Wave Heights and Set-up in a Surf Zone

By I . A. Svendsen

Dept. o f 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 o f Delaware

A p r i l 1983

On l e a v e from I n s t , o f Hydrodyn. and Hydraul. Engrg., T e c h n i c a l U n i v e r s i t y o f Denmark.

ABSTRACT

A t h e o r e t i c a l model i s developed f o r wave h e i g h t s and set-up i n a s u r f zone. I n the time averaged equations of energy and momentvmi the energy f l u x , r a d i a t i o n s t r e s s and energy d i s s i p a t i o n a r e determined by s i m p l e

approximations which i n c l u d e the s u r f a c e r o l l e r i n the b r e a k e r . Comparison w i t h measurements shows good agreement. A l s o the t r a n s i t i o n s immediately a f t e r b r e a k i n g a r e a n a l y s e d and shown to be i n accordance w i t h t h e above mentioned xdeas and r e s u l t s .

1. INTRODUCTION

The p r o p e r modeling o f wave motion i n t h e s u r f zone on a l i t t o r a l c o a s t has been the g o a l o f many i n v e s t i g a t i o n s i n p a r t i c u l a r i n t h e l a s t two decades. Y e t i n s p i t e o f p r o g r e s s both i n terms of a growing s t o c k o f r e l i a b l e ejcperimental r e s u l t s and i n t h e o r e t i c a l understanding o f t h e p r o c e s s e s t h e g e n e r a l i m p r e s s i o n o f t h e s i t u a t i o n today i s s t i l l t h a t much remains t o be done.

The- p r e s e n t i n v e s t i g a t i o n i s mainly t h e o r e t i c a l but r e l i e s h e a v i l y on experimental evidence a t a number o f p o i n t s . I t a l s o i n d i r e c t l y l e a n s on t h e many c o n t r i b u t i o n s i n t h e l i t e r a t u r e which have helped t o show t h a t c o n v e n t i o n a l wave t h e o r i e s cannot be used t o d e s c r i b e s u r f zone waves.

As i n most o f those c o n t r i b u t i o n s we only c o n s i d e r i n t e g r a l p r o p e r t i e s o f the waves and work w i t h equations time averaged over a wave p e r i o d . The g o a l i s o n l y t o d e s c r i b e t h e wave h e i g h t and mean water l e v e l (setup) v a r i a -t i o n s . T h i s means -t h a -t -the i n -t e g r a l wave p r o p e r -t i e s we need -t o de-termine f i r s t o f a l l a r e energy f l u x , r a d i a t i o n s t r e s s and energy d i s s i p a t i o n .

I n s p i t e o f t h e f a c t t h a t i n t e g r a l p r o p e r t i e s

should be l e s s s e n s i t i v e t o minor i n a c c u r a c i e s i n t h e d e t a i l s o f t h e wave d e s c r i p t i o n i t i s g e n e r a l l y accepted t h a t l i n e a r wave theory i s too d i f f e r e n t to y i e l d a proper d e s c r i p t i o n o f energy f l u x and r a d i a t i o n s t r e s s i n t h e s u r f zone. Some authors have t r i e d t o use s o l i t a r y wave theory (see e.g., Divoky e t a l . , 1968), c n o i d a l theory or h y p e r b o l i c wave theory (James, 1974, i s an example) which a r e o t h e r e a s i l y a c c e s s i b l e wave t h e o r i e s . F o r reasons t h a t

2

may become c l e a r from the p r e s e n t c o n t r i b u t i o n these p a r t i c u l a r s u g g e s t i o n s do not work, but t h i s c o n c l u s i o n i s o f t e n concealed by t h e f a c t t h a t r e s u l t s obtained f o r the wave h e i g h t s and set-up a l s o depend on how the energy

d i s s i p a t i o n i s modelled.

A l s o t h e i d e a o f a s i m i l a r i t y s o l u t i o n f o r the waves has been pursued. Such a n approach i s more l i k e l y t o be a b l e to l e a d t o a c c e p t a b l e r e s u l t s , although i n the form p r e s e n t e d by Wang s Yang (1980) i t i m p l i c i t l y assumes t h a t the wave h e i g h t t o water depth i s a c o n s t a n t which we know i s n o t t h e c a s e (see e.g., t h e experimental r e s u l t s by Horikawa s Kuo (1966), and a l s o t h i s p a p e r ) .

I n acknowledgement o f the l i m i t e d s u c c e s s o f the t h e o r e t i c a l i n v e s t i g a t i o n s and the a c c e s s t o modern measuring techniques such a s LDA, c o n t r i b u t i o n s have been p u b l i s h e d g i v i n g both v e r y d e t a i l e d measurements o f v e l o c i t y and p r e s s u r e f i e l d s and computations o f p a r t i c u l a r l y r a d i a t i o n s t r e s s and momentum b a l a n c e from the b a s i c d e f i n i t i o n s (see S t i v e , 1980, and S t i v e & Wind, 1982).

The p o s s i b i l i t i e s f o r e v a l u a t i n g the energy d i s s i p a t i o n a r e f a r more r e s t r i c t e d . I n p r i n c i p l e t u r b u l e n c e i s c r e a t e d by two s o u r c e s : t h e bottom boundary l a y e r and the s u r f a c e b r e a k e r . I n p r a c t i c e , however, t h e bottom d i s s i p a t i o n i s t o t a l l y outweighted by t h e d i s s i p a t i o n due t o b r e a k i n g , and i s consequently n e g l e c t e d h e r e .

The i d e a which t u r n s out t o be most f r u i t f u l i n determining t h e energy d i s s i p a t i o n ( i f a d e t a i l e d d e s c r i p t i o n o f the t u r b u l e n t flow i s n o t attempted) i s based on the resemblance o f s u r f zone waves w i t h b o r e s . T h i s

was d i s c u s s e d by LeMehauté (1962) and l a t e r by Divoky e t a l . (1968). To f i t measiirements, however, they used t h e motion o f a non-saturated b r e a k e r suggested by LeMehauté (1962), which i m p l i e s an energy d i s s i p a t i o n s m a l l e r than the d i s s i p a t i o n i n a h y d r a u l i c jtimp o f the same h e i g h t a s t h e wave. T h i s i s the opposite r e s u l t o f t h a t found by Svendsen e t a l . (1978) who concluded t h a t the d i s s i p a t i o n i n a c t u a l measurements were l a r g e r than i n a jump o f t h e same h e i g h t s The. e x p l a n a t i o n f o r t h i s c o n t r o v e r s y probably l i e s i n t h e d i f f e r e n t v a l u e s used f o r t h e energy f l u x . I n s e c t i o n 7 we w i l l see t h a t i n a c n o i d a l •or s o l i t a r y wave (used f o r a s s e s s i n g t h e energy flvix by Divoky e t a l . ) t h e

energy f l u x f o r a g i v e n wave h e i g h t i s much s m a l l e r than i n a s u r f zone wave. Hence t h e s m a l l e r d i s s i p a t i o n r e q u i r e d by Divoky e t a l . f o r a g i v e n (measured) wave h e i g h t d e c r e a s e .

In t h e p r e s e n t paper we simply use t h e h y d r a u l i c jump e x p r e s s i o n f o r

the energy d i s s i p a t i o n . T h i s i s i n r e a l i z a t i o n o f two f a c t s .

F i r s t l y Svendsen S Madsen (1981) e s s e n t i a l l y confirmed t h e c o n c l u s i o n

±n Svendsen e t a l . (1978) tin t h e f o l l o w i n g denoted I ) , but t h e i r r e s u l t s

i n d i c a t e t h a t many f a c t o r s a r e i n v o l v e d , and f u r t h e r s t u d i e s show t h a t t h e d e v i a t i o n from t h e h y d r a u l i c jump d i s s i p a t i o n i n most c a s e s i s l e s s than 20%.

Secondly t h e o t h e r e f f e c t s s t u d i e d i n t h e f o l l o w i n g a r e found t o be more important.

The paper may be c o n s i d e r e d a c o n t i n u a t i o n o f the work r e p o r t e d i n I . To determine t h e wave h e i g h t and set-up v a r i a t i o n we c o n s i d e r the equations of energy and momentum b a l a n c e , both averaged over the wave p e r i o d ( s e c t i o n 2 ) , and i n s e c t i o n 3 d e r i v e a c l o s e d form s o l u t i o n t o the energy e q u a t i o n . I n

4

s e c t i o n 4 we i n v e s t i g a t e an a n a l y t i c a l s o l u t i o n f o r a s p e c i a l c a s e . T h i s r e v e a l s t h a t n e g l e c t i n g set-up t h e r e i s only one parameter K (given by t h e wave p r o p e r t i e s a t the s t a r t i n g p o i n t ) which determines the wave h e i g h t s . K i s a combination o f the bottom slope parameter S = h L/h i d e n t i f i e d f o r s h o a l i n g by Svendsen S Hansen (1976) (h i s bottom s l o p e , L wave l e n g t h and h watei: depth), the wave h e i g h t t o water depth r a t i o H/h and the d i m e n s i o n l e s s energy f l u x B.

The t h r e e wave p r o p e r t i e s mentioned e a r l i e r (the energy f l u x ,

r a d i a t i o n s t r e s s and energy d i s s i p a t i o n ) a r e determined i n s e c t i o n 5. Here i t becomes n e c e s s a r y t o concentrate on the i n n e r region (defined i n I ) o f the s u r f zone where t h e waves have become b o r e - l i k e ( F i g . 1 ) . I t i s shown t h a t the most important f e a t u r e i s t h e e x i s t e n c e o f a s u r f a c e r o l l e r which t o t h e f i r s t approximation can be c o n s i d e r e d a s a voltime o f water c a r r i e d w i t h the wave. The r o l l e r almost doubles both energy flxxx. and r a d i a t i o n s t r e s s r e l a t i v e t o a s h a l l o w water wave otherwise o f the same shape.

F i g . 1 ' .

S e c t i o n 6 shows comparison w i t h experimental r e s u l t s i n t h e i n n e r r e g i o n . Both wave h e i g h t s and set-up are w e l l p r e d i c t e d .

When i t comes t o the o u t e r r e g i o n ( F i g . 1 ) , however, some p a r a d o x i c a l f e a t u r e s are i d e n t i f i e d i n the measurements ( s e c t i o n 7 ) . The paradox i s r e -s o l v e d by c o n -s i d e r i n g jump c o n d i t i o n -s analogou-s t o tho-se a p p l i e d f o r bore-s and h y d r a u l i c jiimps. T h i s i m p l i e s u s i n g the momentum and energy e q u a t i o n s f o r the e n t i r e outer r e g i o n . The r e s u l t s show t h a t the e x p r e s s i o n s d e r i v e d

i n S e c t i o n 5 f o r waves i n the i n n e r r e g i o n are c o n s i s t e n t w i t h r e s u l t s f o r waves before b r e a k i n g .

Throughout the d e r i v a t i o n s the s i m p l e s t and lowest order approximations have been used. Thus many d e t a i l s such a s non-uniform v e l o c i t y p r o f i l e s , the e f f e c t o f the s t r o n g txirbulence, e t c . a r e simply n e g l e c t e d . The reason i s t h a t t h e s e a s p e c t s , however important, t u r n out to be minor c o r r e c t i o n s

r e l a t i v e to the e f f e c t s i n c l u d e d . Hence the f o l l o w i n g i s meant a s an attempt to show t h a t i t i s p o s s i b l e f o r the wave motion i n a s u r f zone to formulate a crude model which reproduces a l l the major f e a t u r e s observed i n the

Fig. 2

2. THE BASIC EQUATIONS

We c o n s i d e r the two dimensional problem sketched i n F i g . 2 which a l s o shows t h e d e f i n i t i o n o f v a r i a b l e s .

The t h r e e b a s i c equations t o be s a t i s f i e d r e p r e s e n t t h e c o n s e r v a t i o n o f mass, momentum and energy, i n t e g r a t e d over depth and averaged over a wave p e r i o d T .

The conseirvation o f mass w i l l n o t be invoked e x p l i c i t l y but used i n t h e way t h e p a r t i c l e v e l o c i t i e s i n the wave a r e e v a l u a t e d .

We c o n s i d e r r e g u l a r p r o g r e s s i v e waves only and hence t h e momentum equation simply reads

a s

X X

9x = - p g ( h + n)

an

ax (2.1)

where S i s the r a d i a t i o n s t r e s s d e f i n e d ( e x a c t l y ) by ( denoting average

X X over a wave p e r i o d ) S = P + F„ X X m p P = m 2^ pu dz P _-h (2.2) w i t h t h e dynamic p r e s s u r e p^^ given by = pgz + p (2.3) 2-1

Using f o r mean energy f l u x and V f o r energy d i s s i p a t i o n , the energy equation ( a l s o averaged over a wave p e r i o d ) becomes

9E

= V

I n both (2.3) and (2.5) v e l o c i t i e s and p r e s s u r e s a r e t h e i n s t a n t a n e o u s v a l u e s , so t h a t t h e s e d e f i n i t i o n s a l s o cover the t u r b u l e n t flow s i t u a t i o n s i n a s u r f zone. S i n c e , however, any ordered mechanical energy t h a t i s t u r n e d i n t o t u r b u l e n c e w i l l be reduced t o h e a t w i t h i n roughly-the f o l l o w i n g wave p e r i o d we s h a l l choose i n (2.4) t o c o n s i d e r o n l y the f l u x of (ordered) wave energy, t h a t i s , c o n s i d e r energy l o s t a s soon as i t has been changed t o t u r b u l e n c e . To i l l u s t r a t e t h i s we w r i t e

^ f = ^ f , w - ^ ^ f ..

Where E ^ i s the t o t a l f l u x of t u r b u l e n t energy (by a l l means, i n c l u d i n g d i f f u s i o n ) and (with v = 0)

"^f ,w - J

1 2 2

(Pj3 + J p(u + w ) u dz (2.7) -h

(~ denoting ensemble averaged v a l u e s ) i s the f l u x of "wave energy."

Hence (2.4) becomes

9E, 9 E ;

- f ' - ^ = p _ f = (2.8) 3x 9x t

2-3

But i n the steady wave motion c o n s i d e r e d a l s o e q u a l s minus t h e

energy d i s s i p a t i o n V .

Hence i n (2.4) we simply use (2.7) f o r and t h i s means c o n s i d e r i n g o n l y ordered wave energy, which i s a c h o i c e , not an approximation.

I t i s convenient f o r the f o l l o w i n g a n a l y s i s t o i n t r o d u c e non-dimensional

measures o f both t h e wave energy f l u x E , t h e r a d i a t i o n s t r e s s and t h e I ,w

energy d i s s i p a t i o n . The f o l l o w i n g d e f i n i t i o n s a r e used.

B = E ^ Y( p g c H ^ ) (2=9)

P = S ^ p g H 2 (2.10)

D = P^.(4hT/pgH^) (2.11)

I t t u m s out t h a t t h e energy equation can be s o l v e d i n c l o s e d form f o r a f a i r l y g e n e r a l type o f problem. With (2.11) and (2.7) s u b s t i t u t e d

C2.4) r e a d s

f>w ^ H D (3.1) • 3x 4HT

The v a r i a t i o n o f t h e wave h e i g h t i s a combination o f t h r e e e f f e c t s : t h e change i n water depth (which on a beach causes a wave h e i g h t i n c r e a s e ) , the d i s s i p a t i o n o f energy (tending t o reduce t h e wave h e i g h t ) and t h e change i n shape o f t h e wave (taken i n a r a t h e r g e n e r a l sense) which i s r e f l e c t e d i n t h e v a r i a t i o n o f R, P and D.

Whereas t h e f i r s t two o f t h e s e e f f e c t s a r e r e p r e s e n t e d i n t h e time averaged e q u a t i o n s , t h e same time averaging has excluded t h e p o s s i b i l i t y o f a s s e s s i n g t h e change i n wave shape ( n , u, p, e t c . ) d i r e c t l y from the e q u a t i o n s and t h e v a l u e s o f B, P and D must be e v a l u a t e d s e p a r a t e l y (see s e c t . 5) . , Hence i n t h e f o l l o w i n g we assume these parameters a r e known.

To o b t a i n a s o l u t i o n t o (3.1) we i n t r o d u c e a s h o a l i n g c o e f f i c i e n t K d e f i n e d so t h a t a t any depth s 2 2 c B ( K H ) = c B H = c o n s t (3.2) ^ s r r r r

where index ^ r e f e r s t o some chosen r e f e r e n c e p o i n t . T h i s equation would g i v e t h e H v a r i a t i o n i n the absence o f d i s s i p a t i o n . The a c t u a l wave h e i g h t i s then expressed as

H = K K^H (3.3) s d r

3-2

which d e f i n e s the d i s s i p a t i o n c o e f f i c i e n t K^. Thus we get u s i n g (3.2) which i m p l i e s a/8x(cBK ^) = 0: s = 2pg K ^ K ^ C B K / H ^ = 2 E ^ ^ ^ (' denoting 3/9x) which s u b s t i t u t e d i n t o (3.1) y i e l d s o r 4cBhT d K' K H D d _ s r 2 ~ ScBhT o r K H D s r (3.4) 8cBhT

So f a r no assiomptions have been introduced n e i t h e r about the t y p e of wave c o n s i d e r e d (except t h a t i t i s r e g u l a r and p r o g r e s s i v e ) nor about the nature o f the energy d i s s i p a t i o n . Thus (3.4) a p p l i e s to waves i n the s u r f zone as w e l l a s to the a t t e n u a t i o n of waves due to bottom f r i c t i o n . (3.4) even a p p l i e s t o the growth of waves due to wind energy b e i n g added i f we l e t n > 0. I n the f o l l o w i n g , however, we concentrate on the s u r f zone where D < 0.

We now make the assumption (which w i l l l a t e r be j u s t i f i e d ) t h a t the c o e f f i c i e n t s on the r i g h t hand s i d e only depend on the water depth h. (For a n u m e r i c a l e v a l u a t i o n of the f o l l o w i n g s o l u t i o n t h i s assumption may be r e l a x e d t o i n c l u d e a weak dependence on H as w e l l . ) Hence (3.4) may be i n t e g r a t e d d i r e c t l y g i v i n g

where t h e i n t e g r a t i o n constant i s K^^ = 1 f o r x^ x^. That i s ^d = 1 -•X K DH s r ScBhT - r - l dx (3.6) R e c a l l i n g t h a t (3.2) i m p l i e s — '*;r— s c B (3.7) we t h e r e f o r e g e t f o r H by s u b s t i t u t i o n i n t o (3.3) r H 1 -8c B T r r X DK 3 ^ - 1 dx (3.8)

We see t h a t t h i s c l o s e d form s o l u t i o n depends on one combination o f t h e wave p r o p e r t i e s a t t h e r e f e r e n c e p o i n t , namely H , H h 1 r r 8c B T 8B h L r r r r r (3.9)

C3-8), however, corresponds t o a c e r t a i n change i n x. And s i n c e K m a i n l y s

depends on h t h i s means t h a t t h e bottom slope h i s a c t u a l l y a parameter a s w e l l . F o r monotonously changing depth we can e x p r e s s t h i s e x p l i c i t l y by changing t o h a s i n t e g r a t i o n v a r i a b l e . T h i s y i e l d r 1 - K f X DK h s x r h h ' X X r dh w i t h (3.10)

3-4

and h i s h a t the r e f e r e n c e p o i n t . Hence the wave h e i g h t v a r i a t i o n i n

x r X

a s u r f zone w i l l be the same f o r a l l wave c o n d i t i o n s having t h e same v a l u e o f K a t t h e s t a r t i n g p o i n t o f the computation (which, i n p r i n c i p l e , may be any p o i n t from the b r e a k i n g p o i n t where D becomes non-zero and shorewards) .

I t a l s o shows t h a t the bottom s l o p e i t s e l f i s not a proper measure o f the s t e e p n e s s o f the beach. The r e l e v a n t parameter i s h L/h, which i s the same, s l o p e peirameter. as was found f o r t h e s h o a l i n g o f waves by Svendsen

& Hansen (1976).

N o t i c e t h a t h i n c l u d e s the set-up, t h a t i s

h ^ h g ^ + ïï (3.11

4. ANALYTICAL SOLUTION FOR A SPECIAL CASE

One s p e c i a l c a s e i s of p a r t i c u l a r i n t e r e s t as i t r e n d e r s an a n a l y t i c a l

s o l u t i o n p o s s i b l e . I t corresponds to a plane beach w i t h D and B c o n s t a n t and

c = c i/qh. The l a t t e r assxmption w i t h c s l i g h t l y l a r g e r than /gh - and hence c s l i g h t l y more than Vgh but v a r y i n g l i k e h. ' - was found i n I t o be a good

approximation.

I n t h i s c a s e the i n t e g r a l i n (3.10) may r e a d i l y be s o l v e d and we get

w i t h h- = 5 ^ (4.1)

I =s • t ^ •••• "• W X U H IJ. i _

j , . V 4 ^ , ^ 4 ^ ( ^ . - 3 / 4 _ ^ j 3 h

I n t h i s s o l u t i o n i s i n c l u d e d the set-up n determined from the s o l u t i o n o f t h e momentum e q u a t i o n . T h i s , however, i s n o n l i n e a r i n n and not s o l v a b l e except i f H/(h+n') i s assumed c o n s t a n t (Bowen e t a l . ( 1 9 6 8 ) ) . Although t h i s - i s not r e a l i s t i c ^ f o r the major p a r t of the s u r f zone , n « h and hence i n e v a l u a t i o n (4.1) we may e i t h e r completely n e g l e c t n ( v i z means h' - (h/h ) ).or use the above mentioned approximation H/h = a = c o n s t .

The l a t t e r g i v e s the approximation

^ - '1 ~ " i ^ ^ s w L - \,sm) a, = l+2a2p o r ( i f we n e g l e c t i n comparison to h^) (4.2) ^ r ^ SWL (1 + a^) - (4.3) 4-1

4-2

which d i f f e r from Bowen e t a l . ' s r e s u l t i n t h a t a i s not ass\amed e q u a l t o 0.8.

F i g s . 3

& 4 F i g s . 3 and 4 show t h e v a r i a t i o n o f H/H^ and H/h*(H^/h^) v e r s u s h' = Ch/h f o r d i f f e r e n t v a l u e s o f KD and n = 0.

SWL r'SWL

I n t h e s e f i g u r e s we could, f o r example, c o n s i d e r the r e f e r e n c e p o i n t taken a t the b r e a k i n g p o i n t .

Of p a r t i c u l a r i n t e r e s t i s the v a r i a t i o n o f t h e wave h e i g h t t o water depth r a t i o H/h. As a n t i c i p a t e d H/h i s f a r from c o n s t a n t f o r any v a l u e o f KD. For a wide range o f KD, however, (which a l s o t u r n s out t o be p r a c t i c a l l y

r e a l i s t i c ) t h e H/h v a r i a t i o n shows a minimum which may be r e c o g n i z e d a l s o i n many experimental r e s u l t s (see e.g., Horikawa s Kuo ( 1 9 6 6 ) ) .

C l e a r l y t h i s phenomenon r e f l e c t s t h e b a s i c f e a t u r e mentioned above t h a t t h e wave h e i g h t v a r i a t i o n i s a balance between s h o a l i n g and d i s s i p a t i o n . T h i s can more r e a d i l y be seen by w r i t i n g (3.1) i n the form ( f o r d e r i v a t i o n see I )

h h c B h 2c 2B H + _ D _ h 8cTB 2 (4.4) The f i r s t b r a c k e t on t h e r i g h t s i d e r e p r e s e n t s t h e s h o a l i n g , t h e l a s t term the d i s s i p a t i o n . S i n c e a t t h e r e f e r e n c e (breaker) p o i n t H/h w i l l u s u a l l y be q u i t e l a r g e , t h e second term w i l l dominate provided [D| i s s u f f i c i e n t l y l a r g e as i n a b r e a k e r . Hence a t a s t a r t (H/h)^ i s n e g a t i v e . As (H/h)^ however d e c r e a s e s f a s t e r w i t h H/h than does (H/h)""" the d i f f e r e n c e between the two terms d e c r e a s e s t i l l (H/h) ~ 0. Hence were i t not f o r o t h e r e f f e c t s H/h would a s y i f i p t o t i c a l l y approach t h e v a l u e obtained by equating the r i g h t s i d e t o z e r o .

F i g . 5

At the same time, however, as h d e c r e a s e s f o r c o n s t a n t h ^ both b^/h and c^/c i n c r e a s e s and near the shore l i n e t h i s e f f e c t becomes completely dominating r e s u l t i n g i n the shown i n c r e a s e i n H/h.

The p o s i t i o n of the minimum f o r H/h can be found from (4.4) t o be (with c^/c = h ^ 2 h )

h D

(Notice t h a t both h ^ and D are < 0.) I f we i n t r o d u c e K t h i s becomes

(H/h) . ^ T

"^"^ = I (KD>4r^)"^ ( 4 . 6 )

(H/h)^ 4

which may be s o l v e d i n combination w i t h ( 4 . 1 ) . The r e s u l t i s t h a t the minimum f o r H/h o c c u r s a t 1 4 / 3 h' . mxn 8 KD 20 KD - 15

T h i s v a l u e and the corresponding v a l u e of H/h a r e shown i n F i g . 5.

We a l s o see t h a t f o r s u f f i c i e n t l y s m a l l KD(<' 1.25 i t t u r n s out) t h e wave h e i g h t w i l l be i n c r e a s i n g a l r e a d y from the r e f e r e n c e p o i n t . T h i s i s not l i k e l y t o happen f o r b r e a k e r s but w i l l sometimes be the s i t u a t i o n where the d i s s i p a t i o n i s caused by bottom f r i c t i o n ( t h a t i s seawards of the b r e a k i n g p o i n t ) .

On the o t h e r hand, the s i t u a t i o n may a l s o occur t h a t a wave approaching a shore w i l l not break a t a l l because i t s energy i s being d i s s i p a t e d by bottom f r i c t i o n f a s t e r than the h e i g h t can i n c r e a s e due to s h o a l i n g , so t h a t i t never

lb

4-4

reaches a h e i g h t s u f f i c i e n t f o r b r e a k i n g t o o c c u r . Obviously t h i s r e q u i r e s a v e r y s m a l l bottom slope and we see from (3.10) t h a t h^ s m a l l l e a d s to K l a r g e so t h a t even w i t h the s m a l l D from bottom f r i c t i o n KD becomes l a r g e r than 1.2S. ( P i g . 4 a l s o i n c i d a t e s t h a t i f b r e a k i n g i s t o be avoided e n t i r e l y then h must decrease shorewards. With c o n s t a n t h (however s m a l l ) the wave h e i g h t w i l l s t a r t t o i n c r e a s e sooner o r l a t e r and hence the wave w i l l r e a c h breaking.)

The q u e s t i o n , however, remains whether B, D and P can be c o n s i d e r e d c o n s t a n t s through the s u r f zone (as assumed i n t h i s s e c t i o n ) , and what t h e i r v a l u e s w i l l be.

5. ENERGY FLUX, RADIATION STRESS AND ENERGY DISSIPATION IN THE SURF ZONE

The s o l u t i o n s d e r i v e d above can t h e o r e t i c a l l y be used r i g h t from t h e b r e a k i n g p o i n t and shorewards, but i t has o f t e n been p o i n t e d out (see e.g., I ) t h a t the flow i n the s o - c a l l e d "outer r e g i o n " immediately shoreward o f the b r e a k i n g p o i n t i s s i g n i f i c a n t l y d i f f e r e n t from the c o n d i t i o n s i n t h e i n n e r r e g i o n . The t r a m s i t i o n s i n t h e o u t e r r e g i o n w i l l be d i s c u s s e d i n S e c t i o n 7.

F i r s t , however, v a l u e s o f the d i m e n s i o n l e s s parameters B, P and D a r e detejnnined i n t h e i n n e r r e g i o n .

The important f e a t u r e dominating the wave motion i n t h i s r e g i o n i s t h e s u r f a c e r o l l e r , which i n essence i s a volume o f water c a r r i e d shorewards w i t h t h e b r e a k e r . F i g . 6 shows a t y p i c a l s i t u a t i o n , and a l s o i n d i c a t e s a t y p i c a l v e l o c i t y d i s t r i b u t i o n along a v e r t i c a l a t the f r o n t o f t h e wave. F i g . 6

The r o l l e r i s d e f i n e d a s t h e r e c i r c u l a t i n g p a r t of the flow above the d i v i d i n g s t r e a m l i n e ( i n a c o o r d i n a t e system f o l l o w i n g the wave). S i n c e i t i s r e s t i n g on t h e f r o n t o f t h e wave t h e a b s o l u t e mean v e l o c i t y i n t h e r o l l e r e q u a l s t h e propagation speed c f o r t h e wave, and i n the f o l l o w i n g we use t h i s v a l u e f o r t h e v e l o c i t y i n the r o l l e r , n e g l e c t i n g t h e z - v a r i a t i o n .

From t h i s • i t f o l l o w s t h a t t h e r o l l e r r e p r e s e n t s a s i g n i f i c a n t

enhancement o f the o r d i n a r y Stokes d r i f t Q . Thus a s u r f zone wave p o t e n t i a l l y 5

r e p r e s e n t s a much b i g g e r mass t r a n s p o r t than non-breaking waves. The a c t u a l

n e t mass f l u x , however, i s i n any s i t u a t i o n determined by t h e boundary c o n d i t i o n s i n t h e x - d i r e c t i o n , and i n the g e n e r a l t h r e e dimensional case i t w i l l a l s o

5-2

F i g . 7

depend s t r o n g l y on the longshore v a r i a t i o n s o f bottom topography and wave h e i g h t s .

I n the p r e s e n t two dimensional study we assume a zero n e t mass f l u x (Q = 0 ) , which o f course i m p l i e s t h a t t h e r e i s a r e t u r n flow compensating f o r the s u r f a c e d r i f t .

From o b s e r v a t i o n s we know t h a t i n the i n n e r r e g i o n the change i n wave shape i s slow so the instantaneous volume f l u x

rn

u ( x , z , t ) d z (5.1) -h

may b a determined a s

Q = c n + Q = U d + Q (5.2)

where t h e s u r f a c e p r o f i l e i s s p e c i f i e d so t h a t n" = 0. U i s the wave p a r t i c l e v e l o c i t y averaged o v e r depth. As mentioned we s h a l l f u r t h e r assume t h a t Q = 0.

Although i t i s a crude s i m p l i f i c a t i o n the v e l o c i t y d i s t r i b u t i o n shown i n F i g . 7 w i l l c o n t a i n a l l the primary i n f o r m a t i o n o u t l i n e d above and we w i l l use t h a t . The t h i c k n e s s e o f the s u r f a c e r o l l e r w i l l be zero except i n the f r o n t which i s i m p l i c i t l y understood i n t h e f o l l o w i n g d e r i v a t i o n s . Hence we g e t

Q = cn = ce + u^(d-e)

Outside the r o l l e r we have

u = c(ri - e ) / ( d - e) o

(5.3)

and i n the r o l l e r

u = c (5.4b)

The p r e s s u r e i n the wave motion i s o f course not s t a t i c . I n combination w i t h t h e r a t h e r crude assumptions made above, however, i t i s c o n s i s t e n t t o assxime a s t a t i c p r e s s u r e v a r i a t i o n corresponding to the l o c a l , i n s t a n t a n e o u s p o s i t i o n of t h e f r e e s u r f a c e . That i s , u s i n g (2.3)

Pj3 = pgn (5.5)

The Energy F l u x B

With t h e s e assumptions we f i r s t c a l c u l a t e the non-dimensional energy f l u x B d e f i n e d by ( 2 . 9 ) . S u b s t i t u t e d i n t o (.2.7) t h i s y i e l d s (omitting t h e " which i n d i c a t e s t h e t u r b u l e n t ensemble averaging)

2 pgcH

, | p C u 2 . „ 2 , (5.6) -h ° ^

I n t r o d u c i n g t h e assumptions o u t l i n e d above - which a l s o i m p l i e s n e g l e c t i n g the w ^ - c o n t r i b u t i o n - we f i r s t g e t from (2.7)

^ f , w = R + Ï p^'<3-=Ef,o ^ f ,1 (s-'^) -h

where E i s t h e f i r s t term i n t h e i n t e g r a l and E - t h e second. (5.5) g i v e s r,u r , i Pj^udz = -h rn pgnudz (5.8) -h which by v i r t u e o f (5.2b) becomes ^ f , 0 " P^^"^^ " pgcH^(n/H)^ (5.9)

Thus the r o l l e r does not c o n t r i b u t e to t h i s term. For E we get

x ,x

h

where c - gh and TI .« h has been used i n the f i r s t term. S i n c e n = 0 we w i l l 3 2

f i n d t h a t (ri/H) « (ri/H) , the l a t t e r being an i n t e g r a l of a non-negative q u a n t i t y (see Hansen, l a s a i .

The a r e a A o f the s u r f a c e r o l l e r i s d e f i n e d as

edt =

-c edx = (5.11)

where X i s the l e n g t h of the r o l l e r (see P i g . 6 ) . I n (5.10) t h i s y i e l d s

1 3 A 1 „2 A h ^ f , l = 2 P° 1 = 2

rl

(5.12)

which shows t h a t t h e c o n t r i b u t i o n to the energy f l u x from the s u r f a c e r o l l e r

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

Veiry l i t t l e i n f o r m a t i o n i s a v a i l a b l e about the s i z e o f the s u r f a c e r o l l e r . Duncan (1981) has measured A i n a b r e a k e r behind a towed h y d r o f o i l , and h i s r e s u l t s a r e shown i n F i g . 8. For the p r e s e n t a p p l i c a t i o n we w i l l

approximate t h e s e r e s u l t s w i t h

A ~ 0.9 H (5.13)

and hence we get f o r E_ ^ f ,w ^f,w ^ r \2 1 _{+ 0.45} -IJ (5.14)

and from (5.6) f o r B - I n [ H J "dt + i . A . = 1 2 gL T n [ H J dt + 0.45 - (5.15) L

Here L i s a l o c a l q u a n t i t y d e f i n e d a s cT, not the p h y s i c a l d i s t a n c e between two c o n s e c u t i v e wave c r e s t s .

V a l u e s o f B d e f i n e d a s o O T •T IL •'0 H 2 d t

a r e shown-for waves i n t h e s u r f zone i n F i g . 9. The measurements a r e a l l F i g . 9 taken on a 1/34.3 slope a t ISVA and t h e r e s u l t s o n l y show t h e t r e n d i n the

v a r i a t i o n . C l e a r l y t h e r e i s a s i g n i f i c a n t amount o f s c a t t e r i n g b u t a l s o some s y s t e m a t i c v a r i a t i o n w i t h wave parameters (such as t h e deep water s t e e p n e s s H / L ) which needs f u r t h e r documentation and a n a l y s i s ,

o o

The main tendency, however, i s q u i t e o b v i o u s l y t h a t from t h e p o i n t o f b r e a k i n g B^ i n c r e a s e s r a p i d l y from a r e l a t i v e l y s m a l l v a l u e ( i n d i c a t i n g a r a t h e r peaky wave p r o f i l e r e p r e s e n t i n g a s m a l l energy f l u x f o r a gxven wave h e i g h t ) towards v a l u e s around 0.07 - 0.08. The v a r i a t i o n may be f u r t h e r

i l l u s t r a t e d by t h e examples given i n Table 1 f o r some simple s u r f a c e p r o f i l e s . Of p a r t i c u l a r i n t e r e s t a r e t h e v a l u e 0.083 and 0.089 f o r t r i a n g u l a r and

p a r a b o l i c wave shapes r e s p e c t i v e l y , because s u r f zone wave p r o f i l e s o f t e n resemble such shapes a s P i g . 10 shows.

5-6

Table 1

The Radiation. S t r e s s P

The r a d i a t i o n s t r e s s d e f i n e d by (2.2) r e p r e s e n t s t h e time averaged momentum f l u x . I n (2.2) i s i n c l u d e d the e f f e c t o f t h e t u r b u l e n t normal s t r e s s e s a s w e l l a s any n e t volume f l u x superimposed on the wave.

The l a t t e r , however, we a r e e x c l u d i n g h e r e . The c o n t r i b u t i o n from the t u r b u l e n t s t r e s s e s was a n a l y s e d by S t i v e s Wind (1982) on t h e b a s i s o f e x p e r i m e n t a l d a t a . They concluded t h a t t h i s e f f e c t only i n c r e a s e s t h e r a d i a t i o n s t r e s s by some 5%. P a r t o f t h e r e a s o n f o r t h i s modest e f f e c t o f s t r o n g

t u r b u l e n c e i s t h a t t h e v e r t i c a l v e l o c i t y f l u c t u a t i o n s reduce t h e p r e s s u r e whereas t h e h o r i z o n t a l f l u c t u a t i o n s i n c r e a s e t h e momentum f l u x . Hence i n the p r e s e n t c o n t e x t we may n e g l e c t the t u r b u l e n t c o n t r i b u t i o n t o S ^ .

U s i n g (5.5) we then g e t f o r i n (2.2) F P 1 2 1 2 pgndz - — pgn = -r pgn (5.17) J - h ^ ^

(5.18) F o r F ^ we g e t , u s i n g (5.4a,b) F = m pu dz = p -h 2 2 u (d-e) + c e o (5.19) Here 2,^ X 2 . u (d-e) = c o 2 2 n - 2rie + e d - e (5.20)

which we approximate by u s i n g e « d. We a l s o n o t i c e t h a t s i n c e e 7^ Q almost s y m m e t r i c a l l y around n = 0 t h e v a l u e o f ne must be very s m a l l . Hence we g e t

F - pc m 2 2 ~ p - H 2 c ^2 -e| . e h H (5.21)

From Duncan's r e s u l t s can be found t h a t e/H - 0 . 3 and s i n c e e = 0 2 2 over most o f the wave p r o f i l e i t f o l l o w s t h a t (e/h) « (ri/H) so t h a t we

f o r F g e t (again u s i n g c - v'gh) m 2, A h , ^m~ ( B ^ - ^ - - ) H I n combination w i t h (5.18) and (5.13) t h i s y i e l d s f o r S X X (5.22) (5.23) o r P = I + 0.9 7" 2 o L (5.24)

With the r e s u l t s from F i g . 9 f o r B^ and a t y p i c a l v a l u e o f 0.05 - 0.10 f o r h/L we see t h a t the p r e s e n c e of the s u r f a c e r o l l e r roughly doubles the r a d i a t i o n s t r e s s .

5-8

The Energy D i s s i p a t i o n D

The t h i r d parameter i n the equations i s the d i m e n s i o n l e s s energy d i s s i p a t i o n D. A d e t a i l e d a n a l y s i s o f D i s q u i t e complicated and i s c o n s i d e r e d unnecessary i n combination w i t h the s i m p l i f y i n g assiomptions i n t r o d u c e d a t a number o f other p o i n t s . Svendsen S Madsen (1981) showed t h a t n e g l e c t i n g t h e form change the energy d i s s i p a t i o n i n a s u r f zone wave can be determined by

D = 1 + _i±i E — Ë ° • (5.25) °bore . V - " c

where a , 8 a r e c o e f f i c i e n t s f o r depth averaged v e l o c i t y and p r e s s u r e c o n t r i b u t i o n s i n the momentum and energy e q u a t i o n s , and ^ and ^ r e f e r to trough and c r e s t

r e s p e c t i v e l y . With t h e assumptions o u t l i n e d above, however, t h i s e x p r e s s i o n s i m p l i f i e s to

D = D (5.26)

bore

where D i s the energy d i s s i p a t i o n i n a bore o f the same h e i g h t as the Wave, bore

The dimensional form o f DJ^QJ,Q i s ( C = ^ ^ / ^ ^ )

AE = p g Q d ^ i ^ ^ (5.27)

(see e.g., Henderson ( 1 9 6 6 ) ) . I n a wave Q = u^d^ = ch so t h i s may a l s o be w r i t t e n

AE = pgch (5.28) t c

I f we assume t h a t each b r e a k e r s u f f e r s a s i m i l a r d i s s i p a t i o n per second then the mean d i s s i p a t i o n per m^ bottom a r e a i s A E / L . And s u b s t i t u t i n g

t h i s f o r i n (2.11) then y i e l d s

D = J i — (5.29)

\%

I n t r o d u c i n g the c r e s t e l e v a t i o n i n s t e a d of d^ and d.^. we get

d = h + n ; d. = h + n ^ - H (5.30) C G t C

and t h u s

D = (5.31)

whick shows t h a t f o r f i x e d n /H D does depend s l i g h t l y on H/h. F i g . 11 shows c

t h e v a r i a t i o n and F i g . 12 g i v e s v a l u e s o f from the experiments quoted

above. As was t h e c a s e f o r B the r e s u l t s f o r n^/H show s i g n i f i c a n t s c a t t e r i n g b u t i n the i n n e r r e g i o n of the s u r f zone the v a l u e i s mostly 0.6 - 0.7 w h i c h

from F i g . 11 i s seen t o r e p r e s e n t a D n e a r l y independent of H/h.

F i g . 11 a l s o shows t h a t D o n l y v a r i e s s l i g h t l y w i t h n^/H. I n o t h e r words, t h e primary v a r i a t i o n of the energy d i s s i p a t i o n i s r e p r e s e n t e d by t h e H"^/h dependency a l r e a d y accounted f o r i n the d e f i n i t i o n ( 2 . 1 1 ) .

2^

6. COMPARISON WITH MEASUREMENTS

The outer and t h e i n n e r r e g i o n

The o r i g i n a l concept o f an o u t e r ( t r a n s i t i o n ) r e g i o n and an i n n e r

(bore) r e g i o n was p r i m a r i l y based on the v i s u a l o b s e r v a t i o n s o f wave behaviour a f t e r b r e a k i n g (see I ) . The i m p r e s s i o n i s one o f a gradual change towards t h e bore shape- found^ i n t h e i n n e r r e g i o n . Consequently no attempt was made t o d e f i n e a proper l i m i t between.the two r e g i o n s and wave h e i g h t measurements t r u l y do n o t suggest a n a t u r a l d e f i n i t i o n .

The s i t u a t i o n i s q u i t e d i f f e r e n t when t h e v a r i a t i o n s i n mean water l e v e l a r e c o n s i d e r e d . F i g . 13 shows some examples from Hansen S Svendsen

(1979) c o v e r i n g a wide range o f deep water s t e e p n e s s e s . They a l l e x h i b i t a marked change i n the slope o f t h e mean water l e v e l a t some d i s t a n c e shoreward from t h e b r e a k i n g p o i n t . A s i m i l a r v a r i a t i o n c a n a l s o be; seen i n o t h e r i n v e s t i g a t i o n s such a s Bowen e t a l . (1968) andT S t i v e & Wind ( 1 9 8 2 ) . The mean w a t e r l e v e l i s h o r i z o n t a l o r weakly s l o p i n g over a d i s t a n c e o f 5-8 times t h e b r e a k e r depth and then a r a t h e r sharp i n c r e a s e i n slope o c c u r s . The d i s t a n c e o f n e a r l y h o r i z o n t a l mean water l e v e l i s comparable t o t h e d i s t a n c e o f t h e most obvious t r a n s f o r m a t i o n s o f t h e wave shape f o l l o w i n g a f t e r t h e i n i t i a t i o n o f b r e a k i n g , and so i t w i l l be coherent w i t h the o r i g i n a l concept t o d e f i n e t h e

l i m i t between t h e o u t e r and t h e i n n e r r e g i o n a s t h e p o i n t where t h e s l o p e o f the mean water l e v e l changes. I n the f o l l o w i n g t h i s i s termed t h e t r a n s i t i o n p o i n t .

Wave c o n d i t i o n s i n the i n n e r r e g i o n

P h y s i c a l e x p l a n a t i o n s f o r t h e s e changes a r e sought i n s e c t i o n 7.

F i r s t , however, we n o t i c e t h a t s i n c e t h e r e s u l t s d e r i v e d above f o r t h e p a r a m e t e r s B, P and D a r e based on the wave p r o p e r t i e s i n the i n n e r zone comparisons w i t h e x p e r i m e n t a l d a t a s h o u l d s t a r t i n t h a t r e g i o n . I t i s 'convenient t o u s e t h e same r e f e r e n c e p o i n t i n a l l computations. To s t a r t computations a t the above d e f i n e d t r a n s i t i o n p o i n t (and emphasize t h e a r b i t r a r i n e s s o f t h e r e f e r e n c e p o i n t ) we t h e r e f o r e w r i t e (3.6) a s ^d ' = r V dh + h •— r dh t DK hh (6.1) which upon d e f i n i t i o n o f H ^ t Sc^B^T fh^ DK dh t s h X r (6.2) can be w r i t t e n H rh DK ^d ^ t ^ ° r V , hh dh (6.-3)

(which c a n a l s o ba o b t a i n e d d i r e c t l y from C 3 , 5 ) ) . F o r t h e wave h e i g h t t h i s means fh' . - r - l

Ddh' h^ ( c ' B ' ) ^ / \ ' _

(6.4)

where we i n analogy t o (4.1b) have d e f i n e d

= c / c ^ ; B ' = B / B ^ (6.5)

K i s g i v e n by (3.10b) and K^ by (6.2) which can a l s o be w r i t t e n ( s e e (3.3) and ( 3 . 7 ) )

-

3o

The momentiam e q u a t i o n (2.1) i s s o l v e d s i m u l t a n e o u s l y w i t h t h e d e t e r m i n a t i o n o f t h e wave h e i g h t . Hence (3.11) i s used f o r h i n ( 6 . 4 ) . With (2.10) and (5.24) s u b s t i t u t e d (2.1) becomes

d n ^ J ^ ^ j ( 3 + 0.9 ^)H^] (6.7)

dx , - dx ' ^2 o L '

h+ri

I n t h e computations we n e g l e c t t h e v a r i a t i o n o f B^ and assume t h a t h/L

^'ïP'. I t i s emphasized a g a i n t h a t t h i s means c i s p r o p o r t i o n a l t o /gh, n o t

e q u a l t o (see I ) . F o r B and n i s used B = 0.075 (see P i g . 9) and n /H

^ O C O

V-= 0.6 ( F i g . 1 2 ) . '

D i s c u s s i o n o f r e s u l t s

F i g s . 14, 15 and 16 show a comparison w i t h r e s u l t s f o r t h r e e r a t h e r d i f f e r e n t wave s t e e p n e s s e s , a l l on a p l a n e s l o p e 1/34.3. I n g e n e r a l t h e

agreement i s q u i t e good p a r t i c u l a r l y f o r t h e s e t - u p . The l a t t e r i s o f p a r t i c u l a r i n t e r e s t because t h e c a l c u l a t i o n s show t h a t n i s much more s e n s i t i v e t o t h e

assumptions made than i s t h e wave h e i g h t v a r i a t i o n . As c a n be expected from what was s a i d above t h e H v a r i a t i o n i s v i r t u a l l y independent o f t h e c h o i c e o f

I t i s n o t i c e d t h a t i n some o f the c a s e s the H - v a r i a t i o n i s s l i g h t l y l e s s c u r v e d than c o r r e s p o n d i n g t o the b e s t f i t o f measurements, and the v a l u e s o f H grow a l i t t l e too l a r g e . T h i s tendency w i l l be f u r t h e r a m p l i f i e d i f a l a r g e r v a l u e f o r n /H i s used (see P i g s . 11 and 1 2 ) . Both t h e s e p o i n t s can be a d j u s t e d

c

by u s i n g a v a l u e o f D perhaps 20-30% l a r g e r than g i v e n by ( 5 . 3 1 ) , which i s

q u i t e c o n s i s t e n t w i t h the r e s u l t s r e p o r t e d e a r l i e r (see I and Svendsen S Madsen, 1981) t h a t t h e a c t u a l energy d i s s i p a t i o n i n a s u r f zone wave i s l a r g e r t h a n i n a hydratü-ic jump o f the same hèight.

The e f f e c t of i n c l u d i n g the s u r f a c e r o l l e r i n B can be understood by c o n s i d e r i n g t h e energy e q u a t i o n i n the form ( 4 . 4 ) . I n the f i r s t b r a c k e t r e p r e -s e n t i a q t h e -s h o a l i n g mechani-sm B ^ B i -s much -s m a l l e r than the o t h e r two t e r m -s . So the v a l u e o f B m a i n l y e n t e r s the. l a s t term,. Hence, the,• i n c i j e a s e i n B due t o the r o l l e r i s e q u i v a l e n t t o a s i m i l a r d e c r e a s e i n D , and the o b s e r v a t i o n above t h a t D i s too s m a l l c o u l d a l s o be due t o an o v e r e s t i m a t i o n o f B.

i n F i g s . 14-16 a r e a l s o i n c l u d e d r e s u l t s obtained by o m i t t i n g the sxirface r o l l e r (.dqtted curve c o r r e s p o n d i n g t o B = B^ and P = 3/2 B^) . The e f f e c t i s q u i t e a p p r e c i a b l e . On the o t h e r hand, c o n s i d e r i n g t h a t the p r e s e n c e o f the s u r f a c e r o l l e r n e a r l y doubles energy f l u x and r a d i a t i o n s t r e s s t h e d i f f e r e n c e between the f u l l and the d o t t e d l i n e s i n t h e s e f i g u r e s i n d i c a t e s t h a t the e f f e c t o f a l s o i n c l u d i n g t u r b u l e n c e , d e v i a t i o n from s t a t i c p r e s s u r e , e t c . would h a r d l y be d i s c e r n i b l e .

The s t r o n g e s t j u s t i f i c a t i o n , however, f o r the importance of the s u r f a c e r o l l e r i s o b t a i n e d by c o n s i d e r i n g the motion i n the t r a n s i t i o n r e g i o n .

7. THE WAVE MOTION IMMEDIATELY AFTER BREAKING

I t i s tempting and i l l u s t r a t i v e f i r s t t o t r y i f t h e s o l u t i o n p r e s e n t e d i n the p r e v i o u s c h a p t e r s a l s o a p p l i e s t o t h e r e g i o n o f r a p i d t r a n s i t i o n r i g h t a f t e r t h e i n i t i a t i o n o f b r e a k i n g .

F i g . 17 shows a computation o f t h e wave h e i g h t v a r i a t i o n , s t a r t i n g a t t h e b r e a k i n g p o i n t . The agreement i s s u r p r i s i n g l y good. (Again e i t h e r D i s . s l i g h t l y t o o s m a l l o r B too l a r g e . ) T h i s , however, does n o t a p p l y t o F i g . 18, w h i c h g i v e s a s i m i l a r comparison f o r t h e s e t - u p n/h^^g. The two f i g u r e s

together- show t h e • p a r a d o x i c a l f a c t a l r e a d y h i n t e d a t e a r l i e r t h a t t h e

r a d i a t i o n s t r e s s i n t h e t r a n s i t i o n r e g i o n s t a y s n e a r l y c o n s t a n t even w i t h a 3 0 - 4 0 % decrease i n wave h e i g h t . R e c a l l i n g ( 2 . 1 0 ) t h i s c a n o n l y be t r u e i f P i s i n c r e a s i n g , roughly--as; H. ,.

By c o n s i d e r i n g what happens when t h e b r e a k i n g s t a r t s , i t becomes c l e a r t h a t t h e c o l l a p s e o f t h e wave cannot immediately be matched by d i s s i p a t i o n o f a ' ' s i m i l a r amount o f energy. I n t h e f i r s t t r a n s f o r m a t i o n a l a r g e amount o f t h e l o s t p o t e n t i a l energy i s c o n v e r t e d i n t o forward momentum f l u x which e v e n t u a l l y i s c o n c e n t r a t e d m a i n l y i n t h e r o l l e r , and t h i s must be t h e r e a s o n f o r t h e s i m u l t a n e o u s i n c r e a s e i n P.

T h i s i s a l s o c o n s i s t e n t w i t h t h e f a c t t h a t P f o r v e r y h i g h waves i s r a t h e r s m a l l . There a r e no r e s u l t s a v a i l a b l e f o r t h e skew waves a t t h e b r e a k i n g p o i n t , b u t t h e h i g h o r d e r r e s u l t s f o r Stokes waves p r e s e n t e d by C o k e l e t (1977) can be used t o determine P f o r v e r y h i g h , s y m m e t r i c a l waves. F i g . 19 shows the v a r i a t i o n o f P w i t h H/h f o r two v a l u e s o f h/L. Notice t h a t f o r t h e h i g h e s t

waves P i s l e s s than h a l f t h e v a l u e o f 3/15 f o r l i n e a r long waves and c o n s i d e r a b l y l e s s than P f o r c n o i d a l waves o f t h e same h e i g h t .

The i n c r e a s e i n P, however, i s i n e v i t a b l y a s s o c i a t e d w i t h a s i m i l a r i n c r e a s e i n B, t h e energy f l u x f o r a wave o f u n i t h e i g h t and p r o p a g a t i o n speed. The problem i s t h e same a s f o r P: v e r y s t e e p waves w i t h peaky c r e s t s r e p r e s e n t a v e r y s m a l l energy f l u x r e l a t i v e t o t h e i r h e i g h t ( F i g . 20 shows r e s u l t s s i m i l a r t o those f o r P) and t h e c o l l a p s e o f t h e c r e s t i n t h e i n i t i a l s t a g e o f b r e a k i n g l e a d s t o a s i g n i f i c a n t i n c r e a s e i n B.

As we w i l l s e e s h o r t l y thèse.shifts i n P and B a r e a l s o c o n s i s t e n t w i t h t h e . r e s u l t found i n Chapter 5, t h a t waves i n t h e i n n e r r e g i o n r e p r e s e n t r a t h e r - h i g h v a l u e s o f - r a d i a t i o n s t r e s s and energy f l u x r e l a t i v e t o - t h e i r h e i g h t and speed.

B u t even w i t h no energy d i s s i p a t i o n a n i n c r e a s e i n - B w i l l i n i t s e l f r e q u i r e d e c r e a s i n g wave h e i g h t . - Hence -fche' q u e s t i o n a r i s e s : hov; much o f t h e wave h e i g h t d e c r e a s e i n t h e o u t e r t r a n s i t i o n r e g i o n i s a c t u a l l y due t o r e -d i s t r i b u t i o n o f energy ( r e p r e s e n t e -d by t h e changes i n P an-d B) an-d how much i s r e a l energy d i s s i p a t i o n ?

T h i s problem and t h e change i n B and P can be a n a l y s e d by c o n s i d e r i n g t h e c o n s e r v a t i o n o f momentum and energy over t h e t r a n s i t i o n r e g i o n a s a whole i n analogy t o t h e jump c o n d i t i o n s t h a t a p p l i e s t o bores and h y d r a u l i c jumps i n open c h a n n e l flow and t o shocks i n c o m p r e s s i b l e f l o w s .

2>f

Between t h e b r e a k i n g p o i n t (denoted w i t h s u f f i x g) and t h e t r a n s i t i o n p o i n t we have from (2.8)

^ f , t - ^ f , B I?dx = (7.1)

where I i s t h e w i d t h o f the t r a n s i t i o n r e g i o n . We a r e i n t e r e s t e d i n d e t e r m i n i n g how l a r g e a f r a c t i o n i s o f t h e energy d i s s i p a t i o n we would have had, had B s t a y e d c o n s t a n t and e q u a l t o B^. T h i s energy d i s s i p a t i o n would o b v i o u s l y have been ^m = P^Bg^o^H^' - C ^ H / ) = E ^ ^ ( ^ J [ ^ ] - 1) by v i r t u e o f ( 2 . 9 ) . U s i n g (2.9) i n (7.1) a s w e l l y i e l d s 2 B. 1) (7.2) (7.3) so t h a t t h e r a t i o we a r e l o o k i n g f o r i s A = v.

t

I n t h e momentum e q u a t i o n (2.1) we have 3n/3x = 0 and hence. u s i n g C2.10)

\ = ^ B ( W '

(7.5)

F i n a l l y s i n c e P. and B a r e g i v e n by (5.24) and ( 5 . 1 5 ) , r e s p e c t i v e l y . we have by e l i m i n a t i o n o f B between t h o s e two

o

= f ^ t - V ^ B

Thus i f we know the p r o p e r t i e s of the wave a t b r e a k i n g we can determine B^, and A- I n need o f more c o r r e c t i n f o r m a t i o n we use (from F i g s . 19 and 20) f o r an example w i t h h /L_ = 0.057

= 0.05 , Pg = 0.07

and assume t h e t r a n s i t i o n p o i n t i s a t h^ = 0.85 w i t h H^/Hg = 0 . 6 5 . We then g e t from (7.4)

A = 0.33

which means t h a t o n l y 33% o f t h e energy t h a t corresponds t o t h e d e c r e a s e i n wave h e i g h t i s a c t u a l l y l o s t - The i n c r e a s e i n B accovints f o r t h e r e s t . We f u r t h e r g e t

C7.5): P = 0 . 1 6 6 a g a i n s t (.5.24): P = 0.152 t

( 7 . 6 ) : B^ - 0.103 a g a i n s t ( 5 . 1 5 ) : B^ = 0.094

Thus the v a l u e o f P and B. r e q u i r e d to a c c o m t f o r t h e g r o s s change

l l t —

i n the r a d i a t i o n s t r e s s / m e a n water l e v e l and the wave h e i g h t o v e r the t r a n s i t i o n r e g i o n a g r e e s w e l l w i t h the v a l u e s we can determine f o r a wave a t tha s t a r t o f t h e bore r e g i o n u s i n g t h e i d e a s from s e c t i o n 5.

T h i s i s taken a s another i n d i c a t i o n t h a t the i d e a s p r e s e n t e d i n t h a t

s e c t i o n f o r the p r o p e r t i e s o f waves i n the i n n e r r e g i o n a r e a t l e a s t q u a l i t a t i v e l y c o r r e c t .

As mentioned above the c o n s t a n t r a d i a t i o n s t r e s s i n t h e t r a n s i t i o n r e g i o n must imply

7-5

P « H " ^ ( V . 7 )

Most measurements f u r t h e r show t h a t H v a r i e s approximately l i n e a r l y , i . e . ,

H a - ax ( 7 . 8 )

a f t e r b r e a k i n g . S i n c e the v a r i a t i o n o f H i s much f a s t e r than t h e v a r i a t i o n 2

i n depth, and s i n c e E _ H we may n e g l e c t c i n (2.8) and w r i t e

1 ^ f - °x 2a DH ^ y ^ g j dx B " H 4hLB

However i f we i n t h i s e q u a t i o n - i n analogy w i t h the s i t u a t i o n i n the bore r e g i o n r e p r e s e n t e d by ( 7 . 6 ) - assume t h a t B « P + c o n s t , then ( 7 . 9 ) y i e l d s D = 0 which c l e a r l y i s not c o r r e c t . Hence ( 7 . 9 ) i s not s u i t a b l e f o r a s s e s s i n g D on the b a s i s of some r e a s o n a b l e c o n j e c t u r e f o r B.

So i f one wants t o be a b l e t o extend the method d e s c r i b e d i n t h i s paper t o t h e t r a n s i t i o n r e g i o n t h e r e i s room f o r both f u r t h e r e x p e r i m e n t a l i n v e s t i g a t i o n s and f o r some e m p i r i c a l i n t e r p o l a t i o n formulas f o r t h e d e v e l o p

-ment o f D from zero a t the b r e a k i n g p o i n t t o the v a l u e g i v e n by (5.31)

a t the t r a n s i t i o n p o i n t . >

I n the absence o f such r e s u l t s t h e r e i s always the p o s s i b i l i t y o f u s i n g ( 7 . 7 ) f o r P i n the t r a n s i t i o n r e g i o n t o get the c o r r e c t n - v a r i a t i o n and l e t B and D be g i v e n by the bore v a l u e s (5.15) and (5.31) r i g h t from t h e b r e a k i n g p o i n t (which we have seen w i l l g i v e approximately the c o r r e c t H - v a r i a t i o n ) ,

I t i s j u s t t h a t t h i s means B i s d i s c o n t i n u o u s a t t h e b r e a k i n g p o i n t (which, o f c o u r s e i s not t r u e ) and i t i s a l s o an u n s a t i s f a c t o r y procedure from a p h y s i c a l p o i n t of view because i t o b v i o u s l y does not model the r e a l p r o c e s s e s .

8. SUMMARY AND CONCLUDING REMARKS

The p h y s i c a l mechanisms behind the v a r i a t i o n of wave h e i g h t s and set-up i n the s u r f zone have been a n a l y s e d and a t h e o r e t i c a l model has been suggested. I t i s based on r a t h e r s i m p l e approximations f o r the i n t e g r a l p r o p e r t i e s o f the wave motion. Comparison i n the i n n e r r e g i o n o f t h e s u r f zone w i t h ejqperiments shows a c c e p t a b l e agreement.

An attempt h a s been made t o e x p l a i n t h e n a t u r e o f the t r a n s f o r m a t i o n s o f the waves i n t h e t r a n s i t i o n zone r i g h t a f t e r b r e a k i n g ( S e c t i o n 7) .

The methods f o r determining the non-dimensional energy f l u x B,

r a d i a t i o n s t r e s s P and energy d i s s i p a t i o n D can e a s i l y be r e f i n e d , t h e c r e s t e l e v a t i o n rj /H may perhaps more c o r r e c t l y be determined by a l i n e a r l y de¬

c

c r e a s i n g f u n c t i o n , e t c . Such improvements, however, a r e not l i k e l y t o change t h e b a s i c c o n c l u s i o n t h a t the major d i f f e r e n c e between s u r f zone waves and o r d i n a r y waves i s r e p r e s e n t e d by the s u r f a c e r o l l e r . And a s a f i r s t a p p r o x i -mation the r o l l e r can be c o n s i d e r e d a s a volume o f water c a r r i e d shoreward w i t h t h e wave. T h i s p i c t u r e i s found t o be i n accordance both w i t h t h e motion. a l l e g e d f o r the i n n e r r e g i o n and w i t h t h e changes o c c u r r i n g over t h e t r a n s i t i o n r e g i o n ,

The author g r a t e f u l l y acknowledges a c c e s s t o the u n p u b l i s h e d d a t a s e n t by J . Biihr Hansen and used i n some o f the f i g u r e s .

52

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COKELET, E . D, (19771. Steep g r a v i t y waves i n water o f a r b i t r a r y u n i f o r m depth.. P h i l . T r a n s . Roy. S o c . Lond. , 286, 183-230.

DIVOKY, D., B„ LEMEHAUTÉ S A. L I N ( 1 9 6 8 ) . B r e a k i n g waves on g e n t l e s l o p e s . J a u r . Geophys. R e s . , 75_, 1681-1692.

DUNCAN, J . -H. (19^81) . An e x p e r i m e n t a l i n v e s t i g a t i o n o f b r e a k i n g waves

produced by a towed h y d r o f o i l . P r o c . Roy. Soc. Lond., A 377, 331-348. HANSEN, J . B. ( 1 9 8 0 ) . E x p e r i m e n t a l i n v e s t i g a t i o n s o f p e r i o d i c waves n e a r

b r e a k i n g . P r o c . 17 C o a s t . Engrg. Conf., Sydney, 260-277.

HANSEN, J . B. (1982). Wave measurements i n t h e s u r f zone. P r i v a t e communi-c a t i o n o f unpxiblished d a t a .

HANSEN, J . B. & I . A. SVENDSEN ( 1 9 7 9 ) . R e g u l a r waves i n s h o a l i n g w a t e r , e x p e r i m e n t a l d a t a . I n s t . Hydrodyn'. & Hydr. Engrg. S e r i e s Paper 21. HENDERSON, F . M. ( 1 9 6 6 ) . Open c h a n n e l f l o w . Macmillan P u b l . Co., N.Y.,

522 pp.

HORIKAWA, K & C. KUO ( 1 9 6 6 ) , Wave t r a n s f o r m a t i o n a f t e r b r e a k i n g p o i n t . 10th Conf. C o a s t . Engrg., Tokyo, 217-233.

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L I S T OF CAPTIONS

1. Regions i n the s u r f zone Cf rora I ) ,

2. D e f i n i t i o n s . 3 . H/H from a n a l y t i c s o l u t i o n (4.1) v e r s u s (h/h_)^,„. f 1 H r h r from a n a l y t i c s o l u t i o n v e r s u s C h / h ^ l ^ g .

Minimum v a l u e s o f H/h^^/(H^/h^) and the v a l u e h^^^ o f h' a t w h i c h t h e y occvir.

6. The r o l l e r o f a s u r f zone wave.

7. The approximation f o r t h e h o r i z o n t a l v e l o c i t y p r o f i l e .

8. C r o s s s e c t i o n a r e a A f o r a r o l l e r . Measurements by Duncan (1981) .

9. Measured v a l u e s o f B d e f i n e d by ( 5 . 1 6 ) . (Data from Hansen, 1 9 8 2 ) . o

10. Wave p r o f i l e s i n the s u r f zone. D e r i v e d from measurements i n I .

11. The v a r i a t i o n o f D w i t h H/h and n /H a c c o r d i n g to (5.31) . c

12. Measurements of n /H i n the s u r f zone. , (Data from Hansen, 1 9 8 2 ) . c

13. Measurements of t h e mean water l e v e l shoreward of t h a b r e a k i n g p o i n t . A l s o shown i s t h e v a l u e o f l o c a l wave h e i g h t t o b r e a k e r h e i g h t , H/H^. (Measurements from Hansen & Svendsen, 1 9 7 9 ) .

Measurements by Hansen S Svendsen (1979), Case B.

15. Wave h e i g h t s and set-up f o r a wave w i t h deep water s t e e p n e s s H^/L^ = 0.024. Theory u s i n g (6.4) and ( 5 . 7 ) .

Measurements by Hansen ( 1 9 8 2 ) , Case H.

16. Wave h e i g h t s and set-up f o r a wave w i t h deep water s t e e p n e s s H^A^ = 0.0107. Theory u s i n g (6.4) and ( 6 . 7 ) .

Measurements by Hansen & Svendsen (1979), Case N.

17. Wave h e i g h t u s i n g (6.4) and (6.7) from t h e b r e a k i n g p o i n t . Measurements by Hansen (1982), Case H.

13. Set-=-up u s i n g (6.4) and (6.7) from t h e b r e a k i n g p o i n t . Measurements by Hansen (1982).

19. The non-dimensional r a d i a t i o n s t r e s s f o r s y m m e t r i c a l waves.

R e s u l t s from C o k e l e t ( 1 9 7 7 ) , l o w e s t o r d e r c n o i d a l waves.

20. The non-dimensional energy f l u x i n s y m m e t r i c a l waves.

O U T E R R E G I O N I N N E R R E G I O N

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