This document discusses theories for explaining freak waves through stochastic wave groups and nonlinear random sea models. It presents the theory of stochastic wave groups which models waves as a random distribution of wave envelopes. Key points include: nonlinear evolution of wave groups can produce higher waves; successive crests are correlated and bound waves emerge. Comparisons to wave flume data show the theory agrees with observed wave crest and height distributions. The document also discusses using Gram-Charlier approximations to model nonlinear random seas and compares these to measurements from ocean platforms.

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

2. A NATURAL BEAUTY !

3. Rogue waves
Freak waves
Giant waves
Extreme waves

4. Rogue
waves
Freak
waves
Giant
waves
Extreme
waves

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

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. 
 
+d
+d

E() d
E()


   





N
j
j
j
j
j t
x
k
a
t
x
1
cos
, 




j
j
a
d
E 2
2
1
)
( 






0
0
0
0
0 cos
)
(
)
,
(
)
,
(
)
( 




 d
T
E
T
t
x
t
x
T
LINEAR WAVES : GAUSSIAN SEAS

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

11. 0.2H 0.8H
0.38H
0.62H
0.5H+ o(H-1)
T1
T2
T*+o(H-1)
0.5H+ o(H-1)
*Theory of quasi-determinism, Boccotti P. Wave Mechanics 2000 Elsevier
NECESSARY AND SUFFICIENT CONDITIONS
FOR THE OCCURRENCE OF A HIGH WAVE IN TIME*

12.  
?
2
/
)
,
x
(
,
2
/
)
,
x
(
to
d
conditione
,
)
,
X
x
(
that
y
probabilit
the
is
What
*
2
0
0
0
0
0
0
H
T
t
H
t
du
u
u
T
t














H
*Boccotti P. Wave Mechanics 2000 Elsevier
What happens in the neighborhood of a point x0 if a large crest
followed by large trough are recorded in time at x0 ?
  )
,
(
2
,
X
x 0
0 T
X
H
T
t
c 




SPACE-TIME covariance

13. 1
T
0
h 1
h
3
T
0
h 1
h
 
1
1
1
0
*
2 , 

 h
h
O
T



 1
0 , h
h
as
0
h
1
h
* Fedele F., Successive wave crests in a Gaussian sea, Probabilistic Eng. Mechanics 2005 vol. 20, Issue 4, 355-363
0 2 4 6 8 10 12 14 16 18 20
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
T*
*
2
SUCCESSIVE WAVE CRESTS IN TIME*

14.  

















2
*
2
0
0
1
0
0
0
0
)
,
x
(
,
)
,
x
(
,
)
,
X
x
(
Pr
h
T
t
h
t
to
d
conditione
du
u
u
T
t









2
1
,
h
h
)
,
(
)
(
)
0
(
)
(
)
0
(
)
,
(
)
(
)
0
(
)
(
)
0
(
)
,
( *
2
2
*
2
2
*
2
1
2
2
*
2
2
*
2
2
1
T
T
T
T
h
h
T
T
T
h
h
T
c 







 X
X
X









What happens in the neighborhood of a point x0
if two large successive wave crests are recorded in time at x0 ?
EXPECTED SHAPE OF THE SEA LOCALLY TO
TWO SUCCESSIVE WAVE CRESTS *
* Fedele F., 2006. On wave groups in a Gaussian sea. Ocean Engineering 2006 ( in press)
What is hidden “behind” this equation ?

15. -10 -8 -6 -4 -2 0 2
-1
0
1
-10 -8 -6 -4 -2 0 2
-1
0
1
-10 -8 -6 -4 -2 0 2
-1
0
1
-10 -8 -6 -4 -2 0 2
-1
0
1
-10 -8 -6 -4 -2 0 2
-1
0
1
-10 -8 -6 -4 -2 0 2
-1
0
1
-10 -8 -6 -4 -2 0 2
-1
0
1
-10 -8 -6 -4 -2 0 2
-1
0
1
-5T2
*
-2T2
*
T=-10T2
-T2
*
-0.5T
2
*
0
T=-10T
2
*
0.5T2
*
T2
*
X/Lp
a b
b
a
b
b
a
a
* Fedele F., 2006. On wave groups in a Gaussian sea. Ocean Engineering 2006 ( in press)
STOCHASTIC WAVE GROUP
amplitude h random variable
distributed according to Rayeligh
stochastic family of wave groups
A SINGLE WAVE GROUP CAUSES
TWO SUCCESSIVE WAVE CRESTS !*
)
(
)
,
X
(
)
,
( 0
h
O
T
h
T
X
ex 



SPACE-TIME covariance

16. Second order effects:
BOUND WAVES
Third order effects :
FOUR-WAVE RESONANCE
(WEAK WAVE TURBULENCE)
NONLINEAR RANDOM SEAS cont’d
Quartet interaction
r
q
p
n k
k
k
k 


n
k
r
k
p
k
q
k
 





r
q
p
r
q
p
r
q
p
n
npqr
n
n
n
B
B
B
T
i
B
i
dt
dB
,
,
*


 
 


n
n
n
n
n
t
t
B
g
t )
(
cos
)
(
2
1
)
,
( 


 x
k
x


n
n
n t
B
t
B )
(
)
( *
A
Conserved quantities :
Hamiltonian
Wave action
Wave momentum

 




n,p,q,r
r
q
*
p
*
n
r
q
p
n
npqr
n
*
n
n
n (t)
(t)B
(t)B
(t)B
B
δ
T
(t)
(t)B
B
ω
2
1
H


n
n
n
n t
B
t
B )
(
)
( *
k
M
)
,
(
)
,
(
)
,
(
~
2 t
t
t x
x
x 

 

 
 
 
 

















q
p
q
p
q
p
pq
q
p
q
p
q
p
q
p
q
p
pq
q
p
q
p
t
B
t
B
t
B
t
B
t
,
,
2
cos
)
(
)
(
cos
)
(
)
(
)
,
(









x
k
k
x
k
k
x

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

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

19. 4 2
1 


 

nl
 



































4
2
2
2
2
2
2
2
ln
2
1
2
ln
2
arctan
2
1
cos
1
2
ln
4
BFI
T
BFI
T
BFI
p
p





0
/
2
k
k
BFI











 )
(
2
1
exp
]
~
Pr[ 2
y
y
nl 

2
2
~
nl
nl
nl 


 

y
nl 

~
Symmetric
Third order effects
Second order
Bound effects
More precisely after some boring math …..
Self-focusing parameter

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

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
y
+
E
0 1 2 3 4 5 6
10
-4
10
-3
10
-2
10
-1
10
0
+ Onorato et al. (2005)
* ~ 0.075
 30
~ 0.225
 40
~ 0.800
 22
~  30
/ 3
 04
~  40
R
NB-GC
NB
0 1 2 3 4 5 6
10
-4
10
-3
10
-2
10
-1
10
0
y
-
y
+
E
y
-
&
E
y
+
* = 0.083
 = 0.660
 30
= 0.1995
 12
= 0.0665
 40
= 0.0234
 22
= 0.0089
 04
= 0.0290 R
&
NB ~ NB-GC NB NB-GC
. wave crests y +
+ wave troughs y -
TAYFUN
TAYFUN
Rayleigh
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 ~ λ 3 / 3
• Second-order corrections: NB model for wave crests
crests ~ ξ+ = ξ + ( μ / 2 ) ξ 2
• Exceedance probability distribution
E ξ+ = exp{ -[-1 + ( 1 + 2 μ ξ )1/2 ] 2/ 2 μ 2 }

27. MODIFIED THIRD-ORDER MODEL
(NB-GC Model)
Modify Gram-Charlier:
―› replace E R = exp(- ξ 2 / 2)
―› with E ξ+ = exp{ -[-1 + ( 1 + 2 μ ξ )1/2 ] 2/ 2 μ 2 }
Wave Crests: NB - GC model
E+ = E ξ+ [ 1 + Λ ξ2 ( 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. *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)
THE PROBABILITY OF EXCEEDANCE
Wave tank experiments:
unidirectional narrow-band seas ( Onorato et all. 2005)
MODULATION + SECOND ORDER EFFECTS
Wave crest Wave height

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
y
+
E
0 1 2 3 4 5 6
10
-4
10
-3
10
-2
10
-1
10
0
+ Onorato et al. (2005)
* ~ 0.075
 30
~ 0.225
 40
~ 0.800
 22
~  30
/ 3
 04
~  40
R
NB-GC
NB
0 1 2 3 4 5 6
10
-4
10
-3
10
-2
10
-1
10
0
y
-
y
+
E
y
-
&
E
y
+
* = 0.083
 = 0.660
 30
= 0.1995
 12
= 0.0665
 40
= 0.0234
 22
= 0.0089
 04
= 0.0290 R
&
NB ~ NB-GC NB NB-GC
. wave crests y +
+ wave troughs y -
TAYFUN
TAYFUN
Rayleigh
Rayleigh