Douro Hydro-morphological

«OMWnSTIMQAO D M M R T M P O D O U W I L l I X f l M

Hydro-morphological study

Douro Estuary

P a r t 2 L o n g s h o r e t r a n s p o r t c a l c u l a t i o n s 11 n o v « m b a r 1 9 8 2 / P 6 1 3

PORT AND WATERWAY ENGINEERS

h y ö R o n A m i c

ADMINISTRACAO DOS PORTOS DO DOURO E L E I X O E S

Hydro«morpho!ogical study

ouro Estuary

Part 2

Longshore transport calculations

november 1982 / P613

PORT AND WATERWAY ENGINEERS

TABLE Of CONTENTS 1. INTRODUCTION . . 1 2 . LONGSHORE TRANSPORT 3 3. ACTUAL COMPUTATIONS 5 k, SET-UP DIFFERENCE 7 5. RESULTS , 8 6 . CONCLUSIONS e 9 APPENDIX . . 1 1

U INTRODUCTION

The Admin i s t r a c a o dos P o r t o s do Douro i s c o n f r o n t e d w i t h t h e f a c t

t h a t f u t u r e expansion o f t h e p o r t areas o f L e i x o e s i s l i m i t e d by t h e

s u r r o u n d i n g h i l l s and urban i s a t i o n . One o f t h e a l t e r n a t i v e s f o r f u

-t u r e expansion migh-t be -the c r e a -t i o n o f p o r -t f a c i l i -t i e s i n -the Douro

e s t u a r y , where a r e l a t i v e l y l a r g e , f l a t a r e a i s a v a i l a b l e .

However, t h e r i v e r e n t r a n c e i s r a t h e r u n s t a b l e f r o m a m o r p h o l o g i c a l

po i n t o f v i e w , so t h a t r e g u l a r d r e d g i n g i s r e q u i r e d t o m a i n t a i n an

access c h a n n e l .

In o r d e r t o e l i m i n a t e t h e r i s k o f unexpected morpholog i c a l changes

i n the Douro e s t u a r y , due t o harbour works APDL has decided t o s t u d y

t h e m o r p h o l o g i c a l aspects o f t h e Douro bar t h o r o u g h l y , and t o make a

d e s i g n f o r a mathematical model t o i n v e s t i g a t e t h e i n f l u e n c e o f h a r

-bour works i n t h e a r e a .

T h i s s t u d y was o f f e r e d by APDL t o the Sociedade Portuguesa de

Dragagens i n January 1982. A f t e r r e c e i p t o f t h e l e t t e r o f a d j u d i c a

-t i o n f r o m APDL, da-ted J u l y 2 9 -t h , I9B2, Hydronamic bv was asked by

S.P.D. t o s t a r t t h e s t u d y .

The study i s s u b d i v i d e d i n t o s i x phases, v i z :

a l . wave p e n e t r a t i o n c a l c u l a t i o n

a2. c a l c u l a t i o n o f l o n g s h o r e t r a n s p o r t

a3. c a l c u l a t i o n o f t h e t i d a l p r i s m

b m o r p h o l o g i c a l e v a l u a t i o n

c s e t - u p o f t h e mathematical model

As d e s c r i b e d i n our proposal o f January 1982, a r e p o r t f o r each

phase w i l l be presented t o APOL.

T h i s r e p o r t g i v e s t h e r e s u l t s o f phase a2, c a l c u l a t i o n o f t h e

l o n g s h o r e t r a n s p o r t .

2 . LONGSHORE TRANSPORT

The l o n g s h o r e sediment t r a n s p o r t c a p a c i t y has been c a l c u l a t e d at

f o u r l o c a t i o n s along t h e c o a s t (see f i g . 2 ) .

When waves approach a c o a s t l i n e o b l i q u e l y , t h e y g e n e r a t e a l o n g s h o r e

c u r r e n t a t t h e i r b r e a k e r p o i n t . T h i s longshore c u r r e n t , t o g e t h e r

w i t h 'che s t i r r i n g o f t h e waves may t r a n s p o r t sed iment a l o n g t h e

c o a s t . Of c o u r s e , t r a n s p o r t can o n l y occur i f t h e r e i s sed iment

a v a i l a b l e t o t r a n s p o r t .

The e f f e c t i v e t r a n s p o r t e s p e c i a l l y on a r o c k y c o a s t may be l e s s than

t h e t r a n s p o r t c a p a c i t y .

Only t h e wave-induced longshore c u r r e n t i s i m p o r t a n t f o r t h e

s t a b i l i t y o f c o a s t l i n e s . Ocean d r i f t c u r r e n t has h a r d l y no i n

-f l u e n c e . T i d a l c u r r e n t s may be s i g n i -f i c a n t i n s h a l l o w seas, b u t n o t

along ocean c o a s t s .

T h e r e f o r e , t h e s t a b i l i t y o f t h e c o a s t l i n e near Porto i s f u l l y d e t e r

-mined by wave a c t i o n .

The w a v e - c l i m a t e can be d e s c r i b e d using 3^ w a v e - c l i m a t e elements

(see p a r t 1 ) . Using the r e s u l t s o f t h e w a v e p a n e t r a t i o n c a l c u l a

-t i o n , we have de-termined -the w a v e - c l i m a -t e a-t -t h e f o u r l o c a -t i o n s -t o

be c a l c u l a t e d . T h i s was done by a v e r a g i n g t h e r e s u l t s f r o m a small

nuttnber o f p o i n t s near t h e l o c a t i o n s t o be c a l c u l a t e d . (See f i g . 3)«

The r e s u l t i n g w a v e - c l i m a t e f o r each l o c a t i o n i s presented in t a b l e

1-4. These data were e n t e r e d in 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 .

Other d a t a e n t e r e d in t h i s model were t h e g r a i n - s i z e o f t h e beach

m a t e r i a l and t h e c r o s s - s e c t i o n a l p r o f i l e o f t h e beach. T h i s p r o f i l e

has been determined using t h e 1972 c h a r t o f t h e I n s t i t u t o

H i d r o g r a f i c o , because t h e water depth d a t a o f t h e wave- p e n e t r a t i o n

model i s a l s o based on t h i s c h a r t . Since 1972 t h e c r o s s - s e c t i o n a l

p r o f i l e s have changed c o n s i d e r a b l y , b u t we decided t o use t h e 1972

d a t a , t o be c o n s i s t e n t w i t h t h e w a v e - p e n e t r a t i o n c a l c u l a t i o n .

In o r d e r t o d e t e r m i n e t h e g r a i n - s i z e , f o u r samples have been taken

f r o m t h e sand s p i t . For l o c a t i o n s see f i g u r e 4, g r a i n s i z e d i s

-t r i b u -t i o n i s presen-ted in f i g u r e 5. In -t h e c a l c u l a -t i o n s a 050 o f

950 mu and a D90 o f 1600 mu have been used. Because o f t h e c o a r s e

sand a r i p p l e - h e i g h t ( r o u g h n e s s - c o e f f i c i e n t ) o f 0.01 m has been

u s e d .

The c a l c u l a t i o n s have been c a r r i e d out w i t h t h e BYKBAT-program.

T h i s computer program c a l c u l a t e d t h e l o n g s h o r e c u r r e n t u s i n g t h e

r a d i a t i o n s t r e s s method and sediment t r a n s p o r t w i t h t h e B i j k e r ' s

method Some background I n f o r m a t i o n Is g i v e n in an appendix at t h e

3. ACTUAL COMPUTATIONS

Using the BYKBAT program, t h e 3k w a v e - c l i m a t e elements have been run

f o r each o f t h e f o u r p r o f i l e s . The c o m p u t e r - o u t p u t i s g i v e n i n a

s e p e r a t e annex t o t h i s r e p o r t .

On each page t h e t r a n s p o r t due t o one c l i m a t e - e l e m e n t i s g i v e n .

F i r s t some p h y s i c a l and c o a s t a l c o n s t a n t s a r e p r e s e n t e d . G i s t h e

a c c e l e r a t i o n o f g r a v i t y , RHO i s t h e d e n s i t y o f sea water and KAPPA

i s t h e Von Karman-number. D e l t a i s t h e s p e c i f i c w e i g h t o f sediment

r e l a t e d t o s e a - w a t e r . The breaker index i s t h e r e l a t i o n between

b r e a k i n g w a v e - h e i g h t and water d e p t h .

The second b l o c k o f d a t a summarises t h e w a v e - c l i m a t e d a t a .

The m a j o r data b l o c k g i v e s t h e r e f r a c t e d w a v e - h e i g h t ( a t 10 m) and

t h e angle o f i n c i d e n c e a t a depth o f 10 m. In the c o l u n n s t h e f o l

-lowing f i g u r e s are g i v e n :

D - water depth (m)

ALFA - t a n g e n t o f t h e bottom slope

H - o c c u r r i n g w a v e - h e i g h t a t g i v e n water depth (m)

SETUP - average water l e v e l r i s e due t o wave s e t - u p (m)

Ul - maximiOT o r b i t a l v e l o c i t y near t h e bottom ( m / s )

C - Chezy-parameter KSl - B i j k e r - p a r a m e t e r R - bottom roughness (m) L - l o c a l wave l e n g t h (m) V - longshore c u r r e n t v e l o c i t y ( m / s ) S - sediment t r a n s p o r t (m3/5/ni)

F i n a l l y t h e t o t a l t r a n s p o r t due t o t h i s c l i m a t e - e l e m e n t i s g i v e n

b o t h f o r t h e CERC-formula, as w e l l as f o r t h e B i j k e r f o r m u l a . The

r e s u l t f r o m t h e B i j k e r method w i l l be used.

A f t e r c l i m a t e element 3^ a l l t h e r e s u l t s are added t o g e t h e r t o o b

-t a i n -t h e y e a r l y -t r a n s p o r -t .

h, SET-UP DIFFERENCE

Due t o w a v e - b r e a k i n g the average water l e v e l r i s e s . T h i s phenomenon

i s c a l l e d wave s e t - u p . W i t h h i g h waves o f long p e r i o d , t h i s v a l u e

may be c o n s i d e r a b l e ( f o r h.S m waves w i t h a p e r i o d o f 12 sec i t can

be more than 1 m ) . At t h e e n t r a n c e o f t h e Douro r i v e r t h e

wave-b r e a k i n g i s l e s s than on t h e wave-beach. Consequently t h e f o r c e c a u s i n g

wave s e t - u p i n t h e r i v e r e n t r a n c e i s l e s s and because t h e water can

e a s i l y f l o w away, t h e e f f e c t i v e s e t - u p w i l l be n e a r l y z e r o .

T h i s means t h a t t h e r e i s a c o n s i d e r a b l e water l e v e l d i f f e r e n c e along

t h e beach o f t h e Cabedelo. T h e r e f o r e along t h e beach a water l e v e l

slope e x i s t s , and c o n s e q u e n t l y t h e r e i s a d r i v i n g f o r c e on the water

in t h e d i r e c t i o n o f t h e r i v e r .

So t h e r e are two f o r c e s w o r k i n g on t h e w a t e r :

1. Force i n t h e d i r e c t i o n o f t h e r i v e r (due t o s e t - u p d i f f e r e n c e )

2 . Force i n southward d i r e c t i o n (due t o o b l i q u e v<ave-breaking)

The d i r e c t i o n o f t h e c u r r e n t i s t h e r e f o r e n o t d e t e r m i n e d b e f o r e h a n d .

An a d d i t i o n a l sediment t r a n s p o r t c a l c u l a t i o n has been p e r f o r m e d f o r

p r o f i l e 2 w i t h a s e t - u p d i f f e r e n c e d i s t a n c e o f 750 m and f o r p r o f i l e

3 w i t h a s e t - u p d i f f e r e n c e d i s t a n c e o f 250 m.

The s e t - u p d i f f e r e n c e d i s t a n c e i s t h e d i s t a n c e f r o m the p r o f i l e t o

t h e p o i n t where t h e s e t - u p i s z e r o .

5. RESULTS The f o l l o w i n g r e s u l t s were o b t a i n e d : n o r t h w a r d t r a n s p o r t southward t r a n s p o r t net t r a n s p o r t p r o f i l e 1 1.918 0 1.918 southw. p r o f i 1 e 2 899 -21 878 southw. p r o f 1 1 e 3 790 - 7 783 southw. p r o f i 1 e k 2.780 - 2 2.778 southw. p r o f i 1 e 2 ( s e t - -up) 130 742 612 n o r t h w . p r o f i 1 e 3 ( s e t - •up) h3 1.533 1.483 n o r t h w . v a l u e s i n 1000 m3/year

The v a l u e s g i v e n i n the above t a b l e are t r a n s p o r t - c a p a c i t i e s . In

p r o f i l e 1 and 4 e s p e c i a l l y , t h e e f f e c t i v e t r a n s p o r t w i l l be c o n

-s i d e r a b l y l e -s -s , becau-se here t h e beach i -s n o t a l l u v i a l but r o c k y and

t h e r e f o r e n o t f u l l y e r o d i b l e .

The i n f l u e n c e o f s e t - u p i s q u i t e l a r g e , as can be seen in t h e above

f i g u r e s .

6. CONCLUSIONS

The t r a n s p o r t c a p a c i t i e s n o r t h and south o f t h e r i v e r e n t r a n c e are

q u i t e h i g h , and c l e a r l y h i g h e r than the a c t u a l t r a n s p o r t . T h i s

means t h a t a l l ' s e d i m e n t t h a t comes f r o m t h e n o r t h , and i s a b l e t o

pass t h e Leixoes p o r t w i l l be t r a n s p o r t e d towards t h e r i v e r

e n t r a n c e . Also s e d i m e n t , w h i c h i s placed o f f s h o r e by r i v e r a c t i o n ,

and i s thrown back t o t h e beach by waves, w i l l be t r a n s p o r t e d t o the

r i v e r e n t r a n c e ,

To t h e s o u t h , one may c o n c l u d e t h a t a l l s e d i m e n t , which i s a b l e t o

pass t h e Seca do Bacalhau {hSS^ N) w i l l be t r a n s p o r t e d s o u t h w a r d s ,

and w i l l n o t r e t u r n t o t h e r i v e r e n t r a n c e .

Along t h e Cabedelo sediment t r a n s p o r t i s m a i n l y governed by t h e c u r

-r e n t due t o s e t - u p d i f f e -r e n c e . T h i s c u -r -r e n t causes a t -r a n s p o -r t i n

n o r t h w a r d d i r e c t i o n . In t h e n o r t h e r n s e c t i o n o f the Cabedelo the

s e t u p i n f l u e n c e i s m a x i m a l , i n t h e s o u t h e r n s e c t o r i t i s c o n

-s i d e r a b l y l e -s -s .

One may c o n c l u d e f r o m t h e f i g u r e s presented in c h a p t e r 5, t h a t t h e r e

i s a s t r o n g sediment t r a n s p o r t movement along the s p i t , ( a p p r o x .

1.5 m i l l i o n m 3 / y e a r ) . T h i s t r a n s p o r t t r i e s t o c l o s e the e n t r a n c e .

T i d a l c u r r e n t s are c o n t i n u o u s l y opening the e n t r a n c e and t r a n s p o r t

-ing the sediment t o t h e o f f s h o r e b a r .

Wave a c t i o n t r a n s p o r t s t h e sediment f r o m t h e o f f s h o r e bar towards

t h e Cabedelo, and f r o m t h e r e t h e sediment i s a g a i n t r a n s p o r t e d by

l o n g s h o r e c u r r e n t i n t h e d i r e c t i o n o f t h e r i v e r e n t r a n c e .

Consequently t h e r e i s a c l o s e d c i r c u i t o f sediment i n the mouth o f

t h e r i v e r . In p e r i o d s o f low r i v e r d i s c h a r g e the e n t r a n c e w i l l s i l t

up. During h i g h r i v e r d i s c h a r g e the e n t r a n c e i s opened a g a i n .

Only d u r i n g e x t r e m e l y h i g h r i v e r d i s c h a r g e i s t h e sediment f r o m the

r i v e r and f r o m the r i v e r e n t r a n c e b r o u g h t over the o f f s h o r e b a r .

And o n l y i f t h e sediment i s t r a n s p o r t e d f a r out t o sea, w i l l i t be

b r o u g h t (by w a v e - a c t i o n ) t o t h e beach south o f Seca do Bacalhau, and

c o n s e q u e n t l y s u p p l y t h e s o u t h e r n beach.

Because e x t r e m e l y h i g h r i v e r d i s c h a r g e does n o t occur any more due

t o the dams i n the r i v e r , t h e r e i s no longer any s u p p l y t o the

s o u t h e r n beach. Due t o t h i s f a c t the c o a s t i s e r o d i n g ( u n t i l rocks

a p p e a r ) . The consequences o f t h i s e r o s i o n can be c l e a r l y seen at

E s p i n h o .

T h i s c a l c u l a t i o n a f f i r m s the statement o f P i r e s Castanho, Gomes,

Mota O l i v e i r a and P i n t o Simoes i n t h e i r paper presented at t h e I 9 8 I

PIANC-congress i n E d i n b u r g h , t h a t the e r o s i o n at Espinho i s n o t

caused by t h e Leixoes p o r t , b u t by t h e c o n s t r u c t i o n o f dans i n the

Douro r i v e r .

APPENDIX

Some background i n f o r m a t i o n on the BYKBAT program

The computer program BYKBAT c a l c u l a t e s l o n g s h o r e t r a n s p o r t o f s e d i

-ment due t o t h e o b l i q u e b r e a k i n g o f waves. The c a l c u l a t i o n i s s p l i t

i n t o two p a r t s . F i r s t t h e v e l o c i t y o f t h e l o n g s h o r e c u r r e n t i s c a l

-c u l a t e d w i t h t h e t h e o r y o f r a d i a t i o n s t r e s s . In t h e se-cond s t e p ,

t h e q u a n t i t y o f s e d i m e n t a t i o n , s t i r r e d up by t h e waves and

t r a n s p o r t e d by t h e a f o r e mentioned c u r r e n t , i s c a l c u l a t e d .

B è s i d e s t h e beach-parameters ( g r a i n s i z e s , s l o p e s , e t c . ) t h e program

r e q u i r e s wave d a t a on "deep w a t e r " . S t a r t i n g w i t h these d a t a , t h e

program c a l c u l a t e s t h e wave d a t a on the b r e a k e r l i n e , u s i n g l i n e a r

r e f r a c t i o n t h e o r y and p a r a l l e l depth c o n t o u r s . Because i n r e a l i t y

depth c o n t o u r s are n o t p a r a l l e l , a s p e c i a l f e a t u r e i s a v a i l a b l e t o

overcome t h i s p r o b l e m . W i t h a s e p e r a t e wave p e n e t r a t i o n m o d e l , t h e

n e a r s h o r e wave data are c a l c u l a t e d at a p o i n t j u s t o u t s i d e t h e

b r e a k e r l i n e . Then t h e wave data o f t h i s p o i n t and t h e water depth

a t t h i s p o i n t are e n t e r e d in the BYKBAT p r o g r a m .

The p r o g r a n now s t a r t s t h e c a l c u l a t i o n s f o r t h e g i v e n p o i n t w i t h t h e

r e f r a c t i o n c o e f f i c i e n t s f r o m t h e s e p a r a t e wave p e n e t r a t i o n m o d e l .

In t h e program a number o f "seasons" are e n t e r e d . In r e a l i t y t h e s e

a r e n o t r e a l seasons but components o f t h e wave c l i m a t e . Each

"season" Is d e f i n e d by wave h e i g h t , wave p e r i o d , wave d i r e c t i o n and

f r e q u e n c y o f o c c u r r e n c e o f t h a t "season". For each "season" a

complete c a l c u l a t i o n i s c a r r i e d o u t .

The slope o f t h e beach i s n o t a c o n s t a n t i n t h i s program. The r e a l

water depth a t v a r i o u s d i s t a n c e s f r o m the c o a s t i s e n t e r e d i n t o t h e

program. The bottom slope i s then c a l c u l a t e d as a f u n c t i o n o f t h e

1ocat i o n .

The c o a s t a l p r o f i l e i s d i v i d e d i n t o 12 s e c t i o n s . The t r a n s p o r t i s

c a l c u l a t e d f o r each s e c t i o n and f i n a l l y a l l these t r a n s p o r t s a r e

added t o get t h e t o t a l t r a n s p o r t f o r one season.

The d e p t h , t h e bottom slope and the s e t - u p are f i r s t c a l c u l a t e d f o r

each s e c t i o n . The s e t - u p i s c a l c u l a t e d w i t h f o r m u l a e g i v e n by

B a t t j e s ( 1 9 7 4 ) . D e r i v a t i o n s o f these f o r m u l a e are r a t h e r e x t e n s i v e ,

and are n o t repeated h e r e . Then the l o n g s h o r e v e l o c i t y i s c a l c u

-l a t e d a p p -l y i n g r a d i a t i o n s t r e s s t h e o r y (method o f L o n g u e t - H i g g i n s S

Steward, 1 9 6 4 ) .

R a d i a t i o n s t r e s s i s a p r e s s u r e f o r c e in excess o f the h y d r o s t a t i c

p r e s s u r e f o r c e caused by t h e presence o f waves. In r e a l i t y , t h e

r a d i a t i o n s t r e s s i s n e i t h e r a t r u e s t r e s s ( f o r c e / a r e a ) nor a t r u e

f o r c e , but a f o r c e per u n i t l e n g t h . Even so, t r a n s f o r m a t i o n s a p

-p l i c a b l e t o t r u e s t r e s s e s can be a -p -p l i e d t o r a d i a t i o n s t r e s s .

U n l i k e h y d r o s t a t i c p r e s s u r e , t h e r a d i a t i o n s t r e s s i s n o t i s o t r o p i c ;

indeed, i t i s j u s t as f o r s t r e s s e s .

According t o the law o f Newton, a f o r c e i s e q u i v a l e n t t o a r a t e o f

change o f momentun. A s t r e s s i s é q u i v a l e n t t o a momentun-f1ux, and

t h e r a d i a t i o n s t r e s s i s determined by i n t e g r a t i n g t h i s momentum

-f l u x o -f t h e waves over the d e p t h . Thus: -f o r c e s on an element o -f

water w i t h v e r t i c a l planes ( f r o m bottom t o s u r f a c e ) are e q u i v a l e n t

t o the changes i n r a d i a t i o n s t r e s s .

Because o f r e f r a c t i o n , t h e p r i n c i p a l s t r e s s e s are changing d i r e c t i o n

and thus t h e r a d i a t i o n s t r e s s component p a r a l l e l t o t h e c o a s t l i n e i s

changing t o o . T h i s component causes a f o r c e p a r a l l e l t o t h e c o a s t ,

w h i c h p r o p e l s a l o n g s h o r e c u r r e n t . These t r a n s f o r m a t i o n s a r e cai

r i e d out by t h e computer program w i t h t h e mathematical e q u i v a l e n t o f

M o h r ' s C i r c l e a n a l y s i s .

I t can be proved t h a t f o r normal c o a s t s (no changes i n wave h e i g h t s

along the c o a s t ) n e i t h e r " t e n s i o n r a d i a t i o n s t r e s s " or " p r e s s u r e

r a d i a t i o n s t r e s s " are c a u s i n g f o r c e s p a r a l l e l t o t h e c o a s t l i n e .

Only "shear r a d i a t i o n s t r e s s " causes such a f o r c e component.

In f o r m u l a : as xy 3x _a 8 P3 1 2 (n s ln(j)Cos<j) . -g- pgH )

1 sin2<()H^ ~ + ncos2*H^ | | + nsin2(j)H | ^

in wh i c h : H - wave h e i g h t (j) - wave d i r e c t ion n - wave number 9 - a c c e l e r a t i o n o f g r a v i t y p - d e n s i t y o f water

T h i s d i f f e r e n t i a l e q u a t i o n i s solved by t h e BYKBAT program i n a

n u m e r i c a l way.

When, along a c o a s t , t h e wave s e t - u p v a r i e s , t h e average water l e v e l

i s a l s o v a r y i n g , c a u s i n g a h y d r o s t a t i c p r e s s u r e - d i f f e r e n c e i n a

d i r e c t i o n p a r a l l e l t o t h e c o a s t l i n e . T h i s p r e s s u r e - d i f f e r e n c e a l s o

causes a f o r c e p a r a l l e l t o the c o a s t . T h i s f o r c e may e i t h e r speed

up or slow down the l o n g s h o r e c u r r e n t .

To compute the v e l o c i t y o f t h e longshore c u r r e n t , t h e

abovementioned p r o p u l s i v e f o r c e p l u s t h e f o r c e caused by s e t u p d i f

f e r e n c e i s set equal t o the d i s s i p a t i n g f o r c e o f t h e b o t t o m r e s i s

-t a n c e . The bo-t-tom r e s i s -t a n c e i s de-termined u s i n g -the average

v e l o c i t y component o f t h e combined v e l o c i t y o f l o n g s h o r e - c u r r e n t and

v e l o c i t y due t o wave a c t i o n . I t can be proved (see B i j k e r , 1967)

t h a t t h e bottom shear s t r e s s i s g i v e n b y : T _q 2 2 j k P 2 ^ T T c I 1 ' l +• 5^ i L . + 25 ^ sincf) (1 H ^ sin<^) V d t 1n wh1ch: u = u sinujt - o r b i t a l v e l o c i t y at t h e bottom V - longshore v e l o c i t y c - Chezy c o n s t a n t T - wave p e r i o d ^ = £ . 1 / e x p ( - 5 . 9 7 7 . 5.213 i^)''-'^'] / ? a - a m p l i t u d e o f the o r b i t a l motion r - bottom roughness

The f o r m u l a f o r k s i i s based on Jonsson ( 1 9 6 6 ) .

The i n t e g r a t i o n i s c a r r i e d out by t h e BYKBAT program in a n u m e r i c a l

way, w h i l e the e q u a t i o n i s solved f o r v w i t h a r e g u l a - f a l s i method.

A f t e r d e t e r m i n i n g o f t h e l o n g s h o r e v e l o c i t y , t h e sediment t r a n s p o r t

can be c a l c u l a t e d .

F i r s t bottom t r a n s p o r t i s c a l c u l a t e d a c c o r d i n g t o B i j l < e r i - 0 . 2 7 AD^. s = b / g exp yv^

{1

+ i

(57) }

in w h i c i i : A . b - B i j k e r parameter - r e l a t i v e d e n s i t y o f sediment - median g r a i n s i z e - r i p p l e f a c t o r

The bottom t r a n s p o r t v a l u e can be used as boundary v a l u e f o r the

c a l c u l a t i o n o f suspended l o a d . C a l c u l a t i o n o f suspended load Is i n

f a c t t h e i n t e g r a t i o n over the v e r t i c a l o f t h e c o n c e n t r a t i o n . In the

c u r r e n t v e r s i o n o f BYKBAT t h e c o n c e n t r a t i o n i s c a l c u l a t e d w i t h t h e

method o f B a t t a c h a r y a , as d e s c r i b e d in TOW ( 1 9 7 6 ) .

The r i p p l e h e i g h t can be c a l c u l a t e d or e n t e r e d as a c o n s t a n t i n the

p r o g r a n . I f t h e program Is requested t o c a l c u l a t e the r i p p l e

h e i g h t , t h e method o f Swart (1976) i s used. For d e t a i l s o f t h i s

method r e f e r e n c e can be made t o the o r i g i n a l p u b l i c a t i o n .

References

T.O.W. 1976 ( A u t h o r D.H. Swart) Computation o f l o n g s h o r e t r a n s p o r t ;

D e l f t H y d r a u l i c s L a b o r a t o r y , r e p o r t R968I.

B a t t j e s , J . A . 1974; Computation o f s e t u p , l o n g s h o r e c u r r e n t s , r u n

-up and o v e r t o p p i n g due t o w i n d - g e n e r a t e d waves. D e l f t U n i v e r s i t y o f

Technology.

Lonquet-Higgens M.S. & Stewart R.W. 1964; R a d i a t i o n s t r e s s e s i n

water waves. Deep Sea Research, v o l . 11, pp 529-562.

B i j k e r , E.W. 1967; Some c o n s i d e r a t i o n s on s c a l e s f o r models w i t h

movable beds. D e l f t H y d r a u l i c s L a b o r a t o r y .

Jonsson, l . G . 1966; Wave Boundary l a y e r s and f r i c t i o n f a c t o r s , 10th

Coastal E n g i n e e r i n g C o n f e r e n c e , Tokyo.

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