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