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Subsurface currents in the Strait of Georgia, west of Sturgeon Bank Chang, Phillip Yit Kuen 1976

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SUBSURFACE CURRENTS IN THE STRAIT OF GEORGIA, WEST-' OF STURGEON BANK by P H I L L I P YIT KUEN CHANG B.Sc. , U n i v e r s i t y o f V i c t o r i a , 1971 A T h e s i s S u b m i t t e d i n P a r t i a l F u l f i l m e n t o f the R e q u i r e m e n t s f o r t h e Degree of M a s t e r o f S c i e n c e i n t h e Department o f P h y s i c s and t h e I n s t i t u t e o f Oceanography We a c c e p t t h i s t h e s i s as c o n f o r m i n g to t he r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA O c t o b e r , 1976 (§) P h i l l i p Y i t Kuen Chang In presenting th i s thes i s in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i lab le for reference and study. I further agree that permission for extensive copying of th i s thesis for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or p u b l i c a t i o n of th i s thesis fo r f inanc ia l gain sha l l not be allowed without my written permission. Department of P-^Aj&Lfc& $01$/ QrvL&JuJji ^/CJlOAl/O^^P^ The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date DtdrltfA, S i i A B S T R A C T S p e c t r a l c h a r a c t e r i s t i c s of s u b - s u r f a c e c u r r e n t and temperature r e c o r d s spanning 310-533 days from m e S t r a i t of G e o r g i a near S t u r g e o n Bank are examined. Of p a r t i c u l a r i n t e r e s t a re l o w - f r e q u e n c y f l u c t u a t i o n s having, a c h a r a c t e r i s t i c p e r i o d of 30 days. T y p i c a l l y , the energy of trie l o w - f r e q u e n c y and t i d a l c u r r e n t s a r e e q u a l at 50m (each about 40% of the t o t a l ) , and 10% and 10%, r e s p e c t i v e l y , a t 200m. Low-freguency c u r r e n t s a r e observed t o be b a r o c l i n i c and posses a l a t e r a l . l e n g t h s c a l e l e s s than 10km. T h e i r g e n e r a t i o n mechanisms are not i d e n t i f i e d . The i o w - f r e g u s n c y f l u c t u a t i o n s • i n s e a - l e v e l and c u r r e n t have p o o r - t o - f a i r c o h e r e n c e , w h i l e t h o s e i n a i r - p r e s s u r e and c u r r e n t have poor coherence. Low-frequency f l u c t u a t i o n s i n wind and c u r r e n t have f a i r coherence at t h e e a s t e r n mooring, and poor coherence a t the r e m a i n i n g two moorings. F l u c t u a t i o n s i n water t e m p e r a t u r e have a c n a r a c t e r i s t i c p e r i o d of 300 days. Water temperatures at d i f f e r e n t l o c a t i o n s g e n e r a l l y have poor coherence e x c e p t a t the lowe s t f r e q u e n c i e s where they are f a i r - t o - g o o d . Water temperature and c u r r e n t at the same l o c a t i o n have poor coherence. T i d a l c u r r e n t s a r e found t o be e l l i p s e s , w i t h the l a r g e s t c o n s t i t u e n t s b e i n g & z and K t. The magnitude, shape .and o r i e n t a t i o n of the t i d a l c u r r e n t e l l i p s e s ar<= time-dependent. I n t e r n a l t i d e s a r e suggested as an e x p l a n a t i o n . TABLE OF CONTENTS i i i P age ABSTRACT .• . . . i i L I S T OF TABLES • v i LI S T OF FIGURES . . .. v i i ACKNOWLEDGEMENTS x i C h a p t e r 1. INTRODUCTION 1 2. DEFINITIONS AND METHODS 5 2.1 • F r e q u e n c y R e p r e s e n t a t i o n o f T i m e - V a r y i n g F u n c t i o n s 5 C o n t i n u o u s and D i s c r e t e F o u r i e r T r a n s f o r m s 5 The E l l i p t i c a l I n t e r p r e t a t i o n .... 8 Power and C r o s s S p e c t r u m ... 11 2.2 S t o c h a s t i c P r o c e s s e s 14 Random V a r i a b l e s 14 T h e o r e t i c a l S p e c t r a l F u n c t i o n s .' 15 C o n f i d e n c e I n t e r v a l s 15 3. THE DATA 17 3.1 The C u r r e n t s and Water T e m p e r a t u r e s ...... 17 E d i t i n g 21 The H a l f - H o u r l y T i m e - S e r i e s 23 i v C h a p t e r Page 3,2 The S u p p l e m e n t a r y D a t a .27 4. THE FREQUENCY REPRESENTATION - 30 4.1 Power S p e c t r u m and D a t a Gaps 30 4.2 S p e c t r a o f O r i g i n a l and E d i t e d R e c o r d s ... 33 4.3 S p e c t r a o f C u r r e n t s 36 4.4 C o h e r e n c e and P h a s e F u n c t i o n s ............. 46 4.5 C u r r e n t - C u r r e n t R e l a t i o n s 49 C o h e r e n c e .... 49 P h a s e 54 4.6 S p e c t r a of. T e m p e r a t u r e s .56 4.7 T e m p e r a t u r e - T e m p e r a t u r e R e l a t i o n s ......... 59 4.3 C u r r e n t - T e m p e r a t u r e R e l a t i o n s 64 4.9 S p e c t r a o f S e a - L e v e l , A i r - P r e s s u r e , Winds 68 4.10 R e l a t i o n s Between S e a - L e v e l s 73 4.11 R e l a t i o n s Between A i r - P r e s s u r e s . . * .. 78 4.12 R e l a t i o n s Between Winds . ... 81 4.13 R e l a t i o n s Between S e a - L a v a l , A i r - P r e s s u r e , Winds 83 4.14 R e l a t i o n s Between C u r r e n t s and S e a - L e v e l , A i r - P r e s s u r e , Winas ... ., 87 C u r r e n t s and S e a - L e v e l 89 C u r r e n t s and A i r - P r e s s u r e 93 C u r r e n t s and Wind 96 4. 15 Summary 99 5. LOW-FREQUSNCY FLUCTUATIONS 102 5.1 'Low-Pass f i l t e r i n g • -102 V C h a p t e r . . . Page 5.2 L o w - F r e q u e n c y T i m e - S e r i e s 105 5.3 U-V S c a t t e r D i a g r a m s . 113 The B o u n d a r i e s 116 5.4 M o n t h l y Means 118 . 5. 5 M o n t h l y FKE 123 5.6 Long-Term Means 128 C o n f i d e n c e I n t e r v a l s f o r t h e Means 132 5.7 Long-Term V a r i a n c e s 1.36 C o n f i d e n c e I n t e r v a l s f o r t h e FaE 142 6. TIDAL CURRENTS 147 6.1 H a r m o n i c A n a l y s i s .. 147 U n c e r t a i n t i e s 149 6.2 Time V a r i a t i o n s 152 C e n t r a l 50m L o c a t i o n 152 F i v e O t h e r L o c a t i o n s 164 6.3 T i m e - A v e r a g e d P a r a m e t e r s 166 6.4 I n t e r n a l T i d e s 168 7 . SUMMARY 170 BIBLIOGRAPHY 173 APPENDIX I 178 vi LIST OF TABLES Table • . Page I Frequency D i s t r i b u t i o n o f the V a r i a n c e i n the C u r r e n t s 39 I I Frequency D i s t r i b u t i o n o f t h e V a r i a n c e i n the Water Temperatures 58 I I I Frequency D i s t r i b u t i o n of t h e V a r i a n c e i n S e a - L e v e l , A i r - P r e s s u r e and Wind 72 IV Long-Term Means and FKE 117 V V a r i a n c e s ( a c c o r d i n g t o e q u a t i o n 5.5) and 95% C o n f i d e n c e I n t e r v a l s of C u r r e n t Component Means D e r i v e d from T i m e - S e r i e s o f L e n g t h , T 134 VI Comparisons Between th e T h e o r e t i c a l V a r i a n c e s of Table V and the C o r r e s p o n d i n g E m p i r i c a l V a r i a n c e s C a l c u l a t e d from the C u r r e n t Component T i m e - S e r i e s of L e n g t h , T 135 V I I T h e o r e t i c a l V a r i a n c e ( e q u a t i o n 5.6) of Component-FKE's Based on Record Length T 143 V I I I Comparison Between T h e o r e t i c a l and E m p i r i c a l V a r i a n c e s of C u r r e n t Component FKE's• Based on Record Length T 144 IX T i d a l C o n s t i t u e n t s Used f o r Harmonic A n a l y s i s 148 X Average T i d a l C u r r e n t E l l i p s e s 167 v i i LIST OF FIGURES F i g u r e Page 1. Map o f V a n c o u v e r I s l a n d and v i c i n i t y 18 2., (A) Map o f S t r a i t o f G e o r g i a showing p l a c e m e n t o f t h e t h r e e moorings (B) C r o s s - s e c t i o n a l view a l o n g l i n e o f m o o r i n g s 19 3. S c h e m a t i c s h o w i n g t i m e c o v e r a g e o f t h e c u r r e n t and water t e m p e r a t u r e r e c o r d s 20 4. P o r t i o n s o f t h e o r i g i n a l and e d i t e d 50m cu r r e n t , r e c o r d s ' f r o m t h e c e n t r a l mooring 23 5. P o r t i o n s of- t h e o r i g i n a l and e d i t e d water t e m p e r a t u r e r e c o r d s from t h e 140m e a s t e r n l o c a t i o n '24 6. P o r t i o n s o f t h e s u r f a c e wind r e c o r d s 28 7. P o r t i o n s of- t h e s e a - l e v e l and a i r - p r e s s u r e r e c o r d s f r o m v a r i o u s l o c a t i o n s a r o u n d V a n c o u v e r I s l a n d 29 '8. S p e c t r a (f»S) o f o r i g i n a l and e d i t e d c u r r e n t s , and o f o r i g i n a l and e d i t e d t e m p e r a t u r e s 34 9. S p e c t r a o f 50m c u r r e n t s 37 10. S p e c t r a o f 140m and 200m c u r r e n t s 38 11. S p e c t r a (f»S) o f 50m c u r r e n t s 43 12. S p e c t r a (f «S) o f 140m and 200m c u r r e n t s 44 13. The 95% c o n f i d e n c e i n t e r v a l s o f t h e phase s p e c t r u m 50 •14. C o h e r e n c e and phase s p e c t r a between c u r r e n t s from v e r t i c a l l y - s e p a r a t e d l o c a t i o n s 51 15. C o h e r e n c e and phase s p e c t r a between c u r r e n t s from h o r i z o n t a l l y - s e p a r a t e d l o c a t i o n s 52 16. S p e c t r a o f water t e m p e r a t u r e s „ 57 v i i i F i q u r e . Page 17. C o h e r e n c e and phase s p e c t r a between w a t e r t e m p e r a t u r e s from, v e r t i c a l l y - s e p a r a t e d l o c a t i o n s 60 18. C o h e r e n c e and phase s p e c t r a between w a t e r t e m p e r a t u r e s from h o r i z o n t a l l y - s e p a r a t e d l o c a t i o n s 62 19. C o h e r e n c e and phase s p e c t r a between c u r r e n t and t e m p e r a t u r e a t t h e 50m l o c a t i o n s 65 20. C o h e r e n c e a nd phase s p e c t r a between c u r r e n t and t e m p e r a t u r e a t t h e 140m and 200m l o c a t i o n s 66 21. S p e c t r a o f w i n d (Sand H e a d s ) , a i r - p r e s s u r e ( V a n c o u v e r ) and s e a - l e v a l ( P t . A t k i n s o n ) 69 22. S p e c t r a (f«S) o f wind (Sand Heads) , a i r - p r e s s u r e ( V a n c o u v e r ) and s e a - l e v e l ( P t . A t k i n s o n ) 71 23. C o h e r e n c e and phase s p e c t r a between s e a - l e v e l a t P t . A t k i n s o n w i t h t h o s e a t o t h e r l o c a t i o n s 74 24. C o h e r e n c e and phase s p e c t r a between a i r - p r e s s u r e ( a t mean s e a - l e v e l ) a t V a n c o u v e r w i t h t h o s e f r o m o t h e r l o c a t i o n s 79 25. C o h e r e n c e a nd ph a s e s p e c t r a between wind, a t Sand Heads and t h o s e a t E n t r a n c e I s l a n d 82 26. C o h e r e n c e a n d phase s p e c t r a among w i n d (Sand Heads) , a i r - p r e s s u r e ( V a n c o u v e r ) and s e a - l e v e l ( P t . A t k i n s o n ) 84 27. C o m p a r i s o n o f o b s e r v e d c o h e r e n c e between s e a - l e v e l ( P t . A t k i n s o n ) and w i n d (Sand H e a d s ) , and t h e c o h e r e n c e between t h e same p a r a m e t e r s as c a l c u l a t e d by e q u a t i o n . (4.11) . 88 23. C o h e r e n c e and ph a s e s p e c t r a between 50m • c u r r e n t s and s e a - l e v e l a t P t . A t k i n s o n 90 29. C o h e r e n c e and phase s p e c t r a between (140m,200m) c u r r e n t s and s e a - l e v a l a t P t . A t k i n s o n - 91 30. C o h e r e n c e and phase s p e c t r a between 50m c u r r e n t s and a i r - p r e s s u r e a t V a n c o u v e r 94 i x F i g u r e • Page .3 1. C o h e r e n c e and phase s p e c t r a between (140m,200m) c u r r e n t s and a i r - p r e s s u r e a t V a n c o u v e r 95 32. C o h e r e n c e and phase s p e c t r a between 50ui c u r r e n t s and wind a t Sand Heads 97 33. C o h e r e n c e and phase s p e c t r a between (140m,200m) c u r r e n t s and w i n d a t Sand Heads .... • 98 34.. F i l t e r f u n c t i o n e m p l o y e d t o o b t a i n t h e l o w - p a s s e d d a i l y t i m e - s e r i e s (A) , and t h a t c o r r e s p o n d i n g t o a r u n n i n g d a i l y mean (B) ...... 103 35. L o w - p a s s e d d a i l y t i m e - s e r i e s o f t h e 50iu c u r r e n t s 10.6 36. L o w - p a s s e d d a i l y t i m e - s e r i e s o f t h e 140m and 200m c u r r e n t s ,. 107 37. L o w - p a s s e d d a i l y t i m e - s e r i e s o f w a t e r t e m p e r a t u r e 109 38. L o w - p a s s e d d a i l y t i m e - s e r i e s o f a i r - p r e s s u r e and s e a - l e v e l 110 39. Low-passed, d a i l y t i m e - s e r i e s o f w i n d 111 40. D a i l y t i m e - s e r i e s o f F r a s e r R i v e r a t A g a s s i . 2 , B r i t i s h C o l u m b i a (49° 14.7»N, 121°45.5•W) 112 41. U-V s c a t t e r d i a g r a m s o f t h e l o w - p a s s e d c u r r e n t s 114 42. M a g n i t u d e and d i r e c t i o n o f t h e m o n t h l y v e c t o r means f o r t h e l o w - p a s s e d c u r r e n t s v e r s u s t i m e 119 43. M a g n i t u d e and d i r e c t i o n o f t h e m o n t h l y v e c t o r means f o r t h e l o w - p a s s e d wind (Sand H e a d s ) , and t h e m o n t h l y means o f t h e l o w - p a s s e d a i r - p r e s s u r e ( V a n c o u v e r ) and t h e l o w - p a s s e d s e a - l e v e l • ( P t . A t k i n s o n ) , a l l v e r s u s t i m e . .., 122 44. M o n t h l y FKE o f t h e l o w - p a s s e d c u r r e n t c o m p o n e n t s v e r s u s t i m e • 125 45. M o n t h l y v a r i a n c e o f t h e l o w - p a s s e d wind c o i a p o n e n t s (Sand Heads) , l o w - p a s s e d a i r - p r e s s u r e ( V a n c o u v e r ) and l o w - p a s s e d s e a - l e v e l ( P t . A t k i n s o n ) , a l l v e r s u s t i m e ...... ,126 X F i g u r e • Page 46. M a g n i t u d e and d i r e c t i o n o f t h e v e c t o r means f o r t h e l o w - p a s s e d c u r r e n t s v e r s u s a v e r a g i n g t i m e • 130 47. M a g n i t u d e and d i r e c t i o n o f t h e v e c t o r mean f o r t h e l o w - p a s s e d wind (Sand H e a d s ) , a n d t h e means c f t h e l o w - p a s s e d a i r - p r e s s u r e ( V a n c o u v e r ) and l o w - p a s s e d s e a - l e v e l ( P t . A t k i n s o n ) , a l l v e r s u s a v e r a g i n g t i m e i . . . . . 131 48. . L o n g - t e r m t o t a l FKE o f t h e l o w - p a s s e d c u r r e n t s v e r s u s d e p t h ( A ) , and maximum c o r r e l a t i o n • c o e f f i c i e n t (c^) and r a t i o o f v-FK£/u,-r'K£, b o t h v e r s u s d i s t a n c e t o t h e n e a r e s t l a t e r a l b o u n d a r y (B) 137 4.9. FKE o f t h e l o w - p a s s e d c u r r e n t c o m p o n e n t s v e r s u s a v e r a g i n g t i m e 140 50. V a r i a n c e o f t h e l o w - p a s s e d w i n d c o m p o n e n t s (Sand H e a d s ) , l o w - p a s s e d a i r - p r e s s u r e ( V a n c o u v e r ) , and l o w - p a s s e d s e a - l e v e l ( P t . A t k i n s o n ) , a l l v e r s u s a v e r a g i n g t i m e 141 51. Time v a r i a t i o n s o f t h e 0, t i d a l c u r r e n t . e l l i p s e and 95% c o n f i d e n c e . i n t e r v a l s 153 52. Time v a r i a t i o n s o f t h e P, t i d a l c u r r e n t e l l i p s e a n d . 9 5 % c o n f i d e n c e i n t e r v a l s 154 53. Time v a r i a t i o n s o f t h e K, t i d a l c u r r e n t e l l i p s e a n d 95% c o n f i d e n c e i n t e r v a l s 155 54. Time v a r i a t i o n s o f t h e N 2 t i d a l c u r r e n t e l l i p s e a n d 9 5 % c o n f i d e n c e i n t e r v a l s 156 55. Time v a r i a t i o n s o f t h e M 2 t i d a l c u r r e n t e l l i p s e and 95% c o n f i d e n c e i n t e r v a l s 157 56. Time v a r i a t i o n s o f t h e S 2 t i d a l c u r r e n t e l l i p s e and 95% c o n f i d e n c e i n t e r v a l s 158 57. . Time v a r i a t i o n s o f t h e K 2 t i d a l c u r r e n t e l l i p s e and 95% c o n f i d e n c e i n t e r v a l s 159 58. Time, v a r i a t i o n s o f t h e M^ t i d a l c u r r e n t e l l i p s e a n d 95% c o n f i d e n c e i n t e r v a l s 160 59. Time v a r i a t i o n s o f t h e MS 4. t i d a l c u r r e n t e l l i p s e a n d '95% c o n f i d e n c e i n t e r v a l s 161 x i ACKNOWLEDGEMENTS I s i n c e r e l y t h a n k e v e r y o n e who has beau c o n n e c t e d w i t h t h i s t h e s i s i n one way o r a n o t h e r . F o r e m o s t c r e d i t must qo t o Dr. P o n d , my s u p e r v i s o r , who a f t e r s u g g e s t i n g t h e s u b j e c t , p r o v i d e d g e n e r o u s c o n t i n u a l e n c o u r a g e m e n t , and- p a t i e n t l y g u i d e d me w i t h numerous v a l u a b l e s u g g e s t i o n s . T h i s m a n u s c r i p t .has a l s o b e n e f i t e d f r o m . t h e w o r t h y c r i t i c i s m s o f . J I S . L e B l o n d and T a b a t a . I t i s my p l e a s u r e to' a c k n o w l e d g e the c o - o p e r a t i o n o f Dr. T a b a t a , who f r e e l y p r o v i d e d me w i t h t h e c u r r e n t m e t e r d a t a , and who was h e l p f u l i n making a v a i l a b l e t h e w i n d , a i r - p r e s s u r e , and s e a - l e v e l d a t a c o l l e c t e d by t h e A t m o s p h e r i c E n v i r o n m e n t S e r v i c e . D i s c h a r g e s o f t h e F r a s e r R i v e r h a v e been s u p p l i e d by t h e W a t e r S u r v e y o f Canada. .Messrs. D e l S m i t h arm A l D o u q l a s ( I n s t i t u t e o f Ocean S c i e n c e s ) g r e a t l y a s s i s t e d i n t h e o r g a n i z a t i o n o f t h i s l a r g e amount o f d a t a . I am a l s o g r a t e f u l f o r t h e c o - o p e r a t i o n f r o m a l l members o f t h e I n s t i t u t e o f O c e a n o g r a p h y o f 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 . F i n a l l y , t h e N a t i o n a l R e s e a r c h C o u n c i l o f C a n a d a , t h e I n s t i t u t e o f Ocean S c i e n c e s ( D e p a r t m e n t o f E n v i r o n m e n t ) and a U.B.C. Summer R e s e a r c h ' S c h o l a r s h i p have s u p p o r t e d me f i n a n c i a l l y i n 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 . 1 C h a p t e r 1 INTRODUCTION T h i s t h e s i s d e a l s w i t h t h e a n a l y s i s c f s u b s u r f a c e w a t e r c u r r e n t s and t e m p e r a t u r e s west o f S t u r g e o n Bank i n t h e S t r a i t o f G e o r g i a . A s h o r t e n e d v e r s i o n a p p e a r s i n Chang e t _ a l . _ (1976) . The S t r a i t o f G e o r g i a (see F i g u r e 1) i s ' a d j a c e n t t o . a l a r g e c e n t r e o f p o p u l a t i o n and i n d u s t r y . I t i s t h e c c m t r e r c i a l seaway t o t h e P e r t o f V a n c o u v e r and i s an i m p o r t a n t f i s h i n g a r e a . I t s use as a r e c r e a t i o n a l a r e a i s s t i l l g r o w i n g . I-'cr t h e s e r e a s o n s , t h e w a t e r p r o p e r t i e s o f t h e S t r a i t o f G e o r g i a h a v e been r e l a t i v e l y w e l l s t u d i e d . H u t c h i n s o n and L u c a s (1931) d e s c r i b e d t h e w a t e r p r o p e r t i e s , t h e n u t r i e n t c o n c e n t r a t i o n s and p h y t o p l a n k t o n p o p u l a t i o n s i n S t r a i t o f G e o r g i a d u r i n g t h e f o u r s u m a e r s c f 1 926-1929 . A mere e x t e n s i v e s t u d y . w a s c o n d u c t e d by T u l l y and Dcdimead (1957) u s i n g d a t a c o l l e c t e d by N.f. C a r t e r a r d J . P . T u l l y i n 1931 and 1932. W a l d i c h u k (1957) gave a c o m p r e h e n s i v e t r e a t m e n t c f t h e p h y s i c a l o c e a n o g r a p h y c f the S t r a i t o f G e o r g i a , b a s e d m a i n l y on d a t a t a k e n d u r i n g 1 9 4 9 - 1 9 5 3 . Though n o t e x a m i n e d i n d e t a i l , t h e r e a r e o c e a n c g r a p h i c d a t a f o r t h e S t r a i t o f G e o r g i a and J u a n de Fuca S t r a i t c o l l e c t e d r e g u l a r l y d u r i n g 1968 a t i n t e r v a l s o f a p p r o x i m a t e l y 5 weeks ( C r e a n and Ages, 1 9 7 1 ) . .. . . . 2 I n f o r m a t i o n on the c i r c u l a t i o n i n t h e S t r a i t of G e o r q i a i s more s p a r s e , due to t h e f a c t t h a t c u r r e n t , measurements are more d i f f i c u l t t o make than customary o c e a n o g r a p h i c measurements, and r e q u i r e much more t i m e , e f f o r t , and r e s o u r c e s . The s u r f a c e c i r c u l a t i o n , b e i n g more e a s i l y measured, was the f i r s t t o be examined. tfaldichuk (1958) d e s c r i b e d t h e s u r f a c e c i r c u l a t i o n i n the S t r a i t of G e o r g i a based on d r i f t b o t t l e data c o l l e c t e d by W. Clemens d u r i n g the f i v e summers of 1926-1929 and 1931. Much more r e c e n t l y , Tabata (1972) deduced t h e s u r f a c e c i r c u l a t i o n near the F r a s e r R i v e r mouth from a e r i a l photographs c o l l e c t e d i n June, 1950. The b e h a v i o r of t h e F r a s e r R i v e r plume was examined by Giovando and Tabata (1970) f o r the summers of 1966 and 1967, and by Tabata a t _ a l t (1971b) d u r i n g the s p r i n g and summer of 1968. Most r e c e n t l y , Cordes' ( i n p r o g r e s s ) a t the I n s t i t u t e of Oceanography, U.B.C, conducted an i n t e n s i v e s u r v e y of the plume b e h a v i o r i n t h e s p r i n g o f 1974. The s u b s u r f a c e c i r c u l a t i o n i n t h e S t r a i t of G e o r g i a was f i r s t examined i n t h e summer of 1953 by Waldichuk (1957). Measurements were done over' a.25 hour p e r i o d from an anchored v e s s e l u s i n g an Ek-man c u r r e n t meter a t a number of s c a t t e r e d l o c a t i o n s i n the S t r a i t . P i c k a r d (1956) measured t h e s u r f a c e and bottom c u r r e n t s f o r 25 hours at each of s e v e r a l s t a t i o n s on a l i n e from G a l i a n o I s l a n d t o P o i n t R o b e r t s . V e r t i c a l p r o f i l e s o f the c u r r e n t u s i n g a S a vonius r o t o r from an anchored s h i p were o b t a i n e d d u r i n g 1967-1969 by Tabata et„al^ (1970a) , (1970b) , (1970c) , (1971b) . Measurements down t o 50m were 7 made; s t a t i o n s were o c c u p i e d f o r as l o n g a s 90 h o u r s ; a n c h o r a g e s i t e s were i n t h e r e g i o n between E c i n t G r e y and P o i n t R o b e r t s . The d e v e l o p m e n t o f r e l i a b l e s e l f - r e c o r d i n g c u r r e n t m e t e r s r a d i c a l l y i n c r e a s e d t h e t i m e and s p a c e c o v e r a g e o f c u r r e n t m e a s u r e m e n t s . S e l f - r e c o r d i n g c u r r e n t m e t e r s h a v e been a v a i l a b l e f c r seme t i m e , b u t t h e y were e i t h e r u n r e l i a b l e o r t h e d a t a p r o c e s s i n g was v e r y t i m e c o n s u m i n g . H u g g e t t (1966) was p e r h a p s t h e f i r s t t c r e p o r t cn t h e usage o f s u c h e q u i p m e n t i n t h e S t r a i t c f G e o r g i a . He e m p l o y e d t h e s e m e t e r s ,frc m moored c a b l e s , a t t h r e e s t a t i o n s between Nanaimc and Sech-elt f c r p e r i o d s up t o 30 d a y s . The D e p a r t m e n t o f t h e E n v i r o n m e n t (1972a) , ( 1 9 7 2 b ) , ( 1 9 7 2 c ) , (1973) c o n t i n u e d t h e s e m e asurements i n t h e y e a r s 1S69-1S72. M e a s u r e m e n t s were . t a k e n w i t h m o o r i n g s t h a t were p o s i t i o n e d a l o n g l i n e s r u n n i n g a c r o s s t h e S t r a i t c f G e o r g i a . These l i n e s c f m o o r i n g s were p l a c e d as f a r s o u t h as F e i n t R o b e r t s and as f a r n o r t h a s Texada I s l a n d . I n 1968 T a b a t a and S t i c k l a n d (1972d) i n i t i a t e d a p r o g r a m t o p r o v i d e an e x t e n s i v e t i m e c o v e r a g e c f t h e s u b s u r f a c e c u r r e n t s west o f S t u r g e o n Bank. The p r o g r a m a t t e m p t e d t o g a t h e r w a t e r c u r r e n t and t e m p e r a t u r e d a t a e v e r a p e r i o d c f 18 months a t e a c h c f s i x l o c a t i o n s . T a b a t a e t _ a l J L (197 1a) r e p o r t e d c n t h e s u c c e s s o f t h i s program and cn t h e p e r f o r m a n c e o f t h e s e l f - r e c o r d i n g c u r r e n t m e t e r s u s e d . The d a t a t h e n s e l v e s were p r e s e n t e d i n t h e f o r m c f h i s t o g r a m s and p r o g r e s s i v e v e c t o r d i a g r a m s by T a b a t a and S t i c k l a n d <1 9 7 2 a ) , (1972b) , (1972c) . The s u p p l e m e n t a r y a n a l y s i s o f t h i s body o f d a t a i s t h e s u b j e c t o f t h i s t h e s i s . D e f i n i t i o n s and m a t h e m a t i c a l t o o l s 4 u s e d a r e p r e s e n t e d i n t h e n e x t c h a p t e r . E x a m p l e s , d e t a i l s and p e c u l i a r i t i e s o f t h e d a t a a r e shewn and d i s c u s s e d i n C h a p t e r 3. I n t h e s u c c e e d i n g c h a p t e r , t h e f r e q u e n c y r e p r e s e n t a t i o n c f t h e o b s e r v a t i o n s i s e x a m i n e d . A t t e n t i o n i s t h e n f o c u s s e d cn t h e v a r i a t i o n s c o n t a i n e d i n s p e c i f i c f r e g u e n c y b a n d s . C h a p t e r 5 d e a l s w i t h t h e v a r i a t i o n s h a v i n g p e r i o d s f r o m two d a y s t o a b o u t one y e a r . The t i d a l o s c i l l a t i o n s a r e e x a m i n e d s e p a r a t e l y i n C h a p t e r 6. A summary c f t h e r e s u l t s i s g i v e n i n t h e f i n a l c h a p t e r . 5 C h a p t e r 2 ' DEFINITIONS AND METHODS The f u n c t i o n s t o he c a l c u l a t e d f c r t h e s p e c t r a l a n a l y s i s a r e d e f i n e d i n t h i s c h a p t e r . The t h e o r y i s n o t c o m p r e h e n s i v e and t h e r e a d e r s h o u l d t u r n t o t h e r e f e r e n c e s c i t e d f o r more d e t a i l s . The m a t e r i a l i n t h e f i r s t s e c t i o n i s l a r g e l y t a k e n f r o m M c o e r s (1S73) . J e n k i n s and W a t t s (1968) and B e n d a t and E i e r s o l (1971) g i v e a much more c o m p l e t e t r e a t m e n t o f s t o c h a s t i c p r o c e s s e s . A v e r y c u r s o r y p r e s e n t a t i o n of h a r m o n i c a n a l y s i s t y t h e l e a s t - s q u a r e s method i s c o n t a i n e d i n C h a p t e r 6. The t h e o r e t i c a l b a c k g r o u n d o f l e a s t - s g u a r e s e s t i m a t i o n c a n be f o u n d i n C h a p t e r 4 of J e n k i n s and W a t t s ( 1 9 6 8 ) . G o d i n ( 1972) d i s c u s s e s i n d e p t h t h e h a r m o n i c a r a l y s i s by l e a s t - s q u a r e s methods and by o t h e r methods. B r i g h a m (1974) d e s c r i b e s t h e F a s t F o u r i e r T r a n s f o r m a l g o r i t h m . He a l s o g i v e s an e x c e l l e n t e x p l a n a t i o n o f t h e d i f f e r e n c e s between I c u r i e r t r a n s f o r m s c f c o n t i n u o u s f u n c t i o n s d e f i n e d e v e r a l l t i m e , and d i s c r e t e f r e q u e n c y - l i m i t e d F o u r i e r t r a n s f o r m s c f d i s c r e t e f u n c t i o n s d e f i n e d a t a f i n i t e number o f t i m e p o i n t s . 2_. X F r e c[u e ncy__ Per. r e sen t a t i c n _ c f _Time - Va r j ^ i n ^ _ F unc t i o ns The t i m e - v a r y i n q h o r i z o n t a l c u r r e n t v e c t o r i s r e s o l v e d 6 i n t c i t s r e a l - v a l u e d c o m p o n e n t s , u (t) and v ( t ) . P o s i t i v e u and v d e n o t e t he t r u e e a s t and t r u e n o r t h c o m p o n e n t s , r e s p e c t i v e l y . The c o m p o n e n t s a r e r e g a r d e d as s u p e r p o s i t i o n s o f s i n u s o i d s h a v i n g f r e q u e n c i e s f:-CD u (t) = f [ B ( f ) «cos (2-rff t ) + E ( f ) «sin (2if£t) ] .df ° •= fu (f) «exp (i2-rrft) d f -® (2.1) v (t) = $°[ B 3 ( f ) «cos ( 2 i f f t ) + J ^ ( f ) • s i n (2-rrft) ] d f = fl (f) *exp ( i 2 i r f t ) df -oo where' i 2 = -1 and -oo<t<a>. The B . ( f ) ' s a r e r e a l - v a l u e d w h i l e (U(f) /I ( f ) ) a r e c a l l e d t h e F o u r i e r t r a n s f o r m s o f (u ( t ) , v (t) ) :-U ( f ) = J u (t) *exp (- i 2 i T f t) dt (-a><f<oo) -oo (2 . 2) U ( + f ) = [ B , <f)±B 2 ( f ) ]/2 (0<f<oo) and s i m i l a r l y f o r V ( f ) and V(±f). The B - ( f ) 's a r e d e f i n e d o n l y o v e r p o s i t i v e f r e q u e n c i e s ( c n e - s i d e d t r a n s f o r m ) w h i l e U ( f ) and V (f) a r e d e f i n e d o v e r a l l f r e q u e n c i e s ( t w o - s i d e d t r a n s f o r m s ) . I n p r a c t i c e , one i s u s u a l l y i n t e r e s t e d i n f u n c t i o n s w h i c h a r e known o n l y o v e r ,a f i n i t e p e r i o d c f t i m e , and w h i c h a r e d e f i n e d t o re z e r o e l s e w h e r e . To t h i s e n d , t h e f u n c t i o n g ( t ) i s i n t r o d u c e d . g (t) i s u n i t y o v e r t h e r e g i o n |t|<T/2 and z e r o e l s e w h e r e , w h i l e u ( t ) i s d e f i n e d t o be u(t)«g(t). The F o u r i e r t r a n s f o r m , U ( f ) , c f u ( t ) i s U ( f ) * G ( f ) , where t h e sym h o i * d e n o t e s t h e c o n v o l u t i o n c p e r a t o r : -oo U ( f ) *G ( f ) = J U ( f ') »G ( f - f •) df» ,(2.3) -co and G ( f ) = 2 T * s i n (TfTf) / (TTTf) i s t h e F o u r i e r t r a n s f o r m o f g (t) . I n g e n e r a l , U (f)#U ( f ) and one must use c a u t i o n i n t h e c h o i c e o f T so t h a t U ( f ) i s an u s a b l e a p p r o x i m a t i o n o f U ( f ) . As a r e s u l t o f t h e c o n v o l u t i o n w i t h G (f) , t h e (am p i i t ud e , w i d t h) o f p e a k s i n ^  ( f ) a r e ( s m a l l e r , l a r g e r ) r e l a t i v e to t h o s e i n U (f) . One c o n s e q u e n c e i s t h a t two l i n e s c f e q u a l a m p l i t u d e , S ( f - f , ) and S ( f - i z ) ( S d e n o t e s t h e D i r a c - d e l t a f u n c t i o n ) , i n 0 ( f ) w i l l n o t be r e s o l v e d i n U (f) u n l e s s (f, - f z ) < T . I f one o f t h e l i n e s has a much l a r g e r a m p l i t u d e t h a n t h e e t h e r , t h e n T must be much l a r g e r t h a n ( f , - f 2 ) - 1 f o r t h e two l i n e s t o be r e s o l v e d i n U ( f ) . A n o t h e r c o n s e q u e n c e i s t h a t a l e w - f r e q u e n c y l i n e 6 ( f - f 3 ) { f 3 = 0) w i l l n o t be r e s o l v e d i n U (f) u n l e s s f 3 - 1 < l . C o m p u t a t i o n a l l y , one d e a l s w i t h f i n i t e s e t s c f numbers. Thus b o t h t h e t i m e - v a r y i n g f u n c t i o n and i t s t r a n s f o r m a r e d i s c r e t e f u n c t i o n s . . C o n s i d e r t h e f i n i t e d i s c r e t e t i m e - s e r i e s u ( r a t ) o f l e n g t h T = 2.C4t w h i c h i s a' s u p e r p o s i t i o n o f s i n u s o i d s : -u (r A t ) = Z [ E i ( m a f ) » c o s ( . . ) + B 2 ( m A f ) e s i n ( . . ) 1 = 2 u ( m^ f) *e x p [ i (. .) ] where (r = -n ,.. . , 0 ,1 ,. . . , n- 1) , A f = T ~ 1 and (..) = (27r«m A f « r At) . U (mAf) , t h e d i s c r e t e c c r t p l e x - v a l u e d F o u r i e r t r a n s f o r m o f u ( r A t ) , i s d e f i n e d o v e r b o t h p o s i t i v e and n e g a t i v e f r e q u e n c i e s , w h i l e t h e r e a l - v a l u e d Bj^ (mAf) 's a r e d e f i n e d o n l y o v e r p o s i t i v e f r e q u e n c i e s : — -rt, _ U(±mAf)^(2n)- 1 £u(rAt)*exp[+i(..) ] (2.5) = [ 5 , (mAf) + i B 2 (mAf) ]/2 where (m = 1 , . . . , n ) , (..)= (27r«ni4f «rAt) , and U ( C ) = B , ( 0 ) . S i m i l a r r e l a t i o n s h e l d f o r t h e component v ( r A t ) and i t s d i s c r e t e F o u r i e r t r a n s f o r m V (±mAf) = [ B 3 (niA-f) n-iB^ (maf) J / 2 . One dees n o t c h e e s e t h e s a m p l i n g i n t e r v a l A t i n a h a p h a z a r d manner. L e t u ( t ) d e n o t e t h e c o n t i n u o u s t i m e - s e r i e s f r o m w h i c h u ( r A t ) i s d e r i v e d by s a m p l i n g a t t = r A t ( r - - n , . . . , 0 , 1 , . .. ,n- 1) . I f t h e r e a r e v a r i a t i o n s i n u ( t ) w h i c h have f r e q u e n c i e s h i g h e r t h a n f 7 V = ( 2 A t ) _ 1 , a l i a s i n g ( a p p e a r a n c e c f h i g h f r e q u e n c y e n e r g y a t l o w e r f r e q u e n c i e s ) o c c u r s b e c a u s e : -exp[ i2ir(kf u±f) t ]=exp (±i2fff t ) where k=2,4,6,8,... (2.6) f=mAf; m= 1, 2 ,... , n- 1 t = r A t ; r=-n , . . . , 0 , 1 , . ..,n~1 Hence t o p r e v e n t a l i a s i n q , t h e h i g h e s t , f r e g u e r . c y i n u ( t ) must be l e s s t h a n f u = ( 2 A t ) ~ 1 , t h e N y q u i s t f r e q u e n c y . I n e q u a t i o n s (2.4) and ( 2 . 5 ) , i t a p p e a r s t h a t u ( r A t ) i s o n l y d e f i n e d f o r J t | < n A t , and t h a t i t s d i s c r e t e f c u r i e r t r a n s f o r m U( I P A L) i s o n l y d e f i n e d f o r | f | < f ^ . I n f a c t , t h e d i s c r e t e F o u r i e r t r a n s f o r m i m p l i c i t l y assumes t h a t u ( l A t ) i s p e r i o d i c and d e f i n e d o v e r -oo<t<co, and t h a t one c o m p l e t e p e r i o d i s c o n t a i n e d i n -r.At<t< ( n - 1 ) A t . I n f a c t , t h e d i s c r e t e f c u r i e r t r a n s f o r m i s a l s o p e r i o d i c , and d e f i n e d o v e r - o K f S o o , and t h a t one c o m p l e t e p e r i o d i s c o n t a i n e d i n - f u < f < + f . E r i g h a m (1974) e x p l a i n s c l e a r l y the n e c e s s i t y o f s u c h d e f i n i t i o n s . The t i m e - v a r y i n q c u r r e n t v e c t o r may be c o n s i d e r e d as a c c o m p l e x h o r i z o n t a l v e l o c i t y w ( r A t ) : -w ( r a t ) = u ( r A t ) +iv ( r A t ) <*. • (2.7) = £. W (mAf) »exp (i2iT»mAf °r4t) where t h e i m a g i n a r y a x i s p o i n t s t o t r u e n o r t h and where W(inAf). i s t h e c o m p l e x d i s c r e t e F o u r i e r t r a n s f o r m • c f w ( r A t ) : -T l - I W(mAf)-= w ( r A t ) «exp (-i2<f«»mAf* r A t ) • . = (2.8) = U (mAf) +iV (raA f) E x a m i n a t i o n c f e q u a t i o n (2.7) i n d i c a t e s t h a t w ( r A t ) i s a s u p e r p o s i t i o n c f a s t a t i o n a r y v e c t o r W(0), and n r o t a t i n g v e c t o r s E ( r A t , m A f ) , e a c h o f whose t i p t r a c e s c u t an e l l i t s e i n t h e h o r i z o n t a l c o m p l e x p l a n e . Any o f t h e s e e l l i p s e s can he w r i t t e n a s: -E ( r A t , mAf) = W (mAf) »exp[ i {.. ) ] + w (-mAf) «exp[ - i (. .) ] (2.9) where (m= 1 ,. . . ,n) , where ( r - - n , . . . , 0 , 1,... ,n- 1) and where (..) • = (2ir»mAf * r A t ) . Each v e c t o r E ( r A t , m A f ) t r a c e s c u t an e l l i p s e i n a p e r i o d c f ( m A f ) - 1 . Each o f t h e s e v e c t o r s nay he r e g a r d e d as a s u p e r p o s i t i o n c f a c l o c k w i s e and a n t i - c l o c k w i s e r o t a t i n g v e c t o r . T h i s i s e m p h a s i z e d by w r i t i n g :-j A (mAf) «ex p[ i cj> (mAf) ] [»= 1, .. . , n W (mAf) = j " (2. 10) I C (mAf) •exp [ - i e(mAf) ] m = -n 1 The ( a m p l i t u d e , p h a s e ) o f t h e a n t i - c l o c k w i s e a r d c l c c k w i s e v e c t o r s a r e (A,4>) and ( C , 8 ) , r e s p e c t i v e l y . T h e s e a r e d e f i n e d a t p o s i t i v e and n e g a t i v e f r e q u e n c i e s , r e s p e c t i v e l y . 10 .Each v e c t c r E ( r A t , mAf) t r a c a s c u t an e l l i p s e w i t h a s e m i - m a j o r a x i s l e n g t h o f A + C and a s e m i - m i n o r a x i s l e n g t h o f !S-C|. The s i g n c f (A-C) d e t e r m i n e s t h e s e n s e c f r o t a t i o n . E ( r A t , m A f ) r o t a t e s c l o c k w i s e or a n t i - c l o c k w i s e (as s e e n f r o m above) i f t h e s i g n o f _ (A-C) i s n e g a t i v e o r p o s i t i v e , r e s p e c t i v e l y . The s e m i - m a j o r a x i s has an o r i e n t a t i o n c f °t=(4>-e)/2 r e l a t i v e t o t h e r e a l x - a x i s . At r A t - 0 , t h e two r o t a t i n g v e c t o r s w i t h a m p l i t u d e s o f (A,C) have an o r i e n t a t i o n o f ( + r e l a t i v e t o t h e s e m i - m a j o r a x i s , where .^= (4> + e ) / 2 . The c o m p o n e n t s (u,v) o f w ( r A t ) a r e , r e s o l v e d u s i r g t h e c o o r d i n a t e s y s t e m ( i , j ) , where i and j a r e u n i t v e c t o r s d i r e c t e d t o w a r d s t r u e e a s t and t r u e n o r t h , r e s p e c t i v e l y . The p a r a m e t e r s A, C and ji a r e i n v a r i a n t u n d e r r o t a t i o n c f the c o o r d i n a t e s y s t e m ( i , j ) . A l t h o u g h <x i s d e p e n d e n t cn t h e c o o r d i n a t e s y s t e m u s e d t o r e s o l v e w ( r A t ) , t h e s e m i - m a j c r a x i s o r i e n t a t i o n , r e l a t i v e t o t r u e n o r t h , i s i n v a r i a n t u n d e r A A r o t a t i o n o f ( i , j) . I t i s t h e B- L(mAf) 's o f e q u a t i o n (2.4) t h a t a r e c o m p u t e d w i t h t h e E a s t E c u r i e r T r a n s f o r m a l g o r i t h m ( S i n g l e t o n , 1969). From t h e s e A, C, 4> and e a r e c a l c u l a t e d a c c o r d i n g t c : -fl2=[ ( E | +B^) 2 + ( D 3 - B 2 ) 2 J/4 C 2 = [ (B, -B ) 2 + (B +Q )z']/H . (2. 11) t a n (<J>) = ( B 3 - E 2 ) / _ ( B ( +B^) t a n (-6) = (B 3 + B Z ) / (8, -B^) where $ and © a r e r e s t r i c t e d t o t h e r a n g e (-TT,71) / and where t h e a r g u m e n t s (maf) o f t h e p a r a m e t e r s have been d r o p p e d . 11 w e r _ a n d _ C r o s s _ S n e c t r u m The c o m p l e x v e c t o r c r o s s s p e c t r u m b e t w e e n t h e two h o r i z o n t a l c o m p l e x v e l o c i t i e s , w, <t) and w 2 ( t ) , i s S l 2 ( f ) : -S | 2 (f ) = <W* K e / A f > (2. 12) <A, A 2 oexp ( - i A ^ ) /Af> . ( 0 < f < f n ) <C, C z°exp ( i A Q [ z )/At> (-f^<f<0) where w i ( f ) i s t h e d i s c r e t e F o u r i e r t r a n s f o r m o f w, (t) as d e f i n e d i n e q u a t i o n ( 2 . 8 ) , and where * . d e n o t e s t h e c o m p l e x c o n j u g a t e , A f=T~ l= (2n At) ~ 1 , Acp^ = <j>( - ^  and A e 1 2 ~ e , - e 2 . I t i s u n d e r s t o o d t h a t t h e s y m b o l s f• and t s t a n d f o r d i s c r e t e f r e q u e n c i e s and t i m e , r e s p e c t i v e l y . The 0 > o p e r a t o r d e n o t e s an a v e r a g e o v e r a number o f c o n s e c u t i v e f r e q u e n c i e s . E e c a u s e o f t h e < ®> o p e r a t o r , S | Z ( f ) i s d e f i n e d o v e r f r e q u e n c i e s w h i c h a r e f e w e r i n number t h a n t h e o r i g i n a l 2n+1 o n e s . The c r e s s s p e c t r u m may a l s o be w r i t t e n a s a c o m b i n a t i o n o f two r e a l -v a l u e d f u n c t i o n s : -where C o l 2 (f) = Q (f) = s I 5, (f) = c c ) 2 (f) + i Q l 2 (f) <A, A 2»ccs (A<j>)Z ) /Af > ( 0 < f < f w ) <C, C 2 - c o s ( A 6 I X ) /Af> (-f^<f<0) <-A, A 2 * s i n ( A q ) i z ) / A f > ( 0 < f < f n ) <C, c 2«sin ( A 6 | 2 )/Af > (-f^<f<0) (2. 13) C c ( Z ( f ) and Q ^ f f ) a r e t h e c o - s p e c t r u m and q u a d r a t u r e s p e c t r u m , r e s p e c t i v e l y . The f u n c t i o n | S I 2 ( f ) | shows w h e t h e r t h e w, (t) a m p l i t u d e o f m o t i o n s i n a p a r t i c u l a r f r e q u e n c y band i s 12 a s s o c i a t e d w i t h a l a r g e c r - s i r - a l l w 2 ( t ) a m p l i t u d e i n the- same f r e q u e n c y band. T h i s a s s o c i a t i o n c a n be f u r t h e r d i v i d e d i n t o a s s o c i a t i o n s o f an i n - p h a s e c o m p o n e n t , C o i 2 ( f) ' a n c i a q u a d r a t u r e component (90° pha s e d i f f e r e n c e ) , Q( ( f ) . The c c r r p l e x - v a l u e d f u n c t i o n , S ) 2 ( f ) , c a n be a l t e r n a t i v e l y r e p r e s e n t e d by two e t h e r r e a l - v a l u e d f u n c t i o n s , t h e v e c t o r c o h e r e n c y - s q u a r e d s p e c t r u m , y | ( f ) : -(2. 14) ^ ( f ) = | S l 2 <f) S,, ( f ) . S j , 2 ( f ) ] ( - f ^ <f<f^) - [ C o * ( f ) +Qfz (f) J / [ S M (f) » S 2 i ( f ) ] and t h e v e c t o r phase s p e c t r u m , % l 2 ( f ) : ~ t a n [ X[Z(f) ]=-Q, 2 ( f ) / C o ( 1 (f) . (-f 7 v<f<r\) (2.15) whose r a n g e i s (-TT,TT) . The n o r m a l i z e d | S ) 2 ( f ) | s p e c t r u m i s j u s t t h e s p e c t r u m w h i c h i s r e s t r i c t e d t o t h e r a n g e (0., 1 ) . The i n t e r p r e t a t i o n c f . ^n(f) and | S l 2 ( f ) | i s i d e n t i c a l but t h e f o r m e r may t e e a s i e r t o work w i t h b e c a u s e c f i t s l i m i t e d r a n g e . The r e a l - v a l u e d v e c t o r , power s p e c t r u m , S-- ( f ) , o f w- (t) i s: -s ^ ( f ) = <| w- (f) | 2 / 4 f > (-f^<f<f^) <A?(f)/Af> (0<f<f 7 l) 0" (2. 16) < C ? ( f ) / A f > (-f^<f<0) i The S r ; ( f ) s p e c t r u m i s d e f i n e d i n t e r m s c f a |W-(f) | 2 a v e r a g e as a . e v e r b c o n s e c u t i v e . f r e q u e n c i e s ( t h e z e r o f r e q u e n c y i s r e t one o f t h e s e ) . S i n c e W- ( f ) i s d e f i n e d o v e r 2n+1 f r e q u e n c i e s s p a c e d 0 13 A f a p a r t , Sy (f) i s d e f i n e d o v e r 2k + 1= ( 2 n / t ) +1 f r e q u e n c i e s ( f ^ ) s p a c e d bAf a p a r t ( i t i s assumed t h a t n/b i s a l s o an i n t e g e r ) : -f^= ( p b + 1 ) A f / 2 . p = - k ,...,0,1 , ... ,k (2.17) so t h a t : -22 S ,-. (f ^ ) b A f = v a r i a n c e (u-J + v a r i a n c e (v, ) (2. 1£) where i t i s u n d e r s t o o d that , ( t ) =u^ ( t ) + i v ^ ( t ) has a z e r c mean. Hence,. ( f ) i n d i c a t e s how t h e c o m b i n e d v a r i a n c e c f t h e Wj.(t) f c o m p o n e n t s i s d i s t r i b u t e d w i t h f r e q u e n c y . The u l t i m a t e r e s o l u t i o n o f t h e Sj.j, ( f ) s p e c t r u m i s ( z n A t ) - 1 ^ ! - 1 w i t h no a v e r a g i n g ( b - 1 ) . T h i s r e s o l u t i o n o f I - 1 a r i s e s b e c a u s e t h e S ^ ( f ) s p e c t r u m i s d e f i n e d i n t e r m s of the d i s c r e t e F o u r i e r t r a n s f o r m ( f ) , w h i c h a l s c has a r e s o l u t i o n .of T _ 1 . T h e r e a r e . e t h e r methods c f d e f i n i n g t h e (f) s p e c t r u m w h i c h y i e l d b e t t e r r e s o l u t i o n s t h a n T _ 1 . L a c c s s (1971) d i s c u s s e s methods w h i c h have r e s o l u t i o n s p r o p o r t i o n a l t o T - 3 / 2 ' a n d T~2. As s t a t e d p r e v i o u s l y , an h o r i z o n t a l t w c - d i m e n s i c r a l A A c o o r d i n a t e s y s t e m i s used where i t s u n i t v e c t o r s ( i , j ) a r e p o i n t e d ( t r u e e a s t , t r u e n o r t h ) . The f u n c t i o n s ( u,,v,) a r e t h e / \ A c o m p o n e n t s c f t h e c u r r e n t v e c t c r w, r e s o l v e d a l o n g ( i , j ) . The ( r e a l , i m a g i n a r y ) a.xes o f t h e c o m p l e x p l a n e a l s o c o i n c i d e w i t h A A A A ( i , j ) . R o t a t i o n o f ( i , j ) . i m p l i e s an e q u a l r c t a t i c n c f the c o m p l e x p l a n e a x e s . I t i s assumed t h a t t h e same c o o r d i n a t e s y s t e m i s u s e d t o measure (u, ,v, ) and ( u 1 , v i ) . The p a r a m e t e r s A-^, C^, and A&[Z i n e q u a t i o n s (2.12) and (2,13) a r e net A A a f f e c t e d by a r c t a t i c n c f ( i , j ) . Hence C o ( Z , Q {1 , , S;i , 1U X[z and X iZ a r e i n v a r i a n t , u n d e r a r o t a t i o n o f ( i , j ) . 2_.2 S t o c h as t i c _ P r o c e s s e s J s n d o m _ V a r i a b l e s I n t h e m a j o r i t y c f c a s e s , t h e f u n c t i o n s u n d e r s t u d y a r e n e t d e t e r m i n i s t i c . I h a t i s , one i s u n a b l e t o e x a c t l y p r e d i c t t h e i r f u t u r e v a l u e s from t h e i r p a s t v a l u e s . w i t h t h i s i r . m i n d , t h e o b s e r v e d v a l u e o f w- ( t c ) can be r e g a r d e d as one m-ember f r o m an i n f i n i t e s e t o f p o s s i b l e v a l u e s . The o b s e r v a t i o n w• ( t c ) i s a d e s c r i b e d by t h e random v a r i a b l e Wj.(t 0) w h i c h has an a s s o c i a t e d p r o b a b i l i t y d i s t r i b u t i o n g i v i n g t h e p r o b a b i l i t y o f o b s e r v i n g t h e v a l u e w • (t„ ) . The c o m p l e x v e l o c i t y K j ( t ) i s d e s c r i b e d by t h e o r d e r e d i n f i n i t e s e t c f random v a r i a b l e s { W j ( t ) ; -co<t<a>} w h i c h i s c a l l e d a s t o c h a s t i c p r o c e s s . The c o m p l e x v e l o c i t y w-_ <t) i s r e g a r d e d as a r e a l i z a t i o n c f t h e s t o c h a s t i c p r o c e s s { . W j ( t ) } . I f t h e s t o c h a s t i c p r o c e s s [w^. (t) } i s s t a t i o n a r y , t h e n a r e a l i z a t i o n , w j ( t ) , i s i n ' s t a t i s t i c a l e g u i l i b r i u m ' . T h a t i s , t h e a v e r a g e p r o p e r t i e s (mean, v a r i a n c e , power s p e c t r a , e t c . ) c f w-_(t) d u r i n g two d i f f e r e n t t i m e i n t e r v a l s a r e n e t s i g n i f i c a n t l y d i f f e r e n t . S i g n i f i c a n t d i f f e r e n c e s a r e t h e s e t h a t a r e l a r g e r t h a n t h e e x p e c t e d d i f f e r e n c e s i n h e r e n t i n s t a t i s t i c a l s a m p l i n g . F u r t h e r m o r e , w- (t) c o n t a i n s no t r e n d s o v e r t he t i m e i n t e r v a l 6 c o n s i d e r e d . T he o r e t i c a 1_ Sp_e c t r a l _ F u n c t i c n s The s a m p l e ( c a l c u l a t e d ) f u n c t i o n s W- (f) and S-- ( f ) , may a l s o be d e s c r i b e d by t h e o r d e r e d s e t s c f random v a r i a b l e s , {W j . ( f ) } and. { S j j ( f ) } r e s p e c t i v e l y . T h e . e x p e c t e d v a l u e o f t h e random v a r i a b l e { S j j { . f ) } i s d e n o t e d by E{S.j. ( f ) } . I t i s t h e a v e r a g e f e w e r s p e c t r u m , a v e r a g e d o v e r t h e s p e c t r a o f a l l p o s s i b l e r e a l i z a t i o n s w- ( t ) w i t h r e c o r d l e n g t h T. The t h e o r e t i c a l v e c t o r power s p e c t r u m , i " ^ ( f ) , i s d e f i n e d a s : -JT ' ( f ) = l i m S { S T ( f ) } (2. 1S) T-»co I t i s t h e a v e r a g e pewer s p e c t r u m , • a v e r a g e d o v e r t h e s p e c t r a o f a l l p o s s i b l e r e a l i z a t i o n s w ^ ( t ) w i t h i n f i n i t e r e c o r d l e n g t h . The t h e o r e t i c a l s p e c t r u m J ? - r K ( f ) c o r r e s p o n d i n g t o (f) i s d e f i n e d . i n a s i m i l a r manner. The t h e o r e t i c a l s p e c t r a c o r r e s p o n d i n g t o C o ^ ( f ) , (f) / v ? ( f ) and X ^ ( f ) a r e d e f i n e d i n t e r m s o f £ 7 J ( ± ) , F K K ( f ) and f ^ f ) . C o n f i d e n c e _ I n t e r v a l s R e g a r d l e s s o f t h e p r o b a b i l i t y d i s t r i b u t i o n c f t h e s t a t i o n a r y s t o c h a s t i c p r o c e s s (w-j.(t)}, t h e { S j j ( f ) } j r c c e s s does t e n d ( f o r l a r g e -|) ) t o be d i s t r i b u t e d a c c o r d i n g t c a % ^ d i s t r i b u t i o n w i t h i } = 2 b d e g r e e s o f f r e e d o m ( S ^ (f) i s a | K ^ ( f ) | 2 a v e r a g e e v e r b c o n s e c u t i v e f r e q u e n c y b a n d s ) . T h i s i s an i n d i r e c t r e s u l t c f t h e C e n t r a l L i m i t Theorem. Hence t h e c o n f i d e n c e l i m i t s f o r S ( f ) can be c o n s t r u c t e d w i t h t h e a i d c f J J i % - f > t a b l e s . 16 The c o n s t r u c t i o n o f c o n f i d e n c e l i m i t s f o r t h e t h e o r e t i c a l c o h e r e n c y and p h a s e s p e c t r a i s n o t a s s t r a i g h t f o r w a r d . L e t yjK(£) and XJK ( f ) d e n o t e t h e random v a r i a b l e s d e s c r i b i n g t h e r e a l i z a t i o n s , ( f ) and X-^(f) r e s p e c t i v e l y . The c o n f i d e n c e l i m i t s f o r t h e t h e o r e t i c a l c o h e r e n c y s p e c t r u m i s a f u n c t i o n o f f r e q u e n c y . However, t h e random v a r i a b l e Y-j-^  ( f ) = a r c t a n h [ ^  ( f ) •] has a v a r i a n c e o f ' ( 2 b ) - 1 w h i c h i s i n d e p e n d e n t o f f r e q u e n c y . W i t h t h e r e a s o n a b l e • a s s u r n p t i o n t h a t Y J K ( f ) h a s a N o r m a l d i s t r i b u t i o n , c o n f i d e n c e .Limits f o r t h e t h e o r e t i c a l c o h e r e n c e s p e c t r u m c a n be c o n s t r u c t e d . The c o n f i d e n c e l i m i t s f o r • t h e t h e o r e t i c a l phase s p e c t r u m i s d e p e n d e n t on t h e t h e o r e t i c a l c o h e r e n c y s p e c t r u m . The 95% c o n f i d e n c e l i m i t s f o r t h e t h e o r e t i c a l p h a s e s p e c t r u m a r e g i v e n i n F i g u r e 9.3 o f J e n k i n s and W a t t s ( 1 9 6 8 ) , w h i c h i s r e p r o d u c e d l a t e r i n F i g u r e 13 (page 5 0 ) . A c c o r d i n g t o G r a n g e r and H a t a u a k a ( 1 9 6 4 ) , t h e s e c o n f i d e n c e l i m i t s a r e c r u d e a p p r o x i m a t i o n s . 17 C h a t t e r 3 1 HE DATA T h i s c h a p t e r i s c o n c e r n e d w i t h t h e d a t a r e c o r d s , t h e i r p e c u l i a r i t i e s and t h e i r p r e l i m i n a r y t r e a t m e n t . T?he c u r r e n t and w a t e r t e m p e r a t u r e r e c o r d s were c o l l e c t e d by T a b a t a and S t i c k l a n d ( 1 9 7 2 a ) , ( 1 9 7 2 b ) , ( 1 9 7 2 c ) . The w i n d s , s e a e l e v a t i o n and a i r p r e s s u r e were a l s o o b t a i n e d f r o m T a b a t a who o b t a i n e d them f r o m t h e A t m o s p h e r i c E n v i r o n m e n t S e r v i c e . The F r a s e r R i v e r d i s c h a r g e d a t a were s u p p l i e d by t h e Water S u r v e y c f Canada ( I n l a n d W a t e r s D i r e c t o r a t e , P a c i f i c R e g i o n ) . _3_.J The_ Cur r e n t s _ a nd_ Wa ter_Te£j:e r a t u re s The l o c a t i o n s o f t h e c u r r e n t m e t e r s a r e shown i n F i g u r e 2, a nd t h e t i m e c o v e r a g e i n F i g u r e 3. Most o f t h e . d a t a r e c o r d s were c o l l e c t e d w i t h t h e A a n d e r a a ( B e r g e n ) Model 4 s e l f -r e c o r d i n g c u r r e n t m e t e r , and s e v e r a l w i t h t h e Gecdyne M o d e l 650. The A a n d e r a a m e t e r s r e c o r d t h e c u r r e n t and w a t e r t e m p e r a t u r e e v e r y 10 m i n u t e s , cn m a g n e t i c t a p e , w h i l e t h e Geodyne meters' a r e o n l y e q u i p p e d t o s a m p l e t h e c u r r e n t e v e r y 15 rri n u t . e s . F o r t h e Geodyne each v a l u e e v e r y 15 m i n u t e s c o n s i s t s o f an a v e r a g e c f 15 v a l u e s s a m p l e d e v e r y 5 s e c o n d s . A s u b s u r f a c e buoy m o o r i n g was used u n t i l t h e l a t t e r p a r t c f A p r i l 1970, a f t e r w h i c h a s u r f a c e buoy t a u t r o p e m o o r i n g was u s e d so Figure 1. Map of Vancouver Island and v i c i n i t y . 65 19 Figure 2. (A) Map of S t r a i t of Georgia showing placement of the three moorings. Depth contours-are i n fathoms. (B) Cross-sectional view along the l i n e of moorings (after Tabata et. a l . , 1971a). 1969 3 50 _ 140 E j 50 • 200 3 50 200 APR MAY JUN JUL AUG SEP OCT NOV DEC I I I I I I 1 1 STATION H-26 JAN STATION H-16 ? O -STATION H - 0 6 \MOCRING'< \MOORINC DAMAGED \ m 1970 F E B MAR APR MAY JUN JUL AUG SEP "1 1 1 T C o--i —Q INSTRUMCNT ry~ FA IW8£ UW£T£ft*nNEO - 3 - 50 - 140 - 3 Y - 50 - 200 - 3 - 50 -200 - S U B S U R F A C E - B U O Y M O O R I N G " O BERGEN RECORDING CURRENT METER A GEODYNE RECORDING CURRENT METER - T A U T - R O P E MOOR ING-INSTRUMENT FAILURE C CLOCK FAILURE i LEAK IN INSTRUMENT B FAULTY BATTERY CONNECTION ? QUESTIONABLE DATA PRESENT i g u r e 3. S chema t i c show ing t i m e cove r age o f t h e c u r r e n t and r e c o r d e d o n l y t h e c u r r e n t v e c t o r e v e r y 15 m i n u t e s , v e c t o r and w a t e r t e m p e r a t u r e e v e r y 10 m i n u t e s . T a b a t a e t . a l . , 1 9 7 1 a ) . w a t e r t e m p e r a t u r e r e c o r d s . Geodyne c u r r e n t m e t e r s w h i l e Be rgen c u r r e n t mete r s r e c o r d e d b o t h t h e c u r r e n t Types o f m o o r i n g employed a r e a l s o i n d i c a t e d ( a f t e r t h a t t h e n e a r s u r f a c e c u r r e n t s c o u l d be m e a s u r e d . T a b a t a £.£-§±1*. (1971a) g i v e d e t a i l s , cn t h e m o o r i n g s y s t e m s , t h e a c c u r a c y o f t h e m e t e r s and t h e i r p e r f o r m a n c e . E r i e f l y , t h e p r e l i m i n a r y t r e a t m e n t c f t h e d a t a c o n s i s t e d o f t h e f o l l o w i n g s t e p s . S u s p e c t d a t a v a l u e s ' i n t h e o r i g i n a l d a t a r e c o r d s were c o r r e c t e d t o . y i e l d t h e e d i t e d d a t a r e c o r d s . These e d i t e d d a t a r e c o r d s have s a m p l i n g i n t e r v a l s c f 10 o r 15 m i n u t e s . The e d i t e d d a t a r e c o r d s v e r e time-a.vera ged t o y i e l d t h e h a l f - h o u r l y d a t a r e c o r d s , r e c o r d s w i t h a s a m p l i n g i n t e r v a l o f 30 m i n u t e s . A l l t h e h a l f - h o u r l y d a t a r e c o r d s f c r t h e same l o c a t i o n were t h e n j o i n e d t o y i e l d a s i n g l e h a l f -h o u r l y t i m e - s e r i e s o f much g r e a t e r l e n g t h . E d i t i n a The o r i g i n a l d a t a r e c o r d s a r e t h e s e t h a t have a l r e a d y u n d e r g o n e cne s t e p , o f t r a n s l a t i o r. and p r o c e s s i n g t e c o n v e r t them i n t o a c o m p u t e r - c o m p a t i b l e d i g i t a l f o r m a t , E v e r y o r i g i n a l r e c o r d was m a c h i n e p l o t t e d and v i s u a l l y i n s p e c t e d . T h i s p r o c e d u r e i s n e c e s s a r y - t o d e t e r m i n e w h e t h e r t h e i n s t r u m e n t d i d f u n c t i c n p r o p e r l y . F c r seme c f t h e r e c o r d s t a k e n w i t h t h e A a n d e r a a i n s t r u m e n t s , o n l y t h e c u r r e n t o r t e m p e r a t u r e d a t a a r e a v a i l a b l e . I n t h e s e c a s e s , t h e t e m p e r a t u r e c h a n n e l m a l f u n c t i o n e d w h i l e t h e c u r r e n t v e l o c i t y c h a n n e l c o n t i n u e d t o f u n c t i o n , c r v i c e v e r s a . Many o f t h e o r i g i n a l r e c o r d s r e q u i r e d e d i t i n g . E x a m p l e s c f o r i g i n a l and e d i t e d r e c o r d s o f c u r r e n t and t e m p e r a t u r e a r e g i v e n i n F i g u r e . 4 and F i g u r e 5 . P o s i t i v e v and u - c o m p c n e n t v a l u e s ( c f c u r r e n t s and winds) c o r r e s p o n d t o t r u e n o r t h and t r u e e a s t , r e s p e c t i v e l y . . I n t h e p r o c e s s o f e d i t i n g s u s p e c t d a t a v a l u e s were r e p l a c e d by l i n e a r l y i n t e r r e l a t e d o n e s . Twc p o i n t s cn e i t h e r s i d e o f t h e s u s p e c t v a l u e s (or s e v e r a l c o n s e c u t i v e . s u s p e c t v a l u e s ) were used i n t h e l i n e a r . i n t e r p o l a t i o n . • F c r t h e c u r r e n t s , e i t h e r t h e s p e e d o r d i r e c t i o n was e d i t e d when o n l y one o f t h e s e p a r a m e t e r s was s u s p e c t . When b o t h t h e s e p a r a m e t e r s were s u s p e c t , t h e c u r r e n t c o m p o n e n t s were e d i t e d i n s t e a d . The e d i t i n g was done m a n u a l l y ( r a t h e r t h a n w i t h a c o m p u t e r program) by v i s u a l i n s p e c t i o n o f t h e p l o t t e d d a t a r e c o r d s and r e f e r e n c e s t o a l i s t i n g o f t h e i r t a b u l a t e d v a l u e s . D a t a v a l u e s a r e s u s p e c t b e c a u s e t h e y a p p e a r t o be u n r e a s o n a b l y l a r g e v a l u e s , w h i l e e t h e r d a t a v a l u e s a r e s u s p e c t when t h e t i m e - d e r i v a t i v e s c f t h e p a r a m e t e r have u n r e a s o n a b l y l a r g e m a g n i t u d e s (see F i g u r e 4 and 5 ) . A m a g n i t u d e i n t h e t i m e -d e r i v a t i v e i s deemed u n r e a s o n a b l y l a r g e by c o m p a r i s o n t o t h e s e o f t h e n e i g h b o u r i n g p o i n t s . S u s p e c t d a t a v a l u e s a l s o a r i s e f r o m i n t e r m i t t e n t e r r o r s . T h e s e ' e r r o r s most p r o b a b l y o r i g i n a t e a t t h e t i m e t h e v a l u e s a r e r e c o r d e d on m a g n e t i c t a p e , o r a t t h e t i m e t h e t a p e i s r e a d f o r t r a n s l a t i o n i n t o a c o m p u t e r - c o m p a t i b l e d i g i t a l f o r m . CURRENTS 2 3 Figure 4. Portions of the o r i g i n a l and edited 50m current records from the c e n t r a l mooring. The' sampling i n t e r v a l i s 10 minutes. H26-140M WATER TEMPERATURES 1 0 -U J a g -C T c r Z 8 -LU U LD Q r ORIGINAL H8 r-7 "-6 r 10 EDITED 8-1 ^ 8 1 16 1 17 1 18 1 19 1 a r JULY 1969 21 i 55 r 22 1 23 Figure 5. Portions of the o r i g i n a l and edited v/ater temperature records f roni the 140m eastern l o c a t i o n . The sampling interval' i s 10 minutes. ro 25 The n H a l f - h o u r l y . T i m a - s e r i e s • The i n s t r u m e n t s were s t a r t e d a t random t i m e s . Hence t h e o b s e r v a t i o n s do n o t have t i m e s c o r r e s p o n d i n g t o a m u l t i p l e o f 10 or. 15 m i n u t e s . M e c h a n i c a l c l o c k s were u t i l i z e d i n b o t h t h e A a n d e r a a and. Geodyne' i n s t r u m e n t s t o keep t i m e . T h e s e c l o c k s h a v e r e l a t i v e l y l o w a c c u r a c y and o v e r a p e r i o d o f 30 d a y s , t h e - t i m e h a s an e r r o r o f 4 h o u r s , i n t h e w o r s t c a s e . F o r t h e A a n d e r a a i n s t r u m e n t s , t h e s p e c i f i e d s a m p l i n g i n t e r v a l o f 10 m i n u t e s was n o t u s e d . I n s t e a d , t h e s a m p l i n g i n t e r v a l was c a l c u l a t e d f r o m t h e s t a r t and S t o p t i m e s o f t h e i n s t r u m e n t and f r o m t h e 'number o f v a l u e s i n ' t h e • r e c o r d . Hence some o f t h e t i m e i n t e r v a l s ( t h e s e - are- r o u n d e d t o t h e n e a r e s t m i n u t e i n t h e o r i g i n a l r e c o r d s ) b e tween c o n s e c u t i v e v a l u e s m t h e o r i g i n a l and e d i t e d r e c o r d s may be 9 o r 11 m i n u t e s . The s a m p l i n g i n t e r v a l f o r t h e two Geodyne r e c o r d s i s assumed t o be e x a c t l y 15 m i n u t e s . S i n c e t h e Geodyne m e t e r s u s e t h e same b a s i c c l o c k , t h e Geodyne r e c o r d s may a l s o have t i m i n g e r r o r s . The m a g n i t u d e s o f t h e s e e r r o r s were unknown a t t h e t i m e t h e r e c o r d s were e d i t e d . D a t a r e c o r d s w i t h v a l u e s e v e r y h a l f - h o u r were d e r i v e d by a v e r a g i n g t h r e e c o n s e c u t i v e v a l u e s o f t h o s e e d i t e d r e c o r d s h a v i n q a s a m p l i n g i n t e r v a l o f 10 m i n u t e s . These h a l f - h o u r l y d a t a r e c o r d s h a v e v a l u e s whose t i m e s a r e on t h e h o u r o r h a l f -h o u r . The t h r e e c o n s e c u t i v e v a l u e s u s e d f o r a v e r a g i n g were p i c k e d -such t h a t t h e m i d d l e v a l u e h a s a t i m e t h a t i s w i t h i n +5 m i n u t e s f r o m t h e h o u r o r h a l f - h o u r . The c u r r e n t c o m p o n e n t s o f t h e e d i t e d r e c o r d s a r e a v e r a g e d t o y i e l d t h e c u r r e n t c o m p o n e n t s 26 o f t h e h a l f - h o u r l y r e c o r d s . Thus t h e t i m e - a v e r a g e i s a v e c t c r a v e r a g e . Two h a l f - h o u r l y d a t a r e c o r d s f r o m 40m and 50m a t t h e c e n t r a l m o c r i n g were d e r i v e d f r o m e d i t e d r e c o r d s (Geodyne) h a v i n g a s a m p l i n g i n t e r v a l o f 15 m i n u t e s . The h a l f - h o u r l y r e c o r d f r o m t h e e a r l i e r (Dec/1 96 9.) Geodyne r e c o r d a t 4 Gm was d e r i v e d by s i m p l y p i c k i n g e v e r y s e c o n d v a l u e . T h i s was p o s s i b l e b e c a u s e t h e s a m p l i n g t i m e s . c f t h e e d i t e d r e c o r d a r e 14,29,44 and 59 m i n u t e s a f t e r t h e s t a r t c f e a c h h o u r . The o t h e r h a l f - h o u r l y r e c o r d f r o m t h e l a t e r ( J a n - F e b / 1 9 7 0 ) Geodyne r e c o r d a t 50m was d e r i v e d by v e c t o r - a v e r a g i n g two c o n s e c u t i v e v a l u e s . T h i s was done b e c a u s e t h e s a m p l i n g t i m e s c f t h e e d i t e d r e c o r d a r e 9,24,39 and 54 m i n u t e s a f t e r t h e s t a r t o f e a c h h o u r . A l l t h e h a l f - h o u r l y d a t a r e c o r d s f o r t h e same . l o c a t i o n were j o i n e d t o y i e l d a s i n g l e t i m e - s e r i e s c f much g r e a t e r l e n g t h . I n t h e c o n s t r u c t i o n c f t h e t i m e - s e r i e s a t 50m c f t h e c e n t r a l m o o r i n g , t h e h a l f - h o u r l y Geodyne r e c o r d (Dec/1.969) a t 40m was u s e d because- t h e m e t e r a t 50m f a i l e d . The r e l i a b i l i t y c f t h i s m e ter was s u s p e c t and a n o t h e r meter was p l a c e d , a t 40m a s a b a c k - u p . F o r t h o s e p e r i o d s where t h e r e were nc a v a i l a b l e d a t a , t h e a v e r a g e v a l u e ( c f t e m p e r a t u r e and c u r r e n t c o m p o n e n t s ) o v e r a l l t h e a v a i l a b l e d a t a was u s e d i n t h e c o n s t r u c t i o n o f t h e s i n g l e t i m e - s e r i e s ..for e a c h l o c a t i o n . As shown i n F i g u r e 3, t h e s e d a t a gaps may be g u i t e s u b s t a n t i a l . U s u a l l y t h e r e i s a s h o r t gap c f a few h o u r s b e tween t h e end o f one r e c o r d and t h e s t a r t o f t h e n e x t . T hese d a t a gaps c o r r e s p o n d t o t h e i n t e r v a l when t h e i n s t r u m e n t s a r e b r o u g h t cn b o a r d s h i p f o r s e r v i c i n g . 27 The s i n g l e h a ' i f - h c u r l y t i m e - s e r i e s f o r e a c h l o c a t i o n a r e t h e s u b j e c t o f t h e a n a l y s i s i n C h a p t e r 4 and 6. C h a p t e r 5 e x a m i n e s t h e d a i l y l o w - p a s s e d t i m e - s e r i e s f c r each' l e c a t i c n . 3_.2 l i f _S u i ^ l e mer.ta r j _ Da t a P o r t i o n s o f t h e data' r e c o r d s o f t h e w i n d s , sea e l e v a t i o n and a i r p r e s s u r e (at s e a l e v e l ) a r e g i v e n i n F i g u r e 6 and F i g u r e 7. R e f e r t o F i g u r e 1 f c r l o c a t i o n s . A l l t h e s e d a t a r e c o r d s have a s a m p l i n g i n t e r v a l c f one h o u r . E x c e p t f c r t h e w i n d s , t h e s e d a t a r e c o r d s s p a n . t h e two y e a r s o f 1969-70. The w i n d r e c o r d f o r E n t r a n c e I s l a n d c o v e r s t h e p e r i o d s O c t o b e r 10, 1969 t o December 3 1 , 1969 and A p r i l 1, 1970 t o December 31, 1370, w h i l e t h e w i n d d a t a f c r Sand Heads s p a n t h e p e r i o d f r o m J a n u a r y 2 0 , 1969 t o S e p t e m b e r 1, 1970. Figure 6. Portions of the surface wind records. The sampling i n t e r v a l i s one hour. AIR PRESSURE (MBflR) 29 10S0-1040-1030 -1020 -10J0H 12 13 TOFINO PORT HARDY COMflX VICTORIA VANCOUVER 14 15 16 JULY 1970 17 18 19 20 hi 050 1040 •1030 1020 h 3010 SER LEVEL•ELEVATION (METERS) 4H o-J o - J 12 13 11 IS 1 16 JULY 1970 17 1 16 19 TOFINO ALERT BAY POINT ATKINSON VICTORIA Figure 7. Portions of the sea-level and air-pressure records from various loca t i o n s around Vancouver Island. The sea-levels are r e l a t i v e to the reference bench marks while the air-pressures are corrected so that they correspond to those measured at mean sea-l e v e l . The air-pressure axis r e f e r s to that at Vancouver. Air-pressure s i g n a l s are separated from each other by 5mb. The sampling i n t e r v a l i s one hour for a l l records. 30 C h a p t e r 4 THE FREQUENCY REPRESENTATION The f r e q u e n c y r e p r e s e n t a t i o n o f t h e t i m e - s e r i e s i s e x a m i n e d i n t h i s c h a p t e r . A g i v e n t i m e - s e r i e s may he t h e o b s e r v e d r e s u l t c i s e v e r a l p r o c e s s e s o c c u r r i n g s i m u l t a n e o u s l y . However, e a c h o f t h e s e p r o c e s s e s may o c c u r i n s e p a r a t e and d i s t i n c t f r e q u e n c y r e g i o n s . When t h i s i s s o , t h e p r o c e s s e s may be i s o l a t e d i n t h e t i m e domain by f i l t e r i n g , e v e n t h o u g h t h e y a r e p r o c e e d i n g s i m u l t a n e o u s l y . Fo we r _ S t e c t r un!_ajjd_ Cat a_Ga _ps The v e c t o r power s p e c t r u m (S) i s t h e f r e q u e n c y (£) d i s t r i b u t i o n c f t h e v a r i a n c e - i n t h e t i m e - s e r i e s u n d e r e x a m i n a t i o n . T h i s d i s t r i b u t i o n i s p r e s e n t e d i n two g r a p h i c a l f o r m a t s . The f i r s t a r e g r a p h s of S v e r s u s f , and t h e s e c o n d a r e g r a p h s c f f«S v e r s u s l o g f . W i t h b o t h f o r m a t s , t h e a r e a u n d e r t h e g r a p h s i s p r o p o r t i o n a l t o v a r i a n c e , and t h e t o t a l a r e a ( i n t e g r a l ) i s e q u a l t o t h e sum o f t h e u - c c m p c n e n t v a r i a n c e and t h e v - c c m p c n e n t v a r i a n c e . E a ch p o i n t c f a g r a p h has an a s s o c i a t e d V (number o f d e g r e e s c f f r e e d o m ) . The p a r a m e t e r -p i s p r o p o r t i o n a l t o t h e s m o o t h i n g c f S, w h i c h i s e f f e c t e d by a v e r a g i n g i n t h e f r e q u e n c y d o m a i n . The - p o i n t s o f t h e f i r s t g r a p h i c a l f o r m a t have a c o n s t a n t i) , w h i l e t h e s e c f t h e . s e c o n d have a -p t h a t i n c r e a s e s r a p i d l y ' w i t h i n c r e a s i n g | f | , and a c c n f i d e n . e e i n t e r v a l w h i c h d e c r e a s e s w i t h i n c r e a s i n g | f | . The s e c o n d g r a p h i c a l f o r m a t i s used p r i m a r i l y t o p r e s e n t t h e v e c t o r pewer s p e c t r u m o v e r t h e c o m p l e t e f r e q u e n c y r a n g e . A s c a l a r t i m e - s e r i e s ( t e m p e r a t u r e say) i s c o n v e r t e d i n t o a p s e u d o - v e c t o r t i m e - s e r i e s whose u-ccmpenent i s e q u a l t o t h e s c a l a r t i m e - s e r i e s and whose v - c c m p o n e n t i s z e r o . Such a c o n v e r s i o n i s n e c e s s a r y f o r a c r o s s - s p e c t r a l a n a l y s i s b e t w e e n a v e c t o r and s c a l a r t i m e - s e r i e s . The v e c t o r power s p e c t r u m of a p s e u d o - v e c t c r t i m e - s e r i e s i s an e v e n f u n c t i o n c f f r e q u e n c y . B e c a u s e o f t h i s s y m m e t r y , s u c h s p e c t r a a r e f r e q u e n t l y p r e s e n t e d f o r p o s i t i v e f r e q u e n c i e s o n l y . H e n c e f o r t h , t h e v e c t o r power s p e c t r u m w i l l be r e f e r r e d t o as t h e power s p e c t r u m c r more s i m p l y a s t h e s p e c t r u m . The power s p e c t r a p r e s e n t e d i n t h i s c h a p t e r a r e b a s e d cn a t i m e - s e r i e s l e n q t h c f N-N^+N^ p o i n t s . The t i m e - s e r i e s c o n t a i n N. d a t a p c i n t s and a t o t a l o f N p o i n t s t h a t c o r r e s p o n d t c one c r mere gaps i n t h e t i m e - s e r i e s . As d i s c u s s e d i n t h e p r e v i o u s c h a p t e r , t h e s e gaps a r e f i l l e d w i t h t h e mean d e r i v e d f r o m t h e K d d a t a p o i n t s . T h i s i s e q u i v a l e n t t c f i l l i r g t h e gaps i n a t i m e - s e r i e s h a v i n g a z e r o mean w i t h z e r o e s . The t o t a l v a r i a n c e c f t h e v e c t o r t i m e - s e r i e s s h o u l d be c c m p u t e d a c c o r d i n g t c : -v a r d = N d - i £ < u 2 + v2) (4.1) where u and .v a r e t h e components ( h a v i n g z e r o means) c f t h e v e c t o r t i m e - s e r i e s and t h e summation i s t a k e n o n l y e v e r t h e N d d a t a p o i n t s . However,- t h e i n t e g r a l o f t h e power s p e c t r u m g i v e s a t o t a l v a r i a n c e c f : -v a x s = (N4+N )-» E ( u * + v2) < v a r d (4.2) where t h e summation i s now t a k e n o v e r b o t h t h e N d d a t a p o i n t s and t h e N^ p o i n t s , w h i c h a r e z e r o e s . Hence, t h e power s p e c t r u m g i v e s . a v a r i a n c e , v a r s , t h a t i s l e w e r t h a n t h e most a p p r o p r i a t e e s t i m a t e , v a r d . I n t h i s c h a p t e r , a l l t h e power s p e c t r a shc«n have been m u l t i p l i e d by a f a c t o r (Nd_ + N ^ ) / N d b e c a u s e : -v a r d - v a r s » ( N d + N ^ ) / N d (4.3) The h y p o t h e t i c a l t o t a l v a r i a n c e i s d e f i n e d a s : -var R_= (Nd + N. )-» 2 ( u 2 + v2) (4.U) where u and v a r e t h e . c o m p o n e n t s of t h e h y p o t h e t i c a l t i m e -s e r i e s w h i c h w o u l d have been o b s e r v e d had t h e r e been no e q u i p m e n t m a l f u n c t i o n s , e t c . , w h i c h p r o d u c e d t h e d a t a g a p s . In t h i s t h e s i s i t i s i m p l i c i t l y assumed t h a t v a r c l = v a r ^ . T h i s a s s u m p t i o n i s e x a c t o n l y i f : -where t h e l e f t and r i g h t s u m m a t i o n a r e t a k e n e v e r t h e d a t a p o i n t s i n (u,v) t h a t c o r r e s p o n d t o t h e N d d a t a p o i n t s i n ( u , v ) , and t o t h e N^ p c i n t s i n ( u , v ) , r e s p e c t i v e l y . F o r p u r p o s e s o f d i s c u s s i o n r e g a r d as b e i n g f i x e d and on t h e o r d e r o f (N d+N^)/10 ( t h i s i n f a c t i s t h e s i t u a t i o n w i t h t h e d a t a b e i n g a n a l y z e d ) . D e p e n d i n g on t h e n a t u r e o f t h e power s p e c t r u m , e q u a t i o n (4.5) may o r may n o t be a c c c d a p e r c x i m a t i c n t o t h e r e a l s i t u a t i o n . E q u a t i o n (4.5) c l o s e l y a p p r o x i m a t e s t h e r e a l s i t u a t i o n when a l l t h e v a r i a n c e i s c o n c e n t r a t e d i n v e r y n a r r o w f r e q u e n c y b a n d s , and when t h e p e r i o d s c f t h e s e o s c i l l a t i o n s a r e s m a l l when compared t o a p e r i o d c o r r e s p o n d i n g t o N^. V a r i a n c e c o n c e n t r a t e d i n v e r y h a r r o w f r e q u e n c y bands r e p r e s e n t s s i n u s o i d a l o s c i l l a t i o n s , o r a t w o r s t , q u a s i - s t e a d y s i n u s o i d a l - l i k e f l u c t u a t i o n s . Good e x a m p l e s o f s u c h f l u c t u a t i o n s are- t i d a l c u r r e n t s . E q u a t i o n (4.5) i s a l s o a g ood a p p r o x i m a t i o n f o r b r o a d - b a n d v a r i a n c e h a v i n g a c h a r a c t e r i s t i c p e r i o d t h a t i s s m a l l compared t o N^. However, e g u a t i c r (4.5) c a n be a p o t e n t i a l l y p e e r a p p r o x i m a t i o n f o r v a r i a n c e s whose c h a r a c t e r i s t i c p e r i o d s a r e cn t h e o r d e r o f N^ .. G e n e r a l l y , t h e s m a l l e r t h e c h a r a c t e r i s t i c p e r i o d , ( r e l a t i v e t o N^) c f t h e v a r i a n c e u n d e r c o n s i d e r a t i o n , t h e b e t t e r t h e a p p r c x i n a t i c n . T h i s s t a t e m e n t stems from the s t a t i o n a r y a s s u m p t i o n , w h i c h e n s u r e s t h a t t h e v a r i a n c e c f t h e t i m e - s e r i e s , c a l c u l a t e d o v e r an a p p r o p r i a t e l e n g t h , i s i n d e p e n d e n t o f t i m e . An a p p r o p r i a t e l e n g t h i s one t h a t c c n t a i n s many c h a r a c t e r i s t i c p e r i o d s c f t h e f l u c t u a t i o n s u n d e r c o n s i d e r a t i o n . 4_. 2 Sj->e c t r a _ o f _ 0 r i c | i n a l _ a n d _ J d i t e d _ E e c c r d s Shewn i n F i g u r e 8 a r e t h e power s p e c t r a c f t h e o r i g i n a l and e d i t e d (At=10min.) c u r r e n t r e c o r d s w h i c h a p p e a r i n F i g u r e 4, and t h e power s p e c t r a o f t h e o r i g i n a l and e d i t e d t e m p e r a t u r e r e c o r d s w h i c h a p p e a r i n F i g u r e 5. The s p e c t r u m c f t h e e d i t e d 34 ORIGINAL TEMPERATURES r .01 10 •100 -10 , L-T~l 00 o o —, o-100 .01 o in-o EDITED T E M P E R A T U R E S "~1 10 100 ORIGINAL CURRENTS -I -.1 .1 FREQUENCY (CPD) 10 100 _r~i EDITED CURRENTS Figure 8. Spectra (f»S) of o r i g i n a l and edited currents, and of o r i g i n a l and edited temperatures. Based on 8 day time-series of (current, temperature) at (50m, 140m) of the (central, eastern) mooring s t a r t i n g (July 12 1970, July 16 1969), which are shown i n Figure (4, 5). c u r r e n t r e c o r d d i f f e r s c n l y s l i g h t l y f r o m t h a t c f t h e o r i g i n a l c u r r e n t r e c o r d . As e x p e c t e d , t h e h i g h - f r e q u e n c y v a r i a n c e o f t h e e d i t e d r e c o r d i n t h e f r e q u e n c y i n t e r v a l (3,72cpd) i s r e d u c e d f r o m t h a t o f t h e o r i g i n a l r e c o r d . C o m p a r i n g the two power s p e c t r a , t h e r e i s a l s o a s l i g h t t e n d e n c y f o r t h e v a r i a n c e t c be s h i f t e d f r o m one r e g i o n t c a n o t h e r i n t h e f r e q u e n c y d o m a i n . G e n e r a l l y t h o u g h , t h e main f e a t u r e s i n t h e e d i t e d c u r r e n t s p e c t r u m c l o s e l y r e s e m b l e t h o s e i n t h e o r i g i n a l c u r r e n t s p e c t r u m . I n c o n t r a s t , t h e power s p e c t r a o f t h e o r i g i n a l ar.d e d i t e d t e m p e r a t u r e r e c o r d s a r e v e r y d i f f e r e n t . The h i g h -f r e q u e n c y (3,72cpd) v a r i a n c e o f t h e o r i g i n a l r e c o r d i s much g r e a t e r t h a n t h a t o f t h e e d i t e d r e c o r d . The l e w - f r e q u e n c y s p e c t r a l f e a t u r e s c f t h e o r i g i n a l r e c o r d have been s e v e r e l y d i s t o r t e d when compared t o t h o s e o f t h e e d i t e d r e c o r d . The d i f f e r e n c e s between t h e two t e m p e r a t u r e s p e c t r a a r e r e a s o n a b l e , c o n s i d e r i n g t h e s e v e r e d i f f e r e n c e s between t h e o r i g i n a l and e d i t e d t e m p e r a t u r e r e c o r d s . The c o n v e r s i o n o f e d i t e d r e c o r d s h a v i n g a s a m p l i n g i n t e r v a l (At) o f 10min t o t h o s e h a v i n g a A t c f 3 0 u i n i s d i s c u s s e d i n t h e p r e v i o u s c h a p t e r . D u r i n g t h i s c o n v e r s i o n , t h e s p e c t r a c f t h e r e c o r d s w i t h •At=30'min c a n be b a d l y d i s t o r t e d t y t h e h i g h - f r e q u e n c y ( 2 4 ,72cpd) v a r i a n c e i n t h e r e c o r d s w i t h A t = 1 0 m i n . From t h e s p e c t r a c f t h e e d i t e d r e c o r d s shown i n F i g u r e 8, t h i s v a r i a n c e i s 2% and 4% o f t h e t o t a l v a r i a n c e f o r t h e e d i t e d c u r r e n t and e d i t e d t e m p e r a t u r e r e c o r d s , r e s p e c t i v e l y . Ey i n s p e c t i o n o f t h e 95% c o n f i d e n c e i n t e r v a l s f c r a f r e q u e n c y o f 2 4 c p d , i t i s c o n c l u d e d t h a t t h e s e lew p e r c e n t a g e s a r e t y p i c a l c f e t h e r e d i t e d r e c o r d s w i t h A t - 1 0 r a i n . Hence, t h e s p e c t r a l d i s t o r t i o n c f t h e t i m e - s e r i e s ( A t = 3 0 m i n ) , due t o t h e a l i a s i n g a s s o c i a t e d , w i t h t h e c o n v e r s i o n f r o m A t=10min t o A t = 3 0 m i n , i s n e g l i g i b l e . 4_.3 2 p e c t r a _ c f . C u r r e n t s The power s p e c t r a c f t h e c u r r e n t t i m e - s e r i e s a r e shown i n F i g u r e 9 and F i g u r e 10. These r e s u l t s a r e b a s e d on t i r c e -s e r i e s o f 533 1/3 days s t a r t i n g A p r i l 16, 1969. The n o r m a l i z e d 9 5 % c o n f i d e n c e i n t e r v a l (Yz rY^) i-s a l s o i n d i c a t e d . The l o w e r and u p p e r l i m i t s o f t h e c o n f i d e n c e i n t e r v a l f c r a p o i n t S ( f D ) , a r e y ^ o S f f o ) and y u » S ( f 0 ) , r e s p e c t i v e l y . The s i x s p e c t r a h-ave t h e same g e n e r a l f e a t u r e s . A l l have r e l a t i v e l y b r o a d - b a n d v a r i a n c e a t v e r y l e w f r e q u e n c i e s and n a r r o w - b a n d v a r i a n c e a t c r n e a r ±1cpd and ±2cpd. Shewn i n T a b l e I a r e t h e v a r i a n c e s o f t h e c u r r e n t s c a l c u l a t e d f o r s e l e c t e d f r e q u e n c y b a n d s . They a r e g i v e n as a p e r c e n t a g e o f t h e t o t a l v a r i a n c e , t h a t f o u n d i n t h e c o m p l e t e f r e q u e n c y r a n g e ( 0 . 0 0 2 , 2 4 c p d ) . The v a r i a n c e has been s e p a r a t e d i n t o t h e u - v a r i a n c e , v - v a r i a n c e , C - v a r i a n c e and A - v a r i a n c e . The f o r m e r two a r e t h e v e c t o r component v a r i a n c e s , w h i l e t h e l a t t e r 'twe a r e t h e r o t a r y ( c l o c k w i s e , a n t i - c l o c k w i s e ) v a r i a n c e s . I n any f r e q u e n c y band, t h e sum c f t h e component v a r i a n c e s ' i s i n d e n t i c a i to. t h e sum o f t h e r o t a r y v a r i a n c e s . The t o t a l u - v a r i a n c e and t o t a l v - v a r i a n c e a r e v a r i a n c e s Figure 9. Spectra of 50m currents. Based on time-series of 533 days s t a r t i n g A p r i l 16 1969. A l l spectra have 18 degrees of freedom. Numbers above the o f f - s c a l e peaks indi c a t e t h e i r values, 38 o 9 Figure 10. Spectra of 140m and 200m currents. Based on time-series of 533 days s t a r t i n g A p r i l 16 1969. A l l spectra have 18 degrees of freedom. Numbers above the o f f - s c a l e peaks i n d i c a t e t h e i r values. FREQUENCY RANGE (CYCLES/DAY) U TABLE I Frequency d i s t r i b u t i o n of the variance i n the currents * * WESTERN MOORING . CENTRAL MOORING EASTERN MOORING V U+V C A U V U+V C A U V U+V C A .002 -.05 -.10 -.15 -.20 _ .002 -.25 -.50 -.80 -1.1 -1.8 -2.1 -3.0 -4.0 .05 .10 .15 .20 .25 .25 .50 .80 1.1 1.8 2.1 3.0 4.0 24.0 .002 - 24.0 .002 - 24.0 2.59 2.11 1.48 0.92 •0.40 15.08 5.46 2.86 0.90 0.48 17.68 7.57 4.34 1.82 0.89 8.14 2.90 1.85 1.21 0.50 9.60 4.70 2.51 0.61 0.39 7.51 24.78 32.29 14.61 17.81 1.33 0.75 2.87 3.27 4.31 1.70 1.04 2.56 1.13 0.87 14.67 2.87 24. 78 1.78 0.99 2.86 2.46 1.62 17.54 6.15 29.09 3.48 2.03 5.41 1.50 0.93 8.31 5.05 12.24 2.61 1.47 3.37 0.97 0.69 9.29 1.11 16.96 0.89 0.56 2.06 51.27 151.34 202.61 100.67 101.18 .002 - .05 0.27 2.36 2.63 1.14 .05 - .10 0.30 2.43 2.74 1.44 .10 - .15 0.14 0.64 0.78 0.31 .15 - .20 0.13 0.55 0.68 0.37 .20 - .25 0.06 0.31 0.37 0.21 .002 _ .25 0.91 6.29 7.20 3.46 .25 - .50 0.19 0.56 0.75 0.40 .50 - .80 0.34 0.52 0.86 0.47 .80 - 1.1 3.21 22.18 25.39 18.81 1.1 - 1.8 1.50 1.52 3.02 2.36 1.8 - 2.1 17.08 36.53 53.61 43.53 2.1 - 3.0 1.80 1.49 3.28 2.49 3.0 - 4.0 0.74 1.21 1.86 1.30 4.0 - 24.0 1.94 2.16 4.10 2.62 1.50 1.31 0.48 0.31 0.17 0.35 0.39 6.64 0.67 10.22 0.80 0.57 1.49 40.47 105.77 146.24 109.61 36.28 15.87 2.93 1.96 0.72 0.38 6.52 2.43 1.35 0.94 0.42 22.39 5.36 3.31 1.66 0.80 8.64 2.15 1.20 0.94 0.39 13.82 3.23 2.12 0.73 0.41 21.85 11.66 33.51 13.32 20.30 0.96 0.73 4.14 3.20 3.07 1.39 1.05 2.29 1.13 0.86 12.61 2.85 27.45 1.63 0.93 2.25 2.10 1.59 16.75 6.05 30.52 3.02 1.98 4.55. 1.01 1.00 6.70 5.12 15.25 2.27 1.44 2.75 1.10 0.60 10.10 0.95 15.37 0.76 0.54 1.81 72.63 115.26 187.89 91.13 96.15 1.01 0.42 0.37 0.50 0.19 2.49 0.94 0.71 2.38 1.80 9.86 1.29 0.70 4.16 2.40 1.78 1.18 0.35 9.87 1.19 0.74 27.02 2.27 32.60 1.93 0.93 5.17 2.82 2.15 1.67 0.54 12.36 2.13 1.45 29.40 4.06 42.46 3.22 1.62 2.81 1.69 1.17 0.97 0.26 6.89 1.41 0.78 13.52 2.98 25.48 2.31 1.08 2.36 1.13 0.98 0.70 0.29 5.46 0.72 0.67 15.87 1.08 16.95 0.91 0.54 1.53 . 1.87 3.41 2.10 1.30 30.05 103.78 138.83 78.51 60.39 0.94 0.67 0.69 0,24 0.21 21.92 14.26 4.78 1.37 1.02 22.85 14.92 5.47 1.62 1.23 12.30 8.60 3.34 0.90 0.69 0.65 1.77 2.43 1.33 0.47 1.01 1.96 3.71 2.12 1.19 2.41 1.08 9.95 2.20 20.48 1.78 0.99 2.21 1.55 10.96 4.17 24.19 3.90 2.18 4.62 0.81 5.22 3.18 13.18 3.02 1.56 2.90 0.86 11.17 0.60 13.45 0.29 5.46 0.20 1.58 0.10 1.69 2.04 33.34 0.39 1.37 0.30 1.11 2.34 20.12 1.28 2.23 6.27 17.58 2.-79 2.22 1.18 1.47 1.97 2.13 21.37 94.13 12.02 14.06 5.74 1.77 1.78 35.37 1.76 1.42 22.45 3.51 23.85 5.01 2.66 4.10 6. 7. 3. 1. 1. 18. 0. 0. 9. 2. 14'. 3. 1. 2. 30 20 11 04 01 65 99 80 62 21 60 58 73 55 10.55 6.32 2.13 0.72 0.54 2.75 43.34 46.09 25.82 20.25 1.09 .0.74 5.73 0.98 11.00 0.88 0.62 1.71 21.48 110.81 132.29 75.45 56.90 5.71 6.85 2.63 0.73 0.77 16.70 0.77 0.62 12.82 1.2 9 9.24 1.42 0.92 1.55 115.50 63.21 52.36 50m Western, Central 200m Eastern 140m * * Var iances summed over the complete frequency range ( 0^24)cpd are in cm 2sec 2 . A l l other var iances are given as a % of the t o t a l var iance ( t o t a l u + t o t a l v ) . This tab le i s based on t ime-ser ies lengths of 533 days s t a r t i n g A p r i l 16 1969. LO <40 c o n t a i n e d o v e r t h e c o m p l e t e f r e q u e n c y i n t e r v a l ( 0 . 002, 24cpd) , and t h e t c t a l v a r i a n c e i s t h e sum o f t h e t o t a l u - v a r i a n c e and t c t a l v - v a r i a n c e . A l l v a r i a n c e s c o n t a i n e d i n T a b l e I have been w e i g h t e d by t h e f a c t o r ( N ^ + ) / N A d i s c u s s e d e a r l i e r . As T a b l e I s h o w s , t h e l o w - f r e q u e n c y v a r i a n c e i n t h e i n t e r v a l (0. 002,0. 2 5 c p c ) , and t h e d i u r n a l and s e m i - d i u r n a l v a r i a n c e s a c c o u n t f o r t y p i c a l l y 80% of t h e t o t a l v a r i a n c e , w h i l e t h e v a r i a n c e a b o v e 2.1cpd a c c o u n t s f c r o n l y 10% o f t h e t o t a l v a r i a n c e . I t i s s e e n t h a t t h e t c t a l v a r i a n c e v a r i e s f r c m l o c a t i o n t c l o c a t i o n . The t c t a l v a r i a n c e c f ' t h e t h r e e . 50m l o c a t i o n s and t h a t o f t h e t h r e e d e e p e r l o c a t i o n s i n c r e a s e s f r c m e a s t t o w e s t . At e a c h c f t h e t h r e e s t a t i o n s , t h e t o t a l v a r i a n c e d e c r e a s e s w i t h d e p t h . I t i s s e e n t h a t t h e a n t i -c l o c k w i s e and c l o c k w i s e v a r i a n c e s ( 0 . 0 0 2 , 2 4 c p d ) a t f i v e l c c a t i c n s a r e r o u g h l y e q u a l . At t h e 200m w e s t e r n l o c a t i o n , t h e c l o c k w i s e v a r i a n c e (0.002,24cpd) i s a b o u t t h r e e t i m e s t h a t c f t h e a n t i - c i c c k w i s e v a r i a n c e . The l e w - f r e q u e n c y ( 0 .002,0.25cpd) v a r i a n c e a t most l o c a t i o n s c o n t r i b u t e s a s i g n i f i c a n t amount t o t h e ' t o t a l v a r i a n c e . As T a b l e I s h o w s , t h e l o w - f r e q u e n c y v a r i a n c e c f t h e two ,200m l o c a t i o n s a r e 1% and 127c o f t h e t c t a l v a r i a n c e , w h i l e t h a t o f t h e r e m a i n i n g f o u r l e c a t i e n s r a n g e f r c m 22% t o U6n o f t h e t o t a l v a r i a n c e . The l o w - f r e q u e n c y v a r i a n c e s o f t h e t h r e e 50m t i m e - s e r i e s a r e a p p r o x i m a t e l y e q u a l . West c f t h e l e w -f r e q u e r . c y v a r i a n c e a t a l l s i x l o c a t i o n s i s c o n c e n t r a t e d i n t h e r e g i o n | f | < 0 . 2 5 c p d . The l e w - f r e q u e n c y s p e c t r a l d e t a i l s shown i n F i g u r e 9 41 and F i g u r e 10 r e q u i r e c a r e f u l i n t e r p r e t a t i o n . F c r a l l s i x l o c a t i o n s , t h e r e a r e s p e c t r a l p e a k s . bet ween 0 and + 0 . 2 c f d . To 95% c o n f i d e n c e , t h e s e p e a k s a r e n e t s t a t i s t i c a l l y s i g n i f i c a n t , and a r e i g n o r e d . I t i s c o n c l u d e d t h a t t h e s p e c t r a , f c r t h e 140m and 50m l o c a t i o n s , i n c r e a s e s w i t h d e c r e a s i n g f r e q u e n c y i n t h i s l o w - f r e q u e n c y r a n g e , w h i l e t h e s e . c f t h e 200m l o c a t i o n s h ave a t e n d e n c y t o l e v e l o f f as t h e f r e q u e n c y a p p r o a c h e s z e r o . T h e s e c o n c l u s i o n s c a n a l s o be made f r o m i n s p e c t i o n o f t h e l o w -f r e q u e n c y v a r i a n c e s l i s t e d i n T a b l e I . However, t h e s e c o n c l u s i o n s a r e t o v i e w e d w i t h some c a u t i c n . ' E x c e p t f c r t h e 50m c e n t r a l l o c a t i o n , a l l t h e t i m e - s e r i e s c o n t a i n s i z a b l e d a t a g a p s (see F i g u r e 3) w h i c h may have d i s t o r t e d t h e l e w - f r e q u e n c y r e g i o n o f t h e i r s p e c t r a . The n a r r o w - b a n d v a r i a n c e s c o n c e n t r a t e d n e a r c r a t ±1cpd and ±2cpd r e p r e s e n t t h e t i d a l c u r r e n t s . The r r a j o r t i d a l c o n s t i t u e n t s , w h i c h a r e a s s o c i a t e d w i t h t h e s e c u r r e n t s , a r e i d e n t i f i e d i n F i g u r e 9 and F i g u r e .10. B e c a u s e o f l e a k a g e c f v a r i a n c e t o a d j a c e n t f r e q u e n c i e s , t h e s e s p e c t r a l p e a k s a p p e a r t c be much b r c a d e r t h a n t h e y a r e . A t a l l s i x l o c a t i o n s , t h e v a r i a n c e c f t h e (K, +F,) c o n s t i t u e n t i s l a r g e r t h a n t h e 0, c o n s t i t u e n t , and t h e M 2 c o n s t i t u e n t i s t h e l a r g e s t o f t h e t h r e e s e m i - d i u r n a l c o n s t i t u e n t s . At a l l b u t one c f t h e l o c a t i o n s (140m e a s t e r n ' l o c a t i o n n e a r - 2 c p d ) , t h e v a r i a n c e o f t h e ( K 2 + S 2 ) c o n s t i t u e n t i s • l a r g e r t h a n (or a p p r o x i m a t e l y e g u a l to) the N 2 c o n s t i t u e n t . The s p e c t r a f o r t h e 50m l o c a t i o n s r e v e a l t h a t t h e K 2 c o n s t i t u e n t a t - 1 . 9 3 2 c p d a p p e a r s t o have b e e r b r o a d e n e d . T h i s 42 b r o a d e n i n g r e s u l t s ( a t seme l o c a t i o n s ) i n t h e a b s e n c e c f t h e s p e c t r a l v a l l e y between t h e M z and N 2 s p e c t r a l peak's r e a r -2 c p d . The t i m e - d o m a i n d e s c r i p t i o n i s t h a t t h e M 2 c l c c k w i s e r o t a t i n g . v e c t o r , has an a m p l i t u d e w h i c h i s s l o w l y c h a n g i n g w i t h t i m e , i n s t e a d c f one w h i c h i s r e l a t i v e l y c o n s t a n t - w i t h t i m e . I n s p e c t i o n o f T a b l e I shows t h a t , . a t a l l s i x l o c a t i o n s , t h e (u + v) v a r i a n c e o f t h e se ir i - d i u r n a 1 c u r r e n t s i s l a r g e r t h a n t h a t o f t h e d i u r n a l o n e s . E o t h t h e d i u r n a l and s e m i - d i u r n a l c u r r e n t s have a v - v a r i a n c e t h a t i s two t o s e v e r a l t i m e s t h a t o f t h e u - v a r i a n c e . The S - v a r i a n c s i s a p p r o x i m a t e l y e q u a l t c t h e C - v a r i a n c e . f o r b c t h d i u r n a l and s e m i - d i u r n a l c u r r e n t s , at f i v e l o c a t i o n s . Hence a t . t h e m a j o r i t y o f t h e l o c a t i o n s , the t i d a l c u r r e n t s a r e mere or l e s s r e c t i l i n e a r . At t h e 200m w e s t e r n l o c a t i o n , t h e t i d a l m o t i o n s t e n d t c be c l o c k w i s e r o t a t i n g , t h e C - v a r i a n c e a t 1cpd and 2cpd b e i n g 3 and 4 t i m e s t h a t , c f t h e c o r r e s p o n d i n g A - v a r i a n c e . At t h e 200m l o c a t i o n s , t h e t i d a l m o t i o n s a c c o u n t f o r a b o u t 3/4 o f t h e t o t a l v a r i a n c e , w h i l e t h o s e at the. f o u r r e m a i n i n g l o c a t i o n s r a n g e f r o m 367< t o 46/? o f t h e t o t a l v a r i a n c e . The g r a p h s o f f»S v e r s u s l o g f f c r t h e c u r r e n t s a r e shown i n F i g u r e 11 and F i g u r e 12. These g r a p h s show t h e l o w -f r e q u e n c y (|f|<0.7cpd) ' s p e c t r a l s t r u c t u r e a t t h e e x p e n s e c f s t a t i s t i c a l c o n f i d e n c e . The h i g h - f r e q u e n c y (|f|>2cpd) r e q i o n s c f t h e s p e c t r a have lew r e s o l u t i o n and h i g h s t a t i s t i c a l c o n f i d e n c e . F o r a i l s i x l o c a t i o n s , t h e s p e c t r a l v a l l e y between ±0.1cpd and ±1cpd i s s t a t i s t i c a l l y s i g n i f i c a n t t c S5% c o n f i d e n c e . G e n e r a l l y , ' t h e l e w - f r e q u e n c y v a r i a n c e i s 43 Figure 11. Spectra (f-S) of 50m currents. Based on time-series of 533 days s t a r t i n g A p r i l 16 1969. Numbers above the o f f - s c a l e peaks in d i c a t e t h e i r values. gure 12. Spectra (f*S) of 140m and 200m currents. Based on time-series of 533 days s t a r t i n g A p r i l 16 1969. Numbers above the o f f - s c a l e peaks indi c a t e t h e i r values. 45 c o n c e n t r a t e d between ±0,01cpd and ±0.1cpd, w i t h a s p e c t r a l p eak i n o r n e a r t h i s f r e q u e n c y r e g i o n . The v a r i a n c e i s s e e n t o i n c r e a s e a s | f | i n c r e a s e s f r o m . 0 1 c p d u p w a r d s . The v a r i a n c e i n c r e a s e w i t h f r e q u e n c y , t h e l o w - f r e q u e n c y s p e c t r a l p e a k , and t h e s p e c t r a l s t r u c t u r e f o r | f | < . 1 c p d a r e t o be v i e w e d w i t h c a u t i o n . F o r most l o c a t i o n s , t h e s e f e a t u r e s a r e n e t s t a t i s t i c a l l y s i g n i f i c a n t t o 95% c o n f i d e n c e . F c r o t h e r l o c a t i o n s , t h e l e w - f r e q u e n c y s p e c t r a l peak i s s t a t i s t i c a l l y s i g n i f i c a n t , but t h e p r e s e n c e c f t h e d a t a qap.s have i n t r o d u c e d an unknown amount c f l o w - f r e q u e n c y d i s t o r t i o n . ' T h i s d i s t o r t i o n may a f f e c t t h e t o t a l l c w - f r e g u e n c y v a r i a n c e , i t may s h i f t , v a r i a n c e f r o m a f r e q u e n c y r e g i o n t c n e i g h b o u r i n g f r e q u e n c y r e g i o n s , and even f r o m n e g a t i v e f r e q u e n c y r e g i o n s t c p o s i t i v e o n e s (and v i c e - v e r s a ) . S u b j e c t t o t h i s q u a l i f i c a t i o n , i t a p p e a r s t h a t t h e c u r r e n t t i m e - s e r i e s a r e l o n g enough t c r e s o l v e t h e b r o a d l e w - f r e q u e n c y peak, at e a c h c f t h e s i x l o c a t i o n s . Seme o f the. i n d i v i d u a l p e a k s a r e n o t s t a t i s t i c a l l y s i g n i f i c a n t , b u t t h e i r a v e r a g e i s s t a t i s t i c a l l y s i g n i f i c a n t t c 95% c o n f i d e n c e . The two t i d a l p e a k s a t 1cpd and 2cpd a r e s t a t i s t i c a l l y s i g n i f i c a n t . I t i s se e n t h a t t h e v a r i a n c e d e c r e a s e s as | f | i n c r e a s e s f r o m 3cpd u p w a r d s . T h i s f e a t u r e i s s t a t i s t i c a l l y s i g n i f i c a n t t o 95% c o n f i d e n c e . The h i g h - f r e q u e n c y v a r i a n c e (|r'l>3cpd) c o n t r i b u t e s t y p i c a l l y 10% t c t h e t o t a l v a r i a n c e , and t h i s l ow c o n t r i b u t i o n i s a l s o s t a t i s t i c a l l y s i q n i f i c a n t . F o r a l l s i x l o c a t i o n s , t h e h i q h - f r e q u e n c y c l o c k w i s e v a r i a n c e i s l a r q e r t h a n t h e h i g h - f r e q u e n c y a n t i - c l o c k w i s e v a r i a n c e . tie 4 C p h e r e n c e _ a n d_ F h a s e _ F u n c t i o n s The c a l c u l a t i o n c f t h e v e c t o r c o h e r e n c e s q u a r e d f u n c t i o n , , i n v o l v e s a t i m e - s e r i e s 1 and a t i m e - s e r i e s 2, b o t h o f l e n g t h N. O r i g i n a l l y t h e r e a r e d a t a gaps i n b o t h t i m e -s e r i e s . The d a t a gaps i n t i m e - s e r i e s 1 d i f f e r i n number and l e n g t h f r c m t h o s e i n t i m e - s e r i e s 2, and g e n e r a l l y t h e g a p s c f t h e two t i m e - s e r i e s dc n e t c o i n c i d e . B e f o r e • per f o r m i n g t h e c r o s s - s p e c t r a l a n a l y s i s , a d d i t i o n a l gaps a r e . i n t r o d u c e d i n b o t h t i m e - s e r i e s . A f t e r t h i s o p e r a t i o n , t h e d a t a gaps o f b o t h t i m e -s e r i e s a r e i d e n t i c a l i n number and l e n g t h , sc. t h a t t h e g a p s e x a c t l y c o i n c i d e i n t h e t i m e d o m a i n . . The net r e s u l t i s t h a t t h e l i m i t s c f t h e f u n c t i o n a r e s t i l l ( 0 , 1 ) . The v e c t o r c o h e r e n c e s q u a r e d , f u n c t i o n i s h e n c e f o r t h r e f e r r e d t c as t h e c o h e r e n c e f u n c t i o n . I t i s a measure o f t h e r e s e m b l e n c e between t h e f l u c t u a t i o n s o f t i m e - s e r i e s 1 and t i m e -s e r i e s 2, i n a p a r t i c u l a r f r e q u e n c y r e g i o n . I f a l l t h e f l u c t u a t i o n s o f b o t h t i m e - s e r i e s a r e a s s o c i a t e d w i t h a s i n g l e l i n e a r p r o c e s s , t h e c o h e r e n c e f u n c t i o n i s u n i t y . . The c o h e r e n c e f u n c t i o n becomes l e s s t h a n u n i t y i f t h a t s i n g l e p r o c e s s - i s n o n -l i n e a r . C o n s i d e r t h e s i t u a t i o n where a l l t h e f l u c t u a t i o n s ( i n a c e r t a i n f r e q u e n c y , r e g i o n ) o f t i m e - s e r i e s 1 a r e a s s o c i a t e d w i t h a s i n g l e l i n e a r p r o c e s s , and where t h e m a j o r p o r t i o n o f t h e f l u c t u a t i o n s i n t i m e - s e r i e s 2 a r e a l s o a s s o c i a t e d w i t h t h e same l i n e a r p r o c e s s . A s e c o n d l i n e a r p r o c e s s new c o n t r i b u t e s p a r t i a l l y t o t h e f l u c t u a t i o n s i n t i m e - s e r i e s 2. The c o h e r e n c e f u n c t i o n a g a i n becomes l e s s t h a n - u n i t y . The l a r g e r t h e 47 c o n t r i b u t i o n t o t h e f l u c t u a t i o n s c f t i m e - s e r i e s 2 by t h e s e c o n d l i n e a r p r c c e s s , t h e c l o s e r t h e c o h e r e n c e f u n c t i o n a p p r o a c h e s z e r o . I n t h i s c a s e , t h a t p o r t i o n o f t h e f l u c t u a t i o n s i n t i m e -s e r i e s 2 due t c t h e s e c o n d l i n e a r p r o c e s s i s ' n o i s e ' , and t h e r e m a i n i n g p o r t i o n due t o t h e f i r s t l i n e a r p r o c e s s i s t e r m e d t h e c o h e r e n t p o r t i o n . a l i n e a r f r e g u e n c y s c a l e w i t h e a c h and e v e r y p o i n t , h a v i n g a c o n s t a n t -j) ( d e g r e e s o f freedom) . O f t e n t h e c o h e r e n c e f u n c t i o n i s p l o t t e d u s i n g an a r c t a n h ( X l 2 ) v e r t i c a l s c a l e . W i t h such a s c a l e , a s i n g l e c o n f i d e n c e i n t e r v a l s u f f i c e s f o r e a c h and e v e r y p o i n t o f t h e f u n c t i o n . A c c o r d i n g t o J e n k i n s and W a t t s (1966) t h e p a r a m e t e r , Y l 2 = a r c t a n h (t x z ) , has a p p r o x i m a t e l y a n o r m a l d i s t r i b u t i o n w i t h a v a r i a n c e o f V ~ 1 . Thus t h e 95$ c o n f i d e n c e i n t e r v a l s c f . Y | 2 _ a r e ; +1.96/ -fp . A c c o r d i n g t c E e n d a t -and P i e r s c l (1971) h o w e v e r , Y ,z i s n o r m a l l y d i s t r i b u t e d s u b j e c t t c t h e c o n d i t i o n s t h a t 0. 35<yz <0.9 5 and t h a t T>>20. When t h e s e c o n d i t i o n s do n e t h o l d , i t i s assumed i n t h i s . t h e s i s t h a t Y ) 2 i s s t i l l a p p r o x i m a t e l y n o r m a l , .and t h a t t h e c o n f i d e n c e i n t e r v a l s o c a l c u l a t e d i s a c r u d e b u t t o l e r a b l e a p p r o x i m a t i o n . A l i g h t h o r i z o n t a l l i n e a p p e a r s i n a l l t h e g r a p h s c f t h e c o h e r e n c e f u n c t i o n . G r o v e s and Hannan (1968) c a l l i t t h e 95% l e v e l o f t h e n u l l h y p o t h e s i s , but h e n c e f o r t h , i t i s r e f e r r e d t c as t h e 9 5 % n o i s e l e v e l . I t i s g i v e n by:-where 1 - d ^ C 9 5 and M = 2/(i>--2). The 9 5 % n o i s e l e v e l , # 2, i s net I n a l l c a s e s , t h e c o h e r e n c e f u n c t i o n i s p l o t t e d a g a i n s t (4.6) 48 r e l a t e d i n any manner t o t h e c o h e r e n c e f u n c t i o n u n d e r s c r u t i n y , b u t t o a h y p o t h e t i c a l c o h e r e n c e - f u n c t i o n e s t i m a t e b e t w e e n two t i m e - s e r i e s w h i c h have z e r o c o h e r e n c e . The 95% n c i s e l e v e l i s t h e v a l u e t h a t w c u l d be e x c e e d e d .100d=5% o f the t i m e by t h i s h y p o t h e t i c a l c o h e r e n c e - f u n c t i o n e s t i m a t e . I t i s n e t i d e n t i c a l l y z e r o b e c a u s e t h e two t i m e - s e r i e s used f o r a n a l y s i s a r e o f f i n i t e l e n g t h . I n c o n t r a s t . , t h e 95$ c o n f i d e n c e i n t e r v a l i s r e l a t e d t o t h e c o h e r e n c e f u n c t i o n b e i n g e x a m i n e d . The c o h e r e n c e f u n c t i o n u n d e r s c r u t i n y i s com p a r e d w i t h t h e 9 5 % n o i s e l e v e l , and when t h e f o r m e r f a l l s b e l c w t h e l a t t e r , t h e c o h e r e n c e b etween t h e two t i m e - s e r i e s may be j u d g e d t c be e s s e n t i a l l y z e r o . The 95$ c o n f i d e n c e i n t e r v a l i s t c t e u s e d i n d e p e n d e n t l y o f t h e 95% n c i s e l e v e l . The c o n f i d e n c e -i n t e r v a l i s u s e d t o d e t e r m i n e w h e t h e r a p a r t i c u l a r f e a t u r e c f t h e c o h e r e n c e f u n c t i o n u n d e r e x a m i n a t i o n i s s t a t i s t i c a l l y s i g n i f i c a n t . The v e c t o r phase f u n c t i o n , (J) , i s f r e g u e n t l y r e f e r r e d t c as t h e phase f u n c t i o n . A p o s i t i v e v a l u e o f t h e ph a s e f u n c t i o n means t h a t t h e r o t a r y component ( c l o c k w i s e c r a n t i -c l o c k w i s e ) i n t i m e - s e r i e s 1 i s l e a d i n g , i n t i m e , t h a t o f t i m e -s e r i e s 2. I n a l l - d i s c u s s i o n s o f #2 and (j) , t i m e - s e r i e s 1 i s a l w a y s i d e n t i f i e d b e f o r e t i m e - s e r i e s 2. When t h e c o h e r e n c e f u n c t i o n i s much l e s s t h a n u n i t y , t h e phase- f u n c t i o n may s t i l l be m e a n i n g f u l . A c o h e r e n c e f u n c t i o n v a l u e s m a l l e r t h a n u n i t y , i n d i c a t e s t h a t o n l y a p o r t i o n o f t h e f l u c t u a t i o n s i n b o t h t i m e - s e r i e s i s c o h e r e n t , and t h e co h e r e n t , p o r t i o n s have a c o n s t a n t phase d i f f e r e n c e g i v e n by t h e 49 p hase f u n c t i o n . . U n f o r t u n a t e l y , t h e c o n f i d e n c e i n t e r v a l s f c r t h e p h a s e f u n c t i o n a r e d e p e n d e n t cn and on • t h e c o h e r e n c e f u n c t i o n . The 95% c o n f i d e n c e i n t e r v a l s , ± A ( j ) , f c r t h e phase f u n c t i o n a r e g i v e n i n F i g u r e 13, w h i c h shews the. d e p e n d e n c e c f Atj) cn T) and *he c o h e r e n c e f u n c t i o n . F o r a f i x e d T) , i t i s s e e n t h a t t h e c o n f i d e n c e i n t e r v a l i n c r e a s e s t c ±A<| = ±180° as t h e c o h e r e n c e f u n c t i o n a p p r o a c h e s z e r o . Thus i n r e g i o n s where t h e c o h e r e n c e v a l u e s a r e l e w , i n f o r m a t i o n c o n t a i n e d i n t h e phase f u n c t i o n c a n p o t e n t i a l l y .be o b s c u r e d by t h e s c a t t e r i n t h e l a t t e r . T h i s s c a t t e r a r i s e s b e c a u s e o f t h e l a r g e amounts c f n o i s e p r e s e n t i n b o t h t i m e - s e r i e s , t h e l a r g e amounts c f n o i s e b e i n g i n d i c a t e d by. t h e low c o h e r e n c e v a l u e s . 4_.5 C u r r e n t - C u r r e n t _ F e l a t i e n s Shewn i n F i g u r e 14 and F i g u r e 15 a r e t h e c o h e r e n c e and p h a s e f u n c t i o n s between t h e c u r r e n t s f r o m v e r t i c a l l y and h o r i z o n t a l l y s e p a r a t e d l o c a t i o n s , r e s p e c t i v e l y . The r e g i o n s c f g r e a t e s t i n t e r e s t a r e t h e l o w - f r e q u e n c y and t i d a l r e g i o n s , s i n c e t h e m a j o r p o r t i o n c f t h e v a r i a n c e i s f o u n d t h e r e . The p r e s e n c e of t h e d a t a g a p s can be e x p e c t e d t o i n c r e a s e t h e l e w -f r e q u e n c y c o h e r e n c e v a l u e s by a s l i g h t amount, w h i l e t h e i r e f f e c t cn t h e phase f u n c t i o n i s d i f f i c u l t t o p r e d i c t . C o h e r e n c e G e n e - r a l l y , t h e r e a r e p o o r c o h e r e n c e s between c u r r e n t s 50 I 1 1 1 1 1 1 0 10 20 30 4 0 50 6 0 DEGREES OF FREEDOM Figure 13. The 95% confidence i n t e r v a l s of the phase spectrum (after Jenkins and Watts, 1968). 51 (A) E A S T E R N o !95 0 . 9 -0.8 0 . 7 -0.5 ' . . • I 1 • . . • ' - 2 . 5 T " I •2 - 1 . 5 -1 - 0 . 5 FREQUENCY (CPD) j95% r - o.i - i o 0 0 180-, 95% 90-0-- 9 0 --180 -i r n r * — 1 0.5 1 1.5 2 2.5 FREQUENCY (CPD) cp . . . , • 0 . 9 7 - , (B) CENTRAL 0 . 9 5 - | 0 .9 -0 .8 -0 .7 -0 . 5 : . 0-3\ . . . 0 . 1 -- 2 . 5 - 1 . 5 . • . o . -1 J 1 0--0 .5 r o o 180-90-) —I 0.5 1.5 —I 2.5 0 -90 -160 J 0 - 2 . 5 ~"1 - 1 . 5 0.97 , (C) WESTERN 0.9S 0 .9 -0 .8 -1 0.7-1 0.5 0.3=| r r v W ' T T " O-1 ' I * *, -l - 0 . 5 ... .. | 0.5 ^ — 1.5 —I 2.5 180 -q 90^ > • 0 ' -90 - 1 8 0 -Figure 14. Coherence and phase spectra between currents- from v e r t i c a l l y -separated l o c a t i o n s . A l l have 48 degrees of freedom. Parts-(A,B,C) are based on 360 day time-series s t a r t i n g ( A p r i l 17, June 17, July 11) 1969. 52 - 2 . 5 (A) C E N T R A L - E A S T E R N 0 . 9 7 ~ i 5 0 m 0 . 9 5 -0.9-0 . 8 -0 . 7 -0 . 5 -0.35 " 0 . 1 -1 ° J • • • • ~—r~ - 1 . 5 - I F R E Q U E N C Y (CPD) 1 -0.5 T 9 5 % 9-5% • • • r o o loo-q ' . 9 0 -0 -- 9 0 -- i e o J — 1 , a — . ! «. 0.5 1 l.S F R E Q U E N C Y (CPD) CD , • • • » — i 2 . 5 ( B ) C E N T R A L — W E S T E R N 5 0 m ,. . • • 1 - 2 . 5 - 2 ~1 - 1 . 5 —r--l - 0 . 5 A « « • * * • 0 . 9 7 - , 0.95-1 0 . 9 0 . 8 0 . 7 0 . 5 0 . 3 — 0.1-d — 1 0 J r o o 1 8 0 -. 9 0 -0 -0 . 5 r 1.5 —I 2 . 5 - 9 0 --180 -0 (C ) C E N T R A L — W E S T E R N 2 0 0 m • • • » • - 2 . 5 - 2 - 1 . 5 - 0 . 5 0 . 9 7 -0 . 9 5 -0 . 9 -0 . 8 -0 . 7 -0 . 5 : .• 0 . 3 3 — 0.1=) -1 0 I . I • • « • • • —I 0 . 5 ~ T — 1.5 1 0 0 -9 0 -0 -- 9 0 -- 1 8 0 -0 • « « 2 . 5 Figure 15. Coherence and phase spectra between currents from h o r i z o n t a l l y -separated l o c a t i o n s . Parts (A,B,C) are based on (50m, 50m, 200m) currents from the (central-eastern, central-western, central-western) mooring. Spectra i n parts (A,B,C) have (48,64,50) degrees of freedom using (360,432,333) day time-s e r i e s s t a r t i n g ( A p r i l 17, July 11,July 11) 1969. 53 f r o m d i f f e r e n t l o c a t i o n s ' i n t h e l o w - f r e q u e n c y (| f | <0 . 25c pd) r e g i o n , t h e h i g h e s t o b s e r v e d v a l u e o f # 2 b e i n g 0.28.. T h e r e i s no d o u b t t h a t t h e l o w - f r e q u e n c y c u r r e n t f l u c t u a t i o n s h a v e n e t been s p a t i a l l y . w e l l r e s o l v e d by t h e i n s t r u m e n t a r r a y u s e d . I n t h e l e w - f r e q u e n c y r e q i c n , i t i s s e e n t h a t t h e h i g h -e s t c o h e r e n c e v a l u e s a r e o b t a i n e d w i t h c u r r e n t s f r o m t h e v e r t i c a l l y - s e p a r a t e d e a s t e r n l o c a t i o n s . The c o h e r e n c e s between t h e c l o c k w i s e f l u c t u a t i o n s a r e h i g h e r t h a n t h e s e c f t h e a n t i -c l o c k w i s e f l u c t u a t i o n s , but b o t h l i e above t h e 95% n c i s e l e v e l . The c o h e r e n c e f u n c t i o n f c r t h e h c r i z c n t a l l . y - s e p a r a t ed 200m l o c a t i o n s have l e w - f r e q u e n c y f e a t u r e s v e r y s i m i l a r t o t h e c o h e r e n c e f u n c t i o n j u s t d i s c u s s e d . The r e m a i n i n g f o u r c o h e r e n c e f u n c t i o n s a r e o b s e r v e d t c have c o h e r e n c e v a l u e s t h a t , cn t h e a v e r a g e , l i e b e l o w t h e 95% n o i s e l e v e l . A l l s i x c o h e r e n c e f u n c t i o n s l a c k any l e w - f r e q u e n c y f e a t u r e s w h i c h a r e s t a t i s t i c a l l y s i g n i f i c a n t . T h a t i s , t h e 95%, c o n f i d e n c e i n t e r v a l s a l l o w one t o draw a h o r i z o n t a l l i n e c o n n e c t i n g t h e p c i n t s o f t h e c o h e r e n c e f u n c t i o n f r o m t h e l o w e s t f r e q u e n c y t o a b o u t O.Rcpd. The e n l y s t a t i s t i c a l l y s i g n i f i c a n t (95%) p e a k s t h a t a p p e a r i n t h e c o h e r e n c e f u n c t i o n s a r e t h e cnes r e p r e s e n t i n g t h e d i u r n a l and s e m i - d i u r n a l t i d a l c u r r e n t s . The t i d a l f l u c t u a t i o n s f r c m d i f f e r e n t l o c a t i o n s a r e h i g h l y c o h e r e n t , t h e l o w e s t o b s e r v e d v a l u e o f % z b e i n g 0.55. T h e s e h i g h v a l u e s a r e o b t a i n e d b e c a u s e a m a j o r p c r t i c n o f t h e t i d a l f l u c t u a t i o n s a t e a c h l o c a t i o n c a n be a s s o c i a t e d w i t h a s i n g l e p r o c e s s . T h i s p r o c e s s i s t h e f o r c i n g o f t h e f l u i d m o t i o n s i n t h e S t r a i t c f G e o r g i a , by t h e t i d a l . m o t i o n s o f t h e P a c i f i c Ccean s u r f a c e a t t h e mouth c f J u a n de F u c a S t r a i t , and at. t h e n o r t h e r n end c f V a n c o u v e r I s l a n d . C o m p a r i n g t h e s e m i - d i u r n a l c u r r e n t s , i t i s o b s e r v e d t h a t t h e c o h e r e n c e between t h e c l o c k w i s e component i s c o n s i s t e n t l y l o w e r t h a n t h e a n t i - c I c e k w i s e c o m p o n e n t . W i t h one e x c e p t i o n , t h i s i s a l s o t r u e o f t h e - d i u r n a l c u r r e n t s . The e x c e p t i o n o c c u r s w i t h t h e d i u r n a l c u r r e n t s f r o m t h e v e r t i c a l l y -s e p a r a t e d w e s t e r n l o c a t i o n s . Phase I n F i g u r e 1U and F i g u r e 15, t h e h i g h e s t l e w - f r e q u e n c y (|f|<0.25cpd) c o h e r e n c e s a r e o b t a i n e d w i t h t h e c u r r e n t s f r o m t h e v e r t i c a l l y - s e p a r a t e d e a s t e r n , l o c a t i o n s . I t i s o b s e r v e d t h a t t h e 50m l e w - f r e q u e n c y f l u c t u a t i o n s l a g t h o s e a t 1-JOm by 45°. The n e x t h i g h e s t l e w - f r e g u e n c y c o h e r e n c e s a r e c c t a i n e d w i t h ' t h e c u r r e n t s f r o m t h e h o r i z o n t a l l y - s e p a r a t e d 200m l o c a t i o n s . I n t h i s c a s e , t h e f l u c t u a t i o n s o f t h e c e n t r a l l o c a t i o n l e a d t h e s e o f t h e w e s t e r n l o c a t i o n by a b o u t 1 6 0 c . The s h a r p d e c r e a s e c f t h e phase f u n c t i o n a t +0.15cpd i s s t a t i s t i c a l l y s i g n i f i c a n t t c 95% c o n f i d e n c e . I n t h r e e o f t h e r e m a i n i n g f o u r c a s e s , t h e phase f u n c t i o n s i n t h e l o w - f r e q u e n c y r e g i o n have a . r e l a t i v e l y c o n s t a n t v a l u e o r a smooth v a r i a t i o n w i t h f r e q u e n c y . T h i s b e h a v i o u r i s i n c o n s i s t e n t w i t h t h e c o h e r e n c e f u n c t i o n s , w h i c h f a l l b e l c w t h e 95% n o i s e l e v e l i n t h e l o w - f r e q u e n c y r e g i o n , i m p l y i n g t h a t t h e f l u c t u a t i o n s have z e r o c o h e r e n c e s . W i t h two t i m e - s e r i e s h a v i n g a z e r o c o h e r e n c e , a phase f u n c t i o n h a v i n g l a r g e , s c a t t e r i s e x p e c t e d . The p h a s e f u n c t i o n c f two s u c h t i m e - s e r i e s i s n o t e x p e c t e d t o have a r e l a t i v e l y c o n s t a n t v a l u e o r a smooth v a r i a t i o n . I n t h i s s i t u a t i o n , i t i s c o n c l u d e d t h a t t h e c o h e r e n c e s a r e n o n - z e r o , e v e n t h o u g h t h e y l i e fcelcw the 9 5.7. n o i s e l e v e l . I n t h e l e w - f r e q u e n c y (|f|<0.2cpd) r e g i o n , the. phase between t h e f l u c t u a t i o n s f r o m t h e v e r t i c a l l y - s e p a r a t e d e a s t e r n l o c a t i o n s has a r e l a t i v e l y c o n s t a n t v a l u e c f -45°. U s i n g t h e 50m c u r r e n t s f r o m t h e c e n t r a l and e a s t e r n l o c a t i o n s , t h e p h a s e has a smooth v a r i a t i o n a t n e g a t i v e f r e q u e n c i e s (|f|<0.2cpd) and a r e l a t i v e l y c o n s t a n t v a l u e o f -90° a t p o s i t i v e f r e q u e n c i e s . The 50m l o w - f r e q u e n c y c u r r e n t f l u c t u a t i o n s f r o m t h e c e n t r a l and w e s t e r n l o c a t i o n s a r e a l s o o b s e r v e d t o have r e l a t i v e l y c o n s t a n t p hase d i f f e r e n c e s . The s m a l l e s t 95% c o n f i d e n c e i n t e r v a l s f o r the ph ase • f u n c t i o n s a r e f o u n d a t t h e t i d a l f r e q u e n c i e s , s i n c e t h e h i g h e s t c o h e r e n c e v a l u e s a r e f o u n d a t t h e s e f r e q u e n c i e s . The pha s es b etween t h e t i d a l c u r r e n t s from d i f f e r e n t l o c a t i o n s . d c n c t shew a c o n s i s t e n t b e h a v i o u r , and t h i s i n c o n s i s t e n c y i n t h e pha s e s c a n n o t be e x p l a i n e d by t h e i r s t a t i s t i c a l y u n c e r t a i n t y . I t i s o b s e r v e d t h a t ' t h e r o t a r y t i d a l c u r r e n t c o m p o n e n t s a r e i n ph a se t o w i t h i n ±45°. A r e l a t i v e l y c o n s t a n t v a l u e o f 0° i s c b s e r ved f c r t i . c p hase f u n c t i o n s e v e r a wide f r e q u e n c y r a n g e , f r o m | f 1=0. £cpd t o | f | = 2 . 5 c p d . The c o n s t a n t phase d i f f e r e n c e s a r e f o u n d a t p o s i t i v e f r e q u e n c i e s f o r t h e 50m c e n t r a l and w e s t e r n l o c a t i o n ' s , and a t n e g a t i v e f r e q u e n c i e s f o r t h e 200m c e n t r a l and w e s t e r n l o c a t i o n s . The c o r r e s p o n d i n g c o h e r e n c e f u n c t i o n s f a l l b e l c w t h e 95% n c i s e l e v e l a t many p l a c e s i n t h e f r e q u e n c y d o m a i n . However, i t i s c o n c l u d e d (from t h e b e h a v i o u r c f t h e phase f u n c t i o n s ) t h a t t h e two p a i r s c f t i m e - s e r i e s b e i n q e x a m i n e d do i n f a c t have n o n - z e r o c o h e r e n c e s . • • 4_. 6 S e j e c t r a _ o f _ T e m j j e r a t u r e s The g r a p h s o f f * S v e r s u s l o g f f c r the w a t e r t e m p e r a t u r e s " a r e shewn i n F i g u r e 16. These s p e c t r a l r e s u l t s a r e b a s e d cn t i m e - s e r i e s a l l o f l e n g t h 533 1/3 d a y s s t a r t i n g A p r i l 16, 1969. As F i g u r e 3 s h e w s , t h e r e a r e more ( o r as many) gaps i n t h e t e m p e r a t u r e t i m e - s e r i e s , a s t h e r e are- i n t h e c u r r e n t t i m e - s e r i e s . The f r e q u e n c y d i s t r i b u t i o n o f t h e v a r i a n c e i n t h e t e m p e r a t u r e t i m e - s e r i e s i s g i v e n i n T a b l e I I . A l l v a r i a n c e s i n t h i s t a b l e have t e e n m u l t i p l i e d by t h e f a c t o r (K^ + N ^ / N ^ w h i c h a p p e a r s i n e q u a t i o n ( 4 . 3 ) . Frcm T a b l e I I , i t i s s e e n t h a t t h e t c t a l v a r i a n c e d e c r e a s e s w i t h d e p t h a t e a c h o f t h e t h r e e s t a t i o n s . The c e n t r a l 50m l o c a t i o n has t h e l a r g e s t t o t a l v a r i a n c e o f t h e t h r e e 50m l o c a t i o n s , w h i l e t h e c e n t r a l 200m l o c a t i o n has t w i c e t h e t c t a l v a r i a n c e c f t h e w e s t e r n 200m l o c a t i o n . I t i s o b s e r v e d t h a t t y p i c a l l y 95% o f the t o t a l v a r i a n c e i s ' l e w -f r e q u e n c y (f<0.25cpd) v a r i a n c e . The l o w - f r e q u e n c y v a r i a n c e c f t h e t e m p e r a t u r e t i m e - s e r i e s i s c o n c e n t r a t e d i n a much n a r r o w e r f r e q u e n c y r e g i o n t h a n t h a t c f t h e c u r r e n t t i m e - s e r i e s . T a b l e T Figure 16. Spectra of water temperatures. Based on 533 day time-series s t a r t i n g A p r i l 16 1969. Table II - Frequency D i s t r i b u t i o n of the Variance i n the Water Temperatures** Frequency Western Central Eastern Range Station Station Station (Cyc: les/Day) 50m 200m 50m 200m 50m 140m .002 - 0.01 74.2 72.3 65.1 64.1 65.2 66.5 0.01 - 0.02 12.0 7.6-1 19.1 14.7 20.4 15.8 0.02 - 0.03 3.22 3. 72 1.67 3.98 2.16 2.98 0.03 - 0.04 0.77 2.11 2.26 2.21 • 1.18 1.25 0.04 - 0.05 0.89 2.18 0.92 2.83 1.14 2.50 0.05 - 0.10 3.23 6.02 2.17 2.49 3.52 3.01 0.15 - 0.20 1.06 1.50 1.94 1.48 0.90 1.28 0.20 - 0.25 0.60 0.39 1.05 0.84 0.96 1.15 .002 - 0.25 96.5 96.5 94.9 93.4 95.9 94.8 0.25 - 0.50 1.04 0.83 1.65 1.34 1.10 0.82 0.50 - 0.80 0.34 0.43 0.52 0.82 0.48 0.30 0.80 - 1.10 0. 31 0.68 0.68 1.13 0.64 1.44 1.10 - 1.80 0.47 0. 32 0.56 0.85 0.57 0.42 1.80 - 2.10 0.29 0.46 0.32 0.67 0.28 . 1.08 2.10 - 3.00 0.31 0.27 0.44 0.55 0.34 0.52 3.00 - 4.00 0.20 0.21 0.27 ' 0.38 0.23 0.24 4.00 - 24.0 0.63 0.55 0.74 1.08 0.55 0.47 .002 24.0 57.83 8.42 64.23 15.59 44.32 36.29 ** The t o t a l variances (0.002-24 cpd) are given i n units of 10~2 °C^. A l l other variances are given as a % of the t o t a l variance. Based on time-series lengths of 533 days s t a r t i n g A p r i l 16, 1969. 59 and T a b l e I I shews t h a t a m a j o r f r a c t i o n o f t h e l e w - f r e q u e n c y v a r i a n c e i n t h e t e m p e r a t u r e and c u r r e n t t i m e - s e r i e s i s f c u n d i n t h e r e g i o n s , f < 0 . 0 3 c p d and | f | < 0 . 1 5 c p d , r e s p e c t i v e l y . T y p i c a l l y G.5% t c 1% c f t h e t c t a l v a r i a n c e i s f o u n d i n t h e f r e q u e n c y , r a n g e (4,24) c p d . The l e w - f r e q u e n c y s p e c t r a l . s t r u c t u r e s c f t h e g r a p h s i n F i g u r e 16 a r e n e t s t a t i s t i c a l l y s i g n i f i c a n t t o 95% c o n f i d e n c e . F u r t h e r m o r e , i t a p p e a r s t h a t t h e t i m e - s e r i e s used a r e f o e s h c r t t c r e s o l v e t h e l e w - f r e q u e n c y f l u c t u a t i o n s . S p e c t r a l peaks a t 1 cpd and 2 c p d a r e f o u n d i n a l l s i x s p e c t r a . T hese peaks a r e s t a t i s t i c a l l y s i g n i f i c a n t t o 9 5 % c o n f i d e n c e , e x c e p t f o r t h e 1 cpd peak c f t h e c e n t r a l 50m l o c a t i o n . The p e a k s i n t h e t e m p e r a t u r e s p e c t r a a r e a s s o c i a t e d w i t h t h e t i d a l c u r r e n t s , but t h e y a r e much b r o a d e r t h a n t h e t i d a l p e a k s i n t h e c u r r e n t s p e c t r a . At some l o c a t i o n s , t h e p e a k s i n t h e t e m p e r a t u r e s p e c t r a a r e n e t l o c a t e d e x a c t l y at. t h e f r e q u e n c i e s c f t h e t i d a l , c u r r e n t s . 4_. 7 ^SiS£ii§iilISz5Sffi£§Isture_Relations The c o h e r e n c e and ph a s e f u n c t i o n s f o r t e m p e r a t u r e s f r c m v e r t i c a l l y - s e p a r a t e d • l o c a t i o n s a r e shown i n F i q u r e M. The s t a r t i n g d a t e s and l e n g t h s c f t h e t e m p e r a t u r e t i m e - s e r i e s u s e d a r e a l s o i n d i c a t e d . These a r e p i c k e d sc t h e t i m e - s e r i e s c o n t a i n o n l y s h c r t d a t a g a p s . The l e w - f r e q u e n c y ( f < 0 . 2 0 c p d ) r e g i o n s o f t h e c o h e r e n c e and phase f u n c t i o n s a r e t h e r e g i o n s c f g r e a t e s t i n t e r e s t , s i n c e a l m o s t a l l t h e v a r i a n c e i s f c u n d 6 0 0.97-. 0.95-1 0.9-0.8H 0.7-0.5-0.3-0.1-0-100 - i 90-0 -SO -180 -J (A) EASTERN 0.5 - r — 1.5 0.97-0.95-0.9-0.0-0.7-0.5-j . 0.3^| 0.1 -,0-2.5 ISO • 90-0--90--100-1 (B) EASTERN 95% 95% IL CD —, 1 1 r 0.5 1 1.5 2 FREQUENCY (CPD) I 2.5 0.97-0.95-0.9-0.8-0.7-0.5-0.3H 0.1-o-J. 180-90-0--90--180-(C) WESTERN 0.97-0.95-0..9-0.8-0.7 0.5 0.3--0.1-I 0.5 1.5 ,0-J 2.S 180 » 90 0 -90--180-(D) CENTRAL 0.5 1.5 I 2.5 Figure 17. Coherence and phase spectra between water temperatures from v e r t i c a l l y - s e p a r a t e d l o c a t i o n s . Parts (A,B,C,D) have (30,32,52,40) degrees of freedom and are based on (100,107, 260,130) day time-series s t a r t i n g ( A p r i l 16, August 28, July 10, A p r i l 16) 1969. 61 t h e r e . I n a l l f o u r c o h e r e n c e f u n c t i o n s , h i g h v a l u e s a r e o b s e r v e d i n t h e l e w - f r e q u e n c y r e g i o n ( 0 , 0 . 2 ) c p d . The i n c r e a s e c f t h e c o h e r e n c e s . i n t h e l e w - f r e q u e n c y r e g i o n i s a s t a t i s t i c a l l y s i g n i f i c a n t (95%) f e a t u r e . None o f t h e p e a k s f c u n d e l s e w h e r e i n t h e f r e q u e n c y domain a r e s t a t i s t i c a l l y s i g n i f i c a n t . O v e r much c f t h e f r e q u e n c y d o m a i n , t h e c o h e r e n c e v a l u e s l i e b e l o w t h e 95% n o i s e l e v e l . The t e m p e r a t u r e f l u c t u a t i o n s a t 1cpd and 2 c p d a r e n o t c o h e r e n t . The f l u c t u a t i o n s from t h e v e r t i c a l l y - s e p a r a t e d w e s t e r n l o c a t i o n s a r e 180° out. o f p h a s e a t t h e l o w e s t f r e q u e n c y . I n c o n t r a s t , t h e f l u c t u a t i o n s o f l o w e s t f r e q u e n c y f r c m t h e v e r t i c a l l y - s e p a r a t e d c e n t r a l and e a s t e r n l o c a t i o n s t e n d t c t e i n p h a s e . T h e r e a r e two phase f u n c t i o n s f c r t h e t e m p e r a t u r e s f r o m t h e v e r t i c a l l y - s e p a r a t e d e a s t e r n l o c a t i o n s . These a r e b a s e d cn t i m e - s e r i e s w h i c h s p a n d i f f e r e n t p e r i o d s i n t h e t i m e d o m a i n . The phase f u n c t i o n f o r t h e e a r l i e r ( A p r i l 1969) t i m e -s e r i e s p a i r has a r e l a t i v e l y c o n s t a n t v a l u e o f 0° o v e r t h e c o m p l e t e f r e q u e n c y r a n g e . That f c r t h e l a t e r (Auq 1969) t i m e -s e r i e s p a i r shows l a r g e s c a t t e r . Hence t h e phase f u n c t i o n s f c r t h e v e r t i c a l l y - s e p a r a t e d e a s t e r n l o c a t i o n s a r e t i m e - d e p e n d e n t . Cn t h e 50m p l a n e , t h e t e m p e r a t u r e s f r c m t h e c e n t r a l l o c a t i o n a r e compared w i t h t h o s e f r o m t h e e a s t e r n l o c a t i o n d u r i n g two p e r i o d s i n 1969. The r e s u l t s a r e shown i n F i g u r e 18. D u r i n g t h e e a r l i e r p e r i o d , t h e f l u c t u a t i o n s c f l e w e s t f r e q u e n c y a r e h i g h l y , c o h e r e n t and i n p h a s e . D u r i n g t h e l a t e r p e r i o d , t h e c o h e r e n c e i s p o o r a t t h e l e w e s t f r e q u e n c y . Hence, 62 0 . 8 7 -0 . 9 S -0 . 9 -0 . 8 -0 . 7 0 . 5 -0 . 3 -0 . 1 -0 -1 8 0 -9 0 -0 -- 9 0 -- 1 8 0 -0 . 9 7 -0 . 9 5 -0 . 9 -0 . 8 -0 . 7 -0 . 5 -0 . 3 -0 . 1 -0 -1 B 0 -9 0 -0 -- 9 0 -- 1 8 0 -(A) CENTRAL-EASTERN 50m H V 2 95% ' 95% 1 . 5 0 . 5 1 FREQUENCY (CPD) (B) CENTRAL—EASTERN 50m — 0 . 5 1 . 5 9 0 . S 7 -0 . 9 5 -0 . 9 -0 . 0 -0 . 7 -0 . 5 : 0 . 3 : 0 . 1 : 0 -160-~ 9 0 -(C) CENTRAL-WESTERN 50m T . - oH - 9 0 -- 1 8 0 -0 . 9 7 -0 . 9 5 -0 . 9 -0 . 8 -0 . 7 -0 . 5 : 0 . 3 : o.i : 0 -180-9 0 -0 -- 9 0 -- 1 8 0 -0 . 9 7 -_ 0 . 9 5 -0 . 9 -e _ 0 . 8 -0.7H 0 . 5 0 . 3 0 . 1 0 180-1 * 90 0 0 . T . -160 J . 0 . 5 l . S 2.5 (D) CENTRAL-WESTERN. 50m I 1 : 1 1 1 1 0 0 . 5 1 1 . 5 " 2 2 . 5 • e (E) CENTRAL—WESTERN 200m 0 . 5 —T— 1 . 5 cb . • . • • -1 2 . 5 Figure 18. Coherence and phase spectra between water temperatures from horizontally-separated l o c a t i o n s . Parts (A,B,C,D,E) have (30,26,30,42,36) degrees of freedom and are based on (100,90,101,141,120) day time-series s t a r t i n g ( A p r i l 16 1969, August 28 1969, June 14 1970, July 10 1969, November 25 1969) 63 t h e coherence- a t t h e l o w e s t f r e q u e n c y i s t i m e - d e p e n d e n t . In t h e c o m p a r i s o n s o f t e m p e r a t u r e s (50m) f r o m t h e c e n t r a l and w e s t e r n l o c a t i o n , t h e c o h e r e n c e at t h e l o w e s t f r e q u e n c y i s a l s c t i m e — d e p e n d e n t . I n t h e c o m p a r i s o n between t e m p e r a t u r e s , r e l a t i v e l y h i g h c o h e r e n c e s a t t h e l e w s s t f r e q u e n c y a r e t h e o n l y f e a t u r e s w h i c h a p p e a r a b o v e t h e 95% n o i s e l e v e l , and w h i c h a r e s t a t i s t i c a l l y s i g n i f i c a n t t o 95% c o n f i d e n c e . H o w e v e r , t h e 95% c o n f i d e n c e i n t e r v a l s a r e n o t v a l i d when t h e c o h e r e n c e f u n c t i o n i s t i m e -d e p e n d e n t . I t s t i m e - d e p e n d e n c y i n d i c a t e s t h a t t h e t e m p e r a t u r e s t i m e - s e r i e s do r o t r e p r e s e n t s t a t i o n a r y s t o c h a s t i c p r o c e s s e s . The t e m p e r a t u r e t i m e - s e r i e s a n a l y z e d a r e t o o s h o r t t c s a t i s f y t h e s t a t i o n a r y c r i t e r i o n . The s p e c t r a c f t h e t e m p e r a t u r e s i n d i c a t e t h a t t h e v a r i a n c e s a r e f o u n d i n t h e r e g i o n 50-5C0 d a y s , g i v i n g a c h a r a c t e r i s t i c p e r i c d o f 200 d a y s ( s a y ) . The c o h e r e n c e s between t e m p e r a t u r e s a r e b a s e d cn t i m e - s e r i e s l e n g t h s f r o m 90-260 d a y s , w h i c h a r e a p p r o x i m a t e l y t h e d u r a t i o n o f a c h a r a c t e r i s t i c p e r i c d , and w h i c h a r e t o e s h o r t t c s a t i s f y t h e s t a t i o n a r y c r i t e r i o n . A t e m p e r a t u r e t i m e - s e r i e s l e n g t h c f 1000 d a y s (5 c h a r a c t e r i s t i c p e r i o d s ) may be s u f f i c i e n t l y l o n g t c s a t i s f y t h e s t a t i o n a r y c r i t e r i o n f o r t h e l o w - f r e q u e n c y f l u c t u a t i o n s . I n view' o f t h i s demanding r e q u i r e m e n t o f t h e t i m e - s e r i e s l e n g t h , t h e c l a s s i c a l methods ( s u c h as i s e n t r c p i c a n a l y s i s ) c f p h y s i c a l o c e a n o g r a p h y ( W a l d i c h u k , 1957) a p p e a r t o be mere a p p r o p r i a t e f c r t h e a n a l y s i s o f t h e t e m p e r a t u r e t i m e - s e r i e s , t h a n t h e methods o f t i m e - s e r i e s a n a l y s i s . 6U 4_. 8 Cur r e n t rlSi2£^ISiMES_^S 2^iioil§ The c u r r e n t s and t e m p e r a t u r e s f r o m t h e same l o c a t i o n a r e c o m p a r e d and t h e r e s u l t s a r e shown i n F i g u r e 19 and F i g u r e 20. The c o m p a r i s o n s f c r t h e 50m l o c a t i o n s a p p e a r i n t h e f i r s t f i g u r e . G e n e r a l l y , t h e c u r r e n t s and t e m p e r a t u r e s a t a l o c a t i o n a r e i n c o h e r e n t . I n s e v e r a l i n s t a n c e s , t h e c o h e r e n c e d o e s r i s e above t h e 9 5 % n o i s e l e v e l , t u t i n a l l c a s e s t h e s e p e a k s a r e n e t s t a t i s t i c a l l y s i g n i f i c a n t t c '95% c o n f i d e n c e . The c u r r e n t s h a v e n a r r o w - b a n d v a r i a n c e a t t h e t i d a l ' f r e q u e n c i e s , but t h e s e f l u c t u a t i o n s a r e n e t c o h e r e n t w i t h t h o s e i n t h e t e m p e r a t u r e s . At t h e c e n t r a l 50m l o c a t i o n , t h e c u r r e n t and t e m p e r a t u r e f o r two p e r i o d s a r e c o m p a r e d , s h o w i n g t h a t t h e c o h e r e n c e a t t h e l e w e s t f r e q u e n c y i s t i m e - d e p e n d e n t . The c o m p a r i s o n s o f c u r r e n t and t e m p e r a t u r e f c r the d e e p e r l o c a t i o n s a p p e a r i n F i g u r e 2 0 . G e n e r a l l y , t h e c u r r e n t s and t e m p e r a t u r e s a t t h e deep l o c a t i o n s a r e a l s o i n c o h e r e n t o v e r a m a j o r p o r t i o n o f t h e f r e q u e n c y r a n g e . The c o h e r e n c e f c r t h e deep c e n t r a l l o c a t i o n shows s l i g h t l y h i g h e r c o h e r e n c e s t h a n t h o s e a t t h e e t h e r l o c a t i o n s . The peak a t 0.7cpd i s above t h e n o i s e l e v e l , and s t a t i s t i c a l l y s i g n i f i c a n t t c 95% c o n f i d e n c e . P r o b a b l y t h o u g h , t h i s f e a t u r e i s t i m e - d e p e n d e n t , i r a k i n g i n v a l i d t h e a s s u m p t i o n cn w h i c h t h e c o n f i d e n c e i n t e r v a l i s b a s e d . The- c o h e r e n c e between c u r r e n t and t e m p e r a t u r e i s a p a r a m e t e r n o t e a s i l y r e l a t e d t o t h e p h y s i c a l s i t u a t i o n . C o n s i d e r an i n f i n i t e l y l o n g c a n a l c f u n i f o r m w i d t h and d e p t h r u n n i n g i n t h e x - d i r e c t i c n . The f l u i d i n t h e - c h a n n e l mcves as a r i g i d mass w i t h v e l o c i t y 0 (t) w h i c h i s o n l y d e p e n d e n t cn 65 (A) EASTERN "1 1 1 " — I — -2 -1.5 -1 -D.5 FREQUENCY (CP'D) -2.5 (B) CENTRAL . • • • • • • • . -2.5 -1.5 -1 0.97-' 0.95-0.9-0.8-0.7-0.5 : 0.3 : - 0.1--1 0-0 180 -' 90 H 0 -90-) -180 0.97-, 0.95 0.9 0.8 0.7 . 0.5 * 0.3 ~ 0.1 I o -y-2 J95C r 95 % 0.5 . *« 1 1.5 FREQUENCY (CPD) — i 2.5 -0.5 (C) CENTRAL I— -2.5 -1.5 -0.5 (D) WESTERN -2.5 -1.5 - l -0.5 • 180-' 90-0--90--180 -0.97-0.95-0.9-0.0-1 0.7-j 0.5 — 0.33 • O . l l - i .0 0 180-] 90 0 . -90 -180 0.97-, 0.95 0.9 0.8-1 0.7-J 0.5 0.3 — 0.1 "I o J 0 160-q 0 --90 -180 J 0 0.5 I 0.5 0.5 0 - r ~ 1.5 1.5 2 -~1 2.5 "715 I 1 .5 2.5 Figure 19. Coherence and phase spectra between current and temperature at the 50m l o c a t i o n s . Parts (A,B,C,D) have (72,34,16,64) degrees of freedom and are based on (480,225,100,432) day-time-series s t a r t i n g ( A p r i l 16 1969, A p r i l 16 1969, June 14 1970, July 10 1969). 66 -2.5 -2.5 -2.5 (A) EASTERN 1 1 1— -1.5 -1 -0.5 FREQUENCY (CPD) 0.07-' 0.05-0.9-o.a-o.7-j T2 0.5^ 0.3 7 0.1 o-J 0 leo 90 OH -30 0 (B) CENTRAL ~ I — -1.5 -180 0.97 0.35 0.9 o.aH 0.7 C.5H 0.3H -0.1 h 0 AT' r -0.5 0 0 180-1 90 OH -90 0 (C) WESTERN -1.5 — 1 -0.5 -180-1 0.97-0.95-0.9-o.a-0.7-0.3 = -r-r- 0.1-0-0 100-90-0--00--100-0 95% _J ;: 95% — i 1 1 — 1 0.5 1 1.5 2 2.5 FREQUENCY (CPD) 0.5 I 1.5 O.S ~1— l.S • i 0 —1 2.5 -1 2.5 Figure 20. Coherence and phase spectra between current and temperature at the 140m and. 200m lo c a t i o n s . Parts (A,B,C) have (58,48,58) degrees of freedom and are based on (384,320,384) day time-serie s s t a r t i n g ( A p r i l 16, June 18, J u l y 10) 1969. 67 t i m e . The t e m p e r a t u r e o f t h e w a t e r i s a f u n c t i o n c f p (the p-a x i s o r i g i n moves a l o n g w i t h t h e f l u i d ) :-(H) (P)= Z \ F -k (4 . 1) R e l a t i v e t o t h e p - a x i s , © i s t i m e - i n d e p e n d e n t . The o r i g i n o f p mcves a l o n g w i t h t h e f l u i d w i t h v e l o c i t y U ( t ) , so t h a t : -p - x- J U (t«) d t ' = x-d (t) (4 . £) -co i s t h e r e l a t i o n fcetwesn t h e o r i g i n o f p and x. Hence t h e t e m p e r a t u r e m oving w i t h a f l u i d e l e m e n t i s c o n s t a n t b e c a u s e m c l e c u l a r d i f f u s i o n , h e a t s i n k s and s o u r c e s have been n e g l e c t e d . A s e n s c r a t a f i x e d l o c a ' t i o n x=x 0 would measure a t e m p e r a t u r e v a r i a t i o n c f : -2t 2t - -Uct) 2 © C-x.ti 3x (4.9) CO p •U(t) * £ * a j x 0 - d (t) ] a nd ^ f T ( t ) } = i / (2irf) « ^{U ( t ) }* ^{ Z ka £ x Q - d (t) } (4.10) where and * d e n o t e t h e E o u r i e r t r a n s f o r m and c c n v c l u t i c n o p e r a t o r s , r e s p e c t i v e l y . E ven f o r a s i t u a t i o n a s s i m p l e and u n r e a l i s t i c as t h e one d i s c u s s e d , t h e power s p e c t r u m c f t e m p e r a t u r e may be v e r y ee d i f f e r e n t . ( b e c a u s e c f t h e c o n v o l u t i o n w i t h t h e l a s t t r a n s f o r m ) f r o m t h a t c f t h e c u r r e n t s , g i v i n g r i s e t o a lew c o h e r e n c e between c u r r e n t and t e m p e r a t u r e . I t i s d o u b t f u l t h a t a s i m p l e r e l a t i o n b e t w een ^{T} and 3<{U} e x i s t s f o r t h e c a s e c f a mere r e a l i s t i c s i t u a t i o n . 9 S _ p e c t r a _ c f _ S e a - L e v e l J t _ A i r - P r e s s u r e ^ W i _ n d s The power s p e c t r a o f s e a - l e v e l ( E t . A t k i n s o n ) , a i r -p r e s s u r e ( V a n c o u v e r ) and w i n d s (Sand Heads) a r e shown i n F i g u r e 2 1. The s e a - l e v e l has s i g n i f i c a n t l o w - f r e q u e n c y v a r i a n c e i n t h e r e g i o n ( 0 , 0 . 3 ) c p d . T h i s v a r i a n c e d e c r e a s e s s h a r p l y f c r i n c r e a s i n g f r e q u e n c y , a f e a t u r e w h i c h i s s t a t i s t i c a l l y s i g n i f i c a n t t o 95% c o n f i d e n c e . The s m a l l peak a t 0.07cpd i s n e t s t a t i s t i c a l l y s i g n i f i c a n t . The m a j o r p o r t i o n c f t h e s e a -l e v e l v a r i a n c e l i e s a t t h e t i d a l f r e q u e n c i e s n e a r c r a t 1cpd and 2 c p d . The a i r - p r e s s u r e ( F i g u r e 21b) a l s o has l e w - f r e q u e n c y v a r i a n c e but i n a l a r g e r r e g i o n ( 0 , 0 . 5 ) c p d . The f l u c t u a t i o n s i n t h e s p e c t r u m b e l o w 0.5cpd i s n o t s t a t i s t i c a l l y s i g n i f i c a n t . The o n l y s t a t i s t i c a l l y s i g n i f i c a n t f e a t u r e i n t h i s r e g i o n i s t h e i n c r e a s i n g v a r i a n c e as t h e f r e g u e n c y d e c r e a s e s t o .zero. The a i r - p r e s s u r e s p e c t r u m has s h a r p p e a k s a t 1cpd and 2 c p d , b o t h o f w h i c h a r e s t a t i s t i c a l l y s i g n i f i c a n t . The peak a t 2 c p d i s t o o s m a l l t o be s e e n i n t h e g r a p h . A m a j o r p o r t i o n c f t h e w i n d e n e r g y ( F i g u r e 21a) i s c o n t a i n e d i n t h e r e g i o n ( C , 1 . 5 ) c p d . The peak a t 1cpd i s 69 Figure 21. Spectra of wind (Sand Heads), air-pressure (Vancouver) and sea-level (Pt. Atkinson). A l l have 22 degrees of freedom and a l l are based on 600 day time-series s t a r t i n g January 03 1969. Numbers i n brackets i n d i c a t e the values of the o f f - s c a l e peaks. 70 s t a t i s t i c a l l y s i g n i f i c a n t , w h i l e t h e peaks a t l o w e r f r e q u e n c i e s are. n e t . The o n l y s t a t i s t i c a l l y s i g n i f i c a n t l o w - f r e q u e n c y f e a t u r e i s t h e i n c r e a s e o f enexgy-.for d e c r e a s i n g | f | . The d i s t r i b u t i o n c f v a r i a n c e i s p l o t t e d cn a l o g f r e q u e n c y s c a l e i n F i g u r e 22. The s e a - l e v e l c o n t a i n s a l a r g e a n n u a l o s c i l l a t i o n . H o wever, t h e s p e c t r a l f e a t u r e s below 0.1 cpd a r e n o t s t a t i s t i c a l l y s i g n i f i c a n t . The d e c r e a s e c f v a r i a n c e as t h e f r e q u e n c y i s i n c r e a s e d f r o m 0.1cpd t o G.5cpd, and t h e i n c r e a s e o f v a r i a n c e a s t h e f r e q u e n c y i s f u r t h e r i n c r e a s e d , e r e s t a t i s t i c a l l y s i g n i f i c a n t . A s m a l l p e r c e n t a g e o f t h e s e a - l e v e l v a r i a n c e i s f o u n d a t f r e q u e n c i e s g r e a t e r t h a n 2 .1cpd. The peak at 5cpd i s s t a t i s t i c a l l y s i g n i f i c a n t . The l e w - f r e q u e n c y s e a - l e v e l f l u c t u a t i o n s have . p e r i o d s 2-3G0 d a y s . I n F i g u r e 22b, i t i s se e n t h a t t h e a i r - p r e s s u r e has e s s e n t i a l l y z e r o v a r i a n c e a t f r e q u e n c i e s above 2 c p d . I n c o n t r a s t t c t h e s e a - l e v e l , t h e a i r - p r e s s u r e d o e s n o t have a l a r g e a n n u a l v a r i a t i o n . The d e c r e a s e o f v a r i a n c e as t h e f r e q u e n c y i n c r e a s e s upward from 0.2cpd i s s t a t i s t i c a l l y s i g n i f i c a n t , and t h i s r a t e c f d e c r e a s e i s much l a r q e r t h a n t h e c o r r e s p o n d i n g d e c r e a s e i n t h e s e a - l e v e l s p e c t r u m . The a i r -p r e s s u r e f l u c t u a t i o n s have p e r i o d s 2-100 d a y s . The w i n d has a r e l a t i v e l y b r o a d s p e c t r u m ( F i g u r e 2 2 a ) . The wind e n e r g y i s f o u n d f r c m 0.015cpd t o t h e N y g u i s t f r e q u e n c y o f 12 c p d . The p e a k s a t ±1cpd a r e t h e e n l y s t a t i s t i c a l l y s i g n i f i c a n t o n e s . The d e c r e a s e o f v a r i a n c e as | f | i n c r e a s e s upwa'rd f r o m 1 c p d , and a s | f | d e c r e a s e s downward f r c m C . l c p d , a r e s t a t i s t i c a l l y s i g n i f i c a n t f e a t u r e s . I t i s seen t h a t t h e 71 (A) WIND 8^ i o Ul [ A i l L0G l 0 F (F IN CPD) (8) AIR-PRESSURE VI f J l i 1 r~ -1 -2 -3 (C) S E A - L E V E L tr < CD SH +F (I -2 I -1 (10,4001-1 —i Lr A ! • - ( 6 , 9 6 0 ) -3 -1 Figure 22. Spectra (f«S) of wind (Sand Heads), air-pressure (Vancouver) and sea-level (Pt. Atkinson). A l l are based on 600 day time-series s t a r t i n g January 03 1969. Numbers near the o f f - s c a l e peaks ind i c a t e t h e i r values. 72 TABLE III - Frequency D i s t r i b u t i o n of the Variance i n Sea-Level, Air-Pressure, and Wind** Frequency Sea-Level A i r Winds Range Point Pressure . Sand Heads (Cycles/Day) Atkinson Vancouver U V u+v C A .002 - 0.05 1.16 38.50 8.86 3.95 12.80 6.69 6.11 0.05 - 0.10 0.29 19.00 6.93 3.95 10.90 6.18 4.70 0.10 - 0.15 0.11 12.10 6.73 3.72 10.50 6.31 4.14 0.15 - 0.20 0.05 8.22 3.63 2.41 ' 6.04 3.31 2.73 0.20 - 0.25 0.07 7.79 4.07 2.40 •6.48 3.65 2.83 0.25 - 0.30 0.04 3.63 2.52 1.04 3.56 2.07 1.48 0.30 - 0.35 0.02 1.91 2.68 0.97 3.65 1.83 1.82 0.35 - 0.40 0.03 2. 34 2.63 0.88 3.51 2.14 1.36 0.40 - 0.45 0.02 1.65 2.84 0.87 3.70 2.12 1.59 0.45 - 0.50 0.02 1.18 1.98 1.01 2.99 1.72 1.27 0.50 - 0.60 0.02 1.04 2.41 1.11 3.52 2.00 1.52 0.60 - 0.70 0.03 0.90 2.52 0.86 3.38 1.54 1.84 0.70 - 0.80 0.03 0- 38 1.85 0.78 2.63 1.32 1.31 0.80 - 0.90 0.59 0.26 1.70 0.67 2.36 1.28 1.08 0.90 — 1.10 58.10 0.74 3. 34 2.15 5.49 2.88 2.61 1.10 - 1.45 0.10 0.15 2.54 1.64 4.18 2.37 1.81 1.45 - 1.80 0.05 0.05 1.54 1.02 2.56 1.45 1.11 1.80 - 2.10 39. 30 0.09 0.96 0.75 1.71 0.99 0.72 2.10 - 3.00 0.02 0.03 1.48 1.40 2.88 1.72 1.16 3.00 - 4.00 0.01 0.02 0.88 0.98 l.-86 1.08 0.78 4.00 - 12.0 0.02 0.04 2. 36 3.07 5.43 2.92 2.50 .002 - 12.0 11,360 53.31 2.214 1.222 3.436 1.909 1.528 x 10 5 x 10 5 x 10 5 x 10 5 x 10 5 ** The t o t a l variances (0.002-12 cpd) of sea-level and air-pressure are given i n units of cn\2 and mb , r e s p e c t i v e l y . A l l other var-iances of sea-level and air-pressure are given as a % of the t o t a l variance. The wind variances i n the range (0.002-12 cpd) are given i n units of cm^sec -^. A l l other wind variances are given as a % of the U+V variance i n the range (0.002-12 cpd). . Variances of a l l parameters are based on time-series lengths of 600 days, s t a r t i n g January 3, 1969. w i n d s p e c t r u m t e n d s t c l e v e l o f f a t ±10cpd, - i n d i c a t i n g t h e w i n d has e n e r g y above 1 2 c p d , and t h a t seme a l i a s i n g has o c c u r r e d . I t i s c e r t a i n t h a t a p o r t i o n o f t h e h i g h - f r e g u e n c y v a r i a n c e i s d i g i t i z a t i o n n o i s e . The wind d i r e c t i o n s a r e o n l y d i g i t i z e d t c t h e n e a r e s t 22.5°. I t i s j u d g e d t h a t t h e a l i a s i n g h a s net s e r i o u s l y d i s t o r t e d t h e wind s p e c t r u m . The g r o s s d i s t r i b u t i o n o f v a r i a n c e i n t h e s e a - l e v e l , a i r - p r e s s u r e and w i n d s a r e g i v e n i n T a b l e I I I . E x a m i n a t i o n c f t h e s p e c t r a and o f T a b l e I I I i n d i c a t e s t h a t t h e s e t h r e e p a r a m e t e r s have v a r i a n c e i n d i f f e r e n t and o v e r l a p p i n g f r e q u e n c y r e g i o n s . F o r e x a u p l e , h a l f c f t h e w i n d e n e r g y i s f o u n d above and b e l o w 0.3cpd ( a p p r o x i m a t e l y ) . The a p p r o x i m a t e c o r r e s p o n d i n g f r e q u e n c i e s o f t h e a i r - p r e s s u r e and l o w - f r e q u e n c y s e a - l e v e l f l u c t u a t i o n s a r e O . l c p d a nd 0.07cpd,. r e s p e c t i v e l y . H e n c e , t h e wind e n e r q y i s f o u n d i n a h i g h e r and o v e r l a p p i n g f r e q u e n c y r e g i o n t h a n t h e a i r - p r e s s u r e v a r i a n c e , w h i c h i n t u r n i s f o u n d i n a h i q h e r and o v e r l a p p i n g f r e q u e n c y r a n g e t h a n t h e l e w - f r e q u e n c y s e a - l e v e l v a r i a n c e . U_. JO Re l a t i c n s _ E e t we £2_Sea-Ley e l s The power s p e c t r a o f s e a - l e v e l a t V i c t o r i a , T c f i n c and A l e r t . Bay (see F i g u r e 1 f c r l o c a t i o n s ) were a l s o c a l c u l a t e d , b u t n o t shewn s i n c e t h e y a r e v e r y s i m i l a r t o t h a t at I t . A t k i n s o n . The c o m p a r i s o n s o f s e a - l e v e l a t P t . A t k i n s c n w i t h t h o s e a t t h e t h r e e o t h e r l o c a t i o n s a p p e a r " i n F i g u r e 23. N c t e t h a t t h e c o h e r e n c e f u n c t i o n s a r e p l o t t e d , cn a l i n e a r s c a l e . 74 0- 5]T 95% ZL — r | I I 1 = , , = P» 0-5 I 1.5 2 2.5 3 3.5 4 4.5 180 -! FREQUENCY (CPD) 90-0-3(J) • . PT. ATKINSON — VICTORIA -SO4 ,. . -180-1 0.5-160 SO 0--S0--180 Ct) 0.5 1.5 I 2.S PT. ATKINSON — T O F I N O 3.5 4.5 • .. • 0.S-' " I ; I 1 I I , , ! ! > 0 0-5 1 1.5 2 2.5 3 3.5 4 4.5 IfiO-l 90-Q J i . . PT. ATKINSON ALERT BAY -90 -] -180-" Figure 23. Coherence and phase spectra between sea-level at Pt. Atkinson with those at other l o c a t i o n s . A l l have 52 degrees of freedom and a l l are based on 533 day time-series s t a r t i n g A p r i l 16 1969. 75 The c o h e r e n c e s a t some f r e q u e n c i e s a r e c l o s e t o u n i t y , a v a l u e w h i c h i s l o c a t e d a t i n f i n i t y ' o n t h e a r c t a n h s c a l e . The 95% s t a t i s t i c a l l y s i g n i f i c a n t f e a t u r e s c f t h e c o h e r e n c e f u n c t i o n f o r P t . A t k i n s o n and V i c t o r i a a r e new d e s c r i b e d . S t a r t i n g a t the l e w e s t f r e q u e n c y , t h e c o h e r e n c e f u n c t i o n d e c r e a s e s t c a l e c a l minimum a t . 0.8cpd. From t h i s minimum i t i n c r e a s e s t o a b r o a d peak c e n t r e d a t 1 c p d , a f t e r w h i c h i t a g a i n d e c r e a s e s a t 1.15cpd. B e t w e e n 1.15cpd and 1.E c p d , t h e c o h e r e n c e f u n c t i o n has a c o n s t a n t v a l u e , o f 0.75 ( t h e l e c a l minimum c e n t r e d a t 1.45cpd i s n e t s t a t i s t i c a l l y s i g n i f i c a n t ) . The two peaks a t 1.93cpd and 2.02cpd a r e s t a t i s t i c a l l y s i g n i f i c a n t . These p e a k s n e a r o r a t 1cpd and 2 cp d r e p r e s e n t t h e d i u r n a l and s e m i - d i u r n a l t i d a l f l u c t u a t i o n s i n t h e s e a - l e v e l . From t h e s e m i - d i u r n a l peak, t h e c o h e r e n c e f u n c t i o n d e c r e a s e s t o a s m a l l v a l u e a t . 2 . 8 5 c p d . The twe peaks a t 2.95cpd and 3.85cpd a r e s t a t i s t i c a l l y s i g n i f i c a n t w h i l e t h a t a t 3 . 1 4 c pd i s n e t . Between' t h e f o r m e r two p e a k s , t h e c o h e r e n c e f u n c t i o n has a c o n s t a n t v a l u e o f a b o u t 0.15. N e g a t i v e v a l u e s c f t h e phase f u n c t i o n c o r r e s p o n d t o a t i m e l a g o f t h e f l u c t u a t i o n s a t E t . A t k i n s o n r e l a t i v e t c t h e s e a t V i c t o r i a . The phase a p p e a r s t c h a v e a c o n s t a n t v a l u e c f z e r o f r c m t h e l e w e s t f r e q u e n c y t c a b o u t 1.1cpd. Frcm 1.1cpd, t h e p h a s e d e c r e a s e s s m o o t h l y and a l m o s t l i n e a r l y t o a v a l u e c f -155° a t ' 2.95-cpd. The phase i s o b s e r v e d t o be -166° a t 3.85cpd. The b r o a d maximum i n t h e p h a s e f u n c t i o n c e n t r e d a t 3.4cpd i s a s t a t i s t i c a l l y s i g n i f i c a n t f e a t u r e . The h i g h c o h e r e n c e s f o u n d a t 2.95cpd and 3.85cpd a r e 76 a s s o c i a t e d w i t h t h e s h a l l o w w a t e r t i d a l c o n s t i t u e n t s . T hese c o n s t i t u e n t do not a p p e a r i n t h e g r a v i t a t i o n a l p o t e n t i a l . R a t h e r , t h e y a r e p r o d u c e d by n o n - l i n e a r i n t e r a c t i o n s between t h e c o n s t i t u e n t s w h i c h a r e i n t h e g r a v i t a t i o n a l p o t e n t i a l . The peak a t 2.95cpd i s a s s o c i a t e d w i t h t h e c o n s t i t u e n t MK3 and p e r h a p s w i t h SC^. The c o n s t i t u e n t i1K 3 has a f r e q u e n c y w h i c h i s t h e sum c f t h e f r e q u e n c i e s c f M • and K, . The peak i n t h e c o h e r e n c e f u n c t i o n a t 3.85cpd i s . a s s o c i a t e d w i t h t h e s h a l l o w w a t e r c o n s t i t u e n t , f l 4 . The c o h e r e n c e f u n c t i o n f o r P t . A t k i n s o n ' and T c f i n c a r e i n many ways s i m i l a r t c t h a t f c r P t . A t k i n s o n , a n d V i c t o r i a . The 95% s t a t i s t i c a l l y s i g n i f i c a n t f e a t u r e s a r e now d e s c r i b e d . S t a r t i n g f r o m t h e l e w e s t f r e q u e n c y , t h e c o h e r e n c e f u n c t i o n f o r P t . A t k i n s o n and T c f i n - o r e m a i n s a t a r e l a t i v e l y c o n s t a n t v a l u e u n t i l a t 0 . 3 c p d , i t t a k e s a s t e p d e c r e a s e t c a n o t h e r c o n s t a n t v a l u e . From t h i s l a t t e r c o n s t a n t v a l u e , t h e c o h e r e n c e f u n c t i o n d e c r e a s e s t c a l o c a l minimum a t 0-. 8 c p d . A f t e r i n c r e a s i n g t c a maximum n e a r 1 c p d , a n o t h e r l o c a l minimum i s a g a i n r e a c h e d a t 1.15cpd. The v e r y b r e a d peak c e n t r e d a t 1.93cpd i s a l s o s t a t i s t i c a l l y s i g n i f i c a n t . A g a i n , -the p e a k s a t 2.95cpd and 3.85cpd a r e s t a t i s t i c a l l y s i g n i f i c a n t , w h i l e t h e f e a t u r e s 'between- t h e s e two peaks a r e n o t . . S t a r t i n g f r o m a v a l u e c f 10° a t t h e l o w e s t f r e q u e n c y , t h e p h a s e f u n c t i c n d e c r e a s e s a l m o s t l i n e a r l y u n t i l a t 0 . 3 5 c p d , i t r e a c h e s a c o n s t a n t v a l u e o f abo u t -35°. A r e l a t i v e l y s h a r p d e c r e a s e a p p e a r s a t 1.15cpd. The phase f u n c t i o n r e m a i n s a t a c o n s t a n t v a l u e c f -150° a t f r e q u e n c i e s g r e a t e r t h a n 1.15cpd. 77 H c v e v e r , t h e peak i n t h e phase f u n c t i o n a t 2.95cpd i s s t a t i s t i c a l l y s i g n i f i c a n t . The c o h e r e n c e f u n c t i o n f o r P t . A t k i n s o n and A l e r t Eay i s v e r y s i m i l a r t o t h a t f o r F t , A t k i n s o n and T c f i n c . The o n l y d i f f e r e n c e s a r e t h a t t h e f o r m e r d o es n o t have a peak a t j . 8 5 c p d w h i l e t h e 2.95cpd peak now a p p e a r s a t 3 c p d . F u r t h e r m o r e , s t a r t i n g f r o m t h e l e w e s t f r e q u e n c y , t h e c o h e r e n c e f u n c t i o n d e c r e a s e s s m o o t h l y t o a l o c a l minimum a t 0 . 7 5 c p d , k i t h no i n d i c a t i o n o f a s t e p d e c r e a s e . The p h a s e f u n c t i o n f c r P t . A t k i n s o n and A l e r t Eay i s c o n s t a n t a t t h e two l o w e s t f r e q u e n c i e s . A d e c r e a s e f r c m t h e s e c o n d p o i n t t c t h e t h i r d p c i n t (G.12cpd) i s o b s e r v e d . From 0.12cpd upwards t o 1cpd t h e phase f u n c t i o n r e m a i n s a t a c o n s t a n t v a l u e c f -25°. The r e l a t i v e l y s h a r p d e c r e a s e f r o m -25° t c -130° i s o b s e r v e d t c be c e n t r e d a t 1,15cpd. E x c e p t f c r t h e p e r t u r b a t i o n c e n t r e d a t 2.95cpd, t h e phase f u n c t i o n r e m a i n s a t a c o n s t a n t v a l u e o f -13C° f o r f r e q u e n c i e s a b o v e 1.5cpd. I t i s s e e n t h a t t h e phase f u n c t i o n f c r P t . A t k i n s o n and S l e r t Eay i s v e r y s i m i l a r t o t h a t f o r F t . A t k i n s o n and T c f i n c , The c o h e r e n c e f u n c t i o n f o r P t . A t k i n s o n and V i c t o r i a shows h i g h v a l u e s o f 0.5 and l a r q e r between 1cpd" and 2.5c pd. When t h e s e a - l e v e l a t P t . A t k i n s o n i s compared w i t h t h a t a t A l e r t E a y , h i q h v a l u e s a r e o b s e r v e d b e t w e e n 1.4cpd and 2 . 7 c p d . H i g h v a l u e s a r e o b s e r v e d between 1.25cpd and 2.85cpd i n t h e c o h e r e n c e f u n c t i o n f o r P t . A t k i n s o n and T c f i n c . I n t h e s e f r e q u e n c y r e g i o n s ( e x c l u d i n g t i d a l f r e q u e n c i e s ) , t h e power s p e c t r a c f t h e s e a - l e v e l s a r e o b s e r v e d t c r e l a t i v e l y f l a t , 78 i n d i c a t i n g t h a t t h e s e a r e v e r y b r o a d - b a n d f l u c t u a t i o n s . To g i v e ' s u c h h i g h c o h e r e n c e s , t h e p r o c e s s e s a s s o c i a t e d w i t h t h e s e f l u c t u a t i o n s a r e e s s e n t i a l l y t h e same a t e a c h o f t h e f o u r w i d e l y s e p a r a t e d l o c a t i o n s . These n o n - t i d a l s e a - l e v e l f l u c t u a t i o n s a p p e a r t o be b a r o t r o p i c w i t h s p a t i a l s c a l e s as l a r g e a s t h e l e n g t h o f V a n c o u v e r I s l a n d . 4... 11 R s l a t i o n s _ B e t wee n _ R i r - P r e s s u r e s The power s p e c t r a ' o f a i r - p r e s s u r e s a t V i c t o r i a , Comox, T o f i n o and A l e r t Bay ( s e e - F i g u r e 1 f o r l o c a t i o n s ) were c a l c u l a t e d . B e i n g v e r y s i m i l a r t o t h e power s p e c t r u m o f a i r -p r e s s u r e a t V a n c o u v e r , t h e y a r e n o t shown. The a i r - p r e s s u r e a t V a n c o u v e r i s c o m p a r e d w i t h t h o s e a t t h e o t h e r f o u r l o c a t i o n s a n d t h e r e s u l t s e x h i b i t e d i n F i g u r e 24. N o t e t h a t t h e c o h e r e n c e ' f u n c t i o n s ' a r e p l o t t e d on a l i n e a r s c a l e . The c o h e r e n c e f u n c t i o n s a l l have 95% s t a t i s t i c a l l y s i g n i f i c a n t p e a k s a t 2, 3 and 4 c p d . O n l y t h e two c o m p a r i s o n s o f a i r - p r e s s u r e a t V a n c o u v e r w i t h t h a t a t V i c t o r i a a n d w i t h t h a t a t Comox> have 9 5 % s t a t i s t i c a l l y s i g n i f i c a n t p e a k s a t 1cpd . The r e l a t i v e l y h i g h c o h e r e n c e s (peaks) a t 1,2,3 and 4cpd a r e a s s o c i a t e d ' w i t h a t m o s p h e r i c t i d e s , and t h e d i u r n a l h e a t i n g and c o o l i n g 1 o f t h e a i r . The a i r - p r e s s u r e a l s o h a s a d i u r n a l v a r i a t i o n s i n c e i t i s d e p e n d e n t on t h e a i r - t e m p e r a t u r e . T h i s p r o c e s s o c c u r s a t e a c h l o c a t i o n , g i v i n g r i s e t o h i g n c o h e r e n c e s when t h e ' a i r - p r e s s u r e s ' ' a t d i f f e r e n t l o c a t i o n s a r e c o m p a r e d . The d i u r n a l v a r i a t i o n i n a i r - p r e s s u r e i s n o t p u r e l y s i n u s o i d a l . 7 9 0 . 5 -180-90-0-- 9 0 --180-1 -0 95% 0.5 1.5 I 2.5 FREQUENCY (CPD) VANCOUVER —VICTORIA I 3.5 4.5 0 . 5 -1B0-90-0 -- 8 0 --180 -1 -Kb ••• 0.5 1.5 2.5 3.5 4 .5 VANCOUVER — COMOX 0.5-180-90-0 -- 9 0 -- 1 8 0 -1 -0 . 5 -i : i i 0.5 1 1.5 VANCOUVER — TOFINO cb -2.5 3.5 • • • IT' 4.5 0-> 180-S0-0 -- 8 0 --180-0.5 1 1.5 VANCOUVER — P O R T HARDY • « « « - * - - •« 2.5 3.5 4 .5 Figure 24. Coherence and phase spectra between air-pressure (at mean sea-level) at Vancouver with those from other l o c a t i o n s . A l l have 52 degrees of freedom and a l l are based on 533 day time-series s t a r t i n g A p r i l 16 1969. eo g i v i n g r i s e t o t h e h i g h e r h a r m o n i c s a t 2,3 and 4 c p d . I t i s r e t known why seme c f t h e c o h e r e n c e f u n c t i o n s have s t a t i s t i c a l l y s i g n i f i c a n t p e a k s a t 2 , 3 , a n d Ucpd, b u t n o t a t 1 c p d . I n t h e f r e g u e n c y r e g i o n s not a t 1,2,2 and U cpd, t h e a i r - p r e s s u r e f l u c t u a t i o n s a r e a s s o c i a t e d w i t h a n o t h e r p r o c e s s , t h e p a s s a g e o f w e a t h e r s y s t e m s . T h i s p r o c e s s g i v e s r i s e t h e f o l l o w i n g b e h a v i o u r o f t h e c o h e r e n c e , f u n c t i o n s . A g a i n s t the b a c k g r o u n d o f t h e p e a k s , t h e f o u r c o h e r e n c e f u n c t i o n s h a v e r e l a t i v e l y h i g h v a l u e s a t t h e l o w e s t f r e g u e n c y , and w h i c h d e c r e a s e s m o o t h l y w i t h i n c r e a s i n g f r e q u e n c y . The d e c r e a s e w i t h f r e q u e n c y i n d i c a t e s t h a t h i g h f r e q u e n c i e s a r e a s s c c i a t e d w i t h h i g h wavenumbers ( s m a l l s c a l e s ) , w h i c h a r e not as c o h e r e n t as as t h e l o w wave ru it b e r s ( l e w f r e q u e n c i e s , l a r g e s c a l e s ) b e c a u s e c f t h e d i s t a n c e s e p a r a t i n g t h e l o c a t i o n s . I g n o r i n g t h e p e a k s f o r t h e moment, t h e c o h e r e n c e f u n c t i o n s d e c r e a s e s m o o t h l y u n t i l t h e y d r o p b e l o w t h e 95 5 n o i s e l e v e l ; The f r e q u e n c i e s a t w h i c h t h i s o c c u r s a r e 1.3, 2.1, 2.5 and 3.3cpd f o r t h e c o m p a r i s o n s of a i r - p r e s s u r e a t V a n c o u v e r w i t h t h o s e a t P o r t H a r d y , T c f i n o , Comox . and V i c t o r i a , r e s p e c t i v e l y . F c r a f i x e d ' f r e q u e n c y ( l e n g t h s c a l e ) , the f l u c t u a t i o n s a t V a n c o u v e r a r e most c o h e r e n t w i t h t h e s e at V i c t o r i a and l e a s t c o h e r e n t w i t h t h o s e a t P o r t H a r d y . The g e o g r a p h i c a l r e l a t i o n s c f t h e f o u r l o c a t i o n s r e l a t i v e to V a n c o u v e r a r e n o t i 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 made from t h e f o u r c o h e r e n c e f u n c t i o n s . The a i r - p r e s s u r e f l u c t u a t i o n s a t V a n c o u v e r a r e i n phase w i t h t h o s e a t t h e two c l o s e s t l o c a t i o n s , V i c t o r i a and Ccmcx. 81 The a i r - p r e s s u r e f l u c t u a t i o n s a t V a n c o u v e r l a y t h o s e a t the two f a r t h e s t l o c a t i o n s , T o f i n c and P e r t H a r d y . From a v a l u e of 0° a t t h e l e w e s t f r e g u e n c y , t h e s e p h a s e l a g s i n c r e a s e w i t h i n c r e a s i n g f r e q u e n c y . The s l o p e o f t h e phase f u n c t i o n s f o r ( V a n c o u v e r , T o f i n o ) and ( V a n c o u v e r , P o r t H a r d y ) a p p e a r t o he i d e n t i c a l . E v i d e n t l y , t h e p r e s s u r e f i e l d • a r r i v e s a t F o r t Hardy and T c f i n o s i m u l t a n e o u s l y , b e f o r e a r r i v i n g a t V a n c o u v e r , V i c t o r i a and Comcx, s i m u l t a n e o u s l y . . . 4_._12 Ee l a t i c n s _ Eet wsen_W i n d s The w i n d r e c o r d from E n t r a n c e I s l a n d i s net a c o n t i n u o u s one. The wind a t Sand Heads i s compared w i t h t h e s e a t E n t r a n c e I s l a n d f o r two d i f f e r e n t p e r i o d s , and t h e r e s u l t s shown i n F i g u r e 25. F o r ' t h e 80 day c o m p a r i s o n p e r i o d , t h e c o h e r e n c e f u n c t i o n shews an a v e r a g e v a l u e c f 0.7 f c r t h e r e g i o n 0<|f|<0.5. None c f t h e p e a k s a r e s t a t i s t i c a l l y s i g n i f i c a n t t o 95% c o n f i d e n c e . The o n l y s t a t i s t i c a l l y s i g n i f i c a n t f e a t u r e i s t h a t t h e c o h e r e n c e d e c r e a s e s s m o o t h l y a s | f | i n c r e a s e s , r e a c h i n g t h e 95%. n o i s e l e v e l a t ±1cpd. The ph a s e f u n c t i o n shows t h a t t h e c o h e r e n t p o r t i o n o f t h e f l u c t u a t i o n s i s i n p h a s e , when t h e i n h e r e n t s c a t t e r i n t h e phase f u n c t i o n i s i g n o r e d . The c o h e r e n c e f u n c t i o n f o r t h e 15G day c o m p a r i s o n p e r i o d i s s m c c t h e r f c r i t has b e t t e r s t a t i s t i c s . I n t h e r e g i o n ( 0 , - 0 . 5 ) c p d , t h e c o h e r e n c e f u n c t i o n h a s an a v e r a g e v a l u e c f 0.8, w h i l e i n t h e r e g i o n (0,+0.5)'cpd, a l o w e r v a l u e c f 0.75 i s 82 -2.5 (A) 0.87-1 0.95 • 0.9-1 l I I 1 -2 -1.5 -1 -0.5 FREQUENCY (CPD) 0.8-0.7-0.5-0.3-0.1-0-180-95% » 4-95% — i — 0.5 1.5 I 2.5 90 0 -90--180-FREQUENCY (CPD) (B) .97-, .95 .9 .8 .7-.5-.3 -.1 -•y2 • -2.5 -1.5 -0.5 "I 0 I8O-1 0.5 1.5 — I 2.5 90--90 -180^ cp • . . Figure 25. Coherence and phase spectra between wind at Sand Heads and those at Entrance Island. Parts (A,B) have (16,30) degrees of freedom and are based on (80,150) day time-series s t a r t i n g (October 10 1969, A p r i l 1 1970). o b s e r v e d . The peak at. - 2 . 1 5 c p d i s s t a t i s t i c a l l y s i g n i f i c a n t t o 9 5 % c o n f i d e n c e . The d i u r n a l f l u c t u a t i o n s of t h e wind at d i f f e r e n t l o c a t i o n s a r e n o t c o h e r e n t . A g a i n , t h e c o h e r e n c e f u n c t i o n d e c r e a s e s s m o o t h l y a s | f | i n c r e a s e s , b u t r e a c h i n g t h e 95% n c i s e l e v e l a t ±1.5cpd i n s t e a d o f t h e + 1cpd o b s e r v e d i n t h e 80 day c o m p a r i s o n . The p h a s e f u n c t i o n shows t h a t t h e c c h e r e r t p o r t i o n o f t h e f l u c t u a t i o n s f o r t h i s p e r i o d i s a l s o i n p h a s e . The c o h e r e n c e f u n c t i o n d e c r e a s e s f c r i n c r e a s i n g f r e q u e n c i e s , o r f c r i n c r e a s i n g w a v enumters i n t h e w ind f i e l d . The h i g h w a v e n u m t e r s { s m a l l e r l e n g t h s c a l e s ) ' p r o p a g a t i n g • p a s t e a c h l o c a t i o n h ave l e s s r e s e m b l a n c e t h a n t h e l o w e r wave'numbers ( l a r g e r l e n g t h s c a l e s ) , b e c a u s e t h e f o r m e r a r e a f f e c t e d by t h e t o p o g r a p h y t o a g r e a t e r e x t e n t t h a n a r e t h e l a t t e r . U_. 1_3 R§laii21is_ Eet ween_Sea-L The s e a - l e v e l a t P t . A t k i n s o n i s compared v i t h t h e a i r -p r e s s u r e a t V a n c o u v e r and w i t h t h e w ind a t Sand Heads i n F i g u r e 26. The s e a - l e v e l f l u c t u a t i o n s a r e shown t o be c o h e r e n t ( t f ^ - 0 . 7 ) ' w i t h t h e a i r - p r e s s u r e f l u c t u a t i o n s i n t h e r e g i o n ( 0 , 0 . 5 ) c p d ( F i g u r e 2 6 a ) . Upwards f r o m 0 . 5 c p d , t h e c o h e r e n c e f u n c t i o n d e c r e a s e s s m o o t h l y r e a c h i n g t h e 95% n c i s e l e v e l a t 0.8 c p d . The p e a ks a t 1cpd and 2cpd a r e s t a t i s t i c a l l y s i g n i f i c a n t t c 9 5% c o n f i d e n c e . I t i s o b s e r v e d t h a t t h e peak, a t 2cpd i s a b ove t h e n o i s e l e v e l t c 95% c o n f i d e n c e . The s e a - l e v e l and a i r - p r e s s u r e f l u c t u a t i o n s a r e o u t o f p hase by 180° i n t h e r e g i o n ( 0 , 0 . 8 ) c p d . I n t h i s r e g i o n , t h e s e a - l e v e l f l u c t u a t i o n s 84 (A) S E A - L E V E L AIR PRESSURE 0.97 0.95 0.9-t -2.5 -2 -1.5 -1 -0.5 0.5 0.3 3 — o.i-J 0 *•• •** *• o'7--v-2 '* '"• * " j • » ,* m • • ioo 90-| 0 -90 • ••• -leo-1 0 0.5 I 1.5 I 2.5 0 (B) S E A - L E V E L WIND -2.5 -1.5 ~1 -0.5 0.97 0.9S-] 0.9-0.8-0.7-0.5: •* 0.3: — 0.1-- l o J r 0 0 . 180^, • 90-0--90--180-0.5 0 1.5 —I 2.5 (C) A IR -PRESSURE WIND T ' I l l 1 1 — -2.5 -2 -1.5 -1 -0.5 FREQUENCY (CPD) 0.97 0.95-| 0.9-D.B-. 0.53 ° - 3 , -0.1-d - i o J 0 0 180-90 -j [ 9 5 % 9 5 % - o -90 -j -100 I • . ! r 0.5 1 1.5 2 FREQUENCY (CPD) — I 2.5 4> Figure 26'. Coherence and phase spectra among wind (Sand Heads) , a i r -pressure (Vancouver) and sea-level (Pt. Atkinson). A l l have 60 degrees of freedom and are based on 600 day time-serie s s t a r t i n g January 03 1969. £5 a r e c a u s e d by t h e a i r - p r e s s u r e f l u c t u a t i o n s ' ( i n v e r s e b a r o m e t e r e f f e c t ) . Amidst, t h e s c a t t e r i n t h e p h a s e , i t i s s e e n t h a t the f l u c t u a t i o n s a t 1cpd and 2cpd a r e a l s o cut. o f phase by 180°. At each l o c a t i o n , a l a r g e p o r t i o n c f t h e s e a - i e v e l f l u c t u a t i o n s a r e c a u s e d by t h e a i r - p r e s s u r e f l u c t u a t i o n s . The c o m p a r i s o n c f a i r - p r e s s u r e s between f o u r p a i r s o f l o c a t i o n s show t h a t s e a - l e v e l s f r o m a p a i r o f l o c a t i o n s s h o u l d a l s o be c o h e r e n t , as i s o b s e r v e d . I c a f a i r a p p r o x i m a t i o n , t h e a i r -p r e s s u r e f i e l d a c t s as a r i g i d l i d cn t h e w a t e r s a r o u n d V a n c o u v e r I s l a n d i n t h e l o w - f r e q u e n c y r e g i o n ( 0 , 0 . 5 ) c p d . The c o m p a r i s o n between s e a - l e v e l a t E t . A t k i n s o n and w i n d a t Sand Heads i s now d i s c u s s e d ( F i g u r e 26b) . ' Over rruch c f t h e f r e q u e n c y r a n g e , t h e c o h e r e n c e between s e a - l e v e l and w i n d f l u c t u a t i o n s i s b e l o w t h e 9555 n o i s e l e v e l . I n t h e r e g i o n (0, - 0 . 5 ) c p d , t h e c o h e r e n c e f u n c t i o n h a s an a v e r a g e v a l u e c f 0.25. I n t h i s r e g i o n , t h e phase ( s e a - l e v e l i s l e a d i n g ) s t a r t s a t 160° b e f o r e l e v e l l i n g o f f a t a b o u t 90°. The c o h e r e n c e f u n c t i o n has an a v e r a g e v a l u e o f 0.41 i n t h e r e g i o n ( 0 , +0 . 1) cpd , b e f o r e d r o p p i n g v e r y s h a r p l y ( a t 0.1cpd) b e l o w t h e 95% n o i s e l e v e l . The p h a s e s t a r t s a t a v a l u e o f -165° ( w i n d l e a d i n g ) i n t h i s r e g i o n . A s h a r p t r a n s i t i o n o f t h e phase i s " o b s e r v e d (at O . l c p d ) f r o m -165° t o a b o u t 95°, and t h i s c o r r e s p o n d s t o t h e s h a r p d e c r e a s e i n t h e c o h e r e n c e f u n c t i o n . None c f t h e p e a k s a r e s t a t i s t i c a l l y s i g n i f i c a n t t c 95% c o n f i d e n c e e x c e p t f o r t h e peak a t 1 c p d . H e r e , t h e c o h e r e n c e v a l u e i s 0.38 w h i l e the phase i s 165°. The a i r - p r e s s u r e a t V a n c o u v e r i s compared v.ith t h e w i n d 86 a t Sand Heads i n F i g u r e 26c. The coherence f u n c t i o n i s seen t o be f a i r l y s y m m e t r i c a l f o r f r e q u e n c i e s c l o s e t o z e r o . The average coherence v a l u e s are 0.18 and 0.27 f o r the r e g i o n s (0,-0. 1) cpd and (0 , +-0. 1) cpd , r e s p e c t i v e l y . With i n c r e a s i n g | f | , ' t he coherence f u n c t i o n drops below the 95% n o i s e l e v e l a t +0. 1cpd; C o r r e s p o n d i n g t o thre decrease i n coherence a t ±0.1cpd, 'the- phase makes- a sharp t r a n s i t i o n ( a t ±0.15cpd) from 60°' t o -165°. I n c o n t r a s t , t h e - phase f o r n e g a t i v e f r e q u e n c i e s , i s seen to s t a r t at 0° b e f o r e - d e c r e a s i n g smoothly t o a c o n s t a n t v a l u e near -90°. T h i s b e h a v i o u r (smooth v a r i a t i o n -and sh a r p t r a n s i t i o n f o r n e g a t i v e and p o s i t i v e f r e g u e n c y , - r e s p e c t i v e l y ) of the phase f u n c t i o n i s v e r y s i m i l a r t o the phase f u n c t i o n between s e a - l e v e l and wind. The s i m i l a r i t y ' i s e x p l a i n e d by the f a c t t h a t t u e r e i s a h i g h coherence -between s e a - l e v e l and a i r - p r e s s u r e f l u c t u a t i o n s i n the r e g i o n (0,0.5)cpd. The peaks a t ±1cpd i n the coherence f u n c t i o n s a r e s t a t i s t i c a l l y 1 s i g n i f i c a n t t o 95% c o n f i d e n c e . These 'peaks' are a s s o c i a t e d w i t h the sharp peaks at ±1cpd found i n t h e power s p e c t r a of a i r - p r e s s u r e and o f wind. The peak a t -1.7cpd i s c o n s i d e r e d a s p u r i o u s one. C o n s i d e r a l i n e a r system where t a e a i r - p r e s s u r e f l u c t u a t i o n s c o n t r i b u t e to t h e s e a - l e v e l f l u c t u a t i o n s , but where the wind f l u c t u a t i o n s do not. Then:-FLSVV D P W °SP where s , p and w denote s e a - l e v e l , a i r - p r e s s u r e and wind, r e s p e c t i v e l y . The s u b s c r i p t s i n d i c a t e the coherence f u n c t i o n £7 r e s u l t i n g f r c m t h e c o m p a r i s o n o f a p a i r o f t h e s e p a r a n e t s r s . T h i s e g u a l i t y i s n e t o b s e r v e d t o h o l d , i n d i c a t i n g t h a t t h e w i n d f l u c t u a t i o n s a r e c o n t r i b u t i n g t o t h e s e a - l e v e l f l u c t u a t i o n s to seme e x t e n t . The c o m p a r i s o n o f o f e g u a t i o n (4.11) and 2 j s shown i n . F i g u r e 27. The f u n c t i o n s shown have l a r g e s c a t t e r , i n d i c a t i n g t h a t b e t t e r s t a t i s t i c s ( l o n g e r t i m e - s e r i e s ) s h o u l d be u s e d f c r mere c o n f i d e n t c o n c l u s i o n s . C o n t r i b u t i o n s by t h e wind f l u c t u a t i o n s t o t h e s e a - l e v e l f l u c t u a t i o n s a r e i n d i c a t e d by g r o s s d e v i a t i o n s b e t ween V 2 and Y 2 . c f c o u r s e , t h e c o m p a r i s o n i s o n l y v a l i d i n r e g i o n s where a s i g n i f i c a n t p o r t i o n c f t h e s e a - l e v e l f l u c t u a t i o n s a r e c a u s e d by t h e a i r - p r e s s u r e f l u c t u a t i o n s . The c o h e r e n c e f u n c t i o n b e t w e e n s e a - l e v e l and a i r - p r e s s u r e i n d i c a t e t h a t t h e s e r e g i o n s a r e ( 0 , 0 . 6 ) c p d and p o s s i b l y a n a r r o w r e g i o n n e a r l e p d . E x a m i n a t i o n o f F i g u r e 27 i n d i c a t e s t h a t t h e w i n d f l u c t u a t i o n s ( c l o c k w i s e c omponents) a r e c o n t r i b u t i n g t o t h e s e a - l e v e l f l u c t u a t i o n s i n t h e r e g i o n ( 0 , - 0 . 5 ) c p d and n o t a t -1 c p d . I n c o n t r a s t , t h e wind f l u c t u a t i o n s ( a n t i - c l c c k w i s e c o m p o n e n t s ) c o n t r i b u t e t o t h e s e a - l e v e l f l u c t u a t i o n s i n t h e r e g i o n (0, + 0. 1 ) c p d and a l s c a t +1cpd. I t i s s t r e s s e d t h a t t h e c o n f i d e n c e c f t h e s e c o n c l u s i o n s i s o n l y f a i r . jK_1ii R f ^ l a t i c n s _ B e t ween_Cur r e n t s _ a n d _ S e a - L e v e l Winds The c o h e r e n c e and phase f u n c t i o n s of t h e c u r r e n t s w i t h s e a - l e v e l , a i r - p r e s s u r e and wind a r e e x a m i n e d i n t h i s s e c t i o n . 88 FREQUENCY (CPD) Figure 27. Comparison of observed coherence (s o l i d line) between sea-l e v e l (Pt. Atkinson) and wind (Sand Heads), and the coherence between the same parameters (dashed li n e ) as calculated by equation (4.11). Observed, coherence values needed for t h i s comparison come from Figure 26. 89 These may p r o v i d e some i d e a s a b o u t t h e g e n e r a t i o n o f t h e c u r r e n t s . The l e w , d i u r n a l and s e m i - d i u r n a l f r e q u e n c i e s a r e o f t h e g r e a t e s t i n t e r e s t , s i n c e a m a j o r p o r t i o n c f t h e v a r i a n c e i n t h e c u r r e n t s i s f c u n d t h e r e . C u r r e n t s_ a n d_ S e a^ L e v e 1 The c u r r e n t s on t h e 50m p l a n e a r e com p a r e d w i t h t h e s e a - l e v e l a t F t . A t k i n s o n i n F i g u r e 28. The c n l y s t a t i s t i c a l l y s i g n i f i c a n t (95%) p e a k s f c r . t h e c o m p a r i s o n ' c f t h e e a s t e r n l o c a t i o n c u r r e n t s w i t h s e a - l e v e l , a r e t h e o n e s a t t h e t i d a l f r e q u e n c i e s n e a r o r a t ±1cpd and ±2cpd, and t h e n c n l y t h e h i g h e r o f t h e two p e a k s a t ±1cpd i s s t a t i s t i c a l l y s i g n i f i c a n t . Two p e a k s a p p e a r a t ±1cpd b e c a u s e t h e 0, c o n s t i t u e n t • a n d t h e (K,,P ( ) c o m b i n a t i o n , have been r e s o l v e d , w h i l e c n l y one peak a p p e a r a t +2cpd, w h i c h i s a s s o c i a t e d w i t h t h e (ftz , KZ,SZ, c o m b i n a t i o n . The p e a k s a t ±0.05cpd a r e not s t a t i s t i c a l l y s i g n i f i c a n t . The c o h e r e n c e b e t w e e n t h e e a s t e r n 50m l o c a t i o n c u r r e n t s and s e a - l e v e l a r e r a t h e r low i n r e g i o n s away f r c m the t i d a l f r e q u e n c i e s . The c o m p a r i s o n s o f t h e c u r r e n t s a t t h e e t h e r two 50m l o c a t i o n s g i v e s i m i l a r r e s u l t s : no s t a t i s t i c a l l y s i g n i f i c a n t (95%) p e a k s e x c e p t a t t h e t i d a l f r e q u e n c i e s , and p o c r c o h e r e n c e s ' i n r e g i o n s away f r o m t h e t i d a l f r e q u e n c i e s . E e c a u s e c f b e t t e r r e s o l u t i o n f c r t h e c e n t r a l l e c a t i o n c u r r e n t s and s e a -l e v e l , two s t a t i s t i c a l l y s i g n i f i c a n t p e a k s a p p e a r a t t h e s e m i -d i u r n a l f r e q u e n c i e s . I t i s o b s e r v e d t h a t t h e h i g h e s t l o w -90 (A) E A S T E R N ^Jj -2.5 T" T 1 -1.5 -1 -0.5 F R E Q U E N C Y (CPD) 0.9 0.Q 0.7 • 0.33 r 7 o.i o-J T ' J95% | — 9 5 % o 100 90 o -90 •-1Q0 J -1 I — 1 0.5 1 1.5 F R E Q U E N C Y (CPD) • * • • . • 2.5 -2.5 -2 -1.5 (B) C E N T R A L Hll -1 -0.5 0.95 0.9 0.8H 0.7 0.5 0.33 rr- 0.1 —I 0-1 0 . 180 90 0 -90 3 -180 =T 2 I ^ 7 -0.5 I — 1.5 -2.5 (C) W E S T E R N g ; ^ : 0.9-0.8 , 0.73~V~2 -iT" -1 -0.5 • * • • • • * 0.5 0.3 ~ 0.1-d - i o J 0 0 IGO -90-] -90-] -160-T 2 I I — 0.5 1.5 — I 2.5 Figure 28. Coherence and phase spectra between 50m currents and sea-l e v e l at Pt. Atkinson. Parts (A,B,C) have (46,52,44) degrees of freedom and are based on (469,520,432) day time-series s t a r t i n g ( A p r i l 16, A p r i l 16, July 10) 1969. 91 i i 1 — 1 -2.5 -2 -l.s -1 FREQUENCY (CPD) (A) E A S T E R N jj-jg 0.8 0.7 0.5. \ \ . 0.1 ^-1 =, 0-J -0.5 T 2 . J r o o . 100 " i 90 3 0 -90 -leo- 1 cb 95% 9 5 % — i JI * _. v —I I 1 1 "I 0.5 1 1.5 2 2.5 FREQUENCY (CPD) (B) CENTRAL 200m ° 0 H O.fi 0.7 0.53 ^ • 0.3 *"•-.. ' . - 0.1 0 -2.5 -1.5 ~1 -0.5 Y 2 I o . iuo-q 90 03 -90 -180 0.5 1.5 2.5 4> (C) WESTERN H 2 l 200m -2.5 ~I -2 -1.5 -0.5 0.93 0.8 0.7 0.5 0.3 " 0.1 1 » J r o o . I C O - i 90-03c -90' -100-CD 0.5 1.5 —I 2.5 Figure 29. Coherence and phase spectra between (140m,200m) currents and sea-level at Pt. Atkinson. Parts (A,B,C) have (38,42/38) degrees of freedom and are based on (375,417,375) day time-serie s s t a r t i n g ( A p r i l 16, A p r i l 16, J u l y 10) 1969. 92 f r e g u e n c y (±0.05cpd) c o h e r e n c e s a r e o b t a i n e d when c o m p a r i n g t h e c u r r e n t s a t t h e 50m e a s t e r n l o c a t i o n w i t h s e a - l e v e l . Some c o n s i s t e n c y i s o b s e r v e d i n t h e phase f u n c t i o n s f o r t h e 50m c u r r e n t s and s e a - l e v e l . The p h a s e d e c r e a s e s s m o o t h l y f r o m - 0 . 0 2 c p d t o - 0 . 2 c p d . T h i s smooth d e c r e a s e i s a l s o o b s e r v e d i n t h e r e g i o n ( 0 . 0 2 , 0 . 2 ) c p d i n one o f t h e c o m p a r i s o n s . A l t h o u g h t h e t h r e e p h a s e f u n c t i o n s have l a r g e s c a t t e r , some t e n d e n c i e s i n t h e phase a r e o b s e r v e d . The c l o c K w i s e r o t a t i n g c u r r e n t f l u c t u a t i o n s ( t h a t p o r t i o n w h i c h i s c o h e r e n t w i t h s e a -l e v e l ) h a v e a t e n d e n c y t o l a g t h e s e a - l e v e l f l u c t u a t i o n s by 60°, w h i l e t h e a n t i - c l o c k w i s e r o t a t i n g f l u c t u a t i o n s have a t e n d e n c y - t o l e a d t h e s e a - l e v e l f l u c t u a t i o n s by 150°.- I t i s s e e n t h a t t h e s e t e n d e n c i e s a r e o b e y e d a t t h e t i d a l f r e q u e n c i e s where t h e c o h e r e n c e s a r e h i g h e s t , and t h e c o n f i d e n c e i n t e r v a l s on t h e ph a s e e s t i m a t e s a r e s m a l l e s t . The c u r r e n t s a t ' t h e t h r e e d e e p e r l o c a t i o n s a r e compared w i t h t h e s e a - l e v e l a t P t . A t k i n s o n i n F i g u r e 29. G e n e r a l l y , t h e r e s u l t s o f t h e s e t h r e e c o m p a r i s o n s a r e s i m i l a r i n many ways t o t h e t h r e e c o m p a r i s o n s f o r m e r l y d i s c u s s e d . I t i s seen- t h a t t h e o n l y s t a t i s t i c a l l y s i g n i f i c a n t (95%) p e a k s a r e a t t h e t i d a l f r e q u e n c i e s and t h a t away f r o m t h e s e f r e q u e n c i e s , t h e c o h e r e n c e v a l u e s a r e p o o r . The h i g h e s t l o w - f r e q u e n c y (±0.05cpd) c o h e r e n c e s a r e o b t a i n e d when c o m p a r i n g t h e c u r r e n t s a t t h e two s i d e l o c a t i o n s w i t h s e a - l e v e l . The p h a s e f u n c t i o n s i n t h e l o w - f r e q u e n c y r e g i o n ( If|<0.Ucpd) a r e o b s e r v e d t o d e c r e a s e s m o o t h l y . T e n d e n c i e s i n t h e p h a s e f u n c t i o n s a r e a g a i n o b s e r v e d . I n t w o o f t h e c o m p a r i s o n s , t h e c l o c k w i s e - r o t a t i n g deep c u r r e n t f l u c t u a t i o n s ( t h a t p o r t i o n w h i c h i s c o h e r e n t w i t h s e a - l e v e l f l u c t u a t i o n s ) have a t e n d e n c y t c l a g the. s e a - l e v e l f l u c t u a t i o n s by a b o u t 45°, w h i l e t h e a n t i - c l o c k w i s e deep f l u c t u a t i o n s have a t e n d e n c y t c l e a d t h e s e a - l e v e l f l u c t u a t i o n s by a b o u t 140°. The r e m a i n i n g phase f u n c t i o n ( w e s t e r n 200m l o c a t i o n c u r r e n t , s e a - l e v e l ) i s d i s s i m i l a r t o t h e f o r m e r two phase f u n c t i o n s . The c l o c k w i s e -r o t a t i n g c u r r e n t f l u c t u a t i o n s a p p e a r t c l a g t h e s e a - l e v e l f l u c t u a t i o n s by g r e a t e r and g r e a t e r amounts as t h e f r e g u e n c y d e c r e a s e s t c more n e g a t i v e v a l u e s . The a n t i - c l o c k w i s e c u r r e n t f l u c t u a t i o n s have a t e n d e n c y t o be 180° c u t c f phase w i t h t h e s e a - l e v e l f l u c t u a t i o n s . C u r r e n t s _ a n d _ A i r ; P r e s s u r e The c u r r e n t s a t t h e t h r e e 50m l o c a t i o n s a r e c o m p a r e d w i t h t h e a i r - p r e s s u r e a t V a n c o u v e r i n F i g u r e 30. The c o h e r e n c e s between t h e c u r r e n t and a i r - p r e s s u r e f l u c t u a t i o n s a r e s e e n t c be p o o r o v e r much o f t h e f r e q u e n c y r a n g e . The c n l y s t a t i s t i c a l l y s i g n i f i c a n t (95%) peaks a r e a l l a t +2cpd ( i n one c o m p a r i s o n t h e peak a t -2cpd i s a b s e n t ) . I n a l l t h r e e c o m p a r i s o n s , l a r g e r c o h e r e n c e v a l u e s a r e o b s e r v e d a t 2cpd t h a n a t . - 2 c p d . The peaks, a t +1cpd a r e not s t a t i s t i c a l l y s i g n i f i c a n t . The h i g h e s t d i u r n a l p e a k s a r e f o u n d w i t h t h e c e n t r a l 50m l o c a t i o n c u r r e n t and a i r - p r e s s u r e . A l l t h r e e phase f u n c t i o n s a r e o b s e r v e d t c have l a r g e s c a t t e r and no t e n d e n c i e s . The c u r r e n t s a t t h e t h r e e d e e p e r l o c a t i o n s a r e c o m p a r e d 94 (A) E A S T E R N -2.5 ~r ~ i — -2 -1.5 -1 FREQUENCY (CPD) — 1 — -0.5 0.97 0.95 0.9 0.0 0.7 0.5d 0.3 - 0.1 r 0 0 I B O T 90-1 0--90 -180-1 J 9 5 % 9 5 o / o « . « I  ~ i 1 1 — — — 1 0.5 1 1.5 2 2.5 FREQUENCY (CPD) (B) C E N T R A L -2.5 -1.5 -0.5 0.97 0.95 0.9-| 0.8 0.7 0.5 0.3 <•• 0.1 -1 ° J 180 90-) 0 -90 -100 J 3 V 0.5 1.5 — I 2.5 -2.5 (C) W E S T E R N ~ l — -1.5 -0.5 0.97 0.95-1 0.9-0.0 0.7 0.5 0.3 — 0.1-4 ^ 0 0 180 90 0 -90 *-100 J -~r~ 0.5 ~H— 1.5 — I 2.5 Figure 30. Coherence and phase spectra between 50m currents and a i r -pressure at Vancouver. Parts (A,B,C) have (46,52,44) degrees of freedom and are based on (469,520,432) day time-series s t a r t i n g ( A p r i l 16, A p r i l 16, Ju l y 10) 1969. 95 (A) EASTERN 140m I n 1 1 -2.5 -2 -1.5- -1 FREQUENCY (CPD) -0.5 0.97-) 0.95-0.9-0.8-1 0,7 0.5 . 0.3 ~ 0.1 "I 0-> 180 90 , 0 -90 -180 -1 0 T.95% I—95% i I 1— « T -0.5 1 .1.5 2 FREQUENCY (CPD) — I 2.5 (B) CENTRAL 200m « * -2.5 -1.5 -1 0.97-1 0.95 0.9H 0.8-0.7-0.5-0.33 — 0.1 H 1 o T 2 I -0.5 180 90 0 -SO -180 J 0 . 0.5 i 1.5 I 2.5 (0 WESTERN 200m -2.5 0.97 0.95 0.9 0.8 0.7 0.5 0.3=1 i 1 1 = - ! - r , 1 o J -2 -1.5 -1 -0.5 0 . • mo *• . .* . . • • O H o . * -90 -160 0.5 ~ 1 — 1.5 2.5 Figure 31. Coherence and phase spectra between (140m,200m) currents and air-pressure at Vancouver. Parts (A,B,C) have (38,42,38) degrees of freedom and are based on (375,417,375) day time-serie s s t a r t i n g (April 16, A p r i l 16, Ju l y 10) 1969. 96 w i t h t h e a i r - p r e s s u r e a t V a n c o u v e r i n F i g u r e 3 1. ' In t h e c o m p a r i s o n u s i n g t h e c u r r e n t s a t t h e w e s t e r n 200m. l o c a t i o n , s t a t i s t i c a l l y s i g n i f i c a n t p e a k s a r e f o u n d a t ±2cpd.. U s i n g t h e c u r r e n t s a t t h e c e n t r a l and e a s t e r n l o c a t i o n s , s t a t i s t i c a l l y s i g n i f i c a n t p e a k s a r e f o u n d o n l y at +2cpd . and - 2 c p d , r e s p e c t i v e l y . No e t h e r s t a t i s t i c a l l y s i g n i f i c a n t p e a k s a r e f c u n d and t h e c o h e r e n c e s a r e b e l o w t h e 95% n o i s e l e v e l o v e r much o f t h e f r e g u e n c y r a n g e . .A l a r g e amount c f s c a t t e r i s a g a i n o b s e r v e d i n a l l t h r e e phase f u n c t i o n s . C u r r ent_s_ a nd_ Wind The c u r r e n t s a t t h e t h r e e 50m l o c a t i o n s a r e c o m p a r e d w i t h t h e w i n d s a t Sand Heads i n F i g u r e 32. W i t h one e x c e p t i o n , t h e r e a r e no s t a t i s t i c a l l y s i g n i f i c a n t f e a t u r e s i n t h e c o h e r e n c e f u n c t i o n s w h i c h shew p e e r c o h e r e n c e s e v e r y w h e r e . The h i g h e s t l e w - f r e g u e n c y (|f)<0.3cpd) c o h e r e n c e s a r e o b t a i n e d u s i n g t h e c u r r e n t s a t t h e e a s t e r n 50m l o c a t i o n . However t h e h i g h v a l u e s i n t h e r e g i o n ( 0 , - 0 . 2) c p d a r e n o t s t a t i s t i c a l l y s i g n i f i c a n t . The peak a t + 0 . I c p d i s a l s o n o t s t a t i s t i c a l l y s i g n i f i c a n t . What i s s t a t i s t i c a l l y s i g n i f i c a n t (95%) i s t h e smooth d e c r e a s e i n t h e c o h e r e n c e , f u n c t i o n as t h e f r e g u e n c y i n c r e a s e s f r c m z e r c t o mere p o s i t i v e v a l u e s . I t i s a l s o o b s e r v e d t h a t h i g h e r c o h e r e n c e s a r e f o u n d a t +1cpd t h a n a t -1 c p d . A l a r g e amcunt o f s c a t t e r ' i s o b s e r v e d i n t h e t h r e e p h a s e f u n c t i o n s . S i m i l a r r e s u l t s a r e o b t a i n e d when c o m p a r i n g t h e 97 (A) E A S T E R N i 1 1 1 — -2.5 -2 -1.5 -1 -0.5 FREQUENCY (CPD) • • • 0.97-1 0.9S-0.9-0.8-0.7 = 0.5-0.3-r o.i-1 o-J Y 2 -I 95% 95% r o o 180 -, 90 H 0 -90 -180 4> , f ] 0.5 1 1.5 FREQUENCY (CPD) • • • . • , —1 2.5 (B) C E N T R A L -2.5 n — -1.5 -1 • -0.5 0.97-, 0.95-0.9-0.8-o.7H>r2 0.5 0.3--i o-J p o o 180-90-• 0-1/ Y< I -90 -180 -> CD 0.5 1.5 — I 2.5 (C) WESTERN 0.97-, 0.95 0.9-0.8-0.7 = o.5 = 0.3 -2.5 -2 -] -1.5 • —" —77~ 0.1-d Y 2 I -0.5 1 0 100 90 0 -90 3 -180-J 0.5 1.5 1 2.5 Figure 32. Coherence and phase spectra between 50m currents and wind at Sand Heads. Parts (A,B,C) have (46,52,44) degrees of freedom and are. based on (469,520,432) day time-series s t a r t i n g ( A p r i l 16, A p r i l 16, Ju l y 10) 1969. 98 (A) EASTERN ^ J . 140m -2.5 -f- T -2 -1.5 -1 -0.5 FREQUENCY (CPD) O.'JS 0.0-o.a-" 0.7-0.53 • 0.3d '7 o.i--1 o-0 100-90-• 0-1 -30--100-y 2 J95% -rr——j— *w * 95% , r o.s 1 1.5 2 FREQUENCY (CPD) —1 2.5 r — r -2.S (B) CENTRAL 200m -1.5 -o.s Vf2 I 0.97-, 0.95-1 0.9 0.8 . O. I-IY2 0.5 0.3 ^ 0 . 1 3 -1 o-l r 0 0 180-90-1 ^ — 0.5 0--90--180 cb 1.5 1 2.5 (C) WESTERN 200m -2.5 -2 -1.5 -0.5 0.97-, 0.95-0.9-0.8-0.7- -v-2 0.5- I 0 .3-~0.X-- i ° J r-0 0 180-0--90--180-cb - 1 — 0.5 1.5 ~1 2.5 Figure 33. Coherence and phase spectra between (140m,200m) currents and wind at Sand Heads. Parts (A,B,C) have (38,42,38) degrees of freedom and are based on (375,417,375) day time-series s t a r t i n g ( A p r i l .16, A p r i l 16, Ju l y 10) 1969. 99 c u r r e n t s a t t h e t h r e e d e e p e r l o c a t i o n s w i t h t h e wind a t Sand Heads ( f i g u r e 3 3 ) . The h i g h e s t l o w - f r e q u e n c y (|f|<C.3cpd) c o h e r e n c e s a r e f c u n d u s i n g t h e c u r r e n t s a t t h e e a s t e r n 140m l o c a t i o n . I n t h i s c o m p a r i s o n , a s t a t i s t i c a l l y s i g n i f i c a n t (95%) f e a t u r e c f t h e c o h e r e n c e f u n c t i o n i s t h e s m c c t h d e c r e a s e f o u n d i n t h e r e g i c n ( 0 , 0 . 3 ) c p d . The h i g h . c o h e r e n c e s ( u s i n g t h e c u r r e n t s a t t h e e a s t e r n 140m l o c a t i o n ) i n t h e r e g i c n (0,-0.3) cpd a r e not. s t a t i s t i c a l l y s i g n i f i c a n t , -but t h e peak at - 0 . 9 c p d i s s t a t i s t i c a l l y s i g n i f i c a n t . E v e r y w h e r e e l s e , t h e c o h e r e n c e s a r e p o o r f c r t h e e a s t e r n l o c a t i o n c u r r e n t s and w i n d s . No s t a t i s t i c a l l y s i g n i f i c a n t f e a t u r e s i n t h e c o h e r e n c e f u n c t i o n s a r e f c u n d u s i n g t h e c u r r e n t s a t t h e deep c e n t r a l and w e s t e r n l o c a t i o n s . The c o h e r e n c e f u n c t i o n s g e n e r a l l y l i e b e l c w t h e 9 5 % n o i s e l e v e l . R e l a t i v e l y h i g h c o h e r e n c e v a l u e s {%f >0.25) a r e f o u n d n e a r - 0 . 3 5 c p d when t h e c u r r e n t s a t t h e w e s t e r n 200m l o c a t i o n a r e compared w i t h t h e wind a t Sand Heads. The t h r e e phase f u n c t i o n s f o r t h e deep c u r r e n t s and wind a g a i n show a l a r g e d e g r e e o f s c a t t e r . However, two c f t h e t h r e e p h a s e f u n c t i o n s ( t h e two o b t a i n e d w i t h t h e s i d e l o c a t i o n c u r r e n t s ) show c o n s i s t e n t b e h a v i o u r i n t h e l e w - f r e q u e n c y r e g i c n (0,±0.3) c p d . In t h i s r e g i o n , the. two phase f u n c t i o n s show a smooth d e c r e a s e w i t h i n c r e a s i n g | f | . 4_._15 Summary The l o w - f r e g u e n c y (C,0.25cpd) and t i d a l p o r t i o n s o f t h e 50m and 140m c u r r e n t s a r e a p p r o x i m a t e l y e q u a l . T o g e t h e r t h e y 100 a c c o u n t f o r 80% o f t h e t o t a l e n e r g y i n t h e s e c u r r e n t s . The l e w - f r e q u e n c y and t i d a l p o r t i o n s c o n t r i b u t e a b o u t 10% and 7G%, r e s p e c t i v e l y , t c t h e t c t a l e n e r g y o f t h e 200m c u r r e n t s . The c o h e r e n c e f u n c t i o n s r e v e a l t h a t t h e h o r i z o n t a l and v e r t i c a l r e s o l u t i o n s o f t h e l e w - f r e q u e n c y c u r r e n t s a r e r e l a t i v e l y p o o r , w h i l e t h a t o f t h e t i d a l c u r r e n t s a r e r e l a t i v e l y g c o d . A l a r g e p o r t i o n o f t h e t e i r p e r a t u r e v a r i a n c e i s f o u n d i n t h e l o w - f r e q u e n c y r e g i o n ( C , 0 . 0 3 ) c p d . T y p i c a l l y 1% c f t h e t e m p e r a t u r e v a r i a n c e i s f o u n d at f r e q u e n c i e s above Ucpd. The t e m p e r a t u r e and c u r r e n t s p e c t r a a r e q u i t e d i s s i m i l a r . The c o h e r e n c e f u n c t i o n s between t e m p e r a t u r e , t o g e t h e r w i t h t h e t e m p e r a t u r e s p e c t r a , s u q q e s t t h a t t h e , t e m p e r a t u r e • t i m e - s e r i e s a r e t o e s h o r t t o be c o n s i d e r e d s t a t i o n a r y , b e c a u s e t h e y c o n t a i n f l u c t u a t i o n s h a v i n g p e r i o d s on t h e o r d e r c f t h e t i m e - s e r i e s l e n g t h . ' The t i d a l o s c i l l a t i o n s c o n t r i b u t e 97% t c t h e t c t a l v a r i a n c e c f t h e s e a - l e v e l . The. r e m a i n i n g p o r t i o n i s f c u n d i n t h e r e g i o n (0,0.9) c p d . The l o w - f r e q u e n c y (0,0.25cpd) f l u c t u a t i o n s c o n t r i b u t e 85% t o t h e t o t a l v a r i a n c e c f a i r -p r e s s u r e . I t i s f c u n d t h a t t h e l e w - f r e q u e n c y s e a - l e v e l and a i r - p r e s s u r e f l u c t u a t i o n s a r e h i g h l y c o h e r e n t . The wind e n e r g y i s f o u n d i n a r e l a t i v e l y w i d e f r e g u e n c y r e g i c n . O n l y a b c u t 50% c f t h e w i n d e n e r g y i s f c u n d a t f r e q u e n c i e s b e l o w 0 .25cpd. I t i s f o u n d t h a t t h e wind f l u c t u a t i o n s do c o n t r i b u t e t c t h e l e w -f r e q u e n c y s e a - l e v e l f l u c t u a t i o n s . The t i d a l c u r r e n t s a r e f o u n d t o be h i g h l y c o h e r e n t w i t h t h e t i d a l o s c i l l a t i o n s i n t h e s e a - l e v e l . The 2cpd c u r r e n t and 1C1 a i r - p r e s s u r e f l u c t u a t i o n s have f a i r c o h e r e n c e s . P e e r c o h e r e n c e s a r e f c u n d between t h e l e w - f r e q u e n c y c u r r e n t s and t h e l e w - f r e q u s n c y f l u c t u a t i o n s i n s e a - l e v e l and a i r - p r e s s u r e . F c c r - t o - f a i r c o h e r e n c e s a r e o b s e r v e d between t h e l e w - f r e q u e n c y f l u c t u a t i o n s i n t h e . c u r r e n t s and w i n d . The h i g h e s t l o w -f r e q u e n c y c o h e r e n c e s a r e o b s e r v e d wherr c o m p a r i n g t h e e a s t e r n l o c a t i o n c u r r e n t s w i t h t h e w i n d . T h i s s u g g e s t s t h a t an i n v e s t i g a t i o n on t h e g e n e r a t i o n c f . c u r r e n t s a t t h e e a s t e r n l o c a t i o n by t h e w i n d may be f r u i t f u l . 102 C h a p t e r 5 • LOW-FREQUENCY FLUCTUATIONS I n t h i s c h a p t e r , t h e l o w - f r e q u e n c y f l u c t u a t i o n s a r e e x a m i n e d i n more d e t a i l . . The (u,v) v e c t o r c o m p o n e n t s a r e t a k e n t o he p o s i t i v e i n t h e t r u e ( e a s t , n o r t h ) d i r e c t i o n s . The d i r e c t i o n s f e l l o w t h e c o n v e n t i o n , 0° f o r t r u e n o r t h , 90° f c r t r u e e a s t , e t c . The mean s p e e d s and d i r e c t i o n s , u n l e s s o t h e r w i s e s t a t e d , a r e computed f r o m t h e means, o f t h e v e c t o r c o m p o n e n t s . Thus t h e s e mean s p e e d s r e p r e s e n t t h e m a g n i t u d e s c f t h e v e c t o r 'mean' v e l o c i t i e s . F c r t h e o v e r a l l a v e r a g e , they, a r e t h e m a g n i t u d e o f .the v e c t o r mean. I n t h e c a s e where smo o t h e d c r f i l t e r e d c o m p o n e n t s a r e u s e d , t h e y r e p r e s e n t t h e m a g n i t u d e o f what i s e s s e n t i a l l y a r u n n i n g v e c t o r mean. The mean d i r e c t i o n s a r e t h e r e s u l t a n t d i r e c t i o n s c f t h e s e v e c t o r -a v e r a g e d d a t a . 5_. _1 ^L2±'Z i i i S i i U S The l o w - f r e q u e n c y f l u c t u a t i o n s a r e e x a m i n e d i n t h e t i m e d o m a i n . Tc t h i s e n d , t h e d a t a a r e l o w - p a s s e d u s i n g t h e f i l t e r f u n c t i o n , R ( f ) , shewn i n F i g u r e 3 4 a . T h i s f u n c t i o n has a s l o p i n g s i d e . P r o b l e m s a r i s e w i t h , a f i l t e r f u n c t i o n h a v i n g a much s t e e p e r s i d e . A t i m e ^ s e r i e s , l o w - p a s s e d w i t h such a s t e e p - s i d e d f i l t e r f u n c t i o n , w o u l d e x h i b i t u n n a t u r a l Figure 34. F i l t e r function employed to obtain the low-passed d a i l y time-seri e s (A), and that corresponding to a running d a i l y mean (B). 104 o s c i l l a t l e n s c r ' r i n g i n g ' . These o s c i l l a t i o n s i n t h e l o w -p a s s e d t i m e - s e r i e s a r e t h e . s t r o n g e s t i n t h e v i c i n i t y c f f a s t c h a n g e s o r l a r g e 'jumps'. ' W i t h t h e f i l t e r f u n c t i o n , H ( f ) , t h i s r i n g i n g i s e s s e n t i a l l y a b s e n t i n the r e s u l t i n g l o w - p a s s e d t i m e -s e r i e s . The l e w - p a s s f i l t e r i n g o f a t i m e - s e r i e s i n v o l v e s f o u r s t e p s . The t i m e - s e r i e s i s F o u r i e r t r a n s f o r m e d i n t o t h e f r e q u e n c y d o m a i n . Then t h e F o u r i e r c o e f f i c i e n t s a r e m u l t i p l i e d by H ( f ) . B o t h t h e F o u r i e r c o s i n e , and s i n e c o e f f i c i e n t s a r e m u l t i p l i e d by t h e same f a c t o r . Hence the' p h a s e s o f t h e r e s u l t i n g l e w - p a s s e d t i m e - s e r i e s r e m a i n u n c h a n g e d , r e l a t i v e t o t h e s e i n t h e o r i g i n a l t i m e - s e r i e s . T h i r d l y , t h e c o e f f i c i e n t s a r e F o u r i e r t r a n s f o r m e d b a c k i n t o the t i m e d o m a i n . F i n a l l y , t h e t i m e - s e r i e s i s r e s a m p l e d a t t h e s t a r t o f e a c h day. A v e c t o r t i m e - s e r i e s i s l o w - p a s s f i l t e r e d by p e r f o r m i n g .the o p e r a t i o n cn t h e |u,v) c o m p o n e n t s i n d i v i d u a l l y . The, l a s t s t e p above p r o d u c e s a d a i l y t i m e - s e r i e s from, a h a l f - h o u r l y t i m e - s e r i e s . A l i a s i n g w o u l d o c c u r a t t h i s s t a g e i f t h e h a l f - h o u r l y t i m e - s e r i e s c o n t a i n f l u c t u a t i o n s h a v i n g f r e q u e n c i e s g r e a t e r t h a n t h e N y q u i s t f r e g u e n c y (O.Scpd) c f the d a i l y t i m e - s e r i e s . The f i l t e r f u n c t i o n , H ( f ) , i s c h o s e n t o have a c u t - o f f f r e q u e n c y c f 0.44cpd. Thus no a l i a s i n g o c c u r s b e c a u s e f l u c t u a t i o n s h a v i n g f r e q u e n c i e s g r e a t e r t h a n G.5cpd, a r e c o m p l e t e l y removed f r o m t h e h a l f - h o u r l y t i m e - s e r i e s b e f o r e t h e y a r e r e s a m p l e d a t t h e s t a r t c f e a c h day. The l o w - p a s s e d d a i l y t i m e - s e r i e s c f c u r r e n t s , w a t e r t e m p e r a t u r e , w i n d , a i r - p r e s s u r e and s e a - l e v e l , a s d e r i v e d u s i n g 1G5 t h e f i l t e r f u n c t i o n c f F i g u r e 3 4 a , a r e t h e . s u b j e c t o f t h i s c h a p t e r . A i l . r e s u l t s p r e s e n t e d i n t h i s c h a p t e r use t h e s e l o w -p a s s e d d a i l y t i m e - s e r i e s as t h e s t a r t i n g p o i n t c f t h e a n a l y s e s . The l o w - p a s s e d f i l t e r i n g d e s c r i b e d above i s scmewhat s i m i l a r t o t h e method o f d a i l y a v e r a g e s , but t h e f o r m e r has b e t t e r c h a r a c t e r i s t i c s . The l a t t e r method c o n s i s t s o f a v e r a g i n g 48 p o i n t s o f t h e h a l f - h o u r l y t i m e - s e r i e s t c o b t a i n a s i n g l e p o i n t o f t h e d a i l y t i m e - s e r i e s . The f i l t e r f u n c t i o n , H, (f) f.cr t h e method o f d a i l y a v e r a g e s i s : -H, (f ) - s i n ( 4 8 l i f A t ) / {48»sin (iff At) } ( 5 . 1) where a t - ( 1 / 4 8 ) d a y - 1 ( G c d i n , 1 972 , page 62) , and i t s g r a p h a p p e a r s i n F i g u r e 34b. As can be s e e n , t h e c h a r a c t e r i s t i c s o f t h i s f i l t e r {K, ) a r e n o t as good as t h o s e c f the cr.e u s e d . The c u t - o f f i n t h e r a n g e 0-0.5cpd i s not as s h a r p , and a l i a s i n g f r c m f r e q u e n c i e s 0.5cpd t o 24cpd c a n c c c u r b e c a u s e t h e s e - f r e q u e n c i e s a r e n o t c o m p l e t e l y s u p p r e s s e d . I n p a r t i c u l a r , t h e 0, , N z and M z t i d e s a r e n o t w e l l r e m o v e d , and a s i g n i f i c a n t p o r t i o n o f t h e s e t i d e s w ould r e m a i n as a l i a s e s i n t h e d a i l y t i m e - s e r i e s . Shewn i n F i g u r e 35 and F i g u r e 36 a r e t h e l e w - f r e q u e n c y c u r r e n t component f l u c t u a t i o n s , as s e e n t h r o u g h t h e f r e q u e n c y window c f F i g u r e 34a. These l e w - f r e q u e n c y f l u c t u a t i o n s a r e a p e r i o d i c i n c h a r a c t e r . T h i s i s t c be e x p e c t e d s i n c e t h e i r CURRENTS ^p^W^ ^ , ^ # ( ^ ^ 4 ^ , — ^ ^ ^ ^ ^ / u ^ ^ ^ -EASTERN 50m 20- E-W COMPONENT 10 WESTERN 50m V Figure 35. Low-passed daily time-series of the 50m currents. CURRENTS EASTERN 140m "T T -. ^ p ^ r yyww \ M M r CENTRAL 200m ~T 1 2oj E-W COMPONENT WESTERN 200m 1969 1970 s o A ' M ' J r~~J ' A ' S T ) ' N~' D ' J ' F ' M-"* A ' M ' J ""J ' A ^ N-S COMPONENT gure 36. Low-passed d a i l y time-series of the 140m and 200m currents. 108 power s p e c t r a shew them t c h a v e b r e a d - b a n d v a r i a n c e . I t i s see n t h a t t h e c u r r e n t s a t one l o c a t i o n do n o t c l o s e l y r e s e m b l e t h o s e a t any e t h e r l o c a t i o n . T h i s i s n e t s u r p r i s i n g s i n c e t h e c o h e r e n c e f u n c t i o n s b e t ween c u r r e n t s y i e l d a l o w v a l u e when t h e y a r e a v e r a g e d o v e r t h e r e g i o n ( 0 , 0 . 4 ) c p d . The l e w - f r e q u e n c y f l u c t u a t i o n s o f t h e w a t e r t e m p e r a t u r e a r e e x h i b i t e d i n F i g u r e 37. P o o r r e s e m b l e n c e i s o b s e r v e d b e t ween any t i m e - s e r i e s p a i r . The t e m p e r a t u r e f l u c t u a t i o n s a t t h e two '200m l o c a t i o n s a r e much s t i a l l e r t h a n t h o s e a t t h e o t h e r l o c a t i o n s . The t e m p e r a t u r e and c u r r e n t f l u c t u a t i o n s a t a l o c a t i o n a r e d i s s i m i l a r . I t i s seen t h a t t h e f l u c t u a t i o n s , h a v i n g t i m e s c a l e s o f a week t o a few m o n t h s , d o m i n a t e t h e c u r r e n t t i m e - s e r i e s t u t n o t t h e t e m p e r a t u r e t i m e - s e r i e s . I n s p e c t i o n shews t h a t t h e f l u c t u a t i o n s i n t h e t e m p e r a t u r e t i m e -s e r i e s a r e c o n t a i n e d i n a l o w e r f r e q u e n c y r e g i o n t h a n t h o s e i n t h e c u r r e n t t i m e - s e r i e s , a r e s u l t c o n f i r m e d by t h e i r power s p e c t r a . The t e m p e r a t u r e t i m e - s e r i e s a r e d o m i n a t e d by f l u c t u a t i o n s h a v i n g s u c h lew f r e q u e n c i e s t h a t t h e y a p p e a r as t r e n d s i n F i g u r e 37. The l e w - f r e q u e n c y a i r - p r e s s u r e and s e a - l e v e l f l u c t u a t i o n s , as seen t h r o u g h t h e f r e q u e n c y windew c f F i g u r e 34a, a p p e a r i n F i g u r e 38. The a i r - p r e s s u r e f l u c t u a t i o n s a t one l o c a t i o n a r e v e r y s i m i l a r t o t h o s e a t t h e o t h e r l o c a t i o n s . . T h i s i s a l s o t r u e f o r t h e s e a - l e v e l f l u c t u a t i o n s . I t i s se e n t h a t t h e s e a - l e v e l f l u c t u a t i o n s a r e n e t d o m i n a t e d by t h e f o r t n i g h t l y and m o n t h l y t i d a l o s c i l l a t i o n s . The a i r - p r e s s u r e and s e a - l e v e l f l u c t u a t i o n s a r e t h e l a r g e s t d u r i n g f a l l and 109 WATER TEMPERATURE , APR , MAY , JUN , JUL , AUG , S E P , OCT , NOV , DEC , JAN , FES , 1,'AR , APR , MAY , JUN , JUL , AUG , S EP , 1969 1970 Figure 37. Low-passed daily time-series of water temperature. 10 AIR PRESSURE AT SEA LEVEL 5 MILLIBAR/DIV L V A N C O U V E R PORT H A R D Y J w 1 APR ' M A Y 1 J U N ' J U L ' A U G ' S E P ' O C T ' NOV ' OEC ' J A N ' FEB ' MAR ' APR ' MAY ' JUN ' JUL ' A U G ' S E P 1969 1970 SEA LEVEL 10 CM/DIV 160-f PT. ATKINSON 1 — 1— Figure 38. Low-passed daily time-series of air-pressure and sea-level. WINDS O -,0r SAND HEADS E - W C O M P O N E N T ENTRANCE ISLAND N-S C O M P O N E N T SAND H E A D S -I wL 12r E N T R A N C E ISLAND 9 iii 6 I/) 3 0 Af'H MAY JUN JUL AUG SET' OCT NOV DEC JAN T FED MAft APR f/AY ^ JUN ' JUL ^ AUG SEP 1969 1970 Figure 39. Low-passed daily time-series of wind. FRASER RIVER DISCHARGE AT AGASSIZ 113 w i n t e r . At any l o c a t i o n , the. a i r - p r e s s u r e and s e a - l e v e l f l u c t u a t i o n s a r e v e r y s i m i l a r and 180° c u t c f p h a s e , t h e i n v e r s e b a r o m e t e r e f f e c t . Cr t h e open s e a , a r i s e c f 1mb i n a i r - p r e s s u r e d e p r e s s e s t h e s e a - l e v e l by 1cm t h r o u g h t h e i n v e r s e b a r o m e t e r e f f e c t ( w u n s c h , 1 9 7 2 ) . However, i n t h e v i c i n i t y c f V a n c o u v e r I s l a n d a r i s e o f 1nb i n a i r - p r e s s u r e d e p r e s s e s t h e . s e a - l e v e l by a b o u t 2cm, i n d i c a t i n g a m p l i f i c a t i o n c f t h e i n v e r s e b a r o m e t e r e f f e c t . F i n a l l y , t o a f a i r a p p r o x i m a t i o n , t h e a i r -p r e s s u r e a c t s as a r i g i d l i d on t h e w a t e r s a r o u n d V a n c o u v e r I s l a n d . Shewn i n F i g u r e 39 a r e t h e l e w - f r e g u e n c y w i n d f l u c t u a t i o n s , as seen t h r o u g h t h e f r e g u e n c y window c f F i g u r e 3 4 a . , I n s p e c t i o n o f T a b l e I I I shows t h a t t h e l e w - f r e q u e n c y f l u c t u a t i o n s e x h i b i t e d i n F i g u r e 39 r e p r e s e n t s a b o u t 50% c f t h e w i n d v a r i a n c e . The wind f l u c t u a t i o n s a r e , a p e r i o d i c i n c h a r a c t e r . The l e w - f r e g u e n c y f l u c t u a t i o n s a t Sand Heads and E n t r a n c e I s l a n d show a f a i r - t o - g c e d r e s e i u b l e r . e s . The F r a s e r R i v e r d i s c h a r g e ( a t A g a s s i z , ' E r i t i s h C o l u m b i a ) i s p r e s e n t e d i n E i g u r e 4 0. T h i s t i m e - s e r i e s i s o b t a i n e d i n t h e form o f d a i l y a v e r a g e s , so t h a t f u r t h e r l e w -p a s s f i l t e r i n g i s u n n e c e s s a r y . I n b o t h y e a r s shown, t h e f r e s h e t o c c u r r e d d u r i n g Way and J u n e . 5^ .3 U ~ ^ _ S c a t f e r _ D i a a r ams The U-V s c a t t e r d i a g r a m s c f t h e c u r r e n t s a t each o f the s i x l o c a t i o n s a r e shewn i n F i g u r e 4 1 . I n t h e s e d i a g r a m s , e a c h 50m 50m .;-'"V«!^": C **. " • " v"i" 3 ^ 50m WESTERN I 1 iO CM/SEC •.•<$&-: 200 m CENTRAL -*„l?-*r\:-N 200m EASTERN < * 3 \ K'"" '40m Figure 41. U-V scatter diagrams of the low-passed currents. • 115 c u r r e n t o b s e r v a t i o n (u,v) c f t h e d a i l y t i m e - s e r i e s i s p l o t t e d as a p o i n t cn t h e u-v p l a n e . The p o i n t s i n s e r t e d i n t h e g a p s c f t h e t i m e - s e r i e s a r e n o t p l o t t e d . F o r each s c a t t e r d i a g r a m , t h e r e i s an a s s o c i a t e d c o r r e l a t i o n c o e f f i c i e n t , c , d e f i n e d a s : -c= (u-U) (v-V) / V v a r (u) »var (v)' (5.2) where (U,V) a r e t h e means o f t h e d a i l y (u,v) component t i m e -s e r i e s , where v a r («) d e n o t e s t h e v a r i a n c e a b o u t t h e mean o f t h e i n d i c a t e d p a r a m e t e r , and where t h e c v e r b a r i n d i c a t e s a t i m e a v e r a g e . P o i n t s i n s e r t e d i n t h e gaps o f t h e t i m e - s e r i e s a r e n e t used i n t h e c a l c u l a t i o n o f t h e v a r i a n c e and t i i r e a v e r a g e s . The c o e f f i c i e n t , c , has a r a n g e (-1,1). I t h a s a v a l u e o f +1 when a l l t h e p o i n t s cn t h e u-v p l a n e f a l l cn a s t r a i g h t l i n e h a v i n g a (±)slope. The c o e f f i c i e n t s (c) f c r t h e U-V s c a t t e r d i a g r a m s a r e l i s t e d i n T a b l e . I V . I t i s s e e n t h a t t h e c u r r e n t m o t i o n s a r e a l i g n e d i n a SS d i r e c t i o n a t t h e two e a s t e r n l o c a t i o n s . The c u r r e n t s a t t h e deep e a s t e r n l o c a t i o n have a p r e d o m i n a t e l y n o r t h w a r d v-c e m p e n e n t . The c u r r e n t s a t t h e c e n t r a l 50m l o c a t i o n e x h i b i t t h e most s c a t t e r i n g . The o v e r a l l s t r e n g t h o f t h e c u r r e n t s i s seen t o d e c r e a s e w i t h d e p t h . The c u r r e n t s need not be r e s o l v e d i n t o c o m p o n e n t s t h a t a r e p a r a l l e l t c t h e (x,y) a x e s , where t h e y - a x i s i s o r i e n t e d t c w a r d s t r u e n o r t h . The c u r r e n t s may be r e s o l v e d i n t o t h e ( u ' j ' ' ) c o m p o n e n t s t h a t a r e p a r a l l e l t o t h e ( x ' , y ' ) a x e s , so t h a t t h e c o r r e l a t i o n c o e f f i c i e n t , (c') b a s e d on t h e (u',v') c o m p o n e n t s , i s 2 e r o . Hence t h e F i e y n o l d s s t r e s s , . (u'-u*) ( v ' ~ 1 16 V ' ) , as measured i n t h e ( x ' , y ' ) c o o r d i n a t e s y s t e m i s z e r c . T h e r e i s a c o o r d i n a t e s y s t e m ( x " , y " ) where t h e c o r r e l a t i o n c o e f f i c i e n t (and R e y n o l d s s t r e s s ) has a maximum v a l u e , c . The c o o r d i n a t e s y s t e m ( x ' , y ' ) c o i n c i d e s w i t h ( x " , y " ) i f t h e f o r m e r i s r o t a t e d 45° i n an a n t i - c l o c k w i s e d i r e c t i o n . G e n e r a l l y t h e y ' - a x i s i s n o t p a r a l l e l t o t h e mean v e c t o r ( u , V ) . The y ' - a x i s d i r e c t i o n end t h e maximum c o r r e l a t i o n c o e f f i c i e n t , c , f c r e a c h l o c a t i o n a r e g i v e n i n T a b l e I V . I t i s s e e n t h a t t h e y ' - a x i s i s w i t h i n 1.3° c f t r u e n o r t h a t f i v e l o c a t i o n s . The c u r r e n t s a t t h e c e n t r a l 50m l o c a t i o n have a y ' - a x i s whose d i r e c t i o n i s 68°. The d e p t h c o n t o u r s a r e a l i g n e d n o r t h - s o u t h i n t h e r e g i o n . b e i n g s t u d i e d . To t h e n o r t h and s o u t h o f t h i s r e g i o n , t h e d e p t h c o n t o u r s a r e a l i g n e d a p p r o x i m a t e l y n o r t h w e s t -s o u t h e a s t . The h o r i z o n t a l d i s t a n c e s i n b o t h t h e NS and EW d i r e c t i o n s , f r o m e a c h o f t h e s i x l e c a t i c n s t c t h e n e a r e s t b o u n d a r i e s , a r e l i s t e d i n T a b l e . I V . These b o u n d a r i e s a r e r e l e v a n t t o t h e s t u d y c f t h e l e w - f r e q u e n c y c u r r e n t m o t i o n s . Most o f t h e b o u n d a r i e s a r e t h e s i d e b o u n d a r i e s c f the S t r a i t o f G e o r g i a , b u t some a r e n e t . R e l a t i v e t o t h e w e s t e r n 200m l o c a t i o n , t h e r e i s a s u b m a r i n e r i d g e - 4km t c t h e s o u t h . T h i s r i d g e i s a l i g n e d i n a NNW d i r e c t i o n and i t s peak l i e s a t a d e p t h o f 17Cm. R e l a t i v e t o t h e c e n t r a l 200m l o c a t i o n , t h e r e l i e s t h e s t e e p w a l l o f a s u b m a r i n e p l a t e a u 3.5km t c t h e e a s t . TABLE IV - Long-Term Mea H06 (Western) 50m 200m Days of A v a i l a b l e Data Distance to Boundary (E or W) Distance to Boundary (N or S) (km) Mean magnitude of current vector (cm/s) (with 95% confidence l i m i t s ) Mean d i r e c t i o n of current vector (deg.true) (with 95% confidence l i m i t s ) Mean of current speeds (cm/s) Low Frequency F l u c t u a t i n g K i n e t i c Energy per u n i t mass (ergs/gr) (with 95% confidence i n t e r v a l s ) u-component v-component u+v C o r r e l a t i o n c o e f f i c i e n t (c) Maximum c o r r e l a t i o n c o e f f i c i e n t (c n) y-axis o r i e n t a t i o n f o r zero Reynolds s t r e s s (y') (deg. true) 419 5.5 (W) 10.4 (S) 3.0+.5 355 7.65 8. 3 25 33 •«(S:a 0.30 0.57 13 389 2.6 (W) .4.0 (S) 1.6+.8 346 /353N V33o; 16 3. 4.8 7.4 3.4 0. 27 0.76 Mean water temperature CC) 8.47 8.82 and FKE H16 (Central) 50m 200m H26 (Eastern) 50m 140m 533 10.8 (E) 20.3 (N) 4.3+.5 ("0 8.23 20 .5 f 4 5- 7 N) 11.3 31.8 0.30 0. 41 68 8.81 22. 3 6.8 /58.0\ \20.l) 346 3.5 (E) 12.6 (N) 1.4+.5 3. 56 " (") 7.2 9.1 10.8 5.1 -0.29 0.61 349 8.70 412 2.3 (E) 15.8 (N) 3.0+1.1 '(") 6.98 2.1 (1:1) 28.9 48. 5 19.1 30 .9 f 5 ° - 0 N ) \21.0 / 0.11 0.87 2 8.48 319 0.8 (E) 10.6 (S) 7 . 4+. 5 8.40 1.3 19.0 20.3 /2.2\ \0.8f 32.5 12.5 /33.4S \13.6/ -0.64 0.93 10 8.45 I—1 T h i s w a l l , w h i c h i s a l i g n e d i n a NS d i r e c t i o n , e x t e n d s a c o n s i d e r a b l e d i s t a n c e t o t h e s c u t h t u t o n l y a b o u t 2km t o t h e n o r t h . The edge c f t h i s p l a t e a u l i e s T80m b e l o w the. w a t e r s u r f a c e . From t h i s e d g e , t h e p l a t e a u s l o p e s g r a d u a l l y upward t o w a r d s t h e e a s t . 5_.4 H£HlJlll_.Ll52I!S The m o n t h l y v e c t o r means c f t h e l o w - f r e g u e n c y c u r r e n t s a r e shown i n F i g u r e 42. M o n t h l y v e c t o r means a r e d e r i v e d f r c m s e c t i o n s o f t h e l o w - j j a s s e d _ d a i l j t i m e - s e r i e s a l l h a v i n g a d u r a t i o n o f 'one month. The m i d - p o i n t t i m e - c r i c i n c f c o n s e c u t i v e s e c t i o n s ' a r e s e p a r a t e d by 10 d a y s . M o n t h l y v e c t o r means a r e p l o t t e d a g a i n s t t h e m i d - p o i n t t i m e - o r i g i n c f t h e cne month s e c t i o n f r o m w h i c h t h e y a r e d e r i v e d . At t h e e a s t e r n m o o r i n g , t h e 140m c u r r e n t d i r e c t i o n s a r e a l w a y s n o r t h . T h e r e , t h e 50m c u r r e n t d i r e c t i o n s a r e m o s t l y n o r t h , a l t h o u g h s c u t h d i r e c t i o n s a r e o b s e r v e d when t h e c u r r e n t m a g n i t u d e s a r e b e l o w 4 cm/s. The m a g n i t u d e v a r i a t i o n s c f t h e 50m and 140m c u r r e n t s a t t h e e a s t e r n m o o r i n g a r e somewhat s i m i l a r . The r e l e v a n t c o h e r e n c e f u n c t i o n s o f t h e p r e v i o u s c h a p t e r show t h a t t h e h i g h e s t v e r t i c a l c o h e r e n c e a r e f c u n d a t t h e e a s t e r n m o o r i n g . A t ' t h e c e n t r a l m c o r i n g , t h e h i g h e s t 50m c u r r e n t m a g n i t u d e s a r e o b s e r v e d d u r i n g M a r c h , O c t o b e r , and Deceirber c f 1969 and d u r i n g J u l y 1970. D u r i n g t h e t h r e e p e r i o d s i n 1969, t h e 50m c u r r e n t has d i r e c t i o n s w h i c h a r e e a s t . D u r i n g J u l y Figure 42. Magnitude and d i r e c t i o n of the monthly vector means for the low-passed currents versus time. 12.0 1970 h o w e v e r , t h e 50m c u r r e n t has a SW d i r e c t i o n . The .50m c u r r e n t d u r i n g J u l y c f t h e p r e v i o u s y e a r i s d i r e c t e d s o u t h . These s o u t h w a r d d i r e c t i o n s c f t h e 50m c u r r e n t c o i n c i d e w i t h t h e t i m e s c f t h e F r a s e r E i v e r f r e s h e t . However, t h e r e i s no way c f k n o w i n g w i t h o n l y two e v e n t s w h e t h e r c r n e t t h e r e i s any r e l a t i o n s h i p . The 50m c u r r e n t i s o b s e r v e d t o f l o w i n a WKK d i r e c t i o n w i t h a r e l a t i v e l y s m a l l m a g n i t u d e c f 3 cro/s d u r i n g F e b r u a r y 1970. The 20Cm c u r r e n t a t t h e c e n t r a l m o o r i n g has d i r e c t i o n s v a r y i n g f r c m N t o NE d u r i n g p e r i o d s o f r e l a t i v e l y h i g h c u r r e n t m a g n i t u d e s . O t h e r d i r e c t i o n s a r e o b s e r v e d f o r c u r r e n t m a g n i t u d e s l e s s t h a n 1 cm/s. The m a g n i t u d e s o f t h e 5 0m and 200m c u r r e n t s a t t h e c e n t r a l m c c r i n g a r e n e t w e l l c o r r e l a t e d . At t h e w e s t e r n m o o r i n g , t h e h i g h e s t m a g n i t u d e s o f t h e 50m c u r r e n t a r e o b s e r v e d t o be 7 cm/s. At s u c h t i m e s t h e c u r r e n t i s d i r e c t e d N c r NNW. D u r i n g F e b r u a r y 1S70, a r e l a t i v e l y h i g h 5Cm c u r r e n t m a g n i t u d e c f 5 cm/s i s f o u n d t o be d i r e c t e d s o u t h . At t h e w e s t e r n m o o r i n g , t h e 20Cm c u r r e n t d i r e c t i o n i s c f t e n n o r t h . D u r i n g December 1969 and J a n u a r y 1 970, t h e 200m c u r r e n t i s d i r e c t e d t o t h e w and WNw. A 200m c u r r e n t d i r e c t i o n c f SvJ i s o b s e r v e d d u r i n g J u n e and J u l y 197C. The m a g n i t u d e v a r i a t i o n s o f t h e 200m c u r r e n t a t the w e s t e r n l e c a t i e n do n o t c o r r e l a t e w i t h t h e s e o f t h e 50m c u r r e n t a t t h e same m o o r i n g , o r w i t h t h a t o f t h e 200m c u r r e n t a t the c e n t r a l m c c r i nq. The m a g n i t u d e v a r i a t i o n s o f t h e 50m c u r r e n t s a t t h e t h r e e m o o r i n g s t e n d t c be c o r r e l a t e d w i t h . e a c h e t h e r . H i g h 50m c u r r e n t m a g n i t u d e s a r e o b s e r v e d d u r i n g M a r c h , O c t o b e r , and December o f 1969, and d u r i n g J u l y 1970 a t a l l l o c a t i o n s where t h e r e i s a v a i l a b l e d a t a . I t i s s e e n t h a t t h e h i g h e s t 50m c u r r e n t m a g n i t u d e s a t t h e w e s t e r n m o o r i n g t e n d t o o c c u r h a l f a mcnth e a r l i e r t h a n t h o s e a t t h e c e n t r a l m o c r i n g . The m o n t h l y v e c t o r means o f wind a t Sand Heads, t h e m o n t h l y means c f a i r - p r e s s u r e a t V a n c o u v e r , and t h a t o f s e a -l e v e l a t F t . A t k i n s o n a r e g i v e n i n F i g u r e 43. Between May 1969 and " F e b r u a r y 1970, t h e wind d i r e c t i o n s a r e m o s t l y .found i n t h e r a n g e W t c NW. D u r i n g t h i s p e r i o d , t h e l a r g e s t w ind m a g n i t u d e s o f 3 m/s a r e a s s o c i a t e d w i t h a WNW d i r e c t i o n . A r e l a t i v e l y h i g h w i n d m a g n i t u d e o f 2 m/s i s d i r e c t e d a p p r o x i m a t e l y NE i n A p r i l 1970. The i n f l u e n c e c f a i r - p r e s s u r e cn s e a - l e v e l i s e a s i l y d i s c e r n e d . ' Most p e a k s and v a l l e y s i n a i r - p r e s s u r e c o r r e s p o n d to" v a l l e y s and peaks i n s e a - l e v e l . H o wever, t h e a i r - p r e s s u r e v a r i a t i o n s c a n n o t a c c o u n t f o r a l l t h e s e a - l e v e l v a r i a t i o n s . . The l a r g e 20cm d r o p i n s e a - l e v e l d u r i n g March 1970 i s n o t a c c o u n t e d f o r by t h e a i r - p r e s s u r e . T h i s s e a - l e v e l d r o p i s a l s o n e t a s s o c i a t e d w i t h t h e w i n d , w h i c h has r e l a t i v e l y s m a l l m a g n i t u d e s i n March 1370. The i n c r e a s e of s e a - l e v e l i n J u n e 1970 i s a t t r i b u t e d t o t h e F r a s e r R i v e r f r e s h e t . W a l d i c h u k (1964) g i v e s ample e v i d e n c e p e r t a i n i n g t o t h e s e a - l e v e l c h a n g e s i n t h e S t r a i t o f G e o r g i a due t o t h e F r a s e r R i v e r f r e s h e t . D u r i n g J u n e 1969, t h e s e a - l e v e l peak i s c a u s e d by t h e F r a s e r E i v e r f r e s h e t as w e l l as by a d r o p i n a i r - p r e s s u r e . The m o n t h l y means o f s e a - l a v e l and w i n d do n o t c o r r e l a t e t o o w e l l . Figure 43. Magnitude and d i r e c t i o n of the monthly vector means for the low-passed wind (Sand Heads), and the monthly means of the air-pressure (Vancouver) and the low-passed sea-level (Pt. Atkinson), a l l versus time. 123 At t h e two e a s t e r n l o c a t i o n s , t h e r e i s some t e n d e n c y f c r h i g h c u r r e n t m a g n i t u d e s t o o c c u r when t h e w i n d m a g n i t u d e s a r e l o w . However, t h i s t e n d e n c y i s not o b s e r v e d d u r i n g November and December 1969, and t h e d a t a ' r e c o r d i s t o e s h o r t t c be s u r e t h a t t h i s a p p a r e n t p o s s i b l e r e l a t i o n s h i p i s n e t j u s t a c o i n c i d e n c e . The m o n t h l y means o f s e a r l e v e l , a i r - p r e s s u r e , o r w i n d c o r r e l a t e p o o r l y w i t h t h o s e o f t h e c u r r e n t s a t any c f t h e f o u r r e m a i n i n g l o c a t i o n s . T h e s e c o n c l u s i o n s c o u l d haue been p r e d i c t e d f r c m t h e l o w - f r e q u e n c y r e g i o n s o f t h e c o h e r e n c e f u n c t i o n ' s p r e s e n t e d i n t h e p r e v i o u s c h a p t e r . 5^5 K o n t h l y _ F K J The" f l u c t u a t i n g k i n e t i c e n e r g y p e r u n i t mass (ergs/gm) s h a l l be d e n o t e d by FKE. The u-FKE i s h a l f t h e v a r i a n c e o f a u-component t i m e - s e r i e s , and s i m i l a r l y f c r the ,v-.FKE. The t o t a l ' FKE i s t h e sum o f t h e u-FKE and v-FKE. The c a l c u l a t i o n c f FKE p r o c e e d s a f t e r r e m o v a l c f t h e l o n g - t e r m component means f r o m t h e component t i m e - s e r i e s . L o n g - t e r m means a r e t h e means d e r i v e d u s i n g a l l t h e a v a i l a b l e d a t a f r o m a p a r t i c u l a r l o c a t i o n . C o n s e c u t i v e m o n t h l y FKE * s a r e d e r i v e d f r c m c o n s e c u t i v e cne month s e c t i o n s o f t h e lo^zS3£§ed_daily- t i m e - s e r i e s whose m i d - p o i n t t i m e - c r i q i n s a r e s e p a r a t e d by 10 d a y s . A m o n t h l y FKE i s p l o t t e d a g a i n s t t h e m i d - p c i n t t i m e - o r i g i n c f t h e s e c t i o n f r o m w h i c h i t i s d e r i v e d . The m o n t h l y FKE o f t h e c u r r e n t c o m p o n e n t s and t h e 124 m o n t h l y t o t a l FKE a r e p r e s e n t e d i n F i g u r e 44. The r r c n t h l y v a r i a n c e s o f t h e wind a t Sand Heads, of t h e a i r - p r e s s u r e a t V a n c o u v e r , and o f t h e s e a - l e v e l a t P t . A t k i n s o n a r e p r e s e n t e d i n F i g u r e 45. These m o n t h l y v a r i a n c e s a r e a l s o c a l c u l a t e d f r c m t h e l o w - p a s s e d d a i l y t i m e - s e r i e s a f t e r r e m o v a l o f t h e l o n g - t e r m means. The u-FKE's f c r t h e c u r r e n t s f r o m t h e t h r e e deep l o c a t i o n s do n o t e x c e e d 5 e r g s / g m a t any t i m e . Gn the 50m p l a n e , t h e u-FKE a t t h e c e n t r a l l o c a t i o n i s u s u a l l y much l a r g e r t h a n t h o s e a t t h e s i d e l o c a t i o n s . u-FKE's a b c v e 40 e r g s / g m a r e f c u n d d u r i n g F e b r u a r y and J u n e of 1970 a t t h e 50ir c e n t r a l l o c a t i o n . The l a r g e u-FKE o f June 1970 i s not p r e s e n t d u r i n g J u n e o f t h e p r e v i o u s y e a r . The u-FKE's o f t h e 50m c e n t r a l and w e s t e r n l o c a t i o n s a r e n o t w e l l c o r r e l a t e d . Among t h e t h r e e deep l o c a t i o n s , - t h e h i g h e s t v -FKE's a r e a l w a y s f o u n d a t t h e e a s t e r n l o c a t i o n . Among t h e t h r e e 50m l o c a t i o n s , t h e h i g h e s t v -FKE's a t e a l s o f o u n d a t t h e e a s t e r n l o c a t i o n , w h i l e t h e l o w e s t v-FKE's a r e o b s e r v e d a t t h e c e n t r a l 50m l o c a t i o n . At. t h e c e n t r a l 50m l o c a t i o n , h i g h v -FKE's t e n d t c be a s s o c i a t e d w i t h h i g h u - F K E ' s . T h i s i s n e t t r u e c f the EKE's at t h e w e s t e r n 5Cm l o c a t i o n . The v-FKE's o f t h e t h r e e 50m l o c a t . i c n s a r e n o t w e l l c o r r e l a t e d . The v-FKE's c f t h e v e r t i c a l l y - s e p a r a t e d e a s t e r n l o c a t i o n s a l s o show p e e r c o r r e l a t i o n . The t o t a l EKE's c f t h e deep l o c a t i o n s a r e n e t w e l l c o r r e l a t e d w i t h e a c h e t h e r . T h i s i s a l s o t r u e o f t h e t o t a l F KE's a t t h e 50m l o c a t i o n s . D u r i n g J u l y 1S70, l o w v a l u e s c f 125 TOTAL FKE Figure 44. Monthly FKE of the low-passed current components versus time. 126 Figure 45. Monthly variance of the low-passed wind components (Sand Heads), low-passed air-pressure (Vancouver) and low-passed sea-level (Pt. Atkinson), a l l versus time. 127 t h e t o t a l FKE a r e o b s e r v e d a t t h e c e n t r a l 50m l o c a t i o n . One y e a r l a t e r , v e r y - h i g h v a l u e s o f t h e t o t a l FKE a r e f o u n d a t t h i s l o c a t i o n . T h e r e may be s e a s o n a l v a r i a t i o n s i n t h e n c n t h l y t c t a l E K E ' s , but much l o n g e r t i m e - s e r i e s a r e r e q u i r e d t o p e r c e i v e them. T h e r e i s a f a i r c o r r e l a t i o n b e t ween t h e u - v a r i a n c e and . v - v a r i a n c e o f t h e w i n d . The h i q h e s t w i n d v a r i a n c e s a r e o b s e r v e d d u r i n g A p r i l 1S70. The m o n t h l y v a r i a n c e c f a i r -p r e s s u r e show a b r o a d maximum d u r i n g f a l l - w i n t e r . V e r y low v a l u e s c f t h e s e a - l e v e l v a r i a n c e c o i n c i d e w i t h t h e s e c f a i r -p r e s s u r e v a r i a n c e . .The peak i n t h e s e a - l e v e l v a r i a n c e d u r i n g A p r i l 1970 c o n c i d e s w i t h one i n t h e t o t a l w ind v a r i a n c e . However, n e i t h e r t h e t c t a l wind v a r i a n c e n o r t h e a i r - p r e s s u r e v a r i a n c e a p p e a r t o be r e s p o n s i b l e f c r t h e l a r g e peak i n s e a -l e v e l v a r i a n c e d u r i n g December 1969. One may compare t h e t o t a l wind v a r i a n c e , t h e a i r -p r e s s u r e v a r i a n c e , o r t h a t o f s e a - l e v e l w i t h t h e t c t a l FKE c f t h e c u r r e n t s a t any c f t h e s i x l o c a t i o n s . I n a l l c o m p a r i s o n s , p o o r c o r r e l a t i o n s a r e o b t a i n e d . T h i s i s a l s o t r u e when c o m p a r i s o n s a r e made w i t h t h e u-FKE and v-FKE o f t h e c u r r e n t s . The w i n d , a i r - p r e s s u r e , and F r a s e r R i v e r d i s c h a r q e may w e l l be i m p o r t a n t p a r a m e t e r s i n t h e g e n e r a t i o n o f t h e s e l e w - f r e q u e n c y c u r r e n t s . The c o u p l i n g o f t h e s e p a r a m e t e r s t o t h e s u b - s u r f a c e c u r r e n t s may w e l l be a v e r y c o m p l e x c o u p l i n g , e x p l a i n i n g t h e p e e r c o r r e l a t i o n s o b s e r v e d . H o w e v e r , t h e s e p a r a m e t e r s do n e t c o n s t i t u t e a c o m p l e t e l i s t o f i m p o r t a n t p a r a m e t e r s . The w a t e r d e n s i t y i s an i m p o r t a n t p a r a m e t e r ( b e c a u s e t h e c u r r e n t s a r e 128 b a r c c l i n i c ) t h a t i s not. a v a i l a b l e . The l o n g - t e r m means a r e t h o s e d e r i v e d f r c m a l l the a v a i l a b l e d a t a i n a t i m e - s e r i e s ( e x c l u d i n g d a t a g a p s ) . The l o n g - t e r m c u r r e n t component means a r e g i v e n i n T a b l e • IV as m a g n i t u d e s and d i r e c t i o n s . The d i r e c t i o n s o f t h e c u r r e n t means a t t h e f o u r s i d e l o c a t i o n s a r e a l l w i t h i n 16° c f t r u e n o r t h . They a r e e s e n t i a l l y p a r a l l e l t o t h e d e p t h c c n t c u r s . At t h e c e n t r a l m o o r i n g , t h e d i r e c t i o n o f t h e 200m c u r r e n t mean i s NKK, w h i l e t h a t o f t h e 50m c u r r e n t mean i s ESE. The h i g h e s t mean c u r r e n t m a g n i t u d e c f 7.4 cm/s i s f o u n d a t t h e deep e a s t e r n l o c a t i o n . Egua.1 mean c u r r e n t m a g n i t u d e s a r e o b s e r v e d a t t h e two 50m s i d e l o c a t i o n s . The m a g n i t u d e s f o u n d a t t h e p a i r ' c f 200m l o c a t i o n s a r e a l s o a p p r o x i m a t e l y , e g u a l . A r e l a t i v e l y h i g h mean c u r r e n t m a g n i t u d e o f 4.3 cm/s i s o b s e r v e d a t the 50m c e n t r a l l o c a t i o n . T h e s e l o n g - t e r m c u r r e n t means may n o t be used i n t h e c a l c u l a t i o n o f t r a n s p o r t . P r e v i o u s r e s u l t s shew t h a t t h e c u r r e n t .motions a r e p o o r l y r e s o l v e d b o t h i n t h e v e r t i c a l and h o r i z o n t a l . The m o o r i n g s were p l a c e d 10km a p a r t . M o o r i n g s p l a c e d 2km a p a r t m i g h t be s u f f i c i e n t t o y i e l d l o n g - t e r m c u r r e n t means s u i t a b l e ' f c r t r a n s p o r t c a l c u l a t i o n s . The two i n s t r u m e n t s on t h e e a s t e r n m o o r i n g were s e p a r a t e d by 90m, a s e p a r a t i o n t e c l a r g e t o r e s o l v e 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 l o w - f r e q u e n c y c u r r e n t m o t i o n s . 129 The D e p a r t m e n t ' o f t h e E n v i r o n m e n t (1 9721) , 1197 2c) p l a c e d l i n e s c f m o o r i n g s w i t h i n 10-20 km c f t h e p r e s e n t l i n e , b e i n g s t u d i e d . The m o o r i n g s i n t h e s e l i n e s were p l a c e d 4km a p a r t and d a t a c o l l e c t e d f r o m t h e s e i n s t r u m e n t s s p a n a p p r o x i m a t e l y a month. P o s s i b l y , e x a m i n a t i o n o f t h i s d a t a b a s e may y i e l d a b e t t e r i d e a o f t h e h o r i z o n t a l . i n s t r u m e n t s e p a r a t i o n r e q u i r e d , t c r e s o l v e t h e l e w - f r e q u e n c y c u r r e n t m o t i o n s w e l l . One may wonder a b o u t t h e d e p e n d e n c e c f t h e l o n g - t e r m means cn t h e l e n g t h o f t h e t i m e - s e r i e s . The m a g n i t u d e s and d i r e c t i o n s c f t h e c u r r e n t means v e r s u s a v e r a g i n g t i m e a r e p r e s e n t e d i n F i g u r e 46. F c r c o m p a r i s o n , t h e wind v e c t o r mean, t h e a i r - p r e s s u r e mean, and t h e s e a - l e v e l ' mean -are p l o t t e d v e r s u s a v e r a g i n g t i m e i n F i g u r e 4 7 . F o r t h e s e c a l c u l a t i o n s , d a t a g a p s i n a l o w - p a s s e d_dajJ._y_ t i m e - s e r i e s a r e removed and t h e r e m a i n i n g s e c t i o n s a r e s h i f t e d i n t i m e as n e c e s s a r y t c j o i n them end t o end. An a v e r a g e e v e r N d a y s i s t h e n d e r i v e d f r c m t h e f i r s t N p o i n t s i n t h e l o w - p a s s e d d a i l y t i m e - s e r i e s c i t h e c o m p o n e n t s , and t h e r e s u l t a n t m a g n i t u d e and d i r e c t i o n a r e p l o t t e d , a g a i n s t t h e a v e r a g i n g t i m e , N. The mean c u r r e n t m a g n i t u d e s a t t h e t h r e e deep l o c a t i o n s (1-2 cm/s a t 200m) a r e c o n s t a n t t c w i t h i n 0.4 cm/s f c r a v e r a g i n g t i m e s c f 8 months c r l o n g e r . Those a t t h e t h r e e 5Cm l o c a t i o n s ( a b o u t 3 cm/s) v a r y w i t h i n a r a n g e o f 1 cm/s f o r a v e r a g i n g t i m e s c f 10 months or l o n g e r . The mean c u r r e n t d i r e c t i o n s a t t h e twe e a s t e r n l o c a t i o n s and t h a t a t t h e w e s t e r n 200m l o c a t i o n a r e c o n s t a n t t c w i t h i n 5° f o r a v e r a g i n g t i m e s c f 8 months c r l o n g e r . T hose a t t h e r e m a i n i n g l o c a t i o n s a r e l e s s 130 sw-WH NW-NE-DIRECTION EASTERN 50m 140m -W -NW -N -NE -E HO h5 i r i r~ 4 i i 1 1 1 1 1 r 6 8 10 12 AVERAGING TIME (MONTHS) L0 14 16 CENTRAL Inn, 200m---NW -N -NE "> -E -SE -10 i i i i i i i 1 1 1 1 r 1 - — I 1 1 r - — i u0 sw-w-w-N-N E J WESTERN IRJ?, 200m--W -NW -N -NE -E -10 Figure 46. Magnitude and d i r e c t i o n of the vector means for the low-passed currents versus averaging time. Each graph of d i r e c t i o n has an arrow i n d i c a t i n g i t s v e r t i c a l scale. 131 DIRECTION WIND (SAND HEADS) -w -N -E hS -W r 2 MAGNITUDE T 1 1 1 1 1 r-6 8 10 12 AVERAGING TIME (MONTHS) o UJ CO " 14 I 16 "I 1 18 AIR-PRESSURE (VANCOUVER) I 1 1 1 1—;—i 1 1 1 r n 1 1 : 1 1 1 1 1018 1017 1016 1015 M0I4 SEA-LEVEL (PT. ATKINSON) :3I2 h-310 -308 -306 i 1 1 1 1 1 1 1 1 1 1 1 1 r 1 1 1 1 L 3 0 4 F i g u r e 47. Magnitude and d i r e c t i o n o f the v e c t o r mean f o r the low-pa s s e d wind (Sand Heads), and the means o f the low-passed a i r - p r e s s u r e (Vancouver) and low-passed s e a - l e v e l (Pt. A t k i n s o n ) , a l l v e r s u s a v e r a g i n g time. 132 c o n s t a n t . They v a r y w i t h i n a r a n g e o f 16° f c r a v e r a g i n g t i m e s o f 8 months c r l e n g e r . The mean wind d i r e c t i o n i s c o n s t a n t t c w i t h i n 20° w h i l e i t s m a g n i t u d e (a b o u t 100 cm/s) i s c o n s t a n t t o w i t h i n 5C cm/s, f o r a v e r a g i n g t i m e s c f .6 months o r l o n g e r . The mean a i r -p r e s s u r e v a r i e s w i t h i n a r a n g e o f 1mb f o r a v e r a g i n g t i m e s o f 6 months c r l e n g e r . T h e r e i s an i n v e r s e c o r r e l a t i o n , b etween t h e f l u c t u a t i o n s o f mean s e a - l e v e l and a i r - p r e s s u r e b e c a u s e o f t h e i n v e r s e b a r o m e t e r effect... For a v e r a g i n g t i m e s c f 14 months o r l e n g e r , mean s e a - l e v e l has a t r e n d w h i l e mean a i r - p r e s s u r e d o e s n o t . The power s p e c t r u m c f s e a - l e v e l shews t h a t f l u c t u a t i o n s w i t h a p e r i o d o f 300 d a y s have s i g n i f i c a n t v a r i a n c e . Thus t h e b e h a v i o u r c f mean s e a - l e v e l f o r a v e r a g i n g t i m e s c f 14 months o r l e n g e r i s i n t u i t i v e l y e x p e c t e d . O v e r a l l , t h e c u r r e n t v e c t o r means a p p e a r t c c o n v e r g e t o w a r d s s t a b l e v a l u e s w i t h i n c r e a s i n g a v e r a g i n g t i m e . T h e s e m e a n s ' w i l l c o n t i n u e t o c o n v e r g e ( f o r a v e r a g i n g t i m e s c f 2 y e a r s and l o n g e r ) p r o v i d e d t h a t t h e c u r r e n t s a r e t r u l y s t a t i o n a r y . I t i s i m p o s s i b l e w i t h t h e ' a v a i l a b l e d a t a t c c o n c l u d e t h a t t h e c u r r e n t s a r e t r u l y s t a t i o n a r y o r n o n - s t a t i o n a r y . However, i t a p p e a r s t h a t t h e c u r r e n t s a r e s t a t i o n a r y ( w i t h r e s p e c t t c t h e i r means) d u r i n g t h e l i m i t e d p e r i o d 1969-70. Q£H t. i ^ST i£§_ I-SiS 11J ! . §_ 15£_i]i.§_i5j.§lL§ Suppose t h a t an i n f i n i t e s t a t i o n a r y t i m e - s e r i e s , u (t) has mean yu and v a r i a n c e <sz . I t s a u t o c o r r e l a t i o n f u n c t i c n r (t) 133 i s : -r {T)=[u(t)-^]«[u_(t + f ) - / i ] / 0 ' 2 (5.3) where t h e o v e r f c a r i n d i c a t e s a t i m e a v e r a g e . T h i s f u n c t i o n i s an e v e n f u n c t i o n o f T" and i t has t h e l i m i t s (-1,1). Cne can o n l y o b t a i n e s t i m a t e s o f t h e a u t o c o r r e l a t i o n f u n c t i o n , r ( k ) , w i t h a d i s c r e t e , t i m e - s e r i e s , u- L:-N-fe where t h e s u b s c r i p t s a r e t i m e i n d i c e s , N i s ' t h e t i m e - s e r i e s l e n g t h , and ( u , u 2 ) a r e t h e ( m e a n , v a r i a n e e ) o f t h e t i m e - s e r i e s . C o n s i d e r means, U T , t h a t a r e d e r i v e d f r c m f i n i t e s e c t i o n s o f u ( t ) a l l h a v i n g l e n g t h 1. The d i s t r i b u t i o n of D T ( f o r l a r g e T) a b o u t t h e e x p e c t e d v a l u e (^) i s G a u s s i a n w i t h a v a r i a n c e : -T v a r (U T) = (2rf2/T) C [ 1- (T/T) ]«r (f) dr (5.5) - o a r e s u l t g i v e n by T e n n e k e s and Lumley (1S72) ( s e c t i o n 6.4, 6.5) . The 95% c o n f i d e n c e i n t e r v a l s f c r t h e means ( D T , V y ) o f t h e c u r r e n t c o m p o n e n t s a r e ±1.96»[/\/ v a r ( 0 T ) ' , /^/ v a r ( V T ) ' ] , w h i c h a r e l i s t e d i n T a b l e V f o r v a r i o u s v a l u e s o f T. These c o n f i d e n c e i n t e r v a l s a r e t h e m s e l v e s e s t i m a t e s s i n c e t h e c a l c u l a t i o n o f v a r (U T) p r o c e e d s a f t e r r (T-) has been r e p l a c e d by r (k) , and ( J X , 6 z ) have- been r e p l a c e d by ( u , P ) . As e x p e c t e d , t h e c o n f i d e n c e i n t e r v a l s f o r t h e c u r r e n t component means d e c r e a s e w i t h i n c r e a s i n g a v e r a g i n g t i m e , T. I f t h e t i m e - s e r i e s TABLE V - Variances (according to equation 5.5) and 9.5% confidence i n t e r v a l s of current component means derived from time-series of length, T. Location (days) Var(U T) Var(V T) 2 2 (cm/sec) (cm/sec) 95% CONFIDENCE INTERVALS U(cm/sec) V(cm/sec) EASTERN 50m EASTERN 140m CENTRAL 50m CENTRAL 200m WESTERN 50m WESTERN 200m 10 1.09 27.1 2.05 10.2 20 0.587 18.4 1.50 8.40 30' 0.399 13.7 1.24 7.26 40 0.307 10.7 1.09 6.41 50 0.233 8.23 0.946 5.62 60 0.191 6.60 0*856 5.04 370 0.00145 0.317 0.0746 1.10 10 0.984 13.0 1.94 7.06 20 0.738 7.37 1.68 5.32 30 0.667 5. 38 • 1.60 4.55 40 0.632 4.37 1.56 4.10 50 0.600 3.57 1.52 3.70 60 0.566 .3.08 1.48 3.44 180 0.198 0.0283 0.873 0.329 10 26.1 11.1 10.0 6.54 20 17.9 8.08 8.28 5.57 30 12.2 6.65 • 6.85 5.05 40 8.98 5.86 5.87 4. 75 50 6.91 5.16 5.15 4.45 60 5.41 4.57 4.56 4.19 ' 480 '0.0215 0.0874 0.287 0.579 10 1.27 4.80 2.20 4.29 20 0.817 2.20 1.77 2.91 30 0.639 1.29 1.57 2.22 40 0.561 1.01 1.47 1.97 50 0.506 0.722 1.39 1.67 60 0.442 0.580 1.30 1.49 300 0.0628 0.0229 0.491 0.297 10 5.06 26.7 4.41 10.1 20 2.84 17.4 3. 30 8.17 30 1.80 13.7 2.63 7.25 40 1.23 11.6 2.17 6.67' 50 0.918 10.3 1.88 6.29 60 0.755 9.08 1.70 5.91 290 0.0458 0.0511 0.419 0.443 10 0.395 3.15 1.23' 3.48 20 0.269 1.75 1.02 2.59 30 0.230 1.38 0.940 2.31 40 0.205 1.26 0.888 2.20 50 0.188 1.09 0.850 2.05 60 0.171 0.961 0.811 1.92 330 0.00161 0.147 0.0786 0.751 135 TABLE VI - Comparisons between the t h e o r e t i c a l variances of Table V and the corresponding empirical variances c a l c u l a t e d from the current component time-series of length, T. Location T Var(U ) (cm/s) Var(V ) (days) • Empirical Equation 5.5 Empirical (cm/s) Equation 5.5 EASTERN 50m 30 60 90 0.370 0.136: 0.0870 0.399 0.191 0.121 14.1 6.16 3.73 13.7 6.60 4.34 EASTERN 140m 30 60 90 0.749 0.709 0.554 0.667 0.566 0.470 5.75 3. 33 1.99 5.38 3.08 2.05 CENTRAL 50m 30 60 90 12.5 5.70 2.80 12. 2 5.41 2.60 6.83 4.68 3.81 6.65 4.57 3.55 CENTRAL 200m 30 60 90 0.614 0.278 0.172 0.639 0.441 0.290 1.17 0.541 0. 330 1.29 0.580 0.349 WESTERN 50m 30 60 90 2.07 0.864 0.528 1.80 0.755 0.445 14.3 9.41 5. 72 13. 9. 6. 7 08 64 WESTERN 200m 30 60 90 0.244 0.176 0.134 0.230 0.171 0.123 1.49 0.887 0.595 1.38 0.961 0.723 126 i s t r u l y s t a t i o n a r y , - o n e c a n o b t a i n an e s t i m a t e (U T) t h a t i s a r b i t r a r i l y c l c s e t o t h e t h e o r e t i c a l v a l u e (^ ) , b y . c h e e s i n g a s u f f i c i e n t l y l o n g a v e r a g i n g t i m e , T. The 9 5 % c o n f i d e n c e - i n t e r v a l s f o r t h e l o n g - t e r m c u r r e n t means, l i s t e d i n • T a b l e I V , - a r e d e r i v e d f r c m t h e e n t r i e s i n T a b l e V w h i c h have t h e maximum v a l u e o f T. The l a r g e s t 95% c o n f i d e n c e i n t e r v a l s f c r t h e ( m a g n i t u d e , d i r e c t i o n ) o f t h e l o n g -t e r m v e c t o r means a r e a p p r o x i m a t e l y (±1 cm/s, ±25°). H e n c e , i f t h e c u r r e n t s a r e t r u l y s t a t i o n a r y , v e c t o r means b a s e d cn f u t u r e t i m e - s e r i e s c f c o m p a r a b l e l e n g t h w i l l p r o b a b l y be w i t h i n 2 cm/s and 50° o f t h e v a l u e s l i s t e d i n T a b l e I V . As s t a t e d e a r l i e r t h o u g h , i t i s n e t p o s s i b l e w i t h t h e a v a i l a b l e d a t a t c c o n c l u d e t h a t t h e c u r r e n t s a r e t r u l y s t a t i o n a r y o r n o n - s t a t i o n a r y . I n T a b l e V I t h e v a r i a n c e s o f t h e component means, as d e r i v e d f r c m e g u a t i c n (5.5) , a r e compared w i t h t h e e m p i r i c a l l y d e r i v e d v a l u e s , f c r T=2G, 6C, 90 d a y s . The e m p i r i c a l l y d e r i v e d v a l u e s f o r T-30 d a y s a r e c a l c u l a t e d u s i n g the t i m e - s e r i e s c f m o n t h l y c u r r e n t means p r e s e n t e d e a r l i e r . . T h e r e i s s e e n t o be f a i r l y c l o s e a g r e e m e n t . 5_.7 l'2£2~^2£iS._ V a r i a n c e s The l o n g - t e r m t o t a l FKE's ( T a b l e IV) o f t h e - c u r r e n t s d o e s v a r y c o n s i s t e n t l y w i t h d e p t h , a s shown i n F i g u r e 4 8 a . I t i s t e m p t i n g t c s t a t e t h a t t h e t c t a l FKE i n a w a t e r c o l u m n i s d e p e n d e n t o n l y cn d e p t h , p r o v i d e d t h a t t h e t c t a l EKE i s measured o u t s i d e any b o u n d a r y l a y e r s . The 140m e a s t e r n 48. Long-term t o t a l FKE of the low-passed currents versus depth (A), and maximum c o r r e l a t i o n c o e f f i c i e n t (C m) and r a t i o of v-FKE/u-FKE, both versus distance to the nearest l a t e r a l boundary (B). 138 l o c a t i o n i s s i t u a t e d 25rr, f r c m t h e b o t t o m and 0.8km f r c m t h e l a t e r a l b o u n d a r y . A f t e r l o c a t i n g the t o t a l FKE f o r t h i s l o c a t i o n i n F i g u r e 4 8 a , i t i s t e m p t i n g . t o s t a t e t h a t ' t h i s l o c a t i o n i s o u t s i d e t h e b o u n d a r y l a y e r s . W h a t e v e r t h e b e h a v i o u r o f t h e t c t a l FKE i n a w a t e r c o l u m n , i t must be z e r o a t iraximum d e p t h . one. may n a i v e l y e x p e c t t h e t o t a l FKE t o d e c r e a s e w i t h d e p t h . Most c f e n e r g y s o u r c e s a r e l o c a t e d a t t h e s u r f a c e w h i l e an e n e r g y s i n k ( f r i c t i o n ) i s l o c a t e d a t t h e b o t t o m . ' T h a t t h e t o t a l FKE s h o u l d d e c r e a s e l i n e a r l y w i t h d e p t h ( i f i t dees ) , - r e q u i r e s a mere s o p h i s t i c a t e d e x p l a n a t i o n . The p a r t i t i o n o f t h e t o t a l FKE i n t o t h e u-FKE and v-FKE i s i n f l u e n c e d s t r o n g l y by t h e l a t e r a l b o u n d a r i e s . F i g u r e 48b shows t h e d e p e n d e n c e o f v-FKE/u-FKE on t h e s e p a r a t i o n o f t h e l o c a t i o n s f r o m t h e n e a r e s t l a t e r a l b o u n d a r y . A l s c p l o t t e d a r e t h e maximum. c o r r e l a t i o n c o e f f i c i e n t s , c , ( T a b l e IV) c f t h e l e w - f r e q u e n c y c u r r e n t II-V s c a t t e r d i a g r a m s . The c o e f f i c i e n t , c m , may be . i n t e r p r e t e d as one measure of. t h e t e n d e n c y t o w a r d s r e c t i l i n e a r m o t i o n . T h i s p a r a m e t e r i s more u s e f u l i n t h a t i t i s bounded (0,1) , and t h a t i t - i s i n d e p e n d e n t o f any c o o r d i n a t e s y s t e m . A v a l u e o f u n i t y i n d i c a t e s c o m p l e t e l y r e c t i l i n e a r m o t i o n , w h i l e a v a l u e c f z e r o i n d i c a t e s c o m p l e t e l y d i s o r g a n i z e d ( h o r i z o n t a l l y i s o t r o p i c ) m o t i o n . I t - i s i n t u i t i v e l y r e a s o n a b l e t c e x p e c t t h e t e n d e n c y f e w a r d s r e c t i l i n e a r m o t i o n ( c m ) t o i n c r e a s e as one a p p r o a c h e s a l a t e r a l b o u n d a r y . F i g u r e 48b c o n f i r m s t h i s e x p e c t a t i o n . I t i s o b s e r v e d t h a t c ^ v a r i e s most r a p i d l y a t a l c c a t i c n - t c - b c u n d a r y s e p a r a t i o n o f 2.5 km. The e x a c t s i g n i f i c a n c e o f t h i s number i s unknown. P e r h a p s , i t w i l l u l t i m a t e l y be u s e f u l a s a l a t e r a l l e n g t h s c a l e , o r as a c h a r a c t e r i s t i c l e n g t h c f seme b o u n d a r y l a y e r f o r t h e 1 c w - f r e g u e n c y c u r r e n t m o t i o n s . The component and t c t a l c u r r e n t FKE's v e r s u s a v e r a g i n g t i m e a r e p r e s e n t e d i n F i g u r e 49. These a r e d e r i v e d f r c m t h e icw-£assed_dail_y t i m e - s e r i e s . F o r c o m p a r i s o n , t h e w i r d v a r i a n c e a t Sand Heads, t h e a i r - p r e s s u r e v a r i a n c e a t V a n c o u v e r , and t h e s e a - l e v e l v a r i a n c e a t P t . A t k i n s o n a r e p l o t t e d v e r s u s a v e r a g i n g t i m e i n F i g u r e 50. These c a l c u l a t i o n s p r o c e e d o r . l y a f t e r t h e l o n g - t e r m means a r e removed f r o m e a c h l o w - p a s s e d t i m e - s e r i e s . The t o t a l c u r r e n t FKE c f t h e t h r e e deep l e c a t i e n s a r e c o n s t a n t t c w i t h i n 3 ergs/gm f o r a v e r a g i n g t i m e s e v e r 4 months. T h a t o f t h e two 50m s i d e l o c a t i o n s v a r y w i t h i n a r a n g e o f • 3 e r g s / g m f c r a v e r a g i n g t i m e s o v e r 8 months. The t o t a l EKE o f t h e c e n t r a l 50m l o c a t i o n v a r i e s w i t h i n a r a n g e c f 7 ergs/gm f c r a v e r a g i n g t i m e s e v e r 10 months. The c o r r e s p o n d i n g c o m p onent FKE's a r e ' c o n s t a n t t o w i t h i n t h e same s p e c i f i e d r a n g e s ( i f net l e s s ) f o r a v e r a g i n g t i m e s g r e a t e r t h a n t h e s p e c i f i e d t i m e s . The component and t c t a l w i nd v a r i a n c e s a r e c o n s t a n t t c w i t h i n , 15 (m / s) 2 f o r a v e r a g i n g t i m e s e v e r 6 months. The a i r -p r e s s u r e and s e a - l e v e l v a r i a n c e s v a r y w i t h i n a r a n g e o f 5mb 2 and 1 5 c m 2 , ' r e s p e c t i v e l y , f o r a v e r a g i n g t i m e s e v e r 10 months. .The c o i t i f c n e n t FKE's o f t h e 50m c u r r e n t s a t t h e c e n t r a l m o o r i n g do n o t a p p e a r t o c o n v e r g e t o w a r d s s t a b l e v a l u e s e v e n f c r a v e r a g i n g t i m e s o f 16 months and l o n g e r . However, t h i s 140 EASTERN• 50m CENTRAL' WESTERN TOTAL FKE EASTERN 140m — CENTRAL 200m — WESTERN 200m — i 1 1 1 r (-40 -30 s -20 to o cc -10 UJ -0 ~l I 1 I I I I 1 1 1 1 1 1 1 6 8 10 12 14 16 18 AVERAGING TIME (MONTHS) •?0 s 10 § t r 0 w V—FKE i i i i i i 1 1 1 1 1 1 1 1 1 1 r r 4 0 30 20 10 L 0 20 10 L 0 U-FKE i r i r i r r 40 -30 -20 10 L 0 20 Ho 0 gure 49. FKE of the low-passed current components versus averaging time. Figure 50. Variance of the low-passed wind components (Sand Heads), low-passed air-pressure (Vancouver), and low-passed sea-l e v e l (Pt. Atkinson), a l l versus averaging time. 142 b e h a v i o u r d e e s n o t mean t h a t t h e c u r r e n t t i m e - s e r i e s i s n e t -s t a t i o n a r y . The FKE (seccr.9 c r d e r s t a t i s t i c ) c f a t r u l y s t a t i o n a r y t i m e - s e r i e s h a v i n g a G a u s s i a n d i s t r i b u t i o n c o n v e r g e s more s l o w l y t h a n i t s mean ( f i r s t o r d e r s t a t i s t i c ) . T h i s i s r e a d i l y s e e n i f one c o m p a r e s t h e 95% c o n f i d e n c e i n t e r v a l s f c r t h e mean and FKE. I t i s n o t p o s s i b l e t o c o n c l u d e t h a t t h e l e w -f r e q u e n c y c u r r e n t s a r e t r u l y s t a t i o n a r y o r n o n - s t a t d e n a r y . However, i t a p p e a r s t h a t t h e y a r e s t a t i o n a r y w i t h r e s p e c t t o t h e FKE d u r i n g t h e l i m i t e d p e r i o d 1969-70. C c n f i d e n c e _ In+;ery a l s _ f o r _ t h e _ F K E . Suppose t h a t an i n f i n i t e s t a t i o n a r y t i m e - s e r i e s , u ( t ) , has z e r o mean and v a r i a n c e tf2. C o n s i d e r e s t i m a t e s (cr 2) t h a t a r e b a s e d on d i f f e r e n t s e c t i o n s o f u ( t ) w h i c h a l l have l e n g t h T. The e s t i m a t e s (o'2.)-u_ a r e c l u s t e r e d a b o u t t h e t h e o r e t i c a l v a l u e ' (cf 2) w i t h a v a r i a n c e c f : -T v a r ( r f f ) ^ - (4o-Vl) J [ 1- (T7T) ]»r 2 (i) d-f (5.6) o a r e s u l t g i v e n t y E e n d a t and P i e r s c l (1971) ( s e c t i o n 6 . 2 ) . The e s t i m a t e s u-FKE = ( d " 2 ) u / 2 has a v a r i a n c e o f va r (a-2) /4. The p r i m a r y a s s u m p t i o n made i s t h a t u ( t ) i s a s t a t i o n a r y G a u s s i a n s t o c h a s t i c p r o c e s s . The v a r i a n c e s c f <52 f o r t h e c u r r e n t , c o m p o n e n t s ' a r e g i v e n i n T a b l e V I I f o r v a r i o u s v a l u e s o f T. As e x p e c t e d t h e v a r i a n c e i s a d e c r e a s i n g f u n c t i o n o f T. The v a r i a n c e c f t h e p a r a m e t e r (<sl) - (<<2) + (o'2.) i s g i v e n by :-143 TABLE VII - Theoretical Variance (equation 5.6) of Current Component FKE's Based on Record Length, T 2 2 Location T Var(CJ' ) Var(<j ' ) T u T v 4 4 (days) (cm/s) (cm/s) EASTERN 50m EASTERN 140m CENTRAL 50m CENTRAL 200m WESTERN 50m WESTERN 200m 10 5.91 2200. 20 3.11 1240. 30 •2.13 .874. 40 1.63 680. 50 1. 33 571. 60 1.12 494. 420 0.385 182. 10 2.97 771. 20 1.80 429. 30 1.41 ' 303. 40 1.21 235. 50 1.07 194. 60 0.957 166. 320 0.417 83.6 10 1510. 313. 20 917. 187. 30 656. 136. 40 511. 109. 50 419. 90.7 60 356. 77.7 480 97.2 22.3 10 5.36 90.8 • 20 2.93 49.0 30 2.04. 34.2 40 1.62 27.0 50 1. 36 22.6 60 1.18 19.6 300 0.448 7. 35 10 109. 1860. 20 57.3 1040. 30 39.4 733. 40 30.4 572. 50 24.8 473. 60 . 21.1 403. 400 7.58 133. 10 0.788 45.3 20 0.456 24.6 30 0. 341 17.3 40 0.282 13.8 50 0. 245 11.5 60 0.216 9.94 330 0.0985 3.55 TABLE VIII - Comparison Between Theoretical and Empirical Variances of Current Component FKE 1s Based on Record Length, T. Location T (days) Vartfh (cm/s) 4 T u Empirical Equation 5.6 V a r ( < J ^ ) (cm/s) 4 T v Empirical Equation 5.6 Cov L • T u T V EASTERN 50m 30 60 90 4.18 1.81 1.01 2.13 1.12 0.793 1000. 578. 387. 874. 494. 358. 7.81 4.82 2.14 EASTERN 140m 30 60 90 0.761 0.427 0. 532 1.41 0.957 0.743 237. 87.1 87.0 303. 166. 127. 2.29 0.97S 1.57 CENTRAL 50m 30 60 90 1020. 720. 525. 656. 357. 252. 208. 135. 110. 136. 77.7 54.9 92.3 71.2 57.1 CENTRAL 200m 30 60 90 3.98 2.49 1.62 2.04 1.18 0.858 36.8 19.4 14.2 34.2 19.6 14.5 2.22 1.57 1.08 WESTERN 50m 30 60 90 55.9 29.7 23.6 39.4 21.1 15.0 482. 235. 112. 733. 403. 280. 51 77 97 WESTERN 200m 30 60 90 0.654 0. 373 0.281 0.341 0. 216 0.166 29.1 14.1 8.12 17. 3 9.94 5.14 0.244 0.245 0.280 145 v a r C c f 2 ) ^ ^ = v a r (cf*)^ + v a r { d?)v +2*cov[ (d*)^ , U 2 ) v J (5.7) where cov d e n o t e s t h e c o v a r i a n c e between t h e twc i n d i c a t e d p a r a m e t e r s . The e m p i r i c a l v a l u e s o f t h e s e c c v a r i a n c e s (T=30, 60, 90 d a y s ) a r e f o u n d i n T a b l e V I I I . . At t h r e e I c c a t i c r . s , .the c o v a r i a n c e s a r e a b c u t 1% o f [ v a r Ccf-2.) + va r ( c 2 ) v ]. The h i g h e s t c c v a r i a n c e s a r e o b s e r v e d a t t h e c e n t r a l 50m l o c a t i o n where t h e y a r e 10% o f [ v a r (tff) + v a r ( ^ ) ]. . The d i s t r i b u t i o n s c f t h e e s t i m a t e s (ty 2) , (a- 2) , and ( f l ' j ) U : f v a r e unknown. B e i n g p o s i t i v e - v a l u e d v a r i a b l e s , t h e s e e s t i m a t e s a r e f i t t e d t o an a 'X-p d i s t r i b u t i o n where:-•P =2 o r * / v a r ( c 2 ) ^ a-va r ( t f 2 ) ^ / ( 2 c 2 ) (5.6) a p r o c e d u r e u s e d by J e n k i n s and W a t t s (1 968) ( s e c t i o n 3 . 3 . 5 ) . The e s t i m a t e s u-FKE a r e f i t t e d t c (a/2) • d i s t r i b u t i c r . The v a l u e s c f v a r ( d * ) d e r i v e d f r o m e q u a t i o n (5.6) a r e c o mpared t o t h e e m p i r i c a l v a l u e s i n T a b l e V I I I . E m p i r i c a l v a l u e s f o r T-30 d a y s a r e c a l c u l a t e d u s i n g t h e t i m e - s e r i e s c f t h e m o n t h l y c u r r e n t F K E ' s . I t i s s e e n t h a t t h e e m p i r i c a l v a l u e s c o d i s a g r e e , by as much as a f a c t o r o f t w o , w i t h t h e s e v a l u e s d e r i v e d f r c m e q u a t i o n ( 5 . 6 ) . I t i s c o n j e c t u r e d t h a t t h e d i s a g r e e m e n t s t e m s f r o m t h e a s s u m p t i o n o f ' u ( t ) f o l l o w i n g a G a u s s i a n d i s t r i b u t i o n . The 95% c o n f i d e n c e i n t e r v a l s f c r t h e l o n g - t e r m c u r r e n t I K E ' s a r e l i s t e d T a b l e I V . These a r e d e r i v e d u s i n g the e n t r i e s ( h a v i n g t h e l a r g e s t T) f r c m T a b l e V I I , and by f i t t i n g t h e FKE e s t i m a t e s t c (a/2) « d i s t r i b u t i o n . Note t h a t t h e v a r i a n c e s f c r t h e component FKE's f r c m t h i s t a b l e have been d o u b l e d , 146 p r i o r t o t h e c a l c u l a t i o n o f t h e c o n f i d e n c e i n t e r v a l s f o r th.e c e n t r a l and w e s t e r n 50m l o c a t i o n s . T h i s has t e e n dene b e c a u s e T a b l e V I I I i n d i c a t e s t h a t t h e v a r i a n c e s , d e r i v e d f r o m e q u a t i o n ( 5 . 6 ) . f o r t h e s e two. l o c a t i o n s , a r e l e w . The c o n f i d e n c e i n t e r v a l s f o r t h e m o n t h l y FKE's may be s i m i l a r l y d e r i v e d u s i n g t h e e n t r i e s ( w i t h T-30 d a y s ) i n T a b l e V I I ; The v a r i a n c e s o f t h e l o n g - t e r m t c t a l FKE e s t i m a t e s f o r t h e f o u r - s i d e l o c a t i o n s a r e c a l c u l a t e d by a s s u m i n g th a t . c c v [ {<j-2)_^ , ( ^ 2 ) ] i s a p p r o x i m a t e l y z e r o . F o r t h e two c e n t r a l ' l o c a t i o n s , t h e c c v a r i a n c e has been a p p r o x i m a t e d as 10% c f [ v a r (dj) +var ( 6 ^ ) v ]. These p r o c e d u r e s a r e t a k e n w i t h . re f e r e nee t c t h e e m p i r i c a l c c v a r i a n c e v a l u e s i n T a b l e V I I I . 147 C h a p t e r 6 TIDAL CUSFENTS 6_.J HJIJB£nic_ A n a l y s i s The r e s u l t s o f t h i s c h a p t e r a r e b a s e d c r h a r m o n i c a n a l y s e s c f 60 day s e c t i o n s c f t h e h a l f - h c u r l y c u r r e n t t i m e -s e r i e s . A h a r m o n i c a n a l y s i s i s e s s e n t i a l l y a l e a s t - s g u a r e s f i t t o t h e c u r r e n t c o m p o n e n t s u s i n g f u n c t i o n s o f t h e f o r m : -u* ( t ) = ^ T B, ( f . ) c o s ( 2 i r f ^ t ) + B z ( f , ) s i n ( 2 t r f ^ t ) ] +a + bt V ( t ) = 2_,.[B3 ( f ^ ) c o s (2TTf^t) + B.^  ( f f e ) s i n (2-rrf^t) ] +c + d t where a , b, c , d and t h e 8^(f^.) 's a r e t h e unknowns. E a c h o s c i l l a t i o n ( t i d a l c o n s t i t u e n t ) w i t h f r e q u e n c y , f ^ , h a s been g i v e n a name by p r e v i o u s w o r k e r s . These names and t h e i r a s s o c i a t e d f r e q u e n c i e s a p p e a r i n T a b l e I X . K i t h t h e e x c e p t i o n o f t h e Mm c o n s t i t u e n t , t h e s e t o f c o n s t i t u e n t s i n t h i s t a b l e i s i d e n t i c a l t o t h e s e t used by t h e C a n a d i a n P i y d r c g r a p h i c S e r v i c e f o r t h e i r a n a l y s e s on 29 day t i m e - s e r i e s . . I n g e n e r a l , e a c h t i d a l c o n s t i t u e n t i s r e p r e s e n t e d by a v e c t o r whose t i p t r a c e s c u t a c o m p l e t e e l l i p s e ( i n t h e h o r i z o n t a l p l a n e ) i n a t i m e q i v e n by 1/fg,. The e l l i p s e may be d e s c r i b e d by i t s s e m i - m a j o r a x i s l e n g t h ( A ^ ) , i t s s e m i - m i n c i a x i s l e n g t h (a„), and t h e o r i e n t a t i o n o f t h e s e m i - m a j o r a x i s 148 TABLE IX - T i d a l Constituents Used For Harmonic Analysis f k(cpd) Constituent 10 11 1 2 3 4 5 6 7 8 9 0.0362 9165 0.0732 0220 0.9295 3571 0.9972 6209 1.0027 3791 1.8959 8197 1.9322 7361 2.0000 0000 2.0054 7582 3.8645 4723 3.9322 7361 M M, M 4 MS , 2850.61 883.09 879.17 2857.65 2835.00 883.08 878.61 2879.30 2879.30 * Term appears i n equation (6.6) 149 (9^) . These a r e r e l a t e d t c t h e B ^ f f ^ ' s a c c o r d i n g t o (6 . 2 ) where: ACj, = y .(B, +B 4) 2+ ( E 3 - E 2 ) ^ ' / 2 r i ^ ( 6 . 5 ) = y ( E , - E t + ) 2 + ( P 3 + B 2 ) 2 l / 2 n ^ t a n ( ^ ) = ( E 3 - B z ) / ( E , +B 4) • t a n i-p^)= ( B 3 + E i ) / ( E , - B H ) and where t h e a r g u m e n t s c f the B j _ ( f ^ ) ' s" have been d r o p p e d . P o s i t i v e and n e g a t i v e v a l u e s o f a^ c o r r e s p o n d t o a n t i - c l o c k w i s e and c l o c k w i s e , r c t a t i c n , r e s p e c t i v e l y , c f t h e v e c t o r when v i e w e d f r o m a b o v e . The node f a c t o r s , n_^, a r e a c t u a l l y s i n u s o i d a l f u n c t i o n s w i t h a p e r i o d o f 16.67 y e a r s ( S h u r e m a n , 1 95 6 ) . H owever, t h e y a r e t a k e n t o t e c o n s t a n t s o v e r a 60 day p e r i o d . iJD£.££iilil>.|ies The d i s c r e t e F o u r i e r t r a n s f o r m has a r e s o l u t i o n of •1/T where T i s t h e l e n g t h o f t h e t i m e - s e r i e s . U n l i k e t h e F o u r i e r t r a n s f o r m , t h e l e a s t - s g u a r e s f i t c f a h a r m o n i c a n a l y s i s has a r e s o l u t i o n g i v e n a p p r o x i m a t e l y by 1/ (T »VSN' ) , where SN i s t h e s i g n a l - t o - n o i s e r a t i o (Hunk and H a s s e l m a n n , 1 9 6 4 ) . In a c t u a l , f a c t , t h e ' ' r e s o l u t i o n c a p a b i l i t y o f a h a r m o n i c a n a l y s i s i s p h r a s e d i n a d i f f e r e n t . m a n n e r : What i s t h e l a r g e s t u n c e r t a i n t y i n t h e B - L ( f ^ ) ' s t h a t c a n be t o l e r a t e d ? L e t t h e v a r i a n c e o f t h e n o i s e be d e n o t e d b y : -150 N v a r ( u ) = N - i ^ { u (t^)-u« ( t ^ ) } 2 (6.4) and s i m i l a r l y f o r v a r ( v ) . H e r e , N i s t h e t c t a l number c f p o i n t s , i n t h e e a s t - w e s t c u r r e n t component t i m e - s e r i e s u (t- ) , o w h i l e u' (t-) i s t h e d i s c r e t e f orm o f u ' ( t ) a p p e a r i n g i n e g u a t i c n ( 6 . 1 ) . a c c o r d i n g t o G o d i n ( 1 9 7 2 ) , t h e u n c e r t a i n t y c r v a r i a n c e i n E , ( f ^ ) and E ^ f f ^ ) i s : -/ A > ^ ( f | i ) = v a r (u) »M^^/| L l (6,5) and s i m i l a r l y f o r ^ ^ f f ^ ) , t h e v a r i a n c e o r u n c e r t a i n t y i n B 3 ( f ^ ) and B^ ( f ^ ) . H e r e , |L| i s t h e d e t e r m i n a n t o f t h e 9x9 m a t r i x (L) whose e l e m e n t s a r e g i v e n by:-(L) = s i n { i f N * t ( f ^ - f ^ ) } / s i n K A t ( f ^ - f ^ )} (6.6) where t h e f r e q u e n c i e s ( , f : k , .1 = 1 ,2 ,... , 9) a r e i d e n t i f i e d . i n T a b l e I X , where K. . i s t h e d e t e r m i n a n t o f t h e m a t r i x f o r m e d by "n.-7t d e l e t i n g t h e k (th ) row and k ( t h ) c o l u m n f r c m t h e m a t r i x ( L ) , and where £t i s t h e s a m p l i n g i n t e r v a l o f t h e t i m e - s e r i e s . A l s o t a b u l a t e d i n T a b l e I X a r e t h e v a l u e s o f j L I / M ^ f c r . 1=6C d a y s and At=3 0 m i n . I t i s assumed t h a t t h e B - b ( f ^ ) 9 s f c l l c w a G a u s s i a n d i s t r i b u t i c n and t h a t B-L ( f ^ ) and Bj_ ( f ^ ) a r e u n c o r r e l a t e d . T h u s , t e r m s s u c h as ( B ^ E ^ ) i n e q u a t i o n (6.3) a l s c f c l l c w a G a u s s i a n d i s t r i b u t i o n w i t h a v a r i a n c e o f A 2 ( ~ A ^ ( ^ ) + A ^ . ( f ^ ) • The 95% c o n f i d e n c e l i m i t s f o r s u c h t e r m s as (E ( + E^) a r e g i v e n by (B,+B^) ±1.96*^\(f^) . These l i m i t s , i n c o n j u n c t i o n w i t h e q u a t i o n s (6.2) and ( 6 . 3 ) , may be used i n c a l c u l a t i n g t h e 95% c o n f i d e n c e l i m i t s f o r A_^, a^ and 6^. A l l t h e c o n f i d e n c e l i m i t s . 151 g i v e n i n t h i s c h a p t e r a r e t h e 95% c o n f i d e n c e l i m i t s d e r i v e d i n t h i s manner. L e t S ^ f f ) d e n o t e t h e power s p e c t r u m o f t h e n o i s e so t h a t r -ee v a r (u)= .Jj Su{f) d f (6.7) - C D The d e r i v a t i o n , c f e q u a t i o n (6.5) assumes t h a t t h e n o i s e i s w h i t e , t h a t S u ( f ) i s a c o n s t a n t f u n c t i o n o f f r e q u e n c y . I n v a r i a b l y , t h i s w i l l n ot be t h e c a s e . Munk and H a s s e l m a n n (1964) g i v e mere g e n e r a l e x p r e s s i o n s w h i c h r e p l a c e e q u a t i o n (6.5) f c r n o n - w h i t e n o i s e . The t e r m s a,b,c and d i n e q u a t i o n (6.1) t a k e c a r e c f any n o n - z e r o means and t r e n d s w h i c h may be p r e s e n t i n t h e c u r r e n t v e c t o r t i m e - s e r i e s . The p r e s e n c e o f t h e s e t e r m s s e r v e t o d e c r e a s e t h e v a r i a n c e c f t h e n o i s e and the u n c e r t a i n t y • i n t h e c a l c u l a t e d e l l i p s e p a r a m e t e r s . The l e a s t - s q u a r e s f i t c f e q u a t i o n (6.1) i n c l u d e t h e Mm and Hf c o n s t i t u e n t s , t h e m o n t h l y and f o r t n i g h t l y t i d a l o s c i l l a t i o n s i n t h e c u r r e n t . As s e e n i n t h e p r e v i o u s c h a p t e r , t h e s e c s c i l l a t i o n s a r e b u r i e d i n t h e l e w -f r e q u e n c y b r e a d - b a n d f l u c t u a t i o n s whose o r i g i n s a r e o t h e r t h a n t i d a l f o r c i n g . E e c a u s e o f t h i s , t h e r e s u l t s f c r t h e Mm and Mf c o n s t i t u e n t s a r e n o t p r e s e n t e d . A g a i n , t h e p r e s e n c e o f t h e s e c o n s t i t u e n t s i n e q u a t i o n (6.1) s e r v e t c d e c r e a s e v a r ( u ) and v a r (v) , and t c d e c r e a s e s l i g h t l y t h e u n c e r t a i n t y i n t h e c a l c u l a t e d e l l i p s e p a r a m e t e r s . H i n d s i g h t g i v e s an ' a l t e r n a t e and p r e f e r r e d metho'd o f m i n i m i z i n g v a r ( u ) and v a r ( v ) . T h i s i s t c h i g h - p a s s f i l t e r t h e h a l f - h o u r l y t i m e - s e r i e s , t o remove t h e 152 b r o a d - b a n d l e w - f r e q u e n c y e n e r g y b e f o r e p e r f o r m i n g t h e h a r m o n i c a n a l y s i s . 6_.2 T i j n e _ V a r i a t i o n s C ^ n t r a l _ 5 0 m _ L o c a t i o n H a r m o n i c a n a l y s e s were p e r f o r m e d on 60 da,y s e c t i o n s c f t k * 5 h§. l£.zh2HElX c u r r e n t t i m e - s e r i e s a t t h e c e n t r a l 5Cm l o c a t i o n . These were c h o s e n sc t h a t t h e t i m e o r i g i n s ( m i d -p o i n t ) o f c o n s e c u t i v e s e c t i o n s a r e s e p a r a t e d by 15 d a y s . The e l l i p s e p a r a m e t e r s , c a l c u l a t e d f r c m e q u a t i e n s (6.2) and ( 6 . 3 ) , a r e p l o t t e d a g a i n s t t h e t i m e o r i g i n s of t h e 60 day s e c t i o n s . I n a d d i t i o n , t h e 95% c o n f i d e n c e l i m i t s f o r t h e e l l i p s e p a r a m e t e r s a r e . a l s o c a l c u l a t e d (as d e s c r i b e d p r e v i o u s l y ) and p l o t t e d . The t i m e v a r i a t i o n s of t h e e l l i p s e p a r a m e t e r s f o r n i n e c o n s t i t u e n t s a p p e a r i n F i g u r e 51 t h r o u g h t o f i g u r e 59. An e l l i p s e p a r a m e t e r i s c o n s i d e r e d c o n s t a n t w i t h t i m e i f a h o r i z o n t a l l i n e c a n be drawn w h i c h l i e s c o m p l e t e l y between t h e 9 5% c o n f i d e n c e l i m i t s o f t h a t p a r a m e t e r . O t h e r w i s e , t h e e l l i p s e p a r a m e t e r i s s a i d t c be t i m e - d e p e n d e n t . The most i n t e r e s t i n g e l l i p s e p a r a m e t e r i s t h e s e m i -m a j o r a x i s l e n g t h ( A ^ ) . I t i s o b s e r v e d t h a t t h e f c r t h e K 2, and M S ^ c o n s t i t u e n t s a r e t i m e - i n v a r i a n t . The A ^ f c r t h e r e m a i n i n g s i x c o n s t i t u e n t s a r e t i m e - d e p e n d e n t . These v a r i a t i o n s i n . A ^ have p e r i o d s o f 3-6 months. • The M 2 c o n s t i t u e n t h a s t h e g r e a t e s t r a n g e , v a r y i n g f r c m 7 cm/s t o 12 cm/s. The A p v a r i a t i o n s c f one c o n s t i t u e n t do n e t c o r r e l a t e 153 e i i i i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r E -S -w -N -E -S -W r 5 a + + + + + + ' A ' M ' J ' J ' A ' S ' O ' N ' D ' J ' F ' M ' A ' M ' J ' J ' A ' S ' ~ 5 ISS9 1970 + + + + + + ±.tjL+ + + + + + + + + + + + + + ++ t + + + + 15 HO I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ^5 1-0 Figure 51'. Time v a r i a t i o n s of the O-^  t i d a l current e l l i p s e and 95% confidence i n t e r v a l s . Parameters A,a, and 8 are the semi-major length, semi-minor length, and o r i e n t a t i o n of the major e l l i p s e axis, r e s p e c t i v e l y . 154 Figure 52.. Time v a r i a t i o n s of the P-]_ t i d a l current e l l i p s e and 95% confidence i n t e r v a l s . Parameters A,a, and 9 are the semi-major length, semi-minor length, and o r i e n t a t i o n of the major e l l i p s e axis, r e s p e c t i v e l y . 155 8 i i i i i i 1 1 1 1 ^ 1 1 1 1 1 1 1 1 r-E -S w hN E S W a o U J CO \ • o r-?r-r A 1 M ' J ' J ' A ' S ' O ' N ' D . ' J ' F ' M ' A ' M ' J 1969 1970 J 1 A ' S 1 L - 5 r!5 H O A o U l CO o h-5 I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1-0 Figure 53. Time v a r i a t i o n s of the t i d a l current e l l i p s e and 95% confidence i n t e r v a l s . Parameters A,a, and 9 are the semi-major length, semi-minor length, and o r i e n t a t i o n of the major e l l i p s e axis, r e s p e c t i v e l y . 156 r-E -S Figure 54. Time v a r i a t i o n s of the N 2 t i d a l current e l l i p s e and 95% confidence i n t e r v a l s . Parameters A,a, and 9 are the semi-major length, semi-minor length, and o r i e n t a t i o n of the major e l l i p s e axis, r e s p e c t i v e l y . 157 Figure 55. Time v a r i a t i o n s of the M 2 t i d a l current e l l i p s e and 95% confidence i n t e r v a l s . Parameters A,a, and 9 are the semi-major length, semi-minor length, and o r i e n t a t i o n of the major e l l i p s e axis, r e s p e c t i v e l y . 158 r-E -S Figure 56. Time v a r i a t i o n s of the S 2 t i d a l current e l l i p s e and 95% confidence i n t e r v a l s . Parameters A,a, and 9 are the semi-major length, semi-minor length, and o r i e n t a t i o n of the major e l l i p s e axis, r e s p e c t i v e l y . 159 Figure 57.. Time v a r i a t i o n s of the K 2 t i d a l current e l l i p s e and 95% confidence i n t e r v a l s . Parameters A,a, and 9 are the semi-major length, semi-minor length, and o r i e n t a t i o n of the major e l l i p s e axis, r e s p e c t i v e l y . 160 Figure 58. Time v a r i a t i o n s of the M4 t i d a l current e l l i p s e and 95% confidence i n t e r v a l s . Parameters A,a, and 9 are the semi-major length, semi-minor length, and o r i e n t a t i o n of the major e l l i p s e axis, r e s p e c t i v e l y . 161 Figure 59.. Time v a r i a t i o n s of the MS4 t i d a l current e l l i p s e and 95% confidence i n t e r v a l s . Parameters A,a, and 9 are the semi-major length, semi-minor length, and o r i e n t a t i o n of the major e l l i p s e axis, r e s p e c t i v e l y . 162 w e l l w i t h t h o s e o f any o t h e r c o n s t i t u e n t . The s i g n c f t h e s e m i - m i n e r a x i s l e n g t h (a^) d e t e r m i n e s t h e r o t a t i o n a l s e n s e o f t h e c o n s t i t u e n t e l l i p s e . A n t i -c l o c k w i s e and c l o c k w i s e r c t a t i c n (as v i e w e d f r e i r above) a r e r e p r e s e n t e d by p o s i t i v e and n e g a t i v e v a l u e s , r e s p e c t i v e l y . I t i s s e e n t h a t t h e a^ f o r t h e 0, , S^, K z , M^ and MS^ c o n s t i t u e n t s a r e t i m e - i n v a r i a n t . I n a d d i t i o n , t h e r o t a t i o n a l s e n s e c f t h e S z , M and KS^ c o n s t i t u e n t , e l l i p s e s i s a m b i g u o u s . • T h a t i s , h o r i z o n t a l l i n e s can be drawn w h i c h l i e c o m p l e t e l y between t h e 95% c o n f i d e n c e l i m i t s , and which l i e b o t h above and below t h e z e r o v a l u e . Cf t h e n i n e c o n s t i t u e n t s , t h e Nz - c o n s t i t u e n t has t h e l a r g e s t J } v a l u e c f 3.cm/s. The r o t a t i o n a l s e n s e c f t h e N ? and M e l l i p s e s i s t i m e -d e p e n d e n t . C u r i n g J u n e and J u l y c f b o t h y e a r s , t h e a^ o f t h e s e -two e l l i p s e s a t t a i n t h e i r most p o s i t i v e v a l u e s . These e v e n t s c o i n c i d e w i t h t h e F r a s e r F i v e r f r e s h e t . W i t h t h e e x c e p t i o n c f t h e s e two c o n s t i t u e n t s , t h e a^ v a r i a t i o n s c f one c o n s t i t u e n t c o r r e l a t e p o o r l y with, t h o s e c f any o t h e r c o n s t i t u e n t . T h e r e i s an a m b i g u i t y o f 180° i n , t h e o r i e n t a t i o n c f t h e s e m i - m a j o r a x i s . I n most c a s e s , v a l u e s o f 8^ a r e c h o s e n w h i c h l i e i n t h e h a l f - c i r c l e f r o m e a s t t o w e s t , and w h i c h i n c l u d e s n c r t h . As A^ d e c r e a s e s , t h e c o n f i d e n c e l i m i t s f o r 9^ i n c r e a s e . I t i s s e e n t h a t t h e o r i e n t a t i o n s c f t h e 0, , P,, and MS e l l i p s e s a r e t i m e - i n v a r i a n t . T h a t o f t h e r e m a i n i n g f i v e c o n s t i t u e n t s a r e t i m e - d e p e n d e n t . The r a n g e c f t h e 9^ c h a n g e s f c r t h e K, e l l i p s e , w h i c h i s 33°, i s l a r g e r t h a n t h a t f o r t h e L % e l l i p s e , w h i c h i s 19°. P r o b a b l y , the r a n g e f c r t h e 163 K j e l l i p s e i s g r e a t e r b e c a u s e t h e A^ f c r t h i s e l l i p s e i s s m a l l e r . The K, e l l i p s e c r i e n t a t i o n i s p r e d o m i n a n t l y NNW, w h i l e t h a t f o r t h e Nz e l l i p s e i s p r e d o m i n a n t l y n o r t h . F c r t h o s e c o n s t i t u e n t s w i t h a t i m e - v a r y i n g o r i e n t a t i o n , t h e 8^ c h a n g e s o f one e l l i p s e dc n e t c o r r e l a t e w e l l w i t h t h e s e c f any o t h e r e l l i p s e . _ G e n e r a l l y , i t a p p e a r s t h a t i f t h e r e a r e i n t e r a c t i o n s i n t h e c u r r e n t s between t h e t i d a l c o n s t i t u e n t s , t h e s e i n t e r a c t i o n s do n o t a p p e a r as o b v i o u s c o r r e l a t i o n s i n t h e e l l i p s e p a r a m e t e r s c f any two c o n s t i t u e n t s . I n t h e p r e v i o u s c h a p t e r , t h e m o n t h l y means o f t h e c u r r e n t ( a t t h e c e n t r a l 50m l o c a t i o n ) , w i n d , a i r - p r e s s i r e and s e a - l e v e l a r e p r e s e n t e d . The v a r i a t i o n s i n t h e e l l i p s e p a r a m e t e r s a r e p o o r l y c o r r e l a t e d w i t h t h e s e p a r a m e t e r s p r e s e n t e d i n t h e p r e v i o u s c h a p t e r . I n a d d i t i o n , t h e v a r i a t i o n s i n t h e e l l i p s e p a r a m e t e r s a r e a l s o p o o r l y c o r r e l a t e d w i t h t h e v a r i a t i o n s i n t h e m o n t h l y c u r r e n t (at t h e c e n t r a l 50m l o c a t i o n ) and w i t h t h e v a r i a t i o n s i n t h e m o n t h l y w i n d , s e a - l e v e l , and a i r - p r e s s u r e v a r i a n c e s . The s e m i - m i n o r a x i s p a r a m e t e r (a^) c f t h e N and K e l l i p s e s a r e t h e o n l y e x c e p t i o n s t c t h e p r e v i o u s s t a t e m e n t . . I t i s o b s e r v e d t h a t t h e a^ f o r t h e s e two e l l i p s e s a r e most p o s i t i v e d u r i n g J u n e and J u l y o f b o t h y e a r s , and t h e s e f e a t u r e s c o i n c i d e w i t h t h e l e w summer v a r i a n c e s o b s e r v e d i n b o t h a i r - p r e s s u r e and s e a - l e v e l . I t a p p e a r s t h a t i f t h e r e a r e i n t e r a c t i o n s b e t ween t h e l e w - f r e q u e n c y c u r r e n t s ( C h a p t e r 5) and t h e t i d a l c u r r e n t s , t h e s e i n t e r a c t i o n s do n o t a p p e a r a s o b v i o u s c o r r e l a t i o n s 16 4 between t h e p a r a m e t e r s d i s c u s s e d i n t h e p r e c e d i n g p a r a g r a p h . F i y e _ O t h e r _ l o c a t i o n s The c u r r e n t t i m e - s e r i e s f c r c e n t r a l 50m l o c a t i o n i s the l o n g e s t c o n t i n u o u s one. Those f o r t h e f i v e o t h e r l o c a t i o n s c o n t a i n s i g n i f i c a n t g a p s . H a r m o n i c a n a l y s e s were a l s o p e r f o r m e d cn 60 day s e c t i o n s o f t h e h a l f - h c u r l_y c u r r e n t t i m e -s e r i e s at. t h e f i v e o t h e r l o c a t i o n s . The t i m e o r i g i n i s a g a i n t a k e n t o be t h e m i d - p o i n t o f e a c h 60 day s e c t i o n . F o r t h e f i v e o t h e r l o c a t i o n s , t h e t i m e o r i g i n s o f c o n s e c u t i v e 60 day s e c t i o n s a r e s e p a r a t e d by 45 d a y s whenever p o s s i b l e , and n o t by 15 d a y s a s f o r t h e c e n t r a l 50m l o c a t i o n . The 45 day s e p a r a t i o n i s u s e d t c d e c r e a s e t h e amount o f c o m p u t a t i o n a l l a b o u r . The e l l i p s e p a r a m e t e r s a l o n g w i t h t h e i r 95% c o n f i d e n c e l i m i t s a r e t a b u l a t e d i n A p p e n d i x I . I n t h i s a p p e n d i x , the. d i r e c t i o n s c f ( e a s t , n o r t h , w e s t ) a r e r e p r e s e n t e d by (0° , 90° , 180°) , w h i l e s o u t h i s r e p r e s e n t e d by -90°. T h i s c o n v e n t i o n f o r t h e d i r e c t i o n s i s t h e s t a n d a r d m a t h e m a t i c a l one. ' A s . b e f o r e , an e l l i p s e p a r a m e t e r i s c o n s i d e r e d t o be t i m e - i n v a r i a n t i f a h o r i z o n t a l l i r e c a n be drawn (cn a g r a p h c f t h e p a r a m e t e r v e r s u s t i m e ) w h i c h c o m p l e t e l y l i e s between t h e 9 5% c o n f i d e n c e l i m i t s . E x a m i n a t i o n o f A p p e n d i x I shows t h a t t h e s e m i - m a j o r a x i s l e n g t h (A^) o f t h e M^ e l l i p s e a t t h e e a s t e r n 50m l o c a t i o n i s t i m e - i n v a r i a n t . The A^'s c f the K z e l l i p s e a t t h e r e m a i n i n g f o u r l o c a t i o n s a r e t i m e - d e p e n d e n t . I t i s o b s e r v e d t h a t t h e A» c f t h e K, e l l i p s e i s t i m e - d e p e n d e n t 165 o n l y a t t h e deep e a s t e r n and c e n t r a l l o c a t i o n s . The c f the 0, e l l i p s e - i s seen t o v a r y a t t h r e e o f t h e f i v e . l o c a t i o n s . T h o se o f t h e r e m a i n i n g e l l i p s e s a r e s e e n t c v a r y a t 'two c r l e s s c f the- f i v e l o c a t i o n s . The s e m i - m i n e r a x i s l e n g t h (a^) of t h e M 2 e l l i p s e i s t i m e - d e p e n d e n t a t a l l f i v e l o c a t i o n s . However, i t s r o t a t i o n a l s e n s e i s c n l y o b s e r v e d t c v a r y a t t h e deep e a s t e r n l o c a t i o n . The a ^ o f t h e K, e l l i p s e i s t i m e - d e p e n d e n t o n l y a t t h e c e n t r a l 200m l o c a t i o n . At t h r e e o f t h e f i v e l o c a t i o n s , t h e a ^ c f the 0, e l l i p s e i s t i m e - d e p e n d e n t . The a ^ o f t h e r e m a i n i n g s i x c o n s t i t u e n t s a r e t i m e - i n v a r i a n t a t a l l f i v e l o c a t i o n s . The o r i e n t a t i o n (8^) of t h e M 2 e l l i p s e i s t i m e -d e p e n d e n t a t a l l f i v e l o c a t i o n s . T h a t o f t h e K, e l l i p s e i s t i m e - d e p e n d e n t c n l y a t t h e w e s t e r n 50m and e a s t e r n 200m l o c a t i o n s . At two o f t h e f i v e l o c a t i o n s , t h e 8^ , o f t h e - N 2 e l l i p s e i s t i m e - d e p e n d e n t . The 8^ of t h e r e m a i n i n g s i x e l l i p s e s a r e t i m e - d e p e n d e n t a t o n l y one o r none c f t h e f i v e • l o c a t i o n s . The M 2 e l l i p s e p a r a m e t e r s a t any p a i r o f l o c a t i o n s may be p l o t t e d a g a i n s t t i m e f c r c o m p a r i s o n . The s i m u l t a n e o u s t i m e c o v e r a g e o f t h e d a t a f o r a g i v e n p a i r c f l o c a t i o n s iray be s m a l l . However, t h e c o v e r a g e i s s u f f i c i e n t t c s u g g e s t t h a t t h e M2 e l l i p s e p a r a m e t e r s a t one l o c a t i o n c o r r e l a t e . p o o r l y w i t h t h e c o r r e s p o n d i n g K 2 e l l i p s e p a r a m e t e r s a t any o t h e r l o c a t i o n . T h i s i s a l s o t r u e f o r c o m p a r i s o n s b e t w e e n c o r r e s p o n d i n g p a r a m e t e r s o f t h e o t h e r c o n s t i t u e n t s w h i c h a r e t i m e - d e p e n d s n t , The c n l y e x c e p t i o n t c t h e s e p e e r c o r r e l a t i o n s a p p e a r s t o be t h e 166 A^ p a r a m e t e r f o r t h e -Mz c o n s t i t u e n t a t t h e w e s t e r n m o o r i n g . T h e r e , t h e o f t h e e l l i p s e a t t h e 200m l o c a t i o n d e c r e a s e s w i t h t i m e , w h i l e t h a t a t t h e 50m l o c a t i o n i n c r e a s e s w i t h t i m e . I n g e n e r a l h o w e v e r , t h e v a r i a t i o n s o f t h e e l l i p s e p a r a m e t e r s a t any one l o c a t i o n do n e t a p p e a r t o be h i g h l y c o h e r e n t w i t h t h e v a r i a t i o n s - o f t h e c o r r e s p o n d i n g e l l i p s e p a r a m e t e r s a t any o t h e r l o c a t i o n . 6_.3 Iii!iirAliI3SS^ _ l S£a m e t e r s The e l l i p s e p a r a m e t e r s a r e t i m e - a v e r a g e d and t a b u l a t e d i n T a b l e X.. F o r t h e c e n t r a l 50m l o c a t i o n , t h e y a r e d e r i v e d f r c m t h e g r a p h s a p p e a r i n g i n F i g u r e 51 t h r o u g h t c F i g u r e 59. F o r t h e e t h e r l o c a t i o n s , t h e t i m e - a v e r a g e d v a l u e s are d e r i v e d f r o m t h e v a l u e s i n A p p e n d i x I . The c o n f i d e n c e l i m i t s a r e i g n o r e d . At a l l s i x l o c a t i o n s , t h e and K, e l l i p s e s h a v e t h e l a r g e s t and s e c o n d l a r g e s t a m p l i t u d e s ( A ^ ) , r e s p e c t i v e l y . A c c o r d i n g t c e q u i l i b r i u m t i d e t h e o r y , t h e t h r e e d i u r n a l c o n s t i t u e n t s i n o r d e r o f d e c r e a s i n g a m p l i t u d e a r e (K 0 , , P ( ) . T h i s c r d e r i s o b s e r v e d a t a l l t h e l o c a t i o n s e x c e p t t h e e a s t e r n 50m l o c a t i o n . The f o u r s e m i - d i u r n a l c o n s t i t u e n t s i n o r d e r o f d e c r e a s i n g a m p l i t u d e a r e ( M z , 5Z , iN 2 , Kz ) . T h i s c r d e r i s o b s e r v e d a t a l l l o c a t i o n s e x c e p t t h e ' e a s t e r n 140m l o c a t i o n . At a l l s i x l o c a t i o n s , t h e a m p l i t u d e i s l a r g e r t h a n t h e MS H a m p l i t u d e . At a l l t h r e e m o o r i n g s , b o t h t h e 0, and K ( a m p l i t u d e s i n c r e a s e w i t h d e p t h . B o t h t h e Uz and M 2 a m p l i t u d e s i n c r e a s e TABLE X - Average T i d a l Current E l l i p s e s Depth 50m 200m 50ra 200m 50m 200m 50m 200m 50m 200m 50m 200m 50m 200m 50m 200m 50m 200m A (cm/s)-2.91 3.10 2. 55 1. 78 5. 88 5.94 2. 25 2.63 8. 52 11. 10 3.02-2. 64 .95 .96 .36 . 62 Western Station (cm/s) (deg.) .29 .55 . 22 -.97 . 52 -.59 . 53 -1.58 .03 -1. 12 1.67 -3.97 -.04 -.95 -.33 -.35 .12 .15 -.10 -.09 106 97 116 86 110 97 112 120 109 120 86 120 92 104 102 89 105 74 Kl M2 A (cm/s) 83 63 29 47 52 21 90 . 31 .70 70 62 53 53 MS; .61 .47 .47 .31 Central Station a (cm/s) . 37 . 35 . 32 -.18 . 69 .05 . 04 .27 .73 . 39 .23 . 19 .67 . 24 .25 .04 .15 .01 9 (deg.) 119 96 106 98 117 97 95 111 95 123 101 117 112 88 111 91 130 93 A (cm/s) 1. 20 2.90 '2.45 1.94 3.06 5.10 1.60 .97 6. 70 6. 31 1.65 2.19 .93 1.80 .49 .86 .26 . 61 Eastern Station (cm/s) (deg.) .24 .50 . 14 .08 .59 .56 .02 .02 .26 .55 .33 . 31 . 33 .17 .16 .02 .04 .04 87 79 99 109 89 76 92 87 92 111 116 38 94 16 142 50 157 69 MS/, Depth 50m 140m 50m 140m 50m 140m 50m 140m 50m 140m 50m 140m 50m 140m 50m 140m 50ra 140m • 168 w i t h d e p t h a t t h e w e s t e r n m c c r i n g , b u t b o t h d e c r e a s e w i t h d e p t h a t t h e r e m a i n i n g two m o o r i n g s . At a l l t h r e e m o o r i n g s , b o t h t h e t and MS^ a m p l i t u d e s i n c r e a s e w i t h d e p t h . At t h e w e s t e r n 200m l o c a t i o n , a l l c o n s t i t u e n t e l l i p s e s a r e r o t a t i n g c l o c k w i s e , w h i l e a t t h e c e n t r a l 200m l o c a t i o n s e v e n o u t o f n i n e e l l i p s e s a r e r o t a t i n g c l o c k w i s e . I he 0, e l l i p s e t h r o u g h t c t h e M 2 e l l i p s e r o t a t e a n t i - c l o c k w i s e a t t h e w e s t e r n 50m l o c a t i o n . At t h e same l o c a t i o n , t h e r e m a i n i n g e l l i p s e s r o t a t e c l o c k w i s e . I h e p r e c e d i n g two s t a t e m e n t s a l s o a p p l y t o t h e c o n s t i t u e n t e l l i p s e s a t t h e c e n t r a l 50m l e c a t i c n . The r o t a t i o n a l s e n s e o f t h e N a and M 2 e l l i p s e s change w i t h d e p t h a t a l l t h r e e m o o r i n g s . At a l l t h r e e m o o r i n g s , t h e o r i e n t a t i o n c f the M 2 e l l i p s e i n c r e a s e s w i t h d e p t h , w h i l e t h o s e o f t h e 0, and K, e l l i p s e s d e c r e a s e w i t h d e p t h . 0..4 lHi£££al_li^s One may e x p e c t i n t e r n a l ( b a r o c l i n i c ) t i d e s i n a s y s t e m where t h e r e e x i s t s a s i g n i f i c a n t d e n s i t y s t r a t i f i c a t i o n and r a p i d c h a n g e s i n b o t t o m t o p o g r a p h y ( F o r r e s t e r , 1 9 7 4 ) . T h e s e c o n d i t i o n s e x i s t i n t h e v i c i n i t y c f t h e t h r e e m o o r i n g s . The o b s e r v e d d e p t h v a r i a t i o n s i n t h e e l l i p s e a m p l i t u d e (A^) s u p p o r t t h e h y p o t h e s i s o f i n t e r n a l t i d e s . F c r e x a m p l e , a t t h e e a s t e r n m c c r i n g d u r i n g March 1969, t h e K, a m p l i t u d e d e c r e a s e s w i t h d e p t h . T h i s d i f f e r e n c e i n t h e K, a m p l i t u d e s c a n n o t be e x p l a i n e d by t h e 95% c o n f i d e n c e l i m i t s . The same i s 169 t r u e o f t h e ' K, a m p l i t u d e d i f f e r e n c e s a t t h e c e n t r a l m o o r i n g d u r i n g May 1970, and o f t h e M 2 a m p l i t u d e d i f f e r e n c e s a t t h e w e s t e r n m o o r i n g d u r i n g A u g u s t 1969. The t i m e - v a r y i n g c u r r e n t e l l i p s e parameter-s c a n n o t be e x p l a i n e d by t h e v a r i a t i o n s i n t h e c o n s t i t u e n t a m p l i t u d e s f o r t h e s e a s u r f a c e . F o u r i e r t r a n s f o r m s were p e r f o r m e d on n i n e 60 day s e c t i o n s o f s e a - l e v e l a t P t . A t k i n s o n . The m i d - p o i n t t i m e o r i g i n s o f t h e s e s e c t i o n s r a n g e d f r o m May 16, 1969 t o S e p t e m b e r 8, 1970. A f t e r p e r f o r m i n g t h e node f a c t o r c o r r e c t i o n s , t h e M a a m p l i t u d e s a r e o b s e r v e d t o v a r y f r o m 89.35 cm t o 94.60 cm. R e l a t i v e t o t h e a v e r a g e v a l u e o f 91.64 cm, t h i s r e p r e s e n t s a 6% v a r i a t i o n . • H owever, t h e a m p l i t u d e s o f t h e M 2 c u r r e n t e l l i p s e r a n g e f r o m 7 cm/s t o 12 cm/s." R e l a t i v e t o t h e a v e r a g e v a l u e o f 9 cm/s, t h i s r e p r e s e n t s a 5 6 % v a r i a t i o n . I n t e r n a l t i d e s a r e t h e most- p l a u s i b l e e x p l a n a t i o n o f t h e t i m e - v a r y i n g e l l i p s e p a r a m e t e r s . The c h a r a c t e r o f t h e i n t e r n a l t i d e s i s e x p e c t e d t o c h a n g e • w i t h c h a n g e s i n t h e d e n s i t y s t r a t i f i c a t i o n , -and t h e s e a r e known t o e x i s t i n t h e v i c i n i t y o f t h e t h r e e m o o r i n g s ( C r e a n and A g e s , 1 9 7 1 ) . U n f o r t u n a t e l y , t h e r e a r e no t i m e - s e r i e s o f d e n s i t y p r o f i l e s f o r 1969 and 1970, p r o f i l e s w h i c h may-or may n o t ba w e l l c o r r e l a t e d w i t h t h e t i m e - v a r y i n g e l l i p s e p a r a m e t e r s . C h a p t e r 7 SUMMAEY S u b - s u r f a c e c u r r e n t s i n t h e S t r a i t o f G e o r g i a n e a r S t u r g e o n Bank have been a n a l y s e d . The c u r r e n t s , c o l l e c t e d a t s i x l o c a t i o n s cn t h r e e m o o r i n g s , s p a n .319-533 d a y s . T h e r e a r e two main p a r t i t i o n s o f e n e r g y : b r o a d - b a n d l c w - f r e g u e n c y and n a r r o w - b a n d t i d a l m o t i o n s . T o g e t h e r t h e y a c c o u n t f c r 80% c f t h e e n e r g y i n t h e c u r r e n t s . The 1 c w - f r e g u e n c y m o t i o n s a r e l o c a t e d i n peak c e n t r e d a t 30 d a y s . They a r e n e i t h e r p r e d o m i n a n t l y c l o c k w i s e nor p r e d o m i n a n t l y a n t i - c l o c k w i s e . The c u r r e n t s a t t h e t h r e e 50m l o c a t i o n s c o n t a i n e q u a l amounts o f l e w - f r e q u e n c y e n e r q y . T h i s e n e r q y i s a l s o a p p r o x i m a t e l y e q u a l t o t h e t i d a l e n e r q y . The e v i d e n c e s u g g e s t s t h a t t h e l o w - f r e q u e n c y e n e r q y may d e c r e a s e l i n e a r l y w i t h d e p t h . C l o s e t o t h e s i d e b o u n d a r i e s c f t h e S t r a i t , the l o w - f r e q u e n c y m e t i e n s a r e c e n s t r a i n e d t c be h i g h l y r e c t i l i n e a r . St f i v e l o c a t i o n s , t h e c u r r e n t means a r e n o r t h w a r d , p a r a l l e l t c t h e d e p t h c o n t o u r s . The mean c u r r e n t i s d i r e c t e d ESE a t t h e c e n t r a l 50m l o c a t i o n . M o o r i n g s were p o s i t i o n e d l a t e r a l l y a c r o s s t h e S t r a i t a t 10km i n t e r v a l s . The l e w - f r e q u e n c y m o t i o n s a r e e s s e n t i a l l y i n c o h e r e n t , i n d i c a t i n g t h a t t h e y have a l a t e r a l l e n g t h s c a l e much s m a l l e r t h a n 10km.. I n a d d i t i o n , t h e s e m o t i o n s a r e v e r y b a r c c l i n i c . l o w - f r e q u e n c y m o t i o n s a t t h e e a s t e r n m e c r i n q a r e 111 t h e l e a s t b a r o c l i n i c . W ind, a i r - p r e s s u r e and s e a - l e v e l a l s o have - t r e a d - b a n d l e w - f r e q u e n c y f l u c t u a t i o n s w i t h c h a r a c t e r i s t i c t i m e s c a l e s c f 3, 10 and 30 d a y s , r e s p e c t i v e l y . B o t h s e a - l e v e l and a i r -p r e s s u r e have h o r i z o n t a l l e n g t h s c a l e s a s l e a s t as l a r g e as t h e l e n g t h o f V a n c o u v e r I s l a n d . The wind has a s m a l l e r l e n g t h s c a l e , b u t i t i s s t i l l much l a r g e r t h a n t h e s e o f t h e l o w -f r e q u e n c y c u r r e n t s . l o w - f r e q u e n c y s e a - l e v e l and a i r - p r e s s u r e f l u c t u a t i o n s a r e h i g h l y c o h e r e n t . Wind and a i r - p r e s s u r e v a r i a t i o n s a r e much l e s s c o h e r e n t , as a r e t h e wind and s e a -l e v e l v a r i a t i o n s . G e n e r a t i o n m e c hanisms o f t h e l e w - f r e q u e n c y c u r r e n t s a r e not. i d e n t i f i e d . L e w - f r e q u e n c y f l u c t u a t i o n s o f c u r r e n t s and a i r - p r e s s u r e a r e net c o h e r e n t . T hose c f c u r r e n t s and s e a - l e v e l e x h i b i t p e c r - t o - f a i r c o h e r e n c e . A t t h e e a s t e r n m o o r i n g , t h o s e o f c u r r e n t s and wind have f a i r c o h e r e n c e . T h i s l e n d s e n c o u r a g e m e n t t c a f u r t h e r s t u d y c f t h e l e w - f r e q u e n c y c u r r e n t s and w i n d i n t h e v i c i n i t y o f t h i s m o o r i n g . T i d a l c u r r e n t s a t 50m and 200m r e p r e s e n t 40% and 7 0 % c f t h e e n e r g y . The l a r g e s t c o m p o n e n t s a r e t h e and K, c o n s t i t u e n t s . T i d a l c u r r e n t s a r e h i g h l y c o h e r e n t b o t h .in t h e h o r i z o n t a l and v e r t i c a l . The p a r a m e t e r s d e s c r i b i n g t h e t i d a l c u r r e n t e l l i p s e s a r e t i m e - d e p e n d e n t . T i m e - v a r y i n g p a r a m e t e r s f c r d i f f e r e n t c o n s t i t u e n t s a r e p c c r l y c o r r e l a t e d , as. a r e t h o s e f c r d i f f e r e n t l o c a t i o n s . G e n e r a l l y , t h e e l l i p s e p a r a m e t e r v a r i a t i o n s a r e p c c r l y c o r r e l a t e d w i t h t h e s e i n t h e m o n t h l y means and v a r i a n c e s o f t h e l e w - f r e q u e n c y c u r r e n t s , w i n d , s e a -172 l e v e l and a i r - p r e s s u r e . V a r i a t i o n s o f t h e t i d a l e l l i p s e p a r a m e t e r s c a n n o t he e x p l a i n e d by v a r i a t i o n s o f t h e t i d a l c o n s t i t u e n t s f c r s e a -l e v e l , w h i c h a r e r e l a t i v e l y s n a i l . I n t e r n a l t i d e s a r e h y p o t h e s i z e d . The c h a n g i n g e l l i p s e p a r a m e t e r s a r e t h e n c o n n e c t e d t c t h e c h a n g i n g c h a r a c t e r o f t h e i n t e r n a l t i d e , w h i c h i s e f f e c t e d by t h e c h a n g i n g d e n s i t y s t r a t i f i c a t i o n . ' T i m e - s e r i e s o f w a t e r t e m p e r a t u r e s p a n 307-411 d a y s . These t e m p e r a t u r e f l u c t u a t i o n s a r e b r e a d - b a n d w i t h a c h a r a c t e r i s t i c t i m e s c a l e c f 300 d a y s . The t i m e - s e r i e s a r e t o o s h o r t t c be c o n s i d e r e d - s t a t i o n a r y . W ater t e m p e r a t u r e s at d i f f e r e n t l o c a t i o n s g e n e r a l l y have l o w c o h e r e n c e , as have w a t e r t e m p e r a t u r e and c u r r e n t a t t h e same l o c a t i o n . 173 BIBLIOGRAPHY B e n d a t , J . S . and A.G. P i e r s o l (1:971). Random_Data:. Analysis s f i : i _ . ! l e a s u r & m e n r _ P r o c e d u r e s . W i l a y - I r . t e r s c i . e n c e , New York'. 40 7 pp. B r i g h a m , 3.0. ( 1 9 7 4 ) . The . F a s i . F o u r i e r J r a n g f o r u u P r e n t i c e -H a l l I n c . , E n g l e w o o d C l i f f s , New J e r s e y . 252 pp. C o r d e s , E.-2. ( i n p r o g r e s s ) . The B e h a v i o r o f t n e f r a s e r R i v e r Plume D u r i n g S p r i n g 1974. M.Sc. T h e s i s , D e p a r t m e n t o f P h y s i c s and I n s t i t u t e o f O c e a n o g r a p h y , 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 . C hang, P.Y.K., S. Pond and • S. T a b a t a ( 1 9 7 6 ) . S u b s u r f a c e C u r r e n t s i n t h e s t r a i t o f G e o r g i a , West o f • S t u r g e o n Bank. J o u r n a l o f t h e F i s h e r i e s R e s e a r c h B o a r d o r Canada. I n p r e s s . C r e a n , P.B. and A.3. Ages ( 1 9 7 1 ) . O c e a n o g r a p h x c R e c o r d s f r o m T w e l v e C r u i s e s i n t h e S t r a i t o f G e o r g i a and Juan' de' F u c a S t r a i t , 1963. D e p a r t m e n t o f E n e r g y , M i n e s and R e s o u r c e s , M a r i n e S c i e n c e s B r a n c h , V o l . 1-5.. 389 pp. D e p a r t m e n t o f t h e E n v i r o n m e n t ( 1 9 7 2 a ) . D a t a H a c o r d o f C u r r e n t O b s e r v a t i o n s , S t r a i t o f G e o r g i a . , S e c t i o n 2, Cape L a z o t o G r i e f P o i n t , 1970. w a t e r Management S e r v i c e , M a r i n e S c i e n c e s D i r e c t o r a t e , ' P a c i f i c R e g i o n , M a n u s c r i p t R e p o r t S e r i e s , V o l . 9. 83 pp. ( 1 9 7 2 b ) . D a t a R e c o r d o f C u r r e n t O b s e r v a t i o n s , S t r a i t o f G e o r g i a , S e c t i o n 4, G a b r i o l a I s l a n d t o Gower P o i n t , 1969-1 972. Water ..Management S e r v i c e , . c i a r i n e S c i e n c e s D i r e c t o r a t e , P a c i f i c R e g i o n , M a n u s c r i p t R e p o r t S e r i e s , V o l . 1.0. 153 pp. ( 1 9 7 2 c ) . D a t a Re-cord o f C u r r e n t O b s e r v a t i o n s , S t r a i t o f G e o r g i a , S e c t i o n 5, P o r l i e r P a s s t o Sand d e a d s , 1969-1972. Water Management S e r v i c e , M a r i n e S c i e n c e s D i r e c t o r a t e , P a c i f i c R e g i o n , M a n u s c r i p t R e p o r t S e r i e s , V o l . JM. 124 pp. 174 ( 1973). D a t a R e c o r d o f C u r r e n t O b s e r v u t i o i i s , S t r a i t o f G e o r g i a , S e c t i o n 6, Samuel I s l a n d t o P o i n t R o b e r t s , 1969-1970. w a t e r Management S e r v i c e , M a r i u e S c i e n c e s D i r e c t o r a t e , P a c i f i c R e g i o n , M a n u s c r i p t R e p o r t ' S e r i e s , V o l . 12. 96 pp. F o r r e s t e r , W..D. ( 1 9 7 4 ) . I n t e r n a l T i d e s i n S t . L a w r e n c e E s t u a r y . J o u r n a l o f M a r i n e R e s e a r c h , Vox. 3 2 ( 1 ) , pp. 55-66. G i o v a n d o , L.F. and S. T a b a t a ( 1 9 7 0 ) . M e a s u r e m e n t s o f S u r f a c e F l o w i n t h e S t r a i t o f G e o r g i a by Means- o f F r e e - F l o a t i n g C u r r e n t F o l l o w e r s . F i s h e r i e s R e s e a r c h B o a r d o f C a n a d a , T e c h n i c a l R e p o r t No. _16j. 69 pp. G o d i n G. ( 1 9 7 2 ) . T h e _ A n a J . y s j . s _ o _ _ _ _ _ _ _ . U n i v e r s i t y o f T o r o n t o P r e s s , T o r o n t o . 264 pp. Sfiec t r a l _ - A n a l y s i s _ o f U n i v e r s i t y P r e s s , G r a n g e r , C . S . J . and M. H a t a n a k a ( 1 9 6 4 ) . ^£2H2iSi£_Xii!lS_5sries. P r i n c e t o n -P r i n c e t o n , New J e r s e y . 299 pp. G r o v e s , G.W. and E . J . Hannan ( 1 9 6 8 ) . Time S e r i e s R e g r e s s i o n o f Sea L e v e l on We a t h e r . R e v i e w s o f G e o p h y s i c s , V o l . ' 6 ( 2 ) , pp. 129-174. H u g g e t t , W.S. ( 1 9 6 6 ) . Some A n o m a l i e s i n T a x i n g S h o r t Term O b s e r v a t i o n s on t h e West C o a s t . C a n a d i a n i i y d r o g r a p h i c S e r v i c e , M a r i n e S c i e n c e s B r a n c h , D e p a r t m e n t o f M i n e s and T e c h n i c a l S u r v e y s , T e c h n i c a l R e p o r t . 3 3 pp. H u t c h i n s o n , A.H. and C C . L u c a s (19.31) . Tiie S p i t n a l a s s a o f t h e S t r a i t o f G e o r g i a : S a l i n i t y , T e m p e r a t u r e , pH, and P h y t o p l a n k t o n . C a n a d i a n J o u r n a l o f R e s e a r c h , V o l . 5 ( 2 ) , pp. 231-284. J e n k i n s , G.M. and D.G. W a t t s ( 1 5 6 8 ) . S£ectral_Anail§ iS_^a4_ I I § £E2li£ii.i2H§- Holder.-Day, San F r a n c i s c o . 525 pp. L a c o s s , R.T. ( 1 9 7 1 ) . D a t a A d a p t i v e S p e c t r a l A n a l y s i s M e t h o d s . G e o p h y s i c s , V o l . 3 6 ( 4 ) , pp. 66 1 - 6 7 5 . 175 Wooers, C.N.K. ( 1 9 7 3 ) . A t e c h n i g u e f o r t h e c r o s s s p e c t r u m a n a l y s i s o f p a i r s o f c o m p l e x - v a l u e d t i m e s e r i e s , w i t h e m p h a s i s on p r o p e r t i e s o f p o l a r i z e d c o m p o n e n t s and r o t a t i o n a l i n v a r i a n t s . Deep Sea R e s e a r c h , V o l . 2 0 ( 1 2 ) , pp. 1129-1141. Munk, W. and K. H a s s e l m a n n ( 1 9 6 4 ) . 5 u p e r - r e s o l u t ^ o n o f t i d e s . S t _ d i e s _ q n _ o c e a _ o _ r a _ h y , e d i t e d by Kozo Y o s h i d a , U n i v e r s i t y o f W a s h i n g t o n P r e s s , S e a t t l e , pp. 339-344. P i c k a r d , G . l . ( 1 9 5 6 ) . S u r f a c e and B o t t o m C u r r e n t s i n t h e S t r a i t o f G e o r g i a . J o u r n a l o f t h e F i s h e r i e s R e s e a r c h B o a r d o f C a n a d a , V o l . J.3(4), pp. 581-590.-S h u r e i i i a n , P. ( 1 9 5 8 ) . M a n u a l o f l i a r m o n - c A n a l y s i s and P r e d i c t i o n . U.S. D e p a r t m e n t q f Commerce, C o a s t and G e o d e t i c S u r v e y , W a s h i n g t o n D.C., S p e c i a l P u b l i c a t i o n No. 98. • 317 pp. S i n g l e t o n , R.C. ( 1 9 6 9 ) . . An a l g o r i t h m f o r c o m p u t i n g t h e m i x e d r a d i x . F a s t F o u r i e r T r a n s f o r m . I E E E T r a n s a c t i o n s on A u d i o and E l e c t r o a c o u s t i c s , V o l . A'U-_7, pp. 93-10 3. T a b a t a , S., L . F . G i o v a n d o , J.A. S t i c k l a n d and J . Wong ( 1 9 7 0 a ) . C u r r e n t V e l o c i t y M e a s u r e m e n t s i n t h e S t r a i t o r G e o r g i a 1967. F i s h e r i e s R e s e a r c h B o a r d of Canada,. T e c h n i c a l R e p o r t No. _ 6 9 . 24 5 pp. ( 1 9 7 0 b ) . C u r r e n t V e l o c i t y M e a s u r e m e n t s i n t h e S t r a i t o f G e o r g i a - 1958. F i s h e r i e s R e s e a r c h B o a r d of C a n a d a , T e c h n i c a l R e p o r t No. _ 7 8 . 112 pp. . ( 1 9 7 0 c ) . C u r r e n t V e l o c i t y M e a s u r e m e n t s i n t h e S t r a i t o f G e o r g i a - 1969. F i s h e r i e s R e s e a r c h B o a r d o f C a n a d a , T e c h n i c a l R e p o r t No. _ 9 _ . 72 pp. T a b a t a , 3., J.A. S t i c k l a n d and B.E. de Lange Boom ( 1 9 7 1 a ) . The Pr o g r a m o f C u r r e n t V e l o c i t y and Water T e m p e r a t u r e O b s e r v a t i o n s f r o m Moored I n s t r u m e n t s i n t n e S t r a i t o f G e o r g i a - 1963-1970 and E x a m p l e s o f a e c o r d s O b t a i n e d . ' F i s h e r i e s R e s e a r c h B o a r d o f C a n a d a , T e c h n i c a l R e p o r t No. 25 3. 22 2 pp. 176 T a b a t a , S. , L.F. G i o v a n d o a nd D. D e v l i n ( 1 9 7 1 b ) . C u r r e n t V e l o c i t i e s i n t h e V i c i n i t y - o f t h e G r e a t e r V a n c o u v e r S e w erage and D r a i n a g e D i s t r i c t ' s I o n a I s l a n d O u t f a l l -1968. F i s h e r i e s R e s e a r c h B o a r d o f C a n a d a , T e c h n i c a l R e p o r t No. 263. 110 pp. T a b a t a , S. ( 1 9 7 2 ) . The Movement o f F r a s e r R i v e r - I n f l u e n c e d S u r f a c e Water i n t h e S t r a i t o f G e o r g i a a s Deduced f r o m a S e r i e s o f A e r i a l P h o t o g r a p h s . E n v i r o n m e n t C a n a d a , Water Management s e r v i c e , M a r i n e S c i e n c e s B r a n c h , P a c i f i c R e g i o n , P a c i f i c M a r i n e S c i e n c e R e p o r t No. 72^6. 69 pp. T a b a t a , S. a n d J.A. S t i c k l a n d ( 1 9 7 2 a ) . Summary o f Oceano-g r a p h i c R e c o r d s O b t a i n e d f r o m Moored I n s t r u m e n t s i n t h e S t r a i t o f G e o r g i a - 1969-1970: C u r r e n t V e l o c i t y and S e a w a t e r T e m p e r a t u r e f r o m S t a t i o n ' . H-06. E n v i r o n m e n t C a n a d a , W a t e r Management S e r v i c e , M a r i n e S c i e n c e s B r a n c h , P a c i f i c R e g i o n , P a c i f i c M a r i n e S c i e n c e R e p o r t No. 72-7, 132 pp. (1.972b). Summary o f O c e a n o g r a p h i c R e c o r d s O b t a i n e d f r o m Moored I n s t r u m e n t s i n t h e S t r a i t o f G e o r g i a - 1969-1970: C u r r e n t V e l o c i t y and S e a w a t e r T e m p e r a t u r e f r o m S t a t i o n H-16. E n v i r o n m e n t C a n a d a , Water Management S e r v i c e , M a r i n e S c i e n c e s B r a n c h , P a c i f i c R e g i o n , P a c i f i c M a r i n e S c i e n c e R e p o r t No. 72^8. 144 pp. -( 1 9 7 2 c ) . Summary o f O c e a n o g r a p h i c R e c o r d s . O b t a i n e d f r o m Moored I n s t r u m e n t s i n - t h e S t r a i t o f G e o r g i a - 1969-1970: C u r r e n t V e l o c i t y and S e a w a t e r T e m p e r a t u r e f r o m S t a t i o n H-•26. • E n v i r o n m e n t . C a n a d a , Water Management S e r v i c e . , M a r i n e S c i e n c e s B r a n c h , P a c i f i c R e g i o n , P a c i f i c M a r i n e S c i e n c e R e p o r t ' N o . 7 2 - 9 . 141 pp. ( 1 9 7 2 d ) . Summary o f O c e a n o g r a p h i c R e c o r d s O b t a i n e d f r o m Moored I n s t r u m e n t s i n t h e S t r a i t o f G e o r q i a - 1968-1970. C u r r e n t V e l o c i t y f r o m S t a t i o n s F - 11, M-10 and 1-31 E n v i r o n m e n t C a n a d a , Water Management S e r v i c e , M a r i n e S c i e n c e s B r a n c h , P a c i f i c R e g i o n , P a c i f i c M a r i n e S c i e n c e R e p o r t No. 72-1.0. 22 pp. T e n n e k e s , H. and J . L . L u m l e y ( 1 9 7 2 ) . ^ _ F i r s t _ C o u r s e _ i n 2ii£bulence. MIT P r e s s , C a m b r i d g e , M a s s a c h u s e t t s . 300 pp. T u l l y , J . P . and A . J . Dodimead ( 1 9 5 7 ) . P r o p e r i t e s o f t h e Water i n t h e S t r a i t of. G e o r g i a , . B r i t i s h C o l u m b i a , and I n f l u e n c i n g F a c t o r s . J o u r n a l o f t h e F i s h e r i e s R e s e a r c h B o a r d o f C a n a d a , V o l . 1.4(3), pp. 241-319. 177 W a l d i c h u k , 11. ( 1 9 5 7 ) . P h y s i c a l O c e a n o g r a p h y or r u e S t r a i t o f G e o r g i a , B r i t i s h C o l u m b i a . J o u r n a l o f t a e F i s h e r i e s R e s e a r c h B o a r d o f C a n a d a , V o l . .Iii ( 3 ) , pp. 321-466. ( 1 9 5 8 ) . D r i f t B o t t l e O b s e r v a t i o n s i n t h e S t r a i t o f G e o r g i a . J o u r n a l o f t h e F i s h e r i e s R e s e a r c h B o a r d o f C a n a d a , V o l . 1.5(5),. pp. 1065-1 102. (1 9 6 4 ) . D a i l y and s e a s o n a l s e a - l e v e l o s c i l l a t i o n s on the. P a c i f i c C o a s t o f Canada. S t u _ i _ s _ o _ _ O c e a _ o _ r _ _ e d i t e d by Kozo Y c s h i d a , U n i v e r s i t y o f W a s h i n g t o n P r e s s , S e a t t l e , pp. 181-201. 178 APPENDIX I Results of harmonic analyses on 60 day sections of the hal f - h o u r l y current time-series are given here. Dates correspond to the mid-point of each time-series section. E l l i p s e parameters A,a and 9 are the semi-major length, semi-minor length, and o r i e n t a t i o n of the major axis, r e s p e c t i v e l y . Subscripts 1 and u are the lower and upper l i m i t s of the 95% confidence i n t e r v a l s whose c a l c u l a t i o n i s explained i n Chapter 6. E l l i p s e parameters A and a are i n cm/s. Values of 9 of (0,90,180,-90,-180) correspond to true (east, north, west, south, west), r e s p e c t i v e l y . 179 APPENDIX I continued WESTERN MOORING 50m A \ A u a a i a u 9 6 1 e u 1969 Aug 14 2, .97 2, .39 3, . 58 0, .10 -0. .48 0. .71 104. .9 92 .5 115 .8 Sep 28 3. .03 2. .47 3, .62 0 .39 -0. .19 0. .96 102. .7 91 .5 114 .8 Nov 12 2. .54 1. .80 3. . 32 0. .22 -0. .54 0 .98 109, .6 92. .8 126 .9 Dec 27 2. .83 2. .15 3. .55 0 .86 0, .15 1. . 55 109. . 2 92. .4 123 . 7 1970 Feb 10 3. .19 2, .65 3, .75 -0, .49 -1. • 04 0. .06 103. .5 93. .6 114 .6 1969 Aug 14 2. . 20 1. .08 3. .51 0, .36 -0, .85 1. .58 128. .0 91 .0 167 .2 Sep 28 2. . 33 1, .10 3. .67 0, .77 -0, .56 2. ..00 112. .8 69. .6 156. .5 Nov 12 .2. ,18 ' 0. .98 3. .85 1. .37 -0, . 26 2, .60 160. .1 106. .7 244. . 1 Dec 27 3. , 58 2. .21 5. .12 , -0, .10 -1. .53 1. . 37 63. .4 38. .7 87. . 5 1970 Feb 10 2. ,49 1. .29 3. .73 0, .20 -1. .00 1, .49 117. .6 89, .5 147. .5 1969 Aug 14 5. .40 4. .27 6. .59 -0, .02 -1, .20 1. .12 ' 110. .1 98. .4 122. .9 Sep 28 5. , 76 4. .67 6. ,91 0, .80 -0, . 30 • 1. 94 101. . 3 88. .7 112. . 1 Nov 12 6. ,07 4. .55 7. ,61 1, .49 -0, .05 3. .01 94. .4 80 . 1 110, .1 Dec 27 6. ,04 4. .66 7. .48 0. .66 -0, .75 2. .08 132. ,6 120. .2 147. . 1 1970 Feb 10 6. ,14 5. .03 7. ,28 -0, .31 -1. .45 0. .81 111. .3 101. .0 122. .1 1969 Aug 14 2. 33 1. .56 3^  .12 .0. , 24 -0. .54 1. .03 118. ,8 99. .5 137. . 5 Sep 28 2. 14 1. .43 2. ,92 0. .36 -0. .40 1. .09 113. ,9 94. .6 135. .4 Nov 12 1. 99 1, .14 2. .91 0. .03 -0. .89 0. .89 116. ,5 88, .3 143. .2 Dec 27 2. , 38 1. .59 3. , 23 -0. .09 -0. ,90 0, . 74 103. ,1 80. . 1 123. . 7 1970 Feb 10 2. , 39 1. .75 3. .10 ' -0. 39 -1. .05 0. .30 106. .3 89, .5 124. .0 1969 Aug 14 7. ,64 6. .92 . 8. , 37 2. .43 1. .72 3. .16 114. ,9 108, . 7 121 . 3 Sep 28 7. ,78 7. .01 8. . 56 2. .25 1. .48 3. .04 106. , 1 99. .7 112. .4 Nov 12 8. 68 7. , 77 9. ,60 1. ,47 0. ,55 2. .38 112. .3 106, .3- 118. . 7 Dec 27 9. ,44 8. ,61 10. .30 1. .45 0. .59 2, . 28 109. , 3 104. .3 114. .7 1970 Feb 10 9. 05 8. .39 9. , 72 0. .73 0. .06 1. .40 103. .8 99. .5 108, .0 1969 Aug 14 3. 43 2. . 19 4. ,76 -0. .19 -1. .43 1. .14 64. . 1 40. .7 85. . 7 Sep 28 2. 92 1. ,81 4. , 22 0. .17 -1. .05 1. , 36 85. .5 55. .8 109. . 4 Nov 12 3. 03 1. ,68 4. ,60 -0. .52 -1. ,95 0. .97 82. .5 51. .1- 118. .3 Dec 27 2. 71 1. ,27' • .4. ,25 0. .66 -0. . 77 2. .21 108. ,4 73. .9' 141. .0 1970 Feb 10 3. 03 1. .76 4. ,35 -0. ,32 -1. .62 0. .97 87. .9 61. .6 109. . 2 1969 Aug 14 1. 37 0. .56 2. , 37 -0. .59 -1. .49 0. , 33 86. ,3 23. .0 143. .0 Sep 28 1. 06 0. .15 2. ,09 -0. .14 -1. .08 0. .86 62. .0 0. .2 131. . 2 Nov 12 1. 11 0. . 39 2. 39 -0. .74 -1. .55 0, .45 26. ,9 -74. .7 90. . 1 Dec 27 0. 63 0. ..00 1. .86 0. , 18 -0. .85 1. .02 153. 3 44. ,4 262. , 3 1970 Feb 10 0. 60 0. .06 1. ,63 -0. .37 -1. .00 0. . 57 129. 7 10. ,6 211. . 2 1969 Aug 14 0. 36 0. 00 1. .19 -0. ,03 -0. .61 0. ,57 77. .3 -76. ,0 160. .4 Sep 28 0. 58 0. ,13 1. 37 -0. ,29 -0. ,80 0. .45 118. .0 44. ,7 235. . 1 Nov 12 0. 24 0. 00 1. 24 -0. 10 -0. 66 0. ,58 28. 4 -71. ,9 176. . 1 Dec 27 0. 20 0. 00 1. ,16 -0. .01 -0. 58 0. 58 171. 5 48. .9 300. .5 1970 Feb 10 0. 44 0. 00 1. 21 -0. ,16 -0. 68 0. .53 116. .8 9. ,4 219. .9 1969 Aug 14 0. 09 0. 00 0. 89 0. .02 -0. 44 0. ,45 174. 4 49. . 2 309. ,4 Sep 28 0. 35 0. 00 1. 16 -0. ,13 -0. ,65 0. , 52 153. 8 26. .8 261. . 2 Nov 12 0 . 41 0. ,00 1. 36 -0. .11 -0. ,74 0. ,62 132. 4 -2. .0 226. .6 Dec 27 0. 17 " 0. ,00 1. 11 -0. ,06 -0. , 58 0. ,53 31. 0 -69. ,1 186. .2 1970 Feb 10 0. 43 . 0. ,00 1. 19 -0. .21 -0. ,71 0. .49 32. ,4 -113. .8 111. . 7 180 APPENDIX I c o n t i n u e d WESTERN MOORING 200m A A l A u a 3 1 a u e 9 1 e u 1969 Aug 14 3. .24 2. ,94 3. .55 -0. .65 -0. .95 -0. .34 99. .7 94. .2 105. . 5 Sep 28 3. .00 2. 64 3, .36 -0. ,88 -1, .24 -0. .52 87. .6 80 .6 95. . 5 Nov 12. 3. .67 3. 21 4, .13 -0. ,72 -1. .18 -0. .26 90. .8 83 .2 98. .0 Dec 27 3. .19 2. ;82 3. .59 -1. .07 -1. .45 -0. .68 88. .8 80. .1 95. .9 1970 Feb 10 2, .81 2. 43 3. .20 -1. .22 -1, .60 -0. .84 98. . 3 89. . 2 109, .0 Mar 27 2, .68 2, ,26 3. .13 -1. ,26 -1. .69 -0. .81 113. ,6 100. .6 126. .0 1969 Aug 14 2. .21 1. 57 2. .93 -0. .42 • • -1. .10 0. .27 90. .3 74. .1 112. .1 Sep 28 1, .47 0. 79 2. .24 -0. ,60 -1. .30 0. ,15 74. .6 33. .8 120. .1 Nov 12 1. .71 1. 01 2. .54 -0. .68 -1, .43 0. .09 89. .4 46. .1 132. .5 Dec 27 2, .03 1. 17 2. .99 -1. ,07 -1. .93 -0. .11 75. , 5 29, .4 118. , 4 1970 Feb 10 1, .56 0. 89 2. .36 -0. .43 -1. .15 0. ,33 82. ,7 45. .1 _ 121. ,8 Mar 27 1. .72 0. 82 2. .71 . -0. 32 -1. .22 0. ,67 105. ,2 67. . 3 ' 142. , 3 1969 Aug 14 5. .79 5. 09 6. .49 -1. 31 -2. .01 -0. .61 100. ,1 92. .7 107. , 2 Sep 28 5. .62 5. 05 6. .24 -1. 86 -2. .45 -1. ,26 < 91. ,7 84'. .1 98. ,1 Nov 12 6. .63 5. 75 7. .51 -1. .91 -2, .80 -1. ,03 93. ,4 85, .3 101. .8 Dec 27 6. .16 5. 47 6. ,90 -1. .31 -2. .02 -0. ,59 95. ,9 88. .5 103. , 1 1970 Feb 10 5. .76 4. 95 6. .58 -1. ,32 -2. .14 -0. ,51 • 97. ,7 89, .3 106. , 3 Mar 27 5. .70 4. 92 6. .53 -1. 79 -2. .58 -0. .98 102. ,9 94. .4 112. .4 1969 Aug 14 3. .01 2. 60 3. .43 -1. 13 -1. .55 -0. ,71 115. .5 106, .0 124. ,9 Sep 28 2. .87 2. 43 3. .31 -1. 30 -1. .75 -0. ,86 ' 119. ,4 107, .8 130. . 5 Nov 12 2. .81 2. 32 3. . 32 -1. ,03 -1, .54 -0. ,54 116. ,0 103. .6 127. . 7 Dec 27 2. .63 2. 16 3. .14 -0. ,94 -1, .41 -0. .44 123. .0 109, . 7 136. . 1 1970 Feb 10 2. .33 1. 89 2. .81 -0. ,85 -1. .30 -0. .38 107. . 3 95. .4 121. .6 Mar 27 2. .11 1. 68 2. .60 -1. .45 -1, .89 -0. ,97 137. ,0 101. . 7 171. .3 1969 Aug 14 12. .15 11. 72 12. .58 -3. ,74 -4, .17 -3. , 31 120. , 3 118. .2 122. .6 Sep 28 11-.74 11. 31 12. .17 -3. ,94 -.4. . 37 -3. .51 117. . 3 115. .0 119. .6 Nov 12 11. .35 10. 85 11. .85 -4. 05 -4. .55 -3. , 55 116. .4 113. .4 ' 119. .4 Dec 27 10, .85 10. 32 11. . 38 -3. ,98 -4. .52 -3. .45 119. .2 115. .8 122. . 5 1970 Feb 10 10. .70 10. 18 11. .22 -4. ,85 -5. . 38 -4. ,33 126. .2 122. .7 129. , 7 Mar 27 9. .84 9. 25 10. ,43 -3. ,24 -3. .83 -2. .65 123. , 2 119. .5 127. .1 1969 Aug 14 2. .30 1. 73 2. ,95 -0. , 33 -0. .93 0. .28 131. .8 113, .2 150. .2 Sep 28 2. .60 1. 85 3. .41 -0. ,98 -1. .74 -0. .18 99. .3 77, .0 117. . 1 Nov 12 2, .10 . 1. 36 2. ,95 -0. 22 -1. .02 0. ,57 128. . 0 103, .7 155. ,9 Dec 27 3. . 30 2. 45 4. .22 -1. ,40 -2, .26 -0. .49 99. ,5 77, .1 119. .6 1970 Feb 10 2. .45 1. 72 3. .29 -1. .28 -2. .05 -0. .48 141. ,0 112. .6 170. ,5 Mar 27 3. .07 2. 13 4. .09 -1. 50 -2. .44 -0. ,48 118. ,9 94. . 3 145. , 4 1969 Aug 14 0. .97 0. 54 1. ,48 -0. .43 -°. .89 0. .06 92. ,2 40. .9 140. .9 Sep 28 0. .82 0. 33 1. ,41 0. 02 -0. .52 0. . 56 165. ,8 116. .9 213. . 8 Nov 12 1. .24 0. 59 1. .96 -0. 44 -1. ,08 0. .28 155'. ,8 114. . 5 201. , 7 Dec 27 0. .84 0. 40 1. .62 -0. 73 -1. ,18 0. .04 103. ,7 32. .8 188. .0 1970 Feb 10 1. .10 0. 50 1. .77 -0. 09 -0. ,70 0. .57 80. ,1 45, .4 118. . 3 Mar 27 0. , 77 0. 23 1. ,55 -0. 45 -1. ,01 0. .31 25. ,7 -42. . 7 105. .5 1969 Aug 14 0. .75 0. 39 1. .18 -0. 43 -0. ,81 -0. .02 53. ,2 -4. .4 109. . 5 Sep 28 0. ,57 0. 26 1. ,04 -0. 34 -0. ,68 0. ,10 . . 50. 4 -36. .5 107. .9 Nov. 12 0. .56 0. 16 1. ,13 -0. 33 -0. .74 0. .24 102. .7 35. .8 184. ,4 Dec 27 0. .57 0. 05 1. ,17 -0. 06 -0. .61 0. ,51 130. ,5 79. .4 247. ,9 1970 Feb 10 9. ,58 0. 16' 1. . 10 0. ,14 -0. .33 0. .61 114. ,7 32, .8 169. . 7 Mar 27 0. ,70 0. 12 1. .32 0. .12 -0, .50 0. ,70 82. ,0 37. .6 154. .6 1969 Aug 14 0. .34' 0. 06 .0. .78 -0. ,12 -0. .44 0. .28 71. .3 -5. .4 173. .4 Sep 28 0. .38 0. 03 0. .81 0. .03 -0. . 36 0. .42 47. ,9 -30, .0 139. .6 Nov 12 0. ,37 0. 06 0. .90 0. ,17 -0, .31 0. ,53 58. .1 -31. .8 157. ,9 Dec 27 0. .65 0. 31 1. .21 -0. .41 -0. .80 0. ,10 113. , 1 58, .7 207. ,6 1970 Feb 10 0. .81 0. 36 1. .31 -0. ,20 -0. .66 0. .30 69. .8 25. .5 109. .4 Mar 27 0. .73 0. 21 1. .32 • -0. ,02 -0, .59 0. .52 81. .0 35, .6 125. .3 181 A P P E N D I X I c o n t i n u e d C E N T R A L M O O R I N G 200m A A a . 5 , a 9 e 1 u 1 u 1 u 1969 S e p 28 3. .13 •2, .72 3, .55 0. .65 0 .23 1 .06 103, .5 95 .4 111 .9 D e c 27 3 .69 3, .25 4. .15 0 .29 -0 .16 0 .74 ' 102. .2 94 .9 109 .0 1970 F e b 10 4. .04 3, .69 4. .39 -0, .22 -0 .57 0 .13 101 .2 96 .3 106 . 3 M a r 27 3. .88 3. . 38 4. .38 . 0, .30 • -0 .20 0. .80 90. .3 83 .1 98 .0 M a y 11 3, .43 2, .67 4. .25 0, .71 -0. .08 1. .'50 82. .1 66 .0 96 .8 1969 S e p 28 2. .43 1. .57 3, .39 -0, . 33 -1 .26 0. .57 117. .4 93 .8 139 . 2 D e c 27 2. .24 1. . 30 3. .24 -0. .19 -1. .15 0. . 79 89. .6 68 .6 117. .0 1970 F e b 10 2. .54 1, .70 3. .41 -o. .63 -1. .50 0, .21 102. .1 83 .0 122, .6 M a r 27 2. .00 1. .05 3. ,05 -0. .38 -1. .36 0. .64 86. .4 57 . 1 120. .7 M a y 11 5. .01 • 3. .55 6. ,68 0. .66 -0. .92 2, .21 92. .5 69 . 3 115. .0 1969 S e p 28 6. .30 5. .49 7. , 13 0. .75 -0, .06 1, .58 101. .7 93 .5 103. .9 D e c 27 6. . 34 5. .44 7. ,24 -0. ,17 -1. .06 'o. .74 104. ,2 96 .0 112. . 2 1970 F e b 10 8. ,14 7. ,52 8. .79 -0. ,95 -1, . 59 -0, . 31 95. ,4 91 .0 100. . 3 M a r 27 6. ,81 5. ,83 7. ,79 -0. ,44 -1. .42 0. .54 89. ,5 81. . 1 97. .5 M a y 11 10. ,01 8. ,15 11. ,88 0. .54 -1. .33 2. ,41 91. ,8 81. .2 102. .4 1969 S e p 28 1. ,84 1. .35 2. .39 -0. , 30 -0. .82 0. .23 105. 8 89. .4- 123. . 5 D e c 27 2. ,14 1. .67 2. 67 -0. 47 -0. .96 0. ,04 122. 4 107 . . 2 136. ,6 1970 F e b 10 2. 08 1. 65 2. 54 -0. ,06 -0. ,51 0. .39 106. 5 94, .2 119. .0 M a r 27 1. 88 1. 25 2. 52 0. ,51 -0. .13 1. .14 88. .7 68. . 3 108. .0 M a y 11 1. .55 0. .70 2. 73 -1 . 03 -1 . .88 0. ,15 132. 2 40. .6 183. . 4 1969 S e p 28 10. 15 9. 58 10. 71 -1 . 48 -2. ,04 -0. .91 110. 8 107. .5 113. ,9 D e c 27 7. 96 7. 37 8. 55 0. 29 -0. ,30 0. ,87 126. 5 122. . 2 130. .6 1970 F e b 10 7. 90 7. 40 8. 40 0. 17 -0. ,33 0. .67 122. 8 119. .2 ' 126. ,5 M a r 27 8. 50 7. 98 9. 04 -0 . 35 -0. ,88 0. , 19 122. 8 119. , 2 126. .4 M a y 11 9. 02 7. 86 10. 18 -0. 57 -1 . .73 0. .60 130. 2 123. .0 137. ,7 1 9 6 9 S e p 2 8 3 . . 6 5 2. . 7 6 4 . . 6 1 - 0 , . 4 5 - 1 . 3 5 0 . . 5 0 1 1 0 . 1 9 5 . 1 1 2 5 . . 8 D e c 2 7 2 , . 6 3 1 . 7 9 3 . . 5 8 - 0 , . 4 7 ' - 1 . . 3 7 0 . . 4 2 1 2 8 . . 9 1 0 5 . 2 1 4 6 . . 5 1 9 7 0 F e b 1 0 1. . 9 9 1 . r 2 7 2 , . 8 1 0 , . 8 1 0 , . 0 3 1 . . 5 7 1 4 0 , . 0 1 0 9 . . 0 1 6 7 , . 9 M a r 2 7 3 . . 3 2 2, . 4 0 4 . . 2 9 - 0 , . 2 5 - 1 , . 2 1 0 , . 6 8 1 1 9 , . 0 1 0 0 . . 7 1 3 5 . . 8 M a y 1 1 1. . 5 2 0 . . 0 7 3 . . 5 8 - 0 . . 5 9 -2, . 0 8 1 . . 4 3 8 7 . . 6 - 2 4 , . 9 1 5 6 . . 3 1 9 6 9 S e p 2 8 0 . . 7 7 0 , . 1 2 1, . 5 4 - 0 . . 2 5 - 0 , . 9 1 0 . . 5 1 2 3 . . 7 - 3 9 . . 2 9 3 . , 7 D e c 2 7 1 . . 1 6 0 . . 5 0 1. . 9 1 - 0 . , 3 2 - 1 , . 0 0 0 . , 4 1 1 2 2 . . 2 7 6 . . 4 1 6 8 . . 2 1 9 7 0 F e b 1 0 1 . . 6 9 1 . . 0 7 2 . . 3 5 - 0 . . 3 6 - 1 . . 0 0 0 . , 2 8 9 0 . . 5 7 1 . . 3 1 1 5 . . 5 M a r 2 7 1 . . 1 1 0 . . 4 9 1 . . 8 7 - 0 . . 1 1 - 0 . . 7 8 0 . . 5 9 8 2 . , 0 3 3 . . 6 1 3 3 . . 4 M a y 1 1 2 . . 9 1 1 . . 6 8 4 . . 3 6 - 0 . , 1 6 - 1 . . 5 1 1 . , 1 8 1 2 0 . . 1 8 9 . . 0 1 4 9 . . 7 1 9 6 9 S e p 2 8 0 . , 4 6 0 . . 0 1 1 . , 0 4 0 . , 0 1 - 0 . , 5 1 0 . , 5 3 1 4 8 . . 1 4 6 . . 8 2 2 8 . , 2 D e c 2 7 0 . , 5 4 0 . . 0 2 1 . . 1 5 - 0 . 1 2 - 0 . , 6 4 0 . , 4 9 3 3 . , 0 - 5 0 . . 9 9 0 . , 2 1 9 7 0 F e b 1 0 0 . , 5 1 0 . . 1 2 1 . . 0 0 - 0 . 1 5 . - 0 . . 5 8 0 . 3 0 1 7 2 . 4 9 7 . , 0 2 3 9 . 6 M a r - 2 7 0 . , 4 1 0 . . 0 0 1 . , 0 6 0 . 0 3 - 0 . , 5 2 0 . . 5 4 3 6 . , 3 - 6 9 . , 6 1 5 2 . 2 M a y 1 1 0 . 4 4 0 . , 0 0 1 . , 6 3 0 . 0 1 - 0 . . 8 2 0 . 8 1 6 3 . , 9 - 4 2 . , 7 1 9 3 . 9 1 9 6 9 S e p 2 8 0 . 2 9 0 . . 0 0 0 . 8 8 - 0 . 1 0 - 0 . , 5 0 0 . 3 9 1 0 4 . 4 - 3 5 . , 3 2 0 3 . 4 D e c 2 7 0 . 2 5 0 . , 0 0 0 . . 8 3 0 . 0 6 - 0 . , 3 9 0 . 4 4 5 4 . 1 - 3 3 . 1 1 9 6 . , 5 1 9 7 0 F e b 1 0 0 . 4 3 0 . . 1 1 0 . , 9 2 - 0 . 1 8 - 0 . , 5 4 0 . 2 7 2 6 . . 1 - 8 0 . 5 9 0 . , 7 M a r 2 7 0 . 2 8 0 . 0 0 0 . 9 0 0 . 0 1 - 0 . . 4 5 0 . 4 5 1 4 3 . 1 6 2 . . 5 2 9 5 . , 7 M a y 11 0 . 3 0 0 . , 0 0 1. , 4 7 0 . 2 6 - 0 . . 6 1 0 . 8 6 1 3 7 . . 3 - 1 7 . 1 2 3 1 . 0 182 APPENDIX I continued EASTERN MOORING 50m A A l A u a a i a u 9 e. 1 9 u 1969 May 16 0. .97 0. .46 1 .•51 0. :28 -0. .25 0 .80 84 .1 56 .4 117 .9 Sep 28 1. .70 1 .28 2 .17 0 .21 -0, . 24 0 .65 101 .0 84 .0 116 .2 Nov 12 1 .27 0. .74 1 .84 0. .24 -0. . 32 0. .78 97, .4 75 .5 124 . 7 1970 Mar 27 0 .93 0, ,42 1. .48 0. .06 -0, .49 0 .57 69, .3 32 .1 106 . 7 May 11 1. .16 0. .54 1. .88 0 .39' -0, .30 1 .05 80, .8 33 .0 127 .8 1969 May 16 2. .03 1. .02 3. .11 -0. .70 -1, .78 0. . 31 115, .8 82. .8 147. . 5 Sep 28 2. . 30 1. .46 • 3. .30 -0. .14 -1, .06 0. .78 90, .2 63 .4 ' 120. . 3 Nov •12 2. . 39 1. .23 3, .59 -0 .75 -1. .91 0. .45 91. .6 60 .1 117. . 7 1970 Mar 27 1. .30 0, .15 2. .55 0. .21 -1. .03 1. . 38 83. .4 38 .6 174. .6 May 11 4. .26 2. .79 5, .82 0. .70 -0. .83 2. . 21 111. .4 91 .6 132. ,9 1969 May 16 '2. .07 1, .18 3. .08 0. .82 -0. .15 • 1, .75 68, .4 36 .2 106. .3 Sep -28 3. .03 2. .01 4. .06 0, .27 -0. .76 • 1, . 30 94. .0 74 .3 112. .1 Nov 12 3, ,02 2. .10 4. .02 0. .43 -0. .56 1. .37 102. .4 83 . 5 123. . 8 1970 Mar 27 3. .83 2, .94 4, .83 0, .08 -0, .88 1, .01 ' 86. .4 69 .8 99. . 3 May 11 3, .33 2. .21 4. .66 1, .38 0. .13 2, .58 91. .6 59. .0 123. .0 1969 May 16 1. .97 1. .40 2. .59 -0, .69 -1. .27 -0. .08 64. .5 42. .0 86. .9 Sep 28 1. . 74 1. .16 2. .36 o, .49 -0. . 10 1. . 10 149. , 2 126 .4 168. . 8 Nov 12 1. . 55 0. ,89 2, .24 0. .03 --o. .65 0, .70 93. .5 72. .0 119. . 8 1970 Mar 27 . 1. .23 0. .66 1. .89 -0, .03 -0. .66 0. .58 105. . 8 75 .7 140. . 1 May 11 1. .49 0. .58 2 .43 0. .33 -0. .61. 1. .25 44. . 2 . 13. .8 80. • 1 1969 May 16 7. .05 6. ,56 7, . 56 -0. .29 -0. . 79 0, . 21 87. ,8 83 .6 91. .9 Sep 28 6. , 86 6. .25 7. .48 0. .94 0. . 32 1. .55 108. ,4 103. .4 113. . 7 Nov 12 6. .64 5. .99 7. .31. 0. .11 -0. . 55 0, . 76 95. ,5 90 .0 . 101. 4 1970 Mar 27 6. . 29 5. .62 6. .97 -0. .68 -1. , 35 0. .00 92. .9 86 .4 98. , 8 May 11 6. ,67 5. . 76 7. ,59 1, .25 0. .33 2. .16 75. .1 66, .8 83. .0 1969 May 16 0. .91 0. .01 2. ,04 -0. ,03 -1. .04 0. .99 158. , 1 65. .2 242. . 8 Sep 28 2. .47 1. .48 3. . 58 0, .62 -0. .46 1. .64 129. .2 99, .5 154. . 4 Nov 12 1. ,67 0. ,61 2. ,86 0. .02 -1. ,13 1. .12 72. .9 35, .9 113. .0 1970 Mar 27 1. .24 0. 32 2. ,49 0. .63 -0. .61 1. .56 145. .8 53. .1 200. .9 May 11 1. ,93 0. ,57 3. ,49 0. .42 -1. 01 1. .91 • 74. 4 -18. .6 121. .1 1969 May 16 1. , 28 0. ,72 2. ,15. -0. ,97 -1. .55 -0. , 12 54. .6 -43. .6 102. 2 Sep 28 0: ,53 0. ,00 1. , 38 -0. .00 -0. 70 0. ,68 153. 5 61. .6 266. 8 Nov 12 0. .57 0. ,00 1. .50 0. ,08 -0. 71 0. .79 49. 4 -17. .8 199. 5 1970 Mar 27 1. ,09 0. ,65 1. .98 -0. ,90 -1. 40 -0. ,07 146. 8 58. ,9 212. 8 May 11 1. .16 0. 20 2. .34 0. ,15 -0. 94 1. .20 65. .7 4. .7 176. ,3 1969 May 16 0. 41 0. 04 1. .10 -0. ,27 -0. 69 0. ,37 124. 1 9. .2 206. 0 Sep 28 0. , 70 0. 15 1. . 37 -0. ,20 -0. 79 0. .44' 19. 5 -34. .6 98. 5 Nov 12 0. 53 0. 04 1. ,22 -0. 08 -0. 64 0. .54 160. .0 79. .7 265. ,8 1970 Mar 27 0. .42 0. 00 1. 11 0. ,03 -0. ,54 0. .58 125. 3 26. .5 232. ,7 May 11 0. 39 0. 00 1. ,31 -0. ,28 -0. 77 0. ,54 103. 2 -19. , 7 203. 4 1969 May 16 0. 12 0. 00 0. 78 -0. ,05 -0. 41 0. ,36 127. 2 -0. .9 252. 4 Sep 28 0. 42 0. 02 1. .07 -0. ,16 -0. 62 0. ,43 22. 5 -78. . 1 119. 2 Nov 12 0. ,42 0. 00 1. ,08 0. ,01 -0. 55 0. ,54 15. 6 -78. .4 124. ,8 1970 Mar 27 0. ,28 0. 00 . 0. ,96 0. ,03 -0. 46 0. ,50 125. 7 20. . 1 251. .5 May 11 0. 09 0. 00 1. 01 -0. ,01 -0. 51 0. ,50 134. 0 7. . 3 269. 5 183 APPENDIX I c o n t i n u e d EASTERN MOORING 140m A A a a. 1 e e ft 1 u 1 u 1 u 1969 May 16 2 .96 2 .38 3 .58 0 .22 -0 .38 0 .82 75 . 3 63 .9 . 87 .3 Sep 28 2 .73 2 . 32 3 .18 -0 .06 -0 .49 0 .37 87 .6 77 . 3 97 .4 Nov 12 3 .53 2 .97 4 .11 0 .62 0 .06 1 .20 75 .7 66 . 5 85 .8 1970 Feb 10 ' 2 .87 2 .31 3 .44 0 .83 0 .26 1. .39 79 .7 66 . 7 90 .9 Mar 27 2 .41 1 .94 2 .93 0 .88' 0 .38 1 .37 78 .4 65 .6 94 .0 1969 May 16 2 .34 1 .27 3 .62 0 .87 -0 .34 2 .01 93 .4 48 .5 142 . 7 Sep 28 1 .80 0 .77 2 .93 -0 .47 -1 .51 0 .64 107 .4 60 .9 151 . 2 Nov 12 1. .23 0. .26 2, .46 -0. . 15 -1 .24 0.96 171 .0 98 .9 248 .4 1970 Feb 10 ' 1. . 62 0. .76 2 .75 . -0. .81 -1, . 77 0 .23 100 .7 30 . 7 156 .0 Mar 27 2 .73 1 .72 3 .87 0. .97 -0, .11 2, .04 73. .4 46 .9 102. . 2 1969 May 16 ' 6. 89 5 .79 8 .05 0, .59 -0, .56 1, .70 ' 75 .7 . 65. 8 .85 . 4 Sep 28 5, .24 4 .15 6 . 33 0. .68 -0. .41 . 1. .77 93 .5 81 .9 105 .9 Nov 12 4, . 57 3. .64 5'. 60 -0. .06 -1, .04 0. .93 84. .2 72. . 1 98 .5 1970 Feb 10 3. . 73 2. .74 4, .79 0. .38 -0. .68 1, .38 58. .3 43 .7 75. .8 Mar 27 5. .06 4. .02 6, .14 1, .23 0. .16 2. .29 68, .1 54. .8 SO. .2 1969 May 16 0, .89 0. .17 1. .'74 -0. .19 -0. .96 0. .61 120, .4 38. .2 170. . 6 Sep 28 0. .90 0. . 27 1. .60 -0. .18 -0. .83 0. .51 13, .4 -28. .0 70. . 6 Nov 12 0. .81 0, .12 1. .57 -0. .11 -0. .80 0. .65 81. .0 15. .2 133. . 1 1970 Feb 10. 1. . 18 0. . 57 1. .85 0. .06 -0. .61 0. .68 106, .6 72. •8 142. . 3 Mar 27 1, .09 0. .48 1, -.77 0. .32 -0. .32 0. .98 114, .3 77. .1 164 . 0 1969 May 16 5. .44 4. . 72 6. .19 -0. .12 -0. .87 0. ,60 105. . 7 97. .9 113. . 6 Sep 28 6. .55 5. .84 7. .26 -2. .44 -3. .14 -1. .73 95. .2 • 88. . 1 102. . 4 Nov 12 6. .93 6. .21 7. .67 -1. .58 -2. .32 -0. ,86 103. .9 97 , .6 110. . 5 1970 Feb 10 6. . 54 5. .83 7. .25 0. , 38 -0. .33 1. .09 130. .6 124. .5' 137. .0 Mar .27 6. .11 5. .44 6. .79 1. .02 0. .36 1. .70 121. .8 115. .4 128. .6 1969 May 16 3. 39 2. .16 4. .75 0. , 12 -1. .21 1. ,38 53. .4 32. . 3 79. . 2 Sep 28 3. .00 1. .95 4. .15 -1. ,19 -2. ,25 -0. .05 58. ,4 28. .7 86. .9 Nov 12 0. .86 0. .01 2. .20 -0. ,28 -1. ,24 0. .96 165. , 8 60. .0 265. .7 1970 Feb 10 1. .88 0. .97 . 3. ,00 0. ,56 -0. ,47 1. .56 0. ,0 -47. . 2 45. . 1 Mar 27 1. .80 0. .79 3. .00 -0. ,73 -1. ,81 0. .40 89. ,8 42. , 1 150. .2 1969 May 16 2. ,54 1. .49 3. ,64 0. ,22 -0. ,83 1. .31 33. ,6 11. .8 58. .8 Sep 28 1. ,64 0. 81 2. ,55 0. ,38 -0. 53 1. ,21 33. ,4 -2. ,1 68. . 2 Nov 12' 0. ,85 0. .00 1. ,87 0. 08 -0. 88 0. 99 149. ,1 67. ,8 258. i 1970 Feb 10 2. 21 1. 44 3. 07 0. 04 -0. 79 0. 85 30. ,8 6. ,4 53. 1 Mar 27 1. 76 1. ,00 2. .64 0. 12 -0. 70 0. 95 14. .2 -18. ,2 43. .5 1969 May 16 0. 93 0. 25 1. 76 -0. 09 -0. 84 0. 67 61. .5 -1. ,0 123. 8 Sep 28 0. 95 0. 32 . 1. 65 0. 22 -0. 48 0. 85 59. 2 8. 0 107. 6 Nov 12 0. 88 0. 31 1. 57 0. 10 -0. 54 0. 72 45. 0 -16. .5 107. 9 1970 Feb 10 0. 96 0. 27 1. 70 -0. 22 -0. 92 0. 51 45. 8 3. 3 103. 1 Mar 27 0. 60 0. 05 1. 35 -0. 12 -0. 73 0. 57 36. 8 -19. 7 170. 7 1969 May 16 0. 69 0. 05 1. 50 -0. 05 -0. 77 0. 68 63. 1- -25. 5 144. 5 Sep 28 0. 42 0. 00 1. 12 0. 19 • -0. 46 0. 66 81. 5 9. 0 229. 4 Nov 12 0. 91 0. 28 1. 65 -0. 12 -o-. 81 0. 56 67. 4 4. 7 116. 6 1970 Feb 10 0. 64 0. 04 1. 32 0. 06 -0. 61 0. 67 65. 6 -1. •8 170. 0 Mar 27 0. 38 0. 00 1. 09 0. 14 -0. 48 0. 62 66. 6 -45. 1 170. 6 

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