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Low-frequency vorticity waves over strong topography Gratton, Yves 1983

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LOW-FREQUENCY VORTICITY WAVES OVER STRONG TOPOGRAPHY by YVES GRATTON B.A. , U n i v e r s i t e du Quebec a Montreal, 1968 B . S c , U n i v e r s i t e du Quebec a Montreal, 1976 M.Sc., U n i v e r s i t e du Quebec a Rimouski, 1979 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Ph y s i c s and Department of Oceanography) We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA August 1983 © Yves Gratton, 1985' In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department o r by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department o f Physics and department of Oceanography The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date October 17 1983 DE-6 (3/81) i i ABSTRACT T h i s t h e s i s addresses the g e n e r a l problem of v o r t i c i t y waves propagating over s t e e p l y s l o p i n g topography, i n the presence of s t r a t i f i c a t i o n and r o t a t i o n . From the i n v i s c i d u nforced long-wave equations f o r a two-layer f l u i d on an f-plane, i t i s shown t h a t , as long as the r a t i o of the upper to lower l a y e r depths i s s m a l l , semi-enclosed and enclosed basins can s u s t a i n low-frequency, short s c a l e , s u r f a c e - i n t e n s i f i e d motions. Simple a n a l y t i c a l s o l u t i o n s are to be found only i f the upper to lower l a y e r depths r a t i o i s s m a l l . Then, we o b t a i n a set of equations which d e s c r i b e s a b a r o t r o p i c wave which f o r c e s a b a r o c l i n i c response through topographic c o u p l i n g . Two bottom p r o f i l e s are c o n s i d e r e d : l i n e a r and p a r a b o l i c . S o l u t i o n s are found with and without the small slope approximation. I t i s shown that the small slope approximation underestimates a l l the parameters of low-frequency topographic waves, even when the slope i s s m a l l . The theory i s compared with o b s e r v a t i o n s from the S t r a i t of Georgia and with a numerical model of the S a i n t Lawrence e s t u a r y . I t i s found t h a t , f o r bathymetric p r o f i l e s s i m i l a r to those of the S t r a i t of Georgia ( l i n e a r ) and the S a i n t Lawrence ( p a r a b o l i c ) , bur model p r o v i d e s a b e t t e r f i t to the topography, leads to s u r f a c e - i n t e n s i f i e d motions and produces c r o s s - c h a n n e l v e l o c i t i e s very s i m i l a r to those observed i n s i t u . i i i T able of Contents ABSTRACT i i LIST OF TABLES i v LIST OF FIGURES v ACKNOWLEDGEMENT v i i i 1 INTRODUCTION 1 2 LOW-FREQUENCY TOPOGRAPHIC WAVE-MOTION 3 2.1 Low-frequency Motion In The S t r a i t Of Georgia 3 2.2 Low-frequency T o p o g r a p h i c a l l y Trapped Waves 15 2.2.1 C o n t i n e n t a l S h e l f Waves 16 2.2.2 Channel Models 24 2.2.3 E n c l o s e d Basins 26 3 PHYSICAL AND MATHEMATICAL MODEL 30 3.1 M o d e l l i n g The S t r a i t Of Georgia 30 3.2 The P h y s i c a l Model 35 3.3 Basic Assumptions 38 3.4 Governing Equations ...40 3.5 S i m p l i f y i n g Assumptions 49 3.6 Boundary C o n d i t i o n s And I n t e r f a c e Motion 55 4 SOLUTION : TOPOGRAPHIC WAVES 59 4.1 Channel Waves 59 4.1.1 L i n e a r Topography ...59 4.1.1.1 Small Slope Approximation 60 4.1.1.2 Large Bottom Slopes 67 4.1.2 P a r a b o l i c Topography 80 4.1.2.1 Small Slope Approximation 80 4.1.2.2 Large Bottom Slopes 92 4.2 Basin Waves 101 4.2.1 The General S o l u t i o n 104 4.2.2 C o n i c a l And P a r a b o l o i d a l Bottom P r o f i l e s ......108 5 DISCUSSION , 113 5.1 General Assessment Of The Model 113 5.2 In f l u e n c e Of The Bottom Slope 115 5.3 A p p l i c a t i o n To The S t r a i t Of Georgia 124 5.4 A p p l i c a t i o n To The S a i n t Lawrence 130 6 SUMMARY AND CONCLUSION 135 BIBLIOGRAPHY 1 38 i v LIST OF TABLES Table page I D i s c r e t e e i g e n v a l u e s df eq. 4.55, the d i s p e r s i o n r e l a t i o n f o r c i r c u l a r l a k e s 110 II Comparison of the f r e q u e n c i e s computed with the small l i n e a r topography (SLT), the l a r g e l i n e a r topography (LLT), the small p a r a b o l i c topography (SPT), and the l a r g e p a r a b o l i c topography (LPT) models, at the same wavenumber (k=1.6) 118 I I I Frequencies and wavenumbers at which the group v e l o -c i t i e s v a n i s h 121 IV Correspondence between the dimensional and the nondi-mensional wavenumbers and f r e q u e n c i e s 129 V Frequencies computed by L i e and El-Sabh (1983) and by the l a r g e p a r a b o l i c (s=0.60) topography model. The nondimensional wavenumber i s the same i n both i n s -tances (k=1.6). The c o r r e s p o n d i n g dimensional wave-le n g t h i s twice the channel width 134 V LIST OF FIGURES FIGURE page 1 The S t r a i t of Georgia and adjacent waters, showing the four Regions 4 2 D e t a i l e d bathymetry of the s o u t h - c e n t r a l S t r a i t of Georgia, showing the p o s i t i o n s of moorings H06, H16, and H26 analyzed by Chang (1976) and of moorings 20, 21, 22 d e s c r i b e d by Yao et a l . (1982). Adapted from LeBlond (1983) 8 3 Spectra of c u r r e n t s at 50m. Number above the o f f -s c a l e peaks i n d i c a t e t h e i r v a l u e s . Adapted from Chang (1976) 9 4 Spectra of c u r r e n t s at 140 and 200m. Numbers above the o f f - s c a l e peaks i n d i c a t e t h e i r v a l u e s . Adapted from Chang (1976) 10 5 S t i c k diagrams of c u r r e n t s at Cyclesonde mooring GS20 M o d i f i e d from Yao et a l . ( 1982) 12 6 Topographic p r o f i l e s and two-layer models used i n S h e l f waves l i t t e r a t u r e 18 7 T r a n s p o r t s t r e a m l i n e p a t t e r n : a) at t=0, b) at t= one-eight p e r i o d , c) at one-fourth p e r i o d . From Sa y l o r et a l . ( 1980) 28 8 Plan view of the S t r a i t of Georgia showing l i n e s of topographic c r o s s - s e c t i o n (1-10) presented i n f i g u r e 9. From He l b i g ( 1 978) 31 9 Topographic c r o s s - s e c t i o n s : a) upper p a n e l s : 1-9, b) lower p a n e l : 10. From H e l b i g ( l 9 7 8 ) 32 10 Model f i t to the bathymetry of l i n e H. M o d i f i e d from H e l b i g d 978) 34 1 1 Coordinate system, shapes of the basins and topogra-phic p r o f i l e s s t u d i e d i n t h i s t h e s i s 36 12 D i s p e r s i o n curves f o r the f i r s t three modes i n the small l i n e a r (s=0.26) topography case of an i n f i n i t e channel 62 13 Channel waves c r o s s - c h a n n e l dependence of (a) the mass-transport stream f u n c t i o n , and (b) the i n t e r f a c e motion corresponding to the three modes of f i g u r e 12. k=1 .6 . . . 63 v i 14 E u l e r i a n flow p a t t e r n i n the s u r f a c e and bottom l a y e r s , corresponding t d the f i r s t mode df f i g u r e 13. The phase propagates td the r i g h t 65 15 Same as i n f i g u r e 14, but f o r the second mdde 66 16 D i s p e r s i o n curves f o r the f i r s t three modes i n the. l a r g e l i n e a r (s=0.26) tdpdgraphy case df an i n f i n i t e channel 75 17 Channel waves cr o s s - c h a n n e l dependence of (a) the mass-transport stream f u n c t i o n , and (b) the i n t e r f a c e motion, cdrrespdnding td the three modes of f i g u r e 16. k=1.6 76 18 E u l e r i a n flow p a t t e r n i n both l a y e r s , cdrrespdnding td the f i r s t mdde df f i g u r e 17. The phase prdpagates td the r i g h t 78 19 Same as i n f i g u r e 18, but f o r the second mdde 79 20 D i s p e r s i o n curves f o r the f i r s t t h ree modes i n the small p a r a b o l i c (s=0.26) tdpdgraphy case of an i n f i -n i t e channel 87 21 Channel waves cr o s s - c h a n n e l dependence df (a) the mass-transport stream f u n c t i o n , and (b) the i n t e r f a c e mdtidn, cdrrespdnding t d the three mddes df f i g u r e 20. k=1.6 88 22 E u l e r i a n flow p a t t e r n i n both l a y e r s , cdrrespdnding t d the f i r s t mdde df f i g u r e 21. The phase prdpagates td the r i g h t 90 23 Same as i n f i g u r e 22, but f d r the second mdde........ 91 24 D i s p e r s i o n curves fdr the f i r s t three mddes i n the l a r g e p a r a b o l i c (s=0.26) tdpdgraphy case df an i n -f i n i t e channel 96 25 Channel waves cr o s s - c h a n n e l dependence df (a) the mass-transpdrt stream f u n c t i o n , and (b) the i n t e r f a c e mdtidn, cdrrespdnding t d the three mddes df f i g u r e 24. k=1.6 97 26 E u l e r i a n fldw p a t t e r n i n both l a y e r s , cdrrespdnding td the f i r s t mdde df f i g u r e 25. The phase prdpagates td the r i g h t 99 27 Same as i n f i g u r e 26, but f d r the secdnd mode........100 28 E u l e r i a n fldw p a t t e r n i n both l a y e r s f d r Basin waves. m= 1 , k= 1 111 v i i 29 Same as i n f i g u r e 28, but f o r m=2 112 30 I n f i n i t e channel d i s p e r s i o n curves f d r the four bdttom p r o f i l e s s t u d i e d , s=0.26....' 116 31 I n f i n i t e channel d i s p e r s i o n curves f d r the four p r o f i l e s s t u d i e d , s=0.75 117 32 P l o t s df frequency versus sldpe f d r small l i n e a r (SLT), l a r g e l i n e a r (LLT), small p a r a b o l i c (SPT), and l a r g e p a r a b o l i c (LPT) topographies 119 33 Cross-channel s t r u c t u r e df the mass-transpdrt stream f u n c t i o n over a p a r a b o l i c bottom p r o f i l e with the small sldpe approximation (A: s=0.26, C: s=0.75) and without the small sldpe approximation (B: s= 0.26, D: s=0.75) 122 34 Same as i n f i g . 33, but f d r the i n t e r f a c e e l evation..123 35 D i s p e r s i o n curves f d r the right-bounded wave (upper l e f t ) and the left-bounded wave (lower l e f t ) propagating over the tdpdgraphy shown i n f i g u r e 10. The two other d i s p e r s i d n curves are f d r waves over an axi-symmetric channel 126 36 L o c a t i o n and shape df the Matane-Pdinte a l a C r o i x s e c t i o n . The dashed l i n e d i s the f i t used by L i e and El-Sabh(1983) and the s o l i d l i n e (parabola) i s our f i t t d the tdpdgraphy df the s e c t i o n . M o d i f i e d from L i e and El-Sabh ( 1983) 131 37 D i s p e r s i d n curves obtained by L i e and El-Sabh (1983) (lower p a n e l ) , and with the l a r g e p a r a b o l i c tdpd-graphy (s = 0.60) mddel(upper panel) 133 v i i i ACKNOWLEDGEMENT I would l i k e td thank Dr. P.H. LeBldnd, my t h e s i s s u p e r v i s o r , f o r h i s e n t h u s i a s t i c and c d n t i n u a l guidance, and h i s steady encouragement. My g r a t i t u d e a l s o goes td Drs L.A. Mysak, S. Pond and W. J . Emery, the other members df my committee, f d r t h e i r a v a i l a b i l i t y and a d v i c e . In p a r t i c u l a r , I would l i k e t d thank Dr. Mysak f d r h i s guidance, while my su p e r v i s d r was on s a b b a t i c a l . The a s s i s t a n c e df the t e c h n i c a l s t a f f members of the computing c e n t e r df the U n i v e r s i t e du Quebec a Rimduski, e s p e c i a l l y Mr. J . Landry, i s g r a t e f u l l y acknowledged. Much a p p r e c i a t e d f i n a n c i a l support has been pr o v i d e d by the N a t i o n a l Research C o u n c i l df Canada, through a grant t d Dr. LeBldnd, and the Government df Quebec, through a post-graduate f e l l o w s h i p . 1 1 INTRODUCTION T h i s t h e s i s addresses the gen e r a l problem of low-frequency wave-motion over s t e e p l y s l o p i n g topography, i n the presence of s t r a t i f i c a t i o n and r o t a t i o n . The ba s i n s c o n s i d e r e d w i l l be of the e n c l o s e d type, represented by a c i r c u l a r lake , and of the semi-enclosed type, represented by an i n f i n i t e channel. S t r a t i f i c a t i o n i s simulated by a two-layer system, while r o t a t i o n i s taken i n t o account by the f-plane approximation. A r b i t r a r y enclosed (and semi-enclosed) basins can support f r e e o s c i l l a t i o n s of two d i s t i n c t types: o s c i l l a t i o n s of the f i r s t c l a s s ( g r a v i t a t i o n a l modes) and o s c i l l a t i o n s Of the second c l a s s ( r o t a t i o n a l modes). The g r a v i t a t i o n a l modes are produced by the u n d u l a t i o n of the f r e e s u r f a c e of a f l u i d and have f r e q u e n c i e s g e n e r a l l y g r e a t e r than the l o c a l i n e r t i a l frequency (LeBlond and Mysak, 1978, chapter 3 ) . The r o t a t i o n a l modes depend On the e x i s t e n c e of a g r a d i e n t of p o t e n t i a l v o r t i c i t y and correspond to low-frequency o s c i l l a t i o n s slower than the the l o c a l i n e r t i a l frequency. The p o t e n t i a l v o r t i c i t y g r a d i e n t i s produced by e i t h e r a v a r i a t i o n of the C O r i o l i s e f f e c t with l a t i t u d e , by v a r i a t i o n s i n the bathymetry Of the ba s i n or by b a r o c l i n i c i t y (Pedlosky, 1979, chapter 2). The present study w i l l c o n c e n t r a t e on the r o t a t i o n a l modes produced by topographic v a r i a t i o n s i n the presence of r o t a t i o n and s t r a t i f i c a t i o n , A r b i t r a r y topography i n t r o d u c e s inhomogeneities i n the 2 dynamical equations. Even i n the l i n e a r b a r d t r d p i c case, a n a l y t i c a l s o l u t i o n s are to be found only i f the inhomdgeneities are e i t h e r s l i g h t or very abrupt (Rhines and B r e t h e r t d n , 1973). We w i l l c o n c e n t r a t e on depth p r o f i l e s that are f u n c t i o n s df one cd d r d i n a t e o n l y . T h i s w i l l i n s u r e the s e p a r a b i l i t y df the equations and w i l l allow a n a l y t i c a l s d l u t i d n s . T h i s work was mdtivated by db s e r v a t i d n s from the S t r a i t of Georgia, B r i t i s h Columbia, that are s t i l l unexplained. T h e i r main f e a t u r e s are low-frequency, short s c a l e , s u r f a c e i n t e n s i f i e d mdtidns seemingly u n r e l a t e d t d the wind dr pressure systems. I t w i l l be shown t h a t , i n a body df water s i m i l a r i n shape and s i z e t d the S t r a i t df Georgia, f r e e topographic waves possess these p r e v i o u s l y mentioned c h a r a c t e r i s t i c s . The content df the t h e s i s i s as f o l l o w s . The second chapter i s a review of low-frequency topographic wave-mdtidn. The f i r s t p a r t d e a l s with the o b s e r v a t i o n a l b a s i s and the second p a r t with the e x i s t i n g theory. Chapter 3 pres e n t s the p h y s i c a l model and the dynamical equations. Moreover, i t intr o d u c e s s c a l i n g arguments, the c r u c i a l p o i n t df the t h e s i s , that w i l l a l low us t d s i m p l i f y the dynamical equations and thus t d o b t a i n a n a l y t i c a l s d l u t i d n s . In Chapter 4, the s d l u t i d n s are found f d r semi-encldsed and enc l o s e d b a s i n s with l i n e a r and p a r a b o l i c depth p r d f i l e s . These s d l u t i d n s are d i s c u s s e d and cdmpared i n chapter 5. F i n a l l y , the main p d i n t s df the t h e s i s are u n d e r l i n e d i n the C o n c l u s i o n . 3 2 LOW-FREQUENCY TOPOGRAPHIC WAVE-MOTION We are l o o k i n g f o r a n a l y t i c a l s o l u t i o n s Of a d i f f i c u l t problem. Thus, the shapes of the bas i n s we w i l l be studying are "mathematical shapes": i n f i n i t e l y long and s t r a i g h t channels, and p e r f e c t l y round l a k e s . The problem i s a t h e o r e t i c a l One, but i t has i t s r o o t s i n the p u z z l i n g r e s u l t s that came from the a n a l y s i s of measurements taken i n the S t r a i t of Georgia (GS f o r s h o r t ) , B r i t i s h Columbia. In the f i r s t s e c t i o n , we w i l l b r i e f l y review the oceanography Of the GS and d i s c u s s those r e s u l t s . As was s t a t e d i n the i n t r o d u c t i o n , we attempt t o e x p l a i n some of the GS's low-frequency motions with topographic waves. Thus, i n the second s e c t i o n , we w i l l d i s c u s s the t o p i c of topographic waves i n g e n e r a l , i . e . the e x i s t i n g theory. There, we w i l l a l s o d i s c o v e r why Oceandgraphers make so much use of what C.J. Yorath i n h i s foreword of R.E. Thomson's book (Thomson, 1981) c a l l e d the oceanographers' p r i e s t s : the computers. 2.1 Low-frequency Motion In The S t r a i t Of Georgia The S t r a i t Of Georgia (GS) i s the main body Of water between Vancouver I s l a n d , the mainland of B r i t i s h Columbia and the State of Washington ( f i g . 1). The S t r a i t has access to the P a c i f i c Ocean through Johnstone S t r a i t and Queen C h a r l o t t e S t r a i t to the nor t h , and through Juan de Fuca S t r a i t to the F i g u r e 1. The S t r a i t of Georgia and adjacent waters, showing the four r e g i o n s . 5 south. Access to Johnstone S t r a i t i s p e r m i t t e d through f j o r d -l i k e channels with shallow s i l l s , while access to Juan de Fuca i s p a r t i a l l y blocked by the San Juan A r c h i p e l a g o . Waldichuk (1957) argued t h a t , s i n c e the i n f l u e n c e of the deep waters to the n o r t h i s blocked by three major s i l l s and s i n c e the c r o s s -s e c t i o n area of the northern chanels i s Only 7% of that Of the southern ones, water exchange at the northern boundary can be n e g l e c t e d compared to the exchange at the southern boundary. Hence the b a s i n can be c o n s i d e r e d c l o s e d at i t s northern end and o u t s i d e i n f l u e n c e i s mostly due to inflow through the southern end. The GS can be d i v i d e d i n t o four regions ( f i g . 1). Three Of them, the Northern (1), C e n t r a l (2), and Southern (3) are the same as d e s c r i b e d by Waldichuk (1957). The Northern r e g i o n i s both i n f l u e n c e d by the F r a s e r River runoff and the northern t i d a l channels. The major i n f l u e n c e i n the C e n t r a l region i s the F r a s e r R i v e r r u n o f f . The Southern i s the s i t e Of intense t i d a l mixing. The f o u r t h region has been c a l l e d the San Juan region by Samuels (1979). T h i s region i s the s i t e of intense t i d a l mixing, while being i n f l u e n c e d by the waters of the Juan de Fuca S t r a i t . The p h y s i c a l Oceanography of these r e g i o n s , as w e l l as of a l l the B r i t i s h Columbia c o a s t , i s d i s c u s s e d i n Thomson (1981). The p h y s i c a l oceanography Of the GS has been r e c e n t l y reviewed i n depth by LeBlond (1983). In the S t r a i t of Georgia, the t i d e s are p r i m a r i l y of the mixed s e m i - d i u r n a l (M2) and d i u r n a l (K1) v a r i e t y . The s e a - l e v e l v a r i a t i o n s are mainly s e m i - d i u r n a l through the S t r a i t except i n 6 the San Juan region where they are mainly d i u r n a l (Parker, 1977). The t i d e s are b e l i e v e d to enter the system through the Juan de Fuca S t r a i t as a b a r o t r o p i c K e l v i n wave. As the wave proceeds through the t o p o g r a p h i c a l l y c o m p l icated and shallow San Juan A r c h i p e l a g o , i t l o o s e s i t s p r o g r e s s i v e nature. The t i d a l wave i s r e f l e c t e d at the northern end of the GS , thus forming a standing wave p a t t e r n i n the GS. The r e f l e c t e d wave has been completely f r i c t i o n a l l y damped by the time i t reaches Juan de Fuca S t r a i t (Parker, 1977). T i d a l motion i s the s i n g l e most important d r i v i n g mechanism i n the GS ( a l s o i n the Juan de Fuca). Up to 70% of the k i n e t i c energy i s found at t i d a l f r e q u e n c i e s (Schumacher et a l . , 1978). A strong f o r t n i g h t l y modulation of the c u r r e n t s and, i n c e r t a i n i n s t a n c e s , of other p r o p e r t i e s i s found i n a l l r e g i o n s of GS (Johnstone S t r a i t : Thomson, 1976; Regions 2-3: H e l b i g , 1978; Region 4: Webster, 1977; Juan de Fuca S t r a i t : Holbrdok et a l . , 1978, 1980). T h i s modulation i s b e l i e v e d to be of t i d a l o r i g i n . Numerical t i d a l models of the GS have been developed by Crean (1976, 1978). They provide t i d a l e l e v a t i o n s and c u r r e n t s on a 2-km s c a l e f o r s e l e c t e d areas and on a 4-km s c a l e f o r the whole S t r a i t . These models p r e d i c t coherent r e s i d u a l c i r c u l a t i o n p a t t e r n s ( H e l b i g , 1978). T i d a l motion can induce lower frequency r e s i d u a l motions through n o n l i n e a r and f r i c t i o n a l e f f e c t s , and by f l o w i n g around or over topographic i r r e g u l a r i t i e s . I a n i e l l o (1977, 1979) showed that both steady and slowly v a r y i n g r e s i d u a l c u r r e n t s can be generated by no n l i n e a r t i d a l i n t e r a c t i o n s . Loder (1980) used the 7 r e c t i f i c a t i o n Of M2 r o t a r y t i d a l c u r r e n t s On the s l o p i n g s i d e s Of GeOrges Bank (Gulf Of Maine) to account f o r the p e r s i s t e n t c l o c k w i s e c i r c u l a t i o n around the bank. The r e c t i f i c a t i o n mechanism i n v o l v e s c o n t i n u i t y and C O r i o l i s e f f e c t s r e g u l a t e d by bottom f r i c t i o n . The c r o s s - i s o b a t h component of v e l o c i t y i s i n c r e a s e d by c o n t i n u i t y as i t moves i n t o a shallower r e g i o n . The C O r i o l i s e f f e c t i s i n c r e a s e d . Thus, a water column w i l l move through a l a r g e r h a l f - e l l i p s e i n shallower water than i n deeper water. Over a complete t i d a l p e r i o d , there w i l l be a net d r i f t i n the alOng-isObath d i r e c t i o n . Zimmerman (1981) reviewed the dynamics Of such t i d a l r e s i d u a l e d d i e s . The i r r e g u l a r s t r u c t u r e Of the bottom topography Or Of the c o a s t l i n e seems to generate a n o n l i n e a r t r a n s f e r Of v o r t i c i t y from the o s c i l l a t i n g t i d a l f i e l d to the mean f i e l d . LOw-frequency ( s u b t i d a l ) v a r i a n c e was Observed by Chang (1976) and by Chang et a l . ( 1 9 7 6 ) . They analyzed long-term records (319 to 533 days) Of c u r r e n t c o l l e c t e d at s i x l o c a t i o n s On s i x moorings ( l i n e H i n f i g . 2) i n GS. They found that 80% Of the energy in the c u r r e n t s was p a r t i t i o n n e d i n two main s p e c t r a l r e g i o n s : brOad-band lOw-frequencies and narrOw-band t i d a l motions. The lOw-frequency motions are l o c a t e d i n a broad peak centered at 10-25 days. Rotary s p e c t r a Of the subsurface c u r r e n t s are shown i n f i g u r e 3 while r o t a r y s p e c t r a Of the bottom c u r r e n t s are presented i n f i g u r e 4. These are graphs Of frequency times power a g a i n s t l o g Of frequency. They show the 8 Figure 2. Detailed bathymetry of the south-central S t r a i t of Georgia, showing the positions df mddrings H06, H16, H26 analyzed by Chang(l976) and df mddrings 20, 21, 22 described by Yao et a l . ( l 9 8 2 ) . Adapted from LeBldnd(1983). Figure 3. Spectra of currents at 50 m. Number above the of f - s c a l e peaks indicate their values. Adapted from Chang(1976). 10 63.0 36.0 fll if H 26 Ln -2 EASTERN 140m 1= -3 -4 -3 L0Gl0|F| {F IN CPD) i 48.3 138.7 -i -F + F 129 62.5 H16 CENTRAL 200m 83.3 73.4 IPAJ —r--1 -3 -2 226 i94.5 lu L i i — ! — ^ -1 -2 H 0 6 WESTERN 200 m 53.0 32.5 Figure 4. Spectra of currents at 140 and 200 m. Numbers above the of f - s c a l e peaks indicate their values. Adapted from Chang(!976). 11 low-frequency s p e c t r a l s t r u c t u r e at the expense of s t a t i s t i c a l c o n f i d e n c e . The s p e c t r a l v a l l e y between 0.1 cpd and 1 cpd i s s t a t i s t i c a l l y s i g n i f i c a n t , but the d e t a i l e d s t r u c t u r e of the low-frequency regions i s not. The general c h a r a c t e r i s t i c of these s p e c t r a i s t h e i r broad-bandness. Except f o r the two 200m l o c a t i o n s , the low-frequency v a r i a n c e ranges from 30 to 50% of the t o t a l v a r i a n c e . The motion appears to be barOtrOpic i n the east e r n s p e c t r a , while i t 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 . The s u r f a c e i n t e n s i f i c a t i o n Of the c u r r e n t s i s c l e a r l y seen i n f i g u r e 5. These s t i c k diagrams were computed from measurements made between February and May 1981 with p r o f i l i n g cyclesOndes (YaO et a l . , 1982). The moorings were p l a c e d at m i d - s t r a i t and t h e i r l o c a t i o n s are shown as t r i a n g l e s ( s t a t i o n s 20, 21, 22) i n f i g u r e 2.2. YaO et a l . (1982) found that the mean c u r r e n t was c r o s s - s t r a i t and decreased with depth. They a l s o noted that the lOw-frequency c u r r e n t f l u c t u a t i o n s Occured predominantly at p e r i o d s exceeding a week and were a l s o d e c r e a s i n g with depth. However, they found some evidence Of bottom i n t e n s i f i c a t i o n as the amplitude i n the isOpycnal displacements i n c r e a s e d i n magnitude with depth, below 100m. Because the GS i s a major s h i p i n g area, a l l p r e v i o u s l y moored instruments were p o s i t i o n e d at Or belOw 50 m. There are no long-term measurements near the s u r f a c e , where the main p y c n o c l i n e l i e s . The nature and the g e n e r a t i n g mechanisms Of the 1 2 1 1 1 1 1 1 1 1 i 1 i 4 4 ^ 4 4. 7 1 { 1 i i 5 1 C D C D C D C D C D C D C D C D C D C D C D C D — ^ - ^ - ^ - v — C \ I C \ J C \ i C \ J C \ J o CXI o CD O. CM o o CO o o C D O L O co £ o F i g u r e 5. S t i c k s diagrams of c u r r e n t s at GS20 at 20 m i n t e r v a l s . The c u r r e n t s v e c t o r s are d a i l y averaged and p l o t t e d a g a i n s t the day Of the year. North i s to the l e f t Of t h i s page. Adapted from YaO et a l . d 982). 13 low-frequency c u r r e n t s i n GS are, as y e t , u n i d e n t i f i e d . I t appears that these motions are not d i r e c t l y f o r c e d . Chang (1976) analyzed sea l e v e l , atmospheric p r e s s u r e , and winds f o r the 18 month measurement p e r i o d . These q u a n t i t i e s were e s s e n t i a l l y u n c o r r e l a t e d with the c u r r e n t s . The hig h e s t coherence was found between the winds and the c u r r e n t s at the eastern l o c a t i o n . S i m i l a r r e s u l t s are repor t e d by H e l b i g (1978) in the C e n t r a l region as w e l l as by Schumacher et a l . (1978) i n the Southern r e g i o n . Yao et a l . (1982) found that the wind energy i s mainly at higher f r e q u e n c i e s than the c u r r e n t energy. H e l b i g (1978) found that the wind spectrum peaks around 3-5 days. Chang et a l . (1976) a l s o found that coherences between c u r r e n t s f d r both v e r t i c a l and h o r i z d n t a l s e p a r a t i o n s were low at low f r e q u e n c i e s : the c o r r e l a t i o n s c a l e appears t d be s h o r t e r than the s t a t i d n s e p a r a t i o n (10km). However, Yad et a l . (1982) found v i s u a l s i m i l a r i t y between the c u r r e n t s at t h e i r s t a t i o n s 20 and 22 (see f i g . 2.2, f d r l o c a t i o n ) which would suggest a c o r r e l a t i o n s c a l e df at l e a s t 4km. In the v e r t i c a l , they r e p o r t a c o r r e l a t i o n s c a l e df the order df 100m. These r e s u l t s c d n t r a s t with dpen dcean s i t u a t i o n s (MODE Group, 1978) where h d r i z d n t a l c o r r e l a t i o n s c a l e s were found td be 140km at 500m, 70km at 1500m, and 55km at 4000m, while the v e r t i c a l p a t t e r n df coherences was i r r e g u l a r but g e n e r a l l y coherent f d r the m e r i d i d n a l fldw and ndt so s t r o n g l y coherent f d r zonal fldw. Attempts t d provide a t h e o r e t i c a l mddel df such ldw-frequency c u r r e n t f l u c t u a t i o n s have been u n s u c c e s s f u l . H e l b i g and Mysak (1976) showed that an i n f i n i t e twd-layer channel with 1 4 topography l i k e that df l i n e H (slowly eastward s l o p i n g bottom) admits t r a v e l l i n g topographic waves whose f r e q u e n c i e s l i e i n the above dbserved range. T h e i r model however does not p r e d i c t the observed b a r d c l i n i c d i s t r i b u t i d n of k i n e t i c energy, s i n c e t h e i r tdpdgraphic waves are bdttom-trapped. H e l b i g (1978) r u l e s out b a r d t r d p i c - b a r o c l i n i c i n s t a b i l i t y df aldng-channel flow. However, work c u r r e n t l y i n prdgress (Mysak, persdnnal cdmmunicatidn) shows that the eastward cdmpdnent df the mean fldw may be b a r d c l i n a l l y u n s t a b l e at s t a t i o n 20. Schdtt and Mysak (1980) t r i e d td r e s o l v e the h o r i z o n t a l s t r u c t u r e df the low-frequency motidns. D i r e c t i d n a l s p e c t r a c a l c u l a t i o n s d i d not i n d i c a t e ( i n a s t a t i s t i c a l l y s i g n i f i c a n t sense) the presence df a coherent long wave propagating i n the GS, but i t confirmed that low-frequency energy has s h o r t wavelengths ( s h o r t e r than 40km). T h i s l a s t c o n c l u s i o n i s df a negative nature r a t h e r than a p o s i t i v e r e s u l t . I t simply means that any wave-like mdtidn advanced as a model df the low-frequency v a r i a n c e should possess a wavelength s h o r t e r than the s e p a r a t i o n between the c u r r e n t l y a v a i l a b l e r e c o r d s . Yao et a l . (1982) a l s o r e j e c t e d the i n t e r n a l K e l v i n wave as an a l t e r n a t i v e model. Fdr i n t e r n a l K e l v i n waves, the amplitude df the v e r t i c a l v e l d c i t y i n c r e a s e s towards the p y c n d c l i n e . T h e i r d b s e r v a t i d n s i n d i c a t e d that the amplitude of the i s o p y c n a l displacements i n c r e a s e s i n magnitude with depth. H e l b i g (1978) c l a i m s that the dbserved f l u c t u a t i d n s are probably ndt due t d simple wave-like mdtidns. LeBlond (1983) p o i n t s out that the brdad bandwidths and the ldw c d r r e l a t i d n s df 1 5 the observed c u r r e n t s are more suggestive of turbulence than wave motion. F o l l o w i n g Rhines (1977), he suggests that an ex p l a n a t i o n of observed r e s i d u a l c u r r e n t v a r i a b i l i t y and s p a t i a l s t r u c t u r e should be sought i n terms of g e o s t r o p h i c t u r b u l e n c e r a t h e r than i n a wave propagation model. 2.2 Low-frequency T o p o g r a p h i c a l l y Trapped Waves Even i f the pre v i o u s f i n d i n g s tend to q u e s t i o n the p e r t i n e n c y of wave propagation models f o r GS, a l o g i c a l s t e p i s to examine i t s second c l a s s modes of o s c i l l a t i o n . The bathymetry of GS presents l a r g e changes over short s p a t i a l s c a l e s . Current a n a l y t i c a l models do not take i n t o account both l a r g e bathymetric v a r i a t i o n s on a channel or basin s c a l e together with s t r a t i f i c a t i o n . To quote Csanady (1982): "key simple models have not yet c r y s t a l l i z e d " . We now examine some df the low-frequency topographic wave models. Second c l a s s dr v d r t i c i t y waves are waves whose r e s t o r i n g mechanism a c t s through the c o n s e r v a t i o n df p o t e n t i a l v d r t i c i t y . In an hdmogenedus ocean, the c o n s e r v a t i o n df p o t e n t i a l v d r t i c i t y reads: D_ _ + f = 0 , f = 2S2sin(0) (2.1) Dt H + r? where $ i s the v e r t i c a l cdmpdnent df r e l a t i v e v d r t i c i t y , f the C d r i d l i s parameter (a f u n c t i o n df l a t i t u d e , 6), H(x,y) the depth, and rj(x,y,t) the displacement above the g e d p d t e n t i a l 1 6 l e v e l of the Ocean at r e s t . F o l l o w i n g the motion, l o c a l ( r e l a t i v e ) v o r t i c i t y i s generated to counterbalance the v a r i a t i o n s Of f with l a t i t u d e and/Or the depth changes. V a r i a t i o n s Of the C O r i o l i s e f f e c t with l a t i t u d e w i l l s u s t a i n ROssby waves-, while depth v a r i a t i o n s w i l l s u s t a i n topographic ROssby waves. The reader i s r e f e r r e d t o LeBlOnd and Mysak (1978) Or to Pedlosky (1979) f o r more d e t a i l s . The areas where the depth changes are most pronounced are c o a s t a l a r e a s : c o n t i n e n t a l slope and s h e l f , e s t u a r i e s and l a k e s . COastal trapped waves and t h e i r r o l e i n s h e l f dynamics have been r e c e n t l y reviewed by LeBlOnd and Mysak (1977), Mysak (1980a, b), A l l e n (1980), Huthnance (1981). The l a t t e r two reviews are broader i n scope and c o n s i d e r C o n t i n e n t a l Shelf waves as One Of many c o a s t a l p r o c e s s e s . FOr channels, lakes and i n l a n d seas, the reader should c o n s u l t Simons (1980) and Csanady (1981, 1982). We w i l l now review some pre v i o u s r e s u l t s p e r t i n e n t to t h i s work to see how they can or cannot e x p l a i n GS low-frequency motions. 2.2.1 C o n t i n e n t a l Shelf Waves C o n t i n e n t a l Shelf waves represent the simplest type Of lOw-frequency c O a s t a l l y trapped waves. They are the simplest because they have Only One boundary c o n d i t i o n to s a t i s f y : ( u s u a l l y ) at an i n f i n i t e l y long s t r a i g h t c o a s t . T y p i c a l l y , they have amplitudes Of a few ce n t i m e t e r s , wavelengths Of a few thousand k i l o m e t e r s and p e r i o d s Of a few days.. They propagate with the coast to the r i g h t ( l e f t ) i n the northern (southern) 1 7 hemisphere. Many t h e o r e t i c a l e f f o r t s have focused on determining the p r o p e r t i e s of S h e l f waves over d i f f e r e n t types of topography. We w i l l review some of them as we are i n t e r e s t e d i n how those r e s u l t s w i l l be m o d i f i e d by the i n s e r t i o n of a second w a l l (channel model). The a n a l y s i s of C o n t i n e n t a l S h e l f wave motion i s based on the unforced l i n e a r i z e d shallow water equations f o r a u n i f o r m l y r o t a t i n g f l u i d . These equations are given i n s e c t i o n 3.3 f o r a two-layer f l u i d . For an homogeneous f l u i d , they can be manipulated i n t o a s i n g l e equation f d r the s u r f a c e e l e v a t i d n [HV 2T/ + H TJ + g - M 3 + f 2 ) 7? 3 + fH rj = 0 ( 2 . 2 ) x x t t t x y T y p i c a l topographic p r o f i l e s used i n the s t u d i e s df Shelf waves are presented i n f i g u r e 6. The i n f i n i t e s l o p i n g beach ( f i g . 6a) was s t u d i e d by Reid (1958). Reid named the s u b i n e r t i a l s d l u t i d n s " q u a s i - g e d s t r d p h i c waves". T h e i r c r d s s -s h e l f s t r u c t u r e i s d e s c r i b e d by Laguerre polynomials. Robinson (1964) s t u d i e d trapped waves on a s l o p i n g s h e l f df f i n i t e width ( f i g . 6b). The c r o s s - s h e l f s t r u c t u r e df these i s r e l a t e d t d Laguerre f u n c t i o n s . I t i s Robinson who c a l l e d those low-frequency trapped waves " C o n t i n e n t a l S h e l f waves". For very long waves df very low-frequency, the mddal s t r u c t u r e may be expressed i n terms df zerdth-drder B e s s e l f u n c t i o n s . The next p r d f i l e , with upward c o n c a v i t y ( f i g . 6 c ) , was s t u d i e d by B a l l (1967). B a l l fdund that the a p p r d p r i a t e s d l u t i d n s f d r the 18 Figure 6. Topographic p r o f i l e s and two-layer models used in Shelf waves l i t t e r a t u r e . 19 trapped modes are i n terms of J a c d b i polynomials. In ndndimensidnal form, the i n t e g r a t e d c o n s e r v a t i o n of mass equation reads (Hu) + (Hv) = - (L/Re) 2 TJ = -FT? (2.3) x y t t where Re i s the e x t e r n a l Rdssby deformation r a d i u s Re 2 = gD / f 2 (2.4) and where D and L are the t y p i c a l depth and l e n g t h s c a l e s df mdtidn, r e s p e c t i v e l y . Hence, f d r s c a l e s small compared t d the e x t e r n a l Rdssby deformation r a d i u s the s u r f a c e e l e v a t i d n may be n e g l e c t e d . In the s h e l f r e g i d n , a t y p i c a l l e n g t h s c a l e i s the s h e l f width, L = O(lOOkm), while the e x t e r n a l Rdssby r a d i u s i s 0(1000km). So F i s small and the mdtidn can be c o n s i d e r e d h o r i z o n t a l l y non-divergent. The e x t e r n a l Rdssby r a d i u s i s the l e n g t h s c a l e at which the c o n t r i b u t i o n df the f r e e s u r f a c e t d the vortex s t r e t c h i n g phenomenon becomes df the same magnitude, i n the v o r t i c i t y balance , as the c o n t r i b u t i o n s due t d the mdtidn over v a r i a b l e depth (Pedldsky, 1979). With the r i g i d - l i d approximation, equation (2.3) becomes (Hu) + (Hv) = 0 x y and a d e p t h - i n t e g r a t e d mass-transpdrt stream f u n c t i o n , i/>, may be 20 i n t r o d u c e d ty = Hv and ty = -Hu (2.5) x y where u and v are the v e l o c i t i e s i n the x and y d i r e c t i o n s , r e s p e c t i v e l y . The v d r t i c i t y equation (2.1), once l i n e a r i z e d , becomes [HW2ty - H ty - H ty ] + f [ H i / / - H ty ] = 0 (2.6) x x y y t x y y x Buchwald and Adams (1968) invdked the r i g i d - l i d approximation and a l s o i n t r o d u c e d an e x p o n e n t i a l depth p r o f i l e ( f i g . 6e). T h i s p r o f i l e r e s u l t s i n a constant topographic l e n g t h s c a l e L = H / H = constant (2.7) * x T h i s p r o f i l e i s a l s o mathematically a t t r a c t i v e s i n c e equation (2.6) reduces t d a second order d i f f e r e n t i a l equation w i t h constant c o e f f i c i e n t s . The q u a l i t a t i v e theory df c o a s t a l trapped waves i n an homogeneous dcean has been s t u d i e d by Huthnance (1975). He showed that f d r any mdndtdnic and continuous depth p r o f i l e that tends t d a constant as x tends t d i n f i n i t y , there i s an i n f i n i t e d i s c r e t e set df C d n t i n e n t a l S h e l f waves that are trapped at a l l 21 wavenumbers. He a l s o e s t a b l i s h e d the f o l l o w i n g important r e s u l t s r e g a r d i n g the group v e l o c i t y of each Shelf wave mode. F i r s t , long C o n t i n e n t a l S h e l f waves are n o n - d i s p e r s i v e : the group v e l o c i t y tends t o the phase v e l o c i t y as the wavenumber tends to zero. Second, i f the topographic l e n g t h s c a l e (eq. 2.7) i s bounded f o r a l l x then, f o r some wavenumber, the group v e l o c i t y w i l l be ne g a t i v e . Since the d i s p e r s i o n curve f o r each mode i s continuous, t h i s i m p l i e s that the Sh e l f wave has a zero group v e l o c i t y f o r some wavenumber. The presence Of s t r a t i f i c a t i o n , among other t h i n g s (mean c u r r e n t , longshore v a r i a t i o n s of the topography, e t c . ) , can s i g n i f i c a n t l y a l t e r the p r o p e r t i e s of Sh e l f waves. In presence Of s t r a t i f i c a t i o n , i t i s not g e n e r a l l y p o s s i b l e to separate the dependent v a r i a b l e s d e s c r i b i n g the wave motion i n t o h o r i z o n t a l and v e r t i c a l p a r t s , corresponding to h o r i z o n t a l l y propagating waves with an independent v e r t i c a l normal mode s t r u c t u r e . In the c o n t i n u o u s l y s t r a t i f i e d problem, we l o s e the s e p a r a b i l i t y . However, i f the equations can be s o l v e d by other means, we gain a b e t t e r r e p r e s e n t a t i o n of the v e r t i c a l s t r u c t u r e of the motion. Most a n a l y t i c a l s t u d i e s of C o n t i n e n t a l Shelf waves i n presence of s t r a t i f i c a t i o n i n v o l v e two-layer models. Two-layer models f a l l i n t o two c l a s s e s : On-shelf s t r a t i f i c a t i o n ( f i g . St) and deep-sea s t r a t i f i c a t i o n ( f i g . 6e). The deep-sea s t r a t i f i c a t i o n i s e a s i e r to handle s i n c e , over the s h e l f , the problem reduces to one of S h e l f waves i n an homogeneous ocean, while the depth i s constant i n the deep-sea r e g i o n . I t was found (Mysak, 1967; K a j i u r a , 1974) that the frequency of long 22 She l f waves i s s i g n i f i c a n t l y i n c r e a s e d , i n the deep-sea s t r a t i f i c a t i o n model, above that of an u n s t r a t i f i e d model. The most thorough s t u d i e s of the on-shelf s t r a t i f i c a t i o n model were c a r r i e d out independently by Wang (1975), n u m e r i c a l l y , and by A l l e n (1975), a n a l y t i c a l l y . In s e c t i o n 3.3, we w i l l d e r i v e the dynamical equations f o r the on-shelf s t r a t i f i c a t i o n model and A l l e n ' s approach w i l l be d i s c u s s e d i n s e c t i o n 3.4. Because df the combined e f f e c t s df s t r a t i f i c a t i d n and tdpdgraphy and the presence df a v e r t i c a l w a l l , two types df low-frequency c d a s t a l l y trapped waves w i l l e x i s t : Shelf waves m o d i f i e d by the presence df s t r a t i f i c a t i o n and i n t e r n a l K e l v i n waves md d i f i e d by the presence df tdpdgraphy. Surface g r a v i t y waves and the e x t e r n a l K e l v i n waves are f i l t e r e d out by the r i g i d - l i d approximatidn. A l l e n (1975) found that S h e l f waves and i n t e r n a l K e l v i n waves are coupled and the s t r e n g t h df the c d u p l i n g i s represented by the magnitude df the parameter X 2 = R i 2 / L 2 « 1 (2.8) * where L i s the topographic l e n g t h s c a l e d e f i n e d by (2.7) and Ri * i s the i n t e r n a l Rdssby deformation r a d i u s d e f i n e d , f d r a two-l a y e r f l u i d , as R i 2 = g' D T D 2 / f 2'(D, + D 2) (2.9) 23 where g' i s the reduced g r a v i t y and D, and D 2 the t y p i c a l or mean depth of each l a y e r . As the e x t e r n a l ROssby r a d i u s i s the t r a p p i n g s c a l e f o r the e x t e r n a l K e l v i n wave, the i n t e r n a l ROssby r a d i u s i s the t r a p p i n g s c a l e f o r the i n t e r n a l K e l v i n wave. The i n t e r n a l ROssby r a d i u s i s 0(100km) i n the Open Ocean and 0(15km) On the s h e l f . A l l e n (1975) used a r e g u l a r p e r t u r b a t i o n expansion i n powers Of X and s t u d i e d the c o u p l i n g between Sh e l f waves and i n t e r n a l K e l v i n waves f o r long, i n t e r m e d i a t e and very short wavelengths. For long wavelengths, he found that 0(1) K e l v i n waves are accompanied by O(X) b a r o t r o p i c motion and t h a t 0(1) S h e l f waves are accompanied by weak 0 ( X 2 ) b a r O c l i n i c motion. FOr i n t e r m e d i a t e wavelengths, the b a r o t r o p i c and b a r O c l i n i c motions are f u l l y coupled. FOr very short wavelengths, A l l e n (1975) showed t h a t the Shelf wave motion i s bottom trapped. At intermediate wavelengths, he found that there i s a change i n modal s t r u c t u r e Of the K e l v i n wave and S h e l f waves. Because Of t h e i r comparable phase speeds, there are p o i n t s i n the frequency-wavenumber plane where both waves exchange i d e n t i t i e s . T h i s was a l s o noted by Wang (1975). When the s t r a t i f i c a t i o n i s continuous, we step from a system Of coupled d i f f e r e n t i a l equations to a s i n g l e equation which now i n c l u d e s v e r t i c a l d e r i v a t i v e s . The most complete a n a l y s i s Of the dynamical equations i n the case Of continuous s t r a t i f i c a t i o n was performed by Huthnance (1978). Huthnance d i d not invoke the r i g i d - l i d approximation a p r i o r i . However, he showed that the f r e e - s u r f a c e divergence e f f e c t s are g e n e r a l l y very s m a l l . His c a l c u l a t i o n s were done n u m e r i c a l l y , except f o r 24 a h(x) = \/x p r o f i l e f o r which the equations are separable i n p a r a b o l i c c o o r d i n a t e s . Huthnance (1978) showed that over any monotdnically i n c r e a s i n g depth p r o f i l e , there i s a s i n g l e i n f i n i t e f a m i l y of low-frequency (frequency s m a l l e r than f) trapped mddes with f r e q u e n c i e s d e c r e a s i n g t d zero, f d r a cdnstant wavenumber, as the mdde number i n c r e a s e s . The phase t r a v e l s with the cdast t d i t s r i g h t i n the ndrthern hemisphere. In g e n e r a l , the r e s u l t s are f a i r l y complex, but Huthnance fdund that the waves adopt s p e c i a l forms i n three asymptotic l i m i t s . Fdr weak s t r a t i f i c a t i o n , the mdtidn becomes depth-independent and the waves reduce t d b a r d t r d p i c Shelf waves. Fdr strdng s t r a t i f i c a t i o n , the wave c h a r a c t e r i s t i c s are those df i n t e r n a l K e l v i n waves. F i n a l l y , f d r l a r g e longshore wavenumbers, the waves become bdttdm-trapped topographic Rdssby waves df the type d e s c r i b e d by Rhines (1970). Cdntrary t d A l l e n (1975) and Wang (1975), Huthnance (1978) fdund no evidence df the mdde-cdupling between Shelf waves and K e l v i n waves. According t d Mysak (1980a), i t appears that t h i s phenomenon may be due t d the twd-l a y e r approximation. I t c d u l d a l s o be due t d the f a c t that Huthnance never c o n s i d e r e d a p r o f i l e with a v e r t i c a l w a l l at the cdast and a wide f l a t s h e l f . More work i s r e q u i r e d i n t h i s matter. 2.2.2 Channel Models The problem df tdpdgraphic waves t r a v e l l i n g aldng a channel i s b a s i c a l l y the same as that df C o n t i n e n t a l S h e l f waves. There 25 are two added d i f f i c u l t i e s . The f i r s t One comes from the second boundary. Simple f i n i t e polynomial s o l u t i o n s w i l l not be p o s s i b l e anymore, s i n c e we need the second s o l u t i o n Of the d i f f e r e n t i a l equation to s a t i s f y the boundary c o n d i t i o n s . Thus the d i s p e r s i o n r e l a t i o n w i l l be an i m p l i c i t (and sometimes messy) One. The second d i f f i c u l t y cOmes from the d i f f e r e n t i a l equations themselves. In channel models, the e x p o n e n t i a l depth p r o f i l e i s u s u a l l y i n a p p r o p r i a t e and the equations (eqs 2.2 Or 2.6, i n the homogeneous case) w i l l have v a r i a b l e c o e f f i c i e n t s . The c h a r a c t e r i s t i c type Of behaviour Of Channel waves i s best i l l u s t r a t e d by t h e i r c l o s e s t r e l a t i v e s : the Trench waves (Mysak et a l . , 1979). When a t r e n c h f o l l o w s a c o a s t l i n e the depth does not i n c r e a s e monotOnically away from the boundary. The r e v e r s a l i n bottom slope supports another set Of v o r t i c i t y waves propagating t h e i r phase i n a d i r e c t i o n Opposite to that Of S h e l f waves. Since the channel models we w i l l examine present such a slope r e v e r s a l , the s o l u t i o n s should present the same O s c i l l a t o r y - d e c a y i n g behaviour as the Trench and S h e l f waves. That i s , the motion w i l l be O s c i l l a t o r y On the s i d e Of the channel to the r i g h t Of t h e i r d i r e c t i o n Of propagation and decaying on the Other s i d e ( i n the northern hemisphere). FOr homogeneous channels, q u a l i t a t i v e r e s u l t s s i m i l a r to those Of Huthnance (1975) were d e r i v e d by Odulo (1975). Because Of the presence Of two w a l l s , Odulo found that two K e l v i n waves are p o s s i b l e , as w e l l as ROssby waves and topographic ROssby waves. MOst Of the work On i n f i n i t e channels has been d e d i c a t e d to wind-forced steady and unsteady motions (see Simons, 1980, 26 fd r a review). Fdr low-frequency tdpdgraphic mdtidns, channels are o f t e n c o n s i d e r e d wide endugh that the e f f e c t df the second w a l l may be ne g l e c t e d , as i n Csanady (1973). Another approach i s t d intrdduce a second w a l l befdre the bottom sldpe changes s i g n , as i n H e l b i g and Mysak (1976). Recently, L i e and El-Sabh (1983) used a two-layer numerical model t d study the f r e e tdpdgraphic mddes i n a channel s i m i l a r i n shape and s i z e t d the Saint-Lawrence e s t u a r y . T h e i r numerical e i g e n f u n c t i o n s present the d s c i l l a t d r y - d e c a y i n g behaviour d i s c u s s e d i n the l a s t paragraph. 2.2.3 Enclosed Basins Only simple b a s i n shapes allow s d l u t i d n s of the ldw-frequency wave equations i n a n a l y t i c a l fdrm and, sometimes, pr o v i d e simple c l o s e d s d l u t i d n s . Ldnguet-Higgins (1964, 1965) c a l c u l a t e d the r o t a t i o n a l mddes (normal Rdssby mddes) f d r r e c t a n g u l a r and t r i a n g u l a r b a s i n s , but f l a t bottom. The problem with basins df a r b i t r a r y shapes and tdpdgraphies i s that the equations are not separable u n l e s s the tdpdgraphy s a t i s f i e s (Rhines and B r e t h e r t d n , 1973) V • [p - f V l n ( H ) ] = 0 (2.10) If the depth f u n c t i o n , H(x,y), s a t i s f i e s (2.10), dne w i l l be abl e t d f a c t d r out a ca r r i e r - w a v e from (2.6) t d db t a i n an Helmhdtz-1ike equatidn f d r the modulating amplitude. The prdblem i s reduced t d a membrane v i b r a t i o n problem. Ripa (1978) 27 i n t r o d u c e d a v a r i a t i o n a l p r i n c i p l e to approximate the modes df a r e c t a n g u l a r basin which i n c l u d e s a s h e l f dr a r i d g e , and those df a c i r c u l a r b a s i n c o n t a i n i n g a seamdunt. T h i s method i s e s p e c i a l l y e f f e c t i v e f d r strong and i s d l a t e d tdpdgraphic f e a t u r e s f d r which i t y i e l d s a good approximation df the s o l u t i o n as w e l l as a lower bdund t d the f a s t e s t mdde. With a r b i t r a r y tdpdgraphy and shape, the prdblem has td be so l v e d n u m e r i c a l l y . Platzman (1978, 1980) computed the normal modes df the world ocean. He fdund 13 tdpdgraphic mddes sldwer than 30h. Rad and Schwab (1975) c a l c u l a t e d the g r a v i t a t i o n a l and r o t a t i o n a l mddes df Lake O n t a r i o and Lake S u p e r i o r . Fdr both l a k e s , they computed 40 r d t a t i d n a l mddes with p e r i o d s ranging between 52h t d 14 years f d r Lake O n t a r i o , and p e r i o d s between 79h t d s e v e r a l thousand years f d r Lake S u p e r i o r . When the b a s i n has c i r c u l a r (dr e l l i p t i c ) symmetry, the r d t a t i d n a l mddes can be c a l c u l a t e d a n a l y t i c a l l y . B a l l (1965) s t u d i e d the r d t a t i d n a l mddes df an e l l i p t i c a l p a r a b d l d i d . He fdund that the lowest order ndn-cdnstant (frequency d i f f e r e n t than zero) mode c o n s i s t s df two e l l i p t i c a l c e l l s r o t a t i n g i n a counter-clockwise d i r e c t i o n around the b a s i n . The t r a n s p o r t s t r e a m l i n e p a t t e r n df those c e l l s i s shown i n f i g u r e 7. Csanady (1973), with the h e l p df an e n l i g h t e n i n g s c a l i n g argument, showed that i t takes approximately one i n e r t i a l p e r i o d fdr these c e l l s t d develop. Once set up, they r d t a t e at much lower frequency. Sayldr et a l . (1980) a p p l i e d B a l l ' s r e s u l t s t d Lake Michigan. They fdund t h a t , at l e a s t f d r the gravest mdde, the p e r i d d df r d t a t i d n i s much more s e n s i t i v e to the bathymetric p r o f i l e than t d the 28 Figure 7. Transport streamline pattern: a) at t=0, b) at t=6ne-eight period, c) at one-fourth period. From Saylbr et al.(1980). 29 e l l i p t i c i t y of the b a s i n . F i n a l l y , B i r c h f i e l d and H i c k i e (1977) and Huang and SaylOr (1982) looked at wind-forced modes in c i r c u l a r and e l l i p t i c a l b a s i n s , r e s p e c t i v e l y . TO summarize : the bulk Of the p r e v i o u s a n a l y t i c a l work concerning lOw-frequency topographic waves was done f o r an homogeneous Ocean and mostly i n the c o n t i n e n t a l s h e l f r e g i o n . For c o n t i n u o u s l y s t r a t i f i e d media, the problem i s s o l v e d n u m e r i c a l l y . In two-layer models, One u s u a l l y f o l l o w A l l e n (1975) by d e r i v i n g a p a i r Of coupled equations (eqs 3.20 and 3.21 i n the next chapter) f o r the i n t e g r a t e d mass-transport and the i n t e r f a c e motion. A n a l y t i c a l s o l u t i o n s are to be found i f the topographic l e n g t h s c a l e and the i n t e r n a l ROssby r a d i u s are not Of comparable magnitude, Or i f the bathymetry changes are e i t h e r very slow or very abrupt. In channels and b a s i n s , except f o r the Ball(1965) and SaylOr et a l . (1980) approach , topographic waves are u s u a l l y s t u d i e d as s h e l f waves through c r o s s - s e c t i o n models i n which the shores are c o n s i d e r e d f a r a p a r t . The Only model i n which r e v e r s i n g s l o p e s are c o n s i d e r e d i s the Trench wave model. Moreover, most models deal with an homogeneous water column, hence they cannot reproduce the s u r f a c e - i n t e n s i f i e d motion found in the S t r a i t Of Georgia. In the next chapters we w i l l develop a two-layer model based On A l l e n ' s (1975) approach. We w i l l look at the propagation Of topographic v o r t i c i t y waves Over l a r g e r e v e r s i n g slopes when a l l the l e n g t h s c a l e s are Of the same magnitude. 30 3 PHYSICAL AND MATHEMATICAL MODEL The u l t i m a t e aim of t h i s t h e s i s , beyond the development of a t h e o r e t i c a l model of topographic wave propagation i n channels and b a s i n s , i s to reproduce the general behaviour of low-frequency motions observed i n the GS. Hence, we w i l l e s t a b l i s h the g e n e r a l c h a r a c t e r i s t i c s of GS s t r a t i f i c a t i o n and bathymetry t h a t should be represented i n our model. T h i s w i l l be done i n the f i r s t s e c t i o n . In the f o l l o w i n g s e c t i o n s , we w i l l develop our t h e o r e t i c a l model. 3.1 M o d e l l i n g The S t r a i t Of Georgia To o b t a i n a n a l y t i c a l s o l u t i o n s of the dynamical equations, the model of GS must be simple. At the same time i t must reproduce the general f e a t u r e s of geometry, bathymetry and s t r a t i f i c a t i o n . D e t a i l e d bathymetry of the s o u t h - c e n t r a l GS was presented i n f i g u r e 2. The bathymetry e x h i b i t s great i r r e g u l a r i t y . In g e n e r a l , the GS i s c h a r a c t e r i z e d by extremely steep s l o p e s along i t s western boundary with slopes n e a r l y as steep along the e a s t . T h i s can be seen by l o o k i n g at the bathymetric c r o s s - s e c t i o n s presented i n f i g u r e 9. The l o c a t i o n s of the c r o s s - s e c t i o n s are shown i n f i g u r e 8. The a x i a l bathymetry i s smoother than the t r a n s v e r s e bathymetry N e v e r t h e l e s s , i t a l s o possesses a high degree of i r r e g u l a r i t y . 31 125 124 123 W Figure 8. Plan view of the S t r a i t of Georgia showing l i n e s of topographic cross-sect ion (1-10) presented in figure 9. From Helbig(1978). Figure 9. Topographic cross-sections : a) upper panels: 1-9, b) lower panel: 10. From Helbig(1978). 33 The B r u n t - V a i s a l a frequency N 2 = - ( g / p 0 ) p z depends s t r o n g l y on depth, thdughdut GS. N g e n e r a l l y l i e s between 3 X 10" 3 rad/s t d 3 X 10" 2 rad/s: the water column i s thus w e l l s t r a t i f i e d . The d e n s i t y shows a c o n s i s t e n t p a t t e r n through a l l the GS: r e l a t i v e l y l i t t l e change occurs throughout the year below 50m. At shallower depths, strdng seasonal e f f e c t s are present ( H e l b i g and Mysak, 1976). Thus, a two-layer model i s a c c e p t a b l e . Only a numerical model would be able t d i m i t a t e the cdmplex bathymetry df GS. Here, we wish td b u i l d a model based on the o b s e r v a t i o n s that low-frequency motions are s u r f a c e - i n t e n s i f i e d . We w i l l keep dnly the g e n e r a l f e a t u r e s df GS i n the hope that the f r e e second c l a s s mddes df o s c i l l a t i o n s df our model w i l l be more l i k e the o b s e r v a t i o n s than p r e v i d u s l y known s o l u t i o n s . T h e r e f o r e , the GS i s represented as an i n f i n i t e l y long and s t r a i g h t two-layer channel. The t h i c k n e s s df the s u r f a c e l a y e r w i l l be kept c o n s t a n t . The steep slopes w i l l be c o n s i d e r e d l i n e a r and df o p p o s i t e s i g n s dn each s i d e df the p o i n t df maximum depth ( f i g . 10). Fdr convenience i n the mathematical development, t h i s p d i n t w i l l be l o c a t e d i n the middle df the channel. As w i l l be seen i n chapter 4, i t i s a t r i v i a l matter t d t r a n s l a t e the o r i g i n anywhere a c r o s s the channel. A l l a x i a l v a r i a t i d n s are n e g l e c t e d . T h i s i s an improvement over the p r e v i d u s model ( H e l b i g and Mysak, 1976), a l s d shown in f i g u r e 4 0 0 Fiqure 10. Model f i t to the Strait of Georgia (line H). The dash line is the f i t used by Helbig and and Mysak(1976) and the solid line is our f i t . to 35 10, where the p o i n t of maximum depth was l o c a t e d on the western boundary, and where the small slope approximation was used. At t h i s p o i n t , we set a s i d e the s p e c i f i c problem of the GS to c o n c e n t r a t e on the general problem of topographic waves in two-layer channels and basins with symmetrical depth p r o f i l e s . 3.2 The P h y s i c a l Model In s e c t i o n 2.1, we fdund that one df the main unexplained f e a t u r e s df the GS was that the mdtidn i s s u r f a c e - i n t e n s i f i e d . As was noted i n s e c t i o n 2.2, the equations df motion are not separable i n presence df both cdntinuous s t r a t i f i c a t i o n and tdpdgraphic v a r i a t i o n s . To keep a c e r t a i n amount df s t r a t i f i c a t i d n and t d c o n s i d e r l a r g e bottom s l o p e s , we chose a two-layer model. We w i l l c o n s i d e r two sheets df f l u i d df d i f f e r e n t d e n s i t i e s p, and p 2 , the l i g h t e r f l u i d l y i n g on top df the h e a v i e r . The d i f f e r e n c e i n d e n s i t i e s between the two l a y e r s i s c o n s i d e r e d s m a l l , that i s 6 = ( p 2 " P i ) / P 2 « 1 (3.1) The v a r i a b l e s and v e l o c i t y components are chodsen as shown in f i g u r e 11. The body f o r c e i s modeled as a v e c t o r , _ , a n t i p a r a l l e l t d the v e r t i c a l a x i s . The r o t a t i o n a x i s df the f l u i d c o i n c i d e s with the z - a x i s , sd that the C d r i d l i s parameter i s simply equal t d twice the angular speed df r o t a t i o n df the system. F i n a l l y , the mdtidn i s assumed i n v i s c i d , that i s we Figure 11. Coordinate system, shapes Of the basins and topographic p r o f i l e s studied in th i s thesis. 37 c o n s i d e r only flows f o r which v i s c o s i t y i s unimportant. In the next s e c t i o n , the governing equations w i l l be d e r i v e d a c c o r d i n g to these d e f i n i t i o n s . Next, we w i l l be i n t e r e s t e d i n s o l v i n g the equations f o r deep e s t u a r i e s and l a k e s . They are modelled as i n f i g u r e s 11b and 11c, r e s p e c t i v e l y . E s t u a r i e s are represented by an i n f i n i t e l y long and s t r a i g h t channel bf width 2L, while lakes are t y p i f i e d by a c i r c u l a r b a s i n of r a d i u s L. We w i l l c o n s i d e r topographic waves propagating Over v a r i a b l e topography. To Obtain a n a l y t i c a l s o l u t i o n s , the topography w i l l be a f u n c t i o n bf one c o o r d i n a t e o n l y . In the i n f i n i t e channel problem, i t w i l l vary with the c r o s s - c h a n n e l c o o r d i n a t e , x , while i n the c i r c u l a r b a s i n problem, i t w i l l be a f u n c t i o n of the r a d i u s , r , o n l y . Two d i f f e r e n t topographic p r o f i l e s w i l l be c o n s i d e r e d : l i n e a r and p a r a b o l i c . TO i n s u r e wave propagation, we introduce a c a r r i e r - w a v e s u b s t i t u t i o n G(x,y,t) = Real ( G(x) • e x p ( i k y - itot) ) (3.2a) G(r,0,t) = Real ( G(r) • exp(ik<9 - icot) ) (3.2b) and we w i l l study the c r o s s - c h a n n e l and r a d i a l s t r u c t u r e Of the s o l u t i o n s . In the channel problem, we w i l l compare the behaviour Of the s o l u t i o n s f o r l a r g e topographic v a r i a t i o n s with and without the small slope approximation . The p r o f i l e s w i l l be, i n the l i n e a r case ( f i g . l i d ) 38 H(x) = H 0(1 - s|x|) « H 0 , s « 1 (3.3a) H(x) = H 0(1 - s|x|> , s = 0(1) (3.3b) In the p a r a b o l i c case, the form of the p r o f i l e w i l l be ( f i g . 1 l e ) H(x) = H 0(1 - s x 2 ) * H 0 , s « 1 (3.4a) H(x) = H 0(1 - s x 2 ) , s = 0(1) (3.4b) In the c i r c u l a r b a s i n problem, the depth v a r i a t i o n s are repr e s e n t e d by m H(r) = H 0[1 - ( r / r 0 ) ] (3.5) where a c o n i c a l p a r a m e t e r i z a t i o n i s obtained by s e t t i n g m = 1 , while a p a r a b d l d i d a l p a r a m e t e r i z a t i o n i s obtained f d r m = 2. In s e c t i o n 3.4, we w i l l d e r i v e and d i s c u s s the governing equations, s t a r t i n g from the l i n e a r i z e d long wave equations. The f o l l o w i n g assumptions and approximations are used. 3.3 Basic Assumptidns L i n e a r momentum balance. The v e l o c i t i e s are small sd that t d f i r s t order i n the Rdssby number ( U / f L ) , the n o n l i n e a r terms are n e g l e c t e d . Frdm Chang et a l . (1976), the mean v e l o c i t i e s 39 vary from 3 to 8 cm/s. The C O r i o l i s parameter a p p r o p r i a t e to the mean l a t i t u d e Of GS i s 1.2 X 10"" rad/s. Using the h a l f -width of the GS (10km) as the l e n g t h s c a l e , we o b t a i n a ROssby number Of 0.04. The Rossby number would be Of Order One f o r v e l o c i t i e s Of 10 cm/s and a l e n g t h s c a l e Of 1km. Shallow water model. Although the depth Of the f l u i d v a r i e s i n space, we suppose that a c h a r a c t e r i s t i c v a l u e , D , can be chosen to represent the depth. We a l s o assume t h a t t h i s depth c h a r a c t e r i z e s the v e r t i c a l s c a l e Of motion. The motion's aspect r a t i o (D/L) i s 0.03, i f we take the maximum depth (300m) as the c h a r a c t e r i s t i c depth. The aspect r a t i o i s much smal l e r than One. T h i s means th a t the v e r t i c a l v e l o c i t y , w, i s 0(DU/L), by c o n t i n u i t y . Hence the v e r t i c a l v e l o c i t y i s much smal l e r than the h o r i z o n t a l v e l o c i t y . H y d r o s t a t i c approximation. In the absence Of motion, the v e r t i c a l p r e s s u r e v a r i a t i o n s are Only due to the changes i n weight Of the water column. The v e r t i c a l pressure g r a d i e n t due to the motion i s at most Of 0 ( D 2 / L 2 ) times the v e r t i c a l a c c e l e r a t i o n which i s a l s o small as long as the aspect r a t i o i s s m a l l . Thus the v e r t i c a l balance i s assumed to be h y d r o s t a t i c . T h i s , i n t u r n , y i e l d s that the h o r i z o n t a l pressure g r a d i e n t s (as w e l l as the h o r i z o n t a l v e l o c i t i e s ) are depth-independent i n each l a y e r . M i d - l a t i t u d e f-plane approximation. The e f f e c t Of the bottom s l o p e , a, Outweighs any p o s s i b l e j3-effect (Rhines, 1969). The 0 - e f f e c t i s 0((3/f), t h a t i s 1 0 " 1 V 1 0 " * Or 10" 7 nr 1 . The topographic e f f e c t s are 0(VH/H), t h a t i s 10" 2/10 2 Or 10"" nr 1 . 40 Small amplitude motion. In each l a y e r , the depth changes dues to the v e r t i c a l motions are n e g l e c t e d , compared to the depth of the l a y e r . 3.4 Governing Equations The equations governing the motion are the l i n e a r i z e d momentum equations f o r a two-layer f l u i d on a f-plane (LeBlond and Mysak, 1978, page 134) u, - f v , + gi?, = 0 (3.6a) t x v, .+ f u , + gT?! = 0 (3.6b) t y (H,u,) + (H,v,) = ( T?2-77 n ) (3.6c) x y t u 2 - f v 2 + gT?, + g ' ( T ? 2 - T ? 1 ) = 0 (3.7a) t x x v 2 + f u 2 + gT?, + g ' ( i 7 2 - T ? i ) = 0 (3.7b) t y y ( H 2 u 2 ) + ( H 2 v 2 ) = -T? 2 (3.7c) x y t where the numeral and the l e t t e r s u b s c r i p t s represent the l a y e r (1 fdr upper, 2 fdr lower) and p a r t i a l d i f f e r e n t i a t i o n , r e s p e c t i v e l y . The v a r i a b l e s are d e f i n e d as f o l l o w s (see f i g . 11a) (u ,v ) : crdss-channel and aldng-channel v e l d c i t i e s i i 41 f : C o r i o l i s parameter (taken as 1.2 X 10"" rad/s) g : 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 (taken as 9.8 m/s 2) p : the d e n s i t y i n l a y e r i i g' : the reduced g r a v i t y = ( p 2 ~ P i ) g / P 2 = 5g H (x,y) : the t h i c k n e s s Of the i - t h l a y e r when at r e s t The d e r i v a t i o n s i n t h i s s e c t i o n f o l l o w c l o s e l y that of A l l e n (1975), but i n dimensional form. Equations (3.6c) and (3.7c) are the d e p t h - i n t e g r a t e d c o n t i n u i t y equations f o r each l a y e r . The depth changes (vortex s t r e t c h i n g and compression) are due to two aspects of the same process : changes i n t o t a l depth through v a r i a t i o n s i n the mean depth H(x,y) as w e l l as i n the s u r f a c e and i n t e r f a c e displacement and TJ 2 , r e s p e c t i v e l y . The small amplitude assumption t e l l s us that the depth changes due to the sur f a c e and i n t e r f a c e motions may be neg l e c t e d , compared to the depth i n each l a y e r . Adding and s u b s t r a c t i n g (3.7c) to and from (3.6c) y i e l d s , r e s p e c t i v e l y , (H,u, + H 2u 2) + (H,v, + H 2 v 2 ) = -T?, (3.8a) x y t (H,u, " H 2u 2) + (H,v, - H 2 v 2 ) = (2TJ2-7},) (3.8b) x y t For s c a l e s Of motion small compared to the e x t e r n a l ROssby deformation r a d i u s , Re, the motion may be assumed h o r i z o n t a l l y 42 non-divergent and the s u r f a c e of the top l a y e r may be c o n s i d e r e d r i g i d . Then, the gr?, and grj, terms i n (3.6) represent the x y pressure e x e r t e d on t h i s l i d by the wave motion. We are l o o k i n g at time s c a l e s long compared to the l o c a l i n e r t i a l p e r i o d ( 27r/f). Thus, i n (3.6), the main balance i s between the pre s s u r e g r a d i e n t and the C d r i d l i s term. In the q u a s i - g e d s t r o p h i c paradigm (we keep the l o c a l time v a r i a t i o n term), the r i g h t - h a n d s i d e d f (3.8a) i s 0 ( L 2 / R e 2 ) times the l e f t - h a n d s i d e . In the GS, the l e n g t h s c a l e i s chdsen as 10km (the h a l f - w i d t h ) , while the e x t e r n a l Rdssby r a d i u s i s O(450km). Hence, the 77, term may be s a f e l y n e g l e c t e d : the mdtidn i s t h o r i z o n t a l l y non-divergent. Now, equation (3.7a) may be w r i t t e n as U 2 " f V 2 + g [ TJ ! + 6 ( 77 2 - 7},)] = 0 t x Again, i n a q u a s i - g e d s t r d p h i c paradigm, a l l the terms are r e t a i n e d . Since the s t r a t i f i c a t i o n i s weak (5 = 2.25 X 10" 3, fdr the GS), 77, = 0 ( 5 T 7 2 ) . Otherwise, t d 0 ( 5 ) , the mdtidn w i l l be p u r e l y b a r d t r d p i c . Thus, equations (3.8a) and (3.8b) may be w r i t t e n as (H,u, + H 2u 2) + (H,v, + H 2 v 2 ) = 0 - 0(77, ) (3.9a) x y t (H,u, - H 2u 2) + (H,v, - H 2 v 2 ) = 2T72 - 0(5 T ? 2 ) (3.9b) x y t t 43 Two major consequences f o l l o w from the r i g i d l i d approximation. F i r s t , g r a v i t y waves, edge waves and b a r o t r o p i c K e l v i n waves are f i l t e r e d out. Second, i t allows the i n t r o d u c t i o n of a mass-t r a n s p o r t stream f u n c t i o n \p : H,u, + H 2 u 2 = -\p (3.10a) y H,v, + H 2 v 2 = $ (3.10b) x The i n t r o d u c t i o n bf the depth i n t e g r a t e d mass-transport stream f u n c t i o n enables us to express the set Of governing equations (3.6-3.7) as a system bf two coupled l i n e a r second order d i f f e r e n t i a l equations f o r the v a r i a b l e s \p and 7?2 . T h i s i s done i n the f o l l o w i n g manner. Ex p r e s s i n g u ,v i n terms bf T?, and i ? 2 , we get from (3.6) i i and (3.7) M(u t) = - g [ T j , + fr?, ] (3.11a) xt y M(v, ) = -g[i?, ~ tr), ] (3.11b) yt x M(u 2) = -g[ (, + 677 2 > + f(»Ji + 8TJ 2) ] (3.12a) xt y M(v 2) = - g [ ( T ) , + 8rj 2) - f(i?,'+ ST? 2) ] (3.12b) yt x where M = 3 + f 2 (3.13) t t 44 S u b t r a c t i n g a l t e r n a t e p a i r s i n (3.11) and (3.12), we ob t a i n M(u, - u 2) = -g'(r? 2 + f T ? 2 ) (3.14a) xt y M(v, - v 2 ) = -g'(?j 2 - fr? 2 ) (3.14b) yt x The operator M i s then a p p l i e d to (3.10), g i v i n g M(H 1u 1 + H 2u 2) = -M(ty ) y M(H,v, + H 2 v 2 ) = M(ty ) x The i n d i v i d u a l v e l o c i t y components may be obtained from the l a s t four equations i n terms of ty and T J 2 i n the form M(u,) = -[ M(ty ) - g'H 2 ( 7 } 2 + fr? 2 ) ] / H (3.15a) y xt y M(v,) = -[-Mity ) - g ' H 2 ( T j 2 - f i ? 2 ) ] / H (3.15b) x yt x M(u 2) = -[ M(0 ) + g ' H ^ T j a + f T ? 2 ) ] / H (3.16a) y xt y M(v 2) = -[-MU ) + g fH, ( T J 2 - f T ? 2 ) ] / H (3.16b) x yt x Equations (3.15)-(3.16) w i l l g i ve the v e l o c i t i e s u and v i i in terms of \p and T J 2 once a s o l u t i o n has been found. Next, to o b t a i n the equations f o r ty and T?2 , we combine 45 (3.6) and (3.7) by d e r i v i n g the v o r t i c i t y equations f o r the t o t a l mass-transport and f o r the v e l o c i t y d i f f e r e n c e s . The f i r s t r e l a t i o n i s obtained through the f o l l o w i n g m anipulation [ H,-(3.6a) + H 2.(3.7a) ] - [ H,.(3.6b) + H 2-(3.7b) ] y x Upon u s i n g (3.10), we get V 2 i / / = - g [ H 2 (TJ, + 6T72 ) " H 2 ( T J , + 6 T J 2 ) ] (3.17) t x y y x where V 2 i s the two-dimensional L a p l a c i a n . The second r e l a t i o n i s Obtained by a p p l y i n g the M operator t o [ (3.6a) - (3.7a) ] - [ (3.6b) - (3.7b) ] y x to y i e l d [M(v,-v 2) - M(u,-u 2) ] + f[M(u,-u 2) + M(v,-v 2) ] = 0 (3.18) x y t x y With the h e l p bf (3.6) and (3.7), TJ , may be expressed i n terms bf \p and T J 2 i n the form H g T j , = \p + t\ji - H 2 g ' T j 2 (3.19a) x yt x x H g T j , = -xp + t4> - H 2 g ' T j 2 (3.19b) y xt y y 46 L a s t l y , upon using (3.6c), (3.7c) and (3.19), the set (3.17)-(3.18) becomes (HV2<//-H \\J -H \p ) + f(H \p ~U \p ) = -g' H, (H TJ -H TJ ) (3.20) x x y y t x y y x y x x y [ HV 2TJ + Hj_(H TJ +H TJ ) - H 2M(TJ) ] + f H i ( H TJ -H TJ ) H 2 x x y y g ' H, H 2 t H 2 x y y x = -1 [MU )H - M(i// )H ] (3.21) g'H 2 x y y x where TJ has been used f o r TJ 2 , and where, s i n c e H, i s constant, H 2 = H and H 2 = H . Equations (3.20) and (3.21) are x x y y i d e n t i c a l , but i n dimensional form, to equations (2.17a) and (2.17b) bf A l l e n (1975). At t h i s p o i n t , one has two o p t i o n s . The f i r s t One i s to so l v e the system (3.20)-(3.21) n u m e r i c a l l y , f o r given H(x,y), while the other i s to int r o d u c e f u r t h e r assumptions to s i m p l i f y the problem i n the hope bf reaching an a n a l y t i c a l s o l u t i o n . We chose the l a t t e r . S t i l l f o l l o w i n g A l l e n (1975), we note that \p c o n t r i b u t e s a depth-independent component to the v e l o c i t i e s (3.15)-(3.16). We w i l l now r e f e r to the part bf the s o l u t i o n given by </  as the b a r o t r o p i c component. The c o n t r i b u t i o n of TJ to the v e l o c i t y components i s depth-dependent and has a zero depth average. T h i s p a r t w i l l be r e f e r r e d to as the b a r b c l i n i c component. If the bottom i s f l a t , the c o u p l i n g terms i n (3.20)-(3.21) v a n i s h and \p and TJ are the f a m i l i a r constant depth b a r o t r o p i c and b a r b c l i n i c modes (see LeBlbnd and Mysak, 1978, s e c t i o n 16). 47 For a v a r i a b l e depth, however, the c o u p l i n g between the b a r d t r o p i c and the b a r o c l i n i c components i s caused by the bottom topography. Now, we introduce an i d e a l i z e d model geometry. With the assumption that the f u l l depth, H(x,y), i s independent of y (the along-channel c o o r d i n a t e ) , equations (3.20)-(3.21) reduce to (HV2ty - H ty ) + f(H ty ) = g' H T (H r? ) (3.22) x x t x y x y [ HV 2rj + HjH TJ - H 2 f 2 XJ ] + fHjH TJ = f 2 H ty (3.23) H 2 x x g'H!H2 t H 2 x y g'H 2 x y where we a l s o assumed that M = 9 + f 2 = -w2 + f 2 « f 2 t t That i s , we r e s t r i c t o u r s e l v e s to low-frequency, s u b i n e r t i a l mot i o n s . Using the d e f i n i t i o n of the r e l a t i v e v d r t i c i t y fdr l a y e r - i , t h a t i s S = v - u i i x i y we f i n d t h a t the d e p t h - i n t e g r a t e d v d r t i c i t y balance i n each l a y e r w i l l be Hi 5, = -in t t (3.24a) 48 H 2 $ 2 = fr? + f u 2 H 2 (3.24b) t t x and the t o t a l water column d e p t h - i n t e g r a t e d v d r t i c i t y balance i s (H,$, + H 2$ 2) = f u 2 H 2 (3.25) t x The I d e a l time r a t e df change df d e p t h - i n t e g r a t e d v d r t i c i t y i s due s o l e l y t d vortex s t r e t c h i n g and compression by mdtidn up and down the bottom s l o p e s . Since the twd l a y e r s have s l i g h t l y d i f f e r e n t d e n s i t i e s , they w i l l r e a c t s l i g h t l y d i f f e r e n t l y . The v d r t i c i t y balance i n the bottom l a y e r (eq. 3.24b) i s between the l o c a l time rate df change df v o r t i c i t y and s t r e t c h i n g by both the mdtidn over v a r i a b l e depth and the i n t e r f a c e mdtidn. The l e n g t h s c a l e dver which the i n t e r f a c e mdtidn c o n t r i b u t e s t d the p o t e n t i a l v d r t i c i t y balance i s the i n t e r n a l Rdssby r a d i u s . The b a r d c l i n i c mdtidn df both l a y e r s i s coupled by the mdtidn df the i n t e r f a c e which produces vortex s t r e t c h i n g i n one l a y e r and compression i n the dther. When the d e p t h - i n t e g r a t e d v d r t i c i t y equatidn i s manipulated i n t o the d e p t h - i n t e g r a t e d mass-transport v d r t i c i t y equation (eq. 3.20 dr 3.22), p a r t df the s t r e t c h i n g term i s absorbed i n the time r a t e df change df the t r a n s p o r t v d r t i c i t y and pa r t i s exchanged f d r the H 17 term (eq. 3.22). The f i r s t term dn the x y l e f t - h a n d s i d e df (3.22) i s the time r a t e df change df mass-t r a n s p o r t v d r t i c i t y . The t h i r d term represent the change df 49 mass-transport v d r t i c i t y due to s t r e t c h i n g df the water column as the b a r d t r d p i c p a r t df the v e l o c i t y f i e l d advects the water up and down the sldpes (a u«VH term). The l a s t term on the right-hand s i d e df (3.22) i s a pressure torque (p=-pgj?) produced by the r a i s i n g of the i n t e r f a c e along a channel with c r o s s -channel sldpes (a VH x VTJ term) . The equation f o r 7? (3.21 dr 3.23) i s d e r i v e d as the d i f f e r e n c e v d r t i c i t y equatidn. I t i n v o l v e s vdrtex s t r e t c h i n g from i n t e r f a c i a l displacement and depth v a r i a t i o n s ( H 2 $ 2 - H,*,) = 2fr? + f u 2 H 2 (3.26) t t x Since the lower l a y e r v e l o c i t y depends on ty (3.12a), the time Y r a t e df change df A = H 2 $ 2 - H,$, - 2fij a l s o depends on ty . So, the depth-independent component df the y v e l d c i t y i s t h e r e f d r e coupled with the time v a r i a t i o n s df A thrdugh the vdrtex s t r e t c h i n g i n the lower l a y e r caused by mdtidn up and down the s l d p e . 3.5 S i m p l i f y i n g Assumptions A c a r r i e r - w a v e wave t r a n s f o r m a t i o n df the fdrm 50 v H x , y , t ) = Real [ F ( x ) e x p ( i k y - ioot) ] T j ( x , y , t ) = Real [ G(x)exp(iky - ia>t) ] i s i n t r o d u c e d which, upon s u b s t i t u t i o n i n t o (3.22)-(3.23), y i e l d s HF - H F - [k 2H + (fk)H ]F = - g'H,kH G (3.27) XX X X 00 X 00 X HG + H_L(H G ) - [k 2H + H 2 f 2 + Hj_(fk)H ]G = -1 f 2 k H F (3.28) xx H 2 x x g'H,H2 H 2 oo x 9'H2 oo x A l l e n ( l 9 7 5 ) , f o l l o w i n g Buchwald and Adams (1968), i n t r o d u c e d an e x p o n e n t i a l depth p r o f i l e . The e x p o n e n t i a l depth p r o f i l e has the marked advantage of having a constant topographic l e n g t h s c a l e . T h i s l e n g t h s c a l e i s e s s e n t i a l l y the d i s t a n c e over which the change i n depth i s of the same magnitude as the (average) depth i t s e l f . In the c o n t i n e n t a l s h e l f r e g i o n , L i s of the same order as the s h e l f width (100km). The * assumption of an e x p o n e n t i a l depth p r o f i l e s i m p l i f i e s the set of equations by r e s u l t i n g i n constant c o e f f i c i e n t s i n (3.27). In the s h e l f r e g i o n , the i n t e r n a l Rossby r a d i u s i s 0(15km). I n t e r n a l K e l v i n waves are trapped w i t h i n an x - s c a l e of O(Ri) near the c o a s t , while b a r o t r o p i c S h e l f waves have an o f f s h o r e s c a l e which, f o r the lowest modes, i s O(L), the s h e l f width. A l l e n (1975) used a p e r t u r b a t i o n expansion i n powers of a 51 small parameter X, with X 2 = R i 2 / L 2 « 1 * For s c a l e s Of motion Of the Order Of the s h e l f width Or l a r g e r and f o r the i n t e r i o r region (away from the boundaries), he Obtained a. n F ( x ) = Z X F = 0 ( 1 ) n=0 n G(x) = 0 ( X 2 ) T h i s means that an 0(1) Sh e l f wave i s coupled to a r a t h e r weak 0( X 2 ) b a r O c l i n i c motion. At the same time, an 0(1) K e l v i n wave w i l l be coupled to an 0(X) b a r o t r o p i c motion. At intermediate wavelengths, the Shelf wave and the K e l v i n wave are cOupled at the lowest Order. FOr short wavelengths, he found t h a t the motion i s bottom-trapped. In a narrow channel where the slope changes s i g n , X = 0(1) and the e x p o n e n t i a l p r o f i l e i s i n a p p r o p r i a t e . Moreover, the f i n d i n g s Of SchOtt and Mysak (1978) suggest that the wavelengths are s m a l l e r than 40km. That i s Wavelength = O(Ri) = 0(L ) = 0(L) * Hence, i n the geometry Of i n t e r e s t , the system of equations i s 52 f u l l y coupled. I w i l l now develop a new approach, based on some assumptions about the nature bf the motion. F i r s t l y , the v a r i a b l e s are nbndimensionalized i n the f o l l o w i n g manner u = of Lk = k' x = Lx' u = Uu ' i i H = Dh H, = D, H 2 = D 2 h 2 v = Uv ' i i G = NG' F = QF' t = t ' / f where a, h, h 2 , and the primed v a r i a b l e s are d i m e n s i o n l e s s , and D = D, + D 2 (see f i g . 3.4). The s c a l e s Q and N w i l l be s p e c i f i e d l a t e r , but are such that G' = O(F') = 0 ( 1 ) . Upon s u b s t i t u t i o n i n t o (3.27)-(3.28) and a f t e r dropping the primes, we get hF - h F - [ k 2 h + kh ]F = -[q'D 1N]kh G (3.29) xx x x a x f Q a x hh 2G + Djh G - [ k 2 h h 2 + L 2 h 2 + D,kh ]G = [ - f Q]kh F (3.30) xx D 2 x x R i 2 D 2 a x g'D 2N a x The four terms on the l e f t - h a n d s i d e Of (3.29) are Of the same order. The f i r s t and the t h i r d come from the l o c a l time change bf mass-transport v o r t i c i t y . The second and f o u r t h represent the vortex s t r e t c h i n g . To achieve f u l l c o u p l i n g , the r i g h t - h a n d s i d e must be Of the same Order, that i s g'D, / f = Q / N 53 M u l t i p l y i n g by D 2 / D 2 , one gets (g'D 2/f)(D,/D 2) = Q / N S i m i l a r l y , f o r the r i g h t - h a n d s i d e bf (3.30) to be 0(1) g'D 2 / f = Q / N must h o l d . Hence, the c o u p l i n g terms are both of 0(1) together only i f D, = D 2. For channels where the top l a y e r i s a l o t t h i n n e r than the bottom l a y e r , the equations w i l l be uncoupled to 0 ( D 1 / D 2 ) . We have two c h o i c e s bf s c a l i n g : g'D 2/f = Q/N and g'D,/f = Q/N With the f i r s t c h o i c e , the c o u p l i n g term i n (3.29) i s 0(D!/D 2) and we have b a r o t r o p i c wave which f o r c e s a b a r b c l i n i c response through the topographic c o u p l i n g . As mentioned b e f o r e , the vortex s t r e t c h i n g i n the lower l a y e r depends on the lower l a y e r v e l o c i t y u 2 which has a b a r o t r o p i c component (eq. 3.26). The other s c a l i n g , as w i l l be seen s h o r t l y , must be r e j e c t e d . We are now able to s p e c i f y the s c a l i n g s N and Q. From (3.10), the t o t a l mass-transport stream f u n c t i o n must be nondimensibnalized by (depth s c a l e ) ( v e l o c i t y s c a l e ) ( l e n g t h s c a l e ) 54 Thus, we choose Q = D U L (3.31) TwO s c a l i n g s are p o s s i b l e f o r N N = Q (f/g'D 2) (3.32a) N = Q (f/g'D,) (3.32b) Whith (3.32a), the set (3.29)-(3.30) becomes, with the h e l p Of (3.31) , hF - h F - [ k 2 h + (k/a)h ]F = 0 + 0(D,/D 2) (3.33) XX X X X G - (k 2 + L 2 / R i 2 ) G = - [ k / ( h h 2 a ) ] h F + 0(D,/D 2) (3.34) xx x Where we assumed that D, i s much smal l e r than D 2. Using the same s c a l i n g , the nondimensionalized v e l o c i t i e s become u, = h " 1 [ k F - h 2(-aG + k G ) ] s i n ( k y - at) (3.35a) x v, = h-'[F + h 2 ( G - kaG)]cbs(ky - at) (3.35b) x x u 2 = h " 1 [ k F + D (-aG + k G ) ] s i n ( k y - at) (3.36a) * x v 2 = h " 1 [ F - D ( G - kaG)]cbs(ky - at) (3.36b) x * x 55 where D = D, / D 2 . Since h = 0 ( 1 ) , the motion w i l l be * s u r f a c e - i n t e n s i f i e d , that i s u, = ( B a r d t r d p i c - p a r t + B a r d c l i n i c - p a r t ) u 2 = ( B a r d t r d p i c - p a r t - ( D , / D 2 ) B a r d c l i n i c - p a r t ) If the s c a l i n g (3.32b) i s used, a s i m i l a r a n a l y s i s leads t d u t = ( B a r d t r d p i c - p a r t + ( D 2 / D , ) B a r d c l i n i c - p a r t ) u 2 = ( B a r d t r o p i c - p a r t - B a r o c l i n i c - p a r t ) The v e l o c i t y i n the f i r s t l a y e r i s now df 0(D 2/D,) >> 1. The process df n d n d i m e n s i o n a l i z a t i d n i n t r o d u c e s 0(1) v a r i a b l e s , by d e f i n i t i o n . Hence, t h i s s c a l i n g must be r e j e c t e d , s i n c e UT^/D, would the a p p r o p r i a t e v e l o c i t y s c a l e , not U. 3.6 Boundary C o n d i t i o n s And I n t e r f a c e Mdtidn I t i s now cdnvenient t d introduce the t r a n s f o r m a t i o n F(x) = h 2 ( x ) q ( x ) (3.37) A f t e r s u b s t i t u t i n g (3.37) i n t d (3.33) and (3.34), and using the f a c t t h a t , s i n c e D, i s much smal l e r than D 2 then h 2 « h, we dbt a i n the f i n a l set df equations which d e s c r i b e s a b a r d t r d p i c 56 wave which f o r c e s a b a r o c l i n i c response through topographic c o u p l i n g hq + 3h q + (2h - k 2 h - h k/a)q = 0 (3.38) XX X X XX X G - ( k 2 + L 2 / R i 2 ) G = -(k/a)h q (3.39) xx x The s o l u t i o n of (3.38) w i l l depend on the cro s s - c h a n n e l depth p r o f i l e h ( x ) . On the other hand, the general form of the s o l u t i o n of (3.39) may be obtained immediately. Let g(x) = -( k / a ) h (x)q(x) and y2= k 2 + ( L 2 / R i 2 ) (3.40) x By the method of v a r i a t i o n of parameter (see Boyce and DiPrima, 1969, f o r d e t a i l s ) , the general s o l u t i o n of (3.39) i s 7X - 7 X G(x) = (C,/2y)-e + ( C 2 / 2 7 ) . e + 7X x ~7t - 7 X x 7t (\/2y)[e g ( t ) e -dt - e •/ g ( t ) e -dt] Once w r i t t e n i n terms of d e f i n i t e i n t e g r a l s , G(x) becomes ~ 7 X x 7t G(x) = ( l / 2 7 ) e •[R - / g ( t ) e -dt] + o 7X x —7t ( l / 2 7 ) e •[S + / g ( t ) e -dt] (3.41) 0 57 where R ans S w i l l be determined from the f o l l o w i n g boundary c o n d i t i o n s at x = ±1. I f there i s a slope d i s c o n t i n u i t y at x = 0 or anywhere e l s e , then the pressure and the normal t r a n s p o r t must be continuous ac r o s s i t . The normal v e l o c i t i e s must v a n i s h on the boundaries. T h i s means that u,(x) = u 2 ( x ) = 0 at x = ±1 From (3.35a) and (3.36a), we need F(x) = 0 G (x) - (k/a)G(x) = 0 x at x = ±1 (3.42) For the l i n e a r p r o f i l e shown i n f i g u r e 3.4d, h changes s i g n at x x = 0. T h i s r e q u i r e s that F + ( 0 ) = F-(0) and F + (0) = F" (0) x x (3.43) GMO) = G"(0) and G + (0) = G" (0) x x where F + ( x ) , x > 0 G +(x) , x > 0 F(x) = and G(x) = F-(x) , x < 0 G"(x) , x < 0 58 The general s o l u t i o n of (3.39) which f u l f i l s the boundary and matching c o n d i t i o n s i s ~ 7X -yX G(x) = ( 1 / 2 7 ) e »[R - I ( x ) ] + 0 / 2 7 ) e -[S + J ( x ) ] (3.44) where x yt x -yt I(x) = / g ( t ) e dt and J(x) = / g ( t ) e dt and where ~2y 2y R = -1 { ( 1 - k / g 7 ) [ J ( l ) - J ( - D ] + [ I ( 1 ) e - I ( - D e ]} 2 s i n h ( 2 7 ) (1+k/a 7 ) 2y -2y s = -1 { ( 1 + k / a 7 ) [ K l ) - K - 1 ) ] + [ J ( D e - J(-1 ) e ]} 2 s i n h ( 2 7 ) ( 1 - k / a 7 ) The problem now c o n s i s t s bf s o l v i n g (3.38), given h ( x ) , su b j e c t to (3.42) (and (3.43), i f needed). As soon as q(x) i s known, G(x) f o l l o w s d i r e c t l y from (3.44). From now on, we w i l l c o n c e n t r a t e On the b a r o t r o p i c p a r t bf the s o l u t i o n . 5 9 4 SOLUTION : TOPOGRAPHIC WAVES Topographic waves are named a f t e r the f a c t t h a t they owe t h e i r e x i s t e n c e t o the s t r e t c h i n g and c o n t r a c t i n g of water columns as they move a c r o s s depth c o n t o u r s i n the p r e s ence of p l a n e t a r y v o r t i c i t y . There i s no wave motion a s s o c i a t e d w i t h the b a r o t r o p i c p a r t bf the m o t i o n , \p , On a f - p l a n e w i t h c o n s t a n t depth and a r i g i d upper s u r f a c e . When the t o p o g r a p h i c waves t r a v e l a l o n g the c o n t i n e n t a l s h e l f , they a r e c a l l e d C o n t i n e n t a l S h e l f waves. When they t r a v e l a l o n g a t r e n c h , they a r e c a l l e d T r e n c h waves. The t o p o g r a p h i c waves t r a v e l l i n g a l o n g a c h a n n e l w i l l be c a l l e d Channel waves. Those t r a v e l l i n g around an e n c l o s e d b a s i n w i l l be c a l l e d B a s i n waves. 4.1 Channel Waves In the i n f i n i t e l y l o n g and s t r a i g h t c h a n n e l problem, we w i l l l o o k a t the f o u r p r o f i l e s i n t r o d u c e d i n s e c t i o n 3.2. We w i l l c o n s i d e r l i n e a r and p a r a b o l i c bottom p r o f i l e s , both f o r s m a l l and l a r g e bottom s l o p e s . 4.1.1 L i n e a r Topography R e f e r r i n g t o f i g u r e 11d, t h e m a t h e m a t i c a l r e p r e s e n t a t i o n bf the l i n e a r bottom p r o f i l e i s h ( x ) = 1 - s | x | , s = aL/D (3.3a) 60 More p r e c i s e l y , h(x) = 1 - s,x , s, > 0 and x > 0 h(x) = 1 + s 2 x , s 2 > 0 and x < 0 (4.1) The s are the ndndimensidnal s l o p e s , i 4.1.1.1 Small Slope Approximation When s i s much smaller than one, the depth changes may be n e g l e c t e d , compared t d the f u l l depth, and h(x) * 1 (4.2) Upon s u b s t i t u t i o n of (4.1) and using (4.2), equation (3.33) becomes F + + s,F + + ( s,k/a - k 2 ) F + = 0 , x > 0 (4.3a) XX X P- - s 2 F " •+ (-s 2k/a - k 2 ) F " = 0 , x < 0 (4.3b) XX X Td s i m p l i f y the n d t a t i d n , l e t us d e f i n e X 2 = s , k / a - k 2 - s 1 2 / 4 and 5 2 = s 2 k / a + k 2 + s 2 2 / 4 The s o l u t i o n of (4.3) that meets the bdundary c d n d i t i d n s (3.42) and the matching c d n d i t i d n s (3.43) i s F + (x) = -s , x -E«sinh(5)«e T • s i n [ X ( x - l ) ] (4.4a) 61 +s 2x F"(x) = E-sin(X) -e T • sinh[6(x+1)] (4.4b) p r o v i d e d that the d i s p e r s i o n r e l a t i o n (s, + s 2 ) / 2 + XcOtan(X) + 5cotanh(5) = 0 (4.5) i s f u l f i l l e d . As expected, f o r waves t r a v e l l i n g i n the p o s i t i v e y - d i r e c t i b n , F(x) i s O s c i l l a t o r y f o r x p o s i t i v e and e x p o n e n t i a l f o r x n e g a t i v e . T h i s behaviour i s reversed f o r waves t r a v e l l i n g i n the negative y - d i r e c t i b n . The d i s p e r s i o n curves f o r the f i r s t three modes, with s, = s 2 = s = 0.26, are shown i n f i g u r e 12. The Channel waves d i s p e r s i o n curves are s i m i l a r to those bf Shelf waves and Trench waves. F i g u r e 13 shows the cr o s s - c h a n n e l dependence bf the mass-transport stream-function ( f i g . 13a) and the i n t e r f a c e motion ( f i g . 13b), c a l c u l a t e d from (3.44) f o r y = 0 and t = 0, corresp o n d i n g to the modes bf f i g u r e 12. The nbndimensibnal wavenumber 1.6 (wavelength bf twice the channel width) was chosen f o r l a t e r comparison with the numerical model bf L i e and El-Sabh (1983). F i g u r e s 14 and 15 present the E u l e r i a n flow p a t t e r n i n both l a y e r s , f o r the f i r s t two modes. The su r f a c e i n t e n s i f i c a t i o n bf the c u r r e n t s may be observed, as w e l l as t h e i r d i r e c t i o n bf r o t a t i o n . Again, the o s c i l l a t i n g / decaying nature of the motion on the r i g h t / l e f t hand s i d e bf the channel i s e v i d e n t . The r o t a r y c u r r e n t p a t t e r n does not change d i r e c t i o n with depth. 62 Figure 12. Dispersion curves fdr the f i r s t three modes in the small l i n e a r tdpdgraphy case of an i n f i n i t e channel. A l l variables are dimensionless. 63 CHANNEL MODES : SMALL L I N E A R SLOPE BAROTROPIC E IGENFUNCTIONS SLOPE WAVENUMBER FREQUENCY 1. 6 I 1 ( 2 ( 3 4 6 2 E - 0 1 1 2 4 E - 0 1 5 3 5 E - 0 2 Figure 13a. Cross-channel dependence of the mass-transport stream f u n c t i o n , corresponding to the modes df f i g u r e 12. A l l v a r i a b l e s are dimensionless. 64 CHANNEL MODES : SMALL L I N E A R SLOPE B A R O C L I N I C E IGENFUNCTIONS o r -co. CO _| Q ZD o o. tt. 00 CE CD CD O I to 1 o i o o SLOPE WAVENUMBER FREQUENCY : ( 1 ) . 4 6 2 E - 0 1 1 2 4 E - 0 1 5 3 5 E - 0 2 Figure 13b. Cross-channel dependence bf the interface motion, corresponding to the modes bf figure 12. A l l variables are dimensionless. 65 E U L E R I A N F L O W P A T T E R N k=1.6 SIGMA=. 0462 s=0. 26 Hl/H2=0. 08 FIRST MODE «— -s 1 \ ' -s , / . J SURFACE LAYER 1 i > * ; j j » * i i BOTTOM LAYER Figure 14. Eulerian flow pattern in both layers, corres-ponding to the f i r s t mode of figure 13. A l l variables are dimensionless. 66 E U L E R I A N F L O W P A T T E R N k=1.6 SIGMA=. 0124 8=0. 26 Hl/H2=0. 08 SECOND MODE SURFACE LAYER 4 i -V BOTTOM LAYER Figure 15. Same as in figure 14, but for the second mode, 67 4.1.1.2 Large Bottom Slopes When s = 0( 1 ) , the depth, h ( x ) , may not be co n s i d e r e d c o n s t a n t . Upon s u b s t i t u t i o n Of (4.1) i n t o (3.38), we Obtain (1 - s , x ) q + - 3s,q + + [ s,k/o - k 2(1 - s , x ) q + = 0 (4.6a) XX X (1 + s 2 x ) q - + 3s 2q- + [-s 2k/a - k 2 ( l + s 2 x ) q " = 0 (4.6b) XX X Equation (4.6) may be transformed i n t o a Confluent Hypergebmetric equation with the h e l p Of the f o l l o w i n g change Of v a r i a b l e " I 1 f o r x > 0 : q + ( x ) = e -g +(x) , z, = 2 k ( l - s , x ) / s 1 (4.7a) + Z 2 f o r x < 0 : q-(x) = e T -g"(x) , z 2 =-2k(1+s 2x)/s 2 (4.7b) When s, = s 2 = s , then z, = - z 2 = z , and (4.6) becomes z g + + (3 - z ) g + - -(3 - l / o ) g + = 0 , x > 0 (4.8a) zz z 2 zg- + (3 + z)g- - -(3 - 1/a)g" = 0 , x < 0 (4.8b) zz z 2 The general s o l u t i o n Of the Confluent HypergeOmetric equation (4.8a) i s (Abrambwitz and Stegun, 1964, page 504) 68 g + ( z ) = A-M[a;b;z] + B-U[a;b;z] (4.9) where M and U are l i n e a r l y independent Rummer f u n c t i o n s and where a = 3/2 - 1/2a and b = 3 (4.10) Since b i s a p o s i t i v e i n t e g e r , the second s o l u t i o n , U, w i l l be l o g a r i t h m i c . Moreover, t h i s s o l u t i o n w i l l be a n a l y t i c everywhere s i n c e z i s always g r e a t e r than zero : z represent the depth times a c o n s t a n t . The o s c i l l a t o r y and e x p o n e n t i a l c h a r a c t e r bf the M[a;b;z] f u n c t i o n s depends on the si g n bf the f i r s t parameter, a. The f u n c t i o n M[a;b;z] has z e r o - c r o s s i n g s only i f a i s ne g a t i v e . T h i s leads to a c u t - o f f frequency bf tfo = 1 / 3 , s i n c e a = 3/2 - l/2a must be ne g a t i v e . That o0 = 1/3 i s a c u t - o f f frequency may be seen with the h e l p bf the f o l l o w i n g argument. Let us assume that the channel i s homogeneous (no s t r a t i f i c a t i o n ) and l e t the depth be zero On the boundaries. The boundary c o n d i t i o n s are a u t o m a t i c a l l y s a t i s f i e d s i n c e , from (3.37) F(x) = h 2 ( x ) q ( x ) Under these c o n d i t i o n s , the Kummer f u n c t i o n M[a;b;z] reduces to a s s o c i a t e d Laguerre polynomials (Abrambwitz and Stegun, 1964, 69 page 509) g + ( z ) = A•(n!/3)•£ 2(z) and B = 0 n pr o v i d e d that a = 3/2 - l/2a = -n. The d i s p e r s i o n r e l a t i o n becomes a = 1/(2n+3) and the f i r s t mode (n=0) has a frequency of 1/3. The f a c t t h a t M[a;b;z] i s r e q u i r e d t d have z e r d - c r d s s i n g s may be proved e a s i l y . Even i f M and U had nd z e r d - c r d s s i n g s , A and B might be fdund ( i n 4.9) such that g(z) i s zero at the boundaries. In that case, s i n c e M and U are not o s c i l l a t o r y , they dd not c a n c e l anywhere e l s e . Hence no higher mddes. The s i t u a t i o n i s s i m i l a r t d th a t df the p r e v i o u s case df smal l s. C d e f f i c i e n t s were fdund such that the cosh and s i n h f u n c t i o n s c a n c e l at x = -1. But i t i s the o s c i l l a t o r y nature df the s i n and cos f u n c t i o n s that allowed f d r higher mddes. When x i s negat i v e , the s o l u t i o n df (4.6b) i s g-(z) = C-M[a;3;-z] + D-U[a-3;-z] dr with the h e l p df the Kummer t r a n s f d r m a t i d n -z g-(z) = e T-{OM[3-a;3;z] + D»U[ 3-a; 3; z ]} (4.11) 70 When a i s l a r g e (very low frequency) and negat i v e , (3-a) i s la r g e and p o s i t i v e . When (3-a) and z are both l a r g e and p o s i t i v e , there e x i s t no known asymptotic behaviour f o r M and U (Abrambwitz and Stegun, 1974, page 512), and the s e r i e s r e p r e s e n t a t i o n converges slowly when i t i s not n u m e r i c a l l y u n s t a b l e . Luke (1969a, b; 1975) s t u d i e d the e v a l u a t i o n Of s p e c i a l f u n c t i o n s . Luke (1977) g i v e s not Only the alg o r i t h m s (and programs) to compute the r e p r e s e n t a t i o n Of most s p e c i a l f u n c t i o n s i n terms Of Chebyshev polynomials, but he a l s o g i v e s the a l g o r i t h m to compute -rx -u-1 x r t u e 'X . J e -t«F(t)'dt o Once the Chebyshev r e p r e s e n t a t i o n Of F(x) i s known. U n f o r t u n a t e l y , t h i s a l g o r i t h m i s n u m e r i c a l l y unstable f o r 80% Of the parameters found i n t h i s t h e s i s . Topographic wave s o l u t i o n s are c o n s i d e r e d f o r four topographic p r o f i l e s , each at 10 d i f f e r e n t values Of s. Moreover, the a l g o r i t h m to compute M and U i s a l s o unstable f o r some l a r g e values Of (3-a) and z. I t was mentioned that the s o l u t i o n s are always a n a l y t i c i n s i d e the channel boundaries. Thus, i n s t e a d of expanding the s o l u t i o n of (4.8) i n Frobenius s e r i e s (the Rummer f u n c t i o n s ) around the s i n g u l a r p o i n t z = 0, we sol v e d (4.6) d i r e c t l y by expanding the s o l u t i o n i n T a y l o r s e r i e s around the r e g u l a r p o i n t x = 0. Since the ra d i u s Of convergence of such s e r i e s i s One, and s i n c e (1 - sx) i s always sma l l e r than One, the s o l u t i o n 71 converges although the convergence becomes slower as s gets c l o s e r t d one. I t was fdund t h a t , by t a k i n g 300 terms in the T a y l o r s e r i e s , the number df s i g n i f i c a n t d i g i t s was always gr e a t e r dr equal td 8. The convergence may be a c c e l e r a t e d by c o n v e r t i n g the T a y l o r s e r i e s i n t o Chebyshev s e r i e s . The number df s i g n i f i c a n t d i g i t s was set a t 8 because i t was fdund that the frequency must be computed with a p r e c i s i o n df 7 s i g n i f i c a n t d i g i t s . The f r e q u e n c i e s are dbtained by n u m e r i c a l l y computing the zeros df the d i s p e r s i d n r e l a t i d n , once a wavenumber has been chosen. The subroutine ZREAL1 df the IMSL L i b r a r y ( K r a i n e r et a l . , 1975) was used. The d i s p e r s i d n r e l a t i d n i s the determinant df the l i n e a r system r e s u l t i n g from e i t h e r the matching c d n d i t i d n s dr the boundary c o n d i t i o n s . The determinant must be zero and i t i s computed as the d i f f e r e n c e between two l a r g e numbers. In c e r t a i n i n s t a n c e s (very low f r e q u e n c i e s ) i t was found that the e r r o r c o u l d be as l a r g e as 0.1 i f the frequency i s computed with l e s s than 7 s i g n i f i c a n t d i g i t s . Once the s o l u t i o n has been fdund f d r the b a r d t r d p i c p a r t df the mdtidn, equation (3.44) i s e v a l u a t e d n u m e r i c a l l y with the h e l p of the subroutine DCADRE df the IMSL L i b r a r y . Equation (4.6a) and (4.6b) are s i m i l a r . The s i m i l a r i t y may accentuated with the f o l l o w i n g change df v a r i a b l e z = sx where s = s, i f x > 0 (4.12) 72 S u b s t i t u t i o n bf (4.12) i n t o (4.6a) and (4.6b) y i e l d s ( l - z ) q - d q + ( a 2 z + b ) q = 0 (4.13a) zz z where d = 3 , a 2 = k 2 / s 2 and b = a/a - a 2 (4.13b) The T a y l o r s e r i e s s o l u t i o n bf (4.13) i s 0 0 n n q(x) = A- Z C s x + n=0 n q(x) = A • LLC[a;b;d;s;x] . + where LLC[a;b;d;s;x] and LLD[a;b;d;s;x] N n n = Z C s x n=0 n N n n = Z D s x n=0 n w n n B • Z D s x n = 0 n B • LLD[a;b;d;s;x] , N = 300 , N = 300. (4.14a) (4.14b) The two c h a r a c t e r s LL stand f o r Large L i n e a r topography. S i m i l a r l y , SP and LP w i l l stand f o r Small P a r a b o l i c and Large P a r a b o l i c topography, r e s p e c t i v e l y . The C and the D are given by C 0 C, C 2 = 1 = 0 = -b/2 Do D 2 = 0 = 1 = d/2 73 C = {[(n-1)(n-2)+d(n-1)]C n n-1 D = {[(n-1)(n-2)+d(n-1)]D n n-1 bC - a 2C }/n(n-1) n-2 n-3 bD - a 2D }/n(n-1) n-2 n-3 The s o l u t i o n of (4.6) i s q + ( x ) = E.B 0{A 0LLC[a,;b,;3; s,;x]-A,LLD[a,;b,;3; s,;x]} (4.15a) q-(x) = E«A 0{BoLLC[a 2.;b 2?3;-S2;x]-B 1LLD[a2;b 2;3;-s 2;x]} (4.15b) where the a and the b are obtained by s u b s t i t u t i n g s, and - s 2 i i f o r s i n (4.13b). The A and the B are d e r i v e d from the i i boundary c o n d i t i o n s (3.42) and the matching c o n d i t i o n s (3.43). They are A 0 = LLD[a 1;b!;3; s,; 1] A, = LLC[a,;b,;3; s,; 1] B 0 = L L D [ a 2 ; b 2 ; 3 ; - s 2 ; - 1 ] B, = L L C [ a 2 ; b 2 ; 3 ; - s 2 ; - 1 ] p r o v i d e d that the d i s p e r s i o n r e l a t i o n S 2 A 0 ( B , + 2B 0) + s 1 B 0 ( A 1 + 2A 0) = 0 (4.16) i s s a t i s f i e d . The d i s p e r s i o n curves f o r the f i r s t three mddes with s, = s 2 = s = 0.26 are shown in f i g u r e 16. The d i s p e r s i d n 74 curves are s i m i l a r to those with the small slope approximation but, at each wavenumber, the frequency i s h i g h e r . The d i f f e r e n c e s , f o r k = 1.5, vary from 5% f o r s = 0.1 to 40% f o r s = 0.8. The small slope approximation seems to underestimate the f r e q u e n c i e s , even when s i s s m a l l . F i g u r e 17 present the cr o s s - c h a n n e l dependence Of the mass-t r a n s p o r t stream f u n c t i o n ( f i g . 17a) and of the i n t e r f a c e motion ( f i g . 17b) f o r t = 0 and y = 0, corresponding to the modes Of f i g u r e 16. The nondimensiOnal wavenumber i s 1.6. F i g u r e 18 and 19 present the E u l e r i a n flow p a t t e r n i n both l a y e r s , f o r the f i r s t two modes. The s u r f a c e i n t e n s i f i c a t i o n Of the c u r r e n t s may be observed, as w e l l as t h e i r d i r e c t i o n Of r o t a t i o n . Again, the O s c i l l a t o r y / decaying nature Of the motion i s e v i d e n t . The c u r r e n t e l l i p s e s do not change d i r e c t i o n with depth. FOr l a r g e r slopes (Chapter 5), the behaviour i s very s i m i l a r . I t i s mostly the f r e q u e n c i e s that are a f f e c t e d : steeper s l o p e s w i l l allow f a s t e r waves, at the same wavenumber, s i n c e the r e s t o r i n g mechanism i s s t r o n g e r . 75 g D I S P E R S I O N RELAT ION o l WAVENUMBER Figure 16. Dispersion curves for the f i r s t three modes in the large linear topography case of an i n f i n i t e channel. A l l variables are dimensionless. CHANNEL MODES LARGE L I N E A R SLOPE BAROTROPIC E IGENFUNCTIONS 76 SLOPE WAVENUMBER FREQUENCY 6 6 ( 1 ( 2 ( 3 5 1 8 E - 0 1 1 4 3 E - 0 1 6 1 6 E - 0 2 Figure 17a. Cross-channel dependence df the mass-transport stream function, corresponding td the mddes df figure 16. A l l variables are dimensionless. CHANNEL MODES : LARGE L I N E A R SLOPE B A R O C L I N I C E IGENFUNCTIONS o o. to. a o CO co_| Q ZD •—i _ J o tt. 00 CX co co o to • . o i o o -0. 60 SLOPE WAVENUMBER FREQUENCY 5 1 8 E - 0 1 1 4 3 E - 0 1 6 1 6 E - 0 2 F i g u r e 17b. Cross-channel dependence of the i n t e r f a c e motion, corresponding to the modes Of f i g u r e 16. A l l v a r i a b l e s are d i m e n s i o n l e s s . 78 E U L E R I A N F L O W P A T T E R N k=1.6 SIGMA=. 0518 s =0. 26 Hl/H2=0. 08 FIRST MODE 1 1 ' ' J 7 J ' ' 1 1 SURFACE LAYER BOTTOM LAYER Figure 18. Eulerian flow pattern in both layers, corres-ponding to the f i r s t mode of figure 17. A l l variables are dimensionless. 79 E U L E R I A N F L O W P A T T E R N k=1.6 SIGMA=. 0143 s=0. 26 Hl/H2=0. 08 SECOND MODE SURFACE LAYER BOTTOM LAYER Figure 19. Same as in figure 18, but for the second mode, 80 4.1.2 P a r a b o l i c Topography We now look at Channel waves over a p a r a b o l i c depth p r o f i l e . R e f e r r i n g to f i g u r e l i e , the topographic p r o f i l e i s chosen as The general approach w i l l be the same as i n the l i n e a r topography case : small slope approximation and then 0(1) s l o p e s . The main d i f f e r e n c e i s that the depth i s nOw a f u n c t i o n Of x 2 and (4.17) now a p p l i e s to both s i d e s Of the channel, i . e . fo r p o s i t i v e and negative x. As i n the l a r g e l i n e a r topography problem, we w i l l formulate the s o l u t i o n s i n terms Of known higher t r a n s c e n d e n t a l f u n c t i o n s to Obtain an i n s i g h t i n t o the behaviour Of the waves. However, the s o l u t i o n s w i l l u l t i m a t e l y be c a l c u l a t e d i n terms Of T a y l o r s e r i e s around x equal zero 4.1.2.1 Small Slope Approximation When s i s much smal l e r than One, the depth changes may be n e g l e c t e d , compared to the f u l l depth h(x) = 1 - s x 2 -1 < x £ 1 (4.17) h(x) * 1 (4.18a) h (x) = -2sx (4.18b) x Upon s u b s t i t u t i o n Of (4.18), equation (3.33) becomes 81 F + 2sxF + (2ksx/a - k 2 ) F = 0 (4.19) XX X Equation (4.19) may be transformed i n t o a P a r a b o l i c C y l i n d e r equation with the help of the f o l l o w i n g change df v a r i a b l e f (z) F(x) = e «g(z) where z = (2s)*(x - k/sa) and f ( z ) = - s x 2 = - z 2 - z(k 2/2sa 2)' 2" - k 2 / 2 s a 2 T h i s change of v a r i a b l e r e s u l t i n g - [ z 2 - ( k 2 / o 2 - k 2 - s ) / 2 s ] g = 0 (4.20) zz The g e n e r a l s o l u t i o n df the P a r a b o l i c C y l i n d e r equation i s (Abramowitz and Stegun, 1964, page 686) _ ~ 2 2 2 1 1 1 1 Z 1 3 3 1 g(z) = A-e T •M[-a+-;-;-z 2] + B^z-e T -M[-a+-;-;-z 2] (4.21) 2 4 2 2 2 2 2 2 where M i s the Kummer f u n c t i o n encountered i n the pre v i o u s s e c t i o n , and where 82 a = - ( k 2 / a 2 - k 2 - s)/2s (4.22) Equation (4.20) i s c h a r a c t e r i s t i c Of a two-turning p o i n t problem. The behaviour bf the s o l u t i o n , g, i s O s c i l l a t o r y between the t u r n i n g p o i n t s and Of e x p o n e n t i a l c h a r a c t e r Outside. The t u r n i n g p o i n t s are found at z = ±[2(k 2/cr 2 - k 2 - sj/s ] " 2 " 1,2 I f (4.20) i s w r i t t e n i n terms Of x i n s t e a d Of z, we Obtain g - s 2 [ x 2 - 2kx/sa + ( k 2 / s 2 + 1/s)]g = 0 XX The zeros Of the bracketed e x p r e s s i o n are x = k{l ± [1 - a 2(1 + s / k 2 ) ] * } / so 2 J Since k = 0 ( 1 ) and a2 i s much smaller than One, the f i r s t r o o t , x,, i s near the O r i g i n and the second r o o t , x 2 , i s f a r Outside the channel boundary (x = 1). Thus the cros s - c h a n n e l dependence Of the b a r o t r o p i c channel wave w i l l be O s c i l l a t o r y f o r X, < X ^ 1 « x 2 and Of e x p o n e n t i a l c h a r a c t e r f o r 83 -1 < x < x In order to make further progress, we must evaluate M[-a+-;-;-z2 ] = M[c;b;r] 2 4 4 2 Since c i s large and negative, and r i s large and positive we may evaluate the function M through asymptotic approximations. The asymptotic form w i l l depend on the comparative behaviour df r and p, where p i s defined as p = 2b - 4c = ( k 2 / a 2 - k 2 - s)/s = 0(k 2/sa 2) The order df magnitude df r i s r = z 2/2 = s(x - k/sa) 2 = 0(k 2/sa 2) Since, r and p are both df the same order and large, the asymptotic behaviour of M w i l l be (Abramdwitz and Stegun, 1964, page 508) = e T-(p/2) r 2/3-b M [ c ; b; r ] •r(b)«{ Ai ( T) •cds(c7r) + Bi (r)-sin(c7r) + 0[(p/4)"^] } (4.23) where Ai and Bi are the Airy functions, and where T i s given by r = (p/2)^(r/p - 1) 84 U n f o r t u n a t e l y , t h i s approach does not y i e l d enough s i g n i f i c a n t d i g i t s . The T a y l o r s e r i e s computations (300 terms) y i e l d 7 s i g n i f i c a n t d i g i t s . A quick e s t i m a t i o n of the e r r o r inherent to the asymptotic expansion (4.23) y i e l d s (p/4)"* = 0 [ ( 2 s a 2 / k 2 ) ^ ] = 0.9 x 10' 2 when a = 0.05, s = 0.25 and k = 2.4. The e r r o r introduced i n the d i s p e r s i o n curves i s not v i s u a l l y apparent. But the c r o s s -channel s t r u c t u r e Of the motion i s i n f l u e n c e d at higher modes. Spurious z e r o - c r o s s i n g s appear on the s i d e of the channel where the motion should be decaying. To e v a l u a t e the s o l u t i o n s of (4.19) i n term of a T a y l o r s e r i e s , we f i r s t w r i t e i t as F + dF + (ax + b)F = 0 (4.25) XX X where d = 2s , a = 2sk/a , b = k 2 The s o l u t i o n bf (4.25) i s oo n °° n F(x) = A • I C x + B • I D x (4.26a) n=0 n n=0 n F(x) = A • SPC[a;b;d;x] + B • SPD[a;b;d;x] (4.26b) The C and the D are given by i i 85 C 0 = 1 D 0 = 0 C, = 0 D, = 1 C 2 = b/2 D 2 = 0 C = [(b - (n-2)d)C - aC ] / n ( n - l ) n n-2 n-3 D = [(b - (n-2)d)D - aD ] / n ( n - l ) n n-2 n-3 The s o l u t i o n of (4.19) that s a t i s f i e s the boundary c o n d i t i o n s (3.42) w i l l be F(x) = E-{ A-SPC[a;b;d;x] - B.SPD[a;b;d;;x] } (4.27) p r o v i d e d that the d i s p e r s i o n r e l a t i o n B-SPD[a;b;d--1] - A«SPD[a;b;d;-1] = 0 (4.28) i s s a t i s f i e d . The c o e f f i c i e n t s A and B are given by A = SPD[a;b;d;1] and B = SPC[a;b;d;1] The d i s p e r s i o n curves f o r the f i r s t three modes, with s = 0.26, are shown i n f i g u r e 20. The d i s p e r s i o n curves are s i m i l a r i n shape to those obtained i n the l i n e a r case. At each wavenumber however, the f r e q u e n c i e s are higher than those f d r small l i n e a r p r o f i l e s , but ldwer than those fdr l a r g e l i n e a r 86 p r o f i l e s . F i g u r e 21 presents the c r o s s - c h a n n e l dependence Of the mass-transport stream f u n c t i o n ( f i g . 21a) and Of the i n t e r f a c e displacement (fig.21b) f o r t = 0 and y = 0, corresponding to the modes Of f i g u r e 20. The nOndimensiOnal wavenumber i s 1.6 . The l o c a t i o n bf the maxima (and hence the zeros) are s h i f t e d towards the x = 1 boundary, compared t o both l i n e a r cases. Moreover, the amplitudes Of the maxima nearest to x = 1, f o r the second and t h i r d modes, are roughly 20% lower. F i g u r e s 22 and 23 show the E u l e r i a n flow p a t t e r n i n both l a y e r s f o r the two f i r s t modes. The su r f a c e i n t e n s i f i c a t i o n Of the c u r r e n t s may again be observed, as w e l l as t h e i r d i r e c t i o n Of r o t a t i o n . 87 D I S P E R S I O N RELRTION WAVENUMBER Figure 20. Dispersion curves for the f i r s t three modes in the small parabolic topography case bf an i n f i -nite channel. A l l variables are dimensionless. 88 C H A N N E L MODES : S M A L L P A R A B O L I C S L O P E B A R O T R O P I C E I G E N F U N C T I O N S S L O P E : . 2 6 WAVENUMBER : 1.6 F R E Q U E N C Y : [ 1 ) . 4 1 3 E - 0 1 { 2 ) . 1 0 9 E - 0 1 ( 3 ) . 4 7 1 E - 0 2 Figure 2 1 a . Cross-channel dependence of the mass-transport stream function, corresponding to the modes Of figure 20. A l l variables are dimensionless. 89 C H R N N E L MODES : S M A L L P A R A B O L I C S L O P E B A R O C L I N I C E IG-ENFUNCT IONS . oo S L O P E WAVENUMBER F R E Q U E N C Y 4 1 3 E - 0 1 1 0 9 E - 0 1 4 7 1 E - 0 2 Figure 21b. Cross-channel dependence of the interface mo-tion, corresponding to the modes of figure 20 A l l variables are dimensionless. 90 E U L E R I A N F L O W P A T T E R N k=1.6 SIGMA=. 0413 s =0. 26 Hl/H2=0. 08 FIRST MODE > • •* > • 1 i — "S 1 S 1 / \ 1 / \ J \ 1 I \ \ \ N y 1 7 \ X - 7 > - „ , * — —» N \ ^ ^ \ 1 . , i l SURFACE LAYER i J V « , , , - -v V i > > - , , , . / J BOTTOM LAYER Figure 22. Eulerian flow pattern in both layers, corres-ponding to the f i r s t mode bf figure 21. A l l variables are dimensionless. 91 E U L E R I A N F L O W P A T T E R N k = l . B SIGMA=. 0109 s=0. 26 Hl/H2=0. 08 SECOND MODE 1 y — — 1 J " - -\ j 1 N • / J s ^ ^ ^ , 1 / ^ - N , 1 SURFACE LAYER BOTTOM LAYER Figure 23. Same as in figure 22, but for the second mode, 92 4.1.2.2 Large Bottom Slopes When s = 0 ( 1 ) , the depth may not be co n s i d e r e d constant, Upon s u b s t i t u t i o n of (4.17), equation (3.38) becomes (1 - s x 2 ) q - 6sxq + [2skx/a - k 2 ( l - s x 2 ) - 4s]q = 0 (4.29) xx x Equation (4.29) has almost the form of the g e n e r a l i z e d S p h e r o i d a l Wave equation s t u d i e d by Wilson(1928). If we l e t z = s^x, then (4.29) becomes (1 - z 2 ) q - 6zq + ( a 2 z 2 + 2az/a - 4 - a 2 ) q = 0 (4.30) zz z where a 2 = k 2 / s The f u r t h e r t r a n s f o r m a t i o n -aw w = z - 1 and q(z) = e g(w) b r i n g s (4.30) i n t o the standard form s t u d i e d by Wilson (1928) w(w+2)g + 2[3+(3-2a)w-aw 2]g + [4-2a(w+1)(3+1/a)]g = 0 (4.31) WW w The power s e r i e s s o l u t i o n of (4.31) i s s i n g u l a r at w = -2, unle s s i t terminates. If we were i n t e r e s t e d i n the b a r o t r o p i c topographic modes of an homogeneous (no s t r a t i f i c a t i o n ) channel 93 with zero depth On the boundaries, then we would choose 3 + 1/a = -m Or a = -l/(m+3), m = 1, 2, 3,... (4.32) T h i s c h o i c e Of a wOuld le a d to f i n i t e polynomial s o l u t i o n s and to a c u t - o f f frequency Of -1/4 ( f o r m = 1 ) . The c u t - o f f frequency i s thus lower i n the l a r g e p a r a b o l i c case (1/4) than in the l a r g e l i n e a r case (1/3). TO e v a l u a t e the s o l u t i o n Of (4.30) in term Of T a y l o r s e r i e s , we f i r s t w r i t e i t as (1 - z 2 ) q - dzq + ( a 2 z 2 + bz + c ) q = 0 (4.33) zz z where d = 6 , a 2 = k 2 / s , b = 2a/a , c = -4 - a 2 The s o l u t i o n Of (4.33) i s » n n oo n n q(x) = A • L C s T x + B • S D s T x (4.34a) n=0 n n=0 n q(x) = A • LPC[a;b;c;d;s;x] + B • LPD[a;b;c;d;s;x] (4.34b) The c o e f f i c i e n t s C and D are given by i i 94 C 0 = 1 D 0 = 0 C, = 0 D, = 1 C 2 = -c/2 D 2 = 0 C 3 = -b/6 D 3 = (d-c)/6 C = {[(n-2)(n-3)+(n-2)b-c]C - bC - a 2C } / n(n-1) n n-2 n-3 n-4 D = {[(n-2)(n-3)+(n-2)b-c]D - bD - a 2D } / n ( n - l ) n n-2 n-3 n-4 The f u n c t i o n q that s a t i s f i e d (4.29) and the boundary c o n d i t i o n s (3.42) i s q(x) = E•{A•LPC[a;b;c;d;s;x] - B«LPD[a;b;c;d;s;x]} (4.35) p r o v i d e d that the d i s p e r s i o n r e l a t i o n B-LPD[a;b;c;d;s;-1] - A-LPC[a;b;c;d;s;-1] = 0 (4.36) i s s a t i s f i e d . The c o e f f i c i e n t s A and B are given by A = LPD[a;b;c;d;s;1] and B = LPC[a;b;c;d;s;1] The d i s p e r s i o n curves f d r the f i r s t three mddes, with s = 0.26, are shdwn i n f i g u r e 24. The d i s p e r s i d n curves are s i m i l a r i n shape t d a l l the p r e v i o u s c a s e s . At each wavenumber, the frequency i s higher than than those obtained from the small 95 slope approximation. The d i f f e r e n c e s vary from 5% f o r s = 0.1 td 30% f d r s = 0.8. Moreover, at each wavenumber, the f r e q u e n c i e s are lower than thdse df the l a r g e l i n e a r slope mddel. T h i s l a s t r e s u l t i s i n agreement with the f a c t that the c u t - o f f frequency i s lower f d r l a r g e p a r a b d l i c p r o f i l e (1/4) than f d r l a r g e l i n e a r p r o f i l e s (1/3). F i g u r e 25 p r e s e n t s the c r o s s - c h a n n e l dependence df the mass-transport stream f u n c t i o n ( f i g . 25a) and df the i n t e r f a c e mdtidn ( f i g . 25b) f o r t = 0 and y = 0, cdrrespdnding t d the mddes df f i g u r e 24. The ndndimensidnal wavenumber i s 1.6 . As i n the small p a r a b o l i c case, the l o c a t i o n df the maxima (and zeros) are s h i f t e d tdwards the x = 1 boundary. The amplitude df the maxima nearest the boundary are a l s o about 20% sma l l e r than those obtained f d r the l i n e a r bdttdm p r o f i l e s F i g u r e s 26 and 27 show the E u l e r i a n flow p a t t e r n i n both l a y e r s , f d r the two f i r s t mddes. The s u r f a c e i n t e n s i f i c a t i o n df the c u r r e n t s may be dbserved, as w e l l as t h e i r d i r e c t i o n df r o t a t i o n . Again, the o s c i l l a t o r y - d e c a y i n g nature df the c u r r e n t s i s e v i d e n t . The c u r r e n t e l l i p s e s dd not change d i r e c t i o n with depth. Fdr l a r g e r sldpes (Chapter 5), the behavidur i s very s i m i l a r . As i n the l a r g e l i n e a r case, i t i s mostly the f r e q u e n c i e s that are a f f e c t e d : steeper sldpes w i l l allow f a s t e r waves, at the same wavenumber. 96 Figure 24. Dispersion curves for the f i r s t three modes in the large parabolic topography case bf an i n f i -nite channel. A l l variables are dimensionless. 97 C H A N N E L M O D E S : L A R G E P A R A B O L I C S L O P E B A R O T R O P I C E I G E N F U N C T I O N S W A V E N U M B E R : 1 . 6 F R E Q U E N C Y : ( 1 ) ' . 4 5 3 E - 0 1 { 2 ) . 1 2 2 E - 0 1 ( 3 ) . 5 3 1 E - 0 2 Figure 25a. Cross-channel dependence of the mass-transport stream function, corresponding td the modes df figure 24. A l l variables are dimensionless. C H A N N E L MODES : LARG-E P A R A B O L I C S L O P E B A R O C L I N I C E I G E N F U N C T I O N S o CD. oo o o CE cn CO o I co o i o o LU Q Q_ CE S L O P E WAVENUMBER F R E Q U E N C Y 4 5 3 E - 0 1 1 2 2 E - 0 1 5 3 1 E - 0 2 Figure 25b. Cross-channel dependence Of the interface mo-tion, corresponding to the modes of figure 24, A l l variables are dimensionless. E U L E R I A N F L O W P A T T E R N k=1.6 SIGMA=. 0453 s =0. 26 Hl/H2=0. 08 FIRST MODE J J \ J \ \ 7 / 7 / 1 1 7 -N — -» s —• -» N , J J SURFACE LAYER BOTTOM LAYER Figure 26. Eulerian flow pattern in both layers, corres-ponding to the f i r s t mode bf figure 25. A l l variables are dimensionless. 100 E U L E R I A N F L O W P A T T E R N k=1.6 SIGMA=. 0122 s=0. 26 Hl/H2=0. 08 SECOND MODE 1 S r- ~-7 7 SURFACE LAYER BOTTOM LAYER Figure 27. Same as in figure 26, but fdr the second mode 101 4.2 Basin Waves We now look at the propagation of topographic modes i n l a k e s . Lakes are modelled as c i r c u l a r b asins (see f i g . 11c) bf ra d i u s L. In the hope bf reachin g an a n a l y t i c a l s o l u t i o n , the bathymetry c o n s i d e r e d w i l l be a f u n c t i o n bf the r a d i u s , r , on l y . I t i s bf the form m H(r) = D[ 1 - ( r / r 0 ) ] (4.37) When m = 1, we have a c o n i c a l b a s i n , with m = 2, a p a r a b b l b i d a l b a s i n . The a n a l y s i s w i l l be done f o r m = 1,2 , but i t extends to l a r g e r m as w e l l . F i r s t , the set (3 .20)-(3.21) i s transformed i n t o p o l a r c o o r d i n a t e s to y i e l d [rH\p + H \p - Hxp - rH H + tU \p = g'H,H T? (4.38) r r r r r 00 r r t r 0 r 0 [rHr? + H t) - Hr? + rH_,H r? - H 2 f 2 rr?] - fHjH 77 r r r r r 00 H 2 r r g'H TH 2 t H 2 r r = f 2 H xP (4.39) g'H 2 r 6 The same s c a l i n g arguments as i n s e c t i o n 3.4 are advocated. The v a r i a b l e s are nbndimensibnalized with respect to the s c a l e s 1 02 r = L r ' co — of t = t ' / f H = Dh H! = D, H 2 = D 2 h 2 8 r 0 r r? = Nr?' ^ = CM/*' ( u , U ) = U ( V ,V ) i i i i 8 r where o, V , V , and the primed v a r i a b l e s are dimensionless, i i The s c a l e s Q and N are as before, that i s Q = D U L and N = Q f / g' D 2 For p e r i o d i c s o l u t i o n s i n azimuth 8 as w e l l as i n time t , ty{r,8,t) = ReaK F ( r ) exp(ik0 - iat) ) (4.40a) r?(r,0,t) = ReaK G(r) exp(ik0 - i a t ) ) (4.40b) where k = 0,1,2,... because of the p e r i o d i c i t y i n 8. When D, i s much smal l e r than D 2 , then h 2 h and the set (4. 38 ) - (4.39) becomes, once nondimensionalized, h [ r F + F - r - ^ F ] - rh F - kh F = 0 + OCD^/Bz) (4.41) r r r r r a r h h 2 [ r 2 G + rG - ( X 2 r 2 + k 2)G] = -krh F + 0{T>J-D2) (4,42) r r r a r where X 2 = L 2 / R i 2 Equation (4.37) i s w r i t t e n as 103 mm m h( r ) = 1 - ( L / r 0 ) r = 1 - s r , 0 < r < 1 (4.43a) Hence the d e r i v a t i v e of the depth f u n c t i o n w i l l be m-1 h (r) = -msr , s < 1 (4.43b) r Upon s u b s t i t u t i o n bf (4.43) i n t o (4.41)-(4.42) , the f i n a l set bf equations becomes m m-1 m-2 (1-sr )(F + r ' 1 F - r " 2 k 2 F ) + msr F + kmsr F = 0 (4.44) r r r r a m h 2 [ r 2 G + rG - ( X 2 r 2 + k 2)G] = kmsr F (4.45) r r r a The expr e s s i o n s f o r the v e l o c i t i e s (3.35)-(3.36) are, i n p o l a r c o o r d i n a t e s 6 V = h " 1 [ ( h / r ) F - ( G + (ko/r)G)]cbs(kd - at) (4.46a) 1 r r r V = h ' 1 [ ( k / r ) F + (aG - (k/r )G) ] s i n (k6 - at) (4.46b) 1 r 6 V = h " 1 [ ( h / r ) F + D ( G + (ka/r)G)]cbs( k d - at) (4.47a) 2 r * r r V = h " 1 [ ( k / r ) F - D (aG - (k/r )G) ] s i n (kt? - at) (4.47b) 2 * r where D = D, / D 2. The boundary c o n d i t i o n s r e q u i r e that the * r a d i a l v e l o c i t i e s v a n i s h at r = 1. Hence, F(1) = 0 and G (1) - ( k / a r ) G ( l ) = 0 (4.48) 4.2.1 The General S o l u t i o n The i n t r o d u c t i o n of a t r a n s f o r m a t i o n of the form k/m m F ( r ) = z q(z) and z = sr converts (4.44) i n t o (1 - z ) z q + [(2k+m)/m - 2kz/m]q + (k/m)(l + a " 1 ) q = 0 (4.49) zz z Equation (4.49) i s the Hypergeometric d i f f e r e n t i a l e q uation. The s o l u t i o n i s (Abramowitz and Stegun, 1964, page 563) the Gauss Hypergeometric s e r i e s q(z) = A» 2F,[a;b;c;z] + B-(sec6nd s o l u t i o n ) (4.50) where c = (2k + m) / m a + b + 1 = 2k/m ab = -k(1 + l/o)/m (4.51a) (4.51b) (4.51c) 105 The second c o e f f i c i e n t B must be zero, s i n c e the second s o l u t i o n i s always s i n g u l a r at r = 0. ( E r d e l y i et a l . , 1953, chapter 2). Indeed c - a - b = 2 , a n i n t e g e r (4.52) and c i s always a n a t u r a l number ( f o r m = 1,2). The s i n g u l a r i t y i s l o g a r i t h m i c i f a Or b i s not a p o s i t i v e i n t e g e r . I t i s a l g e b r a i c i f a Or b i s a n a t u r a l number smal l e r than c. Because bf (4.52), the s o l u t i o n (4.50) may be w r i t t e n as k/m F(z) = A«z -(1 - z ) 2 • 2 F , [ c - a ; c - b ; c ; z ] (4.53) F i n a l l y , expressed i n terms Of the r a d i a l v a r i a b l e r , the s o l u t i o n bf (4.44) i s k m F ( r ) = A - ( L r / r 0 ) - h 2 • 2 F , [ c - a ; c - b ; c ; ( L r / r 0 ) ] (4.54) The boundary c o n d i t i o n s (4.48) r e q u i r e that m 2 F,[2+b;c-b;c ; ( L / r 0 ) ] = 0 (4.55) s i n c e , from (4.52), c-a = 2+b. The s e r i e s r e p r e s e n t a t i o n Of the HypergeOmetric f u n c t i o n i s 106 2 F , [ a ; b ; c ; z ] = 1 + ab z + a(a+1I )b(b+1) z 2 + ... c 2c(c+1) For a root of (4.55) to e x i s t , ab must be negative s i n c e c and z are p o s i t i v e . Hence, r e f e r r i n g to (4.51a,b) 2 + b < 0 or b < - 2 From (4.51c), a w i l l be a = 2k-m + [(2k-m) 2 + k(1 + J_) I"2" m 4m2 m a where the p o s i t i v e root has been chosen s i n c e a must be p o s i t i v e . When b i s e x a c t l y equal to -2, that i s when a = k/(2m + 3k) (4.56) we f i n d that 2F,[2+b;c-a;c;z] = 1 and the s o l u t i o n (4.54) reduces to k F ( r ) = A - ( L r / r 0 ) h 2 (4.57) To s a t i s f y the boundary c o n d i t i o n s , the depth must be zero on 107 the boundary. Equation (4.56) and (4.57) are the nOndimensiOnal v e r s i o n s Of the d i s p e r s i o n r e l a t i o n and volume t r a n s p o r t stream f u n c t i o n Obtained by SaylOr at a l . (1980) f o r the case Of an homogeneous c i r c u l a r basin with zero depth On the boundary. The equation f o r the f o r c e d i n t e r f a c e motion (4.45) i s a M o d i f i e d B e s s e l equation. The homogeneous s o l u t i o n i s (Abrambwitz and Stegun, 1964, page 374) G(r) = C-I (Xr) + D-K (Xr) k k Since the second s o l u t i o n K (Xr) i s s i n g u l a r at r = 0, the k constant D must be zero. The WrOnskian Of the m o d i f i e d B e s s e l f u n c t i o n s i s (Abrambwitz and Stegun, 1964, page 375) W[K (Xr) , I (Xr)] = ( X r ) ' 1 k k The general s o l u t i o n Of (4.47) i s then r G(r) = C-I (Xr) - K ( X r ) - [ / I ( X t ) - g ( t ) - d t ] k k k r + I ( X r ) • [ / K ( X t ) • g ( t ) • d t ] (4.58) k k where 108 m+1 g(r) = mskXh" 2r -F(r) (4.59) a When w r i t t e n i n terms of d e f i n i t e i n t e g r a l s , G(r) becomes, a f t e r the boundary c o n d i t i o n s are s a t i s f i e d G(r) = [R - / K ( X t ) . g ( t ) - d t M (Xr) -r k k [S - / I ( X t ) . g ( t ) - d t ] - K (Xr) (4.60) r k k where R ans S are given by S = $ \ ( X t ) - g ( t ) - d t 0 k R = -S-[XK (X) + d + l ) k K (X)]-[XI (X) - ( 1 + l ) k l ( X ) ] - 1 k-1 a k k-1 o k 4.2.2 C o n i c a l And P a r a b o l o i d a l Bottom P r o f i l e s The Hypergeometric f u n c t i o n s are eval u a t e d through t h e i r Chebyshev r e p r e s e n t a t i o n , f o l l o w i n g the a l g o r i t h m given i n Luke (1977). The i n t e r f a c e mdtidn i s obtained by n u m e r i c a l l y i n t e g r a t i n g equation (4.60), once the b a r d t r d p i c part df the mdtidn , F, has been fdund. The d i s p e r s i d n r e l a t i d n (eq. 4.55) f o r the f i r s t azimuthal mdde (k = 1) i s eval u a t e d f o r m = 1 ( c o n i c a l p r o f i l e ) and m = 2 109 ( p a r a b o l o i d a l p r o f i l e ) . The r e s u l t s are given i n Table I, f o r the cases where the depth On the boundary, H(L), i s 100 and 210 m. In both cases, the t h i c k n e s s Of the s u r f a c e l a y e r i s 45 m and the maximum depth, H(0), i s 300 m. The c u t - o f f f r e q u e n c i e s are, from the homogeneous problem (eq. 4.56), f/5 when m = 1 and f/7 when m = 2. In the homogeneous problem with zero depth On the boundary, only One mode i s p o s s i b l e f o r each azimuthal wavenumber. When the depth i s nb longer zero On the boundary, higher modes are now p o s s i b l e , at each k. T h i s may be seen i n f i g u r e 28 (m = 1) and i n f i g u r e 29 (m = 2). In each f i g u r e , the upper panel i s the e u l e r i a n flow p a t t e r n f o r an homogeneous basin with zero depth On the boundary , as i n SaylOr et a l . (1980). The Other panels are the e u l e r i a n v e l o c i t i e s at t = 0 f o r a ba s i n where the depth On the boundary i s 210m. These modes c o n s i s t of c e l l s r o t a t i n g in a cOunter-clOckwise d i r e c t i o n around the b a s i n , as i n f i g u r e 7. The f r e q u e n c i e s bf r o t a t i o n are given i n Table I. For m = 1 ( f i g . 28), the c e n t e r s bf the two c e l l s are d i s p l a c e d towards the cente r (r = 0) Of the b a s i n . The main new r e s u l t i s the appearance Of a kidney-shaped second c e l l i n the second mode. FOr m = 2, there i s no evident displacement Of the c e n t e r s Of the c e l l s . There i s a l s o a second kidney-shaped second c e l l i n the second mode. In both cases, the motion i s s u r f a c e i n t e n s i f i e d and the c u r r e n t s do not change d i r e c t i o n with depth. TABLE I D i s c r e t e eigenvalues of eq. (4.55), the d i s p e r s i o n r e l a t i o n f o r c i r c u l a r l a k e s . The depths are i n me-t e r s . A l l other v a r i a b l e s are d i m e n s i o n l e s s . H(L) H(0) k m u/f 0.791X10" 1 100 300 1 1 0.298X10" 1 0.157X10- 1 0.141X10" 1 210 300 1 1 0.524X10- 2 0.275X10- 2 0.659X10- 1 100 300 1 2 0.210X10- 1 0.101XI0 - 1 0.125X10" 1 210 300 1 2 0.376X10" 2 0.179X10- 2 111 EULERIAN FLOW PATTERN K = l HOMOGENEOUS B A S I N M=l F I R S T L A Y E R F I R S T MODE S E C O N D MODE S E C O N D L A Y E R F I R S T L A Y E R S E C O N D L A Y E R Figure 28. Eulerian flow pattern in both layers for Basin waves. The upper pannel i s the eulerian flow pattern for an homogeneous basin with zero dpth on the boundary. A l l variables are dimensionless, EULERIAN FLOW PATTERN 1 / I / 1 I ^ i -~ —. F I R S T L R Y E R F I R S T L R Y E R F I R S T MODE S E C O N D MODE - - • -•'//// S E C O N D L R Y E R S E C O N D L R Y E R Figure 29. Same as in figure 28, but for m=2. 1 1 3 5 DISCUSSION 5 .1 General Assessment Of The Model In the p r e v i o u s c h a p t e r s , new s o l u t i o n s were obtained f o r topographic waves propagating over l a r g e axi-symmetric l i n e a r and p a r a b o l i c bottom s l o p e s . Because of the d e c o u p l i n g between b a r o t r o p i c and b a r o c l i n i c waves, the equation f o r the b a r o t r o p i c p a r t of the motion (eq. 3 . 3 3 ) i s homogeneous. T h e r e f o r e the s o l u t i o n s we found represent the exact s o l u t i o n s f o r waves t r a v e l l i n g i n an homogeneous f l u i d . The behaviour of these s o l u t i o n s was d i s c u s s e d i n terms of known higher t r a n s c e n d e n t a l f u n c t i o n s . However, because no s i n g u l a r i t y i s found i n s i d e the channel boundaries, a l l the s o l u t i o n s can be obtained by a simple T a y l o r expansion around x = 0 . Moreover, t h i s method may be t r i v i a l l y extended, i n the i n f i n i t e channel problem, to bottom p r o f i l e s df the form bx N n H(x) = e (1 - I a x ) ( 5 . 1 ) n = 0 n I t i s now p o s s i b l e t d model much more c l o s e l y the bathymetry of channels and i t i s ndt necessary t d r e l y on the small sldpe apprdximation anymore. Fdr a body df water whose s t r a t i f i c a t i o n may be apprdximated by a two-layer mddel, the s d l u t i d n depends on the r e l a t i v e t h i c k n e s s df both l a y e r s . The e r r d r inherent t d our 1 14 r e g u l a r p e r t u r b a t i o n approach i s 0(D,/D 2). T h i s e r r o r w i l l be small as long as the s u r f a c e l a y e r i s much t h i n n e r than the bottom l a y e r . Obviously, the e r r o r gets worse as we approach the boundaries. In that r e g i o n , the s o l u t i o n i s a l r e a d y at best an approximation s i n c e , as i n many models, a l a t e r a l s l i p c o n d i t i o n was used. On the other hand, the e r r o r decreases with d i s t a n c e from the boundaries. We exchange the small slope approximation fo r the small d e p t h - r a t i o approximation. The advantage of the l a s t approach i s t w o - f o l d . As p r e v i o u s l y mentioned, c o n t r a r y to the small slope approximation, the e r r o r decreases as we p a r t from the boundary. Secondly, t h i s model y i e l d s low-frequency s u r f a c e i n t e n s i f i e d motions of the type observed i n the S t r a i t of Georgia and i n the S a i n t Lawrence est u a r y (El-Sabh et a l . , 1982). Of course, because what we r e a l l y have so l v e d i s the b a r o t r o p i c problem, the f r e q u e n c i e s are those df tdpdgraphic waves propagating i n an homogeneous f l u i d . We would have td proceed td the next drder i n (D,/D 2) td introduce the e f f e c t df s t r a t i f i c a t i o n dn the f r e q u e n c i e s . The zerdth-drder s d l u t i d n g i v e s a lower bound dn the f r e q u e n c i e s . Mysak (1967) and K a j i u r a (1974) fdund a s i g n i f i c a n t i n c r e a s e i n the phase speed df long s h e l f waves with the deep-sea s t r a t i f i c a t i o n model ( f i g . 6e), above t h a t df an u n s t r a t i f i e d model. However, Wang (1975) and Wright and Mysak (1977) fdund that long s h e l f wave speeds i n c r e a s e d by dnly a few percent, when the dn-shelf s t r a t i f i c a t i o n mddel ( f i g . 6f) i s used. L i e and El-Sabh (1983) fdund a d i f f e r e n c e df abdut 10% f d r the ldwer mdde and abdut 20% 1 15 f d r the four higher mddes, in t h e i r numerical model df the S a i n t Lawrence e s t u a r y . Hence, our approach i s a step i n the r i g h t d i r e c t i o n . 5.2 I n f l u e n c e Of The Bottom Sldpe The shape df the bathymetry as w e l l as the s t r e n g t h df the bottom sldpes g r e a t l y i n f l u e n c e the c h a r a c t e r df the low-frequency tdpdgraphic mddes. F i g u r e s 30 and 31 present the d i s p e r s i d n curves f d r the four models df tdpdgraphic p r d f i l e s that were s t u d i e d . In f i g u r e 30, a l l d i s p e r s i d n curves were c a l c u l a t e d with s = 0.26. L i k e w i s e , in f i g u r e 31, a l l d i s p e r s i d n curves were c a l c u l a t e d with s = 0.75. In a l l cases the waves are ndn d i s p e r s i v e at low-frequency. The curves pass through a p o i n t where the group v e l o c i t y i s zero and then decay s l o w l y . The shapes df the d i s p e r s i d n curves dd not change very much. Fdr constant sldpe and wavenumber, the p e r i o d df o s c i l l a t i o n i s s e n s i t i v e t d the bathymetric p r o f i l e . T h i s i s i n agreement with the r e s u l t s df Sayldr et a l (1980) i n the case df an hdmdgenedus c i r c u l a r l a k e . Table II shows that the frequency i s always higher fdr the l i n e a r p r o f i l e than f d r the p a r a b o l i c p r o f i l e . T h i s may a l s o be seen i n f i g u r e 32. Fdr wavenumbers 1.5, 3.0, 6.0, the frequency at each wavenumber i s p l o t t e d a g a i n s t the s l d p e . At each wavenumber, the frequency i s normalized by the value obtained f d r s = 0.1. The second 11 gure 30. I n f i n i t e channel dispersion curves for the four bottom p r o f i l e s studied, s=0.26. A l l variables are dimensionless. igure I n f i n i t e channel dispersion curves for the four bottom p r o f i l e s studied, s=0.75. A l l variables are dimensionless. TABLE II Comparison Of the fre q u e n c i e s computed with the small l i n e a r (SLT), the l a r g e l i n e a r (LLT), the small p a r a b o l i c (SPT), and the l a r g e p a r a b o l i c (LPT) topography models, at the same wavenumber k=1.6. A l l v a r i a b l e s are di m e n s i o n l e s s . S = 0.20 s= 0.50 s= 0.80 SLT 0. 0356 0. 0869 0. 1 352 FIRST LLT 0. 0389 0. 1096 0. 1977 MODE SPT 0. 0320 0. 0768 0. 1 178 LPT 0. 0344 0. 0920 0. 1538 SLT 0. 0096 0. 0237 0. 0376 SECOND LLT 0. 01 06 0. 031 7 0. 0652 MODE SPT 0. 0084 0. 0207 0. 0327 LPT 0. 0091 0. 0263 0. 0508 SLT 0. 0041 0. 01 02 0. 01 63 THIRD LLT 0. 0046 0. 0139 0. 0298 MODE SPT 0. 0036 0. 0090 0. 01 43 LPT 0. 0040 0. 0116 0. 0236 119 t l 1 l ' OL 3 ' Ol 5 l Ol 7' SLOPE : i r o T 3 T o : 5' o: r SLOPE L = l . 5 L=3. 0 L=6. 0 SLOPE 1 l ' Ol 3 l Ol 5' Ol 7' SLOPE F i g u r e 32. P l o t s of frequency versus slope f d r small l i n e a r (SLT), l a r g e l i n e a r (LLT), small p a r a b o l i c (SPT) and l a r g e p a r a b d l i c (LPT) tdpdgraphies. A l l va-r i a b l e s are d i m e n s i o n l e s s . Note that curves f o r k=3.0 and 6.0 are o f f s e t by 1 and 2 u n i t s , r e s -p e c t i v e l y . 120 (k=3.0) and t h i r d (k=6.0) curves are t r a n s l a t e d by one and two u n i t s , r e s p e c t i v e l y , On the v e r t i c a l s c a l e to improve l e g i b i l i t y . Two more c o n c l u s i o n s may be e x t r a c t e d from f i g u r e 32. F i r s t l y , the frequency grows a l i t t l e f a s t e r with l a r g e r s Over l i n e a r p r o f i l e than over p a r a b o l i c p r o f i l e s . Secondly, i t i s now evident that the small slope approximation s y s t e m a t i c a l l y underestimates the frequency. The e r r o r grows with s. I t can exceed 80% when s i s 0.8, depending On the wavenumber. Table III g i v e s the approximate f r e q u e n c i e s at which the group v e l o c i t y Of the f i r s t mode vanishes, f o r d i f f e r e n t v a l u e s Of the nondimensiOnal s l o p e s . Both grow with s. T h i s means that s t ronger slopes w i l l allow the e x i s t e n c e Of motion Of s h o r t e r time s c a l e s ( t h i s i s the h i g h e s t a t t a i n a b l e f requency). For i n s t a n c e , Over a p a r a b o l i c p r o f i l e with s = 0.7, the f a s t e s t wave w i l l have a p e r i o d Of about 4 days and a wavelength Of about 20 km, i f the channel width i s 20 km. Over the GS, the wind spectrum peaks about 3 to 5 days ( H e l b i g , 1978). These waves are more l i k e l y to be generated. Of course a mechanism capable Of t r a n s f e r r i n g energy down the frequency spectrum w i l l have to be found ... but t h i s i s another s t o r y . TO f i n d Out the major e f f e c t s Of l a r g e r slopes On the c r o s s - c h a n n e l s t r u c t u r e Of Channel waves, we p r e s e n t , i n f i g u r e 33, mass-transport stream f u n c t i o n s and, i n f i g u r e 34, i n t e r f a c e e l e v a t i o n s f o r two v a l u e s Of the s l o p e parameter (s = 0.26 and s = 0.75). In both f i g u r e s , we used a p a r a b o l i c bottom p r o f i l e with ( f i g . 33a, c and f i g . 34a, c) and without ( f i g * 121 TABLE III Frequencies and wavenumber at wich the group v e l o c i t i e s vanish. SLT LLL SPT LLT s a k a k a k a k 0.1 0. 0191 2.3 0. 0199 2.4 0 .0180 2.8 0. 0188 2.8 0.2 0. 0381 2.4 0. 0417 2.4 0 .0358 2.8 0. 0389 2.9 0.3 0. 0569 2.4 0. 0654 2.4 0 .0534 2.9' 0. 0605 3.1 0.4 0. 0754 2.4 0. 091 4 2.5 0 .0706 2.9 0. 0840 3.2 0.5 0. 0938 2.4 0. 1203 2.6 0 .0876 3.0 0. 1096 3.4 0.6 0. 1119 2.5 0. 1 526 2.7 0 . 1 044 3.0 0. 1 381 3.7 0.7 0. 1299 2.5 0. 1893 2.8 0 . 1 208 3.0 0. 1699 3.9 0.8 0. 1 476 2.5 0. 2317 3.0 0 . 1 370 3.1 0. 2070 4.7 SLOPE i HflVENUMBER i FREQUENCY « .28 1.8 ( 1 ) ( 2 ) ( 3 ) . USE-01 .109E-01 . 47IE-02 SLOPE HRVENUMBER FREQUENCY .28 1.6 ( 1 ) t 2 ) ( 3 ) . 453E . 122E .531E -01 -01 -02 CHANNEL NODES » SMALL PARABOLIC SLOPE BAROTROPIC EIGENFUNCTIONS CHRNNEL MODES « LARGE PARABOLIC SLOPE BAROTROPIC EIGENFUNCTIONS 8. - - ^ SLOPE WAVENUMBER FREQUENCY 75 1.8 ( 1 ( 2 ( 3 0 _ s 8 & IS •i <f 8 8 • , UlE+OO .307E-01 , 135E-01 . 144E+00 .459E-01 .2UE-01 Figure 33. Cross-channel structure of the mass-transport stream function over a parabolic bottom p r o f i l e with the small slope approximation (A: s=0.26, C: s=0.75) and with-slbpe approximation (B: s=0.26, D: s=0.75). are dimensionless. but A l l the small variables 1 23 HAVENUMBER « 1.6 FREQUENCY i t 1 ) .413E-01 1 2 ) . 109E-01 t 3 ) . 471E-02 NRVENUMBER I 1.6 FREQUENCY « ( 1 ) .453E-01 1 2 ) .122E-01 I 3 ) .531E-02 CHRNNEL MODES : SMALL PARABOLIC SLOPE BAROCLINIC EIGENFUNCTIONS \ SLOPE i .75 HAVENUMBER i 1.6 FREQUENCY i ( 1 ) ( 2 ) ( 3 ) , 111E+00 307E-01 , 135E-01 CHANNEL MODES : LARGE PARABOLIC SLOPE BAROCLINIC EIGENFUNCTIONS SLOPE i .75 HAVENUMBER t 1.6 FREQUENCY i l l ) . H4E+00 ( 2 ) .459E-01 ( 3 ) .211E-01 Figure 34. Same as in figure 33, but for the interface elevat ion. 1 24 33c, d and f i g . 34c, d) the small slope approximation. For small slopes (s = 0.26), no major d i f f e r e n c e s are v i s i b l e i n the cro s s - c h a n n e l mass-transport s t r u c t u r e ( f i g . 33a, b ) . For l a r g e r slopes (s = 0.75), the mass-transport's maxima are s h i f t e d c l o s e r to the c o a s t . The amplitude of the maxima i s reduced by almost 30% when one does not use the small slope approximation ( f i g . 33c, d ) . The most d r a s t i c d i f f e r e n c e s are found i n the i n t e r f a c e e l e v a t i o n s t r u c t u r e ( f i g . 34). Larger slopes s h i f t the maxima of s u r f a c e e l e v a t i o n towards the c o a s t . Moreover, the maxima of su r f a c e e l e v a t i o n are i n c r e a s e d by about 30% f o r small s l o p e s ( f i g . 34a, b) and are more than doubled f o r l a r g e slopes ( f i g . 34c, d ) , when the small sldpe approximation i s not used. The p i c t u r e i s s i m i l a r f dr motions over a l i n e a r bottom p r o f i l e (not shown) . In c o n c l u s i o n , one can say that the small sldpe apprdximatidn g r e a t l y underestimates a l l the parameters df low-frequency tdpdgraphic waves over l a r g e bottom s l d p e s . 5.3 A p p l i c a t i o n Td The S t r a i t Of Georgia Since t h i s work was motivated by data c o l l e c t e d i n the GS, l e t us see how our c a p a c i t y td model both the geometry and the ldw-frequency motions df the GS has been improved. F i g u r e 10 presen t s the- bathymetry df l i n e H (see f i g . 9 f d r l d c a t i d n ) . The dashed l i n e i s the f i t used by H e l b i g and Mysak (1976) and the s o l i d l i n e i s our f i t t d the bathymetry. The o r i g i n df our cd d r d i n a t e system i s set at the p d i n t df maximum depth (D = 125 375m). The boundaries are s i t u a t e d at x = L (Sturgeon Bank) and x = -L/2 (Valdes I s l a n d ) . The nbndimensibnal (with respect to L = 16 Km) slopes are taken as s, = 0.6 (x p o s i t i v e ) and s 2 = 1.0 (x negative) Since the depth i s never zero on the boundaries, the r e s u l t bf s e c t i o n 4.2.1.2 can be a p p l i e d immediately. The d i s p e r s i o n curves f o r the right-bounded waves (upper l e f t ) and left-bounded (lower l e f t ) waves are shown i n f i g u r e 35. The two other panels show the d i s p e r s i o n curves f o r waves propagating over a symmetric bottom p r o f i l e , f o r comparison. The f r e q u e n c i e s bf the left-bounded waves are lower than those bf the right-bounded waves, even i f the dimensional slope i s steeper ( 2.3X10" 2 compared to 1.4X10~ 2) by almost a f a c t o r bf two ! T h i s shows that i t i s the f r a c t i o n a l depth change s = aL = A H L = AH th a t i s the important parameter. The f r a c t i o n a l depth change On the western s i d e i s 0.5, not 1.0. Hence, the f r e q u e n c i e s bf the left-bounded waves are s i m i l a r to those bf waves t r a v e l l i n g over a symmetric, s = 0.5, p r o f i l e . Indeed, s 2 = a 2 L = AH 2 L = 2AH 2 =1.0 , and AH 2 = 0.5 D L D D D D 12 DISPERSION RELATION DISPERSION RELATION SLOPE - 0.60 0.00 4.00 a. 00 12.00 WAVENUMBER 6.00 12.00 NfiVENUMBER 18.00 20.00 Figure 35. Dispersion curves for the right-bounded wave (up-per l e f t ) and the left-bounded wave (lower l e f t ) propagating over the topographic p r o f i l e shown in f i g . 3.3. The two other dispersion curves are for waves over an axi-symmetric channel. A l l variables are dimensionless. 1 27 The left-bounded waves ( f i g . 3 5 , lower l e f t ) seem much more a f f e c t e d by the slope dn the other s i d e df the channel than the right-bounded ( f i g . 35, upper l e f t ) . T h i s phendmendn i s due t d a d i s t o r t i o n of the wavenumber s c a l e . As a matter df f a c t , we used only one l e n g t h s c a l e , L, t d ndndimensidnalize the wavenumbers. The c r o s s - c h a n n e l s t r u c t u r e i s d s c i l l a t d r y dnly dver that s i d e df the channel that can s u s t a i n a v d r t i c i t y wave. Hence, i t i s l o g i c a l t d expect that the d i s p e r s i d n curves w i l l be s h i f t e d towards higher wavenumbers. In our case, one should expect that a l l wavenumbers should be doubled, s i n c e the l e n g t h s c a l e dn the r i g h t s i d e df the channel (L) i s twice the l e n g t h s c a l e dn the l e f t s i d e ( L / 2 ) . Fdr i n s t a n c e , the group v e l o c i t y i s zerd at k 2.6 and a = 0.1203 dver a symmetrical s = 0.5 bottom p r o f i l e ( f i g . 35, lower r i g h t ) . Fdr our GS model the group v e l o c i t y i s zerd at k = 4.9 and -a = 0.1226. Once halved, t h i s y i e l d s a wavenumber df 2.9. The d i f f e r e n c e s are are much l e s s d r a s t i c . The right-bounded wave has a group v e l o c i t y t h a t vanishes at k = 2.8 and a = 0.1449. Over a symmetrical, s = 0.6, p r o f i l e ( f i g . 35, lower r i g h t ) , the e q u i v a l e n t v a l u e s df k and a are 2.7 and 0.1526. The major d i f f e r e n c e s w i l l be fdund i n the c r o s s - c h a n n e l s t r u c t u r e s , s i n c e the d s c i l l a t d r y behaviour i s trapped dver the sldpe that can s u s t a i n tdpdgraphic mddes. Thus, two waves t r a v e l l i n g i n opposite d i r e c t i d n s w i l l be c h a r a c t e r i z e d by twd d i f f e r e n t l e n g t h and time s c a l e s . I t i s reasonable t d b e l i e v e that d b s e r v a t i d n s from both s i d e s df the channel w i l l be p o o r l y 128 c o r r e l a t e d , i f two such waves are present. A quick glance at Table IV t e l l s us that the s m a l l e s t p o s s i b l e p e r i o d s are about 5 days f d r right-bdunded waves and abdut 6 days f d r left-bounded wave They correspond t d wavelengths df abdut 36 and 17 Km, r e a p e c t i v e l y . In c o n c l u s i o n , our model can f i t more c l o s e l y the bathymetry df the GS (at l e a s t t h at df l i n e H). I t can reproduce s u r f a c e - i n t e n s i f i e d mdtidns. Mdredver, the l e n g t h and time s c a l e s we o b t a i n are c l o s e r t d those b e l i e v e d td be present in the GS. I t a l s d p r o v i d e s an hypothesis t d e x p l a i n sdme of the low coherences dbserved i n the S t r a i t . F i n a l l y , the f a s t e s t waves are df the same time s c a l e as the wind systems, the mdst common source df energy f d r S h e l f waves. TABLE IV Correspondence between the dimensional and nondimensibnal wavenumbers and f r e q u e n c i e s . k 27r/k a 2-n/a 2.5 2. 5L km 0. 02 36.4 days 3.0 2. 1L km 0. 07 10.4 days 3.5 1 .8L km 0. 17 6.1 days 4.0 1 .6L km 0. 17 4.3 days 4.5 1 . 4L km 0. 22 3.3 days 130 '5.4 A p p l i c a t i o n To The S a i n t Lawrence We chose to apply Our model to the St-Lawrence estuary f o r comparison with the numerical model Of L i e and El-Sabh (1983). T h e i r numerical model i s p r e s e n t l y the Only One that reproduces the propagation Of topographic waves over r e v e r s i n g bottom s l o p e s . Moreover, the Matane-POinte a l a C r o i x s e c t i o n (see f i g . 36) can be modelled with a p a r a b o l i c (s = 0.60) p r o f i l e . L i e and El-Sabh (1983) used a p r o f i l e bf the form H 2(y) = h o " s i n [ (1 OTT/4) (1+y) ]exp(-1 2y) 0 < y < 0.5 H 2 ( y ) = h 0 - (he - h 0 ) ( l - s i n y) 0.5 < y < 1.0 where he i s the t h i c k n e s s bf the bottom l a y e r at the Matane boundary, and h 0 i s the maximum depth Of the bottom l a y e r (245m). Since they used a l i n e a r i z e d set of equations and s i n c e the shapes bf a parabola and a h a l f - s i n e are c l o s e l y r e l a t e d , Our r e s u l t s should compare with t h e i r s . A value Of s = 0.60 was chOOsen to Obtain the same depth (he) at the Matane boundary as L i e and El-Sabh (1983). T h i s i s not the optimal f i t to the St-Lawrence bathymetry, but i t w i l l be s u f f i c i e n t f o r comparisons purposes. The numerical values Of the d i f f e r e n t parameters are D = 300m f = l 0 - 4 r a d / s D, = 45m L = 46km 6 = 4.38X10" 3 D,/D2 =0.18 131 Figure 36. Location and shape of the Matane-Pointe a l a croix section. The dashed l i n e i s the f i t used by Lie and El-Sabh(1983) and the s o l i d l i n e (parabola) i s our f i t to the topography of the section. Modified from Lie and E l -Sabh(l983). 1 32 The d i s p e r s i o n curves f o r topographic waves over a p a r a b o l i c p r o f i l e (s = 0.60) are shown in f i g u r e 37 (upper p a n e l ) . The wavenumber a x i s has been t r u n c a t e d at k = 5.0. T h i s was done to o b t a i n the same s c a l e as the one used by L i e and El-Sabh (lower panel i n f i g . 37). The behaviour i s the same i n both models. Since the topographic p r o f i l e s are not the same, there w i l l be a d i f f e r e n c e i n the frequency v a l u e s : we a l r e a d y saw that the r e s u l t s are very s e n s i t i v e to the bathymetry. The r e s u l t s from both models are compared i n Table V. L i e and El-Sabh (1983) used the f u l l channel width as the l e n g t h s c a l e . Hence t h e i r wavenumber corresponds to a wavenumber bf 1.6, i n bur n o t a t i o n . Two t h i n g s are n o t i c e d from Table V. F i r s t , f o r the homogeneous channel, the f r e q u e n c i e s obtained from the l a r g e p a r a b o l i c topography model are w i t h i n about 10% bf those computed by L i e and El-Sabh (1983). A b e t t e r f i t to the bathymetry would probably b r i n g them even c l o s e r . Secondly, the f r e q u e n c i e s are between 10% ( f i r s t mode) and 20% (second and t h i r d modes) higher when a two-layer model i s used. Hence, bur model i s s y s t e m a t i c a l l y underestimating the r e a l f r e q u e n c i e s . One would need a slope bf 0.75 to o b t a i n comparable v a l u e s . In the St-Lawrence, the s t r a t i f i c a t i o n has a non n e g l i g e a b l e e f f e c t on topographic waves higher modes. 133 SLOPE - 0.60 3. HAVENUMBER Figure 37. Dispersion curves obtained by Lie and El-Sabh (1983), lower panel, and with the large para-b o l i c (s=0.60) model. TABLE V Frequencies computed by L i e and El-Sabh(1983) and by the l a r g e p a r a b o l i c (s=0.60) topography model. The nOndimensiOnal wavenumber i s the same i n both i n s t a n c e s (k=1.6). The cor r e s p o n -ding dimensional wavenumber i s twice the chan-n e l width. LIE + EL-SABH(1983) LPT MODES ONE-LAYER TWO-LAYER S=0.60 s=0.75 1 0.1248 0.1426 0.1126 0.1438 2 0.0368 0.0456 0.0334 0.0459 3 0.0168 0.0214 0.0149 0.0211 1 35 6 SUMMARY AND CONCLUSION Low-frequency topographic mode s o l u t i o n s have been known to e x i s t s i n c e a long time. Lamb (1932), s e c t i o n s 193 and 212, had al r e a d y d i s c u s s e d t h e i r p r o p e r t i e s . Simple a n a l y t i c a l models are found only f o r waves propagating i n an homogeneous f l u i d . In presence of s t r a t i f i c a t i o n , the equations are u s u a l l y s o l v e d n u m e r i c a l l y . For channels and b a s i n s , there i s no simple a n a l y t i c a l model that d i s c u s s the topographic modes dn a ba s i n s c a l e , i n presence df s t r a t i f i c a t i o n . P revious models c o u l d not e x p l a i n the s u r f a c e - i n t e n s i f i e d motions, c r o s s - c h a n n e l v e l o c i t i e s i n the GS and i n the St~ Lawrence. They d i d not take i n t o account the sldpe r e v e r s a l c h a r a c t e r i s t i c df a l l channel and l a k e s . I t was fdund that f d r bathymetric p r o f i l e s s i m i l a r t d those df the GS and the S t -Lawrence, our model prov i d e s a b e t t e r f i t t d the tdpdgraphy, leads t d s u r f a c e - i n t e n s i f i e d motions and produce c r o s s - c h a n n e l v e l o c i t i e s . The small sldpe apprdximatidn does not have t d be used, s i n c e a simple T a y l o r expansion p r o v i d e s the exact s d l u t i d n s f d r waves propagating i n an homogeneous f l u i d . The r e s u l t s compare very f a v o r a b l y with those of L i e and El-Sabh (1983). A T a y l o r s e r i e s expansidn would have c d n s i d e r a b l y s i m p l i f i e d the a n a l y s i s df Shaw and Neu (1981) who s t u d i e d the propagation df tdpdgraphic waves dver r i d g e s . Fdr twd-layer systems, the uncoupling hypothesis leads td a simple s d l u t i d n df the dynamical equatidns and y i e l d s s u r f a c e -136 i n t e n s i f i e d motions. The e r r o r i s 0(D,/D 2), the r a t i o of the upper to lower l a y e r depths. The model g i v e s a lower bound On the f r e q u e n c i e s . FOr the Saint-Lawrence, the e r r o r i s 0(10%) fo r the lower mode and 0(20%) for the higher modes. FOr the GS, the model leads to l e n g t h and time s c a l e s c l o s e r to those b e l i e v e d to be present than was Obtained by p r e v i o u s l y known models. I t was a l s o found that the c h a r a c t e r i s t i c s Of those waves are s e n s i t i v e to the bathymetry. The e x i s t e n c e and the generating mechanisms Of the topographic modes are s t i l l Open to q u e s t i o n . Since they are b a s i c a l l y S h e l f waves with an added boundary, the c r O s s - s h e l f modal f i t t i n g technique developed by Hsieh (1981) c o u l d be a p p l i e d to Channel waves. Such an a n a l y s i s , apart from i n v e r s e techniques (see Parker, 1977, f o r i n s t a n c e ) , i s probably the only d e t e c t i o n technique that c o u l d b r i n g Out the topographic waves s i g n a l . In the second chapter, we saw that n o n l i n e a r t i d a l i n t e r a c t i o n s with topography i s a p o s s i b l e g e n e r a t i n g mechanism. In the l a s t chapter, we found that the f a s t e s t waves possess time s c a l e s Of the same Order as the wind systems. T h i s a second p o s s i b l e g e n e r a t i n g mechanism. However, the Observed ( i n GS) frequency s p e c t r a are broad-banded. T h i s means that n o n l i n e a r processes are probably at work, r e d i s t r i b u t i n g energy i n the frequency domain. Hence, a n o n l i n e a r theory Of Channel waves w i l l have to be developed, i f we intend to e x p l a i n the brbad-bandness Of the s p e c t r a . F i n a l l y , the mean flow may give r i s e to new e f f e c t s (Mysak, 137 1980a). Channel waves, l i k e Shelf waves, may be s i g n i f i c a n t l y advected by the current and have their properties strongly affected. Moreover, Channel waves may become amplified, extracting kinetic energy from the mean flow, by the process Of barbtrOpic-barbclinic i n s t a b i l i t y . 138 BIBLIOGRAPHY 1. Abramdwitz, M., and I. A. 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