Evolution of random directional wave and rogue/freak wave occurrence

Transcription

1 Evolution of random directional wave and rogue/freak wave occurrence Takui Waseda 1,, Takeshi Kinoshita 1 Hitoshi Tamura 1 University of Tokyo, JAMSTEC

2 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

3 Methodology Tank Experiment (comparison with weakly nonlinear theory as needed) Directional wave maker (3 segments) Benamin-Feir instability (uni-directional D) 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

4 Summary of conclusion Maximum wave height of random directional wave increases with mean steepness, similar to the unstable Stokes wave case Spectral downshifting due to breaking dissipation spectral downshifting rate correlates well with the magnitude of energy loss, including cases with various directionality Extreme wave statistics Kurtosis reduces as frequency bandwidth and directional spreading broadens Spectral parameters in real ocean Broader distribution of directional spreading than frequency bandwidth

5 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

6 Maritime accidents near Japan California-maru (1970) (18m,34000t) Bolivar-maru (1969) (3m,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 - 0m wave, slamming Onomichi-maru maru (1980) (6m,33800t)

7 Case of California-Maru (ERA40) Significant Wave height, period 10m wind 13. m/s 4.17m Cal Cal Period 8.50s Atm. temperature Sea level pressure Cal 8.8 Cal 1003hPa February 9, 1970

8 Background: Relating wave spectrum and freak wave occurrence Evolution of Kurtosis as a result of non-resonant wave-wave interaction, Benamin-Feir-Index (Janssen 003, Onorato et al. 004) Kurtosis as a relevant parameter to modify wave height statistics (Mori & Janssen 006) Reduction of freak wave occurrence in case of directional sea (Socquet-Juglard et al. 005, Onorato et al. 00) BFIak/(df/f)

9 Freak Wave occurrence and wave statistics Janssen, Onorato, Mori, et al. Kurtosis with higher order corrections η( x) 4 3 η( x) 16 M1,,3,4T1,,3,4 N1NN3 δ K 1,,3,1,,3 N 1 N N 3 dk 13 1 Cos( Δt) Δ dk 134 Free Wave Bound Wave Probability of freak wave occurence P freak 1 exp[ α N(1 8κ 40)] κ 40 π BFI ; BFI 3 ε Δ /

10 Background: Instability of Stokes wave, long-term evolution and extension to a random directional wave Benamin-Feir Instability Stokes wave is unstable (Benamin-Feir 196, McLean 198 etc.), most unstable for uni-directional Long-term evolution Recurrence (Lake & Yuen 197), breaking wave and permanent downshifting (Melville 198, Tulin & Waseda 1999) Instability of random directional wave Instability limited to narrow directional spread, 35.6 degree (Alber & Saffman 1978)

11 Benamin Feir instability in a tank ) cos( ) cos( ) cos( π ϕ ϕ ϕ ϕ η Δ Δ b b a a t b t b t a c c Plunger motion BFI 1 1 / Δ k a k a δ Control Parameter Waseda&Tulin JFM 1999 ( ) 1/.0 δ δ ε β Growth rate BF: BFI 1 1 / Δ k a δ 5 > 0. BFI

12 Long-term evolution of the BF Wave train Impact of Breaking Dissipation Tulin & Waseda JFM 1999 ) cos( ) cos( ) cos( 0 0 π ϕ ϕ ϕ ϕ η b b a a t b t b t a c c With breaking downshift Without breaking recurrence

13 Maximum Wave Height Comparison of experiment and weakly nonlinear solutions 0.18 Hmax/ L Strength of the breaker Breaking threshold ak Lines: Dysthe s eqn Open Circles: Zakharov 4-wave reduced eqn Blue Circles: Experiment at U. Tokyo OE Tank

14 Tank Experiment Exp01: August 006 result presented at the 9 th wave workshop (poster) Reduction of Kurtosis with directionality Strong influence of breaking dissipation η( x) η( x) 4 Exp0: April 007: Directional bandwidth wider range of spectral parameters, better control of the wave maker

15 Ocean Engineering Tank Institute of Industrial Science, U. of Tokyo Kinoshita Lab / Rheem Lab 10 m 5 m 50 m Directional wave maker 3 plungers 0.5~5 s wave period

16 Generation of the Directional Spectrum JONSWAP S S ( f ) ( ) ( ) ( f ) α g ( π ) Control Parameters α H 1/3 Bandwidth & Steepness Directional spreading n G( θ ) G( n) cos ( θ ) CosN Hwang Bimodal, θ S f G f,θ 4 5 f exp 5 4 f f p 4 γ exp σ γ δ f / ( f f ) f p p p Spectral Hwang & Wang 001 f Bimodal

17 Generating random wave in a tank DS (Double Summation) method Large number of waves; nonergodic (amplitude nodes) η( x, y, t) a i SS (Single Summation) method Ergodic (no nodes); reduced number of modes η ( x, y, t) an cos( nt n 1 a S( ) Δ n S(, θ ) ΔΔθ n i 1 1 i n I N J a i cos( t k i k n i x e ) x e n i ) 104 waves Discretization: conserve energy per bin Frequency ~ 0..0 Hz 1 hour record (~4000 wave period) No wind -4-5

18 High-frequency tail saturation spectrum: E(f)xf 5 40 m γ 3.0 n 10 5 m Smoothed periodogram 51 degrees of freedom

19 Cross-correlation G n ( θ ) G( n) cos ( θ ) Correlation (3cm apart) Hwang distribution Correlation 3cm apart across tank Directional Bandwidth from G(θ) (degrees)

20 April 007 Experimental Cases: for a fixed frequency spectrum; Mean Wave length L1m ak ( H 4)( π L) 1/3 Strongbreaker ak0.095 Hs ak0.08 ak0.06 No breaker Directional Spreading (degrees)

21 Maximum wave height Stoke s limit Benamin-Feir wave train Hmax/Lp Random directional wave Hmax/ L ak Steepness ak (initial, mean)

22 Spectral evolution (typical cases) γ 3.0; n 15; Hs 4.15 γ 3.0; n 15; Hs ( ) ( ) ( ) S f, θ S f G f,θ 4 5 S( f ) α g ( π ) f exp n G θ G n cos θ ( ) ( ) ( ) 5 4 f f p 4 γ exp σ ( f f ) f p p

23 Spectral downshifting of random directional wave fp(developed)/fp(initial) BF ak

24 Energy Loss ΔE ak

25 Downshifting due to breaking dissipation T b T T M g M c E M M D E E E where and & T p E E E E ( ) b T p b T p T p p D E D M c E dt d Γ & energy decay rate frequency decay rate 1 Γ T b p p E D dt d Tulin&Waseda 1999

26 Downshifting parameter Γ estimated for strong breaker cases T&W range Γ d dt p 1 p D E b T frequency decay rate energy decay rate Γ Directional Spreading (degrees)

27 Evolution of Kurtosis with fetch Initial value Developed stage γ 3.0 n 15 η( x) η( x) 4

28 BFI vs. kurtosis η( x) η( x) 4 BFI Line: κ 40 π BFI ; BFI 3 ε Δ /

29 Kurtosis and spectral bandwidth (γ) Hwang directional distribution Developed stage Hs4.15 Hs5.00 η( x) η( x) 4 Non-resonant Interaction narrow γ broad

30 Freak Wave Occurrence Increases η( x) η( x) 4 Non-resonant Interaction γ 3 Initial value Developed stage Uni-directional Alber & Saffman limit n Directionality large

31 High-resolution wave-current coupled model Wave-JCOPE, Snl(SRIAM) γ n Degree A&S limit Tamura, Waseda, Miyazawa & Komatsu, 11/16 presentation

32 Wave parameter distribution Non-resonant Interaction Frequency Bandwidth γ Non-resonant Interaction Directional Distribution n

33 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