l u e . The c o n t o u r i n t e r v a l s a r e + o r - .01 r e l a t i v e t o t h e 0 p r e s s u r e c o n t o u r . C o m p a r i n g F i g u r e 63 w i t h F i g u r e 19 i l l u s t r a t e s t h e m a i n f e a t u r e s a r g u e d p r e v i o u s l y t h a t a r e a s s o c i a t e d w i t h d e c r e a s i n g 7* f r o m i t s s t a n d a r d v a l u e . I n F i g u r e 19 t h e p ( 0 ) f i e l d h a s a s u r f a c e maximum o f a b o u t 100 +.1 o v e r t h e seamount. I n F i g u r e 6 3 , p ( 0 ) .08 on z = 1 o v e r t h e seamount, r e p r e s e n t i n g a b o u t a 20% d e c r e a s e . I t i s a l s o p o s s i b l e t o s e e t h a t t h e eddy f l o w f i e l d i n F i g u r e 63 i s l e s s i n t e n s e t h a n i n F i g u r e 19 by t h e f a c t t h a t t h e w i d t h o f t h e eddy ( i n t h i s s e c t i o n ) i s a b o u t 1 (400 km) i n F i g u r e 19 a n d a b o u t .75 (350 km) i n F i g u r e 63. The s u r f a c e s p e e d s i n F i g u r e 63 a t ( x , y ) = (.8,.8) a r e a b o u t 26 cm s " 1 c o m p a r e d t o a b o u t 29 cm s " 1 i n F i g u r e 19, so t h a t t h e r e i s s l i g h t l y more t h a n a 10% r e d u c t i o n i n t h e s p e e d s . The t r a n s p o r t s a t t h i s ( x , y ) c o o r d i n a t e i n F i g u r e 63 i s a b o u t .05 (18 Sv) w h i c h i s a b o u t 5% s m a l l e r t h a n t h e 19 Sv c a l c u l a t e d i n F i g u r e 19. F i g u r e s 64 a n d 65 a r e c o n t o u r e d v e r t i c a l s e c t i o n s o f p ( 0 > a l o n g y = .2 and y = .8 r e s p e c t i v e l y f o r ( 7 * , N o ) = ( 2 2 5 " 1 n r 1,.02 s " 1 ) . B a s e d on p r e v i o u s l y made a r g u m e n t s , i n c r e a s i n g N 0 s h o u l d r e s u l t i n an i n c r e a s e d s u r f a c e i n t e n s i f i c a t i o n a n d an i n c r e a s e d v e r t i c a l p e n e t r a t i o n o f t h e ed d y . D e c r e a s i n g ( 7 * ) " 1 f r o m i t s s t a n d a r d v a l u e t o 225 m e t r e s s h o u l d i n c r e a s e t h e s u r f a c e r e s p o n s e a n d d e c r e a s e t h e d e p t h t o w h i c h t h e eddy e x t e n d s . I t i s n o t s u r p r i s i n g , h o w e v e r , t h a t F i g u r e s 64 a n d 65 show, i n a d d i t i o n t o an i n t e n s i f i e d s u r f a c e r e s p o n s e , an i n c r e a s e d d e p t h p e n e t r a t i o n o f t h e e d d y . The c h a n g e s i n ( 7 * ) " 1 and N 0 f r o m t h e i r s t a n d a r d v a l u e s f o r t h e s e f i g u r e s a r e 12% and 90% r e s p e c t i v e l y . T h e r e f o r e e f f e c t o f v a r i a t i o n s i n N 0 c a n be r e a s o n a b l y e x p e c t e d t o d o m i n a t e t h o s e c r e a t e d by v a r y i n g 7 * . C o m p a r i n g F i g u r e 64 t o F i g u r e 16 t h e d r a m a t i c i n c r e a s e i n 101 t h e v e r t i c a l e x t e n t i n t h e eddy c a n be s e e n . I n F i g u r e 16 t h e +.01 c o n t o u r e x t e n d s t o a b o u t z = .6 (1400 m e t r e s d e e p ) , w h e r e a s i n F i g u r e 64 t h i s c o n t o u r e x t e n d s t o t h e b o t t o m . I n t e r e s t i n g l y t h e m a g n i t u d e o f p ( 0 > i s o n l y s l i g h t l y i n c r e a s e d i n F i g u r e 64 c o m p a r e d t o F i g u r e 16. The most s i g n i f i c a n t c h a n g e o c c u r s i n t h e t r a n s p o r t s . I n F i g u r e 64, a t ( x , y ) = (.2,.2) t h e t r a n s p o r t h a s m a g n i t u d e a b o u t .015 (4 S v ) . T h i s r e p r e s e n t s a b o u t a 2 5 % c h a n g e i n t h e t r a n s p o r t (5 Sv) c a l c u l a t e d i n F i g u r e 16. The i n c r e a s e d v e r t i c a l e x t e n t o f t h e eddy i s a l s o s e e n by c o m p a r i n g F i g u r e 65 w i t h F i g u r e s 63 and 19. I n F i g u r e 65 t h e u p s t r e a m 0 p r e s s u r e c o n t o u r o c c u r s a t a b o u t x = -.1 f o r y = .8, w h e r e a s i n F i g u r e 63 i t o c c u r s f o r x = 0. The +.02 c o n t o u r e x t e n d s down t o a b o u t 900 m e t r e s i n F i g u r e 65 and down t o a b o u t 600 m e t r e s i n F i g u r e 63. The i n t e n s i t y o f t h e eddy i s i n c r e a s e d i n F i g u r e 65 c o m p a r e d t o F i g u r e 63. I n F i g u r e 6 5, p ( 0 ' * +.09 on z = 1 c o m p a r e d t o a b o u t +.08 i n F i g u r e 6 3 . The m a g n i t u d e o f p ! 0 ' i n F i g u r e 65 i s a b o u t t h e same a s i n F i g u r e 19. The s u r f a c e s p e e d s a r e i n c r e a s e d f r o m a b o u t 26 cm s ~ 1 i n F i g u r e 63 t o a b o u t 28 cm" 1 i n F i g u r e 6 5 , c o n s i s t e n t w i t h t h o s e i n F i g u r e 19. The t r a n s p o r t s have i n c r e a s e d a s w e l l , i n l i n e w i t h t h e d e e p e n i n g o f t h e e d d y . I n F i g u r e 65 a t ( x , y ) = (.8,.8) t h e t r a n s p o r t i s a b o u t 21 Sv c o m p a r e d t o a b o u t 17 Sv i n F i g u r e s 63 and 19, r e p r e s e n t i n g a 20% i n c r e a s e . F i g u r e 66 i s a c o n t o u r p l o t o f a v e r t i c a l s e c t i o n f o r p ( 0 ' a l o n g y = .8 i n w h i c h b o t h ( 7 * ) " 1 and N 0 h a v e been i n c r e a s e d f r o m t h e i r s t a n d a r d v a l u e s t o 300 m and .02 s " 1 r e s p e c t i v e l y . 102 The c o n t o u r i n t e r v a l s a r e + o r - .01 r e l a t i v e t o t h e 0 p r e s s u r e c o n t o u r . F i g u r e 66 h a s t h e r e s u l t i n g eddy f l o w e x t e n d i n g d e e p e r i n t o t h e w a t e r c o l u m n and a t t h e same t i m e h a s t h e n e a r s u r f a c e m o t i o n d e c r e a s e d . The m a g n i t u d e o f p ( 0 ' on z = 1 i s a b o u t +.06 i n F i g u r e 66 c o m p a r e d t o a b o u t +.08 i n F i g u r e 6 5 , a b o u t +.08 i n F i g u r e 63 and +.1 f o r t h e s t a n d a r d s e t o f p a r a m e t e r s i n F i g u r e 19. The i n c r e a s e d s p a c i n g b e t w e e n t h e s u r f a c e c o n t o u r s i n F i g u r e 66 s u g g e s t s t h a t t h e h o r i z o n t a l v e l o c i t i e s a r e r e d u c e d . A t ( x , y ) = (.8,.8) t h e s u r f a c e s p e e d i s a b o u t .3 cm s " 1 i n F i g u r e 66 c o m p a r e d t o s u r f a c e s p e e d s i n e x c e s s o f 20 cm s " 1 i n F i g u r e s 6 5 , 63 a n d 19. The t r a n s p o r t i s n o t s i g n i f i c a n t l y d e c r e a s e d i n F i g u r e 66 c o m p a r e d t o F i g u r e 65. I n F i g u r e 66 t h e t r a n s p o r t i s a b o u t 21 Sv w h i c h i s a b o u t t h e same a s i n F i g u r e 65. Thus w h i l e t h e s u r f a c e i n t e n s i t y h a s d e c r e a s e d due t o l a r g e r N 0 a n d s m a l l e r 7 * , t h e l a r g e r N 0 h a s a l s o d e e p e n e d t h e eddy so t h a t t h e i n t e g r a t e d e f f e c t i s t o i n c r e a s e t h e t r a n s p o r t . 4.5 H o r i z o n t a l C u r r e n t S h e a r And S u r f a c e C u r r e n t The p r i n c i p l e c o n c e r n o f t h i s s e c t i o n i s t o d e s c r i b e t h e e f f e c t o f v a r i a t i o n s o f t h e b o u n d a r y c o n d i t i o n Z ( 1 ) = a on t h e q u a l i t a t i v e s t r u c t u r e o f t h e eddy f l o w f i e l d . A t t e n t i o n i s g i v e n t o t h e p a r a m e t e r a o n l y i n so f a r a s i t e f f e c t s t h e r o l e p l a y e d by Z ( 1 ) . A c o m p l e t e d i s c u s s i o n o f t h e r e s p o n s e o f t h e f l o w f i e l d t o v a r i a t i o n s i n a i s g i v e n i n S e c t i o n 4.2. The e f f e c t o f v a r i a t i o n s i n Z ( 1 ) i s a g a i n b e s t u n d e r s t o o d w i t h i n t h e f r a m e work o f v o r t i c i t y a r g u m e n t s . The u p s t r e a m 103 v o r t i c i t y i s g i v e n by - 9 2 u 0 = a e x p ( - a y ) Z ( z ) . C o n s e q u e n t l y i n c r e a s i n g Z(1), h o l d i n g a f i x e d , w i l l i n c r e a s e t h e r e l a t i v e v o r t i c i t y o f t h e s u r f a c e f l o w . M o r e o v e r , s i n c e Z ( z ) s m o o t h l y assumes i t s b o u n d a r y v a l u e a t z = 1, i n c r e a s i n g Z(1) w i l l a l s o t e n d t o i n c r e a s e t h e v o r t i c i t y o f t h e n e a r s u r f a c e f l o w . D e c r e a s i n g Z(1) w i l l o b v i o u s l y d e c r e a s e t h e r e l a t i v e v o r t i c i t y o f t h e n e a r s u r f a c e u p s t r e a m c u r r e n t . C o n s i d e r t h e e f f e c t o f i n c r e a s i n g Z(1) on t h e i n t e r a c t i o n b e t w een t h e u p s t r e a m c u r r e n t a n d t h e t o p o g r a p h y . The c o n s e r v a t i o n o f p o t e n t i a l v o r t i c i t y a n d t h e f i x e d c h a n g e i n t h e c o m p r e s s i o n o f t h e i s o p c y n a l s i m p l i e s t h a t i n c r e a s i n g Z(1) w i l l i n c r e a s e t h e r e l a t i v e v o r t i c i t y o f t h e eddy f i e l d . C o n s e q u e n t l y t h e a n t i c y c l o n i c m o t i o n i n t h e eddy i s r e d u c e d . I n f a c t , f o r , a s u f f i c i e n t l y l a r g e s u r f a c e c u r r e n t no eddy i s p r o d u c e d . I f a l l o t h e r p a r a m e t e r s a r e h e l d t o t h e i r s t a n d a r d v a l u e s , n u m e r i c a l e x p e r i m e n t s s u g g e s t t h a t t h i s s o - c a l l e d c u t o f f s p e e d i s a b o u t 30 cm s " 1 . I n c r e a s i n g ( d e c r e a s i n g ) t h e h o r i z o n t a l c u r r e n t s h e a r p a r a m e t e r a, d e c r e a s e s ( i n c r e a s e s ) t h e n e c e s s a r y c u t o f f s p e e d . I n t h e e x t r e m e s i t u a t i o n w i t h Z(1) = 0 t h e r e i s no s u r f a c e e x p r e s s i o n o f any i n t e r a c t i o n b e t w e e n t h e u p s t r e a m c u r r e n t a n d t h e t o p o g r a p h y . T h i s c a n be s e e n by e x a m i n i n g t h e b o u n d a r y c o n d i t i o n 3.3. I f Z(1) = 0 t h e n 3.3 r e q u i r e s t h a t p D 3Z = 0 on z = 1. I f D 3Z = 0 on z = 1 t h e n ( p r o v i d i n g Z ( z ) i s a n a l y t i c ) Z ( z ) = 0 e v e r y w h e r e i n t h e w a t e r c o l u m n . T h e r e f o r e Z(1) = 0 must i m p l y t h a t p = 0 on z = 1, w h i c h h a s t h e p h y s i c a l i m p l i c a t i o n t h a t no i n t e r a c t i o n o c c u r s on t h e s u r f a c e . 104 The e x i s t e n c e o f a c u t o f f s u r f a c e s p e e d a n d t h e f a c t t h a t Z ( 1 ) = 0 l e a d s t o p ( 0 1 = 0 on z = 1 i m p l i e s t h e e x i s t e n c e o f a v a l u e o f Z ( 1 ) f o r w h i c h t h e s u r f a c e i n t e r a c t i o n i s m a x i m i z e d . F o r t h e s t a n d a r d s e t o f p a r a m e t e r s t h i s v a l u e i s a b o u t 5 cm s " 1 . D e c r e a s i n g ( i n c r e a s i n g ) Z ( 1 ) b e l o w t h i s e x t r e m a l v a l u e w i l l t e n d t o d e c r e a s e ( i n c r e a s e ) t h e s u r f a c e eddy c i r c u l a t i o n . D e c r e a s i n g ( i n c r e a s i n g ) Z ( 1 ) a b o v e t h i s v a l u e w i l l t e n d t o i n c r e a s e ( d e c r e a s e ) t h e s u r f a c e r e s p o n s e . The s o l u t i o n s o b t a i n e d i n C h a p t e r I I I were c o m p u t e d f o r a = Z ( 1 ) t a k i n g on t h e r a n g e o f v a l u e s 0, . 1 , .25, .5, and 1.0 c o r r e s p o n d i n g t o 0 cm s " 1 , 10 cm s " 1 , 25 cm s " 1 , 50 cm s " 1 and 1 m s " 1 . The h o r i z o n t a l c u r r e n t s h e a r was v a r i e d a s i n S e c t i o n 4.2, t h u s 25 n u m e r i c a l e x p e r i m e n t s were p r e f o r m e d v a r y i n g a and Z(1 ) . F i g u r e 67 i s a c o n t o u r p l o t o f p ( 0 ) on z = 1 f o r t h e e x t r e m a l c a s e where Z ( 1 ) = .05 (5 cm s " 1 ) . A l l o t h e r p a r a m e t e r s a r e h e l d t o t h e i r s t a n d a r d v a l u e s . F i g u r e s 68 a n d 69 aire c o n t o u r p l o t s o f v e r t i c a l s e c t i o n s o f p ( 0 ) a l o n g y = .2 and y = .8, r e s p e c t i v e l y . I n a l l t h r e e f i g u r e s t h e c o n t o u r i n g i n t e r v a l i s + o r - .01 r e l a t i v e t o t h e z e r o p r e s s u r e c o n t o u r . C o m p a r i n g F i g u r e 67 w i t h F i g u r e 11 t h e i n c r e a s e d eddy f l o w f e a t u r e s c a n be s e e n . The c o a s t a l s t a g n a t i o n p o i n t s i n F i g u r e 67 a r e d i s p l a c e d o u t w a r d f r o m x = 0 c o m p a r e d t o F i g u r e 11, i m p l y i n g t h a t t h e i n t e r a c t i o n b e t w e e n t h e t o p o g r a p h y and t h e u p s t r e a m c u r r e n t b e g i n s f u r t h e r u p s t r e a m f o r Z ( 1 ) = .05. The a r e a o f eddy c i r c u l a t i o n i s a l s o l a r g e r i n F i g u r e 67 t h a n i n F i g u r e 11. 105 The v a l u e o f p ( 0 ) h a s i n c r e a s e d i n t h e i n t e r i o r o f t h e e d d y . O v e r t h e seamount, p ( 0 ' i s on t h e o r d e r o f .13 i n F i g u r e 67. T h i s r e p r e s e n t s an i n c r e a s e o f a b o u t 30% f r o m p ( 0 ' .1 c a l c u l a t e d i n F i g u r e 11. S i m i l i a r i n c r e a s e s a r e o b s e r v e d o v e r t h e s l o p e ' p r o t r u s i o n w i t h p l 0 > i n c r e a s i n g f r o m a b o u t .08 i n F i g u r e 11 t o a b o u t .1 i n F i g u r e 67. The r e d u c e d s p a c i n g b e t w e e n t h e s t r e a m l i n e c o n t o u r s i n F i g u r e 67 c o m p a r e d t o F i g u r e 11 m a n i f e s t s i t s e l f i n an i n c r e a s e d c l o c k w i s e c i r c u l a t i o n . O v e r t h e s e a m o u n t , t h e s u r f a c e s p e e d s c o m p u t e d i n F i g u r e 67 a r e t y p i c a l l y on t h e o r d e r o f .5 (50 cm s " 1 ) . T h i s r e p r e s e n t s an i n c r e a s e o f a b o u t 25% o v e r t h e .4 (40 cm s " 1 ) c o m p u t e d i n F i g u r e 11. T h e r e i s a s i m i l i a r i n c r e a s e o v e r t h e s l o p e p r o t r u s i o n . A t ( x , y ) = ( 0 , 0 ) , t h e s p e e d s i n F i g u r e 67 a r e a b o u t .75 (75 cm s " 1 ) c o m p a r e d t o t h e .5 (50 cm s " 1 ) i n F i g u r e 11. F u r t h e r o u t , s a y a t ( x , y ) = ( 0 , . 1 ) , t h e s p e e d s a r e a b o u t .45 (45 cm s ~ 1 ) i n F i g u r e 67 c o m p a r e d t o a b o u t .3 (30 cm s " 1 ) i n F i g u r e 11. C o m p a r i n g F i g u r e s 68 a n d 69 w i t h F i g u r e s 16 a n d 19 r e s p e c t i v e l y , t h e i n c r e a s e d v e r t i c a l p e n e t r a t i o n o f t h e eddy f l o w f i e l d c a n be s e e n . I n F i g u r e 68, t h e s l o p e p r o t r u s i o n i n d u c e s a t y p i c a l c u r r e n t s p e e d o f a b o u t .05 (5 cm s " 1 ) i n t h e d e e p i n t e r i o r o f t h e w a t e r c o l u m n , c o m p a r e d t o a n e g l i g i b l e f l o w i n F i g u r e 16. A s i m i l i a r s i t u a t i o n i s f o u n d i n t h e s e c t i o n a l o n g y = .8. I n F i g u r e 69, t h e d e e p i n t e r i o r s p e e d s a r e on t h e o r d e r o f .1 (10 cm s " 1 ) c o m p a r e d t o a b o u t .01 cm s " 1 i n F i g u r e 19. The i n c r e a s e i n t h e m a g n i t u d e o f t h e v e l o c i t y t h r o u g h o u t 106 t h e w a t e r c o l u m n r e s u l t s i n i n c r e a s i n g t h e t r a n s p o r t s i n t h e e d d y . O v e r t h e s e a m o u n t , t h e t r a n s p o r t i s a b o u t .12 (40 Sv) c o m p a r e d t o a b o u t t h e 20 Sv c o m p u t e d f o r t h e s t a n d a r d s e t of p a r a m e t e r s . I n t h e r e t u r n c o a s t a l f l o w t h e t r a n s p o r t h a s i n c r e a s e d f r o m 20 Sv, f o r t h e s t a n d a r d p a r a m e t e r s , t o a b o u t 70 Sv w i t h Z ( 1 ) = .05. F i g u r e 70 i s a c o n t o u r p l o t o f p ( 0 ' on z = 1 f o r Z ( 1 ) = 1. The c o n t o u r i n t e r v a l s a r e + o r - .02 r e l a t i v e t o t h e z e r o p r e s s u r e c o n t o u r . T h i s v a l u e o f Z ( 1 ) i s w e l l a b o v e t h e n u m e r i c a l l y d e t e r m i n e d c u t o f f v a l u e o f .3. C o n s e q u e n t l y , no s u r f a c e e x p r e s s i o n o f an eddy i s e x p e c t e d t o o c c u r . T h e r e i s a s e a w a r d d e f l e c t i o n o f t h e s t r e a m l i n e s o v e r t h e s e a m o u n t . O v e r t h e s l o p e p r o t r u s i o n t h e r e i s a l e s s n o t i c e a b l e s e a w a r d d e f l e c t i o n o f t h e s t r e a m l i n e s . V e r t i c a l s e c t i o n s o f . p ( 0 ) show t h a t t h e r e i s no s u b m e r g e d eddy o v e r t h e s l o p e p r o t r u s i o n o r o v e r t h e seamount. F i g u r e 71 i s a c o n t o u r map o f p ( 0 ' on z = 1 w i t h Z ( 1 ) = .01. The c o n t o u r i n t e r v a l s a r e + o r - .01 r e l a t i v e t o t h e z e r o p< 0 > c o n t o u r . T h i s v a l u e o f Z ( 1 ) i s s m a l l e r t h a n t h e e x t r e m a l v a l u e o f .05, c o n s e q u e n t l y t h e eddy c i r c u l a t i o n shown i n F i g u r e 71 i s w e a k e r t h a n i n F i g u r e 67. O v e r t h e s e a m o u n t , p ( 0 ) =* .04 c o r r e s p o n d i n g t o a b o u t 70% d e c r e a s e f r o m t h a t i n F i g u r e 67. The p ( 0 ' f i e l d o v e r t h e s l o p e p r o t r u s i o n i s s i m i l i a r l y r e d u c e d , w i t h p t 0 ) .03 i n F i g u r e 71 c o m p a r e d t o p ( 0 ' = .1 i n F i g u r e 67. The s u r f a c e s p e e d s h a v e been r e d u c e d a s w e l l . T y p i c a l s p e e d s o v e r t h e seamount a r e a b o u t .12 (12 cm s " 1 ) and o v e r t h e 107 s l o p e p r o t r u s i o n a r e a b o u t .2 (20 cm s " 1 ) , c o r r e s p o n d i n g t o a b o u t a 75% c o m p a r e d t o t h o s e i n F i g u r e 67. F i g u r e s 72 and 73 a r e v e r t i c a l s e c t i o n s o f p ( 0 ) a l o n g y = .2 a n d y = .8 r e s p e c t i v e l y , f o r Z ( 1 ) = .01. The c o n t o u r i n g i n t e r v a l i s + o r - .01 r e l a t i v e t o t h e z e r o p < 0 ) c o n t o u r . T h e s e two f i g u r e s show t h a t , e x c e p t n e a r t h e s u r f a c e , t h e r e s p o n s e i s n e a r l y d e p t h i n d e p e n d e n t . N u m e r i c a l e x p e r i m e n t s show t h a t t h i s i s t h e c a s e w h e n e v e r Z ( 1 ) = Z ( 0 ) . I f t h e b o u n d a r y c o n d i t i o n s on Z ( z ) a r e e q u a l t h e n t h e f o r m o f S ( z ) w i l l i m p l y t h a t Z ( z ) w i l l o n l y s l i g h t l y d e v i a t e f r o m t h e c o n s t a n t v a l u e Z ( 0 ) ( = Z ( 1 ) ) . S i n c e G 0 ( z ) i s p r o p o r t i o n a l t o Z ( z ) , t h e p r i n c i p l e r e s p o n s e o f t h e f l o w t o t o p o g r a p h i c e x c i t m e n t w i l l be o n l y m a r g i n a l l y d e p t h d e p e n d e n t . However i f Z ( z ) i s t r u l y b a r o t r o p i c t h e n f o r m o f t h e s o l u t i o n shown i n F i g u r e s 7 1 , 72 a n d 73 i s i n c o r r e c t . I n S e c t i o n 3.2 i t was shown t h a t i f Z ( z ) i s b a r o t r o p i c t h e n K = -a2 i m p l y i n g t h a t X 0 = 0 w h i c h i n t u r n i m p l i e s t h a t t h e f o r m o f t h e s o l u t i o n u s e d t o c ompute F i g u r e s 7 1 , 72 a n d 73 i s i n v a l i d . 4.6 H o r i z o n t a l C u r r e n t S h e a r And B o t t o m C u r r e n t V a r i a t i o n s i n t h e b o t t o m b o u n d a r y c o n d i t i o n Z ( 0 ) = b h a v e a s i g n i f i c a n t e f f e c t on t h e eddy f l o w f i e l d . V o r t i c i t y a r g u m e n t s a r e a g a i n t h e most p h y s i c a l l y r e l e v a n t way t o u n d e r s t a n d t h e e f f e c t o f t h e p a r a m e t e r Z ( 0 ) . F i g u r e 6 shows t h a t f r o m z = 0 t o z =* .5 Z ( z ) i s a p p r o x i m a t e l y d e p t h i n d e p e n d e n t . T h i s i s a c o n s e q u e n c e o f a B r u n t - V a i s a l a f r e q u e n c y w h i c h i s a p p r o x i m a t e l y z e r o a t t h i s d e p t h . V a r i a t i o n s i n Z ( 0 ) w i l l l e a d t o more o r l e s s u n i f o r m 108 c h a n g e s i n t h e v a l u e o f Z ( z ) i n t h e l o w e r w a t e r c o l u m n . C o n s e q u e n t l y , i n c r e a s i n g Z ( 0 ) w i l l i n c r e a s e t h e r e l a t i v e v o r t i c i t y o f t h e u p s t r e a m c u r r e n t t h r o u g h o u t t h e l o w e r h a l f o f t h e w a t e r c o l u m n . The c o n s e r v a t i o n o f p o t e n t i a l v o r t i c i t y a n d t h e c o m p r e s s i o n o f t h e mean s t a t e i s o p c y n a l s a s t h e f l o w e n c o u n t e r s t h e s u p p o r t o f h ( x , y ) w i l l i m p l y t h a t t h a t t h e a n t i c y c l o n i c m o t i o n o v e r t h e t o p o g r a p h y w i l l be r e d u c e d f o r i n c r e a s e d Z ( 0 ) . I n f a c t , f o r s u f f i c i e n t l y l a r g e Z ( 0 ) no eddy i s p r o d u c e d a t a l l . T h i s c u t o f f v a l u e f o r t h e b o t t o m b o u n d a r y c u r r e n t was n u m e r i c a l l y d e t e r m i n e d t o be a b o u t 50 cm s " 1 , a s s u m i n g a l l o t h e r p a r a m e t e r s a r e h e l d t o t h e i r s t a n d a r d v a l u e s . The a b o v e v o r t i c i t y a r g u ment i m p l i e s t h a t d e c r e a s i n g Z ( 0 ) w i l l t e n d t o i n c r e a s e t h e a n t i c y c l o n i c m o t i o n o v e r t h e t o p o g r a p h y . T h i s s c e n a r i o must be m o d i f i e d by t h e f a c t t h a t Z ( 0 ) = 0 w i l l r e s u l t no i n t e r a c t i o n o c c u r i n g a t a l l . From 3.18 a n d 3.19 i t i s c l e a r t h a t e a c h P ( x , y ) i s p r o p o r t i o n a l t o G ( 0 ) . n n However, f r o m 3.8 i t f o l l o w s t h a t i f Z ( 0 ) = 0 t h e n G (0) = 0 f o r n e a c h mode. T h e r e f o r e no b o t t o m c u r r e n t w i l l i m p l y no i n t e r a c t i o n . The p r e c e d i n g a r g ument a n d t h e p r e v i o u s v o r t i c i t y a r g u m e n t a r e c o m p a t i b l e i f t h e r e e x i s t s some v a l u e o f Z ( 0 ) f o r w h i c h t h e i n t e r a c t i o n f i e l d i s m a x i m i z e d . F o r t h e s t a n d a r d s e t o f p a r a m e t e r s t h i s v a l u e i s Z ( 0 ) = .02. N u m e r i c a l e x p e r i m e n t s seem t o i n d i c a t e t h a t t h e eddy c i r c u l a t i o n i s m a x i m i z e d i f 5«Z(0) = Z ( 1 ) , p r o v i d e d t h e o t h e r p a r a m e t e r s a r e h e l d t o t h e i r s t a n d a r d v a l u e s . I n c r e a s i n g ( d e c r e a s i n g ) Z ( 0 ) o r Z ( 1 ) ab o v e 109 t h i s e q u i l i b r i u m r e l a t i o n s h i p w i l l d e c r e a s e ( i n c r e a s e ) t h e m a g n i t u d e o f t h e i n t e r a c t i o n f l o w f i e l d . I n c r e a s i n g ( d e c r e a s i n g ) Z ( 0 ) o r Z ( 1 ) b e l o w t h i s e q u i l i b r i u m r e l a t i o n s h i p w i l l i n c r e a s e ( d e c r e a s e ) t h e m a g n i t u d e o f t h e i n t e r a c t i o n f l o w f i e l d . F i g u r e 74 i s a c o n t o u r p l o t o f p ( 0 ' on z = 1 f o r Z ( 0 ) = .001. T h i s v a l u e o f Z ( 0 ) c o r r e s p o n d s t o a u p s t r e a m c o a s t a l b o t t o m c u r r e n t o f .1 cm s " 1 . A l l o t h e r p a r a m e t e r s a r e h e l d a t t h e i r s t a n d a r d v a l u e s . The c o n t o u r i n t e r v a l i s + o r -.002 r e l a t i v e t o t h e z e r o s t r e a m l i n e . F o r t h i s v a l u e o f Z ( 0 ) t h e r e i s no s u r f a c e e x p r e s s i o n o f an eddy o v e r t h e s l o p e p r o t r u s i o n . However, c l o s e d s t r e a m l i n e c i r c u l a t i o n d o e s o c c u r o v e r t h e s e a m o u n t . The r e d u c t i o n o f Z ( 0 ) t o .001 f r o m i t s s t a n d a r d v a l u e o f .01 h a s a s i g n i f i c a n t e f f e c t on t h e p r o p e r t i e s o f t h e f l o w f i e l d . O v e r t h e s l o p e p r o t r u s i o n t y p i c a l v e l o c i t i e s were c o m p u t e d t o be a b o u t .02 (2 cm s " 1 ) . Over t h e seamount t h e v e l o c i t i e s a r e a b o u t .03 (3 cm s ~ 1 ) . T r a n s p o r t s a r e a b o u t .001 Sv and .002 Sv o v e r t h e s l o p e p r o t r u s i o n a n d seamount r e s p e c t i v e l y . F i g u r e 75 i s a c o n t o u r p l o t o f a v e r t i c a l s e c t i o n o f p ( 0 ' a l o n g y = .2. The c o n t o u r i n t e r v a l i s + o r - .002 r e l a t i v e t o t h e z e r o p ( 0 ) c o n t o u r . The most i n t e r e s t i n g f e a t u r e o f F i g u r e 75 i s t h e r e g i o n o f p o s i t i v e p ( 0 ) bounded by t h e z e r o c o n t o u r a n d z = 0. T h i s r e g i o n c o r r e s p o n d s t o an a n t i c y c l o n i c eddy e x t e n d i n g f r o m t h e o c e a n f l o o r t o a b o u t z = .75. The m a g n i t u d e o f t h e v e l o c i t y f i e l d i n i t s i n t e r i o r i s v e r y weak, on t h e o r d e r o f .005 cm s " 1 . 110 The near surface contours turn upward toward the surface over the submerged eddy as a consequence of the turning of the coasta l flow seaward as i t encounters the slope pro trus ion . 111 F i g u r e 10 - L o c a t i o n of computed v e r t i c a l p r o f i l e s r e l a t i v e o t o the topography • r pj 1 1 2 113 F i g u r e 12 - H o r i z o n t a l contour p l o t of the s tream f u n c t i o n o on z = .9 i - p j o rsi a a II SIXti A 114 F i g u r e 13 - H o r i z o n t a l c o n t o u r p l o t of the stream f u n c t i o n 0"Z £" I G' I S' C CO 1 1 5 116 117 118 F i g u r e 17 - V e r t i c a l c o n t o u r p l o t of p ( 0 1 on y = .4 o a CM a a It o CC 119 120 F i g u r e 19 - V e r t i c a l c o n t o u r p l o t o f p ( 0 ' o n y = .8 o 121 F i g u r e 20 - V e r t i c a l c o n t o u r p l o t of p ( 0 ) on y = 1 a a II Q cc m ii CD a r—i X a I! in m C M II cr j r j r cr a n o a a a cr a_ o_ cr a a in il cr x 0_ _ l cr. O O >-cn o U J i—i L i _ U J c r CO CO U J c r \ 0' I s r o i £' 0 SIXb Z r S?' 0 a ' C M CO X IT) a i <—i i o C M 0" 0 122 F i g u r e 21 - S t i c k p l o t of the h o r i z o n t a l v e l o c i t y f i e l d on z = 1 o CM O O II o CC m II C\J :r C D o Ji_ 5E ll o 2 in rsj ii cr cr o a ii * — o o ll o o o o ii cr Q _ a a o L P II cr n Q _ _ J cr C D C D c r o L L CD I LL J >-o L U CD 0'2 0' 1 SlXti X 0' o 123 F i g u r e 22 - S t i c k p l o t o f t h e h o r i z o n t a l v e l o c i t y f i e l d o n CD z = .9 o a Q II o CC m II CM :r cn o ji_ X n o LO LO il c r cr a il o II o o a o l! cr a. o_ cr Q O LO II d X 0_ CD cn o c r o _ j L U >-r H C J O UJ \ 1 x \ / \ 1 \ \ 0*2 S" I 1 0 ' 1 SIXd k S' 0 C O 124 F i g u r e 23 - S t i c k p l o t o f t h e h o r i z o n t a l v e l o c i t y f i e l d on z = .8 o CM O O II o CC •tf-ro ll CM cn o J!_ o ll o in LO CM :i cc cr o o II o a a a II cr a. o_ cr ^ o a LD II CX X C D CO C D t r o Q _ J LU >-L J O LU \ ! o ' CM LP. a CO X a i LP I o LCM 0' I SlXd A S" 0 0' 0 125 F i g u r e 24 - S t i c k p l o t o f t h e h o r i z o n t a l v e l o c i t y f i e l d o n Q CM o Q 11 o CC n CM cn o o II o LO LD rM II cr cr o o JL O o II o r--j CD o o II cr •_ cr o o in ii cr zr. z = .7 o ' CM C D o c r o L i _ Q I LU >-o LU un CO ° X a i a CM 0*2 S" I 0' I SIXd A S' 0 0 0 126 F i g u r e 25 - S t i c k p l o t o f t h e h o r i z o n t a l v e l o c i t y f i e l d on z = . 6 o C M a a II II CM cn X CD II o 2-. in C M II cr 5Z cr o a :i a :i o M o CD cr CL Q_ cr a CD L P II CX o CD O c r o i >-C J o U J o C M LT! CD CO ° x X L O CD 1 CD , i — i I L O CD . C M 0'2 0' I SIXbi A s' a 0' o 127 F i g u r e 26 - S t i c k p l o t o f t h e mass t r a n s p o r t f i e l d o a a :i o cc m II C M -XL cn o 1! X ll o in in C M c r sr cr o a il o CD cr a. o_ cr ^ a a m II cr Cl-ef 1 ! c r o a . c o ~z: C E c r CO CO CX o CNJ CO ° X ° C E X in i i L P C'2 0' I —r~ a _ C M C' 0 128 129 130 131 F i g u r e 30 - V e r t i c a l c o n t o u r p l o t o f p ( 0 > o n y = .2 H r i n i 0 " I S L ' O S ' O S J ' O O - O S'lXU Z 132 F i g u r e 31 - V e r t i c a l c o n t o u r p l o t of p < 0 ) on y = Q CM CD M o CC ro n CM cn o EE CD II a un LT) CM II cr rz n cr a n_ a I! Q a a o n cr o_ o_ cr ^ a a LO II cr x Q_ cr. O O c r o L U H H L L _ >-h H C O LU CD -0.00-\ 0' I SL' 0 S " 0 SIXU Z .4 o fM Ln Q CO ° x X CD I a i in i a CM 0" 0 133 F i g u r e 32 - V e r t i c a l c o n t o u r p l o t o f p < 0 ) o n y = .6 o a II a 0"I SL'O S'O S?. '0 O'O srxu z 134 135 F i g u r e 34 - V e r t i c a l c o n t o u r p l o t o f p ( o ) o n y = 1 o CM o a n ro II CM zc CD a n_ x a II a in \r in CM n cc o o \ cr a JI_ O II O (VI a a o II cx Q_ Q_ cr o a LO II cr X a. _J cr c r o CD, I L U >-C O LU 0' I r i r SL'D S'O S?.'0 s r x u Z 0' 0 136 F i g u r e 35 - V e r t i c a l p r o f i l e o f p ( 0 ) a t ( x , y ) = ( - 2 , . 1 ) o CM o o II no II CM CD o II a LO CM i! cr T. cr a 11 a a a a a :i cr a. a a a in n cr x Q_ o *—I o o o >-X c r o L L Q L U I—I L L L U c r L O L O L U c r C L . i SO" 0 —r 0' 0 SO'O- i "0-137 F i g u r e 36 - V e r t i c a l p r o f i l e of p t 0 > a t (x,y) = o rsi o o 11 o cr m n CM DZ CD CD II II o LO CM I! cr TZ sr cr o a n a II o CD a a II cr cr CD CD CD C\J X o i LU (-2, .1) a a in CM in in r-n CO in 1 1 X QCX in rM CO in i— >-in r-OD o a in II cr s C O LU a 0 ' I S' o 0 ' 0 S'O-138 F i g u r e 37 - V e r t i c a l p r o f i l e of u ( 0 > a t (x,y) = (-2,.1) o CM o a ii a CC n ii CM on o Jl o II o in T in ro II cr cr CO o o i ! Q o a a 11 cr a. 0-cr a cn in II cr zz 0_ o CD CD CD C\J X c r o L i . >-i— i—i o i L l J in CM in rs i in m CO un1 1 X <=>cx in CM in in i— a in CD C I S" 0 0' 0 S 'C- 0' [-139 F i g u r e 38 - V e r t i c a l p r o f i l e o f v ( 0 ) a t ( x , y ) = (-2,.1) o rM o o ll Q CC ro II CM zn cn a x X O r—v a O CD :i o to M " L O rM II cn sr. i z c r o o I! cn i i o o o C D II cr CL. cr C D a L O II cr x o C D C\J c r o i—i C J o i U J a a L H CM L O L O L O ' 1 X L O CM C O L O CD L O C O G' I 5' C C D S'C-140 F i g u r e 39 - V e r t i c a l p r o f i l e of w ( 0 > a t (x,y) = (-2,.1) o rsi o o ii o CC "3" ii nr. o o LO \T LO ii CX XT C cr o a II o o o a n cn a_ o_ c; a a m ll cr I E CL. o *—I CD CD O CM x c r o >-L J O LLJ a o in CNJ i n rsi m m CO in1 ' X a d r-vj in CNJ I D in m ZG'C :c* o [O'G- 2C"0-141 F i g u r e 40 - V e r t i c a l p r o f i l e of p ( 0 ) a t (x,y) = o o a II ii cn a x o n a •z. in CM II cr 5Z cr CD a a II o a a a n cr Q_ Q_ cr '^i a a u i n cr x Q_ o V — I CD CD >-X cr: o U_ Q _ J LU L L LU c r CO CO LU r r c r c~ i • o i sc 0 0' 0 r S O ' O -(-.4, . 1 ) a CM LO CS) L D r-ro O CO ^ X o a r LO OJ LO LO o LO r~ CO 142 F i g u r e 41 - V e r t i c a l p r o f i l e of p ( 0 ) a t (x,y) = (-.4, .1) o a CM C3 a II M CN cn o X a I! o in LD CM II cr cr o cn a a a a a I! cr CL. cr C D CD CD CD > -X c r o U _ Q L U I—I i n CM rQ LO r-CO in' 1 X LO CM IO in LO CO a o LO II a x o_ _J cr C O UJ CD 0' I S" 0 I o' a S ' C- 0' [-143 F i g u r e 42 - V e r t i c a l p r o f i l e o f u < 0 ) a t ( x , y ) = ( - . 4 , . 1 ) Q - I s' o era s'o- o'i-144 F i g u r e 43 - V e r t i c a l p r o f i l e of .( 0 ) a t (x,y) = (-.4,.1) o C M o a ll o m cn J l o 5 a II LO M " LT) C M 1! CC cr o a a a li a a o Q II cr o_ D_ cr Q a un n cr x O O o >-X c r o L u >-I—I CJ o UJ G' I S" 0 —I 0' 0 S'ti- er r-145 F i g u r e 44 - V e r t i c a l p r o f i l e of w < 0 ) a t ( x , y ) = (-.4,.1) 2 0 ' 0 1 0 " 0 c o - r o ' o - ZO'O-146 F i g u r e 45 - V e r t i c a l p r o f i l e o f p ( 0 > a t ( x , y ) = a CM o a n Q cr ro II CM CO • EE a I! a LO T " LD CM II c r cr a CD II a II Q O a a u cr o_ Q_ cr a a LO II CX zn Q_ LO JL CD-CD CD CD > -X c r o Q L U h H U _ L U C O C O L U c r ( 0 , . 7 5 ) a a CM LO CM LO r-rn CO LO CM CD LO a LO r -<n a a T' 0 SO' 0 0" 0 SO'O- I ' 0 -147 F i g u r e 46 - V e r t i c a l p r o f i l e of p ( 0 ) a t (x,y) = a o o II o CC CO II CM zn a 5E a 2". in I D CM II cr C D a a a II o o o o n cr a. o_ cr ^ a a m :i cr x CL. LO C D CD CD CD X c r o i L U >-L O L U a (0,.75) a a 0' I S' 0 0' 0 S"C-148 F i g u r e 47 - V e r t i c a l p r o f i l e of u ( 0 ) a t ( x , y ) = ( 0 , . 7 5 ) o CM o o cr. ro II CM o x CD II a LO in CM CX cr o II a 1! Q M o a II cr o_ Cu cr. a o IT 11 cr x o_ _j cr. LO o CD CD CD >-x~ cr: o >-CJ o U J a • in CM in CNI in r-cn CO ^ X O C X in CM LO CD in m r-co C T S' 0 0' 0 S ' O - G' [• 1 4 9 F i g u r e 48 - V e r t i c a l p r o f i l e o f v ( 0 ) a t ( x , y ) = o o cc •<a-ro II CM zn CD ll LO \T LD CM II CC s r n az o a ii Q iv; o o o II cr a. a. cr vr o a in ' i cr zc a. LO o C D C D C D c r o c r o L U ( 0 , . 7 5 ) o a un CM un CM un m CO un— 1 •X OCX un CM LD un Q r -OQ 0" I s' a C D S"0- c t-150 F i g u r e 49 - V e r t i c a l p r o f i l e o f w ( 0 > a t ( x , y ) = (0,.75) o CM o D Nf m n CNJ cn CD i l o II o LO NT LO CNJ II CX CX o a Jl O CD O II o M O o o II CX 0_ Q_ cx CD CD LO : i cr Q_ I CX L O o C D C D C D x~ cn o >-L J O U J SO' 0 IO' 0 co- t O ' O - ZG'C-151 F i g u r e 50 - V e r t i c a l p r o f i l e o f p ( 0 ' a t ( x , y ) = (.75,.75) a CM CD a n o cc m n CM CD a !j O L D CM CX CD ll a o ii a a a a . n cx Q_ cr a o m n cr zr Q_ L O o LO o X c r o L L . Q _J L U L L ) c r Z D C O c n LU c r Q_. i • 0 s o - 0 o- o S O ' O - i 'o-152 153 F i g u r e 52 - V e r t i c a l p r o f i l e o f u ( 0 ) a t ( x , y ) = (.75,.75) o rM a o !l o CC *3-ro II CM CD O EE a ii LO M"' LO CM II cr zz r: cr o a jl r^ : a o o II o a a a II cr. CL cr ^ a o LO II cr rr. CL L O o o >-x" o r—I C J o U J ZD a o in CM in CM in r-m CO i n 1 — 1 X OCT r-si in CM LO m a LO r-co a 0' I S' 0 0' 0 0' [-154 F i g u r e 53 - V e r t i c a l p r o f i l e o f v ( 0 ) a t ( x , y ) = ( . 7 5 , . 7 5 ) o rM o rr ro CM cn o !! X a \T LO CM I! cr zz n a o Q I! a u o o • u cr a. Cu cr a o LO 11 cr x o_ _ J cr o O i! >-x~ c r o C J O _ 1 U J a o LO rM LO CM LO r-CO LO — •X LO CM LD LO a LO r-00 G' 1 S' 0 0' D S ' O - 0" [-155 F i g u r e 54 - V e r t i c a l p r o f i l e o f w ( o > a t (x,y) = (.75,.75) o CM o CD II o cr m II CM zr. ID O f—1 X o II o LO NT LO CM II CX zz zz cr o a a o it o M a a CD II CX CL CL cr CD LO 1! cr CL cr L O o L O o x c r : o X h H L J O U J 20' 0 IG' C O'G- ro'G- 2 C 0 -156 F i g u r e . 5 5 - H o r i z o n t a l c o n t o u r p l o t o f t h e s t r e a m f u n c t i o n C'Z EJ ' T G ' T D C O SIXb X 157 158 F i g u r e 57 - H o r i z o n t a l c o n t o u r p l o t o f t h e s t r e a m f u n c t i o n SIXd JL 159 Figure 58 - Horizonta l contour plot of the stream function o on z - 1 with (a,e) = (1,.01) r r J C"Z S' I D'l 5' 0 C O SIXb A 160 161 162 163 F i g u r e 62 - H o r i z o n t a l c o n t o u r p l o t o f t h e s t r e a m f u n c t i o n CD o n z = 1 w i t h n o s l o p e p r o t r u s i o n r-rsi SIXti JL 164 F i g u r e 63 - V e r t i c a l c o n t o u r p l o t of p ( 0 > on y = .8 w i t h (N 0,7*) = ( . 0 1 s - 1 , 3 0 0 " 1 m - 1 ) I R CD a I! a or. ro il CM a : cr CD % — i x a CD a n a CD o 8 I! cr 5 CD • I! v—• C D CD II Q o CD cr a. a. cr a o LO I! C L 1 cr. O C O o or o _ J LLl L L L U c r C O C O L U c r CL. i \ — re; -0.31-C I SL.' D &"0 S c " D 9IXU Z G' D 165 F i g u r e 64 - V e r t i c a l c o n t o u r p l o t o f p ( 0 ) o n y = R tn a I! a CC ro ll CM X a i—i JL X R C7 a I! a 2*. a c? LO Ri I! CC o a a I! a o a a iv a a a I! cr CL cr a o L P II a, _ J cr. i o i! CC o L L LL J I—I LU CC CO CO LU c r CL ( N 0 , 7 * ) = ( . 0 2 s - 1 , 2 2 5 " 1 m - 1 ) 2 w i t h a 'rsi L O L n C J a x a ! c i I LO i—i ! a 0* I S C O C D S C O srxb z CO 166 F i g u r e 6 5 - V e r t i c a l c o n t o u r p l o t o f p ( 0 ' o n y ( N 0 , 7 * ) = ( . 0 2 s - 1 , 2 2 5 " 1 n r 1 ) I a a I! a CC ro I! CM X C P CJ 1—I I! X CM a a LO Ri I! c r s a i l i— 1 a «—i cn a II a a ii ct a. a. cr u< a a LP I) a, cr. o C O I o > -o L r L U I—I U _ U J c r CO CO L U cr C L . -P.00-0' I — r ~ si' o ,8 w i t h a ' CM E>" 0 S?'0 srxb z D' 0 167 F i g u r e 66 - V e r t i c a l c o n t o u r p l o t o f p ( 0 ' o n y = .8 w i t h o il a CC n II CM I! zr. a I! a a cr? a (3 II cr a Ji a «—i C J a n a cr a, a cn ^ a a m I! CU I cr CD CO CD >-CC o LL a LU L L L U CC CO CO LU CC CL. \ 1 \ r ( N 0 , 7 * ) = ( . 0 2 s - 1 , 3 0 0 - 1 n c 1 ) 1 1 r -s/'o s -o s ro SIXU Z D ' 0 168 F i g u r e 67 - H o r i z o n t a l c o n t o u r p l o t o f t h e s t r e a m f u n c t i o n o n z = 1 w i t h Z ( 1 ) = .05 PrJ . D'Z S - I C I S'O G' 0 SIXH JL 1 6 9 170 171 CD CD a LP T. (X o a a a a l; a a a cp cr. Q.. Q. cr. a a LH cL" ex. on z = 1 with Z(1) = 1 O O O X J LL. CX UJ c r ex. cn Figure 70 '- Horizonta l contour plot of the stream function CD I I I IIIMII CD CM CD o H a CC. cn 1! CM nr. CX a IJ EE CD CO L C T x a r r X LP t- CD CD T CD L rJ 0*2 ST I G ' I SIX'b' A c 172 F i g u r e 71 - H o r i z o n t a l c o n t o u r p l o t of the stream f u n c t i o n o on z = 1 w i t h Z(1) = .01 p r s i o 1! 9IXU A 173 F i g u r e 72 - V e r t i c a l c o n t o u r p l o t o f p l 0 > on y = .2 w i t h Z ( 1 ) = .01 • (M a o t i a cc II CM tr, cn a •—i Ji. X t; Ul to fM l| CX o •—i a a a CD CD it a CD CD H cx a. a. cx a a LT7 I) o o I! o LLl LU c r co co LU c r D M 51.' G b ' G SIXH Z S r " C CD f M C D 174 F i g u r e 73 - V e r t i c a l c o n t o u r p l o t of p ( 0 ' on y = .8 w i t h Z(1) = .01 o CM a a l! a cc - t f cn I! CM rc CD a JL X a a a in rM cx Cl a a o a n o a a cu a. a a C D m i) cr. X cu cr. o C O o c r o L r a _ j L U L L . L U c r ZD C O L O L U c r CL. -s.a.-- O . M --ZC ' t --CC ' t-C" I 5/ ' 0 s i x y z C D 175 176 Figure 75 - V e r t i c a l contour plot of p l 0 ) on y - 2 with a !! a cc ro II CM rr cn a i—i I! T—• T. Sails* 8 8 f f 77 f 1 Z(0) = .001 CM I! Q LO w cn a a a a a I! a rv c j a cr a. a. cc LO i a, cr O C\J » o c r o Lu Q , i U J I—! LL. L U c r ZD C O C O L U c r a . L O C J X LO a i C J '—i I o CM C I S/.' 0 ET G s i x u z. Sc" 0 0- 0 177 V. APPLICATION TO THE SITKA EDDY The m a t h e m a t i c a l m o d e l d e v e l o p e d i n C h a p t e r I I I i l l u s t r a t e s t h a t a b a r o c l i n i c c o a s t a l c u r r e n t f o r c e d by t o p o g r a p h y c a n g e n e r a t e a n t i c y c l o n i c b a r o c l i n i c e d d i e s . The m o d e l c o n t a i n s a number o f p a r a m e t e r s . E s t i m a t e s o f t h e s e p a r a m e t e r s b a s e d on t h e a v a i l a b l e d a t a r e c o r d f o r t h e n o r t h e a s t P a c i f i c Ocean s u g g e s t s t h a t t h e p r o m i n e n t r e g i o n a l b a t h y m e t r y and t h e l o c a l mean f l o w c a n i n t e r a c t t o p r o d u c e e d d i e s w h i c h a r e q u a l i t a t i v e l y s i m i l i a r t o t h e eddy o b s e r v e d by T a b a t a ( 1 9 8 2 ) . T h i s c h a p t e r c o n c e r n s i t s e l f w i t h q u a l i t a t i v e l y c o m p a r i n g t h e n u m e r i c a l c a l c u l a t i o n s d e s c r i b e d i n C h a p t e r I V w i t h T a b a t a ' s ( 1 9 8 2 ) a n d B e n n e t t ' s ( 1 9 5 9 ) o b s e r v a t i o n s . The l a t e r a l s c a l e o f t h e c a l c u l a t e d l a r g e s c a l e a n t i c y c l o n i c c i r c u l a t i o n a g r e e s f a v o r a b l y w i t h t h e o b s e r v e d r a d i u s o f t h e S i t k a e d d y . F i g u r e 11, c o m p u t e d f o r t h e s t a n d a r d s e t o f p a r a m e t e r s , h a s a l a r g e s c a l e a n t i c y c l o n i c c i r c u l a t i o n w i t h n o n d i m e n s i o n a l r a d i u s .5, c o r r e s p o n d i n g t o a d i m e n s i o n a l r a d i u s o f 200 km. T a b a t a ' s e s t i m a t e o f t h e eddy r a d i u s was on t h e o r d e r o f 100 t o 150 km. The c e n t e r o f t h e c o m p u t e d l a r g e s c a l e c i r c u l a t i o n i s more o r l e s s l o c a t e d a t ( x , y ) c o o r d i n a t e s (.25,.5) w h i c h when c o n v e r t e d t o a p p r o x i m a t e l o n g i t u d e a nd l a t i t u d e i s c o n s i s t e n t w i t h T a b a t a ' s l o c a t i o n o f t h e S i t k a eddy g i v e n a s 57°N 138°W. I n S e c t i o n 4.3 t h e n u m e r i c a l c a l c u l a t i o n s o f t h e r e s u l t i n g f l o w f i e l d i n t h e a b s e n c e o f t h e seamount ( F i g u r e 61) o r t h e s l o p e p r o t r u s i o n ( F i g u r e 62) were p r e s e n t e d . I n e i t h e r c a s e t h e 178 r e s u l t i n g eddy h a s a r a d i u s on t h e o r d e r o f 100 km. The l o c a t i o n o f t h e S i t k a eddy s u g g e s t s t h a t i t i s u n l i k e l y t h a t t h e S i t k a eddy i s s i m p l y p r o d u c e d by t h e i n d i v i d u a l i n t e r a c t i o n o f t h e seamount o r t h e s l o p e p r o t r u s i o n on t h e c o a s t a l c u r r e n t . The c a l c u l a t i o n s o f S e c t i o n 4.3 show t h a t t h e s e a w a r d d e f l e c t i o n o f t h e c o a s t a l c u r r e n t by t h e s l o p e p r o t r u s i o n e x t e n d s o u t t o t h e v i n c i n t y o f t h e s e a m o u n t . T h i s i m p l i e s t h a t t h e e f f e c t o f t h e seamount on t h e f l o w f i e l d c a n n o t be i g n o r e d . T h i s s u g g e s t s t h a t u p s t r e a m f r o m t h e S i t k a eddy t h e c u r r e n t s w o u l d show a t e n d e n c y t o w a r d t h e r e g i o n c o n t a i n i n g t h e P r a t t s e a m o u n t . T h i s t o p o g r a p h i c s t e e r i n g o f t h e c o a s t a l c u r r e n t h a s been d e t e c t e d i n t h e o c e a n o g r a p h i c d a t a ( T a b a t a ; p e r s o n a l c o m m u n i c a t i o n ) . I g n o r i n g t h e e f f e c t s o f t h e s l o p e p r o t r u s i o n d o e s n o t l e a d t o r e a l i s t i c r e s u l t s . F i g u r e 62, i n w h i c h h, = 0, h a s t h e r e s u l t i n g eddy l o c a t e d s u b s t a n t i a l l y s e a w a r d o f i t s o b s e r v e d l o c a t i o n . F u r t h e r m o r e , t h e s t r o n g s e a w a r d d e f l e c t i o n o f t h e o b s e r v e d c o a s t a l c u r r e n t i s a b s e n t , a s i s any s i g n i f i c a n t r e t u r n c o a s t a l f l o w . O b s e r v a t i o n s o f t h e i s o p c y n a l d e p r e s s i o n p l a c e t h e v e r t i c a l e x t e n t o f t h e eddy on t h e o r d e r o f a k i l o m e t e r ( T a b a t a ; 1 9 8 2 ) . D u r i n g M a r c h 1958 and J a n u a r y 1960, t h e d e f l e c t i o n s o f t h e i s p c y n a l s p e r s i s t e d a s d e e p a s 2000 m e t r e s . S t r i c t l y s p e a k i n g t h e m o d e l p r e d i c t s a n t i c y c l o n i c m o t i o n t h r o u g h o u t t h e e n t i r e w a t e r c o l u m n . T h e r e a r e , h o w e v e r , l a r g e v e r t i c a l g r a d i e n t s i n p<°> f u ( 0 ) , v ( 0 ) and p ( 0 > n e a r t h e s u r f a c e . V e r t i c a l s e c t i o n s o f t h e d e n s i t y f i e l d ( F i g u r e s 30 t h r o u g h 34) s u g g e s t t h a t t h e 179 s h a r p e s t g r a d i e n t s i n p ( 0 ) o c c u r i n u p p e r 900 m e t r e s o f t h e o c e a n . C o n t o u r p l o t s o f t h e i s o b a r s ( F i g u r e s 16 t h r o u g h 20) show t h a t t h e s i g n i f i c a n t v e r t i c a l v a r i a t i o n i n p ( 0> i s c o n s t r a i n e d t o t h e u p p e r 1100 m e t r e s o f o c e a n . Thus e s t i m a t e s o f t h e v e r t i c a l e x t e n t o f t h e c a l c u l a t e d eddy b a s e d on t h e o p r e s s u r e and d e n s i t y f i e l d s a r e on t h e o r d e r o f 1000 m e t r e s , w h i c h i s e n t i r e l y 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 . The o b s e r v e d s u r f a c e c u r r e n t s i n t h e S i t k a e ddy a r e a l s o c o n s i s t e n t w i t h t h o s e c o m p u t e d i n t h e m o d e l . T a b a t a ( l 9 8 2 ) d e s c r i b e s t h r e e d r i f t i n g b o u y s t h a t i n 1977 e n t e r e d t h e n o r t h w e s t a r e a o f t h e S i t k a e d d y . T h i s r e g i o n w o u l d c o r r e s p o n d t o t h e a r e a s u r r o u n d i n g ( x , y ) c o o r d i n a t e s (.75,1.25) i n t h e h o r i z o n t a l s e c t i o n s c o n t a i n e d i n t h i s t h e s i s . The a v e r a g e d r i f t r a t e f o r t h e s e b o u y s was c o m p u t e d t o be 62 cm s - 1 , 91 cm s ~ 1 a n d 47 cm s " 1 , w i t h t h e l a t e r a v e r a g e o b t a i n e d f r o m a d r o g u e d b ouy. The m o d e l c o m p u t e s a s u r f a c e d r i f t s p e e d i n t h i s r e g i o n , f o r t h e s t a n d a r d s e t o f p a r a m e t e r s , b e t w e e n 30 a n d 40 cm s " 1 . Thus t h e c a l c u l a t i o n s a r e c o n s i s t e n t w i t h t h e d r o g u e bouy r e s u l t s b u t a r e a b o u t 50% o f v a l u e o b t a i n e d f r o m t h e u n d r o g u e d b o u y s . T a b a t a ' s e s t i m a t e s o f t h e s o u t h w a r d d r i f t r a t e i n t h e n o r t h e a s t s e c t o r o f t h e S i t k a eddy i s r a n g e o f v a l u e s b e t w e e n 48 and 64 cm s " 1 . A s s u m i n g t h e n o r t h e a s t s e c t o r t o be i n t h e g e n e r a l n e i g h b o u r h o o d o f ( x , y ) c o o r d i n a t e s ( . 5 , . 2 5 ) , t h e m o d el p r e d i c t s s u r f a c e s p e e d s on t h e o r d e r o f 50 cm s - 1 . C u r r e n t s p e e d s i n t h e ' i n t e r i o r o f t h e w a t e r c o l u m n i n t h e eddy a r e 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 t h e S i t k a e d d y . R e l a t i v e t o t h e 2 5 0 0 - d e c i b a r l e v e l , T a b a t a e s t i m a t e s a 180 1.5 cm s " 1 c u r r e n t a t t h e 2 0 0 0 - d e c i b a r l e v e l . The m o d el c o m p u t e s c u r r e n t s a t t h e 1750 m e t r e d e p t h l e v e l a s 2 cm s ~ 1 o v e r t h e s l o p e p r o t r u s i o n and 3 cm s ' 1 o v e r t h e s e a m o u n t . The u p s t r e a m c u r r e n t was e f f e c t i v e l y m o d e l l e d a s e x p ( - a y ) Z ( z ) . B e n n e t t ' s ( 1 9 5 9 ) a n a l y s i s o f t h e c o a s t a l c u r r e n t r e v e a l e d t h a t t h e c o a s t a l c u r r e n t was s i g n i f i c a n t l y s h e a r e d , b o t h h o r i z o n t a l l y a n d v e r t i c a l l y . The a ssumed h o r i z o n t a l s t r u c t u r e o f t h e u p s t r e a m c u r r e n t , m o d e l l e d w i t h an e x p o n e n t i a l f u n c t i o n w i t h a d i s t a n c e o f 80 km, was an a c c u r a t e i d e a l i z a t i o n o f t h e a c t u a l c o a s t a l c u r r e n t s h o r i z o n t a l s t r u c t u r e . The v e r t i c a l s t r u c t u r e o f t h e u p s t r e a m c u r r e n t was d e s c r i b e d by t h e f u n c t i o n Z ( z ) . A s s u m i n g t h a t t h e p o t e n t i a l v o r t i c i t y was c o n s e r v e d t h r o u g h o u t t h e f l o w f i e l d f o r c e d Z ( z ) t o be f o r m u l a t e d i n a p a r t i c u l a r way. The r e s u l t i n g v e r t i c a l s t r u c t u r e o f t h e u p s t r e a m c u r r e n t i s c o n s i s t e n t w i t h t h e p r o f i l e s shown i n B e n n e t t ( 1 9 5 9 ) . The m o n t o n i c d e c a y of t h e c u r r e n t w i t h i n c r e a s i n g d e p t h was o b t a i n e d . The u p s t r e a m a l o n g s h o r e s u r f a c e c u r r e n t was assumed t o be a b o u t 10 cm s ~ 1 a n d t h e c u r r e n t s p e e d i n t h e d e e p i n t e r i o r o f t h e w a t e r c o l u m n a b o u t 1 cm s " 1 , a s d e s c r i b e d i n B e n n e t t ( 1 9 5 9 ) . T a b a t a ( l 9 8 2 ) r e p o r t s t h a t t h e u p s t r e a m c o a s t a l c u r r e n t h a s a t r a n s p o r t on t h e o r d e r o f 6 Sv. The m o d e l l e d u p s t r e a m c u r r e n t h a d a t r a n s p o r t o f 5.9 Sv. T h i s s u g g e s t s t h a t t h e l i n e a r i z a t i o n o f t h e p o t e n t i a l v o r t i c i t y e q u a t i o n a s m a n i f e s t e d i n 2.10 was an e f f e c t i v e a n d a c c u r a t e m o d e l o f t h e u p s t r e a m c u r r e n t . The t r a n s p o r t s c o m p u t e d i n t h e eddy were i n t h e m a i n c o n s i s t e n t w i t h t h o s e o b s e r v e d by T a b a t a ( 1 9 8 2 ) . The l a r g e s c a l e 181 a n t i c y c l o n i c c i r c u l a t i o n (shown i n F i g u r e 11 as e x t e r i o r t o the s m a l l e r l o c a l e d d i e s ) has a t r a n s p o r t of about 4 Sv. T h i s compares f a v o r a b l y w i t h Tabata's o b s e r v a t i o n of 5 Sv. The l o c a l e d d i e s produced over the s l o p e p r o t r u s i o n and the seamount had computed t r a n s p o r t s on the o r d e r of 20 Sv. These e s t i m a t e s are somewhat l a r g e r than the the o b s e r v a t i o n s . Over the s l o p e p r o t r u s i o n , Tabata e s t i m a t e s a southward t r a n s p o r t of about 8 Sv. Tabata e s t i m a t e s the t r a n s p o r t over the P r a t t seamount as between 5 and 6 Sv. 182 VI . CONCLUSIONS T h i s t h e s i s h a s e x a m i n e d t h e f o l l o w i n g c o n j e c t u r e : t h a t b a t h y m e t r y a n d t h e l o c a l mean c o a s t a l f l o w o f t h e n o r t h e a s t P a c i f i c Ocean c a n i n t e r a c t t o p r o d u c e b a r o c l i n i c a n t i c y c l o n i c e d d i e s . A m a t h e m a t i c a l m o del was d e v e l o p e d t o e x a m i n e t h i s p o s s i b i l i t y . T h i s m o d e l d e m o n s t r a t e s t h a t t h e r e g i o n a l b a t h m e t r y a nd l o c a l mean f l o w c a n i n t e r a c t t o p r o d u c e m e s o s c a l e a n t i c y c l o n i c e d d i e s . T h e s e e d d i e s a r e g e n e r a t e d f o r p a r a m e t e r v a l u e s w h i c h a r e o b t a i n e d f r o m e s t i m a t e s o f t h e g e o m e t r y , b a t h y m e t r y a nd o c e a n o g r a p h i c d a t a f o r t h e n o r t h e a s t P a c i f i c O c e a n . The s o l u t i o n o f t h e m a t h e m a t i c a l m o d e l i s c o n s i s t e n t w i t h t h e o c e a n o g r a p h i c d a t a o f t h i s r e g i o n a s r e p o r t e d by B e n n e t t ( 1 9 5 9 ) a nd T a b a t a ( 1 9 8 2 ) . The l a t e r a l a n d v e r t i c a l s c a l e s o f t h e a n t i c y c l o n i c c i r c u l a t i o n p r e d i c t e d by t h e m o d e l a g r e e s c l o s e l y w i t h t h e o b s e r v a t i o n s . The eddy i s o b s e r v e d a n d c o m p u t e d t o have a r a d i u s o f a b o u t 200 km a n d t o e x t e n d a t l e a s t t o 1000 m e t r e s i n d e p t h . The d e c a y o f t h e v e l o c i t y f i e l d w i t h d e p t h i s 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 . The c o m p u t e d t r a n s p o r t s a r e i n good a g r e e m e n t w i t h t h e o b s e r v a t i o n s . The s o l u t i o n o f t h e f i e l d e q u a t i o n s i s r o b u s t i n t h e s e n s e t h a t f o r v a r i a t i o n s i n t h e p a r a m e t e r s t h e b a s i c q u a l i t a t i v e s t r u c t u r e o f t h e s o l u t i o n v a r i e s l i t t l e . The m a t h e m a t i c a l m o del i s d e r i v e d f r o m t h e s t e a d y , i n v i s c i d , i n c o m p r e s s i b l e , s t r a t i f i e d , f - p l a n e a n d B o u s s i n e s q e q u a t i o n s o f m o t i o n . The o c e a n i s assumed t o l o w e s t o r d e r t o be 183 a t r e s t a n d i n h y d r o s t a t i c b a l a n c e . The B r u n t - V a i s a l a f r e q u e n c y , d e r i v e d f r o m t h e mean s t a t e d e n s i t y f i e l d , i s a l e a s t s q u a r e s f i t ( W i l l m o t t a n d M y s a k ; 1980) o f a t y p i c a l l y o b s e r v e d B r u n t - V a i s a l a f r e q u e n c y f o r t h e n o r t h e a s t P a c i f i c Ocean (Emery e t a l . ; 1983) w i t h an e x p o n e n t i a l f u n c t i o n . The d y n a m i c p r e s s u r e a n d d e n s i t y f i e l d s a s s o c i a t e d w i t h f l u i d m o t i o n a r e i n h y d r o s t a t i c b a l a n c e . The h o r i z o n t a l v e l o c i t y f i e l d i s g e o s t r o p h i c a l l y s c a l e d r e l a t i v e t o t h e d y n a m i c p r e s s u r e f i e l d . T h i s s c a l i n g r e s u l t e d i n t h e n o n d i m e n s i o n a l p a r a m e t e r s e , F, and s 0 w h i c h a r e t h e R o s s b y number, t h e s q u a r e d r a t i o o f t h e l e n g t h s c a l e t o t h e e x t e r n a l R o s s b y r a d i u s a n d t h e B u r g e r number r e s p e c t i v e l y . E s t i m a t e s o f t h e s e p a r a m e t e r s b a s e d on s c a l e s o b t a i n e d f r o m t h e n o r t h e a s t P a c i f i c s u g g e s t t h a t s 0 - 0 ( 1 ) , F 0 ( e ) and 0 ( e ) =* 1 0 " 2 . T h e s e p a r a m e t e r s s u g g e s t t h a t t h e m o t i o n i s p r i m a r i l y g e o s t r o p h i c a n d t h e r e f o r e h o r i z o n t a l . The s m a l l n e s s o f F i m p l i e s t h a t t h e v o r t i c i t y a s s o c i a t e d w i t h t h e d e f o r m a t i o n o f t h e s e a s u r f a c e i s an o r d e r o f m a g n i t u d e s m a l l e r t h a n t h e r e l a t i v e v o r t i c i t y . H o w e v e r , s i n c e t h e B u r g e r number i s o r d e r u n i t y t h e n t h e b a r o c l i n i c c o m p r e s s i o n o f t h e i s o p c y n a l s makes an e q u a l c o n t r i b u t i o n t o t h e v o r t i c i t y o f a v o r t e x t u b e a s d o e s i t s a n g u l a r v e l o c i t y . The s m a l l n e s s o f e i s e x p l o i t e d by c o n s t r u c t i n g t h e l e a d t e r m s f o r t h e p r e s s u r e , d e n s i t y , v e l o c i t y a nd mass t r a n s p o r t f i e l d s i n an a s y m p t o t i c e x p a n s i o n i n t h e p a r a m e t e r e . The e s s e n t i a l p h y s i c a l f e a t u r e c a p t u r e d i n t h i s l e a d o r d e r s o l u t i o n i s t h a t t h e o r d e r one d y n a m i c s i s t h e r e s u l t o f t h e 184 c o n s e r v a t i o n o f p o t e n t i a l v o r t i c i t y a l o n g s t r e a m l i n e s . The p o t e n t i a l v o r t i c i t y i s t h e b a l a n c e b e t w e e n t h e r e l a t i v e v o r t i c i t y a nd t h e v o r t i c i t y i n d u c e d by t h e c o m p r e s s i o n o f v o r t e x t u b e s . The q u a s i - g e o s t r o p h i c p o t e n t i a l v o r t i c i t y e q u a t i o n ( i e . t h e J a c o b i a n b e t w e e n t h e s t r e a m f u n c t i o n a n d t h e p o t e n t i a l v o r t i c i t y must v a n i s h ) was s o l v e d by a s s u m i n g t h a t t h e p o t e n t i a l v o r t i c i t y was a l i n e a r f u n c t i o n o f t h e p r e s s u r e f i e l d . T h i s p r o c e d u r e i m p l i e d t h a t t h e u p s t r e a m v e r t i c a l c u r r e n t s t r u c t u r e was t h e s o l u t i o n o f a s e c o n d o r d e r o r d i n a r y d i f f e r e n t i a l e q u a t i o n . The h o r i z o n t a l s t r u c t u r e o f t h e u p s t r e a m c u r r e n t was a s sumed t o e x p o n e n t i a l l y d e c a y away f r o m t h e c o a s t . The o b s e r v e d u p s t r e a m c u r r e n t and t h e m o d e l l e d u p s t r e a m c u r r e n t were i n v e r y good a g r e e m e n t . The v e r t i c a l s t r u c t u r e o f t h e c o m p u t e d u p s t r e a m c u r r e n t was i n g ood a g r e e m e n t w i t h t h e o b s e r v a t i o n s c o n t a i n e d i n B e n n e t t ( 1 9 5 9 ) . T a b a t a ( l 9 8 2 ) s u g g e s t s t h a t t h i s c u r r e n t t r a n s p o r t s a b o u t 6 Sv o f w a t e r n o r t h w a r d . The m o d e l l e d c u r r e n t t r a n s p o r t e d a b o u t 5.9 Sv n o r t h w a r d . The t o p o g r a p h y o f t h e o c e a n f l o o r i n t h e n o r t h e a s t P a c i f i c O cean was i d e a l i z e d a s an a b y s s a l p l a i n w i t h two o r o g r a p h i c f e a t u r e s . The s e a m o u n t s i n t h e v i n c i n t y o f t h e P r a t t seamount were m o d e l l e d a s a smooth o r o g r a p h i c f e a t u r e w i t h t h e maximum h e i g h t o f t h e P r a t t s e a m o u n t . I n a d d i t i o n , t h e c o n t i n e n t a l s h e l f h a s a p r o m i n e n t h o r i z o n t a l s e a w a r d p r o t r u s i o n i n t h e c o a s t a l r e g i o n n e a r t h e S i t k a e d d y . T h i s f e a t u r e was m o d e l l e d a s a smooth c o s i n e - l i k e o r o g r a p h i c f e a t u r e p r o t r u d i n g f r o m an 185 o t h e r w i s e s t r a i g h t c h a n n e l w a l l . E s t i m a t e s o f t h e h e i g h t s o f t h e t o p o g r a p h y s u g g e s t e d t h a t t h e h e i g h t s o f t h e seamount a nd s l o p e p r o t r u s i o n was o r d e r R o s s b y number w i t h r e s p e c t t o t h e mean d e p t h o f t h e o c e a n . Thus t h e no n o r m a l b o u n d a r y c o n d i t i o n on t h e v e l o c i t y f i e l d a t t h e b o t t o m c o u l d be e x p a n d e d i n a T a y l o r s e r i e s a b o u t a s t a t e o f no t o p o g r a p h y . The s o l u t i o n f o r t h e o r d e r one s t r e a m f u n c t i o n was o b t a i n e d a s a l i n e a r sum o f t h e u p s t r e a m s t r e a m f u n c t i o n a n d an i n t e r a c t i o n p r e s s u r e f i e l d . The i n t e r a c t i o n p r e s s u r e f i e l d was o b t a i n e d v i a a n o r m a l mode a n a l y s i s d e s c r i b e d i n Chao e_t a l . ( 1 9 8 0 ) . The b o u n d a r y c o n d i t i o n s on t h e i n t e r a c t i o n p r e s s u r e f i e l d w ere i n t e g r a t e d u p s t r e a m i n t o t h e f o r m s u g g e s t e d by H o g g ( l 9 8 0 ) . The v e r t i c a l modes and u p s t r e a m v e r t i c a l s t r u c t u r e were o b t a i n e d u s i n g a t e c h n i q u e i l l u s t r a t e d i n B r y a n a n d R i p a ( l 9 7 8 ) i n t h e i r a n a l y s i s o f t h e v e r t i c a l s t r u c t u r e o f t e m p e r a t u r e a n o m a l i e s i n t h e n o r t h e a s t P a c i f i c O c e a n . The h o r i z o n t a l a m p l i t u d e f u n c t i o n , a s s o c i a t e d w i t h t h e n o r m a l mode a n a l y s i s , were o b t a i n e d u s i n g G r e e n ' s f u n c t i o n s . The s o l u t i o n o b t a i n e d i n t h i s t h e s i s s u g g e s t s t h a t S i t k a eddy i s e s s e n t i a l l y p r o d u c e d i n t h e f o l l o w i n g manner. The n o r t h w a r d f l o w i n g c o a s t a l c u r r e n t e n c o u n t e r s t h e s o u t h e r n edge o f t h e s l o p e p r o t r u s i o n . The c o m p r e s s i o n o f t h e i s o p c y n a l s a n d t h e c o n s e r v a t i o n o f p o t e n t i a l v o r t i c i t y i m p l i e s t h a t t h e r e l a t i v e v o r t i c i t y must d e c r e a s e . T h i s d e c r e a s e i n t h e r e l a t i v e v o r t i c i t y i s o b t a i n e d by i n c r e a s i n g t h e a n t i c y c l o n i c m o t i o n o f a v o r t e x t u b e , i m p l y i n g t h a t t h e c u r r e n t t u r n s s e a w a r d . Some o f 186 the d e f l e c t e d c u r r e n t e n c o u n t e r s the P r a t t seamount, the r e s t c o n t i n u i n g downstream. For those s t r e a m l i n e s p a s s i n g over the P r a t t seamount the i s o p c y n a l s a r e a g a i n compressed which in d u c e s an a n t i c y c l o n i c r o t a t i o n toward the c o a s t l i n e . Some of thes e s t r e a m l i n e s s u b s e q u e n t l y i n t e r a c t w i t h the s l o p e p r o t r u s i o n , o t h e r s a re d e f l e c t e d downstream by the c o a s t l i n e . Those t h a t i n t e r a c t w i t h the s l o p e p o r t r u s i o n a r e as a consequence of the c o n s e r v a t i o n of p o t e n t i a l v o r t i c i t y t u r n e d upstream. The c o n s t r a i n t of the c o a s t l i n e and the upstream s t r e a m l i n e s t h e r e f o r e s e t s up a c l o s e d a n t i c y c l o n i c c i r c u l a t i o n . T h i s l a r g e s c a l e c i r c u l a t i o n i s c e n t e r e d and has c h a r a t e r i s t i c s which agree c l o s e l y w i t h the o b s e r v a t i o n s made of the S i t k a eddy c o n t a i n e d i n T a b a t a ( 1 9 8 2 ) . 187 BIBLIOGRAPHY 1. B a k u n , A., 1978: M o n t h l y T r a n s p o r t P a r a m e t e r s Computed From M o n t h l y Mean P r e s s u r e F i e l d s On A 3 - d e g r e e G r i d . U n p u b l i s h e d M a n u s c r i p t , U. S. D e p t . Commerce, N.O.A.A., N a t . M ar. S e r . , P a c i f i c E n v i r o n . G r o u p , M o n t e r e y , C a l . , 15 pp. 2. B e n n e t t , E. B., 1959: Some O c e a n o g r a p h i c F e a t u r e s Of The N o r t h e a s t P a c i f i c Ocean D u r i n g A u g u s t 1955. J . F i s h . R e s . Bd. C a n . , 2 1 , 56 5 - 6 3 3 . 3. B y r a n , K. a n d P. R i p a , 1978: The V e r t i c a l S t r u c t u r e Of N o r t h P a c i f i c T e m p e r a t u r e A n o m a l i e s . J_j_ G e o p h y s . R e s . , 83 , 2 4 1 9 - 2 4 2 9 . 4. Chao, S., L. J . P i e t r a f e s a a n d G. S. J a n o w i t z , 1980: On The D y n a m i c s Of A B a r o c l i n i c J e t Ov e r S h a l l o w T o p o g r a p h y . U n p u b l i s h e d M a n u s c r i p t . 5. Emery, W. J . , W. G. L e e a n d L. M a g a a r d , 1983: G e o g r a p h i c D i s t r i b u t i o n s Of D e n s i t y , B r u n t - V a i s a l a F r e q u e n c y And R o s s b y R a d i i I n The N o r t h A t l a n t i c a n d N o r t h P a c i f i c . S u b m i t t e d t o J_^_ P h y s . O c e a n o g . . 6. Hogg, N. G., 1980: E f f e c t s Of B o t t o m T o p o g r a p h y On Ocean C u r r e n t s , O r o g r a p h i c E f f e c t s I n P l a n e t a r y F l o w s . G a r p P u b l i c a t i o n S e r i e s No. 2 3 . , 167-265. 7. H u p p e r t , H. E., 1975: Some Remarks On The I n i t i a t i o n Of I n e r t i a l T a y l o r C o l u m n s . J_j_ F l u i d Mech., 67, 39 7 - 4 1 2 . 8. L e B l o n d , P. H. a n d L. A. Mys a k , 1978: Waves I n The O c e a n . E l s e v i e r , 602 pp. 9. T a b a t a , S., 1982: The A n t i c y c l o n i c , B a r o c l i n i c Eddy O f f S i t k a , A l a s k a , I n The N o r t h e a s t P a c i f i c O c e a n . J . P h y s . O c e a n o g . , 12, 1260-1282. 10. Thomson, R. E., 1972: On The A l a s k a n S t r e a m . J . P h y s . O c e a n o g . , 2, 3 6 3 - 3 7 1 . 11. W i l l m o t t , A. J . and L. A. M y s a k , 1980: A t m o s p h e r i c a l l y F o r c e d E d d i e s I n The N o r t h e a s t P a c i f i c . J . P h y s . O c e a n o g . , 10, 1769- 1 7 9 1 . 

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