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UBC Theses and Dissertations

Internal solitary waves in Davis Strait Cummins, Patrick 1983

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INTERNAL SOLITARY WAVES IN DAVIS STRAIT by PATRICK CUMMINS B. Eng., Concordia U n i v e r s i t y , 1980 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n 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 June 1983 © P a t r i c k Cummins, 1983 V In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date DE-6 (3/81) i i A b s t r a c t Current meters and a t h e r m i s t o r c h a i n deployed i n p r o x i m i t y of a d r i l l s h i p over the c o n t i n e n t a l s h e l f o f f B a f f i n I s l a n d r e v e a l e d the presence of l a r g e amplitude i n t e r n a l waves. T h i s t h e s i s reviews p r o p e r t i e s of the i n t e r n a l waves, observed to propagate away from the coast and to c o i n c i d e with the l o c a l low water phase of the t i d e , at the d r i l l s h i p . The waves were c h a r a c t e r i z e d by a sudden p u l s e - l i k e i n c r e a s e i n c u r r e n t speed and a r a p i d d e p r e s s i o n of isotherms suggestive of i n t e r n a l s o l i t a r y waves. Measurements of propagation time i n d i c a t e phase speeds of 1.1 m/s are t y p i c a l with h o r i z o n t a l l e n g t h s c a l e s comparable to the t o t a l f l u i d depth. An a n a l y s i s based on l i n e a r i n t e r n a l wave theory and i n v o l v i n g the s o l u t i o n of the v e r t i c a l e i g e n f u n c t i o n was found to f i t the measured wave c h a r a c t e r i s t i c s p o o r l y . A n o n l i n e a r wave a n a l y s i s , based on the Korteweg-de V r i e s equation and employing the v e r t i c a l e i g e n f u n c t i o n , gave b e t t e r r e s u l t s . Wave p r o f i l e s are c o r r e c t l y given by the s o l i t a r y wave s o l u t i o n of the KdV e q u a t i o n . The r e l a t i o n between amplitude and phase speed i n d i c a t e s that the a d v e c t i o n due to the t i d a l flow i s important. The l e n g t h of the d i s t u r b a n c e s i s u n d e r p r e d i c t e d by the theory at l a r g e amplitudes. C a l c u l a t i o n s of Richardson number u s i n g the wave-induced shear show that shear i n s t a b i l i t i e s are l i k e l y to occur. i i i T able of Contents A b s t r a c t i i L i s t of T a b l e s v L i s t of F i g u r e s v i Acknowledgements ix Chapter I INTRODUCTION 1 1. INSTRUMENT DEPLOYMENT 2 2. THE INTERNAL WAVE DATA . 4 3. PHYSICAL OCEANOGRAPHY OF DAVIS STRAIT AND THE NORTH LABRADOR SEA 8 3.1 Water P r o p e r t i e s And R e s i d u a l C i r c u l a t i o n 8 3 . 2 T i d a l C urrents 10 Chapter II WAVE PARAMETERS AND CHARACTERISTICS OF THE MEDIUM 20 1. INTERNAL WAVE PARAMETERS 20 1.1 F i l t e r i n g Of The Current Data 20 1.2 The I n t e r n a l Wave Phase Speed 21 1.3 The H o r i z o n t a l Length S c a l e 23 1.4 Maximum Currents 24 1.5 Wave Amplitude ' 24 2. THE DENSITY STRUCTURE 2 5 2.1 A Note On The Determination Of S t r a t i f i c a t i o n 28 3. THE SHEARED BACKGROUND FLOW 29 Chapter III LINEAR WAVE ANALYSIS : 56 1. LINEAR INTERNAL WAVE MODEL 56 2. VERTICAL NORMAL MODES 58 2.1 Shapes In The Absence Of A Flow 59 2.2 E f f e c t s Of Sheared Flow And F i n i t e Frequency 60 3. THE MODE OF OSCILLATION 61 3.1 Isotherm Displacements 63 3.2 Phase Of O s c i l l a t i o n With Depth 64 4. DISPERSION RELATION 65 Chapter IV NONLINEAR WAVE ANALYSIS 78 1. SOLITARY WAVES 78 i v 1.1 H i s t o r i c a l Review 78 1.2 Modern Approach 82 1.2.1 Shallow Water Theory ..' 83 1.2.2 Deep Water Theory 83 1.2.3 F i n i t e - D e p t h Theory 84 2. CLASSIFICATION 85 3. THE LEE AND BEARDSLEY MODEL 87 4. COMPARISON WITH THE DATA 95 4.1 Korteweg-de V r i e s C o e f f i c i e n t s 95 4.2 T e s t i n g Of The Model 96 4.2.1 The Wave Shape 97 4.2.2 Thermistor Data Comparisons: V e r t i c a l Displacements 98 4.2.3 Current Meter Comparisons: Maximum Currents ....100 4.2.4 Richardson Number C a l c u l a t i o n 101 5. DISCUSSION 103 6. INTERNAL WAVE GENERATION 106 Chapter V SUMMARY 123 BIBLIOGRAPHY 125 V I . Measured d i r e c t i o n and phase speed of each wave. ...35 I I . Length s c a l e and maximum c u r r e n t f o r s e l e c t e d waves. 36 I I I . Displacement and l e n g t h s c a l e at 30 meters depth determined from isotherm contours of t h e r m i s t o r d a t a . L i s t of Tables 37 IV. F l u i d p r o p e r t i e s a s s o c i a t e d with each CTD c a s t 38 V. Time-averaged flow along 72°T f o r each r i p event. ..39 VI. Time-averaged c u r r e n t f o r Flow 1 and Flow 2 at four depths 40 V I I . C o r r e l a t i o n c o e f f i c i e n t s of displacements with the lowest mode e i g e n f u n c t i o n s obtained from CTD 1-16 with no background flow 68 V I I I . D i s p e r s i o n of plane waves under v a r i o u s environmental c o n d i t i o n s 69 IX. Korteweg-de V r i e s c o e f f i c i e n t s under v a r i o u s environmental c o n d i t i o n s 110 v i L i s t of F i g u r e s 1. Bathymetry map of Davis S t r a i t showing the p o s i t i o n of HEKJA and RALEGH d r i l l s i t e s 12 2. (A) I n i t i a l deployment c o n f i g u r a t i o n of RCW system from August 1 to August 25. (B) Redeployment c o n f i g u r a t i o n from August 31 to October 5 13 3. Raw c u r r e n t speed time s e r i e s of Rip 216-1. Two wave s i g n a l s (A) and (B) are i d e n t i f i a b l e 14 4. Raw c u r r e n t d i r e c t i o n time s e r i e s of Rip 216-1 15 5. Isotherm contours at 0.5°C i n t e r v a l s f o r Rip 216-1. ..16 6. Map showing instrument deployment f o r the 1977 survey by Esso Resources L t d . , Calgary, A l b e r t a 17 7. Isotherm s e c t i o n deduced from CTD l i n e 2 shown in F i g u r e 6 18 8. Contours of dynamic height anomaly i n dyn cm r e f e r r e d t o 205 dbars 19 9. F i l t e r e d c u r r e n t time s e r i e s of Rip 215-2 r e s o l v e d onto 74°T 41 10. Temperature p r o f i l e o b t ained from CTD 1-1 42 11. S a l i n i t y p r o f i l e obtained from CTD 1-1 43 12. Sigma-T p r o f i l e obtained from CTD 1-1 and the s p l i n e f i t to that p r o f i l e 44 13. S t r a t i f i c a t i o n p r o f i l e o b t a i n e d from the s p l i n e f i t of the d e n s i t y d i s t r i b u t i o n of F i g u r e 12 45 14. Comparison of two d e n s i t y d i s t r i b u t i o n s : (A) CTD 1-1, (B) CTD 1-15 46 15. Sigma-T p r o f i l e of CTD 1-23 47 16. S t r a t i f i c a t i o n p r o f i l e o b t a i n e d from a s p l i n e f i t of the CTD 1-23 d e n s i t y d i s t r i b u t i o n 48 17. S t i c k diagram of the smoothed c u r r e n t t i m e - s e r i e s of Rip 215-2 49 18. Flow 1 p r o f i l e 50 v i i 19. Flow 2 p r o f i l e 51 20. F i r s t d e r i v a t i v e (shear) of Flow 1 52 21. Second d e r i v a t i v e of Flow 1 53 22. F i r s t d e r i v a t i v e (shear) of Flow 2 54 23. Second d e r i v a t i v e of Flow 2 55 24. F i r s t and second mode e i g e n f u n c t i o n s obtained from CTD 1-1 70 25. F i r s t and second mode e i g e n f u n c t i o n s obtained from CTD 1-16 71 26. F i r s t and second mode e i g e n f u n c t i o n s obtained from CTD 1-16 and Flow 2 72 27. F i r s t and second mode e i g e n f u n c t i o n s obtained from CTD 1-16 and Flow 2 f o r o s c i l l a t i o n s with a ten minute p e r i o d 73 28. S c a t t e r diagram of the displacement of isotherms a s s o c i a t e d with Rip 215-2 74 29. One hour time s e r i e s of c u r r e n t , r e s o l v e d onto the propagation d i r e c t i o n , f o r Rip 216-1 (b) 75 30. C r o s s - c o v a r i a n c e between 20 minutes of data at CM6 and CM7 cen t e r e d about 9:50 GMT f o r Rip 216-2(b) 76 31. D i s p e r s i o n r e l a t i o n o b tained using CTD 1-16 77 32. Comparison of wave p r o f i l e s of n o n l i n e a r theory with t h i r t e e n wave r e a l i z a t i o n s 111 33. Same comparison as i n F i g u r e 32 but with the recorded p r o f i l e s l e f t unconnected 112 34. R e l a t i o n between phase speed and displacement using the e i g e n f u n c t i o n s obtained from (A) CTD 1-1, (B) CTD 1-16 with Flow 1, and (C) CTD 1-16 with Flow 2 113 35. R e l a t i o n between h a l f - a m p l i t u d e width and displacement using the e i g e n f u n c t i o n s obtained from (A) CTD 1-29, (B) CTD 1-16 with Flow 2, and (C) CTD 1-16. 114 36. R e l a t i o n between phase speed and s c a l e d maximum c u r r e n t using CTD 1-16 115 37. R e l a t i o n between phase speed and s c a l e d maximum c u r r e n t using CTD 1-16 and Flow 2 116 v i i i 38. R e l a t i o n between haIf-amplitude width and s c a l e d maximum cu r r e n t u s i n g CTD 1-16 117 39. R e l a t i o n between haIf-amplitude width and s c a l e d maximum c u r r e n t u s i n g CTD 1-16 and Flow 2 118 40. Wave-induced shear u s i n g CTD 1-16 with no background shear 119 41. S t r a t i f i c a t i o n obtained from a s p l i n e f i t smoothing of CTD 1-23 120 42. Richardson Number obtained from the wave-induced shear of F i g u r e 40 and the s t r a t i f i c a t i o n of F i g u r e 41. ...121 43. R e l a t i o n between maximum c u r r e n t and amplitude using CTD 1-16 122 i x Acknowledgement I wish to express my deepest g r a t i t u d e to my s u p e r v i s o r , Dr. Paul LeBlond, who has guided my s t u d i e s with p a t i e n c e and wisdom. As w e l l he generously p r o v i d e d f i n a n c i a l support d u r i n g the year i n which t h i s t h e s i s was i n p r e p a r a t i o n . I a l s o wish to thank Don Hodgins of Seaconsult Marine Research and Can t e r r a Energy L t d . f o r p r o v i d i n g me with the i n t e r n a l wave data. During my course of study at the U n i v e r s i t y of B r i t i s h Columbia I have r e c e i v e d f i n a n c i a l support from t e a c h i n g a s s i s t a n t s h i p s and from NSERC grant A-7490 to Dr. LeBlond. 1 I. INTRODUCTION An oceanographic program was conducted i n the summer of 1980 by A q u i t a i n e Company of Canada L t d . (now C a n t e r r a Ltd.) to measure n e a r - s u r f a c e c u r r e n t s i n the v i c i n i t y of a d r i l l s h i p o p e r a t i n g at the HEKJA 0-71 s i t e (62°11'N,62°59'W) in Davis S t r a i t . There were two reasons f o r having t h i s program: to p r o v i d e r e a l - t i m e i n f o r m a t i o n f o r o i l s p i l l countermeasures and to warn the d r i l l s h i p of l a r g e c u r r e n t s a s s o c i a t e d with approaching i n t e r n a l waves. These c u r r e n t s had the u n d e s i r a b l e e f f e c t of d e f l e c t i n g the d y n a m i c a l l y p o s i t i o n e d d r i l l s h i p . To meet the o b j e c t i v e s of the program s e v e r a l c u r r e n t meters and a t h e r m i s t o r c h a i n were deployed near the d r i l l i n g s i t e . In a d d i t i o n a hydrographic survey of the r e g i o n took p l a c e . The data c o l l e c t e d by these instruments r e v e a l e d the presence of l a r g e i n t e r n a l waves t r a v e l i n g away from the c o a s t , past HEKJA, towards open water. F i g u r e 1 shows the l o c a t i o n of HEKJA with r e s p e c t to major topographic f e a t u r e s . Note that the s i t e i s l o c a t e d on the c o n t i n e n t a l s h e l f . The ocean f l o o r i s r e l a t i v e l y f l a t near HEKJA and the water depth i s approximately 355m. The focus of t h i s t h e s i s i s on the a p p l i c a t i o n of a s u i t a b l e model to d e s c r i b e the i n t e r n a l wave o b s e r v a t i o n s . In order to t e s t the model, s e v e r a l wave parameters were determined from the d a t a . The r e s u l t s are presented i n Chapter II along with a d i s c u s s i o n of the p r o p e r t i e s of the f l u i d medium. Hodgins and Hodgins(1981 a) have examined the data and c l a s s i f i e d the o b s e r v a t i o n s as i n t e r n a l s o l i t a r y waves or 2 s o l i t o n s . As a f i r s t approximation to the governing dynamics, a l i n e a r i n t e r n a l wave model i s i n v e s t i g a t e d i n Chapter I I I . The a n a l y s i s depends p r i m a r i l y upon the s o l u t i o n of the v e r t i c a l normal mode equation. In Chapter IV n o n l i n e a r i n t e r n a l wave models are c o n s i d e r e d and the a n a l y s i s leads to the a p p l i c a t i o n of a s o l i t a r y wave model i n v o l v i n g use of the v e r t i c a l mode s o l u t i o n presented i n Chapter I I I . The theory permits a d e t a i l e d comparison of the s o l i t a r y wave s o l u t i o n with the measurements of Chapter I I . The remainder of t h i s i n t r o d u c t i o n presents a summary of the instrument deployment and a d e s c r i p t i o n of the data set that was o b t a i n e d at HEKJA i n the summer of 1980. A b r i e f review of the p h y s i c a l oceanography of Davis S t r a i t concludes the chapter. 1. INSTRUMENT DEPLOYMENT A review of the instrument deployment and performance i s presented i n t h i s s e c t i o n . Greater d e t a i l on t h i s matter may be found i n Hodgins and Hodgins(1981 a ) . During the summer months of 1980 a Rip Current Warning System (RCWS) was deployed near the HEKJA d r i l l i n g s t a t i o n i n Davis S t r a i t i n support of a d r i l l i n g program of A q u i t a i n e Company of Canada. The RCW system c o n s i s t e d of three c u r r e n t meter moorings p l a c e d i n a southwesterly d i r e c t i o n at 4, 2 and, 1 n a u t i c a l m i l e s (nm) from the d r i l l s h i p at HEKJA. These moorings are designated r e s p e c t i v e l y as moorings 1/2, 3/4, and 5/6/7. The two moorings nearest the d r i l l s h i p were at 250°T with r e s p e c t to HEKJA whereas mooring 1/2 was at 236°T. The numbers, 1-7, r e f e r to 3 the c u r r e n t meters that are suspended on the mooring l i n e s . The seven instruments were I0195RX v e c t o r a v e raging, e l e c t r o m a g n e t i c c u r r e n t meters (VACM) manufactured by InterOcean Systems Inc. of San Diego, C a l i f o r n i a . These instruments had an estimated accuracy of ±2.0 cm/s f o r speed and ±2° f o r d i r e c t i o n . Each meter was e l e c t r i c a l l y connected to a spar buoy at the s u r f a c e which t r a n s m i t t e d data to the d r i l l s h i p at one minute i n t e r v a l s . The data were logged at the s h i p on magnetic c a s s e t t e tape f o r l a t e r a n a l y s i s . Current meters (CM) 1, 3 and, 5 were deployed at a nominal depth of 15m while CM 2, 4 and, 6 were at 30m. CM 7 was deeply moored at 120m. In a d d i t i o n , an Aanderaa RCM-4 i n t e r n a l l y r e c o r d i n g c u r r e n t meter was p l a c e d on mooring 5/6/7 at 200m depth. T h i s instrument sampled the speed and d i r e c t i o n of the flow at 10 minute i n t e r v a l s . The RCW system was i n p l a c e from August 1 t o August 29, 1980. The meters were redeployed, i n a m o d i f i e d c o n f i g u r a t i o n , on September 5 and recovered on October 1, 1980. In the m o d i f i e d c o n f i g u r a t i o n , mooring 1/2 was absent, CM 2 r e p l a c e d CM 5 on mooring 5/6/7, and CM 3 and 4 were suspended from the d r i l l s h i p . The nominal depths of the instruments were unchanged. F i g u r e 2 i l l u s t r a t e s the two deployment c o n f i g u r a t i o n s . Although CM 2 p h y s i c a l l y r e p l a c e d CM 5 a f t e r September 5, we w i l l c o ntinue to r e f e r to data c o l l e c t e d from t h i s p o s i t i o n as data of CM 5. The RCW system performed w e l l and most of the data were of high q u a l i t y . Two exc e p t i o n s w i l l be noted here. F i r s t l y , 4 there were d i f f i c u l t i e s with mooring 1/2. CM 1 f a i l e d very s h o r t l y a f t e r deployment and no data was ob t a i n e d from i t . A l s o CM 2 d i d not gi v e u s e f u l data u n t i l August 3. Secondly, c a l i b r a t i o n t e s t s on CM 7 showed that t h i s instrument s y s t e m a t i c a l l y underestimated the c u r r e n t speed. To make t h i s data u s e f u l , a c a l i b r a t i o n f a c t o r of 1.34 was a p p l i e d to the CM 7 speeds. T h i s f a c t o r i s an average based an predeployment and postdeployment t e s t s . To complement the RCW system, data were a l s o c o l l e c t e d from a t h e r m i s t o r c h a i n , CTD c a s t s and a water l e v e l r e c o r d e r . The t h e r m i s t o r chain was deployed 1 nm from HEKJA at 206°T. T h i s c h a i n h e l d eleven t h e r m i s t o r s , spaced at 10m i n t e r v a l s , s t a r t i n g a t 12m depth. Temperature was d i g i t i z e d at,2 minute i n t e r v a l s and each sensor had an estimated accuracy of ±0.05°C. The t h e r m i s t o r c h a i n remained i n p l a c e from J u l y 28 to August 4, 1980. S e v e r a l CTD s t a t i o n s were occupied about the HEKJA s i t e from J u l y 28 to August 12, 1980 i n order to determine water p r o p e r t i e s . F i n a l l y a water l e v e l r e corder (WLR) was i n p l a c e at HEKJA d u r i n g the e n t i r e sampling p e r i o d . T h i s instrument measures pressure and i s u s e f u l f o r determining v a r i a t i o n s i n f l u i d depth due to t i d a l motions. 2. THE INTERNAL WAVE DATA An e x t e n s i v e subset of the data recorded near HEKJA was assembled onto a s i n g l e magnetic tape by Seaconsult Marine Research L t d . to form the data a r c h i v e . T h i s i n c l u d e d the complete despiked t h e r m i s t o r , RCM, and WLR data; as w e l l as 5 complete despiked and decimated CTD data. From the RCW system ten 3 to 4 hour r e c o r d s , each c o n t a i n i n g a ' r i p e v e n t 1 , were i n c l u d e d . A r i p event i s d e f i n e d here as a group of one or more l a r g e i n t e r n a l waves propagating past HEKJA on a given low t i d e . From one to a maximum of four waves were c o n t a i n e d w i t h i n a r i p event. The c r i t e r i a f o r the i d e n t i f i c a t i o n of a wave s i g n a l w i t h i n a r i p event i s that i t be observed at more than one mooring although not n e c e s s a r i l y by every instrument. A t o t a l of 18 l a r g e i n t e r n a l waves were i d e n t i f i e d i n the data a r c h i v e . We w i l l r e f e r to each event by the J u l i a n date and the low t i d e on which i t was observed p a s s i n g the d r i l l s h i p . Thus Rip 216-2 r e f e r s to the r i p event observed d u r i n g the second low t i d e of J u l i a n day 216 of 1980. P a r t i c u l a r waves w i t h i n a r i p event are r e f e r r e d to with'an a l p h a b e t i c a l l a b e l . Thus Rip 216-2(b) r e f e r s to the second (of the three) waves of R i p 216-2. T y p i c a l time s e r i e s t r a c e s are presented i n F i g u r e s 3 and 4 and g i v e the raw c u r r e n t speed and d i r e c t i o n data of R i p 216-1. Two wave s i g n a l s are evident here and are c h a r a c t e r i z e d by a p u l s e - l i k e i n c r e a s e i n speed over the background l e v e l l a s t i n g about 15 minutes. The propagation time between moorings 3/4 and 5/6/7 i s about 30 minutes. The e l a p s e d time between the appearance of s u c c e s s i v e waves a t a mooring was h i g h l y v a r i a b l e . Extremes of 8 minutes to 2 hours were observed. I n t e r n a l wave s i g n a l s were observed i n the t h e r m i s t o r data 6 which corresponded to the f i r s t 3 r i p events of the RCW system data. In t h i s case the i n t e r n a l waves are c h a r a c t e r i z e d by a sudden and pronounced t h i c k e n i n g of the upper l a y e r s of the water column. C o n d i t i o n s r e t u r n back to normal a f t e r about 12 minutes. To i l l u s t r a t e t h i s , isotherm contours at 0.5°C i n t e r v a l s were prepared using a c o n t o u r i n g program. F i g u r e 5 i l l u s t r a t e s the r e s u l t f o r Rip 216-1. Two i n t e r n a l waves are e v i d e n t from the l a r g e d e p r e s s i o n s of the isotherms. The time i n t e r v a l between the two i s about 40 minutes. Note that the s m a l l e r wave leads the l a r g e r and that a l l isotherms r e t u r n to about t h e i r former depth. A q u a l i t a t i v e examination of the records r e v e a l e d s e v e r a l f a c t s which are important f o r i n t e r p r e t a t i o n of these data w i t h i n the e x i s t i n g knowledge of i n t e r n a l waves. These are summarized by the f o l l o w i n g . 1. The i n t e r n a l waves propagate past HEKJA away from the coast towards deeper water i n a predominantly e a s t e r l y d i r e c t i o n . Hodgins and Hodgins(1981b) s t a t e t h a t the average d i r e c t i o n of a l l wave o b s e r v a t i o n s made i n 1980 i s 72°T. 2. The appearance of the r i p s always c o i n c i d e d with the l o c a l low t i d e . N early every low t i d e was marked by one or more of these waves. 3. The r e g i o n of g e n e r a t i o n i s unknown. Since the events are l i n k e d with the t i d e , i t i s reasonable to suspect that the i n t e r a c t i o n of t i d a l c u r r e n t s with topography, p o s s i b l y at the entrance of G a b r i e l S t r a i t (between R e s o l u t i o n I s l a n d and B a f f i n I s l a n d ) or F r o b i s h e r Bay i s r e s p o n s i b l e . 7 4. A band of t u r b u l e n t w h i t e water was o b s e r v e d a t the s u r f a c e and c o i n c i d e d w i t h the appearance of the i n t e r n a l waves. The choppy water was s e v e r a l t e n s of meters wide, many k i l o m e t e r s l o n g , and was o r i e n t e d p a r a l l e l t o the c r e s t s of the waves. The term ' r i p event' was c o i n e d i n r e f e r e n c e t o t h i s phenomena which resembles the rough water produced by . r i p t i d e s . R i p t i d e s a r e an u n r e l a t e d phenomena and o c c u r when t h e r e a r e s t r o n g t i d a l c u r r e n t s over s h a l l o w b a r s or o t h e r topography. 5. The i n t e r n a l waves appeared as s t r o n g c u r r e n t p u l s e s l a s t i n g 10-15 minutes each. They were not u s u a l l y o r d e r e d i n a m p l i t u d e and the time between each p u l s e was v a r i a b l e . These f e a t u r e s suggest t h a t the l a r g e c u r r e n t and t e m p e r a t u r e p e r t u r b a t i o n s e v i d e n t i n the d a t a a r e due t o i n t e r n a l s o l i t a r y waves or s o l i t o n s . A s o l i t a r y wave i s a p r o p a g a t i n g l o c a l i z e d p u l s e of permanent form t h a t e x i s t s t h r o u g h a b a l a n c e between a m p l i t u d e and l i n e a r d i s p e r s i o n . S o l i t o n s a r e s o l i t a r y waves t h a t i n t e r a c t i n a way t h a t i s s i m i l a r t o e l e m e n t a r y p a r t i c l e s (see Chapter I V ) . There have been a number of documented o b s e r v a t i o n s of i n t e r n a l s o l i t a r y waves i n the oceans. Osborne and B u r c h ( l 9 8 0 ) have r e p o r t e d on l a r g e i n t e r n a l s o l i t o n s i n the Andaman Sea. Sandstrom and E l l i o t t ( 1 9 8 2 ) have d i s c o v e r e d such waves over the c o n t i n e n t a l s h e l f o f f Nova S c o t i a . I n t e r n a l s o l i t a r y waves have a l s o been o b s e r v e d i n a f j o r d (Farmer and Smith,1978), i n a s h a l l o w bay ( H a l p e r n , 1 9 7 1 ) , and i n a l a k e (Hunkins and F l i e g e l , 1 9 7 3 ) . The athmosphere has a l s o been shown t o s u p p o r t t h e s e waves 8 ( C h r i s t i e , Muirhead, and Hales,1978). S o l i t a r y wave t h e o r i e s and the a p p l i c a t i o n of a model are c o n s i d e r e d i n Chapter IV. 3. PHYSICAL OCEANOGRAPHY OF DAVIS STRAIT AND THE NORTH  LABRADOR SEA 3.1 Water P r o p e r t i e s And R e s i d u a l C i r c u l a t i o n In order to p l a c e the i n t e r n a l wave data w i t h i n the context of the l a r g e r s c a l e oceanographic s t r u c t u r e s , a b r i e f review of the p h y s i c a l oceanography of the region i s presented here. Knowledge of the oceanographic p r o p e r t i e s of the North Labrador Sea was f i r s t o b t ained from the work of S m i t h ( l 9 3 7 ) . D u n b a r ( l 9 5 l ) has given a review of the e a r l y e x p l o r a t i o n s of Canadian A r c t i c waters. One of the most s i g n i f i c a n t recent c o n t r i b u t i o n s to the knowledge of the c i r c u l a t i o n of t h i s r e g i o n d e r i v e s from the survey conducted f o r Esso Resources L t d . , C a l g a r y , A l b e r t a (LeBlond et a l . , 1 9 8 1 ) . Note that the d e s c r i p t i o n given below i s based on h i s t o r i c a l data c o l l e c t e d p r i m a r i l y d u r i n g summer c o n d i t i o n s . The i n s e t map of F i g u r e 6 shows the major c u r r e n t systems which c i r c u l a t e around the p e r i p h e r y of the Labrador Sea i n a c y c l o n i c gyre. The West Greenland Current flows northward, along the west coast of Greenland. The p r i n c i p a l water mass t r a n s p o r t e d by t h i s c u r r e n t i s warm (>4.0°C), s a l i n e (=34.92 °/J) Irminger Sea Water. In a d d i t i o n to t h i s warm water the West Greenland Current i s composed of a c o a s t a l c u r r e n t with water p r o p e r t i e s d e r i v e d from the East Greenland C u r r e n t . The West Greenland c u r r e n t b i f u r c a t e s near 61°N with a component fl o w i n g 9 westwards a c r o s s the S t r a i t . On the western s i d e of the b a s i n we f i n d the B a f f i n and Labrador Currents f l o w i n g to the south over the c o n t i n e n t a l s h e l f and s l o p e . The B a f f i n Current o r i g i n a t e s from B a f f i n Bay and t r a n s p o r t s a very c o l d (<-1.0°C), low s a l i n i t y (^33.5 water mass i n a s o u t h e a s t e r l y d i r e c t i o n f o l l o w i n g the coast of B a f f i n I s l a n d . F i g u r e 7 was d e r i v e d from CTD s e c t i o n 2 of F i g u r e 6 and shows the summer temperature f i e l d a cross western Davis S t r a i t . The c o l d core of the B a f f i n Current i s e v i d e n t over the s h e l f at 100m depth. F i g u r e 8 d i s p l a y s contours of dynamic height anomaly r e f e r e n c e d to 205 dbars. On t h i s f i g u r e we can i d e n t i f y the B a f f i n Current f l o w i n g south along the coast of B a f f i n I s l a n d , over the c o n t i n e n t a l s h e l f and s l o p e . Current meter measurements ob t a i n e d from moorings l o c a t e d i n the B a f f i n C urrent have shown that the c u r r e n t s at the r e f e r e n c e depth f o r the contours of F i g u r e 8 are non-zero. The v e l o c i t i e s of r e s i d u a l s u r f a c e c u r r e n t as determined from drogue t r a c k s (LeBlond et al.,1981) are two to three times the value determined from the g e o s t r o p h i c c a l c u l a t i o n . However they are i n the same g e n e r a l d i r e c t i o n as the g e o s t r o p h i c balance i n d i c a t e s . Hence a strong shear i s present i n the upper l a y e r s of the water column. As the B a f f i n Current approaches the mouth of Hudson S t r a i t i t branches i n t o two d i s t i n c t f lows. Par t of the c u r r e n t c o n t i n u e s along a s o u t h e r l y course to j o i n the Labrador C u r r e n t . The other branch of the B a f f i n Current t u r n s westwards f o l l o w i n g the c o a s t l i n e and flows i n t o Hudson S t r a i t on both the north and south shore of R e s o l u t i o n I s l a n d . 10 Drogues r e l e a s e d i n the B a f f i n Current entered Hudson S t r a i t and d r i f t e d as f a r west as 68°W before they c i r c l e d around and began to flow eastwards. The i n f l o w of the B a f f i n Current i n t o Hudson S t r a i t i s s t r o n g e s t t o the south of R e s o l u t i o n I s l a n d . A s t r o n g r e t u r n flow out of Hudson S t r a i t , n o r t h of Cape C h i d l e y , i s d i s p l a y e d on F i g u r e 8. The c o l d f r e s h water of t h i s flow forms the core of the Labrador C u r r e n t . T h i s water mass a c q u i r e s i t s p r o p e r t i e s through mixing of the B a f f i n Current Water with the low s a l i n i t y water f l o w i n g out of of Hudson's Bay. Drogue t r a c k s agree with the g e o s t r o p h i c computations and show the s w i f t e s t flow to be over the c o n t i n e n t a l s l o p e . T y p i c a l l y , s u r f a c e c u r r e n t s flow at 30 km/day and exceed the g e o s t r o p h i c flow (20 km/day). Kollmeyer, M c G i l l , and Oorwin(1967) have r e p o r t e d that the e a s t e r l y flow of the f r e s h c o l d water out. of Hudson S t r a i t occurs- d u r i n g the ebb t i d e so t h a t the outflow e f f e c t i v e l y has a p u l s e - l i k e nature. 3.2 T i d a l C u r r e n t s The t i d a l frequency which predominates i n Davis S t r a i t and the North Labrador Sea i s the s e m i - d i u r n a l (M 2) component (Osborn, LeBlond, and Hodgins,1978). There i s a l s o a s m a l l e r d i u r n a l component and these t i d e s have a s p r i n g and neap t i d e modulation. The e f f e c t s of the t i d e which are of most i n t e r e s t here are the v a r i a t i o n s i n sea l e v e l , f o r which there are many data, and t i d a l c u r r e n t s , which are l e s s w e l l documented. Do h l e r ( l 9 6 4 ) has c o n s t r u c t e d a map of corange and cophase l i n e s f o r B a f f i n Bay and Davis S t r a i t . The mean s p r i n g range v a r i e s 11 from zero at an amphidromic p o i n t i n B a f f i n Bay to a maximum of 9m i n F r o b i s h e r Bay. The t i d a l range i s 6m at the mouth of Hudson S t r a i t which i s a l s o the s i t e of very l a r g e t i d a l c u r r e n t s (^I.Sm/s). Osborn, LeBlond, and Hodgins(1978) have presented evidence f o r the e x i s t e n c e of i n t e r n a l t i d e s i n Davis S t r a i t . At HEKJA, as elsewhere, we f i n d predominantly the M 2 t i d e with a s p r i n g and neap t i d e modulation. The WLR data showed the s p r i n g t i d a l range to be about 4.-5m. Low t i d e at HEKJA leads low t i d e at F r o b i s h e r Bay by 1.0 to 2.5 hours. Hodgins and Hodgins(1981 a) used the RCM data, c o l l e c t e d at 200m depth, to examine the t i d a l c u r r e n t s at HEKJA. They found c u r r e n t s i n the E-W , d i r e c t i o n t o be much l a r g e r than c u r r e n t s i n the N-S d i r e c t i o n . Current d i r e c t i o n s t a t i s t i c s compiled from the RCM showed the flow to be o r i e n t e d i n two main d i r e c t i o n s . During the f l o o d t i d e the flow was o r i e n t e d i n i t s p r i n c i p a l d i r e c t i o n , 255°T, while d u r i n g ebb t i d e i t was d i r e c t e d at 70°T, the secondary d i r e c t i o n . The westwards c u r r e n t to 255°T was s l i g h t l y s t r o n g e r than the one to 70°T. Maximum t i d a l c u r r e n t s d u r i n g the s p r i n g t i d e were 40-50 cm/s. I n t e r n a l waves propagating away from the coast i n Davis S t r a i t w i l l be advected by these t i d a l c u r r e n t s . Note that the average d i r e c t i o n of wave propagation (72°T) i s very n e a r l y that of the secondary d i r e c t i o n of the flow (70°T). F i g u r e 1 - Bathymetry map of Davis S t r a i t showing th p o s i t i o n of HEKJA and RALEGH d r i l l s i t e s 13 F i g u r e 2 - (A) I n i t i a l deployment c o n f i g u r a t i o n of RCW system from August 1 to August 25. (B) Redeployment c o n f i g u r a t i o n from August 31 to October 5. 15 DIRECTION METER 7 METER 6 METER 5 METER 4 METER 3 0 3G0 D 3 6 0 0 3GD 0 3G0 0 3G0 F i g u r e 5 - Isotherm contours at 0.5°C i n t e r v a l s f o r Rip 216-1. 17 F i g u r e 6 - Map showing instrument deployment f o r the 1977 survey by Esso Resources L t d . , Ca l g a r y , A l b e r t a . The i n s e t map shows f e a t u r e s of the r e s i d u a l s u r f a c e c i r c u l a t i o n . (From LeBlond, p e r s o n a l communication) 18 6 2 ' W LONG. 59* W LONG. 14 13 12 II 10 9 8 7 6 5 4 3 2 I H 5 0 0 m F i g u r e 7 - Isotherm s e c t i o n deduced from CTD l i n e 2 shown in F i g u r e 6. B a f f i n Current Water i s i n d i c a t e d by the shaded area. (From LeBlond, p e r s o n a l communication) 19 F i g u r e 8 - Contours of dynamic height anomaly i n dyn cm r e f e r r e d to 205 dbars. The g e o s t r o p h i c c u r r e n t s c a l e i s given i n the i n s e t f i g u r e . Surface d r i f t c u r r e n t s as determined from drogue t r a c k s are given by the s o l i d arrows with speed i n u n i t s of km/day. (From LeBlond, p e r s o n a l communication) 20 I I . WAVE PARAMETERS AND CHARACTERISTICS OF THE MEDIUM 1. INTERNAL WAVE PARAMETERS In order to compare the o b s e r v a t i o n s of l a r g e amplitude i n t e r n a l waves with t h e o r e t i c a l r e s u l t s s e v e r a l parameters of these waves were determined from the v a r i o u s types of da t a . Of most i n t e r e s t are the i n t e r n a l wave phase speed, amplitude, h o r i z o n t a l l e n g t h s c a l e , and the maximum c u r r e n t . Values of these parameters are shown i n Tab l e s 1 to 3 f o r each r i p event i d e n t i f i e d i n the data s e t . In some i n s t a n c e s i t was not p o s s i b l e to ob t a i n a r e l i a b l e estimate of a given parameter. T h i s was due to v a r i o u s f a c t o r s such as a n o i s y s i g n a l or an i n o p e r a t i o n a l instrument and i s i n d i c a t e d by a blank i n the t a b l e s . A d e s c r i p t i o n of how the i n t e r n a l wave parameters were determined i s given below. 1.1 F i l t e r i n g Of The Current Data The data from the c u r r e n t meters were f i l t e r e d p r i o r to any a n a l y s i s . The data c o n t a i n e d both h i g h frequency f l u c t u a t i o n s , and low frequency trends due to t i d a l c u r r e n t s . The h i g h frequency n o i s e was p a r t i c u l a r l y e v i d e n t at mooring 5/6/7 as can be seen i n F i g u r e 3. I t i s suspected that the high frequency n o i s e at t h i s mooring i s a consequence of the mooring being deployed at a s i t e that was s l i g h t l y shallower than expected so that meters were su b j e c t e d to the f o r c i n g of sur f a c e waves and c u r r e n t s . The f i l t e r i n g adopted here i s i d e n t i c a l to that d e s c r i b e d i n Hodgins and Hodgins(1981 a ) . The raw data were f i l t e r e d with 21 a 5 minute running mean and a 60 minute running mean thus y i e l d i n g a smoothed s i g n a l (u,v), and a* low frequency s i g n a l , (u , v L ) . The i n t e r n a l wave s i g n a l was taken to be (u',v')=(u-u L , v - v L ) . In most cases t h i s f i l t e r i n g helped g r e a t l y to r e s o l v e the i n t e r n a l wave s i g n a l , o f t e n y i e l d i n g smooth shapes suggestive of s o l i t a r y wave p r o f i l e s . F i g u r e 9 shows the f i l t e r e d wave s i g n a l , ( u ' , v ' ) f i n the d i r e c t i o n of wave propagation f o r Rip 215-2. There were s e v e r a l i n s t a n c e s however i n which no c l e a r wave s i g n a l c o u l d be i d e n t i f i e d at a given meter. In some cases a s i g n a l c o u l d be i d e n t i f i e d but was m o d i f i e d f o r some reason so that the r e s u l t i n g waveform had an i r r e g u l a r shape be a r i n g no resemblence to a s o l i t a r y wave. 1 .2 The I n t e r n a l Wave Phase Speed The phase speed of the i n t e r n a l waves was determined from the c u r r e n t meter data by measurement of the propagation time of a s i g n a l between p a i r s of c u r r e n t meters, i . e . , between meters 2 and 4, 4-6, and 3-5. The phase speed i s given by c =L cos(0 i -6)/Atf (2.1) L and 8 are the d i s t a n c e and d i r e c t i o n r e s p e c t i v e l y between adjacent moorings. For moorings 1/2 and 3/4 we have L=2.07 nm and 0=42.5°T, while between moorings 3/4 and 5/6/7, L=1 nm and 0=7O°T. <p| i s the d i r e c t i o n of wave propagation of a given wave. T h i s angle was determined f o r each wave to be the d i r e c t i o n of the v e c t o r average of the i n t e r n a l wave s i g n a l (u',v') at meters 4 and 6 at the time of a wave maximum. An assumption of t h i s method i s that the l a r g e amplitude i n t e r n a l 22 wave i s e n t i r e l y r e s p o n s i b l e f o r the high frequency s i g n a l ( u ' , v ' ) . Furthermore the p e r i o d of the wave i s assumed to be s u f f i c i e n t l y s h o r t e r than a pendulum day to n e g l e c t the d e f l e c t i o n of flow from the d i r e c t i o n of propagation by the C o r i o l i s f o r c e . The <f>. presented i n Table 1 were ob t a i n e d from Hodgins and Hodgins(1 981 b ) . The mean of the <p • i s 77°T which i s c o n s i s t e n t with propagation away from the c o a s t . The u n c e r t a i n t y on t h i s parameter has been taken to be 11.7° which i s the standard d e v i a t i o n of the <p • f o r the waves c o n t a i n e d i n the data s e t . At ; i s the propagation time f o r a wave to t r a v e l between two moorings. T h i s q u a n t i t y was obtained by r e s o l v i n g the c u r r e n t v e c t o r onto the d i r e c t i o n of propag a t i o n . The time of propagat i o n was taken as the time between peak c u r r e n t s at meters of the same nominal depth. To o b t a i n an estimate of the u n c e r t a i n t y on A t j i t i s assumed t h a t , at a sampling r a t e of one minute, an event o c c u r r i n g i n the c u r r e n t data can be r e s o l v e d to w i t h i n ±30 seconds. Consequently the u n c e r t a i n t y on At, has been taken t o be ±1.0 minute. Using t h i s u n c e r t a i n t y and the one on <j> • , u n c e r t a i n t i e s have been computed f o r the phase speed between moorings. These speeds have been averaged to y i e l d a mean phase speed f o r each wave. R e s u l t s are presented i n Table I. 23 1.3 The H o r i z o n t a l Length S c a l e From the f i l t e r e d c u r r e n t meter data and the computed phase speeds we are a l s o a b l e to i n f e r a measure of the h o r i z o n t a l extent (or wavelength) of these d i s t u r b a n c e s . However, due to the presence of background c u r r e n t s and high frequency n o i s e there were d i f f i c u l t i e s i n determining p r e c i s e l y the onset of a given wave. To overcome t h i s d i f f i c u l t y only the h a l f - a m p l i t u d e p e r i o d (and hence wavelength) of the i n t e r n a l waves was measured. The h a l f -amplitude p e r i o d i s the time i n t e r v a l d u r i n g which the c u r r e n t i n the wave d i r e c t i o n <f> ^  exceeded one h a l f of i t s maximum va l u e . The u n c e r t a i n t y on t h i s measurement was taken to be ±1.0 minute. The h a l f - a m p l i t u d e width was o b t a i n e d from the h a l f - a m p l i t u d e p e r i o d through m u l t i p l i c a t i o n by the a p p r o p r i a t e average phase speed. An u n c e r t a i n t y on these l e n g t h s c a l e s was determined u s i n g the u n c e r t a i n t y on the average phase speed and on the measurement of the h a l f - a m p l i t u d e p e r i o d . The l e n g t h s c a l e was c a l c u l a t e d only f o r waves whose waveform resembled the shape of a t h e o r e t i c a l s o l i t a r y wave ( i . e . a smooth r e g u l a r p u l s e ) . There were 13 o b s e r v a t i o n s from the c u r r e n t meter r e c o r d s f o r which t h i s parameter was determined. Table II c o n t a i n s the l e n g t h s c a l e f o r the s e l e c t e d o b s e r v a t i o n s . U n c e r t a i n t i e s on t h i s parameter are given g r a p h i c a l l y i n the f i g u r e s of Chapter IV. Note that there are v a r i a t i o n s between measurements of width taken from d i f f e r e n t c u r r e n t meters, even when these are on the same mooring. 24 1.4 Maximum C u r r e n t s The maximum c u r r e n t s , i n the d i r e c t i o n of wave prop a g a t i o n , f o r s e l e c t e d waves are given i n Table I I . The v a l u e s g i v e n i n the t a b l e were obtained a f t e r the f i l t e r i n g o p e r a t i o n . The low frequency f i l t e r removed the s u b s t a n t i a l t i d a l c u r r e n t s ; thus these maximum c u r r e n t s are not equal to those observed at the s i t e . They are i n s t e a d the maxima of ( u ' , v ' ) f the assumed i n t e r n a l wave s i g n a l . Note that again we have v a r i a t i o n s i n the measured value f o r the same wave event. 1.5 Wave Amplitude The t h e r m i s t o r c h a i n data p e r m i t t e d the d e t e r m i n a t i o n of the displacement amplitudes and of the h a l f - a m p l i t u d e widths f o r the f i r s t t h r e e r i p events. The measurements were ob t a i n e d from isotherm contours s i m i l a r to F i g u r e 5. The r e s u l t i n g data are presented i n Table I I I . The wave amplitude i s based on the v e r t i c a l displacement of the isotherm nominally l o c a t e d at 30m depth. The displacement was taken as the d i f f e r e n c e between the depth of maximum displacement and a r e f e r e n c e l e v e l f o r the isotherm. T h i s r e f e r e n c e l e v e l was taken as the depth of the isotherm immediately p r i o r t o the onset of the v e r t i c a l motions. The average phase speeds o b t a i n e d from the c u r r e n t meter data were used to get the l e n g t h s c a l e from the h a l f - a m p l i t u d e p e r i o d estimated from the isotherm displacements. The h a l f -amplitude p e r i o d was d e f i n e d here as i t was f o r the c u r r e n t meter data. However s i n c e temperature was d i g i t i z e d at only 25 two minute i n t e r v a l s the u n c e r t a i n t y on the h a l f - a m p l i t u d e p e r i o d as measured from the temperature data i s c o n s i d e r e d to be ±2.0 minutes. The u n c e r t a i n t y on the h o r i z o n t a l l e n g t h s c a l e i n c o r p o r a t e d t h i s and the u n c e r t a i n t y on the phase speed. These are shown i n the f i g u r e s of Chapter IV. There i s agreement w i t h i n the estimated u n c e r t a i n t i e s between the two measures of h a l f - a m p l i t u d e width. 2. THE DENSITY STRUCTURE S e v e r a l CTD s t a t i o n s were occupied i n the v i c i n i t y of Hekja d u r i n g J u l y and August of 1980. Data from 35 of these s t a t i o n s were despiked and t r a n s c r i b e d onto magnetic tape by Seaconsult Marine Research to form p a r t of the data a r c h i v e . Of the 35 s t a t i o n s , 16 were 'shallow' c a s t s i n which the CTD was c y c l e d up and down f o r s e v e r a l hours with the hope of sampling water p r o p e r t i e s d u r i n g the passage of a wave. These e f f o r t s were u n s u c c e s s f u l and these data have not been c o n s i d e r e d . The p o s i t i o n and time of the other 19 s t a t i o n s , along with other p r o p e r t i e s of i n t e r e s t , are given i n Table 4. The temperature and s a l i n i t y of CTD 1-1 are p l o t t e d a g a i n s t depth i n F i g u r e s 10 and 11. We see that there i s a t h i n i s othermal l a y e r at the s u r f a c e . Beneath t h i s l a y e r the temperature decreases by n e a r l y 5.5°C over no more than 30m depth. U n d e r l y i n g t h i s we f i n d a very c o l d (*-1.7°C), n e a r l y i s o t h e r m a l l a y e r of about 125m t h i c k n e s s . T h i s c o l d core marks the B a f f i n C u r r e n t . The temperature beneath t h i s water mass i n c r e a s e s g r a d u a l l y with depth. In c o n t r a s t to the temperature p r o f i l e , the s a l i n i t y 26 i n c r e a s e s m o n o t o n i c a l l y w i t h depth from a s u r f a c e value of 32.7 t to 34.2°/* at depth. A s t e p - l i k e s t r u c t u r e with small but numerous i n v e r s i o n s of l i m i t e d v e r t i c a l extent i s present i n the s a l i n i t y p r o f i l e at the h a l o c l i n e (30-40m below the s u r f a c e ) . D e n s i t y was i n f e r r e d from the temperature and s a l i n i t y measurements thus p e r m i t t i n g the c o n s t r u c t i o n of o T p r o f i l e s with depth. The d e t a i l s of the s a l i n i t y are almost e x a c t l y reproduced i n the d e n s i t y p r o f i l e as i s t y p i c a l of the water of p o l a r seas, where s a l i n i t y i s the dominant parameter determining d e n s i t y . F i g u r e s 12 and 13 show the oT p r o f i l e , a c u b i c s p l i n e f i t to that p r o f i l e , and the s t r a t i f i c a t i o n p r o f i l e ( N 2 ( z ) = -g p 0 " 1 dp/dz) f o r CTD 1-1. A d i s c u s s i o n of the s p l i n e f i t t i n g and of the de t e r m i n a t i o n of N 2 i s presented i n s e c t i o n 2.1. These f i g u r e s i l l u s t r a t e the gen e r a l and p e r s i s t e n t f e a t u r e s of the r e g i o n d u r i n g summer months. We have a t h i n mixed upper r e g i o n o v e r l y i n g a denser and much deeper lower l a y e r . A sharp p y c n o c l i n e separates the two. The N 2 p r o f i l e t y p i c a l l y has a w e l l d e f i n e d maximum at the l e v e l of the p y c n o c l i n e . Table IV g i v e s the value of the maximum of N 2 and the cor r e s p o n d i n g depth of t h i s maximum f o r each s t a t i o n where these parameters c o u l d be d e f i n e d . The d e n s i t y p r o f i l e s are i n g e n e r a l c h a r a c t e r i z e d by many f i n e s t r u c t u r e f e a t u r e s such as patches of w e l l mixed water or small i n v e r s i o n s . These are t r a n s i e n t f e a t u r e s t h a t occur most f r e q u e n t l y at the p y c n o c l i n e depth. V a r i a t i o n s i n the 27 f i n e s t r u c t u r e present can l e a d v a r i a t i o n s i n the shape of the N 2 p r o f i l e so that o c c a s i o n a l l y there were two adjacent maxima i n the s t r a t i f i c a t i o n obtained from a given CTD c a s t . I t i s due to e f f e c t s of t h i s s o r t that a c l e a r l y d e f i n a b l e maximum N 2 c o u l d not be e s t a b l i s h e d f o r each s t a t i o n . Some v a r i a b l i t y i n the d e n s i t y was evident on a somewhat l a r g e r s c a l e than f i n e s t r u c t u r e . F i g u r e 14 may be used to compare the d e n s i t y p r o f i l e of CTD 1-1 with that of CTD 1-15. A i n t e r v a l of s i x days separates the two s t a t i o n s . We note that at the l a t t e r s t a t i o n the water i s c o n s i d e r a b l y l i g h t e r between 50 and 250 m. In the mixed region i t i s a l s o l i g h t e r . However the depth of the p y c n o c l i n e and i t s slope do not change g r e a t l y . T h i s f i g u r e i s i l l u s t r a t i v e of a t r e n d i n the data. S t a t i o n s 1-1 to 1-5 show s i m i l a r d e n s i t y p r o f i l e s while those of s t a t i o n s 1-15 to 1-23 bear a c l o s e resemblence to each other. CTD 1-6 i s i n t e r m e d i a t e between these two groups. Since d e n s i t y i s c o n t r o l l e d by s a l i n i t y , i t i s p o s s i b l e that the r e l e a s e of f r e s h water from m e l t i n g i c e i s r e s p o n s i b l e f o r the observed decrease i n d e n s i t y with time near the s u r f a c e of the water column. Two CTD c a s t s , CTD 1-18 and CTD 1-23, were taken d u r i n g the passage of separate s o l i t a r y wave events (Hodgins and Hodgins,1981 a ) . The d e n s i t y d i s t r i b u t i o n of one of these, CTD 1-23, and the corresponding s t r a t i f i c a t i o n p r o f i l e are given i n F i g u r e s 15 and 16. What i s most s t r i k i n g here i s the deepening of the upper mixed l a y e r . I t i s more than double i n t h i c k n e s s over q u i e s c e n t c o n d i t i o n s . A l s o i n t e r e s t i n g i s that the 28 i n v e r s i o n and mixed patches commonly seen i n the other CTD c a s t s are l a r g e r and t h i c k e r . C o r r e s p o n d i n g l y , the s t r a t i f i c a t i o n p r o f i l e has many l a r g e negative peaks and i s i r r e g u l a r over a c o n s i d e r a b l e t h i c k n e s s . These anomalous reg i o n s of an otherwise s t a b l e d e n s i t y s t r u c t u r e may be the r e s u l t of l o c a l shear i n s t a b i l i t i e s , induced by the wave, that are mixing p a r c e l s of f l u i d . Note that the same degree of s p l i n e f i t smoothing has been employed here as f o r F i g u r e 13. The d e n s i t y p r o f i l e d e r i v e d from CTD 1-18 a l s o i n d i c a t e s t h a t mixing i s t a k i n g p l a c e at the p y c n o c l i n e although the t h i c k n e s s of the anomalous patches i s somewhat smal l e r than f o r CTD 1-23. 2.1 A Note On The Determination Of S t r a t i f i c a t i o n The s t r a t i f i c a t i o n was obtained by f i t t i n g p(z) with a s p l i n e r o u t i n e and a d j u s t i n g the t e n s i o n on the s p l i n e so that the f i t matched the data c l o s e l y . A comparison of the two p r o f i l e s of F i g u r e 10 shows that the f i t i s n e a r l y i d e n t i c a l t o the o r i g i n a l . Only f i n e s t r u c t u r e f e a t u r e s are not reproduced e x a c t l y . Rather the s p l i n e r o u t i n e smooths over these s m a l l s c a l e v a r i a t i o n s . The r o u t i n e a l s o p e r m i t t e d the e v a l u a t i o n of the f i r s t d e r i v a t i v e to the f i t and from t h i s the s t r a t i f i c a t i o n was computed at 2m i n t e r v a l s . The smoothing of the s p l i n e f i t i s a d e s i r a b l e f e a t u r e . The small i n v e r s i o n s or s t e p - l i k e s t r u c t u r e s , present at the p y c n o c l i n e f o r most of the s t a t i o n s , would otherwise l e a d to negative (and at times q u i t e l a r g e ) v a l u e s of N 2. P r o f i l e s with negative v a l u e s of N 2 are u n d e s i r a b l e f o r use with i n t e r n a l wave models and are not r e p r e s e n t a t i v e of the average s t r a t i f i c a t i o n of the f l u i d . The 29 i n v e r s i o n s seen i n the CTD c a s t s are, as d i s c u s s e d above, t r a n s i e n t and unstable f e a t u r e s . The s t r a t i f i c a t i o n p r o f i l e s were extended down, to the bottom depth, from maximum depth of each CTD c a s t . T h i s was done by t a k i n g the l a s t value of N 2 computed from the CTD c a s t and assuming the s t r a t i f i c a t i o n decreased l i n e a r l y from t h i s v a lue to zero at the bottom. G e n e r a l l y the s t r a t i f i c a t i o n near the end of the CTD c a s t s was very sm a l l so i t i s assumed that t h i s h olds a l l the way to the bottom. Other CTD data from Davis S t r a i t (Osborn, LeBlond, and Hodgins,1978) show that no s i g n i f i c a n t d e n s i t y g r a d i e n t s are present at depths c l o s e to the bottom. Furthermore the r e s u l t s presented l a t e r are i n s e n s i t i v e to the p r e c i s e s p e c i f i c a t i o n of the s t r a t i f i c a t i o n i n the lower depths. 3. THE SHEARED BACKGROUND FLOW In t h i s s e c t i o n we w i l l examine the s t r u c t u r e of the time-averaged background flow i n the v i c i n i t y of Hekja at the time of passage of the s o l i t a r y waves. R e c a l l from the d i s c u s s i o n of the p h y s i c a l oceanography of t h i s region that there e x i s t c u r r e n t s which are of low frequency compared with the i n t e r n a l wave s i g n a l . These are the B a f f i n Current, which i s a r e s i d u a l flow, and the t i d a l c u r r e n t s . Since motions a s s o c i a t e d with the l a r g e amplitude i n t e r n a l waves have short p e r i o d s (=10 minutes) the t i d a l flow w i l l appear as a q u a s i - s t e a d y mean flow to the waves. As mentioned p r e v i o u s l y , the waves passed Hekja d u r i n g low t i d e . The c u r r e n t meter records c o n t a i n e d i n the data set are 30 s h o r t , 3-4 hours each. Consequently the sampling occurs only d u r i n g the low water phase of the t i d e . Although the records are short they should be s u f f i c i e n t t o g i v e us an estimate of the background c o n d i t i o n s at low t i d e . For the computation of these temporal averages we have used the data of mooring 5/6/7. T h i s was the mooring with the two deep c u r r e n t meters. We w i l l use i n f o r m a t i o n obtained from the three VACMs and a l s o from the i n t e r n a l l y r e c o r d i n g c u r r e n t meter (RCM) at 200m depth. We w i l l c o n s i d e r only the mean flow i n the d i r e c t i o n of wave propag a t i o n . I t i s shown i n s e c t i o n 3.1 that only the flow i n t h i s d i r e c t i o n has an e f f e c t upon the v e r t i c a l s t r u c t u r e of an i n t e r n a l wave. The average d i r e c t i o n of propagation of a l l the r i p events observed d u r i n g the summer and autumn of 1980 at Hekja ( i n c l u d i n g events not recorded i n t h i s data set) was 72°T (Hodgins and Hodgins,1981b). The mean flows r e p o r t e d i n t h i s t h e s i s are a l l i n t h i s d i r e c t i o n . F i g u r e 17 shows a s t i c k p l o t of the c u r r e n t data at four depths a s s o c i a t e d with Rip 215-2. In the c o n s t r u c t i o n of t h i s p l o t a f i v e minute running mean was a p p l i e d to the RCW system r e c o r d s . Note that the d i g i t i z a t i o n r a t e of the VACM instruments was one minute while t h a t of the RCM was 10 minutes. The low water s l a c k t i d e was at 21:10 GMT while the s o l i t a r y wave i s observed to pass t h i s mooring at 21:25 GMT. The f i g u r e shows the near s u r f a c e c u r r e n t s to be r e l a t i v e l y s t r o n g l y sheared i n the East-West d i r e c t i o n . In a d d i t i o n t h e r e must be some shear present between 120 and 200m s i n c e there i s a r e v e r s a l i n the East-West d i r e c t i o n at depth. As we s h a l l 31 see the shears that are i n d i c a t e d by t h i s p l o t are c h a r a c t e r i s t i c of one type of flow c o n d i t i o n . To c h a r a c t e r i z e the average flow f o r each r i p event, the c u r r e n t s recorded at each meter were r e s o l v e d onto 72°T. The r e s u l t i n g components were averaged over each r e c o r d y i e l d i n g an average c u r r e n t at four depths f o r the ten r i p events. These are g iven i n Table V. An examination of these averages suggests that there are s i m i l a r i t i e s between groups of s u c c e s s i v e p r o f i l e s . For example the f i r s t t h ree r i p events show a f a i r l y s t r o n g negative c u r r e n t at 15 and 200m. The next four r i p events have n e a r l y no flow near the s u r f a c e and a p o s i t i v e c u r r e n t at 200m. S u b s t a n t i a l p o s i t i v e c u r r e n t s are present i n n e a r l y a l l cases at 30m. The c u r r e n t s at 120m are p o s i t i v e i n a l l but the l a s t case and are somewhat weaker than the those at 30m. F u r t h e r averaging i s needed to reduce the amount of data we must handle and a l s o to gain some s t a t i s t i c a l s i g n i f i c a n c e f o r our estimate of the average flow c o n d i t i o n . I t seems i n a p p r o p r i a t e however to simply lump a l l of the samples at each l e v e l t o g e t h e r . The r e s u l t i n g mean would not resemble any of the sample p r o f i l e s . In order to preserve the d e t a i l s seen i n these averages, the ten p r o f i l e s were d i v i d e d i n t o two groups. The f i r s t three r i p events comprise the f i r s t group while the next four comprise the second group. Since the l a s t three events d i d not y i e l d any c o n s i s t e n t p r o f i l e the averages o b t a i n e d from these were not c o n s i d e r e d . The r e s u l t s of t h i s d i v i s i o n i s presented i n Table VI. In any f u t u r e r e f e r e n c e we 32 w i l l r e f e r to these averages as Flow 1 and Flow 2 r e s p e c t i v e l y . T h i s way of grouping the averages has been made on the b a s i s of having s i m i l a r flow c o n d i t i o n s at each depth. A l s o t h i s method does not d i s t u r b the c h r o n o l o g i c a l sequence of events. Note that l a s t t hree time s e r i e s are based on events which o c c u r r e d approximately one month a f t e r the i n i t i a l seven. I t i s s i g n i f i c a n t t h a t the grouping used i s c o n s i s t e n t with the r e l a t i o n of the phase of the t i d e to the c u r r e n t time s e r i e s r e c o r d s . For example, low s l a c k t i d e (as determined from the water l e v e l data) o c c u r r e d at a time approximately i n the middle of the time s e r i e s of the f i r s t three events. However the time s e r i e s of the next four events were a l l taken p r i o r to low water s l a c k . T h i s suggests t h a t the averages are samples of d i f f e r e n t phases of the t i d a l flow. Given a measure of the c u r r e n t at four f l u i d depths, the next step i s to c o n s t r u c t smooth p r o f i l e s of flow from the s u r f a c e to the bottom. In a d d i t i o n we r e q u i r e , f o r l a t e r c o n s i d e r a t i o n , the mean shear U z ( z ) and the shear g r a d i e n t U z z ( z ) . To get t h i s , the v a l u e s of Tab l e VI were p l o t t e d a g a i n s t depth axes and were connected with a smooth curve drawn by eye. The n o - s l i p c o n d i t i o n at the bottom was used to continue the curves below 200m. S e v e r a l p o i n t s (10-15) were p i c k e d o f f these curves and were f i t t e d with a s p l i n e f i t r o u t i n e . The t e n s i o n on the s p l i n e f i t s was kept h i g h to have smooth p r o f i l e s that f i t the data p o i n t s c l o s e l y . The r e s u l t i n g f i t s , were e v a l u a t e d at 2m depth i n t e r v a l s . In a d d i t i o n i t was p o s s i b l e to e v a l u a t e the f i r s t and second 33 d e r i v a t i v e s of U ( z ) . F i g u r e s 18 and 19 show p r o f i l e s of Flow 1 and 2 p l o t t e d a g a i n s t depth. The mean background flow, due to t i d a l f o r c i n g , v a r i e s with depth and has r e v e r s a l s i n the flow d i r e c t i o n . T h i s p i c t u r e of the t i d a l flow i s c o n s i s t e n t with the t i d a l c u r r e n t data r e p o r t e d by Vandall(1982) f o r the r e g i o n near the RALEGH w e l l s i t e , 26 km from HEKJA ( F i g u r e 1). The t i d a l flow there was a l s o found t o be complex and nonuniform with depth. The f i r s t and second d e r i v a t i v e s of Flow 1 and 2 are given i n F i g u r e s 20 to 23. These p l o t s show that Flows 1 and 2 are r a t h e r s t r o n g l y sheared near the s u r f a c e . Flow 1 has a s i g n i f i c a n t shear near 160m. We may c o n s i d e r the background flow to be the sum of a depth independent ( b a r o t r o p i c ) c u r r e n t , given by o U = H"1 J U(z)dz (2.2) -H and a depth v a r i a b l e ( b a r o c l i n i c ) c u r r e n t . The b a r o t r o p i c component w i l l advect a wave d i s t u r b a n c e so that the observed phase speed d e v i a t e s from the i n t r i n s i c phase speed of pro p a g a t i o n . To determine the b a r o t r o p i c c u r r e n t of Flows 1 and 2, (2.2) was e v a l u a t e d using the s p l i n e f i t to s p e c i f y U ( z ) . For Flow 1 0=0.00 m/s while f o r Flow 2 U=0.12 m/s. These are estimates of the b a r o t r o p i c f i e l d f o r a group of events; consequently these averages may not apply to any wave event i n p a r t i c u l a r . N e v e r t h e l e s s i t i s s i g n i f i c a n t t h a t the c a l c u l a t i o n i n d i c a t e s that the b a r o t r o p i c c u r r e n t s are q u i t e small and t h a t a d v e c t i o n seems to accounts f o r onl y 10 to 15% of the observed phase speeds. T h i s i s somewhat s u r p r i s i n g c o n s i d e r i n g the s t r e n g t h of the t i d a l c u r r e n t s at Hekja. 34 However i t i s important to keep i n mind that we are s p e c i f y i n g the flow f i e l d f o r the p a r t i c u l a r phase of the t i d e d u r i n g which the c u r r e n t data were c o l l e c t e d . I t i s probable that a more s i g n i f i c a n t b a r o t r o p i c f i e l d advects the i n t e r n a l waves du r i n g a d i f f e r e n t phase of the t i d e . WAVE (°T) 2-4 (m/s) J 4 - 6 (m/s) ' 3 -5 (m/s) C (m/s) 215-2 74 2 16- 1 (a) 66 216-1(b ) 85 2 16-2(a) 59 216-2(b ) 78 216-2 (c ) 95 217-2(a ) 53 217-2(b ) 69 217-2 (c ) 82 217-2(d ) 80 218-2(a ) 73 218-2(b ) 92 219-1 92 2 19-2 77 251-1 66 251-2 80 256- 1(a) 76 256- K b ) 91 1 18.+0 06, -0 12 1 02,+0 15, -0 18 0 86,+0 23, -0 26 1 07,+0 03, -0 08 0 94,+0 15, -0 18 1 06,+0 13, -0 17 0 77,+0 18, -0 21 1 01 .+0 25, -0 28 1 05,+0 15, -o 18 1 10,+0 04 , -0 08 1 03,+0 0 3 , -o 07 1 03,+0 0 7 , -0 1 1 0 98,+0 0 5 , -0 09 1 13.+0 06 , -0 09 0 90,+0 10, -0 13 0 95,+0 0 7 , -0 10 1 27,+0 06 , -0 09 0 97,+0 0 6 , -0 07 1 23,+0 06 , -0 08 1 10,+0 1 1 , -0 15 1 02 ,+0 10, -0 13 1 28,+0 06 , -0 1 1 1 18,+0 06 , -0 08 1 13.+0 06 . -0 1 1 1 28,+0 0 6 , -0 10 1 20,+0 12, -0 16 1 10,+0 04, -0 08 1 30,+0 10, -0 15 1 17.+0 07 , -0 1 1 0 98,+0 08 , -0 1 1 1 26,+0 06 , -0 07 1 01 ,+0 0 5 , -o 08 1 10,+0 04 , -0 07 1 02,+0 10, -0 13 1 13,+0 06 , -0 09 0 99,+0 04 , -0 06 1 17.+0 06 , -0 11 1 23,+0 06 , -0 10 1 15,+0 12, -o 15 1 10,+0 04 , -0 08 1 03,+0 0 3 , -0 07 1 17.+0 08 , -0 14 1 1 1 ,+0 6 , -0 11 1 08,+0 1 1 . -0 14 0 88 ,+0 17. -0 19 1 00,+0 0 6 , -0 10 1 32,+0 06 . -0 08 0 99,+0 0 5 , -0 08 0 94,+0 15, -0 18 1 13.+0 0 8 , -0 1 1 0 96,+0 13, -0 16 1 02,+0 17, -0 21 1 15.+0 0 9 , -o 12 1 09.+0 0 5 , -0 07 1 15.+0 0 6 , -0 1 1 1 26.+0 0 6 , -0 10 1 18,+0 12, -0 . 16 Table I - Measured d i r e c t i o n and phase speed of each wave. 36 WAVE EVENT HALF-AMPLITUDE MAX. CURRENT AND INSTRUMENT WIDTH (M) (M/S) 215-2, CM 3 550 0.44 215-2, CM 4 555 0.59 215-2, CM 6 480 0.85 216-2(b), CM 3 500 0.43 2 l 6 - 2 ( b ) , CM 4 590 0.42 2 l 8 - 2 ( a ) , CM 6 575 0.49 219-1, CM 2 510 0.38 219-2, CM 2 645 0.39 251-1, CM 4 480 0.45 251-2, CM 4 475 ' 0.24 256-1(a), CM 3 735 0.27 256-1(a), CM 4 615 0.59 256-1(a), CM 6 500 0.45 Table II - Length s c a l e and maximum c u r r e n t f o r s e l e c t e d waves. 37 WAVE DISP. 1/2 AMP. C 1/2 AMP. (M) PERIOD (MIN.) (M/S) WIDTH (M) 215-2 43 6.5 1.10 430 216-2U) 22 7.2 1 .03 445 216-1(b) 36 8.0 1.17 560 2l6-2(a) 30 8.3 1.11 555 216-2(b) 48 7.8 1 .08 505 216-2(C) 17 12.0 0.88 635 Table I I I - Displacement and l e n g t h s c a l e at 30 meters depth determined from isotherm contours of t h e r m i s t o r d ata. STATION NO. RANGE (nm) & BEARING (°T) FROM HEKJA DATE (1980) TIME (GMT ) |N | MAX. ( S - 2 X 1 0 - 3 ) D E P T H OF |N2| M A X . ( m ) MODE 1 P H A S E S P E E D ( m / s ) MODE 2 P H A S E S P E E D ( m / s ) CTD 1-1 1 • 0, 206 JULY 2 8 16 0 0 0 43 3 4 0 6 0 0 3 1 1-2 1 • 0, 206 28 17 0 0 0 6 1 0 3 2 1-3 0. 75, 180 28 19 2 1 0 5 6 3 6 0 5 5 0 3 3 1-4 0. 75, 180 28 20 40 0 4 6 3 6 0 5 6 0 3 4 1-5 0. 75. 180 29 12 30 0 4 3 3 0 0 5 7 0 3 3 1-6 1 .0, 250 30 18 18 0 5 2 3 6 0 5 8 0 3 4 1-15 2 .0, 250 AUG 3 17 0 5 0 5 7 3 6 0 6 5 0 3 8 1-16 6 5, 250 3 17 5 0 0 46 3 8 0 6 2 0 3 5 1-17 10 .0, 250 3 18 29 0 52 2 6 0 5 6 0 3 3 1-18 8 .0, 280 3 19 16 0 7 6 0 3 5 1-19 8 .0, 310 3 19 5 6 0 5 8 0 3 2 1-20 10 .0, 340 3 20 50 0 6 6 0 3 5 1-21 5 .0, 340 3 21 37 0 6 3 0 37 1-22 5 0, 160 3 22 5 0 0 97 4 8 0 7 0 0 3 6 1-23 10 .0, 160 3 23 3 8 0 9 2 0 31 1-29 20 .0, 225 6 22 45 0 8 2 2 4 0 59 0 3 6 1-30 15 • 0, 225 6 23 45 1 27 2 6 0 6 2 0 37 1-31 10 • 0, 225 7 00 42 0 6 7 0 3 7 1-32 5 0, 225 7 01 24 0 6 0 0 35 Table IV - F l u i d p r o p e r t i e s a s s o c i a t e d with each CTD c a s t . 39 RIP EVENT U AT 15M U AT 30M U AT 120M U AT 200M (M/S) (M/S) (M/S) (M/S) 215-1 -0.23 0.15 0.09 -0.37 216-1 -0.18 0.26 0.17 -0.23 216-2 -0.18 0.15 0.21 -0.20 217-2 0.09 0.25 0.18 0.06 218-2 -0.02 0.28 0.16 0.11 219-1 -0.05 0.37 0.27 0.10 219-2 -0.01 0.35 0.23 0.12 251-1 0.17 0.04 0.12 0.07 251-2 0.21 0.18 0.06 -0.07 256-1 0.40 0.39 -0.21 -0.10 Table V - Time-averaged flow along 72°T f o r each r i p event. 40 DEPTH FLOW 1 FLOW 2 (M) U (M/S) U (M/S) 15 -0.20 0.00 30 0.19 0.31 120 0.16 0.21 200 -0.27 0.09 Table VI - Time-averaged c u r r e n t f o r Flow 1 and Flow 2 at four depths. F i g u r e 9 - F i l t e r e d c u r r e n t time s e r i e s of Rip 215-2 r e s o l v e d onto 74°T. Note that the propagation time of the s i g n a l from mooring 3/4 to 5/6/7 i s approximately 27 minutes. F i g u r e 10 - Temperature p r o f i l e o b t ained from CTD 1-1. The c o l d core of the B a f f i n Current i s evident at 100m depth. SALINITY (PPT) 32.5 34.5 Q ) _ J CD C D F i g u r e 11 - S a l i n i t y p r o f i l e o b t a i n e d from CTD 1-1. F i g u r e 12 - Sigma-T p r o f i l e obtained from CTD 1-1 and the s p l i n e f i t to that p r o f i l e . *»5 ) ( X i t r 2 ) 0 . 0 2 5 0 . 0 5 J I I CD CD F i g u r e 13 - S t r a t i f i c a t i o n p r o f i l e obtained from the s p l i n e f i t of the d e n s i t y d i s t r i b u t i o n of F i g u r e 12. e 14 - Comparison of two d e n s i t y d i s t r i b u t i o n s : (A) CTD 1 - 1 , (B) CTD 1-15. F i g u r e 15 - Sigma-T p r o f i l e of CTD 1-23. 4 8 N 2 (S~ 2 ) ( X l O " 2 ) 0.0 0 . 0 7 5 0 . 1 5 00 _ J cn CD F i g u r e 16 - S t r a t i f i c a t i o n p r o f i l e o b t a i n e d from a s p l i n e f i t of the CTD 1-23 d e n s i t y d i s t r i b u t i o n . LO C D co -3" CD CM C D C D CM LU to n u a —, a in. i " 7 1 — - ^ l ^ \ —^• l I^ —I —— I —" I ^ — I —" 1 —" I — - U 1 1 T 1 1 1 1 r 2100 AUG 2 1 1 1 I I I I I I 2200 2300 F i g u r e 17 - S t i c k Diagram of the smoothed c u r r e n t time s e r i e s of Rip 215-2. The wave o c c u r r e d at 21:27 GMT. Note the r e v e r s a l of flow at d i f f e r e n t l e v e l s . F i g u r e 18 - Flow 1 p r o f i l e . F i g u r e 19 - Flow 2 p r o f i l e . F i g u r e 2 0 - F i r s t d e r i v a t i v e (shear) of Flow 1 . 53 (Z) - 0 . 0 3 I CO CD CD F i g u r e 21 - Second d e r i v a t i v e of Flow 1. (Z) ( M ^ S - 1 ) CXI 0" -0.03 0.03 F i g u r e 21 - Second d e r i v a t i v e of Flow 1. F i g u r e 2 2 - F i r s t d e r i v a t i v e (shear) of Flow 55 Uzz(Z) (M -0.03 ^ S " 1 ) ( X l O " 1 ) ro o 00 J cn o F i g u r e 23 - Second d e r i v a t i v e of Flow 2. 56 I I I . LINEAR WAVE ANALYSIS In t h i s chapter we w i l l c o n s i d e r a wave model which i s l i n e a r , i . e . one which n e g l e c t s the n o n l i n e a r a d v e c t i v e terms i n the equations of motion, on the assumption that these terms are s m a l l . The equations may be l i n e a r i z e d when rjk<<1, where TJ i s a displacement s c a l e and k~ 1 i s a h o r i z o n t a l l e n g t h s c a l e . The v a l i d i t y of the l i n e a r i t y assumption i s c e r t a i n l y q u e s t i o n a b l e here; however a l i n e a r model i n c o r p o r a t i n g the e f f e c t s of a depth v a r i a b l e s t r a t i f i c a t i o n and shear flow i s of va l u e . The l i n e a r model should h e l p r e v e a l the s i g n i f i c a n c e of these environmental parameters upon propagating i n t e r n a l waves. So, while a d m i t t e d l y i n c a p a b l e of reproducing the d e t a i l s of the d i s t u r b a n c e s , a l i n e a r model may provide i n f o r m a t i o n of use in the a p p l i c a t i o n of more s o p h i s t i c a t e d models. In a d d i t i o n the l i n e a r e i g e n f u n c t i o n s that are presented below are a l s o r e q u i r e d i n the n o n l i n e a r a n a l y s i s . The s e c t i o n s that f o l l o w d i s c u s s the v e r t i c a l e i g e n f u n c t i o n equation and i t s s o l u t i o n . A l s o i n c l u d e d i s an a n a l y s i s of the mode of o s c i l l a t i o n and a d i s c u s s i o n of the d i s p e r s i v e p r o p e r t i e s of the medium. 1. LINEAR INTERNAL WAVE MODEL We w i l l examine the motion of an i n v i s c i d , i n c o m p r e s s i b l e , n o n r o t a t i n g , Boussinesq f l u i d with a time-averaged, depth _^ A A ^ A independent flow U(z)=iU(z)+jV(z) where i and j are u n i t v e c t o r s d i r e c t e d to the East and North r e s p e c t i v e l y . R o t a t i o n e f f e c t s have been assumed to be n e g l i b l e due to the r e l a t i v e l y 57 short time s c a l e a s s o c i a t e d with the r i p events i n comparison to the i n e r t i a l p e r i o d . The v e r t i c a l a x i s i s p o s i t i v e upwards and zero at the s u r f a c e . Under the assumptions given above, the l i n e a r i z e d equations of motion reduce to L(u)+wU z=-p x/p 0 (3.1) L(v)+wV z =-p y/p 0 (3.2) L(w)=(-p z-pg)/p 0 (3.3) L(p)+wp z=0 (3.4) u x+v y+w z=0 (3.5) L i s the operator (3 t+U»3g) with 3* = i 3 x + j 3 y . p 0 i s an average f l u i d d e n s i t y . Through manipulation of (3.1) to (3.5) a r e l a t i o n governing w, the v e r t i c a l v e l o c i t y , i s ob t a i n e d : L,(V 2w) + L [ ( L ( w 2 ) ) z ] - L[ (wxUz+wy V z ) z ] = 0 (3.6) We have d e f i n e d the operator L,=(L 2+N 2(z)) where N 2 ( z ) = - g p 0 - 1 3p/3z (3.7) i s the B r u n t - V a i s a l l a frequency of an inco m p r e s s i b l e f l u i d . V 2 i s the operator ( 3 X X + 3 y y ) . We now make the assumption of plane waves with a s e p a r a b l e v e r t i c a l dependence. That i s we l e t w=W(z)exp(-i ( K ' i - o i t ) ). (3.8) A l s o we have K»$=k,x+k 2y where and k 2 are h o r i z o n t a l wavenumbers i n the x and y d i r e c t i o n r e s p e c t i v e l y . Upon s u b s t i t u t i o n we have f o r the v e r t i c a l dependence ( C J - K . U ) 2 ( W Z Z - | K | 2W) + N 2|»c| 2W + (cj-ic-U) U-U Z 2W)=0. (3.9) I t i s e v i d e n t from t h i s equation that o n l y the component of the mean flow which i s i n the d i r e c t i o n of wave propagation (ic 58 d i r e c t i o n ) has an e f f e c t upon the v e r t i c a l s t r u c t u r e , W(z). So l e t us d e f i n e t h i s time averaged flow U ( Z ) = K « U ( Z ) / | K \ . (3.9) then reduces to W" + [ N 2 / ( c - U * ) 2 - ( u / c ) 2 +U z"z/(c-U*)]W = 0 (3.10) where C=CO/|K| i s the phase speed and primes i n d i c a t e d i f f r e n t i a t i o n with respect to z. (3.10) i s the w e l l known T a y l o r - G o l d s t e i n equation with the Boussinesq approximation (LeBlond and Mysak,1978,Section 41). The c o n d i t i o n of no normal flow through the bottom and the r i g i d l i d approximation g i v e the boundary c o n d i t i o n s W(0)=W(-H)=0 (3.11) (3.10) together with these boundary c o n d i t i o n s c o n s t i t u t e s a S t u r m - L i o u v i l i e problem with eigenvalue c. The s o l u t i o n i s an i n f i n i t e set of modes W n(Z) with an a s s o c i a t e d set of eig e n v a l u e s c n , n=1,2,3, ... . The eige n v a l u e s are ordered such that c, i s the l a r g e s t , c 2 the second l a r g e s t and so f o r t h . With each mode n we have n-1 t u r n i n g p o i n t s or nodes i n the v e r t i c a l . A numerical s o l u t i o n of (3.10) i s presented i n the next s e c t i o n f o r v a r i o u s d e n s i t y and flow c o n d i t i o n s . 2. VERTICAL NORMAL MODES A numerical a l g o r i t h m was used to s o l v e equation (3.10) with boundary c o n d i t i o n s (3.11). The a l g o r i t h m employed a 'shooting' method to o b t a i n the s o l u t i o n . T h i s method b a s i c a l l y i n v o l v e s t r e a t i n g the boundary value problem as an i n i t i a l value problem. Two i n i t i a l c o n d i t i o n s are s p e c i f i e d : W=0 and W'=X at z=-H, with X a r b i t a r y . (3.10) was then s o l v e d with these i n i t i a l c o n d i t i o n s u s i n g a Runge-Kutta a l g o r i t h m . 59 The e i g e n v a l u e c was v a r i e d a c c o r d i n g to an i n t e r v a l h a l v i n g procedure u n t i l the boundary c o n d i t i o n at the s u r f a c e ( r i g i d l i d approximation) was s a t i s f i e d . The method p e r m i t t e d the s o l u t i o n of s u c c e s s i v e l y higher modes but only the lowest two were c o n s i d e r e d i n the a n a l y s i s . As v a r i a b l e input parameters to the v e r t i c a l equation we have: an N 2 ( z ) s t r a t i f i c a t i o n p r o f i l e , U (z) and U z z (z) p r o f i l e s p e r t a i n i n g t o the mean flow, and f i n a l l y u> the angular frequency. Since the d e p t h - v a r i a b l e parameters were computed at 2m i n t e r v a l s a l l the e i g e n f u n c t i o n s presented i n t h i s t h e s i s were computed with t h i s s p a c i n g . In a d d i t i o n i t i s important to note that none of the terms of the c o e f f i c i e n t of W(z) i n (3.10) are n e g l i g i b l e . From an examination of the U z z curves shown i n F i g u r e s 21 and 23, one may a s c e r t a i n that t h i s term, over at l e a s t some p a r t of the depth range, i s comparable t o , or even g r e a t e r than N 2 at the same depth. The importance of the u>/c term depends on the frequency of o s c i l l a t i o n . At higher f r e q u e n c i e s t h i s term may a l s o be l a r g e r than N 2 over a p o r t i o n of the f l u i d depth. 2.1 Shapes In The Absence Of A Flow Numerical experiments were run using v a r i o u s s t r a t i f i c a t i o n and shear p r o f i l e s i n order to determine t h e i r e f f e c t on the shape of the normal modes. F i g u r e s 24 and 25 show the f i r s t two modes of an i n f i n i t e l y long wave (CJ=0) using d e n s i t y data from CTD 1-1 and CTD 1-16. These e i g e n f u n c t i o n s a l o n g with those to f o l l o w have been normalized t o have a maximum value of one. The v e r t i c a l d i s t r i b u t i o n s f o r these two 60 cases are q u a l i t a t i v e l y s i m i l a r . The s t r a t i f i c a t i o n of CTD 1-1 leads t o an e i g e n f u n c t i o n that i n the f i r s t mode i s s l i g h t l y more skewed to the s u r f a c e with a shallower maximum. In the second mode we see that the zero c r o s s i n g i s about 10m shallower f o r CTD 1-1. 2.2 E f f e c t s Of Sheared Flow And F i n i t e Frequency F i g u r e 26 shows the shapes of the f i r s t and second mode e i g e n f u n c t i o n s o btained by s p e c i f y i n g the shear to be that p e r t a i n i n g to the Flow 2 p r o f i l e . The frequency CJ i s set to zero here. The shapes we o b t a i n are s i m i l a r to those without a mean flow except near the s u r f a c e where the most s i g n i f i c a n t shears are prese n t . The zero c r o s s i n g of the second mode i s about 10m .shallower than i t would otherwise be without the shear flow. To determine the e f f e c t of a nonzero frequency, CJ was set to a value of 0.0105 rad/sec c o r r e s p o n d i n g to a p e r i o d of 10 minutes. T h i s i s roughly equal to the time s c a l e of the s o l i t a r y wave events. The e i g e n f u n c t i o n s are shown i n F i g u r e 27 f o r the case of CTD 1-16 with Flow 2. Here we see that the shapes become skewed to the s u r f a c e . What we o b t a i n i s e s s e n t i a l l y a curve which r i s e s e x p o n e n t i a l l y from the bottom to a depth of about 50m. Above t h i s the curve peaks and then decreases n e a r l y l i n e a r l y to zero. T h i s e x p o n e n t i a l behaviour may be a n t i c i p a t e d i f we observe t h a t , at the frequency s p e c i f i e d , the c o e f f i c i e n t of W(z) i n (3.10) w i l l be negative over most of the lower l a y e r s of the f l u i d depth. Only i n the v i c i n i t y of the p y c n o c l i n e are va l u e s of N 2 s u f f i c i e n t l y l a r g e 61 to make t h i s c o e f f i c i e n t p o s i t i v e . Thus the s o l u t i o n of W(z) w i l l d i s p l a y o s c i l l a t o r y behaviour only near the s u r f a c e at t h i s frequency. 3. THE MODE OF OSCILLATION Upon the d e t e r m i n a t i o n of the normal modes of o s c i l l a t i o n , the q u e s t i o n a r i s e s : i n which mode or modes do the s o l i t a r y wave events propagate? There are s e v e r a l t h e o r e t i c a l reasons to expect t h a t only the lowest order mode i s p r e s e n t . These are summarized i n Gargett(1976): " F i r s t , the lowest eigenvalue corresponds to the h i g h e s t eigenspeed and f a r from an i n i t i a l d i s t u r b a n c e we expect to see f i r s t the mode which propagates f a s t e s t . Second, each s u c c e s s i v e mode has an a d d i t i o n a l node i n the v e l o c i t y d i s t r i b u t i o n , l e a d i n g to i n c r e a s e d shears and g r e a t e r a t t e n u a t i o n by l o c a l shear i n s t a b i l i t y . " I t i s i n t e r e s t i n g t o see i f the data v e r i f y ' t h i s e x p e c t a t i o n . There are two sources of data a v a i l a b l e which are p o t e n t i a l l y u s e f u l i n d i s c r i m i n a t i n g between modes: the t h e r m i s t o r c h a i n data and the c u r r e n t meter data from mooring 5/6/7. From the t h e r m i s t o r records we have seen t h a t isotherm contours can be drawn t h a t r e v e a l the v e r t i c a l displacements a s s o c i a t e d with the i n t e r n a l wave motions. We expect that the d i f f e r e n t i a l v e r t i c a l displacements of a set of isotherms be d i s t r i b u t e d a c c o r d i n g to one of the computed normal modes. To show t h i s we assume the l i n e a r i z e d kinematic c o n d i t i o n w=97j/9t i s v a l i d , where r? i s the displacement of the f l u i d p a r t i c l e s . L e t t i n g rj and w be plane waves of the form 0=0(z)exp(i(kx-wt)) (3.12) 62 we f i n d that T j(z)=-iW(z)/«. T h i s i n d i c a t e s that each F o u r i e r component of TJ w i l l be d i s p l a c e d i n the v e r t i c a l a c c o r d i n g t o W(z), that i s , a c c o r d i n g to one of the e i g e n f u n c t i o n s . S e c t i o n 3.1 p r e s e n t s the r e s u l t s of a modal a n a l y s i s of the temperature data i n which the displacements of isotherms have been assumed to be i d e n t i c a l to the f l u i d d i s p l a c e m e n t s . In order to make use of the c u r r e n t meter data we r e c a l l t h a t f o r a two dimensional d i s t u r b a n c e the c o n t i n u i t y equation i s 9u/9x=-9w/9z. Assuming the plane wave form (3.12) f o r u and w we have u ( z ) = - i k " 1 W'. Thus each s p a t i a l component of u w i l l be d i s t r i b u t e d i n the v e r t i c a l a c c o r d i n g to the d e r i v a t i v e of W(z). From mooring 5/6/7 we have c u r r e n t meter data at 3 l e v e l s : 15, 30, and 120m. The f i r s t mode e i g e n f u n c t i o n u s u a l l y has a maximum l o c a t e d s l i g h t l y shallower than 120m. The slope at t h i s depth i s small and p o s s i b l y of d i f f e r e n t s i g n from i t s near s u r f a c e v a l u e . In c o n t r a s t the second mode e i g e n f u n c t i o n has a d e r i v a t i v e of the same s i g n a t 120m as near the s u r f a c e . We may d i s t i n g u i s h between the two modes by comparing the f i l t e r e d c u r r e n t s observed near the s u r f a c e with those at 120m d u r i n g the passage of a wave. I f the f i r s t mode i s pre s e n t , as we expect, we should f i n d that the c u r r e n t s t r e n g t h i s much reduced at depth with p o s s i b l y a change of phase. S e c t i o n 3.2 pre s e n t s the r e s u l t s of t h i s a n a l y s i s . 63 3.1 Isotherm Displacements Thermistor data was c o l l e c t e d f o r the f i r s t three events documented i n the data s e t . The isotherms contours that were c o n s t r u c t e d r e v e a l e d the v e r t i c a l displacements a s s o c i a t e d with each wave. For each event i t was p o s s i b l e to draw ten isotherms ranging from 3.5°C to -1.0°C at 0.5° i n t e r v a l s . Thus we can measure the displacement at the ten d i f f e r e n t depths c o r r e s p o n d i n g to the u n d i s t u r b e d depths of each isotherm. We expect t h a t these displacements w i l l be d i s t r i b u t e d a c c o r d i n g to the v e r t i c a l dependence of some low order e i g e n f u n c t i o n . The r e f e r e n c e l e v e l f o r each isotherm was taken as the depth of that isotherm immediately preceeding the passage of the wave. T h i s procedure gave r e s u l t s which i n g e n e r a l were c o n s i s t e n t with our e x p e c t a t i o n that the deeper isotherms undergo g r e a t e r v e r t i c a l motions than those that l i e c l o s e r to the s u r f a c e . The displacement was taken as the d i f f e r e n c e i n depth between the peak of the wave and the r e f e r e n c e l e v e l . The data obtained from t h i s procedure were compared to the e i g e n f u n c t i o n s of F i g u r e 25. Using a s p l i n e f i t r o u t i n e , the e i g e n f u n c t i o n s were e v a l u a t e d at the depths c o r r e s p o n d i n g to the r e f e r e n c e isotherm l e v e l s . F i g u r e 28 shows a p l o t of the displacement data of R i p 215-2 versus the two lowest mode e i g e n f u n c t i o n s as w e l l as the l e a s t squares r e g r e s s i o n l i n e f o r each. There i s some s c a t t e r about both l i n e s but there i s l e s s about the r e g r e s s i o n l i n e f o r the second mode. A c o r r e l a t i o n c o e f f i c i e n t , which i s measure of the s c a t t e r about a r e g r e s s i o n l i n e , has been c a l c u l a t e d f o r the lowest two modes f o r the s i x 64 waves observed i n the t h e r m i s t o r r e c o r d s . These r e s u l t s are given i n Table V I I . We can see that both modes c o r r e l a t e with the data q u i t e w e l l . The second mode i n a l l cases has l e s s s c a t t e r but the improvement over the gravest mode i s m a r g i n a l . We w i l l not a t t a c h any s i g n i f i c a n c e to t h i s d i f f e r e n c e . The only c o n c l u s i o n that can be reached from c o n s i d e r a t i o n of the temperature data i s that i t p r o v i d e s i n s u f f i c i e n t i n f o r m a t i o n to enable us to decide between the two lowest modes. T h i s i s not due to a l a c k of data. Rather i t i s due to the i s o t h e r m a l nature of the water column below 50m. Consequently i t was not p o s s i b l e to draw isotherms beneath t h i s depth. The f i r s t and second modes are very s i m i l a r near the s u r f a c e so that e i t h e r mode can reasonably d e s c r i b e the v e r t i c a l displacements i n f e r r e d from the r e c o r d s . 3.2 Phase Of O s c i l l a t i o n With Depth F i g u r e 29 shows a one hour time s e r i e s of the c u r r e n t at 30 and 120m r e s o l v e d onto 78°T, the d i r e c t i o n of propagation of R i p 216-1(b). T h i s wave i s e v i d e n t i n the CM6 r e c o r d and has a maximum at 9:50 GMT. The CM7 r e c o r d however shows the flow at 120m to be l e s s i n t e n s e with only small f l u c t u a t i o n s . The f l u c t u a t i o n s a t the two depths appear to be out of phase; that i s an i n c r e a s e at 30m i s accompanied by a decrease, a l b e i t q u i t e small at 120m. In order to v e r i f y t h i s more c a r e f u l l y the time-lagged c r o s s - c o v a r i a n c e of the two time s e r i e s was computed, f o r 20 minutes of r e c o r d c e n t e r e d about 9:50 GMT, and i s given i n F i g u r e 30. The negative peak at zero l a g confirms that the two records do vary out of phase thus suggesting that 65 the o s c i l l a t i o n i s of the f i r s t mode. T h i s a n a l y s i s was c a r r i e d out f o r s e v e r a l wave r e c o r d s . A negative peak i n the c o v a r i a n c e , at small time l a g , was f r e q u e n t l y observed although not f o r a l l cases. The s i g n a l at the lower meter was always very weak i n comparison with the n e a r - s u r f a c e c u r r e n t s . T h i s i s as expected f o r waves of the f i r s t mode. From the examination of the c u r r e n t meter and temperature data we have, to some degree, a c o n f i r m a t i o n that the i n t e r n a l waves are of mode one. Thus only t h i s mode w i l l be c o n s i d e r e d i n the a p p l i c a t i o n of a n o n l i n e a r wave model. T h i s c h o i c e i s based as much upon the 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 summarized e a r l i e r as upon the data d i s c u s s e d here. 4. DISPERSION RELATION The s o l u t i o n of (3.10) g i v e s the normal mode e i g e n f u n c t i o n s W n(z) and the a s s o c i a t e d e i g e n v a l u e s c n which are the phase speeds of each corresponding mode. I t i s p o s s i b l e to compare the v a r i a t i o n of t h i s phase speed with frequency f o r each mode by s o l v i n g (3.10) f o r a s e r i e s of values of to. In t h i s way we can determine the d i s p e r s i o n of the f l u i d under v a r y i n g c o n d i t i o n s of s t r a t i f i c a t i o n and shear flow. The c a l c u l a t i o n permits the comparison of p r e d i c t e d phase speeds and wavelengths f o r a set of f r e q u e n c i e s . In Table IV the long wave speed (CJ=0) f o r modes one and two was given with each of the CTD c a s t s used to s p e c i f y the s t r a t i f i c a t i o n . No shear flow was i n c l u d e d i n the c a l c u l a t i o n . The average phase speed i s about 0.61 m/s i f we d i s r e g a r d the r e s u l t f o r CTD c a s t s 1-18 and 1-23. The s t r a t i f i c a t i o n of 66 these s t a t i o n s was d i s t u r b e d by wave motions and t h i s caused the speeds to be anomalously h i g h . Note that the long wave speeds are much s m a l l e r than the measured phase speeds. F i g u r e 31 shows the v a r i a t i o n of the eigenspeed and wavenumber with frequency f o r the two lowest modes us i n g CTD 1-16 without a mean shear flow. The f i r s t mode speeds are c o n s i d e r a b l y s m a l l e r than the measured speed of propagation even i n the case of an i n f i n i t e l y long wave. The second mode phase speeds are s m a l l e r s t i l l and c o n f i r m that -we may e l i m i n a t e t h i s and any higher mode from f u r t h e r c o n s i d e r a t i o n . The d i s p e r s i o n of the f i r s t mode e i g e n f u n c t i o n s c o r r e s p o n d i n g to v a r i o u s environmental parameters i s presented i n T able V I I I . As these r e s u l t s show, the phase speed i s reduced s i g n i f i c a n t l y with frequency. The presence of the shear flow tends t o reduce the degree of d i s p e r s i o n . In a d d i t i o n the long wave speed i s augmented somewhat. T h i s i n c r e a s e depends mostly on the b a r o t r o p i c component of U * ( z ) . In the case of Flow 1 t h i s component was zero and the long wave speed was i n c r e a s e d o n l y s l i g h t l y , due only t o the shear of flow. However, Flow 2 i n c o r p o r a t e s a b a r o t r o p i c flow of 0.12 m/s and t h i s was r e f l e c t e d i n a more s i g n i f i c a n t i n c r e a s e of the l o n g wave speed. I t remains however that t h i s i n c r e a s e i s i n s u f f i c i e n t to account f o r the observed phase speeds of the r i p events. At 10 minutes, a p e r i o d corresponding approximately to the time s c a l e of the waves, the wavelength p r e d i c t e d by the l i n e a r theory i s much s m a l l e r than the observed l e n g t h s c a l e i r r e s p e c t i v e of whether a shear flow i s 67 i n c l u d e d or not. The a n a l y s i s that has been based on the l i n e a r i z e d equations (3.1) to (3.4) thus r e v e a l s i t s l i m i t a t i o n s . The observed phase speeds exceed the l i n e a r long wave speed and the d i s p e r s i o n r e l a t i o n can not account f o r the observed l e n g t h s c a l e . F i n a l l y the assumed plane wave form given by (3.8) i s u n s a t i s f a c t o r y . The Davis S t r a i t i n t e r n a l waves are of f i n i t e amplitude and a s u c c e s s f u l model r e q u i r e s the i n c l u s i o n of the a d v e c t i v e terms n e g l e c t e d i n (3.1) to (3.4). 68 WAVE EVENT MODE 1 MODE 2 215-2 0.88 0.96 216-1(a) 0.86 0.92 216-1(b) 0.87 0.94 216-2(a) 0.78 0.84 216-2(b) 0.87 0.-93 216-2(c) 0.69 0.77 Table VII - C o r r e l a t i o n c o e f f i c i e n t s of displacements with the lowest mode e i g e n f u n c t i o n s o b t a i n e d from CTD 1-16 with no background flow. CTD 1-1 CTD 1-16 CTD 1-16; FLOW 1 | CTD 1-16; FLOW 2 PERIOD PHASE WAVE- PHASE WAVE PHASE WAVE- PHASE WAVE-(min . ) SPEED LENGTH SPEED LENGTH SPEED LENGTH SPEED LENGTH (m/s) (m) (m/s) (m) (m/s) (m) (m/s) (m) oo 0 60 00 0 62 00 0 66 00 0 76 00 120 0 60 4298 0 61 4385 0 65 4677 0 76 5436 60 0 58 2092 0 59 2124 0 63 3400 0 75 2681 30 0 53 1005 0 52 936 0 58 1045 0 71 1277 20 0 46 545 0 44 528 0 54 653 0 67 802 15 0 39 349 0 37 333 0 52 471 6 64 572 12 0 32 232 0 31 223 0 51 366 0 61 441 10 0 26 154 0 25 150 0 50 297 0 60 358 8 0 16 74 0 17 79 0 48 231 0 58 278 Table VIII - D i s p e r s i o n of plane waves under v a r i o u s environmental c o n d i t i o n s . C O cn o F i g u r e 24 - F i r s t and second mode e i g e n f u n c t i o n s obtained from CTD 1 -1. F i g u r e 25 - F i r s t and second mode e i g e n f u n c t i o n s obtained from CTD 1-16. F i g u r e 26 - F i r s t and second mode e i g e n f u n c t i o n s obtained from CTD 1-16 and Flow 2. F i g u r e 27 - F i r s t and second mode e i g e n f u n c t i o n s obtained from CTD 1-16 and Flow 2 f o r o s c i l l a t i o n s with a ten minute p e r i o d . o L U m C J CE _ J CL CO i — i Q 0 0 .0 0 . 2 0.4 W N(Z) 0.G 0 .8 F i g u r e 28 - S c a t t e r diagram of the displacement of isotherms a s s o c i a t e d with Rip 215-2. Using the e i g e n f u n c t i o n s of Fig u r e 25 the f i r s t ( c i r c l e s ) and second ( t r i a n g l e s ) modes were p l o t t e d a g a i n s t displacements f o r a given depth. The r e g r e s s i o n l i n e s about the two sets of p o i n t s are a l s o shown. I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3:30 10:30 AUG 3 F i g u r e 29 - One hour time s e r i e s of c u r r e n t , r e s o l v e d onto the propagation d i r e c t i o n , f o r Rip 216-1(b). The wave peak occ u r r e d at 9:50 GMT. Note that the increased speed at CM 6 i s accompanied by a decrease at CM 7. F i g u r e 30 - C r o s s - c o v a r i a n c e between 20 minutes of d a t a a t CM6 and CM7 c e n t e r e d about 9:50 GMT f o r R i p 216-2(b). F i q u r e 31 - D i s p e r s i o n r e l a t i o n obtained using CTD 1-16. The eigenspeed (A) and wavenumber (B) f o r the f i r s t ( s o l i d l i n e ) and second (dashed l i n e ) modes are shown. 78 IV. NONLINEAR WAVE ANALYSIS In t h i s chapter we w i l l c o n s i d e r models which are no n l i n e a r and which admit s o l i t a r y wave s o l u t i o n s . U n l i k e the l i n e a r a n a l y s i s presented above the models c o n s i d e r e d here h o l d the p o s s i b i l i t y of a c c u r a t e l y reproducing the shape of the di s t u r b a n c e as w e l l as the other s a l i e n t c h a r a c t e r i s t i c s such as phase speed and wavelength that were c o n s i d e r e d i n Chapter I I I . A s h o r t h i s t o r i c a l review of s u r f a c e and i n t e r n a l s o l i t a r y wave theory i s presented i n i t i a l l y . T h i s i s fo l l o w e d by a more r i g o r o u s c l a s s i f i c a t i o n of the governing e v o l u t i o n equations through a u n i f i e d approach. T h i s d i s c u s s i o n f o l l o w s c l o s e l y the scheme pre s e n t e d by Koop and B u t l e r ( 1 9 8 1 ) . The Davis S t r a i t o b s e r v a t i o n s are c o n s i d e r e d i n l i g h t of these c l a s s i f i c a t i o n arguments. The remainder of the chapter i s concerned with the e x p o s i t i o n of a s u i t a b l e n o n l i n e a r wave model and the a p p l i c a t i o n of the cor r e s p o n d i n g s o l i t a r y wave s o l u t i o n . 1. SOLITARY WAVES 1.1 H i s t o r i c a l Review The study of s o l i t a r y waves began i n 1834 when the S c o t t i s h engineer J . S c o t t R u s s e l l f i r s t observed a s i n g l e smooth heap of water propagate as a wave without change of form f o r a long d i s t a n c e down a c a n a l . R u s s e l l ( 1 8 4 5 ) produced s u r f a c e s o l i t a r y waves i n l a b o r a t o r y and observed that the wave speed depended on amplitude and exceeded the speed of a long 79 s i n u s o i d a l wave (/gH). R u s s e l l ' s work s t i m u l a t e d i n t e r e s t which e v e n t u a l l y l e d to the long wave t h e o r i e s of Rayleigh(1876) and Boussinesq(1872) i n which the waveform was found to have a s e c h 2 ( x ) shape. These were shallow water a n a l y s e s , r e s t i n g on the assumption that the waves are much longer than the t o t a l f l u i d depth.. L a t e r Korteweg and de V r i e s ( l 8 9 5 ) showed t h a t an equation, of the same form as (4.3), given below, governed the e v o l u t i o n of long small amplitude waves. T h i s was found to have as s o l u t i o n p e r i o d i c waves, of the form F ( x - c t ) , known as c n o i d a l waves. The s o l i t a r y wave i s recovered from the p e r i o d i c s o l u t i o n i n the l i m i t of i n f i n i t e wavelength. The e s s e n t i a l f e a t u r e of (4.3) i s the c o m p e t i t i o n between the n o n l i n e a r term which tends to steepen a d i s t u r b a n c e and the t r i p l e d e r i v a t i v e term which tends to smooth out g r a d i e n t s . The balance between the two enables waves of permanent form to e x i s t . Since s o l i t a r y waves were c o n s i d e r e d an o d d i t y f o r a long time, the hydrodynamic a p p l i c a t i o n s of the Korteweg-de V r i e s (KdV) theory remained c o n f i n e d to s u r f a c e g r a v i t y waves f o r many y e a r s . The theory of i n t e r n a l s o l i t a r y waves was i n i t i a t e d by Keulegan(1953) who c o n s i d e r e d a two l a y e r system with r i g i d boundaries. He found, i n analogy with s u r f a c e waves, that long waves of permanent form having a s e c h 2 ( x ) p r o f i l e were p o s s i b l e . Note that 'long' i n t h i s context has the i d e n t i c a l meaning as f o r s u r f a c e waves. In a d d i t i o n i t was shown that t h i s was a wave of d e p r e s s i o n i n the case where the lower l a y e r i s t h i c k e r than the upper one. Conversely only a wave of 80 e l e v a t i o n can e x i s t f o r a t h i c k e r upper l a y e r . P e t e r s and S t o k e r ( l 9 6 0 ) c o n s i d e r e d l o n g waves i n a f l u i d w i t h an e x p o n e n t i a l d e n s i t y g r a d i e n t and a f r e e s u r f a c e . T h i s case was shown t o s u p p o r t an i n f i n i t e number of i n t e r n a l wave modes and an a s s o c i a t e d i n f i n i t e spectrum of i n t e r n a l wave speeds. Benjamin(1966) i n a more g e n e r a l t h e o r y demonstrated the p o s s i b i l i t y of l o n g s o l i t a r y and c n o i d a l i n t e r n a l waves i n a she a r e d f l u i d w i t h an a r b i t r a r y d e n s i t y g r a d i e n t and a f r e e s u r f a c e . B enney(l966) c o n s i d e r e d t h e unsteady problem of l o n g waves i n a f l u i d w i t h a f i x e d upper s u r f a c e and an a r b i t r a r y d e n s i t y g r a d i e n t . U s i n g a p e r t u r b a t i o n e x p a n s i o n i n two s m a l l p a rameters measuring wave a m p l i t u d e and l e n g t h he found the KdV e q u a t i o n t o govern the e v o l u t i o n of s m a l l but f i n i t e a m p l i t u d e d i s t u r b a n c e s . In c o n t r a s t t o t h e s e l o n g i n t e r n a l waves t h e o r i e s , d i f f e r e n t k i n d s of s o l i t a r y waves were found p o s s i b l e f o r an i n f i n i t e l y deep f l u i d . The t h e o r y assumes a t h i n r e g i o n of nonuniform d e n s i t y o v e r l y i n g a deep homogeneous r e g i o n . A g a i n the waves a r e ' l o n g ' i n some sense, here w i t h r e s p e c t t o the p y c n o c l i n e t h i c k n e s s and not w i t h r e s p e c t t o t h e t o t a l d e p t h . T h i s s i t u a t i o n was f i r s t c o n s i d e r e d by Benjamin(1967) f o r the st e a d y wave case i n which he found a s o l i t a r y wave s o l u t i o n h a v i n g t h e l o r e n z t i a n p r o f i l e 1/(X 2+1). D a v i s and A c r i v o s ( 1 9 6 7 ) c o n d u c t e d wavetank e x p e r i m e n t s and c o n f i r m e d t h e e x i s t e n c e of t h i s type of wave. Ono(l975) c o n s i d e r e d the unsteady v e r s i o n of t h i s problem and o b t a i n e d an e v o l u t i o n e q u a t i o n , (4.7) below, i n which t h e d i s p e r s i v e term i s a 81 H i l b e r t t r a nsform. The equation admitted the s t a t i o n a r y s o l u t i o n found by Benjamin. U n t i l r e c e n t l y i t was commonly thought that s p e c i a l i z e d i n i t i a l c o n d i t i o n s were r e q u i r e d f o r the g e n e r a t i o n of s o l i t a r y waves from the KdV equation. However Zabusky and Kruskal(1965) d i s c o v e r e d through numerical s i m u l a t i o n of the KdV equation t h a t " s o l i t a r y - w a v e p u l s e s " or " s o l i t o n s " were produced from a wide v a r i e t y of i n i t i a l c o n d i t i o n s . The p u l s e s which emerge from the e v o l u t i o n of an i n i t i a l d i s t u r b a n c e are ordered i n amplitude and phase speed with the l a r g e s t (and f a s t e s t ) f i r s t . F u r t h e r i t was found that n o n l i n e a r i n t e r a c t i o n s between s o l i t o n s d i d not a f f e c t the shape or phase speed of the i n t e r a c t i n g p u l s e s . E f f e c t i v e l y s o l i t o n s 'pass through' each other with only a change of phase. Gardner et a l . ( l 9 6 7 ) confirmed these r e s u l t s a n a l y t i c a l l y and showed that s u f f i c i e n t l y l o c a l i z e d i n i t i a l data w i l l evolve a s s y m t o t i c a l l y i n t o a s e r i e s of rank-ordered s o l i t o n s f o l l o w e d by a d i s p e r s i v e t a i l . L a t e r , s o l u t i o n s of the Benjamin-Ono (B-O) equation were a l s o shown to possess s o l i t o n p r o p e r t i e s (Chen, Lee, and P e r e i r a , 1979). The development of s o l i t o n theory has had an important impact on the p h y s i c a l s c i e n c e s . In the context of oceanography s o l i t o n models have been f r e q u e n t l y invoked to e x p l a i n the g e n e r a t i o n of i n t e r n a l s o l i t a r y waves observed i n the ocean (Maxworthy,1979). 82 1.2 Modern Approach More r e c e n t l y a u n i f i e d approach to i n t e r n a l s o l i t a r y wave theory has been developed by Joseph(l977) and Kubota, Ko, and Dobbs(l978). These authors have shown that the regimes governed by the KdV and B-0 equations are s p e c i f i c l i m i t s of the g e n e r a l i z e d e v o l u t i o n e q u a t i o n , governing n o n l i n e a r d i s p e r s i v e waves, known as the Whitham equ a t i o n : rjt +c 1 7j7} x+9 x J K(x - £ ) T ?U,t)d£ = 0 (4.1) where K ( X ) = ( 2 T T ) - 1 J c ( k)exp( ikx)dk (4.2) and c(k) i s the l i n e a r d i s p e r s i o n law. r?(x,t) measures the f l u i d displacement a s s o c i a t e d with the i n t e r n a l wave motions. Depending on the form that c(k) assumes f o r small wavenumbers we recover the e v o l u t i o n equations d i s c u s s e d p r e v i o u s l y . In a d d i t i o n to t h i s a new e v o l u t i o n equation has been d e r i v e d to d e s c r i b e the propagation of waves with a h o r i z o n t a l s c a l e comparable to the t o t a l f l u i d depth, i . e . i n t e r m e d i a t e i n l e n g t h between the two cases d i s c u s s e d above. The r e l e v a n t l e n g t h s c a l e s f o r the d i s c u s s i o n that f o l l o w s a r e : H - the t o t a l f l u i d depth, h - the s c a l e of the d e n s i t y s t r a t i f i c a t i o n ( p y c n o c l i n e t h i c k n e s s ) , L - a s u i t a b l e h o r i z o n t a l s c a l e f o r the d i s t u r b a n c e , and K - an amplitude s c a l e f o r v e r t i c a l d i s p l a c e m e n t s . 83 1.2.1 Shallow Water Theory We c o n s i d e r here long waves of small amplitude; that i s 6 = ( H / L ) 2 « 1 and e = ( K / H ) « 1 . Fur t h e r we assume that there i s a balance between the d i s p e r s i v e and n o n l i n e a r e f f e c t s as measured by 6 and e r e s p e c t i v e l y . Thus the U r s e l l number, e/8, i s 0 ( 1 ) . Under these c o n d i t i o n s the l i n e a r d i s p e r s i o n at low wavenumbers i s given by ( c ( k ) - c 0 ) / c 0 ~ k 2 . For t h i s form of d i s p e r s i o n the Whitham equation reduces to the KdV equation nt +c 0*? x+c 1 T ? 7 J X + C 2 T J x x x = 0 (4.3) where c 0 i s the l i n e a r long wave speed and c, and c 2 are f u n c t i o n a l s of the v e r t i c a l e i g e n f u n c t i o n at k=0. Equation (4.3) has a s o l i t a r y wave s o l u t i o n Tj(x-ct)=Tj 0 s e c h 2 ( ( x - c t ) / X K V ) (4.4) with phase speed c = c 0 + * ? o C i/3 (4.5) and where X 2 V = 1 2 C 2 / T J 0 C 1 . (4.6) It i s worth n o t i n g that both the phase speed and the h o r i z o n t a l s i z e of the wave are dependent upon i t s amplitude. In gene r a l s o l i t a r y waves i n c r e a s e i n phase speed and become narrower as the amplitude i s i n c r e a s e d . 1.2.2 Deep Water Theory For t h i s regime we have h / H « 1 , L/H<<1 and the balance K/h=0(L/h). In t h i s l i m i t the d i s p e r s i o n law f o r small wavenumbers i s ( c ( k ) - c 0 ) / c 0 ~ k and the Whitham equation reduces to the B-0 equation V + C 0 T ? X + C I T ? T J X + C 2 3 x x / r?U,t) (x-£)" 1 d£ (4.7) where the d i s p e r s i v e term i s now the H i l b e r t Transform of 7 ? X X . 84 (4.7) has the s o l i t a r y wave s o l u t i o n found by Benjamin(1967) T?(x-ct) = T 7 0 \ ^ / ( (x-ct) 2 + X | Q ) (4.8) where c=c 0+»} 0c 1/4 (4.9) and X b o = - 4 C 2 / T J 0 C ! . (4.10) 1.2.3 F i n i t e - D e p t h Theory The s c a l i n g here (Segur and Hammack,1982) r e q u i r e s that the wave be of small amplitude, e=K/h<<1, and have a h o r i z o n t a l s c a l e comparable to the f l u i d t h i c k n e s s , L/H=0(1). Furt h e r we r e q u i r e a t h i n s t r a t i f i e d l a y e r , h/H<<1, and f i n a l l y that the balance, e=0(h/H) , h o l d . Under these c o n d i t i o n s the d i s p e r s i o n i s given by (c ( k ) - c 0 )/c 0'-kcoth(kh) and the Whitham equation becomes oo n+ + c 0 n y + cin n + c 28 / n U , t ) { c o t h IT (x - s ) - sqn(x-0)d5 = 0 (4.11) -°° 2rT "TP (4.11) has a s o l i t a r y wave s o l u t i o n g i v e n by n ( x - c t ) = n 0 / { c o s h 2 ( x - c t ) + (tan Vp_ s i n h ( x - c t ) ) 2 } (4.12) X F D H X F D with c = c 0 - 2c 2{l - 2H cot 2H } (4.13) X F D X F D and no^ F Dcot H = -8c ? (4.14) X F D C l 85 Kubota, Ko and Dobbs(l978) have shown that s o l u t i o n s of (4.11) possess s o l i t o n p r o p e r t i e s . I t i s i n t e r e s t i n g t o note that although the s o l i t o n s o l u t i o n s of (4.11) tend to KdV s o l i t o n s i n the shallow water l i m i t , they do not tend to B-0 s o l i t o n s i n the deep water l i m i t as d i s c u s s e d by Chen and L e e ( l 9 7 9 ) . 2. CLASSIFICATION The c l a s s i f i c a t i o n arguments presented above may be a p p l i e d t o the Davis S t r a i t o b s e r v a t i o n s to determine a s u i t a b l e model. Based on the "data of Chapter II we may take as t y p i c a l v a l u e s : K=40m, L=550m, H=350m, and h=40m. We can draw some immediate c o n c l u s i o n s . E v i d e n t l y these are not deep f l u i d waves and the B-0 theory i s i n a p p r o p r i a t e . Rather the wavelength s c a l e and the t o t a l depth are of the same • order as r e q u i r e d by the f i n i t e - d e p t h a n a l y s i s . However the observed amplitudes are of the same s c a l e as the thermocline t h i c k n e s s so that K/h i s an 0(1) q u a n t i t y . F u r t h e r the balance h/HK)(K/h) i s not met. Thus on l y one of the assumptions of the theory holds here. F i n a l l y s i n c e these are r e a l l y not shallow water waves t h i s assumption of KdV theory i s not met. However the U r s e l l number of shallow water a n a l y s i s i s about 1/3.5 i n d i c a t i n g t h a t the r e q u i r e d balance, between the parameters measuring n o n l i n e a r i t y and d i s p e r s i o n , approximately h o l d s . Thus, to summarize, i t seems that the wave amplitudes are too l a r g e f o r f i n i t e - d e p t h theory o r , c o n v e r s e l y , that the waves are of i n s u f f i c i e n t l e n g t h f o r shallow water theory. I t i s c l e a r from these arguments that the Davis S t r a i t 86 o b s e r v a t i o n s can not be e a s i l y c l a s s i f i e d i n t o a regime governed by one of the extant s o l i t a r y wave t h e o r i e s . However we may look to recent l a b o r a t o r y experiments f o r d i r e c t i o n . Koop and Butler(1981) and Segur and Hammack(1982) conducted independent wavetank experiments with v a r i o u s f l u i d depths and compared t h e i r data with the waveforms and l e n g t h s c a l e s p r e d i c t e d by the a p p r o p r i a t e t h e o r y . The r e s u l t s from each set of experiments showed that the a p p l i c a b i l i t y of f i n i t e - d e p t h theory i s r e s t r i c t e d t o waves of very small amplitude. Segur and Hammack(1982) have taken a two l a y e r f i n i t e - d e p t h model to second order i n amplitude and found that the range of v a l i d i t y i n c r e a s e d but was s t i l l q u i t e r e s t r i c t e d . S p e c i f i c a l l y , with h/H=1/lO and K/h=*0.094 one would expect the assumptions of the f i r s t order theory to be reasonably s a t i s f i e d but n e v e r t h e l e s s the authors found r e l a t i v e l y poor agreement with the data when comparing wave shapes and l e n g t h s . At l a r g e r amplitudes, K/h=0.l8, the f i r s t order theory f a i l e d completely while the second order theory was beginning to f a i l . In c o n t r a s t to t h i s the KdV theory was found to be v a l i d over a l a r g e range of amplitudes. In the experimental c o n f i g u r a t i o n s mentioned above KdV theory was always found to model the data b e t t e r than the f i r s t order f i n i t e - d e p t h theory and at l e a s t as w e l l as the second order f i n i t e - d e p t h expansion. Koop and B u t l e r u s i n g a shallow water c o n f i g u r a t i o n found the two l a y e r KdV theory to be accurate to e=K/h=*0.2 and, when taken to second order i n amplitude, to e^0.8. While v a l i d f o r f a i r l y l a r g e amplitude waves, the shallow 87 water a n a l y s i s has a l s o adequately d e s c r i b e d waves of h o r i z o n t a l s c a l e comparable to the f l u i d depth. The l a r g e i n t e r n a l waves observed i n the Andaman Sea by Osborne and Burch(l980) were s u i t a b l y modeled as KdV s o l i t o n s , yet the approximate l e n g t h s c a l e r a t i o here was L/H=*2 . The p r e d i c t i v e value of long wave theory i n such circumstances has been re c o g n i z e d . Segur and Hammack (1982) s t a t e t h a t : " I t i s w e l l known that the KdV equation d e s c r i b e s the slow e v o l u t i o n of i n t e r n a l waves of f a i r l y s m a l l amplitude that are long i n comparison with the t o t a l f l u i d depth (see e.g. Benney,1966). However, t h i s meaning of 'long' i s o v e r l y r e s t r i c t i v e because i t excludes i n t e r n a l waves where wavelengths may be comparable or even l e s s than the t o t a l f l u i d depth, but which are much longer than the t h i c k n e s s of an a p p r o p r i a t e t h i n l a y e r d e f i n e d by the background d e n s i t y d i s t r i b u t i o n . " 3. THE LEE AND BEARDSLEY MODEL These c o n s i d e r a t i o n s i n d i c a t e that KdV theory may yet be the most a p p r o p r i a t e although the long wave assumption i s not s t r i c t l y c o r r e c t . The Davis S t r a i t o b s e r v a t i o n s have p r e v i o u s l y been c o n s i d e r e d from the p o i n t of view of a two l a y e r KdV model by Hodgins and Hodgins(1981 a ) . To extend t h i s f u r t h e r we wish to c o n s i d e r here the e f f e c t s of a continuous s t r a t i f i c a t i o n and a sheared background flow. A s u i t a b l e model fo r t h i s was developed by Lee and Beardsley(1974) to understand the g e n e r a t i o n and e v o l u t i o n of the f i n i t e amplitude i n t e r n a l waves observed by Halpern(1971) i n Massachusetts Bay. T h i s model, which i s an ex t e n s i o n of Benney 1s method, i n v o l v e s the expansion of the strea m f u n c t i o n and d e n s i t y f i e l d s i n terms of three small parameters measuring n o n l i n e a r , d i s p e r s i v e , and 88 non-Boussinesq e f f e c t s . D e t a i l s of the d e r i v a t i o n are reproduced here f o r completeness. We c o n s i d e r a n o n r o t a t i n g , i n c o m p r e s s i b l e , i n v i s c i d f l u i d with a background shear flow and a weak d e n s i t y s t r a t i f i c a t i o n . We assume that t h i s b a s i c s t a t e i s s t a b l e , that i s the Richardson number Ri=N2(z)/u|(Z)>1/4, f o r a l l z. Assuming motion i n two dimensions the momentum and c o n t i n u i t y equations are p 0(u,+uu x+wu z)=-p x (4.15) p 0(w, +uwx+wwz)=-pz-gp (4.16) u x+w z=0 (4.17) p t+up x+wp z=0 (4.18) where g i s the g r a v i t a t i o n a l c o n s t a n t . E l i m i n a t i n g the pres s u r e term from (1) and (2) we have the v o r t i c i t y e q uation: [p(u,+uu x+wu z)] z -[p(w t+uw x+ww z)] x -gp x =0. (4.19) We form the f o l l o w i n g d i m e n s i o n l e s s parameters e=K/D, 6=(D/L) 2 , and o=Ap/p 0 where K i s a r e p r e s e n t a t i v e displacement amplitude, L i s a c h a r a c t e r i s t i c h o r i z o n t a l s c a l e , p 0 i s a con s t a n t , and Ap i s a r e p r e s e n t a t i v e change i n d e n s i t y over a depth D=H/nir. H i s the t o t a l f l u i d depth and n i s the v e r t i c a l mode number. We w i l l take n=1, c o n s i d e r i n g only the lowest mode to d e s c r i b e the o b s e r v a t i o n s . e, 5 , and a are assumed to be small parameters; thus we have small amplitude, long d i s t u r b a n c e s on a weak s t r a t i f i c a t i o n . The background d e n s i t y s t a t e i s given by p(z) = p 0(1 +op(z)). Representing nondimensional q u a n t i t i e s with primes, the dimensional v a r i a b l e s are s c a l e d such that 89 -1/2 x=Lx', z=Dz', t=(L(agD) ) t ' , -1/2 u=(agD) [ u ( z ) + e u ' ( x , z , t ) ] , -1/2. T-1/2 , w=e8 (agD) w , P=Po[1+ap(z)+aep'(x,z,t)]. Thus the p e r t u r b a t i o n d e n s i t y and h o r i z o n t a l v e l o c i t y f i e l d are co n s i d e r e d 0(c) q u a n t i t i e s . p(z) and u(z) are the background nondimensional d e n s i t y and flow f i e l d s r e s p e c t i v e l y . S u b s t i t u t i n g the nondimensional v a r i a b l e s i n t o the v o r t i c i t y and d e n s i t y equations and dropping primes we o b t a i n p t +up x-p z^* +e\ftz p x-e\p x p z=0 (4.20) and [ ( 1 +op+oep) (\£zt +u\pKZ -u z^* + et//z ~ e ^ x ^ Z Z ) 3 Z p x + 5[ (1+ap+aep) (i//x t+u\//xx +e«//z i//xx -e\//x i//x z ) ] x =0 (4.21) where yp, the p e r t u r b a t i o n s t r e a m f u n c t i o n i s d e f i n e d such that \px =-w and \pz=u. (4.22) We make the r i g i d l i d approximation so that the v e r t i c a l v e l o c i t y vanishes at the su r f a c e and bottom. The boundary c o n d i t i o n s on ^ are then i//x=0 at z=0, 7r. (4.23) The stre a m f u n c t i o n and d e n s i t y f i e l d s are expanded i n a three parameter expansion u s i n g e, 6, and a as the small parameters. Thus we l e t co co co • : , (i i k) i//(x,t,z)= I Z L e'6Jakv// ( x , t , z ) (4.24) i=0 j=0 k=o co co co : ; Lr fi i k ) p(x,t,z)= Z Z Z e'6 Ja p ( x , t , z ) . (4.25) i=o j=o k=o We now assume that the lowest order term has a separable v e r t i c a l dependence such that 90 ^ ( 0 0 0 ) ( x , t r z ) = A ( x f t ) / o 0 , ( Z ) (4.26) and that there i s a steady t r a n s l a t i o n A t=-c 0A x. (4.27) S u b s t i t u t i n g we o b t a i n at lowest order the f o l l o w i n g e i g e n f u n c t i o n problem with c 0 as the ei g e n v a l u e " ( P z / U 2 + uzz/U)sVooo)=0 (4.28) with * C O O O , ( 0 ) = 0 l o o o )( f f)=O (4.29) where U=u(z)-c 0. Note that (4.28) i s simply the nondimensional form of (3.10) with CJ=0 and that the form of 4>°om , under v a r i o u s environmental c o n d i t i o n s , i s given by the mode one e i g e n f u n c t i o n s shown i n F i g u r e s 23 to 25. A(x,t) i s a r b i t r a r y t o lowest o r d e r . To proceed to the next order we must in t r o d u c e c o r r e c t i o n s to (4.27) such that At =-c 0A x+erAA x+6sA xxx +aqA x (4.30) where r , s, and q are c o n s t a n t s to be determined. In order to have separable f i r s t order terms i t i s necessary that ^0.0.0, . A 2 0 ™ . « f ( 4 . 3 1 ) ^ , , 0 , = A x x ^ o , , o ) ^ U 3 2 ) (0,0,1) C0,0,1) and \p =A# . (4.33) C a r r y i n g out the s u b s t i t u t i o n f o r terms i n e, 6, and o s e p a r a t e l y we ob t a i n three inhomogeneous e i g e n f u n c t i o n problems; each with f o r c i n g terms dependent on the z e r o t h order e i g e n f u n c t i o n . These are ^•OO5_ (IE. + ^ iL)§m = _L_[r(_2U(j><o'o,o,+ u ro)) -2uro)ro) zz u 2 u 2 U 2 zz zz zz z (0,00) (0,0,0) <0,0j0)._ (0,0X» _ (0,00) v , + 2U<f> * z z z + * (uz<() - uzzz<£ )] (4.34) (1,0,0) . . (1,0,0) <t> ( 0 ) = cf> (nir) = 0 91 (!?_ + ^£)^ 1' 0 )= - i l f™\ SJlL - f»» ( 4 . 3 5 ) z z U2 U U z z U2 (0,1.0) . . (0.1,0) 4. (0) = $ (mr) = 0 U rfi(0'0,0) r , . ( f z + !zz co.o,= _ 9 ( 2 r , 0 ) + n _ ,o.oy + _ _ ( ( 4 . 3 6 ) z z u 2 u u z z u z z u (J) (0) = cb (nir) = 0 For the inhomogenous boundary value problems to be s o l v a b l e the inhomogenous terms must be orthogonal to the e i g e n f u n c t i o n s of the corresponding homogenous problem, that i s to c6(0'0,01 (Boyce and DiPr ima , 1 9 6 9 ,p. 5 0 6 ). Applying t h i s c o n d i t i o n leads to the d e t e r m i n a t i o n of the co n s t a n t s r, s, and q. These c o e f f i c i e n t s r e f l e c t the environmental c o n d i t i o n s and are given by mr (o.o.o)3 p u mr. (0,o,o) 2 r = ( J 4^_<_£ + _») DZ + / * [G ,<°.°P>_ - jo,oP) ] d z } 0 U U2 U 0 2U2 z z z mr io,o,0)2 p u ( 4 3 7 ) • [ J * < - £ + - ^ ) d z ]- l ^ ' } 0 U2 U 2 2 0 s r 1 J *to'op) dz i I o n J [ 0 ^ (^  + -^)dZ ] 0 U2 U 2 q = 2 / PC dz + / * (pu ) dz 0 z 0 u z nir ? -r r ,(o,o,o)^  p u -, 1 J i ( - ^ + - ^ ) d z 1 0 U2 U 2 ( 4 . 3 9 ) 92 Equation (4.30) f o r the st r e a m f u n c t i o n can be w r i t t e n as A t+CA x-erAA x - 6 s A x x « =0 (4.40) with C=c 0~crq so that the non-Boussinesq e f f e c t s amount t o a re d u c t i o n of the l i n e a r long wave speed. (4.40) i s the KdV equation having r and s as c o e f f i c i e n t s of the n o n l i n e a r and d i s p e r s i v e terms r e s p e c t i v e l y . The s o l i t a r y wave s o l u t i o n of the h o r i z o n t a l dependence of the st r e a m f u n c t i o n i s A(x,t)=a s e c h 2 ( b ( x - c t ) ) (4.41) with a=12(6s)b 2/(er) (4.42) and c=C-aer/3. (4.43) To get the displacement at a depth z 0 we use the kinematic c o n d i t i o n ^«=-ui ? x - T 7 t . (4.44) , .(0.0.0) 10.0.0). . , , , . We have ^ x =A*c6 ( z 0 ) and t o the order r e q u i r e d : r)x=-c0V% » I n t e g r a t i n g with respect t o x we have Tj(x,t;z 0)=A(x,t)c6 ( 0 0 0 ) ( z 0 ) / ( c 0 - u ( z 0 ) ) . (4.45) Le t the nondimensional displacement be T J ' = T J 0 / K where K i s a t y p i c a l displacement. (4.45) i m p l i e s t h a t 7 } ' = a 0 t O O O ) ( z o ) / ( C o - u ( z o ) ) (4.46) and t h a t T?(x-ct,z 0) = T ? ' s e c h 2 ( b ( x - c t ) ) . (4.47) I t i s u s e f u l to express the s o l u t i o n i n terms of dimensional c o o r d i n a t e s and in terms of T J 0 , the dimensional displacement. To do t h i s we wr i t e the argument of the s e c h 2 f u n c t i o n , (b(x-c t ) ) , e x p l i c i t l y as T ) ,e(c 0-u(z 0)) r x n ' E ( c 0 - u ( z 0 ) ) r »/c7gD + * v u 4 8 ) r } l / 2 (_ _ ( r ) I K'i.'iO) , „ „ 10.0.0). . 1 \ V U - ' 0 . 0 - 0 ' . . ' 126 * ( Z 0 ) S L 3* (Z 0 ) L 9 3 w h e r e t h e s t a r r e d v a r i a b l e s i n d i c a t e d i m e n s i o n a l c o o r d i n a t e s . S u b s t i t u t i n g f o r c a n d 6 we h a v e n o ( c o - u ( z 0 ) ) r , * r , 0 ( c 0 - u ( z 0 ) ) r « ( ° > 1 / 2 (x - I C - > ^ 9 D t ) ( 4 . 4 9 ) 12DV ( z 0 ) s 3D* ( z 0 ) T h e d i m e n s i o n a l p h a s e s p e e d i s T i 0 ( c o - u ( z 0 ) ) r C r , « = / ° 9 D { C < o 3 5 T — T } ( 4 . 5 0 ) 1 , 0 3D* ( z 0 ) a n d i s p r o p o r t i o n a l t o t h e a m p l i t u d e 7 j 0 . T h e s e c h 2 ( y ) f u n c t i o n h a s a v a l u e o f o n e h a l f w h e n t h e a r g u m e n t i s s u c h t h a t | y | = 0 . 8 8 1 . S e t t i n g t = 0 we r e q u i r e t h e c o o r d i n a t e x t o s a t i s f y n 0 ( c 0 - u ( z 0 ) ) r < 1000>, . > W 2 M • ° ' 8 8 1 ( 4 . 5 1 ) 1 2 D 3 * ( z 0 ) s a t t h e h a l f - a m p l i t u d e p o i n t s . S o t h e h a l f - a m p l i t u d e w i d t h w i l l b e g i v e n i n t e r m s o f w a v e a m p l i t u d e -by 12D3< P O ) ( z 0 ) s , L - 1 .762 { ) 1 / 2 ( 4 . 5 2 ) r , 0 ( c 0 - u ( z 0 ) ) r T h e d e f i n i t i o n o f w a v e l e n g t h f o r s o l i t a r y w a v e s i s , t o s o m e e x t e n t , a r b i t r a r y . H e r e i s h a s b e e n i d e n t i f i e d w i t h t h e h a l f -a m p l i t u d e w i d t h s o a s t o f a c i l i t a t e c o m p a r i s o n w i t h t h e m e a s u r e m e n t s . N o t e t h a t L s c a l e s a s T J O - 1 ' 2 a n d t h e q u a n t i t y 0 ( z 0 ) / v o ( c 0 ~ u ( z 0 ) ) i s d e p t h i n d e p e n d e n t i n t h e o r y . We a l s o r e q u i r e t h e h o r i z o n t a l c u r r e n t f i e l d . T o g e t t h i s we n o t e t h a t t h e p e r t u r b a t i o n c u r r e n t w a s g i v e n b y 94 (0,0 0) (0,0,0) u=v&z =A(x,t)0 z . (4.53) z *• The t o t a l dimensional c u r r e n t , u*, i s given by u* = (agD)" 1 , 2u(z)+U msech 2 ( b ( x - c t ) )0 Z°' O O T (4.54) - 1 / 2 where U m=(agD) ea (4.55) I t i s u s e f u l to use dimensional c o o r d i n a t e s here as w e l l . In t h i s case the argument of the s e c h 2 f u n c t i o n i n terms of U m i s U_ r * U r * I yio (* - (C - ) (4.56) 126v^gD" s L 3/cTgD L S u b s t i t u t i n g f o r 6 as above we have r Um r ii 17 i * ,j—n r U T,*, (4.57) { — } 1 / z (x - (/ogD C - m )t ) 12D2/c7gD s ~1~ Thus we f i n d the dimensional phase speed i s , U r Cu = ZcJgD C - (4.58) m and the h a l f - a m p l i t u d e width i s 12D2/cTgtT s L = 1.762{ 1 1' 2 ( 4 . 5 9 ) m U r m - 1 / 2 Note that L n s c a l e s as U m . um To compare the measurements to the model we need to r e l a t e * U m to the maximum c u r r e n t which we w i l l c a l l u m . I t i s the maximum of u * which corresponds to the maxima measured i n the da t a . The peak value of u , u m , occurs at x=t=0 and i s r e l a t e d to U m by 95 (4.60) Note t h a t U m i s , i n t h e o r y , a depth independent q u a n t i t y . F i n a l l y we r e q u i r e a r e l a t i o n between t h e d i s p l a c e m e n t T J 0 a t a depth z 0 and the maximum c u r r e n t a t z 0 . U s i n g ( 4 . 6 0 ) , ( 4 . 5 5 ) , and (4.46) we have 4. COMPARISON WITH THE DATA 4.1 Korteweg-de V r i e s C o e f f i c i e n t s The a p p l i c a t i o n of the t h e o r y o u t l i n e d above r e q u i r e s the d e t e r m i n a t i o n of the c o n s t a n t s r , s, q. N u m e r i c a l i n t e g r a t i o n was employed t o s o l v e the i n t e g r a l s g i v e n by (4.37) t o (4.39) w i t h v a r i o u s d e n s i t y and shear p r o f i l e s . These r e s u l t s a r e g i v e n i n T a b l e IX. c 0 i n t h i s t a b l e i s the g r a v e s t mode e i g e n v a l u e of the n o n d i m e n s i o n a l e i g e n f u n c t i o n problem g i v e n by ( 4 . 2 8 ) . These speeds a r e i d e n t i c a l t o the phase speeds g i v e n i n T a b l e IV once the l a t t e r a r e n o n d i m e n s i o n a l i z e d by (agD) w i t h a=20/l000 and D=356/ 7 r . From t h e r e s u l t s of T a b l e IX we can see i m m e d i a t e l y t h a t c0>>o-q so t h a t the no n - B o u s s i n e s q e f f e c t s a r e n e g l i g i b l e . A p a r t from t h i s we f i n d t h a t t h e r e a r e v a r i a t i o n s i n s and p a r t i c u l a r l y i n r w i t h the envir o n m e n t . The p r e s ence of a shear f l o w tends t o reduce r w h i l e t h a t of a s h a l l o w N 2 maximum (CTD 1-29) has the o p p o s i t e tendency. The v a l u e of r i s an i n d i c a t i o n of the s t r e n g t h of n o n l i n e a r * (0.0.0) . (u - jfi$5 u ( z 0 ) H (z 0)D (4.61) no = «GgU ( c 0 - u ( z 0 ) ) * z (z 0) -1/2 96 e f f e c t s . An i n c r e a s e i n r i n d i c a t e s t h a t waves w i l l be narrower and propagate at g r e a t e r speed. D i s p e r s i v e e f f e c t s a r e measured by the magnitude of s, w i t h an i n c r e a s e i n t h i s q u a n t i t y t e n d i n g t o l e n g t h e n a d i s t u r b a n c e . 4.2 T e s t i n g Of The Model The i n t e r n a l wave d a t a w i l l be compared w i t h the model i n t h r e e ways; t h a t i s we w i l l compare the c u r r e n t and t emperature r e c o r d s w i t h the p r e d i c t e d wave shape, a m p l i t u d e - s p e e d r e l a t i o n , and a m p l i t u d e - w a v e l e n g t h r e l a t i o n . The f i l t e r e d c u r r e n t d a t a y i e l d e d the smoothest and most r e g u l a r wave p r o f i l e s ; c o n s e q u e n t l y o n l y t h e s e d a t a were c o n s i d e r e d f o r the c omparison of wave shapes. However both t h e r m i s t o r and c u r r e n t d a t a were c o n s i d e r e d i n the o t h e r t y p e s of c o m p a r i s o n . From the d i f f e r e n t ways t h a t can be d e v i s e d t o t e s t a s o l i t a r y wave model, Koop and B u t l e r ( 1 9 8 1 ) found the a m p l i t u d e -w a v e l e n g t h s c a l i n g t o be the most s i g n i f i c a n t . The wave p r o f i l e s and phase speeds p r e d i c t e d by the f i n i t e d e p t h , KdV, and deep water t h e o r i e s a r e a l l q u i t e s i m i l a r and t h u s * r e q u i r e p r e c i s e measurements f o r v e r i f i c a t i o n . However the w a v e l e n g t h p r e d i c t e d a t a g i v e n a m p l i t u d e d i f f e r s g r e a t l y between t h e o r i e s . As w e l l , w i t h i n KdV t h e o r y we have t h a t the w a v e l e n g t h g i v e n by (4.52) or (4.59) i s q u i t e s e n s i t i v e t o t h e a m p l i t u d e s p e c i f i e d . For t h e s e r e a s o n s the w a v e l e n g t h s c a l i n g w i l l be r e g a r d e d as the most c r i t i c a l t e s t i n t h e assessment of the model. The o t h e r t e s t s , a l t h o u g h l e s s r i g o r o u s , a r e n o n e t h e l e s s of v a l u e and a r e i n c l u d e d below. F i n a l l y b e f o r e p r e s e n t i n g t h e d a t a i t i s i m p o r t a n t t o keep 97 i n mind t h a t some of the d a t a p o i n t s a r e measures of an i d e n t i c a l p h y s i c a l e v e n t . T h i s s i t u a t i o n a r o s e f o r example when comparing measurements of a g i v e n wave but t a k e n from d i f f e r e n t c u r r e n t meter moorings. Thus the s e measurements s p e c i f y the wave a t d i f f e r e n t l o c a t i o n s and a t d i f f e r e n t t i m e s . T h i s f a c t i s i m p o r t a n t t o the e x t e n t t h a t i t reduces the number of c o m p l e t e l y independent comparisons t h a t a r e b e i n g made. The models t h a t have been c o n s i d e r e d do not a l l o w f o r any v a r i a t i o n of wave c h a r a c t e r i s t i c s s i n c e t h e s e assume a h o r i z o n t a l l y homogeneous medium. However s i n c e the o b s e r v e d waves propagate i n a medium t h a t i s not p e r f e c t l y homogeneous we can expect t h a t t h e r e w i l l a slow e v o l u t i o n of t h e s e waves w i t h time and d i s t a n c e . T h i s i n s u r e s t h a t t h e r e i s some degree of independence i n the d a t a s i n c e we a r e not measuring e x a c t l y the same wave t w i c e . 4.2.1 The Wave Shape The comparisons of wave p r o f i l e s w i t h t h e o r e t i c a l shapes ar e shown i n F i g u r e s 32 and 33 f o r the 13 waves g i v e n i n T a b l e I I . I n the f i r s t f i g u r e the d a t a p o i n t s a r e c o n n e c t e d w h i l e i n the second f i g u r e they a r e l e f t unconnected so t h a t the c a l c u l a t e d shapes may be e v i d e n t . The c o n s t r u c t i o n of t h e s e f i g u r e s r e q u i r e d t h a t the c u r r e n t meter time s e r i e s be c o n v e r t e d t o a h o r i z o n t a l l e n g t h s e r i e s t h r o u g h m u l t i p l i c a t i o n by t h e a p p r o p r i a t e average phase speed g i v e n i n T a b l e 1. The h o r i z o n t a l l e n g t h s were s c a l e d by L/2 , t h a t i s the h o r i z o n t a l d i s t a n c e from the peak t o a h a l f - a m p l i t u d e p o i n t . R e c a l l t h a t the h o r i z o n t a l l e n g t h s were g i v e n i n T a b l e I I . The wave shapes 98 were thus normalized so that each had a value of 1 at 2x/L=0 and of 0.5 at |2x/L|=1 In t h i s way the wave shapes are f o r c e d to agree with the t h e o r e t i c a l shapes a t |2x/L|=0,1 and the amplitude-wavelength s c a l i n g i s e f f e c t i v e l y decoupled from t h i s simple comparison of waveforms. The KdV, f i n i t e - d e p t h , and l o r e n z t i a n p r o f i l e s were drawn using (4.4), (4.8), and (4.12) r e s p e c t i v e l y . I t should be noted that (4.12) r e q u i r e s that X F D be s p e c i f i e d so that i n the f i n i t e - d e p t h case the wave shape i s not completely independent of the l e n g t h s c a l i n g . The curve shown i n the f i g u r e s was drawn using X F D=420m. With t h i s value the curve has an amplitude of one h a l f at x=277m which i s the average value of L/2 f o r the d a t a . The two f i g u r e s show that there i s v i r t u a l l y no d i f f e r e n c e i n the t h e o r e t i c a l shapes f o r |2X/L|<1 and the agreement with the r e c o r d s , d e s p i t e some s c a t t e r , i s very good. For |2X/L|>1 the KdV shape l i e s c l o s e s t to the recorded p r o f i l e s . In t h i s range, however, the data have steeper s l o p e s than any p r e d i c t e d shape. In summary i t seems that the s e c h 2 shape, although not completely s u c c e s s f u l , i s the best a v a i l a b l e . Note that no p o i n t s were i n c l u d e d beyond |2X/L|>2 s i n c e i n t h i s range the wave s i g n a l i s at a low l e v e l and can not be d i s t i n g u i s h e d from the n o i s y s i g n a l s that remained a f t e r f i l t e r i n g . 4.2.2 Thermistor Data Comparisons: V e r t i c a l Displacements In t h i s s e c t i o n and the f o l l o w i n g we w i l l be making comparisons with the Lee and Beardsley model. The i n t e r m e d i a t e and deep water t h e o r i e s have not been c o n s i d e r e d . F i g u r e s 34 and 35 present, r e s p e c t i v e l y , phase speed and wavelength vs. 99 v e r t i c a l displacement at 30m. (4.50) and (4.52) were used to draw the curves p r e d i c t e d by the model. V a r i a t i o n s i n the s p e c i f i e d b a s i c d e n s i t y and flow s t a t e account f o r the d i f f e r e n t t h e o r e t i c a l c u r v e s . The data are based on the s i x waves observed i n the t h e r m i s t o r r e c o r d . C o n s i d e r i n g f i r s t the phase speed p l o t we f i n d that the KdV theory i s s u c c e s s f u l to some extent, p a r t i c u l a r l y the curve f o r CTD 1-16 with Flow 2 T h i s i s probably due to the f a c t that t h i s curve has i n c o r p o r a t e d the e f f e c t of an a d v e c t i o n of about 0.12 m/s. The i n c r e a s e i n phase speed with amplitude i s e v i d e n t and appears to be c o r r e c t l y given by the slope of the model cu r v e s . The agreement with the curve obtained from CTD 1-16 with Flow 2 would be enhanced i f the long wave speed was augmented to perhaps 0.84 m/s. T h i s i m p l i e s a mean flow a d v e c t i o n of about 0.2 m/s, i . e . i s somewhat g r e a t e r that the 0.12 m/s i n c o r p o r a t e d i n t o Flow 2. I t i s worth n o t i n g that the presence of the sheared flow does not g r e a t l y a f f e c t the slope of the l i n e . Some degree of success i s a l s o obtained i n the case of the wavelength s c a l i n g . However the l e n g t h seems to be u n d e r p r e d i c t e d somewhat at l a r g e amplitudes. Although the l a r g e e r r o r bars make i t d i f f i c u l t to draw d e f i n i t e c o n c l u s i o n s , i t seems that the marked r e d u c t i o n i n wavelength with amplitude that i s p r e d i c t e d i s not observed. 100 4.2.3 Current Meter Comparisons: Maximum Cu r r e n t s F i g u r e s 36 to 39 show p l o t s of average phase speed and l e n g t h s c a l e versus the s c a l e d maximum c u r r e n t , - U m , f o r the cases of CTD 1-16 without a mean flow and with Flow 2. Note that these two cases, u n l i k e the t h e r m i s t o r data comparisons, c o u l d not be shown on the same axes s i n c e each i n v o l v e s a s c a l i n g of the maximum c u r r e n t with separate v a l u e s of ,(00,0) / \ _ , . . . , -, r , . ,(0.0,0) <t> ( z 0 ) . The negative s i g n i s i n c l u d e d on - U m s i n c e <6Z i s negative near the s u r f a c e . For the i n c l u s i o n of a mean flow . —112 _ the theory r e q u i r e s that (ogD) u ( z 0 ) be s u b t r a c t e d from the maximum c u r r e n t as i n d i c a t e d i n (4.58). However t h i s o p e r a t i o n was ignored s i n c e the peak c u r r e n t s were obtained a f t e r low frequency f i l t e r i n g of the d a t a . The f i l t e r i n g e f f e c t i v e l y removed t h i s low frequency component. The t h e o r e t i c a l curves were c o n s t r u c t e d using (4.58) and (4.59). An examination of the phase speed data shows that there i s poor agreement with the model f o r both cases and i n p a r t i c u l a r for t h a t of CTD 1-16. Even with the i n c l u s i o n of a shear flow the data l i e 0.1 to 0.2 m/s above the t h e o r e t i c a l c urve. The i n c r e a s e i n phase speed with amplitude i s not e v i d e n t with these measurements i n c o n t r a s t to the t h e r m i s t o r data. The model does somewhat b e t t e r with the l e n g t h s c a l i n g , p a r t i c u l a r l y i n the case of CTD 1-16. In a d d i t i o n , a t s m a l l e r v a l u e s of -Um the model does n o t i c e a b l y b e t t e r than at l a r g e r v a l u e s . In F i g u r e 37 the peak c u r r e n t s are s c a l e d by a s m a l l e r value of c6 t 0 0 0 ) ( z 0 ) , due to the d i f f e r e n t e igenf u n c t i o n being used, so that t h e - U m are g r e a t e r f o r t h i s case. Note that i t 101 i s c o n s i s t e n t with the t h e r m i s t o r data comparison that the model i s more s u c c e s s f u l at smaller wave amplitudes. The KdV model i s most s u c c e s s f u l i n a range of amplitudes that i s s m a l l e r than many of the o b s e r v a t i o n s . T h i s i s perhaps not s u r p r i s i n g i n l i g h t of the i n i t i a l assumption of s m a l l , but f i n i t e waves. F i n a l l y the f i g u r e s show that the i n c l u s i o n of a sheared flow improves the agreement i n phase speed, l a r g e l y due to the i n c o r p o r a t i o n of an a d v e c t i o n e f f e c t , but does not improve the l e n g t h s c a l i n g . 4.2.4 Richardson Number C a l c u l a t i o n As mentioned i n Chapter II there i s some evidence that the i n t e r n a l wave motions may be r e s p o n s i b l e f o r l o c a l i n s t a b i l i t y of the water column. I t i s of i n t e r e s t to determine whether the KdV model presented above can c o n f i r m t h i s . For s t r a t i f i e d f l u i d s a s u f f i c i e n t c o n d i t i o n f o r i n s t a b i l i t y i s R i = N 2 ( z ) / u 2 ( z ) < 1/4 (4.62) where Ri i s the Richardson number. The wave-induced shear u*(z) at the apex (x=t=0) of the s o l i t a r y wave i s ob t a i n e d through d i f f e r e n t i a t i o n of (4.54) with r e s p e c t to z and n e g l e c t i n g the background flow. I t i s given i n terms of wave amplitude by * ; F T / —/ \ \ (0.0.0) 3um . (C Q - U ( Z Q ) ) * Z Z { 4 > 6 3 ) * „ o (0.0.0)/ , 3 Z D2 $ ( z 0 ) where (4.46) and (4.55) have been invoked. F i g u r e 40 shows the v a r i a t i o n of wave induced shear with depth that a r i s e s from a s o l i t a r y wave of 35m amplitude a t a depth of 30m. The 102 e i g e n f u n c t i o n obtained from CTD 1-16 with no background flow was employed. The p l o t g i v e s the maximum of u at 40m depth. To determine the Richardson number f i e l d i t i s necessary to s p e c i f y the perturbed s t r a t i f i c a t i o n of the f l u i d . T h i s was done e m p i r i c a l l y through a s p l i n e f i t smoothing of the d e n s i t y p r o f i l e o b t a i n e d from CTD 1-23. The r e s u l t i n g N 2 p r o f i l e i s shown i n F i g u r e 41 and has a maximum d i s p l a c e d down to z=80m. For the purpose of the c a l c u l a t i o n of Ri we w i l l assume t h i s CTD c a s t was taken through the apex of the wave. The v a r i a t i o n of Richardson number with depth i s given i n F i g u r e 42 using the wave-induced shear of F i g u r e 40 and the s t r a t i f i c a t i o n obtained from CTD 1-23. Note that f o r t h i s f i g u r e Ri=l0 was s u b s t i t u t e d when Ri>l0 was o b t a i n e d . The c a l c u l a t i o n shows that over a s i g n i f i c a n t n e a r - s u r f a c e l a y e r Ri i s l e s s than u n i t y . Imbedded w i t h i n t h i s i s a t h i n n e r l a y e r of Ri<l/4. Thus t h i s c a l c u l a t i o n i n d i c a t e s t hat the wave induced shear i s s u f f i c i e n t t o d e s t a b i l i z e to f l u i d . The depths of minimum Ri however do not correspond p r e c i s e l y with the depth of the observed mixing l a y e r and i n v e r s i o n of CTD 1-29. The l a t t e r i s about 40m deeper than otherwise p r e d i c t e d . The reason f o r t h i s l a c k of correspondence between the minimum Richardson number and the mixing l a y e r i s that the former has a minimum where the wave-induced shear i s g r e a t e s t . (0,0,0) . T h i s occurs where <6Z i s maximum, that i s at the p y c n o c l m e . I f we compare the maximum shear of F i g u r e 40 (=0.035 s _ 1 ) with the maximum of N 2 (=0.0004 s" 2) we get Ri=0.44. T h i s i s p r a c t i c a l l y unstable and i t i s easy to imagine the presence of 103 background shears or perhaps a more intense wave d r i v i n g Ri below 0.25. So i t seems that even i f there was a correspondence between the depth of the maximum shear of F i g u r e 40 and the s t r a t i f i c a t i o n peak of F i g u r e 41 the shear i n s t a b i l i t y c r i t e r i o n would s t i l l be s a t i s f i e d . 5. DISCUSSION The shallow water model of Lee and Beardsley(1974) i s s u c c e s s f u l i n reproducing some of the f e a t u r e s of the Davis S t r a i t i n t e r n a l waves. The c u r r e n t meter records y i e l d e d waveforms that l i e remarkably c l o s e to the s e c h 2 shape. In a d d i t i o n , the amplitude phase speed r e l a t i o n i s i n f a i r l y c l o s e agreement with the data when an a d v e c t i o n e f f e c t i s i n c l u d e d . However, as has been s t a t e d e a r l i e r , the most important c r i t e r i a f o r v e r i f i c a t i o n i s the wavelength s c a l i n g . The comparison made in F i g u r e 35 i n d i c a t e d some degree of disagreement as the l a r g e amplitude waves tended to be wider than p r e d i c t e d . I t seems l i k e l y that t h i s i s due to higher order n o n l i n e a r e f f e c t s . Gear and Grimshaw(1982) have shown that the i n c l u s i o n of second order terms i n amplitude i n a KdV-type expansion leads to s o l u t i o n s which are wider at l a r g e r amplitudes than the f i r s t order s o l u t i o n . Koop and B u t l e r ( l 9 8 l ) v e r i f i e d t h i s same c o n c l u s i o n with l a b o r a t o r y experiments. The r e l a t i o n s between phase speed and wavelength with maximum c u r r e n t were somewhat more ambiguous than those with the wave amplitude. One source of d i f f i c u l t y may l i e with the estimates of maximum c u r r e n t . For a given wave there were 1 04 s i g n i f i c a n t v a r i a t i o n s i n the maxima measured at c u r r e n t meters of the same depth. For example, r e f e r r i n g t o Table II we have that Rip 215-2 at CM 4 had a maximum of 0.59 m/s while at CM 6 the maximum was 0.85 m/s. In c o n t r a s t the l e n g t h s c a l e s measured at the two meters were v i r t u a l l y unchanged. The source of these v a r i a t i o n s i n maximum c u r r e n t i s not known however; they may be due to problems in s e p a r a t i n g the wave-induced c u r r e n t from other h i g h frequency motions or from low frequency background c u r r e n t s . Another r e v e a l i n g comparison i n v o l v e s the r e l a t i o n between the displacement and c u r r e n t as given by (4.61). The curves shown i n F i g u r e 43 were ev a l u a t e d at 30 and 15 meters depth u s i n g the e i g e n f u n c t i o n obtained from CTD 1-16 with no shear flow. The data p o i n t s are from the f i r s t f i v e waves of Table II ( i . e . those f o r which an amplitude measure c o u l d be o b t a i n e d ) . Only a s i n g l e datum i s c o r r e c t l y given by (4.61) while the remaining four are a l l above the p r e d i c t e d l i n e . In l i g h t of F i g u r e 43 the source of the d i s c r e p a n c y between the t h e r m i s t o r and c u r r e n t meter data comparisons i s e v i d e n t . I f we assume that the amplitude measurements are more r e l i a b l e than the estimates of maximum c u r r e n t then the l a t t e r appear to be too small to f i t the l i n e a r r e l a t i o n given by (4.61). T h i s e x p l a i n s the tendency f o r the model, when compared with c u r r e n t data, to u n d e r p r e d i c t phase speed but to adequately p r e d i c t l e n g t h s c a l e . The maximum c u r r e n t comparisons would have been more c o n s i s t e n t with r e s u l t s of the t h e r m i s t o r comparisons had the maximum c u r r e n t s been g r e a t e r . T h i s c o n c l u s i o n i s 105 c o n s i s t e n t with that reached be Hodgins and Hodgins(198la) who found that the maximum c u r r e n t s p r e d i c t e d by a two-layer KdV model were more intense than the observed maxima. 'In t h i s chapter we have been concerned with the a p p l i c a t i o n of a n o n l i n e a r wave model to the Davis S t r a i t i n t e r n a l waves. Although the h o r i z o n t a l s c a l e of the waves was comparable to the t o t a l depth the shallow water model i s s t i l l q u i t e s u c c e s s f u l . The wave shape i s c o r r e c t l y given by t h i s model. The model wavelength i s broader than the observed wavelengths at l a r g e amplitudes; an e f f e c t which seems l i k e l y to be due to t r u n c a t i o n of the expansion at f i r s t order i n amplitude. V a r i a t i o n s i n s t r a t i f i c a t i o n and shear were i n v e s t i g a t e d . The a d d i t i o n of a sheared background flow does not s i g n i f i c a n t l y change the problem. The a d v e c t i o n caused by the b a r o t r o p i c component of t h i s flow must be i n c l u d e d f o r the phase speed to be c o r r e c t l y g i v e n . V a r i a t i o n s i n the s t r a t i f i c a t i o n are a l s o important as changes i n the depth of the p y c n o c l i n e can l e a d to v a r i a t i o n s i n the r e l a t i v e s t r e n g t h of n o n l i n e a r i t y to d i s p e r s i o n . We have a l s o seen that wave-induced shear i s l i k e l y to d e s t a b i l i z e the f l u i d and l e a d to mixing l a y e r s l i k e those observed i n CTD 1-23. The Davis S t r a i t waves are s i m i l a r i n many r e s p e c t s to the i n t e r n a l waves found i n the Andaman Sea(Osborne and Burch,l980) and over the S c o t i a n Shelf(Sandstrom and E l l i o t t , 1 9 8 2 ) . In each case we have l a r g e waves,i.e. K / h ~ 0 ( l ) , with the l e n g t h s c a l e roughly equal to the f l u i d depth. In a l l cases a KdV model i s p a r t i a l l y s u c c e s s f u l i n f i t t i n g the observed wave 106 p r o p e r t i e s . T h i s p o i n t s to the p r a c t i c a l value of KdV theory as Segur and Hammack(1982) have noted. However the assumptions of the theory are not s a t i s f i e d and i t i s c l e a r t h a t a t r u l y a p p r o p r i a t e i n t e r n a l s o l i t a r y wave theory i s l a c k i n g . I t may be t h a t the s i t u a t i o n found i n Davis S t r a i t and these other l o c a t i o n s occurs f r e q u e n t l y i n the oceans. H o p e f u l l y a t h e o r e t i c a l e x p o s i t i o n that i s a p p r o p r i a t e to the very l a r g e waves found i n the Andaman Sea or Davis S t r a i t w i l l be a v a i l a b l e i n the f u t u r e . 6. INTERNAL WAVE GENERATION Before c l o s i n g a few comments w i l l be made on the g e n e r a t i o n of the Davis S t r a i t s o l i t a r y waves. There are many q u e s t i o n s that can be r a i s e d concerning t h i s problem, however due to l i m i t e d s p a c i a l coverage the 1980 measurements o f f e r no d e f i n i t e answers. D i r e c t o b s e r v a t i o n s of wave ge n e r a t i o n do not e x i s t and the s i t e of o r i g i n i s a matter f o r s p e c u l a t i o n and debate. The measurements of propagation d i r e c t i o n given i n Table 1 suggest that t h i s s i t e should l i e to the west, that i s near the c o a s t . The average d i r e c t i o n i s 72°T as s t a t e d i n Chapter I I . E x t r a p o l a t i n g back towards the coast along 252 0T(=72°+180 0) d i r e c t s us t o the shore of the s m a l l e r i s l a n d n o r t h of R e s o l u t i o n I s l a n d (on F i g u r e 1). Thus, assuming no wave r e f r a c t i o n , the area l o c a t e d near the entrance of G a b r i e l S t r a i t and the mouth F r o b i s h e r Bay seems to be a l i k e l y s i t e of o r i g i n . Since the r i p events i n v a r i a b l y appeared at HEKJA duri n g low t i d e i t i s reasonable to expect a connection between the M 2 107 t i d a l flow and the formation of i n t e r n a l waves. Hodgins and Hodgins(1981a) have suggested that the model of Maxworthy(1979) may be a p p l i c a b l e . Osborne and Burch(1980) p o s i t e d t h i s same ge n e r a t i o n model f o r the Andaman Sea s o l i t o n s . The proposed mechanism i n v o l v e s the i n t e r a c t i o n of a s t r o n g t i d a l flow with .submarine topography. In h i s l a b o r a t o r y experiments Maxworthy(1979) found that the s i m u l a t i o n of the the t i d a l flow of a s t r a t i f i e d f l u i d over a t h r e e - d i m e n s i o n a l o b s t a c l e produced an i s o p y c n a l d e p r e s s i o n (lee wave) behind the o b s t a c l e . As the t i d e slackens t h i s l e e wave moves over the o b s t a c l e i n the d i r e c t i o n of the ebbing t i d e and undergoes a n o n l i n e a r e v o l u t i o n i n t o a number of rank-ordered s o l i t o n s . The c r i t e r i o n f o r the formation of a l e e wave depends on a Froude number, F r n = U/c n , where 0 i s the b a r o t r o p i c t i d a l c u r r e n t and c n i s the l i n e a r long wave speed of the nth mode. Farmer and Smith(l980) observed the formation of f i r s t mode lee waves downstream of a s i l l when the t i d a l flow was s u p e r c r i t i c a l with r e s p e c t to the f i r s t mode, that i s Fr-, >1. I t i s d i f f i c u l t to apply these r e s u l t s to the g e n e r a t i o n r e g i o n we have i d e n t i f i e d because of the complex topography of t h i s l o c a t i o n . However i t i s c l e a r t h a t the very l a r g e t i d a l c u r r e n t , 0 ( l m / s ) , w i l l be s u p e r c r i t i c a l f o r the f i r s t mode and that some wave-generating i n t e r a c t i o n with topography i s p o s s i b l e . One d i f f i c u l t y with t h i s simple model i s that i t can not account f o r the o b s e r v a t i o n that the s o l i t o n s were o f t e n not rank-ordered at HEKJA. For example, r e f e r r i n g to Table III we 108 see t h a t the l a r g e s t waves of Rips 216-1 and 216-2 were the second of the group i n each case. There i s no obvious way to e x p l a i n t h i s d i s c r e p a n c y from s o l i t o n theory. Having noted that the propagation d i r e c t i o n f o r waves w i t h i n a r i p event can vary to a s i g n i f i c a n t degree, Hodgins and Hodgins(1981b) s p e c u l a t e d that the waves may evolve i n a complex manner w i t h i n a group before reaching HEKJA. Another p o s s i b l i t y i s that waves observed at a given low t i d e o r i g i n a t e from separate but c l o s e l y spaced s i t e s . Once the s o l i t o n s leave t h i s g e n e r a t i o n r e g i o n , assumed here to be i n the v i c i n i t y of R e s o l u t i o n I s l a n d , they propagate i n a g e n e r a l l y n o r t h east d i r e c t i o n and must s u r v i v e a f l o o d t i d e b efore r e a c h i n g HEKJA. One would expect a gradual d i s s i p a t i o n of energy as they continue i n t o deeper water. A RCW system was i n s t a l l e d d u r i n g the summer of 1982 about the RALEGH d r i l l s i t e i n a n t i c i p a t i o n of the r i p events observed so r e g u l a r l y at HEKJA. Data were c o l l e c t e d d u r i n g a s i n g l e week in J u l y and f o r another week i n August. During t h i s p e r i o d only a s i n g l e group of l a r g e amplitude waves was observed i n the c u r r e n t data with the l a r g e s t wave estimated to about about 40m i n amplitude (Vandall,1982). The d i r e c t i o n of propagation was to the north e a s t . Another l a r g e amplitude wave (=*40m) was observed i n l a t e August from continuous CTD p r o f i l i n g . The t h e r m i s t o r c h a i n deployed near RALEGH i n 1982 was l o s t to i c e b e r g s so no temperature data was recovered. The r e l a t i v e absence of s o l i t o n s d u r i n g these two weeks of o b s e r v a t i o n s i s not e a s i l y accounted f o r . Vandall(1982) has 109 suggested that the unusually e x t e n s i v e pack i c e coverage d u r i n g 1982 a l t e r e d oceanographic c o n d i t i o n s , i n p a r t i c u l a r the s t r a t i f i c a t i o n , at the s i t e of g e n e r a t i o n . Another p o s s i b i l i t y i s that the waves d i s s i p a t e d much of t h e i r energy before r e a c h i n g RALEGH. S e v e r a l s m a l l e r amplitude s o l i t a r y wave-like d i s t u r b a n c e s were observed to propagate to the northeast i n the data c o l l e c t e d i n J u l y . In summary however, the lack of c o r r o b o r a t i v e data at HEKJA and the b r e v i t y of the 1982 RALEGH records r e s t r i c t s our a b i l i t y to speak with confidence on the time e v o l u t i o n of the these waves. 110 ENVIRONMENTAL PARAMETER R S ( x l O " 2 ) Co Q CTD 1-1 2.44 -4.33 0.129 0.087 CTD 1-5 1 .93 -4.72 0. 122 0.083 CTD 1-6 2.01 -4.82 0. 125 0.084 CTD 1-15 1 .84 -5.43 0.1 40 0.094 CTD 1-16 1 .81 -5.20 0. 1 32 0.089 CTD 1-29 3.47 -3.03 0.1 27 0.085 CTD 1-16 & FLOW 1 0.83 -6.14 0.141 0.078 CTD 1-16 & FLOW 2 1 .47 -4.45 0. 1 62 0.079 Table IX - Korteweg-de V r i e s c o e f f i c i e n t s under v a r i o u s environmental c o n d i t i o n s . 2X/L F i g u r e 32 - Comparison of wave p r o f i l e s of nonlinear theory with t h i r t e e n wave r e a l i z a t i o n s . The KdV ( s o l i d l i n e ) , , f i n i t e - d e p t h (long dashed l i n e ) , and B (short dashed l i n e ) s o l i t a r y waveforms are shown. F i g u r e 33 - Same comparison as i n F i g u r e 32 but with the recorded p r o f i l e s l e f t unconnected. F i g u r e 34 - R e l a t i o n between phase speed and displacement using the e i g e n f u n c t i o n s obtained from (A) CTD 1-1, (B) CTD 1-16 with Flow 1, and (C) CTD 1-16 with Flow 2. F i g u r e 35 - R e l a t i o n between h a l f - a m p l i t u d e width and displacement u s i n g the e i g e n f u n c t i o n s obtained from (A) CTD 1-29, (B) CTD 1-16 with Flow 2, and (C) CTD 1-16. F i g u r e 36 - R e l a t i o n between phase speed and s c a l e d maximum cu r r e n t using CTD 1-16. F i g u r e 37 - R e l a t i o n between phase speed and s c a l e d maximum c u r r e n t using CTD 1-16 and Flow 2. F i g u r e 39 - R e l a t i o n between ha l f - a m p l i t u d e width and s c a l e d maximum c u r r e n t using CTD 1-16 and Flow 2. U Z ( Z ) ( S " 1 ) 0.0 0.04 CD F i g u r e 40 - Wave-induced shear u s i n g CTD 1-16 with no background shear. 120 o 2 i s - 2 ) ( x i c r 2 ) 0.0 0.025 0.05 I I 1 I I ro CD GO CO CD F i g u r e 41 - S t r a t i f i c a t i o n o b t ained from a s p l i n e f i t smoothing of CTD 1-23. F i g u r e 42 - Richardson Number obt a i n e d from the wave-induced shear of F i g u r e 40 and the s t r a t i f i c a t i o n of F i g u r e 41. F i g u r e 43 - R e l a t i o n between maximum cu r r e n t and amplitude using CTD 1-16. The dashed l i n e and t r i a n g l e s apply to measurements at 15m while the s o l i d l i n e and c i r c l e s are f o r 30m. 123 V. SUMMARY Obser v a t i o n s of l a r g e amplitude i n t e r n a l waves were obt a i n e d from a Rip Current Warning System and other instruments deployed near a d r i l l s h i p i n Davis S t r a i t . The important p r o p e r t i e s of s e v e r a l waves were determined and compared with t h e o r e t i c a l r e s u l t s of both l i n e a r and n o n l i n e a r wave theory. The l i n e a r a n a l y s i s i n v o l v e d numerical s o l u t i o n of an e i g e n f u n c t i o n problem to o b t a i n the v e r t i c a l normal modes and a d i s p e r s i o n r e l a t i o n . Comparison with the observed p r o p e r t i e s demonstrated the inadequacy of t h i s theory and the assumption of l i n e a r i t y . To overcome these d e f i c i e n c i e s v a r i o u s n o n l i n e a r wave t h e o r i e s were c o n s i d e r e d . A c l a s s i f i c a t i o n of the o b s e r v a t i o n s suggested t h a t a long wave ( i . e . KdV) theory was the most a p p r o p r i a t e i n s p i t e of the f a c t t h at the h o r i z o n t a l l ength s c a l e was of the order of the t o t a l f l u i d depth. The s o l i t a r y wave s o l u t i o n of a KdV e q u a t i o n , i n c o r p o r a t i n g the e f f e c t s of continuous s t r a t i f i c a t i o n and shear, was compared with the o b s e r v a t i o n s . The s o l i t a r y wave p r o f i l e d e s c r i b e s the waveforms obtained from c u r r e n t meter records remarkably w e l l . Phase speed comparisons suggest that the a d v e c t i o n of the background flow i s important and accounts up to 2 0 % of the observed phase speeds. Comparison of the amplitude-wavelength s c a l i n g showed some disagreement at l a r g e amplitudes thus i n d i c a t i n g the need to i n c l u d e second order n o n l i n e a r e f f e c t s . The wave-induced shear based on the s o l i t a r y wave model was compared to the perturbed d e n s i t y . T h i s showed the p o s s i b i l i t y of wave-induced shear 124 i n s t a b i l i t y f o r l a r g e amplitude waves. The u s e f u l n e s s of KdV theory even when i t s u n d e r l y i n g assumptions are s a t i s f i e d only m a r g i n a l l y or not at a l l has been reco g n i z e d by other r e s e a r c h e r s and has been v e r i f i e d once a g a i n . Questions concerned with the g e n e r a t i o n and e v e n t u a l d i s s i p a t i o n of the Davis S t r a i t s o l i t o n s have been r a i s e d but the l a c k of d i r e c t o b s e r v a t i o n s p r e c l u d e s any d e f i n i t e answers. I t i s recommended that f u t u r e s t u d i e s address these problems. 125 BIBLIOGRAPHY 1. Benjamin,T.B.,1966. I n t e r n a l waves of f i n i t e amplitude and permanant form. J . F l u i d Mech. 2^ 5, 241. 2. Benjamin,T.B.,1967. I n t e r n a l waves of permanent form i n f l u i d s of great depth. J . F l u i d Mech. 25, 559. 3. Benney,C.J.,1966. Long n o n l i n e a r i n t e r n a l waves i n f l u i d f l o ws. J . Math. Phys. 45, 52. 4. Boussinesq,M.J.,1872. The o r i e des ondes et des remous qui se propagent l e long d'un c a n a l r e c t a n g u l a i r e h o r i z o n t a l , en communiquant au l i q u i d e contenu dans ce c a n a l des v i t e s s e s sensiblement p a r e i l l e s de l a s u r f a c e au fond. J . Math. Pures Appl. (2) J_7, 55. 5. Boyce,W.E. and R.C. DiPrima,1969. Elementary  D i f f e r e n t i a l Equations and Boundary Value Problems, 2nd ed. Wiley, 533pp. 6. Chen,H.H. and Y.C. Lee,1979. Internal-wave s o l i t o n s of f l u i d s with f i n i t e depth. Phys. Rev. L e t t . 4_3, 264. 7. Chen,H.H., L e e , Y . C , and N.R. P e r e i r a , 1 979. A l g e b r a i c i n t e r n a l wave s o l i t o n s and the i n t e g r a b l e Calegero-Moser-S u t h e r l a n d N-body problem. Phys. F l u i d s 22, 187. 8. C h r i s t i e , D . R . , Muirhead,K. and A. Hales,1978. On s o l i t a r y waves i n the athmosphere. J . Atm. S c i . 35, 805. 9. Davis,R.E. and A. Acrivos,1967. S o l i t a r y i n t e r n a l waves i n deep water. J . F l u i d Mech. 2j3, 593. 10. Dohler,G.,1964. T i d e s i n Canadian Waters. Can Hydrog. S e r v i c e , Ottawa, 14 pp. 11. Dunbar,M.J.,1951. Ea s t e r n A r c t i c waters. F i s h e r i e s Research Board of Canada, B u l l e t i n No. 88, 131pp. 12. Farmer,D.M. and J.D. Smith,1977. Non l i n e a r i n t e r n a l waves i n a f j o r d i n Hydrodynamics of E s t u a r i e s and F j o r d s , e d i t e d by J . N i h o u l , E l s e v i e r , 465. 13. Farmer,D.M. and J.D. Smith,1980. T i d a l i n t e r a c t i o n of s t r a t i f i e d flow with a s i l l i n Knight I n l e t . Deep-Sea Res. 27A, 239. 14. Gardner,C.S., Greene,J.M., Kruskal,M.D. and R.M. Muira,l967. Method f o r s o l v i n g the Korteweg-de V r i e s e q u a t i o n . Phys. Rev. L e t t . , J_9, 1095. 126 15. Gargett,A.E., 1 976. Generation of i n t e r n a l waves i n the S t r a i t of Georgia, B r i t i s h Columbia. Deep-Sea Res. 23, 17. 16. Gear,J.A. and R. Grimshaw,1982. A second-order theory f o r s o l i t a r y waves i n shallow f l u i d s . Univ. of Melbourne, Dept. of Mathematics, Research Report No. 8., 45pp. 17. Halpern,D.,1971. Observations on short p e r i o d i n t e r n a l waves i n Massachusetts Bay., J . Mar. Res., 29, 116. 18. Hodgins,D.O. and S. Hodgins,1981(a). Rip c u r r e n t warning system, Davis S t r a i t . Prepared f o r A q u i t a i n e Company of Canada L t d . by Seaconsult Marine Research L t d . , 197pp. 19. Hodgins,D.O. and S. Hodgins,1981(b). The i n t e r n a l wave data a r c h i v e . Prepared f o r A q u i t a i n e Company of Canada L t d . by Seaconsult Marine Research L t d . , 59pp. 20. Hunkins,K. and M. F l i e g e l , 1 9 7 3 . I n t e r n a l undular surges i n Seneca Lake: a n a t u r a l occurence of s o l i t o n s . , J . Geophys. Res. 78(3) , 539. 21. Joseph,R.I.,1977. S o l i t a r y waves i n a f i n i t e depth f l u i d . J . Phys. A, Math. General J_0, L225. 22. Kollmeyer,R.C., McGill,D.A. and N. Corwin,1967. Oceanography of the Labrador Sea i n the v i c i n i t y of Hudson S t r a i t i n 1965. U.S. Coast Guard Oceanographic Report No. 12., 34pp. 23. Koop,C.G. and G. Butler,1981. An i n v e s t i g a t i o n of i n t e r n a l s o l i t a r y waves i n a t w o - f l u i d system., J . F l u i d Mech. 112, 225. 24. Korteweg,D.J. and G. de V r i e s , l 8 9 5 . On the change of form of long waves advancing i n a r e c t a n g u l a r c a n a l and on a new type of long s t a t i o n a r y waves., P h i l . Mag. 39, 422. 25. Kubota,T., Ko,D.R.S. and L.D. Dobbs ,1978. Propagation of weakly n o n l i n e a r i n t e r n a l waves in a s t r a t i f i e d f l u i d of f i n i t e depth., A.I.A.A. J . Hydronautics j_2, 157. 26. Keulegan,G.H.,1953. C h a r a c t e r i s t i c s of i n t e r n a l s o l i t a r y waves., J . Res. N a t l . Bur. Stand. 5J_, 133. 27. LeBlond,P.H. and L.A. Mysak , l978. Waves i n the Ocean. E l s e v i e r , Amsterdam, pp. 602. 28. LeBlond,P.H., Osborn,T.R., Hodgins,D.O., Goodman,R. and M.Metge, 1981. Surface C i r c u l a t i o n i n the western 127 Labrador Sea. Deep-Sea Res. 28A, 683. 29. Lee,C.Y. and R. Beardsley,1974. The g e n e r a t i o n of long n o n l i n e a r i n t e r n a l waves i n a weakly s t r a t i f i e d shear flow. J . Geophys. Res. 79, 453. 30. Maxworthy,T.,1979. A note on the i n t e r n a l s o l i t a r y waves produced by t i d a l flow over a t h r e e - d i m e n s i o n a l r i d g e . J . Geophys. Res. 8_4, 338. 31. Ono,H.,l975. A l g e b r a i c s o l i t a r y waves i n s t r a t i f i e d f l u i d s . J . Phys. Soc. Japan 39, 1082. 32. Osborn,T.R., LeBlond,P.H. and D.O. Hodgins,1978. A n a l y s i s of Ocean Currents Davis S t r a i t - 1 9 7 7 . Prepared f o r I m perial O i l L t d . P r o d u c t i o n Research D i v i s i o n by Seaconsult Marine Research L t d . , 264pp. 33. Osborne,A. and T.Burch,1980. I n t e r n a l s o l i t o n s i n the Andaman Sea. Science 208, 451. 34. Peters,A.S. and J . J . Stoker,1960. S o l i t a r y waves i n l i q u i d s having non-constant d e n s i t y . Comm. Pure Appl. Math. J_3, 115. 35. R a y l e i g h , L o r d , 1876. On waves. P h i l . Mag. J_, 257. S c i . Pap. j_, 251. 36. Russell,J.S.,1845. Report on waves. 14th Meet. B r i t . Assoc. Adv. S c i . , York, 1844. 37. Sandstrom,H. and J.A. E l l i o t t , 1 9 8 2 . I n t e r n a l t i d e and s o l i t o n s on the S c o t i a n s h e l f - a n u t r i e n t pump at work. Unpublished manuscript ( d r a f t ) . 38. Segur,H. and J.L. Hammack,1982. S o l i t o n models of long i n t e r n a l waves. J . F l u i d Mech. 118, 285. 39. Smith,E.H.,1937. The Marion e x p e d i t i o n to Davis S t r a i t and B a f f i n Bay. U.S. Coast Guard B u l l e t i n No. 19, Part 2. Report of the I n t e r n a t i o n a l Ice P a t r o l S e r v i c e i n the North A t l a n t i c Ocean, 259 pp. 40. Vandall,P.E.,1982. A n a l y s i s of oceanographic c o n d i t i o n s at Ralegh N-18 1982. Prepared f o r C a n t e r r a Energy L t d . by Seaconsult Marine Research L t d . , 238pp. 41. Zabusky,N.J. and M.D.,Kruskal,1965. I n t e r a c t i o n of " s o l i t o n s " i n a c o l l i s i o n l e s s plasma and the recur r e n c e of i n i t i a l s t a t e s . Phys. Rev. L e t t . J_5, 240. 

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