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Numerical study of Rogue waves as nonlinear Schrödinger breather solutions under finite water depth
Wave Motion (
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Wave Motion journal homepage: www.elsevier.com/locate/wavemoti
Numerical study of Rogue waves as nonlinear Schrödinger breather solutions under finite water depth Zhe Hu a,b , Wenyong Tang a,b,∗ , Hongxiang Xue a,b , Xiaoying Zhang c a
State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China
b
Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai, 200240, China
c
Bureau Veritas Marine (China) Co., Ltd, Shanghai, 200240, China
highlights • • • •
Peregrine breather of cubic Schrödinger equation is applied to numerical tanks. Nonlinear rogue waves are generated without introducing any linear effect. Meaningful results are found by using the wavelet analysis. Realistic rogue waves in the ocean are generated using Peregrine breather.
article
info
Article history: Received 2 April 2014 Received in revised form 26 August 2014 Accepted 4 September 2014 Available online xxxx Keywords: Nonlinear Schrödinger equation Breather solutions Numerical wave flume Wavelet analysis
abstract Following the existing experimental investigations, in this paper, we perform a series of simulations on rogue waves based on a breather solution of the cubic Schrödinger equation, in a numerical wave tank coded in Visual Basic language. The numerical wave tank solves 2-D Navier–Stokes equations and constructs free surfaces with a volume of fluid method. The simulated results are compared against theoretical solutions, and further analyzed by the Fast Fourier Transformation in the frequency domain. Besides, the wavelet analysis is employed, to obtain the temporal-frequency spectrum of the simulated rogue waves. Finally a realistic rogue wave measured in the Sea of Japan is simulated to validate breather solutions under realistic sea states. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Nowadays, there is a common consensus on the hazards of rogue waves, on the safety of ships and marine structures. During decades, many efforts in the community of physicists and oceanographers have been devoted to the understanding of their formations, which could be divided into two categories, i.e., the linear and nonlinear models [1–3]. The most comprehensive linear model is the superposition of waves, and wave propagations influenced by currents or the nonuniform sea bottom. Kharif and Pelinovsky [4] gave an overall review of the existing mechanisms of rogue waves. Recently, attention is focused on the more sophisticated nonlinear formation of rogue waves, which is described by the nonlinear
∗ Corresponding author at: State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China. Tel.: +86 21 34206642; fax: +86 21 34206642. E-mail address: [email protected] (W. Tang). http://dx.doi.org/10.1016/j.wavemoti.2014.09.002 0165-2125/© 2014 Elsevier B.V. All rights reserved.
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Schrödinger (NLS) equation for finite and deep water [5], and the Korteweg–de Vries (KdV) equation for shallow water [6]. Apart from water waves, the NLS equation is also found in different physical branches, such as optics and quantum physics. In the oceanographic context, it describes the slow modulation of wave envelopes of the carrier waves. It is well known that a uniform train of constant amplitude is unstable to the Benjamin–Feir instability (or modulational instability) corresponding to long disturbances of wave number of the wave envelope. The instability may lead to the formation of periodical giant waves, which is one explanation of the rogue waves. For now most theoretical investigations are carried out based on the NLS model. Osborne et al. [7] studied the nonlinear dynamics of rogue waves under the Benjamin–Feir instability of the 1 + 1 and 2 + 1 dimensional NLS equations. Zakharov et al. [5] numerically simulated freak waves with both the modulation instability of wave train and the evolution of NLS solitons. Didenkulova et al. [8] studied the properties of rogue waves in intermediate depth in comparison with deep waters and found the Peregrine breather contained more individual waves in intermediate depth than in deep waters. Akhmediev et al. [9] presented a method for finding the hierarchy of rational solutions of the self-focusing NLS equation. Xu et al. [10] studied the rogue wave solutions of the derivative NLS equation using the Darboux transformation. He et al. [11] introduced a mechanism for generating higher-order rogue waves (HRWs) of the NLS equation. Kedziora et al. [12] presented a systematic classification for HRW solutions of the NLS equation, constructed as the nonlinear superposition of first-order breathers via the recursive Darboux transformation scheme. In addition to the NLS model, researchers are also studying rogue waves in other nonlinear systems. Shan et al. [13] constructed rogue waves by using the Taylor expansion of the breather solution, on the foundation of the Kundu-DNLS equation. Guo et al. [14] constructed higher order rogue wave solutions from the Gerdjikov–Ivanov equation. Li et al. [15] studied rogue waves as high-order rational solutions for the Hirota equation. In order to investigate the interaction between rogue waves and ships or marine structures, rogue waves are also adequately studied in laboratories and numerical wave flumes. However, despite the nonlinear theories being more reasonable than the linear models, the latter is still the mainstream method for rogue wave generation in realistic and numerical wave tanks. The most adopted method is to gather wave energies at a specific location by the superposition of a number of waves with phases which have a focusing effect. Cui et al. [16] generated rogue waves using the superposition method, and studied the nonlinear effects on their speed. Liu et al. [17] improved the superposition model with a modified phase modulation method, and generated rogue waves recorded under realistic sea states. Sergeeva and Slunyaev [18] conducted numerous simulations of unidirectional wave evolutions under the potential theory, and studied the spatio-temporal wave data in a statistical sense. Fochesato et al. [19] employed a three-dimensional (3D) directional wave focusing mechanism, to simulate rogue waves in a 3D numerical wave tank (NWT). In contrast with the widely adopted linear model, the nonlinear mechanisms are still mainly discussed theoretically. Though there are applications of nonlinear models on the wave generation, most of them are carried out in laboratories. Chabchoub et al. [1,2] generated rogue waves based on the breather solutions of the cubic NLS equation under deep water in the laboratory. Onorato et al. [3] employed the breather solution of the cubic NLS equation under finite water depth in the laboratory, and investigated the response of a scaled chemical tanker hit by the generated rogue waves. Chabchoub et al. [20] observed a hierarchy of up to fifth-order breather-type rogue waves in a water tank under deep water. Slunyaev et al. [21] studied the existence of stable strongly nonlinear groups of deep-water waves by means of laboratory and numerical simulations, and obtained very good agreement. Slunyaev et al. [22] tested rational multi-breathers of NLS in numerical simulations of weakly and fully nonlinear model, and made comparisons with laboratory results. The studies of NLS-based rogue waves in NWTs solving N–S or RANS equations are still not adequate. In this article, we utilize the nonlinear model based on the cubic NLS equation under finite water depth, and generate rogue waves in a NWT. A Peregrine breather solution of the cubic NLS equation is adopted as the envelope of the target rogue wave. The numerical results are compared with the analytical solutions, and investigated with a fast Fourier transformation (FFT) to reveal their energy distribution. To obtain the temporal-frequency characteristics of rogue waves, we perform a wavelet analysis which reflects the energy variation during wave evolution. At last we simulate a realistic rogue wave recorded in the Sea of Japan with the Peregrine breather, to investigate its practicability on rogue waves under natural sea states. 2. The nonlinear Schrödinger equation and Peregrine breather solution The dynamics of surface gravity waves is generally described by the Euler equations for water waves, under some basic hypothesis. However, sometimes the fully nonlinear Euler equations are not easy to solve, therefore a simplified model is required which reserves some nonlinearities of the equations and is relatively easy to give analytical or numerical solutions. As to the problems of rogue waves, it is obvious that the nonlinear theory is more suitable. For rogue waves under deep or finite water depth, the nonlinear Schrödinger equation is often adopted to describe the evolution of wave trains with slow modulation [4]. Such an equation is derived from the Euler equation, under the hypothesis of a relatively small steepness [23]. The nonlinear Schrödinger equation for finite-depth water can be deduced in a variety of ways, for instance Ref. [24]. In this article, we quote the discussions given by Mei [23] and only list some useful formulas which can be further employed by numerical simulations. The cubic NLS equation is deduced by a multiple scale expansion, performed on both the velocity potential and the surface elevation. The velocity potential ϕ and surface elevation ζ are obtained by performing a straightforward perturbation
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expansion, and preserving the leading order as
g cosh Q iψ iAe + c .c . , ϕ = ε ϕ10 − 2ω cosh q ε iψ ζ = Ae + c .c . ,
(1) (2)
2
where ε = kA0 is a small constant, Q = k (z + h), q = kh, ψ = kx − ωt, g is the gravitational acceleration and c .c. represents complex conjugate. It should be noted that the coordinate is set on the still water surface, thus z = 0 is the still free surface and z = −h the water bottom. The wave number k and angular frequency ω satisfy the dispersion relationship ω2 = gk tanh (kh). ϕ10 represents the mean flow and A is the complex wave envelope of the sinusoidal wave train. Both ϕ10 and A are the functions of ε x and ε t, and are governed by the equations below
ω2 2ω cosh2 q + kCg ∂ϕ10 |A|2 , =− ∂ξ 4k sinh2 q gh − Cg −i
(3)
∂A ∂ 2A + α 2 + β |A|2 A = 0, ∂τ ∂ξ
(4)
where Cg is the group velocity of the wave train, i.e., Cg = ∂ω/∂ k. It should be mentioned that the two equations above are established on a scaled coordinate which moves with the group velocity Cg . The new coordinate is characterized by ξ = ε x − Cg t and τ = ε2 t. Eq. (4) is the so called cubic NLS equation, and its solutions have been widely investigated [25]. Parameters α and β in Eq. (4) are formulated as
α=− β=
1 ∂ 2ω 2 ∂ k2
,
(5)
ωk2
16 sinh2 q
4
2
ω
cosh q + 8 − 2 tanh q +
2 sinh2 2q
2ω cosh2 q + kCg
gh − Cg2
2 .
(6)
Peregrine [25] studied the cubic NLS equation with a standard form as iηT + ηXX + 2 |η|2 η = 0.
(7)
Peregrine gave a solution of Eq. (7) by conducting a double Taylor series expansion over the amplitude peak, on the foundation of Ma’s [26] work. The solution, also called the Peregrine breather, is written as
η = η0 e
2iη02 T
4 1 + 4iη02 T
1−
1 + 4 (η0 X )2 + 16 η02 T
2
.
(8)
Until now, the Peregrine breather Eq. (8) is a solution of the standard NLS equation (7). To obtain the Peregrine breather for Eq. (4), a simple transformation is required written as T = −τ , X = α −1/2 ξ and η = (β/2)1/2 A. Note that both η and −η are the solutions of Eq. (7), the Peregrine breather for Eq. (4) adopted in this article is
−iβ A20 τ
A = A0 e
4α 1 − 2iβ A20 τ
−1 . 2 α + α 2β A20 τ + 2β A20 ξ 2
(9)
Fig. 1 illustrates a 3-D non-dimensional contour of solution Eq. (9), where dimensionless parameter kh = 1.112. From Fig. 1 we can observe that the wave envelope is characterized by a peak of 3 times the mean value, and two troughs which reach 0. 3. Numerical methods For the purpose of investigating the viability of the breather solution in numerical tanks, we establish a 2-D numerical wave flume using the Visual Basic Compiler. The overall arrangement of the flume is displayed in Fig. 2. The flume solves the 2-D incompressible Navier–Stokes equations, and computes the water–air interface with a geometric reconstruction scheme proposed by Youngs [27]. Partial cell parameter λ is employed to describe the irregular calculation boundaries which represent the ratio of the volume occupied by the fluid and the solid in one grid. The governing equations of the numerical tank are written as the continuity equation
∂λui = 0, ∂ xi
(10)
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Fig. 1. Absolute value of the complex wave envelope of the Peregrine breather.
Fig. 2. A sketch of the numerical wave tank used in this paper for rogue wave generation.
the momentum equation
∂ ui ∂ ui 1 ∂p ∂ + uj =− + gi + υ ∂t ∂ xj ρ ∂ xi ∂ xj
∂ ui ∂ uj + ∂ xj ∂ xi
,
(11)
the F transport equation
∂λF ∂λui F + = 0, ∂t ∂ xi
(12)
where ( )i and ( )j represent index (tensor) notation, and the summation convention is adopted in the three equations above. Parameter ui is the velocity vector component, p is pressure, gi and υ are the gravitational acceleration and the coefficient of kinematic viscosity, respectively. The SIMPLE algorithm is employed to solve Eqs. (10) and (11), and a simple central scheme is used to compute F through Eq. (12). We adopt the finite difference method for discretion andstaggered grids ∂p
to mesh the computational domain. The pressure gradient term ∂ x and the diffusion term υ ∂∂x i j with the central difference scheme (CDS); the convection term
∂u uj ∂ x i j
∂ ui ∂ xj
+
∂ uj ∂ xi
are discretized
is discretized with a mixing scheme of the upwind
scheme (UDS) and the central difference scheme (CDS), with a blending factor of 0.4. The surface tension is not taken into account in this paper, and the free surface boundary condition is a linear interpolation for pressures and shearing stress condition for velocities. The wall and bottom are considered smooth. A sponge layer is located at the right end of the tank to absorb incident waves (see Fig. 2), and the absorption effect can be written as u1 = Cu0 v1 = C v0 ξ1 = C ξ0 ,
(13)
where the subscript 0 denotes quantities before wave absorption, and the subscript 1 after wave absorption. u and v are the horizontal and vertical components of velocity, respectively. C is the damping parameter which varies within a range of 1–0 from the beginning to the end of the absorption region. Generally, a translational or rotational paddle is placed at the front of the numerical wave tank, to imitate physical wave makers. However, problems occur while generating rogue waves with the traditional method. Since the widely adopted transformation function for the moving paddle is derived from a linear theory, errors may be induced when a linear theory is adopted to generate nonlinear rogue waves. To avoid this issue, we employ another strategy to simulate the NLS-based rogue wavers. Instead of setting the wave-making paddle, we move the fluid by setting the velocity, pressure and surface elevation of the wave-inflowing boundary, according to the analytical solutions given in Section 2. Apart from the wave elevations, fluid velocities and pressures are also required for wave generations in this article. From discussions in Section 2, we can
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see only the leading order is kept for the velocity potential and wave surface, therefore, to accord with the simplification above, we preserve the leading order of velocities and pressures and abandon higher orders. Based on the coordinate transformation relationship between (x, t ) and (ξ , τ ) mentioned in Section 2, the partial derivatives satisfy ∂/∂ x = ε · ∂/∂ξ and ∂/∂ t = ε 2 · ∂/∂τ − Cg ε · ∂/∂ξ . So the ϕ10 in Eq. (1) is ignored while computing velocities as it leads to higher order. The horizontal and vertical velocity components are given as u=ε
gk cosh Q iψ Ae + c .c . ,
(14)
2ω cosh q
v = −ε
gk sinh Q iψ iAe + c .c . .
(15)
2ω cosh q
The pressure is acquired from the Bernoulli equation. Noticing that |∇ϕ|2 and ϕ10 lead to higher order, the pressure can be written as p = −ρ gz + ρε
g cosh Q 2 cosh q
Aeiψ + c .c . .
(16)
The length of the numerical wave tank is 18 m, and water depth is 0.5 m. During simulation, the time interval is taken as 0.003 s which is less than 1/100 of the carrier wave period, the x-directional grid size corresponds to 1/70 of the carrier wavelength, and the y-directional size 1/33 of the water depth. The length of wave absorption region is about one time the wavelength of the carrier wave. The water surface displacements of specific locations selected are recorded and exported every 0.003 s (i.e., a sampling rate of 333 Hz). CPU of the machine used for the numerical experiments is Intel Core i5-3470 3.20 GHz with 3.87 GB available RAM. 4. Simulation results In this section, we generate a series of rogue waves, based on the Peregrine breather and the numerical method mentioned above. To generate rogue waves at the given position and time, coordinate x and t are replaced by x − x0 and t − t0 , where constant x0 and t0 correspond to the focusing position and time of the rogue waves, respectively. During the first 2 periods of the carrier wave, the wave maker boundary settings grow gradually to make the fluid field vary smoothly. Two numerical tests are conducted. The first test simulates given rogue waves and makes comparisons with analytical results, and the second generates rogue waves measured under realistic sea states. 4.1. Experimental rogue wave generation The wave number and period of the carrier wave are k = 2.223 m−1 and Tp = 1.5 s, respectively. Parameter A0 of Eq. (9) is 0.1 m (i.e., ε = 0.22), which corresponds to a mean wave height of 0.045 m for the carrier wave and a crest height of 0.067 m for the peak of the wave envelope. 4 gauges are installed in the tank at 0.5 m, 4.5 m, 8.5 m and 12.5 m from the wave maker boundary. Rogue waves are set to occur at 0.5 m, 4.5 m and 8.5 m from the wave maker boundary, and the focusing moment is 20 s for all the three cases. The numerical and theoretical time histories of the gauges above are illustrated in Figs. 3–5. As is shown in the time histories, the incipient stable traveling waves become focused at a certain position and then spread out into regular waves again, which is driven by a typical nonlinear superposition effect. From Fig. 3a, we could observe the main peak at 19.95 s, with a height of 0.0701 m measured from the still water surface. This is close to the theoretical maximum height (3 times the asymptotic height of the carrier, i.e., 0.0675 m), which means the numerical method is appropriate and valid. From Fig. 3 to Fig. 5, we can see the numerical results are in relatively good agreement with the theoretical ones, though the discrepancies grow when the gauge distances from the wave maker become longer. By comparing rogue waves with different focusing positions, we can find the distinctions between numerical and analytical results at the focusing positions (i.e., Fig. 3a, Fig. 4b and Fig. 5c) increase, when the positions become further away from the wave-making boundary. Both phenomena mentioned above are caused by the dispersion and dissipation of the numerical method. The dissipation becomes more evident when the wave crest is higher or trough is deeper, whereas, this effect is negligible for the mild carrier waves, i.e., the regular-liked waves before and after the rogue wave focusing. To investigate the energy constituent of the generated waves, a fast Fourier transformation (FFT) is performed on the recorded time histories at the gauge nearest to the wave generation boundary. It should be noted that the value of FFT is technically not the wave energy in a physical meaning, yet it reflects the changes of energy, i.e., the FFT value is bigger for higher wave energy and smaller for lower wave energy. Therefore, we use the term ‘‘energy’’ to represent FFTs of the wave histories. The obtained frequency-spectrums are described in Fig. 6. From Fig. 6, we can observe that apart from the first frequency peak, which corresponds to the frequency of the carrier wave (around 0.67 Hz), another frequency component is found nearby 1.3 Hz. This phenomenon means that during the formation of rogue waves, the wave energy splits into two parts: the majority stays as the carrier wave train, and the rest shifts to the higher frequency. This energy split obviously happens only nearby the position where rogue waves occur, as waves are almost stable (i.e., the wave envelope is a constant)
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a
b
c
d
Fig. 3. Numerical and analytical series of rogue waves with x0 = 0.5 m and t0 = 20 s, recorded at (a) 0.5 m, (b) 4.5 m, (c) 8.5 m, and (d) 12.5 m from the wave maker boundary.
a
b
c
d
Fig. 4. Numerical and analytical series of rogue waves with x0 = 4.5 m and t0 = 20 s, recorded at (a) 0.5 m, (b) 4.5 m, (c) 8.5 m, and (d) 12.5 m from the wave maker boundary.
on locations far away from the focusing point, and under this circumstance the FFT will appear only in one peak. For cases studied in this article, we can find that the frequency under the second peak (1.3 Hz) is about twice the corresponding frequency of the first peak (0.67 Hz). Further research may be conducted to investigate whether this relationship is still right or not for other rogue waves. When we further investigate the time records in Figs. 3–5, it can be found the surface elevation decreases just before and after rogue waves appear (see Fig. 3a, Fig. 4b and Fig. 5c), which, however, cannot be reflected by the FFT in Fig. 6. Therefore we perform a wavelet analysis to reveal the temporal-frequency variation of the measured rogue wave histories. We employ the Gabor transformation, in order to study the time-local energy distribution and to eliminate the disturbance
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a
b
c
d
Fig. 5. Numerical and analytical series of rogue waves with x0 = 8.5 m and t0 = 20 s, recorded at (a) 0.5 m, (b) 4.5 m, (c) 8.5 m, and (d) 12.5 m from the wave maker boundary.
a
b
c
Fig. 6. FFT on time series recorded by the first gauge, with focusing sites at (a) 0.5 m, (b) 4.5 m and (c) 8.5 m from the wave-making boundary.
from records of other time. The Gauss function is selected as the window of Gabor transformation. The Gabor transformation is formulated as [28]
(Gf ) (b, ω) =
f (t ) g ∗ (t − b) e−iωt dt ,
(17)
where f (t )√ is the signal and ∗ denotes a complex conjugate. Function g (t ) is the window of Gabor transformation and is taken as 1/ 4π α0 · exp −t 2 /4α0 in this article. The Gabor transformation of the three rogue waves generated above is shown in Fig. 7 (note that the x- and y- coordinates represent time t and angular frequency ω, respectively).
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9
16
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10
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20 25 time(s)
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Fig. 7. Gabor transformation of rogue waves with focusing positions at (a) 0.5 m, (b) 4.5 m and (c) 8.5 m from the wave-making boundary.
From Fig. 7 we can find most energy is distributed around 4.2 rad/s, which corresponds to the angular frequency of the carrier wave train, before and after the formation of rogue waves. However, the energy amplitude becomes bigger and the band wider when rogue waves occur. From the illustrated color bar, we can find that the peak energy (or technically amplitude) value is about twice the mean value of the carrier wave. Unlike the FFT results, from the fringes above, we can clearly observe that there are gaps (deep blue color) between the rogue waves and stable carrier waves, which mean there is a temporally low energy moment just before and after the occurrence of rogue waves. During the low energy time, the surface movement is placid due to a nonlinear superposition effect, which cannot be observed within the linear theory. The provisional low energy may explain some observations that the giant waves occur on a relatively mild sea surface. 4.2. Yura wave generation In this section, we generate a rogue wave recorded 3 km off Yura fishing harbor facing the Sea of Japan [29], where the depth is 43 m, and the significant wave height is 4.6 m. It should be declared that the Yura wave is simulated in the numerical wave flume under a laboratory scale, rather than a realistic sea scale. However, the parameters of the simulated waves are compared with the measured statistics. The generation settings are k = 2.223 m−1 , Tp = 1.5 s and A0 = 0.13 m. The parameters to be compared are: (1) α1 = Hj /Hs , (2) β1 = Hj /Hj−1 and β2 = Hj /Hj+1 , and (3) µ1 = ηj /Hj , where Hj , Hj−1 and Hj+1 are heights of rogue wave, the fore-going wave and the following wave, and ηj is the crest height of the rogue wave. The parameters above were proposed by Sand et al. [30] to give a definition of rogue waves. To evaluate the Peregrine breather with a realistic rogue wave, one issue must be solved. Rogue waves under actual sea states are irregular and random as is shown in Fig. 8, whereas the breather-type rogue waves are ordered and fixed when all control parameters are given. To overcome this problem, one must append random components to the breather solutions, or remove the randomness from realistic rogue waves. In this paper, we take a simple assumption that the randomness is negligible at the moment when rogue waves take place. This simplification will not induce much error as the nonlinear focusing is the main factor that leads to the formation of the giant wave crest. Based on the assumption above, the localized time plots of the Yura rogue wave are illustrated in Fig. 9. As only time histories during the occurrence of rogue waves are considered in this paper, we define: Hj−1 = zAB , Hj = (zBC + zCD ) /2 and Hj+1 = zDE , where z represents vertical distances measured from the still surface. The significant wave height is defined as the mean wave height of the carrier wave train of the Peregrine breather. Fig. 10 shows the simulated truncated rogue wave history on the foundation of the Peregrine breather in this paper, where we can see a similar wave contour to the recorded
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Fig. 8. Time records of the Yura rogue wave [29].
Fig. 9. Truncated time plots of the measured Yura rogue wave.
Fig. 10. Truncated time plots of the simulated Yura rogue wave. Table 1 Parameters of the recorded rogue wave [16] and the simulated wave. Parameter
α1
β1
β2
µ1
Recorded Numerical
2.22 2.32
2.01 2.21
1.86 2.53
0.69 0.65
one shown in Fig. 9. Table 1 lists the value of parameters α1 , β1 , β2 and µ1 . It can be found except for β2 , other simulated parameters are in quite good agreement with the recorded ones. From the comparisons above we can conclude that the Peregrine breather can be introduced to simulate the Yura rogue wave. 5. Conclusions In this article, we first discussed the Peregrine breather solution of the cubic nonlinear Schrödinger equation, and gave the velocities and pressure of the breather solution with a leading-order precision. Then, based on the Peregrine breather, a numerical wave flume was established using a boundary-setting wave generation method. Utilizing the proposed
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numerical flume, we generated rogue waves with different focusing positions under an experimental scale. A fast Fourier transformation and wavelet analysis were performed to investigate the energy characteristics of the simulated rogue waves. To validate the Peregrine breather with realistic rogue waves, we simulated the Yura rogue wave [29], and compared numerical results with realistic data. In our research, we found the Peregrine breather can well reflect the characteristics of rogue waves, and can be conveniently simulated in numerical wave tanks. The FFT analysis showed the whole energy split into 2 parts, and a small part of energy went to the high frequency during the formation of rogue waves. The wavelet analysis further revealed there was a low energy moment before and after the rogue wave occurrence, and the energy grew significantly when the rogue wave appeared. From the comparison between the numerical and recorded Yura wave, it was proved that the Peregrine breather could well describe certain realistic rogue waves. The article has studied the rogue wave on the foundation of the cubic NLS equation. Further researches can be carried out based on a higher-order model, e.g., the 4-order NLS equation. Acknowledgment This work was financed by the National Natural Science Foundation of China (Grant No. 51239007). References [1] A. Chabchoub, N. Hoffmann, N. Akhmediev, Rogue wave observation in a water wave tank, Phys. Rev. Lett. 106 (2011) 204502. [2] A. Chabchoub, N. Akhmediev, N. Hoffmann, Experimental study of spatiotemporally localized surface gravity water waves, Phys. Rev. E 86 (2012) 016311. [3] M. Onorato, D. Proment, G. Clauss, M. Klein, Rogue waves: From Nonlinear Schrödinger breather solutions to sea-keeping test, PLoS ONE 8 (2013) e54629. [4] C. Kharif, E. Pelinovsky, Physical mechanisms of the rogue wave phenomenon, Eur. J. Mech. B Fluids 22 (2003) 603–634. [5] V.E. Zakharov, A.I. Dyachenko, A.O. Prokofiev, Freak waves as nonlinear stage of Stokes wave modulation instability, Eur. J. Mech. B Fluids 25 (2006) 677–692. [6] E. Kit, L. Shemer, E. Pelinovsky, T. Talipova, O. Eitan, H. Jiao, Nonlinear wave group evolution in shallow water, J. Waterway Port Coast. Ocean Eng. 126 (2000) 221–228. [7] A.R. Osborne, M. Onorato, M. Serio, The nonlinear dynamics of rogue waves and holes in deep-water gravity wave trains, Phys. Lett. Sec. A Gen. At. Solid State Phys. 275 (2000) 386–393. [8] I.I. Didenkulova, I.F. Nikolkina, E.N. Pelinovsky, Rogue waves in the basin of intermediate depth and the possibility of their formation due to the modulational instability, JETP Lett. 97 (2013) 194–198. [9] N. Akhmediev, A. Ankiewicz, J.M. Soto-Crespo, Rogue waves and rational solutions of the nonlinear Schrödinger equation, Phys. Rev. E 80 (2009) 026601. [10] S. Xu, J. He, L. Wang, The Darboux transformation of the derivative nonlinear Schrödinger equation, J. Phys. A 44 (2011) 305203. [11] J.S. He, H.R. Zhang, L.H. Wang, K. Porsezian, A.S. Fokas, Generating mechanism for higher-order rogue waves, Phy. Rev. E 87 (2013) 052914. [12] D.J. Kedziora, A. Ankiewicz, N. Akhmediev, Classifying the hierarchy of nonlinear-Schrodinger-equation rogue-wave solutions, Phy. Rev. E 88 (2013) 013207. [13] S. Shan, C. Li, J. He, On rogue wave in the Kundu-DNLS equation, Commun. Nonlinear Sci. Numer. Simul. 18 (2013) 3337–3349. [14] L. Guo, Y. Zhang, S. Xu, Z. Wu, J. He, The higher order rogue wave solutions of the Gerdjikov-Ivanov equation, Phys. Scr. 89 (2014) 035501. [15] L. Li, Z. Wu, L. Wang, J. He, High-order rogue waves for the Hirota equation, Ann. Physics 334 (2013) 198–211. [16] C. Cui, N.C. Zhang, H.G. Kang, Y.X. Yu, An experimental and numerical study of the freak wave speed, Acta Oceanol. Sin. 32 (2013) 51–56. [17] Z.Q. Liu, N.C. Zhang, Y.X. Yu, An efficient focusing model for generation of freak waves, Acta Oceanol. Sin. 30 (2011) 19–26. [18] A. Sergeeva, A. Slunyaev, Rogue waves, rogue events and extreme wave kinematics in spatio-temporal fields of simulated sea states, Nat. Hazards Earth Syst. Sci. 13 (2013) 1759–1771. [19] C. Fochesato, S. Grilli, F. Dias, Numerical modeling of extreme rogue waves generated by directional energy focusing, Wave Motion 44 (2007) 395–416. [20] A. Chabchoub, N. Hoffmann, M. Onorato, A. Slunyaev, A. Sergeeva, E. Pelinovsky, N. Akhmediev, Observation of a hierarchy of up to fifth-order rogue waves in a water tank, Phys. Rev. E 86 (2012) 056601. [21] A. Slunyaev, G.F. Clauss, M. Klein, M. Onorato, Simulations and experiments of short intense envelope solitons of surface water waves, Phys. Fluids 25 (2013) 067105. [22] A. Slunyaev, E. Pelinovsky, A. Sergeeva, A. Chabchoub, N. Hoffmann, M. Onorato, N. Akhmediev, Super-rogue waves in simulations based on weakly nonlinear and fully nonlinear hydrodynamic equations, Phys. Rev. E 88 (2013) 012909. [23] C.C. Mei, The Applied Dynamics of Ocean Surface Waves, World Scientific Pub Co Inc, 1989. [24] A.V. Slunyaev, A high-order nonlinear envelope equation for gravity waves in finite-depth water, J. Exp. Theor. Phys. 101 (2005) 926–941. [25] D.H. Peregrine, Water waves, nonlinear Schrödinger equations and their solutions, J. Austral. Math. Soc. Ser. B 25 (1983) 16–43. [26] Y.C. Ma, The perturbed plane-wave solution of the cubic Schrödinger equation, Stud. Appl. Math 60 (1979) 43–58. [27] D.L. Youngs, Time-dependent multi-material flow with large fluid distortion, in: K.W. Morton, M.J. Baines (Eds.), Numerical Methods for Fluid Dynamics, Academic Press, 1982. [28] I. Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics, 1992. [29] N. Mori, P.C. Liu, T. Yasuda, Analysis of freak wave measurements in the Sea of Japan, Ocean Eng. 29 (2002) 1399–1414. [30] S.E. Sand, N.E.O. Hansen, P. Klinting, O.T. Gudmestad, M.J. Sterndorff, Freak wave kinematics, in: O. Torum, O.T. Gudmestad (Eds.), Water Wave Kinematics, Kluwer Academic Publishers, Dordrecht, 1990, pp. 535–549.
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Contents lists available at ScienceDirect
Wave Motion journal homepage: www.elsevier.com/locate/wavemoti
Numerical study of Rogue waves as nonlinear Schrödinger breather solutions under finite water depth Zhe Hu a,b , Wenyong Tang a,b,∗ , Hongxiang Xue a,b , Xiaoying Zhang c a
State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China
b
Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai, 200240, China
c
Bureau Veritas Marine (China) Co., Ltd, Shanghai, 200240, China
highlights • • • •
Peregrine breather of cubic Schrödinger equation is applied to numerical tanks. Nonlinear rogue waves are generated without introducing any linear effect. Meaningful results are found by using the wavelet analysis. Realistic rogue waves in the ocean are generated using Peregrine breather.
article
info
Article history: Received 2 April 2014 Received in revised form 26 August 2014 Accepted 4 September 2014 Available online xxxx Keywords: Nonlinear Schrödinger equation Breather solutions Numerical wave flume Wavelet analysis
abstract Following the existing experimental investigations, in this paper, we perform a series of simulations on rogue waves based on a breather solution of the cubic Schrödinger equation, in a numerical wave tank coded in Visual Basic language. The numerical wave tank solves 2-D Navier–Stokes equations and constructs free surfaces with a volume of fluid method. The simulated results are compared against theoretical solutions, and further analyzed by the Fast Fourier Transformation in the frequency domain. Besides, the wavelet analysis is employed, to obtain the temporal-frequency spectrum of the simulated rogue waves. Finally a realistic rogue wave measured in the Sea of Japan is simulated to validate breather solutions under realistic sea states. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Nowadays, there is a common consensus on the hazards of rogue waves, on the safety of ships and marine structures. During decades, many efforts in the community of physicists and oceanographers have been devoted to the understanding of their formations, which could be divided into two categories, i.e., the linear and nonlinear models [1–3]. The most comprehensive linear model is the superposition of waves, and wave propagations influenced by currents or the nonuniform sea bottom. Kharif and Pelinovsky [4] gave an overall review of the existing mechanisms of rogue waves. Recently, attention is focused on the more sophisticated nonlinear formation of rogue waves, which is described by the nonlinear
∗ Corresponding author at: State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China. Tel.: +86 21 34206642; fax: +86 21 34206642. E-mail address: [email protected] (W. Tang). http://dx.doi.org/10.1016/j.wavemoti.2014.09.002 0165-2125/© 2014 Elsevier B.V. All rights reserved.
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Schrödinger (NLS) equation for finite and deep water [5], and the Korteweg–de Vries (KdV) equation for shallow water [6]. Apart from water waves, the NLS equation is also found in different physical branches, such as optics and quantum physics. In the oceanographic context, it describes the slow modulation of wave envelopes of the carrier waves. It is well known that a uniform train of constant amplitude is unstable to the Benjamin–Feir instability (or modulational instability) corresponding to long disturbances of wave number of the wave envelope. The instability may lead to the formation of periodical giant waves, which is one explanation of the rogue waves. For now most theoretical investigations are carried out based on the NLS model. Osborne et al. [7] studied the nonlinear dynamics of rogue waves under the Benjamin–Feir instability of the 1 + 1 and 2 + 1 dimensional NLS equations. Zakharov et al. [5] numerically simulated freak waves with both the modulation instability of wave train and the evolution of NLS solitons. Didenkulova et al. [8] studied the properties of rogue waves in intermediate depth in comparison with deep waters and found the Peregrine breather contained more individual waves in intermediate depth than in deep waters. Akhmediev et al. [9] presented a method for finding the hierarchy of rational solutions of the self-focusing NLS equation. Xu et al. [10] studied the rogue wave solutions of the derivative NLS equation using the Darboux transformation. He et al. [11] introduced a mechanism for generating higher-order rogue waves (HRWs) of the NLS equation. Kedziora et al. [12] presented a systematic classification for HRW solutions of the NLS equation, constructed as the nonlinear superposition of first-order breathers via the recursive Darboux transformation scheme. In addition to the NLS model, researchers are also studying rogue waves in other nonlinear systems. Shan et al. [13] constructed rogue waves by using the Taylor expansion of the breather solution, on the foundation of the Kundu-DNLS equation. Guo et al. [14] constructed higher order rogue wave solutions from the Gerdjikov–Ivanov equation. Li et al. [15] studied rogue waves as high-order rational solutions for the Hirota equation. In order to investigate the interaction between rogue waves and ships or marine structures, rogue waves are also adequately studied in laboratories and numerical wave flumes. However, despite the nonlinear theories being more reasonable than the linear models, the latter is still the mainstream method for rogue wave generation in realistic and numerical wave tanks. The most adopted method is to gather wave energies at a specific location by the superposition of a number of waves with phases which have a focusing effect. Cui et al. [16] generated rogue waves using the superposition method, and studied the nonlinear effects on their speed. Liu et al. [17] improved the superposition model with a modified phase modulation method, and generated rogue waves recorded under realistic sea states. Sergeeva and Slunyaev [18] conducted numerous simulations of unidirectional wave evolutions under the potential theory, and studied the spatio-temporal wave data in a statistical sense. Fochesato et al. [19] employed a three-dimensional (3D) directional wave focusing mechanism, to simulate rogue waves in a 3D numerical wave tank (NWT). In contrast with the widely adopted linear model, the nonlinear mechanisms are still mainly discussed theoretically. Though there are applications of nonlinear models on the wave generation, most of them are carried out in laboratories. Chabchoub et al. [1,2] generated rogue waves based on the breather solutions of the cubic NLS equation under deep water in the laboratory. Onorato et al. [3] employed the breather solution of the cubic NLS equation under finite water depth in the laboratory, and investigated the response of a scaled chemical tanker hit by the generated rogue waves. Chabchoub et al. [20] observed a hierarchy of up to fifth-order breather-type rogue waves in a water tank under deep water. Slunyaev et al. [21] studied the existence of stable strongly nonlinear groups of deep-water waves by means of laboratory and numerical simulations, and obtained very good agreement. Slunyaev et al. [22] tested rational multi-breathers of NLS in numerical simulations of weakly and fully nonlinear model, and made comparisons with laboratory results. The studies of NLS-based rogue waves in NWTs solving N–S or RANS equations are still not adequate. In this article, we utilize the nonlinear model based on the cubic NLS equation under finite water depth, and generate rogue waves in a NWT. A Peregrine breather solution of the cubic NLS equation is adopted as the envelope of the target rogue wave. The numerical results are compared with the analytical solutions, and investigated with a fast Fourier transformation (FFT) to reveal their energy distribution. To obtain the temporal-frequency characteristics of rogue waves, we perform a wavelet analysis which reflects the energy variation during wave evolution. At last we simulate a realistic rogue wave recorded in the Sea of Japan with the Peregrine breather, to investigate its practicability on rogue waves under natural sea states. 2. The nonlinear Schrödinger equation and Peregrine breather solution The dynamics of surface gravity waves is generally described by the Euler equations for water waves, under some basic hypothesis. However, sometimes the fully nonlinear Euler equations are not easy to solve, therefore a simplified model is required which reserves some nonlinearities of the equations and is relatively easy to give analytical or numerical solutions. As to the problems of rogue waves, it is obvious that the nonlinear theory is more suitable. For rogue waves under deep or finite water depth, the nonlinear Schrödinger equation is often adopted to describe the evolution of wave trains with slow modulation [4]. Such an equation is derived from the Euler equation, under the hypothesis of a relatively small steepness [23]. The nonlinear Schrödinger equation for finite-depth water can be deduced in a variety of ways, for instance Ref. [24]. In this article, we quote the discussions given by Mei [23] and only list some useful formulas which can be further employed by numerical simulations. The cubic NLS equation is deduced by a multiple scale expansion, performed on both the velocity potential and the surface elevation. The velocity potential ϕ and surface elevation ζ are obtained by performing a straightforward perturbation
Z. Hu et al. / Wave Motion (
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expansion, and preserving the leading order as
g cosh Q iψ iAe + c .c . , ϕ = ε ϕ10 − 2ω cosh q ε iψ ζ = Ae + c .c . ,
(1) (2)
2
where ε = kA0 is a small constant, Q = k (z + h), q = kh, ψ = kx − ωt, g is the gravitational acceleration and c .c. represents complex conjugate. It should be noted that the coordinate is set on the still water surface, thus z = 0 is the still free surface and z = −h the water bottom. The wave number k and angular frequency ω satisfy the dispersion relationship ω2 = gk tanh (kh). ϕ10 represents the mean flow and A is the complex wave envelope of the sinusoidal wave train. Both ϕ10 and A are the functions of ε x and ε t, and are governed by the equations below
ω2 2ω cosh2 q + kCg ∂ϕ10 |A|2 , =− ∂ξ 4k sinh2 q gh − Cg −i
(3)
∂A ∂ 2A + α 2 + β |A|2 A = 0, ∂τ ∂ξ
(4)
where Cg is the group velocity of the wave train, i.e., Cg = ∂ω/∂ k. It should be mentioned that the two equations above are established on a scaled coordinate which moves with the group velocity Cg . The new coordinate is characterized by ξ = ε x − Cg t and τ = ε2 t. Eq. (4) is the so called cubic NLS equation, and its solutions have been widely investigated [25]. Parameters α and β in Eq. (4) are formulated as
α=− β=
1 ∂ 2ω 2 ∂ k2
,
(5)
ωk2
16 sinh2 q
4
2
ω
cosh q + 8 − 2 tanh q +
2 sinh2 2q
2ω cosh2 q + kCg
gh − Cg2
2 .
(6)
Peregrine [25] studied the cubic NLS equation with a standard form as iηT + ηXX + 2 |η|2 η = 0.
(7)
Peregrine gave a solution of Eq. (7) by conducting a double Taylor series expansion over the amplitude peak, on the foundation of Ma’s [26] work. The solution, also called the Peregrine breather, is written as
η = η0 e
2iη02 T
4 1 + 4iη02 T
1−
1 + 4 (η0 X )2 + 16 η02 T
2
.
(8)
Until now, the Peregrine breather Eq. (8) is a solution of the standard NLS equation (7). To obtain the Peregrine breather for Eq. (4), a simple transformation is required written as T = −τ , X = α −1/2 ξ and η = (β/2)1/2 A. Note that both η and −η are the solutions of Eq. (7), the Peregrine breather for Eq. (4) adopted in this article is
−iβ A20 τ
A = A0 e
4α 1 − 2iβ A20 τ
−1 . 2 α + α 2β A20 τ + 2β A20 ξ 2
(9)
Fig. 1 illustrates a 3-D non-dimensional contour of solution Eq. (9), where dimensionless parameter kh = 1.112. From Fig. 1 we can observe that the wave envelope is characterized by a peak of 3 times the mean value, and two troughs which reach 0. 3. Numerical methods For the purpose of investigating the viability of the breather solution in numerical tanks, we establish a 2-D numerical wave flume using the Visual Basic Compiler. The overall arrangement of the flume is displayed in Fig. 2. The flume solves the 2-D incompressible Navier–Stokes equations, and computes the water–air interface with a geometric reconstruction scheme proposed by Youngs [27]. Partial cell parameter λ is employed to describe the irregular calculation boundaries which represent the ratio of the volume occupied by the fluid and the solid in one grid. The governing equations of the numerical tank are written as the continuity equation
∂λui = 0, ∂ xi
(10)
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Fig. 1. Absolute value of the complex wave envelope of the Peregrine breather.
Fig. 2. A sketch of the numerical wave tank used in this paper for rogue wave generation.
the momentum equation
∂ ui ∂ ui 1 ∂p ∂ + uj =− + gi + υ ∂t ∂ xj ρ ∂ xi ∂ xj
∂ ui ∂ uj + ∂ xj ∂ xi
,
(11)
the F transport equation
∂λF ∂λui F + = 0, ∂t ∂ xi
(12)
where ( )i and ( )j represent index (tensor) notation, and the summation convention is adopted in the three equations above. Parameter ui is the velocity vector component, p is pressure, gi and υ are the gravitational acceleration and the coefficient of kinematic viscosity, respectively. The SIMPLE algorithm is employed to solve Eqs. (10) and (11), and a simple central scheme is used to compute F through Eq. (12). We adopt the finite difference method for discretion andstaggered grids ∂p
to mesh the computational domain. The pressure gradient term ∂ x and the diffusion term υ ∂∂x i j with the central difference scheme (CDS); the convection term
∂u uj ∂ x i j
∂ ui ∂ xj
+
∂ uj ∂ xi
are discretized
is discretized with a mixing scheme of the upwind
scheme (UDS) and the central difference scheme (CDS), with a blending factor of 0.4. The surface tension is not taken into account in this paper, and the free surface boundary condition is a linear interpolation for pressures and shearing stress condition for velocities. The wall and bottom are considered smooth. A sponge layer is located at the right end of the tank to absorb incident waves (see Fig. 2), and the absorption effect can be written as u1 = Cu0 v1 = C v0 ξ1 = C ξ0 ,
(13)
where the subscript 0 denotes quantities before wave absorption, and the subscript 1 after wave absorption. u and v are the horizontal and vertical components of velocity, respectively. C is the damping parameter which varies within a range of 1–0 from the beginning to the end of the absorption region. Generally, a translational or rotational paddle is placed at the front of the numerical wave tank, to imitate physical wave makers. However, problems occur while generating rogue waves with the traditional method. Since the widely adopted transformation function for the moving paddle is derived from a linear theory, errors may be induced when a linear theory is adopted to generate nonlinear rogue waves. To avoid this issue, we employ another strategy to simulate the NLS-based rogue wavers. Instead of setting the wave-making paddle, we move the fluid by setting the velocity, pressure and surface elevation of the wave-inflowing boundary, according to the analytical solutions given in Section 2. Apart from the wave elevations, fluid velocities and pressures are also required for wave generations in this article. From discussions in Section 2, we can
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see only the leading order is kept for the velocity potential and wave surface, therefore, to accord with the simplification above, we preserve the leading order of velocities and pressures and abandon higher orders. Based on the coordinate transformation relationship between (x, t ) and (ξ , τ ) mentioned in Section 2, the partial derivatives satisfy ∂/∂ x = ε · ∂/∂ξ and ∂/∂ t = ε 2 · ∂/∂τ − Cg ε · ∂/∂ξ . So the ϕ10 in Eq. (1) is ignored while computing velocities as it leads to higher order. The horizontal and vertical velocity components are given as u=ε
gk cosh Q iψ Ae + c .c . ,
(14)
2ω cosh q
v = −ε
gk sinh Q iψ iAe + c .c . .
(15)
2ω cosh q
The pressure is acquired from the Bernoulli equation. Noticing that |∇ϕ|2 and ϕ10 lead to higher order, the pressure can be written as p = −ρ gz + ρε
g cosh Q 2 cosh q
Aeiψ + c .c . .
(16)
The length of the numerical wave tank is 18 m, and water depth is 0.5 m. During simulation, the time interval is taken as 0.003 s which is less than 1/100 of the carrier wave period, the x-directional grid size corresponds to 1/70 of the carrier wavelength, and the y-directional size 1/33 of the water depth. The length of wave absorption region is about one time the wavelength of the carrier wave. The water surface displacements of specific locations selected are recorded and exported every 0.003 s (i.e., a sampling rate of 333 Hz). CPU of the machine used for the numerical experiments is Intel Core i5-3470 3.20 GHz with 3.87 GB available RAM. 4. Simulation results In this section, we generate a series of rogue waves, based on the Peregrine breather and the numerical method mentioned above. To generate rogue waves at the given position and time, coordinate x and t are replaced by x − x0 and t − t0 , where constant x0 and t0 correspond to the focusing position and time of the rogue waves, respectively. During the first 2 periods of the carrier wave, the wave maker boundary settings grow gradually to make the fluid field vary smoothly. Two numerical tests are conducted. The first test simulates given rogue waves and makes comparisons with analytical results, and the second generates rogue waves measured under realistic sea states. 4.1. Experimental rogue wave generation The wave number and period of the carrier wave are k = 2.223 m−1 and Tp = 1.5 s, respectively. Parameter A0 of Eq. (9) is 0.1 m (i.e., ε = 0.22), which corresponds to a mean wave height of 0.045 m for the carrier wave and a crest height of 0.067 m for the peak of the wave envelope. 4 gauges are installed in the tank at 0.5 m, 4.5 m, 8.5 m and 12.5 m from the wave maker boundary. Rogue waves are set to occur at 0.5 m, 4.5 m and 8.5 m from the wave maker boundary, and the focusing moment is 20 s for all the three cases. The numerical and theoretical time histories of the gauges above are illustrated in Figs. 3–5. As is shown in the time histories, the incipient stable traveling waves become focused at a certain position and then spread out into regular waves again, which is driven by a typical nonlinear superposition effect. From Fig. 3a, we could observe the main peak at 19.95 s, with a height of 0.0701 m measured from the still water surface. This is close to the theoretical maximum height (3 times the asymptotic height of the carrier, i.e., 0.0675 m), which means the numerical method is appropriate and valid. From Fig. 3 to Fig. 5, we can see the numerical results are in relatively good agreement with the theoretical ones, though the discrepancies grow when the gauge distances from the wave maker become longer. By comparing rogue waves with different focusing positions, we can find the distinctions between numerical and analytical results at the focusing positions (i.e., Fig. 3a, Fig. 4b and Fig. 5c) increase, when the positions become further away from the wave-making boundary. Both phenomena mentioned above are caused by the dispersion and dissipation of the numerical method. The dissipation becomes more evident when the wave crest is higher or trough is deeper, whereas, this effect is negligible for the mild carrier waves, i.e., the regular-liked waves before and after the rogue wave focusing. To investigate the energy constituent of the generated waves, a fast Fourier transformation (FFT) is performed on the recorded time histories at the gauge nearest to the wave generation boundary. It should be noted that the value of FFT is technically not the wave energy in a physical meaning, yet it reflects the changes of energy, i.e., the FFT value is bigger for higher wave energy and smaller for lower wave energy. Therefore, we use the term ‘‘energy’’ to represent FFTs of the wave histories. The obtained frequency-spectrums are described in Fig. 6. From Fig. 6, we can observe that apart from the first frequency peak, which corresponds to the frequency of the carrier wave (around 0.67 Hz), another frequency component is found nearby 1.3 Hz. This phenomenon means that during the formation of rogue waves, the wave energy splits into two parts: the majority stays as the carrier wave train, and the rest shifts to the higher frequency. This energy split obviously happens only nearby the position where rogue waves occur, as waves are almost stable (i.e., the wave envelope is a constant)
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a
b
c
d
Fig. 3. Numerical and analytical series of rogue waves with x0 = 0.5 m and t0 = 20 s, recorded at (a) 0.5 m, (b) 4.5 m, (c) 8.5 m, and (d) 12.5 m from the wave maker boundary.
a
b
c
d
Fig. 4. Numerical and analytical series of rogue waves with x0 = 4.5 m and t0 = 20 s, recorded at (a) 0.5 m, (b) 4.5 m, (c) 8.5 m, and (d) 12.5 m from the wave maker boundary.
on locations far away from the focusing point, and under this circumstance the FFT will appear only in one peak. For cases studied in this article, we can find that the frequency under the second peak (1.3 Hz) is about twice the corresponding frequency of the first peak (0.67 Hz). Further research may be conducted to investigate whether this relationship is still right or not for other rogue waves. When we further investigate the time records in Figs. 3–5, it can be found the surface elevation decreases just before and after rogue waves appear (see Fig. 3a, Fig. 4b and Fig. 5c), which, however, cannot be reflected by the FFT in Fig. 6. Therefore we perform a wavelet analysis to reveal the temporal-frequency variation of the measured rogue wave histories. We employ the Gabor transformation, in order to study the time-local energy distribution and to eliminate the disturbance
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a
b
c
d
Fig. 5. Numerical and analytical series of rogue waves with x0 = 8.5 m and t0 = 20 s, recorded at (a) 0.5 m, (b) 4.5 m, (c) 8.5 m, and (d) 12.5 m from the wave maker boundary.
a
b
c
Fig. 6. FFT on time series recorded by the first gauge, with focusing sites at (a) 0.5 m, (b) 4.5 m and (c) 8.5 m from the wave-making boundary.
from records of other time. The Gauss function is selected as the window of Gabor transformation. The Gabor transformation is formulated as [28]
(Gf ) (b, ω) =
f (t ) g ∗ (t − b) e−iωt dt ,
(17)
where f (t )√ is the signal and ∗ denotes a complex conjugate. Function g (t ) is the window of Gabor transformation and is taken as 1/ 4π α0 · exp −t 2 /4α0 in this article. The Gabor transformation of the three rogue waves generated above is shown in Fig. 7 (note that the x- and y- coordinates represent time t and angular frequency ω, respectively).
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Fig. 7. Gabor transformation of rogue waves with focusing positions at (a) 0.5 m, (b) 4.5 m and (c) 8.5 m from the wave-making boundary.
From Fig. 7 we can find most energy is distributed around 4.2 rad/s, which corresponds to the angular frequency of the carrier wave train, before and after the formation of rogue waves. However, the energy amplitude becomes bigger and the band wider when rogue waves occur. From the illustrated color bar, we can find that the peak energy (or technically amplitude) value is about twice the mean value of the carrier wave. Unlike the FFT results, from the fringes above, we can clearly observe that there are gaps (deep blue color) between the rogue waves and stable carrier waves, which mean there is a temporally low energy moment just before and after the occurrence of rogue waves. During the low energy time, the surface movement is placid due to a nonlinear superposition effect, which cannot be observed within the linear theory. The provisional low energy may explain some observations that the giant waves occur on a relatively mild sea surface. 4.2. Yura wave generation In this section, we generate a rogue wave recorded 3 km off Yura fishing harbor facing the Sea of Japan [29], where the depth is 43 m, and the significant wave height is 4.6 m. It should be declared that the Yura wave is simulated in the numerical wave flume under a laboratory scale, rather than a realistic sea scale. However, the parameters of the simulated waves are compared with the measured statistics. The generation settings are k = 2.223 m−1 , Tp = 1.5 s and A0 = 0.13 m. The parameters to be compared are: (1) α1 = Hj /Hs , (2) β1 = Hj /Hj−1 and β2 = Hj /Hj+1 , and (3) µ1 = ηj /Hj , where Hj , Hj−1 and Hj+1 are heights of rogue wave, the fore-going wave and the following wave, and ηj is the crest height of the rogue wave. The parameters above were proposed by Sand et al. [30] to give a definition of rogue waves. To evaluate the Peregrine breather with a realistic rogue wave, one issue must be solved. Rogue waves under actual sea states are irregular and random as is shown in Fig. 8, whereas the breather-type rogue waves are ordered and fixed when all control parameters are given. To overcome this problem, one must append random components to the breather solutions, or remove the randomness from realistic rogue waves. In this paper, we take a simple assumption that the randomness is negligible at the moment when rogue waves take place. This simplification will not induce much error as the nonlinear focusing is the main factor that leads to the formation of the giant wave crest. Based on the assumption above, the localized time plots of the Yura rogue wave are illustrated in Fig. 9. As only time histories during the occurrence of rogue waves are considered in this paper, we define: Hj−1 = zAB , Hj = (zBC + zCD ) /2 and Hj+1 = zDE , where z represents vertical distances measured from the still surface. The significant wave height is defined as the mean wave height of the carrier wave train of the Peregrine breather. Fig. 10 shows the simulated truncated rogue wave history on the foundation of the Peregrine breather in this paper, where we can see a similar wave contour to the recorded
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Fig. 8. Time records of the Yura rogue wave [29].
Fig. 9. Truncated time plots of the measured Yura rogue wave.
Fig. 10. Truncated time plots of the simulated Yura rogue wave. Table 1 Parameters of the recorded rogue wave [16] and the simulated wave. Parameter
α1
β1
β2
µ1
Recorded Numerical
2.22 2.32
2.01 2.21
1.86 2.53
0.69 0.65
one shown in Fig. 9. Table 1 lists the value of parameters α1 , β1 , β2 and µ1 . It can be found except for β2 , other simulated parameters are in quite good agreement with the recorded ones. From the comparisons above we can conclude that the Peregrine breather can be introduced to simulate the Yura rogue wave. 5. Conclusions In this article, we first discussed the Peregrine breather solution of the cubic nonlinear Schrödinger equation, and gave the velocities and pressure of the breather solution with a leading-order precision. Then, based on the Peregrine breather, a numerical wave flume was established using a boundary-setting wave generation method. Utilizing the proposed
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numerical flume, we generated rogue waves with different focusing positions under an experimental scale. A fast Fourier transformation and wavelet analysis were performed to investigate the energy characteristics of the simulated rogue waves. To validate the Peregrine breather with realistic rogue waves, we simulated the Yura rogue wave [29], and compared numerical results with realistic data. In our research, we found the Peregrine breather can well reflect the characteristics of rogue waves, and can be conveniently simulated in numerical wave tanks. The FFT analysis showed the whole energy split into 2 parts, and a small part of energy went to the high frequency during the formation of rogue waves. The wavelet analysis further revealed there was a low energy moment before and after the rogue wave occurrence, and the energy grew significantly when the rogue wave appeared. From the comparison between the numerical and recorded Yura wave, it was proved that the Peregrine breather could well describe certain realistic rogue waves. The article has studied the rogue wave on the foundation of the cubic NLS equation. Further researches can be carried out based on a higher-order model, e.g., the 4-order NLS equation. Acknowledgment This work was financed by the National Natural Science Foundation of China (Grant No. 51239007). References [1] A. Chabchoub, N. Hoffmann, N. Akhmediev, Rogue wave observation in a water wave tank, Phys. Rev. Lett. 106 (2011) 204502. [2] A. Chabchoub, N. Akhmediev, N. Hoffmann, Experimental study of spatiotemporally localized surface gravity water waves, Phys. Rev. E 86 (2012) 016311. [3] M. Onorato, D. Proment, G. Clauss, M. Klein, Rogue waves: From Nonlinear Schrödinger breather solutions to sea-keeping test, PLoS ONE 8 (2013) e54629. [4] C. Kharif, E. Pelinovsky, Physical mechanisms of the rogue wave phenomenon, Eur. J. Mech. 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