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Ship Stability Study in the Coastal Region: New Coastal
Wave Model Coupled with a Dynamic Stability Model
R.-Q. Lin, W. Thomas (Naval Surface Warfare Center, Carclerock Division, USA)
ABSTRACT
Dynamic Stability assessments of Navy ships can
readily be performed for deep ocean regions. Recent em-
phasis on joint operations in littoral regions has identified
the need to include the effects of coastal wave dynamics in
the stability assessments because the waves are significantly
different from deep ocean waves. In coastal regions, the
major source function, nonlinear wave-wave interactions
significantly increase as water depth decreases, significantly
reducing the fetch limit, causing the wind generated waves
to grow faster and steeper than in the deep ocean. Coast
region wave dynamics are further complicated by the very
low frequency waves, such as, edge waves, bottom topo-
graphic waves, as well as solitons etc. Wave-current inter-
actions become very important in littoral regions, and it
can generate coastal-trapped waves and seamount trapped
waves.
Waves in coastal regions can grow faster and
steeper than in the deep ocean. This raises the possibility
that dynamic capsize is more likely in coastal waters in
comparison to deep-water. In this paper we will conduct a
comparative assessment between deep ocean capsize pre-
dictions and coastal waters capsize predictions. Wave pre-
dictions using the New Coastal Wave Model (tin et al)
will be coupled with the FREDYN dynamic stability model
(De Kat et al) in the capsize assessment of a frigate-type
vessel. The New Coastal Wave Model won the competi-
tion in both accuracy and efficiency in the international
conference "Base Enhancement Wave Prediction" in 1998.
I. INTRODUCTION
One of the most critical factors for dynamic sta-
bility assessments of navy ships in the coastal region is the
correct estimation of the environment including wind,
waves, currents, and storm surge events. Hull form geom-
etry, load configuration, heading and speed determine a
ship's dynamic stability performance. In this study, we will
use a naval frigate to highlight differences in dynamic cap-
size behavior during the transition from the deep ocean to
the coastal region in the same wind environment. This na-
val frigate is displayed in Figure 1.
Figure 1. Isometric sketch of Naval Frigate.
A growth of the surface wave is based on the wind,
wave breaking, wave-wave interactions, and wave-current
interactions. The last two mechanisms are especially im-
portant in shallow water. The nonlinear wave-wave inter-
actions in shallow water are order of magnitude greater
than those in deep water (tin and Perrie, 1999~. Wave
amplitudes can substantially increased when the high fre-
quency waves downshift to the lower frequency though
the wave-wave interactions. Therefore, the wind-waves
grow faster and steeper in shallow water than those in deep
water. In addition, wave-current interactions increase with
u
c, increases, where u is current velocity and cg is group
g
velocity. Current velocity, u usually increases with water
depth decreases due to the continuity, but group velocity
decreases with water depth as cg=
1 |g tanh kh | gk h
~ I + 11 · 2 ~ . Therefore the
2 ~ g v tanhkh sech kh '
wave-current interactions are far more important in shal-
low water than those in deep water. Often the coastal
trapped waves can be observed when the water is shallow
and the water depth varies rapidly (tin and Huang, 19964.
Therefore, the presence of larger and steeper waves in the
littoral regions, can expose a ship to a higher level of cap-
size risk in comparison to deep water. Furthermore, even
in deep ocean, if the bottom topography rapidly varies,
and the free wave frequency is similar to the forced fre-
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Representative terms from entire chapter:
dynamic stability
quency, resonant seamount trapped waves may also cause
a ship to capsize.
In this study, we couple the New Coastal Wave
Model (tin and Huang, 1996a and b, Lin and Perrie, 1997;
Lin and Perrie, 1999) with time domain dynamic stability
simulations to predict the implications of rapid changes in
bottom topography in two scenarios:
1) Gulf of St. Lawrence (east coast of Canada);
2) Fieberling Guyot in the eastern North Pacific
Ocean (32°25' N. 127°47' W).
Unlike the State-of-Art Wave Model (WAM,
SWAM) which uses parameterization to account for the
wave-current and wave-wave interactions, the New Coastal
Wave Model is physics based on the first principle, which
is suitable for both deep ocean and shallow water. FREDYN
is a time domain quasi-nonlinear dynamic stability model
that can simulate the capsize and broaching of intact and
damage ships.
Dynamic stability assessments of surface ships
usually rely on the standard equilibrium deep sea spectra,
such as JONSWAP, Pierson-Moskowitz, Neumann, Fisher
and Roll, Darbyshire gravity wave spectra, etc. All these
spectra formulations are proportional to W-5, where w is
intrinsic frequency. These formulations were based on the
unit analysis of Phillips ~ 1958~. Later Zakharov and
Filonenko (1966) applied a conformal transformation to
obtain the Kolmogorov solution and demonstrated that the
equilibrium of deep sea spectra are proportional to UW-4,
where u is wind speed. These were confirmed with obser-
vations (e.g. Toba, 1973; Donelan et al., 1985; as well as
Phillips, 1985~. However, variations of the equilibrium
range spectrum of wind generated waves in finite water
are much complicated. The equilibrium energy spectra are
proportional to the range of W-4 to we. In this study, the
ship stability analysis will be tested by the later spectra
formulations.
II. A COUPLED SURFACE WAVE MODEL AND
SHIP DYNAMIC STABILITY MODEL
A. Surface Wave Model:
New Coastal wave model (tin and Huang, 1996b) is
based on the action conservation equation, which truly
calculates the wave-current interactions. The action
conservation equation follows: is
aA + J(Cg~A) +cOs lit 3(Cg; cos MA)+ 3(C0A)+ 3(CmA)
at 37 3˘ 30 3m
= Sin +Sd5 +SHl'
where A is action density, ~ is longitude, ~ is latitude, ~
wave propagated direction angle, ~ is intrinsic fre-
quency, t is time, Sin, So, and Sn1 are wind input function,
wave dispersion, and nonlinear wave-wave interactions,
respectively.
The terms in left-hand side of the Equation (1)
are based on the nonlinear kinematics (tin and Huang,
1996b). This is because the normalized wave steepness,
3 + tanh2(Kh)
£ (ak 4 tanh3
The hull form is modeled from the keel to the edge of the
main deck and the motions are solved in the time domain
for six degrees of freedom. The model consists of a non-
linear strip theory approach, where linear and non-linear
potential flow forces are added to maneuvering and vis-
cous drag forces. The architecture of FREDYN is based
on the De Kat and Pauling model and in essence, superim-
poses the relevant force contributions in the equations of
motion as follows tdeKat and deKat1:
INERTIAL FORCE = FROUDE-KRYLOV FORCES
+ WAVE DIFFRACTION FORCES
+ WAVE RADIATION FORCES
+ VISCOUS FORCES
+ HULL RESISTANCE FORCES
+ PROPELLER FORCES
+ RUDDER AND SKEG FORCES
+ WIND FORCES
+ FORCES DUE TO INTERNAL LIQUID
OR DAMAGE FLUID
The Froude-Krylov forces are evaluated up to the
instantaneous free surface and include hydrostatic effects.
Linear theory is used in the time domain to estimate the
diffraction and radiation forces where a correction is
made to the convolution integrals to account for large am-
plitude motions. Viscous effects comprise roll damping
due to hull and bilge keels, wave-induced drag due to or-
bital velocities, and calm water-maneuvering forces. Vis-
cous drag due to cross flow velocities is estimated empiri-
cally, using section-dependent drag coefficients derived
from segmented model test results. Propeller and rudder
interaction is also modeled, including the effect of orbital
velocities.
FREDYN has the capability of simulating the
motion behavior of an intact or damaged monohull vessel
in extreme waves and wind. The ship motions, including
capsizing, broaching, and surfriding events, can be dis-
played in real time animation on the screen of a Pentium
III computer.
III. CAPSIZE PREDICTIONS:
Capsize assessment methodologies often focus on
dynamic capsize predictions in deepwater. The need to
incorporate hazard scenarios in capsize assessments has
recently been identified by LAlman et al. 19991. A logical
extension to capsize assessment is littoral regions. As pre-
viously stated, we will investigate two scenarios represent-
ing two coastal cases:
1) Gulf of St Lawrence (east coast of Canada);
2) Guyot in the eastern North Pacific Ocean com
pared with the deep water case.
A. Case I: Saint Lawrence Gulf
The first scenario involves the transit of a naval
frigate in the Gulf of Saint Lawrence southeastward from
the mouth of the Saint Lawrence River Island to the Atlan-
tic Ocean at a speed of 15 knots. In this limited-fetch re-
gion, 40-knot winds have been blowing from the North
east over the past 48 hours.
The bottom topography for the Gulf of Saint
Lawrence is displayed in Figure 2.. In this figure, the yel-
low color is land, and the contour lines in ocean are at 100
A
meters intervals. Near Iles de la Madeleine is a shallow
water area, where the 400-meter water depth from 400
meters rapidly decreases to less than 100 meters, from north
of Iles de la Madeleine.
A semi diurnal tidal current (period is 12 hours)
dominates this region. Figure 3 shows the observed cur-
rent distribution in the highest tidal current case (Bedford
Oceanography Institute of Canada). The wind driving cur-
rent (NE) is greater than the tidal current (E-W, 0.2 meters),
but the tidal current and its associated water depth varia-
tion still significantly effect the wave height and wave pe-
riods. The tidal current contributions are about 30% of the
total forcing (tin et al, 2000~.
Figure 4 illustrates significant wave heights, simu-
lated by New Coastal Wave Model (tin et al, 1996a, b,
1997, and 1999~. The maximum wave height is 11.8 m,
which occurs along the north boundary of Iles de la
Madeleine, and the contour line is 2 meters. The wave pe-
riod distribution (New Coastal Wave Model by Lin et al)
is displayed in Figure 5. The maximum wave period of
16.5 second appears along the north boundary of Iles de
la Madeleine, and the contour line is 2 second. However,
the most probable period is about 12 second. For the same
wind conditions, the wave height is about 8.7 meters and
the wave period is about 12 second in deep water. These
phenomenon along the north boundary of Iles de la
Madeleine is called coastal trapped wave (tin et al, 20004.
The fate of the imaginary naval frigate is outlined
in the track in Figure 6. In the deepwater area, at Location
"A", waves are coming predominantly from 30 degrees
forward of the port beam with roll angles approaching 40
degrees. Heading control is challenging, with occasional
broaching, yet the ship is able to handle the situation. (See
Figure 7.) As the ship enters the coastal trapped wave
area, steep 11.8-meter waves are encountered from the port
b? ~y
- ~ ~ ~
s
As ~ N sat
.
. ~ ~~ i-- ~ -- -my- ~ J'-- t:
.. . ... '^ ~ . .. ..
/~S ~ ..~...._
7~) ~
;~ s 1
~ ~ _
60 'I 55 N.Y
~ IT by ~3 ~~s~ ~ ~ ~~ ~y
Figure 2. Bottom topography of St Lawrence River and
Gulf in Canada. Contour line is 100 meters. Intended
transit route of ship included.
cast
b . . , .... , .. ........ .. .... , . ~ .
.. .3 .-. =. mu-
..... . . .~ ~ ~i; y i
~ ~ fig 3.1~ ~ ~; Bray ~ ~ If
r 1 ~ ~ > ~ / PJ1 f ./ i,
is } f 3 i- Fox
. . .~ ::~/s ~ ~ i s; i ,s ~7 1/
· ~! j ~ ~ ~ ~ ~ ~ '' ~ ~ ~ ~ ~t Or-
i~ ~ i ,,, ,,,, , , In ,! !~l,.~if~j~t/~;? I' ' ' ' - ' ' ' ' ' ' - - S
my. ,, · f~ ' ''''1
~ . . - - ~ /~ _ //~/lf)~ //p,f ,/if! ~ ' ' ' ' ' - ' ' ' ' ' - ' ' S
/' ' ' ~~ - ' - - ' ' ' i~-~~A~/ //fI47 1 iiF/~/ ~
~ Fj ~ ~7~7~77~~,~.i,4 ~,v~ `0 / Ala; i{~/ ri
; ~ - 1~ ~~#~ 1~ ~ CAY ~ Fly I
f - F - i; ~ ~~ ; ~ ~ f -at ~
., ,.~. ..... , ~ .
,, , ~ . , .? hi, . ~ . ~ i
~~ off
70 ~
. . . . . . . . . . . . . . .
u~ // s ' - ~ (~˘
~ {;J ~ ~ ~ :14 ~ :E l: :E :L
Figure 3. St Lawrence River and Gulf current
distribution in the highest current case.
(Bedford Oceanography Institute of Canada)
45 ~
....' . ~ ~ . ~ . . ~ ~ ~ ~
::':::':.' :.:::..:~-~:~i::':.':,,'.':
,~/ ~ ~ s ~ ~ I/
s a-'
3 ~ of - ~ i'
3 ;/ hi/ ~ - - ~ - - . - . ....
~ ''a'
s ' Yf~N
s )/
r -
i
3 ~
-,,
7~
a' ~
~\~- _~7 j/i'_~ i2~0 ~
.... ~ . ..~- .. I..
40 ~ ~ ..,
~ ~ Fit ~ ti
use ~
NN ~iviE PERIOD
Figure 5. New coastal model wave period distribution
for St. Lawrence River and Gulf. (Lin, 2000)
~ s
....3. 3
HA
-.:. :'~'..: :;'/:-.:::/'.:/.'
--- '~ .-.f... Jo
~ iF ~ If
;~ :
; r ~ ~ ~ / ~ 3
fit 3 ~ i~ ~ ~
~ F ~ ;
e ~
JO
1 ~ · ~ . . ~ . . . . . . . . . . . . ~ .
; ~ a;
-f me/ !7 ~ ; , ,, , ~ = ~
70 ~ 60 ~ 55 ~
WAVE HEIGHT
Figure 4. New coastal wave model significant wave
height for St. Lawrence River and Gulf. (Lin, 2000)
5: ~ ~
Ha: ~
. . . . ~ . . ..... ~ . T , - ~~ A ~~ ..
..... ~ , : Location ~ ~7
. . . - - - F - - , , . I . . .~. .
.......... : . ~.::,,,,,/
· ~~ - /- -t ~ l .J
! ~ ~ ~ ~ ~ i, . . .
., . ., ~ . / : ~
~.~ ~ ~
....... F§§ 9_
........
i: .....
:........
. ~ ........
. . ~ .... .
. ~ ... ....
~ .
. . . . . . .
. ...7 . ~
~ ' '''~''~
/if,,, ,, ~i- ~~N,~',',,'-..'''~
~ :~ ~ ~ , .~. ~
i~ ~ ~ · ~ ~ ~ . 1 . . . . ~ ,, , ~
~ ~ \ N - ~ 1> ' ~ i' 'I '- '' ;'; '
... . ~ W~ / t ~ ~
. , . . . ... ~ Be ~ ~ it, ~ . ~ ....
60 W 35 ~
WAVE HE tG'~T
Figure 6. Track of naval frigate with capsize location.
1n ~
- ~
-10 J
50
30
-10
-30
-50
20
15
10
e, 5
o
5
-10
-15
-20
--------77~"'''
lo - AN clear - ~A~IIAA~A~A^AAt~AA A,
50 100 150 200 V 250 300 350 400 450
~ ~ ~ A ~ A HA A . ~ fl ~ 1 ~~ . \~ ~ ~ ~ 4~ A A ~ _ Q A n ~ . nil . . . ~ n . ~
~V~4U~VU~- US - q2~U~U~o~V~3~50V4~O0U 450 V too
90
60
V 11 ~ O' _
\~ =-30 )
-60
-90
/
9~ 30
an
-6
-8
On
20 -
~o 15- ~1 ~~
50 1no 150 200 250 300 350 400 450 5no
Figure 7. Naval frigate motion prediction at
Location "A".
Figure 8. Capsize in region of trapped coastal waves.
beam. The ship capsizes after the passage of a sleep 12- ~ N2 = -dZ ~ P is density of the waler), W is width of
meter wave. (See Figure 8.)
B. Case II: Pacific Ocean Seamount
In the eastern North Pacific Ocean stands a sea-
mount, which extends above the 5000-meter depth ocean
floor. (See Figure 9.) The Fieberling Guyot is 4500 meters
in height and 40 km in width with the top 500 meters be-
low the surface. To the typical navigator aboard ship, the
nearly 500 meter clearance between the underside of a ship
and the top of the seamount would provide no basis for
safety concerns, as the ship would not be exposed a ground-
ing hazard. There would be no hesitation in driving a ship
right over the seamount if it was on the ship's transit route.
Haidvogal et al (1991) pointed out that due to the
free wave frequency of Fieberling Guyot is equivalent to
the forced frequency, significant seamount trapped waves
occur. The free wave frequency depends on the Burger
Number, Br = f h ' where N is buoyancy force,
the seamount, h is the height of the seamount, andfc is the
Coriolis force. The forced frequency is tidal current in this
case. The up welling and down welling are more than 5
times greater than the far field.
In this scenario, a 20 m/s wind blows from the
west for 48 hours continuously. The surface current distri-
bution surrounding Fieberling Guyot is displayed in the
vector diagram to the left of the corresponding time series
plot of current on top of the seamount. (See Figure 10.)
The tidal current is 0.2 m/s in far field, but the current is
1.05 meters on top of the seamount. Figure 10 is predicted
by the Coastal Current Model (Haidvogal et al, l991~. Fig-
ure 11 shows the significant wave height distribution in
this area (New Coastal Wave Model, Lin et al.), with con-
tour lines of 0.5 meters. The significant wave height in far
field (deep water) is about 9.05 meters and on top of the
Fieberling Guyot is about 11.48 meters. The far field
energy density spectrum is displayed in Figure 12. The
wave energy sharply increases on top of the seamount, as
displayed in Figure 13. The peak amplitude of the energy
Figure 10. Surface currents present at the top of the seamount.
~ `~}:pUt: ~ surface c:~{P~~t
In- ~~ (rom currerlt '~l
(:~e current ,~ generated
Me to ~ Tic ti~
stri bins the se-err oust)
Figure 9. Bottom topography of Fieberling Guyot.
u`~o, t-~' a~n's~ ;< bare ~~,
1. . _'. __ ~ . . ~ ~ ~ - . . . 1 - ~ ~ ~
10 12 14
:x
~ IME t OR'? ~
density spectrum on the top of the seamount is 1.7 times of
those in the far field (New Coastal Wave Model by Lin
et al.~.
In the FREDYN simulation, the frigate ap-
proaches the seamount from the northwest at a speed of 15
knots. (See Figure 14.) The frigate experiences roll ampli-
tudes has high as 45 degrees with several broaching events,
producing yaw amplitudes as high as 15 degrees. (See
Figure 15.~. As the frigate transits over the seamount, in
stern quartering seas, the presence of larger, (11.48 meter)
and steeper waves induce a capsize due to loss of stability
on the wave crest, as shown in Figure 16.
IV. CONCLUSIONS:
Rapid changes in bottom contours can signifi-
cantly influence wave fields, as illustrated in the two sce-
narios described in this paper. Even when the water is fairly
deep, changes in water depth, coupled with surface cur-
rents and wind interactions can produce waves locally
which can be substantially larger and steeper, increasing
the likelihood of capsize for some ships. Because in the
coastal region the current significantly increases, the wave-
current interactions become very important. The coast
trapped waves and seamount trapped waves often occur
and can cause ships to capsize. Furthermore, the nonlinear
wave-wave interactions are order of magnitude increase
with water depth decreases. A wave growth needs a much
less fetch in shallow water than those do in deep water.
V. ACKNOWLEDGEMENT
This work is sunnorted TV grants from the ()ffice
2$e
440
-try - -A =--
of Naval Research under ILIR program though the David
Taylor Model Basin, Naval Surface Warfare Center,
Carderock Division.
60
NYKE HEIGHT
.~3
. ~
. ~ ~ :\J"
O~ Q 300~
-~m~~~
-
i, ~
I_
~ ~~ .
~,`\~\~n'
\ \ ~
Figure 11. Coast wave model height prediction at the
Fieberling Guyot.
OFNST TY SPECTRUM
.~r~ r----~----r---~--- ~ ~ ~ T lL T ~ r--~-----l---~r~-~ ~r~r~r~r
-
~ . ~
-
~, I A ; ~. ~
.~6 .~S .~0 .12 .14 .lb .~S .~0 .22 .24 .~6 .~S 30 ~2
If, HZ]' York, M*~Rh.~lANiHZ]
Figure 12. Deep water farfield wave spectra.
240
160:
14et
. 120 Y
100
4$ .
20
E NS ~ ~ ~ SPEC ~ FIRM
- r ~ ~ aim ~ T ~ T ~ t~ ~ AT
~ Deepwater Approach to Seamount
Ill ~ ~ ~ ::
~ . 02 . 04 . 06
it,
IS TO .~2 ~4 ~ 16 .~6 ~0 ~2 ~4 ^~6 .~8 .30 .~2
x
= Hz], YAMS *1
Figure 13. Energy spectrum at the top of the seamount.
[~ted track of ship ~ ~ Deep water location ~
STAVE HEIGHT /
.o
::m
~ .
20 ~
~ 10 ~
~ s -
c~
Figure 14. Track of naval frigate with capsize at the
seamount.
Figure 15. Motion predictions for naval frigate
approaching seamount.
10 Capsize during transit over Seamount
, s ~
am\
Y ~ ~ _
Capsize at
Seamount
w"
_
>
~ ', ~. ~ ~ :
~~ ~~ s
0.-
O s 10 15 20 25
10
in. 1 ~ 25
Figure 16. Capsize off naval frigate passing over the
seamount.
VI. REFERENCES
Alman, P. R., Minnick, P. V., Sheinberg, R. Thomas
III, W. L., "Dynamic Capsize Vulnerability: Reducing
the Hidden Operational Risk", SNAME Transactions,
1999, Vol. 107. In Print.
Donelan, M. A., Hamilton, ,1. and Hui, W. H., "Direc-
tional Spectra of Wind-generated Waves". Phil. Trans.
R. Soc. Lends A. Vol. 315, 1985, pp. 509-562.
De Kat, J. O., R. Brouwer, K., A. McTaggart, W. L.
Thomas, "Intact Ship Survivability in Extreme Waves:
New Criteria from a Research and Navy Perspective".
Fifth International Conference on Stability of Ships and
Lin, R.-Q. and W. Perrie, "A New Coastal Wave
Model, Part III. Nonlinear Wave-wave Interaction." J. of
Physical Oceano., Vol. 27, 1997, pp. 1813-1826.
Lin, R.-Q. and W. Perrie, "Wave-wave Interactions in
Finite Depth Water". J. of Geophv. Res., Vol. 104, No
C5, 1999, pp. 11193-11213.
Lin, R.-Q., W. Perrie, and B. Bash, "Sea State Fore-
casting in the St. Lawrence River and Gluf: Preliminary
Operational Implementation." Technical Report of
Bedford Oceanography Institute. 2000, In press.
Lin, R.-Q. and W. Kuang, "A Pesudo-spectrum Wave
Model." To be submitted J. of Physical Ocean.
Perrie, W. and R.-Q. Lin, " Relating Nonlinear Energy
Cascades to Wind Input and Wave Breaking Dissipa-
tion". Nonlinear Ocean Wave Advances in Fluid
Mechanics. Computational Mechanics Publications.
1997, pp. 61-88.
Phillips, O. M., "The Equilibrium Range in the Spec-
trum of Wind-generated Waves." J. of Fluid Mech., Vol.
4, 1958, pp. 426-434.
Phillips, O. M., "Spectral and Statistical Properties of
the Equilibrium Range in Wind-Generated Gravity
Waves." J. of Fluid Mech., Vol. 156, 1985, pp. 505-531.
Toba, Y., "Local Balance in Air-Sea Boundary pro-
cesses. III. On the Spectrum of Wind-Wave." J. of
Oceano. Soc. Japan Vol. 29, 1973, 209-220.
Zakharov, V. E. and Filonenko, N. N., " The Energy
Spectrum for Stochastic Oscillation of a Fluid's Sur-
face". Dokladv Akad. Nauk, 170, 1968, pp. 1292-1295.
DISCUSSION
J. de Kilt
Maritime Research h~stit te
I would like to commend the mthors for presenting c
paper that for She fi st time couples c wind-wave-
cunent model to c large amplitude, time domain
simulation tool for capsize assessment purposes
As the mthors point out c orrem 1:, wave conditions
c m become more onerous in shallow water or in the
presence of current The way the date are presemed
e g in figures 11, 12 Ed 13 provides c comparison of
the statistical wave properties, including spectral
densities To assess the difference in probabilities of
occurrence of critical .. a ~ e. betw en the deep water
Ed shallow water case, it would be of interest to
compare the joint probability density functions of
wave height Ed period of She individual waves
From such plots it shouldbecome apparent to what
extent waves with critical length Ed teepness are
mme I kely to occur in non-deep water conditions
In the case of the seamount She Increased surface
(tidal) current velocity has c signif c mt influence on
wave teepening effects This effect Held be
highest when the current opposes She wind Ed wave
dinection; presumably for this study the tidal cunent
runs in east we t du ection et Nat location, although it
isnotq iteclearfiomfigure 10 Iffhecunentw re
to run et m oblique Ogle with respect to She
incoming wave system, His would result m short-
crested wa ves clove the seam ount
The program FRdDYN does not account for wave-
cunent Interaction Ed uses She principle of Imear
superposition of wave components in deep water
conditions Would the mthors recommend further
erJkmcement to the simulation model to account for
shallow water or certain nonlinear .. 3 ve effects?
Lastly, the predicted cspsi es show are associated
with one wave realization for each case; have He
mthors pe formed simulations in different wave
realization conditions for the same critical see state,
which also resulted in capsize events?
AUTHOR'S REPLY
are following:
Thmk vou for : our comments The replies
1) The t sditiorLtl critical conditions for She non-
deep water is t mhkh < I, wheee k is wave
mmmber, h is water depth However, in f is study,
w introduce c new concept of non-deep water
conditions: the redo mt mren It waves h msfer
subst mticl energy from the bottom to the see
surface Ed surface .. a ~ es are strongly effected
by this bottom energy The forcing depends on
the bottom topography, buoyancy force (water
density varies with water depth), Coriolis truce,
es w 11 es extermtl forcing, such es tidal cunent
The haditiork~l deep water limit is
signific mtly Educed under the new concept of
the non-deep conditions For example, the
Fieberling GO or se amount trapped .. ares are
effected by the bottom, which is 500 meters
below He see surface
2) in the case of the seamount, the increcsmg
intermtlwaveshave c significant influence on
wave steepening, es w ll es the su face tidal
current Therefore, the effect should be highest
when She tide is hi -h t Ed the tidal current is
equal zero in far field, but on She top of the
seamount, Here are sig if ic mt cunents, which is
due to the reso mt imtemal wave effects es
show d in Figune 1 0c The cunent on top of She
se tmount in Fig 1 0c aim ost propagated in She
same direction of the sw 11, so w should not
observed ah on ~ re at e d .. as e s
3) The current usually signific mt increasing with
water depth decrecsmg This is due to She
continuity principles, es w ll es nonlinear wa ve-
cunent Interactions Sometimes, the wave-
cunent ma. cant She cunent signify mt
increasing, for example, the se tmount tmpped
wa ve case in this study, especially during She
-- m surges The huge current may du ect effect
the ship motion, so ffw c m consider the duect
current effects in Fredy, the simulations may be
more accurate
4) Yes, w used different wave realization
conditions for the same critical see state, which
