Monbaliu, J.

Osuna, P.

Yu, C.S.

Open Marine Archive

i oes ori ?
177
S e n s i t iv i ty  o f  w in d  wave s im u la t io n  to  
co u p l in g  w i th  a  t i d e / s u r g e  m o d e l  w i th  
a p p l ic a t io n  to  t h e  S o u th e r n  N o r t h  S ea
J. Monbaliu?  C.S. Yu* a n  cl P. Osuna*
A b s t r a c t
T h e  w a v e - c u r r e n t  in te rac t ion  process  in a  one  way coupled  s y s t e m  for th e  
S o u t h e r n  N o r t h  S e a  region w as  s tu d ie d .  T o  th i s  end.  a  modif ied version o f  th e  
t h i r d - g e n e r a t i o n  s p e c t r a l  wave mode l  W A M  was  run  in a  th r e e  level nes ted  
gr id  s y s t e m ,  t a k i n g  in to  a c c o u n t  t h e  h y d r o d y n a m i c  fields (coupled version)  
c o m p u t e d  by  a  t i d e / s u r g e  model .  T h e  reso lut ion in t he se  t h r e e  gr ids  c o r r e ­
s p o n d s  r o u g h ly  t o  35 k m  (coarse) ,  5 km  (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  on ly  ava ilab le  a t  t h e  resolu t ion  of th e  local grid.  R esu l t s  a re  
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  cons ider ing  th e  in te r a c t i o n  
w i t h  t i d e  a n d  s u r g e s  (uncoup led  vers ion).  T h e  e m p h a s is  is on t h e  re su l t s  f rom 
t h e  local g r id  c a lc u l a t io ns .  T h e  sens i t iv i ty  to  t h e  sou rce  of  b o u n d a r y  c o n d i ­
t i on s  ( coup led  vs .  unc oupled )  a n d  t o  t h e  f requency  o f  in format ion  e x c h a n g e  
is in v e s t i g a t e d .  Also t h e  s pe c t r a l  evolu t ion  is s t ud ied .
N u m e r i c a l  r esu l t s  in t h e  local gr id a r e  in goo d  a g r e e m e n t  wi t h  bu oy  d a t a .  
T h e  p h a s e  a n d  a m p l i t u d e  of  t h e  m o d u l a t i o n s  o f  wave per iod  observed  f rom t h e  
buo y  d a t a  a r e  q u i t e  well re p ro d u c e d  by th e  coupled  version.  T h e  m od e l  re su l t s  
in t h e  local  g r id  d i d  n o t  show m uch sens i t iv i ty  to  t h e  source  of  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 ,  no r  were t h e y  very  sens i t ive  to  th e  in fo rmat i on  u p d a t e  
f r e q u e n c y  of  t h e  h y d r o d y n a m i c  fields. T h e  d i r ec t iona l  s p e c t r a  c o m p u t e d  in the  
co u p le d  m o d e  s h o w  a  br o ad e r  en e rg y  d i s t r ib u t io n  a n d  a  more rap id  g r o w t h .  
T h e  fine gr id  r e su l t s  on ly  differ m ar g in a l ly  f rom th e  local grid re su l t s  d u e  to 
t h e  l imi t ed  re so lu t io n  o f  t h e  h y d r o d y n a m i c  fields used.
• voor de Z mhM det* m&Hm* h u m *  m
'L a b o r a to r y  o f  H y d ra u l ic s ,  K .U .Leuven, de C roylaan  2, B-3001 Heverlee, Belgium . 
f Dept.  o f  M a rin e  E nv .  an d  Eng.,  NSY SU, L ien-H ai Road 70, Kaoshiung 804, T a iw an .
2 T h e  n u m e r ic a l  m odels
2.1 T h e  w ave  m o d e l
T h e  wave model  used is the third genera t ion  W A M  Cycle-4  m odel  ( G ün the r  et 
ah,  1994). T h e  m odel  solves an energy ba lance  equa t ion  for t h e  s p e c t r u m  F ( ƒ ,0 )  
as a  funct ion  of t h e  wave frequency, ƒ,  wave di rec t ion,  #, a n d  the  geographical  
posi t ion, X ,  a n d  t im e,  t. T h e  s tanda rd  vers ion of  W A M  pro p ag a te s  the  energy  over 
a  calcu lat ional  grid in Car tes ian  coordinates  x ( : r , y )  for smal l  a rea  appl icat ions or 
in spherical  coord ina tes  x(d>, A.r )  for model  appl icat ions  over  large areas as to  take 
into accoun t  the  swell p ropaga t ion over g rea t  circles. For this work the  las t  opt ion 
is used. T h e  W A M  model  pe rm i ts  the  inclusion of a  s t a t io n a r y  c u r ren t  background 
and uses the  relative frequency, a ,  as a  coordina te .  Under  these  condi t ions,  the  
energy t r a n s p o r t  equa t ions  solved in the W A M  Cyclc-4  model  code is equivalent  to 
the  act ion densi ty  t r a n s p o r t  equat ion.  T h e  t r a n s p o r t  equa t ion  for the  evolut ion of 
the wave sp e c t r u m  F( t ,  </>, A, cu 0) then reads
7 7 + = 5 '“ ’ (1)
where the  expressions <j), A, à , and 0 r epresen t  t h e  ra te  of  change  of  energy in the 
space (</>, A, <j, 0). At t h e  r ight  hand side of  (1),  S tot is the  funct ion represen t ing  the  
source and  sink funct ions,  and the  conservat ive non-l inear  t ransfe r  of  energy between 
wave components .  In the  present appl icat ion,  it  includes wind i n p u t  S in, non-linear 
q u a d ru p le t  wave-wave interac t ions  S n/, wh i t ecapp ing  di ss ipa t ion S¿s and  bo t tom  
friction diss ipa t ion Sbj- S ta ndard  values were used for the  empi r ica l  coefficients in 
the  source terms.  T h e  model  has been used in a  qu a s i - s t e a d y  app roach ,  as suming  
t h a t  the  cu r ren t  a n d  t h e  water  dep th  vary  only slowly. A de ta i led  descr ip t ion of 
the  physics  in co rpora ted  in WAM Cycle-4  m odel  and  its num er ica l  im p lem en ta t ion  
can be  found in K o m e n  et  ah,  (1994). Detai ls  on t h e  im provem en ts ,  a l t e ra t ions  and 
addi t ions  for app l icat ion  in nearshore regions can be  found in Luo a n d  Sclavo (1997) 
and  Monbal iu  e t  al. (1998).
2.2 T h e  t i d e / s u r g e  m o d e l
T h e  h y d r o d y n a m i c  (u,?;,?7) fields are  c o m p u te d  wi th  a  m odel  based on the  shallow 
wa te r  equa t ions .  T h e  spherical  coord ina te  expressions for this  set  of equa t ion  are 
(Washing ton  and  Pa rk inson ,  1986)
d u
~dt
u du
R  eos ô  d \  
9  d> 1
V d u
Rd4>
1
u v  t a n  4  
R ~  
d  Pa
— 2zj sin f a
'b A A  A
R  eos (j) d \  p R  eos (j) dX
+ A í
u
V 2n +  —  (1 — t a n  2cf)) — 2
p{h +  rj) p{h +  rj) 
t an  (j) dv
eos ô  d  A. (2 )
C O A R S E  G R I D  D O M A I N70
>600
65
C S M  G R I D  D O M A I N
CD
P .
55
-1 4 -1 0 - 6 - 2 2 6 10 14
l o n g i t u d e  [ d e g ]
F i g u r e  1: Coarse and CSM  grid dom ain .  T h e  coarse grid b a t h y m e t r y  is included.  
Boundary  condit ion for a  nested  local grid (region ind ica ted  by the  squa re )  are  
gene ra ted.  D ep th  values a re  in mete rs .
L O C A L  G R I D  -  1 / 2 4 °  L A T .x  1 /1 2 °  LON .  R E S O L U T I O N
51.5
cncuT> Z E E B R U G G E
O O S T E N D E
ra
50.5
0 0.5 1.5 2 2.5 31 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  wi th  locat ion indication of  the  W E H  and  A2B 
buoys.  Bounda ry  condi t ion for a  fine grid (region indicated by  the  squa re )  are 
gene ra ted.  D e p th  values a re  in mete rs .
In w h a t  follows, t im e  series a t  t h e  locat ions W E H  (W es th inde r )  a n d  A2B from 
bo th  coupled and uncoupled W A M  results are  compared  w i t h  the  available  buoy 
da ta .  T h e  sensi t ivi ty to  ‘coupled' '  in formation  in the  b o u n d a r y  condit ions,  and  to 
t h e  frequency of information exchange  is addressed for the local  gr id  appl icat ion.  
T h e  evolut ion of  l D - s p e c t r a  and 2 D - s p e c t r a  in ‘coupled '  and ‘u n c o u p le d ’ m ode  are 
discussed.
4.2  T i m e  series
T i m e  series calculated by W AM a t  s ta t ion W E H  (30m dep th )  show a good ag reem en t  
with buoy data ,  especial ly for significant  wave height  ( l is)  values dur ing  high-wave 
events  (see Figure 3). T h e  small  m odu la t ions  visible in the  buoy  d a t a  a t  W E H  seem 
well reproduced by the  coupled version of WAM. However,  t h e  presence of some 
m od u la t i o n s  in the  uncoupled  run  results suggests  t h a t  p a r t  of  the  m odu la t ions  
in the  signal m us t  come from wind variabil ity.  T h e  bias for Hs in the  coupled 
and  uncoupled  resul ts  is -0.072m and -0.143m. respectively. T h e  in te rcompari son  
be tw een  resul ts  from coupled and uncoupled  models  show differences (coupled - 
uncoupled)  in Hs smaller  than  0.25m. with a  m ean  difference of 0 .02m. M odu la t io ns  
of  Tnio2 : corresponding to  the d o m in a n t  semid iurnal  t ide pe r iod ,  are qu i te  clear  
in the buoy d a t a  (Figure  3). T h e  im p o r ta n c e  of  including the h y d r o d y n a m ic  fields 
is highlighted qual i tat ively by the  good ag reem en t  between buoy  d a t a  and  WAM- 
coupled  resul ts  for T m o 2 - This  is not  direct ly  reflected in t h e  value for the  bias 
for T m o 2 , which were 0.032s for the  coupled and 0.034s for the  uncoupled  run,  
respect ively.  T h e  model  resul ts  themselves  showed differences up to  1.0s, with a  
m e a n  difference of -0.1.5s.
In the  more shallow A2B stat ion (11m d e p th ) ,  overes t imat ion  of  model  results 
wi th  respec t  to buoy d a t a  is observed,  bo th  in Hs and in T m 0 2 (F igure  4). A lthough  
it is possible to reduce  the  observed differences by tun ing  of the  empir ical  coefficient 
in the  b o t to m  friction te rm  (see Luo and  Monbal iu ,  1994, Luo et  al. ,  1996), this 
was no t  done since it  was not  considered im p o r t a n t  for the  scope of this  work. 
At  this  s ta t ion ,  the  bias  was -0.128m and  -0.116m for Hs and  -0.846s and  -0.965s 
for T m 02 in the  coupled and uncoupled  mode,  respectively.  T h e  im p o r ta n c e  of  t idal 
m odu la t ions  is again observed at  this  location and  it is qua l i ta t ive ly  well r eproduced  
in the  coupled run.  Differences be tween  coupled and uncoupled  model  results do 
no t  exceed 0.25m in Hs and 1.0s in T m 02, whereas  the  mean  difference in the  Hs 
and  Tn io 2 value was 0.014m and -0.217s, respectively.
4.3  S en s i t iv i ty  to  b o u n d a r y  c o n d i t io n s
In order  to  explore t h e  necessi ty to  run in coupled m ode  on a  coarse  grid in order  
to  supp ly  good bounda ry  condi tions for a  subsequen t  nested run,  the  appl icat ion  on 
t h e  local gr id  was run  once using bou n d a ry  condi t ions from the  coupled and  once 
from  the  uncoupled coarse grid run.  In bo th  cases, resul ts  showed clearly the  t idal  
m o d u la t i o n  effect on T m 0 2 (Figure 5a).  T h e  observed  differences a re  small  and  m os t  
of  the  t i m e  do not  exceed 5%, which seem to suggest  t h a t  running  the  coarse grid
appl icat ion  in coupled m ode ,  does not  have a  d r a m a t i c  influence on the  local  grid 
runs.  T id e - in d u ce d  m odu la t ions ,  a t  least  on the  space scales u sed  here,  are main ly  
a  local effect-.
8
B u o y  D ata 
B.C. U ne 
B.C. C ou
7 4+
6 +
E
4
++ +3
2 16 18 2 0  22  : 
e l a p s e d  d a y s  o f  f e b ru a ry  1993
24 26 28
STATION WEH (LOCAL GRID)
8
B u o y  D ata
4/20/60
4/20/20
7
6
EH 4
++
3
2 16 18 20 22 : 
e l a p s e d  d a y s  o f  f e b r u a r y  1993
24 26 28
F i g u r e  5 :  T i m e  s e r i e s  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 en s i t iv i ty  t o  f r e q u e n c y  o f  i n f o r m a t i o n  ex c h an g e
In a  coupled sys te m ,  an i m p o r t a n t  factor  to define is t h e  frequency of  in fo rm a ­
tion t ransfer  be tween  m ode l  components .  In this  work,  hyd rodynam ic  fields were 
u p d a te d  every 20 m in u t e s  ( s t a n d a rd  coupled mode) ,  which seems su i t ab le  for the  
tempora l  scale of t ide  variabi li ty.  In order  to invest iga te  t h e  sensi t ivi ty of t h e  model  
resul ts  to the frequency  of  in fo rmat ion exchange,  the  coupled model  on t h e  local 
grid was also run wi th  an  u p d a t e  of the  hyd ro d y n am ic  fields every 60min.  T h e  
resul ts  were c o m p a red  wi th  those of the  s t a n d a rd  run.  T h e  t ime  series of  T m 0 2 
presented in Figure 5b,  show differences smaller  t h a n  5%. Some detai ls  in t h e  m o d ­
ulat ion are missed. For  th i s  spa t ia l  scale and  with c u r ren t  and de p th  fields which 
vary only slowly in t i m e  a n d  space,  the main  var ia t ion in the spectral  per iods  are  
well reproduced by t h e  D opp le r  shift.  As one can observe from the  t im e  series, the  
t idal  m odu la t ions  r e s p o n d  m ain ly  to  the  semidiurna l  t ida l  cons t i tuent  M2. F u r ­
ther  decrease of t he  f r equency  of information exchange  will lead to increased loss of 
information.
have  a  l inear  ‘Ph i l l ip s ’ t e r m  in i ts  wind inpu t  source funct ion.  I t  was no t  invest igated 
in how far this is r espons ib le  for the  lack of growth in t h e  u n c o u p le d  version.
Couplei Couplei
93022:93022:
Uncoup]Uncoup]
93022.93022:
C o u p le d  -  S ta t io n  A2B
F i g u r e  7: 2D s p e c t r a  a t  s ta t ion A2B ca lculated by W A M  coupled  and uncoupled .  
Significant  wave he igh t  (Hs),  wind direct ion (dark line) and  d a t e  a re  always indi ­
cated.  For  coupled resu l ts ,  curren t  m ag n i tu d e  (U) and  cu r ren t  di rect ion ( l ight line) 
are  indica ted .  T h e  s a m e  con tour  labels were used in all figures ( 1 , 5 ,  and  from 10 
to 100% every 10% of  0 . l m 2/H z /d e g ) .
4.6 F u r t h e r  w o r k
T h e  fine grid b a t h y m e t r y  for the  Flemish  Coast  is much m ore  compl ica ted th a n  can 
be an t i c ipa ted  from F igu re  2. Many  sand  banks more or less paral lel  with t h e  coast  
are  present .  T h e i r  spa t i a l  scale is small  such t h a t  they  dissa.pear f rom the  local grid 
resolut ion.  A fine g r id  wave model  was nested in the local gr id  (see sect ion 3.1). For 
the  h y d r o d y n a m ic  fields l inear  interpola t ion from the CSM-grid  (equal  to  the  local 
wave model  gr id)  to  t h e  fine grid was used. Comparison  of t im e  series a t  A2B from 
the  local and  fine gr ids  showed only negligible differences. Th is  is not unexpec ted  
since all t h e  var iabi l i ty  in the cu rren t  field induced by the local b a t h y m e t r y  is 
not  rep resen ted  in t h e  in terpo lated  cu r ren t  field. If the  details  of the b a t h y m e t r y  
are no t  taken  in to  a c c o u n t  in the  calculat ion of the hydrodynam ica!  variables,  the 
directions of  the  waves and  the  cu r ren ts  become m ore  a n d  more  pe rpend icu la r  as
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of a  wave model  a n d  a  s to rm  surge m ode l  th rough  the  a tm osphe r ic  bou n d a ry  la,ver. 
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Sensitivity of wind wave simulation to coupling with a tide/surge model with application to the Southern North Sea

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