UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Seiches in coastal bays Wang, Lichen 1984

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1985_A6_7 W36.pdf [ 8.45MB ]
Metadata
JSON: 831-1.0053195.json
JSON-LD: 831-1.0053195-ld.json
RDF/XML (Pretty): 831-1.0053195-rdf.xml
RDF/JSON: 831-1.0053195-rdf.json
Turtle: 831-1.0053195-turtle.txt
N-Triples: 831-1.0053195-rdf-ntriples.txt
Original Record: 831-1.0053195-source.json
Full Text
831-1.0053195-fulltext.txt
Citation
831-1.0053195.ris

Full Text

SEICHES IN COASTAL BAYS by LICHEN WANG THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE THE FACULTY OF GRADUATE STUDIES Department of Oceanography We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA December 1984 © LICHEN WANG, 1984 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the The U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree that permission f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of Oceanography The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: December 1984 ABSTRACT Resonant water waves i n c l o s e d or semi-closed f l u i d systems are r e f e r r e d to as " s e i c h e s " . L i t e r a t u r e on t h i s t o p i c i n recent years can be found i n t h i s t h e s i s . T h i s r e s e a r c h i s based on wave data i n three c o a s t a l bays at the southern coast of Vancouver I s l a n d opening to Juan de Fuca S t r a i t . P r e l i m i n a r y aim i s to look f o r the nature and cause of the observed wave motions of these bays. Time s e r i e s a n a l y s i s and numerical modeling are performed to r e v e a l seiche nature and develop a method to p r e d i c t frequency response of bays with a r b i t r a r y shapes and v a r i a b l e depths. The method i s evaluated through i t s a p p l i c a t i o n i n the three c o a s t a l bays. R e s u l t s of s p e c t r a l a n a l y s i s and numerical model agree w e l l . The p o s s i b l e energy sources of observed s e i c h e s are d i s c u s s e d . Edge waves were the major source of observed s e i c h e s . The s t r o n g e s t s e i c h e s observed were a t t r i b u t e d to Proudman c o u p l i n g between pressure waves and water waves. Oceanic i n t e r n a l t i d e s may c o n t r i b u t e energy to a f o r t n i g h t l y p a t t e r n of s e i c h e a c t i v i t y . i i t ACKNOWLEDGEMENT Among a l l the people who helped t h i s t h e s i s p r o j e c t , I wish, i n p a r t i c u l a r , to acknowledge my s u p e r v i s o r , Dr. P.H. LeBlond, f o r h i s e n t h u s i a s t i c encouragement and knowledgeable guidance throughout the development of t h i s t h e s i s . He o f f e r e d me most important data to look i n t o e x c i t i n g sources of the observed s e i c h e s , and was very p a t i e n t to c o r r e c t my p r e l i m i n a r y d r a f t s . My deep g r a t i t u t e a l s o goes to Dr. R.E. Thomson who generously p r o v i d e d a l l the data he had concerning t h i s r e s e a r c h . I would l i k e to take t h i s o p p o r t u n i t y to express my h e a r t f e l t • t h a n k s to my teachers Drs. R.W. B u r l i n g , L.A. Mysak, S. Pond and W.J. Emery f o r t h e i r c o n s i d e r a t e h e l p f u l n e s s and seemingly i n f i n i t e p a t i e n c e , which i n t r o d u c e d me i n t o the promising f i e l d of oceanography. Drs. B u r l i n g , Pond and Thomson read my p r e l i m i n a r y d r a f t s and gave me h e l p f u l d i r e c t i o n s . I h i g h l y a p p r e c i a t e the Oceanography community at UBC whose members' s c i e n t i f i c z e a l enourages me to devote myself to Ocean s c i e n c e s . I have been f e e l i n g indebted to my f r i e n d Mr. D.K. Lee and medical doctor of UBC h o s p i t a l Ms. R. Ree f o r t h e i r s e l f l e s s a s s i s t a n c e d u r i n g my two years study. F i n a l l y , I would never f o r g e t the unending moral support from my f a t h e r P r o f . Bin-hua Wang and my fiancee , Miss Yu-de Pan. in T a b l e of C o n t e n t s I . INTRODUCTION 1 I I . A BRIEF REVIEW OF SEICHE STUDIES 4 A. Examples of Observed S e i c h e A c t i v i t y 4 B. Theory and Method of S e i c h e Study 8 C. The E x c i t i n g Sources of S e i c h e s 14 I I I . SPECTRAL ANALYSIS OF SEA SURFACE FLUCTUATIONS IN COASTAL BAYS 20 A. S y n o p t i c D e s c r i p t i o n of P h y s i c a l Oceanography i n Juan de Fuca S t r a i t 20 B. The Sea S u r f a c e F l u c t u a t i o n Data 21 C. Removal of T i d a l Components 23 D. Method of S p e c t r a l A n a l y s i s 29 E. S p e c t r a l A n a l y s i s ( f i r s t d a t a s e t ) 31 F. S p e c t r a l A n a l y s i s (second d a t a s e t ) 34 G. D e t a i l e d E x a m i n a t i o n of t y p i c a l s e i c h e s 47 H. The s h i f t of resonant p e r i o d between h i g h and low t i d e s t a g e s 54 IV. NUMERICAL MODELLING OF SEICHES IN COASTAL BAYS 73 A. The F o r m u l a t i o n of the B a s i c N u m e r i c a l Model ...73 B. A p p l i c a t i o n of the B a s i c Model and P r e d i c t i o n of Resonant F r e q u e n c i e s i n C o a s t a l Bays 81 C. I n f l u e n c e of T i d a l L e v e l 90 D. The E f f e c t of Bottom F r i c t i o n 101 V. COMPARISON OF THE RESULTS BETWEEN NUMERICAL MODEL AND SPECTRAL ANALYSIS 114 V I . INVESTIGATION OF EXCITING SOURCES FOR OBSERVED SEICHES 121 A. S e i s m i c A c t i v i t y 127 B. Incoming Waves 128 I V 1 . Edge Waves 128 2. Swell 144 3. I n t e r n a l Waves 146 C. Winds 153 D. Barometric V a r i a t i o n 154 E. C o n c l u s i o n 166 BIBLIOGRAPHY 1 68 v L i s t of F i g u r e s F i g u r e Page 1. Map of J u a n de F u c a S t r a i t i n c l u d i n g t h e c o a s t a l bays i n t h i s r e s e a r c h 21 2. An example of p o l y n o m i a l c u r v e f i t t i n g 26 3. P o l y n o m i a l f i l t e r e d s e a s u r f a c e o s c i l l a t i o n . ...26 4. S p e c t r a l d e n s i t y of p o l y n o m i a l - f i l t e r e d d a t a . ..27 5. The p h a s e s o f F o u r i e r c o e f f i c i e n t s o f p o l y n o m i a l - f i l t e r e d d a t a 27 6. C o m p a r i s o n o f s p e c t r a o f P o l y n o m i a l and B u t t e r w o r t h f i l t e r e d d a t a 28 7. The f r e q u e n c y r e s p o n s e of t h e 7 t h o r d e r B u t t e r w o r t h f i l t e r 30 8. 8-hour wave s p e c t r a of B e c h e r Bay, J a n u a r y 33 9. 8-hour wave s p e c t r a of P o r t San J u a n , J a n u a r y . .33 10. Power s p e c t r a o f B e c h e r Bay, J a n u a r y 35 11. Power s p e c t r a o f P o r t San J u a n , J a n u a r y 35 12. C o h e r e n c e and phase o f wave time s e r i e s between B e c h e r Bay and P o r t San J u a n , J a n u a r y . .36 13. 8-hour wave s p e c t r a of Ped d e r Bay, F e b r u a r y . ...38 14. 8-hour wave s p e c t r a o f B e c h e r Bay, F e b r u a r y . ...39 15. 8-hour wave s p e c t r a of P o r t San J u a n , F e b r u a r y 40 16. D a i l y s p e c t r a o f Ped d e r Bay, F e b r u a r y 41 17. D a i l y s p e c t r a o f B e c h e r Bay, F e b r u a r y 41 18. D a i l y s p e c t r a o f P o r t San J u a n , F e b r u a r y 42 19. 20-day a v e r a g e d power s p e c t r a o f Ped d e r Bay. ...42 20. 20-day a v e r a g e d power s p e c t r a of B e c h e r Bay. ...43 21. 20-day a v e r a g e d power s p e c t r a of P o r t San J u a n 4 3 22. C o h e r e n c e and ph a s e o f wave time s e r i e s between Pedder Bay and B e c h e r Bay, F e b r u a r y . ...44 23. C o h e r e n c e and phase of wave time s e r i e s between Pedder Bay and P o r t San J u a n , F e b r u a r y 45 24. C o h e r e n c e and ph a s e o f wave time s e r i e s between B e c h e r Bay and P o r t San J u a n , F e b r u a r y 46 v i > 25. Wave time s e r i e s beginning at 00:00 Feb. 18, 1980 i n Pedder Bay 49 26. Wave time s e r i e s beginning at 09:20 Feb. 2, 1980 i n Becher Bay 49 27. Wave time s e r i e s beginning at 18:40 Jan. 29, 1980 in Port San Juan 50 28. Wave time s e r i e s beginning at 10:40 Feb. 11, 1980 i n Port San Juan 50 29. High frequency components of wave time s e r i e s beginning at 00:00 Feb. 18, 1980 i n Pedder Bay 51 30. High frequency components of wave time s e r i e s beginning at 09:20 Feb. 2, 1980 i n Becher Bay. .51 31. High frequency components of wave time s e r i e s beginning at 18:40 Jan. 29, 1980 i n Port San Juan 52 32. High frequency components of wave time s e r i e s beginning at 10:40 Feb. 11, 1980 i n Port San Juan 52 33. Power spectrum of strong seiche of Pedder Bay. .55 34. Power spectrum of strong seiche of Becher Bay. .55 35. Power spectrum of strong s e i c h e of Port San Juan (1 ) 56 36. Power spectrum of strong seiche of Port San Juan (2) 56 37. Ti d e s i n Pedder Bay 57 38. Ti d e s i n Becher Bay 57 39. Ti d e s i n Port San Juan 58 40. Power s p e c t r a of s e l e c t e d low t i d e s of Pedder Bay, becher Bay and Port San Juan 59 41. Averaged power spectrum of s e l e c t e d low t i d e s of Pedder Bay 60 42. Averaged power spectrum of s e l e c t e d low t i d e s of Becher Bay 60 43. Averaged power spectrum of s e l e c t e d low t i d e s of Port San Juan. 61 44. Coherence of s e l e c t e d low t i d e s , Pedder Bay versus Becher Bay 62 45. Coherence of s e l e c t e d low t i d e s , Pedder Bay versus Port San Juan 62 46. Coherence of s e l e c t e d low t i d e s , Becher Bay versus Port San Juan 63 v i i 1 47. Power s p e c t r a of s e l e c t e d high t i d e s of Pedder Bay, Becher Bay and Port San Juan 64 48. Averaged power spectrum of s e l e c t e d high t i d e s of Pedder Bay 66 49. Averaged power spectrum of s e l e c t e d high t i d e s of Becher Bay 66 50. Averaged power spectrum of s e l e c t e d high t i d e s of Port San Juan 67 51. Coherence of s e l e c t e d high t i d e s , Pedder Bay versus Becher Bay 67 52. Coherence of s e l e c t e d high t i d e s , Pedder Bay versus Port San Juan 68 53. Coherence of s e l e c t e d high t i d e s , Becher Bay versus Port San Juan 68 54. Comparison of averaged power s p e c t r a between low and high t i d e s of Pedder Bay 69 55. Comparison of averaged power s p e c t r a between low and high t i d e s of Becher Bay 69 56. Comparison of averaged power s p e c t r a between low and high t i d e s of Port San Juan 70 57. Comparison of coherences between low and high t i d e s , Pedder-Becher 70 58. Comparison of coherences between low and high t i d e s , Pedder-San Juan 71 59. Comparison of coherences between low and high t i d e s , Becher-San Juan 71 60. The shape of Pedder Bay and i t s numerical g r i d 82 61. The shape of Becher Bay 82 62. The numerical g r i d of Becher Bay 83 63. The shape of Port San Juan 84 64. The numerical g r i d of Port San Juan 84 65. The m o d i f i e d numerical g r i d of Port San Juan. ..85 66. A m p l i f i c a t i o n f a c t o r of g r i d 161 i n Pedder Bay 87 67. A m p l i f i c a t i o n f a c t o r at 5 p o i n t s of Pedder Bay 87 68. A m p l i f i c a t i o n f a c t o r at g r i d 175 i n Becher Bay 88 69. A m p l i f i c a t i o n f a c t o r at 5 p o i n t s of Becher Bay 88 70. A m p l i f i c a t i o n f a c t o r at g r i d 187 of Port San Juan 89 71. A m p l i f i c a t i o n f a c t o r at 5 p o i n t s of Port San Juan 89 72. Contour map-of a m p l i f i c a t i o n f a c t o r : f i r s t mode of Pedder Bay 91 73. Contour map of a m p l i f i c a t i o n f a c t o r : second mode of Pedder Bay 91 74. ' Contour map of a m p l i f i c a t i o n f a c t o r : t h i r d mode of Pedder Bay 92 75. Contour map of a m p l i f i c a t i o n f a c t o r : f i r s t mode of Becher Bay 92 76. Contour map of a m p l i f i c a t i o n f a c t o r : second mode of Becher Bay 93 77. Contour map of a m p l i f i c a t i o n f a c t o r : f i r s t mode of Port San Jaun 93 78. Contour map of a m p l i f i c a t i o n f a c t o r : second mode of Port San Juan 94 79. Contour map of a m p l i f i c a t i o n f a c t o r : t h i r d mode of Port San Juan 94 80. Comparison of a m p l i f i c a t i o n f a c t o r s of Pedder Bay between c h a r t depth and mean sea l e v e l 96 81. Augmented g r i d of Port San Juan 98 82. Comparison of a m p l i f i c a t i o n s (absolute v a l u e s ) between b a s i c g r i d and augmented g r i d , Port San Juan 99 83. Comparison of a m p l i f i c a t i o n s (true v alues) between b a s i c g r i d and augmented g r i d , Port San Juan 100 84. The e f f e c t of f r i c t i o n c o e f f i c i e n t s i n the numerical model of Pedder Bay 110 85. The e f f e c t of f r i c t i o n c o e f f i c i e n t s i n the numerical model of Becher Bay 111 86. The e f f e c t of f r i c t i o n c o e f f i c i e n t s i n the numerical model of Port San Juan 112 87. Comparison between observed spectrum and a m p l i f i c a t i o n f a c t o r s i n Pedder Bay 115 88. Comparison between observed spectrum and a m p l i f i c a t i o n f a c t o r s i n Becher Bay 115 89. Comparison between observed spectrum and a m p l i f i c a t i o n f a c t o r s i n Port San Juan. 117 90. Comparison between 20-day averaged observed spectrum and a m p l i f i c a t i o n f a c t o r s i n Pedder Bay 117 i x 91. Comparison between 20-day averaged observed spectrum and a m p l i f i c a t i o n f a c t o r s i n Becher Bay 118 92. Comparison between 20-day averaged observed spectrum and a m p l i f i c a t i o n f a c t o r s i n Port San Juan 118 93. Comparison of observed spectrum and a m p l i f i c a t i o n f a c t o r s f o r deep water in Pedder Bay 1 20 94. Strong s e i c h e s and corresponding t i d e s 122 95. Coherence and phase between Becher Bay and Port San Juan, January 123 96. Coherence and phase between Pedder Bay and Becher Bay, February 124 97. Coherence and phase between Pedder Bay and Port San Juan, February 125 98. Coherence and phase between Becher Bay and Port San Juan, February 126 99. E x p o n e n t i a l f i t t i n g f o r bottom p r o f i l e o f f Pedder Bay 129 100. D i s p e r s i o n r e l a t i o n of edge waves with e x p o n e n t i a l depth p r o f i l e o f f Pedder Bay. 130 101. E x p o n e n t i a l f i t t i n g f o r bottom p r o f i l e o f f Becher Bay 130 102. D i s p e r s i o n r e l a t i o n of edge waves with e x p o n e n t i a l depth p r o f i l e o f f Becher Bay 131 103. E x p o n e n t i a l f i t t i n g f o r bottom p r o f i l e o f f Port San Juan 131 104. D i s p e r s i o n r e l a t i o n of edge waves with e x p o n e n t i a l depth p r o f i l e o f f Port San Juan. ..132 105. Sea s u r f a c e displacement i n p e r i o d 11 min. o f f Pedder Bay 132 106. Sea s u r f a c e displacement i n p e r i o d 13 min. o f f Becher Bay 133 107. Sea s u r f a c e displacement i n p e r i o d 33 min. o f f Port San Juan ( e x p o n e n t i a l p r o f i l e ) 133 108. Sea s u r f a c e displacement i n p e r i o d 16 min. o f f Port San Juan (e x p o n e n t i a l p r o f i l e ) 134 109. L i n e a r f i t t i n g f o r bottom p r o f i l e of Port San Juan , 135 110. D i s p e r s i o n r e l a t i o n of edge waves with l i n e a r depth p r o f i l e o f f Port San Juan 135 x 111. Sea surface displacement i n p e r i o d 33 min. o f f Port San Juan ( l i n e a r p r o f i l e ) 136 112. Sea surface displacement i n p e r i o d 16 min. o f f Port San Juan ( l i n e a r p r o f i l e ) 136 113. Surface displacement at 6,000 m o f f Port San Juan ( l i n e a r p r o f i l e ) 137 114. S p l i n e f i t of the depth p r o f i l e o f f Port San Juan 139 115. D i s p e r s i o n r e l a t i o n of edge waves from numerical model ( o f f Port San Juan) 140 116. D i s p e r s i o n r e l a t i o n of edge waves from a n a l y t i c model with f l a t bottom ( o f f Port San Juan) 1 40 117. S p l i n e f i t t i n g of depth p r o f i l e o f f Becher Bay 142 118. D i s p e r s i o n r e l a t i o n of edge waves from numerical model ( o f f Becher Bay) 142 119. D i s p e r s i o n r e l a t i o n of edge waves from a n a l y t i c model with f l a t bottom ( o f f Becher Bay) 143 120. Wave data at T o f i n o , Jan. 1980 ..145 121. Wave data at T o f i n o , Feb. 1980 145 122. I n t e r n a l waves from s a t e l l i t e 147 123. A t r a c i n g of part of the s a t e l l i t e image 148 124. T i d e s i n Port San Juan. 151 125. S t i c k diagram of the winds at Race Rocks 155 126. S t i c k diagram of the winds at Cape Beale 156 127. The comparison of seiche energy and wind data at Race Rocks 157 128. The comparison of seiche energy and wind data at Cape Beale 1 58 129. Atmospheric pressure data at s t a t i o n s T o f i n o and V i c t o r i a 159 130. Rate of change of pressure at s t a t i o n s T o f i n o and V i c t o r i a 160 131. Barometric maps (1). 162 132. Barometric maps (2) 163 x i L i s t of Tables Table Page 1. O r i g i n a l Data of Sea Surface F l u c t u a t i o n s 22 2. Data of T y p i c a l Seiches (500 min.) 48 3. Data of T y p i c a l Seiches (4096 min.) 53 4. CPU Time f o r subroutines SLE and INV 108 5. Seismic A c t i v i t i e s , Jan. 20, 1980 1 27 6. Seismic A c t i v i t i e s , Jan. 25 to Feb.4, 1980 ....127 x i i I . INTRODUCTION Water waves with e x c e p t i o n a l l y l a r g e amplitudes and w e l l d e f i n e d frequencies have o f t e n been observed, e s p e c i a l l y i n c l o s e d or semi-closed water bodies, such as l a k e s , c o a s t a l bays e t c . . The cause of these p e r i o d i c a l o s c i l l a t i o n s i s a t t r i b u t e d to the resonant f u n c t i o n of the f l u i d systems. These waves are r e f e r e d to " s e i c h e " . "Seiche" i s a French word to d e s c r i b e t h i s n a t u r a l phenomenon. In recent p u b l i c a t i o n s " A b i k i " , a Japanese word with s i m i l a r meaning, a l s o appears i n the oceanographic l i t e r a t u r e . Oceanic c o u n t r i e s with numerous c o a s t a l bays, i n l e t s , f j o r d s , e s t u a r i e s and s t r a i t s are f a c i n g the c h a l l e n g e s from the ocean. The c r e a t i o n of these words r e f l e c t e d the i n t e r e s t i n t h i s phenomenon. In e a r l y t h i s century t h i s phenomenon was observed mostly in lakes and compared with the t h e o r e t i c a l l y computed resonant p e r i o d s by d i f f e r e n t methods. But there was not as much l i t e r a t u r e on t h i s s u b j e c t as on some other t o p i c s of p h y s i c a l oceanography. Within the past two decades t h i s t o p i c has a t t r a c t e d more s c i e n t i s t s ' a t t e n t i o n and i s s t u d i e d by means of s p e c t r a l a n a l y s i s , a n a l y t i c a l models and numerical models. But there are s t i l l many f e a t u r e s f o r oceanographers to e x p l o r e . T h i s t h e s i s s t a r t s from a review of s e i c h e study. Due to the lack of seiche review i n recent years t h i s review i n c l u d e s as many important papers as p o s s i b l e . The review i n c l u d e s a d i s c u s s i o n of s e i c h e r e p o r t s , r e s e a r c h methods 1 2 and energy sources, and w i l l serve as a guide and a source of comparisons f o r t h i s t h e s i s r e s e a r c h . Sea s u r f a c e o s c i l l a t i o n s from three c o a s t a l bays on the north shore of Juan de Fuca S t r a i t , B r i t i s h Columbia, are analyse d . The d i f f e r e n t procedures to remove low frequency components are d i s c u s s e d . S p e c t r a l a n a l y s i s r e v e a l s seiche a c t i v i t i e s i n these three bays, some of them having s i g n i f i c a n t energy. F u r t h e r attempts are undertaken to study the stronger s e i c h e s and the r e l a t i o n between s e i c h e s o c c u r i n g and d i f f e r e n t t i d a l l e v e l s . The coherences of these data between every p a i r of bays throw l i g h t on the common energy source of observed s e i c h e s . In order to study the wave response c h a r a c t e r i s t i c s , a numerical model i s developed. T h i s model works i n the frequency domain to t e s t the frequency response of the bays with a r b i t r a r y shapes and v a r i a b l e depths. The model-predicted resonant p e r i o d s agree w e l l with the r e s u l t of s p e c t r a l a n a l y s i s . Some f e a t u r e s r e v e a l e d by the model are the same as those r e v e a l e d by the r e a l wave s p e c t r a . T h i s research uses s e i s m i c , wave and m e t e o r o l o g i c a l data t o ex p l o r e the p o s s i b l e energy sources of the observed s e i c h e s . T r a v e l l i n g edge waves along Juan de Fuca S t r a i t are s t u d i e d by d i f f e r e n t t h e o r i e s . The r e s u l t of the edge wave numerical model seems to be a c c e p t a b l e . I t i s d i f f i c u l t to v e r i f y t h i s r e s u l t , because there are no wave data a v a i l a b l e in the S t r a i t . At l a s t a p r e l i m i n a r y c o n c l u s i o n as to the energy source i s reached, i t g e n e r a l l y a t t r i b u t e s the source 3 of the observed s e i c h e s to edge waves in the S t r a i t , and the source of the strongest s e i c h e s observed to t r a v e l l i n g barometric waves, which e x c i t e d edge waves i n the S t r a i t through a i r - s e a c o u p l i n g , t h e r e f o r e the s e i c h e s i n s i d e bays were e x c i t e d by edge waves passin g by t h e i r mouths. I n t e r n a l waves may be other e x c i t i n g source, which i s d i s c u s s e d and supported by a v a i l a b l e data. I I . A BRIEF REVIEW OF SEICHE STUDIES T h i s review i s organized i n three p a r t s . The f i r s t p a r t i s a d e s c r i p t i o n of observed s e i c h e s r e p o r t e d a l l over the world from e a r l y 1960's u n t i l the end of l a s t year. The second p a r t i s about t h e o r i e s and methods used in seiche s t u d i e s . The t h i r d part d e s c r i b e s the p o s s i b l e sources of seiche energy a c c o r d i n g to d i f f e r e n t r e s e a r c h e r s ' o p i n i o n . In each part the m a t e r i a l are mentioned c h r o n o l o g i c a l l y . A. EXAMPLES OF OBSERVED SEICHE ACTIVITY Defant (1961) gave a b r i e f survey of observed s e i c h e s i n v a r i o u s l a k e s , i n c l u d i n g the s t r i k i n g data of Bergsten (1926) who observed a uninodal, one-dimensional seiche i n Lake V a t t e r n , Sweden. Waves with l a r g e amplitude i n s i d e harbours were f r e q u e n t l y r e p o r t e d to cause c o n s i d e r a b l e damage, such as breaking anchors, d e s t r o y i n g c o a s t a l p r o j e c t s , e t c . , even a f t e r c o n s t r u c t i n g breakwaters. Many harbours with narrow entrances are l i m i t e d in t h e i r u s e f u l n e s s not so much by the p e n e t r a t i n g sea s w e l l as by s u r g i n g a s s o c i a t e d with harbour resonances. T h i s phenomenon was r e f e r e d to by M i l e s and Munk (1961) as a harbour paradox, that i s , the wave a g i t a t i o n of resonance w i t h i n the harbour may i n c r e a s e when the opening decreases, thus causing a narrowing of the harbour mouth to d i m i n i s h the p r o t e c t i o n from s e i c h i n g . Olsen and Hwang (1971) r e p o r t e d ocean waves having p e r i o d s between four minutes and about one hour, which were 4 5 measured simultaneously at s e v e r a l l o c a t i o n s i n s i d e Keauhou Bay, Hawaii and at two deep-water s t a t i o n s approximately 4 km o f f s h o r e . Barber and T a y l o r (1977) repo r t e d the measured data on the d i s t r i b u t i o n of c u r r e n t i n Lennox Passage, which i n d i c a t e d the s e l f - o s c i l l a t i o n of Chedabucto Bay, i n e a s t e r n Canada. Lemon et al (1979) reported seiche motions observed with a bottom-mounted pressure gauge in Port San Juan, western Canada. Besides the s e i c h e s i n c l o s e d basins such as l a k e s , and h a l f c l o s e d basins such as harbours, which are commonly observed, P e t r u s e v i c s et al (1979) repo r t e d o b s e r v a t i o n s of a c o n t i n e n t a l s h e l f s e i c h e o f f south-west A u s t r a l i a , even though the near shore topography of the c o n t i n e n t a l s h e l f does not form a c l o s e d b a s i n , nor does i t have a s i n g l e s i d e f u l l y open to a deep ocean. Wiibber and Krauss ( 1 979) v e r i f i e d the r e s u l t s of t h e i r numerical model by comparison with t i d e gauge records a l l along the B a l t i c coast and found that e i g e n - o s c i l l a t i o n s play a major r o l e i n the B a l t i c Sea. T h e i r s p e c t r a l a n a l y s i s d i d not show any p r e f e r e n c e f o r s p e c i a l peaks. A wide v a r i e t y of p e r i o d s seems to be p o s s i b l e . O r l i c (1980) re p o r t e d an e x c e p t i o n a l s e a - l e v e l o s c i l l a t i o n i n V e l a Luka, a small town on the i s l a n d of K o r c u l a , Y u g o s l a v i a , i n the morning of June 21,1978. The peak amplitudes of sea l e v e l changes were evaluated at about 3 meters, while the p e r i o d was equal to some 15 minutes; the o s c i l l a t i o n l a s t e d f o r three hours. At the same time as w e l l as before and a f t e r 6 the date mentioned s i m i l a r s e a - l e v e l changes o c c u r r e d i n a few other p l a c e s i n the A d r i a t i c . Great damage was repor t e d to have been caused by the phenomenon. F a i s and M i c h e l a t o (1980) r e p o r t e d on f r e e o s c i l l a t i o n s of C a g l i a r i G u l f , which i s a semi-enclosed basin l o c a t e d at the southern end of S a r d i n i a , I t a l y , with p e r i o d s of 73.1, 35.3, 23.2 and 18.9 minutes. To study the i n t e r n a l s e i c h e , Dyer (1981) s p e c t r a l l y analyzed the s a l i n i t y f l u c t u a t i o n s at an STD near the h a l o c l i n e i n Southampton Water, England. The predominant e n e r g y - c o n t a i n i n g p e r i o d s had values of 7-9 and 3.5-4.5 minutes. His c a l c u l a t e d wavelengths of the standing wave agree w e l l with echo-sounding surveys of the e l e v a t i o n of the h a l o c l i n e . By c a r e f u l v i s u a l i n s p e c t i o n of the sea l e v e l r e c o r ds at the t i d a l s t a t i o n of Genoa, which i s l o c a t e d in the inner p a r t of the L i g u r i a n Sea, Papa (1981) r e p o r t e d a n o n t i d a l o s c i l l a t i o n with a mean p e r i o d of about 2.6 hours and amplitude f l u c t u a t i o n s from a few m i l l i m e t e r s to some c e n t i m e t e r s . The records cover the i n t e r v a l Oct. 1977 to March 1978 with sampling r a t e of 30 minutes. The s p e c t r a l a n a l y s i s r e v e a l e d that two peaks, 3.36 hours and 2.56 hours, showed up above the background n o i s e . Ben-Menahem and Vered (1982) F o u r i e r analysed the mareograms at H a i f a Bay, I s r a e l , which r e v e a l e d a sharp l i n e spectrum with fundamental peaks at p e r i o d s near 54 minutes. Lanyon et al (1982) c o l l e c t e d data r e l e v a n t to s h e l f waves and bay se i c h e s with p e r i o d s between 10-60 minutes both from the open ocean and from the beach groundwater w e l l s at South Beach, Wollongong, 7 A u s t r a l i a . Giese et al (1982) repo r t e d c o a s t a l s e i c h e s at Magueyes I s l a n d , Puerto Rico, i n d i s t i n c t f o r t n i g h t l y groups with maximum seiche a c t i v i t y f o l l o w i n g new and f u l l moon by 7 days. The r e s u l t they presented were drawn from a many-faceted study which had spanned 14 y e a r s . In the study of bay s e i c h e s at South Beach, A u s t r a l i a , Lanyon et al (1982) adopted data obtained both from the open ocean and beach groundwater w e l l s . They f o l l o w e d Waddell's c o n c l u s i o n (1976) that the beach f r o n t a c t s as a low-pass f i l t e r , damping high-frequency waves and a l l o w i n g low-frequency waves to be propagated onto the groundwater s u r f a c e . The observed o s c i l l a t i o n s with p e r i o d s between 10 to 60 minutes were r e f e r e d to s h e l f waves and bay s e i c h e s . In the e a r l y 1980's the Japanese c a r r i e d out syn o p t i c s t u d i e s of the seiche a c t i v i t y , s t i m u l a t e d by a d i s a s t r o u s s e i c h e which occured i n Nagasaki Bay on March 31, 1979. The se i c h e was the l a r g e s t one in the h i s t o r y of t i d a l o b s e r v a t i o n s i n Nagasaki Bay. The maximum amplitude of the se i c h e observed at Matsugae Quay, l o c a t e d at the middle p a r t of the bay, was 278 cm and the p e r i o d was about 35 minutes, while i t was 478 cm at the mouth of the Urakami-river l o c a t e d at the northern end of the bay (Akamatsu, 1982). The l a r g e s e i c h e s i n Nagasaki Bay are most f r e q u e n t l y observed i n winter (H i b i y a and K a j i u r a , 1982). Besides the r e s e a r c h work mentioned above, I s o z a k i (1979) r e p o r t e d observed s e i c h e s i n Habu-harbor on Sept. 17, 1976 and Oct. 8, 1977. Kosuge and S a i t o (1981) examined the 8 n a t u r a l modes of harbour o s c i l l a t i o n s and e f f e c t s of man-made s t r u c t u r e s on the o s c i l l a t i o n i n Shimizu Harbour. They compared maregrams from three l o c a t i o n s in the harbour, recorded before and a f t e r the breakwaters' c o n s t r u c t i o n , and found that i n t h i s case the "harbour paradox" d i d not play any important r o l e . A f t e r the c o n s t r u c t i o n of the two breakwaters, the amplitudes of the o s c i l l a t i o n s are decreased by one-half, although the p e r i o d s of the n a t u r a l o s c i l l a t i o n s remain almost about the same. Nakamura and Serizawa (1983) s t u d i e d the l o c a l c l i m a t e of s h e l f - s e i c h e s on the b a s i s of maregrams at Susami where the sea l e v e l v a r i a t i o n with 12 minutes of p e r i o d was s i g n i f i c a n t through a year. B. THEORY AND METHOD OF SEICHE STUDY E a r l y seiche t h e o r i e s were s y s t e m a t i c a l l y d e s c r i b e d by Defant (1961). Afterwards Rao (1966) gave a b r i e f survey about t h i s t o p i c which made numerous attempts to sol v e Helmholtz's equation f o r the b a r o t r o p i c mode in a r e c t a n g u l a r basin of constant depth. A f t e r the "harbour paradox" theory was advanced by M i l e s and Munk in the e a r l y 1960's, the development of c o a s t a l survey and p r o j e c t i o n i s bound to be followed by an a c t i v e s c i e n t i f i c i n v e s t i g a t i o n , which c h a l l e n g e s p h y s i c a l oceanographers i n t o t h i s r e s e a r c h f i e l d to develop d i f f e r e n t methods i n seiche study. 9 A n a l y t i c a l s t u d i e s of seiche motions have only been c a r r i e d out i n s p e c i a l cases. C l a r k e ( 1 9 6 8 ) a p p l i e d the G a l e r k i n method to the one-dimensional flow problem. Afterwards he extended the method i n order to t r e a t the two-dimensional flow problem in a basin whose s u r f a c e shape i s r e c t a n g u l a r and whose depth i s a r b i t r a r y although s u b j e c t to the l i n e a r theory. Johns and Hamzah ( 1 9 6 9 ) s o l v e d the seiche motion problem i n a curved l a k e , and found that the pe r i o d s of the f i r s t two modes are not e f f e c t i v e l y changed in comparison with those f o r a s t r a i g h t lake having a l e n g t h equal to that of the median l i n e along the curved l a k e . Based on the s o l u t i o n of a s i n g u l a r i n t e g r a l equation, Hwang and Tuck ( 1 9 7 0 ) developed a theory f o r c a l c u l a t i n g o s c i l l a t i o n s of harbours of constant depth and a r b i t r a r y shape. Lee ( 1 9 7 1 ) i n v e s t i g a t e d both t h e o r e t i c a l l y and e x p e r i m e n t a l l y the wave-induced o s c i l l a t i o n s i n harbours of constant depth but a r b i t r a r y shape i n the h o r i z o n t a l plane connected to the open-sea. He developed a theory c a l l e d the " a r b i t r a r y - s h a p e harbour" theory to so l v e Helmholtz' equation f o r a r b i t r a r y - s h a p e d harbours by a p p l y i n g Green's i d e n t i t y formula and choosing the Hankel f u n c t i o n of the f i r s t kind and z e r o t h order. Olsen and Hwang's ( 1 9 7 1 ) numerical e v a l u a t i o n of t h e i r theory c o u l d account f o r most of the a m p l i f i c a t i o n f a c t o r s observed in Keauhou Bay, Hawaii, which were as high as 1 0 3 ( i n power s p e c t r a l d e n s i t y ) at s e l e c t e d f r e q u e n c i e s . 10 M i l e s (1971) c o n s t r u c t e d an e q u i v a l e n t e l e c t r i c a l c i r c u i t , which c o n t a i n s a r a d i a t i o n impedance and a harbour impedance, to simulate the response of a harbour as a Helmholtz r e s o n a t o r . G a r r e t t (1975) extended the harbour theory of M i l e s to take i n t o account the t i d a l f o r c e s and pr o v i d e d a b e t t e r d e s c r i p t i o n of r e a l i t y . In h i s paper G a r r e t t focused h i s a t t e n t i o n on the p r o p e r t i e s of the f o r c e d o s c i l l a t i o n i n the g u l f induced by the t i d a l f o r c i n g . Afterwards G a r r e t t ' s theory was developed by P i e r i n i (1981) so that i t s formalism i n c l u d e s the f r e e o s c i l a t i o n s ( s e i c h e s ) of the g u l f . Meanwhile, progress has come about through the development of the growth of d i g i t a l computing machines and techniques f o r the p r e s e n t a t i o n and a n a l y s i s of i n f o r m a t i o n . The improved methods are approaching an e n g i n e e r i n g l e v e l of p r e d i c t a b i l i t y . T h e r e f o r e we c o n f i n e t h i s review to recent years' work, mainly i n l a t e 1970's and e a r l y 1980's and with emphasis on numerical modeling. For the l a r g e - s c a l e numerical study of the world oceans, Platzman (1975) c a l c u l a t e d the normal modes f o r a homogeneous ocean occupying a connected domain c o n s i s t i n g of the North A t l a n t i c , South A t l a n t i c and Indian Oceans. Based on the Lanczos process and with a g r i d of 675 six-degree Mercator squares, Platzman found 26 g r a v i t y modes with p e r i o d s from 8 to 67 hours. The North A t l a n t i c c o - o s c i l l a t e s with the South A t l a n t i c at a p e r i o d of about 42 hours, and has s t r o n g resonances at 23, 21, 14.4, 12.8, 8.6 and 8.3 11 hours. Barber and T a y l o r (1977) employed maximum entropy (ME) method ( U l r y c h and Bishop, 1975) to analyze the c u r r e n t f l u c t u a t i o n s i n Chedabucto Bay, e a s t e r n Canada. The r e s u l t agreed f a i r l y w e l l with the spectrum of the output of t h e i r time-dependent numerical model. M a t t i o l i and T i n t i (1978) n u m e r i c a l l y i n v e s t i g a t e d the response of a r e c t a n g u l a r large-mouthed harbour to e x c i t i n g monochromatic plane waves, p a r t i c u l a r l y to waves with rays not orthogonal to the s t r a i g h t c o a s t l i n e . In t h e i r f u r t h e r work (1979) they concluded that the r e l i a b i l i t y of the r e s u l t s obtained from an i n t e g r a l approach i s s t r o n g l y dependent on how the k e r n e l s are d i s c r e t i z e d , e s p e c i a l l y i f r a t h e r complicated geometries are i n v o l v e d . F o l l o w i n g the a n a l y s i s of edge waves over an e x p o n e n t i a l bathymetric p r o f i l e presented by B a l l (1967), Lemon et al (1979) c o n s t r u c t e d a t h e o r e t i c a l model of o s c i l l a t i o n s i n a r e c t a n g u l a r b a s i n with an e x p o n e n t i a l depth p r o f i l e . I s o z a k i (1979) estimated the resonant p e r i o d of Habu-harbor by Merian's formula (LeBlond and Mysak, 1978. p.293), f i n d i n g that the r e s u l t was not c o n s i s t e n t with the observed predominant p e r i o d of 10.5 minutes. So he examined some c h a r a c t e r i s t i c s of the bay water o s c i l l a t i o n s by means of a numerical model and obtained p e r i o d s of 10.5, 2.4 and 1.4 minutes f o r the u n i - , b i - , and t r i - n o d a l o s c i l l a t i o n s . I t was confirmed from t h e o r e t i c a l c o n s i d e r a t i o n s that the 1 2 s e i c h e s of that harbor would be e x p l a i n e d by the coupled o s c i l l a t i o n of the system of the harbor part and the canal p a r t . Wiibber and Krauss (1979) c o n s t r u c t e d a numerical model to study the eigen-response of the B a l t i c Sea. The c a l c u l a t i o n s took i n t o account the bathymetry and shape of the B a l t i c and the earth's r o t a t i o n , but no f r i c t i o n terms, though the authors knew that i t i s e s s e n t i a l f o r an accurate s p e c t r a l d e t e r m i n a t i o n of the e i g e n f u n c t i o n s . The s e i c h e s were produced by a f o r c i n g f u n c t i o n which acted on the water body during an i n i t i a l time. The r e s u l t i n g o s c i l l a t i o n s a f t e r the f o r c e was switched o f f were analyzed at each g r i d p o i n t and d i s p l a y e d i n form of co-range and c o - t i d a l l i n e s . I t was shown that the e i g e n o s c i l l a t i o n s of the B a l t i c Sea were s t r o n g l y m o d i f i e d by the C o r i o l i s f o r c e . R o t a t i o n converts a l l modes i n t o p o s i t i v e amphidromic waves; only the i n t e r i o r p a r t s of the G u l f s and of the Western B a l t i c e x h i b i t e d standing waves. The p e r i o d s of the o s c i l l a t i o n s were reduced by the e a r t h ' s r o t a t i o n , when they were longer than the i n e r t i a l p e r i o d . F a i s and M i c h e l a t o ' s two-dimensional model (1980) r e s u l t e d in the c o n f i r m a t i o n of the p e r i o d s of the f i r s t two l o n g i t u d i n a l modes and the f i r s t two t r a n s v e r s a l modes, which had been i d e n t i f i e d with the four main resonant p e r i o d s of the observed data i n the Gulf of C a g l i a r i , I t a l y . O r l i c (1980) used the Proudman's equations to analyze the source of s e i c h e s i n the A d r i a t i c . But he suggested that 1 3 a hydrodynamical numerical model would be needed, s i n c e the e x i s t i n g a n a l y t i c s o l u t i o n s c o u l d not be s a t i s f a c t o r i l y a p p l i e d to the basins i n the A d r i a t i c . G o t l i b and Kagan (1980) c a l c u l a t e d the corresponding p a i r s of resonance p e r i o d and a m p l i f i c a t i o n f a c t o r i n the A t l a n t i c , Indian and P a c i f i c Oceans. Papa (1981) employed the l e a p - f r o g scheme (Sundermann, 1966) to simulate n u m e r i c a l l y the water movement of the L i g u r i a n Sea, i n a s s o c i a t i o n with a regime of s o u t h e a s t e r l y winds. The r e s u l t agreed well with the observed data. Afterwards (1983) h i s model r e v e a l e d a 5.7 hour p e r i o d seiche which was i d e n t i f i e d in the observed data. M i l e s (1981) s t u d i e d f r e e and f o r c e d o s c i l l a t i o n s i n a basin connected through a narrow c a n a l to e i t h e r the open sea or a second b a s i n . The r e s u l t i n g model y i e l d s a Hamiltonian p a i r of phase-plane equations f o r the f r e e o s c i l l a t i o n s , which are i n t e g r a t e d i n terms of e l l i p t i c f u n c t i o n s . The corresponding model f o r f o r c e d o s c i l l a t i o n s that are l i m i t e d by r a d i a t i o n damping y i e l d s a g e n e r a l i z a t i o n of D u f f i n g ' s equation f o r an o s c i l l a t o r with a s o f t s p r i n g , the s o l u t i o n of which i s obtained as an expansion i n the amplitude of the fundamental term i n a F o u r i e r expansion. Ben-Menahem and Vered (1982) s i m p l i f i e d the c o n f i g u r a t i o n of H a i f a Bay as a s e m i - e l l i p t i c embayment with a s e m i - p a r a b o l o i d a l bathymetric p r o f i l e and s o l v e d the shallow water equations i n t h i s domain to get the eig e n p e r i o d s of the bay. 1 4 H i b i y a and Kajiura(1982) c a r r i e d out a numerical study of s e i c h e s i n Nagasaki bay with the shallow water equations of motion and c o n t i n u i t y i n c l u d i n g the e f f e c t on sea l e v e l of the atmospheric pressure d e v i a t i o n . Akamatsu (1982) adopted more accurate equations i n c l u d i n g the e f f e c t s of bottom f r i c t i o n and wind s t r e s s on sea s u r f a c e . Odamaki et al (1983) developed a numerical model to study the water movement i n a l a r g e sea area. They added terms to represent the C o r i o l i s f o r c e , n o n - l i n e a r and t u r b u l e n t v i s c o u s e f f e c t s . In summary, along with the development of computer techniques numerical m o d e l l i n g has r e p l a c e d the t r a d i t i o n a l a n a l y t i c a l method, and t h e r e f o r e the c o m p l i c a t i o n of topography i s no longer the l e a d i n g o b s t a c l e to studying s e i c h e motions i n r e a l bays. C. THE EXCITING SOURCES OF SEICHES In the o p i n i o n of M i l e s and Munk (1961), which i s supported by many documented i n s t a n c e s , the surges f o l l o w i n g the passage of an atmospheric p r e s s u r e jump and a sudden s h i f t i n the winds appear to have been r e s p o n s i b l e f o r many sei c h e e x c i t a t i o n s . Random v a r i a t i o n s , which occur at a l l times i n the normal and t a n g e n t i a l s t r e s s e s on the water s u r f a c e , a l s o c o n t r i b u t e to the s u r g i n g . Of i n c i d e n t a l i n t e r e s t i s the e x c i t a t i o n of the normal modes by movements of the harbour bottom a s s o c i a t e d with the a r r i v a l of earthquake waves. Seiches can a l s o be e x c i t e d through the 15 harbour entrance. If there i s a broad spectrum of waves e x t e r i o r to the harbour, then those p a r t i c u l a r f r e q u e n c i e s that correspond to the resonant f r e q u e n c i e s w i l l e x c i t e the i n t e r i o r resonances; in some l o c a l i t i e s the spectrum i s even peaked at f r e q u e n c i e s t y p i c a l of harbour s e i c h e s . More f r e q u e n t l y s e i c h e s are generated by long waves reaching the coast from the open sea ( M i l e s , 1974). Such seiche-producing long waves can be generated by d i s t a n t m e t e o r o l o g i c a l d i s t u r b a n c e s at the sea s u r f a c e (Munk, 1962, pp647-663) and by seismic d i s t u r b a n c e s at the sea bed (Matuzawa el al, 1933). Heaps and Ramsbottom (1966) d i d an i n t e r e s t i n g study of wind-generated i n t e r n a l s e i c h e s i n Lake Windermere, England. Olsen and Hwang (1971) suggested that s h e l f resonance and edge-wave e f f e c t s seemed to p l a y a s i g n i f i c a n t and i n t e r e s t i n g r o l e f o r p e r i o d s somewhat longer than the fundamental bay resonance p e r i o d . Bowers (1977) showed t h e o r e t i c a l l y that the n a t u r a l o s c i l l a t i o n s of a harbour c o u l d be e x c i t e d d i r e c t l y , without breaking of the primary wave system, by set-down beneath wave groups, which i s a l o n g - p e r i o d d i s t u r b a n c e t r a v e l l i n g towards the shore l i n e at the group v e l o c i t y . A f t e r studying the c o n t i n e n t a l s h e l f s e i c h e s of south-west A u s t r a l i a , P e t r u s e v i c s et al (1979) concluded that long ocean waves are the most probable source of seiche e x c i t a t i o n , because d u r i n g high seiche a c t i v i t y there were no other apparent sources of e x c i t a t i o n , such as s i g n i f i c a n t 1 6 changes i n wind, or in barometric p r e s s u r e , and no l o c a l seismic a c t i v i t y . S i m i l a r c o n c l u s i o n s were made by Munk (1962) f o r other c o n t i n e n t a l s h e l v e s . Wubber and Krauss (1979) suggested that the development of s e i c h e s of the e n t i r e B a l t i c Sea r e q u i r e s l a r g e - s c a l e m e t e o r o l o g i c a l f o r c i n g . I s o z a k i (1979) a t t r i b u t e d the s e i c h e s i n Habu-harbor on Sept.17, 1976 and Oct.8, 1977 to the s w e l l s from Typhoon No.7619 and Typhoon No.7714 r e s p e c t i v e l y . F a i s and M i c h e l a t o ' s study of sei c h e s i n the Gulf of C a g l i a r i (1980) suggested that seiche s t i m u l a t i o n i s mainly a s s o c i a t e d with the i n s t a b i l i t y l i n e s or m e t e o r o l o g i c a l f r o n t s c r o s s i n g the g u l f at speeds c l o s e to the speeds of long waves on the g u l f . T r a n s v e r s a l s e i c h e s g e n e r a l l y occur f o l l o w i n g the passage of atmospheric p e r t u r b a t i o n s moving eastward or northeastward across the g u l f , while l o n g i t u d i n a l s e i c h e s are u s u a l l y generated by p e r t u r b a t i o n s p a s s i n g northward or northwestward over the g u l f . O r l i c (1980) s t u d i e d the source of the s e i c h e s i n the A d r i a t i c . He gave an e x p l a n a t i o n connected with a f o r c e d p r o g r e s s i v e wave i n the sea, i n s t e a d of a f r e e wave, as the s e i c h e - e x c i t i n g source. A g r a v i t y wave i n the atmosphere caused a f o r c e d p r o g r e s s i v e wave i n the sea, through the a c t i n g of the atmospheric pressure over the sea. The f o r c e d wave was of small amplitude u n t i l i t met a b a s i n whose p o s i t i o n and c o n f i g u r a t i o n are s u i t a b l e f o r an occurrence of the Proudman resonance, i . e . u n t i l the v e l o c i t y of the 17 atmospheric d i s t u r b a n c e (and the f o r c e d p r o g r e s s i v e wave) became equal to the v e l o c i t y of the long wave in the sea. Dyer (1982) co n s i d e r e d that the l a t e r a l i n t e r n a l s e i c h i n g i n Southampton Water was produced by i n t e r a c t i o n of the s u r f a c e seiche with the shallow s i d e of the e s t u a r y . I n t e r a c t i o n of a c u r r e n t with a s i l l c o u l d produce i n t e r n a l s e i c h i n g i f the waves were c o n t i n u a l l y r e f l e c t e d . T h i s has been e x p e r i m e n t a l l y observed at t i d a l f r e q u e n c i e s i n Oslo f j o r d and e x p l a i n e d by a l i n e a r mechanism by St i g e b r a n d t (1976). A n o n - l i n e a r mechanism has been expl o r e d by B l a c k f o r d (1978). Papa (1981) d i s c u s s e d the hypothesis of s o u t h e a s t e r l y winds as the f o r c i n g mechanism of the o s c i l l a t i o n r e v e a l e d at t i d e gauges i n L i g u r i a n Sea. Ben-Menahem and Vered (1982) concluded that the dominant source of the s e i c h e s at H a i f a Bay i s m e t e o r o l o g i c a l and the c o a s t a l s e i c h e s may be both f r e e and f o r c e d . The a n a l y t i c a l study showed that the shape and s i z e of the r e s o n a t i n g water body i s dependent to some degree on the d i r e c t i o n and i n t e n s i t y of the e x c i t i n g m e t e o r o l o g i c a l f r o n t . Giese et al (1982) suggested that some i n t e r n a l waves were generated by t i d e s i n the southeastern Caribbean Sea and they were r e s p o n s i b l e f o r e x c i t i n g the f o r t n i g h t l y groups of c o a s t a l s e i c h e s at Magueyes I s l a n d , Puerto R i c o . To support t h e i r hypothesis they analysed the c h a r a c t e r i s t i c s of c o a s t a l s e i c h i n g at Puerto P r i n c e s a on 1 8 Palawan I s l a n d in the P h i l i p p i n e s and i n d i c a t e d that the s e i c h i n g was e x c i t e d by the Sulu Sea i n t e r n a l waves. Apel et al (1980) observed the g e n e r a t i o n of these i n t e r n a l waves by strong t i d e s i n Sulu Sea. H i b i y a and K a j i u r a ' s numerical model (1982) confirmed that the e x c e p t i o n a l l y l a r g e range of o s c i l l a t i o n s i n Nagasaki Bay was indeed produced by a pressure d i s t u r b a n c e t r a v e l l i n g eastward with an averaged speed of about 110 kmh"1. I t was concluded that the e x c i t i n g mechanism was due to the l e a d i n g part of shallow water waves induced by the atmospheric pressure d i s t u r b a n c e being a m p l i f i e d over a broad c o n t i n e n t a l s h e l f because of near-resonant c o u p l i n g . A f t e r l e a v i n g t h i s c o n t i n e n t a l s h e l f r e g i o n , the a m p l i f i e d water wave converged i n t o the s h e l f r e g i o n . A t r a i n of waves thus formed with a p e r i o d of about 35 minutes entered Nagasaki Bay and was r e s o n a n t l y a m p l i f i e d at p e r i o d s of 36 minutes and 23 minutes which are the e i g e n - p e r i o d s of the Bay. Besides the resonance, the combined e f f e c t s of s h o a l i n g and r e f l e c t i o n i n s i d e Nagasaki Bay a l s o enhanced the a m p l i f i c a t i o n . Akamatsu's a n a l y s i s (1982) l e d to the c o n c l u s i o n that t h i s s e i c h e may be due to the severe barometric pressure jump a s s o c i a t e d with the c o l d a i r f r o n t . A f t e r examining the response of Nagasaki Bay to the f o r c i n g at the mouth of the Bay by means of a two-dimensional numerical model, i t was c o n s i d e r e d reasonable that the s e i c h e s of Nagasaki Bay were caused by the resonance to the long waves due to the barometric pressure jump. Using a 19 s i m i l a r numerical model, Odamaki et al (1983) confirmed the r e s u l t s above. They summed up the e x c i t i n g procedure as three e s s e n t i a l processes: a i r - s e a c o u p l i n g , r e f r a c t i o n and r e f l e c t i o n of the i n c i d e n t water wave by topographic e f f e c t s i n s h e l f and the a m p l i f i c a t i o n by the harbour resonance. Furthermore i t was i n d i c a t e d that the most important process was the second, i . e . the water waves i n c i d e n t onto s h e l f generated the l o c a l o s c i l l a t i o n systems by r e f l e c t i o n and r e f r a c t i o n , and caused the s e i c h e s in the a d j o i n i n g bay or p o r t . Summing up the above researches, s e i c h e s may r e s u l t from d i r e c t d i s t u r b a n c e s by m e t e o r o l o g i c a l f o r c e s at the s u r f a c e of a c o a s t a l bay. More f r e q u e n t l y they are generated by long waves reaching the coast from the open sea. Such se i c h e - p r o d u c i n g long waves can be generated by d i s t a n t m e t e o r o l o g i c a l d i s t u r b a n c e s at the sea s u r f a c e . The mechanism of the l a t t e r i s more complicated. In some cases i n t e r n a l waves may be the cause of the se i c h e e x c i t a t i o n . Sea bottom movements are c o n s i d e r e d as an i n c i d e n t a l source too. I I I . SPECTRAL ANALYSIS OF SEA SURFACE FLUCTUATIONS IN COASTAL BAYS T h i s Chapter analyses the sea s u r f a c e o s c i l l a t i o n data i n Pedder Bay, Becher Bay and Port San Juan on the north shore of Juan de Fuca S t r a i t . The r e s u l t i d e n t i f i e s s e iche a c t i v i t i e s , even very strong at times. The strong s e i c h e s and the i n f l u e n c e of t i d a l phase are s t u d i e d i n more d e t a i l in the f i n a l s e c t i o n s of t h i s c hapter. A. SYNOPTIC DESCRIPTION OF PHYSICAL OCEANOGRAPHY IN JUAN DE  FUCA STRAIT Juan de Fuca S t r a i t i s a long narrow submarine v a l l e y . Port San Juan near the P a c i f i c entrance i s the only major f j o r d - l i k e opening along the no r t h c o a s t . Becher Bay and Pedder Bay are small bays at the e a s t e r n end of the s t r a i t near Race Rocks (Figure 1). Water temperatures throughout the S t r a i t remain c o l d e r than 12°C at depths of more than 10 meters. S a l i n i t y i n c r e a s e s from top to bottom and from east to west. The wind p a t t e r n s w i l l be mentioned in chapter 6. Since d i r e c t measurements of wind waves have not been made in the S t r a i t , e m p i r i c a l r e l a t i o n s h i p s are r e l i e d upon to estimate t h e i r height d i s t r i b u t i o n s . F u l l y developed sea in the S t r a i t can a t t a i n s i g n i f i c a n t wave he i g h t s of 1.5 m, or most probable maximum he i g h t s of 2.7 m, f o r a 10 m/s wind that blows f o r at l e a s t 10 hours over a f e t c h of 140 km. (Thomson, 1981). 20 2 1 F i g u r e 1 Map of Juan de Fuca S t r a i t i n c l u d i n g the c o a s t a l bays i n t h i s r e s e a r c h . The oceanic t i d e t r a v e l s northward along the west coast of North America and enters the S t r a i t as a long p r o g r e s s i v e wave whose speed and range vary eastward as a r e s u l t of changes i n the depth and geometry of the S t r a i t . The s e m i d i u r n a l wave component (M 2 t i d e ) and the d i u r n a l wave component (K, t i d e ) are the two main c o n s t i t u e n t s . T i d e s , freshwater r u n o f f , winds and along-channel atmospheric pressure d i f f e r e n c e s are major f a c t o r s a f f e c t i n g c u r r e n t s i n the S t r a i t . The c u r r e n t d i s t r i b u t i o n s from numerical modeling by Crean (1983) give a d e t a i l e d i l l u s t r a t i o n . B. THE SEA SURFACE FLUCTUATION DATA The data- of sea s u r f a c e f l u c t u a t i o n adopted i n t h i s r e s e a r c h were obtained from Dr. R.E.Thomson, of the I n s t i t u t e of Ocean Sciences (IOS), Sidney ,- B.C. These data ( 22 were measured with pressure gauges. The gauges i n Pedder Bay and Becher Bay were model 2A, manufactured by Aanderaa Instruments L t d . , the gauge i n Port San Juan was model 12A, manufactured by A p p l i e d Micro System L t d . . O r i g i n a l data were d i g i t i z e d and s t o r e d on a computer tape. Part of data were f i l t e r e d by 7th order Butterworth d i g i t a l f i l t e r with a cut o f f p e r i o d of two hours which i s d e s c r i b e d i n the f o l l o w i n g s e c t i o n . Table 1 g i v e s a simple d e s c r i p t i o n of these data: Table 1 O r i g i n a l Data of Sea Surface F l u c t u a t i o n s bays date pre. gauge i n t e r v a l p o s i t i o n s sampling r a t e unf i l t e r e d data p o i n t s f i l t e r e d data p o i n t s Ped. seti Jan. Jan. 1 980 1 5 26 N 48°20.3' W123°33. 1' g r i d #161 2 min. 8490 0 Bee. seti Jan. Jan. 1 980 1 5 26 N 48°20.1' Wl23°36.0' g r i d #175 2 min. 8439 8435 San. seti Jan. Jan. 1 980 1 5 26 N 48°33.7' W124°24.8' g r i d #187 2 min. 7933 7933 Ped. s et 2 Jan. Feb. 1980 27 20 N 48°20.3' W123°33. 1' g r i d #161 2 min. 17475 17000 Bee . s et 2 Jan. Feb. 1980 27 20 N 48°20.1' W123°36.0' g r i d #175 2 min. 1 7444 1 7000 San. s el 2 Jan. Feb. 1 980 27 20 N 48°33.7' W124°24.8' g r i d #187 2 min. 17304 17000 2 3 where the g r i d #'s denote the corresponding p o s i t i o n s i n the numerical model g r i d s . C. REMOVAL OF TIDAL COMPONENTS Our i n t e r e s t i n t h i s r e s e a r c h i s to study the high frequency sea sur f a c e o s c i l l a t i o n with the range of pe r i o d s from s e v e r a l minutes to about one hour. Before a n a l y s i n g the data to i n v e s t i g a t e the seiche p r o p e r t i e s one must remove the low frequency t i d a l c o n s t i t u e n t s . There are many methods to reach t h i s g o a l . The widely adopted procedure i s to s u b t r a c t the p r e d i c t e d t i d a l components from time s e r i e s data. If the amplitudes and phases of a l l the c o n s t i t u e n t s of the t i d e are a v a i l a b l e , the sea su r f a c e v a r i a t i o n a s s o c i a t e d with t i d a l stage can be c a l c u l a t e d from the formula h = H 0 + Z fHcos[at+(V 0+u)-k] where I i s the summation sign f o r a l l the t i d a l c o n s t i t u e n t s h = the t i d e height at time t H 0 = the mean water height above datum H = the amplitude of a t i d a l c o n s t i t u e n t f = a f a c t o r reducing the mean amplitude H to the year p r e d i c t i o n a = the angular frequency of the c o n s t i t u e n t of amplitude H t = the time measured from s t a r t i n g p o i n t of 24 p r e d i c t i o n s (V 0+u) = the v a l u e of e q u i l i b r i u m argument of the c o n s t i t u e n t of amplitude H when t=0 , at Greenwich k = the phase of the c o n s t i t u e n t with respect to Greenwich ( i n the Canadian System) The maximum t i d a l l e v e l and i t ' s corresponding time in Juan de Fuca S t r a i t can be found in Canadian T i d e and Current Tables (1980). C o n s i d e r i n g the f i l t e r f u n c t i o n of the c o a s t a l bays and the adjacent sea area to t i d a l waves, the p r e d i c t e d t i d e s at a s p e c i f i e d p o i n t of a bay may be d i s t o r t e d at other p o i n t s , such as the p o s i t i o n s where the pressure gauges were i n s t a l l e d , so a f t e r s u b t r a c t i n g the p r e d i c t e d t i d e s , there s t i l l remains a r e s i d u a l . Even though the parameters of t i d a l c o n s t i t u e n t s are adopted at the p o s i t i o n where the data f o r a n a l y s i s were measured, t h i s method s t i l l c o u l d not remove a l l the low frequency components. An example co u l d be found in Lemon's t h e s i s (1975). The simplest way to remove t i d e s and other low frequency components i s to f i t a curve to the low frequency part of the measured time s e r i e s i n the sense of l e a s t square d i f f e r e n c e s . If the data set has only one extremum in low frequency component, a polynomial f i t i s s u f f i c i e n t . If there are two or more extrema, the s p l i n e f i t need to be invoked. I chose one p i e c e of time s e r i e s to demonstrate t h i s method. The s e l e c t e d time s e r i e s i s h a l f a t i d a l c y c l e 25 from Port San Juan f i r s t data s e t , and i n c l u d e s 256 data p o i n t s with the f i r s t p o i n t measured at 10:51 Jan. 18 , 1980. The 10th order polynomial curve f i t t e d to t h i s time s e r i e s i s the optimum polynomial curve under 16th order i n the sense of l e a s t square d i f f e r e n c e s which was s e l e c t e d a u t o m a t i c a l l y by the computer subroutine OLSF ( d e s c r i b e d i n "UBC Curve", computing Center, UBC). F i g u r e 2 shows the time s e r i e s and the f i t t i n g curve. They c o i n c i d e p r e t t y w e l l with each other. F i g u r e 3 shows the d i f f e r e n c e between o r i g i n a l time s e r i e s and the f i t t i n g curve, i t i s obvious that low frequency components have been removed. F i g u r e 4 shows the s p e c t r a l d e n s i t y of the f i l t e r e d data which was c a l c u l a t e d by Fast F o u r i e r T r a n s f o r m a t i o n . The spectrum r e v e a l e d that most of the energy i s c o n c e n t r a t e d w i t h i n the frequency band ce n t e r e d at f r e q u e n c i e s 0.03 and 0.075 c y c l e per minute, with corresponding p e r i o d s of 33 and 13.3 minutes. F i g u r e 5 shows the phases of the F o u r i e r c o e f f i c i e n t s which looks l i k e p r e t t y random. The Butterworth f i l t e r e d data at same time i n t e r v a l were processed by Fast F o u r i e r Transformation f o r comparison purpose. Both of these s p e c t r a l d e n s i t y curves are shown in F i g u r e 6. These two curves g i v e q u i t e the same sp e c t r a and r e v e a l q u i t e the same se i c h e phenomenon. It c o u l d be concluded that the f i l t e r s perform t h e i r e f f e c t i v e f u n c t i o n to e l i m i n a t e the t i d a l e f f e c t . The i n t e r e s t i n g p o i n t to be mentioned here i s the d i f f e r e n c e of the two curves at lowest frequency band. The polynomial f i l t e r performs b e t t e r f u n c t i o n i n reducing the lowest 2 6 O D E - 0 0 9 " (roe- oazr-3 o n o a s t - troar-An example of polynomial curve f i t t i n g . J 40.0 2J0.D T J ME — i 1 2BD.0 350 .0 (MJN . ) —I 1 420 .0 490 . 0 5 6 0 . D Polynomial f i l t e r e d sea s u r f a c e o s c i l l a t i o n . 27 31 UJ OL CD 0.0 0.03 D.06 ' 1 " 1 I 1 0-09 0.12 0.15 0.18 0 2] FREQUENCY (CYC./MIN.) —i 1 0.24 0.2"? F i g u r e 4 S p e c t r a l d e n s i t y of p o l y n o m i a l - f i l t e r e d data. UJ CO cc X . 1 1 - , , , . 0 0 D Q J 0.36 0.09 0.12 D .1S 0 IB FREQUENCY [CYC./MJN.l i r— 0.2! 0.24 0.27 F i g u r e 5 The phases of F o u r i e r c o e f f i c i e n t s of p o l y n o m i a l - f i l t e r e d d ata. 28 "1 1 1 1 — 1 1 1 1 1 1 0 . 0 D .03 0 . 0 6 0 .09 0 . 1 2 0 .15 0 . 1 8 0.2J 0 . 2 1 0 . 2 7 FREQUENCY (CYC./MJN.) F i g u r e 6 Comparison of s p e c t r a of P o l y n o m i a l and B u t t e r w o r t h f i l t e r e d d a t a . Dashed c u r v e : p o l y n o m i a l f i l t e r e d . S o l i d c u r v e : B u t t e r w o r t h f i l t e r e d . f r e q u e n c y components. D i g i t a l f i l t e r s a r e w i d e l y a p p l i e d i n communication t e c h n o l o g y , g e o p h y s i c a l r e s e a c h , e t c . . The advantages of thes e f i l t e r s a r e t h a t they can be a p p l i e d and s t u d i e d i n bo t h time and fr e q u e n c y domains, and the c h a r a c t e r i s t i c s of time s e r i e s can be i n t e r p r e t e d i n term of s t a t i s t i c s by f u l l y d e v e l o p e d t h e o r i e s . The f i l t e r e d d a t a used i n t h i s r e s e a r c h work were o b t a i n e d from a 7th o r d e r B u t t e r w o r t h f i l t e r . T h i s h i e r a r c h y of f i l t e r s i s d e f i n e d i n the f r e q u e n c y domain by the e q u a l i t y where 29 1 |H_ (co) | 2 = (3.3-1) 1 + ( *L_ ) 2 n Cc) c 6j i s the s p e c i f i e d cut o f f frequency n i s the order of f i l t e r F i g u r e 7 shows the frequency response of the 7th order Butterworth f i l t e r with cut o f f p e r i o d 2 hours. The mathematical f o r m u l a t i o n , design, c h a r a c t e r i s t i c s and a p p l i c a t i o n of t h i s f i l t e r to t i d a l time s e r i e s are f u l l y d e s c r i b e d by Thomson and Chow (1980). D. METHOD OF SPECTRAL ANALYSIS S p e c t r a l A n a l y s i s i s a very common method. There i s much l i t e r a t u r e about t h i s f i e l d . Only few p o i n t s need to be mentioned i n t h i s t h e s i s . The averaged power spectrum, or the esti m a t o r of power spectrum, may be obtained from ensemble average or frequency band average of s p e c t r a . These two procedures make the estimates c o n s i s t e n t . Futhermore, i f i n the f i r s t case the time s e r i e s i s d i v i d e d i n t o / s u b s e r i e s and i n the second case the spectrum i s smoothed over / f r e q u e n c i e s , both of them w i l l y i e l d same degrees of freedom n=2/ (Bendat and P i e r s o l , 1971). The procedure to compute the 95% confidence i n t e r v a l s of power s p e c t r a was adopted from Jenkins and Watts (1968, p239); the val u e s of c h i - s q u a r e d i s t r i b u t i o n , used f o r computation of c o n f i d e n c e i n t e r v a l s , came from a chi- s q u a r e p l o t (Jenkins and Watts 1968. p82) and a 30 '0.00 0-10 0-15 D-20 NBRMRLIZED (BY 0.25 FSRMP) 0-30 0.35 FREQUENCY 0-40 0-45 0-50 F i g u r e 7 The frequency response of the 7th order Butterworth f i l t e r . c h i - s q u a r e Table (Bendat and P i e r s o l , 1971. p388). The coherence e s t i m a t o r s suggested by Bendat and P i e r s o l (1971, p193) are frequency band averaged; those of Jenkins and Watts (1968, p374) and B l o o m f i e l d (1976, p214) are s p e c t r a l window smoothed. T h i s r e s e a r c h mainly uses the former. The confidence i n t e r v a l s were computed a c c o r d i n g to Jenkins and Watts (1968, p379) and i n d i c a t e d by a v e r t i c a l bar i n each graph. A dashed h o r i z o n t a l l i n e appears i n the graph of coherence too. I t i s the 95% l e v e l of the n u l l h y p o t h e s i s . I t r e f e r s to that value exceeded with 5% p r o b a b i l i t y by randomly r e l a t e d r e c o rds (Groves and Hannan, 1968). The c a l c u l a t i o n of these values f o l l o w s B l o o m f i e l d (1976, p227); the observed v a l u e s of coherence l e s s than those l i n e s should be regarded as not s i g n i f i c a n t l y 31 d i f f e r e n t from zero, and confidence i n t e r v a l should be used only i f the coherences exceed those l i n e s . Examples of the computation c o u l d be found i n Chang's t h e s i s (1976). T h i s procedure to o b t a i n 95% s i g n i f i c a n c e l e v e l s and c o n f i d e n c e i n t e r v a l s were performed throughout t h i s t h e s i s . Dashed l i n e s i n the graphs of phase are zero phase l i n e s . The coherence d e f i n i t i o n given by Godin (1972) i s : « * l n \ - \<Z(0)V(0)>\ 2 . n 1 { o ) - < l z ( a ) l a > < l V ( a ) l a > (3.4-1) where < > i n d i c a t e s ensemble averaging. Z(a)V(o) i s the cross-spectrum of the o b s e r v a t i o n s Z(t) with the v a l u e s of V ( t ) at v a r i o u s times. | Z ( o ) | 2 and | V ( a ) | 2 are power s p e c t r a . T h i s method may be used f o r d i s c o n t i n u o u s time s e r i e s , and performed i n the f o l l o w i n g s e c t i o n to study s e i c h e s i n d i f f e r e n t t i d a l l e v e l s . As f a r as I know, there i s no way to estimate the s i g n i f i c a n c e l e v e l and c o n f i d e n c e i n t e r v a l in t h i s case. E. SPECTRAL ANALYSIS (FIRST DATA SET) The B u t t e r w o r t h - f i l t e r e d time s e r i e s were a n a l y s e d i n order to i n v e s t i g a t e the seiche a c t i v i t i e s . The sampling p e r i o d of these data i s two minutes, so the c u t - o f f frequency (highest frequency o b t a i n a b l e by F o u r i e r a n a l y s i s ) i s corresponding to a p e r i o d of four minutes. F i r s t I analysed the f i r s t data s e t , i n c l u d i n g the data from Becher Bay and Port San Juan. The measured time s e r i e s 32 s t a r t s from 15:10 Jan. 15 , 1980. The data were input to a computer program with 256 p o i n t s per set which stands f o r a 8 hours 32 minutes time i n t e r v a l , and fed i n t o a Fast F o u r i e r Transform to get power s p e c t r a . Every next c a l c u l a t i o n s t a r t e d at a data p o i n t 8 hours l a t e r , so there i s a 32 minute o v e r l a p between s u c c e s s i v e input data s e t s . Each power spectrum represents the frequency c h a r a c t e r i s t i c s of sea s u r f a c e f l u c t u a t i o n w i t h i n the corresponding 8 hours. A l t o g e t h e r 33 s e t s of data were processed to look i n t o the seiche a c t i v i t i e s of 11 days i n these two bays. F i g u r e 8 and F i g u r e 9 i l l u s t r a t e the s u c c e s s i v e 33 power s p e c t r a of Becher Bay and Port San Juan. In the s p e c t r a of Becher Bay two columns of peaks are i n p e r i o d s about 100 minutes and 13 minutes r e s p e c t i v e l y . The 13 minutes p e r i o d f l u c t u a t i o n i s i d e n t i f i e d with e x c i t e d resonant waves i n the bay by numerical r e s u l t i n Chapter 4. The highest energy was found on Jan. 20. The 100 minutes p e r i o d f l u c t u a t i o n may be caused by e x t e r n a l f o r c i n g or data p r o c e s s i n g procedure. For some s p e c t r a the 100 minutes p e r i o d peaks are absent which supports the f i r s t h y p o t h e s i s . The s p e c t r a of Port San Juan give us a much more random p i c t u r e , e s p e c i a l l y i n the high frequency band. Large peaks can be seen at p e r i o d s of about 30 to 45 minutes. It i s i n t e r e s t i n g to note that the high frequency band o s c i l l a t i o n s only appears i n Port San Juan. Some ex p l a n a t i o n s may be invoked. One i s the f i l t e r f u n c t i o n of bays, that means the geometry and bottom topography of Port 33 _7 JflN.25 J JflN.24 _7 JHN .23 _7 JRN.22 J JAN.21 _7 JAN.20 ~_7 JHN.19 J JAN. 18 _7 JRN.17 "_7 JAN. 16 I 1 1 1 1 1 1 1 I I I 0 . 0 0 . 0 5 0 . 1 0 . 1 5 0 . 2 0 . 2 5 FREQUENCY CPM F i g u r e 8 8-hour wave s p e c t r a of Becher Bay, January. . ~_7 JflN.25 0 .0 T ~ T r 0 . 0 5 0 . 1 FREQUENCY " i 1 1— 0.15 0 .2 CPM 0.25 Figu r e 9 8-hour wave s p e c t r a of Port San Juan, January. 34 San Juan a m p l i f y or maintain the high frequency incoming waves i n s i d e the bay, but those of Becher Bay do not. The others are the wind momentum input through a i r - s e a i n t e r a c t i o n and the s w e l l s from P a c i f i c Ocean. The wind p a t t e r n s are d i f f e r e n t i n the v i c i n i t y of Port San Juan from that of Pedder Bay and Becher Bay. The l a t t e r i s an area of calm a i r s . The swel l s of ocean c o u l d not maintain t h e i r energy to t r a v e l to as f a r as the v i c i n i t y of Becher Bay. The ensemble averages of 33 power sp e c t r a of Becher Bay and Port San Juan are shown i n F i g u r e 10 and F i g u r e 11 r e s p e c t i v e l y . The main f e a t u r e s of the spe c t r a d i s c u s s e d above, are made c l e a r i n these two graphs. F i g u r e 12 shows the frequency band-averaged coherence of these two se t s of time s e r i e s ; the main peak occurs at p e r i o d range about 30 to 70 minutes, other peaks above confidence l e v e l have p e r i o d 5 to 6 minutes, which i m p l i e s that these two time s e r i e s were c o r r e l a t e d i n t h i s range of p e r i o d s , furthermore that common sources e x c i t e d s e i c h e s with these p e r i o d s i n the two bays. F. SPECTRAL ANALYSIS (SECOND DATA SET) The measurement of second data set s t a r t e d at 10:40 Jan. 27 , 1980 and l a s t e d a l i t t l e more than 24 days. The time s e r i e s i n the three bays beginning from 16:00 Jan. 27 , 1980 were inputed to a computer program to draw seventy 8-hour power s p e c t r a f o r every bay. The p e r s p e c t i v e views of these s u c c e s s i v e s p e c t r a are shown i n Fi g u r e 13 to Fi g u r e 15 35 o — 0.0 0.03 0.06 0.03 0.12 0.1S 0.1S 0.2! 0.24 0.2"? FREQUENCY ( C Y C . / M I N . ) F i g u r e 10 Power s p e c t r a of Becher Bay, January. 95% co n f i d e n c e i n t e r v a l : (0.72,1.45)*P(f) a 0.0 0.03 0.06 O.OS 0.12 0.1S 0.1B 0.2] 0.24 0.2"? FREQUENCY ( C Y C . / M J N . ) F i g u r e 11 Power s p e c t r a of Port San Juan, January. 95% c o n f i d e n c e i n t e r v a l : (0.72,1.45)*P(f) 36 <x <_)<c x z a <_> U J UJo" CK X o 0.0 0.03 0.06 0.09 0 )2 FREQUENCY 0.15 I C Y C . / M 1 N J o.ie o.2i 0.24 0.27 x ° " a 1 1 1 1 1 1 — 0.0 0.03 0.06 0.09 0.12 0.15 0.18 FREQUENCY ( C Y C . / M I N . ) 0.2! 0.24 0.27 F i g u r e 1 2 Coherence and phase of wave time s e r i e s between Becher Bay and Port San Juan, January. E r r o r bars: 95% confidence i n t e r v a l s . Dashed l i n e : 95% s i g n i f i c a n c e l e v e l 37 fo r the three bays. The one-day ensemble-averaged s p e c t r a beginning from Jan. 28, 1980 are shown i n Fig u r e 16 to F i g u r e 18. The s p e c t r a l s e r i e s of Pedder Bay and Becher Bay have obvious peaks at the resonant f r e q u e n c i e s (confirmed by numerical model i n Chapter 4), the corresponding p e r i o d ranges are centered at 11 and 13 minutes, The s p e c t r a l s e r i e s of Port San Juan d i s p l a y s more d i s o r d e r ; only in the f i r s t and l a s t three d a i l y s p e c t r a does the energy appear con c e n t r a t e d i n w e l l - d e f i n e d peaks w i t h i n the p e r i o d range about 30 to 40 minutes. There were s e v e r a l days centered on Feb. 2 when the s p e c t r a of a l l three bays were much more e n e r g e t i c than d u r i n g the r e l a t i v e l y calmer days, but high frequency band peaks c o u l d be seen only i n Port San Juan. F i g u r e 19 to F i g u r e 21 show the ensemble averages of s p e c t r a f o r the complete 20-day p e r i o d , i . e . the averages of 60 s p e c t r a of every bay r e s p e c t i v e l y , with the corresponding 60 time s e r i e s beginning Jan. 28, 1980 and l a s t i n g 20 days, every set c o n t a i n s 256 data p o i n t s r e p r e s e n t i n g 8 hours 32 minutes, In Pedder Bay and Becher Bay the 11 minutes and 13 minutes p e r i o d peaks are very remarkable, i n Port San Juan the frequency band over 0,04 c y c l e per minute (25 minutes period) c o n t a i n s a l a r g e q u a n t i t y of energy. F i g u r e 22 shows the coherence and phase of time s e r i e s of Pedder Bay and Becher Bay, F i g u r e 23, Pedder Bay and Port San Juan, F i g u r e 24, Becher Bay and Port San Juan. The coherences were estimated by t a k i n g frequency band-average. F i g u r e 22 r e v e a l s that the c o r r e l a t e d p e r i o d s are more than one hour 38 3tz XK2Z 0 . 0 i 1 1 r 0 . 0 5 0 . 1 FREQUENCY C . 1 5 0 . 2 CPM —J FEB.II Z = Z r J FEB.17 — 7 FEB-16 -_7 FEB. 15 :7 FEB.14 7 FEB.13 ""7 FEB.12 "7 FEB.11 "7 FEB.10 : FEB.9 = : J FEB.8 — —_7 FEB.7 ~J FEB-6 _7 FEB.5 - J F E B - 4 7 FEB.3 -_7FEB-2 XZXZTj FEB.1 i r z r j JRN.31 Hr_7 JflN.30 =T_7 JflN.29 :_7 JfiN.28 0 . 2 5 F i g u r e 13 8-hour wave s p e c t r a of Pedder Bay, February. ^ 1 7 FEB.13 ~_7 FEB.12 p_7 FEB.11 '• J FEB.10 EEr7 FEB.9 ErJ FEB.8 FEB.7 •7 FEB.6 EE7.7FEB-5 Er-7FEB-4 ^7 F E B- 3 •J FEB.2 ^ ^ 7 FEB.l 7 J ° N - 3 1 ~ 7 J ° N -30 =^7 J ^ - 2 9 ~7 i 1 1 1 1 r 0 . 0 0 . 0 5 0 . 1 FREQUENCY 0 . 1 5 0 . 2 CPM 0 . 2 5 F i g u r e 14 8-hour wave s p e c t r a of Becher Bay, February. 4 0 J FEB.18 J FEB.17 J FEB.16 J FEB.15 _7 FEB.14 _7 FEB.13 r_7 FEB. 12 'J JflN.29 ~ J JHN.28 r~ 0 . 0 0 . 0 ^REQUt i lCY , 1— 0.15 0.2 CPM 0.25 F i g u r e 15 8-hour wave s p e c t r a of Port San Juan, February. 41 _ A — FEB.16 - FEB.15 FEB.14 FEB.13 - FEB.12 FEB.11 FEB.10 — FEB.9 - FEB.8 FEB.7 FEB.6 FEB.5 FEB.4 FEB.3 FEB.2 — FEB.1 — JRN.31 — JflN.30 • JHK29 JHN.28 i ^ i I i r 0.0 0.05 0.1 FREQUENCY n i i 0.15 0.2 CPM 0.25 F i g u r e 16 D a i l y s p e c t r a of Pedder Bay, February - FEB.16 FEB.15 FEB.14 - FEB.13 FEB.12 FEB.11 - FEB.10 FEB.9 FEB.8 - FEB.7 FEB.6 FEB.5 FEB.4 FEB.3 FEB.2 — FEB.1 — JAN.31 — JftN.30 Jf lK29 JPHJ2S i i r I i i i I I i i 0.0 0.05 0.1 0.15 0.2 0.25 FREQUENCY CPM F i g u r e 17 D a i l y s p e c t r a of Becher Bay, Febr F i g u r e 18 D a i l y s p e c t r a of Port San Juan, February. F i g u r e 19 20-day averaged power s p e c t r a of Pedder Bay. 95% co n f i d e n c e i n t e r v a l : (0 .79,1.31)*P(f) 43 F i g u r e 21 20-day averaged power s p e c t r a of Port San Juan. 95% co n f i d e n c e i n t e r v a l : (0.79,1.31)*P(f) 44 F i g u r e 22 Coherence and phase of wave time s e r i e s between Pedder Bay and Becher Bay, February. E r r o r bars: 95% c o n f i d e n c e i n t e r v a l s . Dashed l i n e : 95% s i g n i f i c a n c e l e v e l 45 Z cr t— <_> cr J IDo' CK D CJ 0.0 FREQUENCY (CYC / M I N . ) 8 0 2 1 0.0 0.03 0.06 i " 1 1 r 0 0 9 0 1 ? 0 1 5 n i R FREQUENCY (CYC / M ] N . ) 0.21 1 1 0.24 0.27 F i g u r e 23 Coherence and phase of wave time s e r i e s between Pedder Bay and Port San Juan, February. E r r o r bars: 95% c o n f i d e n c e . i n t e r v a l s . Dashed l i n e : 95% s i g n i f i c a n c e l e v e l F i g u r e 24 Coherence and phase of wave time s e r i e s between Becher Bay and Port San Juan, February. E r r o r bars: 95% confidence i n t e r v a l s . Dashed l i n e : 95% s i g n i f i c a n c e l e v e l 47 and 15, 10-12, 7.5 minutes, the range of p e r i o d s 10 to 15 covers the resonant p e r i o d s of the two bays. F i g u r e 23 shows that the c o r r e l a t e d p e r i o d s cover a wide range. F i g u r e 24 shows predominant peaks at low frequency band with corresponding p e r i o d s 30 to 80 minutes; i t may be compared with F i g u r e 21: the peaks occur at same frequency band in the power spectrum of Port San Juan as i n the coherence between Becher Bay and Port San Juan, there i s random noise throughout high frequency band in power spectrum, meanwhile i t i s g r e a t l y reduced i n coherence. Keeping i t i n mind that the mouths of these two bays open on the same s t r a i t at the same coast l i n e , i t i s l o g i c a l to assume that the low frequency f l u c t u a t i o n s of the two bays may be e x c i t e d by common e x t e r n a l f o r c i n g sources, such as edge waves, t r a v e l l i n g along the coast l i n e and p a s s i n g a c r o s s the mouths. The high frequency f l u c t u a t i o n of the two bays are not c o r r e l a t e d . The above a n a l y s i s d e s c r i b e s the general f e a t u r e s of the s p e c t r a i n those three bays. The ensemble averages throw l i g h t on these general f e a t u r e s . The coherences imply common e x c i t i n g sources, which w i l l be d i s c u s s e d i n Chapter 6. G. DETAILED EXAMINATION OF TYPICAL SEICHES By v i s u a l examination of a l l the data i t was found that the s e i c h e occurs at any stage of t i d e c y c l e ; sometimes i t i s very strong, sometimes i t d i s a p p e a r s . The t y p i c a l s e i c h e s had w e l l d e f i n e d f r e q u e n c i e s and o f t e n observed i n these 48 three bays, which r e v e a l e d the d i f f e r e n t resonant fr e q u e n c i e s of these three bays. The t y p i c a l s e i c h e s were found in a l l the three bays. These o s c i l l a t i o n s are t y p i c a l to r e v e a l the wave response c h a r a c t e r i s t i c s of every bay. Four s e t s of 500 minutes time s e r i e s were s e l e c t e d to look i n t o t h i s f e a t u r e ; they are d e s c r i b e d i n Table 2 Table 2 Data of T y p i c a l Seiches (500 min.) s t a r t time and date bays t y p i c a l p e r i o d 1st s e t 00:00 18 Feb., 1 980 Pedder 1 1 2nd set. 09:20 2 Feb., 1980 Becher 1 3 3rd s e t 18:40 29 Jan. , 1980 San Juan 33 41 h set 1 0:40 11 Feb., 1980 San Juan 13-15 F i g u r e 25 to F i g u r e 28 are these u n f i l t e r e d time s e r i e s . The time s e r i e s were chosen from stages of both high t i d e and low t i d e . F i g u r e 29 to F i g u r e 32 are the same time s e r i e s , from which the low frequency components a s s o c i a t e d with t i d e s were removed by a Butterworth f i l t e r . F i g u r e 29 shows the t y p i c a l seiche i n Pedder Bay with p e r i o d about 11 minutes. F i g u r e 30 shows the t y p i c a l seiche i n Becher Bay with p e r i o d about 13 minutes. F i g u r e 31 and F i g u r e 32 show the t y p i c a l s e i c h e i n Port San Juan. The f i r s t to s i x t h peaks i n F i g u r e 31 have average p e r i o d about 33 to 34 minutes, the other peaks have roughly same p e r i o d s . F i g u r e 32 shows s e i c h e s with p e r i o d range about 8 to 15 minutes. f-\ 1 1 1 1 1 1 1 1 D . D 6 2 . 5 125.0 JB7.5 2 5 0 . 0 332.5 3 7 5 . 0 4 3 7 . 5 500 TIME (MIN.) F i g u r e 25 Wave time s e r i e s beginning at 00:00 Feb. 18, 1980 i n Pedder Bay. S i i i i i i i i i 0.0 6 2 . 5 125.0 1B7.5 2 5 0 . 0 332.5 3 7 5 . 0 437.5 500 TIME (MIN.) F i g u r e 26 Wave time s e r i e s beginning at 09:20 Feb. 2, 1980 i n Becher Bay. 50 X LU' a i i -1 1 1 1 6 2 . 5 J25.3 itO . 5 2 5 3 . 0 332 .5 3 7 5 . 0 T]ME (MIN.) 437.5 F i g u r e 27 Wave time s e r i e s beginning at 18:40 Jan. 29, 1980 i n Port San Juan. F i g u r e 28 Wave time s e r i e s 11, 1980 i n Port beginning at 10:40 Feb. San Juan. 3 3 2 . 5 (MJN.) 437 .5 5 0 0 . 0 F i g u r e 29 High frequency components of wave time s e r i e s beginning at 00:00 Feb. 18, 1980 in Pedder Bay. 0 . 0 -i i r 6 2 . 5 125.0 1B7.5 TIME 250. 1 3 ) 2 . 5 (MIN . ) T 437. 5 0 0 . 0 F i g u r e 30 High frequency components of wave time s e r i e s beginning at 09:20 Feb. 2, 1980 in Becher Bay. 437.5 5 0 0 . D F i g u r e 31 High frequency components of wave time s e r i e s beginning at 18:40 Jan. 29, 1980 in Port San Juan. UJ 1 3 3 2 . 5 (MIN . ) —I 375.0 ~1 437.5 O.D 62.5 125.0 187.5 TIME 250 .0 I 5 0 0 . 0 F i g u r e 32 High frequency components of wave time s e r i e s beginning at 10:40 Feb. 11, 1980 in Port San Juan. 53 Longer time s e r i e s around those of t a b l e 2 were subject to s p e c t r a l a n a l y s i s to i n s p e c t the frequency s i g n a t u r e of the strong s e i c h e s . Four s e t s of time s e r i e s from three bays which c o n t a i n the strong s e i c h e s of t a b l e 2 were s e l e c t e d . Each time s e r i e s c o n s i s t s of continuous 2,048 sampling p o i n t s , or 68 hours 16 minutes, and i n c l u d e s the time s e r i e s shown i n F i g u r e 29 to F i g u r e 32 which i l l u s t r a t e the t y p i c a l f e a t u r e of the 68-hour time s e r i e s . These 68-hour data are d e s c r i b e d i n Table 3 Table 3 Data of T y p i c a l Seiches (4096 min.) s t a r t time and date bays t y p i c a l p e r i o d 1st set 00:00 16 Feb., 1980 Pedder 1 1 2nd set 00:00 31 Jan., 1980 Becher 1 3 3rd s e t 00:00 28 Jan., 1980 San Juan 33 4t h s e I 00:00 9 Feb., 1980 San Juan 13-15 The four s e l e c t e d time s e r i e s were processed as f o l l o w s . F i r s t , power s p e c t r a were c a l c u l a t e d by Fast F o u r i e r Transformation and 1,025 s p e c t r a l c o e f f i c i e n t s were obtained f o r each s e r i e s at the f r e q u e n c i e s : F n = 47M6 n = 0 ' 1 ' 2 1 ' 0 2 4 Then from F,to F, 0 2 1 1 every 8 s u c c e s s i v e c o e f f i c i e n t s were averaged and set to the c e n t r a l frequency, i . e . the coef f i c i e n t 54 1 8k p = — L F. ~ k 8 i=6k-7 1 was a t t r i b u t e d t o the f r e q u e n c y f _ k 3.5 1 k 512 8.0 512 These band-averaged f r e q u e n c i e s g i v e e q u i v a l e n t r e s u l t t o the ensemble a v e r a g i n g method (Bendat and P i e r s o l , 1971) F i g u r e 33 t o F i g u r e 36 show the f o u r band-averaged f r e q u e n c y s p e c t r a . H. THE SHIFT OF RESONANT PERIOD BETWEEN HIGH AND LOW TIDE  STAGES 1 A c c o r d i n g t o M e r i a n ' s f o r m u l a (LeBlond and Mysak, 1978. p.293) the resonant f r e q u e n c y changes w i t h the v a r i a t i o n of water depth of a bay, namely w i t h the stage of t i d e l e v e l s . T h e r e f o r e I examined s e i c h e a c t i v i t y at h i g h and low t i d e s r e s p e c t i v e l y . Twelve time s e r i e s were chosen from h i g h and low t i d e l e v e l s r e s p e c t i v e l y i n each bay. F i g u r e 37 t o F i g u r e 39 show a t what t i d a l s t a g e the time s e r i e s were p i c k e d up f o r a n a l y s i s . Each time s e r i e s l a s t s f o u r hours and 16 m i n u t e s . The o r i g i n of the X - a x i s s t a n d s f o r the time 00:00 J a n . 27, 1980. At h i g h t i d e l e v e l s the s e l e c t e d d a t a s e t s i n Becher Bay a r e 16 minutes e a r l i e r than those i n Pedder Bay; i n P o r t San Ju a n , 32 minutes e a r l i e r than those i n Pedder bay. At low t i d e l e v e l s the da t a s e t s i n Becher Bay a r e 32 minutes e a r l i e r than those i n Pedder Bay; i n P o r t 55 F i g u r e 33 Power spectrum of strong seiche of Pedder Bay. 95% confidence i n t e r v a l : (0.55,2.32)*P(f) F i g u r e 34 Power spectrum of strong seiche of Becher Bay. 95% con f i d e n c e i n t e r v a l : (0.55,2.32)*P(f) 56 o.os FREQUENCY o.is (CYC/MIN. 0.27 F i g u r e 35 Power spectrum of strong s e i c h e of Port San Juan ( 1 ) . 95% c o n f i d e n c e i n t e r v a l : (0.55,2.32)*P(f) 0.0 0.03 0.06 0 .09 FREQUENCY 0.12 T r O . ' S D . l f i (CYC/MIN.) 0 .21 0.24 0.27 F i g u r e 36 Power spectrum of strong s e i c h e of Port San Juan ( 2 ) . 95% c o n f i d e n c e i n t e r v a l : (0.55,2.32)*P(f) F i g u r e 37 Tides i n Pedder Bay, dotted l i n e : whole t i d a l data; t h i c k l i n e : s e l e c t e d time s e r i e s . a' F i g u r e 38 Tides in Becher Bay, dotted l i n e : whole t i d a l data; t h i c k l i n e : s e l e c t e d time s e r i e s . 58 i i i i i i i i 0.0 3.3 6.0 9.0 12.0 15.0 18.0 21.0 21.0 TIME (DRY! F i g u r e 39 Tides i n Port San Juan, dotted l i n e : whole t i d a l data; t h i c k l i n e : s e l e c t e d time s e r i e s . San Juan, 80 minutes e a r l i e r than those i n Pedder bay. T h i s treatment c o u l d p o s i t i o n the i n d i v i d u a l time s e r i e s at c e n t r a l high or low t i d e l e v e l s and maintain constant time la g s i n d i f f e r e n t bays at high and low t i d e l e v e l s r e s p e c t i v e l y . The time d i f f e r e n c e s are the rough estimates of time i n t e r v a l f o r t i d e to t r a v e l from one bay to the other, which were obtained from v i s u a l examination of o r i g i n a l time s e r i e s in F i g u r e 37 to F i g u r e 39. F i g u r e 40 shows the power s p e c t r a of each time s e r i e s at low t i d e l e v e l in the three bays. F i g u r e 41 to F i g u r e 43 show ensemble averages of 12 time s e r i e s i n the three bays. In Pedder Bay the 11 minutes p e r i o d peak of F i g u r e 33 i s not prominent i n F i g u r e 41; r a t h e r , a low frequency peak predominates. Looking back to F i g u r e 40 a l a r g e low " I — I — I I — I — I — I — I 0.1 0 . 1 5 :RE0UENCY CPM 0.0 0.05 .  .  0.2 0.25 §^ ^ K S B • ttm- a • • » .mm. mmm. m . ^ F i g u r e 40 Power s p e c t r a of s e l e c t e d low t i d e s of Pedder Bay ( u p p e r ) , B e c h e r Bay ( m i d d l e ) and P o r t San J u a n ( l o w e r ) . 60 c o s o FREQUENCY 0.)S ( C Y C / M I N . ) 0.77 F i g u r e 41 Averaged power spectrum of s e l e c t e d low t i d e s of Pedder Bay. 95% c o n f i d e n c e i n t e r v a l : (0.61,1.94)*P(f) F i g u r e 42 Averaged power spectrum of s e l e c t e d low t i d e s of Becher Bay. 95% c o n f i d e n c e i n t e r v a l : (0.61 ,1 . 94)*P(f) 61 -1 i i 1 1 1 1 1 1 s O .D 0 . 0 3 0 . 0 6 0 . 0 9 0 . i " 2 0 . 1 5 DAB 0.2) 0 . 2 4 0 21 FREQUENCY ( C Y C . / M J N . ) F i g u r e 43 Averaged power spectrum of s e l e c t e d low t i d e s of Port San Juan. 95% con f i d e n c e i n t e r v a l : (0.61,1.94)*P(f) frequency peak appears i n s p e c t r a of Pedder Bay i n the s i x t h spectum, which may be e x c i t e d by e x t e r n a l f o r c i n g . F i g u r e 42 shows a predominant peak at about 13 minutes p e r i o d . F i g u r e 43 shows a somewhat random spectrum i n Port San Juan, s i m i l a r to the s p e c t r a of t h i s bay we d e a l t with before. F i g u r e 44 to F i g u r e 46 show the coherences of Pedder Bay versus Becher Bay, Pedder Bay versus Port San Juan and Becher Bay versus Port San Juan r e s p e c t i v e l y . Some l a r g e peaks appear i n high frequency band which may be caused by same high frequency e x t e r n a l f o r c i n g . At h i g h t i d e s , the p e r s p e c t i v e view of s p e c t r a l s e r i e s i s shown i n F i g u r e 47. Comparing with F i g u r e 40 the peaks at the resonant f r e q u e n c i e s of Pedder Bay and Becher Bay are much more obvious. F i g u r e s 48 to 50 show ensemble averaged F i g u r e 44 Coherence of s e l e c t e d low t i d e s , Pedder Bay versus Becher Bay. FREQUENCY (CYC./MJNJ F i g u r e 45 Coherence of s e l e c t e d low t i d e s , Pedder Bay versus Port San Juan. F i g u r e 46 Coherence of s e l e c t e d low t i d e s , Becher . Bay versus Port San Juan. F i g u r e 4 7 Power s p e c t r a of s e l e c t e d high t i d e s of Pedder Bay (upper), becher Bay (middle) and Port San Juan ( l o w e r ) . 65 s p e c t r a of the 12 time s e r i e s i n the three bays r e s p e c t i v e l y . The l a r g e peaks at the resonant f r e q u e n c i e s of Pedder Bay and Becher Bay are p r e t t y n o t i c e a b l e . F i g u r e 51 to F i g u r e 53 show the coherences of Pedder-Becher, Pedder-San Juan and Becher-San Juan. F i g u r e 54 to F i g u r e 56 show the comparisons of ensemble averaged power s p e c t r a between low and high t i d e l e v e l s i n the three bays r e s p e c t i v e l y . In Pedder Bay and Becher Bay the peaks at resonant f r e q u e n c i e s i n high t i d e l e v e l s are much l a r g e r than those i n low t i d e l e v e l s . The l o g i c a l c o n j e c t u r e i s that s e i c h e s are more e a s i l y e x c i t e d at high t i d e s ; in other words, the resonant a m p l i f i c a t i o n f a c t o r s at high t i d e s may be l a r g e r than those at low t i d e s . F i g u r e s 57 to 59 compare the coherences between low and high t i d e s of Pedder-Becher, Pedder-San Juan and Becher-San Juan. F i g u r e 57 show a wide peak at high t i d e s i n frequency range 0.08 to 0.105 cpm or p e r i o d s 12.5 to 9.5 minutes, which g e n e r a l l y covers the range of resonant f r e q u e n c i e s of Pedder Bay and Becher Bay. T h i s peak suggests a common source of resonant o s c i l l a t i o n s i n these two bays. But at low t i d e s coherence i s very low i n same frequency range. A general f e a t u r e i s that low f r e q u e n c i e s seem to be c o r r e l a t e d a t high t i d e s , high f r e q u e n c i e s , at low t i d e s . F i g u r e 58 shows s e v e r a l peaks at low t i d e s , two of them are at p e r i o d s 10 and 14 minutes. F i g u r e 59 shows many sharp peaks at h i g h t i d e s . The o s c i l l a t i o n s at these f r e q u e n c i e s may be generated by waves propagating i n Juan de Fuca 3.03 0.06 - i — r o.oa o . i 2 FREQUENCY 0 . 1 5 (CYC./MJN. F i g u r e 48 Averaged power spectrum of s e l e c t e d high t i d e s of Pedder Bay. 95% confidence i n t e r v a l : (0.61,1.94)*P(f) F i g u r e 49 Averaged power spectrum of s e l e c t e d high t i d e s of Becher Bay. 95% confidence i n t e r v a l : (0.61,1.94)*P(f) T 1 1 1 r 0 . 0 0 . 0 3 0 . 0 6 fl.09 0 . 1 2 0 . 1 5 0 . 1 8 FREQUENCY (CYC./MIN.) 0.2J 0.24 0.21 F i g u r e 50 Averaged power spectrum of s e l e c t e d high t i d e s of Port San Juan. 95% confidence i n t e r v a l : (0.61,1.94)*P(f) i 0 . 0 0 .03 i r 0 0 6 0 . 0 S 0 . ! 2 0 . 1 5 0 . 1 8 FREQUENCY (CYC./MIN.) 0.21 F i g u r e 51 Coherence of s e l e c t e d high t i d e s , Pedder Bay versus Becher Bay. 68 F i g u r e 52 Coherence of s e l e c t e d high t i d e s , Pedder Bay versus Port San Juan. F i g u r e 53 Coherence of s e l e c t e d high t i d e s , Becher Bay versus Port San Juan. 69 . 2 1 FREQUENCY ( C Y C / M I N . ) F i g u r e 54 Comparison of averaged power s p e c t r a between low ( s o l i d l i n e ) and hig h (dashed l i n e ) t i d e s of Pedder Bay. F i g u r e 55 Comparison of averaged power s p e c t r a between low ( s o l i d l i n e ) and hig h (dashed l i n e ) t i d e s of Becher Bay. F i g u r e 56 Comparison of averaged power s p e c t r a between low ( s o l i d l i n e ) and high (dashed l i n e ) t i d e s of Port San Juan. F i g u r e 57 Comparison of coherences between low ( s o l i d l i n e ) and high (dashed l i n e ) t i d e s , Pedder-Becher. 7 1 F i g u r e 58 Comparison of coherences between low ( s o l i d l i n e ) and high (dashed l i n e ) t i d e s , Pedder-San Juan. 0 . 2 7 FREQUENCY ( C Y C / M I N . ) F i g u r e 59 Comparison of coherences between low ( s o l i d l i n e ) and high (dashed l i n e ) t i d e s , Becher-San Juan. 72 S t r a i t . The above three F i g u r e s d e s c r i b e an i n t e r e s t i n g f e a t u r e . I t seems that Pedder Bay and Becher Bay are c o r r e l a t e d at both low and high t i d e s f o r d i f f e r e n t frequency bands, Pedder Bay and Port San Juan are more c o r r e l a t e d at low t i d e s , Becher Bay and Port San Juan are more c o r r e l a t e d at high t i d e s . IV. NUMERICAL MODELLING OF SEICHES IN COASTAL BAYS As a d i r e c t r e s u l t of r a p i d progress of computer techniques i n l a s t decade, p h y s i c a l oceanographers move t h e i r a t t e n t i o n from seeking s p e c i a l f u n c t i o n s s a t i s f y i n g the f l u i d dynamical equations to numerical m o d e l l i n g , which i s a more powerful t o o l i n s o l v i n g c h a l l e n g i n g problems. The work d e s c r i b e d i n t h i s chapter t r i e s to f i n d a simple method to model the seiche a c t i v i t y i n c o a s t a l bays. Due to the c o n s i d e r a t i o n of f u r t h e r extending the model, the s i m p l i c i t y and computing cost of t h i s b a s i c model has been a s e n s i t i v e f a c t o r i n i t s development. A. THE FORMULATION OF THE BASIC NUMERICAL MODEL To d e s c r i b e the c h a r a c t e r i s t i c s of the wave response of a bay, we s t a r t from the shallow water equations: DU Dt + 2ftXU + gVr? + gU ? T = 0 (4.1-1) + V- ( H + T J ) U = 0 where U h o r i z o n t a l v e l o c i t y v e c t o r l o c a l v e r t i c a l component of angular v e l o c i t y of the e a r t h 9 g r a v i t a t i o n a l a c c e l e r a t i o n e l e v a t i o n of sea s u r f a c e above mean sea l e v e l H depth of sea 73 74 C : Chezy c o e f f i c i e n t For s i m p l i c i t y the n o n - l i n e a r e f f e c t i s ignored. For seiche s t u d i e s of t h i s p r o j e c t we c o n f i n e our a t t e n t i o n to small bays which are much smaller than the dimension of the e a r t h and to high f r e q u e n c i e s which are much higher than the angular v e l o c i t y of the e a r t h , so that the C o r i o l i s f o r c e may be ignored. F i r s t i n the numerical model, the f r i c t i o n term i s dropped i n t h i s s e c t i o n . Since the r a t i o 77/H i s normally of the order 0.001 to 0.01, we re p l a c e H+77 by H. Then we get the s i m p l i f i e d shallow water equations: 9u , n dr\ n at + g di = 0 f^glf = 0 (4.1-2) l-Hu + -l-Hv + I? = 0 9x 9y 9t Assuming the time harmonic f a c t o r e1"^" f o r the v a r i a b l e s u,v and 77, a f t e r d i f f e r e n t i a t i n g with r e s p e c t to t and c a n c e l l i n g the f a c t o r e l w t , equations (4.1-2) may be w r i t t e n i n the form of icou + qr> = 0 icov + gr?y = 0 (4.1-3) (Hu) x + (Hv) + itoti = 0 75 S u b s t i t u t i n g u,v from the f i r s t two equations i n t o the l a s t , we get an equation f o r the s i n g l e v a r i a b l e TJ: VH-VT} + H V 2 7 j + |^-77 = 0 ( 4 . 1 - 4 ) T h i s equation governs the s p a t i a l d i s t r i b u t i o n of the sea s u r f a c e w i t h i n a bay. The sea s u r f a c e behaves as a f i e l d of s t a n d i n g waves when the input source to the mouth boundary of the bay i s p e r i o d i c . The l a n d boundary of the bay i s t r e a t e d as a r i g i d v e r t i c a l w a l l . Then the normal v e l o c i t y at that boundary v a n i s h e s ; from equation (4.1-3) we get VT7=0 at the land boundary. For s i n u s o i d a l s o l u t i o n s , at the land boundary the amplitude of the v e l o c i t y reaches i t s minimum and the e l e v a t i o n of sea s u r f a c e reaches i t s l o c a l maximum because of the 7r/2 phase d i f f e r e n c e between the v e l o c i t y and the e l e v a t i o n of sea s u r f a c e . The sea surface f l u c t u a t i o n at the mouth of the bay i s t r e a t e d as a p e r i o d i c f u n c t i o n which i s represented by r? m e l a ) t , where 7?m i s c o n s t a n t . Assuming the s o l u t i o n i n the bay i s 77^  ( x , y ) e l c o t , the a m p l i f i c a t i o n f a c t o r i s i c o t A = ^ — = — (4.1-5) l u ) t 7} m Because of the l i n e a r i t y of equation (4.1-4), A i s the s o l u t i o n of the equation w i t h boundary c o n d i t i o n A=1 at the mouth. Then' we set boundary c o n d i t i o n at mouth to be 1 and 76 s o l v e e q u a t i o n (4.1-4) t o c o n s t r u c t the a m p l i f i c a t i o n f i e l d . I t s h o u l d be p o i n t e d out t h a t rj i n e q u a t i o n (4.1-4) i s t r e a t e d as a complex v a r i a b l e , the a m p l i t u d e and phase of rj r e p r e s e n t r e s p e c t i v e l y the a m p l i t u d e a m p l i f i c a t i o n and phase s h i f t between waves i n the bay and a t i t s mouth. There i s no i m a g i n a r y s i g n ' i ' i n e q u a t i o n (4.1-4) i n the absence of f r i c t i o n , so t h a t the r e a l and i m a g i n a r y p a r t s of the e q u a t i o n a r e i d e n t i c a l . Due t o the z e r o boundary c o n d i t i o n at the mouth i n the i m a g i n a r y p a r t of the e q u a t i o n , which l e a d s t o a homogeneous l i n e a r e q u a t i o n system i n a n u m e r i c a l model, o n l y z e r o s o l u t i o n s e x i s t f o r the i m a g i n a r y p a r t of e q u a t i o n . Based on the a n a l y s i s above we can t h u s t r e a t 77 as a r e a l v a r i a b l e . The method of f i n i t e d i f f e r e n c e l e a d s t o a system of l i n e a r a l g e b r a i c e q u a t i o n s . The o r d e r of the m a t r i x of t h i s system i s the number of g r i d p o i n t s . The computing time i n c r e a s e s c u b i c a l l y w i t h the number. T h i s work r e s t r i c t s the number of g r i d p o i n t s t o l e s s than 200 g e n e r a l l y . In c o n s i d e r a t i o n of the same o r d e r v a r i a t i o n , b oth i n x - d i r e c t i o n and y - d i r e c t i o n , of the bottom topography i n the c o n c e r n e d bays, square g r i d s a r e adopted. The s i m p l i f i e d s h a l l o w water e q u a t i o n (4.1-4) i s d i s c r e t i z e d t o the d i f f e r e n c e e q u a t i o n H. - H. 2A 2A H . - H. i , j - 1 + 2A 77 + H. . • - H i ^ l i ^ L i l (4.1-6) ! » 3 A 2 + T7 . • = 0 The s u b s c r i p t ( i , j ) i n d i c a t e the p o s i t i o n (x^ , y j ), x i = iA 0<i<N y^ = jA 0<j<M A i s the g r i d l e n g t h . In t h i s d i s c r e t i z a t i o n procedure, f i r s t order c e n t r a l d i f f e r e n c e s and second order c e n t r a l d i f f e r e n c e s are adopted to r e p l a c e the corresponding d e r i v a t i v e s , which in t r o d u c e only second order e r r o r s , i . e . 0 ( A 2 ) . The f i r s t order c e n t r a l d i f f e r e n c e c o u l d be w r i t t e n i n the form of ^^+1/2 j - H ^ _ i / 2 j e t c . , but c o n s i d e r i n g that the values H • J . - . /-, -and H. . / 0 . are l i n e a r i n t e r p o l a t i o n s , we have 1+1/2,] 1-1/2,] H i + 1 , j " H i - 1 , j = H i + 1 / 2 , J " H i - 1 / 2 , J 2A A so the d i s c r e t i z a t i o n procedure i s only operated at the a c t u a l g r i d p o i n t s . Another method to d i s c r e t i z e equation (4.1-4) was used by Olsen and Hwang (1971); t h e i r method took the d i f f e r e n c e forms of HV77 f i r s t through both forward and backward d i f f e r e n c e s , then took c e n t r a l d i f f e r e n c e s to estimate V ' ( H V T J ) . T h i s method in t r o d u c e s f i r s t order e r r o r 0(A), and performs the d i s c r e t i z i n g procedure twice. In 78 s p i t e of the c o n s i d e r a t i o n above, i t i s s t i l l d i f f i c u l t to judge which one i s more reasonable. Using the n o t a t i o n s H • , . - H • . H X = -i±U IZI^J. 4A 2 HY = H i ^ + 1 " H i - J ' 1 (4.1-7) 4A 2 H. . H = - L t l , A 2 the d i f f e r e n c e equation (4.1-6) has the form (H+HX ) T J i + 1 ^ .. + (H+HY ) T 7 i f j + 1 + ( H - H X ) r j . _ U j + (H-HY) Vi f j _ , + ( | ^ - 4 H ) 7 ? i . = 0 (4.1-8) At the land boundary the bay shore i s approximated by segments of s t r a i g h t l i n e i n the g r i d system; thus the normal d e r i v a t i v e s 3?i/3n are taken i n e i t h e r the x or y d i r e c t i o n s . If a p o i n t ( i , j ) has a r i g h t boundary, we have 377- • 7 ? . , . . - T J . 3n A SO T? • , , . = 77 • at the boundary, the term rj. . . i n (4.1-8) disappears and I + I , j i t s c o e f f i c i e n t i s added to that of 77. • . The boundaries at 1 /1 79 other s i d e s can be d e a l t with i n the same way. If ( i , j ) i s adjacent to a mouth po i n t ( i , j - l ) , the term r). . i s r e p l a c e d by 1 ; then the constant (H-HY) i s moved to the r i g h t hand s i d e of the a l g e b r a i c equation system. Through the procedure d e s c r i b e d above the corresponding a l g e b r a i c equation c o u l d be c o n s t r u c t e d at every g r i d p o i n t , which formed an a l g e b r a i c equation system. For convenience the equation system i s w r i t t e n i n matrix n o t a t i o n A7j=b (4.1-9) Throughout t h i s t h e s i s the bo l d Roman l e t t e r s and greek l e t t e r s with t i l d e denote matrix or vecto r without f u r t h e r n o t a t i o n . Examples: A, b, D, U, rj, rj^. and 77^  . The c o e f f i c i e n t matrix A and the r i g h t hand s i d e column ve c t o r b are formed through the procedure mentioned above. In order to apply t h i s to a r b i t r a r y shaped bays, the computer program has a s p e c i a l treatment to detect the r e l a t i v e p o s i t i o n between a c u r r e n t g r i d p o i n t and land boundary and mouth p o i n t s . Equation (4.1-8) i n d i c a t e s that C J only appears i n the di a g o n a l of matrix A, which needs to be c a l c u l a t e d only once at w=0; when CJ i s changed, the value w 2/g i s added to the di a g o n a l of A to o b t a i n the amplitude f i e l d f o r the s p e c i f i e d frequency C J . For t h i s program the main computing c o s t i s r e l a t e d to the subroutine SLE which s o l v e s the l i n e a r a l g e b r a i c a l 80 equation system. To guarantee the accuracy of the numerical model output the double p r e c i s i o n subroutine SLE was chosen i n s t e a d of s i n g l e p r e c i s i o n subroutine FSLE (See "UBC M a t r i x " ,Computing Center, UBC, f o r d e s c r i p t i o n of these two subrouti n e s SLE and FSLE). Double p r e c i s i o n computing uses double storage space and much more computing time; the major drawback i s the l i m i t a t i o n i n the number of g r i d p o i n t s ; as a compromise, double p r e c i s i o n was d e c l a r e d only f o r the v a r i a b l e s d i r e c t l y used i n the subroutine SLE. Let x be the c a l c u l a t e d value i n SLE or FSLE and Sx be the d i f f e r e n c e of c a l c u l a t e d and t r u e v a l u e s . The f r a c t i o n a l e r r o r i s 6x I | < n»C-10 - t (4.1-10) x where n and C are the order and c o n d i t i o n number of c o e f f i c i e n t matrix A and 15-16 double p r e c i s i o n SLE t-{ (4.1-11) 6-7 s i n g l e p r e c i s i o n FSLE The e s t i m a t i o n (4.1-10) was given by Conte and Boor (1972). 81 B. APPLICATION OF THE BASIC MODEL AND PREDICTION OF RESONANT  FREQUENCIES IN COASTAL BAYS We f i r s t d e s c r i b e the g r i d s used i n Pedder Bay, Becher Bay and Port San Juan r e s p e c t i v e l y . Pedder Bay i s a t r i a n g u l a r bay with a wide mouth (Figure 60). I t i s d i f f i c u l t to decide e x a c t l y where the mouth i s . The resonant f r e q u e n c i e s are dominated by the geometry and bottom topography, so that the p o s i t i o n of the mouth a f f e c t s the values of these f r e q u e n c i e s . Under rough e s t i m a t i o n the main resonant p e r i o d i s the time w i t h i n which the wave t r a v e l s a quarter of wave leng t h , i . e . , / i s the the i n t e g r a l path from mouth to the head of bay. In Pedder Bay the value of h at mouth i s much l a r g e r than that at the head, so that the resonant p e r i o d s h i f t i s not very s e n s i t i v e to the p o s i t i o n of the mouth adopted by the numerical g r i d . For t h i s work the mouth i s chosen as a s t r a i g h t l i n e from Cape Ca l v e r to W i l l i a m Hd ( F i g u r e 60). Numerical r e s u l t s f o r other mouth p o s i t i o n s have been compared and c o n f i r m the i n s e n s i t i v i t y of the resonant p e r i o d s to a change of mouth p o s i t i o n . Becher Bay i s a roughly square bay with i s l a n d s i n s i d e i t ( F i gure 61). The g r i d mouth i s chosen as a s t r a i g h t l i n e from A l l d r i d g e P t. to Church Pt.. Two i s l a n d s are drawn i n the g r i d ( F igure 62). F i g u r e 61 The shape of Becher Bay. 83 1 7 4 -• 1 8 5 1 7 5 - 1 6 2 8 7 F i g u r e 62 The numerical g r i d of Becher Bay. The case of Port San Juan i s a l i t t l e troublesome. T h i s port i s roughly r e c t a n g u l a r , but the c e n t r a l l i n e i s not p e r p e n d i c u l a r to the coast l i n e ( F i g u r e 63). F i r s t I took the extended coast l i n e as i t s mouth which r e s u l t e d i n a ragged lan d boundary (Figure 64). At the sharp convex co r n e r s the v a r i a b l e s should be s i n g l e - v a l u e d as anywhere e l s e , but a c t u a l l y m u l t i - v a l u e d n e s s i s i n t r o d u c e d by the boundary c o n d i t i o n 3rj/3n = 0 which l e a d i n g to the e q u a l i t i e s T? . . = r j . ^ . . and 77. . = 77. . , i n the case that 77 i s at the '1,] + j '1,3 'i,D"1 ' land boundary (Roache, 1976). Numerical r e s u l t s from the g r i d shown i n F i g u r e 64 were c h a o t i c i n s t e a d of showing resonance modes. Then I changed to the g r i d shown i n F i g u r e 65. The shortcoming of t h i s g r i d i s that the mouth l i n e does not l i n e up with the coast l i n e . But the numerical r e s u l t F i g u r e 63 The shape of Port San Juan. F i g u r e 64 The numerical g r i d of Port San Juan. F i g u r e 65 The m o d i f i e d numerical g r i d of Port San Juan. from t h i s g r i d i s more accep t a b l e a f t e r comparison with the s p e c t r a l a n a l y s i s of measured data. I n t e r p o l a t i o n s of chart depths were d i r e c t l y adopted as input to the numerical model; these are water depths at lower low water of l a r g e t i d e s . The i d e a l input depth should be that at mean sea l e v e l . At mean t i d e s , the rims of the bays w i l l be drowned by incoming t i d e s , r e s u l t i n g i n changes of area, and i n the r e p r e s e n t a t i o n of the bays' geometry. The output of the numerical model a p p l i e d to the three bays i s a f u n c t i o n of 3-dimensional v a r i a b l e : rj(x,y,o)), For convenience of p r e s e n t a t i o n i n 2-dimensional graphs, only 5 g r i d p o i n t s were s e l e c t e d at s e l e c t e d p o s i t i o n s i n each bay (Figure 60 , F i g u r e 62 and F i g u r e 65). 86 F i g u r e 66 shows the a m p l i f i c a t i o n f a c t o r versus p e r i o d at g r i d p o i n t #161 of Pedder Bay (pressure gauge p o s i t i o n ) from the output of numerical model. Here we use a b s o l u t e v a l u e s of the a m p l i f i c a t i o n f a c t o r s i n the graph and w i l l do so i n subsequent graphs i f there i s no s p e c i a l e x p l a n a t i o n . In t h i s F i g u r e there are three obvious peaks at p e r i o d s 18, 11 and 7 minutes. F i g u r e 67 shows the a m p l i f i c a t i o n f a c t o r at a l l 5 p o i n t s i n Pedder Bay. At a l l 5 p o i n t s the same resonant p e r i o d s are r e v e a l e d , but the values of the f a c t o r s have some d i f f e r e n c e s . G e n e r a l l y speaking, f o r the f i r s t mode the c l o s e r to the head of the bay the l a r g e r the a m p l i f i c a t i o n f a c t o r s a r e . F i g u r e 68 shows the curve of a m p l i f i c a t i o n f a c t o r at g r i d p o i n t #175 i n Becher Bay (pressure gauge p o s i t i o n ) . Two peaks are r e v e a l e d : the p e r i o d s at 13 and 7 minutes. F i g u r e 69 shows the a m p l i f i c a t i o n f a c t o r s at a l l of the 5 p o i n t s i n Becher Bay. The same peaks are r e v e a l e d ; the g r i d p o i n t s c l o s e r to the head have l a r g e r a m p l i f i c a t i o n f a c t o r s . F i g u r e 70 and F i g u r e 71 show the a m p l i f i c a t i o n f a c t o r s i n Port San Juan. A l a r g e resonant peak appears at p e r i o d 33 minutes, and there i s a l s o a small resonant peak at p e r i o d 11 minutes. For a more d e t a i l e d p r e s e n t a t i o n , contour graphs are used to d e s c r i b e the d i s t r i b u t i o n s of the a m p l i f i c a t i o n f a c t o r s in d i f f e r e n t bays at d i f f e r e n t t y p i c a l resonant per i o d s . F i g u r e s 72 to 74 show the a m p l i f i c a t i o n f a c t o r f i e l d s i n Pedder Bay. F i g u r e 72 g i v e s the f a c t o r f i e l d at p e r i o d 18 87 QUI . u cc cc c r O.D ' ~ 1 —r J6-25 3 2 . 5 7 5 6 5 0 PERIOD ( M I N . i B1.25 ai. — i 133.75 ~ l 130:0 F i g u r e 66 A m p l i f i c a t i o n f a c t o r of g r i d 161 i n Pedder Bay. g r i d 161 n g r i d 084 A g r i d 128 X+ g r i d 139 X+ g r i d 180 16.25 32.5 1 T ' IB-IS €5.0 PERIOD ( M I N . I 83.25 — i — 97.5 ~ l 133.75 130.0 F i g u r e 67 A m p l i f i c a t i o n f a c t o r a t 5 p o i n t s of Pedder Bay. 88 OS o cc cn C J 1 1 1 3 . 0 1 5 . 0 3 0 . 0 . 4 5 . 0 6 0 . 0 PERIOD (MIN.) 7 5 . 0 ~ T " 3 3 . 105 1VD.3 F i g u r e 68 A m p l i f i c a t i o n f a c t o r at g r i d 175 i n Becher Bay. g r i d 175 n g r i d 087 A g r i d 1 62 + g r i d 174 X g r i d 185 4 5 . 0 6 0 . 0 PERIOD (MIN.) 7 5 . 0 9 0 . 0 1 0 5 . 0 1 2 0 . 0 F i g u r e 69 A m p l i f i c a t i o n f a c t o r at 5 p o i n t s of Becher Bay. F i g u r e 70 A m p l i f i c a t i o n f a c t o r at g r i d 187 of Port San Juan. g r i d 187 n g r i d 029 A g r i d 099 \ g r i d 1 62 X g r i d 182 22.5 30.0 PERIOD ( M I N . 6 0 . 0 F i g u r e 71 A m p l i f i c a t i o n f a c t o r at 5 p o i n t s of Port San Juan. 90 minutes which i s the resonant p e r i o d of f i r s t mode. There i s no nodal l i n e a cross the bay, the a m p l i f i c a t i o n f a c t o r gets l a r g e r from mouth to head. The a m p l i f i c a t i o n f a c t o r v a l u e s i n c r e a s e g r e a t l y along the i n l e t to the head of the bay. F i g u r e 73 shows the a m p l i f i c a t i o n f i e l d at p e r i o d 11 minutes, i t i n d i c a t e s that t h i s p e r i o d i s a second mode resonant p e r i o d with a s i n g l e nodal l i n e a c r o s s the bay; the values of a m p l i f i c a t i o n i n the inner part become ne g a t i v e , which means that the f l u c t u a t i o n of sea s u r f a c e i n t h i s area has i n v e r s e phase. F i g u r e 74 shows the d i s t r i b u t i o n of a m p l i f i c a t i o n f a c t o r at the t h i r d mode, with resonant p e r i o d 7 minutes. Two nodal l i n e s can be seen from the contour graph. F i g u r e 75 and F i g u r e 76 are a m p l i f i c a t i o n f a c t o r f i e l d s in Becher Bay, r e p r e s e n t i n g a f i r s t mode with resonant p e r i o d 13 minutes and a second mode with resonant p e r i o d 7 minutes r e s p e c t i v e l y . F i g u r e 77 to F i g u r e 79 show a m p l i f i c a t i o n f i e l d s i n Port San Juan, the corresponding p e r i o d s are 33,13 and 11 minutes s u c c e s s i v e l y . Periods 33 and 13 minutes are resonant p e r i o d s . The a m p l i f i c a t i o n i s small at p e r i o d 11 minutes, t h i s p e r i o d i s chosen to show the d i f f e r e n t nodal l i n e s . C. INFLUENCE OF TIDAL LEVEL To i n s p e c t the e f f e c t s of a change of water depth the c a l c u l a t i o n was done at mean sea l e v e l i n Pedder Bay, i . e . adding 1.8 meters to c h a r t depth. F i g u r e 80 shows the 91 F i g u r e 72 Contour map of a m p l i f i c a t i o n f a c t o r : f i r s t mode of Pedder Bay. Fi g u r e 73 Contour map of a m p l i f i c a t i o n f a c t o r : second mode of Pedder Bay. 92 F i g u r e 75 Contour map of a m p l i f i c a t i o n f a c t o r : f i r s t mode of Becher Bay. F i g u r e 77 Contour map of a m p l i f i c a t i o n f a c t o r : f i r s t mode of Port San Jaun. 94 F i g u r e 79 Contour map of a m p l i f i c a t i o n f a c t o r : t h i r d mode of Port San Juan. 95 comparison in the 5 g r i d p o i n t s of Pedder Bay. The s o l i d l i n e s represent a m p l i f i c a t i o n f a c t o r at c h a r t depth, the dashed l i n e s represent the f a c t o r at mean sea l e v e l . When the depth i s i n c r e a s e d , the peaks of the dashed l i n e s s h i f t l e f t , i . e . the corresponding resonant f r e q u e n c i e s become hig h e r . T h i s phenomenon agrees with the rough e s t i m a t i o n of resonant f r e q u e n c i e s i n an open-ended r e c t a n g u l a r bay of uniform depth: f r e s = T^n^TTT n = 1 , 2 (4.3-1) where / i s the length of the bay. A f t e r adding 1.8 meters, the t h i r d mode peaks become s m a l l e r . The r e s u l t represented by dashed l i n e s c o u l d not be t r e a t e d as the a c t u a l a m p l i f i c a t i o n f a c t o r at mean sea l e v e l , because the area of the r e a l bay changes due to the r i s i n g t i d e . G e n e r a l l y speaking, the p o s i t i o n of land boundary p l a y s a s i g n i f i c a n t r o l e i n governing the resonant f r e q u e n c i e s . The water depth of the rim of bays i s much shallower than that of bays' c e n t e r s or bay mouths, so waves t r a v e l very slowly, which i n c r e a s e s the wave p e r i o d g r e a t l y . Therefore the p o s i t i o n of the land boundary has a s i g n i f i c a n t i n f l u e n c e to the resonant p e r i o d . T h i s f e a t u r e i s not considered above. The reasonableness of adopting c h a r t depth may be e x p l a i n e d by rough e s t i m a t i o n . Based on (4.2-1) g r i d «B4 0.0 5.0 10.0 15.0 20.0 P E R I O D ( M J N.l F i g u r e 80 Comparison of a m p l i f i c a t i o n f a c t o r s of Pedder Bay. S o l i d l i n e : c h a r t depth. Dashed l i n e : mean sea l e v e l . 97 we have | £ < 0 and JJ- > 0 (4.3-2) so when the t i d e r i s e s , both h and / i n c r e a s e , and t h e i r e f f e c t on the resonant frequency of bay i s to o f f s e t each o t h e r . We w i l l use the output of the numerical model at c h a r t depth f o r comparison purposes in f u r t h e r work. The r e a l s i t u a t i o n i s much more complicated because of the a r b i t r a r y shape of a bay which may not be s i m p l i f i e d to one-dimensional. In order to t e s t the e f f e c t of the change of a bay's area, the g r i d of Port San Juan was augmented as shown i n F i g u r e 81, where the water depths of augmented area are assumed to be 0 . 5 meter. Using the b a s i c numerical model the a m p l i f i c a t i o n s from the augmented g r i d are shown i n F i g u r e 82. We see the resonant peaks of augmented g r i d s h i f t to s h o r t e r p e r i o d s i n s t e a d of longer p e r i o d s . The a m p l i f i c a t i o n s i n F i g u r e 82 are a b s o l u t e values as mentioned b e f o r e . F i g u r e 83 show valu e s of these a m p l i f i c a t i o n s themselves. We may n o t i c e the changes of si g n s of a m p l i f i c a t i o n s , which i m p l i e s that the dominant peaks from b a s i c g r i d and augmented g r i d belong to d i f f e r e n t modes. There i s no way to separate l o n g i t u d i n a l modes and t r a n s v e r s a l modes i n a r b i t r a r y domain, t h e r e f o r e there i s no way to i d e n t i f y e x p l i c i t l y the same order modes of d i f f e r e n t g r i d s . The f u r t h e r e x p l o r a t i o n of t h i s problem i s beyond the scope of t h i s r e s e a r c h . F i g u r e 81 Augmented g r i d of Port San Juan. Thick l i n e s are p r i m i t i v e and augmented boundaries. g r i d 2 9 g r i d 9 9 - i — " — i - i ••' 15.0 JO.O ?5.0 30.0 35.0 PERIOD IMIN.) 0.0 5 0 10.0 — I 45.0 50 0 — I 55.0 . I* 10.0 ="1 15.0 g r i d 162 0 . 0 so —I 1 J5.0 30.0 35.0 PERIOD IMIN.) 40 0 45.0 50.0 55.0 — I ' 10.0 g r i d 182 0.0 5.0 T* 1 1 ?5.0 30.0 35.0 PERIOD (MIN.) - 1 1 1 1 1 40.0 45.0 50.0 55.0 SO. 0 0.0 5.0 10.0 15.0 JO.O 1 1 75.0 30.0 35.0 PERIOD IMIN.) g r i d 187 I I I I I 40.0 45.0 50.0 55.0 60.0 F i g u r e 82 Comparison of a m p l i f i c a t i o n s (absolute values) between b a s i c g r i d and augmented g r i d , Port San Juan. S o l i d l i n e : augmented g r i d . Dashed l i n e : b a s i c g r i d . 0.0 vo g r i d 29 RIOD (MIN.) g r i d 99 ~"i 1 — 10.0 ISO -1 1 1 1 1 1 1 0.0 5.0 25.0 30.0 35.0 PERIOD IM1N.) 40.0 45.0 50 0 55.0 60.0 1 r 0.0 5.0 g r i d 162 — i — 2 0 . 0 I 1 • 1 R.I 90.0 35.0 PERIOD (MIN.) — i — 40.0 — I 1 1 1 45.0 50.0 55.0 BOO - 1 10.0 1 15.0 g r i d 182 1 — 0.0 5.0 D I 1 no 3 0 . 0 PERIOD (MIN. - 1 — 35.0 ~1 1 0 . 0 15.0 20.0 40.0 45.0 50 0 55.0 60.0 g r i d 187 — i — 5.0 — I 10.0 I 15.0 - 1 2 0 . 0 — i r 25.0 30.0 35.0 PERIOD (MIN.) — I — 40.0 45.0 50.0 55.0 60.0 F i g u r e 83 Comparison of a m p l i f i c a t i o n s ( t r u e v a l u e s ) between b a s i c g r i d and augmented g r i d , Port San Juan. S o l i d l i n e : augmented g r i d . Dashed l i n e : b a s i c g r i d . 101 D. THE EFFECT OF BOTTOM FRICTION To c o n s i d e r the e f f e c t of f r i c t i o n we s t i l l s t a r t from the l i n e a r i z e d shallow water equations, f o r s i m p l i c i t y , adopting the l i n e a r f r i c t i o n terms Xu and Xv : u t + qr>x + X u = 0 v f c + g r j y + Xv = 0 (4.4-1 ) (Hu) x + (Hv) + r?t = 0 Time harmonic dependence e l u > ^ i s assumed to e l i m i n a t e the v a r i a b l e t i n equation (4.4-1) ; thus u=u 0 (x , y) e l c J t , v=v 0 ( x , y ) e l a , t , r?=7?0 (x,y ) e l w t . A f t e r the e l c J t i s c a n c e l l e d and dropping the ' 0 ' we o b t a i n iwu + g?7x + Xu = 0 icov + g r j y + Xv = 0 (4.4-2) (Hu) x + (Hv) + itori = 0 From the f i r s t two equations of (4.4-2) we have 9VK u = -icv+X 9V, v = - —L-icj+X 1 02 s u b s t i t u t i n g i n t o the t h i r d equation of (4.4-2) I C J + X 'x x ito+X 'y y ' which may a l s o be w r i t t e n , a f t e r m u l t i p l i c a t i o n by (ico+X) [VH-V + HV 2 + — - i — ] T J = 0 g g I n t r o d u c i n g L = VH«V+HV 2+co 2/g-icoX/g as a l i n e a r o p e rator, then we have Lr) = 0 (4.4-3) The r e s u l t we are i n t e r e s t e d i n i s only the r a t i o of TJ at a s p e c i f i e d p o i n t i n the bay to the TJ at mouth : T? *(x,y,co) A<*'Y.'»> = TJ *(x,y,c) ( 4 ' 4 " 4 ) 'm ' 2 ' where TJ£ stands for TJ i n the bay 'b TJ* stands f o r T? at the mouth m We set boundary c o n d i t i o n at mouth TJ* = TJ e m as a J m 'm uniform p e r i o d i c o s c i l l a t i o n with maximum amplitude j?m and phase <p , and assume the s o l u t i o n of boundary value problem i (b (4.4-3) i s TJ£ = TJ^ e b , i . e . standing wave with maximum amplitude rj^ (x,y,u>) and phase <p^ (x,y,co). T h e r e f o r e we get 103 A = ^ ei{<pb -*m } (4.4-5) 'm s u b s t i t u t i n g (4.4-5) i n t o (4.4-3), we have r » r r ^ b i(<p, -<j> ) •, L A = L I — e b m 1 = L [ T 7 b ( x , y , a ; ) e i 0 ( x ' y ' a ; ) • ( % e^m ] (4.4-6) = (,m e 1 ^ )" 1 L [ % ( x , y , W ) e i ^ , ( x ' y ' c j ) ] = 0 Then the boundary value problem becomes LA = 0 i n s i d e the bay (4.4-7) A = 1 at the mouth (4.4-8) For the f o l l o w i n g work we j u s t c o n s i d e r equation (4.4-3) p l u s open boundary c o n d i t i o n TJ=1 at mouth. T h i s i s e q u i v a l e n t to the problem f o r the a m p l i f i c a t i o n f a c t o r A. N o t a t i n g TJ = TJ^ + ir)^ where 7? r r e a l p a r t of TJ r?^  imaginary p a r t of r? , equation (4.4-3) i s d i v i d e d i n t o two p a r t s : Arjv + Dr\i = 0 -DTjr + Ar}i = 0 (4.4-9) 104 with the o p e r a t o r s A = VH«V + HV 2 + — 9 The open boundary c o n d i t i o n i s TJ = 1 and 77 ^ = 0 , i . e . <t>, =1 and (j) =0 . b m A i s an operator of the f r i c t i o n l e s s shallow water equation (4.1-4) l e a d i n g to matrix A i n (4.1-9), D represents the e f f e c t of f r i c t i o n . To o b t a i n numerical s o l u t i o n of [(4.4-7)+(4.4-8)], I c o n s i d e r e d four p o s s i b l e methods. The f i r s t method i s i d e n t i c a l to that used i n f r i c t i o n l e s s model, f o r which we had (4.1-9) from Ar\ = 0 i n s i d e the bay (4.4-10) 7? = 1 at the mouth The d i s c r e t e f i n i t e d i f f e r e n c e form of (4.4-10) leads to a system of l i n e a r a l g e b r a i c equations A 7 7 0 = b 0 (4.4-11) here A i s the matrix of c o e f f i c i e n t s , i d e n t i c a l to the A i n (4.1-9), 7 7 0 i s the vector of s o l u t i o n and b 0 i s the v e c t o r 105 introduced by the open boundary v a l u e . The open boundary value i s a constant f a c t o r of b 0 ; i t doesn't c o n t r i b u t e anything to matrix A . Now TJ denotes a column v e c t o r formed by rjr and rj^ , b, a column v e c t o r formed by b 0 and a same s i z e zero v e c t o r and D i s the product of — and an i d e n t i t y matrix: 9 V L J n X ! (4.4-12.1) L J 2 n X l b b 0 L J n X l (4.4-12.2) 0 L J 2 n X l *- J n X l g 00 -, 0 D (4.4-12.3) 0 L 00 The equation system [(4.4-7) + (4.4-8) ] leads to a system of l i n e a r a l g e b r a i c e q u a t i o n s : 106 A D V — b - D A (4.4-13) With t h i s method we have to de a l with l a r g e s i z e matrix, i t not only i n c r e a s e s computing expenses, but a l s o decreases the accuracy. From the handl i n g of the equation system i t i s obvious that the bottom f r i c t i o n not only changes the amplitude of the a m p l i f i c a t i o n f a c t o r , but a l s o causes a phase s h i f t . System (4.4-13) can be s o l v e d by subroutine SLE as d e s c r i b e d b e f o r e . A second method s t a r t s from (4.4-13), which we s p l i t i n t o r e a l and imaginary p a r t s : hrjr +DT? 1 =b 0 (4.4-14) hrji - D 7 7 R =0 (4.4-15) from (4.4-14) 9?i =D- 1b 0-D" 1 A ? ? R = D - 1 ( b 0 - A 7 7 R ) . S u b s t i t u t i n g i n t o (4.4-15) A D " 'bo-AD- ' A T J J . - D 7 ? R = 0 107 i . e . D" 1 (Ab 0-A 2 T 7 r -D 2r? r ) = 0 Ab 0-A 2 77r ~D 2 77r = 0 l\v = (A 2+D 2)" 1Ab 0 Because D i s the product of a constant and the i d e n t i t y matrix, we need only one matrix m u l t i p l i c a t i o n to get A*=A 2+D 2 and one m u l t i p l i c a t i o n of a matrix and a v e c t o r to get b*=Ab 0 In order to o b t a i n 77^  , we so l v e the a l g e b r a i c equation system A* r j j . =b* i n s t e a d of c a l c u l a t i n g the i n v e r s e matrix ( A * ) " 1 i n c o n s i d e r a t i o n of saving computing time. Table 4 shows the comparison of computing times of l i n e a r a l g e b r a i c equation system and i n v e r s e matrix, where N i s the order of matrix, namely the number of g r i d p o i n t s . 1 08 Table 4 CPU Time f o r subrout i n e s SLE and INV N System of L i n e a r Equations (SLE) Inverse Matrix (INV) 20 0.003 0.012 50 0.013 0. 154 100 0.051 1 . 1 77 1 50 0.114 3.973 200 0.201 '9.355 From Table 4 we know that computing time i n c r e a s e s g r e a t l y along with the order of matrix, and the computing of inv e r s e matrix should be r e p l a c e d by s o l v i n g the corresponding system o f . l i n e a r e quations. The t h i r d method s t a r t s from (4.4-14) and (4.4-15). From (4.4-15), s u b s t i t u t i n g i n t o (4.4-14), we get (AD" 1A+D)?7i = b To proceed t h i s way, we have to c a l c u l a t e 77^  f i r s t , but 77^  i s smaller than 77^. , and as a r e s u l t small round o f f e r r o r s i n 77^  maybe causes l a r g e e r r o r s i n 7}r . A f o u r t h method was c o n s i d e r e d . From (4.4-15) 77i = A' 1D??r • 1 09 s u b s t i t u t i n g i n t o (4.4-14), we get A 7 ? r + D A - 1 D 7 ? r = b 0 (A + DA' 1D ) 7 7 r = b 0 But the c a l c u l a t i o n of A" 1 c o s t s much more computing time as shown i n Table 4. A f t e r performing computation with the f i r s t three methods with one value of X in Pedder Bay, I found that methods 2 and 3 saved about 3/4 computer storage space and 1/3 computing time compared with method 1. There was no s u b s t a n t i a l d i f f e r e n c e of computed r e s u l t s among the 3 methods when double p r e c i s i o n was invoked, but f o r s i n g l e p r e c i s i o n the d i f f e r e n c e reached up to 10% i n some cases. Taking X=3.O X l0 _ 5•s - 1 (Proudman 1953; Jamart and Winter 1978) and computing with method 3, the r e s u l t shows that the bottom f r i c t i o n doesn't cause any meaningful change in the a m p l i f i c a t i o n f a c t o r s . Two other v a l u e s , X=3.0X10 _ U and X=1.0X10 - 3 were taken i n t o the same computing procedure i n the three bays. The r e s u l t s are shown i n F i g u r e 84 to F i g u r e 86, where the s o l i d l i n e s are f r i c t i o n l e s s output of numerical model, "A" f o r X=3.0X10" 5, "-f" f o r X=3.0X10-", " X " f o r X=1.0X10" 3. For the l a r g e r a m p l i f i c a t i o n f a c t o r , I s o z a k i (1979) p o i n t e d out that the l i n e a r model produced a s l i g h t l y l a r g e r amplitude of sea l e v e l o s c i l l a t i o n s than the n o n l i n e a r 1 1 0 • r i d 8 4 31.5 14.0 17.5 PERIOD (MIN.) g r i d 139 14.0 PERIOD 17.5 (MIN. 31.5 g r i d 161 14.0 PERIOD 17.5 (MIN . 31.5 31.5 F i g u r e 84 The e f f e c t of f r i c t i o n c o e f f i c i e n t s i n the numerical model of Pedder Bay. grid 87 - i 1 1 15.0 20.0 25.0 PERIOD (MIN.) Hi oo 30.0 35.0 40.0 45.0 Si grid 162 45 0.0 1 I 15.0 ?0.0 25.0 PERIOD IMIN.) — I "1 30.0 35.0 40.0 0.0 T 5.0 grid 174 10.0 15.0 1 1 20 0 25 D PERIOD IMIN.) 30.D 35.0 40.0 45 0.0 5.0 10 0 grid 175 i 1 1 15.0 20.0 25.0 PERIOD (MIN.) 30.0 35.0 40.0 grid 185 , | | 0.0 5.0 10.0 15.0 20.0 25 0 30.0 PERIOD IMIN.) 35.0 40.0 ure 85 The e f f e c t of f r i c t i o n c o e f f i c i e n t s i n the numerical model of Becher Bay. a. r a 4.5 9 0 13.5 IB .O PERIOD gr id 182 22.5 I M I N . ) 27. a c i _> . Q . z: a 1.5 ^ — 9.0 gr id 187 13.5 1 18.0 PERIOD 22.5 (MIN F i g u r e 86 The e f f e c t of f r i c t i o n c o e f f i c i e n t s i n the numerical model of Port San Juan. 1 1 3 model, but the d i f f e r e n c e s are not s i g n i f i c a n t . Another p o s s i b l e damping f a c t o r i s the e f f e c t of ocean impedance through the boundary c o n d i t i o n . In the case of l a r g e a m p l i f i c a t i o n the water communication between open ocean and a bay becomes an important f a c t o r , the ocean impedance c o u l d not be ignored. In s p i t e of the reasons given above the range of a m p l i f i c a t i o n i s s t i l l an argueable problem. Olsen and Hwang (1971) r e p o r t e d the simultaneously measured wave data i n s i d e and o u t s i d e Keauhou Bay, Hawaii, which do show a m p l i c a t i o n s of as much as 33 at s e l e c t e d f r e q u e n c i e s . V. COMPARISON OF THE RESULTS BETWEEN NUMERICAL MODEL AND SPECTRAL ANALYSIS The a m p l i f i c a t i o n f a c t o r s p r e d i c t e d by the numerical model are compared here with the power s p e c t r a of measured data. The seiche does not always occur due to the absence of e x c i t i n g sources, so f i r s t I compared the numerical r e s u l t s with the t y p i c a l resonant s p e c t r a , i . e . with band averaged s p e c t r a d e s c r i b e d i n Chapter 3 and presented i n F i g u r e 33 to F i g u r e 35. These s p e c t r a were c a l c u l a t e d from s e l e c t e d time s e r i e s when t y p i c a l s e i c h e f l u c t u a t i o n occured. F i g u r e 87 shows the comparison in Pedder Bay. The s o l i d l i n e i s the measured spectrum, the symbol n r e p r e s e n t s the a m p l i f i c a t i o n f a c t o r i n g r i d #161, where the pressure gauge was i n s t a l l e d to measure the sea l e v e l f l u c t u a t i o n s . The a m p l i f i c a t i o n f a c t o r s were m u l t i p l i e d by 80 to make the maxima of spectrum and the f a c t o r s have the same s c a l e . The peaks at 11 minutes p e r i o d agree very w e l l with each other. The n u m e r i c a l l y p r e d i c t e d resonant p e r i o d s at 18 and 7 minutes peaks, or 0.06 and 0.14 cpm i n frequency, are not present i n the observed spectrum. The low frequency peaks of s p e c t r a may not be due to resonance. F i g u r e 88 shows the comparison in Becher Bay at g r i d #175 which i s the gauge p o s i t i o n . The a m p l i f i c a t i o n f a c t o r s were enlarged 50 times. The peaks at p e r i o d 13 minutes (0.077 cpm i n frequency) agree very w e l l . In the observed s p e c t r a there i s no peak at the p r e d i c t e d second resonant 114 1 1 5 F i g u r e 87 Comparison between observed spectrum ( s o l i d l i n e ) and a m p l i f i c a t i o n f a c t o r s (n) i n Pedder Bay. F i g u r e 88 Comparison between observed spectrum ( s o l i d l i n e ) and a m p l i f i c a t i o n f a c t o r s (n) i n Becher Bay. 1 1 6 p e r i o d 7 minutes. The low frequency peak of the spectrum at p e r i o d about 100 minutes should be a t t r i b u t e d to e x t e r n a l f o r c i n g . F i g u r e 89 shows the comparison i n Port San Juan at g r i d #187 (pressure gauge p o s i t i o n ) . The resonant p e r i o d f o r the p r e d i c t e d f i r s t mode (33 minutes) agrees very w e l l with the observed spectrum. Another peak of spectrum i s in p e r i o d about 15 minutes which may be a t t r i b u t e d to the second mode with p e r i o d 13 minutes. Summarizing the above comparisons we c o u l d conclude that the main a m p l i f i c a t i o n f a c t o r s of numerical model are always i d e n t i f i e d i n the observed s p e c t r a , even though the numerical model i s c o n s t r u c t e d on the b a s i s of a very s i m p l i f i e d equation system. In other words, the resonant f r e q u e n c i e s of a bay are c o n t r o l l e d only by g r a v i t a t i o n a l a c c e l e r a t i o n , p r e s s u r e g r a d i e n t and water mass c o n t i n u i t y . T h e r e f o r e i t i s p r a c t i c a l and r e a l i s t i c to p r e d i c t the resonant f r e q u e n c i e s of these bays by the numerical model on the b a s i s of only these three terms. F i g u r e 90 to F i g u r e 92 show the same comparisons i n Pedder Bay, Becher Bay and Port San Juan r e s p e c t i v e l y , but the observed s p e c t r a are the 20-day ensemble averaged s p e c t r a shown i n Chapter 3 Fig u r e 19 to f i g u r e 21. Features s i m i l a r to those seen i n the l a s t three f i g u r e s are observed. I t i s worth p o i n t i n g out that i n Port San Juan there i s s u b s t a n t i a l h i g h frequency energy which i s not r e l a t e d to the a m p l i f i c a t i o n f a c t o r p r e d i c t e d by numerical 1 1 7 0.03 0 .06 0 .09 0 .! 2 FREQUENCY O.JS O.JB CYC./MJN.J F i g u r e 89 Comparison between observed spectrum ( s o l i d l i n e ) and a m p l i f i c a t i o n f a c t o r s (n) i n Port San Juan. F i g u r e 90 Comparison between 20-day averaged observed spectrum ( s o l i d l i n e ) and a m p l i f i c a t i o n f a c t o r s (n) i n Pedder Bay. 1 1 8 F i g u r e 91 Comparison between 20-day averaged observed spectrum ( s o l i d l i n e ) and a m p l i f i c a t i o n f a c t o r s (n) i n Becher Bay. F i g u r e 92 Comparison between 20-day averaged observed spectrum ( s o l i d l i n e ) and a m p l i f i c a t i o n f a c t o r s (n) i n Port San Juan. 1 19 model. F i g u r e 93 compares the numerical model and the observed spectrum i n Pedder Bay under the circumstance of deeper water. The dashed l i n e i s the a m p l i f i c a t i o n f a c t o r when a depth of 1.8 meters was added to c h a r t depth at g r i d #161 shown i n Chapter 4 F i g u r e 80 f o u r t h graph by a dashed l i n e . The s o l i d l i n e i s the high t i d e l e v e l spectrum of Pedder Bay shown in Chapter 3 F i g u r e 48. Because of the reason d i s c u s s e d i n Chapter 4 the depth-added numerical p r e d i c t i o n i s not r e a l l y r e l i a b l e , t h e r e f o r e the comparison shows poor agreement. In the sp e c t r a of Port San Juan l a r g e peaks are of t e n seen at p e r i o d s of about 30 to 45 minutes, the mean of t h i s p e r i o d range being a l i t t l e l a r g e r than the resonant p e r i o d of 33 minutes p r e d i c t e d by the numerical model. T h i s d i s c r e p a n c y c o u l d be i n t e r p r e t e d by the g r i d of Port San Juan: part of the bay c l o s e to the mouth i s not in c l u d e d i n the g r i d of numerical model because of the l i m i t a t i o n of the c h a r t used in t h i s work, which undoubtedly reduces the p r e d i c t e d resonant p e r i o d of the numerical model. 1 20 F i g u r e 93 Comparison of observed spectrum ( s o l i d l i n e ) and a m p l i f i c a t i o n f a c t o r s (n) f o r deep water i n Pedder Bay. VI. INVESTIGATION OF EXCITING SOURCES FOR OBSERVED SEICHES In my summary of seiche e x c i t i n g sources i n Chapter 2, I concluded that the most probable causes of s e i c h e s are seismic a c t i v i t y on the ocean bottom, incoming waves from adjacent waters and wind s t r e s s or barometric jump. In the other words, s e i c h i n g c o u l d be e x c i t e d through the i n v a s i o n from "army, navy and a i r f o r c e " . T h i s Chapter d i s c u s s e s these p o s s i b i l i t i e s . The coherences from f i r s t data set and second data set showd very obvious c o r r e l a t i o n w i t h i n the range of p e r i o d s 30 minutes to about 1 hour between Becher Bay and Port San Juan. The coherence peaks w i t h i n t h i s range are w e l l above confidence l e v e l s . We may conclude that those o s c i l l a t i o n s were e x c i t e d by sources which t r a v e l e d along,Juan de Fuca S t r a i t . From the coherences of second data set we have found that the o s c i l l a t i o n s of Pedder Bay were c o r r e l a t e d with those of other bays at p e r i o d more than 30 minutes and many sh o r t e r p e r i o d s , i n c l u d i n g i t s resonant p e r i o d 11 minutes,, which suggests the common e x c i t i n g sources had wide range of p e r i o d s . Looking back to F i g u r e s - 8 to 9 and F i g u r e s 13 to 18 in Chapter 3, we may f i n d that around Jan. 20 and Feb. 2 there were o b v i o u s l y s t r o n g s e i c h e s whose d e t a i l i s shown in F i g u r e 94. In order to r e v e a l the common sources of these strong s e i c h e s , coherences and phases were computed and shown in F i g u r e s 95 to 98. The data used span one day and ten hours, 121 1 22 8-r8-- O C O -I 8. I 8-.8-"8J Tidts in Pedder Bay Tides in Becher Bay Tides in Port San Juan Hi-freq. osciliaiion in Pedder Bay Hi-freq. osciJiaiion in Becher Bay Hi-freq. osciliaiion in PorT San Juan J F H 2d JBN. 71 FEB. 2 FEB. 3 F i g u r e 94 Strong s e i c h e s and corres p o n d i n g t i d e s . 123 F i g u r e 95 Coherence and phase between Becher Bay and Port San Juan, January. E r r o r bars: 95% c o n f i d e n c e i n t e r v a l s . Dashed l i n e : 95% s i g n i f i c a n c e l e v e l F i g u r e 96 Coherence and phase between Pedder Bay and Becher Bay, February. E r r o r b ars: 95% c o n f i d e n c e i n t e r v a l s . Dashed l i n e : 95% s i g n i f i c a n c e l e v e l t_>u> F i g u r e 97 Coherence and phase between Pedder Bay and Port San Juan, February. E r r o r b ars: 95% c o n f i d e n c e i n t e r v a l s . Dashed l i n e : 95% s i g n i f i c a n c e l e v e l 1 26 ' 1 1 T r 0 0 3 ° 0 6 0.09 0.1? 0 15 0 18 FREQUENCY (CYC / M I N . ) 0.21 0.24 0.27 LU c r o -0 0 ° - 0 3 0.06 0.09 0 1? 0 15 DIR FREQUENCY (CYC /M]N.) 0.2.1 0.24 0.27 F i g u r e 98 Coherence and phase between Becher Bay and Port San Juan, February. E r r o r b ars: 95% confidence i n t e r v a l s . Dashed l i n e : 95% s i g n i f i c a n c e l e v e l 127 beginning from 00:00 Jan. 20 and 00:00 Feb. 2 r e s p e c t i v e l y . The o s c i l l a t i o n s of Becher Bay and Port San Juan were c o r r e l a t e d on both Jan. 20 and Feb. 2 at low frequency band. But Pedder Bay shows poor coherences with o t h e r s . We now concentrate on these dates to r e v e a l the s e i c h e - e x c i t i n g sources. A. SEISMIC ACTIVITY By c a r e f u l reading of the "Canadian Earthquakes N a t i o n a l Summary", there was no recorded seismic a c t i v i t y on January 18 and 19, 1980. The recorded earthquakes of January 20 are d e s c r i b e d in Table 5: Table 5 Seismic A c t i v i t i e s , Jan. 20, 1980 DATE(1980) H-TIME(UT) LATITUDE (NORTH) LONGITUDE (WEST) MAGNITUDE JAN. 20 11:33 (1) 59.40 134.65 ML=3.5 JAN. 20 13:01 (2) 59.45 134.61 ML=2.0 From Jan. 25 to Feb. 4, 1980, a l l earthquake records are shown in Table 6: Table 6 Seismic A c t i v i t i e s , Jan. 25 to Feb.4, 1980 DATE(1980) H-TIME(UT) LATITUDE (NORTH) LONGITUDE (WEST) MAGNITUDE JAN. 28 00:41 (3) 46.67 122.20 ML=2.9 JAN. 29 00:41 (1) 46.68 122.09 ML=2.8 1 28 JAN. 29 06:07 (1) 47.92 122.49 ML=1.3 JAN. 31 00:00 (2) 49.35 115.18 ML=2.9 JAN. 31 01:41 (2) 48.72 122.06 ML=1.2 The above earthquakes occured i n l a n d f a r from Juan de Fuca S t r a i t . The magnitudes are comparatively s m a l l . T h e r e f o r e we conclude that the observed s e i c h e s are not r e l a t e d to the seismic a c t i v i t y . B. INCOMING WAVES The p o s s i b l e incoming waves which e x c i t e s e i c h e s i n c o a s t a l bays are edge waves, s w e l l and i n t e r n a l waves. 1. EDGE WAVES The f i r s t candidate of s e i c h e - e x c i t i n g waves i s edge waves pa s s i n g along the coast by the mouths of bays. The bottom p r o f i l e of Juan de Fuca S t r a i t j u s t o f f the three bays may be represented by e x p o n e n t i a l curve. Then we s e l e c t parameters H 0 and a to f i t the e x p o n e n t i a l f u n c t i o n s in the form of H=H 0[1-exp(-ax)] to estimate the s e c t i o n of the S t r a i t . Afterwards we c o u l d e x a c t l y f o l l o w B a l l ' s method (1967) to c a l c u l a t e the d i s p e r s i o n r e l a t i o n of edge waves i n term of hypergeometric f u n c t i o n (Abramowitz and Stegun, 1970). F i g u r e 99 shows the s e l e c t e d e x p o n e n t i a l f u n c t i o n by the s o l i d curve and sounding depths by the c i r c l e s f o r the bottom p r o f i l e o f f Pedder Bay. Fi g u r e 100 i s the r e s u l t i n g 129 S H E L F DEPTH P R O F I L E 0-5. UJ l Q OFF Pedder Bay I 1 1 1 ~ 1 1 r — 2-0 ' 1.0 2.0 3.0 1.0 5.0 6.0 7 0 DISTANCE FROM COAST (KM) B.O F i g u r e 99 E x p o n e n t i a l f i t t i n g f o r bottom p r o f i l e o f f Pedder B a y . C i r c l e s are sounding depths. d i s p e r s i o n r e l a t i o n from B a l l ' s method. The dashed l i n e s i n d i c a t e modes which do not s a t i s f y the boundedness c o n d i t i o n s as x goes to » imposed by B a l l ' s theory. Bottom p r o f i l e s and d i s p e r s i o n r e l a t i o n s are shown i n F i g u r e 101 and F i g u r e 102 f o r Becher Bay, F i g u r e 103 and F i g u r e 104 f o r Port San Juan. Based on the c a l c u l a t e d modes the corresponding sea s u r f a c e s are shown in F i g u r e 105 f o r Pedder Bay, where the maximum wave amplitude at coast i s assumed to be 1. F i g u r e 106 f o r Becher Bay, F i g u r e 107 and F i g u r e 108 f o r Port San Juan f o r d i f f e r e n t p e r i o d s . The width of the S t r a i t of Juan de Fuca i s only about 20 km, but the sea s u r f a c e displacement does not change s i g n i f i c a n t l y w i t h i n a d i s t a n c e of the same order. B a l l ' s theory assumes an i n f i n i t e s h e l f , 0 3 0 2 0 3 0 4 0 J I I I i i I | _ 5 0 6 0 _ J I L _ 1 0 BO SO -J I 1 i i i 3 0 0 i I 1 30 3 30 3 2 0 _ J L _ DISPERSION RELATION OF EDGE UflVES U J t t i E x e o n t n t i i l D w * P r o f i l e O f f P e d d e r B a y N-0 N-1 N-2 I l I I I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 — 3 0 20 3 0 40 50 60 10 AO 90 300 3 30 PERIOD MJN. F i g u r e 100 D i s p e r s i o n r e l a t i o n of edge waves with e x p o n e n t i a l depth p r o f i l e o f f Pedder Bay. 3 2 0 S H E L F DEPTH P R O F I L E OFF Becher Bay 1 1 1 1— 3.0 4.0 5.0 €.0 DISTANCE FROM COAST IKMJ —\— "i.O i B.O F i g u r e 101 E x p o n e n t i a l f i t t i n g f o r bottom p r o f i l e o f f Becher B a y . C i r c l e s are sounding depths. F i g u r e 103 E x p o n e n t i a l f i t t i n g f o r bottom p r o f i l e o f f P o r t San J u a n . C i r c l e s are sounding depths. JO 2 0 30 4 0 J 1 1 I I I i i 5 0 6 0 -I L_ 7 0 1 _ 1 32 8 0 9 0 1 0 0 JJO 1 2 0 —1 1 I I I I I i o 11 DISPERSION RELATION OF EDCE UflVES U i * E x p o n e n t i a l Dmtfi P r o f i l e O f f P o r t S a n J u a n £*. CD CM 2 U J U J o N=0 N=! N=2 I JO I 20 30 I 40 1 1 50 PERJ0D i 60 — I 70 M J N . i 80 ao ~ ~ i — 100 JJO F i g u r e 104 D i s p e r s i o n r e l a t i o n of edge waves with e x p o n e n t i a l depth p r o f i l e o f f Port San Juan, . s i CO 2 ' U J CE J20 S e a S u r f a c e D i s p l a c e m e n t of Edge Veres With Exponential Slope H 0 = 1 0 0 . 5 7 T = 1 I m i n . MODE n=0 « = 0 . 6 D 7 3 5 E - E 1 ! ! ! ! ! 0.0 50.0 100.0 150.0 200.0 250.0 300.0 DISTANCE FROM COAST KM 350.0 400.0 F i g u r e 105 Sea s u r f a c e displacement i n p e r i o d 11 min. o f f Pedder Bay. 1 33 Sea Surface Displacement oi Edge Waves With Exponential Slope H0=177.31 T=13 m i n . MODE n=0 et=0.10939E-E T " T " —1 100.0 150.0 200.0 DISTANCE FROM COAST i 250.0 0.0 50.0 KM 300.0 l 350.0 400 .0 F i g u r e 106 Sea s u r f a c e displacement i n p e r i o d min. o f f Becher Bay. 13 Sea Surface Displacement oi Edge Wavei With Exponential Slope H0=536.83 T=33 min. MODE n=0 o=0.77536E-4 —1 1 1 1 - "I 100.0 150.0 200.0 250.0 300.0 DISTANCE FROM COAST KM 0.0 I 50.0 350.0 F i g u r e 107 Sea s u r f a c e displacement i n p e r i o d 33 min. o f f Port San Juan ( e x p o n e n t i a l p r o f i l e ) . 400 .0 1 34 Sea Surface Displacement al Edge Wnv-t With Exponential Slope H0=536.83 T=16 min. MODE n=0 a=0.77536E-4 r i i 1 1 1 : 1 0.0 50.0 100.0 150.0 200.0 250.0 300 0 350 0 DISTANCE FROM COAST KM F i g u r e 108 Sea s u r f a c e displacement i n p e r i o d 16 min. o f f Port San Juan ( e x p o n e n t i a l p r o f i l e ) . •wo.o t h e r e f o r e the c a l c u l a t e d modes do not decay r a p i d l y enough to f i t i n t o the S t r a i t of Juan de Fuca. More attempts were made i n Port San Juan. The bottom p r o f i l e was set as a s t r a i g h t l i n e H=-ax o f f the Port as shown i n F i g u r e 109, where the c i r c l e s i n d i c a t e sounding depths. The modes of d i s p e r s i o n r e l a t i o n s were c a l c u l a t e d i n term of Laguerre f u n c t i o n (LeBlond and Mysak, 1978. p.231). F i g u r e 110 shows the f i r s t four modes. F i g u r e 111 and F i g u r e 112 are the sea s u r f a c e displacement with wave p e r i o d s 33 and 16 minutes. F i g u r e 113 shows the sea s u r f a c e displacement at 6,000 meters o f f the coast a g a i n s t d i f f e r e n t p e r i o d s . The a n a l y s i s mentioned above i s not s a t i s f a c t o r y f o r t h e o r e t i c a l or p r a c t i c a l c o n s i d e r a t i o n s . These edge wave 1 35 SHELF DEPTH P R O F I L E < L i n e a r S l o p e ) Off Port San Juan 1— ] . 0 i 2 . 0 1 DISTANCE FROM*COAST (KM! ~ i — 6 . 0 7 . 0 e.o 109 L i n e a r f i t t i n g f o r bottom p r o f i l e of Port San J u a n . C i r c l e s are sounding depth. 110 D i s p e r s i o n r e l a t i o n of edge waves with l i n e a r depth p r o f i l e o f f Port San Juan. Sea Surface Displacement ol Edge Waves With Linear Slope « = 0 . 0 3 5 2 7 T=33 m i n . H O S E n=0.1...5 (n=0 t h e u p p e r l i n e ) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 1.0 8.0 DISTANCE FROM COAST KM F i g u r e 111 Sea s u r f a c e displacement i n p e r i o d 33 min. o f f Port San Juan ( l i n e a r p r o f i l e ) . F i g u r e 112 Sea s u r f a c e displacement i n p e r i o d 16 min. o f f Port San Juan ( l i n e a r p r o f i l e ) . 1 37 F i g u r e 113 Surface displacement at 6,000 m o f f Port San Juan ( l i n e a r p r o f i l e ) . t h e o r i e s can only be a p p l i e d i n the case of a s i n g l e s i d e boundary. For our problem we need to o b t a i n s o l u t i o n s from both s i d e s of the s t r a i t and match the s o l u t i o n s (sea s u r f a c e displacement and i t s d e r i v a t i v e ) at one p o i n t of the s t r a i t . From the computing r e s u l t s presented above i t seems f u t i l e to t r y to match the s o l u t i o n s . Thereby numerical modeling was invoked again to seek the a c c e p t a b l e modes of d i s p e r s i o n r e l a t i o n i n the S t r a i t of Juan de Fuca. We s t a r t from the equation (4.1-4) for the s i n g l e v a r i a b l e TJ as shown i n the d e r i v a t i o n of b a s i c model: VH-VT? + H V 2 T J + ^ - T J = 0 The depth H changes only i n the c r o s s - s t r a i t d i r e c t i o n and 1 38 the plane waves propagate along the s t r a i t , i . e . H=H(x) and 77=77* (x ) e ^ C J t . A f t e r d i f f e r e n t i a t i o n with respect to y and dropping the "*", we get gH77xx + gH x 77x + (a;2 - g H * 2 ) 7 } = 0 s u b s t i t u t i n g U=T?x the second order equation becomes two f i r s t order e q u a t i o n s : 77 = u x H 2 U X = " r T U ~ ( f l i " with u=0 on both boundaries. F i g u r e 114 i s a s p l i n e f i t of the bottom p r o f i l e o f f Port San Juan. The v a r i a b l e s H, u and 77 take 100 values r e s p e c t i v e l y at equal i n t e r v a l s along the x d i r e c t i o n , i . e . , u^ and 77^  , i = 1 , 2 , . . . 1 00 . Using the corresponding d i f f e r e n c e equations we are able to c a l c u l a t e the H^ + 1 , u^ + 1 and T?^  + 1 from r i \ , u^ and 77^  , i = 1,2,...99. The H, and H 1 0 0 take the value 0.5 m i n s t e a d of 0 to a v o i d s i n g u l a r i t i e s on both boundaries. The u, takes the value of 0 and 77, takes the value of 0.1. The co's take values corresponding to p e r i o d s of every h a l f minute from 1 minute to 40 minutes. The jfc's take v a l u e s c o r r e s p o n d i n g to wave le n g t h s of every h a l f km from 1 km to 120 km. A f t e r computation we get the values of u 1 0 o on the g r i d of u>-k. In 1 39 Depth Profile of Juan de Fuca Strait off Port Son Juan s p l i n t i n t e r p o l a t i o n eurv* chart d'pth x 8 . i — < 0-UJ O 5.0 7 5 DISTANCE FROM 10 0 CORST 20.0 F i g u r e 114 S p l i n e f i t of the depth p r o f i l e o f f Port San Juan. c o n s i d e r a t i o n of c o n t i n u i t y of u as a f u n c t i o n of CJ and k, the 0-contours are drawn on t h i s g r i d which i s shown i n F i g u r e 115. These 0-contours are t r e a t e d as the d i s p e r s i o n r e l a t i o n curves of the along s t r a i t p ropagating waves. For comparison purpose the a n a l y t i c a l d i s p e r s i o n r e l a t i o n i s c a l c u l a t e d i n the case of f l a t bottom. The a n a l y t i c a l d i s p e r s i o n r e l a t i o n i s u>2 n 2 7 r 2 — - = k2 gH L 2 where L i s the width of the s t r a i t and n i s the mode number. Taking the averaged depth of the s t r a i t as H, the c a l c u l a t e d d i s p e r s i o n r e l a t i o n curves are shown in F i g u r e 116. These two s e t s of curves are q u i t e resemblant and c o n f i r m the ft S8-5 f J 8-10 —i r-1S 20 PERIOD H1N. —r— 25 — I — 35 F i g u r e 115 D i s p e r s i o n r e l a t i o n of edge waves from numerical model ( o f f Port San Juan). 8. 8. x8-X S 8 i cr 8-—r~ so 95 10 1 1 i IS 20 25 PER100 MIN. F i g u r e 116 D i s p e r s i o n r e l a t i o n of edge waves from a n a l y t i c model with f l a t bottom ( o f f Port San Juan). 141' c o r r e c t n e s s of the numerical r e s u l t s . The corresponding r e s u l t s of the s t r a i t o f f Becher Bay are shown i n F i g u r e 117 to F i g u r e 119. When these edge waves with p e r i o d s c l o s e to the ei g e n - p e r i o d s of bays propagate by the mouths of the bays, s e i c h e s i n s i d e the bays w i l l be exc i t e d . Looking back to F i g u r e 12 and Figu r e 24 we f i n d that at low frequency band (corresponding p e r i o d s 20 to 60 minutes) the coherences are w e l l above the confidence l e v e l s . I t suggests that the o s c i l l a t i o n s of Becher Bay and Port San Juan are h i g h l y c o r r e l a t e d w i t h i n t h i s range of p e r i o d s , furthermore these o s c i l l a t i o n s may be e x c i t e d by common source. We hypothesize that the common source i s i n the form of plane waves t r a v e l l i n g along the S t r a i t and the phase d i f f e r e n c e s c o n t i n u o u s l y change with f r e q u e n c i e s . A f t e r adding 2n to c e r t a i n values of phases i n the p e r i o d range of 20 to 60 minutes, we f i n d that the phases Ac/>'s i n c r e a s e approximately l i n e a r l y with f r e q u e n c i e s . The approximate value of C=Ac/»/f i s 160 minutes. The d i s t a n c e between Port San Juan and Becher Bay i s L=68km. The source at the mouths i ( kx —tot) of these two bays may be w r i t t e n as e and r so that A0=kL=cjL/c, and the phase speed c may be c a l c u l a t e d : 2 7 T f L A<t> 2TTL C =* 2.67 km/min. 160 km/hr. From the c a l c u l a t e d d i s p e r s i o n r e l a t i o n s of edge waves we Depth Profile of Juan de Fuca Strait off Becher Bay wplin* i n U r p o l o t t o n e u r v t + chart dtpth 1 1 T 1 1 1 1 1 0.0 2.5 5 0 7.5 10.0 12.5 15.0 17.5 20.0 DISTANCE FROM COPST KM ure 117 S p l i n e f i t t i n g of depth p r o f i l e o f f Becher Bay. Symbols -f- are sounding depth. gure 118 D i s p e r s i o n r e l a t i o n of edge waves from numerical model ( o f f Becher Bay). 143 F i g u r e 119 D i s p e r s i o n r e l a t i o n of edge waves from a n a l y t i c model with f l a t bottom ( o f f Becher Bay). know that the phase speed of edge waves i s approximastely 160 km/hr. w i t h i n the p e r i o d range of 20 to 40 minutes. T h i s r e s u l t v e r i f i e s the above h y p o t h e s i s . Based on t h i s a n a l y s i s we conclude that the common source was edge waves. These edge waves o f t e n occur i n Juan de Fuca S t r a i t and are most important common sources throughout the observed s e i c h e s i n t h i s r e s e a r c h . T h i s t h e s i s only r e v e a l e d a l i t t l e of t h i s f a c t , mainly at low frequency band. The c a l c u l a t e d d i s p e r s i o n r e l a t i o n s of edge waves, coherences and phases of observed o s c i l l a t i o n s imply that the o s c i l l a t i o n s i n three bays were o f t e n generated by edge waves with wide range of p e r i o d s . The f u r t h e r a n a l y s i s , e s p e c i a l l y the a n a l y s i s of phases, has not been completed i n t h i s t h e s i s . 1 44 2. SWELL Data of waves at T o f i n o (west coast of Vancouver Island) are a v a i l a b l e and presented i n F i g u r e 120 and F i g u r e 121. In January the peak p e r i o d of waves reaches i t s maximum (about 20 seconds) on the 20th. On February the 2nd, peak p e r i o d reaches i t s l o c a l maximum. On the other days of February the longer peak p e r i o d s appeared again, but the cor r e s p o n d i n g s i g n i f i c a n t wave h e i g h t s are low. During these two days i n t e n s i v e s e i c h e energy was found. The longer peak p e r i o d s (15 to 20 seconds) a s s o c i a t e d with l a r g e r s i g n i f i c a n t height of waves seem to be c o r r e l a t e d with occurrences of s e i c h e s . Within the days Jan. 16 to Feb. 18, the high frequency band of power s p e c t r a i n Port San Juan contained more energy on Jan. 16 to 17, Jan. 20 to 24, Jan. 31 to Feb. 4 and Feb. 7 to 8 than that on the other days. The corresponding s i g n i f i c a n t height of s w e l l s were o b v i o u s l y high. We may conclude that the high frequency peaks i n the s p e c t r a of Port San Juan were r e l a t e d to l a r g e s w e l l s observed at T o f i n o . The f r e q u e n c i e s of those s w e l l s (periods 10 to 20 seconds) were much higher than the cut o f f frequency of the power s p e c t r a , they c o n t r i b u t e d power to the s p e c t r a through a l i a s i n g . Those s w e l l s were not r e l a t e d to the o s c i l l a t i o n s i n Pedder Bay and Becher Bay. •ft Cu MEDS 103 JAN 80 V 0 a ~* 1 ' r—i 1 1 1 1 ( ( 1 1 1 1 1 1 r — , 1 1 , , 1 10 15 20 25 30 Day of Month F i g u r e 120 Wave data at T o f i n o , Jan. 1980, cu MEDS 103 FEB 80 K \ cu o S3 a-- I — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — r 5 10 15 20 25 - i — i — i 30 V lyj °~i—I—I—I—I—I—I—l—I—I—I—I—l—I—I—I—I—I—I—I 1—I—I I I I—I—I—I I—I—I 5 10 15 20 25 30 Day of Month F i g u r e 121 Wave data at T o f i n o , Feb. 1980. 1 46 3. INTERNAL WAVES I n t e r n a l waves may be one of the sources of the e x c i t e d s e i c h e s . Deep ocean i n t e r n a l waves commonly o r i g i n a t e from the i n t e r a c t i o n between t i d a l c u r r e n t s and bottom topography, such as a b y s s a l h i l l s , submarine mountains, or other sea bottom d i s c o n t i n u i t i e s . I t i s g e n e r a l l y accepted that oceanic i n t e r n a l t i d e s are generated at the edge of a c o n t i n e n t a l s h e l f . Recently Baines (1983) estimated the g l o b a l i n t e r n a l t i d e energy gene r a t i o n at a l l c o n t i n e n t a l s h e l f - s l o p e j u n c t i o n s f o r M 2 and S 2 f r e q u e n c i e s . Observations have shown that i n t e r n a l t i d e s have l a r g e amplitudes i n the v i c i n i t y of c o n t i n e n t a l s l o p e s , and c o n t r i b u t e a s i g n i f i c a n t f r a c t i o n of the h o r i z o n t a l v e l o c i t y f i e l d at t i d a l f r e q u e n c i e s (LeBlond and Mysak, 1978. p60). A p o r t i o n of the waves t r a v e l onto the s h e l f . Observations o f f the coast of B r i t i s h Columbia and Washington v e r i f y the e x i s t e n c e of l a r g e i n t e r n a l o s c i l l a t i o n s . Measurement of bottom pressure and temperature v a r i a t i o n o f f the B.C. coast suggest that i n t e r n a l t i d e s o r i g i n a t e with t i d a l flow over seamounts, a b y s s a l h i l l s and rough s e a f l o o r t e r r a i n . Within c e r t a i n B.C. c o a s t a l waters groups of i n t e r n a l waves cover a wide range of p e r i o d s and amplitudes. (Thomson, 1981) F i g u r e 122 shows a s a t e l l i t e image of s u r f a c e s l i c k bands at the mouth of Juan de Fuca S t r a i t , which i l l u s t r a t e s the i n t e r n a l waves i n t h i s r e g i o n . F i g u r e 123 i s a t r a c i n g of p a r t of t h i s image. The troughs of i n t e r n a l waves produce 1 4 C R L V D N D BEABRT-H L-BRND ERR TOFINO, B C BH? ORBIT 6 B I r R U G 13.197B 2SH RR X Z9H RZ RESOLUTION - 4 U BCRLE 1:164BBB HriV \ ,«V^ ;iV \M '.if V DICITRLLV PROCESSED BV THE COMMUNICATIONS RESEARCH CENTRE F i g u r e 122 I n t e r n a l waves from s a t e l l i t e . F i g u r e 123 A t r a c i n g of p a r t of the s a t e l l i t e image. 1 49 convergences of s u r f a c e waters over them, the c r e s t s produce d i v e r g e n c e s . When these waves approach the mouth of a bay, the convergences and divergences work as a pump to dis c h a r g e and charge the bay. In a two-layer system, f o r example, where the upper l a y e r o f f s h o r e i s t h i c k e r than the depth of the bay, the h o r i z o n t a l flow i n the upper l a y e r w i t h i n the bay may not be completely balanced by an opposite flow i n the lower l a y e r , and sea l e v e l o s c i l l a t i o n s w i l l r e s u l t . T h i s procedure may t r a n s f e r energy from i n t e r n a l waves to s e i c h e s . Giese et al (1982) put forward an hypothesis about the r e l a t i o n s h i p between c o a s t a l s e i c h e s and oceanic i n t e r n a l t i d e s . They examined t i d e records from 1955 through 1971 at Magueyes I s l a n d , Puerto Rico, and found that the seiche a c t i v i t y was h i g h l y c o r r e l a t e d with lunar phase. The maximum a c t i v i t y occured approximately 7 days a f t e r new and f u l l moon; the mean number of o s c i l l a t i o n s per f o r t n i g h t v a r i e d i n v e r s e l y with the ab s o l u t e d i f f e r e n c e i n time between the preceding new and f u l l moon and the perigee nearest to i t , i . e . the degree of seiche a c t i v i t y depended upon the magnitude of the preceding s p r i n g t i d e s . The authors concluded that the i n t e r n a l waves were formed by t i d a l c u r r e n t s along the southeastern margin of the Caribbean Sea where the s e m i d i u r n a l t i d e reached i t s g r e a t e s t range; i t took f i v e days f o r these waves to t r a v e l to Puerto R i c o . Since the age of the t i d e i n the southeastern Caribbean was about two days, the l a r g e s t i n t e r n a l waves would reach 1 50 Puerto Rico approximately 7 days a f t e r new and f u l l moon. T h i s hypothesis was v e r i f i e d again by the observed i n t e r n a l waves and s e i c h e s i n the Sulu Sea. The o b s e r v a t i o n of i n t e r n a l waves i n Sulu Sea was c a r r i e d out by Apel et al (1980). In t h i s r e s e a r c h the a v a i l a b l e data cover the days from Jan. 15 to Feb. 20. Within t h i s p e r i o d there were two prominent seiche a c t i v i t i e s around Jan. 20 and Feb. 2, on Feb. 15 to 18 small s e i c h e s were observed i n the three bays. The time i n t e r v a l between these seiche a c t i v i t i e s was a f o r t n i g h t which suggests a p o s s i b l e c o r r e l a t i o n between s e i c h e s and lunar phase. F i g u r e 124 shows the t i d e r e c o r d at Port San Juan, which i s the smoothed curve of the o r i g i n a l data of t h i s r e s e a r c h . Time la g s between s p r i n g t i d e s and the f o l l o w i n g s t r o n g s e i c h e s were approximately one to two days. If we assume these strong s e i c h e s were generated by t i d e - i n d u c e d i n t e r n a l waves, these one to two days were the time d i f f e r e n c e between the su r f a c e t i d e s and the t i d e - i n d u c e d i n t e r n a l waves t r a v e l l i n g from the place where the i n t e r n a l waves were generated by t i d e s (probably at the edge of a c o n t i n e n t a l s h e l f ) to Juan de Fuca S t r a i t . We may estimate the speed of i n t e r n a l waves (Pond and P i c k a r d , 1983. p238): 151 1 52 and take values of the water depths of c o n t i n e n t a l s h e l f o f f Juan de Fuca S t r a i t , and of the winter d e n s i t y p r o f i l e i n that area (Werner and Hickey, 1982): h^SOm, h 2 = l50m, p!=1-025, p 2=1.026. The estimated speed C=2.16 km/hr.. The width of the c o n t i n e n t a l s h e l f o f f Juan de Fuca S t r a i t i s about 100 km; i f i n t e r n a l waves were generated at the edge of the c o n t i n e n t a l s h e l f , i t would take them 46 hours to t r a v e l to Juan de Fuca S t r a i t . Thus the i n t e r n a l waves would a r r i v e i n Juan de Fuca S t r a i t a l i t t l e l e s s than two days a f t e r the preceding s p r i n g t i d e s . If the i n t e r n a l waves were generated at the edges of banks 40 to 80 km from Juan de Fuca S t r a i t on the c o n t i n e n t a l s h e l f , i t would take them about 19 to 37 hours to t r a v e l to Juan de Fuca S t r a i t . From F i g u r e 124 we may see the s p r i n g t i d e of middle January was l a r g e r than that of l a t e January, but degrees of succedent s e i c h e s were i n v e r s e , e s p e c i a l l y i n Port San Juan ( l o o k i n g back to F i g u r e 94). On the other hand the observed s e i c h e s a f t e r s p r i n g t i d e of mid-February were small (see F i g u r e s 13 to 18). The above f a c t s may not d i s p r o v e the hypothesis that t i d e - i n d u c e d i n t e r n a l waves were r e s p o n s i b l e f o r the observed strong s e i c h e s , s i n c e the gener a t i o n of i n t e r n a l waves not only depends upon the producing f o r c e s , but a l s o upon the s t r a t i f i c a t i o n of sea water. The source which may have enhanced the s e i c h e s of Feb. 2 w i l l be d i s c u s s e d i n f o l l o w i n g s e c t i o n . I t i s suggested that long-term (eg. more than one year) o b s e r v a t i o n of sea s u r f a c e o s c i l l a t i o n should be c a r r i e d out 153 to t e s t the f o r t n i g h t l y p a t t e r n , which may improve our understanding not only of s e i c h e s , but a l s o of i n t e r n a l waves o f f B.C. c o a s t . C. WINDS P r e v a i l i n g oceanic winds o f f the outer B r i t i s h Columbia-Washington coast are from the southwest in winter. The flow of a i r w i t h i n Juan de Fuca S t r a i t i s s t r o n g l y i n f l u e n c e d by the a d j o i n i n g mountainous t e r r a i n ; the c o r r e s p o n d i n g wind d i r e c t i o n s are e a s t e r l y in winter. Because of the funneling. e f f e e t of the topography, these e a s t e r l y winds undergo a seaward a c c e l e r a t i o n along the S t r a i t . Wind p a t t e r n s i n the e a s t e r n s e c t o r of the S t r a i t are f u r t h e r complicated by the i n e r t i a of l o c a l a i r flow. In the l e e of the mountains the winds overshoot and leave an area of calm a i r s i n the v i c i n i t y of Port Angeles (Thomson, 1981). From the s p e c t r a l a n a l y s i s we found energy at h i g h frequency only i n Port San Juan. Besides the s w e l l s d i s c u s s e d before, the strong winds i n the eastern s e c t o r of Juan de Fuca S t r a i t may a l s o c o n t r i b u t e energy to the high frequency o s c i l l a t i o n s of Port San Juan. Pedder Bay and Becher Bay are l o c a t e d in an area of calm a i r s , t h e r e f o r e i t ' s very r a r e to f i n d high-frequency o s c i l l a t i o n s . The wind data are a v a i l a b l e at Race Rocks ( c l o s e to Pedder Bay and Becher Bay) and Cape Beale ( c l o s e to Port San Juan, see F i g u r e 1). These data are presented by s t i c k 1 54 diagrams in F i g u r e 125 and F i g u r e 126. The same data are presented by s p e e d - d i r e c t i o n time s e r i e s and compared with seiche energy in F i g u r e 127 and F i g u r e 128. The winds at Race Rocks may a f f e c t Pedder Bay and Becher Bay. The winds at Cape Beale may a f f e c t Port San Juan; w i t h i n Juan de Fuca S t r a i t the winds are i n f l u e n c e d by mountainous t e r r a i n . For the peak of s e i c h e energy on Jan. 20 the wind speeds are g e n e r a l l y small at both s t a t i o n s . The d i r e c t i o n of winds does not change s h a r p l y . Therefore i t i s hard to f i n d any obvious c o r r e l a t i o n between the seiche energy peak and wind d a t a . For the peak of wave energy on Feb. 2 the wind speeds are comparatively l a r g e at both s t a t i o n s . By v i s u a l examination of these wind data F i g u r e s , the seiche a c t i v i t i e s observed i n the three bays were n e i t h e r r e l a t e d to s t r o n g winds, nor c e r t a i n d i r e c t i o n s of winds. I t i s f a i r l y obvious that the observed s e i c h e s were not caused d i r e c t l y by winds. D. BAROMETRIC VARIATION The other p o s s i b l e source of the l a r g e amplitude seiches i s the barometric pressure v a r i a t i o n s . The atmospheric pressure data w i t h i n that p e r i o d are given i n F i g u r e 129. F i g u r e 130 i s the r a t e of change of the p r e s s u r e . There was no s i g n i f i c a n t pressure jump on Jan. 20. On Feb. 1 to Feb. 2 there was peak on the p r e s s u r e curves and the r a t e of change of the pressure dropped a b r u p t l y . S i m i l a r barometric jumps c o u l d be found Feb. 3 through 8. 1 55 Hie Scale nr. Speed Scale k«/hr. DjrecUon l 1 1 1 1 1 1 1 1 1 1 1 1 l 1 1 0 1? 24 36 48 . 0 40 80 JAN. 16 JAN. 17 JAN. 18 JAN. IS ^^^^^^^^^^^^ JAN. 20 JAN. 21 JAN. 22 JAN. 23 JAN. 24 - — l l f > -"-^^y JAN. 25 JAN. 28 JAN. 29 7 ^ JAN. 30 JAN. 31 FEB. 1 FEB. 2 FEB. 3 FEB. 4 in Jill I ^ FEB. 5 FEB. 6 imp I il lim FEB. 7 FEB. 8 FEB. 9 FEB. 10 ,~- , FEB. 11 FEB. 12 FEB. 13 FEB. 14 '77= FEB. 15 FEB. 17 FEB. 18 F i g u r e 125 S t i c k diagram of the Rocks. X : data m i s s i n g . winds at Race 1 56 \ Tine Scale hr. Speed Scale k«i/hr. Direction I 1 1 1 1 1 1 1 1 1 1 1 1 l 1 1 0 12 24 36 48 0 40 60 JRN. 16 JAN. 17 JON. 18 JAN. 19 >\ \ \ - _ - - - i . , \ \ \ \ . • . ' N " " — - -JAN. 20 JAN. 21 JAN. 22 JAN. 23 TT 1—I I < ' ' 1 -JAN. 24 JAN. 25 JAN. 28 JAN. 29 JAN. 30 JAN. 31 FEB. 1 FEB. 2 -. . - ,\\\\\v ,\\\\\V \\\\\\\\V^  B. NN\\ lliiiM,\^\$m\ i/miiiiiiii FEB. 3 FEB. 4 FEB. 5 FEB. 6 l l l l . J ^ . ^ ., ^ \ , » n , v \\\\X\\\\ FEB. 7 FEB. 8 FEB. 9 FEB. 10 t * * v . . . . v \ . t i • \ r—r~-FEB. 11 FEB. 12 FEB. 13 FEB. 14 r - i — ~ ' — i V W f I ' !•>... , , y . . \ . . i • . I VN \ j e x ^ ^ ^ ^ ^ j ^ ^ ^ V - . FEB. 15 FEB. 16 FEB. 17 FEB. 18 , ,,,, . . . . x\\\ wwwww v ^XXXwXX III F i g u r e 126 S t i c k diagram of the winds at Cape Beale. 1 57 - i — — r ^ T ^ n — r y i — i r^—f—i 1— t — n— n 1 1 r JflN.28JflN.23JBN.30.JflN.3I FEE.1 FEB.2 FEB.3 FEB.4 FEB.S FE6.6 FEB.7 FEB.8 FEB.9 FEB.10 FEB.11 FEB.12 FEB.13 FEB.M FEfi.tS FEB.1S FEB.17 FEB.18 F i g u r e 127 The comparison of seiche energy and wind data at Race Rocks. X : data m i s s i n g . F i g u r e 128 The comparison of s e i c h e data at Cape Beale. energy and wind STfil P (Victoria i m . P.) JJK?8JB»L79JIK3(IJBN.3I FEB.1 HBJ FEB.3 FEB.4 FEBi FEB.6 FEB.1 FEBJ3 FEB.9 FEB. ID FEB.11 FEB. 12 FEB.13 FEB.M FEB. IS FEB.16 FEB.11 FEB. 18 F i g u r e 129 Atmospheric pressure data at s t a t i o n s T o f i n o and V i c t o r i a . 160 c r I o CL ~ c c I o a . -STfiT P llofino R) ' S.L. P (Tofino fi) or I o C L - : t h STAT P (Victoria Int. R) S.L. P (Victoria Im. R)' i 1 1 1 1 1 1 1 1 1 1 1 1 1 I 1 i i I I i JfKK JRN.17 JRN.18 JRN.19 JWL20 JAK.21 JRH-2? JPH.23 J*i3t JBK.25 Xo CL" Cc Xo cr= CL" STRT P (Victoria Int. R) S.L. P (Victoria Im. R) I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 JFK.P6JBNJSJJW.30.fW.3l FEB.1 FEB.? FEB.3 FEB.4 FEB.5 FEB.6 FEB.7 FEB.8 FEB.9 FEB.10 HB I I FEB UFEB.Uf E8.W FEB.1SFEB.l6fEB.17FEB.lS F i g u r e 130 Rate of change of pressure at s t a t i o n s T o f i n o and V i c t o r i a . 161 The notable f e a t u r e of the p r e s s u r e on Feb. 1 to Feb. 2 i s that the pressure i n c r e a s e d to the maximum and dropped to minimum w i t h i n about one day, but the s i m i l a r process on Feb. 3 to 8 took four days. F i g u r e 131 shows the s e a - l e v e l pressure maps of Jan. 20. The time on these maps i s Greenwich time. On Jan. 20 the pressure g r a d i e n t i s not along the S t r a i t of Juan de Fuca. The 1028 mb contour moved about 50 km from 00:20 to 15:00 along the S t r a i t at speed l e s s than 5 km. F i g u r e 132 shows the s e a - l e v e l pressure maps of Feb. 2. On Feb. 2 the pressure contours are p e r p e n d i c u l a r to the middle l i n e of the s t r a i t , so that the p r e s s u r e g r a d i e n t i s al o n g the s t r a i t . Taking 1008 mb contour as an example, on Feb. 2, 9 o'clock a c r o s s northern end of Vancouver I s l a n d , 15 o'clock a c r o s s the southern end, 21 o' c l o c k s t i l l a c r o s s the southern end, then back to no r t h e r n end at 3 o' c l o c k of Feb. 3. Then we estimate that the pr e s s u r e f r o n t moved at a speed about 100 km/hr. on the f i r s t h a l f day of Feb. 2 i n d i r e c t i o n SSE and on the second h a l f day of Feb. 2 i n the opp o s i t e d i r e c t i o n . Proudman (1953) d i s c u s s e d the problem of c o u p l i n g between moving a i r - p r e s s u r e d i s t u r b a n c e s and sea l e v e l v a r i a t i o n . Platzman (1958) performed a numerical s i m u l a t i o n of the surge of 26 June 1954 on Lake Michigan caused by an intense and fast-moving s q u a l - l i n e . O r l i c (1980) and H i b i y a (1982) e x p l a i n e d the cause of observed s e i c h e s by Proudman c o u p l i n g . Our problem i s to f i n d a q u a n t i t a t i v e r e l a t i o n 1 62 F i g u r e 131 Barometric maps (1). Upper: Jan. 20 0200. L o w e r - l e f t : Jan. 20 0900. Lower-right: Jan. 20 1500. F i g u r e 132 Barometric maps (2) . U p p e r - l e f t : - F e b . 2 0900. Up p e r - r i g h t : Feb. 2 1500. L o w e r - l e f t : Feb. 2 2100. Lower-right: Feb. 3 0200. 1 64 between the t r a v e l l i n g atmospheric pressure d i s t u r b a n c e and the generated water waves, and then how the generated water waves e x c i t e d s e i c h e s i n the c o a s t a l bays. In the case of wave propagation i n x - d i r e c t i o n , a f o r c e d wave rj(x-vt) with phase v e l o c i t y v c o u l d be generated by a t r a v e l i n g atmospheric pressure d i s t u r b a n c e with the v e l o c i t y v of the form r j * ( x - v t ) . I f 77 i s the water s u r f a c e e l e v a t i o n r e l a t i v e to mean sea l e v e l , then 7j* = -p/(p«g) re p r e s e n t s the barometric response, where p i s the atmospheric pressure d e v i a t i o n at sea s u r f a c e , g the g r a v i t y a c c e l e r a t i o n and p the mean d e n s i t y of sea water. Then TJ i s given by TJ=7J*/( 1 - v 2 / c 2 ) where c i s the phase speed of shallow water waves (Proudman, 1953). For example, i f c=*120 km/hr. and v=O00 km/hr., c > v , the water wave t r a v e l s i n phase with the pressure d i s t u r b a n c e . The s u r f a c e e l e v a t i o n i s a m p l i f i e d by 77/77* =3.3. When v approaches c, the resonance w i l l generate l a r g e amplitude propagating waves. A s i n g u l a r i t y v=c i n d i c a t e s a maximum a m p l i f i c a t i o n . Taking v=80~120 km/hr, i . e . v=22~33 m/s, i f v=c=v/gh, we have h^50~1l0 m. Therefore f o r the occurrence of Proudman resonance the depths from 50 to 110 m are needed. These depths can be found i n f r o n t of those c o a s t a l bays. The speed of the pressure d i s t u r b a n c e changes w i t h i n some range, and the speed of shallow water waves changes due to water depth i n d i f f e r e n t r e g i o n s . The averaged c and v were c l o s e enough to make sure t h i s resonance would probably happen. 1 65 The c a l c u l a t e d d i s p e r s i o n r e l a t i o n of edge waves i n Juan de Fuca S t r a i t shows that the water waves with p e r i o d 10 to 40 minutes have speed about 120 to 170 km/hr., which i s comparable to the speed of the barometric f r o n t . In the v i c i n i t y of resonant c o n d i t i o n v=c, we have to t r e a t the problem as an i n i t i a l value problem, i n which the amplitude of the water waves i n c r e a s e l i n e a r l y with time t . Thus edge waves along the S t r a i t may be generated by the t r a n s i e n t f o r c i n g of the impulsive pressure d i s t u r b a n c e , then e x c i t e the seiches i n s i d e the bays. In c o n t r a s t f o r the case of a pure f o r c e d wave, the maximum e l e v a t i o n Ar? of water l e v e l appears i n the f r o n t p a r t of the f o r c i n g r e g i o n , and i s given by Arj=- (Ar?*/L) -x/2 where x i s the d i s t a n c e t r a v e l l e d by the f r o n t , L i s the d i s t a n c e from the pressure maximum to the f r o n t (Hibiya and K a j i u r a , 1983). I t i s noted that AT? i s p r o p o r t i o n a l to x and i n v e r s e l y p r o p o r t i o n a l to L. So the high p ressure gr a d i e n t and long d i s t a n c e t r a v e l l e d by the pressure d i s t u r b a n c e c o u l d e x c i t e l a r g e amplitude f o r c e d water waves. T h i s s i t u a t i o n can be found on the barometric maps of Feb. 2. E. CONCLUSION A f t e r examining every p o s s i b i l i t y of s e i c h e - e x c i t i n g sources, we may approach our p r e l i m i n a r y c o n c l u s i o n . Edge waves i n Juan de Fuca S t r a i t are the major source of s e i c h e s i n these three bays. The study of coherences and 1 66 phases r e v e a l s the phase speed of the e x c i t i n g source with d i f f e r e n t p e r i o d s , which agrees very w e l l with the computed speed of edge waves i n the S t r a i t . The s t r o n g e s t s e i c h e s on Feb. 2 may be e x p l a i n e d by Proudman resonance of the observed t r a v e l i n g atmospheric pressure wave. The pressure wave may have generated long waves i n Juan de Fuca S t r a i t . When these waves passed by the mouths of the c o a s t a l bays, the s e i c h e s were e x c i t e d at the resonant f r e q u e n c i e s of the corresponding bays. Looking back to F i g u r e s 96 to 98 we may f i n d that o s c i l l a t i o n s i n Becher Bay and Port San Juan were c o r r e l a t e d at d i f f e r e n t f r e q u e n c i e s , which supports the hypothesis that these o s c i l l a t i o n s were e x c i t e d by waves with wide range of p e r i o d s t r a v e l i n g i n the S t r a i t . The o s c i l l a t i o n s of Pedder Bay showed poor coherences with those of other two bays, the strong s e i c h e s i n Pedder Bay may not be generated by the same t r a v e l i n g waves, but by same barometric d i s t u r b a n c e through Proudman resonance. As f o r the e x c i t i n g source of s e i c h e s of Jan. 20, i t i s f a i r l y obvious that those s e i c h e s were not r e l a t e d to seismic a c t i v i t y or wind and barometric p r e s s u r e , as shown by the a v a i l a b l e data. F i g u r e 95 shows that o s c i l l a t i o n s of Becher Bay and Port San Juan were c o r r e l a t e d at low frequency band. A p o s s i b l e candidate i s waves ( s u r f a c e or i n t e r n a l ) propagating from the ocean, t r a v e l i n g i n t o the S t r a i t and e x c i t i n g the s e i c h e s i n the bays. 1 67 Ti d e - i n d u c e d i n t e r n a l waves may p l a y the l e a d i n g r o l e i n the g e n e r a t i o n of these s e i c h e s . T h i s hypothesis may e x p l a i n the f o r t n i g h t p a t t e r n of strong s e i c h e s observed on January 20 and February 2. The s a t e l l i t e image shows that i n t e r n a l waves reach only the mouth of Juan de Fuca S t r a i t , which may e x p l a i n why Port San Juan always possessed more wave energy. U n f o r t u n a t e l y the necessary data of displacement of i s o p y c n a l s are u n a v a i l a b l e , we c o u l d not reach our c o n c l u s i o n through an a n a l y s i s of t i d a l data whose d u r a t i o n was not g r e a t l y longer than a lunar month. The s w e l l data of T o f i n o were o b v i o u s l y c o r r e l a t e d with the high frequency o s c i l l a t i o n s of Port San Juan, they c o n t r i b u t e d power to s p e c t r a through a l i a s i n g . I n t e r n a l waves and edge waves damped when they t r a v e l l e d to the east end of Juan de Fuca S t r a i t , and winds were g e n e r a l l y strong i n the v i c i n i t y of Port San Juan, that i s why o s c i l l a t i o n s i n Port San Juan possessed more energy. BIBLIOGRAPHY Abramowitz,M. and Stegun,I.E., 1970. Handbook of mathematical f u n c t i o n s . 7th ed. , Dover, New York, N.Y. 1046p. Akamatsu,Hideo, 1982. Seiche i n Nagasaki Bay. Pap. Met .Geophys . , Tokyo, 3 3 ( 2) : 9 5- 115. (In Japanese) Apel,J.R., Holbrook,J.R. and T s a i , J . , 1980. The Sulu Sea i n t e r n a l s o l i t o n experiment, p a r t A: background and overview ( a b s . ) . EOS, Trans.Am. Geophy.Un., 61,1009, 1980. B a l l , F . K . , 1967. Edge waves i n an ocean of f i n i t e depth. Deep-sea Res. 14: 79-88. Barber,F.G. and T a y l o r , J . , 1977. A note on f r e e o s c i l l a t i o n of Chedabucto Bay. Manuscript report series No. 47, Published by Fisheries and Environment Canada, Ottawa, Ont . . Bendat,J.S. and P i e r s o l , A . G . , 1971. Ramdom Data: A n a l y s i s and Measurement Procedure. New York, Wiley-Interscience; 407 pps. Ben-Menahem,A. and Vered,M. 1982. Seiches i n the ea s t e r n Mediterranean at H a i f a B a y ( I s r a e l ) . BolI.Geofi s. t eor. appl. , 24(93):17-29. 169 Bergsten,F., 1926. The seiche of Lake V a t t e r n . Geogr. Ann., St ockh. 8: nos . 1,2. Blackford,R.L., 1978. On the gen e r a t i o n of i n t e r n a l waves over a s i l l — a p o s s i b l e n o n - l i n e a r mechanism. Journal of Marine Research 36: 529-549. B l o o m f i e l d , P e t e r , 1976. F o u r i e r a n a l y s i s of time s e r i e s : an i n t r o d u c t i o n . Jhon Wiley & Sons. Bowers,E.C., 1977. Harbour resonance due to set-down beneath wave groups. J. Fl ui d Mech. (1977), Vol.79, part 1, pp. 71-92. Chang, P h i l l i p Y i t Kuen, 1976. Subsurface c u r r e n t s i n the S t r a i t of Georgia, west of Sturgeon Bank. Master' s Thesis, UBC. Clarke,D.J., 1968. Seiche motions f o r one-dimensional flow. / . Mar . Res . , 26: 202-207 . Conte,S.D. and C a r l de Boor, 1972. Elementary Numerical A n a l y s i s . McGraw-Hill, Inc. pp.153 Crean,P.B., 1983. Current A t l a s , Juan de Fuca S t r a i t to S t r a i t of Geogia. Canadi an Government. Publishing Cent er. Defant,A., 1961. P h y s i c a l Oceanography. Vol.2, p a r t 1, Chap.6. Pergamon Press,New York. 170 Dyer,K.R., 1982. Mixing caused by l a t e r a l i n t e r n a l s e i c h i n g w i t h i n a p a r t i a l l y mixed e s t u a r y . Estuar. coast . Shelf S c i . , 15(4): 443-457. F a i r , S . and Michelato,A., 1980. Surface s e i c h e s i n the Gulf of C a g l i a r i (south S a r d i n i a ) . Bo I I .Geofi s . t eor . appl . , 22(85): 6 9-80. G a r r e t t , C , 1975. T i d e s i n g u l f s . Deep-Sea Research, 1975, Vol.22, pp. 23-35. Giese,G.S., Hollander,R.B., Fancher,J.E. and Giese,B.S., 1982. Evidence of c o a s t a l seiche e x c i t a t i o n by ti d e - g e n e r a t e d i n t e r n a l s o l i t a r y waves. Geophysical Reseach L e t t e r s , Dec. 1982, Vol.9, No. 12, pp. 1305-1308. Godin,G., 1972. The a n a l y s i s of t i d e s . University of Toronto Press, 1972. G o t l i b , V . I u and B.A.Kagan, 1980. Resonance p e r i o d s of the World Ocean. Dokl.Nauk SSSR, 252(3): 725-728. (In Rus s i an) Groves,G.W. and Hannan,E.J., 1968.Time s e r i e s r e g r e s s i o n of sea l e v e l on weather. Rev.Geophys . , 6:129-174. Heaps,N.S. and Ramsbottom,A.E., 1966. Wind e f f e c t s on the water i n a narrow two-layered l a k e . Phi I os .Trans . R. Soc. Lond. , A, 259: 391-430. 171 H i b i y a , T o s h i y u k i and K i n j i r o K a j i u r a , 1982. O r i g i n of the A b i k i phenomenon (a kind of seiche) i n Nagasaki Bay. J. oceanogr. Soc. Japan, 38(3):172-182. Hwang,L.S. and Tuck,E.O., 1970. On the o s c i l l a t i o n s of harbours of a r b i t r a r y shape. J . F l u i d Mech. Vol.42, part3, pp. 447-464. I s o z a k i , I c h i r o , 1979. On the s e i c h e s of Habu-harbor (Oshima I s l a n d , Japan). Oceanogrl Mag., 30(I/2):31-46. Jamart,B.M. and Winter,D.F., 1978. A new approach to the computation of t i d a l motions i n e s t u a r i e s . In J. Ni houl [ ed. ] Hydr odynami cs of est uar i es and fjords, E l s e v i e r , Amsterdam, 546p. Jenkins,G.M. and Watts D.G., 1968. S p e c t r a l A n a l y s i s and I t s A p p l i c a t i o n . Hoi den-Day, Inc. San Francisco, Cal if. . Johns,B. and Hamzah,A.M., 1969. On the seiche motion. Proc.Camb. Phil .Soc. (1969), 66,607. Kosuge, Susumu and A k i r a S a i t o , 1981. A study of harbor o s c i l l a t i o n s i n Shimizu Harbor (Japan). J. Fac. mar . Sci .Te c hnol . , Tokai Univ., 14: 275-286 (In Japanese). Lanyon,J.A., E l i o t , I . G . and Clarke,D.J., 1982. Observations of s h e l f waves and bay seic h e s from t i d a l and beach groundwater-level r e c o r d s . Mar.Geol., 172 49(1/2): 23-42. LeBlond,P.H. and Mysak,L.A., 1978. Waves i n the Ocean. E l s e v i e r S c i e n t i f i c Publishing Company, 1978. L e e , J . J . , 1971. Wave-induced o s c i l l a t i o n i n harbours of a r b i t r a r y geometry. J . F l u i d Mech. Vol.45, part 2, pp. 375-394. Lemon,D.D. , 1975.Seiche e x c i t a t i o n i n a c o a s t a l bay by edge waves t r a v e l l i n g on the c o n t i n e n t a l s h e l f . Master's thesis, University of B r i t i s h Columbia. Lemon,D.D., LeBlond,P.H. and Osborn,T.R., 1979. Seiche e x c i t a t i o n i n Port San Juan, B r i t i s h Columbia. /. Fish. Res. Bd Can. , 36(10): 1223-1227. Matuzawa, T., K.Kanbara and T. Minakami, 1933. H o r i z o n t a l movement of water i n the tsunami of March 3, 1933. Jan. J. Astr. Geophys., 11, No.l, 1933. M a t t i o l i , F and T i n t i , S . , 1978. Wave-induced o s c i l l a t i o n s i n large-mouthed harbours. Nuovo Cim., (C) 1(1): 18-30. M a t t i o l i , F and T i n t i , S . , 1979. D i s c r e t i z a t i o n of the harbour resonance problem. J.Wat Way Port coast .Ocean Di v . , Am. Soc. ci v . E n g r s , 105(WW4) : 464-469. M i l e s , J . and Munk,W., 1961. Harbour paradox. J o u r n a l of t h e Waterways and Harbour D i v i s i o n , August 1961, 1 73 pp. 111-130. Miles,J.W., 1971. Resonant response of harbours: an e q u i v a l e n t - c i r c u i t a n a l y s i s . J . F l u i d Mech. (1971), Vol.46, part 2, pp.241-165. Miles,J.W., 1974. Harbour s e i c h i n g . An. Review Fluid Mech., 6, 17-35, 1974. Miles,J.W., 1981. Nonl i n e a r Helmholtz o s c i l l a t i o n s i n harbours and coupled b a s i n s . J. Fluid Mech. (1981), Vol.104, pp. 407-418. Munk,W.H., 1962. The Sea. Interscience P u b l . l , New York Nakamura, S h i g e h i s a and Shi g e a t s u Serizawa, 1983. S h e l f - s e i c h e s o f f Susami,southern Japan. Mer, Tokyo, 21(2): 89-94 (In Japanese). Noiseux,C.F., 1983. Resonance i n open harbours. J . F l u i d Mech. (1983), Vol.126, pp. 219-135. Odamaki,Minoru, Yuko Yano and R i y o s h i N i t t a , 1983. ' A b i k i ' the l a r g e s e i c h e on the west coast of Kyusyu (Japan). Rept hydrogr.Res.,Tokyo,18:83-103 (In Japanese). Maritime Safety Agency, Tokyo, Japan. Olsen K. and Hwang L.S., 1971. O s c i l l a t i o n s i n a Bay of A r b i t r a r y Shape and V a r i a b l e Depth. J.Geop.Res., Vol . 76, No. 21, 5048-5064, O r l i c , M . , 1980. About a p o s s i b l e occurrence of the Proudman 174 resonance in the A d r i a t i c . Thai as si a J u g o s l . , 16(1): 79-88. Papa,Lorenzo, 1981. A numerical and s t a t i s t i c a l i n v e s t i g a t i o n of a se i c h e o s c i l l a t i o n of the L i g u r i a n Sea. Dt . hydrogr.Z., 34(1): 15-25. Papa,Lorenzo, 1983. A numerical computation of a seiche o s c i l l a t i o n of the L i g u r i a n Sea. Geophys. J.R. a s t r . S o c . , (1983) 75, 659-667. P e t r u s e v i c s , P . , H . A l l i s o n and B.Carbon, 1979. Seiches on the western A u s t r a l i a n i n n e r c o n t i n e n t a l s h e l f . Search, 10(11) : 399-400. P i e r i n i , S . , 1981. Tid e s and Seiches i n G u l f s . IL NUOVO CIMENTO, Vol.4C, N.4, pp. 458-470. Platzman,G.W., 1975. Normal Modes of the A t l a n t i c and Indian Ocean. Journal of Physical Oceanography, Vol.5, No. 2: 201-221. Pond, S. and Pickard,G.L. (1983). I n t r o d u c t a r y Dynamical Oceanography. Pergamon Press. Proudman, J . , 1953. Dynamical oceanography. Methuen, London, 409p. Rao,D.B., 1966. Free g r a v i t a t i o n a l o s c i l l a t i o n s i n r o t a t i n g r e c t a n g u l a r b a s i n s . J. Fluid Mech. , 25: 523-555. Roache,P.J., 1976. Computational F l u i d Dynamics. Hermosa 1 75 Publishers, 1976. pp.195. Stigebrandt,A., 1976. V e r t i c a l d i f f u s i o n d r i v e n by i n t e r n a l waves i n a s i l l f j o r d . Journal of Physical Oceanography 6, 634-649. Siindermann, J . , 1966. E i n V e r g l e i c h zwischen der a n a l y t i s c h e n und der numerischen Berechnung winderzeugter Stromungen und Wasserstande i n einem Modellmeer mit Anwendungen auf d i e Nordsee. Mitt. Inst. Meeresk. Univ. Hamburg. Nr. 4, pp.77. Thomson,R.E. and Chow,K.Y., 1980. Butterworth and Lanczos-window c o s i n e d i g i t a l f i l t e r s : with a p p l i c a t i o n to data p r o c e s s i n g on the UNIVAC 1106 computer. Pacific Marine Science Report 80-9. Thomson,R.E., 1981. Oceanography of the B r i t i s h Columbia co a s t . Ottawa , Out. : Canada Dept. of F i s h e r i e s and Oceans, cl981. U l r y c h , T . J . and Bishop,T.N., 1975. Maximum entropy s p e c t r a l a n a l y s i s and a u t o r e g r e s s i v e decomposition. Rev.Geophys. and Space Phys. 13(1):183-200. Waddell,E., 1976. Swash-groundwater-beach p r o f i l e i n t e r a c t i o n s . In: R.A.Davies and R. L. El hi ngt on (Editor), Beach and Nearshore Sedimentation. Soc. Econ. Pal aeont ol .Mi ne r a I . , pp. 115-125. 1 76 Werner,F.E. and Hickey,B.M. (1982). The Role of a Longshore Gradient i n P a c i f i c Northwest C o a s t a l Dynamics. Journal of Physical Oceanography, March 1983. Wubber,Ch. and Krauss,W., 1979. The two-dimensional seiches of the B a l t i c Sea. Oceanol.Act a, 2(4): 435-446. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0053195/manifest

Comment

Related Items