evolution of random directional wave and …...evolution of random directional wave and rogue/freak...
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Evolution of random directional wave and rogue/freak wave
occurrenceTakuji Waseda1,2, Takeshi Kinoshita1
Hitoshi Tamura2
1University of Tokyo, 2JAMSTEC
Motivation• Hypothesis
– Meteorological conditions and wave-current interaction preconditions the wave spectrum (dispersive and geometrical focusing)
– Non-resonant interaction kicks in when the wave spectrum satisfies certain conditions (BFI, Kurtosis, directional spreading)
• Observations do not necessarily support the hypothesis
• Study the evolution of realistic directional waves including impact of breaking dissipation
Methodology• Tank Experiment (comparison with weakly nonlinear
theory as needed)– Directional wave maker (32 segments)– Benjamin-Feir instability (uni-directional 2D)
• Initial instability• long-term evolution: impact of breaking dissipation• maximum wave height
– Random Directional Wave• long-term statistics (~4000 wave periods)• steepness, spectral bandwidth, directional spreading• special spectral shape• maximum wave height, impact of breaking dissipation
• Wave hindcasting/forecasting– Diagnose influence of wind and current on spectral shape
Summary of conclusion•• Maximum wave height of random directional waveMaximum wave height of random directional wave
– increases with mean steepness, similar to the unstable Stokes wave case
•• Spectral downshifting due to breaking dissipationSpectral downshifting due to breaking dissipation– spectral downshifting rate correlates well with the magnitude of
energy loss, including cases with various directionality•• Extreme wave statisticsExtreme wave statistics
– Kurtosis reduces as frequency bandwidth and directional spreading broadens
•• Spectral parameters in real oceanSpectral parameters in real ocean– Broader distribution of directional spreading than frequency
bandwidth
Introduction
• Safe navigation of ships near Japan– Group effort (U. Tokyo, NMRI, JAMSTEC) to
understand generation mechanism of freak wave, its impact on ship, and prediction/avoidance
– Tank experiment, numerical simulation, observation (radar, buoy)
• Tank experiment– Ship model test, radar observation– Generation mechanism
Maritime accidents near Japan
BolivarBolivar--marumaru (1969)(1969)(223m,33800t)
CaliforniaCalifornia--marumaru (1970)(1970) (218m,34000t)
OnomichiOnomichi--marumaru (1980)(1980)(226m,33800t)
Marine Accident Inquiry Agency Report•Bolivar-maru
- Hull damaged, lack of strength of ship•California-maru
- Two large waves merged, mixed sea•Onomichi-maru
- 20m wave, slamming
8.8℃ 1003hPa
Period 8.50s
4.17m Cal Cal
13.2 m/s
Cal Cal
Case of California-Maru (ERA40)
February 9, 1970
Significant Wave height, period 10m wind
Atm. temperature Sea level pressure
Background: Relating wave spectrum and freak wave occurrence
• Evolution of Kurtosis as a result of non-resonant wave-wave interaction, Benjamin-Feir-Index (Janssen 2003, Onorato et al. 2004)
• Kurtosis as a relevant parameter to modify wave height statistics (Mori & Janssen 2006)
• Reduction of freak wave occurrence in case of directional sea (Socquet-Juglard et al. 2005, Onorato et al. 2002)
BFI=ak/(df/f)
Freak Wave occurrence and wave statistics Janssen, Onorato, Mori, et al.
12332131,2,3,1,2,
123443213214,3,2,14,3,2,1
224
K12
)(116)(3)(
dkNNN
dktCosNNNTMxx
∫
∫+
ΔΔ−
+= −−+δω
ωηη
Kurtosis with higher order corrections
Free Wave
Bound Wave
2/
;3
240 ωω
επκΔ
== BFIBFI
)]81(exp[1 40freak κα +−−= NPProbability of freak wave occurence
Background: Instability of Stokes wave, long-term evolution and extension to a random directional wave
• Benjamin-Feir Instability– Stokes wave is unstable (Benjamin-Feir 1962,
McLean 1982 etc.), most unstable for uni-directional• Long-term evolution
– Recurrence (Lake & Yuen 1972), breaking wave and permanent downshifting (Melville 1982, Tulin & Waseda 1999)
• Instability of random directional wave– Instability limited to narrow directional spread, 35.26
degree (Alber & Saffman 1978)
Benjamin-Feir instability in a tank
2
)cos()cos()cos(
0
0
22220
0
πϕϕ
ωωωωωω
ϕωϕωωη
−=+
Δ−=Δ+=
++=
++++=
−+
−
+
−+
−−−+++
bbaa
tbtbta
c
c
Plunger motion
BFI1
21 /
00
0
00
=Δ
=ka
kaωωδ
Control Parameter
Waseda&Tulin JFM 1999
( ) 2/122 0.2 δδεβ −=Growth rate BF:
BFI1
21 /
00
0 =Δ
=kaωωδ 5.0>BFI
Long-term evolution of the BF Wave train – Impact of Breaking Dissipation
Tulin & Waseda JFM 19992
)cos()cos()cos(2222
0
0
πϕϕ
ϕωϕωωη
−=+
++=
++++=
−+
−+
−−−+++
bbaa
tbtbta
c
c
With breakingdownshift
Without breakingrecurrence
0
0.020.04
0.060.080.10.12
0.140.16
0.18
0 0.05 0.1 0.15 0.2 0.25 0.3ak
Hmax/L
Maximum Wave HeightComparison of experiment and weakly nonlinear solutions
Breaking threshold
Lines: Dysthe’s eqnOpen Circles: Zakharov 4-wave reduced eqnBlue Circles: Experiment at U. Tokyo OE Tank
Strength of the breaker
Tank ExperimentExp01: August 2006
result presented at the 9th wave workshop (poster)
Reduction of Kurtosis with directionalityStrong influence of breaking dissipation
Exp02: April 2007: wider range of spectral parameters, better control of the wave maker
Directional bandwidth
22
4
)(
)(
x
x
η
η
Ocean Engineering Tank Institute of Industrial Science, U. of Tokyo
Kinoshita Lab / Rheem Lab
50 m
5 m
10 m
Directional wave maker 32 plungers0.5~5 s wave period
Generation of the Directional Spectrum• JONSWAP
• Control Parameters&
Steepness Spectral Bandwidth
• Directional spreading– CosN– Hwang– Bimodal
3/1H⇔α pff /δγ ⇔
( ) ( )( )
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧ −
−−
−−
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛−= p
p
f
ff
pfffgfS
22
2
2exp4
542
45exp2 σγπα
( ) ( ) ( )θθ ,, fGfSfS ⋅=
( ) ( ) ( )θθ nnGG cos=
Hwang & Wang 2001
Bimodal
Generating random wave in a tank• DS (Double Summation) method
– Large number of waves; non- ergodic (amplitude nodes)
• SS (Single Summation) method– Ergodic (no nodes); reduced
number of modes
ωω
ωη
Δ=
+⋅−= ∑=
=
)(2
)cos(),,(1
nn
nn
Nn
nn
Sa
etatyx xkn
θωθω
ωη
ΔΔ=
+⋅−= ∑∑= =
),(2
)cos(),,(1 1
jiij
I
iijiji
J
jij
Sa
etatyx xk
1024 wavesDiscretization: conserve energy per binFrequency ~ 0.2 – 2.0 Hz1 hour record (~4000 wave period)No wind
-4
-5
High-frequency tail saturation spectrum: E(f)xf5
100.3
==
nγ
5 m
40 m
Smoothed periodogram512 degrees of freedom
Cross-correlation
Hwang distribution
Correlation32cm apartacross tank
Directional Bandwidth from G(θ) (degrees)
Cor
rela
tion
(32c
m a
part
)
( ) ( ) ( )θθ nnGG cos=
April 2007 Experimental Cases:for a fixed frequency spectrum; Mean Wave length L=1m
Directional Spreading (degrees)
Hs
Strongbreaker
No breaker
ak=0.08
ak=0.095
ak=0.06
( )( )LHak π243/1≡
Maximum wave height
0
0.02
0.04
0.06
0.080.1
0.120.14
0.16
0.18
0 0.05 0.1 0.15 0.2 0.25 0.3ak
Hmax
/L
Random directional wave
Benjamin-Feir wave train
Steepness ak (initial, mean)
Hm
ax/L
p
Stoke’s limit
Spectral evolution (typical cases)
( ) ( )( )
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧ −
−−
−−
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛−= p
p
f
ff
pfffgfS
22
2
2exp4
542
45exp2 σγπα
( ) ( ) ( )θθ ,, fGfSfS ⋅=
( ) ( ) ( )θθ nnGG cos=
4.15Hs ;125 ;0.3 === nγ 00.5Hs ;125 ;0.3 === nγ
BF
ak
fp(d
evel
oped
)/fp(
initi
al)
Spectral downshifting of random directional wave
Energy Loss
ak
ΔE
Downshifting due to breaking dissipation
jj
jjjj
jT
bTj
jT
MgMcEMM
DEEE
ω===
−==
∑
∑
where
and &
T
j jj
j j
j jjp E
E
E
E ∑∑
∑==
ωωω
( ) bT
pbTp
T
pp DE
DMcEdt
dΓ−≈−=
ωωω &
ratedecay energy ratedecay frequency 1
≡⎟⎟⎠
⎞⎜⎜⎝
⎛=Γ
T
b
p
p
ED
dtd
ωω
Tulin&Waseda 1999
Downshifting parameter Γ estimated for strong breaker cases
Γ
T&W range
Directional Spreading (degrees)
ratedecay energy ratedecay frequency 1
≡⎟⎟⎠
⎞⎜⎜⎝
⎛=Γ
T
b
p
p
ED
dtd
ωω
Evolution of Kurtosis with fetch
22
4
)(
)(
x
x
η
η
Initial value Developed stage
1250.3
==
nγ
2/
;3
240 ωω
επκΔ
== BFIBFILine:
BFI
22
4
)(
)(
x
x
η
η
BFI vs. kurtosis
Kurtosis and spectral bandwidth (γ) Hwang directional distribution
γ 30 10 5
3 1
broadnarrow
22
4
)(
)(
x
x
η
η
Non-resonantInteraction
Developed stage
Hs=4.15Hs=5.00
□Initial value○Developed stage
Directionality large
Freak WaveOccurrenceIncreases
Uni-directional
Non-resonantInteraction
22
4
)(
)(
x
x
η
η
n 250 50 25 10 5 3 1
γ =3
Alber & Saffman limit
High-resolution wave-current coupled model Wave-JCOPE, Snl(SRIAM)
γ 40 5 3 2
Tamura, Waseda, Miyazawa & Komatsu, 11/16 presentation
A&S limit
n 125 25 10 5 1Degree 7.7 16.3 30
Wave parameter distributionNon-resonantInteraction
Non-resonantInteraction
γ 40 5 3 2
n 125 25 10 5 1
Frequency Bandwidth
Directional Distribution
Concluding remarks• Evolution of random directional wave in the laboratory tank
was studied and revealed that the non-resonant interaction becomes significant as the spectrum narrows, including cases with energetic breakers
• As a result of breaking dissipation, the spectrum downshifts suggesting that the combination of non-resonant interaction and wave breaking is the relevant downshifting mechanism for narrow spectrum
• Analysis of high-resolution wind-wave coupled model suggests that favorable condition for freak wave occurrence, i.e. narrow frequency bandwidth and narrow directional spread, is rare.
Hypothesis: Freak wave is an expected realization for an abnormal wave condition when the wave spectrum is narrow