Applied Ocean Research 31 (2009) 90-100
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Applied Ocean Research
E L S E V I E R
journaI homepage: www.elsevier.com/locate/apor
O C E A N
R E S E A R C H
Probability distributions of wave beights in bimodal seas in an offshore basin
Keywords:
Bimodal sea Sea-swell energy ratio Distribution of wave heights Abnormality index Abnormal waves Freak waves Article history;
Received 23 November 2008 Received in revised form 19 June 2009
Accepted 20 June 2009 Available online 16 July 2009
swell and w i n d sea energy. ^ ^009 Elsevier Ltd. All riglits reserved.
1. Introduction
D i f f e r e n t p r o b a b i l i s t i c m o d e l s f o r i n d i v i d u a l w a v e h e i g h t s h a v e b e e n p r o p o s e d so f a r . T h e y r e p r e s e n t g e n e r a l i z a t i o n s o f the R a y l e i g h d i s t r i b u t i o n , w h i c h w a s a p p l i e d f o r the f i r s t t i m e to ocean w a v e h e i g h t s b y L o n g u e t - H i g g i n s [ 1 ] . This m o d e l assumes t h a t t h e w a v e f i e l d has Gaussian p r o p e r t i e s a n d is d e s c r i b e d b y t h e n a r r o w b a n d s p e c t r u m . The e x t e n d e d m o d e l s , o n the o t h e r h a n d , a c c o u n t f o r t h e f i n i t e b a n d w i d t h e f f e c t s a n d t h e w a v e n o n l i n e a r i t i e s [ 2 ] . Naess V i n j e a n d T a y f u n [ 3 5 ] d e v e l o p e d a s y m p t o t i c e x p r e s sions, i n w h i c h t h e p a r a m e t e r s d e p e n d o n t h e f o r m o f t h e a u t o c o r -r e l a t i o n f u n c t i o n , t h u s a c c o u n t i n g f o -r t h e s p e c t -r a l shape. H o w e v e -r , T a y f u n ' s a p p r o x i m a t i o n is a p p l i c a b l e o n l y to w a v e h e i g h t s larger t h a n t h e m e a n w a v e h e i g h t , w h i c h gives rise t o large p r o b a b i l i t y e s t i m a t e s at t h e l o w range o f values. V i n j e ' s f o r m u l a leads t o a s i m i l a r e f f e c t d u e t o a m u l t i p l i c a t i o n f a c t o r i n c l u d e d . B o c c o t t i f o r m u -l a t i o n [ 6 ] uses the c o n c e p t o f t h e -l i n e a r q u a s i - d e t e r m i n i s t i c t h e o r y a n d is also d e v e l o p e d f o r large w a v e s . These f o u r d i s t r i b u t i o n s r e -duce t o t h e R a y l e i g h f o r m at t h e l i m i t o f a v e r y n a r r o w s p e c t r u m . D e v i a t i o n s b e t w e e n the t h e o r e t i c a l p r o b a b i l i t y e s t i m a t e s arise d u e t o t h e c o n s i d e r e d exact f o r m o f t h e d i s t r i b u t i o n s ' p a r a m e t e r s . A n o t h e r f a c t o r t h a t i n f l u e n c e s t h e w a v e h e i g h t statistics is t h e n o n l i n e a r i t y o f w a v e s . I t has b e e n s h o w n i n a series o f s t u d i e s t h a t t h e second o r d e r b o u n d w a v e e f f e c t s a f f e c t n e g l i g i b l y t h e
* Corresponding author. Tel.: +351 218417957.
E-mail address: guedess®mar.ist.utl.pt (C. Guedes Soares).
w a v e h e i g h t s , w h i l e h a v i n g s i g n i f i c a n t i n f l u e n c e o n t h e w a v e crests a n d t r o u e h s (e g. [ 2 , 4 ] , etc.). O n the o t h e r h a n d , t h e t h i r d o r d e r w a v e - w a v e i n t e r a c t i o n s , expressed q u a n t i t a t i v e l y b y m e a n s o f t h e c o e f f i c i e n t o f k u i t o s i s , X40, have b e e n r e l a t e d t o s o m e e x t r e m e y large w a v e events. Reference can be m a d e t o [ 7 ] f o r f u l l - s c a l e d a t a t o (8 91 f o r l a b o r a t o r y m e a s u r e m e n t s and to [ 1 0 ] f o r n u m e r i c a l s i m u i a i i o n s . C o n s e q u e n t l y , X40 can be used t o i n d i c a t e t h e presence o f an a b n o r m a l or f r e a k w a v e , o r t h e e x p e c t e d large p r o b a b i h t y o f o c c u r r e n c e o f such a w a v e . . , . ^ M o r i a n d Janssen [ 9 ] d e v e l o p e d a d i s t r i b u t i o n based o n he m o d i f i e d E d g e w o r t h f o r m o f t h e G r a m - C h a r l i e r series, c a l l e d m o d i f i e d E d g e w o r t h - R a y l e i g h d i s t r i b u t i o n . It is a p p l i c a b l e t o w e a k l y n o n l i n e a r r a n d o m w a v e s w i t h n a r r o w b a n d s p e c t r u m a n d s p e c i f i c r e l a t i o n s b e t w e e n t h e n o r m a l i z e d f o u r t h o r d e r j o i n t c u m u l a n t s ; A04, and The a n a l y t i c a l f o r m u l a is a s i m p l e
^ " T h e ' d i l t r i b u t i o n o f M o r i a n d Janssen b a s i c a l l y r e p r e s e n t s a s i m p l i f i c a t i o n o f t h e m o d e l o f [ 1 1 ] , w h i c h is less r e s t r i c t i v e , since n o r e q u i r e m e n t s are i m p o s e d o n t h e f o r m o f t h e s p e c t r u m o r t h e s t a t i s t i c a l c u m u l a n t s .
The m o d e l s discussed so f a r have b e e n d e v e l o p e d f o r s i n g l e w a v e systems. H o w e v e r , i t has b e e n o b s e r v e d t h a t t w o - p e a k s p e c t r a o c c u r f r e q u e n t l y at sea. For e x a m p e, t h e r e p o r t e d p r o b a b i l i t y o f o c c u r r e n c e f o r t h e N o r t h Sea is 16% o n average [ 1 2 ] and f o r t h e N o r t h A t l a n t i c is 25% o n average [ 1 3 ) . The f r e q u e n c y o f o c c u r r e n c e w a s f o u n d t o d e p e n d u p o n t h e s e v e r i t y o f t h e sea states, s u c h t h a t t h e large p r o b a b i l i t i e s are associated w i t h l o w sea states a n d l o w p r o b a b i l i t i e s are associated w i t h h i g h w a v e c o n d i t i o n s .
Some s t a t i s t i c a l p r o p e r t i e s o f c o m b i n e d sea states have b e e n i n v e s t i g a t e d i n a series o f studies. R o d r i g u e z a n d Guedes
0141-1187/$ - see front matter doi:10.1016/j.apor.2009.06.005
P.G. Petrova. C. Guedes Soares / AppUed Ocean Research 31 (2009) 90-100 91
Soares [ 1 4 ] m a d e a s t u d y o n t h e b i v a r i a t e w a v e h e i g h t - p e r i o d d i s t r i b u t i o n u s i n g b u o y data a n d l a t e r i n v e s t i g a t e d the c o r r e l a t i o n b e t w e e n c o n s e c u t i v e w a v e h e i g h t s i n n u m e r i c a l s i m u l a t i o n s [ 1 5 ] . F u r t h e r m o r e , [16] s t u d i e d t h e w a v e g r o u p statistics i n n u m e r i c a l l y s i m u l a t e d b i m o d a l sea states a n d [ 1 7 ] s h o w e d results o n t h e m a r g i n a l w a v e h e i g h t d i s t r i b u t i o n using, again, a n u m e r i c a l a p p r o a c h . A r e n a a n d Guedes Soares [ 1 8 ] also i n v e s t i g a t e d t h e p r o b a b i l i s t i c s t r u c t u r e o f n o n l i n e a r w a v e h e i g h t s , crests a n d t r o u g h s o f large w a v e s i n d e e p w a t e r u s i n g m e a s u r e d t w o - p e a k e d spectra f r o m t h e A t l a n t i c Ocean, a n d t h e results have b e e n f u r t h e r v a l i d a t e d w i t h n u m e r i c a l s i m u l a t i o n s .
C o n s i d e r i n g t h e need f o r k n o w l e d g e a b o u t t h e p r o p e r t i e s o f w a v e f i e l d s o f c o m b i n e d w i n d sea a n d s w e l l , t h e p r e s e n t s t u d y addresses t h e s p e c t r a l a n d p r o b a b i l i s t i c characteristics o f c o m b i n e d sea states g e n e r a t e d i n an o f f s h o r e test basin. The a i m is t o d e r i v e c o n c l u s i o n s a b o u t t h e r e l a t i o n b e t w e e n t h e s p e c t r a l shape a n d t h e o b s e r v e d d i s t r i b u t i o n o f w a v e h e i g h t s , a n d also b e t w e e n t h e n a t u r e o f t h e spectra a n d s o m e s t a t i s t i c a l p a r a m e t e r s r e f l e c t i n g t h e w a v e n o n l i n e a r i t i e s , such as t h e c o e f f i c i e n t s o f skewness a n d k u r t o s i s . F u r t h e r m o r e , v a r i o u s a n a l y t i c a l m o d e l s have b e e n u s e d i n t h e c o m p a r i s o n s , w i t h the data f r o m sea states w i t h t w o - p e a k e d spectra.
This paper f o l l o w s [ 1 9 , 2 0 ] , w h i c h have a n a l y z e d some s t a t i s t i c a l p r o p e r d e s o f t h e m a x i m u m w a v e crests a n d h e i g h t s i n single sea states f r o m t h e s a m e o f f s h o r e test basin.
The paper is o r g a n i z e d as f o l l o w s . A b r i e f d e s c r i p 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 m o d e l s is g i v e n i n S e c t i o n 2. The l a b o r a t o r y c o n d i t i o n s f o r t h e p e r f o r m e d test runs are d e s c r i b e d a n d t h e b i m o d a l d a t a sets are i n t r o d u c e d i n Section 3. The o b t a i n e d r e s u l t s are p r e s e n t e d a n d discussed i n Section 4 a n d t h e d e r i v e d c o n c l u s i o n s are s u m m a r i z e d i n Section 5.
2. Distribution models of wave heiglits
L o n g u e t - H i g g i n s [ 1 ] s h o w e d t h a t t h e h e i g h t s o f l i n e a r Gaussian w a v e s w i t h n a r r o w s p e c t r u m f o l l o w t h e R a y l e i g h d i s t r i b u t i o n . A s s u m i n g t h a t t h e w a v e h e i g h t s H are scaled b y t h e s t a n d a r d d e v i a d o n o f t h e f r e e surface d i s p l a c e m e n t a, t h e d i s t r i b u t i o n takes t h e f o r m : P ( H > (xh) = e x p 0 < / i < o o . ( 1 ) H o w e v e r , t h e increase i n s p e c t r a l b a n d w i d t h m a k e s t h e e m p i r i c a l d i s t r i b u d o n n a r r o w e r a n d , s u b s e q u e n t l y , t h e o b s e r v e d w a v e h e i g h t s b e c o m e o v e r p r e d i c t e d b y Eq. ( 1 ) . The e f f e c t o f t h e s p e c t r a l b a n d w i d t h w a s t a k e n i n t o a c c o u n t b y Naess [ 3 ] , w h o p r o p o s e d a t h e o r e d c a l f o r m u l a f o r a Gaussian n a r r o w b a n d process w h i c h , i n t h e l i m i t o f i n f i n i t e l y n a r r o w s p e c t r u m , reduces t o Eq. ( 1 ) . He used t h e g e n e r a l f o r m o f t h e R a y l e i g h e x p r e s s i o n , as s h o w n i n Eq. ( 1 ) , a n d d e v e l o p e d a n a s y m p t o t i c e x p a n s i o n f o r 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 w a v e h e i g h t s : P ( H > ah) = e x p 4 ( 1 - r ) 0 < h < o o ( 2 ) w h e r e r = pit*) is t h e v a l u e o f t h e n o r m a l i z e d a u t o c o r r e l a t i o n f u n c d o n P (t) o f t h e f r e e surface e l e v a t i o n at its g l o b a l m i n i m u m ; T * is t h e t i m e o f o c c u r r e n c e o f t h e g l o b a l m i n i m u m . T h e m o d e l i n Eq. ( 2 ) is g e n e r a l l y a p p l i c a b l e t o a n a r r o w b a n d process f o r w h i c h r - 1 . The advantage o f t h e m o d e l is t h e e x p l i c i t f o r m o f t h e p a r a m e t e r r . A n a s y m p t o t i c m o d i f i c a d o n o f t h e d i s t r i b u t i o n o f Naess [ 3 ] is p r o p o s e d b y V i n j e [ 4 ] . The d i f f e r e n c e w i t h Eq. ( 2 ) o r i g i n a t e s f r o m the i n t r o d u c e d m u l t i p l i e r ( ( 1 -f- q ) / 2 q ) ' ' ' ^ > 1, w h i c h makes t h e f o r m u l a t i o n a p p l i c a b l e o n l y to h i g h w a v e s : P(H > ah) 1 1 1 2q 1/2 e x p 4 ( l + < ? ) h CO ( 3 ) w h e r e q = (p^ -\- p a n d p r e p r e s e n t t h e n o r m a l i z e d a u t o c o r r e l a t i o n f u n c t i o n a n d its H i l b e r t t r a n s f o r m , r e s p e c t i v e l y , c a l c u l a t e d at r * . Thus, q is larger t h a n r . H o w e v e r , f o r a n a r r o w b a n d processes p(t) 0 at t h e e x t r e m e s o f p(r) a n d t h e n q can be a p p r o x i m a t e l y r e p r e s e n t e d b y r = p(r*). The e f f e c t o f the a d d e d f a c t o r o n t h e e s t i m a t i o n o f e x t r e m e h e i g h t statistics, as discussed b y V i n j e , is o f m i n o r i m p o r t a n c e , i n c r e a s i n g o n l y t h e n u m b e r o f w a v e s e f f e c t i v e l y i n v o l v e d i n t h e p r e d i c t i o n s . B o c c o t t i [ 6 ] uses a n u m e r i c a l a p p r o a c h t o f o r m u l a t e t h e d i s t r i b u t i o n f o r a s p e c t r u m o f a r b i t r a r y f o r m a n d b a n d w i d t h . O n t h e basis o f t h e l i n e a r q u a s i - d e t e r m i n i s t i c t h e o r y , t h e f o l l o w i n g e x p r e s s i o n f o r large w a v e h e i g h t s f r o m a Gaussian process is o b t a i n e d : F(H > ah) 1 + b V 2 b ( l - r ) exp 4 ( 1 - r ) ( 4 ) w h e r e the p a r a m e t e r b c o r r e s p o n d s t o t h e second d e r i v a t i v e o f t h e n o r m a l i z e d a u t o c o r r e l a t i o n f u n c d o n at d m e r*, p{r*). T a y f u n [ 5 ] o b t a i n e d closed f o r m a s y m p t o t i c a p p r o x i m a d o n s as s o l u d o n s t o t h e c o m p l i c a t e d e x p r e s s i o n p r o p o s e d e a r i i e r b y h i m [ 2 1 ] f o r i r r e g u l a r w e a k l y - n o n l i n e a r w a v e s , a l t h o u g h t h e y are a p p l i c a b l e o n l y t o w a v e h e i g h t s l a r g e r t h a n t h e m e a n w a v e h e i g h t . The a d o p t e d a s s u m p t i o n is t h a t , f o r a s p e c t r u m o f f i n i t e b a n d w i d t h , t h e w a v e crests a n d t r o u g h s d o n o t o c c u r s i m u l t a n e o u s l y . C o n s e q u e n t l y , t h e t o t a l c r e s t - t o - t r o u g h w a v e h e i g h t c a n n o t be s i m p l y expressed as a d o u b l e w a v e a m p l i t u d e , b u t i t s h o u l d be r e p r e s e n t e d as t h e s u m o f t w o a m p l i t u d e s o f t h e w a v e e n v e l o p e , s h i f t e d w i t h respect t o each o t h e r b y h a l f o f t h e d o m i n a n t w a v e p e r i o d . The l o w e r - b o u n d a p p r o x i m a d o n to t h e d i s t r i b u t i o n o f [ 2 1 ] is g i v e n as: F ; ( H > ah) = 1 + r „ 2rm 1/2 e x p h^ 4 ( 1 + r „ ) h>H ( 5 ) w h e r e H denotes t h e m e a n w a v e h e i g h t ; r™ = p{xm) is t h e v a l u e o f t h e n o r m a l i z e d a u t o c o r r e l a t i o n f u n c t i o n at t i m e l a g t^ , w h e r e Tm does n o t c o i n c i d e w i t h r * , b u t is c o n s i d e r e d b e i n g e q u a l t o TmR f o r s i m p l i c i t y . The p e r i o d Tm, o n t h e o t h e r h a n d , is t h e average p e r i o d associated w i t h t h e average s p e c t r a l f r e q u e n c y , com = m\/mQ, w h e r e m, r e p r e s e n t s t h e i t h o r d e r s p e c t r a l m o m e n t , g i v e n :
f
Jo a ) ' S ( w ) d f t ) , 1 = 0 , 1 , 2 (6) F u r t h e r m o r e , T a y f u n [ 5 ] s h o w e d t h a t t h e p a r a m e t e r r i n t h e d i s t r i b u d o n o f Naess does n o t g i v e a n a d e q u a t e d e s c r i p d o n o f t h e s p e c t r u m c o m p a r e d t o rm used i n Eq. ( 5 ) . For w a v e s p e c t r a t y p i c a l l y o b s e r v e d at sea, i t is v a l i d t h a t | r | < a n d o n l y i n t h e l i m i t o f v e r y n a r r o w s p e c t r u m ( v 0 ) , i t is seen t h a t | r | ^ r ^ ^ 1. The s p e c t r a l b a n d w i d t h v is expressed u s i n g t h e l o w o r d e r s p e c t r a l m o m e n t s [ 2 2 ] := (mom2/m]) - 1. ( 7 )
The p r o b a b i l i s t i c m o d e l s i n Eqs. ( 1 ) a n d ( 2 ) are used t o describe t h e w h o l e range o f values f o r t h e w a v e h e i g h t s , h > 0. O n t h e o t h e r h a n d , Eqs. ( 3 ) - ( 5 ) are d e v e l o p e d f o r w a v e h e i g h t s l a r g e r t h a n a r e f e r e n c e w a v e h e i g h t ho, h > ho. C o n t r a r y t o t h e e f f e c t d u e t o f i n i t e s p e c t r u m , t h e n o n l i n e a r i t i e s i n t h e w a v e f i e l d b r o a d e n t h e e m p i r i c a l t a i l t o w a r d h i g h e r p r o b a b i l i t y levels. The i n f l u e n c e o f t h i r d o r d e r w a v e - w a v e
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p e r i o d s Tp = 7 a n d 2 0 s, a n d s i g n i f i c a n t w a v e h e i g l i t s H , 99 1= BEACH 1 2 3 4 5 6 7 8 9 10 - GAUGE
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M U L T I - F L A P WAVEMAKER, BM3Fig. 1. Location of ttie measuring devices and wave generators in tlie basin. i n t e r a c t i o n s is t a k e n i n t o a c c o u n t i n p r o b a b i l i s t i c m o d e l s based o n t h e G r a m - C h a r l i e r series r e p r e s e n t a t i o n o f t h e d e n s i t y f u n c t i o n o f t h e e l e v a t i o n process. M o r i a n d Janssen [ 9 ] d e v e l o p e d a d i s t r i b u t i o n c a l l e d a m o d i f i e d E d g e w o r t h - R a y l e i g h d i s t r i b u t i o n ( M E R ) e x p r e s s e d as: l + ^ . ^ ( , . ^ - 1 6 ) ( 8 ) T h e a n a l y t i c a l f o r m u l a i n Eq. ( 8 ) r e p r e s e n t s a s i m p l e f u n c t i o n o f X40. T h e M E R assumes w e a k l y n o n - l i n e a r r a n d o m w a v e s w i t h n a r r o w b a n d s p e c t r u m a n d f u r t h e r r e q u i r e s s p e c i f i c r e l a t i o n s h i p s b e t w e e n t h e n o r m a l i z e d f o u r t h o r d e r j o i n t c u m u l a n t s X04, .^40 a n d ^04 = ^^40 a n d X40 = 3 X 2 2 . T a y f u n a n d Fedele [ 1 1 ] d e v e l o p e d a n o t h e r e x p r e s s i o n c o n s i d e r -i n g w e a k n o n l -i n e a r -i t y a n d s p e c t r u m o f a n y f-in-ite b a n d w -i d t h : E c c ( H > ah) = e x p — - 1 + A 1 Ö 2 4 h^(h^ - 1 6 ) ( 9 ) w h e r e A = ^ 4 0 + 2 X 2 2 + X04. W h e n y ^ 0, i t is v a l i d t h a t : Xo4 = X 4 0 . X 4 0 = 3 X 2 2 , A = 8 X 4 0 / 3 a n d t h e n Eq. ( 9 ) r e d u c e s t o Eq. ( 8 ) . The p r e s e n t s t u d y c o m p a r e s t h e o b s e r v e d p r o b a b i l i t i e s o f exceedance o f w a v e h e i g h t s w i t h t h e m o d e l s g i v e n b y Eqs. ( l ) - ( 5 ) a n d ( 9 ) . T h e d i s t r i b u t i o n s a p p e a r i n t h e t e x t w i t h t h e f o l l o w i n g a b b r e v i a t i o n s : Eq. ( 1 ) - R; Eq. ( 2 ) - N ; E q . ( 3 ) - V ; Eq. ( 4 ) - B; E q . ( 5 ) - T L a n d E q . ( 9 ) - G C .
3. Laboratory data
The d a t a u s e d f o r t h e s t u d y has b e e n c o l l e c t e d d u r i n g a n e x p e r i m e n t p e r f o r m e d i n t h e o f f s h o r e b a s i n o f M A R I N T E K . The test b a s i n has t h e f o l l o w i n g d i m e n s i o n s : 8 0 m l e n g t h , 5 0 m w i d t h a n d 2 m d e p t h [ 2 3 ] . T e n gauges u n i f o r m l y spaced a l o n g t h e b a s i n a t 5 m distance ( F i g . 1) w e r e m e a s u r i n g t h e s u r f a c e e l e v a t i o n s . A d o u b l e -flap w a v e - m a k e r ( B M 2 ) i n s t a l l e d a t o n e o f t h e s h o r t w a l l s o f t h e b a s i n w a s used t o g e n e r a t e t h e s h o r t - p e r i o d w a v e s y s t e m f o r each t e s t r u n . T h e d i s t a n c e b e t w e e n t h e first gauge a n d B M 2 w a s 10 m . A second m u l t i f l a p w a v e - m a k e r ( B M 3 ) i n s t a l l e d a t t h e l o n g side o f t h e b a s i n w a s g e n e r a t i n g t h e l o n g - p e r i o d w a v e s y s t e m s . A b e a c h o n t h e o p p o s i t e w a l l i n f r o n t o f each g e n e r a t o r w a s a b s o r b i n g t h e e n e r g y o f t h e i n c i d e n t w a v e s .The l a b o r a t o r y e x p e r i m e n t w a s r u n i n scale 1:50. The g e n e r a t e d r a n d o m seas r e p r e s e n t coexistence o f s w e l l a n d w i n d sea a n d are c h a r a c t e r i z e d b y a t w o - p e a k e d s p e c t r u m c o m p o s e d o f t w o JONSWAP w a v e s y s t e m s [ 1 2 ] . F u r t h e r m o r e , t h e i n d i v i d u a l c o m p o n e n t s have been m o d e l e d b y a u n i d i r e c t i o n a l JONSWAP s p e c t r u m w i t h peak e n h a n c e m e n t f a c t o r y = 3, f u l l - s c a l e peak
= 3 . 6 a n d 3 . 6 r n . Each e x p e r i m e n t a l r e a l i z a t i o n o f t h e sea surface u s e d d i f f e r e n t , r a n d o m l y c h o s e n a m p l i t u d e s a n d phases. The d u r a t i o n o f t h e r e c o r d e d t i m e series at f u l l - s c a l e is m o r e t h a n 3 h at s a m p l i n g f r e q u e n c y d t = 0 . 1 7 6 8 s a n d c o r r e s p o n d s a p p r o x i m a t e l y t o 1 5 0 0 d o w n - c r o s s i n g w a v e s . The e x p e r i m e n t a l c o n d i t i o n s r e p r e s e n t d e e p w a t e r w a v e s p r o p -a g -a t i n g o n w -a t e r o f c o n s t -a n t d e p t h d = 1 0 0 m . T h e s h o r t - p e r i o d w a v e s y s t e m f u l f i l l s t h e d e e p w a t e r i n e q u a l i t y , d / L p „ , s > 1 / 2 , w h e r e Lp„,s = 7 6 . 5 m is t h e w a v e l e n g t h at t h e s p e c t r a l peak f r e -q u e n c y o f t h e w i n d w a v e s c a l c u l a t e d f r o m t h e l i n e a r d i s p e r s i o n r e l a t i o n s h i p . H o w e v e r , t h e i n e q u a l i t y 1 / 2 0 < d/Lps,„ < 1 / 2 is v a l i d f o r t h e w a v e s w i t h t h e l a r g e r p e r i o d (Lpsu, = 6 2 4 . 5 m ) , t h u s t h e y c a n be r e f e r r e d t o as i n t e r m e d i a t e w a t e r w a v e s . T h e a b b r e -v i a t i o n s w s a n d s w i n t h e subscripts d e n o t e w i n d sea a n d s w e l l , r e s p e c t i v e l y .
The m u l t i d i r e c t i o n a l sea states are d e s i g n a t e d as 8 2 3 3 , 8 2 3 4 a n d 8 2 3 5 f o r angles o f p r o p a g a t i o n 9 = 6 0 ° , 1 2 0 ° a n d 9 0 ° , respec-t i v e l y .
4. Results and discussion
The p e r f o r m e d s t u d y addresses t h e r e l a t i o n s h i p b e t w e e n t h e s t a t i s t i c a l r e p r e s e n t a t i o n o f t h e w a v e h e i g h t s a n d t h e s p e c t r a l p r o p e r t i e s o f m u l t i d i r e c t i o n a l seas g e n e r a t e d i n l a b o r a t o r y c o n d i t i o n s .
The b i m o d a l spectra m e a s u r e d a t each o f t h e w a v e gauges have b e e n c l a s s i f i e d i n t o g r o u p s w i t h respect t o t h e d i m e n s i o n l e s s p a r a m e t e r s e a - s w e l l e n e r g y r a t i o (SSER) p r o p o s e d b y Guedes Soares [ 1 2 ] a n d u s e d i n a series o f s t u d i e s o n t h e s t a t i s t i c a l p r o p e r t i e s o f m i x e d sea states ( [ 1 4 ] - f o r b u o y data o f f t h e P o r t u g u e s e coast; [ 1 6 , 1 7 ] - f o r n u m e r i c a l l y s i m u l a t e d d a t a ) .
The SSER p a r a m e t e r is d e f i n e d as t h e r a t i o b e t w e e n t h e energies o f t h e w i n d w a v e s a n d s w e l l r e p r e s e n t e d b y t h e z e r o - m o m e n t s o f t h e c o r r e s p o n d i n g s p e c t r a l d e n s i t i e s :
SSER mow
most,
( 1 0 )
w h e r e mosu, stands f o r t h e e n e r g y o f t h e l o w - f r e q u e n c y range o f t h e s p e c t r u m associated w i t h t h e s w e l l a n d mows stands f o r t h e e n e r g y o f t h e h i g h - f r e q u e n c y range, associated w i t h t h e w i n d w a v e s . T h e e s t i m a t e d z e r o - m o m e n t s o f t h e spectra o f t h e c o m p o n e n t w a v e s y s t e m s f o r a l l b i m o d a l spectra are g i v e n i n Table 1. The t a b u l a t e d d a t a are f u r t h e r o r g a n i z e d i n g r a p h s (Fig. 2 ) , w h i c h i l l u s t r a t e t h e change o f t h e energies o f t h e w a v e s y s t e m s w i t h t h e d i s t a n c e a l o n g t h e b a s i n .
Fig. 2 a refers t o test 8 2 3 3 , w h e r e t h e spectra at t h e first t w o gauges d e m o n s t r a t e v e r y s m a l l c o n t r i b u t i o n d u e t o s w e l l , t h u s t h e y are n o t c l a s s i f i e d as b i m o d a l a n d , s u b s e q u e n t l y , t h e y are n o t c o n s i d e r e d i n t h e s t a t i s t i c a l analysis o f w a v e h e i g h t s . I t is seen f r o m Fig. 2a t h a t , f o r t e s t 8 2 3 3 , t h e e n e r g y o f v ^ i n d w a v e s g e n e r a t e d b y B M 2 is s i g n i f i c a n t l y l a r g e r t h a n t h e e n e r g y o f t h e s w e l l c o m i n g f r o m B M 3 . H o w e v e r , e q u a l energies are e s t i m a t e d at t h e f a r t h e s t gauge. , . Fig. 2 b r e f e r s t o test 8 2 3 4 . S w e l l e n e r g y c o n t r i b u t i o n is r e g i s t e r e d f r o m t h e b e g i n n i n g o f t h e t a n k , w h e r e i t p r e v a i l s o v e r t h e w i n d sea, c o n t r a r y t o t h e last gauges, w h e r e t h e o p p o s i t e t e n d e n c y is o b s e r v e d .
C o m p a r i n g Fig. 2a a n d b, i t becomes clear t h a t t h e l o n g i t u d i n a l d i s t r i b u t i o n o f e n e r g y o f the s o - c a l l e d s w e l l c o m p o n e n t s ( t h e o n e w i t h l a r g e r p e r i o d ) is o p p o s i t e , i n c r e a s i n g f r o m t h e i n i t i a l gauge t o t h e final one, o r decreasing, w h i c h r e f l e c t s t h e d i r e c t i o n i n w h i c h t h e s e w a v e systems are b e i n g g e n e r a t e d . I n Fig. 2c, w h i c h is t h e case o f 9 0 ° i n c i d e n c e , t h e value o f t h e e n e r g y b e t w e e n gauges 3 a n d 8 is s o m e w h a t c o n s t a n t , t a k i n g i n t o c o n s i d e r a t i o n t h e s t a t i s t i c a l v a r i a b i l i t y t o be e x p e c t e d .
0.8
0.6
0.4
0.2
P.G. Petrova, C. Guedes Soares / AppUed Ocean Research 31 (2009) 90-100
2 I , , , 93 Osw 1.5 0.5 1 2 3 4 5 6 Gauges (a) 8233 (e = 6 0 ° ) . 7 8 9 10 1 2 3 4 5 6 Gauges (b) 8234 (0 = 1 2 0 ° ) . 7 a 9 10 Table 1
Spectral energies of the wind seas and swell
(c) 8235 (9 = 9 0 ° ) .
Fig. 2. Change in the spectral energy of wind sea and swell with the distance.
Table 2
Classirication of the sea states with respect to SSER. Test/Gauge 8 2 3 3 / 3 0.16 0.93 8 2 3 3 / 4 0.30 0.91 8233/5 0.60 0.92 8 2 3 3 / 6 0.76 0.82 8233/7 0.81 0.96 8 2 3 3 / 8 0.66 0.76 8 2 3 3 / 9 0.70 0.80 8233/10 0.85 0.84 8234/1 1.72 1.07 8234/2 1.16 0.90 8234/3 1.49 0.86 8 2 3 4 / 4 1.01 0.87 8234/5 0.80 0.89 8 2 3 4 / 6 0.51 0.81 8234/7 0.32 0.81 8 2 3 4 / 8 0.23 0.72 8 2 3 4 / 9 0.15 0.80 8234/10 0.11 0.74 8235/1 0.32 0.97 8 2 3 5 / 2 0.77 0.97 8 2 3 5 / 3 1.00 0.93 8 2 3 5 / 4 0.66 0.96 8235/5 0.54 0.84 8 2 3 5 / 6 0.86 0.84 8235/7 0.82 0.78 8235/8 0.69 0.78 8 2 3 5 / 9 0.26 0.77 8235/10 0.13 0 7 6
Test SSER » 1 SSER RS 1 SSER « 1 Total
8233 3 5 0 8
8234 5 2 3 10
8235 5 5 0 10
Total 13 12 3 28
T h e r e f o r e , those f i g u r e s a l l o w one t o see t h e e f f e c t o f t h e d i r e c t i o n o f g e n e r a t i o n o f t h e w a v e s y s t e m f r o m t h e side B M 3 .
T h e graphs i n Fig. 2 d e m o n s t r a t e a s m a l l v a r i a t i o n i n t h e w i n d sea e n e r g y , w h i c h can be e s t i m a t e d w i t h i n one s t a n d a r d d e v i a t i o n . H o w e v e r , since t h e s w e l l e n e r g y changes s i g n i f i c a n t l y a l o n g t h e basin, i t is e x p e c t e d t h a t t h e e s t i m a t e d SSER values w i l l also v a r y .
D e p e n d i n g o n t h e v a l u e o f SSER, t h e b i m o d a l w a v e s y s t e m s can be d i v i d e d i n t o t h r e e g r o u p s : s w e l l d o m i n a t e d sea states f o r SSER s m a l l e r t h a n o n e ; w i n d sea d o m i n a t e d w a v e f i e l d s f o r SSER l a r g e r t h a n o n e a n d sea states w i t h c o m p a r a b l e energies o f t h e s p e c t r a l c o m p o n e n t s f o r SSER close t o o n e .
T h e p r e s e n t set o f b i m o d a l sea states is c l a s s i f i e d w i t h respect t o SSER as s h o w n i n Table 2. I t is seen t h a t , m o s t f r e q u e n t l y , t h e e s t i m a t e d s p e c t r a b e l o n g t o t h e s e c o n d a n d t h i r d g r o u p . T h e p r e v a l e n c e o f s w e l l is v i s i b l e o n l y at t h e f i r s t t h r e e gauges o f test 8 2 3 4 , w h i c h c o u l d be e x p e c t e d h a v i n g i n m i n d t h a t t h e w a v e s g e n e r a t e d b y B M 3 p r o p a g a t e at 1 2 0 ° w i t h respect to t h e w a v e s c o m i n g f r o m B M 2 .
E x a m p l e s o f t h e m e a s u r e d spectra are g i v e n i n Fig. 3. A case f o r w h i c h t h e l o w - f r e q u e n c y w a v e c o m p o n e n t has n e g l i g i b l e c o n t r i b u t i o n t o t h e s p e c t r a l e n e r g y d e n s i t y is s h o w n i n Fig. 3a. O n l y t w o cases l i k e t h i s w e r e f o u n d , c o r r e s p o n d i n g t o t h e f i r s t t w o gauges o f t e s t 8 2 3 3 (0 = 6 0 ° ) . Spectra w i t h d e f i n i t e t w o peaks are s h o w n i n Fig. 3 b - d , c o r r e s p o n d i n g t o t h e t h r e e possible classes discussed above d u e t o SSER, i.e. s p e c t r a o f d o m i n a t i n g
94 P.G. Petrova, C. Guedes Soares/ Applied Ocean Research 31 (2009)90-100 Test 8233/ Gauge 2/ 9 = 6 0 ° / nfft = 512
b
10 Test 8234/ Gauge 1/ e = 1 2 0 ° / nfft = 512 CO C 5 Test 8233/ Gauge 7/ 8 = 6 0 ° / nfft = 512 4- ojpj = 0.83293[cad/s]/ Tp2 = 7.5435[s] d 3 T e s t 8235/ Gauge 10/ 8 = 9 0 ° / nfft = 5 1 2 -'r Bp, = 0.34705[rad/sl/ T^, = 18.1043[s] +Fig. 3. Examples of spectra estimated in the mixed sea states: (a) bimodal with negligible swell contribution; (b) bimodal swell dominated; (c) bimodal with equivalent sea-swell energies; (d) bimodal wind wave dominated.
s w e l l e n e r g y (Fig. 3 ( b ) ) ; spectra o f e q u i v a l e n t s e a - s w e l l energies (Fig. 3(c)) a n d spectra o f d o m i n a t i n g w i n d sea e n e r g y (Fig. 3 ( d ) ) . T h e n u m b e r o f Fourier c o m p o n e n t s c o m p u t e d b y t h e FFT t e c h n i q u e i s n / / f = 5 1 2 .
4.1. Influence of the spectral energy ratio (SSER)
Fig. 4 s h o w s the estimates o f the n o r m a l i z e d t h i r d a n d f o u r t h o r d e r c u m u l a n t s - t h e c o e f f i c i e n t s o f skewness a n d k u r t o s i s , X30 a n d X40, r e s p e c t i v e l y , w h i c h are p l o t t e d versus t h e values o f SSER. T h e o b s e r v a t i o n s are a p p r o x i m a t e d w e l l , w i t h p o l y n o m i a l s o f s e c o n d degree s h o w i n g m o d e r a t e R-squared values, w h i c h r e f l e c t s l i m i t e d d a t a p o i n t s f o r t h e h i g h values o f SSER. Some o f t h e sea states e x h i b i t characteristics close t o Gaussian, i.e. t h e s t a t i s t i c a l p a r a m e t e r s have values close t o zero. Large discrepancies f r o m the Gaussian values are associated w i t h SSER > 2, i.e. w i t h s i g n i f i c a n t w i n d sea c o n t r i b u t i o n t o t h e s p e c t r u m .
F u r t h e r m o r e , p l o t t i n g t o g e t h e r t h e e s t i m a t e s o f SSER, X30 a n d X40 at each gauge a l o n g the b a s i n , as s h o w n i n Fig. 5, i t can be seen t h a t t h e changes i n t h e c u m u l a n t statistics g e n e r a l l y f o l l o w t h e v a r i a t i o n s o f SSER.
Fig. 5a s h o w s a general d e c r e a s i n g t r e n d f o r SSER a n d X30, i.e. t h e w a v e s are e x p e c t e d t o get less a s y m m e t r i c w i t h t h e distance. H o w e v e r , X40 m a i n t a i n s its s t a b i l i t y w i t h the distance. A t t w o l o c a t i o n s a l o n g t h e b a s i n (gauges 5 a n d 8), s e p a r a t e d a l m o s t one l e n g t h o f t h e l o n g - w a v e c o m p o n e n t , A30 a n d X40 s h o w b e h a v i o r w h i c h d i f f e r s f r o m t h e o t h e r p o i n t s , b u t t h e v a r i a t i o n r e m a i n s w i t h i n t h e c o n f i d e n c e l i m i t s o f s t a t i s t i c a l v a r i a b i l i t y . The test r u n 8 2 3 4 is s h o w n i n Fig. 5 b . The p a r a m e t e r s are c o n s t a n t l y i n c r e a s i n g .
h a v i n g a local m i n i m u m at gauge 3. The results f o r test 8 2 3 5 are g i v e n i n Fig. 5c a n d s h o w t w o local m i n i m a i n t h e w i n d w a v e e n e r g y a r o u n d the c e n t r a l gauge (gauge 5 ) , w h i c h are also seen i n t h e v a r i a t i o n s o f X30 a n d X40.
So far, t h e c o e f f i c i e n t s o f skewness a n d k u r t o s i s have b e e n r e l a t e d t o t h e c a l c u l a t e d SSER, a n d d e p e n d e n c e has b e e n f o u n d b e t w e e n t h e l a r g e r e s t i m a t e d statistics a n d t h e p r e d o m i n a n c e o f t h e w i n d sea e n e r g y i n t h e t w o - p e a k e d spectra. I n t h e f o l l o w i n g , t h e o b s e r v e d s t a t i s t i c a l d i s t r i b u t i o n s o f t h e i n d i v i d u a l w a v e h e i g h t s are discussed a n d c o m p a r e d w i t h s o m e a n a l y t i c a l m o d e l s , d e v e l o p e d f o r u n i m o d a l sea state. The k n o w l e d g e o f the d i s t r i b u t i o n m o d e l s o f i n d i v i d u a l w a v e h e i g h t s f o r c e r t a i n c o n d i t i o n s is o f large i m p o r t a n c e f o r p r a c t i c a l p u r p o s e s . M o r e o v e r , e x i s t i n g p r o b a b i l i s t i c m o d e l s need t o be v e r i f i e d against t h e data. In the present case, t h e v e r i f i c a t i o n is d o n e f o r b i m o d a l sea states. A b r i e f d e s c r i p t i o n o f t h e a d o p t e d p r o b a b i l i s t i c m o d e l s has a l r e a d y been g i v e n i n S e c t i o n 2. Results f o r each o f t h e t h r e e classes o f sea states discussed above are p r e s e n t e d i n t h e f o l l o w i n g .
For b e t t e r r e a d a b i l i t y o f the d i s t r i b u t i o n curves, t h e abscissas i n t h e f i g u r e s , w h i c h r e p r e s e n t t h e n o r m a l i z e d b y t h e s t a n d a r d d e v i a t i o n i n d i v i d u a l w a v e heights, are s h o w n f o r values l a r g e r t h a n t h e m e a n w a v e h e i g h t .
4.2. Swell dominated wave flelds (SSER <^ 1)
T h e o n l y s w e l l d o m i n a t e d w a v e f i e l d s have b e e n e n c o u n t e r e d i n test 8 2 3 4 (Table 2) a n d are r e c o r d e d at t h e f i r s t t h r e e gauges o f t h e b a s i n . E x a m p l e s o f t h e e m p i r i c a l exceedance d i s t r i b u t i o n f u n c t i o n s
P.G. Petrova, C. Guedes Soares/ Applied Ocean Researcti 31 (2009) 90-100 95
b
1 0.8 0.6 0 2 4 6 !: S S E R 0 8 2 3 3 - 6 0 » • 8 2 3 4 - 1 2 0 ° A 8235 - 90°•
0 0
A y = -0.0126x^ + 0.1648x + 0.0467 • = 0.53 S S E RFig. 4. Effect of increasing wind sea energy contribution on: (a) X30 and (b) X40.
(EDFs) c o m p a r e d w i t h t h e a n a l y t i c a l m o d e l s are g i v e n i n Fig. 6 f o r gauges 1,2 a n d 3.
A l l m o d e l s e x c e p t TL, B a n d V, w h i c h are v a l i d f o r large w a v e h e i g h t s , f i t w e l l t h e d a t a i n t h e range o f values s m a l l e r t h a n t h e t n e a n w a v e h e i g h t , w h i c h is e s t i m a t e d as a p p r o x i m a t e l y 2.4a. The R a n d GC m o d e l s o v e r p r e d i c t t h e o b s e r v e d EDFs f o r w a v e h e i g h t s l a r g e r t h a n t h e m e a n (Fig. 6a, b ) . The r e s u l t f o r R agrees w i t h t h e c o n c l u s i o n s f o r n u m e r i c a l l y s i m u l a t e d Gaussian m i x e d sea states o f [ 1 7 ] , w h e r e t h e f i n i t e s p e c t r a l b a n d w i d t h e x p l a i n e d t h e o b s e r v e d d i s c r e p a n c y b e t w e e n data a n d t h e o r y . The s p e c t r a l b a n d w i d t h , g i v e n b y t h e p a r a m e t e r v, is e s t i m a t e d at a p p r o x i m a t e l y 0.6, p r o v i d e d t h a t , i n t y p i c a l s t o r m c o n d i t i o n s , t h e s p e c t r a l w i d t h p a r a m e t e r is a p p r o x i m a t e l y 0.3. So, t h e e s t i m a t e d spectra c o u l d be c o n s i d e r e d b r o a d - b a n d e d . O n t h e o t h e r h a n d , GC gives close p r e d i c t i o n s t o R, s h o w i n g o n l y a s m a l l increase at t h e l o w e s t p r o b a b i l i t y levels w i t h larger X4Q. This c o u l d be due t o t h e f a c t t h a t X40 is v e r y s m a l l i n t h e c o n s i d e r e d cases. A t gauge 1 (Fig. 6 a ) t h e c o e f f i c i e n t o f k u r t o s i s e v e n takes a s m a l l n e g a t i v e v a l u e (X40 = - 0 . 0 2 ) a n d , c o n s e q u e n t l y , t h e CG c u r v e is l o c a t e d l o w e r t h a n t h e R a y l e i g h c u r v e . As e x p e c t e d , GC r e d u c e s t o R w h e n X4Q t e n d s to zero.
The a p p r o x i m a t i o n TL gives, f o r all cases i n t h e g r o u p , t h e l o w e s t p r e d i c t i o n s f o r t h e i n t e r m e d i a t e a n d large data range, c o m p a r e d t o t h e o t h e r m o d e l s . I t fits w e l l t h e i n t e r m e d i a t e w a v e h e i g h t s u p t o a p p r o x i m a t e l y 5.5a (Fig. 6a, b ) .
The sea state i n Fig. 6c, h o w e v e r , has a w a v e , w h i c h can be c o n s i d e r e d a b n o r m a l , since t h e a b n o r m a l i t y r a t i o AI = Hm^x/Hs is l a r g e r t h a n 2 [ 2 4 ] , w h e r e Hmax is t h e m a x i m u m d o w n - c r o s s i n g w a v e h e i g h t i n t h e r e c o r d a n d Hs is t h e s i g n i f i c a n t w a v e h e i g h t , e s t i m a t e d f r o m t h e s p e c t r u m . A s h i f t i n t h e e m p i r i c a l d i s t r i b u t i o n t o w a r d s N f o r h e i g h t s l a r g e r t h a n t h e m e a n is o b s e r v e d a n d N serves as a g o o d fit, a l t h o u g h t h e p r e d i c t i o n s o f B a n d especially t h e r e s u l t s f o r V m o d e l , are v e r y close t o t h o s e o f N. E v e n t u a l l y , t h e t w o largest values are u n d e r e s t i m a t e d b y all m o d e l s , e v e n b y GC.
G a u g e s - e - S S E R ho T40 10-1 i 1 1 1 1 i i =ii 1 2 3 4 5 6 7 8 9 10 Gauges
Fig. 5. Variations of SSER, A30 and X40 with the distance along the basin for tests: (a) 8233; (b) 8234 and (c) 8235. It m u s t be n o t e d t h a t t h e results f o r N , B , TL a n d V are e x p e c t e d t o be close. H o w e v e r , t h e d e v i a t i o n o f TL is large, d u e t o t h e p a r a m e t e r a d o p t e d i n Eq. ( 5 ) . C o m p a r i s o n b e t w e e n t h e c a l c u l a t e d values o f r„, s h o w n i n t h e p l o t s , a n d t h e values o f r r e p r e s e n t i n g t h e a u t o c o r r e l a t i o n f u n c t i o n at its first m i n i m u m , d e m o n s t r a t e t h a t takes c o n s t a n t l y m u c h l o w e r values t h a n j r ] . H o w e v e r , f o r t y p i c a l s t o r m w a v e spectra, t h e o p p o s i t e i n e q u a l i t y is o b s e r v e d , i.e. ] r ] < r^, a n d f o r y 0 i t is seen t h a t ] r | 1. O n t h e o t h e r h a n d , N is closer t o V t h a n t o TL, since t h e c a l c u l a t e d p a r a m e t e r q a l m o s t coincides b y a b s o l u t e v a l u e w i t h r , hence q is also l a r g e r t h a n r ^ .
4.3. Wind sea dominated wave fields (SSER » 1)
The exceedance d i s t r i b u t i o n s f o r t h i s class o f s p e c t r a are r e p r e s e n t e d i n Fig. 7 f o r several w a v e gauges.
It is seen t h a t B a l w a y s p r o d u c e s v e r y l o w e s t i m a t e s f o r t h e i n t e r m e d i a t e a n d h i g h w a v e s , c o m p a r e d w i t h t h e o t h e r m o d e l s a n d w i t h t h e o b s e r v a t i o n s . O n t h e c o n t r a r y , GC a l w a y s s i g n i f i c a n t l y
RG. Petrova. C. Guedes Soares/ AppUed Ocean Research 31 (2009) 90-100
Wave Height, H / a Wave Height, H / a
TANK/test8234/ gS /1261 waves 10' 10' • Data R N ... B V - T L GC ci = 0.61163
' ^ ^ ^ ^ X
= 0.31977 X ^ ^ J x a= 1.5257 \ ^ V =0.60653 \ ci = 0.61163' ^ ^ ^ ^ X
= 0.31977 X ^ ^ J x a= 1.5257 \ ^ V =0.60653 \ = 0.052947 SSER -0.58073^ \ \ \
\ \ \
4 5 6 7 Wave Height, H / aFig. 6. Exceedance probability of the normalized wave heights, H/cr, in the swell dominated wave fields.
o v e r e s t i m a t e s t i i e o t l i e r t h e o r e t i c a l p r e d i c t i o n s a n d t h e data f o r H >
4CT.
V a n d N g i v e v e r y close p r e d i c t i o n s w h i c h , i n s o m e cases, are also close t o TL, n a m e l y , w h e n t h e d i f f e r e n c e b e t w e e n t h e c a l c u l a t e d p a r a m e t e r s r, q a n d is n o t large. These t h r e e m o d e l s f i t w e l l t h e o b s e r v a t i o n s w h e n X40 is s m a l l (Fig. 7a, b ) . The s m a l l e s t X40 belongs t o t h e t i m e series at gauge 1 o f test 8 2 3 5 (X40 = 0 . 1 0 ; Fig. 7a). H o w e v e r , N f i t s w e l l t h e l o w w a v e h e i g h t s , b u t i t s l i g h t l y u n d e r e s t i m a t e s t h e i n t e r m e d i a t e a n d large data. On t h e c o n t r a r y , TJL a n d V m o d e l s c a n n o t be a p p l i e d to p r e d i c t l o w w a v e s , b u t t h e y f i t w e l l t h e i n t e r m e d i a t e a n d large w a v e s . It can be seen i n Fig. 7a w h e r e V s h o w s a s m a l l increase i n the p r o b a b i l i t y estimates as c o m p a r e d t o TL. H o w e v e r , t h e e m p i r i c a l t a i l b e n d s d o w n f a s t e r t h a n t h e t h e o r e t i c a l curves, a n d t h e largest o b s e r v a t i o n s stay o v e r e s t i m a t e d .
Fig. 7c s h o w s a sea state w h e r e the m a x i m u m w a v e h e i g h t has Al i n t h e o r d e r o f 2.3, w h i c h classifies t h e associated w a v e as b e i n g
a b n o r m a l [ 2 4 ] . M o r e o v e r , t h e n o r m a l i z e d crest o f t h e m a x i m u m w a v e h e i g h t ( t h e crest a m p l i f i c a t i o n i n d e x ) is also v e r y h i g h -a b o u t 1.5 f o r t h e d o w n - c r o s s i n g , -as w e l l -as f o r t h e u p - c r o s s i n g w a v e d e f i n i t i o n s . Consequently, t h i s sea state c o n t a i n s the w a v e w i t h the largest characteristics used t o d e f i n e a w a v e as a b n o r m a l , c o m p a r e d t o all b i m o d a l sea states c o n s i d e r e d . O b v i o u s l y , f o r t h i s p a r t i c u l a r case, R f i t s w e l l the m a j o r i t y o f t h e data, e x c e p t f o r t h e i n t e r m e d i a t e range w h e r e i t o v e r e s t i m a t e s the p r o b a b i l i t i e s , a n d a t the t a i l w h e r e i t s i g n i f i c a n t l y u n d e r e s t i m a t e s t h e e x t r e m e s . The largest w a v e h e i g h t is u n d e r e s t i m a t e d b y all m o d e l s , i n c l u d i n g GC. It m u s t be n o t e d t h a t the w a v e f i e l d c o n t a i n i n g t h e a b n o r m a l w a v e does n o t c o r r e s p o n d t o t h e m a x i m u m e s t i m a t e d A40.
Clearly, t h e increase o f X40 makes t h e e m p i r i c a l c u r v e s h i f t u p t o w a r d s t h e n o n l i n e a r GC m o d e l (Fig. 7 c - d ) . I t m u s t be n o t e d t h a t
t h e largest X40 a m o n g a l l b i m o d a l tests p e r t a i n s to t h i s g r o u p o f sea states {X40 = 0 . 7 7 ) . Fig. 7 d s h o w s t h e r e s u l t f o r t h e sea state w i t h t h e second largest A40 (X40 = 0 . 7 4 ) . A l t h o u g h some data f o l l o w GC, as a w h o l e t h e m o d e l fails t o p r e d i c t w e l l t h e w a v e h e i g h t s , especially t h e largest ones w h i c h are t h e m o s t i m p o r t a n t f o r e n g i n e e r i n g p r a c t i c e .
F u r t h e r m o r e , t h e t i m e series w i t h t h e largest A40 are associated w i t h s o m e o f t h e largest values o f SSER. C o m p a r i s o n b e t w e e n t h e t h e o r e t i c a l p r e d i c t i o n s w h e n SSER is g e t t i n g larger s h o w s a t e n d e n c y f o r R, TL, V a n d N t o c o i n c i d e . T h i s t e n d e n c y is w e l l seen w h e n c o m p a r i n g t h e results i n Fig. 7a-b, associated w i t h large d i f f e r e n c e b e t w e e n the values o f SSER.
Fig. 7e, f are r e p r e s e n t a t i v e f o r sea states w i t h SSER b e i n g r e l a t i v e l y s m a l l , i n the o r d e r o f 1.4-1.6, c o m p a r e d to t h e m a j o r i t y o f w i n d seas, w h e r e t y p i c a l l y SSER > 2.5. For these cases, TL < N < V s h o w t h e largest discrepancies b e t w e e n each o t h e r at the tails. The w a v e h e i g h t s u p t o 3CT are w e l l f i t t e d b y R a n d GC. The V m o d e l can be c o n s i d e r e d as a g o o d a p p r o x i m a t i o n o v e r t h e e n t i r e data range, except f o r the l o w e s t h e i g h t s . H o w e v e r , as s h o w n i n Fig. 7e, t h e e x t r e m e tail is s i g n i f i c a n t l y o v e r e s t i m a t e d b y a l l d i s t r i b u t i o n s . A c h a r a c t e r i s t i c o b s e r v e d f o r these t y p e s o f d i s t r i b u t i o n s is t h e p r o b a b i l i s t i c h u m p i n the e m p i r i c a l c u r v e b e t w e e n h e i g h t s o f a p p r o x i m a t e l y 4.5CT a n d 6CT, w h e n t h e o b s e r v a t i o n s c u r v e t o w a r d s R a n d s u b s e q u e n t l y go d o w n . I t is w e l l seen i n Fig. 7f.
One general c o n c l u s i o n can be d e r i v e d , r e g a r d i n g t h e a b i l i t y o f B t o p r e d i c t t h e o b s e r v a t i o n s i n w i n d sea d o m i n a t e d c o n d i t i o n s . The m o d e l s i g n i f i c a n t l y u n d e r e s t i m a t e s t h e data w i t h f e w e x c e p t i o n s (Fig. 7a, b, e), d e v i a t i n g also s i g n i f i c a n t l y f r o m t h e o t h e r t h e o r e t i c a l p r e d i c t i o n s s h o w i n g l o w e r p r o b a b i l i t i e s . The r e s u l t disagrees w i t h p r e v i o u s conclusions o b t a i n e d b y A r e n a a n d Guedes Soares [ 1 8 ] . T h e y s t u d i e d t h e c r e s t - t o - t r o u g h w a v e h e i g h t d i s t r i b u t i o n s i n
10 10 10" 10 . 10 10"' 10' 10 10 . 10' 10' 10
10-P.G. Petrova, C. Guedes Soares / Applied Ocean Researcli 31 (2009) 90-100 TANK/test8235/g1 /1658 waves 97 • Data " ^ " S S ^ w ^ N " ^ ^ ^ ... B r = ^).50281 — TL GC : q = 0.53233 =0.46602 (, = 1.13 v = 0.45361 X,, = 0.10362 SSER "3.0736
•
4 5 6 Wave Height, H / a TANK/lest8235/ g9 /1573 waves • Data R —-. N ... B ~ V — TL GC r = .0.54503 '••C^k^^^. q= 0.56743 V ^ v \ 0.50526 \ ^ \ \ -. 0 = 1.0062 \ \.\
V = 0.42285 ' \ ^ \ 1 „ = 0.54986 \\ \
SSER «=2.9765 \ 3 4 5 6 7 8 9 10 Wave Height, H / a TANK/lesl8235/ g4 /1487 waves • Data R . . . N B V . — TL GC ; r = ^).47577 q = 0.47579 =0.25838 0 = 1.2365 v = 0.52189 X j , = 0.24806 X j , = 0.24806•
SSER = 1.4412 4 5 6 Wave Height, H / a 10 10' 10" 10 10' 10" 10' 10 TANK/lest8233/ g3 /1703 waves • Dala . — N ""'-7:-.^ B •"•'-•ÏSK. ^ V r = 0.63118 '"-s^Stt-. q = 0.6412 \ — TL GG tt"*".. -. =0.61882 ~ \ 0 = 1.0502 V = 0.38205 \ „ = 033786 SSER = 5.6851\
4 5 6 Wave Height, H / a TANK/test8234/ g9 /1666 waves 4 5 6 7 Wave Height, H / a TANK/test8233/ g5 /1437 waves • Data R — - N ... B — V — T L GC r = .0.47111 q = 0.47286 =0.31441'\s5>j\
0 = 1.234'\s5>j\
v = 0.49022 \ = 0.36998 SSER > 1.5384 ' • 4 5 6 7 Wave Height, H / aFig. 7. Exceedance probability ofthe normalized wave heights, H / o , in the wind sea dominated wave fields.
n u m e r i c a l l y s i m u l a t e d deep w a t e r m i x e d sea states, r e p r e s e n t i n g sea states r e c o r d e d i n the N o r t h A t l a n t i c . I t has been f o u n d t h a t t h e o b s e r v e d p r o b a b i l i t i e s are w e l l f i t t e d b y Boccotti's d i s t r i b u t i o n f o r t h e c o n s i d e r e d w i n d w a v e d o m i n a t e d seas, as w e l l as f o r t h e s w e l l d o m i n a t e d seas.
4.4. Sea-swell energy equivalent wave field (SSER « 1)
Results f o r t h e coexistence o f w i n d w a v e s a n d s w e l l w i t h c o m p a r a b l e e n e r g y c o n t e n t s are s h o w n i n Fig. 8. The graphs s h o w t h a t GC a l w a y s o v e r e s t i m a t e s , t o a g r e a t e x t e n t , t h e w a v e h e i g h t s larger t h a n the m e a n w a v e h e i g h t , a n d t h e d i s c r e p a n c y g e n e r a l l y increases t o w a r d s t h e t a i l o f t h e d i s t r i b u t i o n i f n o a b n o r m a l w a v e is i d e n t i f i e d (Fig. 8a, b ) . O n t h e o t h e r h a n d , t h e l o w e s t p r e d i c t i o n s b e l o n g e i t h e r to TL, or t o B, b u t , i n b o t h cases, t h e o b s e r v a t i o n s r e m a i n u n d e r p r e d i c t e d , w i t h o n e e x c e p t i o n s h o w n i n Fig. 8a.
The case i n Fig. 8a r e p r e s e n t s t h e sea state w i t h t h e s m a l l e s t c o e f f i c i e n t o f k u r t o s i s i n t h e e n e r g y e q u i v a l e n t g r o u p , X40 = 0.08, b u t also i t is one o f t h e s m a l l e s t X40 i n t h e w h o l e set o f b i m o d a l sea states. S u b s e q u e n t l y , t h e GC c u r v e a l m o s t coincides w i t h R. As can b e seen, t h e m o d e l s , e x c e p t f o r R a n d GC, g i v e t h e closest e s t i m a t e s . The data are a d e q u a t e l y f i t t e d b y N f o r a l m o s t t h e e n t i r e data range, e x c e p t f o r t h e i n t e r v a l b e t w e e n 2CT a n d 4cx, w h e r e i t s l i g h t l y u n d e r p r e d i c t s t h e e m p i r i c a l e s t i m a t e s . The largest w a v e s r e m a i n o v e r p r e d i c t e d b y a l l m o d e l s .
The increase o f A40 is g e n e r a l l y associated w i t h b r o a d e n i n g o f the o b s e r v e d EDFs t o w a r d s R (Fig. 8c) a n d CG (Fig. 8 d ) . The r e s u l t
98 RC. Petrova, C. Guedes Soares / Applied Ocean Researcli 31 (2009) 90-100
a 10
TANK/tesl8235/g2 /1485 waves TANK/test8233/ g7 /1362 waves
C 10° 10' 10' 11°-^ 10 4 5 6 Wave Height, H / a TANK/lest8233/ g8 /1362 waves 4 5 6 7 Wave Height, H / a TANK/test8233/ g6 /1343 waves • Data R . . . N ... B V — TL GC r = .0.51203 q= 0.51284 0.22099 (, = 1.2269 V =0.53116 >.„ = 0.3555 SSER = 1.1535 10 UJ 10' 4 5 6 Wave Height, H / a TAM</tesl8234/g4 /1332 wares 4 5 6 7 Wave Height, H / a 10" 10" 10'" 10 O 10' 10" 10" 10 Data R N B V TL GC j- =-0.4858 " " ^ ^ q = 0.4865 ^ =0.23612 o = 1.2851 V = 0.53569 ?.^g = 0.33066 SSER =1.0788 » 3 4 5 6 7 8 9 Wave Height, H/o
TANK/test8233/ g 10 /1328 waves
4 5 6 Wave Height, H / a
Fig. 8. Exceedance probability ofthe normalized wave heights, H/a. in the sea-swell energy equivalent wave fields.
• Data R —- N . Q r = .0.50572 ^ - ~ V - — T L GC q = 0-50621 r ^ = 0.23319 a = 1.2686
\ ^
V = 052224 X^, = 022986* \
V
-SSER =0.9797f o r the sea state w i t h possible a b n o r i n a l w a v e is d e m o n s t r a t e d i n Fig. 7 d . I n t h i s case, GC s l i g h t l y o v e r p r e d i c t s t h e e x t r e m e w a v e h e i g h t s .
Four sea states have SSER close to one, b u t s m a l l e r t h a n i t . The g r a p h s i n Fig. 8 e - f r e p r e s e n t t w o o f t h e m . O b v i o u s l y , t h e N m o d e l f i t s t h e d a t a q u i t e w e l l .
The o b s e r v e d u n d e r e s t i m a t i o n b y N i n s o m e sea states o v e r t h e m i d r a n g e o f data was a l r e a d y c o m m e n t e d o n [ 17] w h e n d i s c u s s i n g results f o r n u m e r i c a l l y s i m u l a t e d m i x e d seas w i t h e q u i v a l e n t sea-s w e l l e n e r g y c o n t e n t sea-s . The e f f e c t w a sea-s e x p l a i n e d w i t h the sea-s m a l l c a l c u l a t e d r, c o n s i d e r i n g its a b s o l u t e v a l u e . The t y p i c a l values o f | r | f o r t h e o r e t i c a l u n i m o d a l s p e c t r u m are e x p e c t e d to v a r y b e t w e e n 0.65 a n d 0.75 [ 3 ] . H o w e v e r , t h e m a x i m u m v a l u e o f I r i w h i c h was
f o u n d is o n l y 0.55 i n t h e set o f e n e r g y e q u i v a l e n t records (Fig. 8 b ) , w h i c h falls c l e a r l y o u t o f t h e e x p e c t a t i o n s .
This e f f e c t has b e e n also e n c o u n t e r e d i n t h e w i n d sea d o m i n a t e d m i x e d sea states (Fig. 7a, e, f ) f o r r b e i n g c l e a r l y o u t o f t h e i n t e r v a l , a n d i t is n o t p r e s e n t w h e n r is w i t h i n the e x p e c t e d l i m i t s (Fig. 7b, d ) .
As i t w a s seen, s o m e w a v e events have b e e n c l a s s i f i e d as b e i n g a b n o r m a l waves, a d o p t i n g t h e d e f i n i t i o n o f [ 2 4 ] f o r t h e a b n o r m a l i t y i n d e x Al = H m a x / H j b e i n g l a r g e r t h a n a f a c t o r o f 2 . H o w e v e r , t h e d e f i n i t i o n specifies 2 0 m i n r e c o r d s w i t h R a y l e i g h d i s t r i b u t e d w a v e h e i g h t s . In the p r e s e n t case, d u e t o t h e l o n g e r t i m e series w i t h a p p r o x i m a t e l y 1500 w a v e s , the p r o b a b i l i t y o f e n c o u n t e r i n g an e v e n t e x c e e d i n g the t h r e s h o l d o f 2 is e x p e c t e d t o
P.C. Petrova, C. Guedes Soares/ AppUed Ocean Research 31 (2009) 90-100 99
Fig. 9. Dependence of the abnormality index on the relative energy.
Table 3
Cases of sea states w i t h / i ; > 2.
Class Test/Gauge SSER Al
sw 8234/3 0.58 2.10 8234/8 3.19 2.01 8234/9 5.46 2.01 ws 8234/10 7.03 2.10 8235/9 2.98 2.30 8235/10 5.68 2.11 eq 8233/6 1.08 2.15
increase. Hence, t l i e d e f i n i t i o n a p p l i e d here can o n l y serve as an i n d i c a t o r f o r possible a b n o r m a l w a v e s .
Fig. 9 r e p r e s e n t s t h e r e l a t i o n b e t w e e n A I e s t i m a t e d f r o m t h e d o w n - c r o s s i n g w a v e s a n d SSER. The largest SSER is associated w i t h AI s l i g h t l y h i g h e r t h a n t h e t h r e s h o l d 2. The l a r g e s t n u m b e r o f sea states vjhhAI > 2 (5 cases) are f o u n d i n t h e w i n d sea d o m i n a t e d sea states; one case belongs t o t h e g r o u p o f e q u i v a l e n t s e a - s w e l l e n e r g y sea states a n d one belongs t o t h e s w e l l d o m i n a t e d seas. These sea states are s h o w n i n Table 3 a l o n g w i t h t h e c o r r e s p o n d i n g values o f SSER a n d A I . T h e presence o f e x t r e m e l y large w a v e h e i g h t s is e x p l a i n e d w i t h t h e n o n l i n e a r p r o p e r t i e s o f t h e sea, since t h e b r o a d e n i n g o f t h e s p e c t r u m m a k e s t h e w a v e h e i g h t d i s t r i b u d o n n a r r o w e r . S u b s e q u e n t l y , as has b e e n discussed b y [ 3 ] , t h e e x p e c t e d e x t r e m e v a l u e w i l l decrease. This r e s u l t is i n a g r e e m e n t w i t h t h e d i s c u s s i o n o n t h e j o i n t d i s t r i b u t i o n o f c o n s e c u t i v e w a v e h e i g h t s i n m i x e d sea states p r e s e n t e d b y R o d r i g u e z a n d Guedes Soares [ 1 5 ] . T h e r e s u l t s w e r e o b t a i n e d o n t h e basis o f n u m e r i c a l l y s i m u l a t e d seas w i t h t w o - p e a k e d spectra. The s t u d y c o n c l u d e d t h a t t h e r e exists a large c o r r e l a r i o n b e t w e e n c o n s e c u t i v e w a v e h e i g h t s i n w i n d sea d o m i n a t e d b i m o d a l sea, c o n t r a r y t o t h e s w e l l e n e r g y d o m i n a t e d sea a n d , especially, t o s e a - s w e l l e n e r g y e q u i v a l e n t sea.
5. Conclusions
The d e p e n d e n c i e s b e t w e e n t h i r d a n d f o u r t h o r d e r n o r m a l i z e d c u m u l a n t s a n d SSER are expressed as second o r d e r p o l y n o m i a l s . Generally, t h e s w e l l d o m i n a t e d sea states are f o u n d t o be a l m o s t Gaussian a n d t h e associated c u m u l a n t s t e n d t o z e r o . The l a r g e s t n o n - G a u s s i a n statistics are t y p i c a l f o r w i n d sea d o m i n a t e d w a v e f i e l d s . F u r t h e r m o r e , t h e change o f A30 a n d X40 a l o n g t h e basin g e n e r a l l y f o l l o w s t h e t r e n d i n SSER.
T h e s w e l l d o m i n a t e d seas d e m o n s t r a t e t h e b r o a d e s t spectra a n d , c o n s e q u e n t l y , the e m p i r i c a l d i s t r i b u t i o n s are m u c h n a r r o w e r t h a n R. The increase o f SSER reduces t h e d i s t r i b u t i o n s t o w a r d s TL, w h i c h gives t h e best f i t t o t h e data i n t h e i n t e r m e d i a t e range o f values. N o n e o f t h e m o d e l s , h o w e v e r , is able t o p r e d i c t t h e largest w a v e s c l a s s i f i e d as a b n o r m a l .
The m o d e l s a c c o u n t i n g f o r t h e s p e c t r u m f r e q u e n c y w i d t h f i t a d e q u a t e l y the o b s e r v a t i o n s i n w i n d sea d o m i n a t e d sea states o v e r a l m o s t t h e e n t i r e data range f o r s m a l l X40, e x c e p t f o r t h e e x t r e m e s . H o w e v e r , N fits w e l l the l o w w a v e s , w h i l e i t s l i g h t l y u n d e r e s t i m a t e s the i n t e r m e d i a t e a n d large w a v e s . TL a n d V, o n t h e o t h e r h a n d , c a n n o t be a p p l i e d t o l o w w a v e s , b u t t h e y fit w e l l t h e i n t e r m e d i a t e a n d large waves. T h e t h i r d o r d e r n o n l i n e a r e f f e c t s are r e f l e c t e d i n large X40 a n d s u b s e q u e n t s h i f t o f t h e t a i l t o w a r d s GC. T h e rime series w i t h t h e largest ;i4o are associated also w i t h t h e largest SSER. W h e n SSER increases, t h e m o d e l s R, TL and N c o i n c i d e . O n t h e c o n t r a r y , w h e n SSER is r e l a t i v e l y s m a l l , a h u m p i n t h e e m p i r i c a l c u r v e is o b s e r v e d f o r data b e t w e e n 4.5a a n d 6a. For t h e case o f s m a l l SSER, t h e largest d i f f e r e n c e b e t w e e n TL, N a n d V is o b s e r v e d . The V m o d e l can be c o n s i d e r e d as a g o o d a p p r o x i m a t i o n over the e n t i r e data range, e x c e p t f o r t h e l o w e s t h e i g h t s . H o w e v e r , t h e o b s e r v e d t a i l can be s i g n i f i c a n t l y o v e r e s t i m a t e d b y a l l d i s t r i b u t i o n s .
The e q u i v a l e n t e n e r g y sea states d e m o n s t r a t e t h a t , f o r s m a l l A40 N is a g o o d fit, e x c e p t f o r the largest w a v e h e i g h t s , w h i c h stay o v e r e s r i m a t e d b y a l l m o d e l s . The increase o f X40 is associated w i t h b r o a d e n i n g o f t h e e m p i r i c a l d i s t r i b u t i o n . For sea states w i t h SSER close t o one, b u t s m a l l e r t h a n i t , t h e N m o d e l fits t h e data q u i t e w e l l .
The N m o d e l is f o u n d to u n d e r e s t i m a t e t h e l o w a n d i n t e r -m e d i a t e data i n s o -m e e n e r g y e q u i v a l e n t sea states a n d w i n d sea d o m i n a t e d sea states, w h e n the e s t i m a t e d d i s t r i b u t i o n p a r a m e t e r c l e a r l y f a l l s o u t o f t h e e x p e c t e d range f o r s t o r m w a v e s .
The a b n o r m a l events i n t h e s w e l l a n d w i n d sea d o m i n a t e d sea states are u n d e r e s t i m a t e d b y a l l m o d e l s , w h i l e o n l y s l i g h t l y o v e r e s t i m a t e d b y GC i n t h e e n e r g y e q u i v a l e n t sea states, w h i c h is i n l i n e w i t h e a r i i e r results f r o m m e a s u r e m e n t s at sea [ 2 5 ] .
The largest n u m b e r o f sea states w i t h possible a b n o r m a l w a v e s ( 5 ) are f o u n d i n t h e w i n d sea d o m i n a t e d sea states; one case belongs t o t h e g r o u p o f sea-swell e n e r g y e q u i v a l e n t sea states a n d one belongs to t h e s w e l l e n e r g y d o m i n a t e d seas.
Aclmowledgments
T h e w o r k has b e e n p e r f o r m e d i n t h e scope o f t h e p r o j e c t MARSTRUCT, N e t w o r k o f Excellence o n M a r i n e S t r u c t u r e s ( h t t p : / / w w w . m a r . i s t . u t l . p t / m a r s t r u c t / ) , w h i c h has b e e n financed b y t h e EU t h r o u g h t h e GROWTH P r o g r a m m e u n d e r c o n t r a c t T N E 3 C T 2 0 0 3 -5 0 6 1 4 1 .The data used i n t h e p a p e r has been o b t a i n e d at MARINTEK i n the scope o f t h e p r o j e c t : Large Scale Facilities " I n t e r a c d o n s B e t w e e n W a v e s and C u r r e n t s " , w h i c h has b e e n p a r t i a l l y f u n d e d b y t h e E u r o p e a n U n i o n u n d e r c o n t r a c t ERBFMGECT980135.
The first a u t h o r has been financed b y t h e Portuguese F o u n d a -tion o f Science a n d T e c h n o l o g y (ECT) u n d e r the g r a n t SFRH/BD/ 3 0 1 0 9 / 2 0 0 6 .
Part o f these r e s u l t s has b e e n p r e s e n t e d at t h e D e e p w a t e r O f f s h o r e T e c h n o l o g y S y m p o s i u m (DTEC 2 0 0 8 ) , h e l d at t h e Shanghai Jiao T o n g U n i v e r s i t y f r o m 17 t o 19 N o v e m b e r 2 0 0 8 .
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