UBC Theses and Dissertations

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UBC Theses and Dissertations

A shoreline prediction model Har, Boon Cher 1984

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A SHORELINE PREDICTION MODEL B . S c . ( H o n s ) , The Queen's U n i v e r s i t y Of B e l f a s t , 1981 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE THE FACULTY OF GRADUATE STUDIES D e p a r t m e n t Of C i v i l E n g i n e e r i n g We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o / t h e r e a u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA J a n u a r y 1984 by BOON CHER/HAR i n © Boon Cher Har, 1984 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 at the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying 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 or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department o f (A^\ The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date 2 F&h f DE-6 (3/81) i i ABSTRACT A n u m e r i c a l model i s d e v e l o p e d t o p r e d i c t s h o r e l i n e c h a n g e s as a f u n c t i o n o f deep w a t e r wave c o n d i t i o n s and n e a r s h o r e b a t h y m e t r y . The model c o n s i s t s of t h r e e components, the wave r e f r a c t i o n and s h o a l i n g component, t h e l o n g s h o r e t r a n s p o r t component, and t h e o n - o f f s h o r e component. The r e f r a c t i o n and s h o a l i n g component i s b a s e d on t h e i r r o t a t i o n a l i t y of t h e wave number K, and t h e c o n s e r v a t i o n of e n e r g y e q u a t i o n s s u g g e s t e d by Noda ( 1 9 7 4 ) , w h i l e t h e l o n g s h o r e t r a n s p o r t component i s b a s e d on t h e CERC l o n g s h o r e t r a n s p o r t e q u a t i o n . A f i n i t e d i f f e r e n c e scheme i s a d o p t e d t o s o l v e b o t h t h e wave r e f r a c t i o n and t h e l o n g s h o r e t r a n s p o r t g o v e r n i n g e q u a t i o n s . The t h i r d component, t h e o n - o f f s h o r e t r a n s p o r t ( b each p r o f i l e change) component i s b a s e d on a s e t of e q u a t i o n s d e v e l o p e d from some o f t h e known a s p e c t s o f b e h a v i o u r o b s e r v e d by o t h e r i n v e s t i g a t o r s . I t i s a l s o b a s e d on t h e r e s u l t s o b t a i n e d from a s e t of e x p e r i m e n t s c o n d u c t e d on e q u i l i b r i u m p r o f i l e c h a n g e s , and on a p r o p o s e d c r i t e r i a f o r on or o f f s h o r e s e d i m e n t m o t i o n . The model i s u s e d t o s i m u l a t e t h e s h o r e l i n e c h a n g e s due t o a l i t t o r a l b a r r i e r o r i e n t a t e d n o r m a l l y t o t h e s h o r e l i n e , and t o p r e d i c t t h e c h a n g e s of a beach n o u r i s h m e n t p l a n . R e s u l t s of t h e s e s i m u l a t i o n s show t h a t , i n a d d i t i o n t o t h e l o n g s h o r e t r a n s p o r t i n f l u e n c e s , t h e o n - o f f s h o r e b e a c h m o d i f i c a t i o n has a l a r g e i n f l u e n c e on s h o r e l i n e r e t r e a t under more s e v e r e wave a t t a c k . ACKNOWLEDGEMENTS The a u t h o r i s v e r y g r a t e f u l t o t h e f i n a n c i a l s u p p o r t , encouragement and g u i d a n c e g i v e n by h i s s u p e r v i s o r , P r o f e s s o r M. C. Q u i c k . The a u t h o r would a l s o l i k e t o thank Dr. M. I s s a c s o n f o r h i s v a l u a b l e s u g g e s t i o n s . i v TABLE OF CONTENTS Page A b s t r a c t i i Acknowledgement i i i T a b l e of C o n t e n t i v L i s t of F i g u r e s v i i L i s t of T a b l e s x CHAPTER 1 INTRODUCTION 1 CHAPTER 2 NEARSHORE CURRENT AND TRANSPORT EQUATION REVIEW 4 2. 1 I n t r o d u c t i o n 4 2.2 N e a r s h o r e C u r r e n t s 4 2.3 L o n g s h o r e C u r r e n t s 5 2.3.1 R a d i a t i o n S t r e s s A p p r o a c h 6 2.3.2 S e m i - E m p i r i c a l A p p r o a c h 10 2.4 L o n g s h o r e T r a n s p o r t 12 2.4.1 Wave F l u x M o d e l s 12 2.4.2 Sediment E q u a t i o n M o d e l s 19 2.5 Summary 24 CHAPTER 3 WAVE REFRACTION AND SHOALING ROUTINE 25 3.1 I n t r o d u c t i o n 25 3.2 G e n e r a l D e s c r i p t i o n 25 3.3 G o v e r n i n g E q u a t i o n s 27 3.4 N u m e r i c a l A p p r o a c h 31 3.5 C o m p u t a t i o n 31 3.6 R e s u l t s 36 V Page CHAPTER 4 LONGSHORE TRANSPORT MODEL 4 3 4.1 I n t r o d u c t i o n •••• 43 4.2 Model Review 43 4.2.1 C o n t i n u i t y E q u a t i o n 45 4.3 N u m e r i c a l M o d e l l i n g 48 4.3.1 G o v e r n i n g E q u a t i o n s 48 4.3.2 C o m p u t a t i o n 53 4.3.3- Boundary C o n d i t i o n s 54 4.4 Model A p p l i c a t i o n s 56 4.5 R e s u l t s 56 4.6 D i s c u s s i o n 74 CHAPTER 5 ON-OFFSHORE TRANSPORT INVESTIGATION 7 6 5.1 I n t r o d u c t i o n 76 5.2 P r e v i o u s I n v e s t i g a t i o n s 76 5.3 E q u i l i b r i u m P r o f i l e 78 5.4 Beach S l o p e 79 5.5 E x p e r i m e n t a l d e s i g n and p r o c e d u r e s 80 5.6 R e s u l t s 84 CHAPTER 6 ON-OFFSHORE TRANSPORT MODEL 98 6.1 I n t r o d u c t i o n 98 6.2 Mo d e l O u t l i n e s and A s s u m p t i o n s 98 6.3 A l g o r i t h m s of t h e Mo d e l 104 v i Page 6.3.1 C a l i b r a t i n g t h e I n i t i a l P r o f i l e 104 6.3.2 F i n a l P r o f i l e 107 6.4 Model T e s t i n g 108 6.5 C o m b i n i n g t h e L o n g s h o r e and O n - o f f s h o r e t r a n s p o r t r o u t i n e s 113 6.6 D i s c u s s i o n 119 CHAPTER 7 SUMMARY AND CONCLUSIONS 121 BIBLIOGRAPHY 128 APPENDICES A - F i n i t e d i f f e r e n c e f o r m u l a t i o n 134 B - F l o w c h a r t f o r wave r e f r a c t i o n and s h o a l i n g r o u t i n e • 136 C - F l o w c h a r t f o r l o n g s h o r e t r a n s p o r t model 137 D - A c c r e t i o n c u r v e s w i t h d i f f e r e n t v a l u e s o f Tf .... 138 E - C r i t e r i a f o r s e l e c t i n g Time R a t i o 139 F - G r a i n s i z e d i s t r i b u t i o n 140 G - Program d e s c r i p t i o n s 141 v i i L I S T OF FIGURES F i g u r e T i t l e Page 2.1 D i f f e r e n t l o n g s h o r e c u r r e n t v e l o c i t y models 7 2.2 ( A f t e r L o n g u e t - H i g g i n s , 1970b) A c o m p a r i s o n between measured l o n g s h o r e c u r r e n t v e l o c i t y and t h e o r e c t i c a l d i s t r i b u t i o n s 7 2.3 C l a s s i f i c a t i o n of l o n g s h o r e t r a n s p o r t models 13 3.1 D e f i n i t i o n s k e t c h 29 3.2 O v e r a l l g r i d scheme 34 3.3 L o c a l g r i d scheme 35 3.4 R e s u l t s o f t e s t No 1 showing t h e wave c r e s t s 38 3.5 R e s u l t s of t e s t No 2 showing t h e wave o r t h o g o n a l s ... 38 3.6 R e s u l t s of t e s t No 3 showing t h e wave o r t h o g o n a l s ... 39 3.7 V a r i a t i o n of r e l a t i v e wave h e i g h t w i t h d e p t h ' 40 4.1 D e f i n i t i o n s k e t c h f o r e q u a t i o n [4.1] 46 4.2 D e f i n i t i o n s k e t c h f o r e q u a t i o n [4.4] 46 4.3 ' Assumed b e a c h p r o f i l e change 51 4.4 ( A f t e r Komar, 1973) D i s t r i b u t i o n of sand t r a n s p o r t r a t e a c r o s s b r e a k e r zone 51 4.5 A p p r o x i m a t e d s a n d t r a n s p o r t r a t e d i s t r i b u t i o n 52 4.6 D i s c r e t i z a t i o n of s t u d y a r e a 55 4.7 Boundary c o n d i t i o n s 55 4.8 P l a n view of t h e s h o r e l i n e b u i l d - u p due t o i n f i n i t e l y l o n g b a r r i e r 58 4.9 D e f i n i t i o n s k e t c h f o r n o n - d i m e n s i o n l i s i n g p r o c e d u r e s 60 F i g u r e T i t l e Page 4.10 N o n - d i m e n s i o n a l i s e d a c c r e t i o n c u r v e s f o r i n f i n i t e l y l o n g b a r r i e r 61 4.11 P l a n view of t h e s h o r e l i n e b u i l d - u p due t o f i n i t e l e n g t h b a r r i e r 64 4.12 N o n - d i m e n s i o n a l i s e d a c c r e t i o n c u r v e s f o r f i n i t e l e n g t h b a r r i e r 65 4.13 Two ways t o a c h i e v e t h e r e q u i r e d a c c r e t i o n 67 4.14 Beach n o u r i s h m e n t p l a n c h a n g e s ( w i t h r e f r a c t i o n and s h o a l i n g r o u t i n e ) 70 4.15 ( A f t e r W a l t o n e t a l , 1974) A n a l y t i c a l r e s u l t of b e a c h n o u r i s h m e n t p l a n c h a n g e s 71 4.16 Beach n o u r i s h m e n t p l a n c h a n g e s ( w i t h o u t wave r e f r a c t i o n and s h o a l i n g r o u t i n e ) 72 4.17 D i m e n s i o n l e s s v e r s i o n of F i g u r e 4.16 73 5.1 O v e r a l l e x p e r i m e n t a l s e t - u p 81 5.2 P r o f i l e c h a n g e s - T e s t 1 ( I n i t i a l and f i n a l p r o f i l e s ) 85 5.3 P r o f i l e c h a n g e s - T e s t 1 (At 2 h o u r s i n t e r v a l ) 86 5.4 P r o f i l e c h a n g e s - T e s t 2 87 5.5 P r o f i l e c h a n g e s - T e s t 3 88 5.6 Cummulative c h a n g e s of t h e p r o f i l e a g a i n s t t i m e 91 5.7 ( A f t e r D a l r y m p l e , 1976) E q u i l i b r i u m s l o p e a g a i n s t H 0 /V f T 95 6.1 D e f i n i t i o n s k e t c h f o r dc 100 6.2 T y p i c a l p r o f i l e c h a n g e s 102 6.3 M o d e l l i n g a s s u m p t i o n s 102 6.4 I n i t i a l p r o f i l e c a l i b r a t i o n 106 6.5 I n i t i a l and f i n a l p r o f i l e s 106 i x F i g u r e T i t l e Page 6.6 C o m p a r i s o n between measured p r o f i l e s and model's r e s u l t s 110 6.7 V a r i a t i o n of s h o r e l i n e r e t r e a t w i t h t i m e C o m p a r i s o n between e x p e r i m e n t and model's r e s u l t s ... 111 6.8 Beach p l a n - R e t r e a t of s h o r e l i n e due t o wave a p p r o a c h i n g b each o r t h o g o n a l l y 116 6.9 Beach P l a n - Combined o n - o f f s h o r e and l o n g s h o r e s i m u l a t i o n s 117 6.10 Beach p l a n - Storm waves s i m u l a t i o n 118 X L I S T OF TABLES T a b l e T i t l e Page 2.1 C o e f f i c i e n t s A and B f o r e q u a t i o n [2.7] 16 2.2 S e d i m e n t e q u a t i o n models 21 5.1 Imposed wave c o n d i t i o n and s c h e d u l e f o r d a t a c o l l e c t i o n 83 5.2 E q u a t i o n s f o r t h e r a t e of p r o f i l e c h a n g e s 94 1 CHAPTER 1 INTRODUCTION Over t h e p a s t d e c a d e , t h e r e have been d r a m a t i c i n c r e a s e s i n t h e demand f o r c o a s t a l d e v e l o p m e n t s . T h i s new demand i s p a r t l y due t o t h e i n c r e a s i n g a f f l u e n c e of t h e p e o p l e a s k i n g f o r more c o a s t a l r e c r e a t i o n a l f a c i l i t i e s , and p a r t l y due t o t h e e v e r i n c r e a s i n g p o p u l a t i o n t r y i n g t o g e t away from t h e crowded c i t i e s . In c o u n t r i e s where l a n d s h o r t a g e i s an a c u t e p r o b l e m , c o a s t a l d e v e l o p m e n t l i k e s ea r e c l a i m a t i o n may p r o v e t o be t h e o n l y v i a b l e s o l u t i o n . A good example i s S i n g a p o r e where r e c l a i m e d l a n d p r o v i d e s not o n l y s p a c e f o r h o u s i n g d e v e l o p m e n t but r e c r e a t i o n a l d e v e l o p m e n t a s w e l l . A s s o c i a t e d w i t h t h i s t r e n d i s t h e i n c r e a s i n g demand f o r s o l u t i o n s on how t o p r o t e c t and manage t h i s new economic r e s o u r c e . T h e r e a r e many q u e s t i o n s r e g a r d i n g t h e p r o t e c t i o n of t h e s e s h o r e l i n e s . Among t h e more p r e s s i n g ones a r e : 1. What a r e t h e c a u s e s of b e a c h e r o s i o n , and under what s i t u a t i o n do we e x p e c t i t t o o c c u r ? 2. I f s h o r e p r o t e c t i o n s t r u c t u r e s l i k e g r o y n e s and j e t t i e s a r e b u i l t , what a r e t h e i m p a c t s t h e s e s t r u c t u r e s have on t h e be a c h ? 2 3. What c h a n g e s t o e x p e c t when s h o r e l i n e i s a l t e r e d f o r a new usag e , s a y , b u i l d i n g an a r t i f i c i a l b e ach? 4. How t o p r e d i c t t h e e x t e n t of damage on t h e be a c h c a u s e d by s t o r m o r s i m i l a r d e s t r u c t i v e waves? T h i s r e p o r t w i l l n ot t r y t o p r o v i d e t h e answers t o a l l t h e q u e s t i o n s above, as t h e n e a r s h o r e p r o c e s s e s a r e complex and i n t e r c o n n e c t e d , and many a r e s t i l l p o o r l y u n d e r s t o o d . However w i t h t h e e x i s t i n g knowledge on l o n g s h o r e t r a n s p o r t , b e a c h c h a n g e s and n e a r s h o r e c u r r e n t s , an a t t e m p t t o f o r m u l a t e a s i m p l e model on s h o r e l i n e as w e l l as b e a c h p r o f i l e c h a n g e s can be made. The use of s u c h a model w i l l g i v e a ' f i r s t c u t ' s o l u t i o n t o some of t h e above q u e s t i o n s . P r e s e n t e d i n t h i s r e p o r t i s t h e s t a g e by s t a g e d e v e l o p m e n t o f s u c h a mo d e l . C h a p t e r t h r e e shows t h e wave r e f r a c t i o n and s h o a l i n g s u b r o u t i n e w h i c h w i l l be p r o v i d i n g t h e p r e d i c t i o n of n e a r s h o r e wave c l i m a t e from t h e g i v e n o f f s h o r e c l i m a t e . In c h a p t e r f o u r , t h e l o n g s h o r e t r a n s p o r t r o u t i n e i s d e v e l o p e d b a s e d on t h e CERC t r a n s p o r t f o r m u l a . A l s o shown i n t h e same c h a p t e r a r e t h e r e s u l t s o b t a i n e d u s i n g t h i s l o n g s h o r e t r a n s p o r t r o u t i n e on some t y p i c a l s h o r e l i n e s i t u a t i o n s . 3 C h a p t e r f i v e p r e s e n t s an a c c o u n t of t h e p r e s e n t e x p e r i m e n t a l s t u d i e s on b e a c h p r o f i l e c h a n g e s . I t a l s o h i g h l i g h t s some key a s p e c t s of b e h a v i o u r w h i c h a r e u s e d i n t h e o n - o f f s h o r e t r a n s p o r t r o u t i n e w h i c h c a l c u l a t e s t h e b e a c h p r o f i l e c h a n g e . C h a p t e r s i x shows t h e d e v e l o p m e n t of t h i s p r o f i l e change r o u t i n e . The a l g o r i t h m s of t h i s r o u t i n e a r e t h e n t e s t e d u s i n g t h e e x p e r i m e n t a l r e s u l t s of c h a p t e r f i v e . T h i s r o u t i n e i s l a t e r merged w i t h t h e l o n g s h o r e t r a n s p o r t r o u t i n e t o form t h e c o m p l e t e m o d e l . R e s u l t s from t h i s f i n a l model a r e a l s o shown. L a s t l y , c o n c l u s i o n s drawn from a l l t h e m o d e l l i n g r e s u l t s a r e g i v e n i n c h a p t e r s e v e n . 4 CHAPTER 2 NEARSHORE CURRENT AND  TRANSPORT EQUATION REVIEW 2. 1 I n t r o d u c t i o n T h i s c h a p t e r r e v i e w s b r i e f l y t h e t h e o r e t i c a l b a c k g r o u n d on t h e g e n e r a t i o n and e v a l u a t i o n of n e a r s h o r e c u r r e n t s , a s w e l l as t h e r e s u l t i n g sand t r a n s p o r t . The main f o c u s i s on 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 and t h e l o n g s h o r e t r a n s p o r t . A s u r v e y of t h e v a r i o u s s e d i m e n t t r a n s p o r t r a t e f o r m u l a i s a l s o p r e s e n t e d . 2.2 N e a r s h o r e C u r r e n t s N e a r s h o r e c u r r e n t s a r e o f p a r t i c u l a r i n t e r e s t t o c o a s t a l e n g i n e e r s . One of t h e main r e a s o n s i s t h a t t h e s e c u r r e n t s a r e b e l i e v e d t o be t h e c h i e f c a u s e of l i t t o r a l o r l o n g s h o r e t r a n s p o r t . T h e s e n e a r s h o r e c u r r e n t s c a n be g e n e r a t e d by wind, by t i d e s or by wind g e n e r a t e d waves. Of a l l t h e s e , t h e w a v e - i n d u c e d c u r r e n t i s t h e most dominant and t h e most i m p o r t a n t . In t h e n e a r s h o r e r e g i o n , t h e r e e x i s t two main t y p e s of wave i n d u c e d c u r r e n t s . The r i p c u r r e n t w h i c h i s a s s o c i a t e d w i t h t h e c i r c u l a t i o n c e l l and t h e l o n g s h o r e c u r r e n t w h i c h i s due t o o b l i q u e wave a t t a c k s on t h e s h o r e . 5 R i p c u r r e n t s a r e t h e seaward f l o w i n g component o f th e c i r c u l a t i o n c e l l and as s u c h i t s r o l e i n l o n g s h o r e t r a n s p o r t i s m i n i m a l . T h e r e have been many s t u d i e s (Bowen, 1969; Bowen and Inman, 1969; H i n o , 1975) on r i p c u r r e n t s s i n c e i t was f i r s t s t u d i e d by S h e p a r d e t a l ( 1 9 4 1 ) . R e f e r e n c e c an be made t o D a l y r m p l e (1978) f o r a r e v i e w on t h e more r e c e n t t h e o r i e s on r i p c u r r e n t s and t h e i r g e n e r a t i o n . 2.3 L o n g s h o r e C u r r e n t s L o n g s h o r e c u r r e n t s p l a y an i m p o r t a n t p a r t i n t h e l o n g s h o r e movement of m a t e r i a l . I t i s t h i s movement w h i c h i s f u n d a m e n t a l t o t h e f o r m a t i o n of many c o a s t a l f e a t u r e s . T h e r e a r e many e a r l y t h e o r i e s on t h e g e n e r a t i o n o f l o n g s h o r e c u r r e n t s . T h e s e e a r l y t h e o r i e s , can be c a t e g o r i s e d a c c o r d i n g t o t h e b a s i s o f t h e i r a p p r o a c h e s . G a l v i n (1967) r e v i e w e d most of t h e s i g n i f i c a n t ones and t e s t e d some of t h e dozen or more p r e d i c t i v e e q u a t i o n s on l o n g s h o r e c u r r e n t v e l o c i t y . Among h i s c o n c l u s i o n s , G a l v i n f o u n d t h a t none of t h e e q u a t i o n s gave an a c c e p t a b l e p r e d i c t i o n o f t h e l o n g h o r e c u r r e n t v e l o c i t y . S i n c e t h e n , t h e r e have been new i n v e s t i g a t i o n s , two o f w h i c h w i l l be d e s c r i b e d h e r e , f o r t h e y a r r i v e d a t s i m i l a r e q u a t i o n s f r o m d i f f e r e n t a p p r o a c h e s . 6 2.3.1 R a d i a t i o n S t r e s s A p p r o a c h L o n g u e t - H i g g i n s (1970a) r e - i n t r o d u c e d t h e momentum f l u x a p p r o a c h o f Putnam e t a l ( 1 9 4 9 ) . The model o f Putnam was shown t o have been b a s e d on e r r o n e o u s a s s u m p t i o n s , b e c a u s e i n t h e model's c o n s e r v a t i o n of momentum e q u a t i o n , Putnam e q u a t e d t h e l o n g s h o r e component of t h e b r e a k e r v e l o c i t y t o t h e c u r r e n t v e l o c i t y . T h i s i s wrong as i t can be shown t h a t t h e l o n g s h o r e c u r r e n t v e l o c i t y e x c e e d s t h e l o n g s h o r e component of t h e b r e a k e r v e l o c i t y ( G a l v i n , 1967). M o d i f y i n g t h e a p p r o a c h , L o n g u e t - H i g g i n s p r o p o s e d a new method o f c a l c u l a t i n g t h e momentum f l u x of t h e waves u s i n g t h e c o n c e p t s of r a d i a t i o n s t r e s s i n t r o d u c e d by L o n g u e t - H i g g i n s and S t e w a r t ( 1 9 6 4 ) . The r e s u l t i n g e q u a t i o n f o r t h e mean l o n g s h o r e c u r r e n t v e l o c i t y , V, w i t h r e s p e c t t o t i m e i s 5 TT tan? , „ u , s i n B 0 [2.1] V = —Q~ * ——(gh) — - 2 -f LQ where H = wave h e i g h t h = water d e p t h g = g r a v i t a t i o n a l a c c e l e r a t i o n tan/3 = b e a c h s l o p e So ^0 = the wave a n g l e and c e l e r i t y i n d e e p w a t e r and Cf = t h e b e a c h b o t t o m f r i c t i o n c o e f f i c i e n t . From e q u a t i o n [ 2 . 1 ] , i f and tan^ a r e c o n s t a n t , L o n g s h o r e C u r r e n t V e l o c i t y 7 C o n s e r v a t i o n o f e n e r g y Putnam e t a l (1963) E a g l e s o n (1965) C o n s e r v a t i o n o f momentum Putnam e t a l (1963) L o n g u e t - H i g g i n s (1970a) C o n s e r v a t i o n of mass Inman and B a g n o l d (1963) Brunn (1963) E m p i r i c a l c o r r e l a t i o n B r e b n e r and Kamphui s (1963) H a r r i s o n and Krumbein (1964) F i g u r e 2.1 D i f f e r e n t l o n g s h o r e c u r r e n t v e l o c i t y m o d e ls B reake r I ine Equation [ 2 . 2 ] • Expe r imen ta l d a t a V- v/v c / Vo = l ongshore cu r r en t ve loc i ty at breaker l i ne for P=0.0 X= x/x„ x,-= width of breaker zone F i g u r e 2.2 ( A f t e r L o n g u e t - H i g g i n s , 1970b) A c o m p a r i s o n between measured l o n g s h o r e c u r r e n t v e l o c i t y and t h e o r e c t i c a l d i s t r i b u t i o n s 8 th e mean l o n g s h o r e c u r r e n t v e l o c i t y w i l l v a r y l i n e a r l y f r o m z e r o a t t h e s h o r e l i n e t o maximum a t t h e b r e a k e r l i n e . O u t s i d e t h e s u r f zone, i t i s assumed t h a t t h e r e w i l l be no wave b r e a k i n g and hence no e n e r g y d i s s i p a t i o n and no d r i v i n g f o r c e f o r t h e c u r r e n t . Thus t h e d i s t r i b u t i o n of t h e c u r r e n t a c r o s s t h e s u r f zone i s j u s t a t r i a n g u l a r d i s t r i b u t i o n w i t h a d i s c o n t i n u i t y a t t h e b r e a k e r l i n e . But l a b o r a t o r y and f i e l d d a t a i n d i c a t e d no s u c h d i s c o n t i n u i t y a t t h e b r e a k e r l i n e ( f i g u r e 2 . 2 ) . One of t h e r e a s o n s f o r t h i s d i f f e r e n c e i s f o u n d t o be t h e e x c l u s i o n of h o r i z o n t a l m i x i n g w h i c h i s due t o t h e h o r i z o n t a l eddy v i s c o s i t y . In a r e a l f l u i d , a d i s c o n t i n u i t y as shown i n f i g u r e 2.2 c a n n o t e x i s t . The e f f e c t of i n t r o d u c i n g t h e h o r i z o n t a l m i x i n g w i l l be t o f l a t t e n and smoothen t h e d i s t r i b u t i o n p r o f i l e . T h i s w i l l r e s u l t i n a b e t t e r f i t of t h e d a t a . The maximum v a l u e of t h e l o n g s h o r e c u r r e n t i s t h e n no l o n g e r a t t h e b r e a k e r l i n e , b u t a l i t t l e s h o r e w a r d of t h a t l i n e w h i c h i s f o u n d t o be t h e c a s e i n t h e f i e l d . Bowen (1969a) and L o n g u e t - H i g g i n s (1970b) i n t r o d u c e d t h i s h o r i z o n t a l m i x i n g and o b t a i n e d e q u a t i o n [2.2] f o r t h e l o n g s h o r e c u r r e n t v e l o c i t y ( s e e f i g u r e 2 . 2 ) . P i s a p a r a m e t e r u s e d t o i n d i c a t e t h e d e g r e e of m i x i n g . From a c o m p a r i s o n w i t h t h e d a t a of G a l v i n and E a g l e s o n (1965) t h e v a l u e of P was f o u n d t o l i e between 0.1 ( l i t t l e m i x i n g ) and 0.5 ( g r e a t e r m i x i n g ) . 9 [2.2] 0 < X < 1 1 < X<oO where A, B 1 , B 2 , P-| and P 2 a r e a l l c o n s t a n t s d e p e n d i n g on P, [2.3] 1 3 + ) 9 + l A = ( 1- 5p / 2) P l ' P ? ""4 J16 P • P2~ 1 P i " 1 A B = — B = n ] . A P , - P 2 ' P 1 " P 2 U s i n g e q u a t i o n [ 2.2] w i l l be v e r y t e d i o u s and r e q u i r e s t h e v a l u e of P t o be known. As s u c h , t h e Sho r e P r o t e c t i o n M a n u a l (1977) s u g g e s t e d u s i n g e q u a t i o n [ 2 . 4 ] , an e q u a t i o n m o d i f i e d from t h e above a p p r o a c h , t o c a l c u l a t e t h e maximum l o n g s h o r e c u r r e n t v e l o c i t y i n t h e s u r f z o ne: 1/9 [2.4] Vm = 20. 7 tan/3 (ghb) sin 0 b where i s t h e water d e p t h a t t h e b r e a k e r l i n e . T h i s e q u a t i o n w i l l g i v e o n l y t h e maximum c u r r e n t v a l u e , and n o t t h e whole v a r i a t i o n a c r o s s t h e s u r f zone, as 10 d e s c r i b e d by e q u a t i o n [ 2 . 2 ] . The c o n s t a n t 20.7 i n e q u a t i o n [ 2 . 4 ] was o b t a i n e d from t h e c a l i b r a t i o n of t h e e q u a t i o n w i t h t h e l a b o r a t o r y d a t a of G a l v i n e t a l ( 1 9 6 5 ) , and t h e f i e l d d a t a o f Putnam e t a l ( 1 9 4 9 ) . A l s o i n c o r p o r a t e d i n t o e q u a t i o n [2.5] were t h e v a l u e s of P and Cf as 0.2 and 0.01 r e s p e c t i v e l y . The C f v a l u e o f 0.01 was b a s e d on t h e i n v e s t i g a t i o n s by P r a n d t l ( 1 9 5 2 ) , B r e t s c h n e i d e r (1954) and Meyer ( 1 9 6 9 ) . 2.3.2 S e m i - E m p i r i c a l A p p r o a c h From l i t t o r a l t r a n s p o r t s t u d i e s , Komar and Inman (1970) i n t r o d u c e d a l o n g s h o r e c u r r e n t v e l o c i t y e q u a t i o n s i m i l a r t o e q u a t i o n [ 2 . 1 ] : [2.5] V = 2.7 Um Sin 9 b where Um i s t h e 'maximum o r b i t a l v e l o c i t y under t h e wave' e v a l u a t e d a t t h e b r e a k e r l i n e . T h i s e q u a t i o n was o b t a i n e d by a v e r y d i f f e r e n t a p p r o a c h . Komar e t a l i n i t i a l l y d e v e l o p e d two models f o r l o n g s h o r e s a n d t r a n s p o r t r a t e . They f o u n d t h a t b o t h t h e models a g r e e w e l l w i t h f i e l d d a t a . Prompted by t h i s a g r e e m ent, t h e y e q u a t e d t h e two t r a n s p o r t e q u a t i o n s f o r t h e s e a p p a r e n t l y i n d e p e n d e n t m o d e l s . T h i s l e a d s t o e q u a t i o n [2.6] w h i c h t h e y have shown t o be s u p p o r t e d by f i e l d d a t a . 11 The s i m i l a r i t y of e q u a t i o n [ 2 . 1 ] and [ 2.5] can be seen i f b r e a k e r l i n e c o n d i t i o n s , = \fgh and Um - £*/gh a r e s u b s t i t u t e d i n t o e q u a t i o n [ 2 . 1 ] : [2.6] V = - = f - oC yih^  sin eb B o t h t h e s e e q u a t i o n s a r e s i m i l a r , e x c e p t t h a t e q u a t i o n [ 2.5] i m p l i e s t h e r a t i o t a n ^ 3 / C f must be a c o n s t a n t . Komar e t a l p r e s e n t e d f i e l d and l a b o r a t o r y d a t a i n a p l o t of V a g a i n s t UmSin9j-|, and showed t h a t t h e s l o p e of t h e b e s t f i t l i n e i s i n d e e d 2.7. From t h i s t h e y c o n c l u d e d t h a t t a n ^ / C - f i s c o n s t a n t . However, Cp was shown t o have an a p p r o x i m a t e v a l u e of 0 . 0 1 . T h e r e f o r e , - th e r a t i o t a n / 3 / C ^ s h o u l d not be a c o n s t a n t f o r d i f f e r e n t b e a c h s l o p e . L o n g u e t - H i g g i n s s u g g e s t e d t h a t t h i s a p p a r e n t c o n s t a n c y can be e x p l a i n e d . W i t h an i n c r e a s e i n t h e b e a c h s l o p e t a n / 3 , t h e r e would be an i n c r e a s e i n t h e d i s s i p a t i o n of e n e r g y by b r e a k i n g and hence an i n c r e a s e i n t h e l e v e l of t u r b u l e n c e . T h i s would seem l i k e l y t o e f f e c t t h e h o r i z o n t a l m i x i n g w h i c h w i l l i n d i r e c t l y b r i n g t h e v a l u e of V down, and m a i n t a i n s an a p p a r e n t c o n s t a n c y f o r t h e r a t i o tan / 3/C.f i n e q u a t i o n [ 2 . 5 ] . To d a t e , t h e a c t u a l r e l a t i o n s h i p between t h e l o n g s h o r e c u r r e n t v e l o c i t y and t h e b e a c h s l o p e i s s t i l l r e l a t i v e l y unknown. 2 . 4 L o n g s h o r e T r a n s p o r t L o n g s h o r e t r a n s p o r t i s t h e movement of b e a c h m a t e r i a l i n t h e l i t t o r a l zone by waves and c u r r e n t s i n t h e d i r e c t i o n p a r a l l e l t o t h e s h o r e l i n e . T h i s l i t t o r a l zone i s u s u a l l y d e f i n e d as t h e zone between t h e s h o r e l i n e and t h e b r e a k e r l i n e . Most n e a r s h o r e s e d i m e n t a t i o n s t u d i e s showed t h a t t h e most a c t i v e p a r t of s e d i m e n t movement o c c u r s s l i g h t l y s h o r e w a r d of t h e b r e a k e r l i n e . Due t o l a t e r a l m i x i n g as d e s c r i b e d i n t h e e a r l i e r s e c t i o n , t h e r e w i l l a l s o be some s e d i m e n t t r a n s p o r t s e award of t h e b r e a k e r l i n e . But t h e amount of t h i s movement i s much s m a l l e r when compared t o t h e t o t a l amount of movement i n t h e l i t t o r a l z o n e . F i g u r e 2.3 g i v e s a c l a s s i f i c a t i o n of t h e l o n g s h o r e t r a n s p o r t m o d e l s . T h e r e a r e two main c a t e g o r i e s : t h o s e b a s e d on t h e wave f l u x a p p r o a c h and t h o s e b a s e d on t h e s e d i m e n t ' e q u a t i o n a p p r o a c h . The e q u a t i o n s i n e a c h c a t e g o r y w i l l be b r i e f l y d e s c r i b e d . 2.4.1 Wave F l u x M o d e l s The wave f l u x a p p r o a c h r e c o g n i z e s t h a t e n e r g y i s r e q u i r e d t o t r a n s p o r t t h e s e d i m e n t . T h e r e f o r e , b e c a u s e waves p o s s e s s e n e r g y o r wave power, i t i s t h e n p o s s i b l e t o f i n d a c o r r e l a t i o n between t h e s e d i m e n t t r a n s p o r t r a t e and t h e amount of wave power a v a i l a b l e f o r t h i s t r a n s p o r t . The e q u a t i o n f o r t h e t r a n s p o r t r a t e has t h e f o r m : LONGSHORE TRANSPORT MODELS WAVE FLUX Wave Power or  Energy F l u x E s t i m a t i o n From Mean Wave Height SEDIMENT EQUATION I E n e r g e t i c Model Models Based on  Mechanics of T r a n s p o r t S c r i p p I n s t i t u t e of Oceanography (1947) WATTS (1953) CALDWELL (1956) SAVAGE (1959) INMAN AND BAGNOLD (1963) KOMAR AND INMAN (1970) GALVIN (1972) INMAN AND BAGNOLD (1963) BYKER (1971) FLEMING e t . a l . (1976) NIELSEN (1978) •WILLIS (1978) *SWART AND LENHOFF (1980) * A d a p t a t i o n of A c k e r s and White f o r m u l a CO F i g u r e 2.3 C l a s s i f i c a t i o n of l o n g s h o r e t r a n s p o r t models 14 [ 2 . 7 ] St = A P,B S[ i s t h e volume t r a n s p o r t r a t e and P( i s t h e ' l o n g s h o r e component of wave power' o r t h e ' l o n g s h o r e component of wave e n e r g y f l u x ' . T h i s v a l u e of i s d e t e r m i n e d a t t h e b r e a k e r l i n e . T h i s i s b e c a u s e most of t h e l o n g s h o r e t r a n s p o r t o c c u r s w i t h i n t h e s u r f z o n e . By a s s u m i n g c o n s e r v a t i o n of e n e r g y f l u x i n s h o a l i n g waves and u s i n g t h e s m a l l a m p l i t u d e wave t h e o r y , i t can be shown t h a t [ 2 . 8 ] P, = ( E C g )b C o s 8 b S in9 b From t h e A i r y wave t h e o r y , Cg = Cn where [ 2 . 9 ] n = -2 ; 2Kb. 1 + Sin (2Kh) S i n c e n i s a p p r o x i m a t e l y e q u a l s t o 1.0 i n s h a l l o w w ater and 1 2 [ 2 . 1 0 ] E = gj>g H J> , = mass d e n s i t y o f w a t e r E q u a t i o n [ 2 . 8 ] becomes 1 5 [2.11] 16 C Sin9> E q u a t i o n [2.10] can be e x p r e s s e d i n t e r m s of p a r a m e t e r s a t t h e b r e a k e r l i n e by a s s u m i n g r e f r a c t i o n by s t r a i g h t p a r a l l e l b o t t o m c o n t o u r s . A v a r i a t i o n of e q u a t i o n [2.10] i s t h u s o b t a i n e d b a s e d on o f f s h o r e wave p a r a m e t e r s (SPM, 1977). • [2.12] p. =44"^  T { K R H ° ) 2 S i n 9 b 1 6 4 where i s t h e r e f r a c t i o n c o e f f i c i e n t from d eep water t o s h a l l o w w ater and H 0 i s t h e deep water wave h e i g h t . S c r i p p s I n s t i t u t e of O c e a n o g r a p h y (1947) was among t h e f i r s t t o a d o p t t h e e n e r g y f l u x a p p r o a c h i n c a l c u l a t i n g t h e l o n g s h o r e t r a n s p o r t r a t e . S i n c e t h e n t h e r e have been a number of i n v e s t i g a t o r s t a k i n g t h e same a p p r o a c h , but e a c h o b t a i n e d d i f f e r e n t v a l u e s f o r c o e f f i c i e n t s A and B of e q u a t i o n [ 2 . 7 ] . T a b l e 2.1 g i v e s t h e v a l u e s of t h e c o e f f i c i e n t s a s w e l l as t h e u n i t s i n w h i c h t h e y were o b t a i n e d . From t a b l e 2.1, i t can be seen t h a t t h e a p p r o a c h was s l i g h t l y m o d i f i e d by Komar and Inman ( 1 9 7 0 ) . I n s t e a d o f r e l a t i n g volume t r a n s p o r t r a t e S,, t o P, , t h e y s u g g e s t e d t h e S = A P t B COEFFI A CIENTS B UNIT* > OF P, Watts (1953) 0.0 289 0.9 m Vday Watt/metre C a l d w e l l (1956) 0.0626 0.8 m Vday Watt/metre Inman and Bagnold (1963) 0.0046 1.0 m 3/day Watt/metre SPM (1977) 0.0709 1.0 m 3/day Watt/metre Komar and Inman (1970 ) * I l = A P l B 0.7 7 1.0 I( = dyn/sec Pi * dyn/sec = t_y-Js ) g a' S t where a' i s the void r a t i o of beach sand T a b l e 2.1 C o e f f i c i e n t s A and B o f e q u a t i o n [ 2 . 7 ] 17 use o f immersed w e i g h t t r a n s p o r t r a t e Ij . T h i s i d e a was i n i t i a l l y e x p r e s s e d by B a g n o l d ( 1 9 6 3 ) . The a d v a n t a g e s of t h i s a r e f i r s t l y , t h e c o e f f i c i e n t A becomes d i m e n s i o n l e s s g i v i n g a d i r e c t r e l a t i o n s h i p between P ( and 1^  , and s e c o n d l y , t h e e q u a t i o n t a k e s i n t o c o n s i d e r a t i o n t h e s e d i m e n t d e n s i t y . A l s o n o t e d from t h e s e e q u a t i o n s i s t h a t t h e t r a n s p o r t r a t e i s i n d e p e n d e n t of t h e b e a ch s l o p e . T h i s i n d e p e n d e n c e a l s o has s u p p o r t s b e c a u s e some of t h e d a t a (Komar e t a l , 1970) used t o f i n d t h e c o r r e l a t i o n were measured on q u i t e d i f f e r e n t b e a c h s l o p e s . A c o m p a r i s o n between t h e more r e c e n t e q u a t i o n s by Komar e t a l (1970) and SPM (1977) i n d i c a t e s a d i f f e r e n c e i n t h e i r p r e d i c t i o n of t h e t r a n s p o r t r a t e by a b o u t f i v e p e r c e n t . T h i s shows t h a t t h e e m p i r i c a l e q u a t i o n s a r e f a i r l y c o n s i s t e n t w i t h e a c h o t h e r ( Bruno e t a l , 1980). G a l v i n (1972) p r o p o s e d a n o t h e r e m p i r i c a l method i n e s t i m a t i n g t h e l o n g s h o r e t r a n s p o r t r a t e . From a s e r i e s of d a t a on t r a n s p o r t r a t e and o b s e r v e d mean b r e a k e r h e i g h t s , G a l v i n p l o t t e d a c u r v e [2.13] Sg = 1 6.5 H 2 where Sg i s t h e g r o s s l o n g s h o r e t r a n s p o r t r a t e i n u n i t s of 100,000 c u b i c m e t r e s p e r y e a r and H i s t h e mean b r e a k e r 18 h e i g h t i n m e t r e s . The f a c t s t h a t t h i s c u r v e r e p o r t e d by G a l v i n forms an upper l i m i t o v e r a l m o s t a l l t h e d a t a p o i n t s , and t h a t t h e e m p i r i c a l e q u a t i o n does not c o n s i d e r t h e a n g l e o f a t t a c k , s u g g e s t s t h a t e q u a t i o n [2.13] may p r e d i c t t h e l o n g s h o r e t r a n s p o r t c a p a b i l i t y r a t h e r t h e n t h e a c t u a l t r a n s p o r t . G a l v i n gave a p h y s i c a l e x p l a n a t i o n o f t h i s e m p i r i c a l r e l a t i o n s h i p . He assumed t h e l o n g s h o r e t r a n s p o r t o c c u r s m o s t l y as s u s p e n d e d l o a d , and from t h e law of c o n s e r v a t i o n o f mass he o b t a i n e d [2.14] Q g = [ D Kg fit T Sin 9 b ] H2 where c i s t h e s u s p e n s i o n c o n c e n t r a t i o n , K i s t h e r a t i o of a n n u a l mean i n d i v i d u a l H 2 o v e r s q u a r e of t h e a n n u a l mean H, and D an e m p i r i c a l c o e f f i c e n t . In o r d e r t h a t e q u a t i o n [2.14] p r o v i d e s a p l a u s i b l e e x p l a n a t i o n t o e q u a t i o n [ 2 . 1 3 ] , t h o s e terms i n t h e b r a c k e t w i l l have t o e q u a l 16.5. G a l v i n t e s t e d t h i s h y p o t h e s i s and a r r i v e d a t v a l u e s w h i c h were much l e s s t h a n 16.5. T h i s r e s u l t i s n o t t o o c o n v i n c i n g . F u r t h e r m o r e , t h e a s s u m p t i o n t h a t t h e l o n g s h o r e t r a n s p o r t o c c u r s m o s t l y as s u s p e n d e d l o a d i s s u b j e c t t o some d i s p u t e (Inman, 1969). I t i s i n t e r e s t i n g t o n o t e t h a t e q u a t i o n [2.13] can 1 9 a l s o be o b t a i n e d u s i n g a wave power a p p r o a c h . C o m b i n i n g e q u a t i o n s [2.7] and [2.11] and a s s u m i n g s h a l l o w w ater c o n d i t i o n s , Cg = C = \fgh ( a t t h e b r e a k e r p o i n t ) , t h e r e s u l t i n g e q u a t i o n i s [2.15] S t = A I B B H b T a k i n g t h e c o e f f i c i e n t s A and B as g i v e n by C a l d w e l l ( t a b l e 1.1), and a s s u m i n g t o be a p p r o x i m a t e l y 4 d e g r e e s e q u a t i o n [2.15] becomes, [2.16] S t = 14 .7 Hb 2 T h i s shows t h a t G a l v i n ' s e m p i r i c a l e q u a t i o n i s a l s o b a s e d on t h e wave power a p p r o a c h . Inman and B a g n o l d ' s (1963) a p p r o a c h i s a l s o e s s e n t i a l l y a wave power method, but i s d e r i v e d from a s e d i m e n t t r a n s p o r t argument and so w i l l be . p r e s e n t e d i n t h e n e x t s e c t i o n . 2.4.2 Sediment E q u a t i o n M o d e l s U n l i k e t h o s e i n t h e p r e v i o u s s e c t i o n , a l l t h e models i n t h i s c a t e g o r y a r e b a s e d on two r e l a t i o n s h i p s , one f o r t h e l o n g s h o r e c u r r e n t and a s e d i m e n t t r a n s p o r t r e l a t i o n s h i p w h i c h r e q u i r e s t h e l o n g s h o r e c u r r e n t as i n p u t . Most of t h e s e m o d els have t h e i r o r i g i n from work on s e d i m e n t t r a n s p o r t u nder o s c i l l a t o r y waves o r i n r i v e r i n e c o n d i t i o n s . 20 As a l l t h e models have been r e p o r t e d i n t h e l i t e r a t u r e , t h e d e t a i l s of t h e i r f o r m u l a t i o n w i l l not be g i v e n h e r e , t h i s s e c t i o n i s a b r i e f summary of t h e i r a p p r o a c h e s . T a b l e 2.2 shows the v a r i o u s models and t h e i r e q u a t i o n s . From t a b l e 2.2 i t c a n be seen t h a t most of t h e e q u a t i o n s c a l c u l a t e s t h e t r a n s p o r t s e p a r a t e l y i n terms of bed l o a d and s u s p e n d e d l o a d w i t h th e n o t a b l e e x c e p t i o n s of Inman and B a g n o l d (1963) whose e q u a t i o n i s f o r t o t a l l o a d , and N i e l s e n e t a l (1978) where t h e t r a n s p o r t i s assumed t o be t o t a l l y s u s p e n d e d l o a d . Inman e t a l assumed t h a t a p o r t i o n of t h e wave e n e r g y f l u x ( E C n ^ C o s G t , i s d i s s i p a t e d i n p l a c i n g t h e s e d i m e n t i n m o t i o n . Once t h e s e d i m e n t i s i n m o t i o n , i t becomes a v a i l a b l e f o r t r a n s p o r t by t h e c u r r e n t w h i c h i n t h i s c a s e i s t h e l o n g s h o r e c u r r e n t V . B y k e r (1971) and F l e m i n g e t a l ( 1 9 7 6 ) , however, a d o p t e d t h e a p p r o a c h t h a t l o n g s h o r e t r a n s p o r t c o n s i s t s of two components, namely, a bed l o a d component and a s u s p e n d e d l o a d component. B y k e r u s e d t h e F r y l i n k f o r m u l a f o r s e d i m e n t t r a n s p o r t r a t e under r i v e r i n e c o n d i t i o n s f o r h i s bed l o a d component. As f o r t h e s u s p e n d e d l o a d component, Byker u s e d t h e u s u a l method w h i c h i s t h e i n t e g r a t i o n of t h e p r o d u c t o f v e l o c i t y and s u s p e n s i o n c o n c e n t r a t i o n from t h e bed t o t h e f r e e s u r f a c e . The s u s p e n s i o n c o n c e n t r a t i o n i s b a s e d on t h e E i n s t e i n - R o u s e d i s t r i b u t i o n of s u s p e n d e d m a t e r i a l . From Longshore T r a n s p o r t Rate Remarks Inman and Bagnold (1963) I. = K' ( E C n k C o s 9 b ^ -1 u m K' = 0 . 2 8 (Komar and Inman, 1970) U m = ( 2 E / j > h b ) 1 / 2 Byker (1971) S[ = s b + ss Sb = bed l o a d Ss = suspended l o a d F l e m i n g e t . a l . (1976) S L = f e u dy + [ c u c dy C - sediment d i s t r i b u t i o n from the bed t o t h e f r e e s u r f a c e U, Uc ~ c u r r e n t v e l o c i t y d i s t r i b u t i o n T of the bed l o a d and the suspended l o a d r e g i o n e - r e f e r e n c e p o i n t from the bed (bed l o a d t h i c k n e s s ) N i e l s e n (1978) Q(t)= { u ( Z , t ) C ( Z , t ) dZ Q ( t ) - i n s t a n t a n e o u s f l u x the of sediment U ( Z , t ) - v e r t i c a l d i s t r i b u t i o n of the h o r i z o n t a l water v e l o c i t y C ( Z , t ) - v e r t i c a l d i s t r i b u t i o n of the suspended sediment Table 2.2 Sediment e q u a t i o n models 22 B y k e r ' s f o r m u l a t i o n , i t i s n o t e d t h a t t h e r e i s no p h y s i c a l s e p a r a t i o n of r e g i o n s between t h e bed l o a d and t h e s u s p e n d e d l o a d . I n s t e a d b o t h a r e s u p e r i m p o s e d on one a n o t h e r and can be u s e d q u i t e i n d e p e n d e n t l y . As f o r F l e m i n g ' s model, t h e r e a r e two i d e n t i f i e d r e g i o n s of t r a n s p o r t . In t h e bed l o a d r e g i o n , t h e g r a i n s a r e assumed t o be s u p p o r t e d by i n t e r - p a r t i c l e c o l l i s i o n s , and i n t h e s u s p e n d e d l o a d r e g i o n , t h e g r a i n p a r t i c l e s a r e k ept i n s u s p e n s i o n by t h e f l u i d t u r b u l e n c e . F l e m i n g t h e n d e f i n e d a r e f e r e n c e d e p t h 'e' (bed l o a d t h i c k n e s s ) t h a t s e p a r a t e s t h e two r e g i o n s . F l e m i n g a l s o assumed t h e c o n t i n u i t y of s e d i m e n t c o n c e n t r a t i o n and v e l o c i t y between t h e two r e g i o n s . Hence,, t h e r e i s a c t u a l l y no p h y s i c a l d i s c o n t i n u i t y between the two r e g i o n s . F l e m i n g t h e n summed th e two i n t e g r a t i o n s from t h e bed t o t h e r e f e r e n c e d e p t h 'e' and from 'e' t o t h e f r e e s u r f a c e t o g e t t h e t o t a l t r a n s p o r t r a t e . N i e l s e n e t a l (1978) a p p r o a c h e d th e p r o b l e m q u i t e d i f f e r e n t l y . From d i r e c t v i s u a l o b s e r v a t i o n , t h e y c o n c l u d e d t h a t a l l movements of s e d i m e n t o c c u r i n s u s p e n s i o n . They assumed t h a t a p i c k - u p mechanism i s r e s p o n s i b l e f o r s e t t i n g t h e s e d i m e n t i n t o s u s p e n s i o n . Then by a r g u i n g t h a t t h e m a t e r i a l i s kept i n s u s p e n s i o n by a t y p e o f d i f f u s i o n p r o c e s s , N i e l s o n e t . a l s u g g e s t e d t h a t t h e c o n c e n t r a t i o n of s u s p e n d e d s e d i m e n t s a t i s f i e s t h e d i f f u s i o n e q u a t i o n 23 [2.17] where i s t h e d i f f u s i o n c o e f f i c i e n t and 1 i s t h e v e r t i c a l c o - o r d i n a t e . From t h e i r d a t a , N i e l s e n e t a l f o u n d t h a t i s c o n s t a n t f r o m t h e bed t o t h e f r e e s u r f a c e when waves a r e n o n - b r e a k i n g , and £, i n c r e a s e s from t h e bed when waves a r e b r e a k i n g by s p i l l i n g . From t h e above e q u a t i o n , N i e l s e n e t a l were a b l e t o o b t a i n an e x a c t a n a l y t i c a l s o l u t i o n f o r t h e t i m e v a r i a t i o n of t h e c o n c e n t r a t i o n p r o f i l e , t h e i n s t a n t e o u s s e d i m e n t f l u x , and t h e n e t f l u x of s e d i m e n t o v e r a s i n g l e wave p e r i o d . The l a s t g r o u p of m o d els i s b a s i c a l l y v a r i a t i o n s o f t h e a p p r o a c h d e v e l o p e d by A c k e r s and W h i t e (1973) f o r c a l c u l a t i n g s e d i m e n t t r a n s p o r t r a t e under a u n i d i r e c t i o n a l f l o w c o n d i t i o n . A c k e r s and W h i t e ' s method i s b a s e d cn t h e s t r e a m power a p p r o a c h i n w h i c h t h e work done i n moving t h e s e d i m e n t i s t h e p r o d u c t o f e f f i c i e n c y and t h e s t r e a m power a v a i l a b l e f o r t r a n s p o r t . M o d i f i c a t i o n s of t h i s a p p r o a c h have been c a r r i e d out by W i l l i s (1978) and Swart e t a l (1980) f o r use i n c o a s t a l c o n d i t i o n s . The most n o t a b l e m o d i f i c a t i o n i s t h e change of t h e o r i g i n a l s h e a r s t r e s s r e l a t i o n s h i p o f A c k e r s and W h i t e . T h i s change i s t o compensate f o r t h e i n c r e a s e of i n s t a n t a n e o u s c u r r e n t v e l o c i t y and f o r t h e d i f f e r e n c e i n t h e i n i t i a t i o n of t h r e s h o l d o f m o t i o n due t o t h e p r e s e n c e of waves. The 24 r e s u l t s o f t h e s e m o d i f i c a t i o n have shown t o be p r o m i s i n g (Swart e t a l , 1980). 2.5 Summary The above s u r v e y shows t h a t t h e r e a r e many d i f f e r e n t a p p r o a c h e s t o w a r d p r e d i c t i n g l o n g s h o r e t r a n s p o r t r a t e . Swart e t a l (1980) and Bruno e t a l (1980) have compared some of t h e s e e q u a t i o n s t o f i e l d d a t a . They c o n c l u d e d t h a t most of t h e s e e q u a t i o n s a g r e e s w e l l , w i t h t h e d a t a , and a r e f a i r l y c o n s i s t e n t w i t h e a c h o t h e r when t h e same s e t o f d a t a i s u s e d . A l s o , from a l l t h e p r e d i c t i v e e q u a t i o n s p r e s e n t e d , i t seems t h a t no a c c o u n t o f p a r t i c l e s i z e was t a k e n i n t o c o n s i d e r a t i o n . S i n c e most of t h e e q u a t i o n s were b a s e d on b e a ch s a n d , i t i s f e l t t h a t p e r h a p s t h e r e was n o t enough v a r i a t i o n i n t h e p a r t i c l e s i z e f o r i t s e f f e c t t o be r e f l e c t e d i n t h e e q u a t i o n s . In c o n c l u s i o n , t h e r e i s no ' b e s t ' l o n g s h o r e p r e d i c t i v e e q u a t i o n . However, the 'wave f l u x ' t y p e seems e a s i e r t o use t h a n t h e ' s e d i m e n t e q u a t i o n ' t y p e . T h e r e a r e l e s s v a r i a b l e s r e q u i r e d as i n p u t . B e c a u s e of t h i s and t h e f a c t t h a t t h e s e wave f l u x e q u a t i o n s g i v e s good e s t i m a t e s (Bruno e t a l , 1978), i t i s d e c i d e d t h a t t h e CERC 'wave f l u x ' e q u a t i o n w i l l be use i n t h e model d e s c r i b e d h e r e i n . 25 CHAPTER 3 WAVE REFRACTION AND  SHOALING ROUTINE 3 . 1 I n t r o d u c t i o n A model c a p a b l e of p r e d i c t i n g c h a n g e s i n t h e s h o r e l i n e w i l l r e q u i r e t h e b r e a k i n g wave h e i g h t and t h e d i r e c t i o n of t h e wave a l o n g t h e s h o r e l i n e . I f t h e s e s h a l l o w w ater wave p r o p e r t i e s a r e t o be o b t a i n e d , t h e p r e d i c t e d i n c i d e n t d e e p w a t e r waves must f i r s t be s h o a l e d and r e f r a c t e d up t o t h e s h o r e l i n e . In a d d i t i o n , a s t h e s h o r e l i n e c o n f i g u r a t i o n c h a n g e s , t h e wave r e f r a c t i o n p a t t e r n a l s o c h a n g e s . T h i s w i l l r e s u l t i n a c o n t i n o u s change o f n e a r s h o r e wave p r o p e r t i e s . To a c c o u n t f o r t h i s , a wave r e f r a c t i o n and s h o a l i n g r o u t i n e i s d e v e l o p e d and i n c o r p o r a t e d i n t o t h e s h o r e l i n e p r e d i c t i o n m o d e l . T h i s c h a p t e r d e s c r i b e s t h e w o r k i n g of s u c h a r o u t i n e . 3.2 G e n e r a l D e s c r i p t i o n Waves a r e m o d i f i e d as t h e y t r a v e l from deep water t o w a r d s h o r e . In t h i s s e c t i o n , o n l y e f f e c t s due t o r e f r a c t i o n and s h o a l i n g a r e c o n s i d e r e d . I t i s assumed t h a t wave damping, b o t t o m f r i c t i o n , and any o t h e r n o n - l i n e a r e f f e c t s a r e n e g l i g i b l e . 26 At p r e s e n t , t h e r e a r e s e v e r a l methods a v a i l a b l e t o draw a wave r e f r a c t i o n d i a g r a m . The f i r s t method i s known as t h e wave c r e s t method ( J o h n s o n e t a l , 1948) i n w h i c h t h e wave c r e s t s a r e o b t a i n e d by d r a w i n g t h e e n v e l o p e of c i r c l e s f r o m a p r e c e d i n g wave c r e s t . The r a d i u s of t h e c i r c l e s i s p r o p o r t i o n a l t o t h e l o c a l v a l u e of t h e w a v e l e n g t h . The s e c o n d method i n v o l v e s t h e a p p l i c a t i o n of S n e l l ' s law of wave r e f r a c t i o n : [3.1] C-! Sino^ = C 2 Sind 2 where C , <X a r e t h e wave a n g l e and wave' c e l e r i t y a t two d i f f e r e n t d e p t h s d e n o t e d by t h e s u b s c r i p t s 1 and 2. Based on t h i s law, t h e c u r v a t u r e o f t h e r e f r a c t i n g wave o r t h o g o n a l s c an be c a l c u l a t e d and t h e o r t h o g o n a l s c an be t r a c e d from deep water t o s h o r e . T h e r e a r e s e v e r a l v a r i a t i o n s b a s e d on t h i s a p p r o a c h ( H a r r i s o n and W i l s o n , 1964; W i l s o n , 1966). The a c c u r a c y of t h e s e r e f r a c t i o n d i a g r a m s i s l i m i t e d by t h e v a l i d i t y o f t h e t h e o r y of t h e i r c o n s t r u c t i o n and t h e a c c u r a c y of t h e d e p t h d a t a on which t h e y a r e b a s e d . L i t t l e e r r o r i s i n t r o d u c e d i n t r a c i n g o r t h o g o n a l s o v e r r e l a t i v e l y s i m p l e h y d r o g r a p h y , but i t i s d i f f i c u l t t o c a l c u l a t e an o r t h o g o n a l a c c u r a t e l y o v e r complex b o t t o m f e a t u r e s . 27 Once t h e o r t h o g o n a l s have been c a l c u l a t e d , t h e wave h e i g h t s can be e s t i m a t e d . T h i s i s done by a s s u m i n g t h a t t h e f l u x of t r a n s m i t t e d e n e r g y i s c o n s t a n t between a p a i r o f o r t h o g o n a l s . But t h i s a s s u m p t i o n a l s o means t h a t no e n e r g y t r a v e l s l a t e r a l l y a l o n g a wave c r e s t . T h i s i s a r e a s o n a b l e a s s u m p t i o n , but i f t h e o r t h o g o n a l s bend s h a r p l y , t h e a c c u r a c y o f t h e d e r i v e d wave h e i g h t i s q u e s t i o n a b l e . I t s h o u l d be n o t e d t h a t r e f r a c t i o n e f f e c t s r e m a i n s m a l l as l o n g as t h e w a t e r d e p t h i s l a r g e r t h a n L / 3 . T h e r e f o r e , wave r e f r a c t i o n c a l c u l a t i o n s a r e u s u a l l y c a r r i e d o v e r b o t t o m t o p o g r a p h y s h a l l o w e r t h a n s u c h a d e p t h . 3.3 G o v e r n i n g E q u a t i o n s In t h i s s t u d y , wave r e f r a c t i o n and s h o a l i n g c a l c u l a t i o n s w i l l be b a s e d on two g o v e r n i n g e q u a t i o n s d e v e l o p e d by Noda e t a l ( 1 9 7 4 ) . The f i r s t e q u a t i o n , g i v e n i n c a r t e s i a n c o - o r d i n a t e s , i s e q u a t i o n [ 3 . 2 ] . I t i s d e r i v e d from t h e i r r o t a t i o n a l i t y of t h e wave number v e c t o r , K ( P h i l l i p s , 1980). C o s 8 ^ + S in 8 —y [3.2] 4 ( c o s e ^ - s , n e ^ ) 28 The p a r t i a l d e r i v a t i v e s o f K can be d e t e r m i n e d from t h e wave d i s p e r s i o n r e l a t i o n s h i p : [ 3 . 3 ] 6* = gK Tanh Kh where 6 = wave a n g u l a r v e l o c i t y . The s e c o n d e q u a t i o n ( e q u a t i o n [ 3 . 4 ] ) i s o b t a i n e d from t h e s t e a d y - s t a t e e n e r g y c o n s e r v a t i o n e q u a t i o n ( P h i l l i p s , 1980): Cg ( C o s 0 ) | | ^ + ( C g Sin9)-2--|? + [3.4] C o s 8 ^ - Cg SinB^I + Sin 9 ^ + C g C o s B ^ = 0 where Cg i s t h e g r o u p v e l o c i t y . [ 3.5] Cn = ~ ( 1 + ) . C 9 2 Sinh 2Kh By s o l v i n g e q u a t i o n s [ 3 . 2 ] and [ 3 . 4 ] , t h e v a r i a t i o n of wave h e i g h t H and wave d i r e c t i o n 9 o v e r t h e X, Y domain c a n be o b t a i n e d . F i g u r e 3.1 shows t h e d e f i n i t i o n s k e t c h of t h e p r o b l e m . The main r e a s o n f o r c h o o s i n g t h i s a p p r o a c h o v e r t h e o t h e r two d e s c r i b e d e a r l i e r i s t h a t t h i s a p p r o a c h 30 p r e d i c t s wave h e i g h t and wave d i r e c t i o n a t e a c h g r i d p o i n t ( i n x, y c o o r d i n a t e s ) . T h i s i s v e r y u s e f u l b e c a u s e t h i s p r o g r a m w i l l be i n t e g r a t e d w i t h t h e l o n g s h o r e t r a n s p o r t model which i s a l s o b a s e d on a g r i d s y s t e m . 31 3.4 N u m e r i c a l A p p r o a c h E q u a t i o n s [3.2] and [3.4] a r e s o l v e d u s i n g a f i n i t e d i f f e r e n c e n u m e r i c a l method. I t i n v o l v e s t h e t r a n s f o r m a t i o n o f t h e s e t o f p a r t i a l d i f f e r e n t i a l e q u a t i o n s i n t o a f i n i t e d i f f e r e n c e scheme. The f i n i t e d i f f e r e n c e form of e q u a t i o n [ 3 . 2 ] and [3.4] i s g i v e n i n e q u a t i o n s [3.6] and [ 3 . 7 ] ( s e e A p p e n d i x A f o r d e t a i l s of t h e f o r m u l a t i o n ) . The p r o c e d u r e i s t o r e l a x b o t h e q u a t i o n s [ 3 . 6 ] and [ 3 . 7 ] o v e r t h e e n t i r e X, Y domain. The g r i d s y s t e m u s e d i n t h i s p rogram i s a r e c t a n g u l a r / s q u a r e mesh t y p e and i s shown i n f i g u r e 3.2. From t h e o v e r a l l ( g l o b a l ) g r i d , t h e l o c a t i o n of e a c h l o c a l g r i d p o i n t ( i , j ) i s t h e n e s t a b l i s h e d . T h i s i s done w i t h t h e use of i n d i c e s j , j + 1 , j - 1 , e t c . , f o r Y -d i r e c t i o n s p a c e p o i n t s , and i , i + 1 , i - 1 , e t c . , f o r X -d i r e c t i o n s p a c e p o i n t s . F i g u r e 3.3 shows t h e l o c a l f i n i t e g r i d scheme. 3.5 C o m p u t a t i o n A c o m p u t a t i o n r o u t i n e i s d e v e l o p e d and u s e d t o c a l c u l a t e t h e s o l u t i o n f o r a l o c a l g r i d ( s a y a t i = 3, j = 3 ) . A f t e r t h a t t h e same c o m p u t a t i o n p r o c e d u r e i s a p p l i e d t o t h e n e x t l o c a l g r i d ( i = 3, j = 4 ) . The c o m p u t a t i o n t h e n p r o c e e d s s t e p by s t e p as shown i n f i g u r e 3.2. 32 C o s G ( U ) 9 ( M , j ) - 9 ( i - 1 , j ) 2 A X S i n B { i / j ) 9 ( i , j + D ~ 9 ( i , j - D 2 AY [3.6] K ( i j ) L Wg)(K(i^1)"K(iJ-1))-2 AY S i n e . ; ;W hlld} K ( i - ^ J ) 2 A X c9 ( i , j ) C o s 9 ( i , j ) H ( i . j ) L H ( i + 1 , i ) - H ( M , j) 2 AX 4-c 9 ( i . j ) s j n 0 ( i , j ) H ( i . j) H ( i , j - 1 ) ~ H ( i j+1) ' 2AY + I 3.7] Cos 8 ( i f j ) C g ( i + 1 , j ) ~ C g ( M , j ) 2AX 33 Equation [ 3 . 7 ] cont inued , C g ( i J ) S i n 0 ( i j ) e ( M J ) " 0 ( i - 1 J ) 2 A X + S in8 ( i | j ) c 9 ( i , j - 1 ) ~ C g ( j , j + 1) 2 A Y c 9 ( i j ) t o s e ( i | j ) 9 ( i , j - D ~ 9 ( i , j+1 ) 2 AY = 0 X X N [3-X N - H h -E3 Eh A Y - a B B--t3 E3 Eh 34 4 1r-3 i=2 ^ 1=1 t-A X j=1 j=2 3 4 YN-1 YN Shore l i ne Y • - Deepwater b o u n d a r y c o n d i t i o n A - Side boundary cond i t ion Computat ion begins with j = 1,i = XN-1 as cen te r of loca l g r id . The computat ion then p roceeds to j = 2 , i = X N - 1 t i l l j = Y N - 1 f i = 1 . F i g u r e 3.2 O v e r a l l g r i d scheme 6 ( U - D ^ ( i . j ) A X A Y CO F i g u r e 3.3 L o c a l g r i d scheme 36 The X, Y domain of t h e p r o b l e m a r e a i s s u r r o u n d e d by t h r e e s e t s o f b o u n d a r y c o n d i t i o n s . The f i r s t i s t h e o f f s h o r e b o u n d a r y g i v e n by t h e o f f s h o r e wave c o n d i t i o n s . The s e c o n d and t h i r d a r e t h e two s i d e b o u n d a r i e s of t h e domain (shown i n f i g u r e 3 . 2 ) . T h e s e s i d e b o u n d a r y c o n d i t i o n s a r e u s u a l l y not g i v e n i n t h e p r o b l e m , but t o s i m p l i f y t h e c a l c u l a t i o n r o u t i n e , t h e s e b o u n d a r i e s a r e assumed t o be f a r away from the a r e a of i n t e r e s t i n t h e domain, and t h e c o n d i t i o n s of t h e s e b o u n d a r i e s t o be t h e same as t h e i r immediate n e i g h b o u r ( i e s o l u t i o n of column j = 2 would be e q u a t e d t o column j = 1 and l i k e w i s e f o r j = YN -1 t o j = YN). The d e t a i l s of t h e p r o g r a m and t h e f l o w c h a r t a r e g i v e n i n A p p e n d i x B. 3.6 R e s u l t s A few s i m p l e t e s t r u n s were c a r r i e d u s i n g t h e above p r o c e d u r e s . The r e s u l t s o f t h e s e a r e g i v e n h e r e . T e s t 1 t o T e s t 3 were t o c h e c k t h e r e f r a c t i o n segment of t h e program, w h i l e T e s t 4 was t o c h e c k t h e s h o a l i n g segment. T e s t No T e s t c o n d i t i o n s 1 The a n g l e of wave i n c i d e n c e i s 45. (0.8 r a d i a n s ) t o t h e b e a c h n o r m a l . 8 d e g r e e s The b e a c h 37 s l o p e i s 1 i n 10 and t h e wave p e r i o d i s 5 s e c o n d s . F i g u r e 3.4 shows t h e wave c r e s t o r i e n t a t i o n of e a c h node as w e l l as t h r e e wave o r t h o g o n a l s . 2 The waves a p p r o a c h a s i m u l a t e d h e a d l a n d . The d e e p e s t d e p t h i s a t 3 m e t r e s , t h e s l o p e of t h e b e a c h i s 1 i n 10 and t h e wave p e r i o d i s 3.5 s e c o n d s . The r e s u l t of t h e r e f r a c t i o n i s shown i n f i g u r e 3.5. 3 A l a r g e r g r i d o f 30 by 30 i s u s e d , t o s i m u l a t e a s i t u a t i o n much more s i m i l a r t o f i e l d c o n d i t i o n s . The a n g l e of waves a p p r o a c h i s a t 0.8 r a d i a n s and t h e wave p e r i o d i s 5.5 s e c o n d s . The s l o p e i s 1 i n 10 and t h e d e e p e s t d e p t h i s 6 m e t r e s . F i g u r e 3.6 shows t h e r e s u l t s . 4 T h i s t e s t i s d e s i g n e d t o t e s t t h e s h o a l i n g segment o f t h e program; as s u c h , t h e waves a r e a p p r o a c h i n g t h e 1 i n 10 b e a c h n o r m a l l y . The wave p e r i o d i s s e t a t 4 s e c o n d s and t h e wave h e i g h t a t t h e d e p t h of 12 m e t r e s i s 1 m e t r e . 38 R e f r a c t i o n Diagrams F i g u r e 3.4 R e s u l t s o f t e s t No 1 showing t h e wave c r e s t s I I I I I I I I I I I I I 1 k I I i I i * \ / r - H - k Y I /\ / \ K / 3.o \ * \ X \ i / \ I/K • Y \ X \ / M — k / Y I V2.0 \ ^ \ /s h o r e V / V / c o n t o u r s (m ) F i g u r e 3.5 R e s u l t s of t e s t No 2 s h o w i n g t h e wave o r t h o g o n a l s 3 9 Refraction Diagram Contours F i g u r e 3 . 6 R e s u l t s o f t e s t No 3 s h o w i n g t h e wave o r t h o g o n a l s F i g u r e 3.7 V a r i a t i o n o f r e l a t i v e wave h e i g h t w i t h d e p t h 41 The r e s u l t s from T e s t 1 were c h e c k e d a g a i n s t t h e s o l u t i o n o b t a i n e d by an a n a l y t i c a l a p p r o a c h ( e q u a t i o n [ 3 . 1 ] ) . The c o m p a r i s o n showed t h a t t h e r e s u l t s d i f f e r e d by o n l y 0.2 p e r c e n t . I t was c a r r i e d out a t t h e d e p t h of 3 m e t r e s . The a n a l y t i c a l method gave t h e a n g l e as 0.5761 r a d i a n s , w h i l e T e s t 1 r e s u l t s gave a n g l e a t 0.5750 r a d i a n s . T h i s i s c o n s i d e r e d a c c e p t a b l e b e c a u s e t h e u n c e r t a i n t i e s and e r r o r s f r o m o f f s h o r e wave p r e d i c t i o n s w i l l be g r e a t e r t h a n 0.2 p e r c e n t . A l s o , i t must not be f o r g o t t e n t h a t a f i n i t e d i f f e r e n c e method i s an a p p r o x i m a t i o n method so t h a t t h e r e s u l t s can o n l y be c l o s e t o t h e t r u e s o l u t i o n . The wave h e i g h t r e s u l t s from T e s t 4 a l s o compared w e l l w i t h t h e a n a l y t i c a l s o l u t i o n . The wave h e i g h t a t 4 -m e t r e s d e p t h u s i n g an a n a l y t i c a l a p p r o a c h i s 0.9-235 m e t r e , w h i l e t h e p r o g r a m gave a v a l u e of 0.9236 m e t r e . The d i f f e r e n c e i s o n l y 0.01 p e r c e n t . The r e l a t i v e wave h e i g h t v a r i a t i o n w i t h d e p t h was p l o t t e d ( f i g u r e 3.7) as a f i n a l c h e c k on t h e s h o a l i n g segment. t h a t when waves a p p r o a c h a s h o r e , t h e wave h e i g h t s w i l l i n i t i a l l y r e d u c e b e f o r e i n c r e a s i n g t o t h e b r e a k i n g p o i n t . F i g u r e 3.7 shows t h e p l o t of H/H 0 a g a i n s t d / L 0 - The s u b s c r i p t 'o' r e p r e s e n t s d e e p w a t e r v a l u e . When i t was compared w i t h a p l o t o b t a i n e d a n a l y t i c a l l y (Le Mehaute, 1976), an a l m o s t p e r f e c t match was o b t a i n e d . 42 In c o n c l u s i o n , t h e c o m p u t a t i o n p r o c e d u r e s of t h e wave r e f r a c t i o n and s h o a l i n g p r o g r a m a r e c o r r e c t , and t h e r e s u l t s a r e s u f f i c i e n t l y a c c u r a t e f o r use w i t h t h e l o n g s h o r e t r a n s p o r t segment t o be d e s c r i b e d i n t h e n e x t c h a p t e r . 43 CHAPTER 4 LONGSHORE TRANSPORT MODEL 4 . 1 I n t r o d u c t i o n From t h e c o n c e p t of s e d i m e n t b u d g e t , a s e c t i o n of s h o r e l i n e w i l l change i t s shape as a r e s u l t o f a n e t s e d i m e n t movement i n and out o f t h a t s e c t i o n . In a n e a r s h o r e e n v i r o n m e n t , t h e s e s e d i m e n t movements can be d i v i d e d i n t o l o n g s h o r e and o n - o f f s h o r e d i r e c t i o n s . I t i s o b v i o u s t h a t t o have a c o m p r e h e s i v e s h o r e l i n e e v o l u t i o n model, t h e model w i l l r e q u i r e c o n s i d e r a t i o n s of n e t s e d i m e n t movement i n b o t h d i r e c t i o n s . But t h e model p r e s e n t e d i n t h i s - c h a p t e r w i l l c o n s i d e r o n l y t h e l o n g s h o r e component. The o n - o f f s h o r e and t h e t o t a l l o n g s h o r e and o n - o f f s h o r e p r o c e s s e s w i l l be c o n s i d e r e d i n c h a p t e r s i x . The model w i l l be u s e d t o s i m u l a t e a b e a c h b u i l d up due t o t h e c o n s t r u c t i o n of a b a r r i e r ( g r o y n e or j e t t y ) and t o s t u d y t h e c h a n g e s o f a b e a c h n o u r i s h m e n t p l a n shape w i t h t i m e . 4.2 Model Review A l l t h e e x i s t i n g s h o r e l i n e e v o l u t i o n models can be c l a s s i f i e d i n t o two g r o u p s . The f i r s t g r o u p f o l l o w s t h e a p p r o a c h i n w h i c h t h e t r a n s p o r t r a t e e q u a t i o n and t h e 44 c o n t i n u i t y e q u a t i o n a r e s o l v e d s i m u l t a n e o u s l y . The t r a n s p o r t r a t e e q u a t i o n u s e d i n most of t h e models of t h i s g r o u p i s t h e CERC t r a n s p o r t r a t e e q u a t i o n . Hence, models of t h i s g r o u p a r e a l s o known as CERC m o d e l s . The s e c o n d g r o u p has a s i m i l a r a p p r o a c h as t h e f i r s t , b u t w i t h a s l i g h t v a r i a t i o n . I n s t e a d o f a u s i n g p r e d i c t i v e t r a n s p o r t r a t e e q u a t i o n l i k e t h e CERC e q u a t i o n , t h e models i n t h i s g r o u p compute t h e l o n g s h o r e c u r r e n t s and t h e n use a s u i t a b l e t r a n s p o r t t h e o r y (eg A c k e r s and W h i t e ) t o compute t h e s e d i m e n t l o a d . M o d e l s u s i n g t h i s a p p r o a c h a r e known as ' c u r r e n t ' m o d e l s . A c o m p a r i s o n between a ' c u r r e n t ' model and a CERC model was c a r r i e d o ut by W i l l i s ( 1 9 7 7 ) . The c o n c l u s i o n i n d i c a t e d t h a t on t h e whole, t h e CERC models gave a b e t t e r p r e d i c t i o n of t h e r e s u l t s , i n terms o f t h e volume of s e d i m e n t t r a n s p o r t e d and t h e s h o r e l i n e c h a n g e s , t h a n t h e ' c u r r e n t ' m o d e l s . A p o s s i b l e r e a s o n f o r t h e p o o r e r p r e d i c t i o n by t h e ' c u r r e n t ' models i s t h a t t h e r e i s no s a t i s f a c t o r y t h e o r y r e l a t i n g t h e c u r r e n t s and t h e t r a n s p o r t r a t e s i n c o a s t a l c o n d i t i o n s . O n l y w i t h more a d v a n c e s i n t h i s a r e a w i l l t h e ' c u r r e n t ' models be b e t t e r . The g o v e r n i n g e q u a t i o n s i n a l l t h e models can be s o l v e d e i t h e r by an a n a l y t i c a l o r a n u m e r i c a l m o d e l l i n g method.. When t h e a n a l y t i c a l method i s u s e d , t h e r e s u l t i n g d i f f e r e n t i a l e q u a t i o n i s c o m p l i c a t e d . The s o l u t i o n i s i m p o s s i b l e t o o b t a i n even w i t h a s e t of s i m p l e b o u n d a r y 45 c o n d i t i o n s . However, t h e g o v e r n i n g e q u a t i o n s c a n be s i m p l i f i e d and a p p r o x i m a t e a n a l y t i c a l s o l u t i o n s c a n be o b t a i n e d ( P e l n a r d - C o n s i d e r e , 1956; Bakker e t a l , 1970; W a l t o n e t a l , 1979). The n u m e r i c a l m o d e l l i n g method i s a method where t h e d i f f e r e n t i a l e q u a t i o n s a r e m o d e l l e d u s i n g a n u m e r i c a l a p p r o x i m a t i o n t e c h n i q u e . P r i c e e t a l (1972) and Romar (1973) u s e d t h i s a p p r o a c h i n t h e i r m o d e l s . T h i s a p p r o a c h w i l l be d e s c r i b e d l a t e r i n t h i s c h a p t e r . 4.2.1 C o n t i n u i t y E q u a t i o n Of t h e two g o v e r n i n g e q u a t i o n s , t h e t r a n s p o r t r a t e e q u a t i o n and i t s v a r i a t i o n s have been d e s c r i b e d i n c h a p t e r two. The f o l l o w i n g s e c t i o n w i l l r e v i e w b r i e f l y t h e c o n t i n u i t y e q u a t i o n of some of t h e e a r l y m o d e l s . P r i c e e t a l i n 1972 d e v e l o p e d a CERC model f o r p r e d i c t i n g s h o r e l i n e c h a n g e s . E q u a t i o n [4.1] i s t h e c o n t i n u i t y e q u a t i o n of t h a t m o d e l . The d e f i n i t i o n s k e t c h of t h e e q u a t i o n i s g i v e n i n f i g u r e 4.1. [ 4 . , j 10 + _ ^ ^ Y = 0 <3X 2 2> T The d e p t h 'D' o f e q u a t i o n [ 4 . 1 ] has been d e s c r i b e d d i f f e r e n t l y by v a r i o u s a u t h o r s . W i l l i s e t a l (1975) s u g g e s t e d t h a t 'D' be t h e d e p t h b eyond w h i c h l o n g s h o r e t r a n s p o r t no l o n g e r t a k e s p l a c e . From a s e r i e s o f F i g u r e 4.2 D e f i n i t i o n s k e t c h f o r e q u a t i o n [4.4] 47 o b s e r v a t i o n s , t h e y c o n c l u d e d t h a t a good e s t i m a t e o f 'D' i s [4.2] D = 2;0 H( where Hj i s t h e i n c i d e n t wave h e i g h t . W a l t o n e t a l (1979) s u g g e s t e d a s i m i l a r d e s c r i p t i o n f o r 'D'. W a l t o n d e f i n e d 'D' as t h e d e p t h t h a t encompasses t h e e n t i r e zone of l o n g s h o r e sand movement, and t h e r e f o r e , n o t o n l y i n c l u d e s t h e d e p t h below water t o t h e sand movement l i m i t , but a l s o t h e d e p t h above water of t h e b e a c h . The e x p r e s s i o n f o r 'D' i s t h e n [4.3] D 1.3 Hb + 2a0 +' R where 3 0 i s t h e maximum t i d a l a m p l i t u d e and R i s t h e wave run-up above t h e mean h i g h t i d e l e v e l . From a d i f f e r e n t a p p r o a c h , Komar (1976) p r o p o s e d the use of a v a r i a b l e 'd', s u c h t h a t d A y g i v e s t h e s e c t i o n a r e a of t h e b e a c h e r o d e d o r d e p o s i t e d ( s e e f i g u r e 4 . 2 ) . The r e s u l t i n g c o n t i n u i t y e q u a t i o n becomes [4.4] IQ. + D^Y = 0 ax si The v a l u e of 'd' i s o b t a i n e d by c a l i b r a t i n g t h e model t o t h e p r o t o t y p e . T h i s a p p r o a c h g i v e s t h e model an added f l e x i b i l i t y and a l s o a b e t t e r f i t t o t h e p r o t o t y p e . 48 T h i s v a r i a b l e 'd' i s a c o n c e p t u a l d e p t h and does not have any p h y s i c a l s i g n i f i c a n c e . 4.3 N u m e r i c a l M o d e l l i n g The model a d o p t e d i n t h i s r e p o r t i s a CERC model. The r e a s o n f o r a d o p t i n g t h i s model i s t h a t t h e ' c u r r e n t ' m o d e l s a r e s t i l l s u b j e c t e d t o many u n c e r t a i n t i e s a s m e n t i o n e d e a r l i e r . In a d d i t i o n , a s t u d y by Bruno e t a l (1980) on t h e r e l a t i o n s h i p between t h e l o n g s h o r e e n e r g y f l u x a t b r e a k i n g (P| ) and t h e immersed w e i g h t t r a n s p o r t r a t e {!{ ), has r e a f f i r m e d t h e r e l a t i o n I ( = A P ( ( t a b l e 2 . 1 ) . W i t h t h i s , i t i s f e l t t h a t t h e CERC model w i l l g i v e a more r e a s o n a b l e e s t i m a t e of t h e l o n g s h o r e t r a n s p o r t c h a n g e s . 4.3.1 G o v e r n i n g E q u a t i o n s T h e r e a r e o n l y two g o v e r n i n g e q u a t i o n s f o r t h i s m o d e l . The volume t r a n s p o r t r a t e e q u a t i o n and t h e c o n t i n u i t y e q u a t i o n . The t r a n s p o r t r a t e e q u a t i o n u s e d i s e q u a t i o n [4.5] by Komar: [4.5] I, = 0 . 7 7 P, where 49 P, = E Cn Sin 8 b Cos8 b [ 4 . 6 ] Ii = ( j> - fs) g a' Si a' = c o r r e c t i o n f a c t o r f o r t h e p o r e s p a c e of b e a c h sand = 0.6 and = b e a c h sand d e n s i t y = 2.65. I t must be m e n t i o n e d t h a t c o e f f i c i e n t 0.77 of e q u a t i o n [ 4.5] i s b a s e d on d a t a i n t e r m s of RMS b r e a k i n g wave h e i g h t . S i n c e i t i s common t o use s i g n i f i c a n t wave h e i g h t v a l u e s , a c o r r e c t i o n f a c t o r i s u s e d t o c o n v e r t e q u a t i o n [ 4 . 5 ] . The r e s u l t i n g volume t r a n s p o r t r a t e e q u a t i o n i s where S^  i s t h e volume t r a n s p o r t r a t e i n c u b i c m etre p e r day. The assumed beach p r o f i l e change i s g i v e n i n f i g u r e 4.3. The d e p t h 'D' can be c a l c u l a t e d u s i n g e q u a t i o n [ 4 . 2 ] . However, from s t u d i e s by Bakker e t a l (1970) on l o n g s h o r e s e d i m e n t movement, i t was f o u n d t h a t t h e r e i s v i r t u a l l y no s e d i m e n t movement l o n g s h o r e a t a d i s t a n c e of one and a h a l f t i m e s t h e b r e a k e r zone w i d t h away from t h e s h o r e l i n e . They [4.7] S, = ( 3 .85x10" 2 ) H b Cn S i n 8 b Cos9 b The c o n t i n u i t y e q u a t i o n u s e d i s e q u a t i o n [ 4 . 1 ] . 50 t h e n p r o p o s e d t h a t t h i s be t h e l o c a t i o n f o r d e t e r m i n i n g t h e d e p t h 'D'. T h i s a p p r o a c h w i l l be a d o p t e d i n t h e model b e c a u s e s i m i l a r o b s e r v a t i o n s were a l s o r e p o r t e d by Komar (1971) and T h o r n t o n ( 1 9 7 3 ) . F i g u r e 4.4 shows t h e d i s t r i b u t i o n o f sand t r a n s p o r t r a t e a c r o s s t h e w i d t h of t h e b r e a k e r zone. From th e f i g u r e , i t seems t h a t when a b a r r i e r s t o p s t h e s u p p l y of s e d i m e n t downstream of a b e a c h , a l a r g e volume of sand w i l l be e r o d e d a t t h e l o c a t i o n o f t h e maximum t r a n s p o r t r a t e and z e r o volume a t t h e s h o r e l i n e . T h i s would mean no movement of t h e s h o r e l i n e and an i n c r e a s e of b e a c h s l o p e . O b v i o u s l y , t h i s i s not what happens b e c a u s e t h e r e must be some s e d i m e n t movement a l o n g t h e o r i - o f f s h o r e d i r e c t i o n due t o t h e o r b i t a l m o t i o n of t h e wave. T h i s movement w i l l c o n t i n u o u s l y a d j u s t the s l o p e and c a u s e t h e s h o r e l i n e t o r e c e d e as shown i n f i g u r e 4.3. To a c c o u n t f o r t h i s , a d i s t r i b u t i o n of s e d i m e n t t r a n s p o r t r a t e d i f f e r e n t f r o m t h e one i n f i g u r e 4.4 i s a d o p t e d f o r t h e model. A t p r e s e n t , t h e r e a r e two a p p r o a c h e s : 1. L i n e a r d i s t r i b u t i o n v a r y i n g f r o m z e r o a t d e p t h 'D' t o the maximum a t t h e s h o r e l i n e . 2. U n i f o r m d i s t r i b u t i o n f r o m d e p t h 'D' t o t h e s h o r e l i n e . B o t h c a s e s a r e c o n t r a s t e d w i t h Komar's c u r v e i n f i g u r e 4.5. The a r e a under e a c h c u r v e i s t h e same. 51 0.5L Breaker zone width, L New profile at time t + A t B reake r l ine F i g u r e Assumed b e a c h p r o f i l e change F i g u r e 4.4 ( A f t e r Komar, 1973) D i s t r i b u t i o n of s a n d t r a n s p o r t r a t e a c r o s s b r e a k e r zone F i g u r e 4.5 A p p r o x i m a t e d sand t r a n s p o r t r a t e d i s t r i b u t i o n to 53 From t h e f i g u r e , i t can be seen t h a t u s i n g e i t h e r of t h e two d i s t r i b u t i o n s w i l l t a k e i n t o a c c o u n t i m p l i c i t l y some o n - o f f s h o r e movements of s e d i m e n t . However, t h e u n i f o r m d i s t r i b u t i o n w i l l l e a d t o t h e m a i n t a i n e n c e a c o n s t a n t p r o f i l e o f t h e b e a c h . T h i s i s not a l t o g e t h e r t r u e as a c c r e t i o n a t t h e b a r r i e r w i l l t e n d t o change t h e p r o f i l e . On t h e o t h e r hand, t h e l i n e a r d i s t r i b u t i o n seems t o be g r o s s l y r e - d i s t r i b u t i n g t h e c u r v e by Komar. In view of t h e d i s a d v a n t a g e s , a new d i s t r i b u t i o n i s p r o p o s e d and u s e d i n t h e model. The new d i s t r i b u t i o n , shown as a d a s h e d l i n e i n f i g u r e 4.5, i s a c t u a l l y a c o m b i n a t i o n of t h e two e a r l i e r d i s t r i b u t i o n s . I t assumed t h a t a u n i f o r m r a t e of t r a n s p o r t e x i s t s a c r o s s t h e b r e a k e r zone, and i t w i l l d e c r e a s e l i n e a r l y a f t e r t h e b r e a k e r l i n e t o z e r o a t t h e l o c a t i o n of d e p t h 'D'. 4.3.2 Computat i o n The g o v e r n i n g e q u a t i o n s a r e s o l v e d n u m e r i c a l l y u s i n g a f i n i t e d i f f e r e n c e scheme d e s c r i b e d below. The s h o r e l i n e i n t h e s t u d y a r e a i s d i v i d e d i n t o a s e r i e s o f c e l l s of a f i n i t e and u n i f o r m l e n g t h , A X . T h i s d i s c r e t i z a t i o n i s shown i n f i g u r e 4.6. The t r a n s p o r t i n and out of t h e c e l l , and A Clean be c a l c u l a t e d u s i n g e q u a t i o n [4.7] and a s u i t a b l e t i m e i n t e r v a l AT. B e f o r e t h e whole p r o c e d u r e r e p e a t s i t s e l f w i t h 54 e q u a t i o n [4.7] f o r t h e nex t A T , t h e wave r e f r a c t i o n and s h o a l i n g s u b r o u t i n e i s c a l l e d t o p r o v i d e t h e new n e a r s h o r e wave c o n d i t i o n . A f l o w c h a r t o f t h e model i s g i v e n i n A p p e n d i x C. In summary, t h e model p e r f o r m s t h e f o l l o w i n g : 1) S h o a l s and r e f r a c t s deep water waves. 2) C a l c u l a t e s b r e a k i n g wave c o n d i t i o n s . 3) C a l c u l a t e s t h e b r e a k e r t o s h o r e l i n e a n g l e . 4) C a l c u l a t e s t h e volume of a c r e t i o n / e r o s i o n . 5) D i s t r i b u t e s a c c r e t i o n / e r o s i o n o v e r t h e b e a c h p r o f i l e . 6) R e t u r n s t o (1) and r e - c a l c u l a t e s t h e wave s h o a l i n g and t h e r e f r a c t i o n f o r t h e n e x t i t e r a t i o n . 4.3.3 Boundary C o n d i t i o n s The model r e q u i r e s s u i t a b l e t h e b o u n d a r y c o n d i t i o n s a s i n p u t s , and t h e s e b o u n d a r y c o n d i t i o n s w i l l have t o be s p e c i f i e d as d a t a i n a s i m u l a t i o n r u n . T h e r e a r e t h r e e b o u n d a r y c o n d i t i o n s . Two of them a r e t r a n s p o r t b o undary c o n d i t i o n s w h i c h c o n t r o l t h e r a t e o f s e d i m e n t e n t e r i n g and l e a v i n g t h e s t u d y a r e a . The t h i r d i s t h e s h o r e l i n e b o u n d a r y w h i c h under n o r m a l c i r c u m s t a n c e s i s assumed t o be a f r e e b o u n d a r y , meaning t h a t i t i s e r o d i b l e . T h i s c o n d i t i o n i s b u i l t i n t o t h e model and need not be spec i f i e d . 55 Grid points Q(in) O(out) j for cell j -4 4 4 4 4 + + 4 + 4 4 + 4 +• + 4 + + 4 + = 5 = 4 = 3 = 2 Cell j-m H j j + 1 j+2 Shoreline F i g u r e 4.6 D i s c r e t i z a t i o n of s t u d y a r e a Upstream IN Transport Boundary assume no on-off movement STUDY AREA A Downstream OUT /// ? / / / / / / / / / / / / / / / / / / / / / / ' / / / , S h o r e l i n e F i g u r e 4.7 Boundary c o n d i t i o n s 56 By s p e c i f y i n g c e r t a i n c o n d i t i o n s a t t h e two t r a n s p o r t b o u n d a r i e s , t h e model c a n be u s e d t o s i m u l a t e d i f f e r e n t s i t u a t i o n s . F o r example, i n f i g u r e 4.7, by s p e c i f y i n g no t r a n s p o r t , a c r o s s t h e downstream b o u n d a r y , i t i s p o s s i b l e t o c r e a t e a p h y s i c a l b a r r i e r a t r i g h t a n g l e s t o the b e a c h . By c o n t r o l l i n g t h e volume of t r a n s p o r t a c r o s s t h e u p s t r e a m b o u n d a r y , i t i s a l s o p o s s i b l e t o c r e a t e t h e s i t u a t i o n of a r i v e r s u p p l y i n g more ( o r l e s s ) s e d i m e n t t h a n the b e a c h r e q u i r e s . 4.4 M o d e l A p p l i c a t i o n s The model i s u s e d t o s i m u l a t e t h r e e s i t u a t i o n s : a) T h e • s h o r e l i n e b u i l d - u p a f t e r t h e c o n s t r u c t i o n o f an i n f i n i t e l y l o n g impermeable b a r r i e r . b) The s h o r e l i n e b u i l d - u p same as a) but w i t h a b a r r i e r of f i n i t e l e n g t h . c ) The change o f a b e a c h n o u r i s h m e n t p l a n shape w i t h t i m e . 4.5 R e s u l t s A. I n f i n i t e l y l o n g impermeable b a r r i e r The boundary c o n d i t i o n s f o r t h i s c a s e a r e i ) The t r a n s p o r t b o u n d a r y u p s t r e a m o f t h e s t u d y a r e a i s 57 s e t t o 1.0 ( i e 100 p e r c e n t p e r m e a b l e ) , and i i ) t h e downstream boundary i s s e t t o 0.0 ( i e no s e d i m e n t a l l o w s out of t h e s t u d y a r e a ) . R e s u l t s o f t h e b u i l d - u p due t o t h e b a r r i e r a r e shown i n f i g u r e 4.8 f o r t h e p a r t i c u l a r c a s e 9o = 45 d e g r e e s , t a n p = 0.1, wave p e r i o d T = 5.5s and H 0 = 0.6m. (Due t o th e e f f e c t of d i s t o r t i o n , wave o r t h o g o n a l s would a l s o be d i s t o r t e d and a p p e a r t o be l e s s t h a n n o r m a l , so f o r f i g u r e s 4.8 and 4.11, t h e i n c i d e n t wave c r e s t d i r e c t i o n i s shown.) From t h e f i g u r e , t h e f o l l o w i n g were o b s e r v e d . 1. The s h o r e l i n e b e g i n s i t s a c c r e t i o n i n t h e r e g i o n n e a r e s t t h e b a r r i e r . The r a t e of a c c r e t i o n i n t h i s r e g i o n i s n o t i c e d t o d e c r e a s e w i t h t i m e , w h i l e i n r e g i o n s f u r t h e r away from t h e b a r r i e r , t h e r a t e i s f a i r l y c o n s t a n t . I t i s b e l i e v e d t h a t t h i s i s due t o t h e d e c r e a s i n g a n g l e between t h e b r e a k i n g waves and t h e s h o r e l i n e a s i t a p p r o a c h e s t h e b a r r i e r . 2. The p l a n shape of t h e s h o r e l i n e a t any one i n s t a n t i s s i m i l a r t o t h e shape a t any o t h e r . I t i s s u s p e c t e d t h a t t h e a c c r e t i o n h i s t o r y of t h e s h o r e l i n e t o any t i m e T c a n be non-d i m e n s i o n a l i s e d , and a c u r v e c an be f o u n d t o r e p r e s e n t t h e a c c r e t i o n h i s t o r y f o r a s p e c i f i c l o c a t i o n on t h e s h o r e l i n e . The g r o w t h h i s t o r y of t h e s h o r e l i n e i s d e s c r i b e d by t h r e e v a r i a b l e s . They a r e t h e a c c r e t i o n l e n g t h ( Y -Barrier SHORELINE (10M INTERVAL) F i g u r e 4.8 P l a n view of t h e s h o r e l i n e b u i l d - u p due t o i n f i n i t e l y l o n g b a r r i e r 59 a x i s i n f i g u r e 4.8), t h e l o c a t i o n (X - a x i s , g i v e n as t h e d i s t a n c e away from t h e b a r r i e r ) , and t h e t i m e v a r i a b l e T. To n o n - d i m e n s i o n a l i s e , a c u r v e i s c h o s e n from f i g u r e 4.8 so t h a t two c h a r a c t e r i s t i c v a r i a b l e s , t i m e and l e n g t h , c an be d e f i n e d . The ' c h a r a c t e r i s t i c t i m e ' i s t h e f i n a l t i m e T of th e c h o s e n c u r v e , and t h e ' c h a r a c t e r i s t i c l e n g t h ' i s t h e maximum a c c r e t i o n l e n g t h Lm ( s e e f i g u r e 4 . 9 ) . The d e f i n i t i o n s f o r t h e d i m e n s i o n l e s s l o c a t i o n , t i m e , and l e n g t h r a t i o a r e a l s o shown i n t h e f i g u r e . T h us, a t e a c h v a l u e of D, t h e r e i s a r e l a t i o n s h i p between t h e t i m e r a t i o and t h e a c c r e t i o n r a t i o . The r e s u l t of t h e above p r o c e d u r e s i s shown, i n f i g u r e 4.10. The v a r i a b l e X i n t h e f i g u r e i s t h e same a s t h a t d e f i n e d i n f i g u r e 4.9. T h e r e f o r e , t h e c u r v e X = 0 i n f i g u r e 4.9 i s t h e a c c r e t i o n h i s t o r y a t z e r o d i s t a n c e away from t h e b a r r i e r . S i m i l a r l y , t h e c u r v e X = 2.98 would d e p i c t t h e a c c r e t i o n h i s t o r y a t t h e l o c a t i o n 2.98 t i m e s Lm d i s t a n c e away from t h e b a r r i e r . I t must be n o t e d t h a t t h e s i x c u r v e s i n f i g u r e 4.10 a r e from a f a m i l y o f i n f i n i t e c u r v e s , as X r a n g e s from 0 t o i n f i n i t e . T h ese c u r v e s a r e a l s o i n d e p e n d e n t o f t h e v a l u e of Tp. A p p e n d i x D shows t h a t t h e same f a m i l y o f c u r v e s i s o b t a i n e d even when d i f f e r e n t v a l u e s of T a r e us e d i n t h e n o n - d i m e n s i o n a l i s i n g p r o c e d u r e s . F i g u r e 4.10 shows t h a t t h e a c c r e t i o n r a t e s v a r y w i t h l o c a t i o n . I t a l s o shows t h a t e a c h c u r v e ( w i t h t h e e x c e p t i o n o f X = 0 ) , e x h i b i t s a s l o w , f a s t , and slow 09 Location ratio 0.0 0.13 0.26 OA 0.53 0.66 0.3 0.93 TIME RATIO F i g u r e 4.10 N o n - d i m e n s i o n a l i z e d a c c r e t i o n c u r v e s f o r i n f i n i t e l y l o n g b a r r i e r 62 a c c r e t i o n r a t e l i k e a 'S' c u r v e . In a d d i t i o n t o showing t h e a c c r e t i o n h i s t o r y of t h e v a r i o u s l o c a t i o n s , f i g u r e 4.10 has a u s e f u l a p p l i c a t i o n i n f i n d i n g t h e minimum l e n g t h of t h e b a r r i e r r e q u i r e d t o a c h i e v e t h e r e q u i r e d s e d i m e n t a c c u m m u l a t i o n a t a s p e c i f i c l o c a t i o n w i t h i n a minimum t i m e . Example 4.1 F i n d t h e minimum l e n g t h o f b a r r i e r r e q u i r e d t o a c h i e v e a 5 m e t r e s a c r e t i o n a t a d i s t a n c e 30 m e t r e s away from t h e b a r r i e r . S o l u t i o n Assume t h e l e n g t h L = 10 m e t r e s . T h e r e f o r e X = 30/10 = 3. From f i g u r e 4.11, t h e c u r v e o f X = 3 (by i n t e r p o l a t i o n ) shows t h e a c c r e t i o n h i s t o r y of t h e l o c a t i o n 30m away f r o m a 10 metre b a r r i e r . The c u r v e i n d i c a t e s t h a t a t t h e t i m e r a t i o of 1, t h e a c c r e t i o n r a t i o i s a p p r o x i m a t e l y 0.68. T h i s i m p l i e s t h a t f o r L = 10m, a c c r e t i o n a t 30m away i s 6.8m ( 5.0m). To m i n i m i s e t h e b a r r i e r l e n g t h L, assume a l o w e r v a l u e of L and r e p e a t a g a i n . T h i s t r i a l and e r r o r i s c a r r i e d o ut u n t i l a c c r e t i o n v a l u e of 5m i s o b t a i n e d f o r t h e l o c a t i o n 30m away from t h e b a r r i e r . The answer t o t h i s example i s a p p r o x i m a t e l y 8m. Bar r ie r L = ? 63 T h i s s o l u t i o n i s f o r s i t u a t i o n s w i t h t h e t i m e r a t i o o f 1.0. T h i s means t h a t t h e 5m a c c r e t i o n i s a c h i e v e d a t t h e same t i m e as t h e 8m a c c r e t i o n a t t h e b a r r i e r . F o r v a l u e s g r e a t e r t h a n L = 8m, t h e 5m a c c r e t i o n w i l l s t i l l be a c h i e v e d when t h e a c c r e t i o n a t t h e b a r r i e r r e a c h e s 8m. Hence, a l o n g e r b a r r i e r does n o t mean a c h i e v i n g t h e 5m a c c r e t i o n a t a s h o r t e r t i m e . B. F i n i t e l e n g t h impermeable b a r r i e r In most s i t u a t i o n s , t h e b a r r i e r i s not i n f i n i t e l y l o n g ; so a model s i m u l a t i n g a f i n i t e l e n g t h b a r r i e r would be more u s e f u l and i s p r e s e n t e d h e r e i n . Two a s s u m p t i o n s w i l l have t o be made i n t h i s s i m u l a t i o n : 1. S i n c e t h e b a r r i e r has a f i n i t e l e n g t h , i t i s assumed t h a t t h e s e d i m e n t w i l l s t a r t t o p a s s t h e b a r r i e r a f t e r t h e s e d i m e n t f i l l s up t h e l e n g t h . 2. The amount of s e d i m e n t p a s s i n g out of t h e s t u d y a r e a w i l l be e q u a l t o t h e amount p a s s e d i n t o t h e l a s t c e l l i n t h e s t u d y a r e a , so t h a t no a c c r e t i o n w i l l o c c u r a t t h e l a s t c e l l . The r e s u l t of t h e f i n i t e b a r r i e r s i m u l a t i o n i s g i v e n i n f i g u r e 4.11. The m o d e l l i n g p a r a m e t e r s f o r t h i s example i s t h e same as t h o s e o f t h e i n f i n i t e l e n g t h b a r r i e r example. The f i g u r e shows t h a t even when t h e b a r r i e r Bar r i er 0.0 4.0 8.0 12.0 16.0 20.0 24.0 2S.0 S H O R E L I N E (10M INTERVAL ) F i g u r e 4.11 P l a n view of t h e s h o r e l i n e b u i l d - u p due t o f i n i t e l e n g t h b a r r i e r Location ratio X = 0 - X= k TIME RATIO cn F i g u r e 4.12 N o n - d i m e n s i o n a l i z e d a c c r e t i o n c u r v e s f o r f i n i t e l e n g t h b a r r i e r 66 becomes i n e f f e c t i v e i n r e t a i n i n g t h e s e d i m e n t , t h e r e i s s t i l l a c c r e t i o n i n t h e s t u d y a r e a . U s i n g a s i m i l a r p r o c e d u r e s as i n t h e l a s t s e c t i o n , a f a m i l y o f non-d i m e n s i o n a l i s e d c u r v e s ( f i g u r e 4.12) i s o b t a i n e d . The d e f i n i t i o n of Tp i s d i f f e r e n t f r o m b e f o r e ; i t i s now t h e t i m e t a k e n f o r a c c r e t i o n a t t h e b a r r i e r t o r e a c h t h e b a r r i e r ' s l e n g t h . F o r t i m e r a t i o s between T = 0 and T = 1.0, t h e c u r v e s i n f i g u r e 4.10 and 4.12 a r e t h e same. T h i s i s e x p e c t e d as t h e a c c r e t i o n b e h a v i o u r i s t h e same. A f t e r t h e t i m e r a t i o e x c e e d s 1.0, t h e c u r v e X = 0 becomes a h o r i z o n t a l s t r a i g h t l i n e . T h i s i s b e c a u s e t h e r e i s no a d d i t i o n a l a c c r e t i o n a t t h e b a r r i e r . In t h e o t h e r r e g i o n s , t h e a c c r e t i o n c o n t i n u e s but t h e r a t e of t h i s a c c r e t i o n i s n o t i c e a b l y r e d u c e d . By c o m p a r i n g f i g u r e s 4.8 and 4.11, one c a n see t h a t t h e r e a r e two ways t o a c h i e v e a d e s i r e d a c c r e t i o n a t a p o i n t . The two ways a r e e i t h e r u s i n g a l o n g e r b a r r i e r and a c h i e v i n g t h e r e s u l t i n a s h o r t e r t i m e , o r a s h o r t e r b a r r i e r a t a l o n g e r t i m e . F i g u r e 4.13 shows t h e two p o s s i b i l i t i e s . The s o l u t i o n t o t h e s e two p o s s i b i l i t i e s c a n be o b t a i n e d u s i n g t h e f o l l o w i n g a p p r o a c h . S o l u t i o n R e c a l l t h a t t h e s o l u t i o n f o r an i n f i n i t e l y l o n g b a r r i e r i s 8m. U s i n g t h e same a p p r o a c h w i t h f i g u r e 4.12, t h e r e s u l t 67 f o r a f i n i t e l e n g t h b a r r i e r s i m u l a t i o n i s 6.2m. T h i s two l e n g t h s a r e shown as FB and FA r e s p e c t i v e l y i n f i g u r e 4.13. Assume t h e t i m e t a k e n f o r a c c r e t i o n t o r e a c h B f r o m F i s T. Then f r o m f i g u r e 4.10, t h e t i m e t a k e n t o r e a c h A from F w i l l a p p r o x i m a t e l y be 0.6T. At t h i s same i n s t a n t , t h e a c c r e t i o n a t E would have a l s o r e a c h D. From f i g u r e 4.12, t h e t i m e t a k e n f o r E t o r e a c h C i s 4.2 ( s e e A p p e n d i x E) t i m e s l o n g e r t h a n f o r E t o D. Hence t h e t i m e f o r a c c r e t i o n a t E t o r e a c h C w i l l be 4.2 X 0.6T = 2.5T. Barr ier Accret ion CE at E can be achieved by barrier length FB or FA but at d i f f e r e n t time. F i g u r e 4.13 - Two p o s s i b i l i t i e s of a c h i e v i n g a d e s i r e d a c c r e t i o n T h i s i m p l i e s t h a t when u s i n g a s h o r t e r 6.2m b a r r i e r , t h e t i m e t a k e n t o a c h i e v e t h e r e q u i r e d a c c r e t i o n a t t h e l o c a t i o n would be 2.5 t i m e s g r e a t e r t h a n a l o n g e r 8m b a r r i e r . T h e s e two s o l u t i o n s a r e a c t u a l l y two e x t r e m e s o l u t i o n s . A whole r a n g e of s o l u t i o n e x i s t between t h e two. Which t o c h o o s e w i l l have t o depend on t h e r e s t r i c t i o n s 68 imposed on t h e s o l u t i o n . F o r example, i f t h e t i m e f a c t o r i s not i m p o r t a n t as l o n g as t h e r e q u i r e d a c c r e t i o n i s a c h i e v e d , t h e n i t would make economic s e n s e t o use t h e s h o r t e s t b a r r i e r p o s s i b l e . C. Changes o f beach n o u r i s h m e n t p l a n The o b j e c t i v e of t h i s s i m u l a t i o n i s t o s t u d y t h e c h a n g e s of a b e a c h n o u r i s h m e n t p l a n w i t h t i m e . The m o d e l l i n g p a r a m e t e r s a r e t h e same as b e f o r e e x c e p t t h a t t h e waves a p p r o a c h t h e s h o r e o r t h o g o n a l l y . The i n i t i a l b e a c h p l a n i s t r i a n g u l a r i n shape and s i n c e symmetry i s e x p e c t e d from t h e r e s u l t s , o n l y h a l f of t h e p l a n i s m o d e l l e d . The r e s u l t s o f t h e model a r e shown i n f i g u r e 4.14. As e x p e c t e d , t h e s e d i m e n t i s e r o d i n g away from t h e apex of t h e t r i a n g u l a r p l a n . The d e p o s i t i o n o c c u r s a l o n g th e s h o r e l i n e away from the apex. The t i m e i n t e r v a l between e a c h c u r v e shown i s t h e same. From t h i s , i t can be s e e n t h a t t h e r a t e o f change i s s l o w i n g down. W a l t o n e t a l (1979) p r e s e n t e d t h e r e s u l t s o f t h e same p r o b l e m u s i n g an a n a l y t i c a l model by P e l n a r d - C o n s i d e r e . The r e s u l t s o f t h e a n a l y t i c a l model a r e shown i n f i g u r e 4.15. A c o m p a r i s o n of f i g u r e s 4.14 and 4.15 shows many d i f f e r e n c e s . However, one must l o o k a t t h e a s s u m p t i o n s of t h e a n a l y t i c a l model b e f o r e d r a w i n g c o n c l u s i o n s . 69 In o r d e r t o l i n e a r i s e t h e g o v e r n i n g e q u a t i o n s f o r a s i m p l i f i e d s o l u t i o n , P e l n a r d - C o n s i d e r e assumed t h a t t h e b r e a k e r h e i g h t i s e v e r y w h e r e c o n s t a n t , and t h a t t h e d i f f e r e n c e between t h e b r e a k e r a n g l e and t h e d e e p water i n c i d e n t a n g l e must be s m a l l a t a l l t i m e s . T h e s e two a s s u m p t i o n s a r e c l e a r l y v i o l a t e d i n t h e n u m e r i c a l model p r e s e n t e d i n t h i s r e p o r t . T h i s i s b e c a u s e t h e r e f r a c t i o n r o u t i n e i n t h e model w i l l change t h e b r e a k e r a n g l e and t h e wave h e i g h t i n c o m f o r m i t y w i t h t h e n e a r s h o r e t o p o g r a p y . T h e r e f o r e t h e two a s s u m p t i o n s w i l l have t o be imposed on t h e n u m e r i c a l model t o make a p r o p e r c o m p a r i s i o n . The r e s u l t s of t h e m o d i f i e d model w i t h t h e two imposed c o n d i t i o n s a r e shown i n f i g u r e 4.1'6. F o r e a s e of c o m p a r i s o n w i t h f i g u r e 4.15, a n o n - d i m e n s i o n a l i s e d v e r s i o n o f t h e r e s u l t i s p r e s e n t e d i n f i g u r e 4.17. A c o m p a r i s o n of f i g u r e s 4.14 and 4.17 shows a s a t i s f a c t o r y a g r e e m e n t . As s u c h , t h e a n a l y t i c a l model by P e l n a r d - C o n s i d e r e i s e s s e n t i a l l y a model t h a t does not c o n s i d e r t h e e f f e c t s of wave r e f r a c t i o n and wave s h o a l i n g . The e f f e c t of wave r e f r a c t i o n and s h o a l i n g can be seen c l e a r l y f r o m f i g u r e s 4.15 and 4.16. Due t o r e f r a c t i o n , t h e wave b r e a k i n g a n g l e t o t h e s h o r e l i n e w i l l be s m a l l and a p p r o x i m a t e l y c o n s t a n t a l o n g t h e s h o r e l i n e . T h e r e f o r e , t h e l o n g s h o r e t r a n s p o r t w i l l be l e s s and t h e r a t e o f change o f s h o r e l i n e l e s s d r a m a t i c . F i g u r e 4.14 Beach n o u r i s h m e n t p l a n c h a n g e s ( w i t h r e f r a c t i o n and s h o a l i n g r o u t i n e ) F i g u r e 4.15 ( A f t e r W a l t o n e t a l , 1974) A n a l y t i c a l r e s u l t of b e a c h n o u r i s h m e n t p l a n c h a n ges F i g u r e 4.16 Beach n o u r i s h m e n t p l a n c h a n g e s ( w i t h o u t wave r e f r a c t i o n and s h o a l i n g r o u t i n e ) F i g u r e 4.17 D i m e n s i o n l e s s v e r s i o n of F i g u r e 4.16 74 4.6 D i s c u s s i o n The r e s u l t s p r e s e n t e d i n t h i s c h a p t e r w i l l have t o be v i e w e d u n d e r th e l i g h t of t h e a s s u m p t i o n s made i n t h e model, th e most i m p o r t a n t of w h i c h i s e i t h e r t h a t t h e r e i s n e g l i g i b l e o n - o f f s h o r e t r a n s p o r t , o r t h e v a l u e of t h i s t r a n s p o r t i s known and can be t a k e n i n t o a c c o u n t i n t h e c o n t i n u i t y e q u a t i o n . In a d d i t i o n t o t h e a s s u m p t i o n s , t h e model r e s u l t s w i l l a l s o be i n f l u e n c e d by t h e a c c u r a c y and s t a b i l i t y o f t h e c a l c u l a t i o n s . As i n a l l n u m e r i c a l m o d e l s , t h e a c c u r a c y of t h e r e s u l t s w i l l depend on m o d e l l i n g p a r a m e t e r s s u c h as t h e g r i d s i z e , A X, A Y , and model t i m e i n t e r v a l A T . . I t i s e a s y t o see t h a t i f s m a l l e r v a l u e s o f Ax, A Y , and A T a r e u s e d , t h e r e s u l t s of t h e model w i l l be more a c c u r a t e and d e t a i l e d o v e r t h e s t u d y a r e a . However, due t o t h e l e n g t h of t h e c o m p u t a t i o n i n v o l v e d , a f i n e r g r i d would d e f i n i t e l y e n t a i l more c o m p u t a t i o n t i m e and would be more c o s t l y t o r u n . Thus t h e r e i s t h i s b a l a n c e between t h e a c c u r a c y o f t h e r e s u l t s r e q u i r e d and c o s t . In a d d i t i o n t o t h e a c c u r a c y , t h e r e i s t h e q u e s t i o n of c o m p u t a t i o n a l s t a b i l i t y . S i n c e v a r i o u s c o m b i n a t i o n s o f Ax , A Y , a n d A T can be u s e d , t h e r e would e x i s t a wide s p e c t r u m of s o l u t i o n s . T h e s e s o l u t i o n s can be of t h r e e t y p e s : a) I n a c c u r a t e s o l u t i o n 75 b) A c c u r a t e , c l o s e t o t r u e s o l u t i o n c) O s c i l l a t i n g , u n s t a b l e s o l u t i o n . The a c c u r a c y of s o l u t i o n as m e n t i o n e d b e f o r e , w i l l i m p r ove w i t h a f i n e r g r i d mesh and a s m a l l e r t i m e i n t e r v a l . However, t h e r e e x i s t a c r i t e r i a where t h e s o l u t i o n becomes u n s t a b l e and o s c i l l a t e s . F o r t h e c a l c u l a t i o n t o be ' s t a b l e ' , t h e t i m e - s t e p i n t e r v a l , T must be s m a l l enough t o e n s u r e t h a t t h e b e a c h does not o s c i l l a t e w i t h i n c r e a s i n g a m p l i t u d e a b o u t i t s e q u i l i b r i u m p o s i t i o n . The scheme a d o p t e d t o a c h i e v e t h i s ' s t a b l e ' c a l c u l a t i o n i s e m p i r i c a l and i s d e t e r m i n e d by a t r i a l and e r r o r t e c h n i q u e . The c o n d i t i o n w hich must be s a t i s f i e d i s t h a t t h e n e t volume of t r a n s p o r t f o r a c e l l a t t i m e ' i ' must not be g r e a t e r or e q u a l i n m a g n i t u d e t o t h e ne t volume of t r a n s p o r t of t h e same c e l l a t t i m e ' i +A T', I f t h i s c o n d i t i o n i s not met, t h e c a l c u l a t i o n i s u n s t a b l e and w i l l n ot c o n v e r g e . 76 CHAPTER 5 ON-OFFSHORE TRANSPORT  INVESTIGATION 5.1 I n t r o d u c t i o n I t i s known from f i e l d o b s e r v a t i o n s t h a t t h e on-o f f s h o r e movement of s e d i m e n t c a u s e s c h a n ges i n t h e b e a c h p r o f i l e . The u n d e r s t a n d i n g of t h i s movement and t h e r e s u l t i n g p r o f i l e change i s t h e r e f o r e i m p o r t a n t and e s s e n t i a l i n any a t t e m p t t o model t h e o n - o f f s h o r e b e h a v i o u r . A t h o r o u g h d i s c u s s i o n of a l l p r e v i o u s i n v e s t i g a t i o n s on on-o f f s h o r e s e d i m e n t movement would be e x c e e d i n g l y l e n g t h y . So t h i s c h a p t e r w i l l b r i e f l y m e n t i o n some of t h e more i m p o r t a n t i n v e s t i g a t i o n s . A l s o p r e s e n t e d a r e t h e r e s u l t s o f an e x p e r i m e n t a l s t u d y on t h e p r o f i l e change under c o n s t a n t wave a t t a c k . 5.2 P r e v i o u s I n v e s t i g a t i o n s I n v e s t i g a t o r s i n t h e 1950's d e s c i b e d b e a c h p r o f i l e s b r o u g h t about by d i f f e r e n t wave c o n d i t i o n s as e i t h e r w i n t e r or summer p r o f i l e . A t y p i c a l w i n t e r ( o r storm) p r o f i l e has a m i l d e r s l o p e and c o n t a i n s one o r more o f f s h o r e b a r s , w h i l e a summer ( o r s w e l l ) p r o f i l e i s g e n e r a l l y s t e e p e r and s m o o t h e r . T h i s change i s now 77 u n d e r s t o o d t o be t h e e f f e c t of o n - o f f s h o r e s e d i m e n t movement. The change of d i r e c t i o n of t h e s e d i m e n t movement i s g e n e r a l l y b e l i e v e d t o be r e l a t e d t o t h e d e e p w a t e r wave s t e e p n e s s ( H 0 / L 0 ) . However, i n v e s t i g a t o r s ( J o h n s o n , 1949; K i n g and W i l l i a m s , 1949; R e c t o r , 1954) had came up w i t h r d i f f e r e n t c ' i t i c a l v a l u e s of H 0 / L 0 . The d i f f e r e n c e s i n t h e v a l u e s were f o u n d t o be due t o t h e dependence of t h e r a t i o H 0 / L 0 on s e d i m e n t g r a i n s i z e and s c a l e e f f e c t s ( I w a g a k i and Noda, 1963). Dean (1973) p r e s e n t e d a d i f f e r e n t model f o r d e t e r m i n i n g t h e d i r e c t i o n o f movement. Dean c o n s i d e r e d t h e t r a j e c t o r y o f a s u s p e n d e d sa n d p a r t i c l e d u r i n g i t s f a l l t o t h e b o t t o m w h i c h i s a c t e d upon a t t h e same t i m e by t h e h o r i z o n t a l w a ter p a r t i c l e v e l o c i t y . U s i n g t h e d a t a o f R e c t o r (1954) and S a v i l l e ( 1 9 5 7 ) , Dean o b t a i n e d t h e f o l l o w i n g r e l a t i o n s h i p f o r d e t e r m i n i n g t h e p r o b a b l e d i r e c t i o n of s e d i m e n t movement: [5.1] - 0,85 H 0 / V f T i s known as t h e d i m e n s i o n l e s s f a l l v e l o c i t y , where Vf i s t h e f a l l v e l o c i t y c a l c u l a t e d u s i n g t h e median g r a i n s i z e , D^Q . The c o e f f i c i e n t 0.85 i s sometimes c a l l e d t h e f a l l - t i m e p a r a m e t e r . O t h e r i n v e s t i g a t o r s had s u g g e s t e d v a l u e s o t h e r t h a n t h e 0.85 f o u n d e m p i r i c a l l y by Dean. K o h l e r and G a l v i n 78 (1973) recommended t h e v a l u e o f 0.70, w h i l e SPM(1977) s u g g e s t e d 1.0 t o 2.0. Even t h o u g h t h e use of t h i s d i m e n s i o n l e s s f a l l v e l o c i t y has p r o v e d t o be v e r y s u c c e s s f u l i n t h e p r e d i c t i o n o f t h e t y p e of p r o f i l e and t h e d i r e c t i o n o f s e d i m e n t movement, i t w i l l be shown l a t e r t h a t e q u a t i o n [5.1] c a n n o t be u s e d a l o n e t o p r e d i c t t h e d i r e c t i o n of s e d i m e n t movement. 5.3 E q u i l i b r i u m P r o f i l e An e q u i l i b r i u m p r o f i l e i s d e f i n e d by G o u r l a y (1980) as 'the p r o f i l e shape w h i c h when s u b j e c t e d t o a g i v e n wave c o n d i t i o n d i s s i p a t e s a n d / o r r e f l e c t s a l l t h e wave e n e r g y r e a c h i n g i t i n s u c h a manner t h a t no n e t t r a n s p o r t of t h e b e a c h b o t t o m s e d i m e n t o c c u r s anywhere a l o n g t h e p r o f i l e . ' A t r u e e q u i l i b r i u m p r o f i l e c a n n o t e x i t i n t h e f i e l d b e c a u s e of t h e c h a n g i n g wave c o n d i t i o n s . However, a 'dynamic' e q u i l i b r i u m c an be a c h i e v e d i f t h e t i m e c o n s i d e r a t i o n i s e x t e n d e d t o a c o m p l e t e c y c l e of t h e c h a n g i n g s e a s o n . In a l a b o r a t o r y , t h e t r u e s t a t e of e q u i l i b r i u m c an be a c h i e v e d under a c o n s t a n t , one d i r e c t i o n m o n o c h r o m a t i c wave a t t a c k . Even t h o u g h t h i s t r u e s t a t e o f e q u i l i b r i u m i s not a c h i e v a b l e i n t h e f i e l d , t h e c o n c e p t o f an e q u i l i b r i u m p r o f i l e i s e x t r e m e l y u s e f u l : i t w i l l e n a b l e us t o u n d e r s t a n d t h e r e s p o n s e of a p r o f i l e t o d i f f e r e n t wave c o n d i t i o n s . 79 5.4 Beach S l o p e S i n c e a t y p i c a l summer p r o f i l e i s g e n e r a l l y s t e e p e r t h a n a w i n t e r p r o f i l e , i t would t h e n be e x p e c t e d t h a t t h e s l o p e o f an e q u i l i b r i u m summer p r o f i l e w i l l a l s o be s t e e p e r t h a n a w i n t e r e q u i l i b r i u m p r o f i l e . T h i s v a r i a t i o n of e q u i l i b r i u m b e a c h s l o p e i s of i n t e r e s t b e c a u s e i t i s c l o s e l y l i n k e d t o t h e p r o f i l e c h a n g e . Among t h e many f a c t o r s w h i c h a f f e c t t h e e q u i l i b r i u m s l o p e , t h e more i m p o r t a n t ones a r e : a) Mean g r a i n s i z e , D50 b) Wave e n e r g y l e v e l , E c) Deep water.wave p e r i o d , l e n g t h and h e i g h t , T, L 0 and Ho d) F a l l v e l o c i t y , V f e) D e n s i t y o f s e d i m e n t , ^ . U s i n g d i m e n s i o n a l a n a l y s i s , t h e e q u i l i b r i u m s l o p e s h o u l d t h e n be a f u n c t i o n of "_Llo , _Dso vLfs Vfjr " _ L 0 He ' E H 0 ' Ho Most of t h e r e l a t i o n s h i p s r e l a t i n g t h e above v a r i a b l e s and t h e s l o p e were shown by W i e g e l (1964) and K i n g ( 1 9 7 2 ) . An e x p e r i m e n t was c o n d u c t e d t o a f f i r m some of t h e the r e l a t i o n s h i p s and a l s o t o f i n d some b a s i s f o r n u m e r i c a l m o d e l l i n g i n t h e n e x t c h a p t e r . The d e s c r i p t i o n s and t h e r e s u l t s o f t h e e x p e r i m e n t a r e g i v e n i n t h e n e x t s e c t i o n . 80 5.5 E x p e r i m e n t a l d e s i g n and p r o c e d u r e s The o b j e c t i v e s of t h e e x p e r i m e n t a r e t o s t u d y : 1) t h e two t y p e s of b e a c h p r o f i l e 2) t h e r a t e of change from one e q u i l i b r i u m p r o f i l e t o a n o t h e r due t o c h a n g e s i n wave c o n d i t i o n s . A model beach was s e t up i n a flume 28m l o n g 0.6m wide and 0.7m deep. The waves were g e n e r a t e d by a wave p a d d l e c a p a b l e of v a r y i n g t h e wave h e i g h t and t h e wave p e r i o d . ' F r a s e r r i v e r ' s a n d was u s e d as t h e b e a c h s e d i m e n t . The g r a i n s i z e d i s t r i b u t i o n of t h e sand i s shown i n Ap p e n d i x F. The s e d i m e n t was l a i d o v e r a r i g i d wooden b a s e , s l o p i n g a t 1 i n 10, w i t h an a v e r a g e t h i c k n e s s of about 200mm. The i n s t r u m e n t s s e t up f o r t h e e x p e r i m e n t were as f o l l o w s : 1. A p r o f i l e r , t h a t r e a d s t o t h e n e a r e s t 0.001 f e e t , was mounted on a t r a v e l l i n g c a r r i a g e t h a t can be moved a l o n g t h e l o n g i t u d i n a l l e n g t h of t h e f l u m e . 2. Two wave p r o b e s - one f i x e d a t a d e s i g n a t e d l o c a t i o n and t h e o t h e r mounted on t h e same t r a v e l l i n g c a r r i a g e as t h e p r o f i l e r . 3. A two-pen p a p e r r e c o r d e r f o r r e c o r d i n g t h e o u t p u t from t h e wave p r o b e s . 4. An o s c i l l o s c o p e f o r v i s u a l l y c h e c k i n g t h e o u t p u t f r o m t h e wave p r o b e s . -i'r © PLAN VIEW 28m 0 moving carriage e O p r o f i l e r wave f i l ter wave generator oscilloscope VIEW o o 1 I J J o s ^ o paper recorder NOTE Profile depth taken at each X. The interval between each X is 50mm. F i g u r e 5.1 O v e r a l l e x p e r i m e n t a l s e t - u p 82 The o v e r a l l s e t up o f t h e e x p e r i m e n t i s shown i n f i g u r e 5.1. T h r e e t e s t were c o n d u c t e d , e a c h w i t h a d i f f e r e n t i n i t i a l p r o f i l e and wave c o n d i t i o n . The d i f f e r e n t wave c o n d i t i o n s and t h e s c h e d u l e f o r d a t a r e c o r d i n g a r e shown i n t a b l e 5.1. E a c h o f t h e t e s t s was p e r f o r m e d u s i n g t h e same p r o c e d u r e as d e s c r i b e d below. S i n c e the main o b j e c t i v e i s t o s t u d y t h e c h a n g e s between two e q u i l i b r i u m p r o f i l e s , i t i s e s s e n t i a l t h a t e a c h i n i t i a l p r o f i l e i s an e q u i l i b r i u m p r o f i l e . T h e r e f o r e b e f o r e s t a r t i n g t e s t 1, a c e r t a i n wave c o n d i t i o n was s e l e c t e d , and th e b e a c h was s u b j e c t e d t o t h i s i n i t i a l wave a c t i o n f o r a c o n t i n u o u s p e r i o d of a b o u t 24 h o u r s u n t i l no s i g n i f i c a n t c h a n g e s i n t h e p r o f i l e were d e t e c t e d . The r e s u l t i n g p r o f i l e was t h e n d e s i g n a t e d as t h e i n i t i a l e q u i l i b r i u m p r o f i l e f o r t e s t 1. The wave c o n d i t o n f o r t e s t 1 was t h e n s e t and t e s t 1 began. At s c h e d u l e d t i m e i n t e r v a l s , t h e f o l l o w i n g d a t a were r e c o r d e d : a) The p r o f i l e of t h e b e a c h from t h e b a c k s h o r e berm t o th e t o e of t h e s l o p e . b) The e n v e l o p e of t h e s t a n d i n g waves c a u s e d by t h e i n t e r a c t i o n between i n c i d e n t waves and waves r e f l e c t e d f r o m the model b e a c h . c) The i n c i d e n t wave l e n g t h , p e r i o d , h e i g h t and water d e p t h . TEST IMPOSED WAVE CONDITION D i m e n s i o n l e s s * F a l l V e l o c i t y H o / V f T Data r e c o r d i n g i n t e r v a l s Remarks WAVE HEIGHT H(mm) WAVE PERIOD T ( s ) I n i f i a l 4 0 1.5 0 .5312 ( S w e l l p r o f i l e ) 1 58 1.12 1.0227 (Storm p r o f i l e ) 0 - 8 h r . @ 2 h r . 8 - 2 4 h r . @ 4 h r . T o t a l of 9 p r o f i l e s r e c o r d e d 2 30 1.26 0.4743 ( S w e l l p r o f i l e ) 0 - 8 h r . @ 4 h r . T o t a l of 3 p r o f i l e s r e c o r d e d 3 30 2 .11 0.2832 (Storm p r o f i l e ) 0 - 1 2 hr @ 4 h r . T o t a l of 4 p r o f i l e s r e c o r d e d * The f a l l v e l o c i t y of D S o f o r t h i s experiment i s a p p r o x i m a t e l y 50 mm per second. CD Table 5.1 Imposed wave c o n d i t i o n and sc h e d u l e f o r d a t a c o l l e c t i o n 84 At t h e end of t e s t 1, t h e l a s t p r o f i l e of t h e t e s t became t h e i n i t i a l e q u i l i b r i u m p r o f i l e f o r t h e n e x t t e s t . A l l t h e p r o c e d u r e s were t h e n r e p e a t e d . 5.6 R e s u l t s  P r o f i l e c h a n g e s Of t h e n i n e p r o f i l e s r e c o r d e d i n t e s t 1, f o u r a r e shown i n f i g u r e s 5.2 and 5.3. F i g u r e 5.2 shows t h e p r o f i l e of t h e model b e a c h a t t h e b e g i n n i n g and a t t h e end o f t h e t e s t . I t can be seen t h a t a l a r g e amount of s e d i m e n t has moved o f f s h o r e , as e x p e c t e d . T a b l e 5.1 a l r e a d y showed t h a t t h e p r o f i l e w i l l change from s w e l l t o s t o r m b e c a u s e of t h e c hange i n t h e d i m e n s i o n l e s s f a l l v e l o c i t y . Most p a r t of t h i s movement o c c u r r e d e a r l y i n t h e t e s t . F i g u r e 5.3 shows t h e change f o r t h e f i r s t f o u r h o u r s a t two-hour i n t e r v a l s . The f i n a l l o c a t i o n f o r t h e d e p o s i t i o n c a n n o t be d e t e c t e d a t t h e s e e a r l y s t a g e s ; o n l y l a t e r was i t f o u n d t o o c c u r a t a r o u n d X = 48.0 ( s e e f i g u r e 5 . 2 ) . O t h e r f e a t u r e s o f t h e p r o f i l e c h a n g e, l i k e t h e e r o s i o n of t h e b e ach f a c e , and t h e r e t r e a t of t h e s h o r e l i n e , can a l s o be d e t e c t e d f r o m t h e f i g u r e s . T e s t 2 i s e s s e n t i a l l y t h e r e v e r s e o f t e s t 1. The wave c o n d i t i o n s were changed from a s t o r m t o a s w e l l . The s l o p e of t h e b e a c h was e x p e c t e d t o b u i l d up back t o w a r d s i t s o r i g i n a l c o n d i t i o n , and from f i g u r e 5.4 i n g e n e r a l i t F i g u r e 5.2 P r o f i l e c h a n g e s - T e s t 1 ( I n i t i a l and f i n a l p r o f i l e s ) 86 F i g u r e 5.3 P r o f i l e c h a n g e s - T e s t 1 (At 2 h o u r s i n t e r v a l ) F i g u r e 5.4 P r o f i l e c h a n g e s - T e s t 2 F i g u r e 5.5 P r o f i l e c h a n g e s - T e s t 3 89 c l e a r l y d i d . However, t h e p r o c e s s i s not c o m p l e t e l y r e v e r s i b l e b e c a u s e some s e d i m e n t i s moved o f f s h o r e and s m a l l e r waves c a n n o t move i t o n s h o r e a g a i n . ' In t h e p r e s e n t t e s t , a s t e p l i k e f e a t u r e had formed a t X = 70.0 and i t was n o t i c e d t h a t t h e r e was l i t t l e s e d i m e n t movement o f f s h o r e a t X = 60.0. T h i s t e s t was s t o p p e d a f t e r o n l y 8 h o u r s b e c a u s e of i n s i g n i f i c a n t c h a n g e s i n t h e p r o f i l e . The wave c o n d i t i o n s f o r t e s t 3 were s i m i l a r t o t e s t 2 e x c e p t f o r a l o n g e r wave p e r i o d . The r e s u l t s of t h e t e s t - ( f i g u r e 5.5) show s i m i l a r i t y t o t h o s e o f t e s t 2. The be a c h s l o p e became s t e e p e r and t h e s t e p became more p r o m i n e n t . T h e r e was a l s o a c o n s i d e r a b l e s h o r e w a r d movement of s e d i m e n t a t X = 48.0. T h i s i s a t t r i b u t e d t o t h e l o n g e r wave p e r i o d w h i c h a f f e c t s d e e p e r d e p t h s and r e s u l t s i n a w i d e r zone of s e d i m e n t movement. From t h e t h r e e t e s t s , i t can be seen t h a t beach s e d i m e n t c a n move e i t h e r o n s h o r e o r o f f s h o r e d e p e n d i n g on t h e wave c o n d i t i o n but n o t i n a c o m p l e t e l y r e v e r s i b l e manner. Sediment can sometimes be moved so f a r o f f s h o r e by d e s t r u c t i v e waves t h a t even an e x t e n d e d s w e l l c o n d i t i o n c a n n o t b r i n g them back t o w a r d s t h e s h o r e . When t h i s h appens, a permanent r e c e s s i o n of t h e s h o r e l i n e w i l l r e s u l t . T h e r e f o r e i t i s e x p e c t e d t h a t t h e r a t e of s h o r e l i n e r e c o v e r y a f t e r an e x t e n s i v e s t o r m w i l l be slow and i n c o m p l e t e . The r e s u l t s of t e s t 1 a r e us e d t o f i n d t h e r a t e of 90 change of p r o f i l e . F i g u r e 5.6 shows t h e c u m m u l a t i v e c h a n g e s of t h e p r o f i l e ( i n terms of t o t a l n e t volume t r a n s p o r t e d ) a g a i n s t t i m e . The f i g u r e i n d i c a t e s t h a t t h e r a t e of change i s d e c a y i n g w i t h t i m e . So t h a t c h a n g e s i n t h e p r o f i l e a r e most d e t e c t a b l e d u r i n g t h e e a r l y s t a g e s . S c o t t (1954) o b t a i n e d a s i m i l a r r e s u l t and s u g g e s t e d a l o g a r i t h m i c f i t f o r t h e d a t a . F i g u r e 5.6 shows t h i s l o g a r i t h m i c f i t [5 ;2] Y = a + b l n X as t y p e I, t h e c o e f f i c i e n t s 'a' and 'b' of w h i c h a r e shown i n t a b l e 5.2. U n f o r t u n a t e l y t h i s c u r v e g i v e s n e g a t i v e v a l u e s f o r t o t a l volume t r a n s p o r t e d when t i m e ' t ' i s l e s s t h a n 1 hour and t h i s i s p h y s i c a l l y i m p o s s i b l e . A d i f f e r e n t a p p r o a c h f o r f i t t i n g t h e d a t a i s p r o p o s e d h e r e . A s s u m i n g t h e change o f t h e t r a n s p o r t r a t e a t a s p e c i f i c t i m e i s p r o p o r t i o n a l t o t h e t r a n s p o r t r a t e a t t h a t t i m e , t h e f o l l o w i n g e q u a t i o n c an be w r i t t e n , [ 5 . 3 ] H M =- P C T r ) at where T r i s t h e t r a n s p o r t r a t e and P a c o n s t a n t . The n e g a t i v e s i g n s i g n i f i e s t h a t t h e r a t e of change i s d e c r e a s i n g w i t h t i m e . I n t e g r a t i n g e q u a t i o n [ 5 . 3 ] , [ 5 . 4 ] Tr = e < k + p t > 0 CO op UJ S > i—i CD ~7 0 .AT _^— $ i ZD . 1 O m q z: UJ 0 ZD _J O CM O > O o 6 T Y P E 3 1 r-0.0 3.2 T Y P E II ^ T Y P E - 1 1 1 1 1 1 1 6.4- 9.6 12.-8 16.0 TIME (HR) -1 1 1 — 19.2 22.4-F i g u r e 5.6 Cummulative c h a n g e s o f t h e p r o f i l e a g a i n s t t i m e 92 S i n c e t h e t r a n s p o r t r a t e i n t e g r a t e t o g i v e t h e volume t r a n s p o r t o v e r t i m e , e q u a t i o n [5.4] c a n be r e - w r i t t e n as - P t W [ 5 . 5 ] Tr = - r r - = a e r i , where a = eK t h e r e f o r e a _p + [5.6] Q = b - " p e where Q i s t h e volume t r a n s p o r t e d ; a and b a r e t h e i n t e g r a t i n g c o n s t a n t s . To f i t e q u a t i o n [5.6] by a n o n l i n e a r l e a s t s q u a r e method i s d i f f i c u l t . However, u s i n g a computer p r o g r a m d e v e l o p e d by G o l u b e t a l ( 1 9 7 2 ) , t h e c . o e f f i c i e n t s o f a d i f f e r e n t f u n c t i o n , b u t one t h a t i s c l o s e t o e q u a t i o n [5.6] c a n be o b t a i n e d , ( t y p e I I i n t a b l e 5.2 and f i g u r e 5 . 6 ) . From t a b l e 5.2, t h e c o e f f i c i e n t s a and b a r e so c l o s e t h a t t h e f u n c t i o n can be r e d u c e d t o [5.7] f (x) = a1 +( a., +'a 3)e" b x .which i s e s s e n t i a l l y t h e form we want. The c o e f f i c i e n t s of e q u a t i o n [5.7] a r e o b t a i n e d u s i n g a s i m i l a r c u r v e f i t t i n g p r o c e d u r e as a b o v e , and shown i n t a b l e 5.2. The f i t o f t h i s e q u a t i o n , shown as t y p e I I I i n f i g u r e 5.2, i s good. 93 L o o k i n g a g a i n a t e q u a t i o n [ 5 . 6 ] and by i m p o s i n g t h e i n i t i a l c o n d i t i o n of z e r o t r a n s p o r t a t t i m e t = 0, a a „ . p | [ 5 . 8 ] Q = - - ? e From t h i s e q u a t i o n , i t can be d e d u c e d t h a t when t i m e ' t ' a p p r o a c h e s i n f i n i t y ( i . e . a p p r o a c h i n g e q u i l i b r i u m ) , t h e f i n a l t o t a l volume t r a n s p o r t e d i s a [ 5 . 9 ] Q F = p S u b s t i t u t i n g t h i s i n t o e q u a t i o n [ 5 . 8 ] , we o b t a i n a - P + [ 5 . 1 0 ] Q F - Q = — £ The l e f t hand s i d e of t h e e q u a t i o n r e p r e s e n t s t h e d i f f e r e n c e between t h e f i n a l volume and t h e e x i s t i n g volume, By d i f f e r e n t i a t i n g e q u a t i o n [ 5 . 1 0 ] we o b t a i n : [ 5 . 1 1 ] ^ ( Q r ~ Q ) = P ( Q p - Q ) By i n t u i t i v e r e a s o n i n g , i t i s e x p e c t e d t h a t w i t h an i n c r e a s e i n t h e wave a c t i v i t y and hence t h e wave " f l u x , t h e amount o f s e d i m e n t t r a n s p o r t e d w i l l c o r r e s p o n d i n g l y i n c r e a s e . T h i s s u g g e s t s a r e l a t i o n s h i p between Qp and t h e wave f l u x : T Y P E FUNCTION COEFF IC IENTS I f (x) = a + b In x a = -0.0125 b = 0.0395 11 f i x ) = a 1 + a 2 e " b i x + a 3 e ~ b 2 > ( a 1 = 0.1314 a 2 = - 2 1 . 241 a 3 = 21 . 212 b, = 0.09493 b 2 = 0.09498 III f ( x ) = a! + a 2 e ~ D i x a! = 0.1313 a 2 =-0.1310 DT = 0 .086 T a b l e 5.2 E q u a t i o n s f o r t h e r a t e of p r o f i l e c h ange VO 6' 100. 80. 6Q 4Q 2Q -• | , , .. j 1 1 —r " • i 1— A A A * o Normol o * 0 • Bor — Profi lM A A -* " A o ^ ^ S ^ = • --* O A X J J A a *>. • * > v Nayak V • o O • Rtclor (1954) x Author's A Eagloon, OI*nn« and Dracup (1963) O Nayak (1970) -• Raman and Earattupuitia (1973) * van Hljum (1974) • Thompson (1976) t _ • 1 i i I I i i i i i .04 . 06 .08 .10 .2 .4 6 8 1.0 Ho 6. & 10. VO F i g u r e 5.7 ( A f t e r D a l r y m p l e , 1976) E q u i l i b r i u m s l o p e a g a i n s t Ho/vf T 96 [5 .12] Wave e n e r g y f l u x Qp = — The e x a c t r e l a t i o n s h i p between t h e s e q u a n t i t i e s c a n n o t be d e t e r m i n e d . b e c a u s e i n s u f f i c i e n t d a t a i s a v a i l a b l e f r o m t h e p r e s e n t s e t o f e x p e r i m e n t s . F o r a g i v e n wave c o n d i t i o n , t h e f i n a l v a l u e of t h e s l o p e i s a l s o t h e e q u i l i b r i u m s l o p e , and as shown e a r l i e r t h i s e q u i l i b r i u m s l o p e depends on s e v e r a l p a r a m e t e r s . F i g u r e 5.7 shows a p l o t by D a l r y m p l e e t a l (1976) r e l a t i n g one of t h e s e p a r a m e t e r , HQ/V^T, t o t h e e q u i l i b r i u m s l o p e . I n c l u d e d i n t h e f i g u r e a r e f o u r d a t a p o i n t s f r o m t h i s e x p e r i m e n t . T h e s e p o i n t s show good agreement w i t h t h e c u r v e s u g g e s t e d by D a l r y m p l e . T h i s c a n be seen as s u p p o r t t o D a l r y m p l e ' s f i n d i n g t h a t t h e s l o p e i s u n i q u e l y r e l a t e d t o th e d i m e n s i o n l e s s f a l l v e l o c i t y . T h e r e f o r e , i f t h e se d i m e n t p r o p e r t i e s and t h e wave c o n d i t i o n a r e known, t h e v a l u e of th e e q u i l i b r i u m s l o p e c a n be e s t i m a t e d . O t h e r t h a n t h i s , t h e c u r v e by D a l r y m p l e e t a l a l s o p r o v i d e s a new way of p r e d i c t i n g t h e d i r e c t i o n of s e d i m e n t movement, a s s u g g e s t e d by Q u i c k (1983) . I t i s w i d e l y q u o t e d by many i n v e s t i g a t o r s t h a t e q u a t i o n [5 .1 ] can p r e d i c t w h ether t h e t r a n s p o r t i s on s h o r e o r o f f s h o r e (Komar, 1975; S o r e n s e n , 1978). However, by e x a m i n i n g f i g u r e 5.7 and t h e e x p e r i m e n t a l r e s u l t s , i t i s o b v i o u s t h a t t h e d i r e c t i o n of s e d i m e n t movement depends not o n l y on t h e f i n a l wave c o n d i t i o n b u t a l s o on t h e i n i t i a l wave c o n d i t i o n . F o r example, i f t h e d i m e n s i o n l e s s f a l l 97 v e l o c i t y c h a n g e s from a h i g h e r v a l u e t o a l o w e r v a l u e , t h e s l o p e of t h e b e a c h i s l i k e l y t o become s t e e p e r , i n d i c a t i n g a s h o r e w a r d movement of s e d i m e n t and v i c e v e r s a . From t h i s , i t i s a p p a r e n t t h a t a p r e d i c t i o n of t h e d i r e c t i o n of s e d i m e n t movement i s more a p p r o p r i a t e l y b a s e d on t h e f o l l o w i n g r e l a t i o n s h i p : f > 1 for offshore [5.13] (A) 1 /(-A) 1 T ' V f / T • -Vf T h i s new a p p r o a c h removes some of t h e p r o b l e m s e n c o u n t e r e d when u s i n g H 0 / V f T a l o n e , b e c a u s e a g r e a t r a n g e of v a l u e s of H 0 / V f T have been q u o t e d by d i f f e r e n t r e s e a r c h e r s . I t can a l s o be seen from e q u a t i o n [5.13] t h a t i t i s p o s s i b l e - f o r t h e d i m e n s i o n l e s s f a l l v e l o c i t y H 0/V^T t o be g r e a t e r t h a n 0.85 or 2.0 (as i n d i c a t e d by e q u a t i o n [ 5 . 1 ] ) and f o r t h e o n s h o r e movement of s e d i m e n t t o o c c u r a t t h e same t i m e . As s u c h , t h e use o f H 0 / V f T a l o n e i s p o t e n t i a l l y m i s l e a d i n g , u n l e s s t h e p r o f i l e a t i n i t i a l c o n d i t i o n i s a t e q u i l i b r i u m , where ( H 0 / V f T ) o f e q u a t i o n [5.13] i s 1.0. T h e r e f o r e e q u a t i o n [5.2] r e p r e s e n t s j u s t a s p e c i a l c a s e o f e q u a t i o n [ 5 . 1 3 ] , and e q u a t i o n [5.13] i s more u s e f u l as i t can p r e d i c t t h e d i r e c t i o n of s e d i m e n t movement i n t h e p r o f i l e when t h e wave c o n d i t i o n c h a n g e s from one non-e q u i l i b r i u m s t a t e t o a n o t h e r . ) < 1 ' for onshore ( - 1 equil ibrium 98 CHAPTER 6 ON-OFFSHORE TRANSPORT MODEL 6 . 1 I n t r o d u c t i o n The b e h a v i o u r of o n - o f f s h o r e t r a n s p o r t and t h e r e s u l t i n g b e a c h p r o f i l e have been s t u d i e d by many i n v e s t i g a t o r s (Nayak, 1970; Swart, 1976; Dean, 1977). In t h i s c h a p t e r some of t h e s e s t u d i e s a r e d i s c u s s e d , and on t h e b a s i s of some of t h e s e i d e a s a model i s d e v e l o p e d w h i c h p r e d i c t s t h e c h a n g e s of t h e b e a c h p r o f i l e . Thus, u s i n g t h i s m odel, p r e d i c t i o n s of t h e s h o r e l i n e r e t r e a t due t o v a r i o u s wave c o n d i t i o n s c a n be made. T h i s model w i l l be i n c o r p o r a t e d i n t o t h e l o n g s h o r e t r a n s p o r t model p r e s e n t e d i n c h a p t e r f o u r and t h e r e s u l t s o f t h i s combined model w i l l a l s o be p r e s e n t e d . 6.2 Model O u t l i n e s and A s s u m p t i o n s The model i s b a s e d on t h e f o l l o w i n g a s s u m p t i o n s : 1. The e q u i l i b r i u m b e a c h s l o p e can be d e f i n e d i f t h e wave c o n d i t i o n i s g i v e n . As d e s c r i b e d i n c h a p t e r f i v e , u s i n g t h e c u r v e by D a l r y m p l e e t a l (1976) shown i n f i g u r e 5.8, t h e e q u i l i b r i u m b e a c h s l o p e f o r any wave c o n d i t i o n c a n be f o u n d . 99 2. T h e r e i s a p o i n t on t h e e q u i l i b r i u m p r o f i l e where t h e r e i s no n e t a c c u m m u l a t i o n o r e r o s i o n of s e d i m e n t . Raman e t a l (1972) d e f i n e d t h i s p o i n t a s a ' s t a b l e ' p o i n t w h i c h ' a c t s as a f u l c r u m a b o u t which t h e p r o f i l e s w i n g s w h i l e m a t e r i a l i s moved from one s i d e o f t h e p o i n t t o t h e o t h e r . ' Raman, however, d i d not make any a t t e m p t a t p r e d i c t i n g t h e l o c a t i o n of s u c h a p o i n t . H a l l e r m i e r (1978) a l s o c o n d u c t e d a s i m i l a r s t u d y . He p r o p o s e d t h a t t h e b e a c h be d i v i d e d i n t o two ( o f f s h o r e and l i t t o r a l ) z o n e s . The o f f s h o r e s h o r e i s c h a r a c t e r i z e d by r e l a t i v e l y m o d e r a t e bed a g i t a t i o n w h i l e t h e l i t t o r a l zone i s c h a r a c t e r i s e d by i n c r e a s e d bed s t r e s s e s and s e d i m e n t t r a n s p o r t . Then, by a s s u m i n g t h e h y p o t h e t i c a l b o u n d a r y between t h e s e two z o n e s t o be t h e l i m i t of i n t e n s e bed a g i t a t i o n , H a l l e r m i e r o b t a i n e d f i g u r e 6.1. The v a r i a b l e ' d c ' of f i g u r e 6.1 i s d e f i n e d by H a l l e r m i e r a s t h e maximum wat e r d e p t h f o r i n t e n s e bed a g i t a t i o n , and from l a b o r a t o r y t e s t s i t i s a l s o f o u n d t o be t h e l i m i t d e p t h t o t h e e r o s i v e a c t i o n o f t h e s u r f a c e waves. From t h e f i g u r e , i t c a n be seen t h a t t h e l o c a t i o n o f d c i s a l s o t h e ' s t a b l e ' p o i n t a s d e f i n e d by Raman. T h i s p o i n t , even t h o u g h d e f i n e d a s t h e p o i n t o f no n e t a c u m m u l a t i o n o r e r o s i o n , i s a l s o t h e p o i n t of maximum r a t e of t r a n s p o r t . T h i s c a n be shown by c o n s i d e r i n g t h e k i n e m a t i c s of t h e s e d i m e n t t r a n s p o r t e d as t h e p r o f i l e c h a n g e s . 3. T h e r e e x i s t s a f i x e d e q u i l i b r i u m b e a c h p r o f i l e 100 shape c o r r e s p o n d i n g t o a g i v e n i n t e n s i t y of wave a t t a c k . U s i n g t h e r a d i a t i o n s t r e s s t h e o r y and- a s s u m i n g u n i f o r m e n e r g y d i s s i p a t i o n a c r o s s t h e s u r f z one, Dean (1977.) a r r i v e d a t an e x p r e s s i o n , a power c u r v e , f o r t h e e q u i l i b r i u m b e a ch p r o f i l e : [6.1] h = A x m where h i s t h e water d e p t h from s t i l l w a t er l e v e l and x i s t h e s e a w a r d d i s t a n c e f r o m t h e i n t e r s e c t i o n of t h e s h o r e l i n e and t h e w a t e r l i n e . In Dean's t h e o r e t i c a l s t u d y t h e v a l u e of m was f o u n d t o be 2/3. B a s e d on t h e above work and a s i m i l a r i n v e s t i g a t i o n by Bruun ( 1 9 5 4 ) , Hughes e t a l (1978) f i t t e d o v e r 450 b e a c h p r o f i l e s t o t h e above e x p r e s s i o n and o b t a i n e d 101 t h e b e s t ' l e a s t - s q u a r e s ' f i t . The r e s u l t i n g v a l u e s of A and m were a n a l y s e d and t h e v a l u e s of m were f o u n d t o be n o r m a l l y d i s t r i b u t e d a b o u t an a v e r a g e v a l u e of 0.667. T h e i r r e s u l t s showed t h a t 53 p e r c e n t o f t h e v a l u e s l a y between 0.5 and 0.8. B e c a u s e of t h i s , t h e y c o n s i d e r e d t h a t m c o u l d be f i x e d a t 2/3. T h i s v a l u e of m i s a l s o s u p p o r t e d by t h e r e s u l t s of Sayao (1982) and t h e t h e o r e t i c a l work of Dean c i t e d e a r l i e r . On t h e o t h e r hand A was f o u n d t o - b e d i s t r i b u t e d o v e r a w i d e r r a n g e of v a l u e s . From f u r t h e r i n v e s t i g a t i o n s , Hughes e t a l s u g g e s t e d t h a t A i s p r i m a r i l y a f u n c t i o n of t h e g r a i n s i z e d i s t r i b u t i o n , so t h a t i t s v a l u e v a r i e s from c a s e t o c a s e . 4. The r a t e of p r o f i l e change i s assumed t o f o l l o w an e x p o n e n t i a l d e c a y . T h i s a s s u m p t i o n was c o n f i r m e d from t h e e x p e r i m e n t a l s t u d i e s d e s c r i b e d i n c h a p t e r f i v e . Such e x p o n e n t i a l b e h a v i o u r w i l l e x i s t whenever t h e r a t e of d e c a y i s d e p e n d e n t on t h e d i f f e r e n c e between t h e e x i s t i n g c o n d i t i o n and t h e f i n a l c o n d i t i o n . The p r o g r e s s o f s u c h p r o c e s s e s i s o f t e n d e s c r i b e d i n t e r m s of a h a l f - l i f e . The above a s s u m p t i o n s w i l l be c o m b i n e d i n t o a s i n g l e model, and t o do t h i s i t i s n e c e s s a r y t o examine t h e t y p i c a l c h a n g e s o f p r o f i l e ( f i g u r e 6.2) due t o an i n c r e a s e of wave a c t i v i t y . Under t h e a c t i o n of wave a t t a c k , b e a c h p r o f i l e c h a n g e s from p r o f i l e A t o p r o f i l e B, w h i c h a r e b o t h e q u i l i b r i u m p r o f i l e s f o r t h e i r r e s p e c t i v e wave c o n d i t i o n . R - Shoreline retreat S > S1 F i g u r e 6.3 M o d e l l i n g a s s u m p t i o n s 1 03 S i n c e e q u a t i o n [6.1] p r e d i c t s a f i x e d e q u i l i b r i u m p r o f i l e s h ape, i t i s assumed t h a t p r o f i l e A and p r o f i l e B a r e e s s e n t i a l l y d i f f e r e n t p a r t s of t h e same c u r v e . T h i s a s s u m p t i o n i s shown i n f i g u r e 6.3. Even t h o u g h t h i s a s s u m p t i o n i s a s i m p l i f i c a t i o n o f t h e a c t u a l p r o c e s s e s ( i t i g n o r e s o f f s h o r e b a r f o r m a t i o n ) , t h e c h a r a c t e r i s t i c c h a n g e s b a s e d on t h i s s i m p l i f i e d a p p r o a c h a r e s t i l l i n l i n e w i t h t h o s e o b s e r v e d i n t h e f i e l d . F o r example, t h e r e d u c t i o n of b e a c h s l o p e , t h e r e t r e a t of s h o r e l i n e and t h e f o r m a t i o n of a w i d e r s u r f z o n e . F i g u r e 6.2 shows t h e l o c a t i o n of d c as t h e i n t e r s e c t i o n p o i n t of t h e two e q u i l i b r i u m p r o f i l e s . T h i s i s i n a c c o r d a n c e w i t h t h e d e f i n i t i o n of a ' s t a b l e ' p o i n t . The v a l u e of d c as shown i s d e p e n d e n t on t h e imposed wave c o n d i t i o n , i . e . t h e wave c o n d i t i o n t h a t r e s u l t s i n p r o f i l e B. W i t h t h e above m o d e l l i n g c o n c e p t , a s i m p l e model ca n be formed, and t h e f o l l o w i n g s e c t i o n p r e s e n t s t h e model a l g o r i t h m s . L a t e r t h e model i s t e s t e d w i t h t h e r e s u l t s from t h e e x p e r i m e n t s of c h a p t e r f i v e . 104 6.3 A l g o r i t h m s o f t h e Model 6.3.1 C a l i b r a t i n g t h e i n i t i a l p r o f i l e As m e n t i o n e d e a r l i e r , Dean and Hughes e t a l had a r r i v e d a t an e x p r e s s i o n f o r t h e e q u i l i b r i u m p r o f i l e . G i v e n an i n i t i a l e q u i l i b r i u m p r o f i l e shown as i n f i g u r e 6.4, t h e v a r i a b l e s X, Y and A can be d e t e r m i n e d . The v a l u e of t h e d e p t h dc can be o b t a i n e d from t h e i n i t i a l wave c o n d i t i o n u s i n g t h e f o l l o w i n g f o r m u l a t i o n by H a l l e r m i e r (1978) : [6.2 ] ^ n h ^ T a n h ^ M + ^ f f ^ ) (329 ^ ) where H 0 and LQ a r e t h e wave h e i g h t and wave l e n g t h r e s p e c t i v e l y i n deep water and [6.3] S=(2TT-^) , f ! ' ? The v a r i a b l e L c i s t h e l o c a l wave l e n g t h . U s i n g l i n e a r wave t h e o r y , e q u a t i o n [6.3] can be w r i t t e n as [6.4] d c = ^ T a n h ^ {-}=*-) c y y 2 TT The v a r i a b l e , ^ , can be f o u n d by an i t e r a t i v e s o l u t i o n of e q u a t i o n [ 6 . 2 ] , and t h e v a l u e of d c t h e n o b t a i n e d from e q u a t i o n [ 6 . 4 ] . Once d c i s known, t h e v a l u e of X c can be measured from t h e g i v e n i n i t i a l p r o f i l e . 105 T h e r e a r e two ways of d e t e r m i n i n g t h e v a l u e of t h e i n i t i a l b e a c h p r o f i l e s l o p e , S. I t may be measured from t h e g i v e n i n i t i a l p r o f i l e and r e l a t e d t o t h e p r e c e e d i n g wave h i s t o r y . A l t e r n a t i v e l y , t h e c u r v e by D a l r y m p l e e t a l (1976) can be used, and i t i s t h e n n e c e s s a r y t o know t h e v a l u e s of t h e s e d i m e n t f a l l v e l o c i t y Vf , and t h e wave p e r i o d T, which a r e u s u a l l y known. In t h e p a p e r by D a l r y m p l e e t a l , no e q u a t i o n was g i v e n t o r e p r e s e n t t h e c u r v e , so i n t h e p r e s e n t s t u d y a s e t o f a p p r o x i m a t e e q u a t i o n s a r e d e r i v e d f r o m t h e c u r v e and a r e g i v e n h e r e , W H • 0.5928-0.6945(log-^-) r p f 4 0.1548 Vf I Vf i 0.9438- 0.0087(log—°) + [6 5] V 2 0 . 1 5 4 8 < r 7 ^ ^ 0 . 6 1 J 0.3343 ( l o g ) l V f T y V f T 0.9383 - 0.08 2«( log-^- ) 0.6 < Vf T Vf ] I t r e m a i n s t o f i n d t h e v a r i b l e s X, Y and A b e f o r e t h e i n i t i a l p r o f i l e c a n be u n i q u e l y d e f i n e d . I t must be n o t e d t h a t t h e r e s u l t i n g c u r v e must p a s s t h r o u g h d c and have a s l o p e of S a t t h e s t i l l w ater l e v e l ( f i g u r e 6 . 4 ) . U s i n g t h e s e f a c t s t h e f o l l o w i n g t h r e e e q u a t i o n s a r e o b t a i n e d : [ 6 . 6 ] S = _ A ( X) 3 3 F i g u r e 6.5 I n i t i a l and f i n a l p r o f i l e s 1 07 [6.7] Y = A ( x ) ? / 3 [ 6 . 8 ] d c +Y= A ( X c +x) 2/3 S o l v i n g t h e above t h r e e e q u a t i o n s s i m u l t a n e o u s l y w i l l y i e l d t h e unknown v a r i a b l e s X, Y and A. 6.3.2 F i n a l p r o f i l e F i g u r e 6.5 shows t h e change of t h e p r o f i l e from t h e i n i t i a l c o n d i t i o n t o t h e f i n a l c o n d i t i o n . To d e t e r m i n e t h e unknown v a r i a b l e s X 1 f Y, and X c , of t h e f i n a l p r o f i l e , t h e a p p r o a c h f o r c a l i b r a t i n g t h e i n i t i a l p r o f i l e c a n be u s e d , but w i t h a c h a n g e . I n s t e a d o f f i n d i n g t h e c o e f f i c i e n t A, t h i s c o e f f i c i e n t i s now known and X c , i s t h e new unknown v a r i a b l e . The v a l u e of d c , i s o b t a i n e d t h r o u g h t h e same p r o c e d u r e s as m e n t i o n e d b e f o r e , and t h e s l o p e S, must be o b t a i n e d from D a l r y m p l e ' s c u r v e u s i n g t h e new wave c o n d i t i o n . By s o l v i n g t h e same s e t o f e q u a t i o n s ( [ 6 . 6 ] t o [ 6 . 8 ] ) , t h e v a l u e o f X c , i s o b t a i n e d . Once X c 1 i s known, t h e v a l u e f o r t h e s h o r e l i n e r e t r e a t R can be d e t e r m i n e d . T h i s s h o r e l i n e r e t r e a t R i s t h e r e t r e a t from one e q u i l i b r i u m p r o f i l e t o a n o t h e r . A c c o r d i n g t o t h e e x p o n e n t i a l r a t e o f change of p r o f i l e , t h i s w i l l be a c h i e v e d o n l y a t an i n f i n i t e t i m e . T h e r e f o r e t o o b t a i n i n t e r m e d i a t e 108 v a l u e s of t h e s h o r e l i n e r e t r e a t , t h e f o l l o w i n g e x p r e s s i o n i s u s e d : [ 6.9] R ( t ) = [ 1 - e ~ b W ] R where R ^ j i s t h e s h o r e l i n e r e t r e a t a t t i m e t , and b i s t h e c o e f f i c i e n t o f d e c a y . 6.4 Model T e s t i n g The model was c h e c k e d u s i n g t h e r e s u l t s of t h e e x p e r i m e n t s d e s c r i b e d i n c h a p t e r f i v e . The model was f i r s t c a l i b r a t e d t o t h e i n i t i a l c o n d i t i o n o f t e s t 1, and t h e n u s e d t o p r e d i c t t h e r e t r e a t of t h e s h o r e l i n e when a new wave c o n d i t i o n was imposed. The n o t a t i o n of t h e v a r i a b l e s used h e r e w i l l be t h e same as t h a t shown i n f i g u r e 6.5. The r e s u l t s o f t h e c a l i b r a t i o n t o t h e i n i t i a l c o n d i t i o n of t e s t 1 a r e : A = 1.4282 X = 89.875 cm Y = 28.656 cm Thus t h e e q u a t i o n of t h e e q u i l i b r i u m p r o f i l e i s [6.10] h = 1.428 x 2 / 3 where h and x a r e b o t h i n c e n t i m e t r e s . U s i n g t h e g i v e n new wave c o n d i t i o n , t h e new 1 09 c r i t i c a l d e p t h 6r ^ and t h e new b e a c h s l o p e S, a r e c a l c u l a t e d . They a r e 11.44 cm and 0.1522 r e s p e c t i v e l y . U s i n g t h e s e v a l u e s , t h e c a l c u l a t i o n f o r Xj- , i s c a r r i e d out as o u t l i n e d i n t h e a l g o r i t h m s of t h e model. The v a l u e of XQ, i s f o u n d t o be 78.86 cm. From e q u a t i o n [6.10] t h e v a l u e of W i s 58.85 cm. T h e r e f o r e t h e r e t r e a t of t h e s h o r e l i n e R (= Xr_, - W) i s 20.01 cm. As n o t e d e a r l i e r , t h i s v a l u e of s h o r e l i n e r e t r e a t i s a c h i e v e d o n l y when t i m e a p p r o a c h e s i n f i n i t y . To o b t a i n t h e i n t e r m e d i a t e v a l u e s , e q u a t i o n [6.9] i s u s e d . T h e s e r e s u l t s a r e t h e n compared t o t h e r e s u l t s from t h e e x p e r i m e n t . F i g u r e 6.6 shows a c o m p a r i s o n of t h e p r o f i l e s f r o m t h e . e x p e r i m e n t and t h e p r o f i l e s from t h e m o d e l . T h e s e p r o f i l e s show t h a t t h e p r e d i c t i o n of t h e s h o r e l i n e r e t r e a t a f t e r 24 h o u r s a g r e e s w e l l w i t h t h a t of t h e e x p e r i m e n t . The model p r e d i c t e d a r e t r e a t of 17.5 cm.while t h e e x p e r i m e n t a l r e s u l t s gave 18.6 cm, an e r r o r of -6 p e r c e n t . However, t h e r e s u l t s o b t a i n e d from t h e model a r e not t o o s a t i s f a c t o r y when t h e v a r i a t i o n of t h e s h o r e l i n e r e t r e a t w i t h t i m e i s c o n s i d e r e d ( s e e f i g u r e 6.7), In view o f t h e p o s s i b l e d a t a s c a t t e r f r o m t h e e x p e r i m e n t , t h e e r r o r between t h e e x p e r i m e n t and t h e model may not be as g r e a t as shown. From t h e c o m p a r i s o n , i t can be seen t h a t t h e model g i v e s r e a s o n a b l e p r e d i c t i o n s . The model manages t o p r e d i c t t h e t r e n d of t h e p r o f i l e c h a n g e s . I t a l s o shows t h a t t h e v a r i o u s d i f f e r e n t a s p e c t s o f p r o f i l e b e h a v i o u r can be Shoreline r e t r e a t U- *4 F i g u r e 6.6 Comparison between measured p r o f i l e s and model's r e s u l t s SHOREL INE -p~ ON CP RETREAT (cm) C a> cr < (D rr »-« « fD OJ fD rr 3 o (D 3 X T J O fD l-tl •1 M • 01 3 (D o 3 rr fD i—• OJ 3 3 OJ fD 3 f-l O fD QJ rr fD >-t \-> (D - OJ rr 1 fD 01 rr C tr i — • rr rr cn 3 fD I n o 3 T J OJ «. t-«* 0) o 3 3 I I I 1 12 b r o u g h t t o g e t h e r and r e l a t e d t o one a n o t h e r . T h i s s h o u l d be e s p e c i a l l y n o t e d b e c a u s e t h e s l o p e , t h e c r i t i c a l d e p t h and t h e p r o f i l e a r e d e t e r m i n e d i n ways wh i c h a r e d i f f e r e n t and i n d e p e n d e n t o f e a c h o t h e r . I t t h e r e f o r e a p p e a r s t h a t t h i s s i m p l e model works q u i t e w e l l and makes a c c e p t a b l e p r e d i c t i o n s o f s h o r e l i n e and p r o f i l e c h a n g e s . 1 1 3 6.5 C o m b i n i n g t h e L o n g s h o r e and O n - o f f s h o r e T r a n s p o r t R o u t i n e s In t h i s s e c t i o n , t h e p r o f i l e change r o u t i n e i s merged w i t h t h e l o n g s h o r e t r a n s p o r t r o u t i n e . T h i s combined model w i l l be u s e d t o s i m u l a t e f o u r s i m p l e f i e l d s i t u a t i o n s . F o r e a c h s i m u l a t i o n , t h e m o d e l l i n g c o n d i t i o n s w h i c h have t o be s p e c i f i e d a r e , t h e m o d e l l i n g p a r a m e t e r s , t h e i n i t i a l wave c o n d i t i o n and t h e f i n a l wave c o n d i t i o n . The combined model i s then u s e d t o p r e d i c t t h e c o r r e s p o n d i n g a d j u s t m e n t of t h e beach p r o f i l e t o t h e s e d i f f e r e n t wave c o n d i t i o n s . The f o u r s i m u l a t i o n e x a m ples a r e b r i e f l y d e s c r i b e d h e r e . 1. As an i n i t i a l t e s t t o c h e c k t h a t t h e o n - o f f s h o r e r o u t i n e has been c o r r e c t l y i n c o r p o r a t e d , t h e a n g l e of wave a t t a c k i s s e t e q u a l t o z e r o and no l i t t o r a l b a r r i e r i s i n c l u d e d . The i n i t i a l wave h e i g h t i n t h i s c a s e i s s p e c i f i e d t o be lo w e r t h a n t h e f i n a l wave h e i g h t . E s s e n t i a l l y what i s b e i n g m o d e l l e d h e r e i s t h e e f f e c t of d i r e c t wave a t t a c k on a b e a c h . The s h o r e l i n e c h a n g e s t h a t o c c u r s h o u l d be due o n l y t o p r o f i l e c h a n g e s w h i c h r e s u l t e d f r o m o n - o f f s h o r e t r a n s p o r t . F i g u r e 6.8 shows t h e r e s u l t s of t h e s h o r e l i n e r e t r e a t . 2. The p u r p o s e of t h i s example i s t o compare t h e r e s u l t s o b t a i n e d by t h e combined and t h e p u r e l o n g s h o r e t r a n s p o r t m o del. The r e s u l t of t h e p u r e l o n g s h o r e t r a n s p o r t model i s 1 1 4 shown as c u r v e 2 i n f i g u r e 6.9. The i n i t i a l and f i n a l wave c o n d i t i o n s f o r t h e combined model a r e s p e c i f i e d t o be a p p r o x i m a t e l y t h e same as t h a t f o r t h e p u r e l o n g s h o r e t r a n s p o r t m o d e l . The r e s u l t of t h i s c ombined model i s shown as c u r v e 3 i n f i g u r e 6.9. From t h e f i g u r e , i t c a n be seen t h a t t h e i n c l u s i o n of t h e o n - o f f s h o r e r o u t i n e r e s u l t e d i n a s i g n i f i c a n t l y d i f f e r e n t f i n a l s h o r e l i n e . C u r v e 2 ( p u r e l o n g s h o r e model) shows t h a t t h e r e i s o n l y a c c r e t i o n of t h e s h o r e l i n e , nowhere a l o n g t h e s h o r e l i n e can any s h o r e l i n e r e c e s s i o n be f o u n d , b e c a u s e t h e r e i s no o n - o f f s h o r e t r a n s p o r t . T h e r e f o r e , even when a wave c o n d i t i o n e x i s t s w h i c h w i l l r e s u l t i n a change of p r o f i l e and a s h o r e l i n e r e t r e a t , t h e p u r e l o n g s h o r e t r a n s p o r t model c a n n o t p r o d u c e t h i s e f f e c t , and t h e i n c l u s i o n of t h e o n - o f f s h o r e component i s e s s e n t i a l ( c u r v e 3 ) . The d i f f e r e n c e i n t h e two s h o r e l i n e s , c u r v e 2 and 3 i s e s p e c i a l l y n o t i c e a b l e c l o s e r t o t h e b a r r i e r . The r e a s o n f o r t h i s i s b e c a u s e t h e p r o f i l e s n e a r t h e b a r r i e r a r e s t e e p e r , as a r e s u l t of t h e a c c m m u l a t i o n of s e d i m e n t from t h e l o n g s h o r e component of t h e m o d e l . T h e r e f o r e , when t h e o n - o f f s h o r e component i s a l s o c o n s i d e r e d , t h e o n - o f f t r a n s p o r t c h a n g e s i n t h e s e r e g i o n s w i l l be a d d i t i o n a l l y m a g n i f i e d b e c a u s e of t h i s s t e e p e r i n i t i a l p r o f i l e , r e s u l t i n g i n a g r e a t e r f i n a l s h o r e l i n e d i f f e r e n c e as shown i n f i g u r e 6.9. 3. The t h i r d example i s t o s i m u l a t e t h e e f f e c t of an i n c r e a s e of i n c i d e n t wave h e i g h t on t h e b e a c h s h o r e l i n e . 1 1 5 The i n c r e a s e of wave h e i g h t i s kept s m a l l , as t h e i n t e n t i o n h e r e i s t o s i m u l a t e a h e a v i e r sea c o n d i t i o n b u t n o t as extr e m e as f o r s t o r m waves. The r e s u l t i n g change of t h e s h o r e l i n e i s shown as c u r v e 4 i n f i g u r e 6.9. By c o m p a r i n g t h i s c u r v e w i t h c u r v e 3 i n t h e same f i g u r e , i t c a n be seen t h a t more s h o r e l i n e e r o s i o n t a k e s p l a c e a t some r e g i o n s w h i l e more a c c u m u l a t i o n a t o t h e r s . T h i s shows t h a t a s l i g h t i n c r e a s e of t h e wave h e i g h t d o e s n o t n e c e s s a r i l y mean t h a t a d e s t r u c t i v e e f f e c t on t h e s h o r e l i n e w i l l o c c u r when a b a r r i e r i s p r e s e n t . 4. T h i s l a s t example i s i n t e n d e d t o s i m u l a t e t h e e f f e c t of s t o r m waves on an e x i s t i n g a c c u m u l a t i o n of s e d i m e n t . The wave h e i g h t i s i n c r e a s e d by 75 p e r c e n t of t h e i n i t i a l wave h e i g h t . The r e s u l t s show some i n t e r e s t i n g f e a t u r e s ( f i g u r e 6.10). The l e n g t h of s h o r e l i n e where e r o s i o n o c c u r s i n c r e a s e s as compared t o c u r v e 4 o f f i g u r e 6.9. The amount of a c c u m u l a t i o n i s s m a l l and when compared w i t h e r o s i o n a t th e o t h e r r e g i o n s , t h e net e f f e c t on t h e s h o r e l i n e c an be seen as d e s t r u c t i v e . T h i s , i n c o m p a r i s o n w i t h example 3, shows t h a t t h e o v e r a l l e f f e c t on t h e s h o r e l i n e i s d e t e r m i n e d not o n l y by t h e i n c r e a s e of wave h e i g h t , but a l s o by t h e amount of s u c h an i n c r e a s e . Wave a t t a c k P l an view of r e t r e a t i n g shore l ine Time (day) 0.0 0.2 0.4 0.6 0.8 1.0 — i 1 1 1 1 1— 0.0 6.0 1-2.0 ]3.0 S H O R E T 1 2 4-.0 3 0 . 0 ( X 0.5m) F i g u r e 6.8 Beach p l a n - R e t r e a t of s h o r e l i n e due t o wave a p p r o a c h i n g b each o r t h o g o n a l l y o C u r v e 1 - I n i t i a l s h o r e l i n e . C u r v e 2 - F i n a l s h o r e l i n e when o n l y t h e l o n g s h o r e t r a n s p o r t component i s m o d e l l e d . C u r v e 3 - F i n a l s h o r e l i n e when b o t h t h e l o n g s h o r e and o n - o f f s h o r e components a r e m o d e l l e d . C u r v e 4 - F i n a l s h o r e l i n e w i t h c o n d i t i o n s s i m i l a r t o t h a t f o r c u r v e 3, e x c e p t w i t h an i n c r e a s e of i n c i d e n t wave h e i g h t . B a r r i e r 30.0 ( X 0.5m) F i g u r e 6.9 Beach P l a n - Combined o n - o f f s h o r e and l o n g s h o r e s i m u l a t i o n s . C u r v e 1 - I n i t i a l s h o r e l i n e . C u r v e 2 - F i n a l s h o r e l i n e . Eros ion B a r r i e r Accretion o.o i 6.0 i 12.0 1«.0 S H O R E N o t e t h e net d e s t r u c t i v e e f f e c t o f t h e s t o r m wave. 1 1 1 -2.4-.0 3O.0 ( X 0.5m ) F i g u r e 6.10 Beach p l a n - Storm waves s i m u l a t i o n CO 119 6.6 D i s c u s s i o n From t h e above s i m u l a t i o n s , s e v e r a l g e n e r a l o b s e r v a t i o n s can be drawn. I t has been seen t h a t whether t h e l o n g s h o r e o r t h e o n - o f f s h o r e component of t r a n s p o r t i s d ominant depends on t h e s i t u a t i o n . From t h e r e s u l t s of example 1, i t c a n be seen t h a t i f no b a r r i e r i s p r e s e n t t o o b s t r u c t t h e l o n g s h o r e f l o w of s e d i m e n t , t h e n an i n c r e a s e i n wave h e i g h t w i l l o n l y p r o d u c e a s h o r e l i n e r e t r e a t w h i c h i s due t o t h e a d j u s t m e n t of p r o f i l e r e s u l t i n g from t h e on-o f f s h o r e t r a n s p o r t . However, i f a b a r r i e r i s i n t r o d u c e d as i n example 3, t h e r e s u l t s show a c o m b i n a t i o n o f two e f f e c t s . At r e g i o n s f a r away from t h e b a r r i e r where t h e l o n g s h o r e f l o w of s e d i m e n t r e m a i n s r e l a t i v e l y unhampered, t h e r e s u l t i n g e f f e c t i s t h e same as t h o s e of example 1 ( i e an e x p o n e n t i a l r e t r e a t of t h e s h o r e l i n e ) . But a t r e g i o n s n e a r e r t h e b a r r i e r where t h e l o n g s h o r e movement of s e d i m e n t i s r e s t r i c t e d 7 t h e l o n g s h o r e a c c u m u l a t i o n e f f e c t becomes d o m i n a n t . T h i s i s shown by t h e a c c r e t i o n of t h e s h o r e l i n e . In t h e e v e n t o f a l a r g e s t o r m , t h e l a r g e r s t o r m waves w i t h l o n g e r wave p e r i o d and h i g h e r wave h e i g h t w i l l b e g i n t o r e f r a c t f u r t h e r o f f - s h o r e . When t h e s e waves r e a c h t h e s h o r e and b r e a k , t h e wave o r t h o g o n a l s w i l l t e n d t o be more n o r m a l t o t h e b e a c h . T h i s means t h e a n g l e of wave a t t a c k i s v e r y s m a l l and most of t h e e n e r g y w i l l be e x p ended i n c a u s i n g o n - o f f s h o r e t r a n s p o r t r a t h e r t h a n l o n g s h o r e t r a n s p o r t . T h e s e e f f e c t s a r e seen i n t h e r e s u l t s of example 1 20 4 where the dominant t r a n s p o r t i s o n - o f f s h o r e , c a u s i n g a n e t r e t r e a t of s h o r e l i n e . L a s t l y , t h e r e s u l t s o f t h e l o n g s h o r e model ( c h a p t e r 4) and t h e combined model show t h e f o l l o w i n g d i f f e r e n c e s . 1. When u s i n g o n l y t h e l o n g s h o r e t r a n s p o r t model, t h e r e i s no s h o r e l i n e r e t r e a t anywhere a l o n g t h e s h o r e . T h i s i s b e c a u s e th e l o n g s h o r e model does not a c c o u n t f o r s e d i m e n t l o s s t h r o u g h o n - o f f s h o r e movement. 2. W i t h t h e i n c l u s i o n of t h e o n - o f f s h o r e r o u t i n e , an a c c m u l a t i o n b e h i n d t h e b a r r i e r c a n s t i l l be o b s e r v e d . But t h e amount of t h i s a c c m u l a t i o n i s l e s s t h a n t h a t o b t a i n e d from t h e l o n g s h o r e t r a n s p o r t m o d e l . T h e r e f o r e t h e i n c l u s i o n o f t h e o n - o f f s h o r e r o u t i n e r e d u c e s t h e a c c r e t i o n e f f e c t of t h e l o n g s h o r e t r a n s p o r t model. I t can t h e r e f o r e be c o n c l u d e d ' t h a t t o have a c o m p l e t e s i m u l a t i o n of c o a s t a l s e d i m e n t movement, b o t h t h e l o n g s h o r e and t h e o n - o f f s h o r e r o u t i n e s w i l l have t o be t a k e n i n t o c o n s i d e r a t i o n . 121 CHAPTER 7 SUMMARY AND CONCLUSIONS T h i s r e p o r t has p r e s e n t e d t h e s t e p by s t e p d e v e l o p m e n t of a s h o r e l i n e p r e d i c t i o n m odel. The f o l l o w i n g i s a summary of t h i s d e v e l o p m e n t . A b r i e f r e v i e w of t h e l o n g s h o r e c u r r e n t and t r a n s p o r t e q u a t i o n was c o n d u c t e d b e f o r e t h e f o r m u l a t i o n of t h e m o d e l . In t h e r e v i e w , i t was n o t e d t h a t b o t h t h e r a d i a t i o n a p p r o a c h by L o n g u e t - H i g g i n s and t h e s e m i - e m p i r i c a l a p p r o a c h by Komar y i e l d e d s i m i l a r e q u a t i o n s . The e q u a t i o n by L o n g u e t - H i g g i n s i s b a s e d on t h e c o n s e r v a t i o n o f momentum of t h e waves. T h i s a p p r o a c h i s b e l i e v e d t o be t h e most s u i t a b l e as momentum i s c o n s e r v e d when waves b r e a k , whereas t h e e n e r g y i s n o t . B o t h t h e e q u a t i o n s were shown t o be c o n s i s t e n t w i t h e x p e r i m e n t a l d a t a but t h e e q u a t i o n by Komar r e q u i r e s t h e c o n s t a n c y of t h e r a t i o t a n ^ / C f . Even t h o u g h s u p p o r t i n g d a t a were p r e s e n t e d , t h i s c o n s t a n c y i s s t i l l u n e x p l a i n e d . From t h e r e v i e w on t h e t r a n s p o r t r a t e e q u a t i o n , i t was shown t h a t t h e r e i s no ' b e s t ' p r e d i c t i v e e q u a t i o n a v a i l a b l e . I t was f o u n d t h a t most o f t h e 'wave f l u x ' f o r m u l a t i o n s have s i m i l a r a p p r o a c h e s y i e l d i n g a g e n e r a l e q u a t i o n of t h e f o r m : 1 22 [ v . U Si = A pf However, t h e r e i s no s u c h s i m i l a r i t y i n t h e ' s e d i m e n t e q u a t i o n ' a p p r o a c h e s . Even tho u g h p r e d i c t i o n s by t h e s e ' s e d i m e n t e q u a t i o n ' were shown t o be i n agreement w i t h t h e d a t a , t h e r e i s s t i l l an a p p a r e n t l a c k of agreement on t h e mode of t h e l o n g s h o r e t r a n s p o r t . In g e n e r a l , t h e a p p l i c a t i o n o f 'wave f l u x ' e q u a t i o n s a r e e a s i e r and r e q u i r e l e s s i n p u t s t h a n t h e ' s e d i m e n t e q u a t i o n . ' A f t e r t h e r e v i e w , t h e CERC t r a n s p o r t r a t e e q u a t i o n was s e l e c t e d f o r use i n t h e l o n g s h o r e t r a n s p o r t component of t h e m o d e l . A f i n i t e d i f f e r e n c e scheme was a d o p t e d t o s o l v e t h e wave r e f r a c t i o n and s h o a l i n g e q u a t i o n s as w e l l as t h e t r a n s p o r t and c o n t i n u i t y e q u a t i o n s . The model was u s e d t o s t u d y t h e a c c r e t i o n b e h a v i o u r due t o an i n f i n i t e l e n g t h and a f i n i t e l e n g t h b a r r i e r . In a d d i t i o n , t h e model was a l s o u s e d t o p r e d i c t n o u r i s h m e n t b e a c h p l a n c h a n g e s w i t h t i m e . An e x p e r i m e n t a l s t u d y on t h e o n - o f f s h o r e was p r e s e n t e d a f t e r t h e l o n g s h o r e model. The p u r p o s e of t h e s t u d y was t o h i g h l i g h t some of t h e key a s p e c t s of b e h a v i o u r i n t h e p r o f i l e c h a n g e s and t o f i n d t h e r a t e o f change of t h e p r o f i l e s . E q u a t i o n s were d e v e l o p e d w h i c h d e f i n e a c h a r a c t e r i s t i c d e p t h , b e a c h s l o p e and shape f o r a g i v e n s e t of wave p a r a m e t e r s . The model b a s e d on t h e s e e q u a t i o n s was t h e n t e s t e d a g a i n s t t h e e x p e r i m e n t a l r e s u l t s . L a s t l y , t h i s o n - o f f s h o r e component model was c o m b i n e d i n t o t h e l o n g s h o r e 123 n u m e r i c a l model and t h e combined r e s u l t s a r e p r e s e n t e d . The f o l l o w i n g were o b s e r v e d from t h e v a r i o u s m o d e l l i n g r e s u l t s and t h e e x p e r i m e n t a l s t u d y : a) I t was shown t h a t t h e i n c l u s i o n of wave s h o a l i n g and r e f r a c t i o n ( o r wave d e f o r m a t i o n ) r o u t i n e had a d r a m a t i c e f f e c t on t h e t r a n s p o r t and t h e r e s u l t i n g s h o r e l i n e c h a n g e s . ( s e e f i g u r e s 4.14 and 4.16). The r e a s o n f o r t h i s was f o u n d t o be t h e n o n l i n e a r e f f e c t s w h i c h t h i s wave d e f o r m a t i o n r o u t i n e i n t r o d u c e d i n t o t h e r e s u l t s . T h i s i s c o n c l u d e d from t h e c o m p a r i s i o n o f t h e m o d e l l i n g r e s u l t s and t h e r e s u l t s from W a l t o n ' s p a p e r on s i m p l i f i e d a n a l y t i c a l s o l u t i o n . In W a l t o n ' s model, t h e g o v e r n i n g e q u a t i o n s were l i n e a r i s e d f o r ea s e of s o l u t i o n . The r e s u l t s o b t a i n e d were f o u n d t o be v e r y s i m i l a r t o t h o s e o f t h e model w i t h o u t t h e wave d e f o r m a t i o n r o u t i n e , s u g g e s t i n g t h a t t h e n o n l i n e a r e f f e c t s of t h e model a r e due t o t h e wave s h o a l i n g and r e f r a c t i o n r o u t i n e . b) The r e s u l t s of t h e model a l s o i n d i c a t e d t h a t a c c r e t i o n b e h a v i o u r due t o a b a r r i e r can be non-d i m e n s i o n a l i s e d t o form a s e t o f d i m e n s i o n l e s s a c c r e t i o n c u r v e s f o r a g i v e n wave d i r e c t i o n . T h e s e c u r v e s p r o v e d t o be u s e f u l i n t h e e s t i m a t i o n of t h e b a r r i e r l e n g t h r e q u i r e d t o a c h i e v e a d e s i r e d a c c r e t i o n a t a s p e c i f i c l o c a t i o n . T h e s e n o n - d i m e n s i o n a l i s e d c u r v e s a l s o showed t h a t a d e s i r e d a c c r e t i o n can be a c h i e v e d e i t h e r w i t h a l o n g e r l e n g t h b a r r i e r a t a s h o r t e r t i m e or a s h o r t e r l e n g t h b a r r i e r a t a 124 l o n g e r t i m e . Hence, t h e r e i s not a s i n g l e s o l u t i o n , but s e v e r a l , and t h e most s u i t a b l e s o l u t i o n w i l l have t o depend on a d d i t i o n a l f a c t o r s , s u c h as c o s t . c) From t h e e x p e r i m e n t a l s t u d y of o n - o f f s h o r e b e h a v i o u r , i t was seen t h a t t h e b e a c h p r o f i l e w i l l change as a r e s u l t of a change i n wave a c t i v i t y . I f t h e wave h e i g h t i s i n c r e a s e d , t h e b e a c h s l o p e w i l l r e d u c e and t h e b e a c h s h o r e l i n e w i l l r e t r e a t . The r a t e of t h i s change was f o u n d t o be d e c a y i n g e x p o n e n t i a l l y w i t h t i m e , i m p l y i n g t h a t most of t h e e x p e c t e d c h a n g e s w i l l o c c u r i n t h e e a r l y s t a g e s . I t was a l s o shown t h a t t h e r e i s a r e l a t i o n s h i p between t h e wave e n e r g y f l u x of t h e i n c o m i n g wave and t h e t o t a l amount of s e d i m e n t t r a n s p o r t e d . However, due t o i n s u f f i c i e n t d a t a , no d e f i n i t e r e l a t i o n s h i p was g i v e n . d) The e x p e r i m e n t a l r e s u l t s a l s o s u p p o r t e d D a l r y m p l e e t a l ' s f i n d i n g s on t h e u n i q u e r e l a t i o n s h i p between t h e d i m e n s i o n l e s s f a l l p a r a m e t e r H 0 / V f T and t h e e q u i l i b r i u m b e a c h s l o p e . T h i s u n i q u e r e l a t i o n s h i p s u g g e s t e d t h a t t h e d i r e c t i o n of s e d i m e n t movement i n t h e o n - o f f s h o r e d i r e c t i o n i s not p r e d i c t e d by u s i n g o n l y t h e v a l u e of t h e d i m e n s i o n l e s s f a l l p a r a m e t e r d e r i v e d from t h e imposed wave c o n d i t i o n a s s u g g e s t e d by Dean and o t h e r s . D a l r y m p l e e t a l ' s f i n d i n g s showed t h a t f o r e a c h wave c o n d i t i o n , t h e r e i s a u n i q u e e q u i l i b r i u m b e a c h s l o p e a s s o c i a t e d w i t h i t . As s u c h , when wave c o n d i t i o n s change from one s t a t e t o a n o t h e r , t h e d i r e c t i o n of s e d i m e n t 1 25 movement due t o t h i s wave c o n d i t i o n change w i l l have t o depend on t h e r e l a t i v e change of t h e a s s o c i a t e d e q u i l i b r i u m b e a c h s l o p e . I t was s u g g e s t e d by Q u i c k (1983) t h a t t h e more a p p r o p r i a t e e x p r e s s i o n i n d e t e r m i n i n g t h e d i r e c t i o n of s e d i m e n t movement would be t h e r a t i o of t h e i n i t i a l t o t h e f i n a l v a l u e o f t h e d i m e n s i o n l e s s f a l l p a r a m e t e r (shown as e q u a t i o n [ 5 . 1 3 ] ) . T h i s r e l a t i o n s h i p i s s u p p o r t e d by t h e e x p e r i m e n t a l r e s u l t s . e) From t h e r e s u l t s of t h e comb i n e d model, i t was seen t h a t t h e r e i s a d e f i n i t e i n t e r a c t i o n between l o n g s h o r e and o n - o f f s h o r e t r a n s p o r t . The dominant t r a n s p o r t component c a n e i t h e r be l o n g s h o r e o r o n - o f f s h o r e d e p e n d i n g on t h e wave h e i g h t and d i r e c t i o n . P r e d i c t i o n of s h o r e l i n e c h a n g e s w i l l be i n a c c u r a t e u n l e s s b o t h t h e components a r e i n c o r p o r a t e d i n t o t h e model, b e c a u s e t h e o n - o f f s h o r e t r a n s p o r t c a n p r o d u c e l a r g e p r o f i l e c h a n g e s w h i c h t h e p u r e l o n g s h o r e model c a n n o t r e p r o d u c e . T h i s r e p o r t has u s e d t h e l i t t o r a l b a r r i e r example q u i t e e x t e n s i v e l y b e c a u s e d a t a and o t h e r s t u d i e s e x i s t f o r t h i s s i t u a t i o n . Whether a l i t t o r a l b a r r i e r i s an a p p r o p r i a t e s o l u t i o n f o r an e r o d i n g b e a c h depends on t h e n a t u r e and t h e e x t e n t o f t h e p r o b l e m , not t o m e n t i o n t h e economic and e n v i r o m e n t a l c o n s t r a i n t s . The t o t a l c o n s e q u e n c e s of i n s t a l l i n g a l i t t o r a l b a r r i e r must be examined w i t h c a r e , e s p e c i a l l y t h e downstream c o n s e q u e n c e s . In n e a r l y e v e r y i n s t a n c e , l i t t o r a l b a r r i e r s a r e d e s i g n e d t o 1 26 i n t e r r u p t t h e l o n g s h o r e movement o f s e d i m e n t and t h e r e f o r e s t a r v i n g a d j a c e n t d o w n d r i f t b e a c h e s of i t s s u p p l y , and c a u s i n g s h o r e l i n e r e c e s s i o n t o o c c u r . Though t h i s r e p o r t a l s o showed t h a t a b e a c h i s e r o d e d by s t o r m waves and may l a t e r be r e s t o r e d by s w e l l waves, t h i s r e c o v e r y of o f f s h o r e s e d i m e n t i s not a l w a y s c o m p l e t e . E r o s i o n and a c c r e t i o n p a t t e r n s may o c c u r s e a s o n a l l y , but t h e l o n g range c o n d i t i o n o f t h e b e a c h -whether e r o d i n g , s t a b l e o r a c c r e t i n g w i l l depend on t h e n e t movement o f m a t e r i a l . I t i s b e l i e v e d t h a t t h e r a t e o f b e a c h r e s t o r a t i o n d u r i n g s w e l l c o n d i t i o n i s much s l o w e r t h a n b each e r o s i o n d u r i n g s t o r m c o n d i t o n . T h e r e f o r e one l a r g e s t o r m can o f f s e t many s e a s o n s o f s w e l l r e s t o r a t i o n . L a s t l y , i t must be m e n t i o n e d t h a t t h e r e a r e s e v e r a l f a c t o r s w h i c h have not been c o n s i d e r e d i n t h e model. They a r e wave d i f f r a c t i o n , wave s e t u p , wave runup and t i d e a c t i o n . Wave d i f f r a c t i o n i s due t o t h e d i s t r i b u t i o n of e n e r g y a l o n g t h e c r e s t of t h e wave and i s dominant i n t h e shadow of a b a r r i e r . S i n c e t h e model i n t h i s r e p o r t i s u s e d t o s i m u l a t e t h e e f f e c t u p s t r e a m o f a b a r r i e r , t h e wave d i f f r a c t i o n e f f e c t i s assumed t o be i n s i g n i f i c a n t . As f o r wave runup, i t i s f e l t t h a t l i t t l e i s known ab o u t i t s e f f e c t on s e d i m e n t t r a n s p o r t . In most of t h e wave runup s t u d i e s , t h e e x p e r i m e n t s were c o n d u c t e d on s t e e p , p l a n e and impermeable s l o p e s , so t h a t a p p l i c a t i o n o f t h e r e s u l t s on sand b e a c h e s i s i n a p p r o p r i a t e . W h i l e f o r wave s e t u p , even 127 t h o u g h t h e r e a r e t h e o r i e s t o a c c o u n t f o r many of t h e p r i n c i p a l p r o c e s s e s , t h e s e t h e o r i e s c o n t a i n f a c t o r s t h a t a r e o f t e n d i f f i c u l t t o s p e c i f y i n p r a c t i c a l p r o b l e m s . F o r example, wave s e t u p i n a s i m p l i f i e d p l a n e b e a c h c a n be e s t i m a t e d u s i n g r a d i a t i o n s t r e s s t h e o r y ( L o n g u e t - H i g g i n s and S t e w a r t , 1964), b ut f o r a non p l a n e b e a c h , t h e r e i s s t i l l no s o l u t i o n e x c e p t u s i n g e s t i m a t i o n t h a t r e q u i r e s c o n s i d e r a b l e judgement. As t o t h e t i d e a c t i o n , t h e i n c l u s i o n o f i t i n the model r e q u i r e s a more c o m p l i c a t e d a l g o r i t h m s b e c a u s e t h e t i d e l e v e l i n t h e model w i l l have t o v a r y w i t h t i m e . In o r d e r t o s i m p l i f y t h e p r o b l e m , a model u s i n g o n l y one c o n s t a n t t i d e l e v e l i s a d o p t e d . T h i s s i m p l i f i c a t i o n w i l l n ot c a u s e a s i g n i f i c a n t change i n t h e r e s u l t s i f i t i s assumed t h a t t h e s e d i m e n t t r a n s p o r t due t o wave a c t i o n i s s i g n i f i c a n t l y g r e a t e r t h a n t h a t due t o t i d e a c t i o n . S i n c e t h i s c o n s t a n t t i d e model can be u s e d a t any t i d e l e v e l , t h e model can be u s e d t o s i m u l a t e t h e most d e s t r u c t i v e s i t u a t i o n f o r a b e a c h , i n w h i c h t h e d e s t r u c t i v e wave a t t a c k o c c u r s a t th e same t i m e a s t h e h i g h e s t t i d e . The r e s u l t s o b t a i n e d w i l l be c o n s e r v a t i v e compared t o t h o s e of a v a r i a b l e t i d e m odel. T h i s i s b e c a u s e as t h e t i d e l e v e l d r o p s a f t e r r e a c h i n g t h e h i g h e s t l e v e l ( i n a v a r i a b l e m o d e l ) , t h e d e s t r u c t i v e e f f e c t of t h e waves w i l l n o t be a b l e t o r e a c h t h e same d i s t a n c e i n s h o r e as d u r i n g t h e h i g h e s t t i d e l e v e l . T h e r e f o r e , t o keep t h e o v e r a l l model s i m p l e , i t i s assumed t h a t t h e above n e g l e c t e d f a c t o r s a r e i n s i g n i f i c a n t compared t o t h o s e a l r e a d y c o n s i d e r e d i n t h e mo d e l . 1 28 BIBLIOGRAPHY A c k e r s , P., and W h i t e , W. R. ( 1 9 7 3 ) . Sediment t r a n s p o r t : new a p p r o a c h and a n a l y s i s . J o u r n a l of t h e H y d r a u l i c s D i v i s i o n , ASCE, HY11, pp. 2041 - 2060. B a g n o l d , R. A. (1 9 6 3 ) . , M e c h a n i c s of m a r i n e s e d i m e n t a t i o n . In 'The Sea', e d i t e d by M. N. H i l l , V o l . 3, pp. 507 - 528. 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D e p a r t m e n t of C i v i l E n g i n e e r i n g , U n i v e r s i t y of D e l a w a r e , Newark, D e l a w a r e . F l e m i n g , C. A., and Hunt, J . N. ( 1 9 7 6 ) . A p p l i c a t i o n of a s e d i m e n t t r a n s p o r t model. P r o c e e d i n g s , 15th C o a s t a l E n g i n e e r i n g C o n f e r e n c e , pp. 1184 - 1202. G a l v i n , C. J . ( 1 9 6 7 ) . L o n s h o r e c u r r e n t v e l o c i t y : a r e v i e w of t h e o r y and d a t a . R e v i e w s of G e o p h y s i c s , V o l . 5, No. 3, pp. 287 - 304. G a l v i n , C. J . ( 1 9 7 2 ) . A g r o s s l o n g s h o r e t r a n s p o r t r a t e f o r m u l a . P r o c e e d i n g s , 13th C o a s t a l E n g i n e e r i n g C o n f e r e n c e , pp. 953 - 970. G a l v i n , C. J . , and E a g l e s o n , P. E. ( 1 9 6 5 ) . E x p e r i m e n t a l s t u d y o f l o n g s h o r e c u r r e n t s on a p l a n e b e a c h . U. S. Army C o a s t a l E n g i n e e r i n g R e s e a r c h C e n t e r . T e c h n i c a l Memo. No. 10, pp. 1 - 80. G o l u b , G. H., and P e r e y r a , V. ( 1 9 7 2 ) . The d i f f e r e n t i a t i o n of p s e u d o i n v e r s e s and n o n l i n e a r l e a s t s q u a r e s p r o b l e m s whose v a r i a b l e s s e p a r a t e , STAN-CS-72-261, Computer S c i e n c e D e p a r t m e n t , S t a n f o r d U n i v e r s i t y . G o u r l a y , M. R. ( 1 9 8 0 ) . B e a c h e s : P r o f i l e s , p r o c e s s e s and p e r m e a b i l i t y . P r o c e e d i n g s , 17th C o a s t a l E n g i n e e r i n g C o n f e r e n c e , pp. 1320 - 1339. H a l l e r m e i e r , R. J . ( 1 9 7 8 ) . U s e s f o r a c a l c u l a t e d l i m i t d e p t h t o b e a c h e r o s i o n . P r o c e e d i n g s , 16th C o a s t a l E n g i n e e r i n g C o n f e r e n c e , pp. 1493 - 1512. 130 H a r r i s o n , W., and W i l s o n , W. S. ( 1 9 6 4 ) . Development o f a method f o r n u m e r i c a l c a l c u l a t i o n o f wave r e f r a c t i o n . U. S. Army C o a s t a l E n g i n e e r i n g R e s e a r c h C e n t e r , T e c h n i c a l Memo. No. 6. H i n o , M. ( 1 9 7 5 ) . T h e o r y on f o r m a t i o n o f r i p c u r r e n t and c u s p i d a l c o a s t . P r o c e e d i n g s , 14th C o a s t a l E n g i n e e r i n g C o n f e r e n c e , pp. 901 - 919. Hughes, S. A., and C h i u , T. Y. ( 1 9 7 8 ) . The v a r i a t i o n s i n b e a c h p r o f i l e s when a p p r o x i m a t e d by a t h e o r e c t i c a l c u r v e . R e p o r t TR-039, C o a s t a l and O c e a n o g r a p h i c E n g i n e e r i n g L a b o r a t o r y , U n i v e r s i t y of F l o r i d a , G a i n s v i l l e , F l o r i d a . J o h n s o n , J . W. ( 1 9 4 9 ) . S c a l e e f f e c t s i n h y d r a u l i c models i n v o l v i n g wave m o t i o n . T r a n s a c t i o n s of t h e A m e r i c a n G e o p h y s i c a l U n i o n . V o l . 30, No. 4, pp. 517 - 525. J o h n s o n , J . W., O ' B r i e n , M. P., and I s a a c s , J . D. ( 1 9 4 8 ) . G r a p h i c a l c o n s t r u c t i o n of wave r e f r a c t i o n d i a g r a m s , H y d r o g r a p h i c O f f i c e , Navy D e p a r t m e n t , P u b l i c a t i o n No. 605. K i n g , C. A. M. ( 1 9 7 2 ) . B e a c h e s and C o a s t s . 2nd e d i t i o n , S t M a r t i n ' s P r e s s , NY. K i n g , C. A. M. , and W i l l i a m s , W. W. (1949).' The f o r m a t i o n and movement of sand b a r s by wave a c t i o n . J o u r n a l o f G e o l o g y , V o l . 113, pp. 70 - 85. K o h l e r , R. R., and G a l v i n , C. J . ( 1 9 7 3 ) . Berm - b a r c r i t e r i o n . U n p u b l i s h e d memorandom f o r t h e r e c o r d , U. S. Army C o a s t a l E n g i n e e r i n g R e s e a r c h C e n t e r , W a s h i n g t o n . Komar, P. D. ( 1 9 7 1 ) . The m e c h a n i c s of sand t r a n s p o r t on b e a c h e s . J o u r n a l of G e o p h y s i c a l R e s e a r c h , V o l . 76, No. 3, pp. 713 - 721. Komar, P. D. ( 1 9 7 3 ) . Computer models of d e l t a g r o w t h due t o s e d i m e n t i n p u t from r i v e r s and l o n g s h o r e t r a n s p o r t . G e o l o g i c a l S o c i e t y of A m e r i c a , B u l l e t i n , V o l . 84, pp. 2643 -2650. Komar, P. D. ( 1 9 7 6 ) . Beach p r o c e s s e s and s e d i m e n t a t i o n . P r e n t i c e - H a l l , Englewood C l i f f s , N J. Komar, P. D., and Inman, D. L. ( 1 9 7 0 ) . L o n g s h o r e s a n d t r a n s p o r t on b e a c h e s . J o u r n a l o f G e o p h y s i c a l R e s e a r c h , V o l . 75, No. 30, pp. 5914 - 5927. Le Mehaute, M. ( 1 9 7 6 ) . An i n t r o d u c t i o n t o h y d r o d y n a m i c s and water waves. S p r i n g e r - V e r l a g , NY. 131 L o n g u e t - H i g g i n s , M. S. ( 1 9 7 0 a ) . L o n g s h o r e c u r r e n t s g e n e r a t e d by o b l i q u e l y i n c i d e n t s e a waves, 1, V o l . 75, No. 33, pp. 6778 - 6789. L o n g u e t - H i g g i n s , M. S. ( 1 9 7 0 b ) . L o n g s h o r e c u r r e n t s g e n e r a t e d by o b l i q u e l y i n c i d e n t sea waves, 2, V o l . 75, No. 33, pp. 6790 - 6801 . L o n g u e t - H i g g i n s , M. S. ( 1 9 7 2 ) . R e c e n t p r o g r e s s i n t h e s t u d y of l o n g s h o r e c u r r e n t s . In 'Waves on b e a c h e s ' , e d i t e d by R. E. Meyer, pp. 203 - 248. Academic P r e s s , NY. L o n g u e t - H i g g i n s , M. S., and S t e w a r t , R. W. ( 1 9 6 4 ) . R a d i a t i o n s t r e s s e 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 R e s e a r c h , V o l . 11, pp. 529 -562. Meyer, R. E. ( 1 9 6 9 ) . Note on wave run - u p . J o u r n a l of G e o p h y s i c a l R e s e a r c h , V o l . 75, pp. 687 - 690. Nayak, I . V. ( 1 9 7 1 ) . E q u i l i b r i u m p r o f i l e s of model b e a c h e s . P r o c e e d i n g s , 12th C o a s t a l E n g i n e e r i n g C o n f e r e n c e , pp. 1321 -1 339. N i e l s o n , P., S v e n d s e n , I . A., and S t a u b , C. ( 1 9 7 8 ) . O n s h o r e - o f f s h o r e s e d i m e n t movement on a b e a c h . P r o c e e d i n g s , 16th C o a s t a l E n g i n e e r i n g C o n f e r e n c e , pp. 1475 - 1492. Noda, E. V., Sonu, C. J . , R u p e r t , V. C , and C o l l i n s , J . I . ( 1 9 7 4 ) . N e a r s h o r e c i r c u l a t i o n s under s e a b r e e z e c o n d i t i o n s and w a v e - c u r r e n t i n t e r a c t i o n s i n t h e s u r f z o n e . T e t r a T e c h , I n c . , R e p o r t TC-149-4. P e l n a r d - C o n s i d e r e , R. ( 1 9 5 6 ) . E s s a i de T h e o r i e de l ' E v o l u t i o n des Formes de R i v a g e en P l a g e s de S a b l e e t de G a l e t s . 4 t h J o u r n e e de 1 ' H y d r a u l i q u e , L e s E n e r g i e s de l a Mer, Q u e s t i o n I I I , R a p p o r t No. 1. P h i l l i p s , 0. M. ( 1 9 8 0 ) . The d y n a m i c s of t h e upper o c e a n . Cambridge U n i v e r s i t y P r e s s , C a m b r i d g e . P r a n d t l , L. ( 1 9 5 2 ) . E s s e n t i a l s o f f l u i d d y n a m i c s . H a f f n e r , New Y o r k , NY. P r i c e , W. A., T o m l i n s o n , K. W., and W i l l i s , D. H. ( 1 9 7 2 ) . P r e d i c t i n g c h a n g e s i n t h e p l a n shape of b e a c h e s . P r o c e e d i n g s , 13th C o a s t a l E n g i n e e r i n g C o n f e r e n c e , pp. 1321 -1 329. 1 32 Putnam, J . A., Munk, W. H., and T r a y l o r , M. A. ( 1 9 4 9 ) . The p r e d i c t i o n o f l o n g s h o r e c u r r e n t s . T r a n s a c t i o n of t h e A m e r i c a n G e o p h y s i c s U n i o n , V o l . 30, pp. 337 - 345. Q u i c k , M. C. ( 1 9 8 3 ) . P e r s o n a l D i s c u s s i o n . Raman, H., and E a r a t t u p u z h a , J . J . ( 1 9 7 2 ) . E q u i l i b r i u m c o n d i t i o n s i n be a c h wave i n t e r a c t i o n . P r o c e e d i n g s , 13th C o a s t a l E n g i n e e r i n g C o n f e r e n c e , pp. 1237 - 1256. R e c t o r , R. L. ( 1 9 5 4 ) . L a b o r a t o r y s t u d y of t h e e q u i l i b r i u m p r o f i l e s of b e a c h e s . U.- S. Army C o r p s of E n g i n e e r s , Beach E r o s i o n B o a r d , T e c h n i c a l Memo. No. 41. S a v i l l e , T. ( 1 9 5 7 ) . S c a l e e f f e c t s i n two d i m e n s i o n a l b e a c h s t u d i e s . T r a n s a c t i o n s o f t h e 7 t h m e e t i n g of t h e A s s o c i a t i o n of H y d r a u l i c R e s e a r c h , L i s b o n , P o r t u g a l , pp. A3-1 - A3-9. Sayao, 0. S. F. M., and Kamphuis, J . W. ( 1 9 8 2 ) . Wave a c t i o n on b e a c h e s . CE R e s e a r c h R e p o r t No. 77. Department of C i v i l E n g i n e e r i n g , Queen's U n i v e r s i t y , K i n g s t o n , O n t a r i o . S c o t t , T. ( 1 9 5 1 ) . Sand movement by waves. U. S. Army C o r p s of E n g i n e e r s , Beach E r o s i o n B o a r d , T e c h n i c a l Memo. No. 48. S c r i p p s I n s t i t u t i o n of O c e a n o g r a p h y ( 1 9 4 7 ) . A s t a t i s t i c a l s t u d y o f wave c o n d i t i o n s a t f i v e s e a l o c a l i t i e s a l o n g t h e C a l i f o r n i a c o a s t . U n i v e r s i t y o f C a l i f o r n i a Wave R e p o r t No. 68, La J o l l a , C a l i f o r n i a . S h e p a r d , F. P., Emery, K. 0., and La F o n d , E . C. ( 1 9 4 1 ) . R i p c u r r e n t s : a p r o c e s s of g e o l o g i c a l i m p o r t a n c e . J o u r n a l of G e o l o g y , V o l . 49, No. 4, pp. 337 - 369. Swart , D. H. ( 1 9 7 6 ) . P r e d i c t i v e e q u a t i o n s r e g a r d i n g c o a s t a l t r a n s p o r t s . P r o c e e d i n g s , 15th C o a s t a l E n g i n e e r i n g C o n f e r e n c e , pp. 1113 - 1132. Swa r t , D. H., and F l e m i n g , C. A. ( 1 9 8 0 ) . L o n g s h o r e water and s e d i m e n t movement. P r o c e e d i n g s , 17th C o a s t a l E n g i n e e r i n g C o n f e r e n c e , pp. 1275 - 1294. T h o r n t o n , E . B. ( 1 9 7 3 ) . D i s t r i b u t i o n of s e d i m e n t t r a n s p o r t a c r o s s t h e s u r f z o n e . P r o c e e d i n g s , 13th C o a s t a l E n g i n e e r i n g C o n f e r e n c e , pp. 1049 - 1068. W a l t o n , T. L., and C h i u , T. Y. ( 1 9 7 9 ) . A r e v i e w o f a n a l y t i c a l t e c h n i q u e s t o s o l v e t h e sand t r a n s p o r t e q u a t i o n and some s i m p l i f i e d s o l u t i o n s . C o a s t a l S t r u c t u r e s 79, Waterways, P o r t , C o a s t a l and Ocean D i v i s i o n , ASCE, pp. 809 -837. 1 33 W i e g e l , R. L. ( 1 9 6 4 ) . O c e a n o g r a p h i c a l E n g i n e e r i n g . P r e n t i c e - H a l l , Englewood C l i f f s , NJ. W i l l i s , D. H. ( 1 9 7 7 ) . E v a l u a t i o n of a l o n g s h o r e t r a n s p o r t m o d e l s . P r o c e e d i n g s , 5 t h Symposium of t h e Waterway, P o r t , C o a s t a l and Ocean D i v i s i o n , ' C o a s t a l Sediment 77', ASCE, pp. 350 - 365. W i l l i s , D. H. ( 1 9 7 8 ) . Sediment l o a d under waves and c u r r e n t s . P r o c e e d i n g s , 16th C o a s t a l E n g i n e e r i n g C o n f e r e n c e , pp. 1626 - 1637. W i l l i s , D. H., and P r i c e , W. A. ( 1 9 7 5 ) . T r e n d s i n t h e a p p l i c a t i o n of r e s e a r c h t o s o l v e c o a s t a l e n g i n e e r i n g p r o b l e m s . In ' N e a r s h o r e s e d i m e n t d y n a m i c s and s e d i m e n t a t i o n ' , e d i t e d by H a i l s and C a r r , pp. 111 - 121, John W i l e y and Son L t d . W i l s o n , W. S. ( 1 9 6 6 ) . A method f o r c a l c u l a t i n g and p l o t t i n g s u r f a c e wave r a y s . U. S. Army C o a s t a l E n g i n e e r i n g R e s e a r c h C e n t e r . T e c h n i c a l Memo. No. 17. 1 34 A p p e n d i x A F i n i t e d i f f e r e n c e f o r m u l a t i o n The c o n c e p t of f i n i t e d i f f e r e n c e method i s t o assume t h a t t h e r e e x i s t a s o l u t i o n s u r f a c e ( s a y f o r wave h e i g h t , H) o v e r t h e e n t i r e X,Y domain, as shown i n f i g u r e A1 . H so lu t i on s u r f a c e To a p p r o x i m a t e t h e d e r i v a t i v e of ^ / ^ X a t P ° ^ n t ( i , j ) , t h e r e a r e t h r e e b a s i c methods: 1. F o r w a r d d i f f e r e n c e A d ± Hti+ 1,j) ~ H ( i,j) A X 2. Backward d i f f e r e n c e = H ( i , j ) - H ( i - 1 f J ) ^ X A X 3. C e n t r a l d i f f e r e n c e AH ± H(i+1,j) ~ H f j - i j ) 135 A X A X F i g u r e A2 The method c h o s e n f o r t h e r e f r a c t i o n and s h o a l i n g p r o g r a m i s b a s e d on t h e c e n t r a l d i f f e r e n c e method o f a p p r o x i m a t i o n . A p p e n d i x B F l o w c h a r t f o r wave r e f r a c t i o n and s h o a l i n g r o u t i n e READ: GRID S IZE , NX.NY WAVE PERIOD, T DEPTH CONTOURS I DEFINE LOCAL GRID VARIABLES: AN - WAVE DIR. ANGLE DN - DEPTH WH - WAVE HEIGHT K - WAVE NUMBER Cg - GROUP VELOCITY LOCAL GRID CALCULATION FDR WAVE HEIGHT, WAVE DIRECTION AND WAVE ANGLE BOUNDARY CONDITION CONTROL YES PRINT: WAVE HEIGHT AND WAVE ANGLE STOP A p p e n d i x C F l o w c h a r t f o r l o n g s h o r e t r a n s p o r t model INPUT ECHO INPUT f SHOALING ROUTINE CALCULATE WAVE BREAKING CONDITIONS CALCULATE WAVE ENERGY CALCULATE VOL. TPT. ADJUST SHORELINE FOR EACH CELL YES SHOALING NO 1 38 Appendix D : 1 1 1 1 1 1 1 1 1 T o ot go 90 vo zo oo O l l V d NOI13cd0V 1 39 A p p e n d i x E C r i t e r i a f o r s e l e c t i n g Time R a t i o The v a l u e of 4.2 u s e d i n t h e example has t o be c o r r e c t l y i n f e r r e d . From f i g u r e 4.12, i t c a n be seen t h a t as t h e t i m e r a t i o i n c r e a s e s , t h e a c c r e t i o n r a t i o r e a c h e s an a s y m p t o t i c v a l u e . Hence any v a l u e o f t i m e r a t i o c a n be u s e d i n s t e a d o f 4.2. To a c h i e v e some c o n s i s t e n c y i n a p p r o a c h , a r e a s o n a b l e c r i t e r i a w i l l have t o be a d h e r e d t o . A s u g g e s t e d c r i t e r i a i s by d e f i n i n g an a c c e p t a b l e ' t e r m i n a l ' a c c r e t i o n r a t e as s u f f i c e n t a c c u r a c y f o r answer. Then u s i n g t h i s a c c r e t i o n r a t e as c o n t r o l , a u n i q u e v a l u e of t i m e r a t i o c a n be r e a d o f f from any c u r v e i n f i g u r e 4.12. (See f i g u r e E 1 ) . F i g u r e E1 Time ratio A p p e n d i x F GRAIN S IZE DISTRIBUTION FRASER RIVER SAND 1 40 10.0 1.0 0.1 S IZE (mm) 141 A p p e n d i x G Program d e s c r i p t i o n s The maximum d i m e n s i o n s o f t h e s t u d y a r e a r e : NX = 100 NY = 100 I n p u t of t h e p r o g r a m c o n s i s t s of p r o g r a m o p t i o n and d a t a . The u n i t s u s e d i n t h e pr o g r a m a r e i n S I . ( i e m e t r e , k i l o g r a m , s e c o n d ) . C a r d 1 NX,NY,DX,DY,T,SWL,DT NX,NY - a r e two i n t e g e r s s p e c i f y i n g t h e maximum number of c e l l s i n t h e s t u d y a r e a . The d e f i n i t i o n s a r e shown i n f i g u r e 3.2. DX,DY - a r e t h e c e l l w i d t h and l e n g t h . T - t h e wave p e r i o d of t h e imposed wave c o n d i t i o n . SWL - t h e s t i l l w a t er l e v e l f r o m s h o r e datum as shown i n f i g u r e G1. DT - t h e m o d e l l i n g t i m e i n t e r v a l . C a r d 2 PERL,PERR PERL - t h e p e r m e a b i l i t y of t h e b a r r i e r on t h e u p s t r e a m b o u n d a r y . PERR - t h e p e r m e a b i l i t y of t h e b a r r i e r on t h e downstream b o u n d a r y . The v a l u e o f 1.0 would i n d i c a t e f u l l p e r m e a b i l i t y , i e no s e d i m e n t w i l l be r e t a i n e d , w h i l e 0.0 i n d i c a t e s f u l l i m p e r m e a b i l i t y w i t h no s e d i m e n t moving a c r o s s t h e b a r r i e r . 142 C a r d 3 KOUT,END,KALREF,OPTION,KJOFF KOUT END KALREF - t h i s i n t e g e r i n d i c a t e t h e number of i t e r a t i o n s b e f o r e o u t p u t of t h e s h o r e l i n e r e s u l t s i s p r i n t e d . The p u r p o s e o f t h i s o p t i o n i s t o r e d u c e unwanted i n t e r m e d i a t e i t e r a t i v e o u t p u t . - t h i s i s t h e f i n a l 'Time' i n w h i c h t h e model w i l l t e r m i n a t e i t s p r o g r a m . - T h i s i n t e g e r i s f o r c o n t r o l l i n g t h e r o u t i n g of t h e program t h r o u g h t h e wave r e f r a c t i o n and s h o a l i n g r o u t i n e . T h i s i n t e g e r i n d i c a t e s t h e number of i t e r a t i o n s b e f o r e t h e r e f r a c t i o n and s h o a l i n g r o u t i n e i s c a l l e d . The p r i m a r y p u r p o s e o f t h i s i s t o s t u d y t h e e f f e c t of wave r e f r a c t i o n and s h o a l i n g on t h e r e s u l t s . OPTION KJOFF - t h i s i s a p r o g r a m o p t i o n v a r i a b l e where 1.0 i n d i c a t e s a d e t a i l e d o u t p u t w h i c h i n c l u d e s wave r e f r a c t i o n r e s u l t s and s h o r e l i n e change r e s u l t s . The o u t p u t f i l e i s a t t a c h e d t o I/O u n i t 6. Any o t h e r number would y i e l d o n l y t h e s h o r e l i n e c h a n g e r e s u l t s , t h e o u t p u t i s t h e n t o I/O u n i t 7. - t h i s i n t e g e r c o n t r o l s t h e l o o p i n g of t h e o n - o f f s h o r e r o u t i n e i n t h e p r o g r a m . The i n t e g e r f o r t h i s v a r i a b l e w i l l r e s u l t i n c a l l i n g t h e o n - o f f s h o r e r o u t i n e a f t e r 'KJOFF' i n t e r a t i o n s . C a r d 4 ANGLE,WAVEHT ANGLE, WAVHT - t h i s two a r r a y s w i l l d e f i n e t h e o f f s h o r e wave a n g l e and wave h e i g h t . The p r o g r a m i n p u t c a r d i s f o r m a t t e d t o r e a d 10 number p e r c a r d ( i e f i v e p a i r s o f ANGLE and WAVEHT). I f t h e v a l u e of NY i s say 30, t h e n t h e r e s h o u l d be 6 c a r d s t o d e f i n e a l l t h e o f f s h o r e wave a n g l e and wave h e i g h t . 143 C a r d 5 RDEPTH RDEPTH - t h i s i s an a r r a y w h i c h d e f i n e s t h e t o p o g r a p h i c a l d e p t h o f t h e s t u d y a r e a w i t h r e s p e c t t o t h e s h o r e datum ( s e e f i g u r e G 1 ) . F o r e v e r y p o i n t i n t h e o v e r a l l g r i d , t h e r e i s one RDEPTH. The i n p u t o f t h i s RDEPTH b e g i n s a t I = 1, J=1 and ends a t I = XN, J = YN ( s e e f i g u r e 3 . 2 ) . The number o f v a r i a b l e s p e r c a r d i s f o r m a t t e d a t 10 i n t h e p r o g r a m . C a r d 6 WHI,WAI,TI WHI - t h e i n i t i a l o f f s h o r e wave h e i g h t . WAI - t h e i n i t i a l o f f s h o r e wave a n g l e . TI - t h e i n i t i a l wave p e r i o d . T h e s e v a r i a b l e s a r e d i f f e r e n t from t h o s e wave h e i g h t , wave a n g l e and p e r i o d m e n t i o n e d e a r l i e r . T h e s e v a r i a b l e s a r e t h e i n i t i a l c o n d i t i o n and a r e use t o c a l i b r a t e t h e i n i t i a l e q u i l i b r i u m p r o f i l e i n t h e o n - o f f s h o r e r o u t i n e ( s e e c h a p t e r 6 ) . C a r d 7 FV,BD FV - i s t h e f a l l v e l o c i t y of t h e D s i z e of t h e s e d i m e n t . BD - i s t h e d e c a y c o n s t a n t o f t h e p r o f i l e change (se e c h a p t e r 5 and 6 ) . 1 44 The u n c o m p i l e d and c o m p i l e d v e r s i o n s of t h e p r o g r a m a r e a v a i l a b l e i n t h e C i v i l E n g i n e e r i n g d e p a r t m e n t p rogram l i b r a r y , U n i v e r s i t y of B r i t i s h C o l u m b i a . The f i l e names a r e SHOREC - f o r t h e c o m p i l e d v e r s i o n and SHORE - f o r t h e u n c o m p i l e d v e r s i o n . To run t h e p r o g r a m t y p e t h e f o l l o w i n g command: $RUN SHOREC 5=DATA 6=~6 7=-7 DATA - i s t h e d a t a f i l e c o n t a i n i n g a l l t h e d a t a a s i n t h e e a r l i e r s e c t i o n . -6 - i s t h e o u t p u t f i l e when t h e v a r i a b l e ' o p t i o n ' s p e c i f i e d a s 1.0. -7 - i s t h e o u t p u t f i l e when t h e v a r i a b l e ' o p t i o n ' s p e c i f i e d a s o t h e r w i s e . To o b t a i n a l i s t i n g of t h e p r o g r a m o r t h e o u t p u t f i l e , t y p e : d e s c r i b e d i s i s $ L IST ' F i l e n a m e ' ' F i l e n a m e ' i s t h e name o f t h e f i l e , (eg SHORE or - 6 ) . 

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