Harmonic energy transfer for extreme waves in current

Ning et al. (2015) developed a 2D fully nonlinear potential model to investigate the interaction between focused waves and uniform currents. The effects of uniform current on focusing wave crest, focal time and focal position were given. As its extension, harmonic energy transfer for focused waves in uniform current is studied using the proposed model by Ning et al. (2015) and Fast Fourier Transformation (FFT) technique in this study. It shows that the strong opposing currents, inducing partial wave blocking and reducing the extreme wave crest, make the nonlinear energy transfer non-reversible in the focusing and defocusing processes. The numerical results also provide an explanation to address the shifts of focal points in consideration of the combination effects of wave nonlinearity and current.


Introduction
Extreme wave events, i.e., freak waves, rogue waves, giant waves or focused waves, which were known due to their exceptionally large height, steep shape, asymmetric wave form and unpredictability, can pose a serious threat to devastate ships and offshore structures. Lots of events of ship wreckage and platform breakage have been related to extreme waves over the past decades (Mallory, 1974;Lawton, 2001). Several mechanisms have been suggested as possible causes for extreme waves. Mori and Yasuda (2002) considered that the higher-order nonlinearities (more than third order) can be regarded as one cause of extreme waves in deep water. High-order nonlinear interactions can transfer energy among the Fourier modes and excite apparently chaotic mode evolutions, which can generate a single extreme high wave with an outstanding crest height, such as a freak wave. Another nonlinear mechanism of a narrowband, deep-water wave train may undergo the modulation or Benjamin-Feir instability (Benjamin and Feir, 1967) to self-focusing a nonlinear wave train for forming extreme waves, which have been investigated extensively both analytically and numerically (Osborne et al., 2000;Onorato et al., 2001Onorato et al., , 2002. Besides, dispersive spatial-temporal focusing has been verified to effectively induce extreme waves through the superposition of different frequency wave components at a specific time and position (Kharif et al., 2001). Moreover, a combination of the dispersive-focusing and geometric focusing mechanisms to form an extreme wave has also been examined (Wu and Nepf, 2002). Overall, these studies provide a good understanding for the mechanisms of formation of freak waves.
Based on the above mechanisms, many investigations have been performed to study the physical characteristics of the extreme waves and their interaction with structures by the experimental and numerical simulations. Baldock et al. (1996) experimentally examined the effect of unidirectional focused wave groups and suggested that the nonlinear wave-wave interactions produce an extreme wave form. Johannessen and Swan (1997) developed a wave propagation model to predict 2-D extreme wave events and a good agreement with laboratory data was obtained. Borthwick et al. (2006) made measurements of the water particle kinematics under focused wave groups of normal and oblique incidence to a plane beach. Their results showed that the focal kinematics possesses a harmonic structure consisting of a large linear component and second-order low and high frequency bound components. Li et al. (2008) developed a numerical wave tank to study 2D focusing wave group based on High Order Spectral (HOS) method. Ning et al. (2009) presented the extensive experimental and numerical studies of unidirectional steep wave groups and found that the thirdorder resonant components significantly modify the dynamics of the focusing processes, leading to shifts of the focal position and focal time and modification to the main spectral components in the linear range. Hu et al. (2011) developed a numerical tank with free surface capturing and Cartesian cut-cell method to model focusing wave group and its interaction with fixed and floating objects. Zhao and Hu (2012) used a CIP-based Cartesian grid method to model nonlinear interactions between extreme waves and a floating body. However, most of these studies were conducted on quiescent water.
In the real sea state, waves always coexist with current. Actually, the wave-current interaction also plays an important role in the formation of extreme waves (Lavrenov and Porubov, 2006). Toffoli et al. (2011) studied the occurrence of extreme waves in an oblique current field experimentally. Afterwards, Toffoli et al. (2013) experimentally confirmed the generation of rogue waves in an opposite current, which was ever obtained using a one-dimensional current-modified nonlinear Schrödinger equation by Onorato et al. (2011). Thomas (1981) conducted both numerical and experimental studies of the wave-shear current interactions. He found that even for currents with an arbitrary distribution of vorticity, the variations of wave length and amplitude can be accurately predicted simply on the basis of the irrotational wave-current interaction model with a depth averaged mean current. Lai et al. (1989) experimentally investigated the wave-current interaction and confirmed the blockage of the waves by currents when the ratio of current speed to wave group velocity approaches -0.25.
Besides initiating an extreme wave, current can also influence the characteristics of a formed extreme wave. It is perhaps surprising that studies of the interaction between extreme waves and currents are still quite limited. Wu and Yao (2004) adopted both dispersive spatial-temporal focusing and wave-current interaction to generate extreme waves and experimentally studied the limiting freak wave in currents. Touboul et al. (2007) theoretically studied the influence of current on the focusing of wave groups propagating in deep water. Hjelmervik and Trulsen (2009) investigated how nonlinearity modifies both the significant wave height and the occurrence of freak waves, for waves propagating on inhomogeneous stationary currents. MerKoune et al. (2013) investigated the propagation of a wave train with the modulated frequency with and without current by the experimental and numerical methods. However, most investigations have rarely been concerned with the interaction between extreme waves and current from the point of view of energy conversion. Therefore, dispersive focusing of a wave group in the presence of co-existing current remains to be further investigated.
In the present study, the effects of some important parameters, such as current direction, current speed, focused wave amplitude and wave spectrum bandwidth on the formation of focused waves and the corresponding temporalspatial-spectral evolutions are further studied. Apart from this, the shifts of focal position and focal time due to the current are also given in the present study. Following the introduction in Section 1, the remainder of this paper is organized as follows. In Section 2, the numerical model is briefly described. In Section 3, the proposed numerical model is further validated by comparison with the published experimental data in terms of focused wave groups with current. In Section 4, the numerical results are presented, including the focused wave crest, shifts of focused points, evolution of surface displacement, and variation of wave energy spectra under currents. Finally, summaries are given in Section 5.

Numerical model
The interaction between the extreme waves and the uniform current in a 2D fluid domain is considered as shown in Fig. 1. A Cartesian coordinate system Oxz is employed such that its origin is located on the still water level at the left end of the domain, and the z-axis is positive upward. It is assumed that the fluid is incompressible, inviscid and the flow motion irrotational so that a velocity potential exists in the fluid domain. The total velocity potential can then be expressed as , where U is the steady uniform current velocity, and is the perturbation potential. The boundary value problem is governed by the Laplace equation. Both Φ and satisfy the Laplace equation in the computational domain Ω. The fully nonlinear kinematic and dynamic boundary conditions are satisfied on the instantaneous free surface. The so-called mixed Eulerian-Lagrangian method is used to describe a time-varying free surface with moving nodes. The fluid motion is generated by a piston wave maker, whose displacement S and velocity u p are obtained by using the spatial and temporal focusing technique. The proposed boundary value problem is transformed into a boundary integral equation and solved using the three-node quadratic elements. The whole process of the present numerical model is the same with that proposed by Ning et al. (2015).

Validation tests
The proposed numerical model has been verified by comparison with the published numerical and experimental results on monochromatic wave interaction with current, bichromatic waves interaction with current over a plane slope NING De-zhi et al. China Ocean Eng., 2017, Vol. 31, No. 2, P. 160-166 161 and pure focused wave groups by Ning et al. (2015). Further work is carried out to validate the proposed model by modeling focusing wave groups on uniform current in the present study. The experiments conducted by Wu and Yao (2004) are numerically simulated here, in which the static water depth is 0.6 m, each wave group is composed of 32 wave components and the corresponding frequency f is equally distributed in the scope (0.69 Hz, 1.47 Hz). The wave component amplitude is in accord with the linearslope spectrum and can be expressed as follows: where the superscript "0" corresponds to zero current, N is the number of wave components, i.e., N=32, and G 1 is the gain factor for varying the overall intensity of the n-th wave train. Fig. 2 depicts the spatial evolutions of the free surface displacements for three focused wave trains on the following (U=0.1 m/s), opposing (U=-0.1 m/s), and zero (U=0.0 m/s) currents at five positions with the gain factor G 1 = 0.015, respectively. In the numerical model, the linear focal position and focal time are defined as x p =4.8 m and t p =11.35 s respectively. After the convergent tests, the spatial and temporal steps are selected as x=λ min /15 and t=T min /50, respectively (λ min and T min denote the shortest wave length and the smallest wave period of wave components, respectively). Owing to nonlinear wave effects, the real focal position and time, corresponding to the highest crest used to indicate the occurrence of focused waves, will be different from the input ones, and are defined as x f and t f , respectively. A simple time shift is applied to the desired focal time t p so that it is convenient to compare the surface displacement evolution of the numerical results with those of the experimental data. The figure shows a satisfactory agreement between the proposed numerical model and the experiments (Wu and Yao, 2004). From the figures, the processes of waves focusing and defocusing can also be observed clearly with the wave train on the following current leading, followed by that on zero current and finally the one on the opposing current. The current modulation effect on dispersive focused waves is also shown in some degree.

Numerical results and discussion
We now illustrate the method described by undertaking several numerical simulations of the focused wave group interaction with uniform current. The effects of current velocity, wave spectra bandwidth, input wave group amplitude, form of input wave spectra on the energy transfer in the focusing and defocusing processes and the focal points under the uniform current environment, are investigated, respectively.
Under the circumstance of current, the form of the incident wave spectra is also an important factor affecting the behavior of the focused wave group. Three spectra i.e. linearsteepness spectrum, constant-amplitude spectrum and constant-steepness spectrum, are selected here. The experimental conditions of Wu and Yao (2004) are used here, including the frequency bandwidth 0.69 Hz≤ f ≤1.47 Hz and the number of wave components N=32. The input wave group amplitude A=55 mm is used for three input spectra. Fig. 3 shows relations between the frequency and the component amplitude for these three spectra. A linear-steepness spectrum is much steeper than the constant-steepness and constant-amplitude spectra. The corresponding amplitude of high frequencies (f ≥ 1 Hz) in the linear-steepness spectrum case is the smallest. Fig. 4 gives the distribution of focal crest elevation with current velocity. From the figure, it can be observed that, for the following current cases, the maximum wave elevation decreases and tends to a constant value with the current velocity. However, for the opposing current cases, the maximum wave elevation increases to a crest and then quickly decreases with the increasing of current speed in the negative direction. Such phenomena are especially pronounced for the constant wave amplitude.
Thus, the constant-amplitude spectrum is chosen as the input wave spectrum in the following simulations. The case  Ning et al. (2015) is considered with the static water depth h=0.4 m, number of wave components in each focused wave group N=29. The period ranges for two spectrum bandwidths are given as 0.6 s≤T≤1.4 s and 0.8 s≤ T≤1.2 s. The periods of the wave components are equally spaced. Fig. 5 describes the maximum elevation occurring at the focusing point (x f ) as a function of the current velocity for two different input wave amplitudes (A=22 mm and 55 mm) and two spectrum bandwidths, i.e., the wide one (0.6 s≤T≤ 1.4 s) and the narrow one (0.8 s ≤T≤1.2 s). From the figure, it can be observed when the current velocity U is larger than a certain value U c , e.g. U c =-0.015 m/s for the broadband and U c =-0.045 m/s for the narrow-band, the dimensionless maximum elevation (η crest /A) decreases with the current speed U increasing in the range considered in this investigation. It is mainly due to the energy transfer among wave components in currents, i.e., the opposing current increases the energy transfer from primary waves to higher harmonics and increases the maximum crest elevation, whilst it is vice versa for the following current. Because of stronger wave nonlinearity of larger group amplitude A or narrower bandwidth, many higher frequency wave components can be excited, and the decreasing slope is more gentle for smaller group amplitude A or broader bandwidth. However, when the current velocity U is smaller than U c , the dimensionless maximum elevation (η crest /A) decreases with the current speed U increasing in the negative direction. It means that a stronger opposing current may lead to a smaller extreme wave crest. The same phenomena were also observed numerically by Yan et al. (2010) and experimentally by Wu and Yao (2004). To explain the reason, Fig. 6 gives the power spectra estimates of different currents. In this study, wave energy density spectra are estimated from the time series of surface displacements at the focal point using the fast Fourier transform (FFT) (Rapp and Melville, 1990) with 2000 and 3000 points for wide-and narrowbands respectively. For the wide-and narrow-band cases, the total sampling times are chosen 20 s and 30 s, giving a frequency resolution f of 0.050 Hz and 0.033 Hz. From the figures, it can be seen that the contributions from wave components in high-frequency scope on the opposing currents, i.e., f/f c ≥1.56 for wide-band case with A=22 mm and U=-0.1 m/s shown in Fig. 6a, f/f c ≥2.07 for wide-band case with A=55 mm and U=-0.1 m/s shown in Fig. 6b, f/f c ≥1.41 for narrow-band case with A=22 mm and U=-0.15 m/s shown in Fig. 6c and f/f c ≥1.96 for narrow-band case with A=55 mm and U=-0.15 m/s shown in Fig. 6d, disappear, but vice versa for the cases on the following, zero and smaller opposing (U=-0.03 m/s in Fig. 6d) currents. It suggests that not all wave components are focused at the focal point and some high-frequency wave components are blocked by the stronger opposing current. This can also be theoretically derived by using the kinematic conservation equation (Mei, 1983) and reciprocate the above observation that the maximum crest elevation decreases with the decreasing of current velocity on stronger opposing currents, i.e., U<U c , in Figs. 6. Besides these, it is also found that for the smaller input total wave amplitude A=22 mm in Fig. 6a and 6c, the amplitude of the individual wave components is less than 1 mm and there is no significant transfer of energy within the frequency domain. However, for the larger input amplitude A=55 mm in Figs. 6b and 6d, it is noticeable that the formation of a focused wave group in these cases involves a significant transfer of energy into both the higher and lower harmonics, and steeper wave envelopes can be created.
The spectral energy evolution in the processes of waves focusing and defocusing is revealed on four different currents, i.e., U=0, -0.03 m/s, ±0.15 m/s, as shown in Fig. 7. The input group amplitude A=55 mm of the narrow-band case is considered. Five representative spatial points includ-   ing the upstream one(s), the actual focal point x f and the downstream ones are plotted here. The solid line indicates the density spectrum at the first upstream reference point, and the dashed lines denote the ones at the other marked points in the figures. Owing to the stronger nonlinearity, features of energy transfer to the high-frequency can be clearly seen. As the wave group approaches the focal position, the transfer of spectral energy from the primary-frequency components to the higher-frequency ones is clearly identified for the cases on the following current, zero current and smaller opposing current (i.e., U=-0.03 m/s). At these three focal points, there is a distinguished signature of

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NING De-zhi et al. China Ocean Eng., 2017, Vol. 31, No. 2, P. 160-166 energy density spectrum at the high-frequency scope, which is consistent with that wave-wave interactions becoming pronounced at x f due to waves focusing. Then the wave energy is transferred from the high-frequency components to the fundamental ones and the corresponding spectra are recovered to the original reference spectra gradually during the waves defocusing process. It means that the nonlinear energy transfer is reversible in the focusing and defocusing processes and the effects of the wave-wave interactions are gradually diminished. In contrast, for a stronger opposing current, i.e., a blocking current U=-0.15 m/s, the energy transferred to the high-frequency cannot be recovered to its initial reference level as shown in Fig. 7d. Energy transfer to the high-frequency is still observable at the focal point. In the defocusing process, the wave train continuously propagates along downstream, and the spectra shrink gradually due to partial high-frequency components being blocked. The shift of the focal position and time were experimentally and numerically found to be dependent on the higher-order nonlinearity of the wave group (Baldock et al., 1996;Ning et al., 2009). Most of the related researches are limited to the absence of current, but the current effect on the focal points is more obvious as shown in Fig. 7. A further series of tests of shift of focal points, i.e., (x f -x p ) and (t f -t p ), on different currents are carried out. And the results are shown in Fig. 8. The narrower-band spectra (0.8 s≤T≤ 1.2 s) and group amplitude A=22 mm and 55 mm are also considered here. In the figures, the dots represent the present numerical results and the lines are the corresponding fitting ones. Owing to the combination effects of wave nonlinearity and current, it is noticeable that, in the following current, the focal position is shifted forward and the focal time is delayed relative to the input ones (x p and t p ), more obviously for large input wave amplitude. But it is quite opposite in the opposing current, that the focal position is shifted upward and the focal time is advanced relative to the input ones (x p and t p ), more obvious for small input wave amplitude. The shift increases with the growth of the current strength due to the opposing current effect stronger than that of wave nonlinearity when the current speed U is smaller than a certain value, at which a balance between wave nonlinearity and current effects, i.e., zero shift, is achieved. An interesting phenomenon that the trend of the shifts comes into saturation with the further increase of the following current strength, is found, which is completely different from that of the opposing current. It is maybe due to the nonlinearity decreased by the following current.

Conclusions
This paper presents the extensive numerical studies of the extreme wave on uniform current generated in a flume. A fully nonlinear higher-order boundary element method is used in the numerical simulation. The present comparisons further confirm that the numerical calculations are in reasonable agreement with the published experimental data for focused wave propagation in a uniform current. Many numerical experiments on focused waves on uniform currents are performed and several findings from this study are summarized as follows.
Three incident wave spectra (i.e., constant-amplitude spectrum, linear-steepness spectrum and constant-steepness spectrum) of focused wave group on uniform current are studied. The extreme wave crest of constant-amplitude spectrum is most sensitive to the variation of current speed, and that of linear-steepness spectrum is the last. It also states that the constant-amplitude spectrum can be easily used to produce strong nonlinear extreme wave formation.
A following current can reduce the energy transfer from primary waves to higher harmonics, and the peak wave amplitude is decreased, whilst vice versa for the weak opposing current (no wave blocking). A strong opposing current (partial wave-blocking), with a significant amount of wave energy in the high-frequency range being reduced and even disappeared, can lead to the decrease of the maximum crest elevation and reduction in frequency bandwidth. This trend increases with the increase of strength of currents. Such phenomena are especially pronounced for the cases with narrower spectrum bandwidth or larger input wave amplitude. The present numerical results provide an explanation to address the occurrence and characteristics of ex- treme waves in consideration of the concept of wave-wave and wave-current interactions, which also supports the theoretical explanation of abnormal wave formation in a strong opposing current (Smith, 1976;Lavrenov, 1998).
The spectral evolution of focusing wave group on uniform current is examined. At the focal position, the Fourier transformation displays the distinguished feature of energy transfer from fundamental waves to high-frequency ones. The reversible processes of wave focusing and defocusing can be observed for zero, following and weak opposing currents. Opposite for a strong opposing current with partial blocking, the energy transferred to the high-frequency can not be recovered to its initial reference level.
The shift of focal points due to the combination effects of wave nonlinearity and current is presented. In the following current, the focal position is shifted forward and the focal time is delayed relative to the desired ones, more obviously for larger input wave amplitude. It is completely opposite in the opposing current. This is in accordance with the fact that the wave nonlinearity is decreased by the following current and increased by the opposing current, and their effects on focal shift are opposite in the opposing current.