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UBC Theses and Dissertations

Rotating flows around sharp corners and in channel mouths Cherniawsky, Josef Yuri 1985

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ROTATING FLOWS AROUND SHARP CORNERS AND IN CHANNEL MOUTHS By JOSEF YURI CHERNIAWSKY B.Sc, McGill University, 1973 M.Sc, University o-f V i c t o r i a , 1977 THESIS SUBMITTED IN PARTIAL FULFILLMENT THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY In THE FACULTY OF GRADUATE STUDIES Department o-f Oceanography We accept t h i s t h e s i s as con-forming to the required standards THE UNIVERSITY OF BRITISH COLUMBIA July 1985 Jose-f Yuri Cherniawsky, 19S5 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It i s understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department Of Oceanography  The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 supervisor: Dr. Paul H. LeBlond Abstract This t h e s i s examines buoyancy driven steady flows in mouths o-f sea s t r a i t s and around coastal protrusions. At high l a t i t u d e s , the C o r i o l i s -force keeps these currents banked against the coast even around r e l a t i v e l y sharp re—entrant (convex) corners with r a d i i o-f curvature that are comparable to the width o-f the current. On the other hand, i-f the radius of curvature of the corner i s much smaller than the width of the current, the current may leave the coast at the apex of the corner. A central part of the t h e s i s i s the solution of the nonlinear problem of a steady i n v i s c i d reduced gravity flow in a wedge, 0<©<rr/a (with a>l / 2 ) , around a sharp corner on an f-plane. An exponential upper layer upstream depth p r o f i l e , h=Hexp(-x/X) (where x and X are the offshore distance and the current width scale, r e s p e c t i v e l y ) , i s combined with conservation of potential v o r t i c i t y , Bernoulli and transport equations. The r e s u l t i n g nonlinear equations are expanded i n a Rossby number e=V/fX (where f i s the C o r i o l i s parameter and V i s the upstream boundary value of v e l o c i t y ) . The 0(1) and 0(e) equations are solved. F i r s t , they are s i m p l i f i e d v i a transformations of the transport stream-function variables: H lo=p 4 / 3 and H ' 1 =2p 1 / 3q. By — i i i — modifying the r e s u l t s of Bromwich's (1915) and Whipple's (1916) d i f f r a c t i o n theory, the 0(1) solution i s expressed i n a compact integral form, _ 2a sin(aQ) ,°° cos(krsinhu) cosh(au) du  P n 'o sinh 2(au) + sin 2(ae) The 0(e) contribution q i s calculated using an approximate Green's function method. The wedge of an angle 3rr/2 (a=2/-3) i s used as an example to show d e t a i l s of the solution. The r e s u l t s exhibit the r e l a t i v e importance of the c e n t r i f u g a l , C o r i o l i s and pressure gradient forces. Centrifugal upwelling (surfacing) of the interface occurs very close to the apex. For a rounded re—entrant corner, the upwelling i s important only i f the radius of curvature i s much smaller than the l a t e r a l scale X. liorever, for re—entrant corners, the flow i s s u p e r c r i t i c a l within an arc, whose s i z e depends upon the Rossby number and the angle of the wedge. Using two or more corner solutions, plausible flow streamlines can be generated i n more complicated domains, as long as no two corners are closer than the Rossby radius of deformation. This procedure i s i l l u s t r a t e d with two examples: (a) c i r c u l a t i o n in a channel mouth and (b) flow around a square bump in a coastline. F i n a l l y , b a r o c l i n i c c i r c u l a t i o n i s modeled for boundaries that approximate coastlines near the mouth of Hudson S t r a i t . — i v— TABLE QF CONTENTS page ABSTRACT i i LIST OF FIGURES v i ACKNOWLEDGEMENTS ix CHAPTER Is INTRODUCTION 1 CHAPTER 2: MODELS OF SEA STRAITS 8 CHAPTER 3: STATEMENT OF THE PROBLEM 15 CHAPTER 4: THE UPSTREAM PROFILE 19 CHAPTER 5: SOLUTIONS INSIDE A CHANNEL 24 CHAPTER 6: SCALING AND DERIVATION OF THE GOVERNING EQUATIONS 28 CHAPTER 7: ROTATING FLOW AROUND A SHARP BEND 35 THE 0(1) SOLUTION 36 THE 0(e) SOLUTION 42 THE VALIDITY OF THE SOLUTION 45 SUPERCRITICAL FLOW 46 SEPARATION 48 CHAPTER 8: THE CASE OF MORE COMPLICATED GEOMETRIES 64 CIRCULATION IN THE MOUTH OF A CHANNEL 64 FLOW AROUND A SQUARE BUMP 68 CHAPTER 9: BAROCLINIC CIRCULATION IN HUDSON STRAIT 76 THE TRANSPORTS 77 LATERAL SCALES 79 THE CIRCULATION 82 CHAPTER 10: DISCUSSION 93 —v— REFERENCES 98 APPENDIX A: DERIVATION OF THE 0(e) EQUATION FOR s t 102 APPENDIX B: BEHAVIOUR OF s Q FOR SMALL OR LARBE r 104 APPENDIX C: THE EXTENT OF THE CENTRIFUGAL UPWELLING 107 - v i — LIST QF FIGURES -figure # page 1 The western part of the sub—polar gyre of the 4 North A t l a n t i c . 2 Re-entrant c i r c u l a t i o n i n (a) Lancaster Sound 5 and (b> Hudson S t r a i t . 3 <a) Mean currents in Hudson S t r a i t i n Aug.—Oct., 6 1982. (b) Current t r a j e c t o r i e s near the mouth of Hudson S t r a i t at moorings 6 and 7. 4 M a r s i l i ' s (1681) experiment, designed to dup— 12 l i c a t e an estuarine—type c i r c u l a t i o n i n the S t r a i t of Bosphorus. 5 Temperature, s a l i n i t y and sigma—t section ac- 13 ross the mouth of Hudson S t r a i t . 6 Sample surface configuration from G i l l , 1977. 14 7 Schematic representation of a channel mouth. 18 8 3 p r o f i l e s of the upper layer depth: (a) l i - 23 near h=H(l-x/X), <b) exponential h=He-K/x and (c) exponential h=D+He-*'x. 9 P r o f i l e s of h inside a channel for (a) Q_=Q+ 27 and (b) Q_=2Q+, for several values of width, d. 10 The incoming current p r o f i l e as an evanescent 53 "wave" emanating from the boundary along 6=rr/a. D i f f r a c t i o n term 2F(V), with y=ir/a-e and a=2/3: (a) i t s functional behaviour, (b) as a rotating sink of a unit strength. (a) The 0(1) streamfunction s 0=p 2 / 3 and i t s homogeneous counterpart, s h. (b) Contours of M. (a) The ri g h t hand side of (6.23), f. (b) The 0(e) streamfunction variable S t . (C) The t o t a l transport streamfunction 4*=4,o+e4Ji , for e=0.5 (d) Comparison between f o (e=0) and 4* (e=0.5). (a) The 0(e) depth variable h t. (b> Contours and (c) 3—dimensional view of h=ho+eht (e=0.5). Depth integrated k i n e t i c energy hU: (a) e=0, (b) e=0.5. Contours of (a) r t , (b) r 2 and (c) F,. , . superimposed upon a few streamlines of H1 (e=0.5) Conceptual drawing of the boundary streamline f=l (a) just a f t e r separation and (b) afte r reatachment. h 0 inside a channel, compared to the exact solution, h, for several values of width, d: (a) A=0 and (b) A=l. - v i i i — •figure # page 19 C i r c u l a t i o n in the mouth of a channel: contours 72 of the transport streamfunction 4*0 for d=1.0, <i=0.1: (a) A=0, (b) A=l. 20 Location y 0 of the stagnation streamline f e ^ y o ) 73 as functions of A, for 3 values of width: 2d=l, 2d=2 and 2d=3. 21 (a)-(f) Construction of the solution for a flow 74 around a square bump i n a coastline. A square bump i n a channel for (g) A=0 and (h) A=l. 22 (a) The eastern Hudson S t r a i t and Ungava Bay 85 area, (b) Simplified model boundaries. 23 Temperature and s a l i n i t y sections in Ungava Bay. 86 24 An example of thermal surface signature of 88 c i r c u l a t i o n in Hudson S t r a i t and Ungava Bay. 25 Ice cover i n Ungava Bay and Hudson S t r a i t . 89 26 Temperature, s a l i n i t y and sigma—t transects 90 across Hudson S t r a i t , Aug. 1982. 27 Model contours of (a,c) h Q and (b,d) *¥0 for 91 (a,b) X=20km and (c,d) X=30km. 28 (a) An approximation s p and the exact s 0 for 106 3 small values of r. (b) S i and s 0 for large r. 29 Contours of M» and h (e=0.5) for (a) b=-2, 112 (b) b=-3 and (c) b=-4. ACKNOWLEDGEMENTS F i r s t , I extend my g r a t i t u d e t o my s u p e r v i s o r , Dr. Paul LeBlond, -for su g g e s t i n g t h i s i n t e r e s t i n g problem and -for h e l p i n g with i t s completion. I am a l s o g r a t e f u l t o a number of people at UBC and IOS f o r being p a t i e n t and h e l p f u l when I needed t h e i r a d v i c e . These i n c l u d e Drs. Andrew Bennett, Paul Budgel, David Farmer, Michael Forman, Howard F r e e l a n d , F a l c o n e r Henry, Humphrey M e l l i n g and Steve Pond. S p e c i a l thanks a re due t o Drs. Lawrence Mysak and Richard Thomson f o r t h e i r c o n s t r u c t i v e c r i t i q u e of t h i s t h e s i s . I a l s o thank Dr. John G a r r e t t , D i v i s i o n Head of Ocean P h y s i c s at IOS, and Dr. Ric h a r d Thomson f o r ar r a n g i n g my two year s t a y at the I n s t i t u t e (I admit, the view from the window i s superb). F i n a l l y , I h a p p i l y thank my w i f e , E s t h e r , f o r her l o v i n g support, cheer and encouragement and our c h i l d r e n , T a l i and Z e v i , f o r j u s t beeing here. - 1 -C h a p t e r 1  I n t r o d u c t i o n T h e r e h a v e b e e n a n i n c r e a s i n g number o-f o c e a n o g r a p h i c i n v e s t i g a t i o n s o f t h e C a n a d i a n A r c t i c i n t h e p a s t d e c a d e . T h i s i s m a i n l y d u e t o r e c e n t o i l e x p l o r a t i o n a c t i v i t i e s a n d i n c r e a s e d m a r i n e t r a f f i c i n t h e a r e a . D e t a i l s o f c i r c u l a t i o n p a t t e r n s i n A r c t i c s t r a i t s h a v e now b e g u n t o b e d e t e r m i n e d . W h i l e t h e w a t e r movement i n t h e A r c t i c A r c h i p e l a g o i s g e n e r a l l y f r o m n o r t h w e s t t o s o u t h e a s t ( e . g . , C o l l i n , 1 9 6 2 ) , many o f t h e s t r a i t s a r e w i d e e n o u g h t o a c c o m m o d a t e two b a r o c l i n i c c u r r e n t s i n o p p o s i t e d i r e c t i o n s ( L e B l o n d , 1 9 8 0 ) . B e c a u s e o f t h i s , c o a s t a l c u r r e n t s i n t h e a r e a t e n d t o l o o p i n a n d o u t o f c h a n n e l s . I n some i n s t a n c e s , a p o r t i o n o f t h e i n c o m i n g f l o w c o n t i n u e s e v e n f u r t h e r i n t o t h e c h a n n e l , a g a i n s t p r e v a i l i n g e a s t w a r d t r a n s p o r t s . Some e x a m p l e s o f t h e s e r e - e n t r a n t f l o w s may b e s e e n o n F i g u r e 1 , w h i c h s h o w s a p a r t o f t h e s u b — p o l a r g y r e o f t h e N o r t h A t l a n t i c a n d , i n p a r t i c u l a r , t h e movement o f B a f f i n C u r r e n t i n a n d o u t o f L a n c a s t e r S o u n d a n d H u d s o n S t r a i t . T h e u n d e r s t a n d i n g o f t h e p h y s i c a l p r o c e s s e s w h i c h g o v e r n t h e m o t i o n o f w a t e r m a s s e s i n t h e a r e a i s r e q u i r e d f o r t h e o r e t i c a l a n d p r e d i c t i v e m o d e l s o f (a) i c e b e r g s a n d s e a - i c e h a z a r d s t o d r i l l i n g a n d n a v i g a t i o n , (b) p o l l u t a n t t r a n s p o r t f r o m d r i l l i n g a n d s h i p p i n g s o u r c e s a n d (c) b i o l o g i c a l p r o d u c t i o n z o n e s . T h e e x i s t e n c e o f t h e o p p o s i n g c u r r e n t s i n a r c t i c c h a n n e l s was e x p l a i n e d b y L e B l o n d (1980) i n t e r m s o f s i m p l e g e o s t r o p h i c - 2 -d y n a m i c s o f u p p e r l a y e r wedge—type b o u n d a r y f l o w s . H e r e , I c o n c e n t r a t e m a i n l y o n t h e d e t a i l s o f b a r o c l i n i c c u r r e n t s t u r n i n g v a r i o u s c o r n e r s , a s t h e y l o o p i n a n d o u t o f c h a n n e l s ( s u c h a s L a n c a s t e r S o u n d a n d H u d s o n S t r a i t ) , o r n a v i g a t e c o a s t a l p r o t r u s i o n s . T h i s t o p i c o f r o t a t i n g f l o w s a r o u n d s h a r p c o r n e r s , a n d i t s r e l a t i o n t o t h e c i r c u l a t i o n i n a n d o u t o f s e a s t r a i t s , h a s n o t y e t b e e n e x a m i n e d i n d e t a i l . T h e f o r m u l a t i o n i s s i m i l a r t o t h a t u s e d b y W h i t e h e a d e t a l . ( 1 9 7 4 ) , G i l l ( 1 9 7 7 ) , N o f ( 1 9 7 8 a , b ) a n d o t h e r s , t o s t u d y t h e r o t a t i o n a l h y d r a u l i c s o f c h a n n e l f l o w s , b u t i t i s n o t r e s t r i c t e d t o s l o w l y v a r y i n g c o a s t l i n e s . T h e f u l l t w o — d i m e n s i o n a l n o n l i n e a r p r o b l e m i s t r e a t e d h e r e . I n c o n t r a s t t o p r e v i o u s i n v e s t i g a t i o n s , t h e m o d e l p o t e n t i a l v o r t i c i t y i s n o t a s s u m e d t o b e u n i f o r m . I n s t e a d , i t i s d e r i v e d f r o m a c h o s e n u p s t r e a m p r o f i l e . T h e r e — e n t r a n t f l o w s i n L a n c a s t e r S o u n d a n d i n H u d s o n S t r a i t w e r e o b s e r v e d a n d m e a s u r e d v i a t h e d i s t r i b u t i o n o f w a t e r p r o p e r t i e s ( C a m p b e l l , 1958 ; O s b o r n e t a l . , 1978 ; F i s s e l e t a l . , 1 9 8 2 ) , f l o w m e a s u r e m e n t s f r o m d r i f t i n g a n d m o o r e d i n s t r u m e n t s ( L e B l o n d e t a l . , 1981 ; F i s s e l e t a l . , 1982 ; D r i n k w a t e r , 1 9 8 5 ) , a n d b y i c e b e r g d r i f t o b s e r v a t i o n s ( M a r k o e t a l . , 1 9 8 2 ) . S u r f a c e d r o g u e t r a c k s i n t h e m o u t h s o f L a n c a s t e r S o u n d a n d H u d s o n S t r a i t a r e shown o n F i g u r e 2 ( f r o m F i s s e l e t a l . , 1982 ; L e B l o n d e t a l . , 1 9 8 1 ) . Among t h e many d r o g u e s t h a t w e r e r e l e a s e d n e a r t h e e n t r a n c e s o f t h e s e two c h a n n e l s , n o t a s i n g l e o n e was o b s e r v e d t o p e n e t r a t e much f u r t h e r t h a n a d i s t a n c e c o m p a r a b l e t o t h e w i d t h o f t h e c h a n n e l . On t h e o t h e r h a n d , c u r r e n t m e t e r r e c o r d s - 3 -( F i g u r e s 3 a , b , f r o m D r i n k w a t e r , 1985 a n d L e B l o n d , e t a l . , 1 9 8 1 ) , i c e b e r g s i g h t i n g s ( S m i t h , 1 9 3 1 ) , g e o s t r o p h i c c a l c u l a t i o n s ( C a m p b e l l , 1958 ; D r i n k w a t e r , 1985) a n d d r o g u e s r e l e a s e d i n s i d e t h e c h a n n e l s ( F i s s e l a n d M a r k o , 1978) , a l l show t h e p r e s e n c e o-f o p p o s i n g c u r r e n t s much f a r t h e r i n s i d e t h e s e c h a n n e l s . T h i s a p p a r e n t c o n t r a d i c t i o n w i l l a l s o b e a d d r e s s e d . T h e p l a n o f t h i s t h e s i s i s a s f a l l o w s : c e r t a i n r e l e v a n t t h e o r e t i c a l m o d e l s o f f l o w s i n s t r a i t s a r e r e v i e w e d i n C h a p t e r 2 . C h a p t e r 3 c o n t a i n s t h e f o r m u l a t i o n o f t h e p r o b l e m o f b a r o c l i n i c c i r c u l a t i o n n e a r a c h a n n e l m o u t h . C h a p t e r 4 i n c l u d e s d i s c u s s i o n a b o u t t h e u p s t r e a m c o n d i t i o n s . O n e - d i m e n s i o n a l s o l u t i o n s f a r i n s i d e a c h a n n e l a r e d e r i v e d i n C h a p t e r 5 . T h e g o v e r n i n g e q u a t i o n s f o r a t w o — d i m e n s i o n a l r e d u c e d g r a v i t y f l o w a r o u n d a n a r b i t r a r y c o r n e r a r e d e r i v e d i n C h a p t e r 6 v i a a r e g u l a r p e r t u r b a t i o n e x p a n s i o n i n a R o s s b y number e . T h e 0 ( 1 ) a n d 0 ( e ) s o l u t i o n s a r e c a l c u l a t e d i n C h a p t e r 7 . T h e r e — e n t r a n t wedge o f a n a n g l e 3TT/2 i s u s e d a s a n e x a m p l e t o show t h e d e t a i l s o f t h e s o l u t i o n . I n C h a p t e r 8 , t h e m o d e l i s e x t e n d e d t o m o r e c o m p l i c a t e d g e o m e t r i e s , s u c h a s c h a n n e l m o u t h s a n d bumps i n c o a s t l i n e s , w h i l e i n C h a p t e r 9 , t h e m o d e l i s u s e d t o c o n s t r u c t a b a r o c l i n i c c i r c u l a t i o n i n t h e mouth o f H u d s o n S t r a i t . F i n a l l y , an o v e r v i e w o f t h e r e s u l t s a n d s u g g e s t i o n s f o r f u r t h e r work a r e p r e s e n t e d i n t h e l a s t C h a p t e r . i g u r e 1. T h e w e s t e r n p a r t o f t h e s u b - p o l a r g y r e o f t h e N o r t h A t l a n t i c ( a d a p t e d f r o m F e n c o a n d S l a n e y , 1 9 7 8 ) . F i g u r e 2 . R e e n t r a n t c i r c u l a t i o n i n (a) L a n c a s t e r S o u n d ( r e p r o d u c e d w i t h p e r m i s s i o n f r o m F i s s e l e t a l . , 1982) a n d (b) H u d s o n S t r a i t ( r e p r o d u c e d w i t h p e r m i s s i o n f r o m L e B l o n d e t a l . , 1 9 8 1 ) , a s e x h i b i t e d b y n e a r — s u r f a c e d r o g u e t r a c k s . i g u r e 3 a . U p p e r p a n e l : mean c u r r e n t s i n H u d s o n S t r a i t d u r i n g A u g . - O c t . 19B2 . L o w e r p a n e l : m o o r i n g s H S 1 - H S 5 p r o f i l e s ( r e p r o d u c e d w i t h p e r m i s s i o n f r o m D r i n k w a t e r , 1 9 8 5 ) . - 7 -F i g u r e 3b. L o w e r p a n e l : c u r r e n t t r a j e c t o r i e s n e a r t h e mouth o f H u d s o n S t r a i t a t m o o r i n g s 6 a n d 7 , whose l o c a t i o n i s shown o n t h e map i n t h e u p p e r p a n e l ( r e p r o d u c e d w i t h p e r m i s s i o n f r o m L e B l o n d e t a l . , 1981 ; CHA - C a p e H o p e ' s A d v a n c e , A l - A k p a t o k I s l a n d ) . C h a p t e r 2 M o d e l s o-f s e a s t r a i t s T h e e a r l y t h e o r y o-f w a t e r m o v e m e n t s t h r o u g h s e a s t r a i t s i s g i v e n i n c h a p t e r 5 i n G i l l ( 1 9 8 2 ) . M a r s i l i (1681) u s e d a t a u t r o p e t o m e a s u r e c u r r e n t s a t v a r i o u s d e p t h s i n t h e S t r a i t o-f B o s p h o r u s . He f o u n d t h a t b e l o w a c e r t a i n d e p t h t h e w a t e r moves i n o p p o s i t e d i r e c t i o n t o t h e s u r f a c e c u r r e n t s . He t h e n w e i g h e d w a t e r s a m p l e s f r o m t h e B l a c k S e a a n d f r o m t h e M e d i t e r r a n e a n a n d f o u n d t h a t t h e B l a c k S e a w a t e r i s l i g h t e r . F r o m t h i s , h e d e d u c e d t h e c l a s s i c a l p i c t u r e o f a t w o - l a y e r e s t u a r i n e e x c h a n g e b e t w e e n t w o s e a s . N o t y e t s a t i s f i e d , h e a l s o p e r f o r m e d a n e x p e r i m e n t ( f i g u r e 4 ) , i n w h i c h a p a r t i t i o n s e p a r a t e d t h e d y e d u n d e r c u r r e n t w a t e r f r o m t h e l i g h t e r s u r f a c e w a t e r . Two h o l e s , o n e n e a r t h e t o p a n d t h e o t h e r n e a r t h e b o t t o m , w e r e o p e n e d a n d t h e r e s u l t i n g f l o w r e p r o d u c e d t h e c u r r e n t s i n t h e B o s p h o r u s . T h e S t r a i t o f B o s p h o r u s i s n a r r o w a n d h e n c e t h e c r o s s - c h a n n e l i n t e r f a c e s l o p e , r e s u l t i n g f r o m t h e e a r t h ' s r o t a t i o n , i s n o t i m p o r t a n t . On t h e o t h e r h a n d , when a c h a n n e l i s w i d e , t h e s l o p i n g i n t e r f a c e e v e n t u a l l y m e e t s t h e s u r f a c e . T h e o u t g o i n g l i g h t w a t e r f o r m s a c u r r e n t wedge o n t h e r i g h t s i d e ( l o o k i n g d o w n s t r e a m ) , w h i l e t h e i n c o m i n g d e n s e r f l o w i s now m a i n l y c o n c e n t r a t e d o n t h e o p p o s i t e s i d e . A g o o d e x a m p l e o f a w i d e c h a n n e l e s t u a r i n e e x c h a n g e i s t h a t o f H u d s o n S t r a i t . F i g u r e 5 s h o w s t e m p e r a t u r e s a l i n i t y a n d d e n s i t y ( s i g m a - t ) s e c t i o n s a c r o s s t h e m o u t h o f H u d s o n S t r a i t ( f r o m M a c L a r e n A t l a n t i c , 1 9 7 7 , - 9 -r e p r o d u c e d -from O s b o r n e t a l . , 1 9 7 9 ) . N o t e t h a t i n a d d i t i o n t o t h e l i g h t w a t e r wedge n e a r t h e s o u t h s i d e ( s e e t h e s i g m a - t s e c t i o n o n - f i g u r e 5) , t h e r e i s a l s o a s m a l l e r o n e o n t h e n o r t h s i d e . T h e t h e o r y o-f w a t e r e x c h a n g e t h r o u g h s t r a i t s i n a r o t a t i n g f r a m e o f r e f e r e n c e i s q u i t e r e c e n t . A s e m i n a l a n d o f t e n q u o t e d work i s t h a t o f W h i t e h e a d e t a l . ( 1 9 7 4 ) , who c a r r i e d o u t a t h e o r e t i c a l a n d l a b o r a t o r y s t u d y o f a t w o - l a y e r r o t a t i n g c h a n n e l f l o w u n d e r g e o s t r o p h i c b a l a n c e . T h e y s o l v e d t h e p r o b l e m o f a) a b o t t o m l a y e r f l o w a n d b) t w o - l a y e r e d f l o w s i n o p p o s i t e d i r e c t i o n s o v e r a w e i r . S i n c e t h e y a s s u m e d a n i n f i n i t e b o t t o m d e p t h i n t h e u p s t r e a m b a s i n , t h e i r p o t e n t i a l v o r t i c i t y f u n c t i o n was i d e n t i c a l l y z e r o a n d t h e B e r n o u l l i f u n c t i o n a c o n s t a n t . T h e s e a s s u m p t i o n s l e d t o a l i n e a r g e o s t r o p h i c v e l o c i t y p r o f i l e a n d p a r a b o l i c s h a p e f o r t h e i n t e r f a c e . F o r s m a l l u p s t r e a m h e i g h t , t h e y f o u n d t h a t t h e i n t e r f a c e i n t e r c e p t s t h e c h a n n e l f l o o r a t some d i s t a n c e f r o m s h o r e . B e c a u s e o f t h e a s s u m p t i o n s i n v o l v e d ( o n e - d i m e n s i o n a l f l o w , i n f i n i t e d e p t h u p s t r e a m ) , t h e i r m o d e l i s i n a p p r o p r i a t e f o r t h e p r e s e n t s t u d y , a l t h o u g h i t p r o v i d e s a u s e f u l g u i d e f o r u n d e r s t a n d i n g t h e p h y s i c s o f t h e s i t u a t i o n . N o f ( 1 9 7 8 a , b ) , i n h i s s t u d i e s o f o u f l o w s f r o m c h a n n e l s i n t o w i d e r b a s i n s , e x p a n d e d t h e p o t e n t i a l v o r t i c i t y e q u a t i o n a n d t h e B e r n o u l l i e q u a t i o n , i n p o w e r s o f t h e R o s s b y number e = V / 2 f b ( w h e r e f i s t h e C o r i o l i s p a r a m e t e r , V i s t h e v e l o c i t y s c a l e a n d b i s t h e h a l f - w i d t h o f t h e c h a n n e l ) f o r t h e o n e - l a y e r c a s e - 10 -< 1 9 7 8 a ) , a n d i n p o w e r s o-f t h e F r o u d e number F = V 2 / g ' H ( w h e r e H i s t h e u p p e r l a y e r d e p t h i n t h e c h a n n e l ) i n h i s t w o — l a y e r m o d e l ( 1 9 7 8 b ) . T h e i m p o r t a n t a s s u m p t i o n -for t h e l a t t e r c a s e was t h a t F / e < < l ( i . e . , a n a r r o w c h a n n e l ) . He u s e d t h e s e e x p a n s i o n s t o s o l v e t h e one— a n d t w o - l a y e r p r o b l e m s o-f t h e g e o s t r o p h i c a d j u s t m e n t i n o u t - f l o w s o v e r a s t e p f r o m a c h a n n e l i n t o a w i d e r b a s i n . He -found t h a t t h e f l o w s e p a r a t e s f r o m t h e r i g h t o r f r o m t h e l e f t ( d e p e n d i n g o n s tep—up o r step—down i n t h e o n e - l a y e r c a s e a n d o n t h e l e n g t h o f t h e c h a n n e l i n h i s t w o — l a y e r m o d e l ) i f t h e R o s s b y n u m b e r i s s m a l l e r t h a n some c r i t i c a l v a l u e . S i n c e v e l o c i t y p r o f i l e s i n t h e m o u t h o f t h e c h a n n e l w e r e p r e s c r i b e d a p r i o r i , h i s s o l u t i o n s a r e n o t d i r e c t l y a p p l i c a b l e t o t h e p r e s e n t s t u d y . G i l l (1977) was t h e f i r s t t o 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 f o r t h e f l o w o f a h o m o g e n e o u s f l u i d w i t h a n o n z e r o p o t e n t i a l v o r t i c i t y ( i . e . , a f i n i t e d e p t h u p s t r e a m ) down a r o t a t i n g c h a n n e l w i t h a s i l l a n d a s l o w l y v a r y i n g c r o s s — s e c t i o n . He t h e n d e t e r m i n e d t h e p o s i t i o n o f a " c o n t r o l " s e c t i o n , w h e r e t h e long—wave d i s t u r b a n c e s h a v e a z e r o p h a s e s p e e d . F o r c e r t a i n s i l l h e i g h t s a n d u p s t r e a m p o t e n t i a l v o r t i c i t y v a l u e s , h e f o u n d t h a t a p a r t o f t h e f l o w i s r e v e r s e d b e f o r e t h e s i l l , i . e . , a r e — e n t r a n t c i r c u l a t i o n p a t t e r n was e s t a b l i s h e d ( f i g u r e 6 ) . G i l l ' s (1977) m o d e l i s c o n c e r n e d m a i n l y w i t h t h e h y d r a u l i c c o n t r o l o f a d o w n c h a n n e l f l o w b y a s i l l a n d a g r a d u a l l y n a r r o w i n g p r o f i l e , w h i c h d o e s n o t i n c l u d e f l o w s a r o u n d s h a r p c o r n e r s a n d i n c h a n n e l m o u t h s . I n some r e s p e c t s , t h e p r e s e n t work may b e c o m p l e m e n t a r y t o t h a t o f G i l l ( 1 9 7 7 ) , s i n c e h i s s o l u t i o n s f o r a s l o w l y v a r y i n g c h a n n e l c a n b e m a t c h e d t o (and p r e s c r i b e t h e r e l a t i v e t r a n s p o r t s f o r ) t h e m o d e l o f c i r c u l a t i o n i n t h e mouth o f a c h a n n e l . R o e d (1980) p e r f o r m e d a s i m i l a r a n a l y s i s f o r t h e c a s e o f a s i n g l e c o a s t l i n e w i t h a s l o w l y v a r y i n g c u r v a t u r e a n d d e p t h p r o f i l e . I am a w a r e o f o n l y o n e e x a m p l e o f a n a n a l y t i c a l s o l u t i o n f o r t h e f l o w a r o u n d a f a s t c h a n g i n g c o a s t l i n e . H u g h e s ( 1 9 8 1 , 1982) h a s m o d e l e d an u p p e r l a y e r f l o w w i t h s e p a r a t i o n ( a n d a d o w n s t r e a m c o n t r o l b y a n o t h e r c o a s t l i n e ) a r o u n d a s h a r p c o r n e r o n a l o w - l a t i t u d e f - p l a n e ; i . e . , i n t h e l i m i t o f s l o w r o t a t i o n . I n t h e p r e s e n t c a s e , I a t t e m p t t o d o t h e s a m e , b u t f o r t h e c a s e o f a m i d - t o h i g h - l a t i t u d e f - p l a n e , w h e r e t h e s t r o n g e r C o r i o l i s f o r c e t e n d s t o k e e p t h e b o u n d a r y c u r r e n t a t t a c h e d t o t h e c o a s t o n i t s r i g h t , e v e n a r o u n d c o r n e r s w i t h s i g n i f i c a n t c u r v a t u r e s . - 12 F i g u r e 4. M a r s i l i ' s (1681) e x p e r i m e n t t h a t a n e s t u a r i n e — t y p e c i r c u l a t i o n i n ( r e p r o d u c e d w i t h p e r m i s s i o n -from was d e s i g n e d t o m o d e l t h e S t r a i t o-f B o s p h o r u s . G i l l , 1 9 8 2 ) . - 13 -F i g u r e 5 . T e m p e r a t u r e , s a l i n i t y a n d s i g m a - t s e c t i o n a c r o s s t h e m o u t h o-f H u d s o n S t r a i t (-from M a c L a r e n A t l a n t i c , 1 9 7 7 , a s r e p r o d u c e d i n O s b o r n e t a l . , 1 9 7 9 ) . S e e t h e u p p e r p a n e l i n F i g u r e 3b - for t h e l o c a t i o n o-f R e s o l u t i o n I s l a n d . - 14 -F i g u r e 6 . S a m p l e s u r f a c e c o n f i g u r a t i o n ( e x a g g e r a t e d s c a l e ) f r o m G i l l , 1977 ( w i t h p e r m i s s i o n ) . D a w n c h a n n e l i s a t t h e b o t t o m . N o t e how t h e s u r f a c e s l o p e s o n o p p o s i t e s i d e s b e f o r e t h e c h a n n e l n a r r o w s , i n d i c a t i n g g e o s t r o p h i c f l o w a l o n g d i r e c t i o n s shown b y t h e a r r o w s . - 15 -C h a p t e r 5 S t a t e m e n t o-f t h e p r o b l e m I n t h i s c h a p t e r , we p r e s e n t o u r p r o b l e m i n a g e n e r a l f o r m f o r a n a r b i t r a r y d o m a i n . M o r e s p e c i f i c e x a m p l e s a r e g i v e n i n s u b s e q u e n t c h a p t e r s . T h e a n a l y s i s o f s t e a d y i n v i s c i d c o a s t a l f l o w s i s b a s e d o n t h e c o n s e r v a t i o n p r i n c i p l e s t h a t g o v e r n t h e m o t i o n o f a n i n c o m p r e s s i b l e f l u i d i n a r o t a t i n g f r a m e o f r e f e r e n c e . T h e f i r s t two e q u a t i o n s e x p r e s s c o n s e r v a t i o n o f e n e r g y a n d v o r t i c i t y . T h e s e a r e t h e B e r n o u l l i e q u a t i o n ( w h e r e g* = ( A 5 > / ° ) g i s t h e r e d u c e d g r a v i t y ) a n d t h e p o t e n t i a l v o r t i c i t y e q u a t i o n T h e i r d e r i v a t i o n c a n b e f o u n d i n s t a n d a r d t e x t b o o k s ( e . g . , G u t m a n , 1972 ; P e d l o s k y , 1 9 7 9 ) . T h e t r a n s p o r t s t r e a m f u n c t i o n V i s r e l a t e d t o t h e d e p t h o f t h e i n t e r f a c e h a n d t h e h o r i z o n t a l v e l o c i t y c o m p o n e n t s u a n d v t h r o u g h ( u 2 + v 2 ) / 2 + g ' h = G ( 4 « ) , ( 3 . 1 ) ( v » - u y + f ) / h = K C r ) . ( 3 . 2 ) h v = 4-,, a n d - h u = y • ( 3 . 3 ) T h e l a t t e r two e q u a t i o n s f o l l o w f r o m t h e v o l u m e c o n s e r v a t i o n - 16 -p r i n c i p l e f o r a n o n d i v e r g e n t m o t i o n ( h u ) , + ( h v ) y = 0 . ( 3 . 4 ) G(M»> a n d K("r)=G'("r) ( C h a r n e y , 1 9 5 5 ; G u t m a n , 1972) a r e f u n c t i o n s o f i n t e g r a t i o n t h a t a r e d e r i v e d f r o m t h e u p s t r e a m c o n d i t i o n s . T h i s d e r i v a t i o n i s d o n e i n t h e n e x t c h a p t e r f o r a p a r t i c u l a r c a s e o f a n e x p o n e n t i a l wedge p r o f i l e . T h e b o u n d a r y c o n d i t i o n a t t h e c o a s t i s s p e c i f i e d b y a s s u m i n g i t t o b e a s t r e a m l i n e . F o r e x a m p l e , f-or t h e c a s e o f a c h a n n e l m o u t h ( f i g u r e 7) , we n e e d t o s p e c i f y t h e v a l u e o f 4* o n e a c h b o u n d a r y . T h e d i f f e r e n c e b e t w e e n t h e s e two b o u n d a r y v a l u e s i s t h e n e t t r a n s p o r t o u t o f ( o r i n t o ) t h e c h a n n e l a n d i s d e t e r m i n e d b y c o n d i t i o n s f a r i n s i d e t h e c h a n n e l . M o r e o v e r , i f a c h a n n e l h a s a s l o w l y v a r y i n g c r o s s — s e c t i o n a n d a s i l l ( w e i r ) i n s i d e , t h e n G i l l ' s (1977) m o d e l c a n b e u s e d t o f i n d t h e n e t t r a n s p o r t . On t h e o t h e r h a n d , i f t h e c h a n n e l i s w i d e ( a n d / o r t h e s i l l i s t o o f a r i n s i d e , a s i n t h e c a s e o f L a n c a s t e r S o u n d ) t h e n e t b a r o c l i n i c t r a n s p o r t i s a p a r a m e t e r o f t h e p r o b l e m . T h i s i s d i s c u s s e d i n d e t a i l i n C h a p t e r 5 , w h e r e o n e — d i m e n s i o n a l s o l u t i o n s f o r a c h a n n e l a r e g i v e n . T h e p r o b l e m must a l s o b e w e l l - p o s e d . T h i s i s t r u e a s l o n g a s t h e f l o w d o e s n o t s e p a r a t e f r o m a b o u n d a r y . I n t h i s i n v i s c i d c a s e , t h e s e p a r a t i o n may b e c a u s e d e i t h e r b y a d v e r s e p r e s s u r e g r a d i e n t s o r b y a f l o w a r o u n d a v e r y s h a r p r e - e n t r a n t c o r n e r . I n t h e l a t t e r c a s e , t h e s t r o n g c e n t r i f u g a l f o r c e w i l l move t h e - 17 -- f l u i d away -from t h e c o r n e r a n d , a c c o r d i n g t o ( 3 . 1 ) , t h e s e p a r a t i o n w i l l r e s u l t f r o m a v a n i s h i n g u p p e r l a y e r d e p t h . A s we s h a l l s e e i n C h a p t e r 7 ( a n d A p p e n d i x C ) , i n t h e c a s e o f a r i g h t - b o u n d e d wedge—type f l o w , t h i s c e n t r i f u g a l s e p a r a t i o n o c c u r s o n l y a t r e - e n t r a n t c o r n e r s t h a t h a v e r a d i i o f c u r v a t u r e much s m a l l e r t h a n t h e l a t e r a l s c a l e o f t h e f l o w . When t h i s h a p p e n s , t h e p r o b l e m i s n o t w e l l — p o s e d a n d t h e s o l u t i o n s , t h a t a r e d e r i v e d b e l o w , a r e n o l o n g e r v a l i d . - IB -y 1 1 f » I 1 1 I X 1 1 1 F i g u r e 7 S c h e m a t i c r e p r e s e n t a t i o n o-f a c h a n n e l m o u t h . I t i s a s s u m e d t h a t t h e p r o b l e m o-f t h e c i r c u l a t i o n i n t h e m o u t h ( r e g i o n I) c a n b e s e p a r a t e d f r o m t h a t i n s i d e t h e c h a n n e l ( r e g i o n I I ) . - 1 9 -C h a p t e r 4 T h e u p s t r e a m p r o f i l e L e B l o n d (1980) s h o w e d t h a t when a c o a s t a l c u r r e n t i s i n a g e o s t r o p h i c b a l a n c e w i t h t h e p r e s s u r e g r a d i e n t d u e t o a l i n e a r l y s l o p i n g i n t e r f a c e ( F i g u r e 8 a ) , t h e n t h e i n t e r f a c e m e e t s t h e s u r f a c e ( f o r m i n g a f r o n t ) a t . a d i s t a n c e X , g i v e n b y H e r e , R = C / f i s t h e R o s s b y i n t e r n a l d e f o r m a t i o n r a d i u s , F = V / C i s t h e F r o u d e number b a s e d o n t h e v e l o c i t y C = ( g ' H ) 1 / 2 o f l o n g i n t e r n a l g r a v i t y w a v e s , a n d V a n d H a r e , r e s p e c t i v e l y , t h e f l o w s p e e d a n d t h e u p p e r l a y e r t h i c k n e s s a t t h e c o a s t a l b o u n d a r y . I n o r d e r t o a l l o w f o r a s m o o t h t r a n s i t i o n b e t w e e n t h e c o a s t a l c u r r e n t a n d t h e o f f s h o r e r e g i o n we a s s u m e t h a t t h e i n t e r f a c e d e p t h d e c r e a s e s e x p o n e n t i a l l y o f f s h o r e ( t h e r e f o r e e l i m i n a t i n g f r o n t s ; s e e F i g u r e 8b) w i t h t h e same s c a l e X . L e t t i n g k = l / X , we h a v e I f we a s s u m e t h a t t h e v e l o c i t y i s g e o s t r o p h i c a l l y b a l a n c e d , we a l s o h a v e X = R / F = C z / f V ( 4 . 1) h ( x ) = H e - " - . ( 4 . 2 ) v ( x ) = g ' h , , / f = - V e - " " , ( 4 . 3 ) -20-where V=g'H/fX (=FC) i f we use (4.1). When the f i r s t equation i n (3.3) i s integrated with respect to x, we f i n d that the transport streamfunction i s given by • M»(x) = Qe-2k* , (4.4) where the constant of integration was set to zero and Q=H«(0)=VHX/2. Note that Q i s also the t o t a l transport i n a triangular p r o f i l e (Figure 8a), which lends support to using X as the horizontal length scale. It i s worthwhile to note that for an exponential depth p r o f i l e , X remains invariant ( i . e . , i t does not change with the distance offshore) i f we replace C and F with t h e i r l o c a l values c (x) =Cg'h (x) 3 1 / 2 and F (x ) = I v (x ) I/c (x ) . In order to see the connection between the decay scale X and the Rossby radius of deformation, used by many authors as a l a t e r a l scale for coastal flows (e.g., Nof 1979a,b; Stommel and Luyten, 1984), we consider (for a moment) a current thickness p r o f i l e that i s given by h(x) = D + He""" , (4.5) where D i s some nonzero constant reference depth (Figure 8c). If we now assume a uniform potential v o r t i c i t y , then, for the one—dimensional case, equation (3.2) becomes (v* + f ) / h = f / D . ( 4 . 6 ) B u t , v M = k V e - k K a n d k V = F 2 f , s o t h a t f r o m ( 4 . 6 ) we g e t ( f / D ) ( l + F 2 e - k " ) / ( l + H e - k " / D ) = f / D ( 4 . 7 ) w h i c h g i v e s H / D = F 2 . ( 4 . 8 ) T h u s , X = R / F = ( g ' D ) 1 / 2 / f = R D I ( 4 . 9 ) w h i c h means t h a t i n t h i s c a s e (when D?=0), X is t h e R o s s b y r a d i u s b a s e d o n t h e r e f e r e n c e d e p t h D . We n o t e t h a t f o r F < 1 , H<D, s o t h a t f o r a v a n i s h i n g D ( F i g u r e 8b) t h e u n i f o r m p o t e n t i a l v o r t i c i t y e q u a t i o n ( 4 . 6 ) i s n o t a p p l i c a b l e , RD i s m e a n i n g l e s s a n d X a s d e f i n e d b y ( 4 . 1 ) i s t h e n a t u r a l d e c a y s c a l e . By d e f i n i n g a p o t e n t i a l d e p t h H P = H / F 2 , we g e t a m o d i f i e d e x p r e s s i o n , X = ( g ' H p ) 1 / 2 / f , s o t h a t i n a w i d e r s e n s e , we c a n c a l l X a R o s s b y d e f o r m a t i o n r a d i u s ( b a s e d o n H p ) . T h e u s e o f t h e " p o t e n t i a l d e p t h " i n c o n s t a n t p o t e n t i a l v o r t i c i t y m o d e l s , t h o u g h n o t a l w a y s s o n a m e d , i s n o t new ( e . g . , S t o m m e l , 1 9 6 5 ; F l i e r l , 1 9 7 9 ; G i l l a n d S c h u m a n n , 1979 ; L u y t e n a n d S t o m m e l , 1 9 8 4 ) . We now u s e ( 4 . 2 ) - ( 4 . 4 ) i n t h e o n e — d i m e n s i o n a l v e r s i o n s - 2 2 -o f ( 3 . 1 ) a n d ( 3 . 2 ) a n d o b t a i n G(H») = ( g ' f ) l / 2 ( 2 H ' ) 1 / 2 + e ( f / H ) H » , ( 4 . 1 0 ) a n d , K(4*) = ( g ' f ) 1 / 2 ( 2 M J ) - l / 2 + e ( f / H ) , ( 4 . 1 1 ) w h e r e t h e R o s s b y number e i s g i v e n b y e = F 2 = V 2 / ( g ' H ) = V / ( f X ) . ( 4 . 1 2 ) When t h e s e a r e c o m b i n e d w i t h ( 3 . 1 ) a n d ( 3 . 2 ) we g e t a p a i r o f n o n l i n e a r e q u a t i o n s w h i c h , t o g e t h e r w i t h ( 3 . 3 ) a n d a p p r o p r i a t e b o u n d a r y c o n d i t i o n s ( t o b e s p e c i f i e d l a t e r ) , d e f i n e t h e p h y s i c a l p r o b l e m o f i n v i s c i d r o t a t i o n a l f l o w a l o n g a c o a s t . - 2 3 -(a) (b) (c) F i g u r e 8. 3 p r o - F i l e s o-f t h e u p p e r l a y e r d e p t h s h = H ( l - x / X ) , <b> e x p o n e n t i a l h = H e - " / x , <c> e x p o n e n t i a l h = D + H e - " / x . (a) l i n e a r a n d - 2 4 -C h a p t e r 5 S o l u t i o n s i n s i d e a c h a n n e l F o r s i m p l i c i t y , l e t u s a s s u m e i n i t i a l l y t h a t , a s shown o n F i g u r e 7 , t h e c h a n n e l i s o f u n i f o r m w i d t h 2 d . F o r s t r i c t l y a l o n g - c h a n n e l f l o w , t h e c r o s s - c h a n n e l v e l o c i t y v a n d a l l t h e x - d e r i v a t i v e s a r e z e r o . T h e r e f o r e , t h e momentum e q u a t i o n s r e d u c e t o t h e g e o s t r o p h i c r e l a t i o n - f u = g ' h y . ( 5 . 1 ) When c o m b i n e d w i t h ( 3 . 3 ) a n d i n t e g r a t e d o n c e , we f i n d t h a t h 2 = 2 f 4 V g ' , ( 5 . 2 ) i . e . , h i s a l s o a s t r e a m l i n e . F o r a n e x p o n e n t i a l u p s t r e a m p r o f i l e , ( 4 . 1 1 ) g i v e s t h e p o t e n t i a l v o r t i c i t y f u n c t i o n i n t e r m s o f h , K(h) = f / h + e f / H . ( 5 . 3 ) When t h e l a s t e q u a t i o n i s s u b s t i t u t e d i n t o ( 3 . 2 ) , we g e t , a f t e r u s i n g ( 5 . 1 ) , a l i n e a r e q u a t i o n f o r h - 2 5 -H e r e , a s i n C h a p t e r 4 , k = l / X . T h e b o u n d a r y v a l u e s o f h a r e h ( ± d ) = < 2 f Q ± / g * ) 1 / a , ( 5 . 5 ) w h e r e Q ± = H J ( ± d ) a r e t h e t r a n s p o r t s t r e a m f u n c t i o n v a l u e s o n e a c h b o u n d a r y . / F o r c o m p u t a t i o n a l c o n v e n i e n c e , i t was t a c i t l y a s s u m e d h e r e t h a t X , a n d h e n c e a l s o k , i n ( 5 . 4 ) , a r e u n i f o r m a c r o s s t h e c h a n n e l . T h i s i s t h e s i t u a t i o n w h i c h c o r r e s p o n d s t o a u n i f o r m s t r a t i f i c a t i o n . T h e s o l u t i o n o f ( 5 . 4 ) a n d ( 5 . 5 ) i s h ( y ) = h c o s h ( k y ) / c o s h ( k d ) - ftsinh(ky)/sinh(kd), ( 5 . 6 ) w h e r e , h = C h ( - d ) + h ( + d ) 3 / 2 a n d fi = C h ( - d ) - h ( + d ) 3 / 2 . ( 5 . 7 ) F i n a l l y , M1 a n d u c a n b e c a l c u l a t e d u s i n g ( 5 . 1 ) a n d ( 5 . 2 ) . F i g u r e s 9 a , b show p r o f i l e s o f h f o r s e v e r a l v a l u e s o f d / X when Q ± a r e e q u a l ( F i g u r e 9 a , t h e c a s e o f z e r o n e t t r a n s p o r t ) a n d when Q_=2Q+ ( F i g u r e 9 b ) . F o r s m a l l d / X , t h e t w o b o u n d a r i e s i n f l u e n c e e a c h o t h e r s t r o n g l y , d i c t a t i n g e i t h e r a z e r o f l o w c o n d i t i o n ( f o r Q_=Q+) o r u n i d i r e c t i o n a l d o w n — c h a n n e l f l o w ( f o r Q_=2Q+). A s d / X i n c r e a s e s , t h e two s i d e s b e c o m e d e c o u p l e d , a l l o w i n g i n d e p e n d e n t e x i s t e n c e o f c o a s t a l c u r r e n t s o n e a c h s i d e a n d , h e n c e t h e p o s s i b i l i t y o f p e n e t r a t i o n o f f l o w s i n t o t h e c h a n n e l a t t h e m o u t h . E q u a t i o n ( 5 . 6 ) i s t h e same a s G i l l ' s (1977) ( 5 . 2 ) . I t may - 2 6 -o f t e n b e v a l i d t o a s s u m e t h a t d / X ( e i t h e r d , o r X , o r b o t h ) v a r i e s g r a d u a l l y a l o n g t h e a x i s o f t h e c h a n n e l , o n a s c a l e t h a t i s much l a r g e r t h a n X . I n t h i s c a s e , F i g u r e s 9 a , b may a l s o b e t h o u g h t o f a s a v i e w i n t o a c h a n n e l o f a s l o w l y v a r y i n g r e l a t i v e w i d t h d / X , s h o w i n g t h e same r e c i r c u l a t i o n a c r o s s t h e c h a n n e l a s i n F i g u r e 6 ( G i l l ' s , 1 9 7 7 , F i g u r e 9 e ) . We n o t e t h a t i n c o n t r a s t t o m o s t p r e v i o u s i n v e s t i g a t i o n s ( a s i n G i l l , 1 9 7 7 ) , we d i d n o t a s s u m e a u n i f o r m p o t e n t i a l v o r t i c i t y . I n d e e d , e v e n i n s i d e t h e c h a n n e l , K(h) i s n o t u n i f o r m ( e q u a t i o n ( 5 . 3 ) ) . I n c o n t r a s t , t h e q u a n t i t y - u y / h = e f / H = V / X H , ( 5 . 8 ) i s u n i f o r m i n t h i s o n e - d i m e n s i o n a l c a s e . N o t e t h a t f o r a n a r e a e l e m e n t d A = d x d y , t h i s q u a n t i t y i s t h e r a t i o b e t w e e n c i r c u l a t i o n d C = - C u ( y + d y / 2 ) - u ( y - d y / 2 ) l d x = - u y d A a n d v o l u m e dV=hdA o f a m a t e r i a l v o r t e x t u b e e l e m e n t . I t i s e a s y t o m a t c h ( j o i n ) n o n u n i f o r m ( e . g . , n a r r o w i n g ) c h a n n e l s o l u t i o n s ( l i k e t h e o n e a b o v e , o r G i l l ' s , 1977) t o a s o l u t i o n a r o u n d t h e c h a n n e l m o u t h . F o r e x a m p l e , i t c a n b e d o n e ( a s o n F i g u r e 7) o n e X u n i t away f r o m t h e m o u t h , w h e r e , p r e s u m a b l y , t h e c o r n e r s n o l o n g e r i n f l u e n c e t h e f l o w a n d t h e s t r e a m l i n e s a r e a p p r o x i m a t e l y p a r a l l e l t o t h e s i d e s o f t h e c h a n n e l . - 27 -F i g u r e 9. P r o f i l e s o f h i n s i d e a c h a n n e l f o r (a) Q_=GL> a n d <b> Q--2Q+, f o r s e v e r a l v a l u e s o f w i d t h , d . - 28 -C h a p t e r 6 S c a l i n g a n d d e r i v a t i o n o-f t h e g o v e r n i n g e q u a t i o n s . I t i s n o t p o s s i b l e t o t a c k l e d i r e c t l y t h e s e t o f n o n l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s ( 3 . 1 ) - ( 3 . 3 ) i n t h e g e n e r a l t w o - d i m e n s i o n a l c a s e . I n s t e a d , we u s e a r e g u l a r p e r t u r b a t i o n e x p a n s i o n i n t h e R o s s b y number c ( = F 2 ) . F i r s t , i t i s c o n v e n i e n t t o n o n d i m e n s i o n a l i z e t h e v a r i a b l e s a c c o r d i n g t o ( x , y ) = X ( x ' , y ) , ( u , v ) = V ( u ' , v ' ) , h=Hh ' , H*=QH*' , ( 6 . 1 ) w h e r e X , V , H a n d Q a r e t h e l a t e r a l s c a l e a n d t h e u p s t r e a m b o u n d a r y v a l u e s o f v e l o c i t y , i n t e r f a c e d e p t h a n d t r a n s p o r t s t r e a m f u n c t i o n , r e s p e c t i v e l y , a s d e f i n e d i n C h a p t e r 4 . S u b s t i t u t i o n i n t o ( 3 . 1 ) — ( 3 . 3 ) g i v e s ( a f t e r d r o p p i n g t h e p r i m e s ) t h e n o n d i m e n s i o n a l e q u a t i o n s h + ( e /2 ) ( u 2 + v 2 ) = « t " ' 2 + ( e / 2 ) H » , ( 6 . 2 ) 1 + e ( v „ - u y ) . = h f - 1 ' 2 + e h , ( 6 . 3 ) 2 v h = H»H a n d - 2 u h = 4\,. ( 6 . 4 ) Me e l i m i n a t e s q u a r e r o o t s b y c h a n g i n g t o a new s t r e a m f u n c t i o n v a r i a b l e s , a c c o r d i n g t o M» = s 2 ( 6 . 5 ) - 29 -A s a r e s u l t , ( 6 . 2 ) - ( 6 . 4 ) b e c o m e 2h + e ( u 2 + v 2 ) = s ( 2 + es) ( 6 . 2 ) s C l + e<v* - u v ) 3 = h ( l + es ) ( 6 . 3 ) v h = s s K a n d —uh = s s y . <6 .4 ' ) We c o n s i d e r s , h , u a n d v a s r e g u l a r f u n c t i o n s o f t h e R o s s b y number e , e x p r e s s i n g t h e m i n r e g u l a r p e r t u r b a t i o n e x p a n s i o n s o f t h e f o r m ( s , h , u , v ) = 2 e J <Sj , h j ,Uj , vj) ( 6 . 6 ) i -o A s a r e s u l t , we g e t f r o m ( 6 . 2 ' ) t o o r d e r s e ° , E 1 a n d e 2 , r e s p e c t i v e l y h o = S o , ( 6 . 7 a ) 2 h i + U o z + v 0 2 = 2 s i + S o 2 , ( 6 . 7 b ) h 2 + u 0 U i + V o V i = s 2 + s 0 S i . ( 6 . 7 c ) F r o m ( 6 . 3 ' ) , we o b t a i n 5 0 = h o , 51 + S o < V o « - U o y ) = h i + h o S o , 5 2 + S l ( V o K - U o y ) + S o ( V t K - U l y ) ( 6 . 8 a ) ( 6 . 8 b ) = h 2 + h i S o + h 0 S i . ( 6 . 8 c ) - 30 -I F i n a l l y , t h e c o n t i n u i t y e q u a t i o n s ( 6 . 4 ' ) y i e l d V o h o = S o S o x , - U o h o = S o S o y , ( 6 . 9 a ) v 0 h i + V i h o = ( S O S I ) K , — u 0 h i - U i h o = ( s 0 S i ) y . ( 6 . 9 b ) We c a n now u s e t h e s e e q u a t i o n s t o g e t a s i n g l e e q u a t i o n f o r t h e 0 ( 1 ) s t r e a m f u n c t i o n v a r i a b l e s 0 a n d a n o t h e r o n e f o r t h e 0 ( c ) c o n t r i b u t i o n s t . F r o m e i t h e r ( 6 . 7 a ) o r ( 6 . 8 a ) , we h a v e h o = S o , ( 6 . 10) w h i c h s h o w s t h a t t h e 0 ( 1 ) i n t e r f a c e d e p t h i s a s t r e a m l i n e . A s a r e s u l t , f r o m ( 6 . 9 a ) , we o b t a i n t h e 0 ( 1 ) v e l o c i t i e s , v 0 = S o * a n d , - u 0 = 5 o v , ( 6 . 1 1 ) s o t h a t , a t l e a s t t o t h i s o r d e r , t h e f l o w i s g e o s t r o p h i c . C o n s e q u e n t l y , we a l s o g e t t h e 0 ( 1 ) r e l a t i v e v o r t i c i t y Vox — U o y = V 2 s 0 ( 6 . 12) a n d , t h e 0 ( 1 ) k i n e t i c e n e r g y Uo = ( u 0 2 + V o 2 ) / 2 = ( V s 0 ) 2 / 2 . ( 6 . 1 3 ) F r o m ( 6 . 7 b ) i t f o l l o w s t h a t t h e 0 ( c ) l a y e r d e p t h i s - 31 -r e l a t e d t o S i v i a ht = s , + C s o 2 - ( S 7 s 0 ) 2 J V 2 , ( 6 . 1 4 ) w h i l e , f r o m ( 6 . 8 b ) , we f i n d h i = s t + S o ( v 2 s 0 - S o ) . ( 6 . 1 5 ) We c o m b i n e t h e l a s t t w o e q u a t i o n s t o o b t a i n t h e d i f f e r e n t i a l e q u a t i o n f o r S o : v 2 s 0 - so = C s o 2 - ( V s o ) 2 3 / 2 s 0 . ( 6 . 1 6 ) E v e n t o t h i s l e a d i n g o r d e r , t h e e q u a t i o n i s n o n l i n e a r . T h i s n o n l i n e a r i t y c a n b e t r a c e d b a c k t o t h e n o n u n i f o r m i t y o f t h e p o t e n t i a l v o r t i c i t y , w h i c h i s a c o n s e q u e n c e o f ( 4 . 2 ) . F r o m ( 6 . 1 4 ) , we s e e t h a t t h e r i g h t h a n d s i d e o f ( 6 . 1 6 ) , w h i c h l o o k s l i k e t h e d e p a r t u r e o f t h e 0 ( 1 ) k i n e t i c e n e r g y f r o m i t s v a l u e u p s t r e a m ( d i v i d e d b y t h e d e p t h h o , s e e e q u a t i o n s ( 6 . 2 ) a n d ( 6 . 7 b ) ) , i s a l s o t h e 0 ( c ) d e p a r t u r e o f t h e d e p t h h f r o m t h e s t r e a m l i n e s ( a s g i v e n i n ( 6 . 1 4 ) ) , a n d t h i s d i f f e r e n c e c o n t r i b u t e s t o t h e b a l a n c e b e t w e e n t h e r e l a t i v e v o r t i c i t y t e r m , v 2 s 0 , a n d t h e v o r t e x s t r e t c h i n g t e r m s 0 ( =ho) . A s we s h a l l s e e i n t h e n e x t s e c t i o n , t h e r i g h t h a n d s i d e o f ( 6 . 1 6 ) v a n i s h e s w h e n e v e r t h e m o t i o n i s r e c t i l i n e a r , w h i c h i s t h e c a s e u p s t r e a m o r f a r away f r o m b o u n d a r i e s . I n p a r t i c u l a r , u p s t r e a m , w h e r e h=s - 32 -i d e n t i c a l l y , h i = S i = 0 a n d h e n c e , t h e r e l a t i v e v o r t i c i t y i s e q u a l t o s a n d t h e k i n e t i c e n e r g y i s e q u a l t o s 2 / 2 ( w h i c h c a n b e v e r i f i e d d i r e c t l y f r o m t h e e x p o n e n t i a l p r o f i l e ) . F o l l o w i n g t h e same m e t h o d , i t i s n o t d i f f i c u l t t o show t h a t i n t h e c a s e o f a uniform p o t e n t i a l v o r t i c i t y f u n c t i o n K(M») , t h e r i g h t - h a n d s i d e o f ( 6 . 1 6 ) v a n i s h e s a n d h e n c e , t h e 0 ( 1 ) e q u a t i o n f o r s 0 i s l i n e a r . T h e d e r i v a t i o n o f t h e e q u a t i o n f o r S i i s g i v e n i n a p p e n d i x A . T h e r e s u l t i s S o 7 2 5 , + V S o ' V S ! + ( V 2 S o - 3 S o ) 5 i = W, ( 6 . 1 7 ) w h e r e ( s e e a p p e n d i x A) W = E s o 2 - ( 7 S o ) 2 ] 2 / 4 s o + S o 3 - l / 2 ( v s 0 - V ) ( V S o ) 2 . ( 6 . 1 8 ) T h e f u n c t i o n W c a n b e r e w r i t t e n m o r e c o n c i s e l y a s W = s 0 ( M 2 + S o 2 ) - ( v s o ' V ) U o , ( 6 . 1 9 ) w h e r e M = s 0 / 2 - U o / s 0 i s t h e r i g h t - h a n d s i d e o f ( 6 . 1 6 ) a n d U 0 i s t h e 0 ( 1 ) k i n e t i c e n e r g y , g i v e n b y ( 6 . 1 3 ) . I t c a n b e shown t h a t f a r f r o m t h e a p e x , i n r e g i o n s o f r e c t i l i n e a r f l o w , W v a n i s h e s i d e n t i c a l l y , t o g e t h e r w i t h M . T h e q u e s t i o n o f s i n g u l a r i t y n e a r t h e apex o f a r e — e n t r a n t c o r n e r i s c o n s i d e r e d i n t h e n e x t c h a p t e r , w h e r e e x p l i c i t s o l u t i o n s a r e d e r i v e d . We c a n l i n e a r i z e ( 6 . 1 6 ) a n d s i m p l i f y t h e l e f t s i d e o f - 33 -( 6 . 1 7 ) i f we u s e t r a n s f o r m a t i o n s s o = p 2 ' 3 ( 6 . 2 0 ) a n d , s t = p - l / 3 q . ( 6 . 2 1 ) T h i s r e s u l t s i n two m o d i f i e d H e l m h o l t z e q u a t i o n s , a h o m o g e n e o u s o n e f o r p : V 2 p - ( 3 / 2 ) 2 p = 0 ; ( 6 . 2 2 ) a n d a n i n h o m o g e n e o u s o n e f o r q : <72q - ( 3 / 2 ) 2 q = p " l ' 3 W . ( 6 . 2 3 ) A l s o , s i n c e s 2 = 4 , = M » 0 + e H , 1 + . . . , we g e t 4"o = p « ' 3 a n d V i = 2 p l / 3 q . ( 6 . 2 4 ) E q u a t i o n s ( 6 . 2 0 ) - ( 6 . 2 4 ) d e f i n e t h e p h y s i c a l p r o b l e m i f t h e s h a p e o f t h e d o m a i n a n d t h e b o u n d a r y c o n d i t i o n s a r e s p e c i f i e d . I n t h e n e x t c h a p t e r , we s o l v e t h e s e e q u a t i o n s f o r a d o m a i n b o u n d e d b y t w o s t r a i g h t w a l l s , i . e . , a wedge o f a n a r b i t r a r y a n g l e f r / a , w h e r e t h e a n g u l a r p a r a m e t e r a i s n o t s m a l l e r t h a n 1 / 2 . We w i l l t h e n o b t a i n , i n C h a p t e r 8 , a p p r o x i m a t e s o l u t i o n s f o r f l o w s a l o n g c o a s t l i n e s , w h i c h c a n b e r e p r e s e n t e d b y a ( p o s s i b l y d i s c o n t i n u o u s ) c o m b i n a t i o n o-f c o r n e r s , a s l o n g a s n o t w o c o r n e r s a r e c l o s e r t h a n o n e R o s s b y r a d i u s . - 35 -C h a p t e r 7  R o t a t i n g f l o w a r o u n d a s h a r p b e n d Me p r o c e e d now t o s o l v e t h e p r o b l e m , s t a t e d i n t h e p r e v i o u s c h a p t e r , - for t h e c a s e o-f a d o m a i n b o u n d e d b y t w o s t r a i g h t w a l l s , w h i c h a r e l o c a t e d a t 9=0 a n d © = r r / a , w i t h a > l / 2 ( F i g u r e 1 0 ) . A s s t a t e d p r e v i o u s l y , we a s s u m e t h a t a g e o s t r o p h i c c u r r e n t w i t h a n e x p o n e n t i a l d e p t h p r o f i l e , g i v e n b y ( 4 . 2 ) , a p p r o a c h e s t h e b e n d f r o m u p s t r e a m , w i t h t h e © = n V a b o u n d a r y o n i t s r i g h t . We a l s o a s s u m e t h a t , u n l e s s i t s h o u l d a p p e a r e x p l i c i t l y f r o m t h e s o l u t i o n o f t h e p r o b l e m , t h e r e i s n o s e p a r a t i o n o f s t r e a m l i n e s f r o m t h e b o u n d i n g w a l l . C o n s e q u e n t l y , t h e c u r r e n t t u r n s t h e b e n d , e v e n f o r a r e — e n t r a n t c o r n e r , a n d t h e f a r d o w n s t r e a m d e p t h p r o f i l e i s i d e n t i c a l t o t h a t f a r u p s t r e a m o f t h e c o r n e r . T h e e q u a t i o n s t o b e s o l v e d a r e ( 6 . 2 2 ) a n d ( 6 . 2 3 ) , a n d t h e c o r r e s p o n d i n g b o u n d a r y c o n d i t i o n s a r e p = 1 a t 6 = 0 , / r / a ( 7 . 1) a n d q = 0 a t e = 0 , i r / a . ( 7 . 2 ) We b e g i n w i t h t h e f i r s t s e t . - 36 -T h e 0 ( 1 ) s o l u t i o n B e - f o r e we s t a r t , i t s h o u l d b e p o i n t e d o u t t h a t e q u a t i o n s ( 6 . 2 2 ) a n d ( 7 . 1 ) a r e c l o s e l y c o n n e c t e d w i t h t h e p r o b l e m o f d i f f r a c t i o n o f a K e l v i n wave b y a w e d g e . T h e l a t t e r was s o l v e d b y R o s e a u (1967) a n d a l s o b y P a c k h a m a n d W i l l i a m s (1968) f o r a g e n e r a l wedge a n g l e u s i n g a c o m p l e x i n t e g r a l r e p r e s e n t a t i o n . B u c h w a l d (1968) u s e d t h e W i e n e r - H o p f t e c h n i q u e t o s o l v e t h e d i f f r a c t i o n p r o b l e m f o r t h e p a r t i c u l a r c a s e o f a K e l v i n wave i n c i d e n t a t a r i g h t - a n g l e d c o r n e r . I n p r i n c i p l e , t h e s o l u t i o n o f o u r p r o b l e m s h o u l d b e o b t a i n a b l e f r o m t h e l a t t e r b y a l i m i t i n g p r o c e s s , w h e r e i n t h e K e l v i n wave t r a n s f o r m s i n t o a g e o s t r o p h i c c u r r e n t w i t h a n e x p o n e n t i a l p r o f i l e , i n t h e l i m i t o f z e r o f r e q u e n c y . U n f o r t u n a t e l y , d u e t o i t s c o m p l e x i t y , o n l y a s y m p t o t i c f o r m s o f t h e wave s o l u t i o n w e r e p r e s e n t e d i n t h e a b o v e p a p e r s . T h e m e t h o d s e m p l o y e d b y R o s e a u a n d b y P a c k h a m a n d W i l l i a m s a r e r e l a t e d t o t h e " S o m m e r f e l d d i f f r a c t i o n p r o b l e m " , w h i c h d e a l s w i t h t h e d i f f r a c t i o n o f e l e c t r o m a g n e t i c w a v e s b y a c o n d u c t i n g w e d g e . S o m m e r f e l d (1896) s o l v e d t h i s p r o b l e m f o r t h e c a s e o f a h a l f - s c r e e n (wedge o f a n a n g l e 2ir) a n d f o r a n a n g l e w h i c h i s a s u b m u l t i p l e o f 2nir , w h e r e n i s a p o s i t i v e i n t e g e r ( f o r a n i l l u m i n a t i n g d i s c u s s i o n , t h e r e a d e r i s r e f e r e d t o S o m m e r f e l d ' s b o o k " O p t i c s " , 1954 , s e c t i o n 3 8 ) . H i s m e t h o d was g e n e r a l i z e d t o an a r b i t r a r y a n g l e b y M a c d o n a l d ( 1 9 1 5 ) , B r o m w i c h ( 1 9 1 5 ) , W h i p p l e (1916) a n d C a r s l a w ( 1 9 1 9 ) . Some o f t h e e x t e n s i v e l i t e r a t u r e o n t h e s u b j e c t h a s b e e n r e v i e w e d b y O b e r h e t t i n g e r ( 1 9 5 4 ) . - 37 -S i n c e we i n t e n d t o u s e t h e r e s u l t s o f d i f f r a c t i o n t h e o r y , we b r i e f l y r e v i e w i t s f o r m u l a t i o n . We n e e d c o n s i d e r o n l y t h e s p e c i a l c a s e o f a p l a n e wave o f u n i t s t r e n g t h , F 0 = e x p C i k C c t + r c o s ( 0 - © o ) 3 > , i n c i d e n t f r o m t h e d i r e c t i o n ©o a t a r i g h t a n g l e t o a n e d g e o f t h e w e d g e . A f t e r r e m o v a l o f t h e t i m e d e p e n d e n c e , t h e S o m m e r f e l d d i f f r a c t i o n p r o b l e m r e d u c e s t o s o l v i n g t h e H e l m h o l t z e q u a t i o n , V*V + k 2 V = 0 , (7.3) s u b j e c t t o c o n d i t i o n s t h a t t h e s o l u t i o n V i s z e r o o n e a c h b o u n d a r y a n d t h a t i t s a t i s f i e s a s u i t a b l e r a d i a t i o n c o n d i t i o n . B r o m w i c h (1915) u s e d t h e c a s e o f t h e wedge n V n , w h e r e n i s a p o s i t i v e i n t e g e r , a s a s t a r t i n g p o i n t o f t h e f a m i l i a r m e t h o d o f i m a g e s . He r e p l a c e d t h e sum o f t h e i m a g e s b y a c o m p l e x i n t e g r a l a n d t h e n e x t e n d e d h i s f o r m u l a e t o h o l d f o r a n y ( r e a l a n d p o s i t i v e ) v a l u e o f n = a . S u b s e q u e n t l y , b y d e f o r m i n g t h e i n t e g r a t i o n p a t h , W h i p p l e (1916) was a b l e t o show t h a t t h e s o l u t i o n c a n b e w r i t t e n a s V = " s u m o f v i s i b l e i m a g e s " -- F ( ir+e-© 0 ) - F(n—e+eQ) + F ( J T - e - e 0 ) + F(n-+e+eQ), (7.4) w h e r e t h e d i f f r a c t i o n t e r m s F a r e g i v e n b y - 38 -F (V) = a s i n ( a V ) oo e x p < - i k r c o s h u ) d u c o s h ( a u ) — c o s ( a y ) ( 7 . 5 ) o i n t e g r a t e d a l o n g t h e p o s i t i v e u - a x i s . T h e " v i s i b l e i m a g e s " a r e t h e i n c i d e n t wave F 0 a n d i t s i m a g e s t h a t a r e " v i s i b l e " a t ( r , © ) . W h i p p l e a l s o s h o w e d t h a t e a c h d i - f - f r a c t e d t e r m F ( V ) s a t i s f i e s e q u a t i o n ( 7 . 3 ) . I n o r d e r t o s o l v e t h e b o u n d a r y v a l u e p r o b l e m , d e - f i n e d b y ( 6 . 2 2 ) a n d ( 7 . 1 ) , we c h a n g e k t o i k ( k = 3 / 2 i n e q u a t i o n ( 6 . 2 2 ) ) a n d d e s c r i b e t h e i n c o m i n g c u r r e n t a s an e v a n e s c e n t p l a n e " w a v e " e m a n a t i n g f r o m t h e b o u n d a r y a t © = i r / a . T h u s , t h e " w a v e s o u r c e " i s p r e s u m e d t o " r a d i a t e " f r o m t h e d i r e c t i o n ©0=7r/a+7r/2 ( F i g u r e 10) , i n w h i c h c a s e F o = e x p C k r c o s ( © - © 0 ) J = e x p C k r s i n ( © - 7 r / a ) 1, ( 7 . 6 ) A s a r e s u l t , t h e d i f f r a c t e d t e r m ( 7 . 5 ) b e c o m e s ( s e e W h i p p l e , 1 9 1 6 , f o r d e t a i l s ) a s i n ( a V > 2rr oo e x p ( i k r s i n h u ) d u F (V) = c o s h ( a u ) - c o s ( a y ) o s i n ( a y ) 477 5 oo e x p C i k r s i n h ( u / a ) 1 d u s i n h 2 ( u / 2 ) + s i n 2 ( a y / 2 ) ( 7 . 7 a , b ) o Due t o t h e d i f f e r e n c e i n b o u n d a r y c o n d i t i o n s ( p = l , w h i l e V=0 o n b o t h b o u n d a r i e s ) , we c a n n o t u s e (7 .4 ) d i r e c t l y , b u t must s e e k an a l t e r n a t e c o m b i n a t i o n o f t h e F t e r m s . We i n v e s t i g a t e - 39 -f i r s t t h e f u n c t i o n a l b e h a v i o u r o f F ( V ) , w h i c h may b e s u m m a r i z e d a s f o l l o w s ( F i g u r e s 1 1 a , b ) : i ) F(V a ) i s c o n t i n u o u s f o r 0<V*<27r/a a n d h a s a p e r i o d o f 2 / r / a , i i ) F ( V ) — > 0 a s V—>ir/a, a n d i i i ) F ( S * ) — > l / 2 a s V—>+0 a n d F ( W ) — > - l / 2 a s W—>-0 . P r o p e r t i e s i ) a n d i i ) a r e r e a d i l y s e e n b y i n s p e c t i n g ( 7 . 7 a ) . P r o p e r t y i i i ) means t h a t F h a s a jump d i s c o n t i n u i t y a t V*=0. T h i s i s . b e c a u s e a s V*—>0, s i n ( a V * ) m u l t i p l i e d b y t h e i n t e g r a l h a s a f i n i t e l i m i t t h e r e , w h i l e t h e s i n f u n c t i o n c h a n g e s i t s s i g n . I n d e e d , i f we p u t b = s i n ( a W 2 ) , t h e n , a s b — > 0 , m o s t o f t h e c o n t r i b u t i o n t o t h e i n t e g r a l c o m e s f r o m t h e v i c i n i t y o f u=0 , i n w h i c h c a s e s i n h u ^ u , s i n h ( a u / 2 ) * a u / 2 . T h e r e a l p a r t o f ( 7 . 7 b ) g i v e s , f o r b * 0 ( s i n c e s i n (aP) = 2 s i n ( a W 2 ) c o s ( a W 2 ) %2b t h e r e ) , „. ab / ° ° c o s ( k r u ) d u F ( b ) * \ —-2TT ' o ( a u / 2 ) 2 + b 2 C h a n g i n g t o x = a u / 2 , d u = 2 d x / a , we g e t .- v ~ b f c o s ( 2 k r x / a ) dx b , _ , , . . . . . _ . F ( b ) * — S Z — = „ , , , e x p ( - 2 k r ! b ! / a ) . ( 7 . B ) rr ' 0 x 2 + b 2 2 ! b H e n c e , f o r s m a l l V* - 40 -2 F ( V ) % s i g n ( V ) e x p ( - k r i V S ) —> s i g n ( y ) , a s V — > 0 . F r o m t h e a b o v e p r o p e r t i e s o-f F ( V ) , i t - f o l l o w s t h a t t h e s o l u t i o n o-f t h e b o u n d a r y v a l u e p r o b l e m ( 6 . 2 2 ) a n d ( 7 . 1 ) i s g i v e n b y t h e sum o-f t h e f o u r d i f f r a c t i o n t e r m s p ( r , 6 ) = 2 F ( 0 ) + 2 F ( / r / a - e ) , o r , e x p l i c i t l y , _ 2 a s i n ( a Q ) / ° ° e x p ( i k r s i n h u ) c o s h ( a u ) d u _ P n ^ 0 c o s h 2 ( a u ) - c o s 2 ( a 6 ) ~~ 2 s i n ( a © ) / ° ° e x p C i k r s i n h ( u / a ) 3 c o s h u d u TT 'o s i n h 2 u + s i n 2 ( a © ) ( 7 . 9 a , b ) S i n c e e a c h F t e r m s a t i s f i e s ( 6 . 2 2 ) ( w i t h k = 3 / 2 ) , t h e n s o d o e s p . One c a n a l s o show t h i s d i r e c t l y b y t a k i n g d e r i v a t i v e s o f p w i t h r e s p e c t t o r a n d © a n d s u b s t i t u t i n g b a c k i n t o ( 6 . 2 2 ) . W h i l e t r y i n g t h i s o u t , i t i s u s e f u l t o know t h a t t h e v a l u e o f t h e i n t e g r a l i n ( 7 . 9 b ) d o e s n o t c h a n g e when i t s u p p e r l i m i t i s r e p l a c e d b y o o + i d , w i t h t h e o n l y c o n d i t i o n t h a t 0<d<arr. T h i s i s n e e d e d s i n c e when d i f f e r e n t i a t e d w i t h r e s p e c t t o r , ( 7 . 9 b ) g i v e s an a p p a r e n t l y d i v e r g e n t i n t e g r a l , t h a t c a n b e t r a n s f o r m e d i n t o a c o n v e r g e n t o n e b y i n t e g r a t i o n b y p a r t s , a n d t h e i n t e g r a t e d t e r m s v a n i s h when t h e u p p e r l i m i t h a s t h i s e x t r a + i d ( a n a l o g o u s t o a r t i f i c i a l v i s c o c i t y , w h i c h i s o f t e n u s e d a s a c o n v e n i e n t way t o - 41 -g e t c o n v e r g e n c e a t i n f i n i t y ; s e e , f o r e x a m p l e , C a r r i e r e t a l . , 1966 , p . 3 3 7 ) . I t i s a l s o a s i m p l e e x e r c i s e t o show t h a t f o r t h e p a r t i c u l a r c a s e o f a s t r a i g h t c o a s t l i n e , a = l , ( 7 . 9 ) r e d u c e s t o t h e u p s t r e a m p r o f i l e p = e ~ k x , w h e r e x i s t h e n o r m a l d i s t a n c e t o t h e b o u n d a r y a n d k = 3 / 2 ( h e n c e , 5 0 = 6 - " ) . F o r a # l , a n u m e r i c a l q u a d r a t u r e c a n b e u s e d t o e v a l u a t e ( 7 . 9 ) , o r ( 7 . 7 ) . T h e t e r m 2F(rr/a-e>) c a n b e t h o u g h t o f a s r e p r e s e n t i n g a l i n e a r r o t a t i n g f l o w f r o m t h e u p s t r e a m i n t o a s i n k a t t h e apex o f t h e c o r n e r . S i m i l a r l y , 2 F ( © ) w o u l d b e a s o u r c e f l o w f r o m t h e a p e x . T h e i r s u m , r a i s e d t o t h e p o w e r 2 / 3 , a c c o r d i n g t o ( 6 . 2 0 ) , g i v e s s 0 , t h e 0 ( 1 ) s t r e a m f u n c t i o n , w h i c h i s p l o t t e d o n F i g u r e 12a f o r a = 2 / 3 (most o f t h e F i g u r e s a r e d r a w n f o r t h i s c h o i c e o f a ) . s * , t h e s o l u t i o n t o t h e h o m o g e n e o u s c o u n t e r p a r t o f ( 6 . 1 6 ) , i s a l s o shown o n t h i s F i g u r e . T h e d i f f e r e n c e b e t w e e n t h e two i s v e r y s m a l l , < 0 . 0 3 . T h i s means t h a t M, t h e n o n l i n e a r f o r c i n g t e r m o n t h e r i g h t h a n d s i d e o f ( 6 . 1 6 ) , p l o t t e d o n F i g u r e 1 2 b , h a s a s m a l l e f f e c t o n t h e 0 ( 1 ) s o l u t i o n ( i t moves t h e s t r e a m l i n e s somewhat away f r o m t h e c o r n e r ) . A s e x p e c t e d , f o r l a r g e r , s 0 r e d u c e s t o t h e u p s t r e a m p r o f i l e e~K , w h e r e x i s t h e d i s t a n c e f r o m e i t h e r b o u n d a r y , w h i l e f o r s m a l l r , i t i s g i v e n b y t h e p o t e n t i a l f l o w s t r e a m f u n c t i o n p p , r a i s e d t o t h e p o w e r 2 / 3 . T h i s i s shown i n more d e t a i l i n A p p e n d i x B . We s h o u l d a d d t h a t w h i l e l o o k i n g a t F i g u r e s 1 2 a , b a n d t h e f i g u r e s t h a t f o l l o w , we must remember t h a t , s i n c e t h e d i s t a n c e s a r e i n u n i t s o f X = R / F , t h e a c t u a l d i m e n s i o n s o f t h e s e d r a w i n g s a r e i n v e r s e l y p r o p o r t i o n a l t o F = e l / 2 . - 42 -T h e Q(e) s o l u t i o n We now p r o c e e d t o s o l v e t h e s e c o n d b o u n d a r y v a l u e p r o b l e m , e q u a t i o n s ( 6 . 2 3 ) a n d ( 7 . 2 ) . F o r m a l l y , t h e s o l u t i o n may b e w r i t t e n a s q ( x , y ) = | | f (x ' , y ' ) G ( x , y ! x ' , y * ) d x d y ' , ( 7 . 1 0 ) w h e r e t h e i n t e g r a t i o n i s o v e r t h e s p e c i f i e d d o m a i n , f i s t h e r i g h t h a n d s i d e o-f ( 6 . 2 3 ) , f = p-i ' = W , ( 7 . 1 1 ) a n d W i s g i v e n b y ( 6 . 1 9 ) . T h e G r e e n ' s - f u n c t i o n G ( x , y ! x ' , y ' ) s a t i s f i e s v 2 G - k 2 G = ^ ( x - x ' ) o ' ( y - y ' ) ( 7 . 1 2 ) a n d v a n i s h e s o n t h e b o u n d a r y . I t c a n b e w r i t t e n i n t e r m s o f t h e m o d i f i e d B e s s e l f u n c t i o n s ( s e e , f o r e x a m p l e , S t a k g o l d , 1 9 6 8 ) , o o G ( r , © ! r ' , 0 ' ) = - ^ g ) ~ s i n (ma6) s i n (ma8 ' ) Kro. (kr>) I m , (kr< ) , ( 7 . 1 3 ) w i t h r > = m a x ( r , r ' ) a n d r < = m i n ( r , r ' ) . Due t o t h e s y m m e t r y o f t h e wedge p r o b l e m , o n l y t h e o d d t e r m s i n ( 7 . 1 3 ) c o n t r i b u t e t o t h e i n t e g r a l i n ( 7 . 1 0 ) . We w i l l a l s o b e s a t i s f i e d w i t h a p p r o x i m a t i n g - 43 -t h e 0 (e ) t e r m q . C a l c u l a t i o n s show t h a t e v e n i f o n l y t h e l e a d i n g t e r m i n ( 7 . 1 3 ) , G * — (2a/rr) s i n ( a © ) s i n ( a © ' ) K« (kr>) I « (kr< ) , ( 7 . 1 4 ) i s r e t a i n e d , t h e v a l u e o f q i s i n c r e a s e d b y n o m o r e t h a n a f e w p e r c e n t . We c o m p u t e ( 7 . 1 0 ) u s i n g a s i m p l e q u a d r a t u r e r o u t i n e . F i g u r e s 1 3 a , b show c o n t o u r s o f t h e f u n c t i o n f a n d o f S i = s 0 - 1 / 2 q , r e s p e c t i v e l y . Somewhat s u r p r i s i n g l y , t h e maximum v a l u e o f s , i s l e s s t h a n 0 . 2 , a n d a s a r e s u l t , n o s e p a r a t i o n i s e v i d e n t i n t h e s t r e a m l i n e s o f 4 , = s 0 2 + 2 e s0S i , w h i c h a r e shown o n F i g u r e 13c f o r e = 0 . 5 . W h i l e t h i s v a l u e o f e may seem t o b e somewhat l a r g e ( f o r ( 6 . 6 ) t o b e v a l i d ) , i t i s , n e v e r t h e l e s s , u s e d h e r e f o r d e m o n s t r a t i o n p u r p o s e s . T h e o n l y n o t i c e a b l e e f f e c t o f t h e 0 (e ) t e r m on t h e t r a n s p o r t s t r e a m f u n c t i o n M1 i s t o move t h e s t r e a m l i n e s away f r o m t h e c o r n e r ( F i g u r e 1 3 d ) , a n e f f e c t w h i c h may b e a t t r i b u t e d t o t h e c e n t r i f u g a l a c c e l e r a t i o n o f t h e f l u i d . A w o r d o f c a u t i o n : s i n c e we n o n d i m e n s i o n a l i z e d d i s t a n c e s w i t h X = R / e l / 2 , we c a n n o t c o m p a r e d i m e n s i o n a l v e l o c i t i e s f o r t h e two v a l u e s o f e=0 a n d 0 . 5 o n t h e b a s i s o f s t r e a m l i n e s e p a r a t i o n s o n l y ( F i g u r e 1 3 d ) . In f a c t , s i n c e e — > 0 i m p l i e s X——>oo, s t a t e m e n t s c o n c e r n i n g t h e c=0 c a s e c a n o n l y b e i n t e r p r e t e d i n t h i s l i m i t i n g s e n s e . U s i n g ( 6 . 1 4 ) , we c a l c u l a t e t h e 0(e) d e p t h , hj = S i + M s 0 , ( 7 . 15) - 44 -a n d , f r o m ( 6 . 9 b ) a n d ( 7 . 1 5 ) , we o b t a i n t h e 0 (e ) v e l o c i t i e s , U i = - S i y - M u 0 ( 7 . 1 5 a ) a n d V i — S I K — M v 0 . ( 7 . 1 5 b ) C o n t o u r s o f h i a r e p l o t t e d o n F i g u r e 1 4 a , w h i l e F i g u r e s 14b a n d 14c show t h e t o t a l d e p t h o f t h e i n t e r f a c e h = h 0 + e h i . T h e d e p t h i n t e g r a t e d k i n e t i c e n e r g y hU = hCUo + e ( u o U i + v 0 V i ) 1 i s shown i n F i g u r e s 1 5 a , b f o r t h e two v a l u e s o f e=0 a n d e = 0 . 5 . F i g u r e 14c s h o w s t h e e f f e c t o f t h e n o n l i n e a r a d v e c t i o n t e r m s : a centr ifugal u p w e J1ing f o r a r e — e n t r a n t (a< l ) c o r n e r . I n c o n t r a s t , d o w n w e l l i n g w o u l d b e e x p e c t e d a t an i n s i d e ( a > l ) c o r n e r . I t i s a l s o a p p a r e n t t h a t f o r l a r g e e a n d / o r l a r g e c o r n e r c u r v a t u r e , we may e x p e c t a s u r f a c i n g o f t h e p y c n o c l i n e : i . e . , s e p a r a t i o n o f t h e s t r e a m l i n e s f r o m t h e b o u n d a r y j u s t b e f o r e o r a t t h e c o r n e r . T h i s i s o n l y v a g u e l y d i s c e r n i b l e o n F i g u r e 1 3 c . Due t o t h e a s s u m e d p o w e r s e r i e s f o r m , e q u a t i o n ( 6 . 6 ) , t h e r e i s l i t t l e c o n f i d e n c e i n t h e a b o v e d e r i v e d s o l u t i o n ( s e e b e l o w ) t h a t c l o s e t o t h e c o r n e r , w h e r e t h e 0 (e ) t e r m s a r e l a r g e r t h a n t h e 0 ( 1 ) t e r m ( n e x t s e c t i o n ) . T h i s i s e s p e c i a l l y t r u e f o r r e - e n t r a n t c o r n e r s . A s e p a r a t e c a l c u l a t i o n , w h i c h was d o n e f o r s m a l l e r s c a l e s ( s e e A p p e n d i x C ) , s h o w s a s e p a r a t i o n o f s t r e a m l i n e s v e r y - 45 -c l o s e t o t h e a p e x . T h e q u e s t i o n o f t h e v a l i d i t y o f t h e s o l u t i o n i s e x a m i n e d b e l o w , w h i l e s e p a r a t i o n i s d i s c u s s e d i n t h e e n d o f t h e c h a p t e r . T h e v a l i d i t y o f t h e s o l u t i o n F o r a r e — e n t r a n t c o r n e r o f i n f i n i t e c u r v a t u r e , t h e s o l u t i o n i s n o t v a l i d a t t h e a p e x , w h e r e t h e v e l o c i t i e s u a n d v a r e i n f i n i t e . We m u s t a l s o r e q u i r e t h a t e a c h o f t h e f o u r s e r i e s i n ( 6 . 6 ) c o n v e r g e t o a f i n i t e l i m i t . We c a n n o t p r o v e c o n v e r g e n c e o f ( 6 . 6 ) , b u t a s k , a s an a p p r o x i m a t e r e q u i r e m e n t , t h a t t h e 0 (e ) t e r m s b e o f t h e same o r d e r o f m a g n i t u d e , o r s m a l l e r , t h a n t h e 0 ( 1 ) t e r m s . T o b e more d e f i n i t e , we r e q u i r e t h a t I s , ! < s o , ( 7 . 1 6 a ) ! h i ! < ho = s o , ( 7 . 1 6 b ) a n d ! U i ! < U c , ( 7 . 1 6 c ) w h e r e U i = u 0 U j + V o V i i s t h e 0 (e ) k i n e t i c e n e r g y t e r m . U s i n g ( 7 . 1 5 ) a n d ( 6 . 1 1 ) , t h e l a s t i n e q u a l i t y may a l s o b e w r i t t e n a s I V S o V S i - 2 M U o ! < Uo . ( 7 . 1 7 ) F i g u r e s 12a a n d 13b show t h a t ( 7 . 1 6 a ) i s s a t i s f i e d e v e r y w h e r e . F o r m i n g t h e r a t i o s - 46 -= i h i / h o ! = !M + s , / s , o • ( 7 . 1 8 ) a n d r 2 = !2M - v s o ' V S i / U o | ( 7 . 1 9 ) we p l o t c o n t o u r s o-f r t a n d r 2 o n F i g u r e s 1 6 a , b , s u p e r i m p o s e d u p o n a f e w s t r e a m l i n e s 4" ( f o r e = 0 . 5 ) . We n o t i c e t h a t f o r t h e c h o s e n p a r a m e t e r s ( a = 2 / 3 , e = 0 . 5 ) , o u r s o l u t i o n i s v a l i d ( r i a n d r 2 b o t h l e s s t h e n 1) f o r s t r e a m l i n e s 4 K 0 . 9 . F i g u r e s 1 6 a , b show t h a t , d e p e n d i n g on e , t h e s o l u t i o n i s v a l i d f o r r e l a t i v e l y ( b u t , n o t i n f i n i t e l y ) s h a r p r e - e n t r a n t c o r n e r s , a s l o n g a s t h e r o u n d e d b o u n d a r y ( e . g . , t h e VssO.9) s t r e a m l i n e d o e s n o t p e n e t r a t e f a r i n s i d e t h e c u r v e r i = l . S i n c e f o r a r e — e n t r a n t c o r n e r t h e v e l o c i t y o f t h e f l u i d i s h i g h n e a r t h e a p e x , we e x p e c t t h e f l o w t o b e s u p e r c r i t i c a l t h e r e . T h e l o c a l F r o u d e n u m b e r , i s c o n t o u r e d t o g e t h e r w i t h *¥ o n F i g u r e 1 6 c . F o r t h e c h o s e n p a r a m e t e r s ( e = F 2 = 0 . 5 , a=2 /3 ) , h a l f o f t h e t o t a l t r a n s p o r t i s p a s s i n g t h r o u g h t h e s u p e r c r i t i c a l r e g i o n , w h e r e F r > l . T h i s means t h a t d i s t u r b a n c e s g e n e r a t e d d o w n s t r e a m o f t h e c o r n e r c a n n o t p r o p a g a t e u p s t r e a m , w h i c h may c a u s e a h y d r a u l i c jump n e a r t h e S u p e r c r i t i c a l f l o w F t ( u 2 + v 2 ) / h 3 1 / 2 ( 7 . 2 0 ) - 47 -s e c o n d ( d o w n s t r e a m ) F r = l l i n e . I n o u r c a s e , t h e c o n d i t i o n s d o w n s t r e a m a r e d e t e r m i n e d b y t h e u p s t r e a m p a r a m e t e r s s i n c e , i n t h e a b s e n c e o-f a n o t h e r b o u n d a r y c l o s e - b y , t h e K e l v i n wave c a n p r o p a g a t e o n l y w i t h t h e b o u n d a r y o n i t s r i g h t . H e n c e , ( f o r t h i s s t e a d y m o d e l ) n o h y d r a u l i c j u m p s a r e e x p e c t e d a s t h e f l o w r e - e n t e r s t h e s u b c r i t i c a l r e g i m e . I t s h o u l d b e p o i n t e d o u t , h o w e v e r , t h a t w h i l e n o s t a t i o n a r y j u m p s a r e p r e d i c t e d a t a c o r n e r , t r a v e l l i n g d i s t u r b a n c e s i n a f o r m o f s h o c k — w a v e s o r b o r e s a r e q u i t e l i k e l y . T h e s e may a r i s e , f o r e x a m p l e , when t h e r e i s a s u d d e n c h a n g e i n u p s t r e a m c o n d i t i o n s ( l i k e a n i n c r e a s e i n t r a n s p o r t ) , i n w h i c h c a s e a f u l l y n o n l i n e a r shock—wave w i l l b e g e n e r a t e d . A c c o r d i n g t o N o f ( 1 9 8 4 ) , t h i s s h o c k w o u l d t h e n p r o p a g a t e d o w n s t r e a m ( a l o n g a s t r a i t v e r t i c a l b o u n d a r y ) a t a s p e e d l a r g e r t h a n t h a t o f a K e l v i n wave a s s o c i a t e d w i t h b o t h t h e d i s t u r b e d a n d t h e u n d i s t u r b e d f l o w . B e c a u s e t h e s h o c k i s f a s t e r t h a n t h e sum o f t h e d o w n s t r e a m ( w i t h r e s p e c t t o t h e s h o c k ) a d v e c t i o n s p e e d a n d t h e d o w n s t r e a m K e l v i n wave s p e e d b u t s l o w e r t h e n t h e c o r r e s p o n d i n g sum u p s t r e a m o f t h e s h o c k , n o e n e r g y i s l o s t f r o m t h e s h o c k ( e x c e p t f o r s m a l l f r i c t i o n a l l o s s e s ) a n d t h e s h o c k r e t a i n s i t s f o r m ( N o f , 1 9 8 4 ) . I t i s n o t c l e a r h o w e v e r , how s u c h s h o c k i s t r a n s f o r m e d when i t r o u n d s a s h a r p c o r n e r . A s i n t h e c a s e o f m o n o c h r o m a t i c K e l v i n w a v e s ( F a c k h a m a n d W i l l i a m s , 1968 ; B u c h w a l d , 1968 ; M i l e s , 1 9 7 2 ) , o n e may e x p e c t t h a t a c e r t a i n amount o f d i f f r a c t e d e n e r g y l o s s ( P o i n c a r e w a v e s ) w i l l o c c u r i n i t s h i g h e r ( s u p e r i n e r t i a l ) f r e q u e n c y c o m p o n e n t s . T h u s , t h e - 48 -shock—wave d o w n s t r e a m o f t h e c o r n e r w i l l b e l e s s e n e r g e t i c a n d , p o s s i b l y , o f a d i f f e r e n t s h a p e t h a n when i t was u p s t r e a m o f t h e c o r n e r . S e p a r a t i o n T h e p o w e r s e r i e s e x p a n s i o n ( 6 . 6 ) i s n o t v a l i d v e r y c l o s e t o t h e a p e x . We r e w r i t e ( 6 . 2 ' ) - ( 6 . 4 ' ) : h + e w 2 / 2 = s + e s 2 / 2 , ( 7 . 2 1 ) 1 - c (w n + w / r ) = h / s + e h , ( 7 . 2 2 ) - w h = s s „ , ( 7 . 2 3 ) w h e r e w = ( u 2 + v 2 ) 1 / 2 i s t h e s p e e d , r i s t h e r a d i u s o f c u r v a t u r e o f a s t r e a m l i n e a n d t h e s u b s c r i p t n d e n o t e s a n o r m a l d e r i v a t i v e . We now c o n s i d e r a b o u n d a r y s t r e a m l i n e , s = l , a n d a s s u m e r c t o b e t h e r a d i u s o f c u r v a t u r e t h a t c a u s e s i t s s e p a r a t i o n . Upon s e p a r a t i o n , h=0 , f r o m ( 7 . 2 1 ) , t h e s p e e d w=(1+2/e) 1 / 2 i s c o n s t a n t , a n d f r o m ( 7 . 2 2 ) t h e r e l a t i v e a n d t h e p l a n e t a r y v o r t i c i t i e s a r e e q u a l , E(W„ + w / r = ) = 1 . ( 7 . 2 4 ) D i f f e r e n t i a t i n g ( 7 . 2 1 ) i n t h e n o r m a l d i r e c t i o n , a n d s u b s t i t u t i n g i n t o ( 7 . 2 4 ) y i e l d s t h e momentum e q u a t i o n , - h n + e w 2 / r c = w, ( 7 . 2 5 ) - 49 -w h i c h s t a t e s t h a t t h e b a l a n c e b e t w e e n t h e p r e s s u r e g r a d i e n t a n d t h e c e n t r i f u g a l - f o r c e i s h e l d b y t h e C o r i o l i s f o r c e . E q u a t i o n ( 7 . 2 5 ) i s s a t i s f i e d f o r a n y , h o w e v e r s m a l l , r c ( w i t h a c o r r e s p o n d i n g l y s t e e p e r i n t e r f a c e s l o p e h „ ) . H e n c e , o u r m o d e l d o e s n o t g i v e u s t h e c r i t i c a l s e p a r a t i o n c u r v a t u r e . We c a n g e t some i d e a a b o u t t h e s i z e o f t h e u p w e l l i n g r e g i o n a n d , h e n c e , a b o u t t h e c r i t i c a l r a d i u s r c , i f we s o l v e t h e same p r o b l e m ( e q u a t i o n s ( 6 . 2 ' ) - ( 6 . 4 ' ) ) b u t f o r s m a l l e r s c a l e s , o f t h e o r d e r o f e 2 X , n e a r t h e a p e x . T h i s i s d o n e i n A p p e n d i x C . T h e r e s u l t s o f t h e s e c a l c u l a t i o n s show t h a t , f o r e x a m p l e , f o r e=0 .5 a n d a = 2 / 3 , r = < 0 . 0 2 , a n d t h e c e n t r i f u g a l u p w e l l i n g i s n o t i m p o r t a n t i f t h e r a d i u s o f c u r v a t u r e o f t h e r o u n d e d c o r n e r i s l a r g e r t h a n a b o u t 0 . I X . I t may b e o f some i n t e r e s t t o m a t c h t h e t w o s o l u t i o n s ( e . g . , u s i n g t h e m e t h o d o f m u l t i p l e s c a l e s ) . B u t t h i s w o u l d t a k e u s b e y o n d t h e s c o p e o f t h i s wo rk a n d a l s o may b e o f l i t t l e c o n s e q u e n c e , s i n c e a l l o f t h e c o a s t a l r a d i i o f c u r v a t u r e c o n s i d e r e d h e r e ( a s , i n t h e m o u t h o f H u d s o n S t r a i t ) a r e l a r g e r t h a n 0 . 1 X . I n t h i s r e s p e c t , we w o u l d l i k e t o p o i n t o u t t h a t t h e c e n t r i f u g a l u p w e l l i n g a t a s h a r p c a p e , a n d t h e r e s u l t i n g d o u g h n u t - l i k e s h a p e o f t h e i n t e r f a c e ( F i g u r e 1 4 c ) , a r e e x a c t l y a n a l o g o u s t o ( h y p o t h e t i c a l , s i n c e no o n e h a s e v e r o b s e r v e d them) a n o m a l o u s warm e d d i e s w i t h a c o l d c o r e , w h i c h h a v e s u r f a c e d , f o r e x a m p l e , a s a p o s s i b l e s o l u t i o n i n F l i e r l ' s (1979) a n a l y t i c a l t w o - l a y e r m o d e l o f t h e s t r u c t u r e o f warm a n d c o l d c o r e r i n g s . - 50 -T h e b a l a n c e o-f - f o r c e s i s t h e same — g e o s t r o p h y o n t h e o u t s i d e a n d a c y c l o s t r o p h i c b a l a n c e o n t h e i n s i d e . F i g u r e 13 i n F l i e r l (1979) s h o w s a s t r a i g h t l i n e r e l a t i o n b e t w e e n t h e s t r e n g t h o f t h e r i n g , a s i t was d e f i n e d b y F l i e r l : E F = - ( 2 e + e 2 ) 1 / 2 , a n d i t s i n s i d e r a d i u s r 0 - A l t h o u g h t h i s l i n e s t o p s s h o r t o f t h e r o = 0 a x i s ( p r e s u m a b l y d u e t o t h e s i n g u l a r i t y o f h i s g o v e r n i n g e q u a t i o n ) , i t d o e s seem t o i n d i c a t e t h a t f o r E F > — 1 . 5 ( e < 0 . 8 ) , t h e i n n e r ( c o l d ) c o r e o f t h e r i n g d i s a p p e a r s . I n t h a t c a s e , t h e s m a l l e r c e n t r i f u g a l f o r c e i s n o l o n g e r s u f f i c i e n t " to h o l d t h e i n n e r s l o p e f o r t h e u p w e l l i n g t o t a k e p l a c e , a n d t h e r i n g s t o p s b e i n g a n o m a l o u s . D e s p i t e t h e d i f f e r e n c e s , F l i e r l ' s c o n s t a n t p o t e n t i a l v o r t i c i t y ( a n d , h e n c e , d e e p e r u p p e r l a y e r ; s e e C h a p t e r 4) r a d i a l l y s y m m e t r i c a l m o d e l s e e m s t o i n d i c a t e t h a t t h e h o r i z o n t a l e x t e n t o f a c e n t r i f u g a l u p w e l l i n g a r o u n d a s h a r p r e - e n t r a n t c o r n e r s h o u l d b e r a t h e r s m a l l f o r e v a l u e s t h a t a r e c o n s i s t e n t ( e . g . , e < l / 2 ) w i t h t h e p o w e r s e r i e s e x p a n s i o n ( 6 . 6 ) . T h i s a g r e e s w e l l w i t h o u r r e s u l t s , a n d i n p a r t i c u l a r w i t h F i g u r e 14c ( d r a w n f o r t h e a = 2 / 3 , e = l / 2 c a s e ) . B e c a u s e o f i t s l i m i t e d e x t e n t , t h e c e n t r i f u g a l u p w e l l i n g i s n o t t h e m o s t l i k e l y c a u s e o f s e p a r a t i o n o f t h e b o u n d a r y s t r e a m l i n e . O t h e r e f f e c t s may b e more i m p o r t a n t . F o r e x a m p l e , i f t h e r e i s a n a d v e r s e p r e s s u r e g r a d i e n t w h i c h r a i s e s t h e d e p t h o f t h e u p p e r l a y e r t o i t s maximum v a l u e , l + e / 2 , t h e n , f r o m ( 7 . 2 1 ) , we g e t a s t a g n a t i o n p o i n t , w h e r e w=0, a n d h e n c e , s e p a r a t i o n . T h i s p r e s s u r e g r a d i e n t c o u l d b e d u e t o c h a n g e s i n b u o y a n c y , w i n d f o r c i n g , o r d u e t o b a r o t r o p i c e f f e c t s ( e . g . , c h a n g e s i n - 51 -b a t h y m e t r y ) . I n a d d i t i o n , e n h a n c e d Ekman p u m p i n g (due t o h i g h e r v e l o c i t i e s a t t h e c o r n e r ) may c o n t r i b u t e t o u p w e l l i n g a n d , h e n c e , t o s e p a r a t i o n . C o n s e q u e n t l y , a t h r e e - d i m e n s i o n a l f r i c t i o n a l m o d e l may b e r e q u i r e d t o a n s w e r t h i s q u e s t i o n a b o u t t h e s e p a r a t i o n a t a s h a r p c a p e . M e r k i n e a n d S o l a n (1979) h a v e shown t h a t i n t h e c a s e o-f a u n i f o r m s t r e a m ( V c = c o n s t a n t ) o f d e p t h H p a s t a c i r c u l a r c y l i n d e r o f d i a m e t e r D a n d f o r a s m a l l R o s s b y n u m b e r , Ro=v" c / fD<<l ( i . e . , l a r g e D , o r s m a l l V c ) , t h e f l o w s e p a r a t e s a t some p o i n t o n t h e c y l i n d e r , © < r r , i f t h e r a t i o ( E v / 2 ) 1 / 2 / R o = ( i / f ) l ' 2 D / ( 2 H V C X 1 , w h e r e E v i s t h e v e r t i c a l Ekman number a n d v i s t h e e d d y v i s c o s i t y . F o r e ~ l / 2 , t h e c o n d i t i o n RO<<1 i s e q u i v a l e n t t o X/D<<1. A l t h o u g h t h i s s u g g e s t s t h a t M e r k i n e a n d S o l a n ' s (1979) r e s u l t s a r e n o t a p p l i c a b l e t o s t r o n g f l o w s a r o u n d s h a r p c o r n e r s , a n a n a l o g o u s a p p r o a c h may p r o v e f r u i t f u l . W h i l e t h e p r o b l e m o f s e p a r a t i o n r e m a i n s t o b e s o l v e d , l e t u s a s s u m e f o r t h e s a k e o f a r g u m e n t t h a t t h e f l o w d o e s s e p a r a t e . T h i s r e s u l t s i n a n a n t i c y c l o n i c b a r o c l i n i c j e t t h a t w o u l d i m p i n g e o n t h e s t r a i g h t c o a s t l i n e ( d o w n s t r e a m o f t h e c o r n e r ) a t some n o n z e r o a n g l e . B u t , s i n c e t h e s p e e d o n t h e f r e e s t r e a m l i n e CH=1, h=0 f o r t h e i n v i s c i d m o d e l ) i s f i n i t e , i t c a n n o t p a s s t h r o u g h a s t a g n a t i o n p o i n t a n d , h e n c e , i t m u s t t u r n r i g h t , away f r o m t h e b o u n d a r y ( F i g u r e 1 7 a ) . A t r a n s i e n t a d j u s t m e n t p r o c e s s f o l l o w s , w h e r e b y p a r t o f t h e f l o w p o u r s i n t o a c l o s e d g y r e , w h i l e a d i f f e r e n t s t r e a m l i n e , *¥< 1 , p a s s e s t h r o u g h a s t a g n a t i o n p o i n t . A s t h e s i z e a n d t h e d e p t h o f t h e g y r e i n c r e a s e , h i g h e r v a l u e d - 52 -s t r e a m l i n e s move t h r o u g h t h e s t a g n a t i o n p o i n t . An e q u i l i b r i u m i s r e a c h e d , when t h e d e p t h o f t h e s e p a r a t e d s t r e a m l i n e a t t a i n s i t s maximum v a l u e , h = l + e / 2 , w h i c h a l l o w s i t t o p a s s t h r o u g h t h e s t a g n a t i o n p o i n t ( F i g u r e 1 7 b ) . I n t h i s c o n t e x t , t h e work o-f W h i t e h e a d (1985) o n t h e d e f l e c t i o n o f a b a r o c l i n i c j e t b y a w a l l b e c o m e s r e l e v a n t . H i s r e s u l t s i n d i c a t e t h a t u p o n i m p i n g e m e n t t h e f l o w b i f u r c a t e s w i t h a l a r g e r p a r t t u r n i n g t o t h e r i g h t , i n s u p p o r t o f t h e a b o v e d e s c r i p t i o n o f t h e a d j u s t m e n t p r o c e s s . T h e f i n a l s i z e o f t h e g y r e may b e a f u n c t i o n o f t h e wedge a n g l e , t h e R o s s b y n u m b e r , a n d t h e i n t e r f a c i a l f r i c t i o n . T h i s r a t h e r q u a l i t a t i v e d e s c r i p t i o n i s b a s e d o n t h e a s s u m p t i o n t h a t t h e B e r n o u l l i f u n c t i o n a n d t h e p o t e n t i a l v o r t i c i t y o n e a c h s t r e a m l i n e r e m a i n i n v a r i a n t d u r i n g t h e a d j u s t m e n t p r o c e s s . T h i s a s s u m p t i o n may n o t b e e n t i r e l y t r u e . I f g e n e r a t i o n o f some t y p e o f g r a v i t y o r s e c o n d c l a s s w a v e s a c c o m p a n i e s t h e p r o c e s s , t h e n t h e e n e r g y a n d t h e f i n a l s i z e o f t h e g y r e w i l l b e a f f e c t e d . I t s h o u l d b e e m p h a s i z e d t h a t a f t e r s e p a r a t i o n , t h e f r e e s t r e a m l i n e c o u l d t u r n away f r o m t h e b o u n d a r y ( i . e . , t o t h e l e f t ) , i n w h i c h c a s e t h e r e w i l l b e n o r e a t t a c h m e n t . T h e r e f o r e , u n t i l an e x p l i c i t s o l u t i o n i s f o u n d , t h i s p a r a g r a p h s h o u l d b e r e g a r d e d a s a h y p o t h e s i s t h a t may g u i d e f u t u r e s t u d y . I n a d d i t i o n , c o m p a r i s o n s h o u l d b e made w i t h s o l u t i o n s o f an e q u i v a l e n t n o n r o t a t i n g (f=0) p r o b l e m . - 53 -©=77/a F i g u r e I O . T h e i n c o m i n g c u r r e n t p r o - f i l e a s a n e v a n e s c e n t " w a v e e m a n a t i n g -from t h e b o u n d a r y a l o n g © = 7 r / a . - 54 -Figure 11. Dif-fraction term 2F(V>, with *»=ir/a-e and a=2/3: (a) i t s functional behaviour i n the range -2n<6<2n for r=0.1 and r=1.0; <b) as a rotating sink of a unit strength. The domain i s 2Xx2X large and contour spacing i s 0.1. Unless noted otherwise, the same applies to subsequent contour p l o t s . - 55 -a) F i g u r e 12 . (a) T h e 0 ( 1 ) s t r e a m - f u n c t i o n S o ^ p 2 ' 3 ( c o n t i n u o u s c o n t o u r s ) a n d i t s h o m o g e n e o u s c o u n t e r p a r t s h ( d a s h e d c o n t o u r s ) ; s h i s t h e s o l u t i o n o f v 2 sh- S h = 0 . (b) C o n t o u r s o f M = C s 0 2 - ( v s e ) 2 3 / 2 s 0 • - 56 -(a) (b) Figure 13. (a) The r i g h t hand side o-f (6.23), f=p" l / sW. (b) The 0(e) stream-function variable st. Here, contour spacing i s 0.02. (c) The t o t a l transport stream-function H'=M,o+e,(,i, for e=0.5. (d) Comparison between V0 (e=0) and V (e=0.5). - 58 -. »_ (b> Contours Figure 14. £ d i m e n s i o na l view o - 59 -F i g u r e 1 5 . D e p t h i n t e g r a t e d k i n e t i c e n e r g y hUs (a) e=0 , (b) e = 0 . 5 F o r c l a r i t y , t h e i r / 2 c o r n e r was d e l e t e d i n t h i s f i g u r e . - 61 -r , a n d V <e=0.5> l.Ofti 0.80-0.60-0.4ft 0.20-1.00 -0.80 -0.60 -0.40 -0.20 (a) t.Oth r a a n d V <e=0.5> 0.4ft 1.00 -0.80 -0.60 -0.40 -0.20 (b) F i g u r e 16 . C o n t o u r s o-f (a) r , = ! h j / h c ! = 0 . 3 , 1 . 0 , 2 . 0 ; (b) r 2 = : U i / U o ! = 0 . 1 , 1 . 0 , 3 . 0 a n d (c ) F r o u d e number ^ = 0 . 7 , 1 . 0 , 2 . 0 . T h e s e a r e s u p e r i m p o s e d u p o n s t r e a m l i n e s o f M«=0.4 t o 1 . 0 <e=0.5>. - 6 2 -l .OOn F r a n d V <e=o.5> 0.SO-IL60-V 0.20-1.00 -0.80 -0.60 -0.40 -0.20 i i i i • (a) (b) F i g u r e 1 7 . C o n c e p t u a l d r a w i n g o f t h e b o u n d a r y s t r e a m l i n e H»=l (a) j u s t a f t e r s e p a r a t i o n a n d <b> a f t e r r e a t t a c h m e n t . - 6 4 -C h a p t e r 8 T h e c a s e o-f mare c o m p l i c a t e d g e o m e t r i e s T h e c o r n e r s o l u t i o n , g i v e n i n t h e p r e v i o u s s e c t i o n , i s r e a d i l y e x t e n d e d t o c o a s t l i n e s w h i c h a r e c o m p o s e d o-f two o r more c o r n e r s , a s l o n g a s n o two c o r n e r s a r e c l o s e r t h a n a b o u t o n e ( n o n d i m e n s i o n a l ) u n i t . We i n v o k e t h i s r e s t r i c t i o n b e c a u s e o-f t h e n o n l i n e a r b e h a v i o u r o f t h e c o r n e r s o l u t i o n up t o d i s t a n c e s o f 0 . 5 f r o m t h e a p e x ( F i g u r e 1 3 b ) , a n d b e c a u s e , f o r r < l , t h e n o r m a l d e r i v a t i v e o f t h e d i f f r a c t i o n t e r m F ( 8 ) d o e s n o t v a n i s h a t t h e b o u n d a r y 0=0 ( F i g u r e 1 1 a ) . F o r t h e c a s e o f a c h a n n e l m o u t h , t h i s r e s t r i c t i o n o f minimum w i d t h c a n b e r e l a x e d i f t h e n o n l i n e a r t e r m s a r e s m a l l ( e < < l ) . I c h o s e t h e f o l l o w i n g e x a m p l e s t o d e m o n s t r a t e t h e m e t h o d . C i r c u l a t i o n i n t h e mouth o f a c h a n n e l L e B l o n d (1980) s h o w e d t h a t i n t h e c a s e o f w i d e c h a n n e l s , two b a r o c l i n i c j e t s c a n c o e x i s t i n d e p e n d e n t l y o n a p p o s i t e s i d e s o f t h e c h a n n e l . I n t h i s r e s p e c t , i t may b e c o n v e n i e n t t o c l a s s i f y c h a n n e l s i n t o 3 c a t e g o r i e s : i ) n a r r o w c h a n n e l s , w h i c h a r e n a r r o w e r t h a n t h e R o s s b y r a d i u s o f d e f o r m a t i o n ; i i ) i n t e r m e d i a t e c h a n n e l s , whose w i d t h s a r e b e t w e e n 1 a n d 3 R o s s b y r a d i i ; a n d , i i i ) w i d e c h a n n e l s , w i t h w i d t h s l a r g e r t h a n 3 R o s s b y r a d i i . In a c a s e when R o s s b y r a d i i a r e d i f f e r e n t o n o p p o s i t e - 65 -s i d e s o-f t h e c h a n n e l , some a v e r a g e v a l u e c a n be u s e d . In t h e a b s e n c e o f a d d i t i o n a l d y n a m i c a l c o n s t r a i n t s ( e . g . , a d v e r s e p r e s s u r e g r a d i e n t s ) , t h e l a s t c a t e g o r y , t h a t o f a w i d e c h a n n e l i s t r i v i a l , s i n c e t h e two j e t s do n o t i n t e r a c t a t a l l . In o r d e r t o g i v e a s i m p l e e x a m p l e , I c h o s e t h e c a s e o f a c h a n n e l whose s i d e s a r e a t a r i g h t a n g l e t o t h e c o a s t l i n e . We o r i e n t t h e a x e s s o t h a t t h e two s i d e s a r e p a r a l l e l t o t h e n e g a t i v e x a x i s , w i t h t h e o r i g i n h a l f w a y between t h e two c o r n e r s ( a s i n F i g u r e 7 ) . The s o l u t i o n S,0=p4 / 3 i s c o n s t r u c t e d a s a l i n e a r s u p e r p o s i t i o n o f two c o r n e r s o l u t i o n s , p = Btpt + B2P2 - (B. 1) As we h a ve shown i n t h e p r e v i o u s c h a p t e r , o u r s o l u t i o n s a r e n o t v a l i d i n s m a l l a r e a s a r o u n d e a c h s h a r p r e - e n t r a n t c o r n e r . We c a n o vercame t h i s p r o b l e m by r o u n d i n g o f f t h e s e c o r n e r s , s o t h a t t h e a r e a s i n q u e s t i o n a r e removed f r o m £ h e domain o f t h e s o l u t i o n . In o r d e r t o r o u n d o f f t h e apex o f e a c h c o r n e r , pi and p 2 were r e s c a l e d by f a c t o r s w h i c h we c a l l r e c e s s i o n p a r a m e t e r s , exp(1.5o\) and e x p ( 1 . 5 ^ 2 ) , r e s p e c t i v e l y . "Each ( i = l , 2 ) i s a c t u a l l y t h e d i s t a n c e between t h e s i d e o f t h e ( r e c e s s e d ) c o r n e r and t h e b o u n d a r y s t r e a m l i n e , ¥ = 1 (away f r o m t h e apex; f o r e x a m p l e , on F i g u r e s 19a,b, i t i s t h e d i s t a n c e between t h e b o u n d a r y x=0, y>2 and t h e r e s c a l e d s t r e a m l i n e V=l, s i n c e f o r x>0 and y>2 t h e p r e s e n c e o f a n o t h e r c o r n e r h a s n e g l i g i b l e e f f e c t an *¥) . The c o n s t a n t s Bi and B 2 a r e c a l c u l a t e d ( f o r t h i s c a s e o f - 66 -a nondivergent channel) from the requirement t h a t f a r i n s i d e the channel p (x ,d) = i , and, p(x,-d) = <1 + A ) 3 / 4 = 1 + B, (8.2a,b) where A>-1 i s the a d d i t i o n a l t r a n s p o r t out of the channel, 2d i s the width of the channel and B i s d e f i n e d by (8.2b). For l a r g e n e g a t i v e x, pi=expC-l.5(-y+d - M i ) 1 and p 2=expt-l.5(y+d+o^) 1. Assuming f o r s i m p l i c i t y t h a t pi and p 2 have equal r a d i i of deformation and t h a t Li=&*=&, we get from the 2—point matching, equations (8.1) and (8.2), Bi = exp (1.5,0 Ce 3 d - (1 + B) 3/2sinh (3d) , and, B 2 = exp(1.5<0 C (1 + B ) e 3 d - 13/2sinh(3d), (8.3a,b) so t h a t f a r i n s i d e the channel, where p i s independent of x, p=(l+B/2)cosh(1.5y)/cosh(1.5d)-(B/2)sinh(1.5y)/si nh(1.5d). (8.4) The r e s u l t i n g O ( l ) depth of the i n t e r f a c e , h 0=p 2' 3, i s shown f o r 5 d i f f e r e n t v a l u e s of d on F i g u r e s 18a (A=0) and 18b (A=l). I t i s compared t o the exact s o l u t i o n h (dashed l i n e ) from chapter 5. They are not e x a c t l y the same s i n c e the f u n c t i o n M no longer - 67 -v a n i s h e s i n s i d e t h e c h a n n e l a n d h 0 s a t i s f i e s t h e n o n l i n e a r e q u a t i o n ( 6 . 1 6 ) . T h e d i f f e r e n c e i s r e l a t i v e l y s m a l l , a n d i n s i g n i f i c a n t f o r n a r r o w (d<0 .5 ) o r w i d e (d>3) c h a n n e l s . T h e two s o l u t i o n s w o u l d b e t h e s a m e , i f we u s e d s i n ( 8 . 1 ) i n s t e a d o f p . B u t t h e n , i n o r d e r f o r t h e l i n e a r s u p e r p o s i t i o n t o b e v a l i d , we h a v e t o l i m i t o u r s e l v e s t o d > 0 . 5 . N o t e t h a t w h i l e t h i s 2 - p o i n t m a t c h i n g p r o c e d u r e i s n o t e x a c t , i t r e s u l t s i n o n l y m i n o r d i s t o r t i o n s o f b o u n d a r y s t r e a m l i n e s a n d i s v e r y s i m p l e t o i m p l e m e n t . ( O n l y i f d < 0 . 5 a n d A>>1 o r 1+A<<1, t h e d i s t o r t i o n may become l a r g e , i n w h i c h c a s e i t i s a d v i s a b l e t o u s e d i f f e r e n t r e c e s s i o n p a r a m e t e r s , a l a r g e r o n e f o r a c o r n e r w i t h a s m a l l e r t r a n s p o r t . ) N o t e t h a t e q u a t i o n ( 8 . 4 ) i s e s s e n t i a l l y G i l l ' s (1777) e q u a t i o n ( 5 . 2 ) , a n d i s a p p l i c a b l e t o a c h a n n e l o f s l o w l y v a r y i n g c r o s s — s e c t i o n . F i g u r e s 1 9 a , b show t h e r e s u l t i n g c i r c u l a t i o n , 4*o f o r t h e c a s e d = l , o = 0 . 1 , a n d f o r A=0 ( z e r o n e t o u t f l o w ) a n d A = l ( n e t o u t f l o w e q u a l 5 t o t h e u p s t r e a m t r a n s p o r t ) . O n l y a b o u t 30X i n t h e f i r s t c a s e (A=0) a n d a b o u t 40% i n t h e s e c o n d c a s e ( A = l ) o f t h e u p s t r e a m t r a n s p o r t i s r e c i r c u l a t e d a c r o s s t h e m o u t h a n d o u t o f t h e c h a n n e l . T h e r e s t o f t h e i n c o m i n g t r a n s p o r t c o n t i n u e s up t h e c h a n n e l . T h i s i s o f c o u r s e d u e t o t h e d y n a m i c a l c o n t r o l p l a c e d by t h e R o s s b y r a d i u s a n d t h e w i d t h o f t h e c h a n n e l u p o n t h e r e l a t i v e t r a n s p o r t s o n t h e a p p o s i t e s i d e s . D i f f e r e n t i a t i n g ( 8 . 4 ) we f i n d t h a t p h a s a minimum a t y Q , g i v e n b y t a n h ( 1 . 5 y 0 ) = C B / ( 2 + B ) 3 c o t a n h ( 1 . 5 d ) ( 8 . 5 ) - 68 -I n t h a t c a s e , ^ ( y o ) i s , i n e f f e c t , t h e f r a c t i o n o f t h e i n c o m i n g t r a n s p o r t t h a t i s d e f l e c t e d a c r o s s a n d o u t o f t h e c h a n n e l . I n p a r t i c u l a r , when B=0 ( A = 0 ) , y o = 0 a n d H , 0 ( y o ) = = C c o s h < 1 . 5 d ) ] " 3 . I f l + B > c o s h (3d) , t h e n y 0 > d , a n d t h e f l o w i s o u t o f t h e c h a n n e l e v e n o n t h e l e f t b a n k ( l o o k i n g d o w n s t r e a m ) . S i m i l a r l y , when 0 < 1 + B < s e c h ( 3 d ) , t h e n y 0 < - d , a n d t h e f l o w i s u n i f o r m l y i n t o t h e c h a n n e l . F i g u r e 2 0 a s h o w s t h e r e l a t i o n b e t w e e n A a n d y 0 f o r 3 v a l u e s o f d : 0 . 5 , 1 . 0 , a n d 1 . 5 ( w h i c h s p a n t h e i n t e r m e d i a t e w i d t h c a t e g o r y ) , w h i l e F i g u r e 20b d i s p l a y s c o r r e s p o n d i n g v a l u e s o f VQ ( y 0 ) = p ( y o ) * ' 3 , t h e f r a c t i o n o f t h e i n c o m i n g t r a n s p o r t t h a t i s r e c i r c u l a t e d o u t o f t h e c h a n n e l . A f e w r e m a r k s a r e i n o r d e r . F o r t h e c a s e o f a c h a n n e l w i t h d i v e r g i n g c o a s t l i n e s , t h e same 2 — p o i n t m a t c h i n g t e c h n i q u e c a n b e a p p l i e d t o t h e c o r n e r s t h e m s e l v e s , i n w h i c h c a s e 2d i s t h e d i s t a n c e b e t w e e n t h e two a p i c e s . T h i s i s d o n e i n t h e n e x t c h a p t e r , w h e r e we m o d e l t h e b a r o c l i n i c f l o w i n t h e m o u t h o f H u d s o n S t r a i t . I n g e n e r a l , i f i t i s deemed i m p o r t a n t , t h e 0 (e ) c o n t r i b u t i o n S i c a n b e c a l c u l a t e d u s i n g e i t h e r a n a r r o w c h a n n e l G r e e n ' s f u n c t i o n ( s e e B u c h w a l d , 1 7 7 1 ) o r v i a a n u m e r i c a l a p p r o a c h . I t s e f f e c t i s l i m i t e d , a s was s e e n i n c h a p t e r 7 , t o t h e i m m e d i a t e n e i g h b o r h o o d o f e a c h c o r n e r . F l o w a r o u n d a s q u a r e bump In o r d e r t o c o n s t r u c t a s o l u t i o n f o r t h e s e c o n d e x a m p l e , - 69 -we s t a r t w i t h two c o r n e r s o l u t i o n s : o n e -for t h e 3rr/2 <a=2/3) r e — e n t r a n t c o r n e r , w h i c h was d e s c r i b e d i n d e t a i l i n c h a p t e r 7 , a n d t h e o t h e r f o r a n i n s i d e rr /2 <a=2> c o r n e r . I n a d d i t i o n , we u s e s o u r c e a n d s i n k t r a n s p o r t s t r e a m - f u n c t i o n s , 2 F ( 9 ) a n d 2 F ( i r - 9 ) , •for a s t r a i g h t ( a= l ) c o a s t l i n e . T h e p r o c e d u r e i s b e s t d e s c r i b e d g r a p h i c a l l y , a s shown i n F i g u r e s 21a—f. We - f i r s t s u b t r a c t t h e s o u r c e a n d s i n k - f u n c t i o n s , 2F<9) a n d 2 F ( r r - 9 ) , f r o m t h e r e - e n t r a n t c o r n e r s o l u t i o n ( F i g u r e 2 1 a ) , o n e u n i t d i s t a n c e u p s t r e a m a n d d o w n s t r e a m o f t h e a p e x . T h i s g i v e s u s t h e s o u r c e a n d s i n k f l o w , shown o n F i g u r e 2 1 b . D e l e t i n g t h e s i n k f u n c t i o n 2 F ( n - 9 ) f r o m t h e i n s i d e c o r n e r s o l u t i o n ( F i g u r e 2 1 c ) , o n e u n i t d i s t a n c e u p s t r e a m o f t h e a p e x , y i e l d s t h e s o u r c e f l o w , shown o n F i g u r e 2 1 d . C o m b i n i n g t h e l a s t s o u r c e w i t h t h e s i n k o n F i g u r e 21b g i v e s u s t h e s o u r c e f l o w , w h i c h i s shown o n F i g u r e 2 1 e . F i n a l l y , c o m b i n i n g t h i s s o u r c e w i t h i t s i m a g e ( i n t h e x - a x i s ) s i n k a n d r a i s i n g t o t h e power 4 / 3 r e s u l t s i n ¥<>, t h e f l o w a b o u t t h e 2 x 2 s q u a r e b u m p , a s shown o n F i g u r e 2 1 f . H o w e v e r , s u b t r a c t i n g a s o u r c e a n d a s i n k a t a p o i n t o n a s t r a i g h t b o u n d a r y i s e q u i v a l e n t t o s u b t r a c t i n g a n e x p o n e n t i a l p r o f i l e f r o m t h a t b o u n d a r y . I n o t h e r w o r d s , i n s t e a d o f u s i n g s o u r c e s a n d s i n k s i n t h e a b o v e p r o c e d u r e , o n e c a n s i m p l y a d d two c o r n e r s o l u t i o n s w i t h a common b o u n d a r y , a n d t h e n s u b t r a c t t h e r e s u l t i n g s u r p l u s e x p o n e n t i a l c u r r e n t p r o f i l e f r o m t h a t b o u n d a r y . T h e r e s t r i c t i o n o f t h e common b o u n d a r y b e i n g a t l e a s t o n e u n i t l o n g , s t a n d s a s b e f o r e , s i n c e t h e p r o c e d u r e i s n o l o n g e r v a l i d c l o s e t o t h e c o r n e r s , w h e r e t h e p r o f i l e i s n o t e x p o n e n t i a l . - 70 -I t i s t r i v i a l t o a d d a c u r r e n t i n t h e a p p o s i t e d i r e c t i o n , w h i c h i s ( s a y ) b o u n d e d b y a w a l l 2 u n i t s away f r o m t h e b u m p . T h e r e s u l t i n g b a r o c l i n i c c i r c u l a t i o n a r o u n d a s q u a r e bump i n a w i d e c h a n n e l i s shown o n F i g u r e s 21g ( t h e c a s e o f e q u a l t r a n s p o r t s ) a n d 21h ( d o u b l e t r a n s p o r t i n t h e o p p o s i t e d i r e c t i o n ) . N o t e t h a t a b o u t 30% o f t h e i n c o m i n g t r a n s p o r t i n t h e f i r s t c a s e , a n d a b o u t 40% i n t h e s e c o n d c a s e i s d i v e r t e d b a c k u p s t r e a m b y t h e o p p o s i n g c u r r e n t . T h i s i s t h e same r e c i r c u l a t i o n a s i n t h e c a s e o f t h e c h a n n e l m o u t h ( F i g u r e s 1 9 a T b ) a n d i s c a u s e d b y t h e d y n a m i c a l c o n s t r a i n t d u e t o t h e s c a l e o f t h e R o s s b y r a d i u s o f t h e f l o w . - 71 -F i g u r e I B . h o ^ p 2 ' 3 i n s i d e a c h a n n e l ( s o l i d l i n e ) , c o m p a r e d t o t h e e x a c t s o l u t i o n h ( d a s h e d , f r o m c h a p t e r 5 ) , f o r s e v e r a l v a l u e s o f w i d t h , d , a n d f o r two v a l u e s o f t h e e x c e s s t r a n s p o r t s (a) A=0 a n d (b) A = l . F i g u r e 19 . C i r c u l a t i o n i n t h e mouth o f a c h a n n e l s c o n t o u r s o f t h e t r a n s p o r t s t r e a m f u n c t i o n f o f o r d = 1 . 0 , 4=0.Is (a) A=0 , <b> A = l . - 73 -g u r e 2 0 . L o c a t i o n o f t h e s t a g n a t i o n s t r e a m l i n e , y 0 a n d i t s v a l u e t h e r e M»o<yo>, a s f u n c t i o n s o f t h e t r a n s p o r t p a r a m e t e r A , f o r 3 v a l u e s o f c h a n n e l w i d t h : 2 d = l , 2d=2 a n d 2d=3 . - 74 -F i g u r e 2 1 . ( a ) - ( - f ) C o n s t r u c t i o n o f t h e s o l u t i o n f o r a f l o w a r o u n d a s q u a r e bump i n a c o a s t l i n e . A s q u a r e bump i n a c h a n n e l f o r <g> A=0 a n d (h) A = l . - 76 -C h a p t e r 9 B a r o c l i n i c c i r c u l a t i o n i n H u d s o n S t r a i t I n t h i s C h a p t e r we u s e t h e m e t h o d o u t l i n e d i n t h e p r e v i o u s C h a p t e r t o m o d e l t h e b a r o c l i n i c . c i r c u l a t i o n n e a r t h e m o u t h o-f H u d s o n S t r a i t . I t i s e x p e c t e d t h a t s u c h a m o d e l w o u l d p r o v i d e a n a d e q u a t e d e s c r i p t i o n o-f a n u p p e r l a y e r f l o w a b o v e a q u i e s c e n t l o w e r l a y e r , o r w h e n e v e r t h e c h a n g e s i n b a t h y m e t r y a r e s m a l l e n o u g h , s o t h a t a m o d a l d e c o m p o s i t i o n i s v a l i d . S i n c e t h e b a t h y m e t r i c c h a n g e s i n t h e mouth o f H u d s o n S t r a i t a r e n o t s m a l l ( F i g u r e s 3b a n d 2 2 a ) , we may a l t e r n a t i v e l y c o n s i d e r t h e p r e s e n t m o d e l t o r e p r e s e n t o n l y t h e " b a r o c l i n i c p a r t " o f t h e t o t a l f l o w i n a n d o u t o f t h e s t r a i t . F o r t h e p u r p o s e s o f t h i s m o d e l ( i n o r d e r t o p r o v i d e u p s t r e a m c o n d i t i o n s ) , we r e q u i r e e s t i m a t e s o f t r a n s p o r t s a n d o f l a t e r a l s c a l e s f o r v a r i o u s c u r r e n t s i n s i d e a n d u p s t r e a m o f H u d s o n S t r a i t . Where a v a i l a b l e , we r e l y o n h i s t o r i c a l o c e a n o g r a p h i c d a t a t o o b t a i n t h e s e q u a n t i t i e s . O t h e r w i s e , we must make an i n t e l l i g e n t g u e s s . T h e e x i s t i n g h y d r o g r a p h i c a n d s u r f a c e d r o g u e d a t a ( C a m p b e l l , 1959 ; D r i n k w a t e r , 1 9 S 3 , 1985 ; Q s b o r n e t a l . , 1978; L e B l o n d e t a l . , 1 9 8 1 , a n d r e f e r e n c e s t h e r e i n ) a n d s a t e l l i t e i m a g e r y c o m b i n e t o g i v e u s a c o n s i s t e n t p i c t u r e o f t h e l a t e summer r e s i d u a l c i r c u l a t i o n i n t h e s t r a i t . T h e B a f f i n C u r r e n t f l o w s s o u t h a l o n g t h e c o a s t o f B a f f i n I s l a n d ( s e e F i g u r e 2 2 a ) . - 77 -At about 63°N i t branches into two parts. The eastern part continues south past the mouth o-f Hudson S t r a i t , while the western branch enters the s t r a i t on either side o-f Resolution Island. Inside the s t r a i t , one part o-f the -flow i s r e c i r c u l a t e d across and out o-f the channel. The remainder continues along i t s northern boundary and i s gradually drawn across the s t r a i t into an estuarine type c i r c u l a t i o n , driven by the s i g n i f i c a n t melt—water outflow from Hudson Bay. This water forms the current on the south side of the s t r a i t . It i s fresher, colder and more strongly s t r a t i f i e d than the one on the northern side. From the s a l i n i t y and temperature sections across Ungava Bay (see Figure 23, from Campbell, 1959), i t i s apparent that a s i g n i f i c a n t part of t h i s current turns the corner at Cape Hope's Advance and enters Ungava Bay west of Akpatok Island. The remainder joins the above—mentioned c r o s s — s t r a i t flow north of Akpatok Island and e x i t s Hudson S t r a i t past Cape Chidley together with the coastal current from Ungava Bay. Additional evidence for the Ungava Bay cyclonic gyre i s provided by thermal s a t e l l i t e imagery (an example of which i s shown on Figure 24) and also by the motion of i c e into and out of the Bay. An example of the l a t t e r i s shown on Figure 25 (from Bray et a l . , 1985). The transports Drinkwater (1985) made current measurements (Figure 3) and c o l l e c t e d CTD data (Figure 26) i n the eastern Hudson S t r a i t . His - 78 -c a l c u l a t i o n s o f t h e t o t a l t r a n s p o r t show t h a t a b o u t 1 . 2 x l 0 * m 3 / s i s c r o s s i n g t h e c h a n n e l f r o m n o r t h t o s o u t h b e t w e e n h i s m o o r i n g s HS3 a n d HS5 ( s e e F i g u r e 3 ) . S i m i l a r a m o u n t s w e r e g i v e n f o r f l o w s a l o n g t h e n o r t h ( 0 . 8 2 x 1 0 * m 3 / s ) a n d s o u t h ( 0 . 9 3 x 1 0 * m 3 / s ) s i d e s . T h i s i n d i c a t e s t h a t a t o t a l o f a b o u t 2 x l 0 * m 3 / s e n t e r s a n d e x i t s t h e e a s t e r n e n t r a n c e o f t h e s t r a i t , w h i c h i n c l u d e s b o t h t h e b a r o c l i n i c a n d t h e b a r o t r o p i c p a r t s o f t h e f l o w . T h e r e a r e n o t e n o u g h d a t a t o c a l c u l a t e t h e t r a n s p o r t s o f o t h e r c u r r e n t b r a n c h e s i n t h e s t r a i t . N e v e r t h e l e s s , i t may b e l o g i c a l o n t h e b a s i s o f d r o g u e t r a c k s ( F i g u r e 2b) a n d s a t e l l i t e i m a g e r y ( F i g u r e 24) t o a s s u m e t h a t t h e f l o w t h a t c o n t i n u e s w e s t w a r d , o n t h e n o r t h s i d e o f t h e c h a n n e l , c o m e s m a i n l y f r o m t h e s h e l f w a t e r e a s t o f B a f f i n I s l a n d a n d , h e n c e , t h a t m o s t o f i t e n t e r s t h r o u g h G a b r i e l S t r a i t . T h i s w o u l d t h e n i n d i c a t e t h a t t h e a p p r o x i m a t e t r a n s p o r t s on e i t h e r s i d e o f t h e R e s o l u t i o n I s l a n d a r e o f t h e o r d e r o f 1 0 , i m 3 / s e a c h . S i m i l a r l y , we may a s s u m e t h a t a l a r g e p a r t o f t h e 0 . 9 x l 0 < b m 3 / s f l o w i n g e a s t w a r d a t C a p e H o p e ' s A d v a n c e e n t e r s U n g a v a B a y e a s t o f A k p a t o k I s l a n d , w h i l e more t h a n l x l O A m 3 ' / s , m o s t l y f r o m t h e c r o s s - c h a n n e l f l o w , r o u n d s t h i s i s l a n d o n i t s n o r t h e r n s i d e . T h i s c o m p l e t e s t h e t o t a l t r a n s p o r t e s t i m a t e s . S i n c e we a r e a t t e m p t i n g a b a r o c l i n i c m o d e l , we a r e more i n t e r e s t e d i n e s t i m a t e s o f g e o s t r o p h i c t r a n s p o r t s , w h i c h y i e l d t h e b a r o c l i n i c p a r t o f t h e f l o w . T h e s e a r e c a l c u l a t e d f r o m t h e t e m p e r a t u r e a n d s a l i n i t y t r a n s e c t s , a s s u m i n g e x i s t e n c e o f a l e v e l o f n o m o t i o n ( s a y , a t 2 0 0 m ) . C a m p b e l l ' s (1958) g e o s t r o p h i c c a l c u l a t i o n s f o r J u l y 1956 g a v e s o u t h e a s t t r a n s p o r t s a l o n g t h e - 79 -Q u e b e c s i d e o-f 0 . 6 x l 0 * m 3 / s a n d n o r t h e a s t t r a n s p o r t s a l o n g t h e B a f f i n I s l a n d c o a s t o-f 0 . 3 x l 0 6 m 3 / s . He a d m i t t e d t h a t t h e s e m i g h t b e u n d e r e s t i m a t e s , s i n c e h i s s t a t i o n s w e r e n o t c l o s e e n o u g h t o t h e s h o r e . D r i n k w a t e r ' s <1985) c a l c u l a t i o n s y i e l d e d c o m p a r a b l e v a l u e s o-f 0 . 6 6 x 1 0 6 m 3 / s a n d 0 . 2 6 x 1 0 < > m 3 / s , r e s p e c t i v e l y . A f t e r a d j u s t i n g t o c u r r e n t m e t e r r e a d i n g s ( a t 200m a t HS1 a n d a t 100m a t H S 4 ) , D r i n k w a t e r g a v e r e v i s e d t o t a l t r a n s p o r t f i g u r e s o f 0 . 5 9 x l 0 * m 3 f o r t h e n o r t h s i d e a n d v i r t u a l l y u n c h a n g e d 0 . 6 6 x I O ^ I T ^ / S f o r t h e s o u t h s i d e o f t h e s t r a i t . I n o u r b a r o c l i n i c m o d e l , we w i l l a s s u m e t h a t a b o u t 0 . 6 x l 0 6 m 3 / s e n t e r s U n g a v a B a y a r o u n d C a p e H o p e ' s A d v a n c e , w h i l e 0 . 4 x l 0 * m 3 / s r o u n d s A k p a t o k I s l a n d o n i t s n o r t h s i d e , s o t h a t t h e t o t a l b a r o c l i n i c o u t f l o w f r o m t h e s t r a i t i s a b o u t 1 0 6 m 3 / s . I n a d d i t i o n , we " g u e s s " a v a l u e o f 0 . 3 x l 0 * m 3 / s e n t e r i n g t h e s t r a i t a l o n g t h e s o u t h s i d e o f R e s o l u t i o n I s l a n d . . T h e s e v a l u e s o f b a r o c l i n i c t r a n s p o r t s a r e m a r k e d b e s i d e t h e c o r r e s p o n d i n g c u r r e n t a r r o w s o n F i g u r e 2 2 a . L a t e r a l s c a l e s We now t u r n o u r a t t e n t i o n t o t h e l a t e r a l s c a l e s o f t h e c u r r e n t s . We u s e t h e m e a s u r e d o f f s h o r e v a l u e s o f d e p t h , s t r a t i f i c a t i o n a n d v e l o c i t y i n o r d e r t o c a l c u l a t e X a c c o r d i n g t o ( 4 . 1 ) . F o r t h e s o u t h s i d e o f t h e s t r a i t ( n e a r C a p e H o p e ' s A d v a n c e ) , D r i n k w a t e r ' s c a l c u l a t i o n s g a v e C = 0 . 8 m / s a n d R=6km. U s i n g h i s HS1 30m v a l u e o f V = 0 . 3 m / s , we g e t F = 0 . 3 7 a n d , f r o m - so -( 4 . 1 ) , X=18km. We c a n a l s o e s t i m a t e X u s i n g t h e e x p o n e n t i a l p r o f i l e f o r v e l o c i t y v ( x ) ( e q u a t i o n ( 4 . 3 ) ) a n d t h e t w o a l o n g - s t r a i t 30m v e l o c i t y v a l u e s a t HS1 a n d H S 2 , w h i c h a r e 0 . 2 9 m / s a n d 0 . 0 7 m / s , r e s p e c t i v e l y . S i n c e t h e d i s t a n c e b e t w e e n HS2 a n d HS1 m o o r i n g s i s a b o u t 28km ( s e e F i g u r e 3 ) , we g e t 2 8 k m / X = l n ( 0 . 2 9 / 0 . 0 7 ) , w h i c h g i v e s X=20km, i n g o o d a g r e e m e n t w i t h t h e 18km a b o v e . We a l s o n o t e t h a t D r i n k w a t e r ' s m o o r i n g HS1 i s l o c a t e d n e a r t h e C a p e H o p e ' s A d v a n c e w h e r e a t l e a s t p a r t o f t h e f l o w t u r n s t h e c o r n e r i n t o U n g a v a Bay a n d , i t may b e n a r r o w e r t h e r e t h a n t h e f l o w f a r t h e r u p s t r e a m . D e s p i t e d i f f e r e n t w a t e r m a s s e s a n d w e a k e r s t r a t i f i c a t i o n o n t h e n o r t h s i d e o f t h e s t r a i t ( F i g u r e 2 6 ) , t h e l a t e r a l s c a l e t h e r e may b e c o m p a r a b l e t o t h a t o n t h e s o u t h s i d e . T h i s i s b e c a u s e o f t h e i n v e r s e d e p e n d e n c e o n V ( e q u a t i o n ( 4 . 1 ) ) ; i n o r d e r t o c o n s e r v e t h e t o t a l t r a n s p o r t , s l o w e r f l o w s m u s t b e w i d e r a n d d e e p e r . T o s e e t h i s , l e t u s u s e A ° = 0 . 3 k g / m 3 ( c o m p a r e d t o 1 . 5 k g / m 3 o n t h e s o u t h s i d e ; s e e F i g u r e 26) a n d a s s u m e a r e a s o n a b l e v a l u e o f H=90m, w h i c h i s h a l f t h e d e p t h a t H S 4 . T h e n , u s i n g t h e 30m (HS4) v a l u e o f V = 0 . 0 9 m / s , we g e t f r o m ( 4 . 1 ) a c o m p a r a b l e v a l u e , X=25km. We a l s o n o t e t h a t s i n c e HS4 i s a b o u t 20km o f f s h o r e , D r i n k w a t e r ' s m e a s u r e m e n t s may h a v e m i s s e d most o f t h e b a r o c l i n i c wedge a t t h a t l o c a t i o n . B e c a u s e o f l a r g e u n c e r t a i n i t i e s i n t h e m e t h o d u s e d t o c a l c u l a t e t h e s e v a l u e s o f X , a n d i n o r d e r t o k e e p o u r m o d e l s i m p l e , we w i l l a s s u m e t h a t X=20km o n b o t h s i d e s o f t h e s t r a i t , l i o r o v e r , i n o r d e r t o s e e t h e e f f e c t o f l a r g e r s c a l e s , we w i l l - 81 -r e p e a t o u r c a l c u l a t i o n s w i t h a l a r g e r v a l u e , X=30km. T h e f o l l o w i n g a r g u m e n t l e n d s s u p p o r t t o a l s o u s i n g a h i g h e r v a l u e o f X i n o u r m o d e l . T h e t r a n s p o r t i n a m o d e l b a r o c l i n i c wedge p r o f i l e ( 4 . 1 ) i s g i v e n b y Q = V H X / 2 = g ' H 2 / 2 f , ( 9 . 1 ) w h i c h f o r t h e s o u t h e r n s i d e o f t h e s t r a i t , h e a r C a p e H o p e ' s A d v a n c e , g i v e s ( u s i n g D r i n k w a t e r ' s v a l u e s , e x t r a p o l a t e d t o w i t h i n 2km o f f s h o r e , H=75m a n d V = 0 . 4 5 m / s ) Q=0.34x 1 0 * m s / s , w h i c h i s a b o u t h a l f o f D r i n k w a t e r ' s g e o s t r o p h i c ( r e l a t i v e t o 200m) t r a n s p o r t v a l u e o f 0 . 6 6 x 1 0 * m 3 / s . Dn t h e n o r t h s i d e ( n e a r H S 4 ) , a s i m i l a r c a l c u l a t i o n ( w i t h H=90m, X=20km a n d , V = 0 . 1 9 m / s , e x t r a p o l a t e d t o w i t h i n 5km o f f s h o r e , s i n c e t h e b o t t o m i s l e s s s t e e p t h e r e ) a l s o g i v e s Q = 0 . 3 4 x l O ^ m ' / s , w h i c h i s c o m p a r a b l e t o D r i n k w a t e r ' s v a l u e o f 0 . 2 6 x 1 0 * m 3 / s . T h e r e i s a d i s a g r e e m e n t b e t w e e n t h e two m e t h o d s u s e d t o c a l c u l a t e Q o n t h e s o u t h s i d e , n e a r C a p e H o p e ' s A d v a n c e . One p o s s i b l e r e a s o n f o r t h i s i s t h a t we may h a v e u n d e r e s t i m a t e d X . F r o m t h e c r o s s — c h a n n e l d e n s i t y c o n t o u r s ( F i g u r e 2 6 , i f t h e s i g m a - t s u r f a c e o f 2 5 . 7 5 i s t a k e n a s t h e i n t e r f a c e ) we may i n f e r t h a t t h e c r o s s - s t r e a m s c a l e i s c l o s e r t o X=35km. T h e n , u s i n g t h e same e x t r a p o l a t e d v a l u e s o f H=75m a n d V = 0 . 4 5 m / s , we g e t a r e v i s e d f i g u r e o f Q = 0 . 5 9 x 1 0 4 m 3 , w h i c h i s much c l o s e r t o D r i n k w a t e r ' s 0 . 6 6 x l O A m 3 / s , o r t o C a m p b e l l ' s (1958) v a l u e o f 0 . 6 x l 0 * m 3 / s . T h i s l a s t c a l c u l a t i o n l e n d s e x t r a s u p p o r t f o r t r y i n g b o t h v a l u e s , - 82 -X=20km a n d X=30km i n o u r m o d e l . T h e c i r c u l a t i o n We now p r o c e e d t o c o n s t r u c t a n a l y t i c a l s o l u t i o n s o f t h e b a r o c l i n i c c i r c u l a t i o n i n t h e m o u t h o f H u d s o n S t r a i t . S i n c e t h e R o s s b y number o f t h e f l o w u p s t r e a m o f C a p e C h i d l e y i s o f t h e o r d e r o f e=0 .2 ( b a s e d o n F = 0 . 3 7 , f r o m D r i n k w a t e r ' s (1985) v a l u e s ) , t h e e x t e n t o f t h e c e n t r i f u g a l u p w e l l i n g i s q u i t e s m a l l ( l e s s t h e n 0 . 1 X i n t h i s c a s e , w h i c h i s 3km a t m o s t ) , a n d i s e v e n s m a l l e r a t o t h e r c a p e s . H e n c e , f o r c o r n e r s w i t h r a d i i o f c u r v a t u r e o f t h e o r d e r o r g r e a t e r t h a n 0 . 1 X , t h e c e n t r i f u g a l u p w e l l i n g i s n e g l i g i b l e . T h e r a d i u s o f c u r v a t u r e o f t h e C a p e C h i d l e y c o a s t l i n e ( o r e v e n o f t h e B u t t o n I s l a n d s g r o u p ) i s o f t h e o r d e r o f 10km. H e n c e , e x c e p t f o r t h e e x t r a s p r e a d i n g o f s t r e a m l i n e s n e a r t h e a p e x , t h e 0 ( 1 ) s o l u t i o n i s q u i t e a d e q u a t e t o d e s c r i b e t h e b a r o c l i n i c f l o w a r o u n d C a p e C h i d l e y . T h e same a p p l i e s t o o t h e r c a p e s i n o u r m o d e l , s u c h a s t h e s o u t h e r n t i p o f R e s o l u t i o n I s l a n d . U s i n g t h e m e t h o d o f a l i n e a r s u p e r p o s i t i o n o f r e c e s s e d c o r n e r s o l u t i o n s , w h i c h was d e s c r i b e d i n t h e p r e v i o u s c h a p t e r , we c o n s t r u c t t h e c o n t o u r s o f t h e f u n c t i o n p i n t h e 320x320km a r e a , c e n t e r e d a t t h e m i d p o i n t o f t h e mouth o f H u d s o n S t r a i t ( F i g u r e 2 2 b ) . We a p p r o x i m a t e t h e 100m c o n t o u r s ( w h e r e a v a i l a b l e ; o r e l s e , we i n t e r p o l a t e i n s i d e t h e 200m c o n t o u r s ; s e e F i g u r e 22a) w i t h s t r a i g h t l i n e s e g m e n t s , w h i c h a r e c o m b i n e d t o r e p r e s e n t t h e t i p - 83 -o-f R e s o l u t i o n I s l a n d , t h e L a b r a d o r P e n i n s u l a a n d t h e s h a l l o w ( a b o u t lOOm d e e p ) b a n k e a s t o-f A k p a t o k I s l a n d ( s e e F i g u r e 22b) . T h e 100m v a l u e was c h o s e n , s i n c e n o c h a r t s w i t h l e s s e r c o n t o u r s a r e a v a i l a b l e a n d a l s o b e c a u s e t h e b o u n d a r y v a l u e H i s c l o s e t o t h i s v a l u e . T h e r e s u l t i n g c o n t o u r s o-f h = p 2 / 3 a n d "r=h 2 a r e shown f o r X=20km o n F i g u r e s 2 7 a , b , a n d f o r X=30km o n F i g u r e s 2 7 c , d . T h e t r a n s p o r t s w e r e n o r m a l i z e d t o t h e b a r o c l i n i c o u t f l o w a r o u n d C a p e C h i d l e y , w h i c h was a s s u m e d t o b e a b o u t 1 0 6 m 3 / s ( c o m p o s e d o f 0 . 6 x l 0 * m 3 / s f r o m U n g a v a B a y a n d t h e m o r e u n c e r t a i n v a l u e o f 0 . 4 x l 0 6 m 3 / s f r o m t h e n o r t h s i d e o f A k p a t o k I s l a n d ; a l t e r n a t i v e l y , t h e l a s t v a l u e may b e t r e a t e d a s a n o t h e r p a r a m e t e r o f t h e m o d e l ) . We a l s o s e l e c t e d t h e b a r o c l i n i c p a r t o f t h e t r a n s p o r t o n t h e s o u t h s i d e o f R e s o l u t i o n I s l a n d t o b e 0 . 3 x l 0 * m 3 / s , w h i c h i s c o m p a r a b l e t o t h e g i v e n v a l u e o f t h e g e o s t r o p h i c t r a n s p o r t on t h e n o r t h s i d e o f H u d s o n S t r a i t . C o m p a r i s o n b e t w e e n F i g u r e s 27a—d a n d t h e s igma—t s e c t i o n i n F i g u r e 5 s e e m s t o i n d i c a t e t h a t t h e h i g h e r v a l u e o f X=30km f i t s t h e d a t a b e t t e r . T h i s i s e s p e c i a l l y t r u e f o r t h e s o u t h s i d e o f t h e s t r a i t , w h e r e t h e r e i s a more p r o n o u n c e d t w o — l a y e r s t r u c t u r e . We c a n c o n c l u d e t h i s s e c t i o n b y s a y i n g t h a t d e s p i t e o u r a s s u m p t i o n s o f a u n i f o r m s t r a t i f i c a t i o n a n d o f e q u a l s c a l e s f o r t h e two s i d e s o f t h e s t r a i t a n d d e s p i t e " g u e s s i n g " t h e v a l u e o f H* on t h e n o r t h s i d e , we g o t a g o o d q u a l i t a t i v e a g r e e m e n t b e t w e e n t h e m o d e l a n d t h e known c i r c u l a t i o n p a t t e r n n e a r t h e mouth o f H u d s o n S t r a i t . I n o r d e r t o h a v e a n e v e n b e t t e r a g r e e m e n t , i t may b e n e c c e s s a r y t o h a v e a m o d e l w h i c h c o m b i n e s t w o ( o r more) m o v i n g - 84 -l a y e r s w i t h t h e r e a l i s t i c b a t h y m e t r y o-f t h e s t r a i t . S u c h t a s k i s b e s t a c c o m p l i s h e d u s i n g n u m e r i c a l m e t h o d s . (a) (b) F i g u r e 2 2 . (a) T h e e a s t e r n H u d s o n S t r a i t a n d U n g a v a B a y a r e a . O n l y 200m a n d ' 100m (where a v a i l a b l e ) c o n t o u r s a r e s h o w n . T h e d a s h e d box c o n t a i n s t h e a r e a c o v e r e d b y t h e p r e s e n t b a r o c l i n i c m o d e l . A r r o w s w i t h n u m b e r s show d i r e c t i o n s a n d a p p r o x i m a t e v a l u e s <X10*m 3 / s ) o-f b a r o c l i n i c t r a n s p o r t s ( s e e t h e t e x t ) . P l a c e names a b b r e v i a t i o n s : CHA - C a p e H o p e ' s A d v a n c e , A - A k p a t o k I s l a n d , R - R e s o l u t i o n I s l a n d , C C - C a p e C h i d l e y , B - B u t t o n I s l a n d s a n d G - G a b r i e l S t r a i t , (b) S i m p l i f i e d m o d e l b o u n d a r i e s , w i t h a d o p t e d v a l u e s o f H*. 86 -F i g u r e 2 3 . E a s t - w e s t s e c t i o n s o-f t e m p e r a t u r e a n d s a l i n i t y a c r o s s t h e m o u t h o f U n g a v a B a y ( r e p r o d u c e d w i t h p e r m i s s i o n f r o m C a m p b e l l , 1 9 5 8 ) . - 8 7 -F i g u r e 24 ( n e x t p a g e ) . An e x a m p l e o f t h e r m a l s u r f a c e s i g n a t u r e o f c i r c u l a t i o n i n H u d s o n S t r a i t a n d U n g a v a B a y . T h e c o l d e s t w a t e r (—2 d e g C) i s d a r k b l u e , w h i l e t h e w a r m e s t ( a b o u t 5 d e g C) i s d e e p r e d . C l o u d s , f o g a n d i c e a r e a t a b o u t t h e same t e m p e r a t u r e , t h e y w e r e g i v e n h e r e a b r i g h t b l u e s h a d e . L a n d ( b l a c k ) f e a t u r e s c a n b e i d e n t i f i e d f r o m F i g u r e 2 2 . - 89 -PERCENTAGE ICE COVER O > 90X Q 30 - SOX 70 - 90X E 3 to - 30X EH SO - 70X < 10X 6=3 Near short tea pack H Land- last ica 5& Open water laada & flasurea © Ica lloa ,»»" Shear zona ^ Marina currant — Direction ol prevailing wmd Direction ol ice drill on dale hdicated F i g u r e 2 5 . Map o-f t h e i c e c o v e r i n U n g a v a B a y a n d e a s t e r n H u d s o n S t r a i t o n J u l y 2 n d , 1974 ( r e p r o d u c e d w i t h p e r m i s s i o n •from B r a y e t a l . , 1 9 8 5 ) . - 90 -F i g u r e 2 6 . T e m p e r a t u r e , s a l i n i t y a n d s i g m a - t t r a n s e c t s a c r o s s H u d s o S t r a i t i n A u g . 1982 . T h e s e t r a n s e c t s w e r e made a l o n g t h e l i n e o c c u p i e d b y m o o r i n g s H S 1 - H S 4 , shown o n F i g u r e 3 a ( r e p r o d u c e d w i t h p e r m i s s i o n -from D r i n k w a t e r , 1 9 8 5 ) . F i g u r e 2 7 . B a r o c l i n i c model c o n t o u r s o f t h e 0 ( 1 ) d e p t h ho a n d t h e 0<1) t r a n s p o r t s t r e a m f u n c t i o n H»o f o r <a,b) X=20km a n d <c ,d) X=30km. A s b e f o r e , c o n t o u r s p a c i n g i s 0 . 1 , e x c e p t t h a t o n e e x t r a 0 . 0 1 c o n t o u r was a d d e d t o p l o t s o f *P0 ( d a s h e d l i n e ) , i n d i c a t i n g s c h e m a t i c a l l y t h e e d g e o f t h e f l o w , w h e r e h 0 = 0 . 1 . C h a p t e r 10 D i s c u s s i o n F r o m t h e n o n l i n e a r c o n s e r v a t i o n e q u a t i o n s t h a t g o v e r n an i n v i s c i d u p p e r — l a y e r f l o w i n a t w o — l a y e r r o t a t i n g f l u i d , we o b t a i n e d t h e f i r s t two t e r m s o f t h e p o w e r s e r i e s w h i c h r e p r e s e n t s t h e s o l u t i o n t o t h e p r o b l e m o f a f l o w a r o u n d an a r b i t r a r y c o r n e r . I t was f o u n d t h a t t h e 0 ( 1 ) t e r m i s g e o s t r o p h i c a n d , e x c e p t f o r n a r r o w i n g ( w i d e n i n g ) o f t h e f l o w n e a r a r e — e n t r a n t ( i n s i d e ) c o r n e r , i t d o e s n o t d i f f e r q u a l i t a t i v e l y f r o m t h e f l o w u p s t r e a m . T h e e f f e c t s o f t h e n o n l i n e a r ( a d v e c t i o n ) t e r m s i n t h e e q u a t i o n o f m o t i o n b e c o m e q u i t e e v i d e n t when t h e 0 (e ) t e r m i s a d d e d t o t h e f i r s t : t h e f l o w w i d e n s ( f o r r e - e n t r a n t c o r n e r s ) d u e t o t h e c e n t r i f u g a l a c c e l e r a t i o n o f t h e f l u i d a n d , i n a d d i t i o n , t h e d e p t h o f t h e u p p e r l a y e r d e c r e a s e s t o a l m o s t z e r o n e a r t h e apex o f t h e c o r n e r ( a l t h o u g h we d i d n o t c a l c u l a t e t h i s e x p l i c i t l y , we a s s u m e d t h a t n e a r i n s i d e c o r n e r s t h e e f f e c t i s o p p o s i t e ; t h e 0 ( c ) t e r m s c a u s e some i n c r e a s e i n t h e d e p t h ) . T h i s i s a m a n i f e s t a t i o n o f t h e c e n t r i f u g a l u p w e l l i n g ( d o w n w e l 1 i n g ) . T h i s u p w e l l i n g i s o n l y i m p o r t a n t f o r r e l a t i v e l y s h a r p c o r n e r s . T h e v a l i d i t y o f t h e s o l u t i o n a n d t h e c o n t i n u i t y o f t h e b o u n d a r y s t r e a m l i n e ( i . e . , t h e a b s e n c e o f s e p a r a t i o n ) r e q u i r e t h a t t h e s h a r p apex o f t h e r e — e n t r a n t c o r n e r b e b l u n t e d t o a c e r t a i n e x t e n t . T h i s c a n b e s u c c e s f u l l y a c c o m p l i s h e d i f o n e o f t h e n e i g h b o r i n g s t r e a m l i n e s i s a s s u m e d t o b e t h e p h y s i c a l b o u n d a r y , t h e c h o i c e d e p e n d i n g - 94 -u p o n t h e r e q u i r e d r a d i u s o-f c u r v a t u r e . U s i n g t h e s e r e c e s s e d c o r n e r s o l u t i o n s we w e r e a b l e t o g e n e r a t e s t r e a m l i n e s o-f p l a u s i b l e f l o w s i n more g e n e r a l d o m a i n s : i n s i d e c h a n n e l m o u t h s a n d a r o u n d c o a s t a l i n d e n t a t i o n s . T h e r e l a t i v e s i m p l i c i t y o f t h e m o d e l a s w e l l a s t h e s c a r c i t y o f d a t a d o n o t a l l o w f o r more t h a n a q u a l i t a t i v e c o m p a r i s o n b e t w e e n t h e t h e o r y a n d a c t u a l b a r o c l i n i c c o r n e r f l o w s i n t h e v i c i n i t y o f t h e mouth o f H u d s o n S t r a i t . N e v e r t h e l e s s , t h e r e i s f a i r l y g o o d a g r e e m e n t b e t w e e n t h e m o d e l f l o w a n d t h e b a r o c l i n i c c i r c u l a t i o n i n t h e e a s t e r n p a r t o f t h e s t r a i t . T h e r e i s n o c o n t r a d i c t i o n b e t w e e n t h e t r a c k s o f d r o g u e s t h a t w e r e r e l e a s e d n e a r t h e mouth o f H u d s o n S t r a i t ( f i g u r e 2b ) a n d t h e c u r r e n t m e t e r m e a s u r e m e n t s i n s i d e t h i s c h a n n e l ( f i g u r e 3 ) . S u r f a c e d r o g u e s show t h e L a g r a n g i a n c o m p o n e n t o f t h e m o t i o n t h a t i s s u b j e c t t o p r e f e r e n t i a l d i f f u s i o n i n a s h e a r c u r r e n t a n d t o w i n d d r i f t e f f e c t s . On t h e o t h e r h a n d , c u r r e n t m e t e r s a n d g e o s t r o p h i c c a l c u l a t i o n s g i v e E u l e r i a n v e l o c i t i e s a t a f i x e d l o c a t i o n . T h e e x i s t e n c e o f a p p o s i n g c u r r e n t s i s s t r o n g e v i d e n c e o f t h e i m p o r t a n c e o f s t r a t i f i c a t i o n i n t h e s e c h a n n e l s , a n d , h e n c e , j u s t i f i e s t h e u s e o f a b a r o c l i n i c m o d e l . T h e r e v e r s a l s i n t h e s e c u r r e n t s a r e m o s t l i k e l y c a u s e d b y a d v e r s e ( b a r o t r o p i c a n d / o r b a r o c l i n i c ) p r e s s u r e g r a d i e n t s , w h i c h may b e d u e t o t h e i n f l u x o f f r e s h w a t e r o r c h a n g e s i n b a t h y m e t r y . F u r t h e r i m p r o v e m e n t s t o t h e p r e s e n t m o d e l w i l l come f r o m t h e i n c o r p o r a t i o n o f a l o n g s h o r e p r e s s u r e g r a d i e n t s e f f e c t s . T h e s e may b e d u e t o c h a n g e s i n b u o y a n c y ( e . g . , d u e t o c o a s t a l s o u r c e s ) , - 95 -o r d u e t o c o m p r e s s i o n o-f t h e v o r t e x l i n e s b y c h a n g e s i n t h e t o t a l d e p t h . F o r e x a m p l e , i n t h e c a s e o f a " l o n g " b a t h y m e t r y ( h o r i z o n t a l s c a l e L much l a r g e r t h a n X ) , we may u s e a m e t h o d s i m i l a r t o t h a t o f C u s h m a n - R o i s i n a n d O ' B r i e n (1983) a n d a p p l y a p e r t u r b a t i o n e x p a n s i o n i n a s m a l l p a r a m e t e r X / L . F o r X / L < < e < l , t h e f u l l c o r n e r s o l u t i o n i s s t i l l v a l i d l o c a l l y n e a r e a c h c o r n e r , i n c l u d i n g t h e 0 ( e ) t e r m s , e x c e p t t h a t k ( e . g . , i n e q u a t i o n ( 7 . 9 ) ) i s m o d u l a t e d ( i n t h e WKB s e n s e ) b y c h a n g e s i n b a t h y m e t r y . I f X / L ~ e , t h e n t h e 0 ( e ) e q u a t i o n ( 7 . 1 0 ) must a l s o b e m o d i f i e d t o i n c l u d e t h e s e b a r o t r o p i c e f f e c t s . T h e i n f r a - r e d i m a g e o f t h e s u r f a c e t e m p e r a t u r e i n H u d s o n S t r a i t ( s e e f i g u r e 24) i s an e s p e c i a l l y g o o d e x a m p l e o f c o m p l e x i t y o f t h e f l o w a n d , i n p a r t i c u l a r , o f t h e r o l e o f b a t h y m e t r y a n d m i x i n g , w h i c h w e r e n o t i n c l u d e d i n t h e p r e s e n t m o d e l . T h e e f f e c t o f b a t h y m e t r y o n t h e c i r c u l a t i o n i n s i d e t h e s t r a i t c a n b e s e e n f r o m t h e c r o s s — c h a n n e l f l o w n o r t h o f A k p a t o k I s l a n d . I t c a n b e t r a c e d b y t h e i n t r u s i o n o f c o l d ( b l u e c o l o u r ) w a t e r i n t o t h e warmer ( r e d a n d p i n k ) f l o w o n t h e s o u t h s i d e . S i m i l a r , b u t w e a k e r ( b e c a u s e o f s m a l l e r a l o n g - c h a n n e l b o t t o m s l o p e s ) , c u r r e n t r e v e r s a l s c a n b e s e e n f a r t h e r i n s i d e t h e c h a n n e l . I n a d d i t i o n , t h e b a t h y m e t r y h a s a s t r o n g c h a n n e l i n g e f f e c t o n t h e o u t f l o w e a s t o f C a p e C h i d l e y , w h e r e t h e p r e s e n c e o f r e l a t i v e l y s h a l l o w ( w i t h d e p t h o f t h e o r d e r o f 100m) a n d w i d e ( a b o u t 120km) s h e l f s p l i t s t h i s c o l d ( b l u e ) o u t f l o w i n t o two p a r t s . T e m p e r a t u r e i s o n l y a s u r f a c e t r a c e r a n d , h e n c e , i t i s a p o o r i n d i c a t o r o f a c t u a l t r a n s p o r t s . N e v e r t h e l e s s , i t s e e m s t o - 96 -show t h a t a l a r g e r p o r t i o n o-f t h e f l o w f o l l o w s t h e s h e l f - b r e a k e a s t w a r d s i n t o t h e L a b r a d o r S e a , w h i l e t h e r e s t f l o w s a l o n g t h e e a s t e r n c o a s t o f L a b r a d o r P e n i n s u l a . D i f f u s i o n a f f e c t s c o a s t a l c u r r e n t s o n s c a l e s t h a t a r e much l a r g e r t h a n t h e w i d t h <X) o f t h e f l o w ( e . g . , S a n d e r s o n a n d L e B l o n d , 1 9 8 4 ) . I n t h e same s a t e l l i t e i m a g e ( f i g u r e 2 4 ) , t h e c o o l i n g a n d w a r m i n g o f c o a s t a l c u r r e n t s a l o n g t h e e a s t e r n s h o r e s o f Q u e b e c a n d L a b r a d o r a r e m o s t l i k e l y d u e t o t i d a l m i x i n g . M o r e o v e r , i t i s m o s t p r o n o u n c e d i n t h e mouth o f t h e c h a n n e l , w h e r e t h e s i l l a n d t h e c o n s t r i c t i o n p r o d u c e v e r y s t r o n g t i d a l c u r r e n t s , o f t h e o r d e r o f 2 m / s ( O s b o r n e t a l . , 1 9 7 9 ) . A s a r e s u l t o f t h e s t r o n g m i x i n g i n t h e m o u t h , t h e g e n e r a l l y warmer ( p i n k a n d r e d c o l o u r s ) s u r f a c e f l o w e a s t o f R e s o l u t i o n I s l a n d c o o l s down c l o s e r t o a n a v e r a g e t e m p e r a t u r e o f t h e w a t e r c o l u m n ( b l u e c o l o u r ) a s i t e n t e r s t h e s t r a i t s o u t h o f t h e i s l a n d . T h e same t h i n g h a p p e n s t o t h e o u t f l o w a r o u n d C a p e C h i d l e y . T h e warm ( p i n k ) s u r f a c e l a y e r i s m i x e d c l o s e r t o t h e a v e r a g e t e m p e r a t u r e o f t h e w a t e r c o l u m n . I t i s i n t e r e s t i n g t o n o t e t h a t d e s p i t e t h i s v i g o r o u s m i x i n g , t h e s t r o n g r e s i d u a l m o t i o n k e e p s t h e f l o w s t r a t i f i e d i n t h e mouth ( s e e f i g u r e 5 ) . E f f e c t s s t e m m i n g f r o m t h e v a r i a b i l i t y o f t h e f l o w w e r e n o t i n c l u d e d i n t h e p r e s e n t s t u d y . F o r e x a m p l e , t h e c r i t e r i o n f o r s e p a r a t i o n o f t h e f l o w a t a s h a r p c a p e ( c h a p t e r 7 a n d a p p e n d i x C) w o u l d h a v e t o b e m o d i f i e d i f t h e sum o f r e s i d u a l a n d t i d a l c u r r e n t s i s s u f f i c i e n t t o c a u s e s e p a r a t i o n a t some s t a g e o f t h e t i d e . O n c e s e p a r a t e d , t h e f l o w i s l i k e l y t o r e m a i n t h a t w a y , - 97 -e v e n d u r i n g r e v e r s a l s o-f t h e t i d e . T h i s i s d u e t o t h e a d v e r s e p r e s s u r e g r a d i e n t -from t h e c l o s e d g y r e ( s e e f i g u r e 17) t h a t was c r e a t e d o n t h e d o w n s t r e a m s i d e o f t h e c a p e . I t i s t e m p t i n g t o s u g g e s t t h a t t h i s m e c h a n i s m may b e r e s p o n s i b l e f o r t h e s e p a r a t i o n o f t h e G i b r a l t a r s u r f a c e f l o w f r o m t h e r i g h t b a n k a s i t e n t e r s t h e A l b a r a n S e a . A d e t a i l e d i n v e s t i g a t i o n o f t h e a v a i l a b l e d a t a may p r o v i d e a t e s t o f t h i s s u g g e s t i o n . - 9 8 -R e f e r e n c e s A m e s , W.F. 1965 N o n l i n e a r P a r t i a l D i f f e r e n t i a l E q u a t i o n s i n E n g i n e e r i n g . A c a d e m i c P r e s s . B r o m w i c h , T . J . 1915 D i f f r a c t i o n o f w a v e s b y a w e d g e . P r o c . L o n d o n M a t h . S o c . 14 , 4 5 0 - 4 6 3 . B u c h w a l d , V . T . 1968 T h e d i f f r a c t i o n o f K e l v i n w a v e s a t a c o r n e r . J . F l u i d M e c h . 3 ^ , 1 9 3 - 2 0 5 . B u c h w a l d , V . T . 1971 T h e d i f f r a c t i o n o f t i d e s b y a n a r r o w c h a n n e l . J . F l u i d M e c h . 4 6 , 5 0 1 - 5 1 1 . C a m p b e l l , N . J . , 1958 T h e o c e a n o g r a p h y o f H u d s o n S t r a i t . F i s h . R e s . B o a r d C a n . MS R e p . S e r . N o . 12 . C a r r i e r , 6 . F . , K r o o k , M . a n d P e a r s o n , C . E . 1966 F u n c t i o n s o f a c o m p l e x v a r i a b l e : t h e o r y a n d t e c h n i q u e . McSraw H i l l . C a r s l a w , H . S . 1919 D i f f r a c t i o n o f w a v e s b y a wedge o f a n y a n g l e . P r o c . L o n d o n M a t h . S o c . 18_, 2 9 1 - 3 0 6 . C h a r n e y , J . G . 1955 T h e B u l f S t r e a m a s a n i n e r t i a l b o u n d a r y l a y e r . P r o c . N a t . A c a d . S c i . 4 1 , 7 3 1 - 7 4 0 . C h e r n i a w s k y , J . a n d L e B l o n d , P . H . , R o t a t i n g f l o w s a l o n g i n d e n t e d c o a s t l i n e s . J . F l u i d M e c h . , i n p r e s s . C o l l i n , A . E . 1962 T h e w a t e r s o f t h e C a n a d i a n A r c i p e l a g o . B e d f o r d I n s t . O c e a n o g r . R e p o r t N o . 6 2 - 2 . C u s h m a n - R o i s i n , B . & O ' B r i e n , J . J . 1983 T h e i n f l u e n c e o f b o t t o m t o p o g r a p h y o n b a r o c l i n i c t r a n s p o r t s . J . P h y s . O c e a n o g r . 1 3 , 1 6 0 0 - 1 6 1 1 . D r i n k w a t e r , K . F . 1983 M o o r e d c u r r e n t m e t e r d a t a f r o m H u d s o n S t r a i t , 1982 . C a n . D a t a R e p . F i s h . A q u a t . S c i . 3 8 1 : i v + 4 6 p . D r i n k w a t e r , K . F . On t h e mean a n d t i d a l c u r r e n t s i n H u d s o n S t r a i t . A t m o s p h e r e - O c e a n , i n p r e s s . F e n c o C o n s u l t a n t s L t d a n d F . F . S l a n e y & C o . 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N o f , D . 1978b - . P a r t I I — T w o - l a y e r s y s t e m . J . P h y s . O c e a n o g r . 8 , 8 6 1 - 8 7 2 . N o f , D . 1984 S h o c k w a v e s i n c u r r e n t s a n d o u t f l o w s . J . P h y s . O c e a n o g r . 14 , 1 6 8 3 - 1 7 0 2 . O b e r h e t t i n g e r , F . 1954 D i f f r a c t i o n o f w a v e s b y a w e d g e . Comm. A p p l . M a t h . 7 , 5 5 1 - 5 6 3 . O s b o r n , T . R . , L e B l o n d , P . H . a n d H o d g i n s , D . 0 . 1979 A n a l y s i s o f o c e a n c u r r e n t s : D a v i s S t r a i t - 1 9 7 7 , v o l . I a n d I I . AP0A R e p . N o . 138 ( a v a i l a b l e f r o m t h e A r c t i c P e t r o l e u m O p e r a t o r s A s s o c i a t i o n , P . O . Box 1 2 8 1 , S t a t i o n M , C a l g a r y , A l b e r t a , C a n a d a T 2 P 2 L 2 ) . P a c k h a m , B . A . & W i l l i a m s , W . E . 1968 D i f f r a c t i o n o f K e l v i n w a v e s a t a s h a r p b e n d . J . F l u i d M e c h . 3 4 , 5 1 7 - 5 3 0 . P e d l o s k y , J . 1979 G e o p h y s i c a l F l u i d D y n a m i c s . S p r i n g e r - V e r l a g . R o e d , L . P . 1980 C u r v a t u r e e f f e c t s on h y d r a u l i c a l l y d r i v e n i n e r t i a l b o u n d a r y c u r r e n t s . J . F l u i d M e c h . 9 6 ( 2 ) , 3 9 5 - 4 1 2 . R o s e a u , M . 1967 D i f f r a c t i o n b y a wedge i n an a n i s o t r o p i c m e d i u m . A r c h . R a t i o n a l M e c h . A n a l . 2 6 , 1 8 8 - 2 1 8 . S a n d e r s o n , B . G . & L e B l o n d , P . H . 1985 T h e c r o s s - c h a n n e l f l o w a t t h e e n t r a n c e o f L a n c a s t e r S o u n d . A t m o s p h e r e - O c e a n 2 2 ( 4 ) , 4 8 4 - 4 9 7 . S o m m e r f e l d , A . 1896 M a t h e m a t i s c h e t h e o r i e d e r d i f f r a c t i o n . M a t h . A n n . 4 7 , 3 1 7 - 3 4 1 . -101-Sommer-f el d , A. 1954 Optics. Academic Press. Stakgold, I. 1968 Boundary Value Problems of Mathematical Physics, Vol. I and II. Macmillan. Stommel, H. 1965 The Gulf Stream, 2nd ed. University of C a l i f o r n i a Press. Stommel, H. & Luyten, J. 1984 The density jump across L i t t l e Bahama Bank. J. Geophys. Res. 89(C2), 2097-2100. Whipple, F.J.W. 1916 D i f f r a c t i o n by a wedge and kindred problems. Proc. London Math. Soc. 16, 94-111. Whitehead, J.A. 1985 The d e f l e c t i o n of a b a r o c l i n i c j e t by a wall in a rotating f l u i d . J. F l u i d Mech. - in press. Whitehead, J.A., Leetmaa, A. & Knox, R.A. 1974 Rotating hydraulics of s t r a i t and s i l l flows. Geophys. F l u i d Dyn. 6, 101-125. A p p e n d i x A  D e r i v a t i o n o f t h e 0 ( e ) e q u a t i o n -for S i We d e r i v e a n e q u a t i o n f o r s t , t r e a t i n g s 0 a s a known f u n c t i o n o f x a n d y . I f we a d d ( 6 . 7 c ) t o ( 6 . 8 c ) a n d u s e ( 6 . 1 0 ) a n d ( 6 . 1 1 ) , we o b t a i n (SoVi)„ - (soUi)y + S i V 2 s 0 = 2 s 0 s i + s 0 h i . ( A l ) B u t , f r o m ( 6 . 9 b ) ( 5 o V i ) , - ( s o U i ) y = V 2 ( s 0 S t ) - h i V 2 s 0 - V s o ' V h i , (A2) s o t h a t , v 2 ( s 0 s , ) - h , V 2 s 0 - S 7 s 0 * ? h , + s , v - 2 s 0 = 2 s 0 S i + s 0 h i . (A3) We e l i m i n a t e h t i n f a v o u r o f s t u s i n g ( 6 . 1 4 ) , w h i c h a l s o g i v e s V S o - V t l , = V S o ' V S i + S o ( V S o ) 2 - ( V S o - V ) ( ? s 0 ) 2 / 2 , (A4) f r o m w h i c h , ( 6 . 1 7 ) , t h e e q u a t i o n f o r S i , f o l l o w s : s 0 v 2 S i + Vso-Vs, + ( V 2 s 0 - 3 s 0 ) s t = W, ( 6 . 1 7 ) - 103 -w h e r e W = ( V 2 S o + S o > C s o 2 - < V S o ) 2 3 / 2 + S o < V s 0 ) 2 - < V S o - V ) < V s 0 > 2 / 2 . <A5) B u t , f r o m ( 6 . 1 6 ) , we h a v e V 2 s 0 + so = C 5 s 0 2 - < V S o ) 2 3 / 2 s 0 , <A6) s o t h a t , W s i m p l i f i e s t o W = C s o 2 - ( V s 0 ) 2 3 2 / 4 s o + S o 3 - ( 7 S o - v ) ( ? s Q ) 2 / 2 . ( 6 . 1 8 ) - 104 -A p p e n d i x B  B e h a v i o u r o f s Q f o r s m a l l a n d l a r g e r I n t h i s a p p e n d i x we d e r i v e s i m p l e e x p r e s s i o n s f o r S o , w h i c h a r e v a l i d - for s m a l l , o r l a r g e , d i s t a n c e s f r o m t h e a p e x o f t h e c o r n e r . I f we r e p l a c e p w i t h p ' = l — p , t h e n , f o r s m a l l r , we g e t f r o m ( 6 . 2 2 ) V 2 p * % 0 ( B l ) a n d h e n c e , p ' % ( k r ) a s i n ( a © ) , (B2) w h i c h i s a p o t e n t i a l f l o w s t r e a m f u n c t i o n s o l u t i o n i n a wedge r r / a . C o n s e q u e n t l y , p * p p = 1 - ( k r ) a s i n ( a 6 ) (B3) a n d h e n c e , S o * s p = p P 2 ' 3 = C l - ( k r ) a s i n ( a © ) H 2 ' 3 . (B4) s p a n d t h e e x a c t s o l u t i o n s 0 a r e shown o n F i g u r e 2 8 a f o r s e v e r a l s m a l l v a l u e s o f r . - 105 -I n o r d e r t o g e t t h e b e h a v i o u r o-f s 0 - for l a r g e r , we n o t e f i r s t t h a t i n t h i s c a s e , t h e i n t e g r a n d i n ( 7 . 9 ) i s a r a p i d l y o s c i l l a t i n g f u n c t i o n o f u a n d h e n c e , m o s t o f t h e c o n t r i b u t i o n t h e i n t e g r a l c o m e s f r o m t h e v i c i n i t y o f u%0. T h e r e f o r e , s i n c e f o r s m a l l u , s i n h ( u / a ) = s i n h u / a + 0 ( u 3 ) , (B5) we g e t t h e a p p r o x i m a t e e x p r e s s i o n ^ 2b s°° c o s ( k r s i n h u / a ) c o s h u d u . p % p = — ( : — — — (B6) H K IT o s i n h 2 u + b 2 * w h e r e b = s i n ( a © ) , f o r s h o r t . R e p l a c i n g s i n h u w i t h x , we g e t 2b / ° ° c o s ( k r x / a ) dx Pi = — ) , ^ . , = e x p ( - k r b / a ) . (B7) JT Y a X 2 + b 2 T h e f u n c t i o n S i = p i 2 / 3 a n d s 0 a r e p l o t t e d f o r c o m p a r i s o n o n F i g u r e 28b f o r s e v e r a l v a l u e s o f r . T h e y a g r e e b e s t n e a r t h e b o u n d a r i e s , w h e r e m o s t o f t h e t r a n s p o r t t a k e s p l a c e . - 106 -F i g u r e 2 8 . (a) An a p p r o x i m a t i o n Sp ( s o l i d ) a n d t h e e x a c t s 0 ( d a s h e d ) f o r 3 s m a l l v a l u e s o f r . (b) A s i n ( a ) , e x c e p t f o r l a r g e r . - 107 -A p p e n d i x C T h e e x t e n t o f t h e c e n t r i f u g a l u p w e l l i n g I n t h i s a p p e n d i x we f o l l o w t h e m e t h o d o f c h a p t e r s 6 a n d 7 i n d e r i v i n g a n e x p l i c i t s o l u t i o n f o r t h e u p w e l l i n g r e g i o n a t d i s t a n c e s f r o m t h e apex w h i c h a r e o f t h e o r d e r o f e 2 X . We a r e m o s t l y i n t e r e s t e d i n t h e s i z e o f t h e u p w e l l i n g r e g i o n , a n d n o t i n e x a c t d e t a i l s , s o t h a t n o a t t e m p t i s made t o m a t c h t h e new s o l u t i o n t o t h e o n e d e r i v e d i n c h a p t e r 7 . We n o n d i m e n s i o n a l i z e , a s i n c h a p t e r 6 , e q u a t i o n ( 6 . 1 ) , e x c e p t f o r ( x , y ) = e 2 X ( x ' , y ' ) , h = H ( l - h ' ) a n d V = Q (1—2e 24 l ') . (CI ) T h e l a s t s c a l i n g was u s e d , s i n c e H*%Q n e a r t h e a p e x . D r o p p i n g p r i m e s , we g e t ( a s i n c h a p t e r 6) (1 - h) + e ( u 2 + v 2 ) / 2 = (1 - 2 e 2 H , ) 1 / 2 + e ( l - 2e 2 H») (C2) - u y + e = e ( l - h ) C ( l - 2 e 2 , f ) - 1 / 2 + ell (C3) v ( l - h) = - f x a n d u ( l - h) = Vy (C4) E x p a n d i n g i n p o w e r s o f e , w h i l e u s i n g (1 - 2 e 2 " r , ) , l l / 2 = 1 ± e 2 ^ + 0 (e* ) a n d r e t a i n i n i n g t e r m s up t o 0 ( e 2 ) , we g e t - 108 -h o = 0 (C2a) a n d h e n c e , h=ehi + e 2 h 2 + . . . . S i m i l a r l y , - h i + ( u o 2 + v 0 2 ) / 2 = 1 / 2 , <C2b) - h 2 + U o U t + v 0 V i = - f o , (C2c) v 0 » - U o y = 0 , (C3a) V i , - U i y = 0 , (C3b) v 2 „ — u 2 y = 1 — h i , (C3c) w h e r e we h a v e u s e d (C2a) t o d e r i v e t h e l a s t t w o e q u a t i o n s . A l s o , we g e t - V o = f o x » Uo = V 0 y » <C4a> - V i + V o h i = "HIM, U i - u 0 h i = V i y , <C4b) - v 2 + V i h i + v 0 h 2 = ,Ha» , u 2 - U i h i - u 0 h 2 = H^y . (C4c) E q u a t i o n s (C3a) a n d (C4a) g i v e i m m e d i a t e l y V 2«V 0 = 0 , (C5) w h i l e -from (C2b) , t o g e t h e r w i t h ( C 4 a ) , we g e t h i = ( V 4 * 0 ) 2 / 2 - 1 / 2 . <C6) - 109 -D i f - f e r e n t i a t i n g (C4b> w i t h r e s p e c t t o x a n d y , r e s p e c t i v e l y , a n d a d d i n g , u p o n u s i n g (C4a) a n d ( C 3 b > , we o b t a i n v 2 f , = - W o - v h i . <C7) We m u l t i p l y t h e f i r s t o f (C4b) b y - v 0 . , t h e s e c o n d b y u 0 a n d a d d , u s i n g ( C 4 a ) . T h e n , f r o m (C2c) we g e t h 2 = W o - V H i + h i C V f o ) 2 + fo . <C8) T h e s o l u t i o n o f <C5) , w i t h f o = 0 o n e = 0 , i r / a , i s f o = r a s i n ( a 8 ) , <C9) w h e r e t h e m u l t i p l i c a t i v e c o n s t a n t was c h o s e n t o b e u n i t y , f o r c o n s i s t e n c y w i t h t h e s o l u t i o n i n c h a p t e r 7 ( s e e a l s o ( C I ) ; t h i s c a n b e v e r i f i e d i f we e x p a n d t h e n o n d i m e n s i o n a l c o u n t e r p a r t o f ( 4 . 4 ) i n p o w e r s o f s m a l l x ) . We g e t f r o m (C6) h , = C ( a / r l " a ) 2 - I D / 2 . (CIO) N o t e t h a t , c o n s i s t e n t w i t h ( C I ) a n d an e x p a n s i o n o f ( 4 . 2 ) i n s m a l l x , hi=0 i d e n t i c a l l y f o r a = l . U s i n g ( C I O ) , C ( 7 ) b e c o m e s . V 2Vi = - a 3 ( l - a ) s i n ( a e ) / r 4 - 3 3 , ( C l l ) - 110 -w i t h o n e= 0 , r r / a . T h e s o l u t i o n t o ( C l l ) i s 4 \ = - a 3 s i n ( a © ) / C 4 ( 2 a - l ) r 2 - 3 a 3 + C i 4 * 0 (C12) w h i c h i s a sum o f t h e p a r t i c u l a r s o l u t i o n , f i p , a n d t h e s o l u t i o n o f t h e h o m o g e n e o u s p r o b l e m ( C 5 ) , m u l t i p l i e d b y a n a r b i t r a r y c o n s t a n t C t , w h i c h , i n g e n e r a l , i s a f u n c t i o n o f a ( t h e a n g u l a r p a r a m e t e r ) . I n o r d e r t o b e c o n s i s t e n t w i t h ( C l ) a n d t h e e x p a n s i o n o f ( 4 . 4 ) i n s m a l l x , we must h a v e 'Vi=Q f o r a = l a n d h e n c e , H \ P ( a = l ) = r s i n e / 4 = - C t r s i n e , s o t h a t Ci (a) m u s t s a t i s f y C i ( l ) = - l / 4 . T h a t i s a s f a r a s we g e t , w i t h o u t d o i n g a f o r m a l m a t c h i n g w i t h t h e s o l u t i o n f r o m c h a p t e r 7 . T h e r e a r e many f u n c t i o n s o f a t h a t s a t i s f y t h e l a s t c o n d i t i o n . I n p a r t i c u l a r , we c a n u s e t h e p o w e r f u n c t i o n s Ci ( a ) « - a " / 4 , (C12a) w h e r e b i s a n y r e a l n u m b e r . We c a n n a r r o w t h e c h o i c e o f b i f we d e m a n d , f o r e x a m p l e , t h a t V a n d h h a v e a s i m i l a r u p w e l l i n g a r e a . A f t e r some a l g e b r a , we g e t f r o m (C8) a « C 2 ( a - l ) s i n 2 ( a 0 ) - 5 a + 2 3 4 ( 2 a - l ) r 4 t l - a > + r a s i n ( a 6 ) . (C13) - I l l -A g a i n , we n o t e t h a t f o r a = l (Ci=—1/4) , h 2 = r s i n 6 , w h i c h a g r e e s w i t h (CI) a n d t h e p o w e r e x p a n s i o n o-f ( 4 . 2 ) , - for s m a l l x . Me t r i e d s e v e r a l v a l u e s o f b ( i n ( C 1 2 ) ) , b o t h p o s i t i v e a n d n e g a t i v e , a n d -found t h a t t h e b e s t a g r e e m e n t b e t w e e n M* a n d h h a p p e n s f o r b=-3. T h i s i s e v i d e n t f r o m F i g u r e s 29a—c, w h i c h show t h e p l o t s o f t h e t o t a l s t r e a m f u n c t i o n l-ZE.2tV a n d t h e t o t a l d e p t h o f t h e i n t e r f a c e l - h ( s e e ( C D ) f o r t h r e e v a l u e s o f b=-2,-3,-4. We n o t e t h a t t h e e f f e c t o f Ci o n 4* i s much s t r o n g e r t h a n o n h . I n f a c t , f o r b<-4, t h e a r e a o f u p w e l l i n g shown b y M* e x t e n d s b e y o n d t h e a r e a p l o t t e d o n t h e s e f i g u r e s , w h i l e f o r b > - 2 , t h e u p w e l l i n g a r e a shown b y f i s much s m a l l e r t h a n t h e o n e shown b y h . What i s m o r e i m p o r t a n t , we h a v e shown i n t h i s a p p e n d i x t h a t f o r E=0 .5 t h e u p w e l l i n g r e g i o n i s s m a l l e r t h a n a b o u t 0 . 0 5 X . F o r e x a m p l e , i f X=20km, t h i s a m o u n t s o n l y t o a b o u t 1km r a d i u s . - 112 -(a) O.IOi h (e=o.5) o.o» F i g u r e 2 9 . C o n t o u r s o f H» a n d h (e=0.5> f o r (a) b = - 2 , (b) b=-3 a n d (c) b=-4. T h e d o m a i n i s 0 . 2 X b y 0 . 2 X l a r g e a n d c o n t o u r s p a c i n g s a r e 0 . 1 f o r h a n d 0 . 0 1 f o r V . - 113 -(b) - 114 -

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