LDU FREQUENCY CURRENT OSCILLATIONS AND TOPOGRAPHIC LdAUES IN THE STRAIT OF GEORGIA by JAMES ALFRED HELBTG . B.Sc. (Hons), Alma C o l l e g e , 1969 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (The Department o f P h y s i c s and The I n s t i t u t e o f Oceanography) We a c c e p t t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA F e b r u a r y , 1977 © dames A l f r e d H e l b i g , 1977 In p r e s e n t i n g t h i s thesis an advanced degree at the L i b r a r y I further in p a r t i a l fulfilment the U n i v e r s i t y of s h a l l make it freely of the requirements B r i t i s h Columbia, I agree available for 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 o f of this representatives. thesis for It financial this thesis gain s h a l l not be allowed without my James A l f r e d Helhig r, , . Physics and the I n s t i t u t e o f Oceanaqraphy Department or The ? U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 D a t e February k, 1977 or i s understood that copying o r p u b l i c a t i o n written permission. f that r e f e r e n c e and study. f o r s c h o l a r l y purposes may be granted by the Head o f my Department by h i s for 3 r 1 ii ABSTRACT Chang (1976, see a l s o Chang, Tabata, and Pond, 1976) up t o kG% o f t h e k i n e t i c S t r a i t of Georgia, frequency energy a s s o c i a t e d w i t h h o r i z o n t a l motions i n the B r i t i s h Columbia, i s c o n t a i n e d i n broad-banded, current o s c i l l a t i o n s t o o v e r 100 d a y s . The low- c h a r a c t e r i s e d by p e r i o d s r a n g i n g from U The purpose o f t h i s t h e s i s i s t o p r e s e n t a d y n a m i c a l model which may oscillations. has shown t h a t provide a p a r t i a l e x p l a n a t i o n of S t r a i t of Georgia simple these i s modelled by an i n f i n i t e l y l o n g , r e c t a n g u l a r c h a n n e l w i t h a bottom t h a t s l o p e s upward t o the e a s t . c h o i c e s o f the d e n s i t y s t r a t i f i c a t i o n a r e s t u d i e d : II system and, (2) a system w i t h c o n s t a n t II (1) a two-layer II Brunt-Uaisala frequency. models admit n o r t h w a r d - t r a v e l l i n g t o p o g r a p h i c Two Both p l a n e t a r y waves w i t h periods t h a t l i e i n the observed r a n g e . However t h e s e models do not a c c u r a t e l y p r e d i c t the v e r t i c a l d i s t r i b u t i o n of h o r i z o n t a l k i n e t i c energy. I n a d d i t i o n s e v e r a l g e n e r a l theorems r e g a r d i n g phase and propagation energy and the v e r t i c a l s t r u c t u r e o f t h e s e waves i n a system w i t h a r b i t r a r y continuous s t r a t i f i c a t i o n are p r o v e n . In p a r t i c u l a r i t is shown t h a t both phase and energy propagate n o r t h w a r d , and t h a t i n a s t a b l y s t r a t i f i e d system t h e wave amplitude increases with depth. iii TABLE OF CONTENTS n Page Abstract i i L i s t o f TablES iv L i s t of Figures v Acknowledgements vi Chapter 1. Introduction 1 2. P h y s i c a l oceanography o f t h e S t r a i t o f G e o r g i a k 3. The model k. Lou-frequency 5. 6. non-topographic waves i n a tuio-layer f l u i d . Zk k.l The g o v e r n i n g e q u a t i o n s Zk k.Z I n t e r n a l K e l v i n waves 27 k.3 Rossby waves 31 Topographic waves i n a t w o - l a y e r f l u i d 35 5.1 S o l u t i o n of the governing equations 35 5.2 A p p l i c a t i o n to the S t r a i t of Georgia kZ 5.3 U o r t i c i t y dynamics ^6 Topographic waves i n a f l u i d w i t h c o n t i n u o u s l y v a r y i n g density L 5.1 The p r e s s u r e e q u a t i o n k3 6.2 P a r t i a l s o l u t i o n of the pressure equation f o r a r b i t r a r y N 6.3 7. 19 53 2 Constant N & 2 model Summary 58 • 65 Bibliography 67 Appendix A. The t w o - l a y e r e q u a t i o n s o f motion 69 Appendix C. y fo theorems f symbols B. PG rl o osfssa r o 79 72 iv LIST DF TABLES Table I. II. III. IV. V. Page Mean c u r r e n t s 11 Wavelength, f r e q u e n c y , and p e r i o d o f i n t e r n a l K e l v i n waves i n t h e S t r a i t o f Georgia 31 Wavelength, frequency, and p e r i o d o f t h e f i r s t c r o s s channel b a r t o t r o p i c and b a r o c l i n i c Rossby uaves i n t h e S t r a i t o f Georgia . . . . . . . . . . . . 34 Wavelength, p e r i o d , and phase speed o f t h e f i r s t c r o s s channel mode i n t h e S t r a i t o f Georgia ( t w o - l a y e r model) . . . kk Wavelength, p e r i o d , and phase speed o f t h e f i r s t c r o s s channel mode i n t h e S t r a i t o f Georgia (constant N2 model) . . 63 V LIST OF FIGURES Figure Page 1. A plan view of the S t r a i t of Georgia 5 2. Topographic cross sections 6 3. Density p r o f i l e s at s t a t i o n s 1-k 8 k. Brunt-Vaisala frequency IM f o r the p r o f i l e s shown i n F i g . 3 . • 9 5. Cross section H showing placement of current meters 10 S. Current spectrum S ( f ) of 50-m currents at H26 12 7. Smoothed current spectra, f S ( f ) , f o r low-frequency oscillations 13 Coherence"and"phase spectra between v e r t i c a l l y currents IS B. 9. Coherence'and"phase'spectra between h o r i z o n t a l l y .... separated separated currents 17 10. Model of the S t r a i t of Georgia 20 11. Smoothed topographic cross sections 21 12. 13. F i t of model to section K Dispersion r e l a t i o n f o r the f i r s t three cross-channel modes (twD-layer model) 23 k3 lk. V e r t i c a l d i s t r i b u t i o n of h o r i z o n t a l v e l o c i t y f o r the f i r s t three cross-channel modes (two-layer model) ^5 Dispersion r e l a t i o n f o r the f i r s t three cross-channel modes (constant IM model) 62 V e r t i c a l d i s t r i b u t i o n of h o r i z o n t a l v e l o c i t y f o r the f i r s t three cross-channel modes (constant IM model) 64 15. 2 16. 2 vi ACKNOWLEDGEMENTS I t i s my s i n c e r e s t p l e a s u r e t o express my g r a t i t u d e t o Dr. L.A. Mysak i n h i s c a p a c i t y as t h e s i s s u p e r v i s o r f o r h i s p a t i e n c e , and unflagging enthusiasm d u r i n g advice, t h e course o f t h i s work. a l s o due t o Dr. R.E. Thompson f o r h i s remarks concerning good humour, Thanks a r e H e l b i g and Mysak (1976) and t o Drs. P.H. LeBlond and R.kl. B u r l i n g f o r t h e i r comments concerning t h i s t h e s i s . thesis p r i o r to i t s I thank Mr. P. Chang f o r a l l o w i n g me tc- use h i s completion. I would a l s o l i k e t o express my a p p r e c i a t i o n t o t h e U n i v e r s i t y o f B r i t i s h Columbia and N a t i o n a l Research C o u n c i l o f Canada who p r o v i d e d with f i n a n c i a l support during t h e tenure o f t h i s study. me 1. Introduction Chang (1976) (see also Chang, Pond, and Tabata, 1976) has r e c e n t l y shown that up t o kS% of the h o r i z o n t a l k i n e t i c energy i n the S t r a i t of Georgia (see F i g . 1) c o n s i s t s of broad-band low-frequency motions. This r e s u l t was i n f e r r e d from the a n a l y s i s of IS months of continuous current records obtained by Tabata and S t r i c k l a n d (1972, a,b,c) (see also Tabata, S t r i c k l a n d and de Lange Boom, 1971) during the period extending from A p r i l 1969 u n t i l September 1970. These f l u c t u a t i n g currents are not d i r e c t l y t i d a l i n nature and possess periods exceeding four days. The purpose of t h i s t h e s i s i s to present a simple dynamical model which may provide a p a r t i a l explanation of these o s c i l l a t i o n s . The plan of t h i s t h e s i s i s as f a l l o w s . A b r i e f review of hydro- graphic data from the S t r a i t of Georgia (GS) i s presented i n Chapter 2. These data suggest that GS may be modelled by an i n f i n i t e l y long, s t r a i g h t channel with v e r t i c a l walls and a sloping bottom shoaling t o the east (see Chapter 3 ) . These data also i n d i c a t e that a two-layer system i s a s u i t a b l e representation of the density s t r a t i f i c a t i o n , although t h i s choice i s not absolutely compelling. P r i o r to an examination of the possible motions admitted by t h i s model, however, i n t e r n a l Kelvin waves and Rossby waves i n a flat-bottomed channel are considered i n Chapter k to see i f they might provide an explanation f o r the observed motions. They do not; I n t e r n a l Kelvin waves of the proper frequency have wavelengths that are much too long t o f i t i n GS, and Rossby waves have periods that are f a r greater than those observed. In Chapter 5 i t i s shown that the model admits northward-propagating topographic planetary waves as s o l u t i o n s . An a p p l i c a t i o n of the 2 t h e o r e t i c a l r e s u l t s to GS i s made and reveals that these waves have f r e quencies l y i n g i n the observed range. However, the predicted v e r t i c a l d i s t r i b u t i o n of h o r i z o n t a l k i n e t i c energy does not compare favorably with that observed. In Chapter G a model with continuously varying density i s considered, and several general theorems concerning the s t r u c t u r e and propagation of waves admitted by t h i s model are proven. A p p l i c a t i o n of these r e s u l t s t o GS f o r the s p e c i a l case of constant Brunt-Vaisala frequency y i e l d s s i m i l a r r e s u l t s t o those mentioned above. contains three appendices. This t h e s i s The two-layer equations used i n Chapter U and Chapter 5 are developed i n Appendix A; the proofs of the theorems presented i n Chapter G comprise, Appendix B. For the reader's convenience, a glossary of a l l symbols used i n t h i s t h e s i s i s given i n Appendix C. As a study of wave-like motion i n a system with both topography and s t r a t i f i c a t i o n , t h i s t h e s i s f a l l s into the mainstream of a c u r r e n t l y a c t i v e branch of research i n p h y s i c a l oceanography. The major d i f f i c u l t y confronting the i n v e s t i g a t o r i n t h i s f i e l d i s h i s i n a b i l i t y to separate the v e r t i c a l dependence from the h o r i z o n t a l i n the equations of motion. Hence, although there have been numerous studies made of s t r a t i f i e d systems with no topographic v a r i a t i o n s and of homogeneous systems with v a r i a b l e topography, these e x i s t at present only a handful of studies made of systems embodying both f e a t u r e s . Yet i n the r e a l world, they always e x i s t together, although admittedly i n some cases, one or the other may be neglected to a high degree of approximation. The presence of both features i s e s p e c i a l l y apparent i n c o a s t a l regions where the a p p l i c a t i o n of p h y s i c a l oceanography to p r a c t i c a l problems ( f o r example, environmental) i s of increasing importance. with these thoughts i n mind, the s i g n i f i c a n c e of t h i s f i e l d of research i s p l a i n and should 3 not be The underestimated. study D f s t r a t i f i e d systems w i t h v a r i a b l e topography began w i t h R h i n e s (197D) who d i s c o v e r e d t h e e x i s t e n c e o f a wave t r a p p e d in a continuously s t r a t i f i e d fluid t o t h e bottom ( c o n s t a n t !M) w i t h l i n e a r t o p o g r a p h y . Some o f Rhines r e s u l t s a r e reproduced i n Chapter 6.3 a l t h o u g h t h e p r e s e n t a n a l y s i s d i f f e r s i n some r e s p e c t s from h i s . McLJilliams (1974) have c o n s i d e r e d v a r i e t y of simple topographies Suarez (1971) and both f r e e and f o r c e d motions o v e r a i n a constant l\l model. K a j i u r a (1974) and Yoon (1974) have s t u d i e d f r e e waves i n two-layer, systems w i t h a s t e p l i k e and a l i n e a r c o n t i n e n t a l s h e l f , r e s p e c t i v e l y . the p o s s i b l e i n t e r a c t i o n o f s h e l f - w a v e s Both demonstrated w i t h i n t e r n a l K e l v i n waves, and Yoon showed t h i s i n t e r a c t i o n t o be i m p o r t a n t r a d i u s i s e q u a l t o o r exceeds t h e breadth when t h e i n t e r n a l deformation D f the continental s h e l f . A l l a n (1975), u t i l i z i n g a v e r y complex p e r t u r b a t i o n scheme, has s t u d i e d t h i s i n t e r a c t i o n i n a t w o - l a y e r system w i t h e x p o n e n t i a l topography. F i n a l l y , wang and Mooers (1976) have s o l v e d n u m e r i c a l l y t h e e q u a t i o n s o f motion i n a system w i t h a . c o n t i n u o u s l y exponential continental shelf. v a r y i n g d e n s i t y and a f i n i t e They c l e a r l y i d e n t i f i e d t h e r e l a t i v e r o l e s o f t h e two t r a p p i n g mechanisms i n v o l v e d , the s l o p i n g bottom ( f o r topographic p l a n e t a r y waves) and t h e v e r t i c a l s i d e w a l l ( f o r i n t e r n a l K e l v i n waves). 2 * P h y s i c a l Oceanography of the S t r a i t pf Georgia Although the p h y s i c a l oceanography of the S t r a i t of Georgia (GS) has received comprehensive treatment elsewhere ( c f . , Idaldichuck 1957), i t i s important t o summarize here some of the p r i n c i p a l features of bathymetry and s t r a t i f i c a t i o n i n order to motivate a s i m p l i f i e d yet r e a l i s t i c model of GS. In addition we examine the current spectra c a l c u l a t e d by Chang (1976) from measurements made by Tabata and S t r i c k l a n d (1972) and also b r i e f l y consider r e s u l t s of h i s a n a l y s i s of the winds, sea l e v e l , , atmospheric pressure, and water temperature. A plan view of GS i s shown i n F i g . 1. I t reveals that the average width of GS i s about 30 km and i t s length i s s l i g h t l y l e s s than 250 km. Thus, the aspect r a t i o of channel length to width i s approximately 8:1. The S t r a i t of Georgia i s connected to the P a c i f i c Ocean at both ends through c o n s t r i c t e d t i d a l passages i n which strong mixing occurs. Bathymetric cross sections along the l i n e s A - J are presented i n F i g . 2 and. were extracted from a topographic map of GS compiled by Dr. P. Crean (personal communication) g i v i n g average depths over 2-km squares throughout the S t r a i t . Hence, even though small-scale features are i m p l i c i t l y smoothed, the bathymetry e x h i b i t s great i r r e g u l a r i t y , p a r t i c u l a r l y i n the northern s e c t o r . In general, extremely steep slopes characterize GS along i t s western boundary, while slopes nearly as steep (exceeding 10~2) are common along the east. North of l i n e D two channels e x i s t , a narrow one t o the east of Texada Island and a much wider one on the western s i d e . progressively smoother; South of l i n e D the topography becomes l i n e s G and H i l l u s t r a t e the marked e f f e c t of Fraser River sedimentation as extensive banks along the east. The F i g . 1. A p l a n view o f t h e S t r a i t o f G e o r g i a , B.C., showing l i n e s o f t o p o g r a p h i c c r o s s s e c t i o n s (A-K) presented i n F i g . 2 and 5, s t a t i o n s (1-4) at which v e r t i c a l d e n s i t y p r o f i l e s are given i n F i g . 3, and the l o c a t i o n s o f the c u r r e n t meter moorings (HOG, H16, H2G). 6 DISTANCE 50 FROM EASTERN 40 BOUNDARY 30 (Km) 30 10 h-JOO FROM DISTANCE 200 I H Boundary Passaqe I . . G Fraser . R. . . 1S0 F I NORTHERN . . E . . (Km) BOUNDARY 100 O I . . . C Taxada . so IB . . I. ( *A • — i Cap« Mudqe VM J 0 •loo E \ * i\ •200 p Q. UJ 300 Q •400 F i g . 2. Topographic cross s e c t i o n s : (a) Upper panels: A - I ; (b) Lower panel: 0. The v e r t i c a l exaggeration i s 30:1 i n (a) and 150:1 i n ( b ) . 7 l o n g i t u d i n a l cross section 3 reveals that although the a x i a l bathymetry i s smoother than the transverse bathymetry, i t s t i l l possesses a high degree of i r r e g u l a r i t y and e x h i b i t s slopes that often exceed ID . - 2 Seasonal v e r t i c a l density p r o f i l e s f o r s t a t i o n s 1 to h, based on data c o l l e c t e d by Crean and Ages (1971), are p l a t t e d i n F i g . 3. R e l a t i v e l y l i t t l e change occurs throughout the year below 50 m, but strong seasonal e f f e c t s , e s p e c i a l l y near the Fraser River, e x i s t at shallower depths. Corresponding to these p r o f i l e s , the Brunt-Vaisala (or s t a b i l i t y ) frequency l\l was c a l c u l a t e d and i s shown i n F i g . k. hie see that IM f l u c t u a t e s s i g n i f i c a n t l y , depends strongly on depth, and -3 generally l i e s w i t h i n the range of 3 x ID -2 to 3 x 10 -1 rad s ; the water column i s thus w e l l s t r a t i f i e d . Some r e s u l t s of Chang's a n a l y s i s of GS currents are presented i n Table 1 and i n F i g s . 6 and 7. The current records examined by Chang were c o l l e c t e d at s t a t i o n s H06, H16, and H26 as shown i n F i g s . 1 and 5. Meters were positioned at 5 (approximately), 50, and 200 m i n the western (HOG) and c e n t r a l (H1G) i n the east (H26). instruments. l o c a t i o n s and at 5 (approximately), 50 and IkO Chang d i d not analyze records from the near surface Most of the current records were obtained with Aanderaa Model k current meters, but several Geodyne Model 850 meters were employed. The currents were sampled e i t h e r every 10 (Aanderaa) or 15 (Geodyne) minutes. A subsurface buoy mooring was used f o r the i n i t i a l year of the experiment, but was replaced t h e r e a f t e r by a surface buoy, taut-rope mooring. The threshold l e v e l of these meters i s 1.5 cm s but t h i s presented no problem i n the detection of small low-frequency currents since much stronger t i d a l currents were superposed on these fluctuations. m F i g . 3. D e n s i t y p r o f i l e s at s t a t i o n s 1-4. F i g . k. B r u n t - V a i s a l a frequency IM f o r the p r o f i l e s shown i n F i g . 3. The B r u n t - V a i s a l a frequency i s d e f i n e d by IM =- ( g / f , » ) /"bl, where p„ i s the mean d e n s i t y o f the water column (taken as f = 1 . 0 2 3 5 ) . 2 0 U3 400 F i g . 5. Cross section K showing placement of current meters. The moorings are spaced 10 km apart. The deep meters are (from west to east) 50, BO, and 25 m from the bottom (Tabata et a l . 1971). • Table I . Mean currents. The current d i r e c t i o n s are i n degrees measured from true North WEST CENTRAL EAST 50 m 200 m 50 m 200 m 50 m 140 mean v e l o c i t y (cm s" ) 3.0 ± .5 1.6 ± .8 4.3 i .5 1.4 + .5 3.0 + 1.1 7.4 + mean v e l o c i t y d i r e c t i o n 355° 346° 10S 21° 9° 7° rms speed (cm s 7.7 3.2 8.2 3.6 7.0 8.4 1 ) D PERIOD 0.1 I 1.0 • r.l 10 t »l (days) 1000 100 1000 » »il i i il L 100 li I i LLL 10 I'ti 1.0 li52.1i 66.1 20-r eastern 50 m CM 'o "10 + E u I -3 -4 +F '°9io F F i g . 6. Current spectrum S ( f ) c f 50-m currents at H26. The frequency f i s scaled with respect to 1 cpd. With f S ( f ) platted against log^gf, the area under the curve i s proportional to the t o t a l variance of u + v (Chang 1976). 2 2 0.1 i L F i g . 7. Smoothed current spectra, f S ( f ) , f o r lou-frequency oscillations. The frequency f i s scaled with respect to 1 cpd (Chang 1976). A, 5D m; B, 140 or 200 m. 14 In.obtaining data relevant to Table I, Chang passed the o r i g i n a l data set containing a.ID or 15 minute time step through a low-pass f i l t e r thus obtaining a d a i l y time s e r i e s of currents. The mean and rms values were c a l c u l a t e d by averaging current vectors and magnitudes from t h i s time series. Hence, the rms values correspond to the k i n e t i c energy contained i n periods l e s s than two days and contain no t i d a l c o n t r i b u t i o n s . There are two s i g n i f i c a n t features i n Table I . The f i r s t i s the strong, anomalous cross-channel flow at the 50 m c e n t r a l l o c a t i o n , and the second i s the very strong and almost r e c t i l i n e a r current found at 140 m i n the east. The mean speed there i s 5 times greater than that found at the other deep l o c a t i o n s , while the rms v e l o c i t y i s twice as large. In the east and the west both the shallow and deep currents are c l o s e l y aligned with the l o c a l topography. Rotary spectra of the low-frequency currents are shown i n F i g s . 6 and 7. These spectra were obtained by s p l i t t i n g each frequency component of the v e l o c i t y vector u i n t o two other vectors, one of which rotates with a p o s i t i v e frequency frequency (clockwise). (anti-clockwise) and the other with a negative Hence the p o s i t i v e or negative spectrum repre- sents the respective tendency f o r the currents to move i n an a n t i clockwise or clockwise sense. The reader i s r e f e r r e d to Chang (1976) or Moores (1973) f o r a more comprehensive discussion of rotary current spectra. F i g . 6 shows the spectrum of the 50 m currents at H26 as computed by Chang. In order t o reduce the complexity of t h i s and s i m i l a r spectra and to concentrate on low frequencies, Chang's spectra were f u r t h e r smoothed t o examine frequencies l e s s than 0.25 cpd; are shown i n F i g . 7. these Examination of F i g s . 6 and 7 reveals the extremely complex nature of the low-frequency currents i n GS. The 15 g e n e r a l c h a r a c t e r i s t i c s o f t h e s e s p e c t r a i s t h e i r bread handedness, as a p p r e c i a b l e energy i s c o n t a i n e d i n p e r i o d s r a n g i n g from k t o over 100 I t must be emphasized t h a t most o f the f i n e - s t r u c t u r e p r e s e n t i n F i g . 7 i s p r o b a b l y not s t a t i s t i c a l l y s i g n i f i c a n t t o 95%. Df more d e t a i l e d c o n c l u s i o n s may (1) days. N e v e r t h e l e s s , a number be drawn: At the same depth, the shapes of s p e c t r a from the e a s t e r n and western s t a t i o n s are i n b e t t e r agreement w i t h each o t h e r than they are w i t h those from the c e n t r a l (2) location. S p e c t r a from the c e n t r a l s t a t i o n peak at lower f r e q u e n c i e s than do e i t h e r the western o r e a s t e r n c u r r e n t (3) spectra. A b a r o t r o p i c motion of 10 - 25 day p e r i o d i s c l e a r l y i n the e a s t e r n s p e c t r a , w h i l e motion i n t h i s range appears elsewhere, w i t h most of the energy i n the upper (4) visible baroclinic layer. A b a r o c l i n i c o s c i l l a t i o n of 70 - 125 day p e r i o d and one o f near y e a r l y p e r i o d i s apparent a t a l l s t a t i o n s . i s i n the upper l a y e r . annual changes Again most of the energy The l a t t e r motion, however, may be r e l a t e d t o i n s t r a t i f i c a t i o n i n GS. Chang (1976) found t h a t coherences between c u r r e n t s at p o s i t i o n s separated both h o r i z o n t a l l y and v e r t i c a l l y were g e n e r a l l y s m a l l a t low f r e q u e n c i e s as i s shown i n F i g s . 8 and 9. v e r t i c a l l y s e p a r a t e d c u r r e n t s was The h i g h e s t coherence between observed i n the e a s t . There the upper and lower l a y e r v e l o c i t i e s were n e a r l y i n phase which i s i n d i c a t i v e o f a b a r o t r o p i c motion. At the o t h e r l o c a t i o n s the v e r t i c a l coherence v e r y s m a l l and the phases were s c a t t e r e d ; was t h i s suggests l i t t l e o r no c o u p l i n g between the upper and lower l a y e r s and hence i m p l i e s mainly b a r o c l i n i c motions t h e r e . In a l l cases the h o r i z o n t a l coherences were b u r i e d beneath the 95% n o i s e l e v e l . 16 0.97-, 0.95 -J 0.9-| 0.8 0.1 0.5 . 0.3 - O.H 1 0 (A) EASTERN -1 -2 • «• 1 1 1— -1.5 -I -0.5 FREQUENCY (CPO) 1B0900-BO-180 - • • • • •.• — i »'i 1 — ' . •~- -i L 95% —1 0.5 1 "-1 T 2 • • • 2 —i— o.s o-l -I UO-1 90 0 -90 -180 -> 4> . _i • . » • * —1 -1.5 •••• • • • 1 (C) WESTERN T- 1 15 FREQUENCY (CPO) 4> 0.95 0.90.6 0.7 0.5 0.33 Old (B) CENTRAL 95% 0.97-, 0.95 0.9-1 0.1-, , 0.7J-Y-2 0.5 0.33 . . • . ,i»—o.H • ^ o- 180 90 . 0 -so-3 -180 4>... —p— 15 0.5 t •• • F i g . 8. Coherence and phase spectra between v e r t i c a l l y separated currents. A l l have 48 degrees of freedom. Parts (A,B,C) are based Dn 36D day time-series s t a r t i n g ( A p r i l 17, June 17, J u l y 11) 1959. (Chang, 1976) 17 (D) CENTRAL — EASTERN 50m . I—• -2.5 1 -2 0.97»•»• 0.B0.80.70.5: 0.3: 0 ' 0' .• 1 —I—• -1.5 -I FREQUENCY (CPB) — i 1— -0.5 T95% 93% I 0 1 0.5 180 -j "—i—* * ' i— 1 IS FREOOENCr (CPO) r2 0 -ao -180 (E) CENTRAL-WESTERN 50m —I -2.5 — r * - 0.97 0.9 08 0.7 03 0-> # «• • • —I— 1 -O.S I -1.5 • •. • ' —I— 0.5 —I —I— O.S —I 2.5 0 180 90 0 -90-1 -180 J 0.97 0.95-4 0.8-1 o.e —r— -2 I T 0.33 r O.l^ -! 0 . (F| CENTRAL—WESTERN ZOOm J 0.7 0.3d . 0.33 — O.t T 0 180-n • * 90 -j 0 -«H -180 F i g . 9. Coherence and phase spectra between h o r i z o n t a l l y separated currents. Parts (D, E,F) are based on (50m, 50m, 200m) currents" from the ( c e n t r a l - e a s t e r n , central-western, central-western) mooring. Spectra i n Parts (D,E,F) have (48, 64, 50) degrees of freedom using (360, 432, 333) day time-series s t a r t i n g ( A p r i l 17, J u l y 11, July.11) 1969. (Chang, 1976). J.S 18 Chang also analyzed sea l e v e l , atmospheric pressure, uind, and water temperature records f o r the 18 month period. The temperatures were c o l - l e c t e d by the Aanderaa meters which were equipped to sample currents and temperatures concurrently. In a l l cases these q u a n t i t i e s were e s s e n t i a l l y uncorrelated with the currents. The highest coherence was found between the currents and the wind at the eastern l o c a t i o n , which suggests that the surface wind s t r e s s may be a p a s s i b l e f o r c i n g mechanism. c l e a r i f any other f o r c i n g mechanisms are important. I t i s not 19 3. The Model •UT o b j e c t i v e i n choosing a model of GS was to s e l e c t one that f a i t h - f u l l y represented the gross features of geometry, bathymetry, and stratifi- c a t i o n but at the same time was simple enough to admit an a n a l y t i c s o l u t i o n of the dynamical equations. To t h i s end, as an i n i t i a l approach, an open- ended, rectangular channel with v e r t i c a l s i d e w a l l s and a bottom..inclined upwards to the east u i t h a constant slope <* was chosen as i s shown i n F i g . 1D; a two-layer system was adapted f o r the s t r a t i f i c a t i o n . The s e l e c t i o n of a rectangular basin seems reasonable, and while the large aspect r a t i o of S : 1 allows one to neglect the end boundaries as a f i r s t approximation, choice of a s u i t a b l e bottom topography was more d i f f i c u l t . an overlay of cross sections smoothed from F i g . 2. F i g . 11 shows To obtain these, many small s c a l e i r r e g u l a r i t i e s were f i l t e r e d out of the o r i g i n a l p r o f i l e s , but, i n a d d i t i o n , the eastern boundary i n the northern sector ( l i n e s A-D) and i n the southern sector ( l i n e s G and H) was s h i f t e d westward to exclude, r e s p e c t i v e l y , the secondary eastern channel (Malaspina S t r a i t ) and the extensive a l l u v i a l banks contained i n the o r i g i n a l p r o f i l e s . (This was done despite the f a c t that the narrow northern channel f i t s the proposed graphy better than the western channel.) topo- The channel depth varies s i g n i f i - cantly along GS and i s deepest i n the c e n t r a l region ( l i n e s E and F ) . i s the area of current measurements and i s best f i t t e d by the model. This In f a c t , a strong argument may be made that the model i s a p p l i c a b l e only to the sector south of Texada Island, since a northward propagating topographic wave would he e x t e n s i v e l y scattered by bottom i r r e g u l a r i t i e s near Texada Island. The s e l e c t i o n of a model of the s t r a t i f i c a t i o n i s d i f f i c u l t due to the strong depth dependence of the Brunt-VaisSlS frequency. A two-layer model F i g . 10. Model of the S t r a i t of Georgia F i g . 11. Smoothed topographic cross sections PG 22 was chosen as an i n i t i a l approximation because of i t s s i m p l i c i t y permit both barotropic and b a r o c l i n i c motions. and to In Chapter 6 continuously s t r a t i f i e d systems are considered thus providing a p a r t i a l evaluation of the importance of the choice of s t r a t i f i c a t i o n ; A f i t of the two-layer model to the bathymetry and s t r a t i f i c a t i o n i l l u s t r a t e d i n F i g s . 2 and 3 y i e l d s the f o l l o w i n g average parameter values: L = 22.2 km h 1 = 50 m h 2 Q = 300 m <* = k.9 ^ = 1.0215 gm = 1.02375 gm 2 x 10" 3 cm" 3 cm" 3 I f the f i t i s made, however, to l i n e K and s t a t i o n 3, ( F i g . 12) i . e . the l o c a l i t y of current measurements, the s t r a t i f i c a t i o n parameters h^, j ^ , and 5 j° are unchanged, but the topographic parameters become: 2 L h 2 Q (X. = 22 km = 325 m = 9.3 x 10" 3 Thus i n t h i s region, the channel i s s i g n i f i c a n t l y deeper and the slope much steeper than found on average i n GS. F i g . 12. F i t of model to section K. ro 2k k. Low frequency non-topographic waves i n a two-layer f l u i d In t h i s chapter two c l a s s e s c f wave mptipn of p o t e n t i a l impprtance i n the S t r a i t of Georgia are considered, namely, i n t e r n a l K e l v i n waves and Rossby waves. I t i s shown that i n t e r n a l K e l v i n waves e x h i b i t frequencies too high and Rossby waves too low to provide an adequate explanation of the observed low-frequency current s t r u c t u r e . As an i n i t i a l approximation, topographic e f f e c t s are ignored and free waves are studied i n a two-layer f l u i d contained i n a flat-bottomed channel. A f i r s t order c o r r e c t i o n to the . K e l v i n wave frequency a r i s i n g from topography i s c a l c u l a t e d from a formula due to Le LeBlond (1975). However, t h i s c o r r e c t i o n does not s i g n i f i c a n t l y a l t e r the d i s p e r s i o n r e l a t i o n . The plan of t h i s chapter i s to f i r s t derive ( i n section k.2) a set of coupled equations f o r the free surface and i n t e r f a c i a l displacements from the two-layer equations of motion i n c l u d i n g non-uniform r o t a t i o n (which are developed i n Appendix A). In s e c t i o n k.2 K e l v i n wave s o l u t i o n s are obtained on an f plane and dispersion curves are c a l c u l a t e d f o r parameters appropriate to GS. A scale a n a l y s i s i s employed i n section k.3 to give a s i m p l i f i e d set Df equations which r e a d i l y y i e l d Rossby wave s o l u t i o n s . r e l a t i o n i s evaluated f a r GS k.l Again the d i s p e r s i o n parameters. The Governing equations Although the equations of motion f o r a two-layer system are well-known (see e.g. veron'is and Stommel, 1956), they are derived i n Appendix A f o r the sake of completeness. The l i n e a r i z e d v e r s i o n of these equations f o r the h o r i z o n t a l v e l o c i t i e s (u^, v^) and f o r the surface and i n t e r f a c i a l d i s p l a c e ments,^ and^,, r e s p e c t i v e l y , are given by: 25 Upper L a y e r (4.1.1) (4.1.2) Lower L a y e r (4.1.3) (4.1.4) The parameter £ = 'jty/fa T e P ^ T e s e n s ^ h E r e l a t i v e density difference between t h e upper and lower l a y e r s , g i s t h e g r a v i t a t i o n a l a c c e l e r a t i o n , and f = f o + |?» y i s t h e C o r i o l i s parameter. The mean upper l a y e r and .lower l a y e r d e p t h s , h^ and h^g, r e s p e c t i v e l y are taken as c o n s t a n t . the d e r i v a t i o n of these equations, the h y d r o s t a t i c approximation In was i n v o k e d s i n c e o n l y low-frequency motions a r e o f i n t e r e s t . In the development t h a t f o l l o w s i t i s c o n v e n i e n t t o r e c a s t e q u a t i o n s i n t o n o n - d i m e n s i o n a l form. s c a l i n g i s used: these To do so the f o l l o w i n g g e o s t r o p h i c ( u ^ , v^) = U ( u / , v^ ), t ='fo~ t , ( x , y) = L ( x ' y ), = ( f o L U / g ) ^ , and<^ = (foLU/<3 g ) ^ , where a prime denotes a non1 2 d i m e n s i o n a l q u a n t i t y , U i s a h o r i z o n t a l s c a l e v e l o c i t y , and L i s t h e channel width. The n o n - d i m e n s i o n a l e q u a t i o n s a r e (where we have dropped the p r i m e s ) , . Upper L a y e r (4.1.5) C^,.- s^ t (4.1.6) 26 Lower Layer (4.1.7) A (4.1.8) Here the non-dimensional parameter e = L fo./ g h 2 width L to the lower l a y e r i n t e r n a l deformation 2 Q compares the channel radius ( o g h ^ ) / f o , 6 = f y h ^ i s the r a t i o of l a y e r depths, and f = 1 + |S y i s the nondimensional C o r i o l i s parameter w i t h ^ = L coffe/Re, where <h i s the mean l a t t i t u d e and Re i s the radius of the earth. The boundary conditions appropriate to t h i s system require that the v e l o c i t y components normal to the s i d e w a l l s vanish, i . e . (4.1.9) U. {. K * 0 t * U-i C X = 0 •=- o The free surface, i n t e r f a c i a l , and bottom boundary conditions were taken account of i n the d e r i v a t i o n of (4.1.5) - (4.1.8), (see Appendix A). In order to s i m p l i f y the set (4.1.5) - (4.1.8), we eliminate the v e l o c i t i e s u^ and v^ to obtain a p a i r of coupled equations f o r ^ andt^ Let (4.1.10) and the operator (4.1.11) then (4.1.5) and (4.1.7) give the f a l l o w i n g expressions f o r u^ and (4.1.12) (4.1.13) 27 O p e r a t i o n on t h e c o n t i n u i t y (4.1.6) and (4.1.8) by equations and a p p l i c a t i o n of the i d e n t i t y - - A?fv; (4.1.14) gives f[ otl Substitution the f o l l o w i n g f ^ i s v Uv^ U«0* + 2 2 = ^$^(4.1.16) + S S)J T jjTn\($* 4. Laplacian o (4.i.ia) and t h e o p e r a t o r by - al^t - (4.1.19) must be s o l v e d in sections. In terms of $ 4, and t h e boundary c o n d i t i o n s - 4.2 (4.1.15) t (4.1.14) and (4.1.15) y i e l d s i s t h i s s e t o f two c o u p l e d g o v e r n i n g e q u a t i o n s t h a t the f o l l o w i n g $ - £ oC"( i s the two-dimensional ~ It a equations, - €f + dy e / 6 a C v,= - p a i r of coupled C$ defined 1 - al|S o f (4.1.12) and (4.1.13) i n t o sCL V 2 Here"v 1JW* Internal Kelvin In t h i s s e c t i o n , o (4.1.9) become at x=o,l (4.1.20) Waves i n t e r n a l Helvin wave . s o l u t i o n s t o (4.1.17) and A (4.1.18) a r e b r i e f l y c o n s i d e r e d f o r t h e l i m i t i n g c a s e o f ^ = 0. calculated dispersion r e l a t i o n , estimates of the frequency f o r K e l v i n waves l i k e l y t o be f o u n d i n GS. are From t h e determined 28 A A TD proceed then, withjS = 0 and f = 1, (4.1.17) and (4.1.18) reduce to f [ V«$ + * ^ * £*C5 X I O (4.2.1) . *-&?)] t = ° (4.2.2) For t r a v e l l i n g wave s o l u t i o n s of the form r_4 J] = )L £ [ F U ) , G U ) ] l U V'° (^.2.3) and u i t h k chosen p o s i t i v e f o r convenience, (4.2.1) and (4.2.2) s i m p l i f y to F _ k F *• fc/ & U - <r*) fe = O L <jf (F* fc)* - W C X K F V (4.2.4) (r) - fcU -o- ) ( (y * 6 F ) =• O (4.2.5) 1 To obtain K e l v i n wave s o l u t i o n s , i t i s required that u^ must vanish i d e n t i c a l l y f o r a l l x and y. I t follows from (4.1.20) (with f = 1) and (4.2.3) that t h i s implies F* - W<r F - - o Via- Cr ^ (4.2.6) ° (4.2.7) /Since these equations are v a l i d f o r a l l x, we have U.(yW Cft.S) e * W<r (4.2.8) where A and B are a r b i t r a r y constants to be l a t e r determined. S u b s t i t u t i o n of (4.2.8) i n t o (4.2.4) and (4.2.5) shows that f o r n o n - t r i v i a l A and B, k and 6" must s a t i s f y (W/o-V - 6 O < 0 / A (k/<rT + f l * -O (4.2.9) the s o l u t i o n o f which y i e l d s the dispersion r e l a t i o n -! Cr = i i l< [ Here . V) = tl^)U + ( l - 46i^/Cl+6yV' ]|j r C^/rr t i s the square of the r a t i o of the channel width L to the i n t e r n a l (4.2.ID) (4.2.11) 29 deformation radius r. defined by r { £ S^Vv.W. /H ] lS v (4.2.12) where H = h^ + h ^ i s the t o t a l mean channel depth. To 0(6) the two roots of (4.2.10) are (4.2.13) The f i r s t represents an i n t e r n a l K e l v i n wave and the second a barotropic (or external) K e l v i n wave which i s henceforth ignored, since <r Rr V> CT^. In dimensional form the frequencytO of the i n t e r n a l mode i s given by to -iJfiWj ^±&TT$ r f / ^ (4.2.14) A and i s thus determined by the r a t i o of the i n t e r n a l deformation radius to the wavelength X j , (the s u b s c r i p t d denotes a dimensional q u a n t i t y ) . Since k i s p o s i t i v e , the general s o l u t i o n to (4.2.1) - (4.2.2) c o n s i s t s of a superposition of two "right-trapped" waves, one t r a v e l l i n g northward (c">°) with i t s amplitude maximal along the eastern shore and one propagating southward (<T<o ) with i t s amplitude greatest along the western boundary. The r a t i o of the h o r i z o n t a l v e l o c i t y i n the upper layer to that i n the lower l a y e r i s e a s i l y obtained from (4.1.13), (4.2.3), and (4.2.8). Defining Ry = M ^/\! we have, 9 ^ H " 3 - b. (4.2.15) +- 8 E i t h e r (4.2.4) o r (4.2.5) may be used t o r e l a t e the c o e f f i c i e n t s A and B; to •(&) "B - - L b k H k l / « r O K * U+Mft (4.2.16) 30 S u b s t i t u t i o n of (5.2.IS) i n t o (4.2.15) y i e l d s ^ = >/, hi » - 1 /(* = - Wo/W, (4.2.17) Thus the current magnitude i n a given l a y e r i s i n v e r s e l y p r o p o r t i o n a l to the l a y e r depth. Notice that the two currents are ISO out of phase and 0 that Ry i s independent of the wavelength. S i m i l a r l y , one f i n d s that the 2 r a t i o of h o r i z o n t a l k i n e t i c energy i s given by Ry ; hence i n a channel l i k e GS, with a r e l a t i v e l y t h i n • upper layer (A =0.17), the k i n e t i c energy density i s much greater i n the upper l a y e r . For the average values of the p h y s i c a l q u a n t i t i e s l i s t e d i n Chapter 3, the f o l l o w i n g parameter estimates f o r GS are obtained: £ = (^> - p / p * = 2.25 x 10" r. = 8.1 km & = 0.17 £ v = = a 3 l 1.1 (L/r.) 2 = 7.5. The frequency as calculated from (4.2.14) i s l i s t e d i n Table I I f o r selected values of the wavelength. we see that the only waves with periods comparable to those observed i n GS possess wavelengths that are f a r too long to be r e a l i s t i c a l l y important. For example., a 15 day wave has a wavelength of nearly 1200 km or roughly 5 times the length of GS. At t h i s point one should inquire as to how topography might modify a K e l v i n wave. Although, of course, due to the bottom boundary condition a "Kelvin wave" s t r i c t l y can e x i s t only i n a basin with no cross~channel topographic v a r i a t i o n s . Le Blond (1975) has studied t h i s problem and by employing a perturbation expansion s i m i l a r to that used i n Chapter 6 has derived c o r r e c t i o n formulae f o r Kelvin wave frequencies 31 i n a two-layer system with a r b i t r a r y topography. To f i r s t order i n the parameter L/ri , the c o r r e c t i o n f o r a l i n e a r slope i s 01 Table I I . Wavelength, frequency, and period of i n t e r n a l Helvin waves i n the S t r a i t of Georgia A(km) oo(1D' rad s" ) G 1 T(days) 1500 4.1 17.9 1250 4.9 14.9 1000 '6.1 11.9 750 8.2 8.9 500 12.2 5.9 250 24.5 3.0 100 61.1 1.2 For GS parameters i t follows that cT c0 - _ O •© U That i s , the frequency i s lowered by only one percent and thus the e f f e c t is negligible. 4.3 Rossby Waves The p o s s i b l e importance of Rossby waves i n the S t r a i t of Georgia i s now considered. I n t u i t i v e l y , one expects them to be of l i m i t e d importance since the dimensions of GS are small on a planetary s c a l e . Longuet-Higgins (1964) has shown that the frequency of Rossby waves i n a rectangular, homogeneous ocean i s d i r e c t l y p r o p o r t i o n a l to the l a t e r a l dimensions of the basin. A s i m i l a r r e s u l t f o r b a r o c l i n i c waves i s 32 a n t i c i p a t e d and we expect the periods of these waves i n GS to be very long. As i t stands the system (4.1.17), (4.1.18) and (4.1.2D) i s d i f f i c u l t to solve i n f u l l g e n e r a l i t y , and i t i s useful to perform a scale a n a l y s i s on these equations i n order to procure a s i m p l i f i e d s e t . Since only low-frequency motions are of i n t e r e s t we may employ the i n e q u a l i t y to obtain the f o l l o w i n g operator estimates: f 2> » a? ^ A 6~« f ^ (4.3.1) , 7>*it t where f = 0(1) and i t i s assumed that °x and ° y are also D ( l ) . (In f a c t "by should be l e s s than 2 x , but t h i s does not a f f e c t the argument.) I t follows from (4.3.1) that to subsequent QCf/f) (4.3.2) In t h i s approximation, (4.1.17) and (4.1.18) reduce to (4.3.3) the boundary c o n d i t i o n s (4.1.20) become , q: $ (4.3.5) For a s o l u t i o n of the form -c with i n t e g r a t i n g f a c t o r T= cr •fc) (4.3.6) the f a l l o w i n g set of coupled equations p/26~, are obtained C^H ^ ) .(.k+P) - 6(.(r + 5 P) - O (4.3.8) 33 with boundary conditions oJt (4.3.9) Appropriate s o l u t i o n s to (4.3.7) - (4.3.9) are f\ = • • (4.3.10) where An and Bn are a r b i t r a r y constants and n i s the cross-channel mode number. H A For a n o n - t r i v i a l s o l u t i o n , k and 6" must s a t i s f y - fc £/fi0 + Tl**' '**} 1 x - o (4.3.11) S o l u t i o n of (4.3.11) gives the d i s p e r s i o n r e l a t i o n corresponding to the barotropic and b a r o c l i n i c modes r e s p e c t i v e l y . v i s always p o s i t i v e , In Table I I I , <T 6T cT , < 6c Since 6~BT . , G~ c , and the r e s p e c t i v e periods B 'gr and 'at are l i s t e d f o r the parameter values given at the end of the previous s e c t i o n . As expected the c a l c u l a t e d frequencies are f a r too small f o r these waves to possess any dynamical s i g n i f i c a n c e i n the S t r a i t of Georgia. Notice that both <f and (f are almost independent of k f o r waves exceeding aT 250 km i n l e n g t h . In summary then both i n t e r n a l Kelvin waves and Rossby waves have ) been considered and i t was shown that n e i t h e r could provide an explanation of the observed low-frequency current s t r u c t u r e i n GS. forced to turn to other, more s o p h i s t i c a t e d models; Hence we are t h i s i s done i n the next two chapters where models containing v a r i a b l e bathmetry are analysed. 34 Table I I I . Wavelength, frequency, and period of the f i r s t crosschannel mode barotropic and b a r o c l i n i c Rossby waves i n the S t r a i t of Georgia. A (km) A <T (10" rad s" ) 8 &T 1 T (days) <f (10" rad 8 s" ) 1 Todays) 1500 4.53 134G 3.42 1775 750 4.52 1340 3.41 1775 500 4.51 1345 .3.41 17S0 250 4.46 1360 3.39 1790 100 4.14 1460 3.24 1870 35 5. Topographic Waves i n a Two-layer Fluid A c l a s s of small-amplitude, low-frequency, free motions that can e x i s t i n the model formulated i n Chapter 3 i s studied i n t h i s chapter. These motions are i n t i m a t e l y r e l a t e d to v a r i a t i o n s i n topography. The dynamics are based on the long-wave equations f o r a uniformly r o t a t i n g , two-layer f l u i d . The h y d r o s t a t i c approximation i s again invoked, since only very low-frequency motions are of i n t e r e s t . i s assumed to be constant f o r two reasons: The r o t a t i o n frequency (1) the h o r i z o n t a l length scales i n GS are small (at most 250 km), and (2) the e f f e c t of the bottom slope (which exceeds 10 ) outweighs any p o s s i b l e s - e f f e c t s (Rhines, 1969). 3 Indeed i t was shown i n Chapter U that Rossby waves i n a f l a t bottomed channel of the dimensions of GS have periods of years. The plan of t h i s chapter i s to f i r s t derive a set of coupled equations for ^ and and to solve them by a p p l i c a t i o n of a perturbation expansion i n the bottom slope (Section 5.1). In Section 5.2 these r e s u l t s are applied to GS and i n Section 5.3 the dynamics of'\these waves are i n t e r p r e t e d i n terms of the r e l a t i v e v o r t i c i t y 5.1 S o l u t i o n of Equations The l i n e a r i z e d equations of motion f o r the model outlined i n Chapter 3 are given by (Appendix A) Upper Layer (5.1.1) (5.1.2) 36 Louer Layer N > (5.1.3) (5.1.4) With the exception of (5.1.4)these equations are i d e n t i c a l to the s e t (4.1.1) (4.1.4). The mean upper l a y e r depth h^ i s taken as constant and the mean louer layer depth i s given by h,_,(x) = h^rjQ - r t x / h ^ g ) . I t i s again convenient t o work with non-dimensional equations; the same s c a l i n g used i n Section 4.1 i s employed to obtain Upper Layer (5.1.5) U . , * * - ^ l^/^H^i-^t - (5.1.6) Louer Layer (5.1.7) (u 0Here & g =oiL/h^ l A T" ^ ) * 6 S a x - t (5.1.8) i s a slope parameter representing the f r a c t i o n a l change i n the depth of the louer l a y e r across the channel. Again the appropriate boundary conditions require that u^ vanish on the s i d e u a l l s . In order t o solve the system (5.1.5) - (5.1.8), ue proceed as i n Chapter 4 and eliminate the v e l o c i t i e s to obtain a p a i r of coupled equations f o r - v ^ a n d • ^ D r travelling-uave' s o l u t i o n s of the form e-i(ky-<ft\ u i t h cr chosen p o s i t i v e f o r d e f i n i t e n e s s , (5.1.5) and (5.1.7) yield 37 (5.1.9) where ^ = 7^ and S u b s t i t u t i o n of (5.1.9) i n t o (5.1.6) and ^ ^ - S ^ . (5.l.S) gives two coupled equations f o r §> and j£, + - € cr (6/ jCi-<y«-) 6 C\-<f -) 1 (. 9 <5 $ ) - = O (5.1.ID) 0 •_As they stand, ue are unable to solve these a n a l y t i c a l l y ".because of' trie v a r i a b l e c o e f f i c i e n t c^x which i s due to the sloping bottom. To circum- vent t h i s d i f f i c u l t y a procedure introduced by Rhines (197D) i s used and a perturbation s o l u t i o n i n the parameter S^, small, i s sought. To f i r s t order i n 6 S which i s a p r i o r i assumed t h i s i s exactly analogous to the " t r a d i t i o n a l approximation" employed i n the study of planetary waves on the p-plane i n which the C o r i o l i s parameter f i s assumed constant except when d i f f e r e n t i a t e d with respect to l a t i t u d e . To proceed, a l l dependent v a r i a b l e s are expanded as power s e r i e s -in 6 , v i z . , 'By r e s t r i c t i n g 6"to be D(^ )> we n a u e s frequency ( K e l v i n or g r a v i t y ) waves; f i l t e r e d out any possible hight h i s i s consistent with the aim of searching f o r low-frequency o s c i l l a t i o n s ( i . e . , periods exceeding 4 days). To zeroth order i n o" , (5.1.ID) and (5.1.11) reduce to (where primes s have been dropped) 38 I I^- - ^ W A + (€/«^ 4 ~ O (5.1.13) * (5.1.14) uihile the boundary conditions together with (5.1.9) and (5.1.12) imply that <fy ^ l\- - O 0.+ X^O,i (5.1.15) Notice that t h i s places a lower l i m i t of the s i z e of k ( i n view of (5.1.16) below, t h i s r e s t r i c t i o n i s that k >*> wrrcT" , a condition e a s i l y met f o r a l l but very long waves). (5.1.14) In view of (5.1.15), s o l u t i o n s to (5.1.13) and of the form = 1 ^ , 4 ^ (.ftn.B*) siwCwirO n--i,23 ( --(5.1.16) are sought, where An and Bn are a r b i t r a r y constants and n i s the c r o s s channel mode number. S u b s t i t u t i o n of (5.1.16) into (5.1.13) and (5.1.14) implies that f o r a n o n t r i v i a l s o l u t i o n f o r An and Bn, k and <S~ must s a t i s f y the equation K4 where -T- \Ol>/*Hl-^) - (5-1.17) = k^ + (nTO^. S o l u t i o n of (5.1.17) far<T y i e l d s the f a l l o w i n g d i s p e r s i o n r e l a t i o n : <T klK" it**)} (5.1.18) As before the parameter v> i s defined by + CL/O " 0 - 3 (5.1.19) Where the i n t e r n a l deformation radius i s now given by t W,Wo/rO L /f where H = h^ + h^g i s the maximum mean channel depth. (5.1.20) 39 E i t h e r (5.1.10) or (5.1.11) may be used to r e l a t e A - t o B ; we Find n > n K For a - J ^ t l ^ / y ] K" (5.1.21) 1 given 6~ (5.1.17) possesses four s o l u t i o n s f o r k; f r e a l and correspond two of these are to a long and a short wave and two are complex. In an open-ended channel the complex waves must be discarded, hence k i s r e s t r i c t e d to be r e a l . From the d i s p e r s i o n r e l a t i o n (5.1.18), i t follows that k i s p o s i t i v e since 6~ was chosen as such, and hence the phase propagates northward along the channel with the shoaling bottom to i t s r i g h t . In general, Cf a t t a i n s a maximum f o r some intermediate value of k and tends to zero f o r both small and large k. ward, energy may Thus although the phase always propagates north- be transmitted i n e i t h e r d i r e c t i o n along the channel: northward f o r small k and southward f o r large k. of s p e c i a l i n t e r e s t . Two l i m i t i n g cases are Consider f i r s t the s i t u a t i o n i n which (weak s t r a t i f i c a t i o n or wide channel, say,, 0 —*> see (5.1.19) and (5.1.20)) then (5.1.18) can be approximated by \ 6" where r g w i s the external deformation r e - L £ U * l r t l 6 ] V radius defined by L c ^ N f - (5.1.22) In terms of the dimensional frequency LO = Sf<T and wavenumber k^ = k/L, we have L 0 L T x " (5.1.23) d This i s simply the dispersion r e l a t i o n f o r a barotropic topographic planetary wave i n a channel of depth H with a free surface. In the 40 l i m i t 0-*° <r (strong s t r a t i f i c a t i o n or narrow channel, say)' , k / C k ^ — C«M-V3 (5.1.26) or i n terms of dimensional v a r i a b l e s , LO ^ — — — (5.1.25) which, as shown below, i s the dispersion r e l a t i o n f o r a topographic planetary wave that i s e s s e n t i a l l y trapped i n the lower l a y e r . Hence f o r large v, the motion i s barotropic, while f o r small 0 the motion i s baroc l i n i c and bottom-trapped; the l a t t e r motion i s the two-layer analogue of Rhine's (1970) short-wave, bottom-trapped o s c i l l a t i o n i n a continuously stratified fluid. The r a t i o s of the h o r i z o n t a l v e l o c i t i e s i n the upper layer to those i n the lower l a y e r are e a s i l y obtained from (5.1.9)., (5.1.12), (5.1.IS) and (5.1.21). Define the r a t i o s R = V o u zeroth order i n 6 , s' Z 5- K & u ^ an ^ 'WO*^ = C 1+ 1 = v /vni then to C Ci+ftWo-) K 1 -]"' (5.1.26) we see that as the wavenumber or mode number increases, R becomes small and hence the h o r i z o n t a l k i n e t i c energy becomes more confined to the lower layer. For large 0, R i s nearly unity and the motion i s purely barotropic. On the other hand f o r small \) we f i n d that ^ 0/i\^\C « > and-the motion i s e s s e n t i a l l y bottom-trapped. very long waves (k (5.1.27) This may be true even f o r 1) since the motion i s quantized i n the x - d i r e c t i o n . Therefore t o a great extent the v e r t i c a l d i s t r i b u t i o n of h o r i z o n t a l k i n e t i c 2 energy, which i s d i r e c t l y proportional to R , depends on the aspect r a t i o = L / r of the motion. i I f the channel i s r e l a t i v e l y narrow or i f the 41 s t r a t i f i c a t i o n i s s u f f i c i e n t l y strong, v> w i l l be small and the energy u i l l be confined to the lower l a y e r , whereas i f the channel i s wide or the s t r a t i f i c a t i o n weak, the currents w i l l be b a r o t r o p i c . The r a t i o n of i n t e r f a c i a l to surface displacement, 0|,_,A^, i s obtained i n a s i m i l a r fashion to R and R through (5.1.12), (5.1.21) and the . x y d e f i n i t i o n s of 4 and 4 . To zeroth order i n S ^ one f i n d s that = (4*640/64 - 1 M<5t)[ k^+cwft)1-] (5.1.28) Notice the i n t r i g u i n g r e s u l t that f o r a given wavenumber^^/^ i s <*' e x p l i c i t l y independent of the s t r a t i f i c a t i o n (6 i s p r o p o r t i o n a l to o ). 2 2 The quantity A/6fc = gh^/f L i s the square of the r a t i o of the upper l a y e r deformation radius to the channel width and i s a measure of the importance Df the i n c l u s i o n df a f r e e surface i n the model. particular interest. For "large iv/6fc , corresponding Again two cases are of to a narrow channel Dr t h i c k upper l a y e r , \ r J ' " \ ^ i s large and the surface displacement may be r In the case of small &l6fe-, which corresponds to a wide neglected. channel or- t h i n surface l a y e r , ^ ^ l i s D ^ d e r unity and the motion i s o r barotropic. One may also construct a closed-basin s o l u t i o n to the system (5.1.13) and (5.1.14). earlier. To do so one must include the two complex waves mentioned The s i t u a t i o n i s thus s i m i l a r to that of Helvin wave propagation i n a closed basin i n which one must also consider Poincare waves i n order to s a t i s f y the end boundary c o n d i t i o n s . A s o l u t i o n thus c o n s i s t s of a l i n e a r combination of the four waves p o s s i b l e f o r a given 6" subject to the a d d i t i o n a l boundary condition that v ^ i =1,2) vanishes on the end w a l l s . The complete s o l u t i o n of the closed-basin problem i n p a r t i c u l a r requires that the q u a r t i c (5.1.17) be salved f o r k, which i s a tedious task. 42 5.2 A p p l i c a t i o n to GS For the average values of the p h y s i c a l q u a n t i t i e s l i s t e d i n Chapter 3, one obtains the f o l l o w i n g estimates f o r the various parameters: £= ( f - f ) / f 2 6 = 1 L / h s 20 2 = 2.25 x 10* 3 = 0.3S = 490 km r e r. = 8.1 km l & = h/h = 0.17 2 Q £ = 1.1 v = (IVr) Ue see that 6 = 7.5 2 s i s perhaps a b i t large f o r a "small" expansion parameter thus i n d i c a t i n g an extension of the s o l u t i o n to next order i n S y be made. On the other hand, the v a l i d i t y of the l i n e a r i z e d s should dynamical equations i s questionable as i s shown below, and suggests that such an extension might be a moot e x e r c i s e . At any r a t e , these considerations should not deter us from comparing the r e s u l t s with the data. The d i s p e r s i o n r e l a t i o n (5.1.IS) f o r the f i r s t three cross-channel modes i s p l o t t e d i n F i g . 13, and selected values of the wavelength X, period T, and the phase speed c are presented i n Table IU. In a l l cases the frequency i s a broad-banded f u n c t i o n of the wavenumber; f o r n = 1, wavelengths corresponding to periods extending from 11 to 100 days span the range of 2 to 700 km, while f o r n = 2 and 3, the ranges are 2 - 200 km and 3 - 90 km r e s p e c t i v e l y . I t i s c l e a r that any of these modes f i t the observed broadband spectra f o r reasonable values of the wavelength. From Table IU we also see that these waves propagate very WAVELENGTH K F i g . 13. rf (rad (km) km ) 1 Dispersion r e l a t i o n f o r the f i r s t three cross-channel modes. ui 44 slowly as the maximum value of c^ i s only 9.0 cm s . This corresponds to a t r a n s i t time of nearly 31 days f o r a wave t r a v e l l i n g the length of GS. Table IV. A Wavelength, period, and phase speed f o r t h e " f i r s t crass-channel mode i n the S t r a i t of Georgia. The r a t i o of the mean current magnitude U to t h e . phase speed c^ i s also given f o r U = 5 cm s (km) T(days) c (cm s P ) U/c P 1250 128.0 9.0 0.55 500 164.0 9.0 0.55 250 32.8 8.8 0.57 100 15.2 7.6 0.66 75 12.8 6.8 0.74 50 11.3 5.1 1.0 25. 12.9 2.2 2.2 12.5 20.2 0.7 - 7.0 37.4 0.2 6.25 26.0 The theory does not c o r r e c t l y p r e d i c t the observed v e r t i c a l d i s t r i bution of h o r i z o n t a l k i n e t i c energy as can be seen from F i g . 14 which shows the r a t i o R as a f u n c t i o n of wavenumber. I t s maximum value of 0.39 i s obtained f o r the f i r s t mode i n the l i m i t of very long waves. Signifi- c a n t l y smaller values are found f o r shorter wavelengths and higher modes. 2 Since the v e r t i c a l k i n e t i c energy d i s t r i b u t i o n i s p r o p o r t i o n a l t o R , we see that i n contrast to the observed case nearly a l l the energy i n the model i s trapped w i t h i n the lower l a y e r . Table IV also l i s t s values of the r a t i o of the p a r t i c l e speed t o the phase speed of the wave. The smallness of t h i s parameter i s a c r i t e r i o n f o r the v a l i d i t y of the l i n e a r i z a t i o n hypothesis. For an average observed low-frequency , t h i s r a t i o exceeds current magnitude of 5 cm s 0.5 f o r long waves and r a p i d l y increases as the wavelength diminishes. WAVELENGTH Kd(rad (km) km"1) F i g . Ik. V e r t i c a l d i s t r i b u t i o n of h o r i z o n t a l v e l o c i t y f o r the f i r s t three cross-channel modes. 4G Hence, at best the l i n e a r i z a t i o n i s only a f a i r approximation. Unfortu- n a t e l y , no data e x i s t s that suggests appropriate phase speeds or wavelengths for GS. 5.3 U o r t i c i t y dynamics I t i s of considerable i n t e r e s t to examine the p h y s i c a l mechanisms governing t h i s c l a s s of wave motion. L i k e a l l waves of s u b i n e r t i a l frequency, these o s c i l l a t i o n s may be termed v o r t i c i t y waves; i n f a c t the governing equations (5.1.13) and (5.1.14) are j u s t the zeroth order v o r t i c i t y equations. Since the model i s i n v i s c i d and uniformly r o t a t i n g , and since the atmospheric pressure i s assumed constant and the density uniform, the only mechanism capable of a l t e r i n g the d i s t r i b u t i o n of v o r t i c i t y i s one which e f f e c t s a change i n the depth of the f l u i d . may be accomplished i n three ways: through changes i n (1) the surface e l e v a t i o n ^ , or (2) the i n t e r f a c i a l displacement ^ v a r i a t i o n i n the depth of the b a s i n . s i g n i f i c a n c e since 1^ the l a t t e r two. -Si^d This ^d' or (3) through a Of these the f i r s t i s of l i m i t e d ant ^ h e n c E a t t e n t i o n i s focused on Note that the second mechanism serves to t r a n s f e r v o r t i c i t y between the two l a y e r s , and i t i s only the s l o p i n g bottom that can a l t e r the v o r t i c i t y of an e n t i r e water column. In view of t h i s i t i s not at a l l s u r p r i s i n g that these waves should be more intense i n the lower l a y e r . I f we define the r e l a t i v e v o r t i c i t y by ~], = (v - u. ), then i t f o l l o w s from (5.1.5) - (5.1.8) that It " - (5.3.1) W - c\- & *v' i £f]z-t 5 >). a (5.3.2) The r i g h t hand sides of (5.3.1) and (5.3.2) are the respective h o r i z o n t a l 47 Since <S, S << I divergences i n each l a y e r . s , (5.3.1) and (5.3.2) may be approximated by - 1,-t - The f a c t o r A "|it > t fe (5.3.3) - *s u-z.. (5.3.4) m u l t i p l y i n g 'vj^ i n (5.3.3.) accounts f o r the d i f f e r e n c e i n mean depth of the two l a y e r s . Hence i f uere to vanish, the motion would be stronger i n the upper layer by the factorh^g/h^ (as i t i s f o r i n t e r n a l Kelvin waves). While the s i t u a t i o n i n the upper layer i s c l e a r one must examine the importance of the two i n t e r a c t i n g mechanisms i n the lower l a y e r . The degree of bottom trapping should depend on the r e l a t i v e magnitude of these two mechanisms. In order to make these ideas more p r e c i s e , we u t i l i z e the r e s u l t s of Section 5.1. To zeroth order i n S t.lf 1 t61 - i. ^ 6 - f e i^"' s I $s s one f i n d s that [^64-] (5.3.5) i > G-C^+S^) - Vs. ( 4 * 4 ) Consider the r e l a t i v e s i z e of the two terms i n (5.3.6); 6<57k. ] (5.3.6) their ratio i s Thus f o r large k, the second term dominates and most of the v o r t i c i t y change i n the lower layer i s due to the sloping bottom. case of small k i s s l i g h t l y more involved; as k o ; (nV-) (,v\V+v) + 6<5\>/tn-o) The i t follows from (5.1.18) that 48 Hence i f 6 i s near unity (as i t i s f o r GS) the bottom slope again dominates. Notice that t h i s r e s u l t i s due to the quantization of the motion i n the cross-channel " d i r e c t i o n . u To determine the r e l a t i v e magnitude of the o r t i c i t y i n each l a y e r ue compare the second term i n (5.3.6) with (5.3.5). Again as k-°> <x> , For V:-*o _C If £ * \ D n e k^»o gn[ j ^ e m o ^ Q n ^ ^Tangly s bottom trapped. finds - then £tf~/fc»W + ^ ' ^ < ! -[eO^),L^f i s bounded above by unity and the motion i s again bottom enhanced. Hence ue have seen that an examination of the v o r t i c i t y dynamics of these uaves provides a q u a l i t a t i v e understanding of t h e i r nature. Even more i n s i g h t can be gained through study of a system u i t h v a r i a b l e density, and t h i s i s done i n the next s e c t i o n . 49 6. Topographic Waves i n a F l u i d with Continuously Varying Density In t h i s chapter topographic waves i n a system characterised by a continuous but v a r i a b l e density f i e l d are examined. and i n t e r e s t i n g exercise f o r the f o l l o w i n g reason. This i s an important Mathematically, a two-layer system c o n s t i t u t e s a s p e c i a l case because i t embodies a d i s c o n t i n u i t y i n the v e r t i c a l d i s t r i b u t i o n of d e n s i t y . I t i s unclear how t h i s s i n g u l a r i t y might manifest i t s e l f i n the s o l u t i o n s , and one must be c e r t a i n that any p o s s i b l e spurious r e s u l t s are recognized. Hence although the two-layer model provided a good f i t to the data, one i s compelled to consider other density d i s t r i b u t i o n s . The approach i n t h i s chapter i s e s s e n t i a l l y the same as that used i n Chapters 4 and 5. F i r s t a pressure equation i s derived i n Section 6.1, and then p a r t i a l s o l u t i o n s to i t are obtained i n Section 6.2.through an expansion i n the perturbation parameter A . Several general theorems concerned with the v e r t i c a l s t r u c t u r e and propagation of waves admitted by t h i s system are proved i n s e c t i o n 6.2, as w e l l . In s e c t i o n 6.3 the s p e c i a l case of constant Brunt-VSisMlfl frequency and i s considered and some of Rhines' (1970) r e s u l t s are recovered although the a n a l y s i s d i f f e r s somewhat from his. The r e s u l t s of t h i s model are applied to GS y i e l d i n g conclusions s i m i l a r to those obtained i n the previous chapter. 6.1 The pressure equation In t h i s s e c t i o n an equation f o r the pressure i s derived that i s exactly analogous to the systems (4.1.17) - (4.1.18) and (5.1.ID) - (5.1.11). To begin, l e t there e x i s t a h y d r o s t a t i c e q u i l i b r i u m s t a t e described by - O (6.1.1) 50 with ?•* ~ " ?°^ ( 6 * 1 2 ) To describe small departures from e q u i l i b r i u m i t i s convenient to introduce the perturbation pressure p, density , and v e l o c i t y a , which are defined by (6.1.3) The l i n e a r i s e d equations f o r these perturbation q u a n t i t i e s i n an unforced, i n v i s c i d , uniformly r o t a t i n g f l u i d are Vi\ ~ ?„V' = - /fo (6.1.4) 4 + W - - f^ / fo (6.1.5) = y +" <Jj>' (6.1.6) O II'*. V V'^ ^ w ' Pi +- ^ f o t t t a. O " (6.1.7) ° ( 6 ' 1 8 ) Since the motions of i n t e r e s t are of very low frequencies, the h y d r o s t a t i c approximation was made to obtain (6.1.6). In order to s i m p l i f y the ensuing a n a l y s i s , the Boussinesq approximation i s invoked and any v a r i a t i o n s i n the e q u i l i b r i u m density f i e l d are.ignored except when they occur i n a buoyancy term. That i s , unless ^ (z) occurs 0 i n a term m u l t i p l i e d by .g i t i s replaced by a constant density representative of the water column. approximation i s that PotrWp^^l , j* 5 The c r i t e r i o n of v a l i d i t y f o r t h i s or e q u i v a l e n t l y that 1\|H /g « 2 1. For GS the estimates H = 400 m and l\l = 10" rad s" give PJ H/g = 4 x 10" . J E l i m i n a t i o n of from (6.1.6) and (6.l.S) gives 51 tf w' - - l / f* 11 II (6.1.9) II where the Brunt-l/aisala frequency N i s a measure of the g r a v i t a t i o n a l s t a b i l i t y of the e q u i l i b r i u m s t a t e and i s defined by H = L - ^ f o t / P* (6.1.1D) we thus obtain a set of four Boussinesq equations f o r the four unknowns U-'and Y U t ~ ^ ' * ' ? * •£„ a ' - ~ f ^ /f* (6.1.12) f i / f* (6.1.13) ' - o V tf ^' * 2 - U.* + W t ; (6.1.11) / (6.1.14) The boundary conditions appropriate to t h i s system require that the v e l o c i t y components normal to the s i d e w a l l s and bottom vanish, and that the kinematic and dynamic conditions be s a t i s f i e d at the surface \x! - O NM« ^ *U.' l> - T>o X= O ,L GL-V OX ) - ° * ' V\ * * * I L (6.1.15) J flL+ (6.1.16) t * o, The conditions (6.1.17) are extremely complicated (6.1.17) since they are (1) non- l i n e a r and (2) must be evaluated at the unknown surface To avoid these problems we expand (6.1.17) about the mean sea surface z = • and r e t a i n terms l i n e a r i n the perturbation q u a n t i t i e s to obtain ^ E l i m i n a t i o n of W I i - o (6.1.18) from (6.1.18) by time d i f f e r e n t i a t i o n ^ s u b s t i t u t i o n of w from (6.1.13), and a p p l i c a t i o n of the s t a b i l i t y r e l a t i o n (6.1.9) y i e l d s 52 + *LL* W = o at i - o . (6.1.19) In the BDussinesq approximation the second term i n (6.1.19) i s n e g l i g i b l e as compared to the f i r s t , since the r e l a t i v e magnitudes Df the two terms 2 i s just — « 1. The surface boundary condition thus reduces to a g f o r i g i d top c o n d i t i o n . =0 *x+ l=-o. (6.1.2D) As before i t i s convenient to recast the equations of motion i n t o non-dimensional form. t = fo~V The f o l l o w i n g geostrophic s c a l i n g i s used: (x,y) = L(x?y*), z = Hz? (u, v) = ( u f v * ) , w = H/L wf and p' = fULj^-p* where a double prime denotes a non-dimensional quantity. The non-dimensionalised equations and boundary conditions are ( a f t e r dropping the primes) - - :px (6.1.21) \- LX. - - ^ (6.1.22) U_L_ - V \/ T =• - fit 4- + *U (6.1.23) - ° (6.1.24) UL - o <a> x= o,i (6.1.25) fifc-O o-V £=-o (6.1.26) VM ^ Here B 2 5 S ix- OL--V i = - l +6'sX (6.1.27) 2 2 = (MH/fL) = ( r V O compares the i n t e r n a l deformation radius to the channel width and thus i s analogous to I / ^ ) , To derive an equation f o r the pressure we proceed as i n Section 4.1 and Section 5.1. and eliminate the v e l o c i t i e s from the c o n t i n u i t y equation. From (6.1.21) and (6.1.22) i t fallows that dt 1 c - (6.1.28) - 7^ I f we operate on (6.1.24) with<£, s u b s t i t u t e f o r u and ujhEre v from (6.1.28), and use (6.1.23) f o r w, we obtain [ V^J + £U" ~ O 2 (6.1.29) 2 In the Boussinesq approximation the z-dependence of B i n comparison t o that p ; . z may be neglected TD see t h i s consider the f o l l o w i n g argument, we may w r i t e -L [ \ - (1*1 t r \ - ^/6*) T* 5 the r e l a t i v e magnitude of the two terms i s IM V\/q and hence the second term i s unimportant compared- with the f i r s t . Thus to the order of the Boussinesq approximation the pressure equation i s CV + S^Otfc^ffc = O (6.1.3D) -i<rt For harmonic time dependence of the form e , the pressure equation and boundary conditions reduce to . (VH + ^ « ) f Cv-* ) 1 o-V L CT f a - f ^ . - O f t =. O O.V <r<^ S" 2 6.2 ^ O (6.1.31) X=o.l (6.1.32) i ^ o (6.1.33) 5 O-<J*T l ^ f x - i f t ) s l t=-' + <*s* (6.1.34) 2 P a r t i a l s o l u t i o n of the pressure equation f o r a r b i t r a r y IM' The bottom boundary condition (6.1.34) precludes a separation of i n t o h o r i z o n t a l and v e r t i c a l modes. A perturbation expansion i n the slope parameter %> ^ i s again employed to circumvent t h i s d i f f i c u l t y . To zeroth 54 order i n & the h o r i z o n t a l dependence df 'p\ i s evaluated leaving a second s order d i f f e r e n t i a l equation f o r the v e r t i c a l dependence. In a d d i t i o n expressions f o r the v e l o c i t i e s and surface e l e v a t i o n are derived and several general theorems that are concerned with the v e r t i c a l s t r u c t u r e of "p. and with the phase and group v e l o c i t i e s of the s o l u t i o n s are stated; these are proved i n Appendix B. To begin, a l l dependent v a r i a b l e s are expanded as power s e r i e s i n <£ , s viz., ,4) * 6< ^ > cr - (6.2.1) <rc0 uhere, as before, 6~ i s proscribed to be of order & any unwanted high-frequency waves. To zeroth order i n & s CV T fc Z , i n order to f i l t e r out I t i s assumed that B i s 0(1). the system (6.1.31) - (6.1.34) reduces to * +- %~ 'b±i') = O * o s o> ai ^ O (6.2.2) X-o,\ i - ° (6.2.3) (6.2.4) _ r,t * ~-I. (6.2.5) The bottom boundary condition (6.1.34) was expanded about z =-1 i n order to obtain (6.2.5). Notice that i f z had been scaled by BH instead of H, (6.2.2) would simply be Laplace's equation. This suggests that the appropriate scale f o r v e r t i c a l v a r i a t i o n s i s ( r ^ / L ) H rather than H. Also notice that the dispersion r e l a t i o n i s i m p l i c i t i n the bottom boundary condition (6.2.5). For a t r a v e l l i n g wave s o l u t i o n of the form ^ - f U s v v ^ r c x e'H n U) A hi,-'- (6.2.6) 55 we obtain the f o l l o w i n g system f~l ~ t rj- c 0 n = i (6.2.8) C X - t i - o 0 - k 8 Z 0-+ H 4--I. (6.2.9) In a d d i t i o n the f o l l o w i n g normalisation f o r i " ! i s chosen, niC^ - l ; (6.2.1D) t h i s places no r e s t r i c t i o n s of the form of H . To the present order of a n a l y s i s , p^ "* i s a stream function f o r the 0 h o r i z o n t a l v e l o c i t i e s as may be seen through the expansion of (6.1.28). The v e r t i c a l v e l o c i t y f o l l o w s from the expansion of (6.1.23). Ex'plicity one f i n d s - wi\* tf* ~ ^ _ ^ VJ ^ =. 10 , n r _ n u> e (6.2.H) n C oSmTX H Ci) t ^ A (6.2.12) O Even though w ^ (6.2.13) vanishes, the zeroth order surface displacement does not; i t f o l l o w s from the f i r s t of (6.1.18) that T Fortunately w^ li NM t0 - may be derived from p ' to give D ) tt (5-^ ft'^n ^ w w r v , f^U) e C L ^1 " + ^ (6.2.15) and thus I t i s i m p l i c i t i n t h i s discussion that the dimensional surface e l e v a t i o n i s scaled by RoH where the Rossby number Ro = u/fL. w ^ (Ue might note that = • suggests our o r i g i n a l s c a l i n g of w was too generous). i t f o l l o w s from (6.2.11) and (6.2.12) that the r a t i o R of u ( o ) Finally or v ^ 56 taken at tun d i f f e r e n t depths Qtn , \ - and ui'*^) _ is _ nu.) (6.2.17) 2 jAlthaugh one must specify IM (z) to proceed any f u r t h e r , there are a number of i n t e r e s t i n g and s i g n i f i c a n t g e n e r a l i z a t i o n s that can be made about the properties of the system (6.2.7) - (6.2.9) f o r a given n and k; they are stated here as theorems, the proofs of which are contained i n Appendix B. Theorem 1. positive. For p o s i t i v e B ,U(z) i s of one s i g n , which we chgose as With t h i s c h o i c e r | ( z ) i s a s t r i c t l y decreasing function of z. Theorem 2. There e x i s t s one and only s o l u t i o n to (6.2.7) - (6.2.9). Theorem 3. The frequency <T i s always p o s i t i v e . Theorem 4. The group v e l o c i t y i s p o s i t i v e f o r a l l k. As k-*° <r/k. ; as k-=» <x> <f^ i s bounded by <rl U. 2 Theorem 5. I f B (z) a t t a i n s i t s minimum value at z = -1, the 2 frequency i s an increasing function of B (z = - 1 ) . <p Theorem 6. The frequency i s a decreasing function of the cross- channel mode number n. 2 Theorem 7. s o l u t i o n s 11 I f B^z) andH 2 2 < Q^iz) f o r a l l z, then the corresponding of (6.2.7) - (6.2.9) s a t i s f y H ^ z ) <R 2 ( z ) f o r a l l z<0. Theorem B. H (z) i s an increasing function of n. The. s i g n i f i c a n c e of these theorems i s as f o l l o w s . Theorems 1 and 2 i n d i c a t e that only one v e r t i c a l mode e x i s t s , and that t h i s motion i s bottom i n t e n s i f i e d . That i s , u n l i k e other i n t e r n a l motions, separate b a r t o t r o p i c and b a r o c l i n i c modes are nonexistent. This i s due to our choice of small 6~ and of topography however, since de Szoeke (1975) has shown that small v a r i a t i o n s from a constant sloping bottom 57 implies the existence D f a set of v e r t i c a l modes. S i m i l a r l y Wang and Mooers (1976) treated an exponentially sloping bottom and found a denumerable set of v e r t i c a l modes, h a l f of which resembled i n t e r n a l K e l v i n waves, and the other h a l f of which were bottom i n t e n s i f i e d o s c i l l a t i o n s . Theorem 3 shows that the waves admitted by the model always propagate with the shoaling bottom t o t h e i r r i g h t . The most s i g n i f i c a n t theorem i s probably the f o u r t h since i t i n d i c a t e s that the wave number i s a s i n g l e valued function of (T. This implies that energy can be propagated only i n the d i r e c t i o n of the phase v e l o c i t y . <fy.< ^/k Ue note that the condition that -=> o as as k-* o i n p r a c t i c e implies that k-^-oo since <T must be bounded by unity i f the s o l u t i o n s are t o remain v a l i d . That 6T should be an increasing function of l\l intuitively clear. (Theorem 5) i s There are two forces i n t h i s system capable of balancing pressure gradients, namely the C o l i o l i s force and the buoyant force. As N increases, the l a t t e r force increases and compels a f l u i d p a r t i c l e to execute i t s motion i n a shorter time; increases. thus the frequency I t i s d i f f i c u l t tD discuss the dependence of the frequency 6 0 2 2 2 2 2 on H and L (B = 1\) H / f |_ ) since these parameters also enter the expression U "o^<r f r t*> through 6 . 3 a S i s s l i g h t l y more s u b t l e . The r a t i o of h o r i z o n t a l v e l o c i t i e s f o l l o w s from (6.2.11) and (6.2.12); ix** I - - The p h y s i c a l basis of Theorem 6 it is C W UXK) W ^ r r x . As n increases the motion up and down the slope diminishes and hence l e s s v e r t i c a l motion i s induced at the bottom. that i s responsible f o r these o s c i l l a t i o n s . i s lessened and the frequency decreases. But i t i s t h i s very e x c i t a t i o n Thus the r o l e of topography Conversely as k increases, the 58 motion across the bottom increases and i t f o l l o w s that Cf increases as w e l l . This provides a p h y s i c a l basis f o r Theorem 4. I t i s probably f o r these same reasons that the degree of bottom trapping increases as n increases (Theorem 8 ) . For a given h o r i z o n t a l v e l o c i t y , the induced v e l o c i t y at the bottom diminishes as n increases. vertical Given t h i s weaker value, the motion can penetrate upwards only to a l e s s e r depth. Hence although the motion may be weaker i t i s more g r e a t l y confined to the bottom. A s i m i l a r l i n e of reasoning explains Theorem 7, because of the increased buoyant force as the s t r a t i f i c a t i o n strengthens, the motion i s more strongly i n h i b i t e d from penetrating upwards. Lde note that Theorems 1,2,3,6 and the analogues of Theorems 7 and 8 (with B replaced by */•>?'') hold f o r the two l a y e r system studied i n Chapter 5. However, Theorem 4 i s d e f i n i t e l y v i o l a t e d , and we conclude the existence of a negative group v e l o c i t y i n that case i s a d i r e c t r e s u l t of the s i n g u l a r i t y i n d e n s i t y . To summarise the general character of the s o l u t i o n s to (6.1.31) (6.1.34) f o r an a r b i t r a r y but continuous (and p o s i t i v e l\l ), we f i n d o s c i l l a t i o n s f o r which both phase and energy propagate northward. There e x i s t s only one v e r t i c a l mode and i t i s bottom trapped to some degree. This trapping increases as the s t r a t i f i c a t i o n strengthens or as the h o r i z o n t a l mode number increases. The frequency i s an increasing f u n c t i o n 2 of k and IM and decreases with increasing n. 6.3 2 Constant IM model In t h i s s e c t i o n the simplest n o n - t r i v a l choice f o r the density f i e l d 2 f o r which (6.2.7) - (6.2.ID) may be salved, namely constant IM i s con} sidered. As may be seen from (6.1.ID) t h i s corresponds to a l i n e a r density 59 2 p r o f i l e p,(z) = £>(o) - Cp*JVI / g ) z . This model was f i r s t treated by Rhines (1970) and many o f h i s r e s u l t s are recovered here. After deter- mination of s p e c i f i c forms f o r the v e l o c i t i e s and surface e l e v a t i o n , the r e s u l t s of t h i s a n a l y s i s are applied to the S t r a i t of Georgia and compared with those obtained from the two-layer model. 2 For constant IM , the s o l u t i o n to (6.2.7) - (6.2.10) i s simply where jU= BK?0. I t follows from the r e s u l t s of the previous s e c t i o n that CT - & U. V - \. - J l * s k S\v. wtrx (6.3.2) (6.3.3) CosvyA,t & r\w wtr C o s w t v * C o s V l ^ t e ^ H - ' ( \ A ^ Sv^wrc* i . SvwW^t e 5 " ^ ' (6.3.4) c(.Wv, - <rfc' + tr) 1 ' (6.3.6) Similarly, and the motion i s bottom i n t e n s i f i e d . I t i s not d i f f i c u l t to show that (T i s a single-valued function of k, and tends t o zero as k-^> o and to S B =<X IM as k->o° . s The v e r t i c a l s t r u c t u r e of these o s c i l l a t i o n s depends on the parameter YvS- C; ( V V*" which i s the i n t e r n a l radius of deformation ured i n u n i t s df the " t o t a l wavelength", 3.1T / [ V>t +• C*n) J ^ Two 1 l rneas cases are of i n t e r e s t . The f i r s t corresponds to smalljx, (due t o , f o r example, weak s t r a t i f i c a t i o n ) . c r - 6sk/Kz- i n dimensional form t h i s i s In t h i s case (6.3.2) reduces to A -^—r ; (6.3.a) 60 ^ _ ( *g\ (6.3.9) Notice, ( l ) t h a t t h i s i s p r e c i s e l y the same l i m i t as ue found i n the tuol a y e r case, except that the free surface term i s missing ( c . f . , 5.1.23), and (2)that > i s independent-of= N. In t h i s l i m i t the motion i s only weakly bottom i n t e n s i f i e d as (6.3.10) The second l i m i t i n g case corresponds t o largey*. (strong s t r a t i f i c a t i o n , say). In t h i s case we f i n d that (T - 6s S - ^ s • (6.3.11) i n dimensional form t h i s becomes and co i s thus d i r e c t l y p r o p o r t i o n a l to both the bottom slope and IM. Notice the i n t r i g u i n g r e s u l t that to i s e x p l i c i t l y independent of f (although, o f course, these waves could not e x i s t i n a non-rotating f l u i d ) . The motion i s strongly confined to the bottom as nitf ~~ CotV^t * e^* (6.3.13) Unlike the previous l i m i t , t h i s case i s not d i r e c t l y analogous to the corresponding l i m i t i n the two-layer case. limit (|x.«l That the "long wavelength" ) should correspond to the two-layer case while the "short wavelength" l i m i t (yu>">l ) does not i s not s u r p r i s i n g , as short waves w i l l sense d e t a i l s i n the density d i s t r i b u t i o n that long waves w i l l not. should note the d i f f e r e n c e i n nature of "0 andyu. One The l a t t e r i s i n t i m a t e l y connected to the wavelength while 0 i s independent of wavelength. These r e s u l t s are now applied to the S t r a i t of Georgia. f o l l o w i n g values, For the 61 H = 300 m L = 22.2 km ft = 4.9 x 10~ 3 IM = 7.5 x 10" rad s " 3 1 UJE obtain the parameter estimates £ = 0.36 s B 2 = 0.85 ( B ~ = 1.2) 2 r. = 20.3 km l Notice that t h i s model y i e l d s an i n t e r n a l deformation radius which i s 2.5 times l a r g e r than the two-layer model. The d i s p e r s i o n r e l a t i o n f o r the f i r s t three cross-channel modes i s p l o t t e d i n F i g . 15, and selected values D f the wavelength A, period j and phase speed c- are presented i n Table U. P In addition the f i r s t made two- l a y e r d i s p e r s i o n r e l a t i o n i s p l a t t e d i n F i g . 15 f o r comparison. As before the frequency i s a broad-banded f u n c t i o n of the wavelength, but, u n l i k e the two-layer c a s e , ^ i s a s i n g l e valued f u n c t i o n of co. For a given wavelength the frequency i s about an order of magnitude l a r g e r than the corresponding two-layer frequency. A wave of 15 day period possesses a wavelength of 330 km (n = 1), 170 km (n - 2) and 11D km (n = 3); the corresponding two-layer wavelengths f o r the f i r s t mode are 80 and 20 km. As we see from Table IV, the phase speeds i n t h i s case are much greater with a maximum value of 25.7 cm s . This corresponds to a t r a n s i t time of about 11 days f o r a wave t r a v e l l i n g the length of GS. Moreover, due to the increased magnitude of c ^ the assumption of l i n e a r dynamics i s much better i n t h i s case than i n the two-layer case as can be seen from WAVELENGTH (km) Krf (rad km" ) cn ro F i g . 15. Dispersion r e l a t i o n f o r the f i r s t three-cross-channel modes (constant IM model) 63 the reduced s i z e c f the parameter U/c^ which i s also given i n Table \J. Table V. A 0<m) Wavelength, p e r i o d , phase speed, and v e l o c i t y r a t i o s for the f i r s t cross-channel mode. Here R i = fl(50 m)/ n(200 m) and R2 =".(50 m)/n(140 m). The r a t i o of the mean current magnitude U t o the phase speed c i s also given f o r U = 5 cm s~% ^ T(days) Cp(cm s ) U/c V R 1 R 2 1250 56.2 25.7 .19 .36 .59 1000 45.0 25.7 .19 .36 .59 750 33.8 25.7 .19 .36 .59 500 22.6 25.7 .19 .36 .59 250 11.4 25.4 .20 .36 .58 100 4.9 23.5 .21 .32 .54 50 3.0 19.2 .26 .22 ..43 Again, however, the h o r i z o n t a l k i n e t i c energy I s strongly enhanced toward the bottom. F i g . 16 shows the r a t i o R as a f u n c t i o n of wavenumber for the two depths, 50 and 200 m. I t s maximum value of .36 i s obtained f o r the f i r s t mode i n the l i m i t of long wavelengths and i s s l i g h t l y l e s s than the corresponding value of .39 obtained i n the two-layer case. In f a c t , comparison of F i g s . 16 and 14 shows that the two-layer R always exceeds R In the present case. In summary, t h i s model admits topographic waves with frequencies l y i n g i n the observed range f o r r e a l i s t i c choices of the wavelength but, as before, the motion i s confined to the bottom. More importantly, though, the d i s p e r s i o n r e l a t i o n s d i f f e r considerably i n form f o r the two models. This and Theorem 4 suggests that the two-layer r e s u l t s are a very s p e c i a l case: , and perhaps that future work should be confined t o models with a continuously varying density, even though t h i s would generally necessitate a numerical s o l u t i o n of the v e r t i c a l equations, (6.2.7) (6.2.10). WAVELENGTH (km) Fig.-16. V e r t i c a l d i s t r i b u t i o n of h o r i z o n t a l v e l o c i t y f o r the f i r s t three cross-channel modes (constant model) 65 7. Summary In summary, several classes of long wave motion have been examined and the f a l l o w i n g conclusions i n s o f a r as the low-frequency current f l u c t u a t i o n s i n GS are concerned, may. be made: (1) Neither i n t e r n a l K e l v i n waves, nor Rossby waves are important, since f o r r e a l i s t i c values of the wavelength i n t e r n a l K e l v i n waves have frequencies too high and Rossby waves have frequencies too s m a l l . (2) The chosen two-layer model of GS admits t r a v e l l i n g topographic planetary waves whose c a l c u l a t e d frequencies l i e i n the observed range; t h i s model does not accurately p r e d i c t the observed v e r t i c a l d i s t r i b u t i o n of h o r i z o n t a l k i n e t i c energy, however. Nor does i t account f o r the apparent l o s s of the barotropic nature of the 10 - 25 day: fluctuations at the c e n t r a l and western l o c a t i o n s , although t h i s might be r e l a t e d to the presence of the s h e l f which i s v i s i b l e i n F i g . 2 i n sections E and F. In a d d i t i o n , t h i s model does not account f o r the observed low coherence between h o r i z o n t a l l y separated currents. (3) A continuously s t r a t i f i e d model y i e l d s r e s u l t s that resemble those obtained from a two-layer model except that the d i s p e r s i o n r e l a t i o n s d i f f e r i n form. I t i s suggested that the double-valued nature of k as a f u n c t i o n of <S" i n the l a t t e r case and the concomitant p o s s i b i l i t y of b i d i r e c t i o n a l energy propagation i s a r e s u l t due to the s i n g u l a r i t y i n the density f i e l d . (4) Given the dynamics assumed i n t h i s t h e s i s , the motion i s always bottom i n t e n s i f i e d . However, these models represent only an i n i t i a l step i n a more comprehensive t h e o r e t i c a l i n v e s t i g a t i o n of the low-frequency current f l u c t u a t i o n s i n GS c u r r e n t l y being undertaken by the author. It does not encompass any inhomogeneities i n topography or s t r a t i f i c a t i o n or 66 any l o n g i t u d i n a l v a r i a t i o n s i n topography. But perhaps most s i g n i f i c a n t l y , p o s s i b l e i n t e r a c t i o n s with the mean currents or the Fraser River outflow have not yet been examined. baroclinic instability Since, given the mean current s t r u c t u r e , i s a l i k e l y prospect (Pedlosky (1964)), i t i s of the utmost importance that t h i s be done. F i n a l l y , the question of generation of these o s c i l l a t i o n s has yet t o be t a c k l e d . In t h i s con- nection, however, i t should be mentioned that since GS i s e f f e c t i v e l y i s o l a t e d by the narrow t i d a l passes at both ends, any waves would n e c e s s a r i l y be generated w i t h i n the S t r a i t i t s e l f . Although the current f l u c t u a t i o n s are not d i r e c t l y t i d a l i n nature, i t i s p o s s i b l e that a non-linear t i d a l mechanism might account f o r t h e i r existence. Recognizing GS as an estuary, we see that the f o r t n i g h t l y t i d e could, by a l t e r i n g the i n t e n s i t y of mixing i n the southern passages, generate i n t r u s i v e " i n e r t i a l j e t s " . The surface currents would be stronger than the deeper currents i n proportion to the respective latyer depths. Furthermore, the lack of coherence might be associated with the i n t r u s i v e nature of the process. 67 BIBLIOGRAPHY A l l e n , J.S., 1975. Coastal trapped waves i n a s t r a t i f i e d ocean. of P h y s i c a l Oceanography 5, 300-325. Journal Chang, P.Y.K., 1976. Subsurface currents i n the S t r a i t of Georgia, west of Sturgeon Bank. M.Sc. t h e s i s , U n i v e r s i t y of B r i t i s h Columbia. 181 pp. Chang, P., S. Pond, and S. Tabata, 1976. Subsurface currents i n the S t r a i t of Georgia, west of Sturgeon Bank. Journal of the F i s h e r i e s Research Board of Canada 33 : 2218-2241. Crean, P.B., and A.B. Ages, 1971. Oceanographic records from twelve c r u i s e s i n the S t r a i t of Georgia and Juan de Fuca S t r a i t , 1968. Department of Energy, Mines and Resources, Marine Sciences Branch, Vol. 1-5. de Szoeke, R.A., stability. 1975. Some e f f e c t s of bottom topography on b a r o c l i n i c Journal of Marine Research 33 : 93-122. Helbig, J.A., and L.A. Mysak, 1976. S t r a i t of Georgia o s c i l l a t i o n s : lowfrequency waves and topographic planetary waves. Journal of the F i s h e r i e s Research Board of Canada 33 : 2329-2339. H i l l e , E., 1969. Lectures on Ordinary D i f f e r e n t i a l Equations. Addison-LJesley. Huthnance, J.M., 1975. On trapped waves over a c o n t i n e n t a l s h e l f . Journal of F l u i d Mechanics 69 : 689-704. K a j i u r a , K., 1974. Long period waves of s h e l f and Helvin modes i n a twol a y e r f l u i d with a s h e l f . Journal of the Oceanographic Society of Japan 30 : 271-281. LeBlond, P.H., 1975. Long wave propagation i n channels of non-rectangular c r o s s - s e c t i o n . Unpublished manuscript, 18 pp. Longuet-Higgins, M.S., 1964. Planetary waves on a r o t a t i n g sphere. Proceedings of the Royal Society A. 279 : 446-473. Mctililliarns, J . C , 1974. Forced t r a n s i e n t flow and small scale topography. • Geophysical F l u i d Dynamics 6 : 49-79. MaDers, C.W.H., 1973. A technique f o r the cross spectrum a n a l y s i s of p a i r s of complex-valued time s e r i e s , with emphasis on properties of p o l a r i z e d components and r o t a t i o n a l i n v a r i a n t s . Deep-Sea Research 20 : 1129-1141. Pedlosky, J . , 1964. ocean: Part I . The s t a b i l i t y of currents i n the atmosphere and the Journal of the Atmospheric Sciences 21 : 201-219. 68 Rhines, P.El., 1969. Slow o s c i l l a t i o n s i n an ocean of varying depth. Part 1. Abrupt topography. Journal of F l u i d Mechanics 37 : 161-189. 1970. Edge-, bottom-, and Rossby waves i n a r o t a t i n g s t r a t i f i e d f l u i d . Geophysical F l u i d Dynamics 1 : 273-3G2. Suarez, Alfredo A., 1971. The propagation and generation of topographic o s c i l l a t i o n s i n the ocean. Ph.D. t h e s i s , Massachusetts I n s t i t u t e of Technology. Tabata, 5., and J.A. S t r i c k l a n d , 1972a. Summary of oceanographic records obtained from moored instruments i n the S t r a i t of Georgia - 1969-1970: Current v e l o c i t y and seawater temperature from Station H-06. Environment Canada, Water Management Service, Marine Sciences Branch, P a c i f i c Region, P a c i f i c Marine Science Report IMo. 72-7. 132 pp. 1972c. Summary of oceanographic records obtained from moored instruments i n the S t r a i t of Georgia - 1969-1970: Current v e l o c i t y and seawater temperature from Station H-26. Environment Canada, Water Management Service, Marine Sciences Branch, P a c i f i c Region, P a c i f i c Marine Science Report IMo. 72-9. 141 pp. Tabata, S., J.A. S t r i c k l a n d and B.R. de Lange Boom, 1971. The program of current v e l o c i t y and water temperature observations from moored instruments i n the S t r a i t of Georgia - 1969-1970 and examples of records obtained. F i s h e r i e s Research Board of Canada, Technical Report IMo. 253. 368 pp. Ueronis, G., and H. Stommel, 1956. The action of a v a r i a b l e wind s t r e s s on a s t r a t i f i e d ocean. Journal of Marine Research 15 : 43-75. Waldichuk, M., 1957. P h y s i c a l oceanography of the S t r a i t of Georgia, B r i t i s h Columbia. Journal of the F i s h e r i e s Research Board of Canada 14 : 321-486. Wang, D., and C.IM.K. Mooers, 1976. Coastal trapped waves i n a continuously s t r a t i f i e d ocean. Journal of P h y s i c a l Oceanography ( i n p r e s s ) . Yoon, J.H., 1974. Continental s h e l f waves i n a two-layer ocean. t h e s i s , U n i v e r s i t y of Tokyo. M.Sc. 69 Appendix A. The two-layer equations of motion Although the equations of motion f o r a two-layer f l u i d are w e l l known (see e.g., l/eronis and Stommel, 1956), they are derived here f o r the sake of completeness. Ue consider a system of two layers o f incompressible / i n v i s c i d , r o t a t i n g , and homogeneous f l u i d s r e s t i n g one upon the other. Since only low-frequency motions are of i n t e r e s t to us, we invoke the hydrostatic approximation. Then the non-linear dynamical equations governing the motion i n each l a y e r are (A.D } (A.2) Here the s u b s c r i p t i = 1,2 r e f e r s to the upper and lower layers r e s p e c t i v e l y , the C o r i o l i s parameter f i s defined by f = f o i s the g r a v i t a t i o n a l a c c e l e r a t i o n . ^ y , and g Ue assume from the Dutset that the h o r i z o n t a l v e l o c i t i e s are z-independent v a r i a t i o n s are permitted i n the d e n s i t y . throughout each layer as no This system i s subject to kinematic and dynamic boundary conditions at the surface and i n t e r f a c e . In a d d i t i o n the normal component o f the v e l o c i t y must vanish at the s i d e w a l l s (x = D,L) and the bottom (z = -H(x)). In order to s i m p l i f y these equations we w i l l express the pressure p^ i n terms Df the sea surface and i n t e r f a c i a l displacements and o^, r e s p e c t i v e l y , and we w i l l v e r t i c a l l y integrate the c o n t i n u i t y equations. I t f o l l o w s from the h y d r o s t a t i c r e l a t i o n A.2 that the pressure i n each l a y e r i s given by 70 (A.4) where f,, and are a r b i t r a r y functions to be determined. The dynamic boundary conditions require that the pressure be continuous at the i n t e r faces. Let p (x,y,t) be the atmospheric pressure at the sea surface, then 3 we have (A.5) Since we are concerned s o l e l y with free motions we regard p as constant; a s u b s t i t u t i o n of B.4 and B.5 gives Upper Layer UU t \?. u., 4- . 9 UL, RT - - £ V, = - ^ - Y ^ K (A.6) o (A.7) Lower Layer (A.8) (A.9) I t remains to v e r t i c a l l y integrate (A.7) and (A.9). upper l a y e r f i r s t ; Consider the i n t e g r a t i o n of (A.7) gives. (A.10) The kinematic boundary conditions require that ( ^ H , ^ and z ="Ji-.h^ . ) ^i ^i / a at z = Using these r e l a t i o n s h i p s f o r w^ and using Lagrange's i d e n t i t y f o r the d e r i v a t i v e of an i n t e g r a l we obtain ^ 71 Since the h o r i z o n t a l v e l o c i t i e s are assumed depth independent, we have a t MW.+ Y -vlx v C^>1 (A.11) ^v^, The treatment of the lower layers d i f f e r s i n that there i s a s o l i d boundary at z = - H(x,y). Analogous to (A.ID) ue have (A.12) but the appropriate boundary conditions arE (A.13) and 0.4- (A.14) £ = -HCX.^) Using (A.13) and (A.14) and again i g n o r i n g any depth dependence of or uie obtain (A.15) Ue have now obtained our complete set of two-layer equations which are: Upper Layer (A.16) Lower Layer V (A.18) (A.19) 72 Appendix B. Theorem proofs In t h i s appendix we prove the theorems stated i n Chapter 6.2. lile f i r s t r e c a l l , however, the v e r t i c a l eigenvalue problem defined by (6.2.7) • (6.2.9), n" - K"1 6 n'-O n'» r n - o (B.D cxk i ^ O -k/<r (Q.2) v- n CB.3) Here a prime denotes d i f f e r e n t i a t i o n with respect to z. 2 Theorem 1. positive. For p o s i t i v e B , With t h i s choice flC ) Proof: 1 S D n E s i g n , which we choose as i s a s t r i c t l y decreasing f u n c t i o n of z. 1 M u l t i p l y (B.D by H and i n t e g r a t e o v e r t * . ! ; f o l l o w i n g 0 one i n t e g r a t i o n by parts we obtain r° r A p p l i c a t i o n of (B.2) gives 'o nn'ia » - / ( n'+- K S n )<U z Jl 9 2 a A (B.4) / . since B i s p o s i t i v e by hypothesis, H n (£) i s everywhere negative. follows that n has no zeros i n the i n t e r v a l £o x ] 1 t and hence i s everywhere We choose that s i g n as p o s i t i v e so that n ' < o of one s i g n . the theorem f o l l o w s . It and thus This j u s t i f i e s the normalization that we chose i n Chapter 6.2. Theorem 2. There e x i s t s one and only one s o l u t i o n to (6.2.7) - (6.2.9). Proof: This proof i s based on Lemma 7.6.1 of H i l l e (1964) which we Let G(x}>0 r e s t a t e here: be continuous i n (-oo,-t-oa). Then the equation fu-) - (jL<,^U) = O (B.5) 73 has one and only one s o l u t i o n y+AiO passing through ( 0,1) which i s p o s i t i v e and s t r i c t l y decreasing f o r a l l x and one and only one s o l u t i o n "y.OO through (o,|) which i s p o s i t i v e and s t r i c t l y increasing f o r a l l . x. Although t h i s lemma applies to the i n t e r v a l (o,» ) i t i s equally as v a l i d on ( - 1 , 0 ). (B.5) and define Vt^) - y (-*-) To see t h i s replace y b y i n to obtain YCO - CrCOVttt (B.6) Since G(x) i s p o s i t i v e f o r X f c C r ) } , the stated lemma applies to (B.6) 00 and hence to NJUC) on (-1,0 00 ). In Theorem 1, we showed that l~l was p o s i t i v e (by choice) and s t r i c t l y decreasing on ^fct-'.o] , hence we conclude that i t i s the only s o l u t i o n to ( B . l ) - (B.3). The frequency <f i s always p o s i t i v e . Theorem 3. Proof: Take (B.4) with z" = - 1 to obtain _ n ' - K <^ -n > + <n' > i n where 2 t l defines the i n t e g r a l Jt')&£ • Substituting f o r f l 'ii) from (B.3), we obtain fc\-D n V o W<r = < s n " > *• < n ' * > > o CB.?) z 2 2 Since B n i s p o s i t i v e d e f i n i t e and since k was chosen t o be p o s i t i v e 6~ | i s positive also. Theorem 4. (T^-^^Vk ;\ asProof: The group v e l o c i t y (f^ i s p o s i t i v e f o r a l l k. k-»> co i s bounded by As W-^o^ G'/k. For convenience, i n t h i s proof i t i s to be understood that a v a r i a b l e not w i t h i n an i n t e g r a l i s to be taken at z = -1. Ue obtain an expression f o r 6^ by d i f f e r e n t i a t i n g (B.7) with respect t o k; rearrangement we obtain after ^ _ ak<6 n >-^K <6 -nn > l i 1 1 k k s'nnj, - 3L<n'n'> +- (B.8) k Here the notation ilV represents -cLlnC-ft] • B Y using (B.D - (B.3), obtain the i d e n t i t y < n'n J> + x < n> - x < n'n' * n"n^> - < tn'n^'? ? 5 \ - - n ' n , - (B.5 R2. ^ n n ( where ^ represents some v a r i a b l e or parameter i n the argument D f HI. S u b s t i t u t i o n of (B.9) with f = k into (B.8) gives where ^ 8n w k x v i s defined by > 5^ - ric-0{ n (-0 k [ n(H)] J ca.io) k I t f o l l o w s that from the form of ( B . l ) that 5^-° • To see t h i s con- s i d e r that H depends on k only through K as (B.3) serves simply t o determine <T and places no r e s t r a i n t s on the form of 17. a n d that k^~o K enters Hence i t f o l l o u s that n i s of the (B.l) only as a s c a l i n g parameter. form H(Kt) , (MDLJ as long as the d e r i v a t i v e s i n question are not evaluated at z = o. Thus we have \ - °^ - U cr r . < " > P 5. o" W l (B.11) 75 Ue now w i l l show that the second term i n brackets i s bounded by unity. To do so ue require an expression f o r (P. cr = - k 6 From (B.3) ue have n / n ' l A s i n g l e i n t e g r a t i o n of (B.D gives rV- - K" < B*n > 1 And ue thus obtain (B.12) cr S u b s t i t u t i o n of (B.12) i n t o (B.H) gives (B.13) * k wL \cn<8 n> . l l Consider nou the second term i n brackets; sincef~l(z) a t t a i n s i t s maximum value at z = -1 (Theorem 1) ue obtain 1 (B.14) K n<& n> = u l I t f o l l o w s that T<1 since ak _ ak < ^k/Cwrv) ' < Xk/ W <l 1 L \<<<^ a/k <\ - k> iHence tf^ i s aluays p o s i t i v e and i s bounded byff/j^since T> 0. Ue nou consider the l i m i t i n g cases Df k-^o and k—>co, although the v a l i d i t y of the dynamical equations i s doubtful i n the l a t t e r case. 2 2 2 In the f i r s t case, T = 2k/(k + n TT ) = 0(k)as k->o and hence as k -» a. ^/k. In the second case, T = 0 (1/k) and hence i T ^ i s ^bounded by the phase speed ^/fc. Ue are not able to show that C f i s a bounded function of k, however, i f i t were not, our chosen s c a l i n g would be i l l e g i t i m a t e and our perturbation s o l u t i o n i n v a l i d . 2 Theorem 5. I f B (z) a t t a i n s i t s minimum value at z = -1, the 2 frequency i s an increasing function of B (z = - 1 ) . Proof: The proof of t h i s theorem i s s i m i l a r to that of Theorem k. Again i t i s understood that v a r i a b l e s not w i t h i n an i n t e g r a l are taken at z = -1 . We d i f f e r e n t i a t e (B.7) to obtain an expression f o r (T^j. a<n'nV> * ^ - n n c " '6' V using the i d e n t i t y (B.9) with %~ Q we obtain rf 1 s" 8 crn B An analogous argument to that i n Theorem k shows that kgi_-° . With t h i s and the r e l a t i o n (B.12) f o r CT we may write "n <& n>l l cr * 1 | 8 <n >. x (B.15) x 2 Since Hit) a t t a i n s i t s maximum and, by hypothesis, B (z) a t t a i n s i t s minimum at z = - 1 , we estimate the f i r s t term i n parenthesis as n < g> n> z ^ z & <n > x <s -n> G n<n> 2 n L I t f o l l o w s then that (T^2.>0 Theorem 6. < s -n> 1 > ( <&-n> and (T i s an increasing f u n c t i o n of B . The frequency i s a decreasing function of the cross- channel mode number. Proof: As before we take v a r i a b l e s outside an i n t e g r a l sign at z Replace (_A(\) by the continuous v a r i a b l e X. i n (B.7), then holding k f i x e d we d i f f e r e n t i a t e (B.7) with respect to K * " - CVc1"*-£*") to obtain 77 lilse of the i d e n t i t y (B.C) with ? = K - - 2 gives <8 n*> v since an argument l i k e that i n Theorem k shows that ^Y^~~° • I t follows <Tyz.<0 and that (T i s a decreasing function of n. d i r e c t l y that 2 Theorem 7. 2 I f B^Cz) B^Cz) f o r a l l " , then the corresponding s o l u t i o n s H , and f \ of (6.2.7) - (6.2.9) s a t i s f y PI, 111 < f ^ C i ) f o r a l l z < o. Proof: by M u l t i p l i c a t i o n of the f l , equation by PI and the ("Inequation t and i n t e g r a t i o n over the i n t e r v a l (ifciO) gives n/n,. = - Jr\,'n{W n, r y = - jf n,/n;: A - * " K X ^ J j y 8 n, n , * n , n <U' " ^ We subtract the second expression from the f i r s t to obtain •o n,'n which may be r e w r i t t e n as A ( n.'\ _ _ Kn£ /(B^-BJ) n.n»At' l (B .16) 78 A second i n t e g r a t i o n gives Hi I t follows that n,<n t f o r a l l !<<=>. Theorem 8. H ( ) i s an increasing function of n. z Proof: This theorem i s proved i n exactly the same manner as the preceeding one. Lit) By analogy with (B.1S) we may write (C-K^n^l^n.rU.- - - a second i n t e g r a t i o n gives n, n I t follows that n , < f l k, n , < n L i fn r f o r a l l z < O i f K,*< ^ x < ,n . 2 , and thus f o r f i x e d 79 Appendix C. Glossary of symbols B = f NH/f L = + p y " C o r i o l i s parameter f = 1 + ^ y - IMan-dimensional C o r i o l i s parameter g - Gravitational acceleration -• Mean upper layer depth h^Cx) - Mean lower layer depth h^Q = Maximum mean lower l a y e r depth H = k, k , d - IMondimensional and dimensional wave numbers K = k L - Channel width n - Cross-channel mode number l\l - Brunt-Uaisala" frequency p - Pressure L 2 +h 2 Q - Maximum mean depth of water column 2 + (nn) 2 P Q - E q u i l i b r i u m pressure r g = (gH)^/f - External deformation radius (i>4,5) T \ (g ± I f\lH/f D (§k,5) /h^/f (^6) " •'• '' n :e:rna l deformation radius R x = uVu„ R =v./v - Ratios of upper to lower l a y e r i c., y i d. velocities (J4,5) R = 9 R(z z ) = R =R (§5, to zero order i n & ) x y ' s u(z )/u(z ) = v ( z / v ( z ) (§6) t = Time T = wave period .- The upper ( i = 1 ) and'lower ( i = 2) layer v e l o c i t i e s i n the x and y d i r e c t i o n s J r 2 (u., v.) 1 1 /] 2 1 2 80 eastward, northward, upward v e l o c i t y (#6) eastward, northward, and upward d i r e c t i o n s (V 2 + f ) 2 f 2>X 2 (Non-dimensional) -1 x t t Zfo 2 Bottom slope Dimensional and non-dimensional 6-plane parameters lower layer deformation radius The surface and i n t e r f a c i a l displacements 5) The surface displacement ($6) Non-dimensional and dimensional wavelengths K B (L/r^) - Square of the r a t i o of the channel width t D the i n t e r n a l deformation radius V e r t i c a l s t r u c t u r e function ($6) Upper and lower l a y e r d e n s i t i e s Density ($6) E q u i l i b r i u m and representative constant density ($6) Non-dimensional angular frequency 81 $ ^ \ ^ ifi^ ^ - ^ = j t h order terms i n the expansions of $ and - dimensional angular frequency
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Low frequency current oscillations and topographic waves in the Strait of Georgia Helbig, James Alfred 1977
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Title | Low frequency current oscillations and topographic waves in the Strait of Georgia |
Creator |
Helbig, James Alfred |
Date Issued | 1977 |
Description | Chang (1976, see also Chang, Tabata, and Pond, 1976) has shown that up to 46% of the kinetic energy associated with horizontal motions in the Strait of Georgia, British Columbia, is contained in broad-banded, low-frequency current oscillations characterised by periods ranging from 4 to over 100 days. The purpose of this thesis is to present a simple dynamical model which may provide a partial explanation of these oscillations. The Strait of Georgia is modelled by an infinitely long, rectangular channel with a bottom that slopes upward to the east. Two choices of the density stratification are studied: (1) a two-layer system and, (2) a system with constant Brunt-Väisälä frequency. Both models admit northward-travelling topographic planetary waves with periods that lie in the observed range. However these models do not accurately predict the vertical distribution of horizontal kinetic energy. In addition several general theorems regarding phase and energy propagation and the vertical structure of these waves in a system with arbitrary continuous stratification are proven. In particular it is shown that both phase and energy propagate northward, and that in a stably stratified system the wave amplitude increases with depth. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-02-16 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0053209 |
URI | http://hdl.handle.net/2429/20306 |
Degree |
Master of Science - MSc |
Program |
Oceanography |
Affiliation |
Science, Faculty of Earth, Ocean and Atmospheric Sciences, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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