AbstractThe generation of freak waves in a 2-dimensional random sea state characterized by the JONSWAP spectrum are simulated employing a nonlinear fourth-order Schrödinger equation. The evolution of the freak waves in deep water are analyzed. We investigate the effect of initial wave parameters on kurtosis and occurrence of freak waves. The results show that Benjamin-Feir index (BFI) is an important parameter to identify the presence of instability. The kurtosis presents a similar spatial evolution trend with the occurrence probability of freak waves. Freak waves in a random sea state are more likely to occur for narrow spectrum and small values of significant wave height

Dong, Guohai

Ma, Yuxiang

Xia, Weida

Elsevier - Publisher Connector 

 Procedia Engineering  116 ( 2015 )  366 – 372 
1877-7058 © 2015 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license 
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer- Review under responsibility of organizing committee , IIT Madras , and International Steering Committee of APAC 2015
doi: 10.1016/j.proeng.2015.08.300 
ScienceDirect
Available online at www.sciencedirect.com
8th International Conference on Asian and Pacific Coasts (APAC 2015)
Department of Ocean Engineering, IIT Madras, India.
Numerical simulation of freak waves in random sea state
Weida Xia*, Yuxiang Ma, Guohai Dong,
State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology Dalian, 116024, China
Abstract
The generation of freak waves in a 2-dimensional random sea state characterized by the JONSWAP spectrum are simulated
employing a nonlinear fourth-order Schrödinger equation. The evolution of the freak waves in deep water are analyzed. We
investigate the effect of initial wave parameters on kurtosis and occurrence of freak waves. The results show that Benjamin-Feir
index (BFI) is an important parameter to identify the presence of instability. The kurtosis presents a similar spatial evolution
trend with the occurrence probability of freak waves. Freak waves in a random sea state are more likely to occur for narrow
spectrum and small values of significant wave height.
© 2014 The Authors. Published by Elsevier B.V.
Peer-review under responsibility of organizing committee of APAC 2015, Department of Ocean Engineering, IIT Madras.
Keywords: Freak waves; nonlinear Schrödinger equation; Benjamin-Feir instability; JONSWAP spectrum;
1. Introduction
The freak waves, also known as extreme waves or rouge waves, is a catastrophic event in the ocean that appears
from nowhere without alarm and disappears without a trace. The notion of freak waves was first introduced by
Drapper (1965), and in practice freak waves are defined as large water waves whose heights exceed by a factor of
2.0 the significant wave height (Kharif et al., 2009). Due to extraordinary wave height and strong nonlinearity, freak
waves are believed to cause a number of marine accidents and damages to offshore structures (Lawton, 2009).
Recently, freak waves have become an important topic in marine engineering.
* Corresponding author. Tel.: +8615142481566.
E-mail address: wdxia@mail.dlut.edu.cn
© 2015 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license 
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer- Review under responsibility of organizing committee , IIT Madras , and International Steering Committee of APAC 2015
367 Weida Xia et al. /  Procedia Engineering  116 ( 2015 )  366 – 372 
Over the last decade, numerous researches have been done to investigate the mechanism of freak wave generation,
and the knowledge of freak waves have significantly advanced. Often, freak wave events can be explained by the
superstition of linear dispersive waves or the presence of currents (Kharif et al., 2003), and these mechanisms are
easily understood. In addition, nonlinear modulation instability that can describe many of the features of the dynamics
of freak waves seems to be a more appropriate mechanism of freak wave generation.
This modulation instability also known as Benjamin-Feir instability (Benjamin and Feir, 1967) will result in
focusing of wave energy. Benjamin-Feir instability suggests that Stokes wave is unstable to small sideband
perturbations if the frequency difference between the carrier wave and sideband is small enough. The sideband
components will grow exponentially because of the nonlinear energy transfer.
Nonlinear Schrödinger equation that contains the representation of the Benjamin-Feir modulation instability is
considered to be a simple but useful model for the weakly nonlinear evolution of a wave group. Lo and Mei (1985)
found the unequally growth of sideband perturbations and downshift of carrier wave using fourth-order nonlinear
Schrödinger equation in a numerical study. Subsequent studies (Melville, 1982; Su, 1982) also confirmed the
phenomenon. Goullet and Choi (2011) studied the spatial evolution of nonlinear long-crested irregular waves
characterized by the JONSWAP spectrum using a nonlinear wave model based on modified nonlinear Schrödinger
equation (MNLS). The MNLS equation is found to approximate reasonably well for the wave fields, but the phase
error increases as the propagation distance increases.
Most of the research in the past focused on the mechanism of the freak waves, while studies about the probability
of freak wave occurrence and the relationship between wave parameters and freak waves are rare. In this paper, the
generation of freak waves in a 2-dimensional random sea state characterized by the JONSWAP spectrum are
simulated employing a modified nonlinear fourth-order Schrödinger equation. More wave parameters, such as
spectral bandwidth, significant wave height, are considered. The evolution of kurtosis and probability of freak wave
occurrence are the main considerations. The paper is organized as follows. Section 2 introduces the numerical model
and numerical scheme applied in this paper. In section 3, numerical experimental parameters are given. Results are
discussed in section 4. The last section summarizes the conclusions on the basis of the above discussions.
2. Numerical Model
With minor corrections to the Dysthe’s equation (1979), Lo and Mei (1985) derived the modified nonlinear fourth-
order Schrödinger equation which is valid for deep water. This dimensionless equation that describes the nonlinear
evolution of water waves can be expressed in a coordinate system moving at the linear group velocity as follows:
2
2 22
2
0
8 4 0
z
A A Ai i A A A i A φλ ελ ελ
η ξ ξ ξ 
       
   
(1)
2 2
2 24 0   ( 0)h zz
φ φ
ξ
     
 
(2)
2
   ( 0)A z
z
φ
ξ
  
 
(3)
0   ( )z h
z
φ   

(4)
where dimensionless variables have been scaled according to: A= A̕a , φ=φ̕ωa2 , ξ=εωλ(2kx⁄w-t) , z=εkλz̕ ,
η=ε2kx , h=εkλh̕ . λ is a scale factor that renders the computational domain in ξ to 2π. To a fixed observer, ξ is the
368   Weida Xia et al. /  Procedia Engineering  116 ( 2015 )  366 – 372 
negative of time elapsed and η is the distance over which the group has advanced. A̕ is the complex amplitude of the
first harmonic of the Stokes waves, and φ ̕ is the potential of the induced mean current.
As the computations were performed in a periodic domain of period 2π, the boundary conditions are
   0, 2 ,A Aη π η (5)
   0, , 2 , ,z zφ η φ π η (6)
The nondimensionalized surface displacement can be expressed as:
 
22
2
0 2 2 2 2 3 3
3
8
, Re 2
1 32
2 8
i
z i i
AA i A A e
AA i A e A e
ψ
ψ ψ
ελ εξφζ ξ η ε λ ξ
ε ε λ εξ

  
            
  
               
(7)
Where ψ=-ηε2+ξλε is the phase function of the carrier waves.
The numerical scheme to solve the above equation was proposed by Lo and Mei (1985). Equation 1 can be
separated into linear and nonlinear parts. The nonlinear part can be calculated using a midpoint finite-difference
approximation, and the linear part has an exact solution using pseudo-spectral method.
3. Wave parameters in numerical study
In this study, 7 different Joint North Sea Wave Project (JONSWAP) spectra varying Hs and γ are considered. All
of them were characterized by a peak period of 1.0s. The wave parameters are listed in Table 1.
Table 1. Wave parameters.
Case f0(s) γ Hs(m) ε=kpHs2 Δff0 BFI
C01 1 1.0 0.042 0.08 0.29 0.21
C02 1 3.3 0.042 0.08 0.10 0.63
C03 1 3.3 0.080 0.16 0.10 1.20
C04 1 3.3 0.100 0.20 0.10 1.50
C05 1 3.3 0.120 0.24 0.10 1.80
C06 1 5.0 0.080 0.16 0.08 1.43
C07 1 7.0 0.080 0.16 0.07 1.60
In table 1, f0 is the peak wave frequency; γ is the spectrum enhancement factor; Hs is the significant wave height
and ε is the wave steepness. The frequency bandwidth Δf is defined as the half-width at half maximum of the spectrum.
Various frequency bandwidths can be obtained by changing the enhancement factor γ. We have also included the
Benjamin-Feir index defined as:
0
2
2 /
BFI f f
ε

(8)
369 Weida Xia et al. /  Procedia Engineering  116 ( 2015 )  366 – 372 
The initial wave surface elevation ζ can be obtained by the expression:
 
1
,0 cos(2 )
n
i i i
i
t a f tζ π ϕ

 	 (9)
Where φi are uniformly distributed random numbers on the interval (0,2π), ai= 2S(fi)Δfi is the free wave component
with i-th frequency fi and random phase φi, and S(f) is the JONSWAP spectrum. The number of free wave components
n is 200.
According to Rayleigh distribution, the occurrence probability of freak wave is approximately 1/3000, so each
Case needed to be simulated in a long-time series. A Monte-Carlo method is adopted in this paper, and for each case,
we performed the simulation 100 times with different sets of initial random phases.
4. Results discussion
4.1. Evolution of freak waves
In the long time simulation, freak waves were observed, and the process of the formation of freak waves is clearly
displayed in Fig.1. Fig.1 shows the surface elevation in time series for Case C03. There is an evident wave group in
time series which propagates with the linear group velocity, as coordinate system moving at the linear group velocity
as well, and η represents the nondimensional distance from the wavemaker. At location η=1.72, there is no signs of
freak waves. With the wave advance further, nonlinear focusing generates higher waves. At location η=2.31, a deep
trough appears. As nonlinear interaction becomes stronger, a freak wave forms with H/Hs=2.21 at location η=2.40.
A deep trough appears again at η=2.50, and after that defocusing dominates the evolution, and amplitude subsides
until freak wave disappears. From above, the evolution of freak waves can be summarized as the focusing-defocusing
process. During the generation of freak waves, a single wave absorbs energy from neighbor waves, increasing its
amplitude, reaching a maximum and then returns its energy back to other waves.
Fig. 1. Time series recorded at different locations for Case C03.
4.2. Evolution of kurtosis and occurrence probability of freak waves
Wave parameters, such as kurtosis and the occurrence probability of freak waves will change during the evolution
of wave groups. The kurtosis is presented as:
370   Weida Xia et al. /  Procedia Engineering  116 ( 2015 )  366 – 372 
4
4
1
1 1 ( )
N
n
nrms
kurtosis
N
ζ ζζ  	 (10)
Where ζ is the surface elevation, ζrms is the root mean square of the surface elevation, N is the total number of data
points. The kurtosis is a measure of the deviation between the measured surface wave distribution and Gauss
distribution. The kurtosis equals three when the surface displacement is Gaussian distributed. However, due to
nonlinear interactions, the value of kurtosis is always larger than three, which means a fatter tail and more extreme
values.
Fig. 2. (a) variations of the kurtosis; (b) variations of occurrence probability of freak waves
Fig.2 (a) shows the variations of kurtosis and occurrence probability of freak waves along the propagation
direction. It is found that for small BFI, such as Case C01 and C02, the kurtosis decreased during the evolution. For
the case BFI>1.0, kurtosis reaches maximum at 8 wavelengths, which is primarily caused by Benjamin-Feir
instability, and then the kurtosis gradually becomes steady and constant. This observation is consistent with the
previous study of Alber (1978). Fig.2 (b) shows the evolution of occurrence probability of freak waves, and the trend
is almost identical to the variations of kurtosis. Furthermore, for Case C01 and 02, the probability of freak waves
agrees well with what Rayleigh distribution expects. However, for Case C03, the freak wave probability is evidently
larger than the other case. The above discussion suggests that BFI is an an important parameter to identify the
presence of instability.
Fig.3 shows the variations of kurtosis and occurrence probability of freak waves for Cases with different spectrum
width. It is found that spectrum width has appreciable influence on the values of kurtosis. Kurtosis is smaller for the
case with broader spectrum width. Freak wave probability shows a similar trend with the kurtosis, and the location
where freak wave occurs most frequently coincides with the place where kurtosis reach its maximum. Moreover,
freak waves are more likely to occur for narrow spectrum.
371 Weida Xia et al. /  Procedia Engineering  116 ( 2015 )  366 – 372 
Fig. 3. (a) variations of the kurtosis; (b) variations of occurrence probability of freak waves
Fig.4 shows the variations of kurtosis and occurrence probability of freak waves for Cases with different initial
significant wave height. Kurtosis reach its maximum at different locations for different values of Hs. As expected,
kurtosis depend on the initial wave height with positive correlation. In Fig.4 (b), the freak wave probability reaches
its maximum at different location as well. This phenomenon can be attributed to that wave height is associated with
nonlinearity, and instability will grow faster with stronger nonlinearity, so maximum point will arrive earlier with
bigger wave height. Furthermore, it’s clearly shown that freak waves in a random sea state are more likely to occur
for small values of significant wave height. The main reason causes this phenomenon is that: though the increment
of initial wave height makes the instability easier to occur, it also elevates the threshold to become a freak wave, so
it’s more difficult to form a freak wave for bigger initial wave height.
Fig. 4. (a) variations of the kurtosis; (b) variations of occurrence probability of freak waves
5. Conclusions
The generation of freak waves in a 2-dimensional random sea state characterized by the JONSWAP spectrum are
simulated employing a nonlinear fourth-order Schrödinger equation. Seven Joint North Sea Wave Project
(JONSWAP) spectrum varying Hs and γ are considered. The results show that the evolution of freak waves can be
summarized as the focusing-defocusing process. In the focusing period, wave energy gathers to form a freak wave;
in the defocusing period, energy disperse. BFI is an important parameter to identify the presence of instability. The
Benjamin-Feir instability only occurs for large BFI. Furthermore the kurtosis presents a similar spatial evolution
trend with the occurrence probability of freak waves. It’s well known that large kurtosis and Benjamin-Feir
instability could be due to quasi-resonant four-wave interaction (Janssen, 2002; Onorato et al., 2005; Mori and
Janssen, 2006), so it’s confirmed again that freak waves is caused by Benjamin-Feir instability. Kurtosis becomes
larger with the increasing initial Hs and γ. Freak waves in a random sea state are more likely to occur for narrow
spectrum and small values of significant wave height.
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Numerical Simulation of Freak Waves in Random Sea State 

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Numerical Simulation of Freak Waves in Random Sea State

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