Directional spreading in surfbeat: A first assessment

Directional spreading i n surfbeat

A first assessment

Report submitted to R I K Z

A d R e n i e r s a n d A p v a n D o n g e r e n

TUDeift

Delft University of Technology

Section Fluid Mechanics

CONTENTS 2

Contents

LIST OF FIGURES 3

List of Figures

1 L e f t panel: Snapshots of c o m p u t e d surface elevation f o r a = 3 0 ° . R i g h t panel: Normalised long wave surface elevation a m p l i t u d e at t h e shore line (solid line) compared t o a n a l y t i c a l s o l u t i o n given b y Schaffer [1990]

(dashed line) 9 2 L e f t panel: Short wave energy density s p e c t r u m . R i g h t panel: C o m p a

-rison of corresponding b o u n d l o w frequency energy density according

t o eq. 22 (solid Une) and Hasselmann (1963) (dashed line) 10 3 B a t h y m e t r y f o r O c t 10 and p o s i t i o n of i n s t r u m e n t s i n cross-shore array 11

4 U p p e r panel: Short wave energy density s p e c t r u m f o r O c t l O , l O l i - l l h . Lower panel: T o t a l (solid line) and b o u n d (asterisks) low frequency energy density at measurement locations s t a r t i n g offshore (upper l e f t )

t o w a r d s the shore (lower r i g h t 12 5 C o m p u t e d t o t a l (solid) and b o u n d (asterisks) low frequency energy

density s p e c t r u m f o r O c t l O l O h - l l h 13 6 U p p e r nine panels: D i r e c t i o n a l spreading of b o u n d low frequency energy

density O c t l O , l O h l l h . Lower nine panel: D i r e c t i o n a l spreading o f t o

4

T h i s progress r e p o r t refers t o p a r t of t h e w o r k done w i t h i n t h e f r a m e w o r k o f the D u t c h Center f o r Coastal Research ( N C K ) . T h e p r i m a r y o b j e c t i v e of our research is t o develop knowledge and methods f o r t h e p r e d i c t i o n of t h e h y d r o d y n a m i c c o n d i t i o n s f o r the D u t c h coast t a k i n g i n t o account t h e m o r p h o d y n a m i c behaviour i n t h e nearshore zone.

I n general t h e progress reports describe results, developments and new ideas ob-t a i n e d d u r i n g our research. M o s ob-t resulob-ts have a p r e l i m i n a r y sob-taob-tus and should n o ob-t be used w i t h o u t p r i o r consent o f t h e authors. T h e various topics described i n t h e progress r e p o r t s are b r o u g h t together t o gain more insight i n the process t o reach t h e p r i m a r y o b j e c t i v e . M o r e detailed analyses of t h e various topics are or w i l l be given in t h e f o r m of j o u r n a l papers. I n a d d i t i o n a t t e n t i o n is given t o t h e ( p o t e n t i a l ) links w i t h w o r k done by others.

1 INTRODUCTION 5

1 Introduction

111 the f o l l o w i n g we consider the f o r c i n g of long waves by n o r m a l l y or obliquely i n -cident grouped short waves, also k n o w n as surf beat. T w o mechanisms responsible f o r the generation of long waves are considered: the release of the b o u n d l o n g waves associated w i t h changes i n the spatial v a r i a t i o n of the incident short wave energy [Longuet-Higgins and Stewart, 1964] and the t i m e - v a r y i n g position o f the break p o i n t [Symonds et a l . , 1982]. These mechanisms are examined using Hnear shallow water equations on an alongshore u n i f o r m coast, w i t h special a t t e n t i o n f o r the cases of re-sonant i n t e r a c t i o n between the incident short waves and edge waves. T h e effect of directional spreading w i l l be examined i n d e t a i l . T h e reason t o use linearized equa-t i o n s is equa-t w o f o l d . F i r s equa-t i equa-t renders equa-the possibiliequa-ty equa-t o examine equa-the various l o n g wave generation mechanisms separately ( w i t h o u t nonhnear interactions c o m p l i c a t i n g the analysis). Second, i t provides a quick assessment ( c o m p u t a t i o n a l t i m e is an order of m a g n i t u d e smaller) of the conditions w h i c h are interesting (such as edge waves) and can thus provide the necessary b o u n d a r y conditions f o r the more complex nonlinear modelling.

2 IVEodel equations

2.1 W a v e t r a n s f o r m a t i o n

I n the f o l l o w i n g we consider processes w h i c h occur on the t i m e scale of wave groups, w h i c h are t y p i c a l l y i n the o r d e r / o f t h i r t y seconds t o a few minutes. T o o b t a i n the a p p r o p r i a t e equations t h e relevant variables are averaged over a single s h o r t wave p e r i o d , i.e. short-wave averaged. Considering a n o n - s t a t i o n a r y wave f i e l d o f obliquely incident r a n d o m waves on a' variable b a t h y m e t r y u n i f o r m i n the alongshore d i r e c t i o n , the balance f o r the short/Wave averaged wave energy, E^,, is given by:

/ dE^ ^ dE.^c cosia) ^ _^

at dx ^ '

where Cg represents the g r o u p velocity, S the wave energy dissipation, x the distance along the shore n o r m a l (positive onshore) and a the angle of incidence w i t h respect the .T-axis. T o model t h e wave energy dissipation due t o wave breaking the dissipation model according t o R o e l v i n k (1993) is i n t r o d u c e d :

\ (r^---^'^'-'

S = 2adU{l - . e ^ N ^ (2)

where a'd, n a n d - Y ^ r e the breaker parameters and f p the peak frequency and Hrms the r o o t mean square wave height. For values of n ^ 100 the dissipation f o r m u l a t i o n corresponds t o m o n o c h r o m a t i c wave breaking whereas n-values j 20 are t y p i c a l l y used t o a p p r o x i m a t e wave dissipation i n a r a n d o m wave field.

T h e group velocity is o b t a i n e d f r o m linear wave t h e o r y :

_ d a ^ _ / I ksh \

2 MODEL EQUATIONS 6

where ojg is t h e angular frequency o f t h e s h o r t waves, ks t h e wave number, h t h e t o t a l w a t e r d e p t h (including set-up) and c t h e phase speed given by:

G i v e n t h e w a t e r d e p t h t h e wave n u m b e r , kg, can be o b t a i n e d f r o m t h e Hnear wave dispersion r e l a t i o n :

ujp^ = gksta,nh{ksh) (5) g being t h e g r a v i t a t i o n a l acceleration. T h e wave incidence angle, a, is o b t a i n e d f r o m

Snell's law:

s i n ( a ) _ sin(Q'o) c Co

where t h e subscript, 0, denotes a reference p o i n t offshore.

2 . 2 L o n g w a v e e q u a t i o n s

T h e short-wave and depth-averaged Hnearized c o n t i n u i t y equation is given b y :

1

dt dx dy

where d is t h e water d e p t h w i t h o u t set-up. T h e cross-shore m o m e n t u m balance reduces t o :

du dr]

'"^Yt+P''^Tx =

-

^

-

^

and t h e alongshore m o m e n t u m e q u a t i o n :

, 9 u , ,077 dSyy dSya;

where t h e b o t t o m shear stress and viscosity effects have been neglected. C o m b i n i n g these equations results i n a single e q u a t i o n f o r t h e long wave surface elevation:

- I d ^ d ^ dddn ^ d^V 1 ( d ' S , , d2Sy, d'Syy\

g df^ dx'^ dx dx dy'^ pg y dx'^ dxdy dy"^

I n t h e f i e l d the f o r c i n g on t h e r i g h t h a n d side is governed by t h e frequency-d i r e c t i o n a l s p e c t r u m o f t h e short waves. Consifrequency-dering t h e f u l l short wave s p e c t r u m , each c o m b i n a t i o n o f t w o p r i m a r y waves o f d i f f e r e n t f r e q u e n c y w i l l result i n a bichro-m a t i c wave t r a i n w i t h an accobichro-mpanying b o u n d wave. I n t h e f o l l o w i n g we therefore consider a c o m b i n a t i o n o f t w o s h o r t waves w i t h d i f f e r e n t angular frequency, w^, and possibly d i f f e r e n t directions « j . T h e l o n g wave angular frequency is t h e n given by:

U = OJi — L02 (11)

and corresponding alongshore wave n u m b e r :

2 MODEL EQUATIONS 7

respectively t h e cross-shore wave number o f t h e long wave:

fca; = / j i • c o s ( a i ) - A;2 • COS(Q'2) ( 1 3 )

where fc, represents t h e wave number of t h e s h o r t waves.

Analogous t o Schaffer [1990] we assume t h e l o n g wave m o t i o n s t o be p e r i o d i c a l i n b o t h t i m e and alongshore d i r e c t i o n :

Vi^' 2/i t) = ^fi{x)exp[i{ujt - kyy)] + * (14)

where * stands f o r t h e complex conjugate, ky represent the alongshore wave number of the l o n g waves and oj is t h e long wave frequency. A similar assumption f o r t h e f o r c i n g induced b y t h e r a d i a t i o n stress gradients gives:

Sij{x, y,t) = Sij{x) -E{x)exp[i{ojt - kyy)] + * (15)

where:

S . . ( . T ) = ( ^ ( I + C O S 2 « ) - ^ ) ( 1 6 )

Sxy{x) = ( —cosasincv (17)

^ . . ( • ^ • ) = ( ^ ( l + B i n ^ « ) - ^ ) ( 1 8 )

and t h e energy m o d u l a t i o n associated w i t h t h e b i c h r o m a t i c wave g r o u p i n g is given by:

E{x) = Êexp[-i ƒ k^dx] (19)

where E represents t h e a m p l i t u d e of t h e energy m o d u l a t i o n . I n t r o d u c i n g t h i s i n t o eq. 1 0 yields:

g ^ d ^ ^ d ï d i ~ ^ V ~ ^2/2/)] + * = (20)

- 1 d's^^E ds^yE 2 rA\ rv . 1 M ,

T h i s e q u a t i o n has been solved n u m e r i c a l l y using f i n i t e differences t o yield t h e long wave surface elevation under obliquely incident wave groups, where t h e b o u n d a r y conditions are given by a zero flux at t h e shore line and a weakly reflective b o u n d a r y c o n d i t i o n offshore t a k i n g i n t o account t h e i n c i d e n t b o u n d long wave associated w i t h

3 VERIFICATION 8

the wave groups and free obHque o u t g o i n g long waves. O n a h o r i z o n t a l plane the b o u n d long wave elevation,775, can be obtained f r o m eq. 20:

2-;,,

kx'^dfjb - kldfib ex2)[i{u)t - kyy)] + * = (21)

P9

- [-k^'^s^^Ê - 2ikyk^s^yÈ - k'^SyyE^ exp[i{(x>t - kyy)] + *

w h i c h can be r e w r i t t e n t o : where: and: Tï-Sd = - -^s^^E+-^s^yÊ+-^SyyÊ (22) , n / J fJ \ tv At Ay / k = ^kl + kl (23) ^ = - s ( ö ) ^ ^ = cos(ö) s i n ( ö ) , ^ = s i n ( ö ) ^ (24)

where 9 is the incidence angle of the b o u n d long wave. W i t h use of eqs. 19-21 we o b t a i n the general expression f o r the b o u n d long wave a m p l i t u d e on a h o r i z o n t a l plane:

( S - ad)

I n the case the t w o p r i m a r y waves have the same direction the s o l u t i o n of L o n g u e t -Higgins and Stewart (1964) is retained:

(2n - \ ) È

3 Verification

To test the d e r i v a t i o n and i m p l e m e n t a t i o n we have compared the results t o t h e ana-l y t i c a ana-l soana-lutions provided by Shaffer [1990]. I n t h i s case the t w o p r i m a r y waves have d i f f e r e n t frequencies b u t equal directions. T h e angular frequency of the p r i m a r y wa-ves, u>i and 0J2, are given by

3 VERIFICATION 9

where Ug is the mean angular frequency o f t h e p r i m a r y waves and e = 0 . 1 . T h e wave breaking parameters are y = .64, av = 1.0 and n = 5. T h e m o d u l a t i o n of t h e wave energy w i t h respect t o t h e mean short wave energy is given by:

È = pgSa?

where 5 is taken t o be 0.1 and the mean a m p l i t u d e of the short waves, a, equals 0.0248 ( m ) . T h e angle o f incidence, a, ranges between V and 60° w i t h respect t o the coast n o r m a l using a plane beach w i t h a slope m of 0.05 and an offshore d e p t h , / i q of 0.2485 m . N e x t we compare the normalised surface elevation a m p l i t u d e at the shore line as f u n c t i o n of the incidence angle t o the a n a l y t i c a l results (see the r i g h t panel of F i g u r e 1). W h e r e the normalized surface elevation a m p l i t u d e is given by:

F i g u r e 1: L e f t panel: Snapshots of c o m p u t e d surface elevation f o r a = 3 0 ° . R i g h t panel: Normalised long wave surface elevation a m p l i t u d e at the shore line (solid Hne) compared t o a n a l y t i c a l s o l u t i o n given by Schaffer [1990] (dashed Hne).

Small differences occur, w h i c h can at least i n p a r t be ascribed t o t h e a l t e r n a t i v e wave dissipation f o r m u l a t i o n . O v e r a l l t h e comparison shows good agreement. T h e f a c t t h a t f o r a = 53° t h e shore Hne elevation goes t o i n f i n i t y indicates the presence of resonant i n t e r a c t i o n between t h e s h o r t wave groups and the u n d e r l y i n g l o n g waves c q . edge wave.

N e x t we consider t w o p r i m a r y waves w i t h d i f f e r e n t frequencies and directions:

rji = f]isin[u)t — kicos[ai)x — kisin{a\)y) (28)

and

4 TEST CASE 10

where the mean incidence angle of the short waves is given by:

a = atan f , , ^ (30)

and the incidence angle of the energy m o d u l a t i o n and b o u n d long wave:

/kisin{ai) - k2sin{a2)\

9 = atan ^ - ^ )—{ 31)

\kicos[a2) - k2Cos{a2) J

I n t h i s case the b o u n d long wave behaviour changes considerably. N o t o n l y w i l l the p r o p a g a t i o n d i r e c t i o n of the bound l o n g wave be d i f f e r e n t f r o m the mean p r i m a r y wave d i r e c t i o n , also the a m p l i t u d e w i l l change. T h e b o u n d low frequency energy density f o r a synthetic short wave s p e c t r u m (shown i n t h e l e f t panel of F i g u r e 2) at a w a t e r d e p t h of 3 m is c o m p u t e d w i t h eq. 22 and compared t o the non-linear s o l u t i o n f o r three-wave i n t e r a c t i o n given by Hasselmann (1962). T h e comparison is g o o d .

F i g u r e 2: L e f t panel: Short wave energy density s p e c t r u m . R i g h t panel: C o m p a r i s o n of corresponding b o u n d low frequency energy density according t o eq. 22 (solid line) and Hasselmann (1963) (dashed line)

4 Test case

4.1 Introduction

T h e behaviour of t h e model is assessed w i t h a realistic testcase: the generation of low frequency energy by obliquely incident short waves d u r i n g the D E L I L A H f i e l d experiment. T h e D E L I L A H experiment was p e r f o r m e d i n 1990 at the US A r m y Corps o f Engineers F i e l d Research F a c i l i t y at D u c k , N o r t h CaroHna. T h e positions of colocated velocity meters and pressure transducers are shown i n F i g u r e 3. A cross-shore array was deployed t o measure the wave t r a n s f o r m a t i o n and cross-shore d i s t r i b u t i o n of the longshore current velocities.

4 TEST CASE 11 750 700 650 600 560 500 760 800 860 900 960 1000 1050

F i g u r e 3: B a t l i y m e t r y f o r O c t 10 and position of i n s t r u m e n t s i n cross-sliore array

T h e i n p u t short wave s p e c t r u m is obtained by a m a x i m u m entropy frequency-directional analysis of t h e puv-signals (pressure, cross-shore and alongshore velocities) at the measuring p o i n t f u r t h e s t offshore i n the cross-shore array. T h e resulting short wave s p e c t r u m f o r O c t l O f r o m l O h - l l h is shown i n the upper panel of F i g u r e 4.

T h e significant wave height at t h a t t i m e was 1.0 m w i t h a mean d i r e c t i o n of a p r o x i m a t e l y -25 deg w i t h the shore n o r m a l and a peak frequency of .9 H z . T h e measured b o u n d long wave energy, shown i n the lower panels of F i g u r e 4, is estimated w i t h a bispectral analysis (Hasselmann et al., 1963).

4 . 2 L o n g w a v e c o m p u t a t i o n

T h e offshore b o u n d a r y c o n d i t i o n f o r the l o w frec|uency c o m p u t a t i o n s is set at 8 m water d e p t h . T o t h i s end the s p e c t r u m , shown i n the upper panel of F i g u r e 4, is inversely shoaled and r e f r a c t e d t o w a r d s this water d e p t h , t o account f o r the generation of free long waves while p r o p a g a t i n g f r o m 8 m t o w a r d s the first offshore measuring p o i n t . Beyond this water d e p t h the generation of free long waves is expected t o be negligible. T h e c o m p u t e d low frequency energy density f o r the present test case is shown i n F i g u r e 5.

T h e comparison w i t h the b o u n d low frequency energy density, compare w i t h l o -wer panels of F i g u r e 4, is f a v o u r a b l e f o r the offshore locations. F u r t h e r inshore the comparison is less g o o d . We m e n t i o n the f a c t t h a t the c o m p u t e d b o u n d long wave energy density is based on t h e assumption t h a t f o r c i n g and the b o u n d long waves are i n e q u i l i b r i u m (like on a h o r i z o n t a l plane). T h i s is more or less t h e case at deeper wa-ter b u t no so as the w a t e r d e p t h becomes small w i t h respect t o the short wave height. However, i n t h a t case one w o u l d expect t h a t the b o u n d l o n g wave is overpredicted and n o t underpredicted as is the case here. T h e l a t t e r is most likely associated w i t h the f a c t t h a t i n the c o m p u t a t i o n s t h e m o d u l a t i o n of the short wave energy disap-pears as the waves s t a r t breaking whereas i n r e a l i t y t h i s is n o t the case. A s f o r the t o t a l energy density the comparison is less good. T h o u g h t h e n o d a l features seem

4 TEST CASE 12

F i g u r e 4: U p p e r panel: Short wave energy density spectrum f o r O c t l O , l O h - l l h . Lower panel: T o t a l (solid line) and b o u n d (asterisks) low frequency energy density at measurement locations s t a r t i n g offshore (upper l e f t ) towards the shore (lower r i g h t

t o be reproduced i n some sense, t h e i n t e n s i t y is a f a c t o r 10 t o o small. T h i s can be related t o a number of phenomena such as t h e set-up, longshore current and short wave m o d u l a t i o n w i t h i n the surfzone. These effects w i l l be investigated f u r t h e r .

T h e c o m p u t e d d i r e c t i o n a l spreading is shown i n F i g u r e 6 (same sequence as i n previous f i g u r e s ) . T h e b o u n d l o w frequency energy density is shown i n t h e upper panels. T h e energy density is restricted t o a r e l a t i v e l y n a r r o w area w h i c h is governed by t h e d i r e c t i o n a l spreading of the short waves. T h e short wave breaking s t r o n g l y decreases the b o u n d low frequency energy density.

M o r e i n t r i g u i n g is the observed p a t t e r n i n t h e t o t a l low frequency energy density (lower panels of F i g u r e 6 ) , w h i c h shows areas o f low energy density a l t e r n a t e d b y areas o f h i g h energy density associated w i t h s t a n d i n g wave patterns. I t is also clear t h a t t h e d i r e c t i o n a l spreading i n t h i s case is m u c h broader t h a n i n the case o f b o u n d low-frequency energy density only. T h i s is related t o the f a c t t h a t the f r e e l o n g

5 CONCLUSIONS 13

Figure 5: C o m p u t e d t o t a l (solid) and b o u n d (asterisks) low frequency energy density s p e c t r u m f o r O c t l O l O h - l l h

waves are r e f r a c t i n g much stronger t h a n t h e b o u n d l o n g waves. T h i s has i m p o r t a n t consequences f o r t h e long wave decay i n t h e offshore d i r e c t i o n as observed b y Herbers et a l . [1990].

5 Conclusions

A Hnear m o d e l t o examine t h e generation of s u r f b e a t f o r c e d by d i r e c t i o n a l l y spread w i n d waves has been developed. A first assessment o f t h e directional characteristics of t h e s u r f b e a t clearly shows t h e differences between b o u n d long waves and free l o n g waves. A n i m p r o v e d comparison w i t h d a t a nessecitates a closer look at t h e effects of wave b r e a k i n g , set-up, longshore currents and b o t t o m f r i c t i o n .

5 CONCLUSIONS 14

0.06 'S3 ïBBiïiïiSMïS 0.06 ï , 0 . 0 4 Ï.0.D4

0.02 0.02

0.1 -0.1 O 0.1 -0.1 O 0.1

(rad/m) k, (rad/m) (rad/m)

0.06 0.06 0.06 Eo.i '• EO.04 ï , 0 , 0 4 0.02

i l B i l

i ^ B I I

F i g u r e 6: U p p e r nine panels: D i r e c t i o n a l spreading of b o u n d low frequency energy density O c t l O , l O h l l h . Lower nine panel: D i r e c t i o n a l spreading o f t o t a l low f r e -quency energy density.

5 CONCLUSIONS 15

Acknowledgements

T h e present progress r e p o r t results f r o m a c o l l a b o r a t i o n w i t h t h e Naval Postgraduate School w h i c h enabled us t o use t h e t i m e series i n the test case. A d d i t i o n a l d a t a were p r o v i d e d by t h e F i e l d Research Facility of t h e U . S . A r m y Engineer W a t e r w a y s E x p e r i m e n t Station's Coastal Engineering Research Center. Permission t o use these d a t a is appreciated. T h e sponsoring by t h e N a t i o n a l I n s t i t u t e f o r Coastal and M a r i n e M a n a g e m e n t ( R I K Z ) t h r o u g h t h e Netherlands Centre f o r Coastal Research ( N C K ) is also greatly appreciated.

16

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