FRANCESCO FEDELE

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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
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A NATURAL BEAUTY !
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Freak waves
Giant waves
Rogue waves
Extreme waves
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Rogue waves
Extreme waves
Giant waves
Freak waves
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DRAUPNER EVENT JANUARY 1995
Hmax25.6 m ! 1 in 200,000 waves
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Are freak waves RARE EVENTS OF A NORMAL
POPULATION Or TYPICAL EVENTS OF A SPECIAL
POPULATION ?
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• 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
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STOCHASTIC WAVE GROUPS
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LINEAR WAVES GAUSSIAN SEAS
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TYPICAL WAVE SPECTRA OF THE MEDITERRANEAN SEA
Time covariance
Spectrum
from Boccotti P. Wave Mechanics 2000 Elsevier
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NECESSARY AND SUFFICIENT CONDITIONS FOR THE
OCCURRENCE OF A HIGH WAVE IN TIME
Theory of quasi-determinism, Boccotti P. Wave
Mechanics 2000 Elsevier
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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
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SUCCESSIVE WAVE CRESTS IN TIME
Fedele F., Successive wave crests in a Gaussian
sea, Probabilistic Eng. Mechanics 2005 vol. 20,
Issue 4, 355-363
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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)
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A SINGLE WAVE GROUP CAUSES TWO SUCCESSIVE WAVE
CRESTS !
SPACE-TIME covariance
STOCHASTIC WAVE GROUP amplitude 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)
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NONLINEAR RANDOM SEAS contd
Third order effects FOUR-WAVE RESONANCE
(WEAK WAVE TURBULENCE)
Conserved quantities Hamiltonian Wave action
Wave momentum
Second order effects BOUND WAVES
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NONLINEAR EVOLUTION OF A STOCHASTIC WAVE GROUP
Third order effects FOUR-WAVE RESONANCE
Second order effects BOUND WAVES
Crest-trough symmetry kurtosisgt3 Modulation
instability Effects on slow time scale gtgt wave
period DOMINANT ONLY IN UNIDIRECTIONAL
NARROW-BAND SEAS !
Cresttrough asymmetry skewnessgt0 TAYFUN
DISTRIBUTION FOR PDF CREST Effects on Short time
scale wave period
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NONLINEAR EVOLUTION OF A STOCHASTIC WAVE GROUP
NONLINEAR EVOLUTION OF A STOCHASTIC WAVE GROUP
t0
t-t0
(linear wave group)
h
hNLgth
x
wave action and wave momentum Always conserved
identities
x0
Hamiltonian invariant
Symmetric third order effects
Hmaxf(h)a 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.
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More precisely after some boring math ..
Symmetric Third order effects
Second order Bound effects
Self-focusing parameter
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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)
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COMPARISONS with WAVE-FLUME DATA Wave Heights
unidirectional narrow-band waves ( Onorato et
al. 2005 )
Rayleigh
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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
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GRAM-CHARLIER APPROXIMATIONS
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GRAM-CHARLIER APPROXIMATIONS( GC Model)

• Wave Envelope ? h/s
• Prob ? x E? exp(- x2 / 2) 1 ? x2 (
  x2 4)
• ? ( ?40 2?22 ?04 ) / 64
• Narrow-band wave heights H/s 2?
• E2? exp(- x2 / 8) 1 (? / 16) x2 ( x 2
  16)

Tayfun Lo, 1990. Nonlinear effects on wave
envelope and phase. J. Watwerways, Port, Coastal
Ocean
Engg. 116(1), ASCE, 79-100.
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WAVE HEIGHTS 2D wave-flume data from Onorato et
al. (2004)
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WAVE CRESTS(NB Model)

• Wave steepness
• µ s k ? 3 / 3
• Second-order corrections NB model for wave
  crests
• crests ? ? ( µ / 2 ) ? 2
• Exceedance probability distribution
• E ? exp --1 ( 1 2 µ ? )1/2 2/
  2 µ 2

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MODIFIED THIRD-ORDER MODEL(NB-GC Model)

• Modify Gram-Charlier
• ? replace E R exp(- ? 2 / 2)
• ? with E ? exp --1 ( 1 2 µ ? )1/2
  2/ 2 µ 2
• Wave Crests NB - GC model
• E E ? 1 ? ?2 ( ?2 4)

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WAVE CRESTS 3D numerical simulations from
Socquet-Juglard et al. (2005)
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WAVE CRESTS 2D wave-flume data from Onorato et
al. (2005)
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• 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

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Questions ?
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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)
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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