Dissipation in random wave groups incident on a beach

DISSIPATION I N RANEMDM WAVE GROUPS INCIDENT ON A BEACH by J.A. R o e l v i n k DELFT HYDRAULICS p.o. box 152 8300 AD Emmeloord The N e t h e r l a n d s F e b r u a r y 13, 1992¬ 1. I n t r o d u c t i o n

The t r a n s f o r m a t i o n o f c e r t a i n p a r a m e t e r s o f an i n c i d e n t random wave t r a i n a c r o s s t h e s u r f zone has been t h e s u b j e c t o f much s t u d y and m o d e l l i n g e f f o r t . I n r e c e n t l i t e r a t u r e , two c l a s s e s o f models have been d e v e l o p e d , w h i c h a r e b o t h based on t h e wave e n e r g y b a l a n c e o r t h e wave a c t i o n e q u a t i o n , b u t u s e m a r k e d l y d i f f e r e n t approaches.

I n t h e f i r s t , parametric, c l a s s o f models ( B a t t j e s and Janssen, 1978; T h o r n t o n and Guza, 1983), a shape o f t h e b r e a k i n g wave h e i g h t d i s t r i b u t i o n i s assumed, w i t h p a r a m e t e r s t h a t a r e a f u n c t i o n o f l o c a l , t i m e - a v e r a g e d wave p a r a m e t e r s . The d i s s i p a t i o n p e r b r e a k i n g wave i s m o d e l l e d u s i n g t h e a n a l o g y between f u l l y b r e a k i n g waves and b o r e s , w h i c h was f i r s t p o i n t e d o u t by Le Mehaute ( 1 9 6 2 ) . By c o m b i n a t i o n o f t h e b r e a k i n g wave h e i g h t d i s t r i b u t i o n and t h e d i s s i p a t i o n

f u n c t i o n , t h e average d i s s i p a t i o n as a f u n c t i o n o f l o c a l wave p a r a m e t e r s i s o b t a i n e d . By s o l v i n g t h e wave e n e r g y b a l a n c e e q u a t i o n , t h e s e l o c a l wave p a r a m e t e r s can be computed o v e r an a r b i t r a r y p r o f i l e , g i v e n t h e c o n d i t i o n s a t a seaward boundary.

The second, probabilistic, c l a s s o f models t a k e s t h e p r o b a b i l i t y d e n s i t y f u n c t i o n o f wave h e i g h t (and sometimes wave p e r i o d ) a t a seaward boundary, s c h e m a t i z e s i t t o a d i s c r e t e number o f wave h e i g h t ( p e r i o d ) c l a s s e s , and assumes t h a t each c l a s s behaves l i k e a p e r i o d i c sub-group t h a t p r o p a g a t e s i n d e p e n d e n t l y o f t h e o t h e r s . ( M i z u g u c h i , 1982; Mase and I w a g a k i , 1982; D a l l y e t a l , 1984). The wave e n e r g y b a l a n c e e q u a t i o n i s t h e n s o l v e d s e p a r a t e l y f o r a l l waves. As a r e s u l t , a t each p o i n t a l o n g t h e p r o f i l e , t h e wave h e i g h t d i s t r i b u t i o n can be d e t e r m i n e d . A l l models i n t h i s c l a s s s e p a r a t e t h e

B o t h c l a s s e s o f models, when c a l i b r a t e d , may s e r v e w e l l t o p r e d i c t t h e t r a n s f o r m a t i o n o f c e r t a i n p r o p e r t i e s o f t h e wave h e i g h t d i s t r i b u t i o n a c r o s s the s u r f zone. A l s o , wave-averaged p a r a m e t e r s such as r a d i a t i o n s t r e s s and mass f l u x , r e q u i r e d f o r t h e p r e d i c t i o n o f t h e mean s e t - u p and t h e u n d e r t o w a r e p r e d i c t e d s a t i s f a c t o r i l y by b o t h c l a s s e s o f models.

R e c e n t l y , t h e r e has been a g r o w i n g r e c o g n i t i o n o f t h e i m p o r t a n c e o f v a r i a t i o n s i n s h o r t - w a v e p r o p e r t i e s on t h e t i m e - s c a l e o f wave g r o u p s . Such v a r i a t i o n s can f o r c e long-wave m o t i o n s t h a t may be i m p o r t a n t i n t h e m s e l v e s o r t h r o u g h t h e i r i n t e r a c t i o n w i t h wave g r o u p s (Symonds e t a l , 1982; Symonds and Bowen, 1984; L i s t , 1990; S c h a e f f e r e t a l , 1990). A new c l a s s o f dynamic models ( S a t o , 1 9 9 1 ; R o e l v i n k , 1991; Symonds and B l a c k , 1991) t a k e s i n t o a c c o u n t v a r i a t i o n s on t h i s t i m e - s c a l e . The d i s s i p a t i o n o f t h e s h o r t - w a v e m o t i o n i n t h i s c l a s s o f models i s s l o w l y - v a r y i n g on t h e t i m e - s c a l e o f t h e wave g r o u p s . A l t h o u g h t h e

p r o p a g a t i o n and decay o f wave g r o u p s , and hence t h e e x c i t e d long-wave m o t i o n s , o f t e n depend c r i t i c a l l y on t h e f o r m u l a t i o n o f t h i s d i s s i p a t i o n t e r m , a

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

The main g o a l o f t h i s s t u d y i s t o d e v e l o p a s u i t a b l e f o r m u l a t i o n f o r t h e t i m e - v a r y i n g d i s s i p a t i o n due t o wave b r e a k i n g . As i t i s i m p o s s i b l e t o measure t h e t i m e - v a r y i n g d i s s i p a t i o n d i r e c t l y , t h e f o r m u l a t i o n can o n l y be checked by b u i l d i n g i t i n t o models t h a t p r e d i c t measurable p a r a m e t e r s , such as t h e

average d i s s i p a t i o n , t h e f r a c t i o n o f b r e a k i n g waves and t h e mean wave e n e r g y , and by v e r i f y i n g t h e s e models b o t h e x t e r n a l l y and i n t e r n a l l y .

For t h i s p u r p o s e , one wave p r o p a g a t i o n model o f t h e p r o b a b i l i s t i c c l a s s and t h r e e models o f t h e p a r a m e t r i c c l a s s were f o r m u l a t e d , c a l i b r a t e d and v e r i f i e d i n t h i s s t u d y , a l l based on t h e same d i s s i p a t i o n f o r m u l a t i o n . A l t h o u g h i t has n o t been t h e p r i m a r y g o a l o f t h e s t u d y , t h e s e models a r e an i n t e r e s t i n g

b y - p r o d u c t i n t h e m s e l v e s .

The main p r o d u c t , however, i s a c a l i b r a t e d f o r m u l a t i o n f o r t h e d i s s i p a t i o n o f s h o r t - w a v e e n e r g y as a f u n c t i o n o f e n e r g y and w a t e r d e p t h , w h i c h can be e a s i l y implemented i n models t h a t a r e t i m e - d e p e n d e n t on t h e wave-group s c a l e .

2. D i s s i p a t i o n model

B a s i c c o n c e p t

I n a random wave t r a i n , t h e p r o c e s s o f e n e r g y d i s s i p a t i o n due t o wave b r e a k i n g i s e x t r e m e l y complex. I f i t were p o s s i b l e t o p l o t a t i m e s e r i e s o f t h e

i n t e r m i t t e n t peaks w i t h random h e i g h t and s p a c i n g , w h i c h c a n n o t be d e s c r i b e d i n a d e t e r m i n i s t i c way. Even when a moving average i s a p p l i e d o v e r some s h o r t - w a v e p e r i o d s , t h e s l o w l y - v a r y i n g d i s s i p a t i o n r a t e w i l l s t i l l have a random component. However, we can e x p e c t t h i s s l o w l y - v a r y i n g d i s s i p a t i o n r a t e t o a l s o have a s y s t e m a t i c component w h i c h depends on s l o w l y - v a r y i n g

c h a r a c t e r i s t i c s o f t h e s h o r t waves, i n p a r t i c u l a r t h e wave e n e r g y . T h i s

s y s t e m a t i c component, w h i c h i s t h e e x p e c t e d v a l u e o f t h e d i s s i p a t i o n r a t e p e r u n i t a r e a , D, can i t s e l f be seen as t h e p r o d u c t o f two components:

D = P D [ 1 ]

b b

where P i s t h e p r o b a b i l i t y t h a t a wave i s b r e a k i n g and D t h e e x p e c t e d v a l u e

b b o f t h e d i s s i p a t i o n r a t e i n a b r e a k i n g wave, g i v e n t h a t i t s e n e r g y d e n s i t y i s

E. B o t h P and D v a r y on t h e t i m e - s c a l e o f t h e wave groups. b b

D i s s i p a t i o n i n a b r e a k i n g wave

I n o r d e r t o model t h e d i s s i p a t i o n D i n a b r e a k i n g wave, we use t h e w e l l - k n o w n b

a n a l o g y between b r e a k i n g waves and b o r e s , w h i c h r e s u l t s i n t h e f o l l o w i n g a p p r o x i m a t e e x p r e s s i o n ( B a t t j e s & Janssen, 1978):

= I pg f ïï' [21

where f i s t h e f r e q u e n c y , H i s t h e h e i g h t o f t h e b r e a k i n g wave, h t h e w a t e r d e p t h and a a c a l i b r a t i o n c o e f f i c i e n t . B a t t j e s & Janssen assume a l l b r e a k i n g waves t o have t h e maximum wave h e i g h t H ; as t h i s maximum wave h e i g h t i s o f

m

t h e o r d e r o f t h e w a t e r d e p t h , t h e e x p r e s s i o n r e d u c e s t o :

D^ = I pg f [ 3 ]

As i n o u r case t h e h e i g h t o f b r e a k i n g waves i s a l l o w e d t o be c o n s i d e r a b l y s m a l l e r t h a n t h e maximum wave h e i g h t , e x p r e s s i o n [ 2 ] s h o u l d be used i n

p r i n c i p l e . However, i t can be a r g u e d ( S t i v e & Dingemans, 1984), t h a t t h e w a t e r d e p t h i n e q u a t i o n [ 2 ] s h o u l d r a t h e r be seen as a ' p e n e t r a t i o n d e p t h ' , w h i c h i s o f t h e o r d e r o f t h e wave h e i g h t . I n t h i s case, t h e d i s s i p a t i o n can be w r i t t e n as a s i m p l e f u n c t i o n o f t h e e n e r g y o f t h e b r e a k i n g waves:

D = 2 a f E [ 4 ] b p

where t h e peak f r e q u e n c y f has been t a k e n as a c h a r a c t e r i s t i c

p

P r o b a b i l i t y o f b r e a k i n g

I n g e n e r a l , waves b r e a k when l o c a l l y t h e wave f r o n t becomes t o o s t e e p . F o r i r r e g u l a r waves t h i s may be t h e r e s u l t o f s e v e r a l mechanisms, such as i n t e r a c t i o n between s h o r t waves, i n t e r a c t i o n between wave and b o t t o m o r between wave and c u r r e n t o r wind. For s i m p l i c i t y , we s h a l l n o t c o n s i d e r t h e e f f e c t s o f c u r r e n t o r w i n d on wave b r e a k i n g . Even t h e n , t h e p r o c e s s e s i n v o l v e d a r e e x t r e m e l y complex and no a c c u r a t e model i s a v a i l a b l e t o p r e d i c t t h e

p r o b a b i l i t y o f b r e a k i n g i n i r r e g u l a r waves. T h e r e f o r e , a s i m p l e e m p i r i c a l approach i s chosen, based on some c r u d e a s s u m p t i o n s .

These a s s u m p t i o n s a r e :

1. The p r o b a b i l i t y o f b r e a k i n g depends o n l y on l o c a l and i n s t a n t a n e o u s wave p a r a m e t e r s . I n r e a l i t y , i t a l s o depends on t h e h i s t o r y o f t h e i n d i v i d u a l waves, b u t t h e b r e a k i n g p r o c e s s , e s p e c i a l l y i n random waves, has a t i m e - s c a l e w h i c h i s s h o r t compared t o t h e wave g r o u p s c a l e , so t h i s e f f e c t can be

n e g l e c t e d .

2. The b a s i c p a r a m e t e r s g o v e r n i n g t h e p r o b a b i l i t y o f b r e a k i n g a r e t h e l o c a l and i n s t a n t a n e o u s wave e n e r g y and t h e w a t e r d e p t h .

3. I n p r i n c i p l e , waves o f any e n e r g y may be b r e a k i n g o r n o n - b r e a k i n g . However, t h e p r o b a b i l i t y o f b r e a k i n g s h o u l d i n c r e a s e m o n o t o n i c a l l y t o w a r d s 1 f o r i n c r e a s i n g e n e r g y o r d e c r e a s i n g w a t e r d e p t h . T h o r n t o n and Guza (1983) p r o p o s e t h e f o l l o w i n g e m p i r i c a l ' w e i g h t i n g f u n c t i o n ' , w h i c h can be i n t e r p r e t e d as t h e p r o b a b i l i t y o f b r e a k i n g : "H n r m s 1 - exp -" H -" 2-1 y h :£ 1 [ 5 ] A c c o r d i n g t o t h i s e x p r e s s i o n , t h e p r o b a b i l i t y t h a t a p a r t i c u l a r wave i n an i r r e g u l a r wave t r a i n i s b r e a k i n g n o t o n l y depends on t h e h e i g h t o f t h i s wave r e l a t i v e t o t h e w a t e r d e p t h , b u t a l s o on a c h a r a c t e r i s t i c h e i g h t p a r a m e t e r o f t h e whole wave t r a i n ( i . e . Hrms). T h i s w o u l d i m p l y t h a t t h e b r e a k i n g p r o c e s s i n a g i v e n wave g r o u p i s i n f l u e n c e d by e v e n t s on a much g r e a t e r t i m e - s c a l e , w h i c h seems u n l i k e l y and i s i n c o n t r a d i c t i o n w i t h o u r a s s u m p t i o n 1. We

P (E,h) = 1 - exp b w i t h ; E = - pg h^ r e f 8 '^^ n/2 r e f [ 6 ] where y and n a r e c o e f f i c i e n t s . I n F i g u r e 1 t h i s f u n c t i o n i s p l o t t e d f o r s e v e r a l v a l u e s o f n. I t can be seen t h a t t h e s t e e p n e s s o f t h e f u n c t i o n i n c r e a s e s w i t h i n c r e a s i n g n. The two c o e f f i c i e n t s r and n w i l l have t o be d e t e r m i n e d e m p i r i c a l l y .

C o n d i t i o n a l expected d i s s i p a t i o n r a t e f o r waves w i t h g i v e n energy

The e x p e c t e d d i s s i p a t i o n r a t e , g i v e n a s p e c i f i c v a l u e o f E, i s now s i m p l y f o u n d by s u b s t i t u t i n g e q u a t i o n s [ 4 ] and [ 6 ] i n t o e q u a t i o n [ 1 ] , w h i c h l e a d s t o :

• 2 a f E [ 7 ] p

T h i s e q u a t i o n d e s c r i b e s t h e d i s s i p a t i o n r a t e f o r a g i v e n (random) wave e n e r g y and w a t e r d e p t h , as i s t h e t h e main g o a l o f t h i s s t u d y . The c a l i b r a t i o n o f t h e c o e f f i c i e n t s a, y and n and t h e v e r i f i c a t i o n o f t h e f o r m u l a t i o n as such i s d e s c r i b e d i n t h e f o l l o w i n g S e c t i o n s .

n/2

D = 1 - exp -

3 T r a n s f o r m a t i o n o f wave energy d i s t r i b u t i o n

3.1 P r o b a b i l i s t i c model

The f o r m u l a t i o n f o r t h e probabilistic approach can be d e r i v e d r e a d i l y f r o m t h e wave a c t i o n e q u a t i o n by assuming t h a t t h e s l o w l y - v a r y i n g c r o s s - s h o r e v e l o c i t y s m a l l compared t o t h e g r o u p v e l o c i t y C . 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 , e x c e p t f o r a l i m i t e d a r e a near t h e swash zone, where t h e group v e l o c i t y goes t o z e r o and t h e long-wave v e l o c i t i e s i n c r e a s e .

Under t h i s a s s u m p t i o n , t h e wave a c t i o n e q u a t i o n r e d u c e s t o t h e wave e n e r g y b a l a n c e :

§^UeC) =-D [ 8 ]

a t ox g Assuming C t o be c o n s t a n t i n t i m e , t h e r a t e o f change o f t h e e n e r g y f l u x o f a g ( p a r t o f a ) wave g r o u p as i t t r a v e l s t o w a r d s t h e s h o r e i s d e s c r i b e d by: ^ ( E C ) = -D [ 9 ] dx g

The d i s s i p a t i o n r a t e D depends on l o c a l wave p a r a m e t e r s and t h e s l o w l y - v a r y i n g w a t e r d e p t h . E x c e p t , a g a i n , f o r t h e swash zone, t h e slow f l u c t u a t i o n s i n t h e w a t e r l e v e l can be n e g l e c t e d . I n t h i s case, t h e time-dependence v a n i s h e s f r o m t h e e q u a t i o n , so i t can be s o l v e d f o r any g i v e n (seaward) boundary v a l u e o f E. I n o t h e r words, we can f o l l o w any p a r t o f a wave g r o u p t h r o u g h t h e s u r f zone u s i n g t h i s e q u a t i o n . As a r e s u l t , we can a l s o compute t h e t r a n s f o r m a t i o n o f t h e energy d i s t r i b u t i o n t h r o u g h t h e s u r f zone, s t a r t i n g f r o m a g i v e n

d i s t r i b u t i o n o f E i n deep w a t e r .

I n deep w a t e r , i t i s r e a s o n a b l e t o assume a R a y l e i g h - d i s t r i b u t i o n f o r t h e wave h e i g h t ; t h i s i s e q u i v a l e n t t o an e x p o n e n t i a l d i s t r i b u t i o n f o r t h e wave energy:

P (E < E) = 1 - exp (- — ) [ 1 0 ]

where E i s t h e t i m e - a v e r a g e d wave e n e r g y and P i s t h e p r o b a b i l i t y o f non-exceedance.

I n o r d e r t o compute t h e t r a n s f o r m a t i o n o f t h i s d i s t r i b u t i o n , t h e d i s t r i b u t i o n a t deep w a t e r i s g i v e n as a number o f e n e r g y l e v e l s w i t h d e c r e a s i n g

p r o b a b i l i t y o f exceedance. F o r each deep w a t e r e n e r g y l e v e l , e q u a t i o n [ 9 ] i s s o l v e d by e x p l i c i t n u m e r i c a l i n t e g r a t i o n . The r e s u l t i s a number o f wave

e n e r g y decay l i n e s , w h i c h cannot c r o s s each o t h e r . T h i s i s due t o t h e a s s u m p t i o n s made i n t h i s model, namely a c o n s t a n t group v e l o c i t y and a

d i s s i p a t i o n model w h i c h i s m o n o t o n i c a l l y dependent on t h e l o c a l wave energy.

As a r e s u l t , a l i n e w h i c h s t a r t s a t an e n e r g y l e v e l w i t h a c e r t a i n p r o b a b i l i t y o f exceedance w i l l r e p r e s e n t t h i s p r o b a b i l i t y t h r o u g h o u t t h e s u r f zone. A t any c o m p u t a t i o n p o i n t a l o n g t h e p r o f i l e , t h e d i s t r i b u t i o n o f t h e wave e n e r g y can be reassembled f r o m t h e s e l i n e s . The mean wave e n e r g y can be computed f r o m t h i s d i s t r i b u t i o n . A l s o , t h e t o t a l f r a c t i o n o f b r e a k i n g waves can be deduced f r o m t h e model.

The d i s t r i b u t i o n o f wave e n e r g y can be used t o p r e d i c t t h e wave h e i g h t d i s t r i b u t i o n by means o f a s u i t a b l e n o n - l i n e a r l o c a l wave model, w h i c h uses energy, peak f r e q u e n c y and w a t e r d e p t h as i n p u t . Here, we a p p l y t h e h i g h - o r d e r s t r e a m f u n c t i o n method as d e s c r i b e d by R i e n e c k e r & F e n t o n ( 1 9 8 1 ) . R e s u l t s a r e p r e s e n t e d below.

3.2 P a r a m e t r i c models

B a s i c c o n c e p t

I n t h e parametric c l a s s o f models, t h e e n e r g y b a l a n c e e q u a t i o n [ 8 ] i s a v e r a g e d o v e r a t i m e s c a l e w h i c h i s l a r g e compared w i t h t h e wave g r o u p t i m e s c a l e :

i -

= -D [ 1 1 ]

Assuming a s t a t i o n a r y wave f i e l d and no c o r r e l a t i o n between wave e n e r g y and g r o u p v e l o c i t y , t h i s e q u a t i o n reduces t o :

|^(È C^) = -D [ 1 2 ]

The mean d i s s i p a t i o n can be d e s c r i b e d as t h e w e i g h t e d average o f t h e d i s s i p a t i o n f u n c t i o n :

B = S

p ( E ) D(E) dE [ 1 3 ]

0

where p ( E ) i s t h e l o c a l p r o b a b i l i t y d e n s i t y f u n c t i o n ( p d f ) o f t h e wave e n e r g y . I n o r d e r t o c l o s e t h e e q u a t i o n s , an a s s u m p t i o n must be made r e g a r d i n g t h e shape o f t h i s f u n c t i o n , depending on t h e l o c a l wave p a r a m e t e r s . The s c a l i n g o f t h e f u n c t i o n t h e n f o l l o w s f r o m t h e r e q u i r e m e n t s t h a t t h e f u n c t i o n i s a p d f :

S

p ( E ) dE = 1 [ 1 4 ]

0

and t h a t t h e f i r s t moment e q u a l s t h e mean energy:

00

S

p ( E ) E dE = Ë [ 1 5 ] 0 I n t h e f o l l o w i n g , t h r e e p a r a m e t r i c p r o b a b i l i t y d e n s i t y f u n c t i o n s a r e d i s c u s s e d , v i z . a d e p t h l i m i t e d W e i b u l l d i s t r i b u t i o n , t h e R a y l e i g h -d i s t r i b u t i o n an-d t h e c l i p p e -d R a y l e i g h - -d i s t r i b u t i o n a c c o r -d i n g t o B a t t j e s an-d Janssen ( 1 9 7 8 ) . The f o l l o w i n g p a r a m e t e r s w i l l be used i n o r d e r t o s i m p l i f y t h e e q u a t i o n s : 1 2 Ë E " E = 1 pg h'^ , (T = . E = -, Q = J P J E ) p ( E ) dE r e f 8 t, ** =; b b r e f E 0

W e i b u l l d i s t r i b u t i o n

Klopman and S t i v e (1989) propose a wave h e i g h t d i s t r i b u t i o n , based upon a shape o r i g i n a l l y proposed by G l u k h o v s k i y ( 1 9 6 6 ) , w h i c h d e g e n e r a t e s t o a

R a y l e i g h - d i s t r i b u t i o n i n deep w a t e r , b u t has a d e p t h - l i m i t a t i o n r e s u l t i n g i n a g r a d u a l d e f o r m a t i o n o f t h e d i s t r i b u t i o n f o r d e c r e a s i n g w a t e r d e p t h . I n t e r m s o f wave e n e r g y , t h i s d i s t r i b u t i o n can be w r i t t e n as:

P (E < E) = 1 - exp { -A (§)" } [ 1 6 ] E

Here, m i s a f r e e p a r a m e t e r f o r w h i c h Klopman and S t i v e propose a f o r m u l a t i o n , w h i c h i s r e w r i t t e n h e r e i n terms o f energy:

m = 1 + 0.7 t a n ^ (^ i / f ) = 1 + 0 . 7 t a n ^ (J J ) [ 1 7 ]

^ ^ 2 ^ e f ^ ^ 2

The v a l u e o f as g i v e n by Klopman and S t i v e i s t h e t h e o r e t i c a l maximum o f the wave h e i g h t over d e p t h r a t i o , 0.833. The maximum v a l u e f o r t h e

e n e r g y - r e l a t e d cr-value as d e f i n e d above i s i n t h e o r d e r o f 30 % l o w e r , due t o t h e n o n - l i n e a r i t y o f d e p t h - l i m i t e d waves. T h e r e f o r e a v a l u e o f 0.65 has been used here. The parameter A i s l i n k e d t o m t h r o u g h t h e r e q u i r e m e n t g i v e n by e q u a t i o n [ 1 5 ] : [ 1 8 ] A =

r ( 1 + i )

m The p r o b a b i l i t y d e n s i t y f u n c t i o n i s f o u n d by d i f f e r e n t i a t i n g e q u a t i o n [ 1 6 ] p ( E ) = (^) exp { -A (§)"* > [ 1 9 ] E E E

The mean d i s s i p a t i o n i s now f o u n d by i n t e g r a t i o n o f e q u a t i o n [ 1 3 ] :

D = J- p ( E ) P (E) D (E) dE b b 0 0 E E E 1 - exp If E n/2 2 a f E

p mA S E"; exp(-A E^^) 1-exp

r e f <T_ n/2 2 • * 2 a f E dE P dE.

= 2 a f E f ( c . r . n )

p 1

[ 2 0 ]

The r e s u l t I s t h a t t h e mean d i s s i p a t i o n i s t h e d i s s i p a t i o n i n waves w i t h t h e mean energy, t i m e s a f u n c t i o n o f t h e wave energy r e l a t i v e t o t h e w a t e r d e p t h . T h i s f u n c t i o n i s l e s s t h a n o r e q u a l t o 1, and depends on t h e l o c a l wave h e i g h t t o w a t e r d e p t h r a t i o a- and on t h e e m p i r i c a l c o e f f i c i e n t s y and n.

R a y l e i g h d i s t r i b u t i o n

The R a y l e i g h d i s t r i b u t i o n i s a s p e c i a l case o f e q u a t i o n [ 1 6 ] f o r m e q u a l t o 1. I t has been used by T h o r n t o n and Guza ( 1 9 8 3 ) , i n c o m b i n a t i o n w i t h a s l i g h t l y d i f f e r e n t f o r m u l a t i o n f o r t h e d i s s i p a t i o n . The mean d i s s i p a t i o n f o l l o w s i m m e d i a t e l y f r o m e q u a t i o n [ 2 0 ] and i s g i v e n by: 00 D = 2 a f Ë J E, e x p ( - E J = 2 a f Ë f ( ( r , r , n ) [ 2 1 ] P 2 C l i p p e d R a y l e i g h d i s t r i b u t i o n 1-exp dE.

The c l i p p e d R a y l e i g h d i s t r i b u t i o n as proposed by B a t t j e s and Janssen (1978) i s based on t h e a s s u m p t i o n s t h a t t h e wave h e i g h t s a r e R a y l e i g h - d i s t r i b u t e d up t o a maximum wave h e i g h t , t h a t a l l h i g h e r waves a r e s i m p l y c u t o f f t o t h i s

h e i g h t , t h a t a l l waves h a v i n g t h i s maximum h e i g h t a r e b r e a k i n g and t h a t o n l y t h e s e waves a r e b r e a k i n g . T h i s can be t r a n s l a t e d t o o u r c o n c e p t by l e t t i n g t h e v a l u e o f n i n t h e p r o b a b i l i t y o f b r e a k i n g go t o i n f i n i t y , i n w h i c h case t h e f u n c t i o n becomes a s t e p f u n c t i o n : z e r o f o r E/E < y^, u n i t y f o r E/E a

r e f r e f 2

•y .The maximum wave energy i s d e f i n e d by:

E = / E [ 2 2 ] m r e f S i n c e t h e p r o b a b i l i t y d e n s i t y f u n c t i o n has a ' s p i k e ' a t E=E^, w i t h an a r e a e q u a l t o t h e f r a c t i o n o f b r e a k i n g waves Q , and s i n c e t h e p r o b a b i l i t y o f b b r e a k i n g e q u a l s u n i t y a t t h i s energy, we g e t f o r t h e mean d i s s i p a t i o n :

B = S

p(E) P (E) D (E) dE = D (E )

b b b b m 0

= Q 2 a f E = 2 a f E ^ Q ^

b p m p ^ 2 b

I n t h e c l i p p e d R a y l e i g h d i s t r i b u t i o n , t h e f r a c t i o n o f b r e a k i n g waves i s d e f i n e d by t h e i m p l i c i t r e l a t i o n : Q = exp [- ] [ 2 4 ] ^ Ë/E - 2 2 T h i s r e l a t i o n y i e l d s a u n i q u e f u n c t i o n o f E/E = cr /•)• , so: ID D = 2 a f Ë f ((r,y) [ 2 5 ] P 3 For t h e t h r e e p a r a m e t r i c energy d i s t r i b u t i o n s , we g e t s i m i l a r e x p r e s s i o n s f o r t h e mean d i s s i p a t i o n . F o r g i v e n v a l u e s o f t h e c a l i b r a t i o n p a r a m e t e r s y and n t h e f u n c t i o n s f , f and f depend o n l y on cr. T h e r e f o r e i t i s easy t o g e n e r a t e

1 2 3

t a b l e s o f t h e s e f u n c t i o n s and t o i n t e r p o l a t e f r o m t h e s e t a b l e s when s o l v i n g t h e mean e n e r g y b a l a n c e e q u a t i o n .

4. C a l i b r a t i o n o f t h e models

Three d a t a s e t s , c o n t a i n i n g a t o t a l o f 11 t e s t s , were used t o c a l i b r a t e t h e p r o b a b i l i s t i c model and t h e p a r a m e t r i c models, v i z . t h o s e r e p o r t e d i n B a t t j e s & Janssen ( 1 9 7 8 ) , S t i v e (1985) and H o t t a & M i z u g u c h i (1980). A summary o f t h e c h a r a c t e r i s t i c s o f t h e p r o f i l e and i n c i d e n t wave c o n d i t i o n s i s g i v e n i n T a b l e 1. A l l s e t s p e r t a i n t o i r r e g u l a r waves i n c i d e n t p e r p e n d i c u l a r t o a beach. The l e t t e r L under 'Type' s t a n d s f o r l a b o r a t o r y t e s t , F s t a n d s f o r f i e l d t e s t . Table 1 . E x p e r i m e n t a l p a r a m e t e r s c a l i b r a t i o n s e t s T e s t Source Type h (m) 0 E , 0 f (Hz) P MSIO S t i v e ( 1 9 8 5 ) L , p l a n e 0.70 . 142 .341 MS40 S t i v e ( 1 9 8 5 ) L , p l a n e 0.70 . 135 .633 BJ2 B a t t J e s & J a n s s e n (1978) L , p l a n e 0.70 . 144 .511 BJ3 B a t t J e s & J a n s s e n (1978) L , p l a n e 0.70 . 122 . 383 BJ4 B a t t J e s & J a n s s e n (1978) L , p l a n e 0.70 . 143 . 435 B J l l B a t t J e s & J a n s s e n (1978) L , b a r r e d 0.70 . 137 . 450 BJ12 B a t t J e s & J a n s s e n (1978) L , b a r r e d 0.70 . 121 . 443 BJ13 B a t t J e s & J a n s s e n (1978) L , b a r r e d 0.70 . 104 . 467 BJ14 B a t t J e s & J a n s s e n (1978) L , b a r r e d 0.70 . 118 . 481 BJ15 B a t t J e s & J a n s s e n (1978) L , b a r r e d 0.70 . 143 .498 HotMiz H o t t a & M l z u g u c h l (1980) F , b a r r e d 1.65 .527 . 113 The p a r a m e t e r f o r w h i c h t h e c a l i b r a t i o n was p e r f o r m e d i s t h e o v e r a l l

energy-based wave h e i g h t o f t e n ( c o n f u s i n g l y ) r e f e r r e d t o as H . Here i t w i l l

r m s

be termed H :

E

I n deep w a t e r , H i s e q u a l t o t h e root-mean-square wave h e i g h t H ; i n

E rms

s h a l l o w w a t e r , due t o n o n - l i n e a r i t y o f t h e waves, t h e p a r a m e t e r s d e v i a t e f r o m each o t h e r .

The seawardmost d a t a p o i n t , h a v i n g a wave h e i g h t H , i s used as a boundary

E , 0

c o n d i t i o n f o r t h e models. For a g i v e n s e t o f c a l i b r a t i o n p o i n t s , t h e e n e r g y d i s t r i b u t i o n a c r o s s each p r o f i l e i s computed and compared t o t h e measured d i s t r i b u t i o n . Two i n d i c a t o r s o f t h e o v e r a l l a c c u r a c y o f t h e models a r e computed, v i z . t h e root-mean-square r e l a t i v e e r r o r e and t h e r e l a t i v e

rms

b i a s (mean e r r o r ) e :

H E W E,comp l E , 0 H _ E H E,meas E.O . L y E , m e i s N ^ H [27] E , 0 H H E,comp E,meas E , 0 E , 0 H E,meas [ 2 8 ] As i s a p p a r e n t f r o m t h e f o r m u l a e , t h e e r r o r s were s c a l e d w i t h t h e i n c i d e n t wave h e i g h t ; t h i s i s t o g i v e d a t a p o i n t s comparable w e i g h t s r e g a r d l e s s o f t h e s c a l e o f t h e t e s t s o r t h e i n c i d e n t c o n d i t i o n s . From p r e l i m i n a r y c o m p u t a t i o n s , i t t u r n e d o u t t h a t t h e r e s u l t s were n o t v e r y s e n s i t i v e t o t h e v a l u e o f n, w h i c h i n d i c a t e s t h e s t e e p n e s s o f t h e c u r v e w h i c h d e s c r i b e s t h e p r o b a b i l i t y o f b r e a k i n g . R e a l i s t i c r e s u l t s were o b t a i n e d b o t h f o r n=10 and f o r n=20.

The optimum c o m b i n a t i o n o f t h e c o e f f i c i e n t s a and r was o b t a i n e d by d r a w i n g i s o l i n e s o f t h e e r r o r i n d i c a t o r s i n t h e a,y p l a n e , f o r b o t h v a l u e s o f n, and v i s u a l l y d e t e r m i n g t h e a p p r o x i m a t e l o c a t i o n o f z e r o mean e r r o r and minimum rms e r r o r . P l o t s o f t h e s e i s o l i n e s a r e g i v e n i n F i g u r e s 2a t o 2g. By r e f i n i n g t h e a,y g r i d l o c a l l y and l o o k i n g a t t h e n u m e r i c a l o u t p u t , a more a c c u r a t e l o c a t i o n o f t h i s optimum was t h e n f o u n d .

I n a l l cases t h e optimum a , r - c o m b i n a t i o n i s f o u n d c l o s e t o t h e l i n e a = l . As a c o n s t a n t v a l u e o f a f a c i l i t a t e s t h e c o m p a r i s o n o f t h e d i f f e r e n t models, t h e v a l u e o f a was f i x e d a t 1, and optimum ^r-values were d e t e r m i n e d f o r each model and n - v a l u e . The r e s u l t s a r e g i v e n i n T a b l e 2.

T a b l e 2. Optimum y - v a l u e s and r e l a t i v e rms e r r o r f o r a = l and n=10,20 11 d a t a s e t s , 159 p o i n t s Model n a

r

e F i g . r m s P r o b a b i l i s t i c 10 1. 0 0. 55 0.045 2a 20 1. 0 0. 53 0.054 2b W e i b u l l 10 1. 0 0. 54 0.057 2c 20 1. 0 0. 52 0.057 2d R a y l e i g h 10 1. 0 0 57 0.062 2e 20 1. 0 0 57 0.063 2 f C l i p p e d R a y l e i g h - 1. 0 0 66 0.056 2g

A p p a r e n t l y , a l l models can be c a l i b r a t e d t o g i v e r e a s o n a b l y a c c u r a t e

p r e d i c t i o n s o f t h e s p a t i a l wave energy v a r i a t i o n f o r a f i x e d c o m b i n a t i o n o f c a l i b r a t i o n c o e f f i c i e n t s . The p r o b a b i l i s t i c model seems t h e most a c c u r a t e , whereas t h e p a r a m e t r i c R a y l e i g h model w i t h n s e t a t 10 o r 20 g i v e s t h e g r e a t e s t e r r o r .

The c l i p p e d R a y l e i g h model ( n = oo) seems t o do w e l l w i t h a c o n s t a n t

a , ^ - c o m b i n a t i o n . B a t t j e s and S t i v e (1985) used an e x p r e s s i o n f o r t h e maximum wave h e i g h t w h i c h i n c l u d e s t h e e f f e c t o f wave s t e e p n e s s ; c o n s e q u e n t l y t h e y f o u n d t h a t t h e c a l i b r a t i o n c o e f f i c i e n t y showed a dependence o f t h e deep w a t e r steepness. I t seems t h a t u s i n g t h e s i m p l e r r e l a t i o n [ 2 2 ] removes t h i s

dependence. The optimum y - v a l u e s f o r t h e W e i b u l l p a r a m e t r i c model and t h e p r o b a b i l i s t i c model a g r e e c l o s e l y , w h i c h i n d i c a t e s t h a t t h e energy

d i s t r i b u t i o n s r e s u l t i n g f r o m t h e p r o b a b i l i s t i c model a r e s i m i l a r t o t h e shape assumed b e f o r e h a n d i n t h e p a r a m e t r i c model.

I n t h e R a y l e i g h model w i t h f i n i t e n, h i g h e r wave e n e r g y i s p o s s i b l e t h a n i n t h e W e i b u l l model, so t h e p r o b a b i l i t y o f b r e a k i n g f o r a g i v e n energy must d e c r e a s e i n o r d e r t o g e t t h e same mean d i s s i p a t i o n . T h i s r e s u l t s i n a h i g h e r optimum v a l u e o f

I n t h e c l i p p e d R a y l e i g h model, i t i s assumed t h a t a l l b r e a k i n g waves have t h e maximum wave energy. The y - v a l u e i n t h i s case i n d i c a t e s t h e l e v e l where most d i s s i p a t i o n t a k e s p l a c e . The optimum v a l u e o f 0.66 i s n o t i n c o n t r a d i c t i o n w i t h t h e o t h e r models. A v a l u e o f n e q u a l t o 10 g i v e s s l i g h t l y b e t t e r r e s u l t s t h a n n e q u a l t o 20; f o r t h e p r o b a b i l i s t i c model a v a l u e o f 5 was t r i e d b u t p r o d u c e d no b e t t e r r e s u l t s . The v a l u e o f n was k e p t a t 10 i n a l l f u r t h e r c o m p u t a t i o n s . I n F i g u r e s 3a t h r o u g h 3 1 , t h e wave h e i g h t p r o f i l e s as computed w i t h t h e p r o b a b i l i s t i c model, f o r n e q u a l 10, a r e compared w i t h t h e measured wave h e i g h t p r o f i l e s , f o r a l l c a l i b r a t i o n t e s t s . The agreement i s q u i t e good, e s p e c i a l l y c o n s i d e r i n g t h a t a l l c o m p u t a t i o n s were p e r f o r m e d w i t h t h e same s e t o f c o e f f i c i e n t s .

5. V e r i f i c a t i o n

The p r i m a r y g o a l o f t h i s s t u d y i s t o f o r m u l a t e t h e t i m e - v a r y i n g d i s s i p a t i o n as a f u n c t i o n o f l o c a l wave p a r a m e t e r s . As t h i s i s o n l y one o f t h e i n t e r n a l

p a r a m e t e r s i n t h e models d e s c r i b e d above, t h e f a c t t h a t t h e mean wave e n e r g y ( t h e e x t e r n a l p a r a m e t e r ) i s p r e d i c t e d a c c u r a t e l y i s n o t s u f f i c i e n t ; e r r o r s i n i n t e r n a l p a r a m e t e r s may be c a n c e l l e d o u t by each o t h e r .

The dependence scheme i n F i g u r e 4. i n d i c a t e s w h i c h o t h e r i n t e r n a l p a r a m e t e r s must be checked I n o r d e r t o g a i n c o n f i d e n c e i n t h e f o r m u l a t i o n o f t h e e x p e c t e d v a l u e o f t h e s l o w l y - v a r y i n g d i s s i p a t i o n . d i s s i p a t i o n i n b r e a k i n g waves p r o b a b i l i t y o f b r e a k i n g wave e n e r g y d i s t r i b u t i o n e x p e c t e d s l o w l y - v a r y i n g d i s s i p a t i o n f r a c t i o n o f b r e a k i n g waves s t a t i s t i c a l d i s t r i b u t i o n o f wave h e i g h t s 1 1-average d i s s i p a t i o n

F i g u r e 4. Dependence scheme d i s s i p a t i o n model

I n t h e f o l l o w i n g S e c t i o n s , t h e numbered i t e m s i n t h e dependence scheme w i l l be d i s c u s s e d s e p e r a t e l y ; a f t e r w a r d s , c o n c l u s i o n s a r e drawn on t h e a c c u r a c y o f t h e model o f t h e e x p e c t e d s l o w l y - v a r y i n g d i s s i p a t i o n .

5.1 Average d i s s i p a t i o n

As has been shown i n t h e p r e v i o u s S e c t i o n , t h e average d i s s i p a t i o n i s m o d e l l e d a c c u r a t e l y . An i n d e p e n d e n t v e r i f i c a t i o n i s g i v e n by two a d d i t i o n a l d a t a s e t s , v i z . t h o s e r e p o r t e d by E b e r s o l e and Hughes (1987) and by Van d e r Meer ( 1 9 9 0 ) . The I n c i d e n t wave c o n d i t i o n s a r e g i v e n i n T a b l e 3.

Table 3. E x p e r i m e n t a l p a r a m e t e r s v e r i f i c a t i o n s e t s T e s t Source Type h^(m) E , 0 f (Hz) _p D41400 Ebersole&Hughes (1987) F , b a r r e d 1.75 .600 .089 D41510 Ebersole&Hughes (1987) F , b a r r e d 1.49 .706 .089 D50955 Ebersole&Hughes (1987) F , b a r r e d 2.14 .431 .088 D51055 Ebersole&Hughes (1987) F , b a r r e d 1.80 .353 .089 D51352 Ebersole&Hughes (1987) F , b a r r e d 2. 19 .452 .092 D51525 Ebe r s o1e&Hughes (1987) F , b a r r e d 1.94 .374 .090 D60915 Ebersole&Hughes (1987) F , b a r r e d 1.40 .360 .078 D61015 Ebersole&Hughes (1987) F , b a r r e d 2.14 .296 .076 D61300 Ebersole&Hughes (1987) F , b a r r e d 2.43 .346 .099 T007 van d e r Meer (1990) L , s t e p 0.56 .049 .403 T015 van d e r Meer (1990) L , s t e p 0.56 .071 .438 T l l O van d e r Meer (1990) L , s t e p 0.56 .099 .513 T12 van d e r Meer (1990) L , s t e p 0.71 .059 . 488 T13 van d e r Meer (1990) L , s t e p 0.66 . 109 .488 T212 van d e r Meer (1990) L , s t e p 0.61 .072 .645 T216 van d e r Meer (1990) L , s t e p 0.61 .068 .403 T322 van d e r Meer (1990) L , s t e p 0.66 . 121 .513

The d a t a s e t by E b e r s o l e and Hughes was o b t a i n e d i n t h e f i e l d d u r i n g t h e DUCK85 campaign. I t c o n c e r n s l o n g - p e r i o d s w e l l i n c i d e n t p e r p e n d i c u l a r t o an a l m o s t p r i s m a t i c beach. The measurements were c a r r i e d o u t w i t h t h e p h o t o p o l e

t e c h n i q u e ( H o t t a and M i z u g u c h i , 1980). The measurements have been s t u d i e d i n d e t a i l by D a l l y ( 1 9 9 0 ) . A p r o b l e m w i t h h i n d c a s t i n g t h e s e e x p e r i m e n t s w i t h t h e p r e s e n t model i s t h a t t h e wave h e i g h t d i s t r i b u t i o n s a t t h e o u t e r m o s t m e a s u r i n g p o i n t d e v i a t e s i g n i f i c a n t l y f r o m e i t h e r R a y l e i g h o r W e i b u l l d i s t r i b u t i o n s ; t h e r e f o r e we cannot e x p e c t v e r y good agreement. S t i l l , t h e measurements have been i n c l u d e d as a s e v e r e t e s t case. The models were a p p l i e d w i t h t h e i r p r e - c a l i b r a t e d c o e f f i c i e n t v a l u e s : a = l , n=10 and y as i n T a b l e 2. Model p e r f o r m a n c e was r e a s o n a b l e f o r a l l models: f o r t h e 9 e x p e r i m e n t s , t h e mean e r r o r was l e s s t h a n 2% f o r a l l models and t h e rms e r r o r was i n t h e o r d e r o f 13%. The p r o b a b i l i s t i c model was n o t s i g n i f i c a n t l y b e t t e r t h a n t h e p a r a m e t r i c mode1s.

The d a t a s e t by Van d e r Meer c o n c e r n s l a b o r a t o r y cases o f waves i n c i d e n t on a p r o f i l e w i t h a s t e e p s t e p f o l l o w e d by a v e r y g e n t l y s l o p i n g b o t t o m . Here, t h e p a r a m e t r i c models show a mean e r r o r i n t h e o r d e r o f 1 % and a rms e r r o r i n t h e o r d e r o f 1 1 % o v e r a t o t a l o f 8 t e s t s . The p r o b a b i l i s t i c model shows a mean e r r o r o f a l m o s t 5%, b u t a l o w e r rms e r r o r o f 8%. The g e n e r a l shape o f t h e energy d i s t r i b u t i o n s o v e r t h e p r o f i l e i s r e p r e s e n t e d b e s t by t h e p r o b a b i l i s t i c model; hence t h e l o w e r rms e r r o r .

The e r r o r i n d i c a t o r s were a l s o computed over a l l t e s t s c o n s i d e r e d i n t h i s s t u d y ; t h e r e s u l t s a r e g i v e n i n T a b l e 4. Table 4 . R e l a t i v e 28 t e s t s , mean 389 and rms e r r o r p o i n t s Model n a G e m e a n r m s P r o b a b i l ] L s t i c 10 1 0 0.55 0 .013 0 .088 W e i b u l l 10 1 0 0.54 0 .000 0 .099 R a y l e i g h 10 1 0 0.57 0 .011 0 .099 C l i p p e d R a y l e i g h 00 1 0 0.66 0 .000 0 .096

A l l models can be used t o p r e d i c t t h e v a r i a t i o n o f t h e mean wave e n e r g y o v e r t h e p r o f i l e ; t h e p r o b a b i l i s t i c model i s s l i g h t l y more a c c u r a t e i n t h i s

r e s p e c t . A c o m p a r i s o n between t h e measured wave h e i g h t p r o f i l e s and t h o s e computed w i t h t h e p r o b a b i l i s t i c model i s g i v e n i n F i g u r e s 3a t o 3z.

5.2 S t a t i s t i c a l d i s t r i b u t i o n of wave h e i g h t s

At p r e s e n t , no d a t a a r e a v a i l a b l e on t h e p r o b a b i l i t y d i s t r i b u t i o n o f t h e wave energy; d a t a on wave h e i g h t p r o b a b i l i t y d i s t r i b u t i o n s a r e a v a i l a b l e . W i t h t h e h e l p o f n o n - l i n e a r t h e o r y , wave h e i g h t s can be e s t i m a t e d f r o m wave e n e r g y

l e v e l s . I f t h e v a r i a t i o n o f s t a t i s s t i c a l wave h e i g h t p a r a m e t e r s o v e r t h e p r o f i l e i s p r e d i c t e d c o r r e c t l y , t h e u n d e r l y i n g e n e r g y p r o b a b i l i t y

d i s t r i b u t i o n s a r e l i k e l y t o be c o r r e c t as w e l l .

The s t a t i s t i c a l wave h e i g h t p a r a m e t e r s a r e deduced f r o m t h e p r e d i c t e d wave e n e r g y p r o b a b i l i t y d i s t r i b u t i o n by t h e f o l l o w i n g method. The d i s t r i b u t i o n o f t h e l i n e a r e s t i m a t e o f t h e wave h e i g h t , , was d e r i v e d f r o m t h e wave e n e r g y

d i s t r i b u t i o n , where = / (8E/pg). The s t a t i s t i c a l p a r a m e t e r s H^^^^^ and were computed f r o m t h i s d i s t r i b u t i o n , u s i n g t h e u s u a l d e f i n i t i o n s .

The m a t c h i n g n o n - l i n e a r c r e s t - t o - t r o u g h h e i g h t s were t h e n computed w i t h t h e h e l p o f R i e n e c k e r & Fenton's (1981) s t r e a m f u n c t i o n method.

The d a t a s e t used i s f r o m S t i v e ( 1 9 8 5 ) , t e s t s MSIO and MS40. I n F i g u r e s 5a and 5b, t h e d i s t r i b u t i o n s o f t h e rms wave h e i g h t H , t h e s i g n i f i c a n t wave h e i g h t

rms

H and t h e wave h e i g h t exceeded 1 % o f t h e t i m e , H l % , as measured and as

s l g

computed, a r e g i v e n . Q u a l i t a t i v e l y , t h e agreement i s q u i t e good;

t h e model by t h e mean d e p t h , whereas i n r e a l i t y t h e y a r e l i m i t e d by t h e

s l o w l y - f l u c t u a t i n g d e p t h . A l s o w i t h i n t h e s u r f zone, as can be e x p e c t e d , t h e v a l u e s o f t h e rms wave h e i g h t a r e o v e r p r e d i c t e d by n o n - l i n e a r t h e o r y , due t o t h e d e c o u p l i n g o f t h e h i g h e r h a r m o n i c s . S t i l l , t h e wave h e i g h t d i s t r i b u t i o n s a r e p r e d i c t e d w e l l enough t o l e n d some c o n f i d e n c e t o t h e computed e n e r g y d i s t r i b u t i o n s . F u r t h e r s t u d y i s r e q u i r e d t o c o n f i r m t h i s . 5.3 F r a c t i o n o f b r e a k i n g waves The f r a c t i o n o f b r e a k i n g waves, Q , i s t h e i n t e g r a l o f t h e p r o d u c t o f t h e b e n e r g y d i s t r i b u t i o n and t h e p r o b a b i l i t y o f b r e a k i n g a t a g i v e n energy. T h e r e f o r e , i f t h e e n e r g y d i s t r i b u t i o n i s m o d e l l e d c o r r e c t l y , t h e f r a c t i o n o f b r e a k i n g waves can o n l y be c o r r e c t i f t h e p r o b a b i l i t y o f b r e a k i n g i s c o r r e c t t o o . I n t h e same two t e s t s f r o m S t i v e ( 1 9 8 5 ) , t h e f r a c t i o n o f b r e a k i n g waves was c o u n t e d v i s u a l l y ; measured and p r e d i c t e d v a l u e s a r e shown i n F i g u r e s 5a and 5b. For t e s t MSIO, t h e agreement i s q u i t e good, a l t h o u g h w e l l I n s i d e t h e s u r f zone Q i s somewhat u n d e r p r e d i c t e d , by up t o 30%. For t e s t MS40, t h e u n d e r p r e d i c t i o n i s much more s e r i o u s : up t o a f a c t o r 3. T h i s seems s t r a n g e s i n c e t h e d i s s i p a t i o n i s m o d e l l e d so w e l l .

A p o s s i b l e e x p l a n a t i o n i s t h a t t h e peak p e r i o d d u r i n g t h i s t e s t was t w i c e as s h o r t as d u r i n g t e s t MSIO. The waves t h e r e f o r e tended t o be s p i l l i n g , whereas t h e b o r e model o n l y r e a l l y a p p l i e s t o f u l l y b r e a k i n g waves w i t h a r o l l e r o v e r t h e whole wave f r o n t . I n such a case, t h e c r i t e r i o n t h a t a b r e a k i n g wave i s 'a wave w i t h foam on i t ' w i l l o v e r e s t i m a t e t h e f r a c t i o n o f b o r e - l i k e b r e a k i n g waves.

The q u e s t i o n i s , w h e t h e r t h i s a n a l y s i s s h o u l d l e a d t o an a d j u s t m e n t o f t h e d e f i n i t i o n o f ' b r e a k i n g wave' o r t o an a d j u s t m e n t o f t h e model o f t h e

d i s s i p a t i o n i n a b r e a k i n g wave. T h i s s h o u l d be r e s o l v e d i n f u r t h e r s t u d y ; f o r now, t h e computed f r a c t i o n o f b r e a k i n g waves s h o u l d be i n t e r p r e t e d as t h e f r a c t i o n o f f u l l y - b r e a k i n g , b o r e - l i k e waves.

5. 4 P r o b a b i l i t y of b r e a k i n g

The p r o b a b i l i t y o f b r e a k i n g as a f u n c t i o n o f wave h e i g h t has been i n v e s t i g a t e d by some a u t h o r s ( T h o r n t o n and Guza, 1983; D a l l y , 1990). T h i s g i v e s a

q u a l i t a t i v e check on t h e shape o f t h i s p r o b a b i l i t y as a f u n c t i o n o f energy. However, t h e a v a i l a b l e d a t a do n o t e n a b l e a d i r e c t p l o t o f t h e p r o b a b i l i t y o f b r e a k i n g a g a i n s t t h e wave e n e r g y , f o r g i v e n w a t e r d e p t h s , so a d i r e c t

5.5 D i s s i p a t i o n i n b r e a k i n g waves

\n i n t e r e s t i n g v e r i f i c a t i o n o f t h e f o r m u l a t i o n o f t h e d i s s i p a t i o n i n b r e a k i n g waves i s o b t a i n e d f r o m t h e measurements o f S t i v e ( 1 9 8 4 ) , f o r r e g u l a r waves. On page 109 o f h i s paper, he p r e s e n t s graphs o f a n o n - d i m e n s i o n a l d i s s i p a t i o n A , d e f i n e d as t h e r a t i o o f t h e d i s s i p a t i o n r a t e d e r i v e d f r o m measured e n e r g y f l u x g r a d i e n t s and t h e d i s s i p a t i o n a c c o r d i n g t o a h y d r a u l i c Jump:

° b = ^ i f d V f ^ ^ ^

1 2

where d^ i s t h e d e p t h i n f r o n t o f t h e b r e a k e r and i s t h e d e p t h a t t h e c r e s t . The v a l u e s o f A^ a r e i n t h e range o f 1.5 t o 2.5. I f we now assume:

1 2 we g e t a s i m i l a r e x p r e s s i o n t o our e q u a t i o n [ 4 ] i f a = A^ y^. W i t h t h e o r d e r o f magnitude e s t i m a t e s H/h 0.5, d /h 0.8 and d /h 1.3, 9' i s i n t h e 1 2 3 o r d e r o f 0.5, w h i c h l e a d s t o an a - v a l u e i n t h e o r d e r o f 1. T h i s i s i n agreement w i t h t h e optimum v a l u e f o u n d i n t h e c a l i b r a t i o n . 5.6. C o n c l u s i o n s on v e r i f i c a t i o n

From t h e v e r i f i c a t i o n p r e s e n t e d h e r e we may draw t h e c o n c l u s i o n t h a t t h e mean d i s s i p a t i o n i s m o d e l l e d c o r r e c t l y and t h a t t h e r e a r e i n d i c a t i o n s t h a t t h e wave e n e r g y d i s t r i b u t i o n i s a l s o m o d e l l e d c o r r e c t l y ; t h e r e f o r e , t h e e x p e c t e d

t i m e - v a r y i n g d i s s i p a t i o n must be r e a s o n a b l y a c c u r a t e . T h i s t e r m i s a g a i n

composed o f two t e r m s , v i z . t h e t i m e - v a r y i n g d i s s i p a t i o n i n b r e a k i n g waves and t h e p r o b a b i l i t y o f b r e a k i n g . There i s an i n d i c a t i o n t h a t t h e f i r s t o f t h e s e t e r m s i s m o d e l l e d c o r r e c t l y ; on t h e p r o b a b i l i t y o f b r e a k i n g t h e r e i s s t i l l some u n c e r t a i n t y . Q u a l i t a t i v e l y , t h e r e i s agreement between measured and p r e d i c t e d f r a c t i o n s o f b r e a k i n g waves; q u a n t i t a t i v e l y , t h e y a r e somewhat u n d e r p r e d i c t e d , a l t h o u g h o f t h e r i g h t o r d e r o f magnitude.

6. C o n c l u s i o n s

The e x i s t i n g p a r a m e t r i c model a c c o r d i n g t o B a t t j e s and Janssen has been improved i n t h e sense t h a t t h e i n t e r n a l p a r a m e t e r s a r e more r e a l i s t i c ; a l s o , t h e dependence o f t h e c a l i b r a t i o n c o e f f i c i e n t s on wave s t e e p n e s s has v a n i s h e d . A p a r a m e t r i c model based on a W e i b u l l d i s t r i b u t i o n has been added t o t h i s c l a s s o f models, f o r w h i c h t h e d i s t r i b u t i o n s c l o s e l y resemble t h o s e r e s u l t i n g f r o m t h e p r o b a b i l i s t i c model. A l l t h r e e p a r a m e t r i c models can be used t o

p r e d i c t w i t h r e a s o n a b l e a c c u r a c y t h e s p a t i a l d i s t r i b u t i o n o f t h e mean wave energy; t h e one based on a W e i b u l l d i s t r i b u t i o n i s t h e most a c c u r a t e , and t h e model based on t h e R a y l e i g h d i s t r i b u t i o n t h e l e a s t a c c u r a t e .

The p r o b a b i l i s t i c model p r e s e n t e d has been shown t o f o l l o w f r o m t h e wave a c t i o n e q u a t i o n i f t h e g r o u p v e l o c i t y i s assumed t o be c o n s t a n t i n t i m e and e f f e c t s o f s u r f b e a t can be n e g l e c t e d . These r e s t r i c t i o n s a r e l e s s severe t h a n t h o s e f o r t h e e a r l i e r models i n t h i s c l a s s , w h i c h r e q u i r e a n e g l i g i b l e

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

e q u a t i o n s i s used t h r o u g h o u t t h e s h o a l i n g and b r e a k i n g r e g i o n , as opposed t o e a r l i e r models i n t h i s c l a s s . The model can be used t o p r e d i c t t h e

t r a n s f o r m a t i o n o f t h e p r o b a b i l i t y d i s t r i b u t i o n o f t h e wave energy t h r o u g h t h e s u r f zone. W i t h t h e h e l p o f a n o n - l i n e a r wave t h e o r y , wave h e i g h t

c h a r a c t e r i s t i c s can be d e r i v e d f r o m t h e energy d i s t r i b u t i o n s .

The c a l i b r a t e d and v e r i f i e d e q u a t i o n [ 7 ] f o r t h e e x p e c t e d t i m e - v a r y i n g d i s s i p a t i o n can be r e a d i l y used i n wave p r o p a g a t i o n models t h a t t a k e i n t o account v a r i a t i o n s on t h e t i m e - s c a l e o f wave g r o u p s .

Acknow1edgements

T h i s work was u n d e r t a k e n as p a r t o f t h e MAST G6 C o a s t a l Morphodynamics

r e s e a r c h programme. I t was f u n d e d J o i n t l y by t h e C o a s t a l Genesis programme o f The N e t h e r l a n d s ' R i j k s w a t e r s t a a t and by t h e Commission o f t h e European

Communities, D i r e c t o r a t e G e n e r a l f o r S c i e n c e , Research and Development, under MAST c o n t r a c t no. 0035. The a u t h o r would l i k e t o t h a n k h i s c o l l e a g u e M.J.F. S t i v e and P r o f . J.A. B a t t j e s f o r t h e i r c o n s t r u c t i v e remarks on a d r a f t v e r s i o n o f t h i s paper.

R e f e r e n c e s

B a t t j e s , J.A. and J.P.F.M. Janssen ( 1978 )

Energy l o s s and s e t - u p due t o b r e a k i n g o f random waves

Proc. 1 6 t h I n t . Conf. C o a s t a l E n g i n e e r i n g , pp 569-587, ASCE, New York, 1978 B a t t j e s , J.A. and M.J.F. S t i v e ( 1985 )

C a l i b r a t i o n and v e r i f i c a t i o n o f a d i s s i p a t i o n model f o r random b r e a k i n g waves J o u r n a l o f G e o p h y s i c a l Research, v o l . 90, No. C5, pp. 9159-9167, 1985

D a l l y , W.R. ( 1990 )

Random b r e a k i n g waves: F i e l d v e r i f i c a t i o n o f a wave-by-wave a l g o r i t h m f o r e n g i n e e r i n g a p p l i c a t i o n

i n press

D a l l y , W.R., R.G. Dean and R.A. D a l r y m p l e ( 1984 ) A model f o r b r e a k e r decay on beaches

E b e r s o l e , B.A. and S.A. Hughes ( 1987 ) Duck85 p h o t o p o l e e x p e r i m e n t

U.S. Army Waterways e x p e r i m e n t s t a t i o n , misc. paper CERC-87-18, V i c k s b u r g , 1987

G l u k h o v s k i y , B.Kh. ( 1966 )

I s s l e d o v a n i y e morskogo v e t r o v o g o v o l n e n i y a ( i n v e s t i g a t i o n o f sea w i n d waves) G i d r o m e t e o i z d a t , L e n i n g r a d , 1966

H o t t a , S. and M. M i z u g u c h i ( 1980 ) A f i e l d s t u d y o f waves i n t h e s u r f zone

C o a s t a l E n g i n e e r i n g i n Japan, v o l . X X I I I , Japan Soc. o f C i v i l E n g i n e e r s , Tokyo Klopman, G. and M.J.F. S t i v e ( 1989 )

Extreme waves and wave l o a d i n g i n s h a l l o w w a t e r

Paper p r e s e n t e d a t E&P Forum Workshop "Wave and c u r r e n t k i n e m a t i c s and l o a d i n g " , P a r i s , France, 25-26 Oct. 1989

LeMehaute, B. ( 1962 )

On n o n - s a t u r a t e d b r e a k e r s and t h e wave run-up

Proc. 8 t h I n t . Conf. C o a s t a l E n g i n e e r i n g , pp 1178-1191, ASCE,New York, 1962 L i s t , J.H. ( 1990 )

A model f o r t w o - d i m e n s i o n a l s u r f b e a t subm. J o u r n a l o f G e o p h y s i c a l Research, 1990 Mase, H. and Y. I w a g a k i ( 1982 )

Wave h e i g h t d i s t r i b u t i o n s and wave g r o u p i n g i n s u r f zone

Proc. 1 8 t h I n t . Conf. C o a s t a l E n g i n e e r i n g , pp. 58-76, ASCE, New York, 1982 M i z u g u c h i , M. ( 1982 )

I n d i v i d u a l wave a n a l y s i s o f i r r e g u l a r wave d e f o r m a t i o n i n t h e n e a r s h o r e zone Proc. 1 8 t h I n t . Conf. C o a s t a l E n g i n e e r i n g , pp. 485-504, ASCE, New York, 1982 R i e n e c k e r , M.M. and J.D. F e n t o n ( 1981 )

A F o u r i e r a p p r o x i m a t i o n method f o r s t e a d y w a t e r waves J. F l u i d Mech. ( 1 9 8 1 ) , v o l 104, pp. 119-137

R o e l v i n k , J.A. ( 1991 )

Model l i n g o f c r o s s - s h o r e f l o w and morphology

Proc. ASCE S p e c i a l t y Conf. ' C o a s t a l Sediments', S e a t t l e , 1991, pp 603-617 Sato, S. and N. M i t s u n o b u (1991)

A n u m e r i c a l model o f beach p r o f i l e change due t o random waves

Proc. ASCE S p e c i a l t y Conf. ' C o a s t a l Sediments', S e a t t l e , 1991, pp 674-687 S c h a e f f e r , H.A. and I.G. Jonsson ( 1990 )

Theory v e r s u s e x p e r i m e n t s i n t w o - d i m e n s i o n a l s u r f b e a t s Proc. 22nd I n t . Conf. C o a s t a l E n g i n e e r i n g , D e l f t , 1990, pp 1131-1143 S t i v e , M.J.F. ( 1984 ) Energy d i s s i p a t i o n i n waves b r e a k i n g on g e n t l e s l o p e s C o a s t a l E n g i n e e r i n g 8 ( 1 9 8 4 ) pp 99-127 S t i v e , M.J.F. ( 1985 ) A s c a l e c o m p a r i s o n o f waves b r e a k i n g on a beach C o a s t a l E n g i n e e r i n g 9 ( 1 9 8 5 ) pp 151-158 S t i v e , M.J.F. and M.W. Dingemans ( 1984 )

C a l i b r a t i o n and v e r i f i c a t i o n o f a o n e - d i m e n s i o n a l wave e n e r g y decay model 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 on i n v e s t i g a t i o n M 1882, D e l f t , 1984

Symonds, G. and K.P. B l a c k ( 1991 )

N u m e r i c a l s i m u l a t i o n o f i n f r a g r a v i t y response i n t h e n e a r s h o r e

Proc. 1 0 t h A u s t r a l a s i a n C o a s t a l and Ocean Eng. Conf., A u c k l a n d , New Z e a l a n d , Dec. 1991, pp 339-344.

Symonds, G. and A.J. Bowen ( 1984 )

I n t e r a c t i o n s o f n e a r s h o r e b a r s w i t h i n c o m i n g wave groups

J o u r n a l o f G e o p h y s i c a l Research, v o l . 89, no. C2, pp 1953-1959, 1984 Symonds, G., D.A. H u n t l e y and A.J. Bowen ( 1982 )

Two-dimensional s u r f b e a t : l o n g wave g e n e r a t i o n by a t i m e - v a r y i n g b r e a k p o i n t J o u r n a l o f G e o p h y s i c a l Research, v o l . 87, n o . C l , pp 492-498, 1982

T h o r n t o n , E.B. and R.T. Guza ( 1983 ) T r a n s f o r m a t i o n o f wave h e i g h t d i s t r i b u t i o n

J o u r n a l o f G e o p h y s i c a l Research, v o l . 88, pp. 5925-5938, 1983

L i s t of F i g u r e s

1. P l o t o f t h e f u n c t i o n Y = 1 - exp (-x"^^) f o r n = 5, 10, 20

2.a-g I s o l i n e s o f r m s - e r r o r (drawn l i n e s ) and mean e r r o r ( i n t e r r u p t e d l i n e s ) f o r a l l p o i n t s i n c a l i b r a t i o n s e t s .

3.a-z Comparison o f measured and computed s p a t i a l d i s t r i b u t i o n o f H c a l i b r a t i o n s e t s and v e r i f i c a t i o n s e t s

4. Dependence scheme d i s s i p a t i o n model

5.a-b Measured v s . computed v a l u e s o f H , H , H , H and Q; t e s t s f r o m

E rms s l g 1%

P r o b a b i l i s t i c m o d e l n = 10 P r o b a b i l i s t i c m o d e l _{n = 20} F i g . 2a Pig_ 2b W E I B U L L n=:10 0.60 0.55 I '—^ < 10.50 < 0.45 0.4Q 0.0 0.5 1.0 1.5 ALPHA (-) — 2.0 F i g . 2c W E I B U L L n = 20 0.60 0.55 < 1 0 . 5 0 < 0.45 0.40, 0.5 1.0 1.5 ALPHA (-) — 2.0 F i g . 2d

R a y l e i g h n = 10 R a y l e i g h n = 20 C l i p p e d R a y l e i g h ALPHA ( - ) F i g . 2g

0.40 0.30

I

0.20 DATASET: BJ2 0.0 0.20 0.80 F i g . 3 a 20.0 DATASET: BJ3 0.80 F i g . 3 b DATASET; BJ4 F i g . 3 c X X K 0.40 0.30

I

0.20 ,§0.10 X 0.20 0.40 D 5.Ö X ( m ) — -0.80 DATASET: BJll J t ' X — N — X -10.0 15.0 20.0 F i g . 3 d 0.40 0.30

I

0.20 s§0.10 z 0.20 DATASET: BJ12 F i g . 3 e 0.40 0.30

I

0.20 J.0.10 X 0.20 0.40 O.BO DATASET: B J U 5.Ö X ( m ) — F i g . 3 f 10.0 15.0 20.0

I 0.20 £ 0 . 1 0 0.80 DATASET; BJ14 ^ — X X W X'*^ ^< « X £ 0 . 1 0 o.oo„ 0.20 0.80 DATASET; BJ15 5.Ö (m) ^ 10.0 15.0 20.0 F i g . 3 g F i g . 3 h ' 00 T DATASET: HotMiz I 0.50 P 25.0 50.0 75.0 100.0 125.0 150.0 X (m) — 0.50 1.00 0.10 0.00 0.20 _ 0 . 4 0 E DATA^T: tal007 F i g . 3 i F i g . 3 j 0.00 0.20 0.40 DATASET: t5t015 10.0 20.0 30.0 40.0 X ( m ) — 0.10 0.20 DATASET: tBlllO 10.0 X ( m ) 20.0 30.0 40.0 F i g . 3 k F i g . 31

0.00 OArASET; l3tl2 m——xxxxxxxxxxxxxxxxx 10.0 20.0 30.0 40.0 X ( m ) — 0.40 0.60 0.10 0.00 0.20 ^0.40 0.60 DATASET: lst13 C X > < X X X X X X X X 10.0 X ( m ) 20.0 30.0 40.0 F i g . 3 m F i g . 3 n DATASET: t8t2l2 0.10 10.0 X ( m ) 20.0 30.0 40.0 ,0.40 0.60 0.10 0.40 0.60 DATASET: lal216 10.0 X (m) 20.0 30.0 40.0 F i g . 3 o F i g . 3 p DATASET: t s U 2 2 0.10 0.00 0.20 .,0.40 10.0 X ( m ) 20.0 30.0 40.0 100 T DATASO: D41400 I 0.50 0.00 0.50 125.0 150.0 X (m) ^ 175.0 F i g . 3 q F i g . 3 r

T DATASET: D41510 I 0.50 0.00 0.50 2.00 125.0 X (m) — F i g . 3 s 150.0 175,0 ' • ° 0 T DATASET: I 0.50 0.00 0.50 1.00 2.00 125.0 X ( m ) — F i g . 3 t 150.0 175.0 T DATASET: 051055 0.00 0.50 t.00 2.00 X y X 125.0 150.0 175.0 X (m) — -F i g . 3 u T DATASET: D5I352 2.00 X X 125.0 150.0 X (m) — F i g . 3 v 175.0 T DATASET: 051525 I 0.50 0.00 1.00 X X 125.0 X ( m ) — -150.0 175.0 ' 00 T DATAaiT: D60915 I 0.50 0.00 0.50 1.00 X X 125.0 X ( m ) — • X X 150.0 175.0 F i g . 3 w _{F i g . 3 x}

'•00 T DATASET: D61300 0.00 0.50 1.00 2.00 ~5r-y—»< 125.0 150.0 175.0 X (m) — ~ F i g . 3 y 1.00 0.00 0.50 1.00 2.00 DATASET: 061015 X X 125.0 X (m) — X X • X — K 150.0 175.0 F i g . 3 z

DATASET: MSIOa 1.00

X ( m )