Current Effects on the Generation and Evolution of the

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,


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,


Xinliang Tian State Key Laboratory of Ocean Engineering,


Jun Li State Key Laboratory of Ocean Engineering,


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: [email protected]

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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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:



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 .



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.



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



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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Current Effects on the Generation and Evolution of the