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CURRENT EFFECTS ON THE GENERATION AND EVOLUTION OF THE
PEREGRINE BREATHER-TYPE ROGUE WAVE
Wenyue Lu State Key Laboratory of Ocean Engineering,
Shanghai Jiao Tong University Shanghai, 200030 China
Jianmin Yang* State Key Laboratory of Ocean Engineering,
Shanghai Jiao Tong University Shanghai, 200030 China
Longbin Tao
School of Marine Science and Technology, Newcastle University
Newcastle upon Tyne NE1 7RU, United Kingdom
Haining Lu State Key Laboratory of Ocean Engineering,
Shanghai Jiao Tong University Shanghai, 200030 China
Xinliang Tian State Key Laboratory of Ocean Engineering,
Shanghai Jiao Tong University Shanghai, 200030 China
Jun Li State Key Laboratory of Ocean Engineering,
Shanghai Jiao Tong University Shanghai, 200030 China
ABSTRACT Rogue wave is a kind of surface gravity waves with much
larger wave heights than expected in normal sea state. Since this
extreme sea event always occurred in the areas with strong
currents, the wave-current interaction was considered to be one
of the physical mechanics of the formation of the rogue wave.
Some breather type solutions of the NLS equation have been
considered as prototypes of rogue waves in ocean which usually
appears from smooth initial condition only with a certain
disturbance. In this paper, we have numerically studied
evolutionary process of the Peregrine breather rogue wave based
on the current modified fourth order nonlinear Schrödinger
equation (the CmNLS equation). During the generation and
evolution of the Peregrine breather rogue wave, the effects of the
steady current was investigated by comparing with the results
without current. The differences of the focusing position/time
were observed due to the current influence.
Keywords: Rogue Wave, Wave-Current Interaction, Current
Modified Fourth Order Nonlinear Schrödinger Equation,
CmNLS, Peregrine Breather-Type Solution
* Contact Author: jmyang@sjtu.edu.cn
INTRODUCTION
The rogue waves, which is also known as freak waves,
monster waves, killer waves, extreme waves, and abnormal
waves, is a type of very rare and extremely large wave that
possesses powerful concentrated energy and strong nonlinearity.
It can cause enormously devastating damage to ships and
offshore structures and may even cause significant harm to the
on-board staff and valuable property. Studies on the mechanisms
of rogue waves are thus of great significance to the vessel and
platform design and operation.
The problem of the rogue wave events have been
investigated extensively in recent years. Some possible
mechanisms have been summarized in the review papers [1-4].
Among the various physical mechanisms for the formation of
rogue waves in the deep ocean, the focusing due to modulational
instability (nonlinear self-focusing) is one of the most popular
mechanisms to describe the concentration of wave energy in
small local area of open sea field. The evolution of the
modulational instability, which is also known as the Benjamin–
Feir instability, has been extensively studied and is well-known
in [5-7].
As the rogue waves were frequently observed in the areas
with strong currents such as the Agulhas Current, the Gulf
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May 31-June 5, 2015, St. John's, Newfoundland, Canada
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Stream and the Kuroshio Current [8-10], the concentration of
wave energy influenced by the wave-current interaction has
drawn special attention. The effect of the wave-current
interaction on the modulational instability has also been the
attractive subject of a number studies over the decades years [11-
13]. The CmNLS equation was developed by Stocker and
Peregrine [14]. Assuming that the current U is the O(ε)
perturbation, Hjelmervik and Trulsen [15] then derived the one-
dimensional current-modified nonlinear Schrödinger equation to
investigate the statistics of the rogue waves. Compared with the
equation derived by Hjelmervik and Trulsen, the equation in [14]
is in Dysthe order [16] while the current is taken to be order
O(ε2). They found that the rogue wave events should be rarer
than in the open ocean without currents. Onorato and Proment
[17] found that the rogue wave can be excited from a stable wave
train under the effect of the opposing current flow by solving the
current modified NLS equation numerically. Some recent further
developments have been reported, e.g., a new version of current
modified NLS equation without the constraint of a narrow
angular spectra was derived by Shrira and Slunyaev [18, 19]
based on the modal approach and weakly nonlinear asymptotic
expansions. The influence of the opposite current on the rogue
wave formation has also been studied numerically [20-23].
Moreover, laboratory experiments were performed to investigate
the dynamics of rogue wave under the effect of the wave-current
interaction [24-28].
Some breather-type solutions of the NLS equation can
describe the nonlinear process of the modulational instability and
they have been regarded as being the prototype of a class of the
rogue waves developing in a plane wave background [29-31].
The Peregrine breather is the first order among the hierarchy,
which describes the amplification of an initial infinitesimal
disturbance of a plane wave and in which its maximum
amplitude can reach three times that of the initial wave
amplitude. Noticeably, such solutions can exist under the
background of random waves [32] and have also been observed
in the fully nonlinear simulation based on the Euler equation
[33].
In this paper, the generation and evolution of the Peregrine
breather-type rogue waves under the effect of the current are
investigated by solving the CmNLS equation numerically.
MATHEMATICAL MODEL AND NUMERICAL SCHEME
Mathematical Model
In this section, the CmNLS equation was used to simulate
the generation and evolution of the super rogue waves
numerically. Assuming the current velocity to be a constant, the
CmNLS equation can be also obtained from the equation derived
by Stocker & Peregrine [14]. It can be written in the form:
2 42
2 2
3 * 3*
03 3
28 2
6 04 16
g
z
B B i B ikc U B B ikUB
t x k x
k B B BB B B ikB
x x k x x
(1)
where B denotes the first harmonic of the complex velocity
potential, is the mean velocity potential and the mean
surface elevation due to radiation stress caused by the
modulation of a finite amplitude wave, k and ω are the wave
number and the wave frequency of the carrier wave respectively,
cg is defined by cg=ω/2k as the group velocity and U denotes the
current velocity. The complex conjugate is denoted by ‘*’.
The velocity potential of the induced mean current φ̅ is
governed by:
2 0 (2)
satisfying the boundary condition:
22
, 02
Bkz
z x
(3)
0, zz
(4)
In order to solve the spatial version of the CmNLS equation,
the following dimensionless variables are introduced:
2
2 2
22 , 2 ,
', ', ', '2
kx Ut t k x Ut
ak z z B B a U U
k k
(5)
Where ε=ka and γ is a scale factor which renders the
computational domain in ξ to 2π. With the primes omitted for
brevity, the equations (1)-(4) are changed as follows:
222
2
2
0
2 2
2 2
2
8 4
4 0, 0
, 0
0,
z
B Bi iB B iUB
B BU B i B
zz
Bz
z
z hz
(6)
Once the first harmonic of the complex velocity potential B
is obtained by solving the Eq. (6), the surface elevation can be
reconstructed as follows:
22
2
22 2 2 3 3
1 3
2 2 16. .
3
4 16
i
i i
B iiB B B e
c cB i
B i B e B e
(7)
where 2/ / is the phase function of the carrier
waves. The higher order corrections cannot be ignored when
reconstructing the surface elevation [34].
Numerical Scheme
The CmNLS equation can be solved numerically by the
split-step pseudo-spectral method [35]. In this paper, a modified
numerical solution method is developed in which the fourth-
order split-step method and the fourth-order Runge-Kutta
method are adopted for the iteration processes of the linear part
and nonlinear part alternately. For the nonlinear part, the
equation is obtained as follows:
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2 2
8 4B B
iB B B i B
(8)
The fourth-order Runge-Kutta method is used to evaluate
the solution for next step:
1 2 3
, ,
,6
B B
N B N B N B N B
(9)
where,
1
1, ,
2B B N B (10)
2 1
1,
2B B N B (11)
3 2,B B N B (12)
The linear part involves the equation:
2
2
2
B B Bi U iUB
(13)
By solving the linear equation Eq. (13), ,B can be
obtained from ,B :
2 2
1, ,i U U
B F F B e
(14)
Then, the fourth-order splitting solution operator is given in the
form:
4
4
1
= i iL c t N d t
i
e e
(15)
where,
1/3
1 4 2 31/3 1/3
1/3
1 3 2 41/3 1/3
1 1 2, c
2 2 2 2 2 2
1 2, d , 0
2 2 2 2
c c c
d d d
(16)
Following the iteration operator shown in Eq. (15), the
numerical solution for next step can be calculated. In order to
avoid the aliasing in the numerical solution, we introduced the
rectangular window function during the Fast Fourier Transform.
In numerical simulations, 65536 grid points were used in
every large time record corresponding to the sufficiently small
parameter γ and the spatial integration step is set to be 0.5cm
which is sufficiently small to ensure absence of numerical
instabilities. The step size used in the numerical simulation are
given in Table 1.
To reconstruct the surface elevation from the solution of B
and φ̅, the same mesh was used to represent the wave surface
elevation, and the discretisation of Eq.(7) gives:
222 1
0
22 2 1 2 3 3
1 3
2 2 16
3
4 16
. .
i
z
i i
iiB F i F B B B e
iB i BF i F B e B e
c c
(17)
where υ=0,±1,±2,±3……±N/2 is Fourier Mode.
PEREGRINE BREATHER-TYPE SOLUTION The Third Order Nonlinear Schrödinger Equation can be
written as follows:
2 4
2
2 20
8 2g
B B B ki c B B
t x k x
(18)
Using the dimensionless variables, the dimensionless form
of the NLS equation is obtained:
2
2 0T XXiq q q q (19)
where,
2 2
2, 2 ,4 2
a k BT t X k a x t q i
k a
(20)
The Peregrine breather is the first-order rational solution of
the NLS equation which is localized both temporally and
spatially. Besides, it can be understood as the limiting case of the
Ma breather and Akhmediev breather when 0 . By taking
the plane wave solution ( 0
iXq e ) as the seeding solution of the
NLS equation, the first-order rational solution of the NLS
equation can be constructed by the modified Darboux
Transformation [36]:
2 2
1 2, 1 4
1 4 4
iX
p
iXq X T e
T X
(21)
Figure.1 shows the profile of the Peregrine breather,
demonstrating clearly its spatial and temporal localization. It
describes waves that appear from nowhere and disappear without
a trace. At the position X=0, T=0, it generates the high amplitude
rogue wave when the carrier wave reaches the peak value. On
the contrary, it generates a hole when the carrier wave reaches
the minimum value.
Figure.1 Peregrine breather solution (21). The maximum amplitude reaches the 3 times larger than the background
carrier wave at the position X=0, T=0
In this study, using the dimensional variables same as (5),
the ‘initial’ condition of the rogue waves is obtained:
2
22 2 2
2 2
4 1 2
, 12 4
1 4
dd i
d
d d
ii
B i e
(22)
where d is that the boundary condition for wave maker at
d is specified in the form , dB .
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RESULTS AND DISCUSSION
Evolution of wave surface and generation of rogue wave
under the effect of constant current
In this section, the evolution of the Peregrine breather-type
rogue wave under the effect of constant current (U/cg=-4%) was
numerically simulated in the framework of the CmNLS equation
as demonstrated in Figure 2. The choice for the parameters of the
initial wave is selected near the breaking condition cited in [37].
The initial carrier wave amplitude is selected to be a=0.01m. The
wave number of carrier wave is set to be k=11.63 m-1,
corresponding to the wave length of 0.54m. The wave steepness
of the carrier wave is ε=0.1163. The distance between wave
maker and arranged focusing point is 30m. Since the Peregrine
breather solution is defined on the infinite domain, the scale
factor γ is set to be 0.0145375 which is sufficiently small so that
the time intervals can be much wider than the characteristic
localization interval. All the parameters used in the numerical
simulation are listed in Table 2.
Figure 2: The evolution of breather-type rogue wave generation under the effect of current (U/cg=-4%)
The time series of the waves at different position are
demonstrated in Figure 2. The rogue wave with a very large
amplitude is observed to be developed under the effect of the
constant current from a small disturbance as the waves propagate
forwards. The Peregrine breather-type rogue wave appears at the
position of x=28.26m, while the presumed position of focusing
is selected to be 30m. The maximum amplification of the rogue
wave reaches 3.69, which is higher than that predicted by the
analytical solution of the NLS equation. This can be explained
by the effects of the bound waves and the higher order terms
when reconstructing the surface elevation based on Eq. (7).
Besides, the huge wave crest disappears quickly after reaching
its maximum amplitude, which coincides with the shortlived
property of the rogue wave.
Effect of the constant current on the Peregrine breather-
type rogue wave
In this part, the comparison of the generation of the
Peregrine breather-type rogue wave under the constant current
with different velocities based on the CmNLS equation. The
evolution of the maximum amplitude of complex wave velocity
potential |B|, which can be seen as the function of x, is
demonstrated in Figure 3.
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Figure 3: Comparison of the evolution of the maximum amplitude of complex wave velocity potential |B| under the
current with different velocities
Figure 4: The wave height of the rogue wave and the peaks and troughs adjacent to the rogue wave
Looking at the Figure 3 in detail, due to the effect of the
higher order nonlinearity considered in Eq. (1), lower
amplifications are observed in the numerical simulations based
on the CmNLS equation compared to that predicted by the
analytical solution of the NLS equation. Moreover, it
demonstrated that the opposite current shortens the required
distance of the generation of the Peregrine breather-type solution
while the co-current can lengthen it. The increasing velocity can
enhance such effect. It reveals that the counter current can
accelerate the process of the nonlinear evolution of waves while
the co-current can restrain it. A second peak is observed in the
case of U/cg=-8%, which can be connected with the periodicity
due to the periodic condition used in the pseudo-spectral method.
The kinks existing in the Figure 3 show that the higher order
nonlinearity of the CmNLS equation breaks the periodicity of the
analytical solution of the NLS equation.
The wave height of the rogue wave and the peaks and
troughs adjacent to the rogue wave are summarized in Figure 4.
The wave height of the Peregrine breather-type rogue wave
shows an increasing trend from the case of U/cg=-8% to the case
of U/cg=8%. The stronger nonlinearity caused by the counter
current reduce the actual amplification of the rogue wave. The
front and rear peaks under the counter current are observed to be
larger than that of the co-current, while there is not such feature
in the front and rear troughs along the current velocity. In
addition, slightly asymmetry can be observed before and after
the rogue wave appearance and a strong asymmetry between the
crest and trough is clearly recognized by comparing the two
pictures in Figure 4.
CONCLUSIONS In this paper, we modeled the rogue wave event under the
influence of the constant current by solving the CmNLS
equation numerically. The impact of current on the evolution of
the Peregrine breather-type rogue wave solution of the NLS
equation was investigated. The main conclusions of this study
are:
The Peregrine breather-type rogue waves can be generated
under the influence of the constant current;
The higher order nonlinear terms in the CmNLS equation
play an important role in the evolution of the rogue wave;
The counter current enhance the nonlinearity of the waves
which leads to the acceleration of the generation of the rogue
wave, while the co-current can restrain it.
ACKNOWLEDGEMENT This work was financially supported by the National
Natural Science Foundation of China (Grant No. 51239007).
The authors would also like to acknowledge the support
from the Sino-UK Higher Education Research Partnership for
PhD Studies funded by British Council in China and China
Scholarship Council.
NOMENCLATURE φ Velocity Potential [m2/s]
ζ Wave Surface Elevation [m]
φ Velocity Potential of Mean Flow [m2/s]
ζ Wave Surface Elevation of Mean Flow [m]
A,A2,A3 Complex Displacement Amplitude [m]
B,B2,B3 Complex Velocity Potential Amplitude [m2/s]
k Wave Number [m-1]
ω Wave Frequency [s-1]
cg Wave Group Velocity [m/s]
U Current Velocity [m/s]
x Real Space Variable [m]
t Real Time Variable [s]
a Wave Amplitude [m]
γ Scale Factor
ε Wave Steepness
ξ Dimensionless Time Variable
η Dimensionless Space Variable
Ψ Phase Function
υ Fourier Mode
0 5 10 15 20 25 30 35 40 45 501
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
x/m
max
(|B
|)
U/cg=8%
U/cg=4%
U/cg=0%
U/cg=-4%
U/cg=-8%
-10 -5 0 5 100.01
0.015
0.02
0.025
0.03
0.035
0.04
U/cg/%
/m
Peak Front
Peakmax
Peak Rear
-10 -5 0 5 10-0.03
-0.026
-0.022
-0.018
-0.014
-0.01
U/cg/%
/m
Trough Front
Trough Rear
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q Dimensionless Complex Wave Amplitude
X Dimensionless Time Variable
T Dimensionless Space Variable
Ω,p,β Intermediate Variable
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Table 1 Step size for numerical simulations of the CmNLS equation
Case Number of Mesh Points in ξ∈
[0.2π] ε γ
Evolution of the Peregrine breather type rogue wave
based on the CmNLS equation(Case 1~Case 5) 65536 0.1163 0.0145375
Table 2 The parameters for numerical simulations of the CmNLS equation
case a/m ε γ d/m l/m U/cg
1
0.01 0.1163 0.0145375 30 50
-8%
2 -4%
3 0%
4 4%
5 8%
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