ioes
ori
?
177
•
voor de Z m
m
hM det* m&Hm* h u m *
S e n s i t i v i t y o f w i n d w ave s i m u l a t i o n to
co u p lin g w ith a tid e /s u r g e m o d e l w ith
a p p lica tio n to th e S o u th e rn N o r th Sea
J. M on b a liu ? C.S. Yu* a n cl P. Osuna*
A bstract
T h e w a v e - c u r r e n t i n t e r a c t i o n p r oc e s s in a o ne wa y c ou p l ed s y s t e m for t h e
S o u t h e r n N o r t h S e a region w a s s t u d i e d . T o t h i s e nd. a modif ied versi on o f t h e
t h i r d - g e n e r a t i o n s p e c t r a l w av e m o d e l W A M w a s r un in a t h r e e level n e s t ed
g r i d s y s t e m , t a k i n g i n t o a c c o u n t t h e h y d r o d y n a m i c fields ( c oup le d ver si on)
c o m p u t e d b y a t i d e / s u r g e m o de l . T h e r e s o l ut i o n in t h e s e t h r e e g r i d s c o r r e ­
s p o n d s r o u g h l y t o 35 k m ( coa rs e) , 5 k m (local) a n d 1 km (fine). H y d r o d y n a m i c
i n f o r m a t i o n w a s o n l y ava il ab le a t t h e r e s ol u t i on of t h e local grid. R e s u l t s a r e
c o m p a r e d w i t h t h o s e o f t h e s a m e version w i t h o u t c o n s i d e r i n g t h e i n t e r a c t i o n
w i t h t i d e a n d s u r g e s ( u n c o u p l e d v e rs io n ). T h e e m p h a s i s is on t h e r e s u l t s f r o m
t h e local g r i d c a l c u l a t i o n s . T h e s e n s i t i v i t y t o t h e s o u r c e o f b o u n d a r y c o n d i ­
ti on s (coupled vs. unc oupled) an d to t h e frequency of information exchange
is i n v e s t i g a t e d . A l s o t h e s p e c t r a l e v ol u t i o n is s t u d i e d .
N u m e r i c a l r e s u l t s in t h e local gr id a r e in g o o d a g r e e m e n t w i t h b u o y d a t a .
T h e p h a s e a n d a m p l i t u d e o f t h e m o d u l a t i o n s o f wa v e p e r i od o b s e r ve d f r om t h e
b u o y d a t a a r e q u i t e well r e p r o d u c e d by t h e c o u p l ed ve r si on . T h e m o d e l r e s u l t s
in t h e local g r i d d i d n o t s h o w m u c h s e ns i t i v i t y t o t h e s ou r c e o f b o u n d a r y
c o n d i t i o n i n f o r m a t i o n , n o r wer e t h e y v e ry s e n s i t i ve t o t h e i n f o r m a t i o n u p d a t e
f r e q u e n c y o f t h e h y d r o d y n a m i c fields. T h e d i r e c t i o n a l s p e c t r a c o m p u t e d in t h e
c o u p l e d m o d e s h o w a b r o a d e r e n e r g y d i s t r i b u t i o n a n d a mor e r a p i d g r o w t h .
T h e fine g r i d r e s u l t s o n l y differ m a r g i n a l l y f r o m t h e local grid r e s u l t s d u e to
t h e l i m i t e d r e s o l u t i o n o f t h e h y d r o d y n a m i c fields used.
' L a b o r a t o r y o f H y d r a u lic s , K .U .L e u v en , de C ro y la a n 2, B-3001 Heverlee, B e lgium .
f D e p t. o f M a r in e E n v . a n d E n g ., N S Y S U , L ie n -H a i R o a d 70, K aosh iu n g 804, T a i w a n .
2
2.1
T h e n u m e r i c a l m o d e ls
T h e wave m odel
T h e wave m od el used is th e third g e n e ra ti o n W A M C ycle-4 m o d e l ( G ü n t h e r et
ah, 1994). T h e m o d e l solves an energy b a la n c e e q u a t i o n for t h e s p e c t r u m F ( ƒ , 0 )
as a f u n ct io n of t h e wave frequency, ƒ, wave di rec tio n, #, a n d t h e geographical
position, X , a n d ti m e , t. T h e s t a n d a r d version of W A M p r o p a g a t e s t h e ene rgy over
a calcu lation al grid in Car tes ian co ordinates x ( : r , y ) for sm al l a re a ap p lic at io n s or
in spherical co o rd in a te s x(d>, A.r ) for m o d el ap p lic at io ns over larg e areas as to take
into a c c o u n t t h e swell p ro p ag a tio n over g r e a t circles. For this work t h e last option
is used. T h e W A M m o d el p e r m i ts th e inclusion of a s t a t i o n a r y c u r r e n t ba ck gr o un d
an d uses t h e relative frequency, a , as a co or di na te . U n d e r t h e s e conditions, th e
energy t r a n s p o r t e q u a ti o n s solved in th e W A M Cyclc-4 mo del cod e is eq uival ent to
t h e actio n de n sit y t r a n s p o r t equation. T h e t r a n s p o r t e q u a t i o n for t h e evolution of
th e wave s p e c t r u m F ( t , </>, A, cu 0) then read s
7 7 +
= 5 '“ ’
( 1)
where t h e expressions <j), A, à , and 0 r e p r e s e n t t h e ra t e of ch a n g e of energy in the
space (</>, A, <j, 0). At t h e right hand side of (1), S tot is t h e fu nct io n r e p r e s e n t in g t h e
source a n d sink functions, and th e conservative no n -li n e ar t ra n s fe r of en erg y between
wave c o m p o n e n t s . In th e present appli cat io n, it includes wind i n p u t S in, non-linear
q u a d r u p l e t wave-wave interac tio ns S n/, w h i t e c a p p i n g di ssi pa tio n S¿s a nd b o t t o m
friction diss ipa tion Sbj- S t a n d a r d values were used for t h e e m p i r ic a l coefficients in
th e source t er m s. T h e m o d el has been used in a q u a s i - s t e a d y a p p r o a c h , as su m i n g
t h a t t h e c u r r e n t a n d t h e wat er d e p th v ary only slowly. A de ta i le d des crip tion of
th e physics in c o r p o r a t e d in W AM Cycle-4 m o d e l a n d its n u m e r i c a l i m p l e m e n t a t i o n
can b e found in K o m e n et ah, (1994). Details on t h e i m p r o v e m e n t s , a lt e ra t io n s a nd
a d d iti o n s for a p p lic at io n in nearshore regions can be fo und in Luo a n d Sclavo (1997)
a n d M on b ali u et al. (1998).
2.2
T h e tid e /su rg e model
T h e h y d r o d y n a m i c (u,?;,? 7 ) fields are c o m p u t e d w i t h a m o d e l based on t h e shallow
wa te r eq ua tio ns . T h e spherical co o rd i n a te expressions for this set of e q u a tio n are
( W a s hi ng to n a nd P a r k in s o n , 1986)
du
V du
R eos ô d \
Rd4>
1
du
u
~dt
9
d>1
R eos (j) d \
+
A í
uv tan 4
R~
d Pa
p R eos (j) dX
— 2 zj sin f a
'b A
p{h + rj)
u
t an (j) d v
V 2n + — (1 — t a n 2cf)) — 2
eos ô d A.
A A
p {h + rj)
(2)
C O A R S E G R ID D OM A IN
70
>600
65
C S M G R ID DOM A IN
CD
P .
55
-1 4
-1 0
-6
2
-2
6
10
14
lo n g it u d e [deg]
F i g u r e 1: Coarse an d C S M grid d o m a i n . T h e coarse grid b a t h y m e t r y is included.
B o u n d a r y condition for a nested local grid (region ind ica ted by t h e sq ua re ) are
ge ne ra te d. D e p t h values a re in m ete rs.
L O C A L GRID - 1 /2 4° LAT.x 1/12° LON. R E S O L U T IO N
51.5
cn
cu
T>
ZEEBRUGGE
OOSTENDE
ra
50.5
0
0.5
1
1.5
2
2.5
3
3.5
4
l o n g i t u d e [deg]
F i g u r e 2: Local grid b a t h y m e t r y w i t h location indication of t h e W E H a n d A2B
buoys. B o u n d a r y c on di tio n for a fine grid (region in dic ate d b y t h e sq ua re ) are
g e ne ra te d. D e p t h values a r e in mete rs.
In w h a t follows, t i m e series a t t h e locations W E H ( W e s t h i n d e r ) a n d A2B from
b o t h coupled and unc oupled W A M results are c o m p a r e d w i t h th e available buoy
d a t a . T h e sensitivity to ‘coupled'' in fo rm a tio n in th e b o u n d a r y conditions, a n d to
t h e frequency of information ex c ha ng e is ad dressed for th e local grid ap plication.
T h e evolution of l D - s p e c t r a a nd 2 D - s p e c t r a in ‘coupled' and ‘u n c o u p l e d ’ m o d e are
discussed.
4 .2
T i m e se rie s
T i m e series calculated by W A M a t s tation W E H (30m d e p th ) s ho w a good a g r e e m e n t
wit h buoy d ata , especially for significant wave height (lis) values d u r in g high-wave
e ve nt s (see Figure 3). T h e small m o d u l a t i o n s visible in th e bu o y d a t a a t W E H seem
well rep ro du c e d by t h e coupled version of W AM . However, t h e presence of s o m e
m o d u l a t i o n s in th e unc oup led r u n results suggests t h a t p a r t of t h e m o d u l a t i o n s
in th e signal m u s t come from wind variability. T h e bias for Hs in th e coupled
a n d unc oup led results is -0.072m and -0.143m. respectively. T h e in te r co m p a ri s o n
b e tw e e n results from coupled a nd un c ou pl e d models show differences (coupled un c ou pl e d ) in Hs smaller t h a n 0.25m. with a m e a n difference of 0.02m. M o d u la t io n s
of Tni o 2 : corresponding to th e d o m i n a n t s em id iu rn al tide p e ri o d , are q u i t e clear
in th e buoy d a t a (Figu re 3). T h e i m p o r t a n c e of including th e h y d r o d y n a m i c fields
is highlighted qualitatively by t h e good a g r e e m e n t between b u o y d a t a a nd W A M cou pled results for T m o 2 - Thi s is not directly reflected in t h e value for t h e bias
for T m o 2 , which were 0.032s for t h e coupled a nd 0.034s for t h e unc oup led run,
respectively. T h e model results t h em se lve s showed differences up to 1.0s, wit h a
m e a n difference of -0.1.5s.
In t h e more shallow A2B stat ion (11m d e p t h ) , o v e re s t im at i o n of mo del results
w i t h resp ec t to buoy d a t a is observed, b o t h in Hs a nd in T m 0 2 ( F ig u r e 4). A lt h o u g h
it is possible to reduce th e observed differences by t u n in g of t h e empi ric al coefficient
in t h e b o t t o m friction t e r m (see Luo a n d Mon ba liu , 1994, Luo et al., 1996), this
was n o t done since it was not considered i m p o r t a n t for t h e scope of this work.
A t this s ta tio n , t h e bias was -0.128m a n d -0.116m for Hs a nd -0.846s a nd -0.965s
for T m 02 in th e coupled an d un co up led m o d e, respectively. T h e i m p o r t a n c e of tidal
m o d u l a t i o n s is again observed at this location a n d it is q u a lit a tiv e ly well r e p r o d u c e d
in t h e coupled run. Differences b e tw e en coupled an d un co up led m o d el results do
n o t exceed 0.25m in Hs an d 1.0s in T m 02, wher eas th e m e a n difference in t h e Hs
a n d Tn i o 2 value was 0.014m a nd -0.217s, respectively.
4 .3
S e n s i t i v i t y to b o u n d a r y c o n d i t i o n s
In o r d er to explore t h e necessity to ru n in coupled m o d e on a coarse grid in order
to s u p p ly good b o u n d a r y conditions for a s u b s e q u e n t ne sted run, t h e ap pl ic ati on on
t h e local grid was ru n once using b o u n d a r y co nd itions from th e coupled a n d once
fr o m th e un coupled coarse grid run. In b o t h cases, results showed clearly t h e tidal
m o d u l a t i o n effect on T m 0 2 (Fig ur e 5a). T h e ob served differences a re small a n d m o s t
of t h e t i m e do not exceed 5%, which seem to suggest t h a t r u n n in g t h e coarse grid
ap pl ic ati on in coup led m o d e , does not hav e a d r a m a t i c influence on t h e local grid
runs. T i d e - i n d u c e d m o d u l a t i o n s , a t least on th e spa ce scales u s e d here, are m a i n ly
a local effect-.
8
7
B u o y D ata
B.C. U n e
B.C. C o u
4+
6
+
E
4
3
2
++
16
18
20
24
22
:
+
26
28
e l a p s e d d a y s o f f e b r u a r y 1993
S T A T I O N W E H ( L O C A L GRI D)
8
B u o y D ata
4/2 0 /6 0
4 /2 0 /2 0
7
6
E
H
4
++
3
2
16
18
20
22
:24
26
28
e l a p s e d d a y s o f f e b r u a r y 1993
F i g u r e 5: T i m e series s h o w i n g t h e s e n s i t i v i t y to: (a) b o u n d a r y c o n d i t i o n s , a n d ( b )
i n f o r m a t i o n e x c h a n g e o f T m .02 a t s t a t i o n W E H .
4.4
S ensitivity to fre q u en cy of in fo rm a tio n exchange
In a coupled s y s te m , an i m p o r t a n t factor to define is t h e frequency of i n f o r m a ­
tion transfer b e tw e en m o d e l c o m p o n e n ts . In this work, h y d r o d y n a m i c fields were
u p d a t e d every 20 m i n u t e s ( s t a n d a r d coupled m o d e ) , which seems su it a b le for th e
te m p o ra l scale of ti d e variability. In ord er to invest iga te t h e sensitivity of t h e m od el
results to th e fr e q u e n c y of i n fo r m a t io n ex change, th e coup led mo del on t h e local
grid was also r un w i t h a n u p d a t e of th e h y d r o d y n a m i c fields every 60m in. T h e
results were c o m p a r e d w i t h tho se of th e s t a n d a r d run. T h e ti m e series of T m 0 2
pr esented in Fi gur e 5b, s ho w differences smaller t h a n 5%. Som e details in t h e m o d ­
ulation ar e missed. For t h i s spa tia l scale a nd wit h c u r r e n t an d d e p t h fields which
vary only slowly in t i m e a n d space, th e m ai n va riation in th e spectral p e rio d s are
well rep ro du c e d by t h e D o p p l e r shift. As one can observe from th e t i m e series, t h e
tidal m o d u l a t i o n s r e s p o n d m a i n l y to th e s e m i d iu r n a l t id a l c o n s tit u e n t M2. F u r ­
t h e r decrease of t h e f r e q u e n c y of i nfor ma tio n ex c ha ng e will lead to increased loss of
information.
have a linear ‘P h i l l i p s ’ t e r m in its wind i n p u t source function. I t was n o t in vestigated
in how far this is r es p on s ib le for th e lack of gro wt h in t h e u n c o u p l e d version.
C ouplei
Couplei
93022:
93022:
U ncoup]
Uncoup]
93022:
93022.
C o u p le d - S ta tio n A2B
F i g u r e 7: 2D s p e c t r a a t s ta tio n A2B ca lculated by W A M cou pled an d un co upl ed .
Significant wave h e i g h t (Hs), wind direction (dar k line) a n d d a t e a re always in di ­
cated. For coup led r esu lts , c u rr e n t m a g n i t u d e (U) a nd c u r r e n t di rection (light line)
are i ndi c a ted . T h e s a m e c on to ur labels were used in all figures ( 1 , 5 , a n d fro m 10
to 100% every 10% of 0 . l m 2/ H z / d e g ) .
4.6
F u rth er w ork
T h e fine grid b a t h y m e t r y for th e Fle mi sh Coa st is m uch m o r e c om pl ic a te d t h a n can
be a n t i c i p a t e d fro m F i g u r e 2. M a ny sa nd banks more or less parallel with t h e coast
are prese nt. T h e i r s p a t i a l scale is small such t h a t t h e y dissa.pear from th e local grid
resolution. A fine g r id wave model was ne sted in th e local grid (see section 3.1). For
t h e h y d r o d y n a m i c fields linear inter pola tion from th e CSM -g rid (equal to t h e local
wave m od el grid) to t h e fine grid was used. Com pa ris on of t i m e series a t A2B from
t h e local a n d fine grid s showed only negligible differences. T h i s is not u n e x p e c t e d
since all t h e v a ria bi lit y in th e c u rr e n t field induced by th e local b a t h y m e t r y is
not r ep r e s en t e d in t h e in ter po lat ed c u r r e n t field. If t h e details of th e b a t h y m e t r y
are n o t tak en in to a c c o u n t in th e calculation of th e h y d r o d y n a m i c a ! variables, th e
directions of t h e waves a n d th e c u r r e n ts b e c o m e m o re a n d m ore p e r p e n d i c u l a r as
H u b b e r t , K.P.. a n d J. Wolf, 1991: N u m e r i c a l investigation o f d e p t h a nd c u r r e n t
refraction of waves. J . G eophys. R es., 9 6 , 2737-2748.
Holthuijsen. L., a n d H.L. T o l m a n . 1991: Effects of th e Gulf S t r e a m on ocean waves.
J. G eophys. R es., 9 6 . 12755-12771.
Jon sso n. I.G, 1990: W a v e - c u r r e n t i n te r ac tio ns . In B. Le M e h a u t e a n d D.M. Han es
(Eds .). The Sea, O ce an E n gi ne er in g Science, Vol 9(A), p6 5- 12 0. Wiley, New York.
K o m e n . G .J., L. Cavaleri, M. Done lan , K. H a s s e lm a n n , S. H a s s e l m a n n , a nd P . A . E.M.
J a n s s e n . 1994: D y n a m ic s a n d m odelling o f ocean waves. C a m b r i d g e Un iv ers ity
Press.
Luo, W ., and -J. M on b a liu , 1994: Effects of t h e b o t t o m friction form ula tio n on th e
e ne rgy balance for g r a v i t y waves in shallow water. J. G eophys. Res., 9 9 , 1850118511.
Luo, W ., J. Mon ba liu , a n d J. B e r l a m o n t , 1996: B o t t o m friction dissipation in t h e
Belgian coastal region. Pvoc. 25th Int. C o n f. Coastal Eng., Flo rida , 836-849.
Luo, W.. and M. Sclavo, 1997: I m p r o v e m e n t of th e third g e n e ra ti o n W A M model
(Cycle 4) for ap p lic at io n s in n e a rs h o r e regions. Internal re p o r t no. 116, P r o u d m a n
Oce an og rap hi c L ab o r a l o ry.
Mo nba liu . .]., R. Pa dilla, P. O s u n a , R. F l a t h e r , J. Hargreaves, J . C . C a r r e t e r o , S.
Esp in ar, and H. G ü n t h e r , 1998: A sh allow w a t e r version of th e W AM s p ect ral wa.ve
model. Report nr. 52, P r o u d m a n O c e a n o g r a p h i c Laboratory. To be published.
M a st e n b ro e k . C., G. B ur ge rs , an d P . A . E . M . J a n s s e n , 1993: T h e d y n a m i c a l cou pl in g
of a wave model a n d a s t o r m surge m o d e l th r o u g h th e a t m o s p h e r i c b o u n d a r y la,ver.
J. P hys. Océano gr., 23 , 1856-1866.
Mei, C.C., S. Fan, a n d K. J in , 1997: Res us pe ns io n a n d t r a n s p o r t of fine s e d i m e n t s
by waves. J. G eophys. R e s., 1 0 2 , 15807-15821.
S h e m d in . O.H., S.V. Hsiao, H.E. Ca r ls o n , K. Ha s s e lm a n n , a n d K. Schulze, 1980:
M e ch a n i s m s of wave t ra n s f o r m a t i o n in finite- dep th water. J. G eophys. R e s. 8 5 ,
5012-5018.
T o l m a n . H.L., 1990:
W i n d wave p r o p a g a t i o n in tidal seas.
C o m m u n i c a ti o n s on
hydraulic and geotechnical engineering, Delft University of Technology. R e p o r t nr.
90-1.
W a sh in gt on , W . M . , a n d C.L. Pa r k in s o n , 1986: A n int.rodun.tion to th r e e - d im e n s io n a l
clim ate modelling. O xf or d University Press.
Yu, C.S., 1993: M o d e lli n g Shelf Se a D y n a m i c s , Ph. D. thesis, D e p a r t m e n t of Civil
Engineering, K .U .L e u v e n , Belgium.