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Low frequency current oscillations and topographic waves in the Strait of Georgia Helbig, James Alfred 1977

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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 i n THE FACULTY OF GRADUATE STUDIES (The Department of Physics and The I n s t i t u t e of Oceanography) We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA February, 1977 © dames A l f r e d H e l b i g , 1977 In present ing th is thes is in p a r t i a l fu l f i lment o f the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f r ee ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for s c h o l a r l y purposes may be granted by the Head of my Department or by h is representa t ives . It is understood that copying or p u b l i c a t i o n of th is thes is fo r f i n a n c i a l gain sha l l not be allowed without my wr i t ten permission. James Alfred Helhig r, , . f Physics and the I n s t i t u t e of Oceanaqraphy Department or ? 3 r 1 The Un ivers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 D a t e February k, 1977 i i ABSTRACT Chang (1976, see a l s o Chang, Tabata, and Pond, 1976) has shown that up to kG% of the k i n e t i c energy associated with h o r i z o n t a l motions i n the S t r a i t of Georgia, B r i t i s h Columbia, i s contained i n broad-banded, low-frequency c u r r e n t o s c i l l a t i o n s c h a r a c t e r i s e d by periods ranging from U to over 100 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 S t r a i t of Georgia i s modelled by an i n f i n i t e l y long, r e c t a n g u l a r channel with a bottom that slopes upward to the east. Two choices of the d e n s i t y s t r a t i f i c a t i o n are s t u d i e d : (1) a two-layer II II II system and, (2) a system with constant Brunt-Uaisala frequency. Both models admit n o r t h w a r d - t r a v e l l i n g topographic planetary waves with periods t h a t l i e i n the observed range. However these 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. In a d d i t i o n s e v e r a l general theorems regarding phase and energy propagation and the v e r t i c a l s t r u c t u r e of these waves i n a system with 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 proven. In p a r t i c u l a r i t i s shown that both phase and energy propagate northward, and that i n a s t a b l y s t r a t i f i e d system the wave amplitude increases with depth. i i i TABLE OF CONTENTS n Page Abs t r a c t i i L i s t of TablES i v L i s t of Figures v Acknowledgements v i Chapter 1. I n t r o d u c t i o n 1 2. P h y s i c a l oceanography of the S t r a i t of Georgia k 3. The model 19 k. Lou-frequency non-topographic waves i n a tuio-layer f l u i d . Zk k . l The governing equations Zk k.Z I n t e r n a l K e l v i n waves 27 k.3 Rossby waves 31 5. Topographic waves i n a two-layer 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 6. Topographic waves i n a f l u i d with continuously varying d e n s i t y L& 5.1 The pressure equation 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 2 53 6.3 Constant N 2 model 58 7. Summary • 65 Bi b l i o g r a p h y 67 Appendix A. The two-layer equations of motion 69 Appendix B. Proofs of theorems 72 C Gl ssary of symbols 9i v LIST DF TABLES Table Page I. Mean currents 11 I I . Wavelength, frequency, and period of i n t e r n a l Kelvin waves i n the S t r a i t of Georgia 31 I I I . Wavelength, frequency, and period of the f i r s t cross-channel bartotropic and b a r o c l i n i c Rossby uaves i n the S t r a i t of Georgia . . . . . . . . . . . . 34 IV. Wavelength, period, and phase speed of the f i r s t cross-channel mode i n the S t r a i t of Georgia (two-layer model) . . . kk V. Wavelength, period, and phase speed of the f i r s t cross-channel mode i n the S t r a i t of 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 stations 1-k 8 k. Brunt-Vaisala frequency IM for 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 ) , for low-frequency o s c i l l a t i o n s 13 B. Coherence"and"phase spectra between v e r t i c a l l y separated currents IS 9. Coherence'and"phase'spectra between horizontally separated currents 17 10. Model of the S t r a i t of Georgia 20 11. Smoothed topographic cross sections 21 12. F i t of model to section K 23 13. Dispersion r e l a t i o n for the f i r s t three cross-channel modes (twD-layer model) k3 lk. V e r t i c a l d i s t r i b u t i o n of horizontal velocity f o r the f i r s t three cross-channel modes (two-layer model) ^5 15. Dispersion r e l a t i o n for the f i r s t three cross-channel modes (constant IM2 model) 62 16. V e r t i c a l d i s t r i b u t i o n of horizontal velocity f o r the f i r s t three cross-channel modes (constant IM2 model) 64 ACKNOWLEDGEMENTS v i I t i s my sincerest pleasure to express my gratitude to Dr. L.A. Mysak in his capacity as t h e s i s supervisor f o r his patience, advice, good humour, and unflagging enthusiasm during the course of t h i s work. Thanks are also due to Dr. R.E. Thompson f o r his remarks concerning Helbig and Mysak (1976) and to Drs. P.H. LeBlond and R.kl. Burling f o r t h e i r comments con-cerning t h i s t h e s i s . I thank Mr. P. Chang f o r allowing me tc- use his thesis p r i o r to i t s completion. I would also l i k e to express my appreciation to the University of B r i t i s h Columbia and National Research Council of Canada who provided me with f i n a n c i a l support during the tenure of t h i s study. 1. Introduction Chang (1976) (see also Chang, Pond, and Tabata, 1976) has recently shown that up to kS% of the horizontal k i n e t i c energy i n the S t r a i t of Georgia (see F i g . 1) consists of broad-band low-frequency motions. This result was inferred from the analysis of IS months of continuous current records obtained by Tabata and Strickland (1972, a,b,c) (see also Tabata, Strickland and de Lange Boom, 1971) during the period extending from A p r i l 1969 u n t i l September 1970. These fluctuating 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 thesis 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 thesis i s as fallows. A brief 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, straight channel with v e r t i c a l walls and a sloping bottom shoaling to the east (see Chapter 3). These data also indicate that a two-layer system i s a suitable 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, inter 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; Internal Kelvin waves of the proper frequency have wavelengths that are much too long to f i t i n GS, and Rossby waves have periods that are far greater than those observed. In Chapter 5 i t i s shown that the model admits northward-propagating topographic planetary waves as solutions. An application of the 2 theoretical r e s u l t s to GS i s made and reveals that these waves have f r e -quencies lying 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 horizontal 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 structure and propagation of waves admitted by t h i s model are proven. Application of these results to GS for the special case of constant Brunt-Vaisala frequency yields s i m i l a r results to those mentioned above. This thesis contains three appendices. 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 th i s thesis 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 thesis f a l l s into the mainstream of a currently active branch of research i n physical oceanography. The major d i f f i c u l t y confronting the investigator i n t h i s f i e l d i s his i n a b i l i t y to separate the v e r t i c a l dependence from the horizontal 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 variations and of homogeneous systems with variable topography, these exist at present only a handful of studies made of systems embodying both features. Yet i n the r e a l world, they always exist 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 especially apparent i n coastal regions where the application of physical oceanography to p r a c t i c a l problems (for example, environmental) i s of increasing importance. with these thoughts i n mind, the significance of t h i s f i e l d of research i s pl a i n and should 3 not be underestimated. The study D f s t r a t i f i e d systems with v a r i a b l e topography began with Rhines (197D) who discovered the existence of a wave trapped to the bottom i n a continuously s t r a t i f i e d f l u i d (constant !M) with l i n e a r topography. Some of Rhines r e s u l t s are reproduced i n Chapter 6.3 although the present a n a l y s i s d i f f e r s i n some respects from h i s . Suarez (1971) and McLJilliams (1974) have considered both f r e e and forced motions over a v a r i e t y of simple topographies i n a constant l\l model. K a j i u r a (1974) and Yoon (1974) have studied f r e e waves i n two-layer, systems with 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 . Both demonstrated the p o s s i b l e i n t e r a c t i o n of shelf-waves with 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 to be important when the i n t e r n a l deformation r a d i u s i s equal to or exceeds the breadth D f the c o n t i n e n t a l s h e l f . A l l a n (1975), u t i l i z i n g a very complex p e r t u r b a t i o n scheme, has studi e d t h i s i n t e r a c t i o n i n a two-layer system with exponential topography. F i n a l l y , wang and Mooers (1976) have solved numerically the equations of motion i n a system with a.continuously v a r y i n g d e n s i t y and a f i n i t e e x p o n ential c o n t i n e n t a l s h e l f . They c l e a r l y i d e n t i f i e d the r e l a t i v e r o l e s of the two tra 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 the 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* Physical Oceanography of the S t r a i t pf Georgia Although the physical oceanography of the S t r a i t of Georgia (GS) has received comprehensive treatment elsewhere (cf., Idaldichuck 1957), i t i s important to summarize here some of the p r i n c i p a l features of bathy-metry and s t r a t i f i c a t i o n i n order to motivate a sim 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 calculated by Chang (1976) from measurements made by Tabata and Strickland (1972) and also b r i e f l y consider results of his analysis of the winds, sea l e v e l , , atmospheric pressure, and water temperature. A plan view of GS i s shown i n Fig. 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 less 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 constricted 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 Fig . 2 and. were extracted from a topographic map of GS compiled by Dr. P. Crean (personal communication) giving 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 exhibits 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 sector. 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 ex i s t , a narrow one to the east of Texada Island and a much wider one on the western side. South of l i n e D the topography becomes progressively smoother; l i n e s G and H i l l u s t r a t e the marked effect of Fraser River sedimentation as extensive banks along the east. The F i g . 1. A plan view of the S t r a i t of Georgia, B.C., showing l i n e s of topographic cross sections (A-K) presented i n F i g . 2 and 5, stations (1-4) at which v e r t i c a l density p r o f i l e s are given i n F i g . 3, and the locations of the current meter moorings (HOG, H16, H2G). 6 DISTANCE FROM EASTERN BOUNDARY (Km) 50 40 30 30 10 h-JOO DISTANCE 200 I . . . FROM NORTHERN BOUNDARY 1S0 100 . I . . . . I . . . . (Km) so I . . ( * • — i I H Boundary Passaqe G Fraser . R. F E O C Taxada I. B A Cap« Mudqe VM J \ * i\ 0 •loo E •200 p Q. UJ 300 Q •400 F i g . 2. Topographic cross sections: (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 longitudinal 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 exhibits slopes that often exceed I D - 2 . Seasonal v e r t i c a l density p r o f i l e s for stations 1 to h, based on data collected by Crean and Ages (1971), are platted i n F i g . 3. Relatively l i t t l e change occurs throughout the year below 50 m, but strong seasonal effects, especially near the Fraser River, exist 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 calculated and i s shown i n F i g . k. hie see that IM fluctuates s i g n i f i c a n t l y , depends strongly on depth, and -3 -2 -1 generally l i e s within the range of 3 x ID to 3 x 10 rad s ; the water column i s thus well s t r a t i f i e d . Some results of Chang's analysis of GS currents are presented i n Table 1 and i n Figs. 6 and 7. The current records examined by Chang were collected at stations H06, H16, and H26 as shown i n Figs. 1 and 5. Meters were positioned at 5 (approximately), 50, and 200 m i n the western (HOG) and central (H1G) locations and at 5 (approximately), 50 and IkO m i n the east (H26). Chang did not analyze records from the near surface instruments. 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 either every 10 (Aanderaa) or 15 (Geodyne) minutes. A subsurface buoy mooring was used for the i n i t i a l year of the experiment, but was replaced thereafter 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. F i g . 3. Density p r o f i l e s at stations 1-4. F i g . k. Brunt-Vaisala frequency IM for the p r o f i l e s shown i n F i g . 3. The Brunt-Vaisala frequency i s defined by IM2 =- (g/ f , » ) /"bl, where p„ i s the mean density of the water column (taken as f0 =1.0235). U3 4 0 0 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 directions are i n degrees measured from true North WEST CENTRAL EAST 50 m 200 m 50 m 200 m 50 m 140 mean velocity (cm s" 1) 3.0 ± .5 1.6 ± .8 4.3 i .5 1.4 + .5 3.0 + 1.1 7.4 + mean velocity d i r e c t i o n 355° 346° 10SD 21° 9° 7° rms speed (cm s ) 7.7 3.2 8.2 3.6 7.0 8.4 PERIOD (days) 0.1 1.0 I • r . l 10 100 1000 t »l » »il i i il L 1000 100 li I i L L L 10 1.0 0.1 I ' t i li i i L 52.1 66.1 20-r CM 'o "10 + E u eastern 50 m I -3 -4 ' ° 9 i o F + F Fig. 6. Current spectrum S(f) cf 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 2 + v 2 (Chang 1976). F i g . 7. Smoothed current spectra, f S ( f ) , for lou-frequency o s c i l l a t i o n s . 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 da i l y time series of currents. The mean and rms values were calculated by averaging current vectors and magnitudes from t h i s time series. Hence, the rms values correspond to the ki n e t i c energy contained i n periods less than two days and contain no t i d a l contributions. 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 central location, 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 locations, while the rms velocity i s twice as large. In the east and the west both the shallow and deep currents are closely aligned with the l o c a l topography. Rotary spectra of the low-frequency currents are shown i n Figs. 6 and 7. These spectra were obtained by s p l i t t i n g each frequency component of the velocity vector u into two other vectors, one of which rotates with a positive frequency (anti-clockwise) and the other with a negative frequency (clockwise). Hence the positive or negative spectrum repre-sents the respective tendency for the currents to move i n an a n t i -clockwise or clockwise sense. The reader i s referred to Chang (1976) or Moores (1973) for 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 to reduce the complexity of t h i s and sim i l a r spectra and to concentrate on low frequencies, Chang's spectra were further smoothed to examine frequencies less than 0.25 cpd; these are shown i n F i g . 7. Examination of Figs. 6 and 7 reveals the extremely complex nature of the low-frequency currents i n GS. The 15 general c h a r a c t e r i s t i c s of these spectra i s t h e i r bread handedness, as appreciable energy i s contained i n periods ranging from k to over 100 days. It must be emphasized that most of the f i n e - s t r u c t u r e present i n F i g . 7 i s probably 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 to 95%. Nevertheless, a number Df more de t a i l e d conclusions may be drawn: (1) At the same depth, the shapes of spectra from the eastern and western stations are i n better agreement with each other than they are with those from the ce n t r a l l o c a t i o n . (2) Spectra from the c e n t r a l s t a t i o n peak at lower frequencies than do either the western or eastern current spectra. (3) A barotropic motion of 10 - 25 day period i s c l e a r l y v i s i b l e i n the eastern spectra, while motion i n t h i s range appears b a r o c l i n i c elsewhere, with most of the energy i n the upper l a y e r . (4) 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 period and one of near yearly period i s apparent at a l l s t a t i o n s . Again most of the energy i s i n the upper l a y e r . The l a t t e r motion, however, may be related to annual changes i n s t r a t i f i c a t i o n i n GS. Chang (1976) found that coherences between currents at posi 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 generally small at low frequencies as i s shown i n F i g s . 8 and 9. The highest coherence between v e r t i c a l l y separated currents was observed i n the east. There the upper and lower layer v e l o c i t i e s were nearly i n phase which i s i n d i c a t i v e of a barotropic motion. At the other locations the v e r t i c a l coherence was very small and the phases were scattered; t h i s suggests l i t t l e or no coupling between the upper and lower layers and hence implies mainly b a r o c l i n i c motions there. In a l l cases the horizontal coherences were buried beneath the 95% noise l e v e l . 16 (A) EASTERN -1 1 1 1 — -2 -1.5 -I -0.5 FREQUENCY (CPO) • «• • • 0.97-, 0.95 -J 0.9-| 0.8 0.1 0.5 . 0.3 - O.H 1 0 95% 95% 1B0-90-0--BO--180 -4> —1 1 "-1 T-0.5 1 15 2 FREQUENCY (CPO) • • •.• 1 - I (B) CENTRAL 0.95 0.9-0.6 0.7 0.5 . 0.33 — i — ' Old »'i •~-L-i o-l T 2 UO-1 90 0 -90 -180 -> 4> — i — o.s • • • 0.97-, . (C) WESTERN 0.95 0.9-1 _i • . » • * —1 -1.5 0.5 0.33 . . • . ,i»—o.H • ^ o-0.1-, , .7J-Y-2 —p— 0.5 •••• • • 1 • 180 90 . 0 -so-3 -180 4>... 15 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 starting ( A p r i l 17, June 17, July 11) 1959. (Chang, 1976) 17 (D) CENTRAL — EASTERN 0.97-50m » • » • 0.B-0.8-0.7-0.5: . • 0.3: . 0 ' — i 0' I—• 1 1 —I—• 1— -2.5 -2 -1.5 -I -0.5 FREQUENCY (CPB) 180 -j 0 -ao -180 J T95% 93% I 1 "—i—* * ' i— r-0 0.5 1 IS 2 FREOOENCr (CPO) -2.5 (E) CENTRAL-WESTERN 50m —I — r * -T 0.97 0.9 0 8 0.7 0 3 0.33 r O . l ^ - ! 0-> 0 0 180 . 90 0 -90-1 -180 J I ' —I— 0.5 — I 2.5 (F| CENTRAL—WESTERN ZOOm —r— -2 I -1.5 —I—1 -O.S 0.97 0.95-4 0.8-1 o.e 0.7 0.3d . 0.33 — O.t T # «• • • • •. • 0 180-n • * 90 -j 0 -«H -180 —I— O.S — I J.S F i g . 9. Coherence and phase spectra between horizontally separated currents. Parts (D, E,F) are based on (50m, 50m, 200m) currents" from the (central-eastern, 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, July 11, July.11) 1969. (Chang, 1976). 18 Chang also analyzed sea l e v e l , atmospheric pressure, uind, and water temperature records for the 18 month period. The temperatures were c o l -lected by the Aanderaa meters which were equipped to sample currents and temperatures concurrently. In a l l cases these quantities were essenti a l l y uncorrelated with the currents. The highest coherence was found between the currents and the wind at the eastern location, which suggests that the surface wind stress may be a passible forcing mechanism. I t i s not clear i f any other forcing mechanisms are important. 19 3. The Model •UT objective i n choosing a model of GS was to select one that f a i t h -f u l l y represented the gross features of geometry, bathymetry, and s t r a t i f i -cation but at the same time was simple enough to admit an analytic solution 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 sidewalls and a bottom..inclined upwards to the east uith a constant slope <* was chosen as i s shown i n F ig. 1D; a two-layer system was adapted for the s t r a t i f i c a t i o n . The selection 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 suitable bottom topography was more d i f f i c u l t . F i g. 11 shows an overlay of cross sections smoothed from F i g . 2. To obtain these, many small scale 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, in addition, the eastern boundary i n the northern sector (lines A-D) and i n the southern sector (lines G and H) was shifted westward to exclude, respectively, 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 fact that the narrow northern channel f i t s the proposed topo-graphy better than the western channel.) The channel depth varies s i g n i f i -cantly along GS and i s deepest i n the central region ( l i n e s E and F). This i s the area of current measurements and i s best f i t t e d by the model. In f a c t , a strong argument may be made that the model i s applicable only to the sector south of Texada Island, since a northward propagating topographic wave would he extensively scattered by bottom i r r e g u l a r i t i e s near Texada Island. The selection 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 and to permit both barotropic and baroclinic motions. 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 Figs. 2 and 3 yields the following average parameter values: L = 22.2 km h 1 = 50 m h 2 Q = 300 m <* = k.9 x 10" 3 ^ = 1.0215 gm cm"3  2 = 1.02375 gm cm"3 I f the f i t i s made, however, to l i n e K and station 3, (Fig. 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 5 ^ , and j°2 are unchanged, but the topographic parameters become: L = 22 km h 2 Q = 325 m (X. = 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 k. Low frequency non-topographic waves i n a two-layer f l u i d 2k In t h i s chapter two classes cf wave mptipn of potential impprtance i n the S t r a i t of Georgia are considered, namely, internal Kelvin waves and Rossby waves. I t i s shown that int e r n a l Kelvin waves exhibit frequencies too high and Rossby waves too low to provide an adequate explanation of the observed low-frequency current structure. As an i n i t i a l approximation, topographic effects 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 correction to the . Kelvin wave frequency ar i s i n g from topography i s calculated from a formula due to Le LeBlond (1975). However, t h i s correction does not s i g n i f i c a n t l y a l t e r the dispersion 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 (in section k.2) a set of coupled equations for the free surface and i n t e r f a c i a l displacements from the two-layer equations of motion including non-uniform rotation (which are developed i n Appendix A). In section k.2 Kelvin wave solutions are obtained on an f plane and dispersion curves are calculated for parameters appropriate to GS. A scale analysis 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 readily y i e l d Rossby wave solutions. Again the dispersion r e l a t i o n i s evaluated far GS parameters. k.l The Governing equations Although the equations of motion for a two-layer system are well-known (see e.g. veron'is and Stommel, 1956), they are derived i n Appendix A for the sake of completeness. The linearized version of these equations f o r the horizontal v e l o c i t i e s (u^, v^) and for the surface and i n t e r f a c i a l displace-ments,^ and^,, respectively, are given by: 25 Upper Layer Lower Layer (4.1.1) (4.1.2) (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 d e n s i t y d i f f e r e n c e between the upper and lower l a y e r s , g 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 , and f = fo + |?» y i s the C o r i o l i s parameter. The mean upper l a y e r and .lower l a y e r depths, h^ and h^g, r e s p e c t i v e l y are taken as constant. In 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 was invoked s i n c e only low-frequency motions are of i n t e r e s t . In the development that f o l l o w s i t i s convenient to recast these equations i n t o non-dimensional form. To do so the f o l l o w i n g geostrophic s c a l i n g i s used: (u^, v^) = U(u/ , v^ ), t ='fo~ t , (x, y) = L ( x ' y ), = ( f o L U / g ) ^ , and<^ 2 = (foLU/<3 g ) 1 ^ , where a prime denotes a non-dimensional 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 the channel width. The non-dimensional equations are (where we have dropped the primes), . Upper Layer (4.1.5) C^,.- s^t (4.1.6) Lower Layer A 26 (4.1.7) (4.1.8) Here the non-dimensional parameter e = L 2fo./ g h 2 Q compares the channel width L to the lower layer in t e r n a l deformation radius (ogh^) /fo, 6 = f y h ^ i s the r a t i o of layer depths, and f = 1 + |S y i s the non-dimensional C o r i o l i s parameter w i t h ^ = L coffe/Re, where <h i s the mean la 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 velocity components normal to the sidewalls vanish, i . e . (4.1.9) U.t {. K*0 * 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 derivation of (4.1.5) - (4.1.8), (see Appendix A). In order to simplify 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 pair of coupled equations for ^  a n d t ^ Let and the operator (4.1.10) (4.1.11) then (4.1.5) and (4.1.7) give the fallowing expressions for 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 e q u a t i o n s (4.1.6) and (4.1.8) by and a p p l i c a t i o n o f t h e i d e n t i t y - - A ? f v ; (4.1.14) g i v e s f [ 1 J W * v U v ^ 1 - al|S v,= e / 6 a C a $ t (4 .1 . 15) otl U«0* + - - £ oC"( ^ $ ^ ( 4 . 1 . 1 6 ) S u b s t i t u t i o n o f (4.1.12) and (4.1.13) i n t o (4.1.14) and (4.1.15) y i e l d s t h e f o l l o w i n g p a i r o f c o u p l e d e q u a t i o n s , sCL V C$ - € f + S S)J T4. jjTn\($* o ( 4 . i . i a ) 2 2 2 Here"v = + d y i s t h e t w o - d i m e n s i o n a l L a p l a c i a n and t h e o p e r a t o r f ^ i s d e f i n e d by ~ - a l ^ t - (4.1.19) I t 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 must be s o l v e d i n t h e f o l l o w i n g s e c t i o n s . I n terms o f $ and 4, t h e boundary c o n d i t i o n s (4.1.9) become - o 4.2 I n t e r n a l K e l v i n Waves at x=o,l (4.1.20) I n t h i s s e c t i o n , i n t e r n a l H e l v i n 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. From t h e c a l c u l a t e d d i s p e r s i o n r e l a t i o n , e s t i m a t e s o f t h e f r e q u e n c y a r e d e t e r m i n e d f o r K e l v i n waves l i k e l y t o be found i n GS. 28 A A TD proceed then, withjS = 0 and f = 1, (4.1.17) and (4.1.18) reduce to f [ V«$ + * ^ * O (4.2.1) . X I £*C5 * - & ? ) ] t = ° (4.2.2) For t r a v e l l i n g wave solutions of the form r_4)LJ] = [ F U ) , G U ) ] £ l U V ' ° (^.2.3) and uit h k chosen positive for convenience, (4.2.1) and (4.2.2) simplify to F < j f _ kL F *• fc/ & U - <r*) fe = O (4.2.4) ( F * fc)*K - WX C F V (r) - fcU -o-1) ( (y * 6 F ) =• O (4.2.5) To obtain Kelvin wave solutions, i t i s required that u^ must vanish i d e n t i c a l l y for 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 (4.2.6) - Via- Cr ^ ° (4.2.7) /Since these equations are v a l i d for a l l x, we have U. ( yW Cft.S) eW<r* (4.2.8) where A and B are arbitrary constants to be l a t e r determined. Substitution of (4.2.8) into (4.2.4) and (4.2.5) shows that for 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 solution of which yields the dispersion r e l a t i o n -! i Here C r i = l< [ + ( l - 4 6 i ^ / C l + 6 y V ' r ] | j (4.2.ID) . V) = t l ^ ) U C^/rtr (4.2.11) i s the square of the r a t i o of the channel width L to the internal 29 deformation radius r. defined by r{ £ S^Vv.W. / H ] vlS (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 inte r n a l Kelvin wave and the second a barotropic (or external) Kelvin wave which i s henceforth ignored, since <r R r V> CT^. In dimensional form the frequencytO of the i n t e r n a l mode i s given by t o -iJfiWj ^±&TT$ r f / ^ A (4.2.14) and i s thus determined by the ra t i o of the internal deformation radius to the wavelength X j , (the subscript d denotes a dimensional quantity). Since k i s po s i t i v e , the general solution to (4.2.1) - (4.2.2) consists 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 horizontal velocity i n the upper layer to that i n the lower layer i s easily obtained from (4.1.13), (4.2.3), and (4.2.8). Defining Ry = M ^ /\! we have, 9 ^ 3 - b. (4.2.15) H " +- 8 Either (4.2.4) or (4.2.5) may be used to relate the co 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 + M f t (4.2.16) 30 Substitution of (5.2.IS) into (4.2.15) yields ^ = >/, hi » - 1 /(* = - Wo/W, (4.2.17) Thus the current magnitude i n a given layer i s inversely proportional to the layer depth. Notice that the two currents are ISO 0 out of phase and that Ry i s independent of the wavelength. S i m i l a r l y , one finds that the 2 r a t i o of horizontal 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 thin • upper layer (A =0.17), the k i n e t i c energy density i s much greater i n the upper layer. For the average values of the physical quantities l i s t e d i n Chapter 3, the following parameter estimates for GS are obtained: £ = (^ >a - p / p * = 2.25 x 10" 3 r. = 8.1 km l & = 0.17 £ = 1.1 v = ( 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 for selected values of the wavelength. we see that the only waves with periods comparable to those observed i n GS possess wavelengths that are far 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 Kelvin wave. Although, of course, due to the bottom boundary condition a "Kelvin wave" s t r i c t l y can exist only i n a basin with no cross~channel topographic variations. Le Blond (1975) has studied t h i s problem and by employing a perturbation expansion si m i l a r to that used i n Chapter 6 has derived correction formulae for Kelvin wave frequencies 31 i n a two-layer system with arbitrary topography. To f i r s t order i n the parameter 0 1 L/ri , the correction for a l i n e a r slope i s Table I I . Wavelength, frequency, and period of in t e r n a l Helvin waves i n the S t r a i t of Georgia A(km) oo(1D' G rad s" 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 c 0 - _ O • © U That i s , the frequency i s lowered by only one percent and thus the effect i s n e g l i g i b l e . 4.3 Rossby Waves The possible 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 limited importance since the dimensions of GS are small on a planetary scale. 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 proportional to the l a t e r a l dimensions of the basin. A sim i l a r result for baroclinic waves i s 32 anticipated 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 generality, and i t i s useful to perform a scale analysis on these equations i n order to procure a simpl i f i e d set. Since only low-frequency motions are of interest we may employ the inequality 6~« f to obtain the following operator estimates: ^ f 2>A » a? ^ t , 7>*it (4.3.1) (4.3.2) where f = 0(1) and i t i s assumed that °x and ° y are also D ( l ) . (In fact "by should be less than 2 x , but t h i s does not affect the subsequent argument.) I t follows from (4.3.1) that to Q C f / f ) In t h i s approximation, (4.1.17) and (4.1.18) reduce to the boundary conditions (4.1.20) become (4.3.3) $ , q: For a solution of the form - c cr •fc) (4.3.5) (4.3.6) with integrating factor T = p/26~, the fallowing set of coupled equations are obtained C^H ^ ) .(.k+P) - 6(.(r + 5 P) - O (4.3.8) 33 with boundary conditions oJt (4.3.9) Appropriate solutions to (4.3.7) - (4.3.9) are f \ = • • (4.3.10) where An and Bn are arbitrary constants and n i s the cross-channel mode number. For a n o n - t r i v i a l solution, k and 6" must s a t i s f y HA - T l * * ' 1 ' * * } + fcx£/fi0 - o (4.3.11) corresponding to the barotropic and baroclinic modes respectively. Since v i s always p o s i t i v e , cT6c, < 6~BT . l i s t e d for the parameter values given at the end of the previous section. As expected the calculated frequencies are far too small for these waves to possess any dynamical significance i n the S t r a i t of Georgia. Notice that both <faT and (f are almost independent of k for waves exceeding 250 km i n length. 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 neither could provide an explanation of the observed low-frequency current structure i n GS. Hence we are forced to turn to other, more sophisticated models; t h i s i s done i n the next two chapters where models containing variable bathmetry are analysed. Solution of (4.3.11) gives the dispersion r e l a t i o n In Table I I I , <T6T , G~Bc , and the respective periods 'gr and 'at are 34 Table I I I . Wavelength, frequency, and period of the f i r s t cross-channel mode barotropic and baroclinic Rossby waves i n the S t r a i t of Georgia. A A(km) <T&T(10"8rad s" 1) T (days) <f (10"8rad 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 5. Topographic Waves i n a Two-layer Flu i d 35 A class of small-amplitude, low-frequency, free motions that can exist i n the model formulated i n Chapter 3 i s studied i n t h i s chapter. These motions are intimately related to variations i n topography. The dynamics are based on the long-wave equations for a uniformly rotating, two-layer f l u i d . The hydrostatic approximation i s again invoked, since only very low-frequency motions are of i n t e r e s t . The rotation frequency i s assumed to be constant for two reasons: (1) the horizontal length scales i n GS are small (at most 250 km), and (2) the effect of the bottom slope (which exceeds 10 3) outweighs any p o s s i b l e s - e f f e c t s (Rhines, 1969). 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 application of a perturbation expansion in the bottom slope (Section 5.1). In Section 5.2 these results are applied to GS and i n Section 5.3 the dynamics of'\these waves are interpreted i n terms of the r e l a t i v e v o r t i c i t y 5.1 Solution of Equations The li n e a r i z e d equations of motion for 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) (5.1.5) With the exception of (5.1.4)these equations are i d e n t i c a l to the set (4.1.1) (4.1.4). The mean upper layer depth h^ i s taken as constant and the mean louer layer depth i s given by h,_,(x) = h^rjQ -rtx/h^g). I t i s again convenient to work with non-dimensional equations; the same scaling used i n Section 4.1 i s employed to obtain Upper Layer U . , * * - ^ - l ^ / ^ H ^ i - ^ t (5.1.6) Louer Layer 0 - ( u l A T" ^ ) * 6 S a x - t (5.1.8) Here &g =oiL/h^ 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 layer across the channel. Again the appropriate boundary conditions require that u^ vanish on the si d e u a l l s . In order to 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 pair of coupled equations f o r - v ^ a n d • ^ D r travelling-uave' solutions of the form e-i(ky-<ft\ u i t h cr chosen positive for definiteness, (5.1.5) and (5.1.7) y i e l d (5.1.7) 37 (5.1.9) where ^  = 7^ and ^ ^ - S ^ . Substitution of (5.1.9) into (5.1.6) and (5.l.S) gives two coupled equations for §> and j£, + (6/6jCi-<y«-) 9 - O (5.1.ID) - € cr C\-<f1-) (. <5 $ ) = 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 variable 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 solution i n the parameter S^, which i s a p r i o r i assumed small, i s sought. To f i r s t order i n 6S 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 differentiated with respect to l a t i t u d e . To proceed, a l l dependent variables are expanded as power series -in 6 , v i z . , 'By r e s t r i c t i n g 6"to be D(^s)> we n a u e f i l t e r e d out any possible high-frequency (Kelvin or gravity) waves; th i s i s consistent with the aim of searching for 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 + ( € / « ^ 4 ~ O (5.1.13) A * (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 size of k (in 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 easily met for a l l but very long waves). In view of (5.1.15), solutions to (5.1.13) and (5.1.14) of the form 1 ^ , 4 ^ = (.ftn.B*) siwCwirO n--i,23(- --(5.1.16) are sought, where An and Bn are arbitrary constants and n i s the cross-channel mode number. Substitution 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 solution for An and Bn, k and <S~ must s a t i s f y the equation K 4 -T- \Ol>/*Hl-^ ) - (5-1.17) where = k^ + (nTO^. Solution of (5.1.17) far<T yields the fallowing dispersion r e l a t i o n : <T - k l K " + i t * * ) } (5.1.18) As before the parameter v> i s defined by 0 - CL /O 3" (5.1.19) Where the in t e r n a l deformation radius i s now given by t W,Wo/rO L / f (5.1.20) where H = h^ + h^g i s the maximum mean channel depth. 39 Either (5.1.10) or (5.1.11) may be used to relate A -to B ; we Find n > n K - J ^ t l ^ / y ] K 1 " (5.1.21) For a given 6~f (5.1.17) possesses four solutions for k; two of these are r e a l and correspond 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 dispersion r e l a t i o n (5.1.18), i t follows that k i s positive 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 attains a maximum for some intermediate value of k and tends to zero for both small and large k. Thus although the phase always propagates north-ward, energy may be transmitted i n either direction along the channel: northward for small k and southward for large k. Two l i m i t i n g cases are of special i n t e r e s t . Consider f i r s t the situation i n which 0 —*> (weak s t r a t i f i c a t i o n or wide channel, say,, see (5.1.19) and (5.1.20)) then (5.1.18) can be approximated by \ w 6" where r g i s the external deformation radius defined by r e - L £ U * l r t l 6 ] V - 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 " d (5.1.23) This i s simply the dispersion r e l a t i o n for 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-*° (strong s t r a t i f i c a t i o n or narrow channel, say)' , <r — k / C k ^ C«M-V3 (5.1.26) or i n terms of dimensional variables, LO ^ — — — (5.1.25) which, as shown below, i s the dispersion r e l a t i o n for a topographic planetary wave that i s essenti a l l y trapped i n the lower layer. Hence for large v, the motion i s barotropic, while f o r small 0 the motion i s baro-c 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 s t r a t i f i e d f l u i d . The ra t i o s of the horizontal v e l o c i t i e s i n the upper layer to those i n the lower layer are easily obtained from (5.1.9)., (5.1.12), (5.1.IS) and (5.1.21). Define the ratios R = uVuo an^ ^ = v / v n i then to zeroth order i n 6 , s' Z 5- K & 'WO*1^ = C 1+ C Ci+ftWo-) K 1 - ] " ' (5.1.26) we see that as the wavenumber or mode number increases, R becomes small and hence the horizontal 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 for small \) we find that ^ 0/i\^\C « > (5.1.27) and-the motion i s essentially bottom-trapped. This may be true even for very long waves (k 1) since the motion i s quantized i n the x-direction. Therefore to 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 horizontal 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 i of the motion. 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 layer, 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 barotropic. The ration of i n t e r f a c i a l to surface displacement, 0|,_,A^, i s obtained i n a similar fashion to R and R through (5.1.12), (5.1.21) and the . x y defi n i t i o n s of 4 and 4 . To zeroth order i n S ^  one finds 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 result that for 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 proportional 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 layer deformation radius to the channel width and i s a measure of the importance Df the inclusion df a free surface i n the model. Again two cases are of par t i c u l a r i n t e r e s t . For "large iv/6fc , corresponding to a narrow channel Dr thick upper l a y e r , r \ r J ' " \ ^ i s large and the surface displacement may be neglected. In the case of small &l6fe-, which corresponds to a wide channel or- th i n surface layer, ^ ^ l i s D^ o r d e r unity and the motion i s barotropic. One may also construct a closed-basin solution to the system (5.1.13) and (5.1.14). To do so one must include the two complex waves mentioned e a r l i e r . The situation i s thus si 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 conditions. A solution thus consists of a line a r combination of the four waves possible for a given 6" subject to the additional boundary condition that v ^ i =1,2) vanishes on the end walls. The complete solution of the closed-basin problem in pa r t i c u l a r requires that the quartic (5.1.17) be salved for k, which i s a tedious task. 42 5.2 Application to GS For the average values of the physical quantities l i s t e d i n Chapter 3, one obtains the following estimates for the various parameters: £= ( f 2 - f 1 ) / f 2 = 2.25 x 10* 3 6 s = L / h 2 0 = 0.3S r e = 490 km r. l = 8.1 km & = h / h 2 Q = 0.17 £ = 1.1 v = ( I V r ) 2 = 7.5 Ue see that 6 i s perhaps a b i t large for a "small" expansion parameter s thus indicating an extension of the solution to next order i n S should y s be made. On the other hand, the v a l i d i t y of the linearized dynamical equations i s questionable as i s shown below, and suggests that such an extension might be a moot exercise. At any rate, these considerations should not deter us from comparing the results with the data. The dispersion r e l a t i o n (5.1.IS) for the f i r s t three cross-channel modes i s plotted i n Fig. 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 function of the wavenumber; for n = 1, wavelengths corresponding to periods extending from 11 to 100 days span the range of 2 to 700 km, while for n = 2 and 3, the ranges are 2 - 200 km and 3 - 90 km respectively. I t i s clear 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 (km) Krf (rad km 1) F i g . 13. Dispersion r e l a t i o n for 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 for a wave t r a v e l l i n g the length of GS. Table IV. 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 ra t i o of the mean current magnitude U to the. phase speed c^ i s also given for U = 5 cm s A (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 6.25 37.4 0.2 26.0 The theory does not correctly predict the observed v e r t i c a l d i s t r i -bution of horizontal 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 function 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. S i g n i f i -cantly smaller values are found for 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 proportional to R , we see that i n contrast to the observed case nearly a l l the energy i n the model i s trapped within the lower layer. 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 to 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 for 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 current magnitude of 5 cm s , t h i s r a t i o exceeds 0.5 for long waves and rapidly increases as the wavelength diminishes. WAVELENGTH (km) Kd(rad 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 horizontal 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-nately, no data exists 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 interest to examine the physical mechanisms governing t h i s class of wave motion. Like 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 fact the governing equations (5.1.13) and (5.1.14) are just 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 rotating, 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 effects a change i n the depth of the f l u i d . This 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 ^ or (3) through a variation i n the depth of the basin. Of these the f i r s t i s of limited significance since 1^ -Si^d ^ d ' a n t^ h e n c E attention i s focused on the l a t t e r two. Note that the second mechanism serves to transfer v o r t i c i t y between the two layers, and i t i s only the sloping bottom that can a l t e r the v o r t i c i t y of an entire water column. In view of t h i s i t i s not at a l l surprising that these waves should be more intense i n the lower layer. If 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 follows from (5.1 .5) - (5.1.8) that I t " - (5.3.1) W - c\- &5 *v' i £f]z-t - a>). ( 5 . 3 . 2 ) The right hand sides of (5.3.1) and (5.3 .2) are the respective horizontal 47 divergences i n each layer. Since <S, Ss<< I , (5.3.1) and (5.3.2) may be approximated by 1,-t - - "| i t (5.3.3) - fe > t - *s u-z.. (5.3.4) The factor A multiplying 'vj^ i n (5.3.3.) accounts for the difference i n mean depth of the two layers. 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 for inte r n a l Kelvin waves). While the situation i n the upper layer i s clear one must examine the importance of the two interacting mechanisms i n the lower layer. 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 precise, we u t i l i z e the results of Section 5.1. To zeroth order i n S one finds that s t.lf - i . ^ s6-fe i^"' [ ^ 6 4 - ] (5.3.5) 1 t 6 1 I $s i > G-C^+S^) - Vs. (4*4) ] (5.3.6) Consider the r e l a t i v e size of the two terms i n (5.3.6); t h e i r r a t i o i s 6<57k. Thus for 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. The case of small k i s s l i g h t l y more involved; i t follows from (5.1.18) that as k o ; (nV-) (,v\V+v) + 6<5\>/tn-o) 48 Hence i f 6 i s near unity (as i t i s for GS) the bottom slope again dominates. Notice that t h i s result i s due to the quantization of the motion i n the cross-channel "direction. To determine the r e l a t i v e magnitude of the u o r t i c i t y i n each layer ue compare the second term i n (5.3.6) with (5.3.5). Again as k-°> <x> , k^»o gn[j ^ e m o ^ Q n ^s ^Tangly bottom trapped. For V:-*o D n e finds _C - + ^ ' ^ < ! -[eO^),L^f I f £ * \ then £tf~/fc»W 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 qu a l i t a t i v e understanding of t h e i r nature. Even more insight can be gained through study of a system uit h variable density, and t h i s i s done i n the next section. 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 variable density f i e l d are examined. This i s an important and interesting exercise for the following reason. Mathematically, a two-layer system constitutes a special case because i t embodies a d i s -continuity i n the v e r t i c a l d i s t r i b u t i o n of density. 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 solutions, and one must be certain that any possible spurious results 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 essent 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 solutions to i t are obtained i n Section 6.2.through an expan-sion i n the perturbation parameter A . Several general theorems concerned with the v e r t i c a l structure and propagation of waves admitted by t h i s system are proved i n section 6.2, as w e l l . In section 6.3 the special case of constant Brunt-VSisMlfl frequency and i s considered and some of Rhines' (1970) results are recovered although the analysis d i f f e r s somewhat from h i s . The results of t h i s model are applied to GS yielding conclusions sim i l a r to those obtained i n the previous chapter. 6.1 The pressure equation In t h i s section an equation for 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 exist a hydrostatic equilibrium state described by - O (6.1.1) 50 with ? • * ~ " ?°^ ( 6 * 1 - 2 ) To describe small departures from equilibrium i t i s convenient to introduce the perturbation pressure p, density , and velocity a, which are defined by (6.1.3) The linearised equations for these perturbation quantities i n an unforced, i n v i s c i d , uniformly rotating f l u i d are Vi\ ~ ? „ V ' = - / f o (6.1.4) 4 + W - - f^  / fo (6.1.5) O = yt +" <Jj>' (6.1.6) II'*. V V'^  ^  w ' t a. O (6.1.7) Pi +- ^ f o t " ° ( 6 ' 1 - 8 ) Since the motions of interest are of very low frequencies, the hydrostatic approximation was made to obtain (6.1.6). In order to simplify the ensuing analysis, the Boussinesq approximation i s invoked and any variations i n the equilibrium density f i e l d are.ignored except when they occur i n a buoyancy term. That i s , unless ^0(z) occurs i n a term multiplied by .g i t i s replaced by a constant density j5* representative of the water column. The c r i t e r i o n of v a l i d i t y for t h i s approximation i s that PotrWp^ l^ , or equivalently that 1\|2H /g « 1. For GS the estimates H = 400 m and l\l = 10" J rad s" give PJ H/g = 4 x 10" . Elimination of from (6.1.6) and (6.l.S) gives 51 tflw' - - / f * (6.1.9) 11 II II where the Brunt-l/aisala frequency N i s a measure of the gravitational s t a b i l i t y of the equilibrium state and i s defined by HL = - ^ 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 ~ ^ V ' * ' ? ; / (6.1.11) * •£„ a ' - ~ f ^ / f * (6.1.12) tf2^' * - f i / f* (6.1.13) U.* + W ' t - o (6.1.14) The boundary conditions appropriate to t h i s system require that the velocity components normal to the sidewalls 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 GL-V X = O , L (6.1.15) NM« ^ *U.' OX I ' V\ * ** (6.1.16) l> - L ) flL+ t * o, (6.1.17) T>o * - ° J The conditions (6.1.17) are extremely complicated since they are (1) non-li 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 retain terms l i n e a r i n the perturbation quantities to obtain ^ W I i - o (6.1.18) Elimination of 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 application of the s t a b i l i t y r e l a t i o n (6.1.9) yields 52 + *LL* W = o at i - o . (6.1.19) In the BDussinesq approximation the second term i n (6.1.19) i s neg 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 condition. = 0 *x+ l=-o. (6.1.2D) As before i t i s convenient to recast the equations of motion into non-dimensional form. The following geostrophic scaling i s used: t = f o ~ V (x,y) = L(x?y*), z = Hz? (u, v) = (ufv*), 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 (after dropping the primes) U_L_ - V - - :px (6.1.21) \/T \- LX. - - ^ (6.1.22) =• - f i t (6.1.23) 4- + *U - ° (6.1.24) UL - o <a> x= o,i (6.1.25) f i f c - O o-V £=-o (6.1.26) VM ^ 5 S ix- OL--V i = - l +6'sX (6.1.27) 2 2 2 Here B = (MH/fL) = ( r V O compares the internal deformation radius to the channel width and thus i s analogous to I / ^ ) , To derive an equation for 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 continuity equation. From (6.1.21) and (6.1.22) i t fallows that (6.1.28) dt 1 - c - 7^ ujhEre I f we operate on (6.1.24) with<£, substitute f o r u and v from (6.1.28), and use (6.1.23) for w, we obtain [ V^J + £U"2 ~ O (6.1.29) 2 In the Boussinesq approximation the z-dependence of B may be neglected in comparison to that pz; . TD see t h i s consider the following argument, we may write (1*1 t r \ - -L [ \ - ^/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 C V + S ^ O t f c ^ f f c = O (6.1.3D) -i<rt For harmonic time dependence of the form e , the pressure equation and boundary conditions reduce to . (VH + C v - * 1 ) ^ « ) f ^ O (6.1.31) L CT f a - f ^ . - O o-V X=o.l (6.1.32) f t =. O O.V i ^ o (6.1.33) <r<^  S"2- 5 sO-<J*T l l ^ f x - i f t ) t = - ' + <*s* (6.1.34) 2 6.2 P a r t i a l solution of the pressure equation for arbitrary IM' The bottom boundary condition (6.1.34) precludes a separation of into horizontal 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 horizontal dependence df 'p\ i s evaluated leaving a second s order d i f f e r e n t i a l equation for the v e r t i c a l dependence. In addition expressions f o r the v e l o c i t i e s and surface elevation are derived and several general theorems that are concerned with the v e r t i c a l structure of "p. and with the phase and group v e l o c i t i e s of the solutions are stated; these are proved i n Appendix B. To begin, a l l dependent variables are expanded as power series i n <£s, v i z . , , 4 ) * 6< ^ cr - <rc0 > (6.2.1) uhere, as before, 6~ i s proscribed to be of order &s i n order to f i l t e r out any unwanted high-frequency waves. I t i s assumed that B i s 0(1). To zeroth order i n & the system (6.1.31) - (6.1.34) reduces to s * C V +- %~Z'b±i') ^ O (6.2.2) = O o> X-o,\ (6.2.3) T fc * o a i i - ° (6.2.4) , _ r,t * ~ - I . (6.2.5) The bottom boundary condition (6.1.34) was expanded about z =-1 in 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 for v e r t i c a l variations 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 solution of the form ^ - f U s v v ^ r c x e'H n A U ) h i , - ' - (6.2.6) 55 we obtain the following system f~l t ~ 0 C X - t i - o (6.2.8) r j - c 0 n i = - k 8 Z H 0-+ 4 - - I . (6.2.9) In addition the following normalisation f o r i " ! i s chosen, n iC^ - 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 analysis, p^0"* i s a stream function f o r the horizontal 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 velocity follows from the expansion of (6.1.23). Ex'plicity one finds tf* ~ - wi\* nnu> e (6.2.H) ^ _ ^ , n r _ CoSmTX H A C i ) t ^ (6.2.12) VJ 1 0^ =. O (6.2.13) Even though w ^ vanishes, the zeroth order surface displacement does not; i t follows from the f i r s t of (6.1.18) that Fortunately w ^ may be derived from ttpD') to give T li NMt0 - (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 elevation i s scaled by RoH where the Rossby number Ro = u/fL. (Ue might note that ^ = • suggests our o r i g i n a l scaling of w was too generous). F i n a l l y i t follows from (6.2.11) and (6.2.12) that the r a t i o R of u ( o ) or v ^ w 56 taken at tun different depths and i s Qtn , \ - ui'*^) _ _ n u . ) (6.2.17) 2 jAlthaugh one must specify IM (z) to proceed any further, there are a number of interesting and s i g n i f i c a n t generalizations that can be made about the properties of the system (6.2.7) - (6.2.9) for a given n and k; they are stated here as theorems, the proofs of which are contained i n Appendix B. Theorem 1. For positive B ,U(z) i s of one sign, which we chgose as pos i t i v e . 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 exists one and only solution 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 positive for a l l k. As k-*° <p <r/k. ; as k-=» <x> <f^  i s bounded by <rl U. 2 Theorem 5. I f B (z) attains i t s minimum value at z = -1, the 2 frequency i s an increasing function of B (z = -1). Theorem 6. The frequency i s a decreasing function of the cross-channel mode number n. 2 2 Theorem 7. I f B^z) < Q^iz) for a l l z, then the corresponding solutions 11 andH 2 of (6.2.7) - (6.2.9) s a t i s f y H^z) <R 2 ( z ) for a l l z<0. Theorem B. H (z) i s an increasing function of n. The. significance of these theorems i s as follows. Theorems 1 and 2 indicate that only one v e r t i c a l mode exists, and that t h i s motion i s bottom i n t e n s i f i e d . That i s , unlike other in t e r n a l motions, separate bartotropic and baroclinic 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 variations from a constant sloping bottom 57 implies the existence D f a set of v e r t i c a l modes. Si 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, half of which resembled in t e r n a l Kelvin waves, and the other half 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 to 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 fourth since i t indicates that the wave number i s a single -valued function of (T. This implies that energy can be propagated only in the direc t i o n of the phase ve l o c i t y . Ue note that the condition that <fy.< ^ /k as k-* o i n practice implies that -=> o as k-^-oo since <T must be bounded by unity i f the solutions are to remain v a l i d . That 6T should be an increasing function of l\l (Theorem 5) i s i n t u i t i v e l y c l e a r . 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; thus the frequency increases. I t i s d i f f i c u l t tD discuss the dependence of the frequency 60 2 2 2 2 2 on H and L (B = 1\) H / f |_ ) since these parameters also enter the expression U 3"o^<r f a r t*> through 6 S . The physical basis of Theorem 6 i s s l i g h t l y more subtle. The r a t i o of horizontal v e l o c i t i e s follows from (6.2.11) and (6.2.12); i t i s ix** I - - C W UXK) W ^ r r x . As n increases the motion up and down the slope diminishes and hence less v e r t i c a l motion i s induced at the bottom. But i t i s t h i s very excitation that i s responsible for these o s c i l l a t i o n s . Thus the role of topography i s lessened and the frequency decreases. Conversely as k increases, the 58 motion across the bottom increases and i t follows that Cf increases as w e l l . This provides a physical basis for Theorem 4. I t i s probably for these same reasons that the degree of bottom trapping increases as n increases (Theorem 8). For a given horizontal v e l o c i t y , the induced v e r t i c a l velocity at the bottom diminishes as n increases. Given t h i s weaker value, the motion can penetrate upwards only to a lesser depth. Hence although the motion may be weaker i t i s more greatly 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 inhibited from penetrating upwards. Lde note that Theorems 1,2,3,6 and the analogues of Theorems 7 and 8 (with B replaced by */•>?'') hold for the two layer system studied i n Chapter 5. However, Theorem 4 i s d e f i n i t e l y violated, and we conclude the existence of a negative group velocity i n that case i s a direct result of the sin g u l a r i t y i n density. To summarise the general character of the solutions to (6.1.31) -(6.1.34) for an arbitrary but continuous (and positive l\l ), we f i n d o s c i l l a t i o n s for which both phase and energy propagate northward. There exists 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 horizontal mode number increases. The frequency i s an increasing function 2 of k and IM and decreases with increasing n. 2 6.3 Constant IM model In t h i s section the simplest non-trival choice f o r the density f i e l d 2 for 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 linea 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 of his results are recovered here. After deter-mination of s p e c i f i c forms for the v e l o c i t i e s and surface elevation, the results of t h i s analysis 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 solution to (6.2.7) - (6.2.10) i s simply where jU= BK?0. I t follows from the results of the previous section that CT - &s J l * (6.3.2) U. - k S\v. wtrx CosvyA,t & (6.3.3) V - r\w wtr C o s w t v * C o s V l ^ t e ^ H - 5 " ^ ' (6.3.4) ' i . c(.Wv, - <rfc' + tr) \ . - ( \ A ^ S v ^ w r c * S v w W ^ t e 1 ' (6.3.6) S i m i l a r l y , 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 to zero as k-^ > o and to S B =<X IM as k->o° . s The v e r t i c a l structure of these o s c i l l a t i o n s depends on the parameter YvS- C; ( V 1 V*" which i s the inte r n a l radius of deformation rneas ured i n units df the " t o t a l wavelength", 3.1T / [ V>t +• C*n)lJ ^ Two cases are of in t e r e s t . The f i r s t corresponds to small jx , (due to, for example, weak s t r a t i f i c a t i o n ) . In t h i s case (6.3.2) reduces to c r - 6 s k / K z - A -^—r ; (6.3.a) i n dimensional form t h i s i s 60 ^ _ ( *g\ (6.3.9) Notice, ( l ) t h a t t h i s i s precisely the same l i m i t as ue found i n the tuo-layer 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 to largey*. (strong s t r a t i f i c a t i o n , say). In t h i s case we find that (T - 6s - Ss ^ • (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 proportional to both the bottom slope and IM. Notice the i n t r i g u i n g result that to i s e x p l i c i t l y independent of f (although, of course, these waves could not exist 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. That the "long wavelength" l i m i t ( | x . « l ) should correspond to the two-layer case while the "short wavelength" l i m i t (yu>">l ) does not i s not surprising, as short waves w i l l sense de 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. One should note the difference i n nature of "0 andyu. The l a t t e r i s intimately connected to the wavelength while 0 i s independent of wavelength. These results are now applied to the S t r a i t of Georgia. For the following values, 61 H = 300 m L = 22.2 km ft = 4.9 x 10~ 3 IM = 7.5 x 10" 3 rad s" 1 U J E obtain the parameter estimates £ = 0.36 s B 2 = 0.85 (B~ 2 = 1.2) r. = 20.3 km l Notice that t h i s model yields an in t e r n a l deformation radius which i s 2.5 times larger than the two-layer model. The dispersion r e l a t i o n f o r the f i r s t three cross-channel modes i s plotted 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. In addition the f i r s t made two-P layer dispersion r e l a t i o n i s platted i n F i g . 15 for comparison. As before the frequency i s a broad-banded function of the wavelength, but, unlike the two-layer c a s e , ^ i s a single valued function of co. For a given wavelength the frequency i s about an order of magnitude larger 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 for 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 for 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 li 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 Fig. 15. Dispersion r e l a t i o n for the f i r s t three-cross-channel modes (constant IM model) 63 the reduced size cf the parameter U/c^ which i s also given i n Table \J. Table V. Wavelength, period, phase speed, and velocity r a t i o s for the f i r s t cross-channel mode. Here Ri = fl(50 m)/ n(200 m) and R2 =".(50 m)/n(140 m). The ra t i o of the mean current magnitude U to the phase speed c i s also given for U = 5 cm s~% ^ A 0<m) T(days) Cp(cm s ) U/c V R1 R2 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 horizontal k i n e t i c energy Is strongly enhanced toward the bottom. F i g . 16 shows the r a t i o R as a function of wavenumber for the two depths, 50 and 200 m. I t s maximum value of .36 i s obtained for 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 less than the corresponding value of .39 obtained i n the two-layer case. In fa c t , comparison of Figs. 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 lyi n g i n the observed range for 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 dispersion relations d i f f e r considerably i n form for the two models. This and Theorem 4 suggests that the two-layer results are a very special case: , and perhaps that future work should be confined to models with a continuously varying density, even though t h i s would generally necessitate a numerical solution 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 horizontal velocity for 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 fallowing conclusions insofar as the low-frequency current fluctuations i n GS are concerned, may. be made: (1) Neither int e r n a l Kelvin waves, nor Rossby waves are important, since for r e a l i s t i c values of the wavelength int e r n a l Kelvin waves have frequencies too high and Rossby waves have frequencies too small. (2) The chosen two-layer model of GS admits t r a v e l l i n g topographic planetary waves whose calculated frequencies l i e i n the observed range; t h i s model does not accurately predict the observed v e r t i c a l d i s t r i b u t i o n of horizontal k i n e t i c energy, however. Nor does i t account for the apparent loss of the barotropic nature of the 10 - 25 day: fluctuations at the central and western locations, although t h i s might be related to the presence of the shelf which i s v i s i b l e i n F i g . 2 i n sections E and F. In addition, t h i s model does not account f o r the observed low coherence between horizontally separated currents. (3) A continuously s t r a t i f i e d model yields results that resemble those obtained from a two-layer model except that the dispersion relations d i f f e r i n form. I t i s suggested that the double-valued nature of k as a function 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 result due to the sin 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 thesis, 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 theoretical investigation of the low-frequency current fluctuations i n GS currently being undertaken by the author. I t 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 longitudinal variations i n topography. But perhaps most s i g n i f i c a n t l y , possible interactions with the mean currents or the Fraser River outflow have not yet been examined. Since, given the mean current structure, baro c l i n i c i n s t a b i l i t y 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 to be tackled. 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 isolated by the narrow t i d a l passes at both ends, any waves would necessarily be generated within the S t r a i t i t s e l f . Although the current fluctuations are not d i r e c t l y t i d a l i n nature, i t i s possible that a non-linear t i d a l mechanism might account for t h e i r existence. Recognizing GS as an estuary, we see that the fo r t n i g h t l y tide could, by alteri n g the intensity of mixing in the southern passages, generate intru 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 intrusive nature of the process. 67 BIBLIOGRAPHY Allen , J.S., 1975. Coastal trapped waves i n a s t r a t i f i e d ocean. Journal of Physical Oceanography 5, 300-325. Chang, P.Y.K., 1976. Subsurface currents i n the S t r a i t of Georgia, west of Sturgeon Bank. M.Sc. thesis, University 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 Fisheries Research Board of Canada 33 : 2218-2241. Crean, P.B., and A.B. Ages, 1971. Oceanographic records from twelve cruises 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., 1975. Some effects of bottom topography on baroclinic s t a b i l i t y . 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 : low-frequency waves and topographic planetary waves. Journal of the Fisheries 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 continental shelf. Journal of F l u i d Mechanics 69 : 689-704. Kajiura, K., 1974. Long period waves of shelf and Helvin modes i n a two-layer f l u i d with a shelf. Journal of the Oceanographic Society of Japan 30 : 271-281. LeBlond, P.H., 1975. Long wave propagation i n channels of non-rectangular cross-section. Unpublished manuscript, 18 pp. Longuet-Higgins, M.S., 1964. Planetary waves on a rotating sphere. Proceedings of the Royal Society A. 279 : 446-473. Mctililliarns, J.C, 1974. Forced transient flow and small scale topography. • Geophysical F l u i d Dynamics 6 : 49-79. MaDers, C.W.H., 1973. A technique for the cross spectrum analysis of pairs of complex-valued time series, with emphasis on properties of polarized components and ro t a t i o n a l invariants. Deep-Sea Research 20 : 1129-1141. Pedlosky, J . , 1964. The s t a b i l i t y of currents i n the atmosphere and the ocean: Part I. 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 Flu i d Mechanics 37 : 161-189. 1970. Edge-, bottom-, and Rossby waves i n a rotating s t r a t i f i e d f l u i d . Geophysical Fluid 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. thesis, Massachusetts In s t i t u t e of Technology. Tabata, 5., and J.A. Strickland, 1972a. Summary of oceanographic records obtained from moored instruments i n the S t r a i t of Georgia - 1969-1970: Current velocity 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 velocity 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. Strickland and B.R. de Lange Boom, 1971. The program of current velocity 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. Fisheries Research Board of Canada, Technical Report IMo. 253. 368 pp. Ueronis, G., and H. Stommel, 1956. The action of a variable wind stress on a s t r a t i f i e d ocean. Journal of Marine Research 15 : 43-75. Waldichuk, M., 1957. Physical oceanography of the S t r a i t of Georgia, B r i t i s h Columbia. Journal of the Fisheries 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 Physical Oceanography (in press). Yoon, J.H., 1974. Continental shelf waves i n a two-layer ocean. M.Sc. thesis, University of Tokyo. Appendix A. The two-layer equations of motion 69 } (A.D (A.2) Although the equations of motion for a two-layer f l u i d are well known (see e.g., l/eronis and Stommel, 1956), they are derived here for the sake of completeness. Ue consider a system of two layers of incompressible / i n v i s c i d , rotating, and homogeneous f l u i d s resting one upon the other. Since only low-frequency motions are of interest to us, we invoke the hydrostatic approximation. Then the non-linear dynamical equations governing the motion i n each layer are Here the subscript i = 1,2 refers to the upper and lower layers respectively, the C o r i o l i s parameter f i s defined by f = fo ^y, and g i s the gra v i t a t i o n a l acceleration. Ue assume from the Dutset that the horizontal v e l o c i t i e s are z-independent throughout each layer as no variations are permitted i n the density. This system i s subject to kinematic and dynamic boundary conditions at the surface and interface. In addition the normal component of the veloc i t y must vanish at the side-walls (x = D,L) and the bottom (z = -H(x)). In order to simplify 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^, respectively, and we w i l l v e r t i c a l l y integrate the continuity equations. I t follows from the hydrostatic r e l a t i o n A.2 that the pressure i n each layer i s given by 70 (A.4) where f,, and are arbitrary 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 solely with free motions we regard p as constant; a substitution of B.4 and B.5 gives Upper Layer U U t 4- . 9RTUL, - £ V , = - ^ - Y ^ K \?. u., - o Lower Layer (A.6) (A.7) (A.8) (A.9) I t remains to v e r t i c a l l y integrate (A.7) and (A.9). Consider the (A.10) upper layer f i r s t ; integration of (A.7) gives. The kinematic boundary conditions require that ( ^ H , ^ ) / ^ i a ^ i at z = ^ and z ="Ji-.h^  . Using these relationships f o r w^  and using Lagrange's id e n t i t y for the derivative of an integral we obtain 71 Since the horizontal v e l o c i t i e s are assumed depth independent, we have a t MW . + Y - v l x v C^ >1 ^v ,^ (A.11) 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 but the appropriate boundary conditions arE (A.12) and 0.4- £ = -HCX.^ ) (A.13) (A.14) Using (A.13) and (A.14) and again ignoring 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" - K1" 6 r n - o (B.D n ' - O cxk i ^ O (Q.2) n'» -k/<r 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. For positive B , 1 S D n E sign, which we choose as pos i t i v e . With t h i s choice flC1) i s a s t r i c t l y decreasing function of z. Proof: Multiply (B.D by H and integrate o v e r t * . 0 ! ; following one integration by parts we obtain r° r Application of (B.2) gives ' o nn'ia » - / ( n'z+- K2SanA)<U 9 Jl / . (B.4) since B i s positive by hypothesis, H n (£) i s everywhere negative. I t follows that n has no zeros i n the i n t e r v a l £otx1] and hence i s everywhere of one sign. We choose that sign as positive so that n '<o and thus the theorem follows. This j u s t i f i e s the normalization that we chose i n Chapter 6.2. Theorem 2. There exists one and only one solution 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 restate here: Let G(x}>0 be continuous i n (-oo,-t-oa). Then the equation fu-) - ( jL<,^U) = O (B.5) 73 has one and only one solution y+AiO passing through ( 0,1) which i s positive and s t r i c t l y decreasing for a l l x and one and only one solution "y.OO through (o,|) which i s positive and s t r i c t l y increasing for 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 val i d on ( - 1 , 0 ). To see t h i s replace y b y i n (B.5) and define Vt^) - y (-*-) to obtain YCO - CrCOVttt (B.6) Since G(x) i s positive for XfcCr 0 0 ) 0 0 } , the stated lemma applies to (B.6) and hence to NJUC) on (-1,0 ) . In Theorem 1, we showed that l~l was positive (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 solution to (B.l) - (B.3). Theorem 3. The frequency <f i s always po s i t i v e . Proof: Take (B.4) with z" = - 1 to obtain _ n n ' - K i<^ 2-n t> + <n'l> where defines the i n t e g r a l Jt')&£ • Substituting for f l 'ii) from (B.3), we obtain fc\-D n V o W<r = < s z n " > *• < n ' * > > o CB.?) 2 2 Since B n i s positive d e f i n i t e and since k was chosen to be positive | 6~ i s positive also. Theorem 4. The group velocity (f^ i s positive for a l l k. As W-^o^  (T^-^^Vk ;\ as- k-»> co i s bounded by G ' / k . Proof: For convenience, i n t h i s proof i t i s to be understood that a variable not within an int 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 to k; after rearrangement we obtain ^ _ ak<6lni>-^K1<61-nnk> - 3L<n'n'k> +-k s ' n n j , (B.8) Here the notation ilV represents -cLlnC-ft] • BY using (B.D - (B.3), obtain the i d e n t i t y < n'n J> + x x < n?> - < n'n'5 * n"n^> - < tn'n^'? \ R2. (B .5 - - n ' n , - ^ n n ( where ^ represents some variable or parameter i n the argument Df HI. Substitution of (B.9) with f = k into (B.8) gives k w 8xnv ere ^ i s defined by wh > 5^ - ric-0{ nk(-0 - [ n(H)]kJ ca.io) I t follows that from the form of (B.l) that 5^-° • To see t h i s con-sider that H depends on k only through K as (B.3) serves simply to determine <T and places no rest r a i n t s on the form of 17. (MDLJ K enters (B.l) only as a scaling parameter. Hence i t fo l l o u s that n i s of the form H(Kt) , a n d that k^~o as long as the derivatives i n question are not evaluated at z = o. Thus we have \ - °^  - U < P " l > (B .11) cr r . 5. o" W 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. From (B.3) ue have cr = - k 6 l n / n ' A single integration of (B.D gives rV- - K1" < B*n > And ue thus obtain cr - (B.12) Substitution of (B.12) into (B.H) gives k * w Ll \cn<8ln> . Consider nou the second term i n brackets; sincef~l(z) attains i t s maximum value at z = -1 (Theorem 1) ue obtain (B.13) 1 = Kun<& ln> (B.14) It follows that T<1 since ak _ ak < ^k/Cwrv) 1 ' <l \<<<^  < Xk/ WL - a/k <\ k> iHence tf^ i s aluays positive and i s bounded by ff/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 ^/k. as k -» a. 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 scaling would be i l l e g i t i m a t e and our perturbation solution i n v a l i d . 2 Theorem 5. I f B (z) attains 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 sim i l a r to that of Theorem k. Again i t i s understood that variables not within an integral are taken at z = -1 . We di f f e r e n t i a t e (B.7) to obtain an expression for (T^j. - a<n'nV> * ^ n n c " V '6' using the i d e n t i t y (B.9) with %~ Q we obtain rf1- 8 s" c r n 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) for CT we may write " n <&ln>l cr 1 * 8 x<n x>. | (B.15) 2 Since Hit) attains i t s maximum and, by hypothesis, B (z) attains i t s minimum at z = -1, we estimate the f i r s t term i n parenthesis as n < g > z n > ^ n < s 2 - n > > < s1-n> ( & x<n L> G z n < n > <&-n> I t follows then that (T^2.>0 and (T i s an increasing function of B . Theorem 6. The frequency i s a decreasing function of the cross-channel mode number. Proof: As before we take variables outside an integral sign at z Replace (_A(\) by the continuous variable X. i n (B.7), then holding k fixed we di f f e r e n t i a t e (B.7) with respect to K * " - CVc1"*-£*") to obtain 77 lilse of the id e n t i t y (B.C) with ? = K 2 gives - - <8 vn*> since an argument l i k e that i n Theorem k shows that ^Y^~~° • I t follows d i r e c t l y that <Tyz.<0 and that (T i s a decreasing function of n. 2 2 Theorem 7. I f B^Cz) B^Cz) for a l l " , then the corresponding solutions H , and f \ of (6.2.7) - (6.2.9) s a t i s f y PI, 111 < f^Ci ) f o r a l l z < o. Proof: M u l t i p l i c a t i o n of the f l , equation by PI t and the ("Inequation by and integration over the i n t e r v a l (ifciO) gives n/n,. = - Jr\,'n{W - KX j y n, n, <U' n, ry = - jf n , / n ; : A * " ^ J 8 * n , n " ^ We subtract the second expression from the f i r s t to obtain • o n,'n which may be rewritten as A ( n.'\ _ _ Kln£ /(B^-BJ) n.n»At' ( B . 1 6 ) 78 A second integration gives H i I t follows that n,<nt for a l l !<<=>. Theorem 8. H (z) i s an increasing function of n. Proof: This theorem i s proved i n exactly the same manner as the preceeding one. By analogy with (B.1S) we may write Lit) - - ( C - K ^ n ^ l ^ n . r U . -a second integration gives n , n I t follows that n , < f l x for a l l z < O i f K,*< ^ , and thus for fixed k, n,<n L i f n r < ,n 2. 79 Appendix C. Glossary of symbols B = NH/f L f = + 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 layer depth L +h 2 Q H = - Maximum mean depth of water column k, k , - IMondimensional and dimensional wave numbers d K 2 = k 2 + (nn)2 L - Channel width n - Cross-channel mode number l\l - Brunt-Uaisala" frequency p - Pressure P Q - Equilibrium pressure r g = (gH)^/f - External deformation radius (i>4,5) T± \ (g D/h^/f (§k,5) I f\lH/f (^6) " •'• n'' : e : r n al deformation radius R = uVu„ R =v./v 9 - Ratios of upper to lower layer x i c., y i d. v e l o c i t i e s (J4,5) R = R =R (§5, to zero order i n & ) x y J ' s R ( z r z 2 ) = u(z / ])/u(z 2) = v ( z 1 / v ( z 2 ) (§6) t = Time T = wave period (u., v.) .- The upper (i=1 ) and'lower ( i = 2) layer 1 1 v e l o c i t i e s i n the x and y directions 80 eastward, northward, upward velocity (#6) eastward, northward, and upward directions (V 2 + f 2 ) - (Non-dimensional) f22> - Zfo -1 x t t X 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 internal deformation radius V e r t i c a l structure function ($6) Upper and lower layer densities Density ($6) Equilibrium 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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