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Longshore currents in the vicinity of a breakwater Daniel, Peter Edward 1978

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LONGSHORE CURRENTS IN THE VICINITY OF A BREAKWATER by PETER EDWARD DANIEL B . S c , U n i v e r s i t y of B r i t i s h Columbia, 1974 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA fc) August, 19 78 i n the Department of P h y s i c s In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of P h y s i c s  The U n i v e r s i t y of B r i t i s h Columbia Vancouver, Canada i i i ABSTRACT Se v e r a l t h e o r e t i c a l models of the wave-induced c u r r e n t c i r c u l a t i o n i n the v i c i n i t y of a breakwater extending from shore are presented. The models,which are p a t t e r n e d a f t e r a l o c a l f i e l d s i t e , i n c l u d e s e v e r a l numerical models which take i n t o account v a r i a b l e s e a - f l o o r topography and which compare the e f f e c t s of l i n e a r and n o n - l i n e a r bottom f r i c t i o n as w e l l as an a n a l y t i c a l model c h a r a c t e r i z e d by s e m i - i n f i n i t e beaches and uniform s e a - f l o o r topography. In g e n e r a l ( f o r a g i v e n angle of wave i n c i d e n c e ) the c i r c u l a t i o n p a t t e r n s show two c o u n t e r - r o t a t i n g c e l l s d r i v e n by wave-induced longshore c u r r e n t s which flow along both .the breakwater and n a t u r a l beaches toward t h e i r common i n t e r -s e c t i o n corner, with an o f f s h o r e r e t u r n flow i n the form of a r i p c u r r e n t . The q u a l i t a t i v e f e a t u r e s of the models are c o n s i s t e n t with o b s e r v a t i o n s of sediment t r a n s p o r t taken.at the study s i t e . D i f f e r e n c e s i n the l i n e a r and n o n - l i n e a r bottom f r i c t i o n models do not become apparent u n t i l an o f f - s h o r e t r e n c h p a r a l l e l to the breakwater i s i n t r o d u c e d to the s e a - f l o o r topography. The n o n - l i n e a r model shows a d e f l e c t i o n o f the o f f - s h o r e r e t u r n flow i n t o the t r e n c h i n agreement with p r e -l i m i n a r y a n a l y s i s based on a one-dimensional model. The l i n e a r r e s u l t s , however, d i f f e r c o n s i d e r a b l y from those of the n o n - l i n e a r model and are d i f f i c u l t to i n t e r p r e t , showing an i n o r d i n a t e i n c r e a s e i n t r a n s p o r t over the t r e n c h . During the development of the a n a l y t i c a l model d i f f i c u l t -i e s were encountered due to the c o m p l e x i t i e s of the a n a l y s i s which n e c e s s i t a t e d t h a t p a r t of the s o l u t i o n be s o l v e d nume-r i c a l l y . The r e s u l t s , w h i l e showing the same ge n e r a l f e a t u r e s as the numerical models, e x h i b i t a much more s t r o n g l y d i v e r -gent o f f - s h o r e r e t u r n flow. T h i s d i f f e r e n c e , while unresolved, appears to be one of s c a l e r a t h e r than of -form. TABLE OF CONTENTS i v Page ABSTRACT . i i i TABLE OF CONTENTS i v LIST OF FIGURES v i LIST OF SYMBOLS i x ACKNOWLEDGEMENTS x i i Chapter 1. INTRODUCTION 1 2. DISCUSSION OF THE FIELD SITE 4 I General L o c a t i o n 4 I I Geology, Geometry, Oceanography 4 I I I D i s c u s s i o n o f P r e v a i l i n g Winds 6 and L o c a l Topography IV M o r p h o l o g i c a l Evidence of Long-shore Currents 11 3. THEORY 19 I Review of the Theory of Long-shore Currents 19 I I Formulation of the Problem 21 4. NUMERICAL MODELLING . 28 I I n t r o d u c t i o n 28 I I Model Layout 28 I I I A Simple C o n f i g u r a t i o n 38 IV The Corner Geometry 41 i V Acute I n t e r s e c t i o n Angle 44 VI V a r i a b l e Bottom Topography 48 VII The In f l u e n c e of Depth V a r i a t i o n s 58 Chapter Page 5. ANALYTICAL MODEL 66 I P h y s i c a l D e s c r i p t i o n 66 II A n a l y s i s 69 I I I D i s c u s s i o n 90 6. SUMMARY 94 LIST OF REFERENCES 98 APPENDIX A - SUMMARY OF APPROXIMATIONS 100 APPENDIX B - COMPUTATIONAL CONSIDERATIONS 101 APPENDIX C - EVALUATION OF INTEGRALS I,(a) AND I 2 ( p ) • H O APPENDIX D - INTEGRAL ANALYSIS 114 v i LIST OF FIGURES F i g u r e Page 1. General l o c a t i o n of the study s i t e 5 2. D e t a i l e d view of the study s i t e 7 3. Wind c h a r a c t e r i s t i c s a t the Tsawwassen causeway 9 4. S p e c u l a t i v e diagram of the wave-induced longshore c u r r e n t flow a t the study s i t e .. 10 5. A i r - p h o t o comparison of the growth of "bulges" south-side Tsawwassen causeway ... 12 6. Photograph of a groyne on the Tsawwassen beach approximately 200 m. south of the causeway 14 7. Photograph of a groyne on the Tsawwassen beach approximately 400 m. south of the causeway 15 8. A e r i a l photograph of the i n t e r s e c t i o n corner taken i n 1963 16 9. A e r i a l photograph of the i n t e r s e c t i o n corner taken i n 1975 17 10. Plan view of the numerical model geometry ... 29 11. Boundary c o n d i t i o n s f o r the p a r t i c u l a r case of a p e r p e n d i c u l a r beach i n t e r -s e c t i o n 32 12. Current c i r c u l a t i o n f o r p e r p e n d i c u l a r beach i n t e r s e c t i o n , e q u a l c u r r e n t s t r e n g t h s and no o f f - s h o r e depth v a r i a t i o n s 39 13. P e r p e n d i c u l a r beach i n t e r s e c t i o n model wit h the corner r e p l a c e d 42 14. Normalized stream l i n e p r o f i l e taken a t the c r o s s - s e c t i o n qq' i n Fi g u r e 12, together w i t h t h a t of an i n f i n i t e beach model 43 v i i F i g u r e Page 15. Current c i r c u l a t i o n f o r acute beach-breakwater i n t e r s e c t i o n (& = 60*), equal c u r r e n t s t r e n g t h s and no o f f -shore depth v a r i a t i o n s 45 16. C i r c u l a t o r y p a t t e r n f o r a r a t i o of n a t u r a l beach c u r r e n t s t r e n g t h to causeway beach c u r r e n t s t r e n g t h - of 2:1. There are no depth v a r i a t i o n s 46 17. C i r c u l a t o r y p a t t e r n f o r a r a t i o of 1 n a t u r a l beach c u r r e n t s t r e n g t h to causeway beach c u r r e n t s t r e n g t h of 4:1. There are no depth v a r i a t i o n s 47 18. C i r c u l a t o r y p a t t e r n f o r the l i n e a r bottom f r i c t i o n model i n c o r p o r a t i n g a d r o p - o f f to deep water 49 19. C i r c u l a t o r y p a t t e r n f o r the l i n e a r bottom f r i c t i o n model i n c o r p o r a t i n g a d r o p - o f f to deep water and an o f f - s h o r e t r e n c h running p a r a l l e l t o the causeway 51 20. C i r c u l a t o r y p a t t e r n f o r the n o n - l i n e a r bottom f r i c t i o n model w i t h c u r r e n t s -of equal s t r e n g t h and no o f f - s h o r e depth v a r i a t i o n s 53 21. Current c i r c u l a t i o n f o r the n o n - l i n e a r bottom f r i c t i o n model i n c o r p o r a t i n g a d r o p - o f f to deep water 54 22. Current c i r c u l a t i o n f o r the n o n - l i n e a r bottom f r i c t i o n model i n c o r p o r a t i n g a d r o p - o f f to deep water and an o f f -shore t r e n c h running p a r a l l e l to the causeway 55 23. V e l o c i t y p r o f i l e s taken along the c r o s s -s e c t i o n s pp 1 and dd' of F i g u r e 21 and 22 r e s p e c t i v e l y 57 24. V e l o c i t y f i e l d f o r the one-dimensional model 6 0 v i i i F i g u r e Page 25. V a r i a t i o n i n depth as d e f i n e d by «|=y(y-i) 62 26. V a r i a t i o n i n depth as d e f i n e d by /»j=N(y-.%) 62 27. Plan view and boundary c o n d i t i o n s of the a n a l y t i c a l model 67 28. Current p a t t e r n f o r the a n a l y t i c a l model, where the beaches have been t r u n c a t e d f o r comparison to the numerical s o l u t i o n s 92 29. Normalized s t r e a m - l i n e p r o f i l e s taken across the n a t u r a l beach s u r f zone of the a n a l y t i c a l model a t the p o s i t i o n s shown i n F i g u r e 28 93 30. General l a y - o u t and boundary c o n d i t i o n s f o r the numerical models 102 31. An example of the e f f e c t s of v a r y i n g the r e s o l u t i o n of the numerical models 109 i x LIST OF SYMBOLS a: wave amplitude _2 C: (i) a drag c o e f f i c i e n t of 0(10 ) ( i i ) a f r i c t i o n c o e f f i c i e n t defined by C• Qp |u«|M 0 C : =/Ygd where d i s the t o t a l water depth o > o o at the breaker-line d: the t o t a l water depth equal to the sum of the l o c a l s t i l l water depth h, and the mean displacement of the water s u r f a c e from the s t i l l water l e v e l , ^ ; thus d = h + d^: the t o t a l water depth a t the b r e a k e r - l i n e o energy d e n s i t y of the waves (see (3.3)) f : a f r i c t i o n c o e f f i c i e n t which i s a f u n c t i o n of the t o t a l depth 'd' and wave amplitude (see (3.1) and (3.2)) g: a c c e l e r a t i o n due to g r a v i t y h: l o c a l s t i l l water depth h^: a c h a r a c t e r i s t i c depth H: the wave h e i g h t equal to twice the amplitude 'a' J - ( a x ) : a Bessel f u n c t i o n of the f i r s t k i n d (see (5.10)) kc the wave number o slope of the mean s u r f a c e i n s i d e s urfzones I and I I r e s p e c t i v e l y r a d i a t i o n s t r e s s tensor F r e s n e l i n t e g r a l (see (5.42)) f o r c e per u n i t mass due to divergence of r a d i a t i o n s t r e s s tensor (see (3.6)) the maximum o r b i t a l v e l o c i t y of the waves the instantaneous t o t a l v e l o c i t y v e c t o r j u s t o u t s i d e the bottom boundary l a y e r (see (3.18)) the mean v e l o c i t y component i n the longshor d i r e c t i o n the v e l o c i t y v e c t o r , assumed to be p u r e l y h o r i z o n t a l and depth independent the mean v e l o c i t y component i n the o f f s h o r e d i r e c t i o n width of surfzone I I i n a n a l y t i c model (see F i g u r e 27) width of surfzone I i n a n a l y t i c model (see F i g u r e 27) angle of wave i n c i d e n c e r e l a t i y e to the normal to the b r e a k e r - l i n e of surfzone I x i r a t i o of wave amplitude to mean water depth i n the surfzone (see (4.11)) angle of beach-breakwater i n t e r s e c t i o n the mean displacement of the water s u r f a c e from the s t i l l water l e v e l angle of wave i n c i d e n c e r e l a t i v e to the normal to the b r e a k e r - l i n e of surfzone I I the water d e n s i t y bottom shear s t r e s s i n the presence of steady flow or long p e r i o d waves (see (3.18)) the angle of wave i n c i d e n c e r e l a t i v e to the normal to the s h o r e l i n e t r a n s p o r t stream f u n c t i o n (see (3.15)) ACKNOWLEDGEMENTS I would l i k e to thank Dr. P.H. LeBlond f o r d i r e c t i n g me to t h i s t o p i c and f o r many h e l p f u l d i s c u s s i o n s con-c e r n i n g t h i s work. This work was done with the a i d of a N a t i o n a l Research C o u n c i l grant. 1 CHAPTER I INTRODUCTION The need to understand l i t t o r a l p rocesses has become ever more apparent as our c o a s t a l environment i s i n c r e a s -i n g l y s u b j e c t to human e x p l o r a t i o n and e x p l o i t a t i o n . The d i s p e r s a l of p o l l u t a n t s , the e r o s i o n and a c c r e t i o n of shore-l i n e s , and the behavior of waves are but a few of the p r o -cesses and phenomena which a f f e c t and are themselves, i n t u r n , a f f e c t e d by human a c t i v i t y . The consequences of our encroachment upon the c o a s t a l environment need to be w e l l understood, s i n c e man-made s t r u c t u r e s may have important e f f e c t s (James, 1972). I t i s i n t h i s v e i n of understanding our impact upon the c o a s t a l environment, i n l i g h t of the behavior of waves t h a t we have undertaken t h i s study of longshore c u r r e n t s i n the v i c i n i t y of an i s o l a t e d breakwater extending from shore. We s h a l l thus be examining the e f f e c t s upon near-shore c i r c u l a t o r y systems of one of the more common types of s t r u c t u r e s to have been b u i l t along our c o a s t s . The e x e r c i s e s h a l l c o n s i s t of the development of t h e o r e t i c a l models of the c u r r e n t c i r c u l a t i o n near a breakwater and a comparison of t h e i r r e s u l t s , where p r a c t i c a b l e , to ob-s e r v a t i o n s taken a t a f i e l d l o c a t i o n . 2 R e l a t e d s t u d i e s of longshore c u r r e n t s and t h e i r i n t e r -a c t i o n w i t h c o a s t a l s t r u c t u r e s have been conducted, f o r example, by Dalrymple et a l (19 77), who used s e v e r a l model bas i n s to check t h e i r t h e o r e t i c a l p r e d i c t i o n s of the e f f e c t s of p l a c i n g a w a l l i n the path of a longshore c u r r e n t , and Mei and L i u (1976), w i t h t h e i r study of the combined e f f e c t s of r e f r a c t i o n and d i f f r a c t i o n on wave-induced mean c u r r e n t s i n the v i c i n i t y of a breakwater. T h i s study s h a l l i n v e s t i g a t e the c u r r e n t c i r c u l a t i o n r e s u l t i n g from wave-induced longshore c u r r e n t s generated along both breakwater and n a t u r a l beach s u r f z o n e s . I t s h a l l be presented i n the f o l l o w i n g manner. We s h a l l f i r s t d i s c u s s the evidence f o r longshore c u r -r e n t s a t the f i e l d l o c a t i o n as i n d i c a t e d by sediment t r a n s -p o r t along the beaches, i n s p e c t i o n of groynes, a e r i a l photo-graphs, e t c . , and from these surmise as to the p r e v a i l i n g c i r c u l a t i o n p a t t e r n p r e s e n t i n the area. We s h a l l then develop s e v e r a l models of the c u r r e n t c i r c u l a t i o n i n a beach-breakwater c o n f i g u r a t i o n p a t t e r n e d a f t e r the f i e l d s i t e . These w i l l c o n s i s t o f : i / s e v e r a l numerical models u s i n g f i n i t e beaches i n c l o s e d b a s i n s , w i t h c o n s i d e r a t i o n g i v e n to v a r i a b l e s e a - f l o o r topography, as w e l l as l i n e a r and n o n - l i n e a r forms of bottom f r i c t i o n i i / an a n a l y t i c a l model employing s e m i - i n f i n i t e beaches. 3 We s h a l l conclude w i t h a d i s c u s s i o n of the t h e o r e t i c a l r e s u l t s i n l i g h t of the s i m p l i f y i n g assumptions made through-out the a n a l y s i s , and a comparison of the v a r i o u s models' r e s u l t s . I t should be p o i n t e d out t h a t t h i s t h e s i s s h a l l not attempt to r e s o l v e the p r a c t i c a l q u a n t i t a t i v e aspects of sedimentation i n a p a r t i c u l a r area; i t i s r a t h e r a study of the nature of longshore c u r r e n t s i n a type of n a t u r a l beach-breakwater c o n f i g u r a t i o n as i n s p i r e d by a s p e c i f i c f i e l d l o c a t i o n . CHAPTER 2 DISCUSSION OF THE FIELD SITE I General L o c a t i o n The area which has i n s p i r e d the study i s a p o r t i o n of the t i d a l - z o n e of the F r a s e r R i v e r D e l t a near the south end of Roberts Bank (see F i g u r e 1 ) . I t i s bounded to the n o r t h by the Tsawwassen F e r r y causeway - a breakwater extending i n a south-westerly d i r e c t i o n f o r a d i s t a n c e of approximately 3000 m. - and to the east by the Tsawwassen Beach - a narrow beach backed by 100 m. h i g h c l i f f s extending i n a s o u t h e r l y d i r e c t i o n f o r a d i s t a n c e of approximately 6000 m. to the southern t i p of P o i n t Roberts. The angle formed by the i n t e r s e c t i o n of the causeway and Tsawwassen Beach, to the south, i s o approximately 60 . Our i n t e r e s t l i e s i n observing, and subsequently m o d e l l i n g the c u r r e n t c i r c u l a t i o n over t h i s p o r t i o n of Roberts Bank r e s u l t i n g from wave-induced longshore c u r r e n t s along the Tsawwassen and causeway beaches. II Geology, Geometry, Oceanography With the e x c e p t i o n of an o f f - s h o r e t r e n c h - the r e s u l t of dredging o p e r a t i o n s to procure f i l l f o r the causeway -the southern end of Roberts Bank i s a shallow mudflats r e g i o n having base sediments of sand and s i l t y - s a n d . F i g u r e 1. General l o c a t i o n of the study s i t e 6 As can be seen i n F i g u r e 2, the t r e n c h runs p a r a l l e l to the causeway, across most of the width of Roberts Bank. I t i s approximately 150 m. wide, having mean low and h i g h t i d e depths of 10 m. and 5 m. r e s p e c t i v e l y , and stands approximately 250 m. from the southeast s i d e of the cause-way . Our chosen study s i t e w i l l thus a f f o r d us the opportun-i t y of i s o l a t i n g an aspect of v a r i a b l e s e a - f l o o r topography and s t u d y i n g i t s e f f e c t upon the l o c a l c u r r e n t c i r c u l a t i o n . The remainder of the Bank has a mean h i g h t i d e depth of approximately 5 m. and d r a i n s completely a t low t i d e (see F i g u r e 2), thus d i c t a t i n g t h a t longshore c u r r e n t s w i l l be r e l e v a n t to shore e r o s i o n only d u r i n g p e r i o d s of h i g h t i d e . A l s o note t h a t the o f f - s h o r e edge of Roberts Bank, which f a l l s w i t h a s l o p e of approximately 5 i n 1 to a depth of approximately 100 m., does not run p a r a l l e l to Tsawwassen Beach, but r a t h e r i n a more s o u t h - e a s t e r l y d i r e c t i o n , e f f e c -t i v e l y t a p e r i n g from a d i s t a n c e of 3000 m. o f f s h o r e w i t h r e s p e c t to the mean hig h t i d e l i n e a t the causeway, to 100 m. o f f - s h o r e a t P o i n t Roberts. I l l D i s c u s s i o n of P r e v a i l i n g Winds and L o c a l Topography Since longshore c u r r e n t s are the r e s u l t of waves break-i n g a t an angle on a beach, l e t us examine the b a s i c f e a t u r e s a f f e c t i n g the waves which s t r i k e the causeway and Tsawwassen beaches, these being F i g u r e 2. D e t a i l e d view of the study s i t e 8 a/ the nature of the p r e v a i l i n g winds b/ the l o c a l topography Keeping i n mind the o r i e n t a t i o n of the study area and i t s boundaries, i t i s apparent that i t i s open to attack only by wave t r a i n s i n c i d e n t from the southern quarter. Figure 3 i n d i c a t e s that a s i g n i f i c a n t percentage of the l o c a l winds are, indeed, from the southern quarter, p a r t i c -u l a r l y from the south and south-east. Wave t r a i n s approaching from the south w i l l c l e a r l y have components of t h e i r propagation vector p a r a l l e l to both the causeway and Tsawwassen beaches, suggesting the generation of longshore c u r r e n t s , whose d i r e c t i o n of flow, i n each instance, would be toward the i n t e r s e c t i o n apex (see Figure 4). At f i r s t glance i t appears as though Po i n t Roberts s h i e l d s the area from s o u t h e a s t e r l y waves. However, i t has been suggested by Wood (19 70), t h a t southeasterly wavestrains passing the Point are d i f f r a c t e d and then r e -f r a c t e d to approach the study area i n a more sou t h e r l y d i r e c t i o n . Thus, they too w i l l add to the e f f e c t of southern stoiPms, t h e i r c o n t r i b u t i o n , however, being l e s s than i f they had come over an uninterrupted f e t c h . The predominant nature of the winds (from.the south and south-east) together w i t h the topographical e f f e c t s of P o i n t Roberts and Roberts Bank are thus conducive to the generation of longshore currents along both the Tsawwassen and causeway beaches. OCCURENCE IN HOURS/YR DIRECTION ON % TIME BASE OF WINDS OVER 20 MPH OF ALL ANNUAL WINDS Fi g u r e 3. Wind c h a r a c t e r i s t i c s a t the Tsawwassen cause-way (Wood, 1970) (Winds are from the d i r e c t i o n shown.) F i g u r e 4. S p e c u l a t i v e diagram of the wave-induced longshore c u r r e n t flow a t the study s i t e 11 IV M o r p h o l o g i c a l Evidence of Longshore Currents Since longshore movement of beach m a t e r i a l i s due almost e n t i r e l y to longshore c u r r e n t s (Komar, 19 76), l e t us now d i s c u s s the evidence f o r sediment t r a n s p o r t along the Tsawwassen and causeway beaches as i n d i c a t e d by a e r i a l photographs, i n s p e c t i o n of groynes, e t c . Causeway Beach A e r i a l photographs have shown the growth of two bulges i n the beach on the south s i d e of the causeway near the f e r r y t e r m i n a l , w i t h e r o s i o n between them, at times, t h r e a t e n i n g to undercut the highway as shown i n F i g u r e 5 (Hodge, 19 70). In a study conducted by Hodge i n 19 71, the r e g i o n of the bulges was mapped and the movement of beach m a t e r i a l measured. He concludes t h a t " a c t u a l measurement q u a n t i t a t i v e -l y confirms the observed m o r p h o l o g i c a l changes - beach m a t e r i a l s are moving along the causeway to the NNE," t h a t i s , toward the i n t e r s e c t i o n apex. Tsawwassen Beach Although d i r e c t sand t r a n s p o r t measurements along Tsawwassen Beach are u n a v a i l a b l e , we can n e v e r t h e l e s s i n -d i r e c t l y e v a l u a t e the l i t t o r a l d r i f t by examining the shore-l i n e c o n f i g u r a t i o n i n the v i c i n i t y of s e v e r a l groynes b u i l t along the beach. The d i r e c t i o n of the l i t t o r a l d r i f t d u r i n g the immediately p r e c e d i n g p e r i o d can be i n f e r r e d from the entrapment of sand on e i t h e r s i d e of the groynes. foreshore crest (a) May 18, 1963. (from A e r i a l Photograph BC 5073:38) -eroded escarpment foreshore cres •bulges (b) May 31, 1970. (from A e r i a l Photograph BC 5371:108) F i g u r e 5. A i r - p h o t o comparison of the growth of "bulges" s o u t h - s i d e Tsawwassen causeway. (Hodge, 1971) P e r i o d i c i n s p e c t i o n of these groynes has shown, t h a t at a l l of them, there i s a s i g n i f i c a n t drop i n beach l e v e l from south to north, i n d i c a t i v e of a predominantly n o r t h e r n l i t t o r a l d r i f t and accompanying sediment t r a n s p o r t toward the i n t e r s e c t i o n apex of the two beaches. Of p a r t i c u l a r note i s a groyne approximately 400 m. south of the causeway having a drop i n c r o s s - s e c t i o n a l e l e v a t i o n of 0.5 m. w i t h c h a r a c t e r i s t i c d e p o s i t o n .of beach sediment on i t s south (up-current) s i d e and e r o s i o n to the n o r t h (down-current) s i d e (see F i g u r e s 6 and 7). In l i g h t of the evidence f o r sediment t r a n s p o r t along both the beaches (toward the i n t e r s e c t i o n apex) we should, q u i t e reasonably, expect to f i n d a b u i l d - u p of beach mater-i a l s i n the c o r n e r . F i e l d i n s p e c t i o n of the corner shows i t to be an area of s t a g n a t i o n , a g a t h e r i n g p o i n t of sea-s i d e d e b r i s - d r i f t - w o o d , seaweed, e t c . - w h i l e a e r i a l photographs (taken i n the years f o l l o w i n g c o n s t r u c t i o n of the causeway) c l e a r l y show the corner to be f i l l i n g up (see F i g u r e s 8 and 9 ) . M o r p h o l o g i c a l evidence i s then c o n s i s t e n t w i t h e a r l i e r e x p e c t a t i o n s based upon the nature of the p r e v a i l i n g winds together w i t h l o c a l t o p o g r a p h i c a l e f f e c t s t h a t longshore c u r r e n t s are p r e s e n t i n the study area and are r e s p o n s i b l e f o r the t r a n s p o r t of beach m a t e r i a l s along both the Tsawwassen and causeway beaches toward t h e i r i n t e r s e c t i o n apex. Under c o n d i t i o n s of h i g h t i d e s accompanied by winds from the sou-thern q u a r t e r , we can reasonably expect a c i r c u l a t i o n p a t t e r n F i g u r e 6. Shows a p o r t i o n of a groyne on the Tsawwassen beach approximately 200 m. south of the causeway. A south to n o r t h ( r i g h t to l e f t ) drop i n beach e l e v a t i o n of approximately 0.25 m. i s i n d i c a t e d . F i g u r e 7. Taken a t a groyne on the Tsawwassen beach approximately 400 m. south of the causeway. I t shows a drop i n beach e l e v a t i o n from south to n o r t h ( r i g h t to l e f t ) of approximately 0.5 m. ! 16 F i g u r e 8. A e r i a l photograph of the i n t e r s e c t i o n corner taken i n 1963. F i g u r e 9. A e r i a l photograph of the i n t e r s e c t i o n corner i n 1975. to develop c o n s i s t i n g of longshore c u r r e n t s f l o w i n g along both the Tsawwassen and causeway beaches toward t h e i r common corner w i t h some form of r e t u r n flow out over Roberts Bank. I t i s t h i s c i r c u l a t i o n p a t t e r n t h a t we now wish to model. We should, perhaps, make a note here w i t h regards to the nature of the l i t t o r a l p r o c e s s e s a t work a t the Tsawwassen l o c a t i o n . I t i s h i g h l y l i k e l y t h a t a combination of very h i g h t i d e s together w i t h s t r o n g e r than normal winds a c t i n g over a r e l a t i v e l y s h o r t p e r i o d of time w i l l do as much to a l t e r the c o a s t a l c o n f i g u r a t i o n as w i l l months of moderate winds i n c o n j u n c t i o n w i t h normal h i g h t i d e s . Hence, the t r a n s p o r t of sediments a t the Tsawwassen s i t e may be of a h i g h l y n o n - l i n e a r nature and should not be viewed as being s o l e l y the r e s u l t of a continuous p r o c e s s . 19 CHAPTER 3 THEORY I Review of the Theory of Longshore Currents ^Before beginning the d e t a i l e d a n a l y s i s l e t us b r i e f l y review the mechanism by which wave-induced longshore cu-r r e n t s are generated. A longshore c u r r e n t i s d e f i n e d as the depth and time averaged t o t a l v e l o c i t y i n the longshore d i r e c t i o n . I t i s observed to reach a maximum i n the surfzone (James, 1972). The mean wave-induced c u r r e n t i s d r i v e n by the s p a t i a l v a r i a t i o n of the r a d i a t i o n s t r e s s - which i s the excess flow of momentum a s s o c i a t e d w i t h a p r o g r e s s i v e or st a n d i n g wave - a concept developed by Longuet-Higgins and Stewart (1964) and a p p l i e d to longshore c u r r e n t s by Bowen (1969a). T h i s e x t r a " k i n e t i c " p r e s s u r e term i n the presence of waves, i s due' to the components of v e l o c i t y c o r r e s p o n d i n g to the o r b i t a l motion of wave p a r t i c l e s - i t i s a q u a d r a t i c non-l i n e a r q u a n t i t y which a r i s e s from time averaging over the wave o s c i l l a t i o n and i n t e g r a t i n g over the depth of the water. The s u b j e c t has been reviewed r e c e n t l y by M i l l e r and B a r c i l o n (1976). The momentum t r a n s f e r t h a t produces c i r c u l a t i o n (the spa-t i a l v a r i a t i o n of the r a d i a t i o n s t r e s s e s ) i s d i r e c t l y propor-t i o n a l to energy d i s s i p a t i o n . I f there were no d i s s i p a t i o n there would be no currents, s i n c e i t i s the g r a d i e n t s of the r a d i a t i o n s t r e s s term which are r e s p o n s i b l e f o r the f o r c i n g . 20 These g r a d i e n t s w i l l be balanced by set-up or set-down except when d i s s i p a t i o n i s p r e s e n t . Bowen (1969a) and Longuet-Higgins (19 70) both used f i r s t order s i n u s o i d a l (or 'Ai r y ' ) waves as a b a s i s f o r c a l c u l a t i n g momentum and energy f l u x e s i n the near-shore s t r e g i o n . On the b a s i s of 1 order theory and the assump-t i o n of n e g l i g i b l e wave energy d i s s i p a t i o n o u t s i d e the surfzo n e there are no s p a t i a l v a r i a t i o n s i n the r a d i a t i o n s t r e s s and t h e r e f o r e no d r i v i n g o u t s i d e the b r e a k e r - l i n e s . I n s i d e the surfzone, i f the assumption i s made t h a t wave he i g h t i s p r o p o r t i o n a l to depth, there e x i s t v a r i a t i o n s i n the r a d i a t i o n s t r e s s a c r o s s the surfzone and hence a d r i v i n g f o r c e f o r the longshore c u r r e n t ( M i l l e r and B a r c i l o n , 1976). Under steady s t a t e c o n d i t i o n s , the d r i v i n g f o r c e must be balanced by f r i c t i o n . In shallow water, bottom f r i c - ^ t i o n i s most important. However (James, 1972), l a t e r a l mixing a l s o c o n t r i b u t e s and allows the longshore c u r r e n t to spread seawards of the su r f z o n e . In t h i s chapter we s h a l l d e r i v e the s e t of governing equations which are v a l i d f o r a g e n e r a l beach w i t h arb4 i t r a r y bottom topography. Chapter 4 w i l l be devoted to a d i s c u s s i o n of some numerical models w h i l e i n Chapter 5 we s h a l l apply these equations to an a n a l y t i c a l model. 22 I I F ormulation of the Problem We s h a l l b egin our a n a l y s i s w i t h the mean momentum equations f o r a steady wave f i e l d as g i v e n by O'Rourke and LeBlond (1970). 3.2) where; d i s the t o t a l water depth equal to the sum of the l o c a l s t i l l water depth h, and the mean displacement of the water s u r f a c e from the s t i l l water l e v e l , ?|; thus d=h+ f= ^ C u ^ w i t h C a drag c o e f f i c i e n t of OXlO""^) i s the water d e n s i t y U « ^ = tf(<?h* ) h > W i t h 0-3<V<0-6 hj£_ , a c h a r a c t e r i s t i c depth (Longuet-Higgins, 1970, P a r t 1) g i s the a c c e l e r a t i o n due to g r a v i t y U,V are the mean v e l o c i t y components i n the longshore (x) and o f f s h o r e (y) d i r e c t i o n s r e s p e c t i v e l y . They i n c l u d e the mean c u r r e n t as w e l l as the mass t r a n s p o r t of the waves and are assumed to be depth independent. The r a d i a t i o n s t r e s s terms f o r a t r a i n of monochromat small amplitude waves of amplitude a and wave number k, pro p a g a t i n g over a n e a r l y f l a t bottom are g i v e n by (3.3) where; E= \ ' - * ( <j£ i s the angle of i n c i d e n c e r e l a t i v e to the normal to the s h o r e - l i n e Note, t h a t as a means of s i m p l i f y i n g the a n a l y s i s we s h a l l i g n o r e h o r i z o n t a l eddy v i s c o s i t y , the e f f e c t s o f which (with r e s p e c t to the p r o f i l e of a longshore c u r r e n t along an i n f i n i t e beach) have been d i s c u s s e d by Longuet-Hi g g i n s (1970, P a r t 2). In shallow water and f o r sma l l angles of i n c i d e n c e the r a d i a t i o n s t r e s s terms reduce to (Longuet-Higgins and Stewart, 1964) «3 (3.4) W f/T 23 The v e r t i c a l l y i n t e g r a t e d c o n t i n u i t y equation i s ex-p r e s s e d as (3.5) L e t us d e f i n e the v e c t o r s T and F such t h a t (O'Rourke and LeBlond, 19 70) 1*1 (3.6) S u b s t i t u t i n g these i n t o equations (3.1) and (3.2) we get die * 5} (3.7) (3.8) a s : along w i t h the steady s t a t e c o n t i n u i t y equation (3.9) W r i t i n g them i n v e c t o r n o t a t i o n , the momentum equations take the form (3.10) We can e l i m i n a t e the term i n /q by t a k i n g the c u r l of equation (3.10), to get ~- «• -v, ~ (3.11) As a f u r t h e r s i m p l i f i c a t i o n of the a n a l y s i s we s h a l l now n e g l e c t the remaining n o n - l i n e a r terms i n equation (3.11), on the b a s i s of work done by A r t h u r (1962), who showed t h e i r e f f e c t s to be a narrowing of o f f - s h o r e flows and a widening of on-shore flows. Thus, while the non-l i n e a r terms a f f e c t the d e t a i l s of the c u r r e n t s we t r u s t t h a t t h e i r absence w i l l not g r e a t l y a f f e c t t h e i r g e n e r a l form and w i l l a l l o w us to o b t a i n an a n a l y t i c s o l u t i o n . Equation (3.11) then becomes S7*T = (3.12) In the seaward zone (where there i s no energy d i s s -i p a t i o n and hence no s p a t i a l v a r i a t i o n of the r a d i a t i o n s t r e s s ) , the d r i v i n g torques v a n i s h and (3.12) reduces to = O (3.13) In the surfzone, where there i s a dynamic balance between the d r i v e due to. wave s t r e s s and the r e t a r d i n g e f f e c t of bottom f r i c t i o n we have 25 (3.14) Le t us now d e f i n e a t r a n s p o r t stream f u n c t i o n ^ ( x , y ) from the steady s t a t e c o n t i n u i t y equation (3.9) such t h a t (3.15) 1^ S u b s t i t u t i n g these i n t o equations (3.13) and (3.14) the governing equation i n the seaward zone becomes 5 * (3.16) While i n the surf z o n e we get 2L — (3.17) 1 h 4 , ^ We have now to s o l v e equations (3.16) and (3.17) sub-j e c t to the topographic c h a r a c t e r i s t i c s and boundary c o n d i t i o n s 26 r e l e v a n t to our p a r t i c u l a r problem (as w i l l be d i s c u s s e d i n Chapter 4), but f i r s t l e t us comment on some d<6 the assumptions made i n a r r i v i n g a t these equations. (For a f u l l summary of approximations see Appendix A ) . 1/ The use of a bottom shear s t r e s s : : l i n e a r i z e d i n the longshore c u r r e n t v e l o c i t y (see equations (3.1) and (3.2)) i s j u s t i f i a b l e i f t h i s i s s m a l l compared w i t h the wave o r b i t a l v e l o c i t y , which as James (1972) p o i n t s out i s not always t r u e . In the presence of steady flow or long p e r i o d waves, a reasonable assumption appears to be t h a t the bottom shear s t r e s s be g i v e n by (James, 1972) j u s t o u t s i d e the bottom boundary l a y e r C i s a d i m e n s i o n l e s s c o e f f i c i e n t whose mag-nit u d e may be i n c r e a s e d by the presence of o s c i l l a t o r y flow. We s h a l l c o n s i d e r the use of t h i s f r i c t i o n term i n co n n e c t i o n w i t h the numerical models as d i s c u s s e d i n Chapter 4. A r e c e n t e x t e n s i o n to s t r o n g mean c u r r e n t s has been pr e s e n t e d by L i u and Dalrymple (19 78}. 2/ We have n e g l e c t e d to i n c l u d e a h o r i z o n t a l eddy v i s c o s i t y term (whose e f f e c t i s to t r a n s f e r momentum from the surfzone, where the d r i v i n g of the longshore c u r r e n t (3.18) where; u g i s the instantaneous t o t a l v e l o c i t y v e c t o r takes p l a c e , across the b r e a k e r - l i n e where there i s no d r i v i n g ) s i n c e such f r i c t i o n terms l e a d to a 4' order equation when a stream f u n c t i o n i s i n t r o d u c e d (whereas the ncl l i n e a r i z e d f r i c t i o n terms l e a d only to a 2 order equation) ( M i l l e r and B a r c i l o n , 1976). 3/ The remaining n o n - l i n e a r terms i n . e q u a t i o n (3.11) have been n e g l e c t e d on the b a s i s of work done by A r t h u r (196 2) who showed t h a t they p l a y no c a u s a t i v e r o l e i n the dynamics of the c u r r e n t s although they may a f f e c t t h e i r l o c a l c h a r a c t e r i s t i c s . The balance i n the l i n e a r equations i s then taken to be between the r a d i a t i o n s t r e s s terms, the p r e s s u r e g r a d i e n t (set-up) and the bottom f r i c t i o n terms. CHAPTER 4 NUMERICAL MODELLING I I n t r o d u c t i o n T h i s chapter i s devoted to the d i s c u s s i o n of s e v e r a l numerical models of the wave-induced c i r c u l a t i o n i n a beach-breakwater c o n f i g u r a t i o n s i m i l a r to t h a t of the Tsawwassen s i t e . A numerical approach w i l l g i v e us g r e a t e r f l e x i b i l i t y i n choosing beach c o n f i g u r a t i o n s and bottom topography (such as a tr e n c h and d r o p - o f f to deeper water) than an a n a l y t i c a l approach (to be d i s c u s s e d i n Chapter 5) and a l s o allows the comparison of models u s i n g d i f f e r e n t forms ( l i n e a r and non-l i n e a r ) bottom f r i c t i o n . I I Model Layout Each numerical model s h a l l c o n s i s t of two beaches of f i n i t e l e n g t h i n t e r s e c t i n g a t some angle ^ (0°< 90° ) enclosed w i t h i n a r e c t a n g u l a r b a s i n (the w a l l s of which have been put as f a r away as i s p r a c t i c a l so as to l i m i t t h e i r e f f e c t upon the c u r r e n t c i r c u l a t i o n ) as shown i n Fi g u r e 10. The seaward zone has been d i v i d e d i n t o t h r e e r e g i o n s c o n s i s t i n g of two surfzones and an o f f - s h o r e zone beyond the b r e a k e r - l i n e s . The extent of the surfzones, the regions over which the longshore shear s t r e s s e s e x i s t , are s u b j e c t to wave he i g h t changes a t the b r e a k e r - l i n e s . As d i s c u s s e d by O'Rourke and LeBlond (19 70), a c c u r a t e d e t e r m i n a t i o n of the F i g u r e 10. Plan view of the numerical model geometry p o s i t i o n of the shore and b r e a k e r - l i n e s would l e a d to t e d i o u s matching problems a t these boundaries which would do l i t t l e to a l t e r the o v e r a l l c u r r e n t p a t t e r n s . Hence the boundaries of the surfzones s h a l l be taken as the space-averaged values of the shore and b r e a k e r - l i n e s . To conform to the Tsawwassen c o n f i g u r a t i o n the n a t u r a l beach s h a l l be approximately twice as long as t h a t corresponding to the causeway. The r a t i o s of surfzone l e n g t h to width, i n the models, are 10:1 and 20:1 f o r the causeway and n a t u r a l beach r e s p e c -t i v e l y . The models presented as such e x h i b i t a d i s c r e p a n c y w i t h r e s p e c t to s c a l e , when compared d i r e c t l y to f i e l d dimen-s i o n s , s i n c e they then y i e l d h i g h l y u n r e a l i s t i c s urfzone widths o f approximately 300 m. Attempts to p r e s e n t models which are s c a l e d more r e a l i s t i c a l l y would, however, due to r e s t r i c t i o n s of space, r e s u l t i n surfzones of n e g l i g i b l e width. The models then s t r i k e a compromise, a l l o w i n g us to study the c u r r e n t s i n a c o n f i g u r a t i o n f o r which the s u r f -zones are much longer than they are wide y e t s t i l l r e t a i n some degree of d e t a i l . Note t h a t we have removed the s u r f z o n e s ' i n t e r s e c t i o n corner to ease the s p e c i f i c a t i o n of the boundary c o n d i t i o n s there, the r a t i o n a l i z a t i o n being t h a t i f the surfzone widths are s u f f i c i e n t l y small i n comparison to t h e i r l e n g t h s , then removing the corner w i l l have l i t t l e e f f e c t upon the major f e a t u r e s of the c u r r e n t c i r c u l a t i o n , an assumption which s h a l l be j u s t i f i e d a p o s t e r i o r i . 31 The o f f - s h o r e zone s h a l l accomodate v a r i a t i o n s i n bottom topography i n the f o l l o w i n g manner. Immediately seaward of the b r e a k e r - l i n e s i s a shallow s h e l f r e g i o n which i n some models f i l l s the remainder of the b a s i n . In other models the o f f - s h o r e zone i s f u r t h e r s u b - d i v i d e d (see F i g u r e 10) to i n c o r p o r a t e a much deeper r e g i o n beyond the s h e l f - z o n e t y p i c a l of the d r o p - o f f i n t o Georgia S t r a i t found a t the Tsawwassen l o c t i o n ; w h i l e s t i l l o thers w i l l i n c l u d e a sub-marine t r e n c h running p a r a l l e l to the causeway across the width of the s h e l f - z o n e . We s h a l l thus have an o p p o r t u n i t y to study the e f f e c t s of these v a r i a t i o n s i n bottom topo-graphy upon the wave-induced c i r c u l a t i o n . L e t us f i r s t c o n s i d e r the c o n f i g u r a t i o n shown i n F i g u r e 11, f o r which the beach i n t e r s e c t i o n angle i s 90°. We s h a l l assume surfzone I to be approximately uniform i n the y - d i r e c t i o n so t h a t (4.1) t h a t i s 0 (4.2) The s l o p e m, i n t h i s r e g i o n s h a l l be d e f i n e d by (4.3) 32 F i g u r e 11. B o u n d a r y c o n d i t i o n s f o r t h e p a r t i c u l a r c a s e o f a p e r p e n d i c u l a r b e a c h i n t e r -s e c t i o n . 33 S i m i l a r l y , we s h a l l assume surfzone I I to be approx-imately uniform i n the x - d i r e c t i o n , which g i v e s us (4.4) so t h a t ^cL ^ o (4.5) The slope m^, i n t h i s r e g i o n s h a l l be d e f i n e d by »3* 4, (4.6) Subject to the above approximations the governing equations (3.10) and (3.17) f o r the surfzones and o f f - s h o r e zone become 1. Surfzone I (4.7) 1, [ \ ^ df> ) 7 V 1 J " o c i * l 3 v J J Q» 3 * X 2. Surfzone I I (4.8) 3. O f f - s h o r e zone d 3 ^ ^ % H + - a a«s*i = o (4.9) Consider now equation (4.7), the governing equation f o r surfzone I, i n t o which we s h a l l s u b s t i t u t e the expre-s s i o n s f o r the r a d i a t i o n s t r e s s terms (equations (3.4);)), to get .<\<Xt 1 ^di } ^ • ^<x» I d ? I 5V I * (4.10) 35 Taking the wave amplitude i n the surfzone to be pro-p o r t i o n a l to the mean water depth ( Munk, 1949) so t h a t (4.11) g i v e s us (4.12) and 2w ^ (4.13) so t h a t equation (4.10) reduces to or 11 AVO, (4.14) (4.15) Assuming t h a t S n e l l ' s law of r e f r a c t i o n holds i n the surfzone (Longuet-Higgins, 1956) so t h a t o -(4.16) (where j00 and d 0 are the angle of i n c i d e n c e and the depth at the b r e a k e r - l i n e ) equation (4.15) f o r surfzone I becomes S i m i l a r l y f o r surfzone I I we get "V V. ft J  (4.18) In summary then, the governing equations f o r the three zones of our numerical model are 1. Surfzone I (4.19) 2..Surfzone I I (4.20) 3. Off-shore zone ± ( i - - O (4.21) The values of the stream f u n c t i o n , i n each model, s h a l l be c a l c u l a t e d from these governing d i f f e r e n t i a l equations by the Gauss-Seidel i t e r a t i v e technique (as d i s c u s s e d i n Appendix B) s u b j e c t to the c o n s t r a i n t t h a t the stream func-t i o n i s i d e n t i c a l l y zero along the s h o r e - l i n e , the seaward edges of the i n t e r s e c t i o n corner and the boundaries of the seaward zone as shown i n F i g u r e 11. 38 I I I A Simple C o n f i g u r a t i o n The f i r s t model we s h a l l p r e s e n t i s shown i n F i g u r e 12, and s h a l l serve to i l l u s t r a t e those p r o p e r t i e s common to a l l the models. This c o n f i g u r a t i o n shows a p e r p e n d i c u l a r beach-breakwater i n t e r s e c t i o n w i t h no o f f - s h o r e depth v a r i a -tions. a r i d c u r r e n t s of equal s t r e n g t h . Models i n c o r p o r a t i n g f e a t u r e s which conform more c l o s e l y to the f i e l d s i t e w i l l f o l l o w . I l l u s t r a t e d i n F i g u r e 12 i s the e n t i r e f i e l d of ac c o r d i n g to equations (4.19), (4.20) and (4.21). In steady flow, the s t r e a m l i n e s are e q u i v a l e n t to the p a t h l i n e s of the f l u i d p a r t i c l e s . Note, t h a t i n order to a v o i d c l u t t e r i n g the c u r r e n t c i r c u l a t i o n diagrams, the b r e a k e r - l i n e s and d r o p - o f f w i l l not be shown. T h e i r p o s i t i o n s may be determined by r e f e r r i n g to F i g u r e 10 - the b r e a k e r - l i n e s o r i g i n a t e i n the i n t e r s e c -t i o n corner and run p a r a l l e l to the s h o r e - l i n e s f o r the l e n g t h of the surfzones w h i l e , as w i l l be seen, the l o c a t i o n of the d r o p - o f f , i n those models i n c o r p o r a t i n g i t , w i l l be apparent. The p o s i t i o n of the o f f - s h o r e trench, when present, w i l l always be i n d i c a t e d . The c i r c u l a t i o n p a t t e r n c o n s i s t s of two c o u n t e r - r o t a t i n g c e l l s , each c e l l d r i v e n by one of the wave-induced c u r r e n t s generated i n the s u r f z o n e s . The s t r e a m l i n e s of the longshore c u r r e n t s converge toward the i n t e r s e c t i o n apex where they merge to form a r i p c u r r e n t which then flows out over the s h e l f r e g i o n . Having t r a v e l l e d a d i s t a n c e across the s h e l f , F i g u r e 1 2 . Current c i r c u l a t i o n f o r p e r p e n d i c u l a r beach i n t e r s e c t i o n / e q u a l c u r r e n t s t r e n g t h s and no o f f - s h o r e depth v a r i a t i o n s . 40 the r e t u r n flow must d i v i d e - each branch t u r n i n g to com-p l e t e i t s c e l l - due to c o n t i n u i t y . The q u a l i t a t i v e aspects of the c i r c u l a t i o n p a t t e r n are then c o n s i s t e n t w i t h o b s e r v a t i o n s and measurements of sediment t r a n s p o r t taken at the Tsawwassen s i t e . The l o n g -shore c u r r e n t s f l o w i n g toward the i n t e r s e c t i o n apex are conducive to sediment t r a n s p o r t toward the corner, as was found a t the f i e l d s i t e , w i t h the o f f - s h o r e r e t u r n flow now p r e d i c t e d to take the form of a r i p c u r r e n t . The e f f e c t of the c o n v e c t i v e i n e r t i a l terms which we n e g l e c t e d i n Chapter 3 w i l l be to strengthen the r e t u r n flow as i t comes out of the corner because the c u r r e n t there i s f l o w i n g i n t o deeper water, and to weaken the c u r r e n t s which feed back i n t o the ends of the surfzones s i n c e the flow there i s i n t o shallower water. The e x c l u s i o n of h o r i z o n t a l eddy v i s c o s i t y ( t h a t i s , l a t e r a l mixing) i n the governing equations g e n e r a l l y l e a d s to the v e l o c i t y being g r e a t e r i n s i d e the surfzones and f a l l i n g o f f more r a p i d l y seaward of the b r e a k e r - l i n e s than experimental r e s u l t s i n d i c a t e (James, 1972). To i n c l u d e h o r i z o n t a l eddy v i s c o s i t y i n the equations would i / a l t e r the surfzone v e l o c i t y p r o f i l e s so as to de-crease the v e l o c i t y maxima near the b r e a k e r - l i n e s and i n -crease the v e l o c i t y a t p o i n t s c l o s e r to the shore, t h a t i s , to e f f e c t i v e l y f l a t t e n the p r o f i l e s , and i i / r e s u l t i n the v e l o c i t y o u t s i d e the surfzones f a l l i n g o f f l e s s s h a r p l y than l i n e a r theory, w i t h only bottom f r i c t i o n , p r e d i c t s . 41 IV The Corner Geometry In s e c t i o n I I of t h i s chapter we d i s c u s s e d the need to remove the i n t e r s e c t i o n corner i n order to ease the s p e c i f i c a t i o n of the boundary c o n d i t i o n s t h e r e , and j u s t i -f i e d i t on the grounds t h a t i f the widths of the surfzones were s u f f i c i e n t l y s m a l l i n comparison to t h e i r l e n g t h s then removing the corner would have l i t t l e e f f e c t upon the over-a l l c i r c u l a t i o n . F i g u r e 13 shows a model c o n f i g u r a t i o n i d e n -t i c a l to t h a t of F i g u r e 12 w i t h the e x c e p t i o n t h a t the corner has now been i n c l u d e d . The assumption has been made t h a t the zero-valued stream l i n e would emerge from the corner a t the same angle a t which i t c r o s s e s the s h e l f zone (as shown i n F i g u r e 12), hence we have s p e c i f i e d the boundary c o n d i t i o n i n the corner a c c o r d i n g l y - t h a t the stream func-t i o n must be i d e n t i c a l l y zero along the l i n e b i s e c t i n g the i n t e r s e c t i o n angle ( f o r the case of equal c u r r e n t s t r e n g t h s ) . The r e s u l t s , as shown i n F i g u r e 13, j u s t i f y the arguments f o r removing the corner s i n c e the e f f e c t on the o v e r a l l flow p a t t e r n i s indeed n e g l i g i b l e . Note a l s o t h a t r F i g u r e 13 shows a s t a g n a t i o n p o i n t i n the corner thus i n d i c a t i n g a p o s s i b l e l o c a t i o n f o r the d e p o s i t i o n of beach m a t e r i a l s (which have been c a r r i e d along the s u r f z o n e s by the l o n g -shore c u r r e n t s ) . T h i s i s a g a i n c o n s i s t e n t w i t h o b s e r v a t i o n s a t the Tsawwassen l o c a t i o n which show the corner to be f i l l i n g up. A normalized s t r e a m l i n e p r o f i l e taken half-way along the breakwater surfzone of F i g u r e 12 i s shown i n F i g u r e 14, F i g u r e 13. P e r p e n d i c u l a r beach i n t e r s e c t i o n model with the corner r e p l a c e d . (The s m a l l 'corner' appearing i n the s t r e a m l i n e p a t t e r n i s a numerical a r t i f a c t . ) 1.0 .9 .8 -7 .6 -5 .4 .3 .2 .1 F i g u r e 14. Normalized s t r e a m l i n e p r o f i l e taken a c r o s s the s u r f z o n e along c r o s s - s e c t i o n qq' i n F i g u r e 12 (shown as a s o l i d l i n e ) , t o g e t h e r w i t h t h a t of an i n f i n i t e beach model from Dalrymple e t a l (19 77) (shown as O ) : where i s the v a l u e of the stream f u n c t i o n a t the b r e a k e r - l i n e , the width of which i s /•/! . together w i t h t h a t f o r the i n f i n i t e beach model developed by Dalrymple e t a l (1977). They compare very c l o s e l y , the longshore c u r r e n t p r o f i l e half-way along the breakwater s u r f zone,, approaching t h a t of an i n f i n i t e beach. V Acute I n t e r s e c t i o n Angle To conform more c l o s e l y to the c o n f i g u r a t i o n a t the Tsawwassen s i t e , the beach-breakwater i n t e r s e c t i o n angle must be reduced to approximately 60° as has been done i n F i g u r e 15, which shows the c i r c u l a t o r y system f o r equal c u r r e n t s t r e n g t h s and no v a r i a t i o n s i n o f f - s h o r e bottom topography. The g e n e r a l f e a t u r e s are s i m i l a r to those of the 90° beach i n t e r s e c t i o n model: the c r o s s - s u r f z o n e s t r e a m l i n e p r o f i l e i s unchanged and the o f f - s h o r e r e t u r n flow,;again b i s e c t s the beach i n t e r s e c t i o n angle. (The •steps' i n the s t r e a m l i n e s adjacent to the n a t u r a l beach are a computational a r t i f a c t caused by the f a c t t h a t , i n the acute angle models, the n a t u r a l beach runs d i a g o n a l l y to the ( f i n i t e l y spaced) g r i d l a t t i c e used i n the numerical models). The e f f e c t s of v a r y i n g c u r r e n t s t r e n g t h (due perhaps to r e l a t i v e d i f f e r e n c e s i n angle of wave i n c i d e n c e a t the b r e a k e r - l i n e s , beach s l o p e s , surfzone widths e t c . , or com-b i n a t i o n s t h e r e o f ) are shown i n F i g u r e s 16 and 17. We s h a l l not s p e c i f y the exact v a l u e s of f a c t o r s necessary to account f o r the v a r i a t i o n s i n r e l a t i v e c u r r e n t s t r e n g t h but merely p r e s e n t the r e s u l t s f o r r a t i o s of n a t u r a l beach F i g u r e 15. C u r r e n t c i r c u l a t i o n f o r acute beach-breakwater i n t e r s e c t i o n ( = 6 0 ° ) , equal c u r r e n t s t r e n g t h s and no o f f -shore depth v a r i a t i o n s . F i g u r e 16. C i r c u l a t o r y p a t t e r n f o r a r a t i o of n a t u r a l beach c u r r e n t s t r e n g t h to causeway beach c u r r e n t s t r e n g t h of 2:1.. There are no o f f - s h o r e depth v a r i a t i o n s . F i g u r e 17. C i r c u l a t o r y p a t t e r n f o r a r a t i o of n a t u r a l beach c u r r e n t s t r e n g t h to causeway beach c u r r e n t s t r e n g t h of 4:1. There are no o f f - s h o r e depth v a r i a t i o n s . c u r r e n t s t r e n g t h to causeway beach c u r r e n t s t r e n g t h of 2:1 and 4:1 r e s p e c t i v e l y . The r e s u l t s are what we should i n t u i t i v e l y expect, w i t h the c e l l d r i v e n by the s t r o n g e r c u r r e n t dominating the c i r c u l a t i o n p a t t e r n as the r a t i o of the c u r r e n t s t r e n g t h s i n c r e a s e s . The o f f - s h o r e r e t u r n flow i s d e f l e c t e d toward the causeway i n such a way t h a t i t s tangent d i v i d e s the angle of beach i n t e r s e c t i o n ^ , i n t o angles whose r a t i o s vary as the c u r r e n t s t r e n g t h s , t h a t i s , 2:1 and 4:1 r e s p e c t i v e l y . VI V a r i a b l e Bottom Topography The next models to be presented i n c o r p o r a t e v a r i a t i o n s i n the bottom topography of the type to be found a t the Tsawwassen l o c a t i o n , t h a t i s , a d r o p - o f f from the shallow s h e l f r e g i o n to much deeper water as from Roberts Bank to Georgia S t r a i t , and an o f f - s h o r e t r e n c h which runs p a r a l l e l to the causeway across the width of the s h e l f - z o n e (see F i g u r e 10 f o r the b a s i c l a y o u t ) . Shown i n F i g u r e 18 i s the c i r c u l a t i o n p a t t e r n f o r a model f e a t u r i n g the d r o p - o f f to deeper water ( f o r c u r r e n t s of equal s t r e n g t h ) . The deeper zone has been g i v e n a depth of 100 m. compared to the s h e l f depth of 5 m., a r a t i o of 20: 1. The r e s u l t s show a pronounced d e f l e c t i o n of the stream l i n e s as they approach the l a r g e mass of water beyond the d r o p - o f f and a d i s c o n t i n u i t y i n the t a n g e n t i a l v e l o c i t y C i r c u l a t o r y p a t t e r n f o r the l i n e a r bottom f r i c t i o n model i n c o r p o r a t i n g a d r o p - o f f to deep water. as they c r o s s the boundary from shallow to deep water. Such a d i s c o n t i n u i t y i s allowed by the absence of l a t e r a l f r i c t i o n i n the model. An o f f - s h o r e t r e n c h i s next i n t r o d u c e d to the bottom topography. I t i s r e c t a n g u l a r i n shape, w i t h a mean h i g h t i d e depth of 10 m. (twice t h a t of the s h e l f ) and t r a n v e r s e s most of the width of the s h e l f as shown i n F i g u r e 10, i t s o f f - s h o r e end emptying i n t o the deep o f f - s h o r e zone. The r e s u l t s are giv e n i n F i g u r e 19; the i n f l u e n c e of the t r e n c h may be seen by comparing t h i s f i g u r e w i t h F i g u r e 18. The r e s u l t s are s u r p r i s i n g ; i n a d d i t i o n to the chann-e l i n g of the o f f - s h o r e flow i n t o the tr e n c h , which i s what we would i n t u i t i v e l y expect, we see a l a r g e i n c r e a s e i n the t o t a l t r a n s p o r t of the o f f - s h o r e flow. A l a r g e eddy forms wit h seaward flow over the t r e n c h and r e t u r n flow between the t r e n c h and the causeway. Le t us pause here to r e f l e c t upon the v a l i d i t y of these r e s u l t s . R e c a l l t h a t the l i n e a r form of bottom f r i c t i o n , w h i l e simple, i s not r e a l i s t i c , b e i n g dependent upon two r a t h e r severe r e s t r i c t i o n s : i / t h a t the longshore c u r r e n t v e l o c i t y be smal l com-pared to the wave o r b i t a l v e l o c i t y i n the surfzone and i i / t h a t the angle of wave i n c i d e n c e be very s m a l l . As p o i n t e d out by James (1972) and L i u and Dalrymple (1978) these assumptions are not always s a t i s f i e d . F u r t h e r -more, i n the o f f - s h o r e zone, the wave o r b i t a l v e l o c i t y a t F i g u r e 19. C i r c u l a t o r y p a t t e r n f o r the l i n e a r bottom f r i c t i o n model i n c o r p o r a t i n g a d r o p - o f f to deep water and an o f f -shore t r e n c h running p a r a l l e l to the causeway. the bottom i s l i k e l y to be s m a l l e r than the mean c u r r e n t ; a l i n e a r i z a t i o n of the bottom s t r e s s , i n the form g i v e n below i s then c l e a r l y i n a p p r o p r i a t e . Hence, i n l i g h t of the q u e s t i o n a b l e v a l i d i t y of these assumptions and the d i f f i c u l t y i n i n t e r p r e t i n g the r e s u l t s so d e r i v e d , we s h a l l not s p e c u l a t e f u r t h e r upon the l i n e a r r e s u l t s . Rather, l e t us proceed to d i s c u s s the p r o p e r t i e s of models u s i n g the more r e a l i s t i c form of bottom f r i c t i o n i n the o f f - s h o r e zone, as mentioned i n Chapter 3. (The r e l e v a n t f i n i t e d i f f e r e n c e equation i s g i v e n i n Appendix B). The r e s u l t s as d e r i v e d f o r i i / a model f e a t u r i n g only the d r o p - o f f to deeper water are g i v e n i n F i g u r e s 20 and 21, w h i c h ' e x h i b i t n e g l i g i b l e d i f f e r e n c e s i n comparison to t h e i r l i n e a r c o u n t e r p a r t s ( F i g u r e s 15 and 18). However, the flow p a t t e r n f o r the n o n - l i n e a r model i n c o r p o r a t i n g both the d r o p - o f f and the t r e n c h (as shown i n F i g u r e 22) d i f f e r s c o n s i d e r a b l y from the l i n e a r model (Fi g u r e 19). Upon comparison of F i g u r e s 22 and 21, we see a d e f i n i t e c h a n n e l i n g of the o f f - s h o r e r e t u r n flow i n t o the t r e n c h , again as expected, but without the i n -o r d i n a t e a m p l i f i c a t i o n of the flow p a t t e r n . i / a model w i t h no v a r i a t i o n s i n bottom topography and F i g u r e 20. C i r c u l a t o r y p a t t e r n f o r the n o n - l i n e a r bottom f r i c t i o n model wi t h c u r r e n t s of equal s t r e n g t h and no o f f - s h o r e depth v a r i a t i o n s . F i g u r e 21. C u r r e n t c i r c u l a t i o n f o r the n o n - l i n e a r bottom f r i c t i o n model i n c o r p o r a t i n g a d r o p - o f f to deep water. F i g u r e 22. Current c i r c u l a t i o n f o r the n o n - l i n e a r bottom f r i c t i o n model i n c o r p o r a t i n g a d r o p - o f f to deep water and an o f f - s h o r e t r e n c h running p a r a l l e l to the causeway. The d e f l e c t i o n of the c u r r e n t i s most apparent on the causeway s i d e of the t r e n c h , as i s shown by a decrease i n t r a n s p o r t there, w i t h a corresponding i n c r e a s e i n the t r e n c h as water i s drawn i n t o i t . V e l o c i t y p r o f i l e s taken along c r o s s - s e c t i o n s pp" and dd 1 of F i g u r e s 21 and 22 r e s p e c t i v e l y are compared i n F i g u r e 23, and found to be s i m i l a r i n form. The lower p r o f i l e , from F i g u r e 21, shows the v e l o c i t y i n the absence of a t r e n c h . The flow i s s l i g h t l y slower i n the presence of the t r e n c h (upper curve) but the t r a n s p o r t i s n e v e r t h e l e s s i n c r e a s e d i n the t r e n c h because the i n c r e a s e i n depth more than compensates f o r the decrease i n speed. — \ . r \ I \ I F i g u r e 23. V e l o c i t y p r o f i l e s taken along c r o s s s e c t i o n s pp 1 and dd' of F i g u r e s 21 and 22 r e s p e c t i v e l y . The upper pro f i l e i s taken from the trench model (Fi g u r e 22). The p o s i t i o n of the t r e n c h i s shown by the dashed l i n e . 58 VII The I n f l u e n c e of Depth V a r i a t i o n s Let us c o n s i d e r f u r t h e r the e f f e c t s of v a r i a t i o n s i n bottom topography upon the l o c a l c u r r e n t c i r c u l a t i o n . We b e g i n w i t h the x and y components of v e l o c i t y r e s p e c t i v e l y ; (4.22) (4.23) so t h a t i n the case of a f l a t bottom we have \ 7 ( VA =o (4.24) wh i l e , f o r a bottom which i s not f l a t (4.25) Consider now the bottom to be n e a r l y f l a t , so t h a t 4-(4.26) To z e r o t h order, we have from (4.24) and (4.25) (4.27) while to 1 order 7 J ^ = ?<*^ (4.28) where i s a measure of the departure from average a t a p o i n t , so t h a t i f ^ 'H, > 0 then ^ i s i n c r e a s i n g . What then are the e f f e c t s of a v a r i a t i o n i n depth upon the v e l o c i t y and t r a n s p o r t ? Consider the one-dimensional s i t u a t i o n as i l l u s t r a t e d i n F i g u r e 24, where a l l the motion i s i n the x - d i r e c t i o n , so t h a t 04.29) which g i v e s us ^ o = CO, (4.30) <7 o^= L o ^ o V o ^ (4.31) and « " t * » " V * , r . (4.32) ( =0 i f 9nr^ has no y-component) 60 U Figure.24. V e l o c i t y f i e l d f o r the one-dimensional model. 61 Is the v e l o c i t y changed? ( t h a t i s , i s H, ? ) From (4.22) we have or, to 1 order ^ - ^ • " ^ (4.33) To determine , l e t X (r.V so t h a t - M*\Q ^ ' \ s \ l ( v/ - i ) ( a s shown i n F i g u r e 25) t h a t i s i v * * { ^ o u - 3 4 ) givxng us (4.35) F i g u r e 25. V a r i a t i o n i n depth as d e f i n e d by / r ^ = y ( y - l ) F i g u r e 26. V a r i a t i o n i n depth as d e f i n e d by or)=N(y-ig) 63 s t So t h a t to l 0 1 " order, u, =0, hence the speed of the c u r r e n t i s not changed i n t h i s one-dimensional case. ^ A s l i g h t l y d i f f e r e n t case i s as f o l l o w s : c o n s i d e r a v a r i a t i o n i n depth, as shown i n F i g u r e 26, such t h a t the average depth i s not changed, by l e t t i n g (4.36 ) so t h a t we have o o (4.37) Then we have (4.38) and (4.39) Now the t o t a l t r a n s p o r t i s not changed, t h a t i s u , + <^ = or from (4.33) Thus from (4.39) we have S o l v i n g f o r the constants (4.40) (4.41) H - ^ ^ o U o ^ * COA^0L<s"t v (Lon'STtouA't^ (4.42) V>0 = ^ V o « 0 t > | * - ^ ( 4 - 4 3 ) Again, the v e l o c i t y i s not a f f e c t e d , as i s seen upon s u b s t i t u t i o n of H , from (4.43) i n t o Hence, w i t h ^ =0, the t r a n s p o r t l o c a l l y i s (4.44) so t h a t i n both cases considered, the t r a n s p o r t i n c r e a s e s s t l i n e a r l y w i t h the depth of the f l u i d l a y e r because to 1 order, c o n t i n u i t y alone does not r e q u i r e the v e l o c i t y to change w i t h s m a l l changes i n depth. The r e s u l t s then, f o r our simple one-dimensional model are i n agreement wi t h the numerical models which show an i n c r e a s e i n t r a n s p o r t over the t r e n c h (although there i s l e s s of an i n c r e a s e i n the more r e a l i s t i c n o n - l i n e a r case due to the s t r o n g e r e f f e c t s of f r i c t i o n ) . CHAPTER 5 ANALYTICAL MODEL I. P h y s i c a l D e s c r i p t i o n We s h a l l now attempt to develop an a n a l y t i c a l model of the wave-induced c u r r e n t c i r c u l a t i o n i n the v i c i n i t y of a beach-breakwater i n t e r s e c t i o n . During the course of the d i s c u s s i o n we s h a l l encounter a number of d i f f i c u l t i e s ( d e s p i t e f u r t h e r s i m p l i f i c a t i o n s to the a n a l y s i s ) and s h a l l f i n d , i n f a c t , t h a t we must r e s o r t to some numerical means i n order to salvage a s o l u t i o n . We wish then to apply equations (3.16) and (3.17) to our a n a l y t i c a l model which c o n s i s t s of the i n t e r s e c t i o n a t r i g h t angles of two s e m i - i n f i n i t e beaches, as shown i n F i g u r e 27. (The s o l u t i o n we d e r i v e f o r t h i s g e n e r a l con-f i g u r a t i o n need only be r o t a t e d 180° so as to conform more c l o s e l y to t h a t of the f i e l d s i t e . ) As d i s c u s s e d i n Chapter 2, the Tsawwassen and causeway beaches are of g r e a t l e n g t h , hence we have r e p l a c e d them i n the model by s e m i - i n f i n i t e beaches. T h i s allows us to circumvent the need of s p e c i f y i n g the boundary c o n d i t i o n s a t the ends of the surfzones (which we do not know a p r i o r i ) , s i n c e we can now simply a l l o w the stream f u n c t i o n p r o f i l e s to approach those of the i n f i n i t e beach s o l u t i o n s as we get f u r t h e r from the c o r n e r . The quadrant of i n t e r e s t has a g a i n been d i v i d e d i n t o three zones, c o n s i s t i n g of two surfzones and an o f f - s h o r e F i g u r e 27. Plan view and boundary c o n d i t i o n s of the a n a l y t i c a l model. zone seaward of the b r e a k e r - l i n e s . The angles of approach of a wave-train w i t h r e s p e c t to the b r e a k e r - l i n e s of s u r f - ' We s h a l l invoke the f o l l o w i n g assumptions and approx-imations as f o r the numerical model: i / the i n t e r s e c t i o n corner has been removed to ease the s p e c i f i c a t i o n of the boundary conditons there i i / the boundaries of the surfzones s h a l l be taken as the space - averaged v a l u e s of the shore and breaker-l i n e s i i i / the surfzones are assumed uniform i n the l o n g -shore d i r e c t i o n s (equations (4.1) through (4.6)) i v / the wave amplitude i n the surfzones i s taken to be p r o p o r t i o n a l to the mean depth of the water (equations (4.11), (4.12) and (4.13)) v/ S n e l l ' s law of r e f r a c t i o n holds i n the surfzones (equation (4.16) > F u r t h e r , the o f f - s h o r e zone s h a l l now be assumed to have a u n i f o r m l y f l a t bottom, with a depth equal to the water depth a t the b r e a k e r - l i n e s , which g i v e s us Under these assumptions and approximations the governing equations f o r the three zones of our a n a l y t -i c a l model become zones I and I I w i l l be taken as # r e s p e c t i v e l y . <^3 r ^<^3 = O (5.1) 1. Surfzone I (5.2) 2. Surfzone I I 0^ -4 U (5.3) 3. O f f - s h o r e zone (5.4) which d i f f e r from those f o r the numerical model (equations (4.19), (4.20) and (4.21)) o n l y i n the o f f - s h o r e zone, where v a r i a t i o n s i n the bottom topography are now f o r b i d d e n . II A n a l y s i s Surfzone I Le t us begin w i t h the governing equation f o r s u r f z o n e I (the r e g i o n f o r which 0 $ x < , and y > y\, ) a s g i v e n by equation (5.2), which i s to be s o l v e d s u b j e c t to the f o l l o w i n g boundary c o n d i t i o n s : i / the stream f u n c t i o n i s i d e n t i c a l l y zero along the mean s h o r e - l i n e d e f i n e d by x-0 (5.5) t h a t i s , i i / the stream f u n c t i o n tends to the i n f i n i t e beach s o l u t i o n f a r away from the corner, so t h a t y-=> voo i i i / the stream f u n c t i o n i s i d e n t i c a l l y zero a t the end of the surfzone The homogeneous equation can be s o l v e d u s i n g the s e p a r a t i o n of v a r i a b l e s technique. By making the s u b s t i t u t i o n ^ l - * . ^ = ^ C / o V t N p (5.7) equation (5.6) g i v e s the f o l l o w i n g p a i r of o r d i n a r y d i f f e r e n -t i a l equations > / " - a a V - o (5.8) (5.9) 71 where "a" i s a s e p a r a t i o n constant and the primes denote d i f f e r e n t i a t i o n w i t h r e s p e c t to the a p p r o p r i a t e independent v a r i a b l e . The s o l u t i o n of equation (5.8) which vanishes a t x=0 i s (Kamke, 1959) 3/ ——— (5.10) (where J,.(ax) i s a Bes s e l f u n c t i o n of the f i r s t k ind: The g e n e r a l s o l u t i o n of equation (5.9) i s ^ v p - ^ . e ^ . ^ e ^ ( s . i i ) where B, and B^ s h a l l be determined by the boundary c o n d i t i o n s , As a p a r t i c u l a r s o l u t i o n of the non-homogeneous equation (5.5), l e t us take $ 3 (5.12) where B^ s h a l l a l s o be determined by the boundary c o n d i t i o n s . The g e n e r a l s o l u t i o n of the governing equation (5.5), f o r any one va l u e of the s e p a r a t i o n constant "a", i s then g i v e n by 3S<» (5.13) to which we s h a l l now apply the boundary c o n d i t i o n s to determine B^ , B^, and B^. i / H t t 0 . ^ ) ~ ° i i-s i d e n t i c a l l y s a t i s f i e d by e q u a t i (5.13) . i i / l i ^ <4 L*W\* " ^ i 'i^ g i v e s us 8 , . f , . 4 K. We now have, f o r any one g i v e n value of "a" V*,V0 = ^1.***'**' 3 ^ 0 0 + (5.14) A p p l y i n g the f i n a l boundary c o n d i t i o n g i v e s us, a f t e r i n t e g r a t i n g over a l l p o s s i b l e v a l u e s of "a" ^ B ^ e ' ^ ^ J ^ t ^ d o . « -Vx H7'* ( 5 . i 5 ) or 73 (5.16) which i s of the form OO (where (5.17) which allows us to pass to the upper l i m i t of immediately) so t h a t we may apply the f o l l o w i n g Hankel t r a n s f o r m a t i o n (Morse and Feshbach, 1953, p.944) (5.18) to equation (5.16), g i v i n g us (5.19) or -^ 1 + a v / k — (5.20) where (5.21) Thus, the complete s o l u t i o n f o r the stream f u n c t i o n s u b j e c t to the governing equation (5.5) ( i n surfzone I d e f i n e d by 0£ x< x^, y > y ^ ) now takes the form oo (5.22) Surfzone I I In surfzone I I , d e f i n e d by o $ y < y^, x>x^, the governing equation i s (as g i v e n by eq u a t i o n (5.3)), d _ £ vL d = . ^  *4 where ^W,^ i s s u b j e c t to the f o l l o w i n g boundary c o n d i t i o n s i / the stream f u n c t i o n i s i d e n t i c a l l y zero along the mean s h o r e - l i n e d e f i n e d by y=0, t h a t i s i i / the stream f u n c t i o n tends to the i n f i n i t e beach c o n f i g u r a t i o n f a r away from the corner, so t h a t i i i / the stream f u n c t i o n i s i d e n t i c a l l y zero a t the end of the surfzone, t h a t i s In a manner s i m i l a r to the d e r i v a t i o n of equation (5.2 we can a r r i v e a t the f o l l o w i n g e x p r e s s i o n f o r the stream f u n c t i o n ^V.^,\j)(in the surfz o n e d e f i n e d by 0 $ y< y^, x > x ^ J 7 (5.24) "p" being the s e p a r a t i o n constant i n t h i s case 76 Off-shore Zone We have now to d e r i v e an e x p r e s s i o n f o r the stream f u n c t i o n ^ W . H ) i n t h e o f f - s h o r e zone d e f i n e d by x > x^, The governing equation i n t h i s r e g i o n s u b j e c t to the assumptions and approximations d i s c u s s e d e a r l i e r was found to be Laplace 1 s equation; 4 v =0 (5.25) The e x p r e s s i o n f o r the o f f s h o r e stream f u n c t i o n i s to be found by matching i t a c r o s s the b r e a k e r - l i n e s to the r e s p e c t i v e s u r f z o n e s o l u t i o n s , the boundary c o n d i t i o n s being a s it—*-»oo (5.26) iii/ ^ U , ^ f e m a i ^ ^Xtx^C 0.t\<J O S vj —»r * C O L e t us b e g i n by breaking t h i s i n to two s i m p l e r problems, e x p r e s s i n g ^ ^ W i S J ^ n t h e f o l l o w i n g manner V v i V 4v^,f> + ( 5* 2 7 ) where ^ and ^ 3 ^ s a t i s f y equations (5.28) and (5.29) r e s p e c t i v e l y ; t| S7*^ 3 > | = 0 y>1k (5.28) i v / *4 rtmaJo-s bounded o^s —*••*•«» 1 %V/ *4 r«_tv\aiAS b o n d e d as ^ — (5.29) C o n s i d e r i n g f i r s t equations (5.28), l e t us apply s e p a r a t i o n of v a r i a b l e s by making the s u b s t i t u t i o n 3> I (5.30) i n t o equation ( 5.28(D) to get the f o l l o w i n g p a i r of o r d i n a r y d i f f e r e n t i a l equations = A., e + ^ e ( 5 . 3 1 ) where i s a s e p a r a t i o n c o n s t a n t and the A^'s are to be found from the boundary c o n d i t i o n s . A p p l y i n g then, boundary c o n d i t i o n ( 5 . 2 8 ( i i i ) ) and ( 5 . 2 8 ( i v ) ) to equation (5.31) y i e l d s (5.32) I n t e g r a t i n g over a l l p o s s i b l e p o s i t i v e v a l u e s of ^ , equation (5.30) g i v e s us = \ e *<>|->f^dX (5.33) o To determine A ( ^ ) we apply boundary c o n d i t i o n ( 5 . 2 8 ( i D ) to equation (5.33) to get 79 to o or where ( 5 . 3 4 ) oo AUV- i ^ / ^.^^v^' (5-35) S i m i l a r l y for M- W,M)we get CO 1 C - X ^ ' V O ( 5 . 3 6 ) £ ^ ^ U ' ^ ^ - S f n W d*' ( 5 # 3 7 ) o Combining solutions for H^, and gives us the stream function i n the off-shore zone; ( 5 . 3 8 ) oo o 80 The remaining task i s to s o l v e the ex p r e s s i o n s f o r the stream f u n c t i o n s i n the sur f z o n e s and hence a r r i v e a t a s o l u t i o n i n the o f f - s h o r e zone by matching the stream f u n c t i o n s a c r o s s t h e i r r e s p e c t i v e b r e a k e r - l i n e s . In summary, the ex p r e s s i o n s are; 1. Surfzone I e J ^ ^ U d o . (5.39) Hi 2. Surfzone I I (5.40) 3. Offshore zone (5.41) o Before we can begin to e v a l u a t e the i n t e g r a l expre-s s i o n s f o r ^ (^v j " ) and ^ i**,^ ) we must f i r s t a t t e n d to I 4 ( a ) and I ^ ( p ) , where Hi o U The a l g e b r a i s presented i n Appendix C, from which we get - M (5.42) 82 where S(^(W^ ) i s a F r e s n e l i n t e g r a l as g i v e n by Gradshteyn and Rhyzik (1965, p.930, 8.251(2)0; and PC (5.43) The e x p r e s s i o n s f o r the surfzone stream f u n c t i o n s 4,W » S ) a n d are now 00 3/ / (5.44) 1^ §>i<\0h£ _ Cos (W \ dla r oo " (Hlf^HO • % tVitV'^'^VlO (5.45) - J? S \ S»U \1 - Cos V»f \ c l^ In each of equations (5.44) and (5.45) there are e i g h t i n t e g r a l s to be e v a l u a t e d . L e t us f i r s t t u r n our a t t e n t i o n to ^ | ( x , y ) , f o r which the i n t e g r a l s are eP -ay* , . . (5.46) (5.47) (5.48 ) 00 (5.49 ) 84 (5.50) (5.51) (5.52) C O (5.53) (where y a> y - , ) By r e p e a t e d l y a p p l y i n g the technique of i n t e g r a t i o n by p a r t s (see Appendix D), i n t e g r a l s (5.46) through (5.51) can be reduced to m u l t i p l e s of the f o l l o w i n g standard i n t e g r a l s as g i v e n by Gradshteyn and Rhyzik (1965, p.492) (5.54) CP (5.55) CP o • + 4 Wr' l^tji^.i (5.56) I n t e g r a l s (5.46) through (5.51) then g i v e us (5.57) _ 1 WwT' S i m i l a r l y , by a p p l y i n g the f o l l o w i n g i n t e g r a t i o n by p a r t s r e l a t i o n s a' (5.58) n i° w C ^ - » y 0° fc a. A ft 3 a * a * (5.59) a * r a a ('n' a p o s i t i v e i n t e g e r ) i n t e g r a l s (5.52) and (5.53) can be reduced to the f o l l o w i n g form (5.60) 5-CO 87 (where y => y-YL, ) Adding equations (5.57) and (5.60) g i v e s us, f o r i n t e g r a l s (5.46) through (5.53) y r 3-1 , o5" I S\/o*i do. ( 5 . 6 D ' O J l • 1 \ 7* The e x p r e s s i o n f o r the stream f u n c t i o n *| (x,y) (equation (5.44)) then becomes -4 U> s- ^  ) H s (5.62) M i l ^ l o y I da Hp <7T to* { j \ — f c I (The s o l u t i o n f o r ^  (x^y) i s a r r i v e d a t i n a s i m i l a r manner) S e t t i n g x = 0, g i v e s us V0^ * (5.63) s a t i s f y i n g the boundary c o n d i t i o n ( 5 . 5 ( D ) a t the shore l i n e At the end of the surfzone, y = 0 (th a t i s , as y-^-y^), 0 < x < x f o, we get Making the s u b s t i t u t i o n ^ = (W| i n the i n t e g r a l g i v e s us (5.65) or (Gradshteyn and Rhyzik (1965), p. 654, 6.325) . ^ . Mr. ^  (5.66) 89 which then g i v e s us 0 to s a t i s f y boundary c o n d i t i o n ( 5 . 5 ( i i i ) ) . Attempts to f u r t h e r reduce the i n t e g r a l s o (5.67) a y Co a to standard forms d i d not meet w i t h success. An attempt to s o l v e them by T a y l o r s e r i e s expansion of the F r e s n e l i n t e g r a l met w i t h convergence d i f f i c u l t i e s a t both l i m i t s of y (as y-»0 and as y-*-+00), as w e l l as a d i s c o n t i n u i t y a t the b r e a k e r - l i n e , x=x, . ' o I n t e g r a l s (5.67) were, t h e r e f o r e , e v a l u a t e d n u m e r i c a l l y (using the t r a p e z o i d a l r u l e ) and the r e s u l t s thus o b t a i n e d used i n equation (5.62) to c a l c u l a t e v a l u e s f o r the stream f u n c t i o n ^ , (x,y) i n surfzone I. (Values f o r the stream f u n c t i o n i n surfzone I I were c a l c u l a t e d i n a s i m i l a r manner.) The s o l u t i o n seaward of the b r e a k e r - l i n e s was not c a l c u l a t e d from the c l o s e d form (5.41), which i s not of much p r a c t i c a l use. Instead, Laplace's equation was i n -g r a t e d n u m e r i c a l l y u s i n g the f i n i t e d i f f e r e n c e equation g i v e n i n Appendix B (7.b), w i t h v a l u e s a t the b r e a k e r - l i n e s matched to the a n a l y t i c a l s o l u t i o n s . I l l D i s c u s s i o n The f u l l s o l u t i o n , as presented i n F i g u r e 28, has been o r o t a t e d 180 from the c o n f i g u r a t i o n shown i n F i g u r e 27 and the surfzones t r u n c a t e d a t l e n g t h s equal to those of the numerical models f o r comparison w i t h the s o l u t i o n s of Chapter 4. I t i s r e a l i z e d t h a t the surfzones thus p r e s e n t e d are from s e m i - i n f i n i t e , but n e v e r t h e l e s s of s u f f i c i e n t l e n g t h i n comparison to t h e i r widths, to m e r i t p r e s e n t i n g i n t h i s ••• manner. The g e n e r a l c i r c u l a t i o n p a t t e r n again c o n s i s t s of two c o u n t e r - r o t a t i n g c e l l s d r i v e n by t h e wave-induced longshore c u r r e n t s - which flow toward the i n t e r s e c t i o n c o r n e r . The o f f - s h o r e r e t u r n flow i s , however, more s t r o n g l y d i v e r g e n t than i n the numerical s o l u t i o n s , " l e a k i n g " a c r o s s the breaker l i n e s f o r most of the width of the s u r f z o n e s . Presented i n ' F i g u r e 29 are normalized s t r e a m - l i n e p r o f i l e s taken across the n a t u r a l beach surfzone a t the p o s i t i o n s shown i n F i g u r e 28. (The stream f u n c t i o n v a l u e s have been normalized with r e s p e c t to the walue of the stream f u n c t i o n at the b r e a k e r - l i n e of c r o s s s e c t i o n d.) They d i f f e r c o n s i d e r a b l y from the i n f i n i t e beach p r o f i l e shown i n F i g u r e 14 and again i l l u s t r a t e the r a p i d divergence of the longshore c u r r e n t s i n comparison "to the numerical.models. The d i f f e r e n c e between the a n a l y t i c a l and numerical solution's appears to be one of s c a l e r a t h e r than of form. A thorough v e r i f i c a t i o n of the a n a l y s i s l e a d i n g to equation (5.62) has not r e v e a l e d any e r r o r s i n a l g e b r a . A r e p e t i t i o n of the numerical model with doubled r e s o l u t i o n shows the r e l i a b i l i t y of the numerical i n t e g r a t i o n s and the indepen-dence of the r e s u l t s on g r i d s c a l e . The d i f f e r e n c e between numerical and a n a l y t i c a l models remains unresolved; i n view of i t s complexity the a n a l y t i c a l model does not appear to be a u s e f u l t o o l even i n the i d e a l i z e d c o n d i t i o n s s t u d i e d here. F i g u r e 28. C u r r e n t p a t t e r n f o r the a n a l y t i c a l model, where the beaches have been t r u n c a t e d f o r comparison to the numerical s o l u t i o n s . .9 .8 • 7 -6 .5 .4 .3 .2 .1 0.0 F i g u r e 29. N o r m a l i z e d s t r e a m - l i n e p r o f i l e s t a k e n a c r o s s t h e n a t u r a l b e a c h s u r f z o n e o f t h e a n a l y t i c a l m o d e l a t t h e p o s i t i o n s shown i n F i g u r e 28 CHAPTER 6 SUMMARY We h a v e d e v e l o p e d s e v e r a l t h e o r e t i c a l m o d e l s o f t h e w a v e - i n d u c e d c u r r e n t c i r c u l a t i o n i n t h e v i c i n i t y o f an i s o l a t e d b r e a k w a t e r e x t e n d i n g f r o m s h o r e , a s i n s p i r e d by a l o c a l f i e l d s i t e . T h e s e c o n s i s t o f i / s e v e r a l n u m e r i c a l m o d e l s u s i n g f i n i t e b e a c h e s i n e n c l o s e d b a s i n s , w here c o n s i d e r a t i o n h a s b e e n g i v e n t o v a r i a t i o n s i n s e a - f l o o r t o p o g r a p h y a s w e l l as ; t o l i n e a r and n o n - l i n e a r f o r m s o f b o t t o m f r i c t i o n i i / a n a n a l y t i c a l m o d e l c h a r a c t e r i z e d by s e m i - i n f i n i t e b e a c h e s a n d u n i f o r m s e a - f l o o r t o p o g r a p h y . N u m e r i c a l M o d e l s I n g e n e r a l ( f o r a g i v e n a n g l e o f wave i n c i d e n c e ) t h e c i r c u l a t i o n p a t t e r n s show two c o u n t e r - r o t a t i n g c e l l s d r i v e n by t h e w a v e - i n d u c e d l o n g s h o r e c u r r e n t s - w h i c h f l o w a l o n g e a c h b e a c h t o w a r d t h e i n t e r s e c t i o n a p e x - w i t h a r e t u r n f l o w o u t o v e r t h e s h e l f i n t h e f o r m o f a r i p c u r r e n t . The maximum v e l o c i t y i s f o u n d i n t h e r e t u r n f l o w , s h o r t l y a f t e r i t e merges f r o m t h e i n t e r s e c t i o n c o r n e r . D e s p i t e . a h o s t o f s i m p l i f y i n g a s s u m p t i o n s and a p p r o x i m a t i o n s , t h e q u a l i t a t i v e f e a t u r e s o f t h e m o d e l s a r e c o n s i s t e n t w i t h o b s e r v a t i o n s o f s e d i m e n t t r a n s p o r t t a k e n a t t h e s t u d y s i t e w h i c h show b e a c h s e d i m e n t s t o be m o v i n g a l o n g b o t h t h e c a u s e w a y a n d Tsawwassen b e a c h e s t o w a r d t h e i r common c o r n e r w h i c h i s b u i l d i n g o u t a s a r e s u l t . D i f f e r e n c e s i n the l i n e a r and n o n - l i n e a r bottom f r i c t i o n models d i d not become apparent u n t i l an o f f - s h o r e t r e n c h p a r a l l e l to the causeway, was i n t r o d u c e d to the s e a - f l o o r topography. The r e s u l t s f o r the n o n - l i n e a r model show a d e f l e c t i o n of the o f f - s h o r e r e t u r n flow i n t o the t r e n c h w i t h a c o r -responding i n c r e a s e i n t r a n s p o r t and agree with p r e l i m i n a r y a n a l y s i s based on a one-dimensional model. However, the l i n e a r r e s u l t s d i f f e r c o n s i d e r a b l y from those of the non-l i n e a r model, are d i f f i c u l t to i n t e r p r e t and thought to be suspect due to the r e s t r i c t i v e assumptions upon which the l i n e a r form of bottom f r i c t i o n i s d e r i v e d . The t r e n c h model perhaps serves to i l l u s t r a t e another i n s t a n c e i n which the use of the l i n e a r form of bottom f r i c t i o n may not be a p p r o p r i a t e . A n a l y t i c a l Model D i f f i c u l t i e s were encountered i n the development of an a n a l y t i c a l model of the c u r r e n t c i r c u l a t i o n d e s p i t e the l i n e a r i z a t i o n of the governing equations and the use of s e m i - i n f i n i t e beaches. As a r e s u l t i t was found necessary to n u m e r i c a l l y i n t e g r a t e two of the i n t e g r a l s i n the s u r f zone s o l u t i o n as w e l l as the governing equation f o r the r e g i o n seaward of the b r e a k e r - l i n e . While e x h i b i t i n g the same g e n e r a l f e a t u r e s as the numerical models, the a n a l y t i c a l model shows a much broader and weaker o f f - s h o r e r e t u r n flow. The source of t h i s d i f f e r e n c e has been sought i n p o s s i b l e s c a l i n g d i s p a r i t i e s between the numerical and a n a l y t i c a l models, but without success. In. view of i t s a l g e b r a i c complexity, the a n a l y t i c a l model does not appear to be a u s e f u l t o o l , even i n the r e l a t i v e l y simple geometry s t u d i e d here, and a f o r t i o r i i n more r e a l i s t i c geometries. Improvements to the a n a l y s i s would take i n t o account h o r i z o n t a l eddy v i s c o s i t y (which may reduce the s t r e n g t h of the longshore c u r r e n t s ) , c o n v e c t i v e i n e r t i a (which may strengthen the o f f - s h o r e r e t u r n f l o w ) , and wave-current i n t e r a c t i o n , s i n c e the o f f - s h o r e r e t u r n flow may modify the wave f i e l d and cause the waves to break e a r l i e r than otherwise (Mei and L i u , 19 76, p a r t 2). A l s o of i n t e r e s t (at the Tsawwassen l o c a t i o n ) i s the added e f f e c t of the trench upon c u r r e n t c i r c u l a t i o n i n l i g h t of t i d a l a c t i o n , p a r t i c u l a r l y the e f f e c t s of an ebbing t i d e upon nearshore c i r c u l a t i o n as water i s drawn i n t o the t r e n c h . To v e r i f y the models' p r e d i c t i o n s , a i s e r i e s of f i e l d measurements or wave tank experiments should be undertaken. That longshore c u r r e n t s are p r e s e n t at the study s i t e has been e s t a b l i s h e d by evidence of sediment t r a n s p o r t along the beaches. However, d i f f i c u l t i e s i n o b t a i n i n g f i e l d measurements of the t o t a l c i r c u l a t i o n at t h i s p a r t i c u l a r l o c a t i o n a r i s e due to the r e s t r i c t i o n s imposed by the l o c a l bathymetry and geometry as w e l l as the n o n - l i n e a r i t y of the l i t t o r a l p r o c e s s e s themselves. Wave tank experiments may then g i v e a h i g h e r y i e l d of data per input of time and e f f o r t . The r e l a t i o n s h i p between near-shore c u r r e n t s and c o a s t a l sediment t r a n s p o r t i s a d i f f i c u l t and complicated problem. H o p e f u l l y the b a s i c understanding of the c u r r e n t s i n a t y p i c a l beach-breakwater c o n f i g u r a t i o n gained i n t h i s study may serve as a f i r s t step i n r e s o l v i n g the a s s o c i a t e d q u a n t i t a t i v e aspects of sedimentation. LIST OF REFERENCES Ar t h u r , R.S. (1962). A note on the dynamics of r i p c u r r e n t s . J . Geophys. Res., 67, 2777-2779. Abramowitz, M.., and Segun, I.A. (1965). Handbook of Math-e m a t i c a l F u n c t i o n s . New York, Dover P u b l i c a t i o n s Inc. Bowen, A . J . (1969a). The g e n e r a t i o n of longshore c u r r e n t s on a plane beach. J . Mar. Res., 27, 206-215. Dalrymple, R.A., Eubanks, R.A. and Birkmeier, W.A. Wave induced c i r c u l a t i o n i n shallow r e c t a n g u l a r b a s i n s . J o u r n a l of the Waterway, Port, C o a s t a l and Ocean D i v i s i o n , ASCE, v o l . 104, No. wwl, Proc. Paper 12749, February 1977, pp 117-134. Gradshteyn, I.S. and Ryzhik, I.M. (1965). Table of I n t e g r a l s , S e r i e s and Products. New York and London, Academic Press, 1086 pp. Hodge, R.A.L. (1971). The movement of beach m a t e r i a l s ; south s i d e Tsawwassen Causeway, a r e s e a r c h p r o j e c t submitted d u r i n g the t h i r d year of the course i n a p p l i e d s c i e n c e a t the U n i v e r s i t y of B r i t i s h Columbia. James, I.D. (1972). Ph.D. T h e s i s . Cambridge U n i v e r s i t y . James, M.L., Smith, G.M. and Wolford, J.C. (1968). A p p l i e d Numerical Methods f o r D i g i t a l Computation w i t h F o r t r a n . Scranton, Pennsylvania, I n t e r n a t i o n a l Text Book Company. Kamke, E. (1959). D i f f e r e n t i a l - G l e i c h u n g e n , Losungsmethoden und Lbsungen. New York, Chelsea P u b l i s h i n g Company,-3rd e d i t i o n . Komar, P.D. (19 76). Beach Processes and Sedimentation. Englewood C l i f f s , New J ersey, P r e n t i c e - H a l l . L i u , P.L.F. and Dalrymple, R.A. (1978). Bottom f r i c t i o n s t r e s s e s and longshore c u r r e n t s due waves wi t h l a r g e angles of i n c i d e n c e . J . Mar. Res., 36, 357-375. L i u , P.L.F. and Mei, C C . (1976). Water motion on a beach i n the presence of a breakwater. P a r t 2. J . Geoph. Res., 81, 3085-3094. / 99 Longuet-Higgins, M.S. (19 70). Longshore c u r r e n t s generated by o b l i q u e l y i n c i d e n t sea waves, 1 and 2. J . Geophys. Res., 75, 6778-6801. Longuet-Higgins, M.S. (1956). The r e f r a c t i o n of sea waves i n shallow water. J . F l u i d . Mech., 1, 163-176. Longuet-Higgins, M.S., and Stewart, R.W. (1964). R a d i a t i o n s t r e s s i n water waves; a p h y s i c a l d i s c u s s i o n w i t h a p p l i c a t i o n s . Deep Sea Res., 11, 529-562. M i l l e r , CD., and B a r c i l o n , A. The dynamics of the l i t t o r a l zone. Reviews of Geoph. and Space Phys., v o l . 14, no. 1, February 19 76, pp 81-91. Morse, P.M., and Feshbach, H. (19 53). Methods of Theor-e t i c a l P h y s i c s , New York, McGraw-Hill Book Company.: Munk, W.H. (1949). The s o l i t a r y wave theory and i t s app-l i c a t i o n to s u r f problems. Ann. N.Y. Acad. S c i . , 51, 376-424. O'Rourke, J.C.,. and LeBlond, P.H. (1972). Longshore cu-r r e n t s i n a s e m i c i r c u l a r bay. J . Geophys. Res., 77, 444-452. Wood, J.S. (1970). The Tsawwassen and Roberts Bank cause-ways, a paper p r e s e n t e d a t the Seminar on C o a s t a l E n g i n e e r i n g , U n i v e r s i t y of B r i t i s h Columbia. APPENDIX A SUMMARY OF APPROXIMATIONS The governing equations f o r the l i n e a r numerical models have been d e r i v e d s u b j e c t to the f o l l o w i n g approximations. 1. The equations have been formulated f o r mono-chromatic small amplitude wave t r a i n s propag-a t i n g over a m i l d l y s l o p i n g bottom. 2. A l l averaged motion i s p u r e l y h o r i z o n t a l and depth independent. 3. The energy d e n s i t y of the waves E, i s equal 2 to ^ / 3 ga , where 'a' i s the amplitude of the waves. 4. H o r i z o n t a l eddy v i s c o s i t y has been i g n o r e d . 5. Bottom f r i c t i o n has been l i n e a r i z e d . 6 . The i n e r t i a l c o n v e c t i v e terms have been ig n o r e d . 7. The wave amplitude i n the surfzone i s propor-t i o n a l to the mean water depth. 8. Wave-current i n t e r a c t i o n has been ig n o r e d . 9. The waves are assumed plane a t the breaker-l i n e . In a d d i t i o n , f o r the a n a l y t i c a l model, the o f f - s h o r e zone i s assumed to have a u n i f o r m l y f l a t bottom, w i t h a depth equal to the water depth a t the b r e a k e r - l i n e , w h i l e the n o n - l i n e a r numerical models do not assume the bottom f r i c t i o n to be l i n e a r . APPENDIX B COMPUTATIONAL CONSIDERATIONS We wish to develop a computer programme which w i l l n u m e r i c a l l y s o l v e f o r the stream f u n c t i o n ^ ( x , y ) i n the v i c i n i t y of a beach-breakwater j u n c t i o n . Throughout we s h a l l be r e f e r r i n g to the beach con-f i g u r a t i o n as shown i n F i g u r e 30, which d i f f e r s s l i g h t l y from the Tsawwassen c o n f i g u r a t i o n i n angle of i n t e r s e c t i o n and o r i e n t a t i o n w i t h the g e n e r a l surroundings. The r e g i o n i n which we are to s o l v e f o r ^ ( x , y ) s h a l l be d i v i d e d i n t o t hree zones, s u r f z o n e s I and I I and an o f f - s h o r e zone. Surfzone I (corresponding to the Tsawwassen Beach surfzone) i s p a r a l l e l to the y - a x i s from -y^ to -y^, and i s of width j x ^ J . Surfzone I I (corresponding to the causeway surfzone) i s p a r a l l e l to the x - a x i s from -x^ to -xj£, and i s of width j y^ |. The i n t e r s e c t i o n corner has been removed to ease the s p e c i f i c a t i o n of the boundary con-d i t i o n s t h e r e . The o f f - s h o r e zone i s to accomodate v a r i a t i o n s i n bottom topography such as a t r e n c h running p a r a l l e l to the causeway and a d r o p - o f f to much deeper water, as d i s c u s s e d i n Chapter 4. The problem s h a l l be s o l v e d by the Gauss-Seidel i t e r a t i v e technique, the g e n e r a l a p p l i c a t i o n of which i s d e s c r i b e d below. For a more d e t a i l e d d i s c u s s i o n see James, Smith and Wolford (1968). 10 2 F i g u r e 30. General l a y - o u t and boundary con-d i t i o n s f o r the numerical models. 103 A l a t t i c e or g r i d of mesh s i z e &x=Ay=h and of dimension x t by y^. 1 S superimposed upon the r e g i o n i n which a s o l u t i o n i s r e q u i r e d . The s o l u t i o n o f the governing equations con-s i s t s of determining the stream f u n c t i o n v a l u e s a t the f i n i t e l y - s p a c e d g r i d p o i n t s ( i , j ) . The governing equations f o r each r e g i o n o f the l i n e a r model are as g i v e n by equations (4.19), (4.20) and (4.21). 1. Surfzone I y4 . ^ 4 ( l . b ) 2. Surfzone I I (2.b) 3. Off-shore Zone (3.b) The o f f s h o r e stream f u n c t i o n i s i d e n t i c a l l y zero on the w a l l s of the b a s i n , as shown i n F i g u r e 30, and i s matched across the b r e a k e r - l i n e s and the ends of the s u r f -zones to the r e s p e c t i v e surfzone stream f u n c t i o n s . From T a y l o r s e r i e s expansions we have the f o l l o w i n g f i n i t e d i f f e r e n c e approximations to d e r i v a t i v e s : (4.b) which upon s u b s t i t u t i o n i n t o equations ( l . b ) , (2.b) and (3.b) g i v e : 1. Surfzone I "SO • (5.b) 2. Surfzone I I <>?= v-<? • -st> = 4--? S D (6.b) 3. O f f - s h o r e Zone , N - - i v y i w y l ( 7 > b ) S? = 3-eV ^ c ^ / r t i . ^ 106 S L 5 D = where H^ _. i s the depth at g r i d p o i n t ( i , j ) . The problem has now been reduced to o b t a i n i n g the simultaneous s o l u t i o n of a s e t of l i n e a r a l g e b r a i c equa-the t o t a l number of equations depending upon the extent of the boundaries and the number of g r i d p o i n t s used. The steps i n the Gauss-Seidel i t e r a t i v e method used to o b t a i n the s o l u t i o n s are as f o l l o w s (as g i v e n by James, Smith and Wolford, 1968). i / each equation i s f i r s t w r i t t e n i n a form convenient f o r s o l v i n g f o r the unknown wit h the l a r g e s t c o e f f i c i e n t i n t h a t equation. i i / a l l unknown stream f u n c t i o n values a t the g r i d p o i n t s are then assigned i n i t i a l v a l u e s on the b a s i s of the best estimated v a l u e (the b e t t e r the estimate, the more r a p i d the convergence to a s o l u t i o n ) . i i i / a t each p o i n t ( i , j ) i n the l a t t i c e , an approximate value of the stream f u n c t i o n STEMP i s then c a l c u l a t e d u s i n g the a p p r o p r i a t e equation ((5.b), (6.b) or ( 7 . b ) ) . T h i s c a l c u l a t e d v a l u e supercedes the estimated value and i s then used as the stream f u n c t i o n value ^ i , j , a t t h a t g r i d p o i n t u n t i l i t , i n t u r n , i s superceded by a new c a l c u l a t e d value t i o n s i n the unknown 107 i v / i n a l l i t e r a t i o n s , the l a t e s t c a l c u l a t e d stream func-t i o n v a l u e s are always used i n c a l c u l a t i n g newer and b e t t e r p o i n t s , v/ the p o i n t s on the l a t t i c e a t which the STEMP values are c a l c u l a t e d are s e l e c t e d i n some syst e m a t i c way - i n t h i s case by rows v i / one i t e r a t i o n i s completed when an a p p r o p r i a t e v a l u e has been c a l c u l a t e d f o r each l a t t i c e p o i n t whose stream f u n c t i o n value i s sought v i i / when the stream f u n c t i o n change a t a l l g r i d p o i n t s " between s u c c e s s i v e i t e r a t i o n s i s l e s s than or equal to some pre-determined v a l u e or when a c e r t a i n maximum number of i t e r a t i o n s has been reached the computations are stopped. A p l o t t e r i s used to draw the contour graphs of the s t r e a m - l i n e s as f u n c t i o n s of i and j . In some models a n o n - l i n e a r form o f bottom f r i c t i o n (James, 1972) i s used i n the o f f - s h o r e zone (as d i s c u s s e d i n Chapter 3). The r e l e v a n t f i n i t e d i f f e r e n c e e quation then becomes (8.b) (9.b) S D 108 S P = 4-e>- V\^, And f i n a l l y , a note w i t h regards to the mesh s i z e h, used i n the a n a l y s i s . The models presented i n Chapters 4 and 5 have a l a t t i c e spacing of 18 g r i d p o i n t s per i n c h . The e f f e c t s of v a r y i n g the numerical models' r e s o l u t i o n i s c l e a r l y i l l u s t r a t e d upon comparison of the acute angle model of F i g u r e 31 wit h t h a t of F i g u r e 18, where the models d i f f e r o nly i n t h e i r mesh s i z e ; F i g u r e 18 showing the c i r c u l a t i o n p a t t e r n of a model having twice the r e s o l u t i o n of t h a t shown i n F i g u r e 31. The most n o t i c e a b l e e f f e c t o f i n c r e a s i n g the r e s o l u t i o n i s to decrease the s i z e of the 'step' (which i s a d i r e c t r e f l e c t i o n of the mesh s i z e ) i n the s t r e a m - l i n e s adjacent to the n a t u r a l beach which runs d i a g o n a l l y to the g r i d l a t t i c e . In a d d i t i o n , the 'eddies' which are pr e s e n t i n F i g u r e 31 would a l s o appear to be a m a n i f e s t a t i o n of poof r e s o l u t i o n s i n c e they are not ev i d e n t i n the high e r r e s o l u -t i o n ..models of Chapters 4 and 5. F i g u r e 31. An example of the e f f e c t s of v a r y i n g the r e s o l u t i o n of the numerical models n o : APPENDIX C EVALUATION OF INTEGRALS 1^ ( a ) AND I 2 ( J> .) We wish to eva l u a t e X,<V> - ^ X ^ l o V w 3 ^ ( 1 - c ) where (Abramowitz and Stegun (1965), p. 43 7) so t h a t Hi I f we i n t e g r a t e by p a r t s , l e t t i n g (3.c) I l l equation (3.c) g i v e s us _4 z/ O A <WD (4.c) Consider now <wl> Again i n t e g r a t i n g by p a r t s we get o->d (5.c) 4 so t h a t -1 5/o Now c o n s i d e r ^ d<v/: = ^ ( p * ) once again i n t e g r a t i n g by p a r t s so t h a t •A 113 where S(y^ nr(> ) i s a F r e s n e l i n t e g r a l as g i v e n by Gradshetyn and Rhyzik ((1965), p. 930, 8.251(2)). We then have ii ii 1 (7 . c ) S i m i l a r l y f o r -?cHl^e«Vll-cH^Hl < 8 - c ) APPENDIX D INTEGRAL ANALYSIS We wish f i r s t to ev a l u a t e i n t e g r a l s (5.46) and (5.47) 0 0 n ( l . d ) L e t us s t a r t by i n t e g r a t i n g the f o l l o w i n g i n t e g r a l by p a r t s 4 f -°-y , rl\, \ €. Si<v<v/.«a.»AOotl da \ a-(2.d) l e t t i n g e- S ^ f t * • (W = SiftO*^ , AT = - COS CW^ We then have CP \ e SirsQ^ st<\<w^  da 00 (3.d) 00 * \ a.3 J a. J 115 or Equation ( l . d ) now becomes 116 where i n t e g r a l I I I i s \ of I n t e g r a l (5.47), which we s h a l l now i n t e g r a t e by p a r t s to get o - I?wL I + 1 \ <^_a^ co^cyy:-stncxl did (6.d) so t h a t equation (5.d) becomes 2 t> o (7.d) <P M L y \ — * V i C o n s i d e r i n g now I n t e g r a l V , we have S u b s t i t u t i n g i n t o equation (7.d) we get % &t - « y . i ° —7 ( 9 . d ) o F i n a l l y , we a l s o have * ~ a ( 1 0 .d) o c-r) K oft V CO 1-— which then g i v e s us 5 4 y ^covy^&trvo-jCcia/' ( l l . d ) The i n t e g r a l s i n equation ( l l . d ) are to be found i n Gradshteyn and Rhyzik ((1965), p. 492), so t h a t equation ( l . d ) now becomes 1^1 (12.d) In a s i m i l a r manner i n t e g r a l s (5.48) and (5.49) g i v e us while i n t e g r a l s (5.50) and (5.51) can be shown to equal (14.d) Taking the sum of equations (12.d), (13.d) and (14.d) g i v e s us 120 4 1 v } as shown i n equation (5.57). 

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