Explaining Freak Waves by a Theory of Stochastic Wave Groups

1. EXPLAINING FREAK WAVES BY A THEORY OF STOCHASTIC WAVE GROUPS FRANCESCO FEDELE Goddard Earth Sciences Technology center University of Baltimore County, Maryland, USA Global Modeling Assimilation Office NASA Goddard Space Flight Center Maryland, USA

2. A NATURAL BEAUTY !

3. Freak waves Giant waves Rogue waves Extreme waves

4. Rogue waves Extreme waves Giant waves Freak waves

5. DRAUPNER EVENT JANUARY 1995 Hmax=25.6 m ! 1 in 200,000 waves

6. Are freak waves RARE EVENTS OF A NORMAL POPULATION Or TYPICAL EVENTS OF A SPECIAL POPULATION ?

7. OBJECTIVE • Nonlinear statistics on wave heights & crests TWO APPROACHES • Stochastic Wave Groups (theory of quasi-determinism of prof. Boccotti*) Zakharov equation • Gram-Charlier Approximations (with prof. Aziz Tayfun) Gram-Charlier model *from Boccotti P. Wave Mechanics 2000 Elsevier

8. STOCHASTIC WAVE GROUPS

9. LINEAR WAVES : GAUSSIAN SEAS

10. TYPICAL WAVE SPECTRA OF THE MEDITERRANEAN SEA* Time covariance Spectrum *from Boccotti P. Wave Mechanics 2000 Elsevier

11. NECESSARY AND SUFFICIENT CONDITIONS FOR THE OCCURRENCE OF A HIGH WAVE IN TIME* *Theory of quasi-determinism,Boccotti P. Wave Mechanics 2000 Elsevier

12. What happens in the neighborhood of a point x0 if a large crest followed by large trough are recorded in time at x0 ? SPACE-TIME covariance *Boccotti P. Wave Mechanics 2000 Elsevier

13. SUCCESSIVE WAVE CRESTS IN TIME* * Fedele F., Successive wave crests in a Gaussian sea, Probabilistic Eng. Mechanics 2005 vol. 20, Issue 4, 355-363

14. EXPECTED SHAPE OF THE SEA LOCALLY TO TWO SUCCESSIVE WAVE CRESTS * What happens in the neighborhood of a point x0 if two large successive wave crests are recorded in time at x0 ? What is hidden “behind” this equation ? * Fedele F., 2006. On wave groups in a Gaussian sea. Ocean Engineering 2006 ( in press)

15. A SINGLE WAVE GROUP CAUSES TWO SUCCESSIVE WAVE CRESTS !* SPACE-TIME covariance STOCHASTIC WAVE GROUPamplitude h random variabledistributed according to Rayelighstochastic family of wave groups * Fedele F., 2006. On wave groups in a Gaussian sea. Ocean Engineering 2006 ( in press)

16. NONLINEAR RANDOM SEAS cont’d Third order effects : FOUR-WAVE RESONANCE (WEAK WAVE TURBULENCE) Conserved quantities : Hamiltonian Wave action Wave momentum Second order effects: BOUND WAVES

17. NONLINEAR EVOLUTION OF A STOCHASTIC WAVE GROUP Third order effects : FOUR-WAVE RESONANCE Second order effects: BOUND WAVES Crest-trough symmetry kurtosis>3 Modulation instability Effects on slow time scale >> wave period DOMINANT ONLY IN UNIDIRECTIONAL NARROW-BAND SEAS ! Crest–trough asymmetry skewness>0 TAYFUN DISTRIBUTION FOR PDF CREST Effects on Short time scale : wave period

18. NONLINEAR EVOLUTION OF A STOCHASTIC WAVE GROUP* NONLINEAR EVOLUTION OF A STOCHASTIC WAVE GROUP* t=0 t=-t0 (linear wave group) hNL>h h x wave action and wave momentum Always conserved : identities x=0 Hamiltonian invariant Symmetric third order effects Hmax=f(h)+α f(h)2 Asymmetric second order effects h Rayleigh distributed *Fedele F. 2006. Extreme Events in nonlinear random seas. J. of Offshore Mechanics and Arctic Eng., ASME, 128, 11-16.

19. More precisely after some boring math ….. Symmetric Third order effects Second order Bound effects Self-focusing parameter

20. COMPARISONS with WAVE-FLUME DATA : Wave Crests* unidirectional narrow-band waves ( Onorato et al. 2005) Rayleigh *Fedele F., Tayfun A.Explaining extreme waves by a theory of Stochastic wave groups. PROCEEDINGS of OMAE 2006 ( in press)

21. COMPARISONS with WAVE-FLUME DATA : Wave Heights unidirectional narrow-band waves ( Onorato et al. 2005) Rayleigh

22. THE PROBABILITY OF EXCEEDANCE Wave tank experiments: unidirectional narrow-band seas ( Onorato et all. 2005) unrealistic ocean conditions MODULATION + SECOND ORDER EFFECTS TERN platform, time series ( 6,000 waves) realistic ocean conditions SECOND ORDER EFFECTS DOMINANT TAYFUN Rayleigh TAYFUN Rayleigh

23. GRAM-CHARLIER APPROXIMATIONS

24. GRAM-CHARLIER APPROXIMATIONS( GC Model) • Wave Envelope *~ ξ = h/σ Prob{ ξ› x } =Eξ=exp(- x2 / 2)[1+ Λ x2 ( x2 – 4) ] Λ= ( λ40 + 2λ22 + λ04 ) / 64 • Narrow-band wave heights: H/σ ~ 2ξ E2ξ= exp(- x2 / 8) [ 1+ (Λ / 16) x2 ( x 2 – 16) ] * Tayfun & Lo, 1990. Nonlinear effects on wave envelope and phase. J. Watwerways, Port, Coastal & Ocean Eng’g. 116(1), ASCE, 79-100.

25. WAVE HEIGHTS 2D wave-flume data from Onorato et al. (2004)

26. WAVE CRESTS(NB Model) • Wave steepness: μ~ σ k ~ λ3 / 3 • Second-order corrections: NB model for wave crests crests ~ξ+ = ξ + ( μ / 2 ) ξ2 • Exceedance probability distribution E ξ+= exp{ -[-1 + ( 1 + 2 μξ )1/2 ] 2/ 2 μ 2 }

27. MODIFIED THIRD-ORDER MODEL(NB-GC Model) Modify Gram-Charlier: ―›replaceE R = exp(- ξ 2 / 2) ―› with E ξ+= exp{ -[-1 + ( 1 + 2 μξ )1/2 ] 2/ 2 μ 2 } Wave Crests: NB - GC model E+ = Eξ+[ 1 + Λξ2 ( ξ2 – 4) ]

28. WAVE CRESTS 3D numerical simulations from Socquet-Juglard et al. (2005)

29. WAVE CRESTS 2D wave-flume data from Onorato et al. (2005)

30. CONCLUSIONS • Theory of quasi-determinism of Boccotti identifies a “gene” of the Gaussian sea at high energy levels : WAVE GROUP • The statistics of large events in a nonlinear random sea can be related to the nonlinear dynamics of a wave group • new analytical formula for the probability of exceedance of a large wave crest for the case of a Zakharov system is derived

31. Questions ?

32. THE PROBABILITY OF EXCEEDANCE Wave tank experiments: unidirectional narrow-band seas ( Onorato et all. 2005) MODULATION + SECOND ORDER EFFECTS Wave height Wave crest *Tayfun A.,Fedele F., Wave height distributions and nonlinear effects. PROCEEDINGS of OMAE 2006 ( in press) **Fedele F., Tayfun A.Explaining extreme waves by a theory of Stochastic wave groups. PROCEEDINGS of OMAE 2006 ( in press)

33. THE PROBABILITY OF EXCEEDANCE Wave tank experiments: unidirectional narrow-band seas ( Onorato et all. 2005) unrealistic ocean conditions MODULATION + SECOND ORDER EFFECTS TERN platform, time series ( 6,000 waves) realistic ocean conditions SECOND ORDER EFFECTS DOMINANT TAYFUN Rayleigh TAYFUN Rayleigh